src/HOL/Relation.ML
 author paulson Mon Jul 26 16:29:38 1999 +0200 (1999-07-26) changeset 7083 9663eb2bce05 parent 7031 972b5f62f476 child 7822 09aabe6d04b8 permissions -rw-r--r--
three new theorems
```     1 (*  Title:      Relation.ML
```
```     2     ID:         \$Id\$
```
```     3     Authors:    Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4     Copyright   1996  University of Cambridge
```
```     5 *)
```
```     6
```
```     7 (** Identity relation **)
```
```     8
```
```     9 Goalw [Id_def] "(a,a) : Id";
```
```    10 by (Blast_tac 1);
```
```    11 qed "IdI";
```
```    12
```
```    13 val major::prems = Goalw [Id_def]
```
```    14     "[| p: Id;  !!x.[| p = (x,x) |] ==> P  \
```
```    15 \    |] ==>  P";
```
```    16 by (rtac (major RS CollectE) 1);
```
```    17 by (etac exE 1);
```
```    18 by (eresolve_tac prems 1);
```
```    19 qed "IdE";
```
```    20
```
```    21 Goalw [Id_def] "(a,b):Id = (a=b)";
```
```    22 by (Blast_tac 1);
```
```    23 qed "pair_in_Id_conv";
```
```    24 Addsimps [pair_in_Id_conv];
```
```    25
```
```    26 Goalw [refl_def] "reflexive Id";
```
```    27 by Auto_tac;
```
```    28 qed "reflexive_Id";
```
```    29
```
```    30 (*A strange result, since Id is also symmetric.*)
```
```    31 Goalw [antisym_def] "antisym Id";
```
```    32 by Auto_tac;
```
```    33 qed "antisym_Id";
```
```    34
```
```    35 Goalw [trans_def] "trans Id";
```
```    36 by Auto_tac;
```
```    37 qed "trans_Id";
```
```    38
```
```    39
```
```    40 (** Diagonal relation: indentity restricted to some set **)
```
```    41
```
```    42 (*** Equality : the diagonal relation ***)
```
```    43
```
```    44 Goalw [diag_def] "[| a=b;  a:A |] ==> (a,b) : diag(A)";
```
```    45 by (Blast_tac 1);
```
```    46 qed "diag_eqI";
```
```    47
```
```    48 val diagI = refl RS diag_eqI |> standard;
```
```    49
```
```    50 (*The general elimination rule*)
```
```    51 val major::prems = Goalw [diag_def]
```
```    52     "[| c : diag(A);  \
```
```    53 \       !!x y. [| x:A;  c = (x,x) |] ==> P \
```
```    54 \    |] ==> P";
```
```    55 by (rtac (major RS UN_E) 1);
```
```    56 by (REPEAT (eresolve_tac [asm_rl,singletonE] 1 ORELSE resolve_tac prems 1));
```
```    57 qed "diagE";
```
```    58
```
```    59 AddSIs [diagI];
```
```    60 AddSEs [diagE];
```
```    61
```
```    62 Goal "((x,y) : diag A) = (x=y & x : A)";
```
```    63 by (Blast_tac 1);
```
```    64 qed "diag_iff";
```
```    65
```
```    66 Goal "diag(A) <= A Times A";
```
```    67 by (Blast_tac 1);
```
```    68 qed "diag_subset_Times";
```
```    69
```
```    70
```
```    71
```
```    72 (** Composition of two relations **)
```
```    73
```
```    74 Goalw [comp_def]
```
```    75     "[| (a,b):s; (b,c):r |] ==> (a,c) : r O s";
```
```    76 by (Blast_tac 1);
```
```    77 qed "compI";
```
```    78
```
```    79 (*proof requires higher-level assumptions or a delaying of hyp_subst_tac*)
```
```    80 val prems = Goalw [comp_def]
```
