src/HOL/Quotient.thy
author haftmann
Mon Nov 29 13:44:54 2010 +0100 (2010-11-29)
changeset 40815 6e2d17cc0d1d
parent 40814 fa64f6278568
child 40818 b117df72e56b
permissions -rw-r--r--
equivI has replaced equiv.intro
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
haftmann@40466
     8
imports Plain Hilbert_Choice Equiv_Relations
kaliszyk@35222
     9
uses
wenzelm@37986
    10
  ("Tools/Quotient/quotient_info.ML")
wenzelm@37986
    11
  ("Tools/Quotient/quotient_typ.ML")
wenzelm@37986
    12
  ("Tools/Quotient/quotient_def.ML")
wenzelm@37986
    13
  ("Tools/Quotient/quotient_term.ML")
wenzelm@37986
    14
  ("Tools/Quotient/quotient_tacs.ML")
kaliszyk@35222
    15
begin
kaliszyk@35222
    16
kaliszyk@35222
    17
text {*
kaliszyk@35222
    18
  Basic definition for equivalence relations
kaliszyk@35222
    19
  that are represented by predicates.
kaliszyk@35222
    20
*}
kaliszyk@35222
    21
kaliszyk@35222
    22
text {* Composition of Relations *}
kaliszyk@35222
    23
kaliszyk@35222
    24
abbreviation
kaliszyk@35222
    25
  rel_conj (infixr "OOO" 75)
kaliszyk@35222
    26
where
kaliszyk@35222
    27
  "r1 OOO r2 \<equiv> r1 OO r2 OO r1"
kaliszyk@35222
    28
kaliszyk@35222
    29
lemma eq_comp_r:
kaliszyk@35222
    30
  shows "((op =) OOO R) = R"
nipkow@39302
    31
  by (auto simp add: fun_eq_iff)
kaliszyk@35222
    32
huffman@35294
    33
subsection {* Respects predicate *}
kaliszyk@35222
    34
kaliszyk@35222
    35
definition
haftmann@40466
    36
  Respects :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set"
kaliszyk@35222
    37
where
haftmann@40466
    38
  "Respects R x = R x x"
kaliszyk@35222
    39
kaliszyk@35222
    40
lemma in_respects:
haftmann@40466
    41
  shows "x \<in> Respects R \<longleftrightarrow> R x x"
kaliszyk@35222
    42
  unfolding mem_def Respects_def
kaliszyk@35222
    43
  by simp
kaliszyk@35222
    44
huffman@35294
    45
subsection {* Function map and function relation *}
kaliszyk@35222
    46
haftmann@40602
    47
notation map_fun (infixr "--->" 55)
haftmann@40466
    48
haftmann@40602
    49
lemma map_fun_id:
haftmann@40466
    50
  "(id ---> id) = id"
haftmann@40602
    51
  by (simp add: fun_eq_iff)
kaliszyk@35222
    52
kaliszyk@35222
    53
definition
haftmann@40615
    54
  fun_rel :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool" (infixr "===>" 55)
kaliszyk@35222
    55
where
haftmann@40814
    56
  "fun_rel R1 R2 = (\<lambda>f g. \<forall>x y. R1 x y \<longrightarrow> R2 (f x) (g y))"
kaliszyk@35222
    57
kaliszyk@36276
    58
lemma fun_relI [intro]:
haftmann@40814
    59
  assumes "\<And>x y. R1 x y \<Longrightarrow> R2 (f x) (g y)"
haftmann@40814
    60
  shows "(R1 ===> R2) f g"
kaliszyk@36276
    61
  using assms by (simp add: fun_rel_def)
kaliszyk@35222
    62
haftmann@40466
    63
lemma fun_relE:
haftmann@40814
    64
  assumes "(R1 ===> R2) f g" and "R1 x y"
haftmann@40814
    65
  obtains "R2 (f x) (g y)"
haftmann@40466
    66
  using assms by (simp add: fun_rel_def)
kaliszyk@35222
    67
kaliszyk@35222
    68
lemma fun_rel_eq:
kaliszyk@35222
    69
  shows "((op =) ===> (op =)) = (op =)"
haftmann@40466
    70
  by (auto simp add: fun_eq_iff elim: fun_relE)
kaliszyk@35222
    71
kaliszyk@35222
    72
huffman@35294
    73
subsection {* Quotient Predicate *}
kaliszyk@35222
    74
kaliszyk@35222
    75
definition
haftmann@40814
    76
  "Quotient R Abs Rep \<longleftrightarrow>
haftmann@40814
    77
     (\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. R (Rep a) (Rep a)) \<and>
haftmann@40814
    78
     (\<forall>r s. R r s = (R r r \<and> R s s \<and> (Abs r = Abs s)))"
kaliszyk@35222
    79
kaliszyk@35222
    80
lemma Quotient_abs_rep:
haftmann@40814
    81
  assumes a: "Quotient R Abs Rep"
kaliszyk@35222
    82
  shows "Abs (Rep a) = a"
kaliszyk@35222
    83
  using a
kaliszyk@35222
    84
  unfolding Quotient_def
kaliszyk@35222
    85
  by simp
kaliszyk@35222
    86
kaliszyk@35222
    87
lemma Quotient_rep_reflp:
haftmann@40814
    88
  assumes a: "Quotient R Abs Rep"
haftmann@40814
    89
  shows "R (Rep a) (Rep a)"
kaliszyk@35222
    90
  using a
kaliszyk@35222
    91
  unfolding Quotient_def
kaliszyk@35222
    92
  by blast
kaliszyk@35222
    93
kaliszyk@35222
    94
lemma Quotient_rel:
haftmann@40814
    95
  assumes a: "Quotient R Abs Rep"
haftmann@40814
    96
  shows " R r s = (R r r \<and> R s s \<and> (Abs r = Abs s))"
kaliszyk@35222
    97
  using a
kaliszyk@35222
    98
  unfolding Quotient_def
kaliszyk@35222
    99
  by blast
kaliszyk@35222
   100
kaliszyk@35222
   101
lemma Quotient_rel_rep:
kaliszyk@35222
   102
  assumes a: "Quotient R Abs Rep"
kaliszyk@35222
   103
  shows "R (Rep a) (Rep b) = (a = b)"
kaliszyk@35222
   104
  using a
kaliszyk@35222
   105
  unfolding Quotient_def
kaliszyk@35222
   106
  by metis
kaliszyk@35222
