src/HOL/Lifting.thy
author kuncar
Thu May 24 14:20:23 2012 +0200 (2012-05-24)
changeset 47982 7aa35601ff65
parent 47937 70375fa2679d
child 48891 c0eafbd55de3
permissions -rw-r--r--
prove reflexivity also for the quotient composition relation; reflp_preserve renamed to reflexivity_rule
kuncar@47308
     1
(*  Title:      HOL/Lifting.thy
kuncar@47308
     2
    Author:     Brian Huffman and Ondrej Kuncar
kuncar@47308
     3
    Author:     Cezary Kaliszyk and Christian Urban
kuncar@47308
     4
*)
kuncar@47308
     5
kuncar@47308
     6
header {* Lifting package *}
kuncar@47308
     7
kuncar@47308
     8
theory Lifting
huffman@47325
     9
imports Plain Equiv_Relations Transfer
kuncar@47308
    10
keywords
kuncar@47308
    11
  "print_quotmaps" "print_quotients" :: diag and
kuncar@47308
    12
  "lift_definition" :: thy_goal and
kuncar@47308
    13
  "setup_lifting" :: thy_decl
kuncar@47308
    14
uses
kuncar@47698
    15
  ("Tools/Lifting/lifting_util.ML")
kuncar@47308
    16
  ("Tools/Lifting/lifting_info.ML")
kuncar@47308
    17
  ("Tools/Lifting/lifting_term.ML")
kuncar@47308
    18
  ("Tools/Lifting/lifting_def.ML")
kuncar@47308
    19
  ("Tools/Lifting/lifting_setup.ML")
kuncar@47308
    20
begin
kuncar@47308
    21
huffman@47325
    22
subsection {* Function map *}
kuncar@47308
    23
kuncar@47308
    24
notation map_fun (infixr "--->" 55)
kuncar@47308
    25
kuncar@47308
    26
lemma map_fun_id:
kuncar@47308
    27
  "(id ---> id) = id"
kuncar@47308
    28
  by (simp add: fun_eq_iff)
kuncar@47308
    29
kuncar@47308
    30
subsection {* Quotient Predicate *}
kuncar@47308
    31
kuncar@47308
    32
definition
kuncar@47308
    33
  "Quotient R Abs Rep T \<longleftrightarrow>
kuncar@47308
    34
     (\<forall>a. Abs (Rep a) = a) \<and> 
kuncar@47308
    35
     (\<forall>a. R (Rep a) (Rep a)) \<and>
kuncar@47308
    36
     (\<forall>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s) \<and>
kuncar@47308
    37
     T = (\<lambda>x y. R x x \<and> Abs x = y)"
kuncar@47308
    38
kuncar@47308
    39
lemma QuotientI:
kuncar@47308
    40
  assumes "\<And>a. Abs (Rep a) = a"
kuncar@47308
    41
    and "\<And>a. R (Rep a) (Rep a)"
kuncar@47308
    42
    and "\<And>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s"
kuncar@47308
    43
    and "T = (\<lambda>x y. R x x \<and> Abs x = y)"
kuncar@47308
    44
  shows "Quotient R Abs Rep T"
kuncar@47308
    45
  using assms unfolding Quotient_def by blast
kuncar@47308
    46
huffman@47536
    47
context
huffman@47536
    48
  fixes R Abs Rep T
kuncar@47308
    49
  assumes a: "Quotient R Abs Rep T"
huffman@47536
    50
begin
huffman@47536
    51
huffman@47536
    52
lemma Quotient_abs_rep: "Abs (Rep a) = a"
huffman@47536
    53
  using a unfolding Quotient_def
kuncar@47308
    54
  by simp
kuncar@47308
    55
huffman@47536
    56
lemma Quotient_rep_reflp: "R (Rep a) (Rep a)"
huffman@47536
    57
  using a unfolding Quotient_def
kuncar@47308
    58
  by blast
kuncar@47308
    59
kuncar@47308
    60
lemma Quotient_rel:
huffman@47536
    61
  "R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" -- {* orientation does not loop on rewriting *}
huffman@47536
    62
  using a unfolding Quotient_def
kuncar@47308
    63
  by blast
kuncar@47308
    64
huffman@47536
    65
lemma Quotient_cr_rel: "T = (\<lambda>x y. R x x \<and> Abs x = y)"