```    81     "[| xz : r O s;  \
```
```    82 \       !!x y z. [| xz = (x,z);  (x,y):s;  (y,z):r |] ==> P \
```
```    83 \    |] ==> P";
```
```    84 by (cut_facts_tac prems 1);
```
```    85 by (REPEAT (eresolve_tac [CollectE, splitE, exE, conjE] 1
```
```    86      ORELSE ares_tac prems 1));
```
```    87 qed "compE";
```
```    88
```
```    89 val prems = Goal
```
```    90     "[| (a,c) : r O s;  \
```
```    91 \       !!y. [| (a,y):s;  (y,c):r |] ==> P \
```
```    92 \    |] ==> P";
```
```    93 by (rtac compE 1);
```
```    94 by (REPEAT (ares_tac prems 1 ORELSE eresolve_tac [Pair_inject,ssubst] 1));
```
```    95 qed "compEpair";
```
```    96
```
```    97 AddIs [compI, IdI];
```
```    98 AddSEs [compE, IdE];
```
```    99
```
```   100 Goal "R O Id = R";
```
```   101 by (Fast_tac 1);
```
```   102 qed "R_O_Id";
```
```   103
```
```   104 Goal "Id O R = R";
```
```   105 by (Fast_tac 1);
```
```   106 qed "Id_O_R";
```
```   107
```
```   108 Addsimps [R_O_Id,Id_O_R];
```
```   109
```
```   110 Goal "(R O S) O T = R O (S O T)";
```
```   111 by (Blast_tac 1);
```
```   112 qed "O_assoc";
```
```   113
```
```   114 Goal "[| r'<=r; s'<=s |] ==> (r' O s') <= (r O s)";
```
```   115 by (Blast_tac 1);
```
```   116 qed "comp_mono";
```
```   117
```
```   118 Goal "[| s <= A Times B;  r <= B Times C |] ==> (r O s) <= A Times C";
```
```   119 by (Blast_tac 1);
```
```   120 qed "comp_subset_Sigma";
```
```   121
```
```   122 (** Natural deduction for refl(r) **)
```
```   123
```
```   124 val prems = Goalw [refl_def]
```
```   125     "[| r <= A Times A;  !! x. x:A ==> (x,x):r |] ==> refl A r";
```
```   126 by (REPEAT (ares_tac (prems@[ballI,conjI]) 1));
```
```   127 qed "reflI";
```
```   128
```
```   129 Goalw [refl_def] "[| refl A r; a:A |] ==> (a,a):r";
```
```   130 by (Blast_tac 1);
```
```   131 qed "reflD";
```
```   132
```
```   133 (** Natural deduction for antisym(r) **)
```
```   134
```
```   135 val prems = Goalw [antisym_def]
```
```   136     "(!! x y. [| (x,y):r;  (y,x):r |] ==> x=y) ==> antisym(r)";
```
```   137 by (REPEAT (ares_tac (prems@[allI,impI]) 1));
```
```   138 qed "antisymI";
```
```   139
```
```   140 Goalw [antisym_def] "[| antisym(r);  (a,b):r;  (b,a):r |] ==> a=b";
```
```   141 by (Blast_tac 1);
```
```   142 qed "antisymD";
```
```   143
```
```   144 (** Natural deduction for trans(r) **)
```
```   145
```
```   146 val prems = Goalw [trans_def]
```
```   147     "(!! x y z. [| (x,y):r;  (y,z):r |] ==> (x,z):r) ==> trans(r)";
```
```   148 by (REPEAT (ares_tac (prems@[allI,impI]) 1));
```
```   149 qed "transI";
```
```   150
```
```   151 Goalw [trans_def] "[| trans(r);  (a,b):r;  (b,c):r |] ==> (a,c):r";
```
```   152 by (Blast_tac 1);
```
```   153 qed "transD";
```
```   154
```
```   155 (** Natural deduction for r^-1 **)
```
```   156
```
```   157 Goalw [converse_def] "((a,b): r^-1) = ((b,a):r)";
```
```   158 by (Simp_tac 1);
```
```   159 qed "converse_iff";
```
```   160
```
```   161 AddIffs [converse_iff];
