%   ORIGINAL: h4/poly/POLY__DIFF__AUX__ISZERO
% 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/list/list__Axiom: !f1 f0. ?fn. fn h4/list/NIL = f0 /\ (!a0 a1. fn (h4/list/CONS a0 a1) = f1 a0 a1 (fn a1))
% Assm: h4/poly/POLY__ZERO: !p. h4/poly/poly p = h4/poly/poly h4/list/NIL <=> h4/list/EVERY (\c. c = h4/real/real__of__num h4/num/0) p
% Assm: h4/poly/poly__diff__aux__def_c1: !t n h. h4/poly/poly__diff__aux n (h4/list/CONS h t) = h4/list/CONS (h4/realax/real__mul (h4/real/real__of__num n) h) (h4/poly/poly__diff__aux (h4/num/SUC n) t)
% Assm: h4/poly/POLY__DIFF__LEMMA: !x n l. h4/lim/diffl (\x0. h4/realax/real__mul (h4/real/pow x0 (h4/num/SUC n)) (h4/poly/poly l x0)) (h4/realax/real__mul (h4/real/pow x n) (h4/poly/poly (h4/poly/poly__diff__aux (h4/num/SUC n) l) x)) x
% Assm: h4/poly/POLY__DIFF__AUX__MUL__LEMMA: !p n. h4/poly/poly (h4/poly/poly__diff__aux (h4/num/SUC n) p) = h4/poly/poly (h4/poly/poly__add (h4/poly/poly__diff__aux n p) p)
% Assm: h4/bool/TRUTH: T
% Assm: h4/poly/poly__diff__aux__def_c0: !n. h4/poly/poly__diff__aux n h4/list/NIL = h4/list/NIL
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/poly/POLY__DIFF__AUX__CMUL: !p n c. h4/poly/poly (h4/poly/poly__diff__aux n (h4/poly/_23_23 c p)) = h4/poly/poly (h4/poly/_23_23 c (h4/poly/poly__diff__aux n p))
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/poly/poly__def_c1: !x t h. h4/poly/poly (h4/list/CONS h t) x = h4/realax/real__add h (h4/realax/real__mul x (h4/poly/poly t x))
% Assm: h4/list/list__INDUCT: !P. P h4/list/NIL /\ (!t. P t ==> (!h. P (h4/list/CONS h t))) ==> (!l. P l)
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/poly/poly__diff__def: !l. h4/poly/diff l = h4/bool/COND (l = h4/list/NIL) h4/list/NIL (h4/poly/poly__diff__aux (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/list/TL l))
% Assm: h4/poly/POLY__DIFF__CLAUSES_c2: !t h. h4/poly/diff (h4/list/CONS h t) = h4/poly/poly__diff__aux (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) t
% Assm: h4/poly/POLY__DIFF__AUX__NEG: !p n. h4/poly/poly (h4/poly/poly__diff__aux n (h4/poly/poly__neg p)) = h4/poly/poly (h4/poly/poly__neg (h4/poly/poly__diff__aux n p))
% Assm: h4/bool/EQ__EXT: !g f. (!x. f x = g x) ==> f = g
% Assm: h4/poly/POLY__DIFF__AUX__ADD: !p2 p1 n. h4/poly/poly (h4/poly/poly__diff__aux n (h4/poly/poly__add p1 p2)) = h4/poly/poly (h4/poly/poly__add (h4/poly/poly__diff__aux n p1) (h4/poly/poly__diff__aux n p2))
% Assm: h4/poly/poly__def_c0: !x. h4/poly/poly h4/list/NIL x = h4/real/real__of__num h4/num/0
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/real/REAL__MUL__RZERO: !x. h4/realax/real__mul x (h4/real/real__of__num h4/num/0) = h4/real/real__of__num h4/num/0
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/list/TL0: !t h. h4/list/TL (h4/list/CONS h t) = t
% Assm: h4/list/NOT__CONS__NIL: !a1 a0. ~(h4/list/CONS a0 a1 = h4/list/NIL)
% Assm: h4/real/REAL__MUL__SYM: !y x. h4/realax/real__mul x y = h4/realax/real__mul y x
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/real/REAL__MUL__ASSOC: !z y x. h4/realax/real__mul x (h4/realax/real__mul y z) = h4/realax/real__mul (h4/realax/real__mul x y) z
