--- a/src/HOL/MiniML/Type.ML Fri Jan 17 18:35:44 1997 +0100
+++ b/src/HOL/MiniML/Type.ML Fri Jan 17 18:50:04 1997 +0100
@@ -1,37 +1,116 @@
-open Type;
+(* Title: HOL/MiniML/Type.thy
+ ID: $Id$
+ Author: Wolfgang Naraschewski and Tobias Nipkow
+ Copyright 1996 TU Muenchen
+*)
Addsimps [mgu_eq,mgu_mg,mgu_free];
-(* mgu does not introduce new type variables *)
-goalw thy [new_tv_def]
- "!! n. [|mgu t1 t2 = Ok 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 make scheme *)
+
+goal thy "mk_scheme t = sch1 =-> sch2 --> (? t1 t2. sch1 = mk_scheme t1 & sch2 = mk_scheme t2)";
+by (typ.induct_tac "t" 1);
+by (Simp_tac 1);
+by (Asm_full_simp_tac 1);
+by (Fast_tac 1);
+qed_spec_mp "mk_scheme_Fun";
-(* application of id_subst does not change type expression *)
-goalw thy [id_subst_def]
- "$ id_subst = (%t::typ.t)";
-by (rtac ext 1);
+goal Type.thy "!t'.mk_scheme t = mk_scheme t' --> t=t'";
+by (typ.induct_tac "t" 1);
+ br allI 1;
+ by (typ.induct_tac "t'" 1);
+ by(Simp_tac 1);
+ by(Asm_full_simp_tac 1);
+br allI 1;
+by (typ.induct_tac "t'" 1);
+ by(Simp_tac 1);
+by(Asm_full_simp_tac 1);
+by(Fast_tac 1);
+qed_spec_mp "mk_scheme_injective";
+
+goal thy "!!t. free_tv (mk_scheme t) = free_tv t";
+by (typ.induct_tac "t" 1);
+by (ALLGOALS Asm_simp_tac);
+qed "free_tv_mk_scheme";
+
+Addsimps [free_tv_mk_scheme];
+
+goal thy "!!t. $ S (mk_scheme t) = mk_scheme ($ S t)";
by (typ.induct_tac "t" 1);
by (ALLGOALS Asm_simp_tac);
-qed "app_subst_id_te";
-Addsimps [app_subst_id_te];
+qed "subst_mk_scheme";
+
+Addsimps [subst_mk_scheme];
+
-(* application of id_subst does not change list of type expressions *)
+(* constructor laws for app_subst *)
+
+goalw thy [app_subst_list]
+ "$ S [] = []";
+by (Simp_tac 1);
+qed "app_subst_Nil";
+
goalw thy [app_subst_list]
- "$ id_subst = (%ts::typ list.ts)";
-by (rtac ext 1);
-by (list.induct_tac "ts" 1);
-by (ALLGOALS Asm_simp_tac);
-qed "app_subst_id_tel";
-Addsimps [app_subst_id_tel];
+ "$ 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 thy [new_tv_def]
+ "new_tv n (TVar m) = (m<n)";
+by (fast_tac (HOL_cs addss !simpset) 1);
+qed "new_tv_TVar";
+
+goalw thy [new_tv_def]
+ "new_tv n (FVar m) = (m<n)";
+by (fast_tac (HOL_cs addss !simpset) 1);
+qed "new_tv_FVar";
+
+goalw thy [new_tv_def]
+ "new_tv n (BVar m) = True";
+by (Simp_tac 1);
+qed "new_tv_BVar";
+
+goalw thy [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 Type.thy [id_subst_def,o_def] "$s o id_subst = s";
-by (rtac ext 1);
+goalw thy [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 thy [new_tv_def]
+ "new_tv n []";
by (Simp_tac 1);
-qed "o_id_subst";
-Addsimps[o_id_subst];
+qed "new_tv_Nil";
+
+goalw thy [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 Type.thy [new_tv_def] "!!n. 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 Type.thy [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];
+
+
+(* constructor laws for dom and cod *)
goalw thy [dom_def,id_subst_def,empty_def]
"dom id_subst = {}";
@@ -45,14 +124,136 @@
qed "cod_id_subst";
Addsimps [cod_id_subst];
+
+(* lemmata for free_tv *)
+
goalw thy [free_tv_subst]
"free_tv id_subst = {}";
by (Simp_tac 1);
qed "free_tv_id_subst";
Addsimps [free_tv_id_subst];
