(* Title: CCL/types.thy
ID: $Id$
Author: Martin Coen
Copyright 1993 University of Cambridge
Types in CCL are defined as sets of terms.
*)
Type = Term +
consts
Subtype :: "['a set, 'a => o] => 'a set"
Bool :: "i set"
Unit :: "i set"
"+" :: "[i set, i set] => i set" (infixr 55)
Pi :: "[i set, i => i set] => i set"
Sigma :: "[i set, i => i set] => i set"
Nat :: "i set"
List :: "i set => i set"
Lists :: "i set => i set"
ILists :: "i set => i set"
TAll :: "(i set => i set) => i set" (binder "TALL " 55)
TEx :: "(i set => i set) => i set" (binder "TEX " 55)
Lift :: "i set => i set" ("(3[_])")
SPLIT :: "[i, [i, i] => i set] => i set"
"@Pi" :: "[idt, i set, i set] => i set" ("(3PROD _:_./ _)" [] 60)
"@Sigma" :: "[idt, i set, i set] => i set" ("(3SUM _:_./ _)" [] 60)
"@->" :: "[i set, i set] => i set" ("(_ ->/ _)" [54, 53] 53)
"@*" :: "[i set, i set] => i set" ("(_ */ _)" [56, 55] 55)
"@Subtype" :: "[idt, 'a set, o] => 'a set" ("(1{_: _ ./ _})")
translations
"PROD x:A. B" => "Pi(A, %x. B)"
"A -> B" => "Pi(A, _K(B))"
"SUM x:A. B" => "Sigma(A, %x. B)"
"A * B" => "Sigma(A, _K(B))"
"{x: A. B}" == "Subtype(A, %x. B)"
rules
Subtype_def "{x:A.P(x)} == {x.x:A & P(x)}"
Unit_def "Unit == {x.x=one}"
Bool_def "Bool == {x.x=true | x=false}"
Plus_def "A+B == {x. (EX a:A.x=inl(a)) | (EX b:B.x=inr(b))}"
Pi_def "Pi(A,B) == {x.EX b.x=lam x.b(x) & (ALL x:A.b(x):B(x))}"
Sigma_def "Sigma(A,B) == {x.EX a:A.EX b:B(a).x=<a,b>}"
Nat_def "Nat == lfp(% X.Unit + X)"
List_def "List(A) == lfp(% X.Unit + A*X)"
Lists_def "Lists(A) == gfp(% X.Unit + A*X)"
ILists_def "ILists(A) == gfp(% X.{} + A*X)"
Tall_def "TALL X.B(X) == Inter({X.EX Y.X=B(Y)})"
Tex_def "TEX X.B(X) == Union({X.EX Y.X=B(Y)})"
Lift_def "[A] == A Un {bot}"
SPLIT_def "SPLIT(p,B) == Union({A.EX x y.p=<x,y> & A=B(x,y)})"
end
ML
val print_translation =
[("Pi", dependent_tr' ("@Pi", "@->")),
("Sigma", dependent_tr' ("@Sigma", "@*"))];