   107
kaliszyk@35222
   108
lemma Quotient_rep_abs:
kaliszyk@35222
   109
  assumes a: "Quotient R Abs Rep"
kaliszyk@35222
   110
  shows "R r r \<Longrightarrow> R (Rep (Abs r)) r"
kaliszyk@35222
   111
  using a unfolding Quotient_def
kaliszyk@35222
   112
  by blast
kaliszyk@35222
   113
kaliszyk@35222
   114
lemma Quotient_rel_abs:
haftmann@40814
   115
  assumes a: "Quotient R Abs Rep"
haftmann@40814
   116
  shows "R r s \<Longrightarrow> Abs r = Abs s"
kaliszyk@35222
   117
  using a unfolding Quotient_def
kaliszyk@35222
   118
  by blast
kaliszyk@35222
   119
kaliszyk@35222
   120
lemma Quotient_symp:
haftmann@40814
   121
  assumes a: "Quotient R Abs Rep"
haftmann@40814
   122
  shows "symp R"
haftmann@40814
   123
  using a unfolding Quotient_def using sympI by metis
kaliszyk@35222
   124
kaliszyk@35222
   125
lemma Quotient_transp:
haftmann@40814
   126
  assumes a: "Quotient R Abs Rep"
haftmann@40814
   127
  shows "transp R"
haftmann@40814
   128
  using a unfolding Quotient_def using transpI by metis
kaliszyk@35222
   129
kaliszyk@35222
   130
lemma identity_quotient:
kaliszyk@35222
   131
  shows "Quotient (op =) id id"
kaliszyk@35222
   132
  unfolding Quotient_def id_def
kaliszyk@35222
   133
  by blast
kaliszyk@35222
   134
kaliszyk@35222
   135
lemma fun_quotient:
kaliszyk@35222
   136
  assumes q1: "Quotient R1 abs1 rep1"
kaliszyk@35222
   137
  and     q2: "Quotient R2 abs2 rep2"
kaliszyk@35222
   138
  shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
kaliszyk@35222
   139
proof -
haftmann@40466
   140
  have "\<And>a. (rep1 ---> abs2) ((abs1 ---> rep2) a) = a"
haftmann@40466
   141
    using q1 q2 by (simp add: Quotient_def fun_eq_iff)
kaliszyk@35222
   142
  moreover
haftmann@40466
   143
  have "\<And>a. (R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"
haftmann@40466
   144
    by (rule fun_relI)
haftmann@40466
   145
      (insert q1 q2 Quotient_rel_abs [of R1 abs1 rep1] Quotient_rel_rep [of R2 abs2 rep2],
haftmann@40466
   146
        simp (no_asm) add: Quotient_def, simp)
kaliszyk@35222
   147
  moreover
haftmann@40466
   148
  have "\<And>r s. (R1 ===> R2) r s = ((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
kaliszyk@35222
   149
        (rep1 ---> abs2) r  = (rep1 ---> abs2) s)"
haftmann@40466
   150
    apply(auto simp add: fun_rel_def fun_eq_iff)
kaliszyk@35222
   151
    using q1 q2 unfolding Quotient_def
kaliszyk@35222
   152
    apply(metis)
kaliszyk@35222
   153
    using q1 q2 unfolding Quotient_def
kaliszyk@35222
   154
    apply(metis)
kaliszyk@35222
   155
    using q1 q2 unfolding Quotient_def
kaliszyk@35222
   156
    apply(metis)
kaliszyk@35222
   157
    using q1 q2 unfolding Quotient_def
kaliszyk@35222
   158
    apply(metis)
kaliszyk@35222
   159
    done
kaliszyk@35222
   160
  ultimately
kaliszyk@35222
   161
  show "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
kaliszyk@35222
   162
    unfolding Quotient_def by blast
kaliszyk@35222
   163
qed
kaliszyk@35222
   164
kaliszyk@35222
   165
lemma abs_o_rep:
kaliszyk@35222
   166
  assumes a: "Quotient R Abs Rep"
kaliszyk@35222
   167
  shows "Abs o Rep = id"
nipkow@39302
   168
  unfolding fun_eq_iff
kaliszyk@35222
   169
  by (simp add: Quotient_abs_rep[OF a])
kaliszyk@35222
   170
kaliszyk@35222
   171
lemma equals_rsp:
kaliszyk@35222
   172
  assumes q: "Quotient R Abs Rep"
kaliszyk@35222
   173
  and     a: "R xa xb" "R ya yb"
kaliszyk@35222
   174
  shows "R xa ya = R xb yb"
kaliszyk@35222
   175
  using a Quotient_symp[OF q] Quotient_transp[OF q]
haftmann@40814
   176
  by (blast elim: sympE transpE)
kaliszyk@35222
   177
kaliszyk@35222
   178
lemma lambda_prs:
kaliszyk@35222
   179
  assumes q1: "Quotient R1 Abs1 Rep1"
kaliszyk@35222
   180
  and     q2: "Quotient R2 Abs2 Rep2"
kaliszyk@35222
   181
  shows "(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) = (\<lambda>x. f x)"
nipkow@39302
   182
  unfolding fun_eq_iff
kaliszyk@35222
   183
  using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
haftmann@40814
   184
  by simp
kaliszyk@35222
   185
kaliszyk@35222
   186
lemma lambda_prs1:
kaliszyk@35222
   187
  assumes q1: "Quotient R1 Abs1 Rep1"
kaliszyk@35222
   188
  and     q2: "Quotient R2 Abs2 Rep2"
kaliszyk@35222
   189
  shows "(Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x) = (\<lambda>x. f x)"
nipkow@39302
   190
  unfolding fun_eq_iff
kaliszyk@35222
   191
  using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
haftmann@40814
   192
  by simp
kaliszyk@35222
   193
kaliszyk@35222
   194
lemma rep_abs_rsp:
kaliszyk@35222
   195
  assumes q: "Quotient R Abs Rep"
kaliszyk@35222
   196
  and     a: "R x1 x2"
kaliszyk@35222
   197
  shows "R x1 (Rep (Abs x2))"
kaliszyk@35222
   198
  using a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]
kaliszyk@35222
   199
