src/HOL/MetisExamples/Abstraction.thy
 author wenzelm Mon Mar 16 18:24:30 2009 +0100 (2009-03-16) changeset 30549 d2d7874648bd parent 29676 cfa3378decf7 child 31754 b5260f5272a4 permissions -rw-r--r--
simplified method setup;
```     1 (*  Title:      HOL/MetisExamples/Abstraction.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4
```
```     5 Testing the metis method
```
```     6 *)
```
```     7
```
```     8 theory Abstraction
```
```     9 imports Main FuncSet
```
```    10 begin
```
```    11
```
```    12 (*For Christoph Benzmueller*)
```
```    13 lemma "x<1 & ((op=) = (op=)) ==> ((op=) = (op=)) & (x<(2::nat))";
```
```    14   by (metis One_nat_def less_Suc0 not_less0 not_less_eq numeral_2_eq_2)
```
```    15
```
```    16 (*this is a theorem, but we can't prove it unless ext is applied explicitly
```
```    17 lemma "(op=) = (%x y. y=x)"
```
```    18 *)
```
```    19
```
```    20 consts
```
```    21   monotone :: "['a => 'a, 'a set, ('a *'a)set] => bool"
```
```    22   pset  :: "'a set => 'a set"
```
```    23   order :: "'a set => ('a * 'a) set"
```
```    24
```
```    25 ML{*AtpWrapper.problem_name := "Abstraction__Collect_triv"*}
```
```    26 lemma (*Collect_triv:*) "a \<in> {x. P x} ==> P a"
```
```    27 proof (neg_clausify)
```
```    28 assume 0: "(a\<Colon>'a\<Colon>type) \<in> Collect (P\<Colon>'a\<Colon>type \<Rightarrow> bool)"
```
```    29 assume 1: "\<not> (P\<Colon>'a\<Colon>type \<Rightarrow> bool) (a\<Colon>'a\<Colon>type)"
```
```    30 have 2: "(P\<Colon>'a\<Colon>type \<Rightarrow> bool) (a\<Colon>'a\<Colon>type)"
```
```    31   by (metis CollectD 0)
```
```    32 show "False"
```
```    33   by (metis 2 1)
```
```    34 qed
```
```    35
```
```    36 lemma Collect_triv: "a \<in> {x. P x} ==> P a"
```
```    37 by (metis mem_Collect_eq)
```
```    38
```
```    39
```
```    40 ML{*AtpWrapper.problem_name := "Abstraction__Collect_mp"*}
```
```    41 lemma "a \<in> {x. P x --> Q x} ==> a \<in> {x. P x} ==> a \<in> {x. Q x}"
```
```    42   by (metis CollectI Collect_imp_eq ComplD UnE mem_Collect_eq);
```
```    43   --{*34 secs*}
```
```    44
```
```    45 ML{*AtpWrapper.problem_name := "Abstraction__Sigma_triv"*}
```
```    46 lemma "(a,b) \<in> Sigma A B ==> a \<in> A & b \<in> B a"
```
```    47 proof (neg_clausify)
```
```    48 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)"
```
```    49 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"
```
```    50 have 2: "(a\<Colon>'a\<Colon>type) \<in> (A\<Colon>'a\<Colon>type set)"
```
```    51   by (metis SigmaD1 0)
```
```    52 have 3: "(b\<Colon>'b\<Colon>type) \<in> (B\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>type set) (a\<Colon>'a\<Colon>type)"
```
```    53   by (metis SigmaD2 0)
```
```    54 have 4: "(b\<Colon>'b\<Colon>type) \<notin> (B\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>type set) (a\<Colon>'a\<Colon>type)"
```
```    55   by (metis 1 2)
```
```    56 show "False"
```
```    57   by (metis 3 4)
```
```    58 qed
```
```    59
```
```    60 lemma Sigma_triv: "(a,b) \<in> Sigma A B ==> a \<in> A & b \<in> B a"
```
```    61 by (metis SigmaD1 SigmaD2)
```
```    62
```
```    63 ML{*AtpWrapper.problem_name := "Abstraction__Sigma_Collect"*}
```
```    64 lemma "(a,b) \<in> (SIGMA x: A. {y. x = f y}) ==> a \<in> A & a = f b"
```
```    65 (*???metis says this is satisfiable!
