--- a/src/LCF/LCF.thy Thu Jun 01 21:14:54 2006 +0200
+++ b/src/LCF/LCF.thy Thu Jun 01 23:07:51 2006 +0200
@@ -1,4 +1,4 @@
-(* Title: LCF/lcf.thy
+(* Title: LCF/LCF.thy
ID: $Id$
Author: Tobias Nipkow
Copyright 1992 University of Cambridge
@@ -8,7 +8,6 @@
theory LCF
imports FOL
-uses ("LCF_lemmas.ML") ("pair.ML") ("fix.ML")
begin
text {* This theory is based on Lawrence Paulson's book Logic and Computation. *}
@@ -118,18 +117,277 @@
adm_imp: "[| adm(%x.~P(x)); adm(Q) |] ==> adm(%x. P(x)-->Q(x))"
adm_all: "(!!y. adm(P(y))) ==> adm(%x. ALL y. P(y,x))"
-ML {* use_legacy_bindings (the_context ()) *}
+
+lemma eq_imp_less1: "x = y ==> x << y"
+ by (simp add: eq_def)
+
+lemma eq_imp_less2: "x = y ==> y << x"
+ by (simp add: eq_def)
+
+lemma less_refl [simp]: "x << x"
+ apply (rule eq_imp_less1)
+ apply (rule refl)
+ done
+
+lemma less_anti_sym: "[| x << y; y << x |] ==> x=y"
+ by (simp add: eq_def)
+
+lemma ext: "(!!x::'a::cpo. f(x)=(g(x)::'b::cpo)) ==> (%x. f(x))=(%x. g(x))"
+ apply (rule less_anti_sym)
+ apply (rule less_ext)
+ apply simp
+ apply simp
+ done
+
+lemma cong: "[| f=g; x=y |] ==> f(x)=g(y)"
+ by simp
+
+lemma less_ap_term: "x << y ==> f(x) << f(y)"
+ by (rule less_refl [THEN mono])
+
+lemma less_ap_thm: "f << g ==> f(x) << g(x)"
+ by (rule less_refl [THEN [2] mono])
+
+lemma ap_term: "(x::'a::cpo) = y ==> (f(x)::'b::cpo) = f(y)"
+ apply (rule cong [OF refl])
+ apply simp
+ done
+
+lemma ap_thm: "f = g ==> f(x) = g(x)"
+ apply (erule cong)
+ apply (rule refl)
+ done
+
+
+lemma UU_abs: "(%x::'a::cpo. UU) = UU"
+ apply (rule less_anti_sym)
+ prefer 2
+ apply (rule minimal)
+ apply (rule less_ext)
+ apply (rule allI)
+ apply (rule minimal)
+ done
+
+lemma UU_app: "UU(x) = UU"
+ by (rule UU_abs [symmetric, THEN ap_thm])
+
+lemma less_UU: "x << UU ==> x=UU"
+ apply (rule less_anti_sym)
+ apply assumption
+ apply (rule minimal)
+ done
-use "LCF_lemmas.ML"
+lemma tr_induct: "[| P(UU); P(TT); P(FF) |] ==> ALL b. P(b)"
+ apply (rule allI)
+ apply (rule mp)
+ apply (rule_tac [2] p = b in tr_cases)
+ apply blast
+ done
+
+lemma Contrapos: "~ B ==> (A ==> B) ==> ~A"
+ by blast
+
+lemma not_less_imp_not_eq1: "~ x << y \<Longrightarrow> x \<noteq> y"
+ apply (erule Contrapos)
+ apply simp
+ done
+
+lemma not_less_imp_not_eq2: "~ y << x \<Longrightarrow> x \<noteq> y"
+ apply (erule Contrapos)
+ apply simp
+ done
+
+lemma not_UU_eq_TT: "UU \<noteq> TT"
+ by (rule not_less_imp_not_eq2) (rule not_TT_less_UU)
+lemma not_UU_eq_FF: "UU \<noteq> FF"
+ by (rule not_less_imp_not_eq2) (rule not_FF_less_UU)
