Def as @ bs == Case of as; nil bs ; a.as' a.(as' @ bs) (recursive)

Thm* T:Type, as,bs:T*. (as @ bs) T*

Def ij == ji

Thm* i,j:. ij Prop

Def Inj(A; B; f) == a1,a2:A. f(a1) = f(a2) B a1 = a2

Thm* A,B:Type, f:(AB). Inj(A; B; f) Prop

Def {i..j} == {k:| i k < j }

Thm* m,n:. {m..n} Type

Def ||as|| == Case of as; nil 0 ; a.as' ||as'||+1 (recursive)

Thm* A:Type, l:A*. ||l||

Thm* ||nil||

Def == {i:| 0 < i }

Thm* Type

Def as[m..n] == firstn(n-m;nth_tl(m;as))

Thm* T:Type, as:T*, m,n:. (as[m..n]) T*

Def Surj(A; B; f) == b:B. a:A. f(a) = b

Thm* A,B:Type, f:(AB). Surj(A; B; f) Prop

Def AB == B < A

Thm* i,j:. ij Prop

Def nth_tl(n;as) == if n0 as else nth_tl(n-1;tl(as)) fi (recursive)

Thm* A:Type, as:A*, i:. nth_tl(i;as) A*

Def firstn(n;as) == Case of as; nil nil ; a.as' if 0 < n a.firstn(n-1;as') else nil fi (recursive)

Thm* A:Type, as:A*, n:. firstn(n;as) A*

Def A == A False

Thm* A:Prop. (A) Prop

Def tl(l) == Case of l; nil nil ; h.t t

Thm* A:Type, l:A*. tl(l) A*

Def ij == j < i

Thm* i,j:. ij

Def i < j == if i < j true ; false fi

Thm* i,j:. i < j

Def b == if b false else true fi

Thm* b:. b