%   ORIGINAL: h4/arithmetic/TC__eq__NRC
% 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/prim__rec/num__Axiom: !f e. ?fn. fn h4/num/0 = e /\ (!n. fn (h4/num/SUC n) = f n (fn n))
% Assm: h4/arithmetic/NRC0_c1: !y x n R. h4/arithmetic/NRC R (h4/num/SUC n) x y <=> (?z. R x z /\ h4/arithmetic/NRC R n z y)
% Assm: h4/arithmetic/NRC__SUC__RECURSE__LEFT: !y x n R. h4/arithmetic/NRC R (h4/num/SUC n) x y <=> (?z. h4/arithmetic/NRC R n x z /\ R z y)
% Assm: h4/arithmetic/NRC0_c0: !y x R. h4/arithmetic/NRC R h4/num/0 x y <=> x = y
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/arithmetic/NRC__ADD__EQN: !z x n m R. h4/arithmetic/NRC R (h4/arithmetic/_2B m n) x z <=> (?y. h4/arithmetic/NRC R m x y /\ h4/arithmetic/NRC R n y z)
% Assm: h4/arithmetic/NRC__1: !y x R. h4/arithmetic/NRC R (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) x y <=> R x y
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/arithmetic/NRC__ADD__E: !z x n m R. h4/arithmetic/NRC R (h4/arithmetic/_2B m n) x z ==> (?y. h4/arithmetic/NRC R m x y /\ h4/arithmetic/NRC R n y z)
% Assm: h4/arithmetic/NRC__ADD__I: !z y x n m R. h4/arithmetic/NRC R m x y /\ h4/arithmetic/NRC R n y z ==> h4/arithmetic/NRC R (h4/arithmetic/_2B m n) x z
% Assm: h4/arithmetic/NRC__RTC: !y x n R. h4/arithmetic/NRC R n x y ==> h4/relation/RTC R x y
% Assm: h4/arithmetic/RTC__NRC: !y x R. h4/relation/RTC R x y ==> (?n. h4/arithmetic/NRC R n x y)
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/arithmetic/RTC__eq__NRC: !y x R. h4/relation/RTC R x y <=> (?n. h4/arithmetic/NRC R n x y)
% Assm: h4/arithmetic/NRC__0: !y x R. h4/arithmetic/NRC R h4/num/0 x y <=> x = y
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/LEFT__AND__FORALL__THM: !Q P. (!x. P x) /\ Q <=> (!x. P x /\ Q)
% Assm: h4/bool/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/bool/RIGHT__OR__EXISTS__THM: !Q P. P \/ (?x. Q x) <=> (?x. P \/ Q x)
% Assm: h4/relation/TC__RULES_c0: !y x R. R x y ==> h4/relation/TC R x y
% Assm: h4/relation/TC__RULES_c1: !z y x R. h4/relation/TC R x y /\ h4/relation/TC R y z ==> h4/relation/TC R x z
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/relation/TC__INDUCT: !R P. (!x y. R x y ==> P x y) /\ (!x y z. P x y /\ P y z ==> P x z) ==> (!u v. h4/relation/TC R u v ==> P u v)
% Assm: h4/arithmetic/ADD1: !m. h4/num/SUC m = h4/arithmetic/_2B m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/FORALL__AND__THM: !Q P. (!x. P x /\ Q x) <=> (!x. P x) /\ (!x. Q x)
% Assm: h4/bool/EXISTS__OR__THM: !Q P. (?x. P x \/ Q x) <=> (?x. P x) \/ (?x. Q x)
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/relation/TC__DEF: !b a R. h4/relation/TC R a b <=> (!P. (!x y. R x y ==> P x y) /\ (!x y z. P x y /\ P y z ==> P x z) ==> P a b)
% Assm: h4/bool/SKOLEM__THM: !P. (!x. ?y. P x y) <=> (?f. !x. P x (f x))
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/relation/TC__CASES2__E: !z x R. h4/relation/TC R x z ==> R x z \/ (?y. h4/relation/TC R x y /\ R y z)
% Assm: h4/bool/RIGHT__EXISTS__AND__THM: !Q P. (?x. P /\ Q x) <=> P /\ (?x. Q x)
% Assm: h4/relation/TC__CASES1__E: !z x R. h4/relation/TC R x z ==> R x z \/ (?y. R x y /\ h4/relation/TC R y z)
% Assm: h4/bool/LEFT__EXISTS__AND__THM: !Q P. (?x. P x /\ Q) <=> (?x. P x) /\ Q
% Assm: h4/relation/TC__CASES2: !z x R. h4/relation/TC R x z <=> R x z \/ (?y. h4/relation/TC R x y /\ R y z)
% Assm: h4/relation/EXTEND__RTC__TC__EQN: !z x R. h4/relation/TC R x z <=> (?y. R x y /\ h4/relation/RTC R y z)
% Assm: h4/relation/TC__SUBSET: !y x R. R x y ==> h4/relation/TC R x y
% Assm: h4/relation/TC__CASES1: !z x R. h4/relation/TC R x z <=> R x z \/ (?y. R x y /\ h4/relation/TC R y z)
% Assm: h4/relation/WF__noloops: !y x R. h4/relation/WF R ==> h4/relation/TC R x y ==> ~(x = y)
% Assm: h4/relation/EXTEND__RTC__TC: !z y x R. R x y /\ h4/relation/RTC R y z ==> h4/relation/TC R x z
% Assm: h4/arithmetic/ONE: h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) = h4/num/SUC h4/num/0
% Assm: h4/relation/TC__implies__one__step: !y x R. h4/relation/TC R x y /\ ~(x = y) ==> (?z. R x z /\ ~(x = z))
% Assm: h4/relation/RTC__CASES__TC: !y x R. h4/relation/RTC R x y <=> x = y \/ h4/relation/TC R x y
% 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/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/relation/TC__INDUCT__RIGHT1: !R P. (!x y. R x y ==> P x y) /\ (!x y z. P x y /\ R y z ==> P x z) ==> (!x y. h4/relation/TC R x y ==> P x y)
