src/HOL/Library/Quotient_Sum.thy
author haftmann
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(*  Title:      HOL/Library/Quotient_Sum.thy
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    Author:     Cezary Kaliszyk and Christian Urban
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*)
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header {* Quotient infrastructure for the sum type *}
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theory Quotient_Sum
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imports Main Quotient_Syntax
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begin
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fun
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  sum_rel :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a + 'b \<Rightarrow> 'c + 'd \<Rightarrow> bool"
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where
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  "sum_rel R1 R2 (Inl a1) (Inl b1) = R1 a1 b1"
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| "sum_rel R1 R2 (Inl a1) (Inr b2) = False"
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| "sum_rel R1 R2 (Inr a2) (Inl b1) = False"
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| "sum_rel R1 R2 (Inr a2) (Inr b2) = R2 a2 b2"
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declare [[map sum = (sum_map, sum_rel)]]
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lemma sum_rel_unfold:
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  "sum_rel R1 R2 x y = (case (x, y) of (Inl x, Inl y) \<Rightarrow> R1 x y
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    | (Inr x, Inr y) \<Rightarrow> R2 x y
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    | _ \<Rightarrow> False)"
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  by (cases x) (cases y, simp_all)+
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lemma sum_rel_map1:
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  "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"
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  by (simp add: sum_rel_unfold split: sum.split)
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lemma sum_rel_map2:
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  "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"
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  by (simp add: sum_rel_unfold split: sum.split)
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lemma sum_map_id [id_simps]:
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  "sum_map id id = id"
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  by (simp add: id_def sum_map.identity fun_eq_iff)
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lemma sum_rel_eq [id_simps]:
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  "sum_rel (op =) (op =) = (op =)"
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  by (simp add: sum_rel_unfold fun_eq_iff split: sum.split)
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lemma sum_reflp:
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  "reflp R1 \<Longrightarrow> reflp R2 \<Longrightarrow> reflp (sum_rel R1 R2)"
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  by (auto simp add: sum_rel_unfold split: sum.splits intro!: reflpI elim: reflpE)
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lemma sum_symp:
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  "symp R1 \<Longrightarrow> symp R2 \<Longrightarrow> symp (sum_rel R1 R2)"
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  by (auto simp add: sum_rel_unfold split: sum.splits intro!: sympI elim: sympE)
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lemma sum_transp:
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  "transp R1 \<Longrightarrow> transp R2 \<Longrightarrow> transp (sum_rel R1 R2)"
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  by (auto simp add: sum_rel_unfold split: sum.splits intro!: transpI elim: transpE)
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lemma sum_equivp [quot_equiv]:
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  "equivp R1 \<Longrightarrow> equivp R2 \<Longrightarrow> equivp (sum_rel R1 R2)"
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  by (blast intro: equivpI sum_reflp sum_symp sum_transp elim: equivpE)
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lemma sum_quotient [quot_thm]:
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  assumes q1: "Quotient R1 Abs1 Rep1"
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  assumes q2: "Quotient R2 Abs2 Rep2"
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  shows "Quotient (sum_rel R1 R2) (sum_map Abs1 Abs2) (sum_map Rep1 Rep2)"
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  apply (rule QuotientI)
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  apply (simp_all add: sum_map.compositionality comp_def sum_map.identity sum_rel_eq sum_rel_map1 sum_rel_map2
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    Quotient_abs_rep [OF q1] Quotient_rel_rep [OF q1] Quotient_abs_rep [OF q2] Quotient_rel_rep [OF q2])
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  using Quotient_rel [OF q1] Quotient_rel [OF q2]
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  apply (simp add: sum_rel_unfold comp_def split: sum.split)
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  done
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lemma sum_Inl_rsp [quot_respect]:
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  assumes q1: "Quotient R1 Abs1 Rep1"
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  assumes q2: "Quotient R2 Abs2 Rep2"
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  shows "(R1 ===> sum_rel R1 R2) Inl Inl"
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  by auto
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lemma sum_Inr_rsp [quot_respect]:
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  assumes q1: "Quotient R1 Abs1 Rep1"
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  assumes q2: "Quotient R2 Abs2 Rep2"
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  shows "(R2 ===> sum_rel R1 R2) Inr Inr"
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  by auto
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lemma sum_Inl_prs [quot_preserve]:
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  assumes q1: "Quotient R1 Abs1 Rep1"
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  assumes q2: "Quotient R2 Abs2 Rep2"
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  shows "(Rep1 ---> sum_map Abs1 Abs2) Inl = Inl"
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  apply(simp add: fun_eq_iff)
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  apply(simp add: Quotient_abs_rep[OF q1])
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  done
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lemma sum_Inr_prs [quot_preserve]:
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  assumes q1: "Quotient R1 Abs1 Rep1"
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  assumes q2: "Quotient R2 Abs2 Rep2"
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  shows "(Rep2 ---> sum_map Abs1 Abs2) Inr = Inr"
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  apply(simp add: fun_eq_iff)
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  apply(simp add: Quotient_abs_rep[OF q2])
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  done
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end