src/HOL/Library/Quotient_Sum.thy
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(*  Title:      HOL/Library/Quotient_Sum.thy
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    Author:     Cezary Kaliszyk, Christian Urban and Brian Huffman
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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 {* Relator for sum type *}
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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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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, relator_eq]:
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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 split_sum_all: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (Inl x)) \<and> (\<forall>x. P (Inr x))"
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  by (metis sum.exhaust) (* TODO: move to Sum_Type.thy *)
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lemma split_sum_ex: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P (Inl x)) \<or> (\<exists>x. P (Inr x))"
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  by (metis sum.exhaust) (* TODO: move to Sum_Type.thy *)
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lemma sum_rel_mono[relator_mono]:
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  assumes "A \<le> C"
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  assumes "B \<le> D"
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  shows "(sum_rel A B) \<le> (sum_rel C D)"
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using assms by (auto simp: sum_rel_unfold split: sum.splits)
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lemma sum_rel_OO[relator_distr]:
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  "(sum_rel A B) OO (sum_rel C D) = sum_rel (A OO C) (B OO D)"
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by (rule ext)+ (auto simp add: sum_rel_unfold OO_def split_sum_ex split: sum.split)
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lemma sum_reflp[reflexivity_rule]:
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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_left_total[reflexivity_rule]:
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  "left_total R1 \<Longrightarrow> left_total R2 \<Longrightarrow> left_total (sum_rel R1 R2)"
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  apply (intro left_totalI)
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  unfolding split_sum_ex 
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  by (case_tac x) (auto elim: left_totalE)
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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 sum_reflp sum_symp sum_transp elim: equivpE)
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lemma right_total_sum_rel [transfer_rule]:
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  "right_total R1 \<Longrightarrow> right_total R2 \<Longrightarrow> right_total (sum_rel R1 R2)"
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  unfolding right_total_def split_sum_all split_sum_ex by simp
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lemma right_unique_sum_rel [transfer_rule]:
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  "right_unique R1 \<Longrightarrow> right_unique R2 \<Longrightarrow> right_unique (sum_rel R1 R2)"
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  unfolding right_unique_def split_sum_all by simp
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lemma bi_total_sum_rel [transfer_rule]:
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  "bi_total R1 \<Longrightarrow> bi_total R2 \<Longrightarrow> bi_total (sum_rel R1 R2)"
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  using assms unfolding bi_total_def split_sum_all split_sum_ex by simp
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lemma bi_unique_sum_rel [transfer_rule]:
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  "bi_unique R1 \<Longrightarrow> bi_unique R2 \<Longrightarrow> bi_unique (sum_rel R1 R2)"
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  using assms unfolding bi_unique_def split_sum_all by simp
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subsection {* Transfer rules for transfer package *}
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lemma Inl_transfer [transfer_rule]: "(A ===> sum_rel A B) Inl Inl"
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  unfolding fun_rel_def by simp
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lemma Inr_transfer [transfer_rule]: "(B ===> sum_rel A B) Inr Inr"
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  unfolding fun_rel_def by simp
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lemma sum_case_transfer [transfer_rule]:
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  "((A ===> C) ===> (B ===> C) ===> sum_rel A B ===> C) sum_case sum_case"
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  unfolding fun_rel_def sum_rel_unfold by (simp split: sum.split)
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subsection {* Setup for lifting package *}
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lemma Quotient_sum[quot_map]:
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  assumes "Quotient R1 Abs1 Rep1 T1"
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  assumes "Quotient R2 Abs2 Rep2 T2"
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  shows "Quotient (sum_rel R1 R2) (sum_map Abs1 Abs2)
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    (sum_map Rep1 Rep2) (sum_rel T1 T2)"
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  using assms unfolding Quotient_alt_def
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  by (simp add: split_sum_all)
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fun sum_pred :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> 'a + 'b \<Rightarrow> bool"
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where
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  "sum_pred R1 R2 (Inl a) = R1 a"
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| "sum_pred R1 R2 (Inr a) = R2 a"
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lemma sum_invariant_commute [invariant_commute]: 
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  "sum_rel (Lifting.invariant P1) (Lifting.invariant P2) = Lifting.invariant (sum_pred P1 P2)"
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  apply (simp add: fun_eq_iff Lifting.invariant_def)
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  apply (intro allI) 
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  apply (case_tac x rule: sum.exhaust)
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  apply (case_tac xa rule: sum.exhaust)
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  apply auto[2]
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  apply (case_tac xa rule: sum.exhaust)
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  apply auto
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done
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subsection {* Rules for quotient package *}
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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_unfold 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