%   ORIGINAL: h4/prim__rec/SUC__ID
% 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/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/num/INV__SUC: !n m. h4/num/SUC m = h4/num/SUC n ==> m = n
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/prim__rec/LESS__DEF: !n m. h4/prim__rec/_3C m n <=> (?P. (!n0. P (h4/num/SUC n0) ==> P n0) /\ P m /\ ~P n)
% Assm: h4/prim__rec/LESS__THM: !n m. h4/prim__rec/_3C m (h4/num/SUC n) <=> m = n \/ h4/prim__rec/_3C m n
% Assm: h4/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Assm: h4/prim__rec/LESS__LEMMA1: !n m. h4/prim__rec/_3C m (h4/num/SUC n) ==> m = n \/ h4/prim__rec/_3C m n
% Assm: h4/prim__rec/LESS__LEMMA2: !n m. m = n \/ h4/prim__rec/_3C m n ==> h4/prim__rec/_3C m (h4/num/SUC n)
% Assm: h4/num/SUC__DEF: !m. h4/num/SUC m = h4/num/ABS__num (h4/num/SUC__REP (h4/num/REP__num m))
% Assm: h4/prim__rec/LESS__SUC: !n m. h4/prim__rec/_3C m n ==> h4/prim__rec/_3C m (h4/num/SUC n)
% Assm: h4/prim__rec/LESS__SUC__REFL: !n. h4/prim__rec/_3C n (h4/num/SUC n)
% Assm: h4/num/NOT__SUC: !n. ~(h4/num/SUC n = h4/num/0)
% Assm: h4/prim__rec/LESS__MONO: !n m. h4/prim__rec/_3C m n ==> h4/prim__rec/_3C (h4/num/SUC m) (h4/num/SUC n)
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/num/IS__NUM__REP0: !m. h4/num/IS__NUM__REP m <=> (!P. P h4/num/ZERO__REP /\ (!n. P n ==> P (h4/num/SUC__REP n)) ==> P m)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/num/num__ISO__DEF_c0: !a. h4/num/ABS__num (h4/num/REP__num a) = a
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/prim__rec/LESS__SUC__IMP: !n m. h4/prim__rec/_3C m (h4/num/SUC n) ==> ~(m = n) ==> h4/prim__rec/_3C m n
% Assm: h4/prim__rec/SUC__LESS: !n m. h4/prim__rec/_3C (h4/num/SUC m) n ==> h4/prim__rec/_3C m n
% Assm: h4/prim__rec/EQ__LESS: !n m. h4/num/SUC m = n ==> h4/prim__rec/_3C m n
% Assm: h4/prim__rec/INV__SUC__EQ: !n m. h4/num/SUC m = h4/num/SUC n <=> m = n
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/prim__rec/LESS__0: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm: h4/num/num__ISO__DEF_c1: !r. h4/num/IS__NUM__REP r <=> h4/num/REP__num (h4/num/ABS__num r) = r
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/prim__rec/LESS__0__0: h4/prim__rec/_3C h4/num/0 (h4/num/SUC h4/num/0)
% Assm: h4/num/num__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION h4/num/IS__NUM__REP rep
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/num/ZERO__DEF: h4/num/0 = h4/num/ABS__num h4/num/ZERO__REP
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/ABS__REP__THM: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. abs (rep a) = a) /\ (!r. P r <=> rep (abs r) = r))
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/OR__CLAUSES_c1: !t. t \/ T <=> T
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/prim__rec/LESS__REFL: !n. ~h4/prim__rec/_3C n n
% Assm: h4/num/ZERO__REP__DEF: !y. ~(h4/num/ZERO__REP = h4/num/SUC__REP y)
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/prim__rec/NOT__LESS__0: !n. ~h4/prim__rec/_3C n h4/num/0
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/num/SUC__REP__DEF_c0: h4/bool/ONE__ONE h4/num/SUC__REP
% Assm: h4/num/SUC__REP__DEF_c1: ~h4/bool/ONTO h4/num/SUC__REP
% Assm: h4/bool/ONE__ONE__THM: !f. h4/bool/ONE__ONE f <=> (!x1 x2. f x1 = f x2 ==> x1 = x2)
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> t
% Assm: h4/bool/NOT__AND: !t. ~(t /\ ~t)
% Assm: h4/bool/T__DEF: T <=> (\x. x) = (\x. x)
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/NOT__DEF: $not = (\t. t ==> F)
% Assm: h4/bool/EXCLUDED__MIDDLE0: !t. t \/ ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/OR__DEF: $or = (\t1 t2. !t. (t1 ==> t) ==> (t2 ==> t) ==> t)
% Assm: h4/bool/F__DEF: F <=> (!t. t)
% Assm: h4/bool/AND1__THM: !t2 t1. t1 /\ t2 ==> t1
