src/HOL/Library/Comparator.thy
author haftmann
Sun Nov 04 15:00:30 2018 +0000 (9 months ago)
changeset 69246 c1fe9dcc274a
parent 69194 6d514e128a85
child 69251 d240598e8637
permissions -rw-r--r--
concrecte sorting algorithms beyond insertion sort
haftmann@69184
     1
(*  Title:      HOL/Library/Comparator.thy
haftmann@69184
     2
    Author:     Florian Haftmann, TU Muenchen
haftmann@69184
     3
*)
haftmann@69184
     4
haftmann@69184
     5
theory Comparator
haftmann@69184
     6
  imports Main
haftmann@69184
     7
begin
haftmann@69184
     8
haftmann@69184
     9
section \<open>Comparators on linear quasi-orders\<close>
haftmann@69184
    10
haftmann@69184
    11
datatype comp = Less | Equiv | Greater
haftmann@69184
    12
haftmann@69184
    13
locale comparator =
haftmann@69184
    14
  fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> comp"
haftmann@69184
    15
  assumes refl [simp]: "\<And>a. cmp a a = Equiv"
haftmann@69184
    16
    and trans_equiv: "\<And>a b c. cmp a b = Equiv \<Longrightarrow> cmp b c = Equiv \<Longrightarrow> cmp a c = Equiv"
haftmann@69184
    17
  assumes trans_less: "cmp a b = Less \<Longrightarrow> cmp b c = Less \<Longrightarrow> cmp a c = Less"
haftmann@69184
    18
    and greater_iff_sym_less: "\<And>b a. cmp b a = Greater \<longleftrightarrow> cmp a b = Less"
haftmann@69184
    19
begin
haftmann@69184
    20
haftmann@69184
    21
text \<open>Dual properties\<close>
haftmann@69184
    22
haftmann@69184
    23
lemma trans_greater:
haftmann@69184
    24
  "cmp a c = Greater" if "cmp a b = Greater" "cmp b c = Greater"
haftmann@69184
    25
  using that greater_iff_sym_less trans_less by blast
haftmann@69184
    26
haftmann@69184
    27
lemma less_iff_sym_greater:
haftmann@69184
    28
  "cmp b a = Less \<longleftrightarrow> cmp a b = Greater"
haftmann@69184
    29
  by (simp add: greater_iff_sym_less)
haftmann@69184
    30
haftmann@69184
    31
text \<open>The equivalence part\<close>
haftmann@69184
    32
haftmann@69184
    33
lemma sym:
haftmann@69184
    34
  "cmp b a = Equiv \<longleftrightarrow> cmp a b = Equiv"
haftmann@69184
    35
  by (metis (full_types) comp.exhaust greater_iff_sym_less)
haftmann@69184
    36
haftmann@69184
    37
lemma reflp:
haftmann@69184
    38
  "reflp (\<lambda>a b. cmp a b = Equiv)"
haftmann@69184
    39
  by (rule reflpI) simp
haftmann@69184
    40
haftmann@69184
    41
lemma symp:
haftmann@69184
    42
  "symp (\<lambda>a b. cmp a b = Equiv)"
haftmann@69184
    43
  by (rule sympI) (simp add: sym)
haftmann@69184
    44
haftmann@69184
    45
lemma transp:
haftmann@69184
    46
  "transp (\<lambda>a b. cmp a b = Equiv)"
haftmann@69184
    47
  by (rule transpI) (fact trans_equiv)
haftmann@69184
    48
haftmann@69184
    49
lemma equivp:
haftmann@69184
    50
  "equivp (\<lambda>a b. cmp a b = Equiv)"
haftmann@69184
    51
  using reflp symp transp by (rule equivpI)
haftmann@69184
    52
haftmann@69184
    53
text \<open>The strict part\<close>
haftmann@69184
    54
haftmann@69184
    55
lemma irreflp_less:
haftmann@69184
    56
  "irreflp (\<lambda>a b. cmp a b = Less)"
haftmann@69184
    57
  by (rule irreflpI) simp
haftmann@69184
    58
haftmann@69184
    59
lemma irreflp_greater:
haftmann@69184
    60
  "irreflp (\<lambda>a b. cmp a b = Greater)"
haftmann@69184
