author | wenzelm |
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changeset 78209 | 50c5be88ad59 |
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(* Title: HOL/Library/Quotient_Product.thy |
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Author: Cezary Kaliszyk and Christian Urban |
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*) |
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section \<open>Quotient infrastructure for the product type\<close> |
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theory Quotient_Product |
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imports Quotient_Syntax |
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begin |
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subsection \<open>Rules for the Quotient package\<close> |
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lemma map_prod_id [id_simps]: |
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shows "map_prod id id = id" |
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by (simp add: fun_eq_iff) |
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lemma rel_prod_eq [id_simps]: |
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shows "rel_prod (=) (=) = (=)" |
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by (simp add: fun_eq_iff) |
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lemma prod_equivp [quot_equiv]: |
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assumes "equivp R1" |
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assumes "equivp R2" |
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shows "equivp (rel_prod R1 R2)" |
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using assms by (auto intro!: equivpI reflpI sympI transpI elim!: equivpE elim: reflpE sympE transpE) |
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lemma prod_quotient [quot_thm]: |
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assumes "Quotient3 R1 Abs1 Rep1" |
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assumes "Quotient3 R2 Abs2 Rep2" |
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shows "Quotient3 (rel_prod R1 R2) (map_prod Abs1 Abs2) (map_prod Rep1 Rep2)" |
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apply (rule Quotient3I) |
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apply (simp add: map_prod.compositionality comp_def map_prod.identity |
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Quotient3_abs_rep [OF assms(1)] Quotient3_abs_rep [OF assms(2)]) |
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apply (simp add: split_paired_all Quotient3_rel_rep [OF assms(1)] Quotient3_rel_rep [OF assms(2)]) |
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using Quotient3_rel [OF assms(1)] Quotient3_rel [OF assms(2)] |
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apply (auto simp add: split_paired_all) |
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done |
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declare [[mapQ3 prod = (rel_prod, prod_quotient)]] |
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lemma Pair_rsp [quot_respect]: |
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assumes q1: "Quotient3 R1 Abs1 Rep1" |
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assumes q2: "Quotient3 R2 Abs2 Rep2" |
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shows "(R1 ===> R2 ===> rel_prod R1 R2) Pair Pair" |
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by (rule Pair_transfer) |
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lemma Pair_prs [quot_preserve]: |
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assumes q1: "Quotient3 R1 Abs1 Rep1" |
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assumes q2: "Quotient3 R2 Abs2 Rep2" |
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shows "(Rep1 ---> Rep2 ---> (map_prod Abs1 Abs2)) Pair = Pair" |
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apply(simp add: fun_eq_iff) |
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apply(simp add: Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]) |
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done |
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lemma fst_rsp [quot_respect]: |
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assumes "Quotient3 R1 Abs1 Rep1" |
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assumes "Quotient3 R2 Abs2 Rep2" |
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shows "(rel_prod R1 R2 ===> R1) fst fst" |
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by auto |
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lemma fst_prs [quot_preserve]: |
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assumes q1: "Quotient3 R1 Abs1 Rep1" |
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assumes q2: "Quotient3 R2 Abs2 Rep2" |
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shows "(map_prod Rep1 Rep2 ---> Abs1) fst = fst" |
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by (simp add: fun_eq_iff Quotient3_abs_rep[OF q1]) |
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lemma snd_rsp [quot_respect]: |
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assumes "Quotient3 R1 Abs1 Rep1" |
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assumes "Quotient3 R2 Abs2 Rep2" |
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shows "(rel_prod R1 R2 ===> R2) snd snd" |
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by auto |
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lemma snd_prs [quot_preserve]: |
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assumes q1: "Quotient3 R1 Abs1 Rep1" |
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assumes q2: "Quotient3 R2 Abs2 Rep2" |
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shows "(map_prod Rep1 Rep2 ---> Abs2) snd = snd" |
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by (simp add: fun_eq_iff Quotient3_abs_rep[OF q2]) |
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lemma case_prod_rsp [quot_respect]: |
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shows "((R1 ===> R2 ===> (=)) ===> (rel_prod R1 R2) ===> (=)) case_prod case_prod" |
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by (rule case_prod_transfer) |
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lemma split_prs [quot_preserve]: |
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assumes q1: "Quotient3 R1 Abs1 Rep1" |
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and q2: "Quotient3 R2 Abs2 Rep2" |
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shows "(((Abs1 ---> Abs2 ---> id) ---> map_prod Rep1 Rep2 ---> id) case_prod) = case_prod" |
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by (simp add: fun_eq_iff Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]) |
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lemma [quot_respect]: |
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shows "((R2 ===> R2 ===> (=)) ===> (R1 ===> R1 ===> (=)) ===> |
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rel_prod R2 R1 ===> rel_prod R2 R1 ===> (=)) rel_prod rel_prod" |
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by (rule prod.rel_transfer) |
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lemma [quot_preserve]: |
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assumes q1: "Quotient3 R1 abs1 rep1" |
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and q2: "Quotient3 R2 abs2 rep2" |
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shows "((abs1 ---> abs1 ---> id) ---> (abs2 ---> abs2 ---> id) ---> |
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map_prod rep1 rep2 ---> map_prod rep1 rep2 ---> id) rel_prod = rel_prod" |
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by (simp add: fun_eq_iff Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]) |
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lemma [quot_preserve]: |
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shows"(rel_prod ((rep1 ---> rep1 ---> id) R1) ((rep2 ---> rep2 ---> id) R2) |
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(l1, l2) (r1, r2)) = (R1 (rep1 l1) (rep1 r1) \<and> R2 (rep2 l2) (rep2 r2))" |
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by simp |
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declare prod.inject[quot_preserve] |
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end |