%   ORIGINAL: h4/bool/TYPE__DEFINITION__THM
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/bool/TYPE__DEFINITION0: h4/bool/TYPE__DEFINITION = (\P rep. (!x_27 x_27_27. rep x_27 = rep x_27_27 ==> x_27 = x_27_27) /\ (!x. P x <=> (?x_27. x = rep x_27)))
% Goal: !rep P. h4/bool/TYPE__DEFINITION P rep <=> (!x_27 x_27_27. rep x_27 = rep x_27_27 ==> x_27 = x_27_27) /\ (!x. P x <=> (?x_27. x = rep x_27))
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_TYPEu_u_DEFINITION0]: !x x. h4/bool/TYPE__DEFINITION x x <=> (!x_27 x_27_27. happ x x_27 = happ x x_27_27 ==> x_27 = x_27_27) /\ (!x. happ x x <=> (?x_27. x = happ x x_27))
% Goal: !rep P. h4/bool/TYPE__DEFINITION P rep <=> (!x_27 x_27_27. happ rep x_27 = happ rep x_27_27 ==> x_27 = x_27_27) /\ (!x. happ P x <=> (?x_27. x = happ rep x_27))
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q292714,TV_Q292710]: ![V_f, V_g]: (![V_x]: s(TV_Q292710,happ(s(t_fun(TV_Q292714,TV_Q292710),V_f),s(TV_Q292714,V_x))) = s(TV_Q292710,happ(s(t_fun(TV_Q292714,TV_Q292710),V_g),s(TV_Q292714,V_x))) => s(t_fun(TV_Q292714,TV_Q292710),V_f) = s(t_fun(TV_Q292714,TV_Q292710),V_g))).
fof(ah4s_bools_TYPEu_u_DEFINITION0, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_x0]: (p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(TV_u_27a,t_bool),V_x),s(t_fun(TV_u_27b,TV_u_27a),V_x0)))) <=> (![V_xu_27, V_xu_27u_27]: (s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_x0),s(TV_u_27b,V_xu_27))) = s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_x0),s(TV_u_27b,V_xu_27u_27))) => s(TV_u_27b,V_xu_27) = s(TV_u_27b,V_xu_27u_27)) & ![V_x1]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_x1)))) <=> ?[V_xu_27]: s(TV_u_27a,V_x1) = s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_x0),s(TV_u_27b,V_xu_27))))))).
fof(ch4s_bools_TYPEu_u_DEFINITIONu_u_THM, conjecture, ![TV_u_27a,TV_u_27b]: ![V_rep, V_P]: (p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) <=> (![V_xu_27, V_xu_27u_27]: (s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_xu_27))) = s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_xu_27u_27))) => s(TV_u_27b,V_xu_27) = s(TV_u_27b,V_xu_27u_27)) & ![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) <=> ?[V_xu_27]: s(TV_u_27a,V_x) = s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_xu_27))))))).
