src/HOL/Metis_Examples/Abstraction.thy
author hoelzl
Tue Mar 23 16:17:41 2010 +0100 (2010-03-23)
changeset 35928 d31f55f97663
parent 33027 9cf389429f6d
child 36566 f91342f218a9
permissions -rw-r--r--
Generate image for HOL-Probability
wenzelm@33027
     1
(*  Title:      HOL/Metis_Examples/Abstraction.thy
paulson@23449
     2
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
paulson@23449
     3
wenzelm@33027
     4
Testing the metis method.
paulson@23449
     5
*)
paulson@23449
     6
haftmann@27368
     7
theory Abstraction
haftmann@27368
     8
imports Main FuncSet
paulson@23449
     9
begin
paulson@23449
    10
paulson@23449
    11
(*For Christoph Benzmueller*)
paulson@23449
    12
lemma "x<1 & ((op=) = (op=)) ==> ((op=) = (op=)) & (x<(2::nat))";
paulson@23449
    13
  by (metis One_nat_def less_Suc0 not_less0 not_less_eq numeral_2_eq_2)
paulson@23449
    14
paulson@23449
    15
(*this is a theorem, but we can't prove it unless ext is applied explicitly
paulson@23449
    16
lemma "(op=) = (%x y. y=x)"
paulson@23449
    17
*)
paulson@23449
    18
paulson@23449
    19
consts
paulson@23449
    20
  monotone :: "['a => 'a, 'a set, ('a *'a)set] => bool"
paulson@23449
    21
  pset  :: "'a set => 'a set"
paulson@23449
    22
  order :: "'a set => ('a * 'a) set"
paulson@23449
    23
boehmes@32864
    24
declare [[ atp_problem_prefix = "Abstraction__Collect_triv" ]]
paulson@23449
    25
lemma (*Collect_triv:*) "a \<in> {x. P x} ==> P a"
paulson@23449
    26
proof (neg_clausify)
paulson@23449
    27
assume 0: "(a\<Colon>'a\<Colon>type) \<in> Collect (P\<Colon>'a\<Colon>type \<Rightarrow> bool)"
paulson@23449
    28
assume 1: "\<not> (P\<Colon>'a\<Colon>type \<Rightarrow> bool) (a\<Colon>'a\<Colon>type)"
paulson@23449
    29
have 2: "(P\<Colon>'a\<Colon>type \<Rightarrow> bool) (a\<Colon>'a\<Colon>type)"
paulson@23449
    30
  by (metis CollectD 0)
paulson@23449
    31
show "False"
paulson@23449
    32
  by (metis 2 1)
paulson@23449
    33
qed
paulson@23449
    34
paulson@23449
    35
lemma Collect_triv: "a \<in> {x. P x} ==> P a"
berghofe@23756
    36
by (metis mem_Collect_eq)
paulson@23449
    37
paulson@23449
    38
boehmes@32864
    39
declare [[ atp_problem_prefix = "Abstraction__Collect_mp" ]]
paulson@23449
    40
lemma "a \<in> {x. P x --> Q x} ==> a \<in> {x. P x} ==> a \<in> {x. Q x}"
berghofe@23756
    41
  by (metis CollectI Collect_imp_eq ComplD UnE mem_Collect_eq);
paulson@23449
    42
  --{*34 secs*}
paulson@23449
    43
boehmes@32864
    44
declare [[ atp_problem_prefix = "Abstraction__Sigma_triv" ]]
paulson@23449
    45
lemma "(a,b) \<in> Sigma A B ==> a \<in> A & b \<in> B a"
paulson@23449
    46
proof (neg_clausify)
paulson@23449
    47
assume 0: "(a\<Colon>'a\<Colon>type, b\<Colon>'b\<Colon>type) \<in> Sigma (A\<Colon>'a\<Colon>type set) (B\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>type set)"
paulson@23449
    48