```
```   162
```
```   163 Goalw [converse_def] "(a,b):r ==> (b,a): r^-1";
```
```   164 by (Simp_tac 1);
```
```   165 qed "converseI";
```
```   166
```
```   167 Goalw [converse_def] "(a,b) : r^-1 ==> (b,a) : r";
```
```   168 by (Blast_tac 1);
```
```   169 qed "converseD";
```
```   170
```
```   171 (*More general than converseD, as it "splits" the member of the relation*)
```
```   172
```
```   173 val [major,minor] = Goalw [converse_def]
```
```   174     "[| yx : r^-1;  \
```
```   175 \       !!x y. [| yx=(y,x);  (x,y):r |] ==> P \
```
```   176 \    |] ==> P";
```
```   177 by (rtac (major RS CollectE) 1);
```
```   178 by (REPEAT (eresolve_tac [splitE, bexE,exE, conjE, minor] 1));
```
```   179 by (assume_tac 1);
```
```   180 qed "converseE";
```
```   181 AddSEs [converseE];
```
```   182
```
```   183 Goalw [converse_def] "(r^-1)^-1 = r";
```
```   184 by (Blast_tac 1);
```
```   185 qed "converse_converse";
```
```   186 Addsimps [converse_converse];
```
```   187
```
```   188 Goal "(r O s)^-1 = s^-1 O r^-1";
```
```   189 by (Blast_tac 1);
```
```   190 qed "converse_comp";
```
```   191
```
```   192 Goal "Id^-1 = Id";
```
```   193 by (Blast_tac 1);
```
```   194 qed "converse_Id";
```
```   195 Addsimps [converse_Id];
```
```   196
```
```   197 Goal "(diag A) ^-1 = diag A";
```
```   198 by (Blast_tac 1);
```
```   199 qed "converse_diag";
```
```   200 Addsimps [converse_diag];
```
```   201
```
```   202 Goalw [refl_def] "refl A r ==> refl A (converse r)";
```
```   203 by (Blast_tac 1);
```
```   204 qed "refl_converse";
```
```   205
```
```   206 Goalw [antisym_def] "antisym (converse r) = antisym r";
```
```   207 by (Blast_tac 1);
```
```   208 qed "antisym_converse";
```
```   209
```
```   210 Goalw [trans_def] "trans (converse r) = trans r";
```
```   211 by (Blast_tac 1);
```
```   212 qed "trans_converse";
```
```   213
```
```   214 (** Domain **)
```
```   215
```
```   216 Goalw [Domain_def] "a: Domain(r) = (EX y. (a,y): r)";
```
```   217 by (Blast_tac 1);
```
```   218 qed "Domain_iff";
```
```   219
```
```   220 Goal "(a,b): r ==> a: Domain(r)";
```
```   221 by (etac (exI RS (Domain_iff RS iffD2)) 1) ;
```
```   222 qed "DomainI";
```
```   223
```
```   224 val prems= Goal "[| a : Domain(r);  !!y. (a,y): r ==> P |] ==> P";
```
```   225 by (rtac (Domain_iff RS iffD1 RS exE) 1);
```
```   226 by (REPEAT (ares_tac prems 1)) ;
```
```   227 qed "DomainE";
```
```   228
```
```   229 AddIs  [DomainI];
```
```   230 AddSEs [DomainE];
```
```   231
```
```   232 Goal "Domain Id = UNIV";
```
```   233 by (Blast_tac 1);
```
```   234 qed "Domain_Id";
```
```   235 Addsimps [Domain_Id];
```
```   236
```
```   237 Goal "Domain (diag A) = A";
```
```   238 by Auto_tac;
```
```   239 qed "Domain_diag";
```
```   240 Addsimps [Domain_diag];
```
```   241
```
```   242 Goal "Domain(A Un B) = Domain(A) Un Domain(B)";
```
```   243 by (Blast_tac 1);
```
```   244 qed "Domain_Un_eq";
```
```   245
```
```   246 Goal "Domain(A Int B) <= Domain(A) Int Domain(B)";
