--- a/src/Doc/Isar_Ref/Proof.thy Fri Nov 13 15:59:40 2015 +0000
+++ b/src/Doc/Isar_Ref/Proof.thy Fri Nov 13 20:03:27 2015 +0100
@@ -1,3 +1,5 @@
+(*:maxLineLen=78:*)
+
theory Proof
imports Base Main
begin
@@ -5,37 +7,33 @@
chapter \<open>Proofs \label{ch:proofs}\<close>
text \<open>
- Proof commands perform transitions of Isar/VM machine
- configurations, which are block-structured, consisting of a stack of
- nodes with three main components: logical proof context, current
- facts, and open goals. Isar/VM transitions are typed according to
- the following three different modes of operation:
-
- \<^descr> \<open>proof(prove)\<close> means that a new goal has just been
- stated that is now to be \<^emph>\<open>proven\<close>; the next command may refine
- it by some proof method, and enter a sub-proof to establish the
- actual result.
-
- \<^descr> \<open>proof(state)\<close> is like a nested theory mode: the
- context may be augmented by \<^emph>\<open>stating\<close> additional assumptions,
- intermediate results etc.
+ Proof commands perform transitions of Isar/VM machine configurations, which
+ are block-structured, consisting of a stack of nodes with three main
+ components: logical proof context, current facts, and open goals. Isar/VM
+ transitions are typed according to the following three different modes of
+ operation:
- \<^descr> \<open>proof(chain)\<close> is intermediate between \<open>proof(state)\<close> and \<open>proof(prove)\<close>: existing facts (i.e.\ the
- contents of the special @{fact_ref this} register) have been just picked
- up in order to be used when refining the goal claimed next.
-
+ \<^descr> \<open>proof(prove)\<close> means that a new goal has just been stated that is now to
+ be \<^emph>\<open>proven\<close>; the next command may refine it by some proof method, and
+ enter a sub-proof to establish the actual result.
+
+ \<^descr> \<open>proof(state)\<close> is like a nested theory mode: the context may be
+ augmented by \<^emph>\<open>stating\<close> additional assumptions, intermediate results etc.
+
+ \<^descr> \<open>proof(chain)\<close> is intermediate between \<open>proof(state)\<close> and
+ \<open>proof(prove)\<close>: existing facts (i.e.\ the contents of the special
+ @{fact_ref this} register) have been just picked up in order to be used
+ when refining the goal claimed next.
- The proof mode indicator may be understood as an instruction to the
- writer, telling what kind of operation may be performed next. The
- corresponding typings of proof commands restricts the shape of
- well-formed proof texts to particular command sequences. So dynamic
- arrangements of commands eventually turn out as static texts of a
- certain structure.
+ The proof mode indicator may be understood as an instruction to the writer,
+ telling what kind of operation may be performed next. The corresponding
+ typings of proof commands restricts the shape of well-formed proof texts to
+ particular command sequences. So dynamic arrangements of commands eventually
+ turn out as static texts of a certain structure.
- \Appref{ap:refcard} gives a simplified grammar of the (extensible)
- language emerging that way from the different types of proof
- commands. The main ideas of the overall Isar framework are
- explained in \chref{ch:isar-framework}.
+ \Appref{ap:refcard} gives a simplified grammar of the (extensible) language
+ emerging that way from the different types of proof commands. The main ideas
+ of the overall Isar framework are explained in \chref{ch:isar-framework}.
\<close>
@@ -54,9 +52,9 @@
@@{command end}
\<close>}
- \<^descr> @{command "notepad"}~@{keyword "begin"} opens a proof state without
- any goal statement. This allows to experiment with Isar, without producing
- any persistent result. The notepad is closed by @{command "end"}.
+ \<^descr> @{command "notepad"}~@{keyword "begin"} opens a proof state without any
+ goal statement. This allows to experiment with Isar, without producing any
+ persistent result. The notepad is closed by @{command "end"}.
\<close>
@@ -69,32 +67,30 @@
@{command_def "}"} & : & \<open>proof(state) \<rightarrow> proof(state)\<close> \\
\end{matharray}
- While Isar is inherently block-structured, opening and closing
- blocks is mostly handled rather casually, with little explicit
- user-intervention. Any local goal statement automatically opens
- \<^emph>\<open>two\<close> internal blocks, which are closed again when concluding
- the sub-proof (by @{command "qed"} etc.). Sections of different
- context within a sub-proof may be switched via @{command "next"},
- which is just a single block-close followed by block-open again.
- The effect of @{command "next"} is to reset the local proof context;
+ While Isar is inherently block-structured, opening and closing blocks is
+ mostly handled rather casually, with little explicit user-intervention. Any
+ local goal statement automatically opens \<^emph>\<open>two\<close> internal blocks, which are
+ closed again when concluding the sub-proof (by @{command "qed"} etc.).
+ Sections of different context within a sub-proof may be switched via
+ @{command "next"}, which is just a single block-close followed by block-open
+ again. The effect of @{command "next"} is to reset the local proof context;
there is no goal focus involved here!
For slightly more advanced applications, there are explicit block
- parentheses as well. These typically achieve a stronger forward
- style of reasoning.
+ parentheses as well. These typically achieve a stronger forward style of
+ reasoning.
- \<^descr> @{command "next"} switches to a fresh block within a
- sub-proof, resetting the local context to the initial one.
+ \<^descr> @{command "next"} switches to a fresh block within a sub-proof, resetting
+ the local context to the initial one.
- \<^descr> @{command "{"} and @{command "}"} explicitly open and close
- blocks. Any current facts pass through ``@{command "{"}''
- unchanged, while ``@{command "}"}'' causes any result to be
- \<^emph>\<open>exported\<close> into the enclosing context. Thus fixed variables
- are generalized, assumptions discharged, and local definitions
- unfolded (cf.\ \secref{sec:proof-context}). There is no difference
- of @{command "assume"} and @{command "presume"} in this mode of
- forward reasoning --- in contrast to plain backward reasoning with
- the result exported at @{command "show"} time.
+ \<^descr> @{command "{"} and @{command "}"} explicitly open and close blocks. Any
+ current facts pass through ``@{command "{"}'' unchanged, while ``@{command
+ "}"}'' causes any result to be \<^emph>\<open>exported\<close> into the enclosing context. Thus
+ fixed variables are generalized, assumptions discharged, and local
+ definitions unfolded (cf.\ \secref{sec:proof-context}). There is no
+ difference of @{command "assume"} and @{command "presume"} in this mode of
+ forward reasoning --- in contrast to plain backward reasoning with the
+ result exported at @{command "show"} time.
\<close>
@@ -105,23 +101,20 @@
@{command_def "oops"} & : & \<open>proof \<rightarrow> local_theory | theory\<close> \\
\end{matharray}
- The @{command "oops"} command discontinues the current proof
- attempt, while considering the partial proof text as properly
- processed. This is conceptually quite different from ``faking''
- actual proofs via @{command_ref "sorry"} (see
- \secref{sec:proof-steps}): @{command "oops"} does not observe the
- proof structure at all, but goes back right to the theory level.
- Furthermore, @{command "oops"} does not produce any result theorem
- --- there is no intended claim to be able to complete the proof
- in any way.
+ The @{command "oops"} command discontinues the current proof attempt, while
+ considering the partial proof text as properly processed. This is
+ conceptually quite different from ``faking'' actual proofs via @{command_ref
+ "sorry"} (see \secref{sec:proof-steps}): @{command "oops"} does not observe
+ the proof structure at all, but goes back right to the theory level.
+ Furthermore, @{command "oops"} does not produce any result theorem --- there
+ is no intended claim to be able to complete the proof in any way.
A typical application of @{command "oops"} is to explain Isar proofs
- \<^emph>\<open>within\<close> the system itself, in conjunction with the document
- preparation tools of Isabelle described in \chref{ch:document-prep}.
- Thus partial or even wrong proof attempts can be discussed in a
- logically sound manner. Note that the Isabelle {\LaTeX} macros can
- be easily adapted to print something like ``\<open>\<dots>\<close>'' instead of
- the keyword ``@{command "oops"}''.
+ \<^emph>\<open>within\<close> the system itself, in conjunction with the document preparation
+ tools of Isabelle described in \chref{ch:document-prep}. Thus partial or
+ even wrong proof attempts can be discussed in a logically sound manner. Note
+ that the Isabelle {\LaTeX} macros can be easily adapted to print something
+ like ``\<open>\<dots>\<close>'' instead of the keyword ``@{command "oops"}''.
\<close>
@@ -137,40 +130,41 @@
@{command_def "def"} & : & \<open>proof(state) \<rightarrow> proof(state)\<close> \\
\end{matharray}
- The logical proof context consists of fixed variables and
- assumptions. The former closely correspond to Skolem constants, or
- meta-level universal quantification as provided by the Isabelle/Pure
- logical framework. Introducing some \<^emph>\<open>arbitrary, but fixed\<close>
- variable via ``@{command "fix"}~\<open>x\<close>'' results in a local value
- that may be used in the subsequent proof as any other variable or
- constant. Furthermore, any result \<open>\<turnstile> \<phi>[x]\<close> exported from
- the context will be universally closed wrt.\ \<open>x\<close> at the
- outermost level: \<open>\<turnstile> \<And>x. \<phi>[x]\<close> (this is expressed in normal
- form using Isabelle's meta-variables).
+ The logical proof context consists of fixed variables and assumptions. The
+ former closely correspond to Skolem constants, or meta-level universal
+ quantification as provided by the Isabelle/Pure logical framework.
+ Introducing some \<^emph>\<open>arbitrary, but fixed\<close> variable via ``@{command
+ "fix"}~\<open>x\<close>'' results in a local value that may be used in the subsequent
+ proof as any other variable or constant. Furthermore, any result \<open>\<turnstile> \<phi>[x]\<close>
+ exported from the context will be universally closed wrt.\ \<open>x\<close> at the
+ outermost level: \<open>\<turnstile> \<And>x. \<phi>[x]\<close> (this is expressed in normal form using
+ Isabelle's meta-variables).
- Similarly, introducing some assumption \<open>\<chi>\<close> has two effects.
- On the one hand, a local theorem is created that may be used as a
- fact in subsequent proof steps. On the other hand, any result
- \<open>\<chi> \<turnstile> \<phi>\<close> exported from the context becomes conditional wrt.\
- the assumption: \<open>\<turnstile> \<chi> \<Longrightarrow> \<phi>\<close>. Thus, solving an enclosing goal
- using such a result would basically introduce a new subgoal stemming
- from the assumption. How this situation is handled depends on the
- version of assumption command used: while @{command "assume"}
- insists on solving the subgoal by unification with some premise of
- the goal, @{command "presume"} leaves the subgoal unchanged in order
- to be proved later by the user.
+ Similarly, introducing some assumption \<open>\<chi>\<close> has two effects. On the one hand,
+ a local theorem is created that may be used as a fact in subsequent proof
+ steps. On the other hand, any result \<open>\<chi> \<turnstile> \<phi>\<close> exported from the context
+ becomes conditional wrt.\ the assumption: \<open>\<turnstile> \<chi> \<Longrightarrow> \<phi>\<close>. Thus, solving an
+ enclosing goal using such a result would basically introduce a new subgoal
+ stemming from the assumption. How this situation is handled depends on the
+ version of assumption command used: while @{command "assume"} insists on
+ solving the subgoal by unification with some premise of the goal, @{command
+ "presume"} leaves the subgoal unchanged in order to be proved later by the
+ user.
- Local definitions, introduced by ``@{command "def"}~\<open>x \<equiv>
- t\<close>'', are achieved by combining ``@{command "fix"}~\<open>x\<close>'' with
- another version of assumption that causes any hypothetical equation
- \<open>x \<equiv> t\<close> to be eliminated by the reflexivity rule. Thus,
- exporting some result \<open>x \<equiv> t \<turnstile> \<phi>[x]\<close> yields \<open>\<turnstile>
+ Local definitions, introduced by ``@{command "def"}~\<open>x \<equiv> t\<close>'', are achieved
+ by combining ``@{command "fix"}~\<open>x\<close>'' with another version of assumption
+ that causes any hypothetical equation \<open>x \<equiv> t\<close> to be eliminated by the
+ reflexivity rule. Thus, exporting some result \<open>x \<equiv> t \<turnstile> \<phi>[x]\<close> yields \<open>\<turnstile>
\<phi>[t]\<close>.
@{rail \<open>
@@{command fix} @{syntax "fixes"}
;
- (@@{command assume} | @@{command presume}) (@{syntax props} + @'and')
+ (@@{command assume} | @@{command presume}) concl prems @{syntax for_fixes}
+ ;
+ concl: (@{syntax props} + @'and')
+ ;
+ prems: (@'if' (@{syntax props'} + @'and'))?
;
@@{command def} (def + @'and')
;
@@ -178,27 +172,31 @@
@{syntax name} ('==' | '\<equiv>') @{syntax term} @{syntax term_pat}?
\<close>}
- \<^descr> @{command "fix"}~\<open>x\<close> introduces a local variable \<open>x\<close> that is \<^emph>\<open>arbitrary, but fixed\<close>.
+ \<^descr> @{command "fix"}~\<open>x\<close> introduces a local variable \<open>x\<close> that is \<^emph>\<open>arbitrary,
+ but fixed\<close>.
- \<^descr> @{command "assume"}~\<open>a: \<phi>\<close> and @{command
- "presume"}~\<open>a: \<phi>\<close> introduce a local fact \<open>\<phi> \<turnstile> \<phi>\<close> by
- assumption. Subsequent results applied to an enclosing goal (e.g.\
- by @{command_ref "show"}) are handled as follows: @{command
- "assume"} expects to be able to unify with existing premises in the
- goal, while @{command "presume"} leaves \<open>\<phi>\<close> as new subgoals.
+ \<^descr> @{command "assume"}~\<open>a: \<phi>\<close> and @{command "presume"}~\<open>a: \<phi>\<close> introduce a
+ local fact \<open>\<phi> \<turnstile> \<phi>\<close> by assumption. Subsequent results applied to an enclosing
+ goal (e.g.\ by @{command_ref "show"}) are handled as follows: @{command
+ "assume"} expects to be able to unify with existing premises in the goal,
+ while @{command "presume"} leaves \<open>\<phi>\<close> as new subgoals.
+
+ Several lists of assumptions may be given (separated by @{keyword_ref
+ "and"}; the resulting list of current facts consists of all of these
+ concatenated.