    61
  by (rule irreflpI) simp
haftmann@69184
    62
haftmann@69184
    63
lemma asym_less:
haftmann@69184
    64
  "cmp b a \<noteq> Less" if "cmp a b = Less"
haftmann@69184
    65
  using that greater_iff_sym_less by force
haftmann@69184
    66
haftmann@69184
    67
lemma asym_greater:
haftmann@69184
    68
  "cmp b a \<noteq> Greater" if "cmp a b = Greater"
haftmann@69184
    69
  using that greater_iff_sym_less by force
haftmann@69184
    70
haftmann@69184
    71
lemma asymp_less:
haftmann@69184
    72
  "asymp (\<lambda>a b. cmp a b = Less)"
haftmann@69184
    73
  using irreflp_less by (auto intro: asympI dest: asym_less)
haftmann@69184
    74
haftmann@69184
    75
lemma asymp_greater:
haftmann@69184
    76
  "asymp (\<lambda>a b. cmp a b = Greater)"
haftmann@69184
    77
  using irreflp_greater by (auto intro!: asympI dest: asym_greater)
haftmann@69184
    78
haftmann@69246
    79
lemma trans_equiv_less:
haftmann@69246
    80
  "cmp a c = Less" if "cmp a b = Equiv" and "cmp b c = Less"
haftmann@69246
    81
  using that
haftmann@69246
    82
  by (metis (full_types) comp.exhaust greater_iff_sym_less trans_equiv trans_less)
haftmann@69246
    83
haftmann@69246
    84
lemma trans_less_equiv:
haftmann@69246
    85
  "cmp a c = Less" if "cmp a b = Less" and "cmp b c = Equiv"
haftmann@69246
    86
  using that
haftmann@69246
    87
  by (metis (full_types) comp.exhaust greater_iff_sym_less trans_equiv trans_less)
haftmann@69246
    88
haftmann@69246
    89
lemma trans_equiv_greater:
haftmann@69246
    90
  "cmp a c = Greater" if "cmp a b = Equiv" and "cmp b c = Greater"
haftmann@69246
    91
  using that by (simp add: sym [of a b] greater_iff_sym_less trans_less_equiv)
haftmann@69246
    92
haftmann@69246
    93
lemma trans_greater_equiv:
haftmann@69246
    94
  "cmp a c = Greater" if "cmp a b = Greater" and "cmp b c = Equiv"
haftmann@69246
    95
  using that by (simp add: sym [of b c] greater_iff_sym_less trans_equiv_less)
haftmann@69246
    96
haftmann@69184
    97
lemma transp_less:
haftmann@69184
    98
  "transp (\<lambda>a b. cmp a b = Less)"
haftmann@69184
    99
  by (rule transpI) (fact trans_less)
haftmann@69184
   100
haftmann@69184
   101
lemma transp_greater:
haftmann@69184
   102
  "transp (\<lambda>a b. cmp a b = Greater)"
haftmann@69184
   103
  by (rule transpI) (fact trans_greater)
haftmann@69184
   104
haftmann@69184
   105
text \<open>The reflexive part\<close>
haftmann@69184
   106
haftmann@69184
   107
lemma reflp_not_less:
haftmann@69184
   108
  "reflp (\<lambda>a b. cmp a b \<noteq> Less)"
haftmann@69184
   109
  by (rule reflpI) simp
haftmann@69184
   110
haftmann@69184
   111
lemma reflp_not_greater:
haftmann@69184
   112
  "reflp (\<lambda>a b. cmp a b \<noteq> Greater)"
haftmann@69184
   113
  by (rule reflpI) simp
haftmann@69184
   114
haftmann@69184
   115
lemma quasisym_not_less:
haftmann@69184
   116
  "cmp a b = Equiv" if "cmp a b \<noteq> Less" and "cmp b a \<noteq> Less"
haftmann@69184
   117
  using that comp.exhaust greater_iff_sym_less by auto
haftmann@69184
   118
haftmann@69184
   119
lemma quasisym_not_greater:
haftmann@69184
   120
  "cmp a b = Equiv" if "cmp a b \<noteq> Greater" and "cmp b a \<noteq> Greater"
haftmann@69184
   121
  using that comp.exhaust greater_iff_sym_less by auto
haftmann@69184
