--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/MiniML/I.ML Wed Oct 25 09:46:46 1995 +0100
@@ -0,0 +1,123 @@
+open I;
+
+Unify.trace_bound := 45;
+Unify.search_bound := 50;
+
+goal thy
+ "! a m s s' t n. \
+\ (new_tv m a & new_tv m s) --> I e a m s = Ok(s',t,n) --> \
+\ ( ? r. W e ($ s a) m = Ok(r, $ s' t, n) & s' = ($ r o s) )";
+by (expr.induct_tac "e" 1);
+
+(* case Var n *)
+by (simp_tac (!simpset addsimps [app_subst_list]
+ setloop (split_tac [expand_if])) 1);
+by (fast_tac (HOL_cs addss !simpset) 1);
+
+(* case Abs e *)
+by (asm_full_simp_tac (!simpset setloop (split_tac [expand_bind])) 1);
+by (strip_tac 1);
+br conjI 1;
+
+by (strip_tac 1);
+by (REPEAT (etac allE 1));
+be impE 1;
+by (fast_tac (HOL_cs addss (!simpset addsimps [new_tv_subst])) 2);
+by (fast_tac (HOL_cs addIs [new_tv_Suc_list RS mp,new_tv_subst_le,
+ less_imp_le,lessI]) 1);
+
+by (strip_tac 1);
+by (REPEAT (etac allE 1));
+be impE 1;
+by (fast_tac (HOL_cs addss (!simpset addsimps [new_tv_subst])) 2);
+by (fast_tac (HOL_cs addIs [new_tv_Suc_list RS mp,new_tv_subst_le,
+ less_imp_le,lessI]) 1);
+
+(* case App e1 e2 *)
+by (simp_tac (!simpset setloop (split_tac [expand_bind])) 1);
+by (strip_tac 1);
+by (rename_tac "s1' t1 n1 s2' t2 n2 sa" 1);
+br conjI 1;
+by (fast_tac (HOL_cs addss !simpset) 1);
+
+by (strip_tac 1);
+by (rename_tac "s1 t1' n1'" 1);
+by (eres_inst_tac [("x","a")] allE 1);
+by (eres_inst_tac [("x","m")] allE 1);
+by (eres_inst_tac [("x","s")] allE 1);
+by (eres_inst_tac [("x","s1'")] allE 1);
+by (eres_inst_tac [("x","t1")] allE 1);
+by (eres_inst_tac [("x","n1")] allE 1);
+by (eres_inst_tac [("x","a")] allE 1);
+by (eres_inst_tac [("x","n1")] allE 1);
+by (eres_inst_tac [("x","s1'")] allE 1);
+by (eres_inst_tac [("x","s2'")] allE 1);
+by (eres_inst_tac [("x","t2")] allE 1);
+by (eres_inst_tac [("x","n2")] allE 1);
+
+br conjI 1;
+by (strip_tac 1);
+by (mp_tac 1);
+by (mp_tac 1);
+be exE 1;
+be conjE 1;
+be impE 1;
+by ((forward_tac [new_tv_subst_tel] 1) THEN (atac 1));
+by ((dres_inst_tac [("xc","$ s a")] new_tv_W 1) THEN (atac 1));
+by (fast_tac (HOL_cs addDs [sym RS W_var_geD,new_tv_subst_le,new_tv_list_le]
+ addss !simpset) 1);
+by (fast_tac (HOL_cs addss (!simpset addsimps [subst_comp_tel])) 1);
+
+by (strip_tac 1);
+by (rename_tac "s2 t2' n2'" 1);
+br conjI 1;
+by (strip_tac 1);
+by (mp_tac 1);
+by (mp_tac 1);
+be exE 1;
+be conjE 1;
+be impE 1;
+by ((forward_tac [new_tv_subst_tel] 1) THEN (atac 1));
+by ((dres_inst_tac [("xc","$ s a")] new_tv_W 1) THEN (atac 1));
+by (fast_tac (HOL_cs addDs [sym RS W_var_geD,new_tv_subst_le,new_tv_list_le]
+ addss !simpset) 1);
+by (fast_tac (HOL_cs addss (!simpset addsimps [subst_comp_tel,subst_comp_te])) 1);
+
+by (strip_tac 1);
+by (mp_tac 1);
+by (mp_tac 1);
+be exE 1;
+be conjE 1;
+be impE 1;
+by ((forward_tac [new_tv_subst_tel] 1) THEN (atac 1));
+by ((dres_inst_tac [("xc","$ s a")] new_tv_W 1) THEN (atac 1));
+by (fast_tac (HOL_cs addDs [sym RS W_var_geD,new_tv_subst_le,new_tv_list_le]
+ addss !simpset) 1);
+
+by (mp_tac 1);
+by (REPEAT (eresolve_tac [exE,conjE] 1));
+by (REPEAT
+ ((asm_full_simp_tac (!simpset addsimps [subst_comp_tel,subst_comp_te]) 1) THEN
+ (REPEAT (etac conjE 1)) THEN (hyp_subst_tac 1) ));
+br exI 1;
+by (safe_tac HOL_cs);
+by (fast_tac HOL_cs 1);
+by (Simp_tac 2);
+
+by (subgoal_tac "new_tv n2 s & new_tv n2 r & new_tv n2 ra" 1);
+by (asm_full_simp_tac (!simpset addsimps [new_tv_subst]) 1);
+by ((forward_tac [new_tv_subst_tel] 1) THEN (atac 1));
+by ((dres_inst_tac [("xc","$ s a")] new_tv_W 1) THEN (atac 1));
+by (safe_tac HOL_cs);
+by (fast_tac (HOL_cs addDs [sym RS W_var_geD,new_tv_subst_le,new_tv_list_le]
+ addss !simpset) 1);
+by (fast_tac (HOL_cs addDs [sym RS W_var_geD,new_tv_subst_le,new_tv_list_le]
+ addss !simpset) 1);
+by (dres_inst_tac [("e","expr1")] (sym RS W_var_geD) 1);
+by ((dtac new_tv_subst_tel 1) THEN (atac 1));
+by ((dres_inst_tac [("a","$ s a")] new_tv_list_le 1) THEN (atac 1));
+by ((dtac new_tv_subst_tel 1) THEN (atac 1));
+by (fast_tac (HOL_cs addDs [new_tv_W] addss
+ (!simpset addsimps [subst_comp_tel])) 1);
+
+qed "I_correct_wrt_W";
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/MiniML/I.thy Wed Oct 25 09:46:46 1995 +0100
@@ -0,0 +1,24 @@
+(* Title: MiniML.thy
+ Author: Thomas Stauner
+ Copyright 1995 TU Muenchen
+
+ Recursive definition of type inference algorithm I for Mini_ML.
