--- a/src/HOL/MiniML/MiniML.ML Fri Jan 17 18:35:44 1997 +0100
+++ b/src/HOL/MiniML/MiniML.ML Fri Jan 17 18:50:04 1997 +0100
@@ -4,21 +4,246 @@
Copyright 1995 TU Muenchen
*)
-open MiniML;
-
Addsimps has_type.intrs;
Addsimps [Un_upper1,Un_upper2];
+Addsimps [is_bound_typ_instance_closed_subst];
+
+
+goal thy "!!t::typ. $(%n. if n : (free_tv A) Un (free_tv t) then (S n) else (TVar n)) t = $S t";
+by (rtac typ_substitutions_only_on_free_variables 1);
+by (asm_full_simp_tac (!simpset addsimps [Ball_def]) 1);
+qed "s'_t_equals_s_t";
+
+Addsimps [s'_t_equals_s_t];
+
+goal thy "!!A::type_scheme list. $(%n. if n : (free_tv A) Un (free_tv t) then (S n) else (TVar n)) A = $S A";
+by (rtac scheme_list_substitutions_only_on_free_variables 1);
+by (asm_full_simp_tac (!simpset addsimps [Ball_def]) 1);
+qed "s'_a_equals_s_a";
+
+Addsimps [s'_a_equals_s_a];
+
+goal thy "!!S.($ (%n. if n : (free_tv A) Un (free_tv t) then (S n) else (TVar n)) A |- e :: \
+\ $ (%n. if n : (free_tv A) Un (free_tv t) then (S n) else (TVar n)) t) ==> \
+\ $S A |- e :: $S t";
+
+by (res_inst_tac [("P","%A. A |- e :: $S t")] ssubst 1);
+by (rtac (s'_a_equals_s_a RS sym) 1);
+by (res_inst_tac [("P","%t. $ (%n. if n : free_tv A Un free_tv (?t1 S) then S n else TVar n) A |- e :: t")] ssubst 1);
+by (rtac (s'_t_equals_s_t RS sym) 1);
+by (Asm_full_simp_tac 1);
+qed "replace_s_by_s'";
+
+goal thy "!!A::type_scheme list. $ (%x. TVar (if x : free_tv A then x else n + x)) 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 "alpha_A'";
+
+goal thy "!!A::type_scheme list. $ (%x. TVar (if x : free_tv A then x else n + x)) A = A";
+by (rtac (alpha_A' RS ssubst) 1);
+by (Asm_full_simp_tac 1);
+qed "alpha_A";
+
+goal thy "!!alpha. $ (S o alpha) (t::typ) = $ S ($ (%x. TVar (alpha x)) t)";
+by (typ.induct_tac "t" 1);
+by (ALLGOALS (Asm_full_simp_tac));
+qed "S_o_alpha_typ";
+
+goal thy "!!alpha. $ (%u. (S o alpha) u) (t::typ) = $ S ($ (%x. TVar (alpha x)) t)";
+by (typ.induct_tac "t" 1);
+by (ALLGOALS (Asm_full_simp_tac));
+val S_o_alpha_typ' = result ();
+
+goal thy "!!alpha. $ (S o alpha) (sch::type_scheme) = $ S ($ (%x. TVar (alpha x)) sch)";
+by (type_scheme.induct_tac "sch" 1);
+by (ALLGOALS (Asm_full_simp_tac));
+qed "S_o_alpha_type_scheme";
+
+goal thy "!!alpha. $ (S o alpha) (A::type_scheme list) = $ S ($ (%x. TVar (alpha x)) A)";
+by (list.induct_tac "A" 1);
+by (ALLGOALS (Asm_full_simp_tac));
+by (rtac (rewrite_rule [o_def] S_o_alpha_type_scheme) 1);
+qed "S_o_alpha_type_scheme_list";
+
+goal thy "!!A::type_scheme list. \
+\ ($ (%n. if n : free_tv A Un free_tv t \
+\ then S n else TVar n) \
+\ A) = \
+\ ($ ((%x. if x : free_tv A Un free_tv t then S x \
+\ else TVar x) o \
+\ (%x. if x : free_tv A \
+\ then x else n + x)) A)";
+by (rtac (S_o_alpha_type_scheme_list RS ssubst) 1);
+by (rtac (alpha_A RS ssubst) 1);
+by (rtac refl 1);
+qed "S'_A_eq_S'_alpha_A";
+
+goalw thy [free_tv_subst,dom_def]