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/COND__DEF: h4/bool/COND = (\t t1 t2. h4/min/_40 (\x. ((t <=> T) ==> x = t1) /\ ((t <=> F) ==> x = t2)))
% Assm: h4/bool/BOOL__EQ__DISTINCT_c1: ~(F <=> T)
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/BOOL__FUN__CASES__THM: !f. f = (\b. T) \/ f = (\b. F) \/ f = (\b. b) \/ f = (\b. ~b)
% Assm: h4/relation/RDOM__DELETE__DEF: !x v u R. h4/relation/RDOM__DELETE R x u v <=> R u v /\ ~(u = x)
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/bool/OR__CLAUSES_c4: !t. t \/ t <=> t
% Assm: h4/relation/irreflexive__def: !R. h4/relation/irreflexive R <=> (!x. ~R x x)
% Assm: h4/bool/EQ__EXPAND: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/relation/WF__DEF: !R. h4/relation/WF R <=> (!B. (?w. B w) ==> (?min. B min /\ (!b. R b min ==> ~B b)))
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% 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/relation/WF__NOT__REFL: !y x R. h4/relation/WF R ==> R x y ==> ~(x = y)
% Assm: h4/relation/WF__noloops: !y x R. h4/relation/WF R ==> h4/relation/TC R x y ==> ~(x = y)
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> 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/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/relation/WF__TC__EQN: !R. h4/relation/WF (h4/relation/TC R) <=> h4/relation/WF R
% Assm: h4/relation/nf__def: !x R. h4/relation/nf R x <=> (!y. ~R x y)
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/combin/SAME__KEY__UPDATE__DIFFER: !f c b a. ~(b = c) ==> ~(h4/combin/UPDATE a b f = h4/combin/UPDATE a c f)
% Assm: h4/combin/UPDATE__def: !b a. h4/combin/UPDATE a b = (\f c. h4/bool/COND (a = c) b (f c))
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/bool/MONO__NOT__EQ: !y x. y ==> x <=> ~x ==> ~y
% Assm: h4/bool/MONO__NOT: !y x. (y ==> x) ==> ~x ==> ~y
% Assm: h4/bool/NOT__IMP: !B A. ~(A ==> B) <=> A /\ ~B
% Assm: h4/sat/dc__cond: !s r q p. (p <=> h4/bool/COND q r s) <=> (p \/ q \/ ~s) /\ (p \/ ~r \/ ~q) /\ (p \/ ~r \/ ~s) /\ (~q \/ r \/ ~p) /\ (q \/ s \/ ~p)
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/COND__EXPAND__IMP: !t2 t1 b. h4/bool/COND b t1 t2 <=> (b ==> t1) /\ (~b ==> t2)
% Assm: h4/bool/AND__CLAUSES_c4: !t. t /\ t <=> t
% Assm: h4/combin/UPDATE__APPLY_c1: !x f b a. ~(a = b) ==> h4/combin/UPDATE a x f b = f b
% Assm: h4/bool/LEFT__OR__CONG: !Q_27 Q P_27 P. (P <=> P_27) /\ (~P_27 ==> (Q <=> Q_27)) ==> (P \/ Q <=> P_27 \/ Q_27)
% Assm: h4/bool/BOOL__EQ__DISTINCT_c0: ~(T <=> F)
% Assm: h4/bool/COND__EXPAND__OR: !t2 t1 b. h4/bool/COND b t1 t2 <=> b /\ t1 \/ ~b /\ t2
% Assm: h4/relation/IN__RDOM__DELETE: !x k R. h4/bool/IN x (h4/relation/RDOM (h4/relation/RDOM__DELETE R k)) <=> h4/bool/IN x (h4/relation/RDOM R) /\ ~(x = k)
% Assm: h4/bool/IN__DEF: h4/bool/IN = (\x f. f x)
% Assm: h4/relation/RDOM__DEF: !x R. h4/relation/RDOM R x <=> (?y. R x y)
% Assm: h4/bool/NOT__F: !t. ~t ==> (t <=> F)
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/sat/OR__DUAL: !B A. ~(A \/ B) ==> F <=> ~A ==> ~B ==> F
% Assm: h4/relation/RCOMPL0: !y x R. h4/relation/RCOMPL R x y <=> ~R x y
% Assm: h4/sat/AND__INV: !A. ~A /\ A <=> F
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/combin/UPD11__SAME__BASE: !f d c b a. h4/combin/UPDATE a c f = h4/combin/UPDATE b d f <=> a = b /\ c = d \/ ~(a = b) /\ h4/combin/UPDATE a c f = f /\ h4/combin/UPDATE b d f = f
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/relation/STRORD0: !b a R. h4/relation/STRORD R a b <=> R a b /\ ~(a = b)
% 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/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/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/RIGHT__OR__EXISTS__THM: !Q P. P \/ (?x. Q x) <=> (?x. P \/ Q x)
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Goal: !n. ~(h4/num/SUC n = n)