kuncar@47308
    66
  using a unfolding Quotient_def
kuncar@47308
    67
  by blast
kuncar@47308
    68
huffman@47536
    69
lemma Quotient_refl1: "R r s \<Longrightarrow> R r r"
huffman@47536
    70
  using a unfolding Quotient_def
huffman@47536
    71
  by fast
huffman@47536
    72
huffman@47536
    73
lemma Quotient_refl2: "R r s \<Longrightarrow> R s s"
huffman@47536
    74
  using a unfolding Quotient_def
huffman@47536
    75
  by fast
huffman@47536
    76
huffman@47536
    77
lemma Quotient_rel_rep: "R (Rep a) (Rep b) \<longleftrightarrow> a = b"
huffman@47536
    78
  using a unfolding Quotient_def
huffman@47536
    79
  by metis
huffman@47536
    80
huffman@47536
    81
lemma Quotient_rep_abs: "R r r \<Longrightarrow> R (Rep (Abs r)) r"
kuncar@47308
    82
  using a unfolding Quotient_def
kuncar@47308
    83
  by blast
kuncar@47308
    84
kuncar@47937
    85
lemma Quotient_rep_abs_fold_unmap: 
kuncar@47937
    86
  assumes "x' \<equiv> Abs x" and "R x x" and "Rep x' \<equiv> Rep' x'" 
kuncar@47937
    87
  shows "R (Rep' x') x"
kuncar@47937
    88
proof -
kuncar@47937
    89
  have "R (Rep x') x" using assms(1-2) Quotient_rep_abs by auto
kuncar@47937
    90
  then show ?thesis using assms(3) by simp
kuncar@47937
    91
qed
kuncar@47937
    92
kuncar@47937
    93
lemma Quotient_Rep_eq:
kuncar@47937
    94
  assumes "x' \<equiv> Abs x" 
kuncar@47937
    95
  shows "Rep x' \<equiv> Rep x'"
kuncar@47937
    96
by simp
kuncar@47937
    97
huffman@47536
    98
lemma Quotient_rel_abs: "R r s \<Longrightarrow> Abs r = Abs s"
huffman@47536
    99
  using a unfolding Quotient_def
huffman@47536
   100
  by blast
huffman@47536
   101
kuncar@47937
   102
lemma Quotient_rel_abs2:
kuncar@47937
   103
  assumes "R (Rep x) y"
kuncar@47937
   104
  shows "x = Abs y"
kuncar@47937
   105
proof -
kuncar@47937
   106
  from assms have "Abs (Rep x) = Abs y" by (auto intro: Quotient_rel_abs)
kuncar@47937
   107
  then show ?thesis using assms(1) by (simp add: Quotient_abs_rep)
kuncar@47937
   108
qed
kuncar@47937
   109
huffman@47536
   110
lemma Quotient_symp: "symp R"
kuncar@47308
   111
  using a unfolding Quotient_def using sympI by (metis (full_types))
kuncar@47308
   112
huffman@47536
   113
lemma Quotient_transp: "transp R"
kuncar@47308
   114
  using a unfolding Quotient_def using transpI by (metis (full_types))
kuncar@47308
   115
huffman@47536
   116
lemma Quotient_part_equivp: "part_equivp R"
huffman@47536
   117
by (metis Quotient_rep_reflp Quotient_symp Quotient_transp part_equivpI)
huffman@47536
   118
huffman@47536
   119
end
kuncar@47308
   120
kuncar@47308
   121
lemma identity_quotient: "Quotient (op =) id id (op =)"
kuncar@47308
   122
unfolding Quotient_def by simp 
kuncar@47308
   123
huffman@47652
   124
text {* TODO: Use one of these alternatives as the real definition. *}
huffman@47652
   125
kuncar@47308
   126
lemma Quotient_alt_def:
kuncar@47308
   127
  "Quotient R Abs Rep T \<longleftrightarrow>
kuncar@47308
   128
    (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and>
kuncar@47308
   129
    (\<forall>b. T (Rep b) b) \<and>
kuncar@47308
   130
    (\<forall>x y. R x y \<longleftrightarrow> T x (Abs x) \<and> T y (Abs y) \<and> Abs x = Abs y)"
kuncar@47308
   131
apply safe
kuncar@47308
   132
apply (simp (no_asm_use) only: Quotient_def, fast)
