diff -r ee97b6463cb4 -r b8da5f258b04 src/HOL/MiniML/Type.ML
--- a/src/HOL/MiniML/Type.ML	Mon Mar 01 13:51:21 2004 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,722 +0,0 @@
-(* Title:     HOL/MiniML/Type.thy
-   ID:        $Id$
-   Author:    Wolfgang Naraschewski and Tobias Nipkow
-   Copyright  1996 TU Muenchen
-*)
-
-Addsimps [mgu_eq,mgu_mg,mgu_free];
-
-
-(* lemmata for make scheme *)
-
-Goal "mk_scheme t = sch1 =-> sch2 --> (? t1 t2. sch1 = mk_scheme t1 & sch2 = mk_scheme t2)";
-by (induct_tac "t" 1);
-by (Simp_tac 1);
-by (Asm_full_simp_tac 1);
-by (Fast_tac 1);
-qed_spec_mp "mk_scheme_Fun";
-
-Goal "!t'. mk_scheme t = mk_scheme t' --> t=t'";
-by (induct_tac "t" 1);
- by (rtac allI 1);
- by (induct_tac "t'" 1);
-  by (Simp_tac 1);
- by (Asm_full_simp_tac 1);
-by (rtac allI 1);
-by (induct_tac "t'" 1);
- by (Simp_tac 1);
-by (Asm_full_simp_tac 1);
-qed_spec_mp "mk_scheme_injective";
-
-Goal "free_tv (mk_scheme t) = free_tv t";
-by (induct_tac "t" 1);
-by (ALLGOALS Asm_simp_tac);
-qed "free_tv_mk_scheme";
-
-Addsimps [free_tv_mk_scheme];
-
-Goal "$ S (mk_scheme t) = mk_scheme ($ S t)";
-by (induct_tac "t" 1);
-by (ALLGOALS Asm_simp_tac);
-qed "subst_mk_scheme";
-
-Addsimps [subst_mk_scheme];
-
-
-(* constructor laws for app_subst *)
-
-Goalw [app_subst_list]
-  "$ S [] = []";
-by (Simp_tac 1);
-qed "app_subst_Nil";
-
-Goalw [app_subst_list]
-  "$ S (x#l) = ($ S x)#($ S l)";
-by (Simp_tac 1);
-qed "app_subst_Cons";
-
-Addsimps [app_subst_Nil,app_subst_Cons];
-
-
-(* constructor laws for new_tv *)
-
-Goalw [new_tv_def]
-  "new_tv n (TVar m) = (m<n)";
-by (fast_tac (HOL_cs addss simpset()) 1);
-qed "new_tv_TVar";
-
-Goalw [new_tv_def]
-  "new_tv n (FVar m) = (m<n)";
-by (fast_tac (HOL_cs addss simpset()) 1);
-qed "new_tv_FVar";
-
-Goalw [new_tv_def]
-  "new_tv n (BVar m) = True";
-by (Simp_tac 1);
-qed "new_tv_BVar";
-
-Goalw [new_tv_def]
-  "new_tv n (t1 -> t2) = (new_tv n t1 & new_tv n t2)";
-by (fast_tac (HOL_cs addss simpset()) 1);
-qed "new_tv_Fun";
-
-Goalw [new_tv_def]
-  "new_tv n (t1 =-> t2) = (new_tv n t1 & new_tv n t2)";
-by (fast_tac (HOL_cs addss simpset()) 1);
-qed "new_tv_Fun2";
-
-Goalw [new_tv_def]
-  "new_tv n []";
-by (Simp_tac 1);
-qed "new_tv_Nil";
-
-Goalw [new_tv_def]
-  "new_tv n (x#l) = (new_tv n x & new_tv n l)";
-by (fast_tac (HOL_cs addss simpset()) 1);
-qed "new_tv_Cons";
-
-Goalw [new_tv_def] "new_tv n TVar";
-by (strip_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [free_tv_subst,dom_def,cod_def]) 1);
-qed "new_tv_TVar_subst";
-
-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];
-
-Goalw [id_subst_def,new_tv_def,free_tv_subst,dom_def,cod_def]
-  "new_tv n id_subst";
-by (Simp_tac 1);
-qed "new_tv_id_subst";
-Addsimps[new_tv_id_subst];
-
