src/HOL/HOL.thy
author haftmann
Tue Oct 10 10:35:24 2006 +0200 (2006-10-10)
changeset 20944 34b2c1bb7178
parent 20833 4fcf8ddb54f5
child 20973 0b8e436ed071
permissions -rw-r--r--
cleanup basic HOL bootstrap
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(*  Title:      HOL/HOL.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
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*)
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header {* The basis of Higher-Order Logic *}
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theory HOL
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imports CPure
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uses ("cladata.ML") ("blastdata.ML") ("simpdata.ML") "Tools/res_atpset.ML"
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begin
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subsection {* Primitive logic *}
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subsubsection {* Core syntax *}
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classes type
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defaultsort type
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global
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typedecl bool
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arities
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  bool :: type
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  "fun" :: (type, type) type
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judgment
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  Trueprop      :: "bool => prop"                   ("(_)" 5)
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consts
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  Not           :: "bool => bool"                   ("~ _" [40] 40)
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  True          :: bool
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  False         :: bool
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  arbitrary     :: 'a
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  undefined     :: 'a
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  The           :: "('a => bool) => 'a"
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  All           :: "('a => bool) => bool"           (binder "ALL " 10)
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  Ex            :: "('a => bool) => bool"           (binder "EX " 10)
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  Ex1           :: "('a => bool) => bool"           (binder "EX! " 10)
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  Let           :: "['a, 'a => 'b] => 'b"
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  "="           :: "['a, 'a] => bool"               (infixl 50)
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  &             :: "[bool, bool] => bool"           (infixr 35)
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  "|"           :: "[bool, bool] => bool"           (infixr 30)
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  -->           :: "[bool, bool] => bool"           (infixr 25)
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local
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consts
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  If            :: "[bool, 'a, 'a] => 'a"           ("(if (_)/ then (_)/ else (_))" 10)
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subsubsection {* Additional concrete syntax *}
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const_syntax (output)
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  "op ="  (infix "=" 50)
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abbreviation
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  not_equal     :: "['a, 'a] => bool"               (infixl "~=" 50)
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  "x ~= y == ~ (x = y)"
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const_syntax (output)
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  not_equal  (infix "~=" 50)
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const_syntax (xsymbols)
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  Not  ("\<not> _" [40] 40)
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  "op &"  (infixr "\<and>" 35)
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  "op |"  (infixr "\<or>" 30)
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  "op -->"  (infixr "\<longrightarrow>" 25)
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  not_equal  (infix "\<noteq>" 50)
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const_syntax (HTML output)
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  Not  ("\<not> _" [40] 40)
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  "op &"  (infixr "\<and>" 35)
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  "op |"  (infixr "\<or>" 30)
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  not_equal  (infix "\<noteq>" 50)
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abbreviation (iff)
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  iff :: "[bool, bool] => bool"  (infixr "<->" 25)
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  "A <-> B == A = B"
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const_syntax (xsymbols)
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  iff  (infixr "\<longleftrightarrow>" 25)
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nonterminals
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  letbinds  letbind
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  case_syn  cases_syn
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syntax
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  "_The"        :: "[pttrn, bool] => 'a"                 ("(3THE _./ _)" [0, 10] 10)
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  "_bind"       :: "[pttrn, 'a] => letbind"              ("(2_ =/ _)" 10)
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  ""            :: "letbind => letbinds"                 ("_")
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  "_binds"      :: "[letbind, letbinds] => letbinds"     ("_;/ _")
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  "_Let"        :: "[letbinds, 'a] => 'a"                ("(let (_)/ in (_))" 10)
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  "_case_syntax":: "['a, cases_syn] => 'b"               ("(case _ of/ _)" 10)
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  "_case1"      :: "['a, 'b] => case_syn"                ("(2_ =>/ _)" 10)
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  ""            :: "case_syn => cases_syn"               ("_")
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  "_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ | _")
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translations
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  "THE x. P"              == "The (%x. P)"
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  "_Let (_binds b bs) e"  == "_Let b (_Let bs e)"
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  "let x = a in e"        == "Let a (%x. e)"
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print_translation {*
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(* To avoid eta-contraction of body: *)
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[("The", fn [Abs abs] =>
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     let val (x,t) = atomic_abs_tr' abs
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     in Syntax.const "_The" $ x $ t end)]
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*}
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syntax (xsymbols)
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  "ALL "        :: "[idts, bool] => bool"                ("(3\<forall>_./ _)" [0, 10] 10)
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  "EX "         :: "[idts, bool] => bool"                ("(3\<exists>_./ _)" [0, 10] 10)
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  "EX! "        :: "[idts, bool] => bool"                ("(3\<exists>!_./ _)" [0, 10] 10)
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  "_case1"      :: "['a, 'b] => case_syn"                ("(2_ \<Rightarrow>/ _)" 10)
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(*"_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ \<orelse> _")*)
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syntax (HTML output)
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  "ALL "        :: "[idts, bool] => bool"                ("(3\<forall>_./ _)" [0, 10] 10)
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  "EX "         :: "[idts, bool] => bool"                ("(3\<exists>_./ _)" [0, 10] 10)
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  "EX! "        :: "[idts, bool] => bool"                ("(3\<exists>!_./ _)" [0, 10] 10)
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syntax (HOL)
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  "ALL "        :: "[idts, bool] => bool"                ("(3! _./ _)" [0, 10] 10)
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  "EX "         :: "[idts, bool] => bool"                ("(3? _./ _)" [0, 10] 10)
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  "EX! "        :: "[idts, bool] => bool"                ("(3?! _./ _)" [0, 10] 10)
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subsubsection {* Axioms and basic definitions *}
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axioms
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  eq_reflection:  "(x=y) ==> (x==y)"
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  refl:           "t = (t::'a)"
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  ext:            "(!!x::'a. (f x ::'b) = g x) ==> (%x. f x) = (%x. g x)"
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    -- {*Extensionality is built into the meta-logic, and this rule expresses
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         a related property.  It is an eta-expanded version of the traditional
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         rule, and similar to the ABS rule of HOL*}
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  the_eq_trivial: "(THE x. x = a) = (a::'a)"
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  impI:           "(P ==> Q) ==> P-->Q"
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  mp:             "[| P-->Q;  P |] ==> Q"
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defs
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  True_def:     "True      == ((%x::bool. x) = (%x. x))"
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  All_def:      "All(P)    == (P = (%x. True))"
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  Ex_def:       "Ex(P)     == !Q. (!x. P x --> Q) --> Q"
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  False_def:    "False     == (!P. P)"
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  not_def:      "~ P       == P-->False"
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  and_def:      "P & Q     == !R. (P-->Q-->R) --> R"
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  or_def:       "P | Q     == !R. (P-->R) --> (Q-->R) --> R"
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  Ex1_def:      "Ex1(P)    == ? x. P(x) & (! y. P(y) --> y=x)"
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axioms
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  iff:          "(P-->Q) --> (Q-->P) --> (P=Q)"
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  True_or_False:  "(P=True) | (P=False)"
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defs
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  Let_def:      "Let s f == f(s)"
