
wenzelm@47455: (*  Title:      HOL/Library/Quotient_Sum.thy
huffman@47624:     Author:     Cezary Kaliszyk, Christian Urban and Brian Huffman
kaliszyk@35222: *)
wenzelm@35788: 
wenzelm@35788: header {* Quotient infrastructure for the sum type *}
wenzelm@35788: 
kaliszyk@35222: theory Quotient_Sum
kaliszyk@35222: imports Main Quotient_Syntax
kaliszyk@35222: begin
kaliszyk@35222: 
huffman@47624: subsection {* Relator for sum type *}
huffman@47624: 
kaliszyk@35222: fun
haftmann@40542:   sum_rel :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a + 'b \<Rightarrow> 'c + 'd \<Rightarrow> bool"
kaliszyk@35222: where
kaliszyk@35222:   "sum_rel R1 R2 (Inl a1) (Inl b1) = R1 a1 b1"
kaliszyk@35222: | "sum_rel R1 R2 (Inl a1) (Inr b2) = False"
kaliszyk@35222: | "sum_rel R1 R2 (Inr a2) (Inl b1) = False"
kaliszyk@35222: | "sum_rel R1 R2 (Inr a2) (Inr b2) = R2 a2 b2"
kaliszyk@35222: 
haftmann@40820: lemma sum_rel_unfold:
haftmann@40820:   "sum_rel R1 R2 x y = (case (x, y) of (Inl x, Inl y) \<Rightarrow> R1 x y
haftmann@40820:     | (Inr x, Inr y) \<Rightarrow> R2 x y
haftmann@40820:     | _ \<Rightarrow> False)"
haftmann@40820:   by (cases x) (cases y, simp_all)+
kaliszyk@35222: 
haftmann@40820: lemma sum_rel_map1:
haftmann@40820:   "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"
haftmann@40820:   by (simp add: sum_rel_unfold split: sum.split)
haftmann@40820: 
haftmann@40820: lemma sum_rel_map2:
haftmann@40820:   "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"
haftmann@40820:   by (simp add: sum_rel_unfold split: sum.split)
haftmann@40820: 
haftmann@40820: lemma sum_map_id [id_simps]:
haftmann@40820:   "sum_map id id = id"
haftmann@40820:   by (simp add: id_def sum_map.identity fun_eq_iff)
kaliszyk@35222: 
huffman@47624: lemma sum_rel_eq [id_simps, relator_eq]:
haftmann@40820:   "sum_rel (op =) (op =) = (op =)"
haftmann@40820:   by (simp add: sum_rel_unfold fun_eq_iff split: sum.split)
haftmann@40820: 
huffman@47624: lemma split_sum_all: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (Inl x)) \<and> (\<forall>x. P (Inr x))"
huffman@47624:   by (metis sum.exhaust) (* TODO: move to Sum_Type.thy *)
huffman@47624: 
huffman@47624: lemma split_sum_ex: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P (Inl x)) \<or> (\<exists>x. P (Inr x))"
huffman@47624:   by (metis sum.exhaust) (* TODO: move to Sum_Type.thy *)
huffman@47624: 
kuncar@47982: lemma sum_reflp[reflexivity_rule]:
haftmann@40820:   "reflp R1 \<Longrightarrow> reflp R2 \<Longrightarrow> reflp (sum_rel R1 R2)"
huffman@47624:   unfolding reflp_def split_sum_all sum_rel.simps by fast
kaliszyk@35222: 
kuncar@47982: lemma sum_left_total[reflexivity_rule]:
kuncar@47982:   "left_total R1 \<Longrightarrow> left_total R2 \<Longrightarrow> left_total (sum_rel R1 R2)"
kuncar@47982:   apply (intro left_totalI)
kuncar@47982:   unfolding split_sum_ex 
kuncar@47982:   by (case_tac x) (auto elim: left_totalE)