% Assm: h4/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/relation/TC__STRONG__INDUCT__RIGHT1: !R P. (!x y. R x y ==> P x y) /\ (!x y z. P x y /\ h4/relation/TC R x y /\ R y z ==> P x z) ==> (!u v. h4/relation/TC R u v ==> P u v)
% Assm: h4/relation/TC__RTC: !y x R. h4/relation/TC R x y ==> h4/relation/RTC R x y
% Assm: h4/relation/RTC__RULES_c0: !x R. h4/relation/RTC R x x
% Assm: h4/bool/RIGHT__OR__OVER__AND: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm: h4/relation/TC__INDUCT__LEFT1: !R P. (!x y. R x y ==> P x y) /\ (!x y z. R x y /\ P y z ==> P x z) ==> (!x y. h4/relation/TC R x y ==> P x y)
% Assm: h4/relation/TC__lifts__invariants: !R P. (!x y. P x /\ R x y ==> P y) ==> (!x y. P x /\ h4/relation/TC R x y ==> P y)
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/relation/TC__STRONG__INDUCT__LEFT1: !R P. (!x y. R x y ==> P x y) /\ (!x y z. R x y /\ P y z /\ h4/relation/TC R y z ==> P x z) ==> (!u v. h4/relation/TC R u v ==> P u v)
% Assm: h4/arithmetic/ADD_c0: !n. h4/arithmetic/_2B h4/num/0 n = n
% Assm: h4/arithmetic/ADD_c1: !n m. h4/arithmetic/_2B (h4/num/SUC m) n = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm: h4/relation/RTC__TC__RC: !y x R. h4/relation/RTC R x y ==> h4/relation/RC R x y \/ h4/relation/TC R x y
% Assm: h4/relation/RTC__RULES_c1: !z y x R. R x y /\ h4/relation/RTC R y z ==> h4/relation/RTC R x z
% Assm: h4/relation/TC__STRONG__INDUCT: !R P. (!x y. R x y ==> P x y) /\ (!x y z. P x y /\ P y z /\ h4/relation/TC R x y /\ h4/relation/TC R y z ==> P x z) ==> (!u v. h4/relation/TC R u v ==> P u v)
% Assm: h4/relation/TC__MONOTONE: !y x R Q. (!x0 y0. R x0 y0 ==> Q x0 y0) ==> h4/relation/TC R x y ==> h4/relation/TC Q x y
% Assm: h4/relation/TC__lifts__monotonicities: !f R. (!x y. R x y ==> R (f x) (f y)) ==> (!x y. h4/relation/TC R x y ==> h4/relation/TC R (f x) (f y))
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/LEFT__FORALL__OR__THM: !Q P. (!x. P x \/ Q) <=> (!x. P x) \/ Q
% Assm: h4/relation/RTC__INDUCT: !R P. (!x. P x x) /\ (!x y z. R x y /\ P y z ==> P x z) ==> (!x y. h4/relation/RTC R x y ==> P x y)
% Assm: h4/relation/WF__TC__EQN: !R. h4/relation/WF (h4/relation/TC R) <=> h4/relation/WF R
% Assm: h4/bool/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/bool/LEFT__OR__EXISTS__THM: !Q P. (?x. P x) \/ Q <=> (?x. P x \/ Q)
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/relation/transitive__def: !R. h4/relation/transitive R <=> (!x y z. R x y /\ R y z ==> R x z)
% Assm: h4/relation/WF__NOT__REFL: !y x R. h4/relation/WF R ==> R x y ==> ~(x = y)
% Assm: h4/relation/RTC__CASES1: !y x R. h4/relation/RTC R x y <=> x = y \/ (?u. R x u /\ h4/relation/RTC R u y)
% Assm: h4/relation/TC__TRANSITIVE: !R. h4/relation/transitive (h4/relation/TC R)
% Assm: h4/relation/RTC__RULES__RIGHT1_c1: !z y x R. h4/relation/RTC R x y /\ R y z ==> h4/relation/RTC R x z
% Assm: h4/relation/RTC__RULES__RIGHT1_c0: !x R. h4/relation/RTC R x x
% Assm: h4/relation/RTC__TRANSITIVE: !R. h4/relation/transitive (h4/relation/RTC R)
% Assm: h4/relation/IDEM__TC: h4/relation/IDEM h4/relation/TC
% Assm: h4/bool/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm: h4/relation/RTC__RTC: !y x R. h4/relation/RTC R x y ==> (!z. h4/relation/RTC R y z ==> h4/relation/RTC R x z)
% Assm: h4/bool/EXISTS__REFL: !a. ?x. x = a
% Assm: h4/relation/IDEM0: !f. h4/relation/IDEM f <=> (!x. f (f x) = f x)
% Assm: h4/relation/TC__IDEM: !R. h4/relation/TC (h4/relation/TC R) = h4/relation/TC R
% Assm: h4/relation/RTC__STRONG__INDUCT: !R P. (!x. P x x) /\ (!x y z. R x y /\ h4/relation/RTC R y z /\ P y z ==> P x z) ==> (!x y. h4/relation/RTC R x y ==> P x y)
% Assm: h4/relation/RC__DEF: !y x R. h4/relation/RC R x y <=> x = y \/ R x y
% Assm: h4/relation/inv__TC: !R. h4/relation/inv (h4/relation/TC R) = h4/relation/TC (h4/relation/inv R)
% Assm: h4/relation/WF__TC: !R. h4/relation/WF R ==> h4/relation/WF (h4/relation/TC R)
% Assm: h4/relation/WF__SUBSET: !R P. h4/relation/WF R /\ (!x y. P x y ==> R x y) ==> h4/relation/WF P
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/relation/inv__DEF: !y x R. h4/relation/inv R x y <=> R y x
% Goal: !y x R. h4/relation/TC R x y <=> (?n. h4/arithmetic/NRC R (h4/num/SUC n) x y)
