%   ORIGINAL: h4/arithmetic/EVEN__OR__ODD
% 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/EVEN__ODD: !n. h4/arithmetic/EVEN n <=> ~h4/arithmetic/ODD n
% Assm: h4/arithmetic/ODD__EVEN: !n. h4/arithmetic/ODD n <=> ~h4/arithmetic/EVEN n
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/arithmetic/ODD0_c1: !n. h4/arithmetic/ODD (h4/num/SUC n) <=> ~h4/arithmetic/ODD n
% Assm: h4/arithmetic/EVEN0_c1: !n. h4/arithmetic/EVEN (h4/num/SUC n) <=> ~h4/arithmetic/EVEN n
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/arithmetic/ODD0_c0: h4/arithmetic/ODD h4/num/0 <=> F
% Assm: h4/arithmetic/EVEN0_c0: h4/arithmetic/EVEN h4/num/0 <=> T
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/num/NOT__SUC: !n. ~(h4/num/SUC n = h4/num/0)
% Assm: h4/arithmetic/NOT__EXP__0: !n m. ~(h4/arithmetic/EXP (h4/num/SUC n) m = h4/num/0)
% Assm: h4/arithmetic/MULT__CLAUSES_c1: !m. h4/arithmetic/_2A m h4/num/0 = h4/num/0
% Assm: h4/arithmetic/EXP0_c1: !n m. h4/arithmetic/EXP m (h4/num/SUC n) = h4/arithmetic/_2A m (h4/arithmetic/EXP m n)
% Assm: h4/arithmetic/MULT__MONO__EQ: !n m i. h4/arithmetic/_2A (h4/num/SUC n) m = h4/arithmetic/_2A (h4/num/SUC n) i <=> m = i
% Assm: h4/arithmetic/ONE: h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) = h4/num/SUC h4/num/0
% Assm: h4/arithmetic/EXP0_c0: !m. h4/arithmetic/EXP m h4/num/0 = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm: h4/arithmetic/BIT10: !n. h4/arithmetic/BIT1 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (h4/num/SUC h4/num/0))
% Assm: h4/arithmetic/num__case__def_c1: !v n f. h4/arithmetic/num__CASE (h4/num/SUC n) v f = f n
% Assm: h4/prim__rec/PRIM__REC__EQN_c1: !x n m f. h4/prim__rec/PRIM__REC__FUN x f (h4/num/SUC m) n = f (h4/prim__rec/PRIM__REC__FUN x f m (h4/prim__rec/PRE n)) n
% Assm: h4/prim__rec/SIMP__REC__THM_c1: !x m f. h4/prim__rec/SIMP__REC x f (h4/num/SUC m) = f (h4/prim__rec/SIMP__REC x f m)
% Assm: h4/prim__rec/PRIM__REC__FUN0: !x f. h4/prim__rec/PRIM__REC__FUN x f = h4/prim__rec/SIMP__REC (\n. x) (\fun n. f (fun (h4/prim__rec/PRE n)) n)
% Assm: h4/arithmetic/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/LESS__IMP__LESS__OR__EQ: !n m. h4/prim__rec/_3C m n ==> h4/arithmetic/_3C_3D m n
% Assm: h4/arithmetic/RIGHT__SUB__DISTRIB: !p n m. h4/arithmetic/_2A (h4/arithmetic/_2D m n) p = h4/arithmetic/_2D (h4/arithmetic/_2A m p) (h4/arithmetic/_2A n p)
% Assm: h4/arithmetic/SUB__EQ__0: !n m. h4/arithmetic/_2D m n = h4/num/0 <=> h4/arithmetic/_3C_3D m n
% Assm: h4/arithmetic/MULT__CLAUSES_c0: !m. h4/arithmetic/_2A h4/num/0 m = h4/num/0
% Assm: h4/arithmetic/SUB__ADD: !n m. h4/arithmetic/_3C_3D n m ==> h4/arithmetic/_2B (h4/arithmetic/_2D m n) n = m
% Assm: h4/arithmetic/LESS__MONO__MULT: !p n m. h4/arithmetic/_3C_3D m n ==> h4/arithmetic/_3C_3D (h4/arithmetic/_2A m p) (h4/arithmetic/_2A n p)
% Assm: h4/arithmetic/RIGHT__ADD__DISTRIB: !p n m. h4/arithmetic/_2A (h4/arithmetic/_2B m n) p = h4/arithmetic/_2B (h4/arithmetic/_2A m p) (h4/arithmetic/_2A n p)
% Assm: h4/arithmetic/ADD__EQ__SUB: !p n m. h4/arithmetic/_3C_3D n p ==> (h4/arithmetic/_2B m n = p <=> m = h4/arithmetic/_2D p n)
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/arithmetic/LESS__OR__EQ: !n m. h4/arithmetic/_3C_3D m n <=> h4/prim__rec/_3C m n \/ m = n