  by metis
kaliszyk@35222
   200
kaliszyk@35222
   201
lemma rep_abs_rsp_left:
kaliszyk@35222
   202
  assumes q: "Quotient R Abs Rep"
kaliszyk@35222
   203
  and     a: "R x1 x2"
kaliszyk@35222
   204
  shows "R (Rep (Abs x1)) x2"
kaliszyk@35222
   205
  using a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]
kaliszyk@35222
   206
  by metis
kaliszyk@35222
   207
kaliszyk@35222
   208
text{*
kaliszyk@35222
   209
  In the following theorem R1 can be instantiated with anything,
kaliszyk@35222
   210
  but we know some of the types of the Rep and Abs functions;
kaliszyk@35222
   211
  so by solving Quotient assumptions we can get a unique R1 that
kaliszyk@35236
   212
  will be provable; which is why we need to use @{text apply_rsp} and
kaliszyk@35222
   213
  not the primed version *}
kaliszyk@35222
   214
kaliszyk@35222
   215
lemma apply_rsp:
kaliszyk@35222
   216
  fixes f g::"'a \<Rightarrow> 'c"
kaliszyk@35222
   217
  assumes q: "Quotient R1 Abs1 Rep1"
kaliszyk@35222
   218
  and     a: "(R1 ===> R2) f g" "R1 x y"
kaliszyk@35222
   219
  shows "R2 (f x) (g y)"
haftmann@40466
   220
  using a by (auto elim: fun_relE)
kaliszyk@35222
   221
kaliszyk@35222
   222
lemma apply_rsp':
kaliszyk@35222
   223
  assumes a: "(R1 ===> R2) f g" "R1 x y"
kaliszyk@35222
   224
  shows "R2 (f x) (g y)"
haftmann@40466
   225
  using a by (auto elim: fun_relE)
kaliszyk@35222
   226
huffman@35294
   227
subsection {* lemmas for regularisation of ball and bex *}
kaliszyk@35222
   228
kaliszyk@35222
   229
lemma ball_reg_eqv:
kaliszyk@35222
   230
  fixes P :: "'a \<Rightarrow> bool"
kaliszyk@35222
   231
  assumes a: "equivp R"
kaliszyk@35222
   232
  shows "Ball (Respects R) P = (All P)"
kaliszyk@35222
   233
  using a
kaliszyk@35222
   234
  unfolding equivp_def
kaliszyk@35222
   235
  by (auto simp add: in_respects)
kaliszyk@35222
   236
kaliszyk@35222
   237
lemma bex_reg_eqv:
kaliszyk@35222
   238
  fixes P :: "'a \<Rightarrow> bool"
kaliszyk@35222
   239
  assumes a: "equivp R"
kaliszyk@35222
   240
  shows "Bex (Respects R) P = (Ex P)"
kaliszyk@35222
   241
  using a
kaliszyk@35222
   242
  unfolding equivp_def
kaliszyk@35222
   243
  by (auto simp add: in_respects)
kaliszyk@35222
   244
kaliszyk@35222
   245
lemma ball_reg_right:
kaliszyk@35222
   246
  assumes a: "\<And>x. R x \<Longrightarrow> P x \<longrightarrow> Q x"
kaliszyk@35222
   247
  shows "All P \<longrightarrow> Ball R Q"
blanchet@39956
   248
  using a by (metis Collect_def Collect_mem_eq)
kaliszyk@35222
   249
kaliszyk@35222
   250
lemma bex_reg_left:
kaliszyk@35222
   251
  assumes a: "\<And>x. R x \<Longrightarrow> Q x \<longrightarrow> P x"
kaliszyk@35222
   252
  shows "Bex R Q \<longrightarrow> Ex P"
blanchet@39956
   253
  using a by (metis Collect_def Collect_mem_eq)
kaliszyk@35222
   254
kaliszyk@35222
   255
lemma ball_reg_left:
kaliszyk@35222
   256
  assumes a: "equivp R"
kaliszyk@35222
   257
  shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ball (Respects R) Q \<longrightarrow> All P"
kaliszyk@35222
   258
  using a by (metis equivp_reflp in_respects)
kaliszyk@35222
   259
kaliszyk@35222
   260
lemma bex_reg_right:
kaliszyk@35222
   261
  assumes a: "equivp R"
kaliszyk@35222
   262
  shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ex Q \<longrightarrow> Bex (Respects R) P"
kaliszyk@35222
   263
  using a by (metis equivp_reflp in_respects)
kaliszyk@35222
   264
kaliszyk@35222
   265
lemma ball_reg_eqv_range:
kaliszyk@35222
   266
  fixes P::"'a \<Rightarrow> bool"
kaliszyk@35222
   267
  and x::"'a"
kaliszyk@35222
   268
  assumes a: "equivp R2"
kaliszyk@35222
   269
  shows   "(Ball (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = All (\<lambda>f. P (f x)))"
kaliszyk@35222
   270
  apply(rule iffI)
kaliszyk@35222
   271
  apply(rule allI)
kaliszyk@35222
   272
  apply(drule_tac x="\<lambda>y. f x" in bspec)
haftmann@40466
   273
  apply(simp add: in_respects fun_rel_def)
kaliszyk@35222
   274
  apply(rule impI)
kaliszyk@35222
   275
  using a equivp_reflp_symp_transp[of "R2"]
haftmann@40814
   276
  apply (auto elim: equivpE reflpE)
kaliszyk@35222
   277
  done
kaliszyk@35222
   278
kaliszyk@35222
   279
lemma bex_reg_eqv_range:
kaliszyk@35222
   280
  assumes a: "equivp R2"
kaliszyk@35222
   281
  shows   "(Bex (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = Ex (\<lambda>f. P (f x)))"
kaliszyk@35222
   282
  apply(auto)
kaliszyk@35222
   283
  apply(rule_tac x="\<lambda>y. f x" in bexI)
kaliszyk@35222
   284
  apply(simp)
haftmann@40466
   285
  apply(simp add: Respects_def in_respects fun_rel_def)
kaliszyk@35222
   286
  apply(rule impI)
kaliszyk@35222
   287
  using a equivp_reflp_symp_transp[of "R2"]
haftmann@40814
   288
  apply (auto elim: equivpE reflpE)
kaliszyk@35222
   289
  done
kaliszyk@35222
   290
kaliszyk@35222
