(* Title: CTT/ctt.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
Constructive Type Theory
*)
CTT = Pure +
types
i
t
o
arities
i,t,o :: logic
consts
(*Types*)
F,T :: "t" (*F is empty, T contains one element*)
contr :: "i=>i"
tt :: "i"
(*Natural numbers*)
N :: "t"
succ :: "i=>i"
rec :: "[i, i, [i,i]=>i] => i"
(*Unions*)
inl,inr :: "i=>i"
when :: "[i, i=>i, i=>i]=>i"
(*General Sum and Binary Product*)
Sum :: "[t, i=>t]=>t"
fst,snd :: "i=>i"
split :: "[i, [i,i]=>i] =>i"
(*General Product and Function Space*)
Prod :: "[t, i=>t]=>t"
(*Equality type*)
Eq :: "[t,i,i]=>t"
eq :: "i"
(*Judgements*)
Type :: "t => prop" ("(_ type)" [10] 5)
Eqtype :: "[t,t]=>prop" ("(3_ =/ _)" [10,10] 5)
Elem :: "[i, t]=>prop" ("(_ /: _)" [10,10] 5)
Eqelem :: "[i,i,t]=>prop" ("(3_ =/ _ :/ _)" [10,10,10] 5)
Reduce :: "[i,i]=>prop" ("Reduce[_,_]")
(*Types*)
"@PROD" :: "[idt,t,t]=>t" ("(3PROD _:_./ _)" 10)
"@SUM" :: "[idt,t,t]=>t" ("(3SUM _:_./ _)" 10)
"+" :: "[t,t]=>t" (infixr 40)
(*Invisible infixes!*)
"@-->" :: "[t,t]=>t" ("(_ -->/ _)" [31,30] 30)
"@*" :: "[t,t]=>t" ("(_ */ _)" [51,50] 50)
(*Functions*)
lambda :: "(i => i) => i" (binder "lam " 10)
"`" :: "[i,i]=>i" (infixl 60)
(*Natural numbers*)
"0" :: "i" ("0")
(*Pairing*)
pair :: "[i,i]=>i" ("(1<_,/_>)")
translations
"PROD x:A. B" => "Prod(A, %x. B)"
"A --> B" => "Prod(A, _K(B))"
"SUM x:A. B" => "Sum(A, %x. B)"
"A * B" => "Sum(A, _K(B))"
rules
(*Reduction: a weaker notion than equality; a hack for simplification.
Reduce[a,b] means either that a=b:A for some A or else that "a" and "b"
are textually identical.*)
(*does not verify a:A! Sound because only trans_red uses a Reduce premise
No new theorems can be proved about the standard judgements.*)
refl_red "Reduce[a,a]"
red_if_equal "a = b : A ==> Reduce[a,b]"
trans_red "[| a = b : A; Reduce[b,c] |] ==> a = c : A"
(*Reflexivity*)
refl_type "A type ==> A = A"
refl_elem "a : A ==> a = a : A"
(*Symmetry*)
sym_type "A = B ==> B = A"
sym_elem "a = b : A ==> b = a : A"
(*Transitivity*)
trans_type "[| A = B; B = C |] ==> A = C"
trans_elem "[| a = b : A; b = c : A |] ==> a = c : A"
equal_types "[| a : A; A = B |] ==> a : B"
equal_typesL "[| a = b : A; A = B |] ==> a = b : B"
(*Substitution*)
subst_type "[| a : A; !!z. z:A ==> B(z) type |] ==> B(a) type"
subst_typeL "[| a = c : A; !!z. z:A ==> B(z) = D(z) |] ==> B(a) = D(c)"
subst_elem "[| a : A; !!z. z:A ==> b(z):B(z) |] ==> b(a):B(a)"
subst_elemL
"[| a=c : A; !!z. z:A ==> b(z)=d(z) : B(z) |] ==> b(a)=d(c) : B(a)"
(*The type N -- natural numbers*)
NF "N type"
NI0 "0 : N"
NI_succ "a : N ==> succ(a) : N"
NI_succL "a = b : N ==> succ(a) = succ(b) : N"
NE
"[| p: N; a: C(0); !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |]
==> rec(p, a, %u v. b(u,v)) : C(p)"
NEL
"[| p = q : N; a = c : C(0);
!!u v. [| u: N; v: C(u) |] ==> b(u,v) = d(u,v): C(succ(u)) |]
==> rec(p, a, %u v. b(u,v)) = rec(q,c,d) : C(p)"
NC0
"[| a: C(0); !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |]
==> rec(0, a, %u v. b(u,v)) = a : C(0)"
NC_succ
"[| p: N; a: C(0);
!!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |] ==>
rec(succ(p), a, %u v. b(u,v)) = b(p, rec(p, a, %u v. b(u,v))) : C(succ(p))"
(*The fourth Peano axiom. See page 91 of Martin-Lof's book*)
zero_ne_succ
"[| a: N; 0 = succ(a) : N |] ==> 0: F"