% Assm: h4/prim__rec/INV__SUC__EQ: !n m. h4/num/SUC m = h4/num/SUC n <=> m = n
% Assm: h4/arithmetic/LESS__EQ__MONO: !n m. h4/arithmetic/_3C_3D (h4/num/SUC n) (h4/num/SUC m) <=> h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/LESS__MONO__EQ: !n m. h4/prim__rec/_3C (h4/num/SUC m) (h4/num/SUC n) <=> h4/prim__rec/_3C m n
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/arithmetic/FORALL__NUM__THM: !P. (!n. P n) <=> P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n))
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/bool/LEFT__FORALL__OR__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/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/arithmetic/LESS__EQ__ADD: !n m. h4/arithmetic/_3C_3D m (h4/arithmetic/_2B m n)
% Assm: h4/arithmetic/SUB__LESS__EQ: !n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2D n m) n
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/SUB__PLUS: !c b a. h4/arithmetic/_2D a (h4/arithmetic/_2B b c) = h4/arithmetic/_2D (h4/arithmetic/_2D a b) c
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/arithmetic/ADD__CLAUSES_c1: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/arithmetic/ADD__INV__0__EQ: !n m. h4/arithmetic/_2B m n = m <=> n = h4/num/0
% Assm: h4/arithmetic/ADD__INV__0: !n m. h4/arithmetic/_2B m n = m ==> n = h4/num/0
% Assm: h4/arithmetic/NUMERAL__DEF: !x. h4/arithmetic/NUMERAL x = x
% Assm: h4/arithmetic/LESS__EQUAL__ADD: !n m. h4/arithmetic/_3C_3D m n ==> (?p. n = h4/arithmetic/_2B m p)
% Assm: h4/arithmetic/LESS__ADD: !n m. h4/prim__rec/_3C n m ==> (?p. h4/arithmetic/_2B p n = m)
% Assm: h4/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/arithmetic/NOT__ODD__EQ__EVEN: !n m. ~(h4/num/SUC (h4/arithmetic/_2B n n) = h4/arithmetic/_2B m m)
% Assm: h4/arithmetic/ADD__CLAUSES_c3: !n m. h4/arithmetic/_2B m (h4/num/SUC n) = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm: h4/arithmetic/ADD__CLAUSES_c2: !n m. h4/arithmetic/_2B (h4/num/SUC m) n = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/OR__CLAUSES_c1: !t. t \/ T <=> T
% Assm: h4/prim__rec/LESS__SUC__REFL: !n. h4/prim__rec/_3C n (h4/num/SUC n)
% 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/arithmetic/LESS__OR: !n m. h4/prim__rec/_3C m n ==> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm: h4/arithmetic/LESS__SUC__NOT: !n m. h4/prim__rec/_3C m n ==> ~h4/prim__rec/_3C n (h4/num/SUC m)
% Assm: h4/num/INV__SUC: !n m. h4/num/SUC m = h4/num/SUC n ==> m = n
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% 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/num/num__ISO__DEF_c1: !r. h4/num/IS__NUM__REP r <=> h4/num/REP__num (h4/num/ABS__num r) = r
% Assm: h4/num/num__ISO__DEF_c0: !a. h4/num/ABS__num (h4/num/REP__num a) = a
% 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/num/SUC__REP__DEF_c1: ~h4/bool/ONTO h4/num/SUC__REP
% Assm: h4/num/SUC__REP__DEF_c0: h4/bool/ONE__ONE 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/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% 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/prim__rec/LESS__SUC__SUC_c0: !m. h4/prim__rec/_3C m (h4/num/SUC m)
% Assm: h4/prim__rec/PRE0_c1: !m. h4/prim__rec/PRE (h4/num/SUC m) = m
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/SELECT__REFL__2: !x. h4/min/_40 (\y. x = y) = x
