src/HOL/Hilbert_Choice.thy
author haftmann
Tue Jun 02 16:23:43 2009 +0200 (2009-06-02)
changeset 31380 f25536c0bb80
parent 29655 ac31940cfb69
child 31454 2c0959ab073f
permissions -rw-r--r--
added/moved lemmas by Andreas Lochbihler
paulson@11451
     1
(*  Title:      HOL/Hilbert_Choice.thy
paulson@11451
     2
    Author:     Lawrence C Paulson
paulson@11451
     3
    Copyright   2001  University of Cambridge
wenzelm@12023
     4
*)
paulson@11451
     5
paulson@14760
     6
header {* Hilbert's Epsilon-Operator and the Axiom of Choice *}
paulson@11451
     7
nipkow@15131
     8
theory Hilbert_Choice
haftmann@29655
     9
imports Nat Wellfounded Plain
haftmann@16417
    10
uses ("Tools/meson.ML") ("Tools/specification_package.ML")
nipkow@15131
    11
begin
wenzelm@12298
    12
wenzelm@12298
    13
subsection {* Hilbert's epsilon *}
wenzelm@12298
    14
wenzelm@22690
    15
axiomatization
wenzelm@22690
    16
  Eps :: "('a => bool) => 'a"
wenzelm@22690
    17
where
wenzelm@22690
    18
  someI: "P x ==> P (Eps P)"
paulson@11451
    19
wenzelm@14872
    20
syntax (epsilon)
wenzelm@14872
    21
  "_Eps"        :: "[pttrn, bool] => 'a"    ("(3\<some>_./ _)" [0, 10] 10)
paulson@11451
    22
syntax (HOL)
wenzelm@12298
    23
  "_Eps"        :: "[pttrn, bool] => 'a"    ("(3@ _./ _)" [0, 10] 10)
paulson@11451
    24
syntax
wenzelm@12298
    25
  "_Eps"        :: "[pttrn, bool] => 'a"    ("(3SOME _./ _)" [0, 10] 10)
paulson@11451
    26
translations
wenzelm@22690
    27
  "SOME x. P" == "CONST Eps (%x. P)"
nipkow@13763
    28
nipkow@13763
    29
print_translation {*
nipkow@13763
    30
(* to avoid eta-contraction of body *)
wenzelm@22690
    31
[(@{const_syntax Eps}, fn [Abs abs] =>
nipkow@13763
    32
     let val (x,t) = atomic_abs_tr' abs
nipkow@13763
    33
     in Syntax.const "_Eps" $ x $ t end)]
nipkow@13763
    34
*}
paulson@11451
    35
wenzelm@12298
    36
constdefs
wenzelm@12298
    37
  inv :: "('a => 'b) => ('b => 'a)"
wenzelm@12298
    38
  "inv(f :: 'a => 'b) == %y. SOME x. f x = y"
paulson@11454
    39
wenzelm@12298
    40
  Inv :: "'a set => ('a => 'b) => ('b => 'a)"
paulson@14760
    41
  "Inv A f == %x. SOME y. y \<in> A & f y = x"
paulson@14760
    42
paulson@14760
    43
paulson@14760
    44
subsection {*Hilbert's Epsilon-operator*}
paulson@14760
    45
paulson@14760
    46
text{*Easier to apply than @{text someI} if the witness comes from an
paulson@14760
    47
existential formula*}
paulson@14760
    48
lemma someI_ex [elim?]: "\<exists>x. P x ==> P (SOME x. P x)"
paulson@14760
    49
apply (erule exE)
paulson@14760
    50
apply (erule someI)
paulson@14760
    51
done
paulson@14760
    52
paulson@14760
    53
text{*Easier to apply than @{text someI} because the conclusion has only one
paulson@14760
    54
occurrence of @{term P}.*}
paulson@14760
    55
lemma someI2: "[| P a;  !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
paulson@14760
    56
by (blast intro: someI)
paulson@14760
    57
paulson@14760
    58
text{*Easier to apply than @{text someI2} if the witness comes from an
paulson@14760
    59
existential formula*}
paulson@14760
    60
lemma someI2_ex: "[| \<exists>a. P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
paulson@14760
    61
by (blast intro: someI2)
paulson@14760
    62
paulson@14760
    63
lemma some_equality [intro]:
paulson@14760
    64
     "[| P a;  !!x. P x ==> x=a |] ==> (SOME x. P x) = a"
paulson@14760
    65
by (blast intro: someI2)
paulson@14760
    66
paulson@14760
    67
lemma some1_equality: "[| EX!x. P x; P a |] ==> (SOME x. P x) = a"
paulson@14760
    68
by (blast intro: some_equality)
paulson@14760
    69
paulson@14760
    70
lemma some_eq_ex: "P (SOME x. P x) =  (\<exists>x. P x)"
paulson@14760
    71
by (blast intro: someI)
paulson@14760
    72
paulson@14760
    73
lemma some_eq_trivial [simp]: "(SOME y. y=x) = x"
paulson@14760
    74
apply (rule some_equality)
paulson@14760
    75
apply (rule refl, assumption)
paulson@14760
    76
done
paulson@14760
    77
paulson@14760
    78
lemma some_sym_eq_trivial [simp]: "(SOME y. x=y) = x"
paulson@14760
    79
apply (rule some_equality)
paulson@14760
    80
apply (rule refl)
paulson@14760
    81
apply (erule sym)
paulson@14760
    82
done
paulson@14760
    83
paulson@14760
    84
paulson@14760
    85
subsection{*Axiom of Choice, Proved Using the Description Operator*}
paulson@14760
    86
paulson@14760
    87
text{*Used in @{text "Tools/meson.ML"}*}
paulson@14760
    88
lemma choice: "\<forall>x. \<exists>y. Q x y ==> \<exists>f. \<forall>x. Q x (f x)"
paulson@14760
    89
by (fast elim: someI)
paulson@14760
    90
paulson@14760
    91
lemma bchoice: "\<forall>x\<in>S. \<exists>y. Q x y ==> \<exists>f. \<forall>x\<in>S. Q x (f x)"
paulson@14760
    92
