(* Title: Substitutions/UTerm.thy
Author: Martin Coen, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
Simple term structure for unification.
Binary trees with leaves that are constants or variables.
*)
UTerm = Sexp +
types uterm 1
arities
uterm :: (term)term
consts
uterm :: 'a item set => 'a item set
Rep_uterm :: 'a uterm => 'a item
Abs_uterm :: 'a item => 'a uterm
VAR :: 'a item => 'a item
CONST :: 'a item => 'a item
COMB :: ['a item, 'a item] => 'a item
Var :: 'a => 'a uterm
Const :: 'a => 'a uterm
Comb :: ['a uterm, 'a uterm] => 'a uterm
UTerm_rec :: ['a item, 'a item => 'b, 'a item => 'b,
['a item , 'a item, 'b, 'b]=>'b] => 'b
uterm_rec :: ['a uterm, 'a => 'b, 'a => 'b,
['a uterm, 'a uterm,'b,'b]=>'b] => 'b
defs
(*defining the concrete constructors*)
VAR_def "VAR(v) == In0(v)"
CONST_def "CONST(v) == In1(In0(v))"
COMB_def "COMB t u == In1(In1(t $ u))"
inductive "uterm(A)"
intrs
VAR_I "v:A ==> VAR(v) : uterm(A)"
CONST_I "c:A ==> CONST(c) : uterm(A)"
COMB_I "[| M:uterm(A); N:uterm(A) |] ==> COMB M N : uterm(A)"
rules
(*faking a type definition...*)
Rep_uterm "Rep_uterm(xs): uterm(range(Leaf))"
Rep_uterm_inverse "Abs_uterm(Rep_uterm(xs)) = xs"
Abs_uterm_inverse "M: uterm(range(Leaf)) ==> Rep_uterm(Abs_uterm(M)) = M"
defs
(*defining the abstract constructors*)
Var_def "Var(v) == Abs_uterm(VAR(Leaf(v)))"
Const_def "Const(c) == Abs_uterm(CONST(Leaf(c)))"
Comb_def "Comb t u == Abs_uterm (COMB (Rep_uterm t) (Rep_uterm u))"
(*uterm recursion*)
UTerm_rec_def
"UTerm_rec M b c d == wfrec (trancl pred_sexp)
(%g. Case (%x.b(x)) (Case (%y. c(y)) (Split (%x y. d x y (g x) (g y))))) M"
uterm_rec_def
"uterm_rec t b c d ==
UTerm_rec (Rep_uterm t) (%x.b(Inv Leaf x)) (%x.c(Inv Leaf x))
(%x y q r.d (Abs_uterm x) (Abs_uterm y) q r)"
end