src/HOL/Nitpick_Examples/Manual_Nits.thy
 author wenzelm Sun Nov 02 18:21:45 2014 +0100 (2014-11-02) changeset 58889 5b7a9633cfa8 parent 58310 91ea607a34d8 child 60310 932221b62e89 permissions -rw-r--r--
```     1 (*  Title:      HOL/Nitpick_Examples/Manual_Nits.thy
```
```     2     Author:     Jasmin Blanchette, TU Muenchen
```
```     3     Copyright   2009-2014
```
```     4
```
```     5 Examples from the Nitpick manual.
```
```     6 *)
```
```     7
```
```     8 section {* Examples from the Nitpick Manual *}
```
```     9
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```    10 (* The "expect" arguments to Nitpick in this theory and the other example
```
```    11    theories are there so that the example can also serve as a regression test
```
```    12    suite. *)
```
```    13
```
```    14 theory Manual_Nits
```
```    15 imports Real "~~/src/HOL/Library/Quotient_Product"
```
```    16 begin
```
```    17
```
```    18 section {* 2. First Steps *}
```
```    19
```
```    20 nitpick_params [sat_solver = MiniSat_JNI, max_threads = 1, timeout = 240]
```
```    21
```
```    22
```
```    23 subsection {* 2.1. Propositional Logic *}
```
```    24
```
```    25 lemma "P \<longleftrightarrow> Q"
```
```    26 nitpick [expect = genuine]
```
```    27 apply auto
```
```    28 nitpick [expect = genuine] 1
```
```    29 nitpick [expect = genuine] 2
```
```    30 oops
```
```    31
```
```    32
```
```    33 subsection {* 2.2. Type Variables *}
```
```    34
```
```    35 lemma "x \<in> A \<Longrightarrow> (THE y. y \<in> A) \<in> A"
```
```    36 nitpick [verbose, expect = genuine]
```
```    37 oops
```
```    38
```
```    39
```
```    40 subsection {* 2.3. Constants *}
```
```    41
```
```    42 lemma "x \<in> A \<Longrightarrow> (THE y. y \<in> A) \<in> A"
```
```    43 nitpick [show_consts, expect = genuine]
```
```    44 nitpick [dont_specialize, show_consts, expect = genuine]
```
```    45 oops
```
```    46
```
```    47 lemma "\<exists>!x. x \<in> A \<Longrightarrow> (THE y. y \<in> A) \<in> A"
```
```    48 nitpick [expect = none]
```
```    49 nitpick [card 'a = 1-50, expect = none]
```
```    50 (* sledgehammer *)
```
```    51 by (metis the_equality)
```
```    52
```
```    53
```
```    54 subsection {* 2.4. Skolemization *}
```
```    55
```
```    56 lemma "\<exists>g. \<forall>x. g (f x) = x \<Longrightarrow> \<forall>y. \<exists>x. y = f x"
```
```    57 nitpick [expect = genuine]
```
```    58 oops
```
```    59
```
```    60 lemma "\<exists>x. \<forall>f. f x = x"
```
```    61 nitpick [expect = genuine]
```
```    62 oops
```
```    63
```
```    64 lemma "refl r \<Longrightarrow> sym r"
```
```    65 nitpick [expect = genuine]
```
```    66 oops
```
```    67
```
```    68
```
```    69 subsection {* 2.5. Natural Numbers and Integers *}
```
```    70
```
```    71 lemma "\<lbrakk>i \<le> j; n \<le> (m\<Colon>int)\<rbrakk> \<Longrightarrow> i * n + j * m \<le> i * m + j * n"
```
```    72 nitpick [expect = genuine]
```
```    73 nitpick [binary_ints, bits = 16, expect = genuine]
```
```    74 oops
```
```    75
```
```    76 lemma "\<forall>n. Suc n \<noteq> n \<Longrightarrow> P"
```
```    77 nitpick [card nat = 100, check_potential, tac_timeout = 5, expect = genuine]
```
```    78 oops
```
```    79
```
```    80 lemma "P Suc"
```
```    81 nitpick [expect = none]
```
```    82 oops
```
```    83
```
```    84 lemma "P (op +\<Colon>nat\<Rightarrow>nat\<Rightarrow>nat)"