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  if_def:       "If P x y == THE z::'a. (P=True --> z=x) & (P=False --> z=y)"
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finalconsts
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  "op ="
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  "op -->"
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  The
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  arbitrary
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  undefined
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subsubsection {* Generic algebraic operations *}
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class zero =
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  fixes zero :: "'a"                       ("\<^loc>0")
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class one =
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  fixes one  :: "'a"                       ("\<^loc>1")
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hide (open) const zero one
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class plus =
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  fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   (infixl "\<^loc>+" 65)
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class minus =
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  fixes uminus :: "'a \<Rightarrow> 'a" 
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  fixes minus  :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<^loc>-" 65)
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  fixes abs    :: "'a \<Rightarrow> 'a"
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class times =
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  fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "\<^loc>*" 70)
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class inverse = 
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  fixes inverse :: "'a \<Rightarrow> 'a"
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  fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<^loc>'/" 70)
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syntax
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  "_index1"  :: index    ("\<^sub>1")
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translations
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  (index) "\<^sub>1" => (index) "\<^bsub>\<struct>\<^esub>"
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typed_print_translation {*
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let
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  fun tr' c = (c, fn show_sorts => fn T => fn ts =>
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    if T = dummyT orelse not (! show_types) andalso can Term.dest_Type T then raise Match
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    else Syntax.const Syntax.constrainC $ Syntax.const c $ Syntax.term_of_typ show_sorts T);
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in
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  map (tr' o prefix Syntax.constN) ["HOL.one", "HOL.zero"]
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end;
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*} -- {* show types that are presumably too general *}
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const_syntax
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  uminus  ("- _" [81] 80)
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const_syntax (xsymbols)
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  abs  ("\<bar>_\<bar>")
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const_syntax (HTML output)
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  abs  ("\<bar>_\<bar>")
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subsection {* Fundamental rules *}
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subsubsection {*Equality*}
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text {* Thanks to Stephan Merz *}
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lemma subst:
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  assumes eq: "s = t" and p: "P s"
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  shows "P t"
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proof -
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  from eq have meta: "s \<equiv> t"
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    by (rule eq_reflection)
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  from p show ?thesis
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    by (unfold meta)
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qed
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lemma sym: "s = t ==> t = s"
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  by (erule subst) (rule refl)
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lemma ssubst: "t = s ==> P s ==> P t"
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  by (drule sym) (erule subst)
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lemma trans: "[| r=s; s=t |] ==> r=t"
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  by (erule subst)
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lemma def_imp_eq:
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  assumes meq: "A == B"
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  shows "A = B"
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  by (unfold meq) (rule refl)
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(*a mere copy*)
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lemma meta_eq_to_obj_eq: 
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  assumes meq: "A == B"
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  shows "A = B"
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  by (unfold meq) (rule refl)
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text {* Useful with eresolve\_tac for proving equalties from known equalities. *}
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     (* a = b
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        |   |
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        c = d   *)
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lemma box_equals: "[| a=b;  a=c;  b=d |] ==> c=d"
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apply (rule trans)
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apply (rule trans)
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apply (rule sym)
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apply assumption+
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done
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text {* For calculational reasoning: *}
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lemma forw_subst: "a = b ==> P b ==> P a"
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  by (rule ssubst)
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lemma back_subst: "P a ==> a = b ==> P b"
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  by (rule subst)
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subsubsection {*Congruence rules for application*}
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(*similar to AP_THM in Gordon's HOL*)
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lemma fun_cong: "(f::'a=>'b) = g ==> f(x)=g(x)"
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apply (erule subst)
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apply (rule refl)
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done
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(*similar to AP_TERM in Gordon's HOL and FOL's subst_context*)
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lemma arg_cong: "x=y ==> f(x)=f(y)"
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apply (erule subst)
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apply (rule refl)
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done
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lemma arg_cong2: "\<lbrakk> a = b; c = d \<rbrakk> \<Longrightarrow> f a c = f b d"
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apply (erule ssubst)+
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apply (rule refl)
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done
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lemma cong: "[| f = g; (x::'a) = y |] ==> f(x) = g(y)"
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apply (erule subst)+
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apply (rule refl)
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done
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subsubsection {*Equality of booleans -- iff*}
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lemma iffI: assumes prems: "P ==> Q" "Q ==> P" shows "P=Q"
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  by (iprover intro: iff [THEN mp, THEN mp] impI prems)
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lemma iffD2: "[| P=Q; Q |] ==> P"
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  by (erule ssubst)
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lemma rev_iffD2: "[| Q; P=Q |] ==> P"
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  by (erule iffD2)
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lemmas iffD1 = sym [THEN iffD2, standard]
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lemmas rev_iffD1 = sym [THEN [2] rev_iffD2, standard]
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lemma iffE:
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  assumes major: "P=Q"
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      and minor: "[| P --> Q; Q --> P |] ==> R"
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  shows R
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  by (iprover intro: minor impI major [THEN iffD2] major [THEN iffD1])
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subsubsection {*True*}
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lemma TrueI: "True"
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  by (unfold True_def) (rule refl)
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lemma eqTrueI: "P ==> P=True"
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  by (iprover intro: iffI TrueI)
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lemma eqTrueE: "P=True ==> P"
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apply (erule iffD2)
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apply (rule TrueI)
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done
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subsubsection {*Universal quantifier*}
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lemma allI: assumes p: "!!x::'a. P(x)" shows "ALL x. P(x)"
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apply (unfold All_def)
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   347
apply (iprover intro: ext eqTrueI p)
paulson@15411
   348
done
paulson@15411
   349
paulson@15411
   350
lemma spec: "ALL x::'a. P(x) ==> P(x)"
paulson@15411
   351
apply (unfold All_def)
paulson@15411
   352
apply (rule eqTrueE)
paulson@15411
   353
apply (erule fun_cong)
paulson@15411
   354
done
paulson@15411
   355
paulson@15411
   356
lemma allE:
paulson@15411
   357
  assumes major: "ALL x. P(x)"
paulson@15411
   358
      and minor: "P(x) ==> R"
paulson@15411
   359
  shows "R"
nipkow@17589
   360
by (iprover intro: minor major [THEN spec])
paulson@15411
   361
paulson@15411
   362
lemma all_dupE:
paulson@15411
   363
  assumes major: "ALL x. P(x)"
paulson@15411
   364
      and minor: "[| P(x); ALL x. P(x) |] ==> R"
paulson@15411
   365
  shows "R"
nipkow@17589
   366
by (iprover intro: minor major major [THEN spec])
paulson@15411
   367
paulson@15411
   368
haftmann@20944
   369
subsubsection {*False*}
paulson@15411
   370
