--- a/ANNOUNCE Fri Sep 09 06:45:39 2011 +0200
+++ b/ANNOUNCE Fri Sep 09 06:47:14 2011 +0200
@@ -1,34 +1,15 @@
-Subject: Announcing Isabelle2011
+Subject: Announcing Isabelle2011-1
To: isabelle-users@cl.cam.ac.uk
-Isabelle2011 is now available.
-
-This version significantly improves upon Isabelle2009-2, see the NEWS
-file in the distribution for more details. Some notable changes are:
-
-* Experimental Prover IDE based on Isabelle/Scala and jEdit.
-
-* Coercive subtyping (configured in HOL/Complex_Main).
-
-* HOL code generation: Scala as another target language.
-
-* HOL: partial_function definitions.
+Isabelle2011-1 is now available.
-* HOL: various tool enhancements, including Quickcheck, Nitpick,
- Sledgehammer, SMT integration.
-
-* HOL: various additions to theory library, including HOL-Algebra,
- Imperative_HOL, Multivariate_Analysis, Probability.
+This version improves upon Isabelle2011, see the NEWS file in the
+distribution for more details. Some important changes are:
-* HOLCF: reorganization of library and related tools.
-
-* HOL/SPARK: interactive proof environment for verification conditions
- generated by the SPARK Ada program verifier.
-
-* Improved Isabelle/Isar implementation manual (covering Isabelle/ML).
+* FIXME
-You may get Isabelle2011 from the following mirror sites:
+You may get Isabelle2011-1 from the following mirror sites:
Cambridge (UK) http://www.cl.cam.ac.uk/research/hvg/Isabelle/
Munich (Germany) http://isabelle.in.tum.de/
--- a/Admin/CHECKLIST Fri Sep 09 06:45:39 2011 +0200
+++ b/Admin/CHECKLIST Fri Sep 09 06:47:14 2011 +0200
@@ -3,9 +3,7 @@
- test polyml-5.4.0, polyml-5.3.0, polyml-5.2.1, smlnj;
-- test Proof General 4.1, 4.0, 3.7.1.1;
-
-- test Scala wrapper;
+- test Proof General 4.1, 3.7.1.1;
- check HTML header of library;
--- a/Admin/makebundle Fri Sep 09 06:45:39 2011 +0200
+++ b/Admin/makebundle Fri Sep 09 06:47:14 2011 +0200
@@ -75,7 +75,13 @@
)
case "$PLATFORM" in
- x86-cygwin)
+ *-darwin)
+ perl -pi -e "s,lookAndFeel=.*,lookAndFeel=com.apple.laf.AquaLookAndFeel,g;" \
+ "$TMP/$ISABELLE_NAME/src/Tools/jEdit/dist/properties/jEdit.props"
+ ;;
+ *-cygwin)
+ perl -pi -e "s,lookAndFeel=.*,lookAndFeel=com.sun.java.swing.plaf.windows.WindowsLookAndFeel,g;" \
+ "$TMP/$ISABELLE_NAME/src/Tools/jEdit/dist/properties/jEdit.props"
rm "$TMP/$ISABELLE_NAME/contrib/ProofGeneral"
ln -s ProofGeneral-3.7.1.1 "$TMP/$ISABELLE_NAME/contrib/ProofGeneral"
;;
--- a/CONTRIBUTORS Fri Sep 09 06:45:39 2011 +0200
+++ b/CONTRIBUTORS Fri Sep 09 06:47:14 2011 +0200
@@ -3,8 +3,14 @@
who is listed as an author in one of the source files of this Isabelle
distribution.
-Contributions to this Isabelle version
---------------------------------------
+Contributions to Isabelle2011-1
+-------------------------------
+
+* September 2011: Peter Gammie
+ Theory HOL/Libary/Saturated: numbers with saturated arithmetic.
+
+* August 2011: Florian Haftmann, Johannes Hölzl and Lars Noschinski, TUM
+ Refined theory on complete lattices.
Contributions to Isabelle2011
--- a/NEWS Fri Sep 09 06:45:39 2011 +0200
+++ b/NEWS Fri Sep 09 06:47:14 2011 +0200
@@ -1,11 +1,46 @@
Isabelle NEWS -- history user-relevant changes
==============================================
-New in this Isabelle version
-----------------------------
+New in Isabelle2011-1 (October 2011)
+------------------------------------
*** General ***
+* Improved Isabelle/jEdit Prover IDE (PIDE), which can be invoked as
+"isabelle jedit" on the command line.
+
+ - Management of multiple theory files directly from the editor
+ buffer store -- bypassing the file-system (no requirement to save
+ files for checking).
+
+ - Markup of formal entities within the text buffer, with semantic
+ highlighting, tooltips and hyperlinks to jump to defining source
+ positions.
+
+ - Improved text rendering, with sub/superscripts in the source
+ buffer (including support for copy/paste wrt. output panel, HTML
+ theory output and other non-Isabelle text boxes).
+
+ - Refined scheduling of proof checking and printing of results,
+ based on interactive editor view. (Note: jEdit folding and
+ narrowing allows to restrict buffer perspectives explicitly.)
+
+ - Reduced CPU performance requirements, usable on machines with few
+ cores.
+
+ - Reduced memory requirements due to pruning of unused document
+ versions (garbage collection).
+
+See also ~~/src/Tools/jEdit/README.html for further information,
+including some remaining limitations.
+
+* Theory loader: source files are exclusively located via the master
+directory of each theory node (where the .thy file itself resides).
+The global load path (such as src/HOL/Library) has been discontinued.
+Note that the path element ~~ may be used to reference theories in the
+Isabelle home folder -- for instance, "~~/src/HOL/Library/FuncSet".
+INCOMPATIBILITY.
+
* Theory loader: source files are identified by content via SHA1
digests. Discontinued former path/modtime identification and optional
ISABELLE_FILE_IDENT plugin scripts.
@@ -20,13 +55,6 @@
* Discontinued old lib/scripts/polyml-platform, which has been
obsolete since Isabelle2009-2.
-* Theory loader: source files are exclusively located via the master
-directory of each theory node (where the .thy file itself resides).
-The global load path (such as src/HOL/Library) has been discontinued.
-Note that the path element ~~ may be used to reference theories in the
-Isabelle home folder -- for instance, "~~/src/HOL/Library/FuncSet".
-INCOMPATIBILITY.
-
* Various optional external tools are referenced more robustly and
uniformly by explicit Isabelle settings as follows:
@@ -54,29 +82,38 @@
that the result needs to be unique, which means fact specifications
may have to be refined after enriching a proof context.
+* Attribute "case_names" has been refined: the assumptions in each case
+can be named now by following the case name with [name1 name2 ...].
+
* Isabelle/Isar reference manual provides more formal references in
syntax diagrams.
-* Attribute case_names has been refined: the assumptions in each case can
-be named now by following the case name with [name1 name2 ...].
-
*** HOL ***
-* Classes bot and top require underlying partial order rather than preorder:
-uniqueness of bot and top is guaranteed. INCOMPATIBILITY.
+* Theory Library/Saturated provides type of numbers with saturated
+arithmetic.
+
+* Classes bot and top require underlying partial order rather than
+preorder: uniqueness of bot and top is guaranteed. INCOMPATIBILITY.
* Class complete_lattice: generalized a couple of lemmas from sets;
-generalized theorems INF_cong and SUP_cong. New type classes for complete
-boolean algebras and complete linear orders. Lemmas Inf_less_iff,
-less_Sup_iff, INF_less_iff, less_SUP_iff now reside in class complete_linorder.
-Changed proposition of lemmas Inf_bool_def, Sup_bool_def, Inf_fun_def, Sup_fun_def,
-Inf_apply, Sup_apply.
+generalized theorems INF_cong and SUP_cong. New type classes for
+complete boolean algebras and complete linear orders. Lemmas
+Inf_less_iff, less_Sup_iff, INF_less_iff, less_SUP_iff now reside in
+class complete_linorder.
+
+Changed proposition of lemmas Inf_bool_def, Sup_bool_def, Inf_fun_def,
+Sup_fun_def, Inf_apply, Sup_apply.
+
Redundant lemmas Inf_singleton, Sup_singleton, Inf_binary, Sup_binary,
INF_eq, SUP_eq, INF_UNIV_range, SUP_UNIV_range, Int_eq_Inter,
-INTER_eq_Inter_image, Inter_def, INT_eq, Un_eq_Union, UNION_eq_Union_image,
-Union_def, UN_singleton, UN_eq have been discarded.
-More consistent and less misunderstandable names:
+INTER_eq_Inter_image, Inter_def, INT_eq, Un_eq_Union,
+UNION_eq_Union_image, Union_def, UN_singleton, UN_eq have been
+discarded.
+
+More consistent and comprehensive names:
+
INFI_def ~> INF_def
SUPR_def ~> SUP_def
INF_leI ~> INF_lower
@@ -94,30 +131,39 @@
INCOMPATIBILITY.
-* Theorem collections ball_simps and bex_simps do not contain theorems referring
-to UNION any longer; these have been moved to collection UN_ball_bex_simps.
-INCOMPATIBILITY.
-
-* Archimedean_Field.thy:
- floor now is defined as parameter of a separate type class floor_ceiling.
-
-* Finite_Set.thy: more coherent development of fold_set locales:
+* Added syntactic classes "inf" and "sup" for the respective
+constants. INCOMPATIBILITY: Changes in the argument order of the
+(mostly internal) locale predicates for some derived classes.
+
+* Theorem collections ball_simps and bex_simps do not contain theorems
+referring to UNION any longer; these have been moved to collection
+UN_ball_bex_simps. INCOMPATIBILITY.
+
+* Theory Archimedean_Field: floor now is defined as parameter of a
+separate type class floor_ceiling.
+
+* Theory Finite_Set: more coherent development of fold_set locales:
locale fun_left_comm ~> locale comp_fun_commute
locale fun_left_comm_idem ~> locale comp_fun_idem
-
-Both use point-free characterisation; interpretation proofs may need adjustment.
-INCOMPATIBILITY.
+
+Both use point-free characterization; interpretation proofs may need
+adjustment. INCOMPATIBILITY.
* Code generation:
- - theory Library/Code_Char_ord provides native ordering of characters
- in the target language.
- - commands code_module and code_library are legacy, use export_code instead.
- - method evaluation is legacy, use method eval instead.
- - legacy evaluator "SML" is deactivated by default. To activate it, add the following
- line in your theory:
+
+ - Theory Library/Code_Char_ord provides native ordering of
+ characters in the target language.
+
+ - Commands code_module and code_library are legacy, use export_code instead.
+
+ - Method "evaluation" is legacy, use method "eval" instead.
+
+ - Legacy evaluator "SML" is deactivated by default. May be
+ reactivated by the following theory command:
+
setup {* Value.add_evaluator ("SML", Codegen.eval_term) *}
-
+
* Declare ext [intro] by default. Rare INCOMPATIBILITY.
* Nitpick:
@@ -140,51 +186,57 @@
- Removed "metisF" -- use "metis" instead. INCOMPATIBILITY.
- Obsoleted "metisFT" -- use "metis (full_types)" instead. INCOMPATIBILITY.
-* "try":
- - Renamed "try_methods" and added "simp:", "intro:", "dest:", and "elim:"
- options. INCOMPATIBILITY.
- - Introduced "try" that not only runs "try_methods" but also "solve_direct",
- "sledgehammer", "quickcheck", and "nitpick".
+* Command 'try':
+ - Renamed 'try_methods' and added "simp:", "intro:", "dest:", and
+ "elim:" options. INCOMPATIBILITY.
+ - Introduced 'tryÄ that not only runs 'try_methods' but also
+ 'solve_direct', 'sledgehammer', 'quickcheck', and 'nitpick'.
* Quickcheck:
+
- Added "eval" option to evaluate terms for the found counterexample
- (currently only supported by the default (exhaustive) tester)
+ (currently only supported by the default (exhaustive) tester).
+
- Added post-processing of terms to obtain readable counterexamples
- (currently only supported by the default (exhaustive) tester)
+ (currently only supported by the default (exhaustive) tester).
+
- New counterexample generator quickcheck[narrowing] enables
- narrowing-based testing.
- It requires that the Glasgow Haskell compiler is installed and
- its location is known to Isabelle with the environment variable
- ISABELLE_GHC.
+ narrowing-based testing. Requires the Glasgow Haskell compiler
+ with its installation location defined in the Isabelle settings
+ environment as ISABELLE_GHC.
+
- Removed quickcheck tester "SML" based on the SML code generator
- from HOL-Library
+ (formly in HOL/Library).
* Function package: discontinued option "tailrec".
-INCOMPATIBILITY. Use partial_function instead.
-
-* HOL-Probability:
+INCOMPATIBILITY. Use 'partial_function' instead.
+
+* Session HOL-Probability:
- Caratheodory's extension lemma is now proved for ring_of_sets.
- Infinite products of probability measures are now available.
- - Use extended reals instead of positive extended reals.
- INCOMPATIBILITY.
-
-* Old recdef package has been moved to Library/Old_Recdef.thy, where it
-must be loaded explicitly. INCOMPATIBILITY.
-
-* Well-founded recursion combinator "wfrec" has been moved to
-Library/Wfrec.thy. INCOMPATIBILITY.
-
-* Theory Library/Nat_Infinity has been renamed to Library/Extended_Nat.
-The names of the following types and constants have changed:
- inat (type) ~> enat
+ - Use extended reals instead of positive extended
+ reals. INCOMPATIBILITY.
+
+* Old 'recdef' package has been moved to theory Library/Old_Recdef,
+from where it must be imported explicitly. INCOMPATIBILITY.
+
+* Well-founded recursion combinator "wfrec" has been moved to theory
+Library/Wfrec. INCOMPATIBILITY.
+
+* Theory Library/Nat_Infinity has been renamed to
+Library/Extended_Nat, with name changes of the following types and
+constants:
+
+ type inat ~> type enat
Fin ~> enat
Infty ~> infinity (overloaded)
iSuc ~> eSuc
the_Fin ~> the_enat
+
Every theorem name containing "inat", "Fin", "Infty", or "iSuc" has
been renamed accordingly.
-* Limits.thy: Type "'a net" has been renamed to "'a filter", in
+* Theory Limits: Type "'a net" has been renamed to "'a filter", in
accordance with standard mathematical terminology. INCOMPATIBILITY.
* Session Multivariate_Analysis: Type "('a, 'b) cart" has been renamed
@@ -255,10 +307,10 @@
real_abs_sub_norm ~> norm_triangle_ineq3
norm_cauchy_schwarz_abs ~> Cauchy_Schwarz_ineq2
-* Complex_Main: The locale interpretations for the bounded_linear and
-bounded_bilinear locales have been removed, in order to reduce the
-number of duplicate lemmas. Users must use the original names for
-distributivity theorems, potential INCOMPATIBILITY.
+* Theory Complex_Main: The locale interpretations for the
+bounded_linear and bounded_bilinear locales have been removed, in
+order to reduce the number of duplicate lemmas. Users must use the
+original names for distributivity theorems, potential INCOMPATIBILITY.
divide.add ~> add_divide_distrib
divide.diff ~> diff_divide_distrib
@@ -268,14 +320,16 @@
mult_right.setsum ~> setsum_right_distrib
mult_left.diff ~> left_diff_distrib
-* Complex_Main: Several redundant theorems have been removed or
+* Theory Complex_Main: Several redundant theorems have been removed or
replaced by more general versions. INCOMPATIBILITY.
+ real_of_int_real_of_nat ~> real_of_int_of_nat_eq
real_0_le_divide_iff ~> zero_le_divide_iff
realpow_two_disj ~> power2_eq_iff
real_squared_diff_one_factored ~> square_diff_one_factored
realpow_two_diff ~> square_diff_square_factored
reals_complete2 ~> complete_real
+ real_sum_squared_expand ~> power2_sum
exp_ln_eq ~> ln_unique
expi_add ~> exp_add
expi_zero ~> exp_zero
@@ -301,6 +355,7 @@
LIMSEQ_add_minus ~> tendsto_add [OF _ tendsto_minus]
LIMSEQ_add_const ~> tendsto_add [OF _ tendsto_const]
LIMSEQ_diff_const ~> tendsto_diff [OF _ tendsto_const]
+ LIMSEQ_Complex ~> tendsto_Complex
LIM_ident ~> tendsto_ident_at
LIM_const ~> tendsto_const
LIM_add ~> tendsto_add
@@ -334,26 +389,30 @@
*** Document preparation ***
-* Discontinued special treatment of hard tabulators, which are better
-avoided in the first place. Implicit tab-width is 1.
-
-* Antiquotation @{rail} layouts railroad syntax diagrams, see also
-isar-ref manual.
-
-* Antiquotation @{value} evaluates the given term and presents its result.
-
* Localized \isabellestyle switch can be used within blocks or groups
like this:
\isabellestyle{it} %preferred default
{\isabellestylett @{text "typewriter stuff"}}
-* New term style "isub" as ad-hoc conversion of variables x1, y23 into
-subscripted form x\<^isub>1, y\<^isub>2\<^isub>3.
+* Antiquotation @{rail} layouts railroad syntax diagrams, see also
+isar-ref manual, both for description and actual application of the
+same.
+
+* Antiquotation @{value} evaluates the given term and presents its
+result.
+
+* Antiquotations: term style "isub" provides ad-hoc conversion of
+variables x1, y23 into subscripted form x\<^isub>1,
+y\<^isub>2\<^isub>3.
* Predefined LaTeX macros for Isabelle symbols \<bind> and \<then>
(e.g. see ~~/src/HOL/Library/Monad_Syntax.thy).
+* Discontinued special treatment of hard tabulators, which are better
+avoided in the first place (no universally agreed standard expansion).
+Implicit tab-width is now 1.
+
*** ML ***
@@ -412,12 +471,22 @@
INCOMPATIBILITY, classical tactics and derived proof methods require
proper Proof.context.
+
+*** System ***
+
* Scala layer provides JVM method invocation service for static
-methods of type (String)String, see Invoke_Scala.method in ML.
-For example:
+methods of type (String)String, see Invoke_Scala.method in ML. For
+example:
Invoke_Scala.method "java.lang.System.getProperty" "java.home"
+Togeter with YXML.string_of_body/parse_body and XML.Encode/Decode this
+allows to pass structured values between ML and Scala.
+
+* The IsabelleText fonts includes some further glyphs to support the
+Prover IDE. Potential INCOMPATIBILITY: users who happen to have
+installed a local copy (which is normally *not* required) need to
+delete or update it from ~~/lib/fonts/.
New in Isabelle2011 (January 2011)
--- a/README Fri Sep 09 06:45:39 2011 +0200
+++ b/README Fri Sep 09 06:47:14 2011 +0200
@@ -16,8 +16,8 @@
* The Poly/ML compiler and runtime system (version 5.2.1 or later).
* The GNU bash shell (version 3.x or 2.x).
* Perl (version 5.x).
+ * Java 1.6.x from Oracle or Apple -- for Scala and jEdit.
* GNU Emacs (version 23) -- for the Proof General 4.x interface.
- * Java 1.6.x from Oracle/Sun or Apple -- for Scala and jEdit.
* A complete LaTeX installation -- for document preparation.
Installation
@@ -31,17 +31,18 @@
User interface
+ Isabelle/jEdit is an emerging Prover IDE based on advanced
+ technology of Isabelle/Scala. It provides a metaphor of continuous
+ proof checking of a versioned collection of theory sources, with
+ instantaneous feedback in real-time and rich semantic markup
+ associated with the formal text.
+
The classic Isabelle user interface is Proof General by David
Aspinall and others. It is a generic Emacs interface for proof
assistants, including Isabelle. Its most prominent feature is
script management, providing a metaphor of stepwise proof script
editing.
- Isabelle/jEdit is an experimental Prover IDE based on advanced
- technology of Isabelle/Scala. It provides a metaphor of continuous
- proof checking of a versioned collection of theory sources, with
- instantaneous feedback in real-time.
-
Other sources of information
The Isabelle Page
--- a/doc-src/Sledgehammer/sledgehammer.tex Fri Sep 09 06:45:39 2011 +0200
+++ b/doc-src/Sledgehammer/sledgehammer.tex Fri Sep 09 06:47:14 2011 +0200
@@ -883,13 +883,7 @@
Specifies the type encoding to use in ATP problems. Some of the type encodings
are unsound, meaning that they can give rise to spurious proofs
(unreconstructible using Metis). The supported type encodings are listed below,
-with an indication of their soundness in parentheses.
-%
-All the encodings with \textit{guards} or \textit{tags} in their name are
-available in a ``uniform'' and a ``nonuniform'' variant. The nonuniform variants
-are generally more efficient and are the default; the uniform variants are
-identified by the suffix \textit{\_uniform} (e.g.,
-\textit{mono\_guards\_uniform}{?}).
+with an indication of their soundness in parentheses:
\begin{enum}
\item[$\bullet$] \textbf{\textit{erased} (very unsound):} No type information is
@@ -933,12 +927,11 @@
\item[$\bullet$] \textbf{\textit{mono\_simple} (sound):} Exploits simple
first-order types if the prover supports the TFF0 or THF0 syntax; otherwise,
-falls back on \textit{mono\_guards\_uniform}. The problem is monomorphized.
+falls back on \textit{mono\_guards}. The problem is monomorphized.
\item[$\bullet$] \textbf{\textit{mono\_simple\_higher} (sound):} Exploits simple
higher-order types if the prover supports the THF0 syntax; otherwise, falls back
-on \textit{mono\_simple} or \textit{mono\_guards\_uniform}. The problem is
-monomorphized.
+on \textit{mono\_simple} or \textit{mono\_guards}. The problem is monomorphized.
\item[$\bullet$]
\textbf{%
@@ -949,12 +942,29 @@
\textit{raw\_mono\_guards}, \textit{raw\_mono\_tags}, \textit{mono\_guards},
\textit{mono\_tags}, and \textit{mono\_simple} are fully
typed and sound. For each of these, Sledgehammer also provides a lighter,
-virtually sound variant identified by a question mark (`{?}')\ that detects and
-erases monotonic types, notably infinite types. (For \textit{mono\_simple}, the
-types are not actually erased but rather replaced by a shared uniform type of
-individuals.) As argument to the \textit{metis} proof method, the question mark
-is replaced by a \hbox{``\textit{\_query}''} suffix. If the \emph{sound} option
-is enabled, these encodings are fully sound.
+virtually sound variant identified by a question mark (`\hbox{?}')\ that detects
+and erases monotonic types, notably infinite types. (For \textit{mono\_simple},
+the types are not actually erased but rather replaced by a shared uniform type
+of individuals.) As argument to the \textit{metis} proof method, the question
+mark is replaced by a \hbox{``\textit{\_query}''} suffix. If the \emph{sound}
+option is enabled, these encodings are fully sound.
+
+\item[$\bullet$]
+\textbf{%
+\textit{poly\_guards}??, \textit{poly\_tags}??, \textit{raw\_mono\_guards}??, \\
+\textit{raw\_mono\_tags}??, \textit{mono\_guards}??, \textit{mono\_tags}?? \\
+(quasi-sound):} \\
+Even lighter versions of the `\hbox{?}' encodings. As argument to the
+\textit{metis} proof method, the `\hbox{??}' suffix is replaced by
+\hbox{``\textit{\_query\_query}''}.
+
+\item[$\bullet$]
+\textbf{%
+\textit{poly\_guards}@?, \textit{poly\_tags}@?, \textit{raw\_mono\_guards}@?, \\
+\textit{raw\_mono\_tags}@? (quasi-sound):} \\
+Alternative versions of the `\hbox{??}' encodings. As argument to the
+\textit{metis} proof method, the `\hbox{@?}' suffix is replaced by
+\hbox{``\textit{\_at\_query}''}.
\item[$\bullet$]
\textbf{%
@@ -965,13 +975,30 @@
\textit{raw\_mono\_guards}, \textit{raw\_mono\_tags}, \textit{mono\_guards},
\textit{mono\_tags}, \textit{mono\_simple}, and \textit{mono\_simple\_higher}
also admit a mildly unsound (but very efficient) variant identified by an
-exclamation mark (`{!}') that detects and erases erases all types except those
-that are clearly finite (e.g., \textit{bool}). (For \textit{mono\_simple} and
-\textit{mono\_simple\_higher}, the types are not actually erased but rather
+exclamation mark (`\hbox{!}') that detects and erases erases all types except
+those that are clearly finite (e.g., \textit{bool}). (For \textit{mono\_simple}
+and \textit{mono\_simple\_higher}, the types are not actually erased but rather
replaced by a shared uniform type of individuals.) As argument to the
\textit{metis} proof method, the exclamation mark is replaced by the suffix
\hbox{``\textit{\_bang}''}.
+\item[$\bullet$]
+\textbf{%
+\textit{poly\_guards}!!, \textit{poly\_tags}!!, \textit{raw\_mono\_guards}!!, \\
+\textit{raw\_mono\_tags}!!, \textit{mono\_guards}!!, \textit{mono\_tags}!! \\
+(mildly unsound):} \\
+Even lighter versions of the `\hbox{!}' encodings. As argument to the
+\textit{metis} proof method, the `\hbox{!!}' suffix is replaced by
+\hbox{``\textit{\_bang\_bang}''}.
+
+\item[$\bullet$]
+\textbf{%
+\textit{poly\_guards}@!, \textit{poly\_tags}@!, \textit{raw\_mono\_guards}@!, \\
+\textit{raw\_mono\_tags}@! (mildly unsound):} \\
+Alternative versions of the `\hbox{!!}' encodings. As argument to the
+\textit{metis} proof method, the `\hbox{@!}' suffix is replaced by
+\hbox{``\textit{\_at\_bang}''}.
+
\item[$\bullet$] \textbf{\textit{smart}:} The actual encoding used depends on
the ATP and should be the most efficient virtually sound encoding for that ATP.
\end{enum}
--- a/doc-src/System/Thy/Misc.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/doc-src/System/Thy/Misc.thy Fri Sep 09 06:47:14 2011 +0200
@@ -336,8 +336,8 @@
sub-chunks separated by @{text "\<^bold>Y"}. Markup chunks start
with an empty sub-chunk, and a second empty sub-chunk indicates
close of an element. Any other non-empty chunk consists of plain
- text. For example, see @{file "~~/src/Pure/General/yxml.ML"} or
- @{file "~~/src/Pure/General/yxml.scala"}.
+ text. For example, see @{file "~~/src/Pure/PIDE/yxml.ML"} or
+ @{file "~~/src/Pure/PIDE/yxml.scala"}.
YXML documents may be detected quickly by checking that the first
two characters are @{text "\<^bold>X\<^bold>Y"}.
--- a/doc-src/System/Thy/document/Misc.tex Fri Sep 09 06:45:39 2011 +0200
+++ b/doc-src/System/Thy/document/Misc.tex Fri Sep 09 06:47:14 2011 +0200
@@ -376,8 +376,8 @@
sub-chunks separated by \isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E626F6C643E}{}\isactrlbold Y{\isaliteral{22}{\isachardoublequote}}}. Markup chunks start
with an empty sub-chunk, and a second empty sub-chunk indicates
close of an element. Any other non-empty chunk consists of plain
- text. For example, see \verb|~~/src/Pure/General/yxml.ML| or
- \verb|~~/src/Pure/General/yxml.scala|.
+ text. For example, see \verb|~~/src/Pure/PIDE/yxml.ML| or
+ \verb|~~/src/Pure/PIDE/yxml.scala|.
YXML documents may be detected quickly by checking that the first
two characters are \isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E626F6C643E}{}\isactrlbold X\isaliteral{5C3C5E626F6C643E}{}\isactrlbold Y{\isaliteral{22}{\isachardoublequote}}}.%
--- a/doc/Contents Fri Sep 09 06:45:39 2011 +0200
+++ b/doc/Contents Fri Sep 09 06:47:14 2011 +0200
@@ -1,4 +1,4 @@
-Learning and using Isabelle
+Miscellaneous tutorials
tutorial Tutorial on Isabelle/HOL
main What's in Main
isar-overview Tutorial on Isar
--- a/src/HOL/Algebra/IntRing.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Algebra/IntRing.thy Fri Sep 09 06:47:14 2011 +0200
@@ -35,8 +35,8 @@
apply (rule abelian_groupI, simp_all)
defer 1
apply (rule comm_monoidI, simp_all)
- apply (rule zadd_zmult_distrib)
-apply (fast intro: zadd_zminus_inverse2)
+ apply (rule left_distrib)
+apply (fast intro: left_minus)
done
(*
@@ -298,7 +298,7 @@
from this obtain x
where "1 = x * p" by best
from this [symmetric]
- have "p * x = 1" by (subst zmult_commute)
+ have "p * x = 1" by (subst mult_commute)
hence "\<bar>p * x\<bar> = 1" by simp
hence "\<bar>p\<bar> = 1" by (rule abs_zmult_eq_1)
from this and prime
--- a/src/HOL/Algebra/UnivPoly.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Algebra/UnivPoly.thy Fri Sep 09 06:47:14 2011 +0200
@@ -1818,7 +1818,7 @@
lemma INTEG_cring: "cring INTEG"
by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI
- zadd_zminus_inverse2 zadd_zmult_distrib)
+ left_minus left_distrib)
lemma INTEG_id_eval:
"UP_pre_univ_prop INTEG INTEG id"
--- a/src/HOL/Big_Operators.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Big_Operators.thy Fri Sep 09 06:47:14 2011 +0200
@@ -1517,7 +1517,7 @@
proof qed (auto simp add: max_def)
lemma max_lattice:
- "class.semilattice_inf (op \<ge>) (op >) max"
+ "class.semilattice_inf max (op \<ge>) (op >)"
by (fact min_max.dual_semilattice)
lemma dual_max:
@@ -1611,7 +1611,7 @@
assumes "finite A" and "x \<in> A"
shows "x \<le> Max A"
proof -
- interpret semilattice_inf "op \<ge>" "op >" max
+ interpret semilattice_inf max "op \<ge>" "op >"
by (rule max_lattice)
from assms show ?thesis by (simp add: Max_def fold1_belowI)
qed
@@ -1625,7 +1625,7 @@
assumes "finite A" and "A \<noteq> {}"
shows "Max A \<le> x \<longleftrightarrow> (\<forall>a\<in>A. a \<le> x)"
proof -
- interpret semilattice_inf "op \<ge>" "op >" max
+ interpret semilattice_inf max "op \<ge>" "op >"
by (rule max_lattice)
from assms show ?thesis by (simp add: Max_def below_fold1_iff)
qed
--- a/src/HOL/Boogie/Examples/Boogie_Max_Stepwise.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Boogie/Examples/Boogie_Max_Stepwise.thy Fri Sep 09 06:47:14 2011 +0200
@@ -77,10 +77,10 @@
boogie_vc max (examine: L_14_5c L_14_5b L_14_5a)
proof boogie_cases
case L_14_5c
- thus ?case by (metis less_le_not_le zle_add1_eq_le zless_linear)
+ thus ?case by (metis less_le_not_le zle_add1_eq_le less_linear)
next
case L_14_5b
- thus ?case by (metis less_le_not_le xt1(10) zle_linear zless_add1_eq)
+ thus ?case by (metis less_le_not_le xt1(10) linorder_linear zless_add1_eq)
next
case L_14_5a
note max0 = `max0 = array 0`
--- a/src/HOL/Code_Numeral.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Code_Numeral.thy Fri Sep 09 06:47:14 2011 +0200
@@ -274,7 +274,7 @@
then have "int ((nat_of k div 2) * 2 + nat_of k mod 2) = int (nat_of k)"
by simp
then have "int (nat_of k) = int (nat_of k div 2) * 2 + int (nat_of k mod 2)"
- unfolding int_mult zadd_int [symmetric] by simp
+ unfolding of_nat_mult of_nat_add by simp
then show ?thesis by (auto simp add: int_of_def mult_ac)
qed
--- a/src/HOL/Complete_Lattice.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Complete_Lattice.thy Fri Sep 09 06:47:14 2011 +0200
@@ -33,7 +33,7 @@
begin
lemma dual_complete_lattice:
- "class.complete_lattice Sup Inf (op \<ge>) (op >) sup inf \<top> \<bottom>"
+ "class.complete_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
by (auto intro!: class.complete_lattice.intro dual_bounded_lattice)
(unfold_locales, (fact bot_least top_greatest
Sup_upper Sup_least Inf_lower Inf_greatest)+)
@@ -85,7 +85,7 @@
lemma INF_foundation_dual [no_atp]:
"complete_lattice.SUPR Inf = INFI"
proof (rule ext)+
- interpret dual: complete_lattice Sup Inf "op \<ge>" "op >" sup inf \<top> \<bottom>
+ interpret dual: complete_lattice Sup Inf sup "op \<ge>" "op >" inf \<top> \<bottom>
by (fact dual_complete_lattice)
fix f :: "'b \<Rightarrow> 'a" and A
show "complete_lattice.SUPR Inf A f = (\<Sqinter>a\<in>A. f a)"
@@ -95,7 +95,7 @@
lemma SUP_foundation_dual [no_atp]:
"complete_lattice.INFI Sup = SUPR"
proof (rule ext)+
- interpret dual: complete_lattice Sup Inf "op \<ge>" "op >" sup inf \<top> \<bottom>
+ interpret dual: complete_lattice Sup Inf sup "op \<ge>" "op >" inf \<top> \<bottom>
by (fact dual_complete_lattice)
fix f :: "'b \<Rightarrow> 'a" and A
show "complete_lattice.INFI Sup A f = (\<Squnion>a\<in>A. f a)"
@@ -313,7 +313,7 @@
"\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?P)
"\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?Q)
proof -
- interpret dual: complete_lattice Sup Inf "op \<ge>" "op >" sup inf \<top> \<bottom>
+ interpret dual: complete_lattice Sup Inf sup "op \<ge>" "op >" inf \<top> \<bottom>
by (fact dual_complete_lattice)
from dual.Inf_top_conv show ?P and ?Q by simp_all
qed
@@ -407,7 +407,7 @@
by (simp add: SUP_def inf_Sup image_image)
lemma dual_complete_distrib_lattice:
- "class.complete_distrib_lattice Sup Inf (op \<ge>) (op >) sup inf \<top> \<bottom>"
+ "class.complete_distrib_lattice Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
apply (rule class.complete_distrib_lattice.intro)
apply (fact dual_complete_lattice)
apply (rule class.complete_distrib_lattice_axioms.intro)
@@ -458,7 +458,7 @@
begin
lemma dual_complete_boolean_algebra:
- "class.complete_boolean_algebra Sup Inf (op \<ge>) (op >) sup inf \<top> \<bottom> (\<lambda>x y. x \<squnion> - y) uminus"
+ "class.complete_boolean_algebra Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom> (\<lambda>x y. x \<squnion> - y) uminus"
by (rule class.complete_boolean_algebra.intro, rule dual_complete_distrib_lattice, rule dual_boolean_algebra)
lemma uminus_Inf:
@@ -489,7 +489,7 @@
begin
lemma dual_complete_linorder:
- "class.complete_linorder Sup Inf (op \<ge>) (op >) sup inf \<top> \<bottom>"
+ "class.complete_linorder Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder)
lemma Inf_less_iff (*[simp]*):
@@ -537,7 +537,7 @@
lemma Inf_eq_bot_iff (*[simp]*):
"\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)"
proof -
- interpret dual: complete_linorder Sup Inf "op \<ge>" "op >" sup inf \<top> \<bottom>
+ interpret dual: complete_linorder Sup Inf sup "op \<ge>" "op >" inf \<top> \<bottom>
by (fact dual_complete_linorder)
from dual.Sup_eq_top_iff show ?thesis .
qed
--- a/src/HOL/Complex.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Complex.thy Fri Sep 09 06:47:14 2011 +0200
@@ -253,6 +253,14 @@
shows "complex_of_real r * Complex x y = Complex (r*x) (r*y)"
by (simp add: complex_of_real_def)
+lemma complex_eq_cancel_iff2 [simp]:
+ shows "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
+ by (simp add: complex_of_real_def)
+
+lemma complex_split_polar:
+ "\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))"
+ by (simp add: complex_eq_iff polar_Ex)
+
subsection {* Vector Norm *}
@@ -304,10 +312,10 @@
end
-lemma cmod_unit_one [simp]: "cmod (Complex (cos a) (sin a)) = 1"
+lemma cmod_unit_one: "cmod (Complex (cos a) (sin a)) = 1"
by simp
-lemma cmod_complex_polar [simp]:
+lemma cmod_complex_polar:
"cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r"
by (simp add: norm_mult)
@@ -315,20 +323,29 @@
unfolding complex_norm_def
by (rule real_sqrt_sum_squares_ge1)
-lemma complex_mod_minus_le_complex_mod [simp]: "- cmod x \<le> cmod x"
+lemma complex_mod_minus_le_complex_mod: "- cmod x \<le> cmod x"
by (rule order_trans [OF _ norm_ge_zero], simp)
-lemma complex_mod_triangle_ineq2 [simp]: "cmod(b + a) - cmod b \<le> cmod a"
+lemma complex_mod_triangle_ineq2: "cmod(b + a) - cmod b \<le> cmod a"
by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp)
-lemmas real_sum_squared_expand = power2_sum [where 'a=real]
-
lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"
by (cases x) simp
lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"
by (cases x) simp
+text {* Properties of complex signum. *}
+
+lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
+ by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute)
+
+lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
+ by (simp add: complex_sgn_def divide_inverse)
+
+lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
+ by (simp add: complex_sgn_def divide_inverse)
+
subsection {* Completeness of the Complexes *}
@@ -366,10 +383,6 @@
(simp add: dist_norm real_sqrt_sum_squares_less)
qed
-lemma LIMSEQ_Complex:
- "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. Complex (X n) (Y n)) ----> Complex a b"
- by (rule tendsto_Complex)
-
instance complex :: banach
proof
fix X :: "nat \<Rightarrow> complex"
@@ -379,13 +392,13 @@
from Cauchy_Im [OF X] have 2: "(\<lambda>n. Im (X n)) ----> lim (\<lambda>n. Im (X n))"
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
have "X ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"
- using LIMSEQ_Complex [OF 1 2] by simp
+ using tendsto_Complex [OF 1 2] by simp
thus "convergent X"
by (rule convergentI)
qed
-subsection {* The Complex Number @{term "\<i>"} *}
+subsection {* The Complex Number $i$ *}
definition "ii" :: complex ("\<i>")
where i_def: "ii \<equiv> Complex 0 1"
@@ -429,6 +442,9 @@
lemma inverse_i [simp]: "inverse ii = - ii"
by (rule inverse_unique, simp)
+lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
+ by (simp add: mult_assoc [symmetric])
+
subsection {* Complex Conjugation *}
@@ -513,6 +529,12 @@
lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<twosuperior>"
by (simp add: norm_mult power2_eq_square)
+lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
+ by (simp add: cmod_def power2_eq_square)
+
+lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
+ by simp
+
lemma bounded_linear_cnj: "bounded_linear cnj"
using complex_cnj_add complex_cnj_scaleR
by (rule bounded_linear_intro [where K=1], simp)
@@ -524,74 +546,92 @@
bounded_linear.isCont [OF bounded_linear_cnj]
-subsection{*The Functions @{term sgn} and @{term arg}*}
-
-text {*------------ Argand -------------*}
-
-definition arg :: "complex => real" where
- "arg z = (SOME a. Re(sgn z) = cos a & Im(sgn z) = sin a & -pi < a & a \<le> pi)"
-
-lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
- by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute)
-
-lemma i_mult_eq: "ii * ii = complex_of_real (-1)"
- by (simp add: i_def complex_of_real_def)
-
-lemma i_mult_eq2 [simp]: "ii * ii = -(1::complex)"
- by (simp add: i_def complex_one_def)
-
-lemma complex_eq_cancel_iff2 [simp]:
- shows "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
- by (simp add: complex_of_real_def)
-
-lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
- by (simp add: complex_sgn_def divide_inverse)
-
-lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
- by (simp add: complex_sgn_def divide_inverse)
-
-lemma complex_inverse_complex_split:
- "inverse(complex_of_real x + ii * complex_of_real y) =
- complex_of_real(x/(x ^ 2 + y ^ 2)) -
- ii * complex_of_real(y/(x ^ 2 + y ^ 2))"
- by (simp add: complex_of_real_def i_def diff_minus divide_inverse)
-
-(*----------------------------------------------------------------------------*)
-(* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *)
-(* many of the theorems are not used - so should they be kept? *)
-(*----------------------------------------------------------------------------*)
-
-lemma cos_arg_i_mult_zero_pos:
- "0 < y ==> cos (arg(Complex 0 y)) = 0"
-apply (simp add: arg_def abs_if)
-apply (rule_tac a = "pi/2" in someI2, auto)
-apply (rule order_less_trans [of _ 0], auto)
-done
-
-lemma cos_arg_i_mult_zero_neg:
- "y < 0 ==> cos (arg(Complex 0 y)) = 0"
-apply (simp add: arg_def abs_if)
-apply (rule_tac a = "- pi/2" in someI2, auto)
-apply (rule order_trans [of _ 0], auto)
-done
-
-lemma cos_arg_i_mult_zero [simp]:
- "y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0"
-by (auto simp add: linorder_neq_iff cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg)
-
-
subsection{*Finally! Polar Form for Complex Numbers*}
-text {* An abbreviation for @{text "cos a + i sin a"}. *}
+subsubsection {* $\cos \theta + i \sin \theta$ *}
definition cis :: "real \<Rightarrow> complex" where
"cis a = Complex (cos a) (sin a)"
-text {* An abbreviation for @{text "r(cos a + i sin a)"}. *}
+lemma Re_cis [simp]: "Re (cis a) = cos a"
+ by (simp add: cis_def)
+
+lemma Im_cis [simp]: "Im (cis a) = sin a"
+ by (simp add: cis_def)
+
+lemma cis_zero [simp]: "cis 0 = 1"
+ by (simp add: cis_def)
+
+lemma norm_cis [simp]: "norm (cis a) = 1"
+ by (simp add: cis_def)
+
+lemma sgn_cis [simp]: "sgn (cis a) = cis a"
+ by (simp add: sgn_div_norm)
+
+lemma cis_neq_zero [simp]: "cis a \<noteq> 0"
+ by (metis norm_cis norm_zero zero_neq_one)
+
+lemma cis_mult: "cis a * cis b = cis (a + b)"
+ by (simp add: cis_def cos_add sin_add)
+
+lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
+ by (induct n, simp_all add: real_of_nat_Suc algebra_simps cis_mult)
+
+lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
+ by (simp add: cis_def)
+
+lemma cis_divide: "cis a / cis b = cis (a - b)"
+ by (simp add: complex_divide_def cis_mult diff_minus)
+
+lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
+ by (auto simp add: DeMoivre)
+
+lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
+ by (auto simp add: DeMoivre)
+
+subsubsection {* $r(\cos \theta + i \sin \theta)$ *}
definition rcis :: "[real, real] \<Rightarrow> complex" where
"rcis r a = complex_of_real r * cis a"
+lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
+ by (simp add: rcis_def)
+
+lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
+ by (simp add: rcis_def)
+
+lemma rcis_Ex: "\<exists>r a. z = rcis r a"
+ by (simp add: complex_eq_iff polar_Ex)
+
+lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
+ by (simp add: rcis_def norm_mult)
+
+lemma cis_rcis_eq: "cis a = rcis 1 a"
+ by (simp add: rcis_def)
+
+lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
+ by (simp add: rcis_def cis_mult)
+
+lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
+ by (simp add: rcis_def)
+
+lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
+ by (simp add: rcis_def)
+
+lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \<longleftrightarrow> r = 0"
+ by (simp add: rcis_def)
+
+lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
+ by (simp add: rcis_def power_mult_distrib DeMoivre)
+
+lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
+ by (simp add: divide_inverse rcis_def)
+
+lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
+ by (simp add: rcis_def cis_divide [symmetric])
+
+subsubsection {* Complex exponential *}
+
abbreviation expi :: "complex \<Rightarrow> complex"
where "expi \<equiv> exp"
@@ -616,104 +656,11 @@
lemma expi_def: "expi z = complex_of_real (exp (Re z)) * cis (Im z)"
unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by simp
-lemma complex_split_polar:
- "\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))"
-apply (induct z)
-apply (auto simp add: polar_Ex complex_of_real_mult_Complex)
-done
-
-lemma rcis_Ex: "\<exists>r a. z = rcis r a"
-apply (induct z)
-apply (simp add: rcis_def cis_def polar_Ex complex_of_real_mult_Complex)
-done
-
-lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
- by (simp add: rcis_def cis_def)
-
-lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
- by (simp add: rcis_def cis_def)
-
-lemma sin_cos_squared_add2_mult: "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior>"
-proof -
- have "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior> * ((cos a)\<twosuperior> + (sin a)\<twosuperior>)"
- by (simp only: power_mult_distrib right_distrib)
- thus ?thesis by simp
-qed
-
-lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
- by (simp add: rcis_def cis_def sin_cos_squared_add2_mult)
-
-lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
- by (simp add: cmod_def power2_eq_square)
-
-lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
- by simp
-
-lemma cis_rcis_eq: "cis a = rcis 1 a"
- by (simp add: rcis_def)
-
-lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
- by (simp add: rcis_def cis_def cos_add sin_add right_distrib
- right_diff_distrib complex_of_real_def)
-
-lemma cis_mult: "cis a * cis b = cis (a + b)"
- by (simp add: cis_rcis_eq rcis_mult)
-
-lemma cis_zero [simp]: "cis 0 = 1"
- by (simp add: cis_def complex_one_def)
-
-lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
- by (simp add: rcis_def)
+lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)"
+ unfolding expi_def by simp
-lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
- by (simp add: rcis_def)
-
-lemma complex_of_real_minus_one:
- "complex_of_real (-(1::real)) = -(1::complex)"
- by (simp add: complex_of_real_def complex_one_def)
-
-lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
- by (simp add: mult_assoc [symmetric])
-
-
-lemma cis_real_of_nat_Suc_mult:
- "cis (real (Suc n) * a) = cis a * cis (real n * a)"
- by (simp add: cis_def real_of_nat_Suc left_distrib cos_add sin_add right_distrib)
-
-lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
-apply (induct_tac "n")
-apply (auto simp add: cis_real_of_nat_Suc_mult)
-done
-
-lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
- by (simp add: rcis_def power_mult_distrib DeMoivre)
-
-lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
- by (simp add: cis_def complex_inverse_complex_split diff_minus)
-
-lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
- by (simp add: divide_inverse rcis_def)
-
-lemma cis_divide: "cis a / cis b = cis (a - b)"
- by (simp add: complex_divide_def cis_mult diff_minus)
-
-lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
-apply (simp add: complex_divide_def)
-apply (case_tac "r2=0", simp)
-apply (simp add: rcis_inverse rcis_mult diff_minus)
-done
-
-lemma Re_cis [simp]: "Re(cis a) = cos a"
- by (simp add: cis_def)
-
-lemma Im_cis [simp]: "Im(cis a) = sin a"
- by (simp add: cis_def)
-
-lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
- by (auto simp add: DeMoivre)
-
-lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
- by (auto simp add: DeMoivre)
+lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)"
+ unfolding expi_def by simp
lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"
apply (insert rcis_Ex [of z])
@@ -724,6 +671,85 @@
lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"
by (simp add: expi_def cis_def)
+subsubsection {* Complex argument *}
+
+definition arg :: "complex \<Rightarrow> real" where
+ "arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi))"
+
+lemma arg_zero: "arg 0 = 0"
+ by (simp add: arg_def)
+
+lemma of_nat_less_of_int_iff: (* TODO: move *)
+ "(of_nat n :: 'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x"
+ by (metis of_int_of_nat_eq of_int_less_iff)
+
+lemma real_of_nat_less_number_of_iff [simp]: (* TODO: move *)
+ "real (n::nat) < number_of w \<longleftrightarrow> n < number_of w"
+ unfolding real_of_nat_def nat_number_of_def number_of_eq
+ by (simp add: of_nat_less_of_int_iff zless_nat_eq_int_zless)
+
+lemma arg_unique:
+ assumes "sgn z = cis x" and "-pi < x" and "x \<le> pi"
+ shows "arg z = x"
+proof -
+ from assms have "z \<noteq> 0" by auto
+ have "(SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi) = x"
+ proof
+ fix a def d \<equiv> "a - x"
+ assume a: "sgn z = cis a \<and> - pi < a \<and> a \<le> pi"
+ from a assms have "- (2*pi) < d \<and> d < 2*pi"
+ unfolding d_def by simp
+ moreover from a assms have "cos a = cos x" and "sin a = sin x"
+ by (simp_all add: complex_eq_iff)
+ hence "cos d = 1" unfolding d_def cos_diff by simp
+ moreover hence "sin d = 0" by (rule cos_one_sin_zero)
+ ultimately have "d = 0"
+ unfolding sin_zero_iff even_mult_two_ex
+ by (safe, auto simp add: numeral_2_eq_2 less_Suc_eq)
+ thus "a = x" unfolding d_def by simp
+ qed (simp add: assms del: Re_sgn Im_sgn)
+ with `z \<noteq> 0` show "arg z = x"
+ unfolding arg_def by simp
+qed
+
+lemma arg_correct:
+ assumes "z \<noteq> 0" shows "sgn z = cis (arg z) \<and> -pi < arg z \<and> arg z \<le> pi"
+proof (simp add: arg_def assms, rule someI_ex)
+ obtain r a where z: "z = rcis r a" using rcis_Ex by fast
+ with assms have "r \<noteq> 0" by auto
+ def b \<equiv> "if 0 < r then a else a + pi"
+ have b: "sgn z = cis b"
+ unfolding z b_def rcis_def using `r \<noteq> 0`
+ by (simp add: of_real_def sgn_scaleR sgn_if, simp add: cis_def)
+ have cis_2pi_nat: "\<And>n. cis (2 * pi * real_of_nat n) = 1"
+ by (induct_tac n, simp_all add: right_distrib cis_mult [symmetric],
+ simp add: cis_def)
+ have cis_2pi_int: "\<And>x. cis (2 * pi * real_of_int x) = 1"
+ by (case_tac x rule: int_diff_cases,
+ simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)
+ def c \<equiv> "b - 2*pi * of_int \<lceil>(b - pi) / (2*pi)\<rceil>"
+ have "sgn z = cis c"
+ unfolding b c_def
+ by (simp add: cis_divide [symmetric] cis_2pi_int)
+ moreover have "- pi < c \<and> c \<le> pi"
+ using ceiling_correct [of "(b - pi) / (2*pi)"]
+ by (simp add: c_def less_divide_eq divide_le_eq algebra_simps)
+ ultimately show "\<exists>a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi" by fast
+qed
+
+lemma arg_bounded: "- pi < arg z \<and> arg z \<le> pi"
+ by (cases "z = 0", simp_all add: arg_zero arg_correct)
+
+lemma cis_arg: "z \<noteq> 0 \<Longrightarrow> cis (arg z) = sgn z"
+ by (simp add: arg_correct)
+
+lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z"
+ by (cases "z = 0", simp_all add: rcis_def cis_arg sgn_div_norm of_real_def)
+
+lemma cos_arg_i_mult_zero [simp]:
+ "y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0"
+ using cis_arg [of "Complex 0 y"] by (simp add: complex_eq_iff)
+
text {* Legacy theorem names *}
lemmas expand_complex_eq = complex_eq_iff
--- a/src/HOL/Decision_Procs/Approximation.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Decision_Procs/Approximation.thy Fri Sep 09 06:47:14 2011 +0200
@@ -295,7 +295,7 @@
unfolding real_sqrt_mult[of _ "pow2 (?E mod 2)"] real_sqrt_abs2 by auto
also have "\<dots> < pow2 (?E div 2) * 2"
by (rule mult_strict_left_mono, auto intro: E_mod_pow)
- also have "\<dots> = pow2 (?E div 2 + 1)" unfolding zadd_commute[of _ 1] pow2_add1 by auto
+ also have "\<dots> = pow2 (?E div 2 + 1)" unfolding add_commute[of _ 1] pow2_add1 by auto
finally show ?thesis by auto
qed
finally show ?thesis
--- a/src/HOL/Decision_Procs/Commutative_Ring_Complete.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Decision_Procs/Commutative_Ring_Complete.thy Fri Sep 09 06:47:14 2011 +0200
@@ -12,8 +12,7 @@
begin
text {* Formalization of normal form *}
-fun
- isnorm :: "('a::{comm_ring}) pol \<Rightarrow> bool"
+fun isnorm :: "'a::comm_ring pol \<Rightarrow> bool"
where
"isnorm (Pc c) \<longleftrightarrow> True"
| "isnorm (Pinj i (Pc c)) \<longleftrightarrow> False"
@@ -26,35 +25,40 @@
| "isnorm (PX P i Q) \<longleftrightarrow> isnorm P \<and> isnorm Q"
(* Some helpful lemmas *)
-lemma norm_Pinj_0_False:"isnorm (Pinj 0 P) = False"
-by(cases P, auto)
+lemma norm_Pinj_0_False: "isnorm (Pinj 0 P) = False"
+ by (cases P) auto
-lemma norm_PX_0_False:"isnorm (PX (Pc 0) i Q) = False"
-by(cases i, auto)
+lemma norm_PX_0_False: "isnorm (PX (Pc 0) i Q) = False"
+ by (cases i) auto
-lemma norm_Pinj:"isnorm (Pinj i Q) \<Longrightarrow> isnorm Q"
-by(cases i,simp add: norm_Pinj_0_False norm_PX_0_False,cases Q) auto
+lemma norm_Pinj: "isnorm (Pinj i Q) \<Longrightarrow> isnorm Q"
+ by (cases i) (simp add: norm_Pinj_0_False norm_PX_0_False, cases Q, auto)
-lemma norm_PX2:"isnorm (PX P i Q) \<Longrightarrow> isnorm Q"
-by(cases i, auto, cases P, auto, case_tac pol2, auto)
+lemma norm_PX2: "isnorm (PX P i Q) \<Longrightarrow> isnorm Q"
+ by (cases i) (auto, cases P, auto, case_tac pol2, auto)
+
+lemma norm_PX1: "isnorm (PX P i Q) \<Longrightarrow> isnorm P"
+ by (cases i) (auto, cases P, auto, case_tac pol2, auto)
-lemma norm_PX1:"isnorm (PX P i Q) \<Longrightarrow> isnorm P"
-by(cases i, auto, cases P, auto, case_tac pol2, auto)
-
-lemma mkPinj_cn:"\<lbrakk>y~=0; isnorm Q\<rbrakk> \<Longrightarrow> isnorm (mkPinj y Q)"
-apply(auto simp add: mkPinj_def norm_Pinj_0_False split: pol.split)
-apply(case_tac nat, auto simp add: norm_Pinj_0_False)
-by(case_tac pol, auto) (case_tac y, auto)
+lemma mkPinj_cn: "y ~= 0 \<Longrightarrow> isnorm Q \<Longrightarrow> isnorm (mkPinj y Q)"
+ apply (auto simp add: mkPinj_def norm_Pinj_0_False split: pol.split)
+ apply (case_tac nat, auto simp add: norm_Pinj_0_False)
+ apply (case_tac pol, auto)
+ apply (case_tac y, auto)
+ done
lemma norm_PXtrans:
- assumes A:"isnorm (PX P x Q)" and "isnorm Q2"
+ assumes A: "isnorm (PX P x Q)" and "isnorm Q2"
shows "isnorm (PX P x Q2)"
-proof(cases P)
- case (PX p1 y p2) with assms show ?thesis by(cases x, auto, cases p2, auto)
+proof (cases P)
+ case (PX p1 y p2)
+ with assms show ?thesis by (cases x) (auto, cases p2, auto)
next
- case Pc with assms show ?thesis by (cases x) auto
+ case Pc
+ with assms show ?thesis by (cases x) auto
next
- case Pinj with assms show ?thesis by (cases x) auto
+ case Pinj
+ with assms show ?thesis by (cases x) auto
qed
lemma norm_PXtrans2:
@@ -62,7 +66,7 @@
shows "isnorm (PX P (Suc (n+x)) Q2)"
proof (cases P)
case (PX p1 y p2)
- with assms show ?thesis by (cases x, auto, cases p2, auto)
+ with assms show ?thesis by (cases x) (auto, cases p2, auto)
next
case Pc
with assms show ?thesis by (cases x) auto
@@ -83,27 +87,33 @@
with assms show ?thesis by (cases x) (auto simp add: mkPinj_cn mkPX_def)
next
case (PX P1 y P2)
- with assms have Y0: "y>0" by (cases y) auto
+ with assms have Y0: "y > 0" by (cases y) auto
from assms PX have "isnorm P1" "isnorm P2"
by (auto simp add: norm_PX1[of P1 y P2] norm_PX2[of P1 y P2])
from assms PX Y0 show ?thesis
- by (cases x, auto simp add: mkPX_def norm_PXtrans2[of P1 y _ Q _], cases P2, auto)
+ by (cases x) (auto simp add: mkPX_def norm_PXtrans2[of P1 y _ Q _], cases P2, auto)
qed
text {* add conserves normalizedness *}
-lemma add_cn:"isnorm P \<Longrightarrow> isnorm Q \<Longrightarrow> isnorm (P \<oplus> Q)"
-proof(induct P Q rule: add.induct)
- case (2 c i P2) thus ?case by (cases P2, simp_all, cases i, simp_all)
+lemma add_cn: "isnorm P \<Longrightarrow> isnorm Q \<Longrightarrow> isnorm (P \<oplus> Q)"
+proof (induct P Q rule: add.induct)
+ case (2 c i P2)
+ thus ?case by (cases P2) (simp_all, cases i, simp_all)
next
- case (3 i P2 c) thus ?case by (cases P2, simp_all, cases i, simp_all)
+ case (3 i P2 c)
+ thus ?case by (cases P2) (simp_all, cases i, simp_all)
next
case (4 c P2 i Q2)
- then have "isnorm P2" "isnorm Q2" by (auto simp only: norm_PX1[of P2 i Q2] norm_PX2[of P2 i Q2])
- with 4 show ?case by(cases i, simp, cases P2, auto, case_tac pol2, auto)
+ then have "isnorm P2" "isnorm Q2"
+ by (auto simp only: norm_PX1[of P2 i Q2] norm_PX2[of P2 i Q2])
+ with 4 show ?case
+ by (cases i) (simp, cases P2, auto, case_tac pol2, auto)
next
case (5 P2 i Q2 c)
- then have "isnorm P2" "isnorm Q2" by (auto simp only: norm_PX1[of P2 i Q2] norm_PX2[of P2 i Q2])
- with 5 show ?case by(cases i, simp, cases P2, auto, case_tac pol2, auto)
+ then have "isnorm P2" "isnorm Q2"
+ by (auto simp only: norm_PX1[of P2 i Q2] norm_PX2[of P2 i Q2])
+ with 5 show ?case
+ by (cases i) (simp, cases P2, auto, case_tac pol2, auto)
next
case (6 x P2 y Q2)
then have Y0: "y>0" by (cases y) (auto simp add: norm_Pinj_0_False)
@@ -115,14 +125,17 @@
moreover
note 6 X0
moreover
- from 6 have "isnorm P2" "isnorm Q2" by (auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
+ from 6 have "isnorm P2" "isnorm Q2"
+ by (auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
moreover
- from 6 `x < y` y have "isnorm (Pinj d Q2)" by (cases d, simp, cases Q2, auto)
+ from 6 `x < y` y have "isnorm (Pinj d Q2)"
+ by (cases d, simp, cases Q2, auto)
ultimately have ?case by (simp add: mkPinj_cn) }
moreover
{ assume "x=y"
moreover
- from 6 have "isnorm P2" "isnorm Q2" by(auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
+ from 6 have "isnorm P2" "isnorm Q2"
+ by (auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
moreover
note 6 Y0
moreover
@@ -133,30 +146,35 @@
moreover
note 6 Y0
moreover
- from 6 have "isnorm P2" "isnorm Q2" by (auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
+ from 6 have "isnorm P2" "isnorm Q2"
+ by (auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
moreover
- from 6 `x > y` x have "isnorm (Pinj d P2)" by (cases d, simp, cases P2, auto)
- ultimately have ?case by (simp add: mkPinj_cn)}
+ from 6 `x > y` x have "isnorm (Pinj d P2)"
+ by (cases d) (simp, cases P2, auto)
+ ultimately have ?case by (simp add: mkPinj_cn) }
ultimately show ?case by blast
next
case (7 x P2 Q2 y R)
- have "x=0 \<or> (x = 1) \<or> (x > 1)" by arith
+ have "x = 0 \<or> x = 1 \<or> x > 1" by arith
moreover
{ assume "x = 0"
with 7 have ?case by (auto simp add: norm_Pinj_0_False) }
moreover
{ assume "x = 1"
- from 7 have "isnorm R" "isnorm P2" by (auto simp add: norm_Pinj[of _ P2] norm_PX2[of Q2 y R])
+ from 7 have "isnorm R" "isnorm P2"
+ by (auto simp add: norm_Pinj[of _ P2] norm_PX2[of Q2 y R])
with 7 `x = 1` have "isnorm (R \<oplus> P2)" by simp
- with 7 `x = 1` have ?case by (simp add: norm_PXtrans[of Q2 y _]) }
+ with 7 `x = 1` have ?case
+ by (simp add: norm_PXtrans[of Q2 y _]) }
moreover
{ assume "x > 1" hence "EX d. x=Suc (Suc d)" by arith
- then obtain d where X:"x=Suc (Suc d)" ..
+ then obtain d where X: "x=Suc (Suc d)" ..
with 7 have NR: "isnorm R" "isnorm P2"
by (auto simp add: norm_Pinj[of _ P2] norm_PX2[of Q2 y R])
with 7 X have "isnorm (Pinj (x - 1) P2)" by (cases P2) auto
with 7 X NR have "isnorm (R \<oplus> Pinj (x - 1) P2)" by simp
- with `isnorm (PX Q2 y R)` X have ?case by (simp add: norm_PXtrans[of Q2 y _]) }
+ with `isnorm (PX Q2 y R)` X have ?case
+ by (simp add: norm_PXtrans[of Q2 y _]) }
ultimately show ?case by blast
next
case (8 Q2 y R x P2)
@@ -183,7 +201,7 @@
with 9 have X0: "x>0" by (cases x) auto
with 9 have NP1: "isnorm P1" and NP2: "isnorm P2"
by (auto simp add: norm_PX1[of P1 _ P2] norm_PX2[of P1 _ P2])
- with 9 have NQ1:"isnorm Q1" and NQ2: "isnorm Q2"
+ with 9 have NQ1: "isnorm Q1" and NQ2: "isnorm Q2"
by (auto simp add: norm_PX1[of Q1 _ Q2] norm_PX2[of Q1 _ Q2])
have "y < x \<or> x = y \<or> x < y" by arith
moreover
@@ -194,7 +212,7 @@
have "isnorm (PX P1 d (Pc 0))"
proof (cases P1)
case (PX p1 y p2)
- with 9 sm1 sm2 show ?thesis by - (cases d, simp, cases p2, auto)
+ with 9 sm1 sm2 show ?thesis by (cases d) (simp, cases p2, auto)
next
case Pc with 9 sm1 sm2 show ?thesis by (cases d) auto
next
@@ -214,35 +232,37 @@
have "isnorm (PX Q1 d (Pc 0))"
proof (cases Q1)
case (PX p1 y p2)
- with 9 sm1 sm2 show ?thesis by - (cases d, simp, cases p2, auto)
+ with 9 sm1 sm2 show ?thesis by (cases d) (simp, cases p2, auto)
next
case Pc with 9 sm1 sm2 show ?thesis by (cases d) auto
next
case Pinj with 9 sm1 sm2 show ?thesis by (cases d) auto
qed
ultimately have "isnorm (P2 \<oplus> Q2)" "isnorm (PX Q1 (y - x) (Pc 0) \<oplus> P1)" by auto
- with X0 sm1 sm2 have ?case by (simp add: mkPX_cn)}
+ with X0 sm1 sm2 have ?case by (simp add: mkPX_cn) }
ultimately show ?case by blast
qed simp
text {* mul concerves normalizedness *}
-lemma mul_cn :"isnorm P \<Longrightarrow> isnorm Q \<Longrightarrow> isnorm (P \<otimes> Q)"
-proof(induct P Q rule: mul.induct)
+lemma mul_cn: "isnorm P \<Longrightarrow> isnorm Q \<Longrightarrow> isnorm (P \<otimes> Q)"
+proof (induct P Q rule: mul.induct)
case (2 c i P2) thus ?case
- by (cases P2, simp_all) (cases "i",simp_all add: mkPinj_cn)
+ by (cases P2) (simp_all, cases i, simp_all add: mkPinj_cn)
next
case (3 i P2 c) thus ?case
- by (cases P2, simp_all) (cases "i",simp_all add: mkPinj_cn)
+ by (cases P2) (simp_all, cases i, simp_all add: mkPinj_cn)
next
case (4 c P2 i Q2)
- then have "isnorm P2" "isnorm Q2" by (auto simp only: norm_PX1[of P2 i Q2] norm_PX2[of P2 i Q2])
+ then have "isnorm P2" "isnorm Q2"
+ by (auto simp only: norm_PX1[of P2 i Q2] norm_PX2[of P2 i Q2])
with 4 show ?case
- by - (cases "c = 0", simp_all, cases "i = 0", simp_all add: mkPX_cn)
+ by (cases "c = 0") (simp_all, cases "i = 0", simp_all add: mkPX_cn)
next
case (5 P2 i Q2 c)
- then have "isnorm P2" "isnorm Q2" by (auto simp only: norm_PX1[of P2 i Q2] norm_PX2[of P2 i Q2])
+ then have "isnorm P2" "isnorm Q2"
+ by (auto simp only: norm_PX1[of P2 i Q2] norm_PX2[of P2 i Q2])
with 5 show ?case
- by - (cases "c = 0", simp_all, cases "i = 0", simp_all add: mkPX_cn)
+ by (cases "c = 0") (simp_all, cases "i = 0", simp_all add: mkPX_cn)
next
case (6 x P2 y Q2)
have "x < y \<or> x = y \<or> x > y" by arith
@@ -256,7 +276,7 @@
moreover
from 6 have "isnorm P2" "isnorm Q2" by (auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
moreover
- from 6 `x < y` y have "isnorm (Pinj d Q2)" by - (cases d, simp, cases Q2, auto)
+ from 6 `x < y` y have "isnorm (Pinj d Q2)" by (cases d) (simp, cases Q2, auto)
ultimately have ?case by (simp add: mkPinj_cn) }
moreover
{ assume "x = y"
@@ -278,7 +298,7 @@
moreover
from 6 have "isnorm P2" "isnorm Q2" by (auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
moreover
- from 6 `x > y` x have "isnorm (Pinj d P2)" by - (cases d, simp, cases P2, auto)
+ from 6 `x > y` x have "isnorm (Pinj d P2)" by (cases d) (simp, cases P2, auto)
ultimately have ?case by (simp add: mkPinj_cn) }
ultimately show ?case by blast
next
@@ -356,7 +376,7 @@
proof (induct P)
case (Pinj i P2)
then have "isnorm P2" by (simp add: norm_Pinj[of i P2])
- with Pinj show ?case by - (cases P2, auto, cases i, auto)
+ with Pinj show ?case by (cases P2) (auto, cases i, auto)
next
case (PX P1 x P2) note PX1 = this
from PX have "isnorm P2" "isnorm P1"
@@ -364,7 +384,7 @@
with PX show ?case
proof (cases P1)
case (PX p1 y p2)
- with PX1 show ?thesis by - (cases x, auto, cases p2, auto)
+ with PX1 show ?thesis by (cases x) (auto, cases p2, auto)
next
case Pinj
with PX1 show ?thesis by (cases x) auto
@@ -372,15 +392,18 @@
qed simp
text {* sub conserves normalizedness *}
-lemma sub_cn:"isnorm p \<Longrightarrow> isnorm q \<Longrightarrow> isnorm (p \<ominus> q)"
-by (simp add: sub_def add_cn neg_cn)
+lemma sub_cn: "isnorm p \<Longrightarrow> isnorm q \<Longrightarrow> isnorm (p \<ominus> q)"
+ by (simp add: sub_def add_cn neg_cn)
text {* sqr conserves normalizizedness *}
-lemma sqr_cn:"isnorm P \<Longrightarrow> isnorm (sqr P)"
+lemma sqr_cn: "isnorm P \<Longrightarrow> isnorm (sqr P)"
proof (induct P)
+ case Pc
+ then show ?case by simp
+next
case (Pinj i Q)
then show ?case
- by - (cases Q, auto simp add: mkPX_cn mkPinj_cn, cases i, auto simp add: mkPX_cn mkPinj_cn)
+ by (cases Q) (auto simp add: mkPX_cn mkPinj_cn, cases i, auto simp add: mkPX_cn mkPinj_cn)
next
case (PX P1 x P2)
then have "x + x ~= 0" "isnorm P2" "isnorm P1"
@@ -389,20 +412,23 @@
and "isnorm (mkPX (sqr P1) (x + x) (sqr P2))"
by (auto simp add: add_cn mkPX_cn mkPinj_cn mul_cn)
then show ?case by (auto simp add: add_cn mkPX_cn mkPinj_cn mul_cn)
-qed simp
+qed
text {* pow conserves normalizedness *}
-lemma pow_cn:"isnorm P \<Longrightarrow> isnorm (pow n P)"
-proof (induct n arbitrary: P rule: nat_less_induct)
- case (1 k)
+lemma pow_cn: "isnorm P \<Longrightarrow> isnorm (pow n P)"
+proof (induct n arbitrary: P rule: less_induct)
+ case (less k)
show ?case
proof (cases "k = 0")
+ case True
+ then show ?thesis by simp
+ next
case False
then have K2: "k div 2 < k" by (cases k) auto
- from 1 have "isnorm (sqr P)" by (simp add: sqr_cn)
- with 1 False K2 show ?thesis
- by - (simp add: allE[of _ "(k div 2)" _] allE[of _ "(sqr P)" _], cases k, auto simp add: mul_cn)
- qed simp
+ from less have "isnorm (sqr P)" by (simp add: sqr_cn)
+ with less False K2 show ?thesis
+ by (simp add: allE[of _ "(k div 2)" _] allE[of _ "(sqr P)" _], cases k, auto simp add: mul_cn)
+ qed
qed
end
--- a/src/HOL/Decision_Procs/Cooper.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Decision_Procs/Cooper.thy Fri Sep 09 06:47:14 2011 +0200
@@ -1318,7 +1318,7 @@
also have "\<dots> = (j dvd (- (c*x - ?e)))"
by (simp only: dvd_minus_iff)
also have "\<dots> = (j dvd (c* (- x)) + ?e)"
- apply (simp only: minus_mult_right[symmetric] minus_mult_left[symmetric] diff_minus zadd_ac zminus_zadd_distrib)
+ apply (simp only: minus_mult_right[symmetric] minus_mult_left[symmetric] diff_minus add_ac minus_add_distrib)
by (simp add: algebra_simps)
also have "\<dots> = Ifm bbs ((- x)#bs) (Dvd j (CN 0 c e))"
using numbound0_I[OF nb, where bs="bs" and b="x" and b'="- x"]
@@ -1330,7 +1330,7 @@
also have "\<dots> = (j dvd (- (c*x - ?e)))"
by (simp only: dvd_minus_iff)
also have "\<dots> = (j dvd (c* (- x)) + ?e)"
- apply (simp only: minus_mult_right[symmetric] minus_mult_left[symmetric] diff_minus zadd_ac zminus_zadd_distrib)
+ apply (simp only: minus_mult_right[symmetric] minus_mult_left[symmetric] diff_minus add_ac minus_add_distrib)
by (simp add: algebra_simps)
also have "\<dots> = Ifm bbs ((- x)#bs) (Dvd j (CN 0 c e))"
using numbound0_I[OF nb, where bs="bs" and b="x" and b'="- x"]
--- a/src/HOL/Decision_Procs/Ferrack.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Decision_Procs/Ferrack.thy Fri Sep 09 06:47:14 2011 +0200
@@ -676,13 +676,13 @@
{assume nz: "n = 0" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
moreover
{ assume nnz: "n \<noteq> 0"
- {assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simp_num_pair_def simpnum_ci) }
+ {assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simp_num_pair_def) }
moreover
{assume g1:"?g>1" hence g0: "?g > 0" by simp
from g1 nnz have gp0: "?g' \<noteq> 0" by simp
hence g'p: "?g' > 0" using gcd_ge_0_int[where x="n" and y="numgcd ?t'"] by arith
hence "?g'= 1 \<or> ?g' > 1" by arith
- moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simp_num_pair_def simpnum_ci)}
+ moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
moreover {assume g'1:"?g'>1"
from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff ?t' ?g" ..
let ?tt = "reducecoeffh ?t' ?g'"
@@ -800,32 +800,34 @@
proof(induct p rule: simpfm.induct)
case (6 a) hence nb: "numbound0 a" by simp
hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
- thus ?case by (cases "simpnum a", auto simp add: Let_def)
+ thus ?case by (cases "simpnum a") (auto simp add: Let_def)
next
case (7 a) hence nb: "numbound0 a" by simp
hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
- thus ?case by (cases "simpnum a", auto simp add: Let_def)
+ thus ?case by (cases "simpnum a") (auto simp add: Let_def)
next
case (8 a) hence nb: "numbound0 a" by simp
hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
- thus ?case by (cases "simpnum a", auto simp add: Let_def)
+ thus ?case by (cases "simpnum a") (auto simp add: Let_def)
next
case (9 a) hence nb: "numbound0 a" by simp
hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
- thus ?case by (cases "simpnum a", auto simp add: Let_def)
+ thus ?case by (cases "simpnum a") (auto simp add: Let_def)
next
case (10 a) hence nb: "numbound0 a" by simp
hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
- thus ?case by (cases "simpnum a", auto simp add: Let_def)
+ thus ?case by (cases "simpnum a") (auto simp add: Let_def)
next
case (11 a) hence nb: "numbound0 a" by simp
hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
- thus ?case by (cases "simpnum a", auto simp add: Let_def)
+ thus ?case by (cases "simpnum a") (auto simp add: Let_def)
qed(auto simp add: disj_def imp_def iff_def conj_def not_bn)
lemma simpfm_qf: "qfree p \<Longrightarrow> qfree (simpfm p)"
-by (induct p rule: simpfm.induct, auto simp add: disj_qf imp_qf iff_qf conj_qf not_qf Let_def)
- (case_tac "simpnum a",auto)+
+ apply (induct p rule: simpfm.induct)
+ apply (auto simp add: Let_def)
+ apply (case_tac "simpnum a", auto)+
+ done
consts prep :: "fm \<Rightarrow> fm"
recdef prep "measure fmsize"
@@ -854,7 +856,7 @@
"prep p = p"
(hints simp add: fmsize_pos)
lemma prep: "\<And> bs. Ifm bs (prep p) = Ifm bs p"
-by (induct p rule: prep.induct, auto)
+ by (induct p rule: prep.induct) auto
(* Generic quantifier elimination *)
function (sequential) qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm" where
@@ -1037,7 +1039,7 @@
assumes qfp: "qfree p"
shows "(Ifm bs (rlfm p) = Ifm bs p) \<and> isrlfm (rlfm p)"
using qfp
-by (induct p rule: rlfm.induct, auto simp add: lt le gt ge eq neq conj disj conj_lin disj_lin)
+by (induct p rule: rlfm.induct) (auto simp add: lt le gt ge eq neq conj disj conj_lin disj_lin)
(* Operations needed for Ferrante and Rackoff *)
lemma rminusinf_inf:
@@ -1045,9 +1047,11 @@
shows "\<exists> z. \<forall> x < z. Ifm (x#bs) (minusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p")
using lp
proof (induct p rule: minusinf.induct)
- case (1 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto
+ case (1 p q)
+ thus ?case apply auto apply (rule_tac x= "min z za" in exI) apply auto done
next
- case (2 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto
+ case (2 p q)
+ thus ?case apply auto apply (rule_tac x= "min z za" in exI) apply auto done
next
case (3 c e)
from 3 have nb: "numbound0 e" by simp
--- a/src/HOL/Divides.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Divides.thy Fri Sep 09 06:47:14 2011 +0200
@@ -1732,16 +1732,16 @@
1 div z and 1 mod z **)
lemmas div_pos_pos_1_number_of [simp] =
- div_pos_pos [OF int_0_less_1, of "number_of w", standard]
+ div_pos_pos [OF zero_less_one, of "number_of w", standard]
lemmas div_pos_neg_1_number_of [simp] =
- div_pos_neg [OF int_0_less_1, of "number_of w", standard]
+ div_pos_neg [OF zero_less_one, of "number_of w", standard]
lemmas mod_pos_pos_1_number_of [simp] =
- mod_pos_pos [OF int_0_less_1, of "number_of w", standard]
+ mod_pos_pos [OF zero_less_one, of "number_of w", standard]
lemmas mod_pos_neg_1_number_of [simp] =
- mod_pos_neg [OF int_0_less_1, of "number_of w", standard]
+ mod_pos_neg [OF zero_less_one, of "number_of w", standard]
lemmas posDivAlg_eqn_1_number_of [simp] =
@@ -1790,7 +1790,7 @@
apply (subgoal_tac "b*q = r' - r + b'*q'")
prefer 2 apply simp
apply (simp (no_asm_simp) add: right_distrib)
-apply (subst add_commute, rule zadd_zless_mono, arith)
+apply (subst add_commute, rule add_less_le_mono, arith)
apply (rule mult_right_mono, auto)
done
--- a/src/HOL/Finite_Set.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Finite_Set.thy Fri Sep 09 06:47:14 2011 +0200
@@ -486,22 +486,9 @@
end
-instance unit :: finite proof
-qed (simp add: UNIV_unit)
-
-instance bool :: finite proof
-qed (simp add: UNIV_bool)
-
instance prod :: (finite, finite) finite proof
qed (simp only: UNIV_Times_UNIV [symmetric] finite_cartesian_product finite)
-lemma finite_option_UNIV [simp]:
- "finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)"
- by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some)
-
-instance option :: (finite) finite proof
-qed (simp add: UNIV_option_conv)
-
lemma inj_graph: "inj (%f. {(x, y). y = f x})"
by (rule inj_onI, auto simp add: set_eq_iff fun_eq_iff)
@@ -519,9 +506,22 @@
qed
qed
+instance bool :: finite proof
+qed (simp add: UNIV_bool)
+
+instance unit :: finite proof
+qed (simp add: UNIV_unit)
+
instance sum :: (finite, finite) finite proof
qed (simp only: UNIV_Plus_UNIV [symmetric] finite_Plus finite)
+lemma finite_option_UNIV [simp]:
+ "finite (UNIV :: 'a option set) = finite (UNIV :: 'a set)"
+ by (auto simp add: UNIV_option_conv elim: finite_imageD intro: inj_Some)
+
+instance option :: (finite) finite proof
+qed (simp add: UNIV_option_conv)
+
subsection {* A basic fold functional for finite sets *}
--- a/src/HOL/GCD.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/GCD.thy Fri Sep 09 06:47:14 2011 +0200
@@ -485,16 +485,16 @@
done
lemma gcd_add1_int [simp]: "gcd ((m::int) + n) n = gcd m n"
-by (metis gcd_red_int mod_add_self1 zadd_commute)
+by (metis gcd_red_int mod_add_self1 add_commute)
lemma gcd_add2_int [simp]: "gcd m ((m::int) + n) = gcd m n"
-by (metis gcd_add1_int gcd_commute_int zadd_commute)
+by (metis gcd_add1_int gcd_commute_int add_commute)
lemma gcd_add_mult_nat: "gcd (m::nat) (k * m + n) = gcd m n"
by (metis mod_mult_self3 gcd_commute_nat gcd_red_nat)
lemma gcd_add_mult_int: "gcd (m::int) (k * m + n) = gcd m n"
-by (metis gcd_commute_int gcd_red_int mod_mult_self1 zadd_commute)
+by (metis gcd_commute_int gcd_red_int mod_mult_self1 add_commute)
(* to do: differences, and all variations of addition rules
@@ -1030,8 +1030,7 @@
apply (subst mod_div_equality [of m n, symmetric])
(* applying simp here undoes the last substitution!
what is procedure cancel_div_mod? *)
- apply (simp only: field_simps zadd_int [symmetric]
- zmult_int [symmetric])
+ apply (simp only: field_simps of_nat_add of_nat_mult)
done
qed
@@ -1237,7 +1236,7 @@
lemma lcm_altdef_int [code]: "lcm (a::int) b = (abs a) * (abs b) div gcd a b"
by (simp add: lcm_int_def lcm_nat_def zdiv_int
- zmult_int [symmetric] gcd_int_def)
+ of_nat_mult gcd_int_def)
lemma prod_gcd_lcm_nat: "(m::nat) * n = gcd m n * lcm m n"
unfolding lcm_nat_def
@@ -1452,7 +1451,7 @@
by (metis lcm_0_left_nat lcm_0_nat mult_is_0 prod_gcd_lcm_nat)
lemma lcm_0_iff_int[simp]: "lcm (m::int) n = 0 \<longleftrightarrow> m=0 \<or> n=0"
-by (metis lcm_0_int lcm_0_left_int lcm_pos_int zless_le)
+by (metis lcm_0_int lcm_0_left_int lcm_pos_int less_le)
lemma lcm_1_iff_nat[simp]: "lcm (m::nat) n = 1 \<longleftrightarrow> m=1 \<and> n=1"
by (metis gcd_1_nat lcm_unique_nat nat_mult_1 prod_gcd_lcm_nat)
@@ -1464,17 +1463,17 @@
subsubsection {* The complete divisibility lattice *}
-interpretation gcd_semilattice_nat: semilattice_inf "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)" gcd
+interpretation gcd_semilattice_nat: semilattice_inf gcd "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)"
proof
case goal3 thus ?case by(metis gcd_unique_nat)
qed auto
-interpretation lcm_semilattice_nat: semilattice_sup "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)" lcm
+interpretation lcm_semilattice_nat: semilattice_sup lcm "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)"
proof
case goal3 thus ?case by(metis lcm_unique_nat)
qed auto
-interpretation gcd_lcm_lattice_nat: lattice "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)" gcd lcm ..
+interpretation gcd_lcm_lattice_nat: lattice gcd "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)" lcm ..
text{* Lifting gcd and lcm to finite (Gcd/Lcm) and infinite sets (GCD/LCM).
GCD is defined via LCM to facilitate the proof that we have a complete lattice.
@@ -1580,7 +1579,7 @@
qed
interpretation gcd_lcm_complete_lattice_nat:
- complete_lattice GCD LCM "op dvd" "%m n::nat. m dvd n & ~ n dvd m" gcd lcm 1 0
+ complete_lattice GCD LCM gcd "op dvd" "%m n::nat. m dvd n & ~ n dvd m" lcm 1 0
proof
case goal1 show ?case by simp
next
--- a/src/HOL/Groups.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Groups.thy Fri Sep 09 06:47:14 2011 +0200
@@ -596,6 +596,14 @@
"min x y + z = min (x + z) (y + z)"
unfolding min_def by auto
+lemma max_add_distrib_right:
+ "x + max y z = max (x + y) (x + z)"
+ unfolding max_def by auto
+
+lemma min_add_distrib_right:
+ "x + min y z = min (x + y) (x + z)"
+ unfolding min_def by auto
+
end
subsection {* Support for reasoning about signs *}
--- a/src/HOL/HOLCF/Representable.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/HOLCF/Representable.thy Fri Sep 09 06:47:14 2011 +0200
@@ -5,7 +5,7 @@
header {* Representable domains *}
theory Representable
-imports Algebraic Map_Functions Countable
+imports Algebraic Map_Functions "~~/src/HOL/Library/Countable"
begin
default_sort cpo
--- a/src/HOL/Imperative_HOL/Heap_Monad.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Imperative_HOL/Heap_Monad.thy Fri Sep 09 06:47:14 2011 +0200
@@ -628,7 +628,9 @@
val dummy_case_term = IVar NONE;
(*assumption: dummy values are not relevant for serialization*)
val (unitt, unitT) = case lookup_const naming @{const_name Unity}
- of SOME unit' => (IConst (unit', (([], []), [])), the (lookup_tyco naming @{type_name unit}) `%% [])
+ of SOME unit' =>
+ let val unitT = the (lookup_tyco naming @{type_name unit}) `%% []
+ in (IConst (unit', ((([], []), ([], unitT)), false)), unitT) end
| NONE => error ("Must include " ^ @{const_name Unity} ^ " in generated constants.");
fun dest_abs ((v, ty) `|=> t, _) = ((v, ty), t)
| dest_abs (t, ty) =
@@ -645,13 +647,13 @@
val ((v, ty), t) = dest_abs (t2, ty2);
in ICase (((force t1, ty), [(IVar v, tr_bind' t)]), dummy_case_term) end
and tr_bind' t = case unfold_app t
- of (IConst (c, (_, ty1 :: ty2 :: _)), [x1, x2]) => if is_bind c
+ of (IConst (c, ((_, (ty1 :: ty2 :: _, _)), _)), [x1, x2]) => if is_bind c
then tr_bind'' [(x1, ty1), (x2, ty2)]
else force t
| _ => force t;
fun imp_monad_bind'' ts = (SOME dummy_name, unitT) `|=> ICase (((IVar (SOME dummy_name), unitT),
[(unitt, tr_bind'' ts)]), dummy_case_term)
- fun imp_monad_bind' (const as (c, (_, tys))) ts = if is_bind c then case (ts, tys)
+ fun imp_monad_bind' (const as (c, ((_, (tys, _)), _))) ts = if is_bind c then case (ts, tys)
of ([t1, t2], ty1 :: ty2 :: _) => imp_monad_bind'' [(t1, ty1), (t2, ty2)]
| ([t1, t2, t3], ty1 :: ty2 :: _) => imp_monad_bind'' [(t1, ty1), (t2, ty2)] `$ t3
| (ts, _) => imp_monad_bind (eta_expand 2 (const, ts))
--- a/src/HOL/Import/Generate-HOLLight/GenHOLLight.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Import/Generate-HOLLight/GenHOLLight.thy Fri Sep 09 06:47:14 2011 +0200
@@ -145,9 +145,9 @@
GEQ > HOL.eq
GSPEC > Set.Collect
SETSPEC > HOLLightCompat.SETSPEC
- UNION > Lattices.semilattice_sup_class.sup :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
+ UNION > Lattices.sup_class.sup :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
UNIONS > Complete_Lattice.Sup_class.Sup :: "(('a \<Rightarrow> bool) \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
- INTER > Lattices.semilattice_inf_class.inf :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
+ INTER > Lattices.inf_class.inf :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
INTERS > Complete_Lattice.Inf_class.Inf :: "(('a \<Rightarrow> bool) \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
DIFF > Groups.minus_class.minus :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
SUBSET > Orderings.ord_class.less_eq :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
--- a/src/HOL/Import/HOL/HOL4Base.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Import/HOL/HOL4Base.thy Fri Sep 09 06:47:14 2011 +0200
@@ -4,277 +4,225 @@
;setup_theory bool
-definition ARB :: "'a" where
- "ARB == SOME x::'a::type. True"
-
-lemma ARB_DEF: "ARB = (SOME x::'a::type. True)"
- by (import bool ARB_DEF)
-
-definition IN :: "'a => ('a => bool) => bool" where
- "IN == %(x::'a::type) f::'a::type => bool. f x"
-
-lemma IN_DEF: "IN = (%(x::'a::type) f::'a::type => bool. f x)"
- by (import bool IN_DEF)
-
-definition RES_FORALL :: "('a => bool) => ('a => bool) => bool" where
- "RES_FORALL ==
-%(p::'a::type => bool) m::'a::type => bool. ALL x::'a::type. IN x p --> m x"
-
-lemma RES_FORALL_DEF: "RES_FORALL =
-(%(p::'a::type => bool) m::'a::type => bool.
- ALL x::'a::type. IN x p --> m x)"
- by (import bool RES_FORALL_DEF)
-
-definition RES_EXISTS :: "('a => bool) => ('a => bool) => bool" where
- "RES_EXISTS ==
-%(p::'a::type => bool) m::'a::type => bool. EX x::'a::type. IN x p & m x"
-
-lemma RES_EXISTS_DEF: "RES_EXISTS =
-(%(p::'a::type => bool) m::'a::type => bool. EX x::'a::type. IN x p & m x)"
- by (import bool RES_EXISTS_DEF)
-
-definition RES_EXISTS_UNIQUE :: "('a => bool) => ('a => bool) => bool" where
+definition
+ ARB :: "'a" where
+ "ARB == SOME x. True"
+
+lemma ARB_DEF: "ARB = (SOME x. True)"
+ sorry
+
+definition
+ IN :: "'a => ('a => bool) => bool" where
+ "IN == %x f. f x"
+
+lemma IN_DEF: "IN = (%x f. f x)"
+ sorry
+
+definition
+ RES_FORALL :: "('a => bool) => ('a => bool) => bool" where
+ "RES_FORALL == %p m. ALL x. IN x p --> m x"
+
+lemma RES_FORALL_DEF: "RES_FORALL = (%p m. ALL x. IN x p --> m x)"
+ sorry
+
+definition
+ RES_EXISTS :: "('a => bool) => ('a => bool) => bool" where
+ "RES_EXISTS == %p m. EX x. IN x p & m x"
+
+lemma RES_EXISTS_DEF: "RES_EXISTS = (%p m. EX x. IN x p & m x)"
+ sorry
+
+definition
+ RES_EXISTS_UNIQUE :: "('a => bool) => ('a => bool) => bool" where
"RES_EXISTS_UNIQUE ==
-%(p::'a::type => bool) m::'a::type => bool.
- RES_EXISTS p m &
- RES_FORALL p
- (%x::'a::type. RES_FORALL p (%y::'a::type. m x & m y --> x = y))"
+%p m. RES_EXISTS p m &
+ RES_FORALL p (%x. RES_FORALL p (%y. m x & m y --> x = y))"
lemma RES_EXISTS_UNIQUE_DEF: "RES_EXISTS_UNIQUE =
-(%(p::'a::type => bool) m::'a::type => bool.
- RES_EXISTS p m &
- RES_FORALL p
- (%x::'a::type. RES_FORALL p (%y::'a::type. m x & m y --> x = y)))"
- by (import bool RES_EXISTS_UNIQUE_DEF)
-
-definition RES_SELECT :: "('a => bool) => ('a => bool) => 'a" where
- "RES_SELECT ==
-%(p::'a::type => bool) m::'a::type => bool. SOME x::'a::type. IN x p & m x"
-
-lemma RES_SELECT_DEF: "RES_SELECT =
-(%(p::'a::type => bool) m::'a::type => bool. SOME x::'a::type. IN x p & m x)"
- by (import bool RES_SELECT_DEF)
-
-lemma EXCLUDED_MIDDLE: "ALL t::bool. t | ~ t"
- by (import bool EXCLUDED_MIDDLE)
-
-lemma FORALL_THM: "All (f::'a::type => bool) = All f"
- by (import bool FORALL_THM)
-
-lemma EXISTS_THM: "Ex (f::'a::type => bool) = Ex f"
- by (import bool EXISTS_THM)
-
-lemma F_IMP: "ALL t::bool. ~ t --> t --> False"
- by (import bool F_IMP)
-
-lemma NOT_AND: "~ ((t::bool) & ~ t)"
- by (import bool NOT_AND)
-
-lemma AND_CLAUSES: "ALL t::bool.
- (True & t) = t &
- (t & True) = t & (False & t) = False & (t & False) = False & (t & t) = t"
- by (import bool AND_CLAUSES)
-
-lemma OR_CLAUSES: "ALL t::bool.
- (True | t) = True &
- (t | True) = True & (False | t) = t & (t | False) = t & (t | t) = t"
- by (import bool OR_CLAUSES)
-
-lemma IMP_CLAUSES: "ALL t::bool.
- (True --> t) = t &
- (t --> True) = True &
- (False --> t) = True & (t --> t) = True & (t --> False) = (~ t)"
- by (import bool IMP_CLAUSES)
-
-lemma NOT_CLAUSES: "(ALL t::bool. (~ ~ t) = t) & (~ True) = False & (~ False) = True"
- by (import bool NOT_CLAUSES)
+(%p m. RES_EXISTS p m &
+ RES_FORALL p (%x. RES_FORALL p (%y. m x & m y --> x = y)))"
+ sorry
+
+definition
+ RES_SELECT :: "('a => bool) => ('a => bool) => 'a" where
+ "RES_SELECT == %p m. SOME x. IN x p & m x"
+
+lemma RES_SELECT_DEF: "RES_SELECT = (%p m. SOME x. IN x p & m x)"
+ sorry
+
+lemma EXCLUDED_MIDDLE: "t | ~ t"
+ sorry
+
+lemma FORALL_THM: "All f = All f"
+ sorry
+
+lemma EXISTS_THM: "Ex f = Ex f"
+ sorry
+
+lemma F_IMP: "[| ~ t; t |] ==> False"
+ sorry
+
+lemma NOT_AND: "~ (t & ~ t)"
+ sorry
+
+lemma AND_CLAUSES: "(True & t) = t &
+(t & True) = t & (False & t) = False & (t & False) = False & (t & t) = t"
+ sorry
+
+lemma OR_CLAUSES: "(True | t) = True &
+(t | True) = True & (False | t) = t & (t | False) = t & (t | t) = t"
+ sorry
+
+lemma IMP_CLAUSES: "(True --> t) = t &
+(t --> True) = True &
+(False --> t) = True & (t --> t) = True & (t --> False) = (~ t)"
+ sorry
+
+lemma NOT_CLAUSES: "(ALL t. (~ ~ t) = t) & (~ True) = False & (~ False) = True"
+ sorry
lemma BOOL_EQ_DISTINCT: "True ~= False & False ~= True"
- by (import bool BOOL_EQ_DISTINCT)
-
-lemma EQ_CLAUSES: "ALL t::bool.
- (True = t) = t &
- (t = True) = t & (False = t) = (~ t) & (t = False) = (~ t)"
- by (import bool EQ_CLAUSES)
-
-lemma COND_CLAUSES: "ALL (t1::'a::type) t2::'a::type.
- (if True then t1 else t2) = t1 & (if False then t1 else t2) = t2"
- by (import bool COND_CLAUSES)
-
-lemma SELECT_UNIQUE: "ALL (P::'a::type => bool) x::'a::type.
- (ALL y::'a::type. P y = (y = x)) --> Eps P = x"
- by (import bool SELECT_UNIQUE)
-
-lemma BOTH_EXISTS_AND_THM: "ALL (P::bool) Q::bool.
- (EX x::'a::type. P & Q) = ((EX x::'a::type. P) & (EX x::'a::type. Q))"
- by (import bool BOTH_EXISTS_AND_THM)
-
-lemma BOTH_FORALL_OR_THM: "ALL (P::bool) Q::bool.
- (ALL x::'a::type. P | Q) = ((ALL x::'a::type. P) | (ALL x::'a::type. Q))"
- by (import bool BOTH_FORALL_OR_THM)
-
-lemma BOTH_FORALL_IMP_THM: "ALL (P::bool) Q::bool.
- (ALL x::'a::type. P --> Q) =
- ((EX x::'a::type. P) --> (ALL x::'a::type. Q))"
- by (import bool BOTH_FORALL_IMP_THM)
-
-lemma BOTH_EXISTS_IMP_THM: "ALL (P::bool) Q::bool.
- (EX x::'a::type. P --> Q) =
- ((ALL x::'a::type. P) --> (EX x::'a::type. Q))"
- by (import bool BOTH_EXISTS_IMP_THM)
-
-lemma OR_IMP_THM: "ALL (A::bool) B::bool. (A = (B | A)) = (B --> A)"
- by (import bool OR_IMP_THM)
-
-lemma DE_MORGAN_THM: "ALL (A::bool) B::bool. (~ (A & B)) = (~ A | ~ B) & (~ (A | B)) = (~ A & ~ B)"
- by (import bool DE_MORGAN_THM)
-
-lemma IMP_F_EQ_F: "ALL t::bool. (t --> False) = (t = False)"
- by (import bool IMP_F_EQ_F)
-
-lemma EQ_EXPAND: "ALL (t1::bool) t2::bool. (t1 = t2) = (t1 & t2 | ~ t1 & ~ t2)"
- by (import bool EQ_EXPAND)
-
-lemma COND_RATOR: "ALL (b::bool) (f::'a::type => 'b::type) (g::'a::type => 'b::type)
- x::'a::type. (if b then f else g) x = (if b then f x else g x)"
- by (import bool COND_RATOR)
-
-lemma COND_ABS: "ALL (b::bool) (f::'a::type => 'b::type) g::'a::type => 'b::type.
- (%x::'a::type. if b then f x else g x) = (if b then f else g)"
- by (import bool COND_ABS)
-
-lemma COND_EXPAND: "ALL (b::bool) (t1::bool) t2::bool.
- (if b then t1 else t2) = ((~ b | t1) & (b | t2))"
- by (import bool COND_EXPAND)
-
-lemma ONE_ONE_THM: "ALL f::'a::type => 'b::type.
- inj f = (ALL (x1::'a::type) x2::'a::type. f x1 = f x2 --> x1 = x2)"
- by (import bool ONE_ONE_THM)
-
-lemma ABS_REP_THM: "(All::(('a::type => bool) => bool) => bool)
- (%P::'a::type => bool.
- (op -->::bool => bool => bool)
- ((Ex::(('b::type => 'a::type) => bool) => bool)
- ((TYPE_DEFINITION::('a::type => bool)
- => ('b::type => 'a::type) => bool)
- P))
- ((Ex::(('b::type => 'a::type) => bool) => bool)
- (%x::'b::type => 'a::type.
- (Ex::(('a::type => 'b::type) => bool) => bool)
- (%abs::'a::type => 'b::type.
- (op &::bool => bool => bool)
- ((All::('b::type => bool) => bool)
- (%a::'b::type.
- (op =::'b::type => 'b::type => bool) (abs (x a)) a))
- ((All::('a::type => bool) => bool)
- (%r::'a::type.
- (op =::bool => bool => bool) (P r)
- ((op =::'a::type => 'a::type => bool) (x (abs r))
- r)))))))"
- by (import bool ABS_REP_THM)
-
-lemma LET_RAND: "(P::'b::type => bool) (Let (M::'a::type) (N::'a::type => 'b::type)) =
-(let x::'a::type = M in P (N x))"
- by (import bool LET_RAND)
-
-lemma LET_RATOR: "Let (M::'a::type) (N::'a::type => 'b::type => 'c::type) (b::'b::type) =
-(let x::'a::type = M in N x b)"
- by (import bool LET_RATOR)
-
-lemma SWAP_FORALL_THM: "ALL P::'a::type => 'b::type => bool.
- (ALL x::'a::type. All (P x)) = (ALL (y::'b::type) x::'a::type. P x y)"
- by (import bool SWAP_FORALL_THM)
-
-lemma SWAP_EXISTS_THM: "ALL P::'a::type => 'b::type => bool.
- (EX x::'a::type. Ex (P x)) = (EX (y::'b::type) x::'a::type. P x y)"
- by (import bool SWAP_EXISTS_THM)
-
-lemma AND_CONG: "ALL (P::bool) (P'::bool) (Q::bool) Q'::bool.
- (Q --> P = P') & (P' --> Q = Q') --> (P & Q) = (P' & Q')"
- by (import bool AND_CONG)
-
-lemma OR_CONG: "ALL (P::bool) (P'::bool) (Q::bool) Q'::bool.
- (~ Q --> P = P') & (~ P' --> Q = Q') --> (P | Q) = (P' | Q')"
- by (import bool OR_CONG)
-
-lemma COND_CONG: "ALL (P::bool) (Q::bool) (x::'a::type) (x'::'a::type) (y::'a::type)
- y'::'a::type.
- P = Q & (Q --> x = x') & (~ Q --> y = y') -->
- (if P then x else y) = (if Q then x' else y')"
- by (import bool COND_CONG)
-
-lemma MONO_COND: "((x::bool) --> (y::bool)) -->
-((z::bool) --> (w::bool)) -->
-(if b::bool then x else z) --> (if b then y else w)"
- by (import bool MONO_COND)
-
-lemma SKOLEM_THM: "ALL P::'a::type => 'b::type => bool.
- (ALL x::'a::type. Ex (P x)) =
- (EX f::'a::type => 'b::type. ALL x::'a::type. P x (f x))"
- by (import bool SKOLEM_THM)
-
-lemma bool_case_thm: "(ALL (e0::'a::type) e1::'a::type.
- (case True of True => e0 | False => e1) = e0) &
-(ALL (e0::'a::type) e1::'a::type.
- (case False of True => e0 | False => e1) = e1)"
- by (import bool bool_case_thm)
-
-lemma bool_case_ID: "ALL (x::'a::type) b::bool. (case b of True => x | _ => x) = x"
- by (import bool bool_case_ID)
-
-lemma boolAxiom: "ALL (e0::'a::type) e1::'a::type.
- EX x::bool => 'a::type. x True = e0 & x False = e1"
- by (import bool boolAxiom)
-
-lemma UEXISTS_OR_THM: "ALL (P::'a::type => bool) Q::'a::type => bool.
- (EX! x::'a::type. P x | Q x) --> Ex1 P | Ex1 Q"
- by (import bool UEXISTS_OR_THM)
-
-lemma UEXISTS_SIMP: "(EX! x::'a::type. (t::bool)) = (t & (ALL x::'a::type. All (op = x)))"
- by (import bool UEXISTS_SIMP)
+ sorry
+
+lemma EQ_CLAUSES: "(True = t) = t & (t = True) = t & (False = t) = (~ t) & (t = False) = (~ t)"
+ sorry
+
+lemma COND_CLAUSES: "(if True then t1 else t2) = t1 & (if False then t1 else t2) = t2"
+ sorry
+
+lemma SELECT_UNIQUE: "(!!y. P y = (y = x)) ==> Eps P = x"
+ sorry
+
+lemma BOTH_EXISTS_AND_THM: "(EX x::'a. (P::bool) & (Q::bool)) = ((EX x::'a. P) & (EX x::'a. Q))"
+ sorry
+
+lemma BOTH_FORALL_OR_THM: "(ALL x::'a. (P::bool) | (Q::bool)) = ((ALL x::'a. P) | (ALL x::'a. Q))"
+ sorry
+
+lemma BOTH_FORALL_IMP_THM: "(ALL x::'a. (P::bool) --> (Q::bool)) = ((EX x::'a. P) --> (ALL x::'a. Q))"
+ sorry
+
+lemma BOTH_EXISTS_IMP_THM: "(EX x::'a. (P::bool) --> (Q::bool)) = ((ALL x::'a. P) --> (EX x::'a. Q))"
+ sorry
+
+lemma OR_IMP_THM: "(A = (B | A)) = (B --> A)"
+ sorry
+
+lemma DE_MORGAN_THM: "(~ (A & B)) = (~ A | ~ B) & (~ (A | B)) = (~ A & ~ B)"
+ sorry
+
+lemma IMP_F_EQ_F: "(t --> False) = (t = False)"
+ sorry
+
+lemma COND_RATOR: "(if b::bool then f::'a => 'b else (g::'a => 'b)) (x::'a) =
+(if b then f x else g x)"
+ sorry
+
+lemma COND_ABS: "(%x. if b then f x else g x) = (if b then f else g)"
+ sorry
+
+lemma COND_EXPAND: "(if b then t1 else t2) = ((~ b | t1) & (b | t2))"
+ sorry
+
+lemma ONE_ONE_THM: "inj f = (ALL x1 x2. f x1 = f x2 --> x1 = x2)"
+ sorry
+
+lemma ABS_REP_THM: "(op ==>::prop => prop => prop)
+ ((Trueprop::bool => prop)
+ ((Ex::(('b::type => 'a::type) => bool) => bool)
+ ((TYPE_DEFINITION::('a::type => bool)
+ => ('b::type => 'a::type) => bool)
+ (P::'a::type => bool))))
+ ((Trueprop::bool => prop)
+ ((Ex::(('b::type => 'a::type) => bool) => bool)
+ (%x::'b::type => 'a::type.
+ (Ex::(('a::type => 'b::type) => bool) => bool)
+ (%abs::'a::type => 'b::type.
+ (op &::bool => bool => bool)
+ ((All::('b::type => bool) => bool)
+ (%a::'b::type.
+ (op =::'b::type => 'b::type => bool) (abs (x a)) a))
+ ((All::('a::type => bool) => bool)
+ (%r::'a::type.
+ (op =::bool => bool => bool) (P r)
+ ((op =::'a::type => 'a::type => bool) (x (abs r))
+ r)))))))"
+ sorry
+
+lemma LET_RAND: "(P::'b => bool) (Let (M::'a) (N::'a => 'b)) = (let x::'a = M in P (N x))"
+ sorry
+
+lemma LET_RATOR: "Let (M::'a) (N::'a => 'b => 'c) (b::'b) = (let x::'a = M in N x b)"
+ sorry
+
+lemma AND_CONG: "(Q --> P = P') & (P' --> Q = Q') ==> (P & Q) = (P' & Q')"
+ sorry
+
+lemma OR_CONG: "(~ Q --> P = P') & (~ P' --> Q = Q') ==> (P | Q) = (P' | Q')"
+ sorry
+
+lemma COND_CONG: "P = Q & (Q --> x = x') & (~ Q --> y = y')
+==> (if P then x else y) = (if Q then x' else y')"
+ sorry
+
+lemma MONO_COND: "[| x ==> y; z ==> w; if b then x else z |] ==> if b then y else w"
+ sorry
+
+lemma SKOLEM_THM: "(ALL x. Ex (P x)) = (EX f. ALL x. P x (f x))"
+ sorry
+
+lemma bool_case_thm: "(ALL (e0::'a) e1::'a. (case True of True => e0 | False => e1) = e0) &
+(ALL (e0::'a) e1::'a. (case False of True => e0 | False => e1) = e1)"
+ sorry
+
+lemma bool_case_ID: "(case b of True => x | _ => x) = x"
+ sorry
+
+lemma boolAxiom: "EX x. x True = e0 & x False = e1"
+ sorry
+
+lemma UEXISTS_OR_THM: "EX! x. P x | Q x ==> Ex1 P | Ex1 Q"
+ sorry
+
+lemma UEXISTS_SIMP: "(EX! x::'a. (t::bool)) = (t & (ALL x::'a. All (op = x)))"
+ sorry
consts
RES_ABSTRACT :: "('a => bool) => ('a => 'b) => 'a => 'b"
-specification (RES_ABSTRACT) RES_ABSTRACT_DEF: "(ALL (p::'a::type => bool) (m::'a::type => 'b::type) x::'a::type.
+specification (RES_ABSTRACT) RES_ABSTRACT_DEF: "(ALL (p::'a => bool) (m::'a => 'b) x::'a.
IN x p --> RES_ABSTRACT p m x = m x) &
-(ALL (p::'a::type => bool) (m1::'a::type => 'b::type)
- m2::'a::type => 'b::type.
- (ALL x::'a::type. IN x p --> m1 x = m2 x) -->
+(ALL (p::'a => bool) (m1::'a => 'b) m2::'a => 'b.
+ (ALL x::'a. IN x p --> m1 x = m2 x) -->
RES_ABSTRACT p m1 = RES_ABSTRACT p m2)"
- by (import bool RES_ABSTRACT_DEF)
-
-lemma BOOL_FUN_CASES_THM: "ALL f::bool => bool.
- f = (%b::bool. True) |
- f = (%b::bool. False) | f = (%b::bool. b) | f = Not"
- by (import bool BOOL_FUN_CASES_THM)
-
-lemma BOOL_FUN_INDUCT: "ALL P::(bool => bool) => bool.
- P (%b::bool. True) & P (%b::bool. False) & P (%b::bool. b) & P Not -->
- All P"
- by (import bool BOOL_FUN_INDUCT)
+ sorry
+
+lemma BOOL_FUN_CASES_THM: "f = (%b. True) | f = (%b. False) | f = (%b. b) | f = Not"
+ sorry
+
+lemma BOOL_FUN_INDUCT: "P (%b. True) & P (%b. False) & P (%b. b) & P Not ==> P x"
+ sorry
;end_setup
;setup_theory combin
-definition K :: "'a => 'b => 'a" where
- "K == %(x::'a::type) y::'b::type. x"
-
-lemma K_DEF: "K = (%(x::'a::type) y::'b::type. x)"
- by (import combin K_DEF)
-
-definition S :: "('a => 'b => 'c) => ('a => 'b) => 'a => 'c" where
- "S ==
-%(f::'a::type => 'b::type => 'c::type) (g::'a::type => 'b::type)
- x::'a::type. f x (g x)"
-
-lemma S_DEF: "S =
-(%(f::'a::type => 'b::type => 'c::type) (g::'a::type => 'b::type)
- x::'a::type. f x (g x))"
- by (import combin S_DEF)
-
-definition I :: "'a => 'a" where
+definition
+ K :: "'a => 'b => 'a" where
+ "K == %x y. x"
+
+lemma K_DEF: "K = (%x y. x)"
+ sorry
+
+definition
+ S :: "('a => 'b => 'c) => ('a => 'b) => 'a => 'c" where
+ "S == %f g x. f x (g x)"
+
+lemma S_DEF: "S = (%f g x. f x (g x))"
+ sorry
+
+definition
+ I :: "'a => 'a" where
"(op ==::('a::type => 'a::type) => ('a::type => 'a::type) => prop)
(I::'a::type => 'a::type)
((S::('a::type => ('a::type => 'a::type) => 'a::type)
@@ -288,47 +236,46 @@
=> ('a::type => 'a::type => 'a::type) => 'a::type => 'a::type)
(K::'a::type => ('a::type => 'a::type) => 'a::type)
(K::'a::type => 'a::type => 'a::type))"
- by (import combin I_DEF)
-
-definition C :: "('a => 'b => 'c) => 'b => 'a => 'c" where
- "C == %(f::'a::type => 'b::type => 'c::type) (x::'b::type) y::'a::type. f y x"
-
-lemma C_DEF: "C =
-(%(f::'a::type => 'b::type => 'c::type) (x::'b::type) y::'a::type. f y x)"
- by (import combin C_DEF)
-
-definition W :: "('a => 'a => 'b) => 'a => 'b" where
- "W == %(f::'a::type => 'a::type => 'b::type) x::'a::type. f x x"
-
-lemma W_DEF: "W = (%(f::'a::type => 'a::type => 'b::type) x::'a::type. f x x)"
- by (import combin W_DEF)
-
-lemma I_THM: "ALL x::'a::type. I x = x"
- by (import combin I_THM)
-
-lemma I_o_ID: "ALL f::'a::type => 'b::type. I o f = f & f o I = f"
- by (import combin I_o_ID)
+ sorry
+
+definition
+ C :: "('a => 'b => 'c) => 'b => 'a => 'c" where
+ "C == %f x y. f y x"
+
+lemma C_DEF: "C = (%f x y. f y x)"
+ sorry
+
+definition
+ W :: "('a => 'a => 'b) => 'a => 'b" where
+ "W == %f x. f x x"
+
+lemma W_DEF: "W = (%f x. f x x)"
+ sorry
+
+lemma I_THM: "I x = x"
+ sorry
+
+lemma I_o_ID: "I o f = f & f o I = f"
+ sorry
;end_setup
;setup_theory sum
-lemma ISL_OR_ISR: "ALL x::'a::type + 'b::type. ISL x | ISR x"
- by (import sum ISL_OR_ISR)
-
-lemma INL: "ALL x::'a::type + 'b::type. ISL x --> Inl (OUTL x) = x"
- by (import sum INL)
-
-lemma INR: "ALL x::'a::type + 'b::type. ISR x --> Inr (OUTR x) = x"
- by (import sum INR)
-
-lemma sum_case_cong: "ALL (M::'b::type + 'c::type) (M'::'b::type + 'c::type)
- (f::'b::type => 'a::type) g::'c::type => 'a::type.
- M = M' &
- (ALL x::'b::type. M' = Inl x --> f x = (f'::'b::type => 'a::type) x) &
- (ALL y::'c::type. M' = Inr y --> g y = (g'::'c::type => 'a::type) y) -->
- sum_case f g M = sum_case f' g' M'"
- by (import sum sum_case_cong)
+lemma ISL_OR_ISR: "ISL x | ISR x"
+ sorry
+
+lemma INL: "ISL x ==> Inl (OUTL x) = x"
+ sorry
+
+lemma INR: "ISR x ==> Inr (OUTR x) = x"
+ sorry
+
+lemma sum_case_cong: "(M::'b + 'c) = (M'::'b + 'c) &
+(ALL x::'b. M' = Inl x --> (f::'b => 'a) x = (f'::'b => 'a) x) &
+(ALL y::'c. M' = Inr y --> (g::'c => 'a) y = (g'::'c => 'a) y)
+==> sum_case f g M = sum_case f' g' M'"
+ sorry
;end_setup
@@ -345,34 +292,34 @@
(%y::'a::type.
(op =::bool => bool => bool)
((op =::'a::type option => 'a::type option => bool)
- ((Some::'a::type ~=> 'a::type) x)
- ((Some::'a::type ~=> 'a::type) y))
+ ((Some::'a::type => 'a::type option) x)
+ ((Some::'a::type => 'a::type option) y))
((op =::'a::type => 'a::type => bool) x y))))
((op &::bool => bool => bool)
((All::('a::type => bool) => bool)
(%x::'a::type.
(op =::'a::type => 'a::type => bool)
((the::'a::type option => 'a::type)
- ((Some::'a::type ~=> 'a::type) x))
+ ((Some::'a::type => 'a::type option) x))
x))
((op &::bool => bool => bool)
((All::('a::type => bool) => bool)
(%x::'a::type.
- (Not::bool => bool)
- ((op =::'a::type option => 'a::type option => bool)
- (None::'a::type option) ((Some::'a::type ~=> 'a::type) x))))
+ (op ~=::'a::type option => 'a::type option => bool)
+ (None::'a::type option)
+ ((Some::'a::type => 'a::type option) x)))
((op &::bool => bool => bool)
((All::('a::type => bool) => bool)
(%x::'a::type.
- (Not::bool => bool)
- ((op =::'a::type option => 'a::type option => bool)
- ((Some::'a::type ~=> 'a::type) x) (None::'a::type option))))
+ (op ~=::'a::type option => 'a::type option => bool)
+ ((Some::'a::type => 'a::type option) x)
+ (None::'a::type option)))
((op &::bool => bool => bool)
((All::('a::type => bool) => bool)
(%x::'a::type.
(op =::bool => bool => bool)
((IS_SOME::'a::type option => bool)
- ((Some::'a::type ~=> 'a::type) x))
+ ((Some::'a::type => 'a::type option) x))
(True::bool)))
((op &::bool => bool => bool)
((op =::bool => bool => bool)
@@ -399,7 +346,7 @@
(op -->::bool => bool => bool)
((IS_SOME::'a::type option => bool) x)
((op =::'a::type option => 'a::type option => bool)
- ((Some::'a::type ~=> 'a::type)
+ ((Some::'a::type => 'a::type option)
((the::'a::type option => 'a::type) x))
x)))
((op &::bool => bool => bool)
@@ -407,9 +354,9 @@
(%x::'a::type option.
(op =::'a::type option => 'a::type option => bool)
((option_case::'a::type option
- => ('a::type ~=> 'a::type) => 'a::type option ~=> 'a::type)
+ => ('a::type => 'a::type option) => 'a::type option => 'a::type option)
(None::'a::type option)
- (Some::'a::type ~=> 'a::type) x)
+ (Some::'a::type => 'a::type option) x)
x))
((op &::bool => bool => bool)
((All::('a::type option => bool) => bool)
@@ -417,8 +364,8 @@
(op =::'a::type option
=> 'a::type option => bool)
((option_case::'a::type option
- => ('a::type ~=> 'a::type) => 'a::type option ~=> 'a::type)
- x (Some::'a::type ~=> 'a::type) x)
+ => ('a::type => 'a::type option) => 'a::type option => 'a::type option)
+ x (Some::'a::type => 'a::type option) x)
x))
((op &::bool => bool => bool)
((All::('a::type option => bool) => bool)
@@ -449,8 +396,9 @@
((op =::'a::type option
=> 'a::type option => bool)
((option_case::'a::type option
- => ('a::type ~=> 'a::type) => 'a::type option ~=> 'a::type)
-(ea::'a::type option) (Some::'a::type ~=> 'a::type) x)
+ => ('a::type => 'a::type option)
+ => 'a::type option => 'a::type option)
+(ea::'a::type option) (Some::'a::type => 'a::type option) x)
x)))
((op &::bool => bool => bool)
((All::('b::type => bool) => bool)
@@ -475,7 +423,7 @@
((option_case::'b::type
=> ('a::type => 'b::type)
=> 'a::type option => 'b::type)
- u f ((Some::'a::type ~=> 'a::type) x))
+ u f ((Some::'a::type => 'a::type option) x))
(f x)))))
((op &::bool => bool => bool)
((All::(('a::type => 'b::type) => bool)
@@ -484,51 +432,48 @@
(All::('a::type => bool) => bool)
(%x::'a::type.
(op =::'b::type option => 'b::type option => bool)
- ((option_map::('a::type => 'b::type) => 'a::type option ~=> 'b::type)
- f ((Some::'a::type ~=> 'a::type) x))
- ((Some::'b::type ~=> 'b::type) (f x)))))
+ ((Option.map::('a::type => 'b::type)
+ => 'a::type option => 'b::type option)
+ f ((Some::'a::type => 'a::type option) x))
+ ((Some::'b::type => 'b::type option) (f x)))))
((op &::bool => bool => bool)
((All::(('a::type => 'b::type) => bool)
=> bool)
(%f::'a::type => 'b::type.
(op =::'b::type option => 'b::type option => bool)
- ((option_map::('a::type => 'b::type) => 'a::type option ~=> 'b::type) f
- (None::'a::type option))
+ ((Option.map::('a::type => 'b::type)
+ => 'a::type option => 'b::type option)
+ f (None::'a::type option))
(None::'b::type option)))
((op &::bool => bool => bool)
((op =::'a::type option
=> 'a::type option => bool)
- ((OPTION_JOIN::'a::type option option ~=> 'a::type)
+ ((OPTION_JOIN::'a::type option option => 'a::type option)
(None::'a::type option option))
(None::'a::type option))
((All::('a::type option => bool)
=> bool)
(%x::'a::type option.
(op =::'a::type option => 'a::type option => bool)
- ((OPTION_JOIN::'a::type option option ~=> 'a::type)
- ((Some::'a::type option ~=> 'a::type option) x))
+ ((OPTION_JOIN::'a::type option option => 'a::type option)
+ ((Some::'a::type option => 'a::type option option) x))
x))))))))))))))))))))"
- by (import option option_CLAUSES)
-
-lemma option_case_compute: "option_case (e::'b::type) (f::'a::type => 'b::type) (x::'a::type option) =
+ sorry
+
+lemma option_case_compute: "option_case (e::'b) (f::'a => 'b) (x::'a option) =
(if IS_SOME x then f (the x) else e)"
- by (import option option_case_compute)
-
-lemma OPTION_MAP_EQ_SOME: "ALL (f::'a::type => 'b::type) (x::'a::type option) y::'b::type.
- (option_map f x = Some y) = (EX z::'a::type. x = Some z & y = f z)"
- by (import option OPTION_MAP_EQ_SOME)
-
-lemma OPTION_JOIN_EQ_SOME: "ALL (x::'a::type option option) xa::'a::type.
- (OPTION_JOIN x = Some xa) = (x = Some (Some xa))"
- by (import option OPTION_JOIN_EQ_SOME)
-
-lemma option_case_cong: "ALL (M::'a::type option) (M'::'a::type option) (u::'b::type)
- f::'a::type => 'b::type.
- M = M' &
- (M' = None --> u = (u'::'b::type)) &
- (ALL x::'a::type. M' = Some x --> f x = (f'::'a::type => 'b::type) x) -->
- option_case u f M = option_case u' f' M'"
- by (import option option_case_cong)
+ sorry
+
+lemma OPTION_MAP_EQ_SOME: "(Option.map (f::'a => 'b) (x::'a option) = Some (y::'b)) =
+(EX z::'a. x = Some z & y = f z)"
+ sorry
+
+lemma OPTION_JOIN_EQ_SOME: "(OPTION_JOIN x = Some xa) = (x = Some (Some xa))"
+ sorry
+
+lemma option_case_cong: "M = M' & (M' = None --> u = u') & (ALL x. M' = Some x --> f x = f' x)
+==> option_case u f M = option_case u' f' M'"
+ sorry
;end_setup
@@ -538,531 +483,341 @@
stmarker :: "'a => 'a"
defs
- stmarker_primdef: "stmarker == %x::'a::type. x"
-
-lemma stmarker_def: "ALL x::'a::type. stmarker x = x"
- by (import marker stmarker_def)
-
-lemma move_left_conj: "ALL (x::bool) (xa::bool) xb::bool.
- (x & stmarker xb) = (stmarker xb & x) &
- ((stmarker xb & x) & xa) = (stmarker xb & x & xa) &
- (x & stmarker xb & xa) = (stmarker xb & x & xa)"
- by (import marker move_left_conj)
-
-lemma move_right_conj: "ALL (x::bool) (xa::bool) xb::bool.
- (stmarker xb & x) = (x & stmarker xb) &
- (x & xa & stmarker xb) = ((x & xa) & stmarker xb) &
- ((x & stmarker xb) & xa) = ((x & xa) & stmarker xb)"
- by (import marker move_right_conj)
-
-lemma move_left_disj: "ALL (x::bool) (xa::bool) xb::bool.
- (x | stmarker xb) = (stmarker xb | x) &
- ((stmarker xb | x) | xa) = (stmarker xb | x | xa) &
- (x | stmarker xb | xa) = (stmarker xb | x | xa)"
- by (import marker move_left_disj)
-
-lemma move_right_disj: "ALL (x::bool) (xa::bool) xb::bool.
- (stmarker xb | x) = (x | stmarker xb) &
- (x | xa | stmarker xb) = ((x | xa) | stmarker xb) &
- ((x | stmarker xb) | xa) = ((x | xa) | stmarker xb)"
- by (import marker move_right_disj)
+ stmarker_primdef: "stmarker == %x. x"
+
+lemma stmarker_def: "stmarker x = x"
+ sorry
+
+lemma move_left_conj: "(x & stmarker xb) = (stmarker xb & x) &
+((stmarker xb & x) & xa) = (stmarker xb & x & xa) &
+(x & stmarker xb & xa) = (stmarker xb & x & xa)"
+ sorry
+
+lemma move_right_conj: "(stmarker xb & x) = (x & stmarker xb) &
+(x & xa & stmarker xb) = ((x & xa) & stmarker xb) &
+((x & stmarker xb) & xa) = ((x & xa) & stmarker xb)"
+ sorry
+
+lemma move_left_disj: "(x | stmarker xb) = (stmarker xb | x) &
+((stmarker xb | x) | xa) = (stmarker xb | x | xa) &
+(x | stmarker xb | xa) = (stmarker xb | x | xa)"
+ sorry
+
+lemma move_right_disj: "(stmarker xb | x) = (x | stmarker xb) &
+(x | xa | stmarker xb) = ((x | xa) | stmarker xb) &
+((x | stmarker xb) | xa) = ((x | xa) | stmarker xb)"
+ sorry
;end_setup
;setup_theory relation
-definition TC :: "('a => 'a => bool) => 'a => 'a => bool" where
+definition
+ TC :: "('a => 'a => bool) => 'a => 'a => bool" where
"TC ==
-%(R::'a::type => 'a::type => bool) (a::'a::type) b::'a::type.
- ALL P::'a::type => 'a::type => bool.
- (ALL (x::'a::type) y::'a::type. R x y --> P x y) &
- (ALL (x::'a::type) (y::'a::type) z::'a::type.
- P x y & P y z --> P x z) -->
+%R a b.
+ ALL P.
+ (ALL x y. R x y --> P x y) & (ALL x y z. P x y & P y z --> P x z) -->
P a b"
-lemma TC_DEF: "ALL (R::'a::type => 'a::type => bool) (a::'a::type) b::'a::type.
- TC R a b =
- (ALL P::'a::type => 'a::type => bool.
- (ALL (x::'a::type) y::'a::type. R x y --> P x y) &
- (ALL (x::'a::type) (y::'a::type) z::'a::type.
- P x y & P y z --> P x z) -->
- P a b)"
- by (import relation TC_DEF)
-
-definition RTC :: "('a => 'a => bool) => 'a => 'a => bool" where
+lemma TC_DEF: "TC R a b =
+(ALL P.
+ (ALL x y. R x y --> P x y) & (ALL x y z. P x y & P y z --> P x z) -->
+ P a b)"
+ sorry
+
+definition
+ RTC :: "('a => 'a => bool) => 'a => 'a => bool" where
"RTC ==
-%(R::'a::type => 'a::type => bool) (a::'a::type) b::'a::type.
- ALL P::'a::type => 'a::type => bool.
- (ALL x::'a::type. P x x) &
- (ALL (x::'a::type) (y::'a::type) z::'a::type.
- R x y & P y z --> P x z) -->
- P a b"
-
-lemma RTC_DEF: "ALL (R::'a::type => 'a::type => bool) (a::'a::type) b::'a::type.
- RTC R a b =
- (ALL P::'a::type => 'a::type => bool.
- (ALL x::'a::type. P x x) &
- (ALL (x::'a::type) (y::'a::type) z::'a::type.
- R x y & P y z --> P x z) -->
- P a b)"
- by (import relation RTC_DEF)
+%R a b.
+ ALL P. (ALL x. P x x) & (ALL x y z. R x y & P y z --> P x z) --> P a b"
+
+lemma RTC_DEF: "RTC R a b =
+(ALL P. (ALL x. P x x) & (ALL x y z. R x y & P y z --> P x z) --> P a b)"
+ sorry
consts
RC :: "('a => 'a => bool) => 'a => 'a => bool"
defs
- RC_primdef: "RC ==
-%(R::'a::type => 'a::type => bool) (x::'a::type) y::'a::type. x = y | R x y"
-
-lemma RC_def: "ALL (R::'a::type => 'a::type => bool) (x::'a::type) y::'a::type.
- RC R x y = (x = y | R x y)"
- by (import relation RC_def)
+ RC_primdef: "RC == %R x y. x = y | R x y"
+
+lemma RC_def: "RC R x y = (x = y | R x y)"
+ sorry
consts
transitive :: "('a => 'a => bool) => bool"
defs
- transitive_primdef: "transitive ==
-%R::'a::type => 'a::type => bool.
- ALL (x::'a::type) (y::'a::type) z::'a::type. R x y & R y z --> R x z"
-
-lemma transitive_def: "ALL R::'a::type => 'a::type => bool.
- transitive R =
- (ALL (x::'a::type) (y::'a::type) z::'a::type. R x y & R y z --> R x z)"
- by (import relation transitive_def)
-
-definition pred_reflexive :: "('a => 'a => bool) => bool" where
- "pred_reflexive == %R::'a::type => 'a::type => bool. ALL x::'a::type. R x x"
-
-lemma reflexive_def: "ALL R::'a::type => 'a::type => bool.
- pred_reflexive R = (ALL x::'a::type. R x x)"
- by (import relation reflexive_def)
-
-lemma TC_TRANSITIVE: "ALL x::'a::type => 'a::type => bool. transitive (TC x)"
- by (import relation TC_TRANSITIVE)
-
-lemma RTC_INDUCT: "ALL (x::'a::type => 'a::type => bool) xa::'a::type => 'a::type => bool.
- (ALL x::'a::type. xa x x) &
- (ALL (xb::'a::type) (y::'a::type) z::'a::type.
- x xb y & xa y z --> xa xb z) -->
- (ALL (xb::'a::type) xc::'a::type. RTC x xb xc --> xa xb xc)"
- by (import relation RTC_INDUCT)
-
-lemma TC_RULES: "ALL x::'a::type => 'a::type => bool.
- (ALL (xa::'a::type) xb::'a::type. x xa xb --> TC x xa xb) &
- (ALL (xa::'a::type) (xb::'a::type) xc::'a::type.
- TC x xa xb & TC x xb xc --> TC x xa xc)"
- by (import relation TC_RULES)
-
-lemma RTC_RULES: "ALL x::'a::type => 'a::type => bool.
- (ALL xa::'a::type. RTC x xa xa) &
- (ALL (xa::'a::type) (xb::'a::type) xc::'a::type.
- x xa xb & RTC x xb xc --> RTC x xa xc)"
- by (import relation RTC_RULES)
-
-lemma RTC_STRONG_INDUCT: "ALL (R::'a::type => 'a::type => bool) P::'a::type => 'a::type => bool.
- (ALL x::'a::type. P x x) &
- (ALL (x::'a::type) (y::'a::type) z::'a::type.
- R x y & RTC R y z & P y z --> P x z) -->
- (ALL (x::'a::type) y::'a::type. RTC R x y --> P x y)"
- by (import relation RTC_STRONG_INDUCT)
-
-lemma RTC_RTC: "ALL (R::'a::type => 'a::type => bool) (x::'a::type) y::'a::type.
- RTC R x y --> (ALL z::'a::type. RTC R y z --> RTC R x z)"
- by (import relation RTC_RTC)
-
-lemma RTC_TRANSITIVE: "ALL x::'a::type => 'a::type => bool. transitive (RTC x)"
- by (import relation RTC_TRANSITIVE)
-
-lemma RTC_REFLEXIVE: "ALL R::'a::type => 'a::type => bool. pred_reflexive (RTC R)"
- by (import relation RTC_REFLEXIVE)
-
-lemma RC_REFLEXIVE: "ALL R::'a::type => 'a::type => bool. pred_reflexive (RC R)"
- by (import relation RC_REFLEXIVE)
-
-lemma TC_SUBSET: "ALL (x::'a::type => 'a::type => bool) (xa::'a::type) xb::'a::type.
- x xa xb --> TC x xa xb"
- by (import relation TC_SUBSET)
-
-lemma RTC_SUBSET: "ALL (R::'a::type => 'a::type => bool) (x::'a::type) y::'a::type.
- R x y --> RTC R x y"
- by (import relation RTC_SUBSET)
-
-lemma RC_SUBSET: "ALL (R::'a::type => 'a::type => bool) (x::'a::type) y::'a::type.
- R x y --> RC R x y"
- by (import relation RC_SUBSET)
-
-lemma RC_RTC: "ALL (R::'a::type => 'a::type => bool) (x::'a::type) y::'a::type.
- RC R x y --> RTC R x y"
- by (import relation RC_RTC)
-
-lemma TC_INDUCT: "ALL (x::'a::type => 'a::type => bool) xa::'a::type => 'a::type => bool.
- (ALL (xb::'a::type) y::'a::type. x xb y --> xa xb y) &
- (ALL (x::'a::type) (y::'a::type) z::'a::type.
- xa x y & xa y z --> xa x z) -->
- (ALL (xb::'a::type) xc::'a::type. TC x xb xc --> xa xb xc)"
- by (import relation TC_INDUCT)
-
-lemma TC_INDUCT_LEFT1: "ALL (x::'a::type => 'a::type => bool) xa::'a::type => 'a::type => bool.
- (ALL (xb::'a::type) y::'a::type. x xb y --> xa xb y) &
- (ALL (xb::'a::type) (y::'a::type) z::'a::type.
- x xb y & xa y z --> xa xb z) -->
- (ALL (xb::'a::type) xc::'a::type. TC x xb xc --> xa xb xc)"
- by (import relation TC_INDUCT_LEFT1)
-
-lemma TC_STRONG_INDUCT: "ALL (R::'a::type => 'a::type => bool) P::'a::type => 'a::type => bool.
- (ALL (x::'a::type) y::'a::type. R x y --> P x y) &
- (ALL (x::'a::type) (y::'a::type) z::'a::type.
- P x y & P y z & TC R x y & TC R y z --> P x z) -->
- (ALL (u::'a::type) v::'a::type. TC R u v --> P u v)"
- by (import relation TC_STRONG_INDUCT)
-
-lemma TC_STRONG_INDUCT_LEFT1: "ALL (R::'a::type => 'a::type => bool) P::'a::type => 'a::type => bool.
- (ALL (x::'a::type) y::'a::type. R x y --> P x y) &
- (ALL (x::'a::type) (y::'a::type) z::'a::type.
- R x y & P y z & TC R y z --> P x z) -->
- (ALL (u::'a::type) v::'a::type. TC R u v --> P u v)"
- by (import relation TC_STRONG_INDUCT_LEFT1)
-
-lemma TC_RTC: "ALL (R::'a::type => 'a::type => bool) (x::'a::type) y::'a::type.
- TC R x y --> RTC R x y"
- by (import relation TC_RTC)
-
-lemma RTC_TC_RC: "ALL (R::'a::type => 'a::type => bool) (x::'a::type) y::'a::type.
- RTC R x y --> RC R x y | TC R x y"
- by (import relation RTC_TC_RC)
-
-lemma TC_RC_EQNS: "ALL R::'a::type => 'a::type => bool. RC (TC R) = RTC R & TC (RC R) = RTC R"
- by (import relation TC_RC_EQNS)
-
-lemma RC_IDEM: "ALL R::'a::type => 'a::type => bool. RC (RC R) = RC R"
- by (import relation RC_IDEM)
-
-lemma TC_IDEM: "ALL R::'a::type => 'a::type => bool. TC (TC R) = TC R"
- by (import relation TC_IDEM)
-
-lemma RTC_IDEM: "ALL R::'a::type => 'a::type => bool. RTC (RTC R) = RTC R"
- by (import relation RTC_IDEM)
-
-lemma RTC_CASES1: "ALL (x::'a::type => 'a::type => bool) (xa::'a::type) xb::'a::type.
- RTC x xa xb = (xa = xb | (EX u::'a::type. x xa u & RTC x u xb))"
- by (import relation RTC_CASES1)
-
-lemma RTC_CASES2: "ALL (x::'a::type => 'a::type => bool) (xa::'a::type) xb::'a::type.
- RTC x xa xb = (xa = xb | (EX u::'a::type. RTC x xa u & x u xb))"
- by (import relation RTC_CASES2)
-
-lemma RTC_CASES_RTC_TWICE: "ALL (x::'a::type => 'a::type => bool) (xa::'a::type) xb::'a::type.
- RTC x xa xb = (EX u::'a::type. RTC x xa u & RTC x u xb)"
- by (import relation RTC_CASES_RTC_TWICE)
-
-lemma TC_CASES1: "ALL (R::'a::type => 'a::type => bool) (x::'a::type) z::'a::type.
- TC R x z --> R x z | (EX y::'a::type. R x y & TC R y z)"
- by (import relation TC_CASES1)
-
-lemma TC_CASES2: "ALL (R::'a::type => 'a::type => bool) (x::'a::type) z::'a::type.
- TC R x z --> R x z | (EX y::'a::type. TC R x y & R y z)"
- by (import relation TC_CASES2)
-
-lemma TC_MONOTONE: "ALL (R::'a::type => 'a::type => bool) Q::'a::type => 'a::type => bool.
- (ALL (x::'a::type) y::'a::type. R x y --> Q x y) -->
- (ALL (x::'a::type) y::'a::type. TC R x y --> TC Q x y)"
- by (import relation TC_MONOTONE)
-
-lemma RTC_MONOTONE: "ALL (R::'a::type => 'a::type => bool) Q::'a::type => 'a::type => bool.
- (ALL (x::'a::type) y::'a::type. R x y --> Q x y) -->
- (ALL (x::'a::type) y::'a::type. RTC R x y --> RTC Q x y)"
- by (import relation RTC_MONOTONE)
-
-definition WF :: "('a => 'a => bool) => bool" where
- "WF ==
-%R::'a::type => 'a::type => bool.
- ALL B::'a::type => bool.
- Ex B -->
- (EX min::'a::type. B min & (ALL b::'a::type. R b min --> ~ B b))"
-
-lemma WF_DEF: "ALL R::'a::type => 'a::type => bool.
- WF R =
- (ALL B::'a::type => bool.
- Ex B -->
- (EX min::'a::type. B min & (ALL b::'a::type. R b min --> ~ B b)))"
- by (import relation WF_DEF)
-
-lemma WF_INDUCTION_THM: "ALL R::'a::type => 'a::type => bool.
- WF R -->
- (ALL P::'a::type => bool.
- (ALL x::'a::type. (ALL y::'a::type. R y x --> P y) --> P x) -->
- All P)"
- by (import relation WF_INDUCTION_THM)
-
-lemma WF_NOT_REFL: "ALL (x::'a::type => 'a::type => bool) (xa::'a::type) xb::'a::type.
- WF x --> x xa xb --> xa ~= xb"
- by (import relation WF_NOT_REFL)
-
-definition EMPTY_REL :: "'a => 'a => bool" where
- "EMPTY_REL == %(x::'a::type) y::'a::type. False"
-
-lemma EMPTY_REL_DEF: "ALL (x::'a::type) y::'a::type. EMPTY_REL x y = False"
- by (import relation EMPTY_REL_DEF)
+ transitive_primdef: "transitive == %R. ALL x y z. R x y & R y z --> R x z"
+
+lemma transitive_def: "transitive R = (ALL x y z. R x y & R y z --> R x z)"
+ sorry
+
+definition
+ pred_reflexive :: "('a => 'a => bool) => bool" where
+ "pred_reflexive == %R. ALL x. R x x"
+
+lemma reflexive_def: "pred_reflexive R = (ALL x. R x x)"
+ sorry
+
+lemma TC_TRANSITIVE: "transitive (TC x)"
+ sorry
+
+lemma RTC_INDUCT: "[| (ALL x. xa x x) & (ALL xb y z. x xb y & xa y z --> xa xb z);
+ RTC x xb xc |]
+==> xa xb xc"
+ sorry
+
+lemma TC_RULES: "(ALL xa xb. x xa xb --> TC x xa xb) &
+(ALL xa xb xc. TC x xa xb & TC x xb xc --> TC x xa xc)"
+ sorry
+
+lemma RTC_RULES: "(ALL xa. RTC x xa xa) &
+(ALL xa xb xc. x xa xb & RTC x xb xc --> RTC x xa xc)"
+ sorry
+
+lemma RTC_STRONG_INDUCT: "[| (ALL x. P x x) & (ALL x y z. R x y & RTC R y z & P y z --> P x z);
+ RTC R x y |]
+==> P x y"
+ sorry
+
+lemma RTC_RTC: "[| RTC R x y; RTC R y z |] ==> RTC R x z"
+ sorry
+
+lemma RTC_TRANSITIVE: "transitive (RTC x)"
+ sorry
+
+lemma RTC_REFLEXIVE: "pred_reflexive (RTC R)"
+ sorry
+
+lemma RC_REFLEXIVE: "pred_reflexive (RC R)"
+ sorry
+
+lemma TC_SUBSET: "x xa xb ==> TC x xa xb"
+ sorry
+
+lemma RTC_SUBSET: "R x y ==> RTC R x y"
+ sorry
+
+lemma RC_SUBSET: "R x y ==> RC R x y"
+ sorry
+
+lemma RC_RTC: "RC R x y ==> RTC R x y"
+ sorry
+
+lemma TC_INDUCT: "[| (ALL xb y. x xb y --> xa xb y) & (ALL x y z. xa x y & xa y z --> xa x z);
+ TC x xb xc |]
+==> xa xb xc"
+ sorry
+
+lemma TC_INDUCT_LEFT1: "[| (ALL xb y. x xb y --> xa xb y) &
+ (ALL xb y z. x xb y & xa y z --> xa xb z);
+ TC x xb xc |]
+==> xa xb xc"
+ sorry
+
+lemma TC_STRONG_INDUCT: "[| (ALL x y. R x y --> P x y) &
+ (ALL x y z. P x y & P y z & TC R x y & TC R y z --> P x z);
+ TC R u v |]
+==> P u v"
+ sorry
+
+lemma TC_STRONG_INDUCT_LEFT1: "[| (ALL x y. R x y --> P x y) &
+ (ALL x y z. R x y & P y z & TC R y z --> P x z);
+ TC R u v |]
+==> P u v"
+ sorry
+
+lemma TC_RTC: "TC R x y ==> RTC R x y"
+ sorry
+
+lemma RTC_TC_RC: "RTC R x y ==> RC R x y | TC R x y"
+ sorry
+
+lemma TC_RC_EQNS: "RC (TC R) = RTC R & TC (RC R) = RTC R"
+ sorry
+
+lemma RC_IDEM: "RC (RC R) = RC R"
+ sorry
+
+lemma TC_IDEM: "TC (TC R) = TC R"
+ sorry
+
+lemma RTC_IDEM: "RTC (RTC R) = RTC R"
+ sorry
+
+lemma RTC_CASES1: "RTC x xa xb = (xa = xb | (EX u. x xa u & RTC x u xb))"
+ sorry
+
+lemma RTC_CASES2: "RTC x xa xb = (xa = xb | (EX u. RTC x xa u & x u xb))"
+ sorry
+
+lemma RTC_CASES_RTC_TWICE: "RTC x xa xb = (EX u. RTC x xa u & RTC x u xb)"
+ sorry
+
+lemma TC_CASES1: "TC R x z ==> R x z | (EX y. R x y & TC R y z)"
+ sorry
+
+lemma TC_CASES2: "TC R x z ==> R x z | (EX y. TC R x y & R y z)"
+ sorry
+
+lemma TC_MONOTONE: "[| !!x y. R x y ==> Q x y; TC R x y |] ==> TC Q x y"
+ sorry
+
+lemma RTC_MONOTONE: "[| !!x y. R x y ==> Q x y; RTC R x y |] ==> RTC Q x y"
+ sorry
+
+definition
+ WF :: "('a => 'a => bool) => bool" where
+ "WF == %R. ALL B. Ex B --> (EX min. B min & (ALL b. R b min --> ~ B b))"
+
+lemma WF_DEF: "WF R = (ALL B. Ex B --> (EX min. B min & (ALL b. R b min --> ~ B b)))"
+ sorry
+
+lemma WF_INDUCTION_THM: "[| WF R; !!x. (!!y. R y x ==> P y) ==> P x |] ==> P x"
+ sorry
+
+lemma WF_NOT_REFL: "[| WF x; x xa xb |] ==> xa ~= xb"
+ sorry
+
+definition
+ EMPTY_REL :: "'a => 'a => bool" where
+ "EMPTY_REL == %x y. False"
+
+lemma EMPTY_REL_DEF: "EMPTY_REL x y = False"
+ sorry
lemma WF_EMPTY_REL: "WF EMPTY_REL"
- by (import relation WF_EMPTY_REL)
-
-lemma WF_SUBSET: "ALL (x::'a::type => 'a::type => bool) xa::'a::type => 'a::type => bool.
- WF x & (ALL (xb::'a::type) y::'a::type. xa xb y --> x xb y) --> WF xa"
- by (import relation WF_SUBSET)
-
-lemma WF_TC: "ALL R::'a::type => 'a::type => bool. WF R --> WF (TC R)"
- by (import relation WF_TC)
+ sorry
+
+lemma WF_SUBSET: "WF x & (ALL xb y. xa xb y --> x xb y) ==> WF xa"
+ sorry
+
+lemma WF_TC: "WF R ==> WF (TC R)"
+ sorry
consts
inv_image :: "('b => 'b => bool) => ('a => 'b) => 'a => 'a => bool"
defs
inv_image_primdef: "relation.inv_image ==
-%(R::'b::type => 'b::type => bool) (f::'a::type => 'b::type) (x::'a::type)
- y::'a::type. R (f x) (f y)"
-
-lemma inv_image_def: "ALL (R::'b::type => 'b::type => bool) f::'a::type => 'b::type.
- relation.inv_image R f = (%(x::'a::type) y::'a::type. R (f x) (f y))"
- by (import relation inv_image_def)
-
-lemma WF_inv_image: "ALL (R::'b::type => 'b::type => bool) f::'a::type => 'b::type.
- WF R --> WF (relation.inv_image R f)"
- by (import relation WF_inv_image)
-
-definition RESTRICT :: "('a => 'b) => ('a => 'a => bool) => 'a => 'a => 'b" where
- "RESTRICT ==
-%(f::'a::type => 'b::type) (R::'a::type => 'a::type => bool) (x::'a::type)
- y::'a::type. if R y x then f y else ARB"
-
-lemma RESTRICT_DEF: "ALL (f::'a::type => 'b::type) (R::'a::type => 'a::type => bool) x::'a::type.
- RESTRICT f R x = (%y::'a::type. if R y x then f y else ARB)"
- by (import relation RESTRICT_DEF)
-
-lemma RESTRICT_LEMMA: "ALL (x::'a::type => 'b::type) (xa::'a::type => 'a::type => bool)
- (xb::'a::type) xc::'a::type. xa xb xc --> RESTRICT x xa xc xb = x xb"
- by (import relation RESTRICT_LEMMA)
+%(R::'b => 'b => bool) (f::'a => 'b) (x::'a) y::'a. R (f x) (f y)"
+
+lemma inv_image_def: "relation.inv_image R f = (%x y. R (f x) (f y))"
+ sorry
+
+lemma WF_inv_image: "WF (R::'b => 'b => bool) ==> WF (relation.inv_image R (f::'a => 'b))"
+ sorry
+
+definition
+ RESTRICT :: "('a => 'b) => ('a => 'a => bool) => 'a => 'a => 'b" where
+ "RESTRICT == %f R x y. if R y x then f y else ARB"
+
+lemma RESTRICT_DEF: "RESTRICT f R x = (%y. if R y x then f y else ARB)"
+ sorry
+
+lemma RESTRICT_LEMMA: "xa xb xc ==> RESTRICT x xa xc xb = x xb"
+ sorry
consts
approx :: "('a => 'a => bool) => (('a => 'b) => 'a => 'b) => 'a => ('a => 'b) => bool"
defs
- approx_primdef: "approx ==
-%(R::'a::type => 'a::type => bool)
- (M::('a::type => 'b::type) => 'a::type => 'b::type) (x::'a::type)
- f::'a::type => 'b::type.
- f = RESTRICT (%y::'a::type. M (RESTRICT f R y) y) R x"
-
-lemma approx_def: "ALL (R::'a::type => 'a::type => bool)
- (M::('a::type => 'b::type) => 'a::type => 'b::type) (x::'a::type)
- f::'a::type => 'b::type.
- approx R M x f = (f = RESTRICT (%y::'a::type. M (RESTRICT f R y) y) R x)"
- by (import relation approx_def)
+ approx_primdef: "approx == %R M x f. f = RESTRICT (%y. M (RESTRICT f R y) y) R x"
+
+lemma approx_def: "approx R M x f = (f = RESTRICT (%y. M (RESTRICT f R y) y) R x)"
+ sorry
consts
the_fun :: "('a => 'a => bool) => (('a => 'b) => 'a => 'b) => 'a => 'a => 'b"
defs
- the_fun_primdef: "the_fun ==
-%(R::'a::type => 'a::type => bool)
- (M::('a::type => 'b::type) => 'a::type => 'b::type) x::'a::type.
- Eps (approx R M x)"
-
-lemma the_fun_def: "ALL (R::'a::type => 'a::type => bool)
- (M::('a::type => 'b::type) => 'a::type => 'b::type) x::'a::type.
- the_fun R M x = Eps (approx R M x)"
- by (import relation the_fun_def)
-
-definition WFREC :: "('a => 'a => bool) => (('a => 'b) => 'a => 'b) => 'a => 'b" where
+ the_fun_primdef: "the_fun == %R M x. Eps (approx R M x)"
+
+lemma the_fun_def: "the_fun R M x = Eps (approx R M x)"
+ sorry
+
+definition
+ WFREC :: "('a => 'a => bool) => (('a => 'b) => 'a => 'b) => 'a => 'b" where
"WFREC ==
-%(R::'a::type => 'a::type => bool)
- (M::('a::type => 'b::type) => 'a::type => 'b::type) x::'a::type.
- M (RESTRICT
- (the_fun (TC R)
- (%(f::'a::type => 'b::type) v::'a::type. M (RESTRICT f R v) v) x)
- R x)
- x"
-
-lemma WFREC_DEF: "ALL (R::'a::type => 'a::type => bool)
- M::('a::type => 'b::type) => 'a::type => 'b::type.
- WFREC R M =
- (%x::'a::type.
- M (RESTRICT
- (the_fun (TC R)
- (%(f::'a::type => 'b::type) v::'a::type. M (RESTRICT f R v) v)
- x)
- R x)
- x)"
- by (import relation WFREC_DEF)
-
-lemma WFREC_THM: "ALL (R::'a::type => 'a::type => bool)
- M::('a::type => 'b::type) => 'a::type => 'b::type.
- WF R --> (ALL x::'a::type. WFREC R M x = M (RESTRICT (WFREC R M) R x) x)"
- by (import relation WFREC_THM)
-
-lemma WFREC_COROLLARY: "ALL (M::('a::type => 'b::type) => 'a::type => 'b::type)
- (R::'a::type => 'a::type => bool) f::'a::type => 'b::type.
- f = WFREC R M --> WF R --> (ALL x::'a::type. f x = M (RESTRICT f R x) x)"
- by (import relation WFREC_COROLLARY)
-
-lemma WF_RECURSION_THM: "ALL R::'a::type => 'a::type => bool.
- WF R -->
- (ALL M::('a::type => 'b::type) => 'a::type => 'b::type.
- EX! f::'a::type => 'b::type.
- ALL x::'a::type. f x = M (RESTRICT f R x) x)"
- by (import relation WF_RECURSION_THM)
+%R M x. M (RESTRICT (the_fun (TC R) (%f v. M (RESTRICT f R v) v) x) R x) x"
+
+lemma WFREC_DEF: "WFREC R M =
+(%x. M (RESTRICT (the_fun (TC R) (%f v. M (RESTRICT f R v) v) x) R x) x)"
+ sorry
+
+lemma WFREC_THM: "WF R ==> WFREC R M x = M (RESTRICT (WFREC R M) R x) x"
+ sorry
+
+lemma WFREC_COROLLARY: "[| f = WFREC R M; WF R |] ==> f x = M (RESTRICT f R x) x"
+ sorry
+
+lemma WF_RECURSION_THM: "WF R ==> EX! f. ALL x. f x = M (RESTRICT f R x) x"
+ sorry
;end_setup
;setup_theory pair
-lemma CURRY_ONE_ONE_THM: "(curry (f::'a::type * 'b::type => 'c::type) =
- curry (g::'a::type * 'b::type => 'c::type)) =
-(f = g)"
- by (import pair CURRY_ONE_ONE_THM)
-
-lemma UNCURRY_ONE_ONE_THM: "(op =::bool => bool => bool)
- ((op =::('a::type * 'b::type => 'c::type)
- => ('a::type * 'b::type => 'c::type) => bool)
- ((split::('a::type => 'b::type => 'c::type)
- => 'a::type * 'b::type => 'c::type)
- (f::'a::type => 'b::type => 'c::type))
- ((split::('a::type => 'b::type => 'c::type)
- => 'a::type * 'b::type => 'c::type)
- (g::'a::type => 'b::type => 'c::type)))
- ((op =::('a::type => 'b::type => 'c::type)
- => ('a::type => 'b::type => 'c::type) => bool)
- f g)"
- by (import pair UNCURRY_ONE_ONE_THM)
-
-lemma pair_Axiom: "ALL f::'a::type => 'b::type => 'c::type.
- EX x::'a::type * 'b::type => 'c::type.
- ALL (xa::'a::type) y::'b::type. x (xa, y) = f xa y"
- by (import pair pair_Axiom)
-
-lemma UNCURRY_CONG: "ALL (M::'a::type * 'b::type) (M'::'a::type * 'b::type)
- f::'a::type => 'b::type => 'c::type.
- M = M' &
- (ALL (x::'a::type) y::'b::type.
- M' = (x, y) -->
- f x y = (f'::'a::type => 'b::type => 'c::type) x y) -->
- split f M = split f' M'"
- by (import pair UNCURRY_CONG)
-
-lemma ELIM_PEXISTS: "(EX p::'a::type * 'b::type.
- (P::'a::type => 'b::type => bool) (fst p) (snd p)) =
-(EX p1::'a::type. Ex (P p1))"
- by (import pair ELIM_PEXISTS)
-
-lemma ELIM_PFORALL: "(ALL p::'a::type * 'b::type.
- (P::'a::type => 'b::type => bool) (fst p) (snd p)) =
-(ALL p1::'a::type. All (P p1))"
- by (import pair ELIM_PFORALL)
-
-lemma PFORALL_THM: "(All::(('a::type => 'b::type => bool) => bool) => bool)
- (%x::'a::type => 'b::type => bool.
- (op =::bool => bool => bool)
- ((All::('a::type => bool) => bool)
- (%xa::'a::type. (All::('b::type => bool) => bool) (x xa)))
- ((All::('a::type * 'b::type => bool) => bool)
- ((split::('a::type => 'b::type => bool)
- => 'a::type * 'b::type => bool)
- x)))"
- by (import pair PFORALL_THM)
-
-lemma PEXISTS_THM: "(All::(('a::type => 'b::type => bool) => bool) => bool)
- (%x::'a::type => 'b::type => bool.
- (op =::bool => bool => bool)
- ((Ex::('a::type => bool) => bool)
- (%xa::'a::type. (Ex::('b::type => bool) => bool) (x xa)))
- ((Ex::('a::type * 'b::type => bool) => bool)
- ((split::('a::type => 'b::type => bool)
- => 'a::type * 'b::type => bool)
- x)))"
- by (import pair PEXISTS_THM)
-
-lemma LET2_RAND: "(All::(('c::type => 'd::type) => bool) => bool)
- (%x::'c::type => 'd::type.
- (All::('a::type * 'b::type => bool) => bool)
- (%xa::'a::type * 'b::type.
- (All::(('a::type => 'b::type => 'c::type) => bool) => bool)
- (%xb::'a::type => 'b::type => 'c::type.
- (op =::'d::type => 'd::type => bool)
- (x ((Let::'a::type * 'b::type
- => ('a::type * 'b::type => 'c::type) => 'c::type)
- xa ((split::('a::type => 'b::type => 'c::type)
- => 'a::type * 'b::type => 'c::type)
- xb)))
- ((Let::'a::type * 'b::type
- => ('a::type * 'b::type => 'd::type) => 'd::type)
- xa ((split::('a::type => 'b::type => 'd::type)
- => 'a::type * 'b::type => 'd::type)
- (%(xa::'a::type) y::'b::type. x (xb xa y)))))))"
- by (import pair LET2_RAND)
-
-lemma LET2_RATOR: "(All::('a1::type * 'a2::type => bool) => bool)
- (%x::'a1::type * 'a2::type.
- (All::(('a1::type => 'a2::type => 'b::type => 'c::type) => bool)
- => bool)
- (%xa::'a1::type => 'a2::type => 'b::type => 'c::type.
- (All::('b::type => bool) => bool)
- (%xb::'b::type.
- (op =::'c::type => 'c::type => bool)
- ((Let::'a1::type * 'a2::type
- => ('a1::type * 'a2::type => 'b::type => 'c::type)
- => 'b::type => 'c::type)
- x ((split::('a1::type
- => 'a2::type => 'b::type => 'c::type)
- => 'a1::type * 'a2::type
- => 'b::type => 'c::type)
- xa)
- xb)
- ((Let::'a1::type * 'a2::type
- => ('a1::type * 'a2::type => 'c::type) => 'c::type)
- x ((split::('a1::type => 'a2::type => 'c::type)
- => 'a1::type * 'a2::type => 'c::type)
- (%(x::'a1::type) y::'a2::type. xa x y xb))))))"
- by (import pair LET2_RATOR)
-
-lemma pair_case_cong: "ALL (x::'a::type * 'b::type) (xa::'a::type * 'b::type)
- xb::'a::type => 'b::type => 'c::type.
- x = xa &
- (ALL (x::'a::type) y::'b::type.
- xa = (x, y) -->
- xb x y = (f'::'a::type => 'b::type => 'c::type) x y) -->
- split xb x = split f' xa"
- by (import pair pair_case_cong)
-
-definition LEX :: "('a => 'a => bool) => ('b => 'b => bool) => 'a * 'b => 'a * 'b => bool" where
- "LEX ==
-%(R1::'a::type => 'a::type => bool) (R2::'b::type => 'b::type => bool)
- (s::'a::type, t::'b::type) (u::'a::type, v::'b::type).
- R1 s u | s = u & R2 t v"
-
-lemma LEX_DEF: "ALL (R1::'a::type => 'a::type => bool) R2::'b::type => 'b::type => bool.
- LEX R1 R2 =
- (%(s::'a::type, t::'b::type) (u::'a::type, v::'b::type).
- R1 s u | s = u & R2 t v)"
- by (import pair LEX_DEF)
-
-lemma WF_LEX: "ALL (x::'a::type => 'a::type => bool) xa::'b::type => 'b::type => bool.
- WF x & WF xa --> WF (LEX x xa)"
- by (import pair WF_LEX)
-
-definition RPROD :: "('a => 'a => bool) => ('b => 'b => bool) => 'a * 'b => 'a * 'b => bool" where
- "RPROD ==
-%(R1::'a::type => 'a::type => bool) (R2::'b::type => 'b::type => bool)
- (s::'a::type, t::'b::type) (u::'a::type, v::'b::type). R1 s u & R2 t v"
-
-lemma RPROD_DEF: "ALL (R1::'a::type => 'a::type => bool) R2::'b::type => 'b::type => bool.
- RPROD R1 R2 =
- (%(s::'a::type, t::'b::type) (u::'a::type, v::'b::type). R1 s u & R2 t v)"
- by (import pair RPROD_DEF)
-
-lemma WF_RPROD: "ALL (R::'a::type => 'a::type => bool) Q::'b::type => 'b::type => bool.
- WF R & WF Q --> WF (RPROD R Q)"
- by (import pair WF_RPROD)
+lemma CURRY_ONE_ONE_THM: "(curry f = curry g) = (f = g)"
+ sorry
+
+lemma UNCURRY_ONE_ONE_THM: "((%(x, y). f x y) = (%(x, y). g x y)) = (f = g)"
+ sorry
+
+lemma pair_Axiom: "EX x. ALL xa y. x (xa, y) = f xa y"
+ sorry
+
+lemma UNCURRY_CONG: "M = M' & (ALL x y. M' = (x, y) --> f x y = f' x y)
+==> prod_case f M = prod_case f' M'"
+ sorry
+
+lemma ELIM_PEXISTS: "(EX p. P (fst p) (snd p)) = (EX p1. Ex (P p1))"
+ sorry
+
+lemma ELIM_PFORALL: "(ALL p. P (fst p) (snd p)) = (ALL p1. All (P p1))"
+ sorry
+
+lemma PFORALL_THM: "(ALL xa. All (x xa)) = All (%(xa, y). x xa y)"
+ sorry
+
+lemma PEXISTS_THM: "(EX xa. Ex (x xa)) = Ex (%(xa, y). x xa y)"
+ sorry
+
+lemma LET2_RAND: "(x::'c => 'd)
+ (let (x::'a, y::'b) = xa::'a * 'b in (xb::'a => 'b => 'c) x y) =
+(let (xa::'a, y::'b) = xa in x (xb xa y))"
+ sorry
+
+lemma LET2_RATOR: "(let (x::'a1, y::'a2) = x::'a1 * 'a2 in (xa::'a1 => 'a2 => 'b => 'c) x y)
+ (xb::'b) =
+(let (x::'a1, y::'a2) = x in xa x y xb)"
+ sorry
+
+lemma pair_case_cong: "x = xa & (ALL x y. xa = (x, y) --> xb x y = f' x y)
+==> prod_case xb x = prod_case f' xa"
+ sorry
+
+definition
+ LEX :: "('a => 'a => bool) => ('b => 'b => bool) => 'a * 'b => 'a * 'b => bool" where
+ "LEX == %R1 R2 (s, t) (u, v). R1 s u | s = u & R2 t v"
+
+lemma LEX_DEF: "LEX R1 R2 = (%(s, t) (u, v). R1 s u | s = u & R2 t v)"
+ sorry
+
+lemma WF_LEX: "WF x & WF xa ==> WF (LEX x xa)"
+ sorry
+
+definition
+ RPROD :: "('a => 'a => bool) => ('b => 'b => bool) => 'a * 'b => 'a * 'b => bool" where
+ "RPROD == %R1 R2 (s, t) (u, v). R1 s u & R2 t v"
+
+lemma RPROD_DEF: "RPROD R1 R2 = (%(s, t) (u, v). R1 s u & R2 t v)"
+ sorry
+
+lemma WF_RPROD: "WF R & WF Q ==> WF (RPROD R Q)"
+ sorry
;end_setup
@@ -1073,174 +828,113 @@
;setup_theory prim_rec
lemma LESS_0_0: "0 < Suc 0"
- by (import prim_rec LESS_0_0)
-
-lemma LESS_LEMMA1: "ALL (x::nat) xa::nat. x < Suc xa --> x = xa | x < xa"
- by (import prim_rec LESS_LEMMA1)
-
-lemma LESS_LEMMA2: "ALL (m::nat) n::nat. m = n | m < n --> m < Suc n"
- by (import prim_rec LESS_LEMMA2)
-
-lemma LESS_THM: "ALL (m::nat) n::nat. (m < Suc n) = (m = n | m < n)"
- by (import prim_rec LESS_THM)
-
-lemma LESS_SUC_IMP: "ALL (x::nat) xa::nat. x < Suc xa --> x ~= xa --> x < xa"
- by (import prim_rec LESS_SUC_IMP)
-
-lemma EQ_LESS: "ALL n::nat. Suc (m::nat) = n --> m < n"
- by (import prim_rec EQ_LESS)
-
-lemma NOT_LESS_EQ: "ALL (m::nat) n::nat. m = n --> ~ m < n"
- by (import prim_rec NOT_LESS_EQ)
-
-definition SIMP_REC_REL :: "(nat => 'a) => 'a => ('a => 'a) => nat => bool" where
- "(op ==::((nat => 'a::type)
- => 'a::type => ('a::type => 'a::type) => nat => bool)
- => ((nat => 'a::type)
- => 'a::type => ('a::type => 'a::type) => nat => bool)
- => prop)
- (SIMP_REC_REL::(nat => 'a::type)
- => 'a::type => ('a::type => 'a::type) => nat => bool)
- (%(fun::nat => 'a::type) (x::'a::type) (f::'a::type => 'a::type) n::nat.
- (op &::bool => bool => bool)
- ((op =::'a::type => 'a::type => bool) (fun (0::nat)) x)
- ((All::(nat => bool) => bool)
- (%m::nat.
- (op -->::bool => bool => bool) ((op <::nat => nat => bool) m n)
- ((op =::'a::type => 'a::type => bool)
- (fun ((Suc::nat => nat) m)) (f (fun m))))))"
-
-lemma SIMP_REC_REL: "(All::((nat => 'a::type) => bool) => bool)
- (%fun::nat => 'a::type.
- (All::('a::type => bool) => bool)
- (%x::'a::type.
- (All::(('a::type => 'a::type) => bool) => bool)
- (%f::'a::type => 'a::type.
- (All::(nat => bool) => bool)
- (%n::nat.
- (op =::bool => bool => bool)
- ((SIMP_REC_REL::(nat => 'a::type)
- => 'a::type
- => ('a::type => 'a::type) => nat => bool)
- fun x f n)
- ((op &::bool => bool => bool)
- ((op =::'a::type => 'a::type => bool) (fun (0::nat))
- x)
- ((All::(nat => bool) => bool)
- (%m::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) m n)
- ((op =::'a::type => 'a::type => bool)
- (fun ((Suc::nat => nat) m))
- (f (fun m))))))))))"
- by (import prim_rec SIMP_REC_REL)
-
-lemma SIMP_REC_EXISTS: "ALL (x::'a::type) (f::'a::type => 'a::type) n::nat.
- EX fun::nat => 'a::type. SIMP_REC_REL fun x f n"
- by (import prim_rec SIMP_REC_EXISTS)
-
-lemma SIMP_REC_REL_UNIQUE: "ALL (x::'a::type) (xa::'a::type => 'a::type) (xb::nat => 'a::type)
- (xc::nat => 'a::type) (xd::nat) xe::nat.
- SIMP_REC_REL xb x xa xd & SIMP_REC_REL xc x xa xe -->
- (ALL n::nat. n < xd & n < xe --> xb n = xc n)"
- by (import prim_rec SIMP_REC_REL_UNIQUE)
-
-lemma SIMP_REC_REL_UNIQUE_RESULT: "ALL (x::'a::type) (f::'a::type => 'a::type) n::nat.
- EX! y::'a::type.
- EX g::nat => 'a::type. SIMP_REC_REL g x f (Suc n) & y = g n"
- by (import prim_rec SIMP_REC_REL_UNIQUE_RESULT)
+ sorry
+
+lemma LESS_LEMMA1: "x < Suc xa ==> x = xa | x < xa"
+ sorry
+
+lemma LESS_LEMMA2: "m = n | m < n ==> m < Suc n"
+ sorry
+
+lemma LESS_THM: "(m < Suc n) = (m = n | m < n)"
+ sorry
+
+lemma LESS_SUC_IMP: "[| x < Suc xa; x ~= xa |] ==> x < xa"
+ sorry
+
+lemma EQ_LESS: "Suc m = n ==> m < n"
+ sorry
+
+lemma NOT_LESS_EQ: "(m::nat) = (n::nat) ==> ~ m < n"
+ sorry
+
+definition
+ SIMP_REC_REL :: "(nat => 'a) => 'a => ('a => 'a) => nat => bool" where
+ "SIMP_REC_REL == %fun x f n. fun 0 = x & (ALL m<n. fun (Suc m) = f (fun m))"
+
+lemma SIMP_REC_REL: "SIMP_REC_REL fun x f n = (fun 0 = x & (ALL m<n. fun (Suc m) = f (fun m)))"
+ sorry
+
+lemma SIMP_REC_EXISTS: "EX fun. SIMP_REC_REL fun x f n"
+ sorry
+
+lemma SIMP_REC_REL_UNIQUE: "[| SIMP_REC_REL xb x xa xd & SIMP_REC_REL xc x xa xe; n < xd & n < xe |]
+==> xb n = xc n"
+ sorry
+
+lemma SIMP_REC_REL_UNIQUE_RESULT: "EX! y. EX g. SIMP_REC_REL g x f (Suc n) & y = g n"
+ sorry
consts
SIMP_REC :: "'a => ('a => 'a) => nat => 'a"
-specification (SIMP_REC) SIMP_REC: "ALL (x::'a::type) (f'::'a::type => 'a::type) n::nat.
- EX g::nat => 'a::type.
- SIMP_REC_REL g x f' (Suc n) & SIMP_REC x f' n = g n"
- by (import prim_rec SIMP_REC)
-
-lemma LESS_SUC_SUC: "ALL m::nat. m < Suc m & m < Suc (Suc m)"
- by (import prim_rec LESS_SUC_SUC)
-
-lemma SIMP_REC_THM: "ALL (x::'a::type) f::'a::type => 'a::type.
- SIMP_REC x f 0 = x &
- (ALL m::nat. SIMP_REC x f (Suc m) = f (SIMP_REC x f m))"
- by (import prim_rec SIMP_REC_THM)
-
-definition PRE :: "nat => nat" where
- "PRE == %m::nat. if m = 0 then 0 else SOME n::nat. m = Suc n"
-
-lemma PRE_DEF: "ALL m::nat. PRE m = (if m = 0 then 0 else SOME n::nat. m = Suc n)"
- by (import prim_rec PRE_DEF)
-
-lemma PRE: "PRE 0 = 0 & (ALL m::nat. PRE (Suc m) = m)"
- by (import prim_rec PRE)
-
-definition PRIM_REC_FUN :: "'a => ('a => nat => 'a) => nat => nat => 'a" where
- "PRIM_REC_FUN ==
-%(x::'a::type) f::'a::type => nat => 'a::type.
- SIMP_REC (%n::nat. x) (%(fun::nat => 'a::type) n::nat. f (fun (PRE n)) n)"
-
-lemma PRIM_REC_FUN: "ALL (x::'a::type) f::'a::type => nat => 'a::type.
- PRIM_REC_FUN x f =
- SIMP_REC (%n::nat. x) (%(fun::nat => 'a::type) n::nat. f (fun (PRE n)) n)"
- by (import prim_rec PRIM_REC_FUN)
-
-lemma PRIM_REC_EQN: "ALL (x::'a::type) f::'a::type => nat => 'a::type.
- (ALL n::nat. PRIM_REC_FUN x f 0 n = x) &
- (ALL (m::nat) n::nat.
- PRIM_REC_FUN x f (Suc m) n = f (PRIM_REC_FUN x f m (PRE n)) n)"
- by (import prim_rec PRIM_REC_EQN)
-
-definition PRIM_REC :: "'a => ('a => nat => 'a) => nat => 'a" where
- "PRIM_REC ==
-%(x::'a::type) (f::'a::type => nat => 'a::type) m::nat.
- PRIM_REC_FUN x f m (PRE m)"
-
-lemma PRIM_REC: "ALL (x::'a::type) (f::'a::type => nat => 'a::type) m::nat.
- PRIM_REC x f m = PRIM_REC_FUN x f m (PRE m)"
- by (import prim_rec PRIM_REC)
-
-lemma PRIM_REC_THM: "ALL (x::'a::type) f::'a::type => nat => 'a::type.
- PRIM_REC x f 0 = x &
- (ALL m::nat. PRIM_REC x f (Suc m) = f (PRIM_REC x f m) m)"
- by (import prim_rec PRIM_REC_THM)
-
-lemma DC: "ALL (P::'a::type => bool) (R::'a::type => 'a::type => bool) a::'a::type.
- P a & (ALL x::'a::type. P x --> (EX y::'a::type. P y & R x y)) -->
- (EX x::nat => 'a::type.
- x 0 = a & (ALL n::nat. P (x n) & R (x n) (x (Suc n))))"
- by (import prim_rec DC)
-
-lemma num_Axiom_old: "ALL (e::'a::type) f::'a::type => nat => 'a::type.
- EX! fn1::nat => 'a::type.
- fn1 0 = e & (ALL n::nat. fn1 (Suc n) = f (fn1 n) n)"
- by (import prim_rec num_Axiom_old)
-
-lemma num_Axiom: "ALL (e::'a::type) f::nat => 'a::type => 'a::type.
- EX x::nat => 'a::type. x 0 = e & (ALL n::nat. x (Suc n) = f n (x n))"
- by (import prim_rec num_Axiom)
+specification (SIMP_REC) SIMP_REC: "ALL x f' n. EX g. SIMP_REC_REL g x f' (Suc n) & SIMP_REC x f' n = g n"
+ sorry
+
+lemma LESS_SUC_SUC: "m < Suc m & m < Suc (Suc m)"
+ sorry
+
+lemma SIMP_REC_THM: "SIMP_REC x f 0 = x & (ALL m. SIMP_REC x f (Suc m) = f (SIMP_REC x f m))"
+ sorry
+
+definition
+ PRE :: "nat => nat" where
+ "PRE == %m. if m = 0 then 0 else SOME n. m = Suc n"
+
+lemma PRE_DEF: "PRE m = (if m = 0 then 0 else SOME n. m = Suc n)"
+ sorry
+
+lemma PRE: "PRE 0 = 0 & (ALL m. PRE (Suc m) = m)"
+ sorry
+
+definition
+ PRIM_REC_FUN :: "'a => ('a => nat => 'a) => nat => nat => 'a" where
+ "PRIM_REC_FUN == %x f. SIMP_REC (%n. x) (%fun n. f (fun (PRE n)) n)"
+
+lemma PRIM_REC_FUN: "PRIM_REC_FUN x f = SIMP_REC (%n. x) (%fun n. f (fun (PRE n)) n)"
+ sorry
+
+lemma PRIM_REC_EQN: "(ALL n. PRIM_REC_FUN x f 0 n = x) &
+(ALL m n. PRIM_REC_FUN x f (Suc m) n = f (PRIM_REC_FUN x f m (PRE n)) n)"
+ sorry
+
+definition
+ PRIM_REC :: "'a => ('a => nat => 'a) => nat => 'a" where
+ "PRIM_REC == %x f m. PRIM_REC_FUN x f m (PRE m)"
+
+lemma PRIM_REC: "PRIM_REC x f m = PRIM_REC_FUN x f m (PRE m)"
+ sorry
+
+lemma PRIM_REC_THM: "PRIM_REC x f 0 = x & (ALL m. PRIM_REC x f (Suc m) = f (PRIM_REC x f m) m)"
+ sorry
+
+lemma DC: "P a & (ALL x. P x --> (EX y. P y & R x y))
+==> EX x. x 0 = a & (ALL n. P (x n) & R (x n) (x (Suc n)))"
+ sorry
+
+lemma num_Axiom_old: "EX! fn1. fn1 0 = e & (ALL n. fn1 (Suc n) = f (fn1 n) n)"
+ sorry
+
+lemma num_Axiom: "EX x. x 0 = e & (ALL n. x (Suc n) = f n (x n))"
+ sorry
consts
wellfounded :: "('a => 'a => bool) => bool"
defs
- wellfounded_primdef: "wellfounded ==
-%R::'a::type => 'a::type => bool.
- ~ (EX f::nat => 'a::type. ALL n::nat. R (f (Suc n)) (f n))"
-
-lemma wellfounded_def: "ALL R::'a::type => 'a::type => bool.
- wellfounded R =
- (~ (EX f::nat => 'a::type. ALL n::nat. R (f (Suc n)) (f n)))"
- by (import prim_rec wellfounded_def)
-
-lemma WF_IFF_WELLFOUNDED: "ALL R::'a::type => 'a::type => bool. WF R = wellfounded R"
- by (import prim_rec WF_IFF_WELLFOUNDED)
-
-lemma WF_PRED: "WF (%(x::nat) y::nat. y = Suc x)"
- by (import prim_rec WF_PRED)
+ wellfounded_primdef: "wellfounded == %R. ~ (EX f. ALL n. R (f (Suc n)) (f n))"
+
+lemma wellfounded_def: "wellfounded R = (~ (EX f. ALL n. R (f (Suc n)) (f n)))"
+ sorry
+
+lemma WF_IFF_WELLFOUNDED: "WF R = wellfounded R"
+ sorry
+
+lemma WF_PRED: "WF (%x y. y = Suc x)"
+ sorry
lemma WF_LESS: "(WF::(nat => nat => bool) => bool) (op <::nat => nat => bool)"
- by (import prim_rec WF_LESS)
+ sorry
consts
measure :: "('a => nat) => 'a => 'a => bool"
@@ -1249,616 +943,533 @@
measure_primdef: "prim_rec.measure == relation.inv_image op <"
lemma measure_def: "prim_rec.measure = relation.inv_image op <"
- by (import prim_rec measure_def)
-
-lemma WF_measure: "ALL x::'a::type => nat. WF (prim_rec.measure x)"
- by (import prim_rec WF_measure)
-
-lemma measure_thm: "ALL (x::'a::type => nat) (xa::'a::type) xb::'a::type.
- prim_rec.measure x xa xb = (x xa < x xb)"
- by (import prim_rec measure_thm)
+ sorry
+
+lemma WF_measure: "WF (prim_rec.measure x)"
+ sorry
+
+lemma measure_thm: "prim_rec.measure x xa xb = (x xa < x xb)"
+ sorry
;end_setup
;setup_theory arithmetic
-definition nat_elim__magic :: "nat => nat" where
- "nat_elim__magic == %n::nat. n"
-
-lemma nat_elim__magic: "ALL n::nat. nat_elim__magic n = n"
- by (import arithmetic nat_elim__magic)
+definition
+ nat_elim__magic :: "nat => nat" where
+ "nat_elim__magic == %n. n"
+
+lemma nat_elim__magic: "nat_elim__magic n = n"
+ sorry
consts
EVEN :: "nat => bool"
-specification (EVEN) EVEN: "EVEN 0 = True & (ALL n::nat. EVEN (Suc n) = (~ EVEN n))"
- by (import arithmetic EVEN)
+specification (EVEN) EVEN: "EVEN 0 = True & (ALL n. EVEN (Suc n) = (~ EVEN n))"
+ sorry
consts
ODD :: "nat => bool"
-specification (ODD) ODD: "ODD 0 = False & (ALL n::nat. ODD (Suc n) = (~ ODD n))"
- by (import arithmetic ODD)
+specification (ODD) ODD: "ODD 0 = False & (ALL n. ODD (Suc n) = (~ ODD n))"
+ sorry
lemma TWO: "2 = Suc 1"
- by (import arithmetic TWO)
-
-lemma NORM_0: "(op =::nat => nat => bool) (0::nat) (0::nat)"
- by (import arithmetic NORM_0)
-
-lemma num_case_compute: "ALL n::nat.
- nat_case (f::'a::type) (g::nat => 'a::type) n =
- (if n = 0 then f else g (PRE n))"
- by (import arithmetic num_case_compute)
-
-lemma ADD_CLAUSES: "0 + (m::nat) = m &
-m + 0 = m & Suc m + (n::nat) = Suc (m + n) & m + Suc n = Suc (m + n)"
- by (import arithmetic ADD_CLAUSES)
-
-lemma LESS_ADD: "ALL (m::nat) n::nat. n < m --> (EX p::nat. p + n = m)"
- by (import arithmetic LESS_ADD)
-
-lemma LESS_ANTISYM: "ALL (m::nat) n::nat. ~ (m < n & n < m)"
- by (import arithmetic LESS_ANTISYM)
-
-lemma LESS_LESS_SUC: "ALL (x::nat) xa::nat. ~ (x < xa & xa < Suc x)"
- by (import arithmetic LESS_LESS_SUC)
-
-lemma FUN_EQ_LEMMA: "ALL (f::'a::type => bool) (x1::'a::type) x2::'a::type.
- f x1 & ~ f x2 --> x1 ~= x2"
- by (import arithmetic FUN_EQ_LEMMA)
-
-lemma LESS_NOT_SUC: "ALL (m::nat) n::nat. m < n & n ~= Suc m --> Suc m < n"
- by (import arithmetic LESS_NOT_SUC)
-
-lemma LESS_0_CASES: "ALL m::nat. 0 = m | 0 < m"
- by (import arithmetic LESS_0_CASES)
-
-lemma LESS_CASES_IMP: "ALL (m::nat) n::nat. ~ m < n & m ~= n --> n < m"
- by (import arithmetic LESS_CASES_IMP)
-
-lemma LESS_CASES: "ALL (m::nat) n::nat. m < n | n <= m"
- by (import arithmetic LESS_CASES)
-
-lemma LESS_EQ_SUC_REFL: "ALL m::nat. m <= Suc m"
- by (import arithmetic LESS_EQ_SUC_REFL)
-
-lemma LESS_ADD_NONZERO: "ALL (m::nat) n::nat. n ~= 0 --> m < m + n"
- by (import arithmetic LESS_ADD_NONZERO)
-
-lemma LESS_EQ_ANTISYM: "ALL (x::nat) xa::nat. ~ (x < xa & xa <= x)"
- by (import arithmetic LESS_EQ_ANTISYM)
-
-lemma SUB_0: "ALL m::nat. 0 - m = 0 & m - 0 = m"
- by (import arithmetic SUB_0)
-
-lemma SUC_SUB1: "ALL m::nat. Suc m - 1 = m"
- by (import arithmetic SUC_SUB1)
-
-lemma PRE_SUB1: "ALL m::nat. PRE m = m - 1"
- by (import arithmetic PRE_SUB1)
-
-lemma MULT_CLAUSES: "ALL (x::nat) xa::nat.
- 0 * x = 0 &
- x * 0 = 0 &
- 1 * x = x &
- x * 1 = x & Suc x * xa = x * xa + xa & x * Suc xa = x + x * xa"
- by (import arithmetic MULT_CLAUSES)
-
-lemma PRE_SUB: "ALL (m::nat) n::nat. PRE (m - n) = PRE m - n"
- by (import arithmetic PRE_SUB)
-
-lemma ADD_EQ_1: "ALL (m::nat) n::nat. (m + n = 1) = (m = 1 & n = 0 | m = 0 & n = 1)"
- by (import arithmetic ADD_EQ_1)
-
-lemma ADD_INV_0_EQ: "ALL (m::nat) n::nat. (m + n = m) = (n = 0)"
- by (import arithmetic ADD_INV_0_EQ)
-
-lemma PRE_SUC_EQ: "ALL (m::nat) n::nat. 0 < n --> (m = PRE n) = (Suc m = n)"
- by (import arithmetic PRE_SUC_EQ)
-
-lemma INV_PRE_EQ: "ALL (m::nat) n::nat. 0 < m & 0 < n --> (PRE m = PRE n) = (m = n)"
- by (import arithmetic INV_PRE_EQ)
-
-lemma LESS_SUC_NOT: "ALL (m::nat) n::nat. m < n --> ~ n < Suc m"
- by (import arithmetic LESS_SUC_NOT)
-
-lemma ADD_EQ_SUB: "ALL (m::nat) (n::nat) p::nat. n <= p --> (m + n = p) = (m = p - n)"
- by (import arithmetic ADD_EQ_SUB)
-
-lemma LESS_ADD_1: "ALL (x::nat) xa::nat. xa < x --> (EX xb::nat. x = xa + (xb + 1))"
- by (import arithmetic LESS_ADD_1)
-
-lemma NOT_ODD_EQ_EVEN: "ALL (n::nat) m::nat. Suc (n + n) ~= m + m"
- by (import arithmetic NOT_ODD_EQ_EVEN)
-
-lemma MULT_SUC_EQ: "ALL (p::nat) (m::nat) n::nat. (n * Suc p = m * Suc p) = (n = m)"
- by (import arithmetic MULT_SUC_EQ)
-
-lemma MULT_EXP_MONO: "ALL (p::nat) (q::nat) (n::nat) m::nat.
- (n * Suc q ^ p = m * Suc q ^ p) = (n = m)"
- by (import arithmetic MULT_EXP_MONO)
-
-lemma LESS_ADD_SUC: "ALL (m::nat) n::nat. m < m + Suc n"
- by (import arithmetic LESS_ADD_SUC)
-
-lemma LESS_OR_EQ_ADD: "ALL (n::nat) m::nat. n < m | (EX p::nat. n = p + m)"
- by (import arithmetic LESS_OR_EQ_ADD)
-
-lemma WOP: "(All::((nat => bool) => bool) => bool)
- (%P::nat => bool.
- (op -->::bool => bool => bool) ((Ex::(nat => bool) => bool) P)
- ((Ex::(nat => bool) => bool)
- (%n::nat.
- (op &::bool => bool => bool) (P n)
- ((All::(nat => bool) => bool)
- (%m::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) m n)
- ((Not::bool => bool) (P m)))))))"
- by (import arithmetic WOP)
-
-lemma INV_PRE_LESS: "ALL m>0. ALL n::nat. (PRE m < PRE n) = (m < n)"
- by (import arithmetic INV_PRE_LESS)
-
-lemma INV_PRE_LESS_EQ: "ALL n>0. ALL m::nat. (PRE m <= PRE n) = (m <= n)"
- by (import arithmetic INV_PRE_LESS_EQ)
-
-lemma SUB_EQ_EQ_0: "ALL (m::nat) n::nat. (m - n = m) = (m = 0 | n = 0)"
- by (import arithmetic SUB_EQ_EQ_0)
-
-lemma SUB_LESS_OR: "ALL (m::nat) n::nat. n < m --> n <= m - 1"
- by (import arithmetic SUB_LESS_OR)
-
-lemma LESS_SUB_ADD_LESS: "ALL (n::nat) (m::nat) i::nat. i < n - m --> i + m < n"
- by (import arithmetic LESS_SUB_ADD_LESS)
-
-lemma LESS_EQ_SUB_LESS: "ALL (x::nat) xa::nat. xa <= x --> (ALL c::nat. (x - xa < c) = (x < xa + c))"
- by (import arithmetic LESS_EQ_SUB_LESS)
-
-lemma NOT_SUC_LESS_EQ: "ALL (x::nat) xa::nat. (~ Suc x <= xa) = (xa <= x)"
- by (import arithmetic NOT_SUC_LESS_EQ)
-
-lemma SUB_LESS_EQ_ADD: "ALL (m::nat) p::nat. m <= p --> (ALL n::nat. (p - m <= n) = (p <= m + n))"
- by (import arithmetic SUB_LESS_EQ_ADD)
-
-lemma SUB_CANCEL: "ALL (x::nat) (xa::nat) xb::nat.
- xa <= x & xb <= x --> (x - xa = x - xb) = (xa = xb)"
- by (import arithmetic SUB_CANCEL)
-
-lemma NOT_EXP_0: "ALL (m::nat) n::nat. Suc n ^ m ~= 0"
- by (import arithmetic NOT_EXP_0)
-
-lemma ZERO_LESS_EXP: "ALL (m::nat) n::nat. 0 < Suc n ^ m"
- by (import arithmetic ZERO_LESS_EXP)
-
-lemma ODD_OR_EVEN: "ALL x::nat. EX xa::nat. x = Suc (Suc 0) * xa | x = Suc (Suc 0) * xa + 1"
- by (import arithmetic ODD_OR_EVEN)
-
-lemma LESS_EXP_SUC_MONO: "ALL (n::nat) m::nat. Suc (Suc m) ^ n < Suc (Suc m) ^ Suc n"
- by (import arithmetic LESS_EXP_SUC_MONO)
-
-lemma LESS_LESS_CASES: "ALL (m::nat) n::nat. m = n | m < n | n < m"
- by (import arithmetic LESS_LESS_CASES)
-
-lemma LESS_EQUAL_ADD: "ALL (m::nat) n::nat. m <= n --> (EX p::nat. n = m + p)"
- by (import arithmetic LESS_EQUAL_ADD)
-
-lemma MULT_EQ_1: "ALL (x::nat) y::nat. (x * y = 1) = (x = 1 & y = 1)"
- by (import arithmetic MULT_EQ_1)
+ sorry
+
+lemma NORM_0: "(0::nat) = (0::nat)"
+ sorry
+
+lemma num_case_compute: "nat_case f g n = (if n = 0 then f else g (PRE n))"
+ sorry
+
+lemma ADD_CLAUSES: "0 + m = m & m + 0 = m & Suc m + n = Suc (m + n) & m + Suc n = Suc (m + n)"
+ sorry
+
+lemma LESS_ADD: "(n::nat) < (m::nat) ==> EX p::nat. p + n = m"
+ sorry
+
+lemma LESS_ANTISYM: "~ ((m::nat) < (n::nat) & n < m)"
+ sorry
+
+lemma LESS_LESS_SUC: "~ (x < xa & xa < Suc x)"
+ sorry
+
+lemma FUN_EQ_LEMMA: "f x1 & ~ f x2 ==> x1 ~= x2"
+ sorry
+
+lemma LESS_NOT_SUC: "m < n & n ~= Suc m ==> Suc m < n"
+ sorry
+
+lemma LESS_0_CASES: "(0::nat) = (m::nat) | (0::nat) < m"
+ sorry
+
+lemma LESS_CASES_IMP: "~ (m::nat) < (n::nat) & m ~= n ==> n < m"
+ sorry
+
+lemma LESS_CASES: "(m::nat) < (n::nat) | n <= m"
+ sorry
+
+lemma LESS_EQ_SUC_REFL: "m <= Suc m"
+ sorry
+
+lemma LESS_ADD_NONZERO: "(n::nat) ~= (0::nat) ==> (m::nat) < m + n"
+ sorry
+
+lemma LESS_EQ_ANTISYM: "~ ((x::nat) < (xa::nat) & xa <= x)"
+ sorry
+
+lemma SUB_0: "(0::nat) - (m::nat) = (0::nat) & m - (0::nat) = m"
+ sorry
+
+lemma PRE_SUB1: "PRE m = m - 1"
+ sorry
+
+lemma MULT_CLAUSES: "0 * x = 0 &
+x * 0 = 0 &
+1 * x = x & x * 1 = x & Suc x * xa = x * xa + xa & x * Suc xa = x + x * xa"
+ sorry
+
+lemma PRE_SUB: "PRE (m - n) = PRE m - n"
+ sorry
+
+lemma ADD_EQ_1: "((m::nat) + (n::nat) = (1::nat)) =
+(m = (1::nat) & n = (0::nat) | m = (0::nat) & n = (1::nat))"
+ sorry
+
+lemma ADD_INV_0_EQ: "((m::nat) + (n::nat) = m) = (n = (0::nat))"
+ sorry
+
+lemma PRE_SUC_EQ: "0 < n ==> (m = PRE n) = (Suc m = n)"
+ sorry
+
+lemma INV_PRE_EQ: "0 < m & 0 < n ==> (PRE m = PRE n) = (m = n)"
+ sorry
+
+lemma LESS_SUC_NOT: "m < n ==> ~ n < Suc m"
+ sorry
+
+lemma ADD_EQ_SUB: "(n::nat) <= (p::nat) ==> ((m::nat) + n = p) = (m = p - n)"
+ sorry
+
+lemma LESS_ADD_1: "(xa::nat) < (x::nat) ==> EX xb::nat. x = xa + (xb + (1::nat))"
+ sorry
+
+lemma NOT_ODD_EQ_EVEN: "Suc (n + n) ~= m + m"
+ sorry
+
+lemma MULT_SUC_EQ: "(n * Suc p = m * Suc p) = (n = m)"
+ sorry
+
+lemma MULT_EXP_MONO: "(n * Suc q ^ p = m * Suc q ^ p) = (n = m)"
+ sorry
+
+lemma LESS_ADD_SUC: "m < m + Suc n"
+ sorry
+
+lemma LESS_OR_EQ_ADD: "(n::nat) < (m::nat) | (EX p::nat. n = p + m)"
+ sorry
+
+lemma WOP: "Ex (P::nat => bool) ==> EX n::nat. P n & (ALL m<n. ~ P m)"
+ sorry
+
+lemma INV_PRE_LESS: "0 < m ==> (PRE m < PRE n) = (m < n)"
+ sorry
+
+lemma INV_PRE_LESS_EQ: "0 < n ==> (PRE m <= PRE n) = (m <= n)"
+ sorry
+
+lemma SUB_EQ_EQ_0: "((m::nat) - (n::nat) = m) = (m = (0::nat) | n = (0::nat))"
+ sorry
+
+lemma SUB_LESS_OR: "(n::nat) < (m::nat) ==> n <= m - (1::nat)"
+ sorry
+
+lemma LESS_SUB_ADD_LESS: "(i::nat) < (n::nat) - (m::nat) ==> i + m < n"
+ sorry
+
+lemma LESS_EQ_SUB_LESS: "(xa::nat) <= (x::nat) ==> (x - xa < (c::nat)) = (x < xa + c)"
+ sorry
+
+lemma NOT_SUC_LESS_EQ: "(~ Suc x <= xa) = (xa <= x)"
+ sorry
+
+lemma SUB_LESS_EQ_ADD: "(m::nat) <= (p::nat) ==> (p - m <= (n::nat)) = (p <= m + n)"
+ sorry
+
+lemma SUB_CANCEL: "(xa::nat) <= (x::nat) & (xb::nat) <= x ==> (x - xa = x - xb) = (xa = xb)"
+ sorry
+
+lemma NOT_EXP_0: "Suc n ^ m ~= 0"
+ sorry
+
+lemma ZERO_LESS_EXP: "0 < Suc n ^ m"
+ sorry
+
+lemma ODD_OR_EVEN: "EX xa. x = Suc (Suc 0) * xa | x = Suc (Suc 0) * xa + 1"
+ sorry
+
+lemma LESS_EXP_SUC_MONO: "Suc (Suc m) ^ n < Suc (Suc m) ^ Suc n"
+ sorry
+
+lemma LESS_LESS_CASES: "(m::nat) = (n::nat) | m < n | n < m"
+ sorry
consts
FACT :: "nat => nat"
-specification (FACT) FACT: "FACT 0 = 1 & (ALL n::nat. FACT (Suc n) = Suc n * FACT n)"
- by (import arithmetic FACT)
-
-lemma FACT_LESS: "ALL n::nat. 0 < FACT n"
- by (import arithmetic FACT_LESS)
-
-lemma EVEN_ODD: "ALL n::nat. EVEN n = (~ ODD n)"
- by (import arithmetic EVEN_ODD)
-
-lemma ODD_EVEN: "ALL x::nat. ODD x = (~ EVEN x)"
- by (import arithmetic ODD_EVEN)
-
-lemma EVEN_OR_ODD: "ALL x::nat. EVEN x | ODD x"
- by (import arithmetic EVEN_OR_ODD)
-
-lemma EVEN_AND_ODD: "ALL x::nat. ~ (EVEN x & ODD x)"
- by (import arithmetic EVEN_AND_ODD)
-
-lemma EVEN_ADD: "ALL (m::nat) n::nat. EVEN (m + n) = (EVEN m = EVEN n)"
- by (import arithmetic EVEN_ADD)
-
-lemma EVEN_MULT: "ALL (m::nat) n::nat. EVEN (m * n) = (EVEN m | EVEN n)"
- by (import arithmetic EVEN_MULT)
-
-lemma ODD_ADD: "ALL (m::nat) n::nat. ODD (m + n) = (ODD m ~= ODD n)"
- by (import arithmetic ODD_ADD)
-
-lemma ODD_MULT: "ALL (m::nat) n::nat. ODD (m * n) = (ODD m & ODD n)"
- by (import arithmetic ODD_MULT)
-
-lemma EVEN_DOUBLE: "ALL n::nat. EVEN (2 * n)"
- by (import arithmetic EVEN_DOUBLE)
-
-lemma ODD_DOUBLE: "ALL x::nat. ODD (Suc (2 * x))"
- by (import arithmetic ODD_DOUBLE)
-
-lemma EVEN_ODD_EXISTS: "ALL x::nat.
- (EVEN x --> (EX m::nat. x = 2 * m)) &
- (ODD x --> (EX m::nat. x = Suc (2 * m)))"
- by (import arithmetic EVEN_ODD_EXISTS)
-
-lemma EVEN_EXISTS: "ALL n::nat. EVEN n = (EX m::nat. n = 2 * m)"
- by (import arithmetic EVEN_EXISTS)
-
-lemma ODD_EXISTS: "ALL n::nat. ODD n = (EX m::nat. n = Suc (2 * m))"
- by (import arithmetic ODD_EXISTS)
-
-lemma NOT_SUC_LESS_EQ_0: "ALL x::nat. ~ Suc x <= 0"
- by (import arithmetic NOT_SUC_LESS_EQ_0)
-
-lemma NOT_LEQ: "ALL (x::nat) xa::nat. (~ x <= xa) = (Suc xa <= x)"
- by (import arithmetic NOT_LEQ)
-
-lemma NOT_NUM_EQ: "ALL (x::nat) xa::nat. (x ~= xa) = (Suc x <= xa | Suc xa <= x)"
- by (import arithmetic NOT_NUM_EQ)
-
-lemma NOT_GREATER_EQ: "ALL (x::nat) xa::nat. (~ xa <= x) = (Suc x <= xa)"
- by (import arithmetic NOT_GREATER_EQ)
-
-lemma SUC_ADD_SYM: "ALL (m::nat) n::nat. Suc (m + n) = Suc n + m"
- by (import arithmetic SUC_ADD_SYM)
-
-lemma NOT_SUC_ADD_LESS_EQ: "ALL (m::nat) n::nat. ~ Suc (m + n) <= m"
- by (import arithmetic NOT_SUC_ADD_LESS_EQ)
-
-lemma SUB_LEFT_ADD: "ALL (m::nat) (n::nat) p::nat.
- m + (n - p) = (if n <= p then m else m + n - p)"
- by (import arithmetic SUB_LEFT_ADD)
-
-lemma SUB_RIGHT_ADD: "ALL (m::nat) (n::nat) p::nat. m - n + p = (if m <= n then p else m + p - n)"
- by (import arithmetic SUB_RIGHT_ADD)
-
-lemma SUB_LEFT_SUB: "ALL (m::nat) (n::nat) p::nat.
- m - (n - p) = (if n <= p then m else m + p - n)"
- by (import arithmetic SUB_LEFT_SUB)
-
-lemma SUB_LEFT_SUC: "ALL (m::nat) n::nat. Suc (m - n) = (if m <= n then Suc 0 else Suc m - n)"
- by (import arithmetic SUB_LEFT_SUC)
-
-lemma SUB_LEFT_LESS_EQ: "ALL (m::nat) (n::nat) p::nat. (m <= n - p) = (m + p <= n | m <= 0)"
- by (import arithmetic SUB_LEFT_LESS_EQ)
-
-lemma SUB_RIGHT_LESS_EQ: "ALL (m::nat) (n::nat) p::nat. (m - n <= p) = (m <= n + p)"
- by (import arithmetic SUB_RIGHT_LESS_EQ)
-
-lemma SUB_RIGHT_LESS: "ALL (m::nat) (n::nat) p::nat. (m - n < p) = (m < n + p & 0 < p)"
- by (import arithmetic SUB_RIGHT_LESS)
-
-lemma SUB_RIGHT_GREATER_EQ: "ALL (m::nat) (n::nat) p::nat. (p <= m - n) = (n + p <= m | p <= 0)"
- by (import arithmetic SUB_RIGHT_GREATER_EQ)
-
-lemma SUB_LEFT_GREATER: "ALL (m::nat) (n::nat) p::nat. (n - p < m) = (n < m + p & 0 < m)"
- by (import arithmetic SUB_LEFT_GREATER)
-
-lemma SUB_RIGHT_GREATER: "ALL (m::nat) (n::nat) p::nat. (p < m - n) = (n + p < m)"
- by (import arithmetic SUB_RIGHT_GREATER)
-
-lemma SUB_LEFT_EQ: "ALL (m::nat) (n::nat) p::nat. (m = n - p) = (m + p = n | m <= 0 & n <= p)"
- by (import arithmetic SUB_LEFT_EQ)
-
-lemma SUB_RIGHT_EQ: "ALL (m::nat) (n::nat) p::nat. (m - n = p) = (m = n + p | m <= n & p <= 0)"
- by (import arithmetic SUB_RIGHT_EQ)
-
-lemma LE: "(ALL n::nat. (n <= 0) = (n = 0)) &
+specification (FACT) FACT: "FACT 0 = 1 & (ALL n. FACT (Suc n) = Suc n * FACT n)"
+ sorry
+
+lemma FACT_LESS: "0 < FACT n"
+ sorry
+
+lemma EVEN_ODD: "EVEN n = (~ ODD n)"
+ sorry
+
+lemma ODD_EVEN: "ODD x = (~ EVEN x)"
+ sorry
+
+lemma EVEN_OR_ODD: "EVEN x | ODD x"
+ sorry
+
+lemma EVEN_AND_ODD: "~ (EVEN x & ODD x)"
+ sorry
+
+lemma EVEN_ADD: "EVEN (m + n) = (EVEN m = EVEN n)"
+ sorry
+
+lemma EVEN_MULT: "EVEN (m * n) = (EVEN m | EVEN n)"
+ sorry
+
+lemma ODD_ADD: "ODD (m + n) = (ODD m ~= ODD n)"
+ sorry
+
+lemma ODD_MULT: "ODD (m * n) = (ODD m & ODD n)"
+ sorry
+
+lemma EVEN_DOUBLE: "EVEN (2 * n)"
+ sorry
+
+lemma ODD_DOUBLE: "ODD (Suc (2 * x))"
+ sorry
+
+lemma EVEN_ODD_EXISTS: "(EVEN x --> (EX m. x = 2 * m)) & (ODD x --> (EX m. x = Suc (2 * m)))"
+ sorry
+
+lemma EVEN_EXISTS: "EVEN n = (EX m. n = 2 * m)"
+ sorry
+
+lemma ODD_EXISTS: "ODD n = (EX m. n = Suc (2 * m))"
+ sorry
+
+lemma NOT_SUC_LESS_EQ_0: "~ Suc x <= 0"
+ sorry
+
+lemma NOT_NUM_EQ: "(x ~= xa) = (Suc x <= xa | Suc xa <= x)"
+ sorry
+
+lemma SUC_ADD_SYM: "Suc (m + n) = Suc n + m"
+ sorry
+
+lemma NOT_SUC_ADD_LESS_EQ: "~ Suc (m + n) <= m"
+ sorry
+
+lemma SUB_LEFT_ADD: "(m::nat) + ((n::nat) - (p::nat)) = (if n <= p then m else m + n - p)"
+ sorry
+
+lemma SUB_RIGHT_ADD: "(m::nat) - (n::nat) + (p::nat) = (if m <= n then p else m + p - n)"
+ sorry
+
+lemma SUB_LEFT_SUB: "(m::nat) - ((n::nat) - (p::nat)) = (if n <= p then m else m + p - n)"
+ sorry
+
+lemma SUB_LEFT_SUC: "Suc (m - n) = (if m <= n then Suc 0 else Suc m - n)"
+ sorry
+
+lemma SUB_LEFT_LESS_EQ: "((m::nat) <= (n::nat) - (p::nat)) = (m + p <= n | m <= (0::nat))"
+ sorry
+
+lemma SUB_RIGHT_LESS_EQ: "((m::nat) - (n::nat) <= (p::nat)) = (m <= n + p)"
+ sorry
+
+lemma SUB_RIGHT_LESS: "((m::nat) - (n::nat) < (p::nat)) = (m < n + p & (0::nat) < p)"
+ sorry
+
+lemma SUB_RIGHT_GREATER_EQ: "((p::nat) <= (m::nat) - (n::nat)) = (n + p <= m | p <= (0::nat))"
+ sorry
+
+lemma SUB_LEFT_GREATER: "((n::nat) - (p::nat) < (m::nat)) = (n < m + p & (0::nat) < m)"
+ sorry
+
+lemma SUB_RIGHT_GREATER: "((p::nat) < (m::nat) - (n::nat)) = (n + p < m)"
+ sorry
+
+lemma SUB_LEFT_EQ: "((m::nat) = (n::nat) - (p::nat)) = (m + p = n | m <= (0::nat) & n <= p)"
+ sorry
+
+lemma SUB_RIGHT_EQ: "((m::nat) - (n::nat) = (p::nat)) = (m = n + p | m <= n & p <= (0::nat))"
+ sorry
+
+lemma LE: "(ALL n::nat. (n <= (0::nat)) = (n = (0::nat))) &
(ALL (m::nat) n::nat. (m <= Suc n) = (m = Suc n | m <= n))"
- by (import arithmetic LE)
-
-lemma DA: "ALL (k::nat) n::nat. 0 < n --> (EX (x::nat) q::nat. k = q * n + x & x < n)"
- by (import arithmetic DA)
-
-lemma DIV_LESS_EQ: "ALL n>0. ALL k::nat. k div n <= k"
- by (import arithmetic DIV_LESS_EQ)
-
-lemma DIV_UNIQUE: "ALL (n::nat) (k::nat) q::nat.
- (EX r::nat. k = q * n + r & r < n) --> k div n = q"
- by (import arithmetic DIV_UNIQUE)
-
-lemma MOD_UNIQUE: "ALL (n::nat) (k::nat) r::nat.
- (EX q::nat. k = q * n + r & r < n) --> k mod n = r"
- by (import arithmetic MOD_UNIQUE)
-
-lemma DIV_MULT: "ALL (n::nat) r::nat. r < n --> (ALL q::nat. (q * n + r) div n = q)"
- by (import arithmetic DIV_MULT)
-
-lemma MOD_EQ_0: "ALL n>0. ALL k::nat. k * n mod n = 0"
- by (import arithmetic MOD_EQ_0)
-
-lemma ZERO_MOD: "(All::(nat => bool) => bool)
- (%n::nat.
- (op -->::bool => bool => bool) ((op <::nat => nat => bool) (0::nat) n)
- ((op =::nat => nat => bool) ((op mod::nat => nat => nat) (0::nat) n)
- (0::nat)))"
- by (import arithmetic ZERO_MOD)
-
-lemma ZERO_DIV: "(All::(nat => bool) => bool)
- (%n::nat.
- (op -->::bool => bool => bool) ((op <::nat => nat => bool) (0::nat) n)
- ((op =::nat => nat => bool) ((op div::nat => nat => nat) (0::nat) n)
- (0::nat)))"
- by (import arithmetic ZERO_DIV)
-
-lemma MOD_MULT: "ALL (n::nat) r::nat. r < n --> (ALL q::nat. (q * n + r) mod n = r)"
- by (import arithmetic MOD_MULT)
-
-lemma MOD_TIMES: "ALL n>0. ALL (q::nat) r::nat. (q * n + r) mod n = r mod n"
- by (import arithmetic MOD_TIMES)
-
-lemma MOD_PLUS: "ALL n>0. ALL (j::nat) k::nat. (j mod n + k mod n) mod n = (j + k) mod n"
- by (import arithmetic MOD_PLUS)
-
-lemma MOD_MOD: "ALL n>0. ALL k::nat. k mod n mod n = k mod n"
- by (import arithmetic MOD_MOD)
-
-lemma ADD_DIV_ADD_DIV: "ALL x>0. ALL (xa::nat) r::nat. (xa * x + r) div x = xa + r div x"
- by (import arithmetic ADD_DIV_ADD_DIV)
-
-lemma MOD_MULT_MOD: "ALL (m::nat) n::nat.
- 0 < n & 0 < m --> (ALL x::nat. x mod (n * m) mod n = x mod n)"
- by (import arithmetic MOD_MULT_MOD)
-
-lemma DIVMOD_ID: "(All::(nat => bool) => bool)
- (%n::nat.
- (op -->::bool => bool => bool) ((op <::nat => nat => bool) (0::nat) n)
- ((op &::bool => bool => bool)
- ((op =::nat => nat => bool) ((op div::nat => nat => nat) n n)
- (1::nat))
- ((op =::nat => nat => bool) ((op mod::nat => nat => nat) n n)
- (0::nat))))"
- by (import arithmetic DIVMOD_ID)
-
-lemma DIV_DIV_DIV_MULT: "ALL (x::nat) xa::nat.
- 0 < x & 0 < xa --> (ALL xb::nat. xb div x div xa = xb div (x * xa))"
- by (import arithmetic DIV_DIV_DIV_MULT)
-
-lemma DIV_P: "ALL (P::nat => bool) (p::nat) q::nat.
- 0 < q --> P (p div q) = (EX (k::nat) r::nat. p = k * q + r & r < q & P k)"
- by (import arithmetic DIV_P)
-
-lemma MOD_P: "ALL (P::nat => bool) (p::nat) q::nat.
- 0 < q --> P (p mod q) = (EX (k::nat) r::nat. p = k * q + r & r < q & P r)"
- by (import arithmetic MOD_P)
-
-lemma MOD_TIMES2: "ALL n>0. ALL (j::nat) k::nat. j mod n * (k mod n) mod n = j * k mod n"
- by (import arithmetic MOD_TIMES2)
-
-lemma MOD_COMMON_FACTOR: "ALL (n::nat) (p::nat) q::nat.
- 0 < n & 0 < q --> n * (p mod q) = n * p mod (n * q)"
- by (import arithmetic MOD_COMMON_FACTOR)
-
-lemma num_case_cong: "ALL (M::nat) (M'::nat) (b::'a::type) f::nat => 'a::type.
- M = M' &
- (M' = 0 --> b = (b'::'a::type)) &
- (ALL n::nat. M' = Suc n --> f n = (f'::nat => 'a::type) n) -->
- nat_case b f M = nat_case b' f' M'"
- by (import arithmetic num_case_cong)
-
-lemma SUC_ELIM_THM: "ALL P::nat => nat => bool.
