author  Andreas Lochbihler 
Fri, 27 Sep 2013 09:07:45 +0200  
changeset 53944  50c8f7f21327 
parent 53927  abe2b313f0e5 
child 53952  b2781a3ce958 
permissions  rwrr 
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(* Title: HOL/Transfer.thy 
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Author: Brian Huffman, TU Muenchen 

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Author: Ondrej Kuncar, TU Muenchen 
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*) 
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header {* Generic theorem transfer using relations *} 

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theory Transfer 

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imports Hilbert_Choice 
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begin 
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subsection {* Relator for function space *} 

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definition 

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fun_rel :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool" 
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where 
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"fun_rel A B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))" 

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locale lifting_syntax 
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begin 
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notation fun_rel (infixr "===>" 55) 
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notation map_fun (infixr ">" 55) 
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end 
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context 
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begin 
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interpretation lifting_syntax . 
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lemma fun_relI [intro]: 
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assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)" 

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shows "(A ===> B) f g" 

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using assms by (simp add: fun_rel_def) 

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lemma fun_relD: 

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assumes "(A ===> B) f g" and "A x y" 

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shows "B (f x) (g y)" 

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using assms by (simp add: fun_rel_def) 

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lemma fun_relD2: 
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assumes "(A ===> B) f g" and "A x x" 
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shows "B (f x) (g x)" 
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using assms unfolding fun_rel_def by auto 
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lemma fun_relE: 
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assumes "(A ===> B) f g" and "A x y" 

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obtains "B (f x) (g y)" 

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using assms by (simp add: fun_rel_def) 

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lemma fun_rel_eq: 

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shows "((op =) ===> (op =)) = (op =)" 

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by (auto simp add: fun_eq_iff elim: fun_relE) 

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lemma fun_rel_eq_rel: 

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shows "((op =) ===> R) = (\<lambda>f g. \<forall>x. R (f x) (g x))" 

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by (simp add: fun_rel_def) 

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subsection {* Transfer method *} 

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text {* Explicit tag for relation membership allows for 
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backward proof methods. *} 
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definition Rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" 

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where "Rel r \<equiv> r" 

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text {* Handling of equality relations *} 
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definition is_equality :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" 
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where "is_equality R \<longleftrightarrow> R = (op =)" 
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lemma is_equality_eq: "is_equality (op =)" 
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unfolding is_equality_def by simp 
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text {* Reverse implication for monotonicity rules *} 
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definition rev_implies where 
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"rev_implies x y \<longleftrightarrow> (y \<longrightarrow> x)" 
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text {* Handling of metalogic connectives *} 
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definition transfer_forall where 

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"transfer_forall \<equiv> All" 

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definition transfer_implies where 

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"transfer_implies \<equiv> op \<longrightarrow>" 

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definition transfer_bforall :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" 
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where "transfer_bforall \<equiv> (\<lambda>P Q. \<forall>x. P x \<longrightarrow> Q x)" 
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lemma transfer_forall_eq: "(\<And>x. P x) \<equiv> Trueprop (transfer_forall (\<lambda>x. P x))" 
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unfolding atomize_all transfer_forall_def .. 

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lemma transfer_implies_eq: "(A \<Longrightarrow> B) \<equiv> Trueprop (transfer_implies A B)" 

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unfolding atomize_imp transfer_implies_def .. 

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lemma transfer_bforall_unfold: 
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"Trueprop (transfer_bforall P (\<lambda>x. Q x)) \<equiv> (\<And>x. P x \<Longrightarrow> Q x)" 
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unfolding transfer_bforall_def atomize_imp atomize_all .. 
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lemma transfer_start: "\<lbrakk>P; Rel (op =) P Q\<rbrakk> \<Longrightarrow> Q" 
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unfolding Rel_def by simp 
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lemma transfer_start': "\<lbrakk>P; Rel (op \<longrightarrow>) P Q\<rbrakk> \<Longrightarrow> Q" 
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unfolding Rel_def by simp 
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lemma transfer_prover_start: "\<lbrakk>x = x'; Rel R x' y\<rbrakk> \<Longrightarrow> Rel R x y" 
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by simp 
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lemma untransfer_start: "\<lbrakk>Q; Rel (op =) P Q\<rbrakk> \<Longrightarrow> P" 
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unfolding Rel_def by simp 
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lemma Rel_eq_refl: "Rel (op =) x x" 
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unfolding Rel_def .. 