```
```    85 nitpick [card nat = 1, expect = genuine]
```
```    86 nitpick [card nat = 2, expect = none]
```
```    87 oops
```
```    88
```
```    89
```
```    90 subsection {* 2.6. Inductive Datatypes *}
```
```    91
```
```    92 lemma "hd (xs @ [y, y]) = hd xs"
```
```    93 nitpick [expect = genuine]
```
```    94 nitpick [show_consts, show_types, expect = genuine]
```
```    95 oops
```
```    96
```
```    97 lemma "\<lbrakk>length xs = 1; length ys = 1\<rbrakk> \<Longrightarrow> xs = ys"
```
```    98 nitpick [show_types, expect = genuine]
```
```    99 oops
```
```   100
```
```   101
```
```   102 subsection {* 2.7. Typedefs, Records, Rationals, and Reals *}
```
```   103
```
```   104 definition "three = {0\<Colon>nat, 1, 2}"
```
```   105 typedef three = three
```
```   106   unfolding three_def by blast
```
```   107
```
```   108 definition A :: three where "A \<equiv> Abs_three 0"
```
```   109 definition B :: three where "B \<equiv> Abs_three 1"
```
```   110 definition C :: three where "C \<equiv> Abs_three 2"
```
```   111
```
```   112 lemma "\<lbrakk>A \<in> X; B \<in> X\<rbrakk> \<Longrightarrow> c \<in> X"
```
```   113 nitpick [show_types, expect = genuine]
```
```   114 oops
```
```   115
```
```   116 fun my_int_rel where
```
```   117 "my_int_rel (x, y) (u, v) = (x + v = u + y)"
```
```   118
```
```   119 quotient_type my_int = "nat \<times> nat" / my_int_rel
```
```   120 by (auto simp add: equivp_def fun_eq_iff)
```
```   121
```
```   122 definition add_raw where
```
```   123 "add_raw \<equiv> \<lambda>(x, y) (u, v). (x + (u\<Colon>nat), y + (v\<Colon>nat))"
```
```   124
```
```   125 quotient_definition "add\<Colon>my_int \<Rightarrow> my_int \<Rightarrow> my_int" is add_raw
```
```   126 unfolding add_raw_def by auto
```
```   127
```
```   128 lemma "add x y = add x x"
```
```   129 nitpick [show_types, expect = genuine]
```
```   130 oops
```
```   131
```
```   132 ML {*
```
```   133 fun my_int_postproc _ _ _ T (Const _ \$ (Const _ \$ t1 \$ t2)) =
```
```   134     HOLogic.mk_number T (snd (HOLogic.dest_number t1)
```
```   135                          - snd (HOLogic.dest_number t2))
```
```   136   | my_int_postproc _ _ _ _ t = t
```
```   137 *}
```
```   138
```
```   139 declaration {*
```
```   140 Nitpick_Model.register_term_postprocessor @{typ my_int} my_int_postproc
```
```   141 *}
```
```   142
```
```   143 lemma "add x y = add x x"
```
```   144 nitpick [show_types]
```
```   145 oops
```
```   146
```
```   147 record point =
```
```   148   Xcoord :: int
```
```   149   Ycoord :: int
```
```   150
```
```   151 lemma "Xcoord (p\<Colon>point) = Xcoord (q\<Colon>point)"
```
```   152 nitpick [show_types, expect = genuine]
```
```   153 oops
```
```   154
```
```   155 lemma "4 * x + 3 * (y\<Colon>real) \<noteq> 1 / 2"
```
```   156 nitpick [show_types, expect = genuine]
```
```   157 oops
```
```   158
```
```   159
```
```   160 subsection {* 2.8. Inductive and Coinductive Predicates *}
```
```   161
```
```   162 inductive even where
```
```   163 "even 0" |
```
```   164 "even n \<Longrightarrow> even (Suc (Suc n))"
```
```   165
```
```   166 lemma "\<exists>n. even n \<and> even (Suc n)"
```
```   167 nitpick [card nat = 50, unary_ints, verbose, expect = potential]
```
```   168 oops
```
```   169
```
```   170 lemma "\<exists>n \<le> 49. even n \<and> even (Suc n)"
```
```   171 nitpick [card nat = 50, unary_ints, expect = genuine]