- (ALL n::nat. P (Suc n) n) = (ALL n>0. P n (n - 1))"
- by (import arithmetic SUC_ELIM_THM)
+ sorry
+
+lemma DA: "(0::nat) < (n::nat) ==> EX (x::nat) q::nat. (k::nat) = q * n + x & x < n"
+ sorry
+
+lemma DIV_LESS_EQ: "(0::nat) < (n::nat) ==> (k::nat) div n <= k"
+ sorry
+
+lemma DIV_UNIQUE: "EX r::nat. (k::nat) = (q::nat) * (n::nat) + r & r < n ==> k div n = q"
+ sorry
+
+lemma MOD_UNIQUE: "EX q::nat. (k::nat) = q * (n::nat) + (r::nat) & r < n ==> k mod n = r"
+ sorry
+
+lemma DIV_MULT: "(r::nat) < (n::nat) ==> ((q::nat) * n + r) div n = q"
+ sorry
+
+lemma MOD_EQ_0: "(0::nat) < (n::nat) ==> (k::nat) * n mod n = (0::nat)"
+ sorry
+
+lemma ZERO_MOD: "(0::nat) < (n::nat) ==> (0::nat) mod n = (0::nat)"
+ sorry
+
+lemma ZERO_DIV: "(0::nat) < (n::nat) ==> (0::nat) div n = (0::nat)"
+ sorry
+
+lemma MOD_MULT: "(r::nat) < (n::nat) ==> ((q::nat) * n + r) mod n = r"
+ sorry
+
+lemma MOD_TIMES: "(0::nat) < (n::nat) ==> ((q::nat) * n + (r::nat)) mod n = r mod n"
+ sorry
+
+lemma MOD_PLUS: "(0::nat) < (n::nat)
+==> ((j::nat) mod n + (k::nat) mod n) mod n = (j + k) mod n"
+ sorry
+
+lemma MOD_MOD: "(0::nat) < (n::nat) ==> (k::nat) mod n mod n = k mod n"
+ sorry
+
+lemma ADD_DIV_ADD_DIV: "(0::nat) < (x::nat) ==> ((xa::nat) * x + (r::nat)) div x = xa + r div x"
+ sorry
+
+lemma MOD_MULT_MOD: "(0::nat) < (n::nat) & (0::nat) < (m::nat)
+==> (x::nat) mod (n * m) mod n = x mod n"
+ sorry
+
+lemma DIVMOD_ID: "(0::nat) < (n::nat) ==> n div n = (1::nat) & n mod n = (0::nat)"
+ sorry
+
+lemma DIV_DIV_DIV_MULT: "(0::nat) < (x::nat) & (0::nat) < (xa::nat)
+==> (xb::nat) div x div xa = xb div (x * xa)"
+ sorry
+
+lemma DIV_P: "(0::nat) < (q::nat)
+==> (P::nat => bool) ((p::nat) div q) =
+ (EX (k::nat) r::nat. p = k * q + r & r < q & P k)"
+ sorry
+
+lemma MOD_P: "(0::nat) < (q::nat)
+==> (P::nat => bool) ((p::nat) mod q) =
+ (EX (k::nat) r::nat. p = k * q + r & r < q & P r)"
+ sorry
+
+lemma MOD_TIMES2: "(0::nat) < (n::nat)
+==> (j::nat) mod n * ((k::nat) mod n) mod n = j * k mod n"
+ sorry
+
+lemma MOD_COMMON_FACTOR: "(0::nat) < (n::nat) & (0::nat) < (q::nat)
+==> n * ((p::nat) mod q) = n * p mod (n * q)"
+ sorry
+
+lemma num_case_cong: "M = M' & (M' = 0 --> b = b') & (ALL n. M' = Suc n --> f n = f' n)
+==> nat_case b f M = nat_case b' f' M'"
+ sorry
+
+lemma SUC_ELIM_THM: "(ALL n. P (Suc n) n) = (ALL n>0. P n (n - 1))"
+ sorry
lemma SUB_ELIM_THM: "(P::nat => bool) ((a::nat) - (b::nat)) =
-(ALL x::nat. (b = a + x --> P 0) & (a = b + x --> P x))"
- by (import arithmetic SUB_ELIM_THM)
-
-lemma PRE_ELIM_THM: "(P::nat => bool) (PRE (n::nat)) =
-(ALL m::nat. (n = 0 --> P 0) & (n = Suc m --> P m))"
- by (import arithmetic PRE_ELIM_THM)
-
-lemma MULT_INCREASES: "ALL (m::nat) n::nat. 1 < m & 0 < n --> Suc n <= m * n"
- by (import arithmetic MULT_INCREASES)
-
-lemma EXP_ALWAYS_BIG_ENOUGH: "ALL b>1. ALL n::nat. EX m::nat. n <= b ^ m"
- by (import arithmetic EXP_ALWAYS_BIG_ENOUGH)
-
-lemma EXP_EQ_0: "ALL (n::nat) m::nat. (n ^ m = 0) = (n = 0 & 0 < m)"
- by (import arithmetic EXP_EQ_0)
-
-lemma EXP_1: "(All::(nat => bool) => bool)
- (%x::nat.
- (op &::bool => bool => bool)
- ((op =::nat => nat => bool) ((op ^::nat => nat => nat) (1::nat) x)
- (1::nat))
- ((op =::nat => nat => bool) ((op ^::nat => nat => nat) x (1::nat)) x))"
- by (import arithmetic EXP_1)
-
-lemma EXP_EQ_1: "ALL (n::nat) m::nat. (n ^ m = 1) = (n = 1 | m = 0)"
- by (import arithmetic EXP_EQ_1)
-
-lemma MIN_MAX_EQ: "ALL (x::nat) xa::nat. (min x xa = max x xa) = (x = xa)"
- by (import arithmetic MIN_MAX_EQ)
-
-lemma MIN_MAX_LT: "ALL (x::nat) xa::nat. (min x xa < max x xa) = (x ~= xa)"
- by (import arithmetic MIN_MAX_LT)
-
-lemma MIN_MAX_PRED: "ALL (P::nat => bool) (m::nat) n::nat.
- P m & P n --> P (min m n) & P (max m n)"
- by (import arithmetic MIN_MAX_PRED)
-
-lemma MIN_LT: "ALL (x::nat) xa::nat.
- (min xa x < xa) = (xa ~= x & min xa x = x) &
- (min xa x < x) = (xa ~= x & min xa x = xa) &
- (xa < min xa x) = False & (x < min xa x) = False"
- by (import arithmetic MIN_LT)
-
-lemma MAX_LT: "ALL (x::nat) xa::nat.
- (xa < max xa x) = (xa ~= x & max xa x = x) &
- (x < max xa x) = (xa ~= x & max xa x = xa) &
- (max xa x < xa) = False & (max xa x < x) = False"
- by (import arithmetic MAX_LT)
-
-lemma MIN_LE: "ALL (x::nat) xa::nat. min xa x <= xa & min xa x <= x"
- by (import arithmetic MIN_LE)
-
-lemma MAX_LE: "ALL (x::nat) xa::nat. xa <= max xa x & x <= max xa x"
- by (import arithmetic MAX_LE)
-
-lemma MIN_0: "ALL x::nat. min x 0 = 0 & min 0 x = 0"
- by (import arithmetic MIN_0)
-
-lemma MAX_0: "ALL x::nat. max x 0 = x & max 0 x = x"
- by (import arithmetic MAX_0)
-
-lemma EXISTS_GREATEST: "ALL P::nat => bool.
- (Ex P & (EX x::nat. ALL y::nat. x < y --> ~ P y)) =
- (EX x::nat. P x & (ALL y::nat. x < y --> ~ P y))"
- by (import arithmetic EXISTS_GREATEST)
+(ALL x::nat. (b = a + x --> P (0::nat)) & (a = b + x --> P x))"
+ sorry
+
+lemma PRE_ELIM_THM: "P (PRE n) = (ALL m. (n = 0 --> P 0) & (n = Suc m --> P m))"
+ sorry
+
+lemma MULT_INCREASES: "1 < m & 0 < n ==> Suc n <= m * n"
+ sorry
+
+lemma EXP_ALWAYS_BIG_ENOUGH: "(1::nat) < (b::nat) ==> EX m::nat. (n::nat) <= b ^ m"
+ sorry
+
+lemma EXP_EQ_0: "((n::nat) ^ (m::nat) = (0::nat)) = (n = (0::nat) & (0::nat) < m)"
+ sorry
+
+lemma EXP_1: "(1::nat) ^ (x::nat) = (1::nat) & x ^ (1::nat) = x"
+ sorry
+
+lemma MIN_MAX_EQ: "(min (x::nat) (xa::nat) = max x xa) = (x = xa)"
+ sorry
+
+lemma MIN_MAX_LT: "(min (x::nat) (xa::nat) < max x xa) = (x ~= xa)"
+ sorry
+
+lemma MIN_MAX_PRED: "(P::nat => bool) (m::nat) & P (n::nat) ==> P (min m n) & P (max m n)"
+ sorry
+
+lemma MIN_LT: "(min (xa::nat) (x::nat) < xa) = (xa ~= x & min xa x = x) &
+(min xa x < x) = (xa ~= x & min xa x = xa) &
+(xa < min xa x) = False & (x < min xa x) = False"
+ sorry
+
+lemma MAX_LT: "((xa::nat) < max xa (x::nat)) = (xa ~= x & max xa x = x) &
+(x < max xa x) = (xa ~= x & max xa x = xa) &
+(max xa x < xa) = False & (max xa x < x) = False"
+ sorry
+
+lemma MIN_LE: "min (xa::nat) (x::nat) <= xa & min xa x <= x"
+ sorry
+
+lemma MAX_LE: "(xa::nat) <= max xa (x::nat) & x <= max xa x"
+ sorry
+
+lemma MIN_0: "min (x::nat) (0::nat) = (0::nat) & min (0::nat) x = (0::nat)"
+ sorry
+
+lemma MAX_0: "max (x::nat) (0::nat) = x & max (0::nat) x = x"
+ sorry
+
+lemma EXISTS_GREATEST: "(Ex (P::nat => bool) & (EX x::nat. ALL y>x. ~ P y)) =
+(EX x::nat. P x & (ALL y>x. ~ P y))"
+ sorry
;end_setup
;setup_theory hrat
-definition trat_1 :: "nat * nat" where
+definition
+ trat_1 :: "nat * nat" where
"trat_1 == (0, 0)"
lemma trat_1: "trat_1 = (0, 0)"
- by (import hrat trat_1)
-
-definition trat_inv :: "nat * nat => nat * nat" where
- "trat_inv == %(x::nat, y::nat). (y, x)"
-
-lemma trat_inv: "ALL (x::nat) y::nat. trat_inv (x, y) = (y, x)"
- by (import hrat trat_inv)
-
-definition trat_add :: "nat * nat => nat * nat => nat * nat" where
+ sorry
+
+definition
+ trat_inv :: "nat * nat => nat * nat" where
+ "trat_inv == %(x, y). (y, x)"
+
+lemma trat_inv: "trat_inv (x, y) = (y, x)"
+ sorry
+
+definition
+ trat_add :: "nat * nat => nat * nat => nat * nat" where
"trat_add ==
-%(x::nat, y::nat) (x'::nat, y'::nat).
+%(x, y) (x', y').
(PRE (Suc x * Suc y' + Suc x' * Suc y), PRE (Suc y * Suc y'))"
-lemma trat_add: "ALL (x::nat) (y::nat) (x'::nat) y'::nat.
- trat_add (x, y) (x', y') =
- (PRE (Suc x * Suc y' + Suc x' * Suc y), PRE (Suc y * Suc y'))"
- by (import hrat trat_add)
-
-definition trat_mul :: "nat * nat => nat * nat => nat * nat" where
- "trat_mul ==
-%(x::nat, y::nat) (x'::nat, y'::nat).
- (PRE (Suc x * Suc x'), PRE (Suc y * Suc y'))"
-
-lemma trat_mul: "ALL (x::nat) (y::nat) (x'::nat) y'::nat.
- trat_mul (x, y) (x', y') = (PRE (Suc x * Suc x'), PRE (Suc y * Suc y'))"
- by (import hrat trat_mul)
+lemma trat_add: "trat_add (x, y) (x', y') =
+(PRE (Suc x * Suc y' + Suc x' * Suc y), PRE (Suc y * Suc y'))"
+ sorry
+
+definition
+ trat_mul :: "nat * nat => nat * nat => nat * nat" where
+ "trat_mul == %(x, y) (x', y'). (PRE (Suc x * Suc x'), PRE (Suc y * Suc y'))"
+
+lemma trat_mul: "trat_mul (x, y) (x', y') = (PRE (Suc x * Suc x'), PRE (Suc y * Suc y'))"
+ sorry
consts
trat_sucint :: "nat => nat * nat"
specification (trat_sucint) trat_sucint: "trat_sucint 0 = trat_1 &
-(ALL n::nat. trat_sucint (Suc n) = trat_add (trat_sucint n) trat_1)"
- by (import hrat trat_sucint)
-
-definition trat_eq :: "nat * nat => nat * nat => bool" where
- "trat_eq ==
-%(x::nat, y::nat) (x'::nat, y'::nat). Suc x * Suc y' = Suc x' * Suc y"
-
-lemma trat_eq: "ALL (x::nat) (y::nat) (x'::nat) y'::nat.
- trat_eq (x, y) (x', y') = (Suc x * Suc y' = Suc x' * Suc y)"
- by (import hrat trat_eq)
-
-lemma TRAT_EQ_REFL: "ALL p::nat * nat. trat_eq p p"
- by (import hrat TRAT_EQ_REFL)
-
-lemma TRAT_EQ_SYM: "ALL (p::nat * nat) q::nat * nat. trat_eq p q = trat_eq q p"
- by (import hrat TRAT_EQ_SYM)
-
-lemma TRAT_EQ_TRANS: "ALL (p::nat * nat) (q::nat * nat) r::nat * nat.
- trat_eq p q & trat_eq q r --> trat_eq p r"
- by (import hrat TRAT_EQ_TRANS)
-
-lemma TRAT_EQ_AP: "ALL (p::nat * nat) q::nat * nat. p = q --> trat_eq p q"
- by (import hrat TRAT_EQ_AP)
-
-lemma TRAT_ADD_SYM_EQ: "ALL (h::nat * nat) i::nat * nat. trat_add h i = trat_add i h"
- by (import hrat TRAT_ADD_SYM_EQ)
-
-lemma TRAT_MUL_SYM_EQ: "ALL (h::nat * nat) i::nat * nat. trat_mul h i = trat_mul i h"
- by (import hrat TRAT_MUL_SYM_EQ)
-
-lemma TRAT_INV_WELLDEFINED: "ALL (p::nat * nat) q::nat * nat.
- trat_eq p q --> trat_eq (trat_inv p) (trat_inv q)"
- by (import hrat TRAT_INV_WELLDEFINED)
-
-lemma TRAT_ADD_WELLDEFINED: "ALL (p::nat * nat) (q::nat * nat) r::nat * nat.
- trat_eq p q --> trat_eq (trat_add p r) (trat_add q r)"
- by (import hrat TRAT_ADD_WELLDEFINED)
-
-lemma TRAT_ADD_WELLDEFINED2: "ALL (p1::nat * nat) (p2::nat * nat) (q1::nat * nat) q2::nat * nat.
- trat_eq p1 p2 & trat_eq q1 q2 -->
- trat_eq (trat_add p1 q1) (trat_add p2 q2)"
- by (import hrat TRAT_ADD_WELLDEFINED2)
-
-lemma TRAT_MUL_WELLDEFINED: "ALL (p::nat * nat) (q::nat * nat) r::nat * nat.
- trat_eq p q --> trat_eq (trat_mul p r) (trat_mul q r)"
- by (import hrat TRAT_MUL_WELLDEFINED)
-
-lemma TRAT_MUL_WELLDEFINED2: "ALL (p1::nat * nat) (p2::nat * nat) (q1::nat * nat) q2::nat * nat.
- trat_eq p1 p2 & trat_eq q1 q2 -->
- trat_eq (trat_mul p1 q1) (trat_mul p2 q2)"
- by (import hrat TRAT_MUL_WELLDEFINED2)
-
-lemma TRAT_ADD_SYM: "ALL (h::nat * nat) i::nat * nat. trat_eq (trat_add h i) (trat_add i h)"
- by (import hrat TRAT_ADD_SYM)
-
-lemma TRAT_ADD_ASSOC: "ALL (h::nat * nat) (i::nat * nat) j::nat * nat.
- trat_eq (trat_add h (trat_add i j)) (trat_add (trat_add h i) j)"
- by (import hrat TRAT_ADD_ASSOC)
-
-lemma TRAT_MUL_SYM: "ALL (h::nat * nat) i::nat * nat. trat_eq (trat_mul h i) (trat_mul i h)"
- by (import hrat TRAT_MUL_SYM)
-
-lemma TRAT_MUL_ASSOC: "ALL (h::nat * nat) (i::nat * nat) j::nat * nat.
- trat_eq (trat_mul h (trat_mul i j)) (trat_mul (trat_mul h i) j)"
- by (import hrat TRAT_MUL_ASSOC)
-
-lemma TRAT_LDISTRIB: "ALL (h::nat * nat) (i::nat * nat) j::nat * nat.
- trat_eq (trat_mul h (trat_add i j))
- (trat_add (trat_mul h i) (trat_mul h j))"
- by (import hrat TRAT_LDISTRIB)
-
-lemma TRAT_MUL_LID: "ALL h::nat * nat. trat_eq (trat_mul trat_1 h) h"
- by (import hrat TRAT_MUL_LID)
-
-lemma TRAT_MUL_LINV: "ALL h::nat * nat. trat_eq (trat_mul (trat_inv h) h) trat_1"
- by (import hrat TRAT_MUL_LINV)
-
-lemma TRAT_NOZERO: "ALL (h::nat * nat) i::nat * nat. ~ trat_eq (trat_add h i) h"
- by (import hrat TRAT_NOZERO)
-
-lemma TRAT_ADD_TOTAL: "ALL (h::nat * nat) i::nat * nat.
- trat_eq h i |
- (EX d::nat * nat. trat_eq h (trat_add i d)) |
- (EX d::nat * nat. trat_eq i (trat_add h d))"
- by (import hrat TRAT_ADD_TOTAL)
-
-lemma TRAT_SUCINT_0: "ALL n::nat. trat_eq (trat_sucint n) (n, 0)"
- by (import hrat TRAT_SUCINT_0)
-
-lemma TRAT_ARCH: "ALL h::nat * nat.
- EX (n::nat) d::nat * nat. trat_eq (trat_sucint n) (trat_add h d)"
- by (import hrat TRAT_ARCH)
+(ALL n. trat_sucint (Suc n) = trat_add (trat_sucint n) trat_1)"
+ sorry
+
+definition
+ trat_eq :: "nat * nat => nat * nat => bool" where
+ "trat_eq == %(x, y) (x', y'). Suc x * Suc y' = Suc x' * Suc y"
+
+lemma trat_eq: "trat_eq (x, y) (x', y') = (Suc x * Suc y' = Suc x' * Suc y)"
+ sorry
+
+lemma TRAT_EQ_REFL: "trat_eq p p"
+ sorry
+
+lemma TRAT_EQ_SYM: "trat_eq p q = trat_eq q p"
+ sorry
+
+lemma TRAT_EQ_TRANS: "trat_eq p q & trat_eq q r ==> trat_eq p r"
+ sorry
+
+lemma TRAT_EQ_AP: "p = q ==> trat_eq p q"
+ sorry
+
+lemma TRAT_ADD_SYM_EQ: "trat_add h i = trat_add i h"
+ sorry
+
+lemma TRAT_MUL_SYM_EQ: "trat_mul h i = trat_mul i h"
+ sorry
+
+lemma TRAT_INV_WELLDEFINED: "trat_eq p q ==> trat_eq (trat_inv p) (trat_inv q)"
+ sorry
+
+lemma TRAT_ADD_WELLDEFINED: "trat_eq p q ==> trat_eq (trat_add p r) (trat_add q r)"
+ sorry
+
+lemma TRAT_ADD_WELLDEFINED2: "trat_eq p1 p2 & trat_eq q1 q2 ==> trat_eq (trat_add p1 q1) (trat_add p2 q2)"
+ sorry
+
+lemma TRAT_MUL_WELLDEFINED: "trat_eq p q ==> trat_eq (trat_mul p r) (trat_mul q r)"
+ sorry
+
+lemma TRAT_MUL_WELLDEFINED2: "trat_eq p1 p2 & trat_eq q1 q2 ==> trat_eq (trat_mul p1 q1) (trat_mul p2 q2)"
+ sorry
+
+lemma TRAT_ADD_SYM: "trat_eq (trat_add h i) (trat_add i h)"
+ sorry
+
+lemma TRAT_ADD_ASSOC: "trat_eq (trat_add h (trat_add i j)) (trat_add (trat_add h i) j)"
+ sorry
+
+lemma TRAT_MUL_SYM: "trat_eq (trat_mul h i) (trat_mul i h)"
+ sorry
+
+lemma TRAT_MUL_ASSOC: "trat_eq (trat_mul h (trat_mul i j)) (trat_mul (trat_mul h i) j)"
+ sorry
+
+lemma TRAT_LDISTRIB: "trat_eq (trat_mul h (trat_add i j)) (trat_add (trat_mul h i) (trat_mul h j))"
+ sorry
+
+lemma TRAT_MUL_LID: "trat_eq (trat_mul trat_1 h) h"
+ sorry
+
+lemma TRAT_MUL_LINV: "trat_eq (trat_mul (trat_inv h) h) trat_1"
+ sorry
+
+lemma TRAT_NOZERO: "~ trat_eq (trat_add h i) h"
+ sorry
+
+lemma TRAT_ADD_TOTAL: "trat_eq h i |
+(EX d. trat_eq h (trat_add i d)) | (EX d. trat_eq i (trat_add h d))"
+ sorry
+
+lemma TRAT_SUCINT_0: "trat_eq (trat_sucint n) (n, 0)"
+ sorry
+
+lemma TRAT_ARCH: "EX n d. trat_eq (trat_sucint n) (trat_add h d)"
+ sorry
lemma TRAT_SUCINT: "trat_eq (trat_sucint 0) trat_1 &
-(ALL n::nat.
- trat_eq (trat_sucint (Suc n)) (trat_add (trat_sucint n) trat_1))"
- by (import hrat TRAT_SUCINT)
-
-lemma TRAT_EQ_EQUIV: "ALL (p::nat * nat) q::nat * nat. trat_eq p q = (trat_eq p = trat_eq q)"
- by (import hrat TRAT_EQ_EQUIV)
-
-typedef (open) hrat = "{x::nat * nat => bool. EX xa::nat * nat. x = trat_eq xa}"
- by (rule typedef_helper,import hrat hrat_TY_DEF)
+(ALL n. trat_eq (trat_sucint (Suc n)) (trat_add (trat_sucint n) trat_1))"
+ sorry
+
+lemma TRAT_EQ_EQUIV: "trat_eq p q = (trat_eq p = trat_eq q)"
+ sorry
+
+typedef (open) hrat = "{x. EX xa. x = trat_eq xa}"
+ sorry
lemmas hrat_TY_DEF = typedef_hol2hol4 [OF type_definition_hrat]
@@ -1866,227 +1477,213 @@
mk_hrat :: "(nat * nat => bool) => hrat"
dest_hrat :: "hrat => nat * nat => bool"
-specification (dest_hrat mk_hrat) hrat_tybij: "(ALL a::hrat. mk_hrat (dest_hrat a) = a) &
-(ALL r::nat * nat => bool.
- (EX x::nat * nat. r = trat_eq x) = (dest_hrat (mk_hrat r) = r))"
- by (import hrat hrat_tybij)
-
-definition hrat_1 :: "hrat" where
+specification (dest_hrat mk_hrat) hrat_tybij: "(ALL a. mk_hrat (dest_hrat a) = a) &
+(ALL r. (EX x. r = trat_eq x) = (dest_hrat (mk_hrat r) = r))"
+ sorry
+
+definition
+ hrat_1 :: "hrat" where
"hrat_1 == mk_hrat (trat_eq trat_1)"
lemma hrat_1: "hrat_1 = mk_hrat (trat_eq trat_1)"
- by (import hrat hrat_1)
-
-definition hrat_inv :: "hrat => hrat" where
- "hrat_inv == %T1::hrat. mk_hrat (trat_eq (trat_inv (Eps (dest_hrat T1))))"
-
-lemma hrat_inv: "ALL T1::hrat.
- hrat_inv T1 = mk_hrat (trat_eq (trat_inv (Eps (dest_hrat T1))))"
- by (import hrat hrat_inv)
-
-definition hrat_add :: "hrat => hrat => hrat" where
+ sorry
+
+definition
+ hrat_inv :: "hrat => hrat" where
+ "hrat_inv == %T1. mk_hrat (trat_eq (trat_inv (Eps (dest_hrat T1))))"
+
+lemma hrat_inv: "hrat_inv T1 = mk_hrat (trat_eq (trat_inv (Eps (dest_hrat T1))))"
+ sorry
+
+definition
+ hrat_add :: "hrat => hrat => hrat" where
"hrat_add ==
-%(T1::hrat) T2::hrat.
+%T1 T2.
mk_hrat (trat_eq (trat_add (Eps (dest_hrat T1)) (Eps (dest_hrat T2))))"
-lemma hrat_add: "ALL (T1::hrat) T2::hrat.
- hrat_add T1 T2 =
- mk_hrat (trat_eq (trat_add (Eps (dest_hrat T1)) (Eps (dest_hrat T2))))"
- by (import hrat hrat_add)
-
-definition hrat_mul :: "hrat => hrat => hrat" where
+lemma hrat_add: "hrat_add T1 T2 =
+mk_hrat (trat_eq (trat_add (Eps (dest_hrat T1)) (Eps (dest_hrat T2))))"
+ sorry
+
+definition
+ hrat_mul :: "hrat => hrat => hrat" where
"hrat_mul ==
-%(T1::hrat) T2::hrat.
- mk_hrat (trat_eq (trat_mul (Eps (dest_hrat T1)) (Eps (dest_hrat T2))))"
-
-lemma hrat_mul: "ALL (T1::hrat) T2::hrat.
- hrat_mul T1 T2 =
+%T1 T2.
mk_hrat (trat_eq (trat_mul (Eps (dest_hrat T1)) (Eps (dest_hrat T2))))"
- by (import hrat hrat_mul)
-
-definition hrat_sucint :: "nat => hrat" where
- "hrat_sucint == %T1::nat. mk_hrat (trat_eq (trat_sucint T1))"
-
-lemma hrat_sucint: "ALL T1::nat. hrat_sucint T1 = mk_hrat (trat_eq (trat_sucint T1))"
- by (import hrat hrat_sucint)
-
-lemma HRAT_ADD_SYM: "ALL (h::hrat) i::hrat. hrat_add h i = hrat_add i h"
- by (import hrat HRAT_ADD_SYM)
-
-lemma HRAT_ADD_ASSOC: "ALL (h::hrat) (i::hrat) j::hrat.
- hrat_add h (hrat_add i j) = hrat_add (hrat_add h i) j"
- by (import hrat HRAT_ADD_ASSOC)
-
-lemma HRAT_MUL_SYM: "ALL (h::hrat) i::hrat. hrat_mul h i = hrat_mul i h"
- by (import hrat HRAT_MUL_SYM)
-
-lemma HRAT_MUL_ASSOC: "ALL (h::hrat) (i::hrat) j::hrat.
- hrat_mul h (hrat_mul i j) = hrat_mul (hrat_mul h i) j"
- by (import hrat HRAT_MUL_ASSOC)
-
-lemma HRAT_LDISTRIB: "ALL (h::hrat) (i::hrat) j::hrat.
- hrat_mul h (hrat_add i j) = hrat_add (hrat_mul h i) (hrat_mul h j)"
- by (import hrat HRAT_LDISTRIB)
-
-lemma HRAT_MUL_LID: "ALL h::hrat. hrat_mul hrat_1 h = h"
- by (import hrat HRAT_MUL_LID)
-
-lemma HRAT_MUL_LINV: "ALL h::hrat. hrat_mul (hrat_inv h) h = hrat_1"
- by (import hrat HRAT_MUL_LINV)
-
-lemma HRAT_NOZERO: "ALL (h::hrat) i::hrat. hrat_add h i ~= h"
- by (import hrat HRAT_NOZERO)
-
-lemma HRAT_ADD_TOTAL: "ALL (h::hrat) i::hrat.
- h = i | (EX x::hrat. h = hrat_add i x) | (EX x::hrat. i = hrat_add h x)"
- by (import hrat HRAT_ADD_TOTAL)
-
-lemma HRAT_ARCH: "ALL h::hrat. EX (x::nat) xa::hrat. hrat_sucint x = hrat_add h xa"
- by (import hrat HRAT_ARCH)
+
+lemma hrat_mul: "hrat_mul T1 T2 =
+mk_hrat (trat_eq (trat_mul (Eps (dest_hrat T1)) (Eps (dest_hrat T2))))"
+ sorry
+
+definition
+ hrat_sucint :: "nat => hrat" where
+ "hrat_sucint == %T1. mk_hrat (trat_eq (trat_sucint T1))"
+
+lemma hrat_sucint: "hrat_sucint T1 = mk_hrat (trat_eq (trat_sucint T1))"
+ sorry
+
+lemma HRAT_ADD_SYM: "hrat_add h i = hrat_add i h"
+ sorry
+
+lemma HRAT_ADD_ASSOC: "hrat_add h (hrat_add i j) = hrat_add (hrat_add h i) j"
+ sorry
+
+lemma HRAT_MUL_SYM: "hrat_mul h i = hrat_mul i h"
+ sorry
+
+lemma HRAT_MUL_ASSOC: "hrat_mul h (hrat_mul i j) = hrat_mul (hrat_mul h i) j"
+ sorry
+
+lemma HRAT_LDISTRIB: "hrat_mul h (hrat_add i j) = hrat_add (hrat_mul h i) (hrat_mul h j)"
+ sorry
+
+lemma HRAT_MUL_LID: "hrat_mul hrat_1 h = h"
+ sorry
+
+lemma HRAT_MUL_LINV: "hrat_mul (hrat_inv h) h = hrat_1"
+ sorry
+
+lemma HRAT_NOZERO: "hrat_add h i ~= h"
+ sorry
+
+lemma HRAT_ADD_TOTAL: "h = i | (EX x. h = hrat_add i x) | (EX x. i = hrat_add h x)"
+ sorry
+
+lemma HRAT_ARCH: "EX x xa. hrat_sucint x = hrat_add h xa"
+ sorry
lemma HRAT_SUCINT: "hrat_sucint 0 = hrat_1 &
-(ALL x::nat. hrat_sucint (Suc x) = hrat_add (hrat_sucint x) hrat_1)"
- by (import hrat HRAT_SUCINT)
+(ALL x. hrat_sucint (Suc x) = hrat_add (hrat_sucint x) hrat_1)"
+ sorry
;end_setup
;setup_theory hreal
-definition hrat_lt :: "hrat => hrat => bool" where
- "hrat_lt == %(x::hrat) y::hrat. EX d::hrat. y = hrat_add x d"
-
-lemma hrat_lt: "ALL (x::hrat) y::hrat. hrat_lt x y = (EX d::hrat. y = hrat_add x d)"
- by (import hreal hrat_lt)
-
-lemma HRAT_LT_REFL: "ALL x::hrat. ~ hrat_lt x x"
- by (import hreal HRAT_LT_REFL)
-
-lemma HRAT_LT_TRANS: "ALL (x::hrat) (y::hrat) z::hrat. hrat_lt x y & hrat_lt y z --> hrat_lt x z"
- by (import hreal HRAT_LT_TRANS)
-
-lemma HRAT_LT_ANTISYM: "ALL (x::hrat) y::hrat. ~ (hrat_lt x y & hrat_lt y x)"
- by (import hreal HRAT_LT_ANTISYM)
-
-lemma HRAT_LT_TOTAL: "ALL (x::hrat) y::hrat. x = y | hrat_lt x y | hrat_lt y x"
- by (import hreal HRAT_LT_TOTAL)
-
-lemma HRAT_MUL_RID: "ALL x::hrat. hrat_mul x hrat_1 = x"
- by (import hreal HRAT_MUL_RID)
-
-lemma HRAT_MUL_RINV: "ALL x::hrat. hrat_mul x (hrat_inv x) = hrat_1"
- by (import hreal HRAT_MUL_RINV)
-
-lemma HRAT_RDISTRIB: "ALL (x::hrat) (y::hrat) z::hrat.
- hrat_mul (hrat_add x y) z = hrat_add (hrat_mul x z) (hrat_mul y z)"
- by (import hreal HRAT_RDISTRIB)
-
-lemma HRAT_LT_ADDL: "ALL (x::hrat) y::hrat. hrat_lt x (hrat_add x y)"
- by (import hreal HRAT_LT_ADDL)
-
-lemma HRAT_LT_ADDR: "ALL (x::hrat) xa::hrat. hrat_lt xa (hrat_add x xa)"
- by (import hreal HRAT_LT_ADDR)
-
-lemma HRAT_LT_GT: "ALL (x::hrat) y::hrat. hrat_lt x y --> ~ hrat_lt y x"
- by (import hreal HRAT_LT_GT)
-
-lemma HRAT_LT_NE: "ALL (x::hrat) y::hrat. hrat_lt x y --> x ~= y"
- by (import hreal HRAT_LT_NE)
-
-lemma HRAT_EQ_LADD: "ALL (x::hrat) (y::hrat) z::hrat. (hrat_add x y = hrat_add x z) = (y = z)"
- by (import hreal HRAT_EQ_LADD)
-
-lemma HRAT_EQ_LMUL: "ALL (x::hrat) (y::hrat) z::hrat. (hrat_mul x y = hrat_mul x z) = (y = z)"
- by (import hreal HRAT_EQ_LMUL)
-
-lemma HRAT_LT_ADD2: "ALL (u::hrat) (v::hrat) (x::hrat) y::hrat.
- hrat_lt u x & hrat_lt v y --> hrat_lt (hrat_add u v) (hrat_add x y)"
- by (import hreal HRAT_LT_ADD2)
-
-lemma HRAT_LT_LADD: "ALL (x::hrat) (y::hrat) z::hrat.
- hrat_lt (hrat_add z x) (hrat_add z y) = hrat_lt x y"
- by (import hreal HRAT_LT_LADD)
-
-lemma HRAT_LT_RADD: "ALL (x::hrat) (y::hrat) z::hrat.
- hrat_lt (hrat_add x z) (hrat_add y z) = hrat_lt x y"
- by (import hreal HRAT_LT_RADD)
-
-lemma HRAT_LT_MUL2: "ALL (u::hrat) (v::hrat) (x::hrat) y::hrat.
- hrat_lt u x & hrat_lt v y --> hrat_lt (hrat_mul u v) (hrat_mul x y)"
- by (import hreal HRAT_LT_MUL2)
-
-lemma HRAT_LT_LMUL: "ALL (x::hrat) (y::hrat) z::hrat.
- hrat_lt (hrat_mul z x) (hrat_mul z y) = hrat_lt x y"
- by (import hreal HRAT_LT_LMUL)
-
-lemma HRAT_LT_RMUL: "ALL (x::hrat) (y::hrat) z::hrat.
- hrat_lt (hrat_mul x z) (hrat_mul y z) = hrat_lt x y"
- by (import hreal HRAT_LT_RMUL)
-
-lemma HRAT_LT_LMUL1: "ALL (x::hrat) y::hrat. hrat_lt (hrat_mul x y) y = hrat_lt x hrat_1"
- by (import hreal HRAT_LT_LMUL1)
-
-lemma HRAT_LT_RMUL1: "ALL (x::hrat) y::hrat. hrat_lt (hrat_mul x y) x = hrat_lt y hrat_1"
- by (import hreal HRAT_LT_RMUL1)
-
-lemma HRAT_GT_LMUL1: "ALL (x::hrat) y::hrat. hrat_lt y (hrat_mul x y) = hrat_lt hrat_1 x"
- by (import hreal HRAT_GT_LMUL1)
-
-lemma HRAT_LT_L1: "ALL (x::hrat) y::hrat.
- hrat_lt (hrat_mul (hrat_inv x) y) hrat_1 = hrat_lt y x"
- by (import hreal HRAT_LT_L1)
-
-lemma HRAT_LT_R1: "ALL (x::hrat) y::hrat.
- hrat_lt (hrat_mul x (hrat_inv y)) hrat_1 = hrat_lt x y"
- by (import hreal HRAT_LT_R1)
-
-lemma HRAT_GT_L1: "ALL (x::hrat) y::hrat.
- hrat_lt hrat_1 (hrat_mul (hrat_inv x) y) = hrat_lt x y"
- by (import hreal HRAT_GT_L1)
-
-lemma HRAT_INV_MUL: "ALL (x::hrat) y::hrat.
- hrat_inv (hrat_mul x y) = hrat_mul (hrat_inv x) (hrat_inv y)"
- by (import hreal HRAT_INV_MUL)
-
-lemma HRAT_UP: "ALL x::hrat. Ex (hrat_lt x)"
- by (import hreal HRAT_UP)
-
-lemma HRAT_DOWN: "ALL x::hrat. EX xa::hrat. hrat_lt xa x"
- by (import hreal HRAT_DOWN)
-
-lemma HRAT_DOWN2: "ALL (x::hrat) y::hrat. EX xa::hrat. hrat_lt xa x & hrat_lt xa y"
- by (import hreal HRAT_DOWN2)
-
-lemma HRAT_MEAN: "ALL (x::hrat) y::hrat.
- hrat_lt x y --> (EX xa::hrat. hrat_lt x xa & hrat_lt xa y)"
- by (import hreal HRAT_MEAN)
-
-definition isacut :: "(hrat => bool) => bool" where
+definition
+ hrat_lt :: "hrat => hrat => bool" where
+ "hrat_lt == %x y. EX d. y = hrat_add x d"
+
+lemma hrat_lt: "hrat_lt x y = (EX d. y = hrat_add x d)"
+ sorry
+
+lemma HRAT_LT_REFL: "~ hrat_lt x x"
+ sorry
+
+lemma HRAT_LT_TRANS: "hrat_lt x y & hrat_lt y z ==> hrat_lt x z"
+ sorry
+
+lemma HRAT_LT_ANTISYM: "~ (hrat_lt x y & hrat_lt y x)"
+ sorry
+
+lemma HRAT_LT_TOTAL: "x = y | hrat_lt x y | hrat_lt y x"
+ sorry
+
+lemma HRAT_MUL_RID: "hrat_mul x hrat_1 = x"
+ sorry
+
+lemma HRAT_MUL_RINV: "hrat_mul x (hrat_inv x) = hrat_1"
+ sorry
+
+lemma HRAT_RDISTRIB: "hrat_mul (hrat_add x y) z = hrat_add (hrat_mul x z) (hrat_mul y z)"
+ sorry
+
+lemma HRAT_LT_ADDL: "hrat_lt x (hrat_add x y)"
+ sorry
+
+lemma HRAT_LT_ADDR: "hrat_lt xa (hrat_add x xa)"
+ sorry
+
+lemma HRAT_LT_GT: "hrat_lt x y ==> ~ hrat_lt y x"
+ sorry
+
+lemma HRAT_LT_NE: "hrat_lt x y ==> x ~= y"
+ sorry
+
+lemma HRAT_EQ_LADD: "(hrat_add x y = hrat_add x z) = (y = z)"
+ sorry
+
+lemma HRAT_EQ_LMUL: "(hrat_mul x y = hrat_mul x z) = (y = z)"
+ sorry
+
+lemma HRAT_LT_ADD2: "hrat_lt u x & hrat_lt v y ==> hrat_lt (hrat_add u v) (hrat_add x y)"
+ sorry
+
+lemma HRAT_LT_LADD: "hrat_lt (hrat_add z x) (hrat_add z y) = hrat_lt x y"
+ sorry
+
+lemma HRAT_LT_RADD: "hrat_lt (hrat_add x z) (hrat_add y z) = hrat_lt x y"
+ sorry
+
+lemma HRAT_LT_MUL2: "hrat_lt u x & hrat_lt v y ==> hrat_lt (hrat_mul u v) (hrat_mul x y)"
+ sorry
+
+lemma HRAT_LT_LMUL: "hrat_lt (hrat_mul z x) (hrat_mul z y) = hrat_lt x y"
+ sorry
+
+lemma HRAT_LT_RMUL: "hrat_lt (hrat_mul x z) (hrat_mul y z) = hrat_lt x y"
+ sorry
+
+lemma HRAT_LT_LMUL1: "hrat_lt (hrat_mul x y) y = hrat_lt x hrat_1"
+ sorry
+
+lemma HRAT_LT_RMUL1: "hrat_lt (hrat_mul x y) x = hrat_lt y hrat_1"
+ sorry
+
+lemma HRAT_GT_LMUL1: "hrat_lt y (hrat_mul x y) = hrat_lt hrat_1 x"
+ sorry
+
+lemma HRAT_LT_L1: "hrat_lt (hrat_mul (hrat_inv x) y) hrat_1 = hrat_lt y x"
+ sorry
+
+lemma HRAT_LT_R1: "hrat_lt (hrat_mul x (hrat_inv y)) hrat_1 = hrat_lt x y"
+ sorry
+
+lemma HRAT_GT_L1: "hrat_lt hrat_1 (hrat_mul (hrat_inv x) y) = hrat_lt x y"
+ sorry
+
+lemma HRAT_INV_MUL: "hrat_inv (hrat_mul x y) = hrat_mul (hrat_inv x) (hrat_inv y)"
+ sorry
+
+lemma HRAT_UP: "Ex (hrat_lt x)"
+ sorry
+
+lemma HRAT_DOWN: "EX xa. hrat_lt xa x"
+ sorry
+
+lemma HRAT_DOWN2: "EX xa. hrat_lt xa x & hrat_lt xa y"
+ sorry
+
+lemma HRAT_MEAN: "hrat_lt x y ==> EX xa. hrat_lt x xa & hrat_lt xa y"
+ sorry
+
+definition
+ isacut :: "(hrat => bool) => bool" where
"isacut ==
-%C::hrat => bool.
- Ex C &
- (EX x::hrat. ~ C x) &
- (ALL (x::hrat) y::hrat. C x & hrat_lt y x --> C y) &
- (ALL x::hrat. C x --> (EX y::hrat. C y & hrat_lt x y))"
-
-lemma isacut: "ALL C::hrat => bool.
- isacut C =
- (Ex C &
- (EX x::hrat. ~ C x) &
- (ALL (x::hrat) y::hrat. C x & hrat_lt y x --> C y) &
- (ALL x::hrat. C x --> (EX y::hrat. C y & hrat_lt x y)))"
- by (import hreal isacut)
-
-definition cut_of_hrat :: "hrat => hrat => bool" where
- "cut_of_hrat == %(x::hrat) y::hrat. hrat_lt y x"
-
-lemma cut_of_hrat: "ALL x::hrat. cut_of_hrat x = (%y::hrat. hrat_lt y x)"
- by (import hreal cut_of_hrat)
-
-lemma ISACUT_HRAT: "ALL h::hrat. isacut (cut_of_hrat h)"
- by (import hreal ISACUT_HRAT)
+%C. Ex C &
+ (EX x. ~ C x) &
+ (ALL x y. C x & hrat_lt y x --> C y) &
+ (ALL x. C x --> (EX y. C y & hrat_lt x y))"
+
+lemma isacut: "isacut (CC::hrat => bool) =
+(Ex CC &
+ (EX x::hrat. ~ CC x) &
+ (ALL (x::hrat) y::hrat. CC x & hrat_lt y x --> CC y) &
+ (ALL x::hrat. CC x --> (EX y::hrat. CC y & hrat_lt x y)))"
+ sorry
+
+definition
+ cut_of_hrat :: "hrat => hrat => bool" where
+ "cut_of_hrat == %x y. hrat_lt y x"
+
+lemma cut_of_hrat: "cut_of_hrat x = (%y. hrat_lt y x)"
+ sorry
+
+lemma ISACUT_HRAT: "isacut (cut_of_hrat h)"
+ sorry
typedef (open) hreal = "Collect isacut"
- by (rule typedef_helper,import hreal hreal_TY_DEF)
+ sorry
lemmas hreal_TY_DEF = typedef_hol2hol4 [OF type_definition_hreal]
@@ -2094,795 +1691,506 @@
hreal :: "(hrat => bool) => hreal"
cut :: "hreal => hrat => bool"
-specification (cut hreal) hreal_tybij: "(ALL a::hreal. hreal (hreal.cut a) = a) &
-(ALL r::hrat => bool. isacut r = (hreal.cut (hreal r) = r))"
- by (import hreal hreal_tybij)
-
-lemma EQUAL_CUTS: "ALL (X::hreal) Y::hreal. hreal.cut X = hreal.cut Y --> X = Y"
- by (import hreal EQUAL_CUTS)
-
-lemma CUT_ISACUT: "ALL x::hreal. isacut (hreal.cut x)"
- by (import hreal CUT_ISACUT)
-
-lemma CUT_NONEMPTY: "ALL x::hreal. Ex (hreal.cut x)"
- by (import hreal CUT_NONEMPTY)
-
-lemma CUT_BOUNDED: "ALL x::hreal. EX xa::hrat. ~ hreal.cut x xa"
- by (import hreal CUT_BOUNDED)
-
-lemma CUT_DOWN: "ALL (x::hreal) (xa::hrat) xb::hrat.
- hreal.cut x xa & hrat_lt xb xa --> hreal.cut x xb"
- by (import hreal CUT_DOWN)
-
-lemma CUT_UP: "ALL (x::hreal) xa::hrat.
- hreal.cut x xa --> (EX y::hrat. hreal.cut x y & hrat_lt xa y)"
- by (import hreal CUT_UP)
-
-lemma CUT_UBOUND: "ALL (x::hreal) (xa::hrat) xb::hrat.
- ~ hreal.cut x xa & hrat_lt xa xb --> ~ hreal.cut x xb"
- by (import hreal CUT_UBOUND)
-
-lemma CUT_STRADDLE: "ALL (X::hreal) (x::hrat) y::hrat.
- hreal.cut X x & ~ hreal.cut X y --> hrat_lt x y"
- by (import hreal CUT_STRADDLE)
-
-lemma CUT_NEARTOP_ADD: "ALL (X::hreal) e::hrat.
- EX x::hrat. hreal.cut X x & ~ hreal.cut X (hrat_add x e)"
- by (import hreal CUT_NEARTOP_ADD)
-
-lemma CUT_NEARTOP_MUL: "ALL (X::hreal) u::hrat.
- hrat_lt hrat_1 u -->
- (EX x::hrat. hreal.cut X x & ~ hreal.cut X (hrat_mul u x))"
- by (import hreal CUT_NEARTOP_MUL)
-
-definition hreal_1 :: "hreal" where
+specification (cut hreal) hreal_tybij: "(ALL a. hreal (cut a) = a) & (ALL r. isacut r = (cut (hreal r) = r))"
+ sorry
+
+lemma EQUAL_CUTS: "cut X = cut Y ==> X = Y"
+ sorry
+
+lemma CUT_ISACUT: "isacut (cut x)"
+ sorry
+
+lemma CUT_NONEMPTY: "Ex (cut x)"
+ sorry
+
+lemma CUT_BOUNDED: "EX xa. ~ cut x xa"
+ sorry
+
+lemma CUT_DOWN: "cut x xa & hrat_lt xb xa ==> cut x xb"
+ sorry
+
+lemma CUT_UP: "cut x xa ==> EX y. cut x y & hrat_lt xa y"
+ sorry
+
+lemma CUT_UBOUND: "~ cut x xa & hrat_lt xa xb ==> ~ cut x xb"
+ sorry
+
+lemma CUT_STRADDLE: "cut X x & ~ cut X y ==> hrat_lt x y"
+ sorry
+
+lemma CUT_NEARTOP_ADD: "EX x. cut X x & ~ cut X (hrat_add x e)"
+ sorry
+
+lemma CUT_NEARTOP_MUL: "hrat_lt hrat_1 u ==> EX x. cut X x & ~ cut X (hrat_mul u x)"
+ sorry
+
+definition
+ hreal_1 :: "hreal" where
"hreal_1 == hreal (cut_of_hrat hrat_1)"
lemma hreal_1: "hreal_1 = hreal (cut_of_hrat hrat_1)"
- by (import hreal hreal_1)
-
-definition hreal_add :: "hreal => hreal => hreal" where
- "hreal_add ==
-%(X::hreal) Y::hreal.
- hreal
- (%w::hrat.
- EX (x::hrat) y::hrat.
- w = hrat_add x y & hreal.cut X x & hreal.cut Y y)"
-
-lemma hreal_add: "ALL (X::hreal) Y::hreal.
- hreal_add X Y =
- hreal
- (%w::hrat.
- EX (x::hrat) y::hrat.
- w = hrat_add x y & hreal.cut X x & hreal.cut Y y)"
- by (import hreal hreal_add)
-
-definition hreal_mul :: "hreal => hreal => hreal" where
- "hreal_mul ==
-%(X::hreal) Y::hreal.
- hreal
- (%w::hrat.
- EX (x::hrat) y::hrat.
- w = hrat_mul x y & hreal.cut X x & hreal.cut Y y)"
-
-lemma hreal_mul: "ALL (X::hreal) Y::hreal.
- hreal_mul X Y =
- hreal
- (%w::hrat.
- EX (x::hrat) y::hrat.
- w = hrat_mul x y & hreal.cut X x & hreal.cut Y y)"
- by (import hreal hreal_mul)
-
-definition hreal_inv :: "hreal => hreal" where
+ sorry
+
+definition
+ hreal_add :: "hreal => hreal => hreal" where
+ "hreal_add == %X Y. hreal (%w. EX x y. w = hrat_add x y & cut X x & cut Y y)"
+
+lemma hreal_add: "hreal_add X Y = hreal (%w. EX x y. w = hrat_add x y & cut X x & cut Y y)"
+ sorry
+
+definition
+ hreal_mul :: "hreal => hreal => hreal" where
+ "hreal_mul == %X Y. hreal (%w. EX x y. w = hrat_mul x y & cut X x & cut Y y)"
+
+lemma hreal_mul: "hreal_mul X Y = hreal (%w. EX x y. w = hrat_mul x y & cut X x & cut Y y)"
+ sorry
+
+definition
+ hreal_inv :: "hreal => hreal" where
"hreal_inv ==
-%X::hreal.
- hreal
- (%w::hrat.
- EX d::hrat.
- hrat_lt d hrat_1 &
- (ALL x::hrat. hreal.cut X x --> hrat_lt (hrat_mul w x) d))"
-
-lemma hreal_inv: "ALL X::hreal.
- hreal_inv X =
- hreal
- (%w::hrat.
- EX d::hrat.
- hrat_lt d hrat_1 &
- (ALL x::hrat. hreal.cut X x --> hrat_lt (hrat_mul w x) d))"
- by (import hreal hreal_inv)
-
-definition hreal_sup :: "(hreal => bool) => hreal" where
- "hreal_sup ==
-%P::hreal => bool. hreal (%w::hrat. EX X::hreal. P X & hreal.cut X w)"
-
-lemma hreal_sup: "ALL P::hreal => bool.
- hreal_sup P = hreal (%w::hrat. EX X::hreal. P X & hreal.cut X w)"
- by (import hreal hreal_sup)
-
-definition hreal_lt :: "hreal => hreal => bool" where
- "hreal_lt ==
-%(X::hreal) Y::hreal.
- X ~= Y & (ALL x::hrat. hreal.cut X x --> hreal.cut Y x)"
-
-lemma hreal_lt: "ALL (X::hreal) Y::hreal.
- hreal_lt X Y = (X ~= Y & (ALL x::hrat. hreal.cut X x --> hreal.cut Y x))"
- by (import hreal hreal_lt)
-
-lemma HREAL_INV_ISACUT: "ALL X::hreal.
- isacut
- (%w::hrat.
- EX d::hrat.
- hrat_lt d hrat_1 &
- (ALL x::hrat. hreal.cut X x --> hrat_lt (hrat_mul w x) d))"
- by (import hreal HREAL_INV_ISACUT)
-
-lemma HREAL_ADD_ISACUT: "ALL (X::hreal) Y::hreal.
- isacut
- (%w::hrat.
- EX (x::hrat) y::hrat.
- w = hrat_add x y & hreal.cut X x & hreal.cut Y y)"
- by (import hreal HREAL_ADD_ISACUT)
-
-lemma HREAL_MUL_ISACUT: "ALL (X::hreal) Y::hreal.
- isacut
- (%w::hrat.
- EX (x::hrat) y::hrat.
- w = hrat_mul x y & hreal.cut X x & hreal.cut Y y)"
- by (import hreal HREAL_MUL_ISACUT)
-
-lemma HREAL_ADD_SYM: "ALL (X::hreal) Y::hreal. hreal_add X Y = hreal_add Y X"
- by (import hreal HREAL_ADD_SYM)
-
-lemma HREAL_MUL_SYM: "ALL (X::hreal) Y::hreal. hreal_mul X Y = hreal_mul Y X"
- by (import hreal HREAL_MUL_SYM)
-
-lemma HREAL_ADD_ASSOC: "ALL (X::hreal) (Y::hreal) Z::hreal.
- hreal_add X (hreal_add Y Z) = hreal_add (hreal_add X Y) Z"
- by (import hreal HREAL_ADD_ASSOC)
-
-lemma HREAL_MUL_ASSOC: "ALL (X::hreal) (Y::hreal) Z::hreal.
- hreal_mul X (hreal_mul Y Z) = hreal_mul (hreal_mul X Y) Z"
- by (import hreal HREAL_MUL_ASSOC)
-
-lemma HREAL_LDISTRIB: "ALL (X::hreal) (Y::hreal) Z::hreal.
- hreal_mul X (hreal_add Y Z) = hreal_add (hreal_mul X Y) (hreal_mul X Z)"
- by (import hreal HREAL_LDISTRIB)
-
-lemma HREAL_MUL_LID: "ALL X::hreal. hreal_mul hreal_1 X = X"
- by (import hreal HREAL_MUL_LID)
-
-lemma HREAL_MUL_LINV: "ALL X::hreal. hreal_mul (hreal_inv X) X = hreal_1"
- by (import hreal HREAL_MUL_LINV)
-
-lemma HREAL_NOZERO: "ALL (X::hreal) Y::hreal. hreal_add X Y ~= X"
- by (import hreal HREAL_NOZERO)
-
-definition hreal_sub :: "hreal => hreal => hreal" where
- "hreal_sub ==
-%(Y::hreal) X::hreal.
- hreal
- (%w::hrat. EX x::hrat. ~ hreal.cut X x & hreal.cut Y (hrat_add x w))"
-
-lemma hreal_sub: "ALL (Y::hreal) X::hreal.
- hreal_sub Y X =
- hreal
- (%w::hrat. EX x::hrat. ~ hreal.cut X x & hreal.cut Y (hrat_add x w))"
- by (import hreal hreal_sub)
-
-lemma HREAL_LT_LEMMA: "ALL (X::hreal) Y::hreal.
- hreal_lt X Y --> (EX x::hrat. ~ hreal.cut X x & hreal.cut Y x)"
- by (import hreal HREAL_LT_LEMMA)
-
-lemma HREAL_SUB_ISACUT: "ALL (X::hreal) Y::hreal.
- hreal_lt X Y -->
- isacut
- (%w::hrat. EX x::hrat. ~ hreal.cut X x & hreal.cut Y (hrat_add x w))"
- by (import hreal HREAL_SUB_ISACUT)
-
-lemma HREAL_SUB_ADD: "ALL (X::hreal) Y::hreal. hreal_lt X Y --> hreal_add (hreal_sub Y X) X = Y"
- by (import hreal HREAL_SUB_ADD)
-
-lemma HREAL_LT_TOTAL: "ALL (X::hreal) Y::hreal. X = Y | hreal_lt X Y | hreal_lt Y X"
- by (import hreal HREAL_LT_TOTAL)
-
-lemma HREAL_LT: "ALL (X::hreal) Y::hreal. hreal_lt X Y = (EX D::hreal. Y = hreal_add X D)"
- by (import hreal HREAL_LT)
-
-lemma HREAL_ADD_TOTAL: "ALL (X::hreal) Y::hreal.
- X = Y |
- (EX D::hreal. Y = hreal_add X D) | (EX D::hreal. X = hreal_add Y D)"
- by (import hreal HREAL_ADD_TOTAL)
-
-lemma HREAL_SUP_ISACUT: "ALL P::hreal => bool.
- Ex P & (EX Y::hreal. ALL X::hreal. P X --> hreal_lt X Y) -->
- isacut (%w::hrat. EX X::hreal. P X & hreal.cut X w)"
- by (import hreal HREAL_SUP_ISACUT)
-
-lemma HREAL_SUP: "ALL P::hreal => bool.
- Ex P & (EX Y::hreal. ALL X::hreal. P X --> hreal_lt X Y) -->
- (ALL Y::hreal.
- (EX X::hreal. P X & hreal_lt Y X) = hreal_lt Y (hreal_sup P))"
- by (import hreal HREAL_SUP)
+%X. hreal
+ (%w. EX d. hrat_lt d hrat_1 &
+ (ALL x. cut X x --> hrat_lt (hrat_mul w x) d))"
+
+lemma hreal_inv: "hreal_inv X =
+hreal
+ (%w. EX d. hrat_lt d hrat_1 &
+ (ALL x. cut X x --> hrat_lt (hrat_mul w x) d))"
+ sorry
+
+definition
+ hreal_sup :: "(hreal => bool) => hreal" where
+ "hreal_sup == %P. hreal (%w. EX X. P X & cut X w)"
+
+lemma hreal_sup: "hreal_sup P = hreal (%w. EX X. P X & cut X w)"
+ sorry
+
+definition
+ hreal_lt :: "hreal => hreal => bool" where
+ "hreal_lt == %X Y. X ~= Y & (ALL x. cut X x --> cut Y x)"
+
+lemma hreal_lt: "hreal_lt X Y = (X ~= Y & (ALL x. cut X x --> cut Y x))"
+ sorry
+
+lemma HREAL_INV_ISACUT: "isacut
+ (%w. EX d. hrat_lt d hrat_1 &
+ (ALL x. cut X x --> hrat_lt (hrat_mul w x) d))"
+ sorry
+
+lemma HREAL_ADD_ISACUT: "isacut (%w. EX x y. w = hrat_add x y & cut X x & cut Y y)"
+ sorry
+
+lemma HREAL_MUL_ISACUT: "isacut (%w. EX x y. w = hrat_mul x y & cut X x & cut Y y)"
+ sorry
+
+lemma HREAL_ADD_SYM: "hreal_add X Y = hreal_add Y X"
+ sorry
+
+lemma HREAL_MUL_SYM: "hreal_mul X Y = hreal_mul Y X"
+ sorry
+
+lemma HREAL_ADD_ASSOC: "hreal_add X (hreal_add Y Z) = hreal_add (hreal_add X Y) Z"
+ sorry
+
+lemma HREAL_MUL_ASSOC: "hreal_mul X (hreal_mul Y Z) = hreal_mul (hreal_mul X Y) Z"
+ sorry
+
+lemma HREAL_LDISTRIB: "hreal_mul X (hreal_add Y Z) = hreal_add (hreal_mul X Y) (hreal_mul X Z)"
+ sorry
+
+lemma HREAL_MUL_LID: "hreal_mul hreal_1 X = X"
+ sorry
+
+lemma HREAL_MUL_LINV: "hreal_mul (hreal_inv X) X = hreal_1"
+ sorry
+
+lemma HREAL_NOZERO: "hreal_add X Y ~= X"
+ sorry
+
+definition
+ hreal_sub :: "hreal => hreal => hreal" where
+ "hreal_sub == %Y X. hreal (%w. EX x. ~ cut X x & cut Y (hrat_add x w))"
+
+lemma hreal_sub: "hreal_sub Y X = hreal (%w. EX x. ~ cut X x & cut Y (hrat_add x w))"
+ sorry
+
+lemma HREAL_LT_LEMMA: "hreal_lt X Y ==> EX x. ~ cut X x & cut Y x"
+ sorry
+
+lemma HREAL_SUB_ISACUT: "hreal_lt X Y ==> isacut (%w. EX x. ~ cut X x & cut Y (hrat_add x w))"
+ sorry
+
+lemma HREAL_SUB_ADD: "hreal_lt X Y ==> hreal_add (hreal_sub Y X) X = Y"
+ sorry
+
+lemma HREAL_LT_TOTAL: "X = Y | hreal_lt X Y | hreal_lt Y X"
+ sorry
+
+lemma HREAL_LT: "hreal_lt X Y = (EX D. Y = hreal_add X D)"
+ sorry
+
+lemma HREAL_ADD_TOTAL: "X = Y | (EX D. Y = hreal_add X D) | (EX D. X = hreal_add Y D)"
+ sorry
+
+lemma HREAL_SUP_ISACUT: "Ex P & (EX Y. ALL X. P X --> hreal_lt X Y)
+==> isacut (%w. EX X. P X & cut X w)"
+ sorry
+
+lemma HREAL_SUP: "Ex P & (EX Y. ALL X. P X --> hreal_lt X Y)
+==> (EX X. P X & hreal_lt Y X) = hreal_lt Y (hreal_sup P)"
+ sorry
;end_setup
;setup_theory numeral
lemma numeral_suc: "Suc ALT_ZERO = NUMERAL_BIT1 ALT_ZERO &
-(ALL x::nat. Suc (NUMERAL_BIT1 x) = NUMERAL_BIT2 x) &
-(ALL x::nat. Suc (NUMERAL_BIT2 x) = NUMERAL_BIT1 (Suc x))"
- by (import numeral numeral_suc)
-
-definition iZ :: "nat => nat" where
- "iZ == %x::nat. x"
-
-lemma iZ: "ALL x::nat. iZ x = x"
- by (import numeral iZ)
-
-definition iiSUC :: "nat => nat" where
- "iiSUC == %n::nat. Suc (Suc n)"
-
-lemma iiSUC: "ALL n::nat. iiSUC n = Suc (Suc n)"
- by (import numeral iiSUC)
-
-lemma numeral_distrib: "(op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::nat => nat => bool) ((op +::nat => nat => nat) (0::nat) x) x))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::nat => nat => bool) ((op +::nat => nat => nat) x (0::nat))
- x))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op =::nat => nat => bool)
- ((op +::nat => nat => nat) ((NUMERAL::nat => nat) x)
- ((NUMERAL::nat => nat) xa))
- ((NUMERAL::nat => nat)
- ((iZ::nat => nat) ((op +::nat => nat => nat) x xa))))))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::nat => nat => bool)
- ((op *::nat => nat => nat) (0::nat) x) (0::nat)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::nat => nat => bool)
- ((op *::nat => nat => nat) x (0::nat)) (0::nat)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op =::nat => nat => bool)
- ((op *::nat => nat => nat) ((NUMERAL::nat => nat) x)
- ((NUMERAL::nat => nat) xa))
- ((NUMERAL::nat => nat)
- ((op *::nat => nat => nat) x xa)))))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::nat => nat => bool)
- ((op -::nat => nat => nat) (0::nat) x) (0::nat)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::nat => nat => bool)
- ((op -::nat => nat => nat) x (0::nat)) x))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op =::nat => nat => bool)
- ((op -::nat => nat => nat)
- ((NUMERAL::nat => nat) x)
- ((NUMERAL::nat => nat) xa))
- ((NUMERAL::nat => nat)
- ((op -::nat => nat => nat) x xa)))))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::nat => nat => bool)
- ((op ^::nat => nat => nat) (0::nat)
- ((NUMERAL::nat => nat)
- ((NUMERAL_BIT1::nat => nat) x)))
- (0::nat)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::nat => nat => bool)
- ((op ^::nat => nat => nat) (0::nat)
- ((NUMERAL::nat => nat)
- ((NUMERAL_BIT2::nat => nat) x)))
- (0::nat)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::nat => nat => bool)
- ((op ^::nat => nat => nat) x (0::nat))
- (1::nat)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op =::nat => nat => bool)
- ((op ^::nat => nat => nat)
- ((NUMERAL::nat => nat) x) ((NUMERAL::nat => nat) xa))
- ((NUMERAL::nat => nat)
- ((op ^::nat => nat => nat) x xa)))))
- ((op &::bool => bool => bool)
- ((op =::nat => nat => bool)
- ((Suc::nat => nat) (0::nat)) (1::nat))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::nat => nat => bool)
- ((Suc::nat => nat)
-((NUMERAL::nat => nat) x))
- ((NUMERAL::nat => nat)
-((Suc::nat => nat) x))))
- ((op &::bool => bool => bool)
- ((op =::nat => nat => bool)
- ((PRE::nat => nat) (0::nat)) (0::nat))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::nat => nat => bool) ((PRE::nat => nat) ((NUMERAL::nat => nat) x))
- ((NUMERAL::nat => nat) ((PRE::nat => nat) x))))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::bool => bool => bool)
- ((op =::nat => nat => bool) ((NUMERAL::nat => nat) x) (0::nat))
- ((op =::nat => nat => bool) x (ALT_ZERO::nat))))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::bool => bool => bool)
- ((op =::nat => nat => bool) (0::nat) ((NUMERAL::nat => nat) x))
- ((op =::nat => nat => bool) x (ALT_ZERO::nat))))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op =::bool => bool => bool)
- ((op =::nat => nat => bool) ((NUMERAL::nat => nat) x)
- ((NUMERAL::nat => nat) xa))
- ((op =::nat => nat => bool) x xa))))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::bool => bool => bool)
- ((op <::nat => nat => bool) x (0::nat)) (False::bool)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::bool => bool => bool)
- ((op <::nat => nat => bool) (0::nat) ((NUMERAL::nat => nat) x))
- ((op <::nat => nat => bool) (ALT_ZERO::nat) x)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op =::bool => bool => bool)
- ((op <::nat => nat => bool) ((NUMERAL::nat => nat) x)
- ((NUMERAL::nat => nat) xa))
- ((op <::nat => nat => bool) x xa))))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::bool => bool => bool)
- ((op <::nat => nat => bool) x (0::nat)) (False::bool)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::bool => bool => bool)
- ((op <::nat => nat => bool) (0::nat)
- ((NUMERAL::nat => nat) x))
- ((op <::nat => nat => bool) (ALT_ZERO::nat) x)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op =::bool => bool => bool)
- ((op <::nat => nat => bool)
- ((NUMERAL::nat => nat) xa)
- ((NUMERAL::nat => nat) x))
- ((op <::nat => nat => bool) xa x))))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::bool => bool => bool)
- ((op <=::nat => nat => bool) (0::nat) x)
- (True::bool)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::bool => bool => bool)
- ((op <=::nat => nat => bool)
- ((NUMERAL::nat => nat) x) (0::nat))
- ((op <=::nat => nat => bool) x (ALT_ZERO::nat))))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op =::bool => bool => bool)
- ((op <=::nat => nat => bool)
- ((NUMERAL::nat => nat) x)
- ((NUMERAL::nat => nat) xa))
- ((op <=::nat => nat => bool) x xa))))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::bool => bool => bool)
- ((op <=::nat => nat => bool) (0::nat) x)
- (True::bool)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::bool => bool => bool)
- ((op <=::nat => nat => bool) x (0::nat))
- ((op =::nat => nat => bool) x (0::nat))))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op =::bool => bool => bool)
- ((op <=::nat => nat => bool)
- ((NUMERAL::nat => nat) xa) ((NUMERAL::nat => nat) x))
- ((op <=::nat => nat => bool) xa x))))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::bool => bool => bool)
- ((ODD::nat => bool)
- ((NUMERAL::nat => nat) x))
- ((ODD::nat => bool) x)))
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op =::bool => bool => bool)
- ((EVEN::nat => bool)
-((NUMERAL::nat => nat) x))
- ((EVEN::nat => bool) x)))
- ((op &::bool => bool => bool)
- ((Not::bool => bool)
- ((ODD::nat => bool) (0::nat)))
- ((EVEN::nat => bool)
- (0::nat))))))))))))))))))))))))))))))))))))"
- by (import numeral numeral_distrib)
+(ALL x. Suc (NUMERAL_BIT1 x) = NUMERAL_BIT2 x) &
+(ALL x. Suc (NUMERAL_BIT2 x) = NUMERAL_BIT1 (Suc x))"
+ sorry
+
+definition
+ iZ :: "nat => nat" where
+ "iZ == %x. x"
+
+lemma iZ: "iZ x = x"
+ sorry
+
+definition
+ iiSUC :: "nat => nat" where
+ "iiSUC == %n. Suc (Suc n)"
+
+lemma iiSUC: "iiSUC n = Suc (Suc n)"
+ sorry
+
+lemma numeral_distrib: "(ALL x::nat. (0::nat) + x = x) &
+(ALL x::nat. x + (0::nat) = x) &
+(ALL (x::nat) xa::nat. NUMERAL x + NUMERAL xa = NUMERAL (iZ (x + xa))) &
+(ALL x::nat. (0::nat) * x = (0::nat)) &
+(ALL x::nat. x * (0::nat) = (0::nat)) &
+(ALL (x::nat) xa::nat. NUMERAL x * NUMERAL xa = NUMERAL (x * xa)) &
+(ALL x::nat. (0::nat) - x = (0::nat)) &
+(ALL x::nat. x - (0::nat) = x) &
+(ALL (x::nat) xa::nat. NUMERAL x - NUMERAL xa = NUMERAL (x - xa)) &
+(ALL x::nat. (0::nat) ^ NUMERAL (NUMERAL_BIT1 x) = (0::nat)) &
+(ALL x::nat. (0::nat) ^ NUMERAL (NUMERAL_BIT2 x) = (0::nat)) &
+(ALL x::nat. x ^ (0::nat) = (1::nat)) &
+(ALL (x::nat) xa::nat. NUMERAL x ^ NUMERAL xa = NUMERAL (x ^ xa)) &
+Suc (0::nat) = (1::nat) &
+(ALL x::nat. Suc (NUMERAL x) = NUMERAL (Suc x)) &
+PRE (0::nat) = (0::nat) &
+(ALL x::nat. PRE (NUMERAL x) = NUMERAL (PRE x)) &
+(ALL x::nat. (NUMERAL x = (0::nat)) = (x = ALT_ZERO)) &
+(ALL x::nat. ((0::nat) = NUMERAL x) = (x = ALT_ZERO)) &
+(ALL (x::nat) xa::nat. (NUMERAL x = NUMERAL xa) = (x = xa)) &
+(ALL x::nat. (x < (0::nat)) = False) &
+(ALL x::nat. ((0::nat) < NUMERAL x) = (ALT_ZERO < x)) &
+(ALL (x::nat) xa::nat. (NUMERAL x < NUMERAL xa) = (x < xa)) &
+(ALL x::nat. (x < (0::nat)) = False) &
+(ALL x::nat. ((0::nat) < NUMERAL x) = (ALT_ZERO < x)) &
+(ALL (x::nat) xa::nat. (NUMERAL xa < NUMERAL x) = (xa < x)) &
+(ALL x::nat. ((0::nat) <= x) = True) &
+(ALL x::nat. (NUMERAL x <= (0::nat)) = (x <= ALT_ZERO)) &
+(ALL (x::nat) xa::nat. (NUMERAL x <= NUMERAL xa) = (x <= xa)) &
+(ALL x::nat. ((0::nat) <= x) = True) &
+(ALL x::nat. (x <= (0::nat)) = (x = (0::nat))) &
+(ALL (x::nat) xa::nat. (NUMERAL xa <= NUMERAL x) = (xa <= x)) &
+(ALL x::nat. ODD (NUMERAL x) = ODD x) &
+(ALL x::nat. EVEN (NUMERAL x) = EVEN x) & ~ ODD (0::nat) & EVEN (0::nat)"
+ sorry
lemma numeral_iisuc: "iiSUC ALT_ZERO = NUMERAL_BIT2 ALT_ZERO &
-iiSUC (NUMERAL_BIT1 (n::nat)) = NUMERAL_BIT1 (Suc n) &
+iiSUC (NUMERAL_BIT1 n) = NUMERAL_BIT1 (Suc n) &
iiSUC (NUMERAL_BIT2 n) = NUMERAL_BIT2 (Suc n)"
- by (import numeral numeral_iisuc)
-
-lemma numeral_add: "ALL (x::nat) xa::nat.
- iZ (ALT_ZERO + x) = x &
- iZ (x + ALT_ZERO) = x &
- iZ (NUMERAL_BIT1 x + NUMERAL_BIT1 xa) = NUMERAL_BIT2 (iZ (x + xa)) &
- iZ (NUMERAL_BIT1 x + NUMERAL_BIT2 xa) = NUMERAL_BIT1 (Suc (x + xa)) &
- iZ (NUMERAL_BIT2 x + NUMERAL_BIT1 xa) = NUMERAL_BIT1 (Suc (x + xa)) &
- iZ (NUMERAL_BIT2 x + NUMERAL_BIT2 xa) = NUMERAL_BIT2 (Suc (x + xa)) &
- Suc (ALT_ZERO + x) = Suc x &
- Suc (x + ALT_ZERO) = Suc x &
- Suc (NUMERAL_BIT1 x + NUMERAL_BIT1 xa) = NUMERAL_BIT1 (Suc (x + xa)) &
- Suc (NUMERAL_BIT1 x + NUMERAL_BIT2 xa) = NUMERAL_BIT2 (Suc (x + xa)) &
- Suc (NUMERAL_BIT2 x + NUMERAL_BIT1 xa) = NUMERAL_BIT2 (Suc (x + xa)) &
- Suc (NUMERAL_BIT2 x + NUMERAL_BIT2 xa) = NUMERAL_BIT1 (iiSUC (x + xa)) &
- iiSUC (ALT_ZERO + x) = iiSUC x &
- iiSUC (x + ALT_ZERO) = iiSUC x &
- iiSUC (NUMERAL_BIT1 x + NUMERAL_BIT1 xa) = NUMERAL_BIT2 (Suc (x + xa)) &
- iiSUC (NUMERAL_BIT1 x + NUMERAL_BIT2 xa) =
- NUMERAL_BIT1 (iiSUC (x + xa)) &
- iiSUC (NUMERAL_BIT2 x + NUMERAL_BIT1 xa) =
- NUMERAL_BIT1 (iiSUC (x + xa)) &
- iiSUC (NUMERAL_BIT2 x + NUMERAL_BIT2 xa) = NUMERAL_BIT2 (iiSUC (x + xa))"
- by (import numeral numeral_add)
-
-lemma numeral_eq: "ALL (x::nat) xa::nat.
- (ALT_ZERO = NUMERAL_BIT1 x) = False &
- (NUMERAL_BIT1 x = ALT_ZERO) = False &
- (ALT_ZERO = NUMERAL_BIT2 x) = False &
- (NUMERAL_BIT2 x = ALT_ZERO) = False &
- (NUMERAL_BIT1 x = NUMERAL_BIT2 xa) = False &
- (NUMERAL_BIT2 x = NUMERAL_BIT1 xa) = False &
- (NUMERAL_BIT1 x = NUMERAL_BIT1 xa) = (x = xa) &
- (NUMERAL_BIT2 x = NUMERAL_BIT2 xa) = (x = xa)"
- by (import numeral numeral_eq)
-
-lemma numeral_lt: "ALL (x::nat) xa::nat.
- (ALT_ZERO < NUMERAL_BIT1 x) = True &
- (ALT_ZERO < NUMERAL_BIT2 x) = True &
- (x < ALT_ZERO) = False &
- (NUMERAL_BIT1 x < NUMERAL_BIT1 xa) = (x < xa) &
- (NUMERAL_BIT2 x < NUMERAL_BIT2 xa) = (x < xa) &
- (NUMERAL_BIT1 x < NUMERAL_BIT2 xa) = (~ xa < x) &
- (NUMERAL_BIT2 x < NUMERAL_BIT1 xa) = (x < xa)"
- by (import numeral numeral_lt)
-
-lemma numeral_lte: "ALL (x::nat) xa::nat.
- (ALT_ZERO <= x) = True &
- (NUMERAL_BIT1 x <= ALT_ZERO) = False &
- (NUMERAL_BIT2 x <= ALT_ZERO) = False &
- (NUMERAL_BIT1 x <= NUMERAL_BIT1 xa) = (x <= xa) &
- (NUMERAL_BIT1 x <= NUMERAL_BIT2 xa) = (x <= xa) &
- (NUMERAL_BIT2 x <= NUMERAL_BIT1 xa) = (~ xa <= x) &
- (NUMERAL_BIT2 x <= NUMERAL_BIT2 xa) = (x <= xa)"
- by (import numeral numeral_lte)
+ sorry
+
+lemma numeral_add: "iZ (ALT_ZERO + x) = x &
+iZ (x + ALT_ZERO) = x &
+iZ (NUMERAL_BIT1 x + NUMERAL_BIT1 xa) = NUMERAL_BIT2 (iZ (x + xa)) &
+iZ (NUMERAL_BIT1 x + NUMERAL_BIT2 xa) = NUMERAL_BIT1 (Suc (x + xa)) &
+iZ (NUMERAL_BIT2 x + NUMERAL_BIT1 xa) = NUMERAL_BIT1 (Suc (x + xa)) &
+iZ (NUMERAL_BIT2 x + NUMERAL_BIT2 xa) = NUMERAL_BIT2 (Suc (x + xa)) &
+Suc (ALT_ZERO + x) = Suc x &
+Suc (x + ALT_ZERO) = Suc x &
+Suc (NUMERAL_BIT1 x + NUMERAL_BIT1 xa) = NUMERAL_BIT1 (Suc (x + xa)) &
+Suc (NUMERAL_BIT1 x + NUMERAL_BIT2 xa) = NUMERAL_BIT2 (Suc (x + xa)) &
+Suc (NUMERAL_BIT2 x + NUMERAL_BIT1 xa) = NUMERAL_BIT2 (Suc (x + xa)) &
+Suc (NUMERAL_BIT2 x + NUMERAL_BIT2 xa) = NUMERAL_BIT1 (iiSUC (x + xa)) &
+iiSUC (ALT_ZERO + x) = iiSUC x &
+iiSUC (x + ALT_ZERO) = iiSUC x &
+iiSUC (NUMERAL_BIT1 x + NUMERAL_BIT1 xa) = NUMERAL_BIT2 (Suc (x + xa)) &
+iiSUC (NUMERAL_BIT1 x + NUMERAL_BIT2 xa) = NUMERAL_BIT1 (iiSUC (x + xa)) &
+iiSUC (NUMERAL_BIT2 x + NUMERAL_BIT1 xa) = NUMERAL_BIT1 (iiSUC (x + xa)) &
+iiSUC (NUMERAL_BIT2 x + NUMERAL_BIT2 xa) = NUMERAL_BIT2 (iiSUC (x + xa))"
+ sorry
+
+lemma numeral_eq: "(ALT_ZERO = NUMERAL_BIT1 x) = False &
+(NUMERAL_BIT1 x = ALT_ZERO) = False &
+(ALT_ZERO = NUMERAL_BIT2 x) = False &
+(NUMERAL_BIT2 x = ALT_ZERO) = False &
+(NUMERAL_BIT1 x = NUMERAL_BIT2 xa) = False &
+(NUMERAL_BIT2 x = NUMERAL_BIT1 xa) = False &
+(NUMERAL_BIT1 x = NUMERAL_BIT1 xa) = (x = xa) &
+(NUMERAL_BIT2 x = NUMERAL_BIT2 xa) = (x = xa)"
+ sorry
+
+lemma numeral_lt: "(ALT_ZERO < NUMERAL_BIT1 x) = True &
+(ALT_ZERO < NUMERAL_BIT2 x) = True &
+(x < ALT_ZERO) = False &
+(NUMERAL_BIT1 x < NUMERAL_BIT1 xa) = (x < xa) &
+(NUMERAL_BIT2 x < NUMERAL_BIT2 xa) = (x < xa) &
+(NUMERAL_BIT1 x < NUMERAL_BIT2 xa) = (~ xa < x) &
+(NUMERAL_BIT2 x < NUMERAL_BIT1 xa) = (x < xa)"
+ sorry
+
+lemma numeral_lte: "(ALT_ZERO <= x) = True &
+(NUMERAL_BIT1 x <= ALT_ZERO) = False &
+(NUMERAL_BIT2 x <= ALT_ZERO) = False &
+(NUMERAL_BIT1 x <= NUMERAL_BIT1 xa) = (x <= xa) &
+(NUMERAL_BIT1 x <= NUMERAL_BIT2 xa) = (x <= xa) &
+(NUMERAL_BIT2 x <= NUMERAL_BIT1 xa) = (~ xa <= x) &
+(NUMERAL_BIT2 x <= NUMERAL_BIT2 xa) = (x <= xa)"
+ sorry
lemma numeral_pre: "PRE ALT_ZERO = ALT_ZERO &
PRE (NUMERAL_BIT1 ALT_ZERO) = ALT_ZERO &
-(ALL x::nat.
+(ALL x.
PRE (NUMERAL_BIT1 (NUMERAL_BIT1 x)) =
NUMERAL_BIT2 (PRE (NUMERAL_BIT1 x))) &
-(ALL x::nat.
+(ALL x.
PRE (NUMERAL_BIT1 (NUMERAL_BIT2 x)) = NUMERAL_BIT2 (NUMERAL_BIT1 x)) &
-(ALL x::nat. PRE (NUMERAL_BIT2 x) = NUMERAL_BIT1 x)"
- by (import numeral numeral_pre)
-
-lemma bit_initiality: "ALL (zf::'a::type) (b1f::nat => 'a::type => 'a::type)
- b2f::nat => 'a::type => 'a::type.
- EX x::nat => 'a::type.
- x ALT_ZERO = zf &
- (ALL n::nat. x (NUMERAL_BIT1 n) = b1f n (x n)) &
- (ALL n::nat. x (NUMERAL_BIT2 n) = b2f n (x n))"
- by (import numeral bit_initiality)
+(ALL x. PRE (NUMERAL_BIT2 x) = NUMERAL_BIT1 x)"
+ sorry
+
+lemma bit_initiality: "EX x. x ALT_ZERO = zf &
+ (ALL n. x (NUMERAL_BIT1 n) = b1f n (x n)) &
+ (ALL n. x (NUMERAL_BIT2 n) = b2f n (x n))"
+ sorry
consts
iBIT_cases :: "nat => 'a => (nat => 'a) => (nat => 'a) => 'a"
-specification (iBIT_cases) iBIT_cases: "(ALL (zf::'a::type) (bf1::nat => 'a::type) bf2::nat => 'a::type.
+specification (iBIT_cases) iBIT_cases: "(ALL (zf::'a) (bf1::nat => 'a) bf2::nat => 'a.
iBIT_cases ALT_ZERO zf bf1 bf2 = zf) &
-(ALL (n::nat) (zf::'a::type) (bf1::nat => 'a::type) bf2::nat => 'a::type.
+(ALL (n::nat) (zf::'a) (bf1::nat => 'a) bf2::nat => 'a.
iBIT_cases (NUMERAL_BIT1 n) zf bf1 bf2 = bf1 n) &
-(ALL (n::nat) (zf::'a::type) (bf1::nat => 'a::type) bf2::nat => 'a::type.
+(ALL (n::nat) (zf::'a) (bf1::nat => 'a) bf2::nat => 'a.
iBIT_cases (NUMERAL_BIT2 n) zf bf1 bf2 = bf2 n)"
- by (import numeral iBIT_cases)
-
-definition iDUB :: "nat => nat" where
- "iDUB == %x::nat. x + x"
-
-lemma iDUB: "ALL x::nat. iDUB x = x + x"
- by (import numeral iDUB)
+ sorry
+
+definition
+ iDUB :: "nat => nat" where
+ "iDUB == %x. x + x"
+
+lemma iDUB: "iDUB x = x + x"
+ sorry
consts
iSUB :: "bool => nat => nat => nat"
-specification (iSUB) iSUB_DEF: "(ALL (b::bool) x::nat. iSUB b ALT_ZERO x = ALT_ZERO) &
-(ALL (b::bool) (n::nat) x::nat.
+specification (iSUB) iSUB_DEF: "(ALL b x. iSUB b ALT_ZERO x = ALT_ZERO) &
+(ALL b n x.
iSUB b (NUMERAL_BIT1 n) x =
(if b
- then iBIT_cases x (NUMERAL_BIT1 n) (%m::nat. iDUB (iSUB True n m))
- (%m::nat. NUMERAL_BIT1 (iSUB False n m))
- else iBIT_cases x (iDUB n) (%m::nat. NUMERAL_BIT1 (iSUB False n m))
- (%m::nat. iDUB (iSUB False n m)))) &
-(ALL (b::bool) (n::nat) x::nat.
+ then iBIT_cases x (NUMERAL_BIT1 n) (%m. iDUB (iSUB True n m))
+ (%m. NUMERAL_BIT1 (iSUB False n m))
+ else iBIT_cases x (iDUB n) (%m. NUMERAL_BIT1 (iSUB False n m))
+ (%m. iDUB (iSUB False n m)))) &
+(ALL b n x.
iSUB b (NUMERAL_BIT2 n) x =
(if b
- then iBIT_cases x (NUMERAL_BIT2 n)
- (%m::nat. NUMERAL_BIT1 (iSUB True n m))
- (%m::nat. iDUB (iSUB True n m))
- else iBIT_cases x (NUMERAL_BIT1 n) (%m::nat. iDUB (iSUB True n m))
- (%m::nat. NUMERAL_BIT1 (iSUB False n m))))"
- by (import numeral iSUB_DEF)
-
-lemma bit_induction: "ALL P::nat => bool.
- P ALT_ZERO &
- (ALL n::nat. P n --> P (NUMERAL_BIT1 n)) &
- (ALL n::nat. P n --> P (NUMERAL_BIT2 n)) -->
- All P"
- by (import numeral bit_induction)
-
-lemma iSUB_THM: "ALL (xa::bool) (xb::nat) xc::nat.
- iSUB xa ALT_ZERO (x::nat) = ALT_ZERO &
- iSUB True xb ALT_ZERO = xb &
- iSUB False (NUMERAL_BIT1 xb) ALT_ZERO = iDUB xb &
- iSUB True (NUMERAL_BIT1 xb) (NUMERAL_BIT1 xc) = iDUB (iSUB True xb xc) &
- iSUB False (NUMERAL_BIT1 xb) (NUMERAL_BIT1 xc) =
- NUMERAL_BIT1 (iSUB False xb xc) &
- iSUB True (NUMERAL_BIT1 xb) (NUMERAL_BIT2 xc) =
- NUMERAL_BIT1 (iSUB False xb xc) &
- iSUB False (NUMERAL_BIT1 xb) (NUMERAL_BIT2 xc) =
- iDUB (iSUB False xb xc) &
- iSUB False (NUMERAL_BIT2 xb) ALT_ZERO = NUMERAL_BIT1 xb &
- iSUB True (NUMERAL_BIT2 xb) (NUMERAL_BIT1 xc) =
- NUMERAL_BIT1 (iSUB True xb xc) &
- iSUB False (NUMERAL_BIT2 xb) (NUMERAL_BIT1 xc) = iDUB (iSUB True xb xc) &
- iSUB True (NUMERAL_BIT2 xb) (NUMERAL_BIT2 xc) = iDUB (iSUB True xb xc) &
- iSUB False (NUMERAL_BIT2 xb) (NUMERAL_BIT2 xc) =
- NUMERAL_BIT1 (iSUB False xb xc)"
- by (import numeral iSUB_THM)
-
-lemma numeral_sub: "ALL (x::nat) xa::nat.
- NUMERAL (x - xa) = (if xa < x then NUMERAL (iSUB True x xa) else 0)"
- by (import numeral numeral_sub)
-
-lemma iDUB_removal: "ALL x::nat.
- iDUB (NUMERAL_BIT1 x) = NUMERAL_BIT2 (iDUB x) &
- iDUB (NUMERAL_BIT2 x) = NUMERAL_BIT2 (NUMERAL_BIT1 x) &
- iDUB ALT_ZERO = ALT_ZERO"
- by (import numeral iDUB_removal)
-
-lemma numeral_mult: "ALL (x::nat) xa::nat.
- ALT_ZERO * x = ALT_ZERO &
- x * ALT_ZERO = ALT_ZERO &
- NUMERAL_BIT1 x * xa = iZ (iDUB (x * xa) + xa) &
- NUMERAL_BIT2 x * xa = iDUB (iZ (x * xa + xa))"
- by (import numeral numeral_mult)
-
-definition iSQR :: "nat => nat" where
- "iSQR == %x::nat. x * x"
-
-lemma iSQR: "ALL x::nat. iSQR x = x * x"
- by (import numeral iSQR)
-
-lemma numeral_exp: "(ALL x::nat. x ^ ALT_ZERO = NUMERAL_BIT1 ALT_ZERO) &
-(ALL (x::nat) xa::nat. x ^ NUMERAL_BIT1 xa = x * iSQR (x ^ xa)) &
-(ALL (x::nat) xa::nat. x ^ NUMERAL_BIT2 xa = iSQR x * iSQR (x ^ xa))"
- by (import numeral numeral_exp)
-
-lemma numeral_evenodd: "ALL x::nat.
- EVEN ALT_ZERO &
- EVEN (NUMERAL_BIT2 x) &
- ~ EVEN (NUMERAL_BIT1 x) &
- ~ ODD ALT_ZERO & ~ ODD (NUMERAL_BIT2 x) & ODD (NUMERAL_BIT1 x)"
- by (import numeral numeral_evenodd)
-
-lemma numeral_fact: "ALL n::nat. FACT n = (if n = 0 then 1 else n * FACT (PRE n))"
- by (import numeral numeral_fact)
-
-lemma numeral_funpow: "ALL n::nat.
- ((f::'a::type => 'a::type) ^^ n) (x::'a::type) =
- (if n = 0 then x else (f ^^ (n - 1)) (f x))"
- by (import numeral numeral_funpow)
+ then iBIT_cases x (NUMERAL_BIT2 n) (%m. NUMERAL_BIT1 (iSUB True n m))
+ (%m. iDUB (iSUB True n m))
+ else iBIT_cases x (NUMERAL_BIT1 n) (%m. iDUB (iSUB True n m))
+ (%m. NUMERAL_BIT1 (iSUB False n m))))"
+ sorry
+
+lemma bit_induction: "P ALT_ZERO &
+(ALL n. P n --> P (NUMERAL_BIT1 n)) & (ALL n. P n --> P (NUMERAL_BIT2 n))
+==> P x"
+ sorry
+
+lemma iSUB_THM: "iSUB (x::bool) ALT_ZERO (xn::nat) = ALT_ZERO &
+iSUB True (xa::nat) ALT_ZERO = xa &
+iSUB False (NUMERAL_BIT1 xa) ALT_ZERO = iDUB xa &
+iSUB True (NUMERAL_BIT1 xa) (NUMERAL_BIT1 (xb::nat)) =
+iDUB (iSUB True xa xb) &
+iSUB False (NUMERAL_BIT1 xa) (NUMERAL_BIT1 xb) =
+NUMERAL_BIT1 (iSUB False xa xb) &
+iSUB True (NUMERAL_BIT1 xa) (NUMERAL_BIT2 xb) =
+NUMERAL_BIT1 (iSUB False xa xb) &
+iSUB False (NUMERAL_BIT1 xa) (NUMERAL_BIT2 xb) = iDUB (iSUB False xa xb) &
+iSUB False (NUMERAL_BIT2 xa) ALT_ZERO = NUMERAL_BIT1 xa &
+iSUB True (NUMERAL_BIT2 xa) (NUMERAL_BIT1 xb) =
+NUMERAL_BIT1 (iSUB True xa xb) &
+iSUB False (NUMERAL_BIT2 xa) (NUMERAL_BIT1 xb) = iDUB (iSUB True xa xb) &
+iSUB True (NUMERAL_BIT2 xa) (NUMERAL_BIT2 xb) = iDUB (iSUB True xa xb) &
+iSUB False (NUMERAL_BIT2 xa) (NUMERAL_BIT2 xb) =
+NUMERAL_BIT1 (iSUB False xa xb)"
+ sorry
+
+lemma numeral_sub: "NUMERAL (x - xa) = (if xa < x then NUMERAL (iSUB True x xa) else 0)"
+ sorry
+
+lemma iDUB_removal: "iDUB (NUMERAL_BIT1 x) = NUMERAL_BIT2 (iDUB x) &
+iDUB (NUMERAL_BIT2 x) = NUMERAL_BIT2 (NUMERAL_BIT1 x) &
+iDUB ALT_ZERO = ALT_ZERO"
+ sorry
+
+lemma numeral_mult: "ALT_ZERO * x = ALT_ZERO &
+x * ALT_ZERO = ALT_ZERO &
+NUMERAL_BIT1 x * xa = iZ (iDUB (x * xa) + xa) &
+NUMERAL_BIT2 x * xa = iDUB (iZ (x * xa + xa))"
+ sorry
+
+definition
+ iSQR :: "nat => nat" where
+ "iSQR == %x. x * x"
+
+lemma iSQR: "iSQR x = x * x"
+ sorry
+
+lemma numeral_exp: "(ALL x. x ^ ALT_ZERO = NUMERAL_BIT1 ALT_ZERO) &
+(ALL x xa. x ^ NUMERAL_BIT1 xa = x * iSQR (x ^ xa)) &
+(ALL x xa. x ^ NUMERAL_BIT2 xa = iSQR x * iSQR (x ^ xa))"
+ sorry
+
+lemma numeral_evenodd: "EVEN ALT_ZERO &
+EVEN (NUMERAL_BIT2 x) &
+~ EVEN (NUMERAL_BIT1 x) &
+~ ODD ALT_ZERO & ~ ODD (NUMERAL_BIT2 x) & ODD (NUMERAL_BIT1 x)"
+ sorry
+
+lemma numeral_fact: "FACT n = (if n = 0 then 1 else n * FACT (PRE n))"
+ sorry
+
+lemma numeral_funpow: "(f ^^ n) x = (if n = 0 then x else (f ^^ (n - 1)) (f x))"
+ sorry
;end_setup
;setup_theory ind_type
-lemma INJ_INVERSE2: "ALL P::'A::type => 'B::type => 'C::type.
- (ALL (x1::'A::type) (y1::'B::type) (x2::'A::type) y2::'B::type.
- (P x1 y1 = P x2 y2) = (x1 = x2 & y1 = y2)) -->
- (EX (x::'C::type => 'A::type) Y::'C::type => 'B::type.
- ALL (xa::'A::type) y::'B::type. x (P xa y) = xa & Y (P xa y) = y)"
- by (import ind_type INJ_INVERSE2)
-
-definition NUMPAIR :: "nat => nat => nat" where
- "NUMPAIR == %(x::nat) y::nat. 2 ^ x * (2 * y + 1)"
-
-lemma NUMPAIR: "ALL (x::nat) y::nat. NUMPAIR x y = 2 ^ x * (2 * y + 1)"
- by (import ind_type NUMPAIR)
-
-lemma NUMPAIR_INJ_LEMMA: "ALL (x::nat) (xa::nat) (xb::nat) xc::nat.
- NUMPAIR x xa = NUMPAIR xb xc --> x = xb"
- by (import ind_type NUMPAIR_INJ_LEMMA)
-
-lemma NUMPAIR_INJ: "ALL (x1::nat) (y1::nat) (x2::nat) y2::nat.
- (NUMPAIR x1 y1 = NUMPAIR x2 y2) = (x1 = x2 & y1 = y2)"
- by (import ind_type NUMPAIR_INJ)
+lemma INJ_INVERSE2: "(!!(x1::'A) (y1::'B) (x2::'A) y2::'B.
+ ((P::'A => 'B => 'C) x1 y1 = P x2 y2) = (x1 = x2 & y1 = y2))
+==> EX (x::'C => 'A) Y::'C => 'B.
+ ALL (xa::'A) y::'B. x (P xa y) = xa & Y (P xa y) = y"
+ sorry
+
+definition
+ NUMPAIR :: "nat => nat => nat" where
+ "NUMPAIR == %x y. 2 ^ x * (2 * y + 1)"
+
+lemma NUMPAIR: "NUMPAIR x y = 2 ^ x * (2 * y + 1)"
+ sorry
+
+lemma NUMPAIR_INJ_LEMMA: "NUMPAIR x xa = NUMPAIR xb xc ==> x = xb"
+ sorry
+
+lemma NUMPAIR_INJ: "(NUMPAIR x1 y1 = NUMPAIR x2 y2) = (x1 = x2 & y1 = y2)"
+ sorry
consts
NUMSND :: "nat => nat"
NUMFST :: "nat => nat"
-specification (NUMFST NUMSND) NUMPAIR_DEST: "ALL (x::nat) y::nat. NUMFST (NUMPAIR x y) = x & NUMSND (NUMPAIR x y) = y"
- by (import ind_type NUMPAIR_DEST)
-
-definition NUMSUM :: "bool => nat => nat" where
- "NUMSUM == %(b::bool) x::nat. if b then Suc (2 * x) else 2 * x"
-
-lemma NUMSUM: "ALL (b::bool) x::nat. NUMSUM b x = (if b then Suc (2 * x) else 2 * x)"
- by (import ind_type NUMSUM)
-
-lemma NUMSUM_INJ: "ALL (b1::bool) (x1::nat) (b2::bool) x2::nat.
- (NUMSUM b1 x1 = NUMSUM b2 x2) = (b1 = b2 & x1 = x2)"
- by (import ind_type NUMSUM_INJ)
+specification (NUMFST NUMSND) NUMPAIR_DEST: "ALL x y. NUMFST (NUMPAIR x y) = x & NUMSND (NUMPAIR x y) = y"
+ sorry
+
+definition
+ NUMSUM :: "bool => nat => nat" where
+ "NUMSUM == %b x. if b then Suc (2 * x) else 2 * x"
+
+lemma NUMSUM: "NUMSUM b x = (if b then Suc (2 * x) else 2 * x)"
+ sorry
+
+lemma NUMSUM_INJ: "(NUMSUM b1 x1 = NUMSUM b2 x2) = (b1 = b2 & x1 = x2)"
+ sorry
consts
NUMRIGHT :: "nat => nat"
NUMLEFT :: "nat => bool"
-specification (NUMLEFT NUMRIGHT) NUMSUM_DEST: "ALL (x::bool) y::nat. NUMLEFT (NUMSUM x y) = x & NUMRIGHT (NUMSUM x y) = y"
- by (import ind_type NUMSUM_DEST)
-
-definition INJN :: "nat => nat => 'a => bool" where
- "INJN == %(m::nat) (n::nat) a::'a::type. n = m"
-
-lemma INJN: "ALL m::nat. INJN m = (%(n::nat) a::'a::type. n = m)"
- by (import ind_type INJN)
-
-lemma INJN_INJ: "ALL (n1::nat) n2::nat. (INJN n1 = INJN n2) = (n1 = n2)"
- by (import ind_type INJN_INJ)
-
-definition INJA :: "'a => nat => 'a => bool" where
- "INJA == %(a::'a::type) (n::nat) b::'a::type. b = a"
-
-lemma INJA: "ALL a::'a::type. INJA a = (%(n::nat) b::'a::type. b = a)"
- by (import ind_type INJA)
-
-lemma INJA_INJ: "ALL (a1::'a::type) a2::'a::type. (INJA a1 = INJA a2) = (a1 = a2)"
- by (import ind_type INJA_INJ)
-
-definition INJF :: "(nat => nat => 'a => bool) => nat => 'a => bool" where
- "INJF == %(f::nat => nat => 'a::type => bool) n::nat. f (NUMFST n) (NUMSND n)"
-
-lemma INJF: "ALL f::nat => nat => 'a::type => bool.
- INJF f = (%n::nat. f (NUMFST n) (NUMSND n))"
- by (import ind_type INJF)
-
-lemma INJF_INJ: "ALL (f1::nat => nat => 'a::type => bool) f2::nat => nat => 'a::type => bool.
- (INJF f1 = INJF f2) = (f1 = f2)"
- by (import ind_type INJF_INJ)
-
-definition INJP :: "(nat => 'a => bool) => (nat => 'a => bool) => nat => 'a => bool" where
+specification (NUMLEFT NUMRIGHT) NUMSUM_DEST: "ALL x y. NUMLEFT (NUMSUM x y) = x & NUMRIGHT (NUMSUM x y) = y"
+ sorry
+
+definition
+ INJN :: "nat => nat => 'a => bool" where
+ "INJN == %m n a. n = m"
+
+lemma INJN: "INJN m = (%n a. n = m)"
+ sorry
+
+lemma INJN_INJ: "(INJN n1 = INJN n2) = (n1 = n2)"
+ sorry
+
+definition
+ INJA :: "'a => nat => 'a => bool" where
+ "INJA == %a n b. b = a"
+
+lemma INJA: "INJA a = (%n b. b = a)"
+ sorry
+
+lemma INJA_INJ: "(INJA a1 = INJA a2) = (a1 = a2)"
+ sorry
+
+definition
+ INJF :: "(nat => nat => 'a => bool) => nat => 'a => bool" where
+ "INJF == %f n. f (NUMFST n) (NUMSND n)"
+
+lemma INJF: "INJF f = (%n. f (NUMFST n) (NUMSND n))"
+ sorry
+
+lemma INJF_INJ: "(INJF f1 = INJF f2) = (f1 = f2)"
+ sorry
+
+definition
+ INJP :: "(nat => 'a => bool) => (nat => 'a => bool) => nat => 'a => bool" where
"INJP ==
-%(f1::nat => 'a::type => bool) (f2::nat => 'a::type => bool) (n::nat)
- a::'a::type. if NUMLEFT n then f1 (NUMRIGHT n) a else f2 (NUMRIGHT n) a"
-
-lemma INJP: "ALL (f1::nat => 'a::type => bool) f2::nat => 'a::type => bool.
- INJP f1 f2 =
- (%(n::nat) a::'a::type.
- if NUMLEFT n then f1 (NUMRIGHT n) a else f2 (NUMRIGHT n) a)"
- by (import ind_type INJP)
-
-lemma INJP_INJ: "ALL (f1::nat => 'a::type => bool) (f1'::nat => 'a::type => bool)
- (f2::nat => 'a::type => bool) f2'::nat => 'a::type => bool.
- (INJP f1 f2 = INJP f1' f2') = (f1 = f1' & f2 = f2')"
- by (import ind_type INJP_INJ)
-
-definition ZCONSTR :: "nat => 'a => (nat => nat => 'a => bool) => nat => 'a => bool" where
- "ZCONSTR ==
-%(c::nat) (i::'a::type) r::nat => nat => 'a::type => bool.
- INJP (INJN (Suc c)) (INJP (INJA i) (INJF r))"
-
-lemma ZCONSTR: "ALL (c::nat) (i::'a::type) r::nat => nat => 'a::type => bool.
- ZCONSTR c i r = INJP (INJN (Suc c)) (INJP (INJA i) (INJF r))"
- by (import ind_type ZCONSTR)
-
-definition ZBOT :: "nat => 'a => bool" where
- "ZBOT == INJP (INJN 0) (SOME z::nat => 'a::type => bool. True)"
-
-lemma ZBOT: "ZBOT = INJP (INJN 0) (SOME z::nat => 'a::type => bool. True)"
- by (import ind_type ZBOT)
-
-lemma ZCONSTR_ZBOT: "ALL (x::nat) (xa::'a::type) xb::nat => nat => 'a::type => bool.
- ZCONSTR x xa xb ~= ZBOT"
- by (import ind_type ZCONSTR_ZBOT)
-
-definition ZRECSPACE :: "(nat => 'a => bool) => bool" where
+%f1 f2 n a. if NUMLEFT n then f1 (NUMRIGHT n) a else f2 (NUMRIGHT n) a"
+
+lemma INJP: "INJP f1 f2 =
+(%n a. if NUMLEFT n then f1 (NUMRIGHT n) a else f2 (NUMRIGHT n) a)"
+ sorry
+
+lemma INJP_INJ: "(INJP f1 f2 = INJP f1' f2') = (f1 = f1' & f2 = f2')"
+ sorry
+
+definition
+ ZCONSTR :: "nat => 'a => (nat => nat => 'a => bool) => nat => 'a => bool" where
+ "ZCONSTR == %c i r. INJP (INJN (Suc c)) (INJP (INJA i) (INJF r))"
+
+lemma ZCONSTR: "ZCONSTR c i r = INJP (INJN (Suc c)) (INJP (INJA i) (INJF r))"
+ sorry
+
+definition
+ ZBOT :: "nat => 'a => bool" where
+ "ZBOT == INJP (INJN 0) (SOME z. True)"
+
+lemma ZBOT: "ZBOT = INJP (INJN 0) (SOME z. True)"
+ sorry
+
+lemma ZCONSTR_ZBOT: "ZCONSTR x xa xb ~= ZBOT"
+ sorry
+
+definition
+ ZRECSPACE :: "(nat => 'a => bool) => bool" where
"ZRECSPACE ==
-%a0::nat => 'a::type => bool.
- ALL ZRECSPACE'::(nat => 'a::type => bool) => bool.
- (ALL a0::nat => 'a::type => bool.
- a0 = ZBOT |
- (EX (c::nat) (i::'a::type) r::nat => nat => 'a::type => bool.
- a0 = ZCONSTR c i r & (ALL n::nat. ZRECSPACE' (r n))) -->
- ZRECSPACE' a0) -->
- ZRECSPACE' a0"
+%a0. ALL ZRECSPACE'.
+ (ALL a0.
+ a0 = ZBOT |
+ (EX c i r. a0 = ZCONSTR c i r & (ALL n. ZRECSPACE' (r n))) -->
+ ZRECSPACE' a0) -->
+ ZRECSPACE' a0"
lemma ZRECSPACE: "ZRECSPACE =
-(%a0::nat => 'a::type => bool.
- ALL ZRECSPACE'::(nat => 'a::type => bool) => bool.
- (ALL a0::nat => 'a::type => bool.
- a0 = ZBOT |
- (EX (c::nat) (i::'a::type) r::nat => nat => 'a::type => bool.
- a0 = ZCONSTR c i r & (ALL n::nat. ZRECSPACE' (r n))) -->
- ZRECSPACE' a0) -->
- ZRECSPACE' a0)"
- by (import ind_type ZRECSPACE)
+(%a0. ALL ZRECSPACE'.
+ (ALL a0.
+ a0 = ZBOT |
+ (EX c i r. a0 = ZCONSTR c i r & (ALL n. ZRECSPACE' (r n))) -->
+ ZRECSPACE' a0) -->
+ ZRECSPACE' a0)"
+ sorry
lemma ZRECSPACE_rules: "(op &::bool => bool => bool)
((ZRECSPACE::(nat => 'a::type => bool) => bool)
@@ -2904,26 +2212,19 @@
=> (nat => nat => 'a::type => bool)
=> nat => 'a::type => bool)
c i r))))))"
- by (import ind_type ZRECSPACE_rules)
-
-lemma ZRECSPACE_ind: "ALL x::(nat => 'a::type => bool) => bool.
- x ZBOT &
- (ALL (c::nat) (i::'a::type) r::nat => nat => 'a::type => bool.
- (ALL n::nat. x (r n)) --> x (ZCONSTR c i r)) -->
- (ALL a0::nat => 'a::type => bool. ZRECSPACE a0 --> x a0)"
- by (import ind_type ZRECSPACE_ind)
-
-lemma ZRECSPACE_cases: "ALL a0::nat => 'a::type => bool.
- ZRECSPACE a0 =
- (a0 = ZBOT |
- (EX (c::nat) (i::'a::type) r::nat => nat => 'a::type => bool.
- a0 = ZCONSTR c i r & (ALL n::nat. ZRECSPACE (r n))))"
- by (import ind_type ZRECSPACE_cases)
-
-typedef (open) ('a) recspace = "(Collect::((nat => 'a::type => bool) => bool)
- => (nat => 'a::type => bool) set)
- (ZRECSPACE::(nat => 'a::type => bool) => bool)"
- by (rule typedef_helper,import ind_type recspace_TY_DEF)
+ sorry
+
+lemma ZRECSPACE_ind: "[| x ZBOT & (ALL c i r. (ALL n. x (r n)) --> x (ZCONSTR c i r));
+ ZRECSPACE a0 |]
+==> x a0"
+ sorry
+
+lemma ZRECSPACE_cases: "ZRECSPACE a0 =
+(a0 = ZBOT | (EX c i r. a0 = ZCONSTR c i r & (ALL n. ZRECSPACE (r n))))"
+ sorry
+
+typedef (open) ('a) recspace = "Collect ZRECSPACE :: (nat \<Rightarrow> 'a\<Colon>type \<Rightarrow> bool) set"
+ sorry
lemmas recspace_TY_DEF = typedef_hol2hol4 [OF type_definition_recspace]
@@ -2931,110 +2232,85 @@
mk_rec :: "(nat => 'a => bool) => 'a recspace"
dest_rec :: "'a recspace => nat => 'a => bool"
-specification (dest_rec mk_rec) recspace_repfns: "(ALL a::'a::type recspace. mk_rec (dest_rec a) = a) &
-(ALL r::nat => 'a::type => bool. ZRECSPACE r = (dest_rec (mk_rec r) = r))"
- by (import ind_type recspace_repfns)
-
-definition BOTTOM :: "'a recspace" where
+specification (dest_rec mk_rec) recspace_repfns: "(ALL a::'a recspace. mk_rec (dest_rec a) = a) &
+(ALL r::nat => 'a => bool. ZRECSPACE r = (dest_rec (mk_rec r) = r))"
+ sorry
+
+definition
+ BOTTOM :: "'a recspace" where
"BOTTOM == mk_rec ZBOT"
lemma BOTTOM: "BOTTOM = mk_rec ZBOT"
- by (import ind_type BOTTOM)
-
-definition CONSTR :: "nat => 'a => (nat => 'a recspace) => 'a recspace" where
- "CONSTR ==
-%(c::nat) (i::'a::type) r::nat => 'a::type recspace.
- mk_rec (ZCONSTR c i (%n::nat. dest_rec (r n)))"
-
-lemma CONSTR: "ALL (c::nat) (i::'a::type) r::nat => 'a::type recspace.
- CONSTR c i r = mk_rec (ZCONSTR c i (%n::nat. dest_rec (r n)))"
- by (import ind_type CONSTR)
-
-lemma MK_REC_INJ: "ALL (x::nat => 'a::type => bool) y::nat => 'a::type => bool.
- mk_rec x = mk_rec y --> ZRECSPACE x & ZRECSPACE y --> x = y"
- by (import ind_type MK_REC_INJ)
-
-lemma DEST_REC_INJ: "ALL (x::'a::type recspace) y::'a::type recspace.
- (dest_rec x = dest_rec y) = (x = y)"
- by (import ind_type DEST_REC_INJ)
-
-lemma CONSTR_BOT: "ALL (c::nat) (i::'a::type) r::nat => 'a::type recspace.
- CONSTR c i r ~= BOTTOM"
- by (import ind_type CONSTR_BOT)
-
-lemma CONSTR_INJ: "ALL (c1::nat) (i1::'a::type) (r1::nat => 'a::type recspace) (c2::nat)
- (i2::'a::type) r2::nat => 'a::type recspace.
- (CONSTR c1 i1 r1 = CONSTR c2 i2 r2) = (c1 = c2 & i1 = i2 & r1 = r2)"
- by (import ind_type CONSTR_INJ)
-
-lemma CONSTR_IND: "ALL P::'a::type recspace => bool.
- P BOTTOM &
- (ALL (c::nat) (i::'a::type) r::nat => 'a::type recspace.
- (ALL n::nat. P (r n)) --> P (CONSTR c i r)) -->
- All P"
- by (import ind_type CONSTR_IND)
-
-lemma CONSTR_REC: "ALL Fn::nat
- => 'a::type
- => (nat => 'a::type recspace) => (nat => 'b::type) => 'b::type.
- EX f::'a::type recspace => 'b::type.
- ALL (c::nat) (i::'a::type) r::nat => 'a::type recspace.
- f (CONSTR c i r) = Fn c i r (%n::nat. f (r n))"
- by (import ind_type CONSTR_REC)
+ sorry
+
+definition
+ CONSTR :: "nat => 'a => (nat => 'a recspace) => 'a recspace" where
+ "CONSTR == %c i r. mk_rec (ZCONSTR c i (%n. dest_rec (r n)))"
+
+lemma CONSTR: "CONSTR c i r = mk_rec (ZCONSTR c i (%n. dest_rec (r n)))"
+ sorry
+
+lemma MK_REC_INJ: "[| mk_rec x = mk_rec y; ZRECSPACE x & ZRECSPACE y |] ==> x = y"
+ sorry
+
+lemma DEST_REC_INJ: "(dest_rec x = dest_rec y) = (x = y)"
+ sorry
+
+lemma CONSTR_BOT: "CONSTR c i r ~= BOTTOM"
+ sorry
+
+lemma CONSTR_INJ: "(CONSTR c1 i1 r1 = CONSTR c2 i2 r2) = (c1 = c2 & i1 = i2 & r1 = r2)"
+ sorry
+
+lemma CONSTR_IND: "P BOTTOM & (ALL c i r. (ALL n. P (r n)) --> P (CONSTR c i r)) ==> P x"
+ sorry
+
+lemma CONSTR_REC: "EX f. ALL c i r. f (CONSTR c i r) = Fn c i r (%n. f (r n))"
+ sorry
consts
FCONS :: "'a => (nat => 'a) => nat => 'a"
-specification (FCONS) FCONS: "(ALL (a::'a::type) f::nat => 'a::type. FCONS a f 0 = a) &
-(ALL (a::'a::type) (f::nat => 'a::type) n::nat. FCONS a f (Suc n) = f n)"
- by (import ind_type FCONS)
-
-definition FNIL :: "nat => 'a" where
- "FNIL == %n::nat. SOME x::'a::type. True"
-
-lemma FNIL: "ALL n::nat. FNIL n = (SOME x::'a::type. True)"
- by (import ind_type FNIL)
-
-definition ISO :: "('a => 'b) => ('b => 'a) => bool" where
- "ISO ==
-%(f::'a::type => 'b::type) g::'b::type => 'a::type.
- (ALL x::'b::type. f (g x) = x) & (ALL y::'a::type. g (f y) = y)"
-
-lemma ISO: "ALL (f::'a::type => 'b::type) g::'b::type => 'a::type.
- ISO f g =
- ((ALL x::'b::type. f (g x) = x) & (ALL y::'a::type. g (f y) = y))"
- by (import ind_type ISO)
-
-lemma ISO_REFL: "ISO (%x::'a::type. x) (%x::'a::type. x)"
- by (import ind_type ISO_REFL)
-
-lemma ISO_FUN: "ISO (f::'a::type => 'c::type) (f'::'c::type => 'a::type) &
-ISO (g::'b::type => 'd::type) (g'::'d::type => 'b::type) -->
-ISO (%(h::'a::type => 'b::type) a'::'c::type. g (h (f' a')))
- (%(h::'c::type => 'd::type) a::'a::type. g' (h (f a)))"
- by (import ind_type ISO_FUN)
-
-lemma ISO_USAGE: "ISO (f::'a::type => 'b::type) (g::'b::type => 'a::type) -->
-(ALL P::'a::type => bool. All P = (ALL x::'b::type. P (g x))) &
-(ALL P::'a::type => bool. Ex P = (EX x::'b::type. P (g x))) &
-(ALL (a::'a::type) b::'b::type. (a = g b) = (f a = b))"
- by (import ind_type ISO_USAGE)
+specification (FCONS) FCONS: "(ALL (a::'a) f::nat => 'a. FCONS a f (0::nat) = a) &
+(ALL (a::'a) (f::nat => 'a) n::nat. FCONS a f (Suc n) = f n)"
+ sorry
+
+definition
+ FNIL :: "nat => 'a" where
+ "FNIL == %n. SOME x. True"
+
+lemma FNIL: "FNIL n = (SOME x. True)"
+ sorry
+
+definition
+ ISO :: "('a => 'b) => ('b => 'a) => bool" where
+ "ISO == %f g. (ALL x. f (g x) = x) & (ALL y. g (f y) = y)"
+
+lemma ISO: "ISO f g = ((ALL x. f (g x) = x) & (ALL y. g (f y) = y))"
+ sorry
+
+lemma ISO_REFL: "ISO (%x. x) (%x. x)"
+ sorry
+
+lemma ISO_FUN: "ISO (f::'a => 'c) (f'::'c => 'a) & ISO (g::'b => 'd) (g'::'d => 'b)
+==> ISO (%(h::'a => 'b) a'::'c. g (h (f' a')))
+ (%(h::'c => 'd) a::'a. g' (h (f a)))"
+ sorry
+
+lemma ISO_USAGE: "ISO f g
+==> (ALL P. All P = (ALL x. P (g x))) &
+ (ALL P. Ex P = (EX x. P (g x))) & (ALL a b. (a = g b) = (f a = b))"
+ sorry
;end_setup
;setup_theory divides
-lemma ONE_DIVIDES_ALL: "(All::(nat => bool) => bool) ((op dvd::nat => nat => bool) (1::nat))"
- by (import divides ONE_DIVIDES_ALL)
-
-lemma DIVIDES_ADD_2: "ALL (a::nat) (b::nat) c::nat. a dvd b & a dvd b + c --> a dvd c"
- by (import divides DIVIDES_ADD_2)
-
-lemma DIVIDES_FACT: "ALL b>0. b dvd FACT b"
- by (import divides DIVIDES_FACT)
-
-lemma DIVIDES_MULT_LEFT: "ALL (x::nat) xa::nat. (x * xa dvd xa) = (xa = 0 | x = 1)"
- by (import divides DIVIDES_MULT_LEFT)
+lemma DIVIDES_FACT: "0 < b ==> b dvd FACT b"
+ sorry
+
+lemma DIVIDES_MULT_LEFT: "((x::nat) * (xa::nat) dvd xa) = (xa = (0::nat) | x = (1::nat))"
+ sorry
;end_setup
@@ -3044,17 +2320,16 @@
prime :: "nat => bool"
defs
- prime_primdef: "prime.prime == %a::nat. a ~= 1 & (ALL b::nat. b dvd a --> b = a | b = 1)"
-
-lemma prime_def: "ALL a::nat.
- prime.prime a = (a ~= 1 & (ALL b::nat. b dvd a --> b = a | b = 1))"
- by (import prime prime_def)
+ prime_primdef: "prime.prime == %a. a ~= 1 & (ALL b. b dvd a --> b = a | b = 1)"
+
+lemma prime_def: "prime.prime a = (a ~= 1 & (ALL b. b dvd a --> b = a | b = 1))"
+ sorry
lemma NOT_PRIME_0: "~ prime.prime 0"
- by (import prime NOT_PRIME_0)
+ sorry
lemma NOT_PRIME_1: "~ prime.prime 1"
- by (import prime NOT_PRIME_1)
+ sorry
;end_setup
@@ -3063,997 +2338,758 @@
consts
EL :: "nat => 'a list => 'a"
-specification (EL) EL: "(ALL l::'a::type list. EL 0 l = hd l) &
-(ALL (l::'a::type list) n::nat. EL (Suc n) l = EL n (tl l))"
- by (import list EL)
+specification (EL) EL: "(ALL l::'a list. EL (0::nat) l = hd l) &
+(ALL (l::'a list) n::nat. EL (Suc n) l = EL n (tl l))"
+ sorry
lemma NULL: "(op &::bool => bool => bool)
- ((null::'a::type list => bool) ([]::'a::type list))
+ ((List.null::'a::type list => bool) ([]::'a::type list))
((All::('a::type => bool) => bool)
(%x::'a::type.
(All::('a::type list => bool) => bool)
(%xa::'a::type list.
(Not::bool => bool)
- ((null::'a::type list => bool)
+ ((List.null::'a::type list => bool)
((op #::'a::type => 'a::type list => 'a::type list) x xa)))))"
- by (import list NULL)
-
-lemma list_case_compute: "ALL l::'a::type list.
- list_case (b::'b::type) (f::'a::type => 'a::type list => 'b::type) l =
- (if null l then b else f (hd l) (tl l))"
- by (import list list_case_compute)
-
-lemma LIST_NOT_EQ: "ALL (l1::'a::type list) l2::'a::type list.
- l1 ~= l2 --> (ALL (x::'a::type) xa::'a::type. x # l1 ~= xa # l2)"
- by (import list LIST_NOT_EQ)
-
-lemma NOT_EQ_LIST: "ALL (h1::'a::type) h2::'a::type.
- h1 ~= h2 -->
- (ALL (x::'a::type list) xa::'a::type list. h1 # x ~= h2 # xa)"
- by (import list NOT_EQ_LIST)
-
-lemma EQ_LIST: "ALL (h1::'a::type) h2::'a::type.
- h1 = h2 -->
- (ALL (l1::'a::type list) l2::'a::type list.
- l1 = l2 --> h1 # l1 = h2 # l2)"
- by (import list EQ_LIST)
-
-lemma CONS: "ALL l::'a::type list. ~ null l --> hd l # tl l = l"
- by (import list CONS)
-
-lemma MAP_EQ_NIL: "ALL (l::'a::type list) f::'a::type => 'b::type.
- (map f l = []) = (l = []) & ([] = map f l) = (l = [])"
- by (import list MAP_EQ_NIL)
-
-lemma EVERY_EL: "(All::('a::type list => bool) => bool)
- (%l::'a::type list.
- (All::(('a::type => bool) => bool) => bool)
- (%P::'a::type => bool.
- (op =::bool => bool => bool)
- ((list_all::('a::type => bool) => 'a::type list => bool) P l)
- ((All::(nat => bool) => bool)
- (%n::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) n
- ((size::'a::type list => nat) l))
- (P ((EL::nat => 'a::type list => 'a::type) n l))))))"
- by (import list EVERY_EL)
-
-lemma EVERY_CONJ: "ALL l::'a::type list.
- list_all
- (%x::'a::type. (P::'a::type => bool) x & (Q::'a::type => bool) x) l =
- (list_all P l & list_all Q l)"
- by (import list EVERY_CONJ)
-
-lemma EVERY_MEM: "ALL (P::'a::type => bool) l::'a::type list.
- list_all P l = (ALL e::'a::type. e mem l --> P e)"
- by (import list EVERY_MEM)
-
-lemma EXISTS_MEM: "ALL (P::'a::type => bool) l::'a::type list.
- list_ex P l = (EX e::'a::type. e mem l & P e)"
- by (import list EXISTS_MEM)
-
-lemma MEM_APPEND: "ALL (e::'a::type) (l1::'a::type list) l2::'a::type list.
- e mem l1 @ l2 = (e mem l1 | e mem l2)"
- by (import list MEM_APPEND)
-
-lemma EXISTS_APPEND: "ALL (P::'a::type => bool) (l1::'a::type list) l2::'a::type list.
- list_ex P (l1 @ l2) = (list_ex P l1 | list_ex P l2)"
- by (import list EXISTS_APPEND)
-
-lemma NOT_EVERY: "ALL (P::'a::type => bool) l::'a::type list.
- (~ list_all P l) = list_ex (Not o P) l"
- by (import list NOT_EVERY)
-
-lemma NOT_EXISTS: "ALL (P::'a::type => bool) l::'a::type list.
- (~ list_ex P l) = list_all (Not o P) l"
- by (import list NOT_EXISTS)
-
-lemma MEM_MAP: "ALL (l::'a::type list) (f::'a::type => 'b::type) x::'b::type.
- x mem map f l = (EX y::'a::type. x = f y & y mem l)"
- by (import list MEM_MAP)
-
-lemma LENGTH_CONS: "ALL (l::'a::type list) n::nat.
- (length l = Suc n) =
- (EX (h::'a::type) l'::'a::type list. length l' = n & l = h # l')"
- by (import list LENGTH_CONS)
-
-lemma LENGTH_EQ_CONS: "ALL (P::'a::type list => bool) n::nat.
- (ALL l::'a::type list. length l = Suc n --> P l) =
- (ALL l::'a::type list. length l = n --> (ALL x::'a::type. P (x # l)))"
- by (import list LENGTH_EQ_CONS)
-
-lemma LENGTH_EQ_NIL: "ALL P::'a::type list => bool.
- (ALL l::'a::type list. length l = 0 --> P l) = P []"
- by (import list LENGTH_EQ_NIL)
-
-lemma CONS_ACYCLIC: "ALL (l::'a::type list) x::'a::type. l ~= x # l & x # l ~= l"
- by (import list CONS_ACYCLIC)
-
-lemma APPEND_eq_NIL: "(ALL (l1::'a::type list) l2::'a::type list.
- ([] = l1 @ l2) = (l1 = [] & l2 = [])) &
-(ALL (l1::'a::type list) l2::'a::type list.
- (l1 @ l2 = []) = (l1 = [] & l2 = []))"
- by (import list APPEND_eq_NIL)
-
-lemma APPEND_11: "(ALL (l1::'a::type list) (l2::'a::type list) l3::'a::type list.
+ sorry
+
+lemma list_case_compute: "list_case (b::'b) (f::'a => 'a list => 'b) (l::'a list) =
+(if List.null l then b else f (hd l) (tl l))"
+ sorry
+
+lemma LIST_NOT_EQ: "l1 ~= l2 ==> x # l1 ~= xa # l2"
+ sorry
+
+lemma NOT_EQ_LIST: "h1 ~= h2 ==> h1 # x ~= h2 # xa"
+ sorry
+
+lemma EQ_LIST: "[| h1 = h2; l1 = l2 |] ==> h1 # l1 = h2 # l2"
+ sorry
+
+lemma CONS: "~ List.null l ==> hd l # tl l = l"
+ sorry
+
+lemma MAP_EQ_NIL: "(map (f::'a => 'b) (l::'a list) = []) = (l = []) & ([] = map f l) = (l = [])"
+ sorry
+
+lemma EVERY_EL: "list_all P l = (ALL n<length l. P (EL n l))"
+ sorry
+
+lemma EVERY_CONJ: "list_all (%x. P x & Q x) l = (list_all P l & list_all Q l)"
+ sorry
+
+lemma EVERY_MEM: "list_all P l = (ALL e. List.member l e --> P e)"
+ sorry
+
+lemma EXISTS_MEM: "list_ex P l = (EX e. List.member l e & P e)"
+ sorry
+
+lemma MEM_APPEND: "List.member (l1 @ l2) e = (List.member l1 e | List.member l2 e)"
+ sorry
+
+lemma NOT_EVERY: "(~ list_all P l) = list_ex (Not o P) l"
+ sorry
+
+lemma NOT_EXISTS: "(~ list_ex P l) = list_all (Not o P) l"
+ sorry
+
+lemma MEM_MAP: "List.member (map (f::'a => 'b) (l::'a list)) (x::'b) =
+(EX y::'a. x = f y & List.member l y)"
+ sorry
+
+lemma LENGTH_CONS: "(length l = Suc n) = (EX h l'. length l' = n & l = h # l')"
+ sorry
+
+lemma LENGTH_EQ_CONS: "(ALL l. length l = Suc n --> P l) =
+(ALL l. length l = n --> (ALL x. P (x # l)))"
+ sorry
+
+lemma LENGTH_EQ_NIL: "(ALL l. length l = 0 --> P l) = P []"
+ sorry
+
+lemma CONS_ACYCLIC: "l ~= x # l & x # l ~= l"
+ sorry
+
+lemma APPEND_eq_NIL: "(ALL (l1::'a list) l2::'a list. ([] = l1 @ l2) = (l1 = [] & l2 = [])) &
+(ALL (l1::'a list) l2::'a list. (l1 @ l2 = []) = (l1 = [] & l2 = []))"
+ sorry
+
+lemma APPEND_11: "(ALL (l1::'a list) (l2::'a list) l3::'a list.
(l1 @ l2 = l1 @ l3) = (l2 = l3)) &
-(ALL (l1::'a::type list) (l2::'a::type list) l3::'a::type list.
+(ALL (l1::'a list) (l2::'a list) l3::'a list.
(l2 @ l1 = l3 @ l1) = (l2 = l3))"
- by (import list APPEND_11)
-
-lemma EL_compute: "ALL n::nat.
- EL n (l::'a::type list) = (if n = 0 then hd l else EL (PRE n) (tl l))"
- by (import list EL_compute)
-
-lemma WF_LIST_PRED: "WF (%(L1::'a::type list) L2::'a::type list. EX h::'a::type. L2 = h # L1)"
- by (import list WF_LIST_PRED)
-
-lemma list_size_cong: "ALL (M::'a::type list) (N::'a::type list) (f::'a::type => nat)
- f'::'a::type => nat.
- M = N & (ALL x::'a::type. x mem N --> f x = f' x) -->
- list_size f M = list_size f' N"
- by (import list list_size_cong)
-
-lemma FOLDR_CONG: "ALL (l::'a::type list) (l'::'a::type list) (b::'b::type) (b'::'b::type)
- (f::'a::type => 'b::type => 'b::type)
- f'::'a::type => 'b::type => 'b::type.
- l = l' &
- b = b' & (ALL (x::'a::type) a::'b::type. x mem l' --> f x a = f' x a) -->
- foldr f l b = foldr f' l' b'"
- by (import list FOLDR_CONG)
-
-lemma FOLDL_CONG: "ALL (l::'a::type list) (l'::'a::type list) (b::'b::type) (b'::'b::type)
- (f::'b::type => 'a::type => 'b::type)
- f'::'b::type => 'a::type => 'b::type.
- l = l' &
- b = b' & (ALL (x::'a::type) a::'b::type. x mem l' --> f a x = f' a x) -->
- foldl f b l = foldl f' b' l'"
- by (import list FOLDL_CONG)
-
-lemma MAP_CONG: "ALL (l1::'a::type list) (l2::'a::type list) (f::'a::type => 'b::type)
- f'::'a::type => 'b::type.
- l1 = l2 & (ALL x::'a::type. x mem l2 --> f x = f' x) -->
- map f l1 = map f' l2"
- by (import list MAP_CONG)
-
-lemma EXISTS_CONG: "ALL (l1::'a::type list) (l2::'a::type list) (P::'a::type => bool)
- P'::'a::type => bool.
- l1 = l2 & (ALL x::'a::type. x mem l2 --> P x = P' x) -->
- list_ex P l1 = list_ex P' l2"
- by (import list EXISTS_CONG)
-
-lemma EVERY_CONG: "ALL (l1::'a::type list) (l2::'a::type list) (P::'a::type => bool)
- P'::'a::type => bool.
- l1 = l2 & (ALL x::'a::type. x mem l2 --> P x = P' x) -->
- list_all P l1 = list_all P' l2"
- by (import list EVERY_CONG)
-
-lemma EVERY_MONOTONIC: "ALL (P::'a::type => bool) Q::'a::type => bool.
- (ALL x::'a::type. P x --> Q x) -->
- (ALL l::'a::type list. list_all P l --> list_all Q l)"
- by (import list EVERY_MONOTONIC)
-
-lemma LENGTH_ZIP: "ALL (l1::'a::type list) l2::'b::type list.
- length l1 = length l2 -->
- length (zip l1 l2) = length l1 & length (zip l1 l2) = length l2"
- by (import list LENGTH_ZIP)
-
-lemma LENGTH_UNZIP: "ALL pl::('a::type * 'b::type) list.
- length (fst (unzip pl)) = length pl & length (snd (unzip pl)) = length pl"
- by (import list LENGTH_UNZIP)
-
-lemma ZIP_UNZIP: "ALL l::('a::type * 'b::type) list. ZIP (unzip l) = l"
- by (import list ZIP_UNZIP)
-
-lemma UNZIP_ZIP: "ALL (l1::'a::type list) l2::'b::type list.
- length l1 = length l2 --> unzip (zip l1 l2) = (l1, l2)"
- by (import list UNZIP_ZIP)
-
-lemma ZIP_MAP: "ALL (l1::'a::type list) (l2::'b::type list) (f1::'a::type => 'c::type)
- f2::'b::type => 'd::type.
- length l1 = length l2 -->
- zip (map f1 l1) l2 =
- map (%p::'a::type * 'b::type. (f1 (fst p), snd p)) (zip l1 l2) &
- zip l1 (map f2 l2) =
- map (%p::'a::type * 'b::type. (fst p, f2 (snd p))) (zip l1 l2)"
- by (import list ZIP_MAP)
-
-lemma MEM_ZIP: "(All::('a::type list => bool) => bool)
- (%l1::'a::type list.
- (All::('b::type list => bool) => bool)
- (%l2::'b::type list.
- (All::('a::type * 'b::type => bool) => bool)
- (%p::'a::type * 'b::type.
- (op -->::bool => bool => bool)
- ((op =::nat => nat => bool)
- ((size::'a::type list => nat) l1)
- ((size::'b::type list => nat) l2))
- ((op =::bool => bool => bool)
- ((op mem::'a::type * 'b::type
- => ('a::type * 'b::type) list => bool)
- p ((zip::'a::type list
- => 'b::type list => ('a::type * 'b::type) list)
- l1 l2))
- ((Ex::(nat => bool) => bool)
- (%n::nat.
- (op &::bool => bool => bool)
- ((op <::nat => nat => bool) n
- ((size::'a::type list => nat) l1))
- ((op =::'a::type * 'b::type
- => 'a::type * 'b::type => bool)
- p ((Pair::'a::type
- => 'b::type => 'a::type * 'b::type)
- ((EL::nat => 'a::type list => 'a::type) n l1)
- ((EL::nat => 'b::type list => 'b::type) n
- l2)))))))))"
- by (import list MEM_ZIP)
-
-lemma EL_ZIP: "ALL (l1::'a::type list) (l2::'b::type list) n::nat.
- length l1 = length l2 & n < length l1 -->
- EL n (zip l1 l2) = (EL n l1, EL n l2)"
- by (import list EL_ZIP)
-
-lemma MAP2_ZIP: "(All::('a::type list => bool) => bool)
- (%l1::'a::type list.
- (All::('b::type list => bool) => bool)
- (%l2::'b::type list.
- (op -->::bool => bool => bool)
- ((op =::nat => nat => bool) ((size::'a::type list => nat) l1)
- ((size::'b::type list => nat) l2))
- ((All::(('a::type => 'b::type => 'c::type) => bool) => bool)
- (%f::'a::type => 'b::type => 'c::type.
- (op =::'c::type list => 'c::type list => bool)
- ((map2::('a::type => 'b::type => 'c::type)
- => 'a::type list
- => 'b::type list => 'c::type list)
- f l1 l2)
- ((map::('a::type * 'b::type => 'c::type)
- => ('a::type * 'b::type) list => 'c::type list)
- ((split::('a::type => 'b::type => 'c::type)
- => 'a::type * 'b::type => 'c::type)
- f)
- ((zip::'a::type list
- => 'b::type list => ('a::type * 'b::type) list)
- l1 l2))))))"
- by (import list MAP2_ZIP)
-
-lemma MEM_EL: "(All::('a::type list => bool) => bool)
- (%l::'a::type list.
- (All::('a::type => bool) => bool)
- (%x::'a::type.
- (op =::bool => bool => bool)
- ((op mem::'a::type => 'a::type list => bool) x l)
- ((Ex::(nat => bool) => bool)
- (%n::nat.
- (op &::bool => bool => bool)
- ((op <::nat => nat => bool) n
- ((size::'a::type list => nat) l))
- ((op =::'a::type => 'a::type => bool) x
- ((EL::nat => 'a::type list => 'a::type) n l))))))"
- by (import list MEM_EL)
-
-lemma LAST_CONS: "(ALL x::'a::type. last [x] = x) &
-(ALL (x::'a::type) (xa::'a::type) xb::'a::type list.
- last (x # xa # xb) = last (xa # xb))"
- by (import list LAST_CONS)
-
-lemma FRONT_CONS: "(ALL x::'a::type. butlast [x] = []) &
-(ALL (x::'a::type) (xa::'a::type) xb::'a::type list.
+ sorry
+
+lemma EL_compute: "EL n l = (if n = 0 then hd l else EL (PRE n) (tl l))"
+ sorry
+
+lemma WF_LIST_PRED: "WF (%L1 L2. EX h. L2 = h # L1)"
+ sorry
+
+lemma list_size_cong: "M = N & (ALL x. List.member N x --> f x = f' x)
+==> HOL4Compat.list_size f M = HOL4Compat.list_size f' N"
+ sorry
+
+lemma FOLDR_CONG: "l = l' & b = b' & (ALL x a. List.member l' x --> f x a = f' x a)
+==> foldr f l b = foldr f' l' b'"
+ sorry
+
+lemma FOLDL_CONG: "l = l' & b = b' & (ALL x a. List.member l' x --> f a x = f' a x)
+==> foldl f b l = foldl f' b' l'"
+ sorry
+
+lemma MAP_CONG: "l1 = l2 & (ALL x. List.member l2 x --> f x = f' x) ==> map f l1 = map f' l2"
+ sorry
+
+lemma EXISTS_CONG: "l1 = l2 & (ALL x. List.member l2 x --> P x = P' x)
+==> list_ex P l1 = list_ex P' l2"
+ sorry
+
+lemma EVERY_CONG: "l1 = l2 & (ALL x. List.member l2 x --> P x = P' x)
+==> list_all P l1 = list_all P' l2"
+ sorry
+
+lemma EVERY_MONOTONIC: "[| !!x. P x ==> Q x; list_all P l |] ==> list_all Q l"
+ sorry
+
+lemma LENGTH_ZIP: "length l1 = length l2
+==> length (zip l1 l2) = length l1 & length (zip l1 l2) = length l2"
+ sorry
+
+lemma LENGTH_UNZIP: "length (fst (unzip pl)) = length pl & length (snd (unzip pl)) = length pl"
+ sorry
+
+lemma ZIP_UNZIP: "ZIP (unzip l) = l"
+ sorry
+
+lemma UNZIP_ZIP: "length l1 = length l2 ==> unzip (zip l1 l2) = (l1, l2)"
+ sorry
+
+lemma ZIP_MAP: "length l1 = length l2
+==> zip (map f1 l1) l2 = map (%p. (f1 (fst p), snd p)) (zip l1 l2) &
+ zip l1 (map f2 l2) = map (%p. (fst p, f2 (snd p))) (zip l1 l2)"
+ sorry
+
+lemma MEM_ZIP: "length l1 = length l2
+==> List.member (zip l1 l2) p = (EX n<length l1. p = (EL n l1, EL n l2))"
+ sorry
+
+lemma EL_ZIP: "length l1 = length l2 & n < length l1
+==> EL n (zip l1 l2) = (EL n l1, EL n l2)"
+ sorry
+
+lemma MAP2_ZIP: "length l1 = length l2 ==> map2 f l1 l2 = map (%(x, y). f x y) (zip l1 l2)"
+ sorry
+
+lemma MEM_EL: "List.member l x = (EX n<length l. x = EL n l)"
+ sorry
+
+lemma LAST_CONS: "(ALL x::'a. last [x] = x) &
+(ALL (x::'a) (xa::'a) xb::'a list. last (x # xa # xb) = last (xa # xb))"
+ sorry
+
+lemma FRONT_CONS: "(ALL x::'a. butlast [x] = []) &
+(ALL (x::'a) (xa::'a) xb::'a list.
butlast (x # xa # xb) = x # butlast (xa # xb))"
- by (import list FRONT_CONS)
+ sorry
;end_setup
;setup_theory pred_set
-lemma EXTENSION: "ALL (s::'a::type => bool) t::'a::type => bool.
- (s = t) = (ALL x::'a::type. IN x s = IN x t)"
- by (import pred_set EXTENSION)
-
-lemma NOT_EQUAL_SETS: "ALL (x::'a::type => bool) xa::'a::type => bool.
- (x ~= xa) = (EX xb::'a::type. IN xb xa = (~ IN xb x))"
- by (import pred_set NOT_EQUAL_SETS)
-
-lemma NUM_SET_WOP: "ALL s::nat => bool.
- (EX n::nat. IN n s) =
- (EX n::nat. IN n s & (ALL m::nat. IN m s --> n <= m))"
- by (import pred_set NUM_SET_WOP)
+lemma EXTENSION: "(s = t) = (ALL x. IN x s = IN x t)"
+ sorry
+
+lemma NOT_EQUAL_SETS: "(x ~= xa) = (EX xb. IN xb xa = (~ IN xb x))"
+ sorry
+
+lemma NUM_SET_WOP: "(EX n::nat. IN n (s::nat => bool)) =
+(EX n::nat. IN n s & (ALL m::nat. IN m s --> n <= m))"
+ sorry
consts
GSPEC :: "('b => 'a * bool) => 'a => bool"
-specification (GSPEC) GSPECIFICATION: "ALL (f::'b::type => 'a::type * bool) v::'a::type.
- IN v (GSPEC f) = (EX x::'b::type. (v, True) = f x)"
- by (import pred_set GSPECIFICATION)
-
-lemma SET_MINIMUM: "ALL (s::'a::type => bool) M::'a::type => nat.
- (EX x::'a::type. IN x s) =
- (EX x::'a::type. IN x s & (ALL y::'a::type. IN y s --> M x <= M y))"
- by (import pred_set SET_MINIMUM)
-
-definition EMPTY :: "'a => bool" where
- "EMPTY == %x::'a::type. False"
-
-lemma EMPTY_DEF: "EMPTY = (%x::'a::type. False)"
- by (import pred_set EMPTY_DEF)
-
-lemma NOT_IN_EMPTY: "ALL x::'a::type. ~ IN x EMPTY"
- by (import pred_set NOT_IN_EMPTY)
-
-lemma MEMBER_NOT_EMPTY: "ALL x::'a::type => bool. (EX xa::'a::type. IN xa x) = (x ~= EMPTY)"
- by (import pred_set MEMBER_NOT_EMPTY)
-
-consts
- UNIV :: "'a => bool"
-
-defs
- UNIV_def: "pred_set.UNIV == %x::'a::type. True"
-
-lemma UNIV_DEF: "pred_set.UNIV = (%x::'a::type. True)"
- by (import pred_set UNIV_DEF)
-
-lemma IN_UNIV: "ALL x::'a::type. IN x pred_set.UNIV"
- by (import pred_set IN_UNIV)
+specification (GSPEC) GSPECIFICATION: "ALL (f::'b => 'a * bool) v::'a. IN v (GSPEC f) = (EX x::'b. (v, True) = f x)"
+ sorry
+
+lemma SET_MINIMUM: "(EX x::'a. IN x (s::'a => bool)) =
+(EX x::'a. IN x s & (ALL y::'a. IN y s --> (M::'a => nat) x <= M y))"
+ sorry
+
+definition
+ EMPTY :: "'a => bool" where
+ "EMPTY == %x. False"
+
+lemma EMPTY_DEF: "EMPTY = (%x. False)"
+ sorry
+
+lemma NOT_IN_EMPTY: "~ IN x EMPTY"
+ sorry
+
+lemma MEMBER_NOT_EMPTY: "(EX xa. IN xa x) = (x ~= EMPTY)"
+ sorry
+
+definition
+ UNIV :: "'a => bool" where
+ "UNIV == %x. True"
+
+lemma UNIV_DEF: "pred_set.UNIV = (%x. True)"
+ sorry
+
+lemma IN_UNIV: "IN x pred_set.UNIV"
+ sorry
lemma UNIV_NOT_EMPTY: "pred_set.UNIV ~= EMPTY"
- by (import pred_set UNIV_NOT_EMPTY)
+ sorry
lemma EMPTY_NOT_UNIV: "EMPTY ~= pred_set.UNIV"
- by (import pred_set EMPTY_NOT_UNIV)
-
-lemma EQ_UNIV: "(ALL x::'a::type. IN x (s::'a::type => bool)) = (s = pred_set.UNIV)"
- by (import pred_set EQ_UNIV)
-
-definition SUBSET :: "('a => bool) => ('a => bool) => bool" where
- "SUBSET ==
-%(s::'a::type => bool) t::'a::type => bool.
- ALL x::'a::type. IN x s --> IN x t"
-
-lemma SUBSET_DEF: "ALL (s::'a::type => bool) t::'a::type => bool.
- SUBSET s t = (ALL x::'a::type. IN x s --> IN x t)"
- by (import pred_set SUBSET_DEF)
-
-lemma SUBSET_TRANS: "ALL (x::'a::type => bool) (xa::'a::type => bool) xb::'a::type => bool.
- SUBSET x xa & SUBSET xa xb --> SUBSET x xb"
- by (import pred_set SUBSET_TRANS)
-
-lemma SUBSET_REFL: "ALL x::'a::type => bool. SUBSET x x"
- by (import pred_set SUBSET_REFL)
-
-lemma SUBSET_ANTISYM: "ALL (x::'a::type => bool) xa::'a::type => bool.
- SUBSET x xa & SUBSET xa x --> x = xa"
- by (import pred_set SUBSET_ANTISYM)
-
-lemma EMPTY_SUBSET: "All (SUBSET EMPTY)"
- by (import pred_set EMPTY_SUBSET)
-
-lemma SUBSET_EMPTY: "ALL x::'a::type => bool. SUBSET x EMPTY = (x = EMPTY)"
- by (import pred_set SUBSET_EMPTY)
-
-lemma SUBSET_UNIV: "ALL x::'a::type => bool. SUBSET x pred_set.UNIV"
- by (import pred_set SUBSET_UNIV)
-
-lemma UNIV_SUBSET: "ALL x::'a::type => bool. SUBSET pred_set.UNIV x = (x = pred_set.UNIV)"
- by (import pred_set UNIV_SUBSET)
-
-definition PSUBSET :: "('a => bool) => ('a => bool) => bool" where
- "PSUBSET == %(s::'a::type => bool) t::'a::type => bool. SUBSET s t & s ~= t"
-
-lemma PSUBSET_DEF: "ALL (s::'a::type => bool) t::'a::type => bool.
- PSUBSET s t = (SUBSET s t & s ~= t)"
- by (import pred_set PSUBSET_DEF)
-
-lemma PSUBSET_TRANS: "ALL (x::'a::type => bool) (xa::'a::type => bool) xb::'a::type => bool.
- PSUBSET x xa & PSUBSET xa xb --> PSUBSET x xb"
- by (import pred_set PSUBSET_TRANS)
-
-lemma PSUBSET_IRREFL: "ALL x::'a::type => bool. ~ PSUBSET x x"
- by (import pred_set PSUBSET_IRREFL)
-
-lemma NOT_PSUBSET_EMPTY: "ALL x::'a::type => bool. ~ PSUBSET x EMPTY"
- by (import pred_set NOT_PSUBSET_EMPTY)
-
-lemma NOT_UNIV_PSUBSET: "ALL x::'a::type => bool. ~ PSUBSET pred_set.UNIV x"
- by (import pred_set NOT_UNIV_PSUBSET)
-
-lemma PSUBSET_UNIV: "ALL x::'a::type => bool.
- PSUBSET x pred_set.UNIV = (EX xa::'a::type. ~ IN xa x)"
- by (import pred_set PSUBSET_UNIV)
-
-consts
- UNION :: "('a => bool) => ('a => bool) => 'a => bool"
-
-defs
- UNION_def: "pred_set.UNION ==
-%(s::'a::type => bool) t::'a::type => bool.
- GSPEC (%x::'a::type. (x, IN x s | IN x t))"
-
-lemma UNION_DEF: "ALL (s::'a::type => bool) t::'a::type => bool.
- pred_set.UNION s t = GSPEC (%x::'a::type. (x, IN x s | IN x t))"
- by (import pred_set UNION_DEF)
-
-lemma IN_UNION: "ALL (x::'a::type => bool) (xa::'a::type => bool) xb::'a::type.
- IN xb (pred_set.UNION x xa) = (IN xb x | IN xb xa)"
- by (import pred_set IN_UNION)
-
-lemma UNION_ASSOC: "ALL (x::'a::type => bool) (xa::'a::type => bool) xb::'a::type => bool.
- pred_set.UNION x (pred_set.UNION xa xb) =
- pred_set.UNION (pred_set.UNION x xa) xb"
- by (import pred_set UNION_ASSOC)
-
-lemma UNION_IDEMPOT: "ALL x::'a::type => bool. pred_set.UNION x x = x"
- by (import pred_set UNION_IDEMPOT)
-
-lemma UNION_COMM: "ALL (x::'a::type => bool) xa::'a::type => bool.
- pred_set.UNION x xa = pred_set.UNION xa x"
- by (import pred_set UNION_COMM)
-
-lemma SUBSET_UNION: "(ALL (x::'a::type => bool) xa::'a::type => bool.
- SUBSET x (pred_set.UNION x xa)) &
-(ALL (x::'a::type => bool) xa::'a::type => bool.
- SUBSET x (pred_set.UNION xa x))"
- by (import pred_set SUBSET_UNION)
-
-lemma UNION_SUBSET: "ALL (s::'a::type => bool) (t::'a::type => bool) u::'a::type => bool.
- SUBSET (pred_set.UNION s t) u = (SUBSET s u & SUBSET t u)"
- by (import pred_set UNION_SUBSET)
-
-lemma SUBSET_UNION_ABSORPTION: "ALL (x::'a::type => bool) xa::'a::type => bool.
- SUBSET x xa = (pred_set.UNION x xa = xa)"
- by (import pred_set SUBSET_UNION_ABSORPTION)
-
-lemma UNION_EMPTY: "(ALL x::'a::type => bool. pred_set.UNION EMPTY x = x) &
-(ALL x::'a::type => bool. pred_set.UNION x EMPTY = x)"
- by (import pred_set UNION_EMPTY)
-
-lemma UNION_UNIV: "(ALL x::'a::type => bool. pred_set.UNION pred_set.UNIV x = pred_set.UNIV) &
-(ALL x::'a::type => bool. pred_set.UNION x pred_set.UNIV = pred_set.UNIV)"
- by (import pred_set UNION_UNIV)
-
-lemma EMPTY_UNION: "ALL (x::'a::type => bool) xa::'a::type => bool.
- (pred_set.UNION x xa = EMPTY) = (x = EMPTY & xa = EMPTY)"
- by (import pred_set EMPTY_UNION)
-
-consts
- INTER :: "('a => bool) => ('a => bool) => 'a => bool"
-
-defs
- INTER_def: "pred_set.INTER ==
-%(s::'a::type => bool) t::'a::type => bool.
- GSPEC (%x::'a::type. (x, IN x s & IN x t))"
-
-lemma INTER_DEF: "ALL (s::'a::type => bool) t::'a::type => bool.
- pred_set.INTER s t = GSPEC (%x::'a::type. (x, IN x s & IN x t))"
- by (import pred_set INTER_DEF)
-
-lemma IN_INTER: "ALL (x::'a::type => bool) (xa::'a::type => bool) xb::'a::type.
- IN xb (pred_set.INTER x xa) = (IN xb x & IN xb xa)"
- by (import pred_set IN_INTER)
-
-lemma INTER_ASSOC: "ALL (x::'a::type => bool) (xa::'a::type => bool) xb::'a::type => bool.
- pred_set.INTER x (pred_set.INTER xa xb) =
- pred_set.INTER (pred_set.INTER x xa) xb"
- by (import pred_set INTER_ASSOC)
-
-lemma INTER_IDEMPOT: "ALL x::'a::type => bool. pred_set.INTER x x = x"
- by (import pred_set INTER_IDEMPOT)
-
-lemma INTER_COMM: "ALL (x::'a::type => bool) xa::'a::type => bool.
- pred_set.INTER x xa = pred_set.INTER xa x"
- by (import pred_set INTER_COMM)
-
-lemma INTER_SUBSET: "(ALL (x::'a::type => bool) xa::'a::type => bool.
- SUBSET (pred_set.INTER x xa) x) &
-(ALL (x::'a::type => bool) xa::'a::type => bool.
- SUBSET (pred_set.INTER xa x) x)"
- by (import pred_set INTER_SUBSET)
-
-lemma SUBSET_INTER: "ALL (s::'a::type => bool) (t::'a::type => bool) u::'a::type => bool.
- SUBSET s (pred_set.INTER t u) = (SUBSET s t & SUBSET s u)"
- by (import pred_set SUBSET_INTER)
-
-lemma SUBSET_INTER_ABSORPTION: "ALL (x::'a::type => bool) xa::'a::type => bool.
- SUBSET x xa = (pred_set.INTER x xa = x)"
- by (import pred_set SUBSET_INTER_ABSORPTION)
-
-lemma INTER_EMPTY: "(ALL x::'a::type => bool. pred_set.INTER EMPTY x = EMPTY) &
-(ALL x::'a::type => bool. pred_set.INTER x EMPTY = EMPTY)"
- by (import pred_set INTER_EMPTY)
-
-lemma INTER_UNIV: "(ALL x::'a::type => bool. pred_set.INTER pred_set.UNIV x = x) &
-(ALL x::'a::type => bool. pred_set.INTER x pred_set.UNIV = x)"
- by (import pred_set INTER_UNIV)
-
-lemma UNION_OVER_INTER: "ALL (x::'a::type => bool) (xa::'a::type => bool) xb::'a::type => bool.
- pred_set.INTER x (pred_set.UNION xa xb) =
- pred_set.UNION (pred_set.INTER x xa) (pred_set.INTER x xb)"
- by (import pred_set UNION_OVER_INTER)
-
-lemma INTER_OVER_UNION: "ALL (x::'a::type => bool) (xa::'a::type => bool) xb::'a::type => bool.
- pred_set.UNION x (pred_set.INTER xa xb) =
- pred_set.INTER (pred_set.UNION x xa) (pred_set.UNION x xb)"
- by (import pred_set INTER_OVER_UNION)
-
-definition DISJOINT :: "('a => bool) => ('a => bool) => bool" where
- "DISJOINT ==
-%(s::'a::type => bool) t::'a::type => bool. pred_set.INTER s t = EMPTY"
-
-lemma DISJOINT_DEF: "ALL (s::'a::type => bool) t::'a::type => bool.
- DISJOINT s t = (pred_set.INTER s t = EMPTY)"
- by (import pred_set DISJOINT_DEF)
-
-lemma IN_DISJOINT: "ALL (x::'a::type => bool) xa::'a::type => bool.
- DISJOINT x xa = (~ (EX xb::'a::type. IN xb x & IN xb xa))"
- by (import pred_set IN_DISJOINT)
-
-lemma DISJOINT_SYM: "ALL (x::'a::type => bool) xa::'a::type => bool.
- DISJOINT x xa = DISJOINT xa x"
- by (import pred_set DISJOINT_SYM)
-
-lemma DISJOINT_EMPTY: "ALL x::'a::type => bool. DISJOINT EMPTY x & DISJOINT x EMPTY"
- by (import pred_set DISJOINT_EMPTY)
-
-lemma DISJOINT_EMPTY_REFL: "ALL x::'a::type => bool. (x = EMPTY) = DISJOINT x x"
- by (import pred_set DISJOINT_EMPTY_REFL)
-
-lemma DISJOINT_UNION: "ALL (x::'a::type => bool) (xa::'a::type => bool) xb::'a::type => bool.
- DISJOINT (pred_set.UNION x xa) xb = (DISJOINT x xb & DISJOINT xa xb)"
- by (import pred_set DISJOINT_UNION)
-
-lemma DISJOINT_UNION_BOTH: "ALL (s::'a::type => bool) (t::'a::type => bool) u::'a::type => bool.
- DISJOINT (pred_set.UNION s t) u = (DISJOINT s u & DISJOINT t u) &
- DISJOINT u (pred_set.UNION s t) = (DISJOINT s u & DISJOINT t u)"
- by (import pred_set DISJOINT_UNION_BOTH)
-
-definition DIFF :: "('a => bool) => ('a => bool) => 'a => bool" where
- "DIFF ==
-%(s::'a::type => bool) t::'a::type => bool.
- GSPEC (%x::'a::type. (x, IN x s & ~ IN x t))"
-
-lemma DIFF_DEF: "ALL (s::'a::type => bool) t::'a::type => bool.
- DIFF s t = GSPEC (%x::'a::type. (x, IN x s & ~ IN x t))"
- by (import pred_set DIFF_DEF)
-
-lemma IN_DIFF: "ALL (s::'a::type => bool) (t::'a::type => bool) x::'a::type.
- IN x (DIFF s t) = (IN x s & ~ IN x t)"
- by (import pred_set IN_DIFF)
-
-lemma DIFF_EMPTY: "ALL s::'a::type => bool. DIFF s EMPTY = s"
- by (import pred_set DIFF_EMPTY)
-
-lemma EMPTY_DIFF: "ALL s::'a::type => bool. DIFF EMPTY s = EMPTY"
- by (import pred_set EMPTY_DIFF)
-
-lemma DIFF_UNIV: "ALL s::'a::type => bool. DIFF s pred_set.UNIV = EMPTY"
- by (import pred_set DIFF_UNIV)
-
-lemma DIFF_DIFF: "ALL (x::'a::type => bool) xa::'a::type => bool.
- DIFF (DIFF x xa) xa = DIFF x xa"
- by (import pred_set DIFF_DIFF)
-
-lemma DIFF_EQ_EMPTY: "ALL x::'a::type => bool. DIFF x x = EMPTY"
- by (import pred_set DIFF_EQ_EMPTY)
-
-definition INSERT :: "'a => ('a => bool) => 'a => bool" where
- "INSERT ==
-%(x::'a::type) s::'a::type => bool.
- GSPEC (%y::'a::type. (y, y = x | IN y s))"
-
-lemma INSERT_DEF: "ALL (x::'a::type) s::'a::type => bool.
- INSERT x s = GSPEC (%y::'a::type. (y, y = x | IN y s))"
- by (import pred_set INSERT_DEF)
-
-lemma IN_INSERT: "ALL (x::'a::type) (xa::'a::type) xb::'a::type => bool.
- IN x (INSERT xa xb) = (x = xa | IN x xb)"
- by (import pred_set IN_INSERT)
-
-lemma COMPONENT: "ALL (x::'a::type) xa::'a::type => bool. IN x (INSERT x xa)"
- by (import pred_set COMPONENT)
-
-lemma SET_CASES: "ALL x::'a::type => bool.
- x = EMPTY |
- (EX (xa::'a::type) xb::'a::type => bool. x = INSERT xa xb & ~ IN xa xb)"
- by (import pred_set SET_CASES)
-
-lemma DECOMPOSITION: "ALL (s::'a::type => bool) x::'a::type.
- IN x s = (EX t::'a::type => bool. s = INSERT x t & ~ IN x t)"
- by (import pred_set DECOMPOSITION)
-
-lemma ABSORPTION: "ALL (x::'a::type) xa::'a::type => bool. IN x xa = (INSERT x xa = xa)"
- by (import pred_set ABSORPTION)
-
-lemma INSERT_INSERT: "ALL (x::'a::type) xa::'a::type => bool. INSERT x (INSERT x xa) = INSERT x xa"
- by (import pred_set INSERT_INSERT)
-
-lemma INSERT_COMM: "ALL (x::'a::type) (xa::'a::type) xb::'a::type => bool.
- INSERT x (INSERT xa xb) = INSERT xa (INSERT x xb)"
- by (import pred_set INSERT_COMM)
-
-lemma INSERT_UNIV: "ALL x::'a::type. INSERT x pred_set.UNIV = pred_set.UNIV"
- by (import pred_set INSERT_UNIV)
-
-lemma NOT_INSERT_EMPTY: "ALL (x::'a::type) xa::'a::type => bool. INSERT x xa ~= EMPTY"
- by (import pred_set NOT_INSERT_EMPTY)
-
-lemma NOT_EMPTY_INSERT: "ALL (x::'a::type) xa::'a::type => bool. EMPTY ~= INSERT x xa"
- by (import pred_set NOT_EMPTY_INSERT)
-
-lemma INSERT_UNION: "ALL (x::'a::type) (s::'a::type => bool) t::'a::type => bool.
- pred_set.UNION (INSERT x s) t =
- (if IN x t then pred_set.UNION s t else INSERT x (pred_set.UNION s t))"
- by (import pred_set INSERT_UNION)
-
-lemma INSERT_UNION_EQ: "ALL (x::'a::type) (s::'a::type => bool) t::'a::type => bool.
- pred_set.UNION (INSERT x s) t = INSERT x (pred_set.UNION s t)"
- by (import pred_set INSERT_UNION_EQ)
-
-lemma INSERT_INTER: "ALL (x::'a::type) (s::'a::type => bool) t::'a::type => bool.
- pred_set.INTER (INSERT x s) t =
- (if IN x t then INSERT x (pred_set.INTER s t) else pred_set.INTER s t)"
- by (import pred_set INSERT_INTER)
-
-lemma DISJOINT_INSERT: "ALL (x::'a::type) (xa::'a::type => bool) xb::'a::type => bool.
- DISJOINT (INSERT x xa) xb = (DISJOINT xa xb & ~ IN x xb)"
- by (import pred_set DISJOINT_INSERT)
-
-lemma INSERT_SUBSET: "ALL (x::'a::type) (xa::'a::type => bool) xb::'a::type => bool.
- SUBSET (INSERT x xa) xb = (IN x xb & SUBSET xa xb)"
- by (import pred_set INSERT_SUBSET)
-
-lemma SUBSET_INSERT: "ALL (x::'a::type) xa::'a::type => bool.
- ~ IN x xa -->
- (ALL xb::'a::type => bool. SUBSET xa (INSERT x xb) = SUBSET xa xb)"
- by (import pred_set SUBSET_INSERT)
-
-lemma INSERT_DIFF: "ALL (s::'a::type => bool) (t::'a::type => bool) x::'a::type.
- DIFF (INSERT x s) t = (if IN x t then DIFF s t else INSERT x (DIFF s t))"
- by (import pred_set INSERT_DIFF)
-
-definition DELETE :: "('a => bool) => 'a => 'a => bool" where
- "DELETE == %(s::'a::type => bool) x::'a::type. DIFF s (INSERT x EMPTY)"
-
-lemma DELETE_DEF: "ALL (s::'a::type => bool) x::'a::type. DELETE s x = DIFF s (INSERT x EMPTY)"
- by (import pred_set DELETE_DEF)
-
-lemma IN_DELETE: "ALL (x::'a::type => bool) (xa::'a::type) xb::'a::type.
- IN xa (DELETE x xb) = (IN xa x & xa ~= xb)"
- by (import pred_set IN_DELETE)
-
-lemma DELETE_NON_ELEMENT: "ALL (x::'a::type) xa::'a::type => bool. (~ IN x xa) = (DELETE xa x = xa)"
- by (import pred_set DELETE_NON_ELEMENT)
-
-lemma IN_DELETE_EQ: "ALL (s::'a::type => bool) (x::'a::type) x'::'a::type.
- (IN x s = IN x' s) = (IN x (DELETE s x') = IN x' (DELETE s x))"
- by (import pred_set IN_DELETE_EQ)
-
-lemma EMPTY_DELETE: "ALL x::'a::type. DELETE EMPTY x = EMPTY"
- by (import pred_set EMPTY_DELETE)
-
-lemma DELETE_DELETE: "ALL (x::'a::type) xa::'a::type => bool. DELETE (DELETE xa x) x = DELETE xa x"
- by (import pred_set DELETE_DELETE)
-
-lemma DELETE_COMM: "ALL (x::'a::type) (xa::'a::type) xb::'a::type => bool.
- DELETE (DELETE xb x) xa = DELETE (DELETE xb xa) x"
- by (import pred_set DELETE_COMM)
-
-lemma DELETE_SUBSET: "ALL (x::'a::type) xa::'a::type => bool. SUBSET (DELETE xa x) xa"
- by (import pred_set DELETE_SUBSET)
-
-lemma SUBSET_DELETE: "ALL (x::'a::type) (xa::'a::type => bool) xb::'a::type => bool.
- SUBSET xa (DELETE xb x) = (~ IN x xa & SUBSET xa xb)"
- by (import pred_set SUBSET_DELETE)
-
-lemma SUBSET_INSERT_DELETE: "ALL (x::'a::type) (s::'a::type => bool) t::'a::type => bool.
- SUBSET s (INSERT x t) = SUBSET (DELETE s x) t"
- by (import pred_set SUBSET_INSERT_DELETE)
-
-lemma DIFF_INSERT: "ALL (x::'a::type => bool) (xa::'a::type => bool) xb::'a::type.
- DIFF x (INSERT xb xa) = DIFF (DELETE x xb) xa"
- by (import pred_set DIFF_INSERT)
-
-lemma PSUBSET_INSERT_SUBSET: "ALL (x::'a::type => bool) xa::'a::type => bool.
- PSUBSET x xa = (EX xb::'a::type. ~ IN xb x & SUBSET (INSERT xb x) xa)"
- by (import pred_set PSUBSET_INSERT_SUBSET)
-
-lemma PSUBSET_MEMBER: "ALL (s::'a::type => bool) t::'a::type => bool.
- PSUBSET s t = (SUBSET s t & (EX y::'a::type. IN y t & ~ IN y s))"
- by (import pred_set PSUBSET_MEMBER)
-
-lemma DELETE_INSERT: "ALL (x::'a::type) (xa::'a::type) xb::'a::type => bool.
- DELETE (INSERT x xb) xa =
- (if x = xa then DELETE xb xa else INSERT x (DELETE xb xa))"
- by (import pred_set DELETE_INSERT)
-
-lemma INSERT_DELETE: "ALL (x::'a::type) xa::'a::type => bool.
- IN x xa --> INSERT x (DELETE xa x) = xa"
- by (import pred_set INSERT_DELETE)
-
-lemma DELETE_INTER: "ALL (x::'a::type => bool) (xa::'a::type => bool) xb::'a::type.
- pred_set.INTER (DELETE x xb) xa = DELETE (pred_set.INTER x xa) xb"
- by (import pred_set DELETE_INTER)
-
-lemma DISJOINT_DELETE_SYM: "ALL (x::'a::type => bool) (xa::'a::type => bool) xb::'a::type.
- DISJOINT (DELETE x xb) xa = DISJOINT (DELETE xa xb) x"
- by (import pred_set DISJOINT_DELETE_SYM)
+ sorry
+
+lemma EQ_UNIV: "(ALL x. IN x s) = (s = pred_set.UNIV)"
+ sorry
+
+definition
+ SUBSET :: "('a => bool) => ('a => bool) => bool" where
+ "SUBSET == %s t. ALL x. IN x s --> IN x t"
+
+lemma SUBSET_DEF: "SUBSET s t = (ALL x. IN x s --> IN x t)"
+ sorry
+
+lemma SUBSET_TRANS: "SUBSET x xa & SUBSET xa xb ==> SUBSET x xb"
+ sorry
+
+lemma SUBSET_REFL: "SUBSET x x"
+ sorry
+
+lemma SUBSET_ANTISYM: "SUBSET x xa & SUBSET xa x ==> x = xa"
+ sorry
+
+lemma EMPTY_SUBSET: "SUBSET EMPTY x"
+ sorry
+
+lemma SUBSET_EMPTY: "SUBSET x EMPTY = (x = EMPTY)"
+ sorry
+
+lemma SUBSET_UNIV: "SUBSET x pred_set.UNIV"
+ sorry
+
+lemma UNIV_SUBSET: "SUBSET pred_set.UNIV x = (x = pred_set.UNIV)"
+ sorry
+
+definition
+ PSUBSET :: "('a => bool) => ('a => bool) => bool" where
+ "PSUBSET == %s t. SUBSET s t & s ~= t"
+
+lemma PSUBSET_DEF: "PSUBSET s t = (SUBSET s t & s ~= t)"
+ sorry
+
+lemma PSUBSET_TRANS: "PSUBSET x xa & PSUBSET xa xb ==> PSUBSET x xb"
+ sorry
+
+lemma PSUBSET_IRREFL: "~ PSUBSET x x"
+ sorry
+
+lemma NOT_PSUBSET_EMPTY: "~ PSUBSET x EMPTY"
+ sorry
+
+lemma NOT_UNIV_PSUBSET: "~ PSUBSET pred_set.UNIV x"
+ sorry
+
+lemma PSUBSET_UNIV: "PSUBSET x pred_set.UNIV = (EX xa. ~ IN xa x)"
+ sorry
+
+definition
+ UNION :: "('a => bool) => ('a => bool) => 'a => bool" where
+ "UNION == %s t. GSPEC (%x. (x, IN x s | IN x t))"
+
+lemma UNION_DEF: "pred_set.UNION s t = GSPEC (%x. (x, IN x s | IN x t))"
+ sorry
+
+lemma IN_UNION: "IN xb (pred_set.UNION x xa) = (IN xb x | IN xb xa)"
+ sorry
+
+lemma UNION_ASSOC: "pred_set.UNION x (pred_set.UNION xa xb) =
+pred_set.UNION (pred_set.UNION x xa) xb"
+ sorry
+
+lemma UNION_IDEMPOT: "pred_set.UNION x x = x"
+ sorry
+
+lemma UNION_COMM: "pred_set.UNION x xa = pred_set.UNION xa x"
+ sorry
+
+lemma SUBSET_UNION: "(ALL (x::'a => bool) xa::'a => bool. SUBSET x (pred_set.UNION x xa)) &
+(ALL (x::'a => bool) xa::'a => bool. SUBSET x (pred_set.UNION xa x))"
+ sorry
+
+lemma UNION_SUBSET: "SUBSET (pred_set.UNION s t) u = (SUBSET s u & SUBSET t u)"
+ sorry
+
+lemma SUBSET_UNION_ABSORPTION: "SUBSET x xa = (pred_set.UNION x xa = xa)"
+ sorry
+
+lemma UNION_EMPTY: "(ALL x::'a => bool. pred_set.UNION EMPTY x = x) &
+(ALL x::'a => bool. pred_set.UNION x EMPTY = x)"
+ sorry
+
+lemma UNION_UNIV: "(ALL x::'a => bool. pred_set.UNION pred_set.UNIV x = pred_set.UNIV) &
+(ALL x::'a => bool. pred_set.UNION x pred_set.UNIV = pred_set.UNIV)"
+ sorry
+
+lemma EMPTY_UNION: "(pred_set.UNION x xa = EMPTY) = (x = EMPTY & xa = EMPTY)"
+ sorry
+
+definition
+ INTER :: "('a => bool) => ('a => bool) => 'a => bool" where
+ "INTER == %s t. GSPEC (%x. (x, IN x s & IN x t))"
+
+lemma INTER_DEF: "pred_set.INTER s t = GSPEC (%x. (x, IN x s & IN x t))"
+ sorry
+
+lemma IN_INTER: "IN xb (pred_set.INTER x xa) = (IN xb x & IN xb xa)"
+ sorry
+
+lemma INTER_ASSOC: "pred_set.INTER x (pred_set.INTER xa xb) =
+pred_set.INTER (pred_set.INTER x xa) xb"
+ sorry
+
+lemma INTER_IDEMPOT: "pred_set.INTER x x = x"
+ sorry
+
+lemma INTER_COMM: "pred_set.INTER x xa = pred_set.INTER xa x"
+ sorry
+
+lemma INTER_SUBSET: "(ALL (x::'a => bool) xa::'a => bool. SUBSET (pred_set.INTER x xa) x) &
+(ALL (x::'a => bool) xa::'a => bool. SUBSET (pred_set.INTER xa x) x)"
+ sorry
+
+lemma SUBSET_INTER: "SUBSET s (pred_set.INTER t u) = (SUBSET s t & SUBSET s u)"
+ sorry
+
+lemma SUBSET_INTER_ABSORPTION: "SUBSET x xa = (pred_set.INTER x xa = x)"
+ sorry
+
+lemma INTER_EMPTY: "(ALL x::'a => bool. pred_set.INTER EMPTY x = EMPTY) &
+(ALL x::'a => bool. pred_set.INTER x EMPTY = EMPTY)"
+ sorry
+
+lemma INTER_UNIV: "(ALL x::'a => bool. pred_set.INTER pred_set.UNIV x = x) &
+(ALL x::'a => bool. pred_set.INTER x pred_set.UNIV = x)"
+ sorry
+
+lemma UNION_OVER_INTER: "pred_set.INTER x (pred_set.UNION xa xb) =
+pred_set.UNION (pred_set.INTER x xa) (pred_set.INTER x xb)"
+ sorry
+
+lemma INTER_OVER_UNION: "pred_set.UNION x (pred_set.INTER xa xb) =
+pred_set.INTER (pred_set.UNION x xa) (pred_set.UNION x xb)"
+ sorry
+
+definition
+ DISJOINT :: "('a => bool) => ('a => bool) => bool" where
+ "DISJOINT == %s t. pred_set.INTER s t = EMPTY"
+
+lemma DISJOINT_DEF: "DISJOINT s t = (pred_set.INTER s t = EMPTY)"
+ sorry
+
+lemma IN_DISJOINT: "DISJOINT x xa = (~ (EX xb. IN xb x & IN xb xa))"
+ sorry
+
+lemma DISJOINT_SYM: "DISJOINT x xa = DISJOINT xa x"
+ sorry
+
+lemma DISJOINT_EMPTY: "DISJOINT EMPTY x & DISJOINT x EMPTY"
+ sorry
+
+lemma DISJOINT_EMPTY_REFL: "(x = EMPTY) = DISJOINT x x"
+ sorry
+
+lemma DISJOINT_UNION: "DISJOINT (pred_set.UNION x xa) xb = (DISJOINT x xb & DISJOINT xa xb)"
+ sorry
+
+lemma DISJOINT_UNION_BOTH: "DISJOINT (pred_set.UNION s t) u = (DISJOINT s u & DISJOINT t u) &
+DISJOINT u (pred_set.UNION s t) = (DISJOINT s u & DISJOINT t u)"
+ sorry
+
+definition
+ DIFF :: "('a => bool) => ('a => bool) => 'a => bool" where
+ "DIFF == %s t. GSPEC (%x. (x, IN x s & ~ IN x t))"
+
+lemma DIFF_DEF: "DIFF s t = GSPEC (%x. (x, IN x s & ~ IN x t))"
+ sorry
+
+lemma IN_DIFF: "IN x (DIFF s t) = (IN x s & ~ IN x t)"
+ sorry
+
+lemma DIFF_EMPTY: "DIFF s EMPTY = s"
+ sorry
+
+lemma EMPTY_DIFF: "DIFF EMPTY s = EMPTY"
+ sorry
+
+lemma DIFF_UNIV: "DIFF s pred_set.UNIV = EMPTY"
+ sorry
+
+lemma DIFF_DIFF: "DIFF (DIFF x xa) xa = DIFF x xa"
+ sorry
+
+lemma DIFF_EQ_EMPTY: "DIFF x x = EMPTY"
+ sorry
+
+definition
+ INSERT :: "'a => ('a => bool) => 'a => bool" where
+ "INSERT == %x s. GSPEC (%y. (y, y = x | IN y s))"
+
+lemma INSERT_DEF: "INSERT x s = GSPEC (%y. (y, y = x | IN y s))"
+ sorry
+
+lemma IN_INSERT: "IN x (INSERT xa xb) = (x = xa | IN x xb)"
+ sorry
+
+lemma COMPONENT: "IN x (INSERT x xa)"
+ sorry
+
+lemma SET_CASES: "x = EMPTY | (EX xa xb. x = INSERT xa xb & ~ IN xa xb)"
+ sorry
+
+lemma DECOMPOSITION: "IN x s = (EX t. s = INSERT x t & ~ IN x t)"
+ sorry
+
+lemma ABSORPTION: "IN x xa = (INSERT x xa = xa)"
+ sorry
+
+lemma INSERT_INSERT: "INSERT x (INSERT x xa) = INSERT x xa"
+ sorry
+
+lemma INSERT_COMM: "INSERT x (INSERT xa xb) = INSERT xa (INSERT x xb)"
+ sorry
+
+lemma INSERT_UNIV: "INSERT x pred_set.UNIV = pred_set.UNIV"
+ sorry
+
+lemma NOT_INSERT_EMPTY: "INSERT x xa ~= EMPTY"
+ sorry
+
+lemma NOT_EMPTY_INSERT: "EMPTY ~= INSERT x xa"
+ sorry
+
+lemma INSERT_UNION: "pred_set.UNION (INSERT x s) t =
+(if IN x t then pred_set.UNION s t else INSERT x (pred_set.UNION s t))"
+ sorry
+
+lemma INSERT_UNION_EQ: "pred_set.UNION (INSERT x s) t = INSERT x (pred_set.UNION s t)"
+ sorry
+
+lemma INSERT_INTER: "pred_set.INTER (INSERT x s) t =
+(if IN x t then INSERT x (pred_set.INTER s t) else pred_set.INTER s t)"
+ sorry
+
+lemma DISJOINT_INSERT: "DISJOINT (INSERT x xa) xb = (DISJOINT xa xb & ~ IN x xb)"
+ sorry
+
+lemma INSERT_SUBSET: "SUBSET (INSERT x xa) xb = (IN x xb & SUBSET xa xb)"
+ sorry
+
+lemma SUBSET_INSERT: "~ IN x xa ==> SUBSET xa (INSERT x xb) = SUBSET xa xb"
+ sorry
+
+lemma INSERT_DIFF: "DIFF (INSERT x s) t = (if IN x t then DIFF s t else INSERT x (DIFF s t))"
+ sorry
+
+definition
+ DELETE :: "('a => bool) => 'a => 'a => bool" where
+ "DELETE == %s x. DIFF s (INSERT x EMPTY)"
+
+lemma DELETE_DEF: "DELETE s x = DIFF s (INSERT x EMPTY)"
+ sorry
+
+lemma IN_DELETE: "IN xa (DELETE x xb) = (IN xa x & xa ~= xb)"
+ sorry
+
+lemma DELETE_NON_ELEMENT: "(~ IN x xa) = (DELETE xa x = xa)"
+ sorry
+
+lemma IN_DELETE_EQ: "(IN x s = IN x' s) = (IN x (DELETE s x') = IN x' (DELETE s x))"
+ sorry
+
+lemma EMPTY_DELETE: "DELETE EMPTY x = EMPTY"
+ sorry
+
+lemma DELETE_DELETE: "DELETE (DELETE xa x) x = DELETE xa x"
+ sorry
+
+lemma DELETE_COMM: "DELETE (DELETE xb x) xa = DELETE (DELETE xb xa) x"
+ sorry
+
+lemma DELETE_SUBSET: "SUBSET (DELETE xa x) xa"
+ sorry
+
+lemma SUBSET_DELETE: "SUBSET xa (DELETE xb x) = (~ IN x xa & SUBSET xa xb)"
+ sorry
+
+lemma SUBSET_INSERT_DELETE: "SUBSET s (INSERT x t) = SUBSET (DELETE s x) t"
+ sorry
+
+lemma DIFF_INSERT: "DIFF x (INSERT xb xa) = DIFF (DELETE x xb) xa"
+ sorry
+
+lemma PSUBSET_INSERT_SUBSET: "PSUBSET x xa = (EX xb. ~ IN xb x & SUBSET (INSERT xb x) xa)"
+ sorry
+
+lemma PSUBSET_MEMBER: "PSUBSET s t = (SUBSET s t & (EX y. IN y t & ~ IN y s))"
+ sorry
+
+lemma DELETE_INSERT: "DELETE (INSERT x xb) xa =
+(if x = xa then DELETE xb xa else INSERT x (DELETE xb xa))"
+ sorry
+
+lemma INSERT_DELETE: "IN x xa ==> INSERT x (DELETE xa x) = xa"
+ sorry
+
+lemma DELETE_INTER: "pred_set.INTER (DELETE x xb) xa = DELETE (pred_set.INTER x xa) xb"
+ sorry
+
+lemma DISJOINT_DELETE_SYM: "DISJOINT (DELETE x xb) xa = DISJOINT (DELETE xa xb) x"
+ sorry
consts
CHOICE :: "('a => bool) => 'a"
-specification (CHOICE) CHOICE_DEF: "ALL x::'a::type => bool. x ~= EMPTY --> IN (CHOICE x) x"
- by (import pred_set CHOICE_DEF)
-
-definition REST :: "('a => bool) => 'a => bool" where
- "REST == %s::'a::type => bool. DELETE s (CHOICE s)"
-
-lemma REST_DEF: "ALL s::'a::type => bool. REST s = DELETE s (CHOICE s)"
- by (import pred_set REST_DEF)
-
-lemma CHOICE_NOT_IN_REST: "ALL x::'a::type => bool. ~ IN (CHOICE x) (REST x)"
- by (import pred_set CHOICE_NOT_IN_REST)
-
-lemma CHOICE_INSERT_REST: "ALL s::'a::type => bool. s ~= EMPTY --> INSERT (CHOICE s) (REST s) = s"
- by (import pred_set CHOICE_INSERT_REST)
-
-lemma REST_SUBSET: "ALL x::'a::type => bool. SUBSET (REST x) x"
- by (import pred_set REST_SUBSET)
-
-lemma REST_PSUBSET: "ALL x::'a::type => bool. x ~= EMPTY --> PSUBSET (REST x) x"
- by (import pred_set REST_PSUBSET)
-
-definition SING :: "('a => bool) => bool" where
- "SING == %s::'a::type => bool. EX x::'a::type. s = INSERT x EMPTY"
-
-lemma SING_DEF: "ALL s::'a::type => bool. SING s = (EX x::'a::type. s = INSERT x EMPTY)"
- by (import pred_set SING_DEF)
-
-lemma SING: "ALL x::'a::type. SING (INSERT x EMPTY)"
- by (import pred_set SING)
-
-lemma IN_SING: "ALL (x::'a::type) xa::'a::type. IN x (INSERT xa EMPTY) = (x = xa)"
- by (import pred_set IN_SING)
-
-lemma NOT_SING_EMPTY: "ALL x::'a::type. INSERT x EMPTY ~= EMPTY"
- by (import pred_set NOT_SING_EMPTY)
-
-lemma NOT_EMPTY_SING: "ALL x::'a::type. EMPTY ~= INSERT x EMPTY"
- by (import pred_set NOT_EMPTY_SING)
-
-lemma EQUAL_SING: "ALL (x::'a::type) xa::'a::type.
- (INSERT x EMPTY = INSERT xa EMPTY) = (x = xa)"
- by (import pred_set EQUAL_SING)
-
-lemma DISJOINT_SING_EMPTY: "ALL x::'a::type. DISJOINT (INSERT x EMPTY) EMPTY"
- by (import pred_set DISJOINT_SING_EMPTY)
-
-lemma INSERT_SING_UNION: "ALL (x::'a::type => bool) xa::'a::type.
- INSERT xa x = pred_set.UNION (INSERT xa EMPTY) x"
- by (import pred_set INSERT_SING_UNION)
-
-lemma SING_DELETE: "ALL x::'a::type. DELETE (INSERT x EMPTY) x = EMPTY"
- by (import pred_set SING_DELETE)
-
-lemma DELETE_EQ_SING: "ALL (x::'a::type => bool) xa::'a::type.
- IN xa x --> (DELETE x xa = EMPTY) = (x = INSERT xa EMPTY)"
- by (import pred_set DELETE_EQ_SING)
-
-lemma CHOICE_SING: "ALL x::'a::type. CHOICE (INSERT x EMPTY) = x"
- by (import pred_set CHOICE_SING)
-
-lemma REST_SING: "ALL x::'a::type. REST (INSERT x EMPTY) = EMPTY"
- by (import pred_set REST_SING)
-
-lemma SING_IFF_EMPTY_REST: "ALL x::'a::type => bool. SING x = (x ~= EMPTY & REST x = EMPTY)"
- by (import pred_set SING_IFF_EMPTY_REST)
-
-definition IMAGE :: "('a => 'b) => ('a => bool) => 'b => bool" where
- "IMAGE ==
-%(f::'a::type => 'b::type) s::'a::type => bool.
- GSPEC (%x::'a::type. (f x, IN x s))"
-
-lemma IMAGE_DEF: "ALL (f::'a::type => 'b::type) s::'a::type => bool.
- IMAGE f s = GSPEC (%x::'a::type. (f x, IN x s))"
- by (import pred_set IMAGE_DEF)
-
-lemma IN_IMAGE: "ALL (x::'b::type) (xa::'a::type => bool) xb::'a::type => 'b::type.
- IN x (IMAGE xb xa) = (EX xc::'a::type. x = xb xc & IN xc xa)"
- by (import pred_set IN_IMAGE)
-
-lemma IMAGE_IN: "ALL (x::'a::type) xa::'a::type => bool.
- IN x xa --> (ALL xb::'a::type => 'b::type. IN (xb x) (IMAGE xb xa))"
- by (import pred_set IMAGE_IN)
-
-lemma IMAGE_EMPTY: "ALL x::'a::type => 'b::type. IMAGE x EMPTY = EMPTY"
- by (import pred_set IMAGE_EMPTY)
-
-lemma IMAGE_ID: "ALL x::'a::type => bool. IMAGE (%x::'a::type. x) x = x"
- by (import pred_set IMAGE_ID)
-
-lemma IMAGE_COMPOSE: "ALL (x::'b::type => 'c::type) (xa::'a::type => 'b::type)
- xb::'a::type => bool. IMAGE (x o xa) xb = IMAGE x (IMAGE xa xb)"
- by (import pred_set IMAGE_COMPOSE)
-
-lemma IMAGE_INSERT: "ALL (x::'a::type => 'b::type) (xa::'a::type) xb::'a::type => bool.
- IMAGE x (INSERT xa xb) = INSERT (x xa) (IMAGE x xb)"
- by (import pred_set IMAGE_INSERT)
-
-lemma IMAGE_EQ_EMPTY: "ALL (s::'a::type => bool) x::'a::type => 'b::type.
- (IMAGE x s = EMPTY) = (s = EMPTY)"
- by (import pred_set IMAGE_EQ_EMPTY)
-
-lemma IMAGE_DELETE: "ALL (f::'a::type => 'b::type) (x::'a::type) s::'a::type => bool.
- ~ IN x s --> IMAGE f (DELETE s x) = IMAGE f s"
- by (import pred_set IMAGE_DELETE)
-
-lemma IMAGE_UNION: "ALL (x::'a::type => 'b::type) (xa::'a::type => bool) xb::'a::type => bool.
- IMAGE x (pred_set.UNION xa xb) = pred_set.UNION (IMAGE x xa) (IMAGE x xb)"
- by (import pred_set IMAGE_UNION)
-
-lemma IMAGE_SUBSET: "ALL (x::'a::type => bool) xa::'a::type => bool.
- SUBSET x xa -->
- (ALL xb::'a::type => 'b::type. SUBSET (IMAGE xb x) (IMAGE xb xa))"
- by (import pred_set IMAGE_SUBSET)
-
-lemma IMAGE_INTER: "ALL (f::'a::type => 'b::type) (s::'a::type => bool) t::'a::type => bool.
- SUBSET (IMAGE f (pred_set.INTER s t))
- (pred_set.INTER (IMAGE f s) (IMAGE f t))"
- by (import pred_set IMAGE_INTER)
-
-definition INJ :: "('a => 'b) => ('a => bool) => ('b => bool) => bool" where
+specification (CHOICE) CHOICE_DEF: "ALL x. x ~= EMPTY --> IN (CHOICE x) x"
+ sorry
+
+definition
+ REST :: "('a => bool) => 'a => bool" where
+ "REST == %s. DELETE s (CHOICE s)"
+
+lemma REST_DEF: "REST s = DELETE s (CHOICE s)"
+ sorry
+
+lemma CHOICE_NOT_IN_REST: "~ IN (CHOICE x) (REST x)"
+ sorry
+
+lemma CHOICE_INSERT_REST: "s ~= EMPTY ==> INSERT (CHOICE s) (REST s) = s"
+ sorry
+
+lemma REST_SUBSET: "SUBSET (REST x) x"
+ sorry
+
+lemma REST_PSUBSET: "x ~= EMPTY ==> PSUBSET (REST x) x"
+ sorry
+
+definition
+ SING :: "('a => bool) => bool" where
+ "SING == %s. EX x. s = INSERT x EMPTY"
+
+lemma SING_DEF: "SING s = (EX x. s = INSERT x EMPTY)"
+ sorry
+
+lemma SING: "SING (INSERT x EMPTY)"
+ sorry
+
+lemma IN_SING: "IN x (INSERT xa EMPTY) = (x = xa)"
+ sorry
+
+lemma NOT_SING_EMPTY: "INSERT x EMPTY ~= EMPTY"
+ sorry
+
+lemma NOT_EMPTY_SING: "EMPTY ~= INSERT x EMPTY"
+ sorry
+
+lemma EQUAL_SING: "(INSERT x EMPTY = INSERT xa EMPTY) = (x = xa)"
+ sorry
+
+lemma DISJOINT_SING_EMPTY: "DISJOINT (INSERT x EMPTY) EMPTY"
+ sorry
+
+lemma INSERT_SING_UNION: "INSERT xa x = pred_set.UNION (INSERT xa EMPTY) x"
+ sorry
+
+lemma SING_DELETE: "DELETE (INSERT x EMPTY) x = EMPTY"
+ sorry
+
+lemma DELETE_EQ_SING: "IN xa x ==> (DELETE x xa = EMPTY) = (x = INSERT xa EMPTY)"
+ sorry
+
+lemma CHOICE_SING: "CHOICE (INSERT x EMPTY) = x"
+ sorry
+
+lemma REST_SING: "REST (INSERT x EMPTY) = EMPTY"
+ sorry
+
+lemma SING_IFF_EMPTY_REST: "SING x = (x ~= EMPTY & REST x = EMPTY)"
+ sorry
+
+definition
+ IMAGE :: "('a => 'b) => ('a => bool) => 'b => bool" where
+ "IMAGE == %f s. GSPEC (%x. (f x, IN x s))"
+
+lemma IMAGE_DEF: "IMAGE (f::'a => 'b) (s::'a => bool) = GSPEC (%x::'a. (f x, IN x s))"
+ sorry
+
+lemma IN_IMAGE: "IN (x::'b) (IMAGE (xb::'a => 'b) (xa::'a => bool)) =
+(EX xc::'a. x = xb xc & IN xc xa)"
+ sorry
+
+lemma IMAGE_IN: "IN x xa ==> IN (xb x) (IMAGE xb xa)"
+ sorry
+
+lemma IMAGE_EMPTY: "IMAGE (x::'a => 'b) EMPTY = EMPTY"
+ sorry
+
+lemma IMAGE_ID: "IMAGE (%x. x) x = x"
+ sorry
+
+lemma IMAGE_COMPOSE: "IMAGE ((x::'b => 'c) o (xa::'a => 'b)) (xb::'a => bool) =
+IMAGE x (IMAGE xa xb)"
+ sorry
+
+lemma IMAGE_INSERT: "IMAGE (x::'a => 'b) (INSERT (xa::'a) (xb::'a => bool)) =
+INSERT (x xa) (IMAGE x xb)"
+ sorry
+
+lemma IMAGE_EQ_EMPTY: "(IMAGE (x::'a => 'b) (s::'a => bool) = EMPTY) = (s = EMPTY)"
+ sorry
+
+lemma IMAGE_DELETE: "~ IN x s ==> IMAGE f (DELETE s x) = IMAGE f s"
+ sorry
+
+lemma IMAGE_UNION: "IMAGE (x::'a => 'b) (pred_set.UNION (xa::'a => bool) (xb::'a => bool)) =
+pred_set.UNION (IMAGE x xa) (IMAGE x xb)"
+ sorry
+
+lemma IMAGE_SUBSET: "SUBSET x xa ==> SUBSET (IMAGE xb x) (IMAGE xb xa)"
+ sorry
+
+lemma IMAGE_INTER: "SUBSET
+ (IMAGE (f::'a => 'b) (pred_set.INTER (s::'a => bool) (t::'a => bool)))
+ (pred_set.INTER (IMAGE f s) (IMAGE f t))"
+ sorry
+
+definition
+ INJ :: "('a => 'b) => ('a => bool) => ('b => bool) => bool" where
"INJ ==
-%(f::'a::type => 'b::type) (s::'a::type => bool) t::'b::type => bool.
- (ALL x::'a::type. IN x s --> IN (f x) t) &
- (ALL (x::'a::type) y::'a::type. IN x s & IN y s --> f x = f y --> x = y)"
-
-lemma INJ_DEF: "ALL (f::'a::type => 'b::type) (s::'a::type => bool) t::'b::type => bool.
- INJ f s t =
- ((ALL x::'a::type. IN x s --> IN (f x) t) &
- (ALL (x::'a::type) y::'a::type.
- IN x s & IN y s --> f x = f y --> x = y))"
- by (import pred_set INJ_DEF)
-
-lemma INJ_ID: "ALL x::'a::type => bool. INJ (%x::'a::type. x) x x"
- by (import pred_set INJ_ID)
-
-lemma INJ_COMPOSE: "ALL (x::'a::type => 'b::type) (xa::'b::type => 'c::type)
- (xb::'a::type => bool) (xc::'b::type => bool) xd::'c::type => bool.
- INJ x xb xc & INJ xa xc xd --> INJ (xa o x) xb xd"
- by (import pred_set INJ_COMPOSE)
-
-lemma INJ_EMPTY: "ALL x::'a::type => 'b::type.
- All (INJ x EMPTY) &
- (ALL xa::'a::type => bool. INJ x xa EMPTY = (xa = EMPTY))"
- by (import pred_set INJ_EMPTY)
-
-definition SURJ :: "('a => 'b) => ('a => bool) => ('b => bool) => bool" where
+%f s t.
+ (ALL x. IN x s --> IN (f x) t) &
+ (ALL x y. IN x s & IN y s --> f x = f y --> x = y)"
+
+lemma INJ_DEF: "INJ f s t =
+((ALL x. IN x s --> IN (f x) t) &
+ (ALL x y. IN x s & IN y s --> f x = f y --> x = y))"
+ sorry
+
+lemma INJ_ID: "INJ (%x. x) x x"
+ sorry
+
+lemma INJ_COMPOSE: "INJ x xb xc & INJ xa xc xd ==> INJ (xa o x) xb xd"
+ sorry
+
+lemma INJ_EMPTY: "All (INJ (x::'a => 'b) EMPTY) &
+(ALL xa::'a => bool. INJ x xa EMPTY = (xa = EMPTY))"
+ sorry
+
+definition
+ SURJ :: "('a => 'b) => ('a => bool) => ('b => bool) => bool" where
"SURJ ==
-%(f::'a::type => 'b::type) (s::'a::type => bool) t::'b::type => bool.
- (ALL x::'a::type. IN x s --> IN (f x) t) &
- (ALL x::'b::type. IN x t --> (EX y::'a::type. IN y s & f y = x))"
-
-lemma SURJ_DEF: "ALL (f::'a::type => 'b::type) (s::'a::type => bool) t::'b::type => bool.
- SURJ f s t =
- ((ALL x::'a::type. IN x s --> IN (f x) t) &
- (ALL x::'b::type. IN x t --> (EX y::'a::type. IN y s & f y = x)))"
- by (import pred_set SURJ_DEF)
-
-lemma SURJ_ID: "ALL x::'a::type => bool. SURJ (%x::'a::type. x) x x"
- by (import pred_set SURJ_ID)
-
-lemma SURJ_COMPOSE: "ALL (x::'a::type => 'b::type) (xa::'b::type => 'c::type)
- (xb::'a::type => bool) (xc::'b::type => bool) xd::'c::type => bool.
- SURJ x xb xc & SURJ xa xc xd --> SURJ (xa o x) xb xd"
- by (import pred_set SURJ_COMPOSE)
-
-lemma SURJ_EMPTY: "ALL x::'a::type => 'b::type.
- (ALL xa::'b::type => bool. SURJ x EMPTY xa = (xa = EMPTY)) &
- (ALL xa::'a::type => bool. SURJ x xa EMPTY = (xa = EMPTY))"
- by (import pred_set SURJ_EMPTY)
-
-lemma IMAGE_SURJ: "ALL (x::'a::type => 'b::type) (xa::'a::type => bool) xb::'b::type => bool.
- SURJ x xa xb = (IMAGE x xa = xb)"
- by (import pred_set IMAGE_SURJ)
-
-definition BIJ :: "('a => 'b) => ('a => bool) => ('b => bool) => bool" where
- "BIJ ==
-%(f::'a::type => 'b::type) (s::'a::type => bool) t::'b::type => bool.
- INJ f s t & SURJ f s t"
-
-lemma BIJ_DEF: "ALL (f::'a::type => 'b::type) (s::'a::type => bool) t::'b::type => bool.
- BIJ f s t = (INJ f s t & SURJ f s t)"
- by (import pred_set BIJ_DEF)
-
-lemma BIJ_ID: "ALL x::'a::type => bool. BIJ (%x::'a::type. x) x x"
- by (import pred_set BIJ_ID)
-
-lemma BIJ_EMPTY: "ALL x::'a::type => 'b::type.
- (ALL xa::'b::type => bool. BIJ x EMPTY xa = (xa = EMPTY)) &
- (ALL xa::'a::type => bool. BIJ x xa EMPTY = (xa = EMPTY))"
- by (import pred_set BIJ_EMPTY)
-
-lemma BIJ_COMPOSE: "ALL (x::'a::type => 'b::type) (xa::'b::type => 'c::type)
- (xb::'a::type => bool) (xc::'b::type => bool) xd::'c::type => bool.
- BIJ x xb xc & BIJ xa xc xd --> BIJ (xa o x) xb xd"
- by (import pred_set BIJ_COMPOSE)
+%f s t.
+ (ALL x. IN x s --> IN (f x) t) &
+ (ALL x. IN x t --> (EX y. IN y s & f y = x))"
+
+lemma SURJ_DEF: "SURJ f s t =
+((ALL x. IN x s --> IN (f x) t) &
+ (ALL x. IN x t --> (EX y. IN y s & f y = x)))"
+ sorry
+
+lemma SURJ_ID: "SURJ (%x. x) x x"
+ sorry
+
+lemma SURJ_COMPOSE: "SURJ x xb xc & SURJ xa xc xd ==> SURJ (xa o x) xb xd"
+ sorry
+
+lemma SURJ_EMPTY: "(ALL xa::'b => bool. SURJ (x::'a => 'b) EMPTY xa = (xa = EMPTY)) &
+(ALL xa::'a => bool. SURJ x xa EMPTY = (xa = EMPTY))"
+ sorry
+
+lemma IMAGE_SURJ: "SURJ x xa xb = (IMAGE x xa = xb)"
+ sorry
+
+definition
+ BIJ :: "('a => 'b) => ('a => bool) => ('b => bool) => bool" where
+ "BIJ == %f s t. INJ f s t & SURJ f s t"
+
+lemma BIJ_DEF: "BIJ f s t = (INJ f s t & SURJ f s t)"
+ sorry
+
+lemma BIJ_ID: "BIJ (%x. x) x x"
+ sorry
+
+lemma BIJ_EMPTY: "(ALL xa::'b => bool. BIJ (x::'a => 'b) EMPTY xa = (xa = EMPTY)) &
+(ALL xa::'a => bool. BIJ x xa EMPTY = (xa = EMPTY))"
+ sorry
+
+lemma BIJ_COMPOSE: "BIJ x xb xc & BIJ xa xc xd ==> BIJ (xa o x) xb xd"
+ sorry
consts
LINV :: "('a => 'b) => ('a => bool) => 'b => 'a"
-specification (LINV) LINV_DEF: "ALL (f::'a::type => 'b::type) (s::'a::type => bool) t::'b::type => bool.
- INJ f s t --> (ALL x::'a::type. IN x s --> LINV f s (f x) = x)"
- by (import pred_set LINV_DEF)
+specification (LINV) LINV_DEF: "ALL f s t. INJ f s t --> (ALL x. IN x s --> LINV f s (f x) = x)"
+ sorry
consts
RINV :: "('a => 'b) => ('a => bool) => 'b => 'a"
-specification (RINV) RINV_DEF: "ALL (f::'a::type => 'b::type) (s::'a::type => bool) t::'b::type => bool.
- SURJ f s t --> (ALL x::'b::type. IN x t --> f (RINV f s x) = x)"
- by (import pred_set RINV_DEF)
-
-definition FINITE :: "('a => bool) => bool" where
+specification (RINV) RINV_DEF: "ALL f s t. SURJ f s t --> (ALL x. IN x t --> f (RINV f s x) = x)"
+ sorry
+
+definition
+ FINITE :: "('a => bool) => bool" where
"FINITE ==
-%s::'a::type => bool.
- ALL P::('a::type => bool) => bool.
- P EMPTY &
- (ALL s::'a::type => bool.
- P s --> (ALL e::'a::type. P (INSERT e s))) -->
- P s"
-
-lemma FINITE_DEF: "ALL s::'a::type => bool.
- FINITE s =
- (ALL P::('a::type => bool) => bool.
- P EMPTY &
- (ALL s::'a::type => bool.
- P s --> (ALL e::'a::type. P (INSERT e s))) -->
- P s)"
- by (import pred_set FINITE_DEF)
+%s. ALL P. P EMPTY & (ALL s. P s --> (ALL e. P (INSERT e s))) --> P s"
+
+lemma FINITE_DEF: "FINITE s =
+(ALL P. P EMPTY & (ALL s. P s --> (ALL e. P (INSERT e s))) --> P s)"
+ sorry
lemma FINITE_EMPTY: "FINITE EMPTY"
- by (import pred_set FINITE_EMPTY)
-
-lemma FINITE_INDUCT: "ALL P::('a::type => bool) => bool.
- P EMPTY &
- (ALL s::'a::type => bool.
- FINITE s & P s -->
- (ALL e::'a::type. ~ IN e s --> P (INSERT e s))) -->
- (ALL s::'a::type => bool. FINITE s --> P s)"
- by (import pred_set FINITE_INDUCT)
-
-lemma FINITE_INSERT: "ALL (x::'a::type) s::'a::type => bool. FINITE (INSERT x s) = FINITE s"
- by (import pred_set FINITE_INSERT)
-
-lemma FINITE_DELETE: "ALL (x::'a::type) s::'a::type => bool. FINITE (DELETE s x) = FINITE s"
- by (import pred_set FINITE_DELETE)
-
-lemma FINITE_UNION: "ALL (s::'a::type => bool) t::'a::type => bool.
- FINITE (pred_set.UNION s t) = (FINITE s & FINITE t)"
- by (import pred_set FINITE_UNION)
-
-lemma INTER_FINITE: "ALL s::'a::type => bool.
- FINITE s --> (ALL t::'a::type => bool. FINITE (pred_set.INTER s t))"
- by (import pred_set INTER_FINITE)
-
-lemma SUBSET_FINITE: "ALL s::'a::type => bool.
- FINITE s --> (ALL t::'a::type => bool. SUBSET t s --> FINITE t)"
- by (import pred_set SUBSET_FINITE)
-
-lemma PSUBSET_FINITE: "ALL x::'a::type => bool.
- FINITE x --> (ALL xa::'a::type => bool. PSUBSET xa x --> FINITE xa)"
- by (import pred_set PSUBSET_FINITE)
-
-lemma FINITE_DIFF: "ALL s::'a::type => bool.
- FINITE s --> (ALL t::'a::type => bool. FINITE (DIFF s t))"
- by (import pred_set FINITE_DIFF)
-
-lemma FINITE_SING: "ALL x::'a::type. FINITE (INSERT x EMPTY)"
- by (import pred_set FINITE_SING)
-
-lemma SING_FINITE: "ALL x::'a::type => bool. SING x --> FINITE x"
- by (import pred_set SING_FINITE)
-
-lemma IMAGE_FINITE: "ALL s::'a::type => bool.
- FINITE s --> (ALL f::'a::type => 'b::type. FINITE (IMAGE f s))"
- by (import pred_set IMAGE_FINITE)
+ sorry
+
+lemma FINITE_INDUCT: "[| P EMPTY &
+ (ALL s. FINITE s & P s --> (ALL e. ~ IN e s --> P (INSERT e s)));
+ FINITE s |]
+==> P s"
+ sorry
+
+lemma FINITE_INSERT: "FINITE (INSERT x s) = FINITE s"
+ sorry
+
+lemma FINITE_DELETE: "FINITE (DELETE s x) = FINITE s"
+ sorry
+
+lemma FINITE_UNION: "FINITE (pred_set.UNION s t) = (FINITE s & FINITE t)"
+ sorry
+
+lemma INTER_FINITE: "FINITE s ==> FINITE (pred_set.INTER s t)"
+ sorry
+
+lemma SUBSET_FINITE: "[| FINITE s; SUBSET t s |] ==> FINITE t"
+ sorry
+
+lemma PSUBSET_FINITE: "[| FINITE x; PSUBSET xa x |] ==> FINITE xa"
+ sorry
+
+lemma FINITE_DIFF: "FINITE s ==> FINITE (DIFF s t)"
+ sorry
+
+lemma FINITE_SING: "FINITE (INSERT x EMPTY)"
+ sorry
+
+lemma SING_FINITE: "SING x ==> FINITE x"
+ sorry
+
+lemma IMAGE_FINITE: "FINITE s ==> FINITE (IMAGE f s)"
+ sorry
consts
CARD :: "('a => bool) => nat"
@@ -4077,77 +3113,56 @@
((CARD::('a::type => bool) => nat) s)
((Suc::nat => nat)
((CARD::('a::type => bool) => nat) s)))))))"
- by (import pred_set CARD_DEF)
+ sorry
lemma CARD_EMPTY: "CARD EMPTY = 0"
- by (import pred_set CARD_EMPTY)
-
-lemma CARD_INSERT: "ALL s::'a::type => bool.
- FINITE s -->
- (ALL x::'a::type.
- CARD (INSERT x s) = (if IN x s then CARD s else Suc (CARD s)))"
- by (import pred_set CARD_INSERT)
-
-lemma CARD_EQ_0: "ALL s::'a::type => bool. FINITE s --> (CARD s = 0) = (s = EMPTY)"
- by (import pred_set CARD_EQ_0)
-
-lemma CARD_DELETE: "ALL s::'a::type => bool.
- FINITE s -->
- (ALL x::'a::type.
- CARD (DELETE s x) = (if IN x s then CARD s - 1 else CARD s))"
- by (import pred_set CARD_DELETE)
-
-lemma CARD_INTER_LESS_EQ: "ALL s::'a::type => bool.
- FINITE s -->
- (ALL t::'a::type => bool. CARD (pred_set.INTER s t) <= CARD s)"
- by (import pred_set CARD_INTER_LESS_EQ)
-
-lemma CARD_UNION: "ALL s::'a::type => bool.
- FINITE s -->
- (ALL t::'a::type => bool.
- FINITE t -->
- CARD (pred_set.UNION s t) + CARD (pred_set.INTER s t) =
- CARD s + CARD t)"
- by (import pred_set CARD_UNION)
-
-lemma CARD_SUBSET: "ALL s::'a::type => bool.
- FINITE s --> (ALL t::'a::type => bool. SUBSET t s --> CARD t <= CARD s)"
- by (import pred_set CARD_SUBSET)
-
-lemma CARD_PSUBSET: "ALL s::'a::type => bool.
- FINITE s --> (ALL t::'a::type => bool. PSUBSET t s --> CARD t < CARD s)"
- by (import pred_set CARD_PSUBSET)
-
-lemma CARD_SING: "ALL x::'a::type. CARD (INSERT x EMPTY) = 1"
- by (import pred_set CARD_SING)
-
-lemma SING_IFF_CARD1: "ALL x::'a::type => bool. SING x = (CARD x = 1 & FINITE x)"
- by (import pred_set SING_IFF_CARD1)
-
-lemma CARD_DIFF: "ALL t::'a::type => bool.
- FINITE t -->
- (ALL s::'a::type => bool.
- FINITE s --> CARD (DIFF s t) = CARD s - CARD (pred_set.INTER s t))"
- by (import pred_set CARD_DIFF)
-
-lemma LESS_CARD_DIFF: "ALL t::'a::type => bool.
- FINITE t -->
- (ALL s::'a::type => bool.
- FINITE s --> CARD t < CARD s --> 0 < CARD (DIFF s t))"
- by (import pred_set LESS_CARD_DIFF)
-
-lemma FINITE_COMPLETE_INDUCTION: "ALL P::('a::type => bool) => bool.
- (ALL x::'a::type => bool.
- (ALL y::'a::type => bool. PSUBSET y x --> P y) -->
- FINITE x --> P x) -->
- (ALL x::'a::type => bool. FINITE x --> P x)"
- by (import pred_set FINITE_COMPLETE_INDUCTION)
-
-definition INFINITE :: "('a => bool) => bool" where
- "INFINITE == %s::'a::type => bool. ~ FINITE s"
-
-lemma INFINITE_DEF: "ALL s::'a::type => bool. INFINITE s = (~ FINITE s)"
- by (import pred_set INFINITE_DEF)
+ sorry
+
+lemma CARD_INSERT: "FINITE s ==> CARD (INSERT x s) = (if IN x s then CARD s else Suc (CARD s))"
+ sorry
+
+lemma CARD_EQ_0: "FINITE s ==> (CARD s = 0) = (s = EMPTY)"
+ sorry
+
+lemma CARD_DELETE: "FINITE s ==> CARD (DELETE s x) = (if IN x s then CARD s - 1 else CARD s)"
+ sorry
+
+lemma CARD_INTER_LESS_EQ: "FINITE s ==> CARD (pred_set.INTER s t) <= CARD s"
+ sorry
+
+lemma CARD_UNION: "[| FINITE s; FINITE t |]
+==> CARD (pred_set.UNION s t) + CARD (pred_set.INTER s t) = CARD s + CARD t"
+ sorry
+
+lemma CARD_SUBSET: "[| FINITE s; SUBSET t s |] ==> CARD t <= CARD s"
+ sorry
+
+lemma CARD_PSUBSET: "[| FINITE s; PSUBSET t s |] ==> CARD t < CARD s"
+ sorry
+
+lemma CARD_SING: "CARD (INSERT x EMPTY) = 1"
+ sorry
+
+lemma SING_IFF_CARD1: "SING x = (CARD x = 1 & FINITE x)"
+ sorry
+
+lemma CARD_DIFF: "[| FINITE t; FINITE s |]
+==> CARD (DIFF s t) = CARD s - CARD (pred_set.INTER s t)"
+ sorry
+
+lemma LESS_CARD_DIFF: "[| FINITE t; FINITE s; CARD t < CARD s |] ==> 0 < CARD (DIFF s t)"
+ sorry
+
+lemma FINITE_COMPLETE_INDUCTION: "[| !!x. [| !!y. PSUBSET y x ==> P y; FINITE x |] ==> P x; FINITE x |]
+==> P x"
+ sorry
+
+definition
+ INFINITE :: "('a => bool) => bool" where
+ "INFINITE == %s. ~ FINITE s"
+
+lemma INFINITE_DEF: "INFINITE s = (~ FINITE s)"
+ sorry
lemma NOT_IN_FINITE: "(op =::bool => bool => bool)
((INFINITE::('a::type => bool) => bool) (pred_set.UNIV::'a::type => bool))
@@ -4159,23 +3174,19 @@
(%x::'a::type.
(Not::bool => bool)
((IN::'a::type => ('a::type => bool) => bool) x s)))))"
- by (import pred_set NOT_IN_FINITE)
-
-lemma INFINITE_INHAB: "ALL x::'a::type => bool. INFINITE x --> (EX xa::'a::type. IN xa x)"
- by (import pred_set INFINITE_INHAB)
-
-lemma IMAGE_11_INFINITE: "ALL f::'a::type => 'b::type.
