src/HOL/Extraction.thy
author huffman
Sat Jun 06 09:11:12 2009 -0700 (2009-06-06)
changeset 31488 5691ccb8d6b5
parent 30235 58d147683393
child 33704 6aeb8454efc1
permissions -rw-r--r--
generalize tendsto to class topological_space
berghofe@13403
     1
(*  Title:      HOL/Extraction.thy
berghofe@13403
     2
    ID:         $Id$
berghofe@13403
     3
    Author:     Stefan Berghofer, TU Muenchen
berghofe@13403
     4
*)
berghofe@13403
     5
berghofe@13403
     6
header {* Program extraction for HOL *}
berghofe@13403
     7
nipkow@15131
     8
theory Extraction
nipkow@30235
     9
imports Option
haftmann@16417
    10
uses "Tools/rewrite_hol_proof.ML"
nipkow@15131
    11
begin
berghofe@13403
    12
berghofe@13403
    13
subsection {* Setup *}
berghofe@13403
    14
wenzelm@16121
    15
setup {*
wenzelm@16121
    16
let
berghofe@13725
    17
fun realizes_set_proc (Const ("realizes", Type ("fun", [Type ("Null", []), _])) $ r $
berghofe@13725
    18
      (Const ("op :", _) $ x $ S)) = (case strip_comb S of
berghofe@29930
    19
        (Var (ixn, U), ts) => SOME (list_comb (Var (ixn, U), ts @ [x]))
berghofe@29930
    20
      | (Free (s, U), ts) => SOME (list_comb (Free (s, U), ts @ [x]))
skalberg@15531
    21
      | _ => NONE)
berghofe@13725
    22
  | realizes_set_proc (Const ("realizes", Type ("fun", [T, _])) $ r $
berghofe@13725
    23
      (Const ("op :", _) $ x $ S)) = (case strip_comb S of
berghofe@29930
    24
        (Var (ixn, U), ts) => SOME (list_comb (Var (ixn, T --> U), r :: ts @ [x]))
berghofe@29930
    25
      | (Free (s, U), ts) => SOME (list_comb (Free (s, T --> U), r :: ts @ [x]))
skalberg@15531
    26
      | _ => NONE)
skalberg@15531
    27
  | realizes_set_proc _ = NONE;
berghofe@13725
    28
wenzelm@16121
    29
in
wenzelm@18708
    30
  Extraction.add_types
berghofe@29930
    31
      [("bool", ([], NONE))] #>
wenzelm@18708
    32
  Extraction.set_preprocessor (fn thy =>
berghofe@13403
    33
      Proofterm.rewrite_proof_notypes
wenzelm@28797
    34
        ([], RewriteHOLProof.elim_cong :: ProofRewriteRules.rprocs true) o
wenzelm@17203
    35
      Proofterm.rewrite_proof thy
berghofe@13599
    36
        (RewriteHOLProof.rews, ProofRewriteRules.rprocs true) o
haftmann@27982
    37
      ProofRewriteRules.elim_vars (curry Const @{const_name default}))
wenzelm@16121
    38
end
berghofe@13403
    39
*}
berghofe@13403
    40
berghofe@13403
    41
lemmas [extraction_expand] =
berghofe@22281
    42
  meta_spec atomize_eq atomize_all atomize_imp atomize_conj
berghofe@13403
    43
  allE rev_mp conjE Eq_TrueI Eq_FalseI eqTrueI eqTrueE eq_cong2
haftmann@20941
    44
  notE' impE' impE iffE imp_cong simp_thms eq_True eq_False
wenzelm@18456
    45
  induct_forall_eq induct_implies_eq induct_equal_eq induct_conj_eq
wenzelm@18456
    46
  induct_forall_def induct_implies_def induct_equal_def induct_conj_def
wenzelm@18511
    47
  induct_atomize induct_rulify induct_rulify_fallback
berghofe@25424
    48
  True_implies_equals TrueE
berghofe@13403
    49
berghofe@13403
    50
datatype sumbool = Left | Right
berghofe@13403
    51
berghofe@13403
    52
subsection {* Type of extracted program *}
berghofe@13403
    53
berghofe@13403
    54
extract_type
berghofe@13403
    55
  "typeof (Trueprop P) \<equiv> typeof P"
berghofe@13403
    56