```
```   172 oops
```
```   173
```
```   174 inductive even' where
```
```   175 "even' (0\<Colon>nat)" |
```
```   176 "even' 2" |
```
```   177 "\<lbrakk>even' m; even' n\<rbrakk> \<Longrightarrow> even' (m + n)"
```
```   178
```
```   179 lemma "\<exists>n \<in> {0, 2, 4, 6, 8}. \<not> even' n"
```
```   180 nitpick [card nat = 10, unary_ints, verbose, show_consts, expect = genuine]
```
```   181 oops
```
```   182
```
```   183 lemma "even' (n - 2) \<Longrightarrow> even' n"
```
```   184 nitpick [card nat = 10, show_consts, expect = genuine]
```
```   185 oops
```
```   186
```
```   187 coinductive nats where
```
```   188 "nats (x\<Colon>nat) \<Longrightarrow> nats x"
```
```   189
```
```   190 lemma "nats = (\<lambda>n. n \<in> {0, 1, 2, 3, 4})"
```
```   191 nitpick [card nat = 10, show_consts, expect = genuine]
```
```   192 oops
```
```   193
```
```   194 inductive odd where
```
```   195 "odd 1" |
```
```   196 "\<lbrakk>odd m; even n\<rbrakk> \<Longrightarrow> odd (m + n)"
```
```   197
```
```   198 lemma "odd n \<Longrightarrow> odd (n - 2)"
```
```   199 nitpick [card nat = 4, show_consts, expect = genuine]
```
```   200 oops
```
```   201
```
```   202
```
```   203 subsection {* 2.9. Coinductive Datatypes *}
```
```   204
```
```   205 codatatype 'a llist = LNil | LCons 'a "'a llist"
```
```   206
```
```   207 primcorec iterates where
```
```   208 "iterates f a = LCons a (iterates f (f a))"
```
```   209
```
```   210 lemma "xs \<noteq> LCons a xs"
```
```   211 nitpick [expect = genuine]
```
```   212 oops
```
```   213
```
```   214 lemma "\<lbrakk>xs = LCons a xs; ys = iterates (\<lambda>b. a) b\<rbrakk> \<Longrightarrow> xs = ys"
```
```   215 nitpick [verbose, expect = genuine]
```
```   216 oops
```
```   217
```
```   218 lemma "\<lbrakk>xs = LCons a xs; ys = LCons a ys\<rbrakk> \<Longrightarrow> xs = ys"
```
```   219 nitpick [bisim_depth = -1, show_types, expect = quasi_genuine]
```
```   220 nitpick [card = 1-5, expect = none]
```
```   221 sorry
```
```   222
```
```   223
```
```   224 subsection {* 2.10. Boxing *}
```
```   225
```
```   226 datatype tm = Var nat | Lam tm | App tm tm
```
```   227
```
```   228 primrec lift where
```
```   229 "lift (Var j) k = Var (if j < k then j else j + 1)" |
```
```   230 "lift (Lam t) k = Lam (lift t (k + 1))" |
```
```   231 "lift (App t u) k = App (lift t k) (lift u k)"
```
```   232
```
```   233 primrec loose where
```
```   234 "loose (Var j) k = (j \<ge> k)" |
```
```   235 "loose (Lam t) k = loose t (Suc k)" |
```
```   236 "loose (App t u) k = (loose t k \<or> loose u k)"
```
```   237
```
```   238 primrec subst\<^sub>1 where
```
```   239 "subst\<^sub>1 \<sigma> (Var j) = \<sigma> j" |
```
```   240 "subst\<^sub>1 \<sigma> (Lam t) =
```
```   241  Lam (subst\<^sub>1 (\<lambda>n. case n of 0 \<Rightarrow> Var 0 | Suc m \<Rightarrow> lift (\<sigma> m) 1) t)" |
```
```   242 "subst\<^sub>1 \<sigma> (App t u) = App (subst\<^sub>1 \<sigma> t) (subst\<^sub>1 \<sigma> u)"
```
```   243
```
```   244 lemma "\<not> loose t 0 \<Longrightarrow> subst\<^sub>1 \<sigma> t = t"
```
```   245 nitpick [verbose, expect = genuine]
```
```   246 nitpick [eval = "subst\<^sub>1 \<sigma> t", expect = genuine]
```
```   247 (* nitpick [dont_box, expect = unknown] *)
```
```   248 oops
```
```   249
```
```   250 primrec subst\<^sub>2 where
```
```   251 "subst\<^sub>2 \<sigma> (Var j) = \<sigma> j" |
```
```   252 "subst\<^sub>2 \<sigma> (Lam t) =