+*)
+
+I = W +
+
+consts
+ I :: "[expr, type_expr list, nat, subst] => result_W"
+
+primrec I expr
+ I_Var "I (Var i) a n s = (if i < length a then Ok(s, nth i a, n)
+ else Fail)"
+ I_Abs "I (Abs e) a n s = I e ((TVar n)#a) (Suc n) s bind
+ (%(s,t,m). Ok(s, Fun (TVar n) t, m) )"
+ I_App "I (App e1 e2) a n s =
+ I e1 a n s bind
+ (%(s1,t1,m1). I e2 a m1 s1 bind
+ (%(s2,t2,m2). mgu ($s2 t1) (Fun ($s2 t2) (TVar m2)) bind
+ (%u. Ok($u o s2, TVar m2, Suc m2) ) ))"
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/MiniML/Maybe.ML Wed Oct 25 09:46:46 1995 +0100
@@ -0,0 +1,20 @@
+open Maybe;
+
+(* constructor laws for bind *)
+goalw thy [bind_def] "(Ok s) bind f = (f s)";
+by (Simp_tac 1);
+qed "bind_Ok";
+
+goalw thy [bind_def] "Fail bind f = Fail";
+by (Simp_tac 1);
+qed "bind_Fail";
+
+Addsimps [bind_Ok,bind_Fail];
+
+(* expansion of bind *)
+goal thy
+ "P(res bind f) = ((res = Fail --> P Fail) & (!s. res = Ok s --> P(f s)))";
+by (maybe.induct_tac "res" 1);
+by (fast_tac (HOL_cs addss !simpset) 1);
+by (Asm_simp_tac 1);
+qed "expand_bind";
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/MiniML/Maybe.thy Wed Oct 25 09:46:46 1995 +0100
@@ -0,0 +1,18 @@
+(* Title: HOL/MiniML/Maybe.thy
+ ID: $Id$
+ Author: Dieter Nazareth and Tobias Nipkow
+ Copyright 1995 TU Muenchen
+
+Universal error monad.
+*)
+
+Maybe = List +
+
+datatype 'a maybe = Ok 'a | Fail
+
+consts bind :: "['a maybe, 'a => 'b maybe] => 'b maybe" (infixl 60)
+
+defs
+ bind_def "m bind f == case m of Ok r => f r | Fail => Fail"
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/MiniML/MiniML.ML Wed Oct 25 09:46:46 1995 +0100
@@ -0,0 +1,25 @@
+(* Title: HOL/MiniML/MiniML.ML
+ ID: $Id$
+ Author: Dieter Nazareth and Tobias Nipkow
+ Copyright 1995 TU Muenchen
+*)
+
+open MiniML;
+
+Addsimps has_type.intrs;
+Addsimps [Un_upper1,Un_upper2];
+
+
+(* has_type is closed w.r.t. substitution *)
+goal MiniML.thy
+ "!a e t. (a |- e :: t) --> ($ s a |- e :: $ s t)";
+by (rtac has_type.mutual_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);
+bind_thm ("has_type_cl_sub", result() RS spec RS spec RS spec RS mp);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/MiniML/MiniML.thy Wed Oct 25 09:46:46 1995 +0100
@@ -0,0 +1,32 @@
+(* Title: HOL/MiniML/MiniML.thy
+ ID: $Id$
+ Author: Dieter Nazareth and Tobias Nipkow
+ Copyright 1995 TU Muenchen
+
+Mini_ML with type inference rules.