% Assm: h4/poly/poly__neg__def: h4/poly/poly__neg = h4/poly/_23_23 (h4/realax/real__neg (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))
% Assm: h4/real/REAL__ADD__RDISTRIB: !z y x. h4/realax/real__mul (h4/realax/real__add x y) z = h4/realax/real__add (h4/realax/real__mul x z) (h4/realax/real__mul y z)
% Assm: h4/real/REAL__MUL__LID: !x. h4/realax/real__mul (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) x = x
% Assm: h4/poly/poly__add__def_c0: !l2. h4/poly/poly__add h4/list/NIL l2 = l2
% Assm: h4/poly/poly__add__def_c1: !t l2 h. h4/poly/poly__add (h4/list/CONS h t) l2 = h4/bool/COND (l2 = h4/list/NIL) (h4/list/CONS h t) (h4/list/CONS (h4/realax/real__add h (h4/list/HD l2)) (h4/poly/poly__add t (h4/list/TL l2)))
% Assm: h4/list/HD0: !t h. h4/list/HD (h4/list/CONS h t) = h
% Assm: h4/lim/DIFF__POW: !x n. h4/lim/diffl (\x0. h4/real/pow x0 n) (h4/realax/real__mul (h4/real/real__of__num n) (h4/real/pow x (h4/arithmetic/_2D n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))) x
% Assm: h4/lim/DIFF__CMUL: !x l f c. h4/lim/diffl f l x ==> h4/lim/diffl (\x0. h4/realax/real__mul c (f x0)) (h4/realax/real__mul c l) x
% Assm: h4/lim/DIFF__CONST: !x k. h4/lim/diffl (\x0. k) (h4/real/real__of__num h4/num/0) x
% Assm: h4/real/pow0_c1: !x n. h4/real/pow x (h4/num/SUC n) = h4/realax/real__mul x (h4/real/pow x n)
% Assm: h4/real/REAL__LDISTRIB: !z y x. h4/realax/real__mul x (h4/realax/real__add y z) = h4/realax/real__add (h4/realax/real__mul x y) (h4/realax/real__mul x z)
% Assm: h4/lim/DIFF__ADD: !x m l g f. h4/lim/diffl f l x /\ h4/lim/diffl g m x ==> h4/lim/diffl (\x0. h4/realax/real__add (f x0) (g x0)) (h4/realax/real__add l m) x
% Assm: h4/arithmetic/SUC__SUB1: !m. h4/arithmetic/_2D (h4/num/SUC m) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = m
% Assm: h4/real/REAL__OF__NUM__SUC: !n. h4/realax/real__add (h4/real/real__of__num n) (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) = h4/real/real__of__num (h4/num/SUC n)
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% 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/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/real/REAL__LE__NEG2: !y x. h4/real/real__lte (h4/realax/real__neg x) (h4/realax/real__neg y) <=> h4/real/real__lte y x
% Assm: h4/real/REAL__LE__RNEG: !y x. h4/real/real__lte x (h4/realax/real__neg y) <=> h4/real/real__lte (h4/realax/real__add x y) (h4/real/real__of__num h4/num/0)
% Assm: h4/real/real__lt: !y x. h4/realax/real__lt x y <=> ~h4/real/real__lte y x
% Assm: h4/real/REAL__LE__LNEG: !y x. h4/real/real__lte (h4/realax/real__neg x) y <=> h4/real/real__lte (h4/real/real__of__num h4/num/0) (h4/realax/real__add x y)
% Assm: h4/real/REAL__LE__ANTISYM: !y x. h4/real/real__lte x y /\ h4/real/real__lte y x <=> x = y
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/real/REAL__MUL__LZERO: !x. h4/realax/real__mul (h4/real/real__of__num h4/num/0) x = h4/real/real__of__num h4/num/0
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/list/EVERY__DEF_c0: !P. h4/list/EVERY P h4/list/NIL <=> T