+goal thy "!!A. !n. n < length A --> x : free_tv (nth n A) --> x : free_tv A";
+by (list.induct_tac "A" 1);
+by (Asm_full_simp_tac 1);
+by (rtac allI 1);
+by (res_inst_tac [("n","n")] nat_induct 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 thy "!!A. !nat. nat < length A --> x : free_tv (nth nat A) --> x : free_tv A";
+by (list.induct_tac "A" 1);
+by (Asm_full_simp_tac 1);
+by (rtac allI 1);
+by (res_inst_tac [("n","nat")] nat_induct 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 thy "!!S S'. (!x:free_tv t. (S x) = (S' x)) --> $ S (t::typ) = $ S' t";
+by (typ.induct_tac "t" 1);
+by (Simp_tac 1);
+by (Asm_full_simp_tac 1);
+qed_spec_mp "typ_substitutions_only_on_free_variables";
+
+goal thy
+ "!!t. (!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 thy "!!S S'. (!x:free_tv sch. (S x) = (S' x)) --> $ S (sch::type_scheme) = $ S' sch";
+by (type_scheme.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 thy
+ "!!sch. (!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 thy
+ "(!n. n:(free_tv A) --> S1 n = S2 n) --> $S1 (A::type_scheme list) = $S2 A";
+by (list.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 thy "!!P Q. ((!x:A. (P x)) --> Q) ==> ((!x:A Un B. (P x)) --> Q)";
+by (Fast_tac 1);
+val weaken_asm_Un = result ();
+
+goal thy "!!S S'. (!x:free_tv A. (S x) = (S' x)) --> $ S (A::type_scheme list) = $ S' A";
+by (list.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 thy
+ "$ S1 (t::typ) = $ S2 t --> n:(free_tv t) --> S1 n = S2 n";
+by (typ.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 thy
+ "$ S1 (sch::type_scheme) = $ S2 sch --> n:(free_tv sch) --> S1 n = S2 n";
+by (type_scheme.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 thy
+ "$S1 (A::type_scheme list) = $S2 A --> n:(free_tv A) --> S1 n = S2 n";
+by (list.induct_tac "A" 1);
+(* case [] *)
+by (fast_tac (HOL_cs addss !simpset) 1);
+(* case x#xl *)
+by (fast_tac (HOL_cs addIs [eq_subst_type_scheme_eq_free] addss (!simpset)) 1);
+qed_spec_mp "eq_subst_scheme_list_eq_free";
+
+goalw thy [free_tv_subst]
+ "!!v. v : cod S ==> v : free_tv S";
+by( fast_tac set_cs 1);
+qed "codD";
+
+goalw thy [free_tv_subst,dom_def]
+ "!! x. x ~: free_tv S ==> S x = TVar x";
+by( fast_tac set_cs 1);
+qed "not_free_impl_id";
+
+goalw thy [new_tv_def]
+ "!! n. [| new_tv n t; m:free_tv t |] ==> m<n";
+by( fast_tac HOL_cs 1 );
+qed "free_tv_le_new_tv";
+
goalw thy [dom_def,cod_def,UNION_def,Bex_def]
- "!!v. [| v : free_tv(s n); v~=n |] ==> v : cod s";
+ "!!v. [| v : free_tv(S n); v~=n |] ==> v : cod S";
by (Simp_tac 1);
by (safe_tac (empty_cs addSIs [exI,conjI,notI]));
by (assume_tac 2);
@@ -61,286 +262,43 @@
qed "cod_app_subst";
Addsimps [cod_app_subst];
-
-(* composition of substitutions *)
-goal thy
- "$ g ($ f t::typ) = $ (%x. $ g (f x) ) t";
-by (typ.induct_tac "t" 1);
-by (ALLGOALS Asm_simp_tac);
-qed "subst_comp_te";
-
-goalw thy [app_subst_list]
- "$ g ($ f ts::typ list) = $ (%x. $ g (f x)) ts";
-by (list.induct_tac "ts" 1);
-(* case [] *)
-by (Simp_tac 1);
-(* case x#xl *)
-by (asm_full_simp_tac (!simpset addsimps [subst_comp_te]) 1);
-qed "subst_comp_tel";
-
-
-(* constructor laws for app_subst *)
-goalw thy [app_subst_list]
- "$ s [] = []";
-by (Simp_tac 1);
-qed "app_subst_Nil";
-
