src/HOL/MiniML/Type.ML
author paulson
Thu Feb 19 18:24:08 2004 +0100 (2004-02-19)
changeset 14401 477380c74c1d
parent 13630 a013a9dd370f
permissions -rw-r--r--
removal of the legacy ML structure List
     1 (* Title:     HOL/MiniML/Type.thy
     2    ID:        $Id$
     3    Author:    Wolfgang Naraschewski and Tobias Nipkow
     4    Copyright  1996 TU Muenchen
     5 *)
     6 
     7 Addsimps [mgu_eq,mgu_mg,mgu_free];
     8 
     9 
    10 (* lemmata for make scheme *)
    11 
    12 Goal "mk_scheme t = sch1 =-> sch2 --> (? t1 t2. sch1 = mk_scheme t1 & sch2 = mk_scheme t2)";
    13 by (induct_tac "t" 1);
    14 by (Simp_tac 1);
    15 by (Asm_full_simp_tac 1);
    16 by (Fast_tac 1);
    17 qed_spec_mp "mk_scheme_Fun";
    18 
    19 Goal "!t'. mk_scheme t = mk_scheme t' --> t=t'";
    20 by (induct_tac "t" 1);
    21  by (rtac allI 1);
    22  by (induct_tac "t'" 1);
    23   by (Simp_tac 1);
    24  by (Asm_full_simp_tac 1);
    25 by (rtac allI 1);
    26 by (induct_tac "t'" 1);
    27  by (Simp_tac 1);
    28 by (Asm_full_simp_tac 1);
    29 qed_spec_mp "mk_scheme_injective";
    30 
    31 Goal "free_tv (mk_scheme t) = free_tv t";
    32 by (induct_tac "t" 1);
    33 by (ALLGOALS Asm_simp_tac);
    34 qed "free_tv_mk_scheme";
    35 
    36 Addsimps [free_tv_mk_scheme];
    37 
    38 Goal "$ S (mk_scheme t) = mk_scheme ($ S t)";
    39 by (induct_tac "t" 1);
    40 by (ALLGOALS Asm_simp_tac);
    41 qed "subst_mk_scheme";
    42 
    43 Addsimps [subst_mk_scheme];
    44 
    45 
    46 (* constructor laws for app_subst *)
    47 
    48 Goalw [app_subst_list]
    49   "$ S [] = []";
    50 by (Simp_tac 1);
    51 qed "app_subst_Nil";
    52 
    53 Goalw [app_subst_list]
    54   "$ S (x#l) = ($ S x)#($ S l)";
    55 by (Simp_tac 1);
    56 qed "app_subst_Cons";
    57 
    58 Addsimps [app_subst_Nil,app_subst_Cons];
    59 
    60 
    61 (* constructor laws for new_tv *)
    62 
    63 Goalw [new_tv_def]
    64   "new_tv n (TVar m) = (m<n)";
    65 by (fast_tac (HOL_cs addss simpset()) 1);
    66 qed "new_tv_TVar";
    67 
    68 Goalw [new_tv_def]
    69   "new_tv n (FVar m) = (m<n)";
    70 by (fast_tac (HOL_cs addss simpset()) 1);
    71 qed "new_tv_FVar";
    72 
    73 Goalw [new_tv_def]
    74   "new_tv n (BVar m) = True";
    75 by (Simp_tac 1);
    76 qed "new_tv_BVar";
    77 
    78 Goalw [new_tv_def]
    79   "new_tv n (t1 -> t2) = (new_tv n t1 & new_tv n t2)";
    80 by (fast_tac (HOL_cs addss simpset()) 1);
    81 qed "new_tv_Fun";
    82 
    83 Goalw [new_tv_def]
    84   "new_tv n (t1 =-> t2) = (new_tv n t1 & new_tv n t2)";
    85 by (fast_tac (HOL_cs addss simpset()) 1);
    86 qed "new_tv_Fun2";
    87 
    88 Goalw [new_tv_def]
    89   "new_tv n []";
    90 by (Simp_tac 1);
    91 qed "new_tv_Nil";
    92 
    93 Goalw [new_tv_def]
    94   "new_tv n (x#l) = (new_tv n x & new_tv n l)";
    95 by (fast_tac (HOL_cs addss simpset()) 1);
    96 qed "new_tv_Cons";
    97 
    98 Goalw [new_tv_def] "new_tv n TVar";
    99 by (strip_tac 1);
   100 by (asm_full_simp_tac (simpset() addsimps [free_tv_subst,dom_def,cod_def]) 1);
   101 qed "new_tv_TVar_subst";
   102 
   103 Addsimps [new_tv_TVar,new_tv_FVar,new_tv_BVar,new_tv_Fun,new_tv_Fun2,new_tv_Nil,new_tv_Cons,new_tv_TVar_subst];
   104 