%   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_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_nums_INVu_u_SUC]: !n m. h4/num/SUC m = h4/num/SUC n ==> m = n
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_primu_u_recs_LESSu_u_DEF]: !n m. h4/prim__rec/_3C m n <=> (?P. (!n0. happ P (h4/num/SUC n0) ==> happ P n0) /\ happ P m /\ ~happ P n)
% Assm [h4s_primu_u_recs_LESSu_u_THM]: !n m. h4/prim__rec/_3C m (h4/num/SUC n) <=> m = n \/ h4/prim__rec/_3C m n
% 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_primu_u_recs_LESSu_u_LEMMA1]: !n m. h4/prim__rec/_3C m (h4/num/SUC n) ==> m = n \/ h4/prim__rec/_3C m n
% Assm [h4s_primu_u_recs_LESSu_u_LEMMA2]: !n m. m = n \/ h4/prim__rec/_3C m n ==> h4/prim__rec/_3C m (h4/num/SUC n)
% Assm [h4s_nums_SUCu_u_DEF]: !m. h4/num/SUC m = h4/num/ABS__num (happ h4/num/SUC__REP (h4/num/REP__num m))
% Assm [h4s_primu_u_recs_LESSu_u_SUC]: !n m. h4/prim__rec/_3C m n ==> h4/prim__rec/_3C m (h4/num/SUC n)
% Assm [h4s_primu_u_recs_LESSu_u_SUCu_u_REFL]: !n. h4/prim__rec/_3C n (h4/num/SUC n)
% Assm [h4s_nums_NOTu_u_SUC]: !n. ~(h4/num/SUC n = h4/num/0)
% Assm [h4s_primu_u_recs_LESSu_u_MONO]: !n m. h4/prim__rec/_3C m n ==> h4/prim__rec/_3C (h4/num/SUC m) (h4/num/SUC n)
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_nums_ISu_u_NUMu_u_REP0]: !m. happ h4/num/IS__NUM__REP m <=> (!P. happ P h4/num/ZERO__REP /\ (!n. happ P n ==> happ P (happ h4/num/SUC__REP n)) ==> happ P m)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_nums_numu_u_ISOu_u_DEFu_c0]: !a. h4/num/ABS__num (h4/num/REP__num a) = a
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_primu_u_recs_LESSu_u_SUCu_u_IMP]: !n m. h4/prim__rec/_3C m (h4/num/SUC n) ==> ~(m = n) ==> h4/prim__rec/_3C m n
% Assm [h4s_primu_u_recs_SUCu_u_LESS]: !n m. h4/prim__rec/_3C (h4/num/SUC m) n ==> h4/prim__rec/_3C m n
% Assm [h4s_primu_u_recs_EQu_u_LESS]: !n m. h4/num/SUC m = n ==> h4/prim__rec/_3C m n
% Assm [h4s_primu_u_recs_INVu_u_SUCu_u_EQ]: !n m. h4/num/SUC m = h4/num/SUC n <=> m = n
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_primu_u_recs_LESSu_u_0]: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm [h4s_nums_numu_u_ISOu_u_DEFu_c1]: !r. happ h4/num/IS__NUM__REP r <=> h4/num/REP__num (h4/num/ABS__num r) = r
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_primu_u_recs_LESSu_u_0u_u_0]: h4/prim__rec/_3C h4/num/0 (h4/num/SUC h4/num/0)
% Assm [h4s_nums_numu_u_TYu_u_DEF]: ?rep. h4/bool/TYPE__DEFINITION h4/num/IS__NUM__REP rep
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_nums_ZEROu_u_DEF]: h4/num/0 = h4/num/ABS__num h4/num/ZERO__REP
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_ABSu_u_REPu_u_THM]: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. happ abs (happ rep a) = a) /\ (!r. happ P r <=> happ rep (happ abs r) = r))
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_ORu_u_CLAUSESu_c1]: !t. t \/ T <=> T
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_primu_u_recs_LESSu_u_REFL]: !n. ~h4/prim__rec/_3C n n
% Assm [h4s_nums_ZEROu_u_REPu_u_DEF]: !y. ~(h4/num/ZERO__REP = happ h4/num/SUC__REP y)
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_primu_u_recs_NOTu_u_LESSu_u_0]: !n. ~h4/prim__rec/_3C n h4/num/0
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_nums_SUCu_u_REPu_u_DEFu_c0]: h4/bool/ONE__ONE h4/num/SUC__REP
% Assm [h4s_nums_SUCu_u_REPu_u_DEFu_c1]: ~h4/bool/ONTO h4/num/SUC__REP
% Assm [h4s_bools_ONEu_u_ONEu_u_THM]: !f. h4/bool/ONE__ONE f <=> (!x1 x2. happ f x1 = happ f x2 ==> x1 = x2)
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> t
% Assm [h4s_bools_NOTu_u_AND]: !t. ~(t /\ ~t)
% Assm [h4s_bools_Tu_u_DEF]: T <=> (!x. x <=> x)
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_NOTu_u_DEF]: !x. $not x <=> x ==> F
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE0]: !t. t \/ ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_ORu_u_DEF]: !x x'. $or x x' <=> (!t. (x ==> t) ==> (x' ==> t) ==> t)