(*Depends upon spec; it is impossible to do propositional logic before quantifiers!*)
paulson@15411
   371
paulson@15411
   372
lemma FalseE: "False ==> P"
paulson@15411
   373
apply (unfold False_def)
paulson@15411
   374
apply (erule spec)
paulson@15411
   375
done
paulson@15411
   376
paulson@15411
   377
lemma False_neq_True: "False=True ==> P"
paulson@15411
   378
by (erule eqTrueE [THEN FalseE])
paulson@15411
   379
paulson@15411
   380
haftmann@20944
   381
subsubsection {*Negation*}
paulson@15411
   382
paulson@15411
   383
lemma notI:
paulson@15411
   384
  assumes p: "P ==> False"
paulson@15411
   385
  shows "~P"
paulson@15411
   386
apply (unfold not_def)
nipkow@17589
   387
apply (iprover intro: impI p)
paulson@15411
   388
done
paulson@15411
   389
paulson@15411
   390
lemma False_not_True: "False ~= True"
paulson@15411
   391
apply (rule notI)
paulson@15411
   392
apply (erule False_neq_True)
paulson@15411
   393
done
paulson@15411
   394
paulson@15411
   395
lemma True_not_False: "True ~= False"
paulson@15411
   396
apply (rule notI)
paulson@15411
   397
apply (drule sym)
paulson@15411
   398
apply (erule False_neq_True)
paulson@15411
   399
done
paulson@15411
   400
paulson@15411
   401
lemma notE: "[| ~P;  P |] ==> R"
paulson@15411
   402
apply (unfold not_def)
paulson@15411
   403
apply (erule mp [THEN FalseE])
paulson@15411
   404
apply assumption
paulson@15411
   405
done
paulson@15411
   406
paulson@15411
   407
(* Alternative ~ introduction rule: [| P ==> ~ Pa; P ==> Pa |] ==> ~ P *)
paulson@15411
   408
lemmas notI2 = notE [THEN notI, standard]
paulson@15411
   409
paulson@15411
   410
haftmann@20944
   411
subsubsection {*Implication*}
paulson@15411
   412
paulson@15411
   413
lemma impE:
paulson@15411
   414
  assumes "P-->Q" "P" "Q ==> R"
paulson@15411
   415
  shows "R"
nipkow@17589
   416
by (iprover intro: prems mp)
paulson@15411
   417
paulson@15411
   418
(* Reduces Q to P-->Q, allowing substitution in P. *)
paulson@15411
   419
lemma rev_mp: "[| P;  P --> Q |] ==> Q"
nipkow@17589
   420
by (iprover intro: mp)
paulson@15411
   421
paulson@15411
   422
lemma contrapos_nn:
paulson@15411
   423
  assumes major: "~Q"
paulson@15411
   424
      and minor: "P==>Q"
paulson@15411
   425
  shows "~P"
nipkow@17589
   426
by (iprover intro: notI minor major [THEN notE])
paulson@15411
   427
paulson@15411
   428
(*not used at all, but we already have the other 3 combinations *)
paulson@15411
   429
lemma contrapos_pn:
paulson@15411
   430
  assumes major: "Q"
paulson@15411
   431
      and minor: "P ==> ~Q"
paulson@15411
   432
  shows "~P"
nipkow@17589
   433
by (iprover intro: notI minor major notE)
paulson@15411
   434
paulson@15411
   435
lemma not_sym: "t ~= s ==> s ~= t"
paulson@15411
   436
apply (erule contrapos_nn)
paulson@15411
   437
apply (erule sym)
paulson@15411
   438
done
paulson@15411
   439
paulson@15411
   440
(*still used in HOLCF*)
paulson@15411
   441
lemma rev_contrapos:
paulson@15411
   442
  assumes pq: "P ==> Q"
paulson@15411
   443
      and nq: "~Q"
paulson@15411
   444
  shows "~P"
paulson@15411
   445
apply (rule nq [THEN contrapos_nn])
paulson@15411
   446
apply (erule pq)
paulson@15411
   447
done
paulson@15411
   448
haftmann@20944
   449
subsubsection {*Existential quantifier*}
paulson@15411
   450
paulson@15411
   451
lemma exI: "P x ==> EX x::'a. P x"
paulson@15411
   452
apply (unfold Ex_def)
nipkow@17589
   453
apply (iprover intro: allI allE impI mp)
paulson@15411
   454
done
paulson@15411
   455
paulson@15411
   456
lemma exE:
paulson@15411
   457
  assumes major: "EX x::'a. P(x)"
paulson@15411
   458
      and minor: "!!x. P(x) ==> Q"
paulson@15411
   459
  shows "Q"
paulson@15411
   460
apply (rule major [unfolded Ex_def, THEN spec, THEN mp])
nipkow@17589
   461
apply (iprover intro: impI [THEN allI] minor)
paulson@15411
   462
done
paulson@15411
   463
paulson@15411
   464
haftmann@20944
   465
subsubsection {*Conjunction*}
paulson@15411
   466
paulson@15411
   467
lemma conjI: "[| P; Q |] ==> P&Q"
paulson@15411
   468
apply (unfold and_def)
nipkow@17589
   469
apply (iprover intro: impI [THEN allI] mp)
paulson@15411
   470
done
paulson@15411
   471
paulson@15411
   472
lemma conjunct1: "[| P & Q |] ==> P"
paulson@15411
   473
apply (unfold and_def)
nipkow@17589
   474
apply (iprover intro: impI dest: spec mp)
paulson@15411
   475
done
paulson@15411
   476
paulson@15411
   477
lemma conjunct2: "[| P & Q |] ==> Q"
paulson@15411
   478
apply (unfold and_def)
nipkow@17589
   479
apply (iprover intro: impI dest: spec mp)
paulson@15411
   480
done
paulson@15411
   481
paulson@15411
   482
lemma conjE:
paulson@15411
   483
  assumes major: "P&Q"
paulson@15411
   484
      and minor: "[| P; Q |] ==> R"
paulson@15411
   485
  shows "R"
paulson@15411
   486
apply (rule minor)
paulson@15411
   487
apply (rule major [THEN conjunct1])
paulson@15411
   488
apply (rule major [THEN conjunct2])
paulson@15411
   489
done
paulson@15411
   490
paulson@15411
   491
lemma context_conjI:
paulson@15411
   492
  assumes prems: "P" "P ==> Q" shows "P & Q"
nipkow@17589
   493
by (iprover intro: conjI prems)
paulson@15411
   494
paulson@15411
   495
haftmann@20944
   496
subsubsection {*Disjunction*}
paulson@15411
   497
paulson@15411
   498
lemma disjI1: "P ==> P|Q"
paulson@15411
   499
apply (unfold or_def)
nipkow@17589
   500
apply (iprover intro: allI impI mp)
paulson@15411
   501
done
paulson@15411
   502
paulson@15411
   503
lemma disjI2: "Q ==> P|Q"
paulson@15411
   504
apply (unfold or_def)
nipkow@17589
   505
apply (iprover intro: allI impI mp)
paulson@15411
   506
done
paulson@15411
   507
paulson@15411
   508
lemma disjE:
paulson@15411
   509
  assumes major: "P|Q"
paulson@15411
   510
      and minorP: "P ==> R"
paulson@15411
   511
      and minorQ: "Q ==> R"
paulson@15411
   512
  shows "R"
nipkow@17589
   513
by (iprover intro: minorP minorQ impI
paulson@15411
   514
                 major [unfolded or_def, THEN spec, THEN mp, THEN mp])
paulson@15411
   515
paulson@15411
   516
haftmann@20944
   517
subsubsection {*Classical logic*}
paulson@15411
   518
paulson@15411
   519
lemma classical:
paulson@15411
   520
  assumes prem: "~P ==> P"
paulson@15411
   521
  shows "P"
paulson@15411
   522
apply (rule True_or_False [THEN disjE, THEN eqTrueE])
paulson@15411
   523
apply assumption
paulson@15411
   524
apply (rule notI [THEN prem, THEN eqTrueI])
paulson@15411
   525
apply (erule subst)
paulson@15411
   526
apply assumption
paulson@15411
   527
done
paulson@15411
   528
paulson@15411
   529
lemmas ccontr = FalseE [THEN classical, standard]
paulson@15411
   530
paulson@15411
   531
(*notE with premises exchanged; it discharges ~R so that it can be used to
paulson@15411
   532
  make elimination rules*)
paulson@15411
   533
lemma rev_notE:
paulson@15411
   534
  assumes premp: "P"
paulson@15411
   535
      and premnot: "~R ==> ~P"
paulson@15411
   536
  shows "R"
paulson@15411
   537
apply (rule ccontr)
paulson@15411
   538
apply (erule notE [OF premnot premp])
paulson@15411
   539
done
paulson@15411
   540
paulson@15411
   541
(*Double negation law*)
paulson@15411
   542
lemma notnotD: "~~P ==> P"
paulson@15411
   543
apply (rule classical)
paulson@15411
   544
apply (erule notE)
paulson@15411
   545
apply assumption
paulson@15411
   546
done
paulson@15411
   547
paulson@15411
   548
lemma contrapos_pp:
paulson@15411
   549
  assumes p1: "Q"
paulson@15411
   550
      and p2: "~P ==> ~Q"
paulson@15411
   551
  shows "P"
nipkow@17589
   552
by (iprover intro: classical p1 p2 notE)
paulson@15411
   553
paulson@15411
   554
haftmann@20944
   555
subsubsection {*Unique existence*}
paulson@15411
   556
paulson@15411
   557
lemma ex1I:
paulson@15411
   558
  assumes prems: "P a" "!!x. P(x) ==> x=a"
paulson@15411
   559
  shows "EX! x. P(x)"
nipkow@17589
   560
by (unfold Ex1_def, iprover intro: prems exI conjI allI impI)
paulson@15411
   561
paulson@15411
   562
text{*Sometimes easier to use: the premises have no shared variables.  Safe!*}
paulson@15411
   563
lemma ex_ex1I:
paulson@15411
   564
  assumes ex_prem: "EX x. P(x)"
paulson@15411
   565
      and eq: "!!x y. [| P(x); P(y) |] ==> x=y"
paulson@15411
   566
  shows "EX! x. P(x)"
nipkow@17589
   567
by (iprover intro: ex_prem [THEN exE] ex1I eq)
paulson@15411
   568
paulson@15411
   569
lemma ex1E:
paulson@15411
   570
  assumes major: "EX! x. P(x)"
paulson@15411
   571
      and minor: "!!x. [| P(x);  ALL y. P(y) --> y=x |] ==> R"
paulson@15411
   572
  shows "R"
paulson@15411
   573
apply (rule major [unfolded Ex1_def, THEN exE])
paulson@15411
   574
apply (erule conjE)
nipkow@17589
   575
apply (iprover intro: minor)
paulson@15411
   576
done
paulson@15411
   577
paulson@15411
   578
lemma ex1_implies_ex: "EX! x. P x ==> EX x. P x"
paulson@15411
   579
apply (erule ex1E)
paulson@15411
   580
apply (rule exI)
paulson@15411
   581
apply assumption
paulson@15411
   582
done
paulson@15411
   583
paulson@15411
   584
haftmann@20944
   585
subsubsection {*THE: definite description operator*}
paulson@15411
   586
paulson@15411
   587
lemma the_equality:
paulson@15411
   588
  assumes prema: "P a"
paulson@15411
   589
      and premx: "!!x. P x ==> x=a"
paulson@15411
   590
  shows "(THE x. P x) = a"
paulson@15411
   591
apply (rule trans [OF _ the_eq_trivial])
paulson@15411
   592
apply (rule_tac f = "The" in arg_cong)
paulson@15411
   593
apply (rule ext)
paulson@15411
   594
apply (rule iffI)
paulson@15411
   595
 apply (erule premx)
paulson@15411
   596
apply (erule ssubst, rule prema)
paulson@15411
   597
done
paulson@15411
   598
paulson@15411
   599
lemma theI:
paulson@15411
   600
  assumes "P a" and "!!x. P x ==> x=a"
paulson@15411
   601
  shows "P (THE x. P x)"
nipkow@17589
   602
by (iprover intro: prems the_equality [THEN ssubst])
paulson@15411
   603
paulson@15411
   604
lemma theI': "EX! x. P x ==> P (THE x. P x)"
paulson@15411
   605
apply (erule ex1E)
paulson@15411
   606
apply (erule theI)
paulson@15411
   607
apply (erule allE)
paulson@15411
   608
apply (erule mp)
paulson@15411
   609
apply assumption
paulson@15411
   610
done
paulson@15411
   611
paulson@15411
   612
(*Easier to apply than theI: only one occurrence of P*)
paulson@15411
   613
lemma theI2:
paulson@15411
   614
  assumes "P a" "!!x. P x ==> x=a" "!!x. P x ==> Q x"
paulson@15411
   615
  shows "Q (THE x. P x)"
nipkow@17589
   616
by (iprover intro: prems theI)
paulson@15411
   617
wenzelm@18697
   618
lemma the1_equality [elim?]: "[| EX!x. P x; P a |] ==> (THE x. P x) = a"
paulson@15411
   619
apply (rule the_equality)
paulson@15411
   620
apply  assumption
paulson@15411
   621
apply (erule ex1E)
paulson@15411
   622
apply (erule all_dupE)
paulson@15411
   623
apply (drule mp)
paulson@15411
   624
apply  assumption
paulson@15411
   625
apply (erule ssubst)
paulson@15411
   626
apply (erule allE)
paulson@15411
   627
apply (erule mp)
paulson@15411
   628
apply assumption
paulson@15411
   629
done
paulson@15411
   630
paulson@15411
   631
lemma the_sym_eq_trivial: "(THE y. x=y) = x"
paulson@15411
   632
apply (rule the_equality)
paulson@15411
   633
apply (rule refl)
paulson@15411
   634
apply (erule sym)
paulson@15411
   635
done
paulson@15411
   636
paulson@15411
   637
haftmann@20944
   638
subsubsection {*Classical intro rules for disjunction and existential quantifiers*}
paulson@15411
   639
paulson@15411
   640
lemma disjCI:
paulson@15411
   641
  assumes "~Q ==> P" shows "P|Q"
paulson@15411
   642
apply (rule classical)
nipkow@17589
   643
apply (iprover intro: prems disjI1 disjI2 notI elim: notE)
paulson@15411
   644
done
paulson@15411
   645
paulson@15411
   646
lemma excluded_middle: "~P | P"
nipkow@17589
   647
by (iprover intro: disjCI)
paulson@15411
   648
haftmann@20944
   649
text {*
haftmann@20944
   650
  case distinction as a natural deduction rule.
