src/HOL/Lifting_Product.thy
author kuncar
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abstract Domainp in relator_domain rules => more natural statement of the rule
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(*  Title:      HOL/Lifting_Product.thy
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    Author:     Brian Huffman and Ondrej Kuncar
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*)
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header {* Setup for Lifting/Transfer for the product type *}
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theory Lifting_Product
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imports Lifting Basic_BNFs
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begin
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subsection {* Relator and predicator properties *}
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definition pred_prod :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool"
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where "pred_prod R1 R2 = (\<lambda>(a, b). R1 a \<and> R2 b)"
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lemma pred_prod_apply [simp]:
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  "pred_prod P1 P2 (a, b) \<longleftrightarrow> P1 a \<and> P2 b"
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  by (simp add: pred_prod_def)
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lemmas rel_prod_eq[relator_eq] = prod.rel_eq
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lemmas rel_prod_mono[relator_mono] = prod.rel_mono
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lemma rel_prod_OO[relator_distr]:
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  "(rel_prod A B) OO (rel_prod C D) = rel_prod (A OO C) (B OO D)"
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by (rule ext)+ (auto simp: rel_prod_def OO_def)
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lemma Domainp_prod[relator_domain]: 
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  "Domainp (rel_prod T1 T2) = (pred_prod (Domainp T1) (Domainp T2))"
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unfolding rel_prod_def pred_prod_def by blast
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lemma left_total_rel_prod [transfer_rule]:
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  assumes "left_total R1"
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  assumes "left_total R2"
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  shows "left_total (rel_prod R1 R2)"
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  using assms unfolding left_total_def rel_prod_def by auto
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lemma left_unique_rel_prod [transfer_rule]:
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  assumes "left_unique R1" and "left_unique R2"
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  shows "left_unique (rel_prod R1 R2)"
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  using assms unfolding left_unique_def rel_prod_def by auto
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lemma right_total_rel_prod [transfer_rule]:
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  assumes "right_total R1" and "right_total R2"
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  shows "right_total (rel_prod R1 R2)"
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  using assms unfolding right_total_def rel_prod_def by auto
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lemma right_unique_rel_prod [transfer_rule]:
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  assumes "right_unique R1" and "right_unique R2"
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  shows "right_unique (rel_prod R1 R2)"
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  using assms unfolding right_unique_def rel_prod_def by auto
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lemma bi_total_rel_prod [transfer_rule]:
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  assumes "bi_total R1" and "bi_total R2"
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  shows "bi_total (rel_prod R1 R2)"
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  using assms unfolding bi_total_def rel_prod_def by auto
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lemma bi_unique_rel_prod [transfer_rule]:
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  assumes "bi_unique R1" and "bi_unique R2"
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  shows "bi_unique (rel_prod R1 R2)"
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  using assms unfolding bi_unique_def rel_prod_def by auto
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lemma prod_relator_eq_onp [relator_eq_onp]: 
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  "rel_prod (eq_onp P1) (eq_onp P2) = eq_onp (pred_prod P1 P2)"
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  by (simp add: fun_eq_iff rel_prod_def pred_prod_def eq_onp_def) blast
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subsection {* Quotient theorem for the Lifting package *}
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lemma Quotient_prod[quot_map]:
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  assumes "Quotient R1 Abs1 Rep1 T1"
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  assumes "Quotient R2 Abs2 Rep2 T2"
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  shows "Quotient (rel_prod R1 R2) (map_prod Abs1 Abs2) (map_prod Rep1 Rep2) (rel_prod T1 T2)"
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  using assms unfolding Quotient_alt_def by auto
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subsection {* Transfer rules for the Transfer package *}
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context
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begin
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interpretation lifting_syntax .
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lemma Pair_transfer [transfer_rule]: "(A ===> B ===> rel_prod A B) Pair Pair"
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  unfolding rel_fun_def rel_prod_def by simp
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lemma fst_transfer [transfer_rule]: "(rel_prod A B ===> A) fst fst"
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  unfolding rel_fun_def rel_prod_def by simp
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lemma snd_transfer [transfer_rule]: "(rel_prod A B ===> B) snd snd"
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  unfolding rel_fun_def rel_prod_def by simp
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lemma case_prod_transfer [transfer_rule]:
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  "((A ===> B ===> C) ===> rel_prod A B ===> C) case_prod case_prod"
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  unfolding rel_fun_def rel_prod_def by simp
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lemma curry_transfer [transfer_rule]:
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  "((rel_prod A B ===> C) ===> A ===> B ===> C) curry curry"
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  unfolding curry_def by transfer_prover
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lemma map_prod_transfer [transfer_rule]:
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  "((A ===> C) ===> (B ===> D) ===> rel_prod A B ===> rel_prod C D)
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    map_prod map_prod"
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  unfolding map_prod_def [abs_def] by transfer_prover
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lemma rel_prod_transfer [transfer_rule]:
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  "((A ===> B ===> op =) ===> (C ===> D ===> op =) ===>
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    rel_prod A C ===> rel_prod B D ===> op =) rel_prod rel_prod"
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  unfolding rel_fun_def by auto
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end
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end