kuncar@47308
   133
apply (simp (no_asm_use) only: Quotient_def, fast)
kuncar@47308
   134
apply (simp (no_asm_use) only: Quotient_def, fast)
kuncar@47308
   135
apply (simp (no_asm_use) only: Quotient_def, fast)
kuncar@47308
   136
apply (simp (no_asm_use) only: Quotient_def, fast)
kuncar@47308
   137
apply (simp (no_asm_use) only: Quotient_def, fast)
kuncar@47308
   138
apply (rule QuotientI)
kuncar@47308
   139
apply simp
kuncar@47308
   140
apply metis
kuncar@47308
   141
apply simp
kuncar@47308
   142
apply (rule ext, rule ext, metis)
kuncar@47308
   143
done
kuncar@47308
   144
kuncar@47308
   145
lemma Quotient_alt_def2:
kuncar@47308
   146
  "Quotient R Abs Rep T \<longleftrightarrow>
kuncar@47308
   147
    (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and>
kuncar@47308
   148
    (\<forall>b. T (Rep b) b) \<and>
kuncar@47308
   149
    (\<forall>x y. R x y \<longleftrightarrow> T x (Abs y) \<and> T y (Abs x))"
kuncar@47308
   150
  unfolding Quotient_alt_def by (safe, metis+)
kuncar@47308
   151
huffman@47652
   152
lemma Quotient_alt_def3:
huffman@47652
   153
  "Quotient R Abs Rep T \<longleftrightarrow>
huffman@47652
   154
    (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and>
huffman@47652
   155
    (\<forall>x y. R x y \<longleftrightarrow> (\<exists>z. T x z \<and> T y z))"
huffman@47652
   156
  unfolding Quotient_alt_def2 by (safe, metis+)
huffman@47652
   157
huffman@47652
   158
lemma Quotient_alt_def4:
huffman@47652
   159
  "Quotient R Abs Rep T \<longleftrightarrow>
huffman@47652
   160
    (\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and> R = T OO conversep T"
huffman@47652
   161
  unfolding Quotient_alt_def3 fun_eq_iff by auto
huffman@47652
   162
kuncar@47308
   163
lemma fun_quotient:
kuncar@47308
   164
  assumes 1: "Quotient R1 abs1 rep1 T1"
kuncar@47308
   165
  assumes 2: "Quotient R2 abs2 rep2 T2"
kuncar@47308
   166
  shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2) (T1 ===> T2)"
kuncar@47308
   167
  using assms unfolding Quotient_alt_def2
kuncar@47308
   168
  unfolding fun_rel_def fun_eq_iff map_fun_apply
kuncar@47308
   169
  by (safe, metis+)
kuncar@47308
   170
kuncar@47308
   171
lemma apply_rsp:
kuncar@47308
   172
  fixes f g::"'a \<Rightarrow> 'c"
kuncar@47308
   173
  assumes q: "Quotient R1 Abs1 Rep1 T1"
kuncar@47308
   174
  and     a: "(R1 ===> R2) f g" "R1 x y"
kuncar@47308
   175
  shows "R2 (f x) (g y)"
kuncar@47308
   176
  using a by (auto elim: fun_relE)
kuncar@47308
   177
kuncar@47308
   178
lemma apply_rsp':
kuncar@47308
   179
  assumes a: "(R1 ===> R2) f g" "R1 x y"
kuncar@47308
   180
  shows "R2 (f x) (g y)"
kuncar@47308
   181
  using a by (auto elim: fun_relE)
kuncar@47308
   182
kuncar@47308
   183
lemma apply_rsp'':
kuncar@47308
   184
  assumes "Quotient R Abs Rep T"
kuncar@47308
   185
  and "(R ===> S) f f"
kuncar@47308
   186
  shows "S (f (Rep x)) (f (Rep x))"
kuncar@47308
   187
proof -
kuncar@47308
   188
  from assms(1) have "R (Rep x) (Rep x)" by (rule Quotient_rep_reflp)
kuncar@47308
   189
  then show ?thesis using assms(2) by (auto intro: apply_rsp')
kuncar@47308
   190
qed
kuncar@47308
   191
kuncar@47308
   192
subsection {* Quotient composition *}
kuncar@47308
   193
kuncar@47308
   194
lemma Quotient_compose:
kuncar@47308