% Assm: h4/prim__rec/PRE__DEF: !m. h4/prim__rec/PRE m = h4/bool/COND (m = h4/num/0) h4/num/0 (h4/min/_40 (\n. m = h4/num/SUC n))
% Assm: h4/arithmetic/GREATER__OR__EQ: !n m. h4/arithmetic/_3E_3D m n <=> h4/arithmetic/_3E m n \/ m = n
% Assm: h4/arithmetic/ADD__SUC: !n m. h4/num/SUC (h4/arithmetic/_2B m n) = h4/arithmetic/_2B m (h4/num/SUC n)
% Assm: h4/arithmetic/ADD__0: !m. h4/arithmetic/_2B m h4/num/0 = m
% 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/arithmetic/ADD_c0: !n. h4/arithmetic/_2B h4/num/0 n = n
% Assm: h4/arithmetic/NOT__ZERO__LT__ZERO: !n. ~(n = h4/num/0) <=> h4/prim__rec/_3C h4/num/0 n
% Assm: h4/prim__rec/LESS__0: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/arithmetic/num__CASES: !m. m = h4/num/0 \/ (?n. m = h4/num/SUC n)
% Assm: h4/prim__rec/NOT__LESS__0: !n. ~h4/prim__rec/_3C n h4/num/0
% Assm: h4/arithmetic/LESS__LESS__SUC: !n m. ~(h4/prim__rec/_3C m n /\ h4/prim__rec/_3C n (h4/num/SUC m))
% Assm: h4/arithmetic/LESS__TRANS: !p n m. h4/prim__rec/_3C m n /\ h4/prim__rec/_3C n p ==> h4/prim__rec/_3C m p
% Assm: h4/prim__rec/LESS__REFL: !n. ~h4/prim__rec/_3C n n
% Assm: h4/arithmetic/LESS__ADD__SUC: !n m. h4/prim__rec/_3C m (h4/arithmetic/_2B m (h4/num/SUC n))
% Assm: h4/arithmetic/TIMES2: !n. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n = h4/arithmetic/_2B n n
% Assm: h4/arithmetic/MULT__CLAUSES_c4: !n m. h4/arithmetic/_2A (h4/num/SUC m) n = h4/arithmetic/_2B (h4/arithmetic/_2A m n) n
% Assm: h4/arithmetic/BIT20: !n. h4/arithmetic/BIT2 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (h4/num/SUC (h4/num/SUC h4/num/0)))
% Assm: h4/arithmetic/ALT__ZERO: h4/arithmetic/ZERO = h4/num/0
% Assm: h4/arithmetic/COMPLETE__INDUCTION: !P. (!n. (!m. h4/prim__rec/_3C m n ==> P m) ==> P n) ==> (!n. P n)
% Assm: h4/arithmetic/WOP: !P. (?n. P n) ==> (?n. P n /\ (!m. h4/prim__rec/_3C m n ==> ~P m))
% Assm: h4/bool/NOT__IMP: !B A. ~(A ==> B) <=> A /\ ~B
% Assm: h4/prim__rec/SUC__ID: !n. ~(h4/num/SUC n = n)
% Assm: h4/arithmetic/MULT__CLAUSES_c5: !n m. h4/arithmetic/_2A m (h4/num/SUC n) = h4/arithmetic/_2B m (h4/arithmetic/_2A m n)
% Assm: h4/arithmetic/MULT__RIGHT__1: !m. h4/arithmetic/_2A m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = m
% Assm: h4/arithmetic/MULT__LEFT__1: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/MULT__SUC: !n m. h4/arithmetic/_2A m (h4/num/SUC n) = h4/arithmetic/_2B m (h4/arithmetic/_2A m n)
% Goal: !n. h4/arithmetic/EVEN n \/ h4/arithmetic/ODD 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_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_EVENu_u_ODD]: !n. h4/arithmetic/EVEN n <=> ~h4/arithmetic/ODD n
% Assm [h4s_arithmetics_ODDu_u_EVEN]: !n. h4/arithmetic/ODD n <=> ~h4/arithmetic/EVEN n
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_arithmetics_ODD0u_c1]: !n. h4/arithmetic/ODD (h4/num/SUC n) <=> ~h4/arithmetic/ODD n
% Assm [h4s_arithmetics_EVEN0u_c1]: !n. h4/arithmetic/EVEN (h4/num/SUC n) <=> ~h4/arithmetic/EVEN n
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_arithmetics_ODD0u_c0]: h4/arithmetic/ODD h4/num/0 <=> F
% Assm [h4s_arithmetics_EVEN0u_c0]: h4/arithmetic/EVEN h4/num/0 <=> T