-Goal "new_tv n (sch::type_scheme) --> \
-\     $(%k. if k<n then S k else S' k) sch = $S sch";
-by (induct_tac "sch" 1);
-by (ALLGOALS Asm_simp_tac);
-qed "new_if_subst_type_scheme";
-Addsimps [new_if_subst_type_scheme];
-
-Goal "new_tv n (A::type_scheme list) --> \
-\     $(%k. if k<n then S k else S' k) A = $S A";
-by (induct_tac "A" 1);
-by (ALLGOALS Asm_simp_tac);
-qed "new_if_subst_type_scheme_list";
-Addsimps [new_if_subst_type_scheme_list];
-
-
-(* constructor laws for dom and cod *)
-
-Goalw [dom_def,id_subst_def,empty_def]
-  "dom id_subst = {}";
-by (Simp_tac 1);
-qed "dom_id_subst";
-Addsimps [dom_id_subst];
-
-Goalw [cod_def]
-  "cod id_subst = {}";
-by (Simp_tac 1);
-qed "cod_id_subst";
-Addsimps [cod_id_subst];
-
-
-(* lemmata for free_tv *)
-
-Goalw [free_tv_subst]
-  "free_tv id_subst = {}";
-by (Simp_tac 1);
-qed "free_tv_id_subst";
-Addsimps [free_tv_id_subst];
-
-Goal "!n. n < length A --> x : free_tv (A!n) --> x : free_tv A";
-by (induct_tac "A" 1);
-by (Asm_full_simp_tac 1);
-by (rtac allI 1);
-by (induct_thm_tac nat_induct "n" 1);
-by (Asm_full_simp_tac 1);
-by (Asm_full_simp_tac 1);
-qed_spec_mp "free_tv_nth_A_impl_free_tv_A";
-
-Addsimps [free_tv_nth_A_impl_free_tv_A];
-
-Goal "!nat. nat < length A --> x : free_tv (A!nat) --> x : free_tv A";
-by (induct_tac "A" 1);
-by (Asm_full_simp_tac 1);
-by (rtac allI 1);
-by (induct_thm_tac nat_induct "nat" 1);
-by (strip_tac 1);
-by (Asm_full_simp_tac 1);
-by (Simp_tac 1);
-qed_spec_mp "free_tv_nth_nat_A";
-
-
-(* if two substitutions yield the same result if applied to a type
-   structure the substitutions coincide on the free type variables
-   occurring in the type structure *)
-
-Goal "(!x:free_tv t. (S x) = (S' x)) --> $ S (t::typ) = $ S' t";
-by (induct_tac "t" 1);
-by (Simp_tac 1);
-by (Asm_full_simp_tac 1);
-qed_spec_mp "typ_substitutions_only_on_free_variables";
-
-Goal  "(!n. n:(free_tv t) --> S1 n = S2 n) ==> $ S1 (t::typ) = $ S2 t";
-by (rtac typ_substitutions_only_on_free_variables 1);
-by (simp_tac (simpset() addsimps [Ball_def]) 1);
-qed "eq_free_eq_subst_te";
-
-Goal "(!x:free_tv sch. (S x) = (S' x)) --> $ S (sch::type_scheme) = $ S' sch";
-by (induct_tac "sch" 1);
-by (Simp_tac 1);
-by (Simp_tac 1);
-by (Asm_full_simp_tac 1);
-qed_spec_mp "scheme_substitutions_only_on_free_variables";
-
-Goal
-  "(!n. n:(free_tv sch) --> S1 n = S2 n) ==> $ S1 (sch::type_scheme) = $ S2 sch";
-by (rtac scheme_substitutions_only_on_free_variables 1);
-by (simp_tac (simpset() addsimps [Ball_def]) 1);
-qed "eq_free_eq_subst_type_scheme";
-
-Goal
-  "(!n. n:(free_tv A) --> S1 n = S2 n) --> $S1 (A::type_scheme list) = $S2 A";
-by (induct_tac "A" 1); 
-(* case [] *)
-by (fast_tac (HOL_cs addss simpset()) 1);
-(* case x#xl *)