- Several lists of assumptions may be given (separated by
- @{keyword_ref "and"}; the resulting list of current facts consists
- of all of these concatenated.
+ A structured assumption like \<^theory_text>\<open>assume "B x" and "A x" for x\<close> is equivalent
+ to \<^theory_text>\<open>assume "\<And>x. A x \<Longrightarrow> B x"\<close>, but vacuous quantification is avoided: a
+ for-context only effects propositions according to actual use of variables.
- \<^descr> @{command "def"}~\<open>x \<equiv> t\<close> introduces a local
- (non-polymorphic) definition. In results exported from the context,
- \<open>x\<close> is replaced by \<open>t\<close>. Basically, ``@{command
- "def"}~\<open>x \<equiv> t\<close>'' abbreviates ``@{command "fix"}~\<open>x\<close>~@{command "assume"}~\<open>x \<equiv> t\<close>'', with the resulting
- hypothetical equation solved by reflexivity.
+ \<^descr> @{command "def"}~\<open>x \<equiv> t\<close> introduces a local (non-polymorphic) definition.
+ In results exported from the context, \<open>x\<close> is replaced by \<open>t\<close>. Basically,
+ ``@{command "def"}~\<open>x \<equiv> t\<close>'' abbreviates ``@{command "fix"}~\<open>x\<close>~@{command
+ "assume"}~\<open>x \<equiv> t\<close>'', with the resulting hypothetical equation solved by
+ reflexivity.
- The default name for the definitional equation is \<open>x_def\<close>.
- Several simultaneous definitions may be given at the same time.
+ The default name for the definitional equation is \<open>x_def\<close>. Several
+ simultaneous definitions may be given at the same time.
\<close>
@@ -210,31 +208,28 @@
@{keyword_def "is"} & : & syntax \\
\end{matharray}
- Abbreviations may be either bound by explicit @{command
- "let"}~\<open>p \<equiv> t\<close> statements, or by annotating assumptions or
- goal statements with a list of patterns ``\<open>(\<IS> p\<^sub>1 \<dots>
- p\<^sub>n)\<close>''. In both cases, higher-order matching is invoked to
- bind extra-logical term variables, which may be either named
- schematic variables of the form \<open>?x\<close>, or nameless dummies
- ``@{variable _}'' (underscore). Note that in the @{command "let"}
- form the patterns occur on the left-hand side, while the @{keyword
- "is"} patterns are in postfix position.
+ Abbreviations may be either bound by explicit @{command "let"}~\<open>p \<equiv> t\<close>
+ statements, or by annotating assumptions or goal statements with a list of
+ patterns ``\<open>(\<IS> p\<^sub>1 \<dots> p\<^sub>n)\<close>''. In both cases, higher-order matching is
+ invoked to bind extra-logical term variables, which may be either named
+ schematic variables of the form \<open>?x\<close>, or nameless dummies ``@{variable _}''
+ (underscore). Note that in the @{command "let"} form the patterns occur on
+ the left-hand side, while the @{keyword "is"} patterns are in postfix
+ position.
- Polymorphism of term bindings is handled in Hindley-Milner style,
- similar to ML. Type variables referring to local assumptions or
- open goal statements are \<^emph>\<open>fixed\<close>, while those of finished
- results or bound by @{command "let"} may occur in \<^emph>\<open>arbitrary\<close>
- instances later. Even though actual polymorphism should be rarely
- used in practice, this mechanism is essential to achieve proper
- incremental type-inference, as the user proceeds to build up the
- Isar proof text from left to right.
+ Polymorphism of term bindings is handled in Hindley-Milner style, similar to
+ ML. Type variables referring to local assumptions or open goal statements
+ are \<^emph>\<open>fixed\<close>, while those of finished results or bound by @{command "let"}
+ may occur in \<^emph>\<open>arbitrary\<close> instances later. Even though actual polymorphism
+ should be rarely used in practice, this mechanism is essential to achieve
+ proper incremental type-inference, as the user proceeds to build up the Isar
+ proof text from left to right.
\<^medskip>
- Term abbreviations are quite different from local
- definitions as introduced via @{command "def"} (see
- \secref{sec:proof-context}). The latter are visible within the
- logic as actual equations, while abbreviations disappear during the
- input process just after type checking. Also note that @{command
+ Term abbreviations are quite different from local definitions as introduced
+ via @{command "def"} (see \secref{sec:proof-context}). The latter are
+ visible within the logic as actual equations, while abbreviations disappear
+ during the input process just after type checking. Also note that @{command
"def"} does not support polymorphism.
@{rail \<open>
@@ -244,26 +239,24 @@
The syntax of @{keyword "is"} patterns follows @{syntax term_pat} or
@{syntax prop_pat} (see \secref{sec:term-decls}).
- \<^descr> @{command "let"}~\<open>p\<^sub>1 = t\<^sub>1 \<AND> \<dots> p\<^sub>n = t\<^sub>n\<close> binds any
- text variables in patterns \<open>p\<^sub>1, \<dots>, p\<^sub>n\<close> by simultaneous
- higher-order matching against terms \<open>t\<^sub>1, \<dots>, t\<^sub>n\<close>.
+ \<^descr> @{command "let"}~\<open>p\<^sub>1 = t\<^sub>1 \<AND> \<dots> p\<^sub>n = t\<^sub>n\<close> binds any text variables
+ in patterns \<open>p\<^sub>1, \<dots>, p\<^sub>n\<close> by simultaneous higher-order matching against
+ terms \<open>t\<^sub>1, \<dots>, t\<^sub>n\<close>.
- \<^descr> \<open>(\<IS> p\<^sub>1 \<dots> p\<^sub>n)\<close> resembles @{command "let"}, but
- matches \<open>p\<^sub>1, \<dots>, p\<^sub>n\<close> against the preceding statement. Also
- note that @{keyword "is"} is not a separate command, but part of
- others (such as @{command "assume"}, @{command "have"} etc.).
-
+ \<^descr> \<open>(\<IS> p\<^sub>1 \<dots> p\<^sub>n)\<close> resembles @{command "let"}, but matches \<open>p\<^sub>1, \<dots>,
+ p\<^sub>n\<close> against the preceding statement. Also note that @{keyword "is"} is
+ not a separate command, but part of others (such as @{command "assume"},
+ @{command "have"} etc.).
- Some \<^emph>\<open>implicit\<close> term abbreviations\index{term abbreviations}
- for goals and facts are available as well. For any open goal,
- @{variable_ref thesis} refers to its object-level statement,
- abstracted over any meta-level parameters (if present). Likewise,
- @{variable_ref this} is bound for fact statements resulting from
- assumptions or finished goals. In case @{variable this} refers to
- an object-logic statement that is an application \<open>f t\<close>, then
- \<open>t\<close> is bound to the special text variable ``@{variable "\<dots>"}''
- (three dots). The canonical application of this convenience are
- calculational proofs (see \secref{sec:calculation}).
+ Some \<^emph>\<open>implicit\<close> term abbreviations\index{term abbreviations} for goals and
+ facts are available as well. For any open goal, @{variable_ref thesis}
+ refers to its object-level statement, abstracted over any meta-level
+ parameters (if present). Likewise, @{variable_ref this} is bound for fact
+ statements resulting from assumptions or finished goals. In case @{variable
+ this} refers to an object-logic statement that is an application \<open>f t\<close>, then
+ \<open>t\<close> is bound to the special text variable ``@{variable "\<dots>"}'' (three dots).
+ The canonical application of this convenience are calculational proofs (see
+ \secref{sec:calculation}).
\<close>
@@ -279,16 +272,15 @@
@{command_def "unfolding"} & : & \<open>proof(prove) \<rightarrow> proof(prove)\<close> \\
\end{matharray}
- New facts are established either by assumption or proof of local
- statements. Any fact will usually be involved in further proofs,
- either as explicit arguments of proof methods, or when forward
- chaining towards the next goal via @{command "then"} (and variants);
- @{command "from"} and @{command "with"} are composite forms
- involving @{command "note"}. The @{command "using"} elements
- augments the collection of used facts \<^emph>\<open>after\<close> a goal has been
- stated. Note that the special theorem name @{fact_ref this} refers
- to the most recently established facts, but only \<^emph>\<open>before\<close>
- issuing a follow-up claim.
+ New facts are established either by assumption or proof of local statements.
+ Any fact will usually be involved in further proofs, either as explicit
+ arguments of proof methods, or when forward chaining towards the next goal
+ via @{command "then"} (and variants); @{command "from"} and @{command
+ "with"} are composite forms involving @{command "note"}. The @{command
+ "using"} elements augments the collection of used facts \<^emph>\<open>after\<close> a goal has
+ been stated. Note that the special theorem name @{fact_ref this} refers to
+ the most recently established facts, but only \<^emph>\<open>before\<close> issuing a follow-up
+ claim.
@{rail \<open>
@@{command note} (@{syntax thmdef}? @{syntax thmrefs} + @'and')
@@ -297,55 +289,51 @@
(@{syntax thmrefs} + @'and')
\<close>}
- \<^descr> @{command "note"}~\<open>a = b\<^sub>1 \<dots> b\<^sub>n\<close> recalls existing facts
- \<open>b\<^sub>1, \<dots>, b\<^sub>n\<close>, binding the result as \<open>a\<close>. Note that
- attributes may be involved as well, both on the left and right hand
- sides.
+ \<^descr> @{command "note"}~\<open>a = b\<^sub>1 \<dots> b\<^sub>n\<close> recalls existing facts \<open>b\<^sub>1, \<dots>, b\<^sub>n\<close>,
+ binding the result as \<open>a\<close>. Note that attributes may be involved as well,
+ both on the left and right hand sides.
- \<^descr> @{command "then"} indicates forward chaining by the current
- facts in order to establish the goal to be claimed next. The
- initial proof method invoked to refine that will be offered the
- facts to do ``anything appropriate'' (see also
- \secref{sec:proof-steps}). For example, method @{method (Pure) rule}
- (see \secref{sec:pure-meth-att}) would typically do an elimination
- rather than an introduction. Automatic methods usually insert the
- facts into the goal state before operation. This provides a simple
- scheme to control relevance of facts in automated proof search.
+ \<^descr> @{command "then"} indicates forward chaining by the current facts in order
+ to establish the goal to be claimed next. The initial proof method invoked
+ to refine that will be offered the facts to do ``anything appropriate'' (see
+ also \secref{sec:proof-steps}). For example, method @{method (Pure) rule}
+ (see \secref{sec:pure-meth-att}) would typically do an elimination rather
+ than an introduction. Automatic methods usually insert the facts into the
+ goal state before operation. This provides a simple scheme to control
+ relevance of facts in automated proof search.
- \<^descr> @{command "from"}~\<open>b\<close> abbreviates ``@{command
- "note"}~\<open>b\<close>~@{command "then"}''; thus @{command "then"} is
- equivalent to ``@{command "from"}~\<open>this\<close>''.
+ \<^descr> @{command "from"}~\<open>b\<close> abbreviates ``@{command "note"}~\<open>b\<close>~@{command
+ "then"}''; thus @{command "then"} is equivalent to ``@{command
+ "from"}~\<open>this\<close>''.
- \<^descr> @{command "with"}~\<open>b\<^sub>1 \<dots> b\<^sub>n\<close> abbreviates ``@{command
- "from"}~\<open>b\<^sub>1 \<dots> b\<^sub>n \<AND> this\<close>''; thus the forward chaining
- is from earlier facts together with the current ones.
+ \<^descr> @{command "with"}~\<open>b\<^sub>1 \<dots> b\<^sub>n\<close> abbreviates ``@{command "from"}~\<open>b\<^sub>1 \<dots> b\<^sub>n
+ \<AND> this\<close>''; thus the forward chaining is from earlier facts together
+ with the current ones.
- \<^descr> @{command "using"}~\<open>b\<^sub>1 \<dots> b\<^sub>n\<close> augments the facts being
- currently indicated for use by a subsequent refinement step (such as
- @{command_ref "apply"} or @{command_ref "proof"}).
+ \<^descr> @{command "using"}~\<open>b\<^sub>1 \<dots> b\<^sub>n\<close> augments the facts being currently
+ indicated for use by a subsequent refinement step (such as @{command_ref
+ "apply"} or @{command_ref "proof"}).
- \<^descr> @{command "unfolding"}~\<open>b\<^sub>1 \<dots> b\<^sub>n\<close> is structurally
- similar to @{command "using"}, but unfolds definitional equations
- \<open>b\<^sub>1, \<dots> b\<^sub>n\<close> throughout the goal state and facts.
+ \<^descr> @{command "unfolding"}~\<open>b\<^sub>1 \<dots> b\<^sub>n\<close> is structurally similar to @{command
+ "using"}, but unfolds definitional equations \<open>b\<^sub>1, \<dots> b\<^sub>n\<close> throughout the
+ goal state and facts.
- Forward chaining with an empty list of theorems is the same as not
- chaining at all. Thus ``@{command "from"}~\<open>nothing\<close>'' has no
- effect apart from entering \<open>prove(chain)\<close> mode, since
- @{fact_ref nothing} is bound to the empty list of theorems.
+ Forward chaining with an empty list of theorems is the same as not chaining
+ at all. Thus ``@{command "from"}~\<open>nothing\<close>'' has no effect apart from
+ entering \<open>prove(chain)\<close> mode, since @{fact_ref nothing} is bound to the
+ empty list of theorems.
Basic proof methods (such as @{method_ref (Pure) rule}) expect multiple
- facts to be given in their proper order, corresponding to a prefix
- of the premises of the rule involved. Note that positions may be
- easily skipped using something like @{command "from"}~\<open>_
- \<AND> a \<AND> b\<close>, for example. This involves the trivial rule
- \<open>PROP \<psi> \<Longrightarrow> PROP \<psi>\<close>, which is bound in Isabelle/Pure as
- ``@{fact_ref "_"}'' (underscore).
+ facts to be given in their proper order, corresponding to a prefix of the
+ premises of the rule involved. Note that positions may be easily skipped
+ using something like @{command "from"}~\<open>_ \<AND> a \<AND> b\<close>, for example.
+ This involves the trivial rule \<open>PROP \<psi> \<Longrightarrow> PROP \<psi>\<close>, which is bound in
+ Isabelle/Pure as ``@{fact_ref "_"}'' (underscore).
- Automated methods (such as @{method simp} or @{method auto}) just
- insert any given facts before their usual operation. Depending on
- the kind of procedure involved, the order of facts is less
- significant here.
