--- a/CookBook/Tactical.thy	Wed Feb 25 10:09:18 2009 +0000
+++ b/CookBook/Tactical.thy	Wed Feb 25 10:13:10 2009 +0000
@@ -1,5 +1,6 @@
 theory Tactical
 imports Base FirstSteps
+uses "infix_conv.ML"
 begin
 
 chapter {* Tactical Reasoning\label{chp:tactical} *}
@@ -1317,96 +1318,157 @@
 section {* Conversions\label{sec:conversion} *}
 
 text {*
-conversions: core of the simplifier
-
-see: Pure/conv.ML
-*}
-
-ML {* type conv = Thm.cterm -> Thm.thm *}
-
-text {*
-simple example:
-*}
-
-lemma true_conj_1: "True \<and> P \<equiv> P" by simp
-
-ML {*
-val true1_conv = Conv.rewr_conv @{thm true_conj_1}
-*}
-
-ML {*
-true1_conv @{cterm "True \<and> False"}
+Conversions are meta-equalities depending on some input term. Their type is
+as follows:
 *}
 
-text {*
-@{ML_response_fake "true1_conv @{cterm \"True \<and> False\"}"
-  "True \<and> False \<equiv> False"}
-*}
+ML {*type conv = Thm.cterm -> Thm.thm*}
 
 text {*
-how to extend rewriting to more complex cases? e.g.:
+The simplest two conversions are @{ML "Conv.all_conv"}, which maps a term to an instance of the reflexivity theorem, and @{ML "Conv.no_conv"}, which always fails:
+
+@{ML_response_fake "Conv.all_conv @{cterm True}" "True \<equiv> True"}
+
+@{ML_response_fake "Conv.no_conv @{cterm True}" "*** Exception- CTERM (\"no conversion\", []) raised"}
+
+A further basic conversion is, for example, @{ML "Thm.beta_conversion"}:
+
+@{ML_response_fake "Thm.beta_conversion true @{cterm \"(\<lambda>x. x \<or> False) True\"}"
+"(\<lambda>x. x \<or> False) True \<equiv> True \<or> False"}
+
+User-defined rewrite rules can be applied by the conversion
+@{ML "Conv.rewr_conv"}. Consider, for example, the following rule:
 *}
 
-ML {*
-val complex = @{cterm "distinct [1, x] \<longrightarrow> True \<and> 1 \<noteq> (x::int)"}
-*}
-
-text {* conversionals: basically a count-down version of MetaSimplifier.rewrite *}
+lemma true_conj1: "True \<and> P \<equiv> P" by simp
 
-ML {*
-fun all_true1_conv ctxt ct =
-  (case Thm.term_of ct of
-    @{term "op \<and>"} $ @{term True} $ _ => true1_conv ct
-  | _ $ _ => Conv.combination_conv (all_true1_conv ctxt) (all_true1_conv ctxt) ct
-  | Abs _ => Conv.abs_conv (fn (_, cx) => all_true1_conv cx) ctxt ct)
+text {*
+Here is how this rule can be used for rewriting:
+
+@{ML_response_fake "Conv.rewr_conv @{thm true_conj1} @{cterm \"True \<or> False\"}"
+ "True \<and> False \<equiv> False"}
 *}
 
 text {*
-@{ML_response_fake "all_true1_conv @{context} complex"
-  "distinct [1, x] \<longrightarrow> True \<and> 1 \<noteq> x \<equiv> distinct [1, x] \<longrightarrow> 1 \<noteq> x"}
+Basic conversions can be combined with a number of conversionals, i.e.
+conversion combinators:
+
+@{ML_response_fake
+"(Thm.beta_conversion true then_conv Conv.rewr_conv @{thm true_conj1})
+  @{cterm \"(\<lambda>x. x \<and> False) True\"}"
+"(\<lambda>x. x \<and> False) True \<equiv> False"}
+
+@{ML_response_fake
+"(Conv.rewr_conv @{thm true_conj1} else_conv Conv.all_conv)
+  @{cterm \"P \<or> (True \<and> Q)\"}"
+"P \<or> (True \<and> Q) \<equiv> P \<or> (True \<and> Q)"}
+
+@{ML_response_fake
+"Conv.arg_conv (Conv.rewr_conv @{thm true_conj1})
+  @{cterm \"P \<or> (True \<and> Q)\"}"
+"P \<or> (True \<and> Q) \<equiv> P \<or> Q"}
+
+\begin{readmore}
+See @{ML_file "Pure/conv.ML"} for more conversionals. Further basic conversions
+can be found in, for example, @{ML_file "Pure/thm.ML"},
+@{ML_file "Pure/drule.ML"}, and @{ML_file "Pure/meta_simplifier.ML"}.
+\end{readmore}
+
+Conversions are a thin layer on top of Isabelle's inference kernel, and may
+be seen as a controllable, bare-bone version of Isabelle's simplifier. We
+will demonstrate this feature in the following example.
+
+To begin with, let's assumes we want to simplify with the rewrite rule
+@{text true_conj1}. As a conversion, this may look as follows:
 *}
 
+ML {*fun all_true1_conv ctxt ct =
+  (case Thm.term_of ct of
+    @{term "op \<and>"} $ @{term True} $ _ => 
+      (Conv.arg_conv (all_true1_conv ctxt) then_conv
+        Conv.rewr_conv @{thm true_conj1}) ct
+  | _ $ _ => Conv.combination_conv (all_true1_conv ctxt)
+      (all_true1_conv ctxt) ct
+  | Abs _ => Conv.abs_conv (fn (_, cx) => all_true1_conv cx) ctxt ct
+  | _ => Conv.all_conv ct)*}
 
 text {*
-transforming theorems: @{ML "Conv.fconv_rule"} (give type)
-
-example:
-*}
+Here is this conversion in action:
 
