src/HOL/Library/Quickcheck_Types.thy
 author haftmann Tue Feb 19 19:44:10 2013 +0100 (2013-02-19) changeset 51188 9b5bf1a9a710 parent 44890 22f665a2e91c child 57543 36041934e429 permissions -rw-r--r--
dropped spurious left-over from 0a2371e7ced3
```     1 (*  Title:      HOL/Library/Quickcheck_Types.thy
```
```     2     Author:     Lukas Bulwahn
```
```     3     Copyright   2010 TU Muenchen
```
```     4 *)
```
```     5
```
```     6 theory Quickcheck_Types
```
```     7 imports Main
```
```     8 begin
```
```     9
```
```    10 text {*
```
```    11 This theory provides some default types for the quickcheck execution.
```
```    12 In most cases, the default type @{typ "int"} meets the sort constraints
```
```    13 of the proposition.
```
```    14 But for the type classes bot and top, the type @{typ "int"} is insufficient.
```
```    15 Hence, we provide other types than @{typ "int"} as further default types.
```
```    16 *}
```
```    17
```
```    18 subsection {* A non-distributive lattice *}
```
```    19
```
```    20 datatype non_distrib_lattice = Zero | A | B | C | One
```
```    21
```
```    22 instantiation non_distrib_lattice :: order
```
```    23 begin
```
```    24
```
```    25 definition less_eq_non_distrib_lattice
```
```    26 where
```
```    27   "a <= b = (case a of Zero => True | A => (b = A) \<or> (b = One) | B => (b = B) \<or> (b = One) | C => (b = C) \<or> (b = One) | One => (b = One))"
```
```    28
```
```    29 definition less_non_distrib_lattice
```
```    30 where
```
```    31   "a < b = (case a of Zero => (b \<noteq> Zero) | A => (b = One) | B => (b = One) | C => (b = One) | One => False)"
```
```    32
```
```    33 instance proof
```
```    34 qed (auto simp add: less_eq_non_distrib_lattice_def less_non_distrib_lattice_def split: non_distrib_lattice.split non_distrib_lattice.split_asm)
```
```    35
```
```    36 end
```
```    37
```
```    38 instantiation non_distrib_lattice :: lattice
```
```    39 begin
```
```    40
```
```    41
```
```    42 definition sup_non_distrib_lattice
```
```    43 where
```
```    44   "sup a b = (if a = b then a else (if a = Zero then b else (if b = Zero then a else One)))"
```
```    45
```
```    46 definition inf_non_distrib_lattice
```
```    47 where
```
```    48   "inf a b = (if a = b then a else (if a = One then b else (if b = One then a else Zero)))"
```
```    49
```
```    50 instance proof
```
```    51 qed (auto simp add: inf_non_distrib_lattice_def sup_non_distrib_lattice_def less_eq_non_distrib_lattice_def split: split_if non_distrib_lattice.split non_distrib_lattice.split_asm)
```
```    52
```
```    53 end
```
```    54
```
```    55 hide_const Zero A B C One
```
```    56
```
```    57 subsection {* Values extended by a bottom element *}
```
```    58
```
```    59 datatype 'a bot = Value 'a | Bot
```
```    60
```
```    61 instantiation bot :: (preorder) preorder
```
```    62 begin
```
```    63
```
```    64 definition less_eq_bot where
```
```    65   "x \<le> y \<longleftrightarrow> (case x of Bot \<Rightarrow> True | Value x \<Rightarrow> (case y of Bot \<Rightarrow> False | Value y \<Rightarrow> x \<le> y))"
```
```    66
```
```    67 definition less_bot where
```
```    68   "x < y \<longleftrightarrow> (case y of Bot \<Rightarrow> False | Value y \<Rightarrow> (case x of Bot \<Rightarrow> True | Value x \<Rightarrow> x < y))"
```
```    69
```
