src/HOL/Library/Quotient_Product.thy
author wenzelm
Sun, 14 Mar 2010 14:31:24 +0100
changeset 35788 f1deaca15ca3
parent 35222 4f1fba00f66d
child 36695 b434506fb0d4
permissions -rw-r--r--
observe standard header format;
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
35788
f1deaca15ca3 observe standard header format;
wenzelm
parents: 35222
diff changeset
     1
(*  Title:      HOL/Library/Quotient_Product.thy
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
     2
    Author:     Cezary Kaliszyk and Christian Urban
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
     3
*)
35788
f1deaca15ca3 observe standard header format;
wenzelm
parents: 35222
diff changeset
     4
f1deaca15ca3 observe standard header format;
wenzelm
parents: 35222
diff changeset
     5
header {* Quotient infrastructure for the product type *}
f1deaca15ca3 observe standard header format;
wenzelm
parents: 35222
diff changeset
     6
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
     7
theory Quotient_Product
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
     8
imports Main Quotient_Syntax
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
     9
begin
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    10
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    11
fun
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    12
  prod_rel
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    13
where
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    14
  "prod_rel R1 R2 = (\<lambda>(a, b) (c, d). R1 a c \<and> R2 b d)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    15
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    16
declare [[map * = (prod_fun, prod_rel)]]
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    17
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    18
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    19
lemma prod_equivp[quot_equiv]:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    20
  assumes a: "equivp R1"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    21
  assumes b: "equivp R2"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    22
  shows "equivp (prod_rel R1 R2)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    23
  apply(rule equivpI)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    24
  unfolding reflp_def symp_def transp_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    25
  apply(simp_all add: split_paired_all)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    26
  apply(blast intro: equivp_reflp[OF a] equivp_reflp[OF b])
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    27
  apply(blast intro: equivp_symp[OF a] equivp_symp[OF b])
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    28
  apply(blast intro: equivp_transp[OF a] equivp_transp[OF b])
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    29
  done
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    30
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    31
lemma prod_quotient[quot_thm]:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    32
  assumes q1: "Quotient R1 Abs1 Rep1"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    33
  assumes q2: "Quotient R2 Abs2 Rep2"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    34
  shows "Quotient (prod_rel R1 R2) (prod_fun Abs1 Abs2) (prod_fun Rep1 Rep2)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    35
  unfolding Quotient_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    36
  apply(simp add: split_paired_all)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    37
  apply(simp add: Quotient_abs_rep[OF q1] Quotient_rel_rep[OF q1])
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    38
  apply(simp add: Quotient_abs_rep[OF q2] Quotient_rel_rep[OF q2])
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    39
  using q1 q2
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    40
  unfolding Quotient_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    41
  apply(blast)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    42
  done
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    43
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    44
lemma Pair_rsp[quot_respect]:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    45
  assumes q1: "Quotient R1 Abs1 Rep1"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    46
  assumes q2: "Quotient R2 Abs2 Rep2"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    47
  shows "(R1 ===> R2 ===> prod_rel R1 R2) Pair Pair"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    48
  by simp
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    49
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    50
lemma Pair_prs[quot_preserve]:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    51
  assumes q1: "Quotient R1 Abs1 Rep1"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    52
  assumes q2: "Quotient R2 Abs2 Rep2"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    53
  shows "(Rep1 ---> Rep2 ---> (prod_fun Abs1 Abs2)) Pair = Pair"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    54
  apply(simp add: expand_fun_eq)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    55
  apply(simp add: Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    56
  done
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    57
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    58
lemma fst_rsp[quot_respect]:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    59
  assumes "Quotient R1 Abs1 Rep1"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    60
  assumes "Quotient R2 Abs2 Rep2"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    61
  shows "(prod_rel R1 R2 ===> R1) fst fst"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    62
  by simp
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    63
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    64
lemma fst_prs[quot_preserve]:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    65
  assumes q1: "Quotient R1 Abs1 Rep1"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    66
  assumes q2: "Quotient R2 Abs2 Rep2"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    67
  shows "(prod_fun Rep1 Rep2 ---> Abs1) fst = fst"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    68
  apply(simp add: expand_fun_eq)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    69
  apply(simp add: Quotient_abs_rep[OF q1])
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    70
  done
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    71
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    72
lemma snd_rsp[quot_respect]:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    73
  assumes "Quotient R1 Abs1 Rep1"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    74
  assumes "Quotient R2 Abs2 Rep2"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    75
  shows "(prod_rel R1 R2 ===> R2) snd snd"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    76
  by simp
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    77
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    78
lemma snd_prs[quot_preserve]:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    79
  assumes q1: "Quotient R1 Abs1 Rep1"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    80
  assumes q2: "Quotient R2 Abs2 Rep2"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    81
  shows "(prod_fun Rep1 Rep2 ---> Abs2) snd = snd"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    82
  apply(simp add: expand_fun_eq)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    83
  apply(simp add: Quotient_abs_rep[OF q2])
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    84
  done
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    85
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    86
lemma split_rsp[quot_respect]:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    87
  shows "((R1 ===> R2 ===> (op =)) ===> (prod_rel R1 R2) ===> (op =)) split split"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    88
  by auto
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    89
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    90
lemma split_prs[quot_preserve]:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    91
  assumes q1: "Quotient R1 Abs1 Rep1"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    92
  and     q2: "Quotient R2 Abs2 Rep2"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    93
  shows "(((Abs1 ---> Abs2 ---> id) ---> prod_fun Rep1 Rep2 ---> id) split) = split"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    94
  by (simp add: expand_fun_eq Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    95
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    96
lemma prod_fun_id[id_simps]:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    97
  shows "prod_fun id id = id"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    98
  by (simp add: prod_fun_def)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
    99
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   100
lemma prod_rel_eq[id_simps]:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   101
  shows "prod_rel (op =) (op =) = (op =)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   102
  by (simp add: expand_fun_eq)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   103
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   104
end