%   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_primu_u_recs_numu_u_Axiom]: !f e. ?fn. happ fn h4/num/0 = e /\ (!n. happ fn (h4/num/SUC n) = happ (happ f n) (happ fn n))
% Assm [h4s_arithmetics_NRC0u_c1]: !y x n R. h4/arithmetic/NRC R (h4/num/SUC n) x y <=> (?z. happ (happ R x) z /\ h4/arithmetic/NRC R n z y)
% Assm [h4s_arithmetics_NRCu_u_SUCu_u_RECURSEu_u_LEFT]: !y x n R. h4/arithmetic/NRC R (h4/num/SUC n) x y <=> (?z. h4/arithmetic/NRC R n x z /\ happ (happ R z) y)
% Assm [h4s_arithmetics_NRC0u_c0]: !y x R. h4/arithmetic/NRC R h4/num/0 x y <=> x = y
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_arithmetics_NRCu_u_ADDu_u_EQN]: !z x n m R. h4/arithmetic/NRC R (h4/arithmetic/_2B m n) x z <=> (?y. h4/arithmetic/NRC R m x y /\ h4/arithmetic/NRC R n y z)
% Assm [h4s_arithmetics_NRCu_u_1]: !y x R. h4/arithmetic/NRC R (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) x y <=> happ (happ R x) y
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_arithmetics_NRCu_u_ADDu_u_E]: !z x n m R. h4/arithmetic/NRC R (h4/arithmetic/_2B m n) x z ==> (?y. h4/arithmetic/NRC R m x y /\ h4/arithmetic/NRC R n y z)
% Assm [h4s_arithmetics_NRCu_u_ADDu_u_I]: !z y x n m R. h4/arithmetic/NRC R m x y /\ h4/arithmetic/NRC R n y z ==> h4/arithmetic/NRC R (h4/arithmetic/_2B m n) x z
% Assm [h4s_arithmetics_NRCu_u_RTC]: !y x n R. h4/arithmetic/NRC R n x y ==> happ (happ (h4/relation/RTC R) x) y
% Assm [h4s_arithmetics_RTCu_u_NRC]: !y x R. happ (happ (h4/relation/RTC R) x) y ==> (?n. h4/arithmetic/NRC R n x y)
% Assm [h4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. P \/ happ Q x) <=> P \/ (!x. happ Q x)
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_arithmetics_RTCu_u_equ_u_NRC]: !y x R. happ (happ (h4/relation/RTC R) x) y <=> (?n. h4/arithmetic/NRC R n x y)
% Assm [h4s_arithmetics_NRCu_u_0]: !y x R. h4/arithmetic/NRC R h4/num/0 x y <=> x = y
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_bools_LEFTu_u_ANDu_u_FORALLu_u_THM]: !Q P. (!x. happ P x) /\ Q <=> (!x. happ P x /\ Q)
% Assm [h4s_bools_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM]: !Q P. P \/ (?x. happ Q x) <=> (?x. P \/ happ Q x)
% Assm [h4s_relations_TCu_u_RULESu_c0]: !y x R. happ (happ R x) y ==> happ (happ (happ h4/relation/TC R) x) y
% Assm [h4s_relations_TCu_u_RULESu_c1]: !z y x R. happ (happ (happ h4/relation/TC R) x) y /\ happ (happ (happ h4/relation/TC R) y) z ==> happ (happ (happ h4/relation/TC R) x) z
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_relations_TCu_u_INDUCT]: !R P. (!x y. happ (happ R x) y ==> happ (happ P x) y) /\ (!x y z. happ (happ P x) y /\ happ (happ P y) z ==> happ (happ P x) z) ==> (!u v. happ (happ (happ h4/relation/TC R) u) v ==> happ (happ P u) v)
% Assm [h4s_arithmetics_ADD1]: !m. h4/num/SUC m = h4/arithmetic/_2B m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_FORALLu_u_ANDu_u_THM]: !Q P. (!x. happ P x /\ happ Q x) <=> (!x. happ P x) /\ (!x. happ Q x)
% Assm [h4s_bools_EXISTSu_u_ORu_u_THM]: !Q P. (?x. happ P x \/ happ Q x) <=> (?x. happ P x) \/ (?x. happ Q x)
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_relations_TCu_u_DEF]: !b a R. happ (happ (happ h4/relation/TC R) a) b <=> (!P. (!x y. happ (happ R x) y ==> happ (happ P x) y) /\ (!x y z. happ (happ P x) y /\ happ (happ P y) z ==> happ (happ P x) z) ==> happ (happ P a) b)
% Assm [h4s_bools_SKOLEMu_u_THM]: !P. (!x. ?y. happ (happ P x) y) <=> (?f. !x. happ (happ P x) (happ f x))
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_relations_TCu_u_CASES2u_u_E]: !z x R. happ (happ (happ h4/relation/TC R) x) z ==> happ (happ R x) z \/ (?y. happ (happ (happ h4/relation/TC R) x) y /\ happ (happ R y) z)
% Assm [h4s_bools_RIGHTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. P /\ happ Q x) <=> P /\ (?x. happ Q x)
% Assm [h4s_relations_TCu_u_CASES1u_u_E]: !z x R. happ (happ (happ h4/relation/TC R) x) z ==> happ (happ R x) z \/ (?y. happ (happ R x) y /\ happ (happ (happ h4/relation/TC R) y) z)
% Assm [h4s_bools_LEFTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. happ P x /\ Q) <=> (?x. happ P x) /\ Q
% Assm [h4s_relations_TCu_u_CASES2]: !z x R. happ (happ (happ h4/relation/TC R) x) z <=> happ (happ R x) z \/ (?y. happ (happ (happ h4/relation/TC R) x) y /\ happ (happ R y) z)