-goalw thy [app_subst_list]
- "$ s (t#ts) = ($ s t)#($ s ts)";
-by (Simp_tac 1);
-qed "app_subst_Cons";
-
-Addsimps [app_subst_Nil,app_subst_Cons];
-
-(* constructor laws for new_tv *)
-goalw thy [new_tv_def]
- "new_tv n (TVar m) = (m<n)";
-by (fast_tac (HOL_cs addss !simpset) 1);
-qed "new_tv_TVar";
-
-goalw thy [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 thy [new_tv_def]
- "new_tv n []";
-by (Simp_tac 1);
-qed "new_tv_Nil";
-
-goalw thy [new_tv_def]
- "new_tv n (t#ts) = (new_tv n t & new_tv n ts)";
-by (fast_tac (HOL_cs addss !simpset) 1);
-qed "new_tv_Cons";
-
-Addsimps [new_tv_TVar,new_tv_Fun,new_tv_Nil,new_tv_Cons];
-
-goalw Type.thy [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];
-
-goalw thy [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 thy
- "new_tv n = list_all (new_tv n)";
-by (rtac ext 1);
-by (list.induct_tac "x" 1);
-by (ALLGOALS Asm_simp_tac);
-qed "new_tv_list";
-
-(* substitution affects only variables occurring freely *)
-goal thy
- "new_tv n (t::typ) --> $(%x. if x=n then t' else s x) t = $s t";
-by (typ.induct_tac "t" 1);
-by (ALLGOALS(asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
-qed "subst_te_new_tv";
-Addsimps [subst_te_new_tv];
-
-goal thy
- "new_tv n (ts::typ list) --> $(%x. if x=n then t else s x) ts = $s ts";
-by (list.induct_tac "ts" 1);
-by (ALLGOALS Asm_full_simp_tac);
-qed "subst_tel_new_tv";
-Addsimps [subst_tel_new_tv];
-
-(* all greater variables are also new *)
-goal thy
- "n<=m --> new_tv n (t::typ) --> new_tv m t";
-by (typ.induct_tac "t" 1);
-(* case TVar n *)
-by (fast_tac (HOL_cs addIs [less_le_trans] addss !simpset) 1);
-(* case Fun t1 t2 *)
-by (Asm_simp_tac 1);
-qed_spec_mp "new_tv_le";
-Addsimps [lessI RS less_imp_le RS new_tv_le];
-
-goal thy
- "n<=m --> new_tv n (ts::typ list) --> new_tv m ts";
-by (list.induct_tac "ts" 1);
-(* case [] *)
-by (Simp_tac 1);
-(* case a#al *)
-by (fast_tac (HOL_cs addIs [new_tv_le] addss !simpset) 1);
-qed_spec_mp "new_tv_list_le";
-Addsimps [lessI RS less_imp_le RS new_tv_list_le];
-
-goal thy
- "!! n. [| n<=m; new_tv n (s::subst) |] ==> new_tv m s";
-by (asm_full_simp_tac (!simpset addsimps [new_tv_subst]) 1);
-by (rtac conjI 1);
-by (slow_tac (HOL_cs addIs [le_trans]) 1);
-by (safe_tac HOL_cs );
-by (res_inst_tac [("P","l < n"),("Q","n <= l")] disjE 1);
-by (fast_tac (HOL_cs addss (HOL_ss addsimps [le_def])) 1);
-by (ALLGOALS (asm_full_simp_tac (!simpset addsimps [new_tv_le])) );
-qed "new_tv_subst_le";
-Addsimps [lessI RS less_imp_le RS new_tv_subst_le];
-
-(* new_tv property remains if a substitution is applied *)
-goal thy
- "!!n. [| n<m; new_tv m (s::subst) |] ==> new_tv m (s n)";
-by (asm_full_simp_tac (!simpset addsimps [new_tv_subst]) 1);
-qed "new_tv_subst_var";
-
-goal thy
- "new_tv n s --> new_tv n (t::typ) --> new_tv n ($ s t)";
-by (typ.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 thy
- "new_tv n s --> new_tv n (ts::typ list) --> new_tv n ($ s ts)";
-by (simp_tac (!simpset addsimps [new_tv_list]) 1);
-by (list.induct_tac "ts" 1);
-by (ALLGOALS(fast_tac (HOL_cs addss (!simpset addsimps [new_tv_subst]))));
-qed_spec_mp "new_tv_subst_tel";
-Addsimps [new_tv_subst_tel];
-
-(* auxilliary lemma *)
-goal thy