-by (fast_tac (HOL_cs addIs [eq_free_eq_subst_type_scheme] addss (simpset())) 1);
-qed_spec_mp "eq_free_eq_subst_scheme_list";
-
-Goal "((!x:A. (P x)) --> Q) ==> ((!x:A Un B. (P x)) --> Q)";
-by (Fast_tac 1);
-val weaken_asm_Un = result ();
-
-Goal "(!x:free_tv A. (S x) = (S' x)) --> $ S (A::type_scheme list) = $ S' A";
-by (induct_tac "A" 1);
-by (Simp_tac 1);
-by (Asm_full_simp_tac 1);
-by (rtac weaken_asm_Un 1);
-by (strip_tac 1);
-by (etac scheme_substitutions_only_on_free_variables 1);
-qed_spec_mp "scheme_list_substitutions_only_on_free_variables";
-
-Goal
-  "$ S1 (t::typ) = $ S2 t --> n:(free_tv t) --> S1 n = S2 n";
-by (induct_tac "t" 1);
-(* case TVar n *)
-by (fast_tac (HOL_cs addss simpset()) 1);
-(* case Fun t1 t2 *)
-by (fast_tac (HOL_cs addss simpset()) 1);
-qed_spec_mp "eq_subst_te_eq_free";
-
-Goal
-  "$ S1 (sch::type_scheme) = $ S2 sch --> n:(free_tv sch) --> S1 n = S2 n";
-by (induct_tac "sch" 1);
-(* case TVar n *)
-by (Asm_simp_tac 1);
-(* case BVar n *)
-by (strip_tac 1);
-by (etac mk_scheme_injective 1);
-by (Asm_simp_tac 1);
-(* case Fun t1 t2 *)
-by (Asm_full_simp_tac 1);
-qed_spec_mp "eq_subst_type_scheme_eq_free";
-
-Goal
-  "$S1 (A::type_scheme list) = $S2 A --> n:(free_tv A) --> S1 n = S2 n";
-by (induct_tac "A" 1);
-(* case [] *)
-by (fast_tac (HOL_cs addss simpset()) 1);
-(* case x#xl *)
-by (fast_tac (HOL_cs addIs [eq_subst_type_scheme_eq_free] addss (simpset())) 1);
-qed_spec_mp "eq_subst_scheme_list_eq_free";
-
-Goalw [free_tv_subst] 
-      "v : cod S ==> v : free_tv S";
-by (fast_tac set_cs 1);
-qed "codD";
- 
-Goalw [free_tv_subst,dom_def] 
-      "x ~: free_tv S ==> S x = TVar x";
-by (fast_tac set_cs 1);
-qed "not_free_impl_id";
-
-Goalw [new_tv_def] 
-      "[| new_tv n t; m:free_tv t |] ==> m<n";
-by (fast_tac HOL_cs 1 );
-qed "free_tv_le_new_tv";
-
-Goalw [dom_def,cod_def,UNION_def,Bex_def]
-  "[| v : free_tv(S n); v~=n |] ==> v : cod S";
-by (Simp_tac 1);
-by (safe_tac (empty_cs addSIs [exI,conjI,notI]));
-by (assume_tac 2);
-by (Asm_full_simp_tac 1);
-qed "cod_app_subst";
-Addsimps [cod_app_subst];
-
-Goal "free_tv (S (v::nat)) <= insert v (cod S)";
-by (expand_case_tac "v:dom S" 1);
-by (fast_tac (set_cs addss (simpset() addsimps [cod_def])) 1);
-by (fast_tac (set_cs addss (simpset() addsimps [dom_def])) 1);
-qed "free_tv_subst_var";
-
-Goal "free_tv ($ S (t::typ)) <= cod S Un free_tv t";
-by (induct_tac "t" 1);
-(* case TVar n *)
-by (simp_tac (simpset() addsimps [free_tv_subst_var]) 1);
-(* case Fun t1 t2 *)
-by (fast_tac (set_cs addss simpset()) 1);
-qed "free_tv_app_subst_te";     
-
-Goal "free_tv ($ S (sch::type_scheme)) <= cod S Un free_tv sch";
-by (induct_tac "sch" 1);
-(* case FVar n *)
-by (simp_tac (simpset() addsimps [free_tv_subst_var]) 1);
-(* case BVar n *)
-by (Simp_tac 1);
