%   ORIGINAL: h4/toto/cpn2num__thm_c0
% 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/toto/LESS__def: h4/toto/LESS = h4/toto/num2cpn h4/num/0
% Assm: h4/toto/cpn__BIJ_c0: !a. h4/toto/num2cpn (h4/toto/cpn2num a) = a
% Assm: h4/toto/cpn__BIJ_c1: !r. (\n. h4/prim__rec/_3C n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) r <=> h4/toto/cpn2num (h4/toto/num2cpn r) = r
% Assm: h4/toto/num2cpn__thm_c0: h4/toto/num2cpn h4/num/0 = h4/toto/LESS
% Assm: h4/toto/cpn__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION (\n. h4/prim__rec/_3C n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) rep
% 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__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/toto/EQUAL__def: h4/toto/EQUAL = h4/toto/num2cpn (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/toto/GREATER__def: h4/toto/GREATER = h4/toto/num2cpn (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))
% Assm: h4/toto/cpn2num__11: !a_27 a. h4/toto/cpn2num a = h4/toto/cpn2num a_27 <=> a = a_27
% Assm: h4/toto/cpn2num__ONTO: !r. h4/prim__rec/_3C r (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) <=> (?a. r = h4/toto/cpn2num a)
% Assm: h4/toto/num2cpn__cpn2num: !a. h4/toto/num2cpn (h4/toto/cpn2num a) = a
% Assm: h4/toto/cpn2num__num2cpn: !r. h4/prim__rec/_3C r (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) <=> h4/toto/cpn2num (h4/toto/num2cpn r) = r
% Assm: h4/bool/TRUTH: T
% Assm: h4/toto/num2cpn__thm_c1: h4/toto/num2cpn (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = h4/toto/EQUAL
% Assm: h4/toto/num2cpn__ONTO: !a. ?r. a = h4/toto/num2cpn r /\ h4/prim__rec/_3C r (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% Assm: h4/toto/num2cpn__11: !r_27 r. h4/prim__rec/_3C r (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) ==> h4/prim__rec/_3C r_27 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) ==> (h4/toto/num2cpn r = h4/toto/num2cpn r_27 <=> r = r_27)
% Assm: h4/toto/num2cpn__thm_c2: h4/toto/num2cpn (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) = h4/toto/GREATER
% Assm: h4/numeral/numeral__lt_c0: !n. h4/prim__rec/_3C h4/arithmetic/ZERO (h4/arithmetic/BIT1 n) <=> T
% Assm: h4/numeral/numeral__distrib_c21: !n. h4/prim__rec/_3C h4/num/0 (h4/arithmetic/NUMERAL n) <=> h4/prim__rec/_3C h4/arithmetic/ZERO n
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/arithmetic/MODEQ__NONZERO__MODEQUALITY: !n m2 m1. h4/prim__rec/_3C h4/num/0 n ==> (h4/arithmetic/MODEQ n m1 m2 <=> h4/arithmetic/MOD m1 n = h4/arithmetic/MOD m2 n)
% Assm: h4/arithmetic/MODEQ__INTRO__CONG: !n e1 e0. h4/prim__rec/_3C h4/num/0 n ==> h4/arithmetic/MODEQ n e0 e1 ==> h4/arithmetic/MOD e0 n = h4/arithmetic/MOD e1 n
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~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/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/bool/RIGHT__OR__OVER__AND: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/arithmetic/DIVISION: !n. h4/prim__rec/_3C h4/num/0 n ==> (!k. k = h4/arithmetic/_2B (h4/arithmetic/_2A (h4/arithmetic/DIV k n) n) (h4/arithmetic/MOD k n) /\ h4/prim__rec/_3C (h4/arithmetic/MOD k n) n)
% Assm: h4/arithmetic/ADD__ASSOC: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m 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/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/arithmetic/MOD__UNIQUE: !r n k. (?q. k = h4/arithmetic/_2B (h4/arithmetic/_2A q n) r /\ h4/prim__rec/_3C r n) ==> h4/arithmetic/MOD k n = r
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/marker/Abbrev__def: !x. h4/marker/Abbrev x <=> x
% Assm: h4/bool/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm: h4/marker/AC__DEF: !b2 b1. h4/marker/AC b1 b2 <=> b1 /\ b2
% Assm: h4/arithmetic/ADD__COMM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/bool/LCOMM__THM: !f. (!x y z. f x (f y z) = f (f x y) z) ==> (!x y. f x y = f y x) ==> (!x y z. f x (f y z) = f y (f x z))
% Assm: h4/arithmetic/ADD__DIV__ADD__DIV: !n. h4/prim__rec/_3C h4/num/0 n ==> (!x r. h4/arithmetic/DIV (h4/arithmetic/_2B (h4/arithmetic/_2A x n) r) n = h4/arithmetic/_2B x (h4/arithmetic/DIV r n))