```
```    66 by (metis CollectD SigmaD1 SigmaD2)
```
```    67 *)
```
```    68 by (meson CollectD SigmaD1 SigmaD2)
```
```    69
```
```    70
```
```    71 (*single-step*)
```
```    72 lemma "(a,b) \<in> (SIGMA x: A. {y. x = f y}) ==> a \<in> A & a = f b"
```
```    73 by (metis SigmaD1 SigmaD2 insert_def singleton_conv2 Un_empty_right vimage_Collect_eq vimage_def vimage_singleton_eq)
```
```    74
```
```    75
```
```    76 lemma "(a,b) \<in> (SIGMA x: A. {y. x = f y}) ==> a \<in> A & a = f b"
```
```    77 proof (neg_clausify)
```
```    78 assume 0: "(a\<Colon>'a\<Colon>type, b\<Colon>'b\<Colon>type)
```
```    79 \<in> Sigma (A\<Colon>'a\<Colon>type set)
```
```    80    (COMBB Collect (COMBC (COMBB COMBB op =) (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>type)))"
```
```    81 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)"
```
```    82 have 2: "(a\<Colon>'a\<Colon>type) \<in> (A\<Colon>'a\<Colon>type set)"
```
```    83   by (metis 0 SigmaD1)
```
```    84 have 3: "(b\<Colon>'b\<Colon>type)
```
```    85 \<in> COMBB Collect (COMBC (COMBB COMBB op =) (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>type)) (a\<Colon>'a\<Colon>type)"
```
```    86   by (metis 0 SigmaD2)
```
```    87 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))"
```
```    88   by (metis 3)
```
```    89 have 5: "(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>type) (b\<Colon>'b\<Colon>type) \<noteq> (a\<Colon>'a\<Colon>type)"
```
```    90   by (metis 1 2)
```
```    91 have 6: "(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>type) (b\<Colon>'b\<Colon>type) = (a\<Colon>'a\<Colon>type)"
```
```    92   by (metis 4 vimage_singleton_eq insert_def singleton_conv2 Un_empty_right vimage_Collect_eq vimage_def)
```
```    93 show "False"
```
```    94   by (metis 5 6)
```
```    95 qed
```
```    96
```
```    97 (*Alternative structured proof, untyped*)
```
```    98 lemma "(a,b) \<in> (SIGMA x: A. {y. x = f y}) ==> a \<in> A & a = f b"
```
```    99 proof (neg_clausify)
```
```   100 assume 0: "(a, b) \<in> Sigma A (COMBB Collect (COMBC (COMBB COMBB op =) f))"
```
```   101 have 1: "b \<in> Collect (COMBB (op = a) f)"
```
```   102   by (metis 0 SigmaD2)
```
```   103 have 2: "f b = a"
```
```   104   by (metis 1 vimage_Collect_eq singleton_conv2 insert_def Un_empty_right vimage_singleton_eq vimage_def)
```
```   105 assume 3: "a \<notin> A \<or> a \<noteq> f b"
```
```   106 have 4: "a \<in> A"
```
```   107   by (metis 0 SigmaD1)
```
```   108 have 5: "f b \<noteq> a"
```
```   109   by (metis 4 3)
```
```   110 show "False"
```
```   111   by (metis 5 2)
```
```   112 qed
```
```   113
```
```   114
```
```   115 ML{*AtpWrapper.problem_name := "Abstraction__CLF_eq_in_pp"*}
```
```   116 lemma "(cl,f) \<in> CLF ==> CLF = (SIGMA cl: CL.{f. f \<in> pset cl}) ==> f \<in> pset cl"
```