% Assm: h4/list/EVERY__DEF_c1: !t h P. h4/list/EVERY P (h4/list/CONS h t) <=> P h /\ h4/list/EVERY P t
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/real/REAL__ADD: !n m. h4/realax/real__add (h4/real/real__of__num m) (h4/real/real__of__num n) = h4/real/real__of__num (h4/arithmetic/_2B m n)
% Assm: h4/real/REAL__MUL__RID: !x. h4/realax/real__mul x (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) = x
% Assm: h4/real/REAL__ADD__LID: !x. h4/realax/real__add (h4/real/real__of__num h4/num/0) x = x
% Assm: h4/real/REAL__OF__NUM__LE: !n m. h4/real/real__lte (h4/real/real__of__num m) (h4/real/real__of__num n) <=> h4/arithmetic/_3C_3D m n
% Assm: h4/numeral/numeral__distrib_c20: !n. h4/prim__rec/_3C n h4/num/0 <=> F
% Assm: h4/poly/POLY__ZERO__LEMMA: !t h. h4/poly/poly (h4/list/CONS h t) = h4/poly/poly h4/list/NIL ==> h = h4/real/real__of__num h4/num/0 /\ h4/poly/poly t = h4/poly/poly h4/list/NIL
% Assm: h4/arithmetic/LE_c0: !n. h4/arithmetic/_3C_3D n h4/num/0 <=> n = h4/num/0
% Assm: h4/arithmetic/NOT__LESS__EQUAL: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/prim__rec/_3C n m
% Assm: h4/arithmetic/ADD__EQ__0: !n m. h4/arithmetic/_2B m n = h4/num/0 <=> m = h4/num/0 /\ n = h4/num/0
% 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/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/poly/poly__cmul__def_c1: !t h c. h4/poly/_23_23 c (h4/list/CONS h t) = h4/list/CONS (h4/realax/real__mul c h) (h4/poly/_23_23 c t)
% Assm: h4/poly/poly__cmul__def_c0: !c. h4/poly/_23_23 c h4/list/NIL = h4/list/NIL
% Assm: h4/real/eq__ints_c0: !n m. h4/real/real__of__num n = h4/real/real__of__num m <=> 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/poly/poly__exp__def_c1: !p n. h4/poly/poly__exp p (h4/num/SUC n) = h4/poly/poly__mul p (h4/poly/poly__exp p n)
% Assm: h4/poly/POLY__DIFF__EXP: !p n. h4/poly/poly (h4/poly/diff (h4/poly/poly__exp p (h4/num/SUC n))) = h4/poly/poly (h4/poly/poly__mul (h4/poly/_23_23 (h4/real/real__of__num (h4/num/SUC n)) (h4/poly/poly__exp p n)) (h4/poly/diff p))
% Assm: h4/rich__list/EVERY__REPLICATE: !x n f. h4/list/EVERY f (h4/rich__list/REPLICATE n x) <=> n = h4/num/0 \/ f x
% Assm: h4/rich__list/every__count__list: !n P. h4/list/EVERY P (h4/rich__list/COUNT__LIST n) <=> (!m. h4/prim__rec/_3C m n ==> P m)
% Assm: h4/real/REAL__ADD__LDISTRIB: !z y x. h4/realax/real__mul x (h4/realax/real__add y z) = h4/realax/real__add (h4/realax/real__mul x y) (h4/realax/real__mul x z)
% Assm: h4/real/REAL__NEG__NEG: !x. h4/realax/real__neg (h4/realax/real__neg x) = x
% Assm: h4/real/REAL__LE__REFL: !x. h4/real/real__lte x x
% Assm: h4/real/REAL__NEG__ADD: !y x. h4/realax/real__neg (h4/realax/real__add x y) = h4/realax/real__add (h4/realax/real__neg x) (h4/realax/real__neg y)
% Assm: h4/real/REAL__ADD__RINV: !x. h4/realax/real__add x (h4/realax/real__neg x) = h4/real/real__of__num h4/num/0
% Assm: h4/real/REAL__ADD__RID: !x. h4/realax/real__add x (h4/real/real__of__num h4/num/0) = x
% Assm: h4/real/REAL__ADD__LINV: !x. h4/realax/real__add (h4/realax/real__neg x) x = h4/real/real__of__num h4/num/0
