src/HOL/Library/Quotient_Sum.thy
author kuncar
Tue Feb 18 23:03:49 2014 +0100 (2014-02-18)
changeset 55564 e81ee43ab290
parent 53026 e1a548c11845
child 55931 62156e694f3d
permissions -rw-r--r--
delete or move now not necessary reflexivity rules due to 1726f46d2aa8
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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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subsection {* Rules for the Quotient package *}
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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_def 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_def 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_def fun_eq_iff split: sum.split)
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lemma reflp_sum_rel:
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  "reflp R1 \<Longrightarrow> reflp R2 \<Longrightarrow> reflp (sum_rel R1 R2)"
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  unfolding reflp_def split_sum_all sum_rel_simps by fast
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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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  unfolding symp_def split_sum_all sum_rel_simps by fast
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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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  unfolding transp_def split_sum_all sum_rel_simps by fast
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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 reflp_sum_rel sum_symp sum_transp elim: equivpE)
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lemma sum_quotient [quot_thm]:
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  assumes q1: "Quotient3 R1 Abs1 Rep1"
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  assumes q2: "Quotient3 R2 Abs2 Rep2"
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  shows "Quotient3 (sum_rel R1 R2) (sum_map Abs1 Abs2) (sum_map Rep1 Rep2)"
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  apply (rule Quotient3I)
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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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    Quotient3_abs_rep [OF q1] Quotient3_rel_rep [OF q1] Quotient3_abs_rep [OF q2] Quotient3_rel_rep [OF q2])
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  using Quotient3_rel [OF q1] Quotient3_rel [OF q2]
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  apply (simp add: sum_rel_def comp_def split: sum.split)
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  done
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declare [[mapQ3 sum = (sum_rel, sum_quotient)]]
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lemma sum_Inl_rsp [quot_respect]:
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  assumes q1: "Quotient3 R1 Abs1 Rep1"
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  assumes q2: "Quotient3 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: "Quotient3 R1 Abs1 Rep1"
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  assumes q2: "Quotient3 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: "Quotient3 R1 Abs1 Rep1"
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  assumes q2: "Quotient3 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: Quotient3_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: "Quotient3 R1 Abs1 Rep1"
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  assumes q2: "Quotient3 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: Quotient3_abs_rep[OF q2])
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  done
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end