   291
(* Next four lemmas are unused *)
kaliszyk@35222
   292
lemma all_reg:
kaliszyk@35222
   293
  assumes a: "!x :: 'a. (P x --> Q x)"
kaliszyk@35222
   294
  and     b: "All P"
kaliszyk@35222
   295
  shows "All Q"
kaliszyk@35222
   296
  using a b by (metis)
kaliszyk@35222
   297
kaliszyk@35222
   298
lemma ex_reg:
kaliszyk@35222
   299
  assumes a: "!x :: 'a. (P x --> Q x)"
kaliszyk@35222
   300
  and     b: "Ex P"
kaliszyk@35222
   301
  shows "Ex Q"
kaliszyk@35222
   302
  using a b by metis
kaliszyk@35222
   303
kaliszyk@35222
   304
lemma ball_reg:
kaliszyk@35222
   305
  assumes a: "!x :: 'a. (R x --> P x --> Q x)"
kaliszyk@35222
   306
  and     b: "Ball R P"
kaliszyk@35222
   307
  shows "Ball R Q"
blanchet@39956
   308
  using a b by (metis Collect_def Collect_mem_eq)
kaliszyk@35222
   309
kaliszyk@35222
   310
lemma bex_reg:
kaliszyk@35222
   311
  assumes a: "!x :: 'a. (R x --> P x --> Q x)"
kaliszyk@35222
   312
  and     b: "Bex R P"
kaliszyk@35222
   313
  shows "Bex R Q"
blanchet@39956
   314
  using a b by (metis Collect_def Collect_mem_eq)
kaliszyk@35222
   315
kaliszyk@35222
   316
kaliszyk@35222
   317
lemma ball_all_comm:
kaliszyk@35222
   318
  assumes "\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)"
kaliszyk@35222
   319
  shows "(\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y)"
kaliszyk@35222
   320
  using assms by auto
kaliszyk@35222
   321
kaliszyk@35222
   322
lemma bex_ex_comm:
kaliszyk@35222
   323
  assumes "(\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)"
kaliszyk@35222
   324
  shows "(\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y)"
kaliszyk@35222
   325
  using assms by auto
kaliszyk@35222
   326
huffman@35294
   327
subsection {* Bounded abstraction *}
kaliszyk@35222
   328
kaliszyk@35222
   329
definition
haftmann@40466
   330
  Babs :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
kaliszyk@35222
   331
where
kaliszyk@35222
   332
  "x \<in> p \<Longrightarrow> Babs p m x = m x"
kaliszyk@35222
   333
kaliszyk@35222
   334
lemma babs_rsp:
kaliszyk@35222
   335
  assumes q: "Quotient R1 Abs1 Rep1"
kaliszyk@35222
   336
  and     a: "(R1 ===> R2) f g"
kaliszyk@35222
   337
  shows      "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"
haftmann@40466
   338
  apply (auto simp add: Babs_def in_respects fun_rel_def)
kaliszyk@35222
   339
  apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
haftmann@40466
   340
  using a apply (simp add: Babs_def fun_rel_def)
haftmann@40466
   341
  apply (simp add: in_respects fun_rel_def)
kaliszyk@35222
   342
  using Quotient_rel[OF q]
kaliszyk@35222
   343
  by metis
kaliszyk@35222
   344
kaliszyk@35222
   345
lemma babs_prs:
kaliszyk@35222
   346
  assumes q1: "Quotient R1 Abs1 Rep1"
kaliszyk@35222
   347
  and     q2: "Quotient R2 Abs2 Rep2"
kaliszyk@35222
   348
  shows "((Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f))) = f"
kaliszyk@35222
   349
  apply (rule ext)
haftmann@40466
   350
  apply (simp add:)
kaliszyk@35222
   351
  apply (subgoal_tac "Rep1 x \<in> Respects R1")
kaliszyk@35222
   352
  apply (simp add: Babs_def Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])
kaliszyk@35222
   353
  apply (simp add: in_respects Quotient_rel_rep[OF q1])
kaliszyk@35222
   354
  done
kaliszyk@35222
   355
kaliszyk@35222
   356
lemma babs_simp:
kaliszyk@35222
   357
  assumes q: "Quotient R1 Abs Rep"
kaliszyk@35222
   358
  shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"
kaliszyk@35222
   359
  apply(rule iffI)
kaliszyk@35222
   360
  apply(simp_all only: babs_rsp[OF q])
haftmann@40466
   361
  apply(auto simp add: Babs_def fun_rel_def)
kaliszyk@35222
   362
  apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
kaliszyk@35222
   363
  apply(metis Babs_def)
kaliszyk@35222
   364
  apply (simp add: in_respects)
kaliszyk@35222
   365
  using Quotient_rel[OF q]
kaliszyk@35222
   366
  by metis
kaliszyk@35222
   367
kaliszyk@35222
   368
(* If a user proves that a particular functional relation
kaliszyk@35222
   369
   is an equivalence this may be useful in regularising *)
kaliszyk@35222
   370
lemma babs_reg_eqv:
kaliszyk@35222
   371
  shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"
nipkow@39302
   372
  by (simp add: fun_eq_iff Babs_def in_respects equivp_reflp)
kaliszyk@35222
   373
kaliszyk@35222
   374
kaliszyk@35222
   375
(* 3 lemmas needed for proving repabs_inj *)
kaliszyk@35222
   376
lemma ball_rsp:
kaliszyk@35222
   377
  assumes a: "(R ===> (op =)) f g"
kaliszyk@35222
   378
  shows "Ball (Respects R) f = Ball (Respects R) g"
haftmann@40466
   379
  using a by (auto simp add: Ball_def in_respects elim: fun_relE)
kaliszyk@35222
   380
kaliszyk@35222
   381
lemma bex_rsp:
kaliszyk@35222
   382
  assumes a: "(R ===> (op =)) f g"
kaliszyk@35222
   383