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/arithmetic/ADD__0: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/arithmetic/LESS__DIV__EQ__ZERO: !r n. h4/prim__rec/_3C r n ==> h4/arithmetic/DIV r n = h4/num/0
% Assm: h4/arithmetic/DIV__MULT: !r n. h4/prim__rec/_3C r n ==> (!q. h4/arithmetic/DIV (h4/arithmetic/_2B (h4/arithmetic/_2A q n) r) n = q)
% Assm: h4/arithmetic/DA: !n k. h4/prim__rec/_3C h4/num/0 n ==> (?r q. k = h4/arithmetic/_2B (h4/arithmetic/_2A q n) r /\ h4/prim__rec/_3C r n)
% Assm: h4/prim__rec/DC: !a R P. P a /\ (!x. P x ==> (?y. P y /\ R x y)) ==> (?f. f h4/num/0 = a /\ (!n. P (f n) /\ R (f n) (f (h4/num/SUC n))))
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% 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/SIMP__REC__THM_c0: !x f. h4/prim__rec/SIMP__REC x f h4/num/0 = x
% Assm: h4/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Assm: h4/arithmetic/MOD__EQ__0: !n. h4/prim__rec/_3C h4/num/0 n ==> (!k. h4/arithmetic/MOD (h4/arithmetic/_2A k n) n = h4/num/0)
% Assm: h4/arithmetic/ADD__CLAUSES_c1: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/arithmetic/ADD__MODULUS_c0: !x n. h4/prim__rec/_3C h4/num/0 n ==> h4/arithmetic/MOD (h4/arithmetic/_2B x n) n = h4/arithmetic/MOD x n
% Assm: h4/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/bool/LEFT__FORALL__OR__THM: !Q P. (!x. P x \/ Q) <=> (!x. P x) \/ Q
% Assm: h4/arithmetic/DIVMOD__ID: !n. h4/prim__rec/_3C h4/num/0 n ==> h4/arithmetic/DIV n n = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) /\ h4/arithmetic/MOD n n = h4/num/0
% Assm: h4/arithmetic/MOD__PLUS: !n. h4/prim__rec/_3C h4/num/0 n ==> (!j k. h4/arithmetic/MOD (h4/arithmetic/_2B (h4/arithmetic/MOD j n) (h4/arithmetic/MOD k n)) n = h4/arithmetic/MOD (h4/arithmetic/_2B j k) n)
% Assm: h4/arithmetic/MOD__MOD: !n. h4/prim__rec/_3C h4/num/0 n ==> (!k. h4/arithmetic/MOD (h4/arithmetic/MOD k n) n = h4/arithmetic/MOD k n)
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/arithmetic/DIVMOD__CALC_c0: !n m. h4/prim__rec/_3C h4/num/0 n ==> h4/arithmetic/DIV m n = h4/pair/FST (h4/arithmetic/DIVMOD (h4/pair/_2C h4/num/0 (h4/pair/_2C m n)))
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/arithmetic/DIVMOD__CORRECT: !n m a. h4/prim__rec/_3C h4/num/0 n ==> h4/arithmetic/DIVMOD (h4/pair/_2C a (h4/pair/_2C m n)) = h4/pair/_2C (h4/arithmetic/_2B a (h4/arithmetic/DIV m n)) (h4/arithmetic/MOD m n)
% Assm: h4/arithmetic/MOD__TIMES: !n. h4/prim__rec/_3C h4/num/0 n ==> (!q r. h4/arithmetic/MOD (h4/arithmetic/_2B (h4/arithmetic/_2A q n) r) n = h4/arithmetic/MOD r n)
% Assm: h4/arithmetic/MOD__MULT: !r n. h4/prim__rec/_3C r n ==> (!q. h4/arithmetic/MOD (h4/arithmetic/_2B (h4/arithmetic/_2A q n) r) n = r)
% 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/NUMERAL__MULT__EQ__DIV_c0: !z y x. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 x)) y = h4/arithmetic/NUMERAL z <=> y = h4/arithmetic/DIV (h4/arithmetic/NUMERAL z) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 x)) /\ h4/arithmetic/MOD (h4/arithmetic/NUMERAL z) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 x)) = h4/num/0
% Assm: h4/prim__rec/LESS__0: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% 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/BIT10: !n. h4/arithmetic/BIT1 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (h4/num/SUC h4/num/0))
% Assm: h4/arithmetic/NUMERAL__DEF: !x. h4/arithmetic/NUMERAL x = x
% Assm: h4/arithmetic/MULT__EQ__DIV: !z y x. h4/prim__rec/_3C h4/num/0 x ==> (h4/arithmetic/_2A x y = z <=> y = h4/arithmetic/DIV z x /\ h4/arithmetic/MOD z x = h4/num/0)
% Assm: h4/numeral/texp__help__def_c0: !acc. h4/numeral/texp__help h4/num/0 acc = h4/arithmetic/BIT2 acc
% Assm: h4/pred__set/chooser__def_c0: !s. h4/pred__set/chooser s h4/num/0 = h4/pred__set/CHOICE s
% Assm: h4/rich__list/LENGTH__NOT__NULL: !l. h4/prim__rec/_3C h4/num/0 (h4/list/LENGTH l) <=> ~h4/list/NULL l
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__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/FALSITY: !t. F ==> t