% Assm: h4/real/REAL__ADD__ASSOC: !z y x. h4/realax/real__add x (h4/realax/real__add y z) = h4/realax/real__add (h4/realax/real__add x y) z
% Assm: h4/real/REAL__ADD__SYM: !y x. h4/realax/real__add x y = h4/realax/real__add y x
% Assm: h4/real/REAL__MUL__LNEG: !y x. h4/realax/real__mul (h4/realax/real__neg x) y = h4/realax/real__neg (h4/realax/real__mul x y)
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/numeral/numeral__add_c0: !n. h4/numeral/iZ (h4/arithmetic/_2B h4/arithmetic/ZERO n) = n
% Assm: h4/numeral/numeral__add_c2: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT2 (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm: h4/numeral/numeral__distrib_c2: !n m. h4/arithmetic/_2B (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm: h4/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/rich__list/REPLICATE0_c1: !x n. h4/rich__list/REPLICATE (h4/num/SUC n) x = h4/list/CONS x (h4/rich__list/REPLICATE n x)
% Assm: h4/rich__list/REPLICATE0_c0: !x. h4/rich__list/REPLICATE h4/num/0 x = h4/list/NIL
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/num/NOT__SUC: !n. ~(h4/num/SUC n = h4/num/0)
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/real/REAL__OF__NUM__ADD: !n m. h4/realax/real__add (h4/real/real__of__num m) (h4/real/real__of__num n) = h4/real/real__of__num (h4/arithmetic/_2B m n)
% Assm: h4/poly/POLY__DIFF__MUL: !p2 p1. h4/poly/poly (h4/poly/diff (h4/poly/poly__mul p1 p2)) = h4/poly/poly (h4/poly/poly__add (h4/poly/poly__mul p1 (h4/poly/diff p2)) (h4/poly/poly__mul (h4/poly/diff p1) p2))
% Assm: h4/poly/POLY__MUL: !x p2 p1. h4/poly/poly (h4/poly/poly__mul p1 p2) x = h4/realax/real__mul (h4/poly/poly p1 x) (h4/poly/poly p2 x)
% Assm: h4/poly/POLY__CMUL: !x p c. h4/poly/poly (h4/poly/_23_23 c p) x = h4/realax/real__mul c (h4/poly/poly p x)
% Assm: h4/arithmetic/ADD1: !m. h4/num/SUC m = h4/arithmetic/_2B m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/poly/POLY__DIFF__CLAUSES_c1: !c. h4/poly/diff (h4/list/CONS c h4/list/NIL) = h4/list/NIL
% Assm: h4/poly/POLY__ADD: !x p2 p1. h4/poly/poly (h4/poly/poly__add p1 p2) x = h4/realax/real__add (h4/poly/poly p1 x) (h4/poly/poly p2 x)
% Assm: h4/poly/poly__exp__def_c0: !p. h4/poly/poly__exp p h4/num/0 = h4/list/CONS (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/list/NIL
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/rich__list/COUNT__LIST__def_c1: !n. h4/rich__list/COUNT__LIST (h4/num/SUC n) = h4/list/CONS h4/num/0 (h4/list/MAP h4/num/SUC (h4/rich__list/COUNT__LIST n))
% Assm: h4/rich__list/COUNT__LIST__def_c0: h4/rich__list/COUNT__LIST h4/num/0 = h4/list/NIL
% Assm: h4/list/EVERY__MAP: !l f P. h4/list/EVERY P (h4/list/MAP f l) <=> h4/list/EVERY (\x. P (f x)) l
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/numeral/numeral__distrib_c27: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Goal: !p n. h4/list/EVERY (\c. c = h4/real/real__of__num h4/num/0) (h4/poly/poly__diff__aux (h4/num/SUC n) p) <=> h4/list/EVERY (\c. c = h4/real/real__of__num h4/num/0) p
