src/HOL/Library/Quotient_Sum.thy
author wenzelm
Fri Feb 19 22:29:30 2010 +0100 (2010-02-19)
changeset 35243 024fef37a65d
parent 35222 4f1fba00f66d
child 35788 f1deaca15ca3
permissions -rw-r--r--
fixed document;
     1 (*  Title:      Quotient_Sum.thy
     2     Author:     Cezary Kaliszyk and Christian Urban
     3 *)
     4 theory Quotient_Sum
     5 imports Main Quotient_Syntax
     6 begin
     7 
     8 section {* Quotient infrastructure for the sum type. *}
     9 
    10 fun
    11   sum_rel
    12 where
    13   "sum_rel R1 R2 (Inl a1) (Inl b1) = R1 a1 b1"
    14 | "sum_rel R1 R2 (Inl a1) (Inr b2) = False"
    15 | "sum_rel R1 R2 (Inr a2) (Inl b1) = False"
    16 | "sum_rel R1 R2 (Inr a2) (Inr b2) = R2 a2 b2"
    17 
    18 fun
    19   sum_map
    20 where
    21   "sum_map f1 f2 (Inl a) = Inl (f1 a)"
    22 | "sum_map f1 f2 (Inr a) = Inr (f2 a)"
    23 
    24 declare [[map "+" = (sum_map, sum_rel)]]
    25 
    26 
    27 text {* should probably be in @{theory Sum_Type} *}
    28 lemma split_sum_all:
    29   shows "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (Inl x)) \<and> (\<forall>x. P (Inr x))"
    30   apply(auto)
    31   apply(case_tac x)
    32   apply(simp_all)
    33   done
    34 
    35 lemma sum_equivp[quot_equiv]:
    36   assumes a: "equivp R1"
    37   assumes b: "equivp R2"
    38   shows "equivp (sum_rel R1 R2)"
    39   apply(rule equivpI)
    40   unfolding reflp_def symp_def transp_def
    41   apply(simp_all add: split_sum_all)
    42   apply(blast intro: equivp_reflp[OF a] equivp_reflp[OF b])
    43   apply(blast intro: equivp_symp[OF a] equivp_symp[OF b])
    44   apply(blast intro: equivp_transp[OF a] equivp_transp[OF b])
    45   done
    46 
    47 lemma sum_quotient[quot_thm]:
    48   assumes q1: "Quotient R1 Abs1 Rep1"
    49   assumes q2: "Quotient R2 Abs2 Rep2"
    50   shows "Quotient (sum_rel R1 R2) (sum_map Abs1 Abs2) (sum_map Rep1 Rep2)"
    51   unfolding Quotient_def
    52   apply(simp add: split_sum_all)
    53   apply(simp_all add: Quotient_abs_rep[OF q1] Quotient_rel_rep[OF q1])
    54   apply(simp_all add: Quotient_abs_rep[OF q2] Quotient_rel_rep[OF q2])
    55   using q1 q2
    56   unfolding Quotient_def
    57   apply(blast)+
    58   done
    59 
    60 lemma sum_Inl_rsp[quot_respect]:
    61   assumes q1: "Quotient R1 Abs1 Rep1"
    62   assumes q2: "Quotient R2 Abs2 Rep2"
    63   shows "(R1 ===> sum_rel R1 R2) Inl Inl"
    64   by simp
    65 
    66 lemma sum_Inr_rsp[quot_respect]:
    67   assumes q1: "Quotient R1 Abs1 Rep1"
    68   assumes q2: "Quotient R2 Abs2 Rep2"
    69   shows "(R2 ===> sum_rel R1 R2) Inr Inr"
    70   by simp
    71 
    72 lemma sum_Inl_prs[quot_preserve]:
    73   assumes q1: "Quotient R1 Abs1 Rep1"
    74   assumes q2: "Quotient R2 Abs2 Rep2"
    75   shows "(Rep1 ---> sum_map Abs1 Abs2) Inl = Inl"
    76   apply(simp add: expand_fun_eq)
    77   apply(simp add: Quotient_abs_rep[OF q1])
    78   done
    79 
    80 lemma sum_Inr_prs[quot_preserve]:
    81   assumes q1: "Quotient R1 Abs1 Rep1"
    82   assumes q2: "Quotient R2 Abs2 Rep2"
    83   shows "(Rep2 ---> sum_map Abs1 Abs2) Inr = Inr"
    84   apply(simp add: expand_fun_eq)
    85   apply(simp add: Quotient_abs_rep[OF q2])
    86   done
    87 
    88 lemma sum_map_id[id_simps]:
    89   shows "sum_map id id = id"
    90   by (simp add: expand_fun_eq split_sum_all)
    91 
    92 lemma sum_rel_eq[id_simps]:
    93   shows "sum_rel (op =) (op =) = (op =)"
    94   by (simp add: expand_fun_eq split_sum_all)
    95 
    96 end