```   117 by (metis Collect_mem_eq SigmaD2)
```
```   118
```
```   119 lemma "(cl,f) \<in> CLF ==> CLF = (SIGMA cl: CL.{f. f \<in> pset cl}) ==> f \<in> pset cl"
```
```   120 proof (neg_clausify)
```
```   121 assume 0: "(cl, f) \<in> CLF"
```
```   122 assume 1: "CLF = Sigma CL (COMBB Collect (COMBB (COMBC op \<in>) pset))"
```
```   123 assume 2: "f \<notin> pset cl"
```
```   124 have 3: "\<And>X1 X2. X2 \<in> COMBB Collect (COMBB (COMBC op \<in>) pset) X1 \<or> (X1, X2) \<notin> CLF"
```
```   125   by (metis SigmaD2 1)
```
```   126 have 4: "\<And>X1 X2. X2 \<in> pset X1 \<or> (X1, X2) \<notin> CLF"
```
```   127   by (metis 3 Collect_mem_eq)
```
```   128 have 5: "(cl, f) \<notin> CLF"
```
```   129   by (metis 2 4)
```
```   130 show "False"
```
```   131   by (metis 5 0)
```
```   132 qed
```
```   133
```
```   134 ML{*AtpWrapper.problem_name := "Abstraction__Sigma_Collect_Pi"*}
```
```   135 lemma
```
```   136     "(cl,f) \<in> (SIGMA cl: CL. {f. f \<in> pset cl \<rightarrow> pset cl}) ==>
```
```   137     f \<in> pset cl \<rightarrow> pset cl"
```
```   138 proof (neg_clausify)
```
```   139 assume 0: "f \<notin> Pi (pset cl) (COMBK (pset cl))"
```
```   140 assume 1: "(cl, f)
```
```   141 \<in> Sigma CL
```
```   142    (COMBB Collect
```
```   143      (COMBB (COMBC op \<in>) (COMBS (COMBB Pi pset) (COMBB COMBK pset))))"
```
```   144 show "False"
```
```   145 (*  by (metis 0 Collect_mem_eq SigmaD2 1) ??doesn't terminate*)
```
```   146   by (insert 0 1, simp add: COMBB_def COMBS_def COMBC_def)
```
```   147 qed
```
```   148
```
```   149
```
```   150 ML{*AtpWrapper.problem_name := "Abstraction__Sigma_Collect_Int"*}
```
```   151 lemma
```
```   152     "(cl,f) \<in> (SIGMA cl: CL. {f. f \<in> pset cl \<inter> cl}) ==>
```
```   153    f \<in> pset cl \<inter> cl"
```
```   154 proof (neg_clausify)
```
```   155 assume 0: "(cl, f)
```
```   156 \<in> Sigma CL
```
```   157    (COMBB Collect (COMBB (COMBC op \<in>) (COMBS (COMBB op \<inter> pset) COMBI)))"
```
```   158 assume 1: "f \<notin> pset cl \<inter> cl"
```
```   159 have 2: "f \<in> COMBB Collect (COMBB (COMBC op \<in>) (COMBS (COMBB op \<inter> pset) COMBI)) cl"
```
```   160   by (insert 0, simp add: COMBB_def)
```
```   161 (*  by (metis SigmaD2 0)  ??doesn't terminate*)
```
```   162 have 3: "f \<in> COMBS (COMBB op \<inter> pset) COMBI cl"
```
```   163   by (metis 2 Collect_mem_eq)
```
```   164 have 4: "f \<notin> cl \<inter> pset cl"
```
```   165   by (metis 1 Int_commute)
```
```   166 have 5: "f \<in> cl \<inter> pset cl"
```
```   167   by (metis 3 Int_commute)
```
```   168 show "False"
```
```   169   by (metis 5 4)
```
```   170 qed
```
```   171
```
```   172
```
```   173 ML{*AtpWrapper.problem_name := "Abstraction__Sigma_Collect_Pi_mono"*}
```
```   174 lemma
```
