src/HOL/Library/Quotient_Sum.thy
author Cezary Kaliszyk <kaliszyk@in.tum.de>
Fri, 19 Feb 2010 13:54:19 +0100
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(*  Title:      Quotient_Sum.thy
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    Author:     Cezary Kaliszyk and Christian Urban
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*)
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theory Quotient_Sum
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imports Main Quotient_Syntax
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begin
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section {* Quotient infrastructure for the sum type. *}
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fun
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  sum_rel
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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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fun
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  sum_map
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where
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  "sum_map f1 f2 (Inl a) = Inl (f1 a)"
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| "sum_map f1 f2 (Inr a) = Inr (f2 a)"
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declare [[map "+" = (sum_map, sum_rel)]]
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text {* should probably be in Sum_Type.thy *}
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lemma split_sum_all:
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  shows "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (Inl x)) \<and> (\<forall>x. P (Inr x))"
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  apply(auto)
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  apply(case_tac x)
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  apply(simp_all)
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  done
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lemma sum_equivp[quot_equiv]:
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  assumes a: "equivp R1"
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  assumes b: "equivp R2"
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  shows "equivp (sum_rel R1 R2)"
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  apply(rule equivpI)
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  unfolding reflp_def symp_def transp_def
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  apply(simp_all add: split_sum_all)
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  apply(blast intro: equivp_reflp[OF a] equivp_reflp[OF b])
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  apply(blast intro: equivp_symp[OF a] equivp_symp[OF b])
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  apply(blast intro: equivp_transp[OF a] equivp_transp[OF b])
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  done
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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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  unfolding Quotient_def
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  apply(simp add: split_sum_all)
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  apply(simp_all add: Quotient_abs_rep[OF q1] Quotient_rel_rep[OF q1])
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  apply(simp_all add: Quotient_abs_rep[OF q2] Quotient_rel_rep[OF q2])
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  using q1 q2
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  unfolding Quotient_def
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  apply(blast)+
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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 simp
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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 simp
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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: expand_fun_eq)
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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: expand_fun_eq)
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  apply(simp add: Quotient_abs_rep[OF q2])
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  done
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lemma sum_map_id[id_simps]:
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  shows "sum_map id id = id"
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  by (simp add: expand_fun_eq split_sum_all)
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lemma sum_rel_eq[id_simps]:
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  shows "sum_rel (op =) (op =) = (op =)"
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  by (simp add: expand_fun_eq split_sum_all)
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end