src/HOL/Quotient.thy
author Cezary Kaliszyk <kaliszyk@in.tum.de>
Fri May 21 10:40:59 2010 +0200 (2010-05-21)
changeset 37049 ca1c293e521e
parent 36276 92011cc923f5
child 37493 2377d246a631
permissions -rw-r--r--
Let rsp and prs in fun_rel/fun_map format
kaliszyk@35222
     1
(*  Title:      Quotient.thy
kaliszyk@35222
     2
    Author:     Cezary Kaliszyk and Christian Urban
kaliszyk@35222
     3
*)
kaliszyk@35222
     4
huffman@35294
     5
header {* Definition of Quotient Types *}
huffman@35294
     6
kaliszyk@35222
     7
theory Quotient
blanchet@35827
     8
imports Plain Sledgehammer
kaliszyk@35222
     9
uses
kaliszyk@35222
    10
  ("~~/src/HOL/Tools/Quotient/quotient_info.ML")
kaliszyk@35222
    11
  ("~~/src/HOL/Tools/Quotient/quotient_typ.ML")
kaliszyk@35222
    12
  ("~~/src/HOL/Tools/Quotient/quotient_def.ML")
kaliszyk@35222
    13
  ("~~/src/HOL/Tools/Quotient/quotient_term.ML")
kaliszyk@35222
    14
  ("~~/src/HOL/Tools/Quotient/quotient_tacs.ML")
kaliszyk@35222
    15
begin
kaliszyk@35222
    16
kaliszyk@35222
    17
kaliszyk@35222
    18
text {*
kaliszyk@35222
    19
  Basic definition for equivalence relations
kaliszyk@35222
    20
  that are represented by predicates.
kaliszyk@35222
    21
*}
kaliszyk@35222
    22
kaliszyk@35222
    23
definition
kaliszyk@35222
    24
  "equivp E \<equiv> \<forall>x y. E x y = (E x = E y)"
kaliszyk@35222
    25
kaliszyk@35222
    26
definition
kaliszyk@35222
    27
  "reflp E \<equiv> \<forall>x. E x x"
kaliszyk@35222
    28
kaliszyk@35222
    29
definition
kaliszyk@35222
    30
  "symp E \<equiv> \<forall>x y. E x y \<longrightarrow> E y x"
kaliszyk@35222
    31
kaliszyk@35222
    32
definition
kaliszyk@35222
    33
  "transp E \<equiv> \<forall>x y z. E x y \<and> E y z \<longrightarrow> E x z"
kaliszyk@35222
    34
kaliszyk@35222
    35
lemma equivp_reflp_symp_transp:
kaliszyk@35222
    36
  shows "equivp E = (reflp E \<and> symp E \<and> transp E)"
kaliszyk@35222
    37
  unfolding equivp_def reflp_def symp_def transp_def expand_fun_eq
kaliszyk@35222
    38
  by blast
kaliszyk@35222
    39
kaliszyk@35222
    40
lemma equivp_reflp:
kaliszyk@35222
    41
  shows "equivp E \<Longrightarrow> E x x"
kaliszyk@35222
    42
  by (simp only: equivp_reflp_symp_transp reflp_def)
kaliszyk@35222
    43
kaliszyk@35222
    44
lemma equivp_symp:
kaliszyk@35222
    45
  shows "equivp E \<Longrightarrow> E x y \<Longrightarrow> E y x"
kaliszyk@35222
    46
  by (metis equivp_reflp_symp_transp symp_def)
kaliszyk@35222
    47
kaliszyk@35222
    48
lemma equivp_transp:
kaliszyk@35222
    49
  shows "equivp E \<Longrightarrow> E x y \<Longrightarrow> E y z \<Longrightarrow> E x z"
kaliszyk@35222
    50
  by (metis equivp_reflp_symp_transp transp_def)
kaliszyk@35222
    51
kaliszyk@35222
    52
lemma equivpI:
kaliszyk@35222
    53
  assumes "reflp R" "symp R" "transp R"
kaliszyk@35222
    54
  shows "equivp R"
kaliszyk@35222
    55
  using assms by (simp add: equivp_reflp_symp_transp)
kaliszyk@35222
    56
kaliszyk@35222
    57
lemma identity_equivp:
kaliszyk@35222
    58
  shows "equivp (op =)"
kaliszyk@35222
    59
  unfolding equivp_def
kaliszyk@35222
    60
  by auto
kaliszyk@35222
    61
kaliszyk@35222
    62
text {* Partial equivalences: not yet used anywhere *}
kaliszyk@35222
    63
kaliszyk@35222
    64
definition
kaliszyk@35222
    65
  "part_equivp E \<equiv> (\<exists>x. E x x) \<and> (\<forall>x y. E x y = (E x x \<and> E y y \<and> (E x = E y)))"
kaliszyk@35222
    66
kaliszyk@35222
    67
lemma equivp_implies_part_equivp:
kaliszyk@35222
    68
  assumes a: "equivp E"
kaliszyk@35222
    69
  shows "part_equivp E"
kaliszyk@35222
    70
  using a
kaliszyk@35222
    71
  unfolding equivp_def part_equivp_def
kaliszyk@35222
    72
  by auto
kaliszyk@35222
    73
kaliszyk@35222
    74
text {* Composition of Relations *}
kaliszyk@35222
    75
kaliszyk@35222
    76
abbreviation
kaliszyk@35222
    77
  rel_conj (infixr "OOO" 75)
kaliszyk@35222
    78
where
kaliszyk@35222
    79
  "r1 OOO r2 \<equiv> r1 OO r2 OO r1"
kaliszyk@35222
    80
kaliszyk@35222
    81
lemma eq_comp_r:
kaliszyk@35222
    82
  shows "((op =) OOO R) = R"
kaliszyk@35222
    83
  by (auto simp add: expand_fun_eq)
kaliszyk@35222
    84
huffman@35294
    85
subsection {* Respects predicate *}
kaliszyk@35222
    86
kaliszyk@35222
    87
definition
kaliszyk@35222
    88
  Respects
kaliszyk@35222
    89
where
kaliszyk@35222
    90
  "Respects R x \<equiv> R x x"
kaliszyk@35222
    91
kaliszyk@35222
    92
lemma in_respects:
kaliszyk@35222
    93
  shows "(x \<in> Respects R) = R x x"
kaliszyk@35222
    94
  unfolding mem_def Respects_def
kaliszyk@35222
    95
  by simp
kaliszyk@35222
    96
huffman@35294
    97
subsection {* Function map and function relation *}
kaliszyk@35222
    98
kaliszyk@35222
    99
definition
kaliszyk@35222
   100
  fun_map (infixr "--->" 55)
kaliszyk@35222
   101
where
kaliszyk@35222
   102
[simp]: "fun_map f g h x = g (h (f x))"
kaliszyk@35222
   103
kaliszyk@35222
   104
definition
kaliszyk@35222
