src/HOL/Library/Quotient_Sum.thy
author Andreas Lochbihler
Fri Sep 20 10:09:16 2013 +0200 (2013-09-20)
changeset 53745 788730ab7da4
parent 53026 e1a548c11845
child 55564 e81ee43ab290
permissions -rw-r--r--
prefer Code.abort over code_abort
     1 (*  Title:      HOL/Library/Quotient_Sum.thy
     2     Author:     Cezary Kaliszyk and Christian Urban
     3 *)
     4 
     5 header {* Quotient infrastructure for the sum type *}
     6 
     7 theory Quotient_Sum
     8 imports Main Quotient_Syntax
     9 begin
    10 
    11 subsection {* Rules for the Quotient package *}
    12 
    13 lemma sum_rel_map1:
    14   "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"
    15   by (simp add: sum_rel_def split: sum.split)
    16 
    17 lemma sum_rel_map2:
    18   "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"
    19   by (simp add: sum_rel_def split: sum.split)
    20 
    21 lemma sum_map_id [id_simps]:
    22   "sum_map id id = id"
    23   by (simp add: id_def sum_map.identity fun_eq_iff)
    24 
    25 lemma sum_rel_eq [id_simps]:
    26   "sum_rel (op =) (op =) = (op =)"
    27   by (simp add: sum_rel_def fun_eq_iff split: sum.split)
    28 
    29 lemma sum_symp:
    30   "symp R1 \<Longrightarrow> symp R2 \<Longrightarrow> symp (sum_rel R1 R2)"
    31   unfolding symp_def split_sum_all sum_rel_simps by fast
    32 
    33 lemma sum_transp:
    34   "transp R1 \<Longrightarrow> transp R2 \<Longrightarrow> transp (sum_rel R1 R2)"
    35   unfolding transp_def split_sum_all sum_rel_simps by fast
    36 
    37 lemma sum_equivp [quot_equiv]:
    38   "equivp R1 \<Longrightarrow> equivp R2 \<Longrightarrow> equivp (sum_rel R1 R2)"
    39   by (blast intro: equivpI reflp_sum_rel sum_symp sum_transp elim: equivpE)
    40 
    41 lemma sum_quotient [quot_thm]:
    42   assumes q1: "Quotient3 R1 Abs1 Rep1"
    43   assumes q2: "Quotient3 R2 Abs2 Rep2"
    44   shows "Quotient3 (sum_rel R1 R2) (sum_map Abs1 Abs2) (sum_map Rep1 Rep2)"
    45   apply (rule Quotient3I)
    46   apply (simp_all add: sum_map.compositionality comp_def sum_map.identity sum_rel_eq sum_rel_map1 sum_rel_map2
    47     Quotient3_abs_rep [OF q1] Quotient3_rel_rep [OF q1] Quotient3_abs_rep [OF q2] Quotient3_rel_rep [OF q2])
    48   using Quotient3_rel [OF q1] Quotient3_rel [OF q2]
    49   apply (simp add: sum_rel_def comp_def split: sum.split)
    50   done
    51 
    52 declare [[mapQ3 sum = (sum_rel, sum_quotient)]]
    53 
    54 lemma sum_Inl_rsp [quot_respect]:
    55   assumes q1: "Quotient3 R1 Abs1 Rep1"
    56   assumes q2: "Quotient3 R2 Abs2 Rep2"
    57   shows "(R1 ===> sum_rel R1 R2) Inl Inl"
    58   by auto
    59 
    60 lemma sum_Inr_rsp [quot_respect]:
    61   assumes q1: "Quotient3 R1 Abs1 Rep1"
    62   assumes q2: "Quotient3 R2 Abs2 Rep2"
    63   shows "(R2 ===> sum_rel R1 R2) Inr Inr"
    64   by auto
    65 
    66 lemma sum_Inl_prs [quot_preserve]:
    67   assumes q1: "Quotient3 R1 Abs1 Rep1"
    68   assumes q2: "Quotient3 R2 Abs2 Rep2"
    69   shows "(Rep1 ---> sum_map Abs1 Abs2) Inl = Inl"
    70   apply(simp add: fun_eq_iff)
    71   apply(simp add: Quotient3_abs_rep[OF q1])
    72   done
    73 
    74 lemma sum_Inr_prs [quot_preserve]:
    75   assumes q1: "Quotient3 R1 Abs1 Rep1"
    76   assumes q2: "Quotient3 R2 Abs2 Rep2"
    77   shows "(Rep2 ---> sum_map Abs1 Abs2) Inr = Inr"
    78   apply(simp add: fun_eq_iff)
    79   apply(simp add: Quotient3_abs_rep[OF q2])
    80   done
    81 
    82 end