assume 1: "(a\<Colon>'a\<Colon>type) \<notin> (A\<Colon>'a\<Colon>type set) \<or> (b\<Colon>'b\<Colon>type) \<notin> (B\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>type set) a"
paulson@23449
    49
have 2: "(a\<Colon>'a\<Colon>type) \<in> (A\<Colon>'a\<Colon>type set)"
paulson@23449
    50
  by (metis SigmaD1 0)
paulson@23449
    51
have 3: "(b\<Colon>'b\<Colon>type) \<in> (B\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>type set) (a\<Colon>'a\<Colon>type)"
paulson@23449
    52
  by (metis SigmaD2 0)
paulson@23449
    53
have 4: "(b\<Colon>'b\<Colon>type) \<notin> (B\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>type set) (a\<Colon>'a\<Colon>type)"
paulson@23449
    54
  by (metis 1 2)
paulson@23449
    55
show "False"
paulson@23449
    56
  by (metis 3 4)
paulson@23449
    57
qed
paulson@23449
    58
paulson@23449
    59
lemma Sigma_triv: "(a,b) \<in> Sigma A B ==> a \<in> A & b \<in> B a"
paulson@23449
    60
by (metis SigmaD1 SigmaD2)
paulson@23449
    61
boehmes@32864
    62
declare [[ atp_problem_prefix = "Abstraction__Sigma_Collect" ]]
paulson@23449
    63
lemma "(a,b) \<in> (SIGMA x: A. {y. x = f y}) ==> a \<in> A & a = f b"
paulson@29676
    64
(*???metis says this is satisfiable!
paulson@29676
    65
by (metis CollectD SigmaD1 SigmaD2)
paulson@29676
    66
*)
paulson@23449
    67
by (meson CollectD SigmaD1 SigmaD2)
paulson@23449
    68
paulson@23449
    69
paulson@24827
    70
(*single-step*)
paulson@24827
    71
lemma "(a,b) \<in> (SIGMA x: A. {y. x = f y}) ==> a \<in> A & a = f b"
berghofe@26819
    72
by (metis SigmaD1 SigmaD2 insert_def singleton_conv2 Un_empty_right vimage_Collect_eq vimage_def vimage_singleton_eq)
paulson@23449
    73
paulson@24827
    74
paulson@23449
    75
lemma "(a,b) \<in> (SIGMA x: A. {y. x = f y}) ==> a \<in> A & a = f b"
paulson@23449
    76
proof (neg_clausify)
paulson@24827
    77
assume 0: "(a\<Colon>'a\<Colon>type, b\<Colon>'b\<Colon>type)
paulson@24827
    78
\<in> Sigma (A\<Colon>'a\<Colon>type set)
paulson@24827
    79
   (COMBB Collect (COMBC (COMBB COMBB op =) (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>type)))"
paulson@24827
    80
assume 1: "(a\<Colon>'a\<Colon>type) \<notin> (A\<Colon>'a\<Colon>type set) \<or> a \<noteq> (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>type) (b\<Colon>'b\<Colon>type)"
paulson@24827
    81
have 2: "(a\<Colon>'a\<Colon>type) \<in> (A\<Colon>'a\<Colon>type set)"
paulson@24827
    82
  by (metis 0 SigmaD1)
paulson@24827
    83
have 3: "(b\<Colon>'b\<Colon>type)
paulson@24827
    84
\<in> COMBB Collect (COMBC (COMBB COMBB op =) (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>type)) (a\<Colon>'a\<Colon>type)"
paulson@24827
    85
  by (metis 0 SigmaD2) 
paulson@24827
    86
have 4: "(b\<Colon>'b\<Colon>type) \<in> Collect (COMBB (op = (a\<Colon>'a\<Colon>type)) (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>type))"
paulson@24827
    87
  by (metis 3)