   105
  fun_rel (infixr "===>" 55)
kaliszyk@35222
   106
where
kaliszyk@35222
   107
[simp]: "fun_rel E1 E2 f g = (\<forall>x y. E1 x y \<longrightarrow> E2 (f x) (g y))"
kaliszyk@35222
   108
kaliszyk@36276
   109
lemma fun_relI [intro]:
kaliszyk@36276
   110
  assumes "\<And>a b. P a b \<Longrightarrow> Q (x a) (y b)"
kaliszyk@36276
   111
  shows "(P ===> Q) x y"
kaliszyk@36276
   112
  using assms by (simp add: fun_rel_def)
kaliszyk@35222
   113
kaliszyk@35222
   114
lemma fun_map_id:
kaliszyk@35222
   115
  shows "(id ---> id) = id"
kaliszyk@35222
   116
  by (simp add: expand_fun_eq id_def)
kaliszyk@35222
   117
kaliszyk@35222
   118
lemma fun_rel_eq:
kaliszyk@35222
   119
  shows "((op =) ===> (op =)) = (op =)"
kaliszyk@35222
   120
  by (simp add: expand_fun_eq)
kaliszyk@35222
   121
kaliszyk@35222
   122
lemma fun_rel_id:
kaliszyk@35222
   123
  assumes a: "\<And>x y. R1 x y \<Longrightarrow> R2 (f x) (g y)"
kaliszyk@35222
   124
  shows "(R1 ===> R2) f g"
kaliszyk@35222
   125
  using a by simp
kaliszyk@35222
   126
kaliszyk@35222
   127
lemma fun_rel_id_asm:
kaliszyk@35222
   128
  assumes a: "\<And>x y. R1 x y \<Longrightarrow> (A \<longrightarrow> R2 (f x) (g y))"
kaliszyk@35222
   129
  shows "A \<longrightarrow> (R1 ===> R2) f g"
kaliszyk@35222
   130
  using a by auto
kaliszyk@35222
   131
kaliszyk@35222
   132
huffman@35294
   133
subsection {* Quotient Predicate *}
kaliszyk@35222
   134
kaliszyk@35222
   135
definition
kaliszyk@35222
   136
  "Quotient E Abs Rep \<equiv>
kaliszyk@35222
   137
     (\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. E (Rep a) (Rep a)) \<and>
kaliszyk@35222
   138
     (\<forall>r s. E r s = (E r r \<and> E s s \<and> (Abs r = Abs s)))"
kaliszyk@35222
   139
kaliszyk@35222
   140
lemma Quotient_abs_rep:
kaliszyk@35222
   141
  assumes a: "Quotient E Abs Rep"
kaliszyk@35222
   142
  shows "Abs (Rep a) = a"
kaliszyk@35222
   143
  using a
kaliszyk@35222
   144
  unfolding Quotient_def
kaliszyk@35222
   145
  by simp
kaliszyk@35222
   146
kaliszyk@35222
   147
lemma Quotient_rep_reflp:
kaliszyk@35222
   148
  assumes a: "Quotient E Abs Rep"
kaliszyk@35222
   149
  shows "E (Rep a) (Rep a)"
kaliszyk@35222
   150
  using a
kaliszyk@35222
   151
  unfolding Quotient_def
kaliszyk@35222
   152
  by blast
kaliszyk@35222
   153
kaliszyk@35222
   154
lemma Quotient_rel:
kaliszyk@35222
   155
  assumes a: "Quotient E Abs Rep"
kaliszyk@35222
   156
  shows " E r s = (E r r \<and> E s s \<and> (Abs r = Abs s))"
kaliszyk@35222
   157
  using a
kaliszyk@35222
   158
  unfolding Quotient_def
kaliszyk@35222
   159
  by blast
kaliszyk@35222
   160
kaliszyk@35222
   161
lemma Quotient_rel_rep:
kaliszyk@35222
   162
  assumes a: "Quotient R Abs Rep"
kaliszyk@35222
   163
  shows "R (Rep a) (Rep b) = (a = b)"
kaliszyk@35222
   164
  using a
kaliszyk@35222
   165
  unfolding Quotient_def
kaliszyk@35222
   166
  by metis
kaliszyk@35222
   167
kaliszyk@35222
   168
lemma Quotient_rep_abs:
kaliszyk@35222
   169
  assumes a: "Quotient R Abs Rep"
kaliszyk@35222
   170
  shows "R r r \<Longrightarrow> R (Rep (Abs r)) r"
kaliszyk@35222
   171
  using a unfolding Quotient_def
kaliszyk@35222
   172
  by blast
kaliszyk@35222
   173
kaliszyk@35222
   174
lemma Quotient_rel_abs:
kaliszyk@35222
   175
  assumes a: "Quotient E Abs Rep"
kaliszyk@35222
   176
  shows "E r s \<Longrightarrow> Abs r = Abs s"
kaliszyk@35222
   177
  using a unfolding Quotient_def
kaliszyk@35222
   178
  by blast
kaliszyk@35222
   179
kaliszyk@35222
   180
lemma Quotient_symp:
kaliszyk@35222
   181
  assumes a: "Quotient E Abs Rep"
kaliszyk@35222
   182
  shows "symp E"
kaliszyk@35222
   183
  using a unfolding Quotient_def symp_def
kaliszyk@35222
   184
  by metis
kaliszyk@35222
   185
kaliszyk@35222
   186
lemma Quotient_transp:
kaliszyk@35222
   187
  assumes a: "Quotient E Abs Rep"
kaliszyk@35222
   188
  shows "transp E"
kaliszyk@35222
   189
  using a unfolding Quotient_def transp_def
kaliszyk@35222
   190
  by metis
kaliszyk@35222
   191
kaliszyk@35222
   192
lemma identity_quotient:
kaliszyk@35222
   193
  shows "Quotient (op =) id id"
kaliszyk@35222
   194
  unfolding Quotient_def id_def
kaliszyk@35222
   195
  by blast
kaliszyk@35222
   196
kaliszyk@35222
   197
lemma fun_quotient:
kaliszyk@35222
   198
  assumes q1: "Quotient R1 abs1 rep1"
kaliszyk@35222
   199
  and     q2: "Quotient R2 abs2 rep2"
kaliszyk@35222
   200
  shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
kaliszyk@35222
   201
proof -
kaliszyk@35222
   202
  have "\<forall>a. (rep1 ---> abs2) ((abs1 ---> rep2) a) = a"
kaliszyk@35222
   203
    using q1 q2
kaliszyk@35222
   204
    unfolding Quotient_def
kaliszyk@35222
   205
    unfolding expand_fun_eq
kaliszyk@35222
   206
    by simp
kaliszyk@35222
   207
  moreover
kaliszyk@35222
   208
  have "\<forall>a. (R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"
kaliszyk@35222
   209
    using q1 q2
kaliszyk@35222
   210
    unfolding Quotient_def
kaliszyk@35222
   211