+*)
+
+MiniML = Type +
+
+(* expressions *)
+datatype
+ expr = Var nat | Abs expr | App expr expr
+
+(* type inference rules *)
+consts
+ has_type :: "(type_expr list * expr * type_expr)set"
+syntax
+ "@has_type" :: "[type_expr list, expr, type_expr] => bool"
+ ("((_) |-/ (_) :: (_))" 60)
+translations
+ "a |- e :: t" == "(a,e,t):has_type"
+
+inductive "has_type"
+intrs
+ VarI "[| n < length a |] ==> a |- Var n :: nth n a"
+ AbsI "[| t1#a |- e :: t2 |] ==> a |- Abs e :: Fun t1 t2"
+ AppI "[| a |- e1 :: Fun t2 t1; a |- e2 :: t2 |]
+ ==> a |- App e1 e2 :: t1"
+
+end
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/MiniML/ROOT.ML Wed Oct 25 09:46:46 1995 +0100
@@ -0,0 +1,17 @@
+(* Title: HOL/MiniML/ROOT.ML
+ ID: $Id$
+ Author: Tobias Nipkow
+ Copyright 1995 TUM
+
+Type inference for let-free MiniML
+*)
+
+HOL_build_completed; (*Make examples fail if HOL did*)
+
+writeln"Root file for HOL/MiniML";
+loadpath := [".","MiniML"];
+Unify.trace_bound := 20;
+
+time_use_thy "I";
+
+make_chart (); (*make HTML chart*)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/MiniML/Type.ML Wed Oct 25 09:46:46 1995 +0100
@@ -0,0 +1,341 @@
+open Type;
+
+Addsimps [app_subst_TVar,app_subst_Fun];
+Addsimps [mgu_eq,mgu_mg,mgu_free];
+Addsimps [free_tv_TVar,free_tv_Fun,free_tv_Nil,free_tv_Cons];
+
+(* 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";
+
+(* application of id_subst does not change type expression *)
+goalw thy [id_subst_def]
+ "$ id_subst = (%t::type_expr.t)";
+by (rtac ext 1);
+by (type_expr.induct_tac "t" 1);
+by (ALLGOALS Asm_simp_tac);
+qed "app_subst_id_te";
+Addsimps [app_subst_id_te];
+
+(* application of id_subst does not change list of type expressions *)
+goalw thy [app_subst_list]
+ "$ id_subst = (%l::type_expr list.l)";
+by (rtac ext 1);
+by (list.induct_tac "l" 1);
+by (ALLGOALS Asm_simp_tac);
+qed "app_subst_id_tel";
+Addsimps [app_subst_id_tel];
+
+goalw thy [dom_def,id_subst_def,empty_def]
+ "dom id_subst = {}";
+by (Simp_tac 1);
+qed "dom_id_subst";
+Addsimps [dom_id_subst];
+
+goalw thy [cod_def]
+ "cod id_subst = {}";
+by (Simp_tac 1);
+qed "cod_id_subst";
+Addsimps [cod_id_subst];
+
+goalw thy [free_tv_subst]
+ "free_tv id_subst = {}";
+by (Simp_tac 1);
+qed "free_tv_id_subst";
+Addsimps [free_tv_id_subst];
+
+goalw thy [dom_def,cod_def,UNION_def,Bex_def]
+ "!!v. [| v : free_tv(s n); v~=n |] ==> v : cod s";
+by (Simp_tac 1);
+by (safe_tac (empty_cs addSIs [exI,conjI,notI]));
+ba 2;
+by (rotate_tac 1 1);
+by (Asm_full_simp_tac 1);
+qed "cod_app_subst";
+Addsimps [cod_app_subst];
+
+
+(* composition of substitutions *)
+goal thy
+ "$ g ($ f t::type_expr) = $ (%x. $ g (f x) ) t";
+by (type_expr.induct_tac "t" 1);
+by (ALLGOALS Asm_simp_tac);
+qed "subst_comp_te";
+
+goalw thy [app_subst_list]
+ "$ g ($ f l::type_expr list) = $ (%x. $ g (f x)) l";
+by (list.induct_tac "l" 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 (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 (Fun 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 (x#l) = (new_tv n x & new_tv n l)";
+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 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)";
+br 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::type_expr) --> $(%x. if x=n then t' else s x) t = $s t";
+by (type_expr.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 (a::type_expr list) --> $(%x. if x=n then t else s x) a = $s a";
+by (list.induct_tac "a" 1);
+by (ALLGOALS Asm_full_simp_tac);
+qed "subst_tel_new_tv";
+Addsimps [subst_tel_new_tv];
+
+(* all greater variables are also new *)
+goal thy
+ "n<=m --> new_tv n (t::type_expr) --> new_tv m t";
+by (type_expr.induct_tac "t" 1);
+(* case TVar n *)
+by( fast_tac (HOL_cs addIs [less_le_trans] addss !simpset) 1);
+(* case Fun t1 t2 *)
+by (Asm_simp_tac 1);
+bind_thm ("new_tv_le",result() RS mp RS mp);
+Addsimps [lessI RS less_imp_le RS new_tv_le];
+
+goal thy
+ "n<=m --> new_tv n (a::type_expr list) --> new_tv m a";
+by( list.induct_tac "a" 1);
+(* case [] *)
+by(Simp_tac 1);
+(* case a#al *)
+by (fast_tac (HOL_cs addIs [new_tv_le] addss !simpset) 1);
+bind_thm ("new_tv_list_le",result() RS mp RS mp);
+Addsimps [lessI RS less_imp_le RS new_tv_list_le];
+
+goal thy
+ "!! n. [| n<=m; new_tv n (s::subst) |] ==> new_tv m s";