% Assm [h4s_relations_EXTENDu_u_RTCu_u_TCu_u_EQN]: !z x R. happ (happ (happ h4/relation/TC R) x) z <=> (?y. happ (happ R x) y /\ happ (happ (h4/relation/RTC R) y) z)
% Assm [h4s_relations_TCu_u_SUBSET]: !y x R. happ (happ R x) y ==> happ (happ (happ h4/relation/TC R) x) y
% Assm [h4s_relations_TCu_u_CASES1]: !z x R. happ (happ (happ h4/relation/TC R) x) z <=> happ (happ R x) z \/ (?y. happ (happ R x) y /\ happ (happ (happ h4/relation/TC R) y) z)
% Assm [h4s_relations_WFu_u_noloops]: !y x R. h4/relation/WF R ==> happ (happ (happ h4/relation/TC R) x) y ==> ~(x = y)
% Assm [h4s_relations_EXTENDu_u_RTCu_u_TC]: !z y x R. happ (happ R x) y /\ happ (happ (h4/relation/RTC R) y) z ==> happ (happ (happ h4/relation/TC R) x) z
% Assm [h4s_arithmetics_ONE]: h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) = h4/num/SUC h4/num/0
% Assm [h4s_relations_TCu_u_impliesu_u_oneu_u_step]: !y x R. happ (happ (happ h4/relation/TC R) x) y /\ ~(x = y) ==> (?z. happ (happ R x) z /\ ~(x = z))
% Assm [h4s_relations_RTCu_u_CASESu_u_TC]: !y x R. happ (happ (h4/relation/RTC R) x) y <=> x = y \/ happ (happ (happ h4/relation/TC R) x) y
% 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_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_relations_TCu_u_INDUCTu_u_RIGHT1]: !R P. (!x y. happ (happ R x) y ==> happ (happ P x) y) /\ (!x y z. happ (happ P x) y /\ happ (happ R y) z ==> happ (happ P x) z) ==> (!x y. happ (happ (happ h4/relation/TC R) x) y ==> happ (happ P x) y)
% Assm [h4s_nums_INDUCTION]: !P. happ P h4/num/0 /\ (!n. happ P n ==> happ P (h4/num/SUC n)) ==> (!n. happ P n)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_relations_TCu_u_STRONGu_u_INDUCTu_u_RIGHT1]: !R P. (!x y. happ (happ R x) y ==> happ (happ P x) y) /\ (!x y z. happ (happ P x) y /\ happ (happ (happ h4/relation/TC R) x) y /\ happ (happ R y) z ==> happ (happ P x) z) ==> (!u v. happ (happ (happ h4/relation/TC R) u) v ==> happ (happ P u) v)
% Assm [h4s_relations_TCu_u_RTC]: !y x R. happ (happ (happ h4/relation/TC R) x) y ==> happ (happ (h4/relation/RTC R) x) y
% Assm [h4s_relations_RTCu_u_RULESu_c0]: !x R. happ (happ (h4/relation/RTC R) x) x
% Assm [h4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND]: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm [h4s_relations_TCu_u_INDUCTu_u_LEFT1]: !R P. (!x y. happ (happ R x) y ==> happ (happ P x) y) /\ (!x y z. happ (happ R x) y /\ happ (happ P y) z ==> happ (happ P x) z) ==> (!x y. happ (happ (happ h4/relation/TC R) x) y ==> happ (happ P x) y)
% Assm [h4s_relations_TCu_u_liftsu_u_invariants]: !R P. (!x y. happ P x /\ happ (happ R x) y ==> happ P y) ==> (!x y. happ P x /\ happ (happ (happ h4/relation/TC R) x) y ==> happ P y)
% Assm [h4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% Assm [h4s_relations_TCu_u_STRONGu_u_INDUCTu_u_LEFT1]: !R P. (!x y. happ (happ R x) y ==> happ (happ P x) y) /\ (!x y z. happ (happ R x) y /\ happ (happ P y) z /\ happ (happ (happ h4/relation/TC R) y) z ==> happ (happ P x) z) ==> (!u v. happ (happ (happ h4/relation/TC R) u) v ==> happ (happ P u) v)
% Assm [h4s_arithmetics_ADDu_c0]: !n. h4/arithmetic/_2B h4/num/0 n = n
% Assm [h4s_arithmetics_ADDu_c1]: !n m. h4/arithmetic/_2B (h4/num/SUC m) n = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm [h4s_relations_RTCu_u_TCu_u_RC]: !y x R. happ (happ (h4/relation/RTC R) x) y ==> h4/relation/RC R x y \/ happ (happ (happ h4/relation/TC R) x) y
% Assm [h4s_relations_RTCu_u_RULESu_c1]: !z y x R. happ (happ R x) y /\ happ (happ (h4/relation/RTC R) y) z ==> happ (happ (h4/relation/RTC R) x) z
% Assm [h4s_relations_TCu_u_STRONGu_u_INDUCT]: !R P. (!x y. happ (happ R x) y ==> happ (happ P x) y) /\ (!x y z. happ (happ P x) y /\ happ (happ P y) z /\ happ (happ (happ h4/relation/TC R) x) y /\ happ (happ (happ h4/relation/TC R) y) z ==> happ (happ P x) z) ==> (!u v. happ (happ (happ h4/relation/TC R) u) v ==> happ (happ P u) v)
% Assm [h4s_relations_TCu_u_MONOTONE]: !y x R Q. (!x0 y0. happ (happ R x0) y0 ==> happ (happ Q x0) y0) ==> happ (happ (happ h4/relation/TC R) x) y ==> happ (happ (happ h4/relation/TC Q) x) y
% Assm [h4s_relations_TCu_u_liftsu_u_monotonicities]: !f R. (!x y. happ (happ R x) y ==> happ (happ R (happ f x)) (happ f y)) ==> (!x y. happ (happ (happ h4/relation/TC R) x) y ==> happ (happ (happ h4/relation/TC R) (happ f x)) (happ f y))