   195
  assumes 1: "Quotient R1 Abs1 Rep1 T1"
kuncar@47308
   196
  assumes 2: "Quotient R2 Abs2 Rep2 T2"
kuncar@47308
   197
  shows "Quotient (T1 OO R2 OO conversep T1) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2) (T1 OO T2)"
huffman@47652
   198
  using assms unfolding Quotient_alt_def4 by (auto intro!: ext)
kuncar@47308
   199
kuncar@47521
   200
lemma equivp_reflp2:
kuncar@47521
   201
  "equivp R \<Longrightarrow> reflp R"
kuncar@47521
   202
  by (erule equivpE)
kuncar@47521
   203
huffman@47544
   204
subsection {* Respects predicate *}
huffman@47544
   205
huffman@47544
   206
definition Respects :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set"
huffman@47544
   207
  where "Respects R = {x. R x x}"
huffman@47544
   208
huffman@47544
   209
lemma in_respects: "x \<in> Respects R \<longleftrightarrow> R x x"
huffman@47544
   210
  unfolding Respects_def by simp
huffman@47544
   211
kuncar@47308
   212
subsection {* Invariant *}
kuncar@47308
   213
kuncar@47308
   214
definition invariant :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" 
kuncar@47308
   215
  where "invariant R = (\<lambda>x y. R x \<and> x = y)"
kuncar@47308
   216
kuncar@47308
   217
lemma invariant_to_eq:
kuncar@47308
   218
  assumes "invariant P x y"
kuncar@47308
   219
  shows "x = y"
kuncar@47308
   220
using assms by (simp add: invariant_def)
kuncar@47308
   221
kuncar@47308
   222
lemma fun_rel_eq_invariant:
kuncar@47308
   223
  shows "((invariant R) ===> S) = (\<lambda>f g. \<forall>x. R x \<longrightarrow> S (f x) (g x))"
kuncar@47308
   224
by (auto simp add: invariant_def fun_rel_def)
kuncar@47308
   225
kuncar@47308
   226
lemma invariant_same_args:
kuncar@47308
   227
  shows "invariant P x x \<equiv> P x"
kuncar@47308
   228
using assms by (auto simp add: invariant_def)
kuncar@47308
   229
kuncar@47361
   230
lemma UNIV_typedef_to_Quotient:
kuncar@47308
   231
  assumes "type_definition Rep Abs UNIV"
kuncar@47361
   232
  and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
kuncar@47308
   233
  shows "Quotient (op =) Abs Rep T"
kuncar@47308
   234
proof -
kuncar@47308
   235
  interpret type_definition Rep Abs UNIV by fact
kuncar@47361
   236
  from Abs_inject Rep_inverse Abs_inverse T_def show ?thesis 
kuncar@47361
   237
    by (fastforce intro!: QuotientI fun_eq_iff)
kuncar@47308
   238
qed
kuncar@47308
   239
kuncar@47361
   240
lemma UNIV_typedef_to_equivp:
kuncar@47308
   241
  fixes Abs :: "'a \<Rightarrow> 'b"
kuncar@47308
   242
  and Rep :: "'b \<Rightarrow> 'a"
kuncar@47308
   243
  assumes "type_definition Rep Abs (UNIV::'a set)"
kuncar@47308
   244
  shows "equivp (op=::'a\<Rightarrow>'a\<Rightarrow>bool)"
kuncar@47308
   245
by (rule identity_equivp)
kuncar@47308
   246
huffman@47354
   247
lemma typedef_to_Quotient:
kuncar@47361
   248
  assumes "type_definition Rep Abs S"
kuncar@47361
   249
  and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
kuncar@47501
   250
  shows "Quotient (invariant (\<lambda>x. x \<in> S)) Abs Rep T"
kuncar@47361
   251
proof -
kuncar@47361
   252
  interpret type_definition Rep Abs S by fact
kuncar@47361
   253
  from Rep Abs_inject Rep_inverse Abs_inverse T_def show ?thesis
kuncar@47361
   254
    by (auto intro!: QuotientI simp: invariant_def fun_eq_iff)
kuncar@47361
   255
qed
kuncar@47361