- (ALL (x::'a::type) y::'a::type. f x = f y --> x = y) -->
- (ALL s::'a::type => bool. INFINITE s --> INFINITE (IMAGE f s))"
- by (import pred_set IMAGE_11_INFINITE)
-
-lemma INFINITE_SUBSET: "ALL x::'a::type => bool.
- INFINITE x --> (ALL xa::'a::type => bool. SUBSET x xa --> INFINITE xa)"
- by (import pred_set INFINITE_SUBSET)
-
-lemma IN_INFINITE_NOT_FINITE: "ALL (x::'a::type => bool) xa::'a::type => bool.
- INFINITE x & FINITE xa --> (EX xb::'a::type. IN xb x & ~ IN xb xa)"
- by (import pred_set IN_INFINITE_NOT_FINITE)
+ sorry
+
+lemma INFINITE_INHAB: "INFINITE x ==> EX xa. IN xa x"
+ sorry
+
+lemma IMAGE_11_INFINITE: "[| !!x y. f x = f y ==> x = y; INFINITE s |] ==> INFINITE (IMAGE f s)"
+ sorry
+
+lemma INFINITE_SUBSET: "[| INFINITE x; SUBSET x xa |] ==> INFINITE xa"
+ sorry
+
+lemma IN_INFINITE_NOT_FINITE: "INFINITE x & FINITE xa ==> EX xb. IN xb x & ~ IN xb xa"
+ sorry
lemma INFINITE_UNIV: "(op =::bool => bool => bool)
((INFINITE::('a::type => bool) => bool) (pred_set.UNIV::'a::type => bool))
@@ -4193,14 +3204,11 @@
(%y::'a::type.
(All::('a::type => bool) => bool)
(%x::'a::type.
- (Not::bool => bool)
- ((op =::'a::type => 'a::type => bool) (f x) y))))))"
- by (import pred_set INFINITE_UNIV)
-
-lemma FINITE_PSUBSET_INFINITE: "ALL x::'a::type => bool.
- INFINITE x =
- (ALL xa::'a::type => bool. FINITE xa --> SUBSET xa x --> PSUBSET xa x)"
- by (import pred_set FINITE_PSUBSET_INFINITE)
+ (op ~=::'a::type => 'a::type => bool) (f x) y)))))"
+ sorry
+
+lemma FINITE_PSUBSET_INFINITE: "INFINITE x = (ALL xa. FINITE xa --> SUBSET xa x --> PSUBSET xa x)"
+ sorry
lemma FINITE_PSUBSET_UNIV: "(op =::bool => bool => bool)
((INFINITE::('a::type => bool) => bool) (pred_set.UNIV::'a::type => bool))
@@ -4210,362 +3218,283 @@
((FINITE::('a::type => bool) => bool) s)
((PSUBSET::('a::type => bool) => ('a::type => bool) => bool) s
(pred_set.UNIV::'a::type => bool))))"
- by (import pred_set FINITE_PSUBSET_UNIV)
-
-lemma INFINITE_DIFF_FINITE: "ALL (s::'a::type => bool) t::'a::type => bool.
- INFINITE s & FINITE t --> DIFF s t ~= EMPTY"
- by (import pred_set INFINITE_DIFF_FINITE)
-
-lemma FINITE_ISO_NUM: "ALL s::'a::type => bool.
- FINITE s -->
- (EX f::nat => 'a::type.
- (ALL (n::nat) m::nat.
- n < CARD s & m < CARD s --> f n = f m --> n = m) &
- s = GSPEC (%n::nat. (f n, n < CARD s)))"
- by (import pred_set FINITE_ISO_NUM)
-
-lemma FINITE_WEAK_ENUMERATE: "(All::(('a::type => bool) => bool) => bool)
- (%x::'a::type => bool.
- (op =::bool => bool => bool) ((FINITE::('a::type => bool) => bool) x)
- ((Ex::((nat => 'a::type) => bool) => bool)
- (%f::nat => 'a::type.
- (Ex::(nat => bool) => bool)
- (%b::nat.
- (All::('a::type => bool) => bool)
- (%e::'a::type.
- (op =::bool => bool => bool)
- ((IN::'a::type => ('a::type => bool) => bool) e x)
- ((Ex::(nat => bool) => bool)
- (%n::nat.
- (op &::bool => bool => bool)
- ((op <::nat => nat => bool) n b)
- ((op =::'a::type => 'a::type => bool) e
- (f n)))))))))"
- by (import pred_set FINITE_WEAK_ENUMERATE)
-
-definition BIGUNION :: "(('a => bool) => bool) => 'a => bool" where
- "BIGUNION ==
-%P::('a::type => bool) => bool.
- GSPEC (%x::'a::type. (x, EX p::'a::type => bool. IN p P & IN x p))"
-
-lemma BIGUNION: "ALL P::('a::type => bool) => bool.
- BIGUNION P =
- GSPEC (%x::'a::type. (x, EX p::'a::type => bool. IN p P & IN x p))"
- by (import pred_set BIGUNION)
-
-lemma IN_BIGUNION: "ALL (x::'a::type) xa::('a::type => bool) => bool.
- IN x (BIGUNION xa) = (EX s::'a::type => bool. IN x s & IN s xa)"
- by (import pred_set IN_BIGUNION)
+ sorry
+
+lemma INFINITE_DIFF_FINITE: "INFINITE s & FINITE t ==> DIFF s t ~= EMPTY"
+ sorry
+
+lemma FINITE_ISO_NUM: "FINITE s
+==> EX f. (ALL n m. n < CARD s & m < CARD s --> f n = f m --> n = m) &
+ s = GSPEC (%n. (f n, n < CARD s))"
+ sorry
+
+lemma FINITE_WEAK_ENUMERATE: "FINITE (x::'a => bool) =
+(EX (f::nat => 'a) b::nat. ALL e::'a. IN e x = (EX n<b. e = f n))"
+ sorry
+
+definition
+ BIGUNION :: "(('a => bool) => bool) => 'a => bool" where
+ "BIGUNION == %P. GSPEC (%x. (x, EX p. IN p P & IN x p))"
+
+lemma BIGUNION: "BIGUNION P = GSPEC (%x. (x, EX p. IN p P & IN x p))"
+ sorry
+
+lemma IN_BIGUNION: "IN x (BIGUNION xa) = (EX s. IN x s & IN s xa)"
+ sorry
lemma BIGUNION_EMPTY: "BIGUNION EMPTY = EMPTY"
- by (import pred_set BIGUNION_EMPTY)
-
-lemma BIGUNION_SING: "ALL x::'a::type => bool. BIGUNION (INSERT x EMPTY) = x"
- by (import pred_set BIGUNION_SING)
-
-lemma BIGUNION_UNION: "ALL (x::('a::type => bool) => bool) xa::('a::type => bool) => bool.
- BIGUNION (pred_set.UNION x xa) =
- pred_set.UNION (BIGUNION x) (BIGUNION xa)"
- by (import pred_set BIGUNION_UNION)
-
-lemma DISJOINT_BIGUNION: "(ALL (s::('a::type => bool) => bool) t::'a::type => bool.
+ sorry
+
+lemma BIGUNION_SING: "BIGUNION (INSERT x EMPTY) = x"
+ sorry
+
+lemma BIGUNION_UNION: "BIGUNION (pred_set.UNION x xa) = pred_set.UNION (BIGUNION x) (BIGUNION xa)"
+ sorry
+
+lemma DISJOINT_BIGUNION: "(ALL (s::('a => bool) => bool) t::'a => bool.
DISJOINT (BIGUNION s) t =
- (ALL s'::'a::type => bool. IN s' s --> DISJOINT s' t)) &
-(ALL (x::('a::type => bool) => bool) xa::'a::type => bool.
+ (ALL s'::'a => bool. IN s' s --> DISJOINT s' t)) &
+(ALL (x::('a => bool) => bool) xa::'a => bool.
DISJOINT xa (BIGUNION x) =
- (ALL xb::'a::type => bool. IN xb x --> DISJOINT xa xb))"
- by (import pred_set DISJOINT_BIGUNION)
-
-lemma BIGUNION_INSERT: "ALL (x::'a::type => bool) xa::('a::type => bool) => bool.
- BIGUNION (INSERT x xa) = pred_set.UNION x (BIGUNION xa)"
- by (import pred_set BIGUNION_INSERT)
-
-lemma BIGUNION_SUBSET: "ALL (X::'a::type => bool) P::('a::type => bool) => bool.
- SUBSET (BIGUNION P) X = (ALL Y::'a::type => bool. IN Y P --> SUBSET Y X)"
- by (import pred_set BIGUNION_SUBSET)
-
-lemma FINITE_BIGUNION: "ALL x::('a::type => bool) => bool.
- FINITE x & (ALL s::'a::type => bool. IN s x --> FINITE s) -->
- FINITE (BIGUNION x)"
- by (import pred_set FINITE_BIGUNION)
-
-definition BIGINTER :: "(('a => bool) => bool) => 'a => bool" where
- "BIGINTER ==
-%B::('a::type => bool) => bool.
- GSPEC (%x::'a::type. (x, ALL P::'a::type => bool. IN P B --> IN x P))"
-
-lemma BIGINTER: "ALL B::('a::type => bool) => bool.
- BIGINTER B =
- GSPEC (%x::'a::type. (x, ALL P::'a::type => bool. IN P B --> IN x P))"
- by (import pred_set BIGINTER)
-
-lemma IN_BIGINTER: "IN (x::'a::type) (BIGINTER (B::('a::type => bool) => bool)) =
-(ALL P::'a::type => bool. IN P B --> IN x P)"
- by (import pred_set IN_BIGINTER)
-
-lemma BIGINTER_INSERT: "ALL (P::'a::type => bool) B::('a::type => bool) => bool.
- BIGINTER (INSERT P B) = pred_set.INTER P (BIGINTER B)"
- by (import pred_set BIGINTER_INSERT)
+ (ALL xb::'a => bool. IN xb x --> DISJOINT xa xb))"
+ sorry
+
+lemma BIGUNION_INSERT: "BIGUNION (INSERT x xa) = pred_set.UNION x (BIGUNION xa)"
+ sorry
+
+lemma BIGUNION_SUBSET: "SUBSET (BIGUNION P) X = (ALL Y. IN Y P --> SUBSET Y X)"
+ sorry
+
+lemma FINITE_BIGUNION: "FINITE x & (ALL s. IN s x --> FINITE s) ==> FINITE (BIGUNION x)"
+ sorry
+
+definition
+ BIGINTER :: "(('a => bool) => bool) => 'a => bool" where
+ "BIGINTER == %B. GSPEC (%x. (x, ALL P. IN P B --> IN x P))"
+
+lemma BIGINTER: "BIGINTER B = GSPEC (%x. (x, ALL P. IN P B --> IN x P))"
+ sorry
+
+lemma IN_BIGINTER: "IN x (BIGINTER B) = (ALL P. IN P B --> IN x P)"
+ sorry
+
+lemma BIGINTER_INSERT: "BIGINTER (INSERT P B) = pred_set.INTER P (BIGINTER B)"
+ sorry
lemma BIGINTER_EMPTY: "BIGINTER EMPTY = pred_set.UNIV"
- by (import pred_set BIGINTER_EMPTY)
-
-lemma BIGINTER_INTER: "ALL (x::'a::type => bool) xa::'a::type => bool.
- BIGINTER (INSERT x (INSERT xa EMPTY)) = pred_set.INTER x xa"
- by (import pred_set BIGINTER_INTER)
-
-lemma BIGINTER_SING: "ALL x::'a::type => bool. BIGINTER (INSERT x EMPTY) = x"
- by (import pred_set BIGINTER_SING)
-
-lemma SUBSET_BIGINTER: "ALL (X::'a::type => bool) P::('a::type => bool) => bool.
- SUBSET X (BIGINTER P) = (ALL x::'a::type => bool. IN x P --> SUBSET X x)"
- by (import pred_set SUBSET_BIGINTER)
-
-lemma DISJOINT_BIGINTER: "ALL (x::'a::type => bool) (xa::'a::type => bool)
- xb::('a::type => bool) => bool.
- IN xa xb & DISJOINT xa x -->
- DISJOINT x (BIGINTER xb) & DISJOINT (BIGINTER xb) x"
- by (import pred_set DISJOINT_BIGINTER)
-
-definition CROSS :: "('a => bool) => ('b => bool) => 'a * 'b => bool" where
- "CROSS ==
-%(P::'a::type => bool) Q::'b::type => bool.
- GSPEC (%p::'a::type * 'b::type. (p, IN (fst p) P & IN (snd p) Q))"
-
-lemma CROSS_DEF: "ALL (P::'a::type => bool) Q::'b::type => bool.
- CROSS P Q =
- GSPEC (%p::'a::type * 'b::type. (p, IN (fst p) P & IN (snd p) Q))"
- by (import pred_set CROSS_DEF)
-
-lemma IN_CROSS: "ALL (x::'a::type => bool) (xa::'b::type => bool) xb::'a::type * 'b::type.
- IN xb (CROSS x xa) = (IN (fst xb) x & IN (snd xb) xa)"
- by (import pred_set IN_CROSS)
-
-lemma CROSS_EMPTY: "ALL x::'a::type => bool. CROSS x EMPTY = EMPTY & CROSS EMPTY x = EMPTY"
- by (import pred_set CROSS_EMPTY)
-
-lemma CROSS_INSERT_LEFT: "ALL (x::'a::type => bool) (xa::'b::type => bool) xb::'a::type.
- CROSS (INSERT xb x) xa =
- pred_set.UNION (CROSS (INSERT xb EMPTY) xa) (CROSS x xa)"
- by (import pred_set CROSS_INSERT_LEFT)
-
-lemma CROSS_INSERT_RIGHT: "ALL (x::'a::type => bool) (xa::'b::type => bool) xb::'b::type.
- CROSS x (INSERT xb xa) =
- pred_set.UNION (CROSS x (INSERT xb EMPTY)) (CROSS x xa)"
- by (import pred_set CROSS_INSERT_RIGHT)
-
-lemma FINITE_CROSS: "ALL (x::'a::type => bool) xa::'b::type => bool.
- FINITE x & FINITE xa --> FINITE (CROSS x xa)"
- by (import pred_set FINITE_CROSS)
-
-lemma CROSS_SINGS: "ALL (x::'a::type) xa::'b::type.
- CROSS (INSERT x EMPTY) (INSERT xa EMPTY) = INSERT (x, xa) EMPTY"
- by (import pred_set CROSS_SINGS)
-
-lemma CARD_SING_CROSS: "ALL (x::'a::type) s::'b::type => bool.
- FINITE s --> CARD (CROSS (INSERT x EMPTY) s) = CARD s"
- by (import pred_set CARD_SING_CROSS)
-
-lemma CARD_CROSS: "ALL (x::'a::type => bool) xa::'b::type => bool.
- FINITE x & FINITE xa --> CARD (CROSS x xa) = CARD x * CARD xa"
- by (import pred_set CARD_CROSS)
-
-lemma CROSS_SUBSET: "ALL (x::'a::type => bool) (xa::'b::type => bool) (xb::'a::type => bool)
- xc::'b::type => bool.
- SUBSET (CROSS xb xc) (CROSS x xa) =
- (xb = EMPTY | xc = EMPTY | SUBSET xb x & SUBSET xc xa)"
- by (import pred_set CROSS_SUBSET)
-
-lemma FINITE_CROSS_EQ: "ALL (P::'a::type => bool) Q::'b::type => bool.
- FINITE (CROSS P Q) = (P = EMPTY | Q = EMPTY | FINITE P & FINITE Q)"
- by (import pred_set FINITE_CROSS_EQ)
-
-definition COMPL :: "('a => bool) => 'a => bool" where
+ sorry
+
+lemma BIGINTER_INTER: "BIGINTER (INSERT x (INSERT xa EMPTY)) = pred_set.INTER x xa"
+ sorry
+
+lemma BIGINTER_SING: "BIGINTER (INSERT x EMPTY) = x"
+ sorry
+
+lemma SUBSET_BIGINTER: "SUBSET X (BIGINTER P) = (ALL x. IN x P --> SUBSET X x)"
+ sorry
+
+lemma DISJOINT_BIGINTER: "IN xa xb & DISJOINT xa x
+==> DISJOINT x (BIGINTER xb) & DISJOINT (BIGINTER xb) x"
+ sorry
+
+definition
+ CROSS :: "('a => bool) => ('b => bool) => 'a * 'b => bool" where
+ "CROSS == %P Q. GSPEC (%p. (p, IN (fst p) P & IN (snd p) Q))"
+
+lemma CROSS_DEF: "CROSS P Q = GSPEC (%p. (p, IN (fst p) P & IN (snd p) Q))"
+ sorry
+
+lemma IN_CROSS: "IN xb (CROSS x xa) = (IN (fst xb) x & IN (snd xb) xa)"
+ sorry
+
+lemma CROSS_EMPTY: "CROSS x EMPTY = EMPTY & CROSS EMPTY x = EMPTY"
+ sorry
+
+lemma CROSS_INSERT_LEFT: "CROSS (INSERT xb x) xa =
+pred_set.UNION (CROSS (INSERT xb EMPTY) xa) (CROSS x xa)"
+ sorry
+
+lemma CROSS_INSERT_RIGHT: "CROSS x (INSERT xb xa) =
+pred_set.UNION (CROSS x (INSERT xb EMPTY)) (CROSS x xa)"
+ sorry
+
+lemma FINITE_CROSS: "FINITE x & FINITE xa ==> FINITE (CROSS x xa)"
+ sorry
+
+lemma CROSS_SINGS: "CROSS (INSERT x EMPTY) (INSERT xa EMPTY) = INSERT (x, xa) EMPTY"
+ sorry
+
+lemma CARD_SING_CROSS: "FINITE (s::'b => bool) ==> CARD (CROSS (INSERT (x::'a) EMPTY) s) = CARD s"
+ sorry
+
+lemma CARD_CROSS: "FINITE x & FINITE xa ==> CARD (CROSS x xa) = CARD x * CARD xa"
+ sorry
+
+lemma CROSS_SUBSET: "SUBSET (CROSS xb xc) (CROSS x xa) =
+(xb = EMPTY | xc = EMPTY | SUBSET xb x & SUBSET xc xa)"
+ sorry
+
+lemma FINITE_CROSS_EQ: "FINITE (CROSS P Q) = (P = EMPTY | Q = EMPTY | FINITE P & FINITE Q)"
+ sorry
+
+definition
+ COMPL :: "('a => bool) => 'a => bool" where
"COMPL == DIFF pred_set.UNIV"
-lemma COMPL_DEF: "ALL P::'a::type => bool. COMPL P = DIFF pred_set.UNIV P"
- by (import pred_set COMPL_DEF)
-
-lemma IN_COMPL: "ALL (x::'a::type) xa::'a::type => bool. IN x (COMPL xa) = (~ IN x xa)"
- by (import pred_set IN_COMPL)
-
-lemma COMPL_COMPL: "ALL x::'a::type => bool. COMPL (COMPL x) = x"
- by (import pred_set COMPL_COMPL)
-
-lemma COMPL_CLAUSES: "ALL x::'a::type => bool.
- pred_set.INTER (COMPL x) x = EMPTY &
- pred_set.UNION (COMPL x) x = pred_set.UNIV"
- by (import pred_set COMPL_CLAUSES)
-
-lemma COMPL_SPLITS: "ALL (x::'a::type => bool) xa::'a::type => bool.
- pred_set.UNION (pred_set.INTER x xa) (pred_set.INTER (COMPL x) xa) = xa"
- by (import pred_set COMPL_SPLITS)
-
-lemma INTER_UNION_COMPL: "ALL (x::'a::type => bool) xa::'a::type => bool.
- pred_set.INTER x xa = COMPL (pred_set.UNION (COMPL x) (COMPL xa))"
- by (import pred_set INTER_UNION_COMPL)
+lemma COMPL_DEF: "COMPL P = DIFF pred_set.UNIV P"
+ sorry
+
+lemma IN_COMPL: "IN x (COMPL xa) = (~ IN x xa)"
+ sorry
+
+lemma COMPL_COMPL: "COMPL (COMPL x) = x"
+ sorry
+
+lemma COMPL_CLAUSES: "pred_set.INTER (COMPL x) x = EMPTY &
+pred_set.UNION (COMPL x) x = pred_set.UNIV"
+ sorry
+
+lemma COMPL_SPLITS: "pred_set.UNION (pred_set.INTER x xa) (pred_set.INTER (COMPL x) xa) = xa"
+ sorry
+
+lemma INTER_UNION_COMPL: "pred_set.INTER x xa = COMPL (pred_set.UNION (COMPL x) (COMPL xa))"
+ sorry
lemma COMPL_EMPTY: "COMPL EMPTY = pred_set.UNIV"
- by (import pred_set COMPL_EMPTY)
+ sorry
consts
count :: "nat => nat => bool"
defs
- count_primdef: "count == %n::nat. GSPEC (%m::nat. (m, m < n))"
-
-lemma count_def: "ALL n::nat. count n = GSPEC (%m::nat. (m, m < n))"
- by (import pred_set count_def)
-
-lemma IN_COUNT: "ALL (m::nat) n::nat. IN m (count n) = (m < n)"
- by (import pred_set IN_COUNT)
+ count_primdef: "count == %n. GSPEC (%m. (m, m < n))"
+
+lemma count_def: "count n = GSPEC (%m. (m, m < n))"
+ sorry
+
+lemma IN_COUNT: "IN m (count n) = (m < n)"
+ sorry
lemma COUNT_ZERO: "count 0 = EMPTY"
- by (import pred_set COUNT_ZERO)
-
-lemma COUNT_SUC: "ALL n::nat. count (Suc n) = INSERT n (count n)"
- by (import pred_set COUNT_SUC)
-
-lemma FINITE_COUNT: "ALL n::nat. FINITE (count n)"
- by (import pred_set FINITE_COUNT)
-
-lemma CARD_COUNT: "ALL n::nat. CARD (count n) = n"
- by (import pred_set CARD_COUNT)
-
-definition ITSET_tupled :: "('a => 'b => 'b) => ('a => bool) * 'b => 'b" where
+ sorry
+
+lemma COUNT_SUC: "count (Suc n) = INSERT n (count n)"
+ sorry
+
+lemma FINITE_COUNT: "FINITE (count n)"
+ sorry
+
+lemma CARD_COUNT: "CARD (count n) = n"
+ sorry
+
+definition
+ ITSET_tupled :: "('a => 'b => 'b) => ('a => bool) * 'b => 'b" where
"ITSET_tupled ==
-%f::'a::type => 'b::type => 'b::type.
- WFREC
- (SOME R::('a::type => bool) * 'b::type
- => ('a::type => bool) * 'b::type => bool.
- WF R &
- (ALL (b::'b::type) s::'a::type => bool.
- FINITE s & s ~= EMPTY --> R (REST s, f (CHOICE s) b) (s, b)))
- (%(ITSET_tupled::('a::type => bool) * 'b::type => 'b::type)
- (v::'a::type => bool, v1::'b::type).
- if FINITE v
- then if v = EMPTY then v1
- else ITSET_tupled (REST v, f (CHOICE v) v1)
- else ARB)"
-
-lemma ITSET_tupled_primitive_def: "ALL f::'a::type => 'b::type => 'b::type.
- ITSET_tupled f =
- WFREC
- (SOME R::('a::type => bool) * 'b::type
- => ('a::type => bool) * 'b::type => bool.
- WF R &
- (ALL (b::'b::type) s::'a::type => bool.
- FINITE s & s ~= EMPTY --> R (REST s, f (CHOICE s) b) (s, b)))
- (%(ITSET_tupled::('a::type => bool) * 'b::type => 'b::type)
- (v::'a::type => bool, v1::'b::type).
- if FINITE v
- then if v = EMPTY then v1
- else ITSET_tupled (REST v, f (CHOICE v) v1)
- else ARB)"
- by (import pred_set ITSET_tupled_primitive_def)
-
-definition ITSET :: "('a => 'b => 'b) => ('a => bool) => 'b => 'b" where
- "ITSET ==
-%(f::'a::type => 'b::type => 'b::type) (x::'a::type => bool) x1::'b::type.
- ITSET_tupled f (x, x1)"
-
-lemma ITSET_curried_def: "ALL (f::'a::type => 'b::type => 'b::type) (x::'a::type => bool)
- x1::'b::type. ITSET f x x1 = ITSET_tupled f (x, x1)"
- by (import pred_set ITSET_curried_def)
-
-lemma ITSET_IND: "ALL P::('a::type => bool) => 'b::type => bool.
- (ALL (s::'a::type => bool) b::'b::type.
- (FINITE s & s ~= EMPTY -->
- P (REST s) ((f::'a::type => 'b::type => 'b::type) (CHOICE s) b)) -->
- P s b) -->
- (ALL v::'a::type => bool. All (P v))"
- by (import pred_set ITSET_IND)
-
-lemma ITSET_THM: "ALL (s::'a::type => bool) (f::'a::type => 'b::type => 'b::type) b::'b::type.
- FINITE s -->
- ITSET f s b =
- (if s = EMPTY then b else ITSET f (REST s) (f (CHOICE s) b))"
- by (import pred_set ITSET_THM)
-
-lemma ITSET_EMPTY: "ALL (x::'a::type => 'b::type => 'b::type) xa::'b::type.
- ITSET x EMPTY xa = xa"
- by (import pred_set ITSET_EMPTY)
+%f. WFREC
+ (SOME R.
+ WF R &
+ (ALL b s.
+ FINITE s & s ~= EMPTY --> R (REST s, f (CHOICE s) b) (s, b)))
+ (%ITSET_tupled (v, v1).
+ if FINITE v
+ then if v = EMPTY then v1
+ else ITSET_tupled (REST v, f (CHOICE v) v1)
+ else ARB)"
+
+lemma ITSET_tupled_primitive_def: "ITSET_tupled f =
+WFREC
+ (SOME R.
+ WF R &
+ (ALL b s. FINITE s & s ~= EMPTY --> R (REST s, f (CHOICE s) b) (s, b)))
+ (%ITSET_tupled (v, v1).
+ if FINITE v
+ then if v = EMPTY then v1 else ITSET_tupled (REST v, f (CHOICE v) v1)
+ else ARB)"
+ sorry
+
+definition
+ ITSET :: "('a => 'b => 'b) => ('a => bool) => 'b => 'b" where
+ "ITSET == %f x x1. ITSET_tupled f (x, x1)"
+
+lemma ITSET_curried_def: "ITSET (f::'a => 'b => 'b) (x::'a => bool) (x1::'b) = ITSET_tupled f (x, x1)"
+ sorry
+
+lemma ITSET_IND: "(!!(s::'a => bool) b::'b.
+ (FINITE s & s ~= EMPTY
+ ==> (P::('a => bool) => 'b => bool) (REST s)
+ ((f::'a => 'b => 'b) (CHOICE s) b))
+ ==> P s b)
+==> P (v::'a => bool) (x::'b)"
+ sorry
+
+lemma ITSET_THM: "FINITE s
+==> ITSET f s b =
+ (if s = EMPTY then b else ITSET f (REST s) (f (CHOICE s) b))"
+ sorry
+
+lemma ITSET_EMPTY: "ITSET (x::'a => 'b => 'b) EMPTY (xa::'b) = xa"
+ sorry
;end_setup
;setup_theory operator
-definition ASSOC :: "('a => 'a => 'a) => bool" where
- "ASSOC ==
-%f::'a::type => 'a::type => 'a::type.
- ALL (x::'a::type) (y::'a::type) z::'a::type. f x (f y z) = f (f x y) z"
-
-lemma ASSOC_DEF: "ALL f::'a::type => 'a::type => 'a::type.
- ASSOC f =
- (ALL (x::'a::type) (y::'a::type) z::'a::type. f x (f y z) = f (f x y) z)"
- by (import operator ASSOC_DEF)
-
-definition COMM :: "('a => 'a => 'b) => bool" where
- "COMM ==
-%f::'a::type => 'a::type => 'b::type.
- ALL (x::'a::type) y::'a::type. f x y = f y x"
-
-lemma COMM_DEF: "ALL f::'a::type => 'a::type => 'b::type.
- COMM f = (ALL (x::'a::type) y::'a::type. f x y = f y x)"
- by (import operator COMM_DEF)
-
-definition FCOMM :: "('a => 'b => 'a) => ('c => 'a => 'a) => bool" where
- "FCOMM ==
-%(f::'a::type => 'b::type => 'a::type) g::'c::type => 'a::type => 'a::type.
- ALL (x::'c::type) (y::'a::type) z::'b::type. g x (f y z) = f (g x y) z"
-
-lemma FCOMM_DEF: "ALL (f::'a::type => 'b::type => 'a::type)
- g::'c::type => 'a::type => 'a::type.
- FCOMM f g =
- (ALL (x::'c::type) (y::'a::type) z::'b::type. g x (f y z) = f (g x y) z)"
- by (import operator FCOMM_DEF)
-
-definition RIGHT_ID :: "('a => 'b => 'a) => 'b => bool" where
- "RIGHT_ID ==
-%(f::'a::type => 'b::type => 'a::type) e::'b::type.
- ALL x::'a::type. f x e = x"
-
-lemma RIGHT_ID_DEF: "ALL (f::'a::type => 'b::type => 'a::type) e::'b::type.
- RIGHT_ID f e = (ALL x::'a::type. f x e = x)"
- by (import operator RIGHT_ID_DEF)
-
-definition LEFT_ID :: "('a => 'b => 'b) => 'a => bool" where
- "LEFT_ID ==
-%(f::'a::type => 'b::type => 'b::type) e::'a::type.
- ALL x::'b::type. f e x = x"
-
-lemma LEFT_ID_DEF: "ALL (f::'a::type => 'b::type => 'b::type) e::'a::type.
- LEFT_ID f e = (ALL x::'b::type. f e x = x)"
- by (import operator LEFT_ID_DEF)
-
-definition MONOID :: "('a => 'a => 'a) => 'a => bool" where
- "MONOID ==
-%(f::'a::type => 'a::type => 'a::type) e::'a::type.
- ASSOC f & RIGHT_ID f e & LEFT_ID f e"
-
-lemma MONOID_DEF: "ALL (f::'a::type => 'a::type => 'a::type) e::'a::type.
- MONOID f e = (ASSOC f & RIGHT_ID f e & LEFT_ID f e)"
- by (import operator MONOID_DEF)
+definition
+ ASSOC :: "('a => 'a => 'a) => bool" where
+ "ASSOC == %f. ALL x y z. f x (f y z) = f (f x y) z"
+
+lemma ASSOC_DEF: "ASSOC f = (ALL x y z. f x (f y z) = f (f x y) z)"
+ sorry
+
+definition
+ COMM :: "('a => 'a => 'b) => bool" where
+ "COMM == %f. ALL x y. f x y = f y x"
+
+lemma COMM_DEF: "COMM f = (ALL x y. f x y = f y x)"
+ sorry
+
+definition
+ FCOMM :: "('a => 'b => 'a) => ('c => 'a => 'a) => bool" where
+ "FCOMM == %f g. ALL x y z. g x (f y z) = f (g x y) z"
+
+lemma FCOMM_DEF: "FCOMM f g = (ALL x y z. g x (f y z) = f (g x y) z)"
+ sorry
+
+definition
+ RIGHT_ID :: "('a => 'b => 'a) => 'b => bool" where
+ "RIGHT_ID == %f e. ALL x. f x e = x"
+
+lemma RIGHT_ID_DEF: "RIGHT_ID f e = (ALL x. f x e = x)"
+ sorry
+
+definition
+ LEFT_ID :: "('a => 'b => 'b) => 'a => bool" where
+ "LEFT_ID == %f e. ALL x. f e x = x"
+
+lemma LEFT_ID_DEF: "LEFT_ID f e = (ALL x. f e x = x)"
+ sorry
+
+definition
+ MONOID :: "('a => 'a => 'a) => 'a => bool" where
+ "MONOID == %f e. ASSOC f & RIGHT_ID f e & LEFT_ID f e"
+
+lemma MONOID_DEF: "MONOID f e = (ASSOC f & RIGHT_ID f e & LEFT_ID f e)"
+ sorry
lemma ASSOC_CONJ: "ASSOC op &"
- by (import operator ASSOC_CONJ)
+ sorry
lemma ASSOC_DISJ: "ASSOC op |"
- by (import operator ASSOC_DISJ)
-
-lemma FCOMM_ASSOC: "ALL x::'a::type => 'a::type => 'a::type. FCOMM x x = ASSOC x"
- by (import operator FCOMM_ASSOC)
+ sorry
+
+lemma FCOMM_ASSOC: "FCOMM x x = ASSOC x"
+ sorry
lemma MONOID_CONJ_T: "MONOID op & True"
- by (import operator MONOID_CONJ_T)
+ sorry
lemma MONOID_DISJ_F: "MONOID op | False"
- by (import operator MONOID_DISJ_F)
+ sorry
;end_setup
@@ -4574,1371 +3503,995 @@
consts
SNOC :: "'a => 'a list => 'a list"
-specification (SNOC) SNOC: "(ALL x::'a::type. SNOC x [] = [x]) &
-(ALL (x::'a::type) (x'::'a::type) l::'a::type list.
- SNOC x (x' # l) = x' # SNOC x l)"
- by (import rich_list SNOC)
+specification (SNOC) SNOC: "(ALL x::'a. SNOC x [] = [x]) &
+(ALL (x::'a) (x'::'a) l::'a list. SNOC x (x' # l) = x' # SNOC x l)"
+ sorry
consts
SCANL :: "('b => 'a => 'b) => 'b => 'a list => 'b list"
-specification (SCANL) SCANL: "(ALL (f::'b::type => 'a::type => 'b::type) e::'b::type.
- SCANL f e [] = [e]) &
-(ALL (f::'b::type => 'a::type => 'b::type) (e::'b::type) (x::'a::type)
- l::'a::type list. SCANL f e (x # l) = e # SCANL f (f e x) l)"
- by (import rich_list SCANL)
+specification (SCANL) SCANL: "(ALL (f::'b => 'a => 'b) e::'b. SCANL f e [] = [e]) &
+(ALL (f::'b => 'a => 'b) (e::'b) (x::'a) l::'a list.
+ SCANL f e (x # l) = e # SCANL f (f e x) l)"
+ sorry
consts
SCANR :: "('a => 'b => 'b) => 'b => 'a list => 'b list"
-specification (SCANR) SCANR: "(ALL (f::'a::type => 'b::type => 'b::type) e::'b::type.
- SCANR f e [] = [e]) &
-(ALL (f::'a::type => 'b::type => 'b::type) (e::'b::type) (x::'a::type)
- l::'a::type list.
+specification (SCANR) SCANR: "(ALL (f::'a => 'b => 'b) e::'b. SCANR f e [] = [e]) &
+(ALL (f::'a => 'b => 'b) (e::'b) (x::'a) l::'a list.
SCANR f e (x # l) = f x (hd (SCANR f e l)) # SCANR f e l)"
- by (import rich_list SCANR)
-
-lemma IS_EL_DEF: "ALL (x::'a::type) l::'a::type list. x mem l = list_ex (op = x) l"
- by (import rich_list IS_EL_DEF)
-
-definition AND_EL :: "bool list => bool" where
+ sorry
+
+lemma IS_EL_DEF: "List.member l x = list_ex (op = x) l"
+ sorry
+
+definition
+ AND_EL :: "bool list => bool" where
"AND_EL == list_all I"
lemma AND_EL_DEF: "AND_EL = list_all I"
- by (import rich_list AND_EL_DEF)
-
-definition OR_EL :: "bool list => bool" where
+ sorry
+
+definition
+ OR_EL :: "bool list => bool" where
"OR_EL == list_ex I"
lemma OR_EL_DEF: "OR_EL = list_ex I"
- by (import rich_list OR_EL_DEF)
+ sorry
consts
FIRSTN :: "nat => 'a list => 'a list"
-specification (FIRSTN) FIRSTN: "(ALL l::'a::type list. FIRSTN 0 l = []) &
-(ALL (n::nat) (x::'a::type) l::'a::type list.
- FIRSTN (Suc n) (x # l) = x # FIRSTN n l)"
- by (import rich_list FIRSTN)
+specification (FIRSTN) FIRSTN: "(ALL l::'a list. FIRSTN (0::nat) l = []) &
+(ALL (n::nat) (x::'a) l::'a list. FIRSTN (Suc n) (x # l) = x # FIRSTN n l)"
+ sorry
consts
BUTFIRSTN :: "nat => 'a list => 'a list"
-specification (BUTFIRSTN) BUTFIRSTN: "(ALL l::'a::type list. BUTFIRSTN 0 l = l) &
-(ALL (n::nat) (x::'a::type) l::'a::type list.
- BUTFIRSTN (Suc n) (x # l) = BUTFIRSTN n l)"
- by (import rich_list BUTFIRSTN)
+specification (BUTFIRSTN) BUTFIRSTN: "(ALL l::'a list. BUTFIRSTN (0::nat) l = l) &
+(ALL (n::nat) (x::'a) l::'a list. BUTFIRSTN (Suc n) (x # l) = BUTFIRSTN n l)"
+ sorry
consts
SEG :: "nat => nat => 'a list => 'a list"
-specification (SEG) SEG: "(ALL (k::nat) l::'a::type list. SEG 0 k l = []) &
-(ALL (m::nat) (x::'a::type) l::'a::type list.
- SEG (Suc m) 0 (x # l) = x # SEG m 0 l) &
-(ALL (m::nat) (k::nat) (x::'a::type) l::'a::type list.
+specification (SEG) SEG: "(ALL (k::nat) l::'a list. SEG (0::nat) k l = []) &
+(ALL (m::nat) (x::'a) l::'a list.
+ SEG (Suc m) (0::nat) (x # l) = x # SEG m (0::nat) l) &
+(ALL (m::nat) (k::nat) (x::'a) l::'a list.
SEG (Suc m) (Suc k) (x # l) = SEG (Suc m) k l)"
- by (import rich_list SEG)
-
-lemma LAST: "ALL (x::'a::type) l::'a::type list. last (SNOC x l) = x"
- by (import rich_list LAST)
-
-lemma BUTLAST: "ALL (x::'a::type) l::'a::type list. butlast (SNOC x l) = l"
- by (import rich_list BUTLAST)
+ sorry
+
+lemma LAST: "last (SNOC x l) = x"
+ sorry
+
+lemma BUTLAST: "butlast (SNOC x l) = l"
+ sorry
consts
LASTN :: "nat => 'a list => 'a list"
-specification (LASTN) LASTN: "(ALL l::'a::type list. LASTN 0 l = []) &
-(ALL (n::nat) (x::'a::type) l::'a::type list.
+specification (LASTN) LASTN: "(ALL l::'a list. LASTN (0::nat) l = []) &
+(ALL (n::nat) (x::'a) l::'a list.
LASTN (Suc n) (SNOC x l) = SNOC x (LASTN n l))"
- by (import rich_list LASTN)
+ sorry
consts
BUTLASTN :: "nat => 'a list => 'a list"
-specification (BUTLASTN) BUTLASTN: "(ALL l::'a::type list. BUTLASTN 0 l = l) &
-(ALL (n::nat) (x::'a::type) l::'a::type list.
+specification (BUTLASTN) BUTLASTN: "(ALL l::'a list. BUTLASTN (0::nat) l = l) &
+(ALL (n::nat) (x::'a) l::'a list.
BUTLASTN (Suc n) (SNOC x l) = BUTLASTN n l)"
- by (import rich_list BUTLASTN)
-
-lemma EL: "(ALL x::'a::type list. EL 0 x = hd x) &
-(ALL (x::nat) xa::'a::type list. EL (Suc x) xa = EL x (tl xa))"
- by (import rich_list EL)
+ sorry
+
+lemma EL: "(ALL x::'a list. EL (0::nat) x = hd x) &
+(ALL (x::nat) xa::'a list. EL (Suc x) xa = EL x (tl xa))"
+ sorry
consts
ELL :: "nat => 'a list => 'a"
-specification (ELL) ELL: "(ALL l::'a::type list. ELL 0 l = last l) &
-(ALL (n::nat) l::'a::type list. ELL (Suc n) l = ELL n (butlast l))"
- by (import rich_list ELL)
+specification (ELL) ELL: "(ALL l::'a list. ELL (0::nat) l = last l) &
+(ALL (n::nat) l::'a list. ELL (Suc n) l = ELL n (butlast l))"
+ sorry
consts
IS_PREFIX :: "'a list => 'a list => bool"
-specification (IS_PREFIX) IS_PREFIX: "(ALL l::'a::type list. IS_PREFIX l [] = True) &
-(ALL (x::'a::type) l::'a::type list. IS_PREFIX [] (x # l) = False) &
-(ALL (x1::'a::type) (l1::'a::type list) (x2::'a::type) l2::'a::type list.
+specification (IS_PREFIX) IS_PREFIX: "(ALL l::'a list. IS_PREFIX l [] = True) &
+(ALL (x::'a) l::'a list. IS_PREFIX [] (x # l) = False) &
+(ALL (x1::'a) (l1::'a list) (x2::'a) l2::'a list.
IS_PREFIX (x1 # l1) (x2 # l2) = (x1 = x2 & IS_PREFIX l1 l2))"
- by (import rich_list IS_PREFIX)
-
-lemma SNOC_APPEND: "ALL (x::'a::type) l::'a::type list. SNOC x l = l @ [x]"
- by (import rich_list SNOC_APPEND)
-
-lemma REVERSE: "rev [] = [] &
-(ALL (x::'a::type) xa::'a::type list. rev (x # xa) = SNOC x (rev xa))"
- by (import rich_list REVERSE)
-
-lemma REVERSE_SNOC: "ALL (x::'a::type) l::'a::type list. rev (SNOC x l) = x # rev l"
- by (import rich_list REVERSE_SNOC)
-
-lemma SNOC_Axiom: "ALL (e::'b::type) f::'a::type => 'a::type list => 'b::type => 'b::type.
- EX x::'a::type list => 'b::type.
- x [] = e &
- (ALL (xa::'a::type) l::'a::type list. x (SNOC xa l) = f xa l (x l))"
- by (import rich_list SNOC_Axiom)
+ sorry
+
+lemma SNOC_APPEND: "SNOC x l = l @ [x]"
+ sorry
+
+lemma REVERSE: "rev [] = [] & (ALL (x::'a) xa::'a list. rev (x # xa) = SNOC x (rev xa))"
+ sorry
+
+lemma REVERSE_SNOC: "rev (SNOC x l) = x # rev l"
+ sorry
+
+lemma SNOC_Axiom: "EX x. x [] = e & (ALL xa l. x (SNOC xa l) = f xa l (x l))"
+ sorry
consts
IS_SUFFIX :: "'a list => 'a list => bool"
-specification (IS_SUFFIX) IS_SUFFIX: "(ALL l::'a::type list. IS_SUFFIX l [] = True) &
-(ALL (x::'a::type) l::'a::type list. IS_SUFFIX [] (SNOC x l) = False) &
-(ALL (x1::'a::type) (l1::'a::type list) (x2::'a::type) l2::'a::type list.
+specification (IS_SUFFIX) IS_SUFFIX: "(ALL l::'a list. IS_SUFFIX l [] = True) &
+(ALL (x::'a) l::'a list. IS_SUFFIX [] (SNOC x l) = False) &
+(ALL (x1::'a) (l1::'a list) (x2::'a) l2::'a list.
IS_SUFFIX (SNOC x1 l1) (SNOC x2 l2) = (x1 = x2 & IS_SUFFIX l1 l2))"
- by (import rich_list IS_SUFFIX)
+ sorry
consts
IS_SUBLIST :: "'a list => 'a list => bool"
-specification (IS_SUBLIST) IS_SUBLIST: "(ALL l::'a::type list. IS_SUBLIST l [] = True) &
-(ALL (x::'a::type) l::'a::type list. IS_SUBLIST [] (x # l) = False) &
-(ALL (x1::'a::type) (l1::'a::type list) (x2::'a::type) l2::'a::type list.
+specification (IS_SUBLIST) IS_SUBLIST: "(ALL l::'a list. IS_SUBLIST l [] = True) &
+(ALL (x::'a) l::'a list. IS_SUBLIST [] (x # l) = False) &
+(ALL (x1::'a) (l1::'a list) (x2::'a) l2::'a list.
IS_SUBLIST (x1 # l1) (x2 # l2) =
(x1 = x2 & IS_PREFIX l1 l2 | IS_SUBLIST l1 (x2 # l2)))"
- by (import rich_list IS_SUBLIST)
+ sorry
consts
SPLITP :: "('a => bool) => 'a list => 'a list * 'a list"
-specification (SPLITP) SPLITP: "(ALL P::'a::type => bool. SPLITP P [] = ([], [])) &
-(ALL (P::'a::type => bool) (x::'a::type) l::'a::type list.
+specification (SPLITP) SPLITP: "(ALL P::'a => bool. SPLITP P [] = ([], [])) &
+(ALL (P::'a => bool) (x::'a) l::'a list.
SPLITP P (x # l) =
(if P x then ([], x # l) else (x # fst (SPLITP P l), snd (SPLITP P l))))"
- by (import rich_list SPLITP)
-
-definition PREFIX :: "('a => bool) => 'a list => 'a list" where
- "PREFIX == %(P::'a::type => bool) l::'a::type list. fst (SPLITP (Not o P) l)"
-
-lemma PREFIX_DEF: "ALL (P::'a::type => bool) l::'a::type list.
- PREFIX P l = fst (SPLITP (Not o P) l)"
- by (import rich_list PREFIX_DEF)
-
-definition SUFFIX :: "('a => bool) => 'a list => 'a list" where
- "SUFFIX ==
-%P::'a::type => bool.
- foldl (%(l'::'a::type list) x::'a::type. if P x then SNOC x l' else [])
- []"
-
-lemma SUFFIX_DEF: "ALL (P::'a::type => bool) l::'a::type list.
- SUFFIX P l =
- foldl (%(l'::'a::type list) x::'a::type. if P x then SNOC x l' else [])
- [] l"
- by (import rich_list SUFFIX_DEF)
-
-definition UNZIP_FST :: "('a * 'b) list => 'a list" where
- "UNZIP_FST == %l::('a::type * 'b::type) list. fst (unzip l)"
-
-lemma UNZIP_FST_DEF: "ALL l::('a::type * 'b::type) list. UNZIP_FST l = fst (unzip l)"
- by (import rich_list UNZIP_FST_DEF)
-
-definition UNZIP_SND :: "('a * 'b) list => 'b list" where
- "UNZIP_SND == %l::('a::type * 'b::type) list. snd (unzip l)"
-
-lemma UNZIP_SND_DEF: "ALL l::('a::type * 'b::type) list. UNZIP_SND l = snd (unzip l)"
- by (import rich_list UNZIP_SND_DEF)
+ sorry
+
+definition
+ PREFIX :: "('a => bool) => 'a list => 'a list" where
+ "PREFIX == %P l. fst (SPLITP (Not o P) l)"
+
+lemma PREFIX_DEF: "PREFIX P l = fst (SPLITP (Not o P) l)"
+ sorry
+
+definition
+ SUFFIX :: "('a => bool) => 'a list => 'a list" where
+ "SUFFIX == %P. foldl (%l' x. if P x then SNOC x l' else []) []"
+
+lemma SUFFIX_DEF: "SUFFIX P l = foldl (%l' x. if P x then SNOC x l' else []) [] l"
+ sorry
+
+definition
+ UNZIP_FST :: "('a * 'b) list => 'a list" where
+ "UNZIP_FST == %l. fst (unzip l)"
+
+lemma UNZIP_FST_DEF: "UNZIP_FST l = fst (unzip l)"
+ sorry
+
+definition
+ UNZIP_SND :: "('a * 'b) list => 'b list" where
+ "UNZIP_SND == %l. snd (unzip l)"
+
+lemma UNZIP_SND_DEF: "UNZIP_SND (l::('a * 'b) list) = snd (unzip l)"
+ sorry
consts
GENLIST :: "(nat => 'a) => nat => 'a list"
-specification (GENLIST) GENLIST: "(ALL f::nat => 'a::type. GENLIST f 0 = []) &
-(ALL (f::nat => 'a::type) n::nat.
- GENLIST f (Suc n) = SNOC (f n) (GENLIST f n))"
- by (import rich_list GENLIST)
+specification (GENLIST) GENLIST: "(ALL f::nat => 'a. GENLIST f (0::nat) = []) &
+(ALL (f::nat => 'a) n::nat. GENLIST f (Suc n) = SNOC (f n) (GENLIST f n))"
+ sorry
consts
REPLICATE :: "nat => 'a => 'a list"
-specification (REPLICATE) REPLICATE: "(ALL x::'a::type. REPLICATE 0 x = []) &
-(ALL (n::nat) x::'a::type. REPLICATE (Suc n) x = x # REPLICATE n x)"
- by (import rich_list REPLICATE)
-
-lemma LENGTH_MAP2: "ALL (l1::'a::type list) l2::'b::type list.
- length l1 = length l2 -->
- (ALL f::'a::type => 'b::type => 'c::type.
- length (map2 f l1 l2) = length l1 &
- length (map2 f l1 l2) = length l2)"
- by (import rich_list LENGTH_MAP2)
-
-lemma NULL_EQ_NIL: "ALL l::'a::type list. null l = (l = [])"
- by (import rich_list NULL_EQ_NIL)
-
-lemma LENGTH_EQ: "ALL (x::'a::type list) y::'a::type list. x = y --> length x = length y"
- by (import rich_list LENGTH_EQ)
-
-lemma LENGTH_NOT_NULL: "ALL l::'a::type list. (0 < length l) = (~ null l)"
- by (import rich_list LENGTH_NOT_NULL)
-
-lemma SNOC_INDUCT: "ALL P::'a::type list => bool.
- P [] &
- (ALL l::'a::type list. P l --> (ALL x::'a::type. P (SNOC x l))) -->
- All P"
- by (import rich_list SNOC_INDUCT)
-
-lemma SNOC_CASES: "ALL x'::'a::type list.
- x' = [] | (EX (x::'a::type) l::'a::type list. x' = SNOC x l)"
- by (import rich_list SNOC_CASES)
-
-lemma LENGTH_SNOC: "ALL (x::'a::type) l::'a::type list. length (SNOC x l) = Suc (length l)"
- by (import rich_list LENGTH_SNOC)
-
-lemma NOT_NIL_SNOC: "ALL (x::'a::type) xa::'a::type list. [] ~= SNOC x xa"
- by (import rich_list NOT_NIL_SNOC)
-
-lemma NOT_SNOC_NIL: "ALL (x::'a::type) xa::'a::type list. SNOC x xa ~= []"
- by (import rich_list NOT_SNOC_NIL)
-
-lemma SNOC_11: "ALL (x::'a::type) (l::'a::type list) (x'::'a::type) l'::'a::type list.
- (SNOC x l = SNOC x' l') = (x = x' & l = l')"
- by (import rich_list SNOC_11)
-
-lemma SNOC_EQ_LENGTH_EQ: "ALL (x1::'a::type) (l1::'a::type list) (x2::'a::type) l2::'a::type list.
- SNOC x1 l1 = SNOC x2 l2 --> length l1 = length l2"
- by (import rich_list SNOC_EQ_LENGTH_EQ)
-
-lemma SNOC_REVERSE_CONS: "ALL (x::'a::type) xa::'a::type list. SNOC x xa = rev (x # rev xa)"
- by (import rich_list SNOC_REVERSE_CONS)
-
-lemma MAP_SNOC: "ALL (x::'a::type => 'b::type) (xa::'a::type) xb::'a::type list.
- map x (SNOC xa xb) = SNOC (x xa) (map x xb)"
- by (import rich_list MAP_SNOC)
-
-lemma FOLDR_SNOC: "ALL (f::'a::type => 'b::type => 'b::type) (e::'b::type) (x::'a::type)
- l::'a::type list. foldr f (SNOC x l) e = foldr f l (f x e)"
- by (import rich_list FOLDR_SNOC)
-
-lemma FOLDL_SNOC: "ALL (f::'b::type => 'a::type => 'b::type) (e::'b::type) (x::'a::type)
- l::'a::type list. foldl f e (SNOC x l) = f (foldl f e l) x"
- by (import rich_list FOLDL_SNOC)
-
-lemma FOLDR_FOLDL: "ALL (f::'a::type => 'a::type => 'a::type) e::'a::type.
- MONOID f e --> (ALL l::'a::type list. foldr f l e = foldl f e l)"
- by (import rich_list FOLDR_FOLDL)
-
-lemma LENGTH_FOLDR: "ALL l::'a::type list. length l = foldr (%x::'a::type. Suc) l 0"
- by (import rich_list LENGTH_FOLDR)
-
-lemma LENGTH_FOLDL: "ALL l::'a::type list. length l = foldl (%(l'::nat) x::'a::type. Suc l') 0 l"
- by (import rich_list LENGTH_FOLDL)
-
-lemma MAP_FOLDR: "ALL (f::'a::type => 'b::type) l::'a::type list.
- map f l = foldr (%x::'a::type. op # (f x)) l []"
- by (import rich_list MAP_FOLDR)
-
-lemma MAP_FOLDL: "ALL (f::'a::type => 'b::type) l::'a::type list.
- map f l = foldl (%(l'::'b::type list) x::'a::type. SNOC (f x) l') [] l"
- by (import rich_list MAP_FOLDL)
-
-lemma MAP_o: "ALL (f::'b::type => 'c::type) g::'a::type => 'b::type.
- map (f o g) = map f o map g"
- by (import rich_list MAP_o)
-
-lemma FILTER_FOLDR: "ALL (P::'a::type => bool) l::'a::type list.
- filter P l =
- foldr (%(x::'a::type) l'::'a::type list. if P x then x # l' else l') l []"
- by (import rich_list FILTER_FOLDR)
-
-lemma FILTER_SNOC: "ALL (P::'a::type => bool) (x::'a::type) l::'a::type list.
- filter P (SNOC x l) = (if P x then SNOC x (filter P l) else filter P l)"
- by (import rich_list FILTER_SNOC)
-
-lemma FILTER_FOLDL: "ALL (P::'a::type => bool) l::'a::type list.
- filter P l =
- foldl (%(l'::'a::type list) x::'a::type. if P x then SNOC x l' else l')
- [] l"
- by (import rich_list FILTER_FOLDL)
-
-lemma FILTER_COMM: "ALL (f1::'a::type => bool) (f2::'a::type => bool) l::'a::type list.
- filter f1 (filter f2 l) = filter f2 (filter f1 l)"
- by (import rich_list FILTER_COMM)
-
-lemma FILTER_IDEM: "ALL (f::'a::type => bool) l::'a::type list.
- filter f (filter f l) = filter f l"
- by (import rich_list FILTER_IDEM)
-
-lemma LENGTH_SEG: "ALL (n::nat) (k::nat) l::'a::type list.
- n + k <= length l --> length (SEG n k l) = n"
- by (import rich_list LENGTH_SEG)
-
-lemma APPEND_NIL: "(ALL l::'a::type list. l @ [] = l) & (ALL x::'a::type list. [] @ x = x)"
- by (import rich_list APPEND_NIL)
-
-lemma APPEND_SNOC: "ALL (l1::'a::type list) (x::'a::type) l2::'a::type list.
- l1 @ SNOC x l2 = SNOC x (l1 @ l2)"
- by (import rich_list APPEND_SNOC)
-
-lemma APPEND_FOLDR: "ALL (l1::'a::type list) l2::'a::type list. l1 @ l2 = foldr op # l1 l2"
- by (import rich_list APPEND_FOLDR)
-
-lemma APPEND_FOLDL: "ALL (l1::'a::type list) l2::'a::type list.
- l1 @ l2 = foldl (%(l'::'a::type list) x::'a::type. SNOC x l') l1 l2"
- by (import rich_list APPEND_FOLDL)
-
-lemma CONS_APPEND: "ALL (x::'a::type) l::'a::type list. x # l = [x] @ l"
- by (import rich_list CONS_APPEND)
+specification (REPLICATE) REPLICATE: "(ALL x::'a. REPLICATE (0::nat) x = []) &
+(ALL (n::nat) x::'a. REPLICATE (Suc n) x = x # REPLICATE n x)"
+ sorry
+
+lemma LENGTH_MAP2: "length l1 = length l2
+==> length (map2 f l1 l2) = length l1 & length (map2 f l1 l2) = length l2"
+ sorry
+
+lemma LENGTH_EQ: "x = y ==> length x = length y"
+ sorry
+
+lemma LENGTH_NOT_NULL: "(0 < length l) = (~ List.null l)"
+ sorry
+
+lemma SNOC_INDUCT: "P [] & (ALL l. P l --> (ALL x. P (SNOC x l))) ==> P x"
+ sorry
+
+lemma SNOC_CASES: "x' = [] | (EX x l. x' = SNOC x l)"
+ sorry
+
+lemma LENGTH_SNOC: "length (SNOC x l) = Suc (length l)"
+ sorry
+
+lemma NOT_NIL_SNOC: "[] ~= SNOC x xa"
+ sorry
+
+lemma NOT_SNOC_NIL: "SNOC x xa ~= []"
+ sorry
+
+lemma SNOC_11: "(SNOC x l = SNOC x' l') = (x = x' & l = l')"
+ sorry
+
+lemma SNOC_EQ_LENGTH_EQ: "SNOC x1 l1 = SNOC x2 l2 ==> length l1 = length l2"
+ sorry
+
+lemma SNOC_REVERSE_CONS: "SNOC x xa = rev (x # rev xa)"
+ sorry
+
+lemma MAP_SNOC: "map (x::'a => 'b) (SNOC (xa::'a) (xb::'a list)) = SNOC (x xa) (map x xb)"
+ sorry
+
+lemma FOLDR_SNOC: "foldr (f::'a => 'b => 'b) (SNOC (x::'a) (l::'a list)) (e::'b) =
+foldr f l (f x e)"
+ sorry
+
+lemma FOLDL_SNOC: "foldl (f::'b => 'a => 'b) (e::'b) (SNOC (x::'a) (l::'a list)) =
+f (foldl f e l) x"
+ sorry
+
+lemma FOLDR_FOLDL: "MONOID f e ==> foldr f l e = foldl f e l"
+ sorry
+
+lemma LENGTH_FOLDR: "length l = foldr (%x. Suc) l 0"
+ sorry
+
+lemma LENGTH_FOLDL: "length l = foldl (%l' x. Suc l') 0 l"
+ sorry
+
+lemma MAP_FOLDR: "map (f::'a => 'b) (l::'a list) = foldr (%x::'a. op # (f x)) l []"
+ sorry
+
+lemma MAP_FOLDL: "map (f::'a => 'b) (l::'a list) =
+foldl (%(l'::'b list) x::'a. SNOC (f x) l') [] l"
+ sorry
+
+lemma FILTER_FOLDR: "filter P l = foldr (%x l'. if P x then x # l' else l') l []"
+ sorry
+
+lemma FILTER_SNOC: "filter P (SNOC x l) = (if P x then SNOC x (filter P l) else filter P l)"
+ sorry
+
+lemma FILTER_FOLDL: "filter P l = foldl (%l' x. if P x then SNOC x l' else l') [] l"
+ sorry
+
+lemma FILTER_COMM: "filter f1 (filter f2 l) = filter f2 (filter f1 l)"
+ sorry
+
+lemma FILTER_IDEM: "filter f (filter f l) = filter f l"
+ sorry
+
+lemma LENGTH_SEG: "n + k <= length l ==> length (SEG n k l) = n"
+ sorry
+
+lemma APPEND_NIL: "(ALL l::'a list. l @ [] = l) & (ALL x::'a list. [] @ x = x)"
+ sorry
+
+lemma APPEND_SNOC: "l1 @ SNOC x l2 = SNOC x (l1 @ l2)"
+ sorry
+
+lemma APPEND_FOLDR: "l1 @ l2 = foldr op # l1 l2"
+ sorry
+
+lemma APPEND_FOLDL: "l1 @ l2 = foldl (%l' x. SNOC x l') l1 l2"
+ sorry
+
+lemma CONS_APPEND: "x # l = [x] @ l"
+ sorry
lemma ASSOC_APPEND: "ASSOC op @"
- by (import rich_list ASSOC_APPEND)
+ sorry
lemma MONOID_APPEND_NIL: "MONOID op @ []"
- by (import rich_list MONOID_APPEND_NIL)
-
-lemma APPEND_LENGTH_EQ: "ALL (l1::'a::type list) l1'::'a::type list.
- length l1 = length l1' -->
- (ALL (l2::'a::type list) l2'::'a::type list.
- length l2 = length l2' -->
- (l1 @ l2 = l1' @ l2') = (l1 = l1' & l2 = l2'))"
- by (import rich_list APPEND_LENGTH_EQ)
-
-lemma FLAT_SNOC: "ALL (x::'a::type list) l::'a::type list list.
- concat (SNOC x l) = concat l @ x"
- by (import rich_list FLAT_SNOC)
-
-lemma FLAT_FOLDR: "ALL l::'a::type list list. concat l = foldr op @ l []"
- by (import rich_list FLAT_FOLDR)
-
-lemma FLAT_FOLDL: "ALL l::'a::type list list. concat l = foldl op @ [] l"
- by (import rich_list FLAT_FOLDL)
-
-lemma LENGTH_FLAT: "ALL l::'a::type list list. length (concat l) = sum (map size l)"
- by (import rich_list LENGTH_FLAT)
-
-lemma REVERSE_FOLDR: "ALL l::'a::type list. rev l = foldr SNOC l []"
- by (import rich_list REVERSE_FOLDR)
-
-lemma REVERSE_FOLDL: "ALL l::'a::type list.
- rev l = foldl (%(l'::'a::type list) x::'a::type. x # l') [] l"
- by (import rich_list REVERSE_FOLDL)
-
-lemma ALL_EL_SNOC: "ALL (P::'a::type => bool) (x::'a::type) l::'a::type list.
- list_all P (SNOC x l) = (list_all P l & P x)"
- by (import rich_list ALL_EL_SNOC)
-
-lemma ALL_EL_MAP: "ALL (P::'b::type => bool) (f::'a::type => 'b::type) l::'a::type list.
- list_all P (map f l) = list_all (P o f) l"
- by (import rich_list ALL_EL_MAP)
-
-lemma SOME_EL_SNOC: "ALL (P::'a::type => bool) (x::'a::type) l::'a::type list.
- list_ex P (SNOC x l) = (P x | list_ex P l)"
- by (import rich_list SOME_EL_SNOC)
-
-lemma IS_EL_SNOC: "ALL (y::'a::type) (x::'a::type) l::'a::type list.
- y mem SNOC x l = (y = x | y mem l)"
- by (import rich_list IS_EL_SNOC)
-
-lemma SUM_SNOC: "ALL (x::nat) l::nat list. sum (SNOC x l) = sum l + x"
- by (import rich_list SUM_SNOC)
-
-lemma SUM_FOLDL: "ALL l::nat list. sum l = foldl op + 0 l"
- by (import rich_list SUM_FOLDL)
-
-lemma IS_PREFIX_APPEND: "ALL (l1::'a::type list) l2::'a::type list.
- IS_PREFIX l1 l2 = (EX l::'a::type list. l1 = l2 @ l)"
- by (import rich_list IS_PREFIX_APPEND)
-
-lemma IS_SUFFIX_APPEND: "ALL (l1::'a::type list) l2::'a::type list.
- IS_SUFFIX l1 l2 = (EX l::'a::type list. l1 = l @ l2)"
- by (import rich_list IS_SUFFIX_APPEND)
-
-lemma IS_SUBLIST_APPEND: "ALL (l1::'a::type list) l2::'a::type list.
- IS_SUBLIST l1 l2 =
- (EX (l::'a::type list) l'::'a::type list. l1 = l @ l2 @ l')"
- by (import rich_list IS_SUBLIST_APPEND)
-
-lemma IS_PREFIX_IS_SUBLIST: "ALL (l1::'a::type list) l2::'a::type list.
- IS_PREFIX l1 l2 --> IS_SUBLIST l1 l2"
- by (import rich_list IS_PREFIX_IS_SUBLIST)
-
-lemma IS_SUFFIX_IS_SUBLIST: "ALL (l1::'a::type list) l2::'a::type list.
- IS_SUFFIX l1 l2 --> IS_SUBLIST l1 l2"
- by (import rich_list IS_SUFFIX_IS_SUBLIST)
-
-lemma IS_PREFIX_REVERSE: "ALL (l1::'a::type list) l2::'a::type list.
- IS_PREFIX (rev l1) (rev l2) = IS_SUFFIX l1 l2"
- by (import rich_list IS_PREFIX_REVERSE)
-
-lemma IS_SUFFIX_REVERSE: "ALL (l2::'a::type list) l1::'a::type list.
- IS_SUFFIX (rev l1) (rev l2) = IS_PREFIX l1 l2"
- by (import rich_list IS_SUFFIX_REVERSE)
-
-lemma IS_SUBLIST_REVERSE: "ALL (l1::'a::type list) l2::'a::type list.
- IS_SUBLIST (rev l1) (rev l2) = IS_SUBLIST l1 l2"
- by (import rich_list IS_SUBLIST_REVERSE)
-
-lemma PREFIX_FOLDR: "ALL (P::'a::type => bool) x::'a::type list.
- PREFIX P x =
- foldr (%(x::'a::type) l'::'a::type list. if P x then x # l' else []) x []"
- by (import rich_list PREFIX_FOLDR)
-
-lemma PREFIX: "(ALL x::'a::type => bool. PREFIX x [] = []) &
-(ALL (x::'a::type => bool) (xa::'a::type) xb::'a::type list.
+ sorry
+
+lemma APPEND_LENGTH_EQ: "[| length l1 = length l1'; length l2 = length l2' |]
+==> (l1 @ l2 = l1' @ l2') = (l1 = l1' & l2 = l2')"
+ sorry
+
+lemma FLAT_SNOC: "concat (SNOC x l) = concat l @ x"
+ sorry
+
+lemma FLAT_FOLDR: "concat l = foldr op @ l []"
+ sorry
+
+lemma LENGTH_FLAT: "length (concat l) = HOL4Compat.sum (map length l)"
+ sorry
+
+lemma REVERSE_FOLDR: "rev l = foldr SNOC l []"
+ sorry
+
+lemma ALL_EL_SNOC: "list_all P (SNOC x l) = (list_all P l & P x)"
+ sorry
+
+lemma ALL_EL_MAP: "list_all (P::'b => bool) (map (f::'a => 'b) (l::'a list)) =
+list_all (P o f) l"
+ sorry
+
+lemma SOME_EL_SNOC: "list_ex P (SNOC x l) = (P x | list_ex P l)"
+ sorry
+
+lemma IS_EL_SNOC: "List.member (SNOC x l) y = (y = x | List.member l y)"
+ sorry
+
+lemma SUM_SNOC: "HOL4Compat.sum (SNOC x l) = HOL4Compat.sum l + x"
+ sorry
+
+lemma SUM_FOLDL: "HOL4Compat.sum l = foldl op + 0 l"
+ sorry
+
+lemma IS_PREFIX_APPEND: "IS_PREFIX l1 l2 = (EX l. l1 = l2 @ l)"
+ sorry
+
+lemma IS_SUFFIX_APPEND: "IS_SUFFIX l1 l2 = (EX l. l1 = l @ l2)"
+ sorry
+
+lemma IS_SUBLIST_APPEND: "IS_SUBLIST l1 l2 = (EX l l'. l1 = l @ l2 @ l')"
+ sorry
+
+lemma IS_PREFIX_IS_SUBLIST: "IS_PREFIX l1 l2 ==> IS_SUBLIST l1 l2"
+ sorry
+
+lemma IS_SUFFIX_IS_SUBLIST: "IS_SUFFIX l1 l2 ==> IS_SUBLIST l1 l2"
+ sorry
+
+lemma IS_PREFIX_REVERSE: "IS_PREFIX (rev l1) (rev l2) = IS_SUFFIX l1 l2"
+ sorry
+
+lemma IS_SUFFIX_REVERSE: "IS_SUFFIX (rev l1) (rev l2) = IS_PREFIX l1 l2"
+ sorry
+
+lemma IS_SUBLIST_REVERSE: "IS_SUBLIST (rev l1) (rev l2) = IS_SUBLIST l1 l2"
+ sorry
+
+lemma PREFIX_FOLDR: "PREFIX P x = foldr (%x l'. if P x then x # l' else []) x []"
+ sorry
+
+lemma PREFIX: "(ALL x::'a => bool. PREFIX x [] = []) &
+(ALL (x::'a => bool) (xa::'a) xb::'a list.
PREFIX x (xa # xb) = (if x xa then xa # PREFIX x xb else []))"
- by (import rich_list PREFIX)
-
-lemma IS_PREFIX_PREFIX: "ALL (P::'a::type => bool) l::'a::type list. IS_PREFIX l (PREFIX P l)"
- by (import rich_list IS_PREFIX_PREFIX)
-
-lemma LENGTH_SCANL: "ALL (f::'b::type => 'a::type => 'b::type) (e::'b::type) l::'a::type list.
- length (SCANL f e l) = Suc (length l)"
- by (import rich_list LENGTH_SCANL)
-
-lemma LENGTH_SCANR: "ALL (f::'a::type => 'b::type => 'b::type) (e::'b::type) l::'a::type list.
- length (SCANR f e l) = Suc (length l)"
- by (import rich_list LENGTH_SCANR)
-
-lemma COMM_MONOID_FOLDL: "ALL x::'a::type => 'a::type => 'a::type.
- COMM x -->
- (ALL xa::'a::type.
- MONOID x xa -->
- (ALL (e::'a::type) l::'a::type list.
- foldl x e l = x e (foldl x xa l)))"
- by (import rich_list COMM_MONOID_FOLDL)
-
-lemma COMM_MONOID_FOLDR: "ALL x::'a::type => 'a::type => 'a::type.
- COMM x -->
- (ALL xa::'a::type.
- MONOID x xa -->
- (ALL (e::'a::type) l::'a::type list.
- foldr x l e = x e (foldr x l xa)))"
- by (import rich_list COMM_MONOID_FOLDR)
-
-lemma FCOMM_FOLDR_APPEND: "ALL (x::'a::type => 'a::type => 'a::type)
- xa::'b::type => 'a::type => 'a::type.
- FCOMM x xa -->
- (ALL xb::'a::type.
- LEFT_ID x xb -->
- (ALL (l1::'b::type list) l2::'b::type list.
- foldr xa (l1 @ l2) xb = x (foldr xa l1 xb) (foldr xa l2 xb)))"
- by (import rich_list FCOMM_FOLDR_APPEND)
-
-lemma FCOMM_FOLDL_APPEND: "ALL (x::'a::type => 'b::type => 'a::type)
- xa::'a::type => 'a::type => 'a::type.
- FCOMM x xa -->
- (ALL xb::'a::type.
- RIGHT_ID xa xb -->
- (ALL (l1::'b::type list) l2::'b::type list.
- foldl x xb (l1 @ l2) = xa (foldl x xb l1) (foldl x xb l2)))"
- by (import rich_list FCOMM_FOLDL_APPEND)
-
-lemma FOLDL_SINGLE: "ALL (x::'a::type => 'b::type => 'a::type) (xa::'a::type) xb::'b::type.
- foldl x xa [xb] = x xa xb"
- by (import rich_list FOLDL_SINGLE)
-
-lemma FOLDR_SINGLE: "ALL (x::'a::type => 'b::type => 'b::type) (xa::'b::type) xb::'a::type.
- foldr x [xb] xa = x xb xa"
- by (import rich_list FOLDR_SINGLE)
-
-lemma FOLDR_CONS_NIL: "ALL l::'a::type list. foldr op # l [] = l"
- by (import rich_list FOLDR_CONS_NIL)
-
-lemma FOLDL_SNOC_NIL: "ALL l::'a::type list.
- foldl (%(xs::'a::type list) x::'a::type. SNOC x xs) [] l = l"
- by (import rich_list FOLDL_SNOC_NIL)
-
-lemma FOLDR_REVERSE: "ALL (x::'a::type => 'b::type => 'b::type) (xa::'b::type) xb::'a::type list.
- foldr x (rev xb) xa = foldl (%(xa::'b::type) y::'a::type. x y xa) xa xb"
- by (import rich_list FOLDR_REVERSE)
-
-lemma FOLDL_REVERSE: "ALL (x::'a::type => 'b::type => 'a::type) (xa::'a::type) xb::'b::type list.
- foldl x xa (rev xb) = foldr (%(xa::'b::type) y::'a::type. x y xa) xb xa"
- by (import rich_list FOLDL_REVERSE)
-
-lemma FOLDR_MAP: "ALL (f::'a::type => 'a::type => 'a::type) (e::'a::type)
- (g::'b::type => 'a::type) l::'b::type list.
- foldr f (map g l) e = foldr (%x::'b::type. f (g x)) l e"
- by (import rich_list FOLDR_MAP)
-
-lemma FOLDL_MAP: "ALL (f::'a::type => 'a::type => 'a::type) (e::'a::type)
- (g::'b::type => 'a::type) l::'b::type list.
- foldl f e (map g l) = foldl (%(x::'a::type) y::'b::type. f x (g y)) e l"
- by (import rich_list FOLDL_MAP)
-
-lemma ALL_EL_FOLDR: "ALL (P::'a::type => bool) l::'a::type list.
- list_all P l = foldr (%x::'a::type. op & (P x)) l True"
- by (import rich_list ALL_EL_FOLDR)
-
-lemma ALL_EL_FOLDL: "ALL (P::'a::type => bool) l::'a::type list.
- list_all P l = foldl (%(l'::bool) x::'a::type. l' & P x) True l"
- by (import rich_list ALL_EL_FOLDL)
-
-lemma SOME_EL_FOLDR: "ALL (P::'a::type => bool) l::'a::type list.
- list_ex P l = foldr (%x::'a::type. op | (P x)) l False"
- by (import rich_list SOME_EL_FOLDR)
-
-lemma SOME_EL_FOLDL: "ALL (P::'a::type => bool) l::'a::type list.
- list_ex P l = foldl (%(l'::bool) x::'a::type. l' | P x) False l"
- by (import rich_list SOME_EL_FOLDL)
-
-lemma ALL_EL_FOLDR_MAP: "ALL (x::'a::type => bool) xa::'a::type list.
- list_all x xa = foldr op & (map x xa) True"
- by (import rich_list ALL_EL_FOLDR_MAP)
-
-lemma ALL_EL_FOLDL_MAP: "ALL (x::'a::type => bool) xa::'a::type list.
- list_all x xa = foldl op & True (map x xa)"
- by (import rich_list ALL_EL_FOLDL_MAP)
-
-lemma SOME_EL_FOLDR_MAP: "ALL (x::'a::type => bool) xa::'a::type list.
- list_ex x xa = foldr op | (map x xa) False"
- by (import rich_list SOME_EL_FOLDR_MAP)
-
-lemma SOME_EL_FOLDL_MAP: "ALL (x::'a::type => bool) xa::'a::type list.
- list_ex x xa = foldl op | False (map x xa)"
- by (import rich_list SOME_EL_FOLDL_MAP)
-
-lemma FOLDR_FILTER: "ALL (f::'a::type => 'a::type => 'a::type) (e::'a::type)
- (P::'a::type => bool) l::'a::type list.
- foldr f (filter P l) e =
- foldr (%(x::'a::type) y::'a::type. if P x then f x y else y) l e"
- by (import rich_list FOLDR_FILTER)
-
-lemma FOLDL_FILTER: "ALL (f::'a::type => 'a::type => 'a::type) (e::'a::type)
- (P::'a::type => bool) l::'a::type list.
- foldl f e (filter P l) =
- foldl (%(x::'a::type) y::'a::type. if P y then f x y else x) e l"
- by (import rich_list FOLDL_FILTER)
-
-lemma ASSOC_FOLDR_FLAT: "ALL f::'a::type => 'a::type => 'a::type.
- ASSOC f -->
- (ALL e::'a::type.
- LEFT_ID f e -->
- (ALL l::'a::type list list.
- foldr f (concat l) e = foldr f (map (FOLDR f e) l) e))"
- by (import rich_list ASSOC_FOLDR_FLAT)
-
-lemma ASSOC_FOLDL_FLAT: "ALL f::'a::type => 'a::type => 'a::type.
- ASSOC f -->
- (ALL e::'a::type.
- RIGHT_ID f e -->
- (ALL l::'a::type list list.
- foldl f e (concat l) = foldl f e (map (foldl f e) l)))"
- by (import rich_list ASSOC_FOLDL_FLAT)
-
-lemma SOME_EL_MAP: "ALL (P::'b::type => bool) (f::'a::type => 'b::type) l::'a::type list.
- list_ex P (map f l) = list_ex (P o f) l"
- by (import rich_list SOME_EL_MAP)
-
-lemma SOME_EL_DISJ: "ALL (P::'a::type => bool) (Q::'a::type => bool) l::'a::type list.
- list_ex (%x::'a::type. P x | Q x) l =
- (list_ex P l | list_ex Q l)"
- by (import rich_list SOME_EL_DISJ)
-
-lemma IS_EL_FOLDR: "ALL (x::'a::type) xa::'a::type list.
- x mem xa = foldr (%xa::'a::type. op | (x = xa)) xa False"
- by (import rich_list IS_EL_FOLDR)
-
-lemma IS_EL_FOLDL: "ALL (x::'a::type) xa::'a::type list.
- x mem xa = foldl (%(l'::bool) xa::'a::type. l' | x = xa) False xa"
- by (import rich_list IS_EL_FOLDL)
-
-lemma NULL_FOLDR: "ALL l::'a::type list. null l = foldr (%(x::'a::type) l'::bool. False) l True"
- by (import rich_list NULL_FOLDR)
-
-lemma NULL_FOLDL: "ALL l::'a::type list. null l = foldl (%(x::bool) l'::'a::type. False) True l"
- by (import rich_list NULL_FOLDL)
-
-lemma SEG_LENGTH_ID: "ALL l::'a::type list. SEG (length l) 0 l = l"
- by (import rich_list SEG_LENGTH_ID)
-
-lemma SEG_SUC_CONS: "ALL (m::nat) (n::nat) (l::'a::type list) x::'a::type.
- SEG m (Suc n) (x # l) = SEG m n l"
- by (import rich_list SEG_SUC_CONS)
-
-lemma SEG_0_SNOC: "ALL (m::nat) (l::'a::type list) x::'a::type.
- m <= length l --> SEG m 0 (SNOC x l) = SEG m 0 l"
- by (import rich_list SEG_0_SNOC)
-
-lemma BUTLASTN_SEG: "ALL (n::nat) l::'a::type list.
- n <= length l --> BUTLASTN n l = SEG (length l - n) 0 l"
- by (import rich_list BUTLASTN_SEG)
-
-lemma LASTN_CONS: "ALL (n::nat) l::'a::type list.
- n <= length l --> (ALL x::'a::type. LASTN n (x # l) = LASTN n l)"
- by (import rich_list LASTN_CONS)
-
-lemma LENGTH_LASTN: "ALL (n::nat) l::'a::type list. n <= length l --> length (LASTN n l) = n"
- by (import rich_list LENGTH_LASTN)
-
-lemma LASTN_LENGTH_ID: "ALL l::'a::type list. LASTN (length l) l = l"
- by (import rich_list LASTN_LENGTH_ID)
-
-lemma LASTN_LASTN: "ALL (l::'a::type list) (n::nat) m::nat.
- m <= length l --> n <= m --> LASTN n (LASTN m l) = LASTN n l"
- by (import rich_list LASTN_LASTN)
-
-lemma FIRSTN_LENGTH_ID: "ALL l::'a::type list. FIRSTN (length l) l = l"
- by (import rich_list FIRSTN_LENGTH_ID)
-
-lemma FIRSTN_SNOC: "ALL (n::nat) l::'a::type list.
- n <= length l --> (ALL x::'a::type. FIRSTN n (SNOC x l) = FIRSTN n l)"
- by (import rich_list FIRSTN_SNOC)
-
-lemma BUTLASTN_LENGTH_NIL: "ALL l::'a::type list. BUTLASTN (length l) l = []"
- by (import rich_list BUTLASTN_LENGTH_NIL)
-
-lemma BUTLASTN_SUC_BUTLAST: "ALL (n::nat) l::'a::type list.
- n < length l --> BUTLASTN (Suc n) l = BUTLASTN n (butlast l)"
- by (import rich_list BUTLASTN_SUC_BUTLAST)
-
-lemma BUTLASTN_BUTLAST: "ALL (n::nat) l::'a::type list.
- n < length l --> BUTLASTN n (butlast l) = butlast (BUTLASTN n l)"
- by (import rich_list BUTLASTN_BUTLAST)
-
-lemma LENGTH_BUTLASTN: "ALL (n::nat) l::'a::type list.
- n <= length l --> length (BUTLASTN n l) = length l - n"
- by (import rich_list LENGTH_BUTLASTN)
-
-lemma BUTLASTN_BUTLASTN: "ALL (m::nat) (n::nat) l::'a::type list.
- n + m <= length l --> BUTLASTN n (BUTLASTN m l) = BUTLASTN (n + m) l"
- by (import rich_list BUTLASTN_BUTLASTN)
-
-lemma APPEND_BUTLASTN_LASTN: "ALL (n::nat) l::'a::type list.
- n <= length l --> BUTLASTN n l @ LASTN n l = l"
- by (import rich_list APPEND_BUTLASTN_LASTN)
-
-lemma APPEND_FIRSTN_LASTN: "ALL (m::nat) (n::nat) l::'a::type list.
- m + n = length l --> FIRSTN n l @ LASTN m l = l"
- by (import rich_list APPEND_FIRSTN_LASTN)
-
-lemma BUTLASTN_APPEND2: "ALL (n::nat) (l1::'a::type list) l2::'a::type list.
- n <= length l2 --> BUTLASTN n (l1 @ l2) = l1 @ BUTLASTN n l2"
- by (import rich_list BUTLASTN_APPEND2)
-
-lemma BUTLASTN_LENGTH_APPEND: "ALL (l2::'a::type list) l1::'a::type list.
- BUTLASTN (length l2) (l1 @ l2) = l1"
- by (import rich_list BUTLASTN_LENGTH_APPEND)
-
-lemma LASTN_LENGTH_APPEND: "ALL (l2::'a::type list) l1::'a::type list. LASTN (length l2) (l1 @ l2) = l2"
- by (import rich_list LASTN_LENGTH_APPEND)
-
-lemma BUTLASTN_CONS: "ALL (n::nat) l::'a::type list.
- n <= length l -->
- (ALL x::'a::type. BUTLASTN n (x # l) = x # BUTLASTN n l)"
- by (import rich_list BUTLASTN_CONS)
-
-lemma BUTLASTN_LENGTH_CONS: "ALL (l::'a::type list) x::'a::type. BUTLASTN (length l) (x # l) = [x]"
- by (import rich_list BUTLASTN_LENGTH_CONS)
-
-lemma LAST_LASTN_LAST: "ALL (n::nat) l::'a::type list.
- n <= length l --> 0 < n --> last (LASTN n l) = last l"
- by (import rich_list LAST_LASTN_LAST)
-
-lemma BUTLASTN_LASTN_NIL: "ALL (n::nat) l::'a::type list. n <= length l --> BUTLASTN n (LASTN n l) = []"
- by (import rich_list BUTLASTN_LASTN_NIL)
-
-lemma LASTN_BUTLASTN: "ALL (n::nat) (m::nat) l::'a::type list.
- n + m <= length l -->
- LASTN n (BUTLASTN m l) = BUTLASTN m (LASTN (n + m) l)"
- by (import rich_list LASTN_BUTLASTN)
-
-lemma BUTLASTN_LASTN: "ALL (m::nat) (n::nat) l::'a::type list.
- m <= n & n <= length l -->
- BUTLASTN m (LASTN n l) = LASTN (n - m) (BUTLASTN m l)"
- by (import rich_list BUTLASTN_LASTN)
-
-lemma LASTN_1: "ALL l::'a::type list. l ~= [] --> LASTN 1 l = [last l]"
- by (import rich_list LASTN_1)
-
-lemma BUTLASTN_1: "ALL l::'a::type list. l ~= [] --> BUTLASTN 1 l = butlast l"
- by (import rich_list BUTLASTN_1)
-
-lemma BUTLASTN_APPEND1: "ALL (l2::'a::type list) n::nat.
- length l2 <= n -->
- (ALL l1::'a::type list.
- BUTLASTN n (l1 @ l2) = BUTLASTN (n - length l2) l1)"
- by (import rich_list BUTLASTN_APPEND1)
-
-lemma LASTN_APPEND2: "ALL (n::nat) l2::'a::type list.
- n <= length l2 -->
- (ALL l1::'a::type list. LASTN n (l1 @ l2) = LASTN n l2)"
- by (import rich_list LASTN_APPEND2)
-
-lemma LASTN_APPEND1: "ALL (l2::'a::type list) n::nat.
- length l2 <= n -->
- (ALL l1::'a::type list.
- LASTN n (l1 @ l2) = LASTN (n - length l2) l1 @ l2)"
- by (import rich_list LASTN_APPEND1)
-
-lemma LASTN_MAP: "ALL (n::nat) l::'a::type list.
- n <= length l -->
- (ALL f::'a::type => 'b::type. LASTN n (map f l) = map f (LASTN n l))"
- by (import rich_list LASTN_MAP)
-
-lemma BUTLASTN_MAP: "ALL (n::nat) l::'a::type list.
- n <= length l -->
- (ALL f::'a::type => 'b::type.
- BUTLASTN n (map f l) = map f (BUTLASTN n l))"
- by (import rich_list BUTLASTN_MAP)
-
-lemma ALL_EL_LASTN: "(All::(('a::type => bool) => bool) => bool)
- (%P::'a::type => bool.
- (All::('a::type list => bool) => bool)
- (%l::'a::type list.
- (op -->::bool => bool => bool)
- ((list_all::('a::type => bool) => 'a::type list => bool) P l)
- ((All::(nat => bool) => bool)
- (%m::nat.
- (op -->::bool => bool => bool)
- ((op <=::nat => nat => bool) m
- ((size::'a::type list => nat) l))
- ((list_all::('a::type => bool) => 'a::type list => bool) P
- ((LASTN::nat => 'a::type list => 'a::type list) m
- l))))))"
- by (import rich_list ALL_EL_LASTN)
-
-lemma ALL_EL_BUTLASTN: "(All::(('a::type => bool) => bool) => bool)
- (%P::'a::type => bool.
- (All::('a::type list => bool) => bool)
- (%l::'a::type list.
- (op -->::bool => bool => bool)
- ((list_all::('a::type => bool) => 'a::type list => bool) P l)
- ((All::(nat => bool) => bool)
- (%m::nat.
- (op -->::bool => bool => bool)
- ((op <=::nat => nat => bool) m
- ((size::'a::type list => nat) l))
- ((list_all::('a::type => bool) => 'a::type list => bool) P
- ((BUTLASTN::nat => 'a::type list => 'a::type list) m
- l))))))"
- by (import rich_list ALL_EL_BUTLASTN)
-
-lemma LENGTH_FIRSTN: "ALL (n::nat) l::'a::type list. n <= length l --> length (FIRSTN n l) = n"
- by (import rich_list LENGTH_FIRSTN)
-
-lemma FIRSTN_FIRSTN: "(All::(nat => bool) => bool)
- (%m::nat.
- (All::('a::type list => bool) => bool)
- (%l::'a::type list.
- (op -->::bool => bool => bool)
- ((op <=::nat => nat => bool) m ((size::'a::type list => nat) l))
- ((All::(nat => bool) => bool)
- (%n::nat.
- (op -->::bool => bool => bool)
- ((op <=::nat => nat => bool) n m)
- ((op =::'a::type list => 'a::type list => bool)
- ((FIRSTN::nat => 'a::type list => 'a::type list) n
- ((FIRSTN::nat => 'a::type list => 'a::type list) m l))
- ((FIRSTN::nat => 'a::type list => 'a::type list) n
- l))))))"
- by (import rich_list FIRSTN_FIRSTN)
-
-lemma LENGTH_BUTFIRSTN: "ALL (n::nat) l::'a::type list.
- n <= length l --> length (BUTFIRSTN n l) = length l - n"
- by (import rich_list LENGTH_BUTFIRSTN)
-
-lemma BUTFIRSTN_LENGTH_NIL: "ALL l::'a::type list. BUTFIRSTN (length l) l = []"
- by (import rich_list BUTFIRSTN_LENGTH_NIL)
-
-lemma BUTFIRSTN_APPEND1: "ALL (n::nat) l1::'a::type list.
- n <= length l1 -->
- (ALL l2::'a::type list. BUTFIRSTN n (l1 @ l2) = BUTFIRSTN n l1 @ l2)"
- by (import rich_list BUTFIRSTN_APPEND1)
-
-lemma BUTFIRSTN_APPEND2: "ALL (l1::'a::type list) n::nat.
- length l1 <= n -->
- (ALL l2::'a::type list.
- BUTFIRSTN n (l1 @ l2) = BUTFIRSTN (n - length l1) l2)"
- by (import rich_list BUTFIRSTN_APPEND2)
-
-lemma BUTFIRSTN_BUTFIRSTN: "ALL (n::nat) (m::nat) l::'a::type list.
- n + m <= length l --> BUTFIRSTN n (BUTFIRSTN m l) = BUTFIRSTN (n + m) l"
- by (import rich_list BUTFIRSTN_BUTFIRSTN)
-
-lemma APPEND_FIRSTN_BUTFIRSTN: "ALL (n::nat) l::'a::type list.
- n <= length l --> FIRSTN n l @ BUTFIRSTN n l = l"
- by (import rich_list APPEND_FIRSTN_BUTFIRSTN)
-
-lemma LASTN_SEG: "ALL (n::nat) l::'a::type list.
- n <= length l --> LASTN n l = SEG n (length l - n) l"
- by (import rich_list LASTN_SEG)
-
-lemma FIRSTN_SEG: "ALL (n::nat) l::'a::type list. n <= length l --> FIRSTN n l = SEG n 0 l"
- by (import rich_list FIRSTN_SEG)
-
-lemma BUTFIRSTN_SEG: "ALL (n::nat) l::'a::type list.
- n <= length l --> BUTFIRSTN n l = SEG (length l - n) n l"
- by (import rich_list BUTFIRSTN_SEG)
-
-lemma BUTFIRSTN_SNOC: "ALL (n::nat) l::'a::type list.
- n <= length l -->
- (ALL x::'a::type. BUTFIRSTN n (SNOC x l) = SNOC x (BUTFIRSTN n l))"
- by (import rich_list BUTFIRSTN_SNOC)
-
-lemma APPEND_BUTLASTN_BUTFIRSTN: "ALL (m::nat) (n::nat) l::'a::type list.
- m + n = length l --> BUTLASTN m l @ BUTFIRSTN n l = l"
- by (import rich_list APPEND_BUTLASTN_BUTFIRSTN)
-
-lemma SEG_SEG: "ALL (n1::nat) (m1::nat) (n2::nat) (m2::nat) l::'a::type list.
- n1 + m1 <= length l & n2 + m2 <= n1 -->
- SEG n2 m2 (SEG n1 m1 l) = SEG n2 (m1 + m2) l"
- by (import rich_list SEG_SEG)
-
-lemma SEG_APPEND1: "ALL (n::nat) (m::nat) l1::'a::type list.
- n + m <= length l1 -->
- (ALL l2::'a::type list. SEG n m (l1 @ l2) = SEG n m l1)"
- by (import rich_list SEG_APPEND1)
-
-lemma SEG_APPEND2: "ALL (l1::'a::type list) (m::nat) (n::nat) l2::'a::type list.
- length l1 <= m & n <= length l2 -->
- SEG n m (l1 @ l2) = SEG n (m - length l1) l2"
- by (import rich_list SEG_APPEND2)
-
-lemma SEG_FIRSTN_BUTFISTN: "ALL (n::nat) (m::nat) l::'a::type list.
- n + m <= length l --> SEG n m l = FIRSTN n (BUTFIRSTN m l)"
- by (import rich_list SEG_FIRSTN_BUTFISTN)
-
-lemma SEG_APPEND: "ALL (m::nat) (l1::'a::type list) (n::nat) l2::'a::type list.
- m < length l1 & length l1 <= n + m & n + m <= length l1 + length l2 -->
- SEG n m (l1 @ l2) =
- SEG (length l1 - m) m l1 @ SEG (n + m - length l1) 0 l2"
- by (import rich_list SEG_APPEND)
-
-lemma SEG_LENGTH_SNOC: "ALL (x::'a::type list) xa::'a::type. SEG 1 (length x) (SNOC xa x) = [xa]"
- by (import rich_list SEG_LENGTH_SNOC)
-
-lemma SEG_SNOC: "ALL (n::nat) (m::nat) l::'a::type list.
- n + m <= length l --> (ALL x::'a::type. SEG n m (SNOC x l) = SEG n m l)"
- by (import rich_list SEG_SNOC)
-
-lemma ELL_SEG: "ALL (n::nat) l::'a::type list.
- n < length l --> ELL n l = hd (SEG 1 (PRE (length l - n)) l)"
- by (import rich_list ELL_SEG)
-
-lemma SNOC_FOLDR: "ALL (x::'a::type) l::'a::type list. SNOC x l = foldr op # l [x]"
- by (import rich_list SNOC_FOLDR)
-
-lemma IS_EL_FOLDR_MAP: "ALL (x::'a::type) xa::'a::type list.
- x mem xa = foldr op | (map (op = x) xa) False"
- by (import rich_list IS_EL_FOLDR_MAP)
-
-lemma IS_EL_FOLDL_MAP: "ALL (x::'a::type) xa::'a::type list.
- x mem xa = foldl op | False (map (op = x) xa)"
- by (import rich_list IS_EL_FOLDL_MAP)
-
-lemma FILTER_FILTER: "ALL (P::'a::type => bool) (Q::'a::type => bool) l::'a::type list.
- filter P (filter Q l) = [x::'a::type<-l. P x & Q x]"
- by (import rich_list FILTER_FILTER)
-
-lemma FCOMM_FOLDR_FLAT: "ALL (g::'a::type => 'a::type => 'a::type)
- f::'b::type => 'a::type => 'a::type.
- FCOMM g f -->
- (ALL e::'a::type.
- LEFT_ID g e -->
- (ALL l::'b::type list list.
- foldr f (concat l) e = foldr g (map (FOLDR f e) l) e))"
- by (import rich_list FCOMM_FOLDR_FLAT)
-
-lemma FCOMM_FOLDL_FLAT: "ALL (f::'a::type => 'b::type => 'a::type)
- g::'a::type => 'a::type => 'a::type.
- FCOMM f g -->
- (ALL e::'a::type.
- RIGHT_ID g e -->
- (ALL l::'b::type list list.
- foldl f e (concat l) = foldl g e (map (foldl f e) l)))"
- by (import rich_list FCOMM_FOLDL_FLAT)
-
-lemma FOLDR_MAP_REVERSE: "ALL f::'a::type => 'a::type => 'a::type.
- (ALL (a::'a::type) (b::'a::type) c::'a::type.
- f a (f b c) = f b (f a c)) -->
- (ALL (e::'a::type) (g::'b::type => 'a::type) l::'b::type list.
- foldr f (map g (rev l)) e = foldr f (map g l) e)"
- by (import rich_list FOLDR_MAP_REVERSE)
-
-lemma FOLDR_FILTER_REVERSE: "ALL f::'a::type => 'a::type => 'a::type.
- (ALL (a::'a::type) (b::'a::type) c::'a::type.
- f a (f b c) = f b (f a c)) -->
- (ALL (e::'a::type) (P::'a::type => bool) l::'a::type list.
- foldr f (filter P (rev l)) e = foldr f (filter P l) e)"
- by (import rich_list FOLDR_FILTER_REVERSE)
-
-lemma COMM_ASSOC_FOLDR_REVERSE: "ALL f::'a::type => 'a::type => 'a::type.
- COMM f -->
- ASSOC f -->
- (ALL (e::'a::type) l::'a::type list. foldr f (rev l) e = foldr f l e)"
- by (import rich_list COMM_ASSOC_FOLDR_REVERSE)
-
-lemma COMM_ASSOC_FOLDL_REVERSE: "ALL f::'a::type => 'a::type => 'a::type.
- COMM f -->
- ASSOC f -->
- (ALL (e::'a::type) l::'a::type list. foldl f e (rev l) = foldl f e l)"
- by (import rich_list COMM_ASSOC_FOLDL_REVERSE)
-
-lemma ELL_LAST: "ALL l::'a::type list. ~ null l --> ELL 0 l = last l"
- by (import rich_list ELL_LAST)
-
-lemma ELL_0_SNOC: "ALL (l::'a::type list) x::'a::type. ELL 0 (SNOC x l) = x"
- by (import rich_list ELL_0_SNOC)
-
-lemma ELL_SNOC: "ALL n>0.
- ALL (x::'a::type) l::'a::type list. ELL n (SNOC x l) = ELL (PRE n) l"
- by (import rich_list ELL_SNOC)
-
-lemma ELL_SUC_SNOC: "ALL (n::nat) (x::'a::type) xa::'a::type list.
- ELL (Suc n) (SNOC x xa) = ELL n xa"
- by (import rich_list ELL_SUC_SNOC)
-
-lemma ELL_CONS: "ALL (n::nat) l::'a::type list.
- n < length l --> (ALL x::'a::type. ELL n (x # l) = ELL n l)"
- by (import rich_list ELL_CONS)
-
-lemma ELL_LENGTH_CONS: "ALL (l::'a::type list) x::'a::type. ELL (length l) (x # l) = x"
- by (import rich_list ELL_LENGTH_CONS)
-
-lemma ELL_LENGTH_SNOC: "ALL (l::'a::type list) x::'a::type.
- ELL (length l) (SNOC x l) = (if null l then x else hd l)"
- by (import rich_list ELL_LENGTH_SNOC)
-
-lemma ELL_APPEND2: "ALL (n::nat) l2::'a::type list.
- n < length l2 --> (ALL l1::'a::type list. ELL n (l1 @ l2) = ELL n l2)"
- by (import rich_list ELL_APPEND2)
-
-lemma ELL_APPEND1: "ALL (l2::'a::type list) n::nat.
- length l2 <= n -->
- (ALL l1::'a::type list. ELL n (l1 @ l2) = ELL (n - length l2) l1)"
- by (import rich_list ELL_APPEND1)
-
-lemma ELL_PRE_LENGTH: "ALL l::'a::type list. l ~= [] --> ELL (PRE (length l)) l = hd l"
- by (import rich_list ELL_PRE_LENGTH)
-
-lemma EL_LENGTH_SNOC: "ALL (l::'a::type list) x::'a::type. EL (length l) (SNOC x l) = x"
- by (import rich_list EL_LENGTH_SNOC)
-
-lemma EL_PRE_LENGTH: "ALL l::'a::type list. l ~= [] --> EL (PRE (length l)) l = last l"
- by (import rich_list EL_PRE_LENGTH)
-
-lemma EL_SNOC: "ALL (n::nat) l::'a::type list.
- n < length l --> (ALL x::'a::type. EL n (SNOC x l) = EL n l)"
- by (import rich_list EL_SNOC)
-
-lemma EL_ELL: "ALL (n::nat) l::'a::type list.
- n < length l --> EL n l = ELL (PRE (length l - n)) l"
- by (import rich_list EL_ELL)
-
-lemma EL_LENGTH_APPEND: "ALL (l2::'a::type list) l1::'a::type list.
- ~ null l2 --> EL (length l1) (l1 @ l2) = hd l2"
- by (import rich_list EL_LENGTH_APPEND)
-
-lemma ELL_EL: "ALL (n::nat) l::'a::type list.
- n < length l --> ELL n l = EL (PRE (length l - n)) l"
- by (import rich_list ELL_EL)
-
-lemma ELL_MAP: "ALL (n::nat) (l::'a::type list) f::'a::type => 'b::type.
- n < length l --> ELL n (map f l) = f (ELL n l)"
- by (import rich_list ELL_MAP)
-
-lemma LENGTH_BUTLAST: "ALL l::'a::type list. l ~= [] --> length (butlast l) = PRE (length l)"
- by (import rich_list LENGTH_BUTLAST)
-
-lemma BUTFIRSTN_LENGTH_APPEND: "ALL (l1::'a::type list) l2::'a::type list.
- BUTFIRSTN (length l1) (l1 @ l2) = l2"
- by (import rich_list BUTFIRSTN_LENGTH_APPEND)
-
-lemma FIRSTN_APPEND1: "ALL (n::nat) l1::'a::type list.
- n <= length l1 -->
- (ALL l2::'a::type list. FIRSTN n (l1 @ l2) = FIRSTN n l1)"
- by (import rich_list FIRSTN_APPEND1)
-
-lemma FIRSTN_APPEND2: "ALL (l1::'a::type list) n::nat.
- length l1 <= n -->
- (ALL l2::'a::type list.
- FIRSTN n (l1 @ l2) = l1 @ FIRSTN (n - length l1) l2)"
- by (import rich_list FIRSTN_APPEND2)
-
-lemma FIRSTN_LENGTH_APPEND: "ALL (l1::'a::type list) l2::'a::type list. FIRSTN (length l1) (l1 @ l2) = l1"
- by (import rich_list FIRSTN_LENGTH_APPEND)
-
-lemma REVERSE_FLAT: "ALL l::'a::type list list. rev (concat l) = concat (rev (map rev l))"
- by (import rich_list REVERSE_FLAT)
-
-lemma MAP_FILTER: "ALL (f::'a::type => 'a::type) (P::'a::type => bool) l::'a::type list.
- (ALL x::'a::type. P (f x) = P x) -->
- map f (filter P l) = filter P (map f l)"
- by (import rich_list MAP_FILTER)
-
-lemma FLAT_REVERSE: "ALL l::'a::type list list. concat (rev l) = rev (concat (map rev l))"
- by (import rich_list FLAT_REVERSE)
-
-lemma FLAT_FLAT: "ALL l::'a::type list list list. concat (concat l) = concat (map concat l)"
- by (import rich_list FLAT_FLAT)
-
-lemma SOME_EL_REVERSE: "ALL (P::'a::type => bool) l::'a::type list.
- list_ex P (rev l) = list_ex P l"
- by (import rich_list SOME_EL_REVERSE)
-
-lemma ALL_EL_SEG: "ALL (P::'a::type => bool) l::'a::type list.
- list_all P l -->
- (ALL (m::nat) k::nat. m + k <= length l --> list_all P (SEG m k l))"
- by (import rich_list ALL_EL_SEG)
-
-lemma ALL_EL_FIRSTN: "(All::(('a::type => bool) => bool) => bool)
- (%P::'a::type => bool.
- (All::('a::type list => bool) => bool)
- (%l::'a::type list.
- (op -->::bool => bool => bool)
- ((list_all::('a::type => bool) => 'a::type list => bool) P l)
- ((All::(nat => bool) => bool)
- (%m::nat.
- (op -->::bool => bool => bool)
- ((op <=::nat => nat => bool) m
- ((size::'a::type list => nat) l))
- ((list_all::('a::type => bool) => 'a::type list => bool) P
- ((FIRSTN::nat => 'a::type list => 'a::type list) m
- l))))))"
- by (import rich_list ALL_EL_FIRSTN)
-
-lemma ALL_EL_BUTFIRSTN: "(All::(('a::type => bool) => bool) => bool)
- (%P::'a::type => bool.
- (All::('a::type list => bool) => bool)
- (%l::'a::type list.
- (op -->::bool => bool => bool)
- ((list_all::('a::type => bool) => 'a::type list => bool) P l)
- ((All::(nat => bool) => bool)
- (%m::nat.
- (op -->::bool => bool => bool)
- ((op <=::nat => nat => bool) m
- ((size::'a::type list => nat) l))
- ((list_all::('a::type => bool) => 'a::type list => bool) P
- ((BUTFIRSTN::nat => 'a::type list => 'a::type list) m
- l))))))"
- by (import rich_list ALL_EL_BUTFIRSTN)
-
-lemma SOME_EL_SEG: "ALL (m::nat) (k::nat) l::'a::type list.
- m + k <= length l -->
- (ALL P::'a::type => bool. list_ex P (SEG m k l) --> list_ex P l)"
- by (import rich_list SOME_EL_SEG)
-
-lemma SOME_EL_FIRSTN: "ALL (m::nat) l::'a::type list.
- m <= length l -->
- (ALL P::'a::type => bool. list_ex P (FIRSTN m l) --> list_ex P l)"
- by (import rich_list SOME_EL_FIRSTN)
-
-lemma SOME_EL_BUTFIRSTN: "ALL (m::nat) l::'a::type list.
- m <= length l -->
- (ALL P::'a::type => bool.
- list_ex P (BUTFIRSTN m l) --> list_ex P l)"
- by (import rich_list SOME_EL_BUTFIRSTN)
-
-lemma SOME_EL_LASTN: "ALL (m::nat) l::'a::type list.
- m <= length l -->
- (ALL P::'a::type => bool. list_ex P (LASTN m l) --> list_ex P l)"
- by (import rich_list SOME_EL_LASTN)
-
-lemma SOME_EL_BUTLASTN: "ALL (m::nat) l::'a::type list.
- m <= length l -->
- (ALL P::'a::type => bool.
- list_ex P (BUTLASTN m l) --> list_ex P l)"
- by (import rich_list SOME_EL_BUTLASTN)
-
-lemma IS_EL_REVERSE: "ALL (x::'a::type) l::'a::type list. x mem rev l = x mem l"
- by (import rich_list IS_EL_REVERSE)
-
-lemma IS_EL_FILTER: "ALL (P::'a::type => bool) x::'a::type.
- P x --> (ALL l::'a::type list. x mem filter P l = x mem l)"
- by (import rich_list IS_EL_FILTER)
-
-lemma IS_EL_SEG: "ALL (n::nat) (m::nat) l::'a::type list.
- n + m <= length l --> (ALL x::'a::type. x mem SEG n m l --> x mem l)"
- by (import rich_list IS_EL_SEG)
-
-lemma IS_EL_SOME_EL: "ALL (x::'a::type) l::'a::type list. x mem l = list_ex (op = x) l"
- by (import rich_list IS_EL_SOME_EL)
-
-lemma IS_EL_FIRSTN: "ALL (x::nat) xa::'a::type list.
- x <= length xa --> (ALL xb::'a::type. xb mem FIRSTN x xa --> xb mem xa)"
- by (import rich_list IS_EL_FIRSTN)
-
-lemma IS_EL_BUTFIRSTN: "ALL (x::nat) xa::'a::type list.
- x <= length xa -->
- (ALL xb::'a::type. xb mem BUTFIRSTN x xa --> xb mem xa)"
- by (import rich_list IS_EL_BUTFIRSTN)
-
-lemma IS_EL_BUTLASTN: "ALL (x::nat) xa::'a::type list.
- x <= length xa --> (ALL xb::'a::type. xb mem BUTLASTN x xa --> xb mem xa)"
- by (import rich_list IS_EL_BUTLASTN)
-
-lemma IS_EL_LASTN: "ALL (x::nat) xa::'a::type list.
- x <= length xa --> (ALL xb::'a::type. xb mem LASTN x xa --> xb mem xa)"
- by (import rich_list IS_EL_LASTN)
-
-lemma ZIP_SNOC: "ALL (l1::'a::type list) l2::'b::type list.
- length l1 = length l2 -->
- (ALL (x1::'a::type) x2::'b::type.
- zip (SNOC x1 l1) (SNOC x2 l2) = SNOC (x1, x2) (zip l1 l2))"
- by (import rich_list ZIP_SNOC)
-
-lemma UNZIP_SNOC: "ALL (x::'a::type * 'b::type) l::('a::type * 'b::type) list.
- unzip (SNOC x l) =
- (SNOC (fst x) (fst (unzip l)), SNOC (snd x) (snd (unzip l)))"
- by (import rich_list UNZIP_SNOC)
-
-lemma LENGTH_UNZIP_FST: "ALL x::('a::type * 'b::type) list. length (UNZIP_FST x) = length x"
- by (import rich_list LENGTH_UNZIP_FST)
-
-lemma LENGTH_UNZIP_SND: "ALL x::('a::type * 'b::type) list. length (UNZIP_SND x) = length x"
- by (import rich_list LENGTH_UNZIP_SND)
-
-lemma SUM_APPEND: "ALL (l1::nat list) l2::nat list. sum (l1 @ l2) = sum l1 + sum l2"
- by (import rich_list SUM_APPEND)
-
-lemma SUM_REVERSE: "ALL l::nat list. sum (rev l) = sum l"
- by (import rich_list SUM_REVERSE)
-
-lemma SUM_FLAT: "ALL l::nat list list. sum (concat l) = sum (map sum l)"
- by (import rich_list SUM_FLAT)
-
-lemma EL_APPEND1: "ALL (n::nat) (l1::'a::type list) l2::'a::type list.
- n < length l1 --> EL n (l1 @ l2) = EL n l1"
- by (import rich_list EL_APPEND1)
-
-lemma EL_APPEND2: "ALL (l1::'a::type list) n::nat.
- length l1 <= n -->
- (ALL l2::'a::type list. EL n (l1 @ l2) = EL (n - length l1) l2)"
- by (import rich_list EL_APPEND2)
-
-lemma EL_MAP: "ALL (n::nat) l::'a::type list.
- n < length l -->
- (ALL f::'a::type => 'b::type. EL n (map f l) = f (EL n l))"
- by (import rich_list EL_MAP)
-
-lemma EL_CONS: "ALL n>0. ALL (x::'a::type) l::'a::type list. EL n (x # l) = EL (PRE n) l"
- by (import rich_list EL_CONS)
-
-lemma EL_SEG: "ALL (n::nat) l::'a::type list. n < length l --> EL n l = hd (SEG 1 n l)"
- by (import rich_list EL_SEG)
-
-lemma EL_IS_EL: "ALL (n::nat) l::'a::type list. n < length l --> EL n l mem l"
- by (import rich_list EL_IS_EL)
-
-lemma TL_SNOC: "ALL (x::'a::type) l::'a::type list.
- tl (SNOC x l) = (if null l then [] else SNOC x (tl l))"
- by (import rich_list TL_SNOC)
-
-lemma EL_REVERSE: "ALL (n::nat) l::'a::type list.
- n < length l --> EL n (rev l) = EL (PRE (length l - n)) l"
- by (import rich_list EL_REVERSE)
-
-lemma EL_REVERSE_ELL: "ALL (n::nat) l::'a::type list. n < length l --> EL n (rev l) = ELL n l"
- by (import rich_list EL_REVERSE_ELL)
-
-lemma ELL_LENGTH_APPEND: "ALL (l1::'a::type list) l2::'a::type list.
- ~ null l1 --> ELL (length l2) (l1 @ l2) = last l1"
- by (import rich_list ELL_LENGTH_APPEND)
-
-lemma ELL_IS_EL: "ALL (n::nat) l::'a::type list. n < length l --> ELL n l mem l"
- by (import rich_list ELL_IS_EL)
-
-lemma ELL_REVERSE: "ALL (n::nat) l::'a::type list.
- n < length l --> ELL n (rev l) = ELL (PRE (length l - n)) l"
- by (import rich_list ELL_REVERSE)
-
-lemma ELL_REVERSE_EL: "ALL (n::nat) l::'a::type list. n < length l --> ELL n (rev l) = EL n l"
- by (import rich_list ELL_REVERSE_EL)
-
-lemma FIRSTN_BUTLASTN: "ALL (n::nat) l::'a::type list.
- n <= length l --> FIRSTN n l = BUTLASTN (length l - n) l"
- by (import rich_list FIRSTN_BUTLASTN)
-
-lemma BUTLASTN_FIRSTN: "ALL (n::nat) l::'a::type list.
- n <= length l --> BUTLASTN n l = FIRSTN (length l - n) l"
- by (import rich_list BUTLASTN_FIRSTN)
-
-lemma LASTN_BUTFIRSTN: "ALL (n::nat) l::'a::type list.
- n <= length l --> LASTN n l = BUTFIRSTN (length l - n) l"
- by (import rich_list LASTN_BUTFIRSTN)
-
-lemma BUTFIRSTN_LASTN: "ALL (n::nat) l::'a::type list.
- n <= length l --> BUTFIRSTN n l = LASTN (length l - n) l"
- by (import rich_list BUTFIRSTN_LASTN)
-
-lemma SEG_LASTN_BUTLASTN: "ALL (n::nat) (m::nat) l::'a::type list.
- n + m <= length l -->
- SEG n m l = LASTN n (BUTLASTN (length l - (n + m)) l)"
- by (import rich_list SEG_LASTN_BUTLASTN)
-
-lemma BUTFIRSTN_REVERSE: "ALL (n::nat) l::'a::type list.
- n <= length l --> BUTFIRSTN n (rev l) = rev (BUTLASTN n l)"
- by (import rich_list BUTFIRSTN_REVERSE)
-
-lemma BUTLASTN_REVERSE: "ALL (n::nat) l::'a::type list.
- n <= length l --> BUTLASTN n (rev l) = rev (BUTFIRSTN n l)"
- by (import rich_list BUTLASTN_REVERSE)
-
-lemma LASTN_REVERSE: "ALL (n::nat) l::'a::type list.
- n <= length l --> LASTN n (rev l) = rev (FIRSTN n l)"
- by (import rich_list LASTN_REVERSE)
-
-lemma FIRSTN_REVERSE: "ALL (n::nat) l::'a::type list.
- n <= length l --> FIRSTN n (rev l) = rev (LASTN n l)"
- by (import rich_list FIRSTN_REVERSE)
-
-lemma SEG_REVERSE: "ALL (n::nat) (m::nat) l::'a::type list.
- n + m <= length l -->
- SEG n m (rev l) = rev (SEG n (length l - (n + m)) l)"
- by (import rich_list SEG_REVERSE)
-
-lemma LENGTH_GENLIST: "ALL (f::nat => 'a::type) n::nat. length (GENLIST f n) = n"
- by (import rich_list LENGTH_GENLIST)
-
-lemma LENGTH_REPLICATE: "ALL (n::nat) x::'a::type. length (REPLICATE n x) = n"
- by (import rich_list LENGTH_REPLICATE)
-
-lemma IS_EL_REPLICATE: "ALL n>0. ALL x::'a::type. x mem REPLICATE n x"
- by (import rich_list IS_EL_REPLICATE)
-
-lemma ALL_EL_REPLICATE: "ALL (x::'a::type) n::nat. list_all (op = x) (REPLICATE n x)"
- by (import rich_list ALL_EL_REPLICATE)
-
-lemma AND_EL_FOLDL: "ALL l::bool list. AND_EL l = foldl op & True l"
- by (import rich_list AND_EL_FOLDL)
-
-lemma AND_EL_FOLDR: "ALL l::bool list. AND_EL l = foldr op & l True"
- by (import rich_list AND_EL_FOLDR)
-
-lemma OR_EL_FOLDL: "ALL l::bool list. OR_EL l = foldl op | False l"
- by (import rich_list OR_EL_FOLDL)
-
-lemma OR_EL_FOLDR: "ALL l::bool list. OR_EL l = foldr op | l False"
- by (import rich_list OR_EL_FOLDR)
+ sorry
+
+lemma IS_PREFIX_PREFIX: "IS_PREFIX l (PREFIX P l)"
+ sorry
+
+lemma LENGTH_SCANL: "length (SCANL (f::'b => 'a => 'b) (e::'b) (l::'a list)) = Suc (length l)"
+ sorry
+
+lemma LENGTH_SCANR: "length (SCANR (f::'a => 'b => 'b) (e::'b) (l::'a list)) = Suc (length l)"
+ sorry
+
+lemma COMM_MONOID_FOLDL: "[| COMM x; MONOID x xa |] ==> foldl x e l = x e (foldl x xa l)"
+ sorry
+
+lemma COMM_MONOID_FOLDR: "[| COMM x; MONOID x xa |] ==> foldr x l e = x e (foldr x l xa)"
+ sorry
+
+lemma FCOMM_FOLDR_APPEND: "[| FCOMM x xa; LEFT_ID x xb |]
+==> foldr xa (l1 @ l2) xb = x (foldr xa l1 xb) (foldr xa l2 xb)"
+ sorry
+
+lemma FCOMM_FOLDL_APPEND: "[| FCOMM x xa; RIGHT_ID xa xb |]
+==> foldl x xb (l1 @ l2) = xa (foldl x xb l1) (foldl x xb l2)"
+ sorry
+
+lemma FOLDL_SINGLE: "foldl x xa [xb] = x xa xb"
+ sorry
+
+lemma FOLDR_SINGLE: "foldr (x::'a => 'b => 'b) [xb::'a] (xa::'b) = x xb xa"
+ sorry
+
+lemma FOLDR_CONS_NIL: "foldr op # l [] = l"
+ sorry
+
+lemma FOLDL_SNOC_NIL: "foldl (%xs x. SNOC x xs) [] l = l"
+ sorry
+
+lemma FOLDR_REVERSE: "foldr (x::'a => 'b => 'b) (rev (xb::'a list)) (xa::'b) =
+foldl (%(xa::'b) y::'a. x y xa) xa xb"
+ sorry
+
+lemma FOLDL_REVERSE: "foldl x xa (rev xb) = foldr (%xa y. x y xa) xb xa"
+ sorry
+
+lemma FOLDR_MAP: "foldr (f::'a => 'a => 'a) (map (g::'b => 'a) (l::'b list)) (e::'a) =
+foldr (%x::'b. f (g x)) l e"
+ sorry
+
+lemma ALL_EL_FOLDR: "list_all P l = foldr (%x. op & (P x)) l True"
+ sorry
+
+lemma ALL_EL_FOLDL: "list_all P l = foldl (%l' x. l' & P x) True l"
+ sorry
+
+lemma SOME_EL_FOLDR: "list_ex P l = foldr (%x. op | (P x)) l False"
+ sorry
+
+lemma SOME_EL_FOLDL: "list_ex P l = foldl (%l' x. l' | P x) False l"
+ sorry
+
+lemma ALL_EL_FOLDR_MAP: "list_all x xa = foldr op & (map x xa) True"
+ sorry
+
+lemma ALL_EL_FOLDL_MAP: "list_all x xa = foldl op & True (map x xa)"
+ sorry
+
+lemma SOME_EL_FOLDR_MAP: "list_ex x xa = foldr op | (map x xa) False"
+ sorry
+
+lemma SOME_EL_FOLDL_MAP: "list_ex x xa = foldl op | False (map x xa)"
+ sorry
+
+lemma FOLDR_FILTER: "foldr (f::'a => 'a => 'a) (filter (P::'a => bool) (l::'a list)) (e::'a) =
+foldr (%(x::'a) y::'a. if P x then f x y else y) l e"
+ sorry
+
+lemma FOLDL_FILTER: "foldl (f::'a => 'a => 'a) (e::'a) (filter (P::'a => bool) (l::'a list)) =
+foldl (%(x::'a) y::'a. if P y then f x y else x) e l"
+ sorry
+
+lemma ASSOC_FOLDR_FLAT: "[| ASSOC f; LEFT_ID f e |]
+==> foldr f (concat l) e = foldr f (map (FOLDR f e) l) e"
+ sorry
+
+lemma ASSOC_FOLDL_FLAT: "[| ASSOC f; RIGHT_ID f e |]
+==> foldl f e (concat l) = foldl f e (map (foldl f e) l)"
+ sorry
+
+lemma SOME_EL_MAP: "list_ex (P::'b => bool) (map (f::'a => 'b) (l::'a list)) = list_ex (P o f) l"
+ sorry
+
+lemma SOME_EL_DISJ: "list_ex (%x. P x | Q x) l = (list_ex P l | list_ex Q l)"
+ sorry
+
+lemma IS_EL_FOLDR: "List.member xa x = foldr (%xa. op | (x = xa)) xa False"
+ sorry
+
+lemma IS_EL_FOLDL: "List.member xa x = foldl (%l' xa. l' | x = xa) False xa"
+ sorry
+
+lemma NULL_FOLDR: "List.null l = foldr (%x l'. False) l True"
+ sorry
+
+lemma NULL_FOLDL: "List.null l = foldl (%x l'. False) True l"
+ sorry
+
+lemma SEG_LENGTH_ID: "SEG (length l) 0 l = l"
+ sorry
+
+lemma SEG_SUC_CONS: "SEG m (Suc n) (x # l) = SEG m n l"
+ sorry
+
+lemma SEG_0_SNOC: "m <= length l ==> SEG m 0 (SNOC x l) = SEG m 0 l"
+ sorry
+
+lemma BUTLASTN_SEG: "n <= length l ==> BUTLASTN n l = SEG (length l - n) 0 l"
+ sorry
+
+lemma LASTN_CONS: "n <= length l ==> LASTN n (x # l) = LASTN n l"
+ sorry
+
+lemma LENGTH_LASTN: "n <= length l ==> length (LASTN n l) = n"
+ sorry
+
+lemma LASTN_LENGTH_ID: "LASTN (length l) l = l"
+ sorry
+
+lemma LASTN_LASTN: "[| m <= length l; n <= m |] ==> LASTN n (LASTN m l) = LASTN n l"
+ sorry
+
+lemma FIRSTN_LENGTH_ID: "FIRSTN (length l) l = l"
+ sorry
+
+lemma FIRSTN_SNOC: "n <= length l ==> FIRSTN n (SNOC x l) = FIRSTN n l"
+ sorry
+
+lemma BUTLASTN_LENGTH_NIL: "BUTLASTN (length l) l = []"
+ sorry
+
+lemma BUTLASTN_SUC_BUTLAST: "n < length l ==> BUTLASTN (Suc n) l = BUTLASTN n (butlast l)"
+ sorry
+
+lemma BUTLASTN_BUTLAST: "n < length l ==> BUTLASTN n (butlast l) = butlast (BUTLASTN n l)"
+ sorry
+
+lemma LENGTH_BUTLASTN: "n <= length l ==> length (BUTLASTN n l) = length l - n"
+ sorry
+
+lemma BUTLASTN_BUTLASTN: "n + m <= length l ==> BUTLASTN n (BUTLASTN m l) = BUTLASTN (n + m) l"
+ sorry
+
+lemma APPEND_BUTLASTN_LASTN: "n <= length l ==> BUTLASTN n l @ LASTN n l = l"
+ sorry
+
+lemma APPEND_FIRSTN_LASTN: "m + n = length l ==> FIRSTN n l @ LASTN m l = l"
+ sorry
+
+lemma BUTLASTN_APPEND2: "n <= length l2 ==> BUTLASTN n (l1 @ l2) = l1 @ BUTLASTN n l2"
+ sorry
+
+lemma BUTLASTN_LENGTH_APPEND: "BUTLASTN (length l2) (l1 @ l2) = l1"
+ sorry
+
+lemma LASTN_LENGTH_APPEND: "LASTN (length l2) (l1 @ l2) = l2"
+ sorry
+
+lemma BUTLASTN_CONS: "n <= length l ==> BUTLASTN n (x # l) = x # BUTLASTN n l"
+ sorry
+
+lemma BUTLASTN_LENGTH_CONS: "BUTLASTN (length l) (x # l) = [x]"
+ sorry
+
+lemma LAST_LASTN_LAST: "[| n <= length l; 0 < n |] ==> last (LASTN n l) = last l"
+ sorry
+
+lemma BUTLASTN_LASTN_NIL: "n <= length l ==> BUTLASTN n (LASTN n l) = []"
+ sorry
+
+lemma LASTN_BUTLASTN: "n + m <= length l ==> LASTN n (BUTLASTN m l) = BUTLASTN m (LASTN (n + m) l)"
+ sorry
+
+lemma BUTLASTN_LASTN: "m <= n & n <= length l
+==> BUTLASTN m (LASTN n l) = LASTN (n - m) (BUTLASTN m l)"
+ sorry
+
+lemma LASTN_1: "l ~= [] ==> LASTN 1 l = [last l]"
+ sorry
+
+lemma BUTLASTN_1: "l ~= [] ==> BUTLASTN 1 l = butlast l"
+ sorry
+
+lemma BUTLASTN_APPEND1: "length l2 <= n ==> BUTLASTN n (l1 @ l2) = BUTLASTN (n - length l2) l1"
+ sorry
+
+lemma LASTN_APPEND2: "n <= length l2 ==> LASTN n (l1 @ l2) = LASTN n l2"
+ sorry
+
+lemma LASTN_APPEND1: "length l2 <= n ==> LASTN n (l1 @ l2) = LASTN (n - length l2) l1 @ l2"
+ sorry
+
+lemma LASTN_MAP: "n <= length l ==> LASTN n (map f l) = map f (LASTN n l)"
+ sorry
+
+lemma BUTLASTN_MAP: "n <= length l ==> BUTLASTN n (map f l) = map f (BUTLASTN n l)"
+ sorry
+
+lemma ALL_EL_LASTN: "[| list_all P l; m <= length l |] ==> list_all P (LASTN m l)"
+ sorry
+
+lemma ALL_EL_BUTLASTN: "[| list_all P l; m <= length l |] ==> list_all P (BUTLASTN m l)"
+ sorry
+
+lemma LENGTH_FIRSTN: "n <= length l ==> length (FIRSTN n l) = n"
+ sorry
+
+lemma FIRSTN_FIRSTN: "[| m <= length l; n <= m |] ==> FIRSTN n (FIRSTN m l) = FIRSTN n l"
+ sorry
+
+lemma LENGTH_BUTFIRSTN: "n <= length l ==> length (BUTFIRSTN n l) = length l - n"
+ sorry
+
+lemma BUTFIRSTN_LENGTH_NIL: "BUTFIRSTN (length l) l = []"
+ sorry
+
+lemma BUTFIRSTN_APPEND1: "n <= length l1 ==> BUTFIRSTN n (l1 @ l2) = BUTFIRSTN n l1 @ l2"
+ sorry
+
+lemma BUTFIRSTN_APPEND2: "length l1 <= n ==> BUTFIRSTN n (l1 @ l2) = BUTFIRSTN (n - length l1) l2"
+ sorry
+
+lemma BUTFIRSTN_BUTFIRSTN: "n + m <= length l ==> BUTFIRSTN n (BUTFIRSTN m l) = BUTFIRSTN (n + m) l"
+ sorry
+
+lemma APPEND_FIRSTN_BUTFIRSTN: "n <= length l ==> FIRSTN n l @ BUTFIRSTN n l = l"
+ sorry
+
+lemma LASTN_SEG: "n <= length l ==> LASTN n l = SEG n (length l - n) l"
+ sorry
+
+lemma FIRSTN_SEG: "n <= length l ==> FIRSTN n l = SEG n 0 l"
+ sorry
+
+lemma BUTFIRSTN_SEG: "n <= length l ==> BUTFIRSTN n l = SEG (length l - n) n l"
+ sorry
+
+lemma BUTFIRSTN_SNOC: "n <= length l ==> BUTFIRSTN n (SNOC x l) = SNOC x (BUTFIRSTN n l)"
+ sorry
+
+lemma APPEND_BUTLASTN_BUTFIRSTN: "m + n = length l ==> BUTLASTN m l @ BUTFIRSTN n l = l"
+ sorry
+
+lemma SEG_SEG: "n1 + m1 <= length l & n2 + m2 <= n1
+==> SEG n2 m2 (SEG n1 m1 l) = SEG n2 (m1 + m2) l"
+ sorry
+
+lemma SEG_APPEND1: "n + m <= length l1 ==> SEG n m (l1 @ l2) = SEG n m l1"
+ sorry
+
+lemma SEG_APPEND2: "length l1 <= m & n <= length l2
+==> SEG n m (l1 @ l2) = SEG n (m - length l1) l2"
+ sorry
+
+lemma SEG_FIRSTN_BUTFISTN: "n + m <= length l ==> SEG n m l = FIRSTN n (BUTFIRSTN m l)"
+ sorry
+
+lemma SEG_APPEND: "m < length l1 & length l1 <= n + m & n + m <= length l1 + length l2
+==> SEG n m (l1 @ l2) =
+ SEG (length l1 - m) m l1 @ SEG (n + m - length l1) 0 l2"
+ sorry
+
+lemma SEG_LENGTH_SNOC: "SEG 1 (length x) (SNOC xa x) = [xa]"
+ sorry
+
+lemma SEG_SNOC: "n + m <= length l ==> SEG n m (SNOC x l) = SEG n m l"
+ sorry
+
+lemma ELL_SEG: "n < length l ==> ELL n l = hd (SEG 1 (PRE (length l - n)) l)"
+ sorry
+
+lemma SNOC_FOLDR: "SNOC x l = foldr op # l [x]"
+ sorry
+
+lemma IS_EL_FOLDR_MAP: "List.member xa x = foldr op | (map (op = x) xa) False"
+ sorry
+
+lemma IS_EL_FOLDL_MAP: "List.member xa x = foldl op | False (map (op = x) xa)"
+ sorry
+
+lemma FILTER_FILTER: "filter P (filter Q l) = [x<-l. P x & Q x]"
+ sorry
+
+lemma FCOMM_FOLDR_FLAT: "[| FCOMM g f; LEFT_ID g e |]
+==> foldr f (concat l) e = foldr g (map (FOLDR f e) l) e"
+ sorry
+
+lemma FCOMM_FOLDL_FLAT: "[| FCOMM f g; RIGHT_ID g e |]
+==> foldl f e (concat l) = foldl g e (map (foldl f e) l)"
+ sorry
+
+lemma FOLDR_MAP_REVERSE: "(!!(a::'a) (b::'a) c::'a. (f::'a => 'a => 'a) a (f b c) = f b (f a c))
+==> foldr f (map (g::'b => 'a) (rev (l::'b list))) (e::'a) =
+ foldr f (map g l) e"
+ sorry
+
+lemma FOLDR_FILTER_REVERSE: "(!!(a::'a) (b::'a) c::'a. (f::'a => 'a => 'a) a (f b c) = f b (f a c))
+==> foldr f (filter (P::'a => bool) (rev (l::'a list))) (e::'a) =
+ foldr f (filter P l) e"
+ sorry
+
+lemma COMM_ASSOC_FOLDR_REVERSE: "[| COMM f; ASSOC f |] ==> foldr f (rev l) e = foldr f l e"
+ sorry
+
+lemma COMM_ASSOC_FOLDL_REVERSE: "[| COMM f; ASSOC f |] ==> foldl f e (rev l) = foldl f e l"
+ sorry
+
+lemma ELL_LAST: "~ List.null l ==> ELL 0 l = last l"
+ sorry
+
+lemma ELL_0_SNOC: "ELL 0 (SNOC x l) = x"
+ sorry
+
+lemma ELL_SNOC: "0 < n ==> ELL n (SNOC x l) = ELL (PRE n) l"
+ sorry
+
+lemma ELL_SUC_SNOC: "ELL (Suc n) (SNOC x xa) = ELL n xa"
+ sorry
+
+lemma ELL_CONS: "n < length l ==> ELL n (x # l) = ELL n l"
+ sorry
+
+lemma ELL_LENGTH_CONS: "ELL (length l) (x # l) = x"
+ sorry
+
+lemma ELL_LENGTH_SNOC: "ELL (length l) (SNOC x l) = (if List.null l then x else hd l)"
+ sorry
+
+lemma ELL_APPEND2: "n < length l2 ==> ELL n (l1 @ l2) = ELL n l2"
+ sorry
+
+lemma ELL_APPEND1: "length l2 <= n ==> ELL n (l1 @ l2) = ELL (n - length l2) l1"
+ sorry
+
+lemma ELL_PRE_LENGTH: "l ~= [] ==> ELL (PRE (length l)) l = hd l"
+ sorry
+
+lemma EL_LENGTH_SNOC: "EL (length l) (SNOC x l) = x"
+ sorry
+
+lemma EL_PRE_LENGTH: "l ~= [] ==> EL (PRE (length l)) l = last l"
+ sorry
+
+lemma EL_SNOC: "n < length l ==> EL n (SNOC x l) = EL n l"
+ sorry
+
+lemma EL_ELL: "n < length l ==> EL n l = ELL (PRE (length l - n)) l"
+ sorry
+
+lemma EL_LENGTH_APPEND: "~ List.null l2 ==> EL (length l1) (l1 @ l2) = hd l2"
+ sorry
+
+lemma ELL_EL: "n < length l ==> ELL n l = EL (PRE (length l - n)) l"
+ sorry
+
+lemma ELL_MAP: "n < length l ==> ELL n (map f l) = f (ELL n l)"
+ sorry
+
+lemma LENGTH_BUTLAST: "l ~= [] ==> length (butlast l) = PRE (length l)"
+ sorry
+
+lemma BUTFIRSTN_LENGTH_APPEND: "BUTFIRSTN (length l1) (l1 @ l2) = l2"
+ sorry
+
+lemma FIRSTN_APPEND1: "n <= length l1 ==> FIRSTN n (l1 @ l2) = FIRSTN n l1"
+ sorry
+
+lemma FIRSTN_APPEND2: "length l1 <= n ==> FIRSTN n (l1 @ l2) = l1 @ FIRSTN (n - length l1) l2"
+ sorry
+
+lemma FIRSTN_LENGTH_APPEND: "FIRSTN (length l1) (l1 @ l2) = l1"
+ sorry
+
+lemma REVERSE_FLAT: "rev (concat l) = concat (rev (map rev l))"
+ sorry
+
+lemma MAP_FILTER: "(!!x. P (f x) = P x) ==> map f (filter P l) = filter P (map f l)"
+ sorry
+
+lemma FLAT_REVERSE: "concat (rev l) = rev (concat (map rev l))"
+ sorry
+
+lemma FLAT_FLAT: "concat (concat l) = concat (map concat l)"
+ sorry
+
+lemma ALL_EL_SEG: "[| list_all P l; m + k <= length l |] ==> list_all P (SEG m k l)"
+ sorry
+
+lemma ALL_EL_FIRSTN: "[| list_all P l; m <= length l |] ==> list_all P (FIRSTN m l)"
+ sorry
+
+lemma ALL_EL_BUTFIRSTN: "[| list_all P l; m <= length l |] ==> list_all P (BUTFIRSTN m l)"
+ sorry
+
+lemma SOME_EL_SEG: "[| m + k <= length l; list_ex P (SEG m k l) |] ==> list_ex P l"
+ sorry
+
+lemma SOME_EL_FIRSTN: "[| m <= length l; list_ex P (FIRSTN m l) |] ==> list_ex P l"
+ sorry
+
+lemma SOME_EL_BUTFIRSTN: "[| m <= length l; list_ex P (BUTFIRSTN m l) |] ==> list_ex P l"
+ sorry
+
+lemma SOME_EL_LASTN: "[| m <= length l; list_ex P (LASTN m l) |] ==> list_ex P l"
+ sorry
+
+lemma SOME_EL_BUTLASTN: "[| m <= length l; list_ex P (BUTLASTN m l) |] ==> list_ex P l"
+ sorry
+
+lemma IS_EL_REVERSE: "List.member (rev l) x = List.member l x"
+ sorry
+
+lemma IS_EL_FILTER: "P x ==> List.member (filter P l) x = List.member l x"
+ sorry
+
+lemma IS_EL_SEG: "[| n + m <= length l; List.member (SEG n m l) x |] ==> List.member l x"
+ sorry
+
+lemma IS_EL_SOME_EL: "List.member l x = list_ex (op = x) l"
+ sorry
+
+lemma IS_EL_FIRSTN: "[| x <= length xa; List.member (FIRSTN x xa) xb |] ==> List.member xa xb"
+ sorry
+
+lemma IS_EL_BUTFIRSTN: "[| x <= length xa; List.member (BUTFIRSTN x xa) xb |] ==> List.member xa xb"
+ sorry
+
+lemma IS_EL_BUTLASTN: "[| x <= length xa; List.member (BUTLASTN x xa) xb |] ==> List.member xa xb"
+ sorry
+
+lemma IS_EL_LASTN: "[| x <= length xa; List.member (LASTN x xa) xb |] ==> List.member xa xb"
+ sorry
+
+lemma ZIP_SNOC: "length l1 = length l2
+==> zip (SNOC x1 l1) (SNOC x2 l2) = SNOC (x1, x2) (zip l1 l2)"
+ sorry
+
+lemma UNZIP_SNOC: "unzip (SNOC x l) =
+(SNOC (fst x) (fst (unzip l)), SNOC (snd x) (snd (unzip l)))"
+ sorry
+
+lemma LENGTH_UNZIP_FST: "length (UNZIP_FST x) = length x"
+ sorry
+
+lemma LENGTH_UNZIP_SND: "length (UNZIP_SND (x::('a * 'b) list)) = length x"
+ sorry
+
+lemma SUM_APPEND: "HOL4Compat.sum (l1 @ l2) = HOL4Compat.sum l1 + HOL4Compat.sum l2"
+ sorry
+
+lemma SUM_REVERSE: "HOL4Compat.sum (rev l) = HOL4Compat.sum l"
+ sorry
+
+lemma SUM_FLAT: "HOL4Compat.sum (concat l) = HOL4Compat.sum (map HOL4Compat.sum l)"
+ sorry
+
+lemma EL_APPEND1: "n < length l1 ==> EL n (l1 @ l2) = EL n l1"
+ sorry
+
+lemma EL_APPEND2: "length l1 <= n ==> EL n (l1 @ l2) = EL (n - length l1) l2"
+ sorry
+
+lemma EL_MAP: "n < length l ==> EL n (map f l) = f (EL n l)"
+ sorry
+
+lemma EL_CONS: "0 < n ==> EL n (x # l) = EL (PRE n) l"
+ sorry
+
+lemma EL_SEG: "n < length l ==> EL n l = hd (SEG 1 n l)"
+ sorry
+
+lemma EL_IS_EL: "n < length l ==> List.member l (EL n l)"
+ sorry
+
+lemma TL_SNOC: "tl (SNOC x l) = (if List.null l then [] else SNOC x (tl l))"
+ sorry
+
+lemma EL_REVERSE: "n < length l ==> EL n (rev l) = EL (PRE (length l - n)) l"
+ sorry
+
+lemma EL_REVERSE_ELL: "n < length l ==> EL n (rev l) = ELL n l"
+ sorry
+
+lemma ELL_LENGTH_APPEND: "~ List.null l1 ==> ELL (length l2) (l1 @ l2) = last l1"
+ sorry
+
+lemma ELL_IS_EL: "n < length l ==> List.member l (ELL n l)"
+ sorry
+
+lemma ELL_REVERSE: "n < length l ==> ELL n (rev l) = ELL (PRE (length l - n)) l"
+ sorry
+
+lemma ELL_REVERSE_EL: "n < length l ==> ELL n (rev l) = EL n l"
+ sorry
+
+lemma FIRSTN_BUTLASTN: "n <= length l ==> FIRSTN n l = BUTLASTN (length l - n) l"
+ sorry
+
+lemma BUTLASTN_FIRSTN: "n <= length l ==> BUTLASTN n l = FIRSTN (length l - n) l"
+ sorry
+
+lemma LASTN_BUTFIRSTN: "n <= length l ==> LASTN n l = BUTFIRSTN (length l - n) l"
+ sorry
+
+lemma BUTFIRSTN_LASTN: "n <= length l ==> BUTFIRSTN n l = LASTN (length l - n) l"
+ sorry
+
+lemma SEG_LASTN_BUTLASTN: "n + m <= length l ==> SEG n m l = LASTN n (BUTLASTN (length l - (n + m)) l)"
+ sorry
+
+lemma BUTFIRSTN_REVERSE: "n <= length l ==> BUTFIRSTN n (rev l) = rev (BUTLASTN n l)"
+ sorry
+
+lemma BUTLASTN_REVERSE: "n <= length l ==> BUTLASTN n (rev l) = rev (BUTFIRSTN n l)"
+ sorry
+
+lemma LASTN_REVERSE: "n <= length l ==> LASTN n (rev l) = rev (FIRSTN n l)"
+ sorry
+
+lemma FIRSTN_REVERSE: "n <= length l ==> FIRSTN n (rev l) = rev (LASTN n l)"
+ sorry
+
+lemma SEG_REVERSE: "n + m <= length l ==> SEG n m (rev l) = rev (SEG n (length l - (n + m)) l)"
+ sorry
+
+lemma LENGTH_GENLIST: "length (GENLIST f n) = n"
+ sorry
+
+lemma LENGTH_REPLICATE: "length (REPLICATE n x) = n"
+ sorry
+
+lemma IS_EL_REPLICATE: "0 < n ==> List.member (REPLICATE n x) x"
+ sorry
+
+lemma ALL_EL_REPLICATE: "list_all (op = x) (REPLICATE n x)"
+ sorry
+
+lemma AND_EL_FOLDL: "AND_EL l = foldl op & True l"
+ sorry
+
+lemma AND_EL_FOLDR: "AND_EL l = foldr op & l True"
+ sorry
+
+lemma OR_EL_FOLDL: "OR_EL l = foldl op | False l"
+ sorry
+
+lemma OR_EL_FOLDR: "OR_EL l = foldr op | l False"
+ sorry
;end_setup
;setup_theory state_transformer
-definition UNIT :: "'b => 'a => 'b * 'a" where
+definition
+ UNIT :: "'b => 'a => 'b * 'a" where
"(op ==::('b::type => 'a::type => 'b::type * 'a::type)
=> ('b::type => 'a::type => 'b::type * 'a::type) => prop)
(UNIT::'b::type => 'a::type => 'b::type * 'a::type)
(Pair::'b::type => 'a::type => 'b::type * 'a::type)"
-lemma UNIT_DEF: "ALL x::'b::type. UNIT x = Pair x"
- by (import state_transformer UNIT_DEF)
-
-definition BIND :: "('a => 'b * 'a) => ('b => 'a => 'c * 'a) => 'a => 'c * 'a" where
- "(op ==::(('a::type => 'b::type * 'a::type)
- => ('b::type => 'a::type => 'c::type * 'a::type)
- => 'a::type => 'c::type * 'a::type)
- => (('a::type => 'b::type * 'a::type)
- => ('b::type => 'a::type => 'c::type * 'a::type)
- => 'a::type => 'c::type * 'a::type)
- => prop)
- (BIND::('a::type => 'b::type * 'a::type)
- => ('b::type => 'a::type => 'c::type * 'a::type)
- => 'a::type => 'c::type * 'a::type)
- (%(g::'a::type => 'b::type * 'a::type)
- f::'b::type => 'a::type => 'c::type * 'a::type.
- (op o::('b::type * 'a::type => 'c::type * 'a::type)
- => ('a::type => 'b::type * 'a::type)
- => 'a::type => 'c::type * 'a::type)
- ((split::('b::type => 'a::type => 'c::type * 'a::type)
- => 'b::type * 'a::type => 'c::type * 'a::type)
- f)
- g)"
-
-lemma BIND_DEF: "(All::(('a::type => 'b::type * 'a::type) => bool) => bool)
- (%g::'a::type => 'b::type * 'a::type.
- (All::(('b::type => 'a::type => 'c::type * 'a::type) => bool) => bool)
- (%f::'b::type => 'a::type => 'c::type * 'a::type.
- (op =::('a::type => 'c::type * 'a::type)
- => ('a::type => 'c::type * 'a::type) => bool)
- ((BIND::('a::type => 'b::type * 'a::type)
- => ('b::type => 'a::type => 'c::type * 'a::type)
- => 'a::type => 'c::type * 'a::type)
- g f)
- ((op o::('b::type * 'a::type => 'c::type * 'a::type)
- => ('a::type => 'b::type * 'a::type)
- => 'a::type => 'c::type * 'a::type)
- ((split::('b::type => 'a::type => 'c::type * 'a::type)
- => 'b::type * 'a::type => 'c::type * 'a::type)
- f)
- g)))"
- by (import state_transformer BIND_DEF)
-
-definition MMAP :: "('c => 'b) => ('a => 'c * 'a) => 'a => 'b * 'a" where
- "MMAP ==
-%(f::'c::type => 'b::type) m::'a::type => 'c::type * 'a::type.
- BIND m (UNIT o f)"
-
-lemma MMAP_DEF: "ALL (f::'c::type => 'b::type) m::'a::type => 'c::type * 'a::type.
- MMAP f m = BIND m (UNIT o f)"
- by (import state_transformer MMAP_DEF)
-
-definition JOIN :: "('a => ('a => 'b * 'a) * 'a) => 'a => 'b * 'a" where
- "JOIN ==
-%z::'a::type => ('a::type => 'b::type * 'a::type) * 'a::type. BIND z I"
-
-lemma JOIN_DEF: "ALL z::'a::type => ('a::type => 'b::type * 'a::type) * 'a::type.
- JOIN z = BIND z I"
- by (import state_transformer JOIN_DEF)
-
-lemma BIND_LEFT_UNIT: "ALL (k::'a::type => 'b::type => 'c::type * 'b::type) x::'a::type.
- BIND (UNIT x) k = k x"
- by (import state_transformer BIND_LEFT_UNIT)
-
-lemma UNIT_UNCURRY: "ALL x::'a::type * 'b::type. split UNIT x = x"
- by (import state_transformer UNIT_UNCURRY)
-
-lemma BIND_RIGHT_UNIT: "ALL k::'a::type => 'b::type * 'a::type. BIND k UNIT = k"
- by (import state_transformer BIND_RIGHT_UNIT)
-
-lemma BIND_ASSOC: "ALL (x::'a::type => 'b::type * 'a::type)
- (xa::'b::type => 'a::type => 'c::type * 'a::type)
- xb::'c::type => 'a::type => 'd::type * 'a::type.
- BIND x (%a::'b::type. BIND (xa a) xb) = BIND (BIND x xa) xb"
- by (import state_transformer BIND_ASSOC)
+lemma UNIT_DEF: "UNIT x = Pair x"
+ sorry
+
+definition
+ BIND :: "('a => 'b * 'a) => ('b => 'a => 'c * 'a) => 'a => 'c * 'a" where
+ "BIND == %g f. (%(x, y). f x y) o g"
+
+lemma BIND_DEF: "BIND (g::'a => 'b * 'a) (f::'b => 'a => 'c * 'a) =
+(%(x::'b, y::'a). f x y) o g"
+ sorry
+
+definition
+ MMAP :: "('c => 'b) => ('a => 'c * 'a) => 'a => 'b * 'a" where
+ "MMAP == %(f::'c => 'b) m::'a => 'c * 'a. BIND m (UNIT o f)"
+
+lemma MMAP_DEF: "MMAP f m = BIND m (UNIT o f)"
+ sorry
+
+definition
+ JOIN :: "('a => ('a => 'b * 'a) * 'a) => 'a => 'b * 'a" where
+ "JOIN == %z. BIND z I"
+
+lemma JOIN_DEF: "JOIN z = BIND z I"
+ sorry
+
+lemma BIND_LEFT_UNIT: "BIND (UNIT (x::'a)) (k::'a => 'b => 'c * 'b) = k x"
+ sorry
+
+lemma UNIT_UNCURRY: "prod_case UNIT x = x"
+ sorry
+
+lemma BIND_RIGHT_UNIT: "BIND k UNIT = k"
+ sorry
+
+lemma BIND_ASSOC: "BIND (x::'a => 'b * 'a)
+ (%a::'b. BIND ((xa::'b => 'a => 'c * 'a) a) (xb::'c => 'a => 'd * 'a)) =
+BIND (BIND x xa) xb"
+ sorry
lemma MMAP_ID: "MMAP I = I"
- by (import state_transformer MMAP_ID)
-
-lemma MMAP_COMP: "ALL (f::'c::type => 'd::type) g::'b::type => 'c::type.
- MMAP (f o g) = MMAP f o MMAP g"
- by (import state_transformer MMAP_COMP)
-
-lemma MMAP_UNIT: "ALL f::'b::type => 'c::type. MMAP f o UNIT = UNIT o f"
- by (import state_transformer MMAP_UNIT)
-
-lemma MMAP_JOIN: "ALL f::'b::type => 'c::type. MMAP f o JOIN = JOIN o MMAP (MMAP f)"
- by (import state_transformer MMAP_JOIN)
+ sorry
+
+lemma MMAP_COMP: "MMAP ((f::'c => 'd) o (g::'b => 'c)) = MMAP f o MMAP g"
+ sorry
+
+lemma MMAP_UNIT: "MMAP (f::'b => 'c) o UNIT = UNIT o f"
+ sorry
+
+lemma MMAP_JOIN: "MMAP f o JOIN = JOIN o MMAP (MMAP f)"
+ sorry
lemma JOIN_UNIT: "JOIN o UNIT = I"
- by (import state_transformer JOIN_UNIT)
+ sorry
lemma JOIN_MMAP_UNIT: "JOIN o MMAP UNIT = I"
- by (import state_transformer JOIN_MMAP_UNIT)
+ sorry
lemma JOIN_MAP_JOIN: "JOIN o MMAP JOIN = JOIN o JOIN"
- by (import state_transformer JOIN_MAP_JOIN)
-
-lemma JOIN_MAP: "ALL (x::'a::type => 'b::type * 'a::type)
- xa::'b::type => 'a::type => 'c::type * 'a::type.
- BIND x xa = JOIN (MMAP xa x)"
- by (import state_transformer JOIN_MAP)
-
-lemma FST_o_UNIT: "ALL x::'a::type. fst o UNIT x = K x"
- by (import state_transformer FST_o_UNIT)
-
-lemma SND_o_UNIT: "ALL x::'a::type. snd o UNIT x = I"
- by (import state_transformer SND_o_UNIT)
-
-lemma FST_o_MMAP: "ALL (x::'a::type => 'b::type) xa::'c::type => 'a::type * 'c::type.
- fst o MMAP x xa = x o (fst o xa)"
- by (import state_transformer FST_o_MMAP)
+ sorry
+
+lemma JOIN_MAP: "BIND (x::'a => 'b * 'a) (xa::'b => 'a => 'c * 'a) = JOIN (MMAP xa x)"
+ sorry
+
+lemma FST_o_UNIT: "fst o UNIT (x::'a) = K x"
+ sorry
+
+lemma SND_o_UNIT: "snd o UNIT (x::'a) = I"
+ sorry
+
+lemma FST_o_MMAP: "fst o MMAP (x::'a => 'b) (xa::'c => 'a * 'c) = x o (fst o xa)"
+ sorry
;end_setup
--- a/src/HOL/Import/HOL/HOL4Prob.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Import/HOL/HOL4Prob.thy Fri Sep 09 06:47:14 2011 +0200
@@ -4,357 +4,211 @@
;setup_theory prob_extra
-lemma BOOL_BOOL_CASES_THM: "ALL f::bool => bool.
- f = (%b::bool. False) |
- f = (%b::bool. True) | f = (%b::bool. b) | f = Not"
+lemma BOOL_BOOL_CASES_THM: "f = (%b. False) | f = (%b. True) | f = (%b. b) | f = Not"
by (import prob_extra BOOL_BOOL_CASES_THM)
lemma EVEN_ODD_BASIC: "EVEN 0 & ~ EVEN 1 & EVEN 2 & ~ ODD 0 & ODD 1 & ~ ODD 2"
by (import prob_extra EVEN_ODD_BASIC)
-lemma EVEN_ODD_EXISTS_EQ: "ALL n::nat.
- EVEN n = (EX m::nat. n = 2 * m) & ODD n = (EX m::nat. n = Suc (2 * m))"
+lemma EVEN_ODD_EXISTS_EQ: "EVEN n = (EX m. n = 2 * m) & ODD n = (EX m. n = Suc (2 * m))"
by (import prob_extra EVEN_ODD_EXISTS_EQ)
-lemma DIV_THEN_MULT: "ALL (p::nat) q::nat. Suc q * (p div Suc q) <= p"
+lemma DIV_THEN_MULT: "Suc q * (p div Suc q) <= p"
by (import prob_extra DIV_THEN_MULT)
-lemma DIV_TWO_UNIQUE: "ALL (n::nat) (q::nat) r::nat.
- n = 2 * q + r & (r = 0 | r = 1) --> q = n div 2 & r = n mod 2"
+lemma DIV_TWO_UNIQUE: "(n::nat) = (2::nat) * (q::nat) + (r::nat) & (r = (0::nat) | r = (1::nat))
+==> q = n div (2::nat) & r = n mod (2::nat)"
by (import prob_extra DIV_TWO_UNIQUE)
-lemma DIVISION_TWO: "ALL n::nat. n = 2 * (n div 2) + n mod 2 & (n mod 2 = 0 | n mod 2 = 1)"
+lemma DIVISION_TWO: "(n::nat) = (2::nat) * (n div (2::nat)) + n mod (2::nat) &
+(n mod (2::nat) = (0::nat) | n mod (2::nat) = (1::nat))"
by (import prob_extra DIVISION_TWO)
-lemma DIV_TWO: "ALL n::nat. n = 2 * (n div 2) + n mod 2"
+lemma DIV_TWO: "(n::nat) = (2::nat) * (n div (2::nat)) + n mod (2::nat)"
by (import prob_extra DIV_TWO)
-lemma MOD_TWO: "ALL n::nat. n mod 2 = (if EVEN n then 0 else 1)"
+lemma MOD_TWO: "n mod 2 = (if EVEN n then 0 else 1)"
by (import prob_extra MOD_TWO)
-lemma DIV_TWO_BASIC: "(op &::bool => bool => bool)
- ((op =::nat => nat => bool)
- ((op div::nat => nat => nat) (0::nat)
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- (0::nat))
- ((op &::bool => bool => bool)
- ((op =::nat => nat => bool)
- ((op div::nat => nat => nat) (1::nat)
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- (0::nat))
- ((op =::nat => nat => bool)
- ((op div::nat => nat => nat)
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int))))
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- (1::nat)))"
+lemma DIV_TWO_BASIC: "(0::nat) div (2::nat) = (0::nat) &
+(1::nat) div (2::nat) = (0::nat) & (2::nat) div (2::nat) = (1::nat)"
by (import prob_extra DIV_TWO_BASIC)
-lemma DIV_TWO_MONO: "ALL (m::nat) n::nat. m div 2 < n div 2 --> m < n"
+lemma DIV_TWO_MONO: "(m::nat) div (2::nat) < (n::nat) div (2::nat) ==> m < n"
by (import prob_extra DIV_TWO_MONO)
-lemma DIV_TWO_MONO_EVEN: "ALL (m::nat) n::nat. EVEN n --> (m div 2 < n div 2) = (m < n)"
+lemma DIV_TWO_MONO_EVEN: "EVEN n ==> (m div 2 < n div 2) = (m < n)"
by (import prob_extra DIV_TWO_MONO_EVEN)
-lemma DIV_TWO_CANCEL: "ALL n::nat. 2 * n div 2 = n & Suc (2 * n) div 2 = n"
+lemma DIV_TWO_CANCEL: "2 * n div 2 = n & Suc (2 * n) div 2 = n"
by (import prob_extra DIV_TWO_CANCEL)
-lemma EXP_DIV_TWO: "(All::(nat => bool) => bool)
- (%n::nat.
- (op =::nat => nat => bool)
- ((op div::nat => nat => nat)
- ((op ^::nat => nat => nat)
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int))))
- ((Suc::nat => nat) n))
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- ((op ^::nat => nat => nat)
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int))))
- n))"
+lemma EXP_DIV_TWO: "(2::nat) ^ Suc (n::nat) div (2::nat) = (2::nat) ^ n"
by (import prob_extra EXP_DIV_TWO)
-lemma EVEN_EXP_TWO: "ALL n::nat. EVEN (2 ^ n) = (n ~= 0)"
+lemma EVEN_EXP_TWO: "EVEN (2 ^ n) = (n ~= 0)"
by (import prob_extra EVEN_EXP_TWO)
-lemma DIV_TWO_EXP: "ALL (n::nat) k::nat. (k div 2 < 2 ^ n) = (k < 2 ^ Suc n)"
+lemma DIV_TWO_EXP: "((k::nat) div (2::nat) < (2::nat) ^ (n::nat)) = (k < (2::nat) ^ Suc n)"
by (import prob_extra DIV_TWO_EXP)
consts
inf :: "(real => bool) => real"
defs
- inf_primdef: "inf == %P::real => bool. - sup (IMAGE uminus P)"
+ inf_primdef: "prob_extra.inf == %P. - real.sup (IMAGE uminus P)"
-lemma inf_def: "ALL P::real => bool. inf P = - sup (IMAGE uminus P)"
+lemma inf_def: "prob_extra.inf P = - real.sup (IMAGE uminus P)"
by (import prob_extra inf_def)
-lemma INF_DEF_ALT: "ALL P::real => bool. inf P = - sup (%r::real. P (- r))"
+lemma INF_DEF_ALT: "prob_extra.inf P = - real.sup (%r. P (- r))"
by (import prob_extra INF_DEF_ALT)
-lemma REAL_SUP_EXISTS_UNIQUE: "ALL P::real => bool.
- Ex P & (EX z::real. ALL x::real. P x --> x <= z) -->
- (EX! s::real. ALL y::real. (EX x::real. P x & y < x) = (y < s))"
+lemma REAL_SUP_EXISTS_UNIQUE: "Ex (P::real => bool) & (EX z::real. ALL x::real. P x --> x <= z)
+==> EX! s::real. ALL y::real. (EX x::real. P x & y < x) = (y < s)"
by (import prob_extra REAL_SUP_EXISTS_UNIQUE)
-lemma REAL_SUP_MAX: "ALL (P::real => bool) z::real.
- P z & (ALL x::real. P x --> x <= z) --> sup P = z"
+lemma REAL_SUP_MAX: "P z & (ALL x. P x --> x <= z) ==> real.sup P = z"
by (import prob_extra REAL_SUP_MAX)
-lemma REAL_INF_MIN: "ALL (P::real => bool) z::real.
- P z & (ALL x::real. P x --> z <= x) --> inf P = z"
+lemma REAL_INF_MIN: "P z & (ALL x. P x --> z <= x) ==> prob_extra.inf P = z"
by (import prob_extra REAL_INF_MIN)
-lemma HALF_POS: "(op <::real => real => bool) (0::real)
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))"
- by (import prob_extra HALF_POS)
-
-lemma HALF_CANCEL: "(op =::real => real => bool)
- ((op *::real => real => real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int))))
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int))))))
- (1::real)"
+lemma HALF_CANCEL: "(2::real) * ((1::real) / (2::real)) = (1::real)"
by (import prob_extra HALF_CANCEL)
-lemma POW_HALF_POS: "(All::(nat => bool) => bool)
- (%n::nat.
- (op <::real => real => bool) (0::real)
- ((op ^::real => nat => real)
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- n))"
+lemma POW_HALF_POS: "(0::real) < ((1::real) / (2::real)) ^ (n::nat)"
by (import prob_extra POW_HALF_POS)
-lemma POW_HALF_MONO: "(All::(nat => bool) => bool)
- (%m::nat.
- (All::(nat => bool) => bool)
- (%n::nat.
- (op -->::bool => bool => bool) ((op <=::nat => nat => bool) m n)
- ((op <=::real => real => bool)
- ((op ^::real => nat => real)
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- n)
- ((op ^::real => nat => real)
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- m))))"
+lemma POW_HALF_MONO: "(m::nat) <= (n::nat)
+==> ((1::real) / (2::real)) ^ n <= ((1::real) / (2::real)) ^ m"
by (import prob_extra POW_HALF_MONO)
-lemma POW_HALF_TWICE: "(All::(nat => bool) => bool)
- (%n::nat.
- (op =::real => real => bool)
- ((op ^::real => nat => real)
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- n)
- ((op *::real => real => real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int))))
- ((op ^::real => nat => real)
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- ((Suc::nat => nat) n))))"
+lemma POW_HALF_TWICE: "((1::real) / (2::real)) ^ (n::nat) =
+(2::real) * ((1::real) / (2::real)) ^ Suc n"
by (import prob_extra POW_HALF_TWICE)
-lemma X_HALF_HALF: "ALL x::real. 1 / 2 * x + 1 / 2 * x = x"
+lemma X_HALF_HALF: "(1::real) / (2::real) * (x::real) + (1::real) / (2::real) * x = x"
by (import prob_extra X_HALF_HALF)
-lemma REAL_SUP_LE_X: "ALL (P::real => bool) x::real.
- Ex P & (ALL r::real. P r --> r <= x) --> sup P <= x"
+lemma REAL_SUP_LE_X: "Ex P & (ALL r. P r --> r <= x) ==> real.sup P <= x"
by (import prob_extra REAL_SUP_LE_X)
-lemma REAL_X_LE_SUP: "ALL (P::real => bool) x::real.
- Ex P &
- (EX z::real. ALL r::real. P r --> r <= z) &
- (EX r::real. P r & x <= r) -->
- x <= sup P"
+lemma REAL_X_LE_SUP: "Ex P & (EX z. ALL r. P r --> r <= z) & (EX r. P r & x <= r)
+==> x <= real.sup P"
by (import prob_extra REAL_X_LE_SUP)
-lemma ABS_BETWEEN_LE: "ALL (x::real) (y::real) d::real.
- (0 <= d & x - d <= y & y <= x + d) = (abs (y - x) <= d)"
+lemma ABS_BETWEEN_LE: "((0::real) <= (d::real) & (x::real) - d <= (y::real) & y <= x + d) =
+(abs (y - x) <= d)"
by (import prob_extra ABS_BETWEEN_LE)
-lemma ONE_MINUS_HALF: "(op =::real => real => bool)
- ((op -::real => real => real) (1::real)
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int))))))
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))"
+lemma ONE_MINUS_HALF: "(1::real) - (1::real) / (2::real) = (1::real) / (2::real)"
by (import prob_extra ONE_MINUS_HALF)
-lemma HALF_LT_1: "(op <::real => real => bool)
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- (1::real)"
+lemma HALF_LT_1: "(1::real) / (2::real) < (1::real)"
by (import prob_extra HALF_LT_1)
-lemma POW_HALF_EXP: "(All::(nat => bool) => bool)
- (%n::nat.
- (op =::real => real => bool)
- ((op ^::real => nat => real)
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- n)
- ((inverse::real => real)
- ((real::nat => real)
- ((op ^::nat => nat => nat)
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int))))
- n))))"
+lemma POW_HALF_EXP: "((1::real) / (2::real)) ^ (n::nat) = inverse (real ((2::nat) ^ n))"
by (import prob_extra POW_HALF_EXP)
-lemma INV_SUC_POS: "ALL n::nat. 0 < 1 / real (Suc n)"
+lemma INV_SUC_POS: "0 < 1 / real (Suc n)"
by (import prob_extra INV_SUC_POS)
-lemma INV_SUC_MAX: "ALL x::nat. 1 / real (Suc x) <= 1"
+lemma INV_SUC_MAX: "1 / real (Suc x) <= 1"
by (import prob_extra INV_SUC_MAX)
-lemma INV_SUC: "ALL n::nat. 0 < 1 / real (Suc n) & 1 / real (Suc n) <= 1"
+lemma INV_SUC: "0 < 1 / real (Suc n) & 1 / real (Suc n) <= 1"
by (import prob_extra INV_SUC)
-lemma ABS_UNIT_INTERVAL: "ALL (x::real) y::real.
- 0 <= x & x <= 1 & 0 <= y & y <= 1 --> abs (x - y) <= 1"
+lemma ABS_UNIT_INTERVAL: "(0::real) <= (x::real) &
+x <= (1::real) & (0::real) <= (y::real) & y <= (1::real)
+==> abs (x - y) <= (1::real)"
by (import prob_extra ABS_UNIT_INTERVAL)
-lemma MEM_NIL: "ALL l::'a::type list. (ALL x::'a::type. ~ x mem l) = (l = [])"
+lemma MEM_NIL: "(ALL x. ~ List.member l x) = (l = [])"
by (import prob_extra MEM_NIL)
-lemma MAP_MEM: "ALL (f::'a::type => 'b::type) (l::'a::type list) x::'b::type.
- x mem map f l = (EX y::'a::type. y mem l & x = f y)"
+lemma MAP_MEM: "List.member (map (f::'a => 'b) (l::'a list)) (x::'b) =
+(EX y::'a. List.member l y & x = f y)"
by (import prob_extra MAP_MEM)
-lemma MEM_NIL_MAP_CONS: "ALL (x::'a::type) l::'a::type list list. ~ [] mem map (op # x) l"
+lemma MEM_NIL_MAP_CONS: "~ List.member (map (op # x) l) []"
by (import prob_extra MEM_NIL_MAP_CONS)
-lemma FILTER_TRUE: "ALL l::'a::type list. [x::'a::type<-l. True] = l"
+lemma FILTER_TRUE: "[x<-l. True] = l"
by (import prob_extra FILTER_TRUE)
-lemma FILTER_FALSE: "ALL l::'a::type list. [x::'a::type<-l. False] = []"
+lemma FILTER_FALSE: "[x<-l. False] = []"
by (import prob_extra FILTER_FALSE)
-lemma FILTER_MEM: "ALL (P::'a::type => bool) (x::'a::type) l::'a::type list.
- x mem filter P l --> P x"
+lemma FILTER_MEM: "List.member (filter P l) x ==> P x"
by (import prob_extra FILTER_MEM)
-lemma MEM_FILTER: "ALL (P::'a::type => bool) (l::'a::type list) x::'a::type.
- x mem filter P l --> x mem l"
+lemma MEM_FILTER: "List.member (filter P l) x ==> List.member l x"
by (import prob_extra MEM_FILTER)
-lemma FILTER_OUT_ELT: "ALL (x::'a::type) l::'a::type list. x mem l | [y::'a::type<-l. y ~= x] = l"
+lemma FILTER_OUT_ELT: "List.member l x | [y<-l. y ~= x] = l"
by (import prob_extra FILTER_OUT_ELT)
-lemma IS_PREFIX_NIL: "ALL x::'a::type list. IS_PREFIX x [] & IS_PREFIX [] x = (x = [])"
+lemma IS_PREFIX_NIL: "IS_PREFIX x [] & IS_PREFIX [] x = (x = [])"
by (import prob_extra IS_PREFIX_NIL)
-lemma IS_PREFIX_REFL: "ALL x::'a::type list. IS_PREFIX x x"
+lemma IS_PREFIX_REFL: "IS_PREFIX x x"
by (import prob_extra IS_PREFIX_REFL)
-lemma IS_PREFIX_ANTISYM: "ALL (x::'a::type list) y::'a::type list.
- IS_PREFIX y x & IS_PREFIX x y --> x = y"
+lemma IS_PREFIX_ANTISYM: "IS_PREFIX y x & IS_PREFIX x y ==> x = y"
by (import prob_extra IS_PREFIX_ANTISYM)
-lemma IS_PREFIX_TRANS: "ALL (x::'a::type list) (y::'a::type list) z::'a::type list.
- IS_PREFIX x y & IS_PREFIX y z --> IS_PREFIX x z"
+lemma IS_PREFIX_TRANS: "IS_PREFIX x y & IS_PREFIX y z ==> IS_PREFIX x z"
by (import prob_extra IS_PREFIX_TRANS)
-lemma IS_PREFIX_BUTLAST: "ALL (x::'a::type) y::'a::type list. IS_PREFIX (x # y) (butlast (x # y))"
+lemma IS_PREFIX_BUTLAST: "IS_PREFIX (x # y) (butlast (x # y))"
by (import prob_extra IS_PREFIX_BUTLAST)
-lemma IS_PREFIX_LENGTH: "ALL (x::'a::type list) y::'a::type list.
- IS_PREFIX y x --> length x <= length y"
+lemma IS_PREFIX_LENGTH: "IS_PREFIX y x ==> length x <= length y"
by (import prob_extra IS_PREFIX_LENGTH)
-lemma IS_PREFIX_LENGTH_ANTI: "ALL (x::'a::type list) y::'a::type list.
- IS_PREFIX y x & length x = length y --> x = y"
+lemma IS_PREFIX_LENGTH_ANTI: "IS_PREFIX y x & length x = length y ==> x = y"
by (import prob_extra IS_PREFIX_LENGTH_ANTI)
-lemma IS_PREFIX_SNOC: "ALL (x::'a::type) (y::'a::type list) z::'a::type list.
- IS_PREFIX (SNOC x y) z = (IS_PREFIX y z | z = SNOC x y)"
+lemma IS_PREFIX_SNOC: "IS_PREFIX (SNOC x y) z = (IS_PREFIX y z | z = SNOC x y)"
by (import prob_extra IS_PREFIX_SNOC)
-lemma FOLDR_MAP: "ALL (f::'b::type => 'c::type => 'c::type) (e::'c::type)
- (g::'a::type => 'b::type) l::'a::type list.
- foldr f (map g l) e = foldr (%x::'a::type. f (g x)) l e"
+lemma FOLDR_MAP: "foldr (f::'b => 'c => 'c) (map (g::'a => 'b) (l::'a list)) (e::'c) =
+foldr (%x::'a. f (g x)) l e"
by (import prob_extra FOLDR_MAP)
-lemma LAST_MEM: "ALL (h::'a::type) t::'a::type list. last (h # t) mem h # t"
+lemma LAST_MEM: "List.member (h # t) (last (h # t))"
by (import prob_extra LAST_MEM)
-lemma LAST_MAP_CONS: "ALL (b::bool) (h::bool list) t::bool list list.
- EX x::bool list. last (map (op # b) (h # t)) = b # x"
+lemma LAST_MAP_CONS: "EX x::bool list.
+ last (map (op # (b::bool)) ((h::bool list) # (t::bool list list))) =
+ b # x"
by (import prob_extra LAST_MAP_CONS)
-lemma EXISTS_LONGEST: "ALL (x::'a::type list) y::'a::type list list.
- EX z::'a::type list.
- z mem x # y &
- (ALL w::'a::type list. w mem x # y --> length w <= length z)"
+lemma EXISTS_LONGEST: "EX z. List.member (x # y) z &
+ (ALL w. List.member (x # y) w --> length w <= length z)"
by (import prob_extra EXISTS_LONGEST)
-lemma UNION_DEF_ALT: "ALL (s::'a::type => bool) t::'a::type => bool.
- pred_set.UNION s t = (%x::'a::type. s x | t x)"
+lemma UNION_DEF_ALT: "pred_set.UNION s t = (%x. s x | t x)"
by (import prob_extra UNION_DEF_ALT)
-lemma INTER_UNION_RDISTRIB: "ALL (p::'a::type => bool) (q::'a::type => bool) r::'a::type => bool.
- pred_set.INTER (pred_set.UNION p q) r =
- pred_set.UNION (pred_set.INTER p r) (pred_set.INTER q r)"
+lemma INTER_UNION_RDISTRIB: "pred_set.INTER (pred_set.UNION p q) r =
+pred_set.UNION (pred_set.INTER p r) (pred_set.INTER q r)"
by (import prob_extra INTER_UNION_RDISTRIB)
-lemma SUBSET_EQ: "ALL (x::'a::type => bool) xa::'a::type => bool.
- (x = xa) = (SUBSET x xa & SUBSET xa x)"
+lemma SUBSET_EQ: "(x = xa) = (SUBSET x xa & SUBSET xa x)"
by (import prob_extra SUBSET_EQ)
-lemma INTER_IS_EMPTY: "ALL (s::'a::type => bool) t::'a::type => bool.
- (pred_set.INTER s t = EMPTY) = (ALL x::'a::type. ~ s x | ~ t x)"
+lemma INTER_IS_EMPTY: "(pred_set.INTER s t = EMPTY) = (ALL x. ~ s x | ~ t x)"
by (import prob_extra INTER_IS_EMPTY)
-lemma UNION_DISJOINT_SPLIT: "ALL (s::'a::type => bool) (t::'a::type => bool) u::'a::type => bool.
- pred_set.UNION s t = pred_set.UNION s u &
- pred_set.INTER s t = EMPTY & pred_set.INTER s u = EMPTY -->
- t = u"
+lemma UNION_DISJOINT_SPLIT: "pred_set.UNION s t = pred_set.UNION s u &
+pred_set.INTER s t = EMPTY & pred_set.INTER s u = EMPTY
+==> t = u"
by (import prob_extra UNION_DISJOINT_SPLIT)
-lemma GSPEC_DEF_ALT: "ALL f::'a::type => 'b::type * bool.
- GSPEC f = (%v::'b::type. EX x::'a::type. (v, True) = f x)"
+lemma GSPEC_DEF_ALT: "GSPEC (f::'a => 'b * bool) = (%v::'b. EX x::'a. (v, True) = f x)"
by (import prob_extra GSPEC_DEF_ALT)
;end_setup
@@ -365,158 +219,56 @@
alg_twin :: "bool list => bool list => bool"
defs
- alg_twin_primdef: "alg_twin ==
-%(x::bool list) y::bool list.
- EX l::bool list. x = SNOC True l & y = SNOC False l"
+ alg_twin_primdef: "alg_twin == %x y. EX l. x = SNOC True l & y = SNOC False l"
-lemma alg_twin_def: "ALL (x::bool list) y::bool list.
- alg_twin x y = (EX l::bool list. x = SNOC True l & y = SNOC False l)"
+lemma alg_twin_def: "alg_twin x y = (EX l. x = SNOC True l & y = SNOC False l)"
by (import prob_canon alg_twin_def)
-definition alg_order_tupled :: "bool list * bool list => bool" where
- "(op ==::(bool list * bool list => bool)
- => (bool list * bool list => bool) => prop)
- (alg_order_tupled::bool list * bool list => bool)
- ((WFREC::(bool list * bool list => bool list * bool list => bool)
- => ((bool list * bool list => bool)
- => bool list * bool list => bool)
- => bool list * bool list => bool)
- ((Eps::((bool list * bool list => bool list * bool list => bool) => bool)
- => bool list * bool list => bool list * bool list => bool)
- (%R::bool list * bool list => bool list * bool list => bool.
- (op &::bool => bool => bool)
- ((WF::(bool list * bool list => bool list * bool list => bool)
- => bool)
- R)
- ((All::(bool => bool) => bool)
- (%h'::bool.
- (All::(bool => bool) => bool)
- (%h::bool.
- (All::(bool list => bool) => bool)
- (%t'::bool list.
- (All::(bool list => bool) => bool)
- (%t::bool list.
- R ((Pair::bool list
- => bool list => bool list * bool list)
- t t')
- ((Pair::bool list
- => bool list => bool list * bool list)
- ((op #::bool => bool list => bool list) h
- t)
- ((op #::bool => bool list => bool list) h'
- t')))))))))
- (%alg_order_tupled::bool list * bool list => bool.
- (split::(bool list => bool list => bool)
- => bool list * bool list => bool)
- (%(v::bool list) v1::bool list.
- (list_case::bool
- => (bool => bool list => bool) => bool list => bool)
- ((list_case::bool
- => (bool => bool list => bool)
- => bool list => bool)
- (True::bool) (%(v8::bool) v9::bool list. True::bool) v1)
- (%(v4::bool) v5::bool list.
- (list_case::bool
- => (bool => bool list => bool)
- => bool list => bool)
- (False::bool)
- (%(v10::bool) v11::bool list.
- (op |::bool => bool => bool)
- ((op &::bool => bool => bool)
- ((op =::bool => bool => bool) v4 (True::bool))
- ((op =::bool => bool => bool) v10 (False::bool)))
- ((op &::bool => bool => bool)
- ((op =::bool => bool => bool) v4 v10)
- (alg_order_tupled
- ((Pair::bool list
- => bool list => bool list * bool list)
- v5 v11))))
- v1)
- v)))"
+definition
+ alg_order_tupled :: "bool list * bool list => bool" where
+ "alg_order_tupled ==
+WFREC (SOME R. WF R & (ALL h' h t' t. R (t, t') (h # t, h' # t')))
+ (%alg_order_tupled (v, v1).
+ case v of [] => case v1 of [] => True | _ => True
+ | v4 # v5 =>
+ case v1 of [] => False
+ | v10 # v11 =>
+ v4 = True & v10 = False |
+ v4 = v10 & alg_order_tupled (v5, v11))"
-lemma alg_order_tupled_primitive_def: "(op =::(bool list * bool list => bool)
- => (bool list * bool list => bool) => bool)
- (alg_order_tupled::bool list * bool list => bool)
- ((WFREC::(bool list * bool list => bool list * bool list => bool)
- => ((bool list * bool list => bool)
- => bool list * bool list => bool)
- => bool list * bool list => bool)
- ((Eps::((bool list * bool list => bool list * bool list => bool) => bool)
- => bool list * bool list => bool list * bool list => bool)
- (%R::bool list * bool list => bool list * bool list => bool.
- (op &::bool => bool => bool)
- ((WF::(bool list * bool list => bool list * bool list => bool)
- => bool)
- R)
- ((All::(bool => bool) => bool)
- (%h'::bool.
- (All::(bool => bool) => bool)
- (%h::bool.
- (All::(bool list => bool) => bool)
- (%t'::bool list.
- (All::(bool list => bool) => bool)
- (%t::bool list.
- R ((Pair::bool list
- => bool list => bool list * bool list)
- t t')
- ((Pair::bool list
- => bool list => bool list * bool list)
- ((op #::bool => bool list => bool list) h
- t)
- ((op #::bool => bool list => bool list) h'
- t')))))))))
- (%alg_order_tupled::bool list * bool list => bool.
- (split::(bool list => bool list => bool)
- => bool list * bool list => bool)
- (%(v::bool list) v1::bool list.
- (list_case::bool
- => (bool => bool list => bool) => bool list => bool)
- ((list_case::bool
- => (bool => bool list => bool)
- => bool list => bool)
- (True::bool) (%(v8::bool) v9::bool list. True::bool) v1)
- (%(v4::bool) v5::bool list.
- (list_case::bool
- => (bool => bool list => bool)
- => bool list => bool)
- (False::bool)
- (%(v10::bool) v11::bool list.
- (op |::bool => bool => bool)
- ((op &::bool => bool => bool)
- ((op =::bool => bool => bool) v4 (True::bool))
- ((op =::bool => bool => bool) v10 (False::bool)))
- ((op &::bool => bool => bool)
- ((op =::bool => bool => bool) v4 v10)
- (alg_order_tupled
- ((Pair::bool list
- => bool list => bool list * bool list)
- v5 v11))))
- v1)
- v)))"
+lemma alg_order_tupled_primitive_def: "alg_order_tupled =
+WFREC (SOME R. WF R & (ALL h' h t' t. R (t, t') (h # t, h' # t')))
+ (%alg_order_tupled (v, v1).
+ case v of [] => case v1 of [] => True | _ => True
+ | v4 # v5 =>
+ case v1 of [] => False
+ | v10 # v11 =>
+ v4 = True & v10 = False |
+ v4 = v10 & alg_order_tupled (v5, v11))"
by (import prob_canon alg_order_tupled_primitive_def)
consts
alg_order :: "bool list => bool list => bool"
defs
- alg_order_primdef: "alg_order == %(x::bool list) x1::bool list. alg_order_tupled (x, x1)"
+ alg_order_primdef: "alg_order == %x x1. alg_order_tupled (x, x1)"
-lemma alg_order_curried_def: "ALL (x::bool list) x1::bool list. alg_order x x1 = alg_order_tupled (x, x1)"
+lemma alg_order_curried_def: "alg_order x x1 = alg_order_tupled (x, x1)"
by (import prob_canon alg_order_curried_def)
-lemma alg_order_ind: "ALL P::bool list => bool list => bool.
- (ALL (x::bool) xa::bool list. P [] (x # xa)) &
- P [] [] &
- (ALL (x::bool) xa::bool list. P (x # xa) []) &
- (ALL (x::bool) (xa::bool list) (xb::bool) xc::bool list.
- P xa xc --> P (x # xa) (xb # xc)) -->
- (ALL x::bool list. All (P x))"
+lemma alg_order_ind: "(ALL (x::bool) xa::bool list.
+ (P::bool list => bool list => bool) [] (x # xa)) &
+P [] [] &
+(ALL (x::bool) xa::bool list. P (x # xa) []) &
+(ALL (x::bool) (xa::bool list) (xb::bool) xc::bool list.
+ P xa xc --> P (x # xa) (xb # xc))
+==> P (x::bool list) (xa::bool list)"
by (import prob_canon alg_order_ind)
-lemma alg_order_def: "alg_order [] ((v6::bool) # (v7::bool list)) = True &
+lemma alg_order_def: "alg_order [] (v6 # v7) = True &
alg_order [] [] = True &
-alg_order ((v2::bool) # (v3::bool list)) [] = False &
-alg_order ((h::bool) # (t::bool list)) ((h'::bool) # (t'::bool list)) =
+alg_order (v2 # v3) [] = False &
+alg_order (h # t) (h' # t') =
(h = True & h' = False | h = h' & alg_order t t')"
by (import prob_canon alg_order_def)
@@ -525,42 +277,28 @@
defs
alg_sorted_primdef: "alg_sorted ==
-WFREC
- (SOME R::bool list list => bool list list => bool.
- WF R &
- (ALL (x::bool list) (z::bool list list) y::bool list.
- R (y # z) (x # y # z)))
- (%alg_sorted::bool list list => bool.
+WFREC (SOME R. WF R & (ALL x z y. R (y # z) (x # y # z)))
+ (%alg_sorted.
list_case True
- (%v2::bool list.
- list_case True
- (%(v6::bool list) v7::bool list list.
- alg_order v2 v6 & alg_sorted (v6 # v7))))"
+ (%v2. list_case True
+ (%v6 v7. alg_order v2 v6 & alg_sorted (v6 # v7))))"
lemma alg_sorted_primitive_def: "alg_sorted =
-WFREC
- (SOME R::bool list list => bool list list => bool.
- WF R &
- (ALL (x::bool list) (z::bool list list) y::bool list.
- R (y # z) (x # y # z)))
- (%alg_sorted::bool list list => bool.
+WFREC (SOME R. WF R & (ALL x z y. R (y # z) (x # y # z)))
+ (%alg_sorted.
list_case True
- (%v2::bool list.
- list_case True
- (%(v6::bool list) v7::bool list list.
- alg_order v2 v6 & alg_sorted (v6 # v7))))"
+ (%v2. list_case True
+ (%v6 v7. alg_order v2 v6 & alg_sorted (v6 # v7))))"
by (import prob_canon alg_sorted_primitive_def)
-lemma alg_sorted_ind: "ALL P::bool list list => bool.
- (ALL (x::bool list) (y::bool list) z::bool list list.
- P (y # z) --> P (x # y # z)) &
- (ALL v::bool list. P [v]) & P [] -->
- All P"
+lemma alg_sorted_ind: "(ALL (x::bool list) (y::bool list) z::bool list list.
+ (P::bool list list => bool) (y # z) --> P (x # y # z)) &
+(ALL v::bool list. P [v]) & P []
+==> P (x::bool list list)"
by (import prob_canon alg_sorted_ind)
-lemma alg_sorted_def: "alg_sorted ((x::bool list) # (y::bool list) # (z::bool list list)) =
-(alg_order x y & alg_sorted (y # z)) &
-alg_sorted [v::bool list] = True & alg_sorted [] = True"
+lemma alg_sorted_def: "alg_sorted (x # y # z) = (alg_order x y & alg_sorted (y # z)) &
+alg_sorted [v] = True & alg_sorted [] = True"
by (import prob_canon alg_sorted_def)
consts
@@ -568,42 +306,28 @@
defs
alg_prefixfree_primdef: "alg_prefixfree ==
-WFREC
- (SOME R::bool list list => bool list list => bool.
- WF R &
- (ALL (x::bool list) (z::bool list list) y::bool list.
- R (y # z) (x # y # z)))
- (%alg_prefixfree::bool list list => bool.
+WFREC (SOME R. WF R & (ALL x z y. R (y # z) (x # y # z)))
+ (%alg_prefixfree.
list_case True
- (%v2::bool list.
- list_case True
- (%(v6::bool list) v7::bool list list.
- ~ IS_PREFIX v6 v2 & alg_prefixfree (v6 # v7))))"
+ (%v2. list_case True
+ (%v6 v7. ~ IS_PREFIX v6 v2 & alg_prefixfree (v6 # v7))))"
lemma alg_prefixfree_primitive_def: "alg_prefixfree =
-WFREC
- (SOME R::bool list list => bool list list => bool.
- WF R &
- (ALL (x::bool list) (z::bool list list) y::bool list.
- R (y # z) (x # y # z)))
- (%alg_prefixfree::bool list list => bool.
+WFREC (SOME R. WF R & (ALL x z y. R (y # z) (x # y # z)))
+ (%alg_prefixfree.
list_case True
- (%v2::bool list.
- list_case True
- (%(v6::bool list) v7::bool list list.
- ~ IS_PREFIX v6 v2 & alg_prefixfree (v6 # v7))))"
+ (%v2. list_case True
+ (%v6 v7. ~ IS_PREFIX v6 v2 & alg_prefixfree (v6 # v7))))"
by (import prob_canon alg_prefixfree_primitive_def)
-lemma alg_prefixfree_ind: "ALL P::bool list list => bool.
- (ALL (x::bool list) (y::bool list) z::bool list list.
- P (y # z) --> P (x # y # z)) &
- (ALL v::bool list. P [v]) & P [] -->
- All P"
+lemma alg_prefixfree_ind: "(ALL (x::bool list) (y::bool list) z::bool list list.
+ (P::bool list list => bool) (y # z) --> P (x # y # z)) &
+(ALL v::bool list. P [v]) & P []
+==> P (x::bool list list)"
by (import prob_canon alg_prefixfree_ind)
-lemma alg_prefixfree_def: "alg_prefixfree ((x::bool list) # (y::bool list) # (z::bool list list)) =
-(~ IS_PREFIX y x & alg_prefixfree (y # z)) &
-alg_prefixfree [v::bool list] = True & alg_prefixfree [] = True"
+lemma alg_prefixfree_def: "alg_prefixfree (x # y # z) = (~ IS_PREFIX y x & alg_prefixfree (y # z)) &
+alg_prefixfree [v] = True & alg_prefixfree [] = True"
by (import prob_canon alg_prefixfree_def)
consts
@@ -611,60 +335,44 @@
defs
alg_twinfree_primdef: "alg_twinfree ==
-WFREC
- (SOME R::bool list list => bool list list => bool.
- WF R &
- (ALL (x::bool list) (z::bool list list) y::bool list.
- R (y # z) (x # y # z)))
- (%alg_twinfree::bool list list => bool.
+WFREC (SOME R. WF R & (ALL x z y. R (y # z) (x # y # z)))
+ (%alg_twinfree.
list_case True
- (%v2::bool list.
- list_case True
- (%(v6::bool list) v7::bool list list.
- ~ alg_twin v2 v6 & alg_twinfree (v6 # v7))))"
+ (%v2. list_case True
+ (%v6 v7. ~ alg_twin v2 v6 & alg_twinfree (v6 # v7))))"
lemma alg_twinfree_primitive_def: "alg_twinfree =
-WFREC
- (SOME R::bool list list => bool list list => bool.
- WF R &
- (ALL (x::bool list) (z::bool list list) y::bool list.
- R (y # z) (x # y # z)))
- (%alg_twinfree::bool list list => bool.
+WFREC (SOME R. WF R & (ALL x z y. R (y # z) (x # y # z)))
+ (%alg_twinfree.
list_case True
- (%v2::bool list.
- list_case True
- (%(v6::bool list) v7::bool list list.
- ~ alg_twin v2 v6 & alg_twinfree (v6 # v7))))"
+ (%v2. list_case True
+ (%v6 v7. ~ alg_twin v2 v6 & alg_twinfree (v6 # v7))))"
by (import prob_canon alg_twinfree_primitive_def)
-lemma alg_twinfree_ind: "ALL P::bool list list => bool.
- (ALL (x::bool list) (y::bool list) z::bool list list.
- P (y # z) --> P (x # y # z)) &
- (ALL v::bool list. P [v]) & P [] -->
- All P"
+lemma alg_twinfree_ind: "(ALL (x::bool list) (y::bool list) z::bool list list.
+ (P::bool list list => bool) (y # z) --> P (x # y # z)) &
+(ALL v::bool list. P [v]) & P []
+==> P (x::bool list list)"
by (import prob_canon alg_twinfree_ind)
-lemma alg_twinfree_def: "alg_twinfree ((x::bool list) # (y::bool list) # (z::bool list list)) =
-(~ alg_twin x y & alg_twinfree (y # z)) &
-alg_twinfree [v::bool list] = True & alg_twinfree [] = True"
+lemma alg_twinfree_def: "alg_twinfree (x # y # z) = (~ alg_twin x y & alg_twinfree (y # z)) &
+alg_twinfree [v] = True & alg_twinfree [] = True"
by (import prob_canon alg_twinfree_def)
consts
alg_longest :: "bool list list => nat"
defs
- alg_longest_primdef: "alg_longest ==
-FOLDR (%(h::bool list) t::nat. if t <= length h then length h else t) 0"
+ alg_longest_primdef: "alg_longest == FOLDR (%h t. if t <= length h then length h else t) 0"
-lemma alg_longest_def: "alg_longest =
-FOLDR (%(h::bool list) t::nat. if t <= length h then length h else t) 0"
+lemma alg_longest_def: "alg_longest = FOLDR (%h t. if t <= length h then length h else t) 0"
by (import prob_canon alg_longest_def)
consts
alg_canon_prefs :: "bool list => bool list list => bool list list"
-specification (alg_canon_prefs_primdef: alg_canon_prefs) alg_canon_prefs_def: "(ALL l::bool list. alg_canon_prefs l [] = [l]) &
-(ALL (l::bool list) (h::bool list) t::bool list list.
+specification (alg_canon_prefs_primdef: alg_canon_prefs) alg_canon_prefs_def: "(ALL l. alg_canon_prefs l [] = [l]) &
+(ALL l h t.
alg_canon_prefs l (h # t) =
(if IS_PREFIX h l then alg_canon_prefs l t else l # h # t))"
by (import prob_canon alg_canon_prefs_def)
@@ -672,8 +380,8 @@
consts
alg_canon_find :: "bool list => bool list list => bool list list"
-specification (alg_canon_find_primdef: alg_canon_find) alg_canon_find_def: "(ALL l::bool list. alg_canon_find l [] = [l]) &
-(ALL (l::bool list) (h::bool list) t::bool list list.
+specification (alg_canon_find_primdef: alg_canon_find) alg_canon_find_def: "(ALL l. alg_canon_find l [] = [l]) &
+(ALL l h t.
alg_canon_find l (h # t) =
(if alg_order h l
then if IS_PREFIX l h then h # t else h # alg_canon_find l t
@@ -692,8 +400,8 @@
consts
alg_canon_merge :: "bool list => bool list list => bool list list"
-specification (alg_canon_merge_primdef: alg_canon_merge) alg_canon_merge_def: "(ALL l::bool list. alg_canon_merge l [] = [l]) &
-(ALL (l::bool list) (h::bool list) t::bool list list.
+specification (alg_canon_merge_primdef: alg_canon_merge) alg_canon_merge_def: "(ALL l. alg_canon_merge l [] = [l]) &
+(ALL l h t.
alg_canon_merge l (h # t) =
(if alg_twin l h then alg_canon_merge (butlast h) t else l # h # t))"
by (import prob_canon alg_canon_merge_def)
@@ -711,365 +419,308 @@
alg_canon :: "bool list list => bool list list"
defs
- alg_canon_primdef: "alg_canon == %l::bool list list. alg_canon2 (alg_canon1 l)"
+ alg_canon_primdef: "alg_canon == %l. alg_canon2 (alg_canon1 l)"
-lemma alg_canon_def: "ALL l::bool list list. alg_canon l = alg_canon2 (alg_canon1 l)"
+lemma alg_canon_def: "alg_canon l = alg_canon2 (alg_canon1 l)"
by (import prob_canon alg_canon_def)
consts
algebra_canon :: "bool list list => bool"
defs
- algebra_canon_primdef: "algebra_canon == %l::bool list list. alg_canon l = l"
+ algebra_canon_primdef: "algebra_canon == %l. alg_canon l = l"
-lemma algebra_canon_def: "ALL l::bool list list. algebra_canon l = (alg_canon l = l)"
+lemma algebra_canon_def: "algebra_canon l = (alg_canon l = l)"
by (import prob_canon algebra_canon_def)
-lemma ALG_TWIN_NIL: "ALL l::bool list. ~ alg_twin l [] & ~ alg_twin [] l"
+lemma ALG_TWIN_NIL: "~ alg_twin l [] & ~ alg_twin [] l"
by (import prob_canon ALG_TWIN_NIL)
-lemma ALG_TWIN_SING: "ALL (x::bool) l::bool list.
- alg_twin [x] l = (x = True & l = [False]) &
- alg_twin l [x] = (l = [True] & x = False)"
+lemma ALG_TWIN_SING: "alg_twin [x] l = (x = True & l = [False]) &
+alg_twin l [x] = (l = [True] & x = False)"
by (import prob_canon ALG_TWIN_SING)
-lemma ALG_TWIN_CONS: "ALL (x::bool) (y::bool) (z::bool list) (h::bool) t::bool list.
- alg_twin (x # y # z) (h # t) = (x = h & alg_twin (y # z) t) &
- alg_twin (h # t) (x # y # z) = (x = h & alg_twin t (y # z))"
+lemma ALG_TWIN_CONS: "alg_twin (x # y # z) (h # t) = (x = h & alg_twin (y # z) t) &
+alg_twin (h # t) (x # y # z) = (x = h & alg_twin t (y # z))"
by (import prob_canon ALG_TWIN_CONS)
-lemma ALG_TWIN_REDUCE: "ALL (h::bool) (t::bool list) t'::bool list.
- alg_twin (h # t) (h # t') = alg_twin t t'"
+lemma ALG_TWIN_REDUCE: "alg_twin (h # t) (h # t') = alg_twin t t'"
by (import prob_canon ALG_TWIN_REDUCE)
-lemma ALG_TWINS_PREFIX: "ALL (x::bool list) l::bool list.
- IS_PREFIX x l -->
- x = l | IS_PREFIX x (SNOC True l) | IS_PREFIX x (SNOC False l)"
+lemma ALG_TWINS_PREFIX: "IS_PREFIX x l
+==> x = l | IS_PREFIX x (SNOC True l) | IS_PREFIX x (SNOC False l)"
by (import prob_canon ALG_TWINS_PREFIX)
-lemma ALG_ORDER_NIL: "ALL x::bool list. alg_order [] x & alg_order x [] = (x = [])"
+lemma ALG_ORDER_NIL: "alg_order [] x & alg_order x [] = (x = [])"
by (import prob_canon ALG_ORDER_NIL)
-lemma ALG_ORDER_REFL: "ALL x::bool list. alg_order x x"
+lemma ALG_ORDER_REFL: "alg_order x x"
by (import prob_canon ALG_ORDER_REFL)
-lemma ALG_ORDER_ANTISYM: "ALL (x::bool list) y::bool list. alg_order x y & alg_order y x --> x = y"
+lemma ALG_ORDER_ANTISYM: "alg_order x y & alg_order y x ==> x = y"
by (import prob_canon ALG_ORDER_ANTISYM)
-lemma ALG_ORDER_TRANS: "ALL (x::bool list) (y::bool list) z::bool list.
- alg_order x y & alg_order y z --> alg_order x z"
+lemma ALG_ORDER_TRANS: "alg_order x y & alg_order y z ==> alg_order x z"
by (import prob_canon ALG_ORDER_TRANS)
-lemma ALG_ORDER_TOTAL: "ALL (x::bool list) y::bool list. alg_order x y | alg_order y x"
+lemma ALG_ORDER_TOTAL: "alg_order x y | alg_order y x"
by (import prob_canon ALG_ORDER_TOTAL)
-lemma ALG_ORDER_PREFIX: "ALL (x::bool list) y::bool list. IS_PREFIX y x --> alg_order x y"
+lemma ALG_ORDER_PREFIX: "IS_PREFIX y x ==> alg_order x y"
by (import prob_canon ALG_ORDER_PREFIX)
-lemma ALG_ORDER_PREFIX_ANTI: "ALL (x::bool list) y::bool list. alg_order x y & IS_PREFIX x y --> x = y"
+lemma ALG_ORDER_PREFIX_ANTI: "alg_order x y & IS_PREFIX x y ==> x = y"
by (import prob_canon ALG_ORDER_PREFIX_ANTI)
-lemma ALG_ORDER_PREFIX_MONO: "ALL (x::bool list) (y::bool list) z::bool list.
- alg_order x y & alg_order y z & IS_PREFIX z x --> IS_PREFIX y x"
+lemma ALG_ORDER_PREFIX_MONO: "alg_order x y & alg_order y z & IS_PREFIX z x ==> IS_PREFIX y x"
by (import prob_canon ALG_ORDER_PREFIX_MONO)
-lemma ALG_ORDER_PREFIX_TRANS: "ALL (x::bool list) (y::bool list) z::bool list.
- alg_order x y & IS_PREFIX y z --> alg_order x z | IS_PREFIX x z"
+lemma ALG_ORDER_PREFIX_TRANS: "alg_order x y & IS_PREFIX y z ==> alg_order x z | IS_PREFIX x z"
by (import prob_canon ALG_ORDER_PREFIX_TRANS)
-lemma ALG_ORDER_SNOC: "ALL (x::bool) l::bool list. ~ alg_order (SNOC x l) l"
+lemma ALG_ORDER_SNOC: "~ alg_order (SNOC x l) l"
by (import prob_canon ALG_ORDER_SNOC)
-lemma ALG_SORTED_MIN: "ALL (h::bool list) t::bool list list.
- alg_sorted (h # t) --> (ALL x::bool list. x mem t --> alg_order h x)"
+lemma ALG_SORTED_MIN: "[| alg_sorted (h # t); List.member t x |] ==> alg_order h x"
by (import prob_canon ALG_SORTED_MIN)
-lemma ALG_SORTED_DEF_ALT: "ALL (h::bool list) t::bool list list.
- alg_sorted (h # t) =
- ((ALL x::bool list. x mem t --> alg_order h x) & alg_sorted t)"
+lemma ALG_SORTED_DEF_ALT: "alg_sorted (h # t) =
+((ALL x. List.member t x --> alg_order h x) & alg_sorted t)"
by (import prob_canon ALG_SORTED_DEF_ALT)
-lemma ALG_SORTED_TL: "ALL (h::bool list) t::bool list list. alg_sorted (h # t) --> alg_sorted t"
+lemma ALG_SORTED_TL: "alg_sorted (h # t) ==> alg_sorted t"
by (import prob_canon ALG_SORTED_TL)
-lemma ALG_SORTED_MONO: "ALL (x::bool list) (y::bool list) z::bool list list.
- alg_sorted (x # y # z) --> alg_sorted (x # z)"
+lemma ALG_SORTED_MONO: "alg_sorted (x # y # z) ==> alg_sorted (x # z)"
by (import prob_canon ALG_SORTED_MONO)
-lemma ALG_SORTED_TLS: "ALL (l::bool list list) b::bool. alg_sorted (map (op # b) l) = alg_sorted l"
+lemma ALG_SORTED_TLS: "alg_sorted (map (op # b) l) = alg_sorted l"
by (import prob_canon ALG_SORTED_TLS)
-lemma ALG_SORTED_STEP: "ALL (l1::bool list list) l2::bool list list.
- alg_sorted (map (op # True) l1 @ map (op # False) l2) =
- (alg_sorted l1 & alg_sorted l2)"
+lemma ALG_SORTED_STEP: "alg_sorted (map (op # True) l1 @ map (op # False) l2) =
+(alg_sorted l1 & alg_sorted l2)"
by (import prob_canon ALG_SORTED_STEP)
-lemma ALG_SORTED_APPEND: "ALL (h::bool list) (h'::bool list) (t::bool list list) t'::bool list list.
- alg_sorted ((h # t) @ h' # t') =
- (alg_sorted (h # t) & alg_sorted (h' # t') & alg_order (last (h # t)) h')"
+lemma ALG_SORTED_APPEND: "alg_sorted ((h # t) @ h' # t') =
+(alg_sorted (h # t) & alg_sorted (h' # t') & alg_order (last (h # t)) h')"
by (import prob_canon ALG_SORTED_APPEND)
-lemma ALG_SORTED_FILTER: "ALL (P::bool list => bool) b::bool list list.
- alg_sorted b --> alg_sorted (filter P b)"
+lemma ALG_SORTED_FILTER: "alg_sorted b ==> alg_sorted (filter P b)"
by (import prob_canon ALG_SORTED_FILTER)
-lemma ALG_PREFIXFREE_TL: "ALL (h::bool list) t::bool list list.
- alg_prefixfree (h # t) --> alg_prefixfree t"
+lemma ALG_PREFIXFREE_TL: "alg_prefixfree (h # t) ==> alg_prefixfree t"
by (import prob_canon ALG_PREFIXFREE_TL)
-lemma ALG_PREFIXFREE_MONO: "ALL (x::bool list) (y::bool list) z::bool list list.
- alg_sorted (x # y # z) & alg_prefixfree (x # y # z) -->
- alg_prefixfree (x # z)"
+lemma ALG_PREFIXFREE_MONO: "alg_sorted (x # y # z) & alg_prefixfree (x # y # z)
+==> alg_prefixfree (x # z)"
by (import prob_canon ALG_PREFIXFREE_MONO)
-lemma ALG_PREFIXFREE_ELT: "ALL (h::bool list) t::bool list list.
- alg_sorted (h # t) & alg_prefixfree (h # t) -->
- (ALL x::bool list. x mem t --> ~ IS_PREFIX x h & ~ IS_PREFIX h x)"
+lemma ALG_PREFIXFREE_ELT: "[| alg_sorted (h # t) & alg_prefixfree (h # t); List.member t x |]
+==> ~ IS_PREFIX x h & ~ IS_PREFIX h x"
by (import prob_canon ALG_PREFIXFREE_ELT)
-lemma ALG_PREFIXFREE_TLS: "ALL (l::bool list list) b::bool.
- alg_prefixfree (map (op # b) l) = alg_prefixfree l"
+lemma ALG_PREFIXFREE_TLS: "alg_prefixfree (map (op # b) l) = alg_prefixfree l"
by (import prob_canon ALG_PREFIXFREE_TLS)
-lemma ALG_PREFIXFREE_STEP: "ALL (l1::bool list list) l2::bool list list.
- alg_prefixfree (map (op # True) l1 @ map (op # False) l2) =
- (alg_prefixfree l1 & alg_prefixfree l2)"
+lemma ALG_PREFIXFREE_STEP: "alg_prefixfree (map (op # True) l1 @ map (op # False) l2) =
+(alg_prefixfree l1 & alg_prefixfree l2)"
by (import prob_canon ALG_PREFIXFREE_STEP)
-lemma ALG_PREFIXFREE_APPEND: "ALL (h::bool list) (h'::bool list) (t::bool list list) t'::bool list list.
- alg_prefixfree ((h # t) @ h' # t') =
- (alg_prefixfree (h # t) &
- alg_prefixfree (h' # t') & ~ IS_PREFIX h' (last (h # t)))"
+lemma ALG_PREFIXFREE_APPEND: "alg_prefixfree ((h # t) @ h' # t') =
+(alg_prefixfree (h # t) &
+ alg_prefixfree (h' # t') & ~ IS_PREFIX h' (last (h # t)))"
by (import prob_canon ALG_PREFIXFREE_APPEND)
-lemma ALG_PREFIXFREE_FILTER: "ALL (P::bool list => bool) b::bool list list.
- alg_sorted b & alg_prefixfree b --> alg_prefixfree (filter P b)"
+lemma ALG_PREFIXFREE_FILTER: "alg_sorted b & alg_prefixfree b ==> alg_prefixfree (filter P b)"
by (import prob_canon ALG_PREFIXFREE_FILTER)
-lemma ALG_TWINFREE_TL: "ALL (h::bool list) t::bool list list.
- alg_twinfree (h # t) --> alg_twinfree t"
+lemma ALG_TWINFREE_TL: "alg_twinfree (h # t) ==> alg_twinfree t"
by (import prob_canon ALG_TWINFREE_TL)
-lemma ALG_TWINFREE_TLS: "ALL (l::bool list list) b::bool.
- alg_twinfree (map (op # b) l) = alg_twinfree l"
+lemma ALG_TWINFREE_TLS: "alg_twinfree (map (op # b) l) = alg_twinfree l"
by (import prob_canon ALG_TWINFREE_TLS)
-lemma ALG_TWINFREE_STEP1: "ALL (l1::bool list list) l2::bool list list.
- alg_twinfree (map (op # True) l1 @ map (op # False) l2) -->
- alg_twinfree l1 & alg_twinfree l2"
+lemma ALG_TWINFREE_STEP1: "alg_twinfree (map (op # True) l1 @ map (op # False) l2)
+==> alg_twinfree l1 & alg_twinfree l2"
by (import prob_canon ALG_TWINFREE_STEP1)
-lemma ALG_TWINFREE_STEP2: "ALL (l1::bool list list) l2::bool list list.
- (~ [] mem l1 | ~ [] mem l2) & alg_twinfree l1 & alg_twinfree l2 -->
- alg_twinfree (map (op # True) l1 @ map (op # False) l2)"
+lemma ALG_TWINFREE_STEP2: "(~ List.member l1 [] | ~ List.member l2 []) &
+alg_twinfree l1 & alg_twinfree l2
+==> alg_twinfree (map (op # True) l1 @ map (op # False) l2)"
by (import prob_canon ALG_TWINFREE_STEP2)
-lemma ALG_TWINFREE_STEP: "ALL (l1::bool list list) l2::bool list list.
- ~ [] mem l1 | ~ [] mem l2 -->
- alg_twinfree (map (op # True) l1 @ map (op # False) l2) =
- (alg_twinfree l1 & alg_twinfree l2)"
+lemma ALG_TWINFREE_STEP: "~ List.member l1 [] | ~ List.member l2 []
+==> alg_twinfree (map (op # True) l1 @ map (op # False) l2) =
+ (alg_twinfree l1 & alg_twinfree l2)"
by (import prob_canon ALG_TWINFREE_STEP)
-lemma ALG_LONGEST_HD: "ALL (h::bool list) t::bool list list. length h <= alg_longest (h # t)"
+lemma ALG_LONGEST_HD: "length h <= alg_longest (h # t)"
by (import prob_canon ALG_LONGEST_HD)
-lemma ALG_LONGEST_TL: "ALL (h::bool list) t::bool list list. alg_longest t <= alg_longest (h # t)"
+lemma ALG_LONGEST_TL: "alg_longest t <= alg_longest (h # t)"
by (import prob_canon ALG_LONGEST_TL)
-lemma ALG_LONGEST_TLS: "ALL (h::bool list) (t::bool list list) b::bool.
- alg_longest (map (op # b) (h # t)) = Suc (alg_longest (h # t))"
+lemma ALG_LONGEST_TLS: "alg_longest (map (op # b) (h # t)) = Suc (alg_longest (h # t))"
by (import prob_canon ALG_LONGEST_TLS)
-lemma ALG_LONGEST_APPEND: "ALL (l1::bool list list) l2::bool list list.
- alg_longest l1 <= alg_longest (l1 @ l2) &
- alg_longest l2 <= alg_longest (l1 @ l2)"
+lemma ALG_LONGEST_APPEND: "alg_longest l1 <= alg_longest (l1 @ l2) &
+alg_longest l2 <= alg_longest (l1 @ l2)"
by (import prob_canon ALG_LONGEST_APPEND)
-lemma ALG_CANON_PREFS_HD: "ALL (l::bool list) b::bool list list. hd (alg_canon_prefs l b) = l"
+lemma ALG_CANON_PREFS_HD: "hd (alg_canon_prefs l b) = l"
by (import prob_canon ALG_CANON_PREFS_HD)
-lemma ALG_CANON_PREFS_DELETES: "ALL (l::bool list) (b::bool list list) x::bool list.
- x mem alg_canon_prefs l b --> x mem l # b"
+lemma ALG_CANON_PREFS_DELETES: "List.member (alg_canon_prefs l b) x ==> List.member (l # b) x"
by (import prob_canon ALG_CANON_PREFS_DELETES)
-lemma ALG_CANON_PREFS_SORTED: "ALL (l::bool list) b::bool list list.
- alg_sorted (l # b) --> alg_sorted (alg_canon_prefs l b)"
+lemma ALG_CANON_PREFS_SORTED: "alg_sorted (l # b) ==> alg_sorted (alg_canon_prefs l b)"
by (import prob_canon ALG_CANON_PREFS_SORTED)
-lemma ALG_CANON_PREFS_PREFIXFREE: "ALL (l::bool list) b::bool list list.
- alg_sorted b & alg_prefixfree b --> alg_prefixfree (alg_canon_prefs l b)"
+lemma ALG_CANON_PREFS_PREFIXFREE: "alg_sorted b & alg_prefixfree b ==> alg_prefixfree (alg_canon_prefs l b)"
by (import prob_canon ALG_CANON_PREFS_PREFIXFREE)
-lemma ALG_CANON_PREFS_CONSTANT: "ALL (l::bool list) b::bool list list.
- alg_prefixfree (l # b) --> alg_canon_prefs l b = l # b"
+lemma ALG_CANON_PREFS_CONSTANT: "alg_prefixfree (l # b) ==> alg_canon_prefs l b = l # b"
by (import prob_canon ALG_CANON_PREFS_CONSTANT)
-lemma ALG_CANON_FIND_HD: "ALL (l::bool list) (h::bool list) t::bool list list.
- hd (alg_canon_find l (h # t)) = l | hd (alg_canon_find l (h # t)) = h"
+lemma ALG_CANON_FIND_HD: "hd (alg_canon_find l (h # t)) = l | hd (alg_canon_find l (h # t)) = h"
by (import prob_canon ALG_CANON_FIND_HD)
-lemma ALG_CANON_FIND_DELETES: "ALL (l::bool list) (b::bool list list) x::bool list.
