%   ORIGINAL: h4/poset/complete__top
% 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/TRUTH: T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/poset/top__def: !x s r. h4/poset/top (h4/pair/_2C s r) x <=> s x /\ (!y. s y ==> r y x)
% Assm: h4/poset/lub__def: !x s r p. h4/poset/lub (h4/pair/_2C s r) p x <=> s x /\ (!y. s y /\ p y ==> r y x) /\ (!z. s z /\ (!y. s y /\ p y ==> r y z) ==> r x z)
% Assm: h4/poset/complete__def: !p. h4/poset/complete p <=> (!c. (?x. h4/poset/lub p c x) /\ (?x. h4/poset/glb p c x))
% Goal: !p. h4/poset/poset p /\ h4/poset/complete p ==> (?x. h4/poset/top p x)
%   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_TRUTH]: T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_posets_topu_u_def]: !x s r. h4/poset/top (h4/pair/_2C s r) x <=> happ s x /\ (!y. happ s y ==> happ (happ r y) x)
% Assm [h4s_posets_lubu_u_def]: !x s r p. h4/poset/lub (h4/pair/_2C s r) p x <=> happ s x /\ (!y. happ s y /\ happ p y ==> happ (happ r y) x) /\ (!z. happ s z /\ (!y. happ s y /\ happ p y ==> happ (happ r y) z) ==> happ (happ r x) z)
% Assm [h4s_posets_completeu_u_def]: !p. h4/poset/complete p <=> (!c. (?x. h4/poset/lub p c x) /\ (?x. h4/poset/glb p c x))
% Goal: !p. h4/poset/poset p /\ h4/poset/complete p ==> (?x. h4/poset/top p x)
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_Q95763,TV_Q95759]: ![V_f, V_g]: (![V_x]: s(TV_Q95759,happ(s(t_fun(TV_Q95763,TV_Q95759),V_f),s(TV_Q95763,V_x))) = s(TV_Q95759,happ(s(t_fun(TV_Q95763,TV_Q95759),V_g),s(TV_Q95763,V_x))) => s(t_fun(TV_Q95763,TV_Q95759),V_f) = s(t_fun(TV_Q95763,TV_Q95759),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_pairs_ABSu_u_PAIRu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: ?[V_q, V_r]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_q),s(TV_u_27b,V_r)))).
fof(ah4s_posets_topu_u_def, axiom, ![TV_u_27a]: ![V_x, V_s, V_r]: (p(s(t_bool,h4s_posets_top(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_r))),s(TV_u_27a,V_x)))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_x)))) & ![V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_y)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_r),s(TV_u_27a,V_y))),s(TV_u_27a,V_x)))))))).
fof(ah4s_posets_lubu_u_def, axiom, ![TV_u_27a]: ![V_x, V_s, V_r, V_p]: (p(s(t_bool,h4s_posets_lub(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_r))),s(t_fun(TV_u_27a,t_bool),V_p),s(TV_u_27a,V_x)))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_x)))) & (![V_y]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_y)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_r),s(TV_u_27a,V_y))),s(TV_u_27a,V_x))))) & ![V_z]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_z)))) & ![V_y]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_y)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_r),s(TV_u_27a,V_y))),s(TV_u_27a,V_z)))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_r),s(TV_u_27a,V_x))),s(TV_u_27a,V_z))))))))).
fof(ah4s_posets_completeu_u_def, axiom, ![TV_u_27a]: ![V_p]: (p(s(t_bool,h4s_posets_complete(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),V_p)))) <=> ![V_c]: (?[V_x]: p(s(t_bool,h4s_posets_lub(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),V_p),s(t_fun(TV_u_27a,t_bool),V_c),s(TV_u_27a,V_x)))) & ?[V_x]: p(s(t_bool,h4s_posets_glb(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),V_p),s(t_fun(TV_u_27a,t_bool),V_c),s(TV_u_27a,V_x))))))).
fof(ch4s_posets_completeu_u_top, conjecture, ![TV_u_27a]: ![V_p]: ((p(s(t_bool,h4s_posets_poset(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),V_p)))) & p(s(t_bool,h4s_posets_complete(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),V_p))))) => ?[V_x]: p(s(t_bool,h4s_posets_top(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),V_p),s(TV_u_27a,V_x)))))).