```    70 lemma less_eq_bot_Bot [simp]: "Bot \<le> x"
```
```    71   by (simp add: less_eq_bot_def)
```
```    72
```
```    73 lemma less_eq_bot_Bot_code [code]: "Bot \<le> x \<longleftrightarrow> True"
```
```    74   by simp
```
```    75
```
```    76 lemma less_eq_bot_Bot_is_Bot: "x \<le> Bot \<Longrightarrow> x = Bot"
```
```    77   by (cases x) (simp_all add: less_eq_bot_def)
```
```    78
```
```    79 lemma less_eq_bot_Value_Bot [simp, code]: "Value x \<le> Bot \<longleftrightarrow> False"
```
```    80   by (simp add: less_eq_bot_def)
```
```    81
```
```    82 lemma less_eq_bot_Value [simp, code]: "Value x \<le> Value y \<longleftrightarrow> x \<le> y"
```
```    83   by (simp add: less_eq_bot_def)
```
```    84
```
```    85 lemma less_bot_Bot [simp, code]: "x < Bot \<longleftrightarrow> False"
```
```    86   by (simp add: less_bot_def)
```
```    87
```
```    88 lemma less_bot_Bot_is_Value: "Bot < x \<Longrightarrow> \<exists>z. x = Value z"
```
```    89   by (cases x) (simp_all add: less_bot_def)
```
```    90
```
```    91 lemma less_bot_Bot_Value [simp]: "Bot < Value x"
```
```    92   by (simp add: less_bot_def)
```
```    93
```
```    94 lemma less_bot_Bot_Value_code [code]: "Bot < Value x \<longleftrightarrow> True"
```
```    95   by simp
```
```    96
```
```    97 lemma less_bot_Value [simp, code]: "Value x < Value y \<longleftrightarrow> x < y"
```
```    98   by (simp add: less_bot_def)
```
```    99
```
```   100 instance proof
```
```   101 qed (auto simp add: less_eq_bot_def less_bot_def less_le_not_le elim: order_trans split: bot.splits)
```
```   102
```
```   103 end
```
```   104
```
```   105 instance bot :: (order) order proof
```
```   106 qed (auto simp add: less_eq_bot_def less_bot_def split: bot.splits)
```
```   107
```
```   108 instance bot :: (linorder) linorder proof
```
```   109 qed (auto simp add: less_eq_bot_def less_bot_def split: bot.splits)
```
```   110
```
```   111 instantiation bot :: (order) bot
```
```   112 begin
```
```   113
```
```   114 definition "bot = Bot"
```
```   115
```
```   116 instance proof
```
```   117 qed (simp add: bot_bot_def)
```
```   118
```
```   119 end
```
```   120
```
```   121 instantiation bot :: (top) top
```
```   122 begin
```
```   123
```
```   124 definition "top = Value top"
```
```   125
```
```   126 instance proof
```
```   127 qed (simp add: top_bot_def less_eq_bot_def split: bot.split)
```
```   128
```
```   129 end
```
```   130
```
```   131 instantiation bot :: (semilattice_inf) semilattice_inf
```
```   132 begin
```
```   133
```
```   134 definition inf_bot
```
```   135 where
```
```   136   "inf x y = (case x of Bot => Bot | Value v => (case y of Bot => Bot | Value v' => Value (inf v v')))"
```
```   137
```
```   138 instance proof
```
```   139 qed (auto simp add: inf_bot_def less_eq_bot_def split: bot.splits)
```
```   140
```
```   141 end
```
```   142
```
```   143 instantiation bot :: (semilattice_sup) semilattice_sup
```
```   144 begin
```
```   145
```
```   146 definition sup_bot
```
```   147 where
```
```   148   "sup x y = (case x of Bot => y | Value v => (case y of Bot => x | Value v' => Value (sup v v')))"
```
```   149
```
```   150 instance proof
```
```   151 qed (auto simp add: sup_bot_def less_eq_bot_def split: bot.splits)
```
```   152
```
```   153 end
```
```   154
```
```   155 instance bot :: (lattice) bounded_lattice_bot ..
