(* 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
typ = TVar nat | "->" typ typ (infixr 70)
(* type variable substitution *)
types
subst = nat => typ
arities
typ::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 (t1 -> t2) = ($ 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 (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 :: [typ,typ] => 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