(*  Title:      Quotient_Sum.thy

    Author:     Cezary Kaliszyk and Christian Urban

*)

theory Quotient_Sum

imports Quotient Quotient_Syntax

begin

section {* Quotient infrastructure for the sum type. *}

fun

  sum_rel

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"

fun

  sum_map

where

  "sum_map f1 f2 (Inl a) = Inl (f1 a)"

| "sum_map f1 f2 (Inr a) = Inr (f2 a)"

declare [[map "+" = (sum_map, sum_rel)]]

text {* should probably be in Sum_Type.thy *}

lemma split_sum_all:

  shows "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (Inl x)) \<and> (\<forall>x. P (Inr x))"

  apply(auto)

  apply(case_tac x)

  apply(simp_all)

  done

lemma sum_equivp[quot_equiv]:

  assumes a: "equivp R1"

  assumes b: "equivp R2"

  shows "equivp (sum_rel R1 R2)"

  apply(rule equivpI)

  unfolding reflp_def symp_def transp_def

  apply(simp_all add: split_sum_all)

  apply(blast intro: equivp_reflp[OF a] equivp_reflp[OF b])

  apply(blast intro: equivp_symp[OF a] equivp_symp[OF b])

  apply(blast intro: equivp_transp[OF a] equivp_transp[OF b])

  done

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)"

  unfolding Quotient_def

  apply(simp add: split_sum_all)

  apply(simp_all add: Quotient_abs_rep[OF q1] Quotient_rel_rep[OF q1])

  apply(simp_all add: Quotient_abs_rep[OF q2] Quotient_rel_rep[OF q2])

  using q1 q2

  unfolding Quotient_def

  apply(blast)+

  done

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 simp

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 simp

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: expand_fun_eq)

  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: expand_fun_eq)

  apply(simp add: Quotient_abs_rep[OF q2])

  done

lemma sum_map_id[id_simps]:

  shows "sum_map id id = id"

  by (simp add: expand_fun_eq split_sum_all)

lemma sum_rel_eq[id_simps]:

  shows "sum_rel (op =) (op =) = (op =)"

  by (simp add: expand_fun_eq split_sum_all)

end