src/HOL/Library/Wfrec.thy

+(*  Title:      HOL/Library/Wfrec.thy
+    Author:     Tobias Nipkow
+    Author:     Lawrence C Paulson
+    Author:     Konrad Slind
+*)
+
+header {* Well-Founded Recursion Combinator *}
+
+theory Wfrec
+imports Main
+begin
+
+inductive
+  wfrec_rel :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b => bool"
+  for R :: "('a * 'a) set"
+  and F :: "('a => 'b) => 'a => 'b"
+where
+  wfrecI: "ALL z. (z, x) : R --> wfrec_rel R F z (g z) ==>
+            wfrec_rel R F x (F g x)"
+
+definition
+  cut        :: "('a => 'b) => ('a * 'a)set => 'a => 'a => 'b" where
+  "cut f r x == (%y. if (y,x):r then f y else undefined)"
+
+definition
+  adm_wf :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => bool" where
+  "adm_wf R F == ALL f g x.
+     (ALL z. (z, x) : R --> f z = g z) --> F f x = F g x"
+
+definition
+  wfrec :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b" where
+  "wfrec R F == %x. THE y. wfrec_rel R (%f x. F (cut f R x) x) x y"
+
+lemma cuts_eq: "(cut f r x = cut g r x) = (ALL y. (y,x):r --> f(y)=g(y))"
+by (simp add: fun_eq_iff cut_def)
+
+lemma cut_apply: "(x,a):r ==> (cut f r a)(x) = f(x)"
+by (simp add: cut_def)
+
+text{*Inductive characterization of wfrec combinator; for details see:
+John Harrison, "Inductive definitions: automation and application"*}
+
+lemma wfrec_unique: "[| adm_wf R F; wf R |] ==> EX! y. wfrec_rel R F x y"
+apply (simp add: adm_wf_def)
+apply (erule_tac a=x in wf_induct)
+apply (rule ex1I)
+apply (rule_tac g = "%x. THE y. wfrec_rel R F x y" in wfrec_rel.wfrecI)
+apply (fast dest!: theI')
+apply (erule wfrec_rel.cases, simp)
+apply (erule allE, erule allE, erule allE, erule mp)
+apply (fast intro: the_equality [symmetric])
+done
+
+lemma adm_lemma: "adm_wf R (%f x. F (cut f R x) x)"
+apply (simp add: adm_wf_def)
+apply (intro strip)
+apply (rule cuts_eq [THEN iffD2, THEN subst], assumption)
+apply (rule refl)
+done
+
+lemma wfrec: "wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a"
+apply (simp add: wfrec_def)
+apply (rule adm_lemma [THEN wfrec_unique, THEN the1_equality], assumption)
+apply (rule wfrec_rel.wfrecI)
+apply (intro strip)
+apply (erule adm_lemma [THEN wfrec_unique, THEN theI'])
+done
+
+
+text{** This form avoids giant explosions in proofs.  NOTE USE OF ==*}
+lemma def_wfrec: "[| f==wfrec r H;  wf(r) |] ==> f(a) = H (cut f r a) a"
+apply auto
+apply (blast intro: wfrec)
+done
+
+
+subsection {* Nitpick setup *}
+
+axiomatization wf_wfrec :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
+
+definition wf_wfrec' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
+[nitpick_simp]: "wf_wfrec' R F x = F (cut (wf_wfrec R F) R x) x"
+
+definition wfrec' ::  "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
+"wfrec' R F x \<equiv> if wf R then wf_wfrec' R F x
+                else THE y. wfrec_rel R (%f x. F (cut f R x) x) x y"
+
+setup {*
+  Nitpick_HOL.register_ersatz_global
+    [(@{const_name wf_wfrec}, @{const_name wf_wfrec'}),
+     (@{const_name wfrec}, @{const_name wfrec'})]
+*}
+
+hide_const (open) wf_wfrec wf_wfrec' wfrec'
+hide_fact (open) wf_wfrec'_def wfrec'_def
+
+subsection {* Wellfoundedness of @{text same_fst} *}
+
+definition
+ same_fst :: "('a => bool) => ('a => ('b * 'b)set) => (('a*'b)*('a*'b))set"
+where
+    "same_fst P R == {((x',y'),(x,y)) . x'=x & P x & (y',y) : R x}"
+   --{*For @{text rec_def} declarations where the first n parameters
+       stay unchanged in the recursive call. *}
+
+lemma same_fstI [intro!]:
+     "[| P x; (y',y) : R x |] ==> ((x,y'),(x,y)) : same_fst P R"
+by (simp add: same_fst_def)
+
+lemma wf_same_fst:
+  assumes prem: "(!!x. P x ==> wf(R x))"
+  shows "wf(same_fst P R)"
+apply (simp cong del: imp_cong add: wf_def same_fst_def)
+apply (intro strip)
+apply (rename_tac a b)
+apply (case_tac "wf (R a)")
+ apply (erule_tac a = b in wf_induct, blast)
+apply (blast intro: prem)
+done
+
+end