    by (simp (no_asm)) (metis)
kaliszyk@35222
   212
  moreover
kaliszyk@35222
   213
  have "\<forall>r s. (R1 ===> R2) r s = ((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
kaliszyk@35222
   214
        (rep1 ---> abs2) r  = (rep1 ---> abs2) s)"
kaliszyk@35222
   215
    unfolding expand_fun_eq
kaliszyk@35222
   216
    apply(auto)
kaliszyk@35222
   217
    using q1 q2 unfolding Quotient_def
kaliszyk@35222
   218
    apply(metis)
kaliszyk@35222
   219
    using q1 q2 unfolding Quotient_def
kaliszyk@35222
   220
    apply(metis)
kaliszyk@35222
   221
    using q1 q2 unfolding Quotient_def
kaliszyk@35222
   222
    apply(metis)
kaliszyk@35222
   223
    using q1 q2 unfolding Quotient_def
kaliszyk@35222
   224
    apply(metis)
kaliszyk@35222
   225
    done
kaliszyk@35222
   226
  ultimately
kaliszyk@35222
   227
  show "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
kaliszyk@35222
   228
    unfolding Quotient_def by blast
kaliszyk@35222
   229
qed
kaliszyk@35222
   230
kaliszyk@35222
   231
lemma abs_o_rep:
kaliszyk@35222
   232
  assumes a: "Quotient R Abs Rep"
kaliszyk@35222
   233
  shows "Abs o Rep = id"
kaliszyk@35222
   234
  unfolding expand_fun_eq
kaliszyk@35222
   235
  by (simp add: Quotient_abs_rep[OF a])
kaliszyk@35222
   236
kaliszyk@35222
   237
lemma equals_rsp:
kaliszyk@35222
   238
  assumes q: "Quotient R Abs Rep"
kaliszyk@35222
   239
  and     a: "R xa xb" "R ya yb"
kaliszyk@35222
   240
  shows "R xa ya = R xb yb"
kaliszyk@35222
   241
  using a Quotient_symp[OF q] Quotient_transp[OF q]
kaliszyk@35222
   242
  unfolding symp_def transp_def
kaliszyk@35222
   243
  by blast
kaliszyk@35222
   244
kaliszyk@35222
   245
lemma lambda_prs:
kaliszyk@35222
   246
  assumes q1: "Quotient R1 Abs1 Rep1"
kaliszyk@35222
   247
  and     q2: "Quotient R2 Abs2 Rep2"
kaliszyk@35222
   248
  shows "(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) = (\<lambda>x. f x)"
kaliszyk@35222
   249
  unfolding expand_fun_eq
kaliszyk@35222
   250
  using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
kaliszyk@35222
   251
  by simp
kaliszyk@35222
   252
kaliszyk@35222
   253
lemma lambda_prs1:
kaliszyk@35222
   254
  assumes q1: "Quotient R1 Abs1 Rep1"
kaliszyk@35222
   255
  and     q2: "Quotient R2 Abs2 Rep2"
kaliszyk@35222
   256
  shows "(Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x) = (\<lambda>x. f x)"
kaliszyk@35222
   257
  unfolding expand_fun_eq
kaliszyk@35222
   258
  using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
kaliszyk@35222
   259
  by simp
kaliszyk@35222
   260
kaliszyk@35222
   261
lemma rep_abs_rsp:
kaliszyk@35222
   262
  assumes q: "Quotient R Abs Rep"
kaliszyk@35222
   263
  and     a: "R x1 x2"
kaliszyk@35222
   264
  shows "R x1 (Rep (Abs x2))"
kaliszyk@35222
   265
  using a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]
kaliszyk@35222
   266
  by metis
kaliszyk@35222
   267
kaliszyk@35222
   268
lemma rep_abs_rsp_left:
kaliszyk@35222
   269
  assumes q: "Quotient R Abs Rep"
kaliszyk@35222
   270
  and     a: "R x1 x2"
kaliszyk@35222
   271
  shows "R (Rep (Abs x1)) x2"
kaliszyk@35222
   272
  using a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]
kaliszyk@35222
   273
  by metis
kaliszyk@35222
   274
kaliszyk@35222
   275
text{*
kaliszyk@35222
   276
  In the following theorem R1 can be instantiated with anything,
kaliszyk@35222
   277
  but we know some of the types of the Rep and Abs functions;
kaliszyk@35222
   278
  so by solving Quotient assumptions we can get a unique R1 that
kaliszyk@35236
   279
  will be provable; which is why we need to use @{text apply_rsp} and
kaliszyk@35222
   280
  not the primed version *}
kaliszyk@35222
   281
kaliszyk@35222
   282
lemma apply_rsp:
kaliszyk@35222
   283
  fixes f g::"'a \<Rightarrow> 'c"
kaliszyk@35222
   284
  assumes q: "Quotient R1 Abs1 Rep1"
kaliszyk@35222
   285
  and     a: "(R1 ===> R2) f g" "R1 x y"
kaliszyk@35222
   286
  shows "R2 (f x) (g y)"
kaliszyk@35222
   287
  using a by simp
kaliszyk@35222
   288
kaliszyk@35222
   289
lemma apply_rsp':
kaliszyk@35222
   290
  assumes a: "(R1 ===> R2) f g" "R1 x y"
kaliszyk@35222
   291
  shows "R2 (f x) (g y)"
kaliszyk@35222
   292
  using a by simp
kaliszyk@35222
   293
huffman@35294
   294
subsection {* lemmas for regularisation of ball and bex *}
kaliszyk@35222
   295
kaliszyk@35222
   296
lemma ball_reg_eqv:
kaliszyk@35222
   297
  fixes P :: "'a \<Rightarrow> bool"
kaliszyk@35222
   298
  assumes a: "equivp R"
kaliszyk@35222
   299
  shows "Ball (Respects R) P = (All P)"
kaliszyk@35222
   300
  using a
kaliszyk@35222
   301
  unfolding equivp_def
kaliszyk@35222
   302
  by (auto simp add: in_respects)
kaliszyk@35222
   303
kaliszyk@35222
   304
lemma bex_reg_eqv:
kaliszyk@35222
   305
  fixes P :: "'a \<Rightarrow> bool"
kaliszyk@35222
   306
  assumes a: "equivp R"
kaliszyk@35222
   307
  shows "Bex (Respects R) P = (Ex P)"
kaliszyk@35222
   308
  using a
kaliszyk@35222
   309
  unfolding equivp_def
kaliszyk@35222
   310
  by (auto simp add: in_respects)
