author  huffman 
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child 27316  9e74019041d4 
permissions  rwrr 
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(* Title: HOLCF/Fix.thy 
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ID: $Id$ 
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Author: Franz Regensburger 
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Definitions for fixed point operator and admissibility. 
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*) 
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header {* Fixed point operator and admissibility *} 
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theory Fix 

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imports Cfun Cprod Adm 
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begin 
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defaultsort pcpo 
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subsection {* Iteration *} 
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consts 
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iterate :: "nat \<Rightarrow> ('a::cpo \<rightarrow> 'a) \<rightarrow> ('a \<rightarrow> 'a)" 
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primrec 
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"iterate 0 = (\<Lambda> F x. x)" 
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"iterate (Suc n) = (\<Lambda> F x. F\<cdot>(iterate n\<cdot>F\<cdot>x))" 
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text {* Derive inductive properties of iterate from primitive recursion *} 
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lemma iterate_0 [simp]: "iterate 0\<cdot>F\<cdot>x = x" 
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by simp 
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lemma iterate_Suc [simp]: "iterate (Suc n)\<cdot>F\<cdot>x = F\<cdot>(iterate n\<cdot>F\<cdot>x)" 
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by simp 
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declare iterate.simps [simp del] 
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lemma iterate_Suc2: "iterate (Suc n)\<cdot>F\<cdot>x = iterate n\<cdot>F\<cdot>(F\<cdot>x)" 
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by (induct n) simp_all 
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lemma iterate_iterate: 

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"iterate m\<cdot>F\<cdot>(iterate n\<cdot>F\<cdot>x) = iterate (m + n)\<cdot>F\<cdot>x" 

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by (induct m) simp_all 

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text {* 
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The sequence of function iterations is a chain. 
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This property is essential since monotonicity of iterate makes no sense. 
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*} 
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lemma chain_iterate2: "x \<sqsubseteq> F\<cdot>x \<Longrightarrow> chain (\<lambda>i. iterate i\<cdot>F\<cdot>x)" 
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by (rule chainI, induct_tac i, auto elim: monofun_cfun_arg) 
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lemma chain_iterate [simp]: "chain (\<lambda>i. iterate i\<cdot>F\<cdot>\<bottom>)" 
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by (rule chain_iterate2 [OF minimal]) 
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subsection {* Least fixed point operator *} 
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definition 
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"fix" :: "('a \<rightarrow> 'a) \<rightarrow> 'a" where 
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"fix = (\<Lambda> F. \<Squnion>i. iterate i\<cdot>F\<cdot>\<bottom>)" 
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text {* Binder syntax for @{term fix} *} 
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syntax 
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"_FIX" :: "['a, 'a] \<Rightarrow> 'a" ("(3FIX _./ _)" [1000, 10] 10) 
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syntax (xsymbols) 
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"_FIX" :: "['a, 'a] \<Rightarrow> 'a" ("(3\<mu> _./ _)" [1000, 10] 10) 
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translations 
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"\<mu> x. t" == "CONST fix\<cdot>(\<Lambda> x. t)" 
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text {* Properties of @{term fix} *} 
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text {* direct connection between @{term fix} and iteration *} 
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lemma fix_def2: "fix\<cdot>F = (\<Squnion>i. iterate i\<cdot>F\<cdot>\<bottom>)" 
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apply (unfold fix_def) 
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apply (rule beta_cfun) 
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apply (rule cont2cont_lub) 
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apply (rule ch2ch_lambda) 
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apply (rule chain_iterate) 
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apply simp 
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done 
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text {* 
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Kleene's fixed point theorems for continuous functions in pointed 
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omega cpo's 
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*} 
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lemma fix_eq: "fix\<cdot>F = F\<cdot>(fix\<cdot>F)" 
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apply (simp add: fix_def2) 
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apply (subst lub_range_shift [of _ 1, symmetric]) 
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apply (rule chain_iterate) 

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apply (subst contlub_cfun_arg) 
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apply (rule chain_iterate) 
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apply simp 
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done 
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lemma fix_least_less: "F\<cdot>x \<sqsubseteq> x \<Longrightarrow> fix\<cdot>F \<sqsubseteq> x" 
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apply (simp add: fix_def2) 
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apply (rule is_lub_thelub) 
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apply (rule chain_iterate) 
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apply (rule ub_rangeI) 
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apply (induct_tac i) 
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apply simp 