   122
haftmann@69184
   123
lemma trans_not_less:
haftmann@69184
   124
  "cmp a c \<noteq> Less" if "cmp a b \<noteq> Less" "cmp b c \<noteq> Less"
haftmann@69184
   125
  using that by (metis comp.exhaust greater_iff_sym_less trans_equiv trans_less)
haftmann@69184
   126
haftmann@69184
   127
lemma trans_not_greater:
haftmann@69184
   128
  "cmp a c \<noteq> Greater" if "cmp a b \<noteq> Greater" "cmp b c \<noteq> Greater"
haftmann@69184
   129
  using that greater_iff_sym_less trans_not_less by blast
haftmann@69184
   130
haftmann@69184
   131
lemma transp_not_less:
haftmann@69184
   132
  "transp (\<lambda>a b. cmp a b \<noteq> Less)"
haftmann@69184
   133
  by (rule transpI) (fact trans_not_less)
haftmann@69184
   134
haftmann@69184
   135
lemma transp_not_greater:
haftmann@69184
   136
  "transp (\<lambda>a b. cmp a b \<noteq> Greater)"
haftmann@69184
   137
  by (rule transpI) (fact trans_not_greater)
haftmann@69184
   138
haftmann@69246
   139
text \<open>Substitution under equivalences\<close>
haftmann@69246
   140
haftmann@69246
   141
lemma equiv_subst_left:
haftmann@69246
   142
  "cmp z y = comp \<longleftrightarrow> cmp x y = comp" if "cmp z x = Equiv" for comp
haftmann@69246
   143
proof -
haftmann@69246
   144
  from that have "cmp x z = Equiv"
haftmann@69246
   145
    by (simp add: sym)
haftmann@69246
   146
  with that show ?thesis
haftmann@69246
   147
    by (cases comp) (auto intro: trans_equiv trans_equiv_less trans_equiv_greater)
haftmann@69246
   148
qed
haftmann@69246
   149
haftmann@69246
   150
lemma equiv_subst_right:
haftmann@69246
   151
  "cmp x z = comp \<longleftrightarrow> cmp x y = comp" if "cmp z y = Equiv" for comp
haftmann@69246
   152
proof -
haftmann@69246
   153
  from that have "cmp y z = Equiv"
haftmann@69246
   154
    by (simp add: sym)
haftmann@69246
   155
  with that show ?thesis
haftmann@69246
   156
    by (cases comp) (auto intro: trans_equiv trans_less_equiv trans_greater_equiv)
haftmann@69246
   157
qed
haftmann@69246
   158
haftmann@69184
   159
end
haftmann@69184
   160
haftmann@69184
   161
typedef 'a comparator = "{cmp :: 'a \<Rightarrow> 'a \<Rightarrow> comp. comparator cmp}"
haftmann@69184
   162
  morphisms compare Abs_comparator
haftmann@69184
   163
proof -
haftmann@69184
   164
  have "comparator (\<lambda>_ _. Equiv)"
haftmann@69184
   165
    by standard simp_all
haftmann@69184
   166
  then show ?thesis
haftmann@69184
   167
    by auto
haftmann@69184
   168
qed
haftmann@69184
   169
haftmann@69184
   170
setup_lifting type_definition_comparator
haftmann@69184
   171
haftmann@69184
   172
global_interpretation compare: comparator "compare cmp"
haftmann@69184
   173
  using compare [of cmp] by simp
haftmann@69184
   174
haftmann@69184
   175
lift_definition flat :: "'a comparator"
haftmann@69184
   176
  is "\<lambda>_ _. Equiv" by standard simp_all
haftmann@69184
   177
haftmann@69184
   178
instantiation comparator :: (linorder) default
haftmann@69184
   179
begin
haftmann@69184
   180
haftmann@69184
   181
lift_definition default_comparator :: "'a comparator"
haftmann@69184
   182
  is "\<lambda>x y. if x < y then Less else if x > y then Greater else Equiv"
haftmann@69184
   183
  by standard (auto split: if_splits)
haftmann@69184
   184
haftmann@69184
   185
instance ..