   105 Goalw [id_subst_def,new_tv_def,free_tv_subst,dom_def,cod_def]
   106   "new_tv n id_subst";
   107 by (Simp_tac 1);
   108 qed "new_tv_id_subst";
   109 Addsimps[new_tv_id_subst];
   110 
   111 Goal "new_tv n (sch::type_scheme) --> \
   112 \     $(%k. if k<n then S k else S' k) sch = $S sch";
   113 by (induct_tac "sch" 1);
   114 by (ALLGOALS Asm_simp_tac);
   115 qed "new_if_subst_type_scheme";
   116 Addsimps [new_if_subst_type_scheme];
   117 
   118 Goal "new_tv n (A::type_scheme list) --> \
   119 \     $(%k. if k<n then S k else S' k) A = $S A";
   120 by (induct_tac "A" 1);
   121 by (ALLGOALS Asm_simp_tac);
   122 qed "new_if_subst_type_scheme_list";
   123 Addsimps [new_if_subst_type_scheme_list];
   124 
   125 
   126 (* constructor laws for dom and cod *)
   127 
   128 Goalw [dom_def,id_subst_def,empty_def]
   129   "dom id_subst = {}";
   130 by (Simp_tac 1);
   131 qed "dom_id_subst";
   132 Addsimps [dom_id_subst];
   133 
   134 Goalw [cod_def]
   135   "cod id_subst = {}";
   136 by (Simp_tac 1);
   137 qed "cod_id_subst";
   138 Addsimps [cod_id_subst];
   139 
   140 
   141 (* lemmata for free_tv *)
   142 
   143 Goalw [free_tv_subst]
   144   "free_tv id_subst = {}";
   145 by (Simp_tac 1);
   146 qed "free_tv_id_subst";
   147 Addsimps [free_tv_id_subst];
   148 
   149 Goal "!n. n < length A --> x : free_tv (A!n) --> x : free_tv A";
   150 by (induct_tac "A" 1);
   151 by (Asm_full_simp_tac 1);
   152 by (rtac allI 1);
   153 by (induct_thm_tac nat_induct "n" 1);
   154 by (Asm_full_simp_tac 1);
   155 by (Asm_full_simp_tac 1);
   156 qed_spec_mp "free_tv_nth_A_impl_free_tv_A";
   157 
   158 Addsimps [free_tv_nth_A_impl_free_tv_A];
   159 
   160 Goal "!nat. nat < length A --> x : free_tv (A!nat) --> x : free_tv A";
   161 by (induct_tac "A" 1);
   162 by (Asm_full_simp_tac 1);
   163 by (rtac allI 1);
   164 by (induct_thm_tac nat_induct "nat" 1);
   165 by (strip_tac 1);
   166 by (Asm_full_simp_tac 1);
   167 by (Simp_tac 1);
   168 qed_spec_mp "free_tv_nth_nat_A";
   169 
   170 
   171 (* if two substitutions yield the same result if applied to a type
   172    structure the substitutions coincide on the free type variables
   173    occurring in the type structure *)
   174 
   175 Goal "(!x:free_tv t. (S x) = (S' x)) --> $ S (t::typ) = $ S' t";
   176 by (induct_tac "t" 1);
   177 by (Simp_tac 1);
   178 by (Asm_full_simp_tac 1);
   179 qed_spec_mp "typ_substitutions_only_on_free_variables";
   180 
   181 Goal  "(!n. n:(free_tv t) --> S1 n = S2 n) ==> $ S1 (t::typ) = $ S2 t";
   182 by (rtac typ_substitutions_only_on_free_variables 1);
   183 by (simp_tac (simpset() addsimps [Ball_def]) 1);
   184 qed "eq_free_eq_subst_te";
   185 
   186 Goal "(!x:free_tv sch. (S x) = (S' x)) --> $ S (sch::type_scheme) = $ S' sch";
   187 by (induct_tac "sch" 1);
   188 by (Simp_tac 1);
   189 by (Simp_tac 1);
   190 by (Asm_full_simp_tac 1);
   191 qed_spec_mp "scheme_substitutions_only_on_free_variables";
   192 
   193 Goal
   194   "(!n. n:(free_tv sch) --> S1 n = S2 n) ==> $ S1 (sch::type_scheme) = $ S2 sch";
   195 by (rtac scheme_substitutions_only_on_free_variables 1);
   196 by (simp_tac (simpset() addsimps [Ball_def]) 1);
   197 qed "eq_free_eq_subst_type_scheme";
   198 
   199 Goal
   200   "(!n. n:(free_tv A) --> S1 n = S2 n) --> $S1 (A::type_scheme list) = $S2 A";