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% 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_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_nums_NOTu_u_SUC]: !n. ~(h4/num/SUC n = h4/num/0)
% Assm [h4s_arithmetics_NOTu_u_EXPu_u_0]: !n m. ~(h4/arithmetic/EXP (h4/num/SUC n) m = h4/num/0)
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c1]: !m. h4/arithmetic/_2A m h4/num/0 = h4/num/0
% Assm [h4s_arithmetics_EXP0u_c1]: !n m. h4/arithmetic/EXP m (h4/num/SUC n) = h4/arithmetic/_2A m (h4/arithmetic/EXP m n)
% Assm [h4s_arithmetics_MULTu_u_MONOu_u_EQ]: !n m i. h4/arithmetic/_2A (h4/num/SUC n) m = h4/arithmetic/_2A (h4/num/SUC n) i <=> m = i
% Assm [h4s_arithmetics_ONE]: h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) = h4/num/SUC h4/num/0
% Assm [h4s_arithmetics_EXP0u_c0]: !m. h4/arithmetic/EXP m h4/num/0 = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm [h4s_arithmetics_BIT10]: !n. h4/arithmetic/BIT1 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (h4/num/SUC h4/num/0))
% Assm [h4s_arithmetics_numu_u_caseu_u_defu_c1]: !v n f. h4/arithmetic/num__CASE (h4/num/SUC n) v f = happ f n
% Assm [h4s_primu_u_recs_PRIMu_u_RECu_u_EQNu_c1]: !x n m f. happ (happ (h4/prim__rec/PRIM__REC__FUN x f) (h4/num/SUC m)) n = happ (happ f (happ (happ (h4/prim__rec/PRIM__REC__FUN x f) m) (h4/prim__rec/PRE n))) n
% Assm [h4s_primu_u_recs_SIMPu_u_RECu_u_THMu_c1]: !x m f. happ (h4/prim__rec/SIMP__REC x f) (h4/num/SUC m) = happ f (happ (h4/prim__rec/SIMP__REC x f) m)
% Assm [h4s_primu_u_recs_PRIMu_u_RECu_u_FUN0]: !_2. (!f fun n. happ (happ (happ _2 f) fun) n = happ (happ f (happ fun (h4/prim__rec/PRE n))) n) ==> (!_1. (!f fun. happ (happ _1 f) fun = happ (happ _2 f) fun) ==> (!_0. (!x n. happ (happ _0 x) n = x) ==> (!x f. h4/prim__rec/PRIM__REC__FUN x f = h4/prim__rec/SIMP__REC (happ _0 x) (happ _1 f))))
% Assm [h4s_arithmetics_NOTu_u_LESS]: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_LESSu_u_IMPu_u_LESSu_u_ORu_u_EQ]: !n m. h4/prim__rec/_3C m n ==> h4/arithmetic/_3C_3D m n
% Assm [h4s_arithmetics_RIGHTu_u_SUBu_u_DISTRIB]: !p n m. h4/arithmetic/_2A (h4/arithmetic/_2D m n) p = h4/arithmetic/_2D (h4/arithmetic/_2A m p) (h4/arithmetic/_2A n p)
% Assm [h4s_arithmetics_SUBu_u_EQu_u_0]: !n m. h4/arithmetic/_2D m n = h4/num/0 <=> h4/arithmetic/_3C_3D m n
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2A h4/num/0 m = h4/num/0
% Assm [h4s_arithmetics_SUBu_u_ADD]: !n m. h4/arithmetic/_3C_3D n m ==> h4/arithmetic/_2B (h4/arithmetic/_2D m n) n = m
% Assm [h4s_arithmetics_LESSu_u_MONOu_u_MULT]: !p n m. h4/arithmetic/_3C_3D m n ==> h4/arithmetic/_3C_3D (h4/arithmetic/_2A m p) (h4/arithmetic/_2A n p)
% Assm [h4s_arithmetics_RIGHTu_u_ADDu_u_DISTRIB]: !p n m. h4/arithmetic/_2A (h4/arithmetic/_2B m n) p = h4/arithmetic/_2B (h4/arithmetic/_2A m p) (h4/arithmetic/_2A n p)
% Assm [h4s_arithmetics_ADDu_u_EQu_u_SUB]: !p n m. h4/arithmetic/_3C_3D n p ==> (h4/arithmetic/_2B m n = p <=> m = h4/arithmetic/_2D p n)
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_arithmetics_LESSu_u_ORu_u_EQ]: !n m. h4/arithmetic/_3C_3D m n <=> h4/prim__rec/_3C m n \/ 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_arithmetics_LESSu_u_EQu_u_MONO]: !n m. h4/arithmetic/_3C_3D (h4/num/SUC n) (h4/num/SUC m) <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_LESSu_u_MONOu_u_EQ]: !n m. h4/prim__rec/_3C (h4/num/SUC m) (h4/num/SUC n) <=> h4/prim__rec/_3C m n