-(* case Fun t1 t2 *)
-by (fast_tac (set_cs addss simpset()) 1);
-qed "free_tv_app_subst_type_scheme";  
-
-Goal "free_tv ($ S (A::type_scheme list)) <= cod S Un free_tv A";
-by (induct_tac "A" 1);
-(* case [] *)
-by (Simp_tac 1);
-(* case a#al *)
-by (cut_facts_tac [free_tv_app_subst_type_scheme] 1);
-by (fast_tac (set_cs addss simpset()) 1);
-qed "free_tv_app_subst_scheme_list";
-
-Goalw [free_tv_subst,dom_def]
-     "free_tv (%u::nat. $ s1 (s2 u) :: typ) <=   \
-\     free_tv s1 Un free_tv s2";
-by (fast_tac (set_cs addSDs [free_tv_app_subst_te RS subsetD,
-                             free_tv_subst_var RS subsetD] 
-                     addss (simpset() delsimps bex_simps
-                                     addsimps [cod_def,dom_def])) 1);
-qed "free_tv_comp_subst";
-
-Goalw [o_def] 
-     "free_tv ($ S1 o S2) <= free_tv S1 Un free_tv (S2 :: nat => typ)";
-by (rtac free_tv_comp_subst 1);
-qed "free_tv_o_subst";
-
-Goal "n : free_tv t --> free_tv (S n) <= free_tv ($ S t::typ)";
-by (induct_tac "t" 1);
-by (Simp_tac 1);
-by (Simp_tac 1);
-by (Fast_tac 1);
-qed_spec_mp "free_tv_of_substitutions_extend_to_types";
-
-Goal "n : free_tv sch --> free_tv (S n) <= free_tv ($ S sch::type_scheme)";
-by (induct_tac "sch" 1);
-by (Simp_tac 1);
-by (Simp_tac 1);
-by (Simp_tac 1);
-by (Fast_tac 1);
-qed_spec_mp "free_tv_of_substitutions_extend_to_schemes";
-
-Goal "n : free_tv A --> free_tv (S n) <= free_tv ($ S A::type_scheme list)";
-by (induct_tac "A" 1);
-by (Simp_tac 1);
-by (Simp_tac 1);
-by (fast_tac (claset() addDs [free_tv_of_substitutions_extend_to_schemes]) 1);
-qed_spec_mp "free_tv_of_substitutions_extend_to_scheme_lists";
-
-Addsimps [free_tv_of_substitutions_extend_to_scheme_lists];
-
-Goal "!!sch::type_scheme. (n : free_tv sch) = (n: set (free_tv_ML sch))";
-by (induct_tac "sch" 1);
-by (ALLGOALS Asm_simp_tac);
-qed "free_tv_ML_scheme";
-
-Goal "!!A::type_scheme list. (n : free_tv A) = (n: set (free_tv_ML A))";
-by (induct_tac "A" 1);
-by (Simp_tac 1);
-by (asm_simp_tac (simpset() addsimps [free_tv_ML_scheme]) 1);
-qed "free_tv_ML_scheme_list";
-
-
-(* lemmata for bound_tv *)
-
-Goal "bound_tv (mk_scheme t) = {}";
-by (induct_tac "t" 1);
-by (ALLGOALS Asm_simp_tac);
-qed "bound_tv_mk_scheme";
-
-Addsimps [bound_tv_mk_scheme];
-
-Goal "!!sch::type_scheme. bound_tv ($ S sch) = bound_tv sch";
-by (induct_tac "sch" 1);
-by (ALLGOALS Asm_simp_tac);
-qed "bound_tv_subst_scheme";
-
-Addsimps [bound_tv_subst_scheme];
-
-Goal "!!A::type_scheme list. bound_tv ($ S A) = bound_tv A";
-by (rtac (thm"list.induct") 1);
-by (Simp_tac 1);
-by (Asm_simp_tac 1);
-qed "bound_tv_subst_scheme_list";
-
-Addsimps [bound_tv_subst_scheme_list];
-
-
-(* lemmata for new_tv *)
-
-Goalw [new_tv_def]