   201 by (induct_tac "A" 1); 
   202 (* case [] *)
   203 by (fast_tac (HOL_cs addss simpset()) 1);
   204 (* case x#xl *)
   205 by (fast_tac (HOL_cs addIs [eq_free_eq_subst_type_scheme] addss (simpset())) 1);
   206 qed_spec_mp "eq_free_eq_subst_scheme_list";
   207 
   208 Goal "((!x:A. (P x)) --> Q) ==> ((!x:A Un B. (P x)) --> Q)";
   209 by (Fast_tac 1);
   210 val weaken_asm_Un = result ();
   211 
   212 Goal "(!x:free_tv A. (S x) = (S' x)) --> $ S (A::type_scheme list) = $ S' A";
   213 by (induct_tac "A" 1);
   214 by (Simp_tac 1);
   215 by (Asm_full_simp_tac 1);
   216 by (rtac weaken_asm_Un 1);
   217 by (strip_tac 1);
   218 by (etac scheme_substitutions_only_on_free_variables 1);
   219 qed_spec_mp "scheme_list_substitutions_only_on_free_variables";
   220 
   221 Goal
   222   "$ S1 (t::typ) = $ S2 t --> n:(free_tv t) --> S1 n = S2 n";
   223 by (induct_tac "t" 1);
   224 (* case TVar n *)
   225 by (fast_tac (HOL_cs addss simpset()) 1);
   226 (* case Fun t1 t2 *)
   227 by (fast_tac (HOL_cs addss simpset()) 1);
   228 qed_spec_mp "eq_subst_te_eq_free";
   229 
   230 Goal
   231   "$ S1 (sch::type_scheme) = $ S2 sch --> n:(free_tv sch) --> S1 n = S2 n";
   232 by (induct_tac "sch" 1);
   233 (* case TVar n *)
   234 by (Asm_simp_tac 1);
   235 (* case BVar n *)
   236 by (strip_tac 1);
   237 by (etac mk_scheme_injective 1);
   238 by (Asm_simp_tac 1);
   239 (* case Fun t1 t2 *)
   240 by (Asm_full_simp_tac 1);
   241 qed_spec_mp "eq_subst_type_scheme_eq_free";
   242 
   243 Goal
   244   "$S1 (A::type_scheme list) = $S2 A --> n:(free_tv A) --> S1 n = S2 n";
   245 by (induct_tac "A" 1);
   246 (* case [] *)
   247 by (fast_tac (HOL_cs addss simpset()) 1);
   248 (* case x#xl *)
   249 by (fast_tac (HOL_cs addIs [eq_subst_type_scheme_eq_free] addss (simpset())) 1);
   250 qed_spec_mp "eq_subst_scheme_list_eq_free";
   251 
   252 Goalw [free_tv_subst] 
   253       "v : cod S ==> v : free_tv S";
   254 by (fast_tac set_cs 1);
   255 qed "codD";
   256  
   257 Goalw [free_tv_subst,dom_def] 
   258       "x ~: free_tv S ==> S x = TVar x";
   259 by (fast_tac set_cs 1);
   260 qed "not_free_impl_id";
   261 
   262 Goalw [new_tv_def] 
   263       "[| new_tv n t; m:free_tv t |] ==> m<n";
   264 by (fast_tac HOL_cs 1 );
   265 qed "free_tv_le_new_tv";
   266 
   267 Goalw [dom_def,cod_def,UNION_def,Bex_def]
   268   "[| v : free_tv(S n); v~=n |] ==> v : cod S";
   269 by (Simp_tac 1);
   270 by (safe_tac (empty_cs addSIs [exI,conjI,notI]));
   271 by (assume_tac 2);
   272 by (Asm_full_simp_tac 1);
   273 qed "cod_app_subst";
   274 Addsimps [cod_app_subst];
   275 
   276 Goal "free_tv (S (v::nat)) <= insert v (cod S)";
   277 by (expand_case_tac "v:dom S" 1);
   278 by (fast_tac (set_cs addss (simpset() addsimps [cod_def])) 1);
   279 by (fast_tac (set_cs addss (simpset() addsimps [dom_def])) 1);
   280 qed "free_tv_subst_var";
   281 
   282 Goal "free_tv ($ S (t::typ)) <= cod S Un free_tv t";
   283 by (induct_tac "t" 1);
   284 (* case TVar n *)
   285 by (simp_tac (simpset() addsimps [free_tv_subst_var]) 1);
   286 (* case Fun t1 t2 *)
   287 by (fast_tac (set_cs addss simpset()) 1);
   288 qed "free_tv_app_subst_te";     
   289 
   290 Goal "free_tv ($ S (sch::type_scheme)) <= cod S Un free_tv sch";
