src/HOL/Library/Quotient_Sum.thy
 author kuncar Fri, 23 Mar 2012 14:20:09 +0100 changeset 47094 1a7ad2601cb5 parent 45802 b16f976db515 child 47308 9caab698dbe4 permissions -rw-r--r--
store the relational theorem for every relator
```
(*  Title:      HOL/Library/Quotient_Sum.thy
Author:     Cezary Kaliszyk and Christian Urban
*)

header {* Quotient infrastructure for the sum type *}

theory Quotient_Sum
imports Main Quotient_Syntax
begin

fun
sum_rel :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a + 'b \<Rightarrow> 'c + 'd \<Rightarrow> bool"
where
"sum_rel R1 R2 (Inl a1) (Inl b1) = R1 a1 b1"
| "sum_rel R1 R2 (Inl a1) (Inr b2) = False"
| "sum_rel R1 R2 (Inr a2) (Inl b1) = False"
| "sum_rel R1 R2 (Inr a2) (Inr b2) = R2 a2 b2"

lemma sum_rel_unfold:
"sum_rel R1 R2 x y = (case (x, y) of (Inl x, Inl y) \<Rightarrow> R1 x y
| (Inr x, Inr y) \<Rightarrow> R2 x y
| _ \<Rightarrow> False)"
by (cases x) (cases y, simp_all)+

lemma sum_rel_map1:
"sum_rel R1 R2 (sum_map f1 f2 x) y \<longleftrightarrow> sum_rel (\<lambda>x. R1 (f1 x)) (\<lambda>x. R2 (f2 x)) x y"
by (simp add: sum_rel_unfold split: sum.split)

lemma sum_rel_map2:
"sum_rel R1 R2 x (sum_map f1 f2 y) \<longleftrightarrow> sum_rel (\<lambda>x y. R1 x (f1 y)) (\<lambda>x y. R2 x (f2 y)) x y"
by (simp add: sum_rel_unfold split: sum.split)

lemma sum_map_id [id_simps]:
"sum_map id id = id"
by (simp add: id_def sum_map.identity fun_eq_iff)

lemma sum_rel_eq [id_simps]:
"sum_rel (op =) (op =) = (op =)"
by (simp add: sum_rel_unfold fun_eq_iff split: sum.split)

lemma sum_reflp:
"reflp R1 \<Longrightarrow> reflp R2 \<Longrightarrow> reflp (sum_rel R1 R2)"
by (auto simp add: sum_rel_unfold split: sum.splits intro!: reflpI elim: reflpE)

lemma sum_symp:
"symp R1 \<Longrightarrow> symp R2 \<Longrightarrow> symp (sum_rel R1 R2)"
by (auto simp add: sum_rel_unfold split: sum.splits intro!: sympI elim: sympE)

lemma sum_transp:
"transp R1 \<Longrightarrow> transp R2 \<Longrightarrow> transp (sum_rel R1 R2)"
by (auto simp add: sum_rel_unfold split: sum.splits intro!: transpI elim: transpE)

lemma sum_equivp [quot_equiv]:
"equivp R1 \<Longrightarrow> equivp R2 \<Longrightarrow> equivp (sum_rel R1 R2)"
by (blast intro: equivpI sum_reflp sum_symp sum_transp elim: equivpE)

lemma sum_quotient [quot_thm]:
assumes q1: "Quotient R1 Abs1 Rep1"
assumes q2: "Quotient R2 Abs2 Rep2"
shows "Quotient (sum_rel R1 R2) (sum_map Abs1 Abs2) (sum_map Rep1 Rep2)"
apply (rule QuotientI)
apply (simp_all add: sum_map.compositionality comp_def sum_map.identity sum_rel_eq sum_rel_map1 sum_rel_map2
Quotient_abs_rep [OF q1] Quotient_rel_rep [OF q1] Quotient_abs_rep [OF q2] Quotient_rel_rep [OF q2])
using Quotient_rel [OF q1] Quotient_rel [OF q2]
apply (simp add: sum_rel_unfold comp_def split: sum.split)
done

declare [[map sum = (sum_rel, sum_quotient)]]

lemma sum_Inl_rsp [quot_respect]:
assumes q1: "Quotient R1 Abs1 Rep1"
assumes q2: "Quotient R2 Abs2 Rep2"
shows "(R1 ===> sum_rel R1 R2) Inl Inl"
by auto

lemma sum_Inr_rsp [quot_respect]:
assumes q1: "Quotient R1 Abs1 Rep1"
assumes q2: "Quotient R2 Abs2 Rep2"
shows "(R2 ===> sum_rel R1 R2) Inr Inr"
by auto

lemma sum_Inl_prs [quot_preserve]:
assumes q1: "Quotient R1 Abs1 Rep1"
assumes q2: "Quotient R2 Abs2 Rep2"
shows "(Rep1 ---> sum_map Abs1 Abs2) Inl = Inl"