  shows "(Bex (Respects R) f = Bex (Respects R) g)"
haftmann@40466
   384
  using a by (auto simp add: Bex_def in_respects elim: fun_relE)
kaliszyk@35222
   385
kaliszyk@35222
   386
lemma bex1_rsp:
kaliszyk@35222
   387
  assumes a: "(R ===> (op =)) f g"
kaliszyk@35222
   388
  shows "Ex1 (\<lambda>x. x \<in> Respects R \<and> f x) = Ex1 (\<lambda>x. x \<in> Respects R \<and> g x)"
haftmann@40466
   389
  using a by (auto elim: fun_relE simp add: Ex1_def in_respects) 
kaliszyk@35222
   390
kaliszyk@35222
   391
(* 2 lemmas needed for cleaning of quantifiers *)
kaliszyk@35222
   392
lemma all_prs:
kaliszyk@35222
   393
  assumes a: "Quotient R absf repf"
kaliszyk@35222
   394
  shows "Ball (Respects R) ((absf ---> id) f) = All f"
haftmann@40602
   395
  using a unfolding Quotient_def Ball_def in_respects id_apply comp_def map_fun_def
kaliszyk@35222
   396
  by metis
kaliszyk@35222
   397
kaliszyk@35222
   398
lemma ex_prs:
kaliszyk@35222
   399
  assumes a: "Quotient R absf repf"
kaliszyk@35222
   400
  shows "Bex (Respects R) ((absf ---> id) f) = Ex f"
haftmann@40602
   401
  using a unfolding Quotient_def Bex_def in_respects id_apply comp_def map_fun_def
kaliszyk@35222
   402
  by metis
kaliszyk@35222
   403
huffman@35294
   404
subsection {* @{text Bex1_rel} quantifier *}
kaliszyk@35222
   405
kaliszyk@35222
   406
definition
kaliszyk@35222
   407
  Bex1_rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
kaliszyk@35222
   408
where
kaliszyk@35222
   409
  "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
   410
kaliszyk@35222
   411
lemma bex1_rel_aux:
kaliszyk@35222
   412
  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R x\<rbrakk> \<Longrightarrow> Bex1_rel R y"
kaliszyk@35222
   413
  unfolding Bex1_rel_def
kaliszyk@35222
   414
  apply (erule conjE)+
kaliszyk@35222
   415
  apply (erule bexE)
kaliszyk@35222
   416
  apply rule
kaliszyk@35222
   417
  apply (rule_tac x="xa" in bexI)
kaliszyk@35222
   418
  apply metis
kaliszyk@35222
   419
  apply metis
kaliszyk@35222
   420
  apply rule+
kaliszyk@35222
   421
  apply (erule_tac x="xaa" in ballE)
kaliszyk@35222
   422
  prefer 2
kaliszyk@35222
   423
  apply (metis)
kaliszyk@35222
   424
  apply (erule_tac x="ya" in ballE)
kaliszyk@35222
   425
  prefer 2
kaliszyk@35222
   426
  apply (metis)
kaliszyk@35222
   427
  apply (metis in_respects)
kaliszyk@35222
   428
  done
kaliszyk@35222
   429
kaliszyk@35222
   430
lemma bex1_rel_aux2:
kaliszyk@35222
   431
  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R y\<rbrakk> \<Longrightarrow> Bex1_rel R x"
kaliszyk@35222
   432
  unfolding Bex1_rel_def
kaliszyk@35222
   433
  apply (erule conjE)+
kaliszyk@35222
   434
  apply (erule bexE)
kaliszyk@35222
   435
  apply rule
kaliszyk@35222
   436
  apply (rule_tac x="xa" in bexI)
kaliszyk@35222
   437
  apply metis
kaliszyk@35222
   438
  apply metis
kaliszyk@35222
   439
  apply rule+
kaliszyk@35222
   440
  apply (erule_tac x="xaa" in ballE)
kaliszyk@35222
   441
  prefer 2
kaliszyk@35222
   442
  apply (metis)
kaliszyk@35222
   443
  apply (erule_tac x="ya" in ballE)
kaliszyk@35222
   444
  prefer 2
kaliszyk@35222
   445
  apply (metis)
kaliszyk@35222
   446
  apply (metis in_respects)
kaliszyk@35222
   447
  done
kaliszyk@35222
   448
kaliszyk@35222
   449
lemma bex1_rel_rsp:
kaliszyk@35222
   450
  assumes a: "Quotient R absf repf"
kaliszyk@35222
   451
  shows "((R ===> op =) ===> op =) (Bex1_rel R) (Bex1_rel R)"
haftmann@40466
   452
  apply (simp add: fun_rel_def)
kaliszyk@35222
   453
  apply clarify
kaliszyk@35222
   454
  apply rule
kaliszyk@35222
   455
  apply (simp_all add: bex1_rel_aux bex1_rel_aux2)
kaliszyk@35222
   456
  apply (erule bex1_rel_aux2)
kaliszyk@35222
   457
  apply assumption
kaliszyk@35222
   458
  done
kaliszyk@35222
   459
kaliszyk@35222
   460
kaliszyk@35222
   461
lemma ex1_prs:
kaliszyk@35222
   462
  assumes a: "Quotient R absf repf"
kaliszyk@35222
   463
  shows "((absf ---> id) ---> id) (Bex1_rel R) f = Ex1 f"
haftmann@40466
   464
apply (simp add:)
kaliszyk@35222
   465
apply (subst Bex1_rel_def)
kaliszyk@35222
   466
apply (subst Bex_def)
kaliszyk@35222
   467
apply (subst Ex1_def)
kaliszyk@35222
   468
apply simp
kaliszyk@35222
   469
apply rule
kaliszyk@35222
   470
 apply (erule conjE)+
kaliszyk@35222
   471
 apply (erule_tac exE)
kaliszyk@35222
   472
 apply (erule conjE)
kaliszyk@35222
   473
 apply (subgoal_tac "\<forall>y. R y y \<longrightarrow> f (absf y) \<longrightarrow> R x y")
kaliszyk@35222
   474
  apply (rule_tac x="absf x" in exI)
kaliszyk@35222
   475
  apply (simp)
kaliszyk@35222
   476
  apply rule+
kaliszyk@35222
   477
  using a unfolding Quotient_def
kaliszyk@35222
   478
  apply metis
kaliszyk@35222
   479
 apply rule+
kaliszyk@35222
   480
 apply (erule_tac x="x" in ballE)
kaliszyk@35222
   481