   291 by (induct_tac "sch" 1);
   292 (* case FVar n *)
   293 by (simp_tac (simpset() addsimps [free_tv_subst_var]) 1);
   294 (* case BVar n *)
   295 by (Simp_tac 1);
   296 (* case Fun t1 t2 *)
   297 by (fast_tac (set_cs addss simpset()) 1);
   298 qed "free_tv_app_subst_type_scheme";  
   299 
   300 Goal "free_tv ($ S (A::type_scheme list)) <= cod S Un free_tv A";
   301 by (induct_tac "A" 1);
   302 (* case [] *)
   303 by (Simp_tac 1);
   304 (* case a#al *)
   305 by (cut_facts_tac [free_tv_app_subst_type_scheme] 1);
   306 by (fast_tac (set_cs addss simpset()) 1);
   307 qed "free_tv_app_subst_scheme_list";
   308 
   309 Goalw [free_tv_subst,dom_def]
   310      "free_tv (%u::nat. $ s1 (s2 u) :: typ) <=   \
   311 \     free_tv s1 Un free_tv s2";
   312 by (fast_tac (set_cs addSDs [free_tv_app_subst_te RS subsetD,
   313                              free_tv_subst_var RS subsetD] 
   314                      addss (simpset() delsimps bex_simps
   315                                      addsimps [cod_def,dom_def])) 1);
   316 qed "free_tv_comp_subst";
   317 
   318 Goalw [o_def] 
   319      "free_tv ($ S1 o S2) <= free_tv S1 Un free_tv (S2 :: nat => typ)";
   320 by (rtac free_tv_comp_subst 1);
   321 qed "free_tv_o_subst";
   322 
   323 Goal "n : free_tv t --> free_tv (S n) <= free_tv ($ S t::typ)";
   324 by (induct_tac "t" 1);
   325 by (Simp_tac 1);
   326 by (Simp_tac 1);
   327 by (Fast_tac 1);
   328 qed_spec_mp "free_tv_of_substitutions_extend_to_types";
   329 
   330 Goal "n : free_tv sch --> free_tv (S n) <= free_tv ($ S sch::type_scheme)";
   331 by (induct_tac "sch" 1);
   332 by (Simp_tac 1);
   333 by (Simp_tac 1);
   334 by (Simp_tac 1);
   335 by (Fast_tac 1);
   336 qed_spec_mp "free_tv_of_substitutions_extend_to_schemes";
   337 
   338 Goal "n : free_tv A --> free_tv (S n) <= free_tv ($ S A::type_scheme list)";
   339 by (induct_tac "A" 1);
   340 by (Simp_tac 1);
   341 by (Simp_tac 1);
   342 by (fast_tac (claset() addDs [free_tv_of_substitutions_extend_to_schemes]) 1);
   343 qed_spec_mp "free_tv_of_substitutions_extend_to_scheme_lists";
   344 
   345 Addsimps [free_tv_of_substitutions_extend_to_scheme_lists];
   346 
   347 Goal "!!sch::type_scheme. (n : free_tv sch) = (n: set (free_tv_ML sch))";
   348 by (induct_tac "sch" 1);
   349 by (ALLGOALS Asm_simp_tac);
   350 qed "free_tv_ML_scheme";
   351 
   352 Goal "!!A::type_scheme list. (n : free_tv A) = (n: set (free_tv_ML A))";
   353 by (induct_tac "A" 1);
   354 by (Simp_tac 1);
   355 by (asm_simp_tac (simpset() addsimps [free_tv_ML_scheme]) 1);
   356 qed "free_tv_ML_scheme_list";
   357 
   358 
   359 (* lemmata for bound_tv *)
   360 
   361 Goal "bound_tv (mk_scheme t) = {}";
   362 by (induct_tac "t" 1);
   363 by (ALLGOALS Asm_simp_tac);
   364 qed "bound_tv_mk_scheme";
   365 
   366 Addsimps [bound_tv_mk_scheme];
   367 
   368 Goal "!!sch::type_scheme. bound_tv ($ S sch) = bound_tv sch";
   369 by (induct_tac "sch" 1);
   370 by (ALLGOALS Asm_simp_tac);
   371 qed "bound_tv_subst_scheme";
   372 
   373 Addsimps [bound_tv_subst_scheme];
   374 
   375 Goal "!!A::type_scheme list. bound_tv ($ S A) = bound_tv A";
   376 by (rtac (thm"list.induct") 1);
   377 by (Simp_tac 1);
   378 by (Asm_simp_tac 1);