% Assm: h4/list/NULL0_c0: h4/list/NULL h4/list/NIL
% Assm: h4/list/list__induction: !P. P h4/list/NIL /\ (!t. P t ==> (!h. P (h4/list/CONS h t))) ==> (!l. P l)
% Assm: h4/list/LENGTH0_c1: !t h. h4/list/LENGTH (h4/list/CONS h t) = h4/num/SUC (h4/list/LENGTH t)
% Assm: h4/list/LENGTH0_c0: h4/list/LENGTH h4/list/NIL = h4/num/0
% Assm: h4/list/NULL0_c1: !t h. ~h4/list/NULL (h4/list/CONS h t)
% Assm: h4/arithmetic/MOD__TIMES2: !n. h4/prim__rec/_3C h4/num/0 n ==> (!j k. h4/arithmetic/MOD (h4/arithmetic/_2A (h4/arithmetic/MOD j n) (h4/arithmetic/MOD k n)) n = h4/arithmetic/MOD (h4/arithmetic/_2A j k) n)
% Assm: h4/arithmetic/MULT__ASSOC: !p n m. h4/arithmetic/_2A m (h4/arithmetic/_2A n p) = h4/arithmetic/_2A (h4/arithmetic/_2A m n) p
% Assm: h4/arithmetic/LEFT__ADD__DISTRIB: !p n m. h4/arithmetic/_2A p (h4/arithmetic/_2B m n) = h4/arithmetic/_2B (h4/arithmetic/_2A p m) (h4/arithmetic/_2A p n)
% Assm: h4/arithmetic/MULT__COMM: !n m. h4/arithmetic/_2A m n = h4/arithmetic/_2A n m
% Assm: h4/numeral/numeral__distrib_c30: !n. h4/arithmetic/_3E_3D h4/num/0 n <=> n = h4/num/0
% Assm: h4/numeral/iZ0: !x. h4/numeral/iZ x = x
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/arithmetic/MULT__CLAUSES_c0: !m. h4/arithmetic/_2A h4/num/0 m = h4/num/0
% Assm: h4/arithmetic/MULT__CLAUSES_c1: !m. h4/arithmetic/_2A m h4/num/0 = h4/num/0
% Assm: h4/arithmetic/SUB__0_c1: !m. h4/arithmetic/_2D m h4/num/0 = m
% Assm: h4/arithmetic/LESS__0__CASES: !m. h4/num/0 = m \/ h4/prim__rec/_3C h4/num/0 m
% Assm: h4/arithmetic/SUB__0_c0: !m. h4/arithmetic/_2D h4/num/0 m = h4/num/0
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/bool/EXISTS__OR__THM: !Q P. (?x. P x \/ Q x) <=> (?x. P x) \/ (?x. Q x)
% Assm: h4/arithmetic/GREATER__DEF: !n m. h4/arithmetic/_3E m n <=> h4/prim__rec/_3C n m
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% 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/LESS__OR__EQ: !n m. h4/arithmetic/_3C_3D m n <=> h4/prim__rec/_3C m n \/ m = n
% 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/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/arithmetic/GREATER__OR__EQ: !n m. h4/arithmetic/_3E_3D m n <=> h4/arithmetic/_3E m n \/ m = 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/EVEN0_c0: h4/arithmetic/EVEN h4/num/0 <=> T
% Goal: h4/toto/cpn2num h4/toto/LESS = h4/num/0
%   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_totos_LESSu_u_def]: h4/toto/LESS = h4/toto/num2cpn h4/num/0
% Assm [h4s_totos_cpnu_u_BIJu_c0]: !a. h4/toto/num2cpn (h4/toto/cpn2num a) = a
% Assm [h4s_totos_cpnu_u_BIJu_c1]: !r. h4/prim__rec/_3C r (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) <=> h4/toto/cpn2num (h4/toto/num2cpn r) = r
% Assm [h4s_totos_num2cpnu_u_thmu_c0]: h4/toto/num2cpn h4/num/0 = h4/toto/LESS
% Assm [h4s_totos_cpnu_u_TYu_u_DEF]: !_0. (!n. happ _0 n <=> h4/prim__rec/_3C n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) ==> (?rep. h4/bool/TYPE__DEFINITION _0 rep)
% 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_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_totos_EQUALu_u_def]: h4/toto/EQUAL = h4/toto/num2cpn (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_totos_GREATERu_u_def]: h4/toto/GREATER = h4/toto/num2cpn (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))
% Assm [h4s_totos_cpn2numu_u_11]: !a_27 a. h4/toto/cpn2num a = h4/toto/cpn2num a_27 <=> a = a_27
% Assm [h4s_totos_cpn2numu_u_ONTO]: !r. h4/prim__rec/_3C r (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) <=> (?a. r = h4/toto/cpn2num a)
% Assm [h4s_totos_num2cpnu_u_cpn2num]: !a. h4/toto/num2cpn (h4/toto/cpn2num a) = a
% Assm [h4s_totos_cpn2numu_u_num2cpn]: !r. h4/prim__rec/_3C r (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) <=> h4/toto/cpn2num (h4/toto/num2cpn r) = r
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_totos_num2cpnu_u_thmu_c1]: h4/toto/num2cpn (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = h4/toto/EQUAL