```   175     "(cl,f) \<in> (SIGMA cl: CL. {f. f \<in> pset cl \<rightarrow> pset cl & monotone f (pset cl) (order cl)}) ==>
```
```   176    (f \<in> pset cl \<rightarrow> pset cl)  &  (monotone f (pset cl) (order cl))"
```
```   177 by auto
```
```   178
```
```   179 ML{*AtpWrapper.problem_name := "Abstraction__CLF_subset_Collect_Int"*}
```
```   180 lemma "(cl,f) \<in> CLF ==>
```
```   181    CLF \<subseteq> (SIGMA cl: CL. {f. f \<in> pset cl \<inter> cl}) ==>
```
```   182    f \<in> pset cl \<inter> cl"
```
```   183 by auto
```
```   184
```
```   185 (*??no longer terminates, with combinators
```
```   186 by (metis Collect_mem_eq Int_def SigmaD2 UnCI Un_absorb1)
```
```   187   --{*@{text Int_def} is redundant*}
```
```   188 *)
```
```   189
```
```   190 ML{*AtpWrapper.problem_name := "Abstraction__CLF_eq_Collect_Int"*}
```
```   191 lemma "(cl,f) \<in> CLF ==>
```
```   192    CLF = (SIGMA cl: CL. {f. f \<in> pset cl \<inter> cl}) ==>
```
```   193    f \<in> pset cl \<inter> cl"
```
```   194 by auto
```
```   195 (*??no longer terminates, with combinators
```
```   196 by (metis Collect_mem_eq Int_commute SigmaD2)
```
```   197 *)
```
```   198
```
```   199 ML{*AtpWrapper.problem_name := "Abstraction__CLF_subset_Collect_Pi"*}
```
```   200 lemma
```
```   201    "(cl,f) \<in> CLF ==>
```
```   202     CLF \<subseteq> (SIGMA cl': CL. {f. f \<in> pset cl' \<rightarrow> pset cl'}) ==>
```
```   203     f \<in> pset cl \<rightarrow> pset cl"
```
```   204 by auto
```
```   205 (*??no longer terminates, with combinators
```
```   206 by (metis Collect_mem_eq SigmaD2 subsetD)
```
```   207 *)
```
```   208
```
```   209 ML{*AtpWrapper.problem_name := "Abstraction__CLF_eq_Collect_Pi"*}
```
```   210 lemma
```
```   211   "(cl,f) \<in> CLF ==>
```
```   212    CLF = (SIGMA cl: CL. {f. f \<in> pset cl \<rightarrow> pset cl}) ==>
```
```   213    f \<in> pset cl \<rightarrow> pset cl"
```
```   214 by auto
```
```   215 (*??no longer terminates, with combinators
```
```   216 by (metis Collect_mem_eq SigmaD2 contra_subsetD equalityE)
```
```   217 *)
```
```   218
```
```   219 ML{*AtpWrapper.problem_name := "Abstraction__CLF_eq_Collect_Pi_mono"*}
```
```   220 lemma
```
```   221   "(cl,f) \<in> CLF ==>
```
```   222    CLF = (SIGMA cl: CL. {f. f \<in> pset cl \<rightarrow> pset cl & monotone f (pset cl) (order cl)}) ==>
```
```   223    (f \<in> pset cl \<rightarrow> pset cl)  &  (monotone f (pset cl) (order cl))"
```
```   224 by auto
```
```   225
```
```   226 ML{*AtpWrapper.problem_name := "Abstraction__map_eq_zipA"*}
```
```   227 lemma "map (%x. (f x, g x)) xs = zip (map f xs) (map g xs)"
```
```   228 apply (induct xs)
```
```   229 (*sledgehammer*)
```
```   230 apply auto
```
```   231 done
```
```   232
```
```   233 ML{*AtpWrapper.problem_name := "Abstraction__map_eq_zipB"*}
```
```   234 lemma "map (%w. (w -> w, w \<times> w)) xs =
```
