src/HOL/Nitpick_Examples/Core_Nits.thy
 author wenzelm Fri Apr 23 23:35:43 2010 +0200 (2010-04-23) changeset 36319 8feb2c4bef1a parent 35386 45a4e19d3ebd child 37181 23ab9a5c41cf permissions -rw-r--r--
mark schematic statements explicitly;
```     1 (*  Title:      HOL/Nitpick_Examples/Core_Nits.thy
```
```     2     Author:     Jasmin Blanchette, TU Muenchen
```
```     3     Copyright   2009, 2010
```
```     4
```
```     5 Examples featuring Nitpick's functional core.
```
```     6 *)
```
```     7
```
```     8 header {* Examples Featuring Nitpick's Functional Core *}
```
```     9
```
```    10 theory Core_Nits
```
```    11 imports Main
```
```    12 begin
```
```    13
```
```    14 nitpick_params [max_potential = 0, sat_solver = MiniSat_JNI, max_threads = 1,
```
```    15                 timeout = 60 s]
```
```    16
```
```    17 subsection {* Curry in a Hurry *}
```
```    18
```
```    19 lemma "(\<lambda>f x y. (curry o split) f x y) = (\<lambda>f x y. (\<lambda>x. x) f x y)"
```
```    20 nitpick [card = 1\<midarrow>12, expect = none]
```
```    21 by auto
```
```    22
```
```    23 lemma "(\<lambda>f p. (split o curry) f p) = (\<lambda>f p. (\<lambda>x. x) f p)"
```
```    24 nitpick [card = 1\<midarrow>12, expect = none]
```
```    25 by auto
```
```    26
```
```    27 lemma "split (curry f) = f"
```
```    28 nitpick [card = 1\<midarrow>12, expect = none]
```
```    29 by auto
```
```    30
```
```    31 lemma "curry (split f) = f"
```
```    32 nitpick [card = 1\<midarrow>12, expect = none]
```
```    33 by auto
```
```    34
```
```    35 lemma "(split o curry) f = f"
```
```    36 nitpick [card = 1\<midarrow>12, expect = none]
```
```    37 by auto
```
```    38
```
```    39 lemma "(curry o split) f = f"
```
```    40 nitpick [card = 1\<midarrow>12, expect = none]
```
```    41 by auto
```
```    42
```
```    43 lemma "(split o curry) f = (\<lambda>x. x) f"
```
```    44 nitpick [card = 1\<midarrow>12, expect = none]
```
```    45 by auto
```
```    46
```
```    47 lemma "(curry o split) f = (\<lambda>x. x) f"
```
```    48 nitpick [card = 1\<midarrow>12, expect = none]
```
```    49 by auto
```
```    50
```
```    51 lemma "((split o curry) f) p = ((\<lambda>x. x) f) p"
```
```    52 nitpick [card = 1\<midarrow>12, expect = none]
```
```    53 by auto
```
```    54
```
```    55 lemma "((curry o split) f) x = ((\<lambda>x. x) f) x"
```
```    56 nitpick [card = 1\<midarrow>12, expect = none]
```
```    57 by auto
```
```    58
```
```    59 lemma "((curry o split) f) x y = ((\<lambda>x. x) f) x y"
```
```    60 nitpick [card = 1\<midarrow>12, expect = none]
```
```    61 by auto
```
```    62
```
```    63 lemma "split o curry = (\<lambda>x. x)"
```
```    64 nitpick [card = 1\<midarrow>12, expect = none]
```
```    65 apply (rule ext)+
```
```    66 by auto
```
```    67
```
```    68 lemma "curry o split = (\<lambda>x. x)"
```
```    69 nitpick [card = 1\<midarrow>12, expect = none]
```
```    70 apply (rule ext)+
```
```    71 by auto
```
```    72
```
```    73 lemma "split (\<lambda>x y. f (x, y)) = f"
```
```    74 nitpick [card = 1\<midarrow>12, expect = none]
```
```    75 by auto
```
```    76
```
```    77 subsection {* Representations *}
```
```    78
```
```    79 lemma "\<exists>f. f = (\<lambda>x. x) \<and> f y = y"
```
```    80 nitpick [expect = none]
```
```    81 by auto
```
```    82
```
```    83 lemma "(\<exists>g. \<forall>x. g (f x) = x) \<longrightarrow> (\<forall>y. \<exists>x. y = f x)"
```
```    84 nitpick [card 'a = 25, card 'b = 24, expect = genuine]
```
```    85 nitpick [card = 1\<midarrow>10, mono, expect = none]
```
```    86 oops
```
```    87
```
```    88 lemma "\<exists>f. f = (\<lambda>x. x) \<and> f y \<noteq> y"
```
```    89 nitpick [card = 1, expect = genuine]
```
```    90 nitpick [card = 5, expect = genuine]
```
```    91 oops
```
```    92
```
```    93 lemma "P (\<lambda>x. x)"
```
```    94 nitpick [card = 1, expect = genuine]
```
```    95 nitpick [card = 5, expect = genuine]
```
```    96 oops
```
```    97
```
```    98 lemma "{(a\<Colon>'a\<times>'a, b\<Colon>'b)}^-1 = {(b, a)}"
```
```    99 nitpick [card = 1\<midarrow>12, expect = none]
```
```   100 by auto
```
```   101
```
```   102 lemma "fst (a, b) = a"
```
```   103 nitpick [card = 1\<midarrow>20, expect = none]
```
```   104 by auto
```
```   105
```
```   106 lemma "\<exists>P. P = Id"
```
```   107 nitpick [card = 1\<midarrow>20, expect = none]
```
```   108 by auto
```
```   109
```
```   110 lemma "(a\<Colon>'a\<Rightarrow>'b, a) \<in> Id\<^sup>*"
```
```   111 nitpick [card = 1\<midarrow>3, expect = none]
```
```   112 by auto
```
```   113
```
```   114 lemma "(a\<Colon>'a\<times>'a, a) \<in> Id\<^sup>* \<union> {(a, b)}\<^sup>*"
```
```   115 nitpick [card = 1\<midarrow>4, expect = none]
```
```   116 by auto
```
```   117
```
```   118 lemma "Id (a, a)"
```
```   119 nitpick [card = 1\<midarrow>50, expect = none]
```
```   120 by (auto simp: Id_def Collect_def)
```
```   121
```
```   122 lemma "Id ((a\<Colon>'a, b\<Colon>'a), (a, b))"
```
```   123 nitpick [card = 1\<midarrow>10, expect = none]
```
```   124 by (auto simp: Id_def Collect_def)
```
```   125
```
```   126 lemma "UNIV (x\<Colon>'a\<times>'a)"
```
```   127 nitpick [card = 1\<midarrow>50, expect = none]
```
```   128 sorry
```
```   129
```
