src/HOL/Transfer.thy
author kuncar
Thu Apr 10 17:48:14 2014 +0200 (2014-04-10)
changeset 56518 beb3b6851665
parent 56085 3d11892ea537
child 56520 3373f5d1e074
permissions -rw-r--r--
left_total and left_unique rules are now transfer rules (cleaner solution, reflexvity_rule attribute not needed anymore)
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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 Basic_BNFs Metis
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begin
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subsection {* Relator for function space *}
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locale lifting_syntax
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begin
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  notation rel_fun (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 rel_funD2:
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  assumes "rel_fun A B f g" and "A x x"
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  shows "B (f x) (g x)"
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  using assms by (rule rel_funD)
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lemma rel_funE:
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  assumes "rel_fun 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: rel_fun_def)
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lemmas rel_fun_eq = fun.rel_eq
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lemma rel_fun_eq_rel:
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shows "rel_fun (op =) R = (\<lambda>f g. \<forall>x. R (f x) (g x))"
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  by (simp add: rel_fun_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 rel_fun_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 rel_fun_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 rel_fun_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 Domainp_iff: "Domainp T x \<longleftrightarrow> (\<exists>y. T x y)"
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  by auto
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lemma Domaimp_refl[transfer_domain_rule]:
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  "Domainp T = Domainp T" ..
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lemma Domainp_prod_fun_eq[transfer_domain_rule]:
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  assumes "Domainp T = P"
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  shows "Domainp (op= ===> T) = (\<lambda>f. \<forall>x. P (f x))"
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by (auto intro: choice simp: assms[symmetric] Domainp_iff rel_fun_def fun_eq_iff)
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subsection {* Predicates on relations, i.e. ``class constraints'' *}
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definition left_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
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  where "left_total R \<longleftrightarrow> (\<forall>x. \<exists>y. R x y)"
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definition left_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
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  where "left_unique R \<longleftrightarrow> (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y)"
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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 left_uniqueI: "(\<And>x y z. \<lbrakk> A x z; A y z \<rbrakk> \<Longrightarrow> x = y) \<Longrightarrow> left_unique A"
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unfolding left_unique_def by blast
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lemma left_uniqueD: "\<lbrakk> left_unique A; A x z; A y z \<rbrakk> \<Longrightarrow> x = y"
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unfolding left_unique_def by blast
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lemma left_totalI:
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  "(\<And>x. \<exists>y. R x y) \<Longrightarrow> left_total R"
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unfolding left_total_def by blast
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lemma left_totalE:
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  assumes "left_total R"
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  obtains "(\<And>x. \<exists>y. R x y)"
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using assms unfolding left_total_def by blast
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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 fast
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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 fast
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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 rel_fun_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 rel_fun_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 rel_fun_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 rel_fun_def by auto
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lemma [simp]:
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  shows left_unique_conversep: "left_unique A\<inverse>\<inverse> \<longleftrightarrow> right_unique A"
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  and right_unique_conversep: "right_unique A\<inverse>\<inverse> \<longleftrightarrow> left_unique A"
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by(auto simp add: left_unique_def right_unique_def)
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lemma [simp]:
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  shows left_total_conversep: "left_total A\<inverse>\<inverse> \<longleftrightarrow> right_total A"
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  and right_total_conversep: "right_total A\<inverse>\<inverse> \<longleftrightarrow> left_total A"
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by(simp_all add: left_total_def right_total_def)
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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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lemma bi_total_iff: "bi_total A = (right_total A \<and> left_total A)"
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unfolding left_total_def right_total_def bi_total_def by blast
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lemma bi_total_conv_left_right: "bi_total R \<longleftrightarrow> left_total R \<and> right_total R"
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by(simp add: left_total_def right_total_def bi_total_def)
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lemma bi_unique_iff: "bi_unique A  \<longleftrightarrow> right_unique A \<and> left_unique A"
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unfolding left_unique_def right_unique_def bi_unique_def by blast
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lemma bi_unique_conv_left_right: "bi_unique R \<longleftrightarrow> left_unique R \<and> right_unique R"
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by(auto simp add: left_unique_def right_unique_def bi_unique_def)
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lemma bi_totalI: "left_total R \<Longrightarrow> right_total R \<Longrightarrow> bi_total R"
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unfolding bi_total_iff ..
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lemma bi_uniqueI: "left_unique R \<Longrightarrow> right_unique R \<Longrightarrow> bi_unique R"
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unfolding bi_unique_iff ..