```
```   253  Lam (subst\<^sub>2 (\<lambda>n. case n of 0 \<Rightarrow> Var 0 | Suc m \<Rightarrow> lift (\<sigma> m) 0) t)" |
```
```   254 "subst\<^sub>2 \<sigma> (App t u) = App (subst\<^sub>2 \<sigma> t) (subst\<^sub>2 \<sigma> u)"
```
```   255
```
```   256 lemma "\<not> loose t 0 \<Longrightarrow> subst\<^sub>2 \<sigma> t = t"
```
```   257 nitpick [card = 1-5, expect = none]
```
```   258 sorry
```
```   259
```
```   260
```
```   261 subsection {* 2.11. Scope Monotonicity *}
```
```   262
```
```   263 lemma "length xs = length ys \<Longrightarrow> rev (zip xs ys) = zip xs (rev ys)"
```
```   264 nitpick [verbose, expect = genuine]
```
```   265 oops
```
```   266
```
```   267 lemma "\<exists>g. \<forall>x\<Colon>'b. g (f x) = x \<Longrightarrow> \<forall>y\<Colon>'a. \<exists>x. y = f x"
```
```   268 nitpick [mono, expect = none]
```
```   269 nitpick [expect = genuine]
```
```   270 oops
```
```   271
```
```   272
```
```   273 subsection {* 2.12. Inductive Properties *}
```
```   274
```
```   275 inductive_set reach where
```
```   276 "(4\<Colon>nat) \<in> reach" |
```
```   277 "n \<in> reach \<Longrightarrow> n < 4 \<Longrightarrow> 3 * n + 1 \<in> reach" |
```
```   278 "n \<in> reach \<Longrightarrow> n + 2 \<in> reach"
```
```   279
```
```   280 lemma "n \<in> reach \<Longrightarrow> 2 dvd n"
```
```   281 (* nitpick *)
```
```   282 apply (induct set: reach)
```
```   283   apply auto
```
```   284  nitpick [card = 1-4, bits = 1-4, expect = none]
```
```   285  apply (thin_tac "n \<in> reach")
```
```   286  nitpick [expect = genuine]
```
```   287 oops
```
```   288
```
```   289 lemma "n \<in> reach \<Longrightarrow> 2 dvd n \<and> n \<noteq> 0"
```
```   290 (* nitpick *)
```
```   291 apply (induct set: reach)
```
```   292   apply auto
```
```   293  nitpick [card = 1-4, bits = 1-4, expect = none]
```
```   294  apply (thin_tac "n \<in> reach")
```
```   295  nitpick [expect = genuine]
```
```   296 oops
```
```   297
```
```   298 lemma "n \<in> reach \<Longrightarrow> 2 dvd n \<and> n \<ge> 4"
```
```   299 by (induct set: reach) arith+
```
```   300
```
```   301
```
```   302 section {* 3. Case Studies *}
```
```   303
```
```   304 nitpick_params [max_potential = 0]
```
```   305
```
```   306
```
```   307 subsection {* 3.1. A Context-Free Grammar *}
```
```   308
```
```   309 datatype alphabet = a | b
```
```   310
```
```   311 inductive_set S\<^sub>1 and A\<^sub>1 and B\<^sub>1 where
```
```   312   "[] \<in> S\<^sub>1"
```
```   313 | "w \<in> A\<^sub>1 \<Longrightarrow> b # w \<in> S\<^sub>1"
```
```   314 | "w \<in> B\<^sub>1 \<Longrightarrow> a # w \<in> S\<^sub>1"
```
```   315 | "w \<in> S\<^sub>1 \<Longrightarrow> a # w \<in> A\<^sub>1"
```
```   316 | "w \<in> S\<^sub>1 \<Longrightarrow> b # w \<in> S\<^sub>1"
```
```   317 | "\<lbrakk>v \<in> B\<^sub>1; v \<in> B\<^sub>1\<rbrakk> \<Longrightarrow> a # v @ w \<in> B\<^sub>1"
```
```   318
```
```   319 theorem S\<^sub>1_sound:
```
```   320 "w \<in> S\<^sub>1 \<longrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]"
```
```   321 nitpick [expect = genuine]
```
```   322 oops
```
```   323
```
```   324 inductive_set S\<^sub>2 and A\<^sub>2 and B\<^sub>2 where
```
```   325   "[] \<in> S\<^sub>2"
```
```   326 | "w \<in> A\<^sub>2 \<Longrightarrow> b # w \<in> S\<^sub>2"
```
```   327 | "w \<in> B\<^sub>2 \<Longrightarrow> a # w \<in> S\<^sub>2"
```