+ "!!A. dom (%n. if n : free_tv A Un free_tv t then S n else TVar n) <= \
+\ free_tv A Un free_tv t";
+by (simp_tac (!simpset setloop (split_tac [expand_if])) 1);
+by (Fast_tac 1);
+qed "dom_S'";
+
+goalw thy [free_tv_subst,cod_def,subset_def]
+ "!!(A::type_scheme list) (t::typ). \
+\ cod (%n. if n : free_tv A Un free_tv t then S n else TVar n) <= \
+\ free_tv ($ S A) Un free_tv ($ S t)";
+by (rtac ballI 1);
+by (etac UN_E 1);
+by (dtac (dom_S' RS subsetD) 1);
+by (rotate_tac 1 1);
+by (Asm_full_simp_tac 1);
+by (fast_tac (!claset addDs [free_tv_of_substitutions_extend_to_scheme_lists]
+ addIs [free_tv_of_substitutions_extend_to_types RS subsetD]) 1);
+qed "cod_S'";
+
+goalw thy [free_tv_subst]
+ "!!(A::type_scheme list) (t::typ). \
+\ free_tv (%n. if n : free_tv A Un free_tv t then S n else TVar n) <= \
+\ free_tv A Un free_tv ($ S A) Un free_tv t Un free_tv ($ S t)";
+by (fast_tac (!claset addDs [dom_S' RS subsetD,cod_S' RS subsetD]) 1);
+qed "free_tv_S'";
+
+goal thy "!!t1::typ. \
+\ (free_tv ($ (%x. TVar (if x : free_tv A then x else n + x)) t1) - free_tv A) <= \
+\ {x. ? y. x = n + y}";
+by (typ.induct_tac "t1" 1);
+by (simp_tac (!simpset setloop (split_tac [expand_if])) 1);
+by (Fast_tac 1);
+by (Simp_tac 1);
+by (Fast_tac 1);
+qed "free_tv_alpha";
+
+goal thy "!!(k::nat). n <= n + k";
+by (res_inst_tac [("n","k")] nat_induct 1);
+by (Simp_tac 1);
+by (Asm_simp_tac 1);
+val aux_plus = result();
+
+Addsimps [aux_plus];
+
+goal thy "!!t::typ. new_tv n t ==> {x. ? y. x = n + y} Int free_tv t = {}";
+by (step_tac (!claset) 1);
+by (cut_facts_tac [aux_plus] 1);
+by (dtac new_tv_le 1);
+ba 1;
+by (rotate_tac 1 1);
+by (dtac new_tv_not_free_tv 1);
+by (Fast_tac 1);
+val new_tv_Int_free_tv_empty_type = result ();
+
+goal thy "!!sch::type_scheme. new_tv n sch ==> {x. ? y. x = n + y} Int free_tv sch = {}";
+by (step_tac (!claset) 1);
+by (cut_facts_tac [aux_plus] 1);
+by (dtac new_tv_le 1);
+ba 1;
+by (rotate_tac 1 1);
+by (dtac new_tv_not_free_tv 1);
+by (Fast_tac 1);
+val new_tv_Int_free_tv_empty_scheme = result ();
+
+goal thy "!!n. !A::type_scheme list. new_tv n A --> {x. ? y. x = n + y} Int free_tv A = {}";
+by (rtac allI 1);
+by (list.induct_tac "A" 1);
+by (Simp_tac 1);
+by (Simp_tac 1);
+by (fast_tac (!claset addDs [new_tv_Int_free_tv_empty_scheme ]) 1);
+val new_tv_Int_free_tv_empty_scheme_list = result ();
+
+goalw thy [le_type_scheme_def,is_bound_typ_instance]
+ "!!A. new_tv n A --> gen A t <= gen A ($ (%x. TVar (if x : free_tv A then x else n + x)) t)";
+by (strip_tac 1);
+by (etac exE 1);
+by (hyp_subst_tac 1);
+by (res_inst_tac [("x","(%x. S (if n <= x then x - n else x))")] exI 1);
+by (typ.induct_tac "t" 1);
+by (Simp_tac 1);
+by (case_tac "nat : free_tv A" 1);
+by (Asm_simp_tac 1);
+by (subgoal_tac "n <= n + nat" 1);
+by (dtac new_tv_le 1);
+ba 1;
+by (dtac new_tv_not_free_tv 1);
+by (dtac new_tv_not_free_tv 1);
+by (asm_simp_tac (!simpset addsimps [diff_add_inverse ]) 1);
