author  wenzelm 
Tue, 05 Nov 2019 14:16:16 +0100  
changeset 71046  b8aeeedf7e68 
parent 66453  cc19f7ca2ed6 
child 74563  042041c0ebeb 
permissions  rwrr 
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(* Title: LCF/LCF.thy 
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Author: Tobias Nipkow 
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Copyright 1992 University of Cambridge 
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*) 

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section \<open>LCF on top of FirstOrder Logic\<close> 
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theory LCF 
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imports FOL 
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begin 
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text \<open>This theory is based on Lawrence Paulson's book Logic and Computation.\<close> 
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subsection \<open>Natural Deduction Rules for LCF\<close> 
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class cpo = "term" 
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default_sort cpo 
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typedecl tr 

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typedecl void 

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typedecl ('a,'b) prod (infixl "*" 6) 
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typedecl ('a,'b) sum (infixl "+" 5) 

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instance "fun" :: (cpo, cpo) cpo .. 
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instance prod :: (cpo, cpo) cpo .. 
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instance sum :: (cpo, cpo) cpo .. 
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instance tr :: cpo .. 
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instance void :: cpo .. 
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consts 

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UU :: "'a" 
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TT :: "tr" 
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FF :: "tr" 

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FIX :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a" 
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FST :: "'a*'b \<Rightarrow> 'a" 

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SND :: "'a*'b \<Rightarrow> 'b" 

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INL :: "'a \<Rightarrow> 'a+'b" 

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INR :: "'b \<Rightarrow> 'a+'b" 

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WHEN :: "['a\<Rightarrow>'c, 'b\<Rightarrow>'c, 'a+'b] \<Rightarrow> 'c" 

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adm :: "('a \<Rightarrow> o) \<Rightarrow> o" 

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VOID :: "void" ("'(')") 
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PAIR :: "['a,'b] \<Rightarrow> 'a*'b" ("(1<_,/_>)" [0,0] 100) 
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COND :: "[tr,'a,'a] \<Rightarrow> 'a" ("(_ \<Rightarrow>/ (_ / _))" [60,60,60] 60) 

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less :: "['a,'a] \<Rightarrow> o" (infixl "<<" 50) 

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axiomatization where 
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(** DOMAIN THEORY **) 
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eq_def: "x=y == x << y \<and> y << x" and 
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less_trans: "\<lbrakk>x << y; y << z\<rbrakk> \<Longrightarrow> x << z" and 
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less_ext: "(\<forall>x. f(x) << g(x)) \<Longrightarrow> f << g" and 
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mono: "\<lbrakk>f << g; x << y\<rbrakk> \<Longrightarrow> f(x) << g(y)" and 
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minimal: "UU << x" and 

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FIX_eq: "\<And>f. f(FIX(f)) = FIX(f)" 
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axiomatization where 
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(** TR **) 
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tr_cases: "p=UU \<or> p=TT \<or> p=FF" and 
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not_TT_less_FF: "\<not> TT << FF" and 
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not_FF_less_TT: "\<not> FF << TT" and 

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not_TT_less_UU: "\<not> TT << UU" and 

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not_FF_less_UU: "\<not> FF << UU" and 

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COND_UU: "UU \<Rightarrow> x  y = UU" and 
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COND_TT: "TT \<Rightarrow> x  y = x" and 

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COND_FF: "FF \<Rightarrow> x  y = y" 

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axiomatization where 
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(** PAIRS **) 
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surj_pairing: "<FST(z),SND(z)> = z" and 
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FST: "FST(<x,y>) = x" and 
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SND: "SND(<x,y>) = y" 
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axiomatization where 
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(*** STRICT SUM ***) 
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INL_DEF: "\<not>x=UU \<Longrightarrow> \<not>INL(x)=UU" and 
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INR_DEF: "\<not>x=UU \<Longrightarrow> \<not>INR(x)=UU" and 

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INL_STRICT: "INL(UU) = UU" and 
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INR_STRICT: "INR(UU) = UU" and 

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WHEN_UU: "WHEN(f,g,UU) = UU" and 
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WHEN_INL: "\<not>x=UU \<Longrightarrow> WHEN(f,g,INL(x)) = f(x)" and 
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WHEN_INR: "\<not>x=UU \<Longrightarrow> WHEN(f,g,INR(x)) = g(x)" and 

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SUM_EXHAUSTION: 
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"z = UU \<or> (\<exists>x. \<not>x=UU \<and> z = INL(x)) \<or> (\<exists>y. \<not>y=UU \<and> z = INR(y))" 
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axiomatization where 
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(** VOID **) 
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void_cases: "(x::void) = UU" 
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(** INDUCTION **) 

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axiomatization where 
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induct: "\<lbrakk>adm(P); P(UU); \<forall>x. P(x) \<longrightarrow> P(f(x))\<rbrakk> \<Longrightarrow> P(FIX(f))" 
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axiomatization where 
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(** Admissibility / Chain Completeness **) 
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(* All rules can be found on pages 199200 of Larry's LCF book. 