+ Automated methods (such as @{method simp} or @{method auto}) just insert any
+ given facts before their usual operation. Depending on the kind of procedure
+ involved, the order of facts is less significant here.
\<close>
@@ -365,34 +353,31 @@
@{command_def "print_statement"}\<open>\<^sup>*\<close> & : & \<open>context \<rightarrow>\<close> \\
\end{matharray}
- From a theory context, proof mode is entered by an initial goal command
- such as @{command "lemma"}. Within a proof context, new claims may be
- introduced locally; there are variants to interact with the overall proof
- structure specifically, such as @{command have} or @{command show}.
+ From a theory context, proof mode is entered by an initial goal command such
+ as @{command "lemma"}. Within a proof context, new claims may be introduced
+ locally; there are variants to interact with the overall proof structure
+ specifically, such as @{command have} or @{command show}.
- Goals may consist of multiple statements, resulting in a list of
- facts eventually. A pending multi-goal is internally represented as
- a meta-level conjunction (\<open>&&&\<close>), which is usually
- split into the corresponding number of sub-goals prior to an initial
- method application, via @{command_ref "proof"}
+ Goals may consist of multiple statements, resulting in a list of facts
+ eventually. A pending multi-goal is internally represented as a meta-level
+ conjunction (\<open>&&&\<close>), which is usually split into the corresponding number of
+ sub-goals prior to an initial method application, via @{command_ref "proof"}
(\secref{sec:proof-steps}) or @{command_ref "apply"}
- (\secref{sec:tactic-commands}). The @{method_ref induct} method
- covered in \secref{sec:cases-induct} acts on multiple claims
- simultaneously.
+ (\secref{sec:tactic-commands}). The @{method_ref induct} method covered in
+ \secref{sec:cases-induct} acts on multiple claims simultaneously.
- Claims at the theory level may be either in short or long form. A
- short goal merely consists of several simultaneous propositions
- (often just one). A long goal includes an explicit context
- specification for the subsequent conclusion, involving local
- parameters and assumptions. Here the role of each part of the
- statement is explicitly marked by separate keywords (see also
- \secref{sec:locale}); the local assumptions being introduced here
- are available as @{fact_ref assms} in the proof. Moreover, there
- are two kinds of conclusions: @{element_def "shows"} states several
- simultaneous propositions (essentially a big conjunction), while
- @{element_def "obtains"} claims several simultaneous simultaneous
- contexts of (essentially a big disjunction of eliminated parameters
- and assumptions, cf.\ \secref{sec:obtain}).
+ Claims at the theory level may be either in short or long form. A short goal
+ merely consists of several simultaneous propositions (often just one). A
+ long goal includes an explicit context specification for the subsequent
+ conclusion, involving local parameters and assumptions. Here the role of
+ each part of the statement is explicitly marked by separate keywords (see
+ also \secref{sec:locale}); the local assumptions being introduced here are
+ available as @{fact_ref assms} in the proof. Moreover, there are two kinds
+ of conclusions: @{element_def "shows"} states several simultaneous
+ propositions (essentially a big conjunction), while @{element_def "obtains"}
+ claims several simultaneous simultaneous contexts of (essentially a big
+ disjunction of eliminated parameters and assumptions, cf.\
+ \secref{sec:obtain}).
@{rail \<open>
(@@{command lemma} | @@{command theorem} | @@{command corollary} |
@@ -419,70 +404,64 @@
(@{syntax thmdecl}? (@{syntax prop}+) + @'and')
\<close>}
- \<^descr> @{command "lemma"}~\<open>a: \<phi>\<close> enters proof mode with
- \<open>\<phi>\<close> as main goal, eventually resulting in some fact \<open>\<turnstile>
- \<phi>\<close> to be put back into the target context. An additional @{syntax
- context} specification may build up an initial proof context for the
- subsequent claim; this includes local definitions and syntax as
- well, see also @{syntax "includes"} in \secref{sec:bundle} and
- @{syntax context_elem} in \secref{sec:locale}.
+ \<^descr> @{command "lemma"}~\<open>a: \<phi>\<close> enters proof mode with \<open>\<phi>\<close> as main goal,
+ eventually resulting in some fact \<open>\<turnstile> \<phi>\<close> to be put back into the target
+ context. An additional @{syntax context} specification may build up an
+ initial proof context for the subsequent claim; this includes local
+ definitions and syntax as well, see also @{syntax "includes"} in
+ \secref{sec:bundle} and @{syntax context_elem} in \secref{sec:locale}.
- \<^descr> @{command "theorem"}, @{command "corollary"}, and @{command
- "proposition"} are the same as @{command "lemma"}. The different command
- names merely serve as a formal comment in the theory source.
+ \<^descr> @{command "theorem"}, @{command "corollary"}, and @{command "proposition"}
+ are the same as @{command "lemma"}. The different command names merely serve
+ as a formal comment in the theory source.
- \<^descr> @{command "schematic_goal"} is similar to @{command "theorem"},
- but allows the statement to contain unbound schematic variables.
+ \<^descr> @{command "schematic_goal"} is similar to @{command "theorem"}, but allows
+ the statement to contain unbound schematic variables.
- Under normal circumstances, an Isar proof text needs to specify
- claims explicitly. Schematic goals are more like goals in Prolog,
- where certain results are synthesized in the course of reasoning.
- With schematic statements, the inherent compositionality of Isar
- proofs is lost, which also impacts performance, because proof
- checking is forced into sequential mode.
+ Under normal circumstances, an Isar proof text needs to specify claims
+ explicitly. Schematic goals are more like goals in Prolog, where certain
+ results are synthesized in the course of reasoning. With schematic
+ statements, the inherent compositionality of Isar proofs is lost, which also
+ impacts performance, because proof checking is forced into sequential mode.
- \<^descr> @{command "have"}~\<open>a: \<phi>\<close> claims a local goal,
- eventually resulting in a fact within the current logical context.
- This operation is completely independent of any pending sub-goals of
- an enclosing goal statements, so @{command "have"} may be freely
- used for experimental exploration of potential results within a
- proof body.
+ \<^descr> @{command "have"}~\<open>a: \<phi>\<close> claims a local goal, eventually resulting in a
+ fact within the current logical context. This operation is completely
+ independent of any pending sub-goals of an enclosing goal statements, so
+ @{command "have"} may be freely used for experimental exploration of
+ potential results within a proof body.
- \<^descr> @{command "show"}~\<open>a: \<phi>\<close> is like @{command
- "have"}~\<open>a: \<phi>\<close> plus a second stage to refine some pending
- sub-goal for each one of the finished result, after having been
- exported into the corresponding context (at the head of the
- sub-proof of this @{command "show"} command).
+ \<^descr> @{command "show"}~\<open>a: \<phi>\<close> is like @{command "have"}~\<open>a: \<phi>\<close> plus a second
+ stage to refine some pending sub-goal for each one of the finished result,
+ after having been exported into the corresponding context (at the head of
+ the sub-proof of this @{command "show"} command).
- To accommodate interactive debugging, resulting rules are printed
- before being applied internally. Even more, interactive execution
- of @{command "show"} predicts potential failure and displays the
- resulting error as a warning beforehand. Watch out for the
- following message:
+ To accommodate interactive debugging, resulting rules are printed before
+ being applied internally. Even more, interactive execution of @{command
+ "show"} predicts potential failure and displays the resulting error as a
+ warning beforehand. Watch out for the following message:
@{verbatim [display] \<open>Local statement fails to refine any pending goal\<close>}
- \<^descr> @{command "hence"} abbreviates ``@{command "then"}~@{command
- "have"}'', i.e.\ claims a local goal to be proven by forward
- chaining the current facts. Note that @{command "hence"} is also
- equivalent to ``@{command "from"}~\<open>this\<close>~@{command "have"}''.
+ \<^descr> @{command "hence"} abbreviates ``@{command "then"}~@{command "have"}'',
+ i.e.\ claims a local goal to be proven by forward chaining the current
+ facts. Note that @{command "hence"} is also equivalent to ``@{command
+ "from"}~\<open>this\<close>~@{command "have"}''.
- \<^descr> @{command "thus"} abbreviates ``@{command "then"}~@{command
- "show"}''. Note that @{command "thus"} is also equivalent to
- ``@{command "from"}~\<open>this\<close>~@{command "show"}''.
+ \<^descr> @{command "thus"} abbreviates ``@{command "then"}~@{command "show"}''.
+ Note that @{command "thus"} is also equivalent to ``@{command
+ "from"}~\<open>this\<close>~@{command "show"}''.
- \<^descr> @{command "print_statement"}~\<open>a\<close> prints facts from the
- current theory or proof context in long statement form, according to
- the syntax for @{command "lemma"} given above.
+ \<^descr> @{command "print_statement"}~\<open>a\<close> prints facts from the current theory or
+ proof context in long statement form, according to the syntax for @{command
+ "lemma"} given above.
- Any goal statement causes some term abbreviations (such as
- @{variable_ref "?thesis"}) to be bound automatically, see also
- \secref{sec:term-abbrev}.
+ Any goal statement causes some term abbreviations (such as @{variable_ref
+ "?thesis"}) to be bound automatically, see also \secref{sec:term-abbrev}.
Structured goal statements involving @{keyword_ref "if"} or @{keyword_ref
"when"} define the special fact @{fact_ref that} to refer to these
- assumptions in the proof body. The user may provide separate names
- according to the syntax of the statement.
+ assumptions in the proof body. The user may provide separate names according
+ to the syntax of the statement.
\<close>
@@ -500,38 +479,34 @@
@{attribute symmetric} & : & \<open>attribute\<close> \\
\end{matharray}
- Calculational proof is forward reasoning with implicit application
- of transitivity rules (such those of \<open>=\<close>, \<open>\<le>\<close>,
- \<open><\<close>). Isabelle/Isar maintains an auxiliary fact register
- @{fact_ref calculation} for accumulating results obtained by
- transitivity composed with the current result. Command @{command
- "also"} updates @{fact calculation} involving @{fact this}, while
- @{command "finally"} exhibits the final @{fact calculation} by
- forward chaining towards the next goal statement. Both commands
- require valid current facts, i.e.\ may occur only after commands
- that produce theorems such as @{command "assume"}, @{command
- "note"}, or some finished proof of @{command "have"}, @{command
- "show"} etc. The @{command "moreover"} and @{command "ultimately"}
- commands are similar to @{command "also"} and @{command "finally"},
- but only collect further results in @{fact calculation} without
- applying any rules yet.
+ Calculational proof is forward reasoning with implicit application of
+ transitivity rules (such those of \<open>=\<close>, \<open>\<le>\<close>, \<open><\<close>). Isabelle/Isar maintains an
+ auxiliary fact register @{fact_ref calculation} for accumulating results
+ obtained by transitivity composed with the current result. Command @{command
+ "also"} updates @{fact calculation} involving @{fact this}, while @{command
+ "finally"} exhibits the final @{fact calculation} by forward chaining
+ towards the next goal statement. Both commands require valid current facts,
+ i.e.\ may occur only after commands that produce theorems such as @{command
+ "assume"}, @{command "note"}, or some finished proof of @{command "have"},
+ @{command "show"} etc. The @{command "moreover"} and @{command "ultimately"}
+ commands are similar to @{command "also"} and @{command "finally"}, but only
+ collect further results in @{fact calculation} without applying any rules
+ yet.
- Also note that the implicit term abbreviation ``\<open>\<dots>\<close>'' has
- its canonical application with calculational proofs. It refers to
- the argument of the preceding statement. (The argument of a curried
- infix expression happens to be its right-hand side.)
+ Also note that the implicit term abbreviation ``\<open>\<dots>\<close>'' has its canonical
+ application with calculational proofs. It refers to the argument of the
+ preceding statement. (The argument of a curried infix expression happens to
+ be its right-hand side.)
- Isabelle/Isar calculations are implicitly subject to block structure
- in the sense that new threads of calculational reasoning are
- commenced for any new block (as opened by a local goal, for
- example). This means that, apart from being able to nest
- calculations, there is no separate \<^emph>\<open>begin-calculation\<close> command
- required.
+ Isabelle/Isar calculations are implicitly subject to block structure in the
+ sense that new threads of calculational reasoning are commenced for any new
+ block (as opened by a local goal, for example). This means that, apart from
+ being able to nest calculations, there is no separate \<^emph>\<open>begin-calculation\<close>
+ command required.
\<^medskip>
- The Isar calculation proof commands may be defined as
- follows:\<^footnote>\<open>We suppress internal bookkeeping such as proper
- handling of block-structure.\<close>
+ The Isar calculation proof commands may be defined as follows:\<^footnote>\<open>We suppress
+ internal bookkeeping such as proper handling of block-structure.\<close>
\begin{matharray}{rcl}
@{command "also"}\<open>\<^sub>0\<close> & \equiv & @{command "note"}~\<open>calculation = this\<close> \\
@@ -547,49 +522,46 @@
@@{attribute trans} (() | 'add' | 'del')
\<close>}
- \<^descr> @{command "also"}~\<open>(a\<^sub>1 \<dots> a\<^sub>n)\<close> maintains the auxiliary
- @{fact calculation} register as follows. The first occurrence of
- @{command "also"} in some calculational thread initializes @{fact
- calculation} by @{fact this}. Any subsequent @{command "also"} on
- the same level of block-structure updates @{fact calculation} by
- some transitivity rule applied to @{fact calculation} and @{fact
- this} (in that order). Transitivity rules are picked from the
- current context, unless alternative rules are given as explicit
+ \<^descr> @{command "also"}~\<open>(a\<^sub>1 \<dots> a\<^sub>n)\<close> maintains the auxiliary @{fact
+ calculation} register as follows. The first occurrence of @{command "also"}
+ in some calculational thread initializes @{fact calculation} by @{fact
+ this}. Any subsequent @{command "also"} on the same level of block-structure
+ updates @{fact calculation} by some transitivity rule applied to @{fact
+ calculation} and @{fact this} (in that order). Transitivity rules are picked
+ from the current context, unless alternative rules are given as explicit
arguments.