haftmann@20944
   651
  Note that @{term "~P"} is the second case, not the first
haftmann@20944
   652
*}
paulson@15411
   653
lemma case_split_thm:
paulson@15411
   654
  assumes prem1: "P ==> Q"
paulson@15411
   655
      and prem2: "~P ==> Q"
paulson@15411
   656
  shows "Q"
paulson@15411
   657
apply (rule excluded_middle [THEN disjE])
paulson@15411
   658
apply (erule prem2)
paulson@15411
   659
apply (erule prem1)
paulson@15411
   660
done
haftmann@20944
   661
lemmas case_split = case_split_thm [case_names True False]
paulson@15411
   662
paulson@15411
   663
(*Classical implies (-->) elimination. *)
paulson@15411
   664
lemma impCE:
paulson@15411
   665
  assumes major: "P-->Q"
paulson@15411
   666
      and minor: "~P ==> R" "Q ==> R"
paulson@15411
   667
  shows "R"
paulson@15411
   668
apply (rule excluded_middle [of P, THEN disjE])
nipkow@17589
   669
apply (iprover intro: minor major [THEN mp])+
paulson@15411
   670
done
paulson@15411
   671
paulson@15411
   672
(*This version of --> elimination works on Q before P.  It works best for
paulson@15411
   673
  those cases in which P holds "almost everywhere".  Can't install as
paulson@15411
   674
  default: would break old proofs.*)
paulson@15411
   675
lemma impCE':
paulson@15411
   676
  assumes major: "P-->Q"
paulson@15411
   677
      and minor: "Q ==> R" "~P ==> R"
paulson@15411
   678
  shows "R"
paulson@15411
   679
apply (rule excluded_middle [of P, THEN disjE])
nipkow@17589
   680
apply (iprover intro: minor major [THEN mp])+
paulson@15411
   681
done
paulson@15411
   682
paulson@15411
   683
(*Classical <-> elimination. *)
paulson@15411
   684
lemma iffCE:
paulson@15411
   685
  assumes major: "P=Q"
paulson@15411
   686
      and minor: "[| P; Q |] ==> R"  "[| ~P; ~Q |] ==> R"
paulson@15411
   687
  shows "R"
paulson@15411
   688
apply (rule major [THEN iffE])
nipkow@17589
   689
apply (iprover intro: minor elim: impCE notE)
paulson@15411
   690
done
paulson@15411
   691
paulson@15411
   692
lemma exCI:
paulson@15411
   693
  assumes "ALL x. ~P(x) ==> P(a)"
paulson@15411
   694
  shows "EX x. P(x)"
paulson@15411
   695
apply (rule ccontr)
nipkow@17589
   696
apply (iprover intro: prems exI allI notI notE [of "\<exists>x. P x"])
paulson@15411
   697
done
paulson@15411
   698
paulson@15411
   699
wenzelm@12386
   700
subsubsection {* Intuitionistic Reasoning *}
wenzelm@12386
   701
wenzelm@12386
   702
lemma impE':
wenzelm@12937
   703
  assumes 1: "P --> Q"
wenzelm@12937
   704
    and 2: "Q ==> R"
wenzelm@12937
   705
    and 3: "P --> Q ==> P"
wenzelm@12937
   706
  shows R
wenzelm@12386
   707
proof -
wenzelm@12386
   708
  from 3 and 1 have P .
wenzelm@12386
   709
  with 1 have Q by (rule impE)
wenzelm@12386
   710
  with 2 show R .
wenzelm@12386
   711
qed
wenzelm@12386
   712
wenzelm@12386
   713
lemma allE':
wenzelm@12937
   714
  assumes 1: "ALL x. P x"
wenzelm@12937
   715
    and 2: "P x ==> ALL x. P x ==> Q"
wenzelm@12937
   716
  shows Q
wenzelm@12386
   717
proof -
wenzelm@12386
   718
  from 1 have "P x" by (rule spec)
wenzelm@12386
   719
  from this and 1 show Q by (rule 2)
wenzelm@12386
   720
qed
wenzelm@12386
   721
wenzelm@12937
   722
lemma notE':
wenzelm@12937
   723
  assumes 1: "~ P"
wenzelm@12937
   724
    and 2: "~ P ==> P"
wenzelm@12937
   725
  shows R
wenzelm@12386
   726
proof -
wenzelm@12386
   727
  from 2 and 1 have P .
wenzelm@12386
   728
  with 1 show R by (rule notE)
wenzelm@12386
   729
qed
wenzelm@12386
   730
wenzelm@15801
   731
lemmas [Pure.elim!] = disjE iffE FalseE conjE exE
wenzelm@15801
   732
  and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
wenzelm@15801
   733
  and [Pure.elim 2] = allE notE' impE'
wenzelm@15801
   734
  and [Pure.intro] = exI disjI2 disjI1
wenzelm@12386
   735
wenzelm@12386
   736
lemmas [trans] = trans
wenzelm@12386
   737
  and [sym] = sym not_sym
wenzelm@15801
   738
  and [Pure.elim?] = iffD1 iffD2 impE
wenzelm@11750
   739
wenzelm@11438
   740
wenzelm@11750
   741
subsubsection {* Atomizing meta-level connectives *}
wenzelm@11750
   742
wenzelm@11750
   743
lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
wenzelm@12003
   744
proof
wenzelm@9488
   745
  assume "!!x. P x"
wenzelm@10383
   746
  show "ALL x. P x" by (rule allI)
wenzelm@9488
   747
next
wenzelm@9488
   748
  assume "ALL x. P x"
wenzelm@10383
   749
  thus "!!x. P x" by (rule allE)
wenzelm@9488
   750
qed
wenzelm@9488
   751
wenzelm@11750
   752
lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
wenzelm@12003
   753
proof
wenzelm@9488
   754
  assume r: "A ==> B"
wenzelm@10383
   755
  show "A --> B" by (rule impI) (rule r)
wenzelm@9488
   756
next
wenzelm@9488
   757
  assume "A --> B" and A
wenzelm@10383
   758
  thus B by (rule mp)
wenzelm@9488
   759
qed
wenzelm@9488
   760
paulson@14749
   761
lemma atomize_not: "(A ==> False) == Trueprop (~A)"
paulson@14749
   762
proof
paulson@14749
   763
  assume r: "A ==> False"
paulson@14749
   764
  show "~A" by (rule notI) (rule r)
paulson@14749
   765
next
paulson@14749
   766
  assume "~A" and A
paulson@14749
   767
  thus False by (rule notE)
paulson@14749
   768
qed
paulson@14749
   769
wenzelm@11750
   770
lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
wenzelm@12003
   771
proof
wenzelm@10432
   772
  assume "x == y"
wenzelm@10432
   773
  show "x = y" by (unfold prems) (rule refl)
wenzelm@10432
   774
next
wenzelm@10432
   775
  assume "x = y"
wenzelm@10432
   776
  thus "x == y" by (rule eq_reflection)
wenzelm@10432
   777
qed
wenzelm@10432
   778
wenzelm@12023
   779
lemma atomize_conj [atomize]:
wenzelm@19121
   780
  includes meta_conjunction_syntax
wenzelm@19121
   781
  shows "(A && B) == Trueprop (A & B)"
wenzelm@12003
   782
proof
wenzelm@19121
   783
  assume conj: "A && B"
wenzelm@19121
   784
  show "A & B"
wenzelm@19121
   785
  proof (rule conjI)
wenzelm@19121
   786
    from conj show A by (rule conjunctionD1)
wenzelm@19121
   787
    from conj show B by (rule conjunctionD2)
wenzelm@19121
   788
  qed
wenzelm@11953
   789
next
wenzelm@19121
   790
  assume conj: "A & B"
wenzelm@19121
   791
  show "A && B"
wenzelm@19121
   792
  proof -
wenzelm@19121
   793
    from conj show A ..
wenzelm@19121
   794
    from conj show B ..
wenzelm@11953
   795
  qed
wenzelm@11953
   796
qed
wenzelm@11953
   797
wenzelm@12386
   798
lemmas [symmetric, rulify] = atomize_all atomize_imp
wenzelm@18832
   799
  and [symmetric, defn] = atomize_all atomize_imp atomize_eq
wenzelm@12386
   800
wenzelm@11750
   801
haftmann@20944
   802
subsection {* Package setup *}
haftmann@20944
   803
haftmann@20944
   804
subsubsection {* Fundamental ML bindings *}
haftmann@20944
   805
haftmann@20944
   806
ML {*
haftmann@20944
   807
structure HOL =
haftmann@20944
   808
struct
haftmann@20944
   809
  (*FIXME reduce this to a minimum*)
haftmann@20944
   810
  val eq_reflection = thm "eq_reflection";
haftmann@20944
   811
  val def_imp_eq = thm "def_imp_eq";
haftmann@20944
   812
  val meta_eq_to_obj_eq = thm "meta_eq_to_obj_eq";
haftmann@20944
   813
  val ccontr = thm "ccontr";
haftmann@20944
   814
  val impI = thm "impI";
haftmann@20944
   815
  val impCE = thm "impCE";
haftmann@20944
   816
  val notI = thm "notI";
haftmann@20944
   817
  val notE = thm "notE";
haftmann@20944
   818
  val iffI = thm "iffI";
haftmann@20944
   819
  val iffCE = thm "iffCE";
haftmann@20944
   820
  val conjI = thm "conjI";
haftmann@20944
   821
  val conjE = thm "conjE";
haftmann@20944
   822
  val disjCI = thm "disjCI";
haftmann@20944
   823
  val disjE = thm "disjE";
haftmann@20944
   824
  val TrueI = thm "TrueI";
haftmann@20944
   825
  val FalseE = thm "FalseE";
haftmann@20944
   826
  val allI = thm "allI";
haftmann@20944
   827
  val allE = thm "allE";
haftmann@20944
   828
  val exI = thm "exI";
haftmann@20944
   829
  val exE = thm "exE";
haftmann@20944
   830
  val ex_ex1I = thm "ex_ex1I";
haftmann@20944
   831
  val the_equality = thm "the_equality";
haftmann@20944
   832
  val mp = thm "mp";
haftmann@20944
   833
  val rev_mp = thm "rev_mp"
haftmann@20944
   834
  val classical = thm "classical";
haftmann@20944
   835
  val subst = thm "subst";
haftmann@20944
   836
  val refl = thm "refl";
haftmann@20944
   837
  val sym = thm "sym";
haftmann@20944
   838
  val trans = thm "trans";
haftmann@20944
   839
  val arg_cong = thm "arg_cong";
haftmann@20944
   840
  val iffD1 = thm "iffD1";
haftmann@20944
   841
  val iffD2 = thm "iffD2";
haftmann@20944
   842
  val disjE = thm "disjE";
haftmann@20944
   843
  val conjE = thm "conjE";
haftmann@20944
   844
  val exE = thm "exE";
haftmann@20944
   845
  val contrapos_nn = thm "contrapos_nn";
haftmann@20944
   846
  val contrapos_pp = thm "contrapos_pp";
haftmann@20944
   847
  val notnotD = thm "notnotD";
haftmann@20944
   848
  val conjunct1 = thm "conjunct1";
haftmann@20944
   849
  val conjunct2 = thm "conjunct2";
haftmann@20944
   850
  val spec = thm "spec";
haftmann@20944
   851
  val imp_cong = thm "imp_cong";
haftmann@20944
   852
  val the_sym_eq_trivial = thm "the_sym_eq_trivial";
haftmann@20944
   853
  val triv_forall_equality = thm "triv_forall_equality";
haftmann@20944
   854
  val case_split = thm "case_split_thm";
haftmann@20944
   855
end
haftmann@20944
   856
*}
haftmann@20944
   857
haftmann@20944
   858
wenzelm@11750
   859
subsubsection {* Classical Reasoner setup *}
wenzelm@9529
   860
haftmann@20944
   861
lemma thin_refl:
haftmann@20944
   862
  "\<And>X. \<lbrakk> x=x; PROP W \<rbrakk> \<Longrightarrow> PROP W" .
