src/HOL/MiniML/Type.thy

--- /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
+