% Assm [h4s_totos_num2cpnu_u_ONTO]: !a. ?r. a = h4/toto/num2cpn r /\ h4/prim__rec/_3C r (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% Assm [h4s_totos_num2cpnu_u_11]: !r_27 r. h4/prim__rec/_3C r (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) ==> h4/prim__rec/_3C r_27 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) ==> (h4/toto/num2cpn r = h4/toto/num2cpn r_27 <=> r = r_27)
% Assm [h4s_totos_num2cpnu_u_thmu_c2]: h4/toto/num2cpn (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) = h4/toto/GREATER
% Assm [h4s_numerals_numeralu_u_ltu_c0]: !n. h4/prim__rec/_3C h4/arithmetic/ZERO (h4/arithmetic/BIT1 n) <=> T
% Assm [h4s_numerals_numeralu_u_distribu_c21]: !n. h4/prim__rec/_3C h4/num/0 (h4/arithmetic/NUMERAL n) <=> h4/prim__rec/_3C h4/arithmetic/ZERO n
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_arithmetics_MODEQu_u_NONZEROu_u_MODEQUALITY]: !n m2 m1. h4/prim__rec/_3C h4/num/0 n ==> (h4/arithmetic/MODEQ n m1 m2 <=> h4/arithmetic/MOD m1 n = h4/arithmetic/MOD m2 n)
% Assm [h4s_arithmetics_MODEQu_u_INTROu_u_CONG]: !n e1 e0. h4/prim__rec/_3C h4/num/0 n ==> h4/arithmetic/MODEQ n e0 e1 ==> h4/arithmetic/MOD e0 n = h4/arithmetic/MOD e1 n
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~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_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND]: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_arithmetics_DIVISION]: !n. h4/prim__rec/_3C h4/num/0 n ==> (!k. k = h4/arithmetic/_2B (h4/arithmetic/_2A (h4/arithmetic/DIV k n) n) (h4/arithmetic/MOD k n) /\ h4/prim__rec/_3C (h4/arithmetic/MOD k n) n)
% Assm [h4s_arithmetics_ADDu_u_ASSOC]: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m 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_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_arithmetics_MODu_u_UNIQUE]: !r n k. (?q. k = h4/arithmetic/_2B (h4/arithmetic/_2A q n) r /\ h4/prim__rec/_3C r n) ==> h4/arithmetic/MOD k n = r
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_markers_Abbrevu_u_def]: !x. h4/marker/Abbrev x <=> x
% Assm [h4s_bools_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm [h4s_markers_ACu_u_DEF]: !b2 b1. h4/marker/AC b1 b2 <=> b1 /\ b2
% Assm [h4s_arithmetics_ADDu_u_COMM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_bools_LCOMMu_u_THM]: !f. (!x y z. happ (happ f x) (happ (happ f y) z) = happ (happ f (happ (happ f x) y)) z) ==> (!x y. happ (happ f x) y = happ (happ f y) x) ==> (!x y z. happ (happ f x) (happ (happ f y) z) = happ (happ f y) (happ (happ f x) z))
% Assm [h4s_arithmetics_ADDu_u_DIVu_u_ADDu_u_DIV]: !n. h4/prim__rec/_3C h4/num/0 n ==> (!x r. h4/arithmetic/DIV (h4/arithmetic/_2B (h4/arithmetic/_2A x n) r) n = h4/arithmetic/_2B x (h4/arithmetic/DIV r n))
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_arithmetics_ADDu_u_0]: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm [h4s_arithmetics_LESSu_u_DIVu_u_EQu_u_ZERO]: !r n. h4/prim__rec/_3C r n ==> h4/arithmetic/DIV r n = h4/num/0
% Assm [h4s_arithmetics_DIVu_u_MULT]: !r n. h4/prim__rec/_3C r n ==> (!q. h4/arithmetic/DIV (h4/arithmetic/_2B (h4/arithmetic/_2A q n) r) n = q)
% Assm [h4s_arithmetics_DA]: !n k. h4/prim__rec/_3C h4/num/0 n ==> (?r q. k = h4/arithmetic/_2B (h4/arithmetic/_2A q n) r /\ h4/prim__rec/_3C r n)
% Assm [h4s_primu_u_recs_DC]: !a R P. happ P a /\ (!x. happ P x ==> (?y. happ P y /\ happ (happ R x) y)) ==> (?f. happ f h4/num/0 = a /\ (!n. happ P (happ f n) /\ happ (happ R (happ f n)) (happ f (h4/num/SUC n))))
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_primu_u_recs_SIMPu_u_RECu_u_THMu_c1]: !x m f. h4/prim__rec/SIMP__REC x f (h4/num/SUC m) = happ f (h4/prim__rec/SIMP__REC x f m)
% Assm [h4s_primu_u_recs_SIMPu_u_RECu_u_THMu_c0]: !x f. h4/prim__rec/SIMP__REC x f h4/num/0 = 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_arithmetics_MODu_u_EQu_u_0]: !n. h4/prim__rec/_3C h4/num/0 n ==> (!k. h4/arithmetic/MOD (h4/arithmetic/_2A k n) n = h4/num/0)
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c1]: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm [h4s_arithmetics_ADDu_u_MODULUSu_c0]: !x n. h4/prim__rec/_3C h4/num/0 n ==> h4/arithmetic/MOD (h4/arithmetic/_2B x n) n = h4/arithmetic/MOD x n