- "new_tv n ts --> new_tv (Suc n) ((TVar n)#ts)";
-by (simp_tac (!simpset addsimps [new_tv_list]) 1);
-by (list.induct_tac "ts" 1);
-by (ALLGOALS Asm_full_simp_tac);
-qed "new_tv_Suc_list";
-
-
-(* composition of substitutions preserves new_tv proposition *)
-goal thy
- "!! n. [| 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 thy
- "!! n. [| 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 thy [new_tv_def]
- "!!n. new_tv n ts ==> n~:(free_tv ts)";
-by (fast_tac (HOL_cs addEs [less_irrefl]) 1);
-qed "new_tv_not_free_tv";
-Addsimps [new_tv_not_free_tv];
-
-goal thy
- "(t::typ) mem ts --> free_tv t <= free_tv ts";
-by (list.induct_tac "ts" 1);
-(* case [] *)
-by (Simp_tac 1);
-(* case x#xl *)
-by (fast_tac (set_cs addss (!simpset setloop (split_tac [expand_if]))) 1);
-qed_spec_mp "ftv_mem_sub_ftv_list";
-Addsimps [ftv_mem_sub_ftv_list];
-
-
-(* 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 thy
- "$ s1 (t::typ) = $ s2 t --> n:(free_tv t) --> s1 n = s2 n";
-by (typ.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 thy
- "(!n. n:(free_tv t) --> s1 n = s2 n) --> $ s1 (t::typ) = $ s2 t";
-by (typ.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_free_eq_subst_te";
-
-goal thy
- "$s1 (ts::typ list) = $s2 ts --> n:(free_tv ts) --> s1 n = s2 n";
-by (list.induct_tac "ts" 1);
-(* case [] *)
-by (fast_tac (HOL_cs addss !simpset) 1);
-(* case x#xl *)
-by (fast_tac (HOL_cs addIs [eq_subst_te_eq_free] addss (!simpset)) 1);
-qed_spec_mp "eq_subst_tel_eq_free";
-
-goal thy
- "(!n. n:(free_tv ts) --> s1 n = s2 n) --> $s1 (ts::typ list) = $s2 ts";
-by (list.induct_tac "ts" 1);
-(* case [] *)
-by (fast_tac (HOL_cs addss !simpset) 1);
-(* case x#xl *)
-by (fast_tac (HOL_cs addIs [eq_free_eq_subst_te] addss (!simpset)) 1);
-qed_spec_mp "eq_free_eq_subst_tel";
-
-(* some useful theorems *)
-goalw thy [free_tv_subst]
- "!!v. v : cod s ==> v : free_tv s";
-by (fast_tac set_cs 1);
-qed "codD";
-
-goalw thy [free_tv_subst,dom_def]
- "!! x. x ~: free_tv s ==> s x = TVar x";
-by (fast_tac set_cs 1);
-qed "not_free_impl_id";
-
-goalw thy [new_tv_def]
- "!! n. [| new_tv n t; m:free_tv t |] ==> m<n";
-by (fast_tac HOL_cs 1 );
-qed "free_tv_le_new_tv";
-
-goal thy
- "free_tv (s (v::nat)) <= cod s Un {v}";
-by (cut_inst_tac [("P","v:dom s")] excluded_middle 1);
-by (safe_tac (HOL_cs addSIs [subsetI]) );
+ "free_tv (S (v::nat)) <= cod S Un {v}";
+by( cut_inst_tac [("P","v:dom S")] excluded_middle 1);
+by( safe_tac (HOL_cs addSIs [subsetI]) );
by (fast_tac (set_cs addss (!simpset addsimps [dom_def])) 1);
by (fast_tac (set_cs addss (!simpset addsimps [cod_def])) 1);
qed "free_tv_subst_var";
goal thy
- "free_tv ($ s (t::typ)) <= cod s Un free_tv t";
-by (typ.induct_tac "t" 1);
+ "free_tv ($ S (t::typ)) <= cod S Un free_tv t";
+by( typ.induct_tac "t" 1);
(* case TVar n *)
-by (simp_tac (!simpset addsimps [free_tv_subst_var]) 1);
+by( simp_tac (!simpset addsimps [free_tv_subst_var]) 1);
(* case Fun t1 t2 *)
-by (fast_tac (set_cs addss !simpset) 1);
+by( fast_tac (set_cs addss !simpset) 1);
qed "free_tv_app_subst_te";
goal thy
- "free_tv ($ s (ts::typ list)) <= cod s Un free_tv ts";
-by (list.induct_tac "ts" 1);
+ "free_tv ($ S (sch::type_scheme)) <= cod S Un free_tv sch";