paulson@24827
    88
have 5: "(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>type) (b\<Colon>'b\<Colon>type) \<noteq> (a\<Colon>'a\<Colon>type)"
paulson@24827
    89
  by (metis 1 2)
paulson@24827
    90
have 6: "(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>type) (b\<Colon>'b\<Colon>type) = (a\<Colon>'a\<Colon>type)"
berghofe@26819
    91
  by (metis 4 vimage_singleton_eq insert_def singleton_conv2 Un_empty_right vimage_Collect_eq vimage_def)
paulson@23449
    92
show "False"
paulson@24827
    93
  by (metis 5 6)
paulson@24827
    94
qed
paulson@24827
    95
paulson@24827
    96
(*Alternative structured proof, untyped*)
paulson@24827
    97
lemma "(a,b) \<in> (SIGMA x: A. {y. x = f y}) ==> a \<in> A & a = f b"
paulson@24827
    98
proof (neg_clausify)
paulson@24827
    99
assume 0: "(a, b) \<in> Sigma A (COMBB Collect (COMBC (COMBB COMBB op =) f))"
paulson@24827
   100
have 1: "b \<in> Collect (COMBB (op = a) f)"
paulson@24827
   101
  by (metis 0 SigmaD2)
paulson@24827
   102
have 2: "f b = a"
berghofe@26819
   103
  by (metis 1 vimage_Collect_eq singleton_conv2 insert_def Un_empty_right vimage_singleton_eq vimage_def)
paulson@24827
   104
assume 3: "a \<notin> A \<or> a \<noteq> f b"
paulson@24827
   105
have 4: "a \<in> A"
paulson@24827
   106
  by (metis 0 SigmaD1)
paulson@24827
   107
have 5: "f b \<noteq> a"
paulson@24827
   108
  by (metis 4 3)
paulson@24827
   109
show "False"
paulson@24827
   110
  by (metis 5 2)
paulson@24827
   111
qed
paulson@23449
   112
paulson@23449
   113
boehmes@32864
   114
declare [[ atp_problem_prefix = "Abstraction__CLF_eq_in_pp" ]]
paulson@23449
   115
lemma "(cl,f) \<in> CLF ==> CLF = (SIGMA cl: CL.{f. f \<in> pset cl}) ==> f \<in> pset cl"
paulson@24827
   116
by (metis Collect_mem_eq SigmaD2)
paulson@23449
   117
paulson@24742
   118
lemma "(cl,f) \<in> CLF ==> CLF = (SIGMA cl: CL.{f. f \<in> pset cl}) ==> f \<in> pset cl"
paulson@24742
   119
proof (neg_clausify)
paulson@24827
   120
assume 0: "(cl, f) \<in> CLF"
paulson@24827
   121
assume 1: "CLF = Sigma CL (COMBB Collect (COMBB (COMBC op \<in>) pset))"
paulson@24827
   122
assume 2: "f \<notin> pset cl"
paulson@24827
   123
have 3: "\<And>X1 X2. X2 \<in> COMBB Collect (COMBB (COMBC op \<in>) pset) X1 \<or> (X1, X2) \<notin> CLF"
paulson@24827
   124
  by (metis SigmaD2 1)
paulson@24827
   125
have 4: "\<And>X1 X2. X2 \<in> pset X1 \<or> (X1, X2) \<notin> CLF"
paulson@24827
   126
  by (metis 3 Collect_mem_eq)
paulson@24827
   127
have 5: "(cl, f) \<notin> CLF"
paulson@24827
   128
  by (metis 2 4)
paulson@23449
   129
show "False"
paulson@24827
   130
  by (metis 5 0)
paulson@24827
   131
qed
paulson@23449
   132
boehmes@32864
   133
declare [[ atp_problem_prefix = "Abstraction__Sigma_Collect_Pi" ]]
paulson@23449
   134
lemma
paulson@23449
   135
    "(cl,f) \<in> (SIGMA cl: CL. {f. f \<in> pset cl \<rightarrow> pset cl}) ==> 
paulson@23449
   136