kaliszyk@35222
   311
kaliszyk@35222
   312
lemma ball_reg_right:
kaliszyk@35222
   313
  assumes a: "\<And>x. R x \<Longrightarrow> P x \<longrightarrow> Q x"
kaliszyk@35222
   314
  shows "All P \<longrightarrow> Ball R Q"
kaliszyk@35222
   315
  using a by (metis COMBC_def Collect_def Collect_mem_eq)
kaliszyk@35222
   316
kaliszyk@35222
   317
lemma bex_reg_left:
kaliszyk@35222
   318
  assumes a: "\<And>x. R x \<Longrightarrow> Q x \<longrightarrow> P x"
kaliszyk@35222
   319
  shows "Bex R Q \<longrightarrow> Ex P"
kaliszyk@35222
   320
  using a by (metis COMBC_def Collect_def Collect_mem_eq)
kaliszyk@35222
   321
kaliszyk@35222
   322
lemma ball_reg_left:
kaliszyk@35222
   323
  assumes a: "equivp R"
kaliszyk@35222
   324
  shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ball (Respects R) Q \<longrightarrow> All P"
kaliszyk@35222
   325
  using a by (metis equivp_reflp in_respects)
kaliszyk@35222
   326
kaliszyk@35222
   327
lemma bex_reg_right:
kaliszyk@35222
   328
  assumes a: "equivp R"
kaliszyk@35222
   329
  shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ex Q \<longrightarrow> Bex (Respects R) P"
kaliszyk@35222
   330
  using a by (metis equivp_reflp in_respects)
kaliszyk@35222
   331
kaliszyk@35222
   332
lemma ball_reg_eqv_range:
kaliszyk@35222
   333
  fixes P::"'a \<Rightarrow> bool"
kaliszyk@35222
   334
  and x::"'a"
kaliszyk@35222
   335
  assumes a: "equivp R2"
kaliszyk@35222
   336
  shows   "(Ball (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = All (\<lambda>f. P (f x)))"
kaliszyk@35222
   337
  apply(rule iffI)
kaliszyk@35222
   338
  apply(rule allI)
kaliszyk@35222
   339
  apply(drule_tac x="\<lambda>y. f x" in bspec)
kaliszyk@35222
   340
  apply(simp add: in_respects)
kaliszyk@35222
   341
  apply(rule impI)
kaliszyk@35222
   342
  using a equivp_reflp_symp_transp[of "R2"]
kaliszyk@35222
   343
  apply(simp add: reflp_def)
kaliszyk@35222
   344
  apply(simp)
kaliszyk@35222
   345
  apply(simp)
kaliszyk@35222
   346
  done
kaliszyk@35222
   347
kaliszyk@35222
   348
lemma bex_reg_eqv_range:
kaliszyk@35222
   349
  assumes a: "equivp R2"
kaliszyk@35222
   350
  shows   "(Bex (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = Ex (\<lambda>f. P (f x)))"
kaliszyk@35222
   351
  apply(auto)
kaliszyk@35222
   352
  apply(rule_tac x="\<lambda>y. f x" in bexI)
kaliszyk@35222
   353
  apply(simp)
kaliszyk@35222
   354
  apply(simp add: Respects_def in_respects)
kaliszyk@35222
   355
  apply(rule impI)
kaliszyk@35222
   356
  using a equivp_reflp_symp_transp[of "R2"]
kaliszyk@35222
   357
  apply(simp add: reflp_def)
kaliszyk@35222
   358
  done
kaliszyk@35222
   359
kaliszyk@35222
   360
(* Next four lemmas are unused *)
kaliszyk@35222
   361
lemma all_reg:
kaliszyk@35222
   362
  assumes a: "!x :: 'a. (P x --> Q x)"
kaliszyk@35222
   363
  and     b: "All P"
kaliszyk@35222
   364
  shows "All Q"
kaliszyk@35222
   365
  using a b by (metis)
kaliszyk@35222
   366
kaliszyk@35222
   367
lemma ex_reg:
kaliszyk@35222
   368
  assumes a: "!x :: 'a. (P x --> Q x)"
kaliszyk@35222
   369
  and     b: "Ex P"
kaliszyk@35222
   370
  shows "Ex Q"
kaliszyk@35222
   371
  using a b by metis
kaliszyk@35222
   372
kaliszyk@35222
   373
lemma ball_reg:
kaliszyk@35222
   374
  assumes a: "!x :: 'a. (R x --> P x --> Q x)"
kaliszyk@35222
   375
  and     b: "Ball R P"
kaliszyk@35222
   376
  shows "Ball R Q"
kaliszyk@35222
   377
  using a b by (metis COMBC_def Collect_def Collect_mem_eq)
kaliszyk@35222
   378
kaliszyk@35222
   379
lemma bex_reg:
kaliszyk@35222
   380
  assumes a: "!x :: 'a. (R x --> P x --> Q x)"
kaliszyk@35222
   381
  and     b: "Bex R P"
kaliszyk@35222
   382
  shows "Bex R Q"
kaliszyk@35222
   383
  using a b by (metis COMBC_def Collect_def Collect_mem_eq)
kaliszyk@35222
   384
kaliszyk@35222
   385
kaliszyk@35222
   386
lemma ball_all_comm:
kaliszyk@35222
   387
  assumes "\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)"
kaliszyk@35222
   388
  shows "(\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y)"
kaliszyk@35222
   389
  using assms by auto
kaliszyk@35222
   390
kaliszyk@35222
   391
lemma bex_ex_comm:
kaliszyk@35222
   392
  assumes "(\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)"
kaliszyk@35222
   393
  shows "(\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y)"
kaliszyk@35222
   394
  using assms by auto
kaliszyk@35222
   395
huffman@35294
   396
subsection {* Bounded abstraction *}
kaliszyk@35222
   397
kaliszyk@35222
   398
definition
kaliszyk@35222
   399
  Babs :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
kaliszyk@35222
   400
where
kaliszyk@35222
   401
  "x \<in> p \<Longrightarrow> Babs p m x = m x"
kaliszyk@35222
   402
kaliszyk@35222
   403
lemma babs_rsp:
kaliszyk@35222
   404
  assumes q: "Quotient R1 Abs1 Rep1"
kaliszyk@35222
   405
  and     a: "(R1 ===> R2) f g"
kaliszyk@35222
   406
  shows      "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"
kaliszyk@35222
   407