-  "new_tv n S = ((!m. n <= m --> (S m = TVar m)) & \
-\                (! l. l < n --> new_tv n (S l) ))";
-by (safe_tac HOL_cs );
-(* ==> *)
-by (fast_tac (HOL_cs addDs [leD] addss (simpset() addsimps [free_tv_subst,dom_def])) 1);
-by (subgoal_tac "m:cod S | S l = TVar l" 1);
-by (safe_tac HOL_cs );
-by (fast_tac (HOL_cs addDs [UnI2] addss (simpset() addsimps [free_tv_subst])) 1);
-by (dres_inst_tac [("P","%x. m:free_tv x")] subst 1 THEN atac 1);
-by (Asm_full_simp_tac 1);
-by (fast_tac (set_cs addss (simpset() addsimps [free_tv_subst,cod_def,dom_def])) 1);
-(* <== *)
-by (rewrite_goals_tac [free_tv_subst,cod_def,dom_def] );
-by (safe_tac set_cs );
-by (cut_inst_tac [("m","m"),("n","n")] less_linear 1);
-by (fast_tac (HOL_cs addSIs [less_or_eq_imp_le]) 1);
-by (cut_inst_tac [("m","ma"),("n","n")] less_linear 1);
-by (fast_tac (HOL_cs addSIs [less_or_eq_imp_le]) 1);
-qed "new_tv_subst";
-
-Goal "new_tv n x = (!y:set x. new_tv n y)";
-by (induct_tac "x" 1);
-by (ALLGOALS Asm_simp_tac);
-qed "new_tv_list";
-
-(* substitution affects only variables occurring freely *)
-Goal
-  "new_tv n (t::typ) --> $(%x. if x=n then t' else S x) t = $S t";
-by (induct_tac "t" 1);
-by (ALLGOALS Asm_simp_tac);
-qed "subst_te_new_tv";
-Addsimps [subst_te_new_tv];
-
-Goal
-  "new_tv n (sch::type_scheme) --> $(%x. if x=n then sch' else S x) sch = $S sch";
-by (induct_tac "sch" 1);
-by (ALLGOALS Asm_simp_tac);
-qed_spec_mp "subst_te_new_type_scheme";
-Addsimps [subst_te_new_type_scheme];
-
-Goal
-  "new_tv n (A::type_scheme list) --> $(%x. if x=n then t else S x) A = $S A";
-by (induct_tac "A" 1);
-by (ALLGOALS Asm_full_simp_tac);
-qed_spec_mp "subst_tel_new_scheme_list";
-Addsimps [subst_tel_new_scheme_list];
-
-(* all greater variables are also new *)
-Goalw [new_tv_def] 
-  "n<=m ==> new_tv n t ==> new_tv m t";
-by Safe_tac;
-by (dtac spec 1);
-by (mp_tac 1);
-by (Simp_tac 1);
-qed "new_tv_le";
-Addsimps [lessI RS less_imp_le RS new_tv_le];
-
-Goal "n<=m ==> new_tv n (t::typ) ==> new_tv m t";
-by (asm_simp_tac (simpset() addsimps [new_tv_le]) 1);
-qed "new_tv_typ_le";
-
-Goal "n<=m ==> new_tv n (A::type_scheme list) ==> new_tv m A";
-by (asm_simp_tac (simpset() addsimps [new_tv_le]) 1);
-qed "new_scheme_list_le";
-
-Goal "n<=m ==> new_tv n (S::subst) ==> new_tv m S";
-by (asm_simp_tac (simpset() addsimps [new_tv_le]) 1);
-qed "new_tv_subst_le";
-
-(* new_tv property remains if a substitution is applied *)
-Goal
-  "[| n<m; new_tv m (S::subst) |] ==> new_tv m (S n)";
-by (asm_full_simp_tac (simpset() addsimps [new_tv_subst]) 1);
-qed "new_tv_subst_var";
-
-Goal
-  "new_tv n S --> new_tv n (t::typ) --> new_tv n ($ S t)";
-by (induct_tac "t" 1);
-by (ALLGOALS(fast_tac (HOL_cs addss (simpset() addsimps [new_tv_subst]))));
-qed_spec_mp "new_tv_subst_te";