by (fast elim: someI)
paulson@14760
    93
paulson@14760
    94
paulson@14760
    95
subsection {*Function Inverse*}
paulson@14760
    96
paulson@14760
    97
lemma inv_id [simp]: "inv id = id"
paulson@14760
    98
by (simp add: inv_def id_def)
paulson@14760
    99
paulson@14760
   100
text{*A one-to-one function has an inverse.*}
paulson@14760
   101
lemma inv_f_f [simp]: "inj f ==> inv f (f x) = x"
paulson@14760
   102
by (simp add: inv_def inj_eq)
paulson@14760
   103
paulson@14760
   104
lemma inv_f_eq: "[| inj f;  f x = y |] ==> inv f y = x"
paulson@14760
   105
apply (erule subst)
paulson@14760
   106
apply (erule inv_f_f)
paulson@14760
   107
done
paulson@14760
   108
paulson@14760
   109
lemma inj_imp_inv_eq: "[| inj f; \<forall>x. f(g x) = x |] ==> inv f = g"
paulson@14760
   110
by (blast intro: ext inv_f_eq)
paulson@14760
   111
paulson@14760
   112
text{*But is it useful?*}
paulson@14760
   113
lemma inj_transfer:
paulson@14760
   114
  assumes injf: "inj f" and minor: "!!y. y \<in> range(f) ==> P(inv f y)"
paulson@14760
   115
  shows "P x"
paulson@14760
   116
proof -
paulson@14760
   117
  have "f x \<in> range f" by auto
paulson@14760
   118
  hence "P(inv f (f x))" by (rule minor)
paulson@14760
   119
  thus "P x" by (simp add: inv_f_f [OF injf])
paulson@14760
   120
qed
paulson@11451
   121
paulson@11451
   122
paulson@14760
   123
lemma inj_iff: "(inj f) = (inv f o f = id)"
paulson@14760
   124
apply (simp add: o_def expand_fun_eq)
paulson@14760
   125
apply (blast intro: inj_on_inverseI inv_f_f)
paulson@14760
   126
done
paulson@14760
   127
nipkow@23433
   128
lemma inv_o_cancel[simp]: "inj f ==> inv f o f = id"
nipkow@23433
   129
by (simp add: inj_iff)
nipkow@23433
   130
nipkow@23433
   131
lemma o_inv_o_cancel[simp]: "inj f ==> g o inv f o f = g"
nipkow@23433
   132
by (simp add: o_assoc[symmetric])
nipkow@23433
   133
nipkow@23433
   134
lemma inv_image_cancel[simp]:
nipkow@23433
   135
  "inj f ==> inv f ` f ` S = S"
nipkow@23433
   136
by (simp add: image_compose[symmetric])
nipkow@23433
   137
 
paulson@14760
   138
lemma inj_imp_surj_inv: "inj f ==> surj (inv f)"
paulson@14760
   139
by (blast intro: surjI inv_f_f)
paulson@14760
   140
paulson@14760
   141
lemma f_inv_f: "y \<in> range(f) ==> f(inv f y) = y"
paulson@14760
   142
apply (simp add: inv_def)
paulson@14760
   143
apply (fast intro: someI)
paulson@14760
   144
done
paulson@14760
   145
paulson@14760
   146
lemma surj_f_inv_f: "surj f ==> f(inv f y) = y"
paulson@14760
   147
by (simp add: f_inv_f surj_range)
paulson@14760
   148
paulson@14760
   149
lemma inv_injective:
paulson@14760
   150
  assumes eq: "inv f x = inv f y"
paulson@14760
   151
      and x: "x: range f"
paulson@14760
   152
      and y: "y: range f"
paulson@14760
   153
  shows "x=y"
paulson@14760
   154
proof -
paulson@14760
   155
  have "f (inv f x) = f (inv f y)" using eq by simp
paulson@14760
   156
  thus ?thesis by (simp add: f_inv_f x y) 
paulson@14760
   157
qed
paulson@14760
   158
paulson@14760
   159
lemma inj_on_inv: "A <= range(f) ==> inj_on (inv f) A"
paulson@14760
   160
by (fast intro: inj_onI elim: inv_injective injD)
paulson@14760
   161
paulson@14760
   162
lemma surj_imp_inj_inv: "surj f ==> inj (inv f)"
paulson@14760
   163
by (simp add: inj_on_inv surj_range)
paulson@14760
   164
paulson@14760
   165
lemma surj_iff: "(surj f) = (f o inv f = id)"
paulson@14760
   166
apply (simp add: o_def expand_fun_eq)
paulson@14760
   167
apply (blast intro: surjI surj_f_inv_f)
paulson@14760
   168
done
paulson@14760
   169
paulson@14760
   170
lemma surj_imp_inv_eq: "[| surj f; \<forall>x. g(f x) = x |] ==> inv f = g"
paulson@14760
   171
apply (rule ext)
paulson@14760
   172
apply (drule_tac x = "inv f x" in spec)
paulson@14760
   173
apply (simp add: surj_f_inv_f)
paulson@14760
   174
done
paulson@14760
   175
paulson@14760
   176
lemma bij_imp_bij_inv: "bij f ==> bij (inv f)"
paulson@14760
   177
by (simp add: bij_def inj_imp_surj_inv surj_imp_inj_inv)
wenzelm@12372
   178
paulson@14760
   179
lemma inv_equality: "[| !!x. g (f x) = x;  !!y. f (g y) = y |] ==> inv f = g"
paulson@14760
   180
apply (rule ext)
paulson@14760
   181
apply (auto simp add: inv_def)
paulson@14760
   182
done
paulson@14760
   183
paulson@14760
   184
lemma inv_inv_eq: "bij f ==> inv (inv f) = f"
paulson@14760
   185
apply (rule inv_equality)
paulson@14760
   186
apply (auto simp add: bij_def surj_f_inv_f)
paulson@14760
   187
done
paulson@14760
   188
paulson@14760
   189
(** bij(inv f) implies little about f.  Consider f::bool=>bool such that
paulson@14760
   190
    f(True)=f(False)=True.  Then it's consistent with axiom someI that
paulson@14760
   191
    inv f could be any function at all, including the identity function.