% Assm [h4s_bools_Fu_u_DEF]: F <=> (!t. t)
% Assm [h4s_bools_AND1u_u_THM]: !t2 t1. t1 /\ t2 ==> t1
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_CONDu_u_DEF]: !_0. (!x x x' x''. happ (happ (happ (happ _0 x) x) x') x'' <=> ((x <=> T) ==> x'' = x) /\ ((x <=> F) ==> x'' = x')) ==> (!x x x'. h4/bool/COND x x x' = h4/min/_40 (happ (happ (happ _0 x) x) x'))
% Assm [h4s_bools_BOOLu_u_EQu_u_DISTINCTu_c1]: ~(F <=> T)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_BOOLu_u_FUNu_u_CASESu_u_THM]: !f. (!x. happ f x <=> T) \/ (!x. happ f x <=> F) \/ (!x. happ f x <=> x) \/ (!x. happ f x <=> ~x)
% Assm [h4s_relations_RDOMu_u_DELETEu_u_DEF]: !x v u R. happ (happ (h4/relation/RDOM__DELETE R x) u) v <=> happ (happ R u) v /\ ~(u = x)
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_bools_ORu_u_CLAUSESu_c4]: !t. t \/ t <=> t
% Assm [h4s_relations_irreflexiveu_u_def]: !R. h4/relation/irreflexive R <=> (!x. ~happ (happ R x) x)
% Assm [h4s_bools_EQu_u_EXPAND]: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_relations_WFu_u_DEF]: !R. h4/relation/WF R <=> (!B. (?w. happ B w) ==> (?min. happ B min /\ (!b. happ (happ R b) min ==> ~happ B b)))
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% 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_relations_WFu_u_NOTu_u_REFL]: !y x R. h4/relation/WF R ==> happ (happ R x) y ==> ~(x = y)
% Assm [h4s_relations_WFu_u_noloops]: !y x R. h4/relation/WF R ==> happ (happ (h4/relation/TC R) x) y ==> ~(x = y)
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> 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_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_relations_WFu_u_TCu_u_EQN]: !R. h4/relation/WF (h4/relation/TC R) <=> h4/relation/WF R
% Assm [h4s_relations_nfu_u_def]: !x R. h4/relation/nf R x <=> (!y. ~happ (happ R x) y)
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_combins_SAMEu_u_KEYu_u_UPDATEu_u_DIFFER]: !f c b a. ~(b = c) ==> ~(h4/combin/UPDATE a b f = h4/combin/UPDATE a c f)
% Assm [h4s_combins_UPDATEu_u_def]: !b a x x. ?v. (v <=> a = x) /\ happ (h4/combin/UPDATE a b x) x = h4/bool/COND v b (happ x x)
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_bools_MONOu_u_NOTu_u_EQ]: !y x. y ==> x <=> ~x ==> ~y
% Assm [h4s_bools_MONOu_u_NOT]: !y x. (y ==> x) ==> ~x ==> ~y
% Assm [h4s_bools_NOTu_u_IMP]: !B A. ~(A ==> B) <=> A /\ ~B
% Assm [h4s_sats_dcu_u_cond]: !s r q p. (p <=> h4/bool/COND q r s) <=> (p \/ q \/ ~s) /\ (p \/ ~r \/ ~q) /\ (p \/ ~r \/ ~s) /\ (~q \/ r \/ ~p) /\ (q \/ s \/ ~p)
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_CONDu_u_EXPANDu_u_IMP]: !t2 t1 b. h4/bool/COND b t1 t2 <=> (b ==> t1) /\ (~b ==> t2)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c4]: !t. t /\ t <=> t
% Assm [h4s_combins_UPDATEu_u_APPLYu_c1]: !x f b a. ~(a = b) ==> happ (h4/combin/UPDATE a x f) b = happ f b
% Assm [h4s_bools_LEFTu_u_ORu_u_CONG]: !Q_27 Q P_27 P. (P <=> P_27) /\ (~P_27 ==> (Q <=> Q_27)) ==> (P \/ Q <=> P_27 \/ Q_27)
% Assm [h4s_bools_BOOLu_u_EQu_u_DISTINCTu_c0]: ~(T <=> F)
% Assm [h4s_bools_CONDu_u_EXPANDu_u_OR]: !t2 t1 b. h4/bool/COND b t1 t2 <=> b /\ t1 \/ ~b /\ t2
% Assm [h4s_relations_INu_u_RDOMu_u_DELETE]: !x k R. h4/bool/IN x (h4/relation/RDOM (h4/relation/RDOM__DELETE R k)) <=> h4/bool/IN x (h4/relation/RDOM R) /\ ~(x = k)
% Assm [h4s_bools_INu_u_DEF]: !x x. h4/bool/IN x x <=> happ x x
% Assm [h4s_relations_RDOMu_u_DEF]: !x R. happ (h4/relation/RDOM R) x <=> (?y. happ (happ R x) y)
% Assm [h4s_bools_NOTu_u_F]: !t. ~t ==> (t <=> F)
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_sats_ORu_u_DUAL]: !B A. ~(A \/ B) ==> F <=> ~A ==> ~B ==> F
% Assm [h4s_relations_RCOMPL0]: !y x R. h4/relation/RCOMPL R x y <=> ~happ (happ R x) y