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apply simp 

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apply (erule rev_trans_less) 
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apply (erule monofun_cfun_arg) 
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done 
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lemma fix_least: "F\<cdot>x = x \<Longrightarrow> fix\<cdot>F \<sqsubseteq> x" 
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by (rule fix_least_less, simp) 
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lemma fix_eqI: 
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assumes fixed: "F\<cdot>x = x" and least: "\<And>z. F\<cdot>z = z \<Longrightarrow> x \<sqsubseteq> z" 
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shows "fix\<cdot>F = x" 
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apply (rule antisym_less) 
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apply (rule fix_least [OF fixed]) 
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apply (rule least [OF fix_eq [symmetric]]) 
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done 
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lemma fix_eq2: "f \<equiv> fix\<cdot>F \<Longrightarrow> f = F\<cdot>f" 
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by (simp add: fix_eq [symmetric]) 
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lemma fix_eq3: "f \<equiv> fix\<cdot>F \<Longrightarrow> f\<cdot>x = F\<cdot>f\<cdot>x" 
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by (erule fix_eq2 [THEN cfun_fun_cong]) 
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lemma fix_eq4: "f = fix\<cdot>F \<Longrightarrow> f = F\<cdot>f" 
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apply (erule ssubst) 
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apply (rule fix_eq) 
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done 
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lemma fix_eq5: "f = fix\<cdot>F \<Longrightarrow> f\<cdot>x = F\<cdot>f\<cdot>x" 
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by (erule fix_eq4 [THEN cfun_fun_cong]) 

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text {* strictness of @{term fix} *} 
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lemma fix_defined_iff: "(fix\<cdot>F = \<bottom>) = (F\<cdot>\<bottom> = \<bottom>)" 
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apply (rule iffI) 

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apply (erule subst) 

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apply (rule fix_eq [symmetric]) 

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apply (erule fix_least [THEN UU_I]) 

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done 

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lemma fix_strict: "F\<cdot>\<bottom> = \<bottom> \<Longrightarrow> fix\<cdot>F = \<bottom>" 
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by (simp add: fix_defined_iff) 
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lemma fix_defined: "F\<cdot>\<bottom> \<noteq> \<bottom> \<Longrightarrow> fix\<cdot>F \<noteq> \<bottom>" 
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by (simp add: fix_defined_iff) 
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text {* @{term fix} applied to identity and constant functions *} 
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lemma fix_id: "(\<mu> x. x) = \<bottom>" 
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by (simp add: fix_strict) 
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lemma fix_const: "(\<mu> x. c) = c" 
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by (subst fix_eq, simp) 
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subsection {* Fixed point induction *} 
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lemma fix_ind: "\<lbrakk>adm P; P \<bottom>; \<And>x. P x \<Longrightarrow> P (F\<cdot>x)\<rbrakk> \<Longrightarrow> P (fix\<cdot>F)" 
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unfolding fix_def2 
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apply (erule admD) 
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apply (rule chain_iterate) 
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apply (rule nat_induct, simp_all) 
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done 
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166 

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lemma def_fix_ind: 
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"\<lbrakk>f \<equiv> fix\<cdot>F; adm P; P \<bottom>; \<And>x. P x \<Longrightarrow> P (F\<cdot>x)\<rbrakk> \<Longrightarrow> P f" 
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by (simp add: fix_ind) 
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lemma fix_ind2: 
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assumes adm: "adm P" 
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assumes 0: "P \<bottom>" and 1: "P (F\<cdot>\<bottom>)" 
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assumes step: "\<And>x. \<lbrakk>P x; P (F\<cdot>x)\<rbrakk> \<Longrightarrow> P (F\<cdot>(F\<cdot>x))" 
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shows "P (fix\<cdot>F)" 
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unfolding fix_def2 
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apply (rule admD [OF adm chain_iterate]) 
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apply (rule nat_less_induct) 
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apply (case_tac n) 
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apply (simp add: 0) 
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apply (case_tac nat) 
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apply (simp add: 1) 
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apply (frule_tac x=nat in spec) 
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apply (simp add: step) 
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done 
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subsection {* Recursive let bindings *} 
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definition 
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CLetrec :: "('a \<rightarrow> 'a \<times> 'b) \<rightarrow> 'b" where 
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"CLetrec = (\<Lambda> F. csnd\<cdot>(F\<cdot>(\<mu> x. cfst\<cdot>(F\<cdot>x))))" 
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nonterminals 
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recbinds recbindt recbind 
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syntax 
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"_recbind" :: "['a, 'a] \<Rightarrow> recbind" ("(2_ =/ _)" 10) 
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"" :: "recbind \<Rightarrow> recbindt" ("_") 
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"_recbindt" :: "[recbind, recbindt] \<Rightarrow> recbindt" ("_,/ _") 
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"" :: "recbindt \<Rightarrow> recbinds" ("_") 
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"_recbinds" :: "[recbindt, recbinds] \<Rightarrow> recbinds" ("_;/ _") 
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"_Letrec" :: "[recbinds, 'a] \<Rightarrow> 'a" ("(Letrec (_)/ in (_))" 10) 
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translations 
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(recbindt) "x = a, \<langle>y,ys\<rangle> = \<langle>b,bs\<rangle>" == (recbindt) "\<langle>x,y,ys\<rangle> = \<langle>a,b,bs\<rangle>" 
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(recbindt) "x = a, y = b" == (recbindt) "\<langle>x,y\<rangle> = \<langle>a,b\<rangle>" 
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translations 
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"_Letrec (_recbinds b bs) e" == "_Letrec b (_Letrec bs e)" 
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"Letrec xs = a in \<langle>e,es\<rangle>" == "CONST CLetrec\<cdot>(\<Lambda> xs. \<langle>a,e,es\<rangle>)" 
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"Letrec xs = a in e" == "CONST CLetrec\<cdot>(\<Lambda> xs. \<langle>a,e\<rangle>)" 
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text {* 
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Bekic's Theorem: Simultaneous fixed points over pairs 

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can be written in terms of separate fixed points. 

