src/HOL/Library/Quotient_Product.thy
 author haftmann Tue, 30 Nov 2010 15:58:09 +0100 changeset 40820 fd9c98ead9a9 parent 40607 30d512bf47a7 child 41372 551eb49a6e91 permissions -rw-r--r--
more systematic and compact proofs on type relation operators using natural deduction rules
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(*  Title:      HOL/Library/Quotient_Product.thy
Author:     Cezary Kaliszyk and Christian Urban
*)

header {* Quotient infrastructure for the product type *}

theory Quotient_Product
imports Main Quotient_Syntax
begin

definition
prod_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'b \<times> 'd \<Rightarrow> bool"
where
"prod_rel R1 R2 = (\<lambda>(a, b) (c, d). R1 a c \<and> R2 b d)"

declare [[map prod = (map_pair, prod_rel)]]

lemma prod_rel_apply [simp]:
"prod_rel R1 R2 (a, b) (c, d) \<longleftrightarrow> R1 a c \<and> R2 b d"

lemma map_pair_id [id_simps]:
shows "map_pair id id = id"

lemma prod_rel_eq [id_simps]:
shows "prod_rel (op =) (op =) = (op =)"

lemma prod_equivp [quot_equiv]:
assumes "equivp R1"
assumes "equivp R2"
shows "equivp (prod_rel R1 R2)"
using assms by (auto intro!: equivpI reflpI sympI transpI elim!: equivpE elim: reflpE sympE transpE)

lemma prod_quotient [quot_thm]:
assumes "Quotient R1 Abs1 Rep1"
assumes "Quotient R2 Abs2 Rep2"
shows "Quotient (prod_rel R1 R2) (map_pair Abs1 Abs2) (map_pair Rep1 Rep2)"
apply (rule QuotientI)
Quotient_abs_rep [OF assms(1)] Quotient_abs_rep [OF assms(2)])
apply (simp add: split_paired_all Quotient_rel_rep [OF assms(1)] Quotient_rel_rep [OF assms(2)])
using Quotient_rel [OF assms(1)] Quotient_rel [OF assms(2)]
done

lemma Pair_rsp [quot_respect]:
assumes q1: "Quotient R1 Abs1 Rep1"
assumes q2: "Quotient R2 Abs2 Rep2"
shows "(R1 ===> R2 ===> prod_rel R1 R2) Pair Pair"

lemma Pair_prs [quot_preserve]:
assumes q1: "Quotient R1 Abs1 Rep1"
assumes q2: "Quotient R2 Abs2 Rep2"
shows "(Rep1 ---> Rep2 ---> (map_pair Abs1 Abs2)) Pair = Pair"
apply(simp add: Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])
done

lemma fst_rsp [quot_respect]:
assumes "Quotient R1 Abs1 Rep1"
assumes "Quotient R2 Abs2 Rep2"
shows "(prod_rel R1 R2 ===> R1) fst fst"
by auto

lemma fst_prs [quot_preserve]:
assumes q1: "Quotient R1 Abs1 Rep1"
assumes q2: "Quotient R2 Abs2 Rep2"
shows "(map_pair Rep1 Rep2 ---> Abs1) fst = fst"
by (simp add: fun_eq_iff Quotient_abs_rep[OF q1])

lemma snd_rsp [quot_respect]:
assumes "Quotient R1 Abs1 Rep1"
assumes "Quotient R2 Abs2 Rep2"
shows "(prod_rel R1 R2 ===> R2) snd snd"
by auto

lemma snd_prs [quot_preserve]:
assumes q1: "Quotient R1 Abs1 Rep1"
assumes q2: "Quotient R2 Abs2 Rep2"
shows "(map_pair Rep1 Rep2 ---> Abs2) snd = snd"
by (simp add: fun_eq_iff Quotient_abs_rep[OF q2])

lemma split_rsp [quot_respect]:
shows "((R1 ===> R2 ===> (op =)) ===> (prod_rel R1 R2) ===> (op =)) split split"
by (auto intro!: fun_relI elim!: fun_relE)

lemma split_prs [quot_preserve]:
assumes q1: "Quotient R1 Abs1 Rep1"
and     q2: "Quotient R2 Abs2 Rep2"
shows "(((Abs1 ---> Abs2 ---> id) ---> map_pair Rep1 Rep2 ---> id) split) = split"
by (simp add: fun_eq_iff Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])

lemma [quot_respect]:
shows "((R2 ===> R2 ===> op =) ===> (R1 ===> R1 ===> op =) ===>
prod_rel R2 R1 ===> prod_rel R2 R1 ===> op =) prod_rel prod_rel"

lemma [quot_preserve]:
assumes q1: "Quotient R1 abs1 rep1"
and     q2: "Quotient R2 abs2 rep2"
shows "((abs1 ---> abs1 ---> id) ---> (abs2 ---> abs2 ---> id) --->
map_pair rep1 rep2 ---> map_pair rep1 rep2 ---> id) prod_rel = prod_rel"
by (simp add: fun_eq_iff Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])

lemma [quot_preserve]:
shows"(prod_rel ((rep1 ---> rep1 ---> id) R1) ((rep2 ---> rep2 ---> id) R2)
(l1, l2) (r1, r2)) = (R1 (rep1 l1) (rep1 r1) \<and> R2 (rep2 l2) (rep2 r2))"
by simp

declare Pair_eq[quot_preserve]

end
```