+by (asm_full_simp_tac (!simpset addsimps [new_tv_subst]) 1);
+by (rtac conjI 1);
+by (slow_tac (HOL_cs addIs [le_trans]) 1);
+by (safe_tac HOL_cs );
+by (res_inst_tac [("P","l < n"),("Q","n <= l")] disjE 1);
+by (fast_tac (HOL_cs addss (HOL_ss addsimps [le_def])) 1);
+by (ALLGOALS (asm_full_simp_tac (!simpset addsimps [new_tv_le])) );
+qed "new_tv_subst_le";
+Addsimps [lessI RS less_imp_le RS new_tv_subst_le];
+
+(* new_tv property remains if a substitution is applied *)
+goal thy
+ "!!n. [| n<m; new_tv m (s::subst) |] ==> new_tv m (s n)";
+by (asm_full_simp_tac (!simpset addsimps [new_tv_subst]) 1);
+qed "new_tv_subst_var";
+
+goal thy
+ "new_tv n s --> new_tv n (t::type_expr) --> new_tv n ($ s t)";
+by (type_expr.induct_tac "t" 1);
+by (ALLGOALS(fast_tac (HOL_cs addss (!simpset addsimps [new_tv_subst]))));
+bind_thm ("new_tv_subst_te",result() RS mp RS mp);
+Addsimps [new_tv_subst_te];
+
+goal thy
+ "new_tv n s --> new_tv n (a::type_expr list) --> new_tv n ($ s a)";
+by( simp_tac (!simpset addsimps [new_tv_list]) 1);
+by (list.induct_tac "a" 1);
+by (ALLGOALS(fast_tac (HOL_cs addss (!simpset addsimps [new_tv_subst]))));
+bind_thm ("new_tv_subst_tel",result() RS mp RS mp);
+Addsimps [new_tv_subst_tel];
+
+(* auxilliary lemma *)
+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";
+
+
+(* 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_anti_refl]) 1);
+qed "new_tv_not_free_tv";
+Addsimps [new_tv_not_free_tv];
+
+goal thy
+ "(t::type_expr) mem l --> free_tv t <= free_tv l";
+by (list.induct_tac "l" 1);
+(* case [] *)
+by (Simp_tac 1);
+(* case x#xl *)
+by (fast_tac (set_cs addss (!simpset setloop (split_tac [expand_if]))) 1);
+bind_thm ("ftv_mem_sub_ftv_list",result() RS mp);
+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::type_expr) = $ s2 t --> n:(free_tv t) --> s1 n = s2 n";
+by (type_expr.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);
+bind_thm ("eq_subst_te_eq_free",result() RS mp RS mp);
+
+goal thy
+ "(!n. n:(free_tv t) --> s1 n = s2 n) --> $ s1 (t::type_expr) = $ s2 t";
+by (type_expr.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);
+bind_thm ("eq_free_eq_subst_te",result() RS mp);
+
+goal thy
+ "$ s1 (l::type_expr list) = $ s2 l --> n:(free_tv l) --> s1 n = s2 n";
+by (list.induct_tac "l" 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);
+bind_thm ("eq_subst_tel_eq_free",result() RS mp RS mp);
+
+goal thy
+ "(!n. n:(free_tv l) --> s1 n = s2 n) --> $s1 (l::type_expr list) = $s2 l";
+by (list.induct_tac "l" 1);
+(* case [] *)
+by (fast_tac (HOL_cs addss !simpset) 1);
+(* case x#xl *)
+by (fast_tac (HOL_cs addIs [eq_free_eq_subst_te] addss (!simpset)) 1);
+bind_thm ("eq_free_eq_subst_tel",result() RS mp);
+
+(* some useful theorems *)
+goalw thy [free_tv_subst]
+ "!!v. v : cod s ==> v : free_tv s";
+by( fast_tac set_cs 1);
+qed "codD";
+
+goalw thy [free_tv_subst,dom_def]
+ "!! x. x ~: free_tv s ==> s x = TVar x";
+by( fast_tac set_cs 1);
+qed "not_free_impl_id";
+
+goalw thy [new_tv_def]
+ "!! n. [| new_tv n t; m:free_tv t |] ==> m<n";
+by( fast_tac HOL_cs 1 );
+qed "free_tv_le_new_tv";
+
+goal thy
+ "free_tv (s (v::nat)) <= cod s Un {v}";
+by( cut_inst_tac [("P","v:dom s")] excluded_middle 1);
+by( safe_tac (HOL_cs addSIs [subsetI]) );
+by (fast_tac (set_cs addss (!simpset addsimps [dom_def])) 1);
+by (fast_tac (set_cs addss (!simpset addsimps [cod_def])) 1);
+qed "free_tv_subst_var";
+
+goal thy
+ "free_tv ($ s (e::type_expr)) <= cod s Un free_tv e";
+by( type_expr.induct_tac "e" 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 thy
+ "free_tv ($ s (l::type_expr list)) <= cod s Un free_tv l";
+by( list.induct_tac "l" 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";
+
+goalw thy [free_tv_subst,dom_def]
+ "free_tv (%u::nat. $ s1 ($ s2 (s3 u)) :: type_expr ) <= \
+\ free_tv s1 Un free_tv s2 Un free_tv s3";
+by( fast_tac (set_cs addSDs [free_tv_app_subst_te RS subsetD,free_tv_subst_var RS subsetD] addss (!simpset addsimps [cod_def,dom_def])) 1);
+qed "free_tv_comp_subst";
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/MiniML/Type.thy Wed Oct 25 09:46:46 1995 +0100
@@ -0,0 +1,90 @@
+(* Title: HOL/MiniML/Type.thy
+ ID: $Id$
+ Author: Dieter Nazareth and Tobias Nipkow
+ Copyright 1995 TU Muenchen
