(*  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"

   apply(simp add: fun_eq_iff)

   apply(simp add: Quotient_abs_rep[OF q1])

   done

 lemma sum_Inr_prs [quot_preserve]:

   assumes q1: "Quotient R1 Abs1 Rep1"

   assumes q2: "Quotient R2 Abs2 Rep2"

   shows "(Rep2 ---> sum_map Abs1 Abs2) Inr = Inr"

   apply(simp add: fun_eq_iff)

   apply(simp add: Quotient_abs_rep[OF q2])

   done

 end