```   328 | "w \<in> S\<^sub>2 \<Longrightarrow> a # w \<in> A\<^sub>2"
```
```   329 | "w \<in> S\<^sub>2 \<Longrightarrow> b # w \<in> B\<^sub>2"
```
```   330 | "\<lbrakk>v \<in> B\<^sub>2; v \<in> B\<^sub>2\<rbrakk> \<Longrightarrow> a # v @ w \<in> B\<^sub>2"
```
```   331
```
```   332 theorem S\<^sub>2_sound:
```
```   333 "w \<in> S\<^sub>2 \<longrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]"
```
```   334 nitpick [expect = genuine]
```
```   335 oops
```
```   336
```
```   337 inductive_set S\<^sub>3 and A\<^sub>3 and B\<^sub>3 where
```
```   338   "[] \<in> S\<^sub>3"
```
```   339 | "w \<in> A\<^sub>3 \<Longrightarrow> b # w \<in> S\<^sub>3"
```
```   340 | "w \<in> B\<^sub>3 \<Longrightarrow> a # w \<in> S\<^sub>3"
```
```   341 | "w \<in> S\<^sub>3 \<Longrightarrow> a # w \<in> A\<^sub>3"
```
```   342 | "w \<in> S\<^sub>3 \<Longrightarrow> b # w \<in> B\<^sub>3"
```
```   343 | "\<lbrakk>v \<in> B\<^sub>3; w \<in> B\<^sub>3\<rbrakk> \<Longrightarrow> a # v @ w \<in> B\<^sub>3"
```
```   344
```
```   345 theorem S\<^sub>3_sound:
```
```   346 "w \<in> S\<^sub>3 \<longrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]"
```
```   347 nitpick [card = 1-5, expect = none]
```
```   348 sorry
```
```   349
```
```   350 theorem S\<^sub>3_complete:
```
```   351 "length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b] \<longrightarrow> w \<in> S\<^sub>3"
```
```   352 nitpick [expect = genuine]
```
```   353 oops
```
```   354
```
```   355 inductive_set S\<^sub>4 and A\<^sub>4 and B\<^sub>4 where
```
```   356   "[] \<in> S\<^sub>4"
```
```   357 | "w \<in> A\<^sub>4 \<Longrightarrow> b # w \<in> S\<^sub>4"
```
```   358 | "w \<in> B\<^sub>4 \<Longrightarrow> a # w \<in> S\<^sub>4"
```
```   359 | "w \<in> S\<^sub>4 \<Longrightarrow> a # w \<in> A\<^sub>4"
```
```   360 | "\<lbrakk>v \<in> A\<^sub>4; w \<in> A\<^sub>4\<rbrakk> \<Longrightarrow> b # v @ w \<in> A\<^sub>4"
```
```   361 | "w \<in> S\<^sub>4 \<Longrightarrow> b # w \<in> B\<^sub>4"
```
```   362 | "\<lbrakk>v \<in> B\<^sub>4; w \<in> B\<^sub>4\<rbrakk> \<Longrightarrow> a # v @ w \<in> B\<^sub>4"
```
```   363
```
```   364 theorem S\<^sub>4_sound:
```
```   365 "w \<in> S\<^sub>4 \<longrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]"
```
```   366 nitpick [card = 1-5, expect = none]
```
```   367 sorry
```
```   368
```
```   369 theorem S\<^sub>4_complete:
```
```   370 "length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b] \<longrightarrow> w \<in> S\<^sub>4"
```
```   371 nitpick [card = 1-5, expect = none]
```
```   372 sorry
```
```   373
```
```   374 theorem S\<^sub>4_A\<^sub>4_B\<^sub>4_sound_and_complete:
```
```   375 "w \<in> S\<^sub>4 \<longleftrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b]"
```
```   376 "w \<in> A\<^sub>4 \<longleftrightarrow> length [x \<leftarrow> w. x = a] = length [x \<leftarrow> w. x = b] + 1"
```
```   377 "w \<in> B\<^sub>4 \<longleftrightarrow> length [x \<leftarrow> w. x = b] = length [x \<leftarrow> w. x = a] + 1"
```
```   378 nitpick [card = 1-5, expect = none]
```
```   379 sorry
```
```   380
```
```   381
```
```   382 subsection {* 3.2. AA Trees *}
```
```   383
```
```   384 datatype 'a aa_tree = \<Lambda> | N "'a\<Colon>linorder" nat "'a aa_tree" "'a aa_tree"
```
```   385
```
```   386 primrec data where
```
```   387 "data \<Lambda> = undefined" |
```
```   388 "data (N x _ _ _) = x"
```
```   389
```
```   390 primrec dataset where
```
