%   ORIGINAL: h4/hreal/ISACUT__HRAT
% 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/CONJ__SYM: !t2 t1. t1 /\ t2 <=> t2 /\ t1
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/hreal/HRAT__LT__TRANS: !z y x. h4/hreal/hrat__lt x y /\ h4/hreal/hrat__lt y z ==> h4/hreal/hrat__lt x z
% Assm: h4/hreal/HRAT__LT__ANTISYM: !y x. ~(h4/hreal/hrat__lt x y /\ h4/hreal/hrat__lt y x)
% Assm: h4/hreal/HRAT__UP: !x. ?y. h4/hreal/hrat__lt x y
% Assm: h4/hreal/HRAT__DOWN: !x. ?y. h4/hreal/hrat__lt y x
% Assm: h4/hreal/HRAT__MEAN: !y x. h4/hreal/hrat__lt x y ==> (?z. h4/hreal/hrat__lt x z /\ h4/hreal/hrat__lt z y)
% Assm: h4/hreal/isacut0: !C. h4/hreal/isacut C <=> (?x. C x) /\ (?x. ~C x) /\ (!x y. C x /\ h4/hreal/hrat__lt y x ==> C y) /\ (!x. C x ==> (?y. C y /\ h4/hreal/hrat__lt x y))
% Assm: h4/hreal/cut__of__hrat0: !x. h4/hreal/cut__of__hrat x = (\y. h4/hreal/hrat__lt y x)
% Goal: !h. h4/hreal/isacut (h4/hreal/cut__of__hrat h)
%   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_CONJu_u_SYM]: !t2 t1. t1 /\ t2 <=> t2 /\ t1
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_hreals_HRATu_u_LTu_u_TRANS]: !z y x. h4/hreal/hrat__lt x y /\ h4/hreal/hrat__lt y z ==> h4/hreal/hrat__lt x z
% Assm [h4s_hreals_HRATu_u_LTu_u_ANTISYM]: !y x. ~(h4/hreal/hrat__lt x y /\ h4/hreal/hrat__lt y x)
% Assm [h4s_hreals_HRATu_u_UP]: !x. ?y. h4/hreal/hrat__lt x y
% Assm [h4s_hreals_HRATu_u_DOWN]: !x. ?y. h4/hreal/hrat__lt y x
% Assm [h4s_hreals_HRATu_u_MEAN]: !y x. h4/hreal/hrat__lt x y ==> (?z. h4/hreal/hrat__lt x z /\ h4/hreal/hrat__lt z y)
% Assm [h4s_hreals_isacut0]: !C. h4/hreal/isacut C <=> (?x. happ C x) /\ (?x. ~happ C x) /\ (!x y. happ C x /\ h4/hreal/hrat__lt y x ==> happ C y) /\ (!x. happ C x ==> (?y. happ C y /\ h4/hreal/hrat__lt x y))
% Assm [h4s_hreals_cutu_u_ofu_u_hrat0]: !x x'. happ (h4/hreal/cut__of__hrat x) x' <=> h4/hreal/hrat__lt x' x
% Goal: !h. h4/hreal/isacut (h4/hreal/cut__of__hrat h)
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_Q254356,TV_Q254352]: ![V_f, V_g]: (![V_x]: s(TV_Q254352,happ(s(t_fun(TV_Q254356,TV_Q254352),V_f),s(TV_Q254356,V_x))) = s(TV_Q254352,happ(s(t_fun(TV_Q254356,TV_Q254352),V_g),s(TV_Q254356,V_x))) => s(t_fun(TV_Q254356,TV_Q254352),V_f) = s(t_fun(TV_Q254356,TV_Q254352),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_CONJu_u_SYM, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) <=> (p(s(t_bool,V_t2)) & p(s(t_bool,V_t1))))).
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_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_hreals_HRATu_u_LTu_u_TRANS, axiom, ![V_z, V_y, V_x]: ((p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))) & p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_y),s(t_h4s_hrats_hrat,V_z))))) => p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_z)))))).
fof(ah4s_hreals_HRATu_u_LTu_u_ANTISYM, axiom, ![V_y, V_x]: ~ ((p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))) & p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_y),s(t_h4s_hrats_hrat,V_x))))))).
fof(ah4s_hreals_HRATu_u_UP, axiom, ![V_x]: ?[V_y]: p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y))))).
fof(ah4s_hreals_HRATu_u_DOWN, axiom, ![V_x]: ?[V_y]: p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_y),s(t_h4s_hrats_hrat,V_x))))).
fof(ah4s_hreals_HRATu_u_MEAN, axiom, ![V_y, V_x]: (p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))) => ?[V_z]: (p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_z)))) & p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_z),s(t_h4s_hrats_hrat,V_y))))))).
fof(ah4s_hreals_isacut0, axiom, ![V_C]: (p(s(t_bool,h4s_hreals_isacut(s(t_fun(t_h4s_hrats_hrat,t_bool),V_C)))) <=> (?[V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),V_C),s(t_h4s_hrats_hrat,V_x)))) & (?[V_x]: ~ (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),V_C),s(t_h4s_hrats_hrat,V_x))))) & (![V_x, V_y]: ((p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),V_C),s(t_h4s_hrats_hrat,V_x)))) & p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_y),s(t_h4s_hrats_hrat,V_x))))) => p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),V_C),s(t_h4s_hrats_hrat,V_y))))) & ![V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),V_C),s(t_h4s_hrats_hrat,V_x)))) => ?[V_y]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),V_C),s(t_h4s_hrats_hrat,V_y)))) & p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y))))))))))).
fof(ah4s_hreals_cutu_u_ofu_u_hrat0, axiom, ![V_x, V_xi_]: s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cutu_u_ofu_u_hrat(s(t_h4s_hrats_hrat,V_x))),s(t_h4s_hrats_hrat,V_xi_))) = s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_xi_),s(t_h4s_hrats_hrat,V_x)))).
fof(ch4s_hreals_ISACUTu_u_HRAT, conjecture, ![V_h]: p(s(t_bool,h4s_hreals_isacut(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cutu_u_ofu_u_hrat(s(t_h4s_hrats_hrat,V_h))))))).