```   130 lemma "{} = A - A"
```
```   131 nitpick [card = 1\<midarrow>100, expect = none]
```
```   132 by auto
```
```   133
```
```   134 lemma "g = Let (A \<or> B)"
```
```   135 nitpick [card = 1, expect = none]
```
```   136 nitpick [card = 2, expect = genuine]
```
```   137 nitpick [card = 12, expect = genuine]
```
```   138 oops
```
```   139
```
```   140 lemma "(let a_or_b = A \<or> B in a_or_b \<or> \<not> a_or_b)"
```
```   141 nitpick [expect = none]
```
```   142 by auto
```
```   143
```
```   144 lemma "A \<subseteq> B"
```
```   145 nitpick [card = 100, expect = genuine]
```
```   146 oops
```
```   147
```
```   148 lemma "A = {b}"
```
```   149 nitpick [card = 100, expect = genuine]
```
```   150 oops
```
```   151
```
```   152 lemma "{a, b} = {b}"
```
```   153 nitpick [card = 100, expect = genuine]
```
```   154 oops
```
```   155
```
```   156 lemma "(a\<Colon>'a\<times>'a, a\<Colon>'a\<times>'a) \<in> R"
```
```   157 nitpick [card = 1, expect = genuine]
```
```   158 nitpick [card = 20, expect = genuine]
```
```   159 nitpick [card = 5, dont_box, expect = genuine]
```
```   160 oops
```
```   161
```
```   162 lemma "f (g\<Colon>'a\<Rightarrow>'a) = x"
```
```   163 nitpick [card = 3, expect = genuine]
```
```   164 nitpick [card = 3, dont_box, expect = genuine]
```
```   165 nitpick [card = 10, expect = genuine]
```
```   166 oops
```
```   167
```
```   168 lemma "f (a, b) = x"
```
```   169 nitpick [card = 12, expect = genuine]
```
```   170 oops
```
```   171
```
```   172 lemma "f (a, a) = f (c, d)"
```
```   173 nitpick [card = 12, expect = genuine]
```
```   174 oops
```
```   175
```
```   176 lemma "(x\<Colon>'a) = (\<lambda>a. \<lambda>b. \<lambda>c. if c then a else b) x x True"
```
```   177 nitpick [card = 1\<midarrow>12, expect = none]
```
```   178 by auto
```
```   179
```
```   180 lemma "\<exists>F. F a b = G a b"
```
```   181 nitpick [card = 2, expect = none]
```
```   182 by auto
```
```   183
```
```   184 lemma "f = split"
```
```   185 nitpick [card = 1, expect = none]
```
```   186 nitpick [card = 2, expect = genuine]
```
```   187 oops
```
```   188
```
```   189 lemma "(A\<Colon>'a\<times>'a, B\<Colon>'a\<times>'a) \<in> R \<Longrightarrow> (A, B) \<in> R"
```
```   190 nitpick [card = 20, expect = none]
```
```   191 by auto
```
```   192
```
```   193 lemma "(A, B) \<in> R \<or> (\<exists>C. (A, C) \<in> R \<and> (C, B) \<in> R) \<Longrightarrow>
```
```   194        A = B \<or> (A, B) \<in> R \<or> (\<exists>C. (A, C) \<in> R \<and> (C, B) \<in> R)"
```
```   195 nitpick [card = 1\<midarrow>25, expect = none]
```
```   196 by auto
```
```   197
```
```   198 lemma "f = (\<lambda>x\<Colon>'a\<times>'b. x)"
```
```   199 nitpick [card = 8, expect = genuine]
```
```   200 oops
```
```   201
```
```   202 subsection {* Quantifiers *}
```
```   203
```
```   204 lemma "x = y"
```
```   205 nitpick [card 'a = 1, expect = none]
```
```   206 nitpick [card 'a = 2, expect = genuine]
```
```   207 nitpick [card 'a = 200, expect = genuine]
```
```   208 oops
```
```   209
```
```   210 lemma "\<forall>x. x = y"
```
```   211 nitpick [card 'a = 1, expect = none]
```
```   212 nitpick [card 'a = 2, expect = genuine]
```
```   213 nitpick [card 'a = 200, expect = genuine]
```
```   214 oops
```
```   215
```
```   216 lemma "\<forall>x\<Colon>'a \<Rightarrow> bool. x = y"
```
```   217 nitpick [card 'a = 1, expect = genuine]
```
```   218 nitpick [card 'a = 200, expect = genuine]
```
```   219 oops
```
```   220
```
```   221 lemma "\<exists>x\<Colon>'a \<Rightarrow> bool. x = y"
```
```   222 nitpick [card 'a = 1\<midarrow>20, expect = none]
```
```   223 by auto
```
```   224
```
```   225 lemma "\<exists>x y\<Colon>'a \<Rightarrow> bool. x = y"
```
```   226 nitpick [card = 1\<midarrow>20, expect = none]
```
```   227 by auto
```
```   228
```
```   229 lemma "\<forall>x. \<exists>y. f x y = f x (g x)"
```
```   230 nitpick [card = 1\<midarrow>5, expect = none]
```
```   231 by auto
```
```   232
```
```   233 lemma "\<forall>u. \<exists>v. \<forall>w. \<exists>x. f u v w x = f u (g u) w (h u w)"
```
```   234 nitpick [card = 1\<midarrow>4, expect = none]
```
```   235 by auto
```
```   236
```
```   237 lemma "\<forall>u. \<exists>v. \<forall>w. \<exists>x. f u v w x = f u (g u w) w (h u)"
```
```   238 nitpick [card = 3, expect = genuine]
```
```   239 oops
```
```   240
```
```   241 lemma "\<forall>u. \<exists>v. \<forall>w. \<exists>x. \<forall>y. \<exists>z.
```
```   242        f u v w x y z = f u (g u) w (h u w) y (k u w y)"
```
```   243 nitpick [card = 1\<midarrow>2, expect = none]
```
```   244 nitpick [card = 3, expect = none]
```
```   245 sorry
```
```   246
```
```   247 lemma "\<forall>u. \<exists>v. \<forall>w. \<exists>x. \<forall>y. \<exists>z.
```
```   248        f u v w x y z = f u (g u) w (h u w y) y (k u w y)"
```
```   249 nitpick [card = 1\<midarrow>2, expect = genuine]
```
```   250 oops
```
```   251
```
```   252 lemma "\<forall>u. \<exists>v. \<forall>w. \<exists>x. \<forall>y. \<exists>z.
```
```   253        f u v w x y z = f u (g u w) w (h u w) y (k u w y)"
```
```   254 nitpick [card = 1\<midarrow>2, expect = genuine]
```
```   255 oops
```
```   256
```
```   257 lemma "\<forall>u\<Colon>'a \<times> 'b. \<exists>v\<Colon>'c. \<forall>w\<Colon>'d. \<exists>x\<Colon>'e \<times> 'f.
