
diff -r 804927db5311 -r 98acc6d0fab6 src/CCL/Fix.thy
--- a/src/CCL/Fix.thy	Mon Jul 17 18:42:38 2006 +0200
+++ b/src/CCL/Fix.thy	Tue Jul 18 02:22:38 2006 +0200
@@ -22,6 +22,181 @@
   po_INCL:    "INCL(%x. a(x) [= b(x))"
   INCL_subst: "INCL(P) ==> INCL(%x. P((g::i=>i)(x)))"
 
-ML {* use_legacy_bindings (the_context ()) *}
+
+subsection {* Fixed Point Induction *}
+
+lemma fix_ind:
+  assumes base: "P(bot)"
+    and step: "!!x. P(x) ==> P(f(x))"
+    and incl: "INCL(P)"
+  shows "P(fix(f))"
+  apply (rule incl [unfolded INCL_def, rule_format])
+  apply (rule Nat_ind [THEN ballI], assumption)
+   apply simp_all
+   apply (rule base)
+  apply (erule step)
+  done
+
+
+subsection {* Inclusive Predicates *}
+
+lemma inclXH: "INCL(P) <-> (ALL f. (ALL n:Nat. P(f ^ n ` bot)) --> P(fix(f)))"
+  by (simp add: INCL_def)
+
+lemma inclI: "[| !!f. ALL n:Nat. P(f^n`bot) ==> P(fix(f)) |] ==> INCL(%x. P(x))"
+  unfolding inclXH by blast
+
+lemma inclD: "[| INCL(P);  !!n. n:Nat ==> P(f^n`bot) |] ==> P(fix(f))"
+  unfolding inclXH by blast
+
+lemma inclE: "[| INCL(P);  (ALL n:Nat. P(f^n`bot))-->P(fix(f)) ==> R |] ==> R"
+  by (blast dest: inclD)
+
+
+subsection {* Lemmas for Inclusive Predicates *}
+
+lemma npo_INCL: "INCL(%x.~ a(x) [= t)"
+  apply (rule inclI)
+  apply (drule bspec)
+   apply (rule zeroT)
+  apply (erule contrapos)
+  apply (rule po_trans)
+   prefer 2
+   apply assumption
+  apply (subst napplyBzero)
+  apply (rule po_cong, rule po_bot)
+  done
+
+lemma conj_INCL: "[| INCL(P);  INCL(Q) |] ==> INCL(%x. P(x) & Q(x))"
+  by (blast intro!: inclI dest!: inclD)
+
+lemma all_INCL: "[| !!a. INCL(P(a)) |] ==> INCL(%x. ALL a. P(a,x))"
+  by (blast intro!: inclI dest!: inclD)
+
+lemma ball_INCL: "[| !!a. a:A ==> INCL(P(a)) |] ==> INCL(%x. ALL a:A. P(a,x))"
+  by (blast intro!: inclI dest!: inclD)
+
+lemma eq_INCL: "INCL(%x. a(x) = (b(x)::'a::prog))"
+  apply (simp add: eq_iff)
+  apply (rule conj_INCL po_INCL)+
+  done
+
+
+subsection {* Derivation of Reachability Condition *}
+
+(* Fixed points of idgen *)
+
+lemma fix_idgenfp: "idgen(fix(idgen)) = fix(idgen)"
+  apply (rule fixB [symmetric])
+  done
+
+lemma id_idgenfp: "idgen(lam x. x) = lam x. x"
+  apply (simp add: idgen_def)
+  apply (rule term_case [THEN allI])
+      apply simp_all
+  done
+
+(* All fixed points are lam-expressions *)
+
+lemma idgenfp_lam: "idgen(d) = d ==> d = lam x. ?f(x)"
+  apply (unfold idgen_def)
+  apply (erule ssubst)
+  apply (rule refl)
+  done
+
+(* Lemmas for rewriting fixed points of idgen *)
+
+lemma l_lemma: "[| a = b;  a ` t = u |] ==> b ` t = u"
+  by (simp add: idgen_def)
+
+lemma idgen_lemmas:
+  "idgen(d) = d ==> d ` bot = bot"
+  "idgen(d) = d ==> d ` true = true"
+  "idgen(d) = d ==> d ` false = false"
+  "idgen(d) = d ==> d ` <a,b> = <d ` a,d ` b>"
+  "idgen(d) = d ==> d ` (lam x. f(x)) = lam x. d ` f(x)"
+  by (erule l_lemma, simp add: idgen_def)+
+
+
+(* Proof of Reachability law - show that fix and lam x.x both give LEAST fixed points
+  of idgen and hence are they same *)
+
+lemma po_eta:
+  "[| ALL x. t ` x [= u ` x;  EX f. t=lam x. f(x);  EX f. u=lam x. f(x) |] ==> t [= u"
+  apply (drule cond_eta)+
+  apply (erule ssubst)
+  apply (erule ssubst)
+  apply (rule po_lam [THEN iffD2])
+  apply simp
+  done
+
+lemma po_eta_lemma: "idgen(d) = d ==> d = lam x. ?f(x)"
+  apply (unfold idgen_def)
+  apply (erule sym)
+  done
+
+lemma lemma1:
+  "idgen(d) = d ==>
+    {p. EX a b. p=<a,b> & (EX t. a=fix(idgen) ` t & b = d ` t)} <=
+      POgen({p. EX a b. p=<a,b> & (EX t. a=fix(idgen) ` t  & b = d ` t)})"
+  apply clarify
+  apply (rule_tac t = t in term_case)
+      apply (simp_all add: POgenXH idgen_lemmas idgen_lemmas [OF fix_idgenfp])
+   apply blast
+  apply fast
+  done
+
+lemma fix_least_idgen: "idgen(d) = d ==> fix(idgen) [= d"
+  apply (rule allI [THEN po_eta])
+    apply (rule lemma1 [THEN [2] po_coinduct])
+     apply (blast intro: po_eta_lemma fix_idgenfp)+
+  done
+
+lemma lemma2:
+  "idgen(d) = d ==>
+    {p. EX a b. p=<a,b> & b = d ` a} <= POgen({p. EX a b. p=<a,b> & b = d ` a})"
+  apply clarify
+  apply (rule_tac t = a in term_case)
+      apply (simp_all add: POgenXH idgen_lemmas)
+  apply fast
+  done
+
+lemma id_least_idgen: "idgen(d) = d ==> lam x. x [= d"
+  apply (rule allI [THEN po_eta])
+    apply (rule lemma2 [THEN [2] po_coinduct])
+     apply simp
+    apply (fast intro: po_eta_lemma fix_idgenfp)+
+  done
+
+lemma reachability: "fix(idgen) = lam x. x"
+  apply (fast intro: eq_iff [THEN iffD2]
+    id_idgenfp [THEN fix_least_idgen] fix_idgenfp [THEN id_least_idgen])
+  done
+
+(********)
+
+lemma id_apply: "f = lam x. x ==> f`t = t"
+  apply (erule ssubst)
+  apply (rule applyB)
+  done
+
+lemma term_ind:
+  assumes "P(bot)" "P(true)" "P(false)"
+    "!!x y.[| P(x);  P(y) |] ==> P(<x,y>)"
+    "!!u.(!!x. P(u(x))) ==> P(lam x. u(x))"  "INCL(P)"
+  shows "P(t)"
+  apply (rule reachability [THEN id_apply, THEN subst])
+  apply (rule_tac x = t in spec)
+  apply (rule fix_ind)
+    apply (unfold idgen_def)
+    apply (rule allI)
+    apply (subst applyBbot)
+    apply assumption
+   apply (rule allI)
+   apply (rule applyB [THEN ssubst])
+    apply (rule_tac t = "xa" in term_case)
+       apply simp_all
+       apply (fast intro: prems INCL_subst all_INCL)+
+  done
 
 end