+by( type_scheme.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 thy
+ "free_tv ($ S (A::type_scheme list)) <= cod S Un free_tv A";
+by( list.induct_tac "A" 1);
(* case [] *)
by (Simp_tac 1);
(* case a#al *)
-by (cut_facts_tac [free_tv_app_subst_te] 1);
-by (fast_tac (set_cs addss !simpset) 1);
-qed "free_tv_app_subst_tel";
+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 thy [free_tv_subst,dom_def]
"free_tv (%u::nat. $ s1 (s2 u) :: typ) <= \
@@ -351,3 +309,440 @@
addsimps [cod_def,dom_def])) 1);
qed "free_tv_comp_subst";
+goalw thy [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 thy "!!n. n : free_tv t --> free_tv (S n) <= free_tv ($ S t::typ)";
+by (typ.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 thy "!!n. n : free_tv sch --> free_tv (S n) <= free_tv ($ S sch::type_scheme)";
+by (type_scheme.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 thy "!!n. n : free_tv A --> free_tv (S n) <= free_tv ($ S A::type_scheme list)";
+by (list.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 thy "!!sch::type_scheme. (n : free_tv sch) = (n mem free_tv_ML sch)";
+by (type_scheme.induct_tac "sch" 1);
+by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
+by (strip_tac 1);
+by (Fast_tac 1);
+qed "free_tv_ML_scheme";
+
+goal thy "!!A::type_scheme list. (n : free_tv A) = (n mem free_tv_ML A)";
+by (list.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 thy "!!t. bound_tv (mk_scheme t) = {}";
+by (typ.induct_tac "t" 1);
+by (ALLGOALS Asm_simp_tac);
+qed "bound_tv_mk_scheme";
+
+Addsimps [bound_tv_mk_scheme];
+
+goal thy "!!sch::type_scheme. bound_tv ($ S sch) = bound_tv sch";
+by (type_scheme.induct_tac "sch" 1);
+by (ALLGOALS Asm_simp_tac);
+qed "bound_tv_subst_scheme";
+
+Addsimps [bound_tv_subst_scheme];
+
+goal thy "!!A::type_scheme list. bound_tv ($ S A) = bound_tv A";
+by (rtac 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 thy [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 thy
+ "new_tv n = list_all (new_tv n)";
+by (rtac ext 1);
+by(list.induct_tac "x" 1);
+by(ALLGOALS Asm_simp_tac);
+qed "new_tv_list";
+
+(* substitution affects only variables occurring freely *)
+goal thy
+ "new_tv n (t::typ) --> $(%x. if x=n then t' else S x) t = $S t";
+by (typ.induct_tac "t" 1);
+by (ALLGOALS(asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
+qed "subst_te_new_tv";
+Addsimps [subst_te_new_tv];
+
+goal thy
+ "new_tv n (sch::type_scheme) --> $(%x. if x=n then sch' else S x) sch = $S sch";
+by (type_scheme.induct_tac "sch" 1);
+by (ALLGOALS(asm_simp_tac (!simpset setloop (split_tac [expand_if]))));
+qed_spec_mp "subst_te_new_type_scheme";
+Addsimps [subst_te_new_type_scheme];
+
+goal thy
+ "new_tv n (A::type_scheme list) --> $(%x. if x=n then t else S x) A = $S A";
+by (list.induct_tac "A" 1);
+by (ALLGOALS Asm_full_simp_tac);
+qed_spec_mp "subst_tel_new_scheme_list";
+Addsimps [subst_tel_new_scheme_list];
+
+(* all greater variables are also new *)
+goalw thy [new_tv_def]
+ "!!n m. n<=m ==> new_tv n t ==> new_tv m t";
+by (safe_tac (!claset));
+by (dtac spec 1);
+by (mp_tac 1);
+by (trans_tac 1);
+qed "new_tv_le";
+Addsimps [lessI RS less_imp_le RS new_tv_le];
+
+goal thy "!!n m. n<=m ==> new_tv n (t::typ) ==> new_tv m t";