(*The Product of a family of types*)
ProdF "[| A type; !!x. x:A ==> B(x) type |] ==> PROD x:A. B(x) type"
ProdFL
"[| A = C; !!x. x:A ==> B(x) = D(x) |] ==>
PROD x:A. B(x) = PROD x:C. D(x)"
ProdI
"[| A type; !!x. x:A ==> b(x):B(x)|] ==> lam x. b(x) : PROD x:A. B(x)"
ProdIL
"[| A type; !!x. x:A ==> b(x) = c(x) : B(x)|] ==>
lam x. b(x) = lam x. c(x) : PROD x:A. B(x)"
ProdE "[| p : PROD x:A. B(x); a : A |] ==> p`a : B(a)"
ProdEL "[| p=q: PROD x:A. B(x); a=b : A |] ==> p`a = q`b : B(a)"
ProdC
"[| a : A; !!x. x:A ==> b(x) : B(x)|] ==>
(lam x. b(x)) ` a = b(a) : B(a)"
ProdC2
"p : PROD x:A. B(x) ==> (lam x. p`x) = p : PROD x:A. B(x)"
(*The Sum of a family of types*)
SumF "[| A type; !!x. x:A ==> B(x) type |] ==> SUM x:A. B(x) type"
SumFL
"[| A = C; !!x. x:A ==> B(x) = D(x) |] ==> SUM x:A. B(x) = SUM x:C. D(x)"
SumI "[| a : A; b : B(a) |] ==> <a,b> : SUM x:A. B(x)"
SumIL "[| a=c:A; b=d:B(a) |] ==> <a,b> = <c,d> : SUM x:A. B(x)"
SumE
"[| p: SUM x:A. B(x); !!x y. [| x:A; y:B(x) |] ==> c(x,y): C(<x,y>) |]
==> split(p, %x y. c(x,y)) : C(p)"
SumEL
"[| p=q : SUM x:A. B(x);
!!x y. [| x:A; y:B(x) |] ==> c(x,y)=d(x,y): C(<x,y>)|]
==> split(p, %x y. c(x,y)) = split(q, % x y. d(x,y)) : C(p)"
SumC
"[| a: A; b: B(a); !!x y. [| x:A; y:B(x) |] ==> c(x,y): C(<x,y>) |]
==> split(<a,b>, %x y. c(x,y)) = c(a,b) : C(<a,b>)"
fst_def "fst(a) == split(a, %x y. x)"
snd_def "snd(a) == split(a, %x y. y)"
(*The sum of two types*)
PlusF "[| A type; B type |] ==> A+B type"
PlusFL "[| A = C; B = D |] ==> A+B = C+D"
PlusI_inl "[| a : A; B type |] ==> inl(a) : A+B"
PlusI_inlL "[| a = c : A; B type |] ==> inl(a) = inl(c) : A+B"
PlusI_inr "[| A type; b : B |] ==> inr(b) : A+B"
PlusI_inrL "[| A type; b = d : B |] ==> inr(b) = inr(d) : A+B"
PlusE
"[| p: A+B; !!x. x:A ==> c(x): C(inl(x));
!!y. y:B ==> d(y): C(inr(y)) |]
==> when(p, %x. c(x), %y. d(y)) : C(p)"
PlusEL
"[| p = q : A+B; !!x. x: A ==> c(x) = e(x) : C(inl(x));
!!y. y: B ==> d(y) = f(y) : C(inr(y)) |]
==> when(p, %x. c(x), %y. d(y)) = when(q, %x. e(x), %y. f(y)) : C(p)"
PlusC_inl
"[| a: A; !!x. x:A ==> c(x): C(inl(x));
!!y. y:B ==> d(y): C(inr(y)) |]
==> when(inl(a), %x. c(x), %y. d(y)) = c(a) : C(inl(a))"
PlusC_inr
"[| b: B; !!x. x:A ==> c(x): C(inl(x));
!!y. y:B ==> d(y): C(inr(y)) |]
==> when(inr(b), %x. c(x), %y. d(y)) = d(b) : C(inr(b))"
(*The type Eq*)
EqF "[| A type; a : A; b : A |] ==> Eq(A,a,b) type"
EqFL "[| A=B; a=c: A; b=d : A |] ==> Eq(A,a,b) = Eq(B,c,d)"
EqI "a = b : A ==> eq : Eq(A,a,b)"
EqE "p : Eq(A,a,b) ==> a = b : A"
(*By equality of types, can prove C(p) from C(eq), an elimination rule*)
EqC "p : Eq(A,a,b) ==> p = eq : Eq(A,a,b)"
(*The type F*)
FF "F type"
FE "[| p: F; C type |] ==> contr(p) : C"
FEL "[| p = q : F; C type |] ==> contr(p) = contr(q) : C"
(*The type T
Martin-Lof's book (page 68) discusses elimination and computation.
Elimination can be derived by computation and equality of types,
but with an extra premise C(x) type x:T.
Also computation can be derived from elimination. *)
TF "T type"
TI "tt : T"
TE "[| p : T; c : C(tt) |] ==> c : C(p)"
TEL "[| p = q : T; c = d : C(tt) |] ==> c = d : C(p)"
TC "p : T ==> p = tt : T"
end
ML
val print_translation =
[("Prod", dependent_tr' ("@PROD", "@-->")),
("Sum", dependent_tr' ("@SUM", "@*"))];