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm [h4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. happ P x \/ Q) <=> (!x. happ P x) \/ Q
% Assm [h4s_arithmetics_DIVMODu_u_ID]: !n. h4/prim__rec/_3C h4/num/0 n ==> h4/arithmetic/DIV n n = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) /\ h4/arithmetic/MOD n n = h4/num/0
% Assm [h4s_arithmetics_MODu_u_PLUS]: !n. h4/prim__rec/_3C h4/num/0 n ==> (!j k. h4/arithmetic/MOD (h4/arithmetic/_2B (h4/arithmetic/MOD j n) (h4/arithmetic/MOD k n)) n = h4/arithmetic/MOD (h4/arithmetic/_2B j k) n)
% Assm [h4s_arithmetics_MODu_u_MOD]: !n. h4/prim__rec/_3C h4/num/0 n ==> (!k. h4/arithmetic/MOD (h4/arithmetic/MOD k n) n = h4/arithmetic/MOD k n)
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_arithmetics_DIVMODu_u_CALCu_c0]: !n m. h4/prim__rec/_3C h4/num/0 n ==> h4/arithmetic/DIV m n = h4/pair/FST (h4/arithmetic/DIVMOD (h4/pair/_2C h4/num/0 (h4/pair/_2C m n)))
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_arithmetics_DIVMODu_u_CORRECT]: !n m a. h4/prim__rec/_3C h4/num/0 n ==> h4/arithmetic/DIVMOD (h4/pair/_2C a (h4/pair/_2C m n)) = h4/pair/_2C (h4/arithmetic/_2B a (h4/arithmetic/DIV m n)) (h4/arithmetic/MOD m n)
% Assm [h4s_arithmetics_MODu_u_TIMES]: !n. h4/prim__rec/_3C h4/num/0 n ==> (!q r. h4/arithmetic/MOD (h4/arithmetic/_2B (h4/arithmetic/_2A q n) r) n = h4/arithmetic/MOD r n)
% Assm [h4s_arithmetics_MODu_u_MULT]: !r n. h4/prim__rec/_3C r n ==> (!q. h4/arithmetic/MOD (h4/arithmetic/_2B (h4/arithmetic/_2A q n) r) n = r)
% 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_NUMERALu_u_MULTu_u_EQu_u_DIVu_c0]: !z y x. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 x)) y = h4/arithmetic/NUMERAL z <=> y = h4/arithmetic/DIV (h4/arithmetic/NUMERAL z) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 x)) /\ h4/arithmetic/MOD (h4/arithmetic/NUMERAL z) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 x)) = h4/num/0
% Assm [h4s_primu_u_recs_LESSu_u_0]: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% 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_BIT10]: !n. h4/arithmetic/BIT1 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (h4/num/SUC h4/num/0))
% Assm [h4s_arithmetics_NUMERALu_u_DEF]: !x. h4/arithmetic/NUMERAL x = x
% Assm [h4s_arithmetics_MULTu_u_EQu_u_DIV]: !z y x. h4/prim__rec/_3C h4/num/0 x ==> (h4/arithmetic/_2A x y = z <=> y = h4/arithmetic/DIV z x /\ h4/arithmetic/MOD z x = h4/num/0)
% Assm [h4s_numerals_texpu_u_helpu_u_defu_c0]: !acc. h4/numeral/texp__help h4/num/0 acc = h4/arithmetic/BIT2 acc
% Assm [h4s_predu_u_sets_chooseru_u_defu_c0]: !s. h4/pred__set/chooser s h4/num/0 = h4/pred__set/CHOICE s
% Assm [h4s_richu_u_lists_LENGTHu_u_NOTu_u_NULL]: !l. h4/prim__rec/_3C h4/num/0 (h4/list/LENGTH l) <=> ~h4/list/NULL l
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_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_FALSITY]: !t. F ==> t
% Assm [h4s_lists_NULL0u_c0]: h4/list/NULL h4/list/NIL
% Assm [h4s_lists_listu_u_induction]: !P. happ P h4/list/NIL /\ (!t. happ P t ==> (!h. happ P (h4/list/CONS h t))) ==> (!l. happ P l)
% Assm [h4s_lists_LENGTH0u_c1]: !t h. h4/list/LENGTH (h4/list/CONS h t) = h4/num/SUC (h4/list/LENGTH t)
% Assm [h4s_lists_LENGTH0u_c0]: h4/list/LENGTH h4/list/NIL = h4/num/0
% Assm [h4s_lists_NULL0u_c1]: !t h. ~h4/list/NULL (h4/list/CONS h t)
% Assm [h4s_arithmetics_MODu_u_TIMES2]: !n. h4/prim__rec/_3C h4/num/0 n ==> (!j k. h4/arithmetic/MOD (h4/arithmetic/_2A (h4/arithmetic/MOD j n) (h4/arithmetic/MOD k n)) n = h4/arithmetic/MOD (h4/arithmetic/_2A j k) n)
% Assm [h4s_arithmetics_MULTu_u_ASSOC]: !p n m. h4/arithmetic/_2A m (h4/arithmetic/_2A n p) = h4/arithmetic/_2A (h4/arithmetic/_2A m n) p
% Assm [h4s_arithmetics_LEFTu_u_ADDu_u_DISTRIB]: !p n m. h4/arithmetic/_2A p (h4/arithmetic/_2B m n) = h4/arithmetic/_2B (h4/arithmetic/_2A p m) (h4/arithmetic/_2A p n)
% Assm [h4s_arithmetics_MULTu_u_COMM]: !n m. h4/arithmetic/_2A m n = h4/arithmetic/_2A n m
% Assm [h4s_numerals_numeralu_u_distribu_c30]: !n. h4/arithmetic/_3E_3D h4/num/0 n <=> n = h4/num/0