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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 fast
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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 blast
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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 fast
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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 fast
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lemma left_total_OO: "left_total R \<Longrightarrow> left_total S \<Longrightarrow> left_total (R OO S)"
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unfolding left_total_def OO_def by fast
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lemma left_unique_OO: "left_unique R \<Longrightarrow> left_unique S \<Longrightarrow> left_unique (R OO S)"
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unfolding left_unique_def OO_def by blast
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subsection {* Properties of relators *}
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lemma left_total_eq[transfer_rule]: "left_total op=" 
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  unfolding left_total_def by blast
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lemma left_unique_eq[transfer_rule]: "left_unique op=" 
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  unfolding left_unique_def by blast
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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 left_total_fun[transfer_rule]:
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  "\<lbrakk>left_unique A; left_total B\<rbrakk> \<Longrightarrow> left_total (A ===> B)"
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  unfolding left_total_def rel_fun_def
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  apply (rule allI, 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: left_unique_def)
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  done
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lemma left_unique_fun[transfer_rule]:
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  "\<lbrakk>left_total A; left_unique B\<rbrakk> \<Longrightarrow> left_unique (A ===> B)"
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  unfolding left_total_def left_unique_def rel_fun_def
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  by (clarify, rule ext, fast)
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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 rel_fun_def
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   319
  apply (rule allI, rename_tac g)
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   320
  apply (rule_tac x="\<lambda>x. SOME z. B z (g (THE y. A x y))" in exI)
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   321
  apply clarify
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   322
  apply (subgoal_tac "(THE y. A x y) = y", simp)
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   323
  apply (rule someI_ex)
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   324
  apply (simp)
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   325
  apply (rule the_equality)
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   326
  apply assumption
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   327
  apply (simp add: right_unique_def)
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   328
  done
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   329
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   330
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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   332
  unfolding right_total_def right_unique_def rel_fun_def
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   333
  by (clarify, rule ext, fast)
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   334
kuncar@56518
   335
lemma bi_total_fun[transfer_rule]:
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   336
  "\<lbrakk>bi_unique A; bi_total B\<rbrakk> \<Longrightarrow> bi_total (A ===> B)"
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   337
  unfolding bi_unique_iff bi_total_iff
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   338
  by (blast intro: right_total_fun left_total_fun)
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   339
kuncar@56518
   340
lemma bi_unique_fun[transfer_rule]:
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   341
  "\<lbrakk>bi_total A; bi_unique B\<rbrakk> \<Longrightarrow> bi_unique (A ===> B)"
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   342
  unfolding bi_unique_iff bi_total_iff
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   343
  by (blast intro: right_unique_fun left_unique_fun)
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   344
huffman@47635
   345
subsection {* Transfer rules *}
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   346
kuncar@53952
   347
lemma Domainp_forall_transfer [transfer_rule]:
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   348
  assumes "right_total A"
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   349
  shows "((A ===> op =) ===> op =)
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   350
    (transfer_bforall (Domainp A)) transfer_forall"
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   351
  using assms unfolding right_total_def
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  unfolding transfer_forall_def transfer_bforall_def rel_fun_def Domainp_iff
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   353
  by fast
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   354
huffman@47684
   355
text {* Transfer rules using implication instead of equality on booleans. *}
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   356
huffman@52354
   357
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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   359
  "right_total A \<Longrightarrow> ((A ===> op =) ===> implies) transfer_forall transfer_forall"
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   360
  "right_total A \<Longrightarrow> ((A ===> implies) ===> implies) transfer_forall transfer_forall"
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   361
  "bi_total A \<Longrightarrow> ((A ===> op =) ===> rev_implies) transfer_forall transfer_forall"
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   362
  "bi_total A \<Longrightarrow> ((A ===> rev_implies) ===> rev_implies) transfer_forall transfer_forall"
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   363
  unfolding transfer_forall_def rev_implies_def rel_fun_def right_total_def bi_total_def
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   364
  by fast+
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   365
huffman@52354
   366
lemma transfer_implies_transfer [transfer_rule]:
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   367
  "(op =        ===> op =        ===> op =       ) transfer_implies transfer_implies"
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   368
  "(rev_implies ===> implies     ===> implies    ) transfer_implies transfer_implies"
huffman@52354
   369
  "(rev_implies ===> op =        ===> implies    ) transfer_implies transfer_implies"
huffman@52354
   370
  "(op =        ===> implies     ===> implies    ) transfer_implies transfer_implies"
huffman@52354
   371
  "(op =        ===> op =        ===> implies    ) transfer_implies transfer_implies"
huffman@52354
   372
  "(implies     ===> rev_implies ===> rev_implies) transfer_implies transfer_implies"
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   373
  "(implies     ===> op =        ===> rev_implies) transfer_implies transfer_implies"
huffman@52354
   374
  "(op =        ===> rev_implies ===> rev_implies) transfer_implies transfer_implies"
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   375
  "(op =        ===> op =        ===> rev_implies) transfer_implies transfer_implies"
blanchet@55945
   376
  unfolding transfer_implies_def rev_implies_def rel_fun_def by auto
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   377
huffman@47684
   378
lemma eq_imp_transfer [transfer_rule]:
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   379
  "right_unique A \<Longrightarrow> (A ===> A ===> op \<longrightarrow>) (op =) (op =)"
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   380
  unfolding right_unique_alt_def .