- x mem alg_canon_find l b --> x mem l # b"
+lemma ALG_CANON_FIND_DELETES: "List.member (alg_canon_find l b) x ==> List.member (l # b) x"
by (import prob_canon ALG_CANON_FIND_DELETES)
-lemma ALG_CANON_FIND_SORTED: "ALL (l::bool list) b::bool list list.
- alg_sorted b --> alg_sorted (alg_canon_find l b)"
+lemma ALG_CANON_FIND_SORTED: "alg_sorted b ==> alg_sorted (alg_canon_find l b)"
by (import prob_canon ALG_CANON_FIND_SORTED)
-lemma ALG_CANON_FIND_PREFIXFREE: "ALL (l::bool list) b::bool list list.
- alg_sorted b & alg_prefixfree b --> alg_prefixfree (alg_canon_find l b)"
+lemma ALG_CANON_FIND_PREFIXFREE: "alg_sorted b & alg_prefixfree b ==> alg_prefixfree (alg_canon_find l b)"
by (import prob_canon ALG_CANON_FIND_PREFIXFREE)
-lemma ALG_CANON_FIND_CONSTANT: "ALL (l::bool list) b::bool list list.
- alg_sorted (l # b) & alg_prefixfree (l # b) -->
- alg_canon_find l b = l # b"
+lemma ALG_CANON_FIND_CONSTANT: "alg_sorted (l # b) & alg_prefixfree (l # b) ==> alg_canon_find l b = l # b"
by (import prob_canon ALG_CANON_FIND_CONSTANT)
-lemma ALG_CANON1_SORTED: "ALL x::bool list list. alg_sorted (alg_canon1 x)"
+lemma ALG_CANON1_SORTED: "alg_sorted (alg_canon1 x)"
by (import prob_canon ALG_CANON1_SORTED)
-lemma ALG_CANON1_PREFIXFREE: "ALL l::bool list list. alg_prefixfree (alg_canon1 l)"
+lemma ALG_CANON1_PREFIXFREE: "alg_prefixfree (alg_canon1 l)"
by (import prob_canon ALG_CANON1_PREFIXFREE)
-lemma ALG_CANON1_CONSTANT: "ALL l::bool list list. alg_sorted l & alg_prefixfree l --> alg_canon1 l = l"
+lemma ALG_CANON1_CONSTANT: "alg_sorted l & alg_prefixfree l ==> alg_canon1 l = l"
by (import prob_canon ALG_CANON1_CONSTANT)
-lemma ALG_CANON_MERGE_SORTED_PREFIXFREE_TWINFREE: "ALL (l::bool list) b::bool list list.
- alg_sorted (l # b) & alg_prefixfree (l # b) & alg_twinfree b -->
- alg_sorted (alg_canon_merge l b) &
- alg_prefixfree (alg_canon_merge l b) & alg_twinfree (alg_canon_merge l b)"
+lemma ALG_CANON_MERGE_SORTED_PREFIXFREE_TWINFREE: "alg_sorted (l # b) & alg_prefixfree (l # b) & alg_twinfree b
+==> alg_sorted (alg_canon_merge l b) &
+ alg_prefixfree (alg_canon_merge l b) &
+ alg_twinfree (alg_canon_merge l b)"
by (import prob_canon ALG_CANON_MERGE_SORTED_PREFIXFREE_TWINFREE)
-lemma ALG_CANON_MERGE_PREFIXFREE_PRESERVE: "ALL (l::bool list) (b::bool list list) h::bool list.
- (ALL x::bool list. x mem l # b --> ~ IS_PREFIX h x & ~ IS_PREFIX x h) -->
- (ALL x::bool list.
- x mem alg_canon_merge l b --> ~ IS_PREFIX h x & ~ IS_PREFIX x h)"
+lemma ALG_CANON_MERGE_PREFIXFREE_PRESERVE: "[| !!x. List.member (l # b) x ==> ~ IS_PREFIX h x & ~ IS_PREFIX x h;
+ List.member (alg_canon_merge l b) x |]
+==> ~ IS_PREFIX h x & ~ IS_PREFIX x h"
by (import prob_canon ALG_CANON_MERGE_PREFIXFREE_PRESERVE)
-lemma ALG_CANON_MERGE_SHORTENS: "ALL (l::bool list) (b::bool list list) x::bool list.
- x mem alg_canon_merge l b -->
- (EX y::bool list. y mem l # b & IS_PREFIX y x)"
+lemma ALG_CANON_MERGE_SHORTENS: "List.member (alg_canon_merge l b) x
+==> EX y. List.member (l # b) y & IS_PREFIX y x"
by (import prob_canon ALG_CANON_MERGE_SHORTENS)
-lemma ALG_CANON_MERGE_CONSTANT: "ALL (l::bool list) b::bool list list.
- alg_twinfree (l # b) --> alg_canon_merge l b = l # b"
+lemma ALG_CANON_MERGE_CONSTANT: "alg_twinfree (l # b) ==> alg_canon_merge l b = l # b"
by (import prob_canon ALG_CANON_MERGE_CONSTANT)
-lemma ALG_CANON2_PREFIXFREE_PRESERVE: "ALL (x::bool list list) xa::bool list.
- (ALL xb::bool list.
- xb mem x --> ~ IS_PREFIX xa xb & ~ IS_PREFIX xb xa) -->
- (ALL xb::bool list.
- xb mem alg_canon2 x --> ~ IS_PREFIX xa xb & ~ IS_PREFIX xb xa)"
+lemma ALG_CANON2_PREFIXFREE_PRESERVE: "[| !!xb. List.member x xb ==> ~ IS_PREFIX xa xb & ~ IS_PREFIX xb xa;
+ List.member (alg_canon2 x) xb |]
+==> ~ IS_PREFIX xa xb & ~ IS_PREFIX xb xa"
by (import prob_canon ALG_CANON2_PREFIXFREE_PRESERVE)
-lemma ALG_CANON2_SHORTENS: "ALL (x::bool list list) xa::bool list.
- xa mem alg_canon2 x --> (EX y::bool list. y mem x & IS_PREFIX y xa)"
+lemma ALG_CANON2_SHORTENS: "List.member (alg_canon2 x) xa ==> EX y. List.member x y & IS_PREFIX y xa"
by (import prob_canon ALG_CANON2_SHORTENS)
-lemma ALG_CANON2_SORTED_PREFIXFREE_TWINFREE: "ALL x::bool list list.
- alg_sorted x & alg_prefixfree x -->
- alg_sorted (alg_canon2 x) &
- alg_prefixfree (alg_canon2 x) & alg_twinfree (alg_canon2 x)"
+lemma ALG_CANON2_SORTED_PREFIXFREE_TWINFREE: "alg_sorted x & alg_prefixfree x
+==> alg_sorted (alg_canon2 x) &
+ alg_prefixfree (alg_canon2 x) & alg_twinfree (alg_canon2 x)"
by (import prob_canon ALG_CANON2_SORTED_PREFIXFREE_TWINFREE)
-lemma ALG_CANON2_CONSTANT: "ALL l::bool list list. alg_twinfree l --> alg_canon2 l = l"
+lemma ALG_CANON2_CONSTANT: "alg_twinfree l ==> alg_canon2 l = l"
by (import prob_canon ALG_CANON2_CONSTANT)
-lemma ALG_CANON_SORTED_PREFIXFREE_TWINFREE: "ALL l::bool list list.
- alg_sorted (alg_canon l) &
- alg_prefixfree (alg_canon l) & alg_twinfree (alg_canon l)"
+lemma ALG_CANON_SORTED_PREFIXFREE_TWINFREE: "alg_sorted (alg_canon l) &
+alg_prefixfree (alg_canon l) & alg_twinfree (alg_canon l)"
by (import prob_canon ALG_CANON_SORTED_PREFIXFREE_TWINFREE)
-lemma ALG_CANON_CONSTANT: "ALL l::bool list list.
- alg_sorted l & alg_prefixfree l & alg_twinfree l --> alg_canon l = l"
+lemma ALG_CANON_CONSTANT: "alg_sorted l & alg_prefixfree l & alg_twinfree l ==> alg_canon l = l"
by (import prob_canon ALG_CANON_CONSTANT)
-lemma ALG_CANON_IDEMPOT: "ALL l::bool list list. alg_canon (alg_canon l) = alg_canon l"
+lemma ALG_CANON_IDEMPOT: "alg_canon (alg_canon l) = alg_canon l"
by (import prob_canon ALG_CANON_IDEMPOT)
-lemma ALGEBRA_CANON_DEF_ALT: "ALL l::bool list list.
- algebra_canon l = (alg_sorted l & alg_prefixfree l & alg_twinfree l)"
+lemma ALGEBRA_CANON_DEF_ALT: "algebra_canon l = (alg_sorted l & alg_prefixfree l & alg_twinfree l)"
by (import prob_canon ALGEBRA_CANON_DEF_ALT)
-lemma ALGEBRA_CANON_BASIC: "algebra_canon [] &
-algebra_canon [[]] & (ALL x::bool list. algebra_canon [x])"
+lemma ALGEBRA_CANON_BASIC: "algebra_canon [] & algebra_canon [[]] & (ALL x. algebra_canon [x])"
by (import prob_canon ALGEBRA_CANON_BASIC)
-lemma ALG_CANON_BASIC: "alg_canon [] = [] &
-alg_canon [[]] = [[]] & (ALL x::bool list. alg_canon [x] = [x])"
+lemma ALG_CANON_BASIC: "alg_canon [] = [] & alg_canon [[]] = [[]] & (ALL x. alg_canon [x] = [x])"
by (import prob_canon ALG_CANON_BASIC)
-lemma ALGEBRA_CANON_TL: "ALL (h::bool list) t::bool list list.
- algebra_canon (h # t) --> algebra_canon t"
+lemma ALGEBRA_CANON_TL: "algebra_canon (h # t) ==> algebra_canon t"
by (import prob_canon ALGEBRA_CANON_TL)
-lemma ALGEBRA_CANON_NIL_MEM: "ALL l::bool list list. (algebra_canon l & [] mem l) = (l = [[]])"
+lemma ALGEBRA_CANON_NIL_MEM: "(algebra_canon l & List.member l []) = (l = [[]])"
by (import prob_canon ALGEBRA_CANON_NIL_MEM)
-lemma ALGEBRA_CANON_TLS: "ALL (l::bool list list) b::bool.
- algebra_canon (map (op # b) l) = algebra_canon l"
+lemma ALGEBRA_CANON_TLS: "algebra_canon (map (op # b) l) = algebra_canon l"
by (import prob_canon ALGEBRA_CANON_TLS)
-lemma ALGEBRA_CANON_STEP1: "ALL (l1::bool list list) l2::bool list list.
- algebra_canon (map (op # True) l1 @ map (op # False) l2) -->
- algebra_canon l1 & algebra_canon l2"
+lemma ALGEBRA_CANON_STEP1: "algebra_canon (map (op # True) l1 @ map (op # False) l2)
+==> algebra_canon l1 & algebra_canon l2"
by (import prob_canon ALGEBRA_CANON_STEP1)
-lemma ALGEBRA_CANON_STEP2: "ALL (l1::bool list list) l2::bool list list.
- (l1 ~= [[]] | l2 ~= [[]]) & algebra_canon l1 & algebra_canon l2 -->
- algebra_canon (map (op # True) l1 @ map (op # False) l2)"
+lemma ALGEBRA_CANON_STEP2: "(l1 ~= [[]] | l2 ~= [[]]) & algebra_canon l1 & algebra_canon l2
+==> algebra_canon (map (op # True) l1 @ map (op # False) l2)"
by (import prob_canon ALGEBRA_CANON_STEP2)
-lemma ALGEBRA_CANON_STEP: "ALL (l1::bool list list) l2::bool list list.
- l1 ~= [[]] | l2 ~= [[]] -->
- algebra_canon (map (op # True) l1 @ map (op # False) l2) =
- (algebra_canon l1 & algebra_canon l2)"
+lemma ALGEBRA_CANON_STEP: "l1 ~= [[]] | l2 ~= [[]]
+==> algebra_canon (map (op # True) l1 @ map (op # False) l2) =
+ (algebra_canon l1 & algebra_canon l2)"
by (import prob_canon ALGEBRA_CANON_STEP)
-lemma ALGEBRA_CANON_CASES_THM: "ALL l::bool list list.
- algebra_canon l -->
- l = [] |
- l = [[]] |
- (EX (l1::bool list list) l2::bool list list.
- algebra_canon l1 &
- algebra_canon l2 & l = map (op # True) l1 @ map (op # False) l2)"
+lemma ALGEBRA_CANON_CASES_THM: "algebra_canon l
+==> l = [] |
+ l = [[]] |
+ (EX l1 l2.
+ algebra_canon l1 &
+ algebra_canon l2 & l = map (op # True) l1 @ map (op # False) l2)"
by (import prob_canon ALGEBRA_CANON_CASES_THM)
-lemma ALGEBRA_CANON_CASES: "ALL P::bool list list => bool.
- P [] &
+lemma ALGEBRA_CANON_CASES: "[| P [] &
P [[]] &
- (ALL (l1::bool list list) l2::bool list list.
+ (ALL l1 l2.
algebra_canon l1 &
algebra_canon l2 &
algebra_canon (map (op # True) l1 @ map (op # False) l2) -->
- P (map (op # True) l1 @ map (op # False) l2)) -->
- (ALL l::bool list list. algebra_canon l --> P l)"
+ P (map (op # True) l1 @ map (op # False) l2));
+ algebra_canon l |]
+==> P l"
by (import prob_canon ALGEBRA_CANON_CASES)
-lemma ALGEBRA_CANON_INDUCTION: "ALL P::bool list list => bool.
- P [] &
+lemma ALGEBRA_CANON_INDUCTION: "[| P [] &
P [[]] &
- (ALL (l1::bool list list) l2::bool list list.
+ (ALL l1 l2.
algebra_canon l1 &
algebra_canon l2 &
P l1 &
P l2 & algebra_canon (map (op # True) l1 @ map (op # False) l2) -->
- P (map (op # True) l1 @ map (op # False) l2)) -->
- (ALL l::bool list list. algebra_canon l --> P l)"
+ P (map (op # True) l1 @ map (op # False) l2));
+ algebra_canon l |]
+==> P l"
by (import prob_canon ALGEBRA_CANON_INDUCTION)
-lemma MEM_NIL_STEP: "ALL (l1::bool list list) l2::bool list list.
- ~ [] mem map (op # True) l1 @ map (op # False) l2"
+lemma MEM_NIL_STEP: "~ List.member (map (op # True) l1 @ map (op # False) l2) []"
by (import prob_canon MEM_NIL_STEP)
-lemma ALG_SORTED_PREFIXFREE_MEM_NIL: "ALL l::bool list list.
- (alg_sorted l & alg_prefixfree l & [] mem l) = (l = [[]])"
+lemma ALG_SORTED_PREFIXFREE_MEM_NIL: "(alg_sorted l & alg_prefixfree l & List.member l []) = (l = [[]])"
by (import prob_canon ALG_SORTED_PREFIXFREE_MEM_NIL)
-lemma ALG_SORTED_PREFIXFREE_EQUALITY: "ALL (l::bool list list) l'::bool list list.
- (ALL x::bool list. x mem l = x mem l') &
- alg_sorted l & alg_sorted l' & alg_prefixfree l & alg_prefixfree l' -->
- l = l'"
+lemma ALG_SORTED_PREFIXFREE_EQUALITY: "(ALL x. List.member l x = List.member l' x) &
+alg_sorted l & alg_sorted l' & alg_prefixfree l & alg_prefixfree l'
+==> l = l'"
by (import prob_canon ALG_SORTED_PREFIXFREE_EQUALITY)
;end_setup
@@ -1080,34 +731,33 @@
SHD :: "(nat => bool) => bool"
defs
- SHD_primdef: "SHD == %f::nat => bool. f 0"
+ SHD_primdef: "SHD == %f. f 0"
-lemma SHD_def: "ALL f::nat => bool. SHD f = f 0"
+lemma SHD_def: "SHD f = f 0"
by (import boolean_sequence SHD_def)
consts
STL :: "(nat => bool) => nat => bool"
defs
- STL_primdef: "STL == %(f::nat => bool) n::nat. f (Suc n)"
+ STL_primdef: "STL == %f n. f (Suc n)"
-lemma STL_def: "ALL (f::nat => bool) n::nat. STL f n = f (Suc n)"
+lemma STL_def: "STL f n = f (Suc n)"
by (import boolean_sequence STL_def)
consts
SCONS :: "bool => (nat => bool) => nat => bool"
-specification (SCONS_primdef: SCONS) SCONS_def: "(ALL (h::bool) t::nat => bool. SCONS h t 0 = h) &
-(ALL (h::bool) (t::nat => bool) n::nat. SCONS h t (Suc n) = t n)"
+specification (SCONS_primdef: SCONS) SCONS_def: "(ALL h t. SCONS h t 0 = h) & (ALL h t n. SCONS h t (Suc n) = t n)"
by (import boolean_sequence SCONS_def)
consts
SDEST :: "(nat => bool) => bool * (nat => bool)"
defs
- SDEST_primdef: "SDEST == %s::nat => bool. (SHD s, STL s)"
+ SDEST_primdef: "SDEST == %s. (SHD s, STL s)"
-lemma SDEST_def: "SDEST = (%s::nat => bool. (SHD s, STL s))"
+lemma SDEST_def: "SDEST = (%s. (SHD s, STL s))"
by (import boolean_sequence SDEST_def)
consts
@@ -1122,32 +772,32 @@
consts
STAKE :: "nat => (nat => bool) => bool list"
-specification (STAKE_primdef: STAKE) STAKE_def: "(ALL s::nat => bool. STAKE 0 s = []) &
-(ALL (n::nat) s::nat => bool. STAKE (Suc n) s = SHD s # STAKE n (STL s))"
+specification (STAKE_primdef: STAKE) STAKE_def: "(ALL s. STAKE 0 s = []) &
+(ALL n s. STAKE (Suc n) s = SHD s # STAKE n (STL s))"
by (import boolean_sequence STAKE_def)
consts
SDROP :: "nat => (nat => bool) => nat => bool"
-specification (SDROP_primdef: SDROP) SDROP_def: "SDROP 0 = I & (ALL n::nat. SDROP (Suc n) = SDROP n o STL)"
+specification (SDROP_primdef: SDROP) SDROP_def: "SDROP 0 = I & (ALL n. SDROP (Suc n) = SDROP n o STL)"
by (import boolean_sequence SDROP_def)
-lemma SCONS_SURJ: "ALL x::nat => bool. EX (xa::bool) t::nat => bool. x = SCONS xa t"
+lemma SCONS_SURJ: "EX xa t. x = SCONS xa t"
by (import boolean_sequence SCONS_SURJ)
-lemma SHD_STL_ISO: "ALL (h::bool) t::nat => bool. EX x::nat => bool. SHD x = h & STL x = t"
+lemma SHD_STL_ISO: "EX x. SHD x = h & STL x = t"
by (import boolean_sequence SHD_STL_ISO)
-lemma SHD_SCONS: "ALL (h::bool) t::nat => bool. SHD (SCONS h t) = h"
+lemma SHD_SCONS: "SHD (SCONS h t) = h"
by (import boolean_sequence SHD_SCONS)
-lemma STL_SCONS: "ALL (h::bool) t::nat => bool. STL (SCONS h t) = t"
+lemma STL_SCONS: "STL (SCONS h t) = t"
by (import boolean_sequence STL_SCONS)
-lemma SHD_SCONST: "ALL b::bool. SHD (SCONST b) = b"
+lemma SHD_SCONST: "SHD (SCONST b) = b"
by (import boolean_sequence SHD_SCONST)
-lemma STL_SCONST: "ALL b::bool. STL (SCONST b) = SCONST b"
+lemma STL_SCONST: "STL (SCONST b) = SCONST b"
by (import boolean_sequence STL_SCONST)
;end_setup
@@ -1157,16 +807,15 @@
consts
alg_embed :: "bool list => (nat => bool) => bool"
-specification (alg_embed_primdef: alg_embed) alg_embed_def: "(ALL s::nat => bool. alg_embed [] s = True) &
-(ALL (h::bool) (t::bool list) s::nat => bool.
- alg_embed (h # t) s = (h = SHD s & alg_embed t (STL s)))"
+specification (alg_embed_primdef: alg_embed) alg_embed_def: "(ALL s. alg_embed [] s = True) &
+(ALL h t s. alg_embed (h # t) s = (h = SHD s & alg_embed t (STL s)))"
by (import prob_algebra alg_embed_def)
consts
algebra_embed :: "bool list list => (nat => bool) => bool"
specification (algebra_embed_primdef: algebra_embed) algebra_embed_def: "algebra_embed [] = EMPTY &
-(ALL (h::bool list) t::bool list list.
+(ALL h t.
algebra_embed (h # t) = pred_set.UNION (alg_embed h) (algebra_embed t))"
by (import prob_algebra algebra_embed_def)
@@ -1174,157 +823,127 @@
measurable :: "((nat => bool) => bool) => bool"
defs
- measurable_primdef: "measurable ==
-%s::(nat => bool) => bool. EX b::bool list list. s = algebra_embed b"
+ measurable_primdef: "measurable == %s. EX b. s = algebra_embed b"
-lemma measurable_def: "ALL s::(nat => bool) => bool.
- measurable s = (EX b::bool list list. s = algebra_embed b)"
+lemma measurable_def: "measurable s = (EX b. s = algebra_embed b)"
by (import prob_algebra measurable_def)
-lemma HALVES_INTER: "pred_set.INTER (%x::nat => bool. SHD x = True)
- (%x::nat => bool. SHD x = False) =
-EMPTY"
+lemma HALVES_INTER: "pred_set.INTER (%x. SHD x = True) (%x. SHD x = False) = EMPTY"
by (import prob_algebra HALVES_INTER)
-lemma INTER_STL: "ALL (p::(nat => bool) => bool) q::(nat => bool) => bool.
- pred_set.INTER p q o STL = pred_set.INTER (p o STL) (q o STL)"
+lemma INTER_STL: "pred_set.INTER p q o STL = pred_set.INTER (p o STL) (q o STL)"
by (import prob_algebra INTER_STL)
-lemma COMPL_SHD: "ALL b::bool.
- COMPL (%x::nat => bool. SHD x = b) = (%x::nat => bool. SHD x = (~ b))"
+lemma COMPL_SHD: "COMPL (%x. SHD x = b) = (%x. SHD x = (~ b))"
by (import prob_algebra COMPL_SHD)
lemma ALG_EMBED_BASIC: "alg_embed [] = pred_set.UNIV &
-(ALL (h::bool) t::bool list.
- alg_embed (h # t) =
- pred_set.INTER (%x::nat => bool. SHD x = h) (alg_embed t o STL))"
+(ALL h t.
+ alg_embed (h # t) = pred_set.INTER (%x. SHD x = h) (alg_embed t o STL))"
by (import prob_algebra ALG_EMBED_BASIC)
-lemma ALG_EMBED_NIL: "ALL c::bool list. All (alg_embed c) = (c = [])"
+lemma ALG_EMBED_NIL: "All (alg_embed c) = (c = [])"
by (import prob_algebra ALG_EMBED_NIL)
-lemma ALG_EMBED_POPULATED: "ALL b::bool list. Ex (alg_embed b)"
+lemma ALG_EMBED_POPULATED: "Ex (alg_embed b)"
by (import prob_algebra ALG_EMBED_POPULATED)
-lemma ALG_EMBED_PREFIX: "ALL (b::bool list) (c::bool list) s::nat => bool.
- alg_embed b s & alg_embed c s --> IS_PREFIX b c | IS_PREFIX c b"
+lemma ALG_EMBED_PREFIX: "alg_embed b s & alg_embed c s ==> IS_PREFIX b c | IS_PREFIX c b"
by (import prob_algebra ALG_EMBED_PREFIX)
-lemma ALG_EMBED_PREFIX_SUBSET: "ALL (b::bool list) c::bool list.
- SUBSET (alg_embed b) (alg_embed c) = IS_PREFIX b c"
+lemma ALG_EMBED_PREFIX_SUBSET: "SUBSET (alg_embed b) (alg_embed c) = IS_PREFIX b c"
by (import prob_algebra ALG_EMBED_PREFIX_SUBSET)
-lemma ALG_EMBED_TWINS: "ALL l::bool list.
- pred_set.UNION (alg_embed (SNOC True l)) (alg_embed (SNOC False l)) =
- alg_embed l"
+lemma ALG_EMBED_TWINS: "pred_set.UNION (alg_embed (SNOC True l)) (alg_embed (SNOC False l)) =
+alg_embed l"
by (import prob_algebra ALG_EMBED_TWINS)
lemma ALGEBRA_EMBED_BASIC: "algebra_embed [] = EMPTY &
algebra_embed [[]] = pred_set.UNIV &
-(ALL b::bool. algebra_embed [[b]] = (%s::nat => bool. SHD s = b))"
+(ALL b. algebra_embed [[b]] = (%s. SHD s = b))"
by (import prob_algebra ALGEBRA_EMBED_BASIC)
-lemma ALGEBRA_EMBED_MEM: "ALL (b::bool list list) x::nat => bool.
- algebra_embed b x --> (EX l::bool list. l mem b & alg_embed l x)"
+lemma ALGEBRA_EMBED_MEM: "algebra_embed b x ==> EX l. List.member b l & alg_embed l x"
by (import prob_algebra ALGEBRA_EMBED_MEM)
-lemma ALGEBRA_EMBED_APPEND: "ALL (l1::bool list list) l2::bool list list.
- algebra_embed (l1 @ l2) =
- pred_set.UNION (algebra_embed l1) (algebra_embed l2)"
+lemma ALGEBRA_EMBED_APPEND: "algebra_embed (l1 @ l2) =
+pred_set.UNION (algebra_embed l1) (algebra_embed l2)"
by (import prob_algebra ALGEBRA_EMBED_APPEND)
-lemma ALGEBRA_EMBED_TLS: "ALL (l::bool list list) b::bool.
- algebra_embed (map (op # b) l) (SCONS (h::bool) (t::nat => bool)) =
- (h = b & algebra_embed l t)"
+lemma ALGEBRA_EMBED_TLS: "algebra_embed (map (op # b) l) (SCONS h t) = (h = b & algebra_embed l t)"
by (import prob_algebra ALGEBRA_EMBED_TLS)
-lemma ALG_CANON_PREFS_EMBED: "ALL (l::bool list) b::bool list list.
- algebra_embed (alg_canon_prefs l b) = algebra_embed (l # b)"
+lemma ALG_CANON_PREFS_EMBED: "algebra_embed (alg_canon_prefs l b) = algebra_embed (l # b)"
by (import prob_algebra ALG_CANON_PREFS_EMBED)
-lemma ALG_CANON_FIND_EMBED: "ALL (l::bool list) b::bool list list.
- algebra_embed (alg_canon_find l b) = algebra_embed (l # b)"
+lemma ALG_CANON_FIND_EMBED: "algebra_embed (alg_canon_find l b) = algebra_embed (l # b)"
by (import prob_algebra ALG_CANON_FIND_EMBED)
-lemma ALG_CANON1_EMBED: "ALL x::bool list list. algebra_embed (alg_canon1 x) = algebra_embed x"
+lemma ALG_CANON1_EMBED: "algebra_embed (alg_canon1 x) = algebra_embed x"
by (import prob_algebra ALG_CANON1_EMBED)
-lemma ALG_CANON_MERGE_EMBED: "ALL (l::bool list) b::bool list list.
- algebra_embed (alg_canon_merge l b) = algebra_embed (l # b)"
+lemma ALG_CANON_MERGE_EMBED: "algebra_embed (alg_canon_merge l b) = algebra_embed (l # b)"
by (import prob_algebra ALG_CANON_MERGE_EMBED)
-lemma ALG_CANON2_EMBED: "ALL x::bool list list. algebra_embed (alg_canon2 x) = algebra_embed x"
+lemma ALG_CANON2_EMBED: "algebra_embed (alg_canon2 x) = algebra_embed x"
by (import prob_algebra ALG_CANON2_EMBED)
-lemma ALG_CANON_EMBED: "ALL l::bool list list. algebra_embed (alg_canon l) = algebra_embed l"
+lemma ALG_CANON_EMBED: "algebra_embed (alg_canon l) = algebra_embed l"
by (import prob_algebra ALG_CANON_EMBED)
-lemma ALGEBRA_CANON_UNIV: "ALL l::bool list list.
- algebra_canon l --> algebra_embed l = pred_set.UNIV --> l = [[]]"
+lemma ALGEBRA_CANON_UNIV: "[| algebra_canon l; algebra_embed l = pred_set.UNIV |] ==> l = [[]]"
by (import prob_algebra ALGEBRA_CANON_UNIV)
-lemma ALG_CANON_REP: "ALL (b::bool list list) c::bool list list.
- (alg_canon b = alg_canon c) = (algebra_embed b = algebra_embed c)"
+lemma ALG_CANON_REP: "(alg_canon b = alg_canon c) = (algebra_embed b = algebra_embed c)"
by (import prob_algebra ALG_CANON_REP)
-lemma ALGEBRA_CANON_EMBED_EMPTY: "ALL l::bool list list.
- algebra_canon l --> (ALL v::nat => bool. ~ algebra_embed l v) = (l = [])"
+lemma ALGEBRA_CANON_EMBED_EMPTY: "algebra_canon l ==> (ALL v. ~ algebra_embed l v) = (l = [])"
by (import prob_algebra ALGEBRA_CANON_EMBED_EMPTY)
-lemma ALGEBRA_CANON_EMBED_UNIV: "ALL l::bool list list.
- algebra_canon l --> All (algebra_embed l) = (l = [[]])"
+lemma ALGEBRA_CANON_EMBED_UNIV: "algebra_canon l ==> All (algebra_embed l) = (l = [[]])"
by (import prob_algebra ALGEBRA_CANON_EMBED_UNIV)
-lemma MEASURABLE_ALGEBRA: "ALL b::bool list list. measurable (algebra_embed b)"
+lemma MEASURABLE_ALGEBRA: "measurable (algebra_embed b)"
by (import prob_algebra MEASURABLE_ALGEBRA)
lemma MEASURABLE_BASIC: "measurable EMPTY &
-measurable pred_set.UNIV &
-(ALL b::bool. measurable (%s::nat => bool. SHD s = b))"
+measurable pred_set.UNIV & (ALL b. measurable (%s. SHD s = b))"
by (import prob_algebra MEASURABLE_BASIC)
-lemma MEASURABLE_SHD: "ALL b::bool. measurable (%s::nat => bool. SHD s = b)"
+lemma MEASURABLE_SHD: "measurable (%s. SHD s = b)"
by (import prob_algebra MEASURABLE_SHD)
-lemma ALGEBRA_EMBED_COMPL: "ALL l::bool list list.
- EX l'::bool list list. COMPL (algebra_embed l) = algebra_embed l'"
+lemma ALGEBRA_EMBED_COMPL: "EX l'. COMPL (algebra_embed l) = algebra_embed l'"
by (import prob_algebra ALGEBRA_EMBED_COMPL)
-lemma MEASURABLE_COMPL: "ALL s::(nat => bool) => bool. measurable (COMPL s) = measurable s"
+lemma MEASURABLE_COMPL: "measurable (COMPL s) = measurable s"
by (import prob_algebra MEASURABLE_COMPL)
-lemma MEASURABLE_UNION: "ALL (s::(nat => bool) => bool) t::(nat => bool) => bool.
- measurable s & measurable t --> measurable (pred_set.UNION s t)"
+lemma MEASURABLE_UNION: "measurable s & measurable t ==> measurable (pred_set.UNION s t)"
by (import prob_algebra MEASURABLE_UNION)
-lemma MEASURABLE_INTER: "ALL (s::(nat => bool) => bool) t::(nat => bool) => bool.
- measurable s & measurable t --> measurable (pred_set.INTER s t)"
+lemma MEASURABLE_INTER: "measurable s & measurable t ==> measurable (pred_set.INTER s t)"
by (import prob_algebra MEASURABLE_INTER)
-lemma MEASURABLE_STL: "ALL p::(nat => bool) => bool. measurable (p o STL) = measurable p"
+lemma MEASURABLE_STL: "measurable (p o STL) = measurable p"
by (import prob_algebra MEASURABLE_STL)
-lemma MEASURABLE_SDROP: "ALL (n::nat) p::(nat => bool) => bool.
- measurable (p o SDROP n) = measurable p"
+lemma MEASURABLE_SDROP: "measurable (p o SDROP n) = measurable p"
by (import prob_algebra MEASURABLE_SDROP)
-lemma MEASURABLE_INTER_HALVES: "ALL p::(nat => bool) => bool.
- (measurable (pred_set.INTER (%x::nat => bool. SHD x = True) p) &
- measurable (pred_set.INTER (%x::nat => bool. SHD x = False) p)) =
- measurable p"
+lemma MEASURABLE_INTER_HALVES: "(measurable (pred_set.INTER (%x. SHD x = True) p) &
+ measurable (pred_set.INTER (%x. SHD x = False) p)) =
+measurable p"
by (import prob_algebra MEASURABLE_INTER_HALVES)
-lemma MEASURABLE_HALVES: "ALL (p::(nat => bool) => bool) q::(nat => bool) => bool.
- measurable
- (pred_set.UNION (pred_set.INTER (%x::nat => bool. SHD x = True) p)
- (pred_set.INTER (%x::nat => bool. SHD x = False) q)) =
- (measurable (pred_set.INTER (%x::nat => bool. SHD x = True) p) &
- measurable (pred_set.INTER (%x::nat => bool. SHD x = False) q))"
+lemma MEASURABLE_HALVES: "measurable
+ (pred_set.UNION (pred_set.INTER (%x. SHD x = True) p)
+ (pred_set.INTER (%x. SHD x = False) q)) =
+(measurable (pred_set.INTER (%x. SHD x = True) p) &
+ measurable (pred_set.INTER (%x. SHD x = False) q))"
by (import prob_algebra MEASURABLE_HALVES)
-lemma MEASURABLE_INTER_SHD: "ALL (b::bool) p::(nat => bool) => bool.
- measurable (pred_set.INTER (%x::nat => bool. SHD x = b) (p o STL)) =
- measurable p"
+lemma MEASURABLE_INTER_SHD: "measurable (pred_set.INTER (%x. SHD x = b) (p o STL)) = measurable p"
by (import prob_algebra MEASURABLE_INTER_SHD)
;end_setup
@@ -1335,8 +954,7 @@
alg_measure :: "bool list list => real"
specification (alg_measure_primdef: alg_measure) alg_measure_def: "alg_measure [] = 0 &
-(ALL (l::bool list) rest::bool list list.
- alg_measure (l # rest) = (1 / 2) ^ length l + alg_measure rest)"
+(ALL l rest. alg_measure (l # rest) = (1 / 2) ^ length l + alg_measure rest)"
by (import prob alg_measure_def)
consts
@@ -1344,16 +962,12 @@
defs
algebra_measure_primdef: "algebra_measure ==
-%b::bool list list.
- inf (%r::real.
- EX c::bool list list.
- algebra_embed b = algebra_embed c & alg_measure c = r)"
+%b. prob_extra.inf
+ (%r. EX c. algebra_embed b = algebra_embed c & alg_measure c = r)"
-lemma algebra_measure_def: "ALL b::bool list list.
- algebra_measure b =
- inf (%r::real.
- EX c::bool list list.
- algebra_embed b = algebra_embed c & alg_measure c = r)"
+lemma algebra_measure_def: "algebra_measure b =
+prob_extra.inf
+ (%r. EX c. algebra_embed b = algebra_embed c & alg_measure c = r)"
by (import prob algebra_measure_def)
consts
@@ -1361,209 +975,148 @@
defs
prob_primdef: "prob ==
-%s::(nat => bool) => bool.
- sup (%r::real.
- EX b::bool list list.
- algebra_measure b = r & SUBSET (algebra_embed b) s)"
+%s. real.sup (%r. EX b. algebra_measure b = r & SUBSET (algebra_embed b) s)"
-lemma prob_def: "ALL s::(nat => bool) => bool.
- prob s =
- sup (%r::real.
- EX b::bool list list.
- algebra_measure b = r & SUBSET (algebra_embed b) s)"
+lemma prob_def: "prob s =
+real.sup (%r. EX b. algebra_measure b = r & SUBSET (algebra_embed b) s)"
by (import prob prob_def)
-lemma ALG_TWINS_MEASURE: "(All::(bool list => bool) => bool)
- (%l::bool list.
- (op =::real => real => bool)
- ((op +::real => real => real)
- ((op ^::real => nat => real)
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- ((size::bool list => nat)
- ((SNOC::bool => bool list => bool list) (True::bool) l)))
- ((op ^::real => nat => real)
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- ((size::bool list => nat)
- ((SNOC::bool => bool list => bool list) (False::bool) l))))
- ((op ^::real => nat => real)
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- ((size::bool list => nat) l)))"
+lemma ALG_TWINS_MEASURE: "((1::real) / (2::real)) ^ length (SNOC True (l::bool list)) +
+((1::real) / (2::real)) ^ length (SNOC False l) =
+((1::real) / (2::real)) ^ length l"
by (import prob ALG_TWINS_MEASURE)
lemma ALG_MEASURE_BASIC: "alg_measure [] = 0 &
-alg_measure [[]] = 1 & (ALL b::bool. alg_measure [[b]] = 1 / 2)"
+alg_measure [[]] = 1 & (ALL b. alg_measure [[b]] = 1 / 2)"
by (import prob ALG_MEASURE_BASIC)
-lemma ALG_MEASURE_POS: "ALL l::bool list list. 0 <= alg_measure l"
+lemma ALG_MEASURE_POS: "0 <= alg_measure l"
by (import prob ALG_MEASURE_POS)
-lemma ALG_MEASURE_APPEND: "ALL (l1::bool list list) l2::bool list list.
- alg_measure (l1 @ l2) = alg_measure l1 + alg_measure l2"
+lemma ALG_MEASURE_APPEND: "alg_measure (l1 @ l2) = alg_measure l1 + alg_measure l2"
by (import prob ALG_MEASURE_APPEND)
-lemma ALG_MEASURE_TLS: "ALL (l::bool list list) b::bool.
- 2 * alg_measure (map (op # b) l) = alg_measure l"
+lemma ALG_MEASURE_TLS: "2 * alg_measure (map (op # b) l) = alg_measure l"
by (import prob ALG_MEASURE_TLS)
-lemma ALG_CANON_PREFS_MONO: "ALL (l::bool list) b::bool list list.
- alg_measure (alg_canon_prefs l b) <= alg_measure (l # b)"
+lemma ALG_CANON_PREFS_MONO: "alg_measure (alg_canon_prefs l b) <= alg_measure (l # b)"
by (import prob ALG_CANON_PREFS_MONO)
-lemma ALG_CANON_FIND_MONO: "ALL (l::bool list) b::bool list list.
- alg_measure (alg_canon_find l b) <= alg_measure (l # b)"
+lemma ALG_CANON_FIND_MONO: "alg_measure (alg_canon_find l b) <= alg_measure (l # b)"
by (import prob ALG_CANON_FIND_MONO)
-lemma ALG_CANON1_MONO: "ALL x::bool list list. alg_measure (alg_canon1 x) <= alg_measure x"
+lemma ALG_CANON1_MONO: "alg_measure (alg_canon1 x) <= alg_measure x"
by (import prob ALG_CANON1_MONO)
-lemma ALG_CANON_MERGE_MONO: "ALL (l::bool list) b::bool list list.
- alg_measure (alg_canon_merge l b) <= alg_measure (l # b)"
+lemma ALG_CANON_MERGE_MONO: "alg_measure (alg_canon_merge l b) <= alg_measure (l # b)"
by (import prob ALG_CANON_MERGE_MONO)
-lemma ALG_CANON2_MONO: "ALL x::bool list list. alg_measure (alg_canon2 x) <= alg_measure x"
+lemma ALG_CANON2_MONO: "alg_measure (alg_canon2 x) <= alg_measure x"
by (import prob ALG_CANON2_MONO)
-lemma ALG_CANON_MONO: "ALL l::bool list list. alg_measure (alg_canon l) <= alg_measure l"
+lemma ALG_CANON_MONO: "alg_measure (alg_canon l) <= alg_measure l"
by (import prob ALG_CANON_MONO)
-lemma ALGEBRA_MEASURE_DEF_ALT: "ALL l::bool list list. algebra_measure l = alg_measure (alg_canon l)"
+lemma ALGEBRA_MEASURE_DEF_ALT: "algebra_measure l = alg_measure (alg_canon l)"
by (import prob ALGEBRA_MEASURE_DEF_ALT)
lemma ALGEBRA_MEASURE_BASIC: "algebra_measure [] = 0 &
-algebra_measure [[]] = 1 & (ALL b::bool. algebra_measure [[b]] = 1 / 2)"
+algebra_measure [[]] = 1 & (ALL b. algebra_measure [[b]] = 1 / 2)"
by (import prob ALGEBRA_MEASURE_BASIC)
-lemma ALGEBRA_CANON_MEASURE_MAX: "ALL l::bool list list. algebra_canon l --> alg_measure l <= 1"
+lemma ALGEBRA_CANON_MEASURE_MAX: "algebra_canon l ==> alg_measure l <= 1"
by (import prob ALGEBRA_CANON_MEASURE_MAX)
-lemma ALGEBRA_MEASURE_MAX: "ALL l::bool list list. algebra_measure l <= 1"
+lemma ALGEBRA_MEASURE_MAX: "algebra_measure l <= 1"
by (import prob ALGEBRA_MEASURE_MAX)
-lemma ALGEBRA_MEASURE_MONO_EMBED: "ALL (x::bool list list) xa::bool list list.
- SUBSET (algebra_embed x) (algebra_embed xa) -->
- algebra_measure x <= algebra_measure xa"
+lemma ALGEBRA_MEASURE_MONO_EMBED: "SUBSET (algebra_embed x) (algebra_embed xa)
+==> algebra_measure x <= algebra_measure xa"
by (import prob ALGEBRA_MEASURE_MONO_EMBED)
-lemma ALG_MEASURE_COMPL: "ALL l::bool list list.
- algebra_canon l -->
- (ALL c::bool list list.
- algebra_canon c -->
- COMPL (algebra_embed l) = algebra_embed c -->
- alg_measure l + alg_measure c = 1)"
+lemma ALG_MEASURE_COMPL: "[| algebra_canon l; algebra_canon c;
+ COMPL (algebra_embed l) = algebra_embed c |]
+==> alg_measure l + alg_measure c = 1"
by (import prob ALG_MEASURE_COMPL)
-lemma ALG_MEASURE_ADDITIVE: "ALL l::bool list list.
- algebra_canon l -->
- (ALL c::bool list list.
- algebra_canon c -->
- (ALL d::bool list list.
- algebra_canon d -->
- pred_set.INTER (algebra_embed c) (algebra_embed d) = EMPTY &
- algebra_embed l =
- pred_set.UNION (algebra_embed c) (algebra_embed d) -->
- alg_measure l = alg_measure c + alg_measure d))"
+lemma ALG_MEASURE_ADDITIVE: "[| algebra_canon l; algebra_canon c; algebra_canon d;
+ pred_set.INTER (algebra_embed c) (algebra_embed d) = EMPTY &
+ algebra_embed l = pred_set.UNION (algebra_embed c) (algebra_embed d) |]
+==> alg_measure l = alg_measure c + alg_measure d"
by (import prob ALG_MEASURE_ADDITIVE)
-lemma PROB_ALGEBRA: "ALL l::bool list list. prob (algebra_embed l) = algebra_measure l"
+lemma PROB_ALGEBRA: "prob (algebra_embed l) = algebra_measure l"
by (import prob PROB_ALGEBRA)
lemma PROB_BASIC: "prob EMPTY = 0 &
-prob pred_set.UNIV = 1 &
-(ALL b::bool. prob (%s::nat => bool. SHD s = b) = 1 / 2)"
+prob pred_set.UNIV = 1 & (ALL b. prob (%s. SHD s = b) = 1 / 2)"
by (import prob PROB_BASIC)
-lemma PROB_ADDITIVE: "ALL (s::(nat => bool) => bool) t::(nat => bool) => bool.
- measurable s & measurable t & pred_set.INTER s t = EMPTY -->
- prob (pred_set.UNION s t) = prob s + prob t"
+lemma PROB_ADDITIVE: "measurable s & measurable t & pred_set.INTER s t = EMPTY
+==> prob (pred_set.UNION s t) = prob s + prob t"
by (import prob PROB_ADDITIVE)
-lemma PROB_COMPL: "ALL s::(nat => bool) => bool. measurable s --> prob (COMPL s) = 1 - prob s"
+lemma PROB_COMPL: "measurable s ==> prob (COMPL s) = 1 - prob s"
by (import prob PROB_COMPL)
-lemma PROB_SUP_EXISTS1: "ALL s::(nat => bool) => bool.
- EX (x::real) b::bool list list.
- algebra_measure b = x & SUBSET (algebra_embed b) s"
+lemma PROB_SUP_EXISTS1: "EX x b. algebra_measure b = x & SUBSET (algebra_embed b) s"
by (import prob PROB_SUP_EXISTS1)
-lemma PROB_SUP_EXISTS2: "ALL s::(nat => bool) => bool.
- EX x::real.
- ALL r::real.
- (EX b::bool list list.
- algebra_measure b = r & SUBSET (algebra_embed b) s) -->
+lemma PROB_SUP_EXISTS2: "EX x. ALL r.
+ (EX b. algebra_measure b = r & SUBSET (algebra_embed b) s) -->
r <= x"
by (import prob PROB_SUP_EXISTS2)
-lemma PROB_LE_X: "ALL (s::(nat => bool) => bool) x::real.
- (ALL s'::(nat => bool) => bool.
- measurable s' & SUBSET s' s --> prob s' <= x) -->
- prob s <= x"
+lemma PROB_LE_X: "(!!s'. measurable s' & SUBSET s' s ==> prob s' <= x) ==> prob s <= x"
by (import prob PROB_LE_X)
-lemma X_LE_PROB: "ALL (s::(nat => bool) => bool) x::real.
- (EX s'::(nat => bool) => bool.
- measurable s' & SUBSET s' s & x <= prob s') -->
- x <= prob s"
+lemma X_LE_PROB: "EX s'. measurable s' & SUBSET s' s & x <= prob s' ==> x <= prob s"
by (import prob X_LE_PROB)
-lemma PROB_SUBSET_MONO: "ALL (s::(nat => bool) => bool) t::(nat => bool) => bool.
- SUBSET s t --> prob s <= prob t"
+lemma PROB_SUBSET_MONO: "SUBSET s t ==> prob s <= prob t"
by (import prob PROB_SUBSET_MONO)
-lemma PROB_ALG: "ALL x::bool list. prob (alg_embed x) = (1 / 2) ^ length x"
+lemma PROB_ALG: "prob (alg_embed x) = (1 / 2) ^ length x"
by (import prob PROB_ALG)
-lemma PROB_STL: "ALL p::(nat => bool) => bool. measurable p --> prob (p o STL) = prob p"
+lemma PROB_STL: "measurable p ==> prob (p o STL) = prob p"
by (import prob PROB_STL)
-lemma PROB_SDROP: "ALL (n::nat) p::(nat => bool) => bool.
- measurable p --> prob (p o SDROP n) = prob p"
+lemma PROB_SDROP: "measurable p ==> prob (p o SDROP n) = prob p"
by (import prob PROB_SDROP)
-lemma PROB_INTER_HALVES: "ALL p::(nat => bool) => bool.
- measurable p -->
- prob (pred_set.INTER (%x::nat => bool. SHD x = True) p) +
- prob (pred_set.INTER (%x::nat => bool. SHD x = False) p) =
- prob p"
+lemma PROB_INTER_HALVES: "measurable p
+==> prob (pred_set.INTER (%x. SHD x = True) p) +
+ prob (pred_set.INTER (%x. SHD x = False) p) =
+ prob p"
by (import prob PROB_INTER_HALVES)
-lemma PROB_INTER_SHD: "ALL (b::bool) p::(nat => bool) => bool.
- measurable p -->
- prob (pred_set.INTER (%x::nat => bool. SHD x = b) (p o STL)) =
- 1 / 2 * prob p"
+lemma PROB_INTER_SHD: "measurable p
+==> prob (pred_set.INTER (%x. SHD x = b) (p o STL)) = 1 / 2 * prob p"
by (import prob PROB_INTER_SHD)
-lemma ALGEBRA_MEASURE_POS: "ALL l::bool list list. 0 <= algebra_measure l"
+lemma ALGEBRA_MEASURE_POS: "0 <= algebra_measure l"
by (import prob ALGEBRA_MEASURE_POS)
-lemma ALGEBRA_MEASURE_RANGE: "ALL l::bool list list. 0 <= algebra_measure l & algebra_measure l <= 1"
+lemma ALGEBRA_MEASURE_RANGE: "0 <= algebra_measure l & algebra_measure l <= 1"
by (import prob ALGEBRA_MEASURE_RANGE)
-lemma PROB_POS: "ALL p::(nat => bool) => bool. 0 <= prob p"
+lemma PROB_POS: "0 <= prob p"
by (import prob PROB_POS)
-lemma PROB_MAX: "ALL p::(nat => bool) => bool. prob p <= 1"
+lemma PROB_MAX: "prob p <= 1"
by (import prob PROB_MAX)
-lemma PROB_RANGE: "ALL p::(nat => bool) => bool. 0 <= prob p & prob p <= 1"
+lemma PROB_RANGE: "0 <= prob p & prob p <= 1"
by (import prob PROB_RANGE)
-lemma ABS_PROB: "ALL p::(nat => bool) => bool. abs (prob p) = prob p"
+lemma ABS_PROB: "abs (prob p) = prob p"
by (import prob ABS_PROB)
-lemma PROB_SHD: "ALL b::bool. prob (%s::nat => bool. SHD s = b) = 1 / 2"
+lemma PROB_SHD: "prob (%s. SHD s = b) = 1 / 2"
by (import prob PROB_SHD)
-lemma PROB_COMPL_LE1: "ALL (p::(nat => bool) => bool) r::real.
- measurable p --> (prob (COMPL p) <= r) = (1 - r <= prob p)"
+lemma PROB_COMPL_LE1: "measurable p ==> (prob (COMPL p) <= r) = (1 - r <= prob p)"
by (import prob PROB_COMPL_LE1)
;end_setup
@@ -1583,41 +1136,41 @@
pseudo_linear_tl :: "nat => nat => nat => nat => nat"
defs
- pseudo_linear_tl_primdef: "pseudo_linear_tl ==
-%(a::nat) (b::nat) (n::nat) x::nat. (a * x + b) mod (2 * n + 1)"
+ pseudo_linear_tl_primdef: "pseudo_linear_tl == %a b n x. (a * x + b) mod (2 * n + 1)"
-lemma pseudo_linear_tl_def: "ALL (a::nat) (b::nat) (n::nat) x::nat.
- pseudo_linear_tl a b n x = (a * x + b) mod (2 * n + 1)"
+lemma pseudo_linear_tl_def: "pseudo_linear_tl a b n x = (a * x + b) mod (2 * n + 1)"
by (import prob_pseudo pseudo_linear_tl_def)
-lemma PSEUDO_LINEAR1_EXECUTE: "EX x::nat => nat => bool.
- (ALL xa::nat. SHD (x xa) = pseudo_linear_hd xa) &
- (ALL xa::nat.
- STL (x xa) =
- x (pseudo_linear_tl
- (NUMERAL
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT2 (NUMERAL_BIT1 (NUMERAL_BIT2 ALT_ZERO)))))))
- (NUMERAL
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT2 ALT_ZERO)))))))
- (NUMERAL
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1 (NUMERAL_BIT2 (NUMERAL_BIT1 ALT_ZERO)))))))
- xa))"
+lemma PSEUDO_LINEAR1_EXECUTE: "EX x. (ALL xa. SHD (x xa) = pseudo_linear_hd xa) &
+ (ALL xa.
+ STL (x xa) =
+ x (pseudo_linear_tl
+ (NUMERAL
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT2
+ (NUMERAL_BIT1 (NUMERAL_BIT2 ALT_ZERO)))))))
+ (NUMERAL
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1 (NUMERAL_BIT2 ALT_ZERO)))))))
+ (NUMERAL
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT2 (NUMERAL_BIT1 ALT_ZERO)))))))
+ xa))"
by (import prob_pseudo PSEUDO_LINEAR1_EXECUTE)
consts
pseudo_linear1 :: "nat => nat => bool"
-specification (pseudo_linear1_primdef: pseudo_linear1) pseudo_linear1_def: "(ALL x::nat. SHD (pseudo_linear1 x) = pseudo_linear_hd x) &
-(ALL x::nat.
+specification (pseudo_linear1_primdef: pseudo_linear1) pseudo_linear1_def: "(ALL x. SHD (pseudo_linear1 x) = pseudo_linear_hd x) &
+(ALL x.
STL (pseudo_linear1 x) =
pseudo_linear1
(pseudo_linear_tl
@@ -1657,13 +1210,11 @@
defs
indep_set_primdef: "indep_set ==
-%(p::(nat => bool) => bool) q::(nat => bool) => bool.
- measurable p & measurable q & prob (pred_set.INTER p q) = prob p * prob q"
+%p q. measurable p &
+ measurable q & prob (pred_set.INTER p q) = prob p * prob q"
-lemma indep_set_def: "ALL (p::(nat => bool) => bool) q::(nat => bool) => bool.
- indep_set p q =
- (measurable p &
- measurable q & prob (pred_set.INTER p q) = prob p * prob q)"
+lemma indep_set_def: "indep_set p q =
+(measurable p & measurable q & prob (pred_set.INTER p q) = prob p * prob q)"
by (import prob_indep indep_set_def)
consts
@@ -1671,24 +1222,19 @@
defs
alg_cover_set_primdef: "alg_cover_set ==
-%l::bool list list.
- alg_sorted l & alg_prefixfree l & algebra_embed l = pred_set.UNIV"
+%l. alg_sorted l & alg_prefixfree l & algebra_embed l = pred_set.UNIV"
-lemma alg_cover_set_def: "ALL l::bool list list.
- alg_cover_set l =
- (alg_sorted l & alg_prefixfree l & algebra_embed l = pred_set.UNIV)"
+lemma alg_cover_set_def: "alg_cover_set l =
+(alg_sorted l & alg_prefixfree l & algebra_embed l = pred_set.UNIV)"
by (import prob_indep alg_cover_set_def)
consts
alg_cover :: "bool list list => (nat => bool) => bool list"
defs
- alg_cover_primdef: "alg_cover ==
-%(l::bool list list) x::nat => bool.
- SOME b::bool list. b mem l & alg_embed b x"
+ alg_cover_primdef: "alg_cover == %l x. SOME b. List.member l b & alg_embed b x"
-lemma alg_cover_def: "ALL (l::bool list list) x::nat => bool.
- alg_cover l x = (SOME b::bool list. b mem l & alg_embed b x)"
+lemma alg_cover_def: "alg_cover l x = (SOME b. List.member l b & alg_embed b x)"
by (import prob_indep alg_cover_def)
consts
@@ -1696,195 +1242,148 @@
defs
indep_primdef: "indep ==
-%f::(nat => bool) => 'a::type * (nat => bool).
- EX (l::bool list list) r::bool list => 'a::type.
- alg_cover_set l &
- (ALL s::nat => bool.
- f s =
- (let c::bool list = alg_cover l s in (r c, SDROP (length c) s)))"
+%f. EX l r.
+ alg_cover_set l &
+ (ALL s. f s = (let c = alg_cover l s in (r c, SDROP (length c) s)))"
-lemma indep_def: "ALL f::(nat => bool) => 'a::type * (nat => bool).
- indep f =
- (EX (l::bool list list) r::bool list => 'a::type.
- alg_cover_set l &
- (ALL s::nat => bool.
- f s =
- (let c::bool list = alg_cover l s in (r c, SDROP (length c) s))))"
+lemma indep_def: "indep f =
+(EX l r.
+ alg_cover_set l &
+ (ALL s. f s = (let c = alg_cover l s in (r c, SDROP (length c) s))))"
by (import prob_indep indep_def)
-lemma INDEP_SET_BASIC: "ALL p::(nat => bool) => bool.
- measurable p --> indep_set EMPTY p & indep_set pred_set.UNIV p"
+lemma INDEP_SET_BASIC: "measurable p ==> indep_set EMPTY p & indep_set pred_set.UNIV p"
by (import prob_indep INDEP_SET_BASIC)
-lemma INDEP_SET_SYM: "ALL (p::(nat => bool) => bool) q::(nat => bool) => bool.
- indep_set p q = indep_set p q"
+lemma INDEP_SET_SYM: "indep_set p q = indep_set p q"
by (import prob_indep INDEP_SET_SYM)
-lemma INDEP_SET_DISJOINT_DECOMP: "ALL (p::(nat => bool) => bool) (q::(nat => bool) => bool)
- r::(nat => bool) => bool.
- indep_set p r & indep_set q r & pred_set.INTER p q = EMPTY -->
- indep_set (pred_set.UNION p q) r"
+lemma INDEP_SET_DISJOINT_DECOMP: "indep_set p r & indep_set q r & pred_set.INTER p q = EMPTY
+==> indep_set (pred_set.UNION p q) r"
by (import prob_indep INDEP_SET_DISJOINT_DECOMP)
lemma ALG_COVER_SET_BASIC: "~ alg_cover_set [] & alg_cover_set [[]] & alg_cover_set [[True], [False]]"
by (import prob_indep ALG_COVER_SET_BASIC)
-lemma ALG_COVER_WELL_DEFINED: "ALL (l::bool list list) x::nat => bool.
- alg_cover_set l --> alg_cover l x mem l & alg_embed (alg_cover l x) x"
+lemma ALG_COVER_WELL_DEFINED: "alg_cover_set l
+==> List.member l (alg_cover l x) & alg_embed (alg_cover l x) x"
by (import prob_indep ALG_COVER_WELL_DEFINED)
lemma ALG_COVER_UNIV: "alg_cover [[]] = K []"
by (import prob_indep ALG_COVER_UNIV)
-lemma MAP_CONS_TL_FILTER: "ALL (l::bool list list) b::bool.
- ~ [] mem l -->
- map (op # b) (map tl [x::bool list<-l. hd x = b]) =
- [x::bool list<-l. hd x = b]"
+lemma MAP_CONS_TL_FILTER: "~ List.member (l::bool list list) []
+==> map (op # (b::bool)) (map tl [x::bool list<-l. hd x = b]) =
+ [x::bool list<-l. hd x = b]"
by (import prob_indep MAP_CONS_TL_FILTER)
-lemma ALG_COVER_SET_CASES_THM: "ALL l::bool list list.
- alg_cover_set l =
- (l = [[]] |
- (EX (x::bool list list) xa::bool list list.
- alg_cover_set x &
- alg_cover_set xa & l = map (op # True) x @ map (op # False) xa))"
+lemma ALG_COVER_SET_CASES_THM: "alg_cover_set l =
+(l = [[]] |
+ (EX x xa.
+ alg_cover_set x &
+ alg_cover_set xa & l = map (op # True) x @ map (op # False) xa))"
by (import prob_indep ALG_COVER_SET_CASES_THM)
-lemma ALG_COVER_SET_CASES: "ALL P::bool list list => bool.
- P [[]] &
- (ALL (l1::bool list list) l2::bool list list.
+lemma ALG_COVER_SET_CASES: "[| P [[]] &
+ (ALL l1 l2.
alg_cover_set l1 &
alg_cover_set l2 &
alg_cover_set (map (op # True) l1 @ map (op # False) l2) -->
- P (map (op # True) l1 @ map (op # False) l2)) -->
- (ALL l::bool list list. alg_cover_set l --> P l)"
+ P (map (op # True) l1 @ map (op # False) l2));
+ alg_cover_set l |]
+==> P l"
by (import prob_indep ALG_COVER_SET_CASES)
-lemma ALG_COVER_SET_INDUCTION: "ALL P::bool list list => bool.
- P [[]] &
- (ALL (l1::bool list list) l2::bool list list.
+lemma ALG_COVER_SET_INDUCTION: "[| P [[]] &
+ (ALL l1 l2.
alg_cover_set l1 &
alg_cover_set l2 &
P l1 &
P l2 & alg_cover_set (map (op # True) l1 @ map (op # False) l2) -->
- P (map (op # True) l1 @ map (op # False) l2)) -->
- (ALL l::bool list list. alg_cover_set l --> P l)"
+ P (map (op # True) l1 @ map (op # False) l2));
+ alg_cover_set l |]
+==> P l"
by (import prob_indep ALG_COVER_SET_INDUCTION)
-lemma ALG_COVER_EXISTS_UNIQUE: "ALL l::bool list list.
- alg_cover_set l -->
- (ALL s::nat => bool. EX! x::bool list. x mem l & alg_embed x s)"
+lemma ALG_COVER_EXISTS_UNIQUE: "alg_cover_set l ==> EX! x. List.member l x & alg_embed x s"
by (import prob_indep ALG_COVER_EXISTS_UNIQUE)
-lemma ALG_COVER_UNIQUE: "ALL (l::bool list list) (x::bool list) s::nat => bool.
- alg_cover_set l & x mem l & alg_embed x s --> alg_cover l s = x"
+lemma ALG_COVER_UNIQUE: "alg_cover_set l & List.member l x & alg_embed x s ==> alg_cover l s = x"
by (import prob_indep ALG_COVER_UNIQUE)
-lemma ALG_COVER_STEP: "ALL (l1::bool list list) (l2::bool list list) (h::bool) t::nat => bool.
- alg_cover_set l1 & alg_cover_set l2 -->
- alg_cover (map (op # True) l1 @ map (op # False) l2) (SCONS h t) =
- (if h then True # alg_cover l1 t else False # alg_cover l2 t)"
+lemma ALG_COVER_STEP: "alg_cover_set l1 & alg_cover_set l2
+==> alg_cover (map (op # True) l1 @ map (op # False) l2) (SCONS h t) =
+ (if h then True # alg_cover l1 t else False # alg_cover l2 t)"
by (import prob_indep ALG_COVER_STEP)
-lemma ALG_COVER_HEAD: "ALL l::bool list list.
- alg_cover_set l -->
- (ALL f::bool list => bool. f o alg_cover l = algebra_embed (filter f l))"
+lemma ALG_COVER_HEAD: "alg_cover_set l ==> f o alg_cover l = algebra_embed (filter f l)"
by (import prob_indep ALG_COVER_HEAD)
-lemma ALG_COVER_TAIL_STEP: "ALL (l1::bool list list) (l2::bool list list) q::(nat => bool) => bool.
- alg_cover_set l1 & alg_cover_set l2 -->
- q o
- (%x::nat => bool.
- SDROP
- (length (alg_cover (map (op # True) l1 @ map (op # False) l2) x))
- x) =
- pred_set.UNION
- (pred_set.INTER (%x::nat => bool. SHD x = True)
- (q o ((%x::nat => bool. SDROP (length (alg_cover l1 x)) x) o STL)))
- (pred_set.INTER (%x::nat => bool. SHD x = False)
- (q o ((%x::nat => bool. SDROP (length (alg_cover l2 x)) x) o STL)))"
+lemma ALG_COVER_TAIL_STEP: "alg_cover_set l1 & alg_cover_set l2
+==> q o
+ (%x. SDROP
+ (length (alg_cover (map (op # True) l1 @ map (op # False) l2) x))
+ x) =
+ pred_set.UNION
+ (pred_set.INTER (%x. SHD x = True)
+ (q o ((%x. SDROP (length (alg_cover l1 x)) x) o STL)))
+ (pred_set.INTER (%x. SHD x = False)
+ (q o ((%x. SDROP (length (alg_cover l2 x)) x) o STL)))"
by (import prob_indep ALG_COVER_TAIL_STEP)
-lemma ALG_COVER_TAIL_MEASURABLE: "ALL l::bool list list.
- alg_cover_set l -->
- (ALL q::(nat => bool) => bool.
- measurable
- (q o (%x::nat => bool. SDROP (length (alg_cover l x)) x)) =
- measurable q)"
+lemma ALG_COVER_TAIL_MEASURABLE: "alg_cover_set l
+==> measurable (q o (%x. SDROP (length (alg_cover l x)) x)) = measurable q"
by (import prob_indep ALG_COVER_TAIL_MEASURABLE)
-lemma ALG_COVER_TAIL_PROB: "ALL l::bool list list.
- alg_cover_set l -->
- (ALL q::(nat => bool) => bool.
- measurable q -->
- prob (q o (%x::nat => bool. SDROP (length (alg_cover l x)) x)) =
- prob q)"
+lemma ALG_COVER_TAIL_PROB: "[| alg_cover_set l; measurable q |]
+==> prob (q o (%x. SDROP (length (alg_cover l x)) x)) = prob q"
by (import prob_indep ALG_COVER_TAIL_PROB)
-lemma INDEP_INDEP_SET_LEMMA: "ALL l::bool list list.
- alg_cover_set l -->
- (ALL q::(nat => bool) => bool.
- measurable q -->
- (ALL x::bool list.
- x mem l -->
- prob
- (pred_set.INTER (alg_embed x)
- (q o (%x::nat => bool. SDROP (length (alg_cover l x)) x))) =
- (1 / 2) ^ length x * prob q))"
+lemma INDEP_INDEP_SET_LEMMA: "[| alg_cover_set l; measurable q; List.member l x |]
+==> prob
+ (pred_set.INTER (alg_embed x)
+ (q o (%x. SDROP (length (alg_cover l x)) x))) =
+ (1 / 2) ^ length x * prob q"
by (import prob_indep INDEP_INDEP_SET_LEMMA)
-lemma INDEP_SET_LIST: "ALL (q::(nat => bool) => bool) l::bool list list.
- alg_sorted l &
- alg_prefixfree l &
- measurable q &
- (ALL x::bool list. x mem l --> indep_set (alg_embed x) q) -->
- indep_set (algebra_embed l) q"
+lemma INDEP_SET_LIST: "alg_sorted l &
+alg_prefixfree l &
+measurable q & (ALL x. List.member l x --> indep_set (alg_embed x) q)
+==> indep_set (algebra_embed l) q"
by (import prob_indep INDEP_SET_LIST)
-lemma INDEP_INDEP_SET: "ALL (f::(nat => bool) => 'a::type * (nat => bool)) (p::'a::type => bool)
- q::(nat => bool) => bool.
- indep f & measurable q --> indep_set (p o (fst o f)) (q o (snd o f))"
+lemma INDEP_INDEP_SET: "indep f & measurable q ==> indep_set (p o (fst o f)) (q o (snd o f))"
by (import prob_indep INDEP_INDEP_SET)
-lemma INDEP_UNIT: "ALL x::'a::type. indep (UNIT x)"
+lemma INDEP_UNIT: "indep (UNIT x)"
by (import prob_indep INDEP_UNIT)
lemma INDEP_SDEST: "indep SDEST"
by (import prob_indep INDEP_SDEST)
-lemma BIND_STEP: "ALL f::(nat => bool) => 'a::type * (nat => bool).
- BIND SDEST (%k::bool. f o SCONS k) = f"
+lemma BIND_STEP: "BIND SDEST (%k. f o SCONS k) = f"
by (import prob_indep BIND_STEP)
-lemma INDEP_BIND_SDEST: "ALL f::bool => (nat => bool) => 'a::type * (nat => bool).
- (ALL x::bool. indep (f x)) --> indep (BIND SDEST f)"
+lemma INDEP_BIND_SDEST: "(!!x. indep (f x)) ==> indep (BIND SDEST f)"
by (import prob_indep INDEP_BIND_SDEST)
-lemma INDEP_BIND: "ALL (f::(nat => bool) => 'a::type * (nat => bool))
- g::'a::type => (nat => bool) => 'b::type * (nat => bool).
- indep f & (ALL x::'a::type. indep (g x)) --> indep (BIND f g)"
+lemma INDEP_BIND: "indep f & (ALL x. indep (g x)) ==> indep (BIND f g)"
by (import prob_indep INDEP_BIND)
-lemma INDEP_PROB: "ALL (f::(nat => bool) => 'a::type * (nat => bool)) (p::'a::type => bool)
- q::(nat => bool) => bool.
- indep f & measurable q -->
- prob (pred_set.INTER (p o (fst o f)) (q o (snd o f))) =
- prob (p o (fst o f)) * prob q"
+lemma INDEP_PROB: "indep f & measurable q
+==> prob (pred_set.INTER (p o (fst o f)) (q o (snd o f))) =
+ prob (p o (fst o f)) * prob q"
by (import prob_indep INDEP_PROB)
-lemma INDEP_MEASURABLE1: "ALL (f::(nat => bool) => 'a::type * (nat => bool)) p::'a::type => bool.
- indep f --> measurable (p o (fst o f))"
+lemma INDEP_MEASURABLE1: "indep f ==> measurable (p o (fst o f))"
by (import prob_indep INDEP_MEASURABLE1)
-lemma INDEP_MEASURABLE2: "ALL (f::(nat => bool) => 'a::type * (nat => bool)) q::(nat => bool) => bool.
- indep f & measurable q --> measurable (q o (snd o f))"
+lemma INDEP_MEASURABLE2: "indep f & measurable q ==> measurable (q o (snd o f))"
by (import prob_indep INDEP_MEASURABLE2)
-lemma PROB_INDEP_BOUND: "ALL (f::(nat => bool) => nat * (nat => bool)) n::nat.
- indep f -->
- prob (%s::nat => bool. fst (f s) < Suc n) =
- prob (%s::nat => bool. fst (f s) < n) +
- prob (%s::nat => bool. fst (f s) = n)"
+lemma PROB_INDEP_BOUND: "indep f
+==> prob (%s. fst (f s) < Suc n) =
+ prob (%s. fst (f s) < n) + prob (%s. fst (f s) = n)"
by (import prob_indep PROB_INDEP_BOUND)
;end_setup
@@ -1896,47 +1395,36 @@
defs
unif_bound_primdef: "unif_bound ==
-WFREC
- (SOME R::nat => nat => bool. WF R & (ALL v::nat. R (Suc v div 2) (Suc v)))
- (%unif_bound::nat => nat.
- nat_case 0 (%v1::nat. Suc (unif_bound (Suc v1 div 2))))"
+WFREC (SOME R. WF R & (ALL v. R (Suc v div 2) (Suc v)))
+ (%unif_bound. nat_case 0 (%v1. Suc (unif_bound (Suc v1 div 2))))"
lemma unif_bound_primitive_def: "unif_bound =
-WFREC
- (SOME R::nat => nat => bool. WF R & (ALL v::nat. R (Suc v div 2) (Suc v)))
- (%unif_bound::nat => nat.
- nat_case 0 (%v1::nat. Suc (unif_bound (Suc v1 div 2))))"
+WFREC (SOME R. WF R & (ALL v. R (Suc v div 2) (Suc v)))
+ (%unif_bound. nat_case 0 (%v1. Suc (unif_bound (Suc v1 div 2))))"
by (import prob_uniform unif_bound_primitive_def)
-lemma unif_bound_def: "unif_bound 0 = 0 &
-unif_bound (Suc (v::nat)) = Suc (unif_bound (Suc v div 2))"
+lemma unif_bound_def: "unif_bound 0 = 0 & unif_bound (Suc v) = Suc (unif_bound (Suc v div 2))"
by (import prob_uniform unif_bound_def)
-lemma unif_bound_ind: "ALL P::nat => bool.
- P 0 & (ALL v::nat. P (Suc v div 2) --> P (Suc v)) --> All P"
+lemma unif_bound_ind: "P 0 & (ALL v. P (Suc v div 2) --> P (Suc v)) ==> P x"
by (import prob_uniform unif_bound_ind)
-definition unif_tupled :: "nat * (nat => bool) => nat * (nat => bool)" where
+definition
+ unif_tupled :: "nat * (nat => bool) => nat * (nat => bool)" where
"unif_tupled ==
-WFREC
- (SOME R::nat * (nat => bool) => nat * (nat => bool) => bool.
- WF R & (ALL (s::nat => bool) v2::nat. R (Suc v2 div 2, s) (Suc v2, s)))
- (%(unif_tupled::nat * (nat => bool) => nat * (nat => bool)) (v::nat,
- v1::nat => bool).
+WFREC (SOME R. WF R & (ALL s v2. R (Suc v2 div 2, s) (Suc v2, s)))
+ (%unif_tupled (v, v1).
case v of 0 => (0, v1)
- | Suc (v3::nat) =>
- let (m::nat, s'::nat => bool) = unif_tupled (Suc v3 div 2, v1)
+ | Suc v3 =>
+ let (m, s') = unif_tupled (Suc v3 div 2, v1)
in (if SHD s' then 2 * m + 1 else 2 * m, STL s'))"
lemma unif_tupled_primitive_def: "unif_tupled =
-WFREC
- (SOME R::nat * (nat => bool) => nat * (nat => bool) => bool.
- WF R & (ALL (s::nat => bool) v2::nat. R (Suc v2 div 2, s) (Suc v2, s)))
- (%(unif_tupled::nat * (nat => bool) => nat * (nat => bool)) (v::nat,
- v1::nat => bool).
+WFREC (SOME R. WF R & (ALL s v2. R (Suc v2 div 2, s) (Suc v2, s)))
+ (%unif_tupled (v, v1).
case v of 0 => (0, v1)
- | Suc (v3::nat) =>
- let (m::nat, s'::nat => bool) = unif_tupled (Suc v3 div 2, v1)
+ | Suc v3 =>
+ let (m, s') = unif_tupled (Suc v3 div 2, v1)
in (if SHD s' then 2 * m + 1 else 2 * m, STL s'))"
by (import prob_uniform unif_tupled_primitive_def)
@@ -1944,247 +1432,160 @@
unif :: "nat => (nat => bool) => nat * (nat => bool)"
defs
- unif_primdef: "unif == %(x::nat) x1::nat => bool. unif_tupled (x, x1)"
+ unif_primdef: "unif == %x x1. unif_tupled (x, x1)"
-lemma unif_curried_def: "ALL (x::nat) x1::nat => bool. unif x x1 = unif_tupled (x, x1)"
+lemma unif_curried_def: "unif x x1 = unif_tupled (x, x1)"
by (import prob_uniform unif_curried_def)
-lemma unif_def: "unif 0 (s::nat => bool) = (0, s) &
-unif (Suc (v2::nat)) s =
-(let (m::nat, s'::nat => bool) = unif (Suc v2 div 2) s
+lemma unif_def: "unif 0 s = (0, s) &
+unif (Suc v2) s =
+(let (m, s') = unif (Suc v2 div 2) s
in (if SHD s' then 2 * m + 1 else 2 * m, STL s'))"
by (import prob_uniform unif_def)
-lemma unif_ind: "ALL P::nat => (nat => bool) => bool.
- All (P 0) &
- (ALL (v2::nat) s::nat => bool. P (Suc v2 div 2) s --> P (Suc v2) s) -->
- (ALL v::nat. All (P v))"
+lemma unif_ind: "All ((P::nat => (nat => bool) => bool) (0::nat)) &
+(ALL (v2::nat) s::nat => bool. P (Suc v2 div (2::nat)) s --> P (Suc v2) s)
+==> P (v::nat) (x::nat => bool)"
by (import prob_uniform unif_ind)
-definition uniform_tupled :: "nat * nat * (nat => bool) => nat * (nat => bool)" where
+definition
+ uniform_tupled :: "nat * nat * (nat => bool) => nat * (nat => bool)" where
"uniform_tupled ==
WFREC
- (SOME R::nat * nat * (nat => bool) => nat * nat * (nat => bool) => bool.
+ (SOME R.
WF R &
- (ALL (t::nat) (s::nat => bool) (n::nat) (res::nat) s'::nat => bool.
+ (ALL t s n res s'.
(res, s') = unif n s & ~ res < Suc n -->
R (t, Suc n, s') (Suc t, Suc n, s)))
- (%(uniform_tupled::nat * nat * (nat => bool) => nat * (nat => bool))
- (v::nat, v1::nat * (nat => bool)).
- case v of
- 0 => (%(v3::nat, v4::nat => bool).
- case v3 of 0 => ARB | Suc (v5::nat) => (0, v4))
- v1
- | Suc (v2::nat) =>
- (%(v7::nat, v8::nat => bool).
- case v7 of 0 => ARB
- | Suc (v9::nat) =>
- let (res::nat, s'::nat => bool) = unif v9 v8
- in if res < Suc v9 then (res, s')
- else uniform_tupled (v2, Suc v9, s'))
- v1)"
+ (%uniform_tupled (v, v1).
+ case v of 0 => case v1 of (0, v4) => ARB | (Suc v5, v4) => (0, v4)
+ | Suc v2 =>
+ case v1 of (0, v8) => ARB
+ | (Suc v9, v8) =>
+ let (res, s') = unif v9 v8
+ in if res < Suc v9 then (res, s')
+ else uniform_tupled (v2, Suc v9, s'))"
lemma uniform_tupled_primitive_def: "uniform_tupled =
WFREC
- (SOME R::nat * nat * (nat => bool) => nat * nat * (nat => bool) => bool.
+ (SOME R.
WF R &
- (ALL (t::nat) (s::nat => bool) (n::nat) (res::nat) s'::nat => bool.
+ (ALL t s n res s'.
(res, s') = unif n s & ~ res < Suc n -->
R (t, Suc n, s') (Suc t, Suc n, s)))
- (%(uniform_tupled::nat * nat * (nat => bool) => nat * (nat => bool))
- (v::nat, v1::nat * (nat => bool)).
- case v of
- 0 => (%(v3::nat, v4::nat => bool).
- case v3 of 0 => ARB | Suc (v5::nat) => (0, v4))
- v1
- | Suc (v2::nat) =>
- (%(v7::nat, v8::nat => bool).
- case v7 of 0 => ARB
- | Suc (v9::nat) =>
- let (res::nat, s'::nat => bool) = unif v9 v8
- in if res < Suc v9 then (res, s')
- else uniform_tupled (v2, Suc v9, s'))
- v1)"
+ (%uniform_tupled (v, v1).
+ case v of 0 => case v1 of (0, v4) => ARB | (Suc v5, v4) => (0, v4)
+ | Suc v2 =>
+ case v1 of (0, v8) => ARB
+ | (Suc v9, v8) =>
+ let (res, s') = unif v9 v8
+ in if res < Suc v9 then (res, s')
+ else uniform_tupled (v2, Suc v9, s'))"
by (import prob_uniform uniform_tupled_primitive_def)
consts
uniform :: "nat => nat => (nat => bool) => nat * (nat => bool)"
defs
- uniform_primdef: "uniform == %(x::nat) (x1::nat) x2::nat => bool. uniform_tupled (x, x1, x2)"
+ uniform_primdef: "uniform == %x x1 x2. uniform_tupled (x, x1, x2)"
-lemma uniform_curried_def: "ALL (x::nat) (x1::nat) x2::nat => bool.
- uniform x x1 x2 = uniform_tupled (x, x1, x2)"
+lemma uniform_curried_def: "uniform x x1 x2 = uniform_tupled (x, x1, x2)"
by (import prob_uniform uniform_curried_def)
-lemma uniform_ind: "ALL P::nat => nat => (nat => bool) => bool.
- (ALL x::nat. All (P (Suc x) 0)) &
- All (P 0 0) &
- (ALL x::nat. All (P 0 (Suc x))) &
- (ALL (x::nat) (xa::nat) xb::nat => bool.
- (ALL (xc::nat) xd::nat => bool.
- (xc, xd) = unif xa xb & ~ xc < Suc xa --> P x (Suc xa) xd) -->
- P (Suc x) (Suc xa) xb) -->
- (ALL (x::nat) xa::nat. All (P x xa))"
+lemma uniform_ind: "(ALL x. All (P (Suc x) 0)) &
+All (P 0 0) &
+(ALL x. All (P 0 (Suc x))) &
+(ALL x xa xb.
+ (ALL xc xd.
+ (xc, xd) = unif xa xb & ~ xc < Suc xa --> P x (Suc xa) xd) -->
+ P (Suc x) (Suc xa) xb)
+==> P x xa xb"
by (import prob_uniform uniform_ind)
-lemma uniform_def: "uniform 0 (Suc (n::nat)) (s::nat => bool) = (0, s) &
-uniform (Suc (t::nat)) (Suc n) s =
-(let (xa::nat, x::nat => bool) = unif n s
+lemma uniform_def: "uniform 0 (Suc n) s = (0, s) &
+uniform (Suc t) (Suc n) s =
+(let (xa, x) = unif n s
in if xa < Suc n then (xa, x) else uniform t (Suc n) x)"
by (import prob_uniform uniform_def)
-lemma SUC_DIV_TWO_ZERO: "ALL n::nat. (Suc n div 2 = 0) = (n = 0)"
+lemma SUC_DIV_TWO_ZERO: "(Suc n div 2 = 0) = (n = 0)"
by (import prob_uniform SUC_DIV_TWO_ZERO)
-lemma UNIF_BOUND_LOWER: "ALL n::nat. n < 2 ^ unif_bound n"
+lemma UNIF_BOUND_LOWER: "n < 2 ^ unif_bound n"
by (import prob_uniform UNIF_BOUND_LOWER)
-lemma UNIF_BOUND_LOWER_SUC: "ALL n::nat. Suc n <= 2 ^ unif_bound n"
+lemma UNIF_BOUND_LOWER_SUC: "Suc n <= 2 ^ unif_bound n"
by (import prob_uniform UNIF_BOUND_LOWER_SUC)
-lemma UNIF_BOUND_UPPER: "ALL n::nat. n ~= 0 --> 2 ^ unif_bound n <= 2 * n"
+lemma UNIF_BOUND_UPPER: "n ~= 0 ==> 2 ^ unif_bound n <= 2 * n"
by (import prob_uniform UNIF_BOUND_UPPER)
-lemma UNIF_BOUND_UPPER_SUC: "ALL n::nat. 2 ^ unif_bound n <= Suc (2 * n)"
+lemma UNIF_BOUND_UPPER_SUC: "2 ^ unif_bound n <= Suc (2 * n)"
by (import prob_uniform UNIF_BOUND_UPPER_SUC)
lemma UNIF_DEF_MONAD: "unif 0 = UNIT 0 &
-(ALL n::nat.
+(ALL n.
unif (Suc n) =
BIND (unif (Suc n div 2))
- (%m::nat.
- BIND SDEST (%b::bool. UNIT (if b then 2 * m + 1 else 2 * m))))"
+ (%m. BIND SDEST (%b. UNIT (if b then 2 * m + 1 else 2 * m))))"
by (import prob_uniform UNIF_DEF_MONAD)
-lemma UNIFORM_DEF_MONAD: "(ALL x::nat. uniform 0 (Suc x) = UNIT 0) &
-(ALL (x::nat) xa::nat.
+lemma UNIFORM_DEF_MONAD: "(ALL x. uniform 0 (Suc x) = UNIT 0) &
+(ALL x xa.
uniform (Suc x) (Suc xa) =
- BIND (unif xa)
- (%m::nat. if m < Suc xa then UNIT m else uniform x (Suc xa)))"
+ BIND (unif xa) (%m. if m < Suc xa then UNIT m else uniform x (Suc xa)))"
by (import prob_uniform UNIFORM_DEF_MONAD)
-lemma INDEP_UNIF: "ALL n::nat. indep (unif n)"
+lemma INDEP_UNIF: "indep (unif n)"
by (import prob_uniform INDEP_UNIF)
-lemma INDEP_UNIFORM: "ALL (t::nat) n::nat. indep (uniform t (Suc n))"
+lemma INDEP_UNIFORM: "indep (uniform t (Suc n))"
by (import prob_uniform INDEP_UNIFORM)
-lemma PROB_UNIF: "ALL (n::nat) k::nat.
- prob (%s::nat => bool. fst (unif n s) = k) =
- (if k < 2 ^ unif_bound n then (1 / 2) ^ unif_bound n else 0)"
+lemma PROB_UNIF: "prob (%s. fst (unif n s) = k) =
+(if k < 2 ^ unif_bound n then (1 / 2) ^ unif_bound n else 0)"
by (import prob_uniform PROB_UNIF)
-lemma UNIF_RANGE: "ALL (n::nat) s::nat => bool. fst (unif n s) < 2 ^ unif_bound n"
+lemma UNIF_RANGE: "fst (unif n s) < 2 ^ unif_bound n"
by (import prob_uniform UNIF_RANGE)
-lemma PROB_UNIF_PAIR: "ALL (n::nat) (k::nat) k'::nat.
- (prob (%s::nat => bool. fst (unif n s) = k) =
- prob (%s::nat => bool. fst (unif n s) = k')) =
- ((k < 2 ^ unif_bound n) = (k' < 2 ^ unif_bound n))"
+lemma PROB_UNIF_PAIR: "(prob (%s. fst (unif n s) = k) = prob (%s. fst (unif n s) = k')) =
+((k < 2 ^ unif_bound n) = (k' < 2 ^ unif_bound n))"
by (import prob_uniform PROB_UNIF_PAIR)
-lemma PROB_UNIF_BOUND: "ALL (n::nat) k::nat.
- k <= 2 ^ unif_bound n -->
- prob (%s::nat => bool. fst (unif n s) < k) =
- real k * (1 / 2) ^ unif_bound n"
+lemma PROB_UNIF_BOUND: "k <= 2 ^ unif_bound n
+==> prob (%s. fst (unif n s) < k) = real k * (1 / 2) ^ unif_bound n"
by (import prob_uniform PROB_UNIF_BOUND)
-lemma PROB_UNIF_GOOD: "ALL n::nat. 1 / 2 <= prob (%s::nat => bool. fst (unif n s) < Suc n)"
+lemma PROB_UNIF_GOOD: "1 / 2 <= prob (%s. fst (unif n s) < Suc n)"
by (import prob_uniform PROB_UNIF_GOOD)
-lemma UNIFORM_RANGE: "ALL (t::nat) (n::nat) s::nat => bool. fst (uniform t (Suc n) s) < Suc n"
+lemma UNIFORM_RANGE: "fst (uniform t (Suc n) s) < Suc n"
by (import prob_uniform UNIFORM_RANGE)
-lemma PROB_UNIFORM_LOWER_BOUND: "(All::(real => bool) => bool)
- (%b::real.
- (op -->::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%k::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) k ((Suc::nat => nat) (n::nat)))
- ((op <::real => real => bool)
- ((prob::((nat => bool) => bool) => real)
- (%s::nat => bool.
- (op =::nat => nat => bool)
- ((fst::nat * (nat => bool) => nat)
- ((uniform::nat
- => nat
-=> (nat => bool) => nat * (nat => bool))
- (t::nat) ((Suc::nat => nat) n) s))
- k))
- b)))
- ((All::(nat => bool) => bool)
- (%m::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) m ((Suc::nat => nat) n))
- ((op <::real => real => bool)
- ((prob::((nat => bool) => bool) => real)
- (%s::nat => bool.
- (op <::nat => nat => bool)
- ((fst::nat * (nat => bool) => nat)
- ((uniform::nat
- => nat
-=> (nat => bool) => nat * (nat => bool))
- t ((Suc::nat => nat) n) s))
- ((Suc::nat => nat) m)))
- ((op *::real => real => real)
- ((real::nat => real) ((Suc::nat => nat) m)) b)))))"
+lemma PROB_UNIFORM_LOWER_BOUND: "[| !!k. k < Suc n ==> prob (%s. fst (uniform t (Suc n) s) = k) < b;
+ m < Suc n |]
+==> prob (%s. fst (uniform t (Suc n) s) < Suc m) < real (Suc m) * b"
by (import prob_uniform PROB_UNIFORM_LOWER_BOUND)
-lemma PROB_UNIFORM_UPPER_BOUND: "(All::(real => bool) => bool)
- (%b::real.
- (op -->::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%k::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) k ((Suc::nat => nat) (n::nat)))
- ((op <::real => real => bool) b
- ((prob::((nat => bool) => bool) => real)
- (%s::nat => bool.
- (op =::nat => nat => bool)
- ((fst::nat * (nat => bool) => nat)
- ((uniform::nat
- => nat
-=> (nat => bool) => nat * (nat => bool))
- (t::nat) ((Suc::nat => nat) n) s))
- k)))))
- ((All::(nat => bool) => bool)
- (%m::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) m ((Suc::nat => nat) n))
- ((op <::real => real => bool)
- ((op *::real => real => real)
- ((real::nat => real) ((Suc::nat => nat) m)) b)
- ((prob::((nat => bool) => bool) => real)
- (%s::nat => bool.
- (op <::nat => nat => bool)
- ((fst::nat * (nat => bool) => nat)
- ((uniform::nat
- => nat
-=> (nat => bool) => nat * (nat => bool))
- t ((Suc::nat => nat) n) s))
- ((Suc::nat => nat) m)))))))"
+lemma PROB_UNIFORM_UPPER_BOUND: "[| !!k. k < Suc n ==> b < prob (%s. fst (uniform t (Suc n) s) = k);
+ m < Suc n |]
+==> real (Suc m) * b < prob (%s. fst (uniform t (Suc n) s) < Suc m)"
by (import prob_uniform PROB_UNIFORM_UPPER_BOUND)
-lemma PROB_UNIFORM_PAIR_SUC: "ALL (t::nat) (n::nat) (k::nat) k'::nat.
- k < Suc n & k' < Suc n -->
- abs (prob (%s::nat => bool. fst (uniform t (Suc n) s) = k) -
- prob (%s::nat => bool. fst (uniform t (Suc n) s) = k'))
- <= (1 / 2) ^ t"
+lemma PROB_UNIFORM_PAIR_SUC: "k < Suc n & k' < Suc n
+==> abs (prob (%s. fst (uniform t (Suc n) s) = k) -
+ prob (%s. fst (uniform t (Suc n) s) = k'))
+ <= (1 / 2) ^ t"
by (import prob_uniform PROB_UNIFORM_PAIR_SUC)
-lemma PROB_UNIFORM_SUC: "ALL (t::nat) (n::nat) k::nat.
- k < Suc n -->
- abs (prob (%s::nat => bool. fst (uniform t (Suc n) s) = k) -
- 1 / real (Suc n))
- <= (1 / 2) ^ t"
+lemma PROB_UNIFORM_SUC: "k < Suc n
+==> abs (prob (%s. fst (uniform t (Suc n) s) = k) - 1 / real (Suc n))
+ <= (1 / 2) ^ t"
by (import prob_uniform PROB_UNIFORM_SUC)
-lemma PROB_UNIFORM: "ALL (t::nat) (n::nat) k::nat.
- k < n -->
- abs (prob (%s::nat => bool. fst (uniform t n s) = k) - 1 / real n)
- <= (1 / 2) ^ t"
+lemma PROB_UNIFORM: "k < n
+==> abs (prob (%s. fst (uniform t n s) = k) - 1 / real n) <= (1 / 2) ^ t"
by (import prob_uniform PROB_UNIFORM)
;end_setup
--- a/src/HOL/Import/HOL/HOL4Real.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Import/HOL/HOL4Real.thy Fri Sep 09 06:47:14 2011 +0200
@@ -4,858 +4,702 @@
;setup_theory realax
-lemma HREAL_RDISTRIB: "ALL (x::hreal) (y::hreal) z::hreal.
- hreal_mul (hreal_add x y) z = hreal_add (hreal_mul x z) (hreal_mul y z)"
+lemma HREAL_RDISTRIB: "hreal_mul (hreal_add x y) z = hreal_add (hreal_mul x z) (hreal_mul y z)"
by (import realax HREAL_RDISTRIB)
-lemma HREAL_EQ_ADDL: "ALL (x::hreal) y::hreal. x ~= hreal_add x y"
+lemma HREAL_EQ_ADDL: "x ~= hreal_add x y"
by (import realax HREAL_EQ_ADDL)
-lemma HREAL_EQ_LADD: "ALL (x::hreal) (y::hreal) z::hreal.
- (hreal_add x y = hreal_add x z) = (y = z)"
+lemma HREAL_EQ_LADD: "(hreal_add x y = hreal_add x z) = (y = z)"
by (import realax HREAL_EQ_LADD)
-lemma HREAL_LT_REFL: "ALL x::hreal. ~ hreal_lt x x"
+lemma HREAL_LT_REFL: "~ hreal_lt x x"
by (import realax HREAL_LT_REFL)
-lemma HREAL_LT_ADDL: "ALL (x::hreal) y::hreal. hreal_lt x (hreal_add x y)"
+lemma HREAL_LT_ADDL: "hreal_lt x (hreal_add x y)"
by (import realax HREAL_LT_ADDL)
-lemma HREAL_LT_NE: "ALL (x::hreal) y::hreal. hreal_lt x y --> x ~= y"
+lemma HREAL_LT_NE: "hreal_lt x y ==> x ~= y"
by (import realax HREAL_LT_NE)
-lemma HREAL_LT_ADDR: "ALL (x::hreal) y::hreal. ~ hreal_lt (hreal_add x y) x"
+lemma HREAL_LT_ADDR: "~ hreal_lt (hreal_add x y) x"
by (import realax HREAL_LT_ADDR)
-lemma HREAL_LT_GT: "ALL (x::hreal) y::hreal. hreal_lt x y --> ~ hreal_lt y x"
+lemma HREAL_LT_GT: "hreal_lt x y ==> ~ hreal_lt y x"
by (import realax HREAL_LT_GT)
-lemma HREAL_LT_ADD2: "ALL (x1::hreal) (x2::hreal) (y1::hreal) y2::hreal.
- hreal_lt x1 y1 & hreal_lt x2 y2 -->
- hreal_lt (hreal_add x1 x2) (hreal_add y1 y2)"
+lemma HREAL_LT_ADD2: "hreal_lt x1 y1 & hreal_lt x2 y2
+==> hreal_lt (hreal_add x1 x2) (hreal_add y1 y2)"
by (import realax HREAL_LT_ADD2)
-lemma HREAL_LT_LADD: "ALL (x::hreal) (y::hreal) z::hreal.
- hreal_lt (hreal_add x y) (hreal_add x z) = hreal_lt y z"
+lemma HREAL_LT_LADD: "hreal_lt (hreal_add x y) (hreal_add x z) = hreal_lt y z"
by (import realax HREAL_LT_LADD)
-definition treal_0 :: "hreal * hreal" where
+definition
+ treal_0 :: "hreal * hreal" where
"treal_0 == (hreal_1, hreal_1)"
lemma treal_0: "treal_0 = (hreal_1, hreal_1)"
by (import realax treal_0)
-definition treal_1 :: "hreal * hreal" where
+definition
+ treal_1 :: "hreal * hreal" where
"treal_1 == (hreal_add hreal_1 hreal_1, hreal_1)"
lemma treal_1: "treal_1 = (hreal_add hreal_1 hreal_1, hreal_1)"
by (import realax treal_1)
-definition treal_neg :: "hreal * hreal => hreal * hreal" where
- "treal_neg == %(x::hreal, y::hreal). (y, x)"
-
-lemma treal_neg: "ALL (x::hreal) y::hreal. treal_neg (x, y) = (y, x)"
+definition
+ treal_neg :: "hreal * hreal => hreal * hreal" where
+ "treal_neg == %(x, y). (y, x)"
+
+lemma treal_neg: "treal_neg (x, y) = (y, x)"
by (import realax treal_neg)
-definition treal_add :: "hreal * hreal => hreal * hreal => hreal * hreal" where
- "treal_add ==
-%(x1::hreal, y1::hreal) (x2::hreal, y2::hreal).
- (hreal_add x1 x2, hreal_add y1 y2)"
-
-lemma treal_add: "ALL (x1::hreal) (y1::hreal) (x2::hreal) y2::hreal.
- treal_add (x1, y1) (x2, y2) = (hreal_add x1 x2, hreal_add y1 y2)"
+definition
+ treal_add :: "hreal * hreal => hreal * hreal => hreal * hreal" where
+ "treal_add == %(x1, y1) (x2, y2). (hreal_add x1 x2, hreal_add y1 y2)"
+
+lemma treal_add: "treal_add (x1, y1) (x2, y2) = (hreal_add x1 x2, hreal_add y1 y2)"
by (import realax treal_add)
-definition treal_mul :: "hreal * hreal => hreal * hreal => hreal * hreal" where
+definition
+ treal_mul :: "hreal * hreal => hreal * hreal => hreal * hreal" where
"treal_mul ==
-%(x1::hreal, y1::hreal) (x2::hreal, y2::hreal).
+%(x1, y1) (x2, y2).
(hreal_add (hreal_mul x1 x2) (hreal_mul y1 y2),
hreal_add (hreal_mul x1 y2) (hreal_mul y1 x2))"
-lemma treal_mul: "ALL (x1::hreal) (y1::hreal) (x2::hreal) y2::hreal.
- treal_mul (x1, y1) (x2, y2) =
- (hreal_add (hreal_mul x1 x2) (hreal_mul y1 y2),
- hreal_add (hreal_mul x1 y2) (hreal_mul y1 x2))"
+lemma treal_mul: "treal_mul (x1, y1) (x2, y2) =
+(hreal_add (hreal_mul x1 x2) (hreal_mul y1 y2),
+ hreal_add (hreal_mul x1 y2) (hreal_mul y1 x2))"
by (import realax treal_mul)
-definition treal_lt :: "hreal * hreal => hreal * hreal => bool" where
- "treal_lt ==
-%(x1::hreal, y1::hreal) (x2::hreal, y2::hreal).
- hreal_lt (hreal_add x1 y2) (hreal_add x2 y1)"
-
-lemma treal_lt: "ALL (x1::hreal) (y1::hreal) (x2::hreal) y2::hreal.
- treal_lt (x1, y1) (x2, y2) = hreal_lt (hreal_add x1 y2) (hreal_add x2 y1)"
+definition
+ treal_lt :: "hreal * hreal => hreal * hreal => bool" where
+ "treal_lt == %(x1, y1) (x2, y2). hreal_lt (hreal_add x1 y2) (hreal_add x2 y1)"
+
+lemma treal_lt: "treal_lt (x1, y1) (x2, y2) = hreal_lt (hreal_add x1 y2) (hreal_add x2 y1)"
by (import realax treal_lt)
-definition treal_inv :: "hreal * hreal => hreal * hreal" where
+definition
+ treal_inv :: "hreal * hreal => hreal * hreal" where
"treal_inv ==
-%(x::hreal, y::hreal).
+%(x, y).
if x = y then treal_0
else if hreal_lt y x
then (hreal_add (hreal_inv (hreal_sub x y)) hreal_1, hreal_1)
else (hreal_1, hreal_add (hreal_inv (hreal_sub y x)) hreal_1)"
-lemma treal_inv: "ALL (x::hreal) y::hreal.
- treal_inv (x, y) =
- (if x = y then treal_0
- else if hreal_lt y x
- then (hreal_add (hreal_inv (hreal_sub x y)) hreal_1, hreal_1)
- else (hreal_1, hreal_add (hreal_inv (hreal_sub y x)) hreal_1))"
+lemma treal_inv: "treal_inv (x, y) =
+(if x = y then treal_0
+ else if hreal_lt y x
+ then (hreal_add (hreal_inv (hreal_sub x y)) hreal_1, hreal_1)
+ else (hreal_1, hreal_add (hreal_inv (hreal_sub y x)) hreal_1))"
by (import realax treal_inv)
-definition treal_eq :: "hreal * hreal => hreal * hreal => bool" where
- "treal_eq ==
-%(x1::hreal, y1::hreal) (x2::hreal, y2::hreal).
- hreal_add x1 y2 = hreal_add x2 y1"
-
-lemma treal_eq: "ALL (x1::hreal) (y1::hreal) (x2::hreal) y2::hreal.
- treal_eq (x1, y1) (x2, y2) = (hreal_add x1 y2 = hreal_add x2 y1)"
+definition
+ treal_eq :: "hreal * hreal => hreal * hreal => bool" where
+ "treal_eq == %(x1, y1) (x2, y2). hreal_add x1 y2 = hreal_add x2 y1"
+
+lemma treal_eq: "treal_eq (x1, y1) (x2, y2) = (hreal_add x1 y2 = hreal_add x2 y1)"
by (import realax treal_eq)
-lemma TREAL_EQ_REFL: "ALL x::hreal * hreal. treal_eq x x"
+lemma TREAL_EQ_REFL: "treal_eq x x"
by (import realax TREAL_EQ_REFL)
-lemma TREAL_EQ_SYM: "ALL (x::hreal * hreal) y::hreal * hreal. treal_eq x y = treal_eq y x"
+lemma TREAL_EQ_SYM: "treal_eq x y = treal_eq y x"
by (import realax TREAL_EQ_SYM)
-lemma TREAL_EQ_TRANS: "ALL (x::hreal * hreal) (y::hreal * hreal) z::hreal * hreal.
- treal_eq x y & treal_eq y z --> treal_eq x z"
+lemma TREAL_EQ_TRANS: "treal_eq x y & treal_eq y z ==> treal_eq x z"
by (import realax TREAL_EQ_TRANS)
-lemma TREAL_EQ_EQUIV: "ALL (p::hreal * hreal) q::hreal * hreal.
- treal_eq p q = (treal_eq p = treal_eq q)"
+lemma TREAL_EQ_EQUIV: "treal_eq p q = (treal_eq p = treal_eq q)"
by (import realax TREAL_EQ_EQUIV)
-lemma TREAL_EQ_AP: "ALL (p::hreal * hreal) q::hreal * hreal. p = q --> treal_eq p q"
+lemma TREAL_EQ_AP: "p = q ==> treal_eq p q"
by (import realax TREAL_EQ_AP)
lemma TREAL_10: "~ treal_eq treal_1 treal_0"
by (import realax TREAL_10)
-lemma TREAL_ADD_SYM: "ALL (x::hreal * hreal) y::hreal * hreal. treal_add x y = treal_add y x"
+lemma TREAL_ADD_SYM: "treal_add x y = treal_add y x"
by (import realax TREAL_ADD_SYM)
-lemma TREAL_MUL_SYM: "ALL (x::hreal * hreal) y::hreal * hreal. treal_mul x y = treal_mul y x"
+lemma TREAL_MUL_SYM: "treal_mul x y = treal_mul y x"
by (import realax TREAL_MUL_SYM)
-lemma TREAL_ADD_ASSOC: "ALL (x::hreal * hreal) (y::hreal * hreal) z::hreal * hreal.
- treal_add x (treal_add y z) = treal_add (treal_add x y) z"
+lemma TREAL_ADD_ASSOC: "treal_add x (treal_add y z) = treal_add (treal_add x y) z"
by (import realax TREAL_ADD_ASSOC)
-lemma TREAL_MUL_ASSOC: "ALL (x::hreal * hreal) (y::hreal * hreal) z::hreal * hreal.
- treal_mul x (treal_mul y z) = treal_mul (treal_mul x y) z"
+lemma TREAL_MUL_ASSOC: "treal_mul x (treal_mul y z) = treal_mul (treal_mul x y) z"
by (import realax TREAL_MUL_ASSOC)
-lemma TREAL_LDISTRIB: "ALL (x::hreal * hreal) (y::hreal * hreal) z::hreal * hreal.
- treal_mul x (treal_add y z) = treal_add (treal_mul x y) (treal_mul x z)"
+lemma TREAL_LDISTRIB: "treal_mul x (treal_add y z) = treal_add (treal_mul x y) (treal_mul x z)"
by (import realax TREAL_LDISTRIB)
-lemma TREAL_ADD_LID: "ALL x::hreal * hreal. treal_eq (treal_add treal_0 x) x"
+lemma TREAL_ADD_LID: "treal_eq (treal_add treal_0 x) x"
by (import realax TREAL_ADD_LID)
-lemma TREAL_MUL_LID: "ALL x::hreal * hreal. treal_eq (treal_mul treal_1 x) x"
+lemma TREAL_MUL_LID: "treal_eq (treal_mul treal_1 x) x"
by (import realax TREAL_MUL_LID)
-lemma TREAL_ADD_LINV: "ALL x::hreal * hreal. treal_eq (treal_add (treal_neg x) x) treal_0"
+lemma TREAL_ADD_LINV: "treal_eq (treal_add (treal_neg x) x) treal_0"
by (import realax TREAL_ADD_LINV)
lemma TREAL_INV_0: "treal_eq (treal_inv treal_0) treal_0"
by (import realax TREAL_INV_0)
-lemma TREAL_MUL_LINV: "ALL x::hreal * hreal.
- ~ treal_eq x treal_0 --> treal_eq (treal_mul (treal_inv x) x) treal_1"
+lemma TREAL_MUL_LINV: "~ treal_eq x treal_0 ==> treal_eq (treal_mul (treal_inv x) x) treal_1"
by (import realax TREAL_MUL_LINV)
-lemma TREAL_LT_TOTAL: "ALL (x::hreal * hreal) y::hreal * hreal.
- treal_eq x y | treal_lt x y | treal_lt y x"
+lemma TREAL_LT_TOTAL: "treal_eq x y | treal_lt x y | treal_lt y x"
by (import realax TREAL_LT_TOTAL)
-lemma TREAL_LT_REFL: "ALL x::hreal * hreal. ~ treal_lt x x"
+lemma TREAL_LT_REFL: "~ treal_lt x x"
by (import realax TREAL_LT_REFL)
-lemma TREAL_LT_TRANS: "ALL (x::hreal * hreal) (y::hreal * hreal) z::hreal * hreal.
- treal_lt x y & treal_lt y z --> treal_lt x z"
+lemma TREAL_LT_TRANS: "treal_lt x y & treal_lt y z ==> treal_lt x z"
by (import realax TREAL_LT_TRANS)
-lemma TREAL_LT_ADD: "ALL (x::hreal * hreal) (y::hreal * hreal) z::hreal * hreal.
- treal_lt y z --> treal_lt (treal_add x y) (treal_add x z)"
+lemma TREAL_LT_ADD: "treal_lt y z ==> treal_lt (treal_add x y) (treal_add x z)"
by (import realax TREAL_LT_ADD)
-lemma TREAL_LT_MUL: "ALL (x::hreal * hreal) y::hreal * hreal.
- treal_lt treal_0 x & treal_lt treal_0 y -->
- treal_lt treal_0 (treal_mul x y)"
+lemma TREAL_LT_MUL: "treal_lt treal_0 x & treal_lt treal_0 y ==> treal_lt treal_0 (treal_mul x y)"
by (import realax TREAL_LT_MUL)
-definition treal_of_hreal :: "hreal => hreal * hreal" where
- "treal_of_hreal == %x::hreal. (hreal_add x hreal_1, hreal_1)"
-
-lemma treal_of_hreal: "ALL x::hreal. treal_of_hreal x = (hreal_add x hreal_1, hreal_1)"
+definition
+ treal_of_hreal :: "hreal => hreal * hreal" where
+ "treal_of_hreal == %x. (hreal_add x hreal_1, hreal_1)"
+
+lemma treal_of_hreal: "treal_of_hreal x = (hreal_add x hreal_1, hreal_1)"
by (import realax treal_of_hreal)
-definition hreal_of_treal :: "hreal * hreal => hreal" where
- "hreal_of_treal == %(x::hreal, y::hreal). SOME d::hreal. x = hreal_add y d"
-
-lemma hreal_of_treal: "ALL (x::hreal) y::hreal.
- hreal_of_treal (x, y) = (SOME d::hreal. x = hreal_add y d)"
+definition
+ hreal_of_treal :: "hreal * hreal => hreal" where
+ "hreal_of_treal == %(x, y). SOME d. x = hreal_add y d"
+
+lemma hreal_of_treal: "hreal_of_treal (x, y) = (SOME d. x = hreal_add y d)"
by (import realax hreal_of_treal)
-lemma TREAL_BIJ: "(ALL h::hreal. hreal_of_treal (treal_of_hreal h) = h) &
-(ALL r::hreal * hreal.
- treal_lt treal_0 r = treal_eq (treal_of_hreal (hreal_of_treal r)) r)"
+lemma TREAL_BIJ: "(ALL h. hreal_of_treal (treal_of_hreal h) = h) &
+(ALL r. treal_lt treal_0 r = treal_eq (treal_of_hreal (hreal_of_treal r)) r)"
by (import realax TREAL_BIJ)
-lemma TREAL_ISO: "ALL (h::hreal) i::hreal.
- hreal_lt h i --> treal_lt (treal_of_hreal h) (treal_of_hreal i)"
+lemma TREAL_ISO: "hreal_lt h i ==> treal_lt (treal_of_hreal h) (treal_of_hreal i)"
by (import realax TREAL_ISO)
-lemma TREAL_BIJ_WELLDEF: "ALL (h::hreal * hreal) i::hreal * hreal.
- treal_eq h i --> hreal_of_treal h = hreal_of_treal i"
+lemma TREAL_BIJ_WELLDEF: "treal_eq h i ==> hreal_of_treal h = hreal_of_treal i"
by (import realax TREAL_BIJ_WELLDEF)
-lemma TREAL_NEG_WELLDEF: "ALL (x1::hreal * hreal) x2::hreal * hreal.
- treal_eq x1 x2 --> treal_eq (treal_neg x1) (treal_neg x2)"
+lemma TREAL_NEG_WELLDEF: "treal_eq x1 x2 ==> treal_eq (treal_neg x1) (treal_neg x2)"
by (import realax TREAL_NEG_WELLDEF)
-lemma TREAL_ADD_WELLDEFR: "ALL (x1::hreal * hreal) (x2::hreal * hreal) y::hreal * hreal.
- treal_eq x1 x2 --> treal_eq (treal_add x1 y) (treal_add x2 y)"
+lemma TREAL_ADD_WELLDEFR: "treal_eq x1 x2 ==> treal_eq (treal_add x1 y) (treal_add x2 y)"
by (import realax TREAL_ADD_WELLDEFR)
-lemma TREAL_ADD_WELLDEF: "ALL (x1::hreal * hreal) (x2::hreal * hreal) (y1::hreal * hreal)
- y2::hreal * hreal.
- treal_eq x1 x2 & treal_eq y1 y2 -->
- treal_eq (treal_add x1 y1) (treal_add x2 y2)"
+lemma TREAL_ADD_WELLDEF: "treal_eq x1 x2 & treal_eq y1 y2
+==> treal_eq (treal_add x1 y1) (treal_add x2 y2)"
by (import realax TREAL_ADD_WELLDEF)
-lemma TREAL_MUL_WELLDEFR: "ALL (x1::hreal * hreal) (x2::hreal * hreal) y::hreal * hreal.
- treal_eq x1 x2 --> treal_eq (treal_mul x1 y) (treal_mul x2 y)"
+lemma TREAL_MUL_WELLDEFR: "treal_eq x1 x2 ==> treal_eq (treal_mul x1 y) (treal_mul x2 y)"
by (import realax TREAL_MUL_WELLDEFR)
-lemma TREAL_MUL_WELLDEF: "ALL (x1::hreal * hreal) (x2::hreal * hreal) (y1::hreal * hreal)
- y2::hreal * hreal.
- treal_eq x1 x2 & treal_eq y1 y2 -->
- treal_eq (treal_mul x1 y1) (treal_mul x2 y2)"
+lemma TREAL_MUL_WELLDEF: "treal_eq x1 x2 & treal_eq y1 y2
+==> treal_eq (treal_mul x1 y1) (treal_mul x2 y2)"
by (import realax TREAL_MUL_WELLDEF)
-lemma TREAL_LT_WELLDEFR: "ALL (x1::hreal * hreal) (x2::hreal * hreal) y::hreal * hreal.
- treal_eq x1 x2 --> treal_lt x1 y = treal_lt x2 y"
+lemma TREAL_LT_WELLDEFR: "treal_eq x1 x2 ==> treal_lt x1 y = treal_lt x2 y"
by (import realax TREAL_LT_WELLDEFR)
-lemma TREAL_LT_WELLDEFL: "ALL (x::hreal * hreal) (y1::hreal * hreal) y2::hreal * hreal.
- treal_eq y1 y2 --> treal_lt x y1 = treal_lt x y2"
+lemma TREAL_LT_WELLDEFL: "treal_eq y1 y2 ==> treal_lt x y1 = treal_lt x y2"
by (import realax TREAL_LT_WELLDEFL)
-lemma TREAL_LT_WELLDEF: "ALL (x1::hreal * hreal) (x2::hreal * hreal) (y1::hreal * hreal)
- y2::hreal * hreal.
- treal_eq x1 x2 & treal_eq y1 y2 --> treal_lt x1 y1 = treal_lt x2 y2"
+lemma TREAL_LT_WELLDEF: "treal_eq x1 x2 & treal_eq y1 y2 ==> treal_lt x1 y1 = treal_lt x2 y2"
by (import realax TREAL_LT_WELLDEF)
-lemma TREAL_INV_WELLDEF: "ALL (x1::hreal * hreal) x2::hreal * hreal.
- treal_eq x1 x2 --> treal_eq (treal_inv x1) (treal_inv x2)"
+lemma TREAL_INV_WELLDEF: "treal_eq x1 x2 ==> treal_eq (treal_inv x1) (treal_inv x2)"
by (import realax TREAL_INV_WELLDEF)
;end_setup
;setup_theory real
-lemma REAL_0: "(op =::real => real => bool) (0::real) (0::real)"
+lemma REAL_0: "(0::real) = (0::real)"
by (import real REAL_0)
-lemma REAL_1: "(op =::real => real => bool) (1::real) (1::real)"
+lemma REAL_1: "(1::real) = (1::real)"
by (import real REAL_1)
-lemma REAL_ADD_LID_UNIQ: "ALL (x::real) y::real. (x + y = y) = (x = 0)"
+lemma REAL_ADD_LID_UNIQ: "((x::real) + (y::real) = y) = (x = (0::real))"
by (import real REAL_ADD_LID_UNIQ)
-lemma REAL_ADD_RID_UNIQ: "ALL (x::real) y::real. (x + y = x) = (y = 0)"
+lemma REAL_ADD_RID_UNIQ: "((x::real) + (y::real) = x) = (y = (0::real))"
by (import real REAL_ADD_RID_UNIQ)
-lemma REAL_LNEG_UNIQ: "ALL (x::real) y::real. (x + y = 0) = (x = - y)"
- by (import real REAL_LNEG_UNIQ)
-
-lemma REAL_LT_ANTISYM: "ALL (x::real) y::real. ~ (x < y & y < x)"
+lemma REAL_LT_ANTISYM: "~ ((x::real) < (y::real) & y < x)"
by (import real REAL_LT_ANTISYM)
-lemma REAL_LTE_TOTAL: "ALL (x::real) y::real. x < y | y <= x"
+lemma REAL_LTE_TOTAL: "(x::real) < (y::real) | y <= x"
by (import real REAL_LTE_TOTAL)
-lemma REAL_LET_ANTISYM: "ALL (x::real) y::real. ~ (x < y & y <= x)"
+lemma REAL_LET_ANTISYM: "~ ((x::real) < (y::real) & y <= x)"
by (import real REAL_LET_ANTISYM)
-lemma REAL_LTE_ANTSYM: "ALL (x::real) y::real. ~ (x <= y & y < x)"
+lemma REAL_LTE_ANTSYM: "~ ((x::real) <= (y::real) & y < x)"
by (import real REAL_LTE_ANTSYM)
-lemma REAL_LT_NEGTOTAL: "ALL x::real. x = 0 | 0 < x | 0 < - x"
+lemma REAL_LT_NEGTOTAL: "(x::real) = (0::real) | (0::real) < x | (0::real) < - x"
by (import real REAL_LT_NEGTOTAL)
-lemma REAL_LE_NEGTOTAL: "ALL x::real. 0 <= x | 0 <= - x"
+lemma REAL_LE_NEGTOTAL: "(0::real) <= (x::real) | (0::real) <= - x"
by (import real REAL_LE_NEGTOTAL)
-lemma REAL_LT_ADDNEG: "ALL (x::real) (y::real) z::real. (y < x + - z) = (y + z < x)"
+lemma REAL_LT_ADDNEG: "((y::real) < (x::real) + - (z::real)) = (y + z < x)"
by (import real REAL_LT_ADDNEG)
-lemma REAL_LT_ADDNEG2: "ALL (x::real) (y::real) z::real. (x + - y < z) = (x < z + y)"
+lemma REAL_LT_ADDNEG2: "((x::real) + - (y::real) < (z::real)) = (x < z + y)"
by (import real REAL_LT_ADDNEG2)
-lemma REAL_LT_ADD1: "ALL (x::real) y::real. x <= y --> x < y + 1"
+lemma REAL_LT_ADD1: "(x::real) <= (y::real) ==> x < y + (1::real)"
by (import real REAL_LT_ADD1)
-lemma REAL_SUB_ADD2: "ALL (x::real) y::real. y + (x - y) = x"
+lemma REAL_SUB_ADD2: "(y::real) + ((x::real) - y) = x"
by (import real REAL_SUB_ADD2)
-lemma REAL_SUB_LT: "ALL (x::real) y::real. (0 < x - y) = (y < x)"
+lemma REAL_SUB_LT: "((0::real) < (x::real) - (y::real)) = (y < x)"
by (import real REAL_SUB_LT)
-lemma REAL_SUB_LE: "ALL (x::real) y::real. (0 <= x - y) = (y <= x)"
+lemma REAL_SUB_LE: "((0::real) <= (x::real) - (y::real)) = (y <= x)"
by (import real REAL_SUB_LE)
-lemma REAL_ADD_SUB: "ALL (x::real) y::real. x + y - x = y"
+lemma REAL_ADD_SUB: "(x::real) + (y::real) - x = y"
by (import real REAL_ADD_SUB)
-lemma REAL_NEG_EQ: "ALL (x::real) y::real. (- x = y) = (x = - y)"
+lemma REAL_NEG_EQ: "(- (x::real) = (y::real)) = (x = - y)"
by (import real REAL_NEG_EQ)
-lemma REAL_NEG_MINUS1: "ALL x::real. - x = - 1 * x"
- by (import real REAL_NEG_MINUS1)
-
-lemma REAL_LT_LMUL_0: "ALL (x::real) y::real. 0 < x --> (0 < x * y) = (0 < y)"
+lemma REAL_LT_LMUL_0: "(0::real) < (x::real) ==> ((0::real) < x * (y::real)) = ((0::real) < y)"
by (import real REAL_LT_LMUL_0)
-lemma REAL_LT_RMUL_0: "ALL (x::real) y::real. 0 < y --> (0 < x * y) = (0 < x)"
+lemma REAL_LT_RMUL_0: "(0::real) < (y::real) ==> ((0::real) < (x::real) * y) = ((0::real) < x)"
by (import real REAL_LT_RMUL_0)
-lemma REAL_LT_LMUL: "ALL (x::real) (y::real) z::real. 0 < x --> (x * y < x * z) = (y < z)"
- by (import real REAL_LT_LMUL)
-
-lemma REAL_LINV_UNIQ: "ALL (x::real) y::real. x * y = 1 --> x = inverse y"
+lemma REAL_LINV_UNIQ: "(x::real) * (y::real) = (1::real) ==> x = inverse y"
by (import real REAL_LINV_UNIQ)
-lemma REAL_LE_INV: "(All::(real => bool) => bool)
- (%x::real.
- (op -->::bool => bool => bool)
- ((op <=::real => real => bool) (0::real) x)
- ((op <=::real => real => bool) (0::real) ((inverse::real => real) x)))"
+lemma REAL_LE_INV: "(0::real) <= (x::real) ==> (0::real) <= inverse x"
by (import real REAL_LE_INV)
-lemma REAL_LE_ADDR: "ALL (x::real) y::real. (x <= x + y) = (0 <= y)"
+lemma REAL_LE_ADDR: "((x::real) <= x + (y::real)) = ((0::real) <= y)"
by (import real REAL_LE_ADDR)
-lemma REAL_LE_ADDL: "ALL (x::real) y::real. (y <= x + y) = (0 <= x)"
+lemma REAL_LE_ADDL: "((y::real) <= (x::real) + y) = ((0::real) <= x)"
by (import real REAL_LE_ADDL)
-lemma REAL_LT_ADDR: "ALL (x::real) y::real. (x < x + y) = (0 < y)"
+lemma REAL_LT_ADDR: "((x::real) < x + (y::real)) = ((0::real) < y)"
by (import real REAL_LT_ADDR)
-lemma REAL_LT_ADDL: "ALL (x::real) y::real. (y < x + y) = (0 < x)"
+lemma REAL_LT_ADDL: "((y::real) < (x::real) + y) = ((0::real) < x)"
by (import real REAL_LT_ADDL)
-lemma REAL_LT_NZ: "ALL n::nat. (real n ~= 0) = (0 < real n)"
+lemma REAL_LT_NZ: "(real (n::nat) ~= (0::real)) = ((0::real) < real n)"
by (import real REAL_LT_NZ)
-lemma REAL_NZ_IMP_LT: "ALL n::nat. n ~= 0 --> 0 < real n"
+lemma REAL_NZ_IMP_LT: "(n::nat) ~= (0::nat) ==> (0::real) < real n"
by (import real REAL_NZ_IMP_LT)
-lemma REAL_LT_RDIV_0: "ALL (y::real) z::real. 0 < z --> (0 < y / z) = (0 < y)"
+lemma REAL_LT_RDIV_0: "(0::real) < (z::real) ==> ((0::real) < (y::real) / z) = ((0::real) < y)"
by (import real REAL_LT_RDIV_0)
-lemma REAL_LT_RDIV: "ALL (x::real) (y::real) z::real. 0 < z --> (x / z < y / z) = (x < y)"
+lemma REAL_LT_RDIV: "(0::real) < (z::real) ==> ((x::real) / z < (y::real) / z) = (x < y)"
by (import real REAL_LT_RDIV)
-lemma REAL_LT_FRACTION_0: "ALL (n::nat) d::real. n ~= 0 --> (0 < d / real n) = (0 < d)"
+lemma REAL_LT_FRACTION_0: "(n::nat) ~= (0::nat) ==> ((0::real) < (d::real) / real n) = ((0::real) < d)"
by (import real REAL_LT_FRACTION_0)
-lemma REAL_LT_MULTIPLE: "ALL (x::nat) xa::real. 1 < x --> (xa < real x * xa) = (0 < xa)"
+lemma REAL_LT_MULTIPLE: "(1::nat) < (x::nat) ==> ((xa::real) < real x * xa) = ((0::real) < xa)"
by (import real REAL_LT_MULTIPLE)
-lemma REAL_LT_FRACTION: "ALL (n::nat) d::real. 1 < n --> (d / real n < d) = (0 < d)"
+lemma REAL_LT_FRACTION: "(1::nat) < (n::nat) ==> ((d::real) / real n < d) = ((0::real) < d)"
by (import real REAL_LT_FRACTION)
-lemma REAL_LT_HALF2: "ALL d::real. (d / 2 < d) = (0 < d)"
+lemma REAL_LT_HALF2: "((d::real) / (2::real) < d) = ((0::real) < d)"
by (import real REAL_LT_HALF2)
-lemma REAL_DIV_LMUL: "ALL (x::real) y::real. y ~= 0 --> y * (x / y) = x"
+lemma REAL_DIV_LMUL: "(y::real) ~= (0::real) ==> y * ((x::real) / y) = x"
by (import real REAL_DIV_LMUL)
-lemma REAL_DIV_RMUL: "ALL (x::real) y::real. y ~= 0 --> x / y * y = x"
+lemma REAL_DIV_RMUL: "(y::real) ~= (0::real) ==> (x::real) / y * y = x"
by (import real REAL_DIV_RMUL)
-lemma REAL_DOWN: "(All::(real => bool) => bool)
- (%x::real.
- (op -->::bool => bool => bool)
- ((op <::real => real => bool) (0::real) x)
- ((Ex::(real => bool) => bool)
- (%xa::real.
- (op &::bool => bool => bool)
- ((op <::real => real => bool) (0::real) xa)
- ((op <::real => real => bool) xa x))))"
+lemma REAL_DOWN: "(0::real) < (x::real) ==> EX xa>0::real. xa < x"
by (import real REAL_DOWN)
-lemma REAL_SUB_SUB: "ALL (x::real) y::real. x - y - x = - y"
+lemma REAL_SUB_SUB: "(x::real) - (y::real) - x = - y"
by (import real REAL_SUB_SUB)
-lemma REAL_ADD2_SUB2: "ALL (a::real) (b::real) (c::real) d::real. a + b - (c + d) = a - c + (b - d)"
- by (import real REAL_ADD2_SUB2)
-
-lemma REAL_SUB_LNEG: "ALL (x::real) y::real. - x - y = - (x + y)"
+lemma REAL_SUB_LNEG: "- (x::real) - (y::real) = - (x + y)"
by (import real REAL_SUB_LNEG)
-lemma REAL_SUB_NEG2: "ALL (x::real) y::real. - x - - y = y - x"
+lemma REAL_SUB_NEG2: "- (x::real) - - (y::real) = y - x"
by (import real REAL_SUB_NEG2)
-lemma REAL_SUB_TRIANGLE: "ALL (a::real) (b::real) c::real. a - b + (b - c) = a - c"
+lemma REAL_SUB_TRIANGLE: "(a::real) - (b::real) + (b - (c::real)) = a - c"
by (import real REAL_SUB_TRIANGLE)
-lemma REAL_INV_MUL: "ALL (x::real) y::real.
- x ~= 0 & y ~= 0 --> inverse (x * y) = inverse x * inverse y"
+lemma REAL_INV_MUL: "(x::real) ~= (0::real) & (y::real) ~= (0::real)
+==> inverse (x * y) = inverse x * inverse y"
by (import real REAL_INV_MUL)
-lemma REAL_SUB_INV2: "ALL (x::real) y::real.
- x ~= 0 & y ~= 0 --> inverse x - inverse y = (y - x) / (x * y)"
+lemma REAL_SUB_INV2: "(x::real) ~= (0::real) & (y::real) ~= (0::real)
+==> inverse x - inverse y = (y - x) / (x * y)"
by (import real REAL_SUB_INV2)
-lemma REAL_SUB_SUB2: "ALL (x::real) y::real. x - (x - y) = y"
+lemma REAL_SUB_SUB2: "(x::real) - (x - (y::real)) = y"
by (import real REAL_SUB_SUB2)
-lemma REAL_ADD_SUB2: "ALL (x::real) y::real. x - (x + y) = - y"
+lemma REAL_ADD_SUB2: "(x::real) - (x + (y::real)) = - y"
by (import real REAL_ADD_SUB2)
-lemma REAL_LE_MUL2: "ALL (x1::real) (x2::real) (y1::real) y2::real.
- 0 <= x1 & 0 <= y1 & x1 <= x2 & y1 <= y2 --> x1 * y1 <= x2 * y2"
- by (import real REAL_LE_MUL2)
-
-lemma REAL_LE_DIV: "ALL (x::real) xa::real. 0 <= x & 0 <= xa --> 0 <= x / xa"
+lemma REAL_LE_DIV: "(0::real) <= (x::real) & (0::real) <= (xa::real) ==> (0::real) <= x / xa"
by (import real REAL_LE_DIV)
-lemma REAL_LT_1: "ALL (x::real) y::real. 0 <= x & x < y --> x / y < 1"
+lemma REAL_LT_1: "(0::real) <= (x::real) & x < (y::real) ==> x / y < (1::real)"
by (import real REAL_LT_1)
-lemma REAL_POS_NZ: "(All::(real => bool) => bool)
- (%x::real.
- (op -->::bool => bool => bool)
- ((op <::real => real => bool) (0::real) x)
- ((Not::bool => bool) ((op =::real => real => bool) x (0::real))))"
+lemma REAL_POS_NZ: "(0::real) < (x::real) ==> x ~= (0::real)"
by (import real REAL_POS_NZ)
-lemma REAL_EQ_LMUL_IMP: "ALL (x::real) (xa::real) xb::real. x ~= 0 & x * xa = x * xb --> xa = xb"
+lemma REAL_EQ_RMUL_IMP: "(z::real) ~= (0::real) & (x::real) * z = (y::real) * z ==> x = y"
+ by (import real REAL_EQ_RMUL_IMP)
+
+lemma REAL_EQ_LMUL_IMP: "(x::real) ~= (0::real) & x * (xa::real) = x * (xb::real) ==> xa = xb"
by (import real REAL_EQ_LMUL_IMP)
-lemma REAL_FACT_NZ: "ALL n::nat. real (FACT n) ~= 0"
+lemma REAL_FACT_NZ: "real (FACT n) ~= 0"
by (import real REAL_FACT_NZ)
-lemma REAL_DIFFSQ: "ALL (x::real) y::real. (x + y) * (x - y) = x * x - y * y"
- by (import real REAL_DIFFSQ)
-
-lemma REAL_POASQ: "ALL x::real. (0 < x * x) = (x ~= 0)"
+lemma REAL_POASQ: "((0::real) < (x::real) * x) = (x ~= (0::real))"
by (import real REAL_POASQ)
-lemma REAL_SUMSQ: "ALL (x::real) y::real. (x * x + y * y = 0) = (x = 0 & y = 0)"
- by (import real REAL_SUMSQ)
-
-lemma REAL_DIV_MUL2: "ALL (x::real) z::real.
- x ~= 0 & z ~= 0 --> (ALL y::real. y / z = x * y / (x * z))"
+lemma REAL_DIV_MUL2: "(x::real) ~= (0::real) & (z::real) ~= (0::real)
+==> (y::real) / z = x * y / (x * z)"
by (import real REAL_DIV_MUL2)
-lemma REAL_MIDDLE1: "ALL (a::real) b::real. a <= b --> a <= (a + b) / 2"
+lemma REAL_MIDDLE1: "(a::real) <= (b::real) ==> a <= (a + b) / (2::real)"
by (import real REAL_MIDDLE1)
-lemma REAL_MIDDLE2: "ALL (a::real) b::real. a <= b --> (a + b) / 2 <= b"
+lemma REAL_MIDDLE2: "(a::real) <= (b::real) ==> (a + b) / (2::real) <= b"
by (import real REAL_MIDDLE2)
-lemma ABS_LT_MUL2: "ALL (w::real) (x::real) (y::real) z::real.
- abs w < y & abs x < z --> abs (w * x) < y * z"
+lemma ABS_LT_MUL2: "abs (w::real) < (y::real) & abs (x::real) < (z::real)
+==> abs (w * x) < y * z"
by (import real ABS_LT_MUL2)
-lemma ABS_REFL: "ALL x::real. (abs x = x) = (0 <= x)"
+lemma ABS_REFL: "(abs (x::real) = x) = ((0::real) <= x)"
by (import real ABS_REFL)
-lemma ABS_BETWEEN: "ALL (x::real) (y::real) d::real.
- (0 < d & x - d < y & y < x + d) = (abs (y - x) < d)"
+lemma ABS_BETWEEN: "((0::real) < (d::real) & (x::real) - d < (y::real) & y < x + d) =
+(abs (y - x) < d)"
by (import real ABS_BETWEEN)
-lemma ABS_BOUND: "ALL (x::real) (y::real) d::real. abs (x - y) < d --> y < x + d"
+lemma ABS_BOUND: "abs ((x::real) - (y::real)) < (d::real) ==> y < x + d"
by (import real ABS_BOUND)
-lemma ABS_STILLNZ: "ALL (x::real) y::real. abs (x - y) < abs y --> x ~= 0"
+lemma ABS_STILLNZ: "abs ((x::real) - (y::real)) < abs y ==> x ~= (0::real)"
by (import real ABS_STILLNZ)
-lemma ABS_CASES: "ALL x::real. x = 0 | 0 < abs x"
+lemma ABS_CASES: "(x::real) = (0::real) | (0::real) < abs x"
by (import real ABS_CASES)
-lemma ABS_BETWEEN1: "ALL (x::real) (y::real) z::real. x < z & abs (y - x) < z - x --> y < z"
+lemma ABS_BETWEEN1: "(x::real) < (z::real) & abs ((y::real) - x) < z - x ==> y < z"
by (import real ABS_BETWEEN1)
-lemma ABS_SIGN: "ALL (x::real) y::real. abs (x - y) < y --> 0 < x"
+lemma ABS_SIGN: "abs ((x::real) - (y::real)) < y ==> (0::real) < x"
by (import real ABS_SIGN)
-lemma ABS_SIGN2: "ALL (x::real) y::real. abs (x - y) < - y --> x < 0"
+lemma ABS_SIGN2: "abs ((x::real) - (y::real)) < - y ==> x < (0::real)"
by (import real ABS_SIGN2)
-lemma ABS_CIRCLE: "ALL (x::real) (y::real) h::real.
- abs h < abs y - abs x --> abs (x + h) < abs y"
+lemma ABS_CIRCLE: "abs (h::real) < abs (y::real) - abs (x::real) ==> abs (x + h) < abs y"
by (import real ABS_CIRCLE)
-lemma ABS_BETWEEN2: "ALL (x0::real) (x::real) (y0::real) y::real.
- x0 < y0 & abs (x - x0) < (y0 - x0) / 2 & abs (y - y0) < (y0 - x0) / 2 -->
- x < y"
+lemma ABS_BETWEEN2: "(x0::real) < (y0::real) &
+abs ((x::real) - x0) < (y0 - x0) / (2::real) &
+abs ((y::real) - y0) < (y0 - x0) / (2::real)
+==> x < y"
by (import real ABS_BETWEEN2)
-lemma POW_PLUS1: "ALL e>0. ALL n::nat. 1 + real n * e <= (1 + e) ^ n"
+lemma POW_PLUS1: "0 < e ==> 1 + real n * e <= (1 + e) ^ n"
by (import real POW_PLUS1)
-lemma POW_M1: "(All::(nat => bool) => bool)
- (%n::nat.
- (op =::real => real => bool)
- ((abs::real => real)
- ((op ^::real => nat => real) ((uminus::real => real) (1::real)) n))
- (1::real))"
+lemma POW_M1: "abs ((- (1::real)) ^ (n::nat)) = (1::real)"
by (import real POW_M1)
-lemma REAL_LE1_POW2: "(All::(real => bool) => bool)
- (%x::real.
- (op -->::bool => bool => bool)
- ((op <=::real => real => bool) (1::real) x)
- ((op <=::real => real => bool) (1::real)
- ((op ^::real => nat => real) x
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))))"
+lemma REAL_LE1_POW2: "(1::real) <= (x::real) ==> (1::real) <= x ^ (2::nat)"
by (import real REAL_LE1_POW2)
-lemma REAL_LT1_POW2: "(All::(real => bool) => bool)
- (%x::real.
- (op -->::bool => bool => bool)
- ((op <::real => real => bool) (1::real) x)
- ((op <::real => real => bool) (1::real)
- ((op ^::real => nat => real) x
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))))"
+lemma REAL_LT1_POW2: "(1::real) < (x::real) ==> (1::real) < x ^ (2::nat)"
by (import real REAL_LT1_POW2)
-lemma POW_POS_LT: "ALL (x::real) n::nat. 0 < x --> 0 < x ^ Suc n"
+lemma POW_POS_LT: "(0::real) < (x::real) ==> (0::real) < x ^ Suc (n::nat)"
by (import real POW_POS_LT)
-lemma POW_LT: "ALL (n::nat) (x::real) y::real. 0 <= x & x < y --> x ^ Suc n < y ^ Suc n"
+lemma POW_LT: "(0::real) <= (x::real) & x < (y::real) ==> x ^ Suc (n::nat) < y ^ Suc n"
by (import real POW_LT)
-lemma POW_ZERO_EQ: "ALL (n::nat) x::real. (x ^ Suc n = 0) = (x = 0)"
+lemma POW_ZERO: "(x::real) ^ (n::nat) = (0::real) ==> x = (0::real)"
+ by (import real POW_ZERO)
+
+lemma POW_ZERO_EQ: "((x::real) ^ Suc (n::nat) = (0::real)) = (x = (0::real))"
by (import real POW_ZERO_EQ)
-lemma REAL_POW_LT2: "ALL (n::nat) (x::real) y::real. n ~= 0 & 0 <= x & x < y --> x ^ n < y ^ n"
+lemma REAL_POW_LT2: "(n::nat) ~= (0::nat) & (0::real) <= (x::real) & x < (y::real)
+==> x ^ n < y ^ n"
by (import real REAL_POW_LT2)
-lemma REAL_POW_MONO_LT: "ALL (m::nat) (n::nat) x::real. 1 < x & m < n --> x ^ m < x ^ n"
+lemma REAL_POW_MONO_LT: "(1::real) < (x::real) & (m::nat) < (n::nat) ==> x ^ m < x ^ n"
by (import real REAL_POW_MONO_LT)
-lemma REAL_SUP_SOMEPOS: "ALL P::real => bool.
- (EX x::real. P x & 0 < x) & (EX z::real. ALL x::real. P x --> x < z) -->
- (EX s::real. ALL y::real. (EX x::real. P x & y < x) = (y < s))"
+lemma REAL_SUP_SOMEPOS: "(EX x::real. (P::real => bool) x & (0::real) < x) &
+(EX z::real. ALL x::real. P x --> x < z)
+==> EX s::real. ALL y::real. (EX x::real. P x & y < x) = (y < s)"
by (import real REAL_SUP_SOMEPOS)
-lemma SUP_LEMMA1: "ALL (P::real => bool) (s::real) d::real.
- (ALL y::real. (EX x::real. P (x + d) & y < x) = (y < s)) -->
- (ALL y::real. (EX x::real. P x & y < x) = (y < s + d))"
+lemma SUP_LEMMA1: "(!!y::real.
+ (EX x::real. (P::real => bool) (x + (d::real)) & y < x) =
+ (y < (s::real)))
+==> (EX x::real. P x & (y::real) < x) = (y < s + d)"
by (import real SUP_LEMMA1)
-lemma SUP_LEMMA2: "ALL P::real => bool. Ex P --> (EX (d::real) x::real. P (x + d) & 0 < x)"
+lemma SUP_LEMMA2: "Ex (P::real => bool) ==> EX (d::real) x::real. P (x + d) & (0::real) < x"
by (import real SUP_LEMMA2)
-lemma SUP_LEMMA3: "ALL d::real.
- (EX z::real. ALL x::real. (P::real => bool) x --> x < z) -->
- (EX x::real. ALL xa::real. P (xa + d) --> xa < x)"
+lemma SUP_LEMMA3: "EX z::real. ALL x::real. (P::real => bool) x --> x < z
+==> EX x::real. ALL xa::real. P (xa + (d::real)) --> xa < x"
by (import real SUP_LEMMA3)
-lemma REAL_SUP_EXISTS: "ALL P::real => bool.
- Ex P & (EX z::real. ALL x::real. P x --> x < z) -->
- (EX x::real. ALL y::real. (EX x::real. P x & y < x) = (y < x))"
+lemma REAL_SUP_EXISTS: "Ex (P::real => bool) & (EX z::real. ALL x::real. P x --> x < z)
+==> EX x::real. ALL y::real. (EX x::real. P x & y < x) = (y < x)"
by (import real REAL_SUP_EXISTS)
-definition sup :: "(real => bool) => real" where
- "sup ==
-%P::real => bool.
- SOME s::real. ALL y::real. (EX x::real. P x & y < x) = (y < s)"
-
-lemma sup: "ALL P::real => bool.
- sup P = (SOME s::real. ALL y::real. (EX x::real. P x & y < x) = (y < s))"
+consts
+ sup :: "(real => bool) => real"
+
+defs
+ sup_def: "real.sup == %P. SOME s. ALL y. (EX x. P x & y < x) = (y < s)"
+
+lemma sup: "real.sup P = (SOME s. ALL y. (EX x. P x & y < x) = (y < s))"
by (import real sup)
-lemma REAL_SUP: "ALL P::real => bool.
- Ex P & (EX z::real. ALL x::real. P x --> x < z) -->
- (ALL y::real. (EX x::real. P x & y < x) = (y < sup P))"
+lemma REAL_SUP: "Ex P & (EX z. ALL x. P x --> x < z)
+==> (EX x. P x & y < x) = (y < real.sup P)"
by (import real REAL_SUP)
-lemma REAL_SUP_UBOUND: "ALL P::real => bool.
- Ex P & (EX z::real. ALL x::real. P x --> x < z) -->
- (ALL y::real. P y --> y <= sup P)"
+lemma REAL_SUP_UBOUND: "[| Ex P & (EX z. ALL x. P x --> x < z); P y |] ==> y <= real.sup P"
by (import real REAL_SUP_UBOUND)
-lemma SETOK_LE_LT: "ALL P::real => bool.
- (Ex P & (EX z::real. ALL x::real. P x --> x <= z)) =
- (Ex P & (EX z::real. ALL x::real. P x --> x < z))"
+lemma SETOK_LE_LT: "(Ex (P::real => bool) & (EX z::real. ALL x::real. P x --> x <= z)) =
+(Ex P & (EX z::real. ALL x::real. P x --> x < z))"
by (import real SETOK_LE_LT)
-lemma REAL_SUP_LE: "ALL P::real => bool.
- Ex P & (EX z::real. ALL x::real. P x --> x <= z) -->
- (ALL y::real. (EX x::real. P x & y < x) = (y < sup P))"
+lemma REAL_SUP_LE: "Ex P & (EX z. ALL x. P x --> x <= z)
+==> (EX x. P x & y < x) = (y < real.sup P)"
by (import real REAL_SUP_LE)
-lemma REAL_SUP_UBOUND_LE: "ALL P::real => bool.
- Ex P & (EX z::real. ALL x::real. P x --> x <= z) -->
- (ALL y::real. P y --> y <= sup P)"
+lemma REAL_SUP_UBOUND_LE: "[| Ex P & (EX z. ALL x. P x --> x <= z); P y |] ==> y <= real.sup P"
by (import real REAL_SUP_UBOUND_LE)
-lemma REAL_ARCH_LEAST: "ALL y>0. ALL x>=0. EX n::nat. real n * y <= x & x < real (Suc n) * y"
- by (import real REAL_ARCH_LEAST)
-
consts
sumc :: "nat => nat => (nat => real) => real"
-specification (sumc) sumc: "(ALL (n::nat) f::nat => real. sumc n 0 f = 0) &
-(ALL (n::nat) (m::nat) f::nat => real.
- sumc n (Suc m) f = sumc n m f + f (n + m))"
+specification (sumc) sumc: "(ALL n f. sumc n 0 f = 0) &
+(ALL n m f. sumc n (Suc m) f = sumc n m f + f (n + m))"
by (import real sumc)
consts
sum :: "nat * nat => (nat => real) => real"
defs
- sum_def: "(op ==::(nat * nat => (nat => real) => real)
- => (nat * nat => (nat => real) => real) => prop)
- (real.sum::nat * nat => (nat => real) => real)
- ((split::(nat => nat => (nat => real) => real)
- => nat * nat => (nat => real) => real)
- (sumc::nat => nat => (nat => real) => real))"
-
-lemma SUM_DEF: "ALL (m::nat) (n::nat) f::nat => real. real.sum (m, n) f = sumc m n f"
+ sum_def: "real.sum == %(x, y). sumc x y"
+
+lemma SUM_DEF: "real.sum (m, n) f = sumc m n f"
by (import real SUM_DEF)
-lemma sum: "ALL (x::nat => real) (xa::nat) xb::nat.
- real.sum (xa, 0) x = 0 &
- real.sum (xa, Suc xb) x = real.sum (xa, xb) x + x (xa + xb)"
+lemma sum: "real.sum (xa, 0) x = 0 &
+real.sum (xa, Suc xb) x = real.sum (xa, xb) x + x (xa + xb)"
by (import real sum)
-lemma SUM_TWO: "ALL (f::nat => real) (n::nat) p::nat.
- real.sum (0, n) f + real.sum (n, p) f = real.sum (0, n + p) f"
+lemma SUM_TWO: "real.sum (0, n) f + real.sum (n, p) f = real.sum (0, n + p) f"
by (import real SUM_TWO)
-lemma SUM_DIFF: "ALL (f::nat => real) (m::nat) n::nat.
- real.sum (m, n) f = real.sum (0, m + n) f - real.sum (0, m) f"
+lemma SUM_DIFF: "real.sum (m, n) f = real.sum (0, m + n) f - real.sum (0, m) f"
by (import real SUM_DIFF)
-lemma ABS_SUM: "ALL (f::nat => real) (m::nat) n::nat.
- abs (real.sum (m, n) f) <= real.sum (m, n) (%n::nat. abs (f n))"
+lemma ABS_SUM: "abs (real.sum (m, n) f) <= real.sum (m, n) (%n. abs (f n))"
by (import real ABS_SUM)
-lemma SUM_LE: "ALL (f::nat => real) (g::nat => real) (m::nat) n::nat.
- (ALL r::nat. m <= r & r < n + m --> f r <= g r) -->
- real.sum (m, n) f <= real.sum (m, n) g"
+lemma SUM_LE: "(!!r. m <= r & r < n + m ==> f r <= g r)
+==> real.sum (m, n) f <= real.sum (m, n) g"
by (import real SUM_LE)
-lemma SUM_EQ: "ALL (f::nat => real) (g::nat => real) (m::nat) n::nat.
- (ALL r::nat. m <= r & r < n + m --> f r = g r) -->
- real.sum (m, n) f = real.sum (m, n) g"
+lemma SUM_EQ: "(!!r. m <= r & r < n + m ==> f r = g r)
+==> real.sum (m, n) f = real.sum (m, n) g"
by (import real SUM_EQ)
-lemma SUM_POS: "ALL f::nat => real.
- (ALL n::nat. 0 <= f n) --> (ALL (m::nat) n::nat. 0 <= real.sum (m, n) f)"
+lemma SUM_POS: "(!!n. 0 <= f n) ==> 0 <= real.sum (m, n) f"
by (import real SUM_POS)
-lemma SUM_POS_GEN: "ALL (f::nat => real) m::nat.
- (ALL n::nat. m <= n --> 0 <= f n) -->
- (ALL n::nat. 0 <= real.sum (m, n) f)"
+lemma SUM_POS_GEN: "(!!n. m <= n ==> 0 <= f n) ==> 0 <= real.sum (m, n) f"
by (import real SUM_POS_GEN)
-lemma SUM_ABS: "ALL (f::nat => real) (m::nat) x::nat.
- abs (real.sum (m, x) (%m::nat. abs (f m))) =
- real.sum (m, x) (%m::nat. abs (f m))"
+lemma SUM_ABS: "abs (real.sum (m, x) (%m. abs (f m))) = real.sum (m, x) (%m. abs (f m))"
by (import real SUM_ABS)
-lemma SUM_ABS_LE: "ALL (f::nat => real) (m::nat) n::nat.
- abs (real.sum (m, n) f) <= real.sum (m, n) (%n::nat. abs (f n))"
+lemma SUM_ABS_LE: "abs (real.sum (m, n) f) <= real.sum (m, n) (%n. abs (f n))"
by (import real SUM_ABS_LE)
-lemma SUM_ZERO: "ALL (f::nat => real) N::nat.
- (ALL n::nat. N <= n --> f n = 0) -->
- (ALL (m::nat) n::nat. N <= m --> real.sum (m, n) f = 0)"
+lemma SUM_ZERO: "[| !!n. N <= n ==> f n = 0; N <= m |] ==> real.sum (m, n) f = 0"
by (import real SUM_ZERO)
-lemma SUM_ADD: "ALL (f::nat => real) (g::nat => real) (m::nat) n::nat.
- real.sum (m, n) (%n::nat. f n + g n) =
- real.sum (m, n) f + real.sum (m, n) g"
+lemma SUM_ADD: "real.sum (m, n) (%n. f n + g n) = real.sum (m, n) f + real.sum (m, n) g"
by (import real SUM_ADD)
-lemma SUM_CMUL: "ALL (f::nat => real) (c::real) (m::nat) n::nat.
- real.sum (m, n) (%n::nat. c * f n) = c * real.sum (m, n) f"
+lemma SUM_CMUL: "real.sum (m, n) (%n. c * f n) = c * real.sum (m, n) f"
by (import real SUM_CMUL)
-lemma SUM_NEG: "ALL (f::nat => real) (n::nat) d::nat.
- real.sum (n, d) (%n::nat. - f n) = - real.sum (n, d) f"
+lemma SUM_NEG: "real.sum (n, d) (%n. - f n) = - real.sum (n, d) f"
by (import real SUM_NEG)
-lemma SUM_SUB: "ALL (f::nat => real) (g::nat => real) (m::nat) n::nat.
- real.sum (m, n) (%x::nat. f x - g x) =
- real.sum (m, n) f - real.sum (m, n) g"
+lemma SUM_SUB: "real.sum (m, n) (%x. f x - g x) = real.sum (m, n) f - real.sum (m, n) g"
by (import real SUM_SUB)
-lemma SUM_SUBST: "ALL (f::nat => real) (g::nat => real) (m::nat) n::nat.
- (ALL p::nat. m <= p & p < m + n --> f p = g p) -->
- real.sum (m, n) f = real.sum (m, n) g"
+lemma SUM_SUBST: "(!!p. m <= p & p < m + n ==> f p = g p)
+==> real.sum (m, n) f = real.sum (m, n) g"
by (import real SUM_SUBST)
-lemma SUM_NSUB: "ALL (n::nat) (f::nat => real) c::real.
- real.sum (0, n) f - real n * c = real.sum (0, n) (%p::nat. f p - c)"
+lemma SUM_NSUB: "real.sum (0, n) f - real n * c = real.sum (0, n) (%p. f p - c)"
by (import real SUM_NSUB)
-lemma SUM_BOUND: "ALL (f::nat => real) (k::real) (m::nat) n::nat.
- (ALL p::nat. m <= p & p < m + n --> f p <= k) -->
- real.sum (m, n) f <= real n * k"
+lemma SUM_BOUND: "(!!p. m <= p & p < m + n ==> f p <= k) ==> real.sum (m, n) f <= real n * k"
by (import real SUM_BOUND)
-lemma SUM_GROUP: "ALL (n::nat) (k::nat) f::nat => real.
- real.sum (0, n) (%m::nat. real.sum (m * k, k) f) = real.sum (0, n * k) f"
+lemma SUM_GROUP: "real.sum (0, n) (%m. real.sum (m * k, k) f) = real.sum (0, n * k) f"
by (import real SUM_GROUP)
-lemma SUM_1: "ALL (f::nat => real) n::nat. real.sum (n, 1) f = f n"
+lemma SUM_1: "real.sum (n, 1) f = f n"
by (import real SUM_1)
-lemma SUM_2: "ALL (f::nat => real) n::nat. real.sum (n, 2) f = f n + f (n + 1)"
+lemma SUM_2: "real.sum (n, 2) f = f n + f (n + 1)"
by (import real SUM_2)
-lemma SUM_OFFSET: "ALL (f::nat => real) (n::nat) k::nat.
- real.sum (0, n) (%m::nat. f (m + k)) =
- real.sum (0, n + k) f - real.sum (0, k) f"
+lemma SUM_OFFSET: "real.sum (0, n) (%m. f (m + k)) = real.sum (0, n + k) f - real.sum (0, k) f"
by (import real SUM_OFFSET)
-lemma SUM_REINDEX: "ALL (f::nat => real) (m::nat) (k::nat) n::nat.
- real.sum (m + k, n) f = real.sum (m, n) (%r::nat. f (r + k))"
+lemma SUM_REINDEX: "real.sum (m + k, n) f = real.sum (m, n) (%r. f (r + k))"
by (import real SUM_REINDEX)
-lemma SUM_0: "ALL (m::nat) n::nat. real.sum (m, n) (%r::nat. 0) = 0"
+lemma SUM_0: "real.sum (m, n) (%r. 0) = 0"
by (import real SUM_0)
-lemma SUM_PERMUTE_0: "(All::(nat => bool) => bool)
- (%n::nat.
- (All::((nat => nat) => bool) => bool)
- (%p::nat => nat.
- (op -->::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%y::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) y n)
- ((Ex1::(nat => bool) => bool)
- (%x::nat.
- (op &::bool => bool => bool)
- ((op <::nat => nat => bool) x n)
- ((op =::nat => nat => bool) (p x) y)))))
- ((All::((nat => real) => bool) => bool)
- (%f::nat => real.
- (op =::real => real => bool)
- ((real.sum::nat * nat => (nat => real) => real)
- ((Pair::nat => nat => nat * nat) (0::nat) n)
- (%n::nat. f (p n)))
- ((real.sum::nat * nat => (nat => real) => real)
- ((Pair::nat => nat => nat * nat) (0::nat) n) f)))))"
+lemma SUM_PERMUTE_0: "(!!y. y < n ==> EX! x. x < n & p x = y)
+==> real.sum (0, n) (%n. f (p n)) = real.sum (0, n) f"
by (import real SUM_PERMUTE_0)
-lemma SUM_CANCEL: "ALL (f::nat => real) (n::nat) d::nat.
- real.sum (n, d) (%n::nat. f (Suc n) - f n) = f (n + d) - f n"
+lemma SUM_CANCEL: "real.sum (n, d) (%n. f (Suc n) - f n) = f (n + d) - f n"
by (import real SUM_CANCEL)
-lemma REAL_EQ_RDIV_EQ: "ALL (x::real) (xa::real) xb::real. 0 < xb --> (x = xa / xb) = (x * xb = xa)"
+lemma REAL_LE_RNEG: "((x::real) <= - (y::real)) = (x + y <= (0::real))"
+ by (import real REAL_LE_RNEG)
+
+lemma REAL_EQ_RDIV_EQ: "(0::real) < (xb::real) ==> ((x::real) = (xa::real) / xb) = (x * xb = xa)"
by (import real REAL_EQ_RDIV_EQ)
-lemma REAL_EQ_LDIV_EQ: "ALL (x::real) (xa::real) xb::real. 0 < xb --> (x / xb = xa) = (x = xa * xb)"
+lemma REAL_EQ_LDIV_EQ: "(0::real) < (xb::real) ==> ((x::real) / xb = (xa::real)) = (x = xa * xb)"
by (import real REAL_EQ_LDIV_EQ)
;end_setup
;setup_theory topology
-definition re_Union :: "(('a => bool) => bool) => 'a => bool" where
- "re_Union ==
-%(P::('a::type => bool) => bool) x::'a::type.
- EX s::'a::type => bool. P s & s x"
-
-lemma re_Union: "ALL P::('a::type => bool) => bool.
- re_Union P = (%x::'a::type. EX s::'a::type => bool. P s & s x)"
+definition
+ re_Union :: "(('a => bool) => bool) => 'a => bool" where
+ "re_Union == %P x. EX s. P s & s x"
+
+lemma re_Union: "re_Union P = (%x. EX s. P s & s x)"
by (import topology re_Union)
-definition re_union :: "('a => bool) => ('a => bool) => 'a => bool" where
- "re_union ==
-%(P::'a::type => bool) (Q::'a::type => bool) x::'a::type. P x | Q x"
-
-lemma re_union: "ALL (P::'a::type => bool) Q::'a::type => bool.
- re_union P Q = (%x::'a::type. P x | Q x)"
+definition
+ re_union :: "('a => bool) => ('a => bool) => 'a => bool" where
+ "re_union == %P Q x. P x | Q x"
+
+lemma re_union: "re_union P Q = (%x. P x | Q x)"
by (import topology re_union)
-definition re_intersect :: "('a => bool) => ('a => bool) => 'a => bool" where
- "re_intersect ==
-%(P::'a::type => bool) (Q::'a::type => bool) x::'a::type. P x & Q x"
-
-lemma re_intersect: "ALL (P::'a::type => bool) Q::'a::type => bool.
- re_intersect P Q = (%x::'a::type. P x & Q x)"
+definition
+ re_intersect :: "('a => bool) => ('a => bool) => 'a => bool" where
+ "re_intersect == %P Q x. P x & Q x"
+
+lemma re_intersect: "re_intersect P Q = (%x. P x & Q x)"
by (import topology re_intersect)
-definition re_null :: "'a => bool" where
- "re_null == %x::'a::type. False"
-
-lemma re_null: "re_null = (%x::'a::type. False)"
+definition
+ re_null :: "'a => bool" where
+ "re_null == %x. False"
+
+lemma re_null: "re_null = (%x. False)"
by (import topology re_null)
-definition re_universe :: "'a => bool" where
- "re_universe == %x::'a::type. True"
-
-lemma re_universe: "re_universe = (%x::'a::type. True)"
+definition
+ re_universe :: "'a => bool" where
+ "re_universe == %x. True"
+
+lemma re_universe: "re_universe = (%x. True)"
by (import topology re_universe)
-definition re_subset :: "('a => bool) => ('a => bool) => bool" where
- "re_subset ==
-%(P::'a::type => bool) Q::'a::type => bool. ALL x::'a::type. P x --> Q x"
-
-lemma re_subset: "ALL (P::'a::type => bool) Q::'a::type => bool.
- re_subset P Q = (ALL x::'a::type. P x --> Q x)"
+definition
+ re_subset :: "('a => bool) => ('a => bool) => bool" where
+ "re_subset == %P Q. ALL x. P x --> Q x"
+
+lemma re_subset: "re_subset P Q = (ALL x. P x --> Q x)"
by (import topology re_subset)
-definition re_compl :: "('a => bool) => 'a => bool" where
- "re_compl == %(P::'a::type => bool) x::'a::type. ~ P x"
-
-lemma re_compl: "ALL P::'a::type => bool. re_compl P = (%x::'a::type. ~ P x)"
+definition
+ re_compl :: "('a => bool) => 'a => bool" where
+ "re_compl == %P x. ~ P x"
+
+lemma re_compl: "re_compl P = (%x. ~ P x)"
by (import topology re_compl)
-lemma SUBSET_REFL: "ALL P::'a::type => bool. re_subset P P"
+lemma SUBSET_REFL: "re_subset P P"
by (import topology SUBSET_REFL)
-lemma COMPL_MEM: "ALL (P::'a::type => bool) x::'a::type. P x = (~ re_compl P x)"
+lemma COMPL_MEM: "P x = (~ re_compl P x)"
by (import topology COMPL_MEM)
-lemma SUBSET_ANTISYM: "ALL (P::'a::type => bool) Q::'a::type => bool.
- (re_subset P Q & re_subset Q P) = (P = Q)"
+lemma SUBSET_ANTISYM: "(re_subset P Q & re_subset Q P) = (P = Q)"
by (import topology SUBSET_ANTISYM)
-lemma SUBSET_TRANS: "ALL (P::'a::type => bool) (Q::'a::type => bool) R::'a::type => bool.
- re_subset P Q & re_subset Q R --> re_subset P R"
+lemma SUBSET_TRANS: "re_subset P Q & re_subset Q R ==> re_subset P R"
by (import topology SUBSET_TRANS)
-definition istopology :: "(('a => bool) => bool) => bool" where
+definition
+ istopology :: "(('a => bool) => bool) => bool" where
"istopology ==
-%L::('a::type => bool) => bool.
- L re_null &
- L re_universe &
- (ALL (a::'a::type => bool) b::'a::type => bool.
- L a & L b --> L (re_intersect a b)) &
- (ALL P::('a::type => bool) => bool. re_subset P L --> L (re_Union P))"
-
-lemma istopology: "ALL L::('a::type => bool) => bool.
- istopology L =
- (L re_null &
+%L. L re_null &
L re_universe &
- (ALL (a::'a::type => bool) b::'a::type => bool.
- L a & L b --> L (re_intersect a b)) &
- (ALL P::('a::type => bool) => bool. re_subset P L --> L (re_Union P)))"
+ (ALL a b. L a & L b --> L (re_intersect a b)) &
+ (ALL P. re_subset P L --> L (re_Union P))"
+
+lemma istopology: "istopology L =
+(L re_null &
+ L re_universe &
+ (ALL a b. L a & L b --> L (re_intersect a b)) &
+ (ALL P. re_subset P L --> L (re_Union P)))"
by (import topology istopology)
-typedef (open) ('a) topology = "(Collect::((('a::type => bool) => bool) => bool)
- => (('a::type => bool) => bool) set)
- (istopology::(('a::type => bool) => bool) => bool)"
+typedef (open) ('a) topology = "Collect istopology::(('a::type => bool) => bool) set"
by (rule typedef_helper,import topology topology_TY_DEF)
lemmas topology_TY_DEF = typedef_hol2hol4 [OF type_definition_topology]
@@ -864,94 +708,84 @@
topology :: "(('a => bool) => bool) => 'a topology"
"open" :: "'a topology => ('a => bool) => bool"
-specification ("open" topology) topology_tybij: "(ALL a::'a::type topology. topology (open a) = a) &
-(ALL r::('a::type => bool) => bool. istopology r = (open (topology r) = r))"
+specification ("open" topology) topology_tybij: "(ALL a::'a topology. topology (topology.open a) = a) &
+(ALL r::('a => bool) => bool.
+ istopology r = (topology.open (topology r) = r))"
by (import topology topology_tybij)
-lemma TOPOLOGY: "ALL L::'a::type topology.
- open L re_null &
- open L re_universe &
- (ALL (a::'a::type => bool) b::'a::type => bool.
- open L a & open L b --> open L (re_intersect a b)) &
- (ALL P::('a::type => bool) => bool.
- re_subset P (open L) --> open L (re_Union P))"
+lemma TOPOLOGY: "topology.open L re_null &
+topology.open L re_universe &
+(ALL a b.
+ topology.open L a & topology.open L b -->
+ topology.open L (re_intersect a b)) &
+(ALL P. re_subset P (topology.open L) --> topology.open L (re_Union P))"
by (import topology TOPOLOGY)
-lemma TOPOLOGY_UNION: "ALL (x::'a::type topology) xa::('a::type => bool) => bool.
- re_subset xa (open x) --> open x (re_Union xa)"
+lemma TOPOLOGY_UNION: "re_subset xa (topology.open x) ==> topology.open x (re_Union xa)"
by (import topology TOPOLOGY_UNION)
-definition neigh :: "'a topology => ('a => bool) * 'a => bool" where
- "neigh ==
-%(top::'a::type topology) (N::'a::type => bool, x::'a::type).
- EX P::'a::type => bool. open top P & re_subset P N & P x"
-
-lemma neigh: "ALL (top::'a::type topology) (N::'a::type => bool) x::'a::type.
- neigh top (N, x) =
- (EX P::'a::type => bool. open top P & re_subset P N & P x)"
+definition
+ neigh :: "'a topology => ('a => bool) * 'a => bool" where
+ "neigh == %tp (N, x). EX P. topology.open tp P & re_subset P N & P x"
+
+lemma neigh: "neigh (tp::'a::type topology) (N::'a::type => bool, x::'a::type) =
+(EX P::'a::type => bool. topology.open tp P & re_subset P N & P x)"
by (import topology neigh)
-lemma OPEN_OWN_NEIGH: "ALL (S'::'a::type => bool) (top::'a::type topology) x::'a::type.
- open top S' & S' x --> neigh top (S', x)"
+lemma OPEN_OWN_NEIGH: "topology.open (tp::'a::type topology) (S'::'a::type => bool) &
+S' (x::'a::type)
+==> neigh tp (S', x)"
by (import topology OPEN_OWN_NEIGH)
-lemma OPEN_UNOPEN: "ALL (S'::'a::type => bool) top::'a::type topology.
- open top S' =
- (re_Union (%P::'a::type => bool. open top P & re_subset P S') = S')"
+lemma OPEN_UNOPEN: "topology.open (tp::'a::type topology) (S'::'a::type => bool) =
+(re_Union (%P::'a::type => bool. topology.open tp P & re_subset P S') = S')"
by (import topology OPEN_UNOPEN)
-lemma OPEN_SUBOPEN: "ALL (S'::'a::type => bool) top::'a::type topology.
- open top S' =
- (ALL x::'a::type.
- S' x --> (EX P::'a::type => bool. P x & open top P & re_subset P S'))"
+lemma OPEN_SUBOPEN: "topology.open (tp::'a::type topology) (S'::'a::type => bool) =
+(ALL x::'a::type.
+ S' x -->
+ (EX P::'a::type => bool. P x & topology.open tp P & re_subset P S'))"
by (import topology OPEN_SUBOPEN)
-lemma OPEN_NEIGH: "ALL (S'::'a::type => bool) top::'a::type topology.
- open top S' =
- (ALL x::'a::type.
- S' x --> (EX N::'a::type => bool. neigh top (N, x) & re_subset N S'))"
+lemma OPEN_NEIGH: "topology.open (tp::'a::type topology) (S'::'a::type => bool) =
+(ALL x::'a::type.
+ S' x --> (EX N::'a::type => bool. neigh tp (N, x) & re_subset N S'))"
by (import topology OPEN_NEIGH)
-definition closed :: "'a topology => ('a => bool) => bool" where
- "closed == %(L::'a::type topology) S'::'a::type => bool. open L (re_compl S')"
-
-lemma closed: "ALL (L::'a::type topology) S'::'a::type => bool.
- closed L S' = open L (re_compl S')"
+consts
+ closed :: "'a topology => ('a => bool) => bool"
+
+defs
+ closed_def: "topology.closed == %L S'. topology.open L (re_compl S')"
+
+lemma closed: "topology.closed L S' = topology.open L (re_compl S')"
by (import topology closed)
-definition limpt :: "'a topology => 'a => ('a => bool) => bool" where
- "limpt ==
-%(top::'a::type topology) (x::'a::type) S'::'a::type => bool.
- ALL N::'a::type => bool.
- neigh top (N, x) --> (EX y::'a::type. x ~= y & S' y & N y)"
-
-lemma limpt: "ALL (top::'a::type topology) (x::'a::type) S'::'a::type => bool.
- limpt top x S' =
- (ALL N::'a::type => bool.
- neigh top (N, x) --> (EX y::'a::type. x ~= y & S' y & N y))"
+definition
+ limpt :: "'a topology => 'a => ('a => bool) => bool" where
+ "limpt == %tp x S'. ALL N. neigh tp (N, x) --> (EX y. x ~= y & S' y & N y)"
+
+lemma limpt: "limpt (tp::'a::type topology) (x::'a::type) (S'::'a::type => bool) =
+(ALL N::'a::type => bool.
+ neigh tp (N, x) --> (EX y::'a::type. x ~= y & S' y & N y))"
by (import topology limpt)
-lemma CLOSED_LIMPT: "ALL (top::'a::type topology) S'::'a::type => bool.
- closed top S' = (ALL x::'a::type. limpt top x S' --> S' x)"
+lemma CLOSED_LIMPT: "topology.closed (tp::'a::type topology) (S'::'a::type => bool) =
+(ALL x::'a::type. limpt tp x S' --> S' x)"
by (import topology CLOSED_LIMPT)
-definition ismet :: "('a * 'a => real) => bool" where
+definition
+ ismet :: "('a * 'a => real) => bool" where
"ismet ==
-%m::'a::type * 'a::type => real.
- (ALL (x::'a::type) y::'a::type. (m (x, y) = 0) = (x = y)) &
- (ALL (x::'a::type) (y::'a::type) z::'a::type.
- m (y, z) <= m (x, y) + m (x, z))"
-
-lemma ismet: "ALL m::'a::type * 'a::type => real.
- ismet m =
- ((ALL (x::'a::type) y::'a::type. (m (x, y) = 0) = (x = y)) &
- (ALL (x::'a::type) (y::'a::type) z::'a::type.
- m (y, z) <= m (x, y) + m (x, z)))"
+%m. (ALL x y. (m (x, y) = 0) = (x = y)) &
+ (ALL x y z. m (y, z) <= m (x, y) + m (x, z))"
+
+lemma ismet: "ismet m =
+((ALL x y. (m (x, y) = 0) = (x = y)) &
+ (ALL x y z. m (y, z) <= m (x, y) + m (x, z)))"
by (import topology ismet)
-typedef (open) ('a) metric = "(Collect::(('a::type * 'a::type => real) => bool)
- => ('a::type * 'a::type => real) set)
- (ismet::('a::type * 'a::type => real) => bool)"
+typedef (open) ('a) metric = "Collect ismet :: ('a::type * 'a::type => real) set"
by (rule typedef_helper,import topology metric_TY_DEF)
lemmas metric_TY_DEF = typedef_hol2hol4 [OF type_definition_metric]
@@ -960,330 +794,265 @@
metric :: "('a * 'a => real) => 'a metric"
dist :: "'a metric => 'a * 'a => real"
-specification (dist metric) metric_tybij: "(ALL a::'a::type metric. metric (dist a) = a) &
-(ALL r::'a::type * 'a::type => real. ismet r = (dist (metric r) = r))"
+specification (dist metric) metric_tybij: "(ALL a::'a metric. metric (topology.dist a) = a) &
+(ALL r::'a * 'a => real. ismet r = (topology.dist (metric r) = r))"
by (import topology metric_tybij)
-lemma METRIC_ISMET: "ALL m::'a::type metric. ismet (dist m)"
+lemma METRIC_ISMET: "ismet (topology.dist m)"
by (import topology METRIC_ISMET)
-lemma METRIC_ZERO: "ALL (m::'a::type metric) (x::'a::type) y::'a::type.
- (dist m (x, y) = 0) = (x = y)"
+lemma METRIC_ZERO: "(topology.dist m (x, y) = 0) = (x = y)"
by (import topology METRIC_ZERO)
-lemma METRIC_SAME: "ALL (m::'a::type metric) x::'a::type. dist m (x, x) = 0"
+lemma METRIC_SAME: "topology.dist m (x, x) = 0"
by (import topology METRIC_SAME)
-lemma METRIC_POS: "ALL (m::'a::type metric) (x::'a::type) y::'a::type. 0 <= dist m (x, y)"
+lemma METRIC_POS: "0 <= topology.dist m (x, y)"
by (import topology METRIC_POS)
-lemma METRIC_SYM: "ALL (m::'a::type metric) (x::'a::type) y::'a::type.
- dist m (x, y) = dist m (y, x)"
+lemma METRIC_SYM: "topology.dist m (x, y) = topology.dist m (y, x)"
by (import topology METRIC_SYM)
-lemma METRIC_TRIANGLE: "ALL (m::'a::type metric) (x::'a::type) (y::'a::type) z::'a::type.
- dist m (x, z) <= dist m (x, y) + dist m (y, z)"
+lemma METRIC_TRIANGLE: "topology.dist m (x, z) <= topology.dist m (x, y) + topology.dist m (y, z)"
by (import topology METRIC_TRIANGLE)
-lemma METRIC_NZ: "ALL (m::'a::type metric) (x::'a::type) y::'a::type.
- x ~= y --> 0 < dist m (x, y)"
+lemma METRIC_NZ: "x ~= y ==> 0 < topology.dist m (x, y)"
by (import topology METRIC_NZ)
-definition mtop :: "'a metric => 'a topology" where
+definition
+ mtop :: "'a metric => 'a topology" where
"mtop ==
-%m::'a::type metric.
- topology
- (%S'::'a::type => bool.
- ALL x::'a::type.
- S' x --> (EX e>0. ALL y::'a::type. dist m (x, y) < e --> S' y))"
-
-lemma mtop: "ALL m::'a::type metric.
- mtop m =
- topology
- (%S'::'a::type => bool.
- ALL x::'a::type.
- S' x --> (EX e>0. ALL y::'a::type. dist m (x, y) < e --> S' y))"
+%m. topology
+ (%S'. ALL x.
+ S' x --> (EX e>0. ALL y. topology.dist m (x, y) < e --> S' y))"
+
+lemma mtop: "mtop m =
+topology
+ (%S'. ALL x. S' x --> (EX e>0. ALL y. topology.dist m (x, y) < e --> S' y))"
by (import topology mtop)
-lemma mtop_istopology: "ALL m::'a::type metric.
- istopology
- (%S'::'a::type => bool.
- ALL x::'a::type.
- S' x --> (EX e>0. ALL y::'a::type. dist m (x, y) < e --> S' y))"
+lemma mtop_istopology: "istopology
+ (%S'. ALL x. S' x --> (EX e>0. ALL y. topology.dist m (x, y) < e --> S' y))"
by (import topology mtop_istopology)
-lemma MTOP_OPEN: "ALL (S'::'a::type => bool) x::'a::type metric.
- open (mtop x) S' =
- (ALL xa::'a::type.
- S' xa --> (EX e>0. ALL y::'a::type. dist x (xa, y) < e --> S' y))"
+lemma MTOP_OPEN: "topology.open (mtop x) S' =
+(ALL xa. S' xa --> (EX e>0. ALL y. topology.dist x (xa, y) < e --> S' y))"
by (import topology MTOP_OPEN)
-definition B :: "'a metric => 'a * real => 'a => bool" where
- "B ==
-%(m::'a::type metric) (x::'a::type, e::real) y::'a::type. dist m (x, y) < e"
-
-lemma ball: "ALL (m::'a::type metric) (x::'a::type) e::real.
- B m (x, e) = (%y::'a::type. dist m (x, y) < e)"
+definition
+ B :: "'a metric => 'a * real => 'a => bool" where
+ "B == %m (x, e) y. topology.dist m (x, y) < e"
+
+lemma ball: "B m (x, e) = (%y. topology.dist m (x, y) < e)"
by (import topology ball)
-lemma BALL_OPEN: "ALL (m::'a::type metric) (x::'a::type) e::real.
- 0 < e --> open (mtop m) (B m (x, e))"
+lemma BALL_OPEN: "0 < e ==> topology.open (mtop m) (B m (x, e))"
by (import topology BALL_OPEN)
-lemma BALL_NEIGH: "ALL (m::'a::type metric) (x::'a::type) e::real.
- 0 < e --> neigh (mtop m) (B m (x, e), x)"
+lemma BALL_NEIGH: "0 < e ==> neigh (mtop m) (B m (x, e), x)"
by (import topology BALL_NEIGH)
-lemma MTOP_LIMPT: "ALL (m::'a::type metric) (x::'a::type) S'::'a::type => bool.
- limpt (mtop m) x S' =
- (ALL e>0. EX y::'a::type. x ~= y & S' y & dist m (x, y) < e)"
+lemma MTOP_LIMPT: "limpt (mtop m) x S' =
+(ALL e>0. EX y. x ~= y & S' y & topology.dist m (x, y) < e)"
by (import topology MTOP_LIMPT)
-lemma ISMET_R1: "ismet (%(x::real, y::real). abs (y - x))"
+lemma ISMET_R1: "ismet (%(x, y). abs (y - x))"
by (import topology ISMET_R1)
-definition mr1 :: "real metric" where
- "mr1 == metric (%(x::real, y::real). abs (y - x))"
-
-lemma mr1: "mr1 = metric (%(x::real, y::real). abs (y - x))"
+definition
+ mr1 :: "real metric" where
+ "mr1 == metric (%(x, y). abs (y - x))"
+
+lemma mr1: "mr1 = metric (%(x, y). abs (y - x))"
by (import topology mr1)
-lemma MR1_DEF: "ALL (x::real) y::real. dist mr1 (x, y) = abs (y - x)"
+lemma MR1_DEF: "topology.dist mr1 (x, y) = abs (y - x)"
by (import topology MR1_DEF)
-lemma MR1_ADD: "ALL (x::real) d::real. dist mr1 (x, x + d) = abs d"
+lemma MR1_ADD: "topology.dist mr1 (x, x + d) = abs d"
by (import topology MR1_ADD)
-lemma MR1_SUB: "ALL (x::real) d::real. dist mr1 (x, x - d) = abs d"
+lemma MR1_SUB: "topology.dist mr1 (x, x - d) = abs d"
by (import topology MR1_SUB)
-lemma MR1_ADD_POS: "ALL (x::real) d::real. 0 <= d --> dist mr1 (x, x + d) = d"
+lemma MR1_ADD_POS: "0 <= d ==> topology.dist mr1 (x, x + d) = d"
by (import topology MR1_ADD_POS)
-lemma MR1_SUB_LE: "ALL (x::real) d::real. 0 <= d --> dist mr1 (x, x - d) = d"
+lemma MR1_SUB_LE: "0 <= d ==> topology.dist mr1 (x, x - d) = d"
by (import topology MR1_SUB_LE)
-lemma MR1_ADD_LT: "ALL (x::real) d::real. 0 < d --> dist mr1 (x, x + d) = d"
+lemma MR1_ADD_LT: "0 < d ==> topology.dist mr1 (x, x + d) = d"
by (import topology MR1_ADD_LT)
-lemma MR1_SUB_LT: "ALL (x::real) d::real. 0 < d --> dist mr1 (x, x - d) = d"
+lemma MR1_SUB_LT: "0 < d ==> topology.dist mr1 (x, x - d) = d"
by (import topology MR1_SUB_LT)
-lemma MR1_BETWEEN1: "ALL (x::real) (y::real) z::real. x < z & dist mr1 (x, y) < z - x --> y < z"
+lemma MR1_BETWEEN1: "x < z & topology.dist mr1 (x, y) < z - x ==> y < z"
by (import topology MR1_BETWEEN1)
-lemma MR1_LIMPT: "ALL x::real. limpt (mtop mr1) x re_universe"
+lemma MR1_LIMPT: "limpt (mtop mr1) x re_universe"
by (import topology MR1_LIMPT)
;end_setup
;setup_theory nets
-definition dorder :: "('a => 'a => bool) => bool" where
+definition
+ dorder :: "('a => 'a => bool) => bool" where
"dorder ==
-%g::'a::type => 'a::type => bool.
- ALL (x::'a::type) y::'a::type.
- g x x & g y y -->
- (EX z::'a::type. g z z & (ALL w::'a::type. g w z --> g w x & g w y))"
-
-lemma dorder: "ALL g::'a::type => 'a::type => bool.
- dorder g =
- (ALL (x::'a::type) y::'a::type.
- g x x & g y y -->
- (EX z::'a::type. g z z & (ALL w::'a::type. g w z --> g w x & g w y)))"
+%g. ALL x y.
+ g x x & g y y --> (EX z. g z z & (ALL w. g w z --> g w x & g w y))"
+
+lemma dorder: "dorder g =
+(ALL x y.
+ g x x & g y y --> (EX z. g z z & (ALL w. g w z --> g w x & g w y)))"
by (import nets dorder)
-definition tends :: "('b => 'a) => 'a => 'a topology * ('b => 'b => bool) => bool" where
+definition
+ tends :: "('b => 'a) => 'a => 'a topology * ('b => 'b => bool) => bool" where
"tends ==
-%(s::'b::type => 'a::type) (l::'a::type) (top::'a::type topology,
- g::'b::type => 'b::type => bool).
- ALL N::'a::type => bool.
- neigh top (N, l) -->
- (EX n::'b::type. g n n & (ALL m::'b::type. g m n --> N (s m)))"
-
-lemma tends: "ALL (s::'b::type => 'a::type) (l::'a::type) (top::'a::type topology)
- g::'b::type => 'b::type => bool.
- tends s l (top, g) =
- (ALL N::'a::type => bool.
- neigh top (N, l) -->
- (EX n::'b::type. g n n & (ALL m::'b::type. g m n --> N (s m))))"
+%(s::'b => 'a) (l::'a) (tp::'a topology, g::'b => 'b => bool).
+ ALL N::'a => bool.
+ neigh tp (N, l) -->
+ (EX n::'b. g n n & (ALL m::'b. g m n --> N (s m)))"
+
+lemma tends: "tends (s::'b::type => 'a::type) (l::'a::type)
+ (tp::'a::type topology, g::'b::type => 'b::type => bool) =
+(ALL N::'a::type => bool.
+ neigh tp (N, l) -->
+ (EX n::'b::type. g n n & (ALL m::'b::type. g m n --> N (s m))))"
by (import nets tends)
-definition bounded :: "'a metric * ('b => 'b => bool) => ('b => 'a) => bool" where
+definition
+ bounded :: "'a metric * ('b => 'b => bool) => ('b => 'a) => bool" where
"bounded ==
-%(m::'a::type metric, g::'b::type => 'b::type => bool)
- f::'b::type => 'a::type.
- EX (k::real) (x::'a::type) N::'b::type.
- g N N & (ALL n::'b::type. g n N --> dist m (f n, x) < k)"
-
-lemma bounded: "ALL (m::'a::type metric) (g::'b::type => 'b::type => bool)
- f::'b::type => 'a::type.
- bounded (m, g) f =
- (EX (k::real) (x::'a::type) N::'b::type.
- g N N & (ALL n::'b::type. g n N --> dist m (f n, x) < k))"
+%(m, g) f. EX k x N. g N N & (ALL n. g n N --> topology.dist m (f n, x) < k)"
+
+lemma bounded: "bounded (m, g) f =
+(EX k x N. g N N & (ALL n. g n N --> topology.dist m (f n, x) < k))"
by (import nets bounded)
-definition tendsto :: "'a metric * 'a => 'a => 'a => bool" where
- "tendsto ==
-%(m::'a::type metric, x::'a::type) (y::'a::type) z::'a::type.
- 0 < dist m (x, y) & dist m (x, y) <= dist m (x, z)"
-
-lemma tendsto: "ALL (m::'a::type metric) (x::'a::type) (y::'a::type) z::'a::type.
- tendsto (m, x) y z = (0 < dist m (x, y) & dist m (x, y) <= dist m (x, z))"
+consts
+ tendsto :: "'a metric * 'a => 'a => 'a => bool"
+
+defs
+ tendsto_def: "nets.tendsto ==
+%(m, x) y z.
+ 0 < topology.dist m (x, y) &
+ topology.dist m (x, y) <= topology.dist m (x, z)"
+
+lemma tendsto: "nets.tendsto (m, x) y z =
+(0 < topology.dist m (x, y) &
+ topology.dist m (x, y) <= topology.dist m (x, z))"
by (import nets tendsto)
-lemma DORDER_LEMMA: "ALL g::'a::type => 'a::type => bool.
- dorder g -->
- (ALL (P::'a::type => bool) Q::'a::type => bool.
- (EX n::'a::type. g n n & (ALL m::'a::type. g m n --> P m)) &
- (EX n::'a::type. g n n & (ALL m::'a::type. g m n --> Q m)) -->
- (EX n::'a::type. g n n & (ALL m::'a::type. g m n --> P m & Q m)))"
+lemma DORDER_LEMMA: "[| dorder g;
+ (EX n. g n n & (ALL m. g m n --> P m)) &
+ (EX n. g n n & (ALL m. g m n --> Q m)) |]
+==> EX n. g n n & (ALL m. g m n --> P m & Q m)"
by (import nets DORDER_LEMMA)
lemma DORDER_NGE: "dorder nat_ge"
by (import nets DORDER_NGE)
-lemma DORDER_TENDSTO: "ALL (m::'a::type metric) x::'a::type. dorder (tendsto (m, x))"
+lemma DORDER_TENDSTO: "dorder (nets.tendsto (m, x))"
by (import nets DORDER_TENDSTO)
-lemma MTOP_TENDS: "ALL (d::'a::type metric) (g::'b::type => 'b::type => bool)
- (x::'b::type => 'a::type) x0::'a::type.
- tends x x0 (mtop d, g) =
- (ALL e>0.
- EX n::'b::type.
- g n n & (ALL m::'b::type. g m n --> dist d (x m, x0) < e))"
+lemma MTOP_TENDS: "tends (x::'b => 'a) (x0::'a) (mtop (d::'a metric), g::'b => 'b => bool) =
+(ALL e>0::real.
+ EX n::'b. g n n & (ALL m::'b. g m n --> topology.dist d (x m, x0) < e))"
by (import nets MTOP_TENDS)
-lemma MTOP_TENDS_UNIQ: "ALL (g::'b::type => 'b::type => bool) d::'a::type metric.
- dorder g -->
- tends (x::'b::type => 'a::type) (x0::'a::type) (mtop d, g) &
- tends x (x1::'a::type) (mtop d, g) -->
- x0 = x1"
+lemma MTOP_TENDS_UNIQ: "[| dorder (g::'b => 'b => bool);
+ tends (x::'b => 'a) (x0::'a) (mtop (d::'a metric), g) &
+ tends x (x1::'a) (mtop d, g) |]
+==> x0 = x1"
by (import nets MTOP_TENDS_UNIQ)
-lemma SEQ_TENDS: "ALL (d::'a::type metric) (x::nat => 'a::type) x0::'a::type.
- tends x x0 (mtop d, nat_ge) =
- (ALL xa>0. EX xb::nat. ALL xc::nat. xb <= xc --> dist d (x xc, x0) < xa)"
+lemma SEQ_TENDS: "tends x x0 (mtop d, nat_ge) =
+(ALL xa>0. EX xb. ALL xc>=xb. topology.dist d (x xc, x0) < xa)"
by (import nets SEQ_TENDS)
-lemma LIM_TENDS: "ALL (m1::'a::type metric) (m2::'b::type metric) (f::'a::type => 'b::type)
- (x0::'a::type) y0::'b::type.
- limpt (mtop m1) x0 re_universe -->
- tends f y0 (mtop m2, tendsto (m1, x0)) =
- (ALL e>0.
- EX d>0.
- ALL x::'a::type.
- 0 < dist m1 (x, x0) & dist m1 (x, x0) <= d -->
- dist m2 (f x, y0) < e)"
+lemma LIM_TENDS: "limpt (mtop m1) x0 re_universe
+==> tends f y0 (mtop m2, nets.tendsto (m1, x0)) =
+ (ALL e>0.
+ EX d>0.
+ ALL x.
+ 0 < topology.dist m1 (x, x0) &
+ topology.dist m1 (x, x0) <= d -->
+ topology.dist m2 (f x, y0) < e)"
by (import nets LIM_TENDS)
-lemma LIM_TENDS2: "ALL (m1::'a::type metric) (m2::'b::type metric) (f::'a::type => 'b::type)
- (x0::'a::type) y0::'b::type.
- limpt (mtop m1) x0 re_universe -->
- tends f y0 (mtop m2, tendsto (m1, x0)) =
- (ALL e>0.
- EX d>0.
- ALL x::'a::type.
- 0 < dist m1 (x, x0) & dist m1 (x, x0) < d -->
- dist m2 (f x, y0) < e)"
+lemma LIM_TENDS2: "limpt (mtop m1) x0 re_universe
+==> tends f y0 (mtop m2, nets.tendsto (m1, x0)) =
+ (ALL e>0.
+ EX d>0.
+ ALL x.
+ 0 < topology.dist m1 (x, x0) &
+ topology.dist m1 (x, x0) < d -->
+ topology.dist m2 (f x, y0) < e)"
by (import nets LIM_TENDS2)
-lemma MR1_BOUNDED: "ALL (g::'a::type => 'a::type => bool) f::'a::type => real.
- bounded (mr1, g) f =
- (EX (k::real) N::'a::type.
- g N N & (ALL n::'a::type. g n N --> abs (f n) < k))"
+lemma MR1_BOUNDED: "bounded (mr1, g) f = (EX k N. g N N & (ALL n. g n N --> abs (f n) < k))"
by (import nets MR1_BOUNDED)
-lemma NET_NULL: "ALL (g::'a::type => 'a::type => bool) (x::'a::type => real) x0::real.
- tends x x0 (mtop mr1, g) = tends (%n::'a::type. x n - x0) 0 (mtop mr1, g)"
+lemma NET_NULL: "tends x x0 (mtop mr1, g) = tends (%n. x n - x0) 0 (mtop mr1, g)"
by (import nets NET_NULL)
-lemma NET_CONV_BOUNDED: "ALL (g::'a::type => 'a::type => bool) (x::'a::type => real) x0::real.
- tends x x0 (mtop mr1, g) --> bounded (mr1, g) x"
+lemma NET_CONV_BOUNDED: "tends x x0 (mtop mr1, g) ==> bounded (mr1, g) x"
by (import nets NET_CONV_BOUNDED)
-lemma NET_CONV_NZ: "ALL (g::'a::type => 'a::type => bool) (x::'a::type => real) x0::real.
- tends x x0 (mtop mr1, g) & x0 ~= 0 -->
- (EX N::'a::type. g N N & (ALL n::'a::type. g n N --> x n ~= 0))"
+lemma NET_CONV_NZ: "tends x x0 (mtop mr1, g) & x0 ~= 0
+==> EX N. g N N & (ALL n. g n N --> x n ~= 0)"
by (import nets NET_CONV_NZ)
-lemma NET_CONV_IBOUNDED: "ALL (g::'a::type => 'a::type => bool) (x::'a::type => real) x0::real.
- tends x x0 (mtop mr1, g) & x0 ~= 0 -->
- bounded (mr1, g) (%n::'a::type. inverse (x n))"
+lemma NET_CONV_IBOUNDED: "tends x x0 (mtop mr1, g) & x0 ~= 0 ==> bounded (mr1, g) (%n. inverse (x n))"
by (import nets NET_CONV_IBOUNDED)
-lemma NET_NULL_ADD: "ALL g::'a::type => 'a::type => bool.
- dorder g -->
- (ALL (x::'a::type => real) y::'a::type => real.
- tends x 0 (mtop mr1, g) & tends y 0 (mtop mr1, g) -->
- tends (%n::'a::type. x n + y n) 0 (mtop mr1, g))"
+lemma NET_NULL_ADD: "[| dorder g; tends x 0 (mtop mr1, g) & tends y 0 (mtop mr1, g) |]
+==> tends (%n. x n + y n) 0 (mtop mr1, g)"
by (import nets NET_NULL_ADD)
-lemma NET_NULL_MUL: "ALL g::'a::type => 'a::type => bool.
- dorder g -->
- (ALL (x::'a::type => real) y::'a::type => real.
- bounded (mr1, g) x & tends y 0 (mtop mr1, g) -->
- tends (%n::'a::type. x n * y n) 0 (mtop mr1, g))"
+lemma NET_NULL_MUL: "[| dorder g; bounded (mr1, g) x & tends y 0 (mtop mr1, g) |]
+==> tends (%n. x n * y n) 0 (mtop mr1, g)"
by (import nets NET_NULL_MUL)
-lemma NET_NULL_CMUL: "ALL (g::'a::type => 'a::type => bool) (k::real) x::'a::type => real.
- tends x 0 (mtop mr1, g) --> tends (%n::'a::type. k * x n) 0 (mtop mr1, g)"
+lemma NET_NULL_CMUL: "tends x 0 (mtop mr1, g) ==> tends (%n. k * x n) 0 (mtop mr1, g)"
by (import nets NET_NULL_CMUL)
-lemma NET_ADD: "ALL g::'a::type => 'a::type => bool.
- dorder g -->
- (ALL (x::'a::type => real) (x0::real) (y::'a::type => real) y0::real.
- tends x x0 (mtop mr1, g) & tends y y0 (mtop mr1, g) -->
- tends (%n::'a::type. x n + y n) (x0 + y0) (mtop mr1, g))"
+lemma NET_ADD: "[| dorder g; tends x x0 (mtop mr1, g) & tends y y0 (mtop mr1, g) |]
+==> tends (%n. x n + y n) (x0 + y0) (mtop mr1, g)"
by (import nets NET_ADD)
-lemma NET_NEG: "ALL g::'a::type => 'a::type => bool.
- dorder g -->
- (ALL (x::'a::type => real) x0::real.
- tends x x0 (mtop mr1, g) =
- tends (%n::'a::type. - x n) (- x0) (mtop mr1, g))"
+lemma NET_NEG: "dorder g
+==> tends x x0 (mtop mr1, g) = tends (%n. - x n) (- x0) (mtop mr1, g)"
by (import nets NET_NEG)
-lemma NET_SUB: "ALL g::'a::type => 'a::type => bool.
- dorder g -->
- (ALL (x::'a::type => real) (x0::real) (y::'a::type => real) y0::real.
- tends x x0 (mtop mr1, g) & tends y y0 (mtop mr1, g) -->
- tends (%xa::'a::type. x xa - y xa) (x0 - y0) (mtop mr1, g))"
+lemma NET_SUB: "[| dorder g; tends x x0 (mtop mr1, g) & tends y y0 (mtop mr1, g) |]
+==> tends (%xa. x xa - y xa) (x0 - y0) (mtop mr1, g)"
by (import nets NET_SUB)
-lemma NET_MUL: "ALL g::'a::type => 'a::type => bool.
- dorder g -->
- (ALL (x::'a::type => real) (y::'a::type => real) (x0::real) y0::real.
- tends x x0 (mtop mr1, g) & tends y y0 (mtop mr1, g) -->
- tends (%n::'a::type. x n * y n) (x0 * y0) (mtop mr1, g))"
+lemma NET_MUL: "[| dorder g; tends x x0 (mtop mr1, g) & tends y y0 (mtop mr1, g) |]
+==> tends (%n. x n * y n) (x0 * y0) (mtop mr1, g)"
by (import nets NET_MUL)
-lemma NET_INV: "ALL g::'a::type => 'a::type => bool.
- dorder g -->
- (ALL (x::'a::type => real) x0::real.
- tends x x0 (mtop mr1, g) & x0 ~= 0 -->
- tends (%n::'a::type. inverse (x n)) (inverse x0) (mtop mr1, g))"
+lemma NET_INV: "[| dorder g; tends x x0 (mtop mr1, g) & x0 ~= 0 |]
+==> tends (%n. inverse (x n)) (inverse x0) (mtop mr1, g)"
by (import nets NET_INV)
-lemma NET_DIV: "ALL g::'a::type => 'a::type => bool.
- dorder g -->
- (ALL (x::'a::type => real) (x0::real) (y::'a::type => real) y0::real.
- tends x x0 (mtop mr1, g) & tends y y0 (mtop mr1, g) & y0 ~= 0 -->
- tends (%xa::'a::type. x xa / y xa) (x0 / y0) (mtop mr1, g))"
+lemma NET_DIV: "[| dorder g;
+ tends x x0 (mtop mr1, g) & tends y y0 (mtop mr1, g) & y0 ~= 0 |]
+==> tends (%xa. x xa / y xa) (x0 / y0) (mtop mr1, g)"
by (import nets NET_DIV)
-lemma NET_ABS: "ALL (g::'a::type => 'a::type => bool) (x::'a::type => real) x0::real.
- tends x x0 (mtop mr1, g) -->
- tends (%n::'a::type. abs (x n)) (abs x0) (mtop mr1, g)"
+lemma NET_ABS: "tends x x0 (mtop mr1, g) ==> tends (%n. abs (x n)) (abs x0) (mtop mr1, g)"
by (import nets NET_ABS)
-lemma NET_LE: "ALL g::'a::type => 'a::type => bool.
- dorder g -->
- (ALL (x::'a::type => real) (x0::real) (y::'a::type => real) y0::real.
- tends x x0 (mtop mr1, g) &
- tends y y0 (mtop mr1, g) &
- (EX N::'a::type. g N N & (ALL n::'a::type. g n N --> x n <= y n)) -->
- x0 <= y0)"
+lemma NET_LE: "[| dorder g;
+ tends x x0 (mtop mr1, g) &
+ tends y y0 (mtop mr1, g) &
+ (EX N. g N N & (ALL n. g n N --> x n <= y n)) |]
+==> x0 <= y0"
by (import nets NET_LE)
;end_setup
@@ -1294,84 +1063,77 @@
"hol4-->" :: "(nat => real) => real => bool" ("hol4-->")
defs
- "hol4-->_def": "hol4--> == %(x::nat => real) x0::real. tends x x0 (mtop mr1, nat_ge)"
-
-lemma tends_num_real: "ALL (x::nat => real) x0::real. hol4--> x x0 = tends x x0 (mtop mr1, nat_ge)"
+ "hol4-->_def": "hol4--> == %x x0. tends x x0 (mtop mr1, nat_ge)"
+
+lemma tends_num_real: "hol4--> x x0 = tends x x0 (mtop mr1, nat_ge)"
by (import seq tends_num_real)
-lemma SEQ: "ALL (x::nat => real) x0::real.
- hol4--> x x0 =
- (ALL e>0. EX N::nat. ALL n::nat. N <= n --> abs (x n - x0) < e)"
+lemma SEQ: "hol4--> x x0 = (ALL e>0. EX N. ALL n>=N. abs (x n - x0) < e)"
by (import seq SEQ)
-lemma SEQ_CONST: "ALL k::real. hol4--> (%x::nat. k) k"
+lemma SEQ_CONST: "hol4--> (%x. k) k"
by (import seq SEQ_CONST)
-lemma SEQ_ADD: "ALL (x::nat => real) (x0::real) (y::nat => real) y0::real.
- hol4--> x x0 & hol4--> y y0 --> hol4--> (%n::nat. x n + y n) (x0 + y0)"
+lemma SEQ_ADD: "hol4--> x x0 & hol4--> y y0 ==> hol4--> (%n. x n + y n) (x0 + y0)"
by (import seq SEQ_ADD)
-lemma SEQ_MUL: "ALL (x::nat => real) (x0::real) (y::nat => real) y0::real.
- hol4--> x x0 & hol4--> y y0 --> hol4--> (%n::nat. x n * y n) (x0 * y0)"
+lemma SEQ_MUL: "hol4--> x x0 & hol4--> y y0 ==> hol4--> (%n. x n * y n) (x0 * y0)"
by (import seq SEQ_MUL)
-lemma SEQ_NEG: "ALL (x::nat => real) x0::real.
- hol4--> x x0 = hol4--> (%n::nat. - x n) (- x0)"
+lemma SEQ_NEG: "hol4--> x x0 = hol4--> (%n. - x n) (- x0)"
by (import seq SEQ_NEG)
-lemma SEQ_INV: "ALL (x::nat => real) x0::real.
- hol4--> x x0 & x0 ~= 0 --> hol4--> (%n::nat. inverse (x n)) (inverse x0)"
+lemma SEQ_INV: "hol4--> x x0 & x0 ~= 0 ==> hol4--> (%n. inverse (x n)) (inverse x0)"
by (import seq SEQ_INV)
-lemma SEQ_SUB: "ALL (x::nat => real) (x0::real) (y::nat => real) y0::real.
- hol4--> x x0 & hol4--> y y0 --> hol4--> (%n::nat. x n - y n) (x0 - y0)"
+lemma SEQ_SUB: "hol4--> x x0 & hol4--> y y0 ==> hol4--> (%n. x n - y n) (x0 - y0)"
by (import seq SEQ_SUB)
-lemma SEQ_DIV: "ALL (x::nat => real) (x0::real) (y::nat => real) y0::real.
- hol4--> x x0 & hol4--> y y0 & y0 ~= 0 -->
- hol4--> (%n::nat. x n / y n) (x0 / y0)"
+lemma SEQ_DIV: "hol4--> x x0 & hol4--> y y0 & y0 ~= 0 ==> hol4--> (%n. x n / y n) (x0 / y0)"
by (import seq SEQ_DIV)
-lemma SEQ_UNIQ: "ALL (x::nat => real) (x1::real) x2::real.
- hol4--> x x1 & hol4--> x x2 --> x1 = x2"
+lemma SEQ_UNIQ: "hol4--> x x1 & hol4--> x x2 ==> x1 = x2"
by (import seq SEQ_UNIQ)
-definition convergent :: "(nat => real) => bool" where
- "convergent == %f::nat => real. Ex (hol4--> f)"
-
-lemma convergent: "ALL f::nat => real. convergent f = Ex (hol4--> f)"
+consts
+ convergent :: "(nat => real) => bool"
+
+defs
+ convergent_def: "seq.convergent == %f. Ex (hol4--> f)"
+
+lemma convergent: "seq.convergent f = Ex (hol4--> f)"
by (import seq convergent)
-definition cauchy :: "(nat => real) => bool" where
+definition
+ cauchy :: "(nat => real) => bool" where
"cauchy ==
-%f::nat => real.
- ALL e>0.
- EX N::nat.
- ALL (m::nat) n::nat. N <= m & N <= n --> abs (f m - f n) < e"
-
-lemma cauchy: "ALL f::nat => real.
- cauchy f =
- (ALL e>0.
- EX N::nat.
- ALL (m::nat) n::nat. N <= m & N <= n --> abs (f m - f n) < e)"
+%f. ALL e>0. EX N. ALL m n. N <= m & N <= n --> abs (f m - f n) < e"
+
+lemma cauchy: "cauchy f = (ALL e>0. EX N. ALL m n. N <= m & N <= n --> abs (f m - f n) < e)"
by (import seq cauchy)
-definition lim :: "(nat => real) => real" where
- "lim == %f::nat => real. Eps (hol4--> f)"
-
-lemma lim: "ALL f::nat => real. lim f = Eps (hol4--> f)"
+consts
+ lim :: "(nat => real) => real"
+
+defs
+ lim_def: "seq.lim == %f. Eps (hol4--> f)"
+
+lemma lim: "seq.lim f = Eps (hol4--> f)"
by (import seq lim)
-lemma SEQ_LIM: "ALL f::nat => real. convergent f = hol4--> f (lim f)"
+lemma SEQ_LIM: "seq.convergent f = hol4--> f (seq.lim f)"
by (import seq SEQ_LIM)
-definition subseq :: "(nat => nat) => bool" where
- "subseq == %f::nat => nat. ALL (m::nat) n::nat. m < n --> f m < f n"
-
-lemma subseq: "ALL f::nat => nat. subseq f = (ALL (m::nat) n::nat. m < n --> f m < f n)"
+consts
+ subseq :: "(nat => nat) => bool"
+
+defs
+ subseq_def: "seq.subseq == %f. ALL m n. m < n --> f m < f n"
+
+lemma subseq: "seq.subseq f = (ALL m n. m < n --> f m < f n)"
by (import seq subseq)
-lemma SUBSEQ_SUC: "ALL f::nat => nat. subseq f = (ALL n::nat. f n < f (Suc n))"
+lemma SUBSEQ_SUC: "seq.subseq f = (ALL n. f n < f (Suc n))"
by (import seq SUBSEQ_SUC)
consts
@@ -1379,1818 +1141,1414 @@
defs
mono_def: "seq.mono ==
-%f::nat => real.
- (ALL (m::nat) n::nat. m <= n --> f m <= f n) |
- (ALL (m::nat) n::nat. m <= n --> f n <= f m)"
-
-lemma mono: "ALL f::nat => real.
- seq.mono f =
- ((ALL (m::nat) n::nat. m <= n --> f m <= f n) |
- (ALL (m::nat) n::nat. m <= n --> f n <= f m))"
+%f. (ALL m n. m <= n --> f m <= f n) | (ALL m n. m <= n --> f n <= f m)"
+
+lemma mono: "seq.mono f =
+((ALL m n. m <= n --> f m <= f n) | (ALL m n. m <= n --> f n <= f m))"
by (import seq mono)
-lemma MONO_SUC: "ALL f::nat => real.
- seq.mono f =
- ((ALL x::nat. f x <= f (Suc x)) | (ALL n::nat. f (Suc n) <= f n))"
+lemma MONO_SUC: "seq.mono f = ((ALL x. f x <= f (Suc x)) | (ALL n. f (Suc n) <= f n))"
by (import seq MONO_SUC)
-lemma MAX_LEMMA: "(All::((nat => real) => bool) => bool)
- (%s::nat => real.
- (All::(nat => bool) => bool)
- (%N::nat.
- (Ex::(real => bool) => bool)
- (%k::real.
- (All::(nat => bool) => bool)
- (%n::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) n N)
- ((op <::real => real => bool)
- ((abs::real => real) (s n)) k)))))"
+lemma MAX_LEMMA: "EX k::real. ALL n<N::nat. abs ((s::nat => real) n) < k"
by (import seq MAX_LEMMA)
-lemma SEQ_BOUNDED: "ALL s::nat => real.
- bounded (mr1, nat_ge) s = (EX k::real. ALL n::nat. abs (s n) < k)"
+lemma SEQ_BOUNDED: "bounded (mr1, nat_ge) s = (EX k. ALL n. abs (s n) < k)"
by (import seq SEQ_BOUNDED)
-lemma SEQ_BOUNDED_2: "ALL (f::nat => real) (k::real) k'::real.
- (ALL n::nat. k <= f n & f n <= k') --> bounded (mr1, nat_ge) f"
+lemma SEQ_BOUNDED_2: "(!!n. k <= f n & f n <= k') ==> bounded (mr1, nat_ge) f"
by (import seq SEQ_BOUNDED_2)
-lemma SEQ_CBOUNDED: "ALL f::nat => real. cauchy f --> bounded (mr1, nat_ge) f"
+lemma SEQ_CBOUNDED: "cauchy f ==> bounded (mr1, nat_ge) f"
by (import seq SEQ_CBOUNDED)
-lemma SEQ_ICONV: "ALL f::nat => real.
- bounded (mr1, nat_ge) f &
- (ALL (m::nat) n::nat. n <= m --> f n <= f m) -->
- convergent f"
+lemma SEQ_ICONV: "bounded (mr1, nat_ge) f & (ALL m n. n <= m --> f n <= f m)
+==> seq.convergent f"
by (import seq SEQ_ICONV)
-lemma SEQ_NEG_CONV: "ALL f::nat => real. convergent f = convergent (%n::nat. - f n)"
+lemma SEQ_NEG_CONV: "seq.convergent f = seq.convergent (%n. - f n)"
by (import seq SEQ_NEG_CONV)
-lemma SEQ_NEG_BOUNDED: "ALL f::nat => real.
- bounded (mr1, nat_ge) (%n::nat. - f n) = bounded (mr1, nat_ge) f"
+lemma SEQ_NEG_BOUNDED: "bounded (mr1, nat_ge) (%n. - f n) = bounded (mr1, nat_ge) f"
by (import seq SEQ_NEG_BOUNDED)
-lemma SEQ_BCONV: "ALL f::nat => real. bounded (mr1, nat_ge) f & seq.mono f --> convergent f"
+lemma SEQ_BCONV: "bounded (mr1, nat_ge) f & seq.mono f ==> seq.convergent f"
by (import seq SEQ_BCONV)
-lemma SEQ_MONOSUB: "ALL s::nat => real. EX f::nat => nat. subseq f & seq.mono (%n::nat. s (f n))"
+lemma SEQ_MONOSUB: "EX f. seq.subseq f & seq.mono (%n. s (f n))"
by (import seq SEQ_MONOSUB)
-lemma SEQ_SBOUNDED: "ALL (s::nat => real) f::nat => nat.
- bounded (mr1, nat_ge) s --> bounded (mr1, nat_ge) (%n::nat. s (f n))"
+lemma SEQ_SBOUNDED: "bounded (mr1, nat_ge) s ==> bounded (mr1, nat_ge) (%n. s (f n))"
by (import seq SEQ_SBOUNDED)
-lemma SEQ_SUBLE: "ALL f::nat => nat. subseq f --> (ALL n::nat. n <= f n)"
+lemma SEQ_SUBLE: "seq.subseq f ==> n <= f n"
by (import seq SEQ_SUBLE)
-lemma SEQ_DIRECT: "ALL f::nat => nat.
- subseq f --> (ALL (N1::nat) N2::nat. EX x::nat. N1 <= x & N2 <= f x)"
+lemma SEQ_DIRECT: "seq.subseq f ==> EX x>=N1. N2 <= f x"
by (import seq SEQ_DIRECT)
-lemma SEQ_CAUCHY: "ALL f::nat => real. cauchy f = convergent f"
+lemma SEQ_CAUCHY: "cauchy f = seq.convergent f"
by (import seq SEQ_CAUCHY)
-lemma SEQ_LE: "ALL (f::nat => real) (g::nat => real) (l::real) m::real.
- hol4--> f l &
- hol4--> g m & (EX x::nat. ALL xa::nat. x <= xa --> f xa <= g xa) -->
- l <= m"
+lemma SEQ_LE: "hol4--> f l & hol4--> g m & (EX x. ALL xa>=x. f xa <= g xa) ==> l <= m"
by (import seq SEQ_LE)
-lemma SEQ_SUC: "ALL (f::nat => real) l::real. hol4--> f l = hol4--> (%n::nat. f (Suc n)) l"
+lemma SEQ_SUC: "hol4--> f l = hol4--> (%n. f (Suc n)) l"
by (import seq SEQ_SUC)
-lemma SEQ_ABS: "ALL f::nat => real. hol4--> (%n::nat. abs (f n)) 0 = hol4--> f 0"
+lemma SEQ_ABS: "hol4--> (%n. abs (f n)) 0 = hol4--> f 0"
by (import seq SEQ_ABS)
-lemma SEQ_ABS_IMP: "ALL (f::nat => real) l::real.
- hol4--> f l --> hol4--> (%n::nat. abs (f n)) (abs l)"
+lemma SEQ_ABS_IMP: "hol4--> f l ==> hol4--> (%n. abs (f n)) (abs l)"
by (import seq SEQ_ABS_IMP)
-lemma SEQ_INV0: "ALL f::nat => real.
- (ALL y::real. EX N::nat. ALL n::nat. N <= n --> y < f n) -->
- hol4--> (%n::nat. inverse (f n)) 0"
+lemma SEQ_INV0: "(!!y. EX N. ALL n>=N. y < f n) ==> hol4--> (%n. inverse (f n)) 0"
by (import seq SEQ_INV0)
-lemma SEQ_POWER_ABS: "ALL c::real. abs c < 1 --> hol4--> (op ^ (abs c)) 0"
+lemma SEQ_POWER_ABS: "abs c < 1 ==> hol4--> (op ^ (abs c)) 0"
by (import seq SEQ_POWER_ABS)
-lemma SEQ_POWER: "ALL c::real. abs c < 1 --> hol4--> (op ^ c) 0"
+lemma SEQ_POWER: "abs c < 1 ==> hol4--> (op ^ c) 0"
by (import seq SEQ_POWER)
-lemma NEST_LEMMA: "ALL (f::nat => real) g::nat => real.
- (ALL n::nat. f n <= f (Suc n)) &
- (ALL n::nat. g (Suc n) <= g n) & (ALL n::nat. f n <= g n) -->
- (EX (l::real) m::real.