   256
kuncar@47361
   257
lemma typedef_to_part_equivp:
kuncar@47361
   258
  assumes "type_definition Rep Abs S"
kuncar@47501
   259
  shows "part_equivp (invariant (\<lambda>x. x \<in> S))"
kuncar@47361
   260
proof (intro part_equivpI)
kuncar@47361
   261
  interpret type_definition Rep Abs S by fact
kuncar@47501
   262
  show "\<exists>x. invariant (\<lambda>x. x \<in> S) x x" using Rep by (auto simp: invariant_def)
kuncar@47361
   263
next
kuncar@47501
   264
  show "symp (invariant (\<lambda>x. x \<in> S))" by (auto intro: sympI simp: invariant_def)
kuncar@47361
   265
next
kuncar@47501
   266
  show "transp (invariant (\<lambda>x. x \<in> S))" by (auto intro: transpI simp: invariant_def)
kuncar@47361
   267
qed
kuncar@47361
   268
kuncar@47361
   269
lemma open_typedef_to_Quotient:
kuncar@47308
   270
  assumes "type_definition Rep Abs {x. P x}"
huffman@47354
   271
  and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
kuncar@47308
   272
  shows "Quotient (invariant P) Abs Rep T"
huffman@47651
   273
  using typedef_to_Quotient [OF assms] by simp
kuncar@47308
   274
kuncar@47361
   275
lemma open_typedef_to_part_equivp:
kuncar@47308
   276
  assumes "type_definition Rep Abs {x. P x}"
kuncar@47308
   277
  shows "part_equivp (invariant P)"
huffman@47651
   278
  using typedef_to_part_equivp [OF assms] by simp
kuncar@47308
   279
huffman@47376
   280
text {* Generating transfer rules for quotients. *}
huffman@47376
   281
huffman@47537
   282
context
huffman@47537
   283
  fixes R Abs Rep T
huffman@47537
   284
  assumes 1: "Quotient R Abs Rep T"
huffman@47537
   285
begin
huffman@47376
   286
huffman@47537
   287
lemma Quotient_right_unique: "right_unique T"
huffman@47537
   288
  using 1 unfolding Quotient_alt_def right_unique_def by metis
huffman@47537
   289
huffman@47537
   290
lemma Quotient_right_total: "right_total T"
huffman@47537
   291
  using 1 unfolding Quotient_alt_def right_total_def by metis
huffman@47537
   292
huffman@47537
   293
lemma Quotient_rel_eq_transfer: "(T ===> T ===> op =) R (op =)"
huffman@47537
   294
  using 1 unfolding Quotient_alt_def fun_rel_def by simp
huffman@47376
   295
huffman@47538
   296
lemma Quotient_abs_induct:
huffman@47538
   297
  assumes "\<And>y. R y y \<Longrightarrow> P (Abs y)" shows "P x"
huffman@47538
   298
  using 1 assms unfolding Quotient_def by metis
huffman@47538
   299
huffman@47544
   300
lemma Quotient_All_transfer:
huffman@47544
   301
  "((T ===> op =) ===> op =) (Ball (Respects R)) All"
huffman@47544
   302
  unfolding fun_rel_def Respects_def Quotient_cr_rel [OF 1]
huffman@47544
   303
  by (auto, metis Quotient_abs_induct)
huffman@47544
   304
huffman@47544
   305
lemma Quotient_Ex_transfer:
huffman@47544
   306
  "((T ===> op =) ===> op =) (Bex (Respects R)) Ex"
huffman@47544
   307
  unfolding fun_rel_def Respects_def Quotient_cr_rel [OF 1]
huffman@47544
   308
  by (auto, metis Quotient_rep_reflp [OF 1] Quotient_abs_rep [OF 1])
huffman@47544
   309
huffman@47544
   310
lemma Quotient_forall_transfer:
huffman@47544
   311
  "((T ===> op =) ===> op =) (transfer_bforall (\<lambda>x. R x x)) transfer_forall"
huffman@47544
   312
  using Quotient_All_transfer
huffman@47544
   313
  unfolding transfer_forall_def transfer_bforall_def
huffman@47544
   314
    Ball_def [abs_def] in_respects .