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Note that "easiness" of types is not taken into account 

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because it cannot be expressed schematically; flatness could be. *) 

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adm_less: "\<And>t u. adm(\<lambda>x. t(x) << u(x))" and 
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adm_not_less: "\<And>t u. adm(\<lambda>x.\<not> t(x) << u)" and 

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adm_not_free: "\<And>A. adm(\<lambda>x. A)" and 

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adm_subst: "\<And>P t. adm(P) \<Longrightarrow> adm(\<lambda>x. P(t(x)))" and 

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adm_conj: "\<And>P Q. \<lbrakk>adm(P); adm(Q)\<rbrakk> \<Longrightarrow> adm(\<lambda>x. P(x)\<and>Q(x))" and 

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adm_disj: "\<And>P Q. \<lbrakk>adm(P); adm(Q)\<rbrakk> \<Longrightarrow> adm(\<lambda>x. P(x)\<or>Q(x))" and 

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adm_imp: "\<And>P Q. \<lbrakk>adm(\<lambda>x.\<not>P(x)); adm(Q)\<rbrakk> \<Longrightarrow> adm(\<lambda>x. P(x)\<longrightarrow>Q(x))" and 

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adm_all: "\<And>P. (\<And>y. adm(P(y))) \<Longrightarrow> adm(\<lambda>x. \<forall>y. P(y,x))" 

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lemma eq_imp_less1: "x = y \<Longrightarrow> x << y" 
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by (simp add: eq_def) 
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lemma eq_imp_less2: "x = y \<Longrightarrow> y << x" 
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by (simp add: eq_def) 
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lemma less_refl [simp]: "x << x" 

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apply (rule eq_imp_less1) 

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apply (rule refl) 

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done 

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lemma less_anti_sym: "\<lbrakk>x << y; y << x\<rbrakk> \<Longrightarrow> x=y" 
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by (simp add: eq_def) 
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lemma ext: "(\<And>x::'a::cpo. f(x)=(g(x)::'b::cpo)) \<Longrightarrow> (\<lambda>x. f(x))=(\<lambda>x. g(x))" 
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apply (rule less_anti_sym) 
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apply (rule less_ext) 

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apply simp 

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apply simp 

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done 

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lemma cong: "\<lbrakk>f = g; x = y\<rbrakk> \<Longrightarrow> f(x)=g(y)" 
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by simp 
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lemma less_ap_term: "x << y \<Longrightarrow> f(x) << f(y)" 
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by (rule less_refl [THEN mono]) 
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lemma less_ap_thm: "f << g \<Longrightarrow> f(x) << g(x)" 
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by (rule less_refl [THEN [2] mono]) 
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lemma ap_term: "(x::'a::cpo) = y \<Longrightarrow> (f(x)::'b::cpo) = f(y)" 
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apply (rule cong [OF refl]) 
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apply simp 

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done 

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lemma ap_thm: "f = g \<Longrightarrow> f(x) = g(x)" 
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apply (erule cong) 
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apply (rule refl) 

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done 

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lemma UU_abs: "(\<lambda>x::'a::cpo. UU) = UU" 
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apply (rule less_anti_sym) 
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prefer 2 

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apply (rule minimal) 

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apply (rule less_ext) 

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apply (rule allI) 

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apply (rule minimal) 

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done 

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lemma UU_app: "UU(x) = UU" 

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by (rule UU_abs [symmetric, THEN ap_thm]) 

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lemma less_UU: "x << UU \<Longrightarrow> x=UU" 
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apply (rule less_anti_sym) 
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apply assumption 

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apply (rule minimal) 

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done 

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lemma tr_induct: "\<lbrakk>P(UU); P(TT); P(FF)\<rbrakk> \<Longrightarrow> \<forall>b. P(b)" 
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apply (rule allI) 
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apply (rule mp) 

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apply (rule_tac [2] p = b in tr_cases) 

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apply blast 

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done 

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lemma Contrapos: "\<not> B \<Longrightarrow> (A \<Longrightarrow> B) \<Longrightarrow> \<not>A" 
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by blast 
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lemma not_less_imp_not_eq1: "\<not> x << y \<Longrightarrow> x \<noteq> y" 
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apply (erule Contrapos) 
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apply simp 