+lemma NEST_LEMMA: "(ALL n. f n <= f (Suc n)) & (ALL n. g (Suc n) <= g n) & (ALL n. f n <= g n)
+==> EX l m.
l <= m &
- ((ALL n::nat. f n <= l) & hol4--> f l) &
- (ALL n::nat. m <= g n) & hol4--> g m)"
+ ((ALL n. f n <= l) & hol4--> f l) & (ALL n. m <= g n) & hol4--> g m"
by (import seq NEST_LEMMA)
-lemma NEST_LEMMA_UNIQ: "ALL (f::nat => real) g::nat => real.
- (ALL n::nat. f n <= f (Suc n)) &
- (ALL n::nat. g (Suc n) <= g n) &
- (ALL n::nat. f n <= g n) & hol4--> (%n::nat. f n - g n) 0 -->
- (EX x::real.
- ((ALL n::nat. f n <= x) & hol4--> f x) &
- (ALL n::nat. x <= g n) & hol4--> g x)"
+lemma NEST_LEMMA_UNIQ: "(ALL n. f n <= f (Suc n)) &
+(ALL n. g (Suc n) <= g n) & (ALL n. f n <= g n) & hol4--> (%n. f n - g n) 0
+==> EX x. ((ALL n. f n <= x) & hol4--> f x) &
+ (ALL n. x <= g n) & hol4--> g x"
by (import seq NEST_LEMMA_UNIQ)
-lemma BOLZANO_LEMMA: "ALL P::real * real => bool.
- (ALL (a::real) (b::real) c::real.
- a <= b & b <= c & P (a, b) & P (b, c) --> P (a, c)) &
- (ALL x::real.
- EX d>0.
- ALL (a::real) b::real.
- a <= x & x <= b & b - a < d --> P (a, b)) -->
- (ALL (a::real) b::real. a <= b --> P (a, b))"
- by (import seq BOLZANO_LEMMA)
-
-definition sums :: "(nat => real) => real => bool" where
- "sums == %f::nat => real. hol4--> (%n::nat. real.sum (0, n) f)"
-
-lemma sums: "ALL (f::nat => real) s::real.
- sums f s = hol4--> (%n::nat. real.sum (0, n) f) s"
+consts
+ sums :: "(nat => real) => real => bool"
+
+defs
+ sums_def: "seq.sums == %f. hol4--> (%n. real.sum (0, n) f)"
+
+lemma sums: "seq.sums f s = hol4--> (%n. real.sum (0, n) f) s"
by (import seq sums)
-definition summable :: "(nat => real) => bool" where
- "summable == %f::nat => real. Ex (sums f)"
-
-lemma summable: "ALL f::nat => real. summable f = Ex (sums f)"
+consts
+ summable :: "(nat => real) => bool"
+
+defs
+ summable_def: "seq.summable == %f. Ex (seq.sums f)"
+
+lemma summable: "seq.summable f = Ex (seq.sums f)"
by (import seq summable)
-definition suminf :: "(nat => real) => real" where
- "suminf == %f::nat => real. Eps (sums f)"
-
-lemma suminf: "ALL f::nat => real. suminf f = Eps (sums f)"
+consts
+ suminf :: "(nat => real) => real"
+
+defs
+ suminf_def: "seq.suminf == %f. Eps (seq.sums f)"
+
+lemma suminf: "seq.suminf f = Eps (seq.sums f)"
by (import seq suminf)
-lemma SUM_SUMMABLE: "ALL (f::nat => real) l::real. sums f l --> summable f"
+lemma SUM_SUMMABLE: "seq.sums f l ==> seq.summable f"
by (import seq SUM_SUMMABLE)
-lemma SUMMABLE_SUM: "ALL f::nat => real. summable f --> sums f (suminf f)"
+lemma SUMMABLE_SUM: "seq.summable f ==> seq.sums f (seq.suminf f)"
by (import seq SUMMABLE_SUM)
-lemma SUM_UNIQ: "ALL (f::nat => real) x::real. sums f x --> x = suminf f"
+lemma SUM_UNIQ: "seq.sums f x ==> x = seq.suminf f"
by (import seq SUM_UNIQ)
-lemma SER_0: "ALL (f::nat => real) n::nat.
- (ALL m::nat. n <= m --> f m = 0) --> sums f (real.sum (0, n) f)"
+lemma SER_0: "(!!m. n <= m ==> f m = 0) ==> seq.sums f (real.sum (0, n) f)"
by (import seq SER_0)
-lemma SER_POS_LE: "ALL (f::nat => real) n::nat.
- summable f & (ALL m::nat. n <= m --> 0 <= f m) -->
- real.sum (0, n) f <= suminf f"
+lemma SER_POS_LE: "seq.summable f & (ALL m>=n. 0 <= f m) ==> real.sum (0, n) f <= seq.suminf f"
by (import seq SER_POS_LE)
-lemma SER_POS_LT: "ALL (f::nat => real) n::nat.
- summable f & (ALL m::nat. n <= m --> 0 < f m) -->
- real.sum (0, n) f < suminf f"
+lemma SER_POS_LT: "seq.summable f & (ALL m>=n. 0 < f m) ==> real.sum (0, n) f < seq.suminf f"
by (import seq SER_POS_LT)
-lemma SER_GROUP: "ALL (f::nat => real) k::nat.
- summable f & 0 < k --> sums (%n::nat. real.sum (n * k, k) f) (suminf f)"
+lemma SER_GROUP: "seq.summable f & 0 < k
+==> seq.sums (%n. real.sum (n * k, k) f) (seq.suminf f)"
by (import seq SER_GROUP)
-lemma SER_PAIR: "ALL f::nat => real.
- summable f --> sums (%n::nat. real.sum (2 * n, 2) f) (suminf f)"
+lemma SER_PAIR: "seq.summable f ==> seq.sums (%n. real.sum (2 * n, 2) f) (seq.suminf f)"
by (import seq SER_PAIR)
-lemma SER_OFFSET: "ALL f::nat => real.
- summable f -->
- (ALL k::nat. sums (%n::nat. f (n + k)) (suminf f - real.sum (0, k) f))"
+lemma SER_OFFSET: "seq.summable f
+==> seq.sums (%n. f (n + k)) (seq.suminf f - real.sum (0, k) f)"
by (import seq SER_OFFSET)
-lemma SER_POS_LT_PAIR: "ALL (f::nat => real) n::nat.
- summable f & (ALL d::nat. 0 < f (n + 2 * d) + f (n + (2 * d + 1))) -->
- real.sum (0, n) f < suminf f"
+lemma SER_POS_LT_PAIR: "seq.summable f & (ALL d. 0 < f (n + 2 * d) + f (n + (2 * d + 1)))
+==> real.sum (0, n) f < seq.suminf f"
by (import seq SER_POS_LT_PAIR)
-lemma SER_ADD: "ALL (x::nat => real) (x0::real) (y::nat => real) y0::real.
- sums x x0 & sums y y0 --> sums (%n::nat. x n + y n) (x0 + y0)"
+lemma SER_ADD: "seq.sums x x0 & seq.sums y y0 ==> seq.sums (%n. x n + y n) (x0 + y0)"
by (import seq SER_ADD)
-lemma SER_CMUL: "ALL (x::nat => real) (x0::real) c::real.
- sums x x0 --> sums (%n::nat. c * x n) (c * x0)"
+lemma SER_CMUL: "seq.sums x x0 ==> seq.sums (%n. c * x n) (c * x0)"
by (import seq SER_CMUL)
-lemma SER_NEG: "ALL (x::nat => real) x0::real. sums x x0 --> sums (%xa::nat. - x xa) (- x0)"
+lemma SER_NEG: "seq.sums x x0 ==> seq.sums (%xa. - x xa) (- x0)"
by (import seq SER_NEG)
-lemma SER_SUB: "ALL (x::nat => real) (x0::real) (y::nat => real) y0::real.
- sums x x0 & sums y y0 --> sums (%xa::nat. x xa - y xa) (x0 - y0)"
+lemma SER_SUB: "seq.sums x x0 & seq.sums y y0 ==> seq.sums (%xa. x xa - y xa) (x0 - y0)"
by (import seq SER_SUB)
-lemma SER_CDIV: "ALL (x::nat => real) (x0::real) c::real.
- sums x x0 --> sums (%xa::nat. x xa / c) (x0 / c)"
+lemma SER_CDIV: "seq.sums x x0 ==> seq.sums (%xa. x xa / c) (x0 / c)"
by (import seq SER_CDIV)
-lemma SER_CAUCHY: "ALL f::nat => real.
- summable f =
- (ALL e>0.
- EX N::nat.
- ALL (m::nat) n::nat. N <= m --> abs (real.sum (m, n) f) < e)"
+lemma SER_CAUCHY: "seq.summable f =
+(ALL e>0. EX N. ALL m n. N <= m --> abs (real.sum (m, n) f) < e)"
by (import seq SER_CAUCHY)
-lemma SER_ZERO: "ALL f::nat => real. summable f --> hol4--> f 0"
+lemma SER_ZERO: "seq.summable f ==> hol4--> f 0"
by (import seq SER_ZERO)
-lemma SER_COMPAR: "ALL (f::nat => real) g::nat => real.
- (EX x::nat. ALL xa::nat. x <= xa --> abs (f xa) <= g xa) & summable g -->
- summable f"
+lemma SER_COMPAR: "(EX x. ALL xa>=x. abs (f xa) <= g xa) & seq.summable g ==> seq.summable f"
by (import seq SER_COMPAR)
-lemma SER_COMPARA: "ALL (f::nat => real) g::nat => real.
- (EX x::nat. ALL xa::nat. x <= xa --> abs (f xa) <= g xa) & summable g -->
- summable (%k::nat. abs (f k))"
+lemma SER_COMPARA: "(EX x. ALL xa>=x. abs (f xa) <= g xa) & seq.summable g
+==> seq.summable (%k. abs (f k))"
by (import seq SER_COMPARA)
-lemma SER_LE: "ALL (f::nat => real) g::nat => real.
- (ALL n::nat. f n <= g n) & summable f & summable g -->
- suminf f <= suminf g"
+lemma SER_LE: "(ALL n. f n <= g n) & seq.summable f & seq.summable g
+==> seq.suminf f <= seq.suminf g"
by (import seq SER_LE)
-lemma SER_LE2: "ALL (f::nat => real) g::nat => real.
- (ALL n::nat. abs (f n) <= g n) & summable g -->
- summable f & suminf f <= suminf g"
+lemma SER_LE2: "(ALL n. abs (f n) <= g n) & seq.summable g
+==> seq.summable f & seq.suminf f <= seq.suminf g"
by (import seq SER_LE2)
-lemma SER_ACONV: "ALL f::nat => real. summable (%n::nat. abs (f n)) --> summable f"
+lemma SER_ACONV: "seq.summable (%n. abs (f n)) ==> seq.summable f"
by (import seq SER_ACONV)
-lemma SER_ABS: "ALL f::nat => real.
- summable (%n::nat. abs (f n)) -->
- abs (suminf f) <= suminf (%n::nat. abs (f n))"
+lemma SER_ABS: "seq.summable (%n. abs (f n))
+==> abs (seq.suminf f) <= seq.suminf (%n. abs (f n))"
by (import seq SER_ABS)
-lemma GP_FINITE: "ALL x::real.
- x ~= 1 --> (ALL n::nat. real.sum (0, n) (op ^ x) = (x ^ n - 1) / (x - 1))"
+lemma GP_FINITE: "x ~= 1 ==> real.sum (0, n) (op ^ x) = (x ^ n - 1) / (x - 1)"
by (import seq GP_FINITE)
-lemma GP: "ALL x::real. abs x < 1 --> sums (op ^ x) (inverse (1 - x))"
+lemma GP: "abs x < 1 ==> seq.sums (op ^ x) (inverse (1 - x))"
by (import seq GP)
-lemma ABS_NEG_LEMMA: "(All::(real => bool) => bool)
- (%c::real.
- (op -->::bool => bool => bool)
- ((op <=::real => real => bool) c (0::real))
- ((All::(real => bool) => bool)
- (%x::real.
- (All::(real => bool) => bool)
- (%y::real.
- (op -->::bool => bool => bool)
- ((op <=::real => real => bool) ((abs::real => real) x)
- ((op *::real => real => real) c
- ((abs::real => real) y)))
- ((op =::real => real => bool) x (0::real))))))"
- by (import seq ABS_NEG_LEMMA)
-
-lemma SER_RATIO: "ALL (f::nat => real) (c::real) N::nat.
- c < 1 & (ALL n::nat. N <= n --> abs (f (Suc n)) <= c * abs (f n)) -->
- summable f"
+lemma SER_RATIO: "c < 1 & (ALL n>=N. abs (f (Suc n)) <= c * abs (f n)) ==> seq.summable f"
by (import seq SER_RATIO)
;end_setup
;setup_theory lim
-definition tends_real_real :: "(real => real) => real => real => bool" where
- "tends_real_real ==
-%(f::real => real) (l::real) x0::real.
- tends f l (mtop mr1, tendsto (mr1, x0))"
-
-lemma tends_real_real: "ALL (f::real => real) (l::real) x0::real.
- tends_real_real f l x0 = tends f l (mtop mr1, tendsto (mr1, x0))"
+definition
+ tends_real_real :: "(real => real) => real => real => bool" where
+ "tends_real_real == %f l x0. tends f l (mtop mr1, nets.tendsto (mr1, x0))"
+
+lemma tends_real_real: "tends_real_real f l x0 = tends f l (mtop mr1, nets.tendsto (mr1, x0))"
by (import lim tends_real_real)
-lemma LIM: "ALL (f::real => real) (y0::real) x0::real.
- tends_real_real f y0 x0 =
- (ALL e>0.
- EX d>0.
- ALL x::real.
- 0 < abs (x - x0) & abs (x - x0) < d --> abs (f x - y0) < e)"
+lemma LIM: "tends_real_real f y0 x0 =
+(ALL e>0.
+ EX d>0.
+ ALL x. 0 < abs (x - x0) & abs (x - x0) < d --> abs (f x - y0) < e)"
by (import lim LIM)
-lemma LIM_CONST: "ALL k::real. All (tends_real_real (%x::real. k) k)"
+lemma LIM_CONST: "tends_real_real (%x. k) k x"
by (import lim LIM_CONST)
-lemma LIM_ADD: "ALL (f::real => real) (g::real => real) (l::real) (m::real) x::real.
- tends_real_real f l x & tends_real_real g m x -->
- tends_real_real (%x::real. f x + g x) (l + m) x"
+lemma LIM_ADD: "tends_real_real f l x & tends_real_real g m x
+==> tends_real_real (%x. f x + g x) (l + m) x"
by (import lim LIM_ADD)
-lemma LIM_MUL: "ALL (f::real => real) (g::real => real) (l::real) (m::real) x::real.
- tends_real_real f l x & tends_real_real g m x -->
- tends_real_real (%x::real. f x * g x) (l * m) x"
+lemma LIM_MUL: "tends_real_real f l x & tends_real_real g m x
+==> tends_real_real (%x. f x * g x) (l * m) x"
by (import lim LIM_MUL)
-lemma LIM_NEG: "ALL (f::real => real) (l::real) x::real.
- tends_real_real f l x = tends_real_real (%x::real. - f x) (- l) x"
+lemma LIM_NEG: "tends_real_real f l x = tends_real_real (%x. - f x) (- l) x"
by (import lim LIM_NEG)
-lemma LIM_INV: "ALL (f::real => real) (l::real) x::real.
- tends_real_real f l x & l ~= 0 -->
- tends_real_real (%x::real. inverse (f x)) (inverse l) x"
+lemma LIM_INV: "tends_real_real f l x & l ~= 0
+==> tends_real_real (%x. inverse (f x)) (inverse l) x"
by (import lim LIM_INV)
-lemma LIM_SUB: "ALL (f::real => real) (g::real => real) (l::real) (m::real) x::real.
- tends_real_real f l x & tends_real_real g m x -->
- tends_real_real (%x::real. f x - g x) (l - m) x"
+lemma LIM_SUB: "tends_real_real f l x & tends_real_real g m x
+==> tends_real_real (%x. f x - g x) (l - m) x"
by (import lim LIM_SUB)
-lemma LIM_DIV: "ALL (f::real => real) (g::real => real) (l::real) (m::real) x::real.
- tends_real_real f l x & tends_real_real g m x & m ~= 0 -->
- tends_real_real (%x::real. f x / g x) (l / m) x"
+lemma LIM_DIV: "tends_real_real f l x & tends_real_real g m x & m ~= 0
+==> tends_real_real (%x. f x / g x) (l / m) x"
by (import lim LIM_DIV)
-lemma LIM_NULL: "ALL (f::real => real) (l::real) x::real.
- tends_real_real f l x = tends_real_real (%x::real. f x - l) 0 x"
+lemma LIM_NULL: "tends_real_real f l x = tends_real_real (%x. f x - l) 0 x"
by (import lim LIM_NULL)
-lemma LIM_X: "ALL x0::real. tends_real_real (%x::real. x) x0 x0"
+lemma LIM_X: "tends_real_real (%x. x) x0 x0"
by (import lim LIM_X)
-lemma LIM_UNIQ: "ALL (f::real => real) (l::real) (m::real) x::real.
- tends_real_real f l x & tends_real_real f m x --> l = m"
+lemma LIM_UNIQ: "tends_real_real f l x & tends_real_real f m x ==> l = m"
by (import lim LIM_UNIQ)
-lemma LIM_EQUAL: "ALL (f::real => real) (g::real => real) (l::real) x0::real.
- (ALL x::real. x ~= x0 --> f x = g x) -->
- tends_real_real f l x0 = tends_real_real g l x0"
+lemma LIM_EQUAL: "(!!x. x ~= x0 ==> f x = g x)
+==> tends_real_real f l x0 = tends_real_real g l x0"
by (import lim LIM_EQUAL)
-lemma LIM_TRANSFORM: "ALL (f::real => real) (g::real => real) (x0::real) l::real.
- tends_real_real (%x::real. f x - g x) 0 x0 & tends_real_real g l x0 -->
- tends_real_real f l x0"
+lemma LIM_TRANSFORM: "tends_real_real (%x. f x - g x) 0 x0 & tends_real_real g l x0
+==> tends_real_real f l x0"
by (import lim LIM_TRANSFORM)
-definition diffl :: "(real => real) => real => real => bool" where
- "diffl ==
-%(f::real => real) (l::real) x::real.
- tends_real_real (%h::real. (f (x + h) - f x) / h) l 0"
-
-lemma diffl: "ALL (f::real => real) (l::real) x::real.
- diffl f l x = tends_real_real (%h::real. (f (x + h) - f x) / h) l 0"
+definition
+ diffl :: "(real => real) => real => real => bool" where
+ "diffl == %f l x. tends_real_real (%h. (f (x + h) - f x) / h) l 0"
+
+lemma diffl: "diffl f l x = tends_real_real (%h. (f (x + h) - f x) / h) l 0"
by (import lim diffl)
-definition contl :: "(real => real) => real => bool" where
- "contl ==
-%(f::real => real) x::real. tends_real_real (%h::real. f (x + h)) (f x) 0"
-
-lemma contl: "ALL (f::real => real) x::real.
- contl f x = tends_real_real (%h::real. f (x + h)) (f x) 0"
+definition
+ contl :: "(real => real) => real => bool" where
+ "contl == %f x. tends_real_real (%h. f (x + h)) (f x) 0"
+
+lemma contl: "contl f x = tends_real_real (%h. f (x + h)) (f x) 0"
by (import lim contl)
-definition differentiable :: "(real => real) => real => bool" where
- "differentiable == %(f::real => real) x::real. EX l::real. diffl f l x"
-
-lemma differentiable: "ALL (f::real => real) x::real.
- differentiable f x = (EX l::real. diffl f l x)"
+consts
+ differentiable :: "(real => real) => real => bool"
+
+defs
+ differentiable_def: "lim.differentiable == %f x. EX l. diffl f l x"
+
+lemma differentiable: "lim.differentiable f x = (EX l. diffl f l x)"
by (import lim differentiable)
-lemma DIFF_UNIQ: "ALL (f::real => real) (l::real) (m::real) x::real.
- diffl f l x & diffl f m x --> l = m"
+lemma DIFF_UNIQ: "diffl f l x & diffl f m x ==> l = m"
by (import lim DIFF_UNIQ)
-lemma DIFF_CONT: "ALL (f::real => real) (l::real) x::real. diffl f l x --> contl f x"
+lemma DIFF_CONT: "diffl f l x ==> contl f x"
by (import lim DIFF_CONT)
-lemma CONTL_LIM: "ALL (f::real => real) x::real. contl f x = tends_real_real f (f x) x"
+lemma CONTL_LIM: "contl f x = tends_real_real f (f x) x"
by (import lim CONTL_LIM)
-lemma DIFF_CARAT: "ALL (f::real => real) (l::real) x::real.
- diffl f l x =
- (EX g::real => real.
- (ALL z::real. f z - f x = g z * (z - x)) & contl g x & g x = l)"
+lemma DIFF_CARAT: "diffl f l x =
+(EX g. (ALL z. f z - f x = g z * (z - x)) & contl g x & g x = l)"
by (import lim DIFF_CARAT)
-lemma CONT_CONST: "ALL k::real. All (contl (%x::real. k))"
+lemma CONT_CONST: "contl (%x. k) x"
by (import lim CONT_CONST)
-lemma CONT_ADD: "ALL (f::real => real) (g::real => real) x::real.
- contl f x & contl g x --> contl (%x::real. f x + g x) x"
+lemma CONT_ADD: "contl f x & contl g x ==> contl (%x. f x + g x) x"
by (import lim CONT_ADD)
-lemma CONT_MUL: "ALL (f::real => real) (g::real => real) x::real.
- contl f x & contl g x --> contl (%x::real. f x * g x) x"
+lemma CONT_MUL: "contl f x & contl g x ==> contl (%x. f x * g x) x"
by (import lim CONT_MUL)
-lemma CONT_NEG: "ALL (f::real => real) x::real. contl f x --> contl (%x::real. - f x) x"
+lemma CONT_NEG: "contl f x ==> contl (%x. - f x) x"
by (import lim CONT_NEG)
-lemma CONT_INV: "ALL (f::real => real) x::real.
- contl f x & f x ~= 0 --> contl (%x::real. inverse (f x)) x"
+lemma CONT_INV: "contl f x & f x ~= 0 ==> contl (%x. inverse (f x)) x"
by (import lim CONT_INV)
-lemma CONT_SUB: "ALL (f::real => real) (g::real => real) x::real.
- contl f x & contl g x --> contl (%x::real. f x - g x) x"
+lemma CONT_SUB: "contl f x & contl g x ==> contl (%x. f x - g x) x"
by (import lim CONT_SUB)
-lemma CONT_DIV: "ALL (f::real => real) (g::real => real) x::real.
- contl f x & contl g x & g x ~= 0 --> contl (%x::real. f x / g x) x"
+lemma CONT_DIV: "contl f x & contl g x & g x ~= 0 ==> contl (%x. f x / g x) x"
by (import lim CONT_DIV)
-lemma CONT_COMPOSE: "ALL (f::real => real) (g::real => real) x::real.
- contl f x & contl g (f x) --> contl (%x::real. g (f x)) x"
+lemma CONT_COMPOSE: "contl f x & contl g (f x) ==> contl (%x. g (f x)) x"
by (import lim CONT_COMPOSE)
-lemma IVT: "ALL (f::real => real) (a::real) (b::real) y::real.
- a <= b &
- (f a <= y & y <= f b) & (ALL x::real. a <= x & x <= b --> contl f x) -->
- (EX x::real. a <= x & x <= b & f x = y)"
+lemma IVT: "a <= b & (f a <= y & y <= f b) & (ALL x. a <= x & x <= b --> contl f x)
+==> EX x>=a. x <= b & f x = y"
by (import lim IVT)
-lemma IVT2: "ALL (f::real => real) (a::real) (b::real) y::real.
- a <= b &
- (f b <= y & y <= f a) & (ALL x::real. a <= x & x <= b --> contl f x) -->
- (EX x::real. a <= x & x <= b & f x = y)"
+lemma IVT2: "a <= b & (f b <= y & y <= f a) & (ALL x. a <= x & x <= b --> contl f x)
+==> EX x>=a. x <= b & f x = y"
by (import lim IVT2)
-lemma DIFF_CONST: "ALL k::real. All (diffl (%x::real. k) 0)"
+lemma DIFF_CONST: "diffl (%x. k) 0 x"
by (import lim DIFF_CONST)
-lemma DIFF_ADD: "ALL (f::real => real) (g::real => real) (l::real) (m::real) x::real.
- diffl f l x & diffl g m x --> diffl (%x::real. f x + g x) (l + m) x"
+lemma DIFF_ADD: "diffl f l x & diffl g m x ==> diffl (%x. f x + g x) (l + m) x"
by (import lim DIFF_ADD)
-lemma DIFF_MUL: "ALL (f::real => real) (g::real => real) (l::real) (m::real) x::real.
- diffl f l x & diffl g m x -->
- diffl (%x::real. f x * g x) (l * g x + m * f x) x"
+lemma DIFF_MUL: "diffl f l x & diffl g m x ==> diffl (%x. f x * g x) (l * g x + m * f x) x"
by (import lim DIFF_MUL)
-lemma DIFF_CMUL: "ALL (f::real => real) (c::real) (l::real) x::real.
- diffl f l x --> diffl (%x::real. c * f x) (c * l) x"
+lemma DIFF_CMUL: "diffl f l x ==> diffl (%x. c * f x) (c * l) x"
by (import lim DIFF_CMUL)
-lemma DIFF_NEG: "ALL (f::real => real) (l::real) x::real.
- diffl f l x --> diffl (%x::real. - f x) (- l) x"
+lemma DIFF_NEG: "diffl f l x ==> diffl (%x. - f x) (- l) x"
by (import lim DIFF_NEG)
-lemma DIFF_SUB: "ALL (f::real => real) (g::real => real) (l::real) (m::real) x::real.
- diffl f l x & diffl g m x --> diffl (%x::real. f x - g x) (l - m) x"
+lemma DIFF_SUB: "diffl f l x & diffl g m x ==> diffl (%x. f x - g x) (l - m) x"
by (import lim DIFF_SUB)
-lemma DIFF_CHAIN: "ALL (f::real => real) (g::real => real) (l::real) (m::real) x::real.
- diffl f l (g x) & diffl g m x --> diffl (%x::real. f (g x)) (l * m) x"
+lemma DIFF_CHAIN: "diffl f l (g x) & diffl g m x ==> diffl (%x. f (g x)) (l * m) x"
by (import lim DIFF_CHAIN)
-lemma DIFF_X: "All (diffl (%x::real. x) 1)"
+lemma DIFF_X: "diffl (%x. x) 1 x"
by (import lim DIFF_X)
-lemma DIFF_POW: "ALL (n::nat) x::real. diffl (%x::real. x ^ n) (real n * x ^ (n - 1)) x"
+lemma DIFF_POW: "diffl (%x. x ^ n) (real n * x ^ (n - 1)) x"
by (import lim DIFF_POW)
-lemma DIFF_XM1: "ALL x::real. x ~= 0 --> diffl inverse (- (inverse x ^ 2)) x"
+lemma DIFF_XM1: "x ~= 0 ==> diffl inverse (- (inverse x ^ 2)) x"
by (import lim DIFF_XM1)
-lemma DIFF_INV: "ALL (f::real => real) (l::real) x::real.
- diffl f l x & f x ~= 0 -->
- diffl (%x::real. inverse (f x)) (- (l / f x ^ 2)) x"
+lemma DIFF_INV: "diffl f l x & f x ~= 0 ==> diffl (%x. inverse (f x)) (- (l / f x ^ 2)) x"
by (import lim DIFF_INV)
-lemma DIFF_DIV: "ALL (f::real => real) (g::real => real) (l::real) (m::real) x::real.
- diffl f l x & diffl g m x & g x ~= 0 -->
- diffl (%x::real. f x / g x) ((l * g x - m * f x) / g x ^ 2) x"
+lemma DIFF_DIV: "diffl f l x & diffl g m x & g x ~= 0
+==> diffl (%x. f x / g x) ((l * g x - m * f x) / g x ^ 2) x"
by (import lim DIFF_DIV)
-lemma DIFF_SUM: "ALL (f::nat => real => real) (f'::nat => real => real) (m::nat) (n::nat)
- x::real.
- (ALL r::nat. m <= r & r < m + n --> diffl (f r) (f' r x) x) -->
- diffl (%x::real. real.sum (m, n) (%n::nat. f n x))
- (real.sum (m, n) (%r::nat. f' r x)) x"
+lemma DIFF_SUM: "(!!r. m <= r & r < m + n ==> diffl (f r) (f' r x) x)
+==> diffl (%x. real.sum (m, n) (%n. f n x)) (real.sum (m, n) (%r. f' r x)) x"
by (import lim DIFF_SUM)
-lemma CONT_BOUNDED: "ALL (f::real => real) (a::real) b::real.
- a <= b & (ALL x::real. a <= x & x <= b --> contl f x) -->
- (EX M::real. ALL x::real. a <= x & x <= b --> f x <= M)"
+lemma CONT_BOUNDED: "a <= b & (ALL x. a <= x & x <= b --> contl f x)
+==> EX M. ALL x. a <= x & x <= b --> f x <= M"
by (import lim CONT_BOUNDED)
-lemma CONT_HASSUP: "(All::((real => real) => bool) => bool)
- (%f::real => real.
- (All::(real => bool) => bool)
- (%a::real.
- (All::(real => bool) => bool)
- (%b::real.
- (op -->::bool => bool => bool)
- ((op &::bool => bool => bool)
- ((op <=::real => real => bool) a b)
- ((All::(real => bool) => bool)
- (%x::real.
- (op -->::bool => bool => bool)
- ((op &::bool => bool => bool)
- ((op <=::real => real => bool) a x)
- ((op <=::real => real => bool) x b))
- ((contl::(real => real) => real => bool) f x))))
- ((Ex::(real => bool) => bool)
- (%M::real.
- (op &::bool => bool => bool)
- ((All::(real => bool) => bool)
- (%x::real.
- (op -->::bool => bool => bool)
- ((op &::bool => bool => bool)
- ((op <=::real => real => bool) a x)
- ((op <=::real => real => bool) x b))
- ((op <=::real => real => bool) (f x) M)))
- ((All::(real => bool) => bool)
- (%N::real.
- (op -->::bool => bool => bool)
- ((op <::real => real => bool) N M)
- ((Ex::(real => bool) => bool)
- (%x::real.
- (op &::bool => bool => bool)
- ((op <=::real => real => bool) a x)
- ((op &::bool => bool => bool)
- ((op <=::real => real => bool) x b)
- ((op <::real => real => bool) N (f x))))))))))))"
+lemma CONT_HASSUP: "a <= b & (ALL x. a <= x & x <= b --> contl f x)
+==> EX M. (ALL x. a <= x & x <= b --> f x <= M) &
+ (ALL N<M. EX x>=a. x <= b & N < f x)"
by (import lim CONT_HASSUP)
-lemma CONT_ATTAINS: "ALL (f::real => real) (a::real) b::real.
- a <= b & (ALL x::real. a <= x & x <= b --> contl f x) -->
- (EX x::real.
- (ALL xa::real. a <= xa & xa <= b --> f xa <= x) &
- (EX xa::real. a <= xa & xa <= b & f xa = x))"
+lemma CONT_ATTAINS: "a <= b & (ALL x. a <= x & x <= b --> contl f x)
+==> EX x. (ALL xa. a <= xa & xa <= b --> f xa <= x) &
+ (EX xa>=a. xa <= b & f xa = x)"
by (import lim CONT_ATTAINS)
-lemma CONT_ATTAINS2: "ALL (f::real => real) (a::real) b::real.
- a <= b & (ALL x::real. a <= x & x <= b --> contl f x) -->
- (EX x::real.
- (ALL xa::real. a <= xa & xa <= b --> x <= f xa) &
- (EX xa::real. a <= xa & xa <= b & f xa = x))"
+lemma CONT_ATTAINS2: "a <= b & (ALL x. a <= x & x <= b --> contl f x)
+==> EX x. (ALL xa. a <= xa & xa <= b --> x <= f xa) &
+ (EX xa>=a. xa <= b & f xa = x)"
by (import lim CONT_ATTAINS2)
-lemma CONT_ATTAINS_ALL: "ALL (f::real => real) (a::real) b::real.
- a <= b & (ALL x::real. a <= x & x <= b --> contl f x) -->
- (EX (x::real) M::real.
+lemma CONT_ATTAINS_ALL: "a <= b & (ALL x. a <= x & x <= b --> contl f x)
+==> EX x M.
x <= M &
- (ALL y::real.
- x <= y & y <= M --> (EX x::real. a <= x & x <= b & f x = y)) &
- (ALL xa::real. a <= xa & xa <= b --> x <= f xa & f xa <= M))"
+ (ALL y. x <= y & y <= M --> (EX x>=a. x <= b & f x = y)) &
+ (ALL xa. a <= xa & xa <= b --> x <= f xa & f xa <= M)"
by (import lim CONT_ATTAINS_ALL)
-lemma DIFF_LINC: "ALL (f::real => real) (x::real) l::real.
- diffl f l x & 0 < l -->
- (EX d>0. ALL h::real. 0 < h & h < d --> f x < f (x + h))"
+lemma DIFF_LINC: "diffl f l x & 0 < l ==> EX d>0. ALL h. 0 < h & h < d --> f x < f (x + h)"
by (import lim DIFF_LINC)
-lemma DIFF_LDEC: "ALL (f::real => real) (x::real) l::real.
- diffl f l x & l < 0 -->
- (EX d>0. ALL h::real. 0 < h & h < d --> f x < f (x - h))"
+lemma DIFF_LDEC: "diffl f l x & l < 0 ==> EX d>0. ALL h. 0 < h & h < d --> f x < f (x - h)"
by (import lim DIFF_LDEC)
-lemma DIFF_LMAX: "ALL (f::real => real) (x::real) l::real.
- diffl f l x & (EX d>0. ALL y::real. abs (x - y) < d --> f y <= f x) -->
- l = 0"
+lemma DIFF_LMAX: "diffl f l x & (EX d>0. ALL y. abs (x - y) < d --> f y <= f x) ==> l = 0"
by (import lim DIFF_LMAX)
-lemma DIFF_LMIN: "ALL (f::real => real) (x::real) l::real.
- diffl f l x & (EX d>0. ALL y::real. abs (x - y) < d --> f x <= f y) -->
- l = 0"
+lemma DIFF_LMIN: "diffl f l x & (EX d>0. ALL y. abs (x - y) < d --> f x <= f y) ==> l = 0"
by (import lim DIFF_LMIN)
-lemma DIFF_LCONST: "ALL (f::real => real) (x::real) l::real.
- diffl f l x & (EX d>0. ALL y::real. abs (x - y) < d --> f y = f x) -->
- l = 0"
+lemma DIFF_LCONST: "diffl f l x & (EX d>0. ALL y. abs (x - y) < d --> f y = f x) ==> l = 0"
by (import lim DIFF_LCONST)
-lemma INTERVAL_LEMMA: "ALL (a::real) (b::real) x::real.
- a < x & x < b -->
- (EX d>0. ALL y::real. abs (x - y) < d --> a <= y & y <= b)"
- by (import lim INTERVAL_LEMMA)
-
-lemma ROLLE: "ALL (f::real => real) (a::real) b::real.
- a < b &
- f a = f b &
- (ALL x::real. a <= x & x <= b --> contl f x) &
- (ALL x::real. a < x & x < b --> differentiable f x) -->
- (EX z::real. a < z & z < b & diffl f 0 z)"
+lemma ROLLE: "a < b &
+f a = f b &
+(ALL x. a <= x & x <= b --> contl f x) &
+(ALL x. a < x & x < b --> lim.differentiable f x)
+==> EX z>a. z < b & diffl f 0 z"
by (import lim ROLLE)
-lemma MVT_LEMMA: "ALL (f::real => real) (a::real) b::real.
- f a - (f b - f a) / (b - a) * a = f b - (f b - f a) / (b - a) * b"
- by (import lim MVT_LEMMA)
-
-lemma MVT: "ALL (f::real => real) (a::real) b::real.
- a < b &
- (ALL x::real. a <= x & x <= b --> contl f x) &
- (ALL x::real. a < x & x < b --> differentiable f x) -->
- (EX (l::real) z::real.
- a < z & z < b & diffl f l z & f b - f a = (b - a) * l)"
+lemma MVT: "a < b &
+(ALL x. a <= x & x <= b --> contl f x) &
+(ALL x. a < x & x < b --> lim.differentiable f x)
+==> EX l z. a < z & z < b & diffl f l z & f b - f a = (b - a) * l"
by (import lim MVT)
-lemma DIFF_ISCONST_END: "ALL (f::real => real) (a::real) b::real.
- a < b &
- (ALL x::real. a <= x & x <= b --> contl f x) &
- (ALL x::real. a < x & x < b --> diffl f 0 x) -->
- f b = f a"
+lemma DIFF_ISCONST_END: "a < b &
+(ALL x. a <= x & x <= b --> contl f x) &
+(ALL x. a < x & x < b --> diffl f 0 x)
+==> f b = f a"
by (import lim DIFF_ISCONST_END)
-lemma DIFF_ISCONST: "ALL (f::real => real) (a::real) b::real.
- a < b &
- (ALL x::real. a <= x & x <= b --> contl f x) &
- (ALL x::real. a < x & x < b --> diffl f 0 x) -->
- (ALL x::real. a <= x & x <= b --> f x = f a)"
+lemma DIFF_ISCONST: "[| a < b &
+ (ALL x. a <= x & x <= b --> contl f x) &
+ (ALL x. a < x & x < b --> diffl f 0 x);
+ a <= x & x <= b |]
+==> f x = f a"
by (import lim DIFF_ISCONST)
-lemma DIFF_ISCONST_ALL: "ALL f::real => real. All (diffl f 0) --> (ALL (x::real) y::real. f x = f y)"
+lemma DIFF_ISCONST_ALL: "(!!x. diffl f 0 x) ==> f x = f y"
by (import lim DIFF_ISCONST_ALL)
-lemma INTERVAL_ABS: "ALL (x::real) (z::real) d::real.
- (x - d <= z & z <= x + d) = (abs (z - x) <= d)"
+lemma INTERVAL_ABS: "((x::real) - (d::real) <= (z::real) & z <= x + d) = (abs (z - x) <= d)"
by (import lim INTERVAL_ABS)
-lemma CONT_INJ_LEMMA: "ALL (f::real => real) (g::real => real) (x::real) d::real.
- 0 < d &
- (ALL z::real. abs (z - x) <= d --> g (f z) = z) &
- (ALL z::real. abs (z - x) <= d --> contl f z) -->
- ~ (ALL z::real. abs (z - x) <= d --> f z <= f x)"
+lemma CONT_INJ_LEMMA: "0 < d &
+(ALL z. abs (z - x) <= d --> g (f z) = z) &
+(ALL z. abs (z - x) <= d --> contl f z)
+==> ~ (ALL z. abs (z - x) <= d --> f z <= f x)"
by (import lim CONT_INJ_LEMMA)
-lemma CONT_INJ_LEMMA2: "ALL (f::real => real) (g::real => real) (x::real) d::real.
- 0 < d &
- (ALL z::real. abs (z - x) <= d --> g (f z) = z) &
- (ALL z::real. abs (z - x) <= d --> contl f z) -->
- ~ (ALL z::real. abs (z - x) <= d --> f x <= f z)"
+lemma CONT_INJ_LEMMA2: "0 < d &
+(ALL z. abs (z - x) <= d --> g (f z) = z) &
+(ALL z. abs (z - x) <= d --> contl f z)
+==> ~ (ALL z. abs (z - x) <= d --> f x <= f z)"
by (import lim CONT_INJ_LEMMA2)
-lemma CONT_INJ_RANGE: "ALL (f::real => real) (g::real => real) (x::real) d::real.
- 0 < d &
- (ALL z::real. abs (z - x) <= d --> g (f z) = z) &
- (ALL z::real. abs (z - x) <= d --> contl f z) -->
- (EX e>0.
- ALL y::real.
- abs (y - f x) <= e --> (EX z::real. abs (z - x) <= d & f z = y))"
+lemma CONT_INJ_RANGE: "0 < d &
+(ALL z. abs (z - x) <= d --> g (f z) = z) &
+(ALL z. abs (z - x) <= d --> contl f z)
+==> EX e>0. ALL y. abs (y - f x) <= e --> (EX z. abs (z - x) <= d & f z = y)"
by (import lim CONT_INJ_RANGE)
-lemma CONT_INVERSE: "ALL (f::real => real) (g::real => real) (x::real) d::real.
- 0 < d &
- (ALL z::real. abs (z - x) <= d --> g (f z) = z) &
- (ALL z::real. abs (z - x) <= d --> contl f z) -->
- contl g (f x)"
+lemma CONT_INVERSE: "0 < d &
+(ALL z. abs (z - x) <= d --> g (f z) = z) &
+(ALL z. abs (z - x) <= d --> contl f z)
+==> contl g (f x)"
by (import lim CONT_INVERSE)
-lemma DIFF_INVERSE: "ALL (f::real => real) (g::real => real) (l::real) (x::real) d::real.
- 0 < d &
- (ALL z::real. abs (z - x) <= d --> g (f z) = z) &
- (ALL z::real. abs (z - x) <= d --> contl f z) & diffl f l x & l ~= 0 -->
- diffl g (inverse l) (f x)"
+lemma DIFF_INVERSE: "0 < d &
+(ALL z. abs (z - x) <= d --> g (f z) = z) &
+(ALL z. abs (z - x) <= d --> contl f z) & diffl f l x & l ~= 0
+==> diffl g (inverse l) (f x)"
by (import lim DIFF_INVERSE)
-lemma DIFF_INVERSE_LT: "ALL (f::real => real) (g::real => real) (l::real) (x::real) d::real.
- 0 < d &
- (ALL z::real. abs (z - x) < d --> g (f z) = z) &
- (ALL z::real. abs (z - x) < d --> contl f z) & diffl f l x & l ~= 0 -->
- diffl g (inverse l) (f x)"
+lemma DIFF_INVERSE_LT: "0 < d &
+(ALL z. abs (z - x) < d --> g (f z) = z) &
+(ALL z. abs (z - x) < d --> contl f z) & diffl f l x & l ~= 0
+==> diffl g (inverse l) (f x)"
by (import lim DIFF_INVERSE_LT)
-lemma INTERVAL_CLEMMA: "ALL (a::real) (b::real) x::real.
- a < x & x < b -->
- (EX d>0. ALL y::real. abs (y - x) <= d --> a < y & y < b)"
+lemma INTERVAL_CLEMMA: "(a::real) < (x::real) & x < (b::real)
+==> EX d>0::real. ALL y::real. abs (y - x) <= d --> a < y & y < b"
by (import lim INTERVAL_CLEMMA)
-lemma DIFF_INVERSE_OPEN: "ALL (f::real => real) (g::real => real) (l::real) (a::real) (x::real)
- b::real.
- a < x &
- x < b &
- (ALL z::real. a < z & z < b --> g (f z) = z & contl f z) &
- diffl f l x & l ~= 0 -->
- diffl g (inverse l) (f x)"
+lemma DIFF_INVERSE_OPEN: "a < x &
+x < b &
+(ALL z. a < z & z < b --> g (f z) = z & contl f z) & diffl f l x & l ~= 0
+==> diffl g (inverse l) (f x)"
by (import lim DIFF_INVERSE_OPEN)
;end_setup
;setup_theory powser
-lemma POWDIFF_LEMMA: "ALL (n::nat) (x::real) y::real.
- real.sum (0, Suc n) (%p::nat. x ^ p * y ^ (Suc n - p)) =
- y * real.sum (0, Suc n) (%p::nat. x ^ p * y ^ (n - p))"
+lemma POWDIFF_LEMMA: "real.sum (0, Suc n) (%p. x ^ p * y ^ (Suc n - p)) =
+y * real.sum (0, Suc n) (%p. x ^ p * y ^ (n - p))"
by (import powser POWDIFF_LEMMA)
-lemma POWDIFF: "ALL (n::nat) (x::real) y::real.
- x ^ Suc n - y ^ Suc n =
- (x - y) * real.sum (0, Suc n) (%p::nat. x ^ p * y ^ (n - p))"
+lemma POWDIFF: "x ^ Suc n - y ^ Suc n =
+(x - y) * real.sum (0, Suc n) (%p. x ^ p * y ^ (n - p))"
by (import powser POWDIFF)
-lemma POWREV: "ALL (n::nat) (x::real) y::real.
- real.sum (0, Suc n) (%xa::nat. x ^ xa * y ^ (n - xa)) =
- real.sum (0, Suc n) (%xa::nat. x ^ (n - xa) * y ^ xa)"
+lemma POWREV: "real.sum (0, Suc n) (%xa. x ^ xa * y ^ (n - xa)) =
+real.sum (0, Suc n) (%xa. x ^ (n - xa) * y ^ xa)"
by (import powser POWREV)
-lemma POWSER_INSIDEA: "ALL (f::nat => real) (x::real) z::real.
- summable (%n::nat. f n * x ^ n) & abs z < abs x -->
- summable (%n::nat. abs (f n) * z ^ n)"
+lemma POWSER_INSIDEA: "seq.summable (%n. f n * x ^ n) & abs z < abs x
+==> seq.summable (%n. abs (f n) * z ^ n)"
by (import powser POWSER_INSIDEA)
-lemma POWSER_INSIDE: "ALL (f::nat => real) (x::real) z::real.
- summable (%n::nat. f n * x ^ n) & abs z < abs x -->
- summable (%n::nat. f n * z ^ n)"
+lemma POWSER_INSIDE: "seq.summable (%n. f n * x ^ n) & abs z < abs x
+==> seq.summable (%n. f n * z ^ n)"
by (import powser POWSER_INSIDE)
-definition diffs :: "(nat => real) => nat => real" where
- "diffs == %(c::nat => real) n::nat. real (Suc n) * c (Suc n)"
-
-lemma diffs: "ALL c::nat => real. diffs c = (%n::nat. real (Suc n) * c (Suc n))"
+consts
+ diffs :: "(nat => real) => nat => real"
+
+defs
+ diffs_def: "powser.diffs == %c n. real (Suc n) * c (Suc n)"
+
+lemma diffs: "powser.diffs c = (%n. real (Suc n) * c (Suc n))"
by (import powser diffs)
-lemma DIFFS_NEG: "ALL c::nat => real. diffs (%n::nat. - c n) = (%x::nat. - diffs c x)"
+lemma DIFFS_NEG: "powser.diffs (%n. - c n) = (%x. - powser.diffs c x)"
by (import powser DIFFS_NEG)
-lemma DIFFS_LEMMA: "ALL (n::nat) (c::nat => real) x::real.
- real.sum (0, n) (%n::nat. diffs c n * x ^ n) =
- real.sum (0, n) (%n::nat. real n * (c n * x ^ (n - 1))) +
- real n * (c n * x ^ (n - 1))"
+lemma DIFFS_LEMMA: "real.sum (0, n) (%n. powser.diffs c n * x ^ n) =
+real.sum (0, n) (%n. real n * (c n * x ^ (n - 1))) +
+real n * (c n * x ^ (n - 1))"
by (import powser DIFFS_LEMMA)
-lemma DIFFS_LEMMA2: "ALL (n::nat) (c::nat => real) x::real.
- real.sum (0, n) (%n::nat. real n * (c n * x ^ (n - 1))) =
- real.sum (0, n) (%n::nat. diffs c n * x ^ n) -
- real n * (c n * x ^ (n - 1))"
+lemma DIFFS_LEMMA2: "real.sum (0, n) (%n. real n * (c n * x ^ (n - 1))) =
+real.sum (0, n) (%n. powser.diffs c n * x ^ n) -
+real n * (c n * x ^ (n - 1))"
by (import powser DIFFS_LEMMA2)
-lemma DIFFS_EQUIV: "ALL (c::nat => real) x::real.
- summable (%n::nat. diffs c n * x ^ n) -->
- sums (%n::nat. real n * (c n * x ^ (n - 1)))
- (suminf (%n::nat. diffs c n * x ^ n))"
+lemma DIFFS_EQUIV: "seq.summable (%n. powser.diffs c n * x ^ n)
+==> seq.sums (%n. real n * (c n * x ^ (n - 1)))
+ (seq.suminf (%n. powser.diffs c n * x ^ n))"
by (import powser DIFFS_EQUIV)
-lemma TERMDIFF_LEMMA1: "ALL (m::nat) (z::real) h::real.
- real.sum (0, m) (%p::nat. (z + h) ^ (m - p) * z ^ p - z ^ m) =
- real.sum (0, m) (%p::nat. z ^ p * ((z + h) ^ (m - p) - z ^ (m - p)))"
+lemma TERMDIFF_LEMMA1: "real.sum (0, m) (%p. (z + h) ^ (m - p) * z ^ p - z ^ m) =
+real.sum (0, m) (%p. z ^ p * ((z + h) ^ (m - p) - z ^ (m - p)))"
by (import powser TERMDIFF_LEMMA1)
-lemma TERMDIFF_LEMMA2: "ALL (z::real) (h::real) n::nat.
- h ~= 0 -->
- ((z + h) ^ n - z ^ n) / h - real n * z ^ (n - 1) =
- h *
- real.sum (0, n - 1)
- (%p::nat.
- z ^ p *
- real.sum (0, n - 1 - p)
- (%q::nat. (z + h) ^ q * z ^ (n - 2 - p - q)))"
+lemma TERMDIFF_LEMMA2: "h ~= 0
+==> ((z + h) ^ n - z ^ n) / h - real n * z ^ (n - 1) =
+ h *
+ real.sum (0, n - 1)
+ (%p. z ^ p *
+ real.sum (0, n - 1 - p) (%q. (z + h) ^ q * z ^ (n - 2 - p - q)))"
by (import powser TERMDIFF_LEMMA2)
-lemma TERMDIFF_LEMMA3: "ALL (z::real) (h::real) (n::nat) k'::real.
- h ~= 0 & abs z <= k' & abs (z + h) <= k' -->
- abs (((z + h) ^ n - z ^ n) / h - real n * z ^ (n - 1))
- <= real n * (real (n - 1) * (k' ^ (n - 2) * abs h))"
+lemma TERMDIFF_LEMMA3: "h ~= 0 & abs z <= k' & abs (z + h) <= k'
+==> abs (((z + h) ^ n - z ^ n) / h - real n * z ^ (n - 1))
+ <= real n * (real (n - 1) * (k' ^ (n - 2) * abs h))"
by (import powser TERMDIFF_LEMMA3)
-lemma TERMDIFF_LEMMA4: "ALL (f::real => real) (k'::real) k::real.
- 0 < k &
- (ALL h::real. 0 < abs h & abs h < k --> abs (f h) <= k' * abs h) -->
- tends_real_real f 0 0"
+lemma TERMDIFF_LEMMA4: "0 < k & (ALL h. 0 < abs h & abs h < k --> abs (f h) <= k' * abs h)
+==> tends_real_real f 0 0"
by (import powser TERMDIFF_LEMMA4)
-lemma TERMDIFF_LEMMA5: "ALL (f::nat => real) (g::real => nat => real) k::real.
- 0 < k &
- summable f &
- (ALL h::real.
- 0 < abs h & abs h < k -->
- (ALL n::nat. abs (g h n) <= f n * abs h)) -->
- tends_real_real (%h::real. suminf (g h)) 0 0"
+lemma TERMDIFF_LEMMA5: "0 < k &
+seq.summable f &
+(ALL h. 0 < abs h & abs h < k --> (ALL n. abs (g h n) <= f n * abs h))
+==> tends_real_real (%h. seq.suminf (g h)) 0 0"
by (import powser TERMDIFF_LEMMA5)
-lemma TERMDIFF: "ALL (c::nat => real) (k'::real) x::real.
- summable (%n::nat. c n * k' ^ n) &
- summable (%n::nat. diffs c n * k' ^ n) &
- summable (%n::nat. diffs (diffs c) n * k' ^ n) & abs x < abs k' -->
- diffl (%x::real. suminf (%n::nat. c n * x ^ n))
- (suminf (%n::nat. diffs c n * x ^ n)) x"
+lemma TERMDIFF: "seq.summable (%n. c n * k' ^ n) &
+seq.summable (%n. powser.diffs c n * k' ^ n) &
+seq.summable (%n. powser.diffs (powser.diffs c) n * k' ^ n) & abs x < abs k'
+==> diffl (%x. seq.suminf (%n. c n * x ^ n))
+ (seq.suminf (%n. powser.diffs c n * x ^ n)) x"
by (import powser TERMDIFF)
;end_setup
;setup_theory transc
-definition exp :: "real => real" where
- "exp == %x::real. suminf (%n::nat. inverse (real (FACT n)) * x ^ n)"
-
-lemma exp: "ALL x::real. exp x = suminf (%n::nat. inverse (real (FACT n)) * x ^ n)"
+consts
+ exp :: "real => real"
+
+defs
+ exp_def: "transc.exp == %x. seq.suminf (%n. inverse (real (FACT n)) * x ^ n)"
+
+lemma exp: "transc.exp x = seq.suminf (%n. inverse (real (FACT n)) * x ^ n)"
by (import transc exp)
-definition cos :: "real => real" where
- "cos ==
-%x::real.
- suminf
- (%n::nat.
- (if EVEN n then (- 1) ^ (n div 2) / real (FACT n) else 0) * x ^ n)"
-
-lemma cos: "ALL x::real.
- cos x =
- suminf
- (%n::nat.
- (if EVEN n then (- 1) ^ (n div 2) / real (FACT n) else 0) * x ^ n)"
+consts
+ cos :: "real => real"
+
+defs
+ cos_def: "transc.cos ==
+%x. seq.suminf
+ (%n. (if EVEN n then (- 1) ^ (n div 2) / real (FACT n) else 0) * x ^ n)"
+
+lemma cos: "transc.cos x =
+seq.suminf
+ (%n. (if EVEN n then (- 1) ^ (n div 2) / real (FACT n) else 0) * x ^ n)"
by (import transc cos)
-definition sin :: "real => real" where
- "sin ==
-%x::real.
- suminf
- (%n::nat.
- (if EVEN n then 0 else (- 1) ^ ((n - 1) div 2) / real (FACT n)) *
- x ^ n)"
-
-lemma sin: "ALL x::real.
- sin x =
- suminf
- (%n::nat.
- (if EVEN n then 0 else (- 1) ^ ((n - 1) div 2) / real (FACT n)) *
- x ^ n)"
+consts
+ sin :: "real => real"
+
+defs
+ sin_def: "transc.sin ==
+%x. seq.suminf
+ (%n. (if EVEN n then 0 else (- 1) ^ ((n - 1) div 2) / real (FACT n)) *
+ x ^ n)"
+
+lemma sin: "transc.sin x =
+seq.suminf
+ (%n. (if EVEN n then 0 else (- 1) ^ ((n - 1) div 2) / real (FACT n)) *
+ x ^ n)"
by (import transc sin)
-lemma EXP_CONVERGES: "ALL x::real. sums (%n::nat. inverse (real (FACT n)) * x ^ n) (exp x)"
+lemma EXP_CONVERGES: "seq.sums (%n. inverse (real (FACT n)) * x ^ n) (transc.exp x)"
by (import transc EXP_CONVERGES)
-lemma SIN_CONVERGES: "ALL x::real.
- sums
- (%n::nat.
- (if EVEN n then 0 else (- 1) ^ ((n - 1) div 2) / real (FACT n)) *
- x ^ n)
- (sin x)"
+lemma SIN_CONVERGES: "seq.sums
+ (%n. (if EVEN n then 0 else (- 1) ^ ((n - 1) div 2) / real (FACT n)) *
+ x ^ n)
+ (transc.sin x)"
by (import transc SIN_CONVERGES)
-lemma COS_CONVERGES: "ALL x::real.
- sums
- (%n::nat.
- (if EVEN n then (- 1) ^ (n div 2) / real (FACT n) else 0) * x ^ n)
- (cos x)"
+lemma COS_CONVERGES: "seq.sums
+ (%n. (if EVEN n then (- 1) ^ (n div 2) / real (FACT n) else 0) * x ^ n)
+ (transc.cos x)"
by (import transc COS_CONVERGES)
-lemma EXP_FDIFF: "diffs (%n::nat. inverse (real (FACT n))) =
-(%n::nat. inverse (real (FACT n)))"
+lemma EXP_FDIFF: "powser.diffs (%n. inverse (real (FACT n))) = (%n. inverse (real (FACT n)))"
by (import transc EXP_FDIFF)
-lemma SIN_FDIFF: "diffs
- (%n::nat. if EVEN n then 0 else (- 1) ^ ((n - 1) div 2) / real (FACT n)) =
-(%n::nat. if EVEN n then (- 1) ^ (n div 2) / real (FACT n) else 0)"
+lemma SIN_FDIFF: "powser.diffs
+ (%n. if EVEN n then 0 else (- 1) ^ ((n - 1) div 2) / real (FACT n)) =
+(%n. if EVEN n then (- 1) ^ (n div 2) / real (FACT n) else 0)"
by (import transc SIN_FDIFF)
-lemma COS_FDIFF: "diffs (%n::nat. if EVEN n then (- 1) ^ (n div 2) / real (FACT n) else 0) =
-(%n::nat. - (if EVEN n then 0 else (- 1) ^ ((n - 1) div 2) / real (FACT n)))"
+lemma COS_FDIFF: "powser.diffs (%n. if EVEN n then (- 1) ^ (n div 2) / real (FACT n) else 0) =
+(%n. - (if EVEN n then 0 else (- 1) ^ ((n - 1) div 2) / real (FACT n)))"
by (import transc COS_FDIFF)
-lemma SIN_NEGLEMMA: "ALL x::real.
- - sin x =
- suminf
- (%n::nat.
- - ((if EVEN n then 0 else (- 1) ^ ((n - 1) div 2) / real (FACT n)) *
- x ^ n))"
+lemma SIN_NEGLEMMA: "- transc.sin x =
+seq.suminf
+ (%n. - ((if EVEN n then 0 else (- 1) ^ ((n - 1) div 2) / real (FACT n)) *
+ x ^ n))"
by (import transc SIN_NEGLEMMA)
-lemma DIFF_EXP: "ALL x::real. diffl exp (exp x) x"
+lemma DIFF_EXP: "diffl transc.exp (transc.exp x) x"
by (import transc DIFF_EXP)
-lemma DIFF_SIN: "ALL x::real. diffl sin (cos x) x"
+lemma DIFF_SIN: "diffl transc.sin (transc.cos x) x"
by (import transc DIFF_SIN)
-lemma DIFF_COS: "ALL x::real. diffl cos (- sin x) x"
+lemma DIFF_COS: "diffl transc.cos (- transc.sin x) x"
by (import transc DIFF_COS)
-lemma DIFF_COMPOSITE: "(diffl (f::real => real) (l::real) (x::real) & f x ~= 0 -->
- diffl (%x::real. inverse (f x)) (- (l / f x ^ 2)) x) &
-(diffl f l x & diffl (g::real => real) (m::real) x & g x ~= 0 -->
- diffl (%x::real. f x / g x) ((l * g x - m * f x) / g x ^ 2) x) &
-(diffl f l x & diffl g m x --> diffl (%x::real. f x + g x) (l + m) x) &
+lemma DIFF_COMPOSITE: "(diffl f l x & f x ~= 0 --> diffl (%x. inverse (f x)) (- (l / f x ^ 2)) x) &
+(diffl f l x & diffl g m x & g x ~= 0 -->
+ diffl (%x. f x / g x) ((l * g x - m * f x) / g x ^ 2) x) &
+(diffl f l x & diffl g m x --> diffl (%x. f x + g x) (l + m) x) &
(diffl f l x & diffl g m x -->
- diffl (%x::real. f x * g x) (l * g x + m * f x) x) &
-(diffl f l x & diffl g m x --> diffl (%x::real. f x - g x) (l - m) x) &
-(diffl f l x --> diffl (%x::real. - f x) (- l) x) &
-(diffl g m x -->
- diffl (%x::real. g x ^ (n::nat)) (real n * g x ^ (n - 1) * m) x) &
-(diffl g m x --> diffl (%x::real. exp (g x)) (exp (g x) * m) x) &
-(diffl g m x --> diffl (%x::real. sin (g x)) (cos (g x) * m) x) &
-(diffl g m x --> diffl (%x::real. cos (g x)) (- sin (g x) * m) x)"
+ diffl (%x. f x * g x) (l * g x + m * f x) x) &
+(diffl f l x & diffl g m x --> diffl (%x. f x - g x) (l - m) x) &
+(diffl f l x --> diffl (%x. - f x) (- l) x) &
+(diffl g m x --> diffl (%x. g x ^ n) (real n * g x ^ (n - 1) * m) x) &
+(diffl g m x --> diffl (%x. transc.exp (g x)) (transc.exp (g x) * m) x) &
+(diffl g m x --> diffl (%x. transc.sin (g x)) (transc.cos (g x) * m) x) &
+(diffl g m x --> diffl (%x. transc.cos (g x)) (- transc.sin (g x) * m) x)"
by (import transc DIFF_COMPOSITE)
-lemma EXP_0: "exp 0 = 1"
+lemma EXP_0: "transc.exp 0 = 1"
by (import transc EXP_0)
-lemma EXP_LE_X: "ALL x>=0. 1 + x <= exp x"
+lemma EXP_LE_X: "0 <= x ==> 1 + x <= transc.exp x"
by (import transc EXP_LE_X)
-lemma EXP_LT_1: "ALL x>0. 1 < exp x"
+lemma EXP_LT_1: "0 < x ==> 1 < transc.exp x"
by (import transc EXP_LT_1)
-lemma EXP_ADD_MUL: "ALL (x::real) y::real. exp (x + y) * exp (- x) = exp y"
+lemma EXP_ADD_MUL: "transc.exp (x + y) * transc.exp (- x) = transc.exp y"
by (import transc EXP_ADD_MUL)
-lemma EXP_NEG_MUL: "ALL x::real. exp x * exp (- x) = 1"
+lemma EXP_NEG_MUL: "transc.exp x * transc.exp (- x) = 1"
by (import transc EXP_NEG_MUL)
-lemma EXP_NEG_MUL2: "ALL x::real. exp (- x) * exp x = 1"
+lemma EXP_NEG_MUL2: "transc.exp (- x) * transc.exp x = 1"
by (import transc EXP_NEG_MUL2)
-lemma EXP_NEG: "ALL x::real. exp (- x) = inverse (exp x)"
+lemma EXP_NEG: "transc.exp (- x) = inverse (transc.exp x)"
by (import transc EXP_NEG)
-lemma EXP_ADD: "ALL (x::real) y::real. exp (x + y) = exp x * exp y"
+lemma EXP_ADD: "transc.exp (x + y) = transc.exp x * transc.exp y"
by (import transc EXP_ADD)
-lemma EXP_POS_LE: "ALL x::real. 0 <= exp x"
+lemma EXP_POS_LE: "0 <= transc.exp x"
by (import transc EXP_POS_LE)
-lemma EXP_NZ: "ALL x::real. exp x ~= 0"
+lemma EXP_NZ: "transc.exp x ~= 0"
by (import transc EXP_NZ)
-lemma EXP_POS_LT: "ALL x::real. 0 < exp x"
+lemma EXP_POS_LT: "0 < transc.exp x"
by (import transc EXP_POS_LT)
-lemma EXP_N: "ALL (n::nat) x::real. exp (real n * x) = exp x ^ n"
+lemma EXP_N: "transc.exp (real n * x) = transc.exp x ^ n"
by (import transc EXP_N)
-lemma EXP_SUB: "ALL (x::real) y::real. exp (x - y) = exp x / exp y"
+lemma EXP_SUB: "transc.exp (x - y) = transc.exp x / transc.exp y"
by (import transc EXP_SUB)
-lemma EXP_MONO_IMP: "ALL (x::real) y::real. x < y --> exp x < exp y"
+lemma EXP_MONO_IMP: "x < y ==> transc.exp x < transc.exp y"
by (import transc EXP_MONO_IMP)
-lemma EXP_MONO_LT: "ALL (x::real) y::real. (exp x < exp y) = (x < y)"
+lemma EXP_MONO_LT: "(transc.exp x < transc.exp y) = (x < y)"
by (import transc EXP_MONO_LT)
-lemma EXP_MONO_LE: "ALL (x::real) y::real. (exp x <= exp y) = (x <= y)"
+lemma EXP_MONO_LE: "(transc.exp x <= transc.exp y) = (x <= y)"
by (import transc EXP_MONO_LE)
-lemma EXP_INJ: "ALL (x::real) y::real. (exp x = exp y) = (x = y)"
+lemma EXP_INJ: "(transc.exp x = transc.exp y) = (x = y)"
by (import transc EXP_INJ)
-lemma EXP_TOTAL_LEMMA: "ALL y>=1. EX x>=0. x <= y - 1 & exp x = y"
+lemma EXP_TOTAL_LEMMA: "1 <= y ==> EX x>=0. x <= y - 1 & transc.exp x = y"
by (import transc EXP_TOTAL_LEMMA)
-lemma EXP_TOTAL: "ALL y>0. EX x::real. exp x = y"
+lemma EXP_TOTAL: "0 < y ==> EX x. transc.exp x = y"
by (import transc EXP_TOTAL)
-definition ln :: "real => real" where
- "ln == %x::real. SOME u::real. exp u = x"
-
-lemma ln: "ALL x::real. ln x = (SOME u::real. exp u = x)"
+consts
+ ln :: "real => real"
+
+defs
+ ln_def: "transc.ln == %x. SOME u. transc.exp u = x"
+
+lemma ln: "transc.ln x = (SOME u. transc.exp u = x)"
by (import transc ln)
-lemma LN_EXP: "ALL x::real. ln (exp x) = x"
+lemma LN_EXP: "transc.ln (transc.exp x) = x"
by (import transc LN_EXP)
-lemma EXP_LN: "ALL x::real. (exp (ln x) = x) = (0 < x)"
+lemma EXP_LN: "(transc.exp (transc.ln x) = x) = (0 < x)"
by (import transc EXP_LN)
-lemma LN_MUL: "ALL (x::real) y::real. 0 < x & 0 < y --> ln (x * y) = ln x + ln y"
+lemma LN_MUL: "0 < x & 0 < y ==> transc.ln (x * y) = transc.ln x + transc.ln y"
by (import transc LN_MUL)
-lemma LN_INJ: "ALL (x::real) y::real. 0 < x & 0 < y --> (ln x = ln y) = (x = y)"
+lemma LN_INJ: "0 < x & 0 < y ==> (transc.ln x = transc.ln y) = (x = y)"
by (import transc LN_INJ)
-lemma LN_1: "ln 1 = 0"
+lemma LN_1: "transc.ln 1 = 0"
by (import transc LN_1)
-lemma LN_INV: "ALL x>0. ln (inverse x) = - ln x"
+lemma LN_INV: "0 < x ==> transc.ln (inverse x) = - transc.ln x"
by (import transc LN_INV)
-lemma LN_DIV: "ALL (x::real) y::real. 0 < x & 0 < y --> ln (x / y) = ln x - ln y"
+lemma LN_DIV: "0 < x & 0 < y ==> transc.ln (x / y) = transc.ln x - transc.ln y"
by (import transc LN_DIV)
-lemma LN_MONO_LT: "ALL (x::real) y::real. 0 < x & 0 < y --> (ln x < ln y) = (x < y)"
+lemma LN_MONO_LT: "0 < x & 0 < y ==> (transc.ln x < transc.ln y) = (x < y)"
by (import transc LN_MONO_LT)
-lemma LN_MONO_LE: "ALL (x::real) y::real. 0 < x & 0 < y --> (ln x <= ln y) = (x <= y)"
+lemma LN_MONO_LE: "0 < x & 0 < y ==> (transc.ln x <= transc.ln y) = (x <= y)"
by (import transc LN_MONO_LE)
-lemma LN_POW: "ALL (n::nat) x::real. 0 < x --> ln (x ^ n) = real n * ln x"
+lemma LN_POW: "0 < x ==> transc.ln (x ^ n) = real n * transc.ln x"
by (import transc LN_POW)
-lemma LN_LE: "ALL x>=0. ln (1 + x) <= x"
+lemma LN_LE: "0 <= x ==> transc.ln (1 + x) <= x"
by (import transc LN_LE)
-lemma LN_LT_X: "ALL x>0. ln x < x"
+lemma LN_LT_X: "0 < x ==> transc.ln x < x"
by (import transc LN_LT_X)
-lemma LN_POS: "ALL x>=1. 0 <= ln x"
+lemma LN_POS: "1 <= x ==> 0 <= transc.ln x"
by (import transc LN_POS)
-definition root :: "nat => real => real" where
- "root == %(n::nat) x::real. SOME u::real. (0 < x --> 0 < u) & u ^ n = x"
-
-lemma root: "ALL (n::nat) x::real.
- root n x = (SOME u::real. (0 < x --> 0 < u) & u ^ n = x)"
+consts
+ root :: "nat => real => real"
+
+defs
+ root_def: "transc.root == %n x. SOME u. (0 < x --> 0 < u) & u ^ n = x"
+
+lemma root: "transc.root n x = (SOME u. (0 < x --> 0 < u) & u ^ n = x)"
by (import transc root)
-definition sqrt :: "real => real" where
- "sqrt == root 2"
-
-lemma sqrt: "ALL x::real. sqrt x = root 2 x"
+consts
+ sqrt :: "real => real"
+
+defs
+ sqrt_def: "transc.sqrt == transc.root 2"
+
+lemma sqrt: "transc.sqrt x = transc.root 2 x"
by (import transc sqrt)
-lemma ROOT_LT_LEMMA: "ALL (n::nat) x::real. 0 < x --> exp (ln x / real (Suc n)) ^ Suc n = x"
+lemma ROOT_LT_LEMMA: "0 < x ==> transc.exp (transc.ln x / real (Suc n)) ^ Suc n = x"
by (import transc ROOT_LT_LEMMA)
-lemma ROOT_LN: "ALL (n::nat) x::real. 0 < x --> root (Suc n) x = exp (ln x / real (Suc n))"
+lemma ROOT_LN: "0 < x ==> transc.root (Suc n) x = transc.exp (transc.ln x / real (Suc n))"
by (import transc ROOT_LN)
-lemma ROOT_0: "ALL n::nat. root (Suc n) 0 = 0"
+lemma ROOT_0: "transc.root (Suc n) 0 = 0"
by (import transc ROOT_0)
-lemma ROOT_1: "ALL n::nat. root (Suc n) 1 = 1"
+lemma ROOT_1: "transc.root (Suc n) 1 = 1"
by (import transc ROOT_1)
-lemma ROOT_POS_LT: "ALL (n::nat) x::real. 0 < x --> 0 < root (Suc n) x"
+lemma ROOT_POS_LT: "0 < x ==> 0 < transc.root (Suc n) x"
by (import transc ROOT_POS_LT)
-lemma ROOT_POW_POS: "ALL (n::nat) x::real. 0 <= x --> root (Suc n) x ^ Suc n = x"
+lemma ROOT_POW_POS: "0 <= x ==> transc.root (Suc n) x ^ Suc n = x"
by (import transc ROOT_POW_POS)
-lemma POW_ROOT_POS: "ALL (n::nat) x::real. 0 <= x --> root (Suc n) (x ^ Suc n) = x"
+lemma POW_ROOT_POS: "0 <= x ==> transc.root (Suc n) (x ^ Suc n) = x"
by (import transc POW_ROOT_POS)
-lemma ROOT_POS: "ALL (n::nat) x::real. 0 <= x --> 0 <= root (Suc n) x"
+lemma ROOT_POS: "0 <= x ==> 0 <= transc.root (Suc n) x"
by (import transc ROOT_POS)
-lemma ROOT_POS_UNIQ: "ALL (n::nat) (x::real) y::real.
- 0 <= x & 0 <= y & y ^ Suc n = x --> root (Suc n) x = y"
+lemma ROOT_POS_UNIQ: "0 <= x & 0 <= y & y ^ Suc n = x ==> transc.root (Suc n) x = y"
by (import transc ROOT_POS_UNIQ)
-lemma ROOT_MUL: "ALL (n::nat) (x::real) y::real.
- 0 <= x & 0 <= y -->
- root (Suc n) (x * y) = root (Suc n) x * root (Suc n) y"
+lemma ROOT_MUL: "0 <= x & 0 <= y
+==> transc.root (Suc n) (x * y) =
+ transc.root (Suc n) x * transc.root (Suc n) y"
by (import transc ROOT_MUL)
-lemma ROOT_INV: "ALL (n::nat) x::real.
- 0 <= x --> root (Suc n) (inverse x) = inverse (root (Suc n) x)"
+lemma ROOT_INV: "0 <= x ==> transc.root (Suc n) (inverse x) = inverse (transc.root (Suc n) x)"
by (import transc ROOT_INV)
-lemma ROOT_DIV: "ALL (x::nat) (xa::real) xb::real.
- 0 <= xa & 0 <= xb -->
- root (Suc x) (xa / xb) = root (Suc x) xa / root (Suc x) xb"
+lemma ROOT_DIV: "0 <= xa & 0 <= xb
+==> transc.root (Suc x) (xa / xb) =
+ transc.root (Suc x) xa / transc.root (Suc x) xb"
by (import transc ROOT_DIV)
-lemma ROOT_MONO_LE: "ALL (x::real) y::real.
- 0 <= x & x <= y --> root (Suc (n::nat)) x <= root (Suc n) y"
+lemma ROOT_MONO_LE: "0 <= x & x <= y ==> transc.root (Suc n) x <= transc.root (Suc n) y"
by (import transc ROOT_MONO_LE)
-lemma SQRT_0: "sqrt 0 = 0"
+lemma SQRT_0: "transc.sqrt 0 = 0"
by (import transc SQRT_0)
-lemma SQRT_1: "sqrt 1 = 1"
+lemma SQRT_1: "transc.sqrt 1 = 1"
by (import transc SQRT_1)
-lemma SQRT_POS_LT: "ALL x>0. 0 < sqrt x"
+lemma SQRT_POS_LT: "0 < x ==> 0 < transc.sqrt x"
by (import transc SQRT_POS_LT)
-lemma SQRT_POS_LE: "ALL x>=0. 0 <= sqrt x"
+lemma SQRT_POS_LE: "0 <= x ==> 0 <= transc.sqrt x"
by (import transc SQRT_POS_LE)
-lemma SQRT_POW2: "ALL x::real. (sqrt x ^ 2 = x) = (0 <= x)"
+lemma SQRT_POW2: "(transc.sqrt x ^ 2 = x) = (0 <= x)"
by (import transc SQRT_POW2)
-lemma SQRT_POW_2: "ALL x>=0. sqrt x ^ 2 = x"
+lemma SQRT_POW_2: "0 <= x ==> transc.sqrt x ^ 2 = x"
by (import transc SQRT_POW_2)
-lemma POW_2_SQRT: "0 <= (x::real) --> sqrt (x ^ 2) = x"
+lemma POW_2_SQRT: "0 <= x ==> transc.sqrt (x ^ 2) = x"
by (import transc POW_2_SQRT)
-lemma SQRT_POS_UNIQ: "ALL (x::real) xa::real. 0 <= x & 0 <= xa & xa ^ 2 = x --> sqrt x = xa"
+lemma SQRT_POS_UNIQ: "0 <= x & 0 <= xa & xa ^ 2 = x ==> transc.sqrt x = xa"
by (import transc SQRT_POS_UNIQ)
-lemma SQRT_MUL: "ALL (x::real) xa::real.
- 0 <= x & 0 <= xa --> sqrt (x * xa) = sqrt x * sqrt xa"
+lemma SQRT_MUL: "0 <= x & 0 <= xa ==> transc.sqrt (x * xa) = transc.sqrt x * transc.sqrt xa"
by (import transc SQRT_MUL)
-lemma SQRT_INV: "ALL x>=0. sqrt (inverse x) = inverse (sqrt x)"
+lemma SQRT_INV: "0 <= x ==> transc.sqrt (inverse x) = inverse (transc.sqrt x)"
by (import transc SQRT_INV)
-lemma SQRT_DIV: "ALL (x::real) xa::real.
- 0 <= x & 0 <= xa --> sqrt (x / xa) = sqrt x / sqrt xa"
+lemma SQRT_DIV: "0 <= x & 0 <= xa ==> transc.sqrt (x / xa) = transc.sqrt x / transc.sqrt xa"
by (import transc SQRT_DIV)
-lemma SQRT_MONO_LE: "ALL (x::real) xa::real. 0 <= x & x <= xa --> sqrt x <= sqrt xa"
+lemma SQRT_MONO_LE: "0 <= x & x <= xa ==> transc.sqrt x <= transc.sqrt xa"
by (import transc SQRT_MONO_LE)
-lemma SQRT_EVEN_POW2: "ALL n::nat. EVEN n --> sqrt (2 ^ n) = 2 ^ (n div 2)"
+lemma SQRT_EVEN_POW2: "EVEN n ==> transc.sqrt (2 ^ n) = 2 ^ (n div 2)"
by (import transc SQRT_EVEN_POW2)
-lemma REAL_DIV_SQRT: "ALL x>=0. x / sqrt x = sqrt x"
+lemma REAL_DIV_SQRT: "0 <= x ==> x / transc.sqrt x = transc.sqrt x"
by (import transc REAL_DIV_SQRT)
-lemma SQRT_EQ: "ALL (x::real) y::real. x ^ 2 = y & 0 <= x --> x = sqrt y"
+lemma SQRT_EQ: "x ^ 2 = y & 0 <= x ==> x = transc.sqrt y"
by (import transc SQRT_EQ)
-lemma SIN_0: "sin 0 = 0"
+lemma SIN_0: "transc.sin 0 = 0"
by (import transc SIN_0)
-lemma COS_0: "cos 0 = 1"
+lemma COS_0: "transc.cos 0 = 1"
by (import transc COS_0)
-lemma SIN_CIRCLE: "ALL x::real. sin x ^ 2 + cos x ^ 2 = 1"
+lemma SIN_CIRCLE: "transc.sin x ^ 2 + transc.cos x ^ 2 = 1"
by (import transc SIN_CIRCLE)
-lemma SIN_BOUND: "ALL x::real. abs (sin x) <= 1"
+lemma SIN_BOUND: "abs (transc.sin x) <= 1"
by (import transc SIN_BOUND)
-lemma SIN_BOUNDS: "ALL x::real. - 1 <= sin x & sin x <= 1"
+lemma SIN_BOUNDS: "- 1 <= transc.sin x & transc.sin x <= 1"
by (import transc SIN_BOUNDS)
-lemma COS_BOUND: "ALL x::real. abs (cos x) <= 1"
+lemma COS_BOUND: "abs (transc.cos x) <= 1"
by (import transc COS_BOUND)
-lemma COS_BOUNDS: "ALL x::real. - 1 <= cos x & cos x <= 1"
+lemma COS_BOUNDS: "- 1 <= transc.cos x & transc.cos x <= 1"
by (import transc COS_BOUNDS)
-lemma SIN_COS_ADD: "ALL (x::real) y::real.
- (sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +
- (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2 =
- 0"
+lemma SIN_COS_ADD: "(transc.sin (x + y) -
+ (transc.sin x * transc.cos y + transc.cos x * transc.sin y)) ^
+2 +
+(transc.cos (x + y) -
+ (transc.cos x * transc.cos y - transc.sin x * transc.sin y)) ^
+2 =
+0"
by (import transc SIN_COS_ADD)
-lemma SIN_COS_NEG: "ALL x::real. (sin (- x) + sin x) ^ 2 + (cos (- x) - cos x) ^ 2 = 0"
+lemma SIN_COS_NEG: "(transc.sin (- x) + transc.sin x) ^ 2 +
+(transc.cos (- x) - transc.cos x) ^ 2 =
+0"
by (import transc SIN_COS_NEG)
-lemma SIN_ADD: "ALL (x::real) y::real. sin (x + y) = sin x * cos y + cos x * sin y"
+lemma SIN_ADD: "transc.sin (x + y) =
+transc.sin x * transc.cos y + transc.cos x * transc.sin y"
by (import transc SIN_ADD)
-lemma COS_ADD: "ALL (x::real) y::real. cos (x + y) = cos x * cos y - sin x * sin y"
+lemma COS_ADD: "transc.cos (x + y) =
+transc.cos x * transc.cos y - transc.sin x * transc.sin y"
by (import transc COS_ADD)
-lemma SIN_NEG: "ALL x::real. sin (- x) = - sin x"
+lemma SIN_NEG: "transc.sin (- x) = - transc.sin x"
by (import transc SIN_NEG)
-lemma COS_NEG: "ALL x::real. cos (- x) = cos x"
+lemma COS_NEG: "transc.cos (- x) = transc.cos x"
by (import transc COS_NEG)
-lemma SIN_DOUBLE: "ALL x::real. sin (2 * x) = 2 * (sin x * cos x)"
+lemma SIN_DOUBLE: "transc.sin (2 * x) = 2 * (transc.sin x * transc.cos x)"
by (import transc SIN_DOUBLE)
-lemma COS_DOUBLE: "ALL x::real. cos (2 * x) = cos x ^ 2 - sin x ^ 2"
+lemma COS_DOUBLE: "transc.cos (2 * x) = transc.cos x ^ 2 - transc.sin x ^ 2"
by (import transc COS_DOUBLE)
-lemma SIN_PAIRED: "ALL x::real.
- sums (%n::nat. (- 1) ^ n / real (FACT (2 * n + 1)) * x ^ (2 * n + 1))
- (sin x)"
+lemma SIN_PAIRED: "seq.sums (%n. (- 1) ^ n / real (FACT (2 * n + 1)) * x ^ (2 * n + 1))
+ (transc.sin x)"
by (import transc SIN_PAIRED)
-lemma SIN_POS: "ALL x::real. 0 < x & x < 2 --> 0 < sin x"
+lemma SIN_POS: "0 < x & x < 2 ==> 0 < transc.sin x"
by (import transc SIN_POS)
-lemma COS_PAIRED: "ALL x::real.
- sums (%n::nat. (- 1) ^ n / real (FACT (2 * n)) * x ^ (2 * n)) (cos x)"
+lemma COS_PAIRED: "seq.sums (%n. (- 1) ^ n / real (FACT (2 * n)) * x ^ (2 * n)) (transc.cos x)"
by (import transc COS_PAIRED)
-lemma COS_2: "cos 2 < 0"
+lemma COS_2: "transc.cos 2 < 0"
by (import transc COS_2)
-lemma COS_ISZERO: "EX! x::real. 0 <= x & x <= 2 & cos x = 0"
+lemma COS_ISZERO: "EX! x. 0 <= x & x <= 2 & transc.cos x = 0"
by (import transc COS_ISZERO)
-definition pi :: "real" where
- "pi == 2 * (SOME x::real. 0 <= x & x <= 2 & cos x = 0)"
-
-lemma pi: "pi = 2 * (SOME x::real. 0 <= x & x <= 2 & cos x = 0)"
+consts
+ pi :: "real"
+
+defs
+ pi_def: "transc.pi == 2 * (SOME x. 0 <= x & x <= 2 & transc.cos x = 0)"
+
+lemma pi: "transc.pi = 2 * (SOME x. 0 <= x & x <= 2 & transc.cos x = 0)"
by (import transc pi)
-lemma PI2: "pi / 2 = (SOME x::real. 0 <= x & x <= 2 & cos x = 0)"
+lemma PI2: "transc.pi / 2 = (SOME x. 0 <= x & x <= 2 & transc.cos x = 0)"
by (import transc PI2)
-lemma COS_PI2: "cos (pi / 2) = 0"
+lemma COS_PI2: "transc.cos (transc.pi / 2) = 0"
by (import transc COS_PI2)
-lemma PI2_BOUNDS: "0 < pi / 2 & pi / 2 < 2"
+lemma PI2_BOUNDS: "0 < transc.pi / 2 & transc.pi / 2 < 2"
by (import transc PI2_BOUNDS)
-lemma PI_POS: "0 < pi"
+lemma PI_POS: "0 < transc.pi"
by (import transc PI_POS)
-lemma SIN_PI2: "sin (pi / 2) = 1"
+lemma SIN_PI2: "transc.sin (transc.pi / 2) = 1"
by (import transc SIN_PI2)
-lemma COS_PI: "cos pi = - 1"
+lemma COS_PI: "transc.cos transc.pi = - 1"
by (import transc COS_PI)
-lemma SIN_PI: "sin pi = 0"
+lemma SIN_PI: "transc.sin transc.pi = 0"
by (import transc SIN_PI)
-lemma SIN_COS: "ALL x::real. sin x = cos (pi / 2 - x)"
+lemma SIN_COS: "transc.sin x = transc.cos (transc.pi / 2 - x)"
by (import transc SIN_COS)
-lemma COS_SIN: "ALL x::real. cos x = sin (pi / 2 - x)"
+lemma COS_SIN: "transc.cos x = transc.sin (transc.pi / 2 - x)"
by (import transc COS_SIN)
-lemma SIN_PERIODIC_PI: "ALL x::real. sin (x + pi) = - sin x"
+lemma SIN_PERIODIC_PI: "transc.sin (x + transc.pi) = - transc.sin x"
by (import transc SIN_PERIODIC_PI)
-lemma COS_PERIODIC_PI: "ALL x::real. cos (x + pi) = - cos x"
+lemma COS_PERIODIC_PI: "transc.cos (x + transc.pi) = - transc.cos x"
by (import transc COS_PERIODIC_PI)
-lemma SIN_PERIODIC: "ALL x::real. sin (x + 2 * pi) = sin x"
+lemma SIN_PERIODIC: "transc.sin (x + 2 * transc.pi) = transc.sin x"
by (import transc SIN_PERIODIC)
-lemma COS_PERIODIC: "ALL x::real. cos (x + 2 * pi) = cos x"
+lemma COS_PERIODIC: "transc.cos (x + 2 * transc.pi) = transc.cos x"
by (import transc COS_PERIODIC)
-lemma COS_NPI: "ALL n::nat. cos (real n * pi) = (- 1) ^ n"
+lemma COS_NPI: "transc.cos (real n * transc.pi) = (- 1) ^ n"
by (import transc COS_NPI)
-lemma SIN_NPI: "ALL n::nat. sin (real n * pi) = 0"
+lemma SIN_NPI: "transc.sin (real (n::nat) * transc.pi) = (0::real)"
by (import transc SIN_NPI)
-lemma SIN_POS_PI2: "ALL x::real. 0 < x & x < pi / 2 --> 0 < sin x"
+lemma SIN_POS_PI2: "0 < x & x < transc.pi / 2 ==> 0 < transc.sin x"
by (import transc SIN_POS_PI2)
-lemma COS_POS_PI2: "ALL x::real. 0 < x & x < pi / 2 --> 0 < cos x"
+lemma COS_POS_PI2: "0 < x & x < transc.pi / 2 ==> 0 < transc.cos x"
by (import transc COS_POS_PI2)
-lemma COS_POS_PI: "ALL x::real. - (pi / 2) < x & x < pi / 2 --> 0 < cos x"
+lemma COS_POS_PI: "- (transc.pi / 2) < x & x < transc.pi / 2 ==> 0 < transc.cos x"
by (import transc COS_POS_PI)
-lemma SIN_POS_PI: "ALL x::real. 0 < x & x < pi --> 0 < sin x"
+lemma SIN_POS_PI: "0 < x & x < transc.pi ==> 0 < transc.sin x"
by (import transc SIN_POS_PI)
-lemma COS_POS_PI2_LE: "ALL x::real. 0 <= x & x <= pi / 2 --> 0 <= cos x"
+lemma COS_POS_PI2_LE: "0 <= x & x <= transc.pi / 2 ==> 0 <= transc.cos x"
by (import transc COS_POS_PI2_LE)
-lemma COS_POS_PI_LE: "ALL x::real. - (pi / 2) <= x & x <= pi / 2 --> 0 <= cos x"
+lemma COS_POS_PI_LE: "- (transc.pi / 2) <= x & x <= transc.pi / 2 ==> 0 <= transc.cos x"
by (import transc COS_POS_PI_LE)
-lemma SIN_POS_PI2_LE: "ALL x::real. 0 <= x & x <= pi / 2 --> 0 <= sin x"
+lemma SIN_POS_PI2_LE: "0 <= x & x <= transc.pi / 2 ==> 0 <= transc.sin x"
by (import transc SIN_POS_PI2_LE)
-lemma SIN_POS_PI_LE: "ALL x::real. 0 <= x & x <= pi --> 0 <= sin x"
+lemma SIN_POS_PI_LE: "0 <= x & x <= transc.pi ==> 0 <= transc.sin x"
by (import transc SIN_POS_PI_LE)
-lemma COS_TOTAL: "ALL y::real.
- - 1 <= y & y <= 1 --> (EX! x::real. 0 <= x & x <= pi & cos x = y)"
+lemma COS_TOTAL: "- 1 <= y & y <= 1 ==> EX! x. 0 <= x & x <= transc.pi & transc.cos x = y"
by (import transc COS_TOTAL)
-lemma SIN_TOTAL: "ALL y::real.
- - 1 <= y & y <= 1 -->
- (EX! x::real. - (pi / 2) <= x & x <= pi / 2 & sin x = y)"
+lemma SIN_TOTAL: "- 1 <= y & y <= 1
+==> EX! x. - (transc.pi / 2) <= x & x <= transc.pi / 2 & transc.sin x = y"
by (import transc SIN_TOTAL)
-lemma COS_ZERO_LEMMA: "ALL x::real.
- 0 <= x & cos x = 0 --> (EX n::nat. ~ EVEN n & x = real n * (pi / 2))"
+lemma COS_ZERO_LEMMA: "0 <= x & transc.cos x = 0 ==> EX n. ~ EVEN n & x = real n * (transc.pi / 2)"
by (import transc COS_ZERO_LEMMA)
-lemma SIN_ZERO_LEMMA: "ALL x::real.
- 0 <= x & sin x = 0 --> (EX n::nat. EVEN n & x = real n * (pi / 2))"
+lemma SIN_ZERO_LEMMA: "0 <= x & transc.sin x = 0 ==> EX n. EVEN n & x = real n * (transc.pi / 2)"
by (import transc SIN_ZERO_LEMMA)
-lemma COS_ZERO: "ALL x::real.
- (cos x = 0) =
- ((EX n::nat. ~ EVEN n & x = real n * (pi / 2)) |
- (EX n::nat. ~ EVEN n & x = - (real n * (pi / 2))))"
+lemma COS_ZERO: "(transc.cos x = 0) =
+((EX n. ~ EVEN n & x = real n * (transc.pi / 2)) |
+ (EX n. ~ EVEN n & x = - (real n * (transc.pi / 2))))"
by (import transc COS_ZERO)
-lemma SIN_ZERO: "ALL x::real.
- (sin x = 0) =
- ((EX n::nat. EVEN n & x = real n * (pi / 2)) |
- (EX n::nat. EVEN n & x = - (real n * (pi / 2))))"
+lemma SIN_ZERO: "(transc.sin x = 0) =
+((EX n. EVEN n & x = real n * (transc.pi / 2)) |
+ (EX n. EVEN n & x = - (real n * (transc.pi / 2))))"
by (import transc SIN_ZERO)
-definition tan :: "real => real" where
- "tan == %x::real. sin x / cos x"
-
-lemma tan: "ALL x::real. tan x = sin x / cos x"
+consts
+ tan :: "real => real"
+
+defs
+ tan_def: "transc.tan == %x. transc.sin x / transc.cos x"
+
+lemma tan: "transc.tan x = transc.sin x / transc.cos x"
by (import transc tan)
-lemma TAN_0: "tan 0 = 0"
+lemma TAN_0: "transc.tan 0 = 0"
by (import transc TAN_0)
-lemma TAN_PI: "tan pi = 0"
+lemma TAN_PI: "transc.tan transc.pi = 0"
by (import transc TAN_PI)
-lemma TAN_NPI: "ALL n::nat. tan (real n * pi) = 0"
+lemma TAN_NPI: "transc.tan (real (n::nat) * transc.pi) = (0::real)"
by (import transc TAN_NPI)
-lemma TAN_NEG: "ALL x::real. tan (- x) = - tan x"
+lemma TAN_NEG: "transc.tan (- x) = - transc.tan x"
by (import transc TAN_NEG)
-lemma TAN_PERIODIC: "ALL x::real. tan (x + 2 * pi) = tan x"
+lemma TAN_PERIODIC: "transc.tan (x + 2 * transc.pi) = transc.tan x"
by (import transc TAN_PERIODIC)
-lemma TAN_ADD: "ALL (x::real) y::real.
- cos x ~= 0 & cos y ~= 0 & cos (x + y) ~= 0 -->
- tan (x + y) = (tan x + tan y) / (1 - tan x * tan y)"
+lemma TAN_ADD: "transc.cos x ~= 0 & transc.cos y ~= 0 & transc.cos (x + y) ~= 0
+==> transc.tan (x + y) =
+ (transc.tan x + transc.tan y) / (1 - transc.tan x * transc.tan y)"
by (import transc TAN_ADD)
-lemma TAN_DOUBLE: "ALL x::real.
- cos x ~= 0 & cos (2 * x) ~= 0 -->
- tan (2 * x) = 2 * tan x / (1 - tan x ^ 2)"
+lemma TAN_DOUBLE: "transc.cos x ~= 0 & transc.cos (2 * x) ~= 0
+==> transc.tan (2 * x) = 2 * transc.tan x / (1 - transc.tan x ^ 2)"
by (import transc TAN_DOUBLE)
-lemma TAN_POS_PI2: "ALL x::real. 0 < x & x < pi / 2 --> 0 < tan x"
+lemma TAN_POS_PI2: "0 < x & x < transc.pi / 2 ==> 0 < transc.tan x"
by (import transc TAN_POS_PI2)
-lemma DIFF_TAN: "ALL x::real. cos x ~= 0 --> diffl tan (inverse (cos x ^ 2)) x"
+lemma DIFF_TAN: "transc.cos x ~= 0 ==> diffl transc.tan (inverse (transc.cos x ^ 2)) x"
by (import transc DIFF_TAN)
-lemma TAN_TOTAL_LEMMA: "ALL y>0. EX x>0. x < pi / 2 & y < tan x"
+lemma TAN_TOTAL_LEMMA: "0 < y ==> EX x>0. x < transc.pi / 2 & y < transc.tan x"
by (import transc TAN_TOTAL_LEMMA)
-lemma TAN_TOTAL_POS: "ALL y>=0. EX x>=0. x < pi / 2 & tan x = y"
+lemma TAN_TOTAL_POS: "0 <= y ==> EX x>=0. x < transc.pi / 2 & transc.tan x = y"
by (import transc TAN_TOTAL_POS)
-lemma TAN_TOTAL: "ALL y::real. EX! x::real. - (pi / 2) < x & x < pi / 2 & tan x = y"
+lemma TAN_TOTAL: "EX! x. - (transc.pi / 2) < x & x < transc.pi / 2 & transc.tan x = y"
by (import transc TAN_TOTAL)
-definition asn :: "real => real" where
- "asn == %y::real. SOME x::real. - (pi / 2) <= x & x <= pi / 2 & sin x = y"
-
-lemma asn: "ALL y::real.
- asn y = (SOME x::real. - (pi / 2) <= x & x <= pi / 2 & sin x = y)"
+definition
+ asn :: "real => real" where
+ "asn ==
+%y. SOME x. - (transc.pi / 2) <= x & x <= transc.pi / 2 & transc.sin x = y"
+
+lemma asn: "asn y =
+(SOME x. - (transc.pi / 2) <= x & x <= transc.pi / 2 & transc.sin x = y)"
by (import transc asn)
-definition acs :: "real => real" where
- "acs == %y::real. SOME x::real. 0 <= x & x <= pi & cos x = y"
-
-lemma acs: "ALL y::real. acs y = (SOME x::real. 0 <= x & x <= pi & cos x = y)"
+definition
+ acs :: "real => real" where
+ "acs == %y. SOME x. 0 <= x & x <= transc.pi & transc.cos x = y"
+
+lemma acs: "acs y = (SOME x. 0 <= x & x <= transc.pi & transc.cos x = y)"
by (import transc acs)
-definition atn :: "real => real" where
- "atn == %y::real. SOME x::real. - (pi / 2) < x & x < pi / 2 & tan x = y"
-
-lemma atn: "ALL y::real. atn y = (SOME x::real. - (pi / 2) < x & x < pi / 2 & tan x = y)"
+definition
+ atn :: "real => real" where
+ "atn ==
+%y. SOME x. - (transc.pi / 2) < x & x < transc.pi / 2 & transc.tan x = y"
+
+lemma atn: "atn y =
+(SOME x. - (transc.pi / 2) < x & x < transc.pi / 2 & transc.tan x = y)"
by (import transc atn)
-lemma ASN: "ALL y::real.
- - 1 <= y & y <= 1 -->
- - (pi / 2) <= asn y & asn y <= pi / 2 & sin (asn y) = y"
+lemma ASN: "- 1 <= y & y <= 1
+==> - (transc.pi / 2) <= asn y &
+ asn y <= transc.pi / 2 & transc.sin (asn y) = y"
by (import transc ASN)
-lemma ASN_SIN: "ALL y::real. - 1 <= y & y <= 1 --> sin (asn y) = y"
+lemma ASN_SIN: "- 1 <= y & y <= 1 ==> transc.sin (asn y) = y"
by (import transc ASN_SIN)
-lemma ASN_BOUNDS: "ALL y::real. - 1 <= y & y <= 1 --> - (pi / 2) <= asn y & asn y <= pi / 2"
+lemma ASN_BOUNDS: "- 1 <= y & y <= 1 ==> - (transc.pi / 2) <= asn y & asn y <= transc.pi / 2"
by (import transc ASN_BOUNDS)
-lemma ASN_BOUNDS_LT: "ALL y::real. - 1 < y & y < 1 --> - (pi / 2) < asn y & asn y < pi / 2"
+lemma ASN_BOUNDS_LT: "- 1 < y & y < 1 ==> - (transc.pi / 2) < asn y & asn y < transc.pi / 2"
by (import transc ASN_BOUNDS_LT)
-lemma SIN_ASN: "ALL x::real. - (pi / 2) <= x & x <= pi / 2 --> asn (sin x) = x"
+lemma SIN_ASN: "- (transc.pi / 2) <= x & x <= transc.pi / 2 ==> asn (transc.sin x) = x"
by (import transc SIN_ASN)
-lemma ACS: "ALL y::real.
- - 1 <= y & y <= 1 --> 0 <= acs y & acs y <= pi & cos (acs y) = y"
+lemma ACS: "- 1 <= y & y <= 1
+==> 0 <= acs y & acs y <= transc.pi & transc.cos (acs y) = y"
by (import transc ACS)
-lemma ACS_COS: "ALL y::real. - 1 <= y & y <= 1 --> cos (acs y) = y"
+lemma ACS_COS: "- 1 <= y & y <= 1 ==> transc.cos (acs y) = y"
by (import transc ACS_COS)
-lemma ACS_BOUNDS: "ALL y::real. - 1 <= y & y <= 1 --> 0 <= acs y & acs y <= pi"
+lemma ACS_BOUNDS: "- 1 <= y & y <= 1 ==> 0 <= acs y & acs y <= transc.pi"
by (import transc ACS_BOUNDS)
-lemma ACS_BOUNDS_LT: "ALL y::real. - 1 < y & y < 1 --> 0 < acs y & acs y < pi"
+lemma ACS_BOUNDS_LT: "- 1 < y & y < 1 ==> 0 < acs y & acs y < transc.pi"
by (import transc ACS_BOUNDS_LT)
-lemma COS_ACS: "ALL x::real. 0 <= x & x <= pi --> acs (cos x) = x"
+lemma COS_ACS: "0 <= x & x <= transc.pi ==> acs (transc.cos x) = x"
by (import transc COS_ACS)
-lemma ATN: "ALL y::real. - (pi / 2) < atn y & atn y < pi / 2 & tan (atn y) = y"
+lemma ATN: "- (transc.pi / 2) < atn y & atn y < transc.pi / 2 & transc.tan (atn y) = y"
by (import transc ATN)
-lemma ATN_TAN: "ALL x::real. tan (atn x) = x"
+lemma ATN_TAN: "transc.tan (atn x) = x"
by (import transc ATN_TAN)
-lemma ATN_BOUNDS: "ALL x::real. - (pi / 2) < atn x & atn x < pi / 2"
+lemma ATN_BOUNDS: "- (transc.pi / 2) < atn x & atn x < transc.pi / 2"
by (import transc ATN_BOUNDS)
-lemma TAN_ATN: "ALL x::real. - (pi / 2) < x & x < pi / 2 --> atn (tan x) = x"
+lemma TAN_ATN: "- (transc.pi / 2) < x & x < transc.pi / 2 ==> atn (transc.tan x) = x"
by (import transc TAN_ATN)
-lemma TAN_SEC: "ALL x::real. cos x ~= 0 --> 1 + tan x ^ 2 = inverse (cos x) ^ 2"
+lemma TAN_SEC: "transc.cos x ~= 0 ==> 1 + transc.tan x ^ 2 = inverse (transc.cos x) ^ 2"
by (import transc TAN_SEC)
-lemma SIN_COS_SQ: "ALL x::real. 0 <= x & x <= pi --> sin x = sqrt (1 - cos x ^ 2)"
+lemma SIN_COS_SQ: "0 <= x & x <= transc.pi
+==> transc.sin x = transc.sqrt (1 - transc.cos x ^ 2)"
by (import transc SIN_COS_SQ)
-lemma COS_SIN_SQ: "ALL x::real. - (pi / 2) <= x & x <= pi / 2 --> cos x = sqrt (1 - sin x ^ 2)"
+lemma COS_SIN_SQ: "- (transc.pi / 2) <= x & x <= transc.pi / 2
+==> transc.cos x = transc.sqrt (1 - transc.sin x ^ 2)"
by (import transc COS_SIN_SQ)
-lemma COS_ATN_NZ: "ALL x::real. cos (atn x) ~= 0"
+lemma COS_ATN_NZ: "transc.cos (atn x) ~= 0"
by (import transc COS_ATN_NZ)
-lemma COS_ASN_NZ: "ALL x::real. - 1 < x & x < 1 --> cos (asn x) ~= 0"
+lemma COS_ASN_NZ: "- 1 < x & x < 1 ==> transc.cos (asn x) ~= 0"
by (import transc COS_ASN_NZ)
-lemma SIN_ACS_NZ: "ALL x::real. - 1 < x & x < 1 --> sin (acs x) ~= 0"
+lemma SIN_ACS_NZ: "- 1 < x & x < 1 ==> transc.sin (acs x) ~= 0"
by (import transc SIN_ACS_NZ)
-lemma COS_SIN_SQRT: "ALL x::real. 0 <= cos x --> cos x = sqrt (1 - sin x ^ 2)"
+lemma COS_SIN_SQRT: "0 <= transc.cos x ==> transc.cos x = transc.sqrt (1 - transc.sin x ^ 2)"
by (import transc COS_SIN_SQRT)
-lemma SIN_COS_SQRT: "ALL x::real. 0 <= sin x --> sin x = sqrt (1 - cos x ^ 2)"
+lemma SIN_COS_SQRT: "0 <= transc.sin x ==> transc.sin x = transc.sqrt (1 - transc.cos x ^ 2)"
by (import transc SIN_COS_SQRT)
-lemma DIFF_LN: "ALL x>0. diffl ln (inverse x) x"
+lemma DIFF_LN: "0 < x ==> diffl transc.ln (inverse x) x"
by (import transc DIFF_LN)
-lemma DIFF_ASN_LEMMA: "ALL x::real. - 1 < x & x < 1 --> diffl asn (inverse (cos (asn x))) x"
+lemma DIFF_ASN_LEMMA: "- 1 < x & x < 1 ==> diffl asn (inverse (transc.cos (asn x))) x"
by (import transc DIFF_ASN_LEMMA)
-lemma DIFF_ASN: "ALL x::real. - 1 < x & x < 1 --> diffl asn (inverse (sqrt (1 - x ^ 2))) x"
+lemma DIFF_ASN: "- 1 < x & x < 1 ==> diffl asn (inverse (transc.sqrt (1 - x ^ 2))) x"
by (import transc DIFF_ASN)
-lemma DIFF_ACS_LEMMA: "ALL x::real. - 1 < x & x < 1 --> diffl acs (inverse (- sin (acs x))) x"
+lemma DIFF_ACS_LEMMA: "- 1 < x & x < 1 ==> diffl acs (inverse (- transc.sin (acs x))) x"
by (import transc DIFF_ACS_LEMMA)
-lemma DIFF_ACS: "ALL x::real. - 1 < x & x < 1 --> diffl acs (- inverse (sqrt (1 - x ^ 2))) x"
+lemma DIFF_ACS: "- 1 < x & x < 1 ==> diffl acs (- inverse (transc.sqrt (1 - x ^ 2))) x"
by (import transc DIFF_ACS)
-lemma DIFF_ATN: "ALL x::real. diffl atn (inverse (1 + x ^ 2)) x"
+lemma DIFF_ATN: "diffl atn (inverse (1 + x ^ 2)) x"
by (import transc DIFF_ATN)
-definition division :: "real * real => (nat => real) => bool" where
- "(op ==::(real * real => (nat => real) => bool)
- => (real * real => (nat => real) => bool) => prop)
- (division::real * real => (nat => real) => bool)
- ((split::(real => real => (nat => real) => bool)
- => real * real => (nat => real) => bool)
- (%(a::real) (b::real) D::nat => real.
- (op &::bool => bool => bool)
- ((op =::real => real => bool) (D (0::nat)) a)
- ((Ex::(nat => bool) => bool)
- (%N::nat.
- (op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%n::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) n N)
- ((op <::real => real => bool) (D n)
- (D ((Suc::nat => nat) n)))))
- ((All::(nat => bool) => bool)
- (%n::nat.
- (op -->::bool => bool => bool)
- ((op <=::nat => nat => bool) N n)
- ((op =::real => real => bool) (D n) b)))))))"
-
-lemma division: "(All::(real => bool) => bool)
- (%a::real.
- (All::(real => bool) => bool)
- (%b::real.
- (All::((nat => real) => bool) => bool)
- (%D::nat => real.
- (op =::bool => bool => bool)
- ((division::real * real => (nat => real) => bool)
- ((Pair::real => real => real * real) a b) D)
- ((op &::bool => bool => bool)
- ((op =::real => real => bool) (D (0::nat)) a)
- ((Ex::(nat => bool) => bool)
- (%N::nat.
- (op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%n::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) n N)
- ((op <::real => real => bool) (D n)
- (D ((Suc::nat => nat) n)))))
- ((All::(nat => bool) => bool)
- (%n::nat.
- (op -->::bool => bool => bool)
- ((op <=::nat => nat => bool) N n)
- ((op =::real => real => bool) (D n)
- b)))))))))"
+definition
+ division :: "real * real => (nat => real) => bool" where
+ "division ==
+%(a, b) D.
+ D 0 = a & (EX N. (ALL n<N. D n < D (Suc n)) & (ALL n>=N. D n = b))"
+
+lemma division: "division (a, b) D =
+(D 0 = a & (EX N. (ALL n<N. D n < D (Suc n)) & (ALL n>=N. D n = b)))"
by (import transc division)
-definition dsize :: "(nat => real) => nat" where
- "(op ==::((nat => real) => nat) => ((nat => real) => nat) => prop)
- (dsize::(nat => real) => nat)
- (%D::nat => real.
- (Eps::(nat => bool) => nat)
- (%N::nat.
- (op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%n::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) n N)
- ((op <::real => real => bool) (D n)
- (D ((Suc::nat => nat) n)))))
- ((All::(nat => bool) => bool)
- (%n::nat.
- (op -->::bool => bool => bool)
- ((op <=::nat => nat => bool) N n)
- ((op =::real => real => bool) (D n) (D N))))))"
-
-lemma dsize: "(All::((nat => real) => bool) => bool)
- (%D::nat => real.
- (op =::nat => nat => bool) ((dsize::(nat => real) => nat) D)
- ((Eps::(nat => bool) => nat)
- (%N::nat.
- (op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%n::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) n N)
- ((op <::real => real => bool) (D n)
- (D ((Suc::nat => nat) n)))))
- ((All::(nat => bool) => bool)
- (%n::nat.
- (op -->::bool => bool => bool)
- ((op <=::nat => nat => bool) N n)
- ((op =::real => real => bool) (D n) (D N)))))))"
+definition
+ dsize :: "(nat => real) => nat" where
+ "dsize == %D. SOME N. (ALL n<N. D n < D (Suc n)) & (ALL n>=N. D n = D N)"
+
+lemma dsize: "dsize D = (SOME N. (ALL n<N. D n < D (Suc n)) & (ALL n>=N. D n = D N))"
by (import transc dsize)
-definition tdiv :: "real * real => (nat => real) * (nat => real) => bool" where
+definition
+ tdiv :: "real * real => (nat => real) * (nat => real) => bool" where
"tdiv ==
-%(a::real, b::real) (D::nat => real, p::nat => real).
- division (a, b) D & (ALL n::nat. D n <= p n & p n <= D (Suc n))"
-
-lemma tdiv: "ALL (a::real) (b::real) (D::nat => real) p::nat => real.
- tdiv (a, b) (D, p) =
- (division (a, b) D & (ALL n::nat. D n <= p n & p n <= D (Suc n)))"
+%(a, b) (D, p). division (a, b) D & (ALL n. D n <= p n & p n <= D (Suc n))"
+
+lemma tdiv: "tdiv (a, b) (D, p) =
+(division (a, b) D & (ALL n. D n <= p n & p n <= D (Suc n)))"
by (import transc tdiv)
-definition gauge :: "(real => bool) => (real => real) => bool" where
- "gauge == %(E::real => bool) g::real => real. ALL x::real. E x --> 0 < g x"
-
-lemma gauge: "ALL (E::real => bool) g::real => real.
- gauge E g = (ALL x::real. E x --> 0 < g x)"
+definition
+ gauge :: "(real => bool) => (real => real) => bool" where
+ "gauge == %E g. ALL x. E x --> 0 < g x"
+
+lemma gauge: "gauge E g = (ALL x. E x --> 0 < g x)"
by (import transc gauge)
-definition fine :: "(real => real) => (nat => real) * (nat => real) => bool" where
- "(op ==::((real => real) => (nat => real) * (nat => real) => bool)
- => ((real => real) => (nat => real) * (nat => real) => bool)
- => prop)
- (fine::(real => real) => (nat => real) * (nat => real) => bool)
- (%g::real => real.
- (split::((nat => real) => (nat => real) => bool)
- => (nat => real) * (nat => real) => bool)
- (%(D::nat => real) p::nat => real.
- (All::(nat => bool) => bool)
- (%n::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) n
- ((dsize::(nat => real) => nat) D))
- ((op <::real => real => bool)
- ((op -::real => real => real) (D ((Suc::nat => nat) n))
- (D n))
- (g (p n))))))"
-
-lemma fine: "(All::((real => real) => bool) => bool)
- (%g::real => real.
- (All::((nat => real) => bool) => bool)
- (%D::nat => real.
- (All::((nat => real) => bool) => bool)
- (%p::nat => real.
- (op =::bool => bool => bool)
- ((fine::(real => real)
- => (nat => real) * (nat => real) => bool)
- g ((Pair::(nat => real)
- => (nat => real)
- => (nat => real) * (nat => real))
- D p))
- ((All::(nat => bool) => bool)
- (%n::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) n
- ((dsize::(nat => real) => nat) D))
- ((op <::real => real => bool)
- ((op -::real => real => real)
- (D ((Suc::nat => nat) n)) (D n))
- (g (p n))))))))"
+definition
+ fine :: "(real => real) => (nat => real) * (nat => real) => bool" where
+ "fine == %g (D, p). ALL n<dsize D. D (Suc n) - D n < g (p n)"
+
+lemma fine: "fine g (D, p) = (ALL n<dsize D. D (Suc n) - D n < g (p n))"
by (import transc fine)
-definition rsum :: "(nat => real) * (nat => real) => (real => real) => real" where
- "rsum ==
-%(D::nat => real, p::nat => real) f::real => real.
- real.sum (0, dsize D) (%n::nat. f (p n) * (D (Suc n) - D n))"
-
-lemma rsum: "ALL (D::nat => real) (p::nat => real) f::real => real.
- rsum (D, p) f =
- real.sum (0, dsize D) (%n::nat. f (p n) * (D (Suc n) - D n))"
+definition
+ rsum :: "(nat => real) * (nat => real) => (real => real) => real" where
+ "rsum == %(D, p) f. real.sum (0, dsize D) (%n. f (p n) * (D (Suc n) - D n))"
+
+lemma rsum: "rsum (D, p) f = real.sum (0, dsize D) (%n. f (p n) * (D (Suc n) - D n))"
by (import transc rsum)
-definition Dint :: "real * real => (real => real) => real => bool" where
+definition
+ Dint :: "real * real => (real => real) => real => bool" where
"Dint ==
-%(a::real, b::real) (f::real => real) k::real.
+%(a, b) f k.
ALL e>0.
- EX g::real => real.
- gauge (%x::real. a <= x & x <= b) g &
- (ALL (D::nat => real) p::nat => real.
- tdiv (a, b) (D, p) & fine g (D, p) -->
- abs (rsum (D, p) f - k) < e)"
-
-lemma Dint: "ALL (a::real) (b::real) (f::real => real) k::real.
- Dint (a, b) f k =
- (ALL e>0.
- EX g::real => real.
- gauge (%x::real. a <= x & x <= b) g &
- (ALL (D::nat => real) p::nat => real.
+ EX g. gauge (%x. a <= x & x <= b) g &
+ (ALL D p.
+ tdiv (a, b) (D, p) & fine g (D, p) -->
+ abs (rsum (D, p) f - k) < e)"
+
+lemma Dint: "Dint (a, b) f k =
+(ALL e>0.
+ EX g. gauge (%x. a <= x & x <= b) g &
+ (ALL D p.
tdiv (a, b) (D, p) & fine g (D, p) -->
abs (rsum (D, p) f - k) < e))"
by (import transc Dint)
-lemma DIVISION_0: "ALL (a::real) b::real. a = b --> dsize (%n::nat. if n = 0 then a else b) = 0"
+lemma DIVISION_0: "a = b ==> dsize (%n. if n = 0 then a else b) = 0"
by (import transc DIVISION_0)
-lemma DIVISION_1: "ALL (a::real) b::real. a < b --> dsize (%n::nat. if n = 0 then a else b) = 1"
+lemma DIVISION_1: "a < b ==> dsize (%n. if n = 0 then a else b) = 1"
by (import transc DIVISION_1)
-lemma DIVISION_SINGLE: "ALL (a::real) b::real.
- a <= b --> division (a, b) (%n::nat. if n = 0 then a else b)"
+lemma DIVISION_SINGLE: "a <= b ==> division (a, b) (%n. if n = 0 then a else b)"
by (import transc DIVISION_SINGLE)
-lemma DIVISION_LHS: "ALL (D::nat => real) (a::real) b::real. division (a, b) D --> D 0 = a"
+lemma DIVISION_LHS: "division (a, b) D ==> D 0 = a"
by (import transc DIVISION_LHS)
-lemma DIVISION_THM: "(All::((nat => real) => bool) => bool)
- (%D::nat => real.
- (All::(real => bool) => bool)
- (%a::real.
- (All::(real => bool) => bool)
- (%b::real.
- (op =::bool => bool => bool)
- ((division::real * real => (nat => real) => bool)
- ((Pair::real => real => real * real) a b) D)
- ((op &::bool => bool => bool)
- ((op =::real => real => bool) (D (0::nat)) a)
- ((op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%n::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) n
- ((dsize::(nat => real) => nat) D))
- ((op <::real => real => bool) (D n)
- (D ((Suc::nat => nat) n)))))
- ((All::(nat => bool) => bool)
- (%n::nat.
- (op -->::bool => bool => bool)
- ((op <=::nat => nat => bool)
- ((dsize::(nat => real) => nat) D) n)
- ((op =::real => real => bool) (D n) b))))))))"
+lemma DIVISION_THM: "division (a, b) D =
+(D 0 = a & (ALL n<dsize D. D n < D (Suc n)) & (ALL n>=dsize D. D n = b))"
by (import transc DIVISION_THM)
-lemma DIVISION_RHS: "ALL (D::nat => real) (a::real) b::real.
- division (a, b) D --> D (dsize D) = b"
+lemma DIVISION_RHS: "division (a, b) D ==> D (dsize D) = b"
by (import transc DIVISION_RHS)
-lemma DIVISION_LT_GEN: "ALL (D::nat => real) (a::real) (b::real) (m::nat) n::nat.
- division (a, b) D & m < n & n <= dsize D --> D m < D n"
+lemma DIVISION_LT_GEN: "division (a, b) D & m < n & n <= dsize D ==> D m < D n"
by (import transc DIVISION_LT_GEN)
-lemma DIVISION_LT: "(All::((nat => real) => bool) => bool)
- (%D::nat => real.
- (All::(real => bool) => bool)
- (%a::real.
- (All::(real => bool) => bool)
- (%b::real.
- (op -->::bool => bool => bool)
- ((division::real * real => (nat => real) => bool)
- ((Pair::real => real => real * real) a b) D)
- ((All::(nat => bool) => bool)
- (%n::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) n
- ((dsize::(nat => real) => nat) D))
- ((op <::real => real => bool) (D (0::nat))
- (D ((Suc::nat => nat) n))))))))"
+lemma DIVISION_LT: "[| division (a, b) D; n < dsize D |] ==> D 0 < D (Suc n)"
by (import transc DIVISION_LT)
-lemma DIVISION_LE: "ALL (D::nat => real) (a::real) b::real. division (a, b) D --> a <= b"
+lemma DIVISION_LE: "division (a, b) D ==> a <= b"
by (import transc DIVISION_LE)
-lemma DIVISION_GT: "(All::((nat => real) => bool) => bool)
- (%D::nat => real.
- (All::(real => bool) => bool)
- (%a::real.
- (All::(real => bool) => bool)
- (%b::real.
- (op -->::bool => bool => bool)
- ((division::real * real => (nat => real) => bool)
- ((Pair::real => real => real * real) a b) D)
- ((All::(nat => bool) => bool)
- (%n::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) n
- ((dsize::(nat => real) => nat) D))
- ((op <::real => real => bool) (D n)
- (D ((dsize::(nat => real) => nat) D))))))))"
+lemma DIVISION_GT: "[| division (a, b) D; n < dsize D |] ==> D n < D (dsize D)"
by (import transc DIVISION_GT)
-lemma DIVISION_EQ: "ALL (D::nat => real) (a::real) b::real.
- division (a, b) D --> (a = b) = (dsize D = 0)"
+lemma DIVISION_EQ: "division (a, b) D ==> (a = b) = (dsize D = 0)"
by (import transc DIVISION_EQ)
-lemma DIVISION_LBOUND: "ALL (D::nat => real) (a::real) b::real.
- division (a, b) D --> (ALL r::nat. a <= D r)"
+lemma DIVISION_LBOUND: "division (a, b) D ==> a <= D r"
by (import transc DIVISION_LBOUND)
-lemma DIVISION_LBOUND_LT: "ALL (D::nat => real) (a::real) b::real.
- division (a, b) D & dsize D ~= 0 --> (ALL n::nat. a < D (Suc n))"
+lemma DIVISION_LBOUND_LT: "division (a, b) D & dsize D ~= 0 ==> a < D (Suc n)"
by (import transc DIVISION_LBOUND_LT)
-lemma DIVISION_UBOUND: "ALL (D::nat => real) (a::real) b::real.
- division (a, b) D --> (ALL r::nat. D r <= b)"
+lemma DIVISION_UBOUND: "division (a, b) D ==> D r <= b"
by (import transc DIVISION_UBOUND)
-lemma DIVISION_UBOUND_LT: "ALL (D::nat => real) (a::real) (b::real) n::nat.
- division (a, b) D & n < dsize D --> D n < b"
+lemma DIVISION_UBOUND_LT: "division (a, b) D & n < dsize D ==> D n < b"
by (import transc DIVISION_UBOUND_LT)
-lemma DIVISION_APPEND: "ALL (a::real) (b::real) c::real.
- (EX (D1::nat => real) p1::nat => real.
- tdiv (a, b) (D1, p1) & fine (g::real => real) (D1, p1)) &
- (EX (D2::nat => real) p2::nat => real.
- tdiv (b, c) (D2, p2) & fine g (D2, p2)) -->
- (EX (x::nat => real) p::nat => real. tdiv (a, c) (x, p) & fine g (x, p))"
+lemma DIVISION_APPEND: "(EX D1 p1. tdiv (a, b) (D1, p1) & fine g (D1, p1)) &
+(EX D2 p2. tdiv (b, c) (D2, p2) & fine g (D2, p2))
+==> EX x p. tdiv (a, c) (x, p) & fine g (x, p)"
by (import transc DIVISION_APPEND)
-lemma DIVISION_EXISTS: "ALL (a::real) (b::real) g::real => real.
- a <= b & gauge (%x::real. a <= x & x <= b) g -->
- (EX (D::nat => real) p::nat => real. tdiv (a, b) (D, p) & fine g (D, p))"
+lemma DIVISION_EXISTS: "a <= b & gauge (%x. a <= x & x <= b) g
+==> EX D p. tdiv (a, b) (D, p) & fine g (D, p)"
by (import transc DIVISION_EXISTS)
-lemma GAUGE_MIN: "ALL (E::real => bool) (g1::real => real) g2::real => real.
- gauge E g1 & gauge E g2 -->
- gauge E (%x::real. if g1 x < g2 x then g1 x else g2 x)"
+lemma GAUGE_MIN: "gauge E g1 & gauge E g2 ==> gauge E (%x. if g1 x < g2 x then g1 x else g2 x)"
by (import transc GAUGE_MIN)
-lemma FINE_MIN: "ALL (g1::real => real) (g2::real => real) (D::nat => real) p::nat => real.
- fine (%x::real. if g1 x < g2 x then g1 x else g2 x) (D, p) -->
- fine g1 (D, p) & fine g2 (D, p)"
+lemma FINE_MIN: "fine (%x. if g1 x < g2 x then g1 x else g2 x) (D, p)
+==> fine g1 (D, p) & fine g2 (D, p)"
by (import transc FINE_MIN)
-lemma DINT_UNIQ: "ALL (a::real) (b::real) (f::real => real) (k1::real) k2::real.
- a <= b & Dint (a, b) f k1 & Dint (a, b) f k2 --> k1 = k2"
+lemma DINT_UNIQ: "a <= b & Dint (a, b) f k1 & Dint (a, b) f k2 ==> k1 = k2"
by (import transc DINT_UNIQ)
-lemma INTEGRAL_NULL: "ALL (f::real => real) a::real. Dint (a, a) f 0"
+lemma INTEGRAL_NULL: "Dint (a, a) f 0"
by (import transc INTEGRAL_NULL)
-lemma FTC1: "ALL (f::real => real) (f'::real => real) (a::real) b::real.
- a <= b & (ALL x::real. a <= x & x <= b --> diffl f (f' x) x) -->
- Dint (a, b) f' (f b - f a)"
+lemma FTC1: "a <= b & (ALL x. a <= x & x <= b --> diffl f (f' x) x)
+==> Dint (a, b) f' (f b - f a)"
by (import transc FTC1)
-lemma MCLAURIN: "ALL (f::real => real) (diff::nat => real => real) (h::real) n::nat.
- 0 < h &
- 0 < n &
- diff 0 = f &
- (ALL (m::nat) t::real.
- m < n & 0 <= t & t <= h --> diffl (diff m) (diff (Suc m) t) t) -->
- (EX t>0.
+lemma MCLAURIN: "0 < h &
+0 < n &
+diff 0 = f &
+(ALL m t. m < n & 0 <= t & t <= h --> diffl (diff m) (diff (Suc m) t) t)
+==> EX t>0.
t < h &
f h =
- real.sum (0, n) (%m::nat. diff m 0 / real (FACT m) * h ^ m) +
- diff n t / real (FACT n) * h ^ n)"
+ real.sum (0, n) (%m. diff m 0 / real (FACT m) * h ^ m) +
+ diff n t / real (FACT n) * h ^ n"
by (import transc MCLAURIN)
-lemma MCLAURIN_NEG: "ALL (f::real => real) (diff::nat => real => real) (h::real) n::nat.
- h < 0 &
- 0 < n &
- diff 0 = f &
- (ALL (m::nat) t::real.
- m < n & h <= t & t <= 0 --> diffl (diff m) (diff (Suc m) t) t) -->
- (EX t::real.
- h < t &
+lemma MCLAURIN_NEG: "h < 0 &
+0 < n &
+diff 0 = f &
+(ALL m t. m < n & h <= t & t <= 0 --> diffl (diff m) (diff (Suc m) t) t)
+==> EX t>h.
t < 0 &
f h =
- real.sum (0, n) (%m::nat. diff m 0 / real (FACT m) * h ^ m) +
- diff n t / real (FACT n) * h ^ n)"
+ real.sum (0, n) (%m. diff m 0 / real (FACT m) * h ^ m) +
+ diff n t / real (FACT n) * h ^ n"
by (import transc MCLAURIN_NEG)
-lemma MCLAURIN_ALL_LT: "ALL (f::real => real) diff::nat => real => real.
- diff 0 = f &
- (ALL (m::nat) x::real. diffl (diff m) (diff (Suc m) x) x) -->
- (ALL (x::real) n::nat.
- x ~= 0 & 0 < n -->
- (EX t::real.
- 0 < abs t &
- abs t < abs x &
- f x =
- real.sum (0, n) (%m::nat. diff m 0 / real (FACT m) * x ^ m) +
- diff n t / real (FACT n) * x ^ n))"
+lemma MCLAURIN_ALL_LT: "[| diff 0 = f & (ALL m x. diffl (diff m) (diff (Suc m) x) x);
+ x ~= 0 & 0 < n |]
+==> EX t. 0 < abs t &
+ abs t < abs x &
+ f x =
+ real.sum (0, n) (%m. diff m 0 / real (FACT m) * x ^ m) +
+ diff n t / real (FACT n) * x ^ n"
by (import transc MCLAURIN_ALL_LT)
-lemma MCLAURIN_ZERO: "ALL (diff::nat => real => real) (n::nat) x::real.
- x = 0 & 0 < n -->
- real.sum (0, n) (%m::nat. diff m 0 / real (FACT m) * x ^ m) = diff 0 0"
+lemma MCLAURIN_ZERO: "(x::real) = (0::real) & (0::nat) < (n::nat)
+==> real.sum (0::nat, n)
+ (%m::nat.
+ (diff::nat => real => real) m (0::real) / real (FACT m) * x ^ m) =
+ diff (0::nat) (0::real)"
by (import transc MCLAURIN_ZERO)
-lemma MCLAURIN_ALL_LE: "ALL (f::real => real) diff::nat => real => real.
- diff 0 = f &
- (ALL (m::nat) x::real. diffl (diff m) (diff (Suc m) x) x) -->
- (ALL (x::real) n::nat.
- EX t::real.
- abs t <= abs x &
+lemma MCLAURIN_ALL_LE: "diff 0 = f & (ALL m x. diffl (diff m) (diff (Suc m) x) x)
+==> EX t. abs t <= abs x &
f x =
- real.sum (0, n) (%m::nat. diff m 0 / real (FACT m) * x ^ m) +
- diff n t / real (FACT n) * x ^ n)"
+ real.sum (0, n) (%m. diff m 0 / real (FACT m) * x ^ m) +
+ diff n t / real (FACT n) * x ^ n"
by (import transc MCLAURIN_ALL_LE)
-lemma MCLAURIN_EXP_LT: "ALL (x::real) n::nat.
- x ~= 0 & 0 < n -->
- (EX xa::real.
+lemma MCLAURIN_EXP_LT: "x ~= 0 & 0 < n
+==> EX xa.
0 < abs xa &
abs xa < abs x &
- exp x =
- real.sum (0, n) (%m::nat. x ^ m / real (FACT m)) +
- exp xa / real (FACT n) * x ^ n)"
+ transc.exp x =
+ real.sum (0, n) (%m. x ^ m / real (FACT m)) +
+ transc.exp xa / real (FACT n) * x ^ n"
by (import transc MCLAURIN_EXP_LT)
-lemma MCLAURIN_EXP_LE: "ALL (x::real) n::nat.
- EX xa::real.
- abs xa <= abs x &
- exp x =
- real.sum (0, n) (%m::nat. x ^ m / real (FACT m)) +
- exp xa / real (FACT n) * x ^ n"
+lemma MCLAURIN_EXP_LE: "EX xa.
+ abs xa <= abs x &
+ transc.exp x =
+ real.sum (0, n) (%m. x ^ m / real (FACT m)) +
+ transc.exp xa / real (FACT n) * x ^ n"
by (import transc MCLAURIN_EXP_LE)
-lemma DIFF_LN_COMPOSITE: "ALL (g::real => real) (m::real) x::real.
- diffl g m x & 0 < g x -->
- diffl (%x::real. ln (g x)) (inverse (g x) * m) x"
+lemma DIFF_LN_COMPOSITE: "diffl g m x & 0 < g x ==> diffl (%x. transc.ln (g x)) (inverse (g x) * m) x"
by (import transc DIFF_LN_COMPOSITE)
;end_setup
@@ -3200,15 +2558,14 @@
consts
poly :: "real list => real => real"
-specification (poly_primdef: poly) poly_def: "(ALL x::real. poly [] x = 0) &
-(ALL (h::real) (t::real list) x::real. poly (h # t) x = h + x * poly t x)"
+specification (poly_primdef: poly) poly_def: "(ALL x. poly [] x = 0) & (ALL h t x. poly (h # t) x = h + x * poly t x)"
by (import poly poly_def)
consts
poly_add :: "real list => real list => real list"
-specification (poly_add_primdef: poly_add) poly_add_def: "(ALL l2::real list. poly_add [] l2 = l2) &
-(ALL (h::real) (t::real list) l2::real list.
+specification (poly_add_primdef: poly_add) poly_add_def: "(ALL l2. poly_add [] l2 = l2) &
+(ALL h t l2.
poly_add (h # t) l2 =
(if l2 = [] then h # t else (h + hd l2) # poly_add t (tl l2)))"
by (import poly poly_add_def)
@@ -3216,8 +2573,7 @@
consts
"##" :: "real => real list => real list" ("##")
-specification ("##") poly_cmul_def: "(ALL c::real. ## c [] = []) &
-(ALL (c::real) (h::real) t::real list. ## c (h # t) = c * h # ## c t)"
+specification ("##") poly_cmul_def: "(ALL c. ## c [] = []) & (ALL c h t. ## c (h # t) = c * h # ## c t)"
by (import poly poly_cmul_def)
consts
@@ -3232,8 +2588,8 @@
consts
poly_mul :: "real list => real list => real list"
-specification (poly_mul_primdef: poly_mul) poly_mul_def: "(ALL l2::real list. poly_mul [] l2 = []) &
-(ALL (h::real) (t::real list) l2::real list.
+specification (poly_mul_primdef: poly_mul) poly_mul_def: "(ALL l2. poly_mul [] l2 = []) &
+(ALL h t l2.
poly_mul (h # t) l2 =
(if t = [] then ## h l2 else poly_add (## h l2) (0 # poly_mul t l2)))"
by (import poly poly_mul_def)
@@ -3241,514 +2597,351 @@
consts
poly_exp :: "real list => nat => real list"
-specification (poly_exp_primdef: poly_exp) poly_exp_def: "(ALL p::real list. poly_exp p 0 = [1]) &
-(ALL (p::real list) n::nat. poly_exp p (Suc n) = poly_mul p (poly_exp p n))"
+specification (poly_exp_primdef: poly_exp) poly_exp_def: "(ALL p. poly_exp p 0 = [1]) &
+(ALL p n. poly_exp p (Suc n) = poly_mul p (poly_exp p n))"
by (import poly poly_exp_def)
consts
poly_diff_aux :: "nat => real list => real list"
-specification (poly_diff_aux_primdef: poly_diff_aux) poly_diff_aux_def: "(ALL n::nat. poly_diff_aux n [] = []) &
-(ALL (n::nat) (h::real) t::real list.
- poly_diff_aux n (h # t) = real n * h # poly_diff_aux (Suc n) t)"
+specification (poly_diff_aux_primdef: poly_diff_aux) poly_diff_aux_def: "(ALL n. poly_diff_aux n [] = []) &
+(ALL n h t. poly_diff_aux n (h # t) = real n * h # poly_diff_aux (Suc n) t)"
by (import poly poly_diff_aux_def)
-definition diff :: "real list => real list" where
- "diff == %l::real list. if l = [] then [] else poly_diff_aux 1 (tl l)"
-
-lemma poly_diff_def: "ALL l::real list. diff l = (if l = [] then [] else poly_diff_aux 1 (tl l))"
+definition
+ diff :: "real list => real list" where
+ "diff == %l. if l = [] then [] else poly_diff_aux 1 (tl l)"
+
+lemma poly_diff_def: "diff l = (if l = [] then [] else poly_diff_aux 1 (tl l))"
by (import poly poly_diff_def)
-lemma POLY_ADD_CLAUSES: "poly_add [] (p2::real list) = p2 &
-poly_add (p1::real list) [] = p1 &
-poly_add ((h1::real) # (t1::real list)) ((h2::real) # (t2::real list)) =
-(h1 + h2) # poly_add t1 t2"
+lemma POLY_ADD_CLAUSES: "poly_add [] p2 = p2 &
+poly_add p1 [] = p1 &
+poly_add (h1 # t1) (h2 # t2) = (h1 + h2) # poly_add t1 t2"
by (import poly POLY_ADD_CLAUSES)
-lemma POLY_CMUL_CLAUSES: "## (c::real) [] = [] & ## c ((h::real) # (t::real list)) = c * h # ## c t"
+lemma POLY_CMUL_CLAUSES: "## c [] = [] & ## c (h # t) = c * h # ## c t"
by (import poly POLY_CMUL_CLAUSES)
-lemma POLY_NEG_CLAUSES: "poly_neg [] = [] & poly_neg ((h::real) # (t::real list)) = - h # poly_neg t"
+lemma POLY_NEG_CLAUSES: "poly_neg [] = [] & poly_neg (h # t) = - h # poly_neg t"
by (import poly POLY_NEG_CLAUSES)
-lemma POLY_MUL_CLAUSES: "poly_mul [] (p2::real list) = [] &
-poly_mul [h1::real] p2 = ## h1 p2 &
-poly_mul (h1 # (k1::real) # (t1::real list)) p2 =
-poly_add (## h1 p2) (0 # poly_mul (k1 # t1) p2)"
+lemma POLY_MUL_CLAUSES: "poly_mul [] p2 = [] &
+poly_mul [h1] p2 = ## h1 p2 &
+poly_mul (h1 # k1 # t1) p2 = poly_add (## h1 p2) (0 # poly_mul (k1 # t1) p2)"
by (import poly POLY_MUL_CLAUSES)
-lemma POLY_DIFF_CLAUSES: "diff [] = [] &
-diff [c::real] = [] & diff ((h::real) # (t::real list)) = poly_diff_aux 1 t"
+lemma POLY_DIFF_CLAUSES: "diff [] = [] & diff [c] = [] & diff (h # t) = poly_diff_aux 1 t"
by (import poly POLY_DIFF_CLAUSES)
-lemma POLY_ADD: "ALL (t::real list) (p2::real list) x::real.
- poly (poly_add t p2) x = poly t x + poly p2 x"
+lemma POLY_ADD: "poly (poly_add t p2) x = poly t x + poly p2 x"
by (import poly POLY_ADD)
-lemma POLY_CMUL: "ALL (t::real list) (c::real) x::real. poly (## c t) x = c * poly t x"
+lemma POLY_CMUL: "poly (## c t) x = c * poly t x"
by (import poly POLY_CMUL)
-lemma POLY_NEG: "ALL (x::real list) xa::real. poly (poly_neg x) xa = - poly x xa"
+lemma POLY_NEG: "poly (poly_neg x) xa = - poly x xa"
by (import poly POLY_NEG)
-lemma POLY_MUL: "ALL (x::real) (t::real list) p2::real list.
- poly (poly_mul t p2) x = poly t x * poly p2 x"
+lemma POLY_MUL: "poly (poly_mul t p2) x = poly t x * poly p2 x"
by (import poly POLY_MUL)
-lemma POLY_EXP: "ALL (p::real list) (n::nat) x::real. poly (poly_exp p n) x = poly p x ^ n"
+lemma POLY_EXP: "poly (poly_exp p n) x = poly p x ^ n"
by (import poly POLY_EXP)
-lemma POLY_DIFF_LEMMA: "ALL (t::real list) (n::nat) x::real.
- diffl (%x::real. x ^ Suc n * poly t x)
- (x ^ n * poly (poly_diff_aux (Suc n) t) x) x"
+lemma POLY_DIFF_LEMMA: "diffl (%x. x ^ Suc n * poly t x) (x ^ n * poly (poly_diff_aux (Suc n) t) x)
+ x"
by (import poly POLY_DIFF_LEMMA)
-lemma POLY_DIFF: "ALL (t::real list) x::real. diffl (poly t) (poly (diff t) x) x"
+lemma POLY_DIFF: "diffl (poly t) (poly (diff t) x) x"
by (import poly POLY_DIFF)
-lemma POLY_DIFFERENTIABLE: "ALL l::real list. All (differentiable (poly l))"
+lemma POLY_DIFFERENTIABLE: "lim.differentiable (poly l) x"
by (import poly POLY_DIFFERENTIABLE)
-lemma POLY_CONT: "ALL l::real list. All (contl (poly l))"
+lemma POLY_CONT: "contl (poly l) x"
by (import poly POLY_CONT)
-lemma POLY_IVT_POS: "ALL (x::real list) (xa::real) xb::real.
- xa < xb & poly x xa < 0 & 0 < poly x xb -->
- (EX xc::real. xa < xc & xc < xb & poly x xc = 0)"
+lemma POLY_IVT_POS: "xa < xb & poly x xa < 0 & 0 < poly x xb
+==> EX xc>xa. xc < xb & poly x xc = 0"
by (import poly POLY_IVT_POS)
-lemma POLY_IVT_NEG: "ALL (p::real list) (a::real) b::real.
- a < b & 0 < poly p a & poly p b < 0 -->
- (EX x::real. a < x & x < b & poly p x = 0)"
+lemma POLY_IVT_NEG: "a < b & 0 < poly p a & poly p b < 0 ==> EX x>a. x < b & poly p x = 0"
by (import poly POLY_IVT_NEG)
-lemma POLY_MVT: "ALL (p::real list) (a::real) b::real.
- a < b -->
- (EX x::real.
- a < x & x < b & poly p b - poly p a = (b - a) * poly (diff p) x)"
+lemma POLY_MVT: "a < b ==> EX x>a. x < b & poly p b - poly p a = (b - a) * poly (diff p) x"
by (import poly POLY_MVT)
-lemma POLY_ADD_RZERO: "ALL x::real list. poly (poly_add x []) = poly x"
+lemma POLY_ADD_RZERO: "poly (poly_add x []) = poly x"
by (import poly POLY_ADD_RZERO)
-lemma POLY_MUL_ASSOC: "ALL (x::real list) (xa::real list) xb::real list.
- poly (poly_mul x (poly_mul xa xb)) = poly (poly_mul (poly_mul x xa) xb)"
+lemma POLY_MUL_ASSOC: "poly (poly_mul x (poly_mul xa xb)) = poly (poly_mul (poly_mul x xa) xb)"
by (import poly POLY_MUL_ASSOC)
-lemma POLY_EXP_ADD: "ALL (x::nat) (xa::nat) xb::real list.
- poly (poly_exp xb (xa + x)) =
- poly (poly_mul (poly_exp xb xa) (poly_exp xb x))"
+lemma POLY_EXP_ADD: "poly (poly_exp xb (xa + x)) =
+poly (poly_mul (poly_exp xb xa) (poly_exp xb x))"
by (import poly POLY_EXP_ADD)
-lemma POLY_DIFF_AUX_ADD: "ALL (t::real list) (p2::real list) n::nat.
- poly (poly_diff_aux n (poly_add t p2)) =
- poly (poly_add (poly_diff_aux n t) (poly_diff_aux n p2))"
+lemma POLY_DIFF_AUX_ADD: "poly (poly_diff_aux n (poly_add t p2)) =
+poly (poly_add (poly_diff_aux n t) (poly_diff_aux n p2))"
by (import poly POLY_DIFF_AUX_ADD)
-lemma POLY_DIFF_AUX_CMUL: "ALL (t::real list) (c::real) n::nat.
- poly (poly_diff_aux n (## c t)) = poly (## c (poly_diff_aux n t))"
+lemma POLY_DIFF_AUX_CMUL: "poly (poly_diff_aux n (## c t)) = poly (## c (poly_diff_aux n t))"
by (import poly POLY_DIFF_AUX_CMUL)
-lemma POLY_DIFF_AUX_NEG: "ALL (x::real list) xa::nat.
- poly (poly_diff_aux xa (poly_neg x)) =
- poly (poly_neg (poly_diff_aux xa x))"
+lemma POLY_DIFF_AUX_NEG: "poly (poly_diff_aux xa (poly_neg x)) = poly (poly_neg (poly_diff_aux xa x))"
by (import poly POLY_DIFF_AUX_NEG)
-lemma POLY_DIFF_AUX_MUL_LEMMA: "ALL (t::real list) n::nat.
- poly (poly_diff_aux (Suc n) t) = poly (poly_add (poly_diff_aux n t) t)"
+lemma POLY_DIFF_AUX_MUL_LEMMA: "poly (poly_diff_aux (Suc n) t) = poly (poly_add (poly_diff_aux n t) t)"
by (import poly POLY_DIFF_AUX_MUL_LEMMA)
-lemma POLY_DIFF_ADD: "ALL (t::real list) p2::real list.
- poly (diff (poly_add t p2)) = poly (poly_add (diff t) (diff p2))"
+lemma POLY_DIFF_ADD: "poly (diff (poly_add t p2)) = poly (poly_add (diff t) (diff p2))"
by (import poly POLY_DIFF_ADD)
-lemma POLY_DIFF_CMUL: "ALL (t::real list) c::real. poly (diff (## c t)) = poly (## c (diff t))"
+lemma POLY_DIFF_CMUL: "poly (diff (## c t)) = poly (## c (diff t))"
by (import poly POLY_DIFF_CMUL)
-lemma POLY_DIFF_NEG: "ALL x::real list. poly (diff (poly_neg x)) = poly (poly_neg (diff x))"
+lemma POLY_DIFF_NEG: "poly (diff (poly_neg x)) = poly (poly_neg (diff x))"
by (import poly POLY_DIFF_NEG)
-lemma POLY_DIFF_MUL_LEMMA: "ALL (x::real list) xa::real.
- poly (diff (xa # x)) = poly (poly_add (0 # diff x) x)"
+lemma POLY_DIFF_MUL_LEMMA: "poly (diff (xa # x)) = poly (poly_add (0 # diff x) x)"
by (import poly POLY_DIFF_MUL_LEMMA)
-lemma POLY_DIFF_MUL: "ALL (t::real list) p2::real list.
- poly (diff (poly_mul t p2)) =
- poly (poly_add (poly_mul t (diff p2)) (poly_mul (diff t) p2))"
+lemma POLY_DIFF_MUL: "poly (diff (poly_mul t p2)) =
+poly (poly_add (poly_mul t (diff p2)) (poly_mul (diff t) p2))"
by (import poly POLY_DIFF_MUL)
-lemma POLY_DIFF_EXP: "ALL (p::real list) n::nat.
- poly (diff (poly_exp p (Suc n))) =
- poly (poly_mul (## (real (Suc n)) (poly_exp p n)) (diff p))"
+lemma POLY_DIFF_EXP: "poly (diff (poly_exp p (Suc n))) =
+poly (poly_mul (## (real (Suc n)) (poly_exp p n)) (diff p))"
by (import poly POLY_DIFF_EXP)
-lemma POLY_DIFF_EXP_PRIME: "ALL (n::nat) a::real.
- poly (diff (poly_exp [- a, 1] (Suc n))) =
- poly (## (real (Suc n)) (poly_exp [- a, 1] n))"
+lemma POLY_DIFF_EXP_PRIME: "poly (diff (poly_exp [- a, 1] (Suc n))) =
+poly (## (real (Suc n)) (poly_exp [- a, 1] n))"
by (import poly POLY_DIFF_EXP_PRIME)
-lemma POLY_LINEAR_REM: "ALL (t::real list) h::real.
- EX (q::real list) r::real.
- h # t = poly_add [r] (poly_mul [- (a::real), 1] q)"
+lemma POLY_LINEAR_REM: "EX q r. h # t = poly_add [r] (poly_mul [- a, 1] q)"
by (import poly POLY_LINEAR_REM)
-lemma POLY_LINEAR_DIVIDES: "ALL (a::real) t::real list.
- (poly t a = 0) = (t = [] | (EX q::real list. t = poly_mul [- a, 1] q))"
+lemma POLY_LINEAR_DIVIDES: "(poly t a = 0) = (t = [] | (EX q. t = poly_mul [- a, 1] q))"
by (import poly POLY_LINEAR_DIVIDES)
-lemma POLY_LENGTH_MUL: "ALL x::real list. length (poly_mul [- (a::real), 1] x) = Suc (length x)"
+lemma POLY_LENGTH_MUL: "length (poly_mul [- a, 1] x) = Suc (length x)"
by (import poly POLY_LENGTH_MUL)
-lemma POLY_ROOTS_INDEX_LEMMA: "(All::(nat => bool) => bool)
- (%n::nat.
- (All::(real list => bool) => bool)
- (%p::real list.
- (op -->::bool => bool => bool)
- ((op &::bool => bool => bool)
- ((Not::bool => bool)
- ((op =::(real => real) => (real => real) => bool)
- ((poly::real list => real => real) p)
- ((poly::real list => real => real) ([]::real list))))
- ((op =::nat => nat => bool) ((size::real list => nat) p) n))
- ((Ex::((nat => real) => bool) => bool)
- (%i::nat => real.
- (All::(real => bool) => bool)
- (%x::real.
- (op -->::bool => bool => bool)
- ((op =::real => real => bool)
- ((poly::real list => real => real) p x) (0::real))
- ((Ex::(nat => bool) => bool)
- (%m::nat.
- (op &::bool => bool => bool)
- ((op <=::nat => nat => bool) m n)
- ((op =::real => real => bool) x (i m)))))))))"
+lemma POLY_ROOTS_INDEX_LEMMA: "poly p ~= poly [] & length p = n
+==> EX i. ALL x. poly p x = 0 --> (EX m<=n. x = i m)"
by (import poly POLY_ROOTS_INDEX_LEMMA)
-lemma POLY_ROOTS_INDEX_LENGTH: "(All::(real list => bool) => bool)
- (%p::real list.
- (op -->::bool => bool => bool)
- ((Not::bool => bool)
- ((op =::(real => real) => (real => real) => bool)
- ((poly::real list => real => real) p)
- ((poly::real list => real => real) ([]::real list))))
- ((Ex::((nat => real) => bool) => bool)
- (%i::nat => real.
- (All::(real => bool) => bool)
- (%x::real.
- (op -->::bool => bool => bool)
- ((op =::real => real => bool)
- ((poly::real list => real => real) p x) (0::real))
- ((Ex::(nat => bool) => bool)
- (%n::nat.
- (op &::bool => bool => bool)
- ((op <=::nat => nat => bool) n
- ((size::real list => nat) p))
- ((op =::real => real => bool) x (i n))))))))"
+lemma POLY_ROOTS_INDEX_LENGTH: "poly p ~= poly []
+==> EX i. ALL x. poly p x = 0 --> (EX n<=length p. x = i n)"
by (import poly POLY_ROOTS_INDEX_LENGTH)
-lemma POLY_ROOTS_FINITE_LEMMA: "(All::(real list => bool) => bool)
- (%p::real list.
- (op -->::bool => bool => bool)
- ((Not::bool => bool)
- ((op =::(real => real) => (real => real) => bool)
- ((poly::real list => real => real) p)
- ((poly::real list => real => real) ([]::real list))))
- ((Ex::(nat => bool) => bool)
- (%N::nat.
- (Ex::((nat => real) => bool) => bool)
- (%i::nat => real.
- (All::(real => bool) => bool)
- (%x::real.
- (op -->::bool => bool => bool)
- ((op =::real => real => bool)
- ((poly::real list => real => real) p x) (0::real))
- ((Ex::(nat => bool) => bool)
- (%n::nat.
- (op &::bool => bool => bool)
- ((op <::nat => nat => bool) n N)
- ((op =::real => real => bool) x (i n)))))))))"
+lemma POLY_ROOTS_FINITE_LEMMA: "poly (p::real list) ~= poly []
+==> EX (N::nat) i::nat => real.
+ ALL x::real. poly p x = (0::real) --> (EX n<N. x = i n)"
by (import poly POLY_ROOTS_FINITE_LEMMA)
-lemma FINITE_LEMMA: "(All::((nat => real) => bool) => bool)
- (%i::nat => real.
- (All::(nat => bool) => bool)
- (%x::nat.
- (All::((real => bool) => bool) => bool)
- (%xa::real => bool.
- (op -->::bool => bool => bool)
- ((All::(real => bool) => bool)
- (%xb::real.
- (op -->::bool => bool => bool) (xa xb)
- ((Ex::(nat => bool) => bool)
- (%n::nat.
- (op &::bool => bool => bool)
- ((op <::nat => nat => bool) n x)
- ((op =::real => real => bool) xb (i n))))))
- ((Ex::(real => bool) => bool)
- (%a::real.
- (All::(real => bool) => bool)
- (%x::real.
- (op -->::bool => bool => bool) (xa x)
- ((op <::real => real => bool) x a)))))))"
+lemma FINITE_LEMMA: "(!!xb::real. (xa::real => bool) xb ==> EX n<x::nat. xb = (i::nat => real) n)
+==> EX a::real. ALL x::real. xa x --> x < a"
by (import poly FINITE_LEMMA)
-lemma POLY_ROOTS_FINITE: "(All::(real list => bool) => bool)
- (%p::real list.
- (op =::bool => bool => bool)
- ((Not::bool => bool)
- ((op =::(real => real) => (real => real) => bool)
- ((poly::real list => real => real) p)
- ((poly::real list => real => real) ([]::real list))))
- ((Ex::(nat => bool) => bool)
- (%N::nat.
- (Ex::((nat => real) => bool) => bool)
- (%i::nat => real.
- (All::(real => bool) => bool)
- (%x::real.
- (op -->::bool => bool => bool)
- ((op =::real => real => bool)
- ((poly::real list => real => real) p x) (0::real))
- ((Ex::(nat => bool) => bool)
- (%n::nat.
- (op &::bool => bool => bool)
- ((op <::nat => nat => bool) n N)
- ((op =::real => real => bool) x (i n)))))))))"
+lemma POLY_ROOTS_FINITE: "(poly (p::real list) ~= poly []) =
+(EX (N::nat) i::nat => real.
+ ALL x::real. poly p x = (0::real) --> (EX n<N. x = i n))"
by (import poly POLY_ROOTS_FINITE)
-lemma POLY_ENTIRE_LEMMA: "ALL (p::real list) q::real list.
- poly p ~= poly [] & poly q ~= poly [] --> poly (poly_mul p q) ~= poly []"
+lemma POLY_ENTIRE_LEMMA: "poly p ~= poly [] & poly q ~= poly [] ==> poly (poly_mul p q) ~= poly []"
by (import poly POLY_ENTIRE_LEMMA)
-lemma POLY_ENTIRE: "ALL (p::real list) q::real list.
- (poly (poly_mul p q) = poly []) = (poly p = poly [] | poly q = poly [])"
+lemma POLY_ENTIRE: "(poly (poly_mul p q) = poly []) = (poly p = poly [] | poly q = poly [])"
by (import poly POLY_ENTIRE)
-lemma POLY_MUL_LCANCEL: "ALL (x::real list) (xa::real list) xb::real list.
- (poly (poly_mul x xa) = poly (poly_mul x xb)) =
- (poly x = poly [] | poly xa = poly xb)"
+lemma POLY_MUL_LCANCEL: "(poly (poly_mul x xa) = poly (poly_mul x xb)) =
+(poly x = poly [] | poly xa = poly xb)"
by (import poly POLY_MUL_LCANCEL)
-lemma POLY_EXP_EQ_0: "ALL (p::real list) n::nat.
- (poly (poly_exp p n) = poly []) = (poly p = poly [] & n ~= 0)"
+lemma POLY_EXP_EQ_0: "(poly (poly_exp p n) = poly []) = (poly p = poly [] & n ~= 0)"
by (import poly POLY_EXP_EQ_0)
-lemma POLY_PRIME_EQ_0: "ALL a::real. poly [a, 1] ~= poly []"
+lemma POLY_PRIME_EQ_0: "poly [a, 1] ~= poly []"
by (import poly POLY_PRIME_EQ_0)
-lemma POLY_EXP_PRIME_EQ_0: "ALL (a::real) n::nat. poly (poly_exp [a, 1] n) ~= poly []"
+lemma POLY_EXP_PRIME_EQ_0: "poly (poly_exp [a, 1] n) ~= poly []"
by (import poly POLY_EXP_PRIME_EQ_0)
-lemma POLY_ZERO_LEMMA: "ALL (h::real) t::real list.
- poly (h # t) = poly [] --> h = 0 & poly t = poly []"
+lemma POLY_ZERO_LEMMA: "poly (h # t) = poly [] ==> h = 0 & poly t = poly []"
by (import poly POLY_ZERO_LEMMA)
-lemma POLY_ZERO: "ALL t::real list. (poly t = poly []) = list_all (%c::real. c = 0) t"
+lemma POLY_ZERO: "(poly t = poly []) = list_all (%c. c = 0) t"
by (import poly POLY_ZERO)
-lemma POLY_DIFF_AUX_ISZERO: "ALL (t::real list) n::nat.
- list_all (%c::real. c = 0) (poly_diff_aux (Suc n) t) =
- list_all (%c::real. c = 0) t"
+lemma POLY_DIFF_AUX_ISZERO: "list_all (%c. c = 0) (poly_diff_aux (Suc n) t) = list_all (%c. c = 0) t"
by (import poly POLY_DIFF_AUX_ISZERO)
-lemma POLY_DIFF_ISZERO: "ALL x::real list.
- poly (diff x) = poly [] --> (EX h::real. poly x = poly [h])"
+lemma POLY_DIFF_ISZERO: "poly (diff x) = poly [] ==> EX h. poly x = poly [h]"
by (import poly POLY_DIFF_ISZERO)
-lemma POLY_DIFF_ZERO: "ALL x::real list. poly x = poly [] --> poly (diff x) = poly []"
+lemma POLY_DIFF_ZERO: "poly x = poly [] ==> poly (diff x) = poly []"
by (import poly POLY_DIFF_ZERO)
-lemma POLY_DIFF_WELLDEF: "ALL (p::real list) q::real list.
- poly p = poly q --> poly (diff p) = poly (diff q)"
+lemma POLY_DIFF_WELLDEF: "poly p = poly q ==> poly (diff p) = poly (diff q)"
by (import poly POLY_DIFF_WELLDEF)
-definition poly_divides :: "real list => real list => bool" where
- "poly_divides ==
-%(p1::real list) p2::real list.
- EX q::real list. poly p2 = poly (poly_mul p1 q)"
-
-lemma poly_divides: "ALL (p1::real list) p2::real list.
- poly_divides p1 p2 = (EX q::real list. poly p2 = poly (poly_mul p1 q))"
+definition
+ poly_divides :: "real list => real list => bool" where
+ "poly_divides == %p1 p2. EX q. poly p2 = poly (poly_mul p1 q)"
+
+lemma poly_divides: "poly_divides p1 p2 = (EX q. poly p2 = poly (poly_mul p1 q))"
by (import poly poly_divides)
-lemma POLY_PRIMES: "ALL (a::real) (p::real list) q::real list.
- poly_divides [a, 1] (poly_mul p q) =
- (poly_divides [a, 1] p | poly_divides [a, 1] q)"
+lemma POLY_PRIMES: "poly_divides [a, 1] (poly_mul p q) =
+(poly_divides [a, 1] p | poly_divides [a, 1] q)"
by (import poly POLY_PRIMES)
-lemma POLY_DIVIDES_REFL: "ALL p::real list. poly_divides p p"
+lemma POLY_DIVIDES_REFL: "poly_divides p p"
by (import poly POLY_DIVIDES_REFL)
-lemma POLY_DIVIDES_TRANS: "ALL (p::real list) (q::real list) r::real list.
- poly_divides p q & poly_divides q r --> poly_divides p r"
+lemma POLY_DIVIDES_TRANS: "poly_divides p q & poly_divides q r ==> poly_divides p r"
by (import poly POLY_DIVIDES_TRANS)
-lemma POLY_DIVIDES_EXP: "ALL (p::real list) (m::nat) n::nat.
- m <= n --> poly_divides (poly_exp p m) (poly_exp p n)"
+lemma POLY_DIVIDES_EXP: "m <= n ==> poly_divides (poly_exp p m) (poly_exp p n)"
by (import poly POLY_DIVIDES_EXP)
-lemma POLY_EXP_DIVIDES: "ALL (p::real list) (q::real list) (m::nat) n::nat.
- poly_divides (poly_exp p n) q & m <= n --> poly_divides (poly_exp p m) q"
+lemma POLY_EXP_DIVIDES: "poly_divides (poly_exp p n) q & m <= n ==> poly_divides (poly_exp p m) q"
by (import poly POLY_EXP_DIVIDES)
-lemma POLY_DIVIDES_ADD: "ALL (p::real list) (q::real list) r::real list.
- poly_divides p q & poly_divides p r --> poly_divides p (poly_add q r)"
+lemma POLY_DIVIDES_ADD: "poly_divides p q & poly_divides p r ==> poly_divides p (poly_add q r)"
by (import poly POLY_DIVIDES_ADD)
-lemma POLY_DIVIDES_SUB: "ALL (p::real list) (q::real list) r::real list.
- poly_divides p q & poly_divides p (poly_add q r) --> poly_divides p r"
+lemma POLY_DIVIDES_SUB: "poly_divides p q & poly_divides p (poly_add q r) ==> poly_divides p r"
by (import poly POLY_DIVIDES_SUB)
-lemma POLY_DIVIDES_SUB2: "ALL (p::real list) (q::real list) r::real list.
- poly_divides p r & poly_divides p (poly_add q r) --> poly_divides p q"
+lemma POLY_DIVIDES_SUB2: "poly_divides p r & poly_divides p (poly_add q r) ==> poly_divides p q"
by (import poly POLY_DIVIDES_SUB2)
-lemma POLY_DIVIDES_ZERO: "ALL (p::real list) q::real list. poly p = poly [] --> poly_divides q p"
+lemma POLY_DIVIDES_ZERO: "poly p = poly [] ==> poly_divides q p"
by (import poly POLY_DIVIDES_ZERO)
-lemma POLY_ORDER_EXISTS: "ALL (a::real) (d::nat) p::real list.
- length p = d & poly p ~= poly [] -->
- (EX x::nat.
- poly_divides (poly_exp [- a, 1] x) p &
- ~ poly_divides (poly_exp [- a, 1] (Suc x)) p)"
+lemma POLY_ORDER_EXISTS: "length p = d & poly p ~= poly []
+==> EX x. poly_divides (poly_exp [- a, 1] x) p &
+ ~ poly_divides (poly_exp [- a, 1] (Suc x)) p"
by (import poly POLY_ORDER_EXISTS)
-lemma POLY_ORDER: "ALL (p::real list) a::real.
- poly p ~= poly [] -->
- (EX! n::nat.
+lemma POLY_ORDER: "poly p ~= poly []
+==> EX! n.
poly_divides (poly_exp [- a, 1] n) p &
- ~ poly_divides (poly_exp [- a, 1] (Suc n)) p)"
+ ~ poly_divides (poly_exp [- a, 1] (Suc n)) p"
by (import poly POLY_ORDER)
-definition poly_order :: "real => real list => nat" where
+definition
+ poly_order :: "real => real list => nat" where
"poly_order ==
-%(a::real) p::real list.
- SOME n::nat.
- poly_divides (poly_exp [- a, 1] n) p &
- ~ poly_divides (poly_exp [- a, 1] (Suc n)) p"
-
-lemma poly_order: "ALL (a::real) p::real list.
- poly_order a p =
- (SOME n::nat.
- poly_divides (poly_exp [- a, 1] n) p &
- ~ poly_divides (poly_exp [- a, 1] (Suc n)) p)"
+%a p. SOME n.
+ poly_divides (poly_exp [- a, 1] n) p &
+ ~ poly_divides (poly_exp [- a, 1] (Suc n)) p"
+
+lemma poly_order: "poly_order a p =
+(SOME n.
+ poly_divides (poly_exp [- a, 1] n) p &
+ ~ poly_divides (poly_exp [- a, 1] (Suc n)) p)"
by (import poly poly_order)
-lemma ORDER: "ALL (p::real list) (a::real) n::nat.
- (poly_divides (poly_exp [- a, 1] n) p &
- ~ poly_divides (poly_exp [- a, 1] (Suc n)) p) =
- (n = poly_order a p & poly p ~= poly [])"
+lemma ORDER: "(poly_divides (poly_exp [- a, 1] n) p &
+ ~ poly_divides (poly_exp [- a, 1] (Suc n)) p) =
+(n = poly_order a p & poly p ~= poly [])"
by (import poly ORDER)
-lemma ORDER_THM: "ALL (p::real list) a::real.
- poly p ~= poly [] -->
- poly_divides (poly_exp [- a, 1] (poly_order a p)) p &
- ~ poly_divides (poly_exp [- a, 1] (Suc (poly_order a p))) p"
+lemma ORDER_THM: "poly p ~= poly []
+==> poly_divides (poly_exp [- a, 1] (poly_order a p)) p &
+ ~ poly_divides (poly_exp [- a, 1] (Suc (poly_order a p))) p"
by (import poly ORDER_THM)
-lemma ORDER_UNIQUE: "ALL (p::real list) (a::real) n::nat.
- poly p ~= poly [] &
- poly_divides (poly_exp [- a, 1] n) p &
- ~ poly_divides (poly_exp [- a, 1] (Suc n)) p -->
- n = poly_order a p"
+lemma ORDER_UNIQUE: "poly p ~= poly [] &
+poly_divides (poly_exp [- a, 1] n) p &
+~ poly_divides (poly_exp [- a, 1] (Suc n)) p
+==> n = poly_order a p"
by (import poly ORDER_UNIQUE)
-lemma ORDER_POLY: "ALL (p::real list) (q::real list) a::real.
- poly p = poly q --> poly_order a p = poly_order a q"
+lemma ORDER_POLY: "poly p = poly q ==> poly_order a p = poly_order a q"
by (import poly ORDER_POLY)
-lemma ORDER_ROOT: "ALL (p::real list) a::real.
- (poly p a = 0) = (poly p = poly [] | poly_order a p ~= 0)"
+lemma ORDER_ROOT: "(poly p a = 0) = (poly p = poly [] | poly_order a p ~= 0)"
by (import poly ORDER_ROOT)
-lemma ORDER_DIVIDES: "ALL (p::real list) (a::real) n::nat.
- poly_divides (poly_exp [- a, 1] n) p =
- (poly p = poly [] | n <= poly_order a p)"
+lemma ORDER_DIVIDES: "poly_divides (poly_exp [- a, 1] n) p =
+(poly p = poly [] | n <= poly_order a p)"
by (import poly ORDER_DIVIDES)
-lemma ORDER_DECOMP: "ALL (p::real list) a::real.
- poly p ~= poly [] -->
- (EX x::real list.
- poly p = poly (poly_mul (poly_exp [- a, 1] (poly_order a p)) x) &
- ~ poly_divides [- a, 1] x)"
+lemma ORDER_DECOMP: "poly p ~= poly []
+==> EX x. poly p = poly (poly_mul (poly_exp [- a, 1] (poly_order a p)) x) &
+ ~ poly_divides [- a, 1] x"
by (import poly ORDER_DECOMP)
-lemma ORDER_MUL: "ALL (a::real) (p::real list) q::real list.
- poly (poly_mul p q) ~= poly [] -->
- poly_order a (poly_mul p q) = poly_order a p + poly_order a q"
+lemma ORDER_MUL: "poly (poly_mul p q) ~= poly []
+==> poly_order a (poly_mul p q) = poly_order a p + poly_order a q"
by (import poly ORDER_MUL)
-lemma ORDER_DIFF: "ALL (p::real list) a::real.
- poly (diff p) ~= poly [] & poly_order a p ~= 0 -->
- poly_order a p = Suc (poly_order a (diff p))"
+lemma ORDER_DIFF: "poly (diff p) ~= poly [] & poly_order a p ~= 0
+==> poly_order a p = Suc (poly_order a (diff p))"
by (import poly ORDER_DIFF)
-lemma POLY_SQUAREFREE_DECOMP_ORDER: "ALL (p::real list) (q::real list) (d::real list) (e::real list)
- (r::real list) s::real list.
- poly (diff p) ~= poly [] &
- poly p = poly (poly_mul q d) &
- poly (diff p) = poly (poly_mul e d) &
- poly d = poly (poly_add (poly_mul r p) (poly_mul s (diff p))) -->
- (ALL a::real. poly_order a q = (if poly_order a p = 0 then 0 else 1))"
+lemma POLY_SQUAREFREE_DECOMP_ORDER: "poly (diff p) ~= poly [] &
+poly p = poly (poly_mul q d) &
+poly (diff p) = poly (poly_mul e d) &
+poly d = poly (poly_add (poly_mul r p) (poly_mul s (diff p)))
+==> poly_order a q = (if poly_order a p = 0 then 0 else 1)"
by (import poly POLY_SQUAREFREE_DECOMP_ORDER)
-definition rsquarefree :: "real list => bool" where
+definition
+ rsquarefree :: "real list => bool" where
"rsquarefree ==
-%p::real list.
- poly p ~= poly [] &
- (ALL a::real. poly_order a p = 0 | poly_order a p = 1)"
-
-lemma rsquarefree: "ALL p::real list.
- rsquarefree p =
- (poly p ~= poly [] &
- (ALL a::real. poly_order a p = 0 | poly_order a p = 1))"
+%p. poly p ~= poly [] & (ALL a. poly_order a p = 0 | poly_order a p = 1)"
+
+lemma rsquarefree: "rsquarefree p =
+(poly p ~= poly [] & (ALL a. poly_order a p = 0 | poly_order a p = 1))"
by (import poly rsquarefree)
-lemma RSQUAREFREE_ROOTS: "ALL p::real list.
- rsquarefree p = (ALL a::real. ~ (poly p a = 0 & poly (diff p) a = 0))"
+lemma RSQUAREFREE_ROOTS: "rsquarefree p = (ALL a. ~ (poly p a = 0 & poly (diff p) a = 0))"
by (import poly RSQUAREFREE_ROOTS)
-lemma RSQUAREFREE_DECOMP: "ALL (p::real list) a::real.
- rsquarefree p & poly p a = 0 -->
- (EX q::real list. poly p = poly (poly_mul [- a, 1] q) & poly q a ~= 0)"
+lemma RSQUAREFREE_DECOMP: "rsquarefree p & poly p a = 0
+==> EX q. poly p = poly (poly_mul [- a, 1] q) & poly q a ~= 0"
by (import poly RSQUAREFREE_DECOMP)
-lemma POLY_SQUAREFREE_DECOMP: "ALL (p::real list) (q::real list) (d::real list) (e::real list)
- (r::real list) s::real list.
- poly (diff p) ~= poly [] &
- poly p = poly (poly_mul q d) &
- poly (diff p) = poly (poly_mul e d) &
- poly d = poly (poly_add (poly_mul r p) (poly_mul s (diff p))) -->
- rsquarefree q & (ALL x::real. (poly q x = 0) = (poly p x = 0))"
+lemma POLY_SQUAREFREE_DECOMP: "poly (diff p) ~= poly [] &
+poly p = poly (poly_mul q d) &
+poly (diff p) = poly (poly_mul e d) &
+poly d = poly (poly_add (poly_mul r p) (poly_mul s (diff p)))
+==> rsquarefree q & (ALL x. (poly q x = 0) = (poly p x = 0))"
by (import poly POLY_SQUAREFREE_DECOMP)
consts
normalize :: "real list => real list"
specification (normalize) normalize: "normalize [] = [] &
-(ALL (h::real) t::real list.
+(ALL h t.
normalize (h # t) =
(if normalize t = [] then if h = 0 then [] else [h]
else h # normalize t))"
by (import poly normalize)
-lemma POLY_NORMALIZE: "ALL t::real list. poly (normalize t) = poly t"
+lemma POLY_NORMALIZE: "poly (normalize t) = poly t"
by (import poly POLY_NORMALIZE)
-definition degree :: "real list => nat" where
- "degree == %p::real list. PRE (length (normalize p))"
-
-lemma degree: "ALL p::real list. degree p = PRE (length (normalize p))"
+definition
+ degree :: "real list => nat" where
+ "degree == %p. PRE (length (normalize p))"
+
+lemma degree: "degree p = PRE (length (normalize p))"
by (import poly degree)
-lemma DEGREE_ZERO: "ALL p::real list. poly p = poly [] --> degree p = 0"
+lemma DEGREE_ZERO: "poly p = poly [] ==> degree p = 0"
by (import poly DEGREE_ZERO)
-lemma POLY_ROOTS_FINITE_SET: "ALL p::real list.
- poly p ~= poly [] --> FINITE (GSPEC (%x::real. (x, poly p x = 0)))"
+lemma POLY_ROOTS_FINITE_SET: "poly p ~= poly [] ==> FINITE (GSPEC (%x. (x, poly p x = 0)))"
by (import poly POLY_ROOTS_FINITE_SET)
-lemma POLY_MONO: "ALL (x::real) (k::real) xa::real list.
- abs x <= k --> abs (poly xa x) <= poly (map abs xa) k"
+lemma POLY_MONO: "abs x <= k ==> abs (poly xa x) <= poly (map abs xa) k"
by (import poly POLY_MONO)
;end_setup
--- a/src/HOL/Import/HOL/HOL4Vec.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Import/HOL/HOL4Vec.thy Fri Sep 09 06:47:14 2011 +0200
@@ -4,139 +4,96 @@
;setup_theory res_quan
-lemma RES_FORALL_CONJ_DIST: "ALL (P::'a::type => bool) (Q::'a::type => bool) R::'a::type => bool.
- RES_FORALL P (%i::'a::type. Q i & R i) =
- (RES_FORALL P Q & RES_FORALL P R)"
+lemma RES_FORALL_CONJ_DIST: "RES_FORALL P (%i. Q i & R i) = (RES_FORALL P Q & RES_FORALL P R)"
by (import res_quan RES_FORALL_CONJ_DIST)
-lemma RES_FORALL_DISJ_DIST: "ALL (P::'a::type => bool) (Q::'a::type => bool) R::'a::type => bool.
- RES_FORALL (%j::'a::type. P j | Q j) R =
- (RES_FORALL P R & RES_FORALL Q R)"
+lemma RES_FORALL_DISJ_DIST: "RES_FORALL (%j. P j | Q j) R = (RES_FORALL P R & RES_FORALL Q R)"
by (import res_quan RES_FORALL_DISJ_DIST)
-lemma RES_FORALL_UNIQUE: "ALL (x::'a::type => bool) xa::'a::type. RES_FORALL (op = xa) x = x xa"
+lemma RES_FORALL_UNIQUE: "RES_FORALL (op = xa) x = x xa"
by (import res_quan RES_FORALL_UNIQUE)
-lemma RES_FORALL_FORALL: "ALL (P::'a::type => bool) (R::'a::type => 'b::type => bool) x::'b::type.
- (ALL x::'b::type. RES_FORALL P (%i::'a::type. R i x)) =
- RES_FORALL P (%i::'a::type. All (R i))"
+lemma RES_FORALL_FORALL: "(ALL x::'b.
+ RES_FORALL (P::'a => bool) (%i::'a. (R::'a => 'b => bool) i x)) =
+RES_FORALL P (%i::'a. All (R i))"
by (import res_quan RES_FORALL_FORALL)
-lemma RES_FORALL_REORDER: "ALL (P::'a::type => bool) (Q::'b::type => bool)
- R::'a::type => 'b::type => bool.
- RES_FORALL P (%i::'a::type. RES_FORALL Q (R i)) =
- RES_FORALL Q (%j::'b::type. RES_FORALL P (%i::'a::type. R i j))"
+lemma RES_FORALL_REORDER: "RES_FORALL P (%i. RES_FORALL Q (R i)) =
+RES_FORALL Q (%j. RES_FORALL P (%i. R i j))"
by (import res_quan RES_FORALL_REORDER)
-lemma RES_FORALL_EMPTY: "All (RES_FORALL EMPTY)"
+lemma RES_FORALL_EMPTY: "RES_FORALL EMPTY x"
by (import res_quan RES_FORALL_EMPTY)
-lemma RES_FORALL_UNIV: "ALL p::'a::type => bool. RES_FORALL pred_set.UNIV p = All p"
+lemma RES_FORALL_UNIV: "RES_FORALL pred_set.UNIV p = All p"
by (import res_quan RES_FORALL_UNIV)
-lemma RES_FORALL_NULL: "ALL (p::'a::type => bool) m::bool.
- RES_FORALL p (%x::'a::type. m) = (p = EMPTY | m)"
+lemma RES_FORALL_NULL: "RES_FORALL p (%x. m) = (p = EMPTY | m)"
by (import res_quan RES_FORALL_NULL)
-lemma RES_EXISTS_DISJ_DIST: "ALL (P::'a::type => bool) (Q::'a::type => bool) R::'a::type => bool.
- RES_EXISTS P (%i::'a::type. Q i | R i) =
- (RES_EXISTS P Q | RES_EXISTS P R)"
+lemma RES_EXISTS_DISJ_DIST: "RES_EXISTS P (%i. Q i | R i) = (RES_EXISTS P Q | RES_EXISTS P R)"
by (import res_quan RES_EXISTS_DISJ_DIST)
-lemma RES_DISJ_EXISTS_DIST: "ALL (P::'a::type => bool) (Q::'a::type => bool) R::'a::type => bool.
- RES_EXISTS (%i::'a::type. P i | Q i) R =
- (RES_EXISTS P R | RES_EXISTS Q R)"
+lemma RES_DISJ_EXISTS_DIST: "RES_EXISTS (%i. P i | Q i) R = (RES_EXISTS P R | RES_EXISTS Q R)"
by (import res_quan RES_DISJ_EXISTS_DIST)
-lemma RES_EXISTS_EQUAL: "ALL (x::'a::type => bool) xa::'a::type. RES_EXISTS (op = xa) x = x xa"
+lemma RES_EXISTS_EQUAL: "RES_EXISTS (op = xa) x = x xa"
by (import res_quan RES_EXISTS_EQUAL)
-lemma RES_EXISTS_REORDER: "ALL (P::'a::type => bool) (Q::'b::type => bool)
- R::'a::type => 'b::type => bool.
- RES_EXISTS P (%i::'a::type. RES_EXISTS Q (R i)) =
- RES_EXISTS Q (%j::'b::type. RES_EXISTS P (%i::'a::type. R i j))"
+lemma RES_EXISTS_REORDER: "RES_EXISTS P (%i. RES_EXISTS Q (R i)) =
+RES_EXISTS Q (%j. RES_EXISTS P (%i. R i j))"
by (import res_quan RES_EXISTS_REORDER)
-lemma RES_EXISTS_EMPTY: "ALL p::'a::type => bool. ~ RES_EXISTS EMPTY p"
+lemma RES_EXISTS_EMPTY: "~ RES_EXISTS EMPTY p"
by (import res_quan RES_EXISTS_EMPTY)
-lemma RES_EXISTS_UNIV: "ALL p::'a::type => bool. RES_EXISTS pred_set.UNIV p = Ex p"
+lemma RES_EXISTS_UNIV: "RES_EXISTS pred_set.UNIV p = Ex p"
by (import res_quan RES_EXISTS_UNIV)
-lemma RES_EXISTS_NULL: "ALL (p::'a::type => bool) m::bool.
- RES_EXISTS p (%x::'a::type. m) = (p ~= EMPTY & m)"
+lemma RES_EXISTS_NULL: "RES_EXISTS p (%x. m) = (p ~= EMPTY & m)"
by (import res_quan RES_EXISTS_NULL)
-lemma RES_EXISTS_ALT: "ALL (p::'a::type => bool) m::'a::type => bool.
- RES_EXISTS p m = (IN (RES_SELECT p m) p & m (RES_SELECT p m))"
+lemma RES_EXISTS_ALT: "RES_EXISTS p m = (IN (RES_SELECT p m) p & m (RES_SELECT p m))"
by (import res_quan RES_EXISTS_ALT)
-lemma RES_EXISTS_UNIQUE_EMPTY: "ALL p::'a::type => bool. ~ RES_EXISTS_UNIQUE EMPTY p"
+lemma RES_EXISTS_UNIQUE_EMPTY: "~ RES_EXISTS_UNIQUE EMPTY p"
by (import res_quan RES_EXISTS_UNIQUE_EMPTY)
-lemma RES_EXISTS_UNIQUE_UNIV: "ALL p::'a::type => bool. RES_EXISTS_UNIQUE pred_set.UNIV p = Ex1 p"
+lemma RES_EXISTS_UNIQUE_UNIV: "RES_EXISTS_UNIQUE pred_set.UNIV p = Ex1 p"
by (import res_quan RES_EXISTS_UNIQUE_UNIV)
-lemma RES_EXISTS_UNIQUE_NULL: "ALL (p::'a::type => bool) m::bool.
- RES_EXISTS_UNIQUE p (%x::'a::type. m) =
- ((EX x::'a::type. p = INSERT x EMPTY) & m)"
+lemma RES_EXISTS_UNIQUE_NULL: "RES_EXISTS_UNIQUE p (%x. m) = ((EX x. p = INSERT x EMPTY) & m)"
by (import res_quan RES_EXISTS_UNIQUE_NULL)
-lemma RES_EXISTS_UNIQUE_ALT: "ALL (p::'a::type => bool) m::'a::type => bool.
- RES_EXISTS_UNIQUE p m =
- RES_EXISTS p
- (%x::'a::type. m x & RES_FORALL p (%y::'a::type. m y --> y = x))"
+lemma RES_EXISTS_UNIQUE_ALT: "RES_EXISTS_UNIQUE p m =
+RES_EXISTS p (%x. m x & RES_FORALL p (%y. m y --> y = x))"
by (import res_quan RES_EXISTS_UNIQUE_ALT)
-lemma RES_SELECT_EMPTY: "ALL p::'a::type => bool. RES_SELECT EMPTY p = (SOME x::'a::type. False)"
+lemma RES_SELECT_EMPTY: "RES_SELECT EMPTY p = (SOME x. False)"
by (import res_quan RES_SELECT_EMPTY)
-lemma RES_SELECT_UNIV: "ALL p::'a::type => bool. RES_SELECT pred_set.UNIV p = Eps p"
+lemma RES_SELECT_UNIV: "RES_SELECT pred_set.UNIV p = Eps p"
by (import res_quan RES_SELECT_UNIV)
-lemma RES_ABSTRACT: "ALL (p::'a::type => bool) (m::'a::type => 'b::type) x::'a::type.
- IN x p --> RES_ABSTRACT p m x = m x"
+lemma RES_ABSTRACT: "IN x p ==> RES_ABSTRACT p m x = m x"
by (import res_quan RES_ABSTRACT)
-lemma RES_ABSTRACT_EQUAL: "ALL (p::'a::type => bool) (m1::'a::type => 'b::type)
- m2::'a::type => 'b::type.
- (ALL x::'a::type. IN x p --> m1 x = m2 x) -->
- RES_ABSTRACT p m1 = RES_ABSTRACT p m2"
+lemma RES_ABSTRACT_EQUAL: "(!!x. IN x p ==> m1 x = m2 x) ==> RES_ABSTRACT p m1 = RES_ABSTRACT p m2"
by (import res_quan RES_ABSTRACT_EQUAL)
-lemma RES_ABSTRACT_IDEMPOT: "ALL (p::'a::type => bool) m::'a::type => 'b::type.
- RES_ABSTRACT p (RES_ABSTRACT p m) = RES_ABSTRACT p m"
+lemma RES_ABSTRACT_IDEMPOT: "RES_ABSTRACT p (RES_ABSTRACT p m) = RES_ABSTRACT p m"
by (import res_quan RES_ABSTRACT_IDEMPOT)
-lemma RES_ABSTRACT_EQUAL_EQ: "ALL (p::'a::type => bool) (m1::'a::type => 'b::type)
- m2::'a::type => 'b::type.
- (RES_ABSTRACT p m1 = RES_ABSTRACT p m2) =
- (ALL x::'a::type. IN x p --> m1 x = m2 x)"
+lemma RES_ABSTRACT_EQUAL_EQ: "(RES_ABSTRACT p m1 = RES_ABSTRACT p m2) = (ALL x. IN x p --> m1 x = m2 x)"
by (import res_quan RES_ABSTRACT_EQUAL_EQ)
;end_setup
;setup_theory word_base
-typedef (open) ('a) word = "(Collect::('a::type list recspace => bool) => 'a::type list recspace set)
- (%x::'a::type list recspace.
- (All::(('a::type list recspace => bool) => bool) => bool)
- (%word::'a::type list recspace => bool.
- (op -->::bool => bool => bool)
- ((All::('a::type list recspace => bool) => bool)
- (%a0::'a::type list recspace.
- (op -->::bool => bool => bool)
- ((Ex::('a::type list => bool) => bool)
- (%a::'a::type list.
- (op =::'a::type list recspace
- => 'a::type list recspace => bool)
- a0 ((CONSTR::nat
-=> 'a::type list
- => (nat => 'a::type list recspace) => 'a::type list recspace)
- (0::nat) a
- (%n::nat. BOTTOM::'a::type list recspace))))
- (word a0)))
- (word x)))"
+typedef (open) ('a) word = "{x. ALL word.
+ (ALL a0. (EX a. a0 = CONSTR 0 a (%n. BOTTOM)) --> word a0) -->
+ word x} :: ('a::type list recspace set)"
by (rule typedef_helper,import word_base word_TY_DEF)
lemmas word_TY_DEF = typedef_hol2hol4 [OF type_definition_word]
@@ -145,11 +102,11 @@
mk_word :: "'a list recspace => 'a word"
dest_word :: "'a word => 'a list recspace"
-specification (dest_word mk_word) word_repfns: "(ALL a::'a::type word. mk_word (dest_word a) = a) &
-(ALL r::'a::type list recspace.
- (ALL word::'a::type list recspace => bool.
- (ALL a0::'a::type list recspace.
- (EX a::'a::type list. a0 = CONSTR 0 a (%n::nat. BOTTOM)) -->
+specification (dest_word mk_word) word_repfns: "(ALL a::'a word. mk_word (dest_word a) = a) &
+(ALL r::'a list recspace.
+ (ALL word::'a list recspace => bool.
+ (ALL a0::'a list recspace.
+ (EX a::'a list. a0 = CONSTR (0::nat) a (%n::nat. BOTTOM)) -->
word a0) -->
word r) =
(dest_word (mk_word r) = r))"
@@ -159,12 +116,13 @@
word_base0 :: "'a list => 'a word"
defs
- word_base0_primdef: "word_base0 == %a::'a::type list. mk_word (CONSTR 0 a (%n::nat. BOTTOM))"
+ word_base0_primdef: "word_base0 == %a. mk_word (CONSTR 0 a (%n. BOTTOM))"
-lemma word_base0_def: "word_base0 = (%a::'a::type list. mk_word (CONSTR 0 a (%n::nat. BOTTOM)))"
+lemma word_base0_def: "word_base0 = (%a. mk_word (CONSTR 0 a (%n. BOTTOM)))"
by (import word_base word_base0_def)
-definition WORD :: "'a list => 'a word" where
+definition
+ WORD :: "'a list => 'a word" where
"WORD == word_base0"
lemma WORD: "WORD = word_base0"
@@ -173,601 +131,350 @@
consts
word_case :: "('a list => 'b) => 'a word => 'b"
-specification (word_case_primdef: word_case) word_case_def: "ALL (f::'a::type list => 'b::type) a::'a::type list.
- word_case f (WORD a) = f a"
+specification (word_case_primdef: word_case) word_case_def: "ALL f a. word_base.word_case f (WORD a) = f a"
by (import word_base word_case_def)
consts
word_size :: "('a => nat) => 'a word => nat"
-specification (word_size_primdef: word_size) word_size_def: "ALL (f::'a::type => nat) a::'a::type list.
- word_size f (WORD a) = 1 + list_size f a"
+specification (word_size_primdef: word_size) word_size_def: "ALL f a. word_base.word_size f (WORD a) = 1 + HOL4Compat.list_size f a"
by (import word_base word_size_def)
-lemma word_11: "ALL (a::'a::type list) a'::'a::type list. (WORD a = WORD a') = (a = a')"
+lemma word_11: "(WORD a = WORD a') = (a = a')"
by (import word_base word_11)
-lemma word_case_cong: "ALL (M::'a::type word) (M'::'a::type word) f::'a::type list => 'b::type.
- M = M' &
- (ALL a::'a::type list.
- M' = WORD a --> f a = (f'::'a::type list => 'b::type) a) -->
- word_case f M = word_case f' M'"
+lemma word_case_cong: "M = M' & (ALL a. M' = WORD a --> f a = f' a)
+==> word_base.word_case f M = word_base.word_case f' M'"
by (import word_base word_case_cong)
-lemma word_nchotomy: "ALL x::'a::type word. EX l::'a::type list. x = WORD l"
+lemma word_nchotomy: "EX l. x = WORD l"
by (import word_base word_nchotomy)
-lemma word_Axiom: "ALL f::'a::type list => 'b::type.
- EX fn::'a::type word => 'b::type. ALL a::'a::type list. fn (WORD a) = f a"
+lemma word_Axiom: "EX fn. ALL a. fn (WORD a) = f a"
by (import word_base word_Axiom)
-lemma word_induction: "ALL P::'a::type word => bool. (ALL a::'a::type list. P (WORD a)) --> All P"
+lemma word_induction: "(!!a. P (WORD a)) ==> P x"
by (import word_base word_induction)
-lemma word_Ax: "ALL f::'a::type list => 'b::type.
- EX fn::'a::type word => 'b::type. ALL a::'a::type list. fn (WORD a) = f a"
+lemma word_Ax: "EX fn. ALL a. fn (WORD a) = f a"
by (import word_base word_Ax)
-lemma WORD_11: "ALL (x::'a::type list) xa::'a::type list. (WORD x = WORD xa) = (x = xa)"
+lemma WORD_11: "(WORD x = WORD xa) = (x = xa)"
by (import word_base WORD_11)
-lemma word_induct: "ALL x::'a::type word => bool. (ALL l::'a::type list. x (WORD l)) --> All x"
+lemma word_induct: "(!!l. x (WORD l)) ==> x xa"
by (import word_base word_induct)
-lemma word_cases: "ALL x::'a::type word. EX l::'a::type list. x = WORD l"
+lemma word_cases: "EX l. x = WORD l"
by (import word_base word_cases)
consts
WORDLEN :: "'a word => nat"
-specification (WORDLEN) WORDLEN_DEF: "ALL l::'a::type list. WORDLEN (WORD l) = length l"
+specification (WORDLEN) WORDLEN_DEF: "ALL l. WORDLEN (WORD l) = length l"
by (import word_base WORDLEN_DEF)
consts
PWORDLEN :: "nat => 'a word => bool"
defs
- PWORDLEN_primdef: "PWORDLEN == %n::nat. GSPEC (%w::'a::type word. (w, WORDLEN w = n))"
+ PWORDLEN_primdef: "PWORDLEN == %n. GSPEC (%w. (w, WORDLEN w = n))"
-lemma PWORDLEN_def: "ALL n::nat. PWORDLEN n = GSPEC (%w::'a::type word. (w, WORDLEN w = n))"
+lemma PWORDLEN_def: "PWORDLEN n = GSPEC (%w. (w, WORDLEN w = n))"
by (import word_base PWORDLEN_def)
-lemma IN_PWORDLEN: "ALL (n::nat) l::'a::type list. IN (WORD l) (PWORDLEN n) = (length l = n)"
+lemma IN_PWORDLEN: "IN (WORD l) (PWORDLEN n) = (length l = n)"
by (import word_base IN_PWORDLEN)
-lemma PWORDLEN: "ALL (n::nat) w::'a::type word. IN w (PWORDLEN n) = (WORDLEN w = n)"
+lemma PWORDLEN: "IN w (PWORDLEN n) = (WORDLEN w = n)"
by (import word_base PWORDLEN)
-lemma PWORDLEN0: "ALL w::'a::type word. IN w (PWORDLEN 0) --> w = WORD []"
+lemma PWORDLEN0: "IN w (PWORDLEN 0) ==> w = WORD []"
by (import word_base PWORDLEN0)
-lemma PWORDLEN1: "ALL x::'a::type. IN (WORD [x]) (PWORDLEN 1)"
+lemma PWORDLEN1: "IN (WORD [x]) (PWORDLEN 1)"
by (import word_base PWORDLEN1)
consts
WSEG :: "nat => nat => 'a word => 'a word"
-specification (WSEG) WSEG_DEF: "ALL (m::nat) (k::nat) l::'a::type list.
- WSEG m k (WORD l) = WORD (LASTN m (BUTLASTN k l))"
+specification (WSEG) WSEG_DEF: "ALL m k l. WSEG m k (WORD l) = WORD (LASTN m (BUTLASTN k l))"
by (import word_base WSEG_DEF)
-lemma WSEG0: "ALL (k::nat) w::'a::type word. WSEG 0 k w = WORD []"
+lemma WSEG0: "WSEG 0 k w = WORD []"
by (import word_base WSEG0)
-lemma WSEG_PWORDLEN: "ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w::'a::type word.
- ALL (m::nat) k::nat. m + k <= n --> IN (WSEG m k w) (PWORDLEN m))"
+lemma WSEG_PWORDLEN: "RES_FORALL (PWORDLEN n)
+ (%w. ALL m k. m + k <= n --> IN (WSEG m k w) (PWORDLEN m))"
by (import word_base WSEG_PWORDLEN)
-lemma WSEG_WORDLEN: "ALL x::nat.
- RES_FORALL (PWORDLEN x)
- (%xa::'a::type word.
- ALL (xb::nat) xc::nat.
- xb + xc <= x --> WORDLEN (WSEG xb xc xa) = xb)"
+lemma WSEG_WORDLEN: "RES_FORALL (PWORDLEN x)
+ (%xa. ALL xb xc. xb + xc <= x --> WORDLEN (WSEG xb xc xa) = xb)"
by (import word_base WSEG_WORDLEN)
-lemma WSEG_WORD_LENGTH: "ALL n::nat. RES_FORALL (PWORDLEN n) (%w::'a::type word. WSEG n 0 w = w)"
+lemma WSEG_WORD_LENGTH: "RES_FORALL (PWORDLEN n) (%w. WSEG n 0 w = w)"
by (import word_base WSEG_WORD_LENGTH)
consts
bit :: "nat => 'a word => 'a"
-specification (bit) BIT_DEF: "ALL (k::nat) l::'a::type list. bit k (WORD l) = ELL k l"
+specification (bit) BIT_DEF: "ALL k l. bit k (WORD l) = ELL k l"
by (import word_base BIT_DEF)
-lemma BIT0: "ALL x::'a::type. bit 0 (WORD [x]) = x"
+lemma BIT0: "bit 0 (WORD [x]) = x"
by (import word_base BIT0)
-lemma WSEG_BIT: "(All::(nat => bool) => bool)
- (%n::nat.
- (RES_FORALL::('a::type word => bool)
- => ('a::type word => bool) => bool)
- ((PWORDLEN::nat => 'a::type word => bool) n)
- (%w::'a::type word.
- (All::(nat => bool) => bool)
- (%k::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) k n)
- ((op =::'a::type word => 'a::type word => bool)
- ((WSEG::nat => nat => 'a::type word => 'a::type word)
- (1::nat) k w)
- ((WORD::'a::type list => 'a::type word)
- ((op #::'a::type => 'a::type list => 'a::type list)
- ((bit::nat => 'a::type word => 'a::type) k w)
- ([]::'a::type list)))))))"
+lemma WSEG_BIT: "RES_FORALL (PWORDLEN n) (%w. ALL k<n. WSEG 1 k w = WORD [bit k w])"
by (import word_base WSEG_BIT)
-lemma BIT_WSEG: "ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w::'a::type word.
- ALL (m::nat) (k::nat) j::nat.
- m + k <= n --> j < m --> bit j (WSEG m k w) = bit (j + k) w)"
+lemma BIT_WSEG: "RES_FORALL (PWORDLEN n)
+ (%w. ALL m k j.
+ m + k <= n --> j < m --> bit j (WSEG m k w) = bit (j + k) w)"
by (import word_base BIT_WSEG)
consts
MSB :: "'a word => 'a"
-specification (MSB) MSB_DEF: "ALL l::'a::type list. MSB (WORD l) = hd l"
+specification (MSB) MSB_DEF: "ALL l. MSB (WORD l) = hd l"
by (import word_base MSB_DEF)
-lemma MSB: "ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w::'a::type word. 0 < n --> MSB w = bit (PRE n) w)"
+lemma MSB: "RES_FORALL (PWORDLEN n) (%w. 0 < n --> MSB w = bit (PRE n) w)"
by (import word_base MSB)
consts
LSB :: "'a word => 'a"
-specification (LSB) LSB_DEF: "ALL l::'a::type list. LSB (WORD l) = last l"
+specification (LSB) LSB_DEF: "ALL l. LSB (WORD l) = last l"
by (import word_base LSB_DEF)
-lemma LSB: "ALL n::nat.
- RES_FORALL (PWORDLEN n) (%w::'a::type word. 0 < n --> LSB w = bit 0 w)"
+lemma LSB: "RES_FORALL (PWORDLEN n) (%w. 0 < n --> LSB w = bit 0 w)"
by (import word_base LSB)
consts
WSPLIT :: "nat => 'a word => 'a word * 'a word"
-specification (WSPLIT) WSPLIT_DEF: "ALL (m::nat) l::'a::type list.
- WSPLIT m (WORD l) = (WORD (BUTLASTN m l), WORD (LASTN m l))"
+specification (WSPLIT) WSPLIT_DEF: "ALL m l. WSPLIT m (WORD l) = (WORD (BUTLASTN m l), WORD (LASTN m l))"
by (import word_base WSPLIT_DEF)
consts
WCAT :: "'a word * 'a word => 'a word"
-specification (WCAT) WCAT_DEF: "ALL (l1::'a::type list) l2::'a::type list.
- WCAT (WORD l1, WORD l2) = WORD (l1 @ l2)"
+specification (WCAT) WCAT_DEF: "ALL l1 l2. WCAT (WORD l1, WORD l2) = WORD (l1 @ l2)"
by (import word_base WCAT_DEF)
-lemma WORD_PARTITION: "(op &::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%n::nat.
- (RES_FORALL::('a::type word => bool)
- => ('a::type word => bool) => bool)
- ((PWORDLEN::nat => 'a::type word => bool) n)
- (%w::'a::type word.
- (All::(nat => bool) => bool)
- (%m::nat.
- (op -->::bool => bool => bool)
- ((op <=::nat => nat => bool) m n)
- ((op =::'a::type word => 'a::type word => bool)
- ((WCAT::'a::type word * 'a::type word => 'a::type word)
- ((WSPLIT::nat
- => 'a::type word
- => 'a::type word * 'a::type word)
- m w))
- w)))))
- ((All::(nat => bool) => bool)
- (%n::nat.
- (All::(nat => bool) => bool)
- (%m::nat.
- (RES_FORALL::('a::type word => bool)
- => ('a::type word => bool) => bool)
- ((PWORDLEN::nat => 'a::type word => bool) n)
- (%w1::'a::type word.
- (RES_FORALL::('a::type word => bool)
- => ('a::type word => bool) => bool)
- ((PWORDLEN::nat => 'a::type word => bool) m)
- (%w2::'a::type word.
- (op =::'a::type word * 'a::type word
- => 'a::type word * 'a::type word => bool)
- ((WSPLIT::nat
- => 'a::type word
- => 'a::type word * 'a::type word)
- m ((WCAT::'a::type word * 'a::type word
- => 'a::type word)
- ((Pair::'a::type word
- => 'a::type word
- => 'a::type word * 'a::type word)
- w1 w2)))
- ((Pair::'a::type word
- => 'a::type word
- => 'a::type word * 'a::type word)
- w1 w2))))))"
+lemma WORD_PARTITION: "(ALL n::nat.
+ RES_FORALL (PWORDLEN n)
+ (%w::'a word. ALL m<=n. WCAT (WSPLIT m w) = w)) &
+(ALL (n::nat) m::nat.
+ RES_FORALL (PWORDLEN n)
+ (%w1::'a word.
+ RES_FORALL (PWORDLEN m)
+ (%w2::'a word. WSPLIT m (WCAT (w1, w2)) = (w1, w2))))"
by (import word_base WORD_PARTITION)
-lemma WCAT_ASSOC: "ALL (w1::'a::type word) (w2::'a::type word) w3::'a::type word.
- WCAT (w1, WCAT (w2, w3)) = WCAT (WCAT (w1, w2), w3)"
+lemma WCAT_ASSOC: "WCAT (w1, WCAT (w2, w3)) = WCAT (WCAT (w1, w2), w3)"
by (import word_base WCAT_ASSOC)
-lemma WCAT0: "ALL w::'a::type word. WCAT (WORD [], w) = w & WCAT (w, WORD []) = w"
+lemma WCAT0: "WCAT (WORD [], w) = w & WCAT (w, WORD []) = w"
by (import word_base WCAT0)
-lemma WCAT_11: "ALL (m::nat) n::nat.
- RES_FORALL (PWORDLEN m)
- (%wm1::'a::type word.
- RES_FORALL (PWORDLEN m)
- (%wm2::'a::type word.
- RES_FORALL (PWORDLEN n)
- (%wn1::'a::type word.
- RES_FORALL (PWORDLEN n)
- (%wn2::'a::type word.
- (WCAT (wm1, wn1) = WCAT (wm2, wn2)) =
- (wm1 = wm2 & wn1 = wn2)))))"
+lemma WCAT_11: "RES_FORALL (PWORDLEN m)
+ (%wm1. RES_FORALL (PWORDLEN m)
+ (%wm2. RES_FORALL (PWORDLEN n)
+ (%wn1. RES_FORALL (PWORDLEN n)
+ (%wn2. (WCAT (wm1, wn1) = WCAT (wm2, wn2)) =
+ (wm1 = wm2 & wn1 = wn2)))))"
by (import word_base WCAT_11)
-lemma WSPLIT_PWORDLEN: "(All::(nat => bool) => bool)
- (%n::nat.
- (RES_FORALL::('a::type word => bool)
- => ('a::type word => bool) => bool)
- ((PWORDLEN::nat => 'a::type word => bool) n)
- (%w::'a::type word.
- (All::(nat => bool) => bool)
- (%m::nat.
- (op -->::bool => bool => bool)
- ((op <=::nat => nat => bool) m n)
- ((op &::bool => bool => bool)
- ((IN::'a::type word => ('a::type word => bool) => bool)
- ((fst::'a::type word * 'a::type word => 'a::type word)
- ((WSPLIT::nat
- => 'a::type word
- => 'a::type word * 'a::type word)
- m w))
- ((PWORDLEN::nat => 'a::type word => bool)
- ((op -::nat => nat => nat) n m)))
- ((IN::'a::type word => ('a::type word => bool) => bool)
- ((snd::'a::type word * 'a::type word => 'a::type word)
- ((WSPLIT::nat
- => 'a::type word
- => 'a::type word * 'a::type word)
- m w))
- ((PWORDLEN::nat => 'a::type word => bool) m))))))"
+lemma WSPLIT_PWORDLEN: "RES_FORALL (PWORDLEN n)
+ (%w. ALL m<=n.
+ IN (fst (WSPLIT m w)) (PWORDLEN (n - m)) &
+ IN (snd (WSPLIT m w)) (PWORDLEN m))"
by (import word_base WSPLIT_PWORDLEN)
-lemma WCAT_PWORDLEN: "ALL n1::nat.
- RES_FORALL (PWORDLEN n1)
- (%w1::'a::type word.
- ALL n2::nat.
- RES_FORALL (PWORDLEN n2)
- (%w2::'a::type word. IN (WCAT (w1, w2)) (PWORDLEN (n1 + n2))))"
+lemma WCAT_PWORDLEN: "RES_FORALL (PWORDLEN n1)
+ (%w1. ALL n2.
+ RES_FORALL (PWORDLEN n2)
+ (%w2. IN (WCAT (w1, w2)) (PWORDLEN (n1 + n2))))"
by (import word_base WCAT_PWORDLEN)
-lemma WORDLEN_SUC_WCAT: "ALL (n::nat) w::'a::type word.
- IN w (PWORDLEN (Suc n)) -->
- RES_EXISTS (PWORDLEN 1)
- (%b::'a::type word.
- RES_EXISTS (PWORDLEN n) (%w'::'a::type word. w = WCAT (b, w')))"
+lemma WORDLEN_SUC_WCAT: "IN w (PWORDLEN (Suc n))
+==> RES_EXISTS (PWORDLEN 1)
+ (%b. RES_EXISTS (PWORDLEN n) (%w'. w = WCAT (b, w')))"
by (import word_base WORDLEN_SUC_WCAT)
-lemma WSEG_WSEG: "ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w::'a::type word.
- ALL (m1::nat) (k1::nat) (m2::nat) k2::nat.
- m1 + k1 <= n & m2 + k2 <= m1 -->
- WSEG m2 k2 (WSEG m1 k1 w) = WSEG m2 (k1 + k2) w)"
+lemma WSEG_WSEG: "RES_FORALL (PWORDLEN n)
+ (%w. ALL m1 k1 m2 k2.
+ m1 + k1 <= n & m2 + k2 <= m1 -->
+ WSEG m2 k2 (WSEG m1 k1 w) = WSEG m2 (k1 + k2) w)"
by (import word_base WSEG_WSEG)
-lemma WSPLIT_WSEG: "(All::(nat => bool) => bool)
- (%n::nat.
- (RES_FORALL::('a::type word => bool)
- => ('a::type word => bool) => bool)
- ((PWORDLEN::nat => 'a::type word => bool) n)
- (%w::'a::type word.
- (All::(nat => bool) => bool)
- (%k::nat.
- (op -->::bool => bool => bool)
- ((op <=::nat => nat => bool) k n)
- ((op =::'a::type word * 'a::type word
- => 'a::type word * 'a::type word => bool)
- ((WSPLIT::nat
- => 'a::type word
- => 'a::type word * 'a::type word)
- k w)
- ((Pair::'a::type word
- => 'a::type word => 'a::type word * 'a::type word)
- ((WSEG::nat => nat => 'a::type word => 'a::type word)
- ((op -::nat => nat => nat) n k) k w)
- ((WSEG::nat => nat => 'a::type word => 'a::type word) k
- (0::nat) w))))))"
+lemma WSPLIT_WSEG: "RES_FORALL (PWORDLEN n)
+ (%w. ALL k<=n. WSPLIT k w = (WSEG (n - k) k w, WSEG k 0 w))"
by (import word_base WSPLIT_WSEG)
-lemma WSPLIT_WSEG1: "(All::(nat => bool) => bool)
- (%n::nat.
- (RES_FORALL::('a::type word => bool)
- => ('a::type word => bool) => bool)
- ((PWORDLEN::nat => 'a::type word => bool) n)
- (%w::'a::type word.
- (All::(nat => bool) => bool)
- (%k::nat.
- (op -->::bool => bool => bool)
- ((op <=::nat => nat => bool) k n)
- ((op =::'a::type word => 'a::type word => bool)
- ((fst::'a::type word * 'a::type word => 'a::type word)
- ((WSPLIT::nat
- => 'a::type word
- => 'a::type word * 'a::type word)
- k w))
- ((WSEG::nat => nat => 'a::type word => 'a::type word)
- ((op -::nat => nat => nat) n k) k w)))))"
+lemma WSPLIT_WSEG1: "RES_FORALL (PWORDLEN n) (%w. ALL k<=n. fst (WSPLIT k w) = WSEG (n - k) k w)"
by (import word_base WSPLIT_WSEG1)
-lemma WSPLIT_WSEG2: "(All::(nat => bool) => bool)
- (%n::nat.
- (RES_FORALL::('a::type word => bool)
- => ('a::type word => bool) => bool)
- ((PWORDLEN::nat => 'a::type word => bool) n)
- (%w::'a::type word.
- (All::(nat => bool) => bool)
- (%k::nat.
- (op -->::bool => bool => bool)
- ((op <=::nat => nat => bool) k n)
- ((op =::'a::type word => 'a::type word => bool)
- ((snd::'a::type word * 'a::type word => 'a::type word)
- ((WSPLIT::nat
- => 'a::type word
- => 'a::type word * 'a::type word)
- k w))
- ((WSEG::nat => nat => 'a::type word => 'a::type word) k
- (0::nat) w)))))"
+lemma WSPLIT_WSEG2: "RES_FORALL (PWORDLEN n) (%w. ALL k<=n. snd (WSPLIT k w) = WSEG k 0 w)"
by (import word_base WSPLIT_WSEG2)
-lemma WCAT_WSEG_WSEG: "ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w::'a::type word.
- ALL (m1::nat) (m2::nat) k::nat.
- m1 + (m2 + k) <= n -->
- WCAT (WSEG m2 (m1 + k) w, WSEG m1 k w) = WSEG (m1 + m2) k w)"
+lemma WCAT_WSEG_WSEG: "RES_FORALL (PWORDLEN n)
+ (%w. ALL m1 m2 k.
+ m1 + (m2 + k) <= n -->
+ WCAT (WSEG m2 (m1 + k) w, WSEG m1 k w) = WSEG (m1 + m2) k w)"
by (import word_base WCAT_WSEG_WSEG)
-lemma WORD_SPLIT: "ALL (x::nat) xa::nat.
- RES_FORALL (PWORDLEN (x + xa))
- (%w::'a::type word. w = WCAT (WSEG x xa w, WSEG xa 0 w))"
+lemma WORD_SPLIT: "RES_FORALL (PWORDLEN (x + xa)) (%w. w = WCAT (WSEG x xa w, WSEG xa 0 w))"
by (import word_base WORD_SPLIT)
-lemma WORDLEN_SUC_WCAT_WSEG_WSEG: "RES_FORALL (PWORDLEN (Suc (n::nat)))
- (%w::'a::type word. w = WCAT (WSEG 1 n w, WSEG n 0 w))"
+lemma WORDLEN_SUC_WCAT_WSEG_WSEG: "RES_FORALL (PWORDLEN (Suc n)) (%w. w = WCAT (WSEG 1 n w, WSEG n 0 w))"
by (import word_base WORDLEN_SUC_WCAT_WSEG_WSEG)
-lemma WORDLEN_SUC_WCAT_WSEG_WSEG_RIGHT: "RES_FORALL (PWORDLEN (Suc (n::nat)))
- (%w::'a::type word. w = WCAT (WSEG n 1 w, WSEG 1 0 w))"
+lemma WORDLEN_SUC_WCAT_WSEG_WSEG_RIGHT: "RES_FORALL (PWORDLEN (Suc n)) (%w. w = WCAT (WSEG n 1 w, WSEG 1 0 w))"
by (import word_base WORDLEN_SUC_WCAT_WSEG_WSEG_RIGHT)
-lemma WORDLEN_SUC_WCAT_BIT_WSEG: "ALL n::nat.
- RES_FORALL (PWORDLEN (Suc n))
- (%w::'a::type word. w = WCAT (WORD [bit n w], WSEG n 0 w))"
+lemma WORDLEN_SUC_WCAT_BIT_WSEG: "RES_FORALL (PWORDLEN (Suc n)) (%w. w = WCAT (WORD [bit n w], WSEG n 0 w))"
by (import word_base WORDLEN_SUC_WCAT_BIT_WSEG)
-lemma WORDLEN_SUC_WCAT_BIT_WSEG_RIGHT: "ALL n::nat.
- RES_FORALL (PWORDLEN (Suc n))
- (%w::'a::type word. w = WCAT (WSEG n 1 w, WORD [bit 0 w]))"
+lemma WORDLEN_SUC_WCAT_BIT_WSEG_RIGHT: "RES_FORALL (PWORDLEN (Suc n)) (%w. w = WCAT (WSEG n 1 w, WORD [bit 0 w]))"
by (import word_base WORDLEN_SUC_WCAT_BIT_WSEG_RIGHT)
-lemma WSEG_WCAT1: "ALL (n1::nat) n2::nat.
- RES_FORALL (PWORDLEN n1)
- (%w1::'a::type word.
- RES_FORALL (PWORDLEN n2)
- (%w2::'a::type word. WSEG n1 n2 (WCAT (w1, w2)) = w1))"
+lemma WSEG_WCAT1: "RES_FORALL (PWORDLEN n1)
+ (%w1. RES_FORALL (PWORDLEN n2) (%w2. WSEG n1 n2 (WCAT (w1, w2)) = w1))"
by (import word_base WSEG_WCAT1)
-lemma WSEG_WCAT2: "ALL (n1::nat) n2::nat.
