theory QuotProd

imports QuotMain

begin

fun

  prod_rel

where

  "prod_rel r1 r2 = (\<lambda>(a,b) (c,d). r1 a c \<and> r2 b d)"

(* prod_fun is a good mapping function *)

lemma prod_equivp:

  assumes a: "equivp R1"

  assumes b: "equivp R2"

  shows "equivp (prod_rel R1 R2)"

unfolding equivp_reflp_symp_transp reflp_def symp_def transp_def

apply(auto simp add: equivp_reflp[OF a] equivp_reflp[OF b])

apply(simp only: equivp_symp[OF a])

apply(simp only: equivp_symp[OF b])

using equivp_transp[OF a] apply blast

using equivp_transp[OF b] apply blast

done

lemma prod_quotient:

  assumes q1: "Quotient R1 Abs1 Rep1"

  assumes q2: "Quotient R2 Abs2 Rep2"

  shows "Quotient (prod_rel R1 R2) (prod_fun Abs1 Abs2) (prod_fun Rep1 Rep2)"

unfolding Quotient_def

using q1 q2

apply (simp add: Quotient_abs_rep Quotient_abs_rep Quotient_rel_rep Quotient_rel_rep)

using Quotient_rel[OF q1] Quotient_rel[OF q2] by blast

lemma pair_rsp:

  assumes q1: "Quotient R1 Abs1 Rep1"

  assumes q2: "Quotient R2 Abs2 Rep2"

  shows "(R1 ===> R2 ===> prod_rel R1 R2) Pair Pair"

by auto

lemma pair_prs_aux:

  assumes q1: "Quotient R1 Abs1 Rep1"

  assumes q2: "Quotient R2 Abs2 Rep2"

  shows "(prod_fun Abs1 Abs2) (Rep1 l, Rep2 r) \<equiv> (l, r)"

  by (simp add: Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])

lemma pair_prs[quot_preserve]:

  assumes q1: "Quotient R1 Abs1 Rep1"

  assumes q2: "Quotient R2 Abs2 Rep2"

  shows "(Rep1 ---> Rep2 ---> (prod_fun Abs1 Abs2)) Pair = Pair"

apply(simp only: expand_fun_eq fun_map.simps pair_prs_aux[OF q1 q2])

apply(simp)

done

(* TODO: Is the quotient assumption q1 necessary? *)

(* TODO: Aren't there hard to use later? *)

lemma fst_rsp:

  assumes q1: "Quotient R1 Abs1 Rep1"

  assumes q2: "Quotient R2 Abs2 Rep2"

  assumes a: "(prod_rel R1 R2) p1 p2"

  shows "R1 (fst p1) (fst p2)"

  using a

  apply(case_tac p1)

  apply(case_tac p2)

  apply(auto)

  done

lemma snd_rsp:

  assumes q1: "Quotient R1 Abs1 Rep1"

  assumes q2: "Quotient R2 Abs2 Rep2"

  assumes a: "(prod_rel R1 R2) p1 p2"

  shows "R2 (snd p1) (snd p2)"

  using a

  apply(case_tac p1)

  apply(case_tac p2)

  apply(auto)

  done

lemma fst_prs:

  assumes q1: "Quotient R1 Abs1 Rep1"

  assumes q2: "Quotient R2 Abs2 Rep2"

  shows "Abs1 (fst ((prod_fun Rep1 Rep2) p)) = fst p"

by (case_tac p) (auto simp add: Quotient_abs_rep[OF q1])

lemma snd_prs:

  assumes q1: "Quotient R1 Abs1 Rep1"

  assumes q2: "Quotient R2 Abs2 Rep2"

  shows "Abs2 (snd ((prod_fun Rep1 Rep2) p)) = snd p"

by (case_tac p) (auto simp add: Quotient_abs_rep[OF q2])

lemma prod_fun_id[id_simps]: "prod_fun id id \<equiv> id"

  by (rule eq_reflection) (simp add: prod_fun_def)

section {* general setup for products *}

declare [[map * = (prod_fun, prod_rel)]]

lemmas [quot_thm] = prod_quotient

lemmas [quot_respect] = pair_rsp

lemmas [quot_equiv] = prod_equivp

end