haftmann@69184
   186
haftmann@69184
   187
end
haftmann@69184
   188
haftmann@69184
   189
text \<open>A rudimentary quickcheck setup\<close>
haftmann@69184
   190
haftmann@69184
   191
instantiation comparator :: (enum) equal
haftmann@69184
   192
begin
haftmann@69184
   193
haftmann@69184
   194
lift_definition equal_comparator :: "'a comparator \<Rightarrow> 'a comparator \<Rightarrow> bool"
haftmann@69184
   195
  is "\<lambda>f g. \<forall>x \<in> set Enum.enum. f x = g x" .
haftmann@69184
   196
haftmann@69184
   197
instance
haftmann@69184
   198
  by (standard; transfer) (auto simp add: enum_UNIV)
haftmann@69184
   199
haftmann@69184
   200
end
haftmann@69184
   201
haftmann@69184
   202
lemma [code]:
haftmann@69184
   203
  "HOL.equal cmp1 cmp2 \<longleftrightarrow> Enum.enum_all (\<lambda>x. compare cmp1 x = compare cmp2 x)"
haftmann@69184
   204
  by transfer (simp add: enum_UNIV)
haftmann@69184
   205
haftmann@69184
   206
lemma [code nbe]:
haftmann@69184
   207
  "HOL.equal (cmp :: 'a::enum comparator) cmp \<longleftrightarrow> True"
haftmann@69184
   208
  by (fact equal_refl)
haftmann@69184
   209
haftmann@69184
   210
instantiation comparator :: ("{linorder, typerep}") full_exhaustive
haftmann@69184
   211
begin
haftmann@69184
   212
haftmann@69184
   213
definition full_exhaustive_comparator ::
haftmann@69184
   214
  "('a comparator \<times> (unit \<Rightarrow> term) \<Rightarrow> (bool \<times> term list) option)
haftmann@69184
   215
    \<Rightarrow> natural \<Rightarrow> (bool \<times> term list) option"
haftmann@69184
   216
  where "full_exhaustive_comparator f s =
haftmann@69184
   217
    Quickcheck_Exhaustive.orelse
haftmann@69184
   218
      (f (flat, (\<lambda>u. Code_Evaluation.Const (STR ''Comparator.flat'') TYPEREP('a comparator))))
haftmann@69184
   219
      (f (default, (\<lambda>u. Code_Evaluation.Const (STR ''HOL.default_class.default'') TYPEREP('a comparator))))"
haftmann@69184
   220
haftmann@69184
   221
instance ..
haftmann@69184
   222
haftmann@69184
   223
end
haftmann@69184
   224
haftmann@69194
   225
text \<open>Fundamental comparator combinators\<close>
haftmann@69194
   226
haftmann@69184
   227
lift_definition reversed :: "'a comparator \<Rightarrow> 'a comparator"
haftmann@69184
   228
  is "\<lambda>cmp a b. cmp b a"
haftmann@69184
   229
proof -
haftmann@69184
   230
  fix cmp :: "'a \<Rightarrow> 'a \<Rightarrow> comp"
haftmann@69184
   231
  assume "comparator cmp"
haftmann@69184
   232
  then interpret comparator cmp .
haftmann@69184
   233
  show "comparator (\<lambda>a b. cmp b a)"
haftmann@69184
   234
    by standard (auto intro: trans_equiv trans_less simp: greater_iff_sym_less)
haftmann@69184
   235
qed
haftmann@69184
   236
haftmann@69184
   237
lift_definition key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a comparator \<Rightarrow> 'b comparator"
haftmann@69184
   238
  is "\<lambda>f cmp a b. cmp (f a) (f b)"
haftmann@69184
   239
proof -
haftmann@69184
   240
  fix cmp :: "'a \<Rightarrow> 'a \<Rightarrow> comp" and f :: "'b \<Rightarrow> 'a"
haftmann@69184
   241
  assume "comparator cmp"
haftmann@69184
   242
  then interpret comparator cmp .
haftmann@69184
   243
  show "comparator (\<lambda>a b. cmp (f a) (f b))"
haftmann@69184
   244
    by standard (auto intro: trans_equiv trans_less simp: greater_iff_sym_less)
haftmann@69184
   245
qed
haftmann@69184
   246
haftmann@69184
   247
end