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 ij == ji

Thm* i,j:. ij Prop

Def hd(l) == Case of l; nil "?" ; h.t h

Thm* A:Type, l:A*. ||l||1 hd(l) A

Def {i...j} == {k:| ik & kj }

Thm* i,j:. {i...j} Type

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

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

Thm* ||nil||

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 ij == j < i

Thm* i,j:. ij

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

Thm* i,j:. i < j

Def AB == B < A

Thm* i,j:. ij Prop

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

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

Def A == A False

Thm* A:Prop. (A) Prop

Def b == if b false else true fi

Thm* b:. b