author  huffman 
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parent 47789  71a526ee569a 
child 47937  70375fa2679d 
permissions  rwrr 
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(* Title: HOL/Transfer.thy 
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Author: Brian Huffman, 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 Plain Hilbert_Choice 

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uses ("Tools/transfer.ML") 

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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" (infixr "===>" 55) 

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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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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_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 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 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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use "Tools/transfer.ML" 
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setup Transfer.setup 

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declare fun_rel_eq [relator_eq] 
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hide_const (open) Rel 
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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 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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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) 

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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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done 

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lemma bi_unique_fun [transfer_rule]: 

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

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unfolding bi_total_def bi_unique_def fun_rel_def fun_eq_iff 

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by (safe, metis, fast) 

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subsection {* Transfer rules *} 
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text {* Transfer rules using implication instead of equality on booleans. *} 
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lemma eq_imp_transfer [transfer_rule]: 

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"right_unique A \<Longrightarrow> (A ===> A ===> op \<longrightarrow>) (op =) (op =)" 

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unfolding right_unique_alt_def . 

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lemma forall_imp_transfer [transfer_rule]: 

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"right_total A \<Longrightarrow> ((A ===> op \<longrightarrow>) ===> op \<longrightarrow>) transfer_forall transfer_forall" 

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unfolding right_total_alt_def transfer_forall_def . 

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lemma eq_transfer [transfer_rule]: 
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assumes "bi_unique A" 
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shows "(A ===> A ===> op =) (op =) (op =)" 

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using assms unfolding bi_unique_def fun_rel_def by auto 

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lemma All_transfer [transfer_rule]: 
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assumes "bi_total A" 
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shows "((A ===> op =) ===> op =) All All" 

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using assms unfolding bi_total_def fun_rel_def by fast 

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lemma Ex_transfer [transfer_rule]: 
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assumes "bi_total A" 
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shows "((A ===> op =) ===> op =) Ex Ex" 

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using assms unfolding bi_total_def fun_rel_def by fast 

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lemma If_transfer [transfer_rule]: "(op = ===> A ===> A ===> A) If If" 
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unfolding fun_rel_def by simp 
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lemma Let_transfer [transfer_rule]: "(A ===> (A ===> B) ===> B) Let Let" 
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unfolding fun_rel_def by simp 
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lemma id_transfer [transfer_rule]: "(A ===> A) id id" 
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lemma comp_transfer [transfer_rule]: 
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"((B ===> C) ===> (A ===> B) ===> (A ===> C)) (op \<circ>) (op \<circ>)" 
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unfolding fun_rel_def by simp 

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lemma fun_upd_transfer [transfer_rule]: 
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assumes [transfer_rule]: "bi_unique A" 
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shows "((A ===> B) ===> A ===> B ===> A ===> B) fun_upd fun_upd" 

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unfolding fun_upd_def [abs_def] by transfer_prover 
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lemma nat_case_transfer [transfer_rule]: 
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"(A ===> (op = ===> A) ===> op = ===> A) nat_case nat_case" 

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unfolding fun_rel_def by (simp split: nat.split) 

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

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

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lemma funpow_transfer [transfer_rule]: 

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

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unfolding funpow_def by transfer_prover 

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text {* Fallback rule for transferring universal quantifiers over 
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correspondence relations that are not bitotal, and do not have 
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custom transfer rules (e.g. relations between function types). *} 
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lemma Domainp_iff: "Domainp T x \<longleftrightarrow> (\<exists>y. T x y)" 
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by auto 

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lemma Domainp_forall_transfer [transfer_rule]: 
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assumes "right_total A" 
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shows "((A ===> op =) ===> op =) 
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(transfer_bforall (Domainp A)) transfer_forall" 
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using assms unfolding right_total_def 
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unfolding transfer_forall_def transfer_bforall_def fun_rel_def Domainp_iff 
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by metis 
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text {* Preferred rule for transferring universal quantifiers over 
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bitotal correspondence relations (later rules are tried first). *} 
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lemma forall_transfer [transfer_rule]: 
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"bi_total A \<Longrightarrow> ((A ===> op =) ===> op =) transfer_forall transfer_forall" 
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unfolding transfer_forall_def by (rule All_transfer) 
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end 