Thm* A:S. ( x:A. False)  False | [all_false] |
Thm* 'a:S, P:(eq('a) Prop).
Thm* ( f:eq('a). P(f))  P(<eq_pred:( x:'a. y:'a. x = y)>) | [eq_pred_unabstraction] |
Thm* 'a:S. <eq_pred:( x:'a. y:'a. x = y)> eq('a) | [eq_pred_marker_wf] |
Thm* 'a,'b:S.
Thm* ( p:'a 'b. a:'a. b:'b. (a = 1of(p)) (b = 2of(p)))
Thm* =
Thm* (@ rep:'a 'b 'a 'b 
Thm* (@ (( p',p'':'a 'b. ((rep(p')) = (rep(p'')))  (p' = p''))
Thm* (@ ( x:'a 'b 
Thm* (@ ( ((his_pair('a; 'b)(x)) = ( p':'a 'b. (x = (rep(p')))))))) | [rep_prod_axiom] |
Thm* 'a,'b:S, P:('b  ), rep:('a 'b), abs:('b 'a), a:'a, r:'b.
Thm* iso_pair('a;'b;P;rep;abs)  rep(a) = r  a = abs(r) | [iso_pair_rep_to_abs] |
Thm* 'a,'b:S, P:('b  ), rep:('a 'b), abs:('b 'a).
Thm* iso_pair('a;'b;P;rep;abs)
Thm* 
Thm* ( a:'a. abs(rep(a)) = a) & ( r:'b. P(r) = ((rep(abs(r))) = r)) | [iso_pair_char] |
Thm* T:S, P:(T Prop). ( x:T. P(x))  ( x:T. P(x)) | [not_all] |
Thm* T:S, P:(T Prop). ( x:T. P(x))  ( x:T. P(x)) | [not_exists] |
Thm* T:S, P:(T  ).  ( x:T. P(x)) = ( x:T.  P(x)) | [bnot_ball] |
Thm* T:S, P:(T  ).  ( x:T. P(x)) = ( x:T.  P(x)) | [bnot_bexists] |
Thm* 'a,'b:S, P:('b  ), rep:('a 'b), abs:('b 'a).
Thm* iso_pair('a;'b;P;rep;abs)
Thm* 
Thm* ( rep':('a 'b). type_definition('b;'a;P;rep')) | [type_def_iso] |
Thm* 'a,'b:S, P:('b  ), rep:('a 'b), abs:('b 'a).
Thm* iso_pair('a;'b;P;rep;abs) Prop | [iso_pair_wf] |
Thm* 'a,'b:S. 'a+'b S | [union_wf_stype] |
Thm* Unit S | [unit_wf_stype] |
Thm* 'a:S, P,Q:('a Prop).
Thm* ( x:'a. Q(x)  P(x))  ( x:'a. Q(x))  P(@x:'a. Q(x)) | [choose_elim_pos] |
Thm* 'a:S, P,Q:('a Prop).
Thm* ( x:'a. Q(x)  P(x))
Thm* 
Thm* (( x:'a. Q(x))  ( x:'a. P(x)))  P(@x:'a. Q(x)) | [choose_elim_neg] |
Thm* T:S, P,Q:(T Type). ( x:T. P(x)  Q(x))  (@x:T. P(x)) = (@x:T. Q(x)) | [choose_functionality_axiom] |
Thm* T:S, P:(T Type). (@x:T. P(x)) T | [choose_wf] |
Thm* T:S. arb(T) T | [arb_wf] |
Thm* 'a:S, P:('a Prop). lem('a) = lem({x:'a| True }) 'a | [lem_extensionality_axiom] |