- RES_FORALL (PWORDLEN n1)
- (%w1::'a::type word.
- RES_FORALL (PWORDLEN n2)
- (%w2::'a::type word. WSEG n2 0 (WCAT (w1, w2)) = w2))"
+lemma WSEG_WCAT2: "RES_FORALL (PWORDLEN n1)
+ (%w1. RES_FORALL (PWORDLEN n2) (%w2. WSEG n2 0 (WCAT (w1, w2)) = w2))"
by (import word_base WSEG_WCAT2)
-lemma WSEG_SUC: "ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w::'a::type word.
- ALL (k::nat) m1::nat.
- k + Suc m1 < n -->
- WSEG (Suc m1) k w = WCAT (WSEG 1 (k + m1) w, WSEG m1 k w))"
+lemma WSEG_SUC: "RES_FORALL (PWORDLEN n)
+ (%w. ALL k m1.
+ k + Suc m1 < n -->
+ WSEG (Suc m1) k w = WCAT (WSEG 1 (k + m1) w, WSEG m1 k w))"
by (import word_base WSEG_SUC)
-lemma WORD_CONS_WCAT: "ALL (x::'a::type) l::'a::type list. WORD (x # l) = WCAT (WORD [x], WORD l)"
+lemma WORD_CONS_WCAT: "WORD (x # l) = WCAT (WORD [x], WORD l)"
by (import word_base WORD_CONS_WCAT)
-lemma WORD_SNOC_WCAT: "ALL (l::'a::type list) x::'a::type.
- WORD (SNOC x l) = WCAT (WORD l, WORD [x])"
+lemma WORD_SNOC_WCAT: "WORD (SNOC x l) = WCAT (WORD l, WORD [x])"
by (import word_base WORD_SNOC_WCAT)
-lemma BIT_WCAT_FST: "ALL (n1::nat) n2::nat.
- RES_FORALL (PWORDLEN n1)
- (%w1::'a::type word.
- RES_FORALL (PWORDLEN n2)
- (%w2::'a::type word.
- ALL k::nat.
- n2 <= k & k < n1 + n2 -->
- bit k (WCAT (w1, w2)) = bit (k - n2) w1))"
+lemma BIT_WCAT_FST: "RES_FORALL (PWORDLEN n1)
+ (%w1. RES_FORALL (PWORDLEN n2)
+ (%w2. ALL k.
+ n2 <= k & k < n1 + n2 -->
+ bit k (WCAT (w1, w2)) = bit (k - n2) w1))"
by (import word_base BIT_WCAT_FST)
-lemma BIT_WCAT_SND: "(All::(nat => bool) => bool)
- (%n1::nat.
- (All::(nat => bool) => bool)
- (%n2::nat.
- (RES_FORALL::('a::type word => bool)
- => ('a::type word => bool) => bool)
- ((PWORDLEN::nat => 'a::type word => bool) n1)
- (%w1::'a::type word.
- (RES_FORALL::('a::type word => bool)
- => ('a::type word => bool) => bool)
- ((PWORDLEN::nat => 'a::type word => bool) n2)
- (%w2::'a::type word.
- (All::(nat => bool) => bool)
- (%k::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) k n2)
- ((op =::'a::type => 'a::type => bool)
- ((bit::nat => 'a::type word => 'a::type) k
- ((WCAT::'a::type word * 'a::type word
-=> 'a::type word)
- ((Pair::'a::type word
- => 'a::type word => 'a::type word * 'a::type word)
- w1 w2)))
- ((bit::nat => 'a::type word => 'a::type) k
- w2)))))))"
+lemma BIT_WCAT_SND: "RES_FORALL (PWORDLEN n1)
+ (%w1. RES_FORALL (PWORDLEN n2)
+ (%w2. ALL k<n2. bit k (WCAT (w1, w2)) = bit k w2))"
by (import word_base BIT_WCAT_SND)
-lemma BIT_WCAT1: "ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w::'a::type word. ALL b::'a::type. bit n (WCAT (WORD [b], w)) = b)"
+lemma BIT_WCAT1: "RES_FORALL (PWORDLEN n) (%w. ALL b. bit n (WCAT (WORD [b], w)) = b)"
by (import word_base BIT_WCAT1)
-lemma WSEG_WCAT_WSEG1: "ALL (n1::nat) n2::nat.
- RES_FORALL (PWORDLEN n1)
- (%w1::'a::type word.
- RES_FORALL (PWORDLEN n2)
- (%w2::'a::type word.
- ALL (m::nat) k::nat.
- m <= n1 & n2 <= k -->
- WSEG m k (WCAT (w1, w2)) = WSEG m (k - n2) w1))"
+lemma WSEG_WCAT_WSEG1: "RES_FORALL (PWORDLEN n1)
+ (%w1. RES_FORALL (PWORDLEN n2)
+ (%w2. ALL m k.
+ m <= n1 & n2 <= k -->
+ WSEG m k (WCAT (w1, w2)) = WSEG m (k - n2) w1))"
by (import word_base WSEG_WCAT_WSEG1)
-lemma WSEG_WCAT_WSEG2: "ALL (n1::nat) n2::nat.
- RES_FORALL (PWORDLEN n1)
- (%w1::'a::type word.
- RES_FORALL (PWORDLEN n2)
- (%w2::'a::type word.
- ALL (m::nat) k::nat.
- m + k <= n2 --> WSEG m k (WCAT (w1, w2)) = WSEG m k w2))"
+lemma WSEG_WCAT_WSEG2: "RES_FORALL (PWORDLEN n1)
+ (%w1. RES_FORALL (PWORDLEN n2)
+ (%w2. ALL m k.
+ m + k <= n2 --> WSEG m k (WCAT (w1, w2)) = WSEG m k w2))"
by (import word_base WSEG_WCAT_WSEG2)
-lemma WSEG_WCAT_WSEG: "ALL (n1::nat) n2::nat.
- RES_FORALL (PWORDLEN n1)
- (%w1::'a::type word.
- RES_FORALL (PWORDLEN n2)
- (%w2::'a::type word.
- ALL (m::nat) k::nat.
- m + k <= n1 + n2 & k < n2 & n2 <= m + k -->
- WSEG m k (WCAT (w1, w2)) =
- WCAT (WSEG (m + k - n2) 0 w1, WSEG (n2 - k) k w2)))"
+lemma WSEG_WCAT_WSEG: "RES_FORALL (PWORDLEN n1)
+ (%w1. RES_FORALL (PWORDLEN n2)
+ (%w2. ALL m k.
+ m + k <= n1 + n2 & k < n2 & n2 <= m + k -->
+ WSEG m k (WCAT (w1, w2)) =
+ WCAT (WSEG (m + k - n2) 0 w1, WSEG (n2 - k) k w2)))"
by (import word_base WSEG_WCAT_WSEG)
-lemma BIT_EQ_IMP_WORD_EQ: "(All::(nat => bool) => bool)
- (%n::nat.
- (RES_FORALL::('a::type word => bool)
- => ('a::type word => bool) => bool)
- ((PWORDLEN::nat => 'a::type word => bool) n)
- (%w1::'a::type word.
- (RES_FORALL::('a::type word => bool)
- => ('a::type word => bool) => bool)
- ((PWORDLEN::nat => 'a::type word => bool) n)
- (%w2::'a::type word.
- (op -->::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%k::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) k n)
- ((op =::'a::type => 'a::type => bool)
- ((bit::nat => 'a::type word => 'a::type) k w1)
- ((bit::nat => 'a::type word => 'a::type) k w2))))
- ((op =::'a::type word => 'a::type word => bool) w1 w2))))"
+lemma BIT_EQ_IMP_WORD_EQ: "RES_FORALL (PWORDLEN n)
+ (%w1. RES_FORALL (PWORDLEN n)
+ (%w2. (ALL k<n. bit k w1 = bit k w2) --> w1 = w2))"
by (import word_base BIT_EQ_IMP_WORD_EQ)
;end_setup
;setup_theory word_num
-definition LVAL :: "('a => nat) => nat => 'a list => nat" where
- "LVAL ==
-%(f::'a::type => nat) b::nat. foldl (%(e::nat) x::'a::type. b * e + f x) 0"
+definition
+ LVAL :: "('a => nat) => nat => 'a list => nat" where
+ "LVAL == %f b. foldl (%e x. b * e + f x) 0"
-lemma LVAL_DEF: "ALL (f::'a::type => nat) (b::nat) l::'a::type list.
- LVAL f b l = foldl (%(e::nat) x::'a::type. b * e + f x) 0 l"
+lemma LVAL_DEF: "LVAL f b l = foldl (%e x. b * e + f x) 0 l"
by (import word_num LVAL_DEF)
consts
NVAL :: "('a => nat) => nat => 'a word => nat"
-specification (NVAL) NVAL_DEF: "ALL (f::'a::type => nat) (b::nat) l::'a::type list.
- NVAL f b (WORD l) = LVAL f b l"
+specification (NVAL) NVAL_DEF: "ALL f b l. NVAL f b (WORD l) = LVAL f b l"
by (import word_num NVAL_DEF)
-lemma LVAL: "(ALL (x::'a::type => nat) xa::nat. LVAL x xa [] = 0) &
-(ALL (x::'a::type list) (xa::'a::type => nat) (xb::nat) xc::'a::type.
+lemma LVAL: "(ALL (x::'a => nat) xa::nat. LVAL x xa [] = (0::nat)) &
+(ALL (x::'a list) (xa::'a => nat) (xb::nat) xc::'a.
LVAL xa xb (xc # x) = xa xc * xb ^ length x + LVAL xa xb x)"
by (import word_num LVAL)
-lemma LVAL_SNOC: "ALL (l::'a::type list) (h::'a::type) (f::'a::type => nat) b::nat.
- LVAL f b (SNOC h l) = LVAL f b l * b + f h"
+lemma LVAL_SNOC: "LVAL f b (SNOC h l) = LVAL f b l * b + f h"
by (import word_num LVAL_SNOC)
-lemma LVAL_MAX: "ALL (l::'a::type list) (f::'a::type => nat) b::nat.
- (ALL x::'a::type. f x < b) --> LVAL f b l < b ^ length l"
+lemma LVAL_MAX: "(!!x. f x < b) ==> LVAL f b l < b ^ length l"
by (import word_num LVAL_MAX)
-lemma NVAL_MAX: "ALL (f::'a::type => nat) b::nat.
- (ALL x::'a::type. f x < b) -->
- (ALL n::nat.
- RES_FORALL (PWORDLEN n) (%w::'a::type word. NVAL f b w < b ^ n))"
+lemma NVAL_MAX: "(!!x. f x < b) ==> RES_FORALL (PWORDLEN n) (%w. NVAL f b w < b ^ n)"
by (import word_num NVAL_MAX)
-lemma NVAL0: "ALL (x::'a::type => nat) xa::nat. NVAL x xa (WORD []) = 0"
+lemma NVAL0: "NVAL x xa (WORD []) = 0"
by (import word_num NVAL0)
-lemma NVAL1: "ALL (x::'a::type => nat) (xa::nat) xb::'a::type.
- NVAL x xa (WORD [xb]) = x xb"
+lemma NVAL1: "NVAL x xa (WORD [xb]) = x xb"
by (import word_num NVAL1)
-lemma NVAL_WORDLEN_0: "RES_FORALL (PWORDLEN 0)
- (%w::'a::type word. ALL (fv::'a::type => nat) r::nat. NVAL fv r w = 0)"
+lemma NVAL_WORDLEN_0: "RES_FORALL (PWORDLEN 0) (%w. ALL fv r. NVAL fv r w = 0)"
by (import word_num NVAL_WORDLEN_0)
-lemma NVAL_WCAT1: "ALL (w::'a::type word) (f::'a::type => nat) (b::nat) x::'a::type.
- NVAL f b (WCAT (w, WORD [x])) = NVAL f b w * b + f x"
+lemma NVAL_WCAT1: "NVAL f b (WCAT (w, WORD [x])) = NVAL f b w * b + f x"
by (import word_num NVAL_WCAT1)
-lemma NVAL_WCAT2: "ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w::'a::type word.
- ALL (f::'a::type => nat) (b::nat) x::'a::type.
- NVAL f b (WCAT (WORD [x], w)) = f x * b ^ n + NVAL f b w)"
+lemma NVAL_WCAT2: "RES_FORALL (PWORDLEN n)
+ (%w. ALL f b x. NVAL f b (WCAT (WORD [x], w)) = f x * b ^ n + NVAL f b w)"
by (import word_num NVAL_WCAT2)
-lemma NVAL_WCAT: "ALL (n::nat) m::nat.
- RES_FORALL (PWORDLEN n)
- (%w1::'a::type word.
- RES_FORALL (PWORDLEN m)
- (%w2::'a::type word.
- ALL (f::'a::type => nat) b::nat.
- NVAL f b (WCAT (w1, w2)) =
- NVAL f b w1 * b ^ m + NVAL f b w2))"
+lemma NVAL_WCAT: "RES_FORALL (PWORDLEN n)
+ (%w1. RES_FORALL (PWORDLEN m)
+ (%w2. ALL f b.
+ NVAL f b (WCAT (w1, w2)) =
+ NVAL f b w1 * b ^ m + NVAL f b w2))"
by (import word_num NVAL_WCAT)
consts
NLIST :: "nat => (nat => 'a) => nat => nat => 'a list"
-specification (NLIST) NLIST_DEF: "(ALL (frep::nat => 'a::type) (b::nat) m::nat. NLIST 0 frep b m = []) &
-(ALL (n::nat) (frep::nat => 'a::type) (b::nat) m::nat.
+specification (NLIST) NLIST_DEF: "(ALL (frep::nat => 'a) (b::nat) m::nat. NLIST (0::nat) frep b m = []) &
+(ALL (n::nat) (frep::nat => 'a) (b::nat) m::nat.
NLIST (Suc n) frep b m =
SNOC (frep (m mod b)) (NLIST n frep b (m div b)))"
by (import word_num NLIST_DEF)
-definition NWORD :: "nat => (nat => 'a) => nat => nat => 'a word" where
- "NWORD ==
-%(n::nat) (frep::nat => 'a::type) (b::nat) m::nat. WORD (NLIST n frep b m)"
+definition
+ NWORD :: "nat => (nat => 'a) => nat => nat => 'a word" where
+ "NWORD == %n frep b m. WORD (NLIST n frep b m)"
-lemma NWORD_DEF: "ALL (n::nat) (frep::nat => 'a::type) (b::nat) m::nat.
- NWORD n frep b m = WORD (NLIST n frep b m)"
+lemma NWORD_DEF: "NWORD n frep b m = WORD (NLIST n frep b m)"
by (import word_num NWORD_DEF)
-lemma NWORD_LENGTH: "ALL (x::nat) (xa::nat => 'a::type) (xb::nat) xc::nat.
- WORDLEN (NWORD x xa xb xc) = x"
+lemma NWORD_LENGTH: "WORDLEN (NWORD x xa xb xc) = x"
by (import word_num NWORD_LENGTH)
-lemma NWORD_PWORDLEN: "ALL (x::nat) (xa::nat => 'a::type) (xb::nat) xc::nat.
- IN (NWORD x xa xb xc) (PWORDLEN x)"
+lemma NWORD_PWORDLEN: "IN (NWORD x xa xb xc) (PWORDLEN x)"
by (import word_num NWORD_PWORDLEN)
;end_setup
@@ -780,79 +487,49 @@
defs
PBITOP_primdef: "PBITOP ==
GSPEC
- (%oper::'a::type word => 'b::type word.
+ (%oper.
(oper,
- ALL n::nat.
+ ALL n.
RES_FORALL (PWORDLEN n)
- (%w::'a::type word.
- IN (oper w) (PWORDLEN n) &
- (ALL (m::nat) k::nat.
- m + k <= n --> oper (WSEG m k w) = WSEG m k (oper w)))))"
+ (%w. IN (oper w) (PWORDLEN n) &
+ (ALL m k.
+ m + k <= n --> oper (WSEG m k w) = WSEG m k (oper w)))))"
lemma PBITOP_def: "PBITOP =
GSPEC
- (%oper::'a::type word => 'b::type word.
+ (%oper.
(oper,
- ALL n::nat.
+ ALL n.
RES_FORALL (PWORDLEN n)
- (%w::'a::type word.
- IN (oper w) (PWORDLEN n) &
- (ALL (m::nat) k::nat.
- m + k <= n --> oper (WSEG m k w) = WSEG m k (oper w)))))"
+ (%w. IN (oper w) (PWORDLEN n) &
+ (ALL m k.
+ m + k <= n --> oper (WSEG m k w) = WSEG m k (oper w)))))"
by (import word_bitop PBITOP_def)
-lemma IN_PBITOP: "ALL oper::'a::type word => 'b::type word.
- IN oper PBITOP =
- (ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w::'a::type word.
- IN (oper w) (PWORDLEN n) &
- (ALL (m::nat) k::nat.
- m + k <= n --> oper (WSEG m k w) = WSEG m k (oper w))))"
+lemma IN_PBITOP: "IN oper PBITOP =
+(ALL n.
+ RES_FORALL (PWORDLEN n)
+ (%w. IN (oper w) (PWORDLEN n) &
+ (ALL m k. m + k <= n --> oper (WSEG m k w) = WSEG m k (oper w))))"
by (import word_bitop IN_PBITOP)
lemma PBITOP_PWORDLEN: "RES_FORALL PBITOP
- (%oper::'a::type word => 'b::type word.
- ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w::'a::type word. IN (oper w) (PWORDLEN n)))"
+ (%oper. ALL n. RES_FORALL (PWORDLEN n) (%w. IN (oper w) (PWORDLEN n)))"
by (import word_bitop PBITOP_PWORDLEN)
lemma PBITOP_WSEG: "RES_FORALL PBITOP
- (%oper::'a::type word => 'b::type word.
- ALL n::nat.
+ (%oper.
+ ALL n.
RES_FORALL (PWORDLEN n)
- (%w::'a::type word.
- ALL (m::nat) k::nat.
- m + k <= n --> oper (WSEG m k w) = WSEG m k (oper w)))"
+ (%w. ALL m k.
+ m + k <= n --> oper (WSEG m k w) = WSEG m k (oper w)))"
by (import word_bitop PBITOP_WSEG)
-lemma PBITOP_BIT: "(RES_FORALL::(('a::type word => 'b::type word) => bool)
- => (('a::type word => 'b::type word) => bool) => bool)
- (PBITOP::('a::type word => 'b::type word) => bool)
- (%oper::'a::type word => 'b::type word.
- (All::(nat => bool) => bool)
- (%n::nat.
- (RES_FORALL::('a::type word => bool)
- => ('a::type word => bool) => bool)
- ((PWORDLEN::nat => 'a::type word => bool) n)
- (%w::'a::type word.
- (All::(nat => bool) => bool)
- (%k::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) k n)
- ((op =::'b::type word => 'b::type word => bool)
- (oper
- ((WORD::'a::type list => 'a::type word)
- ((op #::'a::type
- => 'a::type list => 'a::type list)
- ((bit::nat => 'a::type word => 'a::type) k w)
- ([]::'a::type list))))
- ((WORD::'b::type list => 'b::type word)
- ((op #::'b::type => 'b::type list => 'b::type list)
- ((bit::nat => 'b::type word => 'b::type) k
- (oper w))
- ([]::'b::type list))))))))"
+lemma PBITOP_BIT: "RES_FORALL PBITOP
+ (%oper.
+ ALL n.
+ RES_FORALL (PWORDLEN n)
+ (%w. ALL k<n. oper (WORD [bit k w]) = WORD [bit k (oper w)]))"
by (import word_bitop PBITOP_BIT)
consts
@@ -861,532 +538,383 @@
defs
PBITBOP_primdef: "PBITBOP ==
GSPEC
- (%oper::'a::type word => 'b::type word => 'c::type word.
+ (%oper.
(oper,
- ALL n::nat.
+ ALL n.
RES_FORALL (PWORDLEN n)
- (%w1::'a::type word.
- RES_FORALL (PWORDLEN n)
- (%w2::'b::type word.
- IN (oper w1 w2) (PWORDLEN n) &
- (ALL (m::nat) k::nat.
- m + k <= n -->
- oper (WSEG m k w1) (WSEG m k w2) =
- WSEG m k (oper w1 w2))))))"
+ (%w1. RES_FORALL (PWORDLEN n)
+ (%w2. IN (oper w1 w2) (PWORDLEN n) &
+ (ALL m k.
+ m + k <= n -->
+ oper (WSEG m k w1) (WSEG m k w2) =
+ WSEG m k (oper w1 w2))))))"
lemma PBITBOP_def: "PBITBOP =
GSPEC
- (%oper::'a::type word => 'b::type word => 'c::type word.
+ (%oper.
(oper,
- ALL n::nat.
+ ALL n.
RES_FORALL (PWORDLEN n)
- (%w1::'a::type word.
- RES_FORALL (PWORDLEN n)
- (%w2::'b::type word.
- IN (oper w1 w2) (PWORDLEN n) &
- (ALL (m::nat) k::nat.
- m + k <= n -->
- oper (WSEG m k w1) (WSEG m k w2) =
- WSEG m k (oper w1 w2))))))"
+ (%w1. RES_FORALL (PWORDLEN n)
+ (%w2. IN (oper w1 w2) (PWORDLEN n) &
+ (ALL m k.
+ m + k <= n -->
+ oper (WSEG m k w1) (WSEG m k w2) =
+ WSEG m k (oper w1 w2))))))"
by (import word_bitop PBITBOP_def)
-lemma IN_PBITBOP: "ALL oper::'a::type word => 'b::type word => 'c::type word.
- IN oper PBITBOP =
- (ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w1::'a::type word.
- RES_FORALL (PWORDLEN n)
- (%w2::'b::type word.
- IN (oper w1 w2) (PWORDLEN n) &
- (ALL (m::nat) k::nat.
- m + k <= n -->
- oper (WSEG m k w1) (WSEG m k w2) =
- WSEG m k (oper w1 w2)))))"
+lemma IN_PBITBOP: "IN oper PBITBOP =
+(ALL n.
+ RES_FORALL (PWORDLEN n)
+ (%w1. RES_FORALL (PWORDLEN n)
+ (%w2. IN (oper w1 w2) (PWORDLEN n) &
+ (ALL m k.
+ m + k <= n -->
+ oper (WSEG m k w1) (WSEG m k w2) =
+ WSEG m k (oper w1 w2)))))"
by (import word_bitop IN_PBITBOP)
lemma PBITBOP_PWORDLEN: "RES_FORALL PBITBOP
- (%oper::'a::type word => 'b::type word => 'c::type word.
- ALL n::nat.
+ (%oper.
+ ALL n.
RES_FORALL (PWORDLEN n)
- (%w1::'a::type word.
- RES_FORALL (PWORDLEN n)
- (%w2::'b::type word. IN (oper w1 w2) (PWORDLEN n))))"
+ (%w1. RES_FORALL (PWORDLEN n) (%w2. IN (oper w1 w2) (PWORDLEN n))))"
by (import word_bitop PBITBOP_PWORDLEN)
lemma PBITBOP_WSEG: "RES_FORALL PBITBOP
- (%oper::'a::type word => 'b::type word => 'c::type word.
- ALL n::nat.
+ (%oper.
+ ALL n.
RES_FORALL (PWORDLEN n)
- (%w1::'a::type word.
- RES_FORALL (PWORDLEN n)
- (%w2::'b::type word.
- ALL (m::nat) k::nat.
- m + k <= n -->
- oper (WSEG m k w1) (WSEG m k w2) =
- WSEG m k (oper w1 w2))))"
+ (%w1. RES_FORALL (PWORDLEN n)
+ (%w2. ALL m k.
+ m + k <= n -->
+ oper (WSEG m k w1) (WSEG m k w2) =
+ WSEG m k (oper w1 w2))))"
by (import word_bitop PBITBOP_WSEG)
-lemma PBITBOP_EXISTS: "ALL f::'a::type => 'b::type => 'c::type.
- EX x::'a::type word => 'b::type word => 'c::type word.
- ALL (l1::'a::type list) l2::'b::type list.
- x (WORD l1) (WORD l2) = WORD (map2 f l1 l2)"
+lemma PBITBOP_EXISTS: "EX x. ALL l1 l2. x (WORD l1) (WORD l2) = WORD (map2 f l1 l2)"
by (import word_bitop PBITBOP_EXISTS)
consts
WMAP :: "('a => 'b) => 'a word => 'b word"
-specification (WMAP) WMAP_DEF: "ALL (f::'a::type => 'b::type) l::'a::type list.
- WMAP f (WORD l) = WORD (map f l)"
+specification (WMAP) WMAP_DEF: "ALL f l. WMAP f (WORD l) = WORD (map f l)"
by (import word_bitop WMAP_DEF)
-lemma WMAP_PWORDLEN: "RES_FORALL (PWORDLEN (n::nat))
- (%w::'a::type word.
- ALL f::'a::type => 'b::type. IN (WMAP f w) (PWORDLEN n))"
+lemma WMAP_PWORDLEN: "RES_FORALL (PWORDLEN n) (%w. ALL f. IN (WMAP f w) (PWORDLEN n))"
by (import word_bitop WMAP_PWORDLEN)
-lemma WMAP_0: "ALL x::'a::type => 'b::type. WMAP x (WORD []) = WORD []"
+lemma WMAP_0: "WMAP (x::'a => 'b) (WORD []) = WORD []"
by (import word_bitop WMAP_0)
-lemma WMAP_BIT: "(All::(nat => bool) => bool)
- (%n::nat.
- (RES_FORALL::('a::type word => bool)
- => ('a::type word => bool) => bool)
- ((PWORDLEN::nat => 'a::type word => bool) n)
- (%w::'a::type word.
- (All::(nat => bool) => bool)
- (%k::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) k n)
- ((All::(('a::type => 'b::type) => bool) => bool)
- (%f::'a::type => 'b::type.
- (op =::'b::type => 'b::type => bool)
- ((bit::nat => 'b::type word => 'b::type) k
- ((WMAP::('a::type => 'b::type)
- => 'a::type word => 'b::type word)
- f w))
- (f ((bit::nat => 'a::type word => 'a::type) k
- w)))))))"
+lemma WMAP_BIT: "RES_FORALL (PWORDLEN n) (%w. ALL k<n. ALL f. bit k (WMAP f w) = f (bit k w))"
by (import word_bitop WMAP_BIT)
-lemma WMAP_WSEG: "ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w::'a::type word.
- ALL (m::nat) k::nat.
- m + k <= n -->
- (ALL f::'a::type => 'b::type.
- WMAP f (WSEG m k w) = WSEG m k (WMAP f w)))"
+lemma WMAP_WSEG: "RES_FORALL (PWORDLEN n)
+ (%w. ALL m k.
+ m + k <= n --> (ALL f. WMAP f (WSEG m k w) = WSEG m k (WMAP f w)))"
by (import word_bitop WMAP_WSEG)
-lemma WMAP_PBITOP: "ALL f::'a::type => 'b::type. IN (WMAP f) PBITOP"
+lemma WMAP_PBITOP: "IN (WMAP f) PBITOP"
by (import word_bitop WMAP_PBITOP)
-lemma WMAP_WCAT: "ALL (w1::'a::type word) (w2::'a::type word) f::'a::type => 'b::type.
- WMAP f (WCAT (w1, w2)) = WCAT (WMAP f w1, WMAP f w2)"
+lemma WMAP_WCAT: "WMAP (f::'a => 'b) (WCAT (w1::'a word, w2::'a word)) =
+WCAT (WMAP f w1, WMAP f w2)"
by (import word_bitop WMAP_WCAT)
-lemma WMAP_o: "ALL (w::'a::type word) (f::'a::type => 'b::type) g::'b::type => 'c::type.
- WMAP g (WMAP f w) = WMAP (g o f) w"
+lemma WMAP_o: "WMAP (g::'b => 'c) (WMAP (f::'a => 'b) (w::'a word)) = WMAP (g o f) w"
by (import word_bitop WMAP_o)
consts
FORALLBITS :: "('a => bool) => 'a word => bool"
-specification (FORALLBITS) FORALLBITS_DEF: "ALL (P::'a::type => bool) l::'a::type list.
- FORALLBITS P (WORD l) = list_all P l"
+specification (FORALLBITS) FORALLBITS_DEF: "ALL P l. FORALLBITS P (WORD l) = list_all P l"
by (import word_bitop FORALLBITS_DEF)
-lemma FORALLBITS: "(All::(nat => bool) => bool)
- (%n::nat.
- (RES_FORALL::('a::type word => bool)
- => ('a::type word => bool) => bool)
- ((PWORDLEN::nat => 'a::type word => bool) n)
- (%w::'a::type word.
- (All::(('a::type => bool) => bool) => bool)
- (%P::'a::type => bool.
- (op =::bool => bool => bool)
- ((FORALLBITS::('a::type => bool) => 'a::type word => bool) P
- w)
- ((All::(nat => bool) => bool)
- (%k::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) k n)
- (P ((bit::nat => 'a::type word => 'a::type) k
- w)))))))"
+lemma FORALLBITS: "RES_FORALL (PWORDLEN n) (%w. ALL P. FORALLBITS P w = (ALL k<n. P (bit k w)))"
by (import word_bitop FORALLBITS)
-lemma FORALLBITS_WSEG: "ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w::'a::type word.
- ALL P::'a::type => bool.
- FORALLBITS P w -->
- (ALL (m::nat) k::nat. m + k <= n --> FORALLBITS P (WSEG m k w)))"
+lemma FORALLBITS_WSEG: "RES_FORALL (PWORDLEN n)
+ (%w. ALL P.
+ FORALLBITS P w -->
+ (ALL m k. m + k <= n --> FORALLBITS P (WSEG m k w)))"
by (import word_bitop FORALLBITS_WSEG)
-lemma FORALLBITS_WCAT: "ALL (w1::'a::type word) (w2::'a::type word) P::'a::type => bool.
- FORALLBITS P (WCAT (w1, w2)) = (FORALLBITS P w1 & FORALLBITS P w2)"
+lemma FORALLBITS_WCAT: "FORALLBITS P (WCAT (w1, w2)) = (FORALLBITS P w1 & FORALLBITS P w2)"
by (import word_bitop FORALLBITS_WCAT)
consts
EXISTSABIT :: "('a => bool) => 'a word => bool"
-specification (EXISTSABIT) EXISTSABIT_DEF: "ALL (P::'a::type => bool) l::'a::type list.
- EXISTSABIT P (WORD l) = list_exists P l"
+specification (EXISTSABIT) EXISTSABIT_DEF: "ALL P l. EXISTSABIT P (WORD l) = list_ex P l"
by (import word_bitop EXISTSABIT_DEF)
-lemma NOT_EXISTSABIT: "ALL (P::'a::type => bool) w::'a::type word.
- (~ EXISTSABIT P w) = FORALLBITS (Not o P) w"
+lemma NOT_EXISTSABIT: "(~ EXISTSABIT P w) = FORALLBITS (Not o P) w"
by (import word_bitop NOT_EXISTSABIT)
-lemma NOT_FORALLBITS: "ALL (P::'a::type => bool) w::'a::type word.
- (~ FORALLBITS P w) = EXISTSABIT (Not o P) w"
+lemma NOT_FORALLBITS: "(~ FORALLBITS P w) = EXISTSABIT (Not o P) w"
by (import word_bitop NOT_FORALLBITS)
-lemma EXISTSABIT: "(All::(nat => bool) => bool)
- (%n::nat.
- (RES_FORALL::('a::type word => bool)
- => ('a::type word => bool) => bool)
- ((PWORDLEN::nat => 'a::type word => bool) n)
- (%w::'a::type word.
- (All::(('a::type => bool) => bool) => bool)
- (%P::'a::type => bool.
- (op =::bool => bool => bool)
- ((EXISTSABIT::('a::type => bool) => 'a::type word => bool) P
- w)
- ((Ex::(nat => bool) => bool)
- (%k::nat.
- (op &::bool => bool => bool)
- ((op <::nat => nat => bool) k n)
- (P ((bit::nat => 'a::type word => 'a::type) k
- w)))))))"
+lemma EXISTSABIT: "RES_FORALL (PWORDLEN n) (%w. ALL P. EXISTSABIT P w = (EX k<n. P (bit k w)))"
by (import word_bitop EXISTSABIT)
-lemma EXISTSABIT_WSEG: "ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w::'a::type word.
- ALL (m::nat) k::nat.
- m + k <= n -->
- (ALL P::'a::type => bool.
- EXISTSABIT P (WSEG m k w) --> EXISTSABIT P w))"
+lemma EXISTSABIT_WSEG: "RES_FORALL (PWORDLEN n)
+ (%w. ALL m k.
+ m + k <= n -->
+ (ALL P. EXISTSABIT P (WSEG m k w) --> EXISTSABIT P w))"
by (import word_bitop EXISTSABIT_WSEG)
-lemma EXISTSABIT_WCAT: "ALL (w1::'a::type word) (w2::'a::type word) P::'a::type => bool.
- EXISTSABIT P (WCAT (w1, w2)) = (EXISTSABIT P w1 | EXISTSABIT P w2)"
+lemma EXISTSABIT_WCAT: "EXISTSABIT P (WCAT (w1, w2)) = (EXISTSABIT P w1 | EXISTSABIT P w2)"
by (import word_bitop EXISTSABIT_WCAT)
-definition SHR :: "bool => 'a => 'a word => 'a word * 'a" where
+definition
+ SHR :: "bool => 'a => 'a word => 'a word * 'a" where
"SHR ==
-%(f::bool) (b::'a::type) w::'a::type word.
+%f b w.
(WCAT
(if f then WSEG 1 (PRE (WORDLEN w)) w else WORD [b],
WSEG (PRE (WORDLEN w)) 1 w),
bit 0 w)"
-lemma SHR_DEF: "ALL (f::bool) (b::'a::type) w::'a::type word.
- SHR f b w =
- (WCAT
- (if f then WSEG 1 (PRE (WORDLEN w)) w else WORD [b],
- WSEG (PRE (WORDLEN w)) 1 w),
- bit 0 w)"
+lemma SHR_DEF: "SHR f b w =
+(WCAT
+ (if f then WSEG 1 (PRE (WORDLEN w)) w else WORD [b],
+ WSEG (PRE (WORDLEN w)) 1 w),
+ bit 0 w)"
by (import word_bitop SHR_DEF)
-definition SHL :: "bool => 'a word => 'a => 'a * 'a word" where
+definition
+ SHL :: "bool => 'a word => 'a => 'a * 'a word" where
"SHL ==
-%(f::bool) (w::'a::type word) b::'a::type.
+%f w b.
(bit (PRE (WORDLEN w)) w,
WCAT (WSEG (PRE (WORDLEN w)) 0 w, if f then WSEG 1 0 w else WORD [b]))"
-lemma SHL_DEF: "ALL (f::bool) (w::'a::type word) b::'a::type.
- SHL f w b =
- (bit (PRE (WORDLEN w)) w,
- WCAT (WSEG (PRE (WORDLEN w)) 0 w, if f then WSEG 1 0 w else WORD [b]))"
+lemma SHL_DEF: "SHL f w b =
+(bit (PRE (WORDLEN w)) w,
+ WCAT (WSEG (PRE (WORDLEN w)) 0 w, if f then WSEG 1 0 w else WORD [b]))"
by (import word_bitop SHL_DEF)
-lemma SHR_WSEG: "ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w::'a::type word.
- ALL (m::nat) k::nat.
- m + k <= n -->
- 0 < m -->
- (ALL (f::bool) b::'a::type.
- SHR f b (WSEG m k w) =
- (if f
- then WCAT (WSEG 1 (k + (m - 1)) w, WSEG (m - 1) (k + 1) w)
- else WCAT (WORD [b], WSEG (m - 1) (k + 1) w),
- bit k w)))"
+lemma SHR_WSEG: "RES_FORALL (PWORDLEN n)
+ (%w. ALL m k.
+ m + k <= n -->
+ 0 < m -->
+ (ALL f b.
+ SHR f b (WSEG m k w) =
+ (if f
+ then WCAT (WSEG 1 (k + (m - 1)) w, WSEG (m - 1) (k + 1) w)
+ else WCAT (WORD [b], WSEG (m - 1) (k + 1) w),
+ bit k w)))"
by (import word_bitop SHR_WSEG)
-lemma SHR_WSEG_1F: "ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w::'a::type word.
- ALL (b::'a::type) (m::nat) k::nat.
- m + k <= n -->
- 0 < m -->
- SHR False b (WSEG m k w) =
- (WCAT (WORD [b], WSEG (m - 1) (k + 1) w), bit k w))"
+lemma SHR_WSEG_1F: "RES_FORALL (PWORDLEN n)
+ (%w. ALL b m k.
+ m + k <= n -->
+ 0 < m -->
+ SHR False b (WSEG m k w) =
+ (WCAT (WORD [b], WSEG (m - 1) (k + 1) w), bit k w))"
by (import word_bitop SHR_WSEG_1F)
-lemma SHR_WSEG_NF: "ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w::'a::type word.
- ALL (m::nat) k::nat.
- m + k < n -->
- 0 < m -->
- SHR False (bit (m + k) w) (WSEG m k w) =
- (WSEG m (k + 1) w, bit k w))"
+lemma SHR_WSEG_NF: "RES_FORALL (PWORDLEN n)
+ (%w. ALL m k.
+ m + k < n -->
+ 0 < m -->
+ SHR False (bit (m + k) w) (WSEG m k w) =
+ (WSEG m (k + 1) w, bit k w))"
by (import word_bitop SHR_WSEG_NF)
-lemma SHL_WSEG: "ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w::'a::type word.
- ALL (m::nat) k::nat.
- m + k <= n -->
- 0 < m -->
- (ALL (f::bool) b::'a::type.
- SHL f (WSEG m k w) b =
- (bit (k + (m - 1)) w,
- if f then WCAT (WSEG (m - 1) k w, WSEG 1 k w)
- else WCAT (WSEG (m - 1) k w, WORD [b]))))"
+lemma SHL_WSEG: "RES_FORALL (PWORDLEN n)
+ (%w. ALL m k.
+ m + k <= n -->
+ 0 < m -->
+ (ALL f b.
+ SHL f (WSEG m k w) b =
+ (bit (k + (m - 1)) w,
+ if f then WCAT (WSEG (m - 1) k w, WSEG 1 k w)
+ else WCAT (WSEG (m - 1) k w, WORD [b]))))"
by (import word_bitop SHL_WSEG)
-lemma SHL_WSEG_1F: "ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w::'a::type word.
- ALL (b::'a::type) (m::nat) k::nat.
- m + k <= n -->
- 0 < m -->
- SHL False (WSEG m k w) b =
- (bit (k + (m - 1)) w, WCAT (WSEG (m - 1) k w, WORD [b])))"
+lemma SHL_WSEG_1F: "RES_FORALL (PWORDLEN n)
+ (%w. ALL b m k.
+ m + k <= n -->
+ 0 < m -->
+ SHL False (WSEG m k w) b =
+ (bit (k + (m - 1)) w, WCAT (WSEG (m - 1) k w, WORD [b])))"
by (import word_bitop SHL_WSEG_1F)
-lemma SHL_WSEG_NF: "ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w::'a::type word.
- ALL (m::nat) k::nat.
- m + k <= n -->
- 0 < m -->
- 0 < k -->
- SHL False (WSEG m k w) (bit (k - 1) w) =
- (bit (k + (m - 1)) w, WSEG m (k - 1) w))"
+lemma SHL_WSEG_NF: "RES_FORALL (PWORDLEN n)
+ (%w. ALL m k.
+ m + k <= n -->
+ 0 < m -->
+ 0 < k -->
+ SHL False (WSEG m k w) (bit (k - 1) w) =
+ (bit (k + (m - 1)) w, WSEG m (k - 1) w))"
by (import word_bitop SHL_WSEG_NF)
-lemma WSEG_SHL: "ALL n::nat.
- RES_FORALL (PWORDLEN (Suc n))
- (%w::'a::type word.
- ALL (m::nat) k::nat.
- 0 < k & m + k <= Suc n -->
- (ALL b::'a::type.
- WSEG m k (snd (SHL (f::bool) w b)) = WSEG m (k - 1) w))"
+lemma WSEG_SHL: "RES_FORALL (PWORDLEN (Suc n))
+ (%w. ALL m k.
+ 0 < k & m + k <= Suc n -->
+ (ALL b. WSEG m k (snd (SHL f w b)) = WSEG m (k - 1) w))"
by (import word_bitop WSEG_SHL)
-lemma WSEG_SHL_0: "ALL n::nat.
- RES_FORALL (PWORDLEN (Suc n))
- (%w::'a::type word.
- ALL (m::nat) b::'a::type.
- 0 < m & m <= Suc n -->
- WSEG m 0 (snd (SHL (f::bool) w b)) =
- WCAT (WSEG (m - 1) 0 w, if f then WSEG 1 0 w else WORD [b]))"
+lemma WSEG_SHL_0: "RES_FORALL (PWORDLEN (Suc n))
+ (%w. ALL m b.
+ 0 < m & m <= Suc n -->
+ WSEG m 0 (snd (SHL f w b)) =
+ WCAT (WSEG (m - 1) 0 w, if f then WSEG 1 0 w else WORD [b]))"
by (import word_bitop WSEG_SHL_0)
;end_setup
;setup_theory bword_num
-definition BV :: "bool => nat" where
- "BV == %b::bool. if b then Suc 0 else 0"
+definition
+ BV :: "bool => nat" where
+ "BV == %b. if b then Suc 0 else 0"
-lemma BV_DEF: "ALL b::bool. BV b = (if b then Suc 0 else 0)"
+lemma BV_DEF: "BV b = (if b then Suc 0 else 0)"
by (import bword_num BV_DEF)
consts
BNVAL :: "bool word => nat"
-specification (BNVAL) BNVAL_DEF: "ALL l::bool list. BNVAL (WORD l) = LVAL BV 2 l"
+specification (BNVAL) BNVAL_DEF: "ALL l. BNVAL (WORD l) = LVAL BV 2 l"
by (import bword_num BNVAL_DEF)
-lemma BV_LESS_2: "ALL x::bool. BV x < 2"
+lemma BV_LESS_2: "BV x < 2"
by (import bword_num BV_LESS_2)
-lemma BNVAL_NVAL: "ALL w::bool word. BNVAL w = NVAL BV 2 w"
+lemma BNVAL_NVAL: "BNVAL w = NVAL BV 2 w"
by (import bword_num BNVAL_NVAL)
lemma BNVAL0: "BNVAL (WORD []) = 0"
by (import bword_num BNVAL0)
-lemma BNVAL_11: "ALL (w1::bool word) w2::bool word.
- WORDLEN w1 = WORDLEN w2 --> BNVAL w1 = BNVAL w2 --> w1 = w2"
+lemma BNVAL_11: "[| WORDLEN w1 = WORDLEN w2; BNVAL w1 = BNVAL w2 |] ==> w1 = w2"
by (import bword_num BNVAL_11)
-lemma BNVAL_ONTO: "ALL w::bool word. Ex (op = (BNVAL w))"
+lemma BNVAL_ONTO: "Ex (op = (BNVAL w))"
by (import bword_num BNVAL_ONTO)
-lemma BNVAL_MAX: "ALL n::nat. RES_FORALL (PWORDLEN n) (%w::bool word. BNVAL w < 2 ^ n)"
+lemma BNVAL_MAX: "RES_FORALL (PWORDLEN n) (%w. BNVAL w < 2 ^ n)"
by (import bword_num BNVAL_MAX)
-lemma BNVAL_WCAT1: "ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w::bool word.
- ALL x::bool. BNVAL (WCAT (w, WORD [x])) = BNVAL w * 2 + BV x)"
+lemma BNVAL_WCAT1: "RES_FORALL (PWORDLEN n)
+ (%w. ALL x. BNVAL (WCAT (w, WORD [x])) = BNVAL w * 2 + BV x)"
by (import bword_num BNVAL_WCAT1)
-lemma BNVAL_WCAT2: "ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w::bool word.
- ALL x::bool. BNVAL (WCAT (WORD [x], w)) = BV x * 2 ^ n + BNVAL w)"
+lemma BNVAL_WCAT2: "RES_FORALL (PWORDLEN n)
+ (%w. ALL x. BNVAL (WCAT (WORD [x], w)) = BV x * 2 ^ n + BNVAL w)"
by (import bword_num BNVAL_WCAT2)
-lemma BNVAL_WCAT: "ALL (n::nat) m::nat.
- RES_FORALL (PWORDLEN n)
- (%w1::bool word.
- RES_FORALL (PWORDLEN m)
- (%w2::bool word.
- BNVAL (WCAT (w1, w2)) = BNVAL w1 * 2 ^ m + BNVAL w2))"
+lemma BNVAL_WCAT: "RES_FORALL (PWORDLEN n)
+ (%w1. RES_FORALL (PWORDLEN m)
+ (%w2. BNVAL (WCAT (w1, w2)) = BNVAL w1 * 2 ^ m + BNVAL w2))"
by (import bword_num BNVAL_WCAT)
-definition VB :: "nat => bool" where
- "VB == %n::nat. n mod 2 ~= 0"
+definition
+ VB :: "nat => bool" where
+ "VB == %n. n mod 2 ~= 0"
-lemma VB_DEF: "ALL n::nat. VB n = (n mod 2 ~= 0)"
+lemma VB_DEF: "VB n = (n mod 2 ~= 0)"
by (import bword_num VB_DEF)
-definition NBWORD :: "nat => nat => bool word" where
- "NBWORD == %(n::nat) m::nat. WORD (NLIST n VB 2 m)"
+definition
+ NBWORD :: "nat => nat => bool word" where
+ "NBWORD == %n m. WORD (NLIST n VB 2 m)"
-lemma NBWORD_DEF: "ALL (n::nat) m::nat. NBWORD n m = WORD (NLIST n VB 2 m)"
+lemma NBWORD_DEF: "NBWORD n m = WORD (NLIST n VB 2 m)"
by (import bword_num NBWORD_DEF)
-lemma NBWORD0: "ALL x::nat. NBWORD 0 x = WORD []"
+lemma NBWORD0: "NBWORD 0 x = WORD []"
by (import bword_num NBWORD0)
-lemma WORDLEN_NBWORD: "ALL (x::nat) xa::nat. WORDLEN (NBWORD x xa) = x"
+lemma WORDLEN_NBWORD: "WORDLEN (NBWORD x xa) = x"
by (import bword_num WORDLEN_NBWORD)
-lemma PWORDLEN_NBWORD: "ALL (x::nat) xa::nat. IN (NBWORD x xa) (PWORDLEN x)"
+lemma PWORDLEN_NBWORD: "IN (NBWORD x xa) (PWORDLEN x)"
by (import bword_num PWORDLEN_NBWORD)
-lemma NBWORD_SUC: "ALL (n::nat) m::nat.
- NBWORD (Suc n) m = WCAT (NBWORD n (m div 2), WORD [VB (m mod 2)])"
+lemma NBWORD_SUC: "NBWORD (Suc n) m = WCAT (NBWORD n (m div 2), WORD [VB (m mod 2)])"
by (import bword_num NBWORD_SUC)
-lemma VB_BV: "ALL x::bool. VB (BV x) = x"
+lemma VB_BV: "VB (BV x) = x"
by (import bword_num VB_BV)
-lemma BV_VB: "(All::(nat => bool) => bool)
- (%x::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) x
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- ((op =::nat => nat => bool) ((BV::bool => nat) ((VB::nat => bool) x))
- x))"
+lemma BV_VB: "x < 2 ==> BV (VB x) = x"
by (import bword_num BV_VB)
-lemma NBWORD_BNVAL: "ALL n::nat. RES_FORALL (PWORDLEN n) (%w::bool word. NBWORD n (BNVAL w) = w)"
+lemma NBWORD_BNVAL: "RES_FORALL (PWORDLEN n) (%w. NBWORD n (BNVAL w) = w)"
by (import bword_num NBWORD_BNVAL)
-lemma BNVAL_NBWORD: "ALL (n::nat) m::nat. m < 2 ^ n --> BNVAL (NBWORD n m) = m"
+lemma BNVAL_NBWORD: "m < 2 ^ n ==> BNVAL (NBWORD n m) = m"
by (import bword_num BNVAL_NBWORD)
-lemma ZERO_WORD_VAL: "RES_FORALL (PWORDLEN (n::nat))
- (%w::bool word. (w = NBWORD n 0) = (BNVAL w = 0))"
+lemma ZERO_WORD_VAL: "RES_FORALL (PWORDLEN n) (%w. (w = NBWORD n 0) = (BNVAL w = 0))"
by (import bword_num ZERO_WORD_VAL)
-lemma WCAT_NBWORD_0: "ALL (n1::nat) n2::nat. WCAT (NBWORD n1 0, NBWORD n2 0) = NBWORD (n1 + n2) 0"
+lemma WCAT_NBWORD_0: "WCAT (NBWORD n1 0, NBWORD n2 0) = NBWORD (n1 + n2) 0"
by (import bword_num WCAT_NBWORD_0)
-lemma WSPLIT_NBWORD_0: "ALL (n::nat) m::nat.
- m <= n --> WSPLIT m (NBWORD n 0) = (NBWORD (n - m) 0, NBWORD m 0)"
+lemma WSPLIT_NBWORD_0: "m <= n ==> WSPLIT m (NBWORD n 0) = (NBWORD (n - m) 0, NBWORD m 0)"
by (import bword_num WSPLIT_NBWORD_0)
-lemma EQ_NBWORD0_SPLIT: "(All::(nat => bool) => bool)
- (%n::nat.
- (RES_FORALL::(bool word => bool) => (bool word => bool) => bool)
- ((PWORDLEN::nat => bool word => bool) n)
- (%w::bool word.
- (All::(nat => bool) => bool)
- (%m::nat.
- (op -->::bool => bool => bool)
- ((op <=::nat => nat => bool) m n)
- ((op =::bool => bool => bool)
- ((op =::bool word => bool word => bool) w
- ((NBWORD::nat => nat => bool word) n (0::nat)))
- ((op &::bool => bool => bool)
- ((op =::bool word => bool word => bool)
- ((WSEG::nat => nat => bool word => bool word)
- ((op -::nat => nat => nat) n m) m w)
- ((NBWORD::nat => nat => bool word)
- ((op -::nat => nat => nat) n m) (0::nat)))
- ((op =::bool word => bool word => bool)
- ((WSEG::nat => nat => bool word => bool word) m
- (0::nat) w)
- ((NBWORD::nat => nat => bool word) m (0::nat))))))))"
+lemma EQ_NBWORD0_SPLIT: "RES_FORALL (PWORDLEN n)
+ (%w. ALL m<=n.
+ (w = NBWORD n 0) =
+ (WSEG (n - m) m w = NBWORD (n - m) 0 & WSEG m 0 w = NBWORD m 0))"
by (import bword_num EQ_NBWORD0_SPLIT)
-lemma NBWORD_MOD: "ALL (n::nat) m::nat. NBWORD n (m mod 2 ^ n) = NBWORD n m"
+lemma NBWORD_MOD: "NBWORD n (m mod 2 ^ n) = NBWORD n m"
by (import bword_num NBWORD_MOD)
-lemma WSEG_NBWORD_SUC: "ALL (n::nat) m::nat. WSEG n 0 (NBWORD (Suc n) m) = NBWORD n m"
+lemma WSEG_NBWORD_SUC: "WSEG n 0 (NBWORD (Suc n) m) = NBWORD n m"
by (import bword_num WSEG_NBWORD_SUC)
-lemma NBWORD_SUC_WSEG: "ALL n::nat.
- RES_FORALL (PWORDLEN (Suc n))
- (%w::bool word. NBWORD n (BNVAL w) = WSEG n 0 w)"
+lemma NBWORD_SUC_WSEG: "RES_FORALL (PWORDLEN (Suc n)) (%w. NBWORD n (BNVAL w) = WSEG n 0 w)"
by (import bword_num NBWORD_SUC_WSEG)
-lemma DOUBL_EQ_SHL: "ALL x>0.
- RES_FORALL (PWORDLEN x)
- (%xa::bool word.
- ALL xb::bool.
- NBWORD x (BNVAL xa + BNVAL xa + BV xb) = snd (SHL False xa xb))"
+lemma DOUBL_EQ_SHL: "0 < x
+==> RES_FORALL (PWORDLEN x)
+ (%xa. ALL xb.
+ NBWORD x (BNVAL xa + BNVAL xa + BV xb) =
+ snd (SHL False xa xb))"
by (import bword_num DOUBL_EQ_SHL)
-lemma MSB_NBWORD: "ALL (n::nat) m::nat. bit n (NBWORD (Suc n) m) = VB (m div 2 ^ n mod 2)"
+lemma MSB_NBWORD: "bit n (NBWORD (Suc n) m) = VB (m div 2 ^ n mod 2)"
by (import bword_num MSB_NBWORD)
-lemma NBWORD_SPLIT: "ALL (n1::nat) (n2::nat) m::nat.
- NBWORD (n1 + n2) m = WCAT (NBWORD n1 (m div 2 ^ n2), NBWORD n2 m)"
+lemma NBWORD_SPLIT: "NBWORD (n1 + n2) m = WCAT (NBWORD n1 (m div 2 ^ n2), NBWORD n2 m)"
by (import bword_num NBWORD_SPLIT)
-lemma WSEG_NBWORD: "ALL (m::nat) (k::nat) n::nat.
- m + k <= n -->
- (ALL l::nat. WSEG m k (NBWORD n l) = NBWORD m (l div 2 ^ k))"
+lemma WSEG_NBWORD: "m + k <= n ==> WSEG m k (NBWORD n l) = NBWORD m (l div 2 ^ k)"
by (import bword_num WSEG_NBWORD)
-lemma NBWORD_SUC_FST: "ALL (n::nat) x::nat.
- NBWORD (Suc n) x = WCAT (WORD [VB (x div 2 ^ n mod 2)], NBWORD n x)"
+lemma NBWORD_SUC_FST: "NBWORD (Suc n) x = WCAT (WORD [VB (x div 2 ^ n mod 2)], NBWORD n x)"
by (import bword_num NBWORD_SUC_FST)
-lemma BIT_NBWORD0: "ALL (k::nat) n::nat. k < n --> bit k (NBWORD n 0) = False"
+lemma BIT_NBWORD0: "k < n ==> bit k (NBWORD n 0) = False"
by (import bword_num BIT_NBWORD0)
-lemma ADD_BNVAL_LEFT: "ALL n::nat.
- RES_FORALL (PWORDLEN (Suc n))
- (%w1::bool word.
- RES_FORALL (PWORDLEN (Suc n))
- (%w2::bool word.
- BNVAL w1 + BNVAL w2 =
- (BV (bit n w1) + BV (bit n w2)) * 2 ^ n +
- (BNVAL (WSEG n 0 w1) + BNVAL (WSEG n 0 w2))))"
+lemma ADD_BNVAL_LEFT: "RES_FORALL (PWORDLEN (Suc n))
+ (%w1. RES_FORALL (PWORDLEN (Suc n))
+ (%w2. BNVAL w1 + BNVAL w2 =
+ (BV (bit n w1) + BV (bit n w2)) * 2 ^ n +
+ (BNVAL (WSEG n 0 w1) + BNVAL (WSEG n 0 w2))))"
by (import bword_num ADD_BNVAL_LEFT)
-lemma ADD_BNVAL_RIGHT: "ALL n::nat.
- RES_FORALL (PWORDLEN (Suc n))
- (%w1::bool word.
- RES_FORALL (PWORDLEN (Suc n))
- (%w2::bool word.
- BNVAL w1 + BNVAL w2 =
- (BNVAL (WSEG n 1 w1) + BNVAL (WSEG n 1 w2)) * 2 +
- (BV (bit 0 w1) + BV (bit 0 w2))))"
+lemma ADD_BNVAL_RIGHT: "RES_FORALL (PWORDLEN (Suc n))
+ (%w1. RES_FORALL (PWORDLEN (Suc n))
+ (%w2. BNVAL w1 + BNVAL w2 =
+ (BNVAL (WSEG n 1 w1) + BNVAL (WSEG n 1 w2)) * 2 +
+ (BV (bit 0 w1) + BV (bit 0 w2))))"
by (import bword_num ADD_BNVAL_RIGHT)
-lemma ADD_BNVAL_SPLIT: "ALL (n1::nat) n2::nat.
- RES_FORALL (PWORDLEN (n1 + n2))
- (%w1::bool word.
- RES_FORALL (PWORDLEN (n1 + n2))
- (%w2::bool word.
- BNVAL w1 + BNVAL w2 =
- (BNVAL (WSEG n1 n2 w1) + BNVAL (WSEG n1 n2 w2)) * 2 ^ n2 +
- (BNVAL (WSEG n2 0 w1) + BNVAL (WSEG n2 0 w2))))"
+lemma ADD_BNVAL_SPLIT: "RES_FORALL (PWORDLEN (n1 + n2))
+ (%w1. RES_FORALL (PWORDLEN (n1 + n2))
+ (%w2. BNVAL w1 + BNVAL w2 =
+ (BNVAL (WSEG n1 n2 w1) + BNVAL (WSEG n1 n2 w2)) * 2 ^ n2 +
+ (BNVAL (WSEG n2 0 w1) + BNVAL (WSEG n2 0 w2))))"
by (import bword_num ADD_BNVAL_SPLIT)
;end_setup
@@ -1396,8 +924,8 @@
consts
ACARRY :: "nat => bool word => bool word => bool => bool"
-specification (ACARRY) ACARRY_DEF: "(ALL (w1::bool word) (w2::bool word) cin::bool. ACARRY 0 w1 w2 cin = cin) &
-(ALL (n::nat) (w1::bool word) (w2::bool word) cin::bool.
+specification (ACARRY) ACARRY_DEF: "(ALL w1 w2 cin. ACARRY 0 w1 w2 cin = cin) &
+(ALL n w1 w2 cin.
ACARRY (Suc n) w1 w2 cin =
VB ((BV (bit n w1) + BV (bit n w2) + BV (ACARRY n w1 w2 cin)) div 2))"
by (import bword_arith ACARRY_DEF)
@@ -1405,185 +933,122 @@
consts
ICARRY :: "nat => bool word => bool word => bool => bool"
-specification (ICARRY) ICARRY_DEF: "(ALL (w1::bool word) (w2::bool word) cin::bool. ICARRY 0 w1 w2 cin = cin) &
-(ALL (n::nat) (w1::bool word) (w2::bool word) cin::bool.
+specification (ICARRY) ICARRY_DEF: "(ALL w1 w2 cin. ICARRY 0 w1 w2 cin = cin) &
+(ALL n w1 w2 cin.
ICARRY (Suc n) w1 w2 cin =
(bit n w1 & bit n w2 | (bit n w1 | bit n w2) & ICARRY n w1 w2 cin))"
by (import bword_arith ICARRY_DEF)
-lemma ACARRY_EQ_ICARRY: "ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w1::bool word.
- RES_FORALL (PWORDLEN n)
- (%w2::bool word.
- ALL (cin::bool) k::nat.
- k <= n --> ACARRY k w1 w2 cin = ICARRY k w1 w2 cin))"
+lemma ACARRY_EQ_ICARRY: "RES_FORALL (PWORDLEN n)
+ (%w1. RES_FORALL (PWORDLEN n)
+ (%w2. ALL cin k.
+ k <= n --> ACARRY k w1 w2 cin = ICARRY k w1 w2 cin))"
by (import bword_arith ACARRY_EQ_ICARRY)
-lemma BNVAL_LESS_EQ: "ALL n::nat. RES_FORALL (PWORDLEN n) (%w::bool word. BNVAL w <= 2 ^ n - 1)"
+lemma BNVAL_LESS_EQ: "RES_FORALL (PWORDLEN n) (%w. BNVAL w <= 2 ^ n - 1)"
by (import bword_arith BNVAL_LESS_EQ)
-lemma ADD_BNVAL_LESS_EQ1: "ALL (n::nat) cin::bool.
- RES_FORALL (PWORDLEN n)
- (%w1::bool word.
- RES_FORALL (PWORDLEN n)
- (%w2::bool word.
- (BNVAL w1 + (BNVAL w2 + BV cin)) div 2 ^ n <= Suc 0))"
+lemma ADD_BNVAL_LESS_EQ1: "RES_FORALL (PWORDLEN n)
+ (%w1. RES_FORALL (PWORDLEN n)
+ (%w2. (BNVAL w1 + (BNVAL w2 + BV cin)) div 2 ^ n <= Suc 0))"
by (import bword_arith ADD_BNVAL_LESS_EQ1)
-lemma ADD_BV_BNVAL_DIV_LESS_EQ1: "ALL (n::nat) (x1::bool) (x2::bool) cin::bool.
- RES_FORALL (PWORDLEN n)
- (%w1::bool word.
- RES_FORALL (PWORDLEN n)
- (%w2::bool word.
- (BV x1 + BV x2 +
- (BNVAL w1 + (BNVAL w2 + BV cin)) div 2 ^ n) div
- 2
- <= 1))"
+lemma ADD_BV_BNVAL_DIV_LESS_EQ1: "RES_FORALL (PWORDLEN n)
+ (%w1. RES_FORALL (PWORDLEN n)
+ (%w2. (BV x1 + BV x2 +
+ (BNVAL w1 + (BNVAL w2 + BV cin)) div 2 ^ n) div
+ 2
+ <= 1))"
by (import bword_arith ADD_BV_BNVAL_DIV_LESS_EQ1)
-lemma ADD_BV_BNVAL_LESS_EQ: "ALL (n::nat) (x1::bool) (x2::bool) cin::bool.
- RES_FORALL (PWORDLEN n)
- (%w1::bool word.
- RES_FORALL (PWORDLEN n)
- (%w2::bool word.
- BV x1 + BV x2 + (BNVAL w1 + (BNVAL w2 + BV cin))
- <= Suc (2 ^ Suc n)))"
+lemma ADD_BV_BNVAL_LESS_EQ: "RES_FORALL (PWORDLEN n)
+ (%w1. RES_FORALL (PWORDLEN n)
+ (%w2. BV x1 + BV x2 + (BNVAL w1 + (BNVAL w2 + BV cin))
+ <= Suc (2 ^ Suc n)))"
by (import bword_arith ADD_BV_BNVAL_LESS_EQ)
-lemma ADD_BV_BNVAL_LESS_EQ1: "ALL (n::nat) (x1::bool) (x2::bool) cin::bool.
- RES_FORALL (PWORDLEN n)
- (%w1::bool word.
- RES_FORALL (PWORDLEN n)
- (%w2::bool word.
- (BV x1 + BV x2 + (BNVAL w1 + (BNVAL w2 + BV cin))) div
- 2 ^ Suc n
- <= 1))"
+lemma ADD_BV_BNVAL_LESS_EQ1: "RES_FORALL (PWORDLEN n)
+ (%w1. RES_FORALL (PWORDLEN n)
+ (%w2. (BV x1 + BV x2 + (BNVAL w1 + (BNVAL w2 + BV cin))) div
+ 2 ^ Suc n
+ <= 1))"
by (import bword_arith ADD_BV_BNVAL_LESS_EQ1)
-lemma ACARRY_EQ_ADD_DIV: "(All::(nat => bool) => bool)
- (%n::nat.
- (RES_FORALL::(bool word => bool) => (bool word => bool) => bool)
- ((PWORDLEN::nat => bool word => bool) n)
- (%w1::bool word.
- (RES_FORALL::(bool word => bool) => (bool word => bool) => bool)
- ((PWORDLEN::nat => bool word => bool) n)
- (%w2::bool word.
- (All::(nat => bool) => bool)
- (%k::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) k n)
- ((op =::nat => nat => bool)
- ((BV::bool => nat)
- ((ACARRY::nat
- => bool word
-=> bool word => bool => bool)
- k w1 w2 (cin::bool)))
- ((op div::nat => nat => nat)
- ((op +::nat => nat => nat)
- ((op +::nat => nat => nat)
- ((BNVAL::bool word => nat)
- ((WSEG::nat => nat => bool word => bool word)
- k (0::nat) w1))
- ((BNVAL::bool word => nat)
- ((WSEG::nat => nat => bool word => bool word)
- k (0::nat) w2)))
- ((BV::bool => nat) cin))
- ((op ^::nat => nat => nat)
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int)
- (Int.Pls \<Colon> int))))
- k)))))))"
+lemma ACARRY_EQ_ADD_DIV: "RES_FORALL (PWORDLEN n)
+ (%w1. RES_FORALL (PWORDLEN n)
+ (%w2. ALL k<n.
+ BV (ACARRY k w1 w2 cin) =
+ (BNVAL (WSEG k 0 w1) + BNVAL (WSEG k 0 w2) + BV cin) div
+ 2 ^ k))"
by (import bword_arith ACARRY_EQ_ADD_DIV)
-lemma ADD_WORD_SPLIT: "ALL (n1::nat) n2::nat.
- RES_FORALL (PWORDLEN (n1 + n2))
- (%w1::bool word.
- RES_FORALL (PWORDLEN (n1 + n2))
- (%w2::bool word.
- ALL cin::bool.
- NBWORD (n1 + n2) (BNVAL w1 + BNVAL w2 + BV cin) =
- WCAT
- (NBWORD n1
- (BNVAL (WSEG n1 n2 w1) + BNVAL (WSEG n1 n2 w2) +
- BV (ACARRY n2 w1 w2 cin)),
- NBWORD n2
- (BNVAL (WSEG n2 0 w1) + BNVAL (WSEG n2 0 w2) + BV cin))))"
+lemma ADD_WORD_SPLIT: "RES_FORALL (PWORDLEN (n1 + n2))
+ (%w1. RES_FORALL (PWORDLEN (n1 + n2))
+ (%w2. ALL cin.
+ NBWORD (n1 + n2) (BNVAL w1 + BNVAL w2 + BV cin) =
+ WCAT
+ (NBWORD n1
+ (BNVAL (WSEG n1 n2 w1) + BNVAL (WSEG n1 n2 w2) +
+ BV (ACARRY n2 w1 w2 cin)),
+ NBWORD n2
+ (BNVAL (WSEG n2 0 w1) + BNVAL (WSEG n2 0 w2) +
+ BV cin))))"
by (import bword_arith ADD_WORD_SPLIT)
-lemma WSEG_NBWORD_ADD: "ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w1::bool word.
- RES_FORALL (PWORDLEN n)
- (%w2::bool word.
- ALL (m::nat) (k::nat) cin::bool.
- m + k <= n -->
- WSEG m k (NBWORD n (BNVAL w1 + BNVAL w2 + BV cin)) =
- NBWORD m
- (BNVAL (WSEG m k w1) + BNVAL (WSEG m k w2) +
- BV (ACARRY k w1 w2 cin))))"
+lemma WSEG_NBWORD_ADD: "RES_FORALL (PWORDLEN n)
+ (%w1. RES_FORALL (PWORDLEN n)
+ (%w2. ALL m k cin.
+ m + k <= n -->
+ WSEG m k (NBWORD n (BNVAL w1 + BNVAL w2 + BV cin)) =
+ NBWORD m
+ (BNVAL (WSEG m k w1) + BNVAL (WSEG m k w2) +
+ BV (ACARRY k w1 w2 cin))))"
by (import bword_arith WSEG_NBWORD_ADD)
-lemma ADD_NBWORD_EQ0_SPLIT: "ALL (n1::nat) n2::nat.
- RES_FORALL (PWORDLEN (n1 + n2))
- (%w1::bool word.
- RES_FORALL (PWORDLEN (n1 + n2))
- (%w2::bool word.
- ALL cin::bool.
- (NBWORD (n1 + n2) (BNVAL w1 + BNVAL w2 + BV cin) =
- NBWORD (n1 + n2) 0) =
- (NBWORD n1
- (BNVAL (WSEG n1 n2 w1) + BNVAL (WSEG n1 n2 w2) +
- BV (ACARRY n2 w1 w2 cin)) =
- NBWORD n1 0 &
- NBWORD n2
- (BNVAL (WSEG n2 0 w1) + BNVAL (WSEG n2 0 w2) + BV cin) =
- NBWORD n2 0)))"
+lemma ADD_NBWORD_EQ0_SPLIT: "RES_FORALL (PWORDLEN (n1 + n2))
+ (%w1. RES_FORALL (PWORDLEN (n1 + n2))
+ (%w2. ALL cin.
+ (NBWORD (n1 + n2) (BNVAL w1 + BNVAL w2 + BV cin) =
+ NBWORD (n1 + n2) 0) =
+ (NBWORD n1
+ (BNVAL (WSEG n1 n2 w1) + BNVAL (WSEG n1 n2 w2) +
+ BV (ACARRY n2 w1 w2 cin)) =
+ NBWORD n1 0 &
+ NBWORD n2
+ (BNVAL (WSEG n2 0 w1) + BNVAL (WSEG n2 0 w2) + BV cin) =
+ NBWORD n2 0)))"
by (import bword_arith ADD_NBWORD_EQ0_SPLIT)
-lemma ACARRY_MSB: "ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w1::bool word.
- RES_FORALL (PWORDLEN n)
- (%w2::bool word.
- ALL cin::bool.
- ACARRY n w1 w2 cin =
- bit n (NBWORD (Suc n) (BNVAL w1 + BNVAL w2 + BV cin))))"
+lemma ACARRY_MSB: "RES_FORALL (PWORDLEN n)
+ (%w1. RES_FORALL (PWORDLEN n)
+ (%w2. ALL cin.
+ ACARRY n w1 w2 cin =
+ bit n (NBWORD (Suc n) (BNVAL w1 + BNVAL w2 + BV cin))))"
by (import bword_arith ACARRY_MSB)
-lemma ACARRY_WSEG: "ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w1::bool word.
- RES_FORALL (PWORDLEN n)
- (%w2::bool word.
- ALL (cin::bool) (k::nat) m::nat.
- k < m & m <= n -->
- ACARRY k (WSEG m 0 w1) (WSEG m 0 w2) cin =
- ACARRY k w1 w2 cin))"
+lemma ACARRY_WSEG: "RES_FORALL (PWORDLEN n)
+ (%w1. RES_FORALL (PWORDLEN n)
+ (%w2. ALL cin k m.
+ k < m & m <= n -->
+ ACARRY k (WSEG m 0 w1) (WSEG m 0 w2) cin =
+ ACARRY k w1 w2 cin))"
by (import bword_arith ACARRY_WSEG)
-lemma ICARRY_WSEG: "ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w1::bool word.
- RES_FORALL (PWORDLEN n)
- (%w2::bool word.
- ALL (cin::bool) (k::nat) m::nat.
- k < m & m <= n -->
- ICARRY k (WSEG m 0 w1) (WSEG m 0 w2) cin =
- ICARRY k w1 w2 cin))"
+lemma ICARRY_WSEG: "RES_FORALL (PWORDLEN n)
+ (%w1. RES_FORALL (PWORDLEN n)
+ (%w2. ALL cin k m.
+ k < m & m <= n -->
+ ICARRY k (WSEG m 0 w1) (WSEG m 0 w2) cin =
+ ICARRY k w1 w2 cin))"
by (import bword_arith ICARRY_WSEG)
-lemma ACARRY_ACARRY_WSEG: "ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w1::bool word.
- RES_FORALL (PWORDLEN n)
- (%w2::bool word.
- ALL (cin::bool) (m::nat) (k1::nat) k2::nat.
- k1 < m & k2 < n & m + k2 <= n -->
- ACARRY k1 (WSEG m k2 w1) (WSEG m k2 w2)
- (ACARRY k2 w1 w2 cin) =
- ACARRY (k1 + k2) w1 w2 cin))"
+lemma ACARRY_ACARRY_WSEG: "RES_FORALL (PWORDLEN n)
+ (%w1. RES_FORALL (PWORDLEN n)
+ (%w2. ALL cin m k1 k2.
+ k1 < m & k2 < n & m + k2 <= n -->
+ ACARRY k1 (WSEG m k2 w1) (WSEG m k2 w2)
+ (ACARRY k2 w1 w2 cin) =
+ ACARRY (k1 + k2) w1 w2 cin))"
by (import bword_arith ACARRY_ACARRY_WSEG)
;end_setup
@@ -1593,27 +1058,24 @@
consts
WNOT :: "bool word => bool word"
-specification (WNOT) WNOT_DEF: "ALL l::bool list. WNOT (WORD l) = WORD (map Not l)"
+specification (WNOT) WNOT_DEF: "ALL l. WNOT (WORD l) = WORD (map Not l)"
by (import bword_bitop WNOT_DEF)
lemma PBITOP_WNOT: "IN WNOT PBITOP"
by (import bword_bitop PBITOP_WNOT)
-lemma WNOT_WNOT: "ALL w::bool word. WNOT (WNOT w) = w"
+lemma WNOT_WNOT: "WNOT (WNOT w) = w"
by (import bword_bitop WNOT_WNOT)
-lemma WCAT_WNOT: "ALL (n1::nat) n2::nat.
- RES_FORALL (PWORDLEN n1)
- (%w1::bool word.
- RES_FORALL (PWORDLEN n2)
- (%w2::bool word. WCAT (WNOT w1, WNOT w2) = WNOT (WCAT (w1, w2))))"
+lemma WCAT_WNOT: "RES_FORALL (PWORDLEN n1)
+ (%w1. RES_FORALL (PWORDLEN n2)
+ (%w2. WCAT (WNOT w1, WNOT w2) = WNOT (WCAT (w1, w2))))"
by (import bword_bitop WCAT_WNOT)
consts
WAND :: "bool word => bool word => bool word"
-specification (WAND) WAND_DEF: "ALL (l1::bool list) l2::bool list.
- WAND (WORD l1) (WORD l2) = WORD (map2 op & l1 l2)"
+specification (WAND) WAND_DEF: "ALL l1 l2. WAND (WORD l1) (WORD l2) = WORD (map2 op & l1 l2)"
by (import bword_bitop WAND_DEF)
lemma PBITBOP_WAND: "IN WAND PBITBOP"
@@ -1622,8 +1084,7 @@
consts
WOR :: "bool word => bool word => bool word"
-specification (WOR) WOR_DEF: "ALL (l1::bool list) l2::bool list.
- WOR (WORD l1) (WORD l2) = WORD (map2 op | l1 l2)"
+specification (WOR) WOR_DEF: "ALL l1 l2. WOR (WORD l1) (WORD l2) = WORD (map2 op | l1 l2)"
by (import bword_bitop WOR_DEF)
lemma PBITBOP_WOR: "IN WOR PBITBOP"
@@ -1632,8 +1093,7 @@
consts
WXOR :: "bool word => bool word => bool word"
-specification (WXOR) WXOR_DEF: "ALL (l1::bool list) l2::bool list.
- WXOR (WORD l1) (WORD l2) = WORD (map2 (%(x::bool) y::bool. x ~= y) l1 l2)"
+specification (WXOR) WXOR_DEF: "ALL l1 l2. WXOR (WORD l1) (WORD l2) = WORD (map2 op ~= l1 l2)"
by (import bword_bitop WXOR_DEF)
lemma PBITBOP_WXOR: "IN WXOR PBITBOP"
--- a/src/HOL/Import/HOL/HOL4Word32.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Import/HOL/HOL4Word32.thy Fri Sep 09 06:47:14 2011 +0200
@@ -8,80 +8,79 @@
DIV2 :: "nat => nat"
defs
- DIV2_primdef: "DIV2 == %n::nat. n div 2"
+ DIV2_primdef: "DIV2 == %n. n div 2"
-lemma DIV2_def: "ALL n::nat. DIV2 n = n div 2"
+lemma DIV2_def: "DIV2 n = n div 2"
by (import bits DIV2_def)
consts
TIMES_2EXP :: "nat => nat => nat"
defs
- TIMES_2EXP_primdef: "TIMES_2EXP == %(x::nat) n::nat. n * 2 ^ x"
+ TIMES_2EXP_primdef: "TIMES_2EXP == %x n. n * 2 ^ x"
-lemma TIMES_2EXP_def: "ALL (x::nat) n::nat. TIMES_2EXP x n = n * 2 ^ x"
+lemma TIMES_2EXP_def: "TIMES_2EXP x n = n * 2 ^ x"
by (import bits TIMES_2EXP_def)
consts
DIV_2EXP :: "nat => nat => nat"
defs
- DIV_2EXP_primdef: "DIV_2EXP == %(x::nat) n::nat. n div 2 ^ x"
+ DIV_2EXP_primdef: "DIV_2EXP == %x n. n div 2 ^ x"
-lemma DIV_2EXP_def: "ALL (x::nat) n::nat. DIV_2EXP x n = n div 2 ^ x"
+lemma DIV_2EXP_def: "DIV_2EXP x n = n div 2 ^ x"
by (import bits DIV_2EXP_def)
consts
MOD_2EXP :: "nat => nat => nat"
defs
- MOD_2EXP_primdef: "MOD_2EXP == %(x::nat) n::nat. n mod 2 ^ x"
+ MOD_2EXP_primdef: "MOD_2EXP == %x n. n mod 2 ^ x"
-lemma MOD_2EXP_def: "ALL (x::nat) n::nat. MOD_2EXP x n = n mod 2 ^ x"
+lemma MOD_2EXP_def: "MOD_2EXP x n = n mod 2 ^ x"
by (import bits MOD_2EXP_def)
consts
DIVMOD_2EXP :: "nat => nat => nat * nat"
defs
- DIVMOD_2EXP_primdef: "DIVMOD_2EXP == %(x::nat) n::nat. (n div 2 ^ x, n mod 2 ^ x)"
+ DIVMOD_2EXP_primdef: "DIVMOD_2EXP == %x n. (n div 2 ^ x, n mod 2 ^ x)"
-lemma DIVMOD_2EXP_def: "ALL (x::nat) n::nat. DIVMOD_2EXP x n = (n div 2 ^ x, n mod 2 ^ x)"
+lemma DIVMOD_2EXP_def: "DIVMOD_2EXP x n = (n div 2 ^ x, n mod 2 ^ x)"
by (import bits DIVMOD_2EXP_def)
consts
SBIT :: "bool => nat => nat"
defs
- SBIT_primdef: "SBIT == %(b::bool) n::nat. if b then 2 ^ n else 0"
+ SBIT_primdef: "SBIT == %b n. if b then 2 ^ n else 0"
-lemma SBIT_def: "ALL (b::bool) n::nat. SBIT b n = (if b then 2 ^ n else 0)"
+lemma SBIT_def: "SBIT b n = (if b then 2 ^ n else 0)"
by (import bits SBIT_def)
consts
BITS :: "nat => nat => nat => nat"
defs
- BITS_primdef: "BITS == %(h::nat) (l::nat) n::nat. MOD_2EXP (Suc h - l) (DIV_2EXP l n)"
+ BITS_primdef: "BITS == %h l n. MOD_2EXP (Suc h - l) (DIV_2EXP l n)"
-lemma BITS_def: "ALL (h::nat) (l::nat) n::nat.
- BITS h l n = MOD_2EXP (Suc h - l) (DIV_2EXP l n)"
+lemma BITS_def: "BITS h l n = MOD_2EXP (Suc h - l) (DIV_2EXP l n)"
by (import bits BITS_def)
-definition bit :: "nat => nat => bool" where
- "bit == %(b::nat) n::nat. BITS b b n = 1"
+definition
+ bit :: "nat => nat => bool" where
+ "bit == %b n. BITS b b n = 1"
-lemma BIT_def: "ALL (b::nat) n::nat. bit b n = (BITS b b n = 1)"
+lemma BIT_def: "bit b n = (BITS b b n = 1)"
by (import bits BIT_def)
consts
SLICE :: "nat => nat => nat => nat"
defs
- SLICE_primdef: "SLICE == %(h::nat) (l::nat) n::nat. MOD_2EXP (Suc h) n - MOD_2EXP l n"
+ SLICE_primdef: "SLICE == %h l n. MOD_2EXP (Suc h) n - MOD_2EXP l n"
-lemma SLICE_def: "ALL (h::nat) (l::nat) n::nat.
- SLICE h l n = MOD_2EXP (Suc h) n - MOD_2EXP l n"
+lemma SLICE_def: "SLICE h l n = MOD_2EXP (Suc h) n - MOD_2EXP l n"
by (import bits SLICE_def)
consts
@@ -96,282 +95,192 @@
consts
BITWISE :: "nat => (bool => bool => bool) => nat => nat => nat"
-specification (BITWISE_primdef: BITWISE) BITWISE_def: "(ALL (oper::bool => bool => bool) (x::nat) y::nat. BITWISE 0 oper x y = 0) &
-(ALL (n::nat) (oper::bool => bool => bool) (x::nat) y::nat.
+specification (BITWISE_primdef: BITWISE) BITWISE_def: "(ALL oper x y. BITWISE 0 oper x y = 0) &
+(ALL n oper x y.
BITWISE (Suc n) oper x y =
BITWISE n oper x y + SBIT (oper (bit n x) (bit n y)) n)"
by (import bits BITWISE_def)
-lemma DIV1: "ALL x::nat. x div 1 = x"
- by (import bits DIV1)
-
-lemma SUC_SUB: "Suc (a::nat) - a = 1"
+lemma SUC_SUB: "Suc a - a = 1"
by (import bits SUC_SUB)
-lemma DIV_MULT_1: "ALL (r::nat) n::nat. r < n --> (n + r) div n = 1"
+lemma DIV_MULT_1: "(r::nat) < (n::nat) ==> (n + r) div n = (1::nat)"
by (import bits DIV_MULT_1)
-lemma ZERO_LT_TWOEXP: "(All::(nat => bool) => bool)
- (%n::nat.
- (op <::nat => nat => bool) (0::nat)
- ((op ^::nat => nat => nat)
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int))))
- n))"
+lemma ZERO_LT_TWOEXP: "(0::nat) < (2::nat) ^ (n::nat)"
by (import bits ZERO_LT_TWOEXP)
-lemma MOD_2EXP_LT: "ALL (n::nat) k::nat. k mod 2 ^ n < 2 ^ n"
+lemma MOD_2EXP_LT: "(k::nat) mod (2::nat) ^ (n::nat) < (2::nat) ^ n"
by (import bits MOD_2EXP_LT)
-lemma TWOEXP_DIVISION: "ALL (n::nat) k::nat. k = k div 2 ^ n * 2 ^ n + k mod 2 ^ n"
+lemma TWOEXP_DIVISION: "(k::nat) = k div (2::nat) ^ (n::nat) * (2::nat) ^ n + k mod (2::nat) ^ n"
by (import bits TWOEXP_DIVISION)
-lemma TWOEXP_MONO: "(All::(nat => bool) => bool)
- (%a::nat.
- (All::(nat => bool) => bool)
- (%b::nat.
- (op -->::bool => bool => bool) ((op <::nat => nat => bool) a b)
- ((op <::nat => nat => bool)
- ((op ^::nat => nat => nat)
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int))))
- a)
- ((op ^::nat => nat => nat)
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int))))
- b))))"
+lemma TWOEXP_MONO: "(a::nat) < (b::nat) ==> (2::nat) ^ a < (2::nat) ^ b"
by (import bits TWOEXP_MONO)
-lemma TWOEXP_MONO2: "(All::(nat => bool) => bool)
- (%a::nat.
- (All::(nat => bool) => bool)
- (%b::nat.
- (op -->::bool => bool => bool) ((op <=::nat => nat => bool) a b)
- ((op <=::nat => nat => bool)
- ((op ^::nat => nat => nat)
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int))))
- a)
- ((op ^::nat => nat => nat)
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int))))
- b))))"
+lemma TWOEXP_MONO2: "(a::nat) <= (b::nat) ==> (2::nat) ^ a <= (2::nat) ^ b"
by (import bits TWOEXP_MONO2)
-lemma EXP_SUB_LESS_EQ: "(All::(nat => bool) => bool)
- (%a::nat.
- (All::(nat => bool) => bool)
- (%b::nat.
- (op <=::nat => nat => bool)
- ((op ^::nat => nat => nat)
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int))))
- ((op -::nat => nat => nat) a b))
- ((op ^::nat => nat => nat)
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int))))
- a)))"
+lemma EXP_SUB_LESS_EQ: "(2::nat) ^ ((a::nat) - (b::nat)) <= (2::nat) ^ a"
by (import bits EXP_SUB_LESS_EQ)
-lemma BITS_THM: "ALL (x::nat) (xa::nat) xb::nat.
- BITS x xa xb = xb div 2 ^ xa mod 2 ^ (Suc x - xa)"
+lemma BITS_THM: "BITS x xa xb = xb div 2 ^ xa mod 2 ^ (Suc x - xa)"
by (import bits BITS_THM)
-lemma BITSLT_THM: "ALL (h::nat) (l::nat) n::nat. BITS h l n < 2 ^ (Suc h - l)"
+lemma BITSLT_THM: "BITS h l n < 2 ^ (Suc h - l)"
by (import bits BITSLT_THM)
-lemma DIV_MULT_LEM: "ALL (m::nat) n::nat. 0 < n --> m div n * n <= m"
+lemma DIV_MULT_LEM: "(0::nat) < (n::nat) ==> (m::nat) div n * n <= m"
by (import bits DIV_MULT_LEM)
-lemma MOD_2EXP_LEM: "ALL (n::nat) x::nat. n mod 2 ^ x = n - n div 2 ^ x * 2 ^ x"
+lemma MOD_2EXP_LEM: "(n::nat) mod (2::nat) ^ (x::nat) = n - n div (2::nat) ^ x * (2::nat) ^ x"
by (import bits MOD_2EXP_LEM)
-lemma BITS2_THM: "ALL (h::nat) (l::nat) n::nat. BITS h l n = n mod 2 ^ Suc h div 2 ^ l"
+lemma BITS2_THM: "BITS h l n = n mod 2 ^ Suc h div 2 ^ l"
by (import bits BITS2_THM)
-lemma BITS_COMP_THM: "ALL (h1::nat) (l1::nat) (h2::nat) (l2::nat) n::nat.
- h2 + l1 <= h1 --> BITS h2 l2 (BITS h1 l1 n) = BITS (h2 + l1) (l2 + l1) n"
+lemma BITS_COMP_THM: "h2 + l1 <= h1 ==> BITS h2 l2 (BITS h1 l1 n) = BITS (h2 + l1) (l2 + l1) n"
by (import bits BITS_COMP_THM)
-lemma BITS_DIV_THM: "ALL (h::nat) (l::nat) (x::nat) n::nat.
- BITS h l x div 2 ^ n = BITS h (l + n) x"
+lemma BITS_DIV_THM: "BITS h l x div 2 ^ n = BITS h (l + n) x"
by (import bits BITS_DIV_THM)
-lemma BITS_LT_HIGH: "ALL (h::nat) (l::nat) n::nat. n < 2 ^ Suc h --> BITS h l n = n div 2 ^ l"
+lemma BITS_LT_HIGH: "n < 2 ^ Suc h ==> BITS h l n = n div 2 ^ l"
by (import bits BITS_LT_HIGH)
-lemma BITS_ZERO: "ALL (h::nat) (l::nat) n::nat. h < l --> BITS h l n = 0"
+lemma BITS_ZERO: "h < l ==> BITS h l n = 0"
by (import bits BITS_ZERO)
-lemma BITS_ZERO2: "ALL (h::nat) l::nat. BITS h l 0 = 0"
+lemma BITS_ZERO2: "BITS h l 0 = 0"
by (import bits BITS_ZERO2)
-lemma BITS_ZERO3: "ALL (h::nat) x::nat. BITS h 0 x = x mod 2 ^ Suc h"
+lemma BITS_ZERO3: "BITS h 0 x = x mod 2 ^ Suc h"
by (import bits BITS_ZERO3)
-lemma BITS_COMP_THM2: "ALL (h1::nat) (l1::nat) (h2::nat) (l2::nat) n::nat.
- BITS h2 l2 (BITS h1 l1 n) = BITS (min h1 (h2 + l1)) (l2 + l1) n"
+lemma BITS_COMP_THM2: "BITS h2 l2 (BITS h1 l1 n) = BITS (min h1 (h2 + l1)) (l2 + l1) n"
by (import bits BITS_COMP_THM2)
-lemma NOT_MOD2_LEM: "ALL n::nat. (n mod 2 ~= 0) = (n mod 2 = 1)"
+lemma NOT_MOD2_LEM: "((n::nat) mod (2::nat) ~= (0::nat)) = (n mod (2::nat) = (1::nat))"
by (import bits NOT_MOD2_LEM)
-lemma NOT_MOD2_LEM2: "ALL (n::nat) a::'a::type. (n mod 2 ~= 1) = (n mod 2 = 0)"
+lemma NOT_MOD2_LEM2: "((n::nat) mod (2::nat) ~= (1::nat)) = (n mod (2::nat) = (0::nat))"
by (import bits NOT_MOD2_LEM2)
-lemma EVEN_MOD2_LEM: "ALL n::nat. EVEN n = (n mod 2 = 0)"
+lemma EVEN_MOD2_LEM: "EVEN n = (n mod 2 = 0)"
by (import bits EVEN_MOD2_LEM)
-lemma ODD_MOD2_LEM: "ALL n::nat. ODD n = (n mod 2 = 1)"
+lemma ODD_MOD2_LEM: "ODD n = (n mod 2 = 1)"
by (import bits ODD_MOD2_LEM)
lemma LSB_ODD: "LSBn = ODD"
by (import bits LSB_ODD)
-lemma DIV_MULT_THM: "ALL (x::nat) n::nat. n div 2 ^ x * 2 ^ x = n - n mod 2 ^ x"
+lemma DIV_MULT_THM: "(n::nat) div (2::nat) ^ (x::nat) * (2::nat) ^ x = n - n mod (2::nat) ^ x"
by (import bits DIV_MULT_THM)
-lemma DIV_MULT_THM2: "ALL x::nat. 2 * (x div 2) = x - x mod 2"
+lemma DIV_MULT_THM2: "(2::nat) * ((x::nat) div (2::nat)) = x - x mod (2::nat)"
by (import bits DIV_MULT_THM2)
-lemma LESS_EQ_EXP_MULT: "ALL (a::nat) b::nat. a <= b --> (EX x::nat. 2 ^ b = x * 2 ^ a)"
+lemma LESS_EQ_EXP_MULT: "(a::nat) <= (b::nat) ==> EX x::nat. (2::nat) ^ b = x * (2::nat) ^ a"
by (import bits LESS_EQ_EXP_MULT)
-lemma SLICE_LEM1: "ALL (a::nat) (x::nat) y::nat.
- a div 2 ^ (x + y) * 2 ^ (x + y) =
- a div 2 ^ x * 2 ^ x - a div 2 ^ x mod 2 ^ y * 2 ^ x"
+lemma SLICE_LEM1: "(a::nat) div (2::nat) ^ ((x::nat) + (y::nat)) * (2::nat) ^ (x + y) =
+a div (2::nat) ^ x * (2::nat) ^ x -
+a div (2::nat) ^ x mod (2::nat) ^ y * (2::nat) ^ x"
by (import bits SLICE_LEM1)
-lemma SLICE_LEM2: "ALL (a::'a::type) (x::nat) y::nat.
- (n::nat) mod 2 ^ (x + y) = n mod 2 ^ x + n div 2 ^ x mod 2 ^ y * 2 ^ x"
+lemma SLICE_LEM2: "(n::nat) mod (2::nat) ^ ((x::nat) + (y::nat)) =
+n mod (2::nat) ^ x + n div (2::nat) ^ x mod (2::nat) ^ y * (2::nat) ^ x"
by (import bits SLICE_LEM2)
-lemma SLICE_LEM3: "ALL (n::nat) (h::nat) l::nat. l < h --> n mod 2 ^ Suc l <= n mod 2 ^ h"
+lemma SLICE_LEM3: "(l::nat) < (h::nat) ==> (n::nat) mod (2::nat) ^ Suc l <= n mod (2::nat) ^ h"
by (import bits SLICE_LEM3)
-lemma SLICE_THM: "ALL (n::nat) (h::nat) l::nat. SLICE h l n = BITS h l n * 2 ^ l"
+lemma SLICE_THM: "SLICE h l n = BITS h l n * 2 ^ l"
by (import bits SLICE_THM)
-lemma SLICELT_THM: "ALL (h::nat) (l::nat) n::nat. SLICE h l n < 2 ^ Suc h"
+lemma SLICELT_THM: "SLICE h l n < 2 ^ Suc h"
by (import bits SLICELT_THM)
-lemma BITS_SLICE_THM: "ALL (h::nat) (l::nat) n::nat. BITS h l (SLICE h l n) = BITS h l n"
+lemma BITS_SLICE_THM: "BITS h l (SLICE h l n) = BITS h l n"
by (import bits BITS_SLICE_THM)
-lemma BITS_SLICE_THM2: "ALL (h::nat) (l::nat) n::nat.
- h <= (h2::nat) --> BITS h2 l (SLICE h l n) = BITS h l n"
+lemma BITS_SLICE_THM2: "h <= h2 ==> BITS h2 l (SLICE h l n) = BITS h l n"
by (import bits BITS_SLICE_THM2)
-lemma MOD_2EXP_MONO: "ALL (n::nat) (h::nat) l::nat. l <= h --> n mod 2 ^ l <= n mod 2 ^ Suc h"
+lemma MOD_2EXP_MONO: "(l::nat) <= (h::nat) ==> (n::nat) mod (2::nat) ^ l <= n mod (2::nat) ^ Suc h"
by (import bits MOD_2EXP_MONO)
-lemma SLICE_COMP_THM: "ALL (h::nat) (m::nat) (l::nat) n::nat.
- Suc m <= h & l <= m --> SLICE h (Suc m) n + SLICE m l n = SLICE h l n"
+lemma SLICE_COMP_THM: "Suc m <= h & l <= m ==> SLICE h (Suc m) n + SLICE m l n = SLICE h l n"
by (import bits SLICE_COMP_THM)
-lemma SLICE_ZERO: "ALL (h::nat) (l::nat) n::nat. h < l --> SLICE h l n = 0"
+lemma SLICE_ZERO: "h < l ==> SLICE h l n = 0"
by (import bits SLICE_ZERO)
-lemma BIT_COMP_THM3: "ALL (h::nat) (m::nat) (l::nat) n::nat.
- Suc m <= h & l <= m -->
- BITS h (Suc m) n * 2 ^ (Suc m - l) + BITS m l n = BITS h l n"
+lemma BIT_COMP_THM3: "Suc m <= h & l <= m
+==> BITS h (Suc m) n * 2 ^ (Suc m - l) + BITS m l n = BITS h l n"
by (import bits BIT_COMP_THM3)
-lemma NOT_BIT: "ALL (n::nat) a::nat. (~ bit n a) = (BITS n n a = 0)"
+lemma NOT_BIT: "(~ bit n a) = (BITS n n a = 0)"
by (import bits NOT_BIT)
-lemma NOT_BITS: "ALL (n::nat) a::nat. (BITS n n a ~= 0) = (BITS n n a = 1)"
+lemma NOT_BITS: "(BITS n n a ~= 0) = (BITS n n a = 1)"
by (import bits NOT_BITS)
-lemma NOT_BITS2: "ALL (n::nat) a::nat. (BITS n n a ~= 1) = (BITS n n a = 0)"
+lemma NOT_BITS2: "(BITS n n a ~= 1) = (BITS n n a = 0)"
by (import bits NOT_BITS2)
-lemma BIT_SLICE: "ALL (n::nat) (a::nat) b::nat.
- (bit n a = bit n b) = (SLICE n n a = SLICE n n b)"
+lemma BIT_SLICE: "(bit n a = bit n b) = (SLICE n n a = SLICE n n b)"
by (import bits BIT_SLICE)
-lemma BIT_SLICE_LEM: "ALL (y::nat) (x::nat) n::nat. SBIT (bit x n) (x + y) = SLICE x x n * 2 ^ y"
+lemma BIT_SLICE_LEM: "SBIT (bit x n) (x + y) = SLICE x x n * 2 ^ y"
by (import bits BIT_SLICE_LEM)
-lemma BIT_SLICE_THM: "ALL (x::nat) xa::nat. SBIT (bit x xa) x = SLICE x x xa"
+lemma BIT_SLICE_THM: "SBIT (bit x xa) x = SLICE x x xa"
by (import bits BIT_SLICE_THM)
-lemma SBIT_DIV: "ALL (b::bool) (m::nat) n::nat. n < m --> SBIT b (m - n) = SBIT b m div 2 ^ n"
+lemma SBIT_DIV: "n < m ==> SBIT b (m - n) = SBIT b m div 2 ^ n"
by (import bits SBIT_DIV)
-lemma BITS_SUC: "ALL (h::nat) (l::nat) n::nat.
- l <= Suc h -->
- SBIT (bit (Suc h) n) (Suc h - l) + BITS h l n = BITS (Suc h) l n"
+lemma BITS_SUC: "l <= Suc h
+==> SBIT (bit (Suc h) n) (Suc h - l) + BITS h l n = BITS (Suc h) l n"
by (import bits BITS_SUC)
-lemma BITS_SUC_THM: "ALL (h::nat) (l::nat) n::nat.
- BITS (Suc h) l n =
- (if Suc h < l then 0 else SBIT (bit (Suc h) n) (Suc h - l) + BITS h l n)"
+lemma BITS_SUC_THM: "BITS (Suc h) l n =
+(if Suc h < l then 0 else SBIT (bit (Suc h) n) (Suc h - l) + BITS h l n)"
by (import bits BITS_SUC_THM)
-lemma BIT_BITS_THM: "ALL (h::nat) (l::nat) (a::nat) b::nat.
- (ALL x::nat. l <= x & x <= h --> bit x a = bit x b) =
- (BITS h l a = BITS h l b)"
+lemma BIT_BITS_THM: "(ALL x. l <= x & x <= h --> bit x a = bit x b) = (BITS h l a = BITS h l b)"
by (import bits BIT_BITS_THM)
-lemma BITWISE_LT_2EXP: "ALL (n::nat) (oper::bool => bool => bool) (a::nat) b::nat.
- BITWISE n oper a b < 2 ^ n"
+lemma BITWISE_LT_2EXP: "BITWISE n oper a b < 2 ^ n"
by (import bits BITWISE_LT_2EXP)
-lemma LESS_EXP_MULT2: "(All::(nat => bool) => bool)
- (%a::nat.
- (All::(nat => bool) => bool)
- (%b::nat.
- (op -->::bool => bool => bool) ((op <::nat => nat => bool) a b)
- ((Ex::(nat => bool) => bool)
- (%x::nat.
- (op =::nat => nat => bool)
- ((op ^::nat => nat => nat)
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int))))
- b)
- ((op *::nat => nat => nat)
- ((op ^::nat => nat => nat)
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int))))
- ((op +::nat => nat => nat) x (1::nat)))
- ((op ^::nat => nat => nat)
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int))))
- a))))))"
+lemma LESS_EXP_MULT2: "(a::nat) < (b::nat)
+==> EX x::nat. (2::nat) ^ b = (2::nat) ^ (x + (1::nat)) * (2::nat) ^ a"
by (import bits LESS_EXP_MULT2)
-lemma BITWISE_THM: "ALL (x::nat) (n::nat) (oper::bool => bool => bool) (a::nat) b::nat.
- x < n --> bit x (BITWISE n oper a b) = oper (bit x a) (bit x b)"
+lemma BITWISE_THM: "x < n ==> bit x (BITWISE n oper a b) = oper (bit x a) (bit x b)"
by (import bits BITWISE_THM)
-lemma BITWISE_COR: "ALL (x::nat) (n::nat) (oper::bool => bool => bool) (a::nat) b::nat.
- x < n -->
- oper (bit x a) (bit x b) --> BITWISE n oper a b div 2 ^ x mod 2 = 1"
+lemma BITWISE_COR: "[| x < n; oper (bit x a) (bit x b) |]
+==> BITWISE n oper a b div 2 ^ x mod 2 = 1"
by (import bits BITWISE_COR)
-lemma BITWISE_NOT_COR: "ALL (x::nat) (n::nat) (oper::bool => bool => bool) (a::nat) b::nat.
- x < n -->
- ~ oper (bit x a) (bit x b) --> BITWISE n oper a b div 2 ^ x mod 2 = 0"
+lemma BITWISE_NOT_COR: "[| x < n; ~ oper (bit x a) (bit x b) |]
+==> BITWISE n oper a b div 2 ^ x mod 2 = 0"
by (import bits BITWISE_NOT_COR)
-lemma MOD_PLUS_RIGHT: "ALL n>0. ALL (j::nat) k::nat. (j + k mod n) mod n = (j + k) mod n"
+lemma MOD_PLUS_RIGHT: "(0::nat) < (n::nat) ==> ((j::nat) + (k::nat) mod n) mod n = (j + k) mod n"
by (import bits MOD_PLUS_RIGHT)
-lemma MOD_PLUS_1: "ALL n>0. ALL x::nat. ((x + 1) mod n = 0) = (x mod n + 1 = n)"
+lemma MOD_PLUS_1: "(0::nat) < (n::nat)
+==> (((x::nat) + (1::nat)) mod n = (0::nat)) = (x mod n + (1::nat) = n)"
by (import bits MOD_PLUS_1)
-lemma MOD_ADD_1: "ALL n>0. ALL x::nat. (x + 1) mod n ~= 0 --> (x + 1) mod n = x mod n + 1"
+lemma MOD_ADD_1: "[| (0::nat) < (n::nat); ((x::nat) + (1::nat)) mod n ~= (0::nat) |]
+==> (x + (1::nat)) mod n = x mod n + (1::nat)"
by (import bits MOD_ADD_1)
;end_setup
@@ -406,63 +315,57 @@
MODw :: "nat => nat"
defs
- MODw_primdef: "MODw == %n::nat. n mod 2 ^ WL"
+ MODw_primdef: "MODw == %n. n mod 2 ^ WL"
-lemma MODw_def: "ALL n::nat. MODw n = n mod 2 ^ WL"
+lemma MODw_def: "MODw n = n mod 2 ^ WL"
by (import word32 MODw_def)
consts
INw :: "nat => bool"
defs
- INw_primdef: "INw == %n::nat. n < 2 ^ WL"
+ INw_primdef: "INw == %n. n < 2 ^ WL"
-lemma INw_def: "ALL n::nat. INw n = (n < 2 ^ WL)"
+lemma INw_def: "INw n = (n < 2 ^ WL)"
by (import word32 INw_def)
consts
EQUIV :: "nat => nat => bool"
defs
- EQUIV_primdef: "EQUIV == %(x::nat) y::nat. MODw x = MODw y"
+ EQUIV_primdef: "EQUIV == %x y. MODw x = MODw y"
-lemma EQUIV_def: "ALL (x::nat) y::nat. EQUIV x y = (MODw x = MODw y)"
+lemma EQUIV_def: "EQUIV x y = (MODw x = MODw y)"
by (import word32 EQUIV_def)
-lemma EQUIV_QT: "ALL (x::nat) y::nat. EQUIV x y = (EQUIV x = EQUIV y)"
+lemma EQUIV_QT: "EQUIV x y = (EQUIV x = EQUIV y)"
by (import word32 EQUIV_QT)
-lemma FUNPOW_THM: "ALL (f::'a::type => 'a::type) (n::nat) x::'a::type.
- (f ^^ n) (f x) = f ((f ^^ n) x)"
- by (import word32 FUNPOW_THM)
-
-lemma FUNPOW_THM2: "ALL (f::'a::type => 'a::type) (n::nat) x::'a::type.
- (f ^^ Suc n) x = f ((f ^^ n) x)"
+lemma FUNPOW_THM2: "(f ^^ Suc n) x = f ((f ^^ n) x)"
by (import word32 FUNPOW_THM2)
-lemma FUNPOW_COMP: "ALL (f::'a::type => 'a::type) (m::nat) (n::nat) a::'a::type.
- (f ^^ m) ((f ^^ n) a) = (f ^^ (m + n)) a"
+lemma FUNPOW_COMP: "(f ^^ m) ((f ^^ n) a) = (f ^^ (m + n)) a"
by (import word32 FUNPOW_COMP)
-lemma INw_MODw: "ALL n::nat. INw (MODw n)"
+lemma INw_MODw: "INw (MODw n)"
by (import word32 INw_MODw)
-lemma TOw_IDEM: "ALL a::nat. INw a --> MODw a = a"
+lemma TOw_IDEM: "INw a ==> MODw a = a"
by (import word32 TOw_IDEM)
-lemma MODw_IDEM2: "ALL a::nat. MODw (MODw a) = MODw a"
+lemma MODw_IDEM2: "MODw (MODw a) = MODw a"
by (import word32 MODw_IDEM2)
-lemma TOw_QT: "ALL a::nat. EQUIV (MODw a) a"
+lemma TOw_QT: "EQUIV (MODw a) a"
by (import word32 TOw_QT)
lemma MODw_THM: "MODw = BITS HB 0"
by (import word32 MODw_THM)
-lemma MOD_ADD: "ALL (a::nat) b::nat. MODw (a + b) = MODw (MODw a + MODw b)"
+lemma MOD_ADD: "MODw (a + b) = MODw (MODw a + MODw b)"
by (import word32 MOD_ADD)
-lemma MODw_MULT: "ALL (a::nat) b::nat. MODw (a * b) = MODw (MODw a * MODw b)"
+lemma MODw_MULT: "MODw (a * b) = MODw (MODw a * MODw b)"
by (import word32 MODw_MULT)
consts
@@ -474,65 +377,62 @@
lemma AONE_def: "AONE = 1"
by (import word32 AONE_def)
-lemma ADD_QT: "(ALL n::nat. EQUIV (0 + n) n) &
-(ALL (m::nat) n::nat. EQUIV (Suc m + n) (Suc (m + n)))"
+lemma ADD_QT: "(ALL n. EQUIV (0 + n) n) & (ALL m n. EQUIV (Suc m + n) (Suc (m + n)))"
by (import word32 ADD_QT)
-lemma ADD_0_QT: "ALL a::nat. EQUIV (a + 0) a"
+lemma ADD_0_QT: "EQUIV (a + 0) a"
by (import word32 ADD_0_QT)
-lemma ADD_COMM_QT: "ALL (a::nat) b::nat. EQUIV (a + b) (b + a)"
+lemma ADD_COMM_QT: "EQUIV (a + b) (b + a)"
by (import word32 ADD_COMM_QT)
-lemma ADD_ASSOC_QT: "ALL (a::nat) (b::nat) c::nat. EQUIV (a + (b + c)) (a + b + c)"
+lemma ADD_ASSOC_QT: "EQUIV (a + (b + c)) (a + b + c)"
by (import word32 ADD_ASSOC_QT)
-lemma MULT_QT: "(ALL n::nat. EQUIV (0 * n) 0) &
-(ALL (m::nat) n::nat. EQUIV (Suc m * n) (m * n + n))"
+lemma MULT_QT: "(ALL n. EQUIV (0 * n) 0) & (ALL m n. EQUIV (Suc m * n) (m * n + n))"
by (import word32 MULT_QT)
-lemma ADD1_QT: "ALL m::nat. EQUIV (Suc m) (m + AONE)"
+lemma ADD1_QT: "EQUIV (Suc m) (m + AONE)"
by (import word32 ADD1_QT)
-lemma ADD_CLAUSES_QT: "(ALL m::nat. EQUIV (0 + m) m) &
-(ALL m::nat. EQUIV (m + 0) m) &
-(ALL (m::nat) n::nat. EQUIV (Suc m + n) (Suc (m + n))) &
-(ALL (m::nat) n::nat. EQUIV (m + Suc n) (Suc (m + n)))"
+lemma ADD_CLAUSES_QT: "(ALL m. EQUIV (0 + m) m) &
+(ALL m. EQUIV (m + 0) m) &
+(ALL m n. EQUIV (Suc m + n) (Suc (m + n))) &
+(ALL m n. EQUIV (m + Suc n) (Suc (m + n)))"
by (import word32 ADD_CLAUSES_QT)
-lemma SUC_EQUIV_COMP: "ALL (a::nat) b::nat. EQUIV (Suc a) b --> EQUIV a (b + (2 ^ WL - 1))"
+lemma SUC_EQUIV_COMP: "EQUIV (Suc a) b ==> EQUIV a (b + (2 ^ WL - 1))"
by (import word32 SUC_EQUIV_COMP)
-lemma INV_SUC_EQ_QT: "ALL (m::nat) n::nat. EQUIV (Suc m) (Suc n) = EQUIV m n"
+lemma INV_SUC_EQ_QT: "EQUIV (Suc m) (Suc n) = EQUIV m n"
by (import word32 INV_SUC_EQ_QT)
-lemma ADD_INV_0_QT: "ALL (m::nat) n::nat. EQUIV (m + n) m --> EQUIV n 0"
+lemma ADD_INV_0_QT: "EQUIV (m + n) m ==> EQUIV n 0"
by (import word32 ADD_INV_0_QT)
-lemma ADD_INV_0_EQ_QT: "ALL (m::nat) n::nat. EQUIV (m + n) m = EQUIV n 0"
+lemma ADD_INV_0_EQ_QT: "EQUIV (m + n) m = EQUIV n 0"
by (import word32 ADD_INV_0_EQ_QT)
-lemma EQ_ADD_LCANCEL_QT: "ALL (m::nat) (n::nat) p::nat. EQUIV (m + n) (m + p) = EQUIV n p"
+lemma EQ_ADD_LCANCEL_QT: "EQUIV (m + n) (m + p) = EQUIV n p"
by (import word32 EQ_ADD_LCANCEL_QT)
-lemma EQ_ADD_RCANCEL_QT: "ALL (x::nat) (xa::nat) xb::nat. EQUIV (x + xb) (xa + xb) = EQUIV x xa"
+lemma EQ_ADD_RCANCEL_QT: "EQUIV (x + xb) (xa + xb) = EQUIV x xa"
by (import word32 EQ_ADD_RCANCEL_QT)
-lemma LEFT_ADD_DISTRIB_QT: "ALL (m::nat) (n::nat) p::nat. EQUIV (p * (m + n)) (p * m + p * n)"
+lemma LEFT_ADD_DISTRIB_QT: "EQUIV (p * (m + n)) (p * m + p * n)"
by (import word32 LEFT_ADD_DISTRIB_QT)
-lemma MULT_ASSOC_QT: "ALL (m::nat) (n::nat) p::nat. EQUIV (m * (n * p)) (m * n * p)"
+lemma MULT_ASSOC_QT: "EQUIV (m * (n * p)) (m * n * p)"
by (import word32 MULT_ASSOC_QT)
-lemma MULT_COMM_QT: "ALL (m::nat) n::nat. EQUIV (m * n) (n * m)"
+lemma MULT_COMM_QT: "EQUIV (m * n) (n * m)"
by (import word32 MULT_COMM_QT)
-lemma MULT_CLAUSES_QT: "ALL (m::nat) n::nat.
- EQUIV (0 * m) 0 &
- EQUIV (m * 0) 0 &
- EQUIV (AONE * m) m &
- EQUIV (m * AONE) m &
- EQUIV (Suc m * n) (m * n + n) & EQUIV (m * Suc n) (m + m * n)"
+lemma MULT_CLAUSES_QT: "EQUIV (0 * m) 0 &
+EQUIV (m * 0) 0 &
+EQUIV (AONE * m) m &
+EQUIV (m * AONE) m &
+EQUIV (Suc m * n) (m * n + n) & EQUIV (m * Suc n) (m + m * n)"
by (import word32 MULT_CLAUSES_QT)
consts
@@ -548,48 +448,36 @@
ONE_COMP :: "nat => nat"
defs
- ONE_COMP_primdef: "ONE_COMP == %x::nat. 2 ^ WL - 1 - MODw x"
+ ONE_COMP_primdef: "ONE_COMP == %x. 2 ^ WL - 1 - MODw x"
-lemma ONE_COMP_def: "ALL x::nat. ONE_COMP x = 2 ^ WL - 1 - MODw x"
+lemma ONE_COMP_def: "ONE_COMP x = 2 ^ WL - 1 - MODw x"
by (import word32 ONE_COMP_def)
consts
TWO_COMP :: "nat => nat"
defs
- TWO_COMP_primdef: "TWO_COMP == %x::nat. 2 ^ WL - MODw x"
+ TWO_COMP_primdef: "TWO_COMP == %x. 2 ^ WL - MODw x"
-lemma TWO_COMP_def: "ALL x::nat. TWO_COMP x = 2 ^ WL - MODw x"
+lemma TWO_COMP_def: "TWO_COMP x = 2 ^ WL - MODw x"
by (import word32 TWO_COMP_def)
-lemma ADD_TWO_COMP_QT: "ALL a::nat. EQUIV (MODw a + TWO_COMP a) 0"
+lemma ADD_TWO_COMP_QT: "EQUIV (MODw a + TWO_COMP a) 0"
by (import word32 ADD_TWO_COMP_QT)
-lemma TWO_COMP_ONE_COMP_QT: "ALL a::nat. EQUIV (TWO_COMP a) (ONE_COMP a + AONE)"
+lemma TWO_COMP_ONE_COMP_QT: "EQUIV (TWO_COMP a) (ONE_COMP a + AONE)"
by (import word32 TWO_COMP_ONE_COMP_QT)
-lemma BIT_EQUIV_THM: "(All::(nat => bool) => bool)
- (%x::nat.
- (All::(nat => bool) => bool)
- (%xa::nat.
- (op =::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%xb::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) xb (WL::nat))
- ((op =::bool => bool => bool)
- ((bit::nat => nat => bool) xb x)
- ((bit::nat => nat => bool) xb xa))))
- ((EQUIV::nat => nat => bool) x xa)))"
+lemma BIT_EQUIV_THM: "(ALL xb<WL. bit xb x = bit xb xa) = EQUIV x xa"
by (import word32 BIT_EQUIV_THM)
-lemma BITS_SUC2: "ALL (n::nat) a::nat. BITS (Suc n) 0 a = SLICE (Suc n) (Suc n) a + BITS n 0 a"
+lemma BITS_SUC2: "BITS (Suc n) 0 a = SLICE (Suc n) (Suc n) a + BITS n 0 a"
by (import word32 BITS_SUC2)
-lemma BITWISE_ONE_COMP_THM: "ALL (a::nat) b::nat. BITWISE WL (%(x::bool) y::bool. ~ x) a b = ONE_COMP a"
+lemma BITWISE_ONE_COMP_THM: "BITWISE WL (%x y. ~ x) a b = ONE_COMP a"
by (import word32 BITWISE_ONE_COMP_THM)
-lemma ONE_COMP_THM: "ALL (x::nat) xa::nat. xa < WL --> bit xa (ONE_COMP x) = (~ bit xa x)"
+lemma ONE_COMP_THM: "xa < WL ==> bit xa (ONE_COMP x) = (~ bit xa x)"
by (import word32 ONE_COMP_THM)
consts
@@ -614,9 +502,9 @@
EOR :: "nat => nat => nat"
defs
- EOR_primdef: "EOR == BITWISE WL (%(x::bool) y::bool. x ~= y)"
+ EOR_primdef: "EOR == BITWISE WL op ~="
-lemma EOR_def: "EOR = BITWISE WL (%(x::bool) y::bool. x ~= y)"
+lemma EOR_def: "EOR = BITWISE WL op ~="
by (import word32 EOR_def)
consts
@@ -628,177 +516,147 @@
lemma COMP0_def: "COMP0 = ONE_COMP 0"
by (import word32 COMP0_def)
-lemma BITWISE_THM2: "(All::(nat => bool) => bool)
- (%y::nat.
- (All::((bool => bool => bool) => bool) => bool)
- (%oper::bool => bool => bool.
- (All::(nat => bool) => bool)
- (%a::nat.
- (All::(nat => bool) => bool)
- (%b::nat.
- (op =::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%x::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) x (WL::nat))
- ((op =::bool => bool => bool)
- (oper ((bit::nat => nat => bool) x a)
- ((bit::nat => nat => bool) x b))
- ((bit::nat => nat => bool) x y))))
- ((EQUIV::nat => nat => bool)
- ((BITWISE::nat
- => (bool => bool => bool)
- => nat => nat => nat)
- (WL::nat) oper a b)
- y)))))"
+lemma BITWISE_THM2: "(ALL x<WL. oper (bit x a) (bit x b) = bit x y) =
+EQUIV (BITWISE WL oper a b) y"
by (import word32 BITWISE_THM2)
-lemma OR_ASSOC_QT: "ALL (a::nat) (b::nat) c::nat. EQUIV (OR a (OR b c)) (OR (OR a b) c)"
+lemma OR_ASSOC_QT: "EQUIV (OR a (OR b c)) (OR (OR a b) c)"
by (import word32 OR_ASSOC_QT)
-lemma OR_COMM_QT: "ALL (a::nat) b::nat. EQUIV (OR a b) (OR b a)"
+lemma OR_COMM_QT: "EQUIV (OR a b) (OR b a)"
by (import word32 OR_COMM_QT)
-lemma OR_ABSORB_QT: "ALL (a::nat) b::nat. EQUIV (AND a (OR a b)) a"
+lemma OR_ABSORB_QT: "EQUIV (AND a (OR a b)) a"
by (import word32 OR_ABSORB_QT)
-lemma OR_IDEM_QT: "ALL a::nat. EQUIV (OR a a) a"
+lemma OR_IDEM_QT: "EQUIV (OR a a) a"
by (import word32 OR_IDEM_QT)
-lemma AND_ASSOC_QT: "ALL (a::nat) (b::nat) c::nat. EQUIV (AND a (AND b c)) (AND (AND a b) c)"
+lemma AND_ASSOC_QT: "EQUIV (AND a (AND b c)) (AND (AND a b) c)"
by (import word32 AND_ASSOC_QT)
-lemma AND_COMM_QT: "ALL (a::nat) b::nat. EQUIV (AND a b) (AND b a)"
+lemma AND_COMM_QT: "EQUIV (AND a b) (AND b a)"
by (import word32 AND_COMM_QT)
-lemma AND_ABSORB_QT: "ALL (a::nat) b::nat. EQUIV (OR a (AND a b)) a"
+lemma AND_ABSORB_QT: "EQUIV (OR a (AND a b)) a"
by (import word32 AND_ABSORB_QT)
-lemma AND_IDEM_QT: "ALL a::nat. EQUIV (AND a a) a"
+lemma AND_IDEM_QT: "EQUIV (AND a a) a"
by (import word32 AND_IDEM_QT)
-lemma OR_COMP_QT: "ALL a::nat. EQUIV (OR a (ONE_COMP a)) COMP0"
+lemma OR_COMP_QT: "EQUIV (OR a (ONE_COMP a)) COMP0"
by (import word32 OR_COMP_QT)
-lemma AND_COMP_QT: "ALL a::nat. EQUIV (AND a (ONE_COMP a)) 0"
+lemma AND_COMP_QT: "EQUIV (AND a (ONE_COMP a)) 0"
by (import word32 AND_COMP_QT)
-lemma ONE_COMP_QT: "ALL a::nat. EQUIV (ONE_COMP (ONE_COMP a)) a"
+lemma ONE_COMP_QT: "EQUIV (ONE_COMP (ONE_COMP a)) a"
by (import word32 ONE_COMP_QT)
-lemma RIGHT_AND_OVER_OR_QT: "ALL (a::nat) (b::nat) c::nat.
- EQUIV (AND (OR a b) c) (OR (AND a c) (AND b c))"
+lemma RIGHT_AND_OVER_OR_QT: "EQUIV (AND (OR a b) c) (OR (AND a c) (AND b c))"
by (import word32 RIGHT_AND_OVER_OR_QT)
-lemma RIGHT_OR_OVER_AND_QT: "ALL (a::nat) (b::nat) c::nat. EQUIV (OR (AND a b) c) (AND (OR a c) (OR b c))"
+lemma RIGHT_OR_OVER_AND_QT: "EQUIV (OR (AND a b) c) (AND (OR a c) (OR b c))"
by (import word32 RIGHT_OR_OVER_AND_QT)
-lemma DE_MORGAN_THM_QT: "ALL (a::nat) b::nat.
- EQUIV (ONE_COMP (AND a b)) (OR (ONE_COMP a) (ONE_COMP b)) &
- EQUIV (ONE_COMP (OR a b)) (AND (ONE_COMP a) (ONE_COMP b))"
+lemma DE_MORGAN_THM_QT: "EQUIV (ONE_COMP (AND a b)) (OR (ONE_COMP a) (ONE_COMP b)) &
+EQUIV (ONE_COMP (OR a b)) (AND (ONE_COMP a) (ONE_COMP b))"
by (import word32 DE_MORGAN_THM_QT)
-lemma BIT_EQUIV: "ALL (n::nat) (a::nat) b::nat. n < WL --> EQUIV a b --> bit n a = bit n b"
+lemma BIT_EQUIV: "[| n < WL; EQUIV a b |] ==> bit n a = bit n b"
by (import word32 BIT_EQUIV)
-lemma LSB_WELLDEF: "ALL (a::nat) b::nat. EQUIV a b --> LSBn a = LSBn b"
+lemma LSB_WELLDEF: "EQUIV a b ==> LSBn a = LSBn b"
by (import word32 LSB_WELLDEF)
-lemma MSB_WELLDEF: "ALL (a::nat) b::nat. EQUIV a b --> MSBn a = MSBn b"
+lemma MSB_WELLDEF: "EQUIV a b ==> MSBn a = MSBn b"
by (import word32 MSB_WELLDEF)
-lemma BITWISE_ISTEP: "ALL (n::nat) (oper::bool => bool => bool) (a::nat) b::nat.
- 0 < n -->
- BITWISE n oper (a div 2) (b div 2) =
- BITWISE n oper a b div 2 + SBIT (oper (bit n a) (bit n b)) (n - 1)"
+lemma BITWISE_ISTEP: "0 < n
+==> BITWISE n oper (a div 2) (b div 2) =
+ BITWISE n oper a b div 2 + SBIT (oper (bit n a) (bit n b)) (n - 1)"
by (import word32 BITWISE_ISTEP)
-lemma BITWISE_EVAL: "ALL (n::nat) (oper::bool => bool => bool) (a::nat) b::nat.
- BITWISE (Suc n) oper a b =
- 2 * BITWISE n oper (a div 2) (b div 2) + SBIT (oper (LSBn a) (LSBn b)) 0"
+lemma BITWISE_EVAL: "BITWISE (Suc n) oper a b =
+2 * BITWISE n oper (a div 2) (b div 2) + SBIT (oper (LSBn a) (LSBn b)) 0"
by (import word32 BITWISE_EVAL)
-lemma BITWISE_WELLDEF: "ALL (n::nat) (oper::bool => bool => bool) (a::nat) (b::nat) (c::nat) d::nat.
- EQUIV a b & EQUIV c d --> EQUIV (BITWISE n oper a c) (BITWISE n oper b d)"
+lemma BITWISE_WELLDEF: "EQUIV a b & EQUIV c d ==> EQUIV (BITWISE n oper a c) (BITWISE n oper b d)"
by (import word32 BITWISE_WELLDEF)
-lemma BITWISEw_WELLDEF: "ALL (oper::bool => bool => bool) (a::nat) (b::nat) (c::nat) d::nat.
- EQUIV a b & EQUIV c d -->
- EQUIV (BITWISE WL oper a c) (BITWISE WL oper b d)"
+lemma BITWISEw_WELLDEF: "EQUIV a b & EQUIV c d ==> EQUIV (BITWISE WL oper a c) (BITWISE WL oper b d)"
by (import word32 BITWISEw_WELLDEF)
-lemma SUC_WELLDEF: "ALL (a::nat) b::nat. EQUIV a b --> EQUIV (Suc a) (Suc b)"
+lemma SUC_WELLDEF: "EQUIV a b ==> EQUIV (Suc a) (Suc b)"
by (import word32 SUC_WELLDEF)
-lemma ADD_WELLDEF: "ALL (a::nat) (b::nat) (c::nat) d::nat.
- EQUIV a b & EQUIV c d --> EQUIV (a + c) (b + d)"
+lemma ADD_WELLDEF: "EQUIV a b & EQUIV c d ==> EQUIV (a + c) (b + d)"
by (import word32 ADD_WELLDEF)
-lemma MUL_WELLDEF: "ALL (a::nat) (b::nat) (c::nat) d::nat.
- EQUIV a b & EQUIV c d --> EQUIV (a * c) (b * d)"
+lemma MUL_WELLDEF: "EQUIV a b & EQUIV c d ==> EQUIV (a * c) (b * d)"
by (import word32 MUL_WELLDEF)
-lemma ONE_COMP_WELLDEF: "ALL (a::nat) b::nat. EQUIV a b --> EQUIV (ONE_COMP a) (ONE_COMP b)"
+lemma ONE_COMP_WELLDEF: "EQUIV a b ==> EQUIV (ONE_COMP a) (ONE_COMP b)"
by (import word32 ONE_COMP_WELLDEF)
-lemma TWO_COMP_WELLDEF: "ALL (a::nat) b::nat. EQUIV a b --> EQUIV (TWO_COMP a) (TWO_COMP b)"
+lemma TWO_COMP_WELLDEF: "EQUIV a b ==> EQUIV (TWO_COMP a) (TWO_COMP b)"
by (import word32 TWO_COMP_WELLDEF)
-lemma TOw_WELLDEF: "ALL (a::nat) b::nat. EQUIV a b --> EQUIV (MODw a) (MODw b)"
+lemma TOw_WELLDEF: "EQUIV a b ==> EQUIV (MODw a) (MODw b)"
by (import word32 TOw_WELLDEF)
consts
LSR_ONE :: "nat => nat"
defs
- LSR_ONE_primdef: "LSR_ONE == %a::nat. MODw a div 2"
+ LSR_ONE_primdef: "LSR_ONE == %a. MODw a div 2"
-lemma LSR_ONE_def: "ALL a::nat. LSR_ONE a = MODw a div 2"
+lemma LSR_ONE_def: "LSR_ONE a = MODw a div 2"
by (import word32 LSR_ONE_def)
consts
ASR_ONE :: "nat => nat"
defs
- ASR_ONE_primdef: "ASR_ONE == %a::nat. LSR_ONE a + SBIT (MSBn a) HB"
+ ASR_ONE_primdef: "ASR_ONE == %a. LSR_ONE a + SBIT (MSBn a) HB"
-lemma ASR_ONE_def: "ALL a::nat. ASR_ONE a = LSR_ONE a + SBIT (MSBn a) HB"
+lemma ASR_ONE_def: "ASR_ONE a = LSR_ONE a + SBIT (MSBn a) HB"
by (import word32 ASR_ONE_def)
consts
ROR_ONE :: "nat => nat"
defs
- ROR_ONE_primdef: "ROR_ONE == %a::nat. LSR_ONE a + SBIT (LSBn a) HB"
+ ROR_ONE_primdef: "ROR_ONE == %a. LSR_ONE a + SBIT (LSBn a) HB"
-lemma ROR_ONE_def: "ALL a::nat. ROR_ONE a = LSR_ONE a + SBIT (LSBn a) HB"
+lemma ROR_ONE_def: "ROR_ONE a = LSR_ONE a + SBIT (LSBn a) HB"
by (import word32 ROR_ONE_def)
consts
RRXn :: "bool => nat => nat"
defs
- RRXn_primdef: "RRXn == %(c::bool) a::nat. LSR_ONE a + SBIT c HB"
+ RRXn_primdef: "RRXn == %c a. LSR_ONE a + SBIT c HB"
-lemma RRXn_def: "ALL (c::bool) a::nat. RRXn c a = LSR_ONE a + SBIT c HB"
+lemma RRXn_def: "RRXn c a = LSR_ONE a + SBIT c HB"
by (import word32 RRXn_def)
-lemma LSR_ONE_WELLDEF: "ALL (a::nat) b::nat. EQUIV a b --> EQUIV (LSR_ONE a) (LSR_ONE b)"
+lemma LSR_ONE_WELLDEF: "EQUIV a b ==> EQUIV (LSR_ONE a) (LSR_ONE b)"
by (import word32 LSR_ONE_WELLDEF)
-lemma ASR_ONE_WELLDEF: "ALL (a::nat) b::nat. EQUIV a b --> EQUIV (ASR_ONE a) (ASR_ONE b)"
+lemma ASR_ONE_WELLDEF: "EQUIV a b ==> EQUIV (ASR_ONE a) (ASR_ONE b)"
by (import word32 ASR_ONE_WELLDEF)
-lemma ROR_ONE_WELLDEF: "ALL (a::nat) b::nat. EQUIV a b --> EQUIV (ROR_ONE a) (ROR_ONE b)"
+lemma ROR_ONE_WELLDEF: "EQUIV a b ==> EQUIV (ROR_ONE a) (ROR_ONE b)"
by (import word32 ROR_ONE_WELLDEF)
-lemma RRX_WELLDEF: "ALL (a::nat) (b::nat) c::bool. EQUIV a b --> EQUIV (RRXn c a) (RRXn c b)"
+lemma RRX_WELLDEF: "EQUIV a b ==> EQUIV (RRXn c a) (RRXn c b)"
by (import word32 RRX_WELLDEF)
lemma LSR_ONE: "LSR_ONE = BITS HB 1"
by (import word32 LSR_ONE)
-typedef (open) word32 = "{x::nat => bool. EX xa::nat. x = EQUIV xa}"
+typedef (open) word32 = "{x::nat => bool. EX xa. x = EQUIV xa}"
by (rule typedef_helper,import word32 word32_TY_DEF)
lemmas word32_TY_DEF = typedef_hol2hol4 [OF type_definition_word32]
@@ -807,9 +665,8 @@
mk_word32 :: "(nat => bool) => word32"
dest_word32 :: "word32 => nat => bool"
-specification (dest_word32 mk_word32) word32_tybij: "(ALL a::word32. mk_word32 (dest_word32 a) = a) &
-(ALL r::nat => bool.
- (EX x::nat. r = EQUIV x) = (dest_word32 (mk_word32 r) = r))"
+specification (dest_word32 mk_word32) word32_tybij: "(ALL a. mk_word32 (dest_word32 a) = a) &
+(ALL r. (EX x. r = EQUIV x) = (dest_word32 (mk_word32 r) = r))"
by (import word32 word32_tybij)
consts
@@ -839,258 +696,237 @@
lemma w_T_def: "w_T = mk_word32 (EQUIV COMP0)"
by (import word32 w_T_def)
-definition word_suc :: "word32 => word32" where
- "word_suc == %T1::word32. mk_word32 (EQUIV (Suc (Eps (dest_word32 T1))))"
+definition
+ word_suc :: "word32 => word32" where
+ "word_suc == %T1. mk_word32 (EQUIV (Suc (Eps (dest_word32 T1))))"
-lemma word_suc: "ALL T1::word32. word_suc T1 = mk_word32 (EQUIV (Suc (Eps (dest_word32 T1))))"
+lemma word_suc: "word_suc T1 = mk_word32 (EQUIV (Suc (Eps (dest_word32 T1))))"
by (import word32 word_suc)
-definition word_add :: "word32 => word32 => word32" where
+definition
+ word_add :: "word32 => word32 => word32" where
"word_add ==
-%(T1::word32) T2::word32.
- mk_word32 (EQUIV (Eps (dest_word32 T1) + Eps (dest_word32 T2)))"
+%T1 T2. mk_word32 (EQUIV (Eps (dest_word32 T1) + Eps (dest_word32 T2)))"
-lemma word_add: "ALL (T1::word32) T2::word32.
- word_add T1 T2 =
- mk_word32 (EQUIV (Eps (dest_word32 T1) + Eps (dest_word32 T2)))"
+lemma word_add: "word_add T1 T2 =
+mk_word32 (EQUIV (Eps (dest_word32 T1) + Eps (dest_word32 T2)))"
by (import word32 word_add)
-definition word_mul :: "word32 => word32 => word32" where
+definition
+ word_mul :: "word32 => word32 => word32" where
"word_mul ==
-%(T1::word32) T2::word32.
- mk_word32 (EQUIV (Eps (dest_word32 T1) * Eps (dest_word32 T2)))"
+%T1 T2. mk_word32 (EQUIV (Eps (dest_word32 T1) * Eps (dest_word32 T2)))"
-lemma word_mul: "ALL (T1::word32) T2::word32.
- word_mul T1 T2 =
- mk_word32 (EQUIV (Eps (dest_word32 T1) * Eps (dest_word32 T2)))"
+lemma word_mul: "word_mul T1 T2 =
+mk_word32 (EQUIV (Eps (dest_word32 T1) * Eps (dest_word32 T2)))"
by (import word32 word_mul)
-definition word_1comp :: "word32 => word32" where
- "word_1comp ==
-%T1::word32. mk_word32 (EQUIV (ONE_COMP (Eps (dest_word32 T1))))"
+definition
+ word_1comp :: "word32 => word32" where
+ "word_1comp == %T1. mk_word32 (EQUIV (ONE_COMP (Eps (dest_word32 T1))))"
-lemma word_1comp: "ALL T1::word32.
- word_1comp T1 = mk_word32 (EQUIV (ONE_COMP (Eps (dest_word32 T1))))"
+lemma word_1comp: "word_1comp T1 = mk_word32 (EQUIV (ONE_COMP (Eps (dest_word32 T1))))"
by (import word32 word_1comp)
-definition word_2comp :: "word32 => word32" where
- "word_2comp ==
-%T1::word32. mk_word32 (EQUIV (TWO_COMP (Eps (dest_word32 T1))))"
+definition
+ word_2comp :: "word32 => word32" where
+ "word_2comp == %T1. mk_word32 (EQUIV (TWO_COMP (Eps (dest_word32 T1))))"
-lemma word_2comp: "ALL T1::word32.
- word_2comp T1 = mk_word32 (EQUIV (TWO_COMP (Eps (dest_word32 T1))))"
+lemma word_2comp: "word_2comp T1 = mk_word32 (EQUIV (TWO_COMP (Eps (dest_word32 T1))))"
by (import word32 word_2comp)
-definition word_lsr1 :: "word32 => word32" where
- "word_lsr1 == %T1::word32. mk_word32 (EQUIV (LSR_ONE (Eps (dest_word32 T1))))"
+definition
+ word_lsr1 :: "word32 => word32" where
+ "word_lsr1 == %T1. mk_word32 (EQUIV (LSR_ONE (Eps (dest_word32 T1))))"
-lemma word_lsr1: "ALL T1::word32.
- word_lsr1 T1 = mk_word32 (EQUIV (LSR_ONE (Eps (dest_word32 T1))))"
+lemma word_lsr1: "word_lsr1 T1 = mk_word32 (EQUIV (LSR_ONE (Eps (dest_word32 T1))))"
by (import word32 word_lsr1)
-definition word_asr1 :: "word32 => word32" where
- "word_asr1 == %T1::word32. mk_word32 (EQUIV (ASR_ONE (Eps (dest_word32 T1))))"
+definition
+ word_asr1 :: "word32 => word32" where
+ "word_asr1 == %T1. mk_word32 (EQUIV (ASR_ONE (Eps (dest_word32 T1))))"
-lemma word_asr1: "ALL T1::word32.
- word_asr1 T1 = mk_word32 (EQUIV (ASR_ONE (Eps (dest_word32 T1))))"
+lemma word_asr1: "word_asr1 T1 = mk_word32 (EQUIV (ASR_ONE (Eps (dest_word32 T1))))"
by (import word32 word_asr1)
-definition word_ror1 :: "word32 => word32" where
- "word_ror1 == %T1::word32. mk_word32 (EQUIV (ROR_ONE (Eps (dest_word32 T1))))"
+definition
+ word_ror1 :: "word32 => word32" where
+ "word_ror1 == %T1. mk_word32 (EQUIV (ROR_ONE (Eps (dest_word32 T1))))"
-lemma word_ror1: "ALL T1::word32.
- word_ror1 T1 = mk_word32 (EQUIV (ROR_ONE (Eps (dest_word32 T1))))"
+lemma word_ror1: "word_ror1 T1 = mk_word32 (EQUIV (ROR_ONE (Eps (dest_word32 T1))))"
by (import word32 word_ror1)
consts
RRX :: "bool => word32 => word32"
defs
- RRX_primdef: "RRX ==
-%(T1::bool) T2::word32. mk_word32 (EQUIV (RRXn T1 (Eps (dest_word32 T2))))"
+ RRX_primdef: "RRX == %T1 T2. mk_word32 (EQUIV (RRXn T1 (Eps (dest_word32 T2))))"
-lemma RRX_def: "ALL (T1::bool) T2::word32.
- RRX T1 T2 = mk_word32 (EQUIV (RRXn T1 (Eps (dest_word32 T2))))"
+lemma RRX_def: "RRX T1 T2 = mk_word32 (EQUIV (RRXn T1 (Eps (dest_word32 T2))))"
by (import word32 RRX_def)
consts
LSB :: "word32 => bool"
defs
- LSB_primdef: "LSB == %T1::word32. LSBn (Eps (dest_word32 T1))"
+ LSB_primdef: "LSB == %T1. LSBn (Eps (dest_word32 T1))"
-lemma LSB_def: "ALL T1::word32. LSB T1 = LSBn (Eps (dest_word32 T1))"
+lemma LSB_def: "LSB T1 = LSBn (Eps (dest_word32 T1))"
by (import word32 LSB_def)
consts
MSB :: "word32 => bool"
defs
- MSB_primdef: "MSB == %T1::word32. MSBn (Eps (dest_word32 T1))"
+ MSB_primdef: "MSB == %T1. MSBn (Eps (dest_word32 T1))"
-lemma MSB_def: "ALL T1::word32. MSB T1 = MSBn (Eps (dest_word32 T1))"
+lemma MSB_def: "MSB T1 = MSBn (Eps (dest_word32 T1))"
by (import word32 MSB_def)
-definition bitwise_or :: "word32 => word32 => word32" where
+definition
+ bitwise_or :: "word32 => word32 => word32" where
"bitwise_or ==
-%(T1::word32) T2::word32.
- mk_word32 (EQUIV (OR (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))"
+%T1 T2. mk_word32 (EQUIV (OR (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))"
-lemma bitwise_or: "ALL (T1::word32) T2::word32.
- bitwise_or T1 T2 =
- mk_word32 (EQUIV (OR (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))"
+lemma bitwise_or: "bitwise_or T1 T2 =
+mk_word32 (EQUIV (OR (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))"
by (import word32 bitwise_or)
-definition bitwise_eor :: "word32 => word32 => word32" where
+definition
+ bitwise_eor :: "word32 => word32 => word32" where
"bitwise_eor ==
-%(T1::word32) T2::word32.
+%T1 T2.
mk_word32 (EQUIV (EOR (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))"
-lemma bitwise_eor: "ALL (T1::word32) T2::word32.
- bitwise_eor T1 T2 =
- mk_word32 (EQUIV (EOR (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))"
+lemma bitwise_eor: "bitwise_eor T1 T2 =
+mk_word32 (EQUIV (EOR (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))"
by (import word32 bitwise_eor)
-definition bitwise_and :: "word32 => word32 => word32" where
+definition
+ bitwise_and :: "word32 => word32 => word32" where
"bitwise_and ==
-%(T1::word32) T2::word32.
+%T1 T2.
mk_word32 (EQUIV (AND (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))"
-lemma bitwise_and: "ALL (T1::word32) T2::word32.
- bitwise_and T1 T2 =
- mk_word32 (EQUIV (AND (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))"
+lemma bitwise_and: "bitwise_and T1 T2 =
+mk_word32 (EQUIV (AND (Eps (dest_word32 T1)) (Eps (dest_word32 T2))))"
by (import word32 bitwise_and)
consts
TOw :: "word32 => word32"
defs
- TOw_primdef: "TOw == %T1::word32. mk_word32 (EQUIV (MODw (Eps (dest_word32 T1))))"
+ TOw_primdef: "TOw == %T1. mk_word32 (EQUIV (MODw (Eps (dest_word32 T1))))"
-lemma TOw_def: "ALL T1::word32. TOw T1 = mk_word32 (EQUIV (MODw (Eps (dest_word32 T1))))"
+lemma TOw_def: "TOw T1 = mk_word32 (EQUIV (MODw (Eps (dest_word32 T1))))"
by (import word32 TOw_def)
consts
n2w :: "nat => word32"
defs
- n2w_primdef: "n2w == %n::nat. mk_word32 (EQUIV n)"
+ n2w_primdef: "n2w == %n. mk_word32 (EQUIV n)"
-lemma n2w_def: "ALL n::nat. n2w n = mk_word32 (EQUIV n)"
+lemma n2w_def: "n2w n = mk_word32 (EQUIV n)"
by (import word32 n2w_def)
consts
w2n :: "word32 => nat"
defs
- w2n_primdef: "w2n == %w::word32. MODw (Eps (dest_word32 w))"
+ w2n_primdef: "w2n == %w. MODw (Eps (dest_word32 w))"
-lemma w2n_def: "ALL w::word32. w2n w = MODw (Eps (dest_word32 w))"
+lemma w2n_def: "w2n w = MODw (Eps (dest_word32 w))"
by (import word32 w2n_def)
-lemma ADDw: "(ALL x::word32. word_add w_0 x = x) &
-(ALL (x::word32) xa::word32.
- word_add (word_suc x) xa = word_suc (word_add x xa))"
+lemma ADDw: "(ALL x. word_add w_0 x = x) &
+(ALL x xa. word_add (word_suc x) xa = word_suc (word_add x xa))"
by (import word32 ADDw)
-lemma ADD_0w: "ALL x::word32. word_add x w_0 = x"
+lemma ADD_0w: "word_add x w_0 = x"
by (import word32 ADD_0w)
-lemma ADD1w: "ALL x::word32. word_suc x = word_add x w_1"
+lemma ADD1w: "word_suc x = word_add x w_1"
by (import word32 ADD1w)
-lemma ADD_ASSOCw: "ALL (x::word32) (xa::word32) xb::word32.
- word_add x (word_add xa xb) = word_add (word_add x xa) xb"
+lemma ADD_ASSOCw: "word_add x (word_add xa xb) = word_add (word_add x xa) xb"
by (import word32 ADD_ASSOCw)
-lemma ADD_CLAUSESw: "(ALL x::word32. word_add w_0 x = x) &
-(ALL x::word32. word_add x w_0 = x) &
-(ALL (x::word32) xa::word32.
- word_add (word_suc x) xa = word_suc (word_add x xa)) &
-(ALL (x::word32) xa::word32.
- word_add x (word_suc xa) = word_suc (word_add x xa))"
+lemma ADD_CLAUSESw: "(ALL x. word_add w_0 x = x) &
+(ALL x. word_add x w_0 = x) &
+(ALL x xa. word_add (word_suc x) xa = word_suc (word_add x xa)) &
+(ALL x xa. word_add x (word_suc xa) = word_suc (word_add x xa))"
by (import word32 ADD_CLAUSESw)
-lemma ADD_COMMw: "ALL (x::word32) xa::word32. word_add x xa = word_add xa x"
+lemma ADD_COMMw: "word_add x xa = word_add xa x"
by (import word32 ADD_COMMw)
-lemma ADD_INV_0_EQw: "ALL (x::word32) xa::word32. (word_add x xa = x) = (xa = w_0)"
+lemma ADD_INV_0_EQw: "(word_add x xa = x) = (xa = w_0)"
by (import word32 ADD_INV_0_EQw)
-lemma EQ_ADD_LCANCELw: "ALL (x::word32) (xa::word32) xb::word32.
- (word_add x xa = word_add x xb) = (xa = xb)"
+lemma EQ_ADD_LCANCELw: "(word_add x xa = word_add x xb) = (xa = xb)"
by (import word32 EQ_ADD_LCANCELw)
-lemma EQ_ADD_RCANCELw: "ALL (x::word32) (xa::word32) xb::word32.
- (word_add x xb = word_add xa xb) = (x = xa)"
+lemma EQ_ADD_RCANCELw: "(word_add x xb = word_add xa xb) = (x = xa)"
by (import word32 EQ_ADD_RCANCELw)
-lemma LEFT_ADD_DISTRIBw: "ALL (x::word32) (xa::word32) xb::word32.
- word_mul xb (word_add x xa) = word_add (word_mul xb x) (word_mul xb xa)"
+lemma LEFT_ADD_DISTRIBw: "word_mul xb (word_add x xa) = word_add (word_mul xb x) (word_mul xb xa)"
by (import word32 LEFT_ADD_DISTRIBw)
-lemma MULT_ASSOCw: "ALL (x::word32) (xa::word32) xb::word32.
- word_mul x (word_mul xa xb) = word_mul (word_mul x xa) xb"
+lemma MULT_ASSOCw: "word_mul x (word_mul xa xb) = word_mul (word_mul x xa) xb"
by (import word32 MULT_ASSOCw)
-lemma MULT_COMMw: "ALL (x::word32) xa::word32. word_mul x xa = word_mul xa x"
+lemma MULT_COMMw: "word_mul x xa = word_mul xa x"
by (import word32 MULT_COMMw)
-lemma MULT_CLAUSESw: "ALL (x::word32) xa::word32.
- word_mul w_0 x = w_0 &
- word_mul x w_0 = w_0 &
- word_mul w_1 x = x &
- word_mul x w_1 = x &
- word_mul (word_suc x) xa = word_add (word_mul x xa) xa &
- word_mul x (word_suc xa) = word_add x (word_mul x xa)"
+lemma MULT_CLAUSESw: "word_mul w_0 x = w_0 &
+word_mul x w_0 = w_0 &
+word_mul w_1 x = x &
+word_mul x w_1 = x &
+word_mul (word_suc x) xa = word_add (word_mul x xa) xa &
+word_mul x (word_suc xa) = word_add x (word_mul x xa)"
by (import word32 MULT_CLAUSESw)
-lemma TWO_COMP_ONE_COMP: "ALL x::word32. word_2comp x = word_add (word_1comp x) w_1"
+lemma TWO_COMP_ONE_COMP: "word_2comp x = word_add (word_1comp x) w_1"
by (import word32 TWO_COMP_ONE_COMP)
-lemma OR_ASSOCw: "ALL (x::word32) (xa::word32) xb::word32.
- bitwise_or x (bitwise_or xa xb) = bitwise_or (bitwise_or x xa) xb"
+lemma OR_ASSOCw: "bitwise_or x (bitwise_or xa xb) = bitwise_or (bitwise_or x xa) xb"
by (import word32 OR_ASSOCw)
-lemma OR_COMMw: "ALL (x::word32) xa::word32. bitwise_or x xa = bitwise_or xa x"
+lemma OR_COMMw: "bitwise_or x xa = bitwise_or xa x"
by (import word32 OR_COMMw)
-lemma OR_IDEMw: "ALL x::word32. bitwise_or x x = x"
+lemma OR_IDEMw: "bitwise_or x x = x"
by (import word32 OR_IDEMw)
-lemma OR_ABSORBw: "ALL (x::word32) xa::word32. bitwise_and x (bitwise_or x xa) = x"
+lemma OR_ABSORBw: "bitwise_and x (bitwise_or x xa) = x"
by (import word32 OR_ABSORBw)
-lemma AND_ASSOCw: "ALL (x::word32) (xa::word32) xb::word32.
- bitwise_and x (bitwise_and xa xb) = bitwise_and (bitwise_and x xa) xb"
+lemma AND_ASSOCw: "bitwise_and x (bitwise_and xa xb) = bitwise_and (bitwise_and x xa) xb"
by (import word32 AND_ASSOCw)
-lemma AND_COMMw: "ALL (x::word32) xa::word32. bitwise_and x xa = bitwise_and xa x"
+lemma AND_COMMw: "bitwise_and x xa = bitwise_and xa x"
by (import word32 AND_COMMw)
-lemma AND_IDEMw: "ALL x::word32. bitwise_and x x = x"
+lemma AND_IDEMw: "bitwise_and x x = x"
by (import word32 AND_IDEMw)
-lemma AND_ABSORBw: "ALL (x::word32) xa::word32. bitwise_or x (bitwise_and x xa) = x"
+lemma AND_ABSORBw: "bitwise_or x (bitwise_and x xa) = x"
by (import word32 AND_ABSORBw)
-lemma ONE_COMPw: "ALL x::word32. word_1comp (word_1comp x) = x"
+lemma ONE_COMPw: "word_1comp (word_1comp x) = x"
by (import word32 ONE_COMPw)
-lemma RIGHT_AND_OVER_ORw: "ALL (x::word32) (xa::word32) xb::word32.
- bitwise_and (bitwise_or x xa) xb =
- bitwise_or (bitwise_and x xb) (bitwise_and xa xb)"
+lemma RIGHT_AND_OVER_ORw: "bitwise_and (bitwise_or x xa) xb =
+bitwise_or (bitwise_and x xb) (bitwise_and xa xb)"
by (import word32 RIGHT_AND_OVER_ORw)
-lemma RIGHT_OR_OVER_ANDw: "ALL (x::word32) (xa::word32) xb::word32.
- bitwise_or (bitwise_and x xa) xb =
- bitwise_and (bitwise_or x xb) (bitwise_or xa xb)"
+lemma RIGHT_OR_OVER_ANDw: "bitwise_or (bitwise_and x xa) xb =
+bitwise_and (bitwise_or x xb) (bitwise_or xa xb)"
by (import word32 RIGHT_OR_OVER_ANDw)
-lemma DE_MORGAN_THMw: "ALL (x::word32) xa::word32.
- word_1comp (bitwise_and x xa) =
- bitwise_or (word_1comp x) (word_1comp xa) &
- word_1comp (bitwise_or x xa) = bitwise_and (word_1comp x) (word_1comp xa)"
+lemma DE_MORGAN_THMw: "word_1comp (bitwise_and x xa) = bitwise_or (word_1comp x) (word_1comp xa) &
+word_1comp (bitwise_or x xa) = bitwise_and (word_1comp x) (word_1comp xa)"
by (import word32 DE_MORGAN_THMw)
lemma w_0: "w_0 = n2w 0"
@@ -1136,433 +972,411 @@
ALT_ZERO)))))))))))))))))))))))))))))))))"
by (import word32 w_T)
-lemma ADD_TWO_COMP: "ALL x::word32. word_add x (word_2comp x) = w_0"
+lemma ADD_TWO_COMP: "word_add x (word_2comp x) = w_0"
by (import word32 ADD_TWO_COMP)
-lemma ADD_TWO_COMP2: "ALL x::word32. word_add (word_2comp x) x = w_0"
+lemma ADD_TWO_COMP2: "word_add (word_2comp x) x = w_0"
by (import word32 ADD_TWO_COMP2)
-definition word_sub :: "word32 => word32 => word32" where
- "word_sub == %(a::word32) b::word32. word_add a (word_2comp b)"
+definition
+ word_sub :: "word32 => word32 => word32" where
+ "word_sub == %a b. word_add a (word_2comp b)"
-lemma word_sub: "ALL (a::word32) b::word32. word_sub a b = word_add a (word_2comp b)"
+lemma word_sub: "word_sub a b = word_add a (word_2comp b)"
by (import word32 word_sub)
-definition word_lsl :: "word32 => nat => word32" where
- "word_lsl == %(a::word32) n::nat. word_mul a (n2w (2 ^ n))"
+definition
+ word_lsl :: "word32 => nat => word32" where
+ "word_lsl == %a n. word_mul a (n2w (2 ^ n))"
-lemma word_lsl: "ALL (a::word32) n::nat. word_lsl a n = word_mul a (n2w (2 ^ n))"
+lemma word_lsl: "word_lsl a n = word_mul a (n2w (2 ^ n))"
by (import word32 word_lsl)
-definition word_lsr :: "word32 => nat => word32" where
- "word_lsr == %(a::word32) n::nat. (word_lsr1 ^^ n) a"
+definition
+ word_lsr :: "word32 => nat => word32" where
+ "word_lsr == %a n. (word_lsr1 ^^ n) a"
-lemma word_lsr: "ALL (a::word32) n::nat. word_lsr a n = (word_lsr1 ^^ n) a"
+lemma word_lsr: "word_lsr a n = (word_lsr1 ^^ n) a"
by (import word32 word_lsr)
-definition word_asr :: "word32 => nat => word32" where
- "word_asr == %(a::word32) n::nat. (word_asr1 ^^ n) a"
+definition
+ word_asr :: "word32 => nat => word32" where
+ "word_asr == %a n. (word_asr1 ^^ n) a"
-lemma word_asr: "ALL (a::word32) n::nat. word_asr a n = (word_asr1 ^^ n) a"
+lemma word_asr: "word_asr a n = (word_asr1 ^^ n) a"
by (import word32 word_asr)
-definition word_ror :: "word32 => nat => word32" where
- "word_ror == %(a::word32) n::nat. (word_ror1 ^^ n) a"
+definition
+ word_ror :: "word32 => nat => word32" where
+ "word_ror == %a n. (word_ror1 ^^ n) a"
-lemma word_ror: "ALL (a::word32) n::nat. word_ror a n = (word_ror1 ^^ n) a"
+lemma word_ror: "word_ror a n = (word_ror1 ^^ n) a"
by (import word32 word_ror)
consts
BITw :: "nat => word32 => bool"
defs
- BITw_primdef: "BITw == %(b::nat) n::word32. bit b (w2n n)"
+ BITw_primdef: "BITw == %b n. bit b (w2n n)"
-lemma BITw_def: "ALL (b::nat) n::word32. BITw b n = bit b (w2n n)"
+lemma BITw_def: "BITw b n = bit b (w2n n)"
by (import word32 BITw_def)
consts
BITSw :: "nat => nat => word32 => nat"
defs
- BITSw_primdef: "BITSw == %(h::nat) (l::nat) n::word32. BITS h l (w2n n)"
+ BITSw_primdef: "BITSw == %h l n. BITS h l (w2n n)"
-lemma BITSw_def: "ALL (h::nat) (l::nat) n::word32. BITSw h l n = BITS h l (w2n n)"
+lemma BITSw_def: "BITSw h l n = BITS h l (w2n n)"
by (import word32 BITSw_def)
consts
SLICEw :: "nat => nat => word32 => nat"
defs
- SLICEw_primdef: "SLICEw == %(h::nat) (l::nat) n::word32. SLICE h l (w2n n)"
+ SLICEw_primdef: "SLICEw == %h l n. SLICE h l (w2n n)"
-lemma SLICEw_def: "ALL (h::nat) (l::nat) n::word32. SLICEw h l n = SLICE h l (w2n n)"
+lemma SLICEw_def: "SLICEw h l n = SLICE h l (w2n n)"
by (import word32 SLICEw_def)
-lemma TWO_COMP_ADD: "ALL (a::word32) b::word32.
- word_2comp (word_add a b) = word_add (word_2comp a) (word_2comp b)"
+lemma TWO_COMP_ADD: "word_2comp (word_add a b) = word_add (word_2comp a) (word_2comp b)"
by (import word32 TWO_COMP_ADD)
-lemma TWO_COMP_ELIM: "ALL a::word32. word_2comp (word_2comp a) = a"
+lemma TWO_COMP_ELIM: "word_2comp (word_2comp a) = a"
by (import word32 TWO_COMP_ELIM)
-lemma ADD_SUB_ASSOC: "ALL (a::word32) (b::word32) c::word32.
- word_sub (word_add a b) c = word_add a (word_sub b c)"
+lemma ADD_SUB_ASSOC: "word_sub (word_add a b) c = word_add a (word_sub b c)"
by (import word32 ADD_SUB_ASSOC)
-lemma ADD_SUB_SYM: "ALL (a::word32) (b::word32) c::word32.
- word_sub (word_add a b) c = word_add (word_sub a c) b"
+lemma ADD_SUB_SYM: "word_sub (word_add a b) c = word_add (word_sub a c) b"
by (import word32 ADD_SUB_SYM)
-lemma SUB_EQUALw: "ALL a::word32. word_sub a a = w_0"
+lemma SUB_EQUALw: "word_sub a a = w_0"
by (import word32 SUB_EQUALw)
-lemma ADD_SUBw: "ALL (a::word32) b::word32. word_sub (word_add a b) b = a"
+lemma ADD_SUBw: "word_sub (word_add a b) b = a"
by (import word32 ADD_SUBw)
-lemma SUB_SUBw: "ALL (a::word32) (b::word32) c::word32.
- word_sub a (word_sub b c) = word_sub (word_add a c) b"
+lemma SUB_SUBw: "word_sub a (word_sub b c) = word_sub (word_add a c) b"
by (import word32 SUB_SUBw)
-lemma ONE_COMP_TWO_COMP: "ALL a::word32. word_1comp a = word_sub (word_2comp a) w_1"
+lemma ONE_COMP_TWO_COMP: "word_1comp a = word_sub (word_2comp a) w_1"
by (import word32 ONE_COMP_TWO_COMP)
-lemma SUBw: "ALL (m::word32) n::word32. word_sub (word_suc m) n = word_suc (word_sub m n)"
+lemma SUBw: "word_sub (word_suc m) n = word_suc (word_sub m n)"
by (import word32 SUBw)
-lemma ADD_EQ_SUBw: "ALL (m::word32) (n::word32) p::word32.
- (word_add m n = p) = (m = word_sub p n)"
+lemma ADD_EQ_SUBw: "(word_add m n = p) = (m = word_sub p n)"
by (import word32 ADD_EQ_SUBw)
-lemma CANCEL_SUBw: "ALL (m::word32) (n::word32) p::word32.
- (word_sub n p = word_sub m p) = (n = m)"
+lemma CANCEL_SUBw: "(word_sub n p = word_sub m p) = (n = m)"
by (import word32 CANCEL_SUBw)
-lemma SUB_PLUSw: "ALL (a::word32) (b::word32) c::word32.
- word_sub a (word_add b c) = word_sub (word_sub a b) c"
+lemma SUB_PLUSw: "word_sub a (word_add b c) = word_sub (word_sub a b) c"
by (import word32 SUB_PLUSw)
-lemma word_nchotomy: "ALL w::word32. EX n::nat. w = n2w n"
+lemma word_nchotomy: "EX n. w = n2w n"
by (import word32 word_nchotomy)
-lemma dest_word_mk_word_eq3: "ALL a::nat. dest_word32 (mk_word32 (EQUIV a)) = EQUIV a"
+lemma dest_word_mk_word_eq3: "dest_word32 (mk_word32 (EQUIV a)) = EQUIV a"
by (import word32 dest_word_mk_word_eq3)
-lemma MODw_ELIM: "ALL n::nat. n2w (MODw n) = n2w n"
+lemma MODw_ELIM: "n2w (MODw n) = n2w n"
by (import word32 MODw_ELIM)
-lemma w2n_EVAL: "ALL n::nat. w2n (n2w n) = MODw n"
+lemma w2n_EVAL: "w2n (n2w n) = MODw n"
by (import word32 w2n_EVAL)
-lemma w2n_ELIM: "ALL a::word32. n2w (w2n a) = a"
+lemma w2n_ELIM: "n2w (w2n a) = a"
by (import word32 w2n_ELIM)
-lemma n2w_11: "ALL (a::nat) b::nat. (n2w a = n2w b) = (MODw a = MODw b)"
+lemma n2w_11: "(n2w a = n2w b) = (MODw a = MODw b)"
by (import word32 n2w_11)
-lemma ADD_EVAL: "word_add (n2w (a::nat)) (n2w (b::nat)) = n2w (a + b)"
+lemma ADD_EVAL: "word_add (n2w a) (n2w b) = n2w (a + b)"
by (import word32 ADD_EVAL)
-lemma MUL_EVAL: "word_mul (n2w (a::nat)) (n2w (b::nat)) = n2w (a * b)"
+lemma MUL_EVAL: "word_mul (n2w a) (n2w b) = n2w (a * b)"
by (import word32 MUL_EVAL)
-lemma ONE_COMP_EVAL: "word_1comp (n2w (a::nat)) = n2w (ONE_COMP a)"
+lemma ONE_COMP_EVAL: "word_1comp (n2w a) = n2w (ONE_COMP a)"
by (import word32 ONE_COMP_EVAL)
-lemma TWO_COMP_EVAL: "word_2comp (n2w (a::nat)) = n2w (TWO_COMP a)"
+lemma TWO_COMP_EVAL: "word_2comp (n2w a) = n2w (TWO_COMP a)"
by (import word32 TWO_COMP_EVAL)
-lemma LSR_ONE_EVAL: "word_lsr1 (n2w (a::nat)) = n2w (LSR_ONE a)"
+lemma LSR_ONE_EVAL: "word_lsr1 (n2w a) = n2w (LSR_ONE a)"
by (import word32 LSR_ONE_EVAL)
-lemma ASR_ONE_EVAL: "word_asr1 (n2w (a::nat)) = n2w (ASR_ONE a)"
+lemma ASR_ONE_EVAL: "word_asr1 (n2w a) = n2w (ASR_ONE a)"
by (import word32 ASR_ONE_EVAL)
-lemma ROR_ONE_EVAL: "word_ror1 (n2w (a::nat)) = n2w (ROR_ONE a)"
+lemma ROR_ONE_EVAL: "word_ror1 (n2w a) = n2w (ROR_ONE a)"
by (import word32 ROR_ONE_EVAL)
-lemma RRX_EVAL: "RRX (c::bool) (n2w (a::nat)) = n2w (RRXn c a)"
+lemma RRX_EVAL: "RRX c (n2w a) = n2w (RRXn c a)"
by (import word32 RRX_EVAL)
-lemma LSB_EVAL: "LSB (n2w (a::nat)) = LSBn a"
+lemma LSB_EVAL: "LSB (n2w a) = LSBn a"
by (import word32 LSB_EVAL)
-lemma MSB_EVAL: "MSB (n2w (a::nat)) = MSBn a"
+lemma MSB_EVAL: "MSB (n2w a) = MSBn a"
by (import word32 MSB_EVAL)
-lemma OR_EVAL: "bitwise_or (n2w (a::nat)) (n2w (b::nat)) = n2w (OR a b)"
+lemma OR_EVAL: "bitwise_or (n2w a) (n2w b) = n2w (OR a b)"
by (import word32 OR_EVAL)
-lemma EOR_EVAL: "bitwise_eor (n2w (a::nat)) (n2w (b::nat)) = n2w (EOR a b)"
+lemma EOR_EVAL: "bitwise_eor (n2w a) (n2w b) = n2w (EOR a b)"
by (import word32 EOR_EVAL)
-lemma AND_EVAL: "bitwise_and (n2w (a::nat)) (n2w (b::nat)) = n2w (AND a b)"
+lemma AND_EVAL: "bitwise_and (n2w a) (n2w b) = n2w (AND a b)"
by (import word32 AND_EVAL)
-lemma BITS_EVAL: "ALL (h::nat) (l::nat) a::nat. BITSw h l (n2w a) = BITS h l (MODw a)"
+lemma BITS_EVAL: "BITSw h l (n2w a) = BITS h l (MODw a)"
by (import word32 BITS_EVAL)
-lemma BIT_EVAL: "ALL (b::nat) a::nat. BITw b (n2w a) = bit b (MODw a)"
+lemma BIT_EVAL: "BITw b (n2w a) = bit b (MODw a)"
by (import word32 BIT_EVAL)
-lemma SLICE_EVAL: "ALL (h::nat) (l::nat) a::nat. SLICEw h l (n2w a) = SLICE h l (MODw a)"
+lemma SLICE_EVAL: "SLICEw h l (n2w a) = SLICE h l (MODw a)"
by (import word32 SLICE_EVAL)
-lemma LSL_ADD: "ALL (a::word32) (m::nat) n::nat.
- word_lsl (word_lsl a m) n = word_lsl a (m + n)"
+lemma LSL_ADD: "word_lsl (word_lsl a m) n = word_lsl a (m + n)"
by (import word32 LSL_ADD)
-lemma LSR_ADD: "ALL (x::word32) (xa::nat) xb::nat.
- word_lsr (word_lsr x xa) xb = word_lsr x (xa + xb)"
+lemma LSR_ADD: "word_lsr (word_lsr x xa) xb = word_lsr x (xa + xb)"
by (import word32 LSR_ADD)
-lemma ASR_ADD: "ALL (x::word32) (xa::nat) xb::nat.
- word_asr (word_asr x xa) xb = word_asr x (xa + xb)"
+lemma ASR_ADD: "word_asr (word_asr x xa) xb = word_asr x (xa + xb)"
by (import word32 ASR_ADD)
-lemma ROR_ADD: "ALL (x::word32) (xa::nat) xb::nat.
- word_ror (word_ror x xa) xb = word_ror x (xa + xb)"
+lemma ROR_ADD: "word_ror (word_ror x xa) xb = word_ror x (xa + xb)"
by (import word32 ROR_ADD)
-lemma LSL_LIMIT: "ALL (w::word32) n::nat. HB < n --> word_lsl w n = w_0"
+lemma LSL_LIMIT: "HB < n ==> word_lsl w n = w_0"
by (import word32 LSL_LIMIT)
-lemma MOD_MOD_DIV: "ALL (a::nat) b::nat. INw (MODw a div 2 ^ b)"
+lemma MOD_MOD_DIV: "INw (MODw a div 2 ^ b)"
by (import word32 MOD_MOD_DIV)
-lemma MOD_MOD_DIV_2EXP: "ALL (a::nat) n::nat. MODw (MODw a div 2 ^ n) div 2 = MODw a div 2 ^ Suc n"
+lemma MOD_MOD_DIV_2EXP: "MODw (MODw a div 2 ^ n) div 2 = MODw a div 2 ^ Suc n"
by (import word32 MOD_MOD_DIV_2EXP)
-lemma LSR_EVAL: "ALL n::nat. word_lsr (n2w (a::nat)) n = n2w (MODw a div 2 ^ n)"
+lemma LSR_EVAL: "word_lsr (n2w a) n = n2w (MODw a div 2 ^ n)"
by (import word32 LSR_EVAL)
-lemma LSR_THM: "ALL (x::nat) n::nat. word_lsr (n2w n) x = n2w (BITS HB (min WL x) n)"
+lemma LSR_THM: "word_lsr (n2w n) x = n2w (BITS HB (min WL x) n)"
by (import word32 LSR_THM)
-lemma LSR_LIMIT: "ALL (x::nat) w::word32. HB < x --> word_lsr w x = w_0"
+lemma LSR_LIMIT: "HB < x ==> word_lsr w x = w_0"
by (import word32 LSR_LIMIT)
-lemma LEFT_SHIFT_LESS: "ALL (n::nat) (m::nat) a::nat. a < 2 ^ m --> 2 ^ n + a * 2 ^ n <= 2 ^ (m + n)"
+lemma LEFT_SHIFT_LESS: "(a::nat) < (2::nat) ^ (m::nat)
+==> (2::nat) ^ (n::nat) + a * (2::nat) ^ n <= (2::nat) ^ (m + n)"
by (import word32 LEFT_SHIFT_LESS)
-lemma ROR_THM: "ALL (x::nat) n::nat.
- word_ror (n2w n) x =
- (let x'::nat = x mod WL
- in n2w (BITS HB x' n + BITS (x' - 1) 0 n * 2 ^ (WL - x')))"
+lemma ROR_THM: "word_ror (n2w n) x =
+(let x' = x mod WL
+ in n2w (BITS HB x' n + BITS (x' - 1) 0 n * 2 ^ (WL - x')))"
by (import word32 ROR_THM)
-lemma ROR_CYCLE: "ALL (x::nat) w::word32. word_ror w (x * WL) = w"
+lemma ROR_CYCLE: "word_ror w (x * WL) = w"
by (import word32 ROR_CYCLE)
-lemma ASR_THM: "ALL (x::nat) n::nat.
- word_asr (n2w n) x =
- (let x'::nat = min HB x; s::nat = BITS HB x' n
- in n2w (if MSBn n then 2 ^ WL - 2 ^ (WL - x') + s else s))"
+lemma ASR_THM: "word_asr (n2w n) x =
+(let x' = min HB x; s = BITS HB x' n
+ in n2w (if MSBn n then 2 ^ WL - 2 ^ (WL - x') + s else s))"
by (import word32 ASR_THM)
-lemma ASR_LIMIT: "ALL (x::nat) w::word32.
- HB <= x --> word_asr w x = (if MSB w then w_T else w_0)"
+lemma ASR_LIMIT: "HB <= x ==> word_asr w x = (if MSB w then w_T else w_0)"
by (import word32 ASR_LIMIT)
-lemma ZERO_SHIFT: "(ALL n::nat. word_lsl w_0 n = w_0) &
-(ALL n::nat. word_asr w_0 n = w_0) &
-(ALL n::nat. word_lsr w_0 n = w_0) & (ALL n::nat. word_ror w_0 n = w_0)"
+lemma ZERO_SHIFT: "(ALL n. word_lsl w_0 n = w_0) &
+(ALL n. word_asr w_0 n = w_0) &
+(ALL n. word_lsr w_0 n = w_0) & (ALL n. word_ror w_0 n = w_0)"
by (import word32 ZERO_SHIFT)
-lemma ZERO_SHIFT2: "(ALL a::word32. word_lsl a 0 = a) &
-(ALL a::word32. word_asr a 0 = a) &
-(ALL a::word32. word_lsr a 0 = a) & (ALL a::word32. word_ror a 0 = a)"
+lemma ZERO_SHIFT2: "(ALL a. word_lsl a 0 = a) &
+(ALL a. word_asr a 0 = a) &
+(ALL a. word_lsr a 0 = a) & (ALL a. word_ror a 0 = a)"
by (import word32 ZERO_SHIFT2)
-lemma ASR_w_T: "ALL n::nat. word_asr w_T n = w_T"
+lemma ASR_w_T: "word_asr w_T n = w_T"
by (import word32 ASR_w_T)
-lemma ROR_w_T: "ALL n::nat. word_ror w_T n = w_T"
+lemma ROR_w_T: "word_ror w_T n = w_T"
by (import word32 ROR_w_T)
-lemma MODw_EVAL: "ALL x::nat.
- MODw x =
- x mod
- NUMERAL
- (NUMERAL_BIT2
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
-(NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- ALT_ZERO))))))))))))))))))))))))))))))))"
- by (import word32 MODw_EVAL)
-
-lemma ADD_EVAL2: "ALL (b::nat) a::nat. word_add (n2w a) (n2w b) = n2w (MODw (a + b))"
- by (import word32 ADD_EVAL2)
-
-lemma MUL_EVAL2: "ALL (b::nat) a::nat. word_mul (n2w a) (n2w b) = n2w (MODw (a * b))"
- by (import word32 MUL_EVAL2)
-
-lemma ONE_COMP_EVAL2: "ALL a::nat.
- word_1comp (n2w a) =
- n2w (2 ^
- NUMERAL
- (NUMERAL_BIT2
- (NUMERAL_BIT1
- (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))) -
- 1 -
- MODw a)"
- by (import word32 ONE_COMP_EVAL2)
-
-lemma TWO_COMP_EVAL2: "ALL a::nat.
- word_2comp (n2w a) =
- n2w (MODw
- (2 ^
- NUMERAL
- (NUMERAL_BIT2
- (NUMERAL_BIT1
- (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))) -
- MODw a))"
- by (import word32 TWO_COMP_EVAL2)
-
-lemma LSR_ONE_EVAL2: "ALL a::nat. word_lsr1 (n2w a) = n2w (MODw a div 2)"
- by (import word32 LSR_ONE_EVAL2)
-
-lemma ASR_ONE_EVAL2: "ALL a::nat.
- word_asr1 (n2w a) =
- n2w (MODw a div 2 +
- SBIT (MSBn a)
- (NUMERAL
+lemma MODw_EVAL: "MODw x =
+x mod
+NUMERAL
+ (NUMERAL_BIT2
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
(NUMERAL_BIT1
(NUMERAL_BIT1
- (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO)))))))"
- by (import word32 ASR_ONE_EVAL2)
-
-lemma ROR_ONE_EVAL2: "ALL a::nat.
- word_ror1 (n2w a) =
- n2w (MODw a div 2 +
- SBIT (LSBn a)
- (NUMERAL
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
(NUMERAL_BIT1
(NUMERAL_BIT1
- (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO)))))))"
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ ALT_ZERO))))))))))))))))))))))))))))))))"
+ by (import word32 MODw_EVAL)
+
+lemma ADD_EVAL2: "word_add (n2w a) (n2w b) = n2w (MODw (a + b))"
+ by (import word32 ADD_EVAL2)
+
+lemma MUL_EVAL2: "word_mul (n2w a) (n2w b) = n2w (MODw (a * b))"
+ by (import word32 MUL_EVAL2)
+
+lemma ONE_COMP_EVAL2: "word_1comp (n2w a) =
+n2w (2 ^
+ NUMERAL
+ (NUMERAL_BIT2
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))) -
+ 1 -
+ MODw a)"
+ by (import word32 ONE_COMP_EVAL2)
+
+lemma TWO_COMP_EVAL2: "word_2comp (n2w a) =
+n2w (MODw
+ (2 ^
+ NUMERAL
+ (NUMERAL_BIT2
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))) -
+ MODw a))"
+ by (import word32 TWO_COMP_EVAL2)
+
+lemma LSR_ONE_EVAL2: "word_lsr1 (n2w a) = n2w (MODw a div 2)"
+ by (import word32 LSR_ONE_EVAL2)
+
+lemma ASR_ONE_EVAL2: "word_asr1 (n2w a) =
+n2w (MODw a div 2 +
+ SBIT (MSBn a)
+ (NUMERAL
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO)))))))"
+ by (import word32 ASR_ONE_EVAL2)
+
+lemma ROR_ONE_EVAL2: "word_ror1 (n2w a) =
+n2w (MODw a div 2 +
+ SBIT (LSBn a)
+ (NUMERAL
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO)))))))"
by (import word32 ROR_ONE_EVAL2)
-lemma RRX_EVAL2: "ALL (c::bool) a::nat.
- RRX c (n2w a) =
- n2w (MODw a div 2 +
- SBIT c
- (NUMERAL
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO)))))))"
+lemma RRX_EVAL2: "RRX c (n2w a) =
+n2w (MODw a div 2 +
+ SBIT c
+ (NUMERAL
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO)))))))"
by (import word32 RRX_EVAL2)
-lemma LSB_EVAL2: "ALL a::nat. LSB (n2w a) = ODD a"
+lemma LSB_EVAL2: "LSB (n2w a) = ODD a"
by (import word32 LSB_EVAL2)
-lemma MSB_EVAL2: "ALL a::nat.
- MSB (n2w a) =
- bit (NUMERAL
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))))
- a"
+lemma MSB_EVAL2: "MSB (n2w a) =
+bit (NUMERAL
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))))
+ a"
by (import word32 MSB_EVAL2)
-lemma OR_EVAL2: "ALL (b::nat) a::nat.
- bitwise_or (n2w a) (n2w b) =
- n2w (BITWISE
- (NUMERAL
- (NUMERAL_BIT2
- (NUMERAL_BIT1
- (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))))
- op | a b)"
+lemma OR_EVAL2: "bitwise_or (n2w a) (n2w b) =
+n2w (BITWISE
+ (NUMERAL
+ (NUMERAL_BIT2
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))))
+ op | a b)"
by (import word32 OR_EVAL2)
-lemma AND_EVAL2: "ALL (b::nat) a::nat.
- bitwise_and (n2w a) (n2w b) =
- n2w (BITWISE
- (NUMERAL
- (NUMERAL_BIT2
- (NUMERAL_BIT1
- (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))))
- op & a b)"
+lemma AND_EVAL2: "bitwise_and (n2w a) (n2w b) =
+n2w (BITWISE
+ (NUMERAL
+ (NUMERAL_BIT2
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))))
+ op & a b)"
by (import word32 AND_EVAL2)
-lemma EOR_EVAL2: "ALL (b::nat) a::nat.
- bitwise_eor (n2w a) (n2w b) =
- n2w (BITWISE
- (NUMERAL
- (NUMERAL_BIT2
- (NUMERAL_BIT1
- (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))))
- (%(x::bool) y::bool. x ~= y) a b)"
+lemma EOR_EVAL2: "bitwise_eor (n2w a) (n2w b) =
+n2w (BITWISE
+ (NUMERAL
+ (NUMERAL_BIT2
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT1 ALT_ZERO))))))
+ op ~= a b)"
by (import word32 EOR_EVAL2)
-lemma BITWISE_EVAL2: "ALL (n::nat) (oper::bool => bool => bool) (x::nat) y::nat.
- BITWISE n oper x y =
- (if n = 0 then 0
- else 2 * BITWISE (n - 1) oper (x div 2) (y div 2) +
- (if oper (ODD x) (ODD y) then 1 else 0))"
+lemma BITWISE_EVAL2: "BITWISE n oper x y =
+(if n = 0 then 0
+ else 2 * BITWISE (n - 1) oper (x div 2) (y div 2) +
+ (if oper (ODD x) (ODD y) then 1 else 0))"
by (import word32 BITWISE_EVAL2)
-lemma BITSwLT_THM: "ALL (h::nat) (l::nat) n::word32. BITSw h l n < 2 ^ (Suc h - l)"
+lemma BITSwLT_THM: "BITSw h l n < 2 ^ (Suc h - l)"
by (import word32 BITSwLT_THM)
-lemma BITSw_COMP_THM: "ALL (h1::nat) (l1::nat) (h2::nat) (l2::nat) n::word32.
- h2 + l1 <= h1 -->
- BITS h2 l2 (BITSw h1 l1 n) = BITSw (h2 + l1) (l2 + l1) n"
+lemma BITSw_COMP_THM: "h2 + l1 <= h1 ==> BITS h2 l2 (BITSw h1 l1 n) = BITSw (h2 + l1) (l2 + l1) n"
by (import word32 BITSw_COMP_THM)
-lemma BITSw_DIV_THM: "ALL (h::nat) (l::nat) (n::nat) x::word32.
- BITSw h l x div 2 ^ n = BITSw h (l + n) x"
+lemma BITSw_DIV_THM: "BITSw h l x div 2 ^ n = BITSw h (l + n) x"
by (import word32 BITSw_DIV_THM)
-lemma BITw_THM: "ALL (b::nat) n::word32. BITw b n = (BITSw b b n = 1)"
+lemma BITw_THM: "BITw b n = (BITSw b b n = 1)"
by (import word32 BITw_THM)
-lemma SLICEw_THM: "ALL (n::word32) (h::nat) l::nat. SLICEw h l n = BITSw h l n * 2 ^ l"
+lemma SLICEw_THM: "SLICEw h l n = BITSw h l n * 2 ^ l"
by (import word32 SLICEw_THM)
-lemma BITS_SLICEw_THM: "ALL (h::nat) (l::nat) n::word32. BITS h l (SLICEw h l n) = BITSw h l n"
+lemma BITS_SLICEw_THM: "BITS h l (SLICEw h l n) = BITSw h l n"
by (import word32 BITS_SLICEw_THM)
-lemma SLICEw_ZERO_THM: "ALL (n::word32) h::nat. SLICEw h 0 n = BITSw h 0 n"
+lemma SLICEw_ZERO_THM: "SLICEw h 0 n = BITSw h 0 n"
by (import word32 SLICEw_ZERO_THM)
-lemma SLICEw_COMP_THM: "ALL (h::nat) (m::nat) (l::nat) a::word32.