```
```   258        f u v w x = f u (g u) w (h u w)"
```
```   259 nitpick [card = 1\<midarrow>2, expect = none]
```
```   260 sorry
```
```   261
```
```   262 lemma "\<forall>u\<Colon>'a \<times> 'b. \<exists>v\<Colon>'c. \<forall>w\<Colon>'d. \<exists>x\<Colon>'e \<times> 'f.
```
```   263        f u v w x = f u (g u w) w (h u)"
```
```   264 nitpick [card = 1\<midarrow>2, dont_box, expect = genuine]
```
```   265 oops
```
```   266
```
```   267 lemma "\<forall>u\<Colon>'a \<Rightarrow> 'b. \<exists>v\<Colon>'c. \<forall>w\<Colon>'d. \<exists>x\<Colon>'e \<Rightarrow> 'f.
```
```   268        f u v w x = f u (g u) w (h u w)"
```
```   269 nitpick [card = 1\<midarrow>2, dont_box, expect = none]
```
```   270 sorry
```
```   271
```
```   272 lemma "\<forall>u\<Colon>'a \<Rightarrow> 'b. \<exists>v\<Colon>'c. \<forall>w\<Colon>'d. \<exists>x\<Colon>'e \<Rightarrow> 'f.
```
```   273        f u v w x = f u (g u w) w (h u)"
```
```   274 nitpick [card = 1\<midarrow>2, dont_box, expect = genuine]
```
```   275 oops
```
```   276
```
```   277 lemma "\<forall>x. if (\<forall>y. x = y) then False else True"
```
```   278 nitpick [card = 1, expect = genuine]
```
```   279 nitpick [card = 2\<midarrow>5, expect = none]
```
```   280 oops
```
```   281
```
```   282 lemma "\<forall>x\<Colon>'a\<times>'b. if (\<forall>y. x = y) then False else True"
```
```   283 nitpick [card = 1, expect = genuine]
```
```   284 nitpick [card = 2, expect = none]
```
```   285 oops
```
```   286
```
```   287 lemma "\<forall>x. if (\<exists>y. x = y) then True else False"
```
```   288 nitpick [expect = none]
```
```   289 sorry
```
```   290
```
```   291 lemma "\<forall>x\<Colon>'a\<times>'b. if (\<exists>y. x = y) then True else False"
```
```   292 nitpick [expect = none]
```
```   293 sorry
```
```   294
```
```   295 lemma "(\<not> (\<exists>x. P x)) \<longleftrightarrow> (\<forall>x. \<not> P x)"
```
```   296 nitpick [expect = none]
```
```   297 by auto
```
```   298
```
```   299 lemma "(\<not> \<not> (\<exists>x. P x)) \<longleftrightarrow> (\<not> (\<forall>x. \<not> P x))"
```
```   300 nitpick [expect = none]
```
```   301 by auto
```
```   302
```
```   303 lemma "(\<exists>x\<Colon>'a. \<forall>y. P x y) \<or> (\<exists>x\<Colon>'a \<times> 'a. \<forall>y. P y x)"
```
```   304 nitpick [card 'a = 1, expect = genuine]
```
```   305 nitpick [card 'a = 5, expect = genuine]
```
```   306 oops
```
```   307
```
```   308 lemma "\<exists>x. if x = y then (\<forall>y. y = x \<or> y \<noteq> x)
```
```   309            else (\<forall>y. y = (x, x) \<or> y \<noteq> (x, x))"
```
```   310 nitpick [expect = none]
```
```   311 by auto
```
```   312
```
```   313 lemma "\<exists>x. if x = y then (\<exists>y. y = x \<or> y \<noteq> x)
```
```   314            else (\<exists>y. y = (x, x) \<or> y \<noteq> (x, x))"
```
```   315 nitpick [expect = none]
```
```   316 by auto
```
```   317
```
```   318 lemma "let x = (\<forall>x. P x) in if x then x else \<not> x"
```
```   319 nitpick [expect = none]
```
```   320 by auto
```
```   321
```
```   322 lemma "let x = (\<forall>x\<Colon>'a \<times> 'b. P x) in if x then x else \<not> x"
```
```   323 nitpick [expect = none]
```
```   324 by auto
```
```   325
```
```   326 subsection {* Schematic Variables *}
```
```   327
```
```   328 schematic_lemma "x = ?x"
```
```   329 nitpick [expect = none]
```
```   330 by auto
```
```   331
```
```   332 schematic_lemma "\<forall>x. x = ?x"
```
```   333 nitpick [expect = genuine]
```
```   334 oops
```
```   335
```
```   336 schematic_lemma "\<exists>x. x = ?x"
```
```   337 nitpick [expect = none]
```
```   338 by auto
```
```   339
```
```   340 schematic_lemma "\<exists>x\<Colon>'a \<Rightarrow> 'b. x = ?x"
```
```   341 nitpick [expect = none]
```
```   342 by auto
```
```   343
```
```   344 schematic_lemma "\<forall>x. ?x = ?y"
```
```   345 nitpick [expect = none]
```
```   346 by auto
```
```   347
```
```   348 schematic_lemma "\<exists>x. ?x = ?y"
```
```   349 nitpick [expect = none]
```
```   350 by auto
```
```   351
```
```   352 subsection {* Known Constants *}
```
```   353
```
```   354 lemma "x \<equiv> all \<Longrightarrow> False"
```
```   355 nitpick [card = 1, expect = genuine]
```
```   356 nitpick [card = 1, box "('a \<Rightarrow> prop) \<Rightarrow> prop", expect = genuine]
```
```   357 nitpick [card = 2, expect = genuine]
```
```   358 nitpick [card = 6, expect = genuine]
```
```   359 oops
```
```   360
```
```   361 lemma "\<And>x. f x y = f x y"
```
```   362 nitpick [expect = none]
```
```   363 oops
```
```   364
```
```   365 lemma "\<And>x. f x y = f y x"
```
```   366 nitpick [expect = genuine]
```
```   367 oops
```
```   368
```
```   369 lemma "all (\<lambda>x. Trueprop (f x y = f x y)) \<equiv> Trueprop True"
```
```   370 nitpick [expect = none]
```
```   371 by auto
```
```   372
```
```   373 lemma "all (\<lambda>x. Trueprop (f x y = f x y)) \<equiv> Trueprop False"
```
```   374 nitpick [expect = genuine]
```
```   375 oops
```
```   376
```
```   377 lemma "I = (\<lambda>x. x) \<Longrightarrow> all P \<equiv> all (\<lambda>x. P (I x))"
```
```   378 nitpick [expect = none]
```
```   379 by auto
```
```   380
```
```   381 lemma "x \<equiv> (op \<equiv>) \<Longrightarrow> False"
```
```   382 nitpick [card = 1, expect = genuine]
