src/CTT/CTT.thy

   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)"
+   ==> 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
     "[| 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"
+  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)"
-
-  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)"
+   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)"
+   (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)"
-
-  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)"
+    "[| 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)"
+    "[| 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>)"
-
-  fst_def   "fst(a) == split(a, %x y.x)"
-  snd_def   "snd(a) == split(a, %x y.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"

   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)"
+    ==> 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*)
 
   EqF    "[| A type;  a : A;  b : A |] ==> Eq(A,a,b) type"