huffman@47544
   315
huffman@47537
   316
end
huffman@47537
   317
huffman@47537
   318
text {* Generating transfer rules for total quotients. *}
huffman@47376
   319
huffman@47537
   320
context
huffman@47537
   321
  fixes R Abs Rep T
huffman@47537
   322
  assumes 1: "Quotient R Abs Rep T" and 2: "reflp R"
huffman@47537
   323
begin
huffman@47376
   324
huffman@47537
   325
lemma Quotient_bi_total: "bi_total T"
huffman@47537
   326
  using 1 2 unfolding Quotient_alt_def bi_total_def reflp_def by auto
huffman@47537
   327
huffman@47537
   328
lemma Quotient_id_abs_transfer: "(op = ===> T) (\<lambda>x. x) Abs"
huffman@47537
   329
  using 1 2 unfolding Quotient_alt_def reflp_def fun_rel_def by simp
huffman@47537
   330
huffman@47575
   331
lemma Quotient_total_abs_induct: "(\<And>y. P (Abs y)) \<Longrightarrow> P x"
huffman@47575
   332
  using 1 2 assms unfolding Quotient_alt_def reflp_def by metis
huffman@47575
   333
huffman@47889
   334
lemma Quotient_total_abs_eq_iff: "Abs x = Abs y \<longleftrightarrow> R x y"
huffman@47889
   335
  using Quotient_rel [OF 1] 2 unfolding reflp_def by simp
huffman@47889
   336
huffman@47537
   337
end
huffman@47376
   338
huffman@47368
   339
text {* Generating transfer rules for a type defined with @{text "typedef"}. *}
huffman@47368
   340
huffman@47534
   341
context
huffman@47534
   342
  fixes Rep Abs A T
huffman@47368
   343
  assumes type: "type_definition Rep Abs A"
huffman@47534
   344
  assumes T_def: "T \<equiv> (\<lambda>(x::'a) (y::'b). x = Rep y)"
huffman@47534
   345
begin
huffman@47534
   346
huffman@47534
   347
lemma typedef_bi_unique: "bi_unique T"
huffman@47368
   348
  unfolding bi_unique_def T_def
huffman@47368
   349
  by (simp add: type_definition.Rep_inject [OF type])
huffman@47368
   350
huffman@47535
   351
lemma typedef_rep_transfer: "(T ===> op =) (\<lambda>x. x) Rep"
huffman@47535
   352
  unfolding fun_rel_def T_def by simp
huffman@47535
   353
kuncar@47545
   354
lemma typedef_All_transfer: "((T ===> op =) ===> op =) (Ball A) All"
huffman@47534
   355
  unfolding T_def fun_rel_def
huffman@47534
   356
  by (metis type_definition.Rep [OF type]
huffman@47534
   357
    type_definition.Abs_inverse [OF type])
huffman@47534
   358
kuncar@47545
   359
lemma typedef_Ex_transfer: "((T ===> op =) ===> op =) (Bex A) Ex"
kuncar@47545
   360
  unfolding T_def fun_rel_def
kuncar@47545
   361
  by (metis type_definition.Rep [OF type]
kuncar@47545
   362
    type_definition.Abs_inverse [OF type])
kuncar@47545
   363
kuncar@47545
   364
lemma typedef_forall_transfer:
huffman@47534
   365
  "((T ===> op =) ===> op =)
huffman@47534
   366
    (transfer_bforall (\<lambda>x. x \<in> A)) transfer_forall"
huffman@47534
   367
  unfolding transfer_bforall_def transfer_forall_def Ball_def [symmetric]
kuncar@47545
   368
  by (rule typedef_All_transfer)
huffman@47534
   369
huffman@47534
   370
end
huffman@47534
   371
huffman@47368
   372
text {* Generating the correspondence rule for a constant defined with
huffman@47368