% Assm [h4s_sats_ANDu_u_INV]: !A. ~A /\ A <=> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_combins_UPD11u_u_SAMEu_u_BASE]: !f d c b a. h4/combin/UPDATE a c f = h4/combin/UPDATE b d f <=> a = b /\ c = d \/ ~(a = b) /\ h4/combin/UPDATE a c f = f /\ h4/combin/UPDATE b d f = f
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_relations_STRORD0]: !b a R. h4/relation/STRORD R a b <=> happ (happ R a) b /\ ~(a = b)
% Assm [h4s_relations_TCu_u_impliesu_u_oneu_u_step]: !y x R. happ (happ (h4/relation/TC R) x) y /\ ~(x = y) ==> (?z. happ (happ R x) z /\ ~(x = z))
% 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 (h4/relation/TC R) u) v ==> happ (happ P u) v)
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM]: !Q P. P \/ (?x. happ Q x) <=> (?x. P \/ happ Q x)
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Goal: !n. ~(h4/num/SUC n = n)
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_Q1277459,TV_Q1277455]: ![V_f, V_g]: (![V_x]: s(TV_Q1277455,happ(s(t_fun(TV_Q1277459,TV_Q1277455),V_f),s(TV_Q1277459,V_x))) = s(TV_Q1277455,happ(s(t_fun(TV_Q1277459,TV_Q1277455),V_g),s(TV_Q1277459,V_x))) => s(t_fun(TV_Q1277459,TV_Q1277455),V_f) = s(t_fun(TV_Q1277459,TV_Q1277455),V_g))).
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_TRUTH, axiom, p(s(t_bool,t))).
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_nums_INVu_u_SUC, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) => s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_primu_u_recs_LESSu_u_DEF, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) <=> ?[V_P]: (![V_n0]: (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_n0)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n0))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_m)))) & ~ (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_primu_u_recs_LESSu_u_THM, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))) <=> (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n) | p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))))).
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_primu_u_recs_LESSu_u_LEMMA1, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))) => (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n) | p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_primu_u_recs_LESSu_u_LEMMA2, axiom, ![V_n, V_m]: ((s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n) | p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))) => p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))).
fof(ah4s_nums_SUCu_u_DEF, axiom, ![V_m]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_nums_absu_u_num(s(t_h4s_mins_ind,happ(s(t_fun(t_h4s_mins_ind,t_h4s_mins_ind),h4s_nums_sucu_u_rep),s(t_h4s_mins_ind,h4s_nums_repu_u_num(s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_primu_u_recs_LESSu_u_SUC, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))).
fof(ah4s_primu_u_recs_LESSu_u_SUCu_u_REFL, axiom, ![V_n]: p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))))).
fof(ah4s_nums_NOTu_u_SUC, axiom, ![V_n]: ~ (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0))).
fof(ah4s_primu_u_recs_LESSu_u_MONO, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))).
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_nums_ISu_u_NUMu_u_REP0, axiom, ![V_m]: (p(s(t_bool,happ(s(t_fun(t_h4s_mins_ind,t_bool),h4s_nums_isu_u_numu_u_rep),s(t_h4s_mins_ind,V_m)))) <=> ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_mins_ind,t_bool),V_P),s(t_h4s_mins_ind,h4s_nums_zerou_u_rep)))) & ![V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_mins_ind,t_bool),V_P),s(t_h4s_mins_ind,V_n)))) => p(s(t_bool,happ(s(t_fun(t_h4s_mins_ind,t_bool),V_P),s(t_h4s_mins_ind,happ(s(t_fun(t_h4s_mins_ind,t_h4s_mins_ind),h4s_nums_sucu_u_rep),s(t_h4s_mins_ind,V_n)))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_mins_ind,t_bool),V_P),s(t_h4s_mins_ind,V_m))))))).