- \<^descr> @{command "finally"}~\<open>(a\<^sub>1 \<dots> a\<^sub>n)\<close> maintaining @{fact
- calculation} in the same way as @{command "also"}, and concludes the
- current calculational thread. The final result is exhibited as fact
- for forward chaining towards the next goal. Basically, @{command
- "finally"} just abbreviates @{command "also"}~@{command
- "from"}~@{fact calculation}. Typical idioms for concluding
+ \<^descr> @{command "finally"}~\<open>(a\<^sub>1 \<dots> a\<^sub>n)\<close> maintaining @{fact calculation} in the
+ same way as @{command "also"}, and concludes the current calculational
+ thread. The final result is exhibited as fact for forward chaining towards
+ the next goal. Basically, @{command "finally"} just abbreviates @{command
+ "also"}~@{command "from"}~@{fact calculation}. Typical idioms for concluding
calculational proofs are ``@{command "finally"}~@{command
- "show"}~\<open>?thesis\<close>~@{command "."}'' and ``@{command
- "finally"}~@{command "have"}~\<open>\<phi>\<close>~@{command "."}''.
+ "show"}~\<open>?thesis\<close>~@{command "."}'' and ``@{command "finally"}~@{command
+ "have"}~\<open>\<phi>\<close>~@{command "."}''.
- \<^descr> @{command "moreover"} and @{command "ultimately"} are
- analogous to @{command "also"} and @{command "finally"}, but collect
- results only, without applying rules.
+ \<^descr> @{command "moreover"} and @{command "ultimately"} are analogous to
+ @{command "also"} and @{command "finally"}, but collect results only,
+ without applying rules.
- \<^descr> @{command "print_trans_rules"} prints the list of transitivity
- rules (for calculational commands @{command "also"} and @{command
- "finally"}) and symmetry rules (for the @{attribute symmetric}
- operation and single step elimination patters) of the current
- context.
+ \<^descr> @{command "print_trans_rules"} prints the list of transitivity rules (for
+ calculational commands @{command "also"} and @{command "finally"}) and
+ symmetry rules (for the @{attribute symmetric} operation and single step
+ elimination patters) of the current context.
\<^descr> @{attribute trans} declares theorems as transitivity rules.
- \<^descr> @{attribute sym} declares symmetry rules, as well as
- @{attribute "Pure.elim"}\<open>?\<close> rules.
+ \<^descr> @{attribute sym} declares symmetry rules, as well as @{attribute
+ "Pure.elim"}\<open>?\<close> rules.
- \<^descr> @{attribute symmetric} resolves a theorem with some rule
- declared as @{attribute sym} in the current context. For example,
- ``@{command "assume"}~\<open>[symmetric]: x = y\<close>'' produces a
- swapped fact derived from that assumption.
+ \<^descr> @{attribute symmetric} resolves a theorem with some rule declared as
+ @{attribute sym} in the current context. For example, ``@{command
+ "assume"}~\<open>[symmetric]: x = y\<close>'' produces a swapped fact derived from that
+ assumption.
- In structured proof texts it is often more appropriate to use an
- explicit single-step elimination proof, such as ``@{command
- "assume"}~\<open>x = y\<close>~@{command "then"}~@{command "have"}~\<open>y = x\<close>~@{command ".."}''.
+ In structured proof texts it is often more appropriate to use an explicit
+ single-step elimination proof, such as ``@{command "assume"}~\<open>x =
+ y\<close>~@{command "then"}~@{command "have"}~\<open>y = x\<close>~@{command ".."}''.
\<close>
@@ -597,16 +569,16 @@
subsection \<open>Proof method expressions \label{sec:proof-meth}\<close>
-text \<open>Proof methods are either basic ones, or expressions composed of
- methods via ``\<^verbatim>\<open>,\<close>'' (sequential composition), ``\<^verbatim>\<open>;\<close>'' (structural
- composition), ``\<^verbatim>\<open>|\<close>'' (alternative
- choices), ``\<^verbatim>\<open>?\<close>'' (try), ``\<^verbatim>\<open>+\<close>'' (repeat at least
- once), ``\<^verbatim>\<open>[\<close>\<open>n\<close>\<^verbatim>\<open>]\<close>'' (restriction to first
- \<open>n\<close> subgoals). In practice, proof methods are usually just a comma
- separated list of @{syntax nameref}~@{syntax args} specifications. Note
- that parentheses may be dropped for single method specifications (with no
- arguments). The syntactic precedence of method combinators is \<^verbatim>\<open>|\<close> \<^verbatim>\<open>;\<close> \<^verbatim>\<open>,\<close>
- \<^verbatim>\<open>[]\<close> \<^verbatim>\<open>+\<close> \<^verbatim>\<open>?\<close> (from low to high).
+text \<open>
+ Proof methods are either basic ones, or expressions composed of methods via
+ ``\<^verbatim>\<open>,\<close>'' (sequential composition), ``\<^verbatim>\<open>;\<close>'' (structural composition),
+ ``\<^verbatim>\<open>|\<close>'' (alternative choices), ``\<^verbatim>\<open>?\<close>'' (try), ``\<^verbatim>\<open>+\<close>'' (repeat at least
+ once), ``\<^verbatim>\<open>[\<close>\<open>n\<close>\<^verbatim>\<open>]\<close>'' (restriction to first \<open>n\<close> subgoals). In practice,
+ proof methods are usually just a comma separated list of @{syntax
+ nameref}~@{syntax args} specifications. Note that parentheses may be dropped
+ for single method specifications (with no arguments). The syntactic
+ precedence of method combinators is \<^verbatim>\<open>|\<close> \<^verbatim>\<open>;\<close> \<^verbatim>\<open>,\<close> \<^verbatim>\<open>[]\<close> \<^verbatim>\<open>+\<close> \<^verbatim>\<open>?\<close> (from low
+ to high).
@{rail \<open>
@{syntax_def method}:
@@ -615,35 +587,34 @@
methods: (@{syntax nameref} @{syntax args} | @{syntax method}) + (',' | ';' | '|')
\<close>}
- Regular Isar proof methods do \<^emph>\<open>not\<close> admit direct goal addressing, but
- refer to the first subgoal or to all subgoals uniformly. Nonetheless,
- the subsequent mechanisms allow to imitate the effect of subgoal
- addressing that is known from ML tactics.
+ Regular Isar proof methods do \<^emph>\<open>not\<close> admit direct goal addressing, but refer
+ to the first subgoal or to all subgoals uniformly. Nonetheless, the
+ subsequent mechanisms allow to imitate the effect of subgoal addressing that
+ is known from ML tactics.
\<^medskip>
- Goal \<^emph>\<open>restriction\<close> means the proof state is wrapped-up in a
- way that certain subgoals are exposed, and other subgoals are ``parked''
- elsewhere. Thus a proof method has no other chance than to operate on the
- subgoals that are presently exposed.
+ Goal \<^emph>\<open>restriction\<close> means the proof state is wrapped-up in a way that
+ certain subgoals are exposed, and other subgoals are ``parked'' elsewhere.
+ Thus a proof method has no other chance than to operate on the subgoals that
+ are presently exposed.
- Structural composition ``\<open>m\<^sub>1\<close>\<^verbatim>\<open>;\<close>~\<open>m\<^sub>2\<close>'' means
- that method \<open>m\<^sub>1\<close> is applied with restriction to the first subgoal,
- then \<open>m\<^sub>2\<close> is applied consecutively with restriction to each subgoal
- that has newly emerged due to \<open>m\<^sub>1\<close>. This is analogous to the tactic
- combinator @{ML_op THEN_ALL_NEW} in Isabelle/ML, see also @{cite
- "isabelle-implementation"}. For example, \<open>(rule r; blast)\<close> applies
- rule \<open>r\<close> and then solves all new subgoals by \<open>blast\<close>.
+ Structural composition ``\<open>m\<^sub>1\<close>\<^verbatim>\<open>;\<close>~\<open>m\<^sub>2\<close>'' means that method \<open>m\<^sub>1\<close> is
+ applied with restriction to the first subgoal, then \<open>m\<^sub>2\<close> is applied
+ consecutively with restriction to each subgoal that has newly emerged due to
+ \<open>m\<^sub>1\<close>. This is analogous to the tactic combinator @{ML_op THEN_ALL_NEW} in
+ Isabelle/ML, see also @{cite "isabelle-implementation"}. For example, \<open>(rule
+ r; blast)\<close> applies rule \<open>r\<close> and then solves all new subgoals by \<open>blast\<close>.
- Moreover, the explicit goal restriction operator ``\<open>[n]\<close>'' exposes
- only the first \<open>n\<close> subgoals (which need to exist), with default
- \<open>n = 1\<close>. For example, the method expression ``\<open>simp_all[3]\<close>'' simplifies the first three subgoals, while ``\<open>(rule r, simp_all)[]\<close>'' simplifies all new goals that emerge from
+ Moreover, the explicit goal restriction operator ``\<open>[n]\<close>'' exposes only the
+ first \<open>n\<close> subgoals (which need to exist), with default \<open>n = 1\<close>. For example,
+ the method expression ``\<open>simp_all[3]\<close>'' simplifies the first three subgoals,
+ while ``\<open>(rule r, simp_all)[]\<close>'' simplifies all new goals that emerge from
applying rule \<open>r\<close> to the originally first one.
\<^medskip>
- Improper methods, notably tactic emulations, offer low-level goal
- addressing as explicit argument to the individual tactic being involved.
- Here ``\<open>[!]\<close>'' refers to all goals, and ``\<open>[n-]\<close>'' to all
- goals starting from \<open>n\<close>.
+ Improper methods, notably tactic emulations, offer low-level goal addressing
+ as explicit argument to the individual tactic being involved. Here ``\<open>[!]\<close>''
+ refers to all goals, and ``\<open>[n-]\<close>'' to all goals starting from \<open>n\<close>.
@{rail \<open>
@{syntax_def goal_spec}:
@@ -665,39 +636,37 @@
@{method_def standard} & : & \<open>method\<close> \\
\end{matharray}
- Arbitrary goal refinement via tactics is considered harmful.
- Structured proof composition in Isar admits proof methods to be
- invoked in two places only.
-
- \<^enum> An \<^emph>\<open>initial\<close> refinement step @{command_ref
- "proof"}~\<open>m\<^sub>1\<close> reduces a newly stated goal to a number
- of sub-goals that are to be solved later. Facts are passed to
- \<open>m\<^sub>1\<close> for forward chaining, if so indicated by \<open>proof(chain)\<close> mode.
+ Arbitrary goal refinement via tactics is considered harmful. Structured
+ proof composition in Isar admits proof methods to be invoked in two places
+ only.
- \<^enum> A \<^emph>\<open>terminal\<close> conclusion step @{command_ref "qed"}~\<open>m\<^sub>2\<close> is intended to solve remaining goals. No facts are
- passed to \<open>m\<^sub>2\<close>.
-
+ \<^enum> An \<^emph>\<open>initial\<close> refinement step @{command_ref "proof"}~\<open>m\<^sub>1\<close> reduces a
+ newly stated goal to a number of sub-goals that are to be solved later.
+ Facts are passed to \<open>m\<^sub>1\<close> for forward chaining, if so indicated by
+ \<open>proof(chain)\<close> mode.
+
+ \<^enum> A \<^emph>\<open>terminal\<close> conclusion step @{command_ref "qed"}~\<open>m\<^sub>2\<close> is intended to
+ solve remaining goals. No facts are passed to \<open>m\<^sub>2\<close>.
- The only other (proper) way to affect pending goals in a proof body
- is by @{command_ref "show"}, which involves an explicit statement of
- what is to be solved eventually. Thus we avoid the fundamental
- problem of unstructured tactic scripts that consist of numerous
- consecutive goal transformations, with invisible effects.
+ The only other (proper) way to affect pending goals in a proof body is by
+ @{command_ref "show"}, which involves an explicit statement of what is to be
+ solved eventually. Thus we avoid the fundamental problem of unstructured
+ tactic scripts that consist of numerous consecutive goal transformations,
+ with invisible effects.
\<^medskip>
- As a general rule of thumb for good proof style, initial
- proof methods should either solve the goal completely, or constitute
- some well-understood reduction to new sub-goals. Arbitrary
- automatic proof tools that are prone leave a large number of badly
- structured sub-goals are no help in continuing the proof document in
- an intelligible manner.
+ As a general rule of thumb for good proof style, initial proof methods
+ should either solve the goal completely, or constitute some well-understood
+ reduction to new sub-goals. Arbitrary automatic proof tools that are prone
+ leave a large number of badly structured sub-goals are no help in continuing
+ the proof document in an intelligible manner.
- Unless given explicitly by the user, the default initial method is
- @{method standard}, which subsumes at least @{method_ref (Pure) rule} or
- its classical variant @{method_ref (HOL) rule}. These methods apply a
- single standard elimination or introduction rule according to the topmost
- logical connective involved. There is no separate default terminal method.
- Any remaining goals are always solved by assumption in the very last step.
+ Unless given explicitly by the user, the default initial method is @{method
+ standard}, which subsumes at least @{method_ref (Pure) rule} or its
+ classical variant @{method_ref (HOL) rule}. These methods apply a single
+ standard elimination or introduction rule according to the topmost logical
+ connective involved. There is no separate default terminal method. Any
+ remaining goals are always solved by assumption in the very last step.
@{rail \<open>
@@{command proof} method?
@@ -709,70 +678,64 @@
(@@{command "."} | @@{command ".."} | @@{command sorry})
\<close>}
- \<^descr> @{command "proof"}~\<open>m\<^sub>1\<close> refines the goal by proof
- method \<open>m\<^sub>1\<close>; facts for forward chaining are passed if so
- indicated by \<open>proof(chain)\<close> mode.
+ \<^descr> @{command "proof"}~\<open>m\<^sub>1\<close> refines the goal by proof method \<open>m\<^sub>1\<close>; facts for
+ forward chaining are passed if so indicated by \<open>proof(chain)\<close> mode.
- \<^descr> @{command "qed"}~\<open>m\<^sub>2\<close> refines any remaining goals by
- proof method \<open>m\<^sub>2\<close> and concludes the sub-proof by assumption.
- If the goal had been \<open>show\<close> (or \<open>thus\<close>), some
- pending sub-goal is solved as well by the rule resulting from the
- result \<^emph>\<open>exported\<close> into the enclosing goal context. Thus \<open>qed\<close> may fail for two reasons: either \<open>m\<^sub>2\<close> fails, or the
- resulting rule does not fit to any pending goal\<^footnote>\<open>This
- includes any additional ``strong'' assumptions as introduced by
- @{command "assume"}.\<close> of the enclosing context. Debugging such a
- situation might involve temporarily changing @{command "show"} into
- @{command "have"}, or weakening the local context by replacing
- occurrences of @{command "assume"} by @{command "presume"}.