```
```   156
```
```   157 section {* Values extended by a top element *}
```
```   158
```
```   159 datatype 'a top = Value 'a | Top
```
```   160
```
```   161 instantiation top :: (preorder) preorder
```
```   162 begin
```
```   163
```
```   164 definition less_eq_top where
```
```   165   "x \<le> y \<longleftrightarrow> (case y of Top \<Rightarrow> True | Value y \<Rightarrow> (case x of Top \<Rightarrow> False | Value x \<Rightarrow> x \<le> y))"
```
```   166
```
```   167 definition less_top where
```
```   168   "x < y \<longleftrightarrow> (case x of Top \<Rightarrow> False | Value x \<Rightarrow> (case y of Top \<Rightarrow> True | Value y \<Rightarrow> x < y))"
```
```   169
```
```   170 lemma less_eq_top_Top [simp]: "x <= Top"
```
```   171   by (simp add: less_eq_top_def)
```
```   172
```
```   173 lemma less_eq_top_Top_code [code]: "x \<le> Top \<longleftrightarrow> True"
```
```   174   by simp
```
```   175
```
```   176 lemma less_eq_top_is_Top: "Top \<le> x \<Longrightarrow> x = Top"
```
```   177   by (cases x) (simp_all add: less_eq_top_def)
```
```   178
```
```   179 lemma less_eq_top_Top_Value [simp, code]: "Top \<le> Value x \<longleftrightarrow> False"
```
```   180   by (simp add: less_eq_top_def)
```
```   181
```
```   182 lemma less_eq_top_Value_Value [simp, code]: "Value x \<le> Value y \<longleftrightarrow> x \<le> y"
```
```   183   by (simp add: less_eq_top_def)
```
```   184
```
```   185 lemma less_top_Top [simp, code]: "Top < x \<longleftrightarrow> False"
```
```   186   by (simp add: less_top_def)
```
```   187
```
```   188 lemma less_top_Top_is_Value: "x < Top \<Longrightarrow> \<exists>z. x = Value z"
```
```   189   by (cases x) (simp_all add: less_top_def)
```
```   190
```
```   191 lemma less_top_Value_Top [simp]: "Value x < Top"
```
```   192   by (simp add: less_top_def)
```
```   193
```
```   194 lemma less_top_Value_Top_code [code]: "Value x < Top \<longleftrightarrow> True"
```
```   195   by simp
```
```   196
```
```   197 lemma less_top_Value [simp, code]: "Value x < Value y \<longleftrightarrow> x < y"
```
```   198   by (simp add: less_top_def)
```
```   199
```
```   200 instance proof
```
```   201 qed (auto simp add: less_eq_top_def less_top_def less_le_not_le elim: order_trans split: top.splits)
```
```   202
```
```   203 end
```
```   204
```
```   205 instance top :: (order) order proof
```
```   206 qed (auto simp add: less_eq_top_def less_top_def split: top.splits)
```
```   207
```
```   208 instance top :: (linorder) linorder proof
```
```   209 qed (auto simp add: less_eq_top_def less_top_def split: top.splits)
```
```   210
```
```   211 instantiation top :: (order) top
```
```   212 begin
```
```   213
```
```   214 definition "top = Top"
```
```   215
```
```   216 instance proof
```
```   217 qed (simp add: top_top_def)
```
```   218
```
```   219 end
```
```   220
```
```   221 instantiation top :: (bot) bot
```
```   222 begin
```
```   223
```
```   224 definition "bot = Value bot"
```
```   225
```
```   226 instance proof
```
```   227 qed (simp add: bot_top_def less_eq_top_def split: top.split)
```
```   228
```
```   229 end
```
```   230
```
```   231 instantiation top :: (semilattice_inf) semilattice_inf
```
```   232 begin
```
```   233
```
```   234 definition inf_top
```
```   235 where
```
```   236   "inf x y = (case x of Top => y | Value v => (case y of Top => x | Value v' => Value (inf v v')))"
```
```   237
```
```   238 instance proof
```
```   239 qed (auto simp add: inf_top_def less_eq_top_def split: top.splits)
```
```   240
```
```   241 end
```
```   242
```
```   243 instantiation top :: (semilattice_sup) semilattice_sup
```
```   244 begin
```
```   245
```
```   246 definition sup_top
```
```   247 where
```
```   248   "sup x y = (case x of Top => Top | Value v => (case y of Top => Top | Value v' => Value (sup v v')))"
```
```   249
```
```   250 instance proof
```
```   251 qed (auto simp add: sup_top_def less_eq_top_def split: top.splits)
```
```   252
```
```   253 end
```
```   254
```
```   255 instance top :: (lattice) bounded_lattice_top ..