```
```   247 by (Blast_tac 1);
```
```   248 qed "Domain_Int_subset";
```
```   249
```
```   250 Goal "Domain(A) - Domain(B) <= Domain(A - B)";
```
```   251 by (Blast_tac 1);
```
```   252 qed "Domain_Diff_subset";
```
```   253
```
```   254 Goal "Domain (Union S) = (UN A:S. Domain A)";
```
```   255 by (Blast_tac 1);
```
```   256 qed "Domain_Union";
```
```   257
```
```   258
```
```   259 (** Range **)
```
```   260
```
```   261 Goalw [Domain_def, Range_def] "a: Range(r) = (EX y. (y,a): r)";
```
```   262 by (Blast_tac 1);
```
```   263 qed "Range_iff";
```
```   264
```
```   265 Goalw [Range_def] "(a,b): r ==> b : Range(r)";
```
```   266 by (etac (converseI RS DomainI) 1);
```
```   267 qed "RangeI";
```
```   268
```
```   269 val major::prems = Goalw [Range_def]
```
```   270     "[| b : Range(r);  !!x. (x,b): r ==> P |] ==> P";
```
```   271 by (rtac (major RS DomainE) 1);
```
```   272 by (resolve_tac prems 1);
```
```   273 by (etac converseD 1) ;
```
```   274 qed "RangeE";
```
```   275
```
```   276 AddIs  [RangeI];
```
```   277 AddSEs [RangeE];
```
```   278
```
```   279 Goal "Range Id = UNIV";
```
```   280 by (Blast_tac 1);
```
```   281 qed "Range_Id";
```
```   282 Addsimps [Range_Id];
```
```   283
```
```   284 Goal "Range (diag A) = A";
```
```   285 by Auto_tac;
```
```   286 qed "Range_diag";
```
```   287 Addsimps [Range_diag];
```
```   288
```
```   289 Goal "Range(A Un B) = Range(A) Un Range(B)";
```
```   290 by (Blast_tac 1);
```
```   291 qed "Range_Un_eq";
```
```   292
```
```   293 Goal "Range(A Int B) <= Range(A) Int Range(B)";
```
```   294 by (Blast_tac 1);
```
```   295 qed "Range_Int_subset";
```
```   296
```
```   297 Goal "Range(A) - Range(B) <= Range(A - B)";
```
```   298 by (Blast_tac 1);
```
```   299 qed "Range_Diff_subset";
```
```   300
```
```   301 Goal "Range (Union S) = (UN A:S. Range A)";
```
```   302 by (Blast_tac 1);
```
```   303 qed "Range_Union";
```
```   304
```
```   305
```
```   306 (*** Image of a set under a relation ***)
```
```   307
```
```   308 overload_1st_set "Relation.op ^^";
```
```   309
```
```   310 Goalw [Image_def] "b : r^^A = (? x:A. (x,b):r)";
```
```   311 by (Blast_tac 1);
```
```   312 qed "Image_iff";
```
```   313
```
```   314 Goalw [Image_def] "r^^{a} = {b. (a,b):r}";
```
```   315 by (Blast_tac 1);
```
```   316 qed "Image_singleton";
```
```   317
```
```   318 Goal "(b : r^^{a}) = ((a,b):r)";
```
```   319 by (rtac (Image_iff RS trans) 1);
```
```   320 by (Blast_tac 1);
```
```   321 qed "Image_singleton_iff";
```
```   322
```
```   323 AddIffs [Image_singleton_iff];
```
```   324
```
```   325 Goalw [Image_def] "[| (a,b): r;  a:A |] ==> b : r^^A";
```
```   326 by (Blast_tac 1);
```
```   327 qed "ImageI";
```
```   328
```
```   329 val major::prems = Goalw [Image_def]
```
```   330     "[| b: r^^A;  !!x.[| (x,b): r;  x:A |] ==> P |] ==> P";
```
```   331 by (rtac (major RS CollectE) 1);
```
```   332 by (Clarify_tac 1);
```