berghofe@13403
    57
  "typeof P \<equiv> Type (TYPE(Null)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow>
berghofe@13403
    58
     typeof (P \<longrightarrow> Q) \<equiv> Type (TYPE('Q))"
berghofe@13403
    59
berghofe@13403
    60
  "typeof Q \<equiv> Type (TYPE(Null)) \<Longrightarrow> typeof (P \<longrightarrow> Q) \<equiv> Type (TYPE(Null))"
berghofe@13403
    61
berghofe@13403
    62
  "typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow>
berghofe@13403
    63
     typeof (P \<longrightarrow> Q) \<equiv> Type (TYPE('P \<Rightarrow> 'Q))"
berghofe@13403
    64
berghofe@13403
    65
  "(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE(Null))) \<Longrightarrow>
berghofe@13403
    66
     typeof (\<forall>x. P x) \<equiv> Type (TYPE(Null))"
berghofe@13403
    67
berghofe@13403
    68
  "(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE('P))) \<Longrightarrow>
berghofe@13403
    69
     typeof (\<forall>x::'a. P x) \<equiv> Type (TYPE('a \<Rightarrow> 'P))"
berghofe@13403
    70
berghofe@13403
    71
  "(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE(Null))) \<Longrightarrow>
berghofe@13403
    72
     typeof (\<exists>x::'a. P x) \<equiv> Type (TYPE('a))"
berghofe@13403
    73
berghofe@13403
    74
  "(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE('P))) \<Longrightarrow>
berghofe@13403
    75
     typeof (\<exists>x::'a. P x) \<equiv> Type (TYPE('a \<times> 'P))"
berghofe@13403
    76
berghofe@13403
    77
  "typeof P \<equiv> Type (TYPE(Null)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE(Null)) \<Longrightarrow>
berghofe@13403
    78
     typeof (P \<or> Q) \<equiv> Type (TYPE(sumbool))"
berghofe@13403
    79
berghofe@13403
    80
  "typeof P \<equiv> Type (TYPE(Null)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow>
berghofe@13403
    81
     typeof (P \<or> Q) \<equiv> Type (TYPE('Q option))"
berghofe@13403
    82
berghofe@13403
    83
  "typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE(Null)) \<Longrightarrow>
berghofe@13403
    84
     typeof (P \<or> Q) \<equiv> Type (TYPE('P option))"
berghofe@13403
    85
berghofe@13403
    86
  "typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow>
berghofe@13403
    87
     typeof (P \<or> Q) \<equiv> Type (TYPE('P + 'Q))"
berghofe@13403
    88
berghofe@13403
    89
  "typeof P \<equiv> Type (TYPE(Null)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow>
berghofe@13403
    90
     typeof (P \<and> Q) \<equiv> Type (TYPE('Q))"
berghofe@13403
    91
berghofe@13403
    92
  "typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE(Null)) \<Longrightarrow>
berghofe@13403
    93
     typeof (P \<and> Q) \<equiv> Type (TYPE('P))"
berghofe@13403
    94
berghofe@13403
    95
  "typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow> typeof Q \<equiv> Type (TYPE('Q)) \<Longrightarrow>
berghofe@13403
    96
     typeof (P \<and> Q) \<equiv> Type (TYPE('P \<times> 'Q))"
berghofe@13403
    97
berghofe@13403
    98
  "typeof (P = Q) \<equiv> typeof ((P \<longrightarrow> Q) \<and> (Q \<longrightarrow> P))"
berghofe@13403
    99
berghofe@13403
   100
  "typeof (x \<in> P) \<equiv> typeof P"
berghofe@13403
   101
berghofe@13403
   102
subsection {* Realizability *}
berghofe@13403
   103
berghofe@13403
   104
realizability
berghofe@13403
   105