```
```   383 nitpick [card = 20, expect = genuine]
```
```   384 oops
```
```   385
```
```   386 lemma "I = (\<lambda>x. x) \<Longrightarrow> (op \<equiv> x) \<equiv> (\<lambda>y. (x \<equiv> I y))"
```
```   387 nitpick [expect = none]
```
```   388 by auto
```
```   389
```
```   390 lemma "P x \<equiv> P x"
```
```   391 nitpick [card = 1\<midarrow>10, expect = none]
```
```   392 by auto
```
```   393
```
```   394 lemma "P x \<equiv> Q x \<Longrightarrow> P x = Q x"
```
```   395 nitpick [card = 1\<midarrow>10, expect = none]
```
```   396 by auto
```
```   397
```
```   398 lemma "P x = Q x \<Longrightarrow> P x \<equiv> Q x"
```
```   399 nitpick [card = 1\<midarrow>10, expect = none]
```
```   400 by auto
```
```   401
```
```   402 lemma "x \<equiv> (op \<Longrightarrow>) \<Longrightarrow> False"
```
```   403 nitpick [expect = genuine]
```
```   404 oops
```
```   405
```
```   406 lemma "I \<equiv> (\<lambda>x. x) \<Longrightarrow> (op \<Longrightarrow> x) \<equiv> (\<lambda>y. (op \<Longrightarrow> x (I y)))"
```
```   407 nitpick [expect = none]
```
```   408 by auto
```
```   409
```
```   410 lemma "P x \<Longrightarrow> P x"
```
```   411 nitpick [card = 1\<midarrow>10, expect = none]
```
```   412 by auto
```
```   413
```
```   414 lemma "True \<Longrightarrow> True" "False \<Longrightarrow> True" "False \<Longrightarrow> False"
```
```   415 nitpick [expect = none]
```
```   416 by auto
```
```   417
```
```   418 lemma "True \<Longrightarrow> False"
```
```   419 nitpick [expect = genuine]
```
```   420 oops
```
```   421
```
```   422 lemma "x = Not"
```
```   423 nitpick [expect = genuine]
```
```   424 oops
```
```   425
```
```   426 lemma "I = (\<lambda>x. x) \<Longrightarrow> Not = (\<lambda>x. Not (I x))"
```
```   427 nitpick [expect = none]
```
```   428 by auto
```
```   429
```
```   430 lemma "x = True"
```
```   431 nitpick [expect = genuine]
```
```   432 oops
```
```   433
```
```   434 lemma "x = False"
```
```   435 nitpick [expect = genuine]
```
```   436 oops
```
```   437
```
```   438 lemma "x = undefined"
```
```   439 nitpick [expect = genuine]
```
```   440 oops
```
```   441
```
```   442 lemma "(False, ()) = undefined \<Longrightarrow> ((), False) = undefined"
```
```   443 nitpick [expect = genuine]
```
```   444 oops
```
```   445
```
```   446 lemma "undefined = undefined"
```
```   447 nitpick [expect = none]
```
```   448 by auto
```
```   449
```
```   450 lemma "f undefined = f undefined"
```
```   451 nitpick [expect = none]
```
```   452 by auto
```
```   453
```
```   454 lemma "f undefined = g undefined"
```
```   455 nitpick [card = 33, expect = genuine]
```
```   456 oops
```
```   457
```
```   458 lemma "\<exists>!x. x = undefined"
```
```   459 nitpick [card = 30, expect = none]
```
```   460 by auto
```
```   461
```
```   462 lemma "x = All \<Longrightarrow> False"
```
```   463 nitpick [card = 1, dont_box, expect = genuine]
```
```   464 nitpick [card = 2, dont_box, expect = genuine]
```
```   465 nitpick [card = 8, dont_box, expect = genuine]
```
```   466 nitpick [card = 10, dont_box, expect = unknown]
```
```   467 oops
```
```   468
```
```   469 lemma "\<forall>x. f x y = f x y"
```
```   470 nitpick [expect = none]
```
```   471 oops
```
```   472
```
```   473 lemma "\<forall>x. f x y = f y x"
```
```   474 nitpick [expect = genuine]
```
```   475 oops
```
```   476
```
```   477 lemma "All (\<lambda>x. f x y = f x y) = True"
```
```   478 nitpick [expect = none]
```
```   479 by auto
```
```   480
```
```   481 lemma "All (\<lambda>x. f x y = f x y) = False"
```
```   482 nitpick [expect = genuine]
```
```   483 oops
```
```   484
```
```   485 lemma "I = (\<lambda>x. x) \<Longrightarrow> All P = All (\<lambda>x. P (I x))"
```
```   486 nitpick [expect = none]
```
```   487 by auto
```
```   488
```
```   489 lemma "x = Ex \<Longrightarrow> False"
```
```   490 nitpick [card = 1, dont_box, expect = genuine]
```
```   491 nitpick [card = 2, dont_box, expect = genuine]
```
```   492 nitpick [card = 6, dont_box, expect = genuine]
```
```   493 nitpick [card = 10, dont_box, expect = unknown]
```
```   494 oops
```
```   495
```
```   496 lemma "\<exists>x. f x y = f x y"
```
```   497 nitpick [expect = none]
```
```   498 oops
```
```   499
```
```   500 lemma "\<exists>x. f x y = f y x"
```
```   501 nitpick [expect = none]
```
```   502 oops
```
```   503
```
```   504 lemma "Ex (\<lambda>x. f x y = f x y) = True"
```
```   505 nitpick [expect = none]
```
```   506 by auto
```
```   507
```
```   508 lemma "Ex (\<lambda>x. f x y = f y x) = True"
```
```   509 nitpick [expect = none]
```
```   510 by auto
```
```   511
```
```   512 lemma "Ex (\<lambda>x. f x y = f x y) = False"
```
```   513 nitpick [expect = genuine]
```
```   514 oops
```
```   515
```
```   516 lemma "Ex (\<lambda>x. f x y = f y x) = False"
```
```   517 nitpick [expect = genuine]
```
```   518 oops
```
```   519
```
```   520 lemma "Ex (\<lambda>x. f x y \<noteq> f x y) = False"
```
```   521 nitpick [expect = none]
```
```   522 by auto
```
```   523
```
```   524 lemma "I = (\<lambda>x. x) \<Longrightarrow> Ex P = Ex (\<lambda>x. P (I x))"
```
```   525 nitpick [expect = none]
```
```   526 by auto
```
```   527
```
```   528 lemma "I = (\<lambda>x. x) \<Longrightarrow> (op =) = (\<lambda>x. (op= (I x)))"