-Addsimps [new_tv_subst_te];
-
-Goal "new_tv n S --> new_tv n (sch::type_scheme) --> new_tv n ($ S sch)";
-by (induct_tac "sch" 1);
-by (ALLGOALS (Asm_full_simp_tac));
-by (rewtac new_tv_def);
-by (simp_tac (simpset() addsimps [free_tv_subst,dom_def,cod_def]) 1);
-by (strip_tac 1);
-by (case_tac "S nat = TVar nat" 1);
-by (Asm_full_simp_tac 1);
-by (dres_inst_tac [("x","m")] spec 1);
-by (Fast_tac 1);
-qed_spec_mp "new_tv_subst_type_scheme";
-Addsimps [new_tv_subst_type_scheme];
-
-Goal
-  "new_tv n S --> new_tv n (A::type_scheme list) --> new_tv n ($ S A)";
-by (induct_tac "A" 1);
-by (ALLGOALS(fast_tac (HOL_cs addss (simpset()))));
-qed_spec_mp "new_tv_subst_scheme_list";
-Addsimps [new_tv_subst_scheme_list];
-
-Goal "new_tv n A --> new_tv (Suc n) ((TVar n)#A)";
-by (simp_tac (simpset() addsimps [new_tv_list]) 1);
-qed "new_tv_Suc_list";
-
-Goalw [new_tv_def] "!!sch::type_scheme. (free_tv sch) = (free_tv sch') --> (new_tv n sch --> new_tv n sch')";
-by (Asm_simp_tac 1);
-qed_spec_mp "new_tv_only_depends_on_free_tv_type_scheme";
-
-Goalw [new_tv_def] "!!A::type_scheme list. (free_tv A) = (free_tv A') --> (new_tv n A --> new_tv n A')";
-by (Asm_simp_tac 1);
-qed_spec_mp "new_tv_only_depends_on_free_tv_scheme_list";
-
-Goalw [new_tv_def]
-  "!nat. nat < length A --> new_tv n A --> (new_tv n (A!nat))";
-by (induct_tac "A" 1);
-by (Asm_full_simp_tac 1);
-by (rtac allI 1);
-by (induct_thm_tac nat_induct "nat" 1);
-by (strip_tac 1);
-by (Asm_full_simp_tac 1);
-by (Simp_tac 1);
-qed_spec_mp "new_tv_nth_nat_A";
-
-
-(* composition of substitutions preserves new_tv proposition *)
-Goal "[| new_tv n (S::subst); new_tv n R |] ==> new_tv n (($ R) o S)";
-by (asm_full_simp_tac (simpset() addsimps [new_tv_subst]) 1);
-qed "new_tv_subst_comp_1";
-
-Goal "[| new_tv n (S::subst); new_tv n R |] ==> new_tv n (%v.$ R (S v))";
-by (asm_full_simp_tac (simpset() addsimps [new_tv_subst]) 1);
-qed "new_tv_subst_comp_2";
-
-Addsimps [new_tv_subst_comp_1,new_tv_subst_comp_2];
-
-(* new type variables do not occur freely in a type structure *)
-Goalw [new_tv_def] 
-      "new_tv n A ==> n~:(free_tv A)";
-by (fast_tac (HOL_cs addEs [less_irrefl]) 1);
-qed "new_tv_not_free_tv";
-Addsimps [new_tv_not_free_tv];
-
-Goalw [new_tv_def] "!!t::typ. ? n. (new_tv n t)";
-by (induct_tac "t" 1);
-by (res_inst_tac [("x","Suc nat")] exI 1);
-by (Asm_simp_tac 1);
-by (REPEAT (etac exE 1));
-by (rename_tac "n'" 1);
-by (res_inst_tac [("x","max n n'")] exI 1);
-by (simp_tac (simpset() addsimps [less_max_iff_disj]) 1);
-by (Blast_tac 1);
-qed "fresh_variable_types";
-
-Addsimps [fresh_variable_types];
-
-Goalw [new_tv_def] "!!sch::type_scheme. ? n. (new_tv n sch)";
-by (induct_tac "sch" 1);
-by (res_inst_tac [("x","Suc nat")] exI 1);
