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