```
```   256
```
```   257
```
```   258 datatype 'a flat_complete_lattice = Value 'a | Bot | Top
```
```   259
```
```   260 instantiation flat_complete_lattice :: (type) order
```
```   261 begin
```
```   262
```
```   263 definition less_eq_flat_complete_lattice where
```
```   264   "x \<le> y == (case x of Bot => True | Value v1 => (case y of Bot => False | Value v2 => (v1 = v2) | Top => True) | Top => (y = Top))"
```
```   265
```
```   266 definition less_flat_complete_lattice where
```
```   267   "x < y = (case x of Bot => \<not> (y = Bot) | Value v1 => (y = Top) | Top => False)"
```
```   268
```
```   269 lemma [simp]: "Bot <= y"
```
```   270 unfolding less_eq_flat_complete_lattice_def by auto
```
```   271
```
```   272 lemma [simp]: "y <= Top"
```
```   273 unfolding less_eq_flat_complete_lattice_def by (auto split: flat_complete_lattice.splits)
```
```   274
```
```   275 lemma greater_than_two_values:
```
```   276   assumes "a ~= aa" "Value a <= z" "Value aa <= z"
```
```   277   shows "z = Top"
```
```   278 using assms
```
```   279 by (cases z) (auto simp add: less_eq_flat_complete_lattice_def)
```
```   280
```
```   281 lemma lesser_than_two_values:
```
```   282   assumes "a ~= aa" "z <= Value a" "z <= Value aa"
```
```   283   shows "z = Bot"
```
```   284 using assms
```
```   285 by (cases z) (auto simp add: less_eq_flat_complete_lattice_def)
```
```   286
```
```   287 instance proof
```
```   288 qed (auto simp add: less_eq_flat_complete_lattice_def less_flat_complete_lattice_def split: flat_complete_lattice.splits)
```
```   289
```
```   290 end
```
```   291
```
```   292 instantiation flat_complete_lattice :: (type) bot
```
```   293 begin
```
```   294
```
```   295 definition "bot = Bot"
```
```   296
```
```   297 instance proof
```
```   298 qed (simp add: bot_flat_complete_lattice_def)
```
```   299
```
```   300 end
```
```   301
```
```   302 instantiation flat_complete_lattice :: (type) top
```
```   303 begin
```
```   304
```
```   305 definition "top = Top"
```
```   306
```
```   307 instance proof
```
```   308 qed (auto simp add: less_eq_flat_complete_lattice_def top_flat_complete_lattice_def split: flat_complete_lattice.splits)
```
```   309
```
```   310 end
```
```   311
```
```   312 instantiation flat_complete_lattice :: (type) lattice
```
```   313 begin
```
```   314
```
```   315 definition inf_flat_complete_lattice
```
```   316 where
```
```   317   "inf x y = (case x of Bot => Bot | Value v1 => (case y of Bot => Bot | Value v2 => if (v1 = v2) then x else Bot | Top => x) | Top => y)"
```
```   318
```
```   319 definition sup_flat_complete_lattice
```
```   320 where
```
```   321   "sup x y = (case x of Bot => y | Value v1 => (case y of Bot => x | Value v2 => if v1 = v2 then x else Top | Top => Top) | Top => Top)"
```
```   322
```
```   323 instance proof
```
```   324 qed (auto simp add: inf_flat_complete_lattice_def sup_flat_complete_lattice_def less_eq_flat_complete_lattice_def split: flat_complete_lattice.splits)
```
```   325
```
```   326 end
```
```   327
```
```   328 instantiation flat_complete_lattice :: (type) complete_lattice
```
```   329 begin
```
```   330
```
```   331 definition Sup_flat_complete_lattice
```
```   332 where
```
```   333   "Sup A = (if (A = {} \<or> A = {Bot}) then Bot else (if (\<exists> v. A - {Bot} = {Value v}) then Value (THE v. A - {Bot} = {Value v}) else Top))"
```
```   334
```
```   335 definition Inf_flat_complete_lattice
```
```   336 where
```
```   337   "Inf A = (if (A = {} \<or> A = {Top}) then Top else (if (\<exists> v. A - {Top} = {Value v}) then Value (THE v. A - {Top} = {Value v}) else Bot))"
```
```   338
```
```   339 instance
```
```   340 proof
```
```   341   fix x A
```
```   342   assume "(x :: 'a flat_complete_lattice) : A"
```
```   343   {
```
```   344     fix v
```
```   345     assume "A - {Top} = {Value v}"
```
```   346     from this have "(THE v. A - {Top} = {Value v}) = v"
```
```   347       by (auto intro!: the1_equality)
```
```   348     moreover
```
```   349     from `x : A` `A - {Top} = {Value v}` have "x = Top \<or> x = Value v"
```
```   350       by auto
```
```   351     ultimately have "Value (THE v. A - {Top} = {Value v}) <= x"
```
```   352       by auto
```
```   353   }
```
```   354   from `x : A` this show "Inf A <= x"
```
```   355     unfolding Inf_flat_complete_lattice_def
```
```   356     by fastforce
```
```   357 next
```
```   358   fix z A
```
