| | Some definitions of interest. |
|
| assert | Def  b == if b True else False fi |
| | | Thm*  b: . b  Prop |
|
| biject | Def Bij(A; B; f) == Inj(A; B; f) & Surj(A; B; f) |
| | | Thm* A,B:Type, f:(A  B). Bij(A; B; f) Prop |
|
| compose | Def (f o g)(x) == f(g(x)) |
| | | Thm* A,B,C:Type, f:(BC), g:(AB). f o g AC |
|
| filter | Def filter(P;l) == reduce( a,v. if P(a) [a / v] else v fi;nil;l) |
| | | Thm* T:Type, P:(T), l:T List. filter(P;l) T List |
|
| identity | Def Id(x) == x |
| | | Thm* A:Type. Id AA |
|
| int_seg | Def {i..j } == {k: | i  k < j } |
| | | Thm* m,n:. {m..n} Type |
|
| length | Def ||as|| == Case of as; nil 0 ; a.as' ||as'||+1 (recursive) |
| | | Thm* A:Type, l:A List. ||l|| |
| | | Thm* ||nil|| |
|
| nat | Def  == {i:| 0i } |
| | | Thm* Type |
|
| pi1 | Def 1of(t) == t.1 |
| | | Thm* A:Type, B:(AType), p:(a:A B(a)). 1of(p) A |
|
| pi2 | Def 2of(t) == t.2 |
| | | Thm* A:Type, B:(AType), p:(a:AB(a)). 2of(p) B(1of(p)) |
|
| select | Def l[i] == hd(nth_tl(i;l)) |
| | | Thm* A:Type, l:A List, n:. 0n   n < ||l|| l[n] A |
|
| upto | Def upto(i;j) == if i <  j [i / upto(i+1;j)] else nil fi (recursive) |
| | | Thm* i,j:. upto(i;j) {i..j} List |
