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(* Title: Substitutions/uterm.thy
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Author: Martin Coen, Cambridge University Computer Laboratory
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Copyright 1993 University of Cambridge
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Simple term structure for unifiation.
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Binary trees with leaves that are constants or variables.
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*)
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UTerm = Sexp +
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types uterm 1
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arities
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uterm :: (term)term
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consts
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UTerm :: "'a node set set => 'a node set set"
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Rep_UTerm :: "'a uterm => 'a node set"
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Abs_UTerm :: "'a node set => 'a uterm"
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VAR :: "'a node set => 'a node set"
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CONST :: "'a node set => 'a node set"
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COMB :: "['a node set, 'a node set] => 'a node set"
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Var :: "'a => 'a uterm"
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Const :: "'a => 'a uterm"
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Comb :: "['a uterm, 'a uterm] => 'a uterm"
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UTerm_rec :: "['a node set, 'a node set => 'b, 'a node set => 'b, \
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\ ['a node set , 'a node set, 'b, 'b]=>'b] => 'b"
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uterm_rec :: "['a uterm, 'a => 'b, 'a => 'b, \
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\ ['a uterm, 'a uterm,'b,'b]=>'b] => 'b"
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rules
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UTerm_def "UTerm(A) == lfp(%Z. A <+> A <+> Z <*> Z)"
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(*faking a type definition...*)
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Rep_UTerm "Rep_UTerm(xs): UTerm(range(Leaf))"
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Rep_UTerm_inverse "Abs_UTerm(Rep_UTerm(xs)) = xs"
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Abs_UTerm_inverse "M: UTerm(range(Leaf)) ==> Rep_UTerm(Abs_UTerm(M)) = M"
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(*defining the concrete constructors*)
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VAR_def "VAR(v) == In0(v)"
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CONST_def "CONST(v) == In1(In0(v))"
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COMB_def "COMB(t,u) == In1(In1(t . u))"
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(*defining the abstract constructors*)
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Var_def "Var(v) == Abs_UTerm(VAR(Leaf(v)))"
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Const_def "Const(c) == Abs_UTerm(CONST(Leaf(c)))"
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Comb_def "Comb(t,u) == Abs_UTerm(COMB(Rep_UTerm(t),Rep_UTerm(u)))"
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(*uterm recursion*)
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UTerm_rec_def
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"UTerm_rec(M,b,c,d) == wfrec(trancl(pred_Sexp), M, \
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\ %z g. Case(z, %x. b(x), \
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\ %w. Case(w, %x. c(x), \
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\ %v. Split(v, %x y. d(x,y,g(x),g(y))))))"
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uterm_rec_def
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"uterm_rec(t,b,c,d) == \
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\ UTerm_rec(Rep_UTerm(t), %x.b(Inv(Leaf,x)), %x.c(Inv(Leaf,x)), \
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\ %x y q r.d(Abs_UTerm(x),Abs_UTerm(y),q,r))"
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end
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