+ \<^descr> @{command "qed"}~\<open>m\<^sub>2\<close> refines any remaining goals by proof method \<open>m\<^sub>2\<close>
+ and concludes the sub-proof by assumption. If the goal had been \<open>show\<close> (or
+ \<open>thus\<close>), some pending sub-goal is solved as well by the rule resulting from
+ the result \<^emph>\<open>exported\<close> into the enclosing goal context. Thus \<open>qed\<close> may fail
+ for two reasons: either \<open>m\<^sub>2\<close> fails, or the resulting rule does not fit to
+ any pending goal\<^footnote>\<open>This includes any additional ``strong'' assumptions as
+ introduced by @{command "assume"}.\<close> of the enclosing context. Debugging such
+ a situation might involve temporarily changing @{command "show"} into
+ @{command "have"}, or weakening the local context by replacing occurrences
+ of @{command "assume"} by @{command "presume"}.
- \<^descr> @{command "by"}~\<open>m\<^sub>1 m\<^sub>2\<close> is a \<^emph>\<open>terminal
- proof\<close>\index{proof!terminal}; it abbreviates @{command
- "proof"}~\<open>m\<^sub>1\<close>~@{command "qed"}~\<open>m\<^sub>2\<close>, but with
- backtracking across both methods. Debugging an unsuccessful
- @{command "by"}~\<open>m\<^sub>1 m\<^sub>2\<close> command can be done by expanding its
- definition; in many cases @{command "proof"}~\<open>m\<^sub>1\<close> (or even
- \<open>apply\<close>~\<open>m\<^sub>1\<close>) is already sufficient to see the
- problem.
+ \<^descr> @{command "by"}~\<open>m\<^sub>1 m\<^sub>2\<close> is a \<^emph>\<open>terminal proof\<close>\index{proof!terminal}; it
+ abbreviates @{command "proof"}~\<open>m\<^sub>1\<close>~@{command "qed"}~\<open>m\<^sub>2\<close>, but with
+ backtracking across both methods. Debugging an unsuccessful @{command
+ "by"}~\<open>m\<^sub>1 m\<^sub>2\<close> command can be done by expanding its definition; in many
+ cases @{command "proof"}~\<open>m\<^sub>1\<close> (or even \<open>apply\<close>~\<open>m\<^sub>1\<close>) is already sufficient
+ to see the problem.
- \<^descr> ``@{command ".."}'' is a \<^emph>\<open>standard
- proof\<close>\index{proof!standard}; it abbreviates @{command "by"}~\<open>standard\<close>.
+ \<^descr> ``@{command ".."}'' is a \<^emph>\<open>standard proof\<close>\index{proof!standard}; it
+ abbreviates @{command "by"}~\<open>standard\<close>.
- \<^descr> ``@{command "."}'' is a \<^emph>\<open>trivial
- proof\<close>\index{proof!trivial}; it abbreviates @{command "by"}~\<open>this\<close>.
+ \<^descr> ``@{command "."}'' is a \<^emph>\<open>trivial proof\<close>\index{proof!trivial}; it
+ abbreviates @{command "by"}~\<open>this\<close>.
- \<^descr> @{command "sorry"} is a \<^emph>\<open>fake proof\<close>\index{proof!fake}
- pretending to solve the pending claim without further ado. This
- only works in interactive development, or if the @{attribute
- quick_and_dirty} is enabled. Facts emerging from fake
- proofs are not the real thing. Internally, the derivation object is
- tainted by an oracle invocation, which may be inspected via the
+ \<^descr> @{command "sorry"} is a \<^emph>\<open>fake proof\<close>\index{proof!fake} pretending to
+ solve the pending claim without further ado. This only works in interactive
+ development, or if the @{attribute quick_and_dirty} is enabled. Facts
+ emerging from fake proofs are not the real thing. Internally, the derivation
+ object is tainted by an oracle invocation, which may be inspected via the
theorem status @{cite "isabelle-implementation"}.
The most important application of @{command "sorry"} is to support
experimentation and top-down proof development.
- \<^descr> @{method standard} refers to the default refinement step of some
- Isar language elements (notably @{command proof} and ``@{command ".."}'').
- It is \<^emph>\<open>dynamically scoped\<close>, so the behaviour depends on the
- application environment.
+ \<^descr> @{method standard} refers to the default refinement step of some Isar
+ language elements (notably @{command proof} and ``@{command ".."}''). It is
+ \<^emph>\<open>dynamically scoped\<close>, so the behaviour depends on the application
+ environment.
In Isabelle/Pure, @{method standard} performs elementary introduction~/
- elimination steps (@{method_ref (Pure) rule}), introduction of type
- classes (@{method_ref intro_classes}) and locales (@{method_ref
- intro_locales}).
+ elimination steps (@{method_ref (Pure) rule}), introduction of type classes
+ (@{method_ref intro_classes}) and locales (@{method_ref intro_locales}).
- In Isabelle/HOL, @{method standard} also takes classical rules into
- account (cf.\ \secref{sec:classical}).
+ In Isabelle/HOL, @{method standard} also takes classical rules into account
+ (cf.\ \secref{sec:classical}).
\<close>
subsection \<open>Fundamental methods and attributes \label{sec:pure-meth-att}\<close>
text \<open>
- The following proof methods and attributes refer to basic logical
- operations of Isar. Further methods and attributes are provided by
- several generic and object-logic specific tools and packages (see
- \chref{ch:gen-tools} and \partref{part:hol}).
+ The following proof methods and attributes refer to basic logical operations
+ of Isar. Further methods and attributes are provided by several generic and
+ object-logic specific tools and packages (see \chref{ch:gen-tools} and
+ \partref{part:hol}).
\begin{matharray}{rcl}
@{command_def "print_rules"}\<open>\<^sup>*\<close> & : & \<open>context \<rightarrow>\<close> \\[0.5ex]
@@ -813,113 +776,103 @@
@@{attribute "where"} @{syntax named_insts} @{syntax for_fixes}
\<close>}
- \<^descr> @{command "print_rules"} prints rules declared via attributes
- @{attribute (Pure) intro}, @{attribute (Pure) elim}, @{attribute
- (Pure) dest} of Isabelle/Pure.
+ \<^descr> @{command "print_rules"} prints rules declared via attributes @{attribute
+ (Pure) intro}, @{attribute (Pure) elim}, @{attribute (Pure) dest} of
+ Isabelle/Pure.
- See also the analogous @{command "print_claset"} command for similar
- rule declarations of the classical reasoner
- (\secref{sec:classical}).
+ See also the analogous @{command "print_claset"} command for similar rule
+ declarations of the classical reasoner (\secref{sec:classical}).
- \<^descr> ``@{method "-"}'' (minus) inserts the forward chaining facts as
- premises into the goal, and nothing else.
+ \<^descr> ``@{method "-"}'' (minus) inserts the forward chaining facts as premises
+ into the goal, and nothing else.
Note that command @{command_ref "proof"} without any method actually
- performs a single reduction step using the @{method_ref (Pure) rule}
- method; thus a plain \<^emph>\<open>do-nothing\<close> proof step would be ``@{command
- "proof"}~\<open>-\<close>'' rather than @{command "proof"} alone.
+ performs a single reduction step using the @{method_ref (Pure) rule} method;
+ thus a plain \<^emph>\<open>do-nothing\<close> proof step would be ``@{command "proof"}~\<open>-\<close>''
+ rather than @{command "proof"} alone.
- \<^descr> @{method "goal_cases"}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close> turns the current subgoals
- into cases within the context (see also \secref{sec:cases-induct}). The
- specified case names are used if present; otherwise cases are numbered
- starting from 1.
+ \<^descr> @{method "goal_cases"}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close> turns the current subgoals into cases
+ within the context (see also \secref{sec:cases-induct}). The specified case
+ names are used if present; otherwise cases are numbered starting from 1.
Invoking cases in the subsequent proof body via the @{command_ref case}
command will @{command fix} goal parameters, @{command assume} goal
premises, and @{command let} variable @{variable_ref ?case} refer to the
conclusion.
- \<^descr> @{method "fact"}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close> composes some fact from
- \<open>a\<^sub>1, \<dots>, a\<^sub>n\<close> (or implicitly from the current proof context)
- modulo unification of schematic type and term variables. The rule
- structure is not taken into account, i.e.\ meta-level implication is
- considered atomic. This is the same principle underlying literal
- facts (cf.\ \secref{sec:syn-att}): ``@{command "have"}~\<open>\<phi>\<close>~@{command "by"}~\<open>fact\<close>'' is equivalent to ``@{command
- "note"}~\<^verbatim>\<open>`\<close>\<open>\<phi>\<close>\<^verbatim>\<open>`\<close>'' provided that
- \<open>\<turnstile> \<phi>\<close> is an instance of some known \<open>\<turnstile> \<phi>\<close> in the
- proof context.
+ \<^descr> @{method "fact"}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close> composes some fact from \<open>a\<^sub>1, \<dots>, a\<^sub>n\<close> (or
+ implicitly from the current proof context) modulo unification of schematic
+ type and term variables. The rule structure is not taken into account, i.e.\
+ meta-level implication is considered atomic. This is the same principle
+ underlying literal facts (cf.\ \secref{sec:syn-att}): ``@{command
+ "have"}~\<open>\<phi>\<close>~@{command "by"}~\<open>fact\<close>'' is equivalent to ``@{command
+ "note"}~\<^verbatim>\<open>`\<close>\<open>\<phi>\<close>\<^verbatim>\<open>`\<close>'' provided that \<open>\<turnstile> \<phi>\<close> is an instance of some known \<open>\<turnstile> \<phi>\<close>
+ in the proof context.
- \<^descr> @{method assumption} solves some goal by a single assumption
- step. All given facts are guaranteed to participate in the
- refinement; this means there may be only 0 or 1 in the first place.
- Recall that @{command "qed"} (\secref{sec:proof-steps}) already
- concludes any remaining sub-goals by assumption, so structured
- proofs usually need not quote the @{method assumption} method at
- all.
+ \<^descr> @{method assumption} solves some goal by a single assumption step. All
+ given facts are guaranteed to participate in the refinement; this means
+ there may be only 0 or 1 in the first place. Recall that @{command "qed"}
+ (\secref{sec:proof-steps}) already concludes any remaining sub-goals by
+ assumption, so structured proofs usually need not quote the @{method
+ assumption} method at all.
- \<^descr> @{method this} applies all of the current facts directly as
- rules. Recall that ``@{command "."}'' (dot) abbreviates ``@{command
- "by"}~\<open>this\<close>''.
+ \<^descr> @{method this} applies all of the current facts directly as rules. Recall
+ that ``@{command "."}'' (dot) abbreviates ``@{command "by"}~\<open>this\<close>''.
- \<^descr> @{method (Pure) rule}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close> applies some rule given as
- argument in backward manner; facts are used to reduce the rule
- before applying it to the goal. Thus @{method (Pure) rule} without facts
- is plain introduction, while with facts it becomes elimination.
+ \<^descr> @{method (Pure) rule}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close> applies some rule given as argument in
+ backward manner; facts are used to reduce the rule before applying it to the
+ goal. Thus @{method (Pure) rule} without facts is plain introduction, while
+ with facts it becomes elimination.
- When no arguments are given, the @{method (Pure) rule} method tries to
- pick appropriate rules automatically, as declared in the current context
- using the @{attribute (Pure) intro}, @{attribute (Pure) elim}, @{attribute
- (Pure) dest} attributes (see below). This is included in the standard
- behaviour of @{command "proof"} and ``@{command ".."}'' (double-dot) steps
- (see \secref{sec:proof-steps}).
+ When no arguments are given, the @{method (Pure) rule} method tries to pick
+ appropriate rules automatically, as declared in the current context using
+ the @{attribute (Pure) intro}, @{attribute (Pure) elim}, @{attribute (Pure)
+ dest} attributes (see below). This is included in the standard behaviour of
+ @{command "proof"} and ``@{command ".."}'' (double-dot) steps (see
+ \secref{sec:proof-steps}).
- \<^descr> @{attribute (Pure) intro}, @{attribute (Pure) elim}, and
- @{attribute (Pure) dest} declare introduction, elimination, and
- destruct rules, to be used with method @{method (Pure) rule}, and similar
- tools. Note that the latter will ignore rules declared with
- ``\<open>?\<close>'', while ``\<open>!\<close>'' are used most aggressively.
+ \<^descr> @{attribute (Pure) intro}, @{attribute (Pure) elim}, and @{attribute
+ (Pure) dest} declare introduction, elimination, and destruct rules, to be
+ used with method @{method (Pure) rule}, and similar tools. Note that the
+ latter will ignore rules declared with ``\<open>?\<close>'', while ``\<open>!\<close>'' are used most
+ aggressively.
- The classical reasoner (see \secref{sec:classical}) introduces its
- own variants of these attributes; use qualified names to access the
- present versions of Isabelle/Pure, i.e.\ @{attribute (Pure)
- "Pure.intro"}.
+ The classical reasoner (see \secref{sec:classical}) introduces its own
+ variants of these attributes; use qualified names to access the present
+ versions of Isabelle/Pure, i.e.\ @{attribute (Pure) "Pure.intro"}.
- \<^descr> @{attribute (Pure) rule}~\<open>del\<close> undeclares introduction,
- elimination, or destruct rules.
+ \<^descr> @{attribute (Pure) rule}~\<open>del\<close> undeclares introduction, elimination, or
+ destruct rules.
- \<^descr> @{attribute OF}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close> applies some theorem to all
- of the given rules \<open>a\<^sub>1, \<dots>, a\<^sub>n\<close> in canonical right-to-left
- order, which means that premises stemming from the \<open>a\<^sub>i\<close>
- emerge in parallel in the result, without interfering with each
- other. In many practical situations, the \<open>a\<^sub>i\<close> do not have
- premises themselves, so \<open>rule [OF a\<^sub>1 \<dots> a\<^sub>n]\<close> can be actually
- read as functional application (modulo unification).
+ \<^descr> @{attribute OF}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close> applies some theorem to all of the given rules
+ \<open>a\<^sub>1, \<dots>, a\<^sub>n\<close> in canonical right-to-left order, which means that premises
+ stemming from the \<open>a\<^sub>i\<close> emerge in parallel in the result, without
+ interfering with each other. In many practical situations, the \<open>a\<^sub>i\<close> do not
+ have premises themselves, so \<open>rule [OF a\<^sub>1 \<dots> a\<^sub>n]\<close> can be actually read as
+ functional application (modulo unification).
