--- a/src/CTT/CTT.thy Fri Oct 10 16:29:41 1997 +0200
+++ b/src/CTT/CTT.thy Fri Oct 10 17:10:12 1997 +0200
@@ -113,21 +113,21 @@
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)"
+ ==> 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)"
+ ==> 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)"
+ ==> 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))"
+ 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
@@ -136,54 +136,54 @@
(*The Product of a family of types*)
- ProdF "[| A type; !!x. x:A ==> B(x) type |] ==> PROD x:A.B(x) type"
+ 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)"
+ 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)"
+ "[| 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)"
+ 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)"
+ 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)"
+ (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)"
+ "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"
+ 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)"
+ "[| 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)"
+ 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)"
+ "[| 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);
+ "[| 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)"
+ ==> 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>)"
+ ==> 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)"
+ fst_def "fst(a) == split(a, %x y. x)"
+ snd_def "snd(a) == split(a, %x y. y)"
(*The sum of two types*)
@@ -200,22 +200,22 @@
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)"
+ ==> 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)"
+ ==> 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))"
+ ==> 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))"
+ ==> when(inr(b), %x. c(x), %y. d(y)) = d(b) : C(inr(b))"
(*The type Eq*)