paulson@14760
   192
    If inv f=id then inv f is a bijection, but inj f, surj(f) and
paulson@14760
   193
    inv(inv f)=f all fail.
paulson@14760
   194
**)
paulson@14760
   195
paulson@14760
   196
lemma o_inv_distrib: "[| bij f; bij g |] ==> inv (f o g) = inv g o inv f"
paulson@14760
   197
apply (rule inv_equality)
paulson@14760
   198
apply (auto simp add: bij_def surj_f_inv_f)
paulson@14760
   199
done
paulson@14760
   200
paulson@14760
   201
paulson@14760
   202
lemma image_surj_f_inv_f: "surj f ==> f ` (inv f ` A) = A"
paulson@14760
   203
by (simp add: image_eq_UN surj_f_inv_f)
paulson@14760
   204
paulson@14760
   205
lemma image_inv_f_f: "inj f ==> (inv f) ` (f ` A) = A"
paulson@14760
   206
by (simp add: image_eq_UN)
paulson@14760
   207
paulson@14760
   208
lemma inv_image_comp: "inj f ==> inv f ` (f`X) = X"
paulson@14760
   209
by (auto simp add: image_def)
paulson@14760
   210
paulson@14760
   211
lemma bij_image_Collect_eq: "bij f ==> f ` Collect P = {y. P (inv f y)}"
paulson@14760
   212
apply auto
paulson@14760
   213
apply (force simp add: bij_is_inj)
paulson@14760
   214
apply (blast intro: bij_is_surj [THEN surj_f_inv_f, symmetric])
paulson@14760
   215
done
paulson@14760
   216
paulson@14760
   217
lemma bij_vimage_eq_inv_image: "bij f ==> f -` A = inv f ` A" 
paulson@14760
   218
apply (auto simp add: bij_is_surj [THEN surj_f_inv_f])
paulson@14760
   219
apply (blast intro: bij_is_inj [THEN inv_f_f, symmetric])
paulson@14760
   220
done
paulson@14760
   221
haftmann@31380
   222
lemma finite_fun_UNIVD1:
haftmann@31380
   223
  assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
haftmann@31380
   224
  and card: "card (UNIV :: 'b set) \<noteq> Suc 0"
haftmann@31380
   225
  shows "finite (UNIV :: 'a set)"
haftmann@31380
   226
proof -
haftmann@31380
   227
  from fin have finb: "finite (UNIV :: 'b set)" by (rule finite_fun_UNIVD2)
haftmann@31380
   228
  with card have "card (UNIV :: 'b set) \<ge> Suc (Suc 0)"
haftmann@31380
   229
    by (cases "card (UNIV :: 'b set)") (auto simp add: card_eq_0_iff)
haftmann@31380
   230
  then obtain n where "card (UNIV :: 'b set) = Suc (Suc n)" "n = card (UNIV :: 'b set) - Suc (Suc 0)" by auto
haftmann@31380
   231
  then obtain b1 b2 where b1b2: "(b1 :: 'b) \<noteq> (b2 :: 'b)" by (auto simp add: card_Suc_eq)
haftmann@31380
   232
  from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1))" by (rule finite_imageI)
haftmann@31380
   233
  moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1)"
haftmann@31380
   234
  proof (rule UNIV_eq_I)
haftmann@31380
   235
    fix x :: 'a
haftmann@31380
   236
    from b1b2 have "x = inv (\<lambda>y. if y = x then b1 else b2) b1" by (simp add: inv_def)
haftmann@31380
   237
    thus "x \<in> range (\<lambda>f\<Colon>'a \<Rightarrow> 'b. inv f b1)" by blast
haftmann@31380
   238
  qed
haftmann@31380
   239
  ultimately show "finite (UNIV :: 'a set)" by simp
haftmann@31380
   240
qed
paulson@14760
   241
paulson@14760
   242
subsection {*Inverse of a PI-function (restricted domain)*}
paulson@14760
   243
paulson@14760
   244
lemma Inv_f_f: "[| inj_on f A;  x \<in> A |] ==> Inv A f (f x) = x"
paulson@14760
   245
apply (simp add: Inv_def inj_on_def)
paulson@14760
   246
apply (blast intro: someI2)
paulson@14760
   247
done
paulson@14760
   248
paulson@14760
   249
lemma f_Inv_f: "y \<in> f`A  ==> f (Inv A f y) = y"
paulson@14760
   250
apply (simp add: Inv_def)
paulson@13585
   251
apply (fast intro: someI2)
paulson@13585
   252
done
paulson@11451
   253
paulson@14760
   254
lemma Inv_injective:
paulson@14760
   255
  assumes eq: "Inv A f x = Inv A f y"
paulson@14760
   256
      and x: "x: f`A"
paulson@14760
   257
      and y: "y: f`A"
paulson@14760
   258
  shows "x=y"
paulson@14760
   259
proof -
paulson@14760
   260
  have "f (Inv A f x) = f (Inv A f y)" using eq by simp
paulson@14760
   261
  thus ?thesis by (simp add: f_Inv_f x y) 
paulson@14760
   262
qed
paulson@14760
   263
paulson@14760
   264
lemma inj_on_Inv: "B <= f`A ==> inj_on (Inv A f) B"
paulson@14760
   265
apply (rule inj_onI)
paulson@14760
   266
apply (blast intro: inj_onI dest: Inv_injective injD)
paulson@14760
   267
done
paulson@14760