  apply (auto simp add: Babs_def in_respects)
kaliszyk@35222
   408
  apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
kaliszyk@35222
   409
  using a apply (simp add: Babs_def)
kaliszyk@35222
   410
  apply (simp add: in_respects)
kaliszyk@35222
   411
  using Quotient_rel[OF q]
kaliszyk@35222
   412
  by metis
kaliszyk@35222
   413
kaliszyk@35222
   414
lemma babs_prs:
kaliszyk@35222
   415
  assumes q1: "Quotient R1 Abs1 Rep1"
kaliszyk@35222
   416
  and     q2: "Quotient R2 Abs2 Rep2"
kaliszyk@35222
   417
  shows "((Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f))) = f"
kaliszyk@35222
   418
  apply (rule ext)
kaliszyk@35222
   419
  apply (simp)
kaliszyk@35222
   420
  apply (subgoal_tac "Rep1 x \<in> Respects R1")
kaliszyk@35222
   421
  apply (simp add: Babs_def Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])
kaliszyk@35222
   422
  apply (simp add: in_respects Quotient_rel_rep[OF q1])
kaliszyk@35222
   423
  done
kaliszyk@35222
   424
kaliszyk@35222
   425
lemma babs_simp:
kaliszyk@35222
   426
  assumes q: "Quotient R1 Abs Rep"
kaliszyk@35222
   427
  shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"
kaliszyk@35222
   428
  apply(rule iffI)
kaliszyk@35222
   429
  apply(simp_all only: babs_rsp[OF q])
kaliszyk@35222
   430
  apply(auto simp add: Babs_def)
kaliszyk@35222
   431
  apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
kaliszyk@35222
   432
  apply(metis Babs_def)
kaliszyk@35222
   433
  apply (simp add: in_respects)
kaliszyk@35222
   434
  using Quotient_rel[OF q]
kaliszyk@35222
   435
  by metis
kaliszyk@35222
   436
kaliszyk@35222
   437
(* If a user proves that a particular functional relation
kaliszyk@35222
   438
   is an equivalence this may be useful in regularising *)
kaliszyk@35222
   439
lemma babs_reg_eqv:
kaliszyk@35222
   440
  shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"
kaliszyk@35222
   441
  by (simp add: expand_fun_eq Babs_def in_respects equivp_reflp)
kaliszyk@35222
   442
kaliszyk@35222
   443
kaliszyk@35222
   444
(* 3 lemmas needed for proving repabs_inj *)
kaliszyk@35222
   445
lemma ball_rsp:
kaliszyk@35222
   446
  assumes a: "(R ===> (op =)) f g"
kaliszyk@35222
   447
  shows "Ball (Respects R) f = Ball (Respects R) g"
kaliszyk@35222
   448
  using a by (simp add: Ball_def in_respects)
kaliszyk@35222
   449
kaliszyk@35222
   450
lemma bex_rsp:
kaliszyk@35222
   451
  assumes a: "(R ===> (op =)) f g"
kaliszyk@35222
   452
  shows "(Bex (Respects R) f = Bex (Respects R) g)"
kaliszyk@35222
   453
  using a by (simp add: Bex_def in_respects)
kaliszyk@35222
   454
kaliszyk@35222
   455
lemma bex1_rsp:
kaliszyk@35222
   456
  assumes a: "(R ===> (op =)) f g"
kaliszyk@35222
   457
  shows "Ex1 (\<lambda>x. x \<in> Respects R \<and> f x) = Ex1 (\<lambda>x. x \<in> Respects R \<and> g x)"
kaliszyk@35222
   458
  using a
kaliszyk@35222
   459
  by (simp add: Ex1_def in_respects) auto
kaliszyk@35222
   460
kaliszyk@35222
   461
(* 2 lemmas needed for cleaning of quantifiers *)
kaliszyk@35222
   462
lemma all_prs:
kaliszyk@35222
   463
  assumes a: "Quotient R absf repf"
kaliszyk@35222
   464
  shows "Ball (Respects R) ((absf ---> id) f) = All f"
kaliszyk@35222
   465
  using a unfolding Quotient_def Ball_def in_respects fun_map_def id_apply
kaliszyk@35222
   466
  by metis
kaliszyk@35222
   467
kaliszyk@35222
   468
lemma ex_prs:
kaliszyk@35222
   469
  assumes a: "Quotient R absf repf"
kaliszyk@35222
   470
  shows "Bex (Respects R) ((absf ---> id) f) = Ex f"
kaliszyk@35222
   471
  using a unfolding Quotient_def Bex_def in_respects fun_map_def id_apply
kaliszyk@35222
   472
  by metis
kaliszyk@35222
   473
huffman@35294
   474
subsection {* @{text Bex1_rel} quantifier *}
kaliszyk@35222
   475
kaliszyk@35222
   476
definition
kaliszyk@35222
   477
  Bex1_rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
kaliszyk@35222
   478
where
kaliszyk@35222
   479
  "Bex1_rel R P \<longleftrightarrow> (\<exists>x \<in> Respects R. P x) \<and> (\<forall>x \<in> Respects R. \<forall>y \<in> Respects R. ((P x \<and> P y) \<longrightarrow> (R x y)))"
kaliszyk@35222
   480
kaliszyk@35222
   481
lemma bex1_rel_aux:
kaliszyk@35222
   482
  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R x\<rbrakk> \<Longrightarrow> Bex1_rel R y"
kaliszyk@35222
   483
  unfolding Bex1_rel_def
kaliszyk@35222
   484
  apply (erule conjE)+
kaliszyk@35222
   485
  apply (erule bexE)
kaliszyk@35222
   486
  apply rule
kaliszyk@35222
   487
  apply (rule_tac x="xa" in bexI)
kaliszyk@35222
   488
  apply metis
kaliszyk@35222
   489
  apply metis
kaliszyk@35222
   490
  apply rule+
kaliszyk@35222
   491
  apply (erule_tac x="xaa" in ballE)
kaliszyk@35222
   492
  prefer 2
kaliszyk@35222
   493
  apply (metis)
kaliszyk@35222
   494
  apply (erule_tac x="ya" in ballE)
kaliszyk@35222
   495
  prefer 2
kaliszyk@35222
   496
  apply (metis)
kaliszyk@35222
   497
  apply (metis in_respects)
kaliszyk@35222
   498
  done
kaliszyk@35222
   499
kaliszyk@35222
   500
lemma bex1_rel_aux2:
kaliszyk@35222