    f \<in> pset cl \<rightarrow> pset cl"
paulson@24827
   137
proof (neg_clausify)
paulson@24827
   138
assume 0: "f \<notin> Pi (pset cl) (COMBK (pset cl))"
paulson@24827
   139
assume 1: "(cl, f)
paulson@24827
   140
\<in> Sigma CL
paulson@24827
   141
   (COMBB Collect
paulson@24827
   142
     (COMBB (COMBC op \<in>) (COMBS (COMBB Pi pset) (COMBB COMBK pset))))"
paulson@24827
   143
show "False"
paulson@24827
   144
(*  by (metis 0 Collect_mem_eq SigmaD2 1) ??doesn't terminate*)
paulson@24827
   145
  by (insert 0 1, simp add: COMBB_def COMBS_def COMBC_def)
paulson@24827
   146
qed
paulson@23449
   147
paulson@23449
   148
boehmes@32864
   149
declare [[ atp_problem_prefix = "Abstraction__Sigma_Collect_Int" ]]
paulson@23449
   150
lemma
paulson@23449
   151
    "(cl,f) \<in> (SIGMA cl: CL. {f. f \<in> pset cl \<inter> cl}) ==>
paulson@23449
   152
   f \<in> pset cl \<inter> cl"
paulson@24827
   153
proof (neg_clausify)
paulson@24827
   154
assume 0: "(cl, f)
paulson@24827
   155
\<in> Sigma CL
paulson@24827
   156
   (COMBB Collect (COMBB (COMBC op \<in>) (COMBS (COMBB op \<inter> pset) COMBI)))"
paulson@24827
   157
assume 1: "f \<notin> pset cl \<inter> cl"
paulson@24827
   158
have 2: "f \<in> COMBB Collect (COMBB (COMBC op \<in>) (COMBS (COMBB op \<inter> pset) COMBI)) cl" 
paulson@24827
   159
  by (insert 0, simp add: COMBB_def) 
paulson@24827
   160
(*  by (metis SigmaD2 0)  ??doesn't terminate*)
paulson@24827
   161
have 3: "f \<in> COMBS (COMBB op \<inter> pset) COMBI cl"
paulson@24827
   162
  by (metis 2 Collect_mem_eq)
paulson@24827
   163
have 4: "f \<notin> cl \<inter> pset cl"
paulson@24827
   164
  by (metis 1 Int_commute)
paulson@24827
   165
have 5: "f \<in> cl \<inter> pset cl"
paulson@24827
   166
  by (metis 3 Int_commute)
paulson@24827
   167
show "False"
paulson@24827
   168
  by (metis 5 4)
paulson@24827
   169
qed
paulson@24827
   170
paulson@23449
   171
boehmes@32864
   172
declare [[ atp_problem_prefix = "Abstraction__Sigma_Collect_Pi_mono" ]]
paulson@23449
   173
lemma
paulson@23449
   174
    "(cl,f) \<in> (SIGMA cl: CL. {f. f \<in> pset cl \<rightarrow> pset cl & monotone f (pset cl) (order cl)}) ==>
paulson@23449
   175
   (f \<in> pset cl \<rightarrow> pset cl)  &  (monotone f (pset cl) (order cl))"
paulson@23449
   176
by auto
paulson@23449
   177
boehmes@32864
   178
declare [[ atp_problem_prefix = "Abstraction__CLF_subset_Collect_Int" ]]
paulson@23449
   179
lemma "(cl,f) \<in> CLF ==> 
paulson@23449
   180
   CLF \<subseteq> (SIGMA cl: CL. {f. f \<in> pset cl \<inter> cl}) ==>
paulson@23449
   181
   f \<in> pset cl \<inter> cl"
paulson@24827
   182
by auto
haftmann@27368
   183
paulson@24827
   184
(*??no longer terminates, with combinators
paulson@23449
   185
by (metis Collect_mem_eq Int_def SigmaD2 UnCI Un_absorb1)
haftmann@27368
   186
  --{*@{text Int_def} is redundant*}
paulson@24827
   187
*)
paulson@23449
   188