+by (Simp_tac 1);
+by (Asm_simp_tac 1);
+qed_spec_mp "gen_t_le_gen_alpha_t";
+
+AddSIs has_type.intrs;
+
+goal thy "!!e. A |- e::t ==> !B. A <= B --> B |- e::t";
+by(etac has_type.induct 1);
+ by(simp_tac (!simpset addsimps [le_env_def]) 1);
+ by(fast_tac (!claset addEs [bound_typ_instance_trans] addss !simpset) 1);
+ by(Asm_full_simp_tac 1);
+ by(Fast_tac 1);
+by(slow_tac (!claset addEs [le_env_free_tv RS free_tv_subset_gen_le]) 1);
+qed_spec_mp "has_type_le_env";
(* has_type is closed w.r.t. substitution *)
-goal MiniML.thy "!!a e t. a |- e :: t ==> $s a |- e :: $s t";
+goal thy "!!a. A |- e :: t ==> !S. $S A |- e :: $S t";
by (etac has_type.induct 1);
(* case VarI *)
-by (asm_full_simp_tac (!simpset addsimps [app_subst_list]) 1);
-by (forw_inst_tac [("f1","$ s")] (nth_map RS sym) 1);
-by ( fast_tac (HOL_cs addIs [has_type.VarI] addss (!simpset delsimps [nth_map])) 1);
-(* case AbsI *)
-by (asm_full_simp_tac (!simpset addsimps [app_subst_list]) 1);
-(* case AppI *)
-by (Asm_full_simp_tac 1);
+ by (rtac allI 1);
+ by (rtac has_type.VarI 1);
+ by (asm_full_simp_tac (!simpset addsimps [app_subst_list]) 1);
+ by (asm_simp_tac (!simpset addsimps [app_subst_list]) 1);
+ (* case AbsI *)
+ by (rtac allI 1);
+ by (Simp_tac 1);
+ by (rtac has_type.AbsI 1);
+ by (Asm_full_simp_tac 1);
+ (* case AppI *)
+ by (rtac allI 1);
+ by (rtac has_type.AppI 1);
+ by (Asm_full_simp_tac 1);
+ by (etac spec 1);
+ by (etac spec 1);
+(* case LetI *)
+by (rtac allI 1);
+by (rtac replace_s_by_s' 1);
+by (cut_inst_tac [("A","$ S A"),
+ ("A'","A"),
+ ("t","t"),
+ ("t'","$ S t")]
+ ex_fresh_variable 1);
+by (etac exE 1);
+by (REPEAT (etac conjE 1));
+by (res_inst_tac [("t1.0","$((%x. if x : free_tv A Un free_tv t then S x else TVar x) o \
+\ (%x. if x : free_tv A then x else n + x)) t1")]
+ has_type.LETI 1);
+ by (dres_inst_tac [("x","(%x. if x : free_tv A Un free_tv t then S x else TVar x) o \
+\ (%x. if x : free_tv A then x else n + x)")] spec 1);
+ val o_apply = prove_goalw HOL.thy [o_def] "(f o g) x = f (g x)"
+ (fn _ => [rtac refl 1]);
+ by (stac (S'_A_eq_S'_alpha_A) 1);
+ ba 1;
+by (stac S_o_alpha_typ 1);
+by (stac gen_subst_commutes 1);
+ by (rtac subset_antisym 1);
+ by (rtac subsetI 1);
+ by (etac IntE 1);
+ by (dtac (free_tv_S' RS subsetD) 1);
+ by (dtac (free_tv_alpha RS subsetD) 1);
+ by (Asm_full_simp_tac 1);
+ by (etac exE 1);
+ by (hyp_subst_tac 1);
+ by (subgoal_tac "new_tv (n + y) ($ S A)" 1);
+ by (subgoal_tac "new_tv (n + y) ($ S t)" 1);
+ by (subgoal_tac "new_tv (n + y) A" 1);
+ by (subgoal_tac "new_tv (n + y) t" 1);
+ by (REPEAT (dtac new_tv_not_free_tv 1));
+ by (Fast_tac 1);
+ by (REPEAT ((rtac new_tv_le 1) THEN (assume_tac 2) THEN (Simp_tac 1)));
+ by (Fast_tac 1);
+by (rtac has_type_le_env 1);
+ by (dtac spec 1);
+ by (dtac spec 1);
+ ba 1;
+by (rtac (app_subst_Cons RS subst) 1);
+by (rtac S_compatible_le_scheme_lists 1);
+by (Asm_simp_tac 1);
qed "has_type_cl_sub";