+lemma not_TT_eq_UU: "TT \<noteq> UU"
+ by (rule not_less_imp_not_eq1) (rule not_TT_less_UU)
+lemma not_TT_eq_FF: "TT \<noteq> FF"
+ by (rule not_less_imp_not_eq1) (rule not_TT_less_FF)
+lemma not_FF_eq_UU: "FF \<noteq> UU"
+ by (rule not_less_imp_not_eq1) (rule not_FF_less_UU)
+lemma not_FF_eq_TT: "FF \<noteq> TT"
+ by (rule not_less_imp_not_eq1) (rule not_FF_less_TT)
+
+
+lemma COND_cases_iff [rule_format]:
+ "ALL b. P(b=>x|y) <-> (b=UU-->P(UU)) & (b=TT-->P(x)) & (b=FF-->P(y))"
+ apply (insert not_UU_eq_TT not_UU_eq_FF not_TT_eq_UU
+ not_TT_eq_FF not_FF_eq_UU not_FF_eq_TT)
+ apply (rule tr_induct)
+ apply (simplesubst COND_UU)
+ apply blast
+ apply (simplesubst COND_TT)
+ apply blast
+ apply (simplesubst COND_FF)
+ apply blast
+ done
+
+lemma COND_cases:
+ "[| x = UU --> P(UU); x = TT --> P(xa); x = FF --> P(y) |] ==> P(x => xa | y)"
+ apply (rule COND_cases_iff [THEN iffD2])
+ apply blast
+ done
+
+lemmas [simp] =
+ minimal
+ UU_app
+ UU_app [THEN ap_thm]
+ UU_app [THEN ap_thm, THEN ap_thm]
+ not_TT_less_FF not_FF_less_TT not_TT_less_UU not_FF_less_UU not_UU_eq_TT
+ not_UU_eq_FF not_TT_eq_UU not_TT_eq_FF not_FF_eq_UU not_FF_eq_TT
+ COND_UU COND_TT COND_FF
+ surj_pairing FST SND
subsection {* Ordered pairs and products *}
-use "pair.ML"
+lemma expand_all_PROD: "(ALL p. P(p)) <-> (ALL x y. P(<x,y>))"
+ apply (rule iffI)
+ apply blast
+ apply (rule allI)
+ apply (rule surj_pairing [THEN subst])
+ apply blast
+ done
+
+lemma PROD_less: "(p::'a*'b) << q <-> FST(p) << FST(q) & SND(p) << SND(q)"
+ apply (rule iffI)
+ apply (rule conjI)
+ apply (erule less_ap_term)
+ apply (erule less_ap_term)
+ apply (erule conjE)
+ apply (rule surj_pairing [of p, THEN subst])
+ apply (rule surj_pairing [of q, THEN subst])
+ apply (rule mono, erule less_ap_term, assumption)
+ done
+
+lemma PROD_eq: "p=q <-> FST(p)=FST(q) & SND(p)=SND(q)"
+ apply (rule iffI)
+ apply simp
+ apply (unfold eq_def)
+ apply (simp add: PROD_less)
+ done
+
+lemma PAIR_less [simp]: "<a,b> << <c,d> <-> a<<c & b<<d"
+ by (simp add: PROD_less)
+
+lemma PAIR_eq [simp]: "<a,b> = <c,d> <-> a=c & b=d"
+ by (simp add: PROD_eq)
+
+lemma UU_is_UU_UU [simp]: "<UU,UU> = UU"
+ by (rule less_UU) (simp add: PROD_less)
+
+lemma FST_STRICT [simp]: "FST(UU) = UU"
+ apply (rule subst [OF UU_is_UU_UU])
+ apply (simp del: UU_is_UU_UU)
+ done
+
+lemma SND_STRICT [simp]: "SND(UU) = UU"
+ apply (rule subst [OF UU_is_UU_UU])
+ apply (simp del: UU_is_UU_UU)
+ done
subsection {* Fixedpoint theory *}
-use "fix.ML"