   268
paulson@14760
   269
lemma Inv_mem: "[| f ` A = B;  x \<in> B |] ==> Inv A f x \<in> A"
paulson@14760
   270
apply (simp add: Inv_def)
paulson@14760
   271
apply (fast intro: someI2)
paulson@14760
   272
done
paulson@14760
   273
paulson@14760
   274
lemma Inv_f_eq: "[| inj_on f A; f x = y; x \<in> A |] ==> Inv A f y = x"
ballarin@14399
   275
  apply (erule subst)
paulson@14760
   276
  apply (erule Inv_f_f, assumption)
ballarin@14399
   277
  done
ballarin@14399
   278
ballarin@14399
   279
lemma Inv_comp:
paulson@14760
   280
  "[| inj_on f (g ` A); inj_on g A; x \<in> f ` g ` A |] ==>
ballarin@14399
   281
  Inv A (f o g) x = (Inv A g o Inv (g ` A) f) x"
ballarin@14399
   282
  apply simp
ballarin@14399
   283
  apply (rule Inv_f_eq)
ballarin@14399
   284
    apply (fast intro: comp_inj_on)
ballarin@14399
   285
   apply (simp add: f_Inv_f Inv_mem)
ballarin@14399
   286
  apply (simp add: Inv_mem)
ballarin@14399
   287
  done
ballarin@14399
   288
nipkow@26105
   289
lemma bij_betw_Inv: "bij_betw f A B \<Longrightarrow> bij_betw (Inv A f) B A"
nipkow@26105
   290
  apply (auto simp add: bij_betw_def inj_on_Inv Inv_mem)
nipkow@26105
   291
  apply (simp add: image_compose [symmetric] o_def)
nipkow@26105
   292
  apply (simp add: image_def Inv_f_f)
nipkow@26105
   293
  done
paulson@14760
   294
paulson@14760
   295
subsection {*Other Consequences of Hilbert's Epsilon*}
paulson@14760
   296
paulson@14760
   297
text {*Hilbert's Epsilon and the @{term split} Operator*}
paulson@14760
   298
paulson@14760
   299
text{*Looping simprule*}
paulson@14760
   300
lemma split_paired_Eps: "(SOME x. P x) = (SOME (a,b). P(a,b))"
haftmann@26347
   301
  by simp
paulson@14760
   302
paulson@14760
   303
lemma Eps_split: "Eps (split P) = (SOME xy. P (fst xy) (snd xy))"
haftmann@26347
   304
  by (simp add: split_def)
paulson@14760
   305
paulson@14760
   306
lemma Eps_split_eq [simp]: "(@(x',y'). x = x' & y = y') = (x,y)"
haftmann@26347
   307
  by blast
paulson@14760
   308
paulson@14760
   309
paulson@14760
   310
text{*A relation is wellfounded iff it has no infinite descending chain*}
paulson@14760
   311
lemma wf_iff_no_infinite_down_chain:
paulson@14760
   312
  "wf r = (~(\<exists>f. \<forall>i. (f(Suc i),f i) \<in> r))"
paulson@14760
   313
apply (simp only: wf_eq_minimal)
paulson@14760
   314
apply (rule iffI)
paulson@14760
   315
 apply (rule notI)
paulson@14760
   316
 apply (erule exE)
paulson@14760
   317
 apply (erule_tac x = "{w. \<exists>i. w=f i}" in allE, blast)
paulson@14760
   318
apply (erule contrapos_np, simp, clarify)
paulson@14760
   319
apply (subgoal_tac "\<forall>n. nat_rec x (%i y. @z. z:Q & (z,y) :r) n \<in> Q")
paulson@14760
   320
 apply (rule_tac x = "nat_rec x (%i y. @z. z:Q & (z,y) :r)" in exI)
paulson@14760
   321
 apply (rule allI, simp)
paulson@14760
   322
 apply (rule someI2_ex, blast, blast)
paulson@14760
   323
apply (rule allI)
paulson@14760
   324
apply (induct_tac "n", simp_all)
paulson@14760
   325
apply (rule someI2_ex, blast+)
paulson@14760
   326
done
paulson@14760
   327
nipkow@27760
   328
lemma wf_no_infinite_down_chainE:
nipkow@27760
   329
  assumes "wf r" obtains k where "(f (Suc k), f k) \<notin> r"
nipkow@27760
   330
using `wf r` wf_iff_no_infinite_down_chain[of r] by blast
nipkow@27760
   331
nipkow@27760
   332
paulson@14760
   333
text{*A dynamically-scoped fact for TFL *}
wenzelm@12298
   334
lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"
wenzelm@12298
   335
  by (blast intro: someI)
paulson@11451
   336
wenzelm@12298
   337
wenzelm@12298
   338
subsection {* Least value operator *}
paulson@11451
   339
paulson@11451
   340
constdefs
wenzelm@12298
   341
  LeastM :: "['a => 'b::ord, 'a => bool] => 'a"
paulson@14760
   342
  "LeastM m P == SOME x. P x & (\<forall>y. P y --> m x <= m y)"
paulson@11451
   343
paulson@11451
   344
syntax
wenzelm@12298
   345
  "_LeastM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"    ("LEAST _ WRT _. _" [0, 4, 10] 10)
paulson@11451
   346
translations
wenzelm@12298
   347
  "LEAST x WRT m. P" == "LeastM m (%x. P)"
paulson@11451
   348
paulson@11451
   349
lemma LeastMI2:
wenzelm@12298
   350