+
+MiniML-types and type substitutions.
+*)
+
+Type = Maybe +
+
+(* new class for structures containing type variables *)
+classes
+ type_struct < term
+
+(* type expressions *)
+datatype
+ type_expr = TVar nat | Fun type_expr type_expr
+
+(* type variable substitution *)
+types
+ subst = "nat => type_expr"
+
+arities
+ type_expr::type_struct
+ list::(type_struct)type_struct
+ fun::(term,type_struct)type_struct
+
+(* substitutions *)
+
+(* identity *)
+consts
+ id_subst :: "subst"
+defs
+ id_subst_def "id_subst == (%n.TVar n)"
+
+(* extension of substitution to type structures *)
+consts
+ app_subst :: "[subst, 'a::type_struct] => 'a::type_struct" ("$")
+
+rules
+ app_subst_TVar "$ s (TVar n) = s n"
+ app_subst_Fun "$ s (Fun t1 t2) = Fun ($ s t1) ($ s t2)"
+defs
+ app_subst_list "$ s == map ($ s)"
+
+(* free_tv s: the type variables occuring freely in the type structure s *)
+consts
+ free_tv :: "['a::type_struct] => nat set"
+
+rules
+ free_tv_TVar "free_tv (TVar m) = {m}"
+ free_tv_Fun "free_tv (Fun t1 t2) = (free_tv t1) Un (free_tv t2)"
+ free_tv_Nil "free_tv [] = {}"
+ free_tv_Cons "free_tv (x#l) = (free_tv x) Un (free_tv l)"
+
+(* domain of a substitution *)
+consts
+ dom :: "subst => nat set"
+defs
+ dom_def "dom s == {n. s n ~= TVar n}"
+
+(* codomain of a substitutions: the introduced variables *)
+consts
+ cod :: "subst => nat set"
+defs
+ cod_def "cod s == (UN m:dom s. free_tv (s m))"
+
+defs
+ free_tv_subst "free_tv s == (dom s) Un (cod s)"
+
+(* new_tv s n computes whether n is a new type variable w.r.t. a type
+ structure s, i.e. whether n is greater than any type variable
+ occuring in the type structure *)
+consts
+ new_tv :: "[nat,'a::type_struct] => bool"
+defs
+ new_tv_def "new_tv n ts == ! m. m:free_tv ts --> m<n"
+
+(* unification algorithm mgu *)
+consts
+ mgu :: "[type_expr,type_expr] => subst maybe"
+rules
+ mgu_eq "mgu t1 t2 = Ok u ==> $ u t1 = $ u t2"
+ mgu_mg "[| (mgu t1 t2) = Ok u; $ s t1 = $ s t2 |] ==>
+ ? r. s = ($ r) o u"
+ mgu_Ok "$ s t1 = $ s t2 ==> ? u. mgu t1 t2 = Ok u"
+ mgu_free "mgu t1 t2 = Ok u ==> free_tv u <= free_tv t1 Un free_tv t2"
+
+end
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/MiniML/W.ML Wed Oct 25 09:46:46 1995 +0100
@@ -0,0 +1,379 @@
+(* Title: HOL/MiniML/W.ML
+ ID: $Id$
+ Author: Dieter Nazareth and Tobias Nipkow
+ Copyright 1995 TU Muenchen
+
+Correctness and completeness of type inference algorithm W
+*)
+
+open W;
+
+
+(* stronger version of Suc_leD *)
+goalw Nat.thy [le_def]
+ "!!m. Suc m <= n ==> m < n";
+by (Asm_full_simp_tac 1);
+by (cut_facts_tac [less_linear] 1);
+by (fast_tac HOL_cs 1);
+qed "Suc_le_lessD";
+Addsimps [Suc_le_lessD];
+
+(* correctness of W with respect to has_type *)
+goal thy
+ "!a s t m n . Ok (s,t,m) = W e a n --> ($ s a |- e :: t)";
+by (expr.induct_tac "e" 1);
+(* case Var n *)
+by (asm_simp_tac (!simpset setloop (split_tac [expand_if])) 1);
+(* case Abs e *)
+by (asm_full_simp_tac (!simpset addsimps [app_subst_list]
+ setloop (split_tac [expand_bind])) 1);
+by (strip_tac 1);
+by (eres_inst_tac [("x","TVar(n) # a")] allE 1);
+by( fast_tac (HOL_cs addss (!simpset addsimps [eq_sym_conv])) 1);
+(* case App e1 e2 *)
+by (simp_tac (!simpset setloop (split_tac [expand_bind])) 1);
+by (strip_tac 1);
+by( rename_tac "sa ta na sb tb nb sc" 1);
+by (res_inst_tac [("t2.0","$ sc tb")] has_type.AppI 1);
+by (res_inst_tac [("s1","sc")] (app_subst_TVar RS subst) 1);
+by (rtac (app_subst_Fun RS subst) 1);
+by (res_inst_tac [("t","$ sc (Fun tb (TVar nb))"),("s","$ sc ($ sb ta)")] subst 1);
+by (Asm_full_simp_tac 1);
+by (simp_tac (HOL_ss addsimps [subst_comp_tel RS sym]) 1);
+by ( (rtac has_type_cl_sub 1) THEN (rtac has_type_cl_sub 1));
+by (asm_full_simp_tac (!simpset addsimps [eq_sym_conv]) 1);
+by (asm_full_simp_tac (!simpset addsimps [subst_comp_tel RS sym,has_type_cl_sub,eq_sym_conv]) 1);
+qed "W_correct";
+
+val has_type_casesE = map(has_type.mk_cases expr.simps)
+ [" s |- Var n :: t"," s |- Abs e :: t","s |- App e1 e2 ::t"];
+
+
+(* the resulting type variable is always greater or equal than the given one *)
+goal thy
+ "!a n s t m. W e a n = Ok (s,t,m) --> n<=m";
+by (expr.induct_tac "e" 1);
+(* case Var(n) *)
+by (fast_tac (HOL_cs addss (!simpset setloop (split_tac [expand_if]))) 1);
