src/HOL/Library/Quotient_Sum.thy
author wenzelm
Fri Apr 13 14:00:26 2012 +0200 (2012-04-13)
changeset 47455 26315a545e26
parent 47308 9caab698dbe4
child 47624 16d433895d2e
permissions -rw-r--r--
updated headers;
     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 fun
    12   sum_rel :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a + 'b \<Rightarrow> 'c + 'd \<Rightarrow> bool"
    13 where
    14   "sum_rel R1 R2 (Inl a1) (Inl b1) = R1 a1 b1"
    15 | "sum_rel R1 R2 (Inl a1) (Inr b2) = False"
    16 | "sum_rel R1 R2 (Inr a2) (Inl b1) = False"
    17 | "sum_rel R1 R2 (Inr a2) (Inr b2) = R2 a2 b2"
    18 
    19 lemma sum_rel_unfold:
    20   "sum_rel R1 R2 x y = (case (x, y) of (Inl x, Inl y) \<Rightarrow> R1 x y
    21     | (Inr x, Inr y) \<Rightarrow> R2 x y
    22     | _ \<Rightarrow> False)"
    23   by (cases x) (cases y, simp_all)+
    24 
    25 lemma sum_rel_map1:
    26   "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"
    27   by (simp add: sum_rel_unfold split: sum.split)
    28 
    29 lemma sum_rel_map2:
    30   "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"
    31   by (simp add: sum_rel_unfold split: sum.split)
    32 
    33 lemma sum_map_id [id_simps]:
    34   "sum_map id id = id"
    35   by (simp add: id_def sum_map.identity fun_eq_iff)
    36 
    37 lemma sum_rel_eq [id_simps]:
    38   "sum_rel (op =) (op =) = (op =)"
    39   by (simp add: sum_rel_unfold fun_eq_iff split: sum.split)
    40 
    41 lemma sum_reflp:
    42   "reflp R1 \<Longrightarrow> reflp R2 \<Longrightarrow> reflp (sum_rel R1 R2)"
    43   by (auto simp add: sum_rel_unfold split: sum.splits intro!: reflpI elim: reflpE)
    44 
    45 lemma sum_symp:
    46   "symp R1 \<Longrightarrow> symp R2 \<Longrightarrow> symp (sum_rel R1 R2)"
    47   by (auto simp add: sum_rel_unfold split: sum.splits intro!: sympI elim: sympE)
    48 
    49 lemma sum_transp:
    50   "transp R1 \<Longrightarrow> transp R2 \<Longrightarrow> transp (sum_rel R1 R2)"
    51   by (auto simp add: sum_rel_unfold split: sum.splits intro!: transpI elim: transpE)
    52 
    53 lemma sum_equivp [quot_equiv]:
    54   "equivp R1 \<Longrightarrow> equivp R2 \<Longrightarrow> equivp (sum_rel R1 R2)"
    55   by (blast intro: equivpI sum_reflp sum_symp sum_transp elim: equivpE)
    56   
    57 lemma sum_quotient [quot_thm]:
    58   assumes q1: "Quotient3 R1 Abs1 Rep1"
    59   assumes q2: "Quotient3 R2 Abs2 Rep2"
    60   shows "Quotient3 (sum_rel R1 R2) (sum_map Abs1 Abs2) (sum_map Rep1 Rep2)"
    61   apply (rule Quotient3I)
    62   apply (simp_all add: sum_map.compositionality comp_def sum_map.identity sum_rel_eq sum_rel_map1 sum_rel_map2
    63     Quotient3_abs_rep [OF q1] Quotient3_rel_rep [OF q1] Quotient3_abs_rep [OF q2] Quotient3_rel_rep [OF q2])
    64   using Quotient3_rel [OF q1] Quotient3_rel [OF q2]
    65   apply (simp add: sum_rel_unfold comp_def split: sum.split)
    66   done
    67 
    68 declare [[mapQ3 sum = (sum_rel, sum_quotient)]]
    69 
    70 lemma sum_Inl_rsp [quot_respect]:
    71   assumes q1: "Quotient3 R1 Abs1 Rep1"
    72   assumes q2: "Quotient3 R2 Abs2 Rep2"
    73   shows "(R1 ===> sum_rel R1 R2) Inl Inl"
    74   by auto
    75 
    76 lemma sum_Inr_rsp [quot_respect]:
    77   assumes q1: "Quotient3 R1 Abs1 Rep1"
    78   assumes q2: "Quotient3 R2 Abs2 Rep2"
    79   shows "(R2 ===> sum_rel R1 R2) Inr Inr"
    80   by auto
    81 
    82 lemma sum_Inl_prs [quot_preserve]:
    83   assumes q1: "Quotient3 R1 Abs1 Rep1"
    84   assumes q2: "Quotient3 R2 Abs2 Rep2"
    85   shows "(Rep1 ---> sum_map Abs1 Abs2) Inl = Inl"
    86   apply(simp add: fun_eq_iff)
    87   apply(simp add: Quotient3_abs_rep[OF q1])
    88   done
    89 
    90 lemma sum_Inr_prs [quot_preserve]:
    91   assumes q1: "Quotient3 R1 Abs1 Rep1"
    92   assumes q2: "Quotient3 R2 Abs2 Rep2"
    93   shows "(Rep2 ---> sum_map Abs1 Abs2) Inr = Inr"
    94   apply(simp add: fun_eq_iff)
    95   apply(simp add: Quotient3_abs_rep[OF q2])
    96   done
    97 
    98 end