  apply (erule_tac x="y" in ballE)
kaliszyk@35222
   482
   apply simp
kaliszyk@35222
   483
  apply (simp add: in_respects)
kaliszyk@35222
   484
 apply (simp add: in_respects)
kaliszyk@35222
   485
apply (erule_tac exE)
kaliszyk@35222
   486
 apply rule
kaliszyk@35222
   487
 apply (rule_tac x="repf x" in exI)
kaliszyk@35222
   488
 apply (simp only: in_respects)
kaliszyk@35222
   489
  apply rule
kaliszyk@35222
   490
 apply (metis Quotient_rel_rep[OF a])
kaliszyk@35222
   491
using a unfolding Quotient_def apply (simp)
kaliszyk@35222
   492
apply rule+
kaliszyk@35222
   493
using a unfolding Quotient_def in_respects
kaliszyk@35222
   494
apply metis
kaliszyk@35222
   495
done
kaliszyk@35222
   496
kaliszyk@38702
   497
lemma bex1_bexeq_reg:
kaliszyk@38702
   498
  shows "(\<exists>!x\<in>Respects R. P x) \<longrightarrow> (Bex1_rel R (\<lambda>x. P x))"
kaliszyk@35222
   499
  apply (simp add: Ex1_def Bex1_rel_def in_respects)
kaliszyk@35222
   500
  apply clarify
kaliszyk@35222
   501
  apply auto
kaliszyk@35222
   502
  apply (rule bexI)
kaliszyk@35222
   503
  apply assumption
kaliszyk@35222
   504
  apply (simp add: in_respects)
kaliszyk@35222
   505
  apply (simp add: in_respects)
kaliszyk@35222
   506
  apply auto
kaliszyk@35222
   507
  done
kaliszyk@35222
   508
kaliszyk@38702
   509
lemma bex1_bexeq_reg_eqv:
kaliszyk@38702
   510
  assumes a: "equivp R"
kaliszyk@38702
   511
  shows "(\<exists>!x. P x) \<longrightarrow> Bex1_rel R P"
kaliszyk@38702
   512
  using equivp_reflp[OF a]
kaliszyk@38702
   513
  apply (intro impI)
kaliszyk@38702
   514
  apply (elim ex1E)
kaliszyk@38702
   515
  apply (rule mp[OF bex1_bexeq_reg])
kaliszyk@38702
   516
  apply (rule_tac a="x" in ex1I)
kaliszyk@38702
   517
  apply (subst in_respects)
kaliszyk@38702
   518
  apply (rule conjI)
kaliszyk@38702
   519
  apply assumption
kaliszyk@38702
   520
  apply assumption
kaliszyk@38702
   521
  apply clarify
kaliszyk@38702
   522
  apply (erule_tac x="xa" in allE)
kaliszyk@38702
   523
  apply simp
kaliszyk@38702
   524
  done
kaliszyk@38702
   525
huffman@35294
   526
subsection {* Various respects and preserve lemmas *}
kaliszyk@35222
   527
kaliszyk@35222
   528
lemma quot_rel_rsp:
kaliszyk@35222
   529
  assumes a: "Quotient R Abs Rep"
kaliszyk@35222
   530
  shows "(R ===> R ===> op =) R R"
urbanc@38317
   531
  apply(rule fun_relI)+
kaliszyk@35222
   532
  apply(rule equals_rsp[OF a])
kaliszyk@35222
   533
  apply(assumption)+
kaliszyk@35222
   534
  done
kaliszyk@35222
   535
kaliszyk@35222
   536
lemma o_prs:
kaliszyk@35222
   537
  assumes q1: "Quotient R1 Abs1 Rep1"
kaliszyk@35222
   538
  and     q2: "Quotient R2 Abs2 Rep2"
kaliszyk@35222
   539
  and     q3: "Quotient R3 Abs3 Rep3"
kaliszyk@36215
   540
  shows "((Abs2 ---> Rep3) ---> (Abs1 ---> Rep2) ---> (Rep1 ---> Abs3)) op \<circ> = op \<circ>"
kaliszyk@36215
   541
  and   "(id ---> (Abs1 ---> id) ---> Rep1 ---> id) op \<circ> = op \<circ>"
kaliszyk@35222
   542
  using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] Quotient_abs_rep[OF q3]
haftmann@40466
   543
  by (simp_all add: fun_eq_iff)
kaliszyk@35222
   544
kaliszyk@35222
   545
lemma o_rsp:
kaliszyk@36215
   546
  "((R2 ===> R3) ===> (R1 ===> R2) ===> (R1 ===> R3)) op \<circ> op \<circ>"
kaliszyk@36215
   547
  "(op = ===> (R1 ===> op =) ===> R1 ===> op =) op \<circ> op \<circ>"
haftmann@40466
   548
  by (auto intro!: fun_relI elim: fun_relE)
kaliszyk@35222
   549
kaliszyk@35222
   550
lemma cond_prs:
kaliszyk@35222
   551
  assumes a: "Quotient R absf repf"
kaliszyk@35222
   552
  shows "absf (if a then repf b else repf c) = (if a then b else c)"
kaliszyk@35222
   553
  using a unfolding Quotient_def by auto
kaliszyk@35222
   554
kaliszyk@35222
   555
lemma if_prs:
kaliszyk@35222
   556
  assumes q: "Quotient R Abs Rep"
kaliszyk@36123
   557
  shows "(id ---> Rep ---> Rep ---> Abs) If = If"
kaliszyk@36123
   558
  using Quotient_abs_rep[OF q]
nipkow@39302
   559
  by (auto simp add: fun_eq_iff)
kaliszyk@35222
   560
kaliszyk@35222
   561
lemma if_rsp:
kaliszyk@35222
   562
  assumes q: "Quotient R Abs Rep"
kaliszyk@36123
   563
  shows "(op = ===> R ===> R ===> R) If If"
haftmann@40466
   564
  by (auto intro!: fun_relI)
kaliszyk@35222
   565
kaliszyk@35222
   566
lemma let_prs:
kaliszyk@35222
   567
  assumes q1: "Quotient R1 Abs1 Rep1"
kaliszyk@35222
   568
  and     q2: "Quotient R2 Abs2 Rep2"
kaliszyk@37049
   569
  shows "(Rep2 ---> (Abs2 ---> Rep1) ---> Abs1) Let = Let"
kaliszyk@37049
   570
  using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
nipkow@39302
   571
  by (auto simp add: fun_eq_iff)
kaliszyk@35222
   572
kaliszyk@35222
   573
lemma let_rsp:
kaliszyk@37049
   574
  shows "(R1 ===> (R1 ===> R2) ===> R2) Let Let"
haftmann@40466
   575
  by (auto intro!: fun_relI elim: fun_relE)
kaliszyk@35222