+(* case Abs e *)
+by (simp_tac (!simpset setloop (split_tac [expand_bind])) 1);
+by (fast_tac (HOL_cs addDs [Suc_leD]) 1);
+(* case App e1 e2 *)
+by (simp_tac (!simpset setloop (split_tac [expand_bind])) 1);
+by (strip_tac 1);
+by (rename_tac "s t na sa ta nb sb sc tb m" 1);
+by (eres_inst_tac [("x","a")] allE 1);
+by (eres_inst_tac [("x","n")] allE 1);
+by (eres_inst_tac [("x","$ s a")] allE 1);
+by (eres_inst_tac [("x","s")] allE 1);
+by (eres_inst_tac [("x","t")] allE 1);
+by (eres_inst_tac [("x","na")] allE 1);
+by (eres_inst_tac [("x","na")] allE 1);
+by (asm_full_simp_tac (!simpset addsimps [eq_sym_conv]) 1);
+by (etac conjE 1);
+by (eres_inst_tac [("x","sa")] allE 1);
+by (eres_inst_tac [("x","ta")] allE 1);
+by (eres_inst_tac [("x","nb")] allE 1);
+by (etac conjE 1);
+by (res_inst_tac [("j","na")] le_trans 1);
+by (Asm_simp_tac 1);
+by (asm_simp_tac (!simpset addsimps [Suc_leD]) 1);
+qed "W_var_ge";
+
+Addsimps [W_var_ge];
+
+goal thy
+ "!! s. Ok (s,t,m) = W e a n ==> n<=m";
+by (asm_full_simp_tac (!simpset addsimps [eq_sym_conv]) 1);
+qed "W_var_geD";
+
+
+(* auxiliary lemma *)
+goal Maybe.thy "(y = Ok x) = (Ok x = y)";
+by( simp_tac (!simpset addsimps [eq_sym_conv]) 1);
+qed "rotate_Ok";
+
+
+(* resulting type variable is new *)
+goal thy
+ "!n a s t m. new_tv n a --> W e a n = Ok (s,t,m) --> \
+\ (new_tv m s & new_tv m t)";
+by (expr.induct_tac "e" 1);
+(* case Var n *)
+by (fast_tac (HOL_cs addss (!simpset
+ addsimps [id_subst_def,list_all_nth,new_tv_list,new_tv_subst]
+ setloop (split_tac [expand_if]))) 1);
+
+(* case Abs e *)
+by (simp_tac (!simpset addsimps [new_tv_subst,new_tv_Suc_list]
+ setloop (split_tac [expand_bind])) 1);
+by (strip_tac 1);
+by (eres_inst_tac [("x","Suc n")] allE 1);
+by (eres_inst_tac [("x","(TVar n)#a")] allE 1);
+by (fast_tac (HOL_cs addss (!simpset
+ addsimps [new_tv_subst,new_tv_Suc_list])) 1);
+
+(* case App e1 e2 *)
+by (simp_tac (!simpset setloop (split_tac [expand_bind])) 1);
+by (strip_tac 1);
+by (rename_tac "s t na sa ta nb sb sc tb m" 1);
+by (eres_inst_tac [("x","n")] allE 1);
+by (eres_inst_tac [("x","a")] allE 1);
+by (eres_inst_tac [("x","s")] allE 1);
+by (eres_inst_tac [("x","t")] allE 1);
+by (eres_inst_tac [("x","na")] allE 1);
+by (eres_inst_tac [("x","na")] allE 1);
+by (asm_full_simp_tac (!simpset addsimps [eq_sym_conv]) 1);
+by (eres_inst_tac [("x","$ s a")] allE 1);
+by (eres_inst_tac [("x","sa")] allE 1);
+by (eres_inst_tac [("x","ta")] allE 1);
+by (eres_inst_tac [("x","nb")] allE 1);
+by( asm_full_simp_tac (!simpset addsimps [rotate_Ok]) 1);
+by (rtac conjI 1);
+by (rtac new_tv_subst_comp_2 1);
+by (rtac new_tv_subst_comp_2 1);
+by (rtac (lessI RS less_imp_le RS new_tv_subst_le) 1);
+by (res_inst_tac [("n","na")] new_tv_subst_le 1);
+by (asm_full_simp_tac (!simpset addsimps [rotate_Ok]) 1);
+by (Asm_simp_tac 1);
+by (fast_tac (HOL_cs addDs [W_var_geD] addIs
+ [new_tv_list_le,new_tv_subst_tel,lessI RS less_imp_le RS new_tv_subst_le])
+ 1);
+be (sym RS mgu_new) 1;
+by (fast_tac (HOL_cs addDs [W_var_geD] addIs [new_tv_subst_te,new_tv_list_le,
+ new_tv_subst_tel,lessI RS less_imp_le RS new_tv_le,lessI RS less_imp_le RS
+ new_tv_subst_le,new_tv_le]) 1);
+by (fast_tac (HOL_cs addDs [W_var_geD] addIs
+ [new_tv_list_le,new_tv_subst_tel,new_tv_le]
+ addss (!simpset)) 1);
+br (lessI RS new_tv_subst_var) 1;
+be (sym RS mgu_new) 1;
+by (fast_tac (HOL_cs addSIs [lessI RS less_imp_le RS new_tv_le,new_tv_subst_te]
+ addDs [W_var_geD] addIs
+ [new_tv_list_le,new_tv_subst_tel,lessI RS less_imp_le RS
+ new_tv_subst_le,new_tv_le] addss !simpset) 1);
+by (fast_tac (HOL_cs addDs [W_var_geD] addIs
+ [new_tv_list_le,new_tv_subst_tel,new_tv_le]
+ addss (!simpset)) 1);
+bind_thm ("new_tv_W",result() RS spec RS spec RS spec RS spec RS spec RS mp RS mp);
+
+
+goal thy
+ "!n a s t m v. W e a n = Ok (s,t,m) --> \
+\ (v:free_tv s | v:free_tv t) --> v<n --> v:free_tv a";
+by (expr.induct_tac "e" 1);
+(* case Var n *)
+by (fast_tac (HOL_cs addIs [nth_mem,subsetD,ftv_mem_sub_ftv_list]
+ addss (!simpset setloop (split_tac [expand_if]))) 1);
+
+(* case Abs e *)
+by (asm_full_simp_tac (!simpset addsimps
+ [free_tv_subst] setloop (split_tac [expand_bind])) 1);
+by (strip_tac 1);
+by (rename_tac "s t na sa ta m v" 1);