% Assm [h4s_numerals_iZ0]: !x. h4/numeral/iZ x = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2A h4/num/0 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_SUBu_u_0u_c1]: !m. h4/arithmetic/_2D m h4/num/0 = m
% Assm [h4s_arithmetics_LESSu_u_0u_u_CASES]: !m. h4/num/0 = m \/ h4/prim__rec/_3C h4/num/0 m
% Assm [h4s_arithmetics_SUBu_u_0u_c0]: !m. h4/arithmetic/_2D h4/num/0 m = h4/num/0
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_bools_EXISTSu_u_ORu_u_THM]: !Q P. (?x. happ P x \/ happ Q x) <=> (?x. happ P x) \/ (?x. happ Q x)
% Assm [h4s_arithmetics_GREATERu_u_DEF]: !n m. h4/arithmetic/_3E m n <=> h4/prim__rec/_3C n m
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% 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_LESSu_u_ORu_u_EQ]: !n m. h4/arithmetic/_3C_3D m n <=> h4/prim__rec/_3C m n \/ m = n
% 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_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% 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_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_EVEN0u_c0]: h4/arithmetic/EVEN h4/num/0 <=> T
% Goal: h4/toto/cpn2num h4/toto/LESS = h4/num/0
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_Q1343017,TV_Q1343013]: ![V_f, V_g]: (![V_x]: s(TV_Q1343013,happ(s(t_fun(TV_Q1343017,TV_Q1343013),V_f),s(TV_Q1343017,V_x))) = s(TV_Q1343013,happ(s(t_fun(TV_Q1343017,TV_Q1343013),V_g),s(TV_Q1343017,V_x))) => s(t_fun(TV_Q1343017,TV_Q1343013),V_f) = s(t_fun(TV_Q1343017,TV_Q1343013),V_g))).
fof(ah4s_totos_LESSu_u_def, axiom, s(t_h4s_totos_cpn,h4s_totos_less) = s(t_h4s_totos_cpn,h4s_totos_num2cpn(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_totos_cpnu_u_BIJu_c0, axiom, ![V_a]: s(t_h4s_totos_cpn,h4s_totos_num2cpn(s(t_h4s_nums_num,h4s_totos_cpn2num(s(t_h4s_totos_cpn,V_a))))) = s(t_h4s_totos_cpn,V_a)).
fof(ah4s_totos_cpnu_u_BIJu_c1, axiom, ![V_r]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_r),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))) <=> s(t_h4s_nums_num,h4s_totos_cpn2num(s(t_h4s_totos_cpn,h4s_totos_num2cpn(s(t_h4s_nums_num,V_r))))) = s(t_h4s_nums_num,V_r))).
fof(ah4s_totos_num2cpnu_u_thmu_c0, axiom, s(t_h4s_totos_cpn,h4s_totos_num2cpn(s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_totos_cpn,h4s_totos_less)).
fof(ah4s_totos_cpnu_u_TYu_u_DEF, axiom, ![V_uu_0]: (![V_n]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))) => ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_h4s_nums_num,t_bool),V_uu_0),s(t_fun(t_h4s_totos_cpn,t_h4s_nums_num),V_rep)))))).
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_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_totos_EQUALu_u_def, axiom, s(t_h4s_totos_cpn,h4s_totos_equal) = s(t_h4s_totos_cpn,h4s_totos_num2cpn(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_totos_GREATERu_u_def, axiom, s(t_h4s_totos_cpn,h4s_totos_greater) = s(t_h4s_totos_cpn,h4s_totos_num2cpn(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))).
fof(ah4s_totos_cpn2numu_u_11, axiom, ![V_au_27, V_a]: (s(t_h4s_nums_num,h4s_totos_cpn2num(s(t_h4s_totos_cpn,V_a))) = s(t_h4s_nums_num,h4s_totos_cpn2num(s(t_h4s_totos_cpn,V_au_27))) <=> s(t_h4s_totos_cpn,V_a) = s(t_h4s_totos_cpn,V_au_27))).
fof(ah4s_totos_cpn2numu_u_ONTO, axiom, ![V_r]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_r),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))) <=> ?[V_a]: s(t_h4s_nums_num,V_r) = s(t_h4s_nums_num,h4s_totos_cpn2num(s(t_h4s_totos_cpn,V_a))))).
fof(ah4s_totos_num2cpnu_u_cpn2num, axiom, ![V_a]: s(t_h4s_totos_cpn,h4s_totos_num2cpn(s(t_h4s_nums_num,h4s_totos_cpn2num(s(t_h4s_totos_cpn,V_a))))) = s(t_h4s_totos_cpn,V_a)).
fof(ah4s_totos_cpn2numu_u_num2cpn, axiom, ![V_r]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_r),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))) <=> s(t_h4s_nums_num,h4s_totos_cpn2num(s(t_h4s_totos_cpn,h4s_totos_num2cpn(s(t_h4s_nums_num,V_r))))) = s(t_h4s_nums_num,V_r))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_totos_num2cpnu_u_thmu_c1, axiom, s(t_h4s_totos_cpn,h4s_totos_num2cpn(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_totos_cpn,h4s_totos_equal)).
