src/HOL/Library/Quotient_Sum.thy
 author haftmann Thu Nov 18 17:01:16 2010 +0100 (2010-11-18) changeset 40610 70776ecfe324 parent 40542 9a173a22771c child 40820 fd9c98ead9a9 permissions -rw-r--r--
mapper for sum type
```     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 declare [[map sum = (sum_map, sum_rel)]]
```
```    20
```
```    21
```
```    22 text {* should probably be in @{theory Sum_Type} *}
```
```    23 lemma split_sum_all:
```
```    24   shows "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (Inl x)) \<and> (\<forall>x. P (Inr x))"
```
```    25   apply(auto)
```
```    26   apply(case_tac x)
```
```    27   apply(simp_all)
```
```    28   done
```
```    29
```
```    30 lemma sum_equivp[quot_equiv]:
```
```    31   assumes a: "equivp R1"
```
```    32   assumes b: "equivp R2"
```
```    33   shows "equivp (sum_rel R1 R2)"
```
```    34   apply(rule equivpI)
```
```    35   unfolding reflp_def symp_def transp_def
```
```    36   apply(simp_all add: split_sum_all)
```
```    37   apply(blast intro: equivp_reflp[OF a] equivp_reflp[OF b])
```
```    38   apply(blast intro: equivp_symp[OF a] equivp_symp[OF b])
```
```    39   apply(blast intro: equivp_transp[OF a] equivp_transp[OF b])
```
```    40   done
```
```    41
```
```    42 lemma sum_quotient[quot_thm]:
```
```    43   assumes q1: "Quotient R1 Abs1 Rep1"
```
```    44   assumes q2: "Quotient R2 Abs2 Rep2"
```
```    45   shows "Quotient (sum_rel R1 R2) (sum_map Abs1 Abs2) (sum_map Rep1 Rep2)"
```
```    46   unfolding Quotient_def
```
```    47   apply(simp add: split_sum_all)
```
```    48   apply(simp_all add: Quotient_abs_rep[OF q1] Quotient_rel_rep[OF q1])
```
```    49   apply(simp_all add: Quotient_abs_rep[OF q2] Quotient_rel_rep[OF q2])
```
```    50   using q1 q2
```
```    51   unfolding Quotient_def
```
```    52   apply(blast)+
```
```    53   done
```
```    54
```
```    55 lemma sum_Inl_rsp[quot_respect]:
```
```    56   assumes q1: "Quotient R1 Abs1 Rep1"
```
```    57   assumes q2: "Quotient R2 Abs2 Rep2"
```
```    58   shows "(R1 ===> sum_rel R1 R2) Inl Inl"
```
```    59   by auto
```
```    60
```
```    61 lemma sum_Inr_rsp[quot_respect]:
```
```    62   assumes q1: "Quotient R1 Abs1 Rep1"
```
```    63   assumes q2: "Quotient R2 Abs2 Rep2"
```
```    64   shows "(R2 ===> sum_rel R1 R2) Inr Inr"
```
```    65   by auto
```
```    66
```
```    67 lemma sum_Inl_prs[quot_preserve]:
```
```    68   assumes q1: "Quotient R1 Abs1 Rep1"
```
```    69   assumes q2: "Quotient R2 Abs2 Rep2"
```
```    70   shows "(Rep1 ---> sum_map Abs1 Abs2) Inl = Inl"
```
```    71   apply(simp add: fun_eq_iff)
```
```    72   apply(simp add: Quotient_abs_rep[OF q1])
```
```    73   done
```
```    74
```
```    75 lemma sum_Inr_prs[quot_preserve]:
```
```    76   assumes q1: "Quotient R1 Abs1 Rep1"
```
```    77   assumes q2: "Quotient R2 Abs2 Rep2"
```
```    78   shows "(Rep2 ---> sum_map Abs1 Abs2) Inr = Inr"
```
```    79   apply(simp add: fun_eq_iff)
```
```    80   apply(simp add: Quotient_abs_rep[OF q2])
```
```    81   done
```
```    82
```
```    83 lemma sum_map_id[id_simps]:
```
```    84   shows "sum_map id id = id"
```
```    85   by (simp add: fun_eq_iff split_sum_all)
```
```    86
```
```    87 lemma sum_rel_eq[id_simps]:
```
```    88   shows "sum_rel (op =) (op =) = (op =)"
```
```    89   by (simp add: fun_eq_iff split_sum_all)
```
```    90
```
```    91 end
```