  "P x ==> (!!y. P y ==> m x <= m y)
wenzelm@12298
   351
    ==> (!!x. P x ==> \<forall>y. P y --> m x \<le> m y ==> Q x)
wenzelm@12298
   352
    ==> Q (LeastM m P)"
paulson@14760
   353
  apply (simp add: LeastM_def)
paulson@14208
   354
  apply (rule someI2_ex, blast, blast)
wenzelm@12298
   355
  done
paulson@11451
   356
paulson@11451
   357
lemma LeastM_equality:
wenzelm@12298
   358
  "P k ==> (!!x. P x ==> m k <= m x)
wenzelm@12298
   359
    ==> m (LEAST x WRT m. P x) = (m k::'a::order)"
paulson@14208
   360
  apply (rule LeastMI2, assumption, blast)
wenzelm@12298
   361
  apply (blast intro!: order_antisym)
wenzelm@12298
   362
  done
paulson@11451
   363
paulson@11454
   364
lemma wf_linord_ex_has_least:
paulson@14760
   365
  "wf r ==> \<forall>x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k
paulson@14760
   366
    ==> \<exists>x. P x & (!y. P y --> (m x,m y):r^*)"
wenzelm@12298
   367
  apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
paulson@14208
   368
  apply (drule_tac x = "m`Collect P" in spec, force)
wenzelm@12298
   369
  done
paulson@11454
   370
paulson@11454
   371
lemma ex_has_least_nat:
paulson@14760
   372
    "P k ==> \<exists>x. P x & (\<forall>y. P y --> m x <= (m y::nat))"
wenzelm@12298
   373
  apply (simp only: pred_nat_trancl_eq_le [symmetric])
wenzelm@12298
   374
  apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
paulson@16796
   375
   apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le, assumption)
wenzelm@12298
   376
  done
paulson@11454
   377
wenzelm@12298
   378
lemma LeastM_nat_lemma:
paulson@14760
   379
    "P k ==> P (LeastM m P) & (\<forall>y. P y --> m (LeastM m P) <= (m y::nat))"
paulson@14760
   380
  apply (simp add: LeastM_def)
wenzelm@12298
   381
  apply (rule someI_ex)
wenzelm@12298
   382
  apply (erule ex_has_least_nat)
wenzelm@12298
   383
  done
paulson@11454
   384
paulson@11454
   385
lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1, standard]
paulson@11454
   386
paulson@11454
   387
lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)"
paulson@14208
   388
by (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp], assumption, assumption)
paulson@11454
   389
paulson@11451
   390
wenzelm@12298
   391
subsection {* Greatest value operator *}
paulson@11451
   392
paulson@11451
   393
constdefs
wenzelm@12298
   394
  GreatestM :: "['a => 'b::ord, 'a => bool] => 'a"
paulson@14760
   395
  "GreatestM m P == SOME x. P x & (\<forall>y. P y --> m y <= m x)"
wenzelm@12298
   396
wenzelm@12298
   397
  Greatest :: "('a::ord => bool) => 'a"    (binder "GREATEST " 10)
wenzelm@12298
   398
  "Greatest == GreatestM (%x. x)"
paulson@11451
   399
paulson@11451
   400
syntax
wenzelm@12298
   401
  "_GreatestM" :: "[pttrn, 'a=>'b::ord, bool] => 'a"
wenzelm@12298
   402
      ("GREATEST _ WRT _. _" [0, 4, 10] 10)
paulson@11451
   403
paulson@11451
   404
translations
wenzelm@12298
   405
  "GREATEST x WRT m. P" == "GreatestM m (%x. P)"
paulson@11451
   406
paulson@11451
   407
lemma GreatestMI2:
wenzelm@12298
   408
  "P x ==> (!!y. P y ==> m y <= m x)
wenzelm@12298
   409
    ==> (!!x. P x ==> \<forall>y. P y --> m y \<le> m x ==> Q x)
wenzelm@12298
   410
    ==> Q (GreatestM m P)"
paulson@14760
   411
  apply (simp add: GreatestM_def)
paulson@14208
   412
  apply (rule someI2_ex, blast, blast)
wenzelm@12298
   413
  done
paulson@11451
   414
paulson@11451
   415
lemma GreatestM_equality:
wenzelm@12298
   416
 "P k ==> (!!x. P x ==> m x <= m k)
wenzelm@12298
   417
    ==> m (GREATEST x WRT m. P x) = (m k::'a::order)"
paulson@14208
   418
  apply (rule_tac m = m in GreatestMI2, assumption, blast)
wenzelm@12298
   419
  apply (blast intro!: order_antisym)
wenzelm@12298
   420
  done
paulson@11451
   421
paulson@11451
   422
lemma Greatest_equality:
wenzelm@12298
   423
  "P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k"
paulson@14760
   424
  apply (simp add: Greatest_def)
paulson@14208
   425
  apply (erule GreatestM_equality, blast)
wenzelm@12298
   426
  done
paulson@11451
   427
paulson@11451
   428
lemma ex_has_greatest_nat_lemma:
paulson@14760
   429
  "P k ==> \<forall>x. P x --> (\<exists>y. P y & ~ ((m y::nat) <= m x))