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lemma Rel_app: 
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assumes "Rel (A ===> B) f g" and "Rel A x y" 
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shows "Rel B (f x) (g y)" 
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using assms unfolding Rel_def fun_rel_def by fast 
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lemma Rel_abs: 
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assumes "\<And>x y. Rel A x y \<Longrightarrow> Rel B (f x) (g y)" 
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shows "Rel (A ===> B) (\<lambda>x. f x) (\<lambda>y. g y)" 
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using assms unfolding Rel_def fun_rel_def by fast 
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end 
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ML_file "Tools/transfer.ML" 
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setup Transfer.setup 
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declare refl [transfer_rule] 
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declare fun_rel_eq [relator_eq] 
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hide_const (open) Rel 
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context 
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begin 
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interpretation lifting_syntax . 
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text {* Handling of domains *} 
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lemma Domaimp_refl[transfer_domain_rule]: 
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"Domainp T = Domainp T" .. 
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subsection {* Predicates on relations, i.e. ``class constraints'' *} 

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definition right_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" 

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where "right_total R \<longleftrightarrow> (\<forall>y. \<exists>x. R x y)" 

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definition right_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" 

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where "right_unique R \<longleftrightarrow> (\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z)" 

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definition bi_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" 

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where "bi_total R \<longleftrightarrow> (\<forall>x. \<exists>y. R x y) \<and> (\<forall>y. \<exists>x. R x y)" 

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definition bi_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" 

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where "bi_unique R \<longleftrightarrow> 

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(\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z) \<and> 

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(\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y)" 

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lemma bi_uniqueDr: "\<lbrakk> bi_unique A; A x y; A x z \<rbrakk> \<Longrightarrow> y = z" 
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by(simp add: bi_unique_def) 

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lemma bi_uniqueDl: "\<lbrakk> bi_unique A; A x y; A z y \<rbrakk> \<Longrightarrow> x = z" 

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by(simp add: bi_unique_def) 

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lemma right_uniqueI: "(\<And>x y z. \<lbrakk> A x y; A x z \<rbrakk> \<Longrightarrow> y = z) \<Longrightarrow> right_unique A" 

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unfolding right_unique_def by blast 

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lemma right_uniqueD: "\<lbrakk> right_unique A; A x y; A x z \<rbrakk> \<Longrightarrow> y = z" 

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unfolding right_unique_def by blast 

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lemma right_total_alt_def: 
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"right_total R \<longleftrightarrow> ((R ===> op \<longrightarrow>) ===> op \<longrightarrow>) All All" 

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unfolding right_total_def fun_rel_def 

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apply (rule iffI, fast) 

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apply (rule allI) 

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apply (drule_tac x="\<lambda>x. True" in spec) 

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apply (drule_tac x="\<lambda>y. \<exists>x. R x y" in spec) 

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apply fast 

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done 

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lemma right_unique_alt_def: 

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"right_unique R \<longleftrightarrow> (R ===> R ===> op \<longrightarrow>) (op =) (op =)" 

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unfolding right_unique_def fun_rel_def by auto 

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lemma bi_total_alt_def: 

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"bi_total R \<longleftrightarrow> ((R ===> op =) ===> op =) All All" 

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unfolding bi_total_def fun_rel_def 

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apply (rule iffI, fast) 

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apply safe 

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apply (drule_tac x="\<lambda>x. \<exists>y. R x y" in spec) 

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apply (drule_tac x="\<lambda>y. True" in spec) 

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apply fast 

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apply (drule_tac x="\<lambda>x. True" in spec) 

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apply (drule_tac x="\<lambda>y. \<exists>x. R x y" in spec) 

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apply fast 

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done 

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lemma bi_unique_alt_def: 

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"bi_unique R \<longleftrightarrow> (R ===> R ===> op =) (op =) (op =)" 

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unfolding bi_unique_def fun_rel_def by auto 

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lemma bi_unique_conversep [simp]: "bi_unique R\<inverse>\<inverse> = bi_unique R" 
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by(auto simp add: bi_unique_def) 

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lemma bi_total_conversep [simp]: "bi_total R\<inverse>\<inverse> = bi_total R" 

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by(auto simp add: bi_total_def) 

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text {* Properties are preserved by relation composition. *} 
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lemma OO_def: "R OO S = (\<lambda>x z. \<exists>y. R x y \<and> S y z)" 

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by auto 

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lemma bi_total_OO: "\<lbrakk>bi_total A; bi_total B\<rbrakk> \<Longrightarrow> bi_total (A OO B)" 

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unfolding bi_total_def OO_def by metis 

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lemma bi_unique_OO: "\<lbrakk>bi_unique A; bi_unique B\<rbrakk> \<Longrightarrow> bi_unique (A OO B)" 

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unfolding bi_unique_def OO_def by metis 

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lemma right_total_OO: 

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"\<lbrakk>right_total A; right_total B\<rbrakk> \<Longrightarrow> right_total (A OO B)" 

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unfolding right_total_def OO_def by metis 

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lemma right_unique_OO: 