haftmann@20944
   863
wenzelm@10383
   864
use "cladata.ML"
haftmann@20944
   865
ML {* val HOL_cs = HOL.cs *}
haftmann@20944
   866
setup Hypsubst.hypsubst_setup
haftmann@20944
   867
setup {* ContextRules.addSWrapper (fn tac => HOL.hyp_subst_tac' ORELSE' tac) *}
wenzelm@10383
   868
setup Classical.setup
wenzelm@20766
   869
setup ResAtpset.setup
haftmann@20944
   870
setup {* fn thy => (Classical.change_claset_of thy (K HOL.cs); thy) *}
mengj@19162
   871
wenzelm@20223
   872
lemma contrapos_np: "~ Q ==> (~ P ==> Q) ==> P"
wenzelm@20223
   873
  apply (erule swap)
wenzelm@20223
   874
  apply (erule (1) meta_mp)
wenzelm@20223
   875
  done
wenzelm@10383
   876
wenzelm@18689
   877
declare ex_ex1I [rule del, intro! 2]
wenzelm@18689
   878
  and ex1I [intro]
wenzelm@18689
   879
wenzelm@12386
   880
lemmas [intro?] = ext
wenzelm@12386
   881
  and [elim?] = ex1_implies_ex
wenzelm@11977
   882
haftmann@20944
   883
(*Better then ex1E for classical reasoner: needs no quantifier duplication!*)
haftmann@20944
   884
lemma alt_ex1E:
haftmann@20944
   885
  assumes major: "\<exists>!x. P x"
haftmann@20944
   886
      and prem: "\<And>x. \<lbrakk> P x; \<forall>y y'. P y \<and> P y' \<longrightarrow> y = y' \<rbrakk> \<Longrightarrow> R"
haftmann@20944
   887
  shows R
haftmann@20944
   888
apply (rule ex1E [OF major])
haftmann@20944
   889
apply (rule prem)
haftmann@20944
   890
apply (tactic "ares_tac [HOL.allI] 1")+
haftmann@20944
   891
apply (tactic "etac (Classical.dup_elim HOL.allE) 1")
haftmann@20944
   892
by iprover
haftmann@20944
   893
wenzelm@9869
   894
use "blastdata.ML"
wenzelm@9869
   895
setup Blast.setup
wenzelm@4868
   896
haftmann@20944
   897
ML {*
haftmann@20944
   898
structure HOL =
haftmann@20944
   899
struct
wenzelm@11750
   900
haftmann@20944
   901
open HOL;
wenzelm@11750
   902
haftmann@20944
   903
fun case_tac a = res_inst_tac [("P", a)] case_split;
haftmann@20944
   904
haftmann@20944
   905
end;
haftmann@20944
   906
*}
haftmann@20944
   907
haftmann@20944
   908
haftmann@20944
   909
subsubsection {* Simplifier *}
wenzelm@12281
   910
wenzelm@12281
   911
lemma eta_contract_eq: "(%s. f s) = f" ..
wenzelm@12281
   912
wenzelm@12281
   913
lemma simp_thms:
wenzelm@12937
   914
  shows not_not: "(~ ~ P) = P"
nipkow@15354
   915
  and Not_eq_iff: "((~P) = (~Q)) = (P = Q)"
wenzelm@12937
   916
  and
berghofe@12436
   917
    "(P ~= Q) = (P = (~Q))"
berghofe@12436
   918
    "(P | ~P) = True"    "(~P | P) = True"
wenzelm@12281
   919
    "(x = x) = True"
haftmann@20944
   920
  and not_True_eq_False: "(\<not> True) = False"
haftmann@20944
   921
  and not_False_eq_True: "(\<not> False) = True"
haftmann@20944
   922
  and
berghofe@12436
   923
    "(~P) ~= P"  "P ~= (~P)"
haftmann@20944
   924
    "(True=P) = P"
haftmann@20944
   925
  and eq_True: "(P = True) = P"
haftmann@20944
   926
  and "(False=P) = (~P)"
haftmann@20944
   927
  and eq_False: "(P = False) = (\<not> P)"
haftmann@20944
   928
  and
wenzelm@12281
   929
    "(True --> P) = P"  "(False --> P) = True"
wenzelm@12281
   930
    "(P --> True) = True"  "(P --> P) = True"
wenzelm@12281
   931
    "(P --> False) = (~P)"  "(P --> ~P) = (~P)"
wenzelm@12281
   932
    "(P & True) = P"  "(True & P) = P"
wenzelm@12281
   933
    "(P & False) = False"  "(False & P) = False"
wenzelm@12281
   934
    "(P & P) = P"  "(P & (P & Q)) = (P & Q)"
wenzelm@12281
   935
    "(P & ~P) = False"    "(~P & P) = False"
wenzelm@12281
   936
    "(P | True) = True"  "(True | P) = True"
wenzelm@12281
   937
    "(P | False) = P"  "(False | P) = P"
berghofe@12436
   938
    "(P | P) = P"  "(P | (P | Q)) = (P | Q)" and
wenzelm@12281
   939
    "(ALL x. P) = P"  "(EX x. P) = P"  "EX x. x=t"  "EX x. t=x"
wenzelm@12281
   940
    -- {* needed for the one-point-rule quantifier simplification procs *}
wenzelm@12281
   941
    -- {* essential for termination!! *} and
wenzelm@12281
   942
    "!!P. (EX x. x=t & P(x)) = P(t)"
wenzelm@12281
   943
    "!!P. (EX x. t=x & P(x)) = P(t)"
wenzelm@12281
   944
    "!!P. (ALL x. x=t --> P(x)) = P(t)"
wenzelm@12937
   945
    "!!P. (ALL x. t=x --> P(x)) = P(t)"
nipkow@17589
   946
  by (blast, blast, blast, blast, blast, iprover+)
wenzelm@13421
   947
wenzelm@12281
   948
lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
nipkow@17589
   949
  by iprover
wenzelm@12281
   950
wenzelm@12281
   951
lemma ex_simps:
wenzelm@12281
   952
  "!!P Q. (EX x. P x & Q)   = ((EX x. P x) & Q)"
wenzelm@12281
   953
  "!!P Q. (EX x. P & Q x)   = (P & (EX x. Q x))"
wenzelm@12281
   954
  "!!P Q. (EX x. P x | Q)   = ((EX x. P x) | Q)"
wenzelm@12281
   955
  "!!P Q. (EX x. P | Q x)   = (P | (EX x. Q x))"
wenzelm@12281
   956
  "!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
wenzelm@12281
   957
  "!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
wenzelm@12281
   958
  -- {* Miniscoping: pushing in existential quantifiers. *}
nipkow@17589
   959
  by (iprover | blast)+
wenzelm@12281
   960
wenzelm@12281
   961
lemma all_simps:
wenzelm@12281
   962
  "!!P Q. (ALL x. P x & Q)   = ((ALL x. P x) & Q)"
wenzelm@12281
   963
  "!!P Q. (ALL x. P & Q x)   = (P & (ALL x. Q x))"
wenzelm@12281
   964
  "!!P Q. (ALL x. P x | Q)   = ((ALL x. P x) | Q)"
wenzelm@12281
   965
  "!!P Q. (ALL x. P | Q x)   = (P | (ALL x. Q x))"
wenzelm@12281
   966
  "!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
wenzelm@12281
   967
  "!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
wenzelm@12281
   968
  -- {* Miniscoping: pushing in universal quantifiers. *}
nipkow@17589
   969
  by (iprover | blast)+
wenzelm@12281
   970
paulson@14201
   971
lemma disj_absorb: "(A | A) = A"
paulson@14201
   972
  by blast
paulson@14201
   973
paulson@14201
   974
lemma disj_left_absorb: "(A | (A | B)) = (A | B)"
paulson@14201
   975
  by blast
paulson@14201
   976
paulson@14201
   977
lemma conj_absorb: "(A & A) = A"
paulson@14201
   978
  by blast
paulson@14201
   979
paulson@14201
   980
lemma conj_left_absorb: "(A & (A & B)) = (A & B)"
paulson@14201
   981
  by blast
paulson@14201
   982
wenzelm@12281
   983
lemma eq_ac:
wenzelm@12937
   984
  shows eq_commute: "(a=b) = (b=a)"
wenzelm@12937
   985
    and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))"
nipkow@17589
   986
    and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (iprover, blast+)
nipkow@17589
   987
lemma neq_commute: "(a~=b) = (b~=a)" by iprover
wenzelm@12281
   988
wenzelm@12281
   989
lemma conj_comms:
wenzelm@12937
   990
  shows conj_commute: "(P&Q) = (Q&P)"
nipkow@17589
   991
    and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by iprover+
nipkow@17589
   992
lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by iprover
wenzelm@12281
   993
paulson@19174
   994
lemmas conj_ac = conj_commute conj_left_commute conj_assoc
paulson@19174
   995
wenzelm@12281
   996
lemma disj_comms:
wenzelm@12937
   997
  shows disj_commute: "(P|Q) = (Q|P)"
nipkow@17589
   998
    and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by iprover+
nipkow@17589
   999
lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by iprover
wenzelm@12281
  1000
paulson@19174
  1001
lemmas disj_ac = disj_commute disj_left_commute disj_assoc
paulson@19174
  1002
nipkow@17589
  1003
lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by iprover
nipkow@17589
  1004
lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by iprover
wenzelm@12281
  1005
nipkow@17589
  1006
lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by iprover
nipkow@17589
  1007
lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by iprover
wenzelm@12281
  1008
nipkow@17589
  1009
lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by iprover
nipkow@17589
  1010
lemma imp_conjL: "((P&Q) -->R)  = (P --> (Q --> R))" by iprover
nipkow@17589
  1011
lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by iprover
wenzelm@12281
  1012
wenzelm@12281
  1013
text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *}
wenzelm@12281
  1014
lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
wenzelm@12281
  1015
lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
wenzelm@12281
  1016
wenzelm@12281
  1017
lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
wenzelm@12281
  1018
lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
wenzelm@12281
  1019
nipkow@17589
  1020
lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by iprover
wenzelm@12281
  1021
lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
wenzelm@12281
  1022
lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
wenzelm@12281
  1023
lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
wenzelm@12281
  1024
lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
wenzelm@12281
  1025
lemma disj_not2: "(P | ~Q) = (Q --> P)"  -- {* changes orientation :-( *}
wenzelm@12281
  1026
  by blast
wenzelm@12281
  1027
lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
wenzelm@12281
  1028
nipkow@17589
  1029
lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by iprover
wenzelm@12281
  1030
wenzelm@12281
  1031
wenzelm@12281
  1032
lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
wenzelm@12281
  1033
  -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
wenzelm@12281
  1034
  -- {* cases boil down to the same thing. *}
wenzelm@12281
  1035
  by blast
wenzelm@12281
  1036
wenzelm@12281
  1037
lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
wenzelm@12281
  1038
lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
nipkow@17589
  1039
lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by iprover
nipkow@17589
  1040
lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by iprover
wenzelm@12281
  1041
nipkow@17589
  1042
lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by iprover
nipkow@17589
  1043
lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by iprover
wenzelm@12281
  1044
wenzelm@12281
  1045
text {*
wenzelm@12281
  1046
  \medskip The @{text "&"} congruence rule: not included by default!
