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(* Title: HOL/MiniML/Type.thy
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ID: $Id$
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Author: Dieter Nazareth and Tobias Nipkow
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Copyright 1995 TU Muenchen
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MiniML-types and type substitutions.
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*)
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Type = Maybe +
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(* new class for structures containing type variables *)
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classes
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type_struct < term
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(* type expressions *)
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datatype
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type_expr = TVar nat | Fun type_expr type_expr
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(* type variable substitution *)
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types
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subst = nat => type_expr
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arities
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type_expr::type_struct
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list::(type_struct)type_struct
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fun::(term,type_struct)type_struct
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(* substitutions *)
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(* identity *)
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consts
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id_subst :: subst
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defs
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id_subst_def "id_subst == (%n.TVar n)"
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(* extension of substitution to type structures *)
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consts
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app_subst :: [subst, 'a::type_struct] => 'a::type_struct ("$")
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rules
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app_subst_TVar "$ s (TVar n) = s n"
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app_subst_Fun "$ s (Fun t1 t2) = Fun ($ s t1) ($ s t2)"
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defs
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app_subst_list "$ s == map ($ s)"
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(* free_tv s: the type variables occuring freely in the type structure s *)
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consts
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1376
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free_tv :: ['a::type_struct] => nat set
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rules
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free_tv_TVar "free_tv (TVar m) = {m}"
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free_tv_Fun "free_tv (Fun t1 t2) = (free_tv t1) Un (free_tv t2)"
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free_tv_Nil "free_tv [] = {}"
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free_tv_Cons "free_tv (x#l) = (free_tv x) Un (free_tv l)"
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(* domain of a substitution *)
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consts
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dom :: subst => nat set
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defs
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dom_def "dom s == {n. s n ~= TVar n}"
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(* codomain of a substitutions: the introduced variables *)
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consts
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cod :: subst => nat set
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defs
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cod_def "cod s == (UN m:dom s. free_tv (s m))"
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defs
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free_tv_subst "free_tv s == (dom s) Un (cod s)"
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(* new_tv s n computes whether n is a new type variable w.r.t. a type
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structure s, i.e. whether n is greater than any type variable
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occuring in the type structure *)
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consts
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new_tv :: [nat,'a::type_struct] => bool
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defs
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new_tv_def "new_tv n ts == ! m. m:free_tv ts --> m<n"
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(* unification algorithm mgu *)
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consts
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mgu :: [type_expr,type_expr] => subst maybe
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rules
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mgu_eq "mgu t1 t2 = Ok u ==> $ u t1 = $ u t2"
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mgu_mg "[| (mgu t1 t2) = Ok u; $ s t1 = $ s t2 |] ==>
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? r. s = ($ r) o u"
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mgu_Ok "$ s t1 = $ s t2 ==> ? u. mgu t1 t2 = Ok u"
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mgu_free "mgu t1 t2 = Ok u ==> free_tv u <= free_tv t1 Un free_tv t2"
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end
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