-lemma foo: "True \<and> (P \<or> \<not>P)" by simp
+@{ML_response_fake
+"all_true1_conv @{context}
+  @{cterm \"distinct [1, x] \<longrightarrow> True \<and> 1 \<noteq> x\"}"
+"distinct [1, x] \<longrightarrow> True \<and> 1 \<noteq> x \<equiv> distinct [1, x] \<longrightarrow> 1 \<noteq> x"}
 
-text {*
-@{ML_response_fake "Conv.fconv_rule (Conv.arg_conv (all_true1_conv @{context})) @{thm foo}" "P \<or> \<not>P"}
+Now, let's complicate the task a bit: Rewrite according to the rule
+@{text true_conj1}, but only in the first arguments of @{term If}:
 *}
 
-text {* now, replace "True and P" with "P" if it occurs inside an If *}
-
-ML {*
-fun if_true1_conv ctxt ct =
+ML {*fun if_true1_conv ctxt ct =
   (case Thm.term_of ct of
-    Const (@{const_name If}, _) $ _ => Conv.arg_conv (all_true1_conv ctxt) ct
-  | _ $ _ => Conv.combination_conv (if_true1_conv ctxt) (if_true1_conv ctxt) ct
-  | Abs _ => Conv.abs_conv (fn (_, cx) => if_true1_conv cx) ctxt ct)
+    Const (@{const_name If}, _) $ _ =>
+      Conv.arg_conv (all_true1_conv ctxt) ct
+  | _ $ _ => Conv.combination_conv (if_true1_conv ctxt)
+      (if_true1_conv ctxt) ct
+  | Abs _ => Conv.abs_conv (fn (_, cx) => if_true1_conv cx) ctxt ct
+  | _ => Conv.all_conv ct)*}
+
+text {*
+Here is an application of this conversion:
+
+@{ML_response_fake
+"if_true1_conv @{context}
+  @{cterm \"if P (True \<and> 1 \<noteq> 2) then True \<and> True else True \<and> False\"}"
+"if P (True \<and> 1 \<noteq> 2) then True \<and> True else True \<and> False \<equiv> if P (1 \<noteq> 2) then True \<and> True else True \<and> False"}
 *}
 
-text {* show example *}
-
 text {*
-conversions inside a tactic: replace "true" in conclusion, "if true" in assumptions
+Conversions can also be turned into a tactic with the @{ML CONVERSION}
+tactical, and there are predefined conversionals to traverse a goal state.
+Given a state @{term "\<And>x. P \<Longrightarrow> Q"},  the conversional
+@{ML Conv.params_conv} applies a conversion to @{term "P \<Longrightarrow> Q"};
+given a state @{term "\<lbrakk> P1; P2 \<rbrakk> \<Longrightarrow> Q"},
+the conversional @{ML Conv.prems_conv} applies a conversion to the premises
+@{term P1} and @{term P2}, and @{ML Conv.concl_conv} applies a conversion to
+the conclusion @{term Q}.
+
+Assume we want to apply @{ML all_true1_conv} only in the conclusion
+of the goal, and @{ML if_true1_conv} should only be applied in the premises.
+Here is a tactic doing exactly that:
 *}
 
-ML {*
-local open Conv 
-in
-fun true1_tac ctxt =
-  CONVERSION (
-    Conv.params_conv ~1 (fn cx =>
-      (Conv.prems_conv ~1 (if_true1_conv cx)) then_conv
-        (Conv.concl_conv ~1 (all_true1_conv cx))) ctxt)
-end
+ML {*val true1_tac = CSUBGOAL (fn (goal, i) =>
+  let val ctxt = ProofContext.init (Thm.theory_of_cterm goal)
+  in
+    CONVERSION (
+      Conv.params_conv ~1 (fn cx =>
+        (Conv.prems_conv ~1 (if_true1_conv cx)) then_conv
+          Conv.concl_conv ~1 (all_true1_conv cx)) ctxt) i
+  end)*}
+
+text {* 
+To demonstrate this tactic, consider the following example:
 *}
 
-text {* give example, show that the conditional rewriting works *}
+lemma
+  "if True \<and> P then P else True \<and> False \<Longrightarrow>
+  (if True \<and> Q then Q else P) \<longrightarrow> True \<and> (True \<and> Q)"
+apply(tactic {* true1_tac 1 *})
+txt {* \begin{minipage}{\textwidth}
+       @{subgoals [display]} 
+       \end{minipage} *}
+(*<*)oops(*>*)
+
+text {*
+Conversions are not restricted to work on certified terms only, they can also
+be lifted to theorem transformers, i.e. functions mapping a theorem to a
+theorem, by the help of @{ML Conv.fconv_rule}. As an example, consider the
+conversion @{ML all_true1_conv} again:
+
+@{ML_response_fake
+"Conv.fconv_rule (all_true1_conv @{context})
+  @{lemma \"P \<or> (True \<and> \<not>P)\" by simp}" "P \<or> \<not>P"}
+*}
 
 
 
@@ -1453,4 +1515,4 @@
 
 
 
-end
\ No newline at end of file
+end

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/CookBook/infix_conv.ML	Wed Feb 25 10:13:10 2009 +0000
@@ -0,0 +1,9 @@
+signature INFIX_CONV =
+sig
+  val then_conv : Thm.conv * Thm.conv -> Thm.conv
+  val else_conv : Thm.conv * Thm.conv -> Thm.conv
+end
+
+structure InfixConv : INFIX_CONV = Conv
+
+open InfixConv

Binary file cookbook.pdf has changed