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_arithmetics_FORALLu_u_NUMu_u_THM]: !P. (!n. happ P n) <=> happ P h4/num/0 /\ (!n. happ P n ==> happ P (h4/num/SUC n))
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_bools_LEFTu_u_FORALLu_u_ORu_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_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_arithmetics_LESSu_u_EQu_u_ADD]: !n m. h4/arithmetic/_3C_3D m (h4/arithmetic/_2B m n)
% Assm [h4s_arithmetics_SUBu_u_LESSu_u_EQ]: !n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2D n m) n
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_SUBu_u_PLUS]: !c b a. h4/arithmetic/_2D a (h4/arithmetic/_2B b c) = h4/arithmetic/_2D (h4/arithmetic/_2D a b) c
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c1]: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm [h4s_arithmetics_ADDu_u_INVu_u_0u_u_EQ]: !n m. h4/arithmetic/_2B m n = m <=> n = h4/num/0
% Assm [h4s_arithmetics_ADDu_u_INVu_u_0]: !n m. h4/arithmetic/_2B m n = m ==> n = h4/num/0
% Assm [h4s_arithmetics_NUMERALu_u_DEF]: !x. h4/arithmetic/NUMERAL x = x
% Assm [h4s_arithmetics_LESSu_u_EQUALu_u_ADD]: !n m. h4/arithmetic/_3C_3D m n ==> (?p. n = h4/arithmetic/_2B m p)
% Assm [h4s_arithmetics_LESSu_u_ADD]: !n m. h4/prim__rec/_3C n m ==> (?p. h4/arithmetic/_2B p n = m)
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm [h4s_arithmetics_NOTu_u_ODDu_u_EQu_u_EVEN]: !n m. ~(h4/num/SUC (h4/arithmetic/_2B n n) = h4/arithmetic/_2B m m)
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c3]: !n m. h4/arithmetic/_2B m (h4/num/SUC n) = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c2]: !n m. h4/arithmetic/_2B (h4/num/SUC m) n = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_ORu_u_CLAUSESu_c1]: !t. t \/ T <=> T
% Assm [h4s_primu_u_recs_LESSu_u_SUCu_u_REFL]: !n. h4/prim__rec/_3C n (h4/num/SUC n)
% 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_arithmetics_LESSu_u_OR]: !n m. h4/prim__rec/_3C m n ==> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm [h4s_arithmetics_LESSu_u_SUCu_u_NOT]: !n m. h4/prim__rec/_3C m n ==> ~h4/prim__rec/_3C n (h4/num/SUC m)
% Assm [h4s_nums_INVu_u_SUC]: !n m. h4/num/SUC m = h4/num/SUC n ==> m = n
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% 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_nums_numu_u_ISOu_u_DEFu_c1]: !r. h4/num/IS__NUM__REP r <=> h4/num/REP__num (h4/num/ABS__num r) = r
% Assm [h4s_nums_numu_u_ISOu_u_DEFu_c0]: !a. h4/num/ABS__num (h4/num/REP__num a) = a
% Assm [h4s_nums_ISu_u_NUMu_u_REP0]: !m. 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_nums_SUCu_u_REPu_u_DEFu_c1]: ~h4/bool/ONTO h4/num/SUC__REP
% Assm [h4s_nums_SUCu_u_REPu_u_DEFu_c0]: h4/bool/ONE__ONE 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_IMPu_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_THM]: !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_SUCu_u_SUCu_c0]: !m. h4/prim__rec/_3C m (h4/num/SUC m)
% Assm [h4s_primu_u_recs_PRE0u_c1]: !m. h4/prim__rec/PRE (h4/num/SUC m) = m
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_SELECTu_u_REFLu_u_2]: !_0. (!x y. happ (happ _0 x) y <=> x = y) ==> (!x. h4/min/_40 (happ _0 x) = x)