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. happ P x \/ Q) <=> (!x. happ P x) \/ Q
% Assm [h4s_relations_RTCu_u_INDUCT]: !R P. (!x. happ (happ P x) x) /\ (!x y z. happ (happ R x) y /\ happ (happ P y) z ==> happ (happ P x) z) ==> (!x y. happ (happ (h4/relation/RTC R) x) y ==> happ (happ P x) y)
% Assm [h4s_relations_WFu_u_TCu_u_EQN]: !R. h4/relation/WF (happ h4/relation/TC R) <=> h4/relation/WF R
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_bools_LEFTu_u_ORu_u_EXISTSu_u_THM]: !Q P. (?x. happ P x) \/ Q <=> (?x. happ P x \/ Q)
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_relations_transitiveu_u_def]: !R. h4/relation/transitive R <=> (!x y z. happ (happ R x) y /\ happ (happ R y) z ==> happ (happ R x) z)
% Assm [h4s_relations_WFu_u_NOTu_u_REFL]: !y x R. h4/relation/WF R ==> happ (happ R x) y ==> ~(x = y)
% Assm [h4s_relations_RTCu_u_CASES1]: !y x R. happ (happ (h4/relation/RTC R) x) y <=> x = y \/ (?u. happ (happ R x) u /\ happ (happ (h4/relation/RTC R) u) y)
% Assm [h4s_relations_TCu_u_TRANSITIVE]: !R. h4/relation/transitive (happ h4/relation/TC R)
% Assm [h4s_relations_RTCu_u_RULESu_u_RIGHT1u_c1]: !z y x R. happ (happ (h4/relation/RTC R) x) y /\ happ (happ R y) z ==> happ (happ (h4/relation/RTC R) x) z
% Assm [h4s_relations_RTCu_u_RULESu_u_RIGHT1u_c0]: !x R. happ (happ (h4/relation/RTC R) x) x
% Assm [h4s_relations_RTCu_u_TRANSITIVE]: !R. h4/relation/transitive (h4/relation/RTC R)
% Assm [h4s_relations_IDEMu_u_TC]: h4/relation/IDEM h4/relation/TC
% Assm [h4s_bools_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm [h4s_relations_RTCu_u_RTC]: !y x R. happ (happ (h4/relation/RTC R) x) y ==> (!z. happ (happ (h4/relation/RTC R) y) z ==> happ (happ (h4/relation/RTC R) x) z)
% Assm [h4s_bools_EXISTSu_u_REFL]: !a. ?x. x = a
% Assm [h4s_relations_IDEM0]: !f. h4/relation/IDEM f <=> (!x. happ f (happ f x) = happ f x)
% Assm [h4s_relations_TCu_u_IDEM]: !R. happ h4/relation/TC (happ h4/relation/TC R) = happ h4/relation/TC R
% Assm [h4s_relations_RTCu_u_STRONGu_u_INDUCT]: !R P. (!x. happ (happ P x) x) /\ (!x y z. happ (happ R x) y /\ happ (happ (h4/relation/RTC R) y) z /\ happ (happ P y) z ==> happ (happ P x) z) ==> (!x y. happ (happ (h4/relation/RTC R) x) y ==> happ (happ P x) y)
% Assm [h4s_relations_RCu_u_DEF]: !y x R. h4/relation/RC R x y <=> x = y \/ happ (happ R x) y
% Assm [h4s_relations_invu_u_TC]: !R. h4/relation/inv (happ h4/relation/TC R) = happ h4/relation/TC (h4/relation/inv R)
% Assm [h4s_relations_WFu_u_TC]: !R. h4/relation/WF R ==> h4/relation/WF (happ h4/relation/TC R)
% Assm [h4s_relations_WFu_u_SUBSET]: !R P. h4/relation/WF R /\ (!x y. happ (happ P x) y ==> happ (happ R x) y) ==> h4/relation/WF P
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_relations_invu_u_DEF]: !y x R. happ (happ (h4/relation/inv R) x) y <=> happ (happ R y) x
% Goal: !y x R. happ (happ (happ h4/relation/TC R) x) y <=> (?n. h4/arithmetic/NRC R (h4/num/SUC n) x y)
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_Q1408272,TV_Q1408268]: ![V_f, V_g]: (![V_x]: s(TV_Q1408268,happ(s(t_fun(TV_Q1408272,TV_Q1408268),V_f),s(TV_Q1408272,V_x))) = s(TV_Q1408268,happ(s(t_fun(TV_Q1408272,TV_Q1408268),V_g),s(TV_Q1408272,V_x))) => s(t_fun(TV_Q1408272,TV_Q1408268),V_f) = s(t_fun(TV_Q1408272,TV_Q1408268),V_g))).
fof(ah4s_primu_u_recs_numu_u_Axiom, axiom, ![TV_u_27a]: ![V_f, V_e]: ?[V_fn]: (s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_e) & ![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_h4s_nums_num,V_n))),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_arithmetics_NRC0u_c1, axiom, ![TV_u_27a]: ![V_y, V_x, V_n, V_R]: (p(s(t_bool,h4s_arithmetics_nrc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_x),s(TV_u_27a,V_y)))) <=> ?[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)))) & p(s(t_bool,h4s_arithmetics_nrc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_h4s_nums_num,V_n),s(TV_u_27a,V_z),s(TV_u_27a,V_y))))))).
fof(ah4s_arithmetics_NRCu_u_SUCu_u_RECURSEu_u_LEFT, axiom, ![TV_u_27a]: ![V_y, V_x, V_n, V_R]: (p(s(t_bool,h4s_arithmetics_nrc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_x),s(TV_u_27a,V_y)))) <=> ?[V_z]: (p(s(t_bool,h4s_arithmetics_nrc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_h4s_nums_num,V_n),s(TV_u_27a,V_x),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_z))),s(TV_u_27a,V_y))))))).