```   359   assume z: "\<And>x. x : A ==> z <= (x :: 'a flat_complete_lattice)"
```
```   360   {
```
```   361     fix v
```
```   362     assume "A - {Top} = {Value v}"
```
```   363     moreover
```
```   364     from this have "(THE v. A - {Top} = {Value v}) = v"
```
```   365       by (auto intro!: the1_equality)
```
```   366     moreover
```
```   367     note z
```
```   368     moreover
```
```   369     ultimately have "z <= Value (THE v::'a. A - {Top} = {Value v})"
```
```   370       by auto
```
```   371   } moreover
```
```   372   {
```
```   373     assume not_one_value: "A ~= {}" "A ~= {Top}" "~ (EX v::'a. A - {Top} = {Value v})"
```
```   374     have "z <= Bot"
```
```   375     proof (cases "A - {Top} = {Bot}")
```
```   376       case True
```
```   377       from this z show ?thesis
```
```   378         by auto
```
```   379     next
```
```   380       case False
```
```   381       from not_one_value
```
```   382       obtain a1 where a1: "a1 : A - {Top}" by auto
```
```   383       from not_one_value False a1
```
```   384       obtain a2 where "a2 : A - {Top} \<and> a1 \<noteq> a2"
```
```   385         by (cases a1) auto
```
```   386       from this a1 z[of "a1"] z[of "a2"] show ?thesis
```
```   387         apply (cases a1)
```
```   388         apply auto
```
```   389         apply (cases a2)
```
```   390         apply auto
```
```   391         apply (auto dest!: lesser_than_two_values)
```
```   392         done
```
```   393     qed
```
```   394   } moreover
```
```   395   note z moreover
```
```   396   ultimately show "z <= Inf A"
```
```   397     unfolding Inf_flat_complete_lattice_def
```
```   398     by auto
```
```   399 next
```
```   400   fix x A
```
```   401   assume "(x :: 'a flat_complete_lattice) : A"
```
```   402   {
```
```   403     fix v
```
```   404     assume "A - {Bot} = {Value v}"
```
```   405     from this have "(THE v. A - {Bot} = {Value v}) = v"
```
```   406       by (auto intro!: the1_equality)
```
```   407     moreover
```
```   408     from `x : A` `A - {Bot} = {Value v}` have "x = Bot \<or> x = Value v"
```
```   409       by auto
```
```   410     ultimately have "x <= Value (THE v. A - {Bot} = {Value v})"
```
```   411       by auto
```
```   412   }
```
```   413   from `x : A` this show "x <= Sup A"
```
```   414     unfolding Sup_flat_complete_lattice_def
```
```   415     by fastforce
```
```   416 next
```
```   417   fix z A
```
```   418   assume z: "\<And>x. x : A ==> x <= (z :: 'a flat_complete_lattice)"
```
```   419   {
```
```   420     fix v
```
```   421     assume "A - {Bot} = {Value v}"
```
```   422     moreover
```
```   423     from this have "(THE v. A - {Bot} = {Value v}) = v"
```
```   424       by (auto intro!: the1_equality)
```
```   425     moreover
```
```   426     note z
```
```   427     moreover
```
```   428     ultimately have "Value (THE v::'a. A - {Bot} = {Value v}) <= z"
```
```   429       by auto
```
```   430   } moreover
```
```   431   {
```
```   432     assume not_one_value: "A ~= {}" "A ~= {Bot}" "~ (EX v::'a. A - {Bot} = {Value v})"
```
```   433     have "Top <= z"
```
```   434     proof (cases "A - {Bot} = {Top}")
```
```   435       case True
```
```   436       from this z show ?thesis
```
```   437         by auto
```
```   438     next
```
```   439       case False
```
```   440       from not_one_value
```
```   441       obtain a1 where a1: "a1 : A - {Bot}" by auto
```
```   442       from not_one_value False a1
```
```   443       obtain a2 where "a2 : A - {Bot} \<and> a1 \<noteq> a2"
```
```   444         by (cases a1) auto
```
```   445       from this a1 z[of "a1"] z[of "a2"] show ?thesis
```
```   446         apply (cases a1)
```
```   447         apply auto
```
```   448         apply (cases a2)
```
```   449         apply (auto dest!: greater_than_two_values)
```
```   450         done
```
```   451     qed
```
```   452   } moreover
```
```   453   note z moreover
```
```   454   ultimately show "Sup A <= z"
```
```   455     unfolding Sup_flat_complete_lattice_def
```
```   456     by auto
```
```   457 qed
```
```   458
```
```   459 end
```
```   460
```
```   461 section {* Quickcheck configuration *}
```
```   462
```
```   463 quickcheck_params[finite_types = false, default_type = ["int", "non_distrib_lattice", "int bot", "int top", "int flat_complete_lattice"]]
```
```   464
```
```   465 hide_type non_distrib_lattice flat_complete_lattice bot top
```
```   466
```
`   467 end`