   501
  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R y\<rbrakk> \<Longrightarrow> Bex1_rel R x"
kaliszyk@35222
   502
  unfolding Bex1_rel_def
kaliszyk@35222
   503
  apply (erule conjE)+
kaliszyk@35222
   504
  apply (erule bexE)
kaliszyk@35222
   505
  apply rule
kaliszyk@35222
   506
  apply (rule_tac x="xa" in bexI)
kaliszyk@35222
   507
  apply metis
kaliszyk@35222
   508
  apply metis
kaliszyk@35222
   509
  apply rule+
kaliszyk@35222
   510
  apply (erule_tac x="xaa" in ballE)
kaliszyk@35222
   511
  prefer 2
kaliszyk@35222
   512
  apply (metis)
kaliszyk@35222
   513
  apply (erule_tac x="ya" in ballE)
kaliszyk@35222
   514
  prefer 2
kaliszyk@35222
   515
  apply (metis)
kaliszyk@35222
   516
  apply (metis in_respects)
kaliszyk@35222
   517
  done
kaliszyk@35222
   518
kaliszyk@35222
   519
lemma bex1_rel_rsp:
kaliszyk@35222
   520
  assumes a: "Quotient R absf repf"
kaliszyk@35222
   521
  shows "((R ===> op =) ===> op =) (Bex1_rel R) (Bex1_rel R)"
kaliszyk@35222
   522
  apply simp
kaliszyk@35222
   523
  apply clarify
kaliszyk@35222
   524
  apply rule
kaliszyk@35222
   525
  apply (simp_all add: bex1_rel_aux bex1_rel_aux2)
kaliszyk@35222
   526
  apply (erule bex1_rel_aux2)
kaliszyk@35222
   527
  apply assumption
kaliszyk@35222
   528
  done
kaliszyk@35222
   529
kaliszyk@35222
   530
kaliszyk@35222
   531
lemma ex1_prs:
kaliszyk@35222
   532
  assumes a: "Quotient R absf repf"
kaliszyk@35222
   533
  shows "((absf ---> id) ---> id) (Bex1_rel R) f = Ex1 f"
kaliszyk@35222
   534
apply simp
kaliszyk@35222
   535
apply (subst Bex1_rel_def)
kaliszyk@35222
   536
apply (subst Bex_def)
kaliszyk@35222
   537
apply (subst Ex1_def)
kaliszyk@35222
   538
apply simp
kaliszyk@35222
   539
apply rule
kaliszyk@35222
   540
 apply (erule conjE)+
kaliszyk@35222
   541
 apply (erule_tac exE)
kaliszyk@35222
   542
 apply (erule conjE)
kaliszyk@35222
   543
 apply (subgoal_tac "\<forall>y. R y y \<longrightarrow> f (absf y) \<longrightarrow> R x y")
kaliszyk@35222
   544
  apply (rule_tac x="absf x" in exI)
kaliszyk@35222
   545
  apply (simp)
kaliszyk@35222
   546
  apply rule+
kaliszyk@35222
   547
  using a unfolding Quotient_def
kaliszyk@35222
   548
  apply metis
kaliszyk@35222
   549
 apply rule+
kaliszyk@35222
   550
 apply (erule_tac x="x" in ballE)
kaliszyk@35222
   551
  apply (erule_tac x="y" in ballE)
kaliszyk@35222
   552
   apply simp
kaliszyk@35222
   553
  apply (simp add: in_respects)
kaliszyk@35222
   554
 apply (simp add: in_respects)
kaliszyk@35222
   555
apply (erule_tac exE)
kaliszyk@35222
   556
 apply rule
kaliszyk@35222
   557
 apply (rule_tac x="repf x" in exI)
kaliszyk@35222
   558
 apply (simp only: in_respects)
kaliszyk@35222
   559
  apply rule
kaliszyk@35222
   560
 apply (metis Quotient_rel_rep[OF a])
kaliszyk@35222
   561
using a unfolding Quotient_def apply (simp)
kaliszyk@35222
   562
apply rule+
kaliszyk@35222
   563
using a unfolding Quotient_def in_respects
kaliszyk@35222
   564
apply metis
kaliszyk@35222
   565
done
kaliszyk@35222
   566
kaliszyk@35222
   567
lemma bex1_bexeq_reg: "(\<exists>!x\<in>Respects R. P x) \<longrightarrow> (Bex1_rel R (\<lambda>x. P x))"
kaliszyk@35222
   568
  apply (simp add: Ex1_def Bex1_rel_def in_respects)
kaliszyk@35222
   569
  apply clarify
kaliszyk@35222
   570
  apply auto
kaliszyk@35222
   571
  apply (rule bexI)
kaliszyk@35222
   572
  apply assumption
kaliszyk@35222
   573
  apply (simp add: in_respects)
kaliszyk@35222
   574
  apply (simp add: in_respects)
kaliszyk@35222
   575
  apply auto
kaliszyk@35222
   576
  done
kaliszyk@35222
   577
huffman@35294
   578
subsection {* Various respects and preserve lemmas *}
kaliszyk@35222
   579
kaliszyk@35222
   580
lemma quot_rel_rsp:
kaliszyk@35222
   581
  assumes a: "Quotient R Abs Rep"
kaliszyk@35222
   582
  shows "(R ===> R ===> op =) R R"
kaliszyk@35222
   583
  apply(rule fun_rel_id)+
kaliszyk@35222
   584
  apply(rule equals_rsp[OF a])
kaliszyk@35222
   585
  apply(assumption)+
kaliszyk@35222
   586
  done
kaliszyk@35222
   587
kaliszyk@35222
   588
lemma o_prs:
kaliszyk@35222
   589
  assumes q1: "Quotient R1 Abs1 Rep1"
kaliszyk@35222
   590
  and     q2: "Quotient R2 Abs2 Rep2"
kaliszyk@35222
   591
  and     q3: "Quotient R3 Abs3 Rep3"
kaliszyk@36215
   592
  shows "((Abs2 ---> Rep3) ---> (Abs1 ---> Rep2) ---> (Rep1 ---> Abs3)) op \<circ> = op \<circ>"
kaliszyk@36215
   593
  and   "(id ---> (Abs1 ---> id) ---> Rep1 ---> id) op \<circ> = op \<circ>"
kaliszyk@35222
   594
  using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] Quotient_abs_rep[OF q3]
kaliszyk@36215
   595
  unfolding o_def expand_fun_eq by simp_all
kaliszyk@35222
   596
kaliszyk@35222
   597
lemma o_rsp:
kaliszyk@36215
   598
  "((R2 ===> R3) ===> (R1 ===> R2) ===> (R1 ===> R3)) op \<circ> op \<circ>"
kaliszyk@36215
   599
  "(op = ===> (R1 ===> op =) ===> R1 ===> op =) op \<circ> op \<circ>"
kaliszyk@36215
   600
  unfolding fun_rel_def o_def expand_fun_eq by auto
kaliszyk@35222
   601
kaliszyk@35222
