(* Title: Cube/Base.thy
ID: $Id$
Author: Tobias Nipkow
Copyright 1993 University of Cambridge
Barendregt's Lambda-Cube.
*)
Base = Pure +
types
term context typing
arities
term :: logic
consts
Abs, Prod :: "[term, term => term] => term"
Trueprop :: "[context, typing] => prop" ("(_/ |- _)")
MT_context :: "context" ("")
Context :: "[typing, context] => context" ("_ _")
star :: "term" ("*")
box :: "term" ("[]")
"^" :: "[term, term] => term" (infixl 20)
Has_type :: "[term, term] => typing" ("(_:/ _)" [0, 0] 5)
syntax
Trueprop1 :: "typing => prop" ("(_)")
"" :: "id => context" ("_ ")
"" :: "var => context" ("_ ")
Lam :: "[idt, term, term] => term" ("(3Lam _:_./ _)" [0, 0, 0] 10)
Pi :: "[idt, term, term] => term" ("(3Pi _:_./ _)" [0, 0] 10)
"->" :: "[term, term] => term" (infixr 10)
translations
(prop) "x:X" == (prop) "|- x:X"
"Lam x:A. B" == "Abs(A, %x. B)"
"Pi x:A. B" => "Prod(A, %x. B)"
"A -> B" => "Prod(A, _K(B))"
syntax (symbols)
Trueprop :: "[context, typing] => prop" ("(_/ \\<turnstile> _)")
box :: "term" ("\\<box>")
Lam :: "[idt, term, term] => term" ("(3\\<Lambda>_:_./ _)" [0, 0, 0] 10)
Pi :: "[idt, term, term] => term" ("(3\\<Pi>_:_./ _)" [0, 0] 10)
"op ->" :: "[term, term] => term" (infixr "\\<rightarrow>" 10)
rules
s_b "*: []"
strip_s "[| A:*; a:A ==> G |- x:X |] ==> a:A G |- x:X"
strip_b "[| A:[]; a:A ==> G |- x:X |] ==> a:A G |- x:X"
app "[| F:Prod(A, B); C:A |] ==> F^C: B(C)"
pi_ss "[| A:*; !!x. x:A ==> B(x):* |] ==> Prod(A, B):*"
lam_ss "[| A:*; !!x. x:A ==> f(x):B(x); !!x. x:A ==> B(x):* |]
==> Abs(A, f) : Prod(A, B)"
beta "Abs(A, f)^a == f(a)"
end
ML
val print_translation = [("Prod", dependent_tr' ("Pi", "op ->"))];