   576
kaliszyk@38861
   577
lemma mem_rsp:
kaliszyk@38861
   578
  shows "(R1 ===> (R1 ===> R2) ===> R2) op \<in> op \<in>"
haftmann@40466
   579
  by (auto intro!: fun_relI elim: fun_relE simp add: mem_def)
kaliszyk@38861
   580
kaliszyk@38861
   581
lemma mem_prs:
kaliszyk@38861
   582
  assumes a1: "Quotient R1 Abs1 Rep1"
kaliszyk@38861
   583
  and     a2: "Quotient R2 Abs2 Rep2"
kaliszyk@38861
   584
  shows "(Rep1 ---> (Abs1 ---> Rep2) ---> Abs2) op \<in> = op \<in>"
nipkow@39302
   585
  by (simp add: fun_eq_iff mem_def Quotient_abs_rep[OF a1] Quotient_abs_rep[OF a2])
kaliszyk@38861
   586
kaliszyk@39669
   587
lemma id_rsp:
kaliszyk@39669
   588
  shows "(R ===> R) id id"
haftmann@40466
   589
  by (auto intro: fun_relI)
kaliszyk@39669
   590
kaliszyk@39669
   591
lemma id_prs:
kaliszyk@39669
   592
  assumes a: "Quotient R Abs Rep"
kaliszyk@39669
   593
  shows "(Rep ---> Abs) id = id"
haftmann@40466
   594
  by (simp add: fun_eq_iff Quotient_abs_rep [OF a])
kaliszyk@39669
   595
kaliszyk@39669
   596
kaliszyk@35222
   597
locale quot_type =
kaliszyk@35222
   598
  fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kaliszyk@35222
   599
  and   Abs :: "('a \<Rightarrow> bool) \<Rightarrow> 'b"
kaliszyk@35222
   600
  and   Rep :: "'b \<Rightarrow> ('a \<Rightarrow> bool)"
kaliszyk@37493
   601
  assumes equivp: "part_equivp R"
kaliszyk@37493
   602
  and     rep_prop: "\<And>y. \<exists>x. R x x \<and> Rep y = R x"
kaliszyk@35222
   603
  and     rep_inverse: "\<And>x. Abs (Rep x) = x"
kaliszyk@37493
   604
  and     abs_inverse: "\<And>c. (\<exists>x. ((R x x) \<and> (c = R x))) \<Longrightarrow> (Rep (Abs c)) = c"
kaliszyk@35222
   605
  and     rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"
kaliszyk@35222
   606
begin
kaliszyk@35222
   607
kaliszyk@35222
   608
definition
haftmann@40466
   609
  abs :: "'a \<Rightarrow> 'b"
kaliszyk@35222
   610
where
haftmann@40466
   611
  "abs x = Abs (R x)"
kaliszyk@35222
   612
kaliszyk@35222
   613
definition
haftmann@40466
   614
  rep :: "'b \<Rightarrow> 'a"
kaliszyk@35222
   615
where
kaliszyk@35222
   616
  "rep a = Eps (Rep a)"
kaliszyk@35222
   617
kaliszyk@37493
   618
lemma homeier5:
kaliszyk@37493
   619
  assumes a: "R r r"
kaliszyk@37493
   620
  shows "Rep (Abs (R r)) = R r"
kaliszyk@37493
   621
  apply (subst abs_inverse)
kaliszyk@37493
   622
  using a by auto
kaliszyk@35222
   623
kaliszyk@37493
   624
theorem homeier6:
kaliszyk@37493
   625
  assumes a: "R r r"
kaliszyk@37493
   626
  and b: "R s s"
kaliszyk@37493
   627
  shows "Abs (R r) = Abs (R s) \<longleftrightarrow> R r = R s"
kaliszyk@37493
   628
  by (metis a b homeier5)
kaliszyk@35222
   629
kaliszyk@37493
   630
theorem homeier8:
kaliszyk@37493
   631
  assumes "R r r"
kaliszyk@37493
   632
  shows "R (Eps (R r)) = R r"
kaliszyk@37493
   633
  using assms equivp[simplified part_equivp_def]
kaliszyk@37493
   634
  apply clarify
kaliszyk@37493
   635
  by (metis assms exE_some)
kaliszyk@35222
   636
kaliszyk@35222
   637
lemma Quotient:
kaliszyk@35222
   638
  shows "Quotient R abs rep"
kaliszyk@37493
   639
  unfolding Quotient_def abs_def rep_def
kaliszyk@37493
   640
  proof (intro conjI allI)
kaliszyk@37493
   641
    fix a r s
kaliszyk@37493
   642
    show "Abs (R (Eps (Rep a))) = a"
kaliszyk@37493
   643
      by (metis equivp exE_some part_equivp_def rep_inverse rep_prop)
kaliszyk@37493
   644
    show "R r s \<longleftrightarrow> R r r \<and> R s s \<and> (Abs (R r) = Abs (R s))"
kaliszyk@37493
   645
      by (metis homeier6 equivp[simplified part_equivp_def])
kaliszyk@37493
   646
    show "R (Eps (Rep a)) (Eps (Rep a))" proof -
kaliszyk@37493
   647
      obtain x where r: "R x x" and rep: "Rep a = R x" using rep_prop[of a] by auto
kaliszyk@37493
   648
      have "R (Eps (R x)) x" using homeier8 r by simp
kaliszyk@37493
   649
      then have "R x (Eps (R x))" using part_equivp_symp[OF equivp] by fast
kaliszyk@37493
   650
      then have "R (Eps (R x)) (Eps (R x))" using homeier8[OF r] by simp
kaliszyk@37493
   651
      then show "R (Eps (Rep a)) (Eps (Rep a))" using rep by simp
kaliszyk@37493
   652
    qed
kaliszyk@37493
   653
  qed
kaliszyk@35222
   654
kaliszyk@35222
   655
end
kaliszyk@35222
   656
kaliszyk@37493
   657
huffman@35294
   658
subsection {* ML setup *}
kaliszyk@35222
   659
kaliszyk@35222
   660
text {* Auxiliary data for the quotient package *}
kaliszyk@35222
   661
wenzelm@37986
   662
use "Tools/Quotient/quotient_info.ML"
kaliszyk@35222
   663
haftmann@40602
   664
declare [[map "fun" = (map_fun, fun_rel)]]
kaliszyk@35222
   665
kaliszyk@35222
   666
lemmas [quot_thm] = fun_quotient
kaliszyk@39669
   667
lemmas [quot_respect] = quot_rel_rsp if_rsp o_rsp let_rsp mem_rsp id_rsp
kaliszyk@39669
   668
lemmas [quot_preserve] = if_prs o_prs let_prs mem_prs id_prs