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_nums_numu_u_ISOu_u_DEFu_c0, axiom, ![V_a]: s(t_h4s_nums_num,h4s_nums_absu_u_num(s(t_h4s_mins_ind,h4s_nums_repu_u_num(s(t_h4s_nums_num,V_a))))) = s(t_h4s_nums_num,V_a)).
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_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_primu_u_recs_LESSu_u_SUCu_u_IMP, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))) => (~ (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n)) => p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_primu_u_recs_SUCu_u_LESS, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_primu_u_recs_EQu_u_LESS, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_n) => p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_primu_u_recs_INVu_u_SUCu_u_EQ, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_primu_u_recs_LESSu_u_0, axiom, ![V_n]: p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))))).
fof(ah4s_nums_numu_u_ISOu_u_DEFu_c1, axiom, ![V_r]: (p(s(t_bool,happ(s(t_fun(t_h4s_mins_ind,t_bool),h4s_nums_isu_u_numu_u_rep),s(t_h4s_mins_ind,V_r)))) <=> s(t_h4s_mins_ind,h4s_nums_repu_u_num(s(t_h4s_nums_num,h4s_nums_absu_u_num(s(t_h4s_mins_ind,V_r))))) = s(t_h4s_mins_ind,V_r))).
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_primu_u_recs_LESSu_u_0u_u_0, axiom, p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))).
fof(ah4s_nums_numu_u_TYu_u_DEF, axiom, ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_h4s_mins_ind,t_bool),h4s_nums_isu_u_numu_u_rep),s(t_fun(t_h4s_nums_num,t_h4s_mins_ind),V_rep))))).
fof(ah4s_bools_IMPu_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_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_nums_ZEROu_u_DEF, axiom, s(t_h4s_nums_num,h4s_nums_0) = s(t_h4s_nums_num,h4s_nums_absu_u_num(s(t_h4s_mins_ind,h4s_nums_zerou_u_rep)))).
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_bools_ABSu_u_REPu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ?[V_rep, V_abs]: (![V_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))) = s(TV_u_27b,V_a) & ![V_r]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_r)))) <=> s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))))) = s(TV_u_27a,V_r))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_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_ORu_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_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_primu_u_recs_LESSu_u_REFL, axiom, ![V_n]: ~ (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_nums_ZEROu_u_REPu_u_DEF, axiom, ![V_y]: ~ (s(t_h4s_mins_ind,h4s_nums_zerou_u_rep) = s(t_h4s_mins_ind,happ(s(t_fun(t_h4s_mins_ind,t_h4s_mins_ind),h4s_nums_sucu_u_rep),s(t_h4s_mins_ind,V_y))))).
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_primu_u_recs_NOTu_u_LESSu_u_0, axiom, ![V_n]: ~ (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0)))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_nums_SUCu_u_REPu_u_DEFu_c0, axiom, p(s(t_bool,h4s_bools_oneu_u_one(s(t_fun(t_h4s_mins_ind,t_h4s_mins_ind),h4s_nums_sucu_u_rep))))).
fof(ah4s_nums_SUCu_u_REPu_u_DEFu_c1, axiom, ~ (p(s(t_bool,h4s_bools_onto(s(t_fun(t_h4s_mins_ind,t_h4s_mins_ind),h4s_nums_sucu_u_rep)))))).
fof(ah4s_bools_ONEu_u_ONEu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_f]: (p(s(t_bool,h4s_bools_oneu_u_one(s(t_fun(TV_u_27a,TV_u_27b),V_f)))) <=> ![V_x1, V_x2]: (s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x1))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x2))) => s(TV_u_27a,V_x1) = s(TV_u_27a,V_x2)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,f))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
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_ANDu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,f))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_EXISTSu_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_NOTu_u_AND, axiom, ![V_t]: ~ ((p(s(t_bool,V_t)) & ~ (p(s(t_bool,V_t)))))).
fof(ah4s_bools_Tu_u_DEF, axiom, (p(s(t_bool,t)) <=> ![V_x]: s(t_bool,V_x) = s(t_bool,V_x))).
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_NOTu_u_DEF, axiom, ![V_x]: (p(s(t_bool,d_not(s(t_bool,V_x)))) <=> (p(s(t_bool,V_x)) => p(s(t_bool,f))))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE0, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ORu_u_DEF, axiom, ![V_x, V_xi_]: (p(s(t_bool,d_or(s(t_bool,V_x),s(t_bool,V_xi_)))) <=> ![V_t]: ((p(s(t_bool,V_x)) => p(s(t_bool,V_t))) => ((p(s(t_bool,V_xi_)) => p(s(t_bool,V_t))) => p(s(t_bool,V_t)))))).
fof(ah4s_bools_Fu_u_DEF, axiom, (p(s(t_bool,f)) <=> ![V_t]: p(s(t_bool,V_t)))).
fof(ah4s_bools_AND1u_u_THM, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t1)))).
