| | Some definitions of interest. |
|
| hall | Def all ==  p:'a   .   x:'a. (p(x)) |
| | | Thm* 'a:S. all  (('a hbool) hbool) |
|
| assert | Def  b == if b True else False fi |
| | | Thm* b:. b Prop |
|
| hequal | Def equal == x:'a. y:'a. x = y |
| | | Thm* 'a:S. equal ('a 'a hbool) |
|
| bequal | Def x = y ==  (x = y T) |
| | | Thm* T:Type, x,y:T. (x = y) |
|
| habs_list | Def abs_list == r:( 'a) . @a:'a List. (r = rep_list('a;a)) |
| | | Thm* 'a:S. abs_list (hprod((hnum 'a); hnum) hlist('a)) |
|
| hcond | Def cond == b:. p:'a. q:'a. if b then p else q fi |
| | | Thm* 'a:S. cond (hbool 'a 'a 'a) |
|
| hpre | Def pre == n:. pre(n) |
| | | Thm* pre (hnum hnum) |
|
| hrep_list | Def rep_list == l:'a List. rep_list('a;l) |
| | | Thm* 'a:S. rep_list (hlist('a) hprod((hnum 'a); hnum)) |
|
| pre | Def pre(n) == if n= 0 then 0 else n-1 fi |
| | | Thm* n:. pre(n) |
|
| rep_list | Def rep_list('a;l) == <n:. if n<||l|| then l[n] else arb('a) fi ,||l||> |
| | | Thm* 'a:S, l:'a List. rep_list('a;l) ('a) |
|
| bif | Def bif(b; bx.x(bx); by.y(by)) == if b x(*) else y(x.x) fi |
| | | Thm* A:Type, b:, x:(bA), y:(( b)A). bif(b; bx.x(bx); by.y(by)) A |
|
| hcons | Def cons == x:'a. l:'a List. cons(x; l) |
| | | Thm* 'a:S. cons ('a hlist('a) hlist('a)) |
|
| hfst | Def fst == p:'a'b. 1of(p) |
| | | Thm* 'a,'b:S. fst (hprod('a; 'b) 'a) |
|
| hfun | Def 'a 'b == 'a'b |
| | | Thm* 'a,'b:S. ('a 'b) S |
|
| hlist | Def hlist('a) == 'a List |
| | | Thm* 'a:S. hlist('a) S |
|
| hnum | Def hnum == |
| | | Thm* hnum S |
|
| hpair | Def pair == x:'a. y:'b. <x,y> |
| | | Thm* 'a,'b:S. pair ('a 'b hprod('a; 'b)) |
|
| hsnd | Def snd == p:'a'b. 2of(p) |
| | | Thm* 'a,'b:S. snd (hprod('a; 'b) 'b) |
|
| hsuc | Def suc == n:. n+1 |
| | | Thm* suc (hnum hnum) |
|
| pi1 | Def 1of(t) == t.1 |
| | | Thm* A:Type, B:(AType), p:(a:AB(a)). 1of(p) A |
|
| pi2 | Def 2of(t) == t.2 |
| | | Thm* A:Type, B:(AType), p:(a:AB(a)). 2of(p) B(1of(p)) |
|
| stype | Def S == {T:Type|  x:T. True } |
| | | Thm* S Type{2} |
|
| tlambda | Def (x:T. b(x))(x) == b(x) |