fof(ah4s_arithmetics_NRC0u_c0, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: (p(s(t_bool,h4s_arithmetics_nrc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,V_x),s(TV_u_27a,V_y)))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, 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))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_sats_dcu_u_disj, 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_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
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_dcu_u_imp, 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_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
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_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_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_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_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_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = 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_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
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_arithmetics_NRCu_u_ADDu_u_EQN, axiom, ![TV_u_27a]: ![V_z, V_x, V_n, V_m, V_R]: (p(s(t_bool,h4s_arithmetics_nrc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_x),s(TV_u_27a,V_z)))) <=> ?[V_y]: (p(s(t_bool,h4s_arithmetics_nrc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_h4s_nums_num,V_m),s(TV_u_27a,V_x),s(TV_u_27a,V_y)))) & p(s(t_bool,h4s_arithmetics_nrc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_h4s_nums_num,V_n),s(TV_u_27a,V_y),s(TV_u_27a,V_z))))))).
fof(ah4s_arithmetics_NRCu_u_1, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: s(t_bool,h4s_arithmetics_nrc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = 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_y)))).
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_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_arithmetics_NRCu_u_ADDu_u_E, axiom, ![TV_u_27a]: ![V_z, V_x, V_n, V_m, V_R]: (p(s(t_bool,h4s_arithmetics_nrc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_x),s(TV_u_27a,V_z)))) => ?[V_y]: (p(s(t_bool,h4s_arithmetics_nrc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_h4s_nums_num,V_m),s(TV_u_27a,V_x),s(TV_u_27a,V_y)))) & p(s(t_bool,h4s_arithmetics_nrc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_h4s_nums_num,V_n),s(TV_u_27a,V_y),s(TV_u_27a,V_z))))))).
fof(ah4s_arithmetics_NRCu_u_ADDu_u_I, axiom, ![TV_u_27a]: ![V_z, V_y, V_x, V_n, V_m, V_R]: ((p(s(t_bool,h4s_arithmetics_nrc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_h4s_nums_num,V_m),s(TV_u_27a,V_x),s(TV_u_27a,V_y)))) & p(s(t_bool,h4s_arithmetics_nrc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_h4s_nums_num,V_n),s(TV_u_27a,V_y),s(TV_u_27a,V_z))))) => p(s(t_bool,h4s_arithmetics_nrc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_x),s(TV_u_27a,V_z)))))).
fof(ah4s_arithmetics_NRCu_u_RTC, axiom, ![TV_u_27a]: ![V_y, V_x, V_n, V_R]: (p(s(t_bool,h4s_arithmetics_nrc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_h4s_nums_num,V_n),s(TV_u_27a,V_x),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)),h4s_relations_rtc(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_y)))))).
fof(ah4s_arithmetics_RTCu_u_NRC, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: (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)),h4s_relations_rtc(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_y)))) => ?[V_n]: p(s(t_bool,h4s_arithmetics_nrc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_h4s_nums_num,V_n),s(TV_u_27a,V_x),s(TV_u_27a,V_y)))))).
fof(ah4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,V_P)) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (p(s(t_bool,V_P)) | ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(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_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_bools_EXISTSu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_x)))))).
fof(ah4s_arithmetics_RTCu_u_equ_u_NRC, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: (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)),h4s_relations_rtc(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_y)))) <=> ?[V_n]: p(s(t_bool,h4s_arithmetics_nrc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_h4s_nums_num,V_n),s(TV_u_27a,V_x),s(TV_u_27a,V_y)))))).
fof(ah4s_arithmetics_NRCu_u_0, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: (p(s(t_bool,h4s_arithmetics_nrc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,V_x),s(TV_u_27a,V_y)))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_sats_pthu_u_no2, 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_no1, 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_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f))))).
fof(ah4s_bools_LEFTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))) <=> ![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))))).
fof(ah4s_bools_LEFTu_u_ORu_u_OVERu_u_AND, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) & p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) & (p(s(t_bool,V_A)) | p(s(t_bool,V_C)))))).
fof(ah4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) | ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ?[V_x]: (p(s(t_bool,V_P)) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_relations_TCu_u_RULESu_c0, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: (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_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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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_y)))))).
fof(ah4s_relations_TCu_u_RULESu_c1, axiom, ![TV_u_27a]: ![V_z, V_y, V_x, V_R]: ((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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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_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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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_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_relations_TCu_u_INDUCT, axiom, ![TV_u_27a]: ![V_R, V_P]: ((![V_x, 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_x))),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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) & ![V_x, V_y, 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_P),s(TV_u_27a,V_x))),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_P),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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_z)))))) => ![V_u, V_v]: (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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_u))),s(TV_u_27a,V_v)))) => 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_P),s(TV_u_27a,V_u))),s(TV_u_27a,V_v))))))).
fof(ah4s_arithmetics_ADD1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))).
fof(ah4s_bools_NOTu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ?[V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_FORALLu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(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_Q),s(TV_u_27a,V_x))))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_EXISTSu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (?[V_x]: (p(s(t_bool,happ(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_Q),s(TV_u_27a,V_x))))) <=> (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_UNWINDu_u_THM2, axiom, ![TV_u_27a]: ![V_a, V_P]: (?[V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & p(s(t_bool,happ(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_P),s(TV_u_27a,V_a)))))).
fof(ah4s_relations_TCu_u_DEF, axiom, ![TV_u_27a]: ![V_b, V_a, V_R]: (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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_a))),s(TV_u_27a,V_b)))) <=> ![V_P]: ((![V_x, 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_x))),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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) & ![V_x, V_y, 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_P),s(TV_u_27a,V_x))),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_P),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_P),s(TV_u_27a,V_x))),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_P),s(TV_u_27a,V_a))),s(TV_u_27a,V_b))))))).
