wenzelm@47455: (*  Title:      HOL/Library/Quotient_Sum.thy
kuncar@53012:     Author:     Cezary Kaliszyk and Christian Urban
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: 
kuncar@53012: subsection {* Rules for the Quotient package *}
kuncar@51956: 
blanchet@55943: lemma rel_sum_map1:
blanchet@55943:   "rel_sum R1 R2 (map_sum f1 f2 x) y \<longleftrightarrow> rel_sum (\<lambda>x. R1 (f1 x)) (\<lambda>x. R2 (f2 x)) x y"
blanchet@55943:   by (simp add: rel_sum_def split: sum.split)
haftmann@40820: 
blanchet@55943: lemma rel_sum_map2:
blanchet@55943:   "rel_sum R1 R2 x (map_sum f1 f2 y) \<longleftrightarrow> rel_sum (\<lambda>x y. R1 x (f1 y)) (\<lambda>x y. R2 x (f2 y)) x y"
blanchet@55943:   by (simp add: rel_sum_def split: sum.split)
haftmann@40820: 
blanchet@55931: lemma map_sum_id [id_simps]:
blanchet@55931:   "map_sum id id = id"
blanchet@55931:   by (simp add: id_def map_sum.identity fun_eq_iff)
kaliszyk@35222: 
blanchet@55943: lemma rel_sum_eq [id_simps]:
blanchet@55943:   "rel_sum (op =) (op =) = (op =)"
blanchet@55943:   by (simp add: rel_sum_def fun_eq_iff split: sum.split)
haftmann@40820: 
blanchet@55943: lemma reflp_rel_sum:
blanchet@55943:   "reflp R1 \<Longrightarrow> reflp R2 \<Longrightarrow> reflp (rel_sum R1 R2)"
blanchet@55943:   unfolding reflp_def split_sum_all rel_sum_simps by fast
kuncar@55564: 
haftmann@40820: lemma sum_symp:
blanchet@55943:   "symp R1 \<Longrightarrow> symp R2 \<Longrightarrow> symp (rel_sum R1 R2)"
blanchet@55943:   unfolding symp_def split_sum_all rel_sum_simps by fast
haftmann@40820: 
haftmann@40820: lemma sum_transp:
blanchet@55943:   "transp R1 \<Longrightarrow> transp R2 \<Longrightarrow> transp (rel_sum R1 R2)"
blanchet@55943:   unfolding transp_def split_sum_all rel_sum_simps by fast
haftmann@40820: 
haftmann@40820: lemma sum_equivp [quot_equiv]:
blanchet@55943:   "equivp R1 \<Longrightarrow> equivp R2 \<Longrightarrow> equivp (rel_sum R1 R2)"
blanchet@55943:   by (blast intro: equivpI reflp_rel_sum sum_symp sum_transp elim: equivpE)
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"
blanchet@55943:   shows "Quotient3 (rel_sum R1 R2) (map_sum Abs1 Abs2) (map_sum Rep1 Rep2)"
kuncar@47308:   apply (rule Quotient3I)
blanchet@55943:   apply (simp_all add: map_sum.compositionality comp_def map_sum.identity rel_sum_eq rel_sum_map1 rel_sum_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]
blanchet@55943:   apply (simp add: rel_sum_def comp_def split: sum.split)
kaliszyk@35222:   done
kaliszyk@35222: 
blanchet@55943: declare [[mapQ3 sum = (rel_sum, 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"
blanchet@55943:   shows "(R1 ===> rel_sum 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"
blanchet@55943:   shows "(R2 ===> rel_sum 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"
blanchet@55931:   shows "(Rep1 ---> map_sum 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"
blanchet@55931:   shows "(Rep2 ---> map_sum 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