kuncar@47982: 
haftmann@40820: lemma sum_symp:
haftmann@40820:   "symp R1 \<Longrightarrow> symp R2 \<Longrightarrow> symp (sum_rel R1 R2)"
huffman@47624:   unfolding symp_def split_sum_all sum_rel.simps by fast
haftmann@40820: 
haftmann@40820: lemma sum_transp:
haftmann@40820:   "transp R1 \<Longrightarrow> transp R2 \<Longrightarrow> transp (sum_rel R1 R2)"
huffman@47624:   unfolding transp_def split_sum_all sum_rel.simps by fast
haftmann@40820: 
haftmann@40820: lemma sum_equivp [quot_equiv]:
haftmann@40820:   "equivp R1 \<Longrightarrow> equivp R2 \<Longrightarrow> equivp (sum_rel R1 R2)"
haftmann@40820:   by (blast intro: equivpI sum_reflp sum_symp sum_transp elim: equivpE)
huffman@47624: 
huffman@47624: lemma right_total_sum_rel [transfer_rule]:
huffman@47624:   "right_total R1 \<Longrightarrow> right_total R2 \<Longrightarrow> right_total (sum_rel R1 R2)"
huffman@47624:   unfolding right_total_def split_sum_all split_sum_ex by simp
huffman@47624: 
huffman@47624: lemma right_unique_sum_rel [transfer_rule]:
huffman@47624:   "right_unique R1 \<Longrightarrow> right_unique R2 \<Longrightarrow> right_unique (sum_rel R1 R2)"
huffman@47624:   unfolding right_unique_def split_sum_all by simp
huffman@47624: 
huffman@47624: lemma bi_total_sum_rel [transfer_rule]:
huffman@47624:   "bi_total R1 \<Longrightarrow> bi_total R2 \<Longrightarrow> bi_total (sum_rel R1 R2)"
huffman@47624:   using assms unfolding bi_total_def split_sum_all split_sum_ex by simp
huffman@47624: 
huffman@47624: lemma bi_unique_sum_rel [transfer_rule]:
huffman@47624:   "bi_unique R1 \<Longrightarrow> bi_unique R2 \<Longrightarrow> bi_unique (sum_rel R1 R2)"
huffman@47624:   using assms unfolding bi_unique_def split_sum_all by simp
huffman@47624: 
huffman@47635: subsection {* Transfer rules for transfer package *}
huffman@47624: 
huffman@47624: lemma Inl_transfer [transfer_rule]: "(A ===> sum_rel A B) Inl Inl"
huffman@47624:   unfolding fun_rel_def by simp
huffman@47624: 
huffman@47624: lemma Inr_transfer [transfer_rule]: "(B ===> sum_rel A B) Inr Inr"
huffman@47624:   unfolding fun_rel_def by simp
huffman@47624: 
huffman@47624: lemma sum_case_transfer [transfer_rule]:
huffman@47624:   "((A ===> C) ===> (B ===> C) ===> sum_rel A B ===> C) sum_case sum_case"
huffman@47624:   unfolding fun_rel_def sum_rel_unfold by (simp split: sum.split)
huffman@47624: 
huffman@47624: subsection {* Setup for lifting package *}
huffman@47624: 
kuncar@47777: lemma Quotient_sum[quot_map]:
huffman@47624:   assumes "Quotient R1 Abs1 Rep1 T1"
huffman@47624:   assumes "Quotient R2 Abs2 Rep2 T2"
huffman@47624:   shows "Quotient (sum_rel R1 R2) (sum_map Abs1 Abs2)
huffman@47624:     (sum_map Rep1 Rep2) (sum_rel T1 T2)"
huffman@47624:   using assms unfolding Quotient_alt_def
huffman@47624:   by (simp add: split_sum_all)
huffman@47624: 
kuncar@47634: fun sum_pred :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> 'a + 'b \<Rightarrow> bool"
