src/HOL/Transfer.thy
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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" (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_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 meta-logic 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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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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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 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 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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  by metis+
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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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  unfolding transfer_implies_def rev_implies_def fun_rel_def by auto
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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 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 Domainp_iff: "Domainp T x \<longleftrightarrow> (\<exists>y. T x y)"
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  by auto
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lemma right_total_Ex_transfer[transfer_rule]:
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  assumes "right_total A"
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  shows "((A ===> op=) ===> op=) (Bex (Collect (Domainp A))) Ex"
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using assms unfolding right_total_def Bex_def fun_rel_def Domainp_iff[abs_def]
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by blast
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lemma right_total_All_transfer[transfer_rule]:
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  assumes "right_total A"
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  shows "((A ===> op =) ===> op =) (Ball (Collect (Domainp A))) All"
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using assms unfolding right_total_def Ball_def fun_rel_def Domainp_iff[abs_def]
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by blast
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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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  unfolding fun_rel_def by simp
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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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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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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