paulson@14760
   430
    ==> \<exists>y. P y & ~ (m y < m k + n)"
paulson@15251
   431
  apply (induct n, force)
wenzelm@12298
   432
  apply (force simp add: le_Suc_eq)
wenzelm@12298
   433
  done
paulson@11451
   434
wenzelm@12298
   435
lemma ex_has_greatest_nat:
paulson@14760
   436
  "P k ==> \<forall>y. P y --> m y < b
paulson@14760
   437
    ==> \<exists>x. P x & (\<forall>y. P y --> (m y::nat) <= m x)"
wenzelm@12298
   438
  apply (rule ccontr)
wenzelm@12298
   439
  apply (cut_tac P = P and n = "b - m k" in ex_has_greatest_nat_lemma)
paulson@14208
   440
    apply (subgoal_tac [3] "m k <= b", auto)
wenzelm@12298
   441
  done
paulson@11451
   442
wenzelm@12298
   443
lemma GreatestM_nat_lemma:
paulson@14760
   444
  "P k ==> \<forall>y. P y --> m y < b
paulson@14760
   445
    ==> P (GreatestM m P) & (\<forall>y. P y --> (m y::nat) <= m (GreatestM m P))"
paulson@14760
   446
  apply (simp add: GreatestM_def)
wenzelm@12298
   447
  apply (rule someI_ex)
paulson@14208
   448
  apply (erule ex_has_greatest_nat, assumption)
wenzelm@12298
   449
  done
paulson@11451
   450
paulson@11451
   451
lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1, standard]
paulson@11451
   452
wenzelm@12298
   453
lemma GreatestM_nat_le:
paulson@14760
   454
  "P x ==> \<forall>y. P y --> m y < b
wenzelm@12298
   455
    ==> (m x::nat) <= m (GreatestM m P)"
berghofe@21020
   456
  apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec, of P])
wenzelm@12298
   457
  done
wenzelm@12298
   458
wenzelm@12298
   459
wenzelm@12298
   460
text {* \medskip Specialization to @{text GREATEST}. *}
wenzelm@12298
   461
paulson@14760
   462
lemma GreatestI: "P (k::nat) ==> \<forall>y. P y --> y < b ==> P (GREATEST x. P x)"
paulson@14760
   463
  apply (simp add: Greatest_def)
paulson@14208
   464
  apply (rule GreatestM_natI, auto)
wenzelm@12298
   465
  done
paulson@11451
   466
wenzelm@12298
   467
lemma Greatest_le:
paulson@14760
   468
    "P x ==> \<forall>y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
paulson@14760
   469
  apply (simp add: Greatest_def)
paulson@14208
   470
  apply (rule GreatestM_nat_le, auto)
wenzelm@12298
   471
  done
wenzelm@12298
   472
wenzelm@12298
   473
wenzelm@12298
   474
subsection {* The Meson proof procedure *}
paulson@11451
   475
wenzelm@12298
   476
subsubsection {* Negation Normal Form *}
wenzelm@12298
   477
wenzelm@12298
   478
text {* de Morgan laws *}
wenzelm@12298
   479
wenzelm@12298
   480
lemma meson_not_conjD: "~(P&Q) ==> ~P | ~Q"
wenzelm@12298
   481
  and meson_not_disjD: "~(P|Q) ==> ~P & ~Q"
wenzelm@12298
   482
  and meson_not_notD: "~~P ==> P"
paulson@14760
   483
  and meson_not_allD: "!!P. ~(\<forall>x. P(x)) ==> \<exists>x. ~P(x)"
paulson@14760
   484
  and meson_not_exD: "!!P. ~(\<exists>x. P(x)) ==> \<forall>x. ~P(x)"
wenzelm@12298
   485
  by fast+
paulson@11451
   486
wenzelm@12298
   487
text {* Removal of @{text "-->"} and @{text "<->"} (positive and
wenzelm@12298
   488
negative occurrences) *}
wenzelm@12298
   489
wenzelm@12298
   490
lemma meson_imp_to_disjD: "P-->Q ==> ~P | Q"
wenzelm@12298
   491
  and meson_not_impD: "~(P-->Q) ==> P & ~Q"
wenzelm@12298
   492
  and meson_iff_to_disjD: "P=Q ==> (~P | Q) & (~Q | P)"
wenzelm@12298
   493
  and meson_not_iffD: "~(P=Q) ==> (P | Q) & (~P | ~Q)"
wenzelm@12298
   494
    -- {* Much more efficient than @{prop "(P & ~Q) | (Q & ~P)"} for computing CNF *}
paulson@18389
   495
  and meson_not_refl_disj_D: "x ~= x | P ==> P"
wenzelm@12298
   496
  by fast+
wenzelm@12298
   497
wenzelm@12298
   498
wenzelm@12298
   499
subsubsection {* Pulling out the existential quantifiers *}
wenzelm@12298
   500
wenzelm@12298
   501
text {* Conjunction *}
wenzelm@12298
   502
paulson@14760
   503
lemma meson_conj_exD1: "!!P Q. (\<exists>x. P(x)) & Q ==> \<exists>x. P(x) & Q"
paulson@14760
   504
  and meson_conj_exD2: "!!P Q. P & (\<exists>x. Q(x)) ==> \<exists>x. P & Q(x)"
wenzelm@12298
   505
  by fast+
wenzelm@12298
   506
paulson@11451
   507
wenzelm@12298
   508
text {* Disjunction *}
wenzelm@12298
   509
paulson@14760
   510
lemma meson_disj_exD: "!!P Q. (\<exists>x. P(x)) | (\<exists>x. Q(x)) ==> \<exists>x. P(x) | Q(x)"