```   333 by (rtac (hd prems) 1);
```
```   334 by (REPEAT (etac bexE 1 ORELSE ares_tac prems 1)) ;
```
```   335 qed "ImageE";
```
```   336
```
```   337 AddIs  [ImageI];
```
```   338 AddSEs [ImageE];
```
```   339
```
```   340
```
```   341 Goal "R^^{} = {}";
```
```   342 by (Blast_tac 1);
```
```   343 qed "Image_empty";
```
```   344
```
```   345 Addsimps [Image_empty];
```
```   346
```
```   347 Goal "Id ^^ A = A";
```
```   348 by (Blast_tac 1);
```
```   349 qed "Image_Id";
```
```   350
```
```   351 Goal "diag A ^^ B = A Int B";
```
```   352 by (Blast_tac 1);
```
```   353 qed "Image_diag";
```
```   354
```
```   355 Addsimps [Image_Id, Image_diag];
```
```   356
```
```   357 Goal "R ^^ (A Int B) <= R ^^ A Int R ^^ B";
```
```   358 by (Blast_tac 1);
```
```   359 qed "Image_Int_subset";
```
```   360
```
```   361 Goal "R ^^ (A Un B) = R ^^ A Un R ^^ B";
```
```   362 by (Blast_tac 1);
```
```   363 qed "Image_Un";
```
```   364
```
```   365 Goal "r <= A Times B ==> r^^C <= B";
```
```   366 by (rtac subsetI 1);
```
```   367 by (REPEAT (eresolve_tac [asm_rl, ImageE, subsetD RS SigmaD2] 1)) ;
```
```   368 qed "Image_subset";
```
```   369
```
```   370 (*NOT suitable for rewriting*)
```
```   371 Goal "r^^B = (UN y: B. r^^{y})";
```
```   372 by (Blast_tac 1);
```
```   373 qed "Image_eq_UN";
```
```   374
```
```   375
```
```   376 section "Univalent";
```
```   377
```
```   378 Goalw [Univalent_def]
```
```   379      "!x y. (x,y):r --> (!z. (x,z):r --> y=z) ==> Univalent r";
```
```   380 by (assume_tac 1);
```
```   381 qed "UnivalentI";
```
```   382
```
```   383 Goalw [Univalent_def]
```
```   384      "[| Univalent r;  (x,y):r;  (x,z):r|] ==> y=z";
```
```   385 by Auto_tac;
```
```   386 qed "UnivalentD";
```
```   387
```
```   388
```
```   389 (** Graphs of partial functions **)
```
```   390
```
```   391 Goal "Domain{(x,y). y = f x & P x} = {x. P x}";
```
```   392 by (Blast_tac 1);
```
```   393 qed "Domain_partial_func";
```
```   394
```
```   395 Goal "Range{(x,y). y = f x & P x} = f``{x. P x}";
```
```   396 by (Blast_tac 1);
```
```   397 qed "Range_partial_func";
```
```   398
```
```   399
```
```   400 (** Composition of function and relation **)
```
```   401
```
```   402 Goalw [fun_rel_comp_def] "A <= B ==> fun_rel_comp f A <= fun_rel_comp f B";
```
```   403 by (Fast_tac 1);
```
```   404 qed "fun_rel_comp_mono";
```
```   405
```
```   406 Goalw [fun_rel_comp_def] "! x. ?! y. (f x, y) : R ==> ?! g. g : fun_rel_comp f R";
```
```   407 by (res_inst_tac [("a","%x. @y. (f x, y) : R")] ex1I 1);
```
```   408 by (rtac CollectI 1);
```
```   409 by (rtac allI 1);
```
```   410 by (etac allE 1);
```
```   411 by (rtac (select_eq_Ex RS iffD2) 1);
```
```   412 by (etac ex1_implies_ex 1);
```
```   413 by (rtac ext 1);
```
```   414 by (etac CollectE 1);
```
```   415 by (REPEAT (etac allE 1));
```
```   416 by (rtac (select1_equality RS sym) 1);
```
```   417 by (atac 1);
```
```   418 by (atac 1);
```
```   419 qed "fun_rel_comp_unique";
```