  "(realizes t (Trueprop P)) \<equiv> (Trueprop (realizes t P))"
berghofe@13403
   106
berghofe@13403
   107
  "(typeof P) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
berghofe@13403
   108
     (realizes t (P \<longrightarrow> Q)) \<equiv> (realizes Null P \<longrightarrow> realizes t Q)"
berghofe@13403
   109
berghofe@13403
   110
  "(typeof P) \<equiv> (Type (TYPE('P))) \<Longrightarrow>
berghofe@13403
   111
   (typeof Q) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
berghofe@13403
   112
     (realizes t (P \<longrightarrow> Q)) \<equiv> (\<forall>x::'P. realizes x P \<longrightarrow> realizes Null Q)"
berghofe@13403
   113
berghofe@13403
   114
  "(realizes t (P \<longrightarrow> Q)) \<equiv> (\<forall>x. realizes x P \<longrightarrow> realizes (t x) Q)"
berghofe@13403
   115
berghofe@13403
   116
  "(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE(Null))) \<Longrightarrow>
berghofe@13403
   117
     (realizes t (\<forall>x. P x)) \<equiv> (\<forall>x. realizes Null (P x))"
berghofe@13403
   118
berghofe@13403
   119
  "(realizes t (\<forall>x. P x)) \<equiv> (\<forall>x. realizes (t x) (P x))"
berghofe@13403
   120
berghofe@13403
   121
  "(\<lambda>x. typeof (P x)) \<equiv> (\<lambda>x. Type (TYPE(Null))) \<Longrightarrow>
berghofe@13403
   122
     (realizes t (\<exists>x. P x)) \<equiv> (realizes Null (P t))"
berghofe@13403
   123
berghofe@13403
   124
  "(realizes t (\<exists>x. P x)) \<equiv> (realizes (snd t) (P (fst t)))"
berghofe@13403
   125
berghofe@13403
   126
  "(typeof P) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
berghofe@13403
   127
   (typeof Q) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
berghofe@13403
   128
     (realizes t (P \<or> Q)) \<equiv>
berghofe@13403
   129
     (case t of Left \<Rightarrow> realizes Null P | Right \<Rightarrow> realizes Null Q)"
berghofe@13403
   130
berghofe@13403
   131
  "(typeof P) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
berghofe@13403
   132
     (realizes t (P \<or> Q)) \<equiv>
berghofe@13403
   133
     (case t of None \<Rightarrow> realizes Null P | Some q \<Rightarrow> realizes q Q)"
berghofe@13403
   134
berghofe@13403
   135
  "(typeof Q) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
berghofe@13403
   136
     (realizes t (P \<or> Q)) \<equiv>
berghofe@13403
   137
     (case t of None \<Rightarrow> realizes Null Q | Some p \<Rightarrow> realizes p P)"
berghofe@13403
   138
berghofe@13403
   139
  "(realizes t (P \<or> Q)) \<equiv>
berghofe@13403
   140
   (case t of Inl p \<Rightarrow> realizes p P | Inr q \<Rightarrow> realizes q Q)"
berghofe@13403
   141
berghofe@13403
   142
  "(typeof P) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
berghofe@13403
   143
     (realizes t (P \<and> Q)) \<equiv> (realizes Null P \<and> realizes t Q)"
berghofe@13403
   144
berghofe@13403
   145
  "(typeof Q) \<equiv> (Type (TYPE(Null))) \<Longrightarrow>
berghofe@13403
   146
     (realizes t (P \<and> Q)) \<equiv> (realizes t P \<and> realizes Null Q)"
berghofe@13403
   147
berghofe@13403
   148
  "(realizes t (P \<and> Q)) \<equiv> (realizes (fst t) P \<and> realizes (snd t) Q)"
berghofe@13403
   149
berghofe@13403
   150
  "typeof P \<equiv> Type (TYPE(Null)) \<Longrightarrow>
berghofe@13403
   151
     realizes t (\<not> P) \<equiv> \<not> realizes Null P"