% Assm [h4s_primu_u_recs_PREu_u_DEF]: !_0. (!m n. happ (happ _0 m) n <=> m = h4/num/SUC n) ==> (!m. ?v. (v <=> m = h4/num/0) /\ h4/prim__rec/PRE m = h4/bool/COND v h4/num/0 (h4/min/_40 (happ _0 m)))
% Assm [h4s_arithmetics_GREATERu_u_ORu_u_EQ]: !n m. h4/arithmetic/_3E_3D m n <=> h4/arithmetic/_3E m n \/ m = n
% Assm [h4s_arithmetics_ADDu_u_SUC]: !n m. h4/num/SUC (h4/arithmetic/_2B m n) = h4/arithmetic/_2B m (h4/num/SUC n)
% Assm [h4s_arithmetics_ADDu_u_0]: !m. h4/arithmetic/_2B m h4/num/0 = m
% 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_arithmetics_ADDu_c0]: !n. h4/arithmetic/_2B h4/num/0 n = n
% Assm [h4s_arithmetics_NOTu_u_ZEROu_u_LTu_u_ZERO]: !n. ~(n = h4/num/0) <=> h4/prim__rec/_3C h4/num/0 n
% Assm [h4s_primu_u_recs_LESSu_u_0]: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_arithmetics_numu_u_CASES]: !m. m = h4/num/0 \/ (?n. m = h4/num/SUC n)
% Assm [h4s_primu_u_recs_NOTu_u_LESSu_u_0]: !n. ~h4/prim__rec/_3C n h4/num/0
% Assm [h4s_arithmetics_LESSu_u_LESSu_u_SUC]: !n m. ~(h4/prim__rec/_3C m n /\ h4/prim__rec/_3C n (h4/num/SUC m))
% Assm [h4s_arithmetics_LESSu_u_TRANS]: !p n m. h4/prim__rec/_3C m n /\ h4/prim__rec/_3C n p ==> h4/prim__rec/_3C m p
% Assm [h4s_primu_u_recs_LESSu_u_REFL]: !n. ~h4/prim__rec/_3C n n
% Assm [h4s_arithmetics_LESSu_u_ADDu_u_SUC]: !n m. h4/prim__rec/_3C m (h4/arithmetic/_2B m (h4/num/SUC n))
% Assm [h4s_arithmetics_TIMES2]: !n. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n = h4/arithmetic/_2B n n
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c4]: !n m. h4/arithmetic/_2A (h4/num/SUC m) n = h4/arithmetic/_2B (h4/arithmetic/_2A m n) n
% Assm [h4s_arithmetics_BIT20]: !n. h4/arithmetic/BIT2 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (h4/num/SUC (h4/num/SUC h4/num/0)))
% Assm [h4s_arithmetics_ALTu_u_ZERO]: h4/arithmetic/ZERO = h4/num/0
% Assm [h4s_arithmetics_COMPLETEu_u_INDUCTION]: !P. (!n. (!m. h4/prim__rec/_3C m n ==> happ P m) ==> happ P n) ==> (!n. happ P n)
% Assm [h4s_arithmetics_WOP]: !P. (?n. happ P n) ==> (?n. happ P n /\ (!m. h4/prim__rec/_3C m n ==> ~happ P m))
% Assm [h4s_bools_NOTu_u_IMP]: !B A. ~(A ==> B) <=> A /\ ~B
% Assm [h4s_primu_u_recs_SUCu_u_ID]: !n. ~(h4/num/SUC n = n)
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c5]: !n m. h4/arithmetic/_2A m (h4/num/SUC n) = h4/arithmetic/_2B m (h4/arithmetic/_2A m n)
% Assm [h4s_arithmetics_MULTu_u_RIGHTu_u_1]: !m. h4/arithmetic/_2A m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = m
% Assm [h4s_arithmetics_MULTu_u_LEFTu_u_1]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_MULTu_u_SUC]: !n m. h4/arithmetic/_2A m (h4/num/SUC n) = h4/arithmetic/_2B m (h4/arithmetic/_2A m n)
% Goal: !n. h4/arithmetic/EVEN n \/ h4/arithmetic/ODD 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_Q1403255,TV_Q1403251]: ![V_f, V_g]: (![V_x]: s(TV_Q1403251,happ(s(t_fun(TV_Q1403255,TV_Q1403251),V_f),s(TV_Q1403255,V_x))) = s(TV_Q1403251,happ(s(t_fun(TV_Q1403255,TV_Q1403251),V_g),s(TV_Q1403255,V_x))) => s(t_fun(TV_Q1403255,TV_Q1403251),V_f) = s(t_fun(TV_Q1403255,TV_Q1403251),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_EVENu_u_ODD, axiom, ![V_n]: (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_n)))) <=> ~ (p(s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,V_n))))))).