- Suc m <= h & l <= m --> SLICEw h (Suc m) a + SLICEw m l a = SLICEw h l a"
+lemma SLICEw_COMP_THM: "Suc m <= h & l <= m ==> SLICEw h (Suc m) a + SLICEw m l a = SLICEw h l a"
by (import word32 SLICEw_COMP_THM)
-lemma BITSw_ZERO: "ALL (h::nat) (l::nat) n::word32. h < l --> BITSw h l n = 0"
+lemma BITSw_ZERO: "h < l ==> BITSw h l n = 0"
by (import word32 BITSw_ZERO)
-lemma SLICEw_ZERO: "ALL (h::nat) (l::nat) n::word32. h < l --> SLICEw h l n = 0"
+lemma SLICEw_ZERO: "h < l ==> SLICEw h l n = 0"
by (import word32 SLICEw_ZERO)
;end_setup
end
+
--- a/src/HOL/Import/HOL/arithmetic.imp Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Import/HOL/arithmetic.imp Fri Sep 09 06:47:14 2011 +0200
@@ -14,19 +14,19 @@
"NUMERAL_BIT2" > "HOL4Compat.NUMERAL_BIT2"
"NUMERAL_BIT1" > "HOL4Compat.NUMERAL_BIT1"
"NUMERAL" > "HOL4Compat.NUMERAL"
+ "MOD" > "Divides.div_class.mod" :: "nat => nat => nat"
"MIN" > "Orderings.ord_class.min" :: "nat => nat => nat"
"MAX" > "Orderings.ord_class.max" :: "nat => nat => nat"
"FUNPOW" > "HOL4Compat.FUNPOW"
"EXP" > "Power.power_class.power" :: "nat => nat => nat"
"DIV" > "Divides.div_class.div" :: "nat => nat => nat"
- "MOD" > "Divides.div_class.mod" :: "nat => nat => nat"
"ALT_ZERO" > "HOL4Compat.ALT_ZERO"
">=" > "HOL4Compat.nat_ge"
">" > "HOL4Compat.nat_gt"
- "<=" > "Orderings.less_eq" :: "nat => nat => bool"
- "-" > "Groups.minus" :: "nat => nat => nat"
- "+" > "Groups.plus" :: "nat => nat => nat"
- "*" > "Groups.times" :: "nat => nat => nat"
+ "<=" > "Orderings.ord_class.less_eq" :: "nat => nat => bool"
+ "-" > "Groups.minus_class.minus" :: "nat => nat => nat"
+ "+" > "Groups.plus_class.plus" :: "nat => nat => nat"
+ "*" > "Groups.times_class.times" :: "nat => nat => nat"
thm_maps
"num_case_def" > "HOL4Compat.num_case_def"
@@ -41,10 +41,10 @@
"ZERO_DIV" > "HOL4Base.arithmetic.ZERO_DIV"
"WOP" > "HOL4Base.arithmetic.WOP"
"TWO" > "HOL4Base.arithmetic.TWO"
- "TIMES2" > "NatSimprocs.nat_mult_2"
- "SUC_SUB1" > "HOL4Base.arithmetic.SUC_SUB1"
+ "TIMES2" > "Int.semiring_mult_2"
+ "SUC_SUB1" > "Nat.diff_Suc_1"
"SUC_ONE_ADD" > "Nat_Numeral.Suc_eq_plus1_left"
- "SUC_NOT" > "Nat.nat.simps_2"
+ "SUC_NOT" > "Nat.Zero_not_Suc"
"SUC_ELIM_THM" > "HOL4Base.arithmetic.SUC_ELIM_THM"
"SUC_ADD_SYM" > "HOL4Base.arithmetic.SUC_ADD_SYM"
"SUB_SUB" > "Nat.diff_diff_right"
@@ -77,8 +77,8 @@
"SUB_ADD" > "Nat.le_add_diff_inverse2"
"SUB_0" > "HOL4Base.arithmetic.SUB_0"
"SUB" > "HOL4Compat.SUB"
- "RIGHT_SUB_DISTRIB" > "Nat.nat_distrib_3"
- "RIGHT_ADD_DISTRIB" > "Nat.nat_distrib_1"
+ "RIGHT_SUB_DISTRIB" > "Nat.diff_mult_distrib"
+ "RIGHT_ADD_DISTRIB" > "Fields.linordered_field_class.sign_simps_26"
"PRE_SUC_EQ" > "HOL4Base.arithmetic.PRE_SUC_EQ"
"PRE_SUB1" > "HOL4Base.arithmetic.PRE_SUB1"
"PRE_SUB" > "HOL4Base.arithmetic.PRE_SUB"
@@ -101,29 +101,29 @@
"NOT_SUC_ADD_LESS_EQ" > "HOL4Base.arithmetic.NOT_SUC_ADD_LESS_EQ"
"NOT_ODD_EQ_EVEN" > "HOL4Base.arithmetic.NOT_ODD_EQ_EVEN"
"NOT_NUM_EQ" > "HOL4Base.arithmetic.NOT_NUM_EQ"
- "NOT_LESS_EQUAL" > "Orderings.linorder_not_le"
- "NOT_LESS" > "Nat.le_def"
- "NOT_LEQ" > "HOL4Base.arithmetic.NOT_LEQ"
- "NOT_GREATER_EQ" > "HOL4Base.arithmetic.NOT_GREATER_EQ"
- "NOT_GREATER" > "Nat.le_def"
+ "NOT_LESS_EQUAL" > "Orderings.linorder_class.not_le"
+ "NOT_LESS" > "Orderings.linorder_class.not_less"
+ "NOT_LEQ" > "Nat.not_less_eq_eq"
+ "NOT_GREATER_EQ" > "Nat.not_less_eq_eq"
+ "NOT_GREATER" > "Orderings.linorder_class.not_less"
"NOT_EXP_0" > "HOL4Base.arithmetic.NOT_EXP_0"
"NORM_0" > "HOL4Base.arithmetic.NORM_0"
- "MULT_SYM" > "Int.zmult_ac_2"
+ "MULT_SYM" > "Fields.linordered_field_class.sign_simps_40"
"MULT_SUC_EQ" > "HOL4Base.arithmetic.MULT_SUC_EQ"
"MULT_SUC" > "Nat.mult_Suc_right"
- "MULT_RIGHT_1" > "Finite_Set.AC_mult.f_e.ident"
+ "MULT_RIGHT_1" > "Divides.arithmetic_simps_43"
"MULT_MONO_EQ" > "Nat.Suc_mult_cancel1"
"MULT_LESS_EQ_SUC" > "Nat.Suc_mult_le_cancel1"
- "MULT_LEFT_1" > "Finite_Set.AC_mult.f_e.left_ident"
+ "MULT_LEFT_1" > "Divides.arithmetic_simps_42"
"MULT_INCREASES" > "HOL4Base.arithmetic.MULT_INCREASES"
"MULT_EXP_MONO" > "HOL4Base.arithmetic.MULT_EXP_MONO"
- "MULT_EQ_1" > "HOL4Base.arithmetic.MULT_EQ_1"
+ "MULT_EQ_1" > "Nat.nat_mult_eq_1_iff"
"MULT_EQ_0" > "Nat.mult_is_0"
"MULT_DIV" > "Divides.div_mult_self_is_m"
- "MULT_COMM" > "Int.zmult_ac_2"
+ "MULT_COMM" > "Fields.linordered_field_class.sign_simps_40"
"MULT_CLAUSES" > "HOL4Base.arithmetic.MULT_CLAUSES"
- "MULT_ASSOC" > "Int.zmult_ac_1"
- "MULT_0" > "Nat.mult_0_right"
+ "MULT_ASSOC" > "Fields.linordered_field_class.sign_simps_41"
+ "MULT_0" > "Divides.arithmetic_simps_41"
"MULT" > "HOL4Compat.MULT"
"MOD_UNIQUE" > "HOL4Base.arithmetic.MOD_UNIQUE"
"MOD_TIMES2" > "HOL4Base.arithmetic.MOD_TIMES2"
@@ -141,19 +141,19 @@
"MIN_MAX_EQ" > "HOL4Base.arithmetic.MIN_MAX_EQ"
"MIN_LT" > "HOL4Base.arithmetic.MIN_LT"
"MIN_LE" > "HOL4Base.arithmetic.MIN_LE"
- "MIN_IDEM" > "Finite_Set.min.f.ACI_4"
+ "MIN_IDEM" > "Big_Operators.linorder_class.Min.idem"
"MIN_DEF" > "HOL4Compat.MIN_DEF"
- "MIN_COMM" > "Finite_Set.min.f.ACI_2"
- "MIN_ASSOC" > "Finite_Set.min.f.ACI_1"
+ "MIN_COMM" > "Lattices.linorder_class.min_max.inf.commute"
+ "MIN_ASSOC" > "Lattices.linorder_class.min_max.inf.assoc"
"MIN_0" > "HOL4Base.arithmetic.MIN_0"
"MAX_LT" > "HOL4Base.arithmetic.MAX_LT"
"MAX_LE" > "HOL4Base.arithmetic.MAX_LE"
- "MAX_IDEM" > "Finite_Set.max.f.ACI_4"
+ "MAX_IDEM" > "Big_Operators.linorder_class.Max.idem"
"MAX_DEF" > "HOL4Compat.MAX_DEF"
- "MAX_COMM" > "Finite_Set.max.f.ACI_2"
- "MAX_ASSOC" > "Finite_Set.max.f.ACI_1"
+ "MAX_COMM" > "Lattices.linorder_class.min_max.inf_sup_aci_5"
+ "MAX_ASSOC" > "Lattices.linorder_class.min_max.inf_sup_aci_6"
"MAX_0" > "HOL4Base.arithmetic.MAX_0"
- "LESS_TRANS" > "Nat.less_trans"
+ "LESS_TRANS" > "Orderings.order_less_trans"
"LESS_SUC_NOT" > "HOL4Base.arithmetic.LESS_SUC_NOT"
"LESS_SUC_EQ_COR" > "Nat.Suc_lessI"
"LESS_SUB_ADD_LESS" > "HOL4Base.arithmetic.LESS_SUB_ADD_LESS"
@@ -162,28 +162,28 @@
"LESS_OR" > "Nat.Suc_leI"
"LESS_NOT_SUC" > "HOL4Base.arithmetic.LESS_NOT_SUC"
"LESS_MULT_MONO" > "Nat.Suc_mult_less_cancel1"
- "LESS_MULT2" > "Rings.mult_pos_pos"
+ "LESS_MULT2" > "Rings.linordered_semiring_strict_class.mult_pos_pos"
"LESS_MONO_REV" > "Nat.Suc_less_SucD"
"LESS_MONO_MULT" > "Nat.mult_le_mono1"
"LESS_MONO_EQ" > "Nat.Suc_less_eq"
- "LESS_MONO_ADD_INV" > "Groups.add_less_imp_less_right"
- "LESS_MONO_ADD_EQ" > "Groups.add_less_cancel_right"
- "LESS_MONO_ADD" > "Nat.add_less_mono1"
+ "LESS_MONO_ADD_INV" > "Groups.ordered_ab_semigroup_add_imp_le_class.add_less_imp_less_right"
+ "LESS_MONO_ADD_EQ" > "Groups.ordered_ab_semigroup_add_imp_le_class.add_less_cancel_right"
+ "LESS_MONO_ADD" > "Groups.ordered_cancel_ab_semigroup_add_class.add_strict_right_mono"
"LESS_MOD" > "Divides.mod_less"
"LESS_LESS_SUC" > "HOL4Base.arithmetic.LESS_LESS_SUC"
- "LESS_LESS_EQ_TRANS" > "Nat.less_le_trans"
+ "LESS_LESS_EQ_TRANS" > "Orderings.order_less_le_trans"
"LESS_LESS_CASES" > "HOL4Base.arithmetic.LESS_LESS_CASES"
- "LESS_IMP_LESS_OR_EQ" > "Nat.le_simps_1"
- "LESS_IMP_LESS_ADD" > "Nat.trans_less_add1"
+ "LESS_IMP_LESS_OR_EQ" > "FunDef.termination_basic_simps_5"
+ "LESS_IMP_LESS_ADD" > "FunDef.termination_basic_simps_1"
"LESS_EXP_SUC_MONO" > "HOL4Base.arithmetic.LESS_EXP_SUC_MONO"
"LESS_EQ_TRANS" > "Nat.le_trans"
"LESS_EQ_SUC_REFL" > "HOL4Base.arithmetic.LESS_EQ_SUC_REFL"
"LESS_EQ_SUB_LESS" > "HOL4Base.arithmetic.LESS_EQ_SUB_LESS"
- "LESS_EQ_REFL" > "Finite_Set.max.f_below.below_refl"
- "LESS_EQ_MONO_ADD_EQ" > "Groups.add_le_cancel_right"
+ "LESS_EQ_REFL" > "Nat.le_refl"
+ "LESS_EQ_MONO_ADD_EQ" > "Groups.ordered_ab_semigroup_add_imp_le_class.add_le_cancel_right"
"LESS_EQ_MONO" > "Nat.Suc_le_mono"
- "LESS_EQ_LESS_TRANS" > "Nat.le_less_trans"
- "LESS_EQ_LESS_EQ_MONO" > "Nat.add_le_mono"
+ "LESS_EQ_LESS_TRANS" > "Orderings.order_le_less_trans"
+ "LESS_EQ_LESS_EQ_MONO" > "Groups.add_mono_thms_linordered_semiring_1"
"LESS_EQ_IMP_LESS_SUC" > "Nat.le_imp_less_Suc"
"LESS_EQ_EXISTS" > "Nat.le_iff_add"
"LESS_EQ_CASES" > "Nat.nat_le_linear"
@@ -192,8 +192,8 @@
"LESS_EQ_ADD" > "Nat.le_add1"
"LESS_EQ_0" > "Nat.le_0_eq"
"LESS_EQUAL_ANTISYM" > "Nat.le_antisym"
- "LESS_EQUAL_ADD" > "HOL4Base.arithmetic.LESS_EQUAL_ADD"
- "LESS_EQ" > "Nat.le_simps_3"
+ "LESS_EQUAL_ADD" > "Series.le_Suc_ex"
+ "LESS_EQ" > "Nat.Suc_le_eq"
"LESS_DIV_EQ_ZERO" > "Divides.div_less"
"LESS_CASES_IMP" > "HOL4Base.arithmetic.LESS_CASES_IMP"
"LESS_CASES" > "HOL4Base.arithmetic.LESS_CASES"
@@ -203,8 +203,8 @@
"LESS_ADD_1" > "HOL4Base.arithmetic.LESS_ADD_1"
"LESS_ADD" > "HOL4Base.arithmetic.LESS_ADD"
"LESS_0_CASES" > "HOL4Base.arithmetic.LESS_0_CASES"
- "LEFT_SUB_DISTRIB" > "Nat.nat_distrib_4"
- "LEFT_ADD_DISTRIB" > "Nat.nat_distrib_2"
+ "LEFT_SUB_DISTRIB" > "Nat.diff_mult_distrib2"
+ "LEFT_ADD_DISTRIB" > "Fields.linordered_field_class.sign_simps_25"
"LE" > "HOL4Base.arithmetic.LE"
"INV_PRE_LESS_EQ" > "HOL4Base.arithmetic.INV_PRE_LESS_EQ"
"INV_PRE_LESS" > "HOL4Base.arithmetic.INV_PRE_LESS"
@@ -216,11 +216,11 @@
"FUNPOW" > "HOL4Compat.FUNPOW"
"FACT_LESS" > "HOL4Base.arithmetic.FACT_LESS"
"FACT" > "HOL4Base.arithmetic.FACT"
- "EXP_INJECTIVE" > "Power.power_inject_exp"
- "EXP_EQ_1" > "HOL4Base.arithmetic.EXP_EQ_1"
+ "EXP_INJECTIVE" > "Power.linordered_semidom_class.power_inject_exp"
+ "EXP_EQ_1" > "Primes.exp_eq_1"
"EXP_EQ_0" > "HOL4Base.arithmetic.EXP_EQ_0"
"EXP_ALWAYS_BIG_ENOUGH" > "HOL4Base.arithmetic.EXP_ALWAYS_BIG_ENOUGH"
- "EXP_ADD" > "Power.power_add"
+ "EXP_ADD" > "Power.monoid_mult_class.power_add"
"EXP_1" > "HOL4Base.arithmetic.EXP_1"
"EXP" > "HOL4Compat.EXP"
"EXISTS_GREATEST" > "HOL4Base.arithmetic.EXISTS_GREATEST"
@@ -233,11 +233,11 @@
"EVEN_AND_ODD" > "HOL4Base.arithmetic.EVEN_AND_ODD"
"EVEN_ADD" > "HOL4Base.arithmetic.EVEN_ADD"
"EVEN" > "HOL4Base.arithmetic.EVEN"
- "EQ_MULT_LCANCEL" > "Nat_Numeral.nat_mult_eq_cancel_disj"
- "EQ_MONO_ADD_EQ" > "Nat.nat_add_right_cancel"
- "EQ_LESS_EQ" > "Orderings.order_eq_iff"
- "EQ_ADD_RCANCEL" > "Nat.nat_add_right_cancel"
- "EQ_ADD_LCANCEL" > "Nat.nat_add_left_cancel"
+ "EQ_MULT_LCANCEL" > "Numeral_Simprocs.nat_mult_eq_cancel_disj"
+ "EQ_MONO_ADD_EQ" > "Groups.cancel_semigroup_add_class.add_right_cancel"
+ "EQ_LESS_EQ" > "Orderings.order_class.eq_iff"
+ "EQ_ADD_RCANCEL" > "Groups.cancel_semigroup_add_class.add_right_cancel"
+ "EQ_ADD_LCANCEL" > "Groups.cancel_semigroup_add_class.add_left_cancel"
"DIV_UNIQUE" > "HOL4Base.arithmetic.DIV_UNIQUE"
"DIV_P" > "HOL4Base.arithmetic.DIV_P"
"DIV_ONE" > "Divides.div_1"
@@ -248,23 +248,23 @@
"DIVMOD_ID" > "HOL4Base.arithmetic.DIVMOD_ID"
"DIVISION" > "HOL4Compat.DIVISION"
"DA" > "HOL4Base.arithmetic.DA"
- "COMPLETE_INDUCTION" > "Nat.less_induct"
+ "COMPLETE_INDUCTION" > "Nat.nat_less_induct"
"CANCEL_SUB" > "Nat.eq_diff_iff"
"ALT_ZERO" > "HOL4Compat.ALT_ZERO_def"
- "ADD_SYM" > "Finite_Set.AC_add.f.AC_2"
+ "ADD_SYM" > "Fields.linordered_field_class.sign_simps_43"
"ADD_SUC" > "Nat.add_Suc_right"
"ADD_SUB" > "Nat.diff_add_inverse2"
- "ADD_MONO_LESS_EQ" > "Nat.nat_add_left_cancel_le"
+ "ADD_MONO_LESS_EQ" > "Groups.ordered_ab_semigroup_add_imp_le_class.add_le_cancel_left"
"ADD_INV_0_EQ" > "HOL4Base.arithmetic.ADD_INV_0_EQ"
"ADD_INV_0" > "Nat.add_eq_self_zero"
"ADD_EQ_SUB" > "HOL4Base.arithmetic.ADD_EQ_SUB"
"ADD_EQ_1" > "HOL4Base.arithmetic.ADD_EQ_1"
"ADD_EQ_0" > "Nat.add_is_0"
"ADD_DIV_ADD_DIV" > "HOL4Base.arithmetic.ADD_DIV_ADD_DIV"
- "ADD_COMM" > "Finite_Set.AC_add.f.AC_2"
+ "ADD_COMM" > "Fields.linordered_field_class.sign_simps_43"
"ADD_CLAUSES" > "HOL4Base.arithmetic.ADD_CLAUSES"
- "ADD_ASSOC" > "Finite_Set.AC_add.f.AC_1"
- "ADD_0" > "Finite_Set.AC_add.f_e.ident"
+ "ADD_ASSOC" > "Fields.linordered_field_class.sign_simps_44"
+ "ADD_0" > "Divides.arithmetic_simps_39"
"ADD1" > "Nat_Numeral.Suc_eq_plus1"
"ADD" > "HOL4Compat.ADD"
--- a/src/HOL/Import/HOL/bits.imp Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Import/HOL/bits.imp Fri Sep 09 06:47:14 2011 +0200
@@ -83,7 +83,7 @@
"DIVMOD_2EXP_def" > "HOL4Word32.bits.DIVMOD_2EXP_def"
"DIV2_primdef" > "HOL4Word32.bits.DIV2_primdef"
"DIV2_def" > "HOL4Word32.bits.DIV2_def"
- "DIV1" > "HOL4Word32.bits.DIV1"
+ "DIV1" > "Divides.semiring_div_class.div_by_1"
"BIT_def" > "HOL4Word32.bits.BIT_def"
"BIT_SLICE_THM" > "HOL4Word32.bits.BIT_SLICE_THM"
"BIT_SLICE_LEM" > "HOL4Word32.bits.BIT_SLICE_LEM"
--- a/src/HOL/Import/HOL/bool.imp Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Import/HOL/bool.imp Fri Sep 09 06:47:14 2011 +0200
@@ -11,10 +11,13 @@
"IN" > "IN_def"
"ARB" > "ARB_def"
+type_maps
+ "bool" > "HOL.bool"
+
const_maps
"~" > "HOL.Not"
- "bool_case" > "Datatype.bool.bool_case"
- "\\/" > HOL.disj
+ "bool_case" > "Product_Type.bool.bool_case"
+ "\\/" > "HOL.disj"
"TYPE_DEFINITION" > "HOL4Setup.TYPE_DEFINITION"
"T" > "HOL.True"
"RES_SELECT" > "HOL4Base.bool.RES_SELECT"
@@ -30,14 +33,14 @@
"ARB" > "HOL4Base.bool.ARB"
"?!" > "HOL.Ex1"
"?" > "HOL.Ex"
- "/\\" > HOL.conj
+ "/\\" > "HOL.conj"
"!" > "HOL.All"
thm_maps
"bool_case_thm" > "HOL4Base.bool.bool_case_thm"
"bool_case_ID" > "HOL4Base.bool.bool_case_ID"
"bool_case_DEF" > "HOL4Compat.bool_case_DEF"
- "bool_INDUCT" > "Datatype.bool.induct_correctness"
+ "bool_INDUCT" > "Product_Type.bool.induct"
"boolAxiom" > "HOL4Base.bool.boolAxiom"
"UNWIND_THM2" > "HOL.simp_thms_39"
"UNWIND_THM1" > "HOL.simp_thms_40"
@@ -49,21 +52,21 @@
"TYPE_DEFINITION_THM" > "HOL4Setup.TYPE_DEFINITION"
"TYPE_DEFINITION" > "HOL4Setup.TYPE_DEFINITION"
"TRUTH" > "HOL.TrueI"
- "SWAP_FORALL_THM" > "HOL4Base.bool.SWAP_FORALL_THM"
- "SWAP_EXISTS_THM" > "HOL4Base.bool.SWAP_EXISTS_THM"
+ "SWAP_FORALL_THM" > "HOL.all_comm"
+ "SWAP_EXISTS_THM" > "HOL.ex_comm"
"SKOLEM_THM" > "HOL4Base.bool.SKOLEM_THM"
"SELECT_UNIQUE" > "HOL4Base.bool.SELECT_UNIQUE"
"SELECT_THM" > "HOL4Setup.EXISTS_DEF"
"SELECT_REFL_2" > "Hilbert_Choice.some_sym_eq_trivial"
"SELECT_REFL" > "Hilbert_Choice.some_eq_trivial"
- "SELECT_AX" > "Hilbert_Choice.tfl_some"
+ "SELECT_AX" > "Hilbert_Choice.someI"
"RIGHT_OR_OVER_AND" > "HOL.disj_conj_distribR"
"RIGHT_OR_EXISTS_THM" > "HOL.ex_simps_4"
"RIGHT_FORALL_OR_THM" > "HOL.all_simps_4"
"RIGHT_FORALL_IMP_THM" > "HOL.all_simps_6"
"RIGHT_EXISTS_IMP_THM" > "HOL.ex_simps_6"
"RIGHT_EXISTS_AND_THM" > "HOL.ex_simps_2"
- "RIGHT_AND_OVER_OR" > "HOL.conj_disj_distribR"
+ "RIGHT_AND_OVER_OR" > "Groebner_Basis.dnf_2"
"RIGHT_AND_FORALL_THM" > "HOL.all_simps_2"
"RES_SELECT_def" > "HOL4Base.bool.RES_SELECT_def"
"RES_SELECT_DEF" > "HOL4Base.bool.RES_SELECT_DEF"
@@ -74,11 +77,11 @@
"RES_EXISTS_UNIQUE_DEF" > "HOL4Base.bool.RES_EXISTS_UNIQUE_DEF"
"RES_EXISTS_DEF" > "HOL4Base.bool.RES_EXISTS_DEF"
"RES_ABSTRACT_DEF" > "HOL4Base.bool.RES_ABSTRACT_DEF"
- "REFL_CLAUSE" > "HOL.simp_thms_6"
+ "REFL_CLAUSE" > "Groebner_Basis.bool_simps_6"
"OR_INTRO_THM2" > "HOL.disjI2"
"OR_INTRO_THM1" > "HOL.disjI1"
"OR_IMP_THM" > "HOL4Base.bool.OR_IMP_THM"
- "OR_ELIM_THM" > "Recdef.tfl_disjE"
+ "OR_ELIM_THM" > "HOL.disjE"
"OR_DEF" > "HOL.or_def"
"OR_CONG" > "HOL4Base.bool.OR_CONG"
"OR_CLAUSES" > "HOL4Base.bool.OR_CLAUSES"
@@ -87,14 +90,14 @@
"ONE_ONE_THM" > "HOL4Base.bool.ONE_ONE_THM"
"ONE_ONE_DEF" > "HOL4Setup.ONE_ONE_DEF"
"NOT_IMP" > "HOL.not_imp"
- "NOT_FORALL_THM" > "Inductive.basic_monos_15"
- "NOT_F" > "HOL.Eq_FalseI"
- "NOT_EXISTS_THM" > "Inductive.basic_monos_16"
- "NOT_DEF" > "HOL.simp_thms_19"
+ "NOT_FORALL_THM" > "HOL.not_all"
+ "NOT_F" > "Groebner_Basis.PFalse_2"
+ "NOT_EXISTS_THM" > "HOL.not_ex"
+ "NOT_DEF" > "Groebner_Basis.bool_simps_19"
"NOT_CLAUSES" > "HOL4Base.bool.NOT_CLAUSES"
"NOT_AND" > "HOL4Base.bool.NOT_AND"
"MONO_OR" > "Inductive.basic_monos_3"
- "MONO_NOT" > "HOL.rev_contrapos"
+ "MONO_NOT" > "HOL.contrapos_nn"
"MONO_IMP" > "Set.imp_mono"
"MONO_EXISTS" > "Inductive.basic_monos_5"
"MONO_COND" > "HOL4Base.bool.MONO_COND"
@@ -104,27 +107,27 @@
"LET_RATOR" > "HOL4Base.bool.LET_RATOR"
"LET_RAND" > "HOL4Base.bool.LET_RAND"
"LET_DEF" > "HOL4Compat.LET_def"
- "LET_CONG" > "Recdef.let_cong"
+ "LET_CONG" > "FunDef.let_cong"
"LEFT_OR_OVER_AND" > "HOL.disj_conj_distribL"
"LEFT_OR_EXISTS_THM" > "HOL.ex_simps_3"
"LEFT_FORALL_OR_THM" > "HOL.all_simps_3"
- "LEFT_FORALL_IMP_THM" > "HOL.imp_ex"
- "LEFT_EXISTS_IMP_THM" > "HOL.imp_all"
+ "LEFT_FORALL_IMP_THM" > "HOL.all_simps_5"
+ "LEFT_EXISTS_IMP_THM" > "HOL.ex_simps_5"
"LEFT_EXISTS_AND_THM" > "HOL.ex_simps_1"
- "LEFT_AND_OVER_OR" > "HOL.conj_disj_distribL"
+ "LEFT_AND_OVER_OR" > "Groebner_Basis.dnf_1"
"LEFT_AND_FORALL_THM" > "HOL.all_simps_1"
"IN_def" > "HOL4Base.bool.IN_def"
"IN_DEF" > "HOL4Base.bool.IN_DEF"
"INFINITY_AX" > "HOL4Setup.INFINITY_AX"
"IMP_F_EQ_F" > "HOL4Base.bool.IMP_F_EQ_F"
"IMP_F" > "HOL.notI"
- "IMP_DISJ_THM" > "Inductive.basic_monos_11"
+ "IMP_DISJ_THM" > "Groebner_Basis.nnf_simps_3"
"IMP_CONG" > "HOL.imp_cong"
"IMP_CLAUSES" > "HOL4Base.bool.IMP_CLAUSES"
- "IMP_ANTISYM_AX" > "HOL4Setup.light_imp_as"
+ "IMP_ANTISYM_AX" > "HOL.iff"
"F_IMP" > "HOL4Base.bool.F_IMP"
"F_DEF" > "HOL.False_def"
- "FUN_EQ_THM" > "Fun.fun_eq_iff"
+ "FUN_EQ_THM" > "HOL.fun_eq_iff"
"FORALL_THM" > "HOL4Base.bool.FORALL_THM"
"FORALL_SIMP" > "HOL.simp_thms_35"
"FORALL_DEF" > "HOL.All_def"
@@ -139,24 +142,24 @@
"EXISTS_OR_THM" > "HOL.ex_disj_distrib"
"EXISTS_DEF" > "HOL4Setup.EXISTS_DEF"
"EXCLUDED_MIDDLE" > "HOL4Base.bool.EXCLUDED_MIDDLE"
- "ETA_THM" > "Presburger.fm_modd_pinf"
- "ETA_AX" > "Presburger.fm_modd_pinf"
- "EQ_TRANS" > "Set.basic_trans_rules_31"
- "EQ_SYM_EQ" > "HOL.eq_sym_conv"
- "EQ_SYM" > "HOL.meta_eq_to_obj_eq"
- "EQ_REFL" > "Presburger.fm_modd_pinf"
+ "ETA_THM" > "HOL.eta_contract_eq"
+ "ETA_AX" > "HOL.eta_contract_eq"
+ "EQ_TRANS" > "HOL.trans"
+ "EQ_SYM_EQ" > "HOL.eq_ac_1"
+ "EQ_SYM" > "HOL.eq_reflection"
+ "EQ_REFL" > "HOL.refl"
"EQ_IMP_THM" > "HOL.iff_conv_conj_imp"
- "EQ_EXT" > "HOL.meta_eq_to_obj_eq"
- "EQ_EXPAND" > "HOL4Base.bool.EQ_EXPAND"
+ "EQ_EXT" > "HOL.eq_reflection"
+ "EQ_EXPAND" > "Groebner_Basis.nnf_simps_4"
"EQ_CLAUSES" > "HOL4Base.bool.EQ_CLAUSES"
- "DISJ_SYM" > "HOL.disj_comms_1"
+ "DISJ_SYM" > "Groebner_Basis.dnf_4"
"DISJ_IMP_THM" > "HOL.imp_disjL"
- "DISJ_COMM" > "HOL.disj_comms_1"
- "DISJ_ASSOC" > "Recdef.tfl_disj_assoc"
+ "DISJ_COMM" > "Groebner_Basis.dnf_4"
+ "DISJ_ASSOC" > "HOL.disj_ac_3"
"DE_MORGAN_THM" > "HOL4Base.bool.DE_MORGAN_THM"
- "CONJ_SYM" > "HOL.conj_comms_1"
- "CONJ_COMM" > "HOL.conj_comms_1"
- "CONJ_ASSOC" > "HOL.conj_assoc"
+ "CONJ_SYM" > "Groebner_Basis.dnf_3"
+ "CONJ_COMM" > "Groebner_Basis.dnf_3"
+ "CONJ_ASSOC" > "HOL.conj_ac_3"
"COND_RATOR" > "HOL4Base.bool.COND_RATOR"
"COND_RAND" > "HOL.if_distrib"
"COND_ID" > "HOL.if_cancel"
@@ -172,8 +175,8 @@
"BOOL_FUN_INDUCT" > "HOL4Base.bool.BOOL_FUN_INDUCT"
"BOOL_FUN_CASES_THM" > "HOL4Base.bool.BOOL_FUN_CASES_THM"
"BOOL_EQ_DISTINCT" > "HOL4Base.bool.BOOL_EQ_DISTINCT"
- "BOOL_CASES_AX" > "Datatype.bool.nchotomy"
- "BETA_THM" > "Presburger.fm_modd_pinf"
+ "BOOL_CASES_AX" > "HOL.True_or_False"
+ "BETA_THM" > "HOL.eta_contract_eq"
"ARB_def" > "HOL4Base.bool.ARB_def"
"ARB_DEF" > "HOL4Base.bool.ARB_DEF"
"AND_INTRO_THM" > "HOL.conjI"
@@ -183,7 +186,7 @@
"AND_CLAUSES" > "HOL4Base.bool.AND_CLAUSES"
"AND2_THM" > "HOL.conjunct2"
"AND1_THM" > "HOL.conjunct1"
- "ABS_SIMP" > "Presburger.fm_modd_pinf"
+ "ABS_SIMP" > "HOL.refl"
"ABS_REP_THM" > "HOL4Base.bool.ABS_REP_THM"
ignore_thms
--- a/src/HOL/Import/HOL/combin.imp Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Import/HOL/combin.imp Fri Sep 09 06:47:14 2011 +0200
@@ -18,8 +18,8 @@
"C" > "HOL4Base.combin.C"
thm_maps
- "o_THM" > "Fun.o_apply"
- "o_DEF" > "Fun.o_apply"
+ "o_THM" > "Fun.comp_def"
+ "o_DEF" > "Fun.comp_def"
"o_ASSOC" > "Fun.o_assoc"
"W_def" > "HOL4Base.combin.W_def"
"W_THM" > "HOL4Base.combin.W_def"
--- a/src/HOL/Import/HOL/divides.imp Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Import/HOL/divides.imp Fri Sep 09 06:47:14 2011 +0200
@@ -3,22 +3,22 @@
import_segment "hol4"
const_maps
- "divides" > Divides.times_class.dvd :: "nat => nat => bool"
+ "divides" > "Rings.dvd_class.dvd" :: "nat => nat => bool"
thm_maps
"divides_def" > "HOL4Compat.divides_def"
- "ONE_DIVIDES_ALL" > "HOL4Base.divides.ONE_DIVIDES_ALL"
- "NOT_LT_DIV" > "NatSimprocs.nat_dvd_not_less"
- "DIVIDES_TRANS" > "Rings.dvd_trans"
- "DIVIDES_SUB" > "Rings.dvd_diff"
- "DIVIDES_REFL" > "Rings.dvd_refl"
+ "ONE_DIVIDES_ALL" > "GCD.gcd_lcm_complete_lattice_nat.bot_least"
+ "NOT_LT_DIV" > "Nat.nat_dvd_not_less"
+ "DIVIDES_TRANS" > "Nat.dvd.order_trans"
+ "DIVIDES_SUB" > "Nat.dvd_diff_nat"
+ "DIVIDES_REFL" > "Nat.dvd.order_refl"
"DIVIDES_MULT_LEFT" > "HOL4Base.divides.DIVIDES_MULT_LEFT"
- "DIVIDES_MULT" > "Divides.dvd_mult2"
- "DIVIDES_LE" > "Divides.dvd_imp_le"
+ "DIVIDES_MULT" > "Rings.comm_semiring_1_class.dvd_mult2"
+ "DIVIDES_LE" > "Nat.dvd_imp_le"
"DIVIDES_FACT" > "HOL4Base.divides.DIVIDES_FACT"
- "DIVIDES_ANTISYM" > "Divides.dvd_antisym"
- "DIVIDES_ADD_2" > "HOL4Base.divides.DIVIDES_ADD_2"
- "DIVIDES_ADD_1" > "Rings.dvd_add"
- "ALL_DIVIDES_0" > "Divides.dvd_0_right"
+ "DIVIDES_ANTISYM" > "Nat.dvd.antisym"
+ "DIVIDES_ADD_2" > "Primes.divides_add_revr"
+ "DIVIDES_ADD_1" > "Rings.comm_semiring_1_class.dvd_add"
+ "ALL_DIVIDES_0" > "GCD.gcd_lcm_complete_lattice_nat.top_greatest"
end
--- a/src/HOL/Import/HOL/lim.imp Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Import/HOL/lim.imp Fri Sep 09 06:47:14 2011 +0200
@@ -24,7 +24,7 @@
"contl_def" > "HOL4Real.lim.contl_def"
"contl" > "HOL4Real.lim.contl"
"ROLLE" > "HOL4Real.lim.ROLLE"
- "MVT_LEMMA" > "HOL4Real.lim.MVT_LEMMA"
+ "MVT_LEMMA" > "Deriv.lemma_MVT"
"MVT" > "HOL4Real.lim.MVT"
"LIM_X" > "HOL4Real.lim.LIM_X"
"LIM_UNIQ" > "HOL4Real.lim.LIM_UNIQ"
@@ -41,7 +41,7 @@
"LIM" > "HOL4Real.lim.LIM"
"IVT2" > "HOL4Real.lim.IVT2"
"IVT" > "HOL4Real.lim.IVT"
- "INTERVAL_LEMMA" > "HOL4Real.lim.INTERVAL_LEMMA"
+ "INTERVAL_LEMMA" > "Deriv.lemma_interval"
"INTERVAL_CLEMMA" > "HOL4Real.lim.INTERVAL_CLEMMA"
"INTERVAL_ABS" > "HOL4Real.lim.INTERVAL_ABS"
"DIFF_XM1" > "HOL4Real.lim.DIFF_XM1"
--- a/src/HOL/Import/HOL/list.imp Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Import/HOL/list.imp Fri Sep 09 06:47:14 2011 +0200
@@ -19,7 +19,7 @@
"REPLICATE" > "List.replicate"
"NULL" > "List.null"
"NIL" > "List.list.Nil"
- "MEM" > "List.op mem"
+ "MEM" > "HOL4Compat.list_mem"
"MAP2" > "HOL4Compat.map2"
"MAP" > "List.map"
"LENGTH" > "Nat.size_class.size"
@@ -30,25 +30,25 @@
"FOLDL" > "List.foldl"
"FLAT" > "List.concat"
"FILTER" > "List.filter"
- "EXISTS" > "HOL4Compat.list_exists"
+ "EXISTS" > "List.list_ex"
"EVERY" > "List.list_all"
"CONS" > "List.list.Cons"
"APPEND" > "List.append"
thm_maps
- "list_size_def" > "HOL4Compat.list_size_def"
+ "list_size_def" > "HOL4Compat.list_size_def'"
"list_size_cong" > "HOL4Base.list.list_size_cong"
"list_nchotomy" > "HOL4Compat.list_CASES"
- "list_induction" > "HOL4Compat.unzip.induct"
- "list_distinct" > "List.list.simps_1"
+ "list_induction" > "HOL4Compat.list_INDUCT"
+ "list_distinct" > "List.list.distinct_1"
"list_case_def" > "HOL4Compat.list_case_def"
"list_case_cong" > "HOL4Compat.list_case_cong"
"list_case_compute" > "HOL4Base.list.list_case_compute"
- "list_INDUCT" > "HOL4Compat.unzip.induct"
+ "list_INDUCT" > "HOL4Compat.list_INDUCT"
"list_CASES" > "HOL4Compat.list_CASES"
"list_Axiom_old" > "HOL4Compat.list_Axiom_old"
"list_Axiom" > "HOL4Compat.list_Axiom"
- "list_11" > "List.list.simps_3"
+ "list_11" > "List.list.inject"
"ZIP_UNZIP" > "HOL4Base.list.ZIP_UNZIP"
"ZIP_MAP" > "HOL4Base.list.ZIP_MAP"
"ZIP" > "HOL4Compat.ZIP"
@@ -62,11 +62,11 @@
"REVERSE_APPEND" > "List.rev_append"
"NULL_DEF" > "HOL4Compat.NULL_DEF"
"NULL" > "HOL4Base.list.NULL"
- "NOT_NIL_CONS" > "List.list.simps_1"
+ "NOT_NIL_CONS" > "List.list.distinct_1"
"NOT_EXISTS" > "HOL4Base.list.NOT_EXISTS"
"NOT_EVERY" > "HOL4Base.list.NOT_EVERY"
"NOT_EQ_LIST" > "HOL4Base.list.NOT_EQ_LIST"
- "NOT_CONS_NIL" > "List.list.simps_2"
+ "NOT_CONS_NIL" > "List.list.distinct_2"
"MEM_ZIP" > "HOL4Base.list.MEM_ZIP"
"MEM_MAP" > "HOL4Base.list.MEM_MAP"
"MEM_EL" > "HOL4Base.list.MEM_EL"
@@ -102,7 +102,7 @@
"EXISTS_MEM" > "HOL4Base.list.EXISTS_MEM"
"EXISTS_DEF" > "HOL4Compat.list_exists_DEF"
"EXISTS_CONG" > "HOL4Base.list.EXISTS_CONG"
- "EXISTS_APPEND" > "HOL4Base.list.EXISTS_APPEND"
+ "EXISTS_APPEND" > "List.list_ex_append"
"EVERY_MONOTONIC" > "HOL4Base.list.EVERY_MONOTONIC"
"EVERY_MEM" > "HOL4Base.list.EVERY_MEM"
"EVERY_EL" > "HOL4Base.list.EVERY_EL"
@@ -115,7 +115,7 @@
"EL_ZIP" > "HOL4Base.list.EL_ZIP"
"EL" > "HOL4Base.list.EL"
"CONS_ACYCLIC" > "HOL4Base.list.CONS_ACYCLIC"
- "CONS_11" > "List.list.simps_3"
+ "CONS_11" > "List.list.inject"
"CONS" > "HOL4Base.list.CONS"
"APPEND_eq_NIL" > "HOL4Base.list.APPEND_eq_NIL"
"APPEND_NIL" > "List.append_Nil2"
--- a/src/HOL/Import/HOL/num.imp Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Import/HOL/num.imp Fri Sep 09 06:47:14 2011 +0200
@@ -7,12 +7,12 @@
const_maps
"SUC" > "Nat.Suc"
- "0" > "0" :: "nat"
+ "0" > "Groups.zero_class.zero" :: "nat"
thm_maps
- "NOT_SUC" > "Nat.nat.simps_1"
+ "NOT_SUC" > "Nat.Suc_not_Zero"
"INV_SUC" > "Nat.Suc_inject"
- "INDUCTION" > "List.lexn.induct"
+ "INDUCTION" > "Fact.fact_nat.induct"
ignore_thms
"num_TY_DEF"
--- a/src/HOL/Import/HOL/option.imp Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Import/HOL/option.imp Fri Sep 09 06:47:14 2011 +0200
@@ -3,34 +3,34 @@
import_segment "hol4"
type_maps
- "option" > "Datatype.option"
+ "option" > "Option.option"
const_maps
- "option_case" > "Datatype.option.option_case"
- "THE" > "Datatype.the"
- "SOME" > "Datatype.option.Some"
- "OPTION_MAP" > "Datatype.option_map"
+ "option_case" > "Option.option.option_case"
+ "THE" > "Option.the"
+ "SOME" > "Option.option.Some"
+ "OPTION_MAP" > "Option.map"
"OPTION_JOIN" > "HOL4Compat.OPTION_JOIN"
- "NONE" > "Datatype.option.None"
+ "NONE" > "Option.option.None"
"IS_SOME" > "HOL4Compat.IS_SOME"
"IS_NONE" > "HOL4Compat.IS_NONE"
thm_maps
- "option_nchotomy" > "Datatype.option.nchotomy"
- "option_induction" > "HOL4Compat.OPTION_JOIN.induct"
+ "option_nchotomy" > "Option.option.nchotomy"
+ "option_induction" > "Option.option.induct"
"option_case_def" > "HOL4Compat.option_case_def"
"option_case_cong" > "HOL4Base.option.option_case_cong"
"option_case_compute" > "HOL4Base.option.option_case_compute"
"option_CLAUSES" > "HOL4Base.option.option_CLAUSES"
- "THE_DEF" > "Datatype.the.simps"
- "SOME_11" > "Datatype.option.simps_3"
+ "THE_DEF" > "Option.the.simps"
+ "SOME_11" > "Option.option.inject"
"OPTION_MAP_EQ_SOME" > "HOL4Base.option.OPTION_MAP_EQ_SOME"
- "OPTION_MAP_EQ_NONE" > "Datatype.option_map_is_None"
+ "OPTION_MAP_EQ_NONE" > "Option.option_map_is_None"
"OPTION_MAP_DEF" > "HOL4Compat.OPTION_MAP_DEF"
"OPTION_JOIN_EQ_SOME" > "HOL4Base.option.OPTION_JOIN_EQ_SOME"
"OPTION_JOIN_DEF" > "HOL4Compat.OPTION_JOIN_DEF"
- "NOT_SOME_NONE" > "Datatype.option.simps_2"
- "NOT_NONE_SOME" > "Datatype.option.simps_1"
+ "NOT_SOME_NONE" > "Option.option.distinct_2"
+ "NOT_NONE_SOME" > "Option.option.distinct_1"
"IS_SOME_DEF" > "HOL4Compat.IS_SOME_DEF"
"IS_NONE_DEF" > "HOL4Compat.IS_NONE_DEF"
--- a/src/HOL/Import/HOL/pair.imp Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Import/HOL/pair.imp Fri Sep 09 06:47:14 2011 +0200
@@ -10,38 +10,38 @@
"prod" > "Product_Type.prod"
const_maps
- "pair_case" > "Product_Type.prod_case"
- "UNCURRY" > "Product_Type.prod_case"
- "FST" > "Product_Type.fst"
+ "pair_case" > "Product_Type.prod.prod_case"
+ "UNCURRY" > "Product_Type.prod.prod_case"
"SND" > "Product_Type.snd"
"RPROD" > "HOL4Base.pair.RPROD"
"LEX" > "HOL4Base.pair.LEX"
+ "FST" > "Product_Type.fst"
"CURRY" > "Product_Type.curry"
"," > "Product_Type.Pair"
- "##" > "map_pair"
+ "##" > "Product_Type.map_pair"
thm_maps
- "pair_induction" > "Datatype.prod.induct_correctness"
- "pair_case_thm" > "Product_Type.split"
+ "pair_induction" > "Product_Type.prod.induct"
+ "pair_case_thm" > "Product_Type.prod.cases"
"pair_case_def" > "HOL4Compat.pair_case_def"
"pair_case_cong" > "HOL4Base.pair.pair_case_cong"
"pair_Axiom" > "HOL4Base.pair.pair_Axiom"
"WF_RPROD" > "HOL4Base.pair.WF_RPROD"
"WF_LEX" > "HOL4Base.pair.WF_LEX"
- "UNCURRY_VAR" > "Product_Type.split_beta"
+ "UNCURRY_VAR" > "Product_Type.prod_case_beta"
"UNCURRY_ONE_ONE_THM" > "HOL4Base.pair.UNCURRY_ONE_ONE_THM"
- "UNCURRY_DEF" > "Product_Type.split"
+ "UNCURRY_DEF" > "Product_Type.prod.cases"
"UNCURRY_CURRY_THM" > "Product_Type.split_curry"
"UNCURRY_CONG" > "HOL4Base.pair.UNCURRY_CONG"
- "UNCURRY" > "Product_Type.split_beta"
+ "UNCURRY" > "Product_Type.prod_case_beta"
"SND" > "Product_Type.snd_conv"
"RPROD_def" > "HOL4Base.pair.RPROD_def"
"RPROD_DEF" > "HOL4Base.pair.RPROD_DEF"
"PFORALL_THM" > "HOL4Base.pair.PFORALL_THM"
"PEXISTS_THM" > "HOL4Base.pair.PEXISTS_THM"
- "PAIR_MAP_THM" > "Product_Type.map_pair"
+ "PAIR_MAP_THM" > "Product_Type.map_pair_simp"
"PAIR_MAP" > "HOL4Compat.PAIR_MAP"
- "PAIR_EQ" > "Datatype.prod.simps_1"
+ "PAIR_EQ" > "Product_Type.Pair_eq"
"PAIR" > "HOL4Compat.PAIR"
"LEX_def" > "HOL4Base.pair.LEX_def"
"LEX_DEF" > "HOL4Base.pair.LEX_DEF"
@@ -51,14 +51,14 @@
"FST" > "Product_Type.fst_conv"
"FORALL_PROD" > "Product_Type.split_paired_All"
"EXISTS_PROD" > "Product_Type.split_paired_Ex"
- "ELIM_UNCURRY" > "Product_Type.split_beta"
+ "ELIM_UNCURRY" > "Product_Type.prod_case_beta"
"ELIM_PFORALL" > "HOL4Base.pair.ELIM_PFORALL"
"ELIM_PEXISTS" > "HOL4Base.pair.ELIM_PEXISTS"
"CURRY_UNCURRY_THM" > "Product_Type.curry_split"
"CURRY_ONE_ONE_THM" > "HOL4Base.pair.CURRY_ONE_ONE_THM"
"CURRY_DEF" > "Product_Type.curry_conv"
- "CLOSED_PAIR_EQ" > "Datatype.prod.simps_1"
- "ABS_PAIR_THM" > "Datatype.prod.nchotomy"
+ "CLOSED_PAIR_EQ" > "Product_Type.Pair_eq"
+ "ABS_PAIR_THM" > "Product_Type.prod.nchotomy"
ignore_thms
"prod_TY_DEF"
--- a/src/HOL/Import/HOL/poly.imp Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Import/HOL/poly.imp Fri Sep 09 06:47:14 2011 +0200
@@ -13,7 +13,7 @@
"poly_add" > "poly_add_primdef"
"poly" > "poly_primdef"
"normalize" > "normalize_def"
- "diff" > "diff_minus"
+ "diff" > "diff_def"
"degree" > "degree_def"
"##" > "##_def"
--- a/src/HOL/Import/HOL/prim_rec.imp Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Import/HOL/prim_rec.imp Fri Sep 09 06:47:14 2011 +0200
@@ -18,7 +18,7 @@
"PRIM_REC_FUN" > "HOL4Base.prim_rec.PRIM_REC_FUN"
"PRIM_REC" > "HOL4Base.prim_rec.PRIM_REC"
"PRE" > "HOL4Base.prim_rec.PRE"
- "<" > "Orderings.less" :: "nat => nat => bool"
+ "<" > "Orderings.ord_class.less" :: "nat => nat => bool"
thm_maps
"wellfounded_primdef" > "HOL4Base.prim_rec.wellfounded_primdef"
@@ -65,7 +65,7 @@
"LESS_DEF" > "HOL4Compat.LESS_DEF"
"LESS_0_0" > "HOL4Base.prim_rec.LESS_0_0"
"LESS_0" > "Nat.zero_less_Suc"
- "INV_SUC_EQ" > "Nat.nat.simps_3"
+ "INV_SUC_EQ" > "Nat.nat.inject"
"EQ_LESS" > "HOL4Base.prim_rec.EQ_LESS"
"DC" > "HOL4Base.prim_rec.DC"
--- a/src/HOL/Import/HOL/prob_extra.imp Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Import/HOL/prob_extra.imp Fri Sep 09 06:47:14 2011 +0200
@@ -23,9 +23,9 @@
"REAL_SUP_LE_X" > "HOL4Prob.prob_extra.REAL_SUP_LE_X"
"REAL_SUP_EXISTS_UNIQUE" > "HOL4Prob.prob_extra.REAL_SUP_EXISTS_UNIQUE"
"REAL_POW" > "RealDef.power_real_of_nat"
- "REAL_LE_INV_LE" > "Rings.le_imp_inverse_le"
- "REAL_LE_EQ" > "Set.basic_trans_rules_26"
- "REAL_INVINV_ALL" > "Rings.inverse_inverse_eq"
+ "REAL_LE_INV_LE" > "Fields.linordered_field_class.le_imp_inverse_le"
+ "REAL_LE_EQ" > "Orderings.order_antisym"
+ "REAL_INVINV_ALL" > "Fields.division_ring_inverse_zero_class.inverse_inverse_eq"
"REAL_INF_MIN" > "HOL4Prob.prob_extra.REAL_INF_MIN"
"RAND_THM" > "HOL.arg_cong"
"POW_HALF_TWICE" > "HOL4Prob.prob_extra.POW_HALF_TWICE"
@@ -59,7 +59,7 @@
"INTER_UNION_COMPL" > "HOL4Base.pred_set.INTER_UNION_COMPL"
"INTER_IS_EMPTY" > "HOL4Prob.prob_extra.INTER_IS_EMPTY"
"INF_DEF_ALT" > "HOL4Prob.prob_extra.INF_DEF_ALT"
- "HALF_POS" > "HOL4Prob.prob_extra.HALF_POS"
+ "HALF_POS" > "Series.half"
"HALF_LT_1" > "HOL4Prob.prob_extra.HALF_LT_1"
"HALF_CANCEL" > "HOL4Prob.prob_extra.HALF_CANCEL"
"GSPEC_DEF_ALT" > "HOL4Prob.prob_extra.GSPEC_DEF_ALT"
@@ -73,7 +73,7 @@
"EVEN_ODD_EXISTS_EQ" > "HOL4Prob.prob_extra.EVEN_ODD_EXISTS_EQ"
"EVEN_ODD_BASIC" > "HOL4Prob.prob_extra.EVEN_ODD_BASIC"
"EVEN_EXP_TWO" > "HOL4Prob.prob_extra.EVEN_EXP_TWO"
- "EQ_EXT_EQ" > "Fun.fun_eq_iff"
+ "EQ_EXT_EQ" > "HOL.fun_eq_iff"
"DIV_TWO_UNIQUE" > "HOL4Prob.prob_extra.DIV_TWO_UNIQUE"
"DIV_TWO_MONO_EVEN" > "HOL4Prob.prob_extra.DIV_TWO_MONO_EVEN"
"DIV_TWO_MONO" > "HOL4Prob.prob_extra.DIV_TWO_MONO"
--- a/src/HOL/Import/HOL/real.imp Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Import/HOL/real.imp Fri Sep 09 06:47:14 2011 +0200
@@ -10,14 +10,14 @@
const_maps
"sup" > "HOL4Real.real.sup"
"sum" > "HOL4Real.real.sum"
- "real_sub" > "Groups.minus" :: "real => real => real"
+ "real_sub" > "Groups.minus_class.minus" :: "real => real => real"
"real_of_num" > "RealDef.real" :: "nat => real"
- "real_lte" > "Orderings.less_eq" :: "real => real => bool"
+ "real_lte" > "Orderings.ord_class.less_eq" :: "real => real => bool"
"real_gt" > "HOL4Compat.real_gt"
"real_ge" > "HOL4Compat.real_ge"
"pow" > "Power.power_class.power" :: "real => nat => real"
- "abs" > "Groups.abs" :: "real => real"
- "/" > "Ring.divide" :: "real => real => real"
+ "abs" > "Groups.abs_class.abs" :: "real => real"
+ "/" > "Fields.inverse_class.divide" :: "real => real => real"
thm_maps
"sup_def" > "HOL4Real.real.sup_def"
@@ -25,13 +25,13 @@
"sumc" > "HOL4Real.real.sumc"
"sum_def" > "HOL4Real.real.sum_def"
"sum" > "HOL4Real.real.sum"
- "real_sub" > "Groups.diff_minus"
+ "real_sub" > "Fields.linordered_field_class.sign_simps_16"
"real_of_num" > "HOL4Compat.real_of_num"
"real_lte" > "HOL4Compat.real_lte"
- "real_lt" > "Orderings.linorder_not_le"
- "real_gt" > "HOL4Compat.real_gt"
+ "real_lt" > "Orderings.linorder_class.not_le"
+ "real_gt" > "HOL4Compat.GREATER_DEF"
"real_ge" > "HOL4Compat.real_ge"
- "real_div" > "Ring.divide_inverse"
+ "real_div" > "Fields.division_ring_class.divide_inverse"
"pow" > "HOL4Compat.pow"
"abs" > "HOL4Compat.abs"
"SUP_LEMMA3" > "HOL4Real.real.SUP_LEMMA3"
@@ -70,40 +70,40 @@
"REAL_SUP_EXISTS" > "HOL4Real.real.REAL_SUP_EXISTS"
"REAL_SUP_ALLPOS" > "HOL4Compat.REAL_SUP_ALLPOS"
"REAL_SUP" > "HOL4Real.real.REAL_SUP"
- "REAL_SUMSQ" > "HOL4Real.real.REAL_SUMSQ"
+ "REAL_SUMSQ" > "Nat_Numeral.linordered_ring_strict_class.sum_squares_eq_zero_iff"
"REAL_SUB_TRIANGLE" > "HOL4Real.real.REAL_SUB_TRIANGLE"
"REAL_SUB_SUB2" > "HOL4Real.real.REAL_SUB_SUB2"
"REAL_SUB_SUB" > "HOL4Real.real.REAL_SUB_SUB"
- "REAL_SUB_RZERO" > "Groups.diff_0_right"
- "REAL_SUB_RNEG" > "Groups.diff_minus_eq_add"
- "REAL_SUB_REFL" > "Groups.diff_self"
- "REAL_SUB_RDISTRIB" > "Rings.ring_eq_simps_3"
+ "REAL_SUB_RZERO" > "Groups.group_add_class.diff_0_right"
+ "REAL_SUB_RNEG" > "Groups.group_add_class.diff_minus_eq_add"
+ "REAL_SUB_REFL" > "Groups.group_add_class.diff_self"
+ "REAL_SUB_RDISTRIB" > "Fields.linordered_field_class.sign_simps_5"
"REAL_SUB_NEG2" > "HOL4Real.real.REAL_SUB_NEG2"
- "REAL_SUB_LZERO" > "Groups.diff_0"
+ "REAL_SUB_LZERO" > "Groups.group_add_class.diff_0"
"REAL_SUB_LT" > "HOL4Real.real.REAL_SUB_LT"
"REAL_SUB_LNEG" > "HOL4Real.real.REAL_SUB_LNEG"
"REAL_SUB_LE" > "HOL4Real.real.REAL_SUB_LE"
- "REAL_SUB_LDISTRIB" > "Rings.ring_eq_simps_4"
+ "REAL_SUB_LDISTRIB" > "Fields.linordered_field_class.sign_simps_6"
"REAL_SUB_INV2" > "HOL4Real.real.REAL_SUB_INV2"
"REAL_SUB_ADD2" > "HOL4Real.real.REAL_SUB_ADD2"
- "REAL_SUB_ADD" > "Groups.diff_add_cancel"
- "REAL_SUB_ABS" > "Groups.abs_triangle_ineq2"
- "REAL_SUB_0" > "Groups.diff_eq_0_iff_eq"
- "REAL_RNEG_UNIQ" > "RealDef.real_add_eq_0_iff"
- "REAL_RINV_UNIQ" > "Rings.inverse_unique"
- "REAL_RDISTRIB" > "Rings.ring_eq_simps_1"
- "REAL_POW_POW" > "Power.power_mult"
+ "REAL_SUB_ADD" > "Groups.group_add_class.diff_add_cancel"
+ "REAL_SUB_ABS" > "Groups.ordered_ab_group_add_abs_class.abs_triangle_ineq2"
+ "REAL_SUB_0" > "Groups.ab_group_add_class.diff_eq_0_iff_eq"
+ "REAL_RNEG_UNIQ" > "Groups.group_add_class.add_eq_0_iff"
+ "REAL_RINV_UNIQ" > "Fields.division_ring_class.inverse_unique"
+ "REAL_RDISTRIB" > "Fields.linordered_field_class.sign_simps_8"
+ "REAL_POW_POW" > "Power.monoid_mult_class.power_mult"
"REAL_POW_MONO_LT" > "HOL4Real.real.REAL_POW_MONO_LT"
"REAL_POW_LT2" > "HOL4Real.real.REAL_POW_LT2"
- "REAL_POW_LT" > "Power.zero_less_power"
+ "REAL_POW_LT" > "Power.linordered_semidom_class.zero_less_power"
"REAL_POW_INV" > "Power.power_inverse"
"REAL_POW_DIV" > "Power.power_divide"
- "REAL_POW_ADD" > "Power.power_add"
- "REAL_POW2_ABS" > "Nat_Numeral.power2_abs"
+ "REAL_POW_ADD" > "Power.monoid_mult_class.power_add"
+ "REAL_POW2_ABS" > "Nat_Numeral.linordered_idom_class.power2_abs"
"REAL_POS_NZ" > "HOL4Real.real.REAL_POS_NZ"
"REAL_POS" > "RealDef.real_of_nat_ge_zero"
"REAL_POASQ" > "HOL4Real.real.REAL_POASQ"
- "REAL_OVER1" > "Rings.divide_1"
+ "REAL_OVER1" > "Fields.division_ring_class.divide_1"
"REAL_OF_NUM_SUC" > "RealDef.real_of_nat_Suc"
"REAL_OF_NUM_POW" > "RealDef.power_real_of_nat"
"REAL_OF_NUM_MUL" > "RealDef.real_of_nat_mult"
@@ -112,239 +112,239 @@
"REAL_OF_NUM_ADD" > "RealDef.real_of_nat_add"
"REAL_NZ_IMP_LT" > "HOL4Real.real.REAL_NZ_IMP_LT"
"REAL_NOT_LT" > "HOL4Compat.real_lte"
- "REAL_NOT_LE" > "Orderings.linorder_not_le"
- "REAL_NEG_SUB" > "Groups.minus_diff_eq"
- "REAL_NEG_RMUL" > "Rings.mult_minus_right"
- "REAL_NEG_NEG" > "Groups.minus_minus"
- "REAL_NEG_MUL2" > "Rings.minus_mult_minus"
- "REAL_NEG_MINUS1" > "HOL4Real.real.REAL_NEG_MINUS1"
- "REAL_NEG_LT0" > "Groups.neg_less_0_iff_less"
- "REAL_NEG_LMUL" > "Rings.mult_minus_left"
- "REAL_NEG_LE0" > "Groups.neg_le_0_iff_le"
- "REAL_NEG_INV" > "Rings.nonzero_inverse_minus_eq"
- "REAL_NEG_GT0" > "Groups.neg_0_less_iff_less"
- "REAL_NEG_GE0" > "Groups.neg_0_le_iff_le"
- "REAL_NEG_EQ0" > "Groups.neg_equal_0_iff_equal"
+ "REAL_NOT_LE" > "Orderings.linorder_class.not_le"
+ "REAL_NEG_SUB" > "Groups.ab_group_add_class.minus_diff_eq"
+ "REAL_NEG_RMUL" > "Int.int_arith_rules_14"
+ "REAL_NEG_NEG" > "Groups.group_add_class.minus_minus"
+ "REAL_NEG_MUL2" > "Rings.ring_class.minus_mult_minus"
+ "REAL_NEG_MINUS1" > "Semiring_Normalization.comm_ring_1_class.normalizing_ring_rules_1"
+ "REAL_NEG_LT0" > "Groups.ordered_ab_group_add_class.neg_less_0_iff_less"
+ "REAL_NEG_LMUL" > "Int.int_arith_rules_13"
+ "REAL_NEG_LE0" > "Groups.ordered_ab_group_add_class.neg_le_0_iff_le"
+ "REAL_NEG_INV" > "Fields.division_ring_class.nonzero_inverse_minus_eq"
+ "REAL_NEG_GT0" > "Groups.ordered_ab_group_add_class.neg_0_less_iff_less"
+ "REAL_NEG_GE0" > "Groups.ordered_ab_group_add_class.neg_0_le_iff_le"
+ "REAL_NEG_EQ0" > "Groups.group_add_class.neg_equal_0_iff_equal"
"REAL_NEG_EQ" > "HOL4Real.real.REAL_NEG_EQ"
- "REAL_NEG_ADD" > "Groups.minus_add_distrib"
- "REAL_NEG_0" > "Groups.minus_zero"
- "REAL_NEGNEG" > "Groups.minus_minus"
- "REAL_MUL_SYM" > "Int.zmult_ac_2"
- "REAL_MUL_RZERO" > "Rings.mult_zero_right"
- "REAL_MUL_RNEG" > "Rings.mult_minus_right"
- "REAL_MUL_RINV" > "Rings.right_inverse"
- "REAL_MUL_RID" > "Finite_Set.AC_mult.f_e.ident"
- "REAL_MUL_LZERO" > "Rings.mult_zero_left"
- "REAL_MUL_LNEG" > "Rings.mult_minus_left"
- "REAL_MUL_LINV" > "HOL4Compat.REAL_MUL_LINV"
- "REAL_MUL_LID" > "Finite_Set.AC_mult.f_e.left_ident"
- "REAL_MUL_ASSOC" > "HOL4Compat.REAL_MUL_ASSOC"
+ "REAL_NEG_ADD" > "Groups.ab_group_add_class.minus_add_distrib"
+ "REAL_NEG_0" > "Groups.group_add_class.minus_zero"
+ "REAL_NEGNEG" > "Groups.group_add_class.minus_minus"
+ "REAL_MUL_SYM" > "Fields.linordered_field_class.sign_simps_21"
+ "REAL_MUL_RZERO" > "Divides.arithmetic_simps_41"
+ "REAL_MUL_RNEG" > "Int.int_arith_rules_14"
+ "REAL_MUL_RINV" > "Fields.division_ring_class.right_inverse"
+ "REAL_MUL_RID" > "Divides.arithmetic_simps_43"
+ "REAL_MUL_LZERO" > "Divides.arithmetic_simps_40"
+ "REAL_MUL_LNEG" > "Int.int_arith_rules_13"
+ "REAL_MUL_LINV" > "Fields.division_ring_class.left_inverse"
+ "REAL_MUL_LID" > "Divides.arithmetic_simps_42"
+ "REAL_MUL_ASSOC" > "Fields.linordered_field_class.sign_simps_22"
"REAL_MUL" > "RealDef.real_of_nat_mult"
"REAL_MIDDLE2" > "HOL4Real.real.REAL_MIDDLE2"
"REAL_MIDDLE1" > "HOL4Real.real.REAL_MIDDLE1"
- "REAL_MEAN" > "Rings.dense"
- "REAL_LT_TRANS" > "Set.basic_trans_rules_21"
+ "REAL_MEAN" > "Orderings.dense_linorder_class.dense"
+ "REAL_LT_TRANS" > "Orderings.order_less_trans"
"REAL_LT_TOTAL" > "HOL4Compat.REAL_LT_TOTAL"
- "REAL_LT_SUB_RADD" > "Groups.compare_rls_6"
- "REAL_LT_SUB_LADD" > "Groups.compare_rls_7"
- "REAL_LT_RMUL_IMP" > "Rings.mult_strict_right_mono"
+ "REAL_LT_SUB_RADD" > "Fields.linordered_field_class.sign_simps_4"
+ "REAL_LT_SUB_LADD" > "Fields.linordered_field_class.sign_simps_3"
+ "REAL_LT_RMUL_IMP" > "Rings.linordered_semiring_strict_class.mult_strict_right_mono"
"REAL_LT_RMUL_0" > "HOL4Real.real.REAL_LT_RMUL_0"
"REAL_LT_RMUL" > "RealDef.real_mult_less_iff1"
"REAL_LT_REFL" > "Orderings.order_less_irrefl"
- "REAL_LT_RDIV_EQ" > "Rings.pos_less_divide_eq"
+ "REAL_LT_RDIV_EQ" > "Fields.linordered_field_class.pos_less_divide_eq"
"REAL_LT_RDIV_0" > "HOL4Real.real.REAL_LT_RDIV_0"
"REAL_LT_RDIV" > "HOL4Real.real.REAL_LT_RDIV"
- "REAL_LT_RADD" > "Groups.add_less_cancel_right"
+ "REAL_LT_RADD" > "Groups.ordered_ab_semigroup_add_imp_le_class.add_less_cancel_right"
"REAL_LT_NZ" > "HOL4Real.real.REAL_LT_NZ"
"REAL_LT_NEGTOTAL" > "HOL4Real.real.REAL_LT_NEGTOTAL"
- "REAL_LT_NEG" > "Groups.neg_less_iff_less"
+ "REAL_LT_NEG" > "Groups.ordered_ab_group_add_class.neg_less_iff_less"
"REAL_LT_MULTIPLE" > "HOL4Real.real.REAL_LT_MULTIPLE"
- "REAL_LT_MUL2" > "Rings.mult_strict_mono'"
- "REAL_LT_MUL" > "Rings.mult_pos_pos"
- "REAL_LT_LMUL_IMP" > "Rings.linordered_comm_semiring_strict_class.mult_strict_mono"
+ "REAL_LT_MUL2" > "Rings.linordered_semiring_strict_class.mult_strict_mono'"
+ "REAL_LT_MUL" > "RealDef.real_mult_order"
+ "REAL_LT_LMUL_IMP" > "RealDef.real_mult_less_mono2"
"REAL_LT_LMUL_0" > "HOL4Real.real.REAL_LT_LMUL_0"
- "REAL_LT_LMUL" > "HOL4Real.real.REAL_LT_LMUL"
- "REAL_LT_LE" > "Orderings.order_class.order_less_le"
- "REAL_LT_LDIV_EQ" > "Rings.pos_divide_less_eq"
- "REAL_LT_LADD" > "Groups.add_less_cancel_left"
- "REAL_LT_INV_EQ" > "Rings.inverse_positive_iff_positive"
- "REAL_LT_INV" > "Rings.less_imp_inverse_less"
- "REAL_LT_IMP_NE" > "Orderings.less_imp_neq"
+ "REAL_LT_LMUL" > "Rings.linordered_ring_strict_class.mult_less_cancel_left_pos"
+ "REAL_LT_LE" > "Orderings.order_class.less_le"
+ "REAL_LT_LDIV_EQ" > "Fields.linordered_field_class.pos_divide_less_eq"
+ "REAL_LT_LADD" > "Groups.ordered_ab_semigroup_add_imp_le_class.add_less_cancel_left"
+ "REAL_LT_INV_EQ" > "Fields.linordered_field_inverse_zero_class.inverse_positive_iff_positive"
+ "REAL_LT_INV" > "Fields.linordered_field_class.less_imp_inverse_less"
+ "REAL_LT_IMP_NE" > "Orderings.order_class.less_imp_neq"
"REAL_LT_IMP_LE" > "Orderings.order_less_imp_le"
- "REAL_LT_IADD" > "Groups.add_strict_left_mono"
+ "REAL_LT_IADD" > "Groups.ordered_cancel_ab_semigroup_add_class.add_strict_left_mono"
"REAL_LT_HALF2" > "HOL4Real.real.REAL_LT_HALF2"
- "REAL_LT_HALF1" > "NatSimprocs.half_gt_zero_iff"
+ "REAL_LT_HALF1" > "Int.half_gt_zero_iff"
"REAL_LT_GT" > "Orderings.order_less_not_sym"
"REAL_LT_FRACTION_0" > "HOL4Real.real.REAL_LT_FRACTION_0"
"REAL_LT_FRACTION" > "HOL4Real.real.REAL_LT_FRACTION"
- "REAL_LT_DIV" > "Rings.divide_pos_pos"
+ "REAL_LT_DIV" > "Fields.linordered_field_class.divide_pos_pos"
"REAL_LT_ANTISYM" > "HOL4Real.real.REAL_LT_ANTISYM"
- "REAL_LT_ADD_SUB" > "Groups.compare_rls_7"
+ "REAL_LT_ADD_SUB" > "Fields.linordered_field_class.sign_simps_3"
"REAL_LT_ADDR" > "HOL4Real.real.REAL_LT_ADDR"
"REAL_LT_ADDNEG2" > "HOL4Real.real.REAL_LT_ADDNEG2"
"REAL_LT_ADDNEG" > "HOL4Real.real.REAL_LT_ADDNEG"
"REAL_LT_ADDL" > "HOL4Real.real.REAL_LT_ADDL"
- "REAL_LT_ADD2" > "Groups.add_strict_mono"
+ "REAL_LT_ADD2" > "Groups.add_mono_thms_linordered_field_5"
"REAL_LT_ADD1" > "HOL4Real.real.REAL_LT_ADD1"
- "REAL_LT_ADD" > "Groups.add_pos_pos"
+ "REAL_LT_ADD" > "Groups.ordered_comm_monoid_add_class.add_pos_pos"
"REAL_LT_1" > "HOL4Real.real.REAL_LT_1"
- "REAL_LT_01" > "Rings.zero_less_one"
- "REAL_LTE_TRANS" > "Set.basic_trans_rules_24"
+ "REAL_LT_01" > "Rings.linordered_semidom_class.zero_less_one"
+ "REAL_LTE_TRANS" > "Orderings.order_less_le_trans"
"REAL_LTE_TOTAL" > "HOL4Real.real.REAL_LTE_TOTAL"
"REAL_LTE_ANTSYM" > "HOL4Real.real.REAL_LTE_ANTSYM"
- "REAL_LTE_ADD2" > "Groups.add_less_le_mono"
- "REAL_LTE_ADD" > "Groups.add_pos_nonneg"
+ "REAL_LTE_ADD2" > "Groups.add_mono_thms_linordered_field_3"
+ "REAL_LTE_ADD" > "Groups.ordered_comm_monoid_add_class.add_pos_nonneg"
"REAL_LT1_POW2" > "HOL4Real.real.REAL_LT1_POW2"
"REAL_LT" > "RealDef.real_of_nat_less_iff"
- "REAL_LNEG_UNIQ" > "HOL4Real.real.REAL_LNEG_UNIQ"
+ "REAL_LNEG_UNIQ" > "Groups.group_add_class.eq_neg_iff_add_eq_0"
"REAL_LINV_UNIQ" > "HOL4Real.real.REAL_LINV_UNIQ"
- "REAL_LE_TRANS" > "Set.basic_trans_rules_25"
- "REAL_LE_TOTAL" > "Orderings.linorder_class.linorder_linear"
- "REAL_LE_SUB_RADD" > "Groups.compare_rls_8"
- "REAL_LE_SUB_LADD" > "Groups.compare_rls_9"
- "REAL_LE_SQUARE" > "Rings.zero_le_square"
- "REAL_LE_RNEG" > "Groups.le_eq_neg"
- "REAL_LE_RMUL_IMP" > "Rings.mult_right_mono"
+ "REAL_LE_TRANS" > "Orderings.order_trans_rules_23"
+ "REAL_LE_TOTAL" > "Orderings.linorder_class.linear"
+ "REAL_LE_SUB_RADD" > "Fields.linordered_field_class.sign_simps_2"
+ "REAL_LE_SUB_LADD" > "Fields.linordered_field_class.sign_simps_1"
+ "REAL_LE_SQUARE" > "Rings.linordered_ring_class.zero_le_square"
+ "REAL_LE_RNEG" > "HOL4Real.real.REAL_LE_RNEG"
+ "REAL_LE_RMUL_IMP" > "Rings.ordered_semiring_class.mult_right_mono"
"REAL_LE_RMUL" > "RealDef.real_mult_le_cancel_iff1"
- "REAL_LE_REFL" > "Finite_Set.max.f_below.below_refl"
- "REAL_LE_RDIV_EQ" > "Rings.pos_le_divide_eq"
- "REAL_LE_RDIV" > "Rings.mult_imp_le_div_pos"
- "REAL_LE_RADD" > "Groups.add_le_cancel_right"
- "REAL_LE_POW2" > "Nat_Numeral.zero_compare_simps_12"
+ "REAL_LE_REFL" > "Orderings.preorder_class.order_refl"
+ "REAL_LE_RDIV_EQ" > "Fields.linordered_field_class.pos_le_divide_eq"
+ "REAL_LE_RDIV" > "Fields.linordered_field_class.mult_imp_le_div_pos"
+ "REAL_LE_RADD" > "Groups.ordered_ab_semigroup_add_imp_le_class.add_le_cancel_right"
+ "REAL_LE_POW2" > "Nat_Numeral.linordered_idom_class.zero_le_power2"
"REAL_LE_NEGTOTAL" > "HOL4Real.real.REAL_LE_NEGTOTAL"
- "REAL_LE_NEGR" > "Groups.le_minus_self_iff"
- "REAL_LE_NEGL" > "Groups.minus_le_self_iff"
- "REAL_LE_NEG2" > "Groups.neg_le_iff_le"
- "REAL_LE_NEG" > "Groups.neg_le_iff_le"
- "REAL_LE_MUL2" > "HOL4Real.real.REAL_LE_MUL2"
- "REAL_LE_MUL" > "Rings.mult_nonneg_nonneg"
- "REAL_LE_LT" > "Orderings.order_le_less"
+ "REAL_LE_NEGR" > "Groups.linordered_ab_group_add_class.le_minus_self_iff"
+ "REAL_LE_NEGL" > "Groups.linordered_ab_group_add_class.minus_le_self_iff"
+ "REAL_LE_NEG2" > "Groups.ordered_ab_group_add_class.neg_le_iff_le"
+ "REAL_LE_NEG" > "Groups.ordered_ab_group_add_class.neg_le_iff_le"
+ "REAL_LE_MUL2" > "Rings.ordered_semiring_class.mult_mono'"
+ "REAL_LE_MUL" > "Rings.mult_sign_intros_1"
+ "REAL_LE_LT" > "Orderings.order_class.le_less"
"REAL_LE_LNEG" > "RealDef.real_0_le_add_iff"
- "REAL_LE_LMUL_IMP" > "Rings.mult_mono"
+ "REAL_LE_LMUL_IMP" > "Rings.ordered_comm_semiring_class.comm_mult_left_mono"
"REAL_LE_LMUL" > "RealDef.real_mult_le_cancel_iff2"
- "REAL_LE_LDIV_EQ" > "Rings.pos_divide_le_eq"
- "REAL_LE_LDIV" > "Rings.mult_imp_div_pos_le"
- "REAL_LE_LADD_IMP" > "Groups.add_left_mono"
- "REAL_LE_LADD" > "Groups.add_le_cancel_left"
- "REAL_LE_INV_EQ" > "Rings.inverse_nonnegative_iff_nonnegative"
+ "REAL_LE_LDIV_EQ" > "Fields.linordered_field_class.pos_divide_le_eq"
+ "REAL_LE_LDIV" > "Fields.linordered_field_class.mult_imp_div_pos_le"
+ "REAL_LE_LADD_IMP" > "Groups.ordered_ab_semigroup_add_class.add_left_mono"
+ "REAL_LE_LADD" > "Groups.ordered_ab_semigroup_add_imp_le_class.add_le_cancel_left"
+ "REAL_LE_INV_EQ" > "Fields.linordered_field_inverse_zero_class.inverse_nonnegative_iff_nonnegative"
"REAL_LE_INV" > "HOL4Real.real.REAL_LE_INV"
- "REAL_LE_DOUBLE" > "Groups.zero_le_double_add_iff_zero_le_single_add"
+ "REAL_LE_DOUBLE" > "Groups.linordered_ab_group_add_class.zero_le_double_add_iff_zero_le_single_add"
"REAL_LE_DIV" > "HOL4Real.real.REAL_LE_DIV"
- "REAL_LE_ANTISYM" > "Orderings.order_eq_iff"
+ "REAL_LE_ANTISYM" > "Orderings.order_class.eq_iff"
"REAL_LE_ADDR" > "HOL4Real.real.REAL_LE_ADDR"
"REAL_LE_ADDL" > "HOL4Real.real.REAL_LE_ADDL"
- "REAL_LE_ADD2" > "Groups.add_mono"
- "REAL_LE_ADD" > "Groups.add_nonneg_nonneg"
- "REAL_LE_01" > "Rings.zero_le_one"
- "REAL_LET_TRANS" > "Set.basic_trans_rules_23"
- "REAL_LET_TOTAL" > "Orderings.linorder_le_less_linear"
+ "REAL_LE_ADD2" > "Groups.add_mono_thms_linordered_semiring_1"
+ "REAL_LE_ADD" > "Groups.ordered_comm_monoid_add_class.add_nonneg_nonneg"
+ "REAL_LE_01" > "Rings.linordered_semidom_class.zero_le_one"
+ "REAL_LET_TRANS" > "Orderings.order_le_less_trans"
+ "REAL_LET_TOTAL" > "Orderings.linorder_class.le_less_linear"
"REAL_LET_ANTISYM" > "HOL4Real.real.REAL_LET_ANTISYM"
- "REAL_LET_ADD2" > "Groups.add_le_less_mono"
- "REAL_LET_ADD" > "Groups.add_nonneg_pos"
+ "REAL_LET_ADD2" > "Groups.add_mono_thms_linordered_field_4"
+ "REAL_LET_ADD" > "Groups.ordered_comm_monoid_add_class.add_nonneg_pos"
"REAL_LE1_POW2" > "HOL4Real.real.REAL_LE1_POW2"
"REAL_LE" > "RealDef.real_of_nat_le_iff"
- "REAL_LDISTRIB" > "Rings.ring_eq_simps_2"
- "REAL_INV_POS" > "Rings.positive_imp_inverse_positive"
- "REAL_INV_NZ" > "Rings.nonzero_imp_inverse_nonzero"
+ "REAL_LDISTRIB" > "Fields.linordered_field_class.sign_simps_7"
+ "REAL_INV_POS" > "Fields.linordered_field_class.positive_imp_inverse_positive"
+ "REAL_INV_NZ" > "Fields.division_ring_class.nonzero_imp_inverse_nonzero"
"REAL_INV_MUL" > "HOL4Real.real.REAL_INV_MUL"
- "REAL_INV_LT1" > "Fields.one_less_inverse"
- "REAL_INV_INV" > "Rings.inverse_inverse_eq"
- "REAL_INV_EQ_0" > "Rings.inverse_nonzero_iff_nonzero"
- "REAL_INV_1OVER" > "Rings.inverse_eq_divide"
- "REAL_INV_0" > "Rings.division_ring_inverse_zero_class.inverse_zero"
- "REAL_INVINV" > "Rings.nonzero_inverse_inverse_eq"
- "REAL_INV1" > "Rings.inverse_1"
+ "REAL_INV_LT1" > "Fields.linordered_field_class.one_less_inverse"
+ "REAL_INV_INV" > "Fields.division_ring_inverse_zero_class.inverse_inverse_eq"
+ "REAL_INV_EQ_0" > "Fields.division_ring_inverse_zero_class.inverse_nonzero_iff_nonzero"
+ "REAL_INV_1OVER" > "Fields.division_ring_class.inverse_eq_divide"
+ "REAL_INV_0" > "Fields.division_ring_inverse_zero_class.inverse_zero"
+ "REAL_INVINV" > "Fields.division_ring_class.nonzero_inverse_inverse_eq"
+ "REAL_INV1" > "Fields.division_ring_class.inverse_1"
"REAL_INJ" > "RealDef.real_of_nat_inject"
"REAL_HALF_DOUBLE" > "RComplete.real_sum_of_halves"
"REAL_FACT_NZ" > "HOL4Real.real.REAL_FACT_NZ"
- "REAL_EQ_SUB_RADD" > "Rings.ring_eq_simps_15"
- "REAL_EQ_SUB_LADD" > "Rings.ring_eq_simps_16"
- "REAL_EQ_RMUL_IMP" > "Rings.field_mult_cancel_right_lemma"
- "REAL_EQ_RMUL" > "Rings.field_mult_cancel_right"
+ "REAL_EQ_SUB_RADD" > "Fields.linordered_field_class.sign_simps_12"
+ "REAL_EQ_SUB_LADD" > "Fields.linordered_field_class.sign_simps_11"
+ "REAL_EQ_RMUL_IMP" > "HOL4Real.real.REAL_EQ_RMUL_IMP"
+ "REAL_EQ_RMUL" > "Rings.mult_compare_simps_13"
"REAL_EQ_RDIV_EQ" > "HOL4Real.real.REAL_EQ_RDIV_EQ"
- "REAL_EQ_RADD" > "Groups.add_right_cancel"
- "REAL_EQ_NEG" > "Groups.neg_equal_iff_equal"
- "REAL_EQ_MUL_LCANCEL" > "Rings.field_mult_cancel_left"
+ "REAL_EQ_RADD" > "Groups.cancel_semigroup_add_class.add_right_cancel"
+ "REAL_EQ_NEG" > "Groups.group_add_class.neg_equal_iff_equal"
+ "REAL_EQ_MUL_LCANCEL" > "Rings.mult_compare_simps_14"
"REAL_EQ_LMUL_IMP" > "HOL4Real.real.REAL_EQ_LMUL_IMP"
"REAL_EQ_LMUL2" > "RealDef.real_mult_left_cancel"
- "REAL_EQ_LMUL" > "Rings.field_mult_cancel_left"
+ "REAL_EQ_LMUL" > "Rings.mult_compare_simps_14"
"REAL_EQ_LDIV_EQ" > "HOL4Real.real.REAL_EQ_LDIV_EQ"
- "REAL_EQ_LADD" > "Groups.add_left_cancel"
+ "REAL_EQ_LADD" > "Groups.cancel_semigroup_add_class.add_left_cancel"
"REAL_EQ_IMP_LE" > "Orderings.order_eq_refl"
- "REAL_ENTIRE" > "Rings.field_mult_eq_0_iff"
+ "REAL_ENTIRE" > "Rings.ring_no_zero_divisors_class.mult_eq_0_iff"
"REAL_DOWN2" > "RealDef.real_lbound_gt_zero"
"REAL_DOWN" > "HOL4Real.real.REAL_DOWN"
"REAL_DOUBLE" > "Int.mult_2"
"REAL_DIV_RMUL" > "HOL4Real.real.REAL_DIV_RMUL"
- "REAL_DIV_REFL" > "Rings.divide_self"
+ "REAL_DIV_REFL" > "Fields.division_ring_class.divide_self"
"REAL_DIV_MUL2" > "HOL4Real.real.REAL_DIV_MUL2"
- "REAL_DIV_LZERO" > "Rings.divide_zero_left"
+ "REAL_DIV_LZERO" > "Fields.division_ring_class.divide_zero_left"
"REAL_DIV_LMUL" > "HOL4Real.real.REAL_DIV_LMUL"
- "REAL_DIFFSQ" > "HOL4Real.real.REAL_DIFFSQ"
- "REAL_ARCH_LEAST" > "HOL4Real.real.REAL_ARCH_LEAST"
+ "REAL_DIFFSQ" > "Rings.comm_ring_class.square_diff_square_factored"
+ "REAL_ARCH_LEAST" > "Transcendental.reals_Archimedean4"
"REAL_ARCH" > "RComplete.reals_Archimedean3"
- "REAL_ADD_SYM" > "Finite_Set.AC_add.f.AC_2"
+ "REAL_ADD_SYM" > "Fields.linordered_field_class.sign_simps_18"
"REAL_ADD_SUB2" > "HOL4Real.real.REAL_ADD_SUB2"
"REAL_ADD_SUB" > "HOL4Real.real.REAL_ADD_SUB"
- "REAL_ADD_RINV" > "Groups.right_minus"
+ "REAL_ADD_RINV" > "Groups.group_add_class.right_minus"
"REAL_ADD_RID_UNIQ" > "HOL4Real.real.REAL_ADD_RID_UNIQ"
- "REAL_ADD_RID" > "Finite_Set.AC_add.f_e.ident"
- "REAL_ADD_RDISTRIB" > "Rings.ring_eq_simps_1"
- "REAL_ADD_LINV" > "HOL4Compat.REAL_ADD_LINV"
+ "REAL_ADD_RID" > "Divides.arithmetic_simps_39"
+ "REAL_ADD_RDISTRIB" > "Fields.linordered_field_class.sign_simps_8"
+ "REAL_ADD_LINV" > "Groups.ab_group_add_class.ab_left_minus"
"REAL_ADD_LID_UNIQ" > "HOL4Real.real.REAL_ADD_LID_UNIQ"
- "REAL_ADD_LID" > "Finite_Set.AC_add.f_e.left_ident"
- "REAL_ADD_LDISTRIB" > "Rings.ring_eq_simps_2"
- "REAL_ADD_ASSOC" > "HOL4Compat.REAL_ADD_ASSOC"
- "REAL_ADD2_SUB2" > "HOL4Real.real.REAL_ADD2_SUB2"
+ "REAL_ADD_LID" > "Divides.arithmetic_simps_38"
+ "REAL_ADD_LDISTRIB" > "Fields.linordered_field_class.sign_simps_7"
+ "REAL_ADD_ASSOC" > "Fields.linordered_field_class.sign_simps_19"
+ "REAL_ADD2_SUB2" > "RealDef.add_diff_add"
"REAL_ADD" > "RealDef.real_of_nat_add"
- "REAL_ABS_TRIANGLE" > "Groups.abs_triangle_ineq"
- "REAL_ABS_POS" > "Groups.abs_ge_zero"
- "REAL_ABS_MUL" > "Rings.abs_mult"
- "REAL_ABS_0" > "Int.bin_arith_simps_28"
+ "REAL_ABS_TRIANGLE" > "Groups.ordered_ab_group_add_abs_class.abs_triangle_ineq"
+ "REAL_ABS_POS" > "Groups.ordered_ab_group_add_abs_class.abs_ge_zero"
+ "REAL_ABS_MUL" > "Rings.linordered_idom_class.abs_mult"
+ "REAL_ABS_0" > "Divides.arithmetic_simps_27"
"REAL_10" > "HOL4Compat.REAL_10"
"REAL_1" > "HOL4Real.real.REAL_1"
"REAL_0" > "HOL4Real.real.REAL_0"
"REAL" > "RealDef.real_of_nat_Suc"
"POW_ZERO_EQ" > "HOL4Real.real.POW_ZERO_EQ"
- "POW_ZERO" > "RealPow.realpow_zero_zero"
+ "POW_ZERO" > "HOL4Real.real.POW_ZERO"
"POW_POS_LT" > "HOL4Real.real.POW_POS_LT"
- "POW_POS" > "Power.zero_le_power"
+ "POW_POS" > "Power.linordered_semidom_class.zero_le_power"
"POW_PLUS1" > "HOL4Real.real.POW_PLUS1"
- "POW_ONE" > "Power.power_one"
- "POW_NZ" > "Power.field_power_not_zero"
- "POW_MUL" > "Power.power_mult_distrib"
- "POW_MINUS1" > "Nat_Numeral.power_minus1_even"
+ "POW_ONE" > "Power.monoid_mult_class.power_one"
+ "POW_NZ" > "Power.ring_1_no_zero_divisors_class.field_power_not_zero"
+ "POW_MUL" > "Power.comm_monoid_mult_class.power_mult_distrib"
+ "POW_MINUS1" > "Nat_Numeral.ring_1_class.power_minus1_even"
"POW_M1" > "HOL4Real.real.POW_M1"
"POW_LT" > "HOL4Real.real.POW_LT"
- "POW_LE" > "Power.power_mono"
- "POW_INV" > "Power.nonzero_power_inverse"
- "POW_EQ" > "Power.power_inject_base"
- "POW_ADD" > "Power.power_add"
- "POW_ABS" > "Power.power_abs"
- "POW_2_LT" > "RealPow.two_realpow_gt"
- "POW_2_LE1" > "RealPow.two_realpow_ge_one"
- "POW_2" > "Nat_Numeral.power2_eq_square"
- "POW_1" > "Power.power_one_right"
+ "POW_LE" > "Power.linordered_semidom_class.power_mono"
+ "POW_INV" > "Power.division_ring_class.nonzero_power_inverse"
+ "POW_EQ" > "Power.linordered_semidom_class.power_inject_base"
+ "POW_ADD" > "Power.monoid_mult_class.power_add"
+ "POW_ABS" > "Power.linordered_idom_class.power_abs"
+ "POW_2_LT" > "RealDef.two_realpow_gt"
+ "POW_2_LE1" > "RealDef.two_realpow_ge_one"
+ "POW_2" > "Nat_Numeral.monoid_mult_class.power2_eq_square"
+ "POW_1" > "Power.monoid_mult_class.power_one_right"
"POW_0" > "Power.power_0_Suc"
- "ABS_ZERO" > "Groups.abs_eq_0"
- "ABS_TRIANGLE" > "Groups.abs_triangle_ineq"
+ "ABS_ZERO" > "Groups.ordered_ab_group_add_abs_class.abs_eq_0"
+ "ABS_TRIANGLE" > "Groups.ordered_ab_group_add_abs_class.abs_triangle_ineq"
"ABS_SUM" > "HOL4Real.real.ABS_SUM"
- "ABS_SUB_ABS" > "Groups.abs_triangle_ineq3"
- "ABS_SUB" > "Groups.abs_minus_commute"
+ "ABS_SUB_ABS" > "Groups.ordered_ab_group_add_abs_class.abs_triangle_ineq3"
+ "ABS_SUB" > "Groups.ordered_ab_group_add_abs_class.abs_minus_commute"
"ABS_STILLNZ" > "HOL4Real.real.ABS_STILLNZ"
"ABS_SIGN2" > "HOL4Real.real.ABS_SIGN2"
"ABS_SIGN" > "HOL4Real.real.ABS_SIGN"
"ABS_REFL" > "HOL4Real.real.ABS_REFL"
- "ABS_POW2" > "Nat_Numeral.abs_power2"
- "ABS_POS" > "Groups.abs_ge_zero"
- "ABS_NZ" > "Groups.zero_less_abs_iff"
- "ABS_NEG" > "Groups.abs_minus_cancel"
+ "ABS_POW2" > "Nat_Numeral.linordered_idom_class.abs_power2"
+ "ABS_POS" > "Groups.ordered_ab_group_add_abs_class.abs_ge_zero"
+ "ABS_NZ" > "Groups.ordered_ab_group_add_abs_class.zero_less_abs_iff"
+ "ABS_NEG" > "Groups.ordered_ab_group_add_abs_class.abs_minus_cancel"
"ABS_N" > "RealDef.abs_real_of_nat_cancel"
- "ABS_MUL" > "Rings.abs_mult"
+ "ABS_MUL" > "Rings.linordered_idom_class.abs_mult"
"ABS_LT_MUL2" > "HOL4Real.real.ABS_LT_MUL2"
- "ABS_LE" > "Groups.abs_ge_self"
- "ABS_INV" > "Rings.nonzero_abs_inverse"
- "ABS_DIV" > "Rings.nonzero_abs_divide"
+ "ABS_LE" > "Groups.ordered_ab_group_add_abs_class.abs_ge_self"
+ "ABS_INV" > "Fields.linordered_field_class.nonzero_abs_inverse"
+ "ABS_DIV" > "Fields.linordered_field_class.nonzero_abs_divide"
"ABS_CIRCLE" > "HOL4Real.real.ABS_CIRCLE"
"ABS_CASES" > "HOL4Real.real.ABS_CASES"
"ABS_BOUNDS" > "RealDef.abs_le_interval_iff"
@@ -352,8 +352,8 @@
"ABS_BETWEEN2" > "HOL4Real.real.ABS_BETWEEN2"
"ABS_BETWEEN1" > "HOL4Real.real.ABS_BETWEEN1"
"ABS_BETWEEN" > "HOL4Real.real.ABS_BETWEEN"
- "ABS_ABS" > "Groups.abs_idempotent"
- "ABS_1" > "Int.bin_arith_simps_29"
- "ABS_0" > "Int.bin_arith_simps_28"
+ "ABS_ABS" > "Groups.ordered_ab_group_add_abs_class.abs_idempotent"
+ "ABS_1" > "Divides.arithmetic_simps_28"
+ "ABS_0" > "Divides.arithmetic_simps_27"
end
--- a/src/HOL/Import/HOL/realax.imp Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Import/HOL/realax.imp Fri Sep 09 06:47:14 2011 +0200
@@ -27,14 +27,18 @@
"treal_add" > "HOL4Real.realax.treal_add"
"treal_1" > "HOL4Real.realax.treal_1"
"treal_0" > "HOL4Real.realax.treal_0"
- "real_neg" > "Groups.uminus" :: "real => real"
- "real_mul" > "Groups.times" :: "real => real => real"
- "real_lt" > "Orderings.less" :: "real => real => bool"
- "real_add" > "Groups.plus" :: "real => real => real"
- "real_1" > "Groups.one" :: "real"
- "real_0" > "Groups.zero" :: "real"
- "inv" > "Ring.inverse" :: "real => real"
+ "real_sub" > "Groups.minus_class.minus" :: "real => real => real"
+ "real_neg" > "Groups.uminus_class.uminus" :: "real => real"
+ "real_mul" > "Groups.times_class.times" :: "real => real => real"
+ "real_lt" > "Orderings.ord_class.less" :: "real => real => bool"
+ "real_div" > "Fields.inverse_class.divide" :: "real => real => real"
+ "real_add" > "Groups.plus_class.plus" :: "real => real => real"
+ "real_1" > "Groups.one_class.one" :: "real"
+ "real_0" > "Groups.zero_class.zero" :: "real"
+ "mk_real" > "HOL.undefined"
+ "inv" > "Fields.inverse_class.inverse" :: "real => real"
"hreal_of_treal" > "HOL4Real.realax.hreal_of_treal"
+ "dest_real" > "HOL.undefined"
thm_maps
"treal_of_hreal_def" > "HOL4Real.realax.treal_of_hreal_def"
@@ -91,21 +95,21 @@
"TREAL_ADD_ASSOC" > "HOL4Real.realax.TREAL_ADD_ASSOC"
"TREAL_10" > "HOL4Real.realax.TREAL_10"
"REAL_SUP_ALLPOS" > "HOL4Compat.REAL_SUP_ALLPOS"
- "REAL_MUL_SYM" > "Int.zmult_ac_2"
- "REAL_MUL_LINV" > "HOL4Compat.REAL_MUL_LINV"
- "REAL_MUL_LID" > "Finite_Set.AC_mult.f_e.left_ident"
- "REAL_MUL_ASSOC" > "HOL4Compat.REAL_MUL_ASSOC"
- "REAL_LT_TRANS" > "Set.basic_trans_rules_21"
+ "REAL_MUL_SYM" > "Fields.linordered_field_class.sign_simps_21"
+ "REAL_MUL_LINV" > "Fields.division_ring_class.left_inverse"
+ "REAL_MUL_LID" > "Divides.arithmetic_simps_42"
+ "REAL_MUL_ASSOC" > "Fields.linordered_field_class.sign_simps_22"
+ "REAL_LT_TRANS" > "Orderings.order_less_trans"
"REAL_LT_TOTAL" > "HOL4Compat.REAL_LT_TOTAL"
"REAL_LT_REFL" > "Orderings.order_less_irrefl"
- "REAL_LT_MUL" > "Rings.mult_pos_pos"
- "REAL_LT_IADD" > "Groups.add_strict_left_mono"
- "REAL_LDISTRIB" > "Rings.ring_eq_simps_2"
- "REAL_INV_0" > "Rings.division_ring_inverse_zero_class.inverse_zero"
- "REAL_ADD_SYM" > "Finite_Set.AC_add.f.AC_2"
- "REAL_ADD_LINV" > "HOL4Compat.REAL_ADD_LINV"
- "REAL_ADD_LID" > "Finite_Set.AC_add.f_e.left_ident"
- "REAL_ADD_ASSOC" > "HOL4Compat.REAL_ADD_ASSOC"
+ "REAL_LT_MUL" > "RealDef.real_mult_order"
+ "REAL_LT_IADD" > "Groups.ordered_cancel_ab_semigroup_add_class.add_strict_left_mono"
+ "REAL_LDISTRIB" > "Fields.linordered_field_class.sign_simps_7"
+ "REAL_INV_0" > "Fields.division_ring_inverse_zero_class.inverse_zero"
+ "REAL_ADD_SYM" > "Fields.linordered_field_class.sign_simps_18"
+ "REAL_ADD_LINV" > "Groups.ab_group_add_class.ab_left_minus"
+ "REAL_ADD_LID" > "Divides.arithmetic_simps_38"
+ "REAL_ADD_ASSOC" > "Fields.linordered_field_class.sign_simps_19"
"REAL_10" > "HOL4Compat.REAL_10"
"HREAL_RDISTRIB" > "HOL4Real.realax.HREAL_RDISTRIB"
"HREAL_LT_REFL" > "HOL4Real.realax.HREAL_LT_REFL"
--- a/src/HOL/Import/HOL/rich_list.imp Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Import/HOL/rich_list.imp Fri Sep 09 06:47:14 2011 +0200
@@ -34,7 +34,7 @@
"AND_EL" > "HOL4Base.rich_list.AND_EL"
thm_maps
- "list_INDUCT" > "HOL4Compat.unzip.induct"
+ "list_INDUCT" > "HOL4Compat.list_INDUCT"
"list_CASES" > "HOL4Compat.list_CASES"
"ZIP_UNZIP" > "HOL4Base.list.ZIP_UNZIP"
"ZIP_SNOC" > "HOL4Base.rich_list.ZIP_SNOC"
@@ -60,7 +60,7 @@
"SPLITP" > "HOL4Base.rich_list.SPLITP"
"SOME_EL_SNOC" > "HOL4Base.rich_list.SOME_EL_SNOC"
"SOME_EL_SEG" > "HOL4Base.rich_list.SOME_EL_SEG"
- "SOME_EL_REVERSE" > "HOL4Base.rich_list.SOME_EL_REVERSE"
+ "SOME_EL_REVERSE" > "List.list_ex_rev"
"SOME_EL_MAP" > "HOL4Base.rich_list.SOME_EL_MAP"
"SOME_EL_LASTN" > "HOL4Base.rich_list.SOME_EL_LASTN"
"SOME_EL_FOLDR_MAP" > "HOL4Base.rich_list.SOME_EL_FOLDR_MAP"
@@ -71,7 +71,7 @@
"SOME_EL_DISJ" > "HOL4Base.rich_list.SOME_EL_DISJ"
"SOME_EL_BUTLASTN" > "HOL4Base.rich_list.SOME_EL_BUTLASTN"
"SOME_EL_BUTFIRSTN" > "HOL4Base.rich_list.SOME_EL_BUTFIRSTN"
- "SOME_EL_APPEND" > "HOL4Base.list.EXISTS_APPEND"
+ "SOME_EL_APPEND" > "List.list_ex_append"
"SOME_EL" > "HOL4Compat.list_exists_DEF"
"SNOC_REVERSE_CONS" > "HOL4Base.rich_list.SNOC_REVERSE_CONS"
"SNOC_INDUCT" > "HOL4Base.rich_list.SNOC_INDUCT"
@@ -100,7 +100,7 @@
"REVERSE_SNOC" > "HOL4Base.rich_list.REVERSE_SNOC"
"REVERSE_REVERSE" > "List.rev_rev_ident"
"REVERSE_FOLDR" > "HOL4Base.rich_list.REVERSE_FOLDR"
- "REVERSE_FOLDL" > "HOL4Base.rich_list.REVERSE_FOLDL"
+ "REVERSE_FOLDL" > "List.rev_foldl_cons"
"REVERSE_FLAT" > "HOL4Base.rich_list.REVERSE_FLAT"
"REVERSE_EQ_NIL" > "List.rev_is_Nil_conv"
"REVERSE_APPEND" > "List.rev_append"
@@ -116,21 +116,21 @@
"OR_EL_DEF" > "HOL4Base.rich_list.OR_EL_DEF"
"NULL_FOLDR" > "HOL4Base.rich_list.NULL_FOLDR"
"NULL_FOLDL" > "HOL4Base.rich_list.NULL_FOLDL"
- "NULL_EQ_NIL" > "HOL4Base.rich_list.NULL_EQ_NIL"
+ "NULL_EQ_NIL" > "List.eq_Nil_null"
"NULL_DEF" > "HOL4Compat.NULL_DEF"
"NULL" > "HOL4Base.list.NULL"
"NOT_SOME_EL_ALL_EL" > "HOL4Base.list.NOT_EXISTS"
"NOT_SNOC_NIL" > "HOL4Base.rich_list.NOT_SNOC_NIL"
"NOT_NIL_SNOC" > "HOL4Base.rich_list.NOT_NIL_SNOC"
- "NOT_NIL_CONS" > "List.list.simps_1"
+ "NOT_NIL_CONS" > "List.list.distinct_1"
"NOT_EQ_LIST" > "HOL4Base.list.NOT_EQ_LIST"
- "NOT_CONS_NIL" > "List.list.simps_2"
+ "NOT_CONS_NIL" > "List.list.distinct_2"
"NOT_ALL_EL_SOME_EL" > "HOL4Base.list.NOT_EVERY"
"MONOID_APPEND_NIL" > "HOL4Base.rich_list.MONOID_APPEND_NIL"
- "MAP_o" > "HOL4Base.rich_list.MAP_o"
+ "MAP_o" > "List.map.comp"
"MAP_SNOC" > "HOL4Base.rich_list.MAP_SNOC"
"MAP_REVERSE" > "List.rev_map"
- "MAP_MAP_o" > "List.map_map"
+ "MAP_MAP_o" > "List.map.compositionality"
"MAP_FOLDR" > "HOL4Base.rich_list.MAP_FOLDR"
"MAP_FOLDL" > "HOL4Base.rich_list.MAP_FOLDL"
"MAP_FLAT" > "List.map_concat"
@@ -230,7 +230,7 @@
"FOLDL_SNOC" > "HOL4Base.rich_list.FOLDL_SNOC"
"FOLDL_SINGLE" > "HOL4Base.rich_list.FOLDL_SINGLE"
"FOLDL_REVERSE" > "HOL4Base.rich_list.FOLDL_REVERSE"
- "FOLDL_MAP" > "HOL4Base.rich_list.FOLDL_MAP"
+ "FOLDL_MAP" > "List.foldl_map"
"FOLDL_FOLDR_REVERSE" > "List.foldl_foldr"
"FOLDL_FILTER" > "HOL4Base.rich_list.FOLDL_FILTER"
"FOLDL_APPEND" > "List.foldl_append"
@@ -238,7 +238,7 @@
"FLAT_SNOC" > "HOL4Base.rich_list.FLAT_SNOC"
"FLAT_REVERSE" > "HOL4Base.rich_list.FLAT_REVERSE"
"FLAT_FOLDR" > "HOL4Base.rich_list.FLAT_FOLDR"
- "FLAT_FOLDL" > "HOL4Base.rich_list.FLAT_FOLDL"
+ "FLAT_FOLDL" > "List.concat_conv_foldl"
"FLAT_FLAT" > "HOL4Base.rich_list.FLAT_FLAT"
"FLAT_APPEND" > "List.concat_append"
"FLAT" > "HOL4Compat.FLAT"
@@ -301,7 +301,7 @@
"ELL" > "HOL4Base.rich_list.ELL"
"EL" > "HOL4Base.rich_list.EL"
"CONS_APPEND" > "HOL4Base.rich_list.CONS_APPEND"
- "CONS_11" > "List.list.simps_3"
+ "CONS_11" > "List.list.inject"
"CONS" > "HOL4Base.list.CONS"
"COMM_MONOID_FOLDR" > "HOL4Base.rich_list.COMM_MONOID_FOLDR"
"COMM_MONOID_FOLDL" > "HOL4Base.rich_list.COMM_MONOID_FOLDL"
--- a/src/HOL/Import/HOL/seq.imp Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Import/HOL/seq.imp Fri Sep 09 06:47:14 2011 +0200
@@ -106,7 +106,7 @@
"MAX_LEMMA" > "HOL4Real.seq.MAX_LEMMA"
"GP_FINITE" > "HOL4Real.seq.GP_FINITE"
"GP" > "HOL4Real.seq.GP"
- "BOLZANO_LEMMA" > "HOL4Real.seq.BOLZANO_LEMMA"
- "ABS_NEG_LEMMA" > "HOL4Real.seq.ABS_NEG_LEMMA"
+ "BOLZANO_LEMMA" > "Deriv.lemma_BOLZANO"
+ "ABS_NEG_LEMMA" > "Series.rabs_ratiotest_lemma"
end
--- a/src/HOL/Import/HOL/sum.imp Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Import/HOL/sum.imp Fri Sep 09 06:47:14 2011 +0200
@@ -3,10 +3,10 @@
import_segment "hol4"
type_maps
- "sum" > "+"
+ "sum" > "Sum_Type.sum"
const_maps
- "sum_case" > "Datatype.sum.sum_case"
+ "sum_case" > "Sum_Type.sum.sum_case"
"OUTR" > "HOL4Compat.OUTR"
"OUTL" > "HOL4Compat.OUTL"
"ISR" > "HOL4Compat.ISR"
@@ -15,24 +15,24 @@
"INL" > "Sum_Type.Inl"
thm_maps
- "sum_distinct1" > "Datatype.sum.simps_2"
- "sum_distinct" > "Datatype.sum.simps_1"
+ "sum_distinct1" > "Sum_Type.Inr_not_Inl"
+ "sum_distinct" > "Sum_Type.Inl_not_Inr"
"sum_case_def" > "HOL4Compat.sum_case_def"
"sum_case_cong" > "HOL4Base.sum.sum_case_cong"
- "sum_INDUCT" > "HOL4Compat.OUTR.induct"
- "sum_CASES" > "Datatype.sum.nchotomy"
+ "sum_axiom" > "HOL4Compat.sum_axiom"
+ "sum_INDUCT" > "Sum_Type.sum.induct"
+ "sum_CASES" > "Sum_Type.sum.nchotomy"
"OUTR" > "HOL4Compat.OUTR"
"OUTL" > "HOL4Compat.OUTL"
"ISR" > "HOL4Compat.ISR"
"ISL_OR_ISR" > "HOL4Base.sum.ISL_OR_ISR"
"ISL" > "HOL4Compat.ISL"
- "INR_neq_INL" > "Datatype.sum.simps_2"
+ "INR_neq_INL" > "Sum_Type.Inr_not_Inl"
"INR_INL_11" > "HOL4Compat.INR_INL_11"
"INR" > "HOL4Base.sum.INR"
"INL" > "HOL4Base.sum.INL"
ignore_thms
- "sum_axiom"
"sum_TY_DEF"
"sum_ISO_DEF"
"sum_Axiom"
--- a/src/HOL/Import/HOL/word32.imp Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Import/HOL/word32.imp Fri Sep 09 06:47:14 2011 +0200
@@ -284,7 +284,7 @@
"HB_primdef" > "HOL4Word32.word32.HB_primdef"
"HB_def" > "HOL4Word32.word32.HB_def"
"FUNPOW_THM2" > "HOL4Word32.word32.FUNPOW_THM2"
- "FUNPOW_THM" > "HOL4Word32.word32.FUNPOW_THM"
+ "FUNPOW_THM" > "Nat.funpow_swap1"
"FUNPOW_COMP" > "HOL4Word32.word32.FUNPOW_COMP"
"EQ_ADD_RCANCELw" > "HOL4Word32.word32.EQ_ADD_RCANCELw"
"EQ_ADD_RCANCEL_QT" > "HOL4Word32.word32.EQ_ADD_RCANCEL_QT"
--- a/src/HOL/Import/HOLLight/hollight.imp Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Import/HOLLight/hollight.imp Fri Sep 09 06:47:14 2011 +0200
@@ -267,7 +267,7 @@
"WF" > "Wellfounded.wfP"
"UNIV" > "Orderings.top_class.top" :: "'a => bool"
"UNIONS" > "Complete_Lattice.Sup_class.Sup" :: "(('a => bool) => bool) => 'a => bool"
- "UNION" > "Lattices.semilattice_sup_class.sup" :: "('a => bool) => ('a => bool) => 'a => bool"
+ "UNION" > "Lattices.sup_class.sup" :: "('a => bool) => ('a => bool) => 'a => bool"
"UNCURRY" > "HOLLight.hollight.UNCURRY"
"TL" > "List.tl"
"T" > "HOL.True"
@@ -317,7 +317,7 @@
"ITLIST" > "List.foldr"
"ISO" > "HOLLight.hollight.ISO"
"INTERS" > "Complete_Lattice.Inf_class.Inf" :: "(('a => bool) => bool) => 'a => bool"
- "INTER" > "Lattices.semilattice_inf_class.inf" :: "('a => bool) => ('a => bool) => 'a => bool"
+ "INTER" > "Lattices.inf_class.inf" :: "('a => bool) => ('a => bool) => 'a => bool"
"INSERT" > "Set.insert"
"INR" > "Sum_Type.Inr"
"INL" > "Sum_Type.Inl"
--- a/src/HOL/Import/HOLLightInt.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Import/HOLLightInt.thy Fri Sep 09 06:47:14 2011 +0200
@@ -11,7 +11,7 @@
lemma DEF_int_coprime:
"int_coprime = (\<lambda>u. \<exists>x y. ((fst u) * x) + ((snd u) * y) = int 1)"
apply (auto simp add: fun_eq_iff)
- apply (metis bezout_int zmult_commute)
+ apply (metis bezout_int mult_commute)
by (metis coprime_divisors_nat dvd_triv_left gcd_1_int gcd_add2_int)
lemma INT_FORALL_POS:
@@ -24,7 +24,7 @@
lemma INT_ABS_MUL_1:
"(abs (x * y) = int 1) = (abs x = int 1 \<and> abs y = int 1)"
- by simp (metis dvd_mult_right zdvd1_eq abs_zmult_eq_1 abs_mult zmult_1_right)
+ by simp (metis dvd_mult_right zdvd1_eq abs_zmult_eq_1 abs_mult mult_1_right)
lemma dest_int_rep:
"\<exists>(n :: nat). real (i :: int) = real n \<or> real i = - real n"
@@ -56,19 +56,19 @@
lemma DEF_int_max:
"max = (\<lambda>u ua. floor (max (real u) (real ua)))"
- by (metis floor_real_of_int real_of_int_le_iff sup_absorb1 sup_commute sup_max zle_linear)
+ by (metis floor_real_of_int real_of_int_le_iff sup_absorb1 sup_commute sup_max linorder_linear)
lemma int_max_th:
"real (max (x :: int) y) = max (real x) (real y)"
- by (metis min_max.le_iff_sup min_max.sup_absorb1 real_of_int_le_iff zle_linear)
+ by (metis min_max.le_iff_sup min_max.sup_absorb1 real_of_int_le_iff linorder_linear)
lemma DEF_int_min:
"min = (\<lambda>u ua. floor (min (real u) (real ua)))"
- by (metis floor_real_of_int inf_absorb1 inf_absorb2 inf_int_def inf_real_def real_of_int_le_iff zle_linear)
+ by (metis floor_real_of_int inf_absorb1 inf_absorb2 inf_int_def inf_real_def real_of_int_le_iff linorder_linear)
lemma int_min_th:
"real (min (x :: int) y) = min (real x) (real y)"
- by (metis inf_absorb1 inf_absorb2 inf_int_def inf_real_def real_of_int_le_iff zle_linear)
+ by (metis inf_absorb1 inf_absorb2 inf_int_def inf_real_def real_of_int_le_iff linorder_linear)
lemma INT_IMAGE:
"(\<exists>n. x = int n) \<or> (\<exists>n. x = - int n)"
@@ -119,7 +119,7 @@
lemma hl_mod_div:
"n \<noteq> (0\<Colon>int) \<Longrightarrow> m = hl_div m n * n + hl_mod m n"
unfolding hl_div_def hl_mod_def
- by auto (metis zmod_zdiv_equality zmult_commute zmult_zminus)
+ by auto (metis zmod_zdiv_equality mult_commute mult_minus_left)
lemma sth:
"(\<forall>(x :: int) y z. x + (y + z) = x + y + z) \<and>
@@ -131,7 +131,7 @@
(\<forall>(x :: int). int 0 * x = int 0) \<and>
(\<forall>(x :: int) y z. x * (y + z) = x * y + x * z) \<and>
(\<forall>(x :: int). x ^ 0 = int 1) \<and> (\<forall>(x :: int) n. x ^ Suc n = x * x ^ n)"
- by (simp_all add: zadd_zmult_distrib2)
+ by (simp_all add: right_distrib)
lemma INT_DIVISION:
"n ~= int 0 \<Longrightarrow> m = hl_div m n * n + hl_mod m n \<and> int 0 \<le> hl_mod m n \<and> hl_mod m n < abs n"
@@ -160,7 +160,7 @@
apply (simp add: hl_mod_def hl_div_def)
apply (case_tac "xa > 0")
apply (simp add: hl_mod_def hl_div_def)
- apply (metis comm_semiring_1_class.normalizing_semiring_rules(24) div_mult_self2 not_less_iff_gr_or_eq order_less_le zadd_0 zdiv_eq_0_iff zmult_commute)
+ apply (metis comm_semiring_1_class.normalizing_semiring_rules(24) div_mult_self2 not_less_iff_gr_or_eq order_less_le add_0 zdiv_eq_0_iff mult_commute)
apply (simp add: hl_mod_def hl_div_def)
by (metis add.comm_neutral add_pos_nonneg div_mult_self1 less_minus_iff minus_add minus_add_cancel minus_minus mult_zero_right not_square_less_zero zdiv_eq_0_iff zdiv_zminus2)
@@ -182,14 +182,14 @@
apply (simp add: hl_mod_def hl_div_def)
apply (metis add_left_cancel mod_div_equality)
apply (simp add: hl_mod_def hl_div_def)
- by (metis minus_mult_right mod_mult_self2 mod_pos_pos_trivial zadd_commute zminus_zmod zmod_zminus2 zmult_commute)
+ by (metis minus_mult_right mod_mult_self2 mod_pos_pos_trivial add_commute zminus_zmod zmod_zminus2 mult_commute)
lemma DEF_int_gcd:
"int_gcd = (SOME d. \<forall>a b. (int 0) \<le> (d (a, b)) \<and> (d (a, b)) dvd a \<and>
(d (a, b)) dvd b \<and> (\<exists>x y. d (a, b) = (a * x) + (b * y)))"
apply (rule some_equality[symmetric])
apply auto
- apply (metis bezout_int zmult_commute)
+ apply (metis bezout_int mult_commute)
apply (auto simp add: fun_eq_iff)
apply (drule_tac x="a" in spec)
apply (drule_tac x="b" in spec)
--- a/src/HOL/IsaMakefile Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/IsaMakefile Fri Sep 09 06:47:14 2011 +0200
@@ -463,10 +463,10 @@
Library/Quotient_Option.thy Library/Quotient_Product.thy \
Library/Quotient_Sum.thy Library/Quotient_Syntax.thy \
Library/Quotient_Type.thy Library/RBT.thy Library/RBT_Impl.thy \
- Library/RBT_Mapping.thy Library/README.html Library/Set_Algebras.thy \
- Library/State_Monad.thy Library/Ramsey.thy Library/Reflection.thy \
- Library/Sublist_Order.thy Library/Sum_of_Squares.thy \
- Library/Sum_of_Squares/sos_wrapper.ML \
+ Library/RBT_Mapping.thy Library/README.html Library/Saturated.thy \
+ Library/Set_Algebras.thy Library/State_Monad.thy Library/Ramsey.thy \
+ Library/Reflection.thy Library/Sublist_Order.thy \
+ Library/Sum_of_Squares.thy Library/Sum_of_Squares/sos_wrapper.ML \
Library/Sum_of_Squares/sum_of_squares.ML \
Library/Transitive_Closure_Table.thy Library/Univ_Poly.thy \
Library/Wfrec.thy Library/While_Combinator.thy Library/Zorn.thy \
--- a/src/HOL/Lattices.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Lattices.thy Fri Sep 09 06:47:14 2011 +0200
@@ -51,15 +51,18 @@
bot ("\<bottom>") and
top ("\<top>")
+class inf =
+ fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70)
-class semilattice_inf = order +
- fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70)
+class sup =
+ fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<squnion>" 65)
+
+class semilattice_inf = order + inf +
assumes inf_le1 [simp]: "x \<sqinter> y \<sqsubseteq> x"
and inf_le2 [simp]: "x \<sqinter> y \<sqsubseteq> y"
and inf_greatest: "x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<sqinter> z"
-class semilattice_sup = order +
- fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<squnion>" 65)
+class semilattice_sup = order + sup +
assumes sup_ge1 [simp]: "x \<sqsubseteq> x \<squnion> y"
and sup_ge2 [simp]: "y \<sqsubseteq> x \<squnion> y"
and sup_least: "y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<squnion> z \<sqsubseteq> x"
@@ -68,7 +71,7 @@
text {* Dual lattice *}
lemma dual_semilattice:
- "class.semilattice_inf greater_eq greater sup"
+ "class.semilattice_inf sup greater_eq greater"
by (rule class.semilattice_inf.intro, rule dual_order)
(unfold_locales, simp_all add: sup_least)
@@ -236,7 +239,7 @@
begin
lemma dual_lattice:
- "class.lattice (op \<ge>) (op >) sup inf"
+ "class.lattice sup (op \<ge>) (op >) inf"
by (rule class.lattice.intro, rule dual_semilattice, rule class.semilattice_sup.intro, rule dual_order)
(unfold_locales, auto)
@@ -307,7 +310,7 @@
lemma less_supI1:
"x \<sqsubset> a \<Longrightarrow> x \<sqsubset> a \<squnion> b"
proof -
- interpret dual: semilattice_inf "op \<ge>" "op >" sup
+ interpret dual: semilattice_inf sup "op \<ge>" "op >"
by (fact dual_semilattice)
assume "x \<sqsubset> a"
then show "x \<sqsubset> a \<squnion> b"
@@ -317,7 +320,7 @@
lemma less_supI2:
"x \<sqsubset> b \<Longrightarrow> x \<sqsubset> a \<squnion> b"
proof -
- interpret dual: semilattice_inf "op \<ge>" "op >" sup
+ interpret dual: semilattice_inf sup "op \<ge>" "op >"
by (fact dual_semilattice)
assume "x \<sqsubset> b"
then show "x \<sqsubset> a \<squnion> b"
@@ -348,7 +351,7 @@
by(simp add: inf_sup_aci inf_sup_distrib1)
lemma dual_distrib_lattice:
- "class.distrib_lattice (op \<ge>) (op >) sup inf"
+ "class.distrib_lattice sup (op \<ge>) (op >) inf"
by (rule class.distrib_lattice.intro, rule dual_lattice)
(unfold_locales, fact inf_sup_distrib1)
@@ -420,7 +423,7 @@
begin
lemma dual_bounded_lattice:
- "class.bounded_lattice greater_eq greater sup inf \<top> \<bottom>"
+ "class.bounded_lattice sup greater_eq greater inf \<top> \<bottom>"
by unfold_locales (auto simp add: less_le_not_le)
end
@@ -432,7 +435,7 @@
begin
lemma dual_boolean_algebra:
- "class.boolean_algebra (\<lambda>x y. x \<squnion> - y) uminus greater_eq greater sup inf \<top> \<bottom>"
+ "class.boolean_algebra (\<lambda>x y. x \<squnion> - y) uminus sup greater_eq greater inf \<top> \<bottom>"
by (rule class.boolean_algebra.intro, rule dual_bounded_lattice, rule dual_distrib_lattice)
(unfold_locales, auto simp add: inf_compl_bot sup_compl_top diff_eq)
@@ -506,7 +509,7 @@
lemma compl_sup [simp]:
"- (x \<squnion> y) = - x \<sqinter> - y"
proof -
- interpret boolean_algebra "\<lambda>x y. x \<squnion> - y" uminus greater_eq greater sup inf \<top> \<bottom>
+ interpret boolean_algebra "\<lambda>x y. x \<squnion> - y" uminus sup greater_eq greater inf \<top> \<bottom>
by (rule dual_boolean_algebra)
then show ?thesis by simp
qed
@@ -591,7 +594,7 @@
subsection {* @{const min}/@{const max} on linear orders as
special case of @{const inf}/@{const sup} *}
-sublocale linorder < min_max!: distrib_lattice less_eq less min max
+sublocale linorder < min_max!: distrib_lattice min less_eq less max
proof
fix x y z
show "max x (min y z) = min (max x y) (max x z)"
--- a/src/HOL/Library/Abstract_Rat.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Library/Abstract_Rat.thy Fri Sep 09 06:47:14 2011 +0200
@@ -10,64 +10,57 @@
type_synonym Num = "int \<times> int"
-abbreviation
- Num0_syn :: Num ("0\<^sub>N")
-where "0\<^sub>N \<equiv> (0, 0)"
+abbreviation Num0_syn :: Num ("0\<^sub>N")
+ where "0\<^sub>N \<equiv> (0, 0)"
-abbreviation
- Numi_syn :: "int \<Rightarrow> Num" ("_\<^sub>N")
-where "i\<^sub>N \<equiv> (i, 1)"
+abbreviation Numi_syn :: "int \<Rightarrow> Num" ("_\<^sub>N")
+ where "i\<^sub>N \<equiv> (i, 1)"
-definition
- isnormNum :: "Num \<Rightarrow> bool"
-where
+definition isnormNum :: "Num \<Rightarrow> bool" where
"isnormNum = (\<lambda>(a,b). (if a = 0 then b = 0 else b > 0 \<and> gcd a b = 1))"
-definition
- normNum :: "Num \<Rightarrow> Num"
-where
- "normNum = (\<lambda>(a,b). (if a=0 \<or> b = 0 then (0,0) else
- (let g = gcd a b
- in if b > 0 then (a div g, b div g) else (- (a div g), - (b div g)))))"
+definition normNum :: "Num \<Rightarrow> Num" where
+ "normNum = (\<lambda>(a,b).
+ (if a=0 \<or> b = 0 then (0,0) else
+ (let g = gcd a b
+ in if b > 0 then (a div g, b div g) else (- (a div g), - (b div g)))))"
-declare gcd_dvd1_int[presburger]
-declare gcd_dvd2_int[presburger]
+declare gcd_dvd1_int[presburger] gcd_dvd2_int[presburger]
+
lemma normNum_isnormNum [simp]: "isnormNum (normNum x)"
proof -
- have " \<exists> a b. x = (a,b)" by auto
- then obtain a b where x[simp]: "x = (a,b)" by blast
- {assume "a=0 \<or> b = 0" hence ?thesis by (simp add: normNum_def isnormNum_def)}
+ obtain a b where x: "x = (a, b)" by (cases x)
+ { assume "a=0 \<or> b = 0" hence ?thesis by (simp add: x normNum_def isnormNum_def) }
moreover
- {assume anz: "a \<noteq> 0" and bnz: "b \<noteq> 0"
+ { assume anz: "a \<noteq> 0" and bnz: "b \<noteq> 0"
let ?g = "gcd a b"
let ?a' = "a div ?g"
let ?b' = "b div ?g"
let ?g' = "gcd ?a' ?b'"
- from anz bnz have "?g \<noteq> 0" by simp with gcd_ge_0_int[of a b]
+ from anz bnz have "?g \<noteq> 0" by simp with gcd_ge_0_int[of a b]
have gpos: "?g > 0" by arith
- have gdvd: "?g dvd a" "?g dvd b" by arith+
- from zdvd_mult_div_cancel[OF gdvd(1)] zdvd_mult_div_cancel[OF gdvd(2)]
- anz bnz
- have nz':"?a' \<noteq> 0" "?b' \<noteq> 0"
- by - (rule notI, simp)+
- from anz bnz have stupid: "a \<noteq> 0 \<or> b \<noteq> 0" by arith
+ have gdvd: "?g dvd a" "?g dvd b" by arith+
+ from zdvd_mult_div_cancel[OF gdvd(1)] zdvd_mult_div_cancel[OF gdvd(2)] anz bnz
+ have nz': "?a' \<noteq> 0" "?b' \<noteq> 0" by - (rule notI, simp)+
+ from anz bnz have stupid: "a \<noteq> 0 \<or> b \<noteq> 0" by arith
from div_gcd_coprime_int[OF stupid] have gp1: "?g' = 1" .
from bnz have "b < 0 \<or> b > 0" by arith
moreover
- {assume b: "b > 0"
- from b have "?b' \<ge> 0"
- by (presburger add: pos_imp_zdiv_nonneg_iff[OF gpos])
- with nz' have b': "?b' > 0" by arith
- from b b' anz bnz nz' gp1 have ?thesis
- by (simp add: isnormNum_def normNum_def Let_def split_def fst_conv snd_conv)}
- moreover {assume b: "b < 0"
- {assume b': "?b' \<ge> 0"
+ { assume b: "b > 0"
+ from b have "?b' \<ge> 0"
+ by (presburger add: pos_imp_zdiv_nonneg_iff[OF gpos])
+ with nz' have b': "?b' > 0" by arith
+ from b b' anz bnz nz' gp1 have ?thesis
+ by (simp add: x isnormNum_def normNum_def Let_def split_def) }
+ moreover {
+ assume b: "b < 0"
+ { assume b': "?b' \<ge> 0"
from gpos have th: "?g \<ge> 0" by arith
from mult_nonneg_nonneg[OF th b'] zdvd_mult_div_cancel[OF gdvd(2)]
have False using b by arith }
- hence b': "?b' < 0" by (presburger add: linorder_not_le[symmetric])
- from anz bnz nz' b b' gp1 have ?thesis
- by (simp add: isnormNum_def normNum_def Let_def split_def)}
+ hence b': "?b' < 0" by (presburger add: linorder_not_le[symmetric])
+ from anz bnz nz' b b' gp1 have ?thesis
+ by (simp add: x isnormNum_def normNum_def Let_def split_def) }
ultimately have ?thesis by blast
}
ultimately show ?thesis by blast
@@ -75,63 +68,55 @@
text {* Arithmetic over Num *}
-definition
- Nadd :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "+\<^sub>N" 60)
-where
- "Nadd = (\<lambda>(a,b) (a',b'). if a = 0 \<or> b = 0 then normNum(a',b')
- else if a'=0 \<or> b' = 0 then normNum(a,b)
+definition Nadd :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "+\<^sub>N" 60) where
+ "Nadd = (\<lambda>(a,b) (a',b'). if a = 0 \<or> b = 0 then normNum(a',b')
+ else if a'=0 \<or> b' = 0 then normNum(a,b)
else normNum(a*b' + b*a', b*b'))"
-definition
- Nmul :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "*\<^sub>N" 60)
-where
- "Nmul = (\<lambda>(a,b) (a',b'). let g = gcd (a*a') (b*b')
+definition Nmul :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "*\<^sub>N" 60) where
+ "Nmul = (\<lambda>(a,b) (a',b'). let g = gcd (a*a') (b*b')
in (a*a' div g, b*b' div g))"
-definition
- Nneg :: "Num \<Rightarrow> Num" ("~\<^sub>N")
-where
- "Nneg \<equiv> (\<lambda>(a,b). (-a,b))"
+definition Nneg :: "Num \<Rightarrow> Num" ("~\<^sub>N")
+ where "Nneg \<equiv> (\<lambda>(a,b). (-a,b))"
-definition
- Nsub :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "-\<^sub>N" 60)
-where
- "Nsub = (\<lambda>a b. a +\<^sub>N ~\<^sub>N b)"
+definition Nsub :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "-\<^sub>N" 60)
+ where "Nsub = (\<lambda>a b. a +\<^sub>N ~\<^sub>N b)"
-definition
- Ninv :: "Num \<Rightarrow> Num"
-where
- "Ninv \<equiv> \<lambda>(a,b). if a < 0 then (-b, \<bar>a\<bar>) else (b,a)"
+definition Ninv :: "Num \<Rightarrow> Num"
+ where "Ninv = (\<lambda>(a,b). if a < 0 then (-b, \<bar>a\<bar>) else (b,a))"
-definition
- Ndiv :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "\<div>\<^sub>N" 60)
-where
- "Ndiv \<equiv> \<lambda>a b. a *\<^sub>N Ninv b"
+definition Ndiv :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "\<div>\<^sub>N" 60)
+ where "Ndiv = (\<lambda>a b. a *\<^sub>N Ninv b)"
lemma Nneg_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (~\<^sub>N x)"
- by(simp add: isnormNum_def Nneg_def split_def)
+ by (simp add: isnormNum_def Nneg_def split_def)
+
lemma Nadd_normN[simp]: "isnormNum (x +\<^sub>N y)"
by (simp add: Nadd_def split_def)
+
lemma Nsub_normN[simp]: "\<lbrakk> isnormNum y\<rbrakk> \<Longrightarrow> isnormNum (x -\<^sub>N y)"
by (simp add: Nsub_def split_def)
-lemma Nmul_normN[simp]: assumes xn:"isnormNum x" and yn: "isnormNum y"
+
+lemma Nmul_normN[simp]:
+ assumes xn: "isnormNum x" and yn: "isnormNum y"
shows "isnormNum (x *\<^sub>N y)"
-proof-
- have "\<exists>a b. x = (a,b)" and "\<exists> a' b'. y = (a',b')" by auto
- then obtain a b a' b' where ab: "x = (a,b)" and ab': "y = (a',b')" by blast
- {assume "a = 0"
- hence ?thesis using xn ab ab'
- by (simp add: isnormNum_def Let_def Nmul_def split_def)}
+proof -
+ obtain a b where x: "x = (a, b)" by (cases x)
+ obtain a' b' where y: "y = (a', b')" by (cases y)
+ { assume "a = 0"
+ hence ?thesis using xn x y
+ by (simp add: isnormNum_def Let_def Nmul_def split_def) }
moreover
- {assume "a' = 0"
- hence ?thesis using yn ab ab'
- by (simp add: isnormNum_def Let_def Nmul_def split_def)}
+ { assume "a' = 0"
+ hence ?thesis using yn x y
+ by (simp add: isnormNum_def Let_def Nmul_def split_def) }
moreover
- {assume a: "a \<noteq>0" and a': "a'\<noteq>0"
- hence bp: "b > 0" "b' > 0" using xn yn ab ab' by (simp_all add: isnormNum_def)
- from mult_pos_pos[OF bp] have "x *\<^sub>N y = normNum (a*a', b*b')"
- using ab ab' a a' bp by (simp add: Nmul_def Let_def split_def normNum_def)
- hence ?thesis by simp}
+ { assume a: "a \<noteq>0" and a': "a'\<noteq>0"
+ hence bp: "b > 0" "b' > 0" using xn yn x y by (simp_all add: isnormNum_def)
+ from mult_pos_pos[OF bp] have "x *\<^sub>N y = normNum (a * a', b * b')"
+ using x y a a' bp by (simp add: Nmul_def Let_def split_def normNum_def)
+ hence ?thesis by simp }
ultimately show ?thesis by blast
qed
@@ -139,89 +124,77 @@
by (simp add: Ninv_def isnormNum_def split_def)
(cases "fst x = 0", auto simp add: gcd_commute_int)
-lemma isnormNum_int[simp]:
+lemma isnormNum_int[simp]:
"isnormNum 0\<^sub>N" "isnormNum ((1::int)\<^sub>N)" "i \<noteq> 0 \<Longrightarrow> isnormNum (i\<^sub>N)"
by (simp_all add: isnormNum_def)
text {* Relations over Num *}
-definition
- Nlt0:: "Num \<Rightarrow> bool" ("0>\<^sub>N")
-where
- "Nlt0 = (\<lambda>(a,b). a < 0)"
+definition Nlt0:: "Num \<Rightarrow> bool" ("0>\<^sub>N")
+ where "Nlt0 = (\<lambda>(a,b). a < 0)"
-definition
- Nle0:: "Num \<Rightarrow> bool" ("0\<ge>\<^sub>N")
-where
- "Nle0 = (\<lambda>(a,b). a \<le> 0)"
+definition Nle0:: "Num \<Rightarrow> bool" ("0\<ge>\<^sub>N")
+ where "Nle0 = (\<lambda>(a,b). a \<le> 0)"
-definition
- Ngt0:: "Num \<Rightarrow> bool" ("0<\<^sub>N")
-where
- "Ngt0 = (\<lambda>(a,b). a > 0)"
+definition Ngt0:: "Num \<Rightarrow> bool" ("0<\<^sub>N")
+ where "Ngt0 = (\<lambda>(a,b). a > 0)"
-definition
- Nge0:: "Num \<Rightarrow> bool" ("0\<le>\<^sub>N")
-where
- "Nge0 = (\<lambda>(a,b). a \<ge> 0)"
+definition Nge0:: "Num \<Rightarrow> bool" ("0\<le>\<^sub>N")
+ where "Nge0 = (\<lambda>(a,b). a \<ge> 0)"
-definition
- Nlt :: "Num \<Rightarrow> Num \<Rightarrow> bool" (infix "<\<^sub>N" 55)
-where
- "Nlt = (\<lambda>a b. 0>\<^sub>N (a -\<^sub>N b))"
+definition Nlt :: "Num \<Rightarrow> Num \<Rightarrow> bool" (infix "<\<^sub>N" 55)
+ where "Nlt = (\<lambda>a b. 0>\<^sub>N (a -\<^sub>N b))"
-definition
- Nle :: "Num \<Rightarrow> Num \<Rightarrow> bool" (infix "\<le>\<^sub>N" 55)
-where
- "Nle = (\<lambda>a b. 0\<ge>\<^sub>N (a -\<^sub>N b))"
+definition Nle :: "Num \<Rightarrow> Num \<Rightarrow> bool" (infix "\<le>\<^sub>N" 55)
+ where "Nle = (\<lambda>a b. 0\<ge>\<^sub>N (a -\<^sub>N b))"
-definition
- "INum = (\<lambda>(a,b). of_int a / of_int b)"
+definition "INum = (\<lambda>(a,b). of_int a / of_int b)"
lemma INum_int [simp]: "INum (i\<^sub>N) = ((of_int i) ::'a::field)" "INum 0\<^sub>N = (0::'a::field)"
by (simp_all add: INum_def)
-lemma isnormNum_unique[simp]:
- assumes na: "isnormNum x" and nb: "isnormNum y"
+lemma isnormNum_unique[simp]:
+ assumes na: "isnormNum x" and nb: "isnormNum y"
shows "((INum x ::'a::{field_char_0, field_inverse_zero}) = INum y) = (x = y)" (is "?lhs = ?rhs")
proof
- have "\<exists> a b a' b'. x = (a,b) \<and> y = (a',b')" by auto
- then obtain a b a' b' where xy[simp]: "x = (a,b)" "y=(a',b')" by blast
- assume H: ?lhs
- {assume "a = 0 \<or> b = 0 \<or> a' = 0 \<or> b' = 0"
+ obtain a b where x: "x = (a, b)" by (cases x)
+ obtain a' b' where y: "y = (a', b')" by (cases y)
+ assume H: ?lhs
+ { assume "a = 0 \<or> b = 0 \<or> a' = 0 \<or> b' = 0"
hence ?rhs using na nb H
- by (simp add: INum_def split_def isnormNum_def split: split_if_asm)}
+ by (simp add: x y INum_def split_def isnormNum_def split: split_if_asm) }
moreover
{ assume az: "a \<noteq> 0" and bz: "b \<noteq> 0" and a'z: "a'\<noteq>0" and b'z: "b'\<noteq>0"
- from az bz a'z b'z na nb have pos: "b > 0" "b' > 0" by (simp_all add: isnormNum_def)
- from H bz b'z have eq:"a * b' = a'*b"
- by (simp add: INum_def eq_divide_eq divide_eq_eq of_int_mult[symmetric] del: of_int_mult)
- from az a'z na nb have gcd1: "gcd a b = 1" "gcd b a = 1" "gcd a' b' = 1" "gcd b' a' = 1"
- by (simp_all add: isnormNum_def add: gcd_commute_int)
- from eq have raw_dvd: "a dvd a'*b" "b dvd b'*a" "a' dvd a*b'" "b' dvd b*a'"
- apply -
+ from az bz a'z b'z na nb have pos: "b > 0" "b' > 0" by (simp_all add: x y isnormNum_def)
+ from H bz b'z have eq: "a * b' = a'*b"
+ by (simp add: x y INum_def eq_divide_eq divide_eq_eq of_int_mult[symmetric] del: of_int_mult)
+ from az a'z na nb have gcd1: "gcd a b = 1" "gcd b a = 1" "gcd a' b' = 1" "gcd b' a' = 1"
+ by (simp_all add: x y isnormNum_def add: gcd_commute_int)
+ from eq have raw_dvd: "a dvd a' * b" "b dvd b' * a" "a' dvd a * b'" "b' dvd b * a'"
+ apply -
apply algebra
apply algebra
apply simp
apply algebra
done
from zdvd_antisym_abs[OF coprime_dvd_mult_int[OF gcd1(2) raw_dvd(2)]
- coprime_dvd_mult_int[OF gcd1(4) raw_dvd(4)]]
+ coprime_dvd_mult_int[OF gcd1(4) raw_dvd(4)]]
have eq1: "b = b'" using pos by arith
with eq have "a = a'" using pos by simp
- with eq1 have ?rhs by simp}
+ with eq1 have ?rhs by (simp add: x y) }
ultimately show ?rhs by blast
next
assume ?rhs thus ?lhs by simp
qed
-lemma isnormNum0[simp]: "isnormNum x \<Longrightarrow> (INum x = (0::'a::{field_char_0, field_inverse_zero})) = (x = 0\<^sub>N)"
+lemma isnormNum0[simp]:
+ "isnormNum x \<Longrightarrow> (INum x = (0::'a::{field_char_0, field_inverse_zero})) = (x = 0\<^sub>N)"
unfolding INum_int(2)[symmetric]
- by (rule isnormNum_unique, simp_all)
+ by (rule isnormNum_unique) simp_all
-lemma of_int_div_aux: "d ~= 0 ==> ((of_int x)::'a::field_char_0) / (of_int d) =
+lemma of_int_div_aux: "d ~= 0 ==> ((of_int x)::'a::field_char_0) / (of_int d) =
of_int (x div d) + (of_int (x mod d)) / ((of_int d)::'a)"
proof -
assume "d ~= 0"
@@ -231,7 +204,7 @@
by auto
then have eq: "of_int x = ?t"
by (simp only: of_int_mult[symmetric] of_int_add [symmetric])
- then have "of_int x / of_int d = ?t / of_int d"
+ then have "of_int x / of_int d = ?t / of_int d"
using cong[OF refl[of ?f] eq] by simp
then show ?thesis by (simp add: add_divide_distrib algebra_simps `d ~= 0`)
qed
@@ -241,25 +214,26 @@
apply (frule of_int_div_aux [of d n, where ?'a = 'a])
apply simp
apply (simp add: dvd_eq_mod_eq_0)
-done
+ done
lemma normNum[simp]: "INum (normNum x) = (INum x :: 'a::{field_char_0, field_inverse_zero})"
-proof-
- have "\<exists> a b. x = (a,b)" by auto
- then obtain a b where x[simp]: "x = (a,b)" by blast
- {assume "a=0 \<or> b = 0" hence ?thesis
- by (simp add: INum_def normNum_def split_def Let_def)}
- moreover
- {assume a: "a\<noteq>0" and b: "b\<noteq>0"
+proof -
+ obtain a b where x: "x = (a, b)" by (cases x)
+ { assume "a = 0 \<or> b = 0"
+ hence ?thesis by (simp add: x INum_def normNum_def split_def Let_def) }
+ moreover
+ { assume a: "a \<noteq> 0" and b: "b \<noteq> 0"
let ?g = "gcd a b"
from a b have g: "?g \<noteq> 0"by simp
from of_int_div[OF g, where ?'a = 'a]
- have ?thesis by (auto simp add: INum_def normNum_def split_def Let_def)}
+ have ?thesis by (auto simp add: x INum_def normNum_def split_def Let_def) }
ultimately show ?thesis by blast
qed
-lemma INum_normNum_iff: "(INum x ::'a::{field_char_0, field_inverse_zero}) = INum y \<longleftrightarrow> normNum x = normNum y" (is "?lhs = ?rhs")
+lemma INum_normNum_iff:
+ "(INum x ::'a::{field_char_0, field_inverse_zero}) = INum y \<longleftrightarrow> normNum x = normNum y"
+ (is "?lhs = ?rhs")
proof -
have "normNum x = normNum y \<longleftrightarrow> (INum (normNum x) :: 'a) = INum (normNum y)"
by (simp del: normNum)
@@ -268,139 +242,157 @@
qed
lemma Nadd[simp]: "INum (x +\<^sub>N y) = INum x + (INum y :: 'a :: {field_char_0, field_inverse_zero})"
-proof-
-let ?z = "0:: 'a"
- have " \<exists> a b. x = (a,b)" " \<exists> a' b'. y = (a',b')" by auto
- then obtain a b a' b' where x[simp]: "x = (a,b)"
- and y[simp]: "y = (a',b')" by blast
- {assume "a=0 \<or> a'= 0 \<or> b =0 \<or> b' = 0" hence ?thesis
- apply (cases "a=0",simp_all add: Nadd_def)
- apply (cases "b= 0",simp_all add: INum_def)
- apply (cases "a'= 0",simp_all)
- apply (cases "b'= 0",simp_all)
+proof -
+ let ?z = "0:: 'a"
+ obtain a b where x: "x = (a, b)" by (cases x)
+ obtain a' b' where y: "y = (a', b')" by (cases y)
+ { assume "a=0 \<or> a'= 0 \<or> b =0 \<or> b' = 0"
+ hence ?thesis
+ apply (cases "a=0", simp_all add: x y Nadd_def)
+ apply (cases "b= 0", simp_all add: INum_def)
+ apply (cases "a'= 0", simp_all)
+ apply (cases "b'= 0", simp_all)
done }
- moreover
- {assume aa':"a \<noteq> 0" "a'\<noteq> 0" and bb': "b \<noteq> 0" "b' \<noteq> 0"
- {assume z: "a * b' + b * a' = 0"
+ moreover
+ { assume aa': "a \<noteq> 0" "a'\<noteq> 0" and bb': "b \<noteq> 0" "b' \<noteq> 0"
+ { assume z: "a * b' + b * a' = 0"
hence "of_int (a*b' + b*a') / (of_int b* of_int b') = ?z" by simp
- hence "of_int b' * of_int a / (of_int b * of_int b') + of_int b * of_int a' / (of_int b * of_int b') = ?z" by (simp add:add_divide_distrib)
- hence th: "of_int a / of_int b + of_int a' / of_int b' = ?z" using bb' aa' by simp
- from z aa' bb' have ?thesis
- by (simp add: th Nadd_def normNum_def INum_def split_def)}
- moreover {assume z: "a * b' + b * a' \<noteq> 0"
+ hence "of_int b' * of_int a / (of_int b * of_int b') +
+ of_int b * of_int a' / (of_int b * of_int b') = ?z"
+ by (simp add:add_divide_distrib)
+ hence th: "of_int a / of_int b + of_int a' / of_int b' = ?z" using bb' aa'
+ by simp
+ from z aa' bb' have ?thesis
+ by (simp add: x y th Nadd_def normNum_def INum_def split_def) }
+ moreover {
+ assume z: "a * b' + b * a' \<noteq> 0"
let ?g = "gcd (a * b' + b * a') (b*b')"
have gz: "?g \<noteq> 0" using z by simp
have ?thesis using aa' bb' z gz
- of_int_div[where ?'a = 'a, OF gz gcd_dvd1_int[where x="a * b' + b * a'" and y="b*b'"]] of_int_div[where ?'a = 'a,
- OF gz gcd_dvd2_int[where x="a * b' + b * a'" and y="b*b'"]]
- by (simp add: Nadd_def INum_def normNum_def Let_def add_divide_distrib)}
- ultimately have ?thesis using aa' bb'
- by (simp add: Nadd_def INum_def normNum_def Let_def) }
+ of_int_div[where ?'a = 'a, OF gz gcd_dvd1_int[where x="a * b' + b * a'" and y="b*b'"]]
+ of_int_div[where ?'a = 'a, OF gz gcd_dvd2_int[where x="a * b' + b * a'" and y="b*b'"]]
+ by (simp add: x y Nadd_def INum_def normNum_def Let_def add_divide_distrib) }
+ ultimately have ?thesis using aa' bb'
+ by (simp add: x y Nadd_def INum_def normNum_def Let_def) }
ultimately show ?thesis by blast
qed
-lemma Nmul[simp]: "INum (x *\<^sub>N y) = INum x * (INum y:: 'a :: {field_char_0, field_inverse_zero}) "
-proof-
+lemma Nmul[simp]: "INum (x *\<^sub>N y) = INum x * (INum y:: 'a :: {field_char_0, field_inverse_zero})"
+proof -
let ?z = "0::'a"
- have " \<exists> a b. x = (a,b)" " \<exists> a' b'. y = (a',b')" by auto
- then obtain a b a' b' where x: "x = (a,b)" and y: "y = (a',b')" by blast
- {assume "a=0 \<or> a'= 0 \<or> b = 0 \<or> b' = 0" hence ?thesis
- apply (cases "a=0",simp_all add: x y Nmul_def INum_def Let_def)
- apply (cases "b=0",simp_all)
- apply (cases "a'=0",simp_all)
+ obtain a b where x: "x = (a, b)" by (cases x)
+ obtain a' b' where y: "y = (a', b')" by (cases y)
+ { assume "a=0 \<or> a'= 0 \<or> b = 0 \<or> b' = 0"
+ hence ?thesis
+ apply (cases "a=0", simp_all add: x y Nmul_def INum_def Let_def)
+ apply (cases "b=0", simp_all)
+ apply (cases "a'=0", simp_all)
done }
moreover
- {assume z: "a \<noteq> 0" "a' \<noteq> 0" "b \<noteq> 0" "b' \<noteq> 0"
+ { assume z: "a \<noteq> 0" "a' \<noteq> 0" "b \<noteq> 0" "b' \<noteq> 0"
let ?g="gcd (a*a') (b*b')"
have gz: "?g \<noteq> 0" using z by simp
- from z of_int_div[where ?'a = 'a, OF gz gcd_dvd1_int[where x="a*a'" and y="b*b'"]]
- of_int_div[where ?'a = 'a , OF gz gcd_dvd2_int[where x="a*a'" and y="b*b'"]]
- have ?thesis by (simp add: Nmul_def x y Let_def INum_def)}
+ from z of_int_div[where ?'a = 'a, OF gz gcd_dvd1_int[where x="a*a'" and y="b*b'"]]
+ of_int_div[where ?'a = 'a , OF gz gcd_dvd2_int[where x="a*a'" and y="b*b'"]]
+ have ?thesis by (simp add: Nmul_def x y Let_def INum_def) }
ultimately show ?thesis by blast
qed
lemma Nneg[simp]: "INum (~\<^sub>N x) = - (INum x ::'a:: field)"
by (simp add: Nneg_def split_def INum_def)
-lemma Nsub[simp]: shows "INum (x -\<^sub>N y) = INum x - (INum y:: 'a :: {field_char_0, field_inverse_zero})"
-by (simp add: Nsub_def split_def)
+lemma Nsub[simp]: "INum (x -\<^sub>N y) = INum x - (INum y:: 'a :: {field_char_0, field_inverse_zero})"
+ by (simp add: Nsub_def split_def)
lemma Ninv[simp]: "INum (Ninv x) = (1::'a :: field_inverse_zero) / (INum x)"
by (simp add: Ninv_def INum_def split_def)
-lemma Ndiv[simp]: "INum (x \<div>\<^sub>N y) = INum x / (INum y ::'a :: {field_char_0, field_inverse_zero})" by (simp add: Ndiv_def)
+lemma Ndiv[simp]: "INum (x \<div>\<^sub>N y) = INum x / (INum y ::'a :: {field_char_0, field_inverse_zero})"
+ by (simp add: Ndiv_def)
-lemma Nlt0_iff[simp]: assumes nx: "isnormNum x"
- shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})< 0) = 0>\<^sub>N x "
-proof-
- have " \<exists> a b. x = (a,b)" by simp
- then obtain a b where x[simp]:"x = (a,b)" by blast
- {assume "a = 0" hence ?thesis by (simp add: Nlt0_def INum_def) }
+lemma Nlt0_iff[simp]:
+ assumes nx: "isnormNum x"
+ shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})< 0) = 0>\<^sub>N x"
+proof -
+ obtain a b where x: "x = (a, b)" by (cases x)
+ { assume "a = 0" hence ?thesis by (simp add: x Nlt0_def INum_def) }
moreover
- {assume a: "a\<noteq>0" hence b: "(of_int b::'a) > 0" using nx by (simp add: isnormNum_def)
+ { assume a: "a \<noteq> 0" hence b: "(of_int b::'a) > 0"
+ using nx by (simp add: x isnormNum_def)
from pos_divide_less_eq[OF b, where b="of_int a" and a="0::'a"]
- have ?thesis by (simp add: Nlt0_def INum_def)}
+ have ?thesis by (simp add: x Nlt0_def INum_def) }
ultimately show ?thesis by blast
qed
-lemma Nle0_iff[simp]:assumes nx: "isnormNum x"
+lemma Nle0_iff[simp]:
+ assumes nx: "isnormNum x"
shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) \<le> 0) = 0\<ge>\<^sub>N x"
-proof-
- have " \<exists> a b. x = (a,b)" by simp
- then obtain a b where x[simp]:"x = (a,b)" by blast
- {assume "a = 0" hence ?thesis by (simp add: Nle0_def INum_def) }
+proof -
+ obtain a b where x: "x = (a, b)" by (cases x)
+ { assume "a = 0" hence ?thesis by (simp add: x Nle0_def INum_def) }
moreover
- {assume a: "a\<noteq>0" hence b: "(of_int b :: 'a) > 0" using nx by (simp add: isnormNum_def)
+ { assume a: "a \<noteq> 0" hence b: "(of_int b :: 'a) > 0"
+ using nx by (simp add: x isnormNum_def)
from pos_divide_le_eq[OF b, where b="of_int a" and a="0::'a"]
- have ?thesis by (simp add: Nle0_def INum_def)}
+ have ?thesis by (simp add: x Nle0_def INum_def) }
ultimately show ?thesis by blast
qed
-lemma Ngt0_iff[simp]:assumes nx: "isnormNum x" shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})> 0) = 0<\<^sub>N x"
-proof-
- have " \<exists> a b. x = (a,b)" by simp
- then obtain a b where x[simp]:"x = (a,b)" by blast
- {assume "a = 0" hence ?thesis by (simp add: Ngt0_def INum_def) }
+lemma Ngt0_iff[simp]:
+ assumes nx: "isnormNum x"
+ shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})> 0) = 0<\<^sub>N x"
+proof -
+ obtain a b where x: "x = (a, b)" by (cases x)
+ { assume "a = 0" hence ?thesis by (simp add: x Ngt0_def INum_def) }
moreover
- {assume a: "a\<noteq>0" hence b: "(of_int b::'a) > 0" using nx by (simp add: isnormNum_def)
+ { assume a: "a \<noteq> 0" hence b: "(of_int b::'a) > 0" using nx
+ by (simp add: x isnormNum_def)
from pos_less_divide_eq[OF b, where b="of_int a" and a="0::'a"]
- have ?thesis by (simp add: Ngt0_def INum_def)}
- ultimately show ?thesis by blast
-qed
-lemma Nge0_iff[simp]:assumes nx: "isnormNum x"
- shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) \<ge> 0) = 0\<le>\<^sub>N x"
-proof-
- have " \<exists> a b. x = (a,b)" by simp
- then obtain a b where x[simp]:"x = (a,b)" by blast
- {assume "a = 0" hence ?thesis by (simp add: Nge0_def INum_def) }
- moreover
- {assume a: "a\<noteq>0" hence b: "(of_int b::'a) > 0" using nx by (simp add: isnormNum_def)
- from pos_le_divide_eq[OF b, where b="of_int a" and a="0::'a"]
- have ?thesis by (simp add: Nge0_def INum_def)}
+ have ?thesis by (simp add: x Ngt0_def INum_def) }
ultimately show ?thesis by blast
qed
-lemma Nlt_iff[simp]: assumes nx: "isnormNum x" and ny: "isnormNum y"
+lemma Nge0_iff[simp]:
+ assumes nx: "isnormNum x"
+ shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) \<ge> 0) = 0\<le>\<^sub>N x"
+proof -
+ obtain a b where x: "x = (a, b)" by (cases x)
+ { assume "a = 0" hence ?thesis by (simp add: x Nge0_def INum_def) }
+ moreover
+ { assume "a \<noteq> 0" hence b: "(of_int b::'a) > 0" using nx
+ by (simp add: x isnormNum_def)
+ from pos_le_divide_eq[OF b, where b="of_int a" and a="0::'a"]
+ have ?thesis by (simp add: x Nge0_def INum_def) }
+ ultimately show ?thesis by blast
+qed
+
+lemma Nlt_iff[simp]:
+ assumes nx: "isnormNum x" and ny: "isnormNum y"
shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) < INum y) = (x <\<^sub>N y)"
-proof-
+proof -
let ?z = "0::'a"
- have "((INum x ::'a) < INum y) = (INum (x -\<^sub>N y) < ?z)" using nx ny by simp
- also have "\<dots> = (0>\<^sub>N (x -\<^sub>N y))" using Nlt0_iff[OF Nsub_normN[OF ny]] by simp
+ have "((INum x ::'a) < INum y) = (INum (x -\<^sub>N y) < ?z)"
+ using nx ny by simp
+ also have "\<dots> = (0>\<^sub>N (x -\<^sub>N y))"
+ using Nlt0_iff[OF Nsub_normN[OF ny]] by simp
finally show ?thesis by (simp add: Nlt_def)
qed
-lemma Nle_iff[simp]: assumes nx: "isnormNum x" and ny: "isnormNum y"
+lemma Nle_iff[simp]:
+ assumes nx: "isnormNum x" and ny: "isnormNum y"
shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})\<le> INum y) = (x \<le>\<^sub>N y)"
-proof-
- have "((INum x ::'a) \<le> INum y) = (INum (x -\<^sub>N y) \<le> (0::'a))" using nx ny by simp
- also have "\<dots> = (0\<ge>\<^sub>N (x -\<^sub>N y))" using Nle0_iff[OF Nsub_normN[OF ny]] by simp
+proof -
+ have "((INum x ::'a) \<le> INum y) = (INum (x -\<^sub>N y) \<le> (0::'a))"
+ using nx ny by simp
+ also have "\<dots> = (0\<ge>\<^sub>N (x -\<^sub>N y))"
+ using Nle0_iff[OF Nsub_normN[OF ny]] by simp
finally show ?thesis by (simp add: Nle_def)
qed
lemma Nadd_commute:
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
shows "x +\<^sub>N y = y +\<^sub>N x"
-proof-
+proof -
have n: "isnormNum (x +\<^sub>N y)" "isnormNum (y +\<^sub>N x)" by simp_all
have "(INum (x +\<^sub>N y)::'a) = INum (y +\<^sub>N x)" by simp
with isnormNum_unique[OF n] show ?thesis by simp
@@ -409,7 +401,7 @@
lemma [simp]:
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
shows "(0, b) +\<^sub>N y = normNum y"
- and "(a, 0) +\<^sub>N y = normNum y"
+ and "(a, 0) +\<^sub>N y = normNum y"
and "x +\<^sub>N (0, b) = normNum x"
and "x +\<^sub>N (a, 0) = normNum x"
apply (simp add: Nadd_def split_def)
@@ -420,14 +412,13 @@
lemma normNum_nilpotent_aux[simp]:
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
- assumes nx: "isnormNum x"
+ assumes nx: "isnormNum x"
shows "normNum x = x"
-proof-
+proof -
let ?a = "normNum x"
have n: "isnormNum ?a" by simp
- have th:"INum ?a = (INum x ::'a)" by simp
- with isnormNum_unique[OF n nx]
- show ?thesis by simp
+ have th: "INum ?a = (INum x ::'a)" by simp
+ with isnormNum_unique[OF n nx] show ?thesis by simp
qed
lemma normNum_nilpotent[simp]:
@@ -445,7 +436,7 @@
lemma Nadd_normNum1[simp]:
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
shows "normNum x +\<^sub>N y = x +\<^sub>N y"
-proof-
+proof -
have n: "isnormNum (normNum x +\<^sub>N y)" "isnormNum (x +\<^sub>N y)" by simp_all
have "INum (normNum x +\<^sub>N y) = INum x + (INum y :: 'a)" by simp
also have "\<dots> = INum (x +\<^sub>N y)" by simp
@@ -455,7 +446,7 @@
lemma Nadd_normNum2[simp]:
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
shows "x +\<^sub>N normNum y = x +\<^sub>N y"
-proof-
+proof -
have n: "isnormNum (x +\<^sub>N normNum y)" "isnormNum (x +\<^sub>N y)" by simp_all
have "INum (x +\<^sub>N normNum y) = INum x + (INum y :: 'a)" by simp
also have "\<dots> = INum (x +\<^sub>N y)" by simp
@@ -465,7 +456,7 @@
lemma Nadd_assoc:
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
shows "x +\<^sub>N y +\<^sub>N z = x +\<^sub>N (y +\<^sub>N z)"
-proof-
+proof -
have n: "isnormNum (x +\<^sub>N y +\<^sub>N z)" "isnormNum (x +\<^sub>N (y +\<^sub>N z))" by simp_all
have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)" by simp
with isnormNum_unique[OF n] show ?thesis by simp
@@ -476,10 +467,10 @@
lemma Nmul_assoc:
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
- assumes nx: "isnormNum x" and ny:"isnormNum y" and nz:"isnormNum z"
+ assumes nx: "isnormNum x" and ny: "isnormNum y" and nz: "isnormNum z"
shows "x *\<^sub>N y *\<^sub>N z = x *\<^sub>N (y *\<^sub>N z)"
-proof-
- from nx ny nz have n: "isnormNum (x *\<^sub>N y *\<^sub>N z)" "isnormNum (x *\<^sub>N (y *\<^sub>N z))"
+proof -
+ from nx ny nz have n: "isnormNum (x *\<^sub>N y *\<^sub>N z)" "isnormNum (x *\<^sub>N (y *\<^sub>N z))"
by simp_all
have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)" by simp
with isnormNum_unique[OF n] show ?thesis by simp
@@ -487,14 +478,15 @@
lemma Nsub0:
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
- assumes x: "isnormNum x" and y:"isnormNum y" shows "(x -\<^sub>N y = 0\<^sub>N) = (x = y)"
-proof-
- { fix h :: 'a
- from isnormNum_unique[where 'a = 'a, OF Nsub_normN[OF y], where y="0\<^sub>N"]
- have "(x -\<^sub>N y = 0\<^sub>N) = (INum (x -\<^sub>N y) = (INum 0\<^sub>N :: 'a)) " by simp
- also have "\<dots> = (INum x = (INum y :: 'a))" by simp
- also have "\<dots> = (x = y)" using x y by simp
- finally show ?thesis . }
+ assumes x: "isnormNum x" and y: "isnormNum y"
+ shows "x -\<^sub>N y = 0\<^sub>N \<longleftrightarrow> x = y"
+proof -
+ fix h :: 'a
+ from isnormNum_unique[where 'a = 'a, OF Nsub_normN[OF y], where y="0\<^sub>N"]
+ have "(x -\<^sub>N y = 0\<^sub>N) = (INum (x -\<^sub>N y) = (INum 0\<^sub>N :: 'a)) " by simp
+ also have "\<dots> = (INum x = (INum y :: 'a))" by simp
+ also have "\<dots> = (x = y)" using x y by simp
+ finally show ?thesis .
qed
lemma Nmul0[simp]: "c *\<^sub>N 0\<^sub>N = 0\<^sub>N" " 0\<^sub>N *\<^sub>N c = 0\<^sub>N"
@@ -502,24 +494,26 @@
lemma Nmul_eq0[simp]:
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
- assumes nx:"isnormNum x" and ny: "isnormNum y"
- shows "(x*\<^sub>N y = 0\<^sub>N) = (x = 0\<^sub>N \<or> y = 0\<^sub>N)"
-proof-
- { fix h :: 'a
- have " \<exists> a b a' b'. x = (a,b) \<and> y= (a',b')" by auto
- then obtain a b a' b' where xy[simp]: "x = (a,b)" "y = (a',b')" by blast
- have n0: "isnormNum 0\<^sub>N" by simp
- show ?thesis using nx ny
- apply (simp only: isnormNum_unique[where ?'a = 'a, OF Nmul_normN[OF nx ny] n0, symmetric] Nmul[where ?'a = 'a])
- by (simp add: INum_def split_def isnormNum_def split: split_if_asm)
- }
+ assumes nx: "isnormNum x" and ny: "isnormNum y"
+ shows "x*\<^sub>N y = 0\<^sub>N \<longleftrightarrow> x = 0\<^sub>N \<or> y = 0\<^sub>N"
+proof -
+ fix h :: 'a
+ obtain a b where x: "x = (a, b)" by (cases x)
+ obtain a' b' where y: "y = (a', b')" by (cases y)
+ have n0: "isnormNum 0\<^sub>N" by simp
+ show ?thesis using nx ny
+ apply (simp only: isnormNum_unique[where ?'a = 'a, OF Nmul_normN[OF nx ny] n0, symmetric]
+ Nmul[where ?'a = 'a])
+ apply (simp add: x y INum_def split_def isnormNum_def split: split_if_asm)
+ done
qed
+
lemma Nneg_Nneg[simp]: "~\<^sub>N (~\<^sub>N c) = c"
by (simp add: Nneg_def split_def)
-lemma Nmul1[simp]:
- "isnormNum c \<Longrightarrow> 1\<^sub>N *\<^sub>N c = c"
- "isnormNum c \<Longrightarrow> c *\<^sub>N (1\<^sub>N) = c"
+lemma Nmul1[simp]:
+ "isnormNum c \<Longrightarrow> 1\<^sub>N *\<^sub>N c = c"
+ "isnormNum c \<Longrightarrow> c *\<^sub>N (1\<^sub>N) = c"
apply (simp_all add: Nmul_def Let_def split_def isnormNum_def)
apply (cases "fst c = 0", simp_all, cases c, simp_all)+
done
--- a/src/HOL/Library/Float.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Library/Float.thy Fri Sep 09 06:47:14 2011 +0200
@@ -719,11 +719,11 @@
shows "real (x div y) \<le> real (x * c div y) * inverse (real c)"
proof -
have "c * (x div y) + 0 \<le> c * x div y" unfolding zdiv_zmult1_eq[of c x y]
- by (rule zadd_left_mono,
+ by (rule add_left_mono,
auto intro!: mult_nonneg_nonneg
simp add: pos_imp_zdiv_nonneg_iff[OF `0 < y`] `0 < c`[THEN less_imp_le] pos_mod_sign[OF `0 < y`])
hence "real (x div y) * real c \<le> real (x * c div y)"
- unfolding real_of_int_mult[symmetric] real_of_int_le_iff zmult_commute by auto
+ unfolding real_of_int_mult[symmetric] real_of_int_le_iff mult_commute by auto
hence "real (x div y) * real c * inverse (real c) \<le> real (x * c div y) * inverse (real c)"
using `0 < c` by auto
thus ?thesis unfolding mult_assoc using `0 < c` by auto
@@ -777,7 +777,7 @@
have "?X = y * (?X div y) + ?X mod y" by auto
also have "\<dots> \<le> y * (?X div y) + y" by (rule add_mono, auto simp add: pos_mod_bound[OF `0 < y`, THEN less_imp_le])
- also have "\<dots> = y * (?X div y + 1)" unfolding zadd_zmult_distrib2 by auto
+ also have "\<dots> = y * (?X div y + 1)" unfolding right_distrib by auto
finally have "real ?X \<le> real y * real (?X div y + 1)" unfolding real_of_int_le_iff real_of_int_mult[symmetric] .
hence "real ?X / (real y * 2^?l) \<le> real y * real (?X div y + 1) / (real y * 2^?l)"
by (rule divide_right_mono, simp only: `0 \<le> real y * 2^?l`)
@@ -1144,7 +1144,7 @@
have "2^(prec - 1) * m \<le> 2^(prec - 1) * 2^?b"
using `m < 2^?b`[THEN less_imp_le] by (rule mult_left_mono) auto
also have "\<dots> = 2 ^ nat (int prec + bitlen m - 1)"
- unfolding pow_split zpower_zadd_distrib by auto
+ unfolding pow_split power_add by auto
finally have "2^(prec - 1) * m div m \<le> 2 ^ nat (int prec + bitlen m - 1) div m"
using `0 < m` by (rule zdiv_mono1)
hence "2^(prec - 1) \<le> 2 ^ nat (int prec + bitlen m - 1) div m"
@@ -1273,7 +1273,7 @@
next
case False
have "m = m div ?p * ?p + m mod ?p" unfolding zdiv_zmod_equality2[where k=0, unfolded monoid_add_class.add_0_right] ..
- also have "\<dots> \<le> (m div ?p + 1) * ?p" unfolding left_distrib zmult_1 by (rule add_left_mono, rule pos_mod_bound[OF `0 < ?p`, THEN less_imp_le])
+ also have "\<dots> \<le> (m div ?p + 1) * ?p" unfolding left_distrib mult_1 by (rule add_left_mono, rule pos_mod_bound[OF `0 < ?p`, THEN less_imp_le])
finally have "real (Float m e) \<le> real (Float (m div ?p + 1) (e + ?d))" unfolding real_of_float_simp add_commute[of e]
unfolding pow2_add mult_assoc[symmetric] real_of_int_le_iff[of m, symmetric]
by (auto intro!: mult_mono simp add: pow2_add `0 < ?d` pow_d)
@@ -1361,7 +1361,7 @@
have "real m \<le> real (m div 2^(nat ?l) + 1) * pow2 ?l"
proof -
have "m mod 2^(nat ?l) < 2^(nat ?l)" by (rule pos_mod_bound) auto
- hence mod_uneq: "real (m mod 2^(nat ?l)) \<le> 1 * 2^(nat ?l)" unfolding zmult_1 real_of_int_less_iff[symmetric] by auto
+ hence mod_uneq: "real (m mod 2^(nat ?l)) \<le> 1 * 2^(nat ?l)" unfolding mult_1 real_of_int_less_iff[symmetric] by auto
have "real m = real (2^(nat ?l) * (m div 2^(nat ?l)) + m mod 2^(nat ?l))" unfolding zmod_zdiv_equality[symmetric] ..
also have "\<dots> = real (m div 2^(nat ?l)) * 2^(nat ?l) + real (m mod 2^(nat ?l))" unfolding real_of_int_add by auto
--- a/src/HOL/Library/Library.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Library/Library.thy Fri Sep 09 06:47:14 2011 +0200
@@ -55,6 +55,7 @@
Ramsey
Reflection
RBT_Mapping
+ Saturated
Set_Algebras
State_Monad
Sum_of_Squares
--- a/src/HOL/Library/Product_Vector.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Library/Product_Vector.thy Fri Sep 09 06:47:14 2011 +0200
@@ -450,7 +450,7 @@
assumes x: "0 \<le> x" and y: "0 \<le> y"
shows "sqrt (x + y) \<le> sqrt x + sqrt y"
apply (rule power2_le_imp_le)
-apply (simp add: real_sum_squared_expand x y)
+apply (simp add: power2_sum x y)
apply (simp add: mult_nonneg_nonneg x y)
apply (simp add: x y)
done
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Saturated.thy Fri Sep 09 06:47:14 2011 +0200
@@ -0,0 +1,246 @@
+(* Author: Brian Huffman *)
+(* Author: Peter Gammie *)
+(* Author: Florian Haftmann *)
+
+header {* Saturated arithmetic *}
+
+theory Saturated
+imports Main "~~/src/HOL/Library/Numeral_Type" "~~/src/HOL/Word/Type_Length"
+begin
+
+subsection {* The type of saturated naturals *}
+
+typedef (open) ('a::len) sat = "{.. len_of TYPE('a)}"
+ morphisms nat_of Abs_sat
+ by auto
+
+lemma sat_eqI:
+ "nat_of m = nat_of n \<Longrightarrow> m = n"
+ by (simp add: nat_of_inject)
+
+lemma sat_eq_iff:
+ "m = n \<longleftrightarrow> nat_of m = nat_of n"
+ by (simp add: nat_of_inject)
+
+lemma Abs_sa_nat_of [code abstype]:
+ "Abs_sat (nat_of n) = n"
+ by (fact nat_of_inverse)
+
+definition Sat :: "nat \<Rightarrow> 'a::len sat" where
+ "Sat n = Abs_sat (min (len_of TYPE('a)) n)"
+
+lemma nat_of_Sat [simp]:
+ "nat_of (Sat n :: ('a::len) sat) = min (len_of TYPE('a)) n"
+ unfolding Sat_def by (rule Abs_sat_inverse) simp
+
+lemma nat_of_le_len_of [simp]:
+ "nat_of (n :: ('a::len) sat) \<le> len_of TYPE('a)"
+ using nat_of [where x = n] by simp
+
+lemma min_len_of_nat_of [simp]:
+ "min (len_of TYPE('a)) (nat_of (n::('a::len) sat)) = nat_of n"
+ by (rule min_max.inf_absorb2 [OF nat_of_le_len_of])
+
+lemma min_nat_of_len_of [simp]:
+ "min (nat_of (n::('a::len) sat)) (len_of TYPE('a)) = nat_of n"
+ by (subst min_max.inf.commute) simp
+
+lemma Sat_nat_of [simp]:
+ "Sat (nat_of n) = n"
+ by (simp add: Sat_def nat_of_inverse)
+
+instantiation sat :: (len) linorder
+begin
+
+definition
+ less_eq_sat_def: "x \<le> y \<longleftrightarrow> nat_of x \<le> nat_of y"
+
+definition
+ less_sat_def: "x < y \<longleftrightarrow> nat_of x < nat_of y"
+
+instance
+by default (auto simp add: less_eq_sat_def less_sat_def not_le sat_eq_iff min_max.le_infI1 nat_mult_commute)
+
+end
+
+instantiation sat :: (len) "{minus, comm_semiring_1}"
+begin
+
+definition
+ "0 = Sat 0"
+
+definition
+ "1 = Sat 1"
+
+lemma nat_of_zero_sat [simp, code abstract]:
+ "nat_of 0 = 0"
+ by (simp add: zero_sat_def)
+
+lemma nat_of_one_sat [simp, code abstract]:
+ "nat_of 1 = min 1 (len_of TYPE('a))"
+ by (simp add: one_sat_def)
+
+definition
+ "x + y = Sat (nat_of x + nat_of y)"
+
+lemma nat_of_plus_sat [simp, code abstract]:
+ "nat_of (x + y) = min (nat_of x + nat_of y) (len_of TYPE('a))"
+ by (simp add: plus_sat_def)
+
+definition
+ "x - y = Sat (nat_of x - nat_of y)"
+
+lemma nat_of_minus_sat [simp, code abstract]:
+ "nat_of (x - y) = nat_of x - nat_of y"
+proof -
+ from nat_of_le_len_of [of x] have "nat_of x - nat_of y \<le> len_of TYPE('a)" by arith
+ then show ?thesis by (simp add: minus_sat_def)
+qed
+
+definition
+ "x * y = Sat (nat_of x * nat_of y)"
+
+lemma nat_of_times_sat [simp, code abstract]:
+ "nat_of (x * y) = min (nat_of x * nat_of y) (len_of TYPE('a))"
+ by (simp add: times_sat_def)
+
+instance proof
+ fix a b c :: "('a::len) sat"
+ show "a * b * c = a * (b * c)"
+ proof(cases "a = 0")
+ case True thus ?thesis by (simp add: sat_eq_iff)
+ next
+ case False show ?thesis
+ proof(cases "c = 0")
+ case True thus ?thesis by (simp add: sat_eq_iff)
+ next
+ case False with `a \<noteq> 0` show ?thesis
+ by (simp add: sat_eq_iff nat_mult_min_left nat_mult_min_right mult_assoc min_max.inf_assoc min_max.inf_absorb2)
+ qed
+ qed
+next
+ fix a :: "('a::len) sat"
+ show "1 * a = a"
+ apply (simp add: sat_eq_iff)
+ apply (metis One_nat_def len_gt_0 less_Suc0 less_zeroE linorder_not_less min_max.le_iff_inf min_nat_of_len_of nat_mult_1_right nat_mult_commute)
+ done
+next
+ fix a b c :: "('a::len) sat"
+ show "(a + b) * c = a * c + b * c"
+ proof(cases "c = 0")
+ case True thus ?thesis by (simp add: sat_eq_iff)
+ next
+ case False thus ?thesis
+ by (simp add: sat_eq_iff nat_mult_min_left add_mult_distrib min_add_distrib_left min_add_distrib_right min_max.inf_assoc min_max.inf_absorb2)
+ qed
+qed (simp_all add: sat_eq_iff mult.commute)
+
+end
+
+instantiation sat :: (len) ordered_comm_semiring
+begin
+
+instance
+by default (auto simp add: less_eq_sat_def less_sat_def not_le sat_eq_iff min_max.le_infI1 nat_mult_commute)
+
+end
+
+lemma Sat_eq_of_nat: "Sat n = of_nat n"
+ by (rule sat_eqI, induct n, simp_all)
+
+instantiation sat :: (len) number_semiring
+begin
+
+definition
+ number_of_sat_def [code del]: "number_of = Sat \<circ> nat"
+
+instance
+ by default (simp add: number_of_sat_def Sat_eq_of_nat)
+
+end
+
+lemma [code abstract]:
+ "nat_of (number_of n :: ('a::len) sat) = min (nat n) (len_of TYPE('a))"
+ unfolding number_of_sat_def by simp
+
+instance sat :: (len) finite
+proof
+ show "finite (UNIV::'a sat set)"
+ unfolding type_definition.univ [OF type_definition_sat]
+ using finite by simp
+qed
+
+instantiation sat :: (len) equal
+begin
+
+definition
+ "HOL.equal A B \<longleftrightarrow> nat_of A = nat_of B"
+
+instance proof
+qed (simp add: equal_sat_def nat_of_inject)
+
+end
+
+instantiation sat :: (len) "{bounded_lattice, distrib_lattice}"
+begin
+
+definition
+ "(inf :: 'a sat \<Rightarrow> 'a sat \<Rightarrow> 'a sat) = min"
+
+definition
+ "(sup :: 'a sat \<Rightarrow> 'a sat \<Rightarrow> 'a sat) = max"
+
+definition
+ "bot = (0 :: 'a sat)"
+
+definition
+ "top = Sat (len_of TYPE('a))"
+
+instance proof
+qed (simp_all add: inf_sat_def sup_sat_def bot_sat_def top_sat_def min_max.sup_inf_distrib1,
+ simp_all add: less_eq_sat_def)
+
+end
+
+instantiation sat :: (len) complete_lattice
+begin
+
+definition
+ "Inf (A :: 'a sat set) = fold min top A"
+
+definition
+ "Sup (A :: 'a sat set) = fold max bot A"
+
+instance proof
+ fix x :: "'a sat"
+ fix A :: "'a sat set"
+ note finite
+ moreover assume "x \<in> A"
+ ultimately have "fold min top A \<le> min x top" by (rule min_max.fold_inf_le_inf)
+ then show "Inf A \<le> x" by (simp add: Inf_sat_def)
+next
+ fix z :: "'a sat"
+ fix A :: "'a sat set"
+ note finite
+ moreover assume z: "\<And>x. x \<in> A \<Longrightarrow> z \<le> x"
+ ultimately have "min z top \<le> fold min top A" by (blast intro: min_max.inf_le_fold_inf)
+ then show "z \<le> Inf A" by (simp add: Inf_sat_def min_def)
+next
+ fix x :: "'a sat"
+ fix A :: "'a sat set"
+ note finite
+ moreover assume "x \<in> A"
+ ultimately have "max x bot \<le> fold max bot A" by (rule min_max.sup_le_fold_sup)
+ then show "x \<le> Sup A" by (simp add: Sup_sat_def)
+next
+ fix z :: "'a sat"
+ fix A :: "'a sat set"
+ note finite
+ moreover assume z: "\<And>x. x \<in> A \<Longrightarrow> x \<le> z"
+ ultimately have "fold max bot A \<le> max z bot" by (blast intro: min_max.fold_sup_le_sup)
+ then show "Sup A \<le> z" by (simp add: Sup_sat_def max_def bot_unique)
+qed
+
+end
+
+end
--- a/src/HOL/Metis_Examples/Proxies.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Metis_Examples/Proxies.thy Fri Sep 09 06:47:14 2011 +0200
@@ -58,206 +58,206 @@
lemma "id (op =) x x"
sledgehammer [type_enc = erased, expect = none] (id_apply)
-sledgehammer [type_enc = poly_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_tags??, expect = some] (id_apply)
sledgehammer [type_enc = poly_tags, expect = some] (id_apply)
-sledgehammer [type_enc = poly_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_guards??, expect = some] (id_apply)
sledgehammer [type_enc = poly_guards, expect = some] (id_apply)
-sledgehammer [type_enc = mono_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_tags??, expect = some] (id_apply)
sledgehammer [type_enc = mono_tags, expect = some] (id_apply)
-sledgehammer [type_enc = mono_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_guards??, expect = some] (id_apply)
sledgehammer [type_enc = mono_guards, expect = some] (id_apply)
by (metis (full_types) id_apply)
lemma "id True"
sledgehammer [type_enc = erased, expect = some] (id_apply)
-sledgehammer [type_enc = poly_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_tags??, expect = some] (id_apply)
sledgehammer [type_enc = poly_tags, expect = some] (id_apply)
-sledgehammer [type_enc = poly_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_guards??, expect = some] (id_apply)
sledgehammer [type_enc = poly_guards, expect = some] (id_apply)
-sledgehammer [type_enc = mono_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_tags??, expect = some] (id_apply)
sledgehammer [type_enc = mono_tags, expect = some] (id_apply)
-sledgehammer [type_enc = mono_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_guards??, expect = some] (id_apply)
sledgehammer [type_enc = mono_guards, expect = some] (id_apply)
by (metis_exhaust id_apply)
lemma "\<not> id False"
sledgehammer [type_enc = erased, expect = some] (id_apply)
-sledgehammer [type_enc = poly_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_tags??, expect = some] (id_apply)
sledgehammer [type_enc = poly_tags, expect = some] (id_apply)
-sledgehammer [type_enc = poly_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_guards??, expect = some] (id_apply)
sledgehammer [type_enc = poly_guards, expect = some] (id_apply)
-sledgehammer [type_enc = mono_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_tags??, expect = some] (id_apply)
sledgehammer [type_enc = mono_tags, expect = some] (id_apply)
-sledgehammer [type_enc = mono_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_guards??, expect = some] (id_apply)
sledgehammer [type_enc = mono_guards, expect = some] (id_apply)
by (metis_exhaust id_apply)
lemma "x = id True \<or> x = id False"
sledgehammer [type_enc = erased, expect = some] (id_apply)
-sledgehammer [type_enc = poly_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_tags??, expect = some] (id_apply)
sledgehammer [type_enc = poly_tags, expect = some] (id_apply)
-sledgehammer [type_enc = poly_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_guards??, expect = some] (id_apply)
sledgehammer [type_enc = poly_guards, expect = some] (id_apply)
-sledgehammer [type_enc = mono_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_tags??, expect = some] (id_apply)
sledgehammer [type_enc = mono_tags, expect = some] (id_apply)
-sledgehammer [type_enc = mono_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_guards??, expect = some] (id_apply)
sledgehammer [type_enc = mono_guards, expect = some] (id_apply)
by (metis_exhaust id_apply)
lemma "id x = id True \<or> id x = id False"
sledgehammer [type_enc = erased, expect = some] (id_apply)
-sledgehammer [type_enc = poly_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_tags??, expect = some] (id_apply)
sledgehammer [type_enc = poly_tags, expect = some] (id_apply)
-sledgehammer [type_enc = poly_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_guards??, expect = some] (id_apply)
sledgehammer [type_enc = poly_guards, expect = some] (id_apply)
-sledgehammer [type_enc = mono_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_tags??, expect = some] (id_apply)
sledgehammer [type_enc = mono_tags, expect = some] (id_apply)
-sledgehammer [type_enc = mono_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_guards??, expect = some] (id_apply)
sledgehammer [type_enc = mono_guards, expect = some] (id_apply)
by (metis_exhaust id_apply)
lemma "P True \<Longrightarrow> P False \<Longrightarrow> P x"
sledgehammer [type_enc = erased, expect = none] ()
sledgehammer [type_enc = poly_args, expect = none] ()
-sledgehammer [type_enc = poly_tags?, expect = some] ()
+sledgehammer [type_enc = poly_tags??, expect = some] ()
sledgehammer [type_enc = poly_tags, expect = some] ()
-sledgehammer [type_enc = poly_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_guards??, expect = some] (id_apply)
sledgehammer [type_enc = poly_guards, expect = some] ()
-sledgehammer [type_enc = mono_tags?, expect = some] ()
+sledgehammer [type_enc = mono_tags??, expect = some] ()
sledgehammer [type_enc = mono_tags, expect = some] ()
-sledgehammer [type_enc = mono_guards?, expect = some] ()
+sledgehammer [type_enc = mono_guards??, expect = some] ()
sledgehammer [type_enc = mono_guards, expect = some] ()
by (metis (full_types))
lemma "id (\<not> a) \<Longrightarrow> \<not> id a"
sledgehammer [type_enc = erased, expect = some] (id_apply)
-sledgehammer [type_enc = poly_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_tags??, expect = some] (id_apply)
sledgehammer [type_enc = poly_tags, expect = some] (id_apply)
-sledgehammer [type_enc = poly_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_guards??, expect = some] (id_apply)
sledgehammer [type_enc = poly_guards, expect = some] (id_apply)
-sledgehammer [type_enc = mono_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_tags??, expect = some] (id_apply)
sledgehammer [type_enc = mono_tags, expect = some] (id_apply)
-sledgehammer [type_enc = mono_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_guards??, expect = some] (id_apply)
sledgehammer [type_enc = mono_guards, expect = some] (id_apply)
by (metis_exhaust id_apply)
lemma "id (\<not> \<not> a) \<Longrightarrow> id a"
sledgehammer [type_enc = erased, expect = some] (id_apply)
-sledgehammer [type_enc = poly_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_tags??, expect = some] (id_apply)
sledgehammer [type_enc = poly_tags, expect = some] (id_apply)
-sledgehammer [type_enc = poly_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_guards??, expect = some] (id_apply)
sledgehammer [type_enc = poly_guards, expect = some] (id_apply)
-sledgehammer [type_enc = mono_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_tags??, expect = some] (id_apply)
sledgehammer [type_enc = mono_tags, expect = some] (id_apply)
-sledgehammer [type_enc = mono_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_guards??, expect = some] (id_apply)
sledgehammer [type_enc = mono_guards, expect = some] (id_apply)
by (metis_exhaust id_apply)
lemma "id (\<not> (id (\<not> a))) \<Longrightarrow> id a"
sledgehammer [type_enc = erased, expect = some] (id_apply)
-sledgehammer [type_enc = poly_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_tags??, expect = some] (id_apply)
sledgehammer [type_enc = poly_tags, expect = some] (id_apply)
-sledgehammer [type_enc = poly_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_guards??, expect = some] (id_apply)
sledgehammer [type_enc = poly_guards, expect = some] (id_apply)
-sledgehammer [type_enc = mono_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_tags??, expect = some] (id_apply)
sledgehammer [type_enc = mono_tags, expect = some] (id_apply)
-sledgehammer [type_enc = mono_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_guards??, expect = some] (id_apply)
sledgehammer [type_enc = mono_guards, expect = some] (id_apply)
by (metis_exhaust id_apply)
lemma "id (a \<and> b) \<Longrightarrow> id a"
sledgehammer [type_enc = erased, expect = some] (id_apply)
-sledgehammer [type_enc = poly_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_tags??, expect = some] (id_apply)
sledgehammer [type_enc = poly_tags, expect = some] (id_apply)
-sledgehammer [type_enc = poly_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_guards??, expect = some] (id_apply)
sledgehammer [type_enc = poly_guards, expect = some] (id_apply)
-sledgehammer [type_enc = mono_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_tags??, expect = some] (id_apply)
sledgehammer [type_enc = mono_tags, expect = some] (id_apply)
-sledgehammer [type_enc = mono_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_guards??, expect = some] (id_apply)
sledgehammer [type_enc = mono_guards, expect = some] (id_apply)
by (metis_exhaust id_apply)
lemma "id (a \<and> b) \<Longrightarrow> id b"
sledgehammer [type_enc = erased, expect = some] (id_apply)
-sledgehammer [type_enc = poly_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_tags??, expect = some] (id_apply)
sledgehammer [type_enc = poly_tags, expect = some] (id_apply)
-sledgehammer [type_enc = poly_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_guards??, expect = some] (id_apply)
sledgehammer [type_enc = poly_guards, expect = some] (id_apply)
-sledgehammer [type_enc = mono_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_tags??, expect = some] (id_apply)
sledgehammer [type_enc = mono_tags, expect = some] (id_apply)
-sledgehammer [type_enc = mono_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_guards??, expect = some] (id_apply)
sledgehammer [type_enc = mono_guards, expect = some] (id_apply)
by (metis_exhaust id_apply)
lemma "id a \<Longrightarrow> id b \<Longrightarrow> id (a \<and> b)"
sledgehammer [type_enc = erased, expect = some] (id_apply)
-sledgehammer [type_enc = poly_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_tags??, expect = some] (id_apply)
sledgehammer [type_enc = poly_tags, expect = some] (id_apply)
-sledgehammer [type_enc = poly_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_guards??, expect = some] (id_apply)
sledgehammer [type_enc = poly_guards, expect = some] (id_apply)
-sledgehammer [type_enc = mono_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_tags??, expect = some] (id_apply)
sledgehammer [type_enc = mono_tags, expect = some] (id_apply)
-sledgehammer [type_enc = mono_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_guards??, expect = some] (id_apply)
sledgehammer [type_enc = mono_guards, expect = some] (id_apply)
by (metis_exhaust id_apply)
lemma "id a \<Longrightarrow> id (a \<or> b)"
sledgehammer [type_enc = erased, expect = some] (id_apply)
-sledgehammer [type_enc = poly_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_tags??, expect = some] (id_apply)
sledgehammer [type_enc = poly_tags, expect = some] (id_apply)
-sledgehammer [type_enc = poly_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_guards??, expect = some] (id_apply)
sledgehammer [type_enc = poly_guards, expect = some] (id_apply)
-sledgehammer [type_enc = mono_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_tags??, expect = some] (id_apply)
sledgehammer [type_enc = mono_tags, expect = some] (id_apply)
-sledgehammer [type_enc = mono_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_guards??, expect = some] (id_apply)
sledgehammer [type_enc = mono_guards, expect = some] (id_apply)
by (metis_exhaust id_apply)
lemma "id b \<Longrightarrow> id (a \<or> b)"
sledgehammer [type_enc = erased, expect = some] (id_apply)
-sledgehammer [type_enc = poly_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_tags??, expect = some] (id_apply)
sledgehammer [type_enc = poly_tags, expect = some] (id_apply)
-sledgehammer [type_enc = poly_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_guards??, expect = some] (id_apply)
sledgehammer [type_enc = poly_guards, expect = some] (id_apply)
-sledgehammer [type_enc = mono_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_tags??, expect = some] (id_apply)
sledgehammer [type_enc = mono_tags, expect = some] (id_apply)
-sledgehammer [type_enc = mono_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_guards??, expect = some] (id_apply)
sledgehammer [type_enc = mono_guards, expect = some] (id_apply)
by (metis_exhaust id_apply)
lemma "id (\<not> a) \<Longrightarrow> id (\<not> b) \<Longrightarrow> id (\<not> (a \<or> b))"
sledgehammer [type_enc = erased, expect = some] (id_apply)
-sledgehammer [type_enc = poly_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_tags??, expect = some] (id_apply)
sledgehammer [type_enc = poly_tags, expect = some] (id_apply)
-sledgehammer [type_enc = poly_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_guards??, expect = some] (id_apply)
sledgehammer [type_enc = poly_guards, expect = some] (id_apply)
-sledgehammer [type_enc = mono_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_tags??, expect = some] (id_apply)
sledgehammer [type_enc = mono_tags, expect = some] (id_apply)
-sledgehammer [type_enc = mono_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_guards??, expect = some] (id_apply)
sledgehammer [type_enc = mono_guards, expect = some] (id_apply)
by (metis_exhaust id_apply)
lemma "id (\<not> a) \<Longrightarrow> id (a \<longrightarrow> b)"
sledgehammer [type_enc = erased, expect = some] (id_apply)
-sledgehammer [type_enc = poly_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_tags??, expect = some] (id_apply)
sledgehammer [type_enc = poly_tags, expect = some] (id_apply)
-sledgehammer [type_enc = poly_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_guards??, expect = some] (id_apply)
sledgehammer [type_enc = poly_guards, expect = some] (id_apply)
-sledgehammer [type_enc = mono_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_tags??, expect = some] (id_apply)
sledgehammer [type_enc = mono_tags, expect = some] (id_apply)
-sledgehammer [type_enc = mono_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_guards??, expect = some] (id_apply)
sledgehammer [type_enc = mono_guards, expect = some] (id_apply)
by (metis_exhaust id_apply)
lemma "id (a \<longrightarrow> b) \<longleftrightarrow> id (\<not> a \<or> b)"
sledgehammer [type_enc = erased, expect = some] (id_apply)
-sledgehammer [type_enc = poly_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_tags??, expect = some] (id_apply)
sledgehammer [type_enc = poly_tags, expect = some] (id_apply)
-sledgehammer [type_enc = poly_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = poly_guards??, expect = some] (id_apply)
sledgehammer [type_enc = poly_guards, expect = some] (id_apply)
-sledgehammer [type_enc = mono_tags?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_tags??, expect = some] (id_apply)
sledgehammer [type_enc = mono_tags, expect = some] (id_apply)
-sledgehammer [type_enc = mono_guards?, expect = some] (id_apply)
+sledgehammer [type_enc = mono_guards??, expect = some] (id_apply)
sledgehammer [type_enc = mono_guards, expect = some] (id_apply)
by (metis_exhaust id_apply)
--- a/src/HOL/Metis_Examples/Type_Encodings.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Metis_Examples/Type_Encodings.thy Fri Sep 09 06:47:14 2011 +0200
@@ -25,44 +25,46 @@
val type_encs =
["erased",
"poly_guards",
- "poly_guards_uniform",
+ "poly_guards?",
+ "poly_guards??",
+ "poly_guards@?",
+ "poly_guards!",
+ "poly_guards!!",
+ "poly_guards@!",
"poly_tags",
- "poly_tags_uniform",
+ "poly_tags?",
+ "poly_tags??",
+ "poly_tags@?",
+ "poly_tags!",
+ "poly_tags!!",
+ "poly_tags@!",
"poly_args",
"raw_mono_guards",
- "raw_mono_guards_uniform",
+ "raw_mono_guards?",
+ "raw_mono_guards??",
+ "raw_mono_guards@?",
+ "raw_mono_guards!",
+ "raw_mono_guards!!",
+ "raw_mono_guards@!",
"raw_mono_tags",
- "raw_mono_tags_uniform",
+ "raw_mono_tags?",
+ "raw_mono_tags??",
+ "raw_mono_tags@?",
+ "raw_mono_tags!",
+ "raw_mono_tags!!",
+ "raw_mono_tags@!",
"raw_mono_args",
"mono_guards",
- "mono_guards_uniform",
+ "mono_guards?",
+ "mono_guards??",
+ "mono_guards!",
+ "mono_guards!!",
"mono_tags",
- "mono_tags_uniform",
- "mono_args",
- "poly_guards?",
- "poly_guards_uniform?",
- "poly_tags?",
- "poly_tags_uniform?",
- "raw_mono_guards?",
- "raw_mono_guards_uniform?",
- "raw_mono_tags?",
- "raw_mono_tags_uniform?",
- "mono_guards?",
- "mono_guards_uniform?",
"mono_tags?",
- "mono_tags_uniform?",
- "poly_guards!",
- "poly_guards_uniform!",
- "poly_tags!",
- "poly_tags_uniform!",
- "raw_mono_guards!",
- "raw_mono_guards_uniform!",
- "raw_mono_tags!",
- "raw_mono_tags_uniform!",
- "mono_guards!",
- "mono_guards_uniform!",
+ "mono_tags??",
"mono_tags!",
- "mono_tags_uniform!"]
+ "mono_tags!!",
+ "mono_args"]
fun metis_exhaust_tac ctxt ths =
let
--- a/src/HOL/Mirabelle/Tools/mirabelle_sledgehammer.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Mirabelle/Tools/mirabelle_sledgehammer.ML Fri Sep 09 06:47:14 2011 +0200
@@ -574,9 +574,10 @@
relevance_override
in
if !reconstructor = "sledgehammer_tac" then
- sledge_tac 0.25 ATP_Systems.z3_tptpN "mono_simple"
- ORELSE' sledge_tac 0.25 ATP_Systems.eN "mono_guards?"
- ORELSE' sledge_tac 0.25 ATP_Systems.spassN "poly_tags_uniform"
+ sledge_tac 0.2 ATP_Systems.z3_tptpN "mono_simple"
+ ORELSE' sledge_tac 0.2 ATP_Systems.eN "mono_guards??"
+ ORELSE' sledge_tac 0.2 ATP_Systems.vampireN "mono_guards??"
+ ORELSE' sledge_tac 0.2 ATP_Systems.spassN "poly_tags"
ORELSE' Metis_Tactic.metis_tac [] ctxt thms
else if !reconstructor = "smt" then
SMT_Solver.smt_tac ctxt thms
--- a/src/HOL/Multivariate_Analysis/Brouwer_Fixpoint.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Multivariate_Analysis/Brouwer_Fixpoint.thy Fri Sep 09 06:47:14 2011 +0200
@@ -93,7 +93,7 @@
have even_minus_odd:"\<And>x y. even x \<Longrightarrow> odd (y::int) \<Longrightarrow> odd (x - y)" using assms by auto
have odd_minus_even:"\<And>x y. odd x \<Longrightarrow> even (y::int) \<Longrightarrow> odd (x - y)" using assms by auto
show ?thesis unfolding even_nat_def unfolding card_eq_setsum and lem4[THEN sym] and *[unfolded card_eq_setsum]
- unfolding card_eq_setsum[THEN sym] apply (rule odd_minus_even) unfolding zadd_int[THEN sym] apply(rule odd_plus_even)
+ unfolding card_eq_setsum[THEN sym] apply (rule odd_minus_even) unfolding of_nat_add apply(rule odd_plus_even)
apply(rule assms(7)[unfolded even_nat_def]) unfolding int_mult by auto qed
subsection {* The odd/even result for faces of complete vertices, generalized. *}
--- a/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy Fri Sep 09 06:47:14 2011 +0200
@@ -4590,7 +4590,7 @@
hence "b : affine hull S1" using rel_interior_subset hull_subset[of S2] ** by auto
hence "b : S1" using T_def b_def by auto
hence False using b_def assms unfolding rel_frontier_def by auto
-} ultimately show ?thesis using zless_le by auto
+} ultimately show ?thesis using less_le by auto
qed
--- a/src/HOL/Multivariate_Analysis/Linear_Algebra.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Multivariate_Analysis/Linear_Algebra.thy Fri Sep 09 06:47:14 2011 +0200
@@ -415,16 +415,6 @@
from power2_le_imp_le[OF th yz] show ?thesis .
qed
-text {* TODO: move to NthRoot *}
-lemma sqrt_add_le_add_sqrt:
- assumes x: "0 \<le> x" and y: "0 \<le> y"
- shows "sqrt (x + y) \<le> sqrt x + sqrt y"
-apply (rule power2_le_imp_le)
-apply (simp add: real_sum_squared_expand x y)
-apply (simp add: mult_nonneg_nonneg x y)
-apply (simp add: x y)
-done
-
subsection {* A generic notion of "hull" (convex, affine, conic hull and closure). *}
definition hull :: "('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "hull" 75) where
--- a/src/HOL/Mutabelle/mutabelle_extra.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Mutabelle/mutabelle_extra.ML Fri Sep 09 06:47:14 2011 +0200
@@ -286,8 +286,8 @@
(@{const_name "Groups.minus_class.minus"}, @{typ "prop => prop => prop"}),
(@{const_name "Groups.times_class.times"}, @{typ "prop => prop => prop"}),
(@{const_name "Fields.inverse_class.divide"}, @{typ "prop => prop => prop"}),
- (@{const_name "Lattices.semilattice_inf_class.inf"}, @{typ "prop => prop => prop"}),
- (@{const_name "Lattices.semilattice_sup_class.sup"}, @{typ "prop => prop => prop"}),
+ (@{const_name "Lattices.inf_class.inf"}, @{typ "prop => prop => prop"}),
+ (@{const_name "Lattices.sup_class.sup"}, @{typ "prop => prop => prop"}),
(@{const_name "Orderings.bot_class.bot"}, @{typ "prop => prop => prop"}),
(@{const_name "Orderings.ord_class.min"}, @{typ "prop => prop => prop"}),
(@{const_name "Orderings.ord_class.max"}, @{typ "prop => prop => prop"}),
--- a/src/HOL/NSA/NSComplex.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/NSA/NSComplex.thy Fri Sep 09 06:47:14 2011 +0200
@@ -163,8 +163,8 @@
apply (simp add: minus_equation_iff [of x y])
done
-lemma hcomplex_i_mult_eq [simp]: "iii * iii = - 1"
-by transfer (rule i_mult_eq2)
+lemma hcomplex_i_mult_eq [simp]: "iii * iii = -1"
+by transfer (rule i_squared)
lemma hcomplex_i_mult_left [simp]: "!!z. iii * (iii * z) = -z"
by transfer (rule complex_i_mult_minus)
@@ -434,12 +434,6 @@
lemma hIm_hsgn [simp]: "!!z. hIm(hsgn z) = hIm(z)/hcmod z"
by transfer (rule Im_sgn)
-lemma hcomplex_inverse_complex_split:
- "!!x y. inverse(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y) =
- hcomplex_of_hypreal(x/(x ^ 2 + y ^ 2)) -
- iii * hcomplex_of_hypreal(y/(x ^ 2 + y ^ 2))"
-by transfer (rule complex_inverse_complex_split)
-
lemma HComplex_inverse:
"!!x y. inverse (HComplex x y) =
HComplex (x/(x ^ 2 + y ^ 2)) (-y/(x ^ 2 + y ^ 2))"
@@ -463,14 +457,6 @@
(* harg *)
(*---------------------------------------------------------------------------*)
-lemma cos_harg_i_mult_zero_pos:
- "!!y. 0 < y ==> ( *f* cos) (harg(HComplex 0 y)) = 0"
-by transfer (rule cos_arg_i_mult_zero_pos)
-
-lemma cos_harg_i_mult_zero_neg:
- "!!y. y < 0 ==> ( *f* cos) (harg(HComplex 0 y)) = 0"
-by transfer (rule cos_arg_i_mult_zero_neg)
-
lemma cos_harg_i_mult_zero [simp]:
"!!y. y \<noteq> 0 ==> ( *f* cos) (harg(HComplex 0 y)) = 0"
by transfer (rule cos_arg_i_mult_zero)
@@ -561,8 +547,7 @@
"!!a. hcis (hypreal_of_nat (Suc n) * a) =
hcis a * hcis (hypreal_of_nat n * a)"
apply transfer
-apply (fold real_of_nat_def)
-apply (rule cis_real_of_nat_Suc_mult)
+apply (simp add: left_distrib cis_mult)
done
lemma NSDeMoivre: "!!a. (hcis a) ^ n = hcis (hypreal_of_nat n * a)"
@@ -574,7 +559,7 @@
lemma hcis_hypreal_of_hypnat_Suc_mult:
"!! a n. hcis (hypreal_of_hypnat (n + 1) * a) =
hcis a * hcis (hypreal_of_hypnat n * a)"
-by transfer (fold real_of_nat_def, simp add: cis_real_of_nat_Suc_mult)
+by transfer (simp add: left_distrib cis_mult)
lemma NSDeMoivre_ext:
"!!a n. (hcis a) pow n = hcis (hypreal_of_hypnat n * a)"
--- a/src/HOL/Nat.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Nat.thy Fri Sep 09 06:47:14 2011 +0200
@@ -657,46 +657,6 @@
by (cases m) simp_all
-subsubsection {* @{term min} and @{term max} *}
-
-lemma mono_Suc: "mono Suc"
-by (rule monoI) simp
-
-lemma min_0L [simp]: "min 0 n = (0::nat)"
-by (rule min_leastL) simp
-
-lemma min_0R [simp]: "min n 0 = (0::nat)"
-by (rule min_leastR) simp
-
-lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
-by (simp add: mono_Suc min_of_mono)
-
-lemma min_Suc1:
- "min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc(min n m'))"
-by (simp split: nat.split)
-
-lemma min_Suc2:
- "min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc(min m' n))"
-by (simp split: nat.split)
-
-lemma max_0L [simp]: "max 0 n = (n::nat)"
-by (rule max_leastL) simp
-
-lemma max_0R [simp]: "max n 0 = (n::nat)"
-by (rule max_leastR) simp
-
-lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
-by (simp add: mono_Suc max_of_mono)
-
-lemma max_Suc1:
- "max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc(max n m'))"
-by (simp split: nat.split)
-
-lemma max_Suc2:
- "max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc(max m' n))"
-by (simp split: nat.split)
-
-
subsubsection {* Monotonicity of Addition *}
lemma Suc_pred [simp]: "n>0 ==> Suc (n - Suc 0) = n"
@@ -753,11 +713,75 @@
fix a::nat and b::nat show "a ~= 0 \<Longrightarrow> b ~= 0 \<Longrightarrow> a * b ~= 0" by auto
qed
-lemma nat_mult_1: "(1::nat) * n = n"
-by simp
+
+subsubsection {* @{term min} and @{term max} *}
+
+lemma mono_Suc: "mono Suc"
+by (rule monoI) simp
+
+lemma min_0L [simp]: "min 0 n = (0::nat)"
+by (rule min_leastL) simp
+
+lemma min_0R [simp]: "min n 0 = (0::nat)"
+by (rule min_leastR) simp
+
+lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
+by (simp add: mono_Suc min_of_mono)
+
+lemma min_Suc1:
+ "min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc(min n m'))"
+by (simp split: nat.split)
+
+lemma min_Suc2:
+ "min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc(min m' n))"
+by (simp split: nat.split)
+
+lemma max_0L [simp]: "max 0 n = (n::nat)"
+by (rule max_leastL) simp
+
+lemma max_0R [simp]: "max n 0 = (n::nat)"
+by (rule max_leastR) simp
+
+lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
+by (simp add: mono_Suc max_of_mono)
-lemma nat_mult_1_right: "n * (1::nat) = n"
-by simp
+lemma max_Suc1:
+ "max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc(max n m'))"
+by (simp split: nat.split)
+
+lemma max_Suc2:
+ "max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc(max m' n))"
+by (simp split: nat.split)
+
+lemma nat_mult_min_left:
+ fixes m n q :: nat
+ shows "min m n * q = min (m * q) (n * q)"
+ by (simp add: min_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
+
+lemma nat_mult_min_right:
+ fixes m n q :: nat
+ shows "m * min n q = min (m * n) (m * q)"
+ by (simp add: min_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
+
+lemma nat_add_max_left:
+ fixes m n q :: nat
+ shows "max m n + q = max (m + q) (n + q)"
+ by (simp add: max_def)
+
+lemma nat_add_max_right:
+ fixes m n q :: nat
+ shows "m + max n q = max (m + n) (m + q)"
+ by (simp add: max_def)
+
+lemma nat_mult_max_left:
+ fixes m n q :: nat
+ shows "max m n * q = max (m * q) (n * q)"
+ by (simp add: max_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
+
+lemma nat_mult_max_right:
+ fixes m n q :: nat
+ shows "m * max n q = max (m * n) (m * q)"
+ by (simp add: max_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
subsubsection {* Additional theorems about @{term "op \<le>"} *}
@@ -1700,6 +1724,15 @@
by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
+subsection {* aliasses *}
+
+lemma nat_mult_1: "(1::nat) * n = n"
+ by simp
+
+lemma nat_mult_1_right: "n * (1::nat) = n"
+ by simp
+
+
subsection {* size of a datatype value *}
class size =
--- a/src/HOL/Nat_Numeral.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Nat_Numeral.thy Fri Sep 09 06:47:14 2011 +0200
@@ -364,7 +364,7 @@
lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
apply (rule sym)
-apply (simp add: nat_eq_iff int_Suc)
+apply (simp add: nat_eq_iff)
done
lemma Suc_nat_number_of_add:
--- a/src/HOL/Nominal/Nominal.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Nominal/Nominal.thy Fri Sep 09 06:47:14 2011 +0200
@@ -50,17 +50,17 @@
perm_nprod \<equiv> "perm :: 'x prm \<Rightarrow> ('a, 'b) nprod \<Rightarrow> ('a, 'b) nprod" (unchecked)
begin
-definition
- perm_fun_def: "perm_fun pi (f::'a\<Rightarrow>'b) = (\<lambda>x. pi\<bullet>f((rev pi)\<bullet>x))"
+definition perm_fun :: "'x prm \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
+ "perm_fun pi f = (\<lambda>x. pi \<bullet> f (rev pi \<bullet> x))"
definition perm_bool :: "'x prm \<Rightarrow> bool \<Rightarrow> bool" where
- perm_bool_def: "perm_bool pi b = b"
+ "perm_bool pi b = b"
primrec perm_unit :: "'x prm \<Rightarrow> unit \<Rightarrow> unit" where
"perm_unit pi () = ()"
primrec perm_prod :: "'x prm \<Rightarrow> ('a\<times>'b) \<Rightarrow> ('a\<times>'b)" where
- "perm_prod pi (x,y) = (pi\<bullet>x,pi\<bullet>y)"
+ "perm_prod pi (x, y) = (pi\<bullet>x, pi\<bullet>y)"
primrec perm_list :: "'x prm \<Rightarrow> 'a list \<Rightarrow> 'a list" where
nil_eqvt: "perm_list pi [] = []"
@@ -71,13 +71,13 @@
| none_eqvt: "perm_option pi None = None"
definition perm_char :: "'x prm \<Rightarrow> char \<Rightarrow> char" where
- perm_char_def: "perm_char pi c = c"
+ "perm_char pi c = c"
definition perm_nat :: "'x prm \<Rightarrow> nat \<Rightarrow> nat" where
- perm_nat_def: "perm_nat pi i = i"
+ "perm_nat pi i = i"
definition perm_int :: "'x prm \<Rightarrow> int \<Rightarrow> int" where
- perm_int_def: "perm_int pi i = i"
+ "perm_int pi i = i"
primrec perm_noption :: "'x prm \<Rightarrow> 'a noption \<Rightarrow> 'a noption" where
nsome_eqvt: "perm_noption pi (nSome x) = nSome (pi\<bullet>x)"
@@ -962,7 +962,7 @@
fixes pi::"'y prm"
and x ::"'x set"
assumes dj: "disjoint TYPE('x) TYPE('y)"
- shows "(pi\<bullet>x)=x"
+ shows "pi\<bullet>x=x"
using dj by (simp_all add: perm_fun_def disjoint_def perm_bool)
lemma dj_perm_perm_forget:
@@ -1028,7 +1028,7 @@
qed
lemma pt_unit_inst:
- shows "pt TYPE(unit) TYPE('x)"
+ shows "pt TYPE(unit) TYPE('x)"
by (simp add: pt_def)
lemma pt_prod_inst:
--- a/src/HOL/Nominal/nominal_permeq.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Nominal/nominal_permeq.ML Fri Sep 09 06:47:14 2011 +0200
@@ -130,7 +130,7 @@
case redex of
(* case pi o f == (%x. pi o (f ((rev pi)o x))) *)
(Const("Nominal.perm",_) $ pi $ f) =>
- (if (applicable_fun f) then SOME perm_fun_def else NONE)
+ (if applicable_fun f then SOME perm_fun_def else NONE)
| _ => NONE
end
--- a/src/HOL/Number_Theory/Primes.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Number_Theory/Primes.thy Fri Sep 09 06:47:14 2011 +0200
@@ -174,7 +174,7 @@
EX m k. n = m * k & 1 < m & m < n & 1 < k & k < n"
unfolding prime_int_altdef dvd_def
apply auto
- by(metis div_mult_self1_is_id div_mult_self2_is_id int_div_less_self int_one_le_iff_zero_less zero_less_mult_pos zless_le)
+ by(metis div_mult_self1_is_id div_mult_self2_is_id int_div_less_self int_one_le_iff_zero_less zero_less_mult_pos less_le)
lemma prime_dvd_power_nat [rule_format]: "prime (p::nat) -->
n > 0 --> (p dvd x^n --> p dvd x)"
@@ -220,7 +220,7 @@
"prime (p::int) \<longleftrightarrow> p > 1 \<and> (\<forall>n \<in> {1<..<p}. ~ n dvd p)" (is "?L = ?R")
proof
assume "?L" thus "?R"
- by (clarsimp simp: prime_gt_1_int) (metis int_one_le_iff_zero_less prime_int_altdef zless_le)
+ by (clarsimp simp: prime_gt_1_int) (metis int_one_le_iff_zero_less prime_int_altdef less_le)
next
assume "?R" thus "?L" by (clarsimp simp:Ball_def) (metis dvdI not_prime_eq_prod_int)
qed
@@ -272,7 +272,7 @@
lemma primes_coprime_int: "prime (p::int) \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q"
apply (rule prime_imp_coprime_int, assumption)
apply (unfold prime_int_altdef)
- apply (metis int_one_le_iff_zero_less zless_le)
+ apply (metis int_one_le_iff_zero_less less_le)
done
lemma primes_imp_powers_coprime_nat: "prime (p::nat) \<Longrightarrow> prime q \<Longrightarrow> p ~= q \<Longrightarrow> coprime (p^m) (q^n)"
--- a/src/HOL/Number_Theory/UniqueFactorization.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Number_Theory/UniqueFactorization.thy Fri Sep 09 06:47:14 2011 +0200
@@ -766,7 +766,7 @@
lemma multiplicity_dvd'_int: "(0::int) < x \<Longrightarrow> 0 <= y \<Longrightarrow>
\<forall>p. prime p \<longrightarrow> multiplicity p x \<le> multiplicity p y \<Longrightarrow> x dvd y"
by (metis eq_imp_le gcd_lcm_complete_lattice_nat.top_greatest int_eq_0_conv multiplicity_dvd_int
- multiplicity_nonprime_int nat_int transfer_nat_int_relations(4) zless_le)
+ multiplicity_nonprime_int nat_int transfer_nat_int_relations(4) less_le)
lemma dvd_multiplicity_eq_nat: "0 < (x::nat) \<Longrightarrow> 0 < y \<Longrightarrow>
(x dvd y) = (ALL p. multiplicity p x <= multiplicity p y)"
--- a/src/HOL/Old_Number_Theory/Chinese.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Old_Number_Theory/Chinese.thy Fri Sep 09 06:47:14 2011 +0200
@@ -241,7 +241,7 @@
"0 \<le> xilin_sol i n kf bf mf \<and> xilin_sol i n kf bf mf < mf i
\<and> [kf i * mhf mf n i * xilin_sol i n kf bf mf = bf i] (mod mf i)")
prefer 6
- apply (simp add: zmult_ac)
+ apply (simp add: mult_ac)
apply (unfold xilin_sol_def)
apply (tactic {* asm_simp_tac @{simpset} 6 *})
apply (rule_tac [6] ex1_implies_ex [THEN someI_ex])
--- a/src/HOL/Old_Number_Theory/Euler.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Old_Number_Theory/Euler.thy Fri Sep 09 06:47:14 2011 +0200
@@ -67,7 +67,7 @@
then have "[j * j = (a * MultInv p j) * j] (mod p)"
by (auto simp add: zcong_scalar)
then have a:"[j * j = a * (MultInv p j * j)] (mod p)"
- by (auto simp add: zmult_ac)
+ by (auto simp add: mult_ac)
have "[j * j = a] (mod p)"
proof -
from assms(1,2,4) have "[MultInv p j * j = 1] (mod p)"
@@ -148,7 +148,7 @@
c = "a * (x * MultInv p x)" in zcong_trans, force)
apply (frule_tac p = p and x = x in MultInv_prop2, auto)
apply (metis StandardRes_SRStar_prop3 mult_1_right mult_commute zcong_sym zcong_zmult_prop1)
- apply (auto simp add: zmult_ac)
+ apply (auto simp add: mult_ac)
done
lemma aux1: "[| 0 < x; (x::int) < a; x \<noteq> (a - 1) |] ==> x < a - 1"
@@ -237,7 +237,7 @@
then have "a ^ (nat p) = a ^ (1 + (nat p - 1))"
by (auto simp add: diff_add_assoc)
also have "... = (a ^ 1) * a ^ (nat(p) - 1)"
- by (simp only: zpower_zadd_distrib)
+ by (simp only: power_add)
also have "... = a * a ^ (nat(p) - 1)"
by auto
finally show ?thesis .