   602
lemma cond_prs:
kaliszyk@35222
   603
  assumes a: "Quotient R absf repf"
kaliszyk@35222
   604
  shows "absf (if a then repf b else repf c) = (if a then b else c)"
kaliszyk@35222
   605
  using a unfolding Quotient_def by auto
kaliszyk@35222
   606
kaliszyk@35222
   607
lemma if_prs:
kaliszyk@35222
   608
  assumes q: "Quotient R Abs Rep"
kaliszyk@36123
   609
  shows "(id ---> Rep ---> Rep ---> Abs) If = If"
kaliszyk@36123
   610
  using Quotient_abs_rep[OF q]
kaliszyk@36123
   611
  by (auto simp add: expand_fun_eq)
kaliszyk@35222
   612
kaliszyk@35222
   613
lemma if_rsp:
kaliszyk@35222
   614
  assumes q: "Quotient R Abs Rep"
kaliszyk@36123
   615
  shows "(op = ===> R ===> R ===> R) If If"
kaliszyk@36123
   616
  by auto
kaliszyk@35222
   617
kaliszyk@35222
   618
lemma let_prs:
kaliszyk@35222
   619
  assumes q1: "Quotient R1 Abs1 Rep1"
kaliszyk@35222
   620
  and     q2: "Quotient R2 Abs2 Rep2"
kaliszyk@37049
   621
  shows "(Rep2 ---> (Abs2 ---> Rep1) ---> Abs1) Let = Let"
kaliszyk@37049
   622
  using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
kaliszyk@37049
   623
  by (auto simp add: expand_fun_eq)
kaliszyk@35222
   624
kaliszyk@35222
   625
lemma let_rsp:
kaliszyk@37049
   626
  shows "(R1 ===> (R1 ===> R2) ===> R2) Let Let"
kaliszyk@37049
   627
  by auto
kaliszyk@35222
   628
kaliszyk@35222
   629
locale quot_type =
kaliszyk@35222
   630
  fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kaliszyk@35222
   631
  and   Abs :: "('a \<Rightarrow> bool) \<Rightarrow> 'b"
kaliszyk@35222
   632
  and   Rep :: "'b \<Rightarrow> ('a \<Rightarrow> bool)"
kaliszyk@35222
   633
  assumes equivp: "equivp R"
kaliszyk@35222
   634
  and     rep_prop: "\<And>y. \<exists>x. Rep y = R x"
kaliszyk@35222
   635
  and     rep_inverse: "\<And>x. Abs (Rep x) = x"
kaliszyk@35222
   636
  and     abs_inverse: "\<And>x. (Rep (Abs (R x))) = (R x)"
kaliszyk@35222
   637
  and     rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"
kaliszyk@35222
   638
begin
kaliszyk@35222
   639
kaliszyk@35222
   640
definition
kaliszyk@35222
   641
  abs::"'a \<Rightarrow> 'b"
kaliszyk@35222
   642
where
kaliszyk@35222
   643
  "abs x \<equiv> Abs (R x)"
kaliszyk@35222
   644
kaliszyk@35222
   645
definition
kaliszyk@35222
   646
  rep::"'b \<Rightarrow> 'a"
kaliszyk@35222
   647
where
kaliszyk@35222
   648
  "rep a = Eps (Rep a)"
kaliszyk@35222
   649
kaliszyk@35222
   650
lemma homeier_lem9:
kaliszyk@35222
   651
  shows "R (Eps (R x)) = R x"
kaliszyk@35222
   652
proof -
kaliszyk@35222
   653
  have a: "R x x" using equivp by (simp add: equivp_reflp_symp_transp reflp_def)
kaliszyk@35222
   654
  then have "R x (Eps (R x))" by (rule someI)
kaliszyk@35222
   655
  then show "R (Eps (R x)) = R x"
kaliszyk@35222
   656
    using equivp unfolding equivp_def by simp
kaliszyk@35222
   657
qed
kaliszyk@35222
   658
kaliszyk@35222
   659
theorem homeier_thm10:
kaliszyk@35222
   660
  shows "abs (rep a) = a"
kaliszyk@35222
   661
  unfolding abs_def rep_def
kaliszyk@35222
   662
proof -
kaliszyk@35222
   663
  from rep_prop
kaliszyk@35222
   664
  obtain x where eq: "Rep a = R x" by auto
kaliszyk@35222
   665
  have "Abs (R (Eps (Rep a))) = Abs (R (Eps (R x)))" using eq by simp
kaliszyk@35222
   666
  also have "\<dots> = Abs (R x)" using homeier_lem9 by simp
kaliszyk@35222
   667
  also have "\<dots> = Abs (Rep a)" using eq by simp
kaliszyk@35222
   668
  also have "\<dots> = a" using rep_inverse by simp
kaliszyk@35222
   669
  finally
kaliszyk@35222
   670
  show "Abs (R (Eps (Rep a))) = a" by simp
kaliszyk@35222
   671
qed
kaliszyk@35222
   672
kaliszyk@35222
   673
lemma homeier_lem7:
kaliszyk@35222
   674
  shows "(R x = R y) = (Abs (R x) = Abs (R y))" (is "?LHS = ?RHS")
kaliszyk@35222
   675
proof -
kaliszyk@35222
   676
  have "?RHS = (Rep (Abs (R x)) = Rep (Abs (R y)))" by (simp add: rep_inject)
kaliszyk@35222
   677
  also have "\<dots> = ?LHS" by (simp add: abs_inverse)
kaliszyk@35222
   678
  finally show "?LHS = ?RHS" by simp
kaliszyk@35222
   679
qed
kaliszyk@35222
   680
kaliszyk@35222
   681
theorem homeier_thm11:
kaliszyk@35222
   682
  shows "R r r' = (abs r = abs r')"
kaliszyk@35222
   683
  unfolding abs_def
kaliszyk@35222
   684
  by (simp only: equivp[simplified equivp_def] homeier_lem7)
kaliszyk@35222
   685
kaliszyk@35222
   686
lemma rep_refl:
kaliszyk@35222
   687
  shows "R (rep a) (rep a)"
kaliszyk@35222
   688
  unfolding rep_def
kaliszyk@35222
   689
  by (simp add: equivp[simplified equivp_def])
kaliszyk@35222
   690
kaliszyk@35222
   691
kaliszyk@35222
   692
lemma rep_abs_rsp:
kaliszyk@35222
   693
  shows "R f (rep (abs g)) = R f g"
kaliszyk@35222
   694
  and   "R (rep (abs g)) f = R g f"
kaliszyk@35222
   695
  by (simp_all add: homeier_thm10 homeier_thm11)
kaliszyk@35222
   696
kaliszyk@35222
   697
lemma Quotient:
kaliszyk@35222
   698
  shows "Quotient R abs rep"
kaliszyk@35222
   699
  unfolding Quotient_def
kaliszyk@35222
   700
  apply(simp add: homeier_thm10)
kaliszyk@35222
   701