berghofe@13403
   152
berghofe@13403
   153
  "typeof P \<equiv> Type (TYPE('P)) \<Longrightarrow>
berghofe@13403
   154
     realizes t (\<not> P) \<equiv> (\<forall>x::'P. \<not> realizes x P)"
berghofe@13403
   155
berghofe@13403
   156
  "typeof (P::bool) \<equiv> Type (TYPE(Null)) \<Longrightarrow>
berghofe@13403
   157
   typeof Q \<equiv> Type (TYPE(Null)) \<Longrightarrow>
berghofe@13403
   158
     realizes t (P = Q) \<equiv> realizes Null P = realizes Null Q"
berghofe@13403
   159
berghofe@13403
   160
  "(realizes t (P = Q)) \<equiv> (realizes t ((P \<longrightarrow> Q) \<and> (Q \<longrightarrow> P)))"
berghofe@13403
   161
berghofe@13403
   162
subsection {* Computational content of basic inference rules *}
berghofe@13403
   163
berghofe@13403
   164
theorem disjE_realizer:
berghofe@13403
   165
  assumes r: "case x of Inl p \<Rightarrow> P p | Inr q \<Rightarrow> Q q"
berghofe@13403
   166
  and r1: "\<And>p. P p \<Longrightarrow> R (f p)" and r2: "\<And>q. Q q \<Longrightarrow> R (g q)"
berghofe@13403
   167
  shows "R (case x of Inl p \<Rightarrow> f p | Inr q \<Rightarrow> g q)"
berghofe@13403
   168
proof (cases x)
berghofe@13403
   169
  case Inl
berghofe@13403
   170
  with r show ?thesis by simp (rule r1)
berghofe@13403
   171
next
berghofe@13403
   172
  case Inr
berghofe@13403
   173
  with r show ?thesis by simp (rule r2)
berghofe@13403
   174
qed
berghofe@13403
   175
berghofe@13403
   176
theorem disjE_realizer2:
berghofe@13403
   177
  assumes r: "case x of None \<Rightarrow> P | Some q \<Rightarrow> Q q"
berghofe@13403
   178
  and r1: "P \<Longrightarrow> R f" and r2: "\<And>q. Q q \<Longrightarrow> R (g q)"
berghofe@13403
   179
  shows "R (case x of None \<Rightarrow> f | Some q \<Rightarrow> g q)"
berghofe@13403
   180
proof (cases x)
berghofe@13403
   181
  case None
berghofe@13403
   182
  with r show ?thesis by simp (rule r1)
berghofe@13403
   183
next
berghofe@13403
   184
  case Some
berghofe@13403
   185
  with r show ?thesis by simp (rule r2)
berghofe@13403
   186
qed
berghofe@13403
   187
berghofe@13403
   188
theorem disjE_realizer3:
berghofe@13403
   189
  assumes r: "case x of Left \<Rightarrow> P | Right \<Rightarrow> Q"
berghofe@13403
   190
  and r1: "P \<Longrightarrow> R f" and r2: "Q \<Longrightarrow> R g"
berghofe@13403
   191
  shows "R (case x of Left \<Rightarrow> f | Right \<Rightarrow> g)"
berghofe@13403
   192
proof (cases x)
berghofe@13403
   193
  case Left
berghofe@13403
   194
  with r show ?thesis by simp (rule r1)
berghofe@13403
   195
next
berghofe@13403
   196
  case Right
berghofe@13403
   197
  with r show ?thesis by simp (rule r2)
berghofe@13403
   198
qed
berghofe@13403
   199
berghofe@13403
   200
theorem conjI_realizer:
berghofe@13403
   201
  "P p \<Longrightarrow> Q q \<Longrightarrow> P (fst (p, q)) \<and> Q (snd (p, q))"
berghofe@13403
   202
  by simp
berghofe@13403
   203
berghofe@13403
   204
theorem exI_realizer:
berghofe@13918
   205
  "P y x \<Longrightarrow> P (snd (x, y)) (fst (x, y))" by simp
berghofe@13918
   206
berghofe@13918
   207
theorem exE_realizer: "P (snd p) (fst p) \<Longrightarrow>
berghofe@15393
   208
  (\<And>x y. P y x \<Longrightarrow> Q (f x y)) \<Longrightarrow> Q (let (x, y) = p in f x y)"
berghofe@15393
   209