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done 

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lemma not_less_imp_not_eq2: "\<not> y << x \<Longrightarrow> x \<noteq> y" 
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apply (erule Contrapos) 
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apply simp 

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done 

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lemma not_UU_eq_TT: "UU \<noteq> TT" 

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by (rule not_less_imp_not_eq2) (rule not_TT_less_UU) 

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lemma not_UU_eq_FF: "UU \<noteq> FF" 

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by (rule not_less_imp_not_eq2) (rule not_FF_less_UU) 

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lemma not_TT_eq_UU: "TT \<noteq> UU" 

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by (rule not_less_imp_not_eq1) (rule not_TT_less_UU) 

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lemma not_TT_eq_FF: "TT \<noteq> FF" 

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by (rule not_less_imp_not_eq1) (rule not_TT_less_FF) 

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lemma not_FF_eq_UU: "FF \<noteq> UU" 

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by (rule not_less_imp_not_eq1) (rule not_FF_less_UU) 

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lemma not_FF_eq_TT: "FF \<noteq> TT" 

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by (rule not_less_imp_not_eq1) (rule not_FF_less_TT) 

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lemma COND_cases_iff [rule_format]: 

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"\<forall>b. P(b\<Rightarrow>xy) \<longleftrightarrow> (b=UU\<longrightarrow>P(UU)) \<and> (b=TT\<longrightarrow>P(x)) \<and> (b=FF\<longrightarrow>P(y))" 
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apply (insert not_UU_eq_TT not_UU_eq_FF not_TT_eq_UU 
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not_TT_eq_FF not_FF_eq_UU not_FF_eq_TT) 

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apply (rule tr_induct) 

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apply (simplesubst COND_UU) 

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apply blast 

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apply (simplesubst COND_TT) 

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apply blast 

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apply (simplesubst COND_FF) 

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apply blast 

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done 

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lemma COND_cases: 

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"\<lbrakk>x = UU \<longrightarrow> P(UU); x = TT \<longrightarrow> P(xa); x = FF \<longrightarrow> P(y)\<rbrakk> \<Longrightarrow> P(x \<Rightarrow> xa  y)" 
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apply (rule COND_cases_iff [THEN iffD2]) 
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apply blast 

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done 

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lemmas [simp] = 

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minimal 

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UU_app 

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UU_app [THEN ap_thm] 

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UU_app [THEN ap_thm, THEN ap_thm] 

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not_TT_less_FF not_FF_less_TT not_TT_less_UU not_FF_less_UU not_UU_eq_TT 

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not_UU_eq_FF not_TT_eq_UU not_TT_eq_FF not_FF_eq_UU not_FF_eq_TT 

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COND_UU COND_TT COND_FF 

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surj_pairing FST SND 

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subsection \<open>Ordered pairs and products\<close> 
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lemma expand_all_PROD: "(\<forall>p. P(p)) \<longleftrightarrow> (\<forall>x y. P(<x,y>))" 
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apply (rule iffI) 
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apply blast 

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apply (rule allI) 

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apply (rule surj_pairing [THEN subst]) 

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apply blast 

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done 

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lemma PROD_less: "(p::'a*'b) << q \<longleftrightarrow> FST(p) << FST(q) \<and> SND(p) << SND(q)" 
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apply (rule iffI) 
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apply (rule conjI) 

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apply (erule less_ap_term) 

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apply (erule less_ap_term) 

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apply (erule conjE) 

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apply (rule surj_pairing [of p, THEN subst]) 

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apply (rule surj_pairing [of q, THEN subst]) 

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apply (rule mono, erule less_ap_term, assumption) 

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done 

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lemma PROD_eq: "p=q \<longleftrightarrow> FST(p)=FST(q) \<and> SND(p)=SND(q)" 
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apply (rule iffI) 
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apply simp 

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apply (unfold eq_def) 

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apply (simp add: PROD_less) 

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done 

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lemma PAIR_less [simp]: "<a,b> << <c,d> \<longleftrightarrow> a<<c \<and> b<<d" 
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by (simp add: PROD_less) 
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lemma PAIR_eq [simp]: "<a,b> = <c,d> \<longleftrightarrow> a=c \<and> b=d" 
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by (simp add: PROD_eq) 
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lemma UU_is_UU_UU [simp]: "<UU,UU> = UU" 

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by (rule less_UU) (simp add: PROD_less) 

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lemma FST_STRICT [simp]: "FST(UU) = UU" 

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apply (rule subst [OF UU_is_UU_UU]) 

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apply (simp del: UU_is_UU_UU) 

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done 

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lemma SND_STRICT [simp]: "SND(UU) = UU" 

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apply (rule subst [OF UU_is_UU_UU]) 