```   391 "dataset \<Lambda> = {}" |
```
```   392 "dataset (N x _ t u) = {x} \<union> dataset t \<union> dataset u"
```
```   393
```
```   394 primrec level where
```
```   395 "level \<Lambda> = 0" |
```
```   396 "level (N _ k _ _) = k"
```
```   397
```
```   398 primrec left where
```
```   399 "left \<Lambda> = \<Lambda>" |
```
```   400 "left (N _ _ t\<^sub>1 _) = t\<^sub>1"
```
```   401
```
```   402 primrec right where
```
```   403 "right \<Lambda> = \<Lambda>" |
```
```   404 "right (N _ _ _ t\<^sub>2) = t\<^sub>2"
```
```   405
```
```   406 fun wf where
```
```   407 "wf \<Lambda> = True" |
```
```   408 "wf (N _ k t u) =
```
```   409  (if t = \<Lambda> then
```
```   410     k = 1 \<and> (u = \<Lambda> \<or> (level u = 1 \<and> left u = \<Lambda> \<and> right u = \<Lambda>))
```
```   411   else
```
```   412     wf t \<and> wf u \<and> u \<noteq> \<Lambda> \<and> level t < k \<and> level u \<le> k \<and> level (right u) < k)"
```
```   413
```
```   414 fun skew where
```
```   415 "skew \<Lambda> = \<Lambda>" |
```
```   416 "skew (N x k t u) =
```
```   417  (if t \<noteq> \<Lambda> \<and> k = level t then
```
```   418     N (data t) k (left t) (N x k (right t) u)
```
```   419   else
```
```   420     N x k t u)"
```
```   421
```
```   422 fun split where
```
```   423 "split \<Lambda> = \<Lambda>" |
```
```   424 "split (N x k t u) =
```
```   425  (if u \<noteq> \<Lambda> \<and> k = level (right u) then
```
```   426     N (data u) (Suc k) (N x k t (left u)) (right u)
```
```   427   else
```
```   428     N x k t u)"
```
```   429
```
```   430 theorem dataset_skew_split:
```
```   431 "dataset (skew t) = dataset t"
```
```   432 "dataset (split t) = dataset t"
```
```   433 nitpick [card = 1-5, expect = none]
```
```   434 sorry
```
```   435
```
```   436 theorem wf_skew_split:
```
```   437 "wf t \<Longrightarrow> skew t = t"
```
```   438 "wf t \<Longrightarrow> split t = t"
```
```   439 nitpick [card = 1-5, expect = none]
```
```   440 sorry
```
```   441
```
```   442 primrec insort\<^sub>1 where
```
```   443 "insort\<^sub>1 \<Lambda> x = N x 1 \<Lambda> \<Lambda>" |
```
```   444 "insort\<^sub>1 (N y k t u) x =
```
```   445  (* (split \<circ> skew) *) (N y k (if x < y then insort\<^sub>1 t x else t)
```
```   446                              (if x > y then insort\<^sub>1 u x else u))"
```
```   447
```
```   448 theorem wf_insort\<^sub>1: "wf t \<Longrightarrow> wf (insort\<^sub>1 t x)"
```
```   449 nitpick [expect = genuine]
```
```   450 oops
```
```   451
```
```   452 theorem wf_insort\<^sub>1_nat: "wf t \<Longrightarrow> wf (insort\<^sub>1 t (x\<Colon>nat))"
```
```   453 nitpick [eval = "insort\<^sub>1 t x", expect = genuine]
```
```   454 oops
```
```   455
```
```   456 primrec insort\<^sub>2 where
```
```   457 "insort\<^sub>2 \<Lambda> x = N x 1 \<Lambda> \<Lambda>" |
```
```   458 "insort\<^sub>2 (N y k t u) x =
```
```   459  (split \<circ> skew) (N y k (if x < y then insort\<^sub>2 t x else t)
```
```   460                        (if x > y then insort\<^sub>2 u x else u))"
```
```   461
```
```   462 theorem wf_insort\<^sub>2: "wf t \<Longrightarrow> wf (insort\<^sub>2 t x)"
```
```   463 nitpick [card = 1-5, expect = none]
```
```   464 sorry
```
```   465
```
```   466 theorem dataset_insort\<^sub>2: "dataset (insort\<^sub>2 t x) = {x} \<union> dataset t"
```
```   467 nitpick [card = 1-5, expect = none]
```
```   468 sorry
```
```   469
```
```   470 end
```