fof(ah4s_bools_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
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_CONDu_u_DEF, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x, V_x0, V_xi_, V_xi_i_]: (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_bool,t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)))),V_uu_0),s(TV_u_27a,V_x))),s(t_bool,V_x0))),s(TV_u_27a,V_xi_))),s(TV_u_27a,V_xi_i_)))) <=> ((s(t_bool,V_x0) = s(t_bool,t) => s(TV_u_27a,V_xi_i_) = s(TV_u_27a,V_x)) & (s(t_bool,V_x0) = s(t_bool,f) => s(TV_u_27a,V_xi_i_) = s(TV_u_27a,V_xi_)))) => ![V_x, V_x0, V_xi_]: s(TV_u_27a,h4s_bools_cond(s(t_bool,V_x),s(TV_u_27a,V_x0),s(TV_u_27a,V_xi_))) = s(TV_u_27a,h4s_mins_u_40(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_bool,t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)))),V_uu_0),s(TV_u_27a,V_x0))),s(t_bool,V_x))),s(TV_u_27a,V_xi_))))))).
fof(ah4s_bools_BOOLu_u_EQu_u_DISTINCTu_c1, axiom, ~ (s(t_bool,f) = s(t_bool,t))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_BOOLu_u_FUNu_u_CASESu_u_THM, axiom, ![V_f]: (![V_x]: s(t_bool,happ(s(t_fun(t_bool,t_bool),V_f),s(t_bool,V_x))) = s(t_bool,t) | (![V_x]: s(t_bool,happ(s(t_fun(t_bool,t_bool),V_f),s(t_bool,V_x))) = s(t_bool,f) | (![V_x]: s(t_bool,happ(s(t_fun(t_bool,t_bool),V_f),s(t_bool,V_x))) = s(t_bool,V_x) | ![V_x]: (p(s(t_bool,happ(s(t_fun(t_bool,t_bool),V_f),s(t_bool,V_x)))) <=> ~ (p(s(t_bool,V_x)))))))).
fof(ah4s_relations_RDOMu_u_DELETEu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_v, V_u, V_R]: (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)),h4s_relations_rdomu_u_delete(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_u))),s(TV_u_27b,V_v)))) <=> (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_R),s(TV_u_27a,V_u))),s(TV_u_27b,V_v)))) & ~ (s(TV_u_27a,V_u) = s(TV_u_27a,V_x))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = 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_bools_ORu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_relations_irreflexiveu_u_def, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_irreflexive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> ![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_x))))))).
fof(ah4s_bools_EQu_u_EXPAND, 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_t1))) & ~ (p(s(t_bool,V_t2))))))).
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_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_relations_WFu_u_DEF, 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)))) <=> ![V_B]: (?[V_w]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_B),s(TV_u_27a,V_w)))) => ?[V_min]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_B),s(TV_u_27a,V_min)))) & ![V_b]: (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_b))),s(TV_u_27a,V_min)))) => ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_B),s(TV_u_27a,V_b)))))))))).
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_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_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_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_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)),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_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_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_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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
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_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_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_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)),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_relations_nfu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_R]: (p(s(t_bool,h4s_relations_nf(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27a,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_R),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_combins_SAMEu_u_KEYu_u_UPDATEu_u_DIFFER, axiom, ![TV_u_27c,TV_u_27d]: ![V_f, V_c, V_b, V_a]: (~ (s(TV_u_27d,V_b) = s(TV_u_27d,V_c)) => ~ (s(t_fun(TV_u_27c,TV_u_27d),h4s_combins_update(s(TV_u_27c,V_a),s(TV_u_27d,V_b),s(t_fun(TV_u_27c,TV_u_27d),V_f))) = s(t_fun(TV_u_27c,TV_u_27d),h4s_combins_update(s(TV_u_27c,V_a),s(TV_u_27d,V_c),s(t_fun(TV_u_27c,TV_u_27d),V_f)))))).
fof(ah4s_combins_UPDATEu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_b, V_a, V_x, V_x0]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(TV_u_27a,V_a) = s(TV_u_27a,V_x0)) & s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_b),s(t_fun(TV_u_27a,TV_u_27b),V_x))),s(TV_u_27a,V_x0))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27b,V_b),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(TV_u_27a,V_x0))))))).