fof(ah4s_arithmetics_ODDu_u_EVEN, axiom, ![V_n]: (p(s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,V_n)))) <=> ~ (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_n))))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
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_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_arithmetics_ODD0u_c1, axiom, ![V_n]: (p(s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))) <=> ~ (p(s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,V_n))))))).
fof(ah4s_arithmetics_EVEN0u_c1, axiom, ![V_n]: (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))) <=> ~ (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_n))))))).
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_arithmetics_ODD0u_c0, axiom, s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,f)).
fof(ah4s_arithmetics_EVEN0u_c0, axiom, s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,t)).
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_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_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,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_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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
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_arithmetics_NOTu_u_EXPu_u_0, axiom, ![V_n, V_m]: ~ (s(t_h4s_nums_num,h4s_arithmetics_exp(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,h4s_nums_0))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_arithmetics_EXP0u_c1, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_exp(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,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_MULTu_u_MONOu_u_EQ, axiom, ![V_n, V_m, V_i]: (s(t_h4s_nums_num,h4s_arithmetics_u_2a(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,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_i))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_i))).
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_arithmetics_EXP0u_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = 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_arithmetics_BIT10, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0)))))))).
fof(ah4s_arithmetics_numu_u_caseu_u_defu_c1, axiom, ![TV_u_27a]: ![V_v, V_n, V_f]: s(TV_u_27a,h4s_arithmetics_numu_u_case(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_v),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n)))).
fof(ah4s_primu_u_recs_PRIMu_u_RECu_u_EQNu_c1, axiom, ![TV_u_27a]: ![V_x, V_n, V_m, V_f]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),h4s_primu_u_recs_primu_u_recu_u_fun(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))),s(t_h4s_nums_num,V_n))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),h4s_primu_u_recs_primu_u_recu_u_fun(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,V_n))))))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_primu_u_recs_SIMPu_u_RECu_u_THMu_c1, axiom, ![TV_u_27a]: ![V_x, V_m, V_f]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_primu_u_recs_simpu_u_rec(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_primu_u_recs_simpu_u_rec(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_primu_u_recs_PRIMu_u_RECu_u_FUN0, axiom, ![TV_u_27a]: ![V_uu_2]: (![V_f, V_fun, V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27a))),V_uu_2),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f))),s(t_fun(t_h4s_nums_num,TV_u_27a),V_fun))),s(t_h4s_nums_num,V_n))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fun),s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,V_n))))))),s(t_h4s_nums_num,V_n))) => ![V_uu_1]: (![V_f, V_fun]: s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27a))),V_uu_1),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f))),s(t_fun(t_h4s_nums_num,TV_u_27a),V_fun))) = s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27a))),V_uu_2),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f))),s(t_fun(t_h4s_nums_num,TV_u_27a),V_fun))) => ![V_uu_0]: (![V_x, V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_uu_0),s(TV_u_27a,V_x))),s(t_h4s_nums_num,V_n))) = s(TV_u_27a,V_x) => ![V_x, V_f]: s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),h4s_primu_u_recs_primu_u_recu_u_fun(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f))) = s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),h4s_primu_u_recs_simpu_u_rec(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_uu_0),s(TV_u_27a,V_x))),s(t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27a))),V_uu_1),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f))))))))).
fof(ah4s_arithmetics_NOTu_u_LESS, 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_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_LESSu_u_IMPu_u_LESSu_u_ORu_u_EQ, 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_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_RIGHTu_u_SUBu_u_DISTRIB, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p))) = s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_SUBu_u_EQu_u_0, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_arithmetics_SUBu_u_ADD, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))) => s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_m))).
fof(ah4s_arithmetics_LESSu_u_MONOu_u_MULT, axiom, ![V_p, V_n, V_m]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))))))).
fof(ah4s_arithmetics_RIGHTu_u_ADDu_u_DISTRIB, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_ADDu_u_EQu_u_SUB, axiom, ![V_p, V_n, V_m]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))) => (s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_p) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_arithmetics_LESSu_u_ORu_u_EQ, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(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)))) | 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_arithmetics_LESSu_u_EQu_u_MONO, axiom, ![V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_LESSu_u_MONOu_u_EQ, axiom, ![V_n, V_m]: 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))))) = 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_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_arithmetics_FORALLu_u_NUMu_u_THM, axiom, ![V_P]: (![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_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)))))))))).