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"\<lbrakk>right_unique A; right_unique B\<rbrakk> \<Longrightarrow> right_unique (A OO B)" 

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unfolding right_unique_def OO_def by metis 

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subsection {* Properties of relators *} 

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lemma right_total_eq [transfer_rule]: "right_total (op =)" 

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unfolding right_total_def by simp 

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lemma right_unique_eq [transfer_rule]: "right_unique (op =)" 

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unfolding right_unique_def by simp 

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lemma bi_total_eq [transfer_rule]: "bi_total (op =)" 

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unfolding bi_total_def by simp 

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lemma bi_unique_eq [transfer_rule]: "bi_unique (op =)" 

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unfolding bi_unique_def by simp 

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lemma right_total_fun [transfer_rule]: 

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"\<lbrakk>right_unique A; right_total B\<rbrakk> \<Longrightarrow> right_total (A ===> B)" 

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unfolding right_total_def fun_rel_def 

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apply (rule allI, rename_tac g) 

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apply (rule_tac x="\<lambda>x. SOME z. B z (g (THE y. A x y))" in exI) 

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apply clarify 

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apply (subgoal_tac "(THE y. A x y) = y", simp) 

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apply (rule someI_ex) 

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apply (simp) 

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apply (rule the_equality) 

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apply assumption 

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apply (simp add: right_unique_def) 

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done 

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lemma right_unique_fun [transfer_rule]: 

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"\<lbrakk>right_total A; right_unique B\<rbrakk> \<Longrightarrow> right_unique (A ===> B)" 

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unfolding right_total_def right_unique_def fun_rel_def 

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by (clarify, rule ext, fast) 

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lemma bi_total_fun [transfer_rule]: 

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"\<lbrakk>bi_unique A; bi_total B\<rbrakk> \<Longrightarrow> bi_total (A ===> B)" 

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unfolding bi_total_def fun_rel_def 

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apply safe 

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apply (rename_tac f) 

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apply (rule_tac x="\<lambda>y. SOME z. B (f (THE x. A x y)) z" in exI) 

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apply clarify 

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apply (subgoal_tac "(THE x. A x y) = x", simp) 

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apply (rule someI_ex) 

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apply (simp) 

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apply (rule the_equality) 

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apply assumption 

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apply (simp add: bi_unique_def) 

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apply (rename_tac g) 

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apply (rule_tac x="\<lambda>x. SOME z. B z (g (THE y. A x y))" in exI) 

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apply clarify 

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apply (subgoal_tac "(THE y. A x y) = y", simp) 

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apply (rule someI_ex) 

281 
apply (simp) 

282 
apply (rule the_equality) 

283 
apply assumption 

284 
apply (simp add: bi_unique_def) 

285 
done 

286 

287 
lemma bi_unique_fun [transfer_rule]: 

288 
"\<lbrakk>bi_total A; bi_unique B\<rbrakk> \<Longrightarrow> bi_unique (A ===> B)" 

289 
unfolding bi_total_def bi_unique_def fun_rel_def fun_eq_iff 

290 
by (safe, metis, fast) 

291 

292 

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subsection {* Transfer rules *} 
47325  294 

47684  295 
text {* Transfer rules using implication instead of equality on booleans. *} 
296 

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lemma transfer_forall_transfer [transfer_rule]: 
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"bi_total A \<Longrightarrow> ((A ===> op =) ===> op =) transfer_forall transfer_forall" 
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"right_total A \<Longrightarrow> ((A ===> op =) ===> implies) transfer_forall transfer_forall" 
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"right_total A \<Longrightarrow> ((A ===> implies) ===> implies) transfer_forall transfer_forall" 
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"bi_total A \<Longrightarrow> ((A ===> op =) ===> rev_implies) transfer_forall transfer_forall" 
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"bi_total A \<Longrightarrow> ((A ===> rev_implies) ===> rev_implies) transfer_forall transfer_forall" 
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unfolding transfer_forall_def rev_implies_def fun_rel_def right_total_def bi_total_def 
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304 
by metis+ 
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305 

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lemma transfer_implies_transfer [transfer_rule]: 
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"(op = ===> op = ===> op = ) transfer_implies transfer_implies" 
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"(rev_implies ===> implies ===> implies ) transfer_implies transfer_implies" 
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"(rev_implies ===> op = ===> implies ) transfer_implies transfer_implies" 
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"(op = ===> implies ===> implies ) transfer_implies transfer_implies" 
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"(op = ===> op = ===> implies ) transfer_implies transfer_implies" 
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"(implies ===> rev_implies ===> rev_implies) transfer_implies transfer_implies" 
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"(implies ===> op = ===> rev_implies) transfer_implies transfer_implies" 
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"(op = ===> rev_implies ===> rev_implies) transfer_implies transfer_implies" 
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"(op = ===> op = ===> rev_implies) transfer_implies transfer_implies" 
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316 
unfolding transfer_implies_def rev_implies_def fun_rel_def by auto 
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317 

47684  318 
lemma eq_imp_transfer [transfer_rule]: 
319 
"right_unique A \<Longrightarrow> (A ===> A ===> op \<longrightarrow>) (op =) (op =)" 

320 
unfolding right_unique_alt_def . 

