doc-src/IsarImplementation/Thy/Eq.thy

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+++ b/doc-src/IsarImplementation/Thy/Eq.thy	Thu Feb 09 19:34:23 2012 +0100
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+theory Eq
+imports Base
+begin
+
+chapter {* Equational reasoning *}
+
+text {* Equality is one of the most fundamental concepts of
+  mathematics.  The Isabelle/Pure logic (\chref{ch:logic}) provides a
+  builtin relation @{text "\<equiv> :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow> prop"} 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.
+
+  Higher-order matching is able to provide suitable instantiations for
+  giving equality rules, which leads to the versatile concept of
+  @{text "\<lambda>"}-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.
+*}
+
+
+section {* Basic equality rules \label{sec:eq-rules} *}
+
+text {* FIXME *}
+
+
+section {* Conversions \label{sec:conv} *}
+
+text {* FIXME *}
+
+
+section {* Rewriting \label{sec:rewriting} *}
+
+text {* Rewriting normalizes a given term (theorem or goal) by
+  replacing instances of given equalities @{text "t \<equiv> u"} in subterms.
+  Rewriting continues until no rewrites are applicable to any subterm.
+  This may be used to unfold simple definitions of the form @{text "f
+  x\<^sub>1 \<dots> x\<^sub>n \<equiv> u"}, but is slightly more general than that.
+*}
+
+text %mlref {*
+  \begin{mldecls}
+  @{index_ML rewrite_goal_tac: "thm list -> int -> tactic"} \\
+  @{index_ML rewrite_goals_tac: "thm list -> tactic"} \\
+  @{index_ML fold_goals_tac: "thm list -> tactic"} \\
+  \end{mldecls}
+
+  \begin{description}
+
+  \item @{ML rewrite_goal_tac}~@{text "rules i"} rewrites subgoal
+  @{text "i"} by the given rewrite rules.
+
+  \item @{ML rewrite_goals_tac}~@{text "rules"} rewrites all subgoals
+  by the given rewrite rules.
+
+  \item @{ML fold_goals_tac}~@{text "rules"} essentially uses @{ML
+  rewrite_goals_tac} with the symmetric form of each member of @{text
+  "rules"}, re-ordered to fold longer expression first.  This supports
+  to idea to fold primitive definitions that appear in expended form
+  in the proof state.
+
+  \end{description}
+*}
+
+end