boehmes@32864
   189
declare [[ atp_problem_prefix = "Abstraction__CLF_eq_Collect_Int" ]]
paulson@23449
   190
lemma "(cl,f) \<in> CLF ==> 
paulson@23449
   191
   CLF = (SIGMA cl: CL. {f. f \<in> pset cl \<inter> cl}) ==>
paulson@23449
   192
   f \<in> pset cl \<inter> cl"
paulson@24827
   193
by auto
paulson@24827
   194
(*??no longer terminates, with combinators
paulson@23449
   195
by (metis Collect_mem_eq Int_commute SigmaD2)
paulson@24827
   196
*)
paulson@23449
   197
boehmes@32864
   198
declare [[ atp_problem_prefix = "Abstraction__CLF_subset_Collect_Pi" ]]
paulson@23449
   199
lemma 
paulson@23449
   200
   "(cl,f) \<in> CLF ==> 
paulson@23449
   201
    CLF \<subseteq> (SIGMA cl': CL. {f. f \<in> pset cl' \<rightarrow> pset cl'}) ==> 
paulson@23449
   202
    f \<in> pset cl \<rightarrow> pset cl"
nipkow@31754
   203
by fast
paulson@24827
   204
(*??no longer terminates, with combinators
paulson@23449
   205
by (metis Collect_mem_eq SigmaD2 subsetD)
paulson@24827
   206
*)
paulson@23449
   207
boehmes@32864
   208
declare [[ atp_problem_prefix = "Abstraction__CLF_eq_Collect_Pi" ]]
paulson@23449
   209
lemma 
paulson@23449
   210
  "(cl,f) \<in> CLF ==> 
paulson@23449
   211
   CLF = (SIGMA cl: CL. {f. f \<in> pset cl \<rightarrow> pset cl}) ==> 
paulson@23449
   212
   f \<in> pset cl \<rightarrow> pset cl"
paulson@24827
   213
by auto
paulson@24827
   214
(*??no longer terminates, with combinators
paulson@23449
   215
by (metis Collect_mem_eq SigmaD2 contra_subsetD equalityE)
paulson@24827
   216
*)
paulson@23449
   217
boehmes@32864
   218
declare [[ atp_problem_prefix = "Abstraction__CLF_eq_Collect_Pi_mono" ]]
paulson@23449
   219
lemma 
paulson@23449
   220
  "(cl,f) \<in> CLF ==> 
paulson@23449
   221
   CLF = (SIGMA cl: CL. {f. f \<in> pset cl \<rightarrow> pset cl & monotone f (pset cl) (order cl)}) ==>
paulson@23449
   222
   (f \<in> pset cl \<rightarrow> pset cl)  &  (monotone f (pset cl) (order cl))"
paulson@23449
   223
by auto
paulson@23449
   224
boehmes@32864
   225
declare [[ atp_problem_prefix = "Abstraction__map_eq_zipA" ]]
paulson@23449
   226
lemma "map (%x. (f x, g x)) xs = zip (map f xs) (map g xs)"
paulson@23449
   227
apply (induct xs)
paulson@23449
   228
(*sledgehammer*)  
paulson@23449
   229
apply auto
paulson@23449
   230
done
paulson@23449
   231
boehmes@32864
   232
declare [[ atp_problem_prefix = "Abstraction__map_eq_zipB" ]]
paulson@23449
   233
lemma "map (%w. (w -> w, w \<times> w)) xs = 
paulson@23449
   234
       zip (map (%w. w -> w) xs) (map (%w. w \<times> w) xs)"
paulson@23449
   235
apply (induct xs)
paulson@23449
   236
(*sledgehammer*)  
paulson@23449
   237
apply auto
paulson@23449
   238
done
paulson@23449
   239
boehmes@32864
   240
declare [[ atp_problem_prefix = "Abstraction__image_evenA" ]]
paulson@23449
   241
lemma "(%x. Suc(f x)) ` {x. even x} <= A ==> (\<forall>x. even x --> Suc(f x) \<in> A)";
paulson@23449