%   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_lists_listu_u_Axiom]: !f1 f0. ?fn. happ fn h4/list/NIL = f0 /\ (!a0 a1. happ fn (h4/list/CONS a0 a1) = happ (happ (happ f1 a0) a1) (happ fn a1))
% Assm [h4s_polys_POLYu_u_ZERO]: !_0. (!c. happ _0 c <=> c = h4/real/real__of__num h4/num/0) ==> (!p. h4/poly/poly p = h4/poly/poly h4/list/NIL <=> h4/list/EVERY _0 p)
% Assm [h4s_polys_polyu_u_diffu_u_auxu_u_defu_c1]: !t n h. h4/poly/poly__diff__aux n (h4/list/CONS h t) = h4/list/CONS (h4/realax/real__mul (h4/real/real__of__num n) h) (h4/poly/poly__diff__aux (happ h4/num/SUC n) t)
% Assm [h4s_polys_POLYu_u_DIFFu_u_LEMMA]: !_0. (!n l x0. happ (happ (happ _0 n) l) x0 = h4/realax/real__mul (h4/real/pow x0 (happ h4/num/SUC n)) (happ (h4/poly/poly l) x0)) ==> (!x n l. h4/lim/diffl (happ (happ _0 n) l) (h4/realax/real__mul (h4/real/pow x n) (happ (h4/poly/poly (h4/poly/poly__diff__aux (happ h4/num/SUC n) l)) x)) x)
% Assm [h4s_polys_POLYu_u_DIFFu_u_AUXu_u_MULu_u_LEMMA]: !p n. h4/poly/poly (h4/poly/poly__diff__aux (happ h4/num/SUC n) p) = h4/poly/poly (h4/poly/poly__add (h4/poly/poly__diff__aux n p) p)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_polys_polyu_u_diffu_u_auxu_u_defu_c0]: !n. h4/poly/poly__diff__aux n h4/list/NIL = h4/list/NIL
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_polys_POLYu_u_DIFFu_u_AUXu_u_CMUL]: !p n c. h4/poly/poly (h4/poly/poly__diff__aux n (happ (h4/poly/_23_23 c) p)) = h4/poly/poly (happ (h4/poly/_23_23 c) (h4/poly/poly__diff__aux n p))
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_polys_polyu_u_defu_c1]: !x t h. happ (h4/poly/poly (h4/list/CONS h t)) x = h4/realax/real__add h (h4/realax/real__mul x (happ (h4/poly/poly t) x))
% Assm [h4s_lists_listu_u_INDUCT]: !P. happ P h4/list/NIL /\ (!t. happ P t ==> (!h. happ P (h4/list/CONS h t))) ==> (!l. happ P l)
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_polys_polyu_u_diffu_u_def]: !l. ?v. (v <=> l = h4/list/NIL) /\ h4/poly/diff l = h4/bool/COND v h4/list/NIL (h4/poly/poly__diff__aux (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/list/TL l))
% Assm [h4s_polys_POLYu_u_DIFFu_u_CLAUSESu_c2]: !t h. h4/poly/diff (h4/list/CONS h t) = h4/poly/poly__diff__aux (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) t
% Assm [h4s_polys_POLYu_u_DIFFu_u_AUXu_u_NEG]: !p n. h4/poly/poly (h4/poly/poly__diff__aux n (happ h4/poly/poly__neg p)) = h4/poly/poly (happ h4/poly/poly__neg (h4/poly/poly__diff__aux n p))
% Assm [h4s_bools_EQu_u_EXT]: !g f. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_polys_POLYu_u_DIFFu_u_AUXu_u_ADD]: !p2 p1 n. h4/poly/poly (h4/poly/poly__diff__aux n (h4/poly/poly__add p1 p2)) = h4/poly/poly (h4/poly/poly__add (h4/poly/poly__diff__aux n p1) (h4/poly/poly__diff__aux n p2))
% Assm [h4s_polys_polyu_u_defu_c0]: !x. happ (h4/poly/poly h4/list/NIL) x = h4/real/real__of__num h4/num/0
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_reals_REALu_u_MULu_u_RZERO]: !x. h4/realax/real__mul x (h4/real/real__of__num h4/num/0) = h4/real/real__of__num h4/num/0
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_lists_TL0]: !t h. h4/list/TL (h4/list/CONS h t) = t