  apply(simp add: rep_refl)
kaliszyk@35222
   702
  apply(subst homeier_thm11[symmetric])
kaliszyk@35222
   703
  apply(simp add: equivp[simplified equivp_def])
kaliszyk@35222
   704
  done
kaliszyk@35222
   705
kaliszyk@35222
   706
end
kaliszyk@35222
   707
huffman@35294
   708
subsection {* ML setup *}
kaliszyk@35222
   709
kaliszyk@35222
   710
text {* Auxiliary data for the quotient package *}
kaliszyk@35222
   711
kaliszyk@35222
   712
use "~~/src/HOL/Tools/Quotient/quotient_info.ML"
kaliszyk@35222
   713
kaliszyk@35222
   714
declare [[map "fun" = (fun_map, fun_rel)]]
kaliszyk@35222
   715
kaliszyk@35222
   716
lemmas [quot_thm] = fun_quotient
kaliszyk@37049
   717
lemmas [quot_respect] = quot_rel_rsp if_rsp o_rsp let_rsp
kaliszyk@37049
   718
lemmas [quot_preserve] = if_prs o_prs let_prs
kaliszyk@35222
   719
lemmas [quot_equiv] = identity_equivp
kaliszyk@35222
   720
kaliszyk@35222
   721
kaliszyk@35222
   722
text {* Lemmas about simplifying id's. *}
kaliszyk@35222
   723
lemmas [id_simps] =
kaliszyk@35222
   724
  id_def[symmetric]
kaliszyk@35222
   725
  fun_map_id
kaliszyk@35222
   726
  id_apply
kaliszyk@35222
   727
  id_o
kaliszyk@35222
   728
  o_id
kaliszyk@35222
   729
  eq_comp_r
kaliszyk@35222
   730
kaliszyk@35222
   731
text {* Translation functions for the lifting process. *}
kaliszyk@35222
   732
use "~~/src/HOL/Tools/Quotient/quotient_term.ML"
kaliszyk@35222
   733
kaliszyk@35222
   734
kaliszyk@35222
   735
text {* Definitions of the quotient types. *}
kaliszyk@35222
   736
use "~~/src/HOL/Tools/Quotient/quotient_typ.ML"
kaliszyk@35222
   737
kaliszyk@35222
   738
kaliszyk@35222
   739
text {* Definitions for quotient constants. *}
kaliszyk@35222
   740
use "~~/src/HOL/Tools/Quotient/quotient_def.ML"
kaliszyk@35222
   741
kaliszyk@35222
   742
kaliszyk@35222
   743
text {*
kaliszyk@35222
   744
  An auxiliary constant for recording some information
kaliszyk@35222
   745
  about the lifted theorem in a tactic.
kaliszyk@35222
   746
*}
kaliszyk@35222
   747
definition
kaliszyk@36116
   748
  "Quot_True (x :: 'a) \<equiv> True"
kaliszyk@35222
   749
kaliszyk@35222
   750
lemma
kaliszyk@35222
   751
  shows QT_all: "Quot_True (All P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   752
  and   QT_ex:  "Quot_True (Ex P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   753
  and   QT_ex1: "Quot_True (Ex1 P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   754
  and   QT_lam: "Quot_True (\<lambda>x. P x) \<Longrightarrow> (\<And>x. Quot_True (P x))"
kaliszyk@35222
   755
  and   QT_ext: "(\<And>x. Quot_True (a x) \<Longrightarrow> f x = g x) \<Longrightarrow> (Quot_True a \<Longrightarrow> f = g)"
kaliszyk@35222
   756
  by (simp_all add: Quot_True_def ext)
kaliszyk@35222
   757
kaliszyk@35222
   758
lemma QT_imp: "Quot_True a \<equiv> Quot_True b"
kaliszyk@35222
   759
  by (simp add: Quot_True_def)
kaliszyk@35222
   760
kaliszyk@35222
   761
kaliszyk@35222
   762
text {* Tactics for proving the lifted theorems *}
kaliszyk@35222
   763
use "~~/src/HOL/Tools/Quotient/quotient_tacs.ML"
kaliszyk@35222
   764
huffman@35294
   765
subsection {* Methods / Interface *}
kaliszyk@35222
   766
kaliszyk@35222
   767
method_setup lifting =
kaliszyk@35222
   768
  {* Attrib.thms >> (fn thms => fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.lift_tac ctxt thms))) *}
kaliszyk@35222
   769
  {* lifts theorems to quotient types *}
kaliszyk@35222
   770
kaliszyk@35222
   771
method_setup lifting_setup =
kaliszyk@35222
   772
  {* Attrib.thm >> (fn thms => fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.procedure_tac ctxt thms))) *}
kaliszyk@35222
   773
  {* sets up the three goals for the quotient lifting procedure *}
kaliszyk@35222
   774
kaliszyk@35222
   775
method_setup regularize =
kaliszyk@35222
   776
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.regularize_tac ctxt))) *}
kaliszyk@35222
   777
  {* proves the regularization goals from the quotient lifting procedure *}
kaliszyk@35222
   778
kaliszyk@35222
   779
method_setup injection =
kaliszyk@35222
   780
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.all_injection_tac ctxt))) *}
kaliszyk@35222
   781
  {* proves the rep/abs injection goals from the quotient lifting procedure *}
kaliszyk@35222
   782
kaliszyk@35222
   783
method_setup cleaning =
kaliszyk@35222
   784
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.clean_tac ctxt))) *}
kaliszyk@35222
   785
  {* proves the cleaning goals from the quotient lifting procedure *}
kaliszyk@35222
   786
kaliszyk@35222
   787
attribute_setup quot_lifted =
kaliszyk@35222
   788
  {* Scan.succeed Quotient_Tacs.lifted_attrib *}
kaliszyk@35222
   789
  {* lifts theorems to quotient types *}
kaliszyk@35222
   790
kaliszyk@35222
   791
no_notation
kaliszyk@35222
   792
  rel_conj (infixr "OOO" 75) and
kaliszyk@35222
   793
  fun_map (infixr "--->" 55) and
kaliszyk@35222
   794
  fun_rel (infixr "===>" 55)
kaliszyk@35222
   795
kaliszyk@35222
   796
end
kaliszyk@35222
   797