```
```   529       "I = (\<lambda>x. x) \<Longrightarrow> (op =) = (\<lambda>x y. x = (I y))"
```
```   530 nitpick [expect = none]
```
```   531 by auto
```
```   532
```
```   533 lemma "x = y \<Longrightarrow> y = x"
```
```   534 nitpick [expect = none]
```
```   535 by auto
```
```   536
```
```   537 lemma "x = y \<Longrightarrow> f x = f y"
```
```   538 nitpick [expect = none]
```
```   539 by auto
```
```   540
```
```   541 lemma "x = y \<and> y = z \<Longrightarrow> x = z"
```
```   542 nitpick [expect = none]
```
```   543 by auto
```
```   544
```
```   545 lemma "I = (\<lambda>x. x) \<Longrightarrow> (op \<and>) = (\<lambda>x. op \<and> (I x))"
```
```   546       "I = (\<lambda>x. x) \<Longrightarrow> (op \<and>) = (\<lambda>x y. x \<and> (I y))"
```
```   547 nitpick [expect = none]
```
```   548 by auto
```
```   549
```
```   550 lemma "(a \<and> b) = (\<not> (\<not> a \<or> \<not> b))"
```
```   551 nitpick [expect = none]
```
```   552 by auto
```
```   553
```
```   554 lemma "a \<and> b \<Longrightarrow> a" "a \<and> b \<Longrightarrow> b"
```
```   555 nitpick [expect = none]
```
```   556 by auto
```
```   557
```
```   558 lemma "\<not> a \<Longrightarrow> \<not> (a \<and> b)" "\<not> b \<Longrightarrow> \<not> (a \<and> b)"
```
```   559 nitpick [expect = none]
```
```   560 by auto
```
```   561
```
```   562 lemma "I = (\<lambda>x. x) \<Longrightarrow> (op \<or>) = (\<lambda>x. op \<or> (I x))"
```
```   563       "I = (\<lambda>x. x) \<Longrightarrow> (op \<or>) = (\<lambda>x y. x \<or> (I y))"
```
```   564 nitpick [expect = none]
```
```   565 by auto
```
```   566
```
```   567 lemma "a \<Longrightarrow> a \<or> b" "b \<Longrightarrow> a \<or> b"
```
```   568 nitpick [expect = none]
```
```   569 by auto
```
```   570
```
```   571 lemma "\<not> (a \<or> b) \<Longrightarrow> \<not> a" "\<not> (a \<or> b) \<Longrightarrow> \<not> b"
```
```   572 nitpick [expect = none]
```
```   573 by auto
```
```   574
```
```   575 lemma "(op \<longrightarrow>) = (\<lambda>x. op\<longrightarrow> x)" "(op\<longrightarrow> ) = (\<lambda>x y. x \<longrightarrow> y)"
```
```   576 nitpick [expect = none]
```
```   577 by auto
```
```   578
```
```   579 lemma "\<not>a \<Longrightarrow> a \<longrightarrow> b" "b \<Longrightarrow> a \<longrightarrow> b"
```
```   580 nitpick [expect = none]
```
```   581 by auto
```
```   582
```
```   583 lemma "\<lbrakk>a; \<not> b\<rbrakk> \<Longrightarrow> \<not> (a \<longrightarrow> b)"
```
```   584 nitpick [expect = none]
```
```   585 by auto
```
```   586
```
```   587 lemma "((if a then b else c) = d) = ((a \<longrightarrow> (b = d)) \<and> (\<not> a \<longrightarrow> (c = d)))"
```
```   588 nitpick [expect = none]
```
```   589 by auto
```
```   590
```
```   591 lemma "(if a then b else c) = (THE d. (a \<longrightarrow> (d = b)) \<and> (\<not> a \<longrightarrow> (d = c)))"
```
```   592 nitpick [expect = none]
```
```   593 by auto
```
```   594
```
```   595 lemma "I = (\<lambda>x. x) \<Longrightarrow> If = (\<lambda>x. If (I x))"
```
```   596       "J = (\<lambda>x. x) \<Longrightarrow> If = (\<lambda>x y. If x (J y))"
```
```   597       "K = (\<lambda>x. x) \<Longrightarrow> If = (\<lambda>x y z. If x y (K z))"
```
```   598 nitpick [expect = none]
```
```   599 by auto
```
```   600
```
```   601 lemma "fst (x, y) = x"
```
```   602 nitpick [expect = none]
```
```   603 by (simp add: fst_def)
```
```   604
```
```   605 lemma "snd (x, y) = y"
```
```   606 nitpick [expect = none]
```
```   607 by (simp add: snd_def)
```
```   608
```
```   609 lemma "fst (x\<Colon>'a\<Rightarrow>'b, y) = x"
```
```   610 nitpick [expect = none]
```
```   611 by (simp add: fst_def)
```
```   612
```
```   613 lemma "snd (x\<Colon>'a\<Rightarrow>'b, y) = y"
```
```   614 nitpick [expect = none]
```
```   615 by (simp add: snd_def)
```
```   616
```
```   617 lemma "fst (x, y\<Colon>'a\<Rightarrow>'b) = x"
```
```   618 nitpick [expect = none]
```
```   619 by (simp add: fst_def)
```
```   620
```
```   621 lemma "snd (x, y\<Colon>'a\<Rightarrow>'b) = y"
```
```   622 nitpick [expect = none]
```
```   623 by (simp add: snd_def)
```
```   624
```
```   625 lemma "fst (x\<Colon>'a\<times>'b, y) = x"
```
```   626 nitpick [expect = none]
```
```   627 by (simp add: fst_def)
```
```   628
```
```   629 lemma "snd (x\<Colon>'a\<times>'b, y) = y"
```
```   630 nitpick [expect = none]
```
```   631 by (simp add: snd_def)
```
```   632
```
```   633 lemma "fst (x, y\<Colon>'a\<times>'b) = x"
```
```   634 nitpick [expect = none]
```
```   635 by (simp add: fst_def)
```
```   636
```
```   637 lemma "snd (x, y\<Colon>'a\<times>'b) = y"
```
```   638 nitpick [expect = none]
```
```   639 by (simp add: snd_def)
```
```   640
```
```   641 lemma "fst p = (THE a. \<exists>b. p = Pair a b)"
```
```   642 nitpick [expect = none]
```
```   643 by (simp add: fst_def)
```
```   644
```
```   645 lemma "snd p = (THE b. \<exists>a. p = Pair a b)"
```
```   646 nitpick [expect = none]
```
```   647 by (simp add: snd_def)
```
```   648
```
```   649 lemma "I = (\<lambda>x. x) \<Longrightarrow> fst = (\<lambda>x. fst (I x))"
```
```   650 nitpick [expect = none]
```
```   651 by auto
```
```   652
```
```   653 lemma "I = (\<lambda>x. x) \<Longrightarrow> snd = (\<lambda>x. snd (I x))"