216 
*} 

217 

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lemma fix_cprod: 

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"fix\<cdot>(F::'a \<times> 'b \<rightarrow> 'a \<times> 'b) = 

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\<langle>\<mu> x. cfst\<cdot>(F\<cdot>\<langle>x, \<mu> y. csnd\<cdot>(F\<cdot>\<langle>x, y\<rangle>)\<rangle>), 

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\<mu> y. csnd\<cdot>(F\<cdot>\<langle>\<mu> x. cfst\<cdot>(F\<cdot>\<langle>x, \<mu> y. csnd\<cdot>(F\<cdot>\<langle>x, y\<rangle>)\<rangle>), y\<rangle>)\<rangle>" 

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(is "fix\<cdot>F = \<langle>?x, ?y\<rangle>") 

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proof (rule fix_eqI) 
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have 1: "cfst\<cdot>(F\<cdot>\<langle>?x, ?y\<rangle>) = ?x" 
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by (rule trans [symmetric, OF fix_eq], simp) 

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have 2: "csnd\<cdot>(F\<cdot>\<langle>?x, ?y\<rangle>) = ?y" 

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by (rule trans [symmetric, OF fix_eq], simp) 

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from 1 2 show "F\<cdot>\<langle>?x, ?y\<rangle> = \<langle>?x, ?y\<rangle>" by (simp add: eq_cprod) 

229 
next 

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fix z assume F_z: "F\<cdot>z = z" 

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then obtain x y where z: "z = \<langle>x,y\<rangle>" by (rule_tac p=z in cprodE) 

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from F_z z have F_x: "cfst\<cdot>(F\<cdot>\<langle>x, y\<rangle>) = x" by simp 

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from F_z z have F_y: "csnd\<cdot>(F\<cdot>\<langle>x, y\<rangle>) = y" by simp 

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let ?y1 = "\<mu> y. csnd\<cdot>(F\<cdot>\<langle>x, y\<rangle>)" 

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have "?y1 \<sqsubseteq> y" by (rule fix_least, simp add: F_y) 

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hence "cfst\<cdot>(F\<cdot>\<langle>x, ?y1\<rangle>) \<sqsubseteq> cfst\<cdot>(F\<cdot>\<langle>x, y\<rangle>)" by (simp add: monofun_cfun) 

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hence "cfst\<cdot>(F\<cdot>\<langle>x, ?y1\<rangle>) \<sqsubseteq> x" using F_x by simp 

238 
hence 1: "?x \<sqsubseteq> x" by (simp add: fix_least_less) 

239 
hence "csnd\<cdot>(F\<cdot>\<langle>?x, y\<rangle>) \<sqsubseteq> csnd\<cdot>(F\<cdot>\<langle>x, y\<rangle>)" by (simp add: monofun_cfun) 

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hence "csnd\<cdot>(F\<cdot>\<langle>?x, y\<rangle>) \<sqsubseteq> y" using F_y by simp 

241 
hence 2: "?y \<sqsubseteq> y" by (simp add: fix_least_less) 

242 
show "\<langle>?x, ?y\<rangle> \<sqsubseteq> z" using z 1 2 by simp 

243 
qed 

244 

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subsection {* Weak admissibility *} 
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definition 
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admw :: "('a \<Rightarrow> bool) \<Rightarrow> bool" where 
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"admw P = (\<forall>F. (\<forall>n. P (iterate n\<cdot>F\<cdot>\<bottom>)) \<longrightarrow> P (\<Squnion>i. iterate i\<cdot>F\<cdot>\<bottom>))" 
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text {* an admissible formula is also weak admissible *} 
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lemma adm_impl_admw: "adm P \<Longrightarrow> admw P" 
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apply (unfold admw_def) 
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apply (intro strip) 
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apply (erule admD) 
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apply (rule chain_iterate) 
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apply (erule spec) 
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done 
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text {* computational induction for weak admissible formulae *} 
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lemma wfix_ind: "\<lbrakk>admw P; \<forall>n. P (iterate n\<cdot>F\<cdot>\<bottom>)\<rbrakk> \<Longrightarrow> P (fix\<cdot>F)" 
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by (simp add: fix_def2 admw_def) 
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lemma def_wfix_ind: 
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"\<lbrakk>f \<equiv> fix\<cdot>F; admw P; \<forall>n. P (iterate n\<cdot>F\<cdot>\<bottom>)\<rbrakk> \<Longrightarrow> P f" 
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by (simp, rule wfix_ind) 
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269 

243
c22b85994e17
Franz Regensburger's HigherOrder Logic of Computable Functions embedding LCF
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end 