wenzelm@12281
  1047
  May slow rewrite proofs down by as much as 50\% *}
wenzelm@12281
  1048
wenzelm@12281
  1049
lemma conj_cong:
wenzelm@12281
  1050
    "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
nipkow@17589
  1051
  by iprover
wenzelm@12281
  1052
wenzelm@12281
  1053
lemma rev_conj_cong:
wenzelm@12281
  1054
    "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
nipkow@17589
  1055
  by iprover
wenzelm@12281
  1056
wenzelm@12281
  1057
text {* The @{text "|"} congruence rule: not included by default! *}
wenzelm@12281
  1058
wenzelm@12281
  1059
lemma disj_cong:
wenzelm@12281
  1060
    "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
wenzelm@12281
  1061
  by blast
wenzelm@12281
  1062
wenzelm@12281
  1063
wenzelm@12281
  1064
text {* \medskip if-then-else rules *}
wenzelm@12281
  1065
wenzelm@12281
  1066
lemma if_True: "(if True then x else y) = x"
wenzelm@12281
  1067
  by (unfold if_def) blast
wenzelm@12281
  1068
wenzelm@12281
  1069
lemma if_False: "(if False then x else y) = y"
wenzelm@12281
  1070
  by (unfold if_def) blast
wenzelm@12281
  1071
wenzelm@12281
  1072
lemma if_P: "P ==> (if P then x else y) = x"
wenzelm@12281
  1073
  by (unfold if_def) blast
wenzelm@12281
  1074
wenzelm@12281
  1075
lemma if_not_P: "~P ==> (if P then x else y) = y"
wenzelm@12281
  1076
  by (unfold if_def) blast
wenzelm@12281
  1077
wenzelm@12281
  1078
lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
wenzelm@12281
  1079
  apply (rule case_split [of Q])
paulson@15481
  1080
   apply (simplesubst if_P)
paulson@15481
  1081
    prefer 3 apply (simplesubst if_not_P, blast+)
wenzelm@12281
  1082
  done
wenzelm@12281
  1083
wenzelm@12281
  1084
lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
paulson@15481
  1085
by (simplesubst split_if, blast)
wenzelm@12281
  1086
wenzelm@12281
  1087
lemmas if_splits = split_if split_if_asm
wenzelm@12281
  1088
wenzelm@12281
  1089
lemma if_cancel: "(if c then x else x) = x"
paulson@15481
  1090
by (simplesubst split_if, blast)
wenzelm@12281
  1091
wenzelm@12281
  1092
lemma if_eq_cancel: "(if x = y then y else x) = x"
paulson@15481
  1093
by (simplesubst split_if, blast)
wenzelm@12281
  1094
wenzelm@12281
  1095
lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))"
wenzelm@19796
  1096
  -- {* This form is useful for expanding @{text "if"}s on the RIGHT of the @{text "==>"} symbol. *}
wenzelm@12281
  1097
  by (rule split_if)
wenzelm@12281
  1098
wenzelm@12281
  1099
lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
wenzelm@19796
  1100
  -- {* And this form is useful for expanding @{text "if"}s on the LEFT. *}
paulson@15481
  1101
  apply (simplesubst split_if, blast)
wenzelm@12281
  1102
  done
wenzelm@12281
  1103
nipkow@17589
  1104
lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) iprover
nipkow@17589
  1105
lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) iprover
wenzelm@12281
  1106
schirmer@15423
  1107
text {* \medskip let rules for simproc *}
schirmer@15423
  1108
schirmer@15423
  1109
lemma Let_folded: "f x \<equiv> g x \<Longrightarrow>  Let x f \<equiv> Let x g"
schirmer@15423
  1110
  by (unfold Let_def)
schirmer@15423
  1111
schirmer@15423
  1112
lemma Let_unfold: "f x \<equiv> g \<Longrightarrow>  Let x f \<equiv> g"
schirmer@15423
  1113
  by (unfold Let_def)
schirmer@15423
  1114
berghofe@16633
  1115
text {*
ballarin@16999
  1116
  The following copy of the implication operator is useful for
ballarin@16999
  1117
  fine-tuning congruence rules.  It instructs the simplifier to simplify
ballarin@16999
  1118
  its premise.
berghofe@16633
  1119
*}
berghofe@16633
  1120
wenzelm@17197
  1121
constdefs
wenzelm@17197
  1122
  simp_implies :: "[prop, prop] => prop"  (infixr "=simp=>" 1)
wenzelm@17197
  1123
  "simp_implies \<equiv> op ==>"
berghofe@16633
  1124
wenzelm@18457
  1125
lemma simp_impliesI:
berghofe@16633
  1126
  assumes PQ: "(PROP P \<Longrightarrow> PROP Q)"
berghofe@16633
  1127
  shows "PROP P =simp=> PROP Q"
berghofe@16633
  1128
  apply (unfold simp_implies_def)
berghofe@16633
  1129
  apply (rule PQ)
berghofe@16633
  1130
  apply assumption
berghofe@16633
  1131
  done
berghofe@16633
  1132
berghofe@16633
  1133
lemma simp_impliesE:
berghofe@16633
  1134
  assumes PQ:"PROP P =simp=> PROP Q"
berghofe@16633
  1135
  and P: "PROP P"
berghofe@16633
  1136
  and QR: "PROP Q \<Longrightarrow> PROP R"
berghofe@16633
  1137
  shows "PROP R"
berghofe@16633
  1138
  apply (rule QR)
berghofe@16633
  1139
  apply (rule PQ [unfolded simp_implies_def])
berghofe@16633
  1140
  apply (rule P)
berghofe@16633
  1141
  done
berghofe@16633
  1142
berghofe@16633
  1143
lemma simp_implies_cong:
berghofe@16633
  1144
  assumes PP' :"PROP P == PROP P'"
berghofe@16633
  1145
  and P'QQ': "PROP P' ==> (PROP Q == PROP Q')"
berghofe@16633
  1146
  shows "(PROP P =simp=> PROP Q) == (PROP P' =simp=> PROP Q')"
berghofe@16633
  1147
proof (unfold simp_implies_def, rule equal_intr_rule)
berghofe@16633
  1148
  assume PQ: "PROP P \<Longrightarrow> PROP Q"
berghofe@16633
  1149
  and P': "PROP P'"
berghofe@16633
  1150
  from PP' [symmetric] and P' have "PROP P"
berghofe@16633
  1151
    by (rule equal_elim_rule1)
berghofe@16633
  1152
  hence "PROP Q" by (rule PQ)
berghofe@16633
  1153
  with P'QQ' [OF P'] show "PROP Q'" by (rule equal_elim_rule1)
berghofe@16633
  1154
next
berghofe@16633
  1155
  assume P'Q': "PROP P' \<Longrightarrow> PROP Q'"
berghofe@16633
  1156
  and P: "PROP P"
berghofe@16633
  1157
  from PP' and P have P': "PROP P'" by (rule equal_elim_rule1)
berghofe@16633
  1158
  hence "PROP Q'" by (rule P'Q')
berghofe@16633
  1159
  with P'QQ' [OF P', symmetric] show "PROP Q"
berghofe@16633
  1160
    by (rule equal_elim_rule1)
berghofe@16633
  1161
qed
berghofe@16633
  1162
wenzelm@17459
  1163
wenzelm@17459
  1164
text {* \medskip Actual Installation of the Simplifier. *}
paulson@14201
  1165
haftmann@20944
  1166
lemma True_implies_equals: "(True \<Longrightarrow> PROP P) \<equiv> PROP P"
haftmann@20944
  1167
proof
haftmann@20944
  1168
  assume prem: "True \<Longrightarrow> PROP P"
haftmann@20944
  1169
  from prem [OF TrueI] show "PROP P" . 