```
```   654 nitpick [expect = none]
```
```   655 by auto
```
```   656
```
```   657 lemma "fst (x, y) = snd (y, x)"
```
```   658 nitpick [expect = none]
```
```   659 by auto
```
```   660
```
```   661 lemma "(x, x) \<in> Id"
```
```   662 nitpick [expect = none]
```
```   663 by auto
```
```   664
```
```   665 lemma "(x, y) \<in> Id \<Longrightarrow> x = y"
```
```   666 nitpick [expect = none]
```
```   667 by auto
```
```   668
```
```   669 lemma "I = (\<lambda>x. x) \<Longrightarrow> Id = (\<lambda>x. Id (I x))"
```
```   670 nitpick [expect = none]
```
```   671 by auto
```
```   672
```
```   673 lemma "I = (\<lambda>x. x) \<Longrightarrow> curry Id = (\<lambda>x y. Id (x, I y))"
```
```   674 nitpick [expect = none]
```
```   675 by (simp add: curry_def)
```
```   676
```
```   677 lemma "{} = (\<lambda>x. False)"
```
```   678 nitpick [expect = none]
```
```   679 by (metis Collect_def empty_def)
```
```   680
```
```   681 lemma "x \<in> {}"
```
```   682 nitpick [expect = genuine]
```
```   683 oops
```
```   684
```
```   685 lemma "{a, b} = {b}"
```
```   686 nitpick [expect = genuine]
```
```   687 oops
```
```   688
```
```   689 lemma "{a, b} \<noteq> {b}"
```
```   690 nitpick [expect = genuine]
```
```   691 oops
```
```   692
```
```   693 lemma "{a} = {b}"
```
```   694 nitpick [expect = genuine]
```
```   695 oops
```
```   696
```
```   697 lemma "{a} \<noteq> {b}"
```
```   698 nitpick [expect = genuine]
```
```   699 oops
```
```   700
```
```   701 lemma "{a, b, c} = {c, b, a}"
```
```   702 nitpick [expect = none]
```
```   703 by auto
```
```   704
```
```   705 lemma "UNIV = (\<lambda>x. True)"
```
```   706 nitpick [expect = none]
```
```   707 by (simp only: UNIV_def Collect_def)
```
```   708
```
```   709 lemma "UNIV x = True"
```
```   710 nitpick [expect = none]
```
```   711 by (simp only: UNIV_def Collect_def)
```
```   712
```
```   713 lemma "x \<notin> UNIV"
```
```   714 nitpick [expect = genuine]
```
```   715 oops
```
```   716
```
```   717 lemma "op \<in> = (\<lambda>x P. P x)"
```
```   718 nitpick [expect = none]
```
```   719 apply (rule ext)
```
```   720 apply (rule ext)
```
```   721 by (simp add: mem_def)
```
```   722
```
```   723 lemma "I = (\<lambda>x. x) \<Longrightarrow> op \<in> = (\<lambda>x. (op \<in> (I x)))"
```
```   724 nitpick [expect = none]
```
```   725 apply (rule ext)
```
```   726 apply (rule ext)
```
```   727 by (simp add: mem_def)
```
```   728
```
```   729 lemma "P x = (x \<in> P)"
```
```   730 nitpick [expect = none]
```
```   731 by (simp add: mem_def)
```
```   732
```
```   733 lemma "I = (\<lambda>x. x) \<Longrightarrow> insert = (\<lambda>x. insert (I x))"
```
```   734 nitpick [expect = none]
```
```   735 by simp
```
```   736
```
```   737 lemma "insert = (\<lambda>x y. insert x (y \<union> y))"
```
```   738 nitpick [expect = none]
```
```   739 by simp
```
```   740
```
```   741 lemma "I = (\<lambda>x. x) \<Longrightarrow> trancl = (\<lambda>x. trancl (I x))"
```
```   742 nitpick [card = 1\<midarrow>2, expect = none]
```
```   743 by auto
```
```   744
```
```   745 lemma "rtrancl = (\<lambda>x. rtrancl x \<union> {(y, y)})"
```
```   746 nitpick [card = 1\<midarrow>3, expect = none]
```
```   747 apply (rule ext)
```
```   748 by auto
```
```   749
```
```   750 lemma "(x, x) \<in> rtrancl {(y, y)}"
```
```   751 nitpick [expect = none]
```
```   752 by auto
```
```   753
```
```   754 lemma "I = (\<lambda>x. x) \<Longrightarrow> rtrancl = (\<lambda>x. rtrancl (I x))"
```
```   755 nitpick [card = 1\<midarrow>2, expect = none]
```
```   756 by auto
```
```   757
```
```   758 lemma "((x, x), (x, x)) \<in> rtrancl {}"
```
```   759 nitpick [card = 1\<midarrow>5, expect = none]
```
```   760 by auto
```
```   761
```
```   762 lemma "I = (\<lambda>x. x) \<Longrightarrow> op \<union> = (\<lambda>x. op \<union> (I x))"
```
```   763 nitpick [card = 1\<midarrow>5, expect = none]
```
```   764 by auto
```
```   765
```
```   766 lemma "I = (\<lambda>x. x) \<Longrightarrow> op \<union> = (\<lambda>x y. op \<union> x (I y))"
```
```   767 nitpick [card = 1\<midarrow>5, expect = none]
```
```   768 by auto
```
```   769
```
```   770 lemma "a \<in> A \<Longrightarrow> a \<in> (A \<union> B)" "b \<in> B \<Longrightarrow> b \<in> (A \<union> B)"
```
```   771 nitpick [expect = none]
```
```   772 by auto
```
```   773
```
```   774 lemma "a \<in> (A \<union> B) \<Longrightarrow> a \<in> A \<or> a \<in> B"
```
```   775 nitpick [expect = none]
```
```   776 by auto
```
```   777
```
```   778 lemma "I = (\<lambda>x. x) \<Longrightarrow> op \<inter> = (\<lambda>x. op \<inter> (I x))"
```
```   779 nitpick [card = 1\<midarrow>5, expect = none]
```
```   780 by auto
```
```   781
```
```   782 lemma "I = (\<lambda>x. x) \<Longrightarrow> op \<inter> = (\<lambda>x y. op \<inter> x (I y))"
```
```   783 nitpick [card = 1\<midarrow>5, expect = none]
```
```   784 by auto
```
```   785
```
```   786 lemma "a \<notin> A \<Longrightarrow> a \<notin> (A \<inter> B)" "b \<notin> B \<Longrightarrow> b \<notin> (A \<inter> B)"
```
```   787 nitpick [card = 1\<midarrow>5, expect = none]
```
```   788 by auto
```
```   789
```
```   790 lemma "a \<notin> (A \<inter> B) \<Longrightarrow> a \<notin> A \<or> a \<notin> B"
```
```   791 nitpick [expect = none]