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apply (simp del: UU_is_UU_UU) 

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done 

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subsection \<open>Fixedpoint theory\<close> 
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lemma adm_eq: "adm(\<lambda>x. t(x)=(u(x)::'a::cpo))" 
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apply (unfold eq_def) 
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apply (rule adm_conj adm_less)+ 

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done 

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lemma adm_not_not: "adm(P) \<Longrightarrow> adm(\<lambda>x. \<not> \<not> P(x))" 
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by simp 
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lemma not_eq_TT: "\<forall>p. \<not>p=TT \<longleftrightarrow> (p=FF \<or> p=UU)" 
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and not_eq_FF: "\<forall>p. \<not>p=FF \<longleftrightarrow> (p=TT \<or> p=UU)" 

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and not_eq_UU: "\<forall>p. \<not>p=UU \<longleftrightarrow> (p=TT \<or> p=FF)" 

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by (rule tr_induct, simp_all)+ 
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lemma adm_not_eq_tr: "\<forall>p::tr. adm(\<lambda>x. \<not>t(x)=p)" 
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apply (rule tr_induct) 
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apply (simp_all add: not_eq_TT not_eq_FF not_eq_UU) 

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apply (rule adm_disj adm_eq)+ 

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done 

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lemmas adm_lemmas = 

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adm_not_free adm_eq adm_less adm_not_less 

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adm_not_eq_tr adm_conj adm_disj adm_imp adm_all 

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method_setup induct = \<open> 
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Scan.lift Args.embedded_inner_syntax >> (fn v => fn ctxt => 
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SIMPLE_METHOD' (fn i => 
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Rule_Insts.res_inst_tac ctxt [((("f", 0), Position.none), v)] [] @{thm induct} i THEN 
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REPEAT (resolve_tac ctxt @{thms adm_lemmas} i))) 
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\<close> 
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lemma least_FIX: "f(p) = p \<Longrightarrow> FIX(f) << p" 
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apply (induct f) 
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apply (rule minimal) 
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apply (intro strip) 

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apply (erule subst) 

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apply (erule less_ap_term) 

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done 

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lemma lfp_is_FIX: 

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assumes 1: "f(p) = p" 

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and 2: "\<forall>q. f(q)=q \<longrightarrow> p << q" 
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shows "p = FIX(f)" 
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apply (rule less_anti_sym) 

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apply (rule 2 [THEN spec, THEN mp]) 

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apply (rule FIX_eq) 

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apply (rule least_FIX) 

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apply (rule 1) 

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done 

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lemma FIX_pair: "<FIX(f),FIX(g)> = FIX(\<lambda>p.<f(FST(p)),g(SND(p))>)" 
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apply (rule lfp_is_FIX) 
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apply (simp add: FIX_eq [of f] FIX_eq [of g]) 

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apply (intro strip) 

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apply (simp add: PROD_less) 

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apply (rule conjI) 

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apply (rule least_FIX) 

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apply (erule subst, rule FST [symmetric]) 

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apply (rule least_FIX) 

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apply (erule subst, rule SND [symmetric]) 

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done 

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lemma FIX1: "FIX(f) = FST(FIX(\<lambda>p. <f(FST(p)),g(SND(p))>))" 
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by (rule FIX_pair [unfolded PROD_eq FST SND, THEN conjunct1]) 
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lemma FIX2: "FIX(g) = SND(FIX(\<lambda>p. <f(FST(p)),g(SND(p))>))" 
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by (rule FIX_pair [unfolded PROD_eq FST SND, THEN conjunct2]) 
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lemma induct2: 

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assumes 1: "adm(\<lambda>p. P(FST(p),SND(p)))" 
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and 2: "P(UU::'a,UU::'b)" 
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and 3: "\<forall>x y. P(x,y) \<longrightarrow> P(f(x),g(y))" 
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shows "P(FIX(f),FIX(g))" 
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apply (rule FIX1 [THEN ssubst, of _ f g]) 

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apply (rule FIX2 [THEN ssubst, of _ f g]) 

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apply (rule induct [where ?f = "\<lambda>x. <f(FST(x)),g(SND(x))>"]) 
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apply (rule 1) 
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apply simp 
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apply (rule 2) 

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apply (simp add: expand_all_PROD) 

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apply (rule 3) 

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done 

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ML \<open> 
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fun induct2_tac ctxt (f, g) i = 
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Rule_Insts.res_inst_tac ctxt 
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[((("f", 0), Position.none), f), ((("g", 0), Position.none), g)] [] @{thm induct2} i THEN 
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REPEAT(resolve_tac ctxt @{thms adm_lemmas} i) 
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\<close> 
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end 