   379 qed "bound_tv_subst_scheme_list";
   380 
   381 Addsimps [bound_tv_subst_scheme_list];
   382 
   383 
   384 (* lemmata for new_tv *)
   385 
   386 Goalw [new_tv_def]
   387   "new_tv n S = ((!m. n <= m --> (S m = TVar m)) & \
   388 \                (! l. l < n --> new_tv n (S l) ))";
   389 by (safe_tac HOL_cs );
   390 (* ==> *)
   391 by (fast_tac (HOL_cs addDs [leD] addss (simpset() addsimps [free_tv_subst,dom_def])) 1);
   392 by (subgoal_tac "m:cod S | S l = TVar l" 1);
   393 by (safe_tac HOL_cs );
   394 by (fast_tac (HOL_cs addDs [UnI2] addss (simpset() addsimps [free_tv_subst])) 1);
   395 by (dres_inst_tac [("P","%x. m:free_tv x")] subst 1 THEN atac 1);
   396 by (Asm_full_simp_tac 1);
   397 by (fast_tac (set_cs addss (simpset() addsimps [free_tv_subst,cod_def,dom_def])) 1);
   398 (* <== *)
   399 by (rewrite_goals_tac [free_tv_subst,cod_def,dom_def] );
   400 by (safe_tac set_cs );
   401 by (cut_inst_tac [("m","m"),("n","n")] less_linear 1);
   402 by (fast_tac (HOL_cs addSIs [less_or_eq_imp_le]) 1);
   403 by (cut_inst_tac [("m","ma"),("n","n")] less_linear 1);
   404 by (fast_tac (HOL_cs addSIs [less_or_eq_imp_le]) 1);
   405 qed "new_tv_subst";
   406 
   407 Goal "new_tv n x = (!y:set x. new_tv n y)";
   408 by (induct_tac "x" 1);
   409 by (ALLGOALS Asm_simp_tac);
   410 qed "new_tv_list";
   411 
   412 (* substitution affects only variables occurring freely *)
   413 Goal
   414   "new_tv n (t::typ) --> $(%x. if x=n then t' else S x) t = $S t";
   415 by (induct_tac "t" 1);
   416 by (ALLGOALS Asm_simp_tac);
   417 qed "subst_te_new_tv";
   418 Addsimps [subst_te_new_tv];
   419 
   420 Goal
   421   "new_tv n (sch::type_scheme) --> $(%x. if x=n then sch' else S x) sch = $S sch";
   422 by (induct_tac "sch" 1);
   423 by (ALLGOALS Asm_simp_tac);
   424 qed_spec_mp "subst_te_new_type_scheme";
   425 Addsimps [subst_te_new_type_scheme];
   426 
   427 Goal
   428   "new_tv n (A::type_scheme list) --> $(%x. if x=n then t else S x) A = $S A";
   429 by (induct_tac "A" 1);
   430 by (ALLGOALS Asm_full_simp_tac);
   431 qed_spec_mp "subst_tel_new_scheme_list";
   432 Addsimps [subst_tel_new_scheme_list];
   433 
   434 (* all greater variables are also new *)
   435 Goalw [new_tv_def] 
   436   "n<=m ==> new_tv n t ==> new_tv m t";
   437 by Safe_tac;
   438 by (dtac spec 1);
   439 by (mp_tac 1);
   440 by (Simp_tac 1);
   441 qed "new_tv_le";
   442 Addsimps [lessI RS less_imp_le RS new_tv_le];
   443 
   444 Goal "n<=m ==> new_tv n (t::typ) ==> new_tv m t";
   445 by (asm_simp_tac (simpset() addsimps [new_tv_le]) 1);
   446 qed "new_tv_typ_le";
   447 
   448 Goal "n<=m ==> new_tv n (A::type_scheme list) ==> new_tv m A";
   449 by (asm_simp_tac (simpset() addsimps [new_tv_le]) 1);
   450 qed "new_scheme_list_le";
   451 
   452 Goal "n<=m ==> new_tv n (S::subst) ==> new_tv m S";
   453 by (asm_simp_tac (simpset() addsimps [new_tv_le]) 1);
   454 qed "new_tv_subst_le";
   455 
   456 (* new_tv property remains if a substitution is applied *)
   457 Goal
   458   "[| n<m; new_tv m (S::subst) |] ==> new_tv m (S n)";
   459 by (asm_full_simp_tac (simpset() addsimps [new_tv_subst]) 1);
   460 qed "new_tv_subst_var";
   461 
   462 Goal
   463   "new_tv n S --> new_tv n (t::typ) --> new_tv n ($ S t)";
   464 by (induct_tac "t" 1);