@@ -286,7 +286,7 @@
apply (auto simp add: zpower_zpower)
apply (rule zcong_trans)
apply (auto simp add: zcong_sym [of "x ^ nat ((p - 1) div 2)"])
- apply (metis Little_Fermat even_div_2_prop2 mult_Bit0 number_of_is_id odd_minus_one_even one_is_num_one zmult_1 aux__2)
+ apply (metis Little_Fermat even_div_2_prop2 mult_Bit0 number_of_is_id odd_minus_one_even one_is_num_one mult_1 aux__2)
done
--- a/src/HOL/Old_Number_Theory/EulerFermat.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Old_Number_Theory/EulerFermat.thy Fri Sep 09 06:47:14 2011 +0200
@@ -257,7 +257,7 @@
apply (subst setprod_insert)
apply (rule_tac [2] Bnor_prod_power_aux)
apply (unfold inj_on_def)
- apply (simp_all add: zmult_ac Bnor_fin Bnor_mem_zle_swap)
+ apply (simp_all add: mult_ac Bnor_fin Bnor_mem_zle_swap)
done
--- a/src/HOL/Old_Number_Theory/EvenOdd.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Old_Number_Theory/EvenOdd.thy Fri Sep 09 06:47:14 2011 +0200
@@ -39,7 +39,7 @@
then have "2 * (a::int) - 2 * (b :: int) = 1"
by arith
then have "2 * (a - b) = 1"
- by (auto simp add: zdiff_zmult_distrib)
+ by (auto simp add: left_diff_distrib)
moreover have "(2 * (a - b)):zEven"
by (auto simp only: zEven_def)
ultimately have False
@@ -66,7 +66,7 @@
then have "2 * a * y - 2 * b = 1"
by arith
then have "2 * (a * y - b) = 1"
- by (auto simp add: zdiff_zmult_distrib)
+ by (auto simp add: left_diff_distrib)
moreover have "(2 * (a * y - b)):zEven"
by (auto simp only: zEven_def)
ultimately have False
@@ -85,7 +85,7 @@
lemma even_plus_even: "[| x \<in> zEven; y \<in> zEven |] ==> x + y \<in> zEven"
apply (auto simp add: zEven_def)
- apply (auto simp only: zadd_zmult_distrib2 [symmetric])
+ apply (auto simp only: right_distrib [symmetric])
done
lemma even_times_either: "x \<in> zEven ==> x * y \<in> zEven"
@@ -93,12 +93,12 @@
lemma even_minus_even: "[| x \<in> zEven; y \<in> zEven |] ==> x - y \<in> zEven"
apply (auto simp add: zEven_def)
- apply (auto simp only: zdiff_zmult_distrib2 [symmetric])
+ apply (auto simp only: right_diff_distrib [symmetric])
done
lemma odd_minus_odd: "[| x \<in> zOdd; y \<in> zOdd |] ==> x - y \<in> zEven"
apply (auto simp add: zOdd_def zEven_def)
- apply (auto simp only: zdiff_zmult_distrib2 [symmetric])
+ apply (auto simp only: right_diff_distrib [symmetric])
done
lemma even_minus_odd: "[| x \<in> zEven; y \<in> zOdd |] ==> x - y \<in> zOdd"
@@ -109,13 +109,13 @@
lemma odd_minus_even: "[| x \<in> zOdd; y \<in> zEven |] ==> x - y \<in> zOdd"
apply (auto simp add: zOdd_def zEven_def)
- apply (auto simp only: zdiff_zmult_distrib2 [symmetric])
+ apply (auto simp only: right_diff_distrib [symmetric])
done
lemma odd_times_odd: "[| x \<in> zOdd; y \<in> zOdd |] ==> x * y \<in> zOdd"
- apply (auto simp add: zOdd_def zadd_zmult_distrib zadd_zmult_distrib2)
+ apply (auto simp add: zOdd_def left_distrib right_distrib)
apply (rule_tac x = "2 * ka * k + ka + k" in exI)
- apply (auto simp add: zadd_zmult_distrib)
+ apply (auto simp add: left_distrib)
done
lemma odd_iff_not_even: "(x \<in> zOdd) = (~ (x \<in> zEven))"
@@ -195,7 +195,7 @@
finally have "(-1::int)^nat x = (-1)^(2 * nat a + 1)"
by simp
also have "... = ((-1::int)^2)^ (nat a) * (-1)^1"
- by (auto simp add: zpower_zpower [symmetric] zpower_zadd_distrib)
+ by (auto simp add: power_mult power_add)
also have "(-1::int)^2 = 1"
by simp
finally show ?thesis
--- a/src/HOL/Old_Number_Theory/Fib.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Old_Number_Theory/Fib.thy Fri Sep 09 06:47:14 2011 +0200
@@ -87,7 +87,7 @@
else fib (Suc n) * fib (Suc n) + 1)"
apply (rule int_int_eq [THEN iffD1])
apply (simp add: fib_Cassini_int)
- apply (subst zdiff_int [symmetric])
+ apply (subst of_nat_diff)
apply (insert fib_Suc_gr_0 [of n], simp_all)
done
--- a/src/HOL/Old_Number_Theory/Finite2.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Old_Number_Theory/Finite2.thy Fri Sep 09 06:47:14 2011 +0200
@@ -38,18 +38,18 @@
lemma setsum_const: "finite X ==> setsum (%x. (c :: int)) X = c * int(card X)"
apply (induct set: finite)
- apply (auto simp add: left_distrib right_distrib int_eq_of_nat)
+ apply (auto simp add: left_distrib right_distrib)
done
lemma setsum_const2: "finite X ==> int (setsum (%x. (c :: nat)) X) =
int(c) * int(card X)"
apply (induct set: finite)
- apply (auto simp add: zadd_zmult_distrib2)
+ apply (auto simp add: right_distrib)
done
lemma setsum_const_mult: "finite A ==> setsum (%x. c * ((f x)::int)) A =
c * setsum f A"
- by (induct set: finite) (auto simp add: zadd_zmult_distrib2)
+ by (induct set: finite) (auto simp add: right_distrib)
subsection {* Cardinality of explicit finite sets *}
--- a/src/HOL/Old_Number_Theory/Gauss.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Old_Number_Theory/Gauss.thy Fri Sep 09 06:47:14 2011 +0200
@@ -344,7 +344,7 @@
apply (elim zcong_trans)
by (simp only: zcong_refl)
also have "y * a + ya * a = a * (y + ya)"
- by (simp add: zadd_zmult_distrib2 zmult_commute)
+ by (simp add: right_distrib mult_commute)
finally have "[a * (y + ya) = 0] (mod p)" .
with p_prime a_nonzero zcong_zprime_prod_zero [of p a "y + ya"]
p_a_relprime
@@ -429,7 +429,7 @@
also have "setprod uminus E = (setprod id E) * (-1)^(card E)"
using finite_E by (induct set: finite) auto
then have "setprod uminus E = (-1) ^ (card E) * (setprod id E)"
- by (simp add: zmult_commute)
+ by (simp add: mult_commute)
with two show ?thesis
by simp
qed
@@ -444,7 +444,7 @@
"[a ^ nat((p - 1) div 2) = (-1) ^ (card E)] (mod p)"
proof -
have "[setprod id A = setprod id F * setprod id D](mod p)"
- by (auto simp add: prod_D_F_eq_prod_A zmult_commute cong del:setprod_cong)
+ by (auto simp add: prod_D_F_eq_prod_A mult_commute cong del:setprod_cong)
then have "[setprod id A = ((-1)^(card E) * setprod id E) *
setprod id D] (mod p)"
apply (rule zcong_trans)
@@ -453,7 +453,7 @@
then have "[setprod id A = ((-1)^(card E) * setprod id C)] (mod p)"
apply (rule zcong_trans)
apply (insert C_prod_eq_D_times_E, erule subst)
- apply (subst zmult_assoc, auto)
+ apply (subst mult_assoc, auto)
done
then have "[setprod id A = ((-1)^(card E) * setprod id B)] (mod p)"
apply (rule zcong_trans)
@@ -474,7 +474,7 @@
then have "[setprod id A = ((-1)^(card E) * a^(card A) *
setprod id A)](mod p)"
apply (rule zcong_trans)
- apply (simp add: zcong_scalar2 zcong_scalar finite_A setprod_constant zmult_assoc)
+ apply (simp add: zcong_scalar2 zcong_scalar finite_A setprod_constant mult_assoc)
done
then have a: "[setprod id A * (-1)^(card E) =
((-1)^(card E) * a^(card A) * setprod id A * (-1)^(card E))](mod p)"
--- a/src/HOL/Old_Number_Theory/Int2.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Old_Number_Theory/Int2.thy Fri Sep 09 06:47:14 2011 +0200
@@ -51,7 +51,7 @@
have "(x div z) * z \<le> (x div z) * z" by simp
then have "(x div z) * z \<le> (x div z) * z + x mod z" using modth by arith
also have "\<dots> = x"
- by (auto simp add: zmod_zdiv_equality [symmetric] zmult_ac)
+ by (auto simp add: zmod_zdiv_equality [symmetric] mult_ac)
also note `x < y * z`
finally show ?thesis
apply (auto simp add: mult_less_cancel_right)
@@ -115,7 +115,7 @@
lemma zcong_zmult_prop2: "[a = b](mod m) ==>
([c = d * a](mod m) = [c = d * b] (mod m))"
- by (auto simp add: zmult_ac zcong_zmult_prop1)
+ by (auto simp add: mult_ac zcong_zmult_prop1)
lemma zcong_zmult_prop3: "[| zprime p; ~[x = 0] (mod p);
~[y = 0] (mod p) |] ==> ~[x * y = 0] (mod p)"
@@ -125,7 +125,7 @@
lemma zcong_less_eq: "[| 0 < x; 0 < y; 0 < m; [x = y] (mod m);
x < m; y < m |] ==> x = y"
- by (metis zcong_not zcong_sym zless_linear)
+ by (metis zcong_not zcong_sym less_linear)
lemma zcong_neg_1_impl_ne_1:
assumes "2 < p" and "[x = -1] (mod p)"
@@ -198,7 +198,7 @@
lemma MultInv_prop2a: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==>
[(MultInv p x) * x = 1] (mod p)"
- by (auto simp add: MultInv_prop2 zmult_ac)
+ by (auto simp add: MultInv_prop2 mult_ac)
lemma aux_1: "2 < p ==> ((nat p) - 2) = (nat (p - 2))"
by (simp add: nat_diff_distrib)
@@ -227,7 +227,7 @@
apply (insert MultInv_prop2 [of p "MultInv p x"], auto)
apply (drule MultInv_prop2, auto)
apply (drule_tac k = x in zcong_scalar2, auto)
- apply (auto simp add: zmult_ac)
+ apply (auto simp add: mult_ac)
done
lemma MultInv_prop4: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==>
@@ -244,10 +244,10 @@
m = p and k = x in zcong_scalar)
apply (insert MultInv_prop2 [of p x], simp)
apply (auto simp only: zcong_sym [of "MultInv p x * x"])
- apply (auto simp add: zmult_ac)
+ apply (auto simp add: mult_ac)
apply (drule zcong_trans, auto)
apply (drule_tac a = "x * MultInv p y" and k = y in zcong_scalar, auto)
- apply (insert MultInv_prop2a [of p y], auto simp add: zmult_ac)
+ apply (insert MultInv_prop2a [of p y], auto simp add: mult_ac)
apply (insert zcong_zmult_prop2 [of "y * MultInv p y" 1 p y x])
apply (auto simp add: zcong_sym)
done
@@ -264,19 +264,19 @@
[j * k = a * MultInv p k * k] (mod p) |] ==> [j * k = a] (mod p)"
apply (insert MultInv_prop2a [of p k] zcong_zmult_prop2
[of "MultInv p k * k" 1 p "j * k" a])
- apply (auto simp add: zmult_ac)
+ apply (auto simp add: mult_ac)
done
lemma aux___3: "[j * k = a] (mod p) ==> [(MultInv p j) * j * k =
(MultInv p j) * a] (mod p)"
- by (auto simp add: zmult_assoc zcong_scalar2)
+ by (auto simp add: mult_assoc zcong_scalar2)
lemma aux___4: "[|2 < p; zprime p; ~([j = 0](mod p));
[(MultInv p j) * j * k = (MultInv p j) * a] (mod p) |]
==> [k = a * (MultInv p j)] (mod p)"
apply (insert MultInv_prop2a [of p j] zcong_zmult_prop1
[of "MultInv p j * j" 1 p "MultInv p j * a" k])
- apply (auto simp add: zmult_ac zcong_sym)
+ apply (auto simp add: mult_ac zcong_sym)
done
lemma MultInv_zcong_prop2: "[| 2 < p; zprime p; ~([k = 0](mod p));
--- a/src/HOL/Old_Number_Theory/IntPrimes.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Old_Number_Theory/IntPrimes.thy Fri Sep 09 06:47:14 2011 +0200
@@ -43,7 +43,7 @@
lemma zrelprime_zdvd_zmult_aux:
"zgcd n k = 1 ==> k dvd m * n ==> 0 \<le> m ==> k dvd m"
- by (metis abs_of_nonneg dvd_triv_right zgcd_greatest_iff zgcd_zmult_distrib2_abs zmult_1_right)
+ by (metis abs_of_nonneg dvd_triv_right zgcd_greatest_iff zgcd_zmult_distrib2_abs mult_1_right)
lemma zrelprime_zdvd_zmult: "zgcd n k = 1 ==> k dvd m * n ==> k dvd m"
apply (case_tac "0 \<le> m")
@@ -93,7 +93,7 @@
apply (simp add: zgcd_greatest_iff)
apply (rule_tac n = k in zrelprime_zdvd_zmult)
prefer 2
- apply (simp add: zmult_commute)
+ apply (simp add: mult_commute)
apply (metis zgcd_1 zgcd_commute zgcd_left_commute)
done
@@ -142,8 +142,8 @@
"[a = b] (mod m) ==> [c = d] (mod m) ==> [a * c = b * d] (mod m)"
apply (rule_tac b = "b * c" in zcong_trans)
apply (unfold zcong_def)
- apply (metis zdiff_zmult_distrib2 dvd_mult zmult_commute)
- apply (metis zdiff_zmult_distrib2 dvd_mult)
+ apply (metis right_diff_distrib dvd_mult mult_commute)
+ apply (metis right_diff_distrib dvd_mult)
done
lemma zcong_scalar: "[a = b] (mod m) ==> [a * k = b * k] (mod m)"
@@ -165,7 +165,7 @@
apply (rule_tac [3] s = "a * a - 1 + p * (1 - a)" in subst)
prefer 4
apply (simp add: zdvd_reduce)
- apply (simp_all add: zdiff_zmult_distrib zmult_commute zdiff_zmult_distrib2)
+ apply (simp_all add: left_diff_distrib mult_commute right_diff_distrib)
done
lemma zcong_cancel:
@@ -179,7 +179,7 @@
apply (subst zcong_sym)
apply (unfold zcong_def)
apply (rule_tac [!] zrelprime_zdvd_zmult)
- apply (simp_all add: zdiff_zmult_distrib)
+ apply (simp_all add: left_diff_distrib)
apply (subgoal_tac "m dvd (-(a * k - b * k))")
apply simp
apply (subst dvd_minus_iff, assumption)
@@ -188,7 +188,7 @@
lemma zcong_cancel2:
"0 \<le> m ==>
zgcd k m = 1 ==> [k * a = k * b] (mod m) = [a = b] (mod m)"
- by (simp add: zmult_commute zcong_cancel)
+ by (simp add: mult_commute zcong_cancel)
lemma zcong_zgcd_zmult_zmod:
"[a = b] (mod m) ==> [a = b] (mod n) ==> zgcd m n = 1
@@ -197,9 +197,9 @@
apply (subgoal_tac "m dvd n * ka")
apply (subgoal_tac "m dvd ka")
apply (case_tac [2] "0 \<le> ka")
- apply (metis zdvd_mult_div_cancel dvd_refl dvd_mult_left zmult_commute zrelprime_zdvd_zmult)
- apply (metis abs_dvd_iff abs_of_nonneg zadd_0 zgcd_0_left zgcd_commute zgcd_zadd_zmult zgcd_zdvd_zgcd_zmult zgcd_zmult_distrib2_abs zmult_1_right zmult_commute)
- apply (metis mult_le_0_iff zdvd_mono zdvd_mult_cancel dvd_triv_left zero_le_mult_iff zle_antisym zle_linear zle_refl zmult_commute zrelprime_zdvd_zmult)
+ apply (metis zdvd_mult_div_cancel dvd_refl dvd_mult_left mult_commute zrelprime_zdvd_zmult)
+ apply (metis abs_dvd_iff abs_of_nonneg add_0 zgcd_0_left zgcd_commute zgcd_zadd_zmult zgcd_zdvd_zgcd_zmult zgcd_zmult_distrib2_abs mult_1_right mult_commute)
+ apply (metis mult_le_0_iff zdvd_mono zdvd_mult_cancel dvd_triv_left zero_le_mult_iff order_antisym linorder_linear order_refl mult_commute zrelprime_zdvd_zmult)
apply (metis dvd_triv_left)
done
@@ -208,7 +208,7 @@
a < m ==> 0 \<le> b ==> b < m ==> [a = b] (mod m) ==> a = b"
apply (unfold zcong_def dvd_def, auto)
apply (drule_tac f = "\<lambda>z. z mod m" in arg_cong)
- apply (metis diff_add_cancel mod_pos_pos_trivial zadd_0 zadd_commute zmod_eq_0_iff mod_add_right_eq)
+ apply (metis diff_add_cancel mod_pos_pos_trivial add_0 add_commute zmod_eq_0_iff mod_add_right_eq)
done
lemma zcong_square_zless:
@@ -253,7 +253,7 @@
lemma zcong_zmod_aux:
"a - b = (m::int) * (a div m - b div m) + (a mod m - b mod m)"
- by(simp add: zdiff_zmult_distrib2 add_diff_eq eq_diff_eq add_ac)
+ by(simp add: right_diff_distrib add_diff_eq eq_diff_eq add_ac)
lemma zcong_zmod: "[a = b] (mod m) = [a mod m = b mod m] (mod m)"
apply (unfold zcong_def)
@@ -261,7 +261,7 @@
apply (rule_tac m = m in zcong_zmod_aux)
apply (rule trans)
apply (rule_tac [2] k = m and m = "a div m - b div m" in zdvd_reduce)
- apply (simp add: zadd_commute)
+ apply (simp add: add_commute)
done
lemma zcong_zmod_eq: "0 < m ==> [a = b] (mod m) = (a mod m = b mod m)"
@@ -282,7 +282,7 @@
apply (erule disjE)
prefer 2 apply (simp add: zcong_zmod_eq)
txt{*Remainding case: @{term "m<0"}*}
- apply (rule_tac t = m in zminus_zminus [THEN subst])
+ apply (rule_tac t = m in minus_minus [THEN subst])
apply (subst zcong_zminus)
apply (subst zcong_zmod_eq, arith)
apply (frule neg_mod_bound [of _ a], frule neg_mod_bound [of _ b])
@@ -324,7 +324,7 @@
apply (case_tac "r' mod r = 0")
prefer 2
apply (frule_tac a = "r'" in pos_mod_sign, auto)
- apply (metis Pair_eq xzgcda.simps zle_refl)
+ apply (metis Pair_eq xzgcda.simps order_refl)
done
lemma xzgcd_correct:
@@ -341,7 +341,7 @@
lemma xzgcda_linear_aux1:
"(a - r * b) * m + (c - r * d) * (n::int) =
(a * m + c * n) - r * (b * m + d * n)"
- by (simp add: zdiff_zmult_distrib zadd_zmult_distrib2 zmult_assoc)
+ by (simp add: left_diff_distrib right_distrib mult_assoc)
lemma xzgcda_linear_aux2:
"r' = s' * m + t' * n ==> r = s * m + t * n
@@ -391,7 +391,7 @@
prefer 2
apply simp
apply (unfold zcong_def)
- apply (simp (no_asm) add: zmult_commute)
+ apply (simp (no_asm) add: mult_commute)
done
lemma zcong_lineq_unique:
@@ -407,7 +407,7 @@
apply (cut_tac a = a and n = n in zcong_lineq_ex, auto)
apply (rule_tac x = "x * b mod n" in exI, safe)
apply (simp_all (no_asm_simp))
- apply (metis zcong_scalar zcong_zmod zmod_zmult1_eq zmult_1 zmult_assoc)
+ apply (metis zcong_scalar zcong_zmod zmod_zmult1_eq mult_1 mult_assoc)
done
end
--- a/src/HOL/Old_Number_Theory/Legacy_GCD.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Old_Number_Theory/Legacy_GCD.thy Fri Sep 09 06:47:14 2011 +0200
@@ -699,7 +699,7 @@
lemma zgcd_zmult_distrib2: "0 \<le> k ==> k * zgcd m n = zgcd (k * m) (k * n)"
by (simp del: minus_mult_right [symmetric]
add: minus_mult_right nat_mult_distrib zgcd_def abs_if
- mult_less_0_iff gcd_mult_distrib2 [symmetric] zmult_int [symmetric])
+ mult_less_0_iff gcd_mult_distrib2 [symmetric] of_nat_mult)
lemma zgcd_zmult_distrib2_abs: "zgcd (k * m) (k * n) = abs k * zgcd m n"
by (simp add: abs_if zgcd_zmult_distrib2)
--- a/src/HOL/Old_Number_Theory/Quadratic_Reciprocity.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Old_Number_Theory/Quadratic_Reciprocity.thy Fri Sep 09 06:47:14 2011 +0200
@@ -22,7 +22,7 @@
from finite_A have "a * setsum id A = setsum (%x. a * x) A"
by (auto simp add: setsum_const_mult id_def)
also have "setsum (%x. a * x) = setsum (%x. x * a)"
- by (auto simp add: zmult_commute)
+ by (auto simp add: mult_commute)
also have "setsum (%x. x * a) A = setsum id B"
by (simp add: B_def setsum_reindex_id[OF inj_on_xa_A])
also have "... = setsum (%x. p * (x div p) + StandardRes p x) B"
@@ -71,7 +71,7 @@
p * (setsum (%x. ((x * a) div p)) A - int(card E)) + 2 * setsum id E"
proof -
have "(a - 1) * setsum id A = a * setsum id A - setsum id A"
- by (auto simp add: zdiff_zmult_distrib)
+ by (auto simp add: left_diff_distrib)
also note QRLemma1
also from QRLemma2 have "p * (\<Sum>x \<in> A. x * a div p) + setsum id D +
setsum id E - setsum id A =
@@ -82,7 +82,7 @@
p * int (card E) + 2 * setsum id E"
by arith
finally show ?thesis
- by (auto simp only: zdiff_zmult_distrib2)
+ by (auto simp only: right_diff_distrib)
qed
lemma QRLemma4: "a \<in> zOdd ==>
@@ -284,7 +284,7 @@
proof -
fix a and b
assume "~ q * a < p * b" and b1: "0 < b" and b2: "b \<le> (q - 1) div 2"
- with zless_linear have "(p * b < q * a) | (p * b = q * a)" by auto
+ with less_linear have "(p * b < q * a) | (p * b = q * a)" by auto
moreover from pb_neq_qa b1 b2 have "(p * b \<noteq> q * a)" by auto
ultimately show "p * b < q * a" by auto
qed
@@ -388,7 +388,7 @@
by (auto simp add: int_distrib)
then have "((p - 1) * q) div 2 < (((q - 1) * p) + (2 * p)) div 2"
apply (rule_tac x = "((p - 1) * q)" in even_div_2_l)
- by (auto simp add: even3, auto simp add: zmult_ac)
+ by (auto simp add: even3, auto simp add: mult_ac)
also have "((p - 1) * q) div 2 = q * ((p - 1) div 2)"
by (auto simp add: even1 even_prod_div_2)
also have "(((q - 1) * p) + (2 * p)) div 2 = (((q - 1) div 2) * p) + p"
@@ -557,11 +557,11 @@
lemma S1_carda: "int (card(S1)) =
setsum (%j. (j * q) div p) P_set"
- by (auto simp add: S1_card zmult_ac)
+ by (auto simp add: S1_card mult_ac)
lemma S2_carda: "int (card(S2)) =
setsum (%j. (j * p) div q) Q_set"
- by (auto simp add: S2_card zmult_ac)
+ by (auto simp add: S2_card mult_ac)
lemma pq_sum_prop: "(setsum (%j. (j * p) div q) Q_set) +
(setsum (%j. (j * q) div p) P_set) = ((p - 1) div 2) * ((q - 1) div 2)"
@@ -610,7 +610,7 @@
by auto
also have "... = (-1::int) ^ (nat(setsum (%x. ((x * p) div q)) Q_set) +
nat(setsum (%x. ((x * q) div p)) P_set))"
- by (auto simp add: zpower_zadd_distrib)
+ by (auto simp add: power_add)
also have "nat(setsum (%x. ((x * p) div q)) Q_set) +
nat(setsum (%x. ((x * q) div p)) P_set) =
nat((setsum (%x. ((x * p) div q)) Q_set) +
--- a/src/HOL/Old_Number_Theory/WilsonBij.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Old_Number_Theory/WilsonBij.thy Fri Sep 09 06:47:14 2011 +0200
@@ -75,7 +75,7 @@
lemma aux: "[a * (p - 1) = 1] (mod p) = [a = p - 1] (mod p)"
-- {* same as @{text WilsonRuss} *}
apply (unfold zcong_def)
- apply (simp add: diff_diff_eq diff_diff_eq2 zdiff_zmult_distrib2)
+ apply (simp add: diff_diff_eq diff_diff_eq2 right_diff_distrib)
apply (rule_tac s = "p dvd -((a + 1) + (p * -a))" in trans)
apply (simp add: algebra_simps)
apply (subst dvd_minus_iff)
@@ -213,7 +213,7 @@
lemma reciP_sym: "zprime p ==> symP (reciR p)"
apply (unfold reciR_def symP_def)
- apply (simp add: zmult_commute)
+ apply (simp add: mult_commute)
apply auto
done
@@ -240,7 +240,7 @@
apply (subst setprod_insert)
apply (auto simp add: fin_bijER)
apply (subgoal_tac "zcong ((a * b) * \<Prod>A) (1 * 1) p")
- apply (simp add: zmult_assoc)
+ apply (simp add: mult_assoc)
apply (rule zcong_zmult)
apply auto
done
--- a/src/HOL/Old_Number_Theory/WilsonRuss.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Old_Number_Theory/WilsonRuss.thy Fri Sep 09 06:47:14 2011 +0200
@@ -80,7 +80,7 @@
lemma inv_not_p_minus_1_aux:
"[a * (p - 1) = 1] (mod p) = [a = p - 1] (mod p)"
apply (unfold zcong_def)
- apply (simp add: diff_diff_eq diff_diff_eq2 zdiff_zmult_distrib2)
+ apply (simp add: diff_diff_eq diff_diff_eq2 right_diff_distrib)
apply (rule_tac s = "p dvd -((a + 1) + (p * -a))" in trans)
apply (simp add: algebra_simps)
apply (subst dvd_minus_iff)
@@ -122,14 +122,14 @@
lemma inv_inv_aux: "5 \<le> p ==>
nat (p - 2) * nat (p - 2) = Suc (nat (p - 1) * nat (p - 3))"
apply (subst int_int_eq [symmetric])
- apply (simp add: zmult_int [symmetric])
- apply (simp add: zdiff_zmult_distrib zdiff_zmult_distrib2)
+ apply (simp add: of_nat_mult)
+ apply (simp add: left_diff_distrib right_diff_distrib)
done
lemma zcong_zpower_zmult:
"[x^y = 1] (mod p) \<Longrightarrow> [x^(y * z) = 1] (mod p)"
apply (induct z)
- apply (auto simp add: zpower_zadd_distrib)
+ apply (auto simp add: power_add)
apply (subgoal_tac "zcong (x^y * x^(y * z)) (1 * 1) p")
apply (rule_tac [2] zcong_zmult, simp_all)
done
@@ -261,7 +261,7 @@
apply (subgoal_tac [5]
"zcong (a * inv p a * (\<Prod>x\<in>wset (a - 1) p. x)) (1 * 1) p")
prefer 5
- apply (simp add: zmult_assoc)
+ apply (simp add: mult_assoc)
apply (rule_tac [5] zcong_zmult)
apply (rule_tac [5] inv_is_inv)
apply (tactic "clarify_tac @{context} 4")
--- a/src/HOL/Presburger.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Presburger.thy Fri Sep 09 06:47:14 2011 +0200
@@ -308,7 +308,7 @@
{fix x
have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast
also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)" using plus[THEN spec, of "x + i * d"]
- by (simp add:int_distrib zadd_ac)
+ by (simp add:int_distrib add_ac)
ultimately have "P x \<longrightarrow> P(x + (i + 1) * d)" by blast}
thus ?case ..
qed
--- a/src/HOL/Quickcheck.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Quickcheck.thy Fri Sep 09 06:47:14 2011 +0200
@@ -183,7 +183,7 @@
where
"union R1 R2 = (\<lambda>s. let
(P1, s') = R1 s; (P2, s'') = R2 s'
- in (semilattice_sup_class.sup P1 P2, s''))"
+ in (sup_class.sup P1 P2, s''))"
definition if_randompred :: "bool \<Rightarrow> unit randompred"
where
--- a/src/HOL/RealDef.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/RealDef.thy Fri Sep 09 06:47:14 2011 +0200
@@ -1411,16 +1411,13 @@
lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x"
by (insert real_of_nat_div2 [of n x], simp)
-lemma real_of_int_real_of_nat: "real (int n) = real n"
-by (simp add: real_of_nat_def real_of_int_def int_eq_of_nat)
-
lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n"
by (simp add: real_of_int_def real_of_nat_def)
lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x"
apply (subgoal_tac "real(int(nat x)) = real(nat x)")
apply force
- apply (simp only: real_of_int_real_of_nat)
+ apply (simp only: real_of_int_of_nat_eq)
done
lemma Nats_real_of_nat [simp]: "real (n::nat) \<in> Nats"
@@ -1534,7 +1531,7 @@
"real (number_of v :: nat) =
(if neg (number_of v :: int) then 0
else (number_of v :: real))"
-by (simp add: real_of_int_real_of_nat [symmetric])
+by (simp add: real_of_int_of_nat_eq [symmetric])
declaration {*
K (Lin_Arith.add_inj_thms [@{thm real_of_nat_le_iff} RS iffD2, @{thm real_of_nat_inject} RS iffD2]
--- a/src/HOL/SMT_Examples/SMT_Tests.certs Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/SMT_Examples/SMT_Tests.certs Fri Sep 09 06:47:14 2011 +0200
@@ -60074,3 +60074,483 @@
#110 := [asserted]: #38
[mp #110 #120]: false
unsat
+8021f8e09eb3e47791aed2bed0dafccd5948187d 69 0
+#2 := false
+decl f4 :: (-> S2 S1)
+decl f5 :: S2
+#16 := f5
+#19 := (f4 f5)
+decl f1 :: S1
+#4 := f1
+#66 := (= f1 #19)
+#70 := (not #66)
+#20 := (= #19 f1)
+#21 := (not #20)
+#71 := (iff #21 #70)
+#68 := (iff #20 #66)
+#69 := [rewrite]: #68
+#72 := [monotonicity #69]: #71
+#65 := [asserted]: #21
+#75 := [mp #65 #72]: #70
+decl f3 :: (-> S2 S1)
+#17 := (f3 f5)
+#61 := (= f1 #17)
+#18 := (= #17 f1)
+#63 := (iff #18 #61)
+#64 := [rewrite]: #63
+#60 := [asserted]: #18
+#67 := [mp #60 #64]: #61
+#8 := (:var 0 S2)
+#9 := (f3 #8)
+#10 := (pattern #9)
+#12 := (f4 #8)
+#45 := (= f1 #12)
+#42 := (= f1 #9)
+#51 := (not #42)
+#52 := (or #51 #45)
+#57 := (forall (vars (?v0 S2)) (:pat #10) #52)
+#85 := (~ #57 #57)
+#83 := (~ #52 #52)
+#84 := [refl]: #83
+#86 := [nnf-pos #84]: #85
+#13 := (= #12 f1)
+#11 := (= #9 f1)
+#14 := (implies #11 #13)
+#15 := (forall (vars (?v0 S2)) (:pat #10) #14)
+#58 := (iff #15 #57)
+#55 := (iff #14 #52)
+#48 := (implies #42 #45)
+#53 := (iff #48 #52)
+#54 := [rewrite]: #53
+#49 := (iff #14 #48)
+#46 := (iff #13 #45)
+#47 := [rewrite]: #46
+#43 := (iff #11 #42)
+#44 := [rewrite]: #43
+#50 := [monotonicity #44 #47]: #49
+#56 := [trans #50 #54]: #55
+#59 := [quant-intro #56]: #58
+#41 := [asserted]: #15
+#62 := [mp #41 #59]: #57
+#74 := [mp~ #62 #86]: #57
+#137 := (not #61)
+#139 := (not #57)
+#226 := (or #139 #137 #66)
+#224 := (or #137 #66)
+#217 := (or #139 #224)
+#229 := (iff #217 #226)
+#157 := [rewrite]: #229
+#228 := [quant-inst #16]: #217
+#230 := [mp #228 #157]: #226
+[unit-resolution #230 #74 #67 #75]: false
+unsat
+51102b6663906c70b84f1c6e3a1a2e429b49188d 112 0
+#2 := false
+decl f5 :: (-> S2 S1)
+decl f6 :: S2
+#19 := f6
+#24 := (f5 f6)
+decl f1 :: S1
+#4 := f1
+#82 := (= f1 #24)
+#86 := (not #82)
+#25 := (= #24 f1)
+#26 := (not #25)
+#87 := (iff #26 #86)
+#84 := (iff #25 #82)
+#85 := [rewrite]: #84
+#88 := [monotonicity #85]: #87
+#81 := [asserted]: #26
+#91 := [mp #81 #88]: #86
+decl f4 :: (-> S2 S1)
+#22 := (f4 f6)
+#77 := (= f1 #22)
+#23 := (= #22 f1)
+#79 := (iff #23 #77)
+#80 := [rewrite]: #79
+#76 := [asserted]: #23
+#83 := [mp #76 #80]: #77
+decl f3 :: (-> S2 S1)
+#20 := (f3 f6)
+#72 := (= f1 #20)
+#21 := (= #20 f1)
+#74 := (iff #21 #72)
+#75 := [rewrite]: #74
+#71 := [asserted]: #21
+#78 := [mp #71 #75]: #72
+#8 := (:var 0 S2)
+#10 := (f4 #8)
+#9 := (f3 #8)
+#11 := (pattern #9 #10)
+#15 := (f5 #8)
+#56 := (= f1 #15)
+#50 := (= f1 #10)
+#105 := (not #50)
+#47 := (= f1 #9)
+#92 := (not #47)
+#112 := (or #92 #105 #56)
+#117 := (forall (vars (?v0 S2)) (:pat #11) #112)
+#53 := (and #47 #50)
+#62 := (not #53)
+#63 := (or #62 #56)
+#68 := (forall (vars (?v0 S2)) (:pat #11) #63)
+#118 := (iff #68 #117)
+#115 := (iff #63 #112)
+#93 := (or #92 #105)
+#109 := (or #93 #56)
+#113 := (iff #109 #112)
+#114 := [rewrite]: #113
+#110 := (iff #63 #109)
+#107 := (iff #62 #93)
+#94 := (not #93)
+#97 := (not #94)
+#96 := (iff #97 #93)
+#106 := [rewrite]: #96
+#98 := (iff #62 #97)
+#99 := (iff #53 #94)
+#100 := [rewrite]: #99
+#95 := [monotonicity #100]: #98
+#108 := [trans #95 #106]: #107
+#111 := [monotonicity #108]: #110
+#116 := [trans #111 #114]: #115
+#119 := [quant-intro #116]: #118
+#103 := (~ #68 #68)
+#101 := (~ #63 #63)
+#102 := [refl]: #101
+#104 := [nnf-pos #102]: #103
+#16 := (= #15 f1)
+#13 := (= #10 f1)
+#12 := (= #9 f1)
+#14 := (and #12 #13)
+#17 := (implies #14 #16)
+#18 := (forall (vars (?v0 S2)) (:pat #11) #17)
+#69 := (iff #18 #68)
+#66 := (iff #17 #63)
+#59 := (implies #53 #56)
+#64 := (iff #59 #63)
+#65 := [rewrite]: #64
+#60 := (iff #17 #59)
+#57 := (iff #16 #56)
+#58 := [rewrite]: #57
+#54 := (iff #14 #53)
+#51 := (iff #13 #50)
+#52 := [rewrite]: #51
+#48 := (iff #12 #47)
+#49 := [rewrite]: #48
+#55 := [monotonicity #49 #52]: #54
+#61 := [monotonicity #55 #58]: #60
+#67 := [trans #61 #65]: #66
+#70 := [quant-intro #67]: #69
+#46 := [asserted]: #18
+#73 := [mp #46 #70]: #68
+#90 := [mp~ #73 #104]: #68
+#120 := [mp #90 #119]: #117
+#178 := (not #77)
+#265 := (not #72)
+#267 := (not #117)
+#258 := (or #267 #265 #178 #82)
+#179 := (or #265 #178 #82)
+#269 := (or #267 #179)
+#198 := (iff #269 #258)
+#271 := [rewrite]: #198
+#270 := [quant-inst #19]: #269
+#268 := [mp #270 #271]: #258
+[unit-resolution #268 #120 #78 #83 #91]: false
+unsat
+1191e209015c2f2f439f124434700d861e089600 149 0
+#2 := false
+decl f3 :: (-> S2 S1)
+decl f6 :: S2
+#21 := f6
+#22 := (f3 f6)
+decl f1 :: S1
+#4 := f1
+#84 := (= f1 #22)
+#264 := (not #84)
+decl f5 :: (-> S2 S1)
+#27 := (f5 f6)
+#95 := (= f1 #27)
+#178 := (or #264 #95)
+decl f4 :: (-> S2 S1)
+#24 := (f4 f6)
+#88 := (= f1 #24)
+#176 := (not #88)
+#268 := (or #176 #95)
+#266 := (not #268)
+#265 := (not #178)
+#586 := (or #265 #266)
+#375 := (not #586)
+#579 := [hypothesis]: #586
+#8 := (:var 0 S2)
+#11 := (f4 #8)
+#12 := (pattern #11)
+#9 := (f3 #8)
+#10 := (pattern #9)
+#65 := (= f1 #11)
+#71 := (not #65)
+#14 := (f5 #8)
+#53 := (= f1 #14)
+#72 := (or #53 #71)
+#116 := (not #72)
+#50 := (= f1 #9)
+#59 := (not #50)
+#60 := (or #59 #53)
+#105 := (not #60)
+#106 := (or #105 #116)
+#107 := (not #106)
+#108 := (forall (vars (?v0 S2)) (:pat #10 #12) #107)
+#77 := (and #60 #72)
+#80 := (forall (vars (?v0 S2)) (:pat #10 #12) #77)
+#109 := (iff #80 #108)
+#110 := (iff #77 #107)
+#111 := [rewrite]: #110
+#117 := [quant-intro #111]: #109
+#114 := (~ #80 #80)
+#112 := (~ #77 #77)
+#113 := [refl]: #112
+#115 := [nnf-pos #113]: #114
+#15 := (= #14 f1)
+#17 := (= #11 f1)
+#18 := (implies #17 #15)
+#13 := (= #9 f1)
+#16 := (implies #13 #15)
+#19 := (and #16 #18)
+#20 := (forall (vars (?v0 S2)) (:pat #10 #12) #19)
+#81 := (iff #20 #80)
+#78 := (iff #19 #77)
+#75 := (iff #18 #72)
+#68 := (implies #65 #53)
+#73 := (iff #68 #72)
+#74 := [rewrite]: #73
+#69 := (iff #18 #68)
+#54 := (iff #15 #53)
+#55 := [rewrite]: #54
+#66 := (iff #17 #65)
+#67 := [rewrite]: #66
+#70 := [monotonicity #67 #55]: #69
+#76 := [trans #70 #74]: #75
+#63 := (iff #16 #60)
+#56 := (implies #50 #53)
+#61 := (iff #56 #60)
+#62 := [rewrite]: #61
+#57 := (iff #16 #56)
+#51 := (iff #13 #50)
+#52 := [rewrite]: #51
+#58 := [monotonicity #52 #55]: #57
+#64 := [trans #58 #62]: #63
+#79 := [monotonicity #64 #76]: #78
+#82 := [quant-intro #79]: #81
+#49 := [asserted]: #20
+#85 := [mp #49 #82]: #80
+#103 := [mp~ #85 #115]: #80
+#118 := [mp #103 #117]: #108
+#255 := (not #108)
+#589 := (or #255 #375)
+#263 := (or #95 #176)
+#177 := (not #263)
+#256 := (or #265 #177)
+#267 := (not #256)
+#590 := (or #255 #267)
+#592 := (iff #590 #589)
+#593 := (iff #589 #589)
+#583 := [rewrite]: #593
+#582 := (iff #267 #375)
+#588 := (iff #256 #586)
+#270 := (iff #177 #266)
+#196 := (iff #263 #268)
+#269 := [rewrite]: #196
+#249 := [monotonicity #269]: #270
+#243 := [monotonicity #249]: #588
+#254 := [monotonicity #243]: #582
+#587 := [monotonicity #254]: #592
+#241 := [trans #587 #583]: #592
+#591 := [quant-inst #21]: #590
+#246 := [mp #591 #241]: #589
+#217 := [unit-resolution #246 #118 #579]: false
+#218 := [lemma #217]: #375
+#574 := (or #586 #178)
+#575 := [def-axiom]: #574
+#580 := [unit-resolution #575 #218]: #178
+#578 := (or #265 #264)
+#99 := (not #95)
+#28 := (= #27 f1)
+#29 := (not #28)
+#100 := (iff #29 #99)
+#97 := (iff #28 #95)
+#98 := [rewrite]: #97
+#101 := [monotonicity #98]: #100
+#94 := [asserted]: #29
+#104 := [mp #94 #101]: #99
+#569 := (or #265 #264 #95)
+#230 := [def-axiom]: #569
+#581 := [unit-resolution #230 #104]: #578
+#567 := [unit-resolution #581 #580]: #264
+#570 := (or #586 #268)
+#576 := [def-axiom]: #570
+#568 := [unit-resolution #576 #218]: #268
+#274 := (or #266 #176)
+#572 := (or #266 #176 #95)
+#573 := [def-axiom]: #572
+#290 := [unit-resolution #573 #104]: #274
+#291 := [unit-resolution #290 #568]: #176
+#91 := (or #84 #88)
+#25 := (= #24 f1)
+#23 := (= #22 f1)
+#26 := (or #23 #25)
+#92 := (iff #26 #91)
+#89 := (iff #25 #88)
+#90 := [rewrite]: #89
+#86 := (iff #23 #84)
+#87 := [rewrite]: #86
+#93 := [monotonicity #87 #90]: #92
+#83 := [asserted]: #26
+#96 := [mp #83 #93]: #91
+[unit-resolution #96 #291 #567]: false
+unsat
+45f8ffe676ed981a383aaab6faaf520b9ff236c9 69 0
+#2 := false
+decl f4 :: (-> S2 S1)
+decl f5 :: S2
+#16 := f5
+#19 := (f4 f5)
+decl f1 :: S1
+#4 := f1
+#66 := (= f1 #19)
+#70 := (not #66)
+#20 := (= #19 f1)
+#21 := (not #20)
+#71 := (iff #21 #70)
+#68 := (iff #20 #66)
+#69 := [rewrite]: #68
+#72 := [monotonicity #69]: #71
+#65 := [asserted]: #21
+#75 := [mp #65 #72]: #70
+decl f3 :: (-> S2 S1)
+#17 := (f3 f5)
+#61 := (= f1 #17)
+#18 := (= #17 f1)
+#63 := (iff #18 #61)
+#64 := [rewrite]: #63
+#60 := [asserted]: #18
+#67 := [mp #60 #64]: #61
+#8 := (:var 0 S2)
+#9 := (f3 #8)
+#10 := (pattern #9)
+#12 := (f4 #8)
+#45 := (= f1 #12)
+#42 := (= f1 #9)
+#51 := (not #42)
+#52 := (or #51 #45)
+#57 := (forall (vars (?v0 S2)) (:pat #10) #52)
+#85 := (~ #57 #57)
+#83 := (~ #52 #52)
+#84 := [refl]: #83
+#86 := [nnf-pos #84]: #85
+#13 := (= #12 f1)
+#11 := (= #9 f1)
+#14 := (implies #11 #13)
+#15 := (forall (vars (?v0 S2)) (:pat #10) #14)
+#58 := (iff #15 #57)
+#55 := (iff #14 #52)
+#48 := (implies #42 #45)
+#53 := (iff #48 #52)
+#54 := [rewrite]: #53
+#49 := (iff #14 #48)
+#46 := (iff #13 #45)
+#47 := [rewrite]: #46
+#43 := (iff #11 #42)
+#44 := [rewrite]: #43
+#50 := [monotonicity #44 #47]: #49
+#56 := [trans #50 #54]: #55
+#59 := [quant-intro #56]: #58
+#41 := [asserted]: #15
+#62 := [mp #41 #59]: #57
+#74 := [mp~ #62 #86]: #57
+#137 := (not #61)
+#139 := (not #57)
+#226 := (or #139 #137 #66)
+#224 := (or #137 #66)
+#217 := (or #139 #224)
+#229 := (iff #217 #226)
+#157 := [rewrite]: #229
+#228 := [quant-inst #16]: #217
+#230 := [mp #228 #157]: #226
+[unit-resolution #230 #74 #67 #75]: false
+unsat
+ceabafba9f0db45264556e8d9525b667725281c7 76 0
+#2 := false
+decl f4 :: (-> S2 S1)
+decl f5 :: S2
+#15 := f5
+#18 := (f4 f5)
+decl f1 :: S1
+#4 := f1
+#65 := (= f1 #18)
+#69 := (not #65)
+#19 := (= #18 f1)
+#20 := (not #19)
+#70 := (iff #20 #69)
+#67 := (iff #19 #65)
+#68 := [rewrite]: #67
+#71 := [monotonicity #68]: #70
+#64 := [asserted]: #20
+#74 := [mp #64 #71]: #69
+decl f3 :: (-> S2 S1)
+#16 := (f3 f5)
+#60 := (= f1 #16)
+#17 := (= #16 f1)
+#62 := (iff #17 #60)
+#63 := [rewrite]: #62
+#59 := [asserted]: #17
+#66 := [mp #59 #63]: #60
+#8 := (:var 0 S2)
+#11 := (f4 #8)
+#555 := (pattern #11)
+#9 := (f3 #8)
+#554 := (pattern #9)
+#44 := (= f1 #11)
+#41 := (= f1 #9)
+#50 := (not #41)
+#51 := (or #50 #44)
+#556 := (forall (vars (?v0 S2)) (:pat #554 #555) #51)
+#56 := (forall (vars (?v0 S2)) #51)
+#559 := (iff #56 #556)
+#557 := (iff #51 #51)
+#558 := [refl]: #557
+#560 := [quant-intro #558]: #559
+#84 := (~ #56 #56)
+#82 := (~ #51 #51)
+#83 := [refl]: #82
+#85 := [nnf-pos #83]: #84
+#12 := (= #11 f1)
+#10 := (= #9 f1)
+#13 := (implies #10 #12)
+#14 := (forall (vars (?v0 S2)) #13)
+#57 := (iff #14 #56)
+#54 := (iff #13 #51)
+#47 := (implies #41 #44)
+#52 := (iff #47 #51)
+#53 := [rewrite]: #52
+#48 := (iff #13 #47)
+#45 := (iff #12 #44)
+#46 := [rewrite]: #45
+#42 := (iff #10 #41)
+#43 := [rewrite]: #42
+#49 := [monotonicity #43 #46]: #48
+#55 := [trans #49 #53]: #54
+#58 := [quant-intro #55]: #57
+#40 := [asserted]: #14
+#61 := [mp #40 #58]: #56
+#73 := [mp~ #61 #85]: #56
+#561 := [mp #73 #560]: #556
+#136 := (not #60)
+#138 := (not #556)
+#225 := (or #138 #136 #65)
+#223 := (or #136 #65)
+#216 := (or #138 #223)
+#228 := (iff #216 #225)
+#156 := [rewrite]: #228
+#227 := [quant-inst #15]: #216
+#229 := [mp #227 #156]: #225
+[unit-resolution #229 #561 #66 #74]: false
+unsat
--- a/src/HOL/SMT_Examples/SMT_Tests.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/SMT_Examples/SMT_Tests.thy Fri Sep 09 06:47:14 2011 +0200
@@ -201,6 +201,9 @@
"(\<forall>x y z. f x y = f z y \<longrightarrow> x = z) \<and> a \<noteq> d \<longrightarrow> f a b \<noteq> f d b"
by smt+
+
+section {* Guidance for quantifier heuristics: patterns and weights *}
+
lemma
assumes "\<forall>x. SMT.trigger [[SMT.pat (f x)]] (f x = x)"
shows "f 1 = 1"
@@ -211,6 +214,38 @@
shows "f 1 = g 2"
using assms by smt
+lemma
+ assumes "ALL x. SMT.trigger [[SMT.pat (P x)]] (P x --> Q x)"
+ and "P t"
+ shows "Q t"
+ using assms by smt
+
+lemma
+ assumes "ALL x. SMT.trigger [[SMT.pat (P x), SMT.pat (Q x)]]
+ (P x & Q x --> R x)"
+ and "P t" and "Q t"
+ shows "R t"
+ using assms by smt
+
+lemma
+ assumes "ALL x. SMT.trigger [[SMT.pat (P x)], [SMT.pat (Q x)]]
+ ((P x --> R x) & (Q x --> R x))"
+ and "P t | Q t"
+ shows "R t"
+ using assms by smt
+
+lemma
+ assumes "ALL x. SMT.trigger [[SMT.pat (P x)]] (SMT.weight 2 (P x --> Q x))"
+ and "P t"
+ shows "Q t"
+ using assms by smt
+
+lemma
+ assumes "ALL x. SMT.weight 1 (P x --> Q x)"
+ and "P t"
+ shows "Q t"
+ using assms by smt
+
section {* Meta logical connectives *}
--- a/src/HOL/Statespace/StateFun.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Statespace/StateFun.thy Fri Sep 09 06:47:14 2011 +0200
@@ -28,7 +28,7 @@
by (simp add: K_statefun_def)
lemma K_statefun_comp [simp]: "(K_statefun c \<circ> f) = K_statefun c"
- by (rule ext) (simp add: K_statefun_apply comp_def)
+ by (rule ext) (simp add: comp_def)
lemma K_statefun_cong [cong]: "K_statefun c x = K_statefun c x"
by (rule refl)
--- a/src/HOL/Tools/ATP/atp_problem.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Tools/ATP/atp_problem.ML Fri Sep 09 06:47:14 2011 +0200
@@ -22,15 +22,15 @@
AFun of 'a ho_type * 'a ho_type |
ATyAbs of 'a list * 'a ho_type
- datatype tff_polymorphism = TFF_Monomorphic | TFF_Polymorphic
- datatype tff_explicitness = TFF_Implicit | TFF_Explicit
+ datatype tptp_polymorphism = TPTP_Monomorphic | TPTP_Polymorphic
+ datatype tptp_explicitness = TPTP_Implicit | TPTP_Explicit
datatype thf_flavor = THF_Without_Choice | THF_With_Choice
- datatype format =
+ datatype tptp_format =
CNF |
CNF_UEQ |
FOF |
- TFF of tff_polymorphism * tff_explicitness |
- THF0 of thf_flavor
+ TFF of tptp_polymorphism * tptp_explicitness |
+ THF of tptp_polymorphism * tptp_explicitness * thf_flavor
datatype formula_kind = Axiom | Definition | Lemma | Hypothesis | Conjecture
datatype 'a problem_line =
@@ -92,9 +92,9 @@
bool option -> (bool option -> 'c -> 'd -> 'd) -> ('a, 'b, 'c) formula
-> 'd -> 'd
val formula_map : ('c -> 'd) -> ('a, 'b, 'c) formula -> ('a, 'b, 'd) formula
- val is_format_thf : format -> bool
- val is_format_typed : format -> bool
- val tptp_lines_for_atp_problem : format -> string problem -> string list
+ val is_format_thf : tptp_format -> bool
+ val is_format_typed : tptp_format -> bool
+ val tptp_lines_for_atp_problem : tptp_format -> string problem -> string list
val ensure_cnf_problem :
(string * string) problem -> (string * string) problem
val filter_cnf_ueq_problem :
@@ -130,16 +130,16 @@
AFun of 'a ho_type * 'a ho_type |
ATyAbs of 'a list * 'a ho_type
-datatype tff_polymorphism = TFF_Monomorphic | TFF_Polymorphic
-datatype tff_explicitness = TFF_Implicit | TFF_Explicit
+datatype tptp_polymorphism = TPTP_Monomorphic | TPTP_Polymorphic
+datatype tptp_explicitness = TPTP_Implicit | TPTP_Explicit
datatype thf_flavor = THF_Without_Choice | THF_With_Choice
-datatype format =
+datatype tptp_format =
CNF |
CNF_UEQ |
FOF |
- TFF of tff_polymorphism * tff_explicitness |
- THF0 of thf_flavor
+ TFF of tptp_polymorphism * tptp_explicitness |
+ THF of tptp_polymorphism * tptp_explicitness * thf_flavor
datatype formula_kind = Axiom | Definition | Lemma | Hypothesis | Conjecture
datatype 'a problem_line =
@@ -224,10 +224,10 @@
| formula_map f (AConn (c, phis)) = AConn (c, map (formula_map f) phis)
| formula_map f (AAtom tm) = AAtom (f tm)
-fun is_format_thf (THF0 _) = true
+fun is_format_thf (THF _) = true
| is_format_thf _ = false
fun is_format_typed (TFF _) = true
- | is_format_typed (THF0 _) = true
+ | is_format_typed (THF _) = true
| is_format_typed _ = false
fun string_for_kind Axiom = "axiom"
@@ -236,6 +236,12 @@
| string_for_kind Hypothesis = "hypothesis"
| string_for_kind Conjecture = "conjecture"
+fun string_for_app format func args =
+ if is_format_thf format then
+ "(" ^ space_implode (" " ^ tptp_app ^ " ") (func :: args) ^ ")"
+ else
+ func ^ "(" ^ commas args ^ ")"
+
fun flatten_type (ATyAbs (tys, ty)) = ATyAbs (tys, flatten_type ty)
| flatten_type (ty as AFun (ty1 as AType _, ty2)) =
(case flatten_type ty2 of
@@ -247,16 +253,17 @@
| flatten_type _ =
raise Fail "unexpected higher-order type in first-order format"
-fun str_for_type ty =
+fun str_for_type format ty =
let
fun str _ (AType (s, [])) = s
| str _ (AType (s, tys)) =
- tys |> map (str false)
- |> (if s = tptp_product_type then
- space_implode (" " ^ tptp_product_type ^ " ")
- #> length tys > 1 ? enclose "(" ")"
- else
- commas #> enclose "(" ")" #> prefix s)
+ let val ss = tys |> map (str false) in
+ if s = tptp_product_type then
+ ss |> space_implode (" " ^ tptp_product_type ^ " ")
+ |> length ss > 1 ? enclose "(" ")"
+ else
+ string_for_app format s ss
+ end
| str rhs (AFun (ty1, ty2)) =
str false ty1 ^ " " ^ tptp_fun_type ^ " " ^ str true ty2
|> not rhs ? enclose "(" ")"
@@ -266,8 +273,8 @@
ss) ^ "]: " ^ str false ty
in str true ty end
-fun string_for_type (THF0 _) ty = str_for_type ty
- | string_for_type (TFF _) ty = str_for_type (flatten_type ty)
+fun string_for_type (format as THF _) ty = str_for_type format ty
+ | string_for_type (format as TFF _) ty = str_for_type format (flatten_type ty)
| string_for_type _ _ = raise Fail "unexpected type in untyped format"
fun string_for_quantifier AForall = tptp_forall
@@ -288,38 +295,33 @@
else
"")
+fun is_format_with_choice (THF (_, _, THF_With_Choice)) = true
+ | is_format_with_choice _ = false
+
fun string_for_term _ (ATerm (s, [])) = s
| string_for_term format (ATerm (s, ts)) =
- if s = tptp_empty_list then
- (* used for lists in the optional "source" field of a derivation *)
- "[" ^ commas (map (string_for_term format) ts) ^ "]"
- else if is_tptp_equal s then
- space_implode (" " ^ tptp_equal ^ " ") (map (string_for_term format) ts)
- |> is_format_thf format ? enclose "(" ")"
- else
- (case (s = tptp_ho_forall orelse s = tptp_ho_exists,
- s = tptp_choice andalso format = THF0 THF_With_Choice, ts) of
- (true, _, [AAbs ((s', ty), tm)]) =>
- (* Use syntactic sugar "!" and "?" instead of "!!" and "??" whenever
- possible, to work around LEO-II 1.2.8 parser limitation. *)
- string_for_formula format
- (AQuant (if s = tptp_ho_forall then AForall else AExists,
- [(s', SOME ty)], AAtom tm))
- | (_, true, [AAbs ((s', ty), tm)]) =>
- (*There is code in ATP_Translate to ensure that Eps is always applied
- to an abstraction*)
- tptp_choice ^ "[" ^ s' ^ " : " ^ string_for_type format ty ^ "]: " ^
- string_for_term format tm ^ ""
- |> enclose "(" ")"
-
- | _ =>
- let val ss = map (string_for_term format) ts in
- if is_format_thf format then
- "(" ^ space_implode (" " ^ tptp_app ^ " ") (s :: ss) ^ ")"
- else
- s ^ "(" ^ commas ss ^ ")"
- end)
- | string_for_term (format as THF0 _) (AAbs ((s, ty), tm)) =
+ (if s = tptp_empty_list then
+ (* used for lists in the optional "source" field of a derivation *)
+ "[" ^ commas (map (string_for_term format) ts) ^ "]"
+ else if is_tptp_equal s then
+ space_implode (" " ^ tptp_equal ^ " ") (map (string_for_term format) ts)
+ |> is_format_thf format ? enclose "(" ")"
+ else case (s = tptp_ho_forall orelse s = tptp_ho_exists,
+ s = tptp_choice andalso is_format_with_choice format, ts) of
+ (true, _, [AAbs ((s', ty), tm)]) =>
+ (* Use syntactic sugar "!" and "?" instead of "!!" and "??" whenever
+ possible, to work around LEO-II 1.2.8 parser limitation. *)
+ string_for_formula format
+ (AQuant (if s = tptp_ho_forall then AForall else AExists,
+ [(s', SOME ty)], AAtom tm))
+ | (_, true, [AAbs ((s', ty), tm)]) =>
+ (* There is code in "ATP_Translate" to ensure that "Eps" is always
+ applied to an abstraction. *)
+ tptp_choice ^ "[" ^ s' ^ " : " ^ string_for_type format ty ^ "]: " ^
+ string_for_term format tm ^ ""
+ |> enclose "(" ")"
+ | _ => string_for_app format s (map (string_for_term format) ts))
+ | string_for_term (format as THF _) (AAbs ((s, ty), tm)) =
"(^[" ^ s ^ " : " ^ string_for_type format ty ^ "]: " ^
string_for_term format tm ^ ")"
| string_for_term _ _ = raise Fail "unexpected term in first-order format"
@@ -352,7 +354,7 @@
| string_for_format CNF_UEQ = tptp_cnf
| string_for_format FOF = tptp_fof
| string_for_format (TFF _) = tptp_tff
- | string_for_format (THF0 _) = tptp_thf
+ | string_for_format (THF _) = tptp_thf
fun string_for_problem_line format (Decl (ident, sym, ty)) =
string_for_format format ^ "(" ^ ident ^ ", type,\n " ^ sym ^ " : " ^
--- a/src/HOL/Tools/ATP/atp_proof.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Tools/ATP/atp_proof.ML Fri Sep 09 06:47:14 2011 +0200
@@ -92,9 +92,6 @@
InternalError |
UnknownError of string
-fun is_ident_char c = Char.isAlphaNum c orelse c = #"_"
-val strip_spaces_except_between_ident_chars = strip_spaces true is_ident_char
-
fun elide_string threshold s =
if size s > threshold then
String.extract (s, 0, SOME (threshold div 2 - 5)) ^ " ...... " ^
@@ -475,7 +472,7 @@
fun parse_line problem spass_names =
parse_tstp_line problem || parse_spass_line spass_names
fun parse_proof problem spass_names tstp =
- tstp |> strip_spaces_except_between_ident_chars
+ tstp |> strip_spaces_except_between_idents
|> raw_explode
|> Scan.finite Symbol.stopper
(Scan.error (!! (fn _ => raise UNRECOGNIZED_ATP_PROOF ())
--- a/src/HOL/Tools/ATP/atp_reconstruct.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Tools/ATP/atp_reconstruct.ML Fri Sep 09 06:47:14 2011 +0200
@@ -21,7 +21,7 @@
datatype play =
Played of reconstructor * Time.time |
- Trust_Playable of reconstructor * Time.time option|
+ Trust_Playable of reconstructor * Time.time option |
Failed_to_Play of reconstructor
type minimize_command = string list -> string
@@ -41,8 +41,8 @@
val make_tvar : string -> typ
val make_tfree : Proof.context -> string -> typ
val term_from_atp :
- Proof.context -> bool -> int Symtab.table -> typ option -> (string, string) ho_term
- -> term
+ Proof.context -> bool -> int Symtab.table -> typ option
+ -> (string, string) ho_term -> term
val prop_from_atp :
Proof.context -> bool -> int Symtab.table
-> (string, string, (string, string) ho_term) formula -> term
@@ -152,7 +152,7 @@
union (op =) (resolve_fact facts_offset fact_names name)
| add_fact ctxt _ _ (Inference (_, _, deps)) =
if AList.defined (op =) deps leo2_ext then
- insert (op =) (ext_name ctxt, Extensionality)
+ insert (op =) (ext_name ctxt, General)
else
I
| add_fact _ _ _ _ = I
@@ -360,10 +360,10 @@
val var_index = if textual then 0 else 1
fun do_term extra_ts opt_T u =
case u of
- ATerm (a, us) =>
- if String.isPrefix simple_type_prefix a then
+ ATerm (s, us) =>
+ if String.isPrefix simple_type_prefix s then
@{const True} (* ignore TPTP type information *)
- else if a = tptp_equal then
+ else if s = tptp_equal then
let val ts = map (do_term [] NONE) us in
if textual andalso length ts = 2 andalso
hd ts aconv List.last ts then
@@ -372,10 +372,11 @@
else
list_comb (Const (@{const_name HOL.eq}, HOLogic.typeT), ts)
end
- else case strip_prefix_and_unascii const_prefix a of
- SOME s =>
+ else case strip_prefix_and_unascii const_prefix s of
+ SOME s' =>
let
- val ((s', s), mangled_us) = s |> unmangled_const |>> `invert_const
+ val ((s', s''), mangled_us) =
+ s' |> unmangled_const |>> `invert_const
in
if s' = type_tag_name then
case mangled_us @ us of
@@ -396,7 +397,7 @@
@{const True} (* ignore type predicates *)
else
let
- val new_skolem = String.isPrefix new_skolem_const_prefix s
+ val new_skolem = String.isPrefix new_skolem_const_prefix s''
val num_ty_args =
length us - the_default 0 (Symtab.lookup sym_tab s)
val (type_us, term_us) =
@@ -422,7 +423,7 @@
| NONE => HOLogic.typeT))
val t =
if new_skolem then
- Var ((new_skolem_var_name_from_const s, var_index), T)
+ Var ((new_skolem_var_name_from_const s'', var_index), T)
else
Const (unproxify_const s', T)
in list_comb (t, term_ts @ extra_ts) end
@@ -432,15 +433,15 @@
val ts = map (do_term [] NONE) us @ extra_ts
val T = map slack_fastype_of ts ---> HOLogic.typeT
val t =
- case strip_prefix_and_unascii fixed_var_prefix a of
- SOME b =>
+ case strip_prefix_and_unascii fixed_var_prefix s of
+ SOME s =>
(* FIXME: reconstruction of lambda-lifting *)
- Free (b, T)
+ Free (s, T)
| NONE =>
- case strip_prefix_and_unascii schematic_var_prefix a of
- SOME b => Var ((b, var_index), T)
+ case strip_prefix_and_unascii schematic_var_prefix s of
+ SOME s => Var ((s, var_index), T)
| NONE =>
- Var ((a |> textual ? repair_variable_name Char.toLower,
+ Var ((s |> textual ? repair_variable_name Char.toLower,
var_index), T)
in list_comb (t, ts) end
in do_term [] end
@@ -627,7 +628,8 @@
| add_desired_line isar_shrink_factor conjecture_shape fact_names frees
(Inference (name, t, deps)) (j, lines) =
(j + 1,
- if is_fact fact_names name orelse is_conjecture conjecture_shape name orelse
+ if is_fact fact_names name orelse
+ is_conjecture conjecture_shape name orelse
(* the last line must be kept *)
j = 0 orelse
(not (is_only_type_information t) andalso
--- a/src/HOL/Tools/ATP/atp_systems.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Tools/ATP/atp_systems.ML Fri Sep 09 06:47:14 2011 +0200
@@ -7,7 +7,7 @@
signature ATP_SYSTEMS =
sig
- type format = ATP_Problem.format
+ type tptp_format = ATP_Problem.tptp_format
type formula_kind = ATP_Problem.formula_kind
type failure = ATP_Proof.failure
@@ -22,7 +22,8 @@
conj_sym_kind : formula_kind,
prem_kind : formula_kind,
best_slices :
- Proof.context -> (real * (bool * (int * format * string * string))) list}
+ Proof.context
+ -> (real * (bool * (int * tptp_format * string * string))) list}
val force_sos : bool Config.T
val e_smartN : string
@@ -41,6 +42,7 @@
val e_tofofN : string
val leo2N : string
val pffN : string
+ val phfN : string
val satallaxN : string
val snarkN : string
val spassN : string
@@ -51,7 +53,7 @@
val remote_atp :
string -> string -> string list -> (string * string) list
-> (failure * string) list -> formula_kind -> formula_kind
- -> (Proof.context -> int * format * string) -> string * atp_config
+ -> (Proof.context -> int * tptp_format * string) -> string * atp_config
val add_atp : string * atp_config -> theory -> theory
val get_atp : theory -> string -> atp_config
val supported_atps : theory -> string list
@@ -79,7 +81,8 @@
conj_sym_kind : formula_kind,
prem_kind : formula_kind,
best_slices :
- Proof.context -> (real * (bool * (int * format * string * string))) list}
+ Proof.context
+ -> (real * (bool * (int * tptp_format * string * string))) list}
(* "best_slices" must be found empirically, taking a wholistic approach since
the ATPs are run in parallel. The "real" components give the faction of the
@@ -105,6 +108,7 @@
val e_tofofN = "e_tofof"
val leo2N = "leo2"
val pffN = "pff"
+val phfN = "phf"
val satallaxN = "satallax"
val snarkN = "snark"
val spassN = "spass"
@@ -218,11 +222,11 @@
let val method = effective_e_weight_method ctxt in
(* FUDGE *)
if method = e_smartN then
- [(0.333, (true, (500, FOF, "mono_tags?", e_fun_weightN))),
- (0.334, (true, (50, FOF, "mono_guards?", e_fun_weightN))),
- (0.333, (true, (1000, FOF, "mono_tags?", e_sym_offset_weightN)))]
+ [(0.333, (true, (500, FOF, "mono_tags??", e_fun_weightN))),
+ (0.334, (true, (50, FOF, "mono_guards??", e_fun_weightN))),
+ (0.333, (true, (1000, FOF, "mono_tags??", e_sym_offset_weightN)))]
else
- [(1.0, (true, (500, FOF, "mono_tags?", method)))]
+ [(1.0, (true, (500, FOF, "mono_tags??", method)))]
end}
val e = (eN, e_config)
@@ -230,6 +234,8 @@
(* LEO-II *)
+val leo2_thf0 = THF (TPTP_Monomorphic, TPTP_Explicit, THF_Without_Choice)
+
val leo2_config : atp_config =
{exec = ("LEO2_HOME", "leo"),
required_execs = [],
@@ -243,10 +249,8 @@
prem_kind = Hypothesis,
best_slices = fn ctxt =>
(* FUDGE *)
- [(0.667, (false, (150, THF0 THF_Without_Choice,
- "mono_simple_higher", sosN))),
- (0.333, (true, (50, THF0 THF_Without_Choice,
- "mono_simple_higher", no_sosN)))]
+ [(0.667, (false, (150, leo2_thf0, "mono_simple_higher", sosN))),
+ (0.333, (true, (50, leo2_thf0, "mono_simple_higher", no_sosN)))]
|> (if Config.get ctxt force_sos then hd #> apfst (K 1.0) #> single
else I)}
@@ -255,6 +259,8 @@
(* Satallax *)
+val satallax_thf0 = THF (TPTP_Monomorphic, TPTP_Explicit, THF_With_Choice)
+
val satallax_config : atp_config =
{exec = ("SATALLAX_HOME", "satallax"),
required_execs = [],
@@ -266,8 +272,8 @@
conj_sym_kind = Axiom,
prem_kind = Hypothesis,
best_slices =
- K [(1.0, (true, (100, THF0 THF_With_Choice, "mono_simple_higher", "")))]
- (* FUDGE *)}
+ (* FUDGE *)
+ K [(1.0, (true, (100, satallax_thf0, "mono_simple_higher", "")))]}
val satallax = (satallaxN, satallax_config)
@@ -298,9 +304,9 @@
prem_kind = Conjecture,
best_slices = fn ctxt =>
(* FUDGE *)
- [(0.333, (false, (150, FOF, "mono_tags", sosN))),
- (0.333, (false, (300, FOF, "poly_tags?", sosN))),
- (0.334, (true, (50, FOF, "mono_tags?", no_sosN)))]
+ [(0.333, (false, (150, FOF, "mono_tags??", sosN))),
+ (0.333, (false, (300, FOF, "poly_tags??", sosN))),
+ (0.334, (true, (50, FOF, "mono_tags??", no_sosN)))]
|> (if Config.get ctxt force_sos then hd #> apfst (K 1.0) #> single
else I)}
@@ -314,7 +320,7 @@
fun is_old_vampire_version () =
string_ord (getenv "VAMPIRE_VERSION", "1.8") <> GREATER
-val vampire_tff = TFF (TFF_Monomorphic, TFF_Implicit)
+val vampire_tff0 = TFF (TPTP_Monomorphic, TPTP_Implicit)
val vampire_config : atp_config =
{exec = ("VAMPIRE_HOME", "vampire"),
@@ -343,13 +349,13 @@
best_slices = fn ctxt =>
(* FUDGE *)
(if is_old_vampire_version () then
- [(0.333, (false, (150, FOF, "poly_guards", sosN))),
- (0.333, (false, (500, FOF, "mono_tags?", sosN))),
- (0.334, (true, (50, FOF, "mono_guards?", no_sosN)))]
+ [(0.333, (false, (150, FOF, "poly_guards??", sosN))),
+ (0.333, (false, (500, FOF, "mono_tags??", sosN))),
+ (0.334, (true, (50, FOF, "mono_guards??", no_sosN)))]
else
- [(0.333, (false, (150, vampire_tff, "poly_guards", sosN))),
- (0.333, (false, (500, vampire_tff, "mono_simple", sosN))),
- (0.334, (true, (50, vampire_tff, "mono_simple", no_sosN)))])
+ [(0.333, (false, (150, vampire_tff0, "poly_guards??", sosN))),
+ (0.333, (false, (500, vampire_tff0, "mono_simple", sosN))),
+ (0.334, (true, (50, vampire_tff0, "mono_simple", no_sosN)))])
|> (if Config.get ctxt force_sos then hd #> apfst (K 1.0) #> single
else I)}
@@ -358,7 +364,7 @@
(* Z3 with TPTP syntax *)
-val z3_tff = TFF (TFF_Monomorphic, TFF_Implicit)
+val z3_tff0 = TFF (TPTP_Monomorphic, TPTP_Implicit)
val z3_tptp_config : atp_config =
{exec = ("Z3_HOME", "z3"),
@@ -377,18 +383,17 @@
prem_kind = Hypothesis,
best_slices =
(* FUDGE *)
- K [(0.5, (false, (250, z3_tff, "mono_simple", ""))),
- (0.25, (false, (125, z3_tff, "mono_simple", ""))),
- (0.125, (false, (62, z3_tff, "mono_simple", ""))),
- (0.125, (false, (31, z3_tff, "mono_simple", "")))]}
+ K [(0.5, (false, (250, z3_tff0, "mono_simple", ""))),
+ (0.25, (false, (125, z3_tff0, "mono_simple", ""))),
+ (0.125, (false, (62, z3_tff0, "mono_simple", ""))),
+ (0.125, (false, (31, z3_tff0, "mono_simple", "")))]}
val z3_tptp = (z3_tptpN, z3_tptp_config)
-(* Not really a prover: Experimental PFF (Polymorphic TFF) output *)
-val poly_tff = TFF (TFF_Polymorphic, TFF_Implicit)
+(* Not really a prover: Experimental Polymorphic TFF and THF output *)
-val pff_config : atp_config =
+fun dummy_config format type_enc : atp_config =
{exec = ("ISABELLE_ATP", "scripts/dummy_atp"),
required_execs = [],
arguments = K (K (K (K (K "")))),
@@ -396,10 +401,16 @@
known_failures = [(GaveUp, "SZS status Unknown")],
conj_sym_kind = Hypothesis,
prem_kind = Hypothesis,
- best_slices = K [(1.0, (false, (200, poly_tff, "poly_simple", "")))]}
+ best_slices = K [(1.0, (false, (200, format, type_enc, "")))]}
+val pff_format = TFF (TPTP_Polymorphic, TPTP_Explicit)
+val pff_config = dummy_config pff_format "poly_simple"
val pff = (pffN, pff_config)
+val phf_format = THF (TPTP_Polymorphic, TPTP_Explicit, THF_With_Choice)
+val phf_config = dummy_config phf_format "poly_simple_higher"
+val phf = (phfN, phf_config)
+
(* Remote ATP invocation via SystemOnTPTP *)
@@ -475,41 +486,40 @@
(remote_prefix ^ name,
remotify_config system_name system_versions best_slice config)
-val dumb_tff = TFF (TFF_Monomorphic, TFF_Explicit)
+val explicit_tff0 = TFF (TPTP_Monomorphic, TPTP_Explicit)
val remote_e =
remotify_atp e "EP" ["1.0", "1.1", "1.2"]
- (K (750, FOF, "mono_tags?") (* FUDGE *))
+ (K (750, FOF, "mono_tags??") (* FUDGE *))
val remote_leo2 =
remotify_atp leo2 "LEO-II" ["1.2.8", "1.2.6"]
- (K (100, THF0 THF_Without_Choice,
- "mono_simple_higher") (* FUDGE *))
+ (K (100, leo2_thf0, "mono_simple_higher") (* FUDGE *))
val remote_satallax =
remotify_atp satallax "Satallax" ["2.1", "2.0", "2"]
- (K (100, THF0 THF_With_Choice,
- "mono_simple_higher") (* FUDGE *))
+ (K (100, satallax_thf0, "mono_simple_higher") (* FUDGE *))
val remote_vampire =
remotify_atp vampire "Vampire" ["1.8"]
- (K (250, FOF, "mono_guards?") (* FUDGE *))
+ (K (250, FOF, "mono_guards??") (* FUDGE *))
val remote_z3_tptp =
- remotify_atp z3_tptp "Z3" ["3.0"] (K (250, z3_tff, "mono_simple") (* FUDGE *))
+ remotify_atp z3_tptp "Z3" ["3.0"]
+ (K (250, z3_tff0, "mono_simple") (* FUDGE *))
val remote_e_sine =
remote_atp e_sineN "SInE" ["0.4"] [] (#known_failures e_config) Axiom
- Conjecture (K (500, FOF, "mono_guards?") (* FUDGE *))
+ Conjecture (K (500, FOF, "mono_guards??") (* FUDGE *))
val remote_snark =
remote_atp snarkN "SNARK" ["20080805r029", "20080805r024"]
[("refutation.", "end_refutation.")] [] Hypothesis Hypothesis
- (K (100, dumb_tff, "mono_simple") (* FUDGE *))
+ (K (100, explicit_tff0, "mono_simple") (* FUDGE *))
val remote_e_tofof =
remote_atp e_tofofN "ToFoF" ["0.1"] [] (#known_failures e_config) Axiom
- Hypothesis (K (150, dumb_tff, "mono_simple") (* FUDGE *))
+ Hypothesis (K (150, explicit_tff0, "mono_simple") (* FUDGE *))
val remote_waldmeister =
remote_atp waldmeisterN "Waldmeister" ["710"]
[("#START OF PROOF", "Proved Goals:")]
[(OutOfResources, "Too many function symbols"),
(Crashed, "Unrecoverable Segmentation Fault")]
Hypothesis Hypothesis
- (K (50, CNF_UEQ, "mono_tags?") (* FUDGE *))
+ (K (50, CNF_UEQ, "mono_tags??") (* FUDGE *))
(* Setup *)
@@ -532,7 +542,7 @@
Synchronized.change systems (fn _ => get_systems ())
val atps =
- [e, leo2, pff, satallax, spass, vampire, z3_tptp, remote_e, remote_leo2,
+ [e, leo2, pff, phf, satallax, spass, vampire, z3_tptp, remote_e, remote_leo2,
remote_satallax, remote_vampire, remote_z3_tptp, remote_e_sine, remote_snark,
remote_e_tofof, remote_waldmeister]
val setup = fold add_atp atps
--- a/src/HOL/Tools/ATP/atp_translate.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Tools/ATP/atp_translate.ML Fri Sep 09 06:47:14 2011 +0200
@@ -11,29 +11,23 @@
type ('a, 'b) ho_term = ('a, 'b) ATP_Problem.ho_term
type connective = ATP_Problem.connective
type ('a, 'b, 'c) formula = ('a, 'b, 'c) ATP_Problem.formula
- type format = ATP_Problem.format
+ type tptp_format = ATP_Problem.tptp_format
type formula_kind = ATP_Problem.formula_kind
type 'a problem = 'a ATP_Problem.problem
datatype locality =
- General | Helper | Induction | Extensionality | Intro | Elim | Simp |
- Local | Assum | Chained
+ General | Helper | Induction | Intro | Elim | Simp | Local | Assum | Chained
- datatype order = First_Order | Higher_Order
datatype polymorphism = Polymorphic | Raw_Monomorphic | Mangled_Monomorphic
datatype soundness = Sound_Modulo_Infiniteness | Sound
+ datatype granularity = All_Vars | Positively_Naked_Vars | Ghost_Type_Arg_Vars
datatype type_level =
All_Types |
- Noninf_Nonmono_Types of soundness |
- Fin_Nonmono_Types |
+ Noninf_Nonmono_Types of soundness * granularity |
+ Fin_Nonmono_Types of granularity |
Const_Arg_Types |
No_Types
- datatype type_uniformity = Uniform | Nonuniform
-
- datatype type_enc =
- Simple_Types of order * polymorphism * type_level |
- Guards of polymorphism * type_level * type_uniformity |
- Tags of polymorphism * type_level * type_uniformity
+ type type_enc
val type_tag_idempotence : bool Config.T
val type_tag_arguments : bool Config.T
@@ -86,13 +80,13 @@
val new_skolem_var_name_from_const : string -> string
val atp_irrelevant_consts : string list
val atp_schematic_consts_of : term -> typ list Symtab.table
- val type_enc_from_string : soundness -> string -> type_enc
val is_type_enc_higher_order : type_enc -> bool
val polymorphism_of_type_enc : type_enc -> polymorphism
val level_of_type_enc : type_enc -> type_level
val is_type_enc_quasi_sound : type_enc -> bool
val is_type_enc_fairly_sound : type_enc -> bool
- val adjust_type_enc : format -> type_enc -> type_enc
+ val type_enc_from_string : soundness -> string -> type_enc
+ val adjust_type_enc : tptp_format -> type_enc -> type_enc
val mk_aconns :
connective -> ('a, 'b, 'c) formula list -> ('a, 'b, 'c) formula
val unmangled_const : string -> string * (string, 'b) ho_term list
@@ -100,7 +94,7 @@
val helper_table : ((string * bool) * thm list) list
val factsN : string
val prepare_atp_problem :
- Proof.context -> format -> formula_kind -> formula_kind -> type_enc
+ Proof.context -> tptp_format -> formula_kind -> formula_kind -> type_enc
-> bool -> string -> bool -> bool -> term list -> term
-> ((string * locality) * term) list
-> string problem * string Symtab.table * int * int
@@ -142,7 +136,7 @@
(* TFF1 is still in development, and it is still unclear whether symbols will be
allowed to have types like "$tType > $o" (i.e., "![A : $tType] : $o"), with
"dummy" type variables. *)
-val exploit_tff1_dummy_type_vars = false
+val avoid_first_order_dummy_type_vars = true
val bound_var_prefix = "B_"
val all_bound_var_prefix = "BA_"
@@ -325,8 +319,8 @@
fun default c = const_prefix ^ lookup_const c
in
fun make_fixed_const _ @{const_name HOL.eq} = tptp_old_equal
- | make_fixed_const (SOME (THF0 THF_With_Choice)) c =
- if c = choice_const then tptp_choice else default c
+ | make_fixed_const (SOME (THF (_, _, THF_With_Choice))) c =
+ if c = choice_const then tptp_choice else default c
| make_fixed_const _ c = default c
end
@@ -408,7 +402,7 @@
fun multi_arity_clause [] = []
| multi_arity_clause ((tcons, ars) :: tc_arlists) =
- arity_clause [] 1 (tcons, ars) @ multi_arity_clause tc_arlists
+ arity_clause [] 1 (tcons, ars) @ multi_arity_clause tc_arlists
(* Generate all pairs (tycon, class, sorts) such that tycon belongs to class in
theory thy provided its arguments have the corresponding sorts. *)
@@ -531,28 +525,76 @@
in (IAbs ((name, T), tm), union (op =) atomic_Ts (atyps_of T)) end
datatype locality =
- General | Helper | Induction | Extensionality | Intro | Elim | Simp |
- Local | Assum | Chained
+ General | Helper | Induction | Intro | Elim | Simp | Local | Assum | Chained
datatype order = First_Order | Higher_Order
datatype polymorphism = Polymorphic | Raw_Monomorphic | Mangled_Monomorphic
datatype soundness = Sound_Modulo_Infiniteness | Sound
+datatype granularity = All_Vars | Positively_Naked_Vars | Ghost_Type_Arg_Vars
datatype type_level =
All_Types |
- Noninf_Nonmono_Types of soundness |
- Fin_Nonmono_Types |
+ Noninf_Nonmono_Types of soundness * granularity |
+ Fin_Nonmono_Types of granularity |
Const_Arg_Types |
No_Types
-datatype type_uniformity = Uniform | Nonuniform
datatype type_enc =
Simple_Types of order * polymorphism * type_level |
- Guards of polymorphism * type_level * type_uniformity |
- Tags of polymorphism * type_level * type_uniformity
+ Guards of polymorphism * type_level |
+ Tags of polymorphism * type_level
+
+fun is_type_enc_higher_order (Simple_Types (Higher_Order, _, _)) = true
+ | is_type_enc_higher_order _ = false
+
+fun polymorphism_of_type_enc (Simple_Types (_, poly, _)) = poly
+ | polymorphism_of_type_enc (Guards (poly, _)) = poly
+ | polymorphism_of_type_enc (Tags (poly, _)) = poly
+
+fun level_of_type_enc (Simple_Types (_, _, level)) = level
+ | level_of_type_enc (Guards (_, level)) = level
+ | level_of_type_enc (Tags (_, level)) = level
+
+fun granularity_of_type_level (Noninf_Nonmono_Types (_, grain)) = grain
+ | granularity_of_type_level (Fin_Nonmono_Types grain) = grain
+ | granularity_of_type_level _ = All_Vars
+
+fun is_type_level_quasi_sound All_Types = true
+ | is_type_level_quasi_sound (Noninf_Nonmono_Types _) = true
+ | is_type_level_quasi_sound _ = false
+val is_type_enc_quasi_sound = is_type_level_quasi_sound o level_of_type_enc
+
+fun is_type_level_fairly_sound (Fin_Nonmono_Types _) = true
+ | is_type_level_fairly_sound level = is_type_level_quasi_sound level
+val is_type_enc_fairly_sound = is_type_level_fairly_sound o level_of_type_enc
+
+fun is_type_level_monotonicity_based (Noninf_Nonmono_Types _) = true
+ | is_type_level_monotonicity_based (Fin_Nonmono_Types _) = true
+ | is_type_level_monotonicity_based _ = false
+
+(* "_query", "_bang", and "_at" are for the ASCII-challenged Metis and
+ Mirabelle. *)
+val queries = ["?", "_query"]
+val bangs = ["!", "_bang"]
+val ats = ["@", "_at"]
fun try_unsuffixes ss s =
fold (fn s' => fn NONE => try (unsuffix s') s | some => some) ss NONE
+fun try_nonmono constr suffixes fallback s =
+ case try_unsuffixes suffixes s of
+ SOME s =>
+ (case try_unsuffixes suffixes s of
+ SOME s => (constr Positively_Naked_Vars, s)
+ | NONE =>
+ case try_unsuffixes ats s of
+ SOME s => (constr Ghost_Type_Arg_Vars, s)
+ | NONE => (constr All_Vars, s))
+ | NONE => fallback s
+
+fun is_incompatible_type_level poly level =
+ poly = Mangled_Monomorphic andalso
+ granularity_of_type_level level = Ghost_Type_Arg_Vars
+
fun type_enc_from_string soundness s =
(case try (unprefix "poly_") s of
SOME s => (SOME Polymorphic, s)
@@ -563,83 +605,55 @@
case try (unprefix "mono_") s of
SOME s => (SOME Mangled_Monomorphic, s)
| NONE => (NONE, s))
- ||> (fn s =>
- (* "_query" and "_bang" are for the ASCII-challenged Metis and
- Mirabelle. *)
- case try_unsuffixes ["?", "_query"] s of
- SOME s => (Noninf_Nonmono_Types soundness, s)
- | NONE =>
- case try_unsuffixes ["!", "_bang"] s of
- SOME s => (Fin_Nonmono_Types, s)
- | NONE => (All_Types, s))
- ||> apsnd (fn s =>
- case try (unsuffix "_uniform") s of
- SOME s => (Uniform, s)
- | NONE => (Nonuniform, s))
- |> (fn (poly, (level, (uniformity, core))) =>
- case (core, (poly, level, uniformity)) of
- ("simple", (SOME poly, _, Nonuniform)) =>
+ ||> (pair All_Types
+ |> try_nonmono Fin_Nonmono_Types bangs
+ |> try_nonmono (curry Noninf_Nonmono_Types soundness) queries)
+ |> (fn (poly, (level, core)) =>
+ case (core, (poly, level)) of
+ ("simple", (SOME poly, _)) =>
(case (poly, level) of
(Polymorphic, All_Types) =>
Simple_Types (First_Order, Polymorphic, All_Types)
| (Mangled_Monomorphic, _) =>
- Simple_Types (First_Order, Mangled_Monomorphic, level)
- | _ => raise Same.SAME)
- | ("simple_higher", (SOME poly, _, Nonuniform)) =>
- (case (poly, level) of
- (_, Noninf_Nonmono_Types _) => raise Same.SAME
- | (Mangled_Monomorphic, _) =>
- Simple_Types (Higher_Order, Mangled_Monomorphic, level)
+ if granularity_of_type_level level = All_Vars then
+ Simple_Types (First_Order, Mangled_Monomorphic, level)
+ else
+ raise Same.SAME
| _ => raise Same.SAME)
- | ("guards", (SOME poly, _, _)) => Guards (poly, level, uniformity)
- | ("tags", (SOME Polymorphic, _, _)) =>
- Tags (Polymorphic, level, uniformity)
- | ("tags", (SOME poly, _, _)) => Tags (poly, level, uniformity)
- | ("args", (SOME poly, All_Types (* naja *), Nonuniform)) =>
- Guards (poly, Const_Arg_Types, Nonuniform)
- | ("erased", (NONE, All_Types (* naja *), Nonuniform)) =>
- Guards (Polymorphic, No_Types, Nonuniform)
+ | ("simple_higher", (SOME poly, _)) =>
+ (case (poly, level) of
+ (Polymorphic, All_Types) =>
+ Simple_Types (Higher_Order, Polymorphic, All_Types)
+ | (_, Noninf_Nonmono_Types _) => raise Same.SAME
+ | (Mangled_Monomorphic, _) =>
+ if granularity_of_type_level level = All_Vars then
+ Simple_Types (Higher_Order, Mangled_Monomorphic, level)
+ else
+ raise Same.SAME
+ | _ => raise Same.SAME)
+ | ("guards", (SOME poly, _)) =>
+ if is_incompatible_type_level poly level then raise Same.SAME
+ else Guards (poly, level)
+ | ("tags", (SOME poly, _)) =>
+ if is_incompatible_type_level poly level then raise Same.SAME
+ else Tags (poly, level)
+ | ("args", (SOME poly, All_Types (* naja *))) =>
+ Guards (poly, Const_Arg_Types)
+ | ("erased", (NONE, All_Types (* naja *))) =>
+ Guards (Polymorphic, No_Types)
| _ => raise Same.SAME)
- handle Same.SAME => error ("Unknown type system: " ^ quote s ^ ".")
-
-fun is_type_enc_higher_order (Simple_Types (Higher_Order, _, _)) = true
- | is_type_enc_higher_order _ = false
-
-fun polymorphism_of_type_enc (Simple_Types (_, poly, _)) = poly
- | polymorphism_of_type_enc (Guards (poly, _, _)) = poly
- | polymorphism_of_type_enc (Tags (poly, _, _)) = poly
+ handle Same.SAME => error ("Unknown type encoding: " ^ quote s ^ ".")
-fun level_of_type_enc (Simple_Types (_, _, level)) = level
- | level_of_type_enc (Guards (_, level, _)) = level
- | level_of_type_enc (Tags (_, level, _)) = level
-
-fun uniformity_of_type_enc (Simple_Types _) = Uniform
- | uniformity_of_type_enc (Guards (_, _, uniformity)) = uniformity
- | uniformity_of_type_enc (Tags (_, _, uniformity)) = uniformity
-
-fun is_type_level_quasi_sound All_Types = true
- | is_type_level_quasi_sound (Noninf_Nonmono_Types _) = true
- | is_type_level_quasi_sound _ = false
-val is_type_enc_quasi_sound =
- is_type_level_quasi_sound o level_of_type_enc
-
-fun is_type_level_fairly_sound level =
- is_type_level_quasi_sound level orelse level = Fin_Nonmono_Types
-val is_type_enc_fairly_sound = is_type_level_fairly_sound o level_of_type_enc
-
-fun is_type_level_monotonicity_based (Noninf_Nonmono_Types _) = true
- | is_type_level_monotonicity_based Fin_Nonmono_Types = true
- | is_type_level_monotonicity_based _ = false
-
-fun adjust_type_enc (THF0 _) (Simple_Types (order, Polymorphic, level)) =
+fun adjust_type_enc (THF (TPTP_Monomorphic, _, _))
+ (Simple_Types (order, _, level)) =
Simple_Types (order, Mangled_Monomorphic, level)
- | adjust_type_enc (THF0 _) type_enc = type_enc
- | adjust_type_enc (TFF (TFF_Monomorphic, _)) (Simple_Types (_, _, level)) =
+ | adjust_type_enc (THF _) type_enc = type_enc
+ | adjust_type_enc (TFF (TPTP_Monomorphic, _)) (Simple_Types (_, _, level)) =
Simple_Types (First_Order, Mangled_Monomorphic, level)
- | adjust_type_enc (TFF (_, _)) (Simple_Types (_, poly, level)) =
+ | adjust_type_enc (TFF _) (Simple_Types (_, poly, level)) =
Simple_Types (First_Order, poly, level)
| adjust_type_enc format (Simple_Types (_, poly, level)) =
- adjust_type_enc format (Guards (poly, level, Uniform))
+ adjust_type_enc format (Guards (poly, level))
| adjust_type_enc CNF_UEQ (type_enc as Guards stuff) =
(if is_type_enc_fairly_sound type_enc then Tags else Guards) stuff
| adjust_type_enc _ type_enc = type_enc
@@ -690,34 +704,28 @@
(* The Booleans indicate whether all type arguments should be kept. *)
datatype type_arg_policy =
Explicit_Type_Args of bool |
- Mangled_Type_Args of bool |
+ Mangled_Type_Args |
No_Type_Args
-fun should_drop_arg_type_args (Simple_Types _) = false
- | should_drop_arg_type_args type_enc =
- level_of_type_enc type_enc = All_Types andalso
- uniformity_of_type_enc type_enc = Uniform
-
fun type_arg_policy type_enc s =
- if s = type_tag_name then
- (if polymorphism_of_type_enc type_enc = Mangled_Monomorphic then
- Mangled_Type_Args
- else
- Explicit_Type_Args) false
- else case type_enc of
- Tags (_, All_Types, Uniform) => No_Type_Args
- | _ =>
- let val level = level_of_type_enc type_enc in
- if level = No_Types orelse s = @{const_name HOL.eq} orelse
- (s = app_op_name andalso level = Const_Arg_Types) then
- No_Type_Args
- else
- should_drop_arg_type_args type_enc
- |> (if polymorphism_of_type_enc type_enc = Mangled_Monomorphic then
- Mangled_Type_Args
- else
- Explicit_Type_Args)
- end
+ let val mangled = (polymorphism_of_type_enc type_enc = Mangled_Monomorphic) in
+ if s = type_tag_name then
+ if mangled then Mangled_Type_Args else Explicit_Type_Args false
+ else case type_enc of
+ Tags (_, All_Types) => No_Type_Args
+ | _ =>
+ let val level = level_of_type_enc type_enc in
+ if level = No_Types orelse s = @{const_name HOL.eq} orelse
+ (s = app_op_name andalso level = Const_Arg_Types) then
+ No_Type_Args
+ else if mangled then
+ Mangled_Type_Args
+ else
+ Explicit_Type_Args
+ (level = All_Types orelse
+ granularity_of_type_level level = Ghost_Type_Arg_Vars)
+ end
+ end
(* Make atoms for sorted type variables. *)
fun generic_add_sorts_on_type (_, []) = I
@@ -746,8 +754,9 @@
fun type_class_formula type_enc class arg =
AAtom (ATerm (class, arg ::
(case type_enc of
- Simple_Types (_, Polymorphic, _) =>
- if exploit_tff1_dummy_type_vars then [] else [ATerm (TYPE_name, [arg])]
+ Simple_Types (First_Order, Polymorphic, _) =>
+ if avoid_first_order_dummy_type_vars then [ATerm (TYPE_name, [arg])]
+ else []
| _ => [])))
fun formulas_for_types type_enc add_sorts_on_typ Ts =
[] |> level_of_type_enc type_enc <> No_Types ? fold add_sorts_on_typ Ts
@@ -896,7 +905,7 @@
(hd ss, map unmangled_type (tl ss))
end
-fun introduce_proxies type_enc =
+fun introduce_proxies_in_iterm type_enc =
let
fun tweak_ho_quant ho_quant T [IAbs _] = IConst (`I ho_quant, T, [])
| tweak_ho_quant ho_quant (T as Type (_, [p_T as Type (_, [x_T, _]), _]))
@@ -945,11 +954,79 @@
| intro _ _ tm = tm
in intro true [] end
-fun iformula_from_prop thy format type_enc eq_as_iff =
+fun mangle_type_args_in_iterm format type_enc =
+ if polymorphism_of_type_enc type_enc = Mangled_Monomorphic then
+ let
+ fun mangle (IApp (tm1, tm2)) = IApp (mangle tm1, mangle tm2)
+ | mangle (tm as IConst (_, _, [])) = tm
+ | mangle (tm as IConst (name as (s, _), T, T_args)) =
+ (case strip_prefix_and_unascii const_prefix s of
+ NONE => tm
+ | SOME s'' =>
+ case type_arg_policy type_enc (invert_const s'') of
+ Mangled_Type_Args =>
+ IConst (mangled_const_name format type_enc T_args name, T, [])
+ | _ => tm)
+ | mangle (IAbs (bound, tm)) = IAbs (bound, mangle tm)
+ | mangle tm = tm
+ in mangle end
+ else
+ I
+
+fun chop_fun 0 T = ([], T)
+ | chop_fun n (Type (@{type_name fun}, [dom_T, ran_T])) =
+ chop_fun (n - 1) ran_T |>> cons dom_T
+ | chop_fun _ T = ([], T)
+
+fun filter_const_type_args _ _ _ [] = []
+ | filter_const_type_args thy s ary T_args =
+ let
+ val U = robust_const_type thy s
+ val arg_U_vars = fold Term.add_tvarsT (U |> chop_fun ary |> fst) []
+ val U_args = (s, U) |> robust_const_typargs thy
+ in
+ U_args ~~ T_args
+ |> map (fn (U, T) =>
+ if member (op =) arg_U_vars (dest_TVar U) then dummyT else T)
+ end
+ handle TYPE _ => T_args
+
+fun filter_type_args_in_iterm thy type_enc =
let
+ fun filt ary (IApp (tm1, tm2)) = IApp (filt (ary + 1) tm1, filt 0 tm2)
+ | filt _ (tm as IConst (_, _, [])) = tm
+ | filt ary (IConst (name as (s, _), T, T_args)) =
+ (case strip_prefix_and_unascii const_prefix s of
+ NONE =>
+ (name,
+ if level_of_type_enc type_enc = No_Types orelse s = tptp_choice then
+ []
+ else
+ T_args)
+ | SOME s'' =>
+ let
+ val s'' = invert_const s''
+ fun filter_T_args false = T_args
+ | filter_T_args true = filter_const_type_args thy s'' ary T_args
+ in
+ case type_arg_policy type_enc s'' of
+ Explicit_Type_Args drop_args => (name, filter_T_args drop_args)
+ | No_Type_Args => (name, [])
+ | Mangled_Type_Args => raise Fail "unexpected (un)mangled symbol"
+ end)
+ |> (fn (name, T_args) => IConst (name, T, T_args))
+ | filt _ (IAbs (bound, tm)) = IAbs (bound, filt 0 tm)
+ | filt _ tm = tm
+ in filt 0 end
+
+fun iformula_from_prop ctxt format type_enc eq_as_iff =
+ let
+ val thy = Proof_Context.theory_of ctxt
fun do_term bs t atomic_types =
iterm_from_term thy format bs (Envir.eta_contract t)
- |>> (introduce_proxies type_enc #> AAtom)
+ |>> (introduce_proxies_in_iterm type_enc
+ #> mangle_type_args_in_iterm format type_enc
+ #> AAtom)
||> union (op =) atomic_types
fun do_quant bs q pos s T t' =
let
@@ -1017,28 +1094,31 @@
t
else
let
- fun aux Ts t =
+ fun trans Ts t =
case t of
- @{const Not} $ t1 => @{const Not} $ aux Ts t1
+ @{const Not} $ t1 => @{const Not} $ trans Ts t1
| (t0 as Const (@{const_name All}, _)) $ Abs (s, T, t') =>
- t0 $ Abs (s, T, aux (T :: Ts) t')
+ t0 $ Abs (s, T, trans (T :: Ts) t')
| (t0 as Const (@{const_name All}, _)) $ t1 =>
- aux Ts (t0 $ eta_expand Ts t1 1)
+ trans Ts (t0 $ eta_expand Ts t1 1)
| (t0 as Const (@{const_name Ex}, _)) $ Abs (s, T, t') =>
- t0 $ Abs (s, T, aux (T :: Ts) t')
+ t0 $ Abs (s, T, trans (T :: Ts) t')
| (t0 as Const (@{const_name Ex}, _)) $ t1 =>
- aux Ts (t0 $ eta_expand Ts t1 1)
- | (t0 as @{const HOL.conj}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
- | (t0 as @{const HOL.disj}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
- | (t0 as @{const HOL.implies}) $ t1 $ t2 => t0 $ aux Ts t1 $ aux Ts t2
+ trans Ts (t0 $ eta_expand Ts t1 1)
+ | (t0 as @{const HOL.conj}) $ t1 $ t2 =>
+ t0 $ trans Ts t1 $ trans Ts t2
+ | (t0 as @{const HOL.disj}) $ t1 $ t2 =>
+ t0 $ trans Ts t1 $ trans Ts t2
+ | (t0 as @{const HOL.implies}) $ t1 $ t2 =>
+ t0 $ trans Ts t1 $ trans Ts t2
| (t0 as Const (@{const_name HOL.eq}, Type (_, [@{typ bool}, _])))
$ t1 $ t2 =>
- t0 $ aux Ts t1 $ aux Ts t2
+ t0 $ trans Ts t1 $ trans Ts t2
| _ =>
if not (exists_subterm (fn Abs _ => true | _ => false) t) then t
else t |> Envir.eta_contract |> do_lambdas ctxt Ts
val (t, ctxt') = Variable.import_terms true [t] ctxt |>> the_single
- in t |> aux [] |> singleton (Variable.export_terms ctxt' ctxt) end
+ in t |> trans [] |> singleton (Variable.export_terms ctxt' ctxt) end
end
fun do_cheaply_conceal_lambdas Ts (t1 $ t2) =
@@ -1076,12 +1156,12 @@
same in Sledgehammer to prevent the discovery of unreplayable proofs. *)
fun freeze_term t =
let
- fun aux (t $ u) = aux t $ aux u
- | aux (Abs (s, T, t)) = Abs (s, T, aux t)
- | aux (Var ((s, i), T)) =
+ fun freeze (t $ u) = freeze t $ freeze u
+ | freeze (Abs (s, T, t)) = Abs (s, T, freeze t)
+ | freeze (Var ((s, i), T)) =
Free (atp_weak_prefix ^ s ^ "_" ^ string_of_int i, T)
- | aux t = t
- in t |> exists_subterm is_Var t ? aux end
+ | freeze t = t
+ in t |> exists_subterm is_Var t ? freeze end
fun presimp_prop ctxt presimp_consts t =
let
@@ -1099,34 +1179,57 @@
end
(* making fact and conjecture formulas *)
-fun make_formula thy format type_enc eq_as_iff name loc kind t =
+fun make_formula ctxt format type_enc eq_as_iff name loc kind t =
let
val (iformula, atomic_types) =
- iformula_from_prop thy format type_enc eq_as_iff (SOME (kind <> Conjecture)) t []
+ iformula_from_prop ctxt format type_enc eq_as_iff
+ (SOME (kind <> Conjecture)) t []
in
{name = name, locality = loc, kind = kind, iformula = iformula,
atomic_types = atomic_types}
end
fun make_fact ctxt format type_enc eq_as_iff ((name, loc), t) =
- let val thy = Proof_Context.theory_of ctxt in
- case t |> make_formula thy format type_enc (eq_as_iff andalso format <> CNF)
- name loc Axiom of
- formula as {iformula = AAtom (IConst ((s, _), _, _)), ...} =>
- if s = tptp_true then NONE else SOME formula
- | formula => SOME formula
- end
+ case t |> make_formula ctxt format type_enc (eq_as_iff andalso format <> CNF)
+ name loc Axiom of
+ formula as {iformula = AAtom (IConst ((s, _), _, _)), ...} =>
+ if s = tptp_true then NONE else SOME formula
+ | formula => SOME formula
fun s_not_trueprop (@{const Trueprop} $ t) = @{const Trueprop} $ s_not t
| s_not_trueprop t = s_not t
-fun make_conjecture thy format type_enc =
+fun make_conjecture ctxt format type_enc =
map (fn ((name, loc), (kind, t)) =>
t |> kind = Conjecture ? s_not_trueprop
- |> make_formula thy format type_enc (format <> CNF) name loc kind)
+ |> make_formula ctxt format type_enc (format <> CNF) name loc kind)
(** Finite and infinite type inference **)
+fun tvar_footprint thy s ary =
+ (case strip_prefix_and_unascii const_prefix s of
+ SOME s =>
+ s |> invert_const |> robust_const_type thy |> chop_fun ary |> fst
+ |> map (fn T => Term.add_tvarsT T [] |> map fst)
+ | NONE => [])
+ handle TYPE _ => []
+
+fun ghost_type_args thy s ary =
+ let
+ val footprint = tvar_footprint thy s ary
+ fun ghosts _ [] = []
+ | ghosts seen ((i, tvars) :: args) =
+ ghosts (union (op =) seen tvars) args
+ |> exists (not o member (op =) seen) tvars ? cons i
+ in
+ if forall null footprint then
+ []
+ else
+ 0 upto length footprint - 1 ~~ footprint
+ |> sort (rev_order o list_ord Term_Ord.indexname_ord o pairself snd)
+ |> ghosts []
+ end
+
type monotonicity_info =
{maybe_finite_Ts : typ list,
surely_finite_Ts : typ list,
@@ -1150,24 +1253,25 @@
fun should_encode_type _ (_ : monotonicity_info) All_Types _ = true
| should_encode_type ctxt {maybe_finite_Ts, surely_infinite_Ts,
maybe_nonmono_Ts, ...}
- (Noninf_Nonmono_Types soundness) T =
- exists (type_intersect ctxt T) maybe_nonmono_Ts andalso
- not (exists (type_instance ctxt T) surely_infinite_Ts orelse
- (not (member (type_aconv ctxt) maybe_finite_Ts T) andalso
- is_type_kind_of_surely_infinite ctxt soundness surely_infinite_Ts T))
+ (Noninf_Nonmono_Types (soundness, grain)) T =
+ grain = Ghost_Type_Arg_Vars orelse
+ (exists (type_intersect ctxt T) maybe_nonmono_Ts andalso
+ not (exists (type_instance ctxt T) surely_infinite_Ts orelse
+ (not (member (type_aconv ctxt) maybe_finite_Ts T) andalso
+ is_type_kind_of_surely_infinite ctxt soundness surely_infinite_Ts
+ T)))
| should_encode_type ctxt {surely_finite_Ts, maybe_infinite_Ts,
maybe_nonmono_Ts, ...}
- Fin_Nonmono_Types T =
- exists (type_intersect ctxt T) maybe_nonmono_Ts andalso
- (exists (type_generalization ctxt T) surely_finite_Ts orelse
- (not (member (type_aconv ctxt) maybe_infinite_Ts T) andalso
- is_type_surely_finite ctxt T))
+ (Fin_Nonmono_Types grain) T =
+ grain = Ghost_Type_Arg_Vars orelse
+ (exists (type_intersect ctxt T) maybe_nonmono_Ts andalso
+ (exists (type_generalization ctxt T) surely_finite_Ts orelse
+ (not (member (type_aconv ctxt) maybe_infinite_Ts T) andalso
+ is_type_surely_finite ctxt T)))
| should_encode_type _ _ _ _ = false
-fun should_guard_type ctxt mono (Guards (_, level, uniformity)) should_guard_var
- T =
- (uniformity = Uniform orelse should_guard_var ()) andalso
- should_encode_type ctxt mono level T
+fun should_guard_type ctxt mono (Guards (_, level)) should_guard_var T =
+ should_guard_var () andalso should_encode_type ctxt mono level T
| should_guard_type _ _ _ _ _ = false
fun is_maybe_universal_var (IConst ((s, _), _, _)) =
@@ -1179,15 +1283,20 @@
datatype tag_site =
Top_Level of bool option |
Eq_Arg of bool option |
+ Arg of string * int |
Elsewhere
fun should_tag_with_type _ _ _ (Top_Level _) _ _ = false
- | should_tag_with_type ctxt mono (Tags (_, level, uniformity)) site u T =
- (case uniformity of
- Uniform => should_encode_type ctxt mono level T
- | Nonuniform =>
+ | should_tag_with_type ctxt mono (Tags (_, level)) site u T =
+ (case granularity_of_type_level level of
+ All_Vars => should_encode_type ctxt mono level T
+ | grain =>
case (site, is_maybe_universal_var u) of
(Eq_Arg _, true) => should_encode_type ctxt mono level T
+ | (Arg (s, j), true) =>
+ grain = Ghost_Type_Arg_Vars andalso
+ member (op =)
+ (ghost_type_args (Proof_Context.theory_of ctxt) s (j + 1)) j
| _ => false)
| should_tag_with_type _ _ _ _ _ _ = false
@@ -1203,11 +1312,27 @@
(** predicators and application operators **)
type sym_info =
- {pred_sym : bool, min_ary : int, max_ary : int, types : typ list}
+ {pred_sym : bool, min_ary : int, max_ary : int, types : typ list,
+ in_conj : bool}
-fun add_iterm_syms_to_table ctxt explicit_apply =
+fun default_sym_tab_entries type_enc =
+ (make_fixed_const NONE @{const_name undefined},
+ {pred_sym = false, min_ary = 0, max_ary = 0, types = [],
+ in_conj = false}) ::
+ ([tptp_false, tptp_true]
+ |> map (rpair {pred_sym = true, min_ary = 0, max_ary = 0, types = [],
+ in_conj = false})) @
+ ([tptp_equal, tptp_old_equal]
+ |> map (rpair {pred_sym = true, min_ary = 2, max_ary = 2, types = [],
+ in_conj = false}))
+ |> not (is_type_enc_higher_order type_enc)
+ ? cons (prefixed_predicator_name,
+ {pred_sym = true, min_ary = 1, max_ary = 1, types = [],
+ in_conj = false})
+
+fun sym_table_for_facts ctxt type_enc explicit_apply conjs facts =
let
- fun consider_var_arity const_T var_T max_ary =
+ fun consider_var_ary const_T var_T max_ary =
let
fun iter ary T =
if ary = max_ary orelse type_instance ctxt var_T T orelse
@@ -1221,10 +1346,10 @@
(can dest_funT T orelse T = @{typ bool}) then
let
val bool_vars' = bool_vars orelse body_type T = @{typ bool}
- fun repair_min_arity {pred_sym, min_ary, max_ary, types} =
+ fun repair_min_ary {pred_sym, min_ary, max_ary, types, in_conj} =
{pred_sym = pred_sym andalso not bool_vars',
- min_ary = fold (fn T' => consider_var_arity T' T) types min_ary,
- max_ary = max_ary, types = types}
+ min_ary = fold (fn T' => consider_var_ary T' T) types min_ary,
+ max_ary = max_ary, types = types, in_conj = in_conj}
val fun_var_Ts' =
fun_var_Ts |> can dest_funT T ? insert_type ctxt I T
in
@@ -1232,83 +1357,76 @@
pointer_eq (fun_var_Ts', fun_var_Ts) then
accum
else
- ((bool_vars', fun_var_Ts'), Symtab.map (K repair_min_arity) sym_tab)
+ ((bool_vars', fun_var_Ts'), Symtab.map (K repair_min_ary) sym_tab)
end
else
accum
- fun add top_level tm (accum as ((bool_vars, fun_var_Ts), sym_tab)) =
- let val (head, args) = strip_iterm_comb tm in
- (case head of
- IConst ((s, _), T, _) =>
- if String.isPrefix bound_var_prefix s orelse
- String.isPrefix all_bound_var_prefix s then
- add_universal_var T accum
- else if String.isPrefix exist_bound_var_prefix s then
- accum
- else
- let val ary = length args in
- ((bool_vars, fun_var_Ts),
- case Symtab.lookup sym_tab s of
- SOME {pred_sym, min_ary, max_ary, types} =>
- let
- val pred_sym =
- pred_sym andalso top_level andalso not bool_vars
- val types' = types |> insert_type ctxt I T
- val min_ary =
- if is_some explicit_apply orelse
- pointer_eq (types', types) then
- min_ary
- else
- fold (consider_var_arity T) fun_var_Ts min_ary
- in
- Symtab.update (s, {pred_sym = pred_sym,
- min_ary = Int.min (ary, min_ary),
- max_ary = Int.max (ary, max_ary),
- types = types'})
- sym_tab
- end
- | NONE =>
- let
- val pred_sym = top_level andalso not bool_vars
- val min_ary =
- case explicit_apply of
- SOME true => 0
- | SOME false => ary
- | NONE => fold (consider_var_arity T) fun_var_Ts ary
- in
- Symtab.update_new (s, {pred_sym = pred_sym,
- min_ary = min_ary, max_ary = ary,
- types = [T]})
+ fun add_fact_syms conj_fact =
+ let
+ fun add_iterm_syms top_level tm
+ (accum as ((bool_vars, fun_var_Ts), sym_tab)) =
+ let val (head, args) = strip_iterm_comb tm in
+ (case head of
+ IConst ((s, _), T, _) =>
+ if String.isPrefix bound_var_prefix s orelse
+ String.isPrefix all_bound_var_prefix s then
+ add_universal_var T accum
+ else if String.isPrefix exist_bound_var_prefix s then
+ accum
+ else
+ let val ary = length args in
+ ((bool_vars, fun_var_Ts),
+ case Symtab.lookup sym_tab s of
+ SOME {pred_sym, min_ary, max_ary, types, in_conj} =>
+ let
+ val pred_sym =
+ pred_sym andalso top_level andalso not bool_vars
+ val types' = types |> insert_type ctxt I T
+ val in_conj = in_conj orelse conj_fact
+ val min_ary =
+ if is_some explicit_apply orelse
+ pointer_eq (types', types) then
+ min_ary
+ else
+ fold (consider_var_ary T) fun_var_Ts min_ary
+ in
+ Symtab.update (s, {pred_sym = pred_sym,
+ min_ary = Int.min (ary, min_ary),
+ max_ary = Int.max (ary, max_ary),
+ types = types', in_conj = in_conj})
sym_tab
- end)
- end
- | IVar (_, T) => add_universal_var T accum
- | IAbs ((_, T), tm) => accum |> add_universal_var T |> add false tm
- | _ => accum)
- |> fold (add false) args
- end
- in add true end
-fun add_fact_syms_to_table ctxt explicit_apply =
- K (add_iterm_syms_to_table ctxt explicit_apply)
- |> formula_fold NONE |> fact_lift
+ end
+ | NONE =>
+ let
+ val pred_sym = top_level andalso not bool_vars
+ val min_ary =
+ case explicit_apply of
+ SOME true => 0
+ | SOME false => ary
+ | NONE => fold (consider_var_ary T) fun_var_Ts ary
+ in
+ Symtab.update_new (s,
+ {pred_sym = pred_sym, min_ary = min_ary,
+ max_ary = ary, types = [T], in_conj = conj_fact})
+ sym_tab
+ end)
+ end
+ | IVar (_, T) => add_universal_var T accum
+ | IAbs ((_, T), tm) =>
+ accum |> add_universal_var T |> add_iterm_syms false tm
+ | _ => accum)
+ |> fold (add_iterm_syms false) args
+ end
+ in K (add_iterm_syms true) |> formula_fold NONE |> fact_lift end
+ in
+ ((false, []), Symtab.empty)
+ |> fold (add_fact_syms true) conjs
+ |> fold (add_fact_syms false) facts
+ |> snd
+ |> fold Symtab.update (default_sym_tab_entries type_enc)
+ end
-val default_sym_tab_entries : (string * sym_info) list =
- (prefixed_predicator_name,
- {pred_sym = true, min_ary = 1, max_ary = 1, types = []})
- (* FIXME: needed? *) ::
- (make_fixed_const NONE @{const_name undefined},
- {pred_sym = false, min_ary = 0, max_ary = 0, types = []}) ::
- ([tptp_false, tptp_true]
- |> map (rpair {pred_sym = true, min_ary = 0, max_ary = 0, types = []})) @
- ([tptp_equal, tptp_old_equal]
- |> map (rpair {pred_sym = true, min_ary = 2, max_ary = 2, types = []}))
-
-fun sym_table_for_facts ctxt explicit_apply facts =
- ((false, []), Symtab.empty)
- |> fold (add_fact_syms_to_table ctxt explicit_apply) facts |> snd
- |> fold Symtab.update default_sym_tab_entries
-
-fun min_arity_of sym_tab s =
+fun min_ary_of sym_tab s =
case Symtab.lookup sym_tab s of
SOME ({min_ary, ...} : sym_info) => min_ary
| NONE =>
@@ -1332,92 +1450,42 @@
pred_sym andalso min_ary = max_ary
| NONE => false
+val app_op = `(make_fixed_const NONE) app_op_name
val predicator_combconst =
IConst (`(make_fixed_const NONE) predicator_name, @{typ "bool => bool"}, [])
-fun predicator tm = IApp (predicator_combconst, tm)
-
-fun introduce_predicators_in_iterm sym_tab tm =
- case strip_iterm_comb tm of
- (IConst ((s, _), _, _), _) =>
- if is_pred_sym sym_tab s then tm else predicator tm
- | _ => predicator tm
fun list_app head args = fold (curry (IApp o swap)) args head
+fun predicator tm = IApp (predicator_combconst, tm)
-val app_op = `(make_fixed_const NONE) app_op_name
-
-fun explicit_app arg head =
+fun firstorderize_fact thy format type_enc sym_tab =
let
- val head_T = ityp_of head
- val (arg_T, res_T) = dest_funT head_T
- val explicit_app = IConst (app_op, head_T --> head_T, [arg_T, res_T])
- in list_app explicit_app [head, arg] end
-fun list_explicit_app head args = fold explicit_app args head
-
-fun introduce_explicit_apps_in_iterm sym_tab =
- let
- fun aux tm =
+ fun do_app arg head =
+ let
+ val head_T = ityp_of head
+ val (arg_T, res_T) = dest_funT head_T
+ val app =
+ IConst (app_op, head_T --> head_T, [arg_T, res_T])
+ |> mangle_type_args_in_iterm format type_enc
+ in list_app app [head, arg] end
+ fun list_app_ops head args = fold do_app args head
+ fun introduce_app_ops tm =
case strip_iterm_comb tm of
(head as IConst ((s, _), _, _), args) =>
- args |> map aux
- |> chop (min_arity_of sym_tab s)
+ args |> map introduce_app_ops
+ |> chop (min_ary_of sym_tab s)
|>> list_app head
- |-> list_explicit_app
- | (head, args) => list_explicit_app head (map aux args)
- in aux end
-
-fun chop_fun 0 T = ([], T)
- | chop_fun n (Type (@{type_name fun}, [dom_T, ran_T])) =
- chop_fun (n - 1) ran_T |>> cons dom_T
- | chop_fun _ _ = raise Fail "unexpected non-function"
-
-fun filter_type_args _ _ _ [] = []
- | filter_type_args thy s arity T_args =
- let
- val U = robust_const_type thy s
- val arg_U_vars = fold Term.add_tvarsT (U |> chop_fun arity |> fst) []
- val U_args = (s, U) |> robust_const_typargs thy
- in
- U_args ~~ T_args
- |> map (fn (U, T) =>
- if member (op =) arg_U_vars (dest_TVar U) then dummyT else T)
- end
- handle TYPE _ => T_args
-
-fun enforce_type_arg_policy_in_iterm ctxt format type_enc =
- let
- val thy = Proof_Context.theory_of ctxt
- fun aux arity (IApp (tm1, tm2)) = IApp (aux (arity + 1) tm1, aux 0 tm2)
- | aux arity (IConst (name as (s, _), T, T_args)) =
- (case strip_prefix_and_unascii const_prefix s of
- NONE =>
- (name, if level_of_type_enc type_enc = No_Types orelse s = tptp_choice
- then [] else T_args)
- | SOME s'' =>
- let
- val s'' = invert_const s''
- fun filter_T_args false = T_args
- | filter_T_args true = filter_type_args thy s'' arity T_args
- in
- case type_arg_policy type_enc s'' of
- Explicit_Type_Args drop_args => (name, filter_T_args drop_args)
- | Mangled_Type_Args drop_args =>
- (mangled_const_name format type_enc (filter_T_args drop_args)
- name, [])
- | No_Type_Args => (name, [])
- end)
- |> (fn (name, T_args) => IConst (name, T, T_args))
- | aux _ (IAbs (bound, tm)) = IAbs (bound, aux 0 tm)
- | aux _ tm = tm
- in aux 0 end
-
-fun repair_iterm ctxt format type_enc sym_tab =
- not (is_type_enc_higher_order type_enc)
- ? (introduce_explicit_apps_in_iterm sym_tab
- #> introduce_predicators_in_iterm sym_tab)
- #> enforce_type_arg_policy_in_iterm ctxt format type_enc
-fun repair_fact ctxt format type_enc sym_tab =
- update_iformula (formula_map (repair_iterm ctxt format type_enc sym_tab))
+ |-> list_app_ops
+ | (head, args) => list_app_ops head (map introduce_app_ops args)
+ fun introduce_predicators tm =
+ case strip_iterm_comb tm of
+ (IConst ((s, _), _, _), _) =>
+ if is_pred_sym sym_tab s then tm else predicator tm
+ | _ => predicator tm
+ val do_iterm =
+ not (is_type_enc_higher_order type_enc)
+ ? (introduce_app_ops #> introduce_predicators)
+ #> filter_type_args_in_iterm thy type_enc
+ in update_iformula (formula_map do_iterm) end
(** Helper facts **)
@@ -1594,7 +1662,7 @@
|>> apfst (map (apsnd (apsnd freeze_term)))
else
((conjs, facts), [])
- val conjs = conjs |> make_conjecture thy format type_enc
+ val conjs = conjs |> make_conjecture ctxt format type_enc
val (fact_names, facts) =
facts
|> map_filter (fn (name, (_, t)) =>
@@ -1618,21 +1686,45 @@
val type_guard = `(make_fixed_const NONE) type_guard_name
-fun type_guard_iterm ctxt format type_enc T tm =
+fun type_guard_iterm format type_enc T tm =
IApp (IConst (type_guard, T --> @{typ bool}, [T])
- |> enforce_type_arg_policy_in_iterm ctxt format type_enc, tm)
+ |> mangle_type_args_in_iterm format type_enc, tm)
fun is_var_positively_naked_in_term _ (SOME false) _ accum = accum
| is_var_positively_naked_in_term name _ (ATerm ((s, _), tms)) accum =
accum orelse (is_tptp_equal s andalso member (op =) tms (ATerm (name, [])))
| is_var_positively_naked_in_term _ _ _ _ = true
-fun should_guard_var_in_formula pos phi (SOME true) name =
- formula_fold pos (is_var_positively_naked_in_term name) phi false
- | should_guard_var_in_formula _ _ _ _ = true
+fun is_var_ghost_type_arg_in_term thy name pos tm accum =
+ is_var_positively_naked_in_term name pos tm accum orelse
+ let
+ val var = ATerm (name, [])
+ fun is_nasty_in_term (ATerm (_, [])) = false
+ | is_nasty_in_term (ATerm ((s, _), tms)) =
+ (member (op =) tms var andalso
+ let val ary = length tms in
+ case ghost_type_args thy s ary of
+ [] => false
+ | ghosts =>
+ exists (fn (j, tm) => tm = var andalso member (op =) ghosts j)
+ (0 upto length tms - 1 ~~ tms)
+ end) orelse
+ exists is_nasty_in_term tms
+ | is_nasty_in_term _ = true
+ in is_nasty_in_term tm end
+
+fun should_guard_var_in_formula thy level pos phi (SOME true) name =
+ (case granularity_of_type_level level of
+ All_Vars => true
+ | Positively_Naked_Vars =>
+ formula_fold pos (is_var_positively_naked_in_term name) phi false
+ | Ghost_Type_Arg_Vars =>
+ formula_fold pos (is_var_ghost_type_arg_in_term thy name) phi false)
+ | should_guard_var_in_formula _ _ _ _ _ _ = true
fun should_generate_tag_bound_decl _ _ _ (SOME true) _ = false
- | should_generate_tag_bound_decl ctxt mono (Tags (_, level, Nonuniform)) _ T =
+ | should_generate_tag_bound_decl ctxt mono (Tags (_, level)) _ T =
+ granularity_of_type_level level <> All_Vars andalso
should_encode_type ctxt mono level T
| should_generate_tag_bound_decl _ _ _ _ _ = false
@@ -1641,33 +1733,35 @@
fun tag_with_type ctxt format mono type_enc pos T tm =
IConst (type_tag, T --> T, [T])
- |> enforce_type_arg_policy_in_iterm ctxt format type_enc
+ |> mangle_type_args_in_iterm format type_enc
|> ho_term_from_iterm ctxt format mono type_enc (Top_Level pos)
|> (fn ATerm (s, tms) => ATerm (s, tms @ [tm])
| _ => raise Fail "unexpected lambda-abstraction")
and ho_term_from_iterm ctxt format mono type_enc =
let
- fun aux site u =
+ fun term site u =
let
val (head, args) = strip_iterm_comb u
val pos =
case site of
Top_Level pos => pos
| Eq_Arg pos => pos
- | Elsewhere => NONE
+ | _ => NONE
val t =
case head of
IConst (name as (s, _), _, T_args) =>
let
- val arg_site = if is_tptp_equal s then Eq_Arg pos else Elsewhere
+ fun arg_site j =
+ if is_tptp_equal s then Eq_Arg pos else Arg (s, j)
in
- mk_aterm format type_enc name T_args (map (aux arg_site) args)
+ mk_aterm format type_enc name T_args
+ (map2 (term o arg_site) (0 upto length args - 1) args)
end
| IVar (name, _) =>
- mk_aterm format type_enc name [] (map (aux Elsewhere) args)
+ mk_aterm format type_enc name [] (map (term Elsewhere) args)
| IAbs ((name, T), tm) =>
AAbs ((name, ho_type_from_typ format type_enc true 0 T),
- aux Elsewhere tm)
+ term Elsewhere tm)
| IApp _ => raise Fail "impossible \"IApp\""
val T = ityp_of u
in
@@ -1676,20 +1770,22 @@
else
I)
end
- in aux end
+ in term end
and formula_from_iformula ctxt format mono type_enc should_guard_var =
let
+ val thy = Proof_Context.theory_of ctxt
+ val level = level_of_type_enc type_enc
val do_term = ho_term_from_iterm ctxt format mono type_enc o Top_Level
val do_bound_type =
case type_enc of
- Simple_Types (_, _, level) => fused_type ctxt mono level 0
+ Simple_Types _ => fused_type ctxt mono level 0
#> ho_type_from_typ format type_enc false 0 #> SOME
| _ => K NONE
fun do_out_of_bound_type pos phi universal (name, T) =
if should_guard_type ctxt mono type_enc
- (fn () => should_guard_var pos phi universal name) T then
+ (fn () => should_guard_var thy level pos phi universal name) T then
IVar (name, T)
- |> type_guard_iterm ctxt format type_enc T
+ |> type_guard_iterm format type_enc T
|> do_term pos |> AAtom |> SOME
else if should_generate_tag_bound_decl ctxt mono type_enc universal T then
let
@@ -1779,29 +1875,32 @@
(** Symbol declarations **)
-fun decl_line_for_class s =
+fun decl_line_for_class order s =
let val name as (s, _) = `make_type_class s in
Decl (sym_decl_prefix ^ s, name,
- if exploit_tff1_dummy_type_vars then
- AFun (atype_of_types, bool_atype)
+ if order = First_Order andalso avoid_first_order_dummy_type_vars then
+ ATyAbs ([tvar_a_name], AFun (a_itself_atype, bool_atype))
else
- ATyAbs ([tvar_a_name], AFun (a_itself_atype, bool_atype)))
+ AFun (atype_of_types, bool_atype))
end
fun decl_lines_for_classes type_enc classes =
case type_enc of
- Simple_Types (_, Polymorphic, _) => map decl_line_for_class classes
+ Simple_Types (order, Polymorphic, _) =>
+ map (decl_line_for_class order) classes
| _ => []
-fun sym_decl_table_for_facts ctxt format type_enc repaired_sym_tab
- (conjs, facts) =
+fun sym_decl_table_for_facts ctxt format type_enc sym_tab (conjs, facts) =
let
- fun add_iterm_syms in_conj tm =
+ fun add_iterm_syms tm =
let val (head, args) = strip_iterm_comb tm in
(case head of
IConst ((s, s'), T, T_args) =>
let
- val pred_sym = is_pred_sym repaired_sym_tab s
+ val (pred_sym, in_conj) =
+ case Symtab.lookup sym_tab s of
+ SOME {pred_sym, in_conj, ...} => (pred_sym, in_conj)
+ | NONE => (false, false)
val decl_sym =
(case type_enc of
Guards _ => not pred_sym
@@ -1815,12 +1914,11 @@
else
I
end
- | IAbs (_, tm) => add_iterm_syms in_conj tm
+ | IAbs (_, tm) => add_iterm_syms tm
| _ => I)
- #> fold (add_iterm_syms in_conj) args
+ #> fold add_iterm_syms args
end
- fun add_fact_syms in_conj =
- K (add_iterm_syms in_conj) |> formula_fold NONE |> fact_lift
+ val add_fact_syms = K add_iterm_syms |> formula_fold NONE |> fact_lift
fun add_formula_var_types (AQuant (_, xs, phi)) =
fold (fn (_, SOME T) => insert_type ctxt I T | _ => I) xs
#> add_formula_var_types phi
@@ -1835,7 +1933,7 @@
val (s, s') =
`(make_fixed_const NONE) @{const_name undefined}
|> (case type_arg_policy type_enc @{const_name undefined} of
- Mangled_Type_Args _ => mangled_const_name format type_enc [T]
+ Mangled_Type_Args => mangled_const_name format type_enc [T]
| _ => I)
in
Symtab.map_default (s, [])
@@ -1850,11 +1948,10 @@
in
Symtab.empty
|> is_type_enc_fairly_sound type_enc
- ? (fold (add_fact_syms true) conjs
- #> fold (add_fact_syms false) facts
+ ? (fold (fold add_fact_syms) [conjs, facts]
#> (case type_enc of
- Simple_Types (_, poly, _) =>
- if poly = Polymorphic then add_TYPE_const () else I
+ Simple_Types (First_Order, Polymorphic, _) => add_TYPE_const ()
+ | Simple_Types _ => I
| _ => fold add_undefined_const (var_types ())))
end
@@ -1865,7 +1962,7 @@
maybe_infinite_Ts = known_infinite_types,
surely_infinite_Ts =
case level of
- Noninf_Nonmono_Types Sound => []
+ Noninf_Nonmono_Types (Sound, _) => []
| _ => known_infinite_types,
maybe_nonmono_Ts = [@{typ bool}]}
@@ -1878,7 +1975,7 @@
surely_infinite_Ts, maybe_nonmono_Ts}) =
if is_tptp_equal s andalso exists is_maybe_universal_var [tm1, tm2] then
case level of
- Noninf_Nonmono_Types soundness =>
+ Noninf_Nonmono_Types (soundness, _) =>
if exists (type_instance ctxt T) surely_infinite_Ts orelse
member (type_aconv ctxt) maybe_finite_Ts T then
mono
@@ -1895,7 +1992,7 @@
maybe_infinite_Ts = maybe_infinite_Ts,
surely_infinite_Ts = surely_infinite_Ts,
maybe_nonmono_Ts = maybe_nonmono_Ts |> insert_type ctxt I T}
- | Fin_Nonmono_Types =>
+ | Fin_Nonmono_Types _ =>
if exists (type_instance ctxt T) surely_finite_Ts orelse
member (type_aconv ctxt) maybe_infinite_Ts T then
mono
@@ -1938,19 +2035,22 @@
fun add_fact_monotonic_types ctxt mono type_enc =
add_iformula_monotonic_types ctxt mono type_enc |> fact_lift
fun monotonic_types_for_facts ctxt mono type_enc facts =
- [] |> (polymorphism_of_type_enc type_enc = Polymorphic andalso
- is_type_level_monotonicity_based (level_of_type_enc type_enc))
- ? fold (add_fact_monotonic_types ctxt mono type_enc) facts
+ let val level = level_of_type_enc type_enc in
+ [] |> (polymorphism_of_type_enc type_enc = Polymorphic andalso
+ is_type_level_monotonicity_based level andalso
+ granularity_of_type_level level <> Ghost_Type_Arg_Vars)
+ ? fold (add_fact_monotonic_types ctxt mono type_enc) facts
+ end
fun formula_line_for_guards_mono_type ctxt format mono type_enc T =
Formula (guards_sym_formula_prefix ^
ascii_of (mangled_type format type_enc T),
Axiom,
IConst (`make_bound_var "X", T, [])
- |> type_guard_iterm ctxt format type_enc T
+ |> type_guard_iterm format type_enc T
|> AAtom
|> formula_from_iformula ctxt format mono type_enc
- (K (K (K (K true)))) (SOME true)
+ (K (K (K (K (K (K true)))))) (SOME true)
|> bound_tvars type_enc (atyps_of T)
|> close_formula_universally type_enc,
isabelle_info introN, NONE)
@@ -2003,39 +2103,49 @@
fun formula_line_for_guards_sym_decl ctxt format conj_sym_kind mono type_enc n s
j (s', T_args, T, _, ary, in_conj) =
let
+ val thy = Proof_Context.theory_of ctxt
val (kind, maybe_negate) =
if in_conj then (conj_sym_kind, conj_sym_kind = Conjecture ? mk_anot)
else (Axiom, I)
val (arg_Ts, res_T) = chop_fun ary T
- val num_args = length arg_Ts
- val bound_names =
- 1 upto num_args |> map (`I o make_bound_var o string_of_int)
+ val bound_names = 1 upto ary |> map (`I o make_bound_var o string_of_int)
val bounds =
bound_names ~~ arg_Ts |> map (fn (name, T) => IConst (name, T, []))
- val sym_needs_arg_types = exists (curry (op =) dummyT) T_args
- fun should_keep_arg_type T =
- sym_needs_arg_types andalso
- should_guard_type ctxt mono type_enc (K true) T
val bound_Ts =
- arg_Ts |> map (fn T => if should_keep_arg_type T then SOME T else NONE)
+ if exists (curry (op =) dummyT) T_args then
+ case level_of_type_enc type_enc of
+ All_Types => map SOME arg_Ts
+ | level =>
+ if granularity_of_type_level level = Ghost_Type_Arg_Vars then
+ let val ghosts = ghost_type_args thy s ary in
+ map2 (fn j => if member (op =) ghosts j then SOME else K NONE)
+ (0 upto ary - 1) arg_Ts
+ end
+ else
+ replicate ary NONE
+ else
+ replicate ary NONE
in
Formula (guards_sym_formula_prefix ^ s ^
(if n > 1 then "_" ^ string_of_int j else ""), kind,
IConst ((s, s'), T, T_args)
|> fold (curry (IApp o swap)) bounds
- |> type_guard_iterm ctxt format type_enc res_T
+ |> type_guard_iterm format type_enc res_T
|> AAtom |> mk_aquant AForall (bound_names ~~ bound_Ts)
|> formula_from_iformula ctxt format mono type_enc
- (K (K (K (K true)))) (SOME true)
+ (K (K (K (K (K (K true)))))) (SOME true)
|> n > 1 ? bound_tvars type_enc (atyps_of T)
|> close_formula_universally type_enc
|> maybe_negate,
isabelle_info introN, NONE)
end
-fun formula_lines_for_nonuniform_tags_sym_decl ctxt format conj_sym_kind mono
- type_enc n s (j, (s', T_args, T, pred_sym, ary, in_conj)) =
+fun formula_lines_for_tags_sym_decl ctxt format conj_sym_kind mono type_enc n s
+ (j, (s', T_args, T, pred_sym, ary, in_conj)) =
let
+ val thy = Proof_Context.theory_of ctxt
+ val level = level_of_type_enc type_enc
+ val grain = granularity_of_type_level level
val ident_base =
tags_sym_formula_prefix ^ s ^
(if n > 1 then "_" ^ string_of_int j else "")
@@ -2043,19 +2153,28 @@
if in_conj then (conj_sym_kind, conj_sym_kind = Conjecture ? mk_anot)
else (Axiom, I)
val (arg_Ts, res_T) = chop_fun ary T
- val bound_names =
- 1 upto length arg_Ts |> map (`I o make_bound_var o string_of_int)
+ val bound_names = 1 upto ary |> map (`I o make_bound_var o string_of_int)
val bounds = bound_names |> map (fn name => ATerm (name, []))
val cst = mk_aterm format type_enc (s, s') T_args
val eq = maybe_negate oo eq_formula type_enc (atyps_of T) pred_sym
- val should_encode =
- should_encode_type ctxt mono (level_of_type_enc type_enc)
+ val should_encode = should_encode_type ctxt mono level
val tag_with = tag_with_type ctxt format mono type_enc NONE
val add_formula_for_res =
if should_encode res_T then
- cons (Formula (ident_base ^ "_res", kind,
- eq (tag_with res_T (cst bounds)) (cst bounds),
- isabelle_info simpN, NONE))
+ let
+ val tagged_bounds =
+ if grain = Ghost_Type_Arg_Vars then
+ let val ghosts = ghost_type_args thy s ary in
+ map2 (fn (j, arg_T) => member (op =) ghosts j ? tag_with arg_T)
+ (0 upto ary - 1 ~~ arg_Ts) bounds
+ end
+ else
+ bounds
+ in
+ cons (Formula (ident_base ^ "_res", kind,
+ eq (tag_with res_T (cst bounds)) (cst tagged_bounds),
+ isabelle_info simpN, NONE))
+ end
else
I
fun add_formula_for_arg k =
@@ -2073,7 +2192,8 @@
end
in
[] |> not pred_sym ? add_formula_for_res
- |> Config.get ctxt type_tag_arguments
+ |> (Config.get ctxt type_tag_arguments andalso
+ grain = Positively_Naked_Vars)
? fold add_formula_for_arg (ary - 1 downto 0)
end
@@ -2084,7 +2204,7 @@
case type_enc of
Simple_Types _ =>
decls |> map (decl_line_for_sym ctxt format mono type_enc s)
- | Guards (_, level, _) =>
+ | Guards (_, level) =>
let
val decls =
case decls of
@@ -2106,15 +2226,15 @@
|-> map2 (formula_line_for_guards_sym_decl ctxt format conj_sym_kind mono
type_enc n s)
end
- | Tags (_, _, uniformity) =>
- (case uniformity of
- Uniform => []
- | Nonuniform =>
- let val n = length decls in
- (0 upto n - 1 ~~ decls)
- |> maps (formula_lines_for_nonuniform_tags_sym_decl ctxt format
- conj_sym_kind mono type_enc n s)
- end)
+ | Tags (_, level) =>
+ if granularity_of_type_level level = All_Vars then
+ []
+ else
+ let val n = length decls in
+ (0 upto n - 1 ~~ decls)
+ |> maps (formula_lines_for_tags_sym_decl ctxt format conj_sym_kind mono
+ type_enc n s)
+ end
fun problem_lines_for_sym_decl_table ctxt format conj_sym_kind mono type_enc
mono_Ts sym_decl_tab =
@@ -2128,11 +2248,10 @@
syms []
in mono_lines @ decl_lines end
-fun needs_type_tag_idempotence ctxt (Tags (poly, level, uniformity)) =
+fun needs_type_tag_idempotence ctxt (Tags (poly, level)) =
Config.get ctxt type_tag_idempotence andalso
- poly <> Mangled_Monomorphic andalso
- ((level = All_Types andalso uniformity = Nonuniform) orelse
- is_type_level_monotonicity_based level)
+ is_type_level_monotonicity_based level andalso
+ poly <> Mangled_Monomorphic
| needs_type_tag_idempotence _ _ = false
fun offset_of_heading_in_problem _ [] j = j
@@ -2149,12 +2268,22 @@
val conjsN = "Conjectures"
val free_typesN = "Type variables"
-val explicit_apply = NONE (* for experiments *)
+val explicit_apply_threshold = 50
fun prepare_atp_problem ctxt format conj_sym_kind prem_kind type_enc exporter
lambda_trans readable_names preproc hyp_ts concl_t facts =
let
+ val thy = Proof_Context.theory_of ctxt
val type_enc = type_enc |> adjust_type_enc format
+ (* Forcing explicit applications is expensive for polymorphic encodings,
+ because it takes only one existential variable ranging over "'a => 'b" to
+ ruin everything. Hence we do it only if there are few facts. *)
+ val explicit_apply =
+ if polymorphism_of_type_enc type_enc <> Polymorphic orelse
+ length facts <= explicit_apply_threshold then
+ NONE
+ else
+ SOME false
val lambda_trans =
if lambda_trans = smartN then
if is_type_enc_higher_order type_enc then lambdasN else combinatorsN
@@ -2185,22 +2314,22 @@
val (fact_names, classes, conjs, facts, class_rel_clauses, arity_clauses) =
translate_formulas ctxt format prem_kind type_enc trans_lambdas preproc
hyp_ts concl_t facts
- val sym_tab = conjs @ facts |> sym_table_for_facts ctxt explicit_apply
+ val sym_tab =
+ sym_table_for_facts ctxt type_enc explicit_apply conjs facts
val mono = conjs @ facts |> mononotonicity_info_for_facts ctxt type_enc
- val repair = repair_fact ctxt format type_enc sym_tab
- val (conjs, facts) = (conjs, facts) |> pairself (map repair)
- val repaired_sym_tab =
- conjs @ facts |> sym_table_for_facts ctxt (SOME false)
+ val firstorderize = firstorderize_fact thy format type_enc sym_tab
+ val (conjs, facts) = (conjs, facts) |> pairself (map firstorderize)
+ val sym_tab = sym_table_for_facts ctxt type_enc (SOME false) conjs facts
val helpers =
- repaired_sym_tab |> helper_facts_for_sym_table ctxt format type_enc
- |> map repair
+ sym_tab |> helper_facts_for_sym_table ctxt format type_enc
+ |> map firstorderize
val mono_Ts =
helpers @ conjs @ facts
|> monotonic_types_for_facts ctxt mono type_enc
val class_decl_lines = decl_lines_for_classes type_enc classes
val sym_decl_lines =
(conjs, helpers @ facts)
- |> sym_decl_table_for_facts ctxt format type_enc repaired_sym_tab
+ |> sym_decl_table_for_facts ctxt format type_enc sym_tab
|> problem_lines_for_sym_decl_table ctxt format conj_sym_kind mono
type_enc mono_Ts
val helper_lines =
@@ -2232,7 +2361,8 @@
CNF => ensure_cnf_problem
| CNF_UEQ => filter_cnf_ueq_problem
| FOF => I
- | TFF (_, TFF_Implicit) => I
+ | TFF (_, TPTP_Implicit) => I
+ | THF (_, TPTP_Implicit, _) => I
| _ => declare_undeclared_syms_in_atp_problem type_decl_prefix
implicit_declsN)
val (problem, pool) = problem |> nice_atp_problem readable_names
@@ -2243,13 +2373,8 @@
else NONE)
((helpers_offset + 1 upto helpers_offset + length helpers)
~~ helpers)
- fun add_sym_arity (s, {min_ary, ...} : sym_info) =
- if min_ary > 0 then
- case strip_prefix_and_unascii const_prefix s of
- SOME s => Symtab.insert (op =) (s, min_ary)
- | NONE => I
- else
- I
+ fun add_sym_ary (s, {min_ary, ...} : sym_info) =
+ min_ary > 0 ? Symtab.insert (op =) (s, min_ary)
in
(problem,
case pool of SOME the_pool => snd the_pool | NONE => Symtab.empty,
@@ -2257,7 +2382,7 @@
offset_of_heading_in_problem factsN problem 0,
fact_names |> Vector.fromList,
typed_helpers,
- Symtab.empty |> Symtab.fold add_sym_arity sym_tab)
+ Symtab.empty |> Symtab.fold add_sym_ary sym_tab)
end
(* FUDGE *)
--- a/src/HOL/Tools/ATP/atp_util.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Tools/ATP/atp_util.ML Fri Sep 09 06:47:14 2011 +0200
@@ -10,6 +10,7 @@
val hash_string : string -> int
val hash_term : term -> int
val strip_spaces : bool -> (char -> bool) -> string -> string
+ val strip_spaces_except_between_idents : string -> string
val nat_subscript : int -> string
val unyxml : string -> string
val maybe_quote : string -> string
@@ -88,6 +89,9 @@
fun strip_spaces skip_comments is_evil =
implode o strip_spaces_in_list skip_comments is_evil o String.explode
+fun is_ident_char c = Char.isAlphaNum c orelse c = #"_"
+val strip_spaces_except_between_idents = strip_spaces true is_ident_char
+
val subscript = implode o map (prefix "\<^isub>") o raw_explode (* FIXME Symbol.explode (?) *)
fun nat_subscript n =
n |> string_of_int |> print_mode_active Symbol.xsymbolsN ? subscript
--- a/src/HOL/Tools/Metis/metis_tactic.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Tools/Metis/metis_tactic.ML Fri Sep 09 06:47:14 2011 +0200
@@ -39,8 +39,8 @@
val partial_typesN = "partial_types"
val no_typesN = "no_types"
-val really_full_type_enc = "mono_tags_uniform"
-val full_type_enc = "poly_guards_uniform_query"
+val really_full_type_enc = "mono_tags"
+val full_type_enc = "poly_guards_query"
val partial_type_enc = "poly_args"
val no_type_enc = "erased"
--- a/src/HOL/Tools/Metis/metis_translate.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Tools/Metis/metis_translate.ML Fri Sep 09 06:47:14 2011 +0200
@@ -59,8 +59,9 @@
((prefixed_predicator_name, 1), (K metis_predicator, false)),
((prefixed_app_op_name, 2), (K metis_app_op, false)),
((prefixed_type_tag_name, 2),
- (fn Tags (_, All_Types, Uniform) => metis_systematic_type_tag
- | _ => metis_ad_hoc_type_tag, true))]
+ (fn type_enc =>
+ if level_of_type_enc type_enc = All_Types then metis_systematic_type_tag
+ else metis_ad_hoc_type_tag, true))]
fun old_skolem_const_name i j num_T_args =
old_skolem_const_prefix ^ Long_Name.separator ^
--- a/src/HOL/Tools/Nitpick/nitrox.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Tools/Nitpick/nitrox.ML Fri Sep 09 06:47:14 2011 +0200
@@ -21,8 +21,7 @@
exception SYNTAX of string
-fun is_ident_char c = Char.isAlphaNum c orelse c = #"_"
-val tptp_explode = raw_explode o strip_spaces true is_ident_char
+val tptp_explode = raw_explode o strip_spaces_except_between_idents
fun parse_file_path (c :: ss) =
if c = "'" orelse c = "\"" then
--- a/src/HOL/Tools/Quickcheck/Narrowing_Engine.hs Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Tools/Quickcheck/Narrowing_Engine.hs Fri Sep 09 06:47:14 2011 +0200
@@ -4,7 +4,7 @@
import Control.Exception;
import System.IO;
import System.Exit;
-import Code;
+import Generated_Code;
type Pos = [Int];
--- a/src/HOL/Tools/Quickcheck/PNF_Narrowing_Engine.hs Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Tools/Quickcheck/PNF_Narrowing_Engine.hs Fri Sep 09 06:47:14 2011 +0200
@@ -8,7 +8,7 @@
import System.Exit
import Maybe
import List (partition, findIndex)
-import Code
+import Generated_Code
type Pos = [Int]
--- a/src/HOL/Tools/Quickcheck/narrowing_generators.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Tools/Quickcheck/narrowing_generators.ML Fri Sep 09 06:47:14 2011 +0200
@@ -235,17 +235,17 @@
if Config.get ctxt overlord then with_overlord_dir else Isabelle_System.with_tmp_dir
fun run in_path =
let
- val code_file = Path.append in_path (Path.basic "Code.hs")
+ val code_file = Path.append in_path (Path.basic "Generated_Code.hs")
val narrowing_engine_file = Path.append in_path (Path.basic "Narrowing_Engine.hs")
val main_file = Path.append in_path (Path.basic "Main.hs")
val main = "module Main where {\n\n" ^
"import System;\n" ^
"import Narrowing_Engine;\n" ^
- "import Code;\n\n" ^
- "main = getArgs >>= \\[size] -> Narrowing_Engine.depthCheck (read size) (Code.value ())\n\n" ^
+ "import Generated_Code;\n\n" ^
+ "main = getArgs >>= \\[size] -> Narrowing_Engine.depthCheck (read size) (Generated_Code.value ())\n\n" ^
"}\n"
- val code' = prefix "module Code where {\n\ndata Typerep = Typerep String [Typerep];\n"
- (unprefix "module Code where {" code)
+ val code' = prefix "module Generated_Code where {\n\ndata Typerep = Typerep String [Typerep];\n"
+ (unprefix "module Generated_Code where {" code)
val _ = File.write code_file code'
val _ = File.write narrowing_engine_file
(if contains_existentials then pnf_narrowing_engine else narrowing_engine)
--- a/src/HOL/Tools/Sledgehammer/sledgehammer_filter.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Tools/Sledgehammer/sledgehammer_filter.ML Fri Sep 09 06:47:14 2011 +0200
@@ -144,7 +144,6 @@
(j + 1,
((nth_name j,
if member Thm.eq_thm_prop chained_ths th then Chained
- else if Thm.eq_thm_prop (th, @{thm ext}) then Extensionality
else General), th) :: rest))
|> snd
end
@@ -576,7 +575,7 @@
| _ => SOME tab
in aux (prop_of th) [] end
-(* FIXME: This is currently only useful for polymorphic type systems. *)
+(* FIXME: This is currently only useful for polymorphic type encodings. *)
fun could_benefit_from_ext is_built_in_const facts =
fold (consider_arities is_built_in_const o snd) facts (SOME Symtab.empty)
|> is_none
@@ -845,8 +844,7 @@
else if global then
case Termtab.lookup clasimpset_table (prop_of th) of
SOME loc => loc
- | NONE => if Thm.eq_thm_prop (th, @{thm ext}) then Extensionality
- else General
+ | NONE => General
else if is_assum th then Assum else Local
fun is_good_unnamed_local th =
not (Thm.has_name_hint th) andalso
--- a/src/HOL/Tools/Sledgehammer/sledgehammer_isar.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Tools/Sledgehammer/sledgehammer_isar.ML Fri Sep 09 06:47:14 2011 +0200
@@ -19,6 +19,7 @@
structure Sledgehammer_Isar : SLEDGEHAMMER_ISAR =
struct
+open ATP_Util
open ATP_Systems
open ATP_Translate
open Sledgehammer_Util
@@ -151,11 +152,9 @@
error ("Unknown parameter: " ^ quote name ^ "."))
-(* Ensure that type systems such as "mono_simple!" and "mono_guards?" are
- handled correctly. *)
-fun implode_param [s, "?"] = s ^ "?"
- | implode_param [s, "!"] = s ^ "!"
- | implode_param ss = space_implode " " ss
+(* Ensures that type encodings such as "mono_simple?" and "poly_guards!!" are
+ read correctly. *)
+val implode_param = strip_spaces_except_between_idents o space_implode " "
structure Data = Theory_Data
(
@@ -376,12 +375,11 @@
|> sort_strings |> cat_lines))
end))
+val parse_query_bang = Parse.$$$ "?" || Parse.$$$ "!" || Parse.$$$ "!!"
val parse_key =
- Scan.repeat1 (Parse.typ_group || Parse.$$$ "?" || Parse.$$$ "!")
- >> implode_param
+ Scan.repeat1 (Parse.typ_group || parse_query_bang) >> implode_param
val parse_value =
- Scan.repeat1 (Parse.xname || Parse.float_number || Parse.$$$ "?"
- || Parse.$$$ "!")
+ Scan.repeat1 (Parse.xname || Parse.float_number || parse_query_bang)
val parse_param = parse_key -- Scan.optional (Parse.$$$ "=" |-- parse_value) []
val parse_params = Scan.optional (Args.bracks (Parse.list parse_param)) []
val parse_fact_refs =
--- a/src/HOL/Transcendental.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Transcendental.thy Fri Sep 09 06:47:14 2011 +0200
@@ -1065,22 +1065,24 @@
using exp_inj_iff [where x=x and y=0] by simp
lemma lemma_exp_total: "1 \<le> y ==> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y"
-apply (rule IVT)
-apply (auto intro: isCont_exp simp add: le_diff_eq)
-apply (subgoal_tac "1 + (y - 1) \<le> exp (y - 1)")
-apply simp
-apply (rule exp_ge_add_one_self_aux, simp)
-done
+proof (rule IVT)
+ assume "1 \<le> y"
+ hence "0 \<le> y - 1" by simp
+ hence "1 + (y - 1) \<le> exp (y - 1)" by (rule exp_ge_add_one_self_aux)
+ thus "y \<le> exp (y - 1)" by simp
+qed (simp_all add: le_diff_eq)
lemma exp_total: "0 < (y::real) ==> \<exists>x. exp x = y"
-apply (rule_tac x = 1 and y = y in linorder_cases)
-apply (drule order_less_imp_le [THEN lemma_exp_total])
-apply (rule_tac [2] x = 0 in exI)
-apply (frule_tac [3] one_less_inverse)
-apply (drule_tac [4] order_less_imp_le [THEN lemma_exp_total], auto)
-apply (rule_tac x = "-x" in exI)
-apply (simp add: exp_minus)
-done
+proof (rule linorder_le_cases [of 1 y])
+ assume "1 \<le> y" thus "\<exists>x. exp x = y"
+ by (fast dest: lemma_exp_total)
+next
+ assume "0 < y" and "y \<le> 1"
+ hence "1 \<le> inverse y" by (simp add: one_le_inverse_iff)
+ then obtain x where "exp x = inverse y" by (fast dest: lemma_exp_total)
+ hence "exp (- x) = y" by (simp add: exp_minus)
+ thus "\<exists>x. exp x = y" ..
+qed
subsection {* Natural Logarithm *}
@@ -1722,19 +1724,29 @@
lemma sin_ge_zero: "[| 0 \<le> x; x \<le> pi |] ==> 0 \<le> sin x"
by (auto simp add: order_le_less sin_gt_zero_pi)
+text {* FIXME: This proof is almost identical to lemma @{text cos_is_zero}.
+ It should be possible to factor out some of the common parts. *}
+
lemma cos_total: "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)"
-apply (subgoal_tac "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y")
-apply (rule_tac [2] IVT2)
-apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos)
-apply (cut_tac x = xa and y = y in linorder_less_linear)
-apply (rule ccontr, auto)
-apply (drule_tac f = cos in Rolle)
-apply (drule_tac [5] f = cos in Rolle)
-apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos
- dest!: DERIV_cos [THEN DERIV_unique]
- simp add: differentiable_def)
-apply (auto dest: sin_gt_zero_pi [OF order_le_less_trans order_less_le_trans])
-done
+proof (rule ex_ex1I)
+ assume y: "-1 \<le> y" "y \<le> 1"
+ show "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y"
+ by (rule IVT2, simp_all add: y)
+next
+ fix a b
+ assume a: "0 \<le> a \<and> a \<le> pi \<and> cos a = y"
+ assume b: "0 \<le> b \<and> b \<le> pi \<and> cos b = y"
+ have [simp]: "\<forall>x. cos differentiable x"
+ unfolding differentiable_def by (auto intro: DERIV_cos)
+ from a b show "a = b"
+ apply (cut_tac less_linear [of a b], auto)
+ apply (drule_tac f = cos in Rolle)
+ apply (drule_tac [5] f = cos in Rolle)
+ apply (auto dest!: DERIV_cos [THEN DERIV_unique])
+ apply (metis order_less_le_trans less_le sin_gt_zero_pi)
+ apply (metis order_less_le_trans less_le sin_gt_zero_pi)
+ done
+qed
lemma sin_total:
"[| -1 \<le> y; y \<le> 1 |] ==> EX! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)"
@@ -1910,21 +1922,9 @@
thus ?thesis by simp
qed
-lemma tan_half: fixes x :: real assumes "- (pi / 2) < x" and "x < pi / 2"
- shows "tan x = sin (2 * x) / (cos (2 * x) + 1)"
-proof -
- from cos_gt_zero_pi[OF `- (pi / 2) < x` `x < pi / 2`]
- have "cos x \<noteq> 0" by auto
-
- have minus_cos_2x: "\<And>X. X - cos (2*x) = X - (cos x) ^ 2 + (sin x) ^ 2" unfolding cos_double by algebra
-
- have "tan x = (tan x + tan x) / 2" by auto
- also have "\<dots> = sin (x + x) / (cos x * cos x) / 2" unfolding add_tan_eq[OF `cos x \<noteq> 0` `cos x \<noteq> 0`] ..
- also have "\<dots> = sin (2 * x) / ((cos x) ^ 2 + (cos x) ^ 2 + cos (2*x) - cos (2*x))" unfolding divide_divide_eq_left numeral_2_eq_2 by auto
- also have "\<dots> = sin (2 * x) / ((cos x) ^ 2 + cos (2*x) + (sin x)^2)" unfolding minus_cos_2x by auto
- also have "\<dots> = sin (2 * x) / (cos (2*x) + 1)" by auto
- finally show ?thesis .
-qed
+lemma tan_half: "tan x = sin (2 * x) / (cos (2 * x) + 1)"
+ unfolding tan_def sin_double cos_double sin_squared_eq
+ by (simp add: power2_eq_square)
lemma DERIV_tan [simp]: "cos x \<noteq> 0 \<Longrightarrow> DERIV tan x :> inverse ((cos x)\<twosuperior>)"
unfolding tan_def
@@ -2196,15 +2196,24 @@
lemma arctan_ubound: "arctan y < pi/2"
by (auto simp only: arctan)
+lemma arctan_unique:
+ assumes "-(pi/2) < x" and "x < pi/2" and "tan x = y"
+ shows "arctan y = x"
+ using assms arctan [of y] tan_total [of y] by (fast elim: ex1E)
+
lemma arctan_tan:
"[|-(pi/2) < x; x < pi/2 |] ==> arctan(tan x) = x"
-apply (unfold arctan_def)
-apply (rule the1_equality)
-apply (rule tan_total, auto)
-done
+ by (rule arctan_unique, simp_all)
lemma arctan_zero_zero [simp]: "arctan 0 = 0"
-by (insert arctan_tan [of 0], simp)
+ by (rule arctan_unique, simp_all)
+
+lemma arctan_minus: "arctan (- x) = - arctan x"
+ apply (rule arctan_unique)
+ apply (simp only: neg_less_iff_less arctan_ubound)
+ apply (metis minus_less_iff arctan_lbound)
+ apply simp
+ done
lemma cos_arctan_not_zero [simp]: "cos (arctan x) \<noteq> 0"
by (intro less_imp_neq [symmetric] cos_gt_zero_pi
@@ -2235,6 +2244,30 @@
mult_assoc power_inverse [symmetric])
done
+lemma arctan_less_iff: "arctan x < arctan y \<longleftrightarrow> x < y"
+ by (metis tan_monotone' arctan_lbound arctan_ubound tan_arctan)
+
+lemma arctan_le_iff: "arctan x \<le> arctan y \<longleftrightarrow> x \<le> y"
+ by (simp only: not_less [symmetric] arctan_less_iff)
+
+lemma arctan_eq_iff: "arctan x = arctan y \<longleftrightarrow> x = y"
+ by (simp only: eq_iff [where 'a=real] arctan_le_iff)
+
+lemma zero_less_arctan_iff [simp]: "0 < arctan x \<longleftrightarrow> 0 < x"
+ using arctan_less_iff [of 0 x] by simp
+
+lemma arctan_less_zero_iff [simp]: "arctan x < 0 \<longleftrightarrow> x < 0"
+ using arctan_less_iff [of x 0] by simp
+
+lemma zero_le_arctan_iff [simp]: "0 \<le> arctan x \<longleftrightarrow> 0 \<le> x"
+ using arctan_le_iff [of 0 x] by simp
+
+lemma arctan_le_zero_iff [simp]: "arctan x \<le> 0 \<longleftrightarrow> x \<le> 0"
+ using arctan_le_iff [of x 0] by simp
+
+lemma arctan_eq_zero_iff [simp]: "arctan x = 0 \<longleftrightarrow> x = 0"
+ using arctan_eq_iff [of x 0] by simp
+
lemma isCont_inverse_function2:
fixes f g :: "real \<Rightarrow> real" shows
"\<lbrakk>a < x; x < b;
@@ -2427,98 +2460,34 @@
subsection {* Machins formula *}
+lemma arctan_one: "arctan 1 = pi / 4"
+ by (rule arctan_unique, simp_all add: tan_45 m2pi_less_pi)
+
lemma tan_total_pi4: assumes "\<bar>x\<bar> < 1"
shows "\<exists> z. - (pi / 4) < z \<and> z < pi / 4 \<and> tan z = x"
-proof -
- obtain z where "- (pi / 2) < z" and "z < pi / 2" and "tan z = x" using tan_total by blast
- have "tan (pi / 4) = 1" and "tan (- (pi / 4)) = - 1" using tan_45 tan_minus by auto
- have "z \<noteq> pi / 4"
- proof (rule ccontr)
- assume "\<not> (z \<noteq> pi / 4)" hence "z = pi / 4" by auto
- have "tan z = 1" unfolding `z = pi / 4` `tan (pi / 4) = 1` ..
- thus False unfolding `tan z = x` using `\<bar>x\<bar> < 1` by auto
- qed
- have "z \<noteq> - (pi / 4)"
- proof (rule ccontr)
- assume "\<not> (z \<noteq> - (pi / 4))" hence "z = - (pi / 4)" by auto
- have "tan z = - 1" unfolding `z = - (pi / 4)` `tan (- (pi / 4)) = - 1` ..
- thus False unfolding `tan z = x` using `\<bar>x\<bar> < 1` by auto
- qed
-
- have "z < pi / 4"
- proof (rule ccontr)
- assume "\<not> (z < pi / 4)" hence "pi / 4 < z" using `z \<noteq> pi / 4` by auto
- have "- (pi / 2) < pi / 4" using m2pi_less_pi by auto
- from tan_monotone[OF this `pi / 4 < z` `z < pi / 2`]
- have "1 < x" unfolding `tan z = x` `tan (pi / 4) = 1` .
- thus False using `\<bar>x\<bar> < 1` by auto
- qed
- moreover
- have "-(pi / 4) < z"
- proof (rule ccontr)
- assume "\<not> (-(pi / 4) < z)" hence "z < - (pi / 4)" using `z \<noteq> - (pi / 4)` by auto
- have "-(pi / 4) < pi / 2" using m2pi_less_pi by auto
- from tan_monotone[OF `-(pi / 2) < z` `z < -(pi / 4)` this]
- have "x < - 1" unfolding `tan z = x` `tan (-(pi / 4)) = - 1` .
- thus False using `\<bar>x\<bar> < 1` by auto
- qed
- ultimately show ?thesis using `tan z = x` by auto
+proof
+ show "- (pi / 4) < arctan x \<and> arctan x < pi / 4 \<and> tan (arctan x) = x"
+ unfolding arctan_one [symmetric] arctan_minus [symmetric]
+ unfolding arctan_less_iff using assms by auto
qed
lemma arctan_add: assumes "\<bar>x\<bar> \<le> 1" and "\<bar>y\<bar> < 1"
shows "arctan x + arctan y = arctan ((x + y) / (1 - x * y))"
-proof -
- obtain y' where "-(pi/4) < y'" and "y' < pi/4" and "tan y' = y" using tan_total_pi4[OF `\<bar>y\<bar> < 1`] by blast
-
- have "pi / 4 < pi / 2" by auto
-
- have "\<exists> x'. -(pi/4) \<le> x' \<and> x' \<le> pi/4 \<and> tan x' = x"
- proof (cases "\<bar>x\<bar> < 1")
- case True from tan_total_pi4[OF this] obtain x' where "-(pi/4) < x'" and "x' < pi/4" and "tan x' = x" by blast
- hence "-(pi/4) \<le> x'" and "x' \<le> pi/4" and "tan x' = x" by auto
- thus ?thesis by auto
- next
- case False
- show ?thesis
- proof (cases "x = 1")
- case True hence "tan (pi/4) = x" using tan_45 by auto
- moreover
- have "- pi \<le> pi" unfolding minus_le_self_iff by auto
- hence "-(pi/4) \<le> pi/4" and "pi/4 \<le> pi/4" by auto
- ultimately show ?thesis by blast
- next
- case False hence "x = -1" using `\<not> \<bar>x\<bar> < 1` and `\<bar>x\<bar> \<le> 1` by auto
- hence "tan (-(pi/4)) = x" using tan_45 tan_minus by auto
- moreover
- have "- pi \<le> pi" unfolding minus_le_self_iff by auto
- hence "-(pi/4) \<le> pi/4" and "-(pi/4) \<le> -(pi/4)" by auto
- ultimately show ?thesis by blast
- qed
- qed
- then obtain x' where "-(pi/4) \<le> x'" and "x' \<le> pi/4" and "tan x' = x" by blast
- hence "-(pi/2) < x'" and "x' < pi/2" using order_le_less_trans[OF `x' \<le> pi/4` `pi / 4 < pi / 2`] by auto
-
- have "cos x' \<noteq> 0" using cos_gt_zero_pi[THEN less_imp_neq] and `-(pi/2) < x'` and `x' < pi/2` by auto
- moreover have "cos y' \<noteq> 0" using cos_gt_zero_pi[THEN less_imp_neq] and `-(pi/4) < y'` and `y' < pi/4` by auto
- ultimately have "cos x' * cos y' \<noteq> 0" by auto
-
- have divide_nonzero_divide: "\<And> A B C :: real. C \<noteq> 0 \<Longrightarrow> (A / C) / (B / C) = A / B" by auto
- have divide_mult_commute: "\<And> A B C D :: real. A * B / (C * D) = (A / C) * (B / D)" by auto
-
- have "tan (x' + y') = sin (x' + y') / (cos x' * cos y' - sin x' * sin y')" unfolding tan_def cos_add ..
- also have "\<dots> = (tan x' + tan y') / ((cos x' * cos y' - sin x' * sin y') / (cos x' * cos y'))" unfolding add_tan_eq[OF `cos x' \<noteq> 0` `cos y' \<noteq> 0`] divide_nonzero_divide[OF `cos x' * cos y' \<noteq> 0`] ..
- also have "\<dots> = (tan x' + tan y') / (1 - tan x' * tan y')" unfolding tan_def diff_divide_distrib divide_self[OF `cos x' * cos y' \<noteq> 0`] unfolding divide_mult_commute ..
- finally have tan_eq: "tan (x' + y') = (x + y) / (1 - x * y)" unfolding `tan y' = y` `tan x' = x` .
-
- have "arctan (tan (x' + y')) = x' + y'" using `-(pi/4) < y'` `-(pi/4) \<le> x'` `y' < pi/4` and `x' \<le> pi/4` by (auto intro!: arctan_tan)
- moreover have "arctan (tan (x')) = x'" using `-(pi/2) < x'` and `x' < pi/2` by (auto intro!: arctan_tan)
- moreover have "arctan (tan (y')) = y'" using `-(pi/4) < y'` and `y' < pi/4` by (auto intro!: arctan_tan)
- ultimately have "arctan x + arctan y = arctan (tan (x' + y'))" unfolding `tan y' = y` [symmetric] `tan x' = x`[symmetric] by auto
- thus "arctan x + arctan y = arctan ((x + y) / (1 - x * y))" unfolding tan_eq .
+proof (rule arctan_unique [symmetric])
+ have "- (pi / 4) \<le> arctan x" and "- (pi / 4) < arctan y"
+ unfolding arctan_one [symmetric] arctan_minus [symmetric]
+ unfolding arctan_le_iff arctan_less_iff using assms by auto
+ from add_le_less_mono [OF this]
+ show 1: "- (pi / 2) < arctan x + arctan y" by simp
+ have "arctan x \<le> pi / 4" and "arctan y < pi / 4"
+ unfolding arctan_one [symmetric]
+ unfolding arctan_le_iff arctan_less_iff using assms by auto
+ from add_le_less_mono [OF this]
+ show 2: "arctan x + arctan y < pi / 2" by simp
+ show "tan (arctan x + arctan y) = (x + y) / (1 - x * y)"
+ using cos_gt_zero_pi [OF 1 2] by (simp add: tan_add)
qed
-lemma arctan1_eq_pi4: "arctan 1 = pi / 4" unfolding tan_45[symmetric] by (rule arctan_tan, auto simp add: m2pi_less_pi)
-
theorem machin: "pi / 4 = 4 * arctan (1/5) - arctan (1 / 239)"
proof -
have "\<bar>1 / 5\<bar> < (1 :: real)" by auto
@@ -2533,8 +2502,9 @@
from arctan_add[OF this]
have "arctan 1 + arctan (1 / 239) = arctan (120 / 119)" by auto
ultimately have "arctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)" by auto
- thus ?thesis unfolding arctan1_eq_pi4 by algebra
+ thus ?thesis unfolding arctan_one by algebra
qed
+
subsection {* Introducing the arcus tangens power series *}
lemma monoseq_arctan_series: fixes x :: real
@@ -2821,71 +2791,40 @@
have "arctan x = y" using arctan_tan low high y_eq by auto
also have "\<dots> = 2 * (arctan (tan (y/2)))" using arctan_tan[OF low2 high2] by auto
- also have "\<dots> = 2 * (arctan (sin y / (cos y + 1)))" unfolding tan_half[OF low2 high2] by auto
+ also have "\<dots> = 2 * (arctan (sin y / (cos y + 1)))" unfolding tan_half by auto
finally show ?thesis unfolding eq `tan y = x` .
qed
lemma arctan_monotone: assumes "x < y"
shows "arctan x < arctan y"
-proof -
- obtain z where "-(pi / 2) < z" and "z < pi / 2" and "tan z = x" using tan_total by blast
- obtain w where "-(pi / 2) < w" and "w < pi / 2" and "tan w = y" using tan_total by blast
- have "z < w" unfolding tan_monotone'[OF `-(pi / 2) < z` `z < pi / 2` `-(pi / 2) < w` `w < pi / 2`] `tan z = x` `tan w = y` using `x < y` .
- thus ?thesis
- unfolding `tan z = x`[symmetric] arctan_tan[OF `-(pi / 2) < z` `z < pi / 2`]
- unfolding `tan w = y`[symmetric] arctan_tan[OF `-(pi / 2) < w` `w < pi / 2`] .
-qed
+ using assms by (simp only: arctan_less_iff)
lemma arctan_monotone': assumes "x \<le> y" shows "arctan x \<le> arctan y"
-proof (cases "x = y")
- case False hence "x < y" using `x \<le> y` by auto from arctan_monotone[OF this] show ?thesis by auto
-qed auto
-
-lemma arctan_minus: "arctan (- x) = - arctan x"
-proof -
- obtain y where "- (pi / 2) < y" and "y < pi / 2" and "tan y = x" using tan_total by blast
- thus ?thesis unfolding `tan y = x`[symmetric] tan_minus[symmetric] using arctan_tan[of y] arctan_tan[of "-y"] by auto
-qed
-
-lemma arctan_inverse: assumes "x \<noteq> 0" shows "arctan (1 / x) = sgn x * pi / 2 - arctan x"
-proof -
- obtain y where "- (pi / 2) < y" and "y < pi / 2" and "tan y = x" using tan_total by blast
- hence "y = arctan x" unfolding `tan y = x`[symmetric] using arctan_tan by auto
-
- { fix y x :: real assume "0 < y" and "y < pi /2" and "y = arctan x" and "tan y = x" hence "- (pi / 2) < y" by auto
- have "tan y > 0" using tan_monotone'[OF _ _ `- (pi / 2) < y` `y < pi / 2`, of 0] tan_zero `0 < y` by auto
- hence "x > 0" using `tan y = x` by auto
-
- have "- (pi / 2) < pi / 2 - y" using `y > 0` `y < pi / 2` by auto
- moreover have "pi / 2 - y < pi / 2" using `y > 0` `y < pi / 2` by auto
- ultimately have "arctan (1 / x) = pi / 2 - y" unfolding `tan y = x`[symmetric] tan_inverse using arctan_tan by auto
- hence "arctan (1 / x) = sgn x * pi / 2 - arctan x" unfolding `y = arctan x` real_sgn_pos[OF `x > 0`] by auto
- } note pos_y = this
-
- show ?thesis
- proof (cases "y > 0")
- case True from pos_y[OF this `y < pi / 2` `y = arctan x` `tan y = x`] show ?thesis .
- next
- case False hence "y \<le> 0" by auto
- moreover have "y \<noteq> 0"
- proof (rule ccontr)
- assume "\<not> y \<noteq> 0" hence "y = 0" by auto
- have "x = 0" unfolding `tan y = x`[symmetric] `y = 0` tan_zero ..
- thus False using `x \<noteq> 0` by auto
- qed
- ultimately have "y < 0" by auto
- hence "0 < - y" and "-y < pi / 2" using `- (pi / 2) < y` by auto
- moreover have "-y = arctan (-x)" unfolding arctan_minus `y = arctan x` ..
- moreover have "tan (-y) = -x" unfolding tan_minus `tan y = x` ..
- ultimately have "arctan (1 / -x) = sgn (-x) * pi / 2 - arctan (-x)" using pos_y by blast
- hence "arctan (- (1 / x)) = - (sgn x * pi / 2 - arctan x)" unfolding arctan_minus[of x] divide_minus_right sgn_minus by auto
- thus ?thesis unfolding arctan_minus neg_equal_iff_equal .
- qed
+ using assms by (simp only: arctan_le_iff)
+
+lemma arctan_inverse:
+ assumes "x \<noteq> 0" shows "arctan (1 / x) = sgn x * pi / 2 - arctan x"
+proof (rule arctan_unique)
+ show "- (pi / 2) < sgn x * pi / 2 - arctan x"
+ using arctan_bounded [of x] assms
+ unfolding sgn_real_def
+ apply (auto simp add: algebra_simps)
+ apply (drule zero_less_arctan_iff [THEN iffD2])
+ apply arith
+ done
+ show "sgn x * pi / 2 - arctan x < pi / 2"
+ using arctan_bounded [of "- x"] assms
+ unfolding sgn_real_def arctan_minus
+ by auto
+ show "tan (sgn x * pi / 2 - arctan x) = 1 / x"
+ unfolding tan_inverse [of "arctan x", unfolded tan_arctan]
+ unfolding sgn_real_def
+ by (simp add: tan_def cos_arctan sin_arctan sin_diff cos_diff)
qed
theorem pi_series: "pi / 4 = (\<Sum> k. (-1)^k * 1 / real (k*2+1))" (is "_ = ?SUM")
proof -
- have "pi / 4 = arctan 1" using arctan1_eq_pi4 by auto
+ have "pi / 4 = arctan 1" using arctan_one by auto
also have "\<dots> = ?SUM" using arctan_series[of 1] by auto
finally show ?thesis by auto
qed
--- a/src/HOL/Word/Examples/WordExamples.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Word/Examples/WordExamples.thy Fri Sep 09 06:47:14 2011 +0200
@@ -7,7 +7,7 @@
header "Examples of word operations"
theory WordExamples
-imports Word
+imports "../Word"
begin
type_synonym word32 = "32 word"
--- a/src/HOL/Word/Misc_Numeric.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Word/Misc_Numeric.thy Fri Sep 09 06:47:14 2011 +0200
@@ -312,7 +312,7 @@
apply safe
apply (simp add: dvd_eq_mod_eq_0 [symmetric])
apply (drule le_iff_add [THEN iffD1])
- apply (force simp: zpower_zadd_distrib)
+ apply (force simp: power_add)
apply (rule mod_pos_pos_trivial)
apply (simp)
apply (rule power_strict_increasing)
--- a/src/HOL/Word/Word.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/Word/Word.thy Fri Sep 09 06:47:14 2011 +0200
@@ -455,12 +455,12 @@
by (simp add: number_of_eq word_number_of_def)
lemma word_no_wi: "number_of = word_of_int"
- by (auto simp: word_number_of_def intro: ext)
+ by (auto simp: word_number_of_def)
lemma to_bl_def':
"(to_bl :: 'a :: len0 word => bool list) =
bin_to_bl (len_of TYPE('a)) o uint"
- by (auto simp: to_bl_def intro: ext)
+ by (auto simp: to_bl_def)
lemmas word_reverse_no_def [simp] = word_reverse_def [of "number_of w", standard]
@@ -1257,7 +1257,7 @@
word_of_int_Ex [of b]
word_of_int_Ex [of c]
by (auto simp: word_of_int_hom_syms [symmetric]
- zadd_0_right add_commute add_assoc add_left_commute
+ add_0_right add_commute add_assoc add_left_commute
mult_commute mult_assoc mult_left_commute
left_distrib right_distrib)
@@ -4219,7 +4219,7 @@
apply (rule rotater_eqs)
apply (simp add: word_size nat_mod_distrib)
apply (rule int_eq_0_conv [THEN iffD1])
- apply (simp only: zmod_int zadd_int [symmetric])
+ apply (simp only: zmod_int of_nat_add)
apply (simp add: rdmods)
done
@@ -4230,8 +4230,6 @@
(* using locale to not pollute lemma namespace *)
locale word_rotate
-
-context word_rotate
begin
lemmas word_rot_defs' = to_bl_rotl to_bl_rotr
--- a/src/HOL/ex/Quickcheck_Lattice_Examples.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/ex/Quickcheck_Lattice_Examples.thy Fri Sep 09 06:47:14 2011 +0200
@@ -125,7 +125,7 @@
by (rule inf_absorb1) simp
lemma inf_eq_top_iff [simp]:
- "x \<sqinter> y = \<top> \<longleftrightarrow> x = \<top> \<and> y = \<top>"
+ "(x :: 'a :: bounded_lattice_top) \<sqinter> y = \<top> \<longleftrightarrow> x = \<top> \<and> y = \<top>"
quickcheck[expect = no_counterexample]
by (simp add: eq_iff)
--- a/src/HOL/ex/RPred.thy Fri Sep 09 06:45:39 2011 +0200
+++ b/src/HOL/ex/RPred.thy Fri Sep 09 06:47:14 2011 +0200
@@ -25,7 +25,7 @@
(* (infixl "\<squnion>" 80) *)
where
"supp RP1 RP2 = (\<lambda>s. let (P1, s') = RP1 s; (P2, s'') = RP2 s'
- in (semilattice_sup_class.sup P1 P2, s''))"
+ in (sup_class.sup P1 P2, s''))"
definition if_rpred :: "bool \<Rightarrow> unit rpred"
where
--- a/src/Pure/General/antiquote.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Pure/General/antiquote.ML Fri Sep 09 06:47:14 2011 +0200
@@ -12,7 +12,7 @@
Open of Position.T |
Close of Position.T
val is_text: 'a antiquote -> bool
- val report: ('a -> unit) -> 'a antiquote -> unit
+ val reports_of: ('a -> Position.report list) -> 'a antiquote list -> Position.report list
val check_nesting: 'a antiquote list -> unit
val scan_antiq: Symbol_Pos.T list -> (Symbol_Pos.T list * Position.range) * Symbol_Pos.T list
val scan: Symbol_Pos.T list -> 'a antiquote * Symbol_Pos.T list
@@ -35,14 +35,14 @@
| is_text _ = false;
-(* report *)
-
-fun report_antiq pos = Position.report pos Markup.antiq;
+(* reports *)
-fun report report_text (Text x) = report_text x
- | report _ (Antiq (_, (pos, _))) = report_antiq pos
- | report _ (Open pos) = report_antiq pos
- | report _ (Close pos) = report_antiq pos;
+fun reports_of text =
+ maps
+ (fn Text x => text x
+ | Antiq (_, (pos, _)) => [(pos, Markup.antiq)]
+ | Open pos => [(pos, Markup.antiq)]
+ | Close pos => [(pos, Markup.antiq)]);
(* check_nesting *)
@@ -97,7 +97,7 @@
fun read (syms, pos) =
(case Scan.read Symbol_Pos.stopper (Scan.repeat scan_text) syms of
- SOME xs => (List.app (report (K ())) xs; check_nesting xs; xs)
+ SOME xs => (Position.reports (reports_of (K []) xs); check_nesting xs; xs)
| NONE => error ("Malformed quotation/antiquotation source" ^ Position.str_of pos));
end;
--- a/src/Pure/General/position.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Pure/General/position.ML Fri Sep 09 06:47:14 2011 +0200
@@ -35,8 +35,9 @@
val reported_text: T -> Markup.T -> string -> string
val report_text: T -> Markup.T -> string -> unit
val report: T -> Markup.T -> unit
- type reports = (T * Markup.T) list
- val reports: reports Unsynchronized.ref -> T list -> ('a -> Markup.T list) -> 'a -> unit
+ type report = T * Markup.T
+ val reports: report list -> unit
+ val store_reports: report list Unsynchronized.ref -> T list -> ('a -> Markup.T list) -> 'a -> unit
val str_of: T -> string
type range = T * T
val no_range: range
@@ -154,10 +155,14 @@
fun report_text pos markup txt = Output.report (reported_text pos markup txt);
fun report pos markup = report_text pos markup "";
-type reports = (T * Markup.T) list;
+type report = T * Markup.T;
-fun reports _ [] _ _ = ()
- | reports (r: reports Unsynchronized.ref) ps markup x =
+val reports =
+ map (fn (pos, m) => if is_reported pos then Markup.markup_only (markup pos m) else "")
+ #> implode #> Output.report;
+
+fun store_reports _ [] _ _ = ()
+ | store_reports (r: report list Unsynchronized.ref) ps markup x =
let val ms = markup x
in Unsynchronized.change r (fold (fn p => fold (fn m => cons (p, m)) ms) ps) end;
--- a/src/Pure/Isar/outer_syntax.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Pure/Isar/outer_syntax.ML Fri Sep 09 06:47:14 2011 +0200
@@ -258,7 +258,7 @@
let
val commands = lookup_commands outer_syntax;
val range_pos = Position.set_range (Token.range toks);
- val _ = List.app Thy_Syntax.report_token toks;
+ val _ = Position.reports (maps Thy_Syntax.reports_of_token toks);
in
(case Source.exhaust (toplevel_source false NONE (K commands) (Source.of_list toks)) of
[tr] =>
--- a/src/Pure/ML/ml_compiler_polyml-5.3.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Pure/ML/ml_compiler_polyml-5.3.ML Fri Sep 09 06:47:14 2011 +0200
@@ -34,31 +34,31 @@
fun report_parse_tree depth space =
let
- val report_text =
+ val reported_text =
(case Context.thread_data () of
- SOME (Context.Proof ctxt) => Context_Position.report_text ctxt
- | _ => Position.report_text);
+ SOME (Context.Proof ctxt) => Context_Position.reported_text ctxt
+ | _ => Position.reported_text);
- fun report_entity kind loc decl =
- report_text (position_of loc)
+ fun reported_entity kind loc decl =
+ reported_text (position_of loc)
(Markup.entityN, (Markup.kindN, kind) :: Position.def_properties_of (position_of decl)) "";
- fun report loc (PolyML.PTtype types) =
- cons (fn () =>
- PolyML.NameSpace.displayTypeExpression (types, depth, space)
- |> pretty_ml |> Pretty.from_ML |> Pretty.string_of
- |> report_text (position_of loc) Markup.ML_typing)
- | report loc (PolyML.PTdeclaredAt decl) =
- cons (fn () => report_entity Markup.ML_defN loc decl)
- | report loc (PolyML.PTopenedAt decl) =
- cons (fn () => report_entity Markup.ML_openN loc decl)
- | report loc (PolyML.PTstructureAt decl) =
- cons (fn () => report_entity Markup.ML_structN loc decl)
- | report _ (PolyML.PTnextSibling tree) = report_tree (tree ())
- | report _ (PolyML.PTfirstChild tree) = report_tree (tree ())
- | report _ _ = I
- and report_tree (loc, props) = fold (report loc) props;
- in fn tree => List.app (fn e => e ()) (report_tree tree []) end;
+ fun reported loc (PolyML.PTtype types) =
+ cons
+ (PolyML.NameSpace.displayTypeExpression (types, depth, space)
+ |> pretty_ml |> Pretty.from_ML |> Pretty.string_of
+ |> reported_text (position_of loc) Markup.ML_typing)
+ | reported loc (PolyML.PTdeclaredAt decl) =
+ cons (reported_entity Markup.ML_defN loc decl)
+ | reported loc (PolyML.PTopenedAt decl) =
+ cons (reported_entity Markup.ML_openN loc decl)
+ | reported loc (PolyML.PTstructureAt decl) =
+ cons (reported_entity Markup.ML_structN loc decl)
+ | reported _ (PolyML.PTnextSibling tree) = reported_tree (tree ())
+ | reported _ (PolyML.PTfirstChild tree) = reported_tree (tree ())
+ | reported _ _ = I
+ and reported_tree (loc, props) = fold (reported loc) props;
+ in fn tree => Output.report (implode (reported_tree tree [])) end;
(* eval ML source tokens *)
--- a/src/Pure/ML/ml_lex.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Pure/ML/ml_lex.ML Fri Sep 09 06:47:14 2011 +0200
@@ -20,7 +20,6 @@
val content_of: token -> string
val check_content_of: token -> string
val flatten: token list -> string
- val report_token: token -> unit
val keywords: string list
val source: (Symbol.symbol, 'a) Source.source ->
(token, (Symbol_Pos.T, Position.T * (Symbol.symbol, 'a) Source.source)
@@ -126,7 +125,7 @@
in
-fun report_token (Token ((pos, _), (kind, x))) = Position.report pos (token_markup kind x);
+fun report_of_token (Token ((pos, _), (kind, x))) = (pos, token_markup kind x);
end;
@@ -293,7 +292,7 @@
|> Source.source Symbol_Pos.stopper (Scan.bulk (!!! "bad input" scan_antiq))
(SOME (false, fn msg => recover msg >> map Antiquote.Text))
|> Source.exhaust
- |> tap (List.app (Antiquote.report report_token))
+ |> tap (Position.reports o Antiquote.reports_of (single o report_of_token))
|> tap Antiquote.check_nesting
|> tap (List.app (fn Antiquote.Text tok => (check_content_of tok; warn tok) | _ => ())))
handle ERROR msg =>
--- a/src/Pure/PIDE/document.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Pure/PIDE/document.ML Fri Sep 09 06:47:14 2011 +0200
@@ -331,7 +331,6 @@
let
val is_init = Toplevel.is_init tr;
val is_proof = Keyword.is_proof (Toplevel.name_of tr);
- val do_print = not is_init andalso (Toplevel.print_of tr orelse is_proof);
val _ = Multithreading.interrupted ();
val _ = Toplevel.status tr Markup.forked;
@@ -343,13 +342,18 @@
in
(case result of
NONE =>
- (if null errs then Exn.interrupt () else ();
- Toplevel.status tr Markup.failed;
- (st, no_print))
+ let
+ val _ = if null errs then Exn.interrupt () else ();
+ val _ = Toplevel.status tr Markup.failed;
+ in (st, no_print) end
| SOME st' =>
- (Toplevel.status tr Markup.finished;
- proof_status tr st';
- (st', if do_print then print_state tr st' else no_print)))
+ let
+ val _ = Toplevel.status tr Markup.finished;
+ val _ = proof_status tr st';
+ val do_print =
+ not is_init andalso
+ (Toplevel.print_of tr orelse (is_proof andalso Toplevel.is_proof st'));
+ in (st', if do_print then print_state tr st' else no_print) end)
end;
end;
--- a/src/Pure/PIDE/xml.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Pure/PIDE/xml.ML Fri Sep 09 06:47:14 2011 +0200
@@ -47,6 +47,7 @@
val parse_element: string list -> tree * string list
val parse_document: string list -> tree * string list
val parse: string -> tree
+ val cache: unit -> tree -> tree
exception XML_ATOM of string
exception XML_BODY of body
structure Encode: XML_DATA_OPS
@@ -228,6 +229,48 @@
end;
+(** cache for substructural sharing **)
+
+fun tree_ord tu =
+ if pointer_eq tu then EQUAL
+ else
+ (case tu of
+ (Text _, Elem _) => LESS
+ | (Elem _, Text _) => GREATER
+ | (Text s, Text s') => fast_string_ord (s, s')
+ | (Elem e, Elem e') =>
+ prod_ord
+ (prod_ord fast_string_ord (list_ord (prod_ord fast_string_ord fast_string_ord)))
+ (list_ord tree_ord) (e, e'));
+
+structure Treetab = Table(type key = tree val ord = tree_ord);
+
+fun cache () =
+ let
+ val strings = Unsynchronized.ref (Symtab.empty: unit Symtab.table);
+ val trees = Unsynchronized.ref (Treetab.empty: unit Treetab.table);
+
+ fun string s =
+ if size s <= 1 then s
+ else
+ (case Symtab.lookup_key (! strings) s of
+ SOME (s', ()) => s'
+ | NONE => (Unsynchronized.change strings (Symtab.update (s, ())); s));
+
+ fun tree t =
+ (case Treetab.lookup_key (! trees) t of
+ SOME (t', ()) => t'
+ | NONE =>
+ let
+ val t' =
+ (case t of
+ Elem ((a, ps), b) => Elem ((string a, map (pairself string) ps), map tree b)
+ | Text s => Text (string s));
+ val _ = Unsynchronized.change trees (Treetab.update (t', ()));
+ in t' end);
+ in tree end;
+
+
(** XML as data representation language **)
--- a/src/Pure/PIDE/xml.scala Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Pure/PIDE/xml.scala Fri Sep 09 06:47:14 2011 +0200
@@ -84,7 +84,8 @@
def content(body: Body): Iterator[String] = content_stream(body).iterator
- /* pipe-lined cache for partial sharing */
+
+ /** cache for partial sharing (weak table) **/
class Cache(initial_size: Int = 131071, max_string: Int = 100)
{
--- a/src/Pure/Syntax/lexicon.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Pure/Syntax/lexicon.ML Fri Sep 09 06:47:14 2011 +0200
@@ -44,7 +44,7 @@
val tvarT: typ
val terminals: string list
val is_terminal: string -> bool
- val report_token: Proof.context -> token -> unit
+ val report_of_token: token -> Position.report
val reported_token_range: Proof.context -> token -> string
val matching_tokens: token * token -> bool
val valued_token: token -> bool
@@ -203,11 +203,11 @@
| Comment => Markup.inner_comment
| EOF => Markup.empty;
-fun report_token ctxt (Token (kind, s, (pos, _))) =
+fun report_of_token (Token (kind, s, (pos, _))) =
let val markup =
if kind = Literal andalso not (is_ascii_identifier s) then Markup.delimiter
else token_kind_markup kind
- in Context_Position.report ctxt pos markup end;
+ in (pos, markup) end;
fun reported_token_range ctxt tok =
if is_proper tok
--- a/src/Pure/Syntax/syntax.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Pure/Syntax/syntax.ML Fri Sep 09 06:47:14 2011 +0200
@@ -99,6 +99,7 @@
val string_of_sort_global: theory -> sort -> string
type syntax
val eq_syntax: syntax * syntax -> bool
+ val join_syntax: syntax -> unit
val lookup_const: syntax -> string -> string option
val is_keyword: syntax -> string -> bool
val tokenize: syntax -> bool -> Symbol_Pos.T list -> Lexicon.token list
@@ -508,6 +509,8 @@
fun eq_syntax (Syntax (_, s1), Syntax (_, s2)) = s1 = s2;
+fun join_syntax (Syntax ({gram, ...}, _)) = ignore (Future.join gram);
+
fun lookup_const (Syntax ({consts, ...}, _)) = Symtab.lookup consts;
fun is_keyword (Syntax ({lexicon, ...}, _)) = Scan.is_literal lexicon o Symbol.explode;
fun tokenize (Syntax ({lexicon, ...}, _)) = Lexicon.tokenize lexicon;
--- a/src/Pure/Syntax/syntax_phases.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Pure/Syntax/syntax_phases.ML Fri Sep 09 06:47:14 2011 +0200
@@ -10,7 +10,7 @@
val term_sorts: term -> (indexname * sort) list
val typ_of_term: (indexname -> sort) -> term -> typ
val decode_term: Proof.context ->
- Position.reports * term Exn.result -> Position.reports * term Exn.result
+ Position.report list * term Exn.result -> Position.report list * term Exn.result
val parse_ast_pattern: Proof.context -> string * string -> Ast.ast
val term_of_typ: Proof.context -> typ -> term
end
@@ -126,8 +126,8 @@
fun parsetree_to_ast ctxt constrain_pos trf parsetree =
let
- val reports = Unsynchronized.ref ([]: Position.reports);
- fun report pos = Position.reports reports [pos];
+ val reports = Unsynchronized.ref ([]: Position.report list);
+ fun report pos = Position.store_reports reports [pos];
fun trans a args =
(case trf a of
@@ -196,7 +196,7 @@
(* decode_term -- transform parse tree into raw term *)
-fun decode_term _ (result as (_: Position.reports, Exn.Exn _)) = result
+fun decode_term _ (result as (_: Position.report list, Exn.Exn _)) = result
| decode_term ctxt (reports0, Exn.Res tm) =
let
fun get_const a =
@@ -207,7 +207,7 @@
val decodeT = typ_of_term (Proof_Context.get_sort ctxt (term_sorts tm));
val reports = Unsynchronized.ref reports0;
- fun report ps = Position.reports reports ps;
+ fun report ps = Position.store_reports reports ps;
fun decode ps qs bs (Const ("_constrain", _) $ t $ typ) =
(case Term_Position.decode_position typ of
@@ -262,12 +262,10 @@
fun proper_results results = map_filter (fn (y, Exn.Res x) => SOME (y, x) | _ => NONE) results;
fun failed_results results = map_filter (fn (y, Exn.Exn e) => SOME (y, e) | _ => NONE) results;
-fun report ctxt = List.app (fn (pos, m) => Context_Position.report ctxt pos m);
-
fun report_result ctxt pos results =
(case (proper_results results, failed_results results) of
- ([], (reports, exn) :: _) => (report ctxt reports; reraise exn)
- | ([(reports, x)], _) => (report ctxt reports; x)
+ ([], (reports, exn) :: _) => (Context_Position.reports ctxt reports; reraise exn)
+ | ([(reports, x)], _) => (Context_Position.reports ctxt reports; x)
| _ => error (ambiguity_msg pos));
@@ -279,7 +277,7 @@
val ast_tr = Syntax.parse_ast_translation syn;
val toks = Syntax.tokenize syn raw syms;
- val _ = List.app (Lexicon.report_token ctxt) toks;
+ val _ = Context_Position.reports ctxt (map Lexicon.report_of_token toks);
val pts = Syntax.parse ctxt syn root (filter Lexicon.is_proper toks)
handle ERROR msg =>
--- a/src/Pure/System/isabelle_charset.scala Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Pure/System/isabelle_charset.scala Fri Sep 09 06:47:14 2011 +0200
@@ -37,14 +37,18 @@
{
override def charsetForName(name: String): Charset =
{
- if (name.equalsIgnoreCase(Isabelle_Charset.name)) Isabelle_Charset.charset
- else null
+ // FIXME inactive
+ // if (name.equalsIgnoreCase(Isabelle_Charset.name)) Isabelle_Charset.charset
+ // else null
+ null
}
override def charsets(): java.util.Iterator[Charset] =
{
import scala.collection.JavaConversions._
- Iterator(Isabelle_Charset.charset)
+ // FIXME inactive
+ // Iterator(Isabelle_Charset.charset)
+ Iterator()
}
}
--- a/src/Pure/System/isabelle_process.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Pure/System/isabelle_process.ML Fri Sep 09 06:47:14 2011 +0200
@@ -66,27 +66,29 @@
fun chunk s = [string_of_int (size s), "\n", s];
-fun message mbox ch raw_props body =
+fun message do_flush mbox ch raw_props body =
let
val robust_props = map (pairself YXML.embed_controls) raw_props;
val header = YXML.string_of (XML.Elem ((ch, robust_props), []));
- in Mailbox.send mbox (chunk header @ chunk body) end;
+ in Mailbox.send mbox (chunk header @ chunk body, do_flush) end;
fun standard_message mbox opt_serial ch body =
if body = "" then ()
else
- message mbox ch
+ message false mbox ch
((case opt_serial of SOME i => cons (Markup.serialN, string_of_int i) | _ => I)
(Position.properties_of (Position.thread_data ()))) body;
fun message_output mbox out_stream =
let
+ fun flush () = ignore (try TextIO.flushOut out_stream);
fun loop receive =
(case receive mbox of
- SOME msg =>
+ SOME (msg, do_flush) =>
(List.app (fn s => TextIO.output (out_stream, s)) msg;
+ if do_flush then flush () else ();
loop (Mailbox.receive_timeout (seconds 0.02)))
- | NONE => (try TextIO.flushOut out_stream; loop (SOME o Mailbox.receive)));
+ | NONE => (flush (); loop (SOME o Mailbox.receive)));
in fn () => loop (SOME o Mailbox.receive) end;
in
@@ -100,7 +102,7 @@
val _ = TextIO.StreamIO.setBufferMode (TextIO.getOutstream TextIO.stdOut, IO.LINE_BUF);
val _ = TextIO.StreamIO.setBufferMode (TextIO.getOutstream TextIO.stdErr, IO.LINE_BUF);
- val mbox = Mailbox.create () : string list Mailbox.T;
+ val mbox = Mailbox.create () : (string list * bool) Mailbox.T;
val _ = Simple_Thread.fork false (message_output mbox out_stream);
in
Output.Private_Hooks.status_fn := standard_message mbox NONE "B";
@@ -109,10 +111,10 @@
Output.Private_Hooks.tracing_fn := (fn s => standard_message mbox (SOME (serial ())) "E" s);
Output.Private_Hooks.warning_fn := (fn s => standard_message mbox (SOME (serial ())) "F" s);
Output.Private_Hooks.error_fn := (fn (i, s) => standard_message mbox (SOME i) "G" s);
- Output.Private_Hooks.raw_message_fn := message mbox "H";
+ Output.Private_Hooks.raw_message_fn := message true mbox "H";
Output.Private_Hooks.urgent_message_fn := ! Output.Private_Hooks.writeln_fn;
Output.Private_Hooks.prompt_fn := ignore;
- message mbox "A" [] (Session.welcome ());
+ message true mbox "A" [] (Session.welcome ());
in_stream
end;
--- a/src/Pure/System/isabelle_process.scala Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Pure/System/isabelle_process.scala Fri Sep 09 06:47:14 2011 +0200
@@ -33,7 +33,7 @@
('H' : Int) -> Markup.RAW)
}
- abstract class Message
+ sealed abstract class Message
class Input(name: String, args: List[String]) extends Message
{
@@ -75,22 +75,21 @@
}
-class Isabelle_Process(timeout: Time, receiver: Actor, args: String*)
+class Isabelle_Process(timeout: Time, receiver: Isabelle_Process.Message => Unit, args: String*)
{
import Isabelle_Process._
/* demo constructor */
- def this(args: String*) =
- this(Time.seconds(10), actor { loop { react { case res => Console.println(res) } } }, args: _*)
+ def this(args: String*) = this(Time.seconds(10), Console.println(_), args: _*)
/* results */
private def system_result(text: String)
{
- receiver ! new Result(XML.Elem(Markup(Markup.SYSTEM, Nil), List(XML.Text(text))))
+ receiver(new Result(XML.Elem(Markup(Markup.SYSTEM, Nil), List(XML.Text(text)))))
}
private val xml_cache = new XML.Cache()
@@ -99,10 +98,10 @@
{
if (kind == Markup.INIT) rm_fifos()
if (kind == Markup.RAW)
- receiver ! new Result(XML.Elem(Markup(kind, props), body))
+ receiver(new Result(XML.Elem(Markup(kind, props), body)))
else {
val msg = XML.Elem(Markup(kind, props), Isar_Document.clean_message(body))
- receiver ! new Result(xml_cache.cache_tree(msg).asInstanceOf[XML.Elem])
+ receiver(new Result(xml_cache.cache_tree(msg).asInstanceOf[XML.Elem]))
}
}
@@ -206,12 +205,6 @@
def join() { process_manager.join() }
- def interrupt()
- {
- try { process.interrupt }
- catch { case e: IOException => system_result("Failed to interrupt Isabelle: " + e.getMessage) }
- }
-
def terminate()
{
close()
@@ -399,7 +392,7 @@
def input(name: String, args: String*): Unit =
{
- receiver ! new Input(name, args.toList)
+ receiver(new Input(name, args.toList))
input_bytes(name, args.map(Standard_System.string_bytes): _*)
}
--- a/src/Pure/System/session.scala Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Pure/System/session.scala Fri Sep 09 06:47:14 2011 +0200
@@ -7,8 +7,11 @@
package isabelle
+
import java.lang.System
+import java.util.{Timer, TimerTask}
+import scala.collection.mutable
import scala.actors.TIMEOUT
import scala.actors.Actor._
@@ -19,7 +22,7 @@
//{{{
case object Global_Settings
- case object Perspective
+ case object Caret_Focus
case object Assignment
case class Commands_Changed(nodes: Set[Document.Node.Name], commands: Set[Command])
@@ -37,6 +40,7 @@
{
/* real time parameters */ // FIXME properties or settings (!?)
+ val message_delay = Time.seconds(0.01) // prover messages
val input_delay = Time.seconds(0.3) // user input (e.g. text edits, cursor movement)
val output_delay = Time.seconds(0.1) // prover output (markup, common messages)
val update_delay = Time.seconds(0.5) // GUI layout updates
@@ -48,11 +52,13 @@
/* pervasive event buses */
val global_settings = new Event_Bus[Session.Global_Settings.type]
- val perspective = new Event_Bus[Session.Perspective.type]
+ val caret_focus = new Event_Bus[Session.Caret_Focus.type]
val assignments = new Event_Bus[Session.Assignment.type]
val commands_changed = new Event_Bus[Session.Commands_Changed]
val phase_changed = new Event_Bus[Session.Phase]
- val raw_messages = new Event_Bus[Isabelle_Process.Message]
+ val syslog_messages = new Event_Bus[Isabelle_Process.Result]
+ val raw_output_messages = new Event_Bus[Isabelle_Process.Result]
+ val raw_messages = new Event_Bus[Isabelle_Process.Message] // potential bottle-neck
@@ -131,7 +137,7 @@
/* actor messages */
private case class Start(timeout: Time, args: List[String])
- private case object Interrupt
+ private case object Cancel_Execution
private case class Init_Node(name: Document.Node.Name,
header: Document.Node_Header, perspective: Text.Perspective, text: String)
private case class Edit_Node(name: Document.Node.Name,
@@ -141,15 +147,45 @@
doc_edits: List[Document.Edit_Command],
previous: Document.Version,
version: Document.Version)
+ private case class Messages(msgs: List[Isabelle_Process.Message])
private val (_, session_actor) = Simple_Thread.actor("session_actor", daemon = true)
{
val this_actor = self
- var prover: Option[Isabelle_Process with Isar_Document] = None
var prune_next = System.currentTimeMillis() + prune_delay.ms
+ /* buffered prover messages */
+
+ object receiver
+ {
+ private var buffer = new mutable.ListBuffer[Isabelle_Process.Message]
+
+ def flush(): Unit = synchronized {
+ if (!buffer.isEmpty) {
+ val msgs = buffer.toList
+ this_actor ! Messages(msgs)
+ buffer = new mutable.ListBuffer[Isabelle_Process.Message]
+ }
+ }
+ def invoke(msg: Isabelle_Process.Message): Unit = synchronized {
+ buffer += msg
+ msg match {
+ case result: Isabelle_Process.Result if result.is_raw => flush()
+ case _ =>
+ }
+ }
+
+ private val timer = new Timer("session_actor.receiver", true)
+ timer.schedule(new TimerTask { def run = flush }, message_delay.ms, message_delay.ms)
+
+ def cancel() { timer.cancel() }
+ }
+
+ var prover: Option[Isabelle_Process with Isar_Document] = None
+
+
/* delayed command changes */
object commands_changed_delay
@@ -371,7 +407,8 @@
case Start(timeout, args) if prover.isEmpty =>
if (phase == Session.Inactive || phase == Session.Failed) {
phase = Session.Startup
- prover = Some(new Isabelle_Process(timeout, this_actor, args:_*) with Isar_Document)
+ prover =
+ Some(new Isabelle_Process(timeout, receiver.invoke _, args:_*) with Isar_Document)
}
case Stop =>
@@ -383,10 +420,11 @@
phase = Session.Inactive
}
finished = true
+ receiver.cancel()
reply(())
- case Interrupt if prover.isDefined =>
- prover.get.interrupt
+ case Cancel_Execution if prover.isDefined =>
+ prover.get.cancel_execution()
case Init_Node(name, header, perspective, text) if prover.isDefined =>
// FIXME compare with existing node
@@ -408,12 +446,17 @@
case change: Change_Node if prover.isDefined =>
handle_change(change)
- case input: Isabelle_Process.Input =>
- raw_messages.event(input)
+ case Messages(msgs) =>
+ msgs foreach {
+ case input: Isabelle_Process.Input =>
+ raw_messages.event(input)
- case result: Isabelle_Process.Result =>
- handle_result(result)
- raw_messages.event(result)
+ case result: Isabelle_Process.Result =>
+ handle_result(result)
+ if (result.is_syslog) syslog_messages.event(result)
+ if (result.is_stdout) raw_output_messages.event(result)
+ raw_messages.event(result)
+ }
case bad => System.err.println("session_actor: ignoring bad message " + bad)
}
@@ -428,7 +471,7 @@
def stop() { commands_changed_buffer !? Stop; session_actor !? Stop }
- def interrupt() { session_actor ! Interrupt }
+ def cancel_execution() { session_actor ! Cancel_Execution }
def init_node(name: Document.Node.Name,
header: Document.Node_Header, perspective: Text.Perspective, text: String)
--- a/src/Pure/Thy/thy_syntax.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Pure/Thy/thy_syntax.ML Fri Sep 09 06:47:14 2011 +0200
@@ -11,7 +11,7 @@
Source.source) Source.source) Source.source) Source.source
val parse_tokens: Scan.lexicon * Scan.lexicon -> Position.T -> string -> Token.T list
val present_token: Token.T -> Output.output
- val report_token: Token.T -> unit
+ val reports_of_token: Token.T -> Position.report list
datatype span_kind = Command of string | Ignored | Malformed
type span
val span_kind: span -> span_kind
@@ -74,17 +74,17 @@
else I;
in props (token_kind_markup kind) end;
-fun report_symbol (sym, pos) =
- if Symbol.is_malformed sym then Position.report pos Markup.malformed else ();
+fun reports_of_symbol (sym, pos) =
+ if Symbol.is_malformed sym then [(pos, Markup.malformed)] else [];
in
fun present_token tok =
Markup.enclose (token_markup tok) (Output.output (Token.unparse tok));
-fun report_token tok =
- (Position.report (Token.position_of tok) (token_markup tok);
- List.app report_symbol (Symbol_Pos.explode (Token.source_position_of tok)));
+fun reports_of_token tok =
+ (Token.position_of tok, token_markup tok) ::
+ maps reports_of_symbol (Symbol_Pos.explode (Token.source_position_of tok));
end;
--- a/src/Pure/context_position.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Pure/context_position.ML Fri Sep 09 06:47:14 2011 +0200
@@ -14,6 +14,7 @@
val reported_text: Proof.context -> Position.T -> Markup.T -> string -> string
val report_text: Proof.context -> Position.T -> Markup.T -> string -> unit
val report: Proof.context -> Position.T -> Markup.T -> unit
+ val reports: Proof.context -> Position.report list -> unit
end;
structure Context_Position: CONTEXT_POSITION =
@@ -35,4 +36,6 @@
fun report_text ctxt pos markup txt = Output.report (reported_text ctxt pos markup txt);
fun report ctxt pos markup = report_text ctxt pos markup "";
+fun reports ctxt reps = if is_visible ctxt then Position.reports reps else ();
+
end;
--- a/src/Pure/theory.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Pure/theory.ML Fri Sep 09 06:47:14 2011 +0200
@@ -147,6 +147,7 @@
|> Sign.local_path
|> Sign.map_naming (Name_Space.set_theory_name name)
|> apply_wrappers wrappers
+ |> tap (Syntax.join_syntax o Sign.syn_of)
end;
fun end_theory thy =
--- a/src/Tools/Code/code_haskell.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Tools/Code/code_haskell.ML Fri Sep 09 06:47:14 2011 +0200
@@ -25,7 +25,7 @@
(** Haskell serializer **)
fun print_haskell_stmt labelled_name class_syntax tyco_syntax const_syntax
- reserved deresolve contr_classparam_typs deriving_show =
+ reserved deresolve deriving_show =
let
fun class_name class = case class_syntax class
of NONE => deresolve class
@@ -75,20 +75,14 @@
then print_case tyvars some_thm vars fxy cases
else print_app tyvars some_thm vars fxy c_ts
| NONE => print_case tyvars some_thm vars fxy cases)
- and print_app_expr tyvars some_thm vars ((c, (_, function_typs)), ts) = case contr_classparam_typs c
- of [] => (str o deresolve) c :: map (print_term tyvars some_thm vars BR) ts
- | fingerprint => let
- val ts_fingerprint = ts ~~ take (length ts) fingerprint;
- val needs_annotation = forall (fn (_, NONE) => true | (t, SOME _) =>
- (not o Code_Thingol.locally_monomorphic) t) ts_fingerprint;
- fun print_term_anno (t, NONE) _ = print_term tyvars some_thm vars BR t
- | print_term_anno (t, SOME _) ty =
- brackets [print_term tyvars some_thm vars NOBR t, str "::", print_typ tyvars NOBR ty];
- in
- if needs_annotation then
- (str o deresolve) c :: map2 print_term_anno ts_fingerprint (take (length ts) function_typs)
- else (str o deresolve) c :: map (print_term tyvars some_thm vars BR) ts
- end
+ and print_app_expr tyvars some_thm vars ((c, ((_, (function_typs, body_typ)), annotate)), ts) =
+ let
+ val ty = Library.foldr (fn (ty1, ty2) => Code_Thingol.fun_tyco `%% [ty1, ty2]) (function_typs, body_typ)
+ fun put_annotation c = brackets [c, str "::", print_typ tyvars NOBR ty]
+ in
+ ((if annotate then put_annotation else I)
+ ((str o deresolve) c)) :: map (print_term tyvars some_thm vars BR) ts
+ end
and print_app tyvars = gen_print_app (print_app_expr tyvars) (print_term tyvars) const_syntax
and print_bind tyvars some_thm fxy p = gen_print_bind (print_term tyvars) some_thm fxy p
and print_case tyvars some_thm vars fxy (cases as ((_, [_]), _)) =
@@ -230,14 +224,14 @@
]
| SOME k =>
let
- val (c, (_, tys)) = const;
+ val (c, ((_, tys), _)) = const; (* FIXME: pass around the need annotation flag here? *)
val (vs, rhs) = (apfst o map) fst
(Code_Thingol.unfold_abs (Code_Thingol.eta_expand k (const, [])));
val s = if (is_some o const_syntax) c
then NONE else (SOME o Long_Name.base_name o deresolve) c;
val vars = reserved
|> intro_vars (map_filter I (s :: vs));
- val lhs = IConst (classparam, (([], []), tys)) `$$ map IVar vs;
+ val lhs = IConst (classparam, ((([], []), tys), false (* FIXME *))) `$$ map IVar vs;
(*dictionaries are not relevant at this late stage*)
in
semicolon [
@@ -304,7 +298,6 @@
labelled_name module_alias module_prefix (Name.make_context reserved) program;
(* print statements *)
- val contr_classparam_typs = Code_Thingol.contr_classparam_typs program;
fun deriving_show tyco =
let
fun deriv _ "fun" = false
@@ -320,7 +313,7 @@
in deriv [] tyco end;
fun print_stmt deresolve = print_haskell_stmt labelled_name
class_syntax tyco_syntax const_syntax (make_vars reserved)
- deresolve contr_classparam_typs
+ deresolve
(if string_classes then deriving_show else K false);
(* print modules *)
--- a/src/Tools/Code/code_ml.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Tools/Code/code_ml.ML Fri Sep 09 06:47:14 2011 +0200
@@ -117,7 +117,7 @@
then print_case is_pseudo_fun some_thm vars fxy cases
else print_app is_pseudo_fun some_thm vars fxy c_ts
| NONE => print_case is_pseudo_fun some_thm vars fxy cases)
- and print_app_expr is_pseudo_fun some_thm vars (app as ((c, ((_, iss), function_typs)), ts)) =
+ and print_app_expr is_pseudo_fun some_thm vars (app as ((c, (((_, iss), (function_typs, _)), _)), ts)) =
if is_cons c then
let val k = length function_typs in
if k < 2 orelse length ts = k
@@ -417,7 +417,7 @@
then print_case is_pseudo_fun some_thm vars fxy cases
else print_app is_pseudo_fun some_thm vars fxy c_ts
| NONE => print_case is_pseudo_fun some_thm vars fxy cases)
- and print_app_expr is_pseudo_fun some_thm vars (app as ((c, ((_, iss), tys)), ts)) =
+ and print_app_expr is_pseudo_fun some_thm vars (app as ((c, (((_, iss), (tys, _)), _)), ts)) =
if is_cons c then
let val k = length tys in
if length ts = k
--- a/src/Tools/Code/code_printer.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Tools/Code/code_printer.ML Fri Sep 09 06:47:14 2011 +0200
@@ -315,7 +315,7 @@
|-> (fn cs' => pair (Complex_const_syntax (n, f literals cs')));
fun gen_print_app print_app_expr print_term const_syntax some_thm vars fxy
- (app as ((c, (_, function_typs)), ts)) =
+ (app as ((c, ((_, (function_typs, _)), _)), ts)) =
case const_syntax c of
NONE => brackify fxy (print_app_expr some_thm vars app)
| SOME (Plain_const_syntax (_, s)) =>
--- a/src/Tools/Code/code_scala.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Tools/Code/code_scala.ML Fri Sep 09 06:47:14 2011 +0200
@@ -72,7 +72,7 @@
else print_app tyvars is_pat some_thm vars fxy c_ts
| NONE => print_case tyvars some_thm vars fxy cases)
and print_app tyvars is_pat some_thm vars fxy
- (app as ((c, ((arg_typs, _), function_typs)), ts)) =
+ (app as ((c, (((arg_typs, _), (function_typs, _)), _)), ts)) =
let
val k = length ts;
val arg_typs' = if is_pat orelse
@@ -265,7 +265,7 @@
let
val tyvars = intro_tyvars vs reserved;
val classtyp = (class, tyco `%% map (ITyVar o fst) vs);
- fun print_classparam_instance ((classparam, const as (_, (_, tys))), (thm, _)) =
+ fun print_classparam_instance ((classparam, const as (_, ((_, (tys, _)), _))), (thm, _)) =
let
val aux_tys = Name.invent_names (snd reserved) "a" tys;
val auxs = map fst aux_tys;
--- a/src/Tools/Code/code_target.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Tools/Code/code_target.ML Fri Sep 09 06:47:14 2011 +0200
@@ -394,7 +394,7 @@
fun check_code_for thy target strict args naming program names_cs =
let
- val module_name = "Code";
+ val module_name = "Generated_Code";
val { env_var, make_destination, make_command } =
(#check o the_fundamental thy) target;
fun ext_check p =
@@ -435,7 +435,7 @@
fun evaluator thy target naming program consts =
let
val (mounted_serializer, prepared_program) = mount_serializer thy
- target NONE "Code" [] naming program consts;
+ target NONE "Generated_Code" [] naming program consts;
in evaluation mounted_serializer prepared_program consts end;
end; (* local *)
--- a/src/Tools/Code/code_thingol.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Tools/Code/code_thingol.ML Fri Sep 09 06:47:14 2011 +0200
@@ -22,7 +22,7 @@
datatype itype =
`%% of string * itype list
| ITyVar of vname;
- type const = string * ((itype list * dict list list) * itype list)
+ type const = string * (((itype list * dict list list) * (itype list * itype)) * bool)
(* f [T1..Tn] {dicts} (_::S1) .. (_..Sm) =^= (f, (([T1..Tn], dicts), [S1..Sm]) *)
datatype iterm =
IConst of const
@@ -55,7 +55,6 @@
val is_IAbs: iterm -> bool
val eta_expand: int -> const * iterm list -> iterm
val contains_dict_var: iterm -> bool
- val locally_monomorphic: iterm -> bool
val add_constnames: iterm -> string list -> string list
val add_tyconames: iterm -> string list -> string list
val fold_varnames: (string -> 'a -> 'a) -> iterm -> 'a -> 'a
@@ -88,7 +87,6 @@
val map_terms_stmt: (iterm -> iterm) -> stmt -> stmt
val is_cons: program -> string -> bool
val is_case: stmt -> bool
- val contr_classparam_typs: program -> string -> itype option list
val labelled_name: theory -> program -> string -> string
val group_stmts: theory -> program
-> ((string * stmt) list * (string * stmt) list
@@ -145,7 +143,8 @@
`%% of string * itype list
| ITyVar of vname;
-type const = string * ((itype list * dict list list) * itype list (*types of arguments*))
+type const = string * (((itype list * dict list list) *
+ (itype list (*types of arguments*) * itype (*body type*))) * bool (*requires type annotation*))
datatype iterm =
IConst of const
@@ -198,7 +197,7 @@
fun add_tycos (tyco `%% tys) = insert (op =) tyco #> fold add_tycos tys
| add_tycos (ITyVar _) = I;
-val add_tyconames = fold_constexprs (fn (_, ((tys, _), _)) => fold add_tycos tys);
+val add_tyconames = fold_constexprs (fn (_, (((tys, _), _), _)) => fold add_tycos tys);
fun fold_varnames f =
let
@@ -240,7 +239,7 @@
val vs_tys = (map o apfst) SOME (Name.invent_names ctxt "a" tys);
in (vs_tys, t `$$ map (IVar o fst) vs_tys) end;
-fun eta_expand k (c as (name, (_, tys)), ts) =
+fun eta_expand k (c as (name, ((_, (tys, _)), _)), ts) =
let
val j = length ts;
val l = k - j;
@@ -256,18 +255,12 @@
fun cont_dict (Dict (_, d)) = cont_plain_dict d
and cont_plain_dict (Dict_Const (_, dss)) = (exists o exists) cont_dict dss
| cont_plain_dict (Dict_Var _) = true;
- fun cont_term (IConst (_, ((_, dss), _))) = (exists o exists) cont_dict dss
+ fun cont_term (IConst (_, (((_, dss), _), _))) = (exists o exists) cont_dict dss
| cont_term (IVar _) = false
| cont_term (t1 `$ t2) = cont_term t1 orelse cont_term t2
| cont_term (_ `|=> t) = cont_term t
| cont_term (ICase (_, t)) = cont_term t;
in cont_term t end;
-
-fun locally_monomorphic (IConst _) = false
- | locally_monomorphic (IVar _) = true
- | locally_monomorphic (t `$ _) = locally_monomorphic t
- | locally_monomorphic (_ `|=> t) = locally_monomorphic t
- | locally_monomorphic (ICase ((_, ds), _)) = exists (locally_monomorphic o snd) ds;
(** namings **)
@@ -480,28 +473,6 @@
(fn (_, (Classinst ((class, _), (_, (param_insts, _))), _)) =>
Option.map (fn ((const, _), _) => (class, const))
(find_first (fn ((_, (inst_const, _)), _) => inst_const = name) param_insts) | _ => NONE)
-
-fun contr_classparam_typs program name =
- let
- fun contr_classparam_typs' (class, name) =
- let
- val Class (_, (_, (_, params))) = Graph.get_node program class;
- val SOME ty = AList.lookup (op =) params name;
- val (tys, res_ty) = unfold_fun ty;
- fun no_tyvar (_ `%% tys) = forall no_tyvar tys
- | no_tyvar (ITyVar _) = false;
- in if no_tyvar res_ty
- then map (fn ty => if no_tyvar ty then NONE else SOME ty) tys
- else []
- end
- in
- case Graph.get_node program name
- of Classparam (_, class) => contr_classparam_typs' (class, name)
- | Fun (c, _) => (case lookup_classparam_instance program name
- of NONE => []
- | SOME (class, name) => the_default [] (try contr_classparam_typs' (class, name)))
- | _ => []
- end;
fun labelled_name thy program name =
let val ctxt = Proof_Context.init_global thy in
@@ -608,6 +579,42 @@
(err_typ ^ "\n" ^ err_class)
end;
+(* inference of type annotations for disambiguation with type classes *)
+
+fun annotate_term (Const (c', T'), Const (c, T)) tvar_names =
+ let
+ val tvar_names' = Term.add_tvar_namesT T' tvar_names
+ in
+ (Const (c, if eq_set (op =) (tvar_names, tvar_names') then T else Type("", [T])), tvar_names')
+ end
+ | annotate_term (t1 $ u1, t $ u) tvar_names =
+ let
+ val (u', tvar_names') = annotate_term (u1, u) tvar_names
+ val (t', tvar_names'') = annotate_term (t1, t) tvar_names'
+ in
+ (t' $ u', tvar_names'')
+ end
+ | annotate_term (Abs (_, _, t1) , Abs (x, T, t)) tvar_names =
+ apfst (fn t => Abs (x, T, t)) (annotate_term (t1, t) tvar_names)
+ | annotate_term (_, t) tvar_names = (t, tvar_names)
+
+fun annotate_eqns thy eqns =
+ let
+ val ctxt = ProofContext.init_global thy |> Config.put Type_Infer_Context.const_sorts false
+ val erase = map_types (fn _ => Type_Infer.anyT [])
+ val reinfer = singleton (Type_Infer_Context.infer_types ctxt)
+ fun add_annotations ((args, (rhs, some_abs)), (SOME th, proper)) =
+ let
+ val (lhs, drhs) = Logic.dest_equals (prop_of (Thm.unvarify_global th))
+ val drhs' = snd (Logic.dest_equals (reinfer (Logic.mk_equals (lhs, erase drhs))))
+ val (rhs', _) = annotate_term (drhs', rhs) []
+ in
+ ((args, (rhs', some_abs)), (SOME th, proper))
+ end
+ | add_annotations eqn = eqn
+ in
+ map add_annotations eqns
+ end;
(* translation *)
@@ -633,11 +640,12 @@
fun stmt_fun cert =
let
val ((vs, ty), eqns) = Code.equations_of_cert thy cert;
+ val eqns' = annotate_eqns thy eqns
val some_case_cong = Code.get_case_cong thy c;
in
fold_map (translate_tyvar_sort thy algbr eqngr permissive) vs
##>> translate_typ thy algbr eqngr permissive ty
- ##>> translate_eqns thy algbr eqngr permissive eqns
+ ##>> translate_eqns thy algbr eqngr permissive eqns'
#>> (fn info => Fun (c, (info, some_case_cong)))
end;
val stmt_const = case Code.get_type_of_constr_or_abstr thy c
@@ -748,15 +756,17 @@
then translation_error thy permissive some_thm
"Abstraction violation" ("constant " ^ Code.string_of_const thy c)
else ()
- val arg_typs = Sign.const_typargs thy (c, ty);
+ val (annotate, ty') = (case ty of Type("", [ty']) => (true, ty') | ty' => (false, ty'))
+ val arg_typs = Sign.const_typargs thy (c, ty');
val sorts = Code_Preproc.sortargs eqngr c;
- val function_typs = Term.binder_types ty;
+ val (function_typs, body_typ) = Term.strip_type ty';
in
ensure_const thy algbr eqngr permissive c
##>> fold_map (translate_typ thy algbr eqngr permissive) arg_typs
##>> fold_map (translate_dicts thy algbr eqngr permissive some_thm) (arg_typs ~~ sorts)
- ##>> fold_map (translate_typ thy algbr eqngr permissive) function_typs
- #>> (fn (((c, arg_typs), dss), function_typs) => IConst (c, ((arg_typs, dss), function_typs)))
+ ##>> fold_map (translate_typ thy algbr eqngr permissive) (body_typ :: function_typs)
+ #>> (fn (((c, arg_typs), dss), body_typ :: function_typs) =>
+ IConst (c, (((arg_typs, dss), (function_typs, body_typ)), annotate)))
end
and translate_app_const thy algbr eqngr permissive some_thm ((c_ty, ts), some_abs) =
translate_const thy algbr eqngr permissive some_thm (c_ty, some_abs)
@@ -801,7 +811,7 @@
val ts_clause = nth_drop t_pos ts;
val clauses = if null case_pats
then mk_clause (fn ([t], body) => (t, body)) [ty] (the_single ts_clause)
- else maps (fn ((constr as IConst (_, (_, tys)), n), t) =>
+ else maps (fn ((constr as IConst (_, ((_, (tys, _)), _)), n), t) =>
mk_clause (fn (ts, body) => (constr `$$ ts, body)) (take n tys) t)
(constrs ~~ ts_clause);
in ((t, ty), clauses) end;
--- a/src/Tools/jEdit/README.html Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Tools/jEdit/README.html Fri Sep 09 06:47:14 2011 +0200
@@ -139,6 +139,16 @@
to a theory ("<b>uses</b>").<br/>
<em>Workaround:</em> Re-use files manually within the prover.</li>
+ <li>Crude management of new Isar commands that are defined within
+ the running session.<br/>
+ <em>Workaround:</em> Force re-parsing of files using such commands
+ via reload menu of jEdit.</li>
+
+ <li>No way to delete document nodes from the overall collection of
+ theories.<br/>
+ <em>Workaround:</em> Restart whole Isabelle/jEdit session in
+ worst-case situation.</li>
+
<li>No support for non-local markup, e.g. commands reporting on
previous commands (proof end on proof head), or markup produced by
loading external files.</li>
--- a/src/Tools/jEdit/src/document_model.scala Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Tools/jEdit/src/document_model.scala Fri Sep 09 06:47:14 2011 +0200
@@ -69,6 +69,8 @@
/* perspective */
+ def buffer_range(): Text.Range = Text.Range(0, (buffer.getLength - 1) max 0)
+
def perspective(): Text.Perspective =
{
Swing_Thread.require()
@@ -136,6 +138,12 @@
pending_edits.update_perspective()
}
+ def full_perspective()
+ {
+ pending_edits.flush()
+ session.edit_node(name, node_header(), Text.Perspective(List(buffer_range())), Nil)
+ }
+
/* snapshot */
--- a/src/Tools/jEdit/src/document_view.scala Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Tools/jEdit/src/document_view.scala Fri Sep 09 06:47:14 2011 +0200
@@ -118,7 +118,7 @@
def perspective(): Text.Perspective =
{
Swing_Thread.require()
- val buffer_range = Text.Range(0, (model.buffer.getLength - 1) max 0)
+ val buffer_range = model.buffer_range()
Text.Perspective(
for {
i <- 0 until text_area.getVisibleLines
@@ -362,7 +362,7 @@
private val caret_listener = new CaretListener {
private val delay = Swing_Thread.delay_last(session.input_delay) {
- session.perspective.event(Session.Perspective)
+ session.caret_focus.event(Session.Caret_Focus)
}
override def caretUpdate(e: CaretEvent) { delay() }
}
--- a/src/Tools/jEdit/src/output_dockable.scala Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Tools/jEdit/src/output_dockable.scala Fri Sep 09 06:47:14 2011 +0200
@@ -106,7 +106,7 @@
react {
case Session.Global_Settings => handle_resize()
case changed: Session.Commands_Changed => handle_update(Some(changed.commands))
- case Session.Perspective => if (follow_caret && handle_perspective()) handle_update()
+ case Session.Caret_Focus => if (follow_caret && handle_perspective()) handle_update()
case bad => System.err.println("Output_Dockable: ignoring bad message " + bad)
}
}
@@ -116,14 +116,14 @@
{
Isabelle.session.global_settings += main_actor
Isabelle.session.commands_changed += main_actor
- Isabelle.session.perspective += main_actor
+ Isabelle.session.caret_focus += main_actor
}
override def exit()
{
Isabelle.session.global_settings -= main_actor
Isabelle.session.commands_changed -= main_actor
- Isabelle.session.perspective -= main_actor
+ Isabelle.session.caret_focus -= main_actor
}
--- a/src/Tools/jEdit/src/raw_output_dockable.scala Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Tools/jEdit/src/raw_output_dockable.scala Fri Sep 09 06:47:14 2011 +0200
@@ -29,8 +29,6 @@
private val main_actor = actor {
loop {
react {
- case input: Isabelle_Process.Input =>
-
case result: Isabelle_Process.Result =>
if (result.is_stdout)
Swing_Thread.now { text_area.append(XML.content(result.message).mkString) }
@@ -40,6 +38,6 @@
}
}
- override def init() { Isabelle.session.raw_messages += main_actor }
- override def exit() { Isabelle.session.raw_messages -= main_actor }
+ override def init() { Isabelle.session.raw_output_messages += main_actor }
+ override def exit() { Isabelle.session.raw_output_messages -= main_actor }
}
--- a/src/Tools/jEdit/src/session_dockable.scala Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Tools/jEdit/src/session_dockable.scala Fri Sep 09 06:47:14 2011 +0200
@@ -61,13 +61,31 @@
session_phase.border = new SoftBevelBorder(BevelBorder.LOWERED)
session_phase.tooltip = "Prover status"
+ private val cancel = new Button("Cancel") {
+ reactions += { case ButtonClicked(_) => Isabelle.session.cancel_execution }
+ }
+ cancel.tooltip = "Cancel current proof checking process"
+
+ private val check = new Button("Check") {
+ reactions +=
+ {
+ case ButtonClicked(_) =>
+ Isabelle.document_model(view.getBuffer) match {
+ case None =>
+ case Some(model) => model.full_perspective()
+ }
+ }
+ }
+ check.tooltip = "Commence full proof checking of current buffer"
+
private val logic = Isabelle.logic_selector(Isabelle.Property("logic"))
logic.listenTo(logic.selection)
logic.reactions += {
case SelectionChanged(_) => Isabelle.Property("logic") = logic.selection.item.name
}
- private val controls = new FlowPanel(FlowPanel.Alignment.Right)(session_phase, logic)
+ private val controls =
+ new FlowPanel(FlowPanel.Alignment.Right)(check, cancel, session_phase, logic)
add(controls.peer, BorderLayout.NORTH)
@@ -105,8 +123,6 @@
private val main_actor = actor {
loop {
react {
- case input: Isabelle_Process.Input =>
-
case result: Isabelle_Process.Result =>
if (result.is_syslog)
Swing_Thread.now {
@@ -127,13 +143,13 @@
}
override def init() {
- Isabelle.session.raw_messages += main_actor
+ Isabelle.session.syslog_messages += main_actor
Isabelle.session.phase_changed += main_actor
Isabelle.session.commands_changed += main_actor
}
override def exit() {
- Isabelle.session.raw_messages -= main_actor
+ Isabelle.session.syslog_messages -= main_actor
Isabelle.session.phase_changed -= main_actor
Isabelle.session.commands_changed -= main_actor
}
--- a/src/Tools/nbe.ML Fri Sep 09 06:45:39 2011 +0200
+++ b/src/Tools/nbe.ML Fri Sep 09 06:47:14 2011 +0200
@@ -315,7 +315,7 @@
let
val (t', ts) = Code_Thingol.unfold_app t
in of_iapp match_cont t' (fold_rev (cons o of_iterm NONE) ts []) end
- and of_iapp match_cont (IConst (c, ((_, dss), _))) ts = constapp c dss ts
+ and of_iapp match_cont (IConst (c, (((_, dss), _), _))) ts = constapp c dss ts
| of_iapp match_cont (IVar v) ts = nbe_apps (nbe_bound_optional v) ts
| of_iapp match_cont ((v, _) `|=> t) ts =
nbe_apps (nbe_abss 1 (ml_abs (ml_list [nbe_bound_optional v]) (of_iterm NONE t))) ts
@@ -425,7 +425,7 @@
val params = Name.invent Name.context "d" (length names);
fun mk (k, name) =
(name, ([(v, [])],
- [([IConst (class, (([], []), [])) `$$ map (IVar o SOME) params],
+ [([IConst (class, ((([], []), ([], ITyVar "")), false)) `$$ map (IVar o SOME) params],
IVar (SOME (nth params k)))]));
in map_index mk names end
| eqns_of_stmt (_, Code_Thingol.Classrel _) =
@@ -433,8 +433,8 @@
| eqns_of_stmt (_, Code_Thingol.Classparam _) =
[]
| eqns_of_stmt (inst, Code_Thingol.Classinst ((class, (_, arity_args)), (super_instances, (classparam_instances, _)))) =
- [(inst, (arity_args, [([], IConst (class, (([], []), [])) `$$
- map (fn (_, (_, (inst, dss))) => IConst (inst, (([], dss), []))) super_instances
+ [(inst, (arity_args, [([], IConst (class, ((([], []), ([], ITyVar "")), false)) `$$
+ map (fn (_, (_, (inst, dss))) => IConst (inst, ((([], dss), ([], ITyVar "")), false))) super_instances
@ map (IConst o snd o fst) classparam_instances)]))];