```   235        zip (map (%w. w -> w) xs) (map (%w. w \<times> w) xs)"
```
```   236 apply (induct xs)
```
```   237 (*sledgehammer*)
```
```   238 apply auto
```
```   239 done
```
```   240
```
```   241 ML{*AtpWrapper.problem_name := "Abstraction__image_evenA"*}
```
```   242 lemma "(%x. Suc(f x)) ` {x. even x} <= A ==> (\<forall>x. even x --> Suc(f x) \<in> A)";
```
```   243 (*sledgehammer*)
```
```   244 by auto
```
```   245
```
```   246 ML{*AtpWrapper.problem_name := "Abstraction__image_evenB"*}
```
```   247 lemma "(%x. f (f x)) ` ((%x. Suc(f x)) ` {x. even x}) <= A
```
```   248        ==> (\<forall>x. even x --> f (f (Suc(f x))) \<in> A)";
```
```   249 (*sledgehammer*)
```
```   250 by auto
```
```   251
```
```   252 ML{*AtpWrapper.problem_name := "Abstraction__image_curry"*}
```
```   253 lemma "f \<in> (%u v. b \<times> u \<times> v) ` A ==> \<forall>u v. P (b \<times> u \<times> v) ==> P(f y)"
```
```   254 (*sledgehammer*)
```
```   255 by auto
```
```   256
```
```   257 ML{*AtpWrapper.problem_name := "Abstraction__image_TimesA"*}
```
```   258 lemma image_TimesA: "(%(x,y). (f x, g y)) ` (A \<times> B) = (f`A) \<times> (g`B)"
```
```   259 (*sledgehammer*)
```
```   260 apply (rule equalityI)
```
```   261 (***Even the two inclusions are far too difficult
```
```   262 ML{*AtpWrapper.problem_name := "Abstraction__image_TimesA_simpler"*}
```
```   263 ***)
```
```   264 apply (rule subsetI)
```
```   265 apply (erule imageE)
```
```   266 (*V manages from here with help: Abstraction__image_TimesA_simpler_1_b.p*)
```
```   267 apply (erule ssubst)
```
```   268 apply (erule SigmaE)
```
```   269 (*V manages from here: Abstraction__image_TimesA_simpler_1_a.p*)
```
```   270 apply (erule ssubst)
```
```   271 apply (subst split_conv)
```
```   272 apply (rule SigmaI)
```
```   273 apply (erule imageI) +
```
```   274 txt{*subgoal 2*}
```
```   275 apply (clarify );
```
```   276 apply (simp add: );
```
```   277 apply (rule rev_image_eqI)
```
```   278 apply (blast intro: elim:);
```
```   279 apply (simp add: );
```
```   280 done
```
```   281
```
```   282 (*Given the difficulty of the previous problem, these two are probably
```
```   283 impossible*)
```
```   284
```
```   285 ML{*AtpWrapper.problem_name := "Abstraction__image_TimesB"*}
```
```   286 lemma image_TimesB:
```
```   287     "(%(x,y,z). (f x, g y, h z)) ` (A \<times> B \<times> C) = (f`A) \<times> (g`B) \<times> (h`C)"
```
```   288 (*sledgehammer*)
```
```   289 by force
```
```   290
```
```   291 ML{*AtpWrapper.problem_name := "Abstraction__image_TimesC"*}
```
```   292 lemma image_TimesC:
```
```   293     "(%(x,y). (x \<rightarrow> x, y \<times> y)) ` (A \<times> B) =
```
```   294      ((%x. x \<rightarrow> x) ` A) \<times> ((%y. y \<times> y) ` B)"
```
```   295 (*sledgehammer*)
```
```   296 by auto
```
```   297
```
```   298 end
```