kuncar@47634: where
kuncar@47634:   "sum_pred R1 R2 (Inl a) = R1 a"
kuncar@47634: | "sum_pred R1 R2 (Inr a) = R2 a"
kuncar@47634: 
kuncar@47634: lemma sum_invariant_commute [invariant_commute]: 
kuncar@47634:   "sum_rel (Lifting.invariant P1) (Lifting.invariant P2) = Lifting.invariant (sum_pred P1 P2)"
kuncar@47634:   apply (simp add: fun_eq_iff Lifting.invariant_def)
kuncar@47634:   apply (intro allI) 
kuncar@47634:   apply (case_tac x rule: sum.exhaust)
kuncar@47634:   apply (case_tac xa rule: sum.exhaust)
kuncar@47634:   apply auto[2]
kuncar@47634:   apply (case_tac xa rule: sum.exhaust)
kuncar@47634:   apply auto
kuncar@47634: done
kuncar@47634: 
huffman@47624: subsection {* Rules for quotient package *}
huffman@47624: 
haftmann@40820: lemma sum_quotient [quot_thm]:
kuncar@47308:   assumes q1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308:   assumes q2: "Quotient3 R2 Abs2 Rep2"
kuncar@47308:   shows "Quotient3 (sum_rel R1 R2) (sum_map Abs1 Abs2) (sum_map Rep1 Rep2)"
kuncar@47308:   apply (rule Quotient3I)
haftmann@41372:   apply (simp_all add: sum_map.compositionality comp_def sum_map.identity sum_rel_eq sum_rel_map1 sum_rel_map2
kuncar@47308:     Quotient3_abs_rep [OF q1] Quotient3_rel_rep [OF q1] Quotient3_abs_rep [OF q2] Quotient3_rel_rep [OF q2])
kuncar@47308:   using Quotient3_rel [OF q1] Quotient3_rel [OF q2]
haftmann@41372:   apply (simp add: sum_rel_unfold comp_def split: sum.split)
kaliszyk@35222:   done
kaliszyk@35222: 
kuncar@47308: declare [[mapQ3 sum = (sum_rel, sum_quotient)]]
kuncar@47094: 
haftmann@40820: lemma sum_Inl_rsp [quot_respect]:
kuncar@47308:   assumes q1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308:   assumes q2: "Quotient3 R2 Abs2 Rep2"
kaliszyk@35222:   shows "(R1 ===> sum_rel R1 R2) Inl Inl"
haftmann@40465:   by auto
kaliszyk@35222: 
haftmann@40820: lemma sum_Inr_rsp [quot_respect]:
kuncar@47308:   assumes q1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308:   assumes q2: "Quotient3 R2 Abs2 Rep2"
kaliszyk@35222:   shows "(R2 ===> sum_rel R1 R2) Inr Inr"
haftmann@40465:   by auto
kaliszyk@35222: 
haftmann@40820: lemma sum_Inl_prs [quot_preserve]:
kuncar@47308:   assumes q1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308:   assumes q2: "Quotient3 R2 Abs2 Rep2"
kaliszyk@35222:   shows "(Rep1 ---> sum_map Abs1 Abs2) Inl = Inl"
nipkow@39302:   apply(simp add: fun_eq_iff)
kuncar@47308:   apply(simp add: Quotient3_abs_rep[OF q1])
kaliszyk@35222:   done
kaliszyk@35222: 
haftmann@40820: lemma sum_Inr_prs [quot_preserve]:
kuncar@47308:   assumes q1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308:   assumes q2: "Quotient3 R2 Abs2 Rep2"
kaliszyk@35222:   shows "(Rep2 ---> sum_map Abs1 Abs2) Inr = Inr"
nipkow@39302:   apply(simp add: fun_eq_iff)
kuncar@47308:   apply(simp add: Quotient3_abs_rep[OF q2])
kaliszyk@35222:   done
kaliszyk@35222: 
kaliszyk@35222: end