- Argument positions may be effectively skipped by using ``\<open>_\<close>''
- (underscore), which refers to the propositional identity rule in the
- Pure theory.
+ Argument positions may be effectively skipped by using ``\<open>_\<close>'' (underscore),
+ which refers to the propositional identity rule in the Pure theory.
- \<^descr> @{attribute of}~\<open>t\<^sub>1 \<dots> t\<^sub>n\<close> performs positional
- instantiation of term variables. The terms \<open>t\<^sub>1, \<dots>, t\<^sub>n\<close> are
- substituted for any schematic variables occurring in a theorem from
- left to right; ``\<open>_\<close>'' (underscore) indicates to skip a
- position. Arguments following a ``\<open>concl:\<close>'' specification
- refer to positions of the conclusion of a rule.
+ \<^descr> @{attribute of}~\<open>t\<^sub>1 \<dots> t\<^sub>n\<close> performs positional instantiation of term
+ variables. The terms \<open>t\<^sub>1, \<dots>, t\<^sub>n\<close> are substituted for any schematic
+ variables occurring in a theorem from left to right; ``\<open>_\<close>'' (underscore)
+ indicates to skip a position. Arguments following a ``\<open>concl:\<close>''
+ specification refer to positions of the conclusion of a rule.
- An optional context of local variables \<open>\<FOR> x\<^sub>1 \<dots> x\<^sub>m\<close> may
- be specified: the instantiated theorem is exported, and these
- variables become schematic (usually with some shifting of indices).
+ An optional context of local variables \<open>\<FOR> x\<^sub>1 \<dots> x\<^sub>m\<close> may be specified:
+ the instantiated theorem is exported, and these variables become schematic
+ (usually with some shifting of indices).
- \<^descr> @{attribute "where"}~\<open>x\<^sub>1 = t\<^sub>1 \<AND> \<dots> x\<^sub>n = t\<^sub>n\<close>
- performs named instantiation of schematic type and term variables
- occurring in a theorem. Schematic variables have to be specified on
- the left-hand side (e.g.\ \<open>?x1.3\<close>). The question mark may
- be omitted if the variable name is a plain identifier without index.
- As type instantiations are inferred from term instantiations,
- explicit type instantiations are seldom necessary.
+ \<^descr> @{attribute "where"}~\<open>x\<^sub>1 = t\<^sub>1 \<AND> \<dots> x\<^sub>n = t\<^sub>n\<close> performs named
+ instantiation of schematic type and term variables occurring in a theorem.
+ Schematic variables have to be specified on the left-hand side (e.g.\
+ \<open>?x1.3\<close>). The question mark may be omitted if the variable name is a plain
+ identifier without index. As type instantiations are inferred from term
+ instantiations, explicit type instantiations are seldom necessary.
- An optional context of local variables \<open>\<FOR> x\<^sub>1 \<dots> x\<^sub>m\<close> may
- be specified as for @{attribute "of"} above.
+ An optional context of local variables \<open>\<FOR> x\<^sub>1 \<dots> x\<^sub>m\<close> may be specified
+ as for @{attribute "of"} above.
\<close>
@@ -934,14 +887,13 @@
@@{command method_setup} @{syntax name} '=' @{syntax text} @{syntax text}?
\<close>}
- \<^descr> @{command "method_setup"}~\<open>name = text description\<close>
- defines a proof method in the current context. The given \<open>text\<close> has to be an ML expression of type
- @{ML_type "(Proof.context -> Proof.method) context_parser"}, cf.\
- basic parsers defined in structure @{ML_structure Args} and @{ML_structure
- Attrib}. There are also combinators like @{ML METHOD} and @{ML
- SIMPLE_METHOD} to turn certain tactic forms into official proof
- methods; the primed versions refer to tactics with explicit goal
- addressing.
+ \<^descr> @{command "method_setup"}~\<open>name = text description\<close> defines a proof method
+ in the current context. The given \<open>text\<close> has to be an ML expression of type
+ @{ML_type "(Proof.context -> Proof.method) context_parser"}, cf.\ basic
+ parsers defined in structure @{ML_structure Args} and @{ML_structure
+ Attrib}. There are also combinators like @{ML METHOD} and @{ML
+ SIMPLE_METHOD} to turn certain tactic forms into official proof methods; the
+ primed versions refer to tactics with explicit goal addressing.
Here are some example method definitions:
\<close>
@@ -977,52 +929,47 @@
@{attribute_def consumes} & : & \<open>attribute\<close> \\
\end{matharray}
- The puristic way to build up Isar proof contexts is by explicit
- language elements like @{command "fix"}, @{command "assume"},
- @{command "let"} (see \secref{sec:proof-context}). This is adequate
- for plain natural deduction, but easily becomes unwieldy in concrete
- verification tasks, which typically involve big induction rules with
- several cases.
+ The puristic way to build up Isar proof contexts is by explicit language
+ elements like @{command "fix"}, @{command "assume"}, @{command "let"} (see
+ \secref{sec:proof-context}). This is adequate for plain natural deduction,
+ but easily becomes unwieldy in concrete verification tasks, which typically
+ involve big induction rules with several cases.
- The @{command "case"} command provides a shorthand to refer to a
- local context symbolically: certain proof methods provide an
- environment of named ``cases'' of the form \<open>c: x\<^sub>1, \<dots>,
- x\<^sub>m, \<phi>\<^sub>1, \<dots>, \<phi>\<^sub>n\<close>; the effect of ``@{command
- "case"}~\<open>c\<close>'' is then equivalent to ``@{command "fix"}~\<open>x\<^sub>1 \<dots> x\<^sub>m\<close>~@{command "assume"}~\<open>c: \<phi>\<^sub>1 \<dots>
- \<phi>\<^sub>n\<close>''. Term bindings may be covered as well, notably
- @{variable ?case} for the main conclusion.
+ The @{command "case"} command provides a shorthand to refer to a local
+ context symbolically: certain proof methods provide an environment of named
+ ``cases'' of the form \<open>c: x\<^sub>1, \<dots>, x\<^sub>m, \<phi>\<^sub>1, \<dots>, \<phi>\<^sub>n\<close>; the effect of
+ ``@{command "case"}~\<open>c\<close>'' is then equivalent to ``@{command "fix"}~\<open>x\<^sub>1 \<dots>
+ x\<^sub>m\<close>~@{command "assume"}~\<open>c: \<phi>\<^sub>1 \<dots> \<phi>\<^sub>n\<close>''. Term bindings may be covered as
+ well, notably @{variable ?case} for the main conclusion.
- By default, the ``terminology'' \<open>x\<^sub>1, \<dots>, x\<^sub>m\<close> of
- a case value is marked as hidden, i.e.\ there is no way to refer to
- such parameters in the subsequent proof text. After all, original
- rule parameters stem from somewhere outside of the current proof
- text. By using the explicit form ``@{command "case"}~\<open>(c
- y\<^sub>1 \<dots> y\<^sub>m)\<close>'' instead, the proof author is able to
- chose local names that fit nicely into the current context.
+ By default, the ``terminology'' \<open>x\<^sub>1, \<dots>, x\<^sub>m\<close> of a case value is marked as
+ hidden, i.e.\ there is no way to refer to such parameters in the subsequent
+ proof text. After all, original rule parameters stem from somewhere outside
+ of the current proof text. By using the explicit form ``@{command
+ "case"}~\<open>(c y\<^sub>1 \<dots> y\<^sub>m)\<close>'' instead, the proof author is able to chose local
+ names that fit nicely into the current context.
\<^medskip>
- It is important to note that proper use of @{command
- "case"} does not provide means to peek at the current goal state,
- which is not directly observable in Isar! Nonetheless, goal
- refinement commands do provide named cases \<open>goal\<^sub>i\<close>
- for each subgoal \<open>i = 1, \<dots>, n\<close> of the resulting goal state.
- Using this extra feature requires great care, because some bits of
- the internal tactical machinery intrude the proof text. In
- particular, parameter names stemming from the left-over of automated
- reasoning tools are usually quite unpredictable.
+ It is important to note that proper use of @{command "case"} does not
+ provide means to peek at the current goal state, which is not directly
+ observable in Isar! Nonetheless, goal refinement commands do provide named
+ cases \<open>goal\<^sub>i\<close> for each subgoal \<open>i = 1, \<dots>, n\<close> of the resulting goal state.
+ Using this extra feature requires great care, because some bits of the
+ internal tactical machinery intrude the proof text. In particular, parameter
+ names stemming from the left-over of automated reasoning tools are usually
+ quite unpredictable.
Under normal circumstances, the text of cases emerge from standard
- elimination or induction rules, which in turn are derived from
- previous theory specifications in a canonical way (say from
- @{command "inductive"} definitions).
+ elimination or induction rules, which in turn are derived from previous
+ theory specifications in a canonical way (say from @{command "inductive"}
+ definitions).
\<^medskip>
- Proper cases are only available if both the proof method
- and the rules involved support this. By using appropriate
- attributes, case names, conclusions, and parameters may be also
- declared by hand. Thus variant versions of rules that have been
- derived manually become ready to use in advanced case analysis
- later.
+ Proper cases are only available if both the proof method and the rules
+ involved support this. By using appropriate attributes, case names,
+ conclusions, and parameters may be also declared by hand. Thus variant
+ versions of rules that have been derived manually become ready to use in
+ advanced case analysis later.
@{rail \<open>
@@{command case} @{syntax thmdecl}? (nameref | '(' nameref (('_' | @{syntax name}) *) ')')
@@ -1036,67 +983,62 @@
@@{attribute consumes} @{syntax int}?
\<close>}
- \<^descr> @{command "case"}~\<open>a: (c x\<^sub>1 \<dots> x\<^sub>m)\<close> invokes a named local
- context \<open>c: x\<^sub>1, \<dots>, x\<^sub>m, \<phi>\<^sub>1, \<dots>, \<phi>\<^sub>m\<close>, as provided by an
- appropriate proof method (such as @{method_ref cases} and @{method_ref
- induct}). The command ``@{command "case"}~\<open>a: (c x\<^sub>1 \<dots> x\<^sub>m)\<close>''
- abbreviates ``@{command "fix"}~\<open>x\<^sub>1 \<dots> x\<^sub>m\<close>~@{command
- "assume"}~\<open>a.c: \<phi>\<^sub>1 \<dots> \<phi>\<^sub>n\<close>''. Each local fact is qualified by the
- prefix \<open>a\<close>, and all such facts are collectively bound to the name
- \<open>a\<close>.
+ \<^descr> @{command "case"}~\<open>a: (c x\<^sub>1 \<dots> x\<^sub>m)\<close> invokes a named local context \<open>c:
+ x\<^sub>1, \<dots>, x\<^sub>m, \<phi>\<^sub>1, \<dots>, \<phi>\<^sub>m\<close>, as provided by an appropriate proof method (such
+ as @{method_ref cases} and @{method_ref induct}). The command ``@{command
+ "case"}~\<open>a: (c x\<^sub>1 \<dots> x\<^sub>m)\<close>'' abbreviates ``@{command "fix"}~\<open>x\<^sub>1 \<dots>
+ x\<^sub>m\<close>~@{command "assume"}~\<open>a.c: \<phi>\<^sub>1 \<dots> \<phi>\<^sub>n\<close>''. Each local fact is qualified by
+ the prefix \<open>a\<close>, and all such facts are collectively bound to the name \<open>a\<close>.
- The fact name is specification \<open>a\<close> is optional, the default is to
- re-use \<open>c\<close>. So @{command "case"}~\<open>(c x\<^sub>1 \<dots> x\<^sub>m)\<close> is the same
- as @{command "case"}~\<open>c: (c x\<^sub>1 \<dots> x\<^sub>m)\<close>.
-
- \<^descr> @{command "print_cases"} prints all local contexts of the
- current state, using Isar proof language notation.
+ The fact name is specification \<open>a\<close> is optional, the default is to re-use
+ \<open>c\<close>. So @{command "case"}~\<open>(c x\<^sub>1 \<dots> x\<^sub>m)\<close> is the same as @{command
+ "case"}~\<open>c: (c x\<^sub>1 \<dots> x\<^sub>m)\<close>.
- \<^descr> @{attribute case_names}~\<open>c\<^sub>1 \<dots> c\<^sub>k\<close> declares names for
- the local contexts of premises of a theorem; \<open>c\<^sub>1, \<dots>, c\<^sub>k\<close>
- refers to the \<^emph>\<open>prefix\<close> of the list of premises. Each of the
- cases \<open>c\<^sub>i\<close> can be of the form \<open>c[h\<^sub>1 \<dots> h\<^sub>n]\<close> where
- the \<open>h\<^sub>1 \<dots> h\<^sub>n\<close> are the names of the hypotheses in case \<open>c\<^sub>i\<close>
- from left to right.
+ \<^descr> @{command "print_cases"} prints all local contexts of the current state,
+ using Isar proof language notation.
+
+ \<^descr> @{attribute case_names}~\<open>c\<^sub>1 \<dots> c\<^sub>k\<close> declares names for the local contexts
+ of premises of a theorem; \<open>c\<^sub>1, \<dots>, c\<^sub>k\<close> refers to the \<^emph>\<open>prefix\<close> of the list
+ of premises. Each of the cases \<open>c\<^sub>i\<close> can be of the form \<open>c[h\<^sub>1 \<dots> h\<^sub>n]\<close> where
+ the \<open>h\<^sub>1 \<dots> h\<^sub>n\<close> are the names of the hypotheses in case \<open>c\<^sub>i\<close> from left to
+ right.
- \<^descr> @{attribute case_conclusion}~\<open>c d\<^sub>1 \<dots> d\<^sub>k\<close> declares
- names for the conclusions of a named premise \<open>c\<close>; here \<open>d\<^sub>1, \<dots>, d\<^sub>k\<close> refers to the prefix of arguments of a logical formula
- built by nesting a binary connective (e.g.\ \<open>\<or>\<close>).
+ \<^descr> @{attribute case_conclusion}~\<open>c d\<^sub>1 \<dots> d\<^sub>k\<close> declares names for the
+ conclusions of a named premise \<open>c\<close>; here \<open>d\<^sub>1, \<dots>, d\<^sub>k\<close> refers to the prefix
+ of arguments of a logical formula built by nesting a binary connective
+ (e.g.\ \<open>\<or>\<close>).