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   381
kuncar@56518
   382
text {* Transfer rules using equality. *}
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   383
kuncar@56518
   384
lemma left_unique_transfer [transfer_rule]:
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   385
  assumes "right_total A"
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   386
  assumes "right_total B"
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   387
  assumes "bi_unique A"
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   388
  shows "((A ===> B ===> op=) ===> implies) left_unique left_unique"
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   389
using assms unfolding left_unique_def[abs_def] right_total_def bi_unique_def rel_fun_def
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   390
by metis
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   391
huffman@47636
   392
lemma eq_transfer [transfer_rule]:
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   393
  assumes "bi_unique A"
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   394
  shows "(A ===> A ===> op =) (op =) (op =)"
blanchet@55945
   395
  using assms unfolding bi_unique_def rel_fun_def by auto
huffman@47325
   396
kuncar@51956
   397
lemma right_total_Ex_transfer[transfer_rule]:
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   398
  assumes "right_total A"
kuncar@51956
   399
  shows "((A ===> op=) ===> op=) (Bex (Collect (Domainp A))) Ex"
blanchet@55945
   400
using assms unfolding right_total_def Bex_def rel_fun_def Domainp_iff[abs_def]
blanchet@56085
   401
by fast
kuncar@51956
   402
kuncar@51956
   403
lemma right_total_All_transfer[transfer_rule]:
kuncar@51956
   404
  assumes "right_total A"
kuncar@51956
   405
  shows "((A ===> op =) ===> op =) (Ball (Collect (Domainp A))) All"
blanchet@55945
   406
using assms unfolding right_total_def Ball_def rel_fun_def Domainp_iff[abs_def]
blanchet@56085
   407
by fast
kuncar@51956
   408
huffman@47636
   409
lemma All_transfer [transfer_rule]:
huffman@47325
   410
  assumes "bi_total A"
huffman@47325
   411
  shows "((A ===> op =) ===> op =) All All"
blanchet@55945
   412
  using assms unfolding bi_total_def rel_fun_def by fast
huffman@47325
   413
huffman@47636
   414
lemma Ex_transfer [transfer_rule]:
huffman@47325
   415
  assumes "bi_total A"
huffman@47325
   416
  shows "((A ===> op =) ===> op =) Ex Ex"
blanchet@55945
   417
  using assms unfolding bi_total_def rel_fun_def by fast
huffman@47325
   418
huffman@47636
   419
lemma If_transfer [transfer_rule]: "(op = ===> A ===> A ===> A) If If"
blanchet@55945
   420
  unfolding rel_fun_def by simp
huffman@47325
   421
huffman@47636
   422
lemma Let_transfer [transfer_rule]: "(A ===> (A ===> B) ===> B) Let Let"
blanchet@55945
   423
  unfolding rel_fun_def by simp
huffman@47612
   424
huffman@47636
   425
lemma id_transfer [transfer_rule]: "(A ===> A) id id"
blanchet@55945
   426
  unfolding rel_fun_def by simp
huffman@47625
   427
huffman@47636
   428
lemma comp_transfer [transfer_rule]:
huffman@47325
   429
  "((B ===> C) ===> (A ===> B) ===> (A ===> C)) (op \<circ>) (op \<circ>)"
blanchet@55945
   430
  unfolding rel_fun_def by simp
huffman@47325
   431
huffman@47636
   432
lemma fun_upd_transfer [transfer_rule]:
huffman@47325
   433
  assumes [transfer_rule]: "bi_unique A"
huffman@47325
   434
  shows "((A ===> B) ===> A ===> B ===> A ===> B) fun_upd fun_upd"
huffman@47635
   435
  unfolding fun_upd_def [abs_def] by transfer_prover
huffman@47325
   436
blanchet@55415
   437
lemma case_nat_transfer [transfer_rule]:
blanchet@55415
   438
  "(A ===> (op = ===> A) ===> op = ===> A) case_nat case_nat"
blanchet@55945
   439