haftmann@20944
  1170
next
haftmann@20944
  1171
  assume "PROP P"
haftmann@20944
  1172
  show "PROP P" .
haftmann@20944
  1173
qed
haftmann@20944
  1174
haftmann@20944
  1175
lemma uncurry:
haftmann@20944
  1176
  assumes "P \<longrightarrow> Q \<longrightarrow> R"
haftmann@20944
  1177
  shows "P \<and> Q \<longrightarrow> R"
haftmann@20944
  1178
  using prems by blast
haftmann@20944
  1179
haftmann@20944
  1180
lemma iff_allI:
haftmann@20944
  1181
  assumes "\<And>x. P x = Q x"
haftmann@20944
  1182
  shows "(\<forall>x. P x) = (\<forall>x. Q x)"
haftmann@20944
  1183
  using prems by blast
haftmann@20944
  1184
haftmann@20944
  1185
lemma iff_exI:
haftmann@20944
  1186
  assumes "\<And>x. P x = Q x"
haftmann@20944
  1187
  shows "(\<exists>x. P x) = (\<exists>x. Q x)"
haftmann@20944
  1188
  using prems by blast
haftmann@20944
  1189
haftmann@20944
  1190
lemma all_comm:
haftmann@20944
  1191
  "(\<forall>x y. P x y) = (\<forall>y x. P x y)"
haftmann@20944
  1192
  by blast
haftmann@20944
  1193
haftmann@20944
  1194
lemma ex_comm:
haftmann@20944
  1195
  "(\<exists>x y. P x y) = (\<exists>y x. P x y)"
haftmann@20944
  1196
  by blast
haftmann@20944
  1197
wenzelm@9869
  1198
use "simpdata.ML"
haftmann@20944
  1199
setup "Simplifier.method_setup Splitter.split_modifiers"
haftmann@20944
  1200
setup simpsetup
haftmann@20944
  1201
setup Splitter.setup
haftmann@20944
  1202
setup Clasimp.setup
wenzelm@18591
  1203
setup EqSubst.setup
paulson@15481
  1204
haftmann@20944
  1205
text {* Simplifies x assuming c and y assuming ~c *}
haftmann@20944
  1206
lemma if_cong:
haftmann@20944
  1207
  assumes "b = c"
haftmann@20944
  1208
      and "c \<Longrightarrow> x = u"
haftmann@20944
  1209
      and "\<not> c \<Longrightarrow> y = v"
haftmann@20944
  1210
  shows "(if b then x else y) = (if c then u else v)"
haftmann@20944
  1211
  unfolding if_def using prems by simp
haftmann@20944
  1212
haftmann@20944
  1213
text {* Prevents simplification of x and y:
haftmann@20944
  1214
  faster and allows the execution of functional programs. *}
haftmann@20944
  1215
lemma if_weak_cong [cong]:
haftmann@20944
  1216
  assumes "b = c"
haftmann@20944
  1217
  shows "(if b then x else y) = (if c then x else y)"
haftmann@20944
  1218
  using prems by (rule arg_cong)
haftmann@20944
  1219
haftmann@20944
  1220
text {* Prevents simplification of t: much faster *}
haftmann@20944
  1221
lemma let_weak_cong:
haftmann@20944
  1222
  assumes "a = b"
haftmann@20944
  1223
  shows "(let x = a in t x) = (let x = b in t x)"
haftmann@20944
  1224
  using prems by (rule arg_cong)
haftmann@20944
  1225
haftmann@20944
  1226
text {* To tidy up the result of a simproc.  Only the RHS will be simplified. *}
haftmann@20944
  1227
lemma eq_cong2:
haftmann@20944
  1228
  assumes "u = u'"
haftmann@20944
  1229
  shows "(t \<equiv> u) \<equiv> (t \<equiv> u')"
haftmann@20944
  1230
  using prems by simp
haftmann@20944
  1231
haftmann@20944
  1232
lemma if_distrib:
haftmann@20944
  1233
  "f (if c then x else y) = (if c then f x else f y)"
haftmann@20944
  1234
  by simp
haftmann@20944
  1235
haftmann@20944
  1236
text {* For expand\_case\_tac *}
haftmann@20944
  1237
lemma expand_case:
haftmann@20944
  1238
  assumes "P \<Longrightarrow> Q True"
haftmann@20944
  1239
      and "~P \<Longrightarrow> Q False"
haftmann@20944
  1240
  shows "Q P"
haftmann@20944
  1241
proof (tactic {* HOL.case_tac "P" 1 *})
haftmann@20944
  1242
  assume P
haftmann@20944
  1243
  then show "Q P" by simp
haftmann@20944
  1244
next
haftmann@20944
  1245
  assume "\<not> P"
haftmann@20944
  1246
  then have "P = False" by simp
haftmann@20944
  1247
  with prems show "Q P" by simp
haftmann@20944
  1248
qed
haftmann@20944
  1249
haftmann@20944
  1250
text {* This lemma restricts the effect of the rewrite rule u=v to the left-hand
haftmann@20944
  1251
  side of an equality.  Used in {Integ,Real}/simproc.ML *}
haftmann@20944
  1252
lemma restrict_to_left:
haftmann@20944
  1253
  assumes "x = y"
haftmann@20944
  1254
  shows "(x = z) = (y = z)"
haftmann@20944
  1255
  using prems by simp
haftmann@20944
  1256
wenzelm@17459
  1257
haftmann@20944
  1258
subsubsection {* Generic cases and induction *}
wenzelm@17459
  1259
haftmann@20944
  1260
text {* Rule projections: *}
berghofe@18887
  1261
haftmann@20944
  1262
ML {*
haftmann@20944
  1263
structure ProjectRule = ProjectRuleFun
haftmann@20944
  1264
(struct
haftmann@20944
  1265
  val conjunct1 = thm "conjunct1";
haftmann@20944
  1266
  val conjunct2 = thm "conjunct2";
haftmann@20944
  1267
  val mp = thm "mp";
haftmann@20944
  1268
end)
wenzelm@17459
  1269
*}
wenzelm@17459
  1270
wenzelm@11824
  1271
constdefs
wenzelm@18457
  1272
  induct_forall where "induct_forall P == \<forall>x. P x"
wenzelm@18457
  1273
  induct_implies where "induct_implies A B == A \<longrightarrow> B"
wenzelm@18457
  1274
  induct_equal where "induct_equal x y == x = y"
wenzelm@18457
  1275
  induct_conj where "induct_conj A B == A \<and> B"
wenzelm@11824
  1276
wenzelm@11989
  1277
lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
wenzelm@18457
  1278
  by (unfold atomize_all induct_forall_def)
wenzelm@11824
  1279
wenzelm@11989
  1280
lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
wenzelm@18457
  1281
  by (unfold atomize_imp induct_implies_def)
wenzelm@11824
  1282
wenzelm@11989
  1283
lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
wenzelm@18457
  1284
  by (unfold atomize_eq induct_equal_def)
wenzelm@18457
  1285
wenzelm@18457
  1286
lemma induct_conj_eq:
wenzelm@18457
  1287
  includes meta_conjunction_syntax
wenzelm@18457
  1288
  shows "(A && B) == Trueprop (induct_conj A B)"
wenzelm@18457
  1289
  by (unfold atomize_conj induct_conj_def)
wenzelm@18457
  1290
wenzelm@18457
  1291
lemmas induct_atomize = induct_forall_eq induct_implies_eq induct_equal_eq induct_conj_eq
wenzelm@18457
  1292
lemmas induct_rulify [symmetric, standard] = induct_atomize
wenzelm@18457
  1293
lemmas induct_rulify_fallback =
wenzelm@18457
  1294
  induct_forall_def induct_implies_def induct_equal_def induct_conj_def
wenzelm@18457
  1295
wenzelm@11824
  1296
wenzelm@11989
  1297
lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
wenzelm@11989
  1298
    induct_conj (induct_forall A) (induct_forall B)"
nipkow@17589
  1299
  by (unfold induct_forall_def induct_conj_def) iprover
wenzelm@11824
  1300
wenzelm@11989
  1301
lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
wenzelm@11989
  1302
    induct_conj (induct_implies C A) (induct_implies C B)"
nipkow@17589
  1303
  by (unfold induct_implies_def induct_conj_def) iprover
wenzelm@11989
  1304
berghofe@13598
  1305
lemma induct_conj_curry: "(induct_conj A B ==> PROP C) == (A ==> B ==> PROP C)"
berghofe@13598
  1306
proof
berghofe@13598
  1307
  assume r: "induct_conj A B ==> PROP C" and A B
wenzelm@18457
  1308
  show "PROP C" by (rule r) (simp add: induct_conj_def `A` `B`)
berghofe@13598
  1309
next
berghofe@13598
  1310
  assume r: "A ==> B ==> PROP C" and "induct_conj A B"
wenzelm@18457
  1311
  show "PROP C" by (rule r) (simp_all add: `induct_conj A B` [unfolded induct_conj_def])
berghofe@13598
  1312
qed
wenzelm@11824
  1313
wenzelm@11989
  1314
lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
wenzelm@11824
  1315
wenzelm@11989
  1316
hide const induct_forall induct_implies induct_equal induct_conj
wenzelm@11824
  1317
wenzelm@11824
  1318
text {* Method setup. *}
wenzelm@11824
  1319
wenzelm@11824
  1320
ML {*
wenzelm@11824
  1321
  structure InductMethod = InductMethodFun
wenzelm@11824
  1322