```
```   792 by auto
```
```   793
```
```   794 lemma "I = (\<lambda>x. x) \<Longrightarrow> op - = (\<lambda>x\<Colon>'a set. op - (I x))"
```
```   795 nitpick [card = 1\<midarrow>5, expect = none]
```
```   796 by auto
```
```   797
```
```   798 lemma "I = (\<lambda>x. x) \<Longrightarrow> op - = (\<lambda>x y\<Colon>'a set. op - x (I y))"
```
```   799 nitpick [card = 1\<midarrow>5, expect = none]
```
```   800 by auto
```
```   801
```
```   802 lemma "x \<in> ((A\<Colon>'a set) - B) \<longleftrightarrow> x \<in> A \<and> x \<notin> B"
```
```   803 nitpick [card = 1\<midarrow>5, expect = none]
```
```   804 by auto
```
```   805
```
```   806 lemma "I = (\<lambda>x. x) \<Longrightarrow> op \<subset> = (\<lambda>x. op \<subset> (I x))"
```
```   807 nitpick [card = 1\<midarrow>5, expect = none]
```
```   808 by auto
```
```   809
```
```   810 lemma "I = (\<lambda>x. x) \<Longrightarrow> op \<subset> = (\<lambda>x y. op \<subset> x (I y))"
```
```   811 nitpick [card = 1\<midarrow>5, expect = none]
```
```   812 by auto
```
```   813
```
```   814 lemma "A \<subset> B \<Longrightarrow> (\<forall>a \<in> A. a \<in> B) \<and> (\<exists>b \<in> B. b \<notin> A)"
```
```   815 nitpick [card = 1\<midarrow>5, expect = none]
```
```   816 by auto
```
```   817
```
```   818 lemma "I = (\<lambda>x. x) \<Longrightarrow> op \<subseteq> = (\<lambda>x. op \<subseteq> (I x))"
```
```   819 nitpick [card = 1\<midarrow>5, expect = none]
```
```   820 by auto
```
```   821
```
```   822 lemma "I = (\<lambda>x. x) \<Longrightarrow> op \<subseteq> = (\<lambda>x y. op \<subseteq> x (I y))"
```
```   823 nitpick [card = 1\<midarrow>5, expect = none]
```
```   824 by auto
```
```   825
```
```   826 lemma "A \<subseteq> B \<Longrightarrow> \<forall>a \<in> A. a \<in> B"
```
```   827 nitpick [card = 1\<midarrow>5, expect = none]
```
```   828 by auto
```
```   829
```
```   830 lemma "A \<subseteq> B \<Longrightarrow> A \<subset> B"
```
```   831 nitpick [card = 5, expect = genuine]
```
```   832 oops
```
```   833
```
```   834 lemma "A \<subset> B \<Longrightarrow> A \<subseteq> B"
```
```   835 nitpick [expect = none]
```
```   836 by auto
```
```   837
```
```   838 lemma "I = (\<lambda>x\<Colon>'a set. x) \<Longrightarrow> uminus = (\<lambda>x. uminus (I x))"
```
```   839 nitpick [card = 1\<midarrow>7, expect = none]
```
```   840 by auto
```
```   841
```
```   842 lemma "A \<union> - A = UNIV"
```
```   843 nitpick [expect = none]
```
```   844 by auto
```
```   845
```
```   846 lemma "A \<inter> - A = {}"
```
```   847 nitpick [expect = none]
```
```   848 by auto
```
```   849
```
```   850 lemma "A = -(A\<Colon>'a set)"
```
```   851 nitpick [card 'a = 10, expect = genuine]
```
```   852 oops
```
```   853
```
```   854 lemma "I = (\<lambda>x. x) \<Longrightarrow> finite = (\<lambda>x. finite (I x))"
```
```   855 nitpick [card = 1\<midarrow>7, expect = none]
```
```   856 oops
```
```   857
```
```   858 lemma "finite A"
```
```   859 nitpick [expect = none]
```
```   860 oops
```
```   861
```
```   862 lemma "finite A \<Longrightarrow> finite B"
```
```   863 nitpick [expect = none]
```
```   864 oops
```
```   865
```
```   866 lemma "All finite"
```
```   867 nitpick [expect = none]
```
```   868 oops
```
```   869
```
```   870 subsection {* The and Eps *}
```
```   871
```
```   872 lemma "x = The"
```
```   873 nitpick [card = 5, expect = genuine]
```
```   874 oops
```
```   875
```
```   876 lemma "\<exists>x. x = The"
```
```   877 nitpick [card = 1\<midarrow>3]
```
```   878 by auto
```
```   879
```
```   880 lemma "P x \<and> (\<forall>y. P y \<longrightarrow> y = x) \<longrightarrow> The P = x"
```
```   881 nitpick [expect = none]
```
```   882 by auto
```
```   883
```
```   884 lemma "P x \<and> P y \<and> x \<noteq> y \<longrightarrow> The P = z"
```
```   885 nitpick [expect = genuine]
```
```   886 oops
```
```   887
```
```   888 lemma "P x \<and> P y \<and> x \<noteq> y \<longrightarrow> The P = x \<or> The P = y"
```
```   889 nitpick [card = 2, expect = none]
```
```   890 nitpick [card = 3\<midarrow>5, expect = genuine]
```
```   891 oops
```
```   892
```
```   893 lemma "P x \<Longrightarrow> P (The P)"
```
```   894 nitpick [card = 1\<midarrow>2, expect = none]
```
```   895 nitpick [card = 3, expect = genuine]
```
```   896 nitpick [card = 8, expect = genuine]
```
```   897 oops
```
```   898
```
```   899 lemma "(\<forall>x. \<not> P x) \<longrightarrow> The P = y"
```
```   900 nitpick [expect = genuine]
```
```   901 oops
```
```   902
```
```   903 lemma "I = (\<lambda>x. x) \<Longrightarrow> The = (\<lambda>x. The (I x))"
```
```   904 nitpick [card = 1\<midarrow>5, expect = none]
```
```   905 by auto
```
```   906
```
```   907 lemma "x = Eps"
```
```   908 nitpick [card = 5, expect = genuine]
```
```   909 oops
```
```   910
```
```   911 lemma "\<exists>x. x = Eps"
```
```   912 nitpick [card = 1\<midarrow>3, expect = none]
```
```   913 by auto
```
```   914
```
```   915 lemma "P x \<and> (\<forall>y. P y \<longrightarrow> y = x) \<longrightarrow> Eps P = x"
```
```   916 nitpick [expect = none]
```
```   917 by auto
```
```   918
```
```   919 lemma "P x \<and> P y \<and> x \<noteq> y \<longrightarrow> Eps P = z"
```
```   920 nitpick [expect = genuine]
```
```   921 apply auto