   373
  @{text "lift_definition"}. *}
huffman@47368
   374
huffman@47351
   375
lemma Quotient_to_transfer:
huffman@47351
   376
  assumes "Quotient R Abs Rep T" and "R c c" and "c' \<equiv> Abs c"
huffman@47351
   377
  shows "T c c'"
huffman@47351
   378
  using assms by (auto dest: Quotient_cr_rel)
huffman@47351
   379
kuncar@47982
   380
text {* Proving reflexivity *}
kuncar@47982
   381
kuncar@47982
   382
definition left_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
kuncar@47982
   383
  where "left_total R \<longleftrightarrow> (\<forall>x. \<exists>y. R x y)"
kuncar@47982
   384
kuncar@47982
   385
lemma left_totalI:
kuncar@47982
   386
  "(\<And>x. \<exists>y. R x y) \<Longrightarrow> left_total R"
kuncar@47982
   387
unfolding left_total_def by blast
kuncar@47982
   388
kuncar@47982
   389
lemma left_totalE:
kuncar@47982
   390
  assumes "left_total R"
kuncar@47982
   391
  obtains "(\<And>x. \<exists>y. R x y)"
kuncar@47982
   392
using assms unfolding left_total_def by blast
kuncar@47982
   393
kuncar@47982
   394
lemma Quotient_to_left_total:
kuncar@47982
   395
  assumes q: "Quotient R Abs Rep T"
kuncar@47982
   396
  and r_R: "reflp R"
kuncar@47982
   397
  shows "left_total T"
kuncar@47982
   398
using r_R Quotient_cr_rel[OF q] unfolding left_total_def by (auto elim: reflpE)
kuncar@47982
   399
kuncar@47982
   400
lemma reflp_Quotient_composition:
kuncar@47982
   401
  assumes lt_R1: "left_total R1"
kuncar@47982
   402
  and r_R2: "reflp R2"
kuncar@47982
   403
  shows "reflp (R1 OO R2 OO R1\<inverse>\<inverse>)"
kuncar@47982
   404
using assms
kuncar@47982
   405
proof -
kuncar@47982
   406
  {
kuncar@47982
   407
    fix x
kuncar@47982
   408
    from lt_R1 obtain y where "R1 x y" unfolding left_total_def by blast
kuncar@47982
   409
    moreover then have "R1\<inverse>\<inverse> y x" by simp
kuncar@47982
   410
    moreover have "R2 y y" using r_R2 by (auto elim: reflpE)
kuncar@47982
   411
    ultimately have "(R1 OO R2 OO R1\<inverse>\<inverse>) x x" by auto
kuncar@47982
   412
  }
kuncar@47982
   413
  then show ?thesis by (auto intro: reflpI)
kuncar@47982
   414
qed
kuncar@47982
   415
kuncar@47982
   416
lemma reflp_equality: "reflp (op =)"
kuncar@47982
   417
by (auto intro: reflpI)
kuncar@47982
   418
kuncar@47308
   419
subsection {* ML setup *}
kuncar@47308
   420
kuncar@47698
   421
use "Tools/Lifting/lifting_util.ML"
kuncar@47308
   422
kuncar@47308
   423
use "Tools/Lifting/lifting_info.ML"
kuncar@47308
   424
setup Lifting_Info.setup
kuncar@47308
   425
kuncar@47777
   426
declare fun_quotient[quot_map]
kuncar@47982
   427
lemmas [reflexivity_rule] = reflp_equality reflp_Quotient_composition
kuncar@47308
   428
kuncar@47308
   429
use "Tools/Lifting/lifting_term.ML"
kuncar@47308
   430
kuncar@47308
   431
use "Tools/Lifting/lifting_def.ML"
kuncar@47308
   432
kuncar@47308
   433
use "Tools/Lifting/lifting_setup.ML"
kuncar@47308
   434
kuncar@47308
   435
hide_const (open) invariant
kuncar@47308
   436
kuncar@47308
   437
end