  by (cases p) (simp add: Let_def)
berghofe@13918
   210
berghofe@13918
   211
theorem exE_realizer': "P (snd p) (fst p) \<Longrightarrow>
berghofe@13918
   212
  (\<And>x y. P y x \<Longrightarrow> Q) \<Longrightarrow> Q" by (cases p) simp
berghofe@13403
   213
haftmann@27982
   214
setup {*
haftmann@27982
   215
  Sign.add_const_constraint (@{const_name "default"}, SOME @{typ "'a::type"})
haftmann@27982
   216
*}
haftmann@27982
   217
berghofe@13403
   218
realizers
berghofe@13725
   219
  impI (P, Q): "\<lambda>pq. pq"
skalberg@14168
   220
    "\<Lambda> P Q pq (h: _). allI \<cdot> _ \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h \<cdot> x))"
berghofe@13403
   221
berghofe@13403
   222
  impI (P): "Null"
skalberg@14168
   223
    "\<Lambda> P Q (h: _). allI \<cdot> _ \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h \<cdot> x))"
berghofe@13403
   224
skalberg@14168
   225
  impI (Q): "\<lambda>q. q" "\<Lambda> P Q q. impI \<cdot> _ \<cdot> _"
berghofe@13403
   226
berghofe@13725
   227
  impI: "Null" "impI"
berghofe@13403
   228
berghofe@13725
   229
  mp (P, Q): "\<lambda>pq. pq"
skalberg@14168
   230
    "\<Lambda> P Q pq (h: _) p. mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> h)"
berghofe@13403
   231
berghofe@13403
   232
  mp (P): "Null"
skalberg@14168
   233
    "\<Lambda> P Q (h: _) p. mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> h)"
berghofe@13403
   234
skalberg@14168
   235
  mp (Q): "\<lambda>q. q" "\<Lambda> P Q q. mp \<cdot> _ \<cdot> _"
berghofe@13403
   236
berghofe@13725
   237
  mp: "Null" "mp"
berghofe@13403
   238
skalberg@14168
   239
  allI (P): "\<lambda>p. p" "\<Lambda> P p. allI \<cdot> _"
berghofe@13403
   240
berghofe@13725
   241
  allI: "Null" "allI"
berghofe@13403
   242
skalberg@14168
   243
  spec (P): "\<lambda>x p. p x" "\<Lambda> P x p. spec \<cdot> _ \<cdot> x"
berghofe@13403
   244
berghofe@13725
   245
  spec: "Null" "spec"
berghofe@13403
   246
skalberg@14168
   247
  exI (P): "\<lambda>x p. (x, p)" "\<Lambda> P x p. exI_realizer \<cdot> P \<cdot> p \<cdot> x"
berghofe@13403
   248
skalberg@14168
   249
  exI: "\<lambda>x. x" "\<Lambda> P x (h: _). h"
berghofe@13403
   250
berghofe@15393
   251
  exE (P, Q): "\<lambda>p pq. let (x, y) = p in pq x y"
skalberg@14168
   252
    "\<Lambda> P Q p (h: _) pq. exE_realizer \<cdot> P \<cdot> p \<cdot> Q \<cdot> pq \<bullet> h"
berghofe@13403
   253
berghofe@13403
   254
  exE (P): "Null"
skalberg@14168
   255
    "\<Lambda> P Q p. exE_realizer' \<cdot> _ \<cdot> _ \<cdot> _"
berghofe@13403
   256
berghofe@13725
   257
  exE (Q): "\<lambda>x pq. pq x"
skalberg@14168
   258
    "\<Lambda> P Q x (h1: _) pq (h2: _). h2 \<cdot> x \<bullet> h1"
berghofe@13403
   259
berghofe@13403
   260
  exE: "Null"
skalberg@14168
   261
    "\<Lambda> P Q x (h1: _) (h2: _). h2 \<cdot> x \<bullet> h1"
berghofe@13403
   262
berghofe@13725
   263
  conjI (P, Q): "Pair"
skalberg@14168
   264
    "\<Lambda> P Q p (h: _) q. conjI_realizer \<cdot> P \<cdot> p \<cdot> Q \<cdot> q \<bullet> h"
berghofe@13403
   265
berghofe@13725
   266
  conjI (P): "\<lambda>p. p"
skalberg@14168
   267
    "\<Lambda> P Q p. conjI \<cdot> _ \<cdot> _"
berghofe@13403
   268
berghofe@13725
   269
  conjI (Q): "\<lambda>q. q"
skalberg@14168
   270
    "\<Lambda> P Q (h: _) q. conjI \<cdot> _ \<cdot> _ \<bullet> h"