```
```   922 oops
```
```   923
```
```   924 lemma "P x \<Longrightarrow> P (Eps P)"
```
```   925 nitpick [card = 1\<midarrow>8, expect = none]
```
```   926 by (metis exE_some)
```
```   927
```
```   928 lemma "\<forall>x. \<not> P x \<longrightarrow> Eps P = y"
```
```   929 nitpick [expect = genuine]
```
```   930 oops
```
```   931
```
```   932 lemma "P (Eps P)"
```
```   933 nitpick [expect = genuine]
```
```   934 oops
```
```   935
```
```   936 lemma "(P\<Colon>nat set) (Eps P)"
```
```   937 nitpick [expect = genuine]
```
```   938 oops
```
```   939
```
```   940 lemma "\<not> P (Eps P)"
```
```   941 nitpick [expect = genuine]
```
```   942 oops
```
```   943
```
```   944 lemma "\<not> (P\<Colon>nat set) (Eps P)"
```
```   945 nitpick [expect = genuine]
```
```   946 oops
```
```   947
```
```   948 lemma "P \<noteq> {} \<Longrightarrow> P (Eps P)"
```
```   949 nitpick [expect = none]
```
```   950 sorry
```
```   951
```
```   952 lemma "(P\<Colon>nat set) \<noteq> {} \<Longrightarrow> P (Eps P)"
```
```   953 nitpick [expect = none]
```
```   954 sorry
```
```   955
```
```   956 lemma "P (The P)"
```
```   957 nitpick [expect = genuine]
```
```   958 oops
```
```   959
```
```   960 lemma "(P\<Colon>nat set) (The P)"
```
```   961 nitpick [expect = genuine]
```
```   962 oops
```
```   963
```
```   964 lemma "\<not> P (The P)"
```
```   965 nitpick [expect = genuine]
```
```   966 oops
```
```   967
```
```   968 lemma "\<not> (P\<Colon>nat set) (The P)"
```
```   969 nitpick [expect = genuine]
```
```   970 oops
```
```   971
```
```   972 lemma "The P \<noteq> x"
```
```   973 nitpick [expect = genuine]
```
```   974 oops
```
```   975
```
```   976 lemma "The P \<noteq> (x\<Colon>nat)"
```
```   977 nitpick [expect = genuine]
```
```   978 oops
```
```   979
```
```   980 lemma "P x \<Longrightarrow> P (The P)"
```
```   981 nitpick [expect = genuine]
```
```   982 oops
```
```   983
```
```   984 lemma "P (x\<Colon>nat) \<Longrightarrow> P (The P)"
```
```   985 nitpick [expect = genuine]
```
```   986 oops
```
```   987
```
```   988 lemma "P = {x} \<Longrightarrow> P (The P)"
```
```   989 nitpick [expect = none]
```
```   990 oops
```
```   991
```
```   992 lemma "P = {x\<Colon>nat} \<Longrightarrow> P (The P)"
```
```   993 nitpick [expect = none]
```
```   994 oops
```
```   995
```
```   996 consts Q :: 'a
```
```   997
```
```   998 lemma "Q (Eps Q)"
```
```   999 nitpick [expect = genuine]
```
```  1000 oops
```
```  1001
```
```  1002 lemma "(Q\<Colon>nat set) (Eps Q)"
```
```  1003 nitpick [expect = none] (* unfortunate *)
```
```  1004 oops
```
```  1005
```
```  1006 lemma "\<not> Q (Eps Q)"
```
```  1007 nitpick [expect = genuine]
```
```  1008 oops
```
```  1009
```
```  1010 lemma "\<not> (Q\<Colon>nat set) (Eps Q)"
```
```  1011 nitpick [expect = genuine]
```
```  1012 oops
```
```  1013
```
```  1014 lemma "(Q\<Colon>'a set) \<noteq> {} \<Longrightarrow> (Q\<Colon>'a set) (Eps Q)"
```
```  1015 nitpick [expect = none]
```
```  1016 sorry
```
```  1017
```
```  1018 lemma "(Q\<Colon>nat set) \<noteq> {} \<Longrightarrow> (Q\<Colon>nat set) (Eps Q)"
```
```  1019 nitpick [expect = none]
```
```  1020 sorry
```
```  1021
```
```  1022 lemma "Q (The Q)"
```
```  1023 nitpick [expect = genuine]
```
```  1024 oops
```
```  1025
```
```  1026 lemma "(Q\<Colon>nat set) (The Q)"
```
```  1027 nitpick [expect = genuine]
```
```  1028 oops
```
```  1029
```
```  1030 lemma "\<not> Q (The Q)"
```
```  1031 nitpick [expect = genuine]
```
```  1032 oops
```
```  1033
```
```  1034 lemma "\<not> (Q\<Colon>nat set) (The Q)"
```
```  1035 nitpick [expect = genuine]
```
```  1036 oops
```
```  1037
```
```  1038 lemma "The Q \<noteq> x"
```
```  1039 nitpick [expect = genuine]
```
```  1040 oops
```
```  1041
```
```  1042 lemma "The Q \<noteq> (x\<Colon>nat)"
```
```  1043 nitpick [expect = genuine]
```
```  1044 oops
```
```  1045
```
```  1046 lemma "Q x \<Longrightarrow> Q (The Q)"
```
```  1047 nitpick [expect = genuine]
```
```  1048 oops
```
```  1049
```
```  1050 lemma "Q (x\<Colon>nat) \<Longrightarrow> Q (The Q)"
```
```  1051 nitpick [expect = genuine]
```
```  1052 oops
```
```  1053
```
```  1054 lemma "Q = {x\<Colon>'a} \<Longrightarrow> (Q\<Colon>'a set) (The Q)"
```
```  1055 nitpick [expect = none]
```
```  1056 sorry
```
```  1057
```
```  1058 lemma "Q = {x\<Colon>nat} \<Longrightarrow> (Q\<Colon>nat set) (The Q)"
```
```  1059 nitpick [expect = none]
```
```  1060 sorry
```
```  1061
```
```  1062 subsection {* Destructors and Recursors *}
```
```  1063
```
```  1064 lemma "(x\<Colon>'a) = (case True of True \<Rightarrow> x | False \<Rightarrow> x)"
```
```  1065 nitpick [card = 2, expect = none]
```
```  1066 by auto
```
```  1067
```
```  1068 lemma "bool_rec x y True = x"
```
```  1069 nitpick [card = 2, expect = none]
```
```  1070 by auto
```
```  1071
```
```  1072 lemma "bool_rec x y False = y"
```
```  1073 nitpick [card = 2, expect = none]
```
```  1074 by auto
```
```  1075
```
```  1076 lemma "(x\<Colon>bool) = bool_rec x x True"
```
```  1077 nitpick [card = 2, expect = none]
```
```  1078 by auto
```
```  1079
```
```  1080 lemma "x = (case (x, y) of (x', y') \<Rightarrow> x')"
```
```  1081 nitpick [expect = none]
```
```  1082 sorry
```
```  1083
```
```  1084 end
```