+by (asm_simp_tac (!simpset addsimps [new_tv_le]) 1);
+qed "new_tv_typ_le";
+
+goal thy "!!n m. n<=m ==> new_tv n (A::type_scheme list) ==> new_tv m A";
+by (asm_simp_tac (!simpset addsimps [new_tv_le]) 1);
+qed "new_scheme_list_le";
+
+goal thy "!!n m. n<=m ==> new_tv n (S::subst) ==> new_tv m S";
+by (asm_simp_tac (!simpset addsimps [new_tv_le]) 1);
+qed "new_tv_subst_le";
+
+(* new_tv property remains if a substitution is applied *)
+goal thy
+ "!!n. [| n<m; new_tv m (S::subst) |] ==> new_tv m (S n)";
+by (asm_full_simp_tac (!simpset addsimps [new_tv_subst]) 1);
+qed "new_tv_subst_var";
+
+goal thy
+ "new_tv n S --> new_tv n (t::typ) --> new_tv n ($ S t)";
+by (typ.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 thy "new_tv n S --> new_tv n (sch::type_scheme) --> new_tv n ($ S sch)";
+by (type_scheme.induct_tac "sch" 1);
+by (ALLGOALS (Asm_full_simp_tac));
+by (rewrite_goals_tac [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 (rotate_tac 3 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 thy
+ "new_tv n S --> new_tv n (A::type_scheme list) --> new_tv n ($ S A)";
+by (list.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 thy
+ "new_tv n A --> new_tv (Suc n) ((TVar n)#A)";
+by( simp_tac (!simpset addsimps [new_tv_list]) 1);
+by (list.induct_tac "A" 1);
+by (ALLGOALS Asm_full_simp_tac);
+qed "new_tv_Suc_list";
+
+goalw thy [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 thy [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 thy [new_tv_def] "!!A. !nat. nat < length A --> new_tv n A --> (new_tv n (nth nat A))";
+by (list.induct_tac "A" 1);
+by (Asm_full_simp_tac 1);
+by (rtac allI 1);
+by (res_inst_tac [("n","nat")] nat_induct 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 thy
+ "!! n. [| 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 thy
+ "!! n. [| 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 thy [new_tv_def]
+ "!!n. 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 thy [max_def] "!!n::nat. m < n ==> m < max n n'";
+by (simp_tac (!simpset setloop (split_tac [expand_if])) 1);
+by (safe_tac (!claset));
+by (trans_tac 1);
+qed "less_maxL";
+
+goalw thy [max_def] "!!n::nat. m < n' ==> m < max n n'";
+by (simp_tac (!simpset setloop (split_tac [expand_if])) 1);
+by (fast_tac (!claset addDs [not_leE] addIs [less_trans]) 1);
+qed "less_maxR";
+
+goalw thy [new_tv_def] "!!t::typ. ? n. (new_tv n t)";
+by (typ.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 1);
+by (fast_tac (!claset addIs [less_maxR,less_maxL]) 1);
+qed "fresh_variable_types";
+
+Addsimps [fresh_variable_types];
+
+goalw thy [new_tv_def] "!!sch::type_scheme. ? n. (new_tv n sch)";
+by (type_scheme.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 1);
+by (fast_tac (!claset addIs [less_maxR,less_maxL]) 1);
+qed "fresh_variable_type_schemes";
+
+Addsimps [fresh_variable_type_schemes];
+
+goalw thy [max_def] "!!n::nat. n <= (max n n')";
+by (simp_tac (!simpset setloop (split_tac [expand_if])) 1);
+qed "le_maxL";
+
+goalw thy [max_def] "!!n'::nat. n' <= (max n n')";
+by (simp_tac (!simpset setloop (split_tac [expand_if])) 1);
+by (fast_tac (!claset addDs [not_leE] addIs [less_or_eq_imp_le]) 1);
+qed "le_maxR";
+