berghofe@13403
   271
berghofe@13725
   272
  conjI: "Null" "conjI"
berghofe@13403
   273
berghofe@13725
   274
  conjunct1 (P, Q): "fst"
skalberg@14168
   275
    "\<Lambda> P Q pq. conjunct1 \<cdot> _ \<cdot> _"
berghofe@13403
   276
berghofe@13725
   277
  conjunct1 (P): "\<lambda>p. p"
skalberg@14168
   278
    "\<Lambda> P Q p. conjunct1 \<cdot> _ \<cdot> _"
berghofe@13403
   279
berghofe@13403
   280
  conjunct1 (Q): "Null"
skalberg@14168
   281
    "\<Lambda> P Q q. conjunct1 \<cdot> _ \<cdot> _"
berghofe@13403
   282
berghofe@13725
   283
  conjunct1: "Null" "conjunct1"
berghofe@13403
   284
berghofe@13725
   285
  conjunct2 (P, Q): "snd"
skalberg@14168
   286
    "\<Lambda> P Q pq. conjunct2 \<cdot> _ \<cdot> _"
berghofe@13403
   287
berghofe@13403
   288
  conjunct2 (P): "Null"
skalberg@14168
   289
    "\<Lambda> P Q p. conjunct2 \<cdot> _ \<cdot> _"
berghofe@13403
   290
berghofe@13725
   291
  conjunct2 (Q): "\<lambda>p. p"
skalberg@14168
   292
    "\<Lambda> P Q p. conjunct2 \<cdot> _ \<cdot> _"
berghofe@13403
   293
berghofe@13725
   294
  conjunct2: "Null" "conjunct2"
berghofe@13725
   295
berghofe@13725
   296
  disjI1 (P, Q): "Inl"
skalberg@14168
   297
    "\<Lambda> P Q p. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sum.cases_1 \<cdot> P \<cdot> _ \<cdot> p)"
berghofe@13403
   298
berghofe@13725
   299
  disjI1 (P): "Some"
skalberg@14168
   300
    "\<Lambda> P Q p. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.cases_2 \<cdot> _ \<cdot> P \<cdot> p)"
berghofe@13403
   301
berghofe@13725
   302
  disjI1 (Q): "None"
skalberg@14168
   303
    "\<Lambda> P Q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.cases_1 \<cdot> _ \<cdot> _)"
berghofe@13403
   304
berghofe@13725
   305
  disjI1: "Left"
skalberg@14168
   306
    "\<Lambda> P Q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sumbool.cases_1 \<cdot> _ \<cdot> _)"
berghofe@13403
   307
berghofe@13725
   308
  disjI2 (P, Q): "Inr"
skalberg@14168
   309
    "\<Lambda> Q P q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sum.cases_2 \<cdot> _ \<cdot> Q \<cdot> q)"
berghofe@13403
   310
berghofe@13725
   311
  disjI2 (P): "None"
skalberg@14168
   312
    "\<Lambda> Q P. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.cases_1 \<cdot> _ \<cdot> _)"
berghofe@13403
   313
berghofe@13725
   314
  disjI2 (Q): "Some"
skalberg@14168
   315
    "\<Lambda> Q P q. iffD2 \<cdot> _ \<cdot> _ \<bullet> (option.cases_2 \<cdot> _ \<cdot> Q \<cdot> q)"
berghofe@13403
   316
berghofe@13725
   317
  disjI2: "Right"
skalberg@14168
   318
    "\<Lambda> Q P. iffD2 \<cdot> _ \<cdot> _ \<bullet> (sumbool.cases_2 \<cdot> _ \<cdot> _)"
berghofe@13403
   319
berghofe@13725
   320
  disjE (P, Q, R): "\<lambda>pq pr qr.
berghofe@13403
   321
     (case pq of Inl p \<Rightarrow> pr p | Inr q \<Rightarrow> qr q)"
skalberg@14168
   322
    "\<Lambda> P Q R pq (h1: _) pr (h2: _) qr.
berghofe@13725
   323
       disjE_realizer \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> pr \<cdot> qr \<bullet> h1 \<bullet> h2"
berghofe@13403
   324
berghofe@13725
   325
  disjE (Q, R): "\<lambda>pq pr qr.
berghofe@13403
   326
     (case pq of None \<Rightarrow> pr | Some q \<Rightarrow> qr q)"
skalberg@14168
   327
    "\<Lambda> P Q R pq (h1: _) pr (h2: _) qr.
