--- a/src/Doc/Implementation/Eq.thy Wed Dec 16 16:31:36 2015 +0100
+++ b/src/Doc/Implementation/Eq.thy Wed Dec 16 17:28:49 2015 +0100
@@ -6,52 +6,52 @@
chapter \<open>Equational reasoning\<close>
-text \<open>Equality is one of the most fundamental concepts of
- mathematics. The Isabelle/Pure logic (\chref{ch:logic}) provides a
- builtin relation \<open>\<equiv> :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> prop\<close> that expresses equality
- of arbitrary terms (or propositions) at the framework level, as
- expressed by certain basic inference rules (\secref{sec:eq-rules}).
+text \<open>
+ Equality is one of the most fundamental concepts of mathematics. The
+ Isabelle/Pure logic (\chref{ch:logic}) provides a builtin relation \<open>\<equiv> :: \<alpha> \<Rightarrow>
+ \<alpha> \<Rightarrow> prop\<close> that expresses equality of arbitrary terms (or propositions) at
+ the framework level, as expressed by certain basic inference rules
+ (\secref{sec:eq-rules}).
- Equational reasoning means to replace equals by equals, using
- reflexivity and transitivity to form chains of replacement steps,
- and congruence rules to access sub-structures. Conversions
- (\secref{sec:conv}) provide a convenient framework to compose basic
- equational steps to build specific equational reasoning tools.
+ Equational reasoning means to replace equals by equals, using reflexivity
+ and transitivity to form chains of replacement steps, and congruence rules
+ to access sub-structures. Conversions (\secref{sec:conv}) provide a
+ convenient framework to compose basic equational steps to build specific
+ equational reasoning tools.
- Higher-order matching is able to provide suitable instantiations for
- giving equality rules, which leads to the versatile concept of
- \<open>\<lambda>\<close>-term rewriting (\secref{sec:rewriting}). Internally
- this is based on the general-purpose Simplifier engine of Isabelle,
- which is more specific and more efficient than plain conversions.
+ Higher-order matching is able to provide suitable instantiations for giving
+ equality rules, which leads to the versatile concept of \<open>\<lambda>\<close>-term rewriting
+ (\secref{sec:rewriting}). Internally this is based on the general-purpose
+ Simplifier engine of Isabelle, which is more specific and more efficient
+ than plain conversions.
- Object-logics usually introduce specific notions of equality or
- equivalence, and relate it with the Pure equality. This enables to
- re-use the Pure tools for equational reasoning for particular
- object-logic connectives as well.
+ Object-logics usually introduce specific notions of equality or equivalence,
+ and relate it with the Pure equality. This enables to re-use the Pure tools
+ for equational reasoning for particular object-logic connectives as well.
\<close>
section \<open>Basic equality rules \label{sec:eq-rules}\<close>
-text \<open>Isabelle/Pure uses \<open>\<equiv>\<close> for equality of arbitrary
- terms, which includes equivalence of propositions of the logical
- framework. The conceptual axiomatization of the constant \<open>\<equiv>
- :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> prop\<close> is given in \figref{fig:pure-equality}. The
- inference kernel presents slightly different equality rules, which
- may be understood as derived rules from this minimal axiomatization.
- The Pure theory also provides some theorems that express the same
- reasoning schemes as theorems that can be composed like object-level
+text \<open>
+ Isabelle/Pure uses \<open>\<equiv>\<close> for equality of arbitrary terms, which includes
+ equivalence of propositions of the logical framework. The conceptual
+ axiomatization of the constant \<open>\<equiv> :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> prop\<close> is given in
+ \figref{fig:pure-equality}. The inference kernel presents slightly different
+ equality rules, which may be understood as derived rules from this minimal
+ axiomatization. The Pure theory also provides some theorems that express the
+ same reasoning schemes as theorems that can be composed like object-level
rules as explained in \secref{sec:obj-rules}.
- For example, @{ML Thm.symmetric} as Pure inference is an ML function
- that maps a theorem \<open>th\<close> stating \<open>t \<equiv> u\<close> to one
- stating \<open>u \<equiv> t\<close>. In contrast, @{thm [source]
- Pure.symmetric} as Pure theorem expresses the same reasoning in
- declarative form. If used like \<open>th [THEN Pure.symmetric]\<close>
- in Isar source notation, it achieves a similar effect as the ML
- inference function, although the rule attribute @{attribute THEN} or
- ML operator @{ML "op RS"} involve the full machinery of higher-order
- unification (modulo \<open>\<beta>\<eta>\<close>-conversion) and lifting of \<open>\<And>/\<Longrightarrow>\<close> contexts.\<close>
+ For example, @{ML Thm.symmetric} as Pure inference is an ML function that
+ maps a theorem \<open>th\<close> stating \<open>t \<equiv> u\<close> to one stating \<open>u \<equiv> t\<close>. In contrast,
+ @{thm [source] Pure.symmetric} as Pure theorem expresses the same reasoning
+ in declarative form. If used like \<open>th [THEN Pure.symmetric]\<close> in Isar source
+ notation, it achieves a similar effect as the ML inference function,
+ although the rule attribute @{attribute THEN} or ML operator @{ML "op RS"}
+ involve the full machinery of higher-order unification (modulo
+ \<open>\<beta>\<eta>\<close>-conversion) and lifting of \<open>\<And>/\<Longrightarrow>\<close> contexts.