  (struct
paulson@15411
  1323
    val cases_default = thm "case_split"
paulson@15411
  1324
    val atomize = thms "induct_atomize"
wenzelm@18457
  1325
    val rulify = thms "induct_rulify"
wenzelm@18457
  1326
    val rulify_fallback = thms "induct_rulify_fallback"
wenzelm@11824
  1327
  end);
wenzelm@11824
  1328
*}
wenzelm@11824
  1329
wenzelm@11824
  1330
setup InductMethod.setup
wenzelm@11824
  1331
wenzelm@18457
  1332
haftmann@20944
  1333
subsubsection {* Code generator setup *}
haftmann@20944
  1334
haftmann@20944
  1335
types_code
haftmann@20944
  1336
  "bool"  ("bool")
haftmann@20944
  1337
attach (term_of) {*
haftmann@20944
  1338
fun term_of_bool b = if b then HOLogic.true_const else HOLogic.false_const;
haftmann@20944
  1339
*}
haftmann@20944
  1340
attach (test) {*
haftmann@20944
  1341
fun gen_bool i = one_of [false, true];
haftmann@20944
  1342
*}
haftmann@20944
  1343
  "prop"  ("bool")
haftmann@20944
  1344
attach (term_of) {*
haftmann@20944
  1345
fun term_of_prop b =
haftmann@20944
  1346
  HOLogic.mk_Trueprop (if b then HOLogic.true_const else HOLogic.false_const);
haftmann@20944
  1347
*}
haftmann@20944
  1348
haftmann@20944
  1349
consts_code
haftmann@20944
  1350
  "Trueprop" ("(_)")
haftmann@20944
  1351
  "True"    ("true")
haftmann@20944
  1352
  "False"   ("false")
haftmann@20944
  1353
  "Not"     ("not")
haftmann@20944
  1354
  "op |"    ("(_ orelse/ _)")
haftmann@20944
  1355
  "op &"    ("(_ andalso/ _)")
haftmann@20944
  1356
  "HOL.If"      ("(if _/ then _/ else _)")
haftmann@20944
  1357
haftmann@20944
  1358
setup {*
haftmann@20944
  1359
let
haftmann@20944
  1360
haftmann@20944
  1361
fun eq_codegen thy defs gr dep thyname b t =
haftmann@20944
  1362
    (case strip_comb t of
haftmann@20944
  1363
       (Const ("op =", Type (_, [Type ("fun", _), _])), _) => NONE
haftmann@20944
  1364
     | (Const ("op =", _), [t, u]) =>
haftmann@20944
  1365
          let
haftmann@20944
  1366
            val (gr', pt) = Codegen.invoke_codegen thy defs dep thyname false (gr, t);
haftmann@20944
  1367
            val (gr'', pu) = Codegen.invoke_codegen thy defs dep thyname false (gr', u);
haftmann@20944
  1368
            val (gr''', _) = Codegen.invoke_tycodegen thy defs dep thyname false (gr'', HOLogic.boolT)
haftmann@20944
  1369
          in
haftmann@20944
  1370
            SOME (gr''', Codegen.parens
haftmann@20944
  1371
              (Pretty.block [pt, Pretty.str " =", Pretty.brk 1, pu]))
haftmann@20944
  1372
          end
haftmann@20944
  1373
     | (t as Const ("op =", _), ts) => SOME (Codegen.invoke_codegen
haftmann@20944
  1374
         thy defs dep thyname b (gr, Codegen.eta_expand t ts 2))
haftmann@20944
  1375
     | _ => NONE);
haftmann@20944
  1376
haftmann@20944
  1377
in
haftmann@20944
  1378
haftmann@20944
  1379
Codegen.add_codegen "eq_codegen" eq_codegen
haftmann@20944
  1380
haftmann@20944
  1381
end
haftmann@20944
  1382
*}
haftmann@20944
  1383
haftmann@20944
  1384
setup {*
haftmann@20944
  1385
let
haftmann@20944
  1386
haftmann@20944
  1387
fun evaluation_tac i = Tactical.PRIMITIVE (Drule.fconv_rule
haftmann@20944
  1388
  (Drule.goals_conv (equal i) Codegen.evaluation_conv));
haftmann@20944
  1389
haftmann@20944
  1390
val evaluation_meth =
haftmann@20944
  1391
  Method.no_args (Method.METHOD (fn _ => evaluation_tac 1 THEN rtac HOL.TrueI 1));
haftmann@20944
  1392
haftmann@20944
  1393
in
haftmann@20944
  1394
haftmann@20944
  1395
Method.add_method ("evaluation", evaluation_meth, "solve goal by evaluation")
haftmann@20944
  1396
haftmann@20944
  1397
end;
haftmann@20944
  1398
*}
haftmann@20944
  1399
haftmann@20944
  1400
haftmann@20590
  1401
text {* itself as a code generator datatype *}
haftmann@18702
  1402
haftmann@19598
  1403
setup {*
haftmann@20590
  1404
let fun add_itself thy =
haftmann@20590
  1405
  let
haftmann@20590
  1406
    val v = ("'a", []);
haftmann@20590
  1407
    val t = Logic.mk_type (TFree v);
haftmann@20590
  1408
    val Const (c, ty) = t;
haftmann@20590
  1409
    val (_, Type (dtco, _)) = strip_type ty;
haftmann@20590
  1410
  in
haftmann@20590
  1411
    thy
haftmann@20590
  1412
    |> CodegenData.add_datatype (dtco, (([v], [(c, [])]), CodegenData.lazy (fn () => [])))
haftmann@20590
  1413
  end
haftmann@20590
  1414
in add_itself end;
haftmann@20590
  1415
*} 
haftmann@18702
  1416
haftmann@20698
  1417
text {* code generation for arbitrary as exception *}
haftmann@20698
  1418
haftmann@20698
  1419
setup {*
haftmann@20833
  1420
  CodegenSerializer.add_undefined "SML" "arbitrary" "(raise Fail \"arbitrary\")"
haftmann@20698
  1421
*}
haftmann@20944
  1422
haftmann@20698
  1423
code_const arbitrary
haftmann@20698
  1424
  (Haskell target_atom "(error \"arbitrary\")")
haftmann@20698
  1425
haftmann@20944
  1426
haftmann@20944
  1427
subsection {* Other simple lemmas and lemma duplicates *}
haftmann@20944
  1428
haftmann@20944
  1429
lemmas eq_sym_conv = eq_commute
haftmann@20944
  1430
lemmas if_def2 = if_bool_eq_conj
haftmann@20944
  1431
haftmann@20944
  1432
lemma ex1_eq [iff]: "EX! x. x = t" "EX! x. t = x"
haftmann@20944
  1433
  by blast+
haftmann@20944
  1434
haftmann@20944
  1435
lemma choice_eq: "(ALL x. EX! y. P x y) = (EX! f. ALL x. P x (f x))"
haftmann@20944
  1436
  apply (rule iffI)
haftmann@20944
  1437
  apply (rule_tac a = "%x. THE y. P x y" in ex1I)
haftmann@20944
  1438
  apply (fast dest!: theI')
haftmann@20944
  1439
  apply (fast intro: ext the1_equality [symmetric])
haftmann@20944
  1440
  apply (erule ex1E)
haftmann@20944
  1441
  apply (rule allI)
haftmann@20944
  1442
  apply (rule ex1I)
haftmann@20944
  1443
  apply (erule spec)
haftmann@20944
  1444
  apply (erule_tac x = "%z. if z = x then y else f z" in allE)
haftmann@20944
  1445
  apply (erule impE)
haftmann@20944
  1446
  apply (rule allI)
haftmann@20944
  1447
  apply (rule_tac P = "xa = x" in case_split_thm)
haftmann@20944
  1448
  apply (drule_tac [3] x = x in fun_cong, simp_all)
haftmann@20944
  1449
  done
haftmann@20944
  1450
haftmann@20944
  1451
text {* Needs only HOL-lemmas *}
haftmann@20944
  1452
lemma mk_left_commute:
haftmann@20944
  1453
  assumes a: "\<And>x y z. f (f x y) z = f x (f y z)" and
haftmann@20944
  1454
          c: "\<And>x y. f x y = f y x"
haftmann@20944
  1455
  shows "f x (f y z) = f y (f x z)"
haftmann@20944
  1456
  by (rule trans [OF trans [OF c a] arg_cong [OF c, of "f y"]])
haftmann@20944
  1457
haftmann@20944
  1458
haftmann@20944
  1459
subsection {* Conclude HOL structure *}
haftmann@20944
  1460
haftmann@20944
  1461
ML {*
haftmann@20944
  1462
structure HOL =
haftmann@20944
  1463
struct
haftmann@20944
  1464
haftmann@20944
  1465
open HOL;
haftmann@20944
  1466
haftmann@20944
  1467
(* combination of (spec RS spec RS ...(j times) ... spec RS mp) *)
haftmann@20944
  1468
local
haftmann@20944
  1469
  fun wrong_prem (Const ("All", _) $ (Abs (_, _, t))) = wrong_prem t
haftmann@20944
  1470
    | wrong_prem (Bound _) = true
haftmann@20944
  1471
    | wrong_prem _ = false;
haftmann@20944
  1472
  val filter_right = filter (not o wrong_prem o HOLogic.dest_Trueprop o hd o Thm.prems_of);
haftmann@20944
  1473
in
haftmann@20944
  1474
  fun smp i = funpow i (fn m => filter_right ([spec] RL m)) ([mp]);
haftmann@20944
  1475
  fun smp_tac j = EVERY'[dresolve_tac (smp j), atac];
haftmann@20944
  1476
end;
haftmann@20944
  1477
haftmann@20944
  1478
fun strip_tac i = REPEAT (resolve_tac [impI, allI] i);
haftmann@20944
  1479
haftmann@20944
  1480
end;
haftmann@20944
  1481
*}
haftmann@20944
  1482
kleing@14357
  1483
end