berghofe@13725
   328
       disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> pr \<cdot> qr \<bullet> h1 \<bullet> h2"
berghofe@13403
   329
berghofe@13725
   330
  disjE (P, R): "\<lambda>pq pr qr.
berghofe@13403
   331
     (case pq of None \<Rightarrow> qr | Some p \<Rightarrow> pr p)"
skalberg@14168
   332
    "\<Lambda> P Q R pq (h1: _) pr (h2: _) qr (h3: _).
berghofe@13725
   333
       disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> qr \<cdot> pr \<bullet> h1 \<bullet> h3 \<bullet> h2"
berghofe@13403
   334
berghofe@13725
   335
  disjE (R): "\<lambda>pq pr qr.
berghofe@13403
   336
     (case pq of Left \<Rightarrow> pr | Right \<Rightarrow> qr)"
skalberg@14168
   337
    "\<Lambda> P Q R pq (h1: _) pr (h2: _) qr.
berghofe@13725
   338
       disjE_realizer3 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> R \<cdot> pr \<cdot> qr \<bullet> h1 \<bullet> h2"
berghofe@13403
   339
berghofe@13403
   340
  disjE (P, Q): "Null"
skalberg@14168
   341
    "\<Lambda> P Q R pq. disjE_realizer \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>x. R) \<cdot> _ \<cdot> _"
berghofe@13403
   342
berghofe@13403
   343
  disjE (Q): "Null"
skalberg@14168
   344
    "\<Lambda> P Q R pq. disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>x. R) \<cdot> _ \<cdot> _"
berghofe@13403
   345
berghofe@13403
   346
  disjE (P): "Null"
skalberg@14168
   347
    "\<Lambda> P Q R pq (h1: _) (h2: _) (h3: _).
berghofe@13725
   348
       disjE_realizer2 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>x. R) \<cdot> _ \<cdot> _ \<bullet> h1 \<bullet> h3 \<bullet> h2"
berghofe@13403
   349
berghofe@13403
   350
  disjE: "Null"
skalberg@14168
   351
    "\<Lambda> P Q R pq. disjE_realizer3 \<cdot> _ \<cdot> _ \<cdot> pq \<cdot> (\<lambda>x. R) \<cdot> _ \<cdot> _"
berghofe@13403
   352
haftmann@27982
   353
  FalseE (P): "default"
skalberg@14168
   354
    "\<Lambda> P. FalseE \<cdot> _"
berghofe@13403
   355
berghofe@13725
   356
  FalseE: "Null" "FalseE"
berghofe@13403
   357
berghofe@13403
   358
  notI (P): "Null"
skalberg@14168
   359
    "\<Lambda> P (h: _). allI \<cdot> _ \<bullet> (\<Lambda> x. notI \<cdot> _ \<bullet> (h \<cdot> x))"
berghofe@13403
   360
berghofe@13725
   361
  notI: "Null" "notI"
berghofe@13403
   362
haftmann@27982
   363
  notE (P, R): "\<lambda>p. default"
skalberg@14168
   364
    "\<Lambda> P R (h: _) p. notE \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> h)"
berghofe@13403
   365
berghofe@13403
   366
  notE (P): "Null"
skalberg@14168
   367
    "\<Lambda> P R (h: _) p. notE \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> h)"
berghofe@13403
   368
haftmann@27982
   369
  notE (R): "default"
skalberg@14168
   370
    "\<Lambda> P R. notE \<cdot> _ \<cdot> _"
berghofe@13403
   371
berghofe@13725
   372
  notE: "Null" "notE"
berghofe@13403
   373
berghofe@13725
   374
  subst (P): "\<lambda>s t ps. ps"
skalberg@14168
   375
    "\<Lambda> s t P (h: _) ps. subst \<cdot> s \<cdot> t \<cdot> P ps \<bullet> h"
berghofe@13403
   376
berghofe@13725
   377
  subst: "Null" "subst"
berghofe@13725
   378
berghofe@13725
   379
  iffD1 (P, Q): "fst"
skalberg@14168
   380
    "\<Lambda> Q P pq (h: _) p.