   465 by (ALLGOALS(fast_tac (HOL_cs addss (simpset() addsimps [new_tv_subst]))));
   466 qed_spec_mp "new_tv_subst_te";
   467 Addsimps [new_tv_subst_te];
   468 
   469 Goal "new_tv n S --> new_tv n (sch::type_scheme) --> new_tv n ($ S sch)";
   470 by (induct_tac "sch" 1);
   471 by (ALLGOALS (Asm_full_simp_tac));
   472 by (rewtac new_tv_def);
   473 by (simp_tac (simpset() addsimps [free_tv_subst,dom_def,cod_def]) 1);
   474 by (strip_tac 1);
   475 by (case_tac "S nat = TVar nat" 1);
   476 by (Asm_full_simp_tac 1);
   477 by (dres_inst_tac [("x","m")] spec 1);
   478 by (Fast_tac 1);
   479 qed_spec_mp "new_tv_subst_type_scheme";
   480 Addsimps [new_tv_subst_type_scheme];
   481 
   482 Goal
   483   "new_tv n S --> new_tv n (A::type_scheme list) --> new_tv n ($ S A)";
   484 by (induct_tac "A" 1);
   485 by (ALLGOALS(fast_tac (HOL_cs addss (simpset()))));
   486 qed_spec_mp "new_tv_subst_scheme_list";
   487 Addsimps [new_tv_subst_scheme_list];
   488 
   489 Goal "new_tv n A --> new_tv (Suc n) ((TVar n)#A)";
   490 by (simp_tac (simpset() addsimps [new_tv_list]) 1);
   491 qed "new_tv_Suc_list";
   492 
   493 Goalw [new_tv_def] "!!sch::type_scheme. (free_tv sch) = (free_tv sch') --> (new_tv n sch --> new_tv n sch')";
   494 by (Asm_simp_tac 1);
   495 qed_spec_mp "new_tv_only_depends_on_free_tv_type_scheme";
   496 
   497 Goalw [new_tv_def] "!!A::type_scheme list. (free_tv A) = (free_tv A') --> (new_tv n A --> new_tv n A')";
   498 by (Asm_simp_tac 1);
   499 qed_spec_mp "new_tv_only_depends_on_free_tv_scheme_list";
   500 
   501 Goalw [new_tv_def]
   502   "!nat. nat < length A --> new_tv n A --> (new_tv n (A!nat))";
   503 by (induct_tac "A" 1);
   504 by (Asm_full_simp_tac 1);
   505 by (rtac allI 1);
   506 by (induct_thm_tac nat_induct "nat" 1);
   507 by (strip_tac 1);
   508 by (Asm_full_simp_tac 1);
   509 by (Simp_tac 1);
   510 qed_spec_mp "new_tv_nth_nat_A";
   511 
   512 
   513 (* composition of substitutions preserves new_tv proposition *)
   514 Goal "[| new_tv n (S::subst); new_tv n R |] ==> new_tv n (($ R) o S)";
   515 by (asm_full_simp_tac (simpset() addsimps [new_tv_subst]) 1);
   516 qed "new_tv_subst_comp_1";
   517 
   518 Goal "[| new_tv n (S::subst); new_tv n R |] ==> new_tv n (%v.$ R (S v))";
   519 by (asm_full_simp_tac (simpset() addsimps [new_tv_subst]) 1);
   520 qed "new_tv_subst_comp_2";
   521 
   522 Addsimps [new_tv_subst_comp_1,new_tv_subst_comp_2];
   523 
   524 (* new type variables do not occur freely in a type structure *)
   525 Goalw [new_tv_def] 
   526       "new_tv n A ==> n~:(free_tv A)";
   527 by (fast_tac (HOL_cs addEs [less_irrefl]) 1);
   528 qed "new_tv_not_free_tv";
   529 Addsimps [new_tv_not_free_tv];
   530 
   531 Goalw [new_tv_def] "!!t::typ. ? n. (new_tv n t)";
   532 by (induct_tac "t" 1);
   533 by (res_inst_tac [("x","Suc nat")] exI 1);
   534 by (Asm_simp_tac 1);
   535 by (REPEAT (etac exE 1));
   536 by (rename_tac "n'" 1);
   537 by (res_inst_tac [("x","max n n'")] exI 1);
   538 by (simp_tac (simpset() addsimps [less_max_iff_disj]) 1);
   539 by (Blast_tac 1);
   540 qed "fresh_variable_types";
   541 
   542 Addsimps [fresh_variable_types];
   543 
   544 Goalw [new_tv_def] "!!sch::type_scheme. ? n. (new_tv n sch)";
   545 by (induct_tac "sch" 1);