+by (eres_inst_tac [("x","Suc n")] allE 1);
+by (eres_inst_tac [("x","TVar n # a")] allE 1);
+by (eres_inst_tac [("x","s")] allE 1);
+by (eres_inst_tac [("x","t")] allE 1);
+by (eres_inst_tac [("x","na")] allE 1);
+by (eres_inst_tac [("x","v")] allE 1);
+by (fast_tac (HOL_cs addIs [cod_app_subst] addss !simpset) 1);
+
+(* case App e1 e2 *)
+by (simp_tac (!simpset setloop (split_tac [expand_bind])) 1);
+by (strip_tac 1);
+by (rename_tac "s t na sa ta nb sb sc tb m v" 1);
+by (eres_inst_tac [("x","n")] allE 1);
+by (eres_inst_tac [("x","a")] allE 1);
+by (eres_inst_tac [("x","s")] allE 1);
+by (eres_inst_tac [("x","t")] allE 1);
+by (eres_inst_tac [("x","na")] allE 1);
+by (eres_inst_tac [("x","na")] allE 1);
+by (eres_inst_tac [("x","v")] allE 1);
+(* second case *)
+by (eres_inst_tac [("x","$ s a")] allE 1);
+by (eres_inst_tac [("x","sa")] allE 1);
+by (eres_inst_tac [("x","ta")] allE 1);
+by (eres_inst_tac [("x","nb")] allE 1);
+by (eres_inst_tac [("x","v")] allE 1);
+by (safe_tac (empty_cs addSIs [conjI,impI] addSEs [conjE]) );
+by (asm_full_simp_tac (!simpset addsimps [rotate_Ok]) 1);
+bd W_var_geD 1;
+bd W_var_geD 1;
+by ( (forward_tac [less_le_trans] 1) THEN (assume_tac 1) );
+by (fast_tac (HOL_cs addDs [free_tv_comp_subst RS subsetD,sym RS mgu_free,
+ codD,free_tv_app_subst_te RS subsetD,free_tv_app_subst_tel RS subsetD,
+ less_le_trans,less_not_refl2,subsetD]
+ addEs [UnE]
+ addss !simpset) 1);
+by (Asm_full_simp_tac 1);
+bd (sym RS W_var_geD) 1;
+bd (sym RS W_var_geD) 1;
+by ( (forward_tac [less_le_trans] 1) THEN (assume_tac 1) );
+by (fast_tac (HOL_cs addDs [mgu_free, codD,free_tv_subst_var RS subsetD,
+ free_tv_app_subst_te RS subsetD,free_tv_app_subst_tel RS subsetD,
+ less_le_trans,subsetD]
+ addEs [UnE]
+ addss !simpset) 1);
+bind_thm ("free_tv_W",result() RS spec RS spec RS spec RS spec RS spec RS spec RS mp RS mp RS mp);
+
+goal HOL.thy "(~(P | Q)) = (~P & ~Q)";
+by( fast_tac HOL_cs 1);
+qed "not_disj";
+
+(* Completeness of W w.r.t. has_type *)
+goal thy
+ "!s' a t' n. ($ s' a |- e :: t') --> new_tv n a --> \
+\ (? s t. (? m. W e a n = Ok (s,t,m) ) & \
+\ (? r. $ s' a = $ r ($ s a) & \
+\ t' = $ r t ) )";
+by (expr.induct_tac "e" 1);
+(* case Var n *)
+by (strip_tac 1);
+by (simp_tac (!simpset addcongs [conj_cong]
+ setloop (split_tac [expand_if])) 1);
+by (eresolve_tac has_type_casesE 1);
+by (asm_full_simp_tac (!simpset addsimps [eq_sym_conv,app_subst_list]) 1);
+by (res_inst_tac [("x","id_subst")] exI 1);
+by (res_inst_tac [("x","nth nat a")] exI 1);
+by (Simp_tac 1);
+by (res_inst_tac [("x","s'")] exI 1);
+by (Asm_simp_tac 1);
+
+(* case Abs e *)
+by (strip_tac 1);
+by (eresolve_tac has_type_casesE 1);
+by (eres_inst_tac [("x","%x.if x=n then t1 else (s' x)")] allE 1);
+by (eres_inst_tac [("x","(TVar n)#a")] allE 1);
+by (eres_inst_tac [("x","t2")] allE 1);
+by (eres_inst_tac [("x","Suc n")] allE 1);
+by (fast_tac (HOL_cs addss (!simpset addcongs [conj_cong]
+ setloop (split_tac [expand_bind]))) 1);
+
+(* case App e1 e2 *)
+by (strip_tac 1);
+by (eresolve_tac has_type_casesE 1);
+by (eres_inst_tac [("x","s'")] allE 1);
+by (eres_inst_tac [("x","a")] allE 1);
+by (eres_inst_tac [("x","Fun t2 t'")] allE 1);
+by (eres_inst_tac [("x","n")] allE 1);
+by (safe_tac HOL_cs);
+by (eres_inst_tac [("x","r")] allE 1);
+by (eres_inst_tac [("x","$ s a")] allE 1);
+by (eres_inst_tac [("x","t2")] allE 1);
+by (eres_inst_tac [("x","m")] allE 1);
+bd asm_rl 1;
+bd asm_rl 1;
+bd asm_rl 1;
+by (Asm_full_simp_tac 1);
+by (safe_tac HOL_cs);
+by (fast_tac HOL_cs 1);
+by (fast_tac (HOL_cs addIs [sym RS W_var_geD,new_tv_W RS
+ conjunct1,new_tv_list_le,new_tv_subst_tel]) 1);
+
+by (subgoal_tac
+ "$ (%x.if x=ma then t' else (if x:(free_tv t - free_tv sa) then r x \
+\ else ra x)) ($ sa t) = \
+\ $ (%x.if x=ma then t' else (if x:(free_tv t - free_tv sa) then r x \
+\ else ra x)) (Fun ta (TVar ma))" 1);
+by (res_inst_tac [("t","$ (%x. if x = ma then t' else \
+\ (if x:(free_tv t - free_tv sa) then r x else ra x)) ($ sa t)"),
+ ("s","Fun ($ ra ta) t'")] ssubst 2);
+by (asm_simp_tac (!simpset addsimps [subst_comp_te]) 2);
+br eq_free_eq_subst_te 2;
+by (strip_tac 2);
+by (subgoal_tac "na ~=ma" 2);
+by (fast_tac (HOL_cs addDs [new_tv_W,sym RS W_var_geD,
+ new_tv_not_free_tv,new_tv_le]) 3);