   242
(*sledgehammer*)  
paulson@23449
   243
by auto
paulson@23449
   244
boehmes@32864
   245
declare [[ atp_problem_prefix = "Abstraction__image_evenB" ]]
paulson@23449
   246
lemma "(%x. f (f x)) ` ((%x. Suc(f x)) ` {x. even x}) <= A 
paulson@23449
   247
       ==> (\<forall>x. even x --> f (f (Suc(f x))) \<in> A)";
paulson@23449
   248
(*sledgehammer*)  
paulson@23449
   249
by auto
paulson@23449
   250
boehmes@32864
   251
declare [[ atp_problem_prefix = "Abstraction__image_curry" ]]
paulson@23449
   252
lemma "f \<in> (%u v. b \<times> u \<times> v) ` A ==> \<forall>u v. P (b \<times> u \<times> v) ==> P(f y)" 
paulson@23449
   253
(*sledgehammer*)  
paulson@23449
   254
by auto
paulson@23449
   255
boehmes@32864
   256
declare [[ atp_problem_prefix = "Abstraction__image_TimesA" ]]
paulson@23449
   257
lemma image_TimesA: "(%(x,y). (f x, g y)) ` (A \<times> B) = (f`A) \<times> (g`B)"
paulson@23449
   258
(*sledgehammer*) 
paulson@23449
   259
apply (rule equalityI)
paulson@23449
   260
(***Even the two inclusions are far too difficult
boehmes@32864
   261
using [[ atp_problem_prefix = "Abstraction__image_TimesA_simpler"]]
paulson@23449
   262
***)
paulson@23449
   263
apply (rule subsetI)
paulson@23449
   264
apply (erule imageE)
paulson@23449
   265
(*V manages from here with help: Abstraction__image_TimesA_simpler_1_b.p*)
paulson@23449
   266
apply (erule ssubst)
paulson@23449
   267
apply (erule SigmaE)
paulson@23449
   268
(*V manages from here: Abstraction__image_TimesA_simpler_1_a.p*)
paulson@23449
   269
apply (erule ssubst)
paulson@23449
   270
apply (subst split_conv)
paulson@23449
   271
apply (rule SigmaI) 
paulson@23449
   272
apply (erule imageI) +
paulson@23449
   273
txt{*subgoal 2*}
paulson@23449
   274
apply (clarify );
paulson@23449
   275
apply (simp add: );  
paulson@23449
   276
apply (rule rev_image_eqI)  
paulson@23449
   277
apply (blast intro: elim:); 
paulson@23449
   278
apply (simp add: );
paulson@23449
   279
done
paulson@23449
   280
paulson@23449
   281
(*Given the difficulty of the previous problem, these two are probably
paulson@23449
   282
impossible*)
paulson@23449
   283
boehmes@32864
   284
declare [[ atp_problem_prefix = "Abstraction__image_TimesB" ]]
paulson@23449
   285
lemma image_TimesB:
paulson@23449
   286
    "(%(x,y,z). (f x, g y, h z)) ` (A \<times> B \<times> C) = (f`A) \<times> (g`B) \<times> (h`C)" 
paulson@23449
   287
(*sledgehammer*) 
paulson@23449
   288
by force
paulson@23449
   289
boehmes@32864
   290
declare [[ atp_problem_prefix = "Abstraction__image_TimesC" ]]
paulson@23449
   291
lemma image_TimesC:
paulson@23449
   292
    "(%(x,y). (x \<rightarrow> x, y \<times> y)) ` (A \<times> B) = 
paulson@23449
   293
     ((%x. x \<rightarrow> x) ` A) \<times> ((%y. y \<times> y) ` B)" 
paulson@23449
   294
(*sledgehammer*) 
paulson@23449
   295
by auto
paulson@23449
   296
paulson@23449
   297
end