+lemma adm_eq: "adm(%x. t(x)=(u(x)::'a::cpo))"
+ apply (unfold eq_def)
+ apply (rule adm_conj adm_less)+
+ done
+
+lemma adm_not_not: "adm(P) ==> adm(%x.~~P(x))"
+ by simp
+
+lemma not_eq_TT: "ALL p. ~p=TT <-> (p=FF | p=UU)"
+ and not_eq_FF: "ALL p. ~p=FF <-> (p=TT | p=UU)"
+ and not_eq_UU: "ALL p. ~p=UU <-> (p=TT | p=FF)"
+ by (rule tr_induct, simp_all)+
+
+lemma adm_not_eq_tr: "ALL p::tr. adm(%x. ~t(x)=p)"
+ apply (rule tr_induct)
+ apply (simp_all add: not_eq_TT not_eq_FF not_eq_UU)
+ apply (rule adm_disj adm_eq)+
+ done
+
+lemmas adm_lemmas =
+ adm_not_free adm_eq adm_less adm_not_less
+ adm_not_eq_tr adm_conj adm_disj adm_imp adm_all
+
+
+ML {*
+local
+ val adm_lemmas = thms "adm_lemmas"
+ val induct = thm "induct"
+in
+ fun induct_tac v i =
+ res_inst_tac[("f",v)] induct i THEN REPEAT (resolve_tac adm_lemmas i)
+end
+*}
+
+lemma least_FIX: "f(p) = p ==> FIX(f) << p"
+ apply (tactic {* induct_tac "f" 1 *})
+ apply (rule minimal)
+ apply (intro strip)
+ apply (erule subst)
+ apply (erule less_ap_term)
+ done
+
+lemma lfp_is_FIX:
+ assumes 1: "f(p) = p"
+ and 2: "ALL q. f(q)=q --> p << q"
+ shows "p = FIX(f)"
+ apply (rule less_anti_sym)
+ apply (rule 2 [THEN spec, THEN mp])
+ apply (rule FIX_eq)
+ apply (rule least_FIX)
+ apply (rule 1)
+ done
+
+
+lemma FIX_pair: "<FIX(f),FIX(g)> = FIX(%p.<f(FST(p)),g(SND(p))>)"
+ apply (rule lfp_is_FIX)
+ apply (simp add: FIX_eq [of f] FIX_eq [of g])
+ apply (intro strip)
+ apply (simp add: PROD_less)
+ apply (rule conjI)
+ apply (rule least_FIX)
+ apply (erule subst, rule FST [symmetric])
+ apply (rule least_FIX)
+ apply (erule subst, rule SND [symmetric])
+ done
+
+lemma FIX1: "FIX(f) = FST(FIX(%p. <f(FST(p)),g(SND(p))>))"
+ by (rule FIX_pair [unfolded PROD_eq FST SND, THEN conjunct1])
+
+lemma FIX2: "FIX(g) = SND(FIX(%p. <f(FST(p)),g(SND(p))>))"
+ by (rule FIX_pair [unfolded PROD_eq FST SND, THEN conjunct2])
+
+lemma induct2:
+ assumes 1: "adm(%p. P(FST(p),SND(p)))"
+ and 2: "P(UU::'a,UU::'b)"
+ and 3: "ALL x y. P(x,y) --> P(f(x),g(y))"
+ shows "P(FIX(f),FIX(g))"
+ apply (rule FIX1 [THEN ssubst, of _ f g])
+ apply (rule FIX2 [THEN ssubst, of _ f g])
+ apply (rule induct [OF 1, where ?f = "%x. <f(FST(x)),g(SND(x))>"])
+ apply simp
+ apply (rule 2)
+ apply (simp add: expand_all_PROD)
+ apply (rule 3)
+ done
+
+ML {*
+local
+ val induct2 = thm "induct2"
+ val adm_lemmas = thms "adm_lemmas"
+in
+
+fun induct2_tac (f,g) i =
+ res_inst_tac[("f",f),("g",g)] induct2 i THEN
+ REPEAT(resolve_tac adm_lemmas i)
end
+*}
+
+end