wenzelm@12298
   511
  -- {* DO NOT USE with forall-Skolemization: makes fewer schematic variables!! *}
wenzelm@12298
   512
  -- {* With ex-Skolemization, makes fewer Skolem constants *}
paulson@14760
   513
  and meson_disj_exD1: "!!P Q. (\<exists>x. P(x)) | Q ==> \<exists>x. P(x) | Q"
paulson@14760
   514
  and meson_disj_exD2: "!!P Q. P | (\<exists>x. Q(x)) ==> \<exists>x. P | Q(x)"
wenzelm@12298
   515
  by fast+
wenzelm@12298
   516
paulson@11451
   517
wenzelm@12298
   518
subsubsection {* Generating clauses for the Meson Proof Procedure *}
wenzelm@12298
   519
wenzelm@12298
   520
text {* Disjunctions *}
wenzelm@12298
   521
wenzelm@12298
   522
lemma meson_disj_assoc: "(P|Q)|R ==> P|(Q|R)"
wenzelm@12298
   523
  and meson_disj_comm: "P|Q ==> Q|P"
wenzelm@12298
   524
  and meson_disj_FalseD1: "False|P ==> P"
wenzelm@12298
   525
  and meson_disj_FalseD2: "P|False ==> P"
wenzelm@12298
   526
  by fast+
paulson@11451
   527
paulson@14760
   528
paulson@14760
   529
subsection{*Lemmas for Meson, the Model Elimination Procedure*}
paulson@14760
   530
paulson@14760
   531
text{* Generation of contrapositives *}
paulson@14760
   532
paulson@14760
   533
text{*Inserts negated disjunct after removing the negation; P is a literal.
paulson@14760
   534
  Model elimination requires assuming the negation of every attempted subgoal,
paulson@14760
   535
  hence the negated disjuncts.*}
paulson@14760
   536
lemma make_neg_rule: "~P|Q ==> ((~P==>P) ==> Q)"
paulson@14760
   537
by blast
paulson@14760
   538
paulson@14760
   539
text{*Version for Plaisted's "Postive refinement" of the Meson procedure*}
paulson@14760
   540
lemma make_refined_neg_rule: "~P|Q ==> (P ==> Q)"
paulson@14760
   541
by blast
paulson@14760
   542
paulson@14760
   543
text{*@{term P} should be a literal*}
paulson@14760
   544
lemma make_pos_rule: "P|Q ==> ((P==>~P) ==> Q)"
paulson@14760
   545
by blast
paulson@14760
   546
paulson@14760
   547
text{*Versions of @{text make_neg_rule} and @{text make_pos_rule} that don't
paulson@14760
   548
insert new assumptions, for ordinary resolution.*}
paulson@14760
   549
paulson@14760
   550
lemmas make_neg_rule' = make_refined_neg_rule
paulson@14760
   551
paulson@14760
   552
lemma make_pos_rule': "[|P|Q; ~P|] ==> Q"
paulson@14760
   553
by blast
paulson@14760
   554
paulson@14760
   555
text{* Generation of a goal clause -- put away the final literal *}
paulson@14760
   556
paulson@14760
   557
lemma make_neg_goal: "~P ==> ((~P==>P) ==> False)"
paulson@14760
   558
by blast
paulson@14760
   559
paulson@14760
   560
lemma make_pos_goal: "P ==> ((P==>~P) ==> False)"
paulson@14760
   561
by blast
paulson@14760
   562
paulson@14760
   563
paulson@14760
   564
subsubsection{* Lemmas for Forward Proof*}
paulson@14760
   565
paulson@14760
   566
text{*There is a similarity to congruence rules*}
paulson@14760
   567
paulson@14760
   568
(*NOTE: could handle conjunctions (faster?) by
paulson@14760
   569
    nf(th RS conjunct2) RS (nf(th RS conjunct1) RS conjI) *)
paulson@14760
   570
lemma conj_forward: "[| P'&Q';  P' ==> P;  Q' ==> Q |] ==> P&Q"
paulson@14760
   571
by blast
paulson@14760
   572
paulson@14760
   573
lemma disj_forward: "[| P'|Q';  P' ==> P;  Q' ==> Q |] ==> P|Q"
paulson@14760
   574
by blast
paulson@14760
   575
paulson@14760
   576
(*Version of @{text disj_forward} for removal of duplicate literals*)
paulson@14760
   577
lemma disj_forward2:
paulson@14760
   578
    "[| P'|Q';  P' ==> P;  [| Q'; P==>False |] ==> Q |] ==> P|Q"
paulson@14760
   579
apply blast 
paulson@14760
   580
done
paulson@14760
   581
paulson@14760
   582
lemma all_forward: "[| \<forall>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<forall>x. P(x)"
paulson@14760
   583
by blast
paulson@14760
   584
paulson@14760
   585
lemma ex_forward: "[| \<exists>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<exists>x. P(x)"
paulson@14760
   586
by blast
paulson@14760
   587
paulson@17420
   588
paulson@17420
   589
text{*Many of these bindings are used by the ATP linkup, and not just by
paulson@17420
   590
legacy proof scripts.*}
paulson@14760
   591
ML
paulson@14760
   592
{*
paulson@14760
   593
val inv_def = thm "inv_def";