fof(ah4s_totos_num2cpnu_u_ONTO, axiom, ![V_a]: ?[V_r]: (s(t_h4s_totos_cpn,V_a) = s(t_h4s_totos_cpn,h4s_totos_num2cpn(s(t_h4s_nums_num,V_r))) & p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_r),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))))).
fof(ah4s_totos_num2cpnu_u_11, axiom, ![V_ru_27, V_r]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_r),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))) => (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_ru_27),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))) => (s(t_h4s_totos_cpn,h4s_totos_num2cpn(s(t_h4s_nums_num,V_r))) = s(t_h4s_totos_cpn,h4s_totos_num2cpn(s(t_h4s_nums_num,V_ru_27))) <=> s(t_h4s_nums_num,V_r) = s(t_h4s_nums_num,V_ru_27))))).
fof(ah4s_totos_num2cpnu_u_thmu_c2, axiom, s(t_h4s_totos_cpn,h4s_totos_num2cpn(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_totos_cpn,h4s_totos_greater)).
fof(ah4s_numerals_numeralu_u_ltu_c0, axiom, ![V_n]: s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))) = s(t_bool,t)).
fof(ah4s_numerals_numeralu_u_distribu_c21, axiom, ![V_n]: s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))))) = s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_arithmetics_zero),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_MODEQu_u_NONZEROu_u_MODEQUALITY, axiom, ![V_n, V_m2, V_m1]: (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)))) => (p(s(t_bool,h4s_arithmetics_modeq(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m1),s(t_h4s_nums_num,V_m2)))) <=> s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_m1),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_m2),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_MODEQu_u_INTROu_u_CONG, axiom, ![V_n, V_e1, V_e0]: (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)))) => (p(s(t_bool,h4s_arithmetics_modeq(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_e0),s(t_h4s_nums_num,V_e1)))) => s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_e0),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_e1),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_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_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_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ANDu_u_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_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_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND, axiom, ![V_C, V_B, V_A]: (((p(s(t_bool,V_B)) & p(s(t_bool,V_C))) | p(s(t_bool,V_A))) <=> ((p(s(t_bool,V_B)) | p(s(t_bool,V_A))) & (p(s(t_bool,V_C)) | p(s(t_bool,V_A)))))).
fof(ah4s_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_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_arithmetics_DIVISION, 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,V_n)))) => ![V_k]: (s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,V_k),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_k),s(t_h4s_nums_num,V_n))))) & p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_k),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_arithmetics_ADDu_u_ASSOC, axiom, ![V_p, 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_arithmetics_u_2b(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_2b(s(t_h4s_nums_num,V_m),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_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_arithmetics_MODu_u_UNIQUE, axiom, ![V_r, V_n, V_k]: (?[V_q]: (s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_q),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_r))) & p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_r),s(t_h4s_nums_num,V_n))))) => s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_k),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_r))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_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_markers_Abbrevu_u_def, axiom, ![V_x]: s(t_bool,h4s_markers_abbrev(s(t_bool,V_x))) = s(t_bool,V_x)).
fof(ah4s_bools_EQu_u_IMPu_u_THM, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) & (p(s(t_bool,V_t2)) => p(s(t_bool,V_t1)))))).
fof(ah4s_markers_ACu_u_DEF, axiom, ![V_b2, V_b1]: (p(s(t_bool,h4s_markers_ac(s(t_bool,V_b1),s(t_bool,V_b2)))) <=> (p(s(t_bool,V_b1)) & p(s(t_bool,V_b2))))).
fof(ah4s_arithmetics_ADDu_u_COMM, 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_bools_LCOMMu_u_THM, axiom, ![TV_u_27a]: ![V_f]: (![V_x, V_y, V_z]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_x))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_y))),s(TV_u_27a,V_z))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))),s(TV_u_27a,V_z))) => (![V_x, V_y]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_y))),s(TV_u_27a,V_x))) => ![V_x, V_y, V_z]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_x))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_y))),s(TV_u_27a,V_z))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_y))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_x))),s(TV_u_27a,V_z)))))))).
fof(ah4s_arithmetics_ADDu_u_DIVu_u_ADDu_u_DIV, 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,V_n)))) => ![V_x, V_r]: s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_r))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,V_r),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_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_LESSu_u_DIVu_u_EQu_u_ZERO, axiom, ![V_r, V_n]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_r),s(t_h4s_nums_num,V_n)))) => s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,V_r),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0))).
fof(ah4s_arithmetics_DIVu_u_MULT, axiom, ![V_r, V_n]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_r),s(t_h4s_nums_num,V_n)))) => ![V_q]: s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_q),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_r))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_q))).
fof(ah4s_arithmetics_DA, axiom, ![V_n, V_k]: (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)))) => ?[V_r, V_q]: (s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_q),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_r))) & p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_r),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_primu_u_recs_DC, axiom, ![TV_u_27a]: ![V_a, V_R, V_P]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_a)))) & ![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_y)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))))) => ?[V_f]: (s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_a) & ![V_n]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n)))))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,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_f),s(t_h4s_nums_num,h4s_nums_suc(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_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_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_primu_u_recs_SIMPu_u_RECu_u_THMu_c1, axiom, ![TV_u_27a]: ![V_x, V_m, V_f]: s(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,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_SIMPu_u_RECu_u_THMu_c0, axiom, ![TV_u_27a]: ![V_x, V_f]: s(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_0))) = 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_arithmetics_MODu_u_EQu_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,V_n)))) => ![V_k]: s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_k),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0))).