-by (Simp_tac 1);
-by (res_inst_tac [("x","Suc nat")] exI 1);
-by (Simp_tac 1);
-by (REPEAT (etac exE 1));
-by (rename_tac "n'" 1);
-by (res_inst_tac [("x","max n n'")] exI 1);
-by (simp_tac (simpset() addsimps [less_max_iff_disj]) 1);
-by (Blast_tac 1);
-qed "fresh_variable_type_schemes";
-
-Addsimps [fresh_variable_type_schemes];
-
-Goal "!!A::type_scheme list. ? n. (new_tv n A)";
-by (induct_tac "A" 1);
-by (Simp_tac 1);
-by (Simp_tac 1);
-by (etac exE 1);
-by (cut_inst_tac [("sch","a")] fresh_variable_type_schemes 1);
-by (etac exE 1);
-by (rename_tac "n'" 1);
-by (res_inst_tac [("x","(max n n')")] exI 1);
-by (subgoal_tac "n <= (max n n')" 1);
-by (subgoal_tac "n' <= (max n n')" 1);
-by (fast_tac (claset() addDs [new_tv_le]) 1);
-by (ALLGOALS (simp_tac (simpset() addsimps [le_max_iff_disj])));
-qed "fresh_variable_type_scheme_lists";
-
-Addsimps [fresh_variable_type_scheme_lists];
-
-Goal "[| ? n1. (new_tv n1 x); ? n2. (new_tv n2 y)|] ==> ? n. (new_tv n x) & (new_tv n y)";
-by (REPEAT (etac exE 1));
-by (rename_tac "n1 n2" 1);
-by (res_inst_tac [("x","(max n1 n2)")] exI 1);
-by (subgoal_tac "n1 <= max n1 n2" 1);
-by (subgoal_tac "n2 <= max n1 n2" 1);
-by (fast_tac (claset() addDs [new_tv_le]) 1);
-by (ALLGOALS (simp_tac (simpset() addsimps [le_max_iff_disj])));
-qed "make_one_new_out_of_two";
-
-Goal "!!(A::type_scheme list) (A'::type_scheme list) (t::typ) (t'::typ). \
-\         ? n. (new_tv n A) & (new_tv n A') & (new_tv n t) & (new_tv n t')" ;
-by (cut_inst_tac [("t","t")] fresh_variable_types 1);
-by (cut_inst_tac [("t","t'")] fresh_variable_types 1);
-by (dtac make_one_new_out_of_two 1);
-by (assume_tac 1);
-by (thin_tac "? n. new_tv n t'" 1);
-by (cut_inst_tac [("A","A")] fresh_variable_type_scheme_lists 1);
-by (cut_inst_tac [("A","A'")] fresh_variable_type_scheme_lists 1);
-by (rotate_tac 1 1);
-by (dtac make_one_new_out_of_two 1);
-by (assume_tac 1);
-by (thin_tac "? n. new_tv n A'" 1);
-by (REPEAT (etac exE 1));
-by (rename_tac "n2 n1" 1);
-by (res_inst_tac [("x","(max n1 n2)")] exI 1);
-by (rewtac new_tv_def);
-by (simp_tac (simpset() addsimps [less_max_iff_disj]) 1);
-by (Blast_tac 1);
-qed "ex_fresh_variable";
-
-(* mgu does not introduce new type variables *)
-Goalw [new_tv_def] 
-      "[|mgu t1 t2 = Some u; new_tv n t1; new_tv n t2|] ==> new_tv n u";
-by (fast_tac (set_cs addDs [mgu_free]) 1);
-qed "mgu_new";
-
-
-(* lemmata for substitutions *)
-
-Goalw [app_subst_list] 
-   "!!A:: ('a::type_struct) list. length ($ S A) = length A";
-by (Simp_tac 1);
-qed "length_app_subst_list";
-Addsimps [length_app_subst_list];
-
-Goal "!!sch::type_scheme. $ TVar sch = sch";
-by (induct_tac "sch" 1);
-by (ALLGOALS Asm_simp_tac);
-qed "subst_TVar_scheme";
-
-Addsimps [subst_TVar_scheme];
-