berghofe@13403
   381
       mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> p \<bullet> (conjunct1 \<cdot> _ \<cdot> _ \<bullet> h))"
berghofe@13403
   382
berghofe@13725
   383
  iffD1 (P): "\<lambda>p. p"
skalberg@14168
   384
    "\<Lambda> Q P p (h: _). mp \<cdot> _ \<cdot> _ \<bullet> (conjunct1 \<cdot> _ \<cdot> _ \<bullet> h)"
berghofe@13403
   385
berghofe@13403
   386
  iffD1 (Q): "Null"
skalberg@14168
   387
    "\<Lambda> Q P q1 (h: _) q2.
berghofe@13403
   388
       mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> q2 \<bullet> (conjunct1 \<cdot> _ \<cdot> _ \<bullet> h))"
berghofe@13403
   389
berghofe@13725
   390
  iffD1: "Null" "iffD1"
berghofe@13403
   391
berghofe@13725
   392
  iffD2 (P, Q): "snd"
skalberg@14168
   393
    "\<Lambda> P Q pq (h: _) q.
berghofe@13403
   394
       mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> q \<bullet> (conjunct2 \<cdot> _ \<cdot> _ \<bullet> h))"
berghofe@13403
   395
berghofe@13725
   396
  iffD2 (P): "\<lambda>p. p"
skalberg@14168
   397
    "\<Lambda> P Q p (h: _). mp \<cdot> _ \<cdot> _ \<bullet> (conjunct2 \<cdot> _ \<cdot> _ \<bullet> h)"
berghofe@13403
   398
berghofe@13403
   399
  iffD2 (Q): "Null"
skalberg@14168
   400
    "\<Lambda> P Q q1 (h: _) q2.
berghofe@13403
   401
       mp \<cdot> _ \<cdot> _ \<bullet> (spec \<cdot> _ \<cdot> q2 \<bullet> (conjunct2 \<cdot> _ \<cdot> _ \<bullet> h))"
berghofe@13403
   402
berghofe@13725
   403
  iffD2: "Null" "iffD2"
berghofe@13403
   404
berghofe@13725
   405
  iffI (P, Q): "Pair"
skalberg@14168
   406
    "\<Lambda> P Q pq (h1 : _) qp (h2 : _). conjI_realizer \<cdot>
berghofe@13725
   407
       (\<lambda>pq. \<forall>x. P x \<longrightarrow> Q (pq x)) \<cdot> pq \<cdot>
berghofe@13725
   408
       (\<lambda>qp. \<forall>x. Q x \<longrightarrow> P (qp x)) \<cdot> qp \<bullet>
skalberg@14168
   409
       (allI \<cdot> _ \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h1 \<cdot> x))) \<bullet>
skalberg@14168
   410
       (allI \<cdot> _ \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h2 \<cdot> x)))"
berghofe@13403
   411
berghofe@13725
   412
  iffI (P): "\<lambda>p. p"
skalberg@14168
   413
    "\<Lambda> P Q (h1 : _) p (h2 : _). conjI \<cdot> _ \<cdot> _ \<bullet>
skalberg@14168
   414
       (allI \<cdot> _ \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h1 \<cdot> x))) \<bullet>
berghofe@13403
   415
       (impI \<cdot> _ \<cdot> _ \<bullet> h2)"
berghofe@13403
   416
berghofe@13725
   417
  iffI (Q): "\<lambda>q. q"
skalberg@14168
   418
    "\<Lambda> P Q q (h1 : _) (h2 : _). conjI \<cdot> _ \<cdot> _ \<bullet>
berghofe@13403
   419
       (impI \<cdot> _ \<cdot> _ \<bullet> h1) \<bullet>
skalberg@14168
   420
       (allI \<cdot> _ \<bullet> (\<Lambda> x. impI \<cdot> _ \<cdot> _ \<bullet> (h2 \<cdot> x)))"
berghofe@13403
   421
berghofe@13725
   422
  iffI: "Null" "iffI"
berghofe@13403
   423
berghofe@13725
   424
(*
berghofe@13403
   425
  classical: "Null"
skalberg@14168
   426
    "\<Lambda> P. classical \<cdot> _"
berghofe@13725
   427
*)
berghofe@13403
   428
haftmann@27982
   429
setup {*
haftmann@27982
   430
  Sign.add_const_constraint (@{const_name "default"}, SOME @{typ "'a::default"})
haftmann@27982
   431
*}
haftmann@27982
   432
berghofe@13403
   433
end