+by (case_tac "na:free_tv sa" 2);
+(* na ~: free_tv sa *)
+by (asm_simp_tac (!simpset addsimps [not_free_impl_id]
+ setloop (split_tac [expand_if])) 3);
+(* na : free_tv sa *)
+by (dres_inst_tac [("l1","$ s a")] (subst_comp_tel RSN (2,trans)) 2);
+bd eq_subst_tel_eq_free 2;
+by (fast_tac (HOL_cs addIs [free_tv_W,free_tv_le_new_tv] addDs [new_tv_W]) 2);
+by (Asm_simp_tac 2);
+by (case_tac "na:dom sa" 2);
+(* na ~: dom sa *)
+by (asm_full_simp_tac (!simpset addsimps [dom_def]
+ setloop (split_tac [expand_if])) 3);
+(* na : dom sa *)
+br eq_free_eq_subst_te 2;
+by (strip_tac 2);
+by (subgoal_tac "nb ~= ma" 2);
+by ((forward_tac [new_tv_W] 3) THEN (atac 3));
+be conjE 3;
+bd new_tv_subst_tel 3;
+by (fast_tac (HOL_cs addIs [new_tv_list_le] addDs [sym RS W_var_geD]) 3);
+by (fast_tac (set_cs addDs [new_tv_W,new_tv_not_free_tv] addss
+ (!simpset addsimps [cod_def,free_tv_subst])) 3);
+by (fast_tac (set_cs addss (!simpset addsimps [cod_def,free_tv_subst]
+ setloop (split_tac [expand_if]))) 2);
+
+by (Simp_tac 2);
+br eq_free_eq_subst_te 2;
+by (strip_tac 2 );
+by (subgoal_tac "na ~= ma" 2);
+by ((forward_tac [new_tv_W] 3) THEN (atac 3));
+be conjE 3;
+bd (sym RS W_var_geD) 3;
+by (fast_tac (HOL_cs addDs [new_tv_list_le,new_tv_subst_tel,new_tv_W,new_tv_not_free_tv]) 3);
+by (case_tac "na: free_tv t - free_tv sa" 2);
+(* case na ~: free_tv t - free_tv sa *)
+by( asm_full_simp_tac (!simpset setloop (split_tac [expand_if])) 3);
+(* case na : free_tv t - free_tv sa *)
+by( asm_full_simp_tac (!simpset setloop (split_tac [expand_if])) 2);
+by (dres_inst_tac [("l1","$ s a")] (subst_comp_tel RSN (2,trans)) 2);
+bd eq_subst_tel_eq_free 2;
+by (fast_tac (HOL_cs addIs [free_tv_W,free_tv_le_new_tv] addDs [new_tv_W]) 2);
+by (asm_full_simp_tac (!simpset addsimps [free_tv_subst,dom_def,not_disj]) 2);
+
+by (asm_simp_tac (!simpset setloop (split_tac [expand_bind])) 1);
+by (safe_tac HOL_cs );
+bd mgu_Ok 1;
+by( fast_tac (HOL_cs addss !simpset) 1);
+by (REPEAT (resolve_tac [exI,conjI] 1));
+by (fast_tac HOL_cs 1);
+by (fast_tac HOL_cs 1);
+by ((dtac mgu_mg 1) THEN (atac 1));
+be exE 1;
+by (res_inst_tac [("x","rb")] exI 1);
+br conjI 1;
+by (dres_inst_tac [("x","ma")] fun_cong 2);
+by( asm_full_simp_tac (!simpset addsimps [eq_sym_conv]) 2);
+by (simp_tac (!simpset addsimps [subst_comp_tel RS sym]) 1);
+by (res_inst_tac [("l2","($ sa ($ s a))")] ((subst_comp_tel RS sym) RSN
+ (2,trans)) 1);
+by( asm_full_simp_tac (!simpset addsimps [o_def,eq_sym_conv]) 1);
+br eq_free_eq_subst_tel 1;
+by( safe_tac HOL_cs );
+by (subgoal_tac "ma ~= na" 1);
+by ((forward_tac [new_tv_W] 2) THEN (atac 2));
+be conjE 2;
+bd new_tv_subst_tel 2;
+by (fast_tac (HOL_cs addIs [new_tv_list_le] addDs [sym RS W_var_geD]) 2);
+by (( forw_inst_tac [("xd","m")] (sym RSN (2,new_tv_W)) 2) THEN (atac 2));
+be conjE 2;
+bd (free_tv_app_subst_tel RS subsetD) 2;
+by (fast_tac (set_cs addDs [W_var_geD,new_tv_list_le,codD,
+ new_tv_not_free_tv]) 2);
+by (case_tac "na: free_tv t - free_tv sa" 1);
+(* case na ~: free_tv t - free_tv sa *)
+by (asm_full_simp_tac (!simpset setloop (split_tac [expand_if])) 2);
+(* case na : free_tv t - free_tv sa *)
+by (asm_full_simp_tac (!simpset setloop (split_tac [expand_if])) 1);
+bd (free_tv_app_subst_tel RS subsetD) 1;
+by (fast_tac (set_cs addDs [codD,subst_comp_tel RSN (2,trans),
+ eq_subst_tel_eq_free] addss ((!simpset addsimps
+ [not_disj,free_tv_subst,dom_def]))) 1);
+qed "W_complete";
+
+
+
+
+
+
+
+
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/MiniML/W.thy Wed Oct 25 09:46:46 1995 +0100
@@ -0,0 +1,28 @@
+(* Title: HOL/MiniML/W.thy
+ ID: $Id$
+ Author: Dieter Nazareth and Tobias Nipkow
+ Copyright 1995 TU Muenchen
+
+Type inference algorithm W
+*)
+
+W = MiniML +
+
+types
+ result_W = "(subst * type_expr * nat)maybe"
+
+(* type inference algorithm W *)
+consts
+ W :: "[expr, type_expr list, nat] => result_W"
+
+primrec W expr
+ W_Var "W (Var i) a n = (if i < length a then Ok(id_subst, nth i a, n)
+ else Fail)"
+ W_Abs "W (Abs e) a n = W e ((TVar n)#a) (Suc n) bind
+ (%(s,t,m). Ok(s, Fun (s n) t, m) )"
+ W_App "W (App e1 e2) a n =
+ W e1 a n bind
+ (%(s1,t1,m1). W e2 ($ s1 a) m1 bind
+ (%(s2,t2,m2). mgu ($ s2 t1) (Fun t2 (TVar m2)) bind
+ (%u. Ok( ($ u) o (($ s2) o s1), $ u (TVar m2), Suc m2) )))"
+end