kaliszyk@35222
   669
lemmas [quot_equiv] = identity_equivp
kaliszyk@35222
   670
kaliszyk@35222
   671
kaliszyk@35222
   672
text {* Lemmas about simplifying id's. *}
kaliszyk@35222
   673
lemmas [id_simps] =
kaliszyk@35222
   674
  id_def[symmetric]
haftmann@40602
   675
  map_fun_id
kaliszyk@35222
   676
  id_apply
kaliszyk@35222
   677
  id_o
kaliszyk@35222
   678
  o_id
kaliszyk@35222
   679
  eq_comp_r
kaliszyk@35222
   680
kaliszyk@35222
   681
text {* Translation functions for the lifting process. *}
wenzelm@37986
   682
use "Tools/Quotient/quotient_term.ML"
kaliszyk@35222
   683
kaliszyk@35222
   684
kaliszyk@35222
   685
text {* Definitions of the quotient types. *}
wenzelm@37986
   686
use "Tools/Quotient/quotient_typ.ML"
kaliszyk@35222
   687
kaliszyk@35222
   688
kaliszyk@35222
   689
text {* Definitions for quotient constants. *}
wenzelm@37986
   690
use "Tools/Quotient/quotient_def.ML"
kaliszyk@35222
   691
kaliszyk@35222
   692
kaliszyk@35222
   693
text {*
kaliszyk@35222
   694
  An auxiliary constant for recording some information
kaliszyk@35222
   695
  about the lifted theorem in a tactic.
kaliszyk@35222
   696
*}
kaliszyk@35222
   697
definition
haftmann@40466
   698
  Quot_True :: "'a \<Rightarrow> bool"
haftmann@40466
   699
where
haftmann@40466
   700
  "Quot_True x \<longleftrightarrow> True"
kaliszyk@35222
   701
kaliszyk@35222
   702
lemma
kaliszyk@35222
   703
  shows QT_all: "Quot_True (All P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   704
  and   QT_ex:  "Quot_True (Ex P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   705
  and   QT_ex1: "Quot_True (Ex1 P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   706
  and   QT_lam: "Quot_True (\<lambda>x. P x) \<Longrightarrow> (\<And>x. Quot_True (P x))"
kaliszyk@35222
   707
  and   QT_ext: "(\<And>x. Quot_True (a x) \<Longrightarrow> f x = g x) \<Longrightarrow> (Quot_True a \<Longrightarrow> f = g)"
kaliszyk@35222
   708
  by (simp_all add: Quot_True_def ext)
kaliszyk@35222
   709
kaliszyk@35222
   710
lemma QT_imp: "Quot_True a \<equiv> Quot_True b"
kaliszyk@35222
   711
  by (simp add: Quot_True_def)
kaliszyk@35222
   712
kaliszyk@35222
   713
kaliszyk@35222
   714
text {* Tactics for proving the lifted theorems *}
wenzelm@37986
   715
use "Tools/Quotient/quotient_tacs.ML"
kaliszyk@35222
   716
huffman@35294
   717
subsection {* Methods / Interface *}
kaliszyk@35222
   718
kaliszyk@35222
   719
method_setup lifting =
urbanc@37593
   720
  {* Attrib.thms >> (fn thms => fn ctxt => 
urbanc@38859
   721
       SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.lift_tac ctxt [] thms))) *}
kaliszyk@35222
   722
  {* lifts theorems to quotient types *}
kaliszyk@35222
   723
kaliszyk@35222
   724
method_setup lifting_setup =
urbanc@37593
   725
  {* Attrib.thm >> (fn thm => fn ctxt => 
urbanc@38859
   726
       SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.lift_procedure_tac ctxt [] thm))) *}
kaliszyk@35222
   727
  {* sets up the three goals for the quotient lifting procedure *}
kaliszyk@35222
   728
urbanc@37593
   729
method_setup descending =
urbanc@38859
   730
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.descend_tac ctxt []))) *}
urbanc@37593
   731
  {* decends theorems to the raw level *}
urbanc@37593
   732
urbanc@37593
   733
method_setup descending_setup =
urbanc@38859
   734
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.descend_procedure_tac ctxt []))) *}
urbanc@37593
   735
  {* sets up the three goals for the decending theorems *}
urbanc@37593
   736
kaliszyk@35222
   737
method_setup regularize =
kaliszyk@35222
   738
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.regularize_tac ctxt))) *}
kaliszyk@35222
   739
  {* proves the regularization goals from the quotient lifting procedure *}
kaliszyk@35222
   740
kaliszyk@35222
   741
method_setup injection =
kaliszyk@35222
   742
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.all_injection_tac ctxt))) *}
kaliszyk@35222
   743
  {* proves the rep/abs injection goals from the quotient lifting procedure *}
kaliszyk@35222
   744
kaliszyk@35222
   745
method_setup cleaning =
kaliszyk@35222
   746
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.clean_tac ctxt))) *}
kaliszyk@35222
   747
  {* proves the cleaning goals from the quotient lifting procedure *}
kaliszyk@35222
   748
kaliszyk@35222
   749
attribute_setup quot_lifted =
kaliszyk@35222
   750
  {* Scan.succeed Quotient_Tacs.lifted_attrib *}
kaliszyk@35222
   751
  {* lifts theorems to quotient types *}
kaliszyk@35222
   752
kaliszyk@35222
   753
no_notation
kaliszyk@35222
   754
  rel_conj (infixr "OOO" 75) and
haftmann@40602
   755
  map_fun (infixr "--->" 55) and
kaliszyk@35222
   756
  fun_rel (infixr "===>" 55)
kaliszyk@35222
   757
kaliszyk@35222
   758
end