% Assm [h4s_lists_NOTu_u_CONSu_u_NIL]: !a1 a0. ~(h4/list/CONS a0 a1 = h4/list/NIL)
% Assm [h4s_reals_REALu_u_MULu_u_SYM]: !y x. h4/realax/real__mul x y = h4/realax/real__mul y x
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_reals_REALu_u_MULu_u_ASSOC]: !z y x. h4/realax/real__mul x (h4/realax/real__mul y z) = h4/realax/real__mul (h4/realax/real__mul x y) z
% Assm [h4s_polys_polyu_u_negu_u_def]: h4/poly/poly__neg = h4/poly/_23_23 (h4/realax/real__neg (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))
% Assm [h4s_reals_REALu_u_ADDu_u_RDISTRIB]: !z y x. h4/realax/real__mul (h4/realax/real__add x y) z = h4/realax/real__add (h4/realax/real__mul x z) (h4/realax/real__mul y z)
% Assm [h4s_reals_REALu_u_MULu_u_LID]: !x. h4/realax/real__mul (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) x = x
% Assm [h4s_polys_polyu_u_addu_u_defu_c0]: !l2. h4/poly/poly__add h4/list/NIL l2 = l2
% Assm [h4s_polys_polyu_u_addu_u_defu_c1]: !t l2 h. ?v. (v <=> l2 = h4/list/NIL) /\ h4/poly/poly__add (h4/list/CONS h t) l2 = h4/bool/COND v (h4/list/CONS h t) (h4/list/CONS (h4/realax/real__add h (h4/list/HD l2)) (h4/poly/poly__add t (h4/list/TL l2)))
% Assm [h4s_lists_HD0]: !t h. h4/list/HD (h4/list/CONS h t) = h
% Assm [h4s_lims_DIFFu_u_POW]: !_0. (!n x0. happ (happ _0 n) x0 = h4/real/pow x0 n) ==> (!x n. h4/lim/diffl (happ _0 n) (h4/realax/real__mul (h4/real/real__of__num n) (h4/real/pow x (h4/arithmetic/_2D n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))) x)
% Assm [h4s_lims_DIFFu_u_CMUL]: !_0. (!c f x0. happ (happ (happ _0 c) f) x0 = h4/realax/real__mul c (happ f x0)) ==> (!x l f c. h4/lim/diffl f l x ==> h4/lim/diffl (happ (happ _0 c) f) (h4/realax/real__mul c l) x)
% Assm [h4s_lims_DIFFu_u_CONST]: !_0. (!k x0. happ (happ _0 k) x0 = k) ==> (!x k. h4/lim/diffl (happ _0 k) (h4/real/real__of__num h4/num/0) x)
% Assm [h4s_reals_pow0u_c1]: !x n. h4/real/pow x (happ h4/num/SUC n) = h4/realax/real__mul x (h4/real/pow x n)
% Assm [h4s_reals_REALu_u_LDISTRIB]: !z y x. h4/realax/real__mul x (h4/realax/real__add y z) = h4/realax/real__add (h4/realax/real__mul x y) (h4/realax/real__mul x z)
% Assm [h4s_lims_DIFFu_u_ADD]: !_0. (!f g x0. happ (happ (happ _0 f) g) x0 = h4/realax/real__add (happ f x0) (happ g x0)) ==> (!x m l g f. h4/lim/diffl f l x /\ h4/lim/diffl g m x ==> h4/lim/diffl (happ (happ _0 f) g) (h4/realax/real__add l m) x)
% Assm [h4s_arithmetics_SUCu_u_SUB1]: !m. h4/arithmetic/_2D (happ h4/num/SUC m) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = m
% Assm [h4s_reals_REALu_u_OFu_u_NUMu_u_SUC]: !n. h4/realax/real__add (h4/real/real__of__num n) (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) = h4/real/real__of__num (happ h4/num/SUC n)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% 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_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_reals_REALu_u_LEu_u_NEG2]: !y x. h4/real/real__lte (h4/realax/real__neg x) (h4/realax/real__neg y) <=> h4/real/real__lte y x
% Assm [h4s_reals_REALu_u_LEu_u_RNEG]: !y x. h4/real/real__lte x (h4/realax