- Note that proof methods such as @{method induct} and @{method
- coinduct} already provide a default name for the conclusion as a
- whole. The need to name subformulas only arises with cases that
- split into several sub-cases, as in common co-induction rules.
+ Note that proof methods such as @{method induct} and @{method coinduct}
+ already provide a default name for the conclusion as a whole. The need to
+ name subformulas only arises with cases that split into several sub-cases,
+ as in common co-induction rules.
- \<^descr> @{attribute params}~\<open>p\<^sub>1 \<dots> p\<^sub>m \<AND> \<dots> q\<^sub>1 \<dots> q\<^sub>n\<close> renames
- the innermost parameters of premises \<open>1, \<dots>, n\<close> of some
- theorem. An empty list of names may be given to skip positions,
- leaving the present parameters unchanged.
+ \<^descr> @{attribute params}~\<open>p\<^sub>1 \<dots> p\<^sub>m \<AND> \<dots> q\<^sub>1 \<dots> q\<^sub>n\<close> renames the innermost
+ parameters of premises \<open>1, \<dots>, n\<close> of some theorem. An empty list of names may
+ be given to skip positions, leaving the present parameters unchanged.
- Note that the default usage of case rules does \<^emph>\<open>not\<close> directly
- expose parameters to the proof context.
+ Note that the default usage of case rules does \<^emph>\<open>not\<close> directly expose
+ parameters to the proof context.
- \<^descr> @{attribute consumes}~\<open>n\<close> declares the number of ``major
- premises'' of a rule, i.e.\ the number of facts to be consumed when
- it is applied by an appropriate proof method. The default value of
- @{attribute consumes} is \<open>n = 1\<close>, which is appropriate for
- the usual kind of cases and induction rules for inductive sets (cf.\
- \secref{sec:hol-inductive}). Rules without any @{attribute
- consumes} declaration given are treated as if @{attribute
+ \<^descr> @{attribute consumes}~\<open>n\<close> declares the number of ``major premises'' of a
+ rule, i.e.\ the number of facts to be consumed when it is applied by an
+ appropriate proof method. The default value of @{attribute consumes} is \<open>n =
+ 1\<close>, which is appropriate for the usual kind of cases and induction rules for
+ inductive sets (cf.\ \secref{sec:hol-inductive}). Rules without any
+ @{attribute consumes} declaration given are treated as if @{attribute
consumes}~\<open>0\<close> had been specified.
- A negative \<open>n\<close> is interpreted relatively to the total number
- of premises of the rule in the target context. Thus its absolute
- value specifies the remaining number of premises, after subtracting
- the prefix of major premises as indicated above. This form of
- declaration has the technical advantage of being stable under more
- morphisms, notably those that export the result from a nested
- @{command_ref context} with additional assumptions.
+ A negative \<open>n\<close> is interpreted relatively to the total number of premises of
+ the rule in the target context. Thus its absolute value specifies the
+ remaining number of premises, after subtracting the prefix of major premises
+ as indicated above. This form of declaration has the technical advantage of
+ being stable under more morphisms, notably those that export the result from
+ a nested @{command_ref context} with additional assumptions.
- Note that explicit @{attribute consumes} declarations are only
- rarely needed; this is already taken care of automatically by the
- higher-level @{attribute cases}, @{attribute induct}, and
- @{attribute coinduct} declarations.
+ Note that explicit @{attribute consumes} declarations are only rarely
+ needed; this is already taken care of automatically by the higher-level
+ @{attribute cases}, @{attribute induct}, and @{attribute coinduct}
+ declarations.
\<close>
@@ -1110,25 +1052,23 @@
@{method_def coinduct} & : & \<open>method\<close> \\
\end{matharray}
- The @{method cases}, @{method induct}, @{method induction},
- and @{method coinduct}
- methods provide a uniform interface to common proof techniques over
- datatypes, inductive predicates (or sets), recursive functions etc.
- The corresponding rules may be specified and instantiated in a
- casual manner. Furthermore, these methods provide named local
- contexts that may be invoked via the @{command "case"} proof command
- within the subsequent proof text. This accommodates compact proof
- texts even when reasoning about large specifications.
+ The @{method cases}, @{method induct}, @{method induction}, and @{method
+ coinduct} methods provide a uniform interface to common proof techniques
+ over datatypes, inductive predicates (or sets), recursive functions etc. The
+ corresponding rules may be specified and instantiated in a casual manner.
+ Furthermore, these methods provide named local contexts that may be invoked
+ via the @{command "case"} proof command within the subsequent proof text.
+ This accommodates compact proof texts even when reasoning about large
+ specifications.
- The @{method induct} method also provides some additional
- infrastructure in order to be applicable to structure statements
- (either using explicit meta-level connectives, or including facts
- and parameters separately). This avoids cumbersome encoding of
- ``strengthened'' inductive statements within the object-logic.
+ The @{method induct} method also provides some additional infrastructure in
+ order to be applicable to structure statements (either using explicit
+ meta-level connectives, or including facts and parameters separately). This
+ avoids cumbersome encoding of ``strengthened'' inductive statements within
+ the object-logic.
- Method @{method induction} differs from @{method induct} only in
- the names of the facts in the local context invoked by the @{command "case"}
- command.
+ Method @{method induction} differs from @{method induct} only in the names
+ of the facts in the local context invoked by the @{command "case"} command.
@{rail \<open>
@@{method cases} ('(' 'no_simp' ')')? \<newline>
@@ -1151,13 +1091,12 @@
taking: 'taking' ':' @{syntax insts}
\<close>}
- \<^descr> @{method cases}~\<open>insts R\<close> applies method @{method
- rule} with an appropriate case distinction theorem, instantiated to
- the subjects \<open>insts\<close>. Symbolic case names are bound according
- to the rule's local contexts.
+ \<^descr> @{method cases}~\<open>insts R\<close> applies method @{method rule} with an
+ appropriate case distinction theorem, instantiated to the subjects \<open>insts\<close>.
+ Symbolic case names are bound according to the rule's local contexts.
- The rule is determined as follows, according to the facts and
- arguments passed to the @{method cases} method:
+ The rule is determined as follows, according to the facts and arguments
+ passed to the @{method cases} method:
\<^medskip>
\begin{tabular}{llll}
@@ -1170,16 +1109,16 @@
\end{tabular}
\<^medskip>
- Several instantiations may be given, referring to the \<^emph>\<open>suffix\<close>
- of premises of the case rule; within each premise, the \<^emph>\<open>prefix\<close>
- of variables is instantiated. In most situations, only a single
- term needs to be specified; this refers to the first variable of the
- last premise (it is usually the same for all cases). The \<open>(no_simp)\<close> option can be used to disable pre-simplification of
- cases (see the description of @{method induct} below for details).
+ Several instantiations may be given, referring to the \<^emph>\<open>suffix\<close> of premises
+ of the case rule; within each premise, the \<^emph>\<open>prefix\<close> of variables is
+ instantiated. In most situations, only a single term needs to be specified;
+ this refers to the first variable of the last premise (it is usually the
+ same for all cases). The \<open>(no_simp)\<close> option can be used to disable
+ pre-simplification of cases (see the description of @{method induct} below
+ for details).
- \<^descr> @{method induct}~\<open>insts R\<close> and
- @{method induction}~\<open>insts R\<close> are analogous to the
- @{method cases} method, but refer to induction rules, which are
+ \<^descr> @{method induct}~\<open>insts R\<close> and @{method induction}~\<open>insts R\<close> are analogous
+ to the @{method cases} method, but refer to induction rules, which are
determined as follows:
\<^medskip>
@@ -1191,51 +1130,44 @@
\end{tabular}
\<^medskip>
- Several instantiations may be given, each referring to some part of
- a mutual inductive definition or datatype --- only related partial
- induction rules may be used together, though. Any of the lists of
- terms \<open>P, x, \<dots>\<close> refers to the \<^emph>\<open>suffix\<close> of variables
- present in the induction rule. This enables the writer to specify
- only induction variables, or both predicates and variables, for
- example.
+ Several instantiations may be given, each referring to some part of a mutual
+ inductive definition or datatype --- only related partial induction rules
+ may be used together, though. Any of the lists of terms \<open>P, x, \<dots>\<close> refers to
+ the \<^emph>\<open>suffix\<close> of variables present in the induction rule. This enables the
+ writer to specify only induction variables, or both predicates and
+ variables, for example.
- Instantiations may be definitional: equations \<open>x \<equiv> t\<close>
- introduce local definitions, which are inserted into the claim and
- discharged after applying the induction rule. Equalities reappear
- in the inductive cases, but have been transformed according to the
- induction principle being involved here. In order to achieve
- practically useful induction hypotheses, some variables occurring in
- \<open>t\<close> need to be fixed (see below). Instantiations of the form
- \<open>t\<close>, where \<open>t\<close> is not a variable, are taken as a
- shorthand for \mbox{\<open>x \<equiv> t\<close>}, where \<open>x\<close> is a fresh
- variable. If this is not intended, \<open>t\<close> has to be enclosed in
- parentheses. By default, the equalities generated by definitional
- instantiations are pre-simplified using a specific set of rules,
- usually consisting of distinctness and injectivity theorems for
- datatypes. This pre-simplification may cause some of the parameters
- of an inductive case to disappear, or may even completely delete
- some of the inductive cases, if one of the equalities occurring in
- their premises can be simplified to \<open>False\<close>. The \<open>(no_simp)\<close> option can be used to disable pre-simplification.
- Additional rules to be used in pre-simplification can be declared
- using the @{attribute_def induct_simp} attribute.
+ Instantiations may be definitional: equations \<open>x \<equiv> t\<close> introduce local
+ definitions, which are inserted into the claim and discharged after applying
+ the induction rule. Equalities reappear in the inductive cases, but have
+ been transformed according to the induction principle being involved here.
+ In order to achieve practically useful induction hypotheses, some variables
+ occurring in \<open>t\<close> need to be fixed (see below). Instantiations of the form
+ \<open>t\<close>, where \<open>t\<close> is not a variable, are taken as a shorthand for \<open>x \<equiv> t\<close>,
+ where \<open>x\<close> is a fresh variable. If this is not intended, \<open>t\<close> has to be
+ enclosed in parentheses. By default, the equalities generated by
+ definitional instantiations are pre-simplified using a specific set of
+ rules, usually consisting of distinctness and injectivity theorems for
+ datatypes. This pre-simplification may cause some of the parameters of an
+ inductive case to disappear, or may even completely delete some of the
+ inductive cases, if one of the equalities occurring in their premises can be
+ simplified to \<open>False\<close>. The \<open>(no_simp)\<close> option can be used to disable
+ pre-simplification. Additional rules to be used in pre-simplification can be
+ declared using the @{attribute_def induct_simp} attribute.
- The optional ``\<open>arbitrary: x\<^sub>1 \<dots> x\<^sub>m\<close>''
- specification generalizes variables \<open>x\<^sub>1, \<dots>,
- x\<^sub>m\<close> of the original goal before applying induction. One can
- separate variables by ``\<open>and\<close>'' to generalize them in other
- goals then the first. Thus induction hypotheses may become
- sufficiently general to get the proof through. Together with
- definitional instantiations, one may effectively perform induction
- over expressions of a certain structure.
+ The optional ``\<open>arbitrary: x\<^sub>1 \<dots> x\<^sub>m\<close>'' specification generalizes variables
+ \<open>x\<^sub>1, \<dots>, x\<^sub>m\<close> of the original goal before applying induction. One can
+ separate variables by ``\<open>and\<close>'' to generalize them in other goals then the
+ first. Thus induction hypotheses may become sufficiently general to get the
+ proof through. Together with definitional instantiations, one may
+ effectively perform induction over expressions of a certain structure.
- The optional ``\<open>taking: t\<^sub>1 \<dots> t\<^sub>n\<close>''
- specification provides additional instantiations of a prefix of
- pending variables in the rule. Such schematic induction rules
- rarely occur in practice, though.
+ The optional ``\<open>taking: t\<^sub>1 \<dots> t\<^sub>n\<close>'' specification provides additional
+ instantiations of a prefix of pending variables in the rule. Such schematic
+ induction rules rarely occur in practice, though.
- \<^descr> @{method coinduct}~\<open>inst R\<close> is analogous to the
- @{method induct} method, but refers to coinduction rules, which are
- determined as follows:
+ \<^descr> @{method coinduct}~\<open>inst R\<close> is analogous to the @{method induct} method,
+ but refers to coinduction rules, which are determined as follows:
\<^medskip>
\begin{tabular}{llll}
@@ -1246,69 +1178,63 @@
\end{tabular}
\<^medskip>
- Coinduction is the dual of induction. Induction essentially
- eliminates \<open>A x\<close> towards a generic result \<open>P x\<close>,
- while coinduction introduces \<open>A x\<close> starting with \<open>B
- x\<close>, for a suitable ``bisimulation'' \<open>B\<close>. The cases of a
- coinduct rule are typically named after the predicates or sets being
- covered, while the conclusions consist of several alternatives being
- named after the individual destructor patterns.
+ Coinduction is the dual of induction. Induction essentially eliminates \<open>A x\<close>
+ towards a generic result \<open>P x\<close>, while coinduction introduces \<open>A x\<close> starting
+ with \<open>B x\<close>, for a suitable ``bisimulation'' \<open>B\<close>. The cases of a coinduct
+ rule are typically named after the predicates or sets being covered, while
+ the conclusions consist of several alternatives being named after the
+ individual destructor patterns.
- The given instantiation refers to the \<^emph>\<open>suffix\<close> of variables
- occurring in the rule's major premise, or conclusion if unavailable.
- An additional ``\<open>taking: t\<^sub>1 \<dots> t\<^sub>n\<close>''
- specification may be required in order to specify the bisimulation
- to be used in the coinduction step.
+ The given instantiation refers to the \<^emph>\<open>suffix\<close> of variables occurring in
+ the rule's major premise, or conclusion if unavailable. An additional
+ ``\<open>taking: t\<^sub>1 \<dots> t\<^sub>n\<close>'' specification may be required in order to specify
+ the bisimulation to be used in the coinduction step.
Above methods produce named local contexts, as determined by the
- instantiated rule as given in the text. Beyond that, the @{method
- induct} and @{method coinduct} methods guess further instantiations
- from the goal specification itself. Any persisting unresolved
- schematic variables of the resulting rule will render the the
- corresponding case invalid. The term binding @{variable ?case} for
- the conclusion will be provided with each case, provided that term
- is fully specified.