   546 by (res_inst_tac [("x","Suc nat")] exI 1);
   547 by (Simp_tac 1);
   548 by (res_inst_tac [("x","Suc nat")] exI 1);
   549 by (Simp_tac 1);
   550 by (REPEAT (etac exE 1));
   551 by (rename_tac "n'" 1);
   552 by (res_inst_tac [("x","max n n'")] exI 1);
   553 by (simp_tac (simpset() addsimps [less_max_iff_disj]) 1);
   554 by (Blast_tac 1);
   555 qed "fresh_variable_type_schemes";
   556 
   557 Addsimps [fresh_variable_type_schemes];
   558 
   559 Goal "!!A::type_scheme list. ? n. (new_tv n A)";
   560 by (induct_tac "A" 1);
   561 by (Simp_tac 1);
   562 by (Simp_tac 1);
   563 by (etac exE 1);
   564 by (cut_inst_tac [("sch","a")] fresh_variable_type_schemes 1);
   565 by (etac exE 1);
   566 by (rename_tac "n'" 1);
   567 by (res_inst_tac [("x","(max n n')")] exI 1);
   568 by (subgoal_tac "n <= (max n n')" 1);
   569 by (subgoal_tac "n' <= (max n n')" 1);
   570 by (fast_tac (claset() addDs [new_tv_le]) 1);
   571 by (ALLGOALS (simp_tac (simpset() addsimps [le_max_iff_disj])));
   572 qed "fresh_variable_type_scheme_lists";
   573 
   574 Addsimps [fresh_variable_type_scheme_lists];
   575 
   576 Goal "[| ? n1. (new_tv n1 x); ? n2. (new_tv n2 y)|] ==> ? n. (new_tv n x) & (new_tv n y)";
   577 by (REPEAT (etac exE 1));
   578 by (rename_tac "n1 n2" 1);
   579 by (res_inst_tac [("x","(max n1 n2)")] exI 1);
   580 by (subgoal_tac "n1 <= max n1 n2" 1);
   581 by (subgoal_tac "n2 <= max n1 n2" 1);
   582 by (fast_tac (claset() addDs [new_tv_le]) 1);
   583 by (ALLGOALS (simp_tac (simpset() addsimps [le_max_iff_disj])));
   584 qed "make_one_new_out_of_two";
   585 
   586 Goal "!!(A::type_scheme list) (A'::type_scheme list) (t::typ) (t'::typ). \
   587 \         ? n. (new_tv n A) & (new_tv n A') & (new_tv n t) & (new_tv n t')" ;
   588 by (cut_inst_tac [("t","t")] fresh_variable_types 1);
   589 by (cut_inst_tac [("t","t'")] fresh_variable_types 1);
   590 by (dtac make_one_new_out_of_two 1);
   591 by (assume_tac 1);
   592 by (thin_tac "? n. new_tv n t'" 1);
   593 by (cut_inst_tac [("A","A")] fresh_variable_type_scheme_lists 1);
   594 by (cut_inst_tac [("A","A'")] fresh_variable_type_scheme_lists 1);
   595 by (rotate_tac 1 1);
   596 by (dtac make_one_new_out_of_two 1);
   597 by (assume_tac 1);
   598 by (thin_tac "? n. new_tv n A'" 1);
   599 by (REPEAT (etac exE 1));
   600 by (rename_tac "n2 n1" 1);
   601 by (res_inst_tac [("x","(max n1 n2)")] exI 1);
   602 by (rewtac new_tv_def);
   603 by (simp_tac (simpset() addsimps [less_max_iff_disj]) 1);
   604 by (Blast_tac 1);
   605 qed "ex_fresh_variable";
   606 
   607 (* mgu does not introduce new type variables *)
   608 Goalw [new_tv_def] 
   609       "[|mgu t1 t2 = Some u; new_tv n t1; new_tv n t2|] ==> new_tv n u";
   610 by (fast_tac (set_cs addDs [mgu_free]) 1);
   611 qed "mgu_new";
   612 
   613 
   614 (* lemmata for substitutions *)
   615 
   616 Goalw [app_subst_list] 
   617    "!!A:: ('a::type_struct) list. length ($ S A) = length A";
   618 by (Simp_tac 1);
   619 qed "length_app_subst_list";
   620 Addsimps [length_app_subst_list];
   621 
   622 Goal "!!sch::type_scheme. $ TVar sch = sch";
   623 by (induct_tac "sch" 1);
   624 by (ALLGOALS Asm_simp_tac);
   625 qed "subst_TVar_scheme";
   626 
   627 Addsimps [subst_TVar_scheme];
   628 