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_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_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_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_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_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_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_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_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_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_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_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_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_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_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_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_LESSu_u_EQu_u_ADD, axiom, ![V_n, V_m]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(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,V_n))))))).
fof(ah4s_arithmetics_SUBu_u_LESSu_u_EQ, axiom, ![V_n, V_m]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_ADDu_u_SYM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_SUBu_u_PLUS, axiom, ![V_c, V_b, V_a]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_a),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_b),s(t_h4s_nums_num,V_c))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_a),s(t_h4s_nums_num,V_b))),s(t_h4s_nums_num,V_c)))).
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_arithmetics_ADDu_u_CLAUSESu_c1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_INVu_u_0u_u_EQ, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_m) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0))).
fof(ah4s_arithmetics_ADDu_u_INVu_u_0, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_m) => s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0))).
fof(ah4s_arithmetics_NUMERALu_u_DEF, axiom, ![V_x]: s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_x))) = s(t_h4s_nums_num,V_x)).
fof(ah4s_arithmetics_LESSu_u_EQUALu_u_ADD, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) => ?[V_p]: s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))).
fof(ah4s_arithmetics_LESSu_u_ADD, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))) => ?[V_p]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_m))).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_NOTu_u_ODDu_u_EQu_u_EVEN, axiom, ![V_n, V_m]: ~ (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_m))))).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c3, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(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,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_arithmetics_ADDu_u_CLAUSESu_c2, 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_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_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_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_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_arithmetics_LESSu_u_OR, 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_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_LESSu_u_SUCu_u_NOT, 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_n),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))))))).
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_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_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_nums_numu_u_ISOu_u_DEFu_c1, axiom, ![V_r]: (p(s(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_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_nums_ISu_u_NUMu_u_REP0, axiom, ![V_m]: (p(s(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_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_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_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_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_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_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_primu_u_recs_LESSu_u_SUCu_u_SUCu_c0, axiom, ![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_m))))))).
fof(ah4s_primu_u_recs_PRE0u_c1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,V_m)).
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_SELECTu_u_REFLu_u_2, axiom, ![TV_u_27a]: ![V_uu_0]: (![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_uu_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)) => ![V_x]: 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)),V_uu_0),s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x))).
fof(ah4s_primu_u_recs_PREu_u_DEF, axiom, ![V_uu_0]: (![V_m, V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))) => ![V_m]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0)) & s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,V_v),s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_mins_u_40(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_h4s_nums_num,V_m)))))))))).
fof(ah4s_arithmetics_GREATERu_u_ORu_u_EQ, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_arithmetics_u_3eu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) <=> (p(s(t_bool,h4s_arithmetics_u_3e(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) | s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_ADDu_u_SUC, axiom, ![V_n, V_m]: 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))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_ADDu_u_0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_m)).
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_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_NOTu_u_ZEROu_u_LTu_u_ZERO, axiom, ![V_n]: (~ (s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) <=> p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n)))))).
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_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
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_arithmetics_numu_u_CASES, axiom, ![V_m]: (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0) | ?[V_n]: 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_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_arithmetics_LESSu_u_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_n),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))))))).
fof(ah4s_arithmetics_LESSu_u_TRANS, axiom, ![V_p, 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_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
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_arithmetics_LESSu_u_ADDu_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,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))))))).
fof(ah4s_arithmetics_TIMES2, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c4, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(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_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_BIT20, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0)))))))))).
fof(ah4s_arithmetics_ALTu_u_ZERO, axiom, s(t_h4s_nums_num,h4s_arithmetics_zero) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_arithmetics_COMPLETEu_u_INDUCTION, axiom, ![V_P]: (![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,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))))) => ![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_arithmetics_WOP, axiom, ![V_P]: (?[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)))) => ?[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)))) & ![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,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_m))))))))).
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_primu_u_recs_SUCu_u_ID, axiom, ![V_n]: ~ (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_n))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c5, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(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,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_MULTu_u_RIGHTu_u_1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(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))))))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_MULTu_u_LEFTu_u_1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(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,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_MULTu_u_SUC, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(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,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ch4s_arithmetics_EVENu_u_ORu_u_ODD, conjecture, ![V_n]: (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_n)))) | p(s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,V_n)))))).
