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