   629 Goal "!!A::type_scheme list. $ TVar A = A";
   630 by (rtac (thm"list.induct") 1);
   631 by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [app_subst_list])));
   632 qed "subst_TVar_scheme_list";
   633 
   634 Addsimps [subst_TVar_scheme_list];
   635 
   636 (* application of id_subst does not change type expression *)
   637 Goalw [id_subst_def]
   638   "$ id_subst = (%t::typ. t)";
   639 by (rtac ext 1);
   640 by (induct_tac "t" 1);
   641 by (ALLGOALS Asm_simp_tac);
   642 qed "app_subst_id_te";
   643 Addsimps [app_subst_id_te];
   644 
   645 Goalw [id_subst_def]
   646   "$ id_subst = (%sch::type_scheme. sch)";
   647 by (rtac ext 1);
   648 by (induct_tac "sch" 1);
   649 by (ALLGOALS Asm_simp_tac);
   650 qed "app_subst_id_type_scheme";
   651 Addsimps [app_subst_id_type_scheme];
   652 
   653 (* application of id_subst does not change list of type expressions *)
   654 Goalw [app_subst_list]
   655   "$ id_subst = (%A::type_scheme list. A)";
   656 by (rtac ext 1); 
   657 by (induct_tac "A" 1);
   658 by (ALLGOALS Asm_simp_tac);
   659 qed "app_subst_id_tel";
   660 Addsimps [app_subst_id_tel];
   661 
   662 Goal "!!sch::type_scheme. $ id_subst sch = sch";
   663 by (induct_tac "sch" 1);
   664 by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [id_subst_def])));
   665 qed "id_subst_sch";
   666 
   667 Addsimps [id_subst_sch];
   668 
   669 Goal "!!A::type_scheme list. $ id_subst A = A";
   670 by (induct_tac "A" 1);
   671 by (Asm_full_simp_tac 1);
   672 by (Asm_full_simp_tac 1);
   673 qed "id_subst_A";
   674 
   675 Addsimps [id_subst_A];
   676 
   677 (* composition of substitutions *)
   678 Goalw [id_subst_def,o_def] "$S o id_subst = S";
   679 by (rtac ext 1);
   680 by (Simp_tac 1);
   681 qed "o_id_subst";
   682 Addsimps[o_id_subst];
   683 
   684 Goal "$ R ($ S t::typ) = $ (%x. $ R (S x) ) t";
   685 by (induct_tac "t" 1);
   686 by (ALLGOALS Asm_simp_tac);
   687 qed "subst_comp_te";
   688 
   689 Goal "$ R ($ S sch::type_scheme) = $ (%x. $ R (S x) ) sch";
   690 by (induct_tac "sch" 1);
   691 by (ALLGOALS Asm_full_simp_tac);
   692 qed "subst_comp_type_scheme";
   693 
   694 Goalw [app_subst_list]
   695   "$ R ($ S A::type_scheme list) = $ (%x. $ R (S x)) A";
   696 by (induct_tac "A" 1);
   697 (* case [] *)
   698 by (Simp_tac 1);
   699 (* case x#xl *)
   700 by (asm_full_simp_tac (simpset() addsimps [subst_comp_type_scheme]) 1);
   701 qed "subst_comp_scheme_list";
   702 
   703 Goal "!!A::type_scheme list. !x : free_tv A. S x = (TVar x) ==> $ S A = $ id_subst A";
   704 by (rtac scheme_list_substitutions_only_on_free_variables 1);
   705 by (asm_full_simp_tac (simpset() addsimps [id_subst_def]) 1);
   706 qed "subst_id_on_type_scheme_list'";
   707 
   708 Goal "!!A::type_scheme list. !x : free_tv A. S x = (TVar x) ==> $ S A = A";
   709 by (stac subst_id_on_type_scheme_list' 1);
   710 by (assume_tac 1);
   711 by (Simp_tac 1);
   712 qed "subst_id_on_type_scheme_list";
   713 
   714 Goal "!n. n < length A --> ($ S A)!n = $S (A!n)";
   715 by (induct_tac "A" 1);
   716 by (Asm_full_simp_tac 1);
   717 by (rtac allI 1);
   718 by (rename_tac "n1" 1);
   719 by (induct_thm_tac nat_induct "n1" 1);
   720 by (Asm_full_simp_tac 1);
   721 by (Asm_full_simp_tac 1);
   722 qed_spec_mp "nth_subst";