-Goal "!!A::type_scheme list. $ TVar A = A";
-by (rtac (thm"list.induct") 1);
-by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [app_subst_list])));
-qed "subst_TVar_scheme_list";
-
-Addsimps [subst_TVar_scheme_list];
-
-(* application of id_subst does not change type expression *)
-Goalw [id_subst_def]
-  "$ id_subst = (%t::typ. t)";
-by (rtac ext 1);
-by (induct_tac "t" 1);
-by (ALLGOALS Asm_simp_tac);
-qed "app_subst_id_te";
-Addsimps [app_subst_id_te];
-
-Goalw [id_subst_def]
-  "$ id_subst = (%sch::type_scheme. sch)";
-by (rtac ext 1);
-by (induct_tac "sch" 1);
-by (ALLGOALS Asm_simp_tac);
-qed "app_subst_id_type_scheme";
-Addsimps [app_subst_id_type_scheme];
-
-(* application of id_subst does not change list of type expressions *)
-Goalw [app_subst_list]
-  "$ id_subst = (%A::type_scheme list. A)";
-by (rtac ext 1); 
-by (induct_tac "A" 1);
-by (ALLGOALS Asm_simp_tac);
-qed "app_subst_id_tel";
-Addsimps [app_subst_id_tel];
-
-Goal "!!sch::type_scheme. $ id_subst sch = sch";
-by (induct_tac "sch" 1);
-by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [id_subst_def])));
-qed "id_subst_sch";
-
-Addsimps [id_subst_sch];
-
-Goal "!!A::type_scheme list. $ id_subst A = A";
-by (induct_tac "A" 1);
-by (Asm_full_simp_tac 1);
-by (Asm_full_simp_tac 1);
-qed "id_subst_A";
-
-Addsimps [id_subst_A];
-
-(* composition of substitutions *)
-Goalw [id_subst_def,o_def] "$S o id_subst = S";
-by (rtac ext 1);
-by (Simp_tac 1);
-qed "o_id_subst";
-Addsimps[o_id_subst];
-
-Goal "$ R ($ S t::typ) = $ (%x. $ R (S x) ) t";
-by (induct_tac "t" 1);
-by (ALLGOALS Asm_simp_tac);
-qed "subst_comp_te";
-
-Goal "$ R ($ S sch::type_scheme) = $ (%x. $ R (S x) ) sch";
-by (induct_tac "sch" 1);
-by (ALLGOALS Asm_full_simp_tac);
-qed "subst_comp_type_scheme";
-
-Goalw [app_subst_list]
-  "$ R ($ S A::type_scheme list) = $ (%x. $ R (S x)) A";
-by (induct_tac "A" 1);
-(* case [] *)
-by (Simp_tac 1);
-(* case x#xl *)
-by (asm_full_simp_tac (simpset() addsimps [subst_comp_type_scheme]) 1);
-qed "subst_comp_scheme_list";
-
-Goal "!!A::type_scheme list. !x : free_tv A. S x = (TVar x) ==> $ S A = $ id_subst A";
-by (rtac scheme_list_substitutions_only_on_free_variables 1);
-by (asm_full_simp_tac (simpset() addsimps [id_subst_def]) 1);
-qed "subst_id_on_type_scheme_list'";
-
-Goal "!!A::type_scheme list. !x : free_tv A. S x = (TVar x) ==> $ S A = A";
-by (stac subst_id_on_type_scheme_list' 1);
-by (assume_tac 1);
-by (Simp_tac 1);
-qed "subst_id_on_type_scheme_list";
-
-Goal "!n. n < length A --> ($ S A)!n = $S (A!n)";
-by (induct_tac "A" 1);
-by (Asm_full_simp_tac 1);
-by (rtac allI 1);
-by (rename_tac "n1" 1);
-by (induct_thm_tac nat_induct "n1" 1);
-by (Asm_full_simp_tac 1);
-by (Asm_full_simp_tac 1);
-qed_spec_mp "nth_subst";