+\<close>
text %mlref \<open>
\begin{mldecls}
@@ -64,11 +64,11 @@
@{index_ML Thm.equal_elim: "thm -> thm -> thm"} \\
\end{mldecls}
- See also @{file "~~/src/Pure/thm.ML" } for further description of
- these inference rules, and a few more for primitive \<open>\<beta>\<close> and
- \<open>\<eta>\<close> conversions. Note that \<open>\<alpha>\<close> conversion is
- implicit due to the representation of terms with de-Bruijn indices
- (\secref{sec:terms}).\<close>
+ See also @{file "~~/src/Pure/thm.ML" } for further description of these
+ inference rules, and a few more for primitive \<open>\<beta>\<close> and \<open>\<eta>\<close> conversions. Note
+ that \<open>\<alpha>\<close> conversion is implicit due to the representation of terms with
+ de-Bruijn indices (\secref{sec:terms}).
+\<close>
section \<open>Conversions \label{sec:conv}\<close>
@@ -76,19 +76,19 @@
text \<open>
%FIXME
- The classic article that introduces the concept of conversion (for
- Cambridge LCF) is @{cite "paulson:1983"}.
+ The classic article that introduces the concept of conversion (for Cambridge
+ LCF) is @{cite "paulson:1983"}.
\<close>
section \<open>Rewriting \label{sec:rewriting}\<close>
-text \<open>Rewriting normalizes a given term (theorem or goal) by
- replacing instances of given equalities \<open>t \<equiv> u\<close> in subterms.
- Rewriting continues until no rewrites are applicable to any subterm.
- This may be used to unfold simple definitions of the form \<open>f
- x\<^sub>1 \<dots> x\<^sub>n \<equiv> u\<close>, but is slightly more general than that.
-\<close>
+text \<open>
+ Rewriting normalizes a given term (theorem or goal) by replacing instances
+ of given equalities \<open>t \<equiv> u\<close> in subterms. Rewriting continues until no
+ rewrites are applicable to any subterm. This may be used to unfold simple
+ definitions of the form \<open>f x\<^sub>1 \<dots> x\<^sub>n \<equiv> u\<close>, but is slightly more general than
+ that. \<close>
text %mlref \<open>
\begin{mldecls}
@@ -99,24 +99,24 @@
@{index_ML fold_goals_tac: "Proof.context -> thm list -> tactic"} \\
\end{mldecls}
- \<^descr> @{ML rewrite_rule}~\<open>ctxt rules thm\<close> rewrites the whole
- theorem by the given rules.
+ \<^descr> @{ML rewrite_rule}~\<open>ctxt rules thm\<close> rewrites the whole theorem by the
+ given rules.
- \<^descr> @{ML rewrite_goals_rule}~\<open>ctxt rules thm\<close> rewrites the
- outer premises of the given theorem. Interpreting the same as a
- goal state (\secref{sec:tactical-goals}) it means to rewrite all
- subgoals (in the same manner as @{ML rewrite_goals_tac}).
+ \<^descr> @{ML rewrite_goals_rule}~\<open>ctxt rules thm\<close> rewrites the outer premises of
+ the given theorem. Interpreting the same as a goal state
+ (\secref{sec:tactical-goals}) it means to rewrite all subgoals (in the same
+ manner as @{ML rewrite_goals_tac}).
- \<^descr> @{ML rewrite_goal_tac}~\<open>ctxt rules i\<close> rewrites subgoal
- \<open>i\<close> by the given rewrite rules.
+ \<^descr> @{ML rewrite_goal_tac}~\<open>ctxt rules i\<close> rewrites subgoal \<open>i\<close> by the given
+ rewrite rules.
- \<^descr> @{ML rewrite_goals_tac}~\<open>ctxt rules\<close> rewrites all subgoals
- by the given rewrite rules.
+ \<^descr> @{ML rewrite_goals_tac}~\<open>ctxt rules\<close> rewrites all subgoals by the given
+ rewrite rules.
- \<^descr> @{ML fold_goals_tac}~\<open>ctxt rules\<close> essentially uses @{ML
- rewrite_goals_tac} with the symmetric form of each member of \<open>rules\<close>, re-ordered to fold longer expression first. This supports
- to idea to fold primitive definitions that appear in expended form
- in the proof state.
+ \<^descr> @{ML fold_goals_tac}~\<open>ctxt rules\<close> essentially uses @{ML rewrite_goals_tac}
+ with the symmetric form of each member of \<open>rules\<close>, re-ordered to fold longer
+ expression first. This supports to idea to fold primitive definitions that
+ appear in expended form in the proof state.
\<close>
end