+goal thy "!!A::type_scheme list. ? n. (new_tv n A)";
+by (list.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_maxR,le_maxL])));
+qed "fresh_variable_type_scheme_lists";
+
+Addsimps [fresh_variable_type_scheme_lists];
+
+goal thy "!!x y. [| ? 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_maxL,le_maxR])));
+qed "make_one_new_out_of_two";
+
+goal thy "!!(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);
+ba 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);
+ba 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 (rewrite_goals_tac [new_tv_def]);
+by (fast_tac (!claset addIs [less_maxL,less_maxR]) 1);
+qed "ex_fresh_variable";
+
+(* mgu does not introduce new type variables *)
+goalw thy [new_tv_def]
+ "!! n. [|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 Type.thy [app_subst_list] "length ($ S A) = length A";
+by(Simp_tac 1);
+qed "length_app_subst_list";
+Addsimps [length_app_subst_list];
+
+goal thy "!!sch::type_scheme. $ TVar sch = sch";
+by (type_scheme.induct_tac "sch" 1);
+by (ALLGOALS Asm_simp_tac);
+qed "subst_TVar_scheme";
+
+Addsimps [subst_TVar_scheme];
+
+goal thy "!!A::type_scheme list. $ TVar A = A";
+by (rtac 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 thy [id_subst_def]
+ "$ id_subst = (%t::typ.t)";
+by (rtac ext 1);
+by (typ.induct_tac "t" 1);
+by (ALLGOALS Asm_simp_tac);
+qed "app_subst_id_te";
+Addsimps [app_subst_id_te];
+
+goalw thy [id_subst_def]
+ "$ id_subst = (%sch::type_scheme.sch)";
+by (rtac ext 1);
+by (type_scheme.induct_tac "t" 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 thy [app_subst_list]
+ "$ id_subst = (%A::type_scheme list.A)";
+by (rtac ext 1);
+by (list.induct_tac "A" 1);
+by (ALLGOALS Asm_simp_tac);
+qed "app_subst_id_tel";
+Addsimps [app_subst_id_tel];
+
+goal thy "!!sch::type_scheme. $ id_subst sch = sch";
+by (type_scheme.induct_tac "sch" 1);
+by (ALLGOALS (asm_full_simp_tac (!simpset addsimps [id_subst_def])));
+qed "id_subst_sch";
+
+Addsimps [id_subst_sch];
+
+goal thy "!!A::type_scheme list. $ id_subst A = A";
+by (list.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 Type.thy [id_subst_def,o_def] "$S o id_subst = S";
+br ext 1;
+by(Simp_tac 1);
+qed "o_id_subst";
+Addsimps[o_id_subst];
+
+goal thy
+ "$ R ($ S t::typ) = $ (%x. $ R (S x) ) t";
+by (typ.induct_tac "t" 1);
+by (ALLGOALS Asm_simp_tac);
+qed "subst_comp_te";
+
+goal thy
+ "$ R ($ S sch::type_scheme) = $ (%x. $ R (S x) ) sch";
+by (type_scheme.induct_tac "sch" 1);
+by (ALLGOALS Asm_full_simp_tac);
+qed "subst_comp_type_scheme";
+
+goalw thy [app_subst_list]
+ "$ R ($ S A::type_scheme list) = $ (%x. $ R (S x)) A";
+by (list.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 thy "!!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 thy "!!A::type_scheme list. !x : free_tv A. S x = (TVar x) ==> $ S A = A";
+by (stac subst_id_on_type_scheme_list' 1);
+ba 1;
+by (Simp_tac 1);
+qed "subst_id_on_type_scheme_list";
+
+goal thy "!!n. !n. n < length A --> nth n ($ S A) = $S (nth n A)";
+by (list.induct_tac "A" 1);
+by (Asm_full_simp_tac 1);
+by (rtac allI 1);
+by (rename_tac "n1" 1);
+by (res_inst_tac[("n","n1")] nat_induct 1);
+by (Asm_full_simp_tac 1);
+by (Asm_full_simp_tac 1);
+qed_spec_mp "nth_subst";