  unfolding rel_fun_def by (simp split: nat.split)
huffman@47627
   440
blanchet@55415
   441
lemma rec_nat_transfer [transfer_rule]:
blanchet@55415
   442
  "(A ===> (op = ===> A ===> A) ===> op = ===> A) rec_nat rec_nat"
blanchet@55945
   443
  unfolding rel_fun_def by (clarsimp, rename_tac n, induct_tac n, simp_all)
huffman@47924
   444
huffman@47924
   445
lemma funpow_transfer [transfer_rule]:
huffman@47924
   446
  "(op = ===> (A ===> A) ===> (A ===> A)) compow compow"
huffman@47924
   447
  unfolding funpow_def by transfer_prover
huffman@47924
   448
kuncar@53952
   449
lemma mono_transfer[transfer_rule]:
kuncar@53952
   450
  assumes [transfer_rule]: "bi_total A"
kuncar@53952
   451
  assumes [transfer_rule]: "(A ===> A ===> op=) op\<le> op\<le>"
kuncar@53952
   452
  assumes [transfer_rule]: "(B ===> B ===> op=) op\<le> op\<le>"
kuncar@53952
   453
  shows "((A ===> B) ===> op=) mono mono"
kuncar@53952
   454
unfolding mono_def[abs_def] by transfer_prover
kuncar@53952
   455
kuncar@53952
   456
lemma right_total_relcompp_transfer[transfer_rule]: 
kuncar@53952
   457
  assumes [transfer_rule]: "right_total B"
kuncar@53952
   458
  shows "((A ===> B ===> op=) ===> (B ===> C ===> op=) ===> A ===> C ===> op=) 
kuncar@53952
   459
    (\<lambda>R S x z. \<exists>y\<in>Collect (Domainp B). R x y \<and> S y z) op OO"
kuncar@53952
   460
unfolding OO_def[abs_def] by transfer_prover
kuncar@53952
   461
kuncar@53952
   462
lemma relcompp_transfer[transfer_rule]: 
kuncar@53952
   463
  assumes [transfer_rule]: "bi_total B"
kuncar@53952
   464
  shows "((A ===> B ===> op=) ===> (B ===> C ===> op=) ===> A ===> C ===> op=) op OO op OO"
kuncar@53952
   465
unfolding OO_def[abs_def] by transfer_prover
huffman@47627
   466
kuncar@53952
   467
lemma right_total_Domainp_transfer[transfer_rule]:
kuncar@53952
   468
  assumes [transfer_rule]: "right_total B"
kuncar@53952
   469
  shows "((A ===> B ===> op=) ===> A ===> op=) (\<lambda>T x. \<exists>y\<in>Collect(Domainp B). T x y) Domainp"
kuncar@53952
   470
apply(subst(2) Domainp_iff[abs_def]) by transfer_prover
kuncar@53952
   471
kuncar@53952
   472
lemma Domainp_transfer[transfer_rule]:
kuncar@53952
   473
  assumes [transfer_rule]: "bi_total B"
kuncar@53952
   474
  shows "((A ===> B ===> op=) ===> A ===> op=) Domainp Domainp"
kuncar@53952
   475
unfolding Domainp_iff[abs_def] by transfer_prover
kuncar@53952
   476
kuncar@53952
   477
lemma reflp_transfer[transfer_rule]: 
kuncar@53952
   478
  "bi_total A \<Longrightarrow> ((A ===> A ===> op=) ===> op=) reflp reflp"
kuncar@53952
   479
  "right_total A \<Longrightarrow> ((A ===> A ===> implies) ===> implies) reflp reflp"
kuncar@53952
   480
  "right_total A \<Longrightarrow> ((A ===> A ===> op=) ===> implies) reflp reflp"
kuncar@53952
   481
  "bi_total A \<Longrightarrow> ((A ===> A ===> rev_implies) ===> rev_implies) reflp reflp"
kuncar@53952
   482
  "bi_total A \<Longrightarrow> ((A ===> A ===> op=) ===> rev_implies) reflp reflp"
blanchet@55945
   483
using assms unfolding reflp_def[abs_def] rev_implies_def bi_total_def right_total_def rel_fun_def 
kuncar@53952
   484
by fast+
kuncar@53952
   485
kuncar@53952
   486
lemma right_unique_transfer [transfer_rule]:
kuncar@53952
   487
  assumes [transfer_rule]: "right_total A"
kuncar@53952
   488
  assumes [transfer_rule]: "right_total B"
kuncar@53952
   489
  assumes [transfer_rule]: "bi_unique B"
kuncar@53952
   490
  shows "((A ===> B ===> op=) ===> implies) right_unique right_unique"
blanchet@55945
   491
using assms unfolding right_unique_def[abs_def] right_total_def bi_unique_def rel_fun_def
kuncar@53952
   492
by metis
huffman@47325
   493
huffman@47325
   494
end
kuncar@53011
   495
kuncar@53011
   496
end