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_bools_MONOu_u_NOTu_u_EQ, axiom, ![V_y, V_x]: ((p(s(t_bool,V_y)) => p(s(t_bool,V_x))) <=> (~ (p(s(t_bool,V_x))) => ~ (p(s(t_bool,V_y)))))).
fof(ah4s_bools_MONOu_u_NOT, axiom, ![V_y, V_x]: ((p(s(t_bool,V_y)) => p(s(t_bool,V_x))) => (~ (p(s(t_bool,V_x))) => ~ (p(s(t_bool,V_y)))))).
fof(ah4s_bools_NOTu_u_IMP, 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_sats_dcu_u_cond, axiom, ![V_s, V_r, V_q, V_p]: (s(t_bool,V_p) = s(t_bool,h4s_bools_cond(s(t_bool,V_q),s(t_bool,V_r),s(t_bool,V_s))) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_s))))) & ((p(s(t_bool,V_p)) | (~ (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_s))))) & ((~ (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_s)) | ~ (p(s(t_bool,V_p))))))))))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_bools_CONDu_u_EXPANDu_u_IMP, axiom, ![V_t2, V_t1, V_b]: (p(s(t_bool,h4s_bools_cond(s(t_bool,V_b),s(t_bool,V_t1),s(t_bool,V_t2)))) <=> ((p(s(t_bool,V_b)) => p(s(t_bool,V_t1))) & (~ (p(s(t_bool,V_b))) => p(s(t_bool,V_t2)))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_combins_UPDATEu_u_APPLYu_c1, axiom, ![TV_u_27d,TV_u_27c]: ![V_x, V_f, V_b, V_a]: (~ (s(TV_u_27c,V_a) = s(TV_u_27c,V_b)) => s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),h4s_combins_update(s(TV_u_27c,V_a),s(TV_u_27d,V_x),s(t_fun(TV_u_27c,TV_u_27d),V_f))),s(TV_u_27c,V_b))) = s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_f),s(TV_u_27c,V_b))))).
fof(ah4s_bools_LEFTu_u_ORu_u_CONG, axiom, ![V_Qu_27, V_Q, V_Pu_27, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Pu_27) & (~ (p(s(t_bool,V_Pu_27))) => s(t_bool,V_Q) = s(t_bool,V_Qu_27))) => ((p(s(t_bool,V_P)) | p(s(t_bool,V_Q))) <=> (p(s(t_bool,V_Pu_27)) | p(s(t_bool,V_Qu_27)))))).
fof(ah4s_bools_BOOLu_u_EQu_u_DISTINCTu_c0, axiom, ~ (s(t_bool,t) = s(t_bool,f))).
fof(ah4s_bools_CONDu_u_EXPANDu_u_OR, axiom, ![V_t2, V_t1, V_b]: (p(s(t_bool,h4s_bools_cond(s(t_bool,V_b),s(t_bool,V_t1),s(t_bool,V_t2)))) <=> ((p(s(t_bool,V_b)) & p(s(t_bool,V_t1))) | (~ (p(s(t_bool,V_b))) & p(s(t_bool,V_t2)))))).
fof(ah4s_relations_INu_u_RDOMu_u_DELETE, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_k, V_R]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_relations_rdom(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),h4s_relations_rdomu_u_delete(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27a,V_k)))))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_relations_rdom(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R)))))) & ~ (s(TV_u_27a,V_x) = s(TV_u_27a,V_k))))).
fof(ah4s_bools_INu_u_DEF, axiom, ![TV_u_27a]: ![V_x, V_x0]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_x0))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x0),s(TV_u_27a,V_x)))).
fof(ah4s_relations_RDOMu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_R]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),h4s_relations_rdom(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R))),s(TV_u_27a,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_R),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))))).
fof(ah4s_bools_NOTu_u_F, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => s(t_bool,V_t) = s(t_bool,f))).
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_sats_ORu_u_DUAL, 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_relations_RCOMPL0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_R]: (p(s(t_bool,h4s_relations_rcompl(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27a,V_x),s(TV_u_27b,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_R),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))))).
fof(ah4s_sats_ANDu_u_INV, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) & p(s(t_bool,V_A))) <=> p(s(t_bool,f)))).
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_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_combins_UPD11u_u_SAMEu_u_BASE, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_d, V_c, V_b, V_a]: (s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_c),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_b),s(TV_u_27b,V_d),s(t_fun(TV_u_27a,TV_u_27b),V_f))) <=> ((s(TV_u_27a,V_a) = s(TV_u_27a,V_b) & s(TV_u_27b,V_c) = s(TV_u_27b,V_d)) | (~ (s(TV_u_27a,V_a) = s(TV_u_27a,V_b)) & (s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_c),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27a,TV_u_27b),V_f) & s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_b),s(TV_u_27b,V_d),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27a,TV_u_27b),V_f)))))).
fof(ah4s_bools_IMPu_u_DISJu_u_THM, 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_relations_STRORD0, axiom, ![TV_u_27a]: ![V_b, V_a, V_R]: (p(s(t_bool,h4s_relations_strord(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)))) <=> (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_a))),s(TV_u_27a,V_b)))) & ~ (s(TV_u_27a,V_a) = s(TV_u_27a,V_b))))).
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)),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_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)),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_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_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_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(ch4s_primu_u_recs_SUCu_u_ID, conjecture, ![V_n]: ~ (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_n))).