321 

47636  322 
lemma eq_transfer [transfer_rule]: 
47325  323 
assumes "bi_unique A" 
324 
shows "(A ===> A ===> op =) (op =) (op =)" 

325 
using assms unfolding bi_unique_def fun_rel_def by auto 

326 

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lemma Domainp_iff: "Domainp T x \<longleftrightarrow> (\<exists>y. T x y)" 
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328 
by auto 
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329 

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lemma right_total_Ex_transfer[transfer_rule]: 
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331 
assumes "right_total A" 
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332 
shows "((A ===> op=) ===> op=) (Bex (Collect (Domainp A))) Ex" 
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333 
using assms unfolding right_total_def Bex_def fun_rel_def Domainp_iff[abs_def] 
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334 
by blast 
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335 

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lemma right_total_All_transfer[transfer_rule]: 
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assumes "right_total A" 
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338 
shows "((A ===> op =) ===> op =) (Ball (Collect (Domainp A))) All" 
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339 
using assms unfolding right_total_def Ball_def fun_rel_def Domainp_iff[abs_def] 
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340 
by blast 
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341 

47636  342 
lemma All_transfer [transfer_rule]: 
47325  343 
assumes "bi_total A" 
344 
shows "((A ===> op =) ===> op =) All All" 

345 
using assms unfolding bi_total_def fun_rel_def by fast 

346 

47636  347 
lemma Ex_transfer [transfer_rule]: 
47325  348 
assumes "bi_total A" 
349 
shows "((A ===> op =) ===> op =) Ex Ex" 

350 
using assms unfolding bi_total_def fun_rel_def by fast 

351 

47636  352 
lemma If_transfer [transfer_rule]: "(op = ===> A ===> A ===> A) If If" 
47325  353 
unfolding fun_rel_def by simp 
354 

47636  355 
lemma Let_transfer [transfer_rule]: "(A ===> (A ===> B) ===> B) Let Let" 
47612  356 
unfolding fun_rel_def by simp 
357 

47636  358 
lemma id_transfer [transfer_rule]: "(A ===> A) id id" 
47625  359 
unfolding fun_rel_def by simp 
360 

47636  361 
lemma comp_transfer [transfer_rule]: 
47325  362 
"((B ===> C) ===> (A ===> B) ===> (A ===> C)) (op \<circ>) (op \<circ>)" 
363 
unfolding fun_rel_def by simp 

364 

47636  365 
lemma fun_upd_transfer [transfer_rule]: 
47325  366 
assumes [transfer_rule]: "bi_unique A" 
367 
shows "((A ===> B) ===> A ===> B ===> A ===> B) fun_upd fun_upd" 

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368 
unfolding fun_upd_def [abs_def] by transfer_prover 
47325  369 

47637  370 
lemma nat_case_transfer [transfer_rule]: 
371 
"(A ===> (op = ===> A) ===> op = ===> A) nat_case nat_case" 

372 
unfolding fun_rel_def by (simp split: nat.split) 

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373 

47924  374 
lemma nat_rec_transfer [transfer_rule]: 
375 
"(A ===> (op = ===> A ===> A) ===> op = ===> A) nat_rec nat_rec" 

376 
unfolding fun_rel_def by (clarsimp, rename_tac n, induct_tac n, simp_all) 

377 

378 
lemma funpow_transfer [transfer_rule]: 

379 
"(op = ===> (A ===> A) ===> (A ===> A)) compow compow" 

380 
unfolding funpow_def by transfer_prover 

381 

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382 
lemma Domainp_forall_transfer [transfer_rule]: 
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383 
assumes "right_total A" 
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384 
shows "((A ===> op =) ===> op =) 
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385 
(transfer_bforall (Domainp A)) transfer_forall" 
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386 
using assms unfolding right_total_def 
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387 
unfolding transfer_forall_def transfer_bforall_def fun_rel_def Domainp_iff 
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388 
by metis 
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389 

47636  390 
lemma forall_transfer [transfer_rule]: 
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391 
"bi_total A \<Longrightarrow> ((A ===> op =) ===> op =) transfer_forall transfer_forall" 
47636  392 
unfolding transfer_forall_def by (rule All_transfer) 
47325  393 

394 
end 

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395 

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396 
end 