+ instantiated rule as given in the text. Beyond that, the @{method induct}
+ and @{method coinduct} methods guess further instantiations from the goal
+ specification itself. Any persisting unresolved schematic variables of the
+ resulting rule will render the the corresponding case invalid. The term
+ binding @{variable ?case} for the conclusion will be provided with each
+ case, provided that term is fully specified.
- The @{command "print_cases"} command prints all named cases present
- in the current proof state.
+ The @{command "print_cases"} command prints all named cases present in the
+ current proof state.
\<^medskip>
- Despite the additional infrastructure, both @{method cases}
- and @{method coinduct} merely apply a certain rule, after
- instantiation, while conforming due to the usual way of monotonic
- natural deduction: the context of a structured statement \<open>\<And>x\<^sub>1 \<dots> x\<^sub>m. \<phi>\<^sub>1 \<Longrightarrow> \<dots> \<phi>\<^sub>n \<Longrightarrow> \<dots>\<close>
- reappears unchanged after the case split.
+ Despite the additional infrastructure, both @{method cases} and @{method
+ coinduct} merely apply a certain rule, after instantiation, while conforming
+ due to the usual way of monotonic natural deduction: the context of a
+ structured statement \<open>\<And>x\<^sub>1 \<dots> x\<^sub>m. \<phi>\<^sub>1 \<Longrightarrow> \<dots> \<phi>\<^sub>n \<Longrightarrow> \<dots>\<close> reappears unchanged after
+ the case split.
- The @{method induct} method is fundamentally different in this
- respect: the meta-level structure is passed through the
- ``recursive'' course involved in the induction. Thus the original
- statement is basically replaced by separate copies, corresponding to
- the induction hypotheses and conclusion; the original goal context
- is no longer available. Thus local assumptions, fixed parameters
- and definitions effectively participate in the inductive rephrasing
- of the original statement.
+ The @{method induct} method is fundamentally different in this respect: the
+ meta-level structure is passed through the ``recursive'' course involved in
+ the induction. Thus the original statement is basically replaced by separate
+ copies, corresponding to the induction hypotheses and conclusion; the
+ original goal context is no longer available. Thus local assumptions, fixed
+ parameters and definitions effectively participate in the inductive
+ rephrasing of the original statement.
In @{method induct} proofs, local assumptions introduced by cases are split
- into two different kinds: \<open>hyps\<close> stemming from the rule and
- \<open>prems\<close> from the goal statement. This is reflected in the
- extracted cases accordingly, so invoking ``@{command "case"}~\<open>c\<close>'' will provide separate facts \<open>c.hyps\<close> and \<open>c.prems\<close>,
- as well as fact \<open>c\<close> to hold the all-inclusive list.
+ into two different kinds: \<open>hyps\<close> stemming from the rule and \<open>prems\<close> from the
+ goal statement. This is reflected in the extracted cases accordingly, so
+ invoking ``@{command "case"}~\<open>c\<close>'' will provide separate facts \<open>c.hyps\<close> and
+ \<open>c.prems\<close>, as well as fact \<open>c\<close> to hold the all-inclusive list.
In @{method induction} proofs, local assumptions introduced by cases are
- split into three different kinds: \<open>IH\<close>, the induction hypotheses,
- \<open>hyps\<close>, the remaining hypotheses stemming from the rule, and
- \<open>prems\<close>, the assumptions from the goal statement. The names are
- \<open>c.IH\<close>, \<open>c.hyps\<close> and \<open>c.prems\<close>, as above.
-
+ split into three different kinds: \<open>IH\<close>, the induction hypotheses, \<open>hyps\<close>,
+ the remaining hypotheses stemming from the rule, and \<open>prems\<close>, the
+ assumptions from the goal statement. The names are \<open>c.IH\<close>, \<open>c.hyps\<close> and
+ \<open>c.prems\<close>, as above.
\<^medskip>
- Facts presented to either method are consumed according to
- the number of ``major premises'' of the rule involved, which is
- usually 0 for plain cases and induction rules of datatypes etc.\ and
- 1 for rules of inductive predicates or sets and the like. The
- remaining facts are inserted into the goal verbatim before the
- actual \<open>cases\<close>, \<open>induct\<close>, or \<open>coinduct\<close> rule is
+ Facts presented to either method are consumed according to the number of
+ ``major premises'' of the rule involved, which is usually 0 for plain cases
+ and induction rules of datatypes etc.\ and 1 for rules of inductive
+ predicates or sets and the like. The remaining facts are inserted into the
+ goal verbatim before the actual \<open>cases\<close>, \<open>induct\<close>, or \<open>coinduct\<close> rule is
applied.
\<close>
@@ -1334,27 +1260,24 @@
spec: (('type' | 'pred' | 'set') ':' @{syntax nameref}) | 'del'
\<close>}
- \<^descr> @{command "print_induct_rules"} prints cases and induct rules
- for predicates (or sets) and types of the current context.
+ \<^descr> @{command "print_induct_rules"} prints cases and induct rules for
+ predicates (or sets) and types of the current context.
- \<^descr> @{attribute cases}, @{attribute induct}, and @{attribute
- coinduct} (as attributes) declare rules for reasoning about
- (co)inductive predicates (or sets) and types, using the
- corresponding methods of the same name. Certain definitional
- packages of object-logics usually declare emerging cases and
+ \<^descr> @{attribute cases}, @{attribute induct}, and @{attribute coinduct} (as
+ attributes) declare rules for reasoning about (co)inductive predicates (or
+ sets) and types, using the corresponding methods of the same name. Certain
+ definitional packages of object-logics usually declare emerging cases and
induction rules as expected, so users rarely need to intervene.
- Rules may be deleted via the \<open>del\<close> specification, which
- covers all of the \<open>type\<close>/\<open>pred\<close>/\<open>set\<close>
- sub-categories simultaneously. For example, @{attribute
- cases}~\<open>del\<close> removes any @{attribute cases} rules declared for
- some type, predicate, or set.
+ Rules may be deleted via the \<open>del\<close> specification, which covers all of the
+ \<open>type\<close>/\<open>pred\<close>/\<open>set\<close> sub-categories simultaneously. For example, @{attribute
+ cases}~\<open>del\<close> removes any @{attribute cases} rules declared for some type,
+ predicate, or set.
- Manual rule declarations usually refer to the @{attribute
- case_names} and @{attribute params} attributes to adjust names of
- cases and parameters of a rule; the @{attribute consumes}
- declaration is taken care of automatically: @{attribute
- consumes}~\<open>0\<close> is specified for ``type'' rules and @{attribute
+ Manual rule declarations usually refer to the @{attribute case_names} and
+ @{attribute params} attributes to adjust names of cases and parameters of a
+ rule; the @{attribute consumes} declaration is taken care of automatically:
+ @{attribute consumes}~\<open>0\<close> is specified for ``type'' rules and @{attribute
consumes}~\<open>1\<close> for ``predicate'' / ``set'' rules.
\<close>
@@ -1368,11 +1291,11 @@
@{command_def "guess"}\<open>\<^sup>*\<close> & : & \<open>proof(state) | proof(chain) \<rightarrow> proof(prove)\<close> \\
\end{matharray}
- Generalized elimination means that hypothetical parameters and premises
- may be introduced in the current context, potentially with a split into
- cases. This works by virtue of a locally proven rule that establishes the
- soundness of this temporary context extension. As representative examples,
- one may think of standard rules from Isabelle/HOL like this:
+ Generalized elimination means that hypothetical parameters and premises may
+ be introduced in the current context, potentially with a split into cases.
+ This works by virtue of a locally proven rule that establishes the soundness
+ of this temporary context extension. As representative examples, one may
+ think of standard rules from Isabelle/HOL like this:
\<^medskip>
\begin{tabular}{ll}
@@ -1383,9 +1306,9 @@
\<^medskip>
In general, these particular rules and connectives need to get involved at
- all: this concept works directly in Isabelle/Pure via Isar commands
- defined below. In particular, the logic of elimination and case splitting
- is delegated to an Isar proof, which often involves automated tools.
+ all: this concept works directly in Isabelle/Pure via Isar commands defined
+ below. In particular, the logic of elimination and case splitting is
+ delegated to an Isar proof, which often involves automated tools.
@{rail \<open>
@@{command consider} @{syntax obtain_clauses}
@@ -1396,21 +1319,21 @@
@@{command guess} (@{syntax "fixes"} + @'and')
\<close>}
- \<^descr> @{command consider}~\<open>(a) \<^vec>x \<WHERE> \<^vec>A \<^vec>x
- | (b) \<^vec>y \<WHERE> \<^vec>B \<^vec>y | \<dots>\<close> states a rule for case
- splitting into separate subgoals, such that each case involves new
- parameters and premises. After the proof is finished, the resulting rule
- may be used directly with the @{method cases} proof method
- (\secref{sec:cases-induct}), in order to perform actual case-splitting of
- the proof text via @{command case} and @{command next} as usual.
+ \<^descr> @{command consider}~\<open>(a) \<^vec>x \<WHERE> \<^vec>A \<^vec>x | (b)
+ \<^vec>y \<WHERE> \<^vec>B \<^vec>y | \<dots>\<close> states a rule for case splitting
+ into separate subgoals, such that each case involves new parameters and
+ premises. After the proof is finished, the resulting rule may be used
+ directly with the @{method cases} proof method (\secref{sec:cases-induct}),
+ in order to perform actual case-splitting of the proof text via @{command
+ case} and @{command next} as usual.
- Optional names in round parentheses refer to case names: in the proof of
- the rule this is a fact name, in the resulting rule it is used as
- annotation with the @{attribute_ref case_names} attribute.
+ Optional names in round parentheses refer to case names: in the proof of the
+ rule this is a fact name, in the resulting rule it is used as annotation
+ with the @{attribute_ref case_names} attribute.
\<^medskip>
- Formally, the command @{command consider} is defined as derived
- Isar language element as follows:
+ Formally, the command @{command consider} is defined as derived Isar
+ language element as follows:
\begin{matharray}{l}
@{command "consider"}~\<open>(a) \<^vec>x \<WHERE> \<^vec>A \<^vec>x | (b) \<^vec>y \<WHERE> \<^vec>B \<^vec>y | \<dots> \<equiv>\<close> \\[1ex]
@@ -1423,25 +1346,24 @@
\end{matharray}
See also \secref{sec:goals} for @{keyword "obtains"} in toplevel goal
- statements, as well as @{command print_statement} to print existing rules
- in a similar format.
+ statements, as well as @{command print_statement} to print existing rules in
+ a similar format.
- \<^descr> @{command obtain}~\<open>\<^vec>x \<WHERE> \<^vec>A \<^vec>x\<close>
- states a generalized elimination rule with exactly one case. After the
- proof is finished, it is activated for the subsequent proof text: the
- context is augmented via @{command fix}~\<open>\<^vec>x\<close> @{command
- assume}~\<open>\<^vec>A \<^vec>x\<close>, with special provisions to export
- later results by discharging these assumptions again.
+ \<^descr> @{command obtain}~\<open>\<^vec>x \<WHERE> \<^vec>A \<^vec>x\<close> states a
+ generalized elimination rule with exactly one case. After the proof is
+ finished, it is activated for the subsequent proof text: the context is
+ augmented via @{command fix}~\<open>\<^vec>x\<close> @{command assume}~\<open>\<^vec>A
+ \<^vec>x\<close>, with special provisions to export later results by discharging
+ these assumptions again.
Note that according to the parameter scopes within the elimination rule,
- results \<^emph>\<open>must not\<close> refer to hypothetical parameters; otherwise the
- export will fail! This restriction conforms to the usual manner of
- existential reasoning in Natural Deduction.
+ results \<^emph>\<open>must not\<close> refer to hypothetical parameters; otherwise the export
+ will fail! This restriction conforms to the usual manner of existential
+ reasoning in Natural Deduction.
\<^medskip>
- Formally, the command @{command obtain} is defined as derived
- Isar language element as follows, using an instrumented variant of
- @{command assume}:
+ Formally, the command @{command obtain} is defined as derived Isar language
+ element as follows, using an instrumented variant of @{command assume}:
\begin{matharray}{l}
@{command "obtain"}~\<open>\<^vec>x \<WHERE> a: \<^vec>A \<^vec>x \<langle>proof\<rangle> \<equiv>\<close> \\[1ex]
@@ -1459,23 +1381,23 @@
\<^emph>\<open>improper\<close>.
A proof with @{command guess} starts with a fixed goal \<open>thesis\<close>. The
- subsequent refinement steps may turn this to anything of the form \<open>\<And>\<^vec>x. \<^vec>A \<^vec>x \<Longrightarrow> thesis\<close>, but without splitting into new
- subgoals. The final goal state is then used as reduction rule for the
- obtain pattern described above. Obtained parameters \<open>\<^vec>x\<close> are
- marked as internal by default, and thus inaccessible in the proof text.
- The variable names and type constraints given as arguments for @{command
- "guess"} specify a prefix of accessible parameters.
+ subsequent refinement steps may turn this to anything of the form
+ \<open>\<And>\<^vec>x. \<^vec>A \<^vec>x \<Longrightarrow> thesis\<close>, but without splitting into new
+ subgoals. The final goal state is then used as reduction rule for the obtain
+ pattern described above. Obtained parameters \<open>\<^vec>x\<close> are marked as
+ internal by default, and thus inaccessible in the proof text. The variable
+ names and type constraints given as arguments for @{command "guess"} specify
+ a prefix of accessible parameters.
- In the proof of @{command consider} and @{command obtain} the local
- premises are always bound to the fact name @{fact_ref that}, according to
- structured Isar statements involving @{keyword_ref "if"}
- (\secref{sec:goals}).
+ In the proof of @{command consider} and @{command obtain} the local premises
+ are always bound to the fact name @{fact_ref that}, according to structured
+ Isar statements involving @{keyword_ref "if"} (\secref{sec:goals}).
- Facts that are established by @{command "obtain"} and @{command "guess"}
- may not be polymorphic: any type-variables occurring here are fixed in the
- present context. This is a natural consequence of the role of @{command
- fix} and @{command assume} in these constructs.
+ Facts that are established by @{command "obtain"} and @{command "guess"} may
+ not be polymorphic: any type-variables occurring here are fixed in the
+ present context. This is a natural consequence of the role of @{command fix}
+ and @{command assume} in these constructs.
\<close>
end