fof(ah4s_bools_SKOLEMu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (![V_x]: ?[V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) <=> ?[V_f]: ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))))))).
fof(ah4s_bools_NOTu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ![V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_relations_TCu_u_CASES2u_u_E, axiom, ![TV_u_27a]: ![V_z, V_x, V_R]: (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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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)))) => (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)))) | ?[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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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_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)))))))).
fof(ah4s_bools_RIGHTu_u_EXISTSu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (?[V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (p(s(t_bool,V_P)) & ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_relations_TCu_u_CASES1u_u_E, axiom, ![TV_u_27a]: ![V_z, V_x, V_R]: (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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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)))) => (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)))) | ?[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_x))),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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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)))))))).
fof(ah4s_bools_LEFTu_u_EXISTSu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (?[V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))) <=> (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))))).
fof(ah4s_relations_TCu_u_CASES2, axiom, ![TV_u_27a]: ![V_z, V_x, V_R]: (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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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)))) <=> (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)))) | ?[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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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_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)))))))).
fof(ah4s_relations_EXTENDu_u_RTCu_u_TCu_u_EQN, axiom, ![TV_u_27a]: ![V_z, V_x, V_R]: (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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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)))) <=> ?[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_x))),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)),h4s_relations_rtc(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))))))).
fof(ah4s_relations_TCu_u_SUBSET, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: (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_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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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_y)))))).
fof(ah4s_relations_TCu_u_CASES1, axiom, ![TV_u_27a]: ![V_z, V_x, V_R]: (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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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)))) <=> (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)))) | ?[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_x))),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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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)))))))).
fof(ah4s_relations_WFu_u_noloops, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => (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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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_y)))) => ~ (s(TV_u_27a,V_x) = s(TV_u_27a,V_y))))).
fof(ah4s_relations_EXTENDu_u_RTCu_u_TC, axiom, ![TV_u_27a]: ![V_z, V_y, V_x, V_R]: ((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_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)),h4s_relations_rtc(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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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_arithmetics_ONE, axiom, s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_relations_TCu_u_impliesu_u_oneu_u_step, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: ((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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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_y)))) & ~ (s(TV_u_27a,V_x) = s(TV_u_27a,V_y))) => ?[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)))) & ~ (s(TV_u_27a,V_x) = s(TV_u_27a,V_z))))).
fof(ah4s_relations_RTCu_u_CASESu_u_TC, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: (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)),h4s_relations_rtc(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_y)))) <=> (s(TV_u_27a,V_x) = 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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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_y))))))).
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_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_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_relations_TCu_u_INDUCTu_u_RIGHT1, axiom, ![TV_u_27a]: ![V_R, V_P]: ((![V_x, 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_x))),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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) & ![V_x, V_y, 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_P),s(TV_u_27a,V_x))),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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_z)))))) => ![V_x, 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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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_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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))))).
fof(ah4s_nums_INDUCTION, axiom, ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_0)))) & ![V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))) => ![V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_relations_TCu_u_STRONGu_u_INDUCTu_u_RIGHT1, axiom, ![TV_u_27a]: ![V_R, V_P]: ((![V_x, 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_x))),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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) & ![V_x, V_y, 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_P),s(TV_u_27a,V_x))),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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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_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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_z)))))) => ![V_u, V_v]: (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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_u))),s(TV_u_27a,V_v)))) => 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_P),s(TV_u_27a,V_u))),s(TV_u_27a,V_v))))))).
fof(ah4s_relations_TCu_u_RTC, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: (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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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_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)),h4s_relations_rtc(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_y)))))).
fof(ah4s_relations_RTCu_u_RULESu_c0, axiom, ![TV_u_27a]: ![V_x, V_R]: 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)),h4s_relations_rtc(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_x))))).
fof(ah4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND, axiom, ![V_C, V_B, V_A]: (((p(s(t_bool,V_B)) & p(s(t_bool,V_C))) | p(s(t_bool,V_A))) <=> ((p(s(t_bool,V_B)) | p(s(t_bool,V_A))) & (p(s(t_bool,V_C)) | p(s(t_bool,V_A)))))).
fof(ah4s_relations_TCu_u_INDUCTu_u_LEFT1, axiom, ![TV_u_27a]: ![V_R, V_P]: ((![V_x, 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_x))),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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) & ![V_x, V_y, 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_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_P),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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_z)))))) => ![V_x, 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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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_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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))))).
fof(ah4s_relations_TCu_u_liftsu_u_invariants, axiom, ![TV_u_27a]: ![V_R, V_P]: (![V_x, V_y]: ((p(s(t_bool,happ(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),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_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_y))))) => ![V_x, V_y]: ((p(s(t_bool,happ(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),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_y))))))).
fof(ah4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ![V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_relations_TCu_u_STRONGu_u_INDUCTu_u_LEFT1, axiom, ![TV_u_27a]: ![V_R, V_P]: ((![V_x, 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_x))),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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) & ![V_x, V_y, 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_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_P),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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_z)))))) => ![V_u, V_v]: (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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_u))),s(TV_u_27a,V_v)))) => 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_P),s(TV_u_27a,V_u))),s(TV_u_27a,V_v))))))).
fof(ah4s_arithmetics_ADDu_c0, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_arithmetics_ADDu_c1, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_relations_RTCu_u_TCu_u_RC, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: (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)),h4s_relations_rtc(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_y)))) => (p(s(t_bool,h4s_relations_rc(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_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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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_y))))))).