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_MODULUSu_c0, axiom, ![V_x, 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,V_n)))) => s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,V_n))))).
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_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_arithmetics_DIVMODu_u_ID, 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,V_n)))) => (s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_n))) = 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_arithmetics_mod(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_arithmetics_MODu_u_PLUS, 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,V_n)))) => ![V_j, V_k]: s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_j),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_k),s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_j),s(t_h4s_nums_num,V_k))),s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_MODu_u_MOD, 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,V_n)))) => ![V_k]: s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_k),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_k),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_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => 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_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_arithmetics_DIVMODu_u_CALCu_c0, axiom, ![V_n, V_m]: (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)))) => s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_pairs_fst(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_divmod(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))))))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_arithmetics_DIVMODu_u_CORRECT, axiom, ![V_n, V_m, V_a]: (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)))) => s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_arithmetics_divmod(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,V_a),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_a),s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_arithmetics_MODu_u_TIMES, 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,V_n)))) => ![V_q, V_r]: s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_q),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_r))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_r),s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_MODu_u_MULT, axiom, ![V_r, V_n]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_r),s(t_h4s_nums_num,V_n)))) => ![V_q]: s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_q),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_r))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_r))).
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_NUMERALu_u_MULTu_u_EQu_u_DIVu_c0, axiom, ![V_z, V_y, V_x]: (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,V_x))))),s(t_h4s_nums_num,V_y))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_z))) <=> (s(t_h4s_nums_num,V_y) = s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_z))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_x))))))) & s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_z))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_x))))))) = s(t_h4s_nums_num,h4s_nums_0)))).
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_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_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_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_MULTu_u_EQu_u_DIV, axiom, ![V_z, V_y, V_x]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_x)))) => (s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,V_y))) = s(t_h4s_nums_num,V_z) <=> (s(t_h4s_nums_num,V_y) = s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,V_z),s(t_h4s_nums_num,V_x))) & s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_z),s(t_h4s_nums_num,V_x))) = s(t_h4s_nums_num,h4s_nums_0))))).
fof(ah4s_numerals_texpu_u_helpu_u_defu_c0, axiom, ![V_acc]: s(t_h4s_nums_num,h4s_numerals_texpu_u_help(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_acc))) = s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_acc)))).
fof(ah4s_predu_u_sets_chooseru_u_defu_c0, axiom, ![TV_u_27a]: ![V_s]: s(TV_u_27a,h4s_predu_u_sets_chooser(s(t_fun(TV_u_27a,t_bool),V_s),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,h4s_predu_u_sets_choice(s(t_fun(TV_u_27a,t_bool),V_s)))).
fof(ah4s_richu_u_lists_LENGTHu_u_NOTu_u_NULL, axiom, ![TV_u_27a]: ![V_l]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l)))))) <=> ~ (p(s(t_bool,h4s_lists_null(s(t_h4s_lists_list(TV_u_27a),V_l))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
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_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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_lists_NULL0u_c0, axiom, ![TV_u_27a]: p(s(t_bool,h4s_lists_null(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))))).
fof(ah4s_lists_listu_u_induction, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))) & ![V_t]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t)))) => ![V_h]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t)))))))) => ![V_l]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_lists_LENGTH0u_c1, axiom, ![TV_u_27a]: ![V_t, V_h]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_t)))))).
fof(ah4s_lists_LENGTH0u_c0, axiom, ![TV_u_27a]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_lists_NULL0u_c1, axiom, ![TV_u_27a]: ![V_t, V_h]: ~ (p(s(t_bool,h4s_lists_null(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t)))))))).
fof(ah4s_arithmetics_MODu_u_TIMES2, 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,V_n)))) => ![V_j, V_k]: s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_j),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_k),s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_j),s(t_h4s_nums_num,V_k))),s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_MULTu_u_ASSOC, axiom, ![V_p, 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_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(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_p)))).
fof(ah4s_arithmetics_LEFTu_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,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,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_MULTu_u_COMM, 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,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_numerals_numeralu_u_distribu_c30, axiom, ![V_n]: (p(s(t_bool,h4s_arithmetics_u_3eu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n)))) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0))).
fof(ah4s_numerals_iZ0, axiom, ![V_x]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,V_x))) = s(t_h4s_nums_num,V_x)).
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_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_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_SUBu_u_0u_c1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(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_LESSu_u_0u_u_CASES, axiom, ![V_m]: (s(t_h4s_nums_num,h4s_nums_0) = s(t_h4s_nums_num,V_m) | p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_SUBu_u_0u_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(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_bools_NOTu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ?[V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_EXISTSu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (?[V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_arithmetics_GREATERu_u_DEF, axiom, ![V_n, V_m]: s(t_bool,h4s_arithmetics_u_3e(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
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_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_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_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_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_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_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_EVEN0u_c0, axiom, s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,t)).
fof(ch4s_totos_cpn2numu_u_thmu_c0, conjecture, s(t_h4s_nums_num,h4s_totos_cpn2num(s(t_h4s_totos_cpn,h4s_totos_less))) = s(t_h4s_nums_num,h4s_nums_0)).
