# HG changeset patch
# User Andreas Lochbihler
# Date 1363695561 -3600
# Node ID bc3651180a09eef8a6ea9cc92307f01faaac1de4
# Parent  67542078fa2159f2e0b5d076d9eb1313142616f0
add induction rule for partial_function (tailrec)

diff -r 67542078fa21 -r bc3651180a09 src/HOL/Partial_Function.thy
--- a/src/HOL/Partial_Function.thy	Mon Mar 18 20:02:37 2013 +0100
+++ b/src/HOL/Partial_Function.thy	Tue Mar 19 13:19:21 2013 +0100
@@ -244,6 +244,47 @@
 by (rule flat_interpretation)
 
 
+abbreviation "tailrec_ord \<equiv> flat_ord undefined"
+abbreviation "mono_tailrec \<equiv> monotone (fun_ord tailrec_ord) tailrec_ord"
+
+lemma tailrec_admissible:
+  "ccpo.admissible (fun_lub (flat_lub undefined)) (fun_ord tailrec_ord)
+     (\<lambda>a. \<forall>x. a x \<noteq> undefined \<longrightarrow> P x (a x))"
+proof(intro ccpo.admissibleI[OF tailrec.ccpo] strip)
+  fix A x
+  assume chain: "Complete_Partial_Order.chain (fun_ord tailrec_ord) A"
+    and P [rule_format]: "\<forall>f\<in>A. \<forall>x. f x \<noteq> undefined \<longrightarrow> P x (f x)"
+    and defined: "fun_lub (flat_lub undefined) A x \<noteq> undefined"
+  from defined obtain f where f: "f \<in> A" "f x \<noteq> undefined"
+    by(auto simp add: fun_lub_def flat_lub_def split: split_if_asm)
+  hence "P x (f x)" by(rule P)
+  moreover from chain f have "\<forall>f' \<in> A. f' x = undefined \<or> f' x = f x"
+    by(auto 4 4 simp add: Complete_Partial_Order.chain_def flat_ord_def fun_ord_def)
+  hence "fun_lub (flat_lub undefined) A x = f x"
+    using f by(auto simp add: fun_lub_def flat_lub_def)
+  ultimately show "P x (fun_lub (flat_lub undefined) A x)" by simp
+qed
+
+lemma fixp_induct_tailrec:
+  fixes F :: "'c \<Rightarrow> 'c" and
+    U :: "'c \<Rightarrow> 'b \<Rightarrow> 'a" and
+    C :: "('b \<Rightarrow> 'a) \<Rightarrow> 'c" and
+    P :: "'b \<Rightarrow> 'a \<Rightarrow> bool" and
+    x :: "'b"
+  assumes mono: "\<And>x. mono_tailrec (\<lambda>f. U (F (C f)) x)"
+  assumes eq: "f \<equiv> C (ccpo.fixp (fun_lub (flat_lub undefined)) (fun_ord tailrec_ord) (\<lambda>f. U (F (C f))))"
+  assumes inverse2: "\<And>f. U (C f) = f"
+  assumes step: "\<And>f x y. (\<And>x y. U f x = y \<Longrightarrow> y \<noteq> undefined \<Longrightarrow> P x y) \<Longrightarrow> U (F f) x = y \<Longrightarrow> y \<noteq> undefined \<Longrightarrow> P x y"
+  assumes result: "U f x = y"
+  assumes defined: "y \<noteq> undefined"
+  shows "P x y"
+proof -
+  have "\<forall>x y. U f x = y \<longrightarrow> y \<noteq> undefined \<longrightarrow> P x y"
+    by(rule tailrec.fixp_induct_uc[of U F C, OF mono eq inverse2])(auto intro: step tailrec_admissible)
+  thus ?thesis using result defined by blast
+qed
+
+
 abbreviation "option_ord \<equiv> flat_ord None"
 abbreviation "mono_option \<equiv> monotone (fun_ord option_ord) option_ord"
 
@@ -316,13 +357,13 @@
   by blast
 
 declaration {* Partial_Function.init "tailrec" @{term tailrec.fixp_fun}
-  @{term tailrec.mono_body} @{thm tailrec.fixp_rule_uc} NONE *}
+  @{term tailrec.mono_body} @{thm tailrec.fixp_rule_uc} 
+  (SOME @{thm fixp_induct_tailrec}) *}
 
 declaration {* Partial_Function.init "option" @{term option.fixp_fun}
   @{term option.mono_body} @{thm option.fixp_rule_uc} 
   (SOME @{thm fixp_induct_option}) *}
 
-
 hide_const (open) chain
 
 end