fof(ah4s_relations_RTCu_u_RULESu_c1, axiom, ![TV_u_27a]: ![V_z, V_y, V_x, V_R]: ((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_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)),h4s_relations_rtc(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)),h4s_relations_rtc(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_relations_TCu_u_STRONGu_u_INDUCT, axiom, ![TV_u_27a]: ![V_R, V_P]: ((![V_x, 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_x))),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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) & ![V_x, V_y, 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_P),s(TV_u_27a,V_x))),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_P),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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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_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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_z)))))) => ![V_u, V_v]: (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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_u))),s(TV_u_27a,V_v)))) => 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_P),s(TV_u_27a,V_u))),s(TV_u_27a,V_v))))))).
fof(ah4s_relations_TCu_u_MONOTONE, axiom, ![TV_u_27a]: ![V_y, V_x, V_R, V_Q]: (![V_x0, V_y0]: (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_x0))),s(TV_u_27a,V_y0)))) => 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_Q),s(TV_u_27a,V_x0))),s(TV_u_27a,V_y0))))) => (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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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_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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_Q))),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))))).
fof(ah4s_relations_TCu_u_liftsu_u_monotonicities, axiom, ![TV_u_27a]: ![V_f, V_R]: (![V_x, 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_x))),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,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_x))))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_y))))))) => ![V_x, 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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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_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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_x))))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_y))))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))))).
fof(ah4s_relations_RTCu_u_INDUCT, axiom, ![TV_u_27a]: ![V_R, V_P]: ((![V_x]: 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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_x)))) & ![V_x, V_y, 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_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_P),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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_z)))))) => ![V_x, 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)),h4s_relations_rtc(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_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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))))).
fof(ah4s_relations_WFu_u_TCu_u_EQN, axiom, ![TV_u_27a]: ![V_R]: s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) = s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))).
fof(ah4s_bools_CONJu_u_ASSOC, 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_DEu_u_MORGANu_u_THMu_c0, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) & p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) | ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_LEFTu_u_ORu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))) <=> ?[V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))))).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_relations_transitiveu_u_def, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_transitive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> ![V_x, V_y, 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_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_relations_WFu_u_NOTu_u_REFL, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => (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_y)))) => ~ (s(TV_u_27a,V_x) = s(TV_u_27a,V_y))))).
fof(ah4s_relations_RTCu_u_CASES1, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: (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)),h4s_relations_rtc(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_y)))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) | ?[V_u]: (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_u)))) & 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)),h4s_relations_rtc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_u))),s(TV_u_27a,V_y)))))))).
fof(ah4s_relations_TCu_u_TRANSITIVE, axiom, ![TV_u_27a]: ![V_R]: p(s(t_bool,h4s_relations_transitive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))))).
fof(ah4s_relations_RTCu_u_RULESu_u_RIGHT1u_c1, axiom, ![TV_u_27a]: ![V_z, V_y, V_x, V_R]: ((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)),h4s_relations_rtc(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_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)),h4s_relations_rtc(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_relations_RTCu_u_RULESu_u_RIGHT1u_c0, axiom, ![TV_u_27a]: ![V_x, V_R]: 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)),h4s_relations_rtc(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_x))))).
fof(ah4s_relations_RTCu_u_TRANSITIVE, axiom, ![TV_u_27a]: ![V_R]: p(s(t_bool,h4s_relations_transitive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rtc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))))).
fof(ah4s_relations_IDEMu_u_TC, axiom, ![TV_u_27a]: p(s(t_bool,h4s_relations_idem(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc))))).
fof(ah4s_bools_EQu_u_IMPu_u_THM, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((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_relations_RTCu_u_RTC, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: (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)),h4s_relations_rtc(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_y)))) => ![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)),h4s_relations_rtc(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)),h4s_relations_rtc(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_bools_EXISTSu_u_REFL, axiom, ![TV_u_27a]: ![V_a]: ?[V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_a)).
fof(ah4s_relations_IDEM0, axiom, ![TV_u_27z]: ![V_f]: (p(s(t_bool,h4s_relations_idem(s(t_fun(TV_u_27z,TV_u_27z),V_f)))) <=> ![V_x]: s(TV_u_27z,happ(s(t_fun(TV_u_27z,TV_u_27z),V_f),s(TV_u_27z,happ(s(t_fun(TV_u_27z,TV_u_27z),V_f),s(TV_u_27z,V_x))))) = s(TV_u_27z,happ(s(t_fun(TV_u_27z,TV_u_27z),V_f),s(TV_u_27z,V_x))))).
fof(ah4s_relations_TCu_u_IDEM, axiom, ![TV_u_27a]: ![V_R]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))).
fof(ah4s_relations_RTCu_u_STRONGu_u_INDUCT, axiom, ![TV_u_27a]: ![V_R, V_P]: ((![V_x]: 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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_x)))) & ![V_x, V_y, 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_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)),h4s_relations_rtc(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_P),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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_z)))))) => ![V_x, 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)),h4s_relations_rtc(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_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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))))).
fof(ah4s_relations_RCu_u_DEF, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: (p(s(t_bool,h4s_relations_rc(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_y)))) <=> (s(TV_u_27a,V_x) = 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_x))),s(TV_u_27a,V_y))))))).
fof(ah4s_relations_invu_u_TC, axiom, ![TV_u_27a]: ![V_R]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_inv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_inv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))).
fof(ah4s_relations_WFu_u_TC, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))))).
fof(ah4s_relations_WFu_u_SUBSET, axiom, ![TV_u_27a]: ![V_R, V_P]: ((p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) & ![V_x, 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_P),s(TV_u_27a,V_x))),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_x))),s(TV_u_27a,V_y)))))) => p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_P)))))).
fof(ah4s_bools_FUNu_u_EQu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g) <=> ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))).
fof(ah4s_relations_invu_u_DEF, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: 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)),h4s_relations_inv(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_y))) = 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(ch4s_arithmetics_TCu_u_equ_u_NRC, conjecture, ![TV_u_27a]: ![V_y, V_x, V_R]: (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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_tc),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_y)))) <=> ?[V_n]: p(s(t_bool,h4s_arithmetics_nrc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_x),s(TV_u_27a,V_y)))))).