paulson@14760
   594
val Inv_def = thm "Inv_def";
paulson@14760
   595
paulson@14760
   596
val someI = thm "someI";
paulson@14760
   597
val someI_ex = thm "someI_ex";
paulson@14760
   598
val someI2 = thm "someI2";
paulson@14760
   599
val someI2_ex = thm "someI2_ex";
paulson@14760
   600
val some_equality = thm "some_equality";
paulson@14760
   601
val some1_equality = thm "some1_equality";
paulson@14760
   602
val some_eq_ex = thm "some_eq_ex";
paulson@14760
   603
val some_eq_trivial = thm "some_eq_trivial";
paulson@14760
   604
val some_sym_eq_trivial = thm "some_sym_eq_trivial";
paulson@14760
   605
val choice = thm "choice";
paulson@14760
   606
val bchoice = thm "bchoice";
paulson@14760
   607
val inv_id = thm "inv_id";
paulson@14760
   608
val inv_f_f = thm "inv_f_f";
paulson@14760
   609
val inv_f_eq = thm "inv_f_eq";
paulson@14760
   610
val inj_imp_inv_eq = thm "inj_imp_inv_eq";
paulson@14760
   611
val inj_transfer = thm "inj_transfer";
paulson@14760
   612
val inj_iff = thm "inj_iff";
paulson@14760
   613
val inj_imp_surj_inv = thm "inj_imp_surj_inv";
paulson@14760
   614
val f_inv_f = thm "f_inv_f";
paulson@14760
   615
val surj_f_inv_f = thm "surj_f_inv_f";
paulson@14760
   616
val inv_injective = thm "inv_injective";
paulson@14760
   617
val inj_on_inv = thm "inj_on_inv";
paulson@14760
   618
val surj_imp_inj_inv = thm "surj_imp_inj_inv";
paulson@14760
   619
val surj_iff = thm "surj_iff";
paulson@14760
   620
val surj_imp_inv_eq = thm "surj_imp_inv_eq";
paulson@14760
   621
val bij_imp_bij_inv = thm "bij_imp_bij_inv";
paulson@14760
   622
val inv_equality = thm "inv_equality";
paulson@14760
   623
val inv_inv_eq = thm "inv_inv_eq";
paulson@14760
   624
val o_inv_distrib = thm "o_inv_distrib";
paulson@14760
   625
val image_surj_f_inv_f = thm "image_surj_f_inv_f";
paulson@14760
   626
val image_inv_f_f = thm "image_inv_f_f";
paulson@14760
   627
val inv_image_comp = thm "inv_image_comp";
paulson@14760
   628
val bij_image_Collect_eq = thm "bij_image_Collect_eq";
paulson@14760
   629
val bij_vimage_eq_inv_image = thm "bij_vimage_eq_inv_image";
paulson@14760
   630
val Inv_f_f = thm "Inv_f_f";
paulson@14760
   631
val f_Inv_f = thm "f_Inv_f";
paulson@14760
   632
val Inv_injective = thm "Inv_injective";
paulson@14760
   633
val inj_on_Inv = thm "inj_on_Inv";
paulson@14760
   634
val split_paired_Eps = thm "split_paired_Eps";
paulson@14760
   635
val Eps_split = thm "Eps_split";
paulson@14760
   636
val Eps_split_eq = thm "Eps_split_eq";
paulson@14760
   637
val wf_iff_no_infinite_down_chain = thm "wf_iff_no_infinite_down_chain";
paulson@14760
   638
val Inv_mem = thm "Inv_mem";
paulson@14760
   639
val Inv_f_eq = thm "Inv_f_eq";
paulson@14760
   640
val Inv_comp = thm "Inv_comp";
paulson@14760
   641
val tfl_some = thm "tfl_some";
paulson@14760
   642
val make_neg_rule = thm "make_neg_rule";
paulson@14760
   643
val make_refined_neg_rule = thm "make_refined_neg_rule";
paulson@14760
   644
val make_pos_rule = thm "make_pos_rule";
paulson@14760
   645
val make_neg_rule' = thm "make_neg_rule'";
paulson@14760
   646
val make_pos_rule' = thm "make_pos_rule'";
paulson@14760
   647
val make_neg_goal = thm "make_neg_goal";
paulson@14760
   648
val make_pos_goal = thm "make_pos_goal";
paulson@14760
   649
val conj_forward = thm "conj_forward";
paulson@14760
   650
val disj_forward = thm "disj_forward";
paulson@14760
   651
val disj_forward2 = thm "disj_forward2";
paulson@14760
   652
val all_forward = thm "all_forward";
paulson@14760
   653
val ex_forward = thm "ex_forward";
paulson@14760
   654
*}
paulson@14760
   655
paulson@14760
   656
paulson@21999
   657
subsection {* Meson package *}
wenzelm@17893
   658
paulson@11451
   659
use "Tools/meson.ML"
paulson@11451
   660
paulson@26562
   661
setup Meson.setup
paulson@26562
   662
wenzelm@17893
   663
wenzelm@17893
   664
subsection {* Specification package -- Hilbertized version *}
wenzelm@17893
   665
wenzelm@17893
   666
lemma exE_some: "[| Ex P ; c == Eps P |] ==> P c"
wenzelm@17893
   667
  by (simp only: someI_ex)
wenzelm@17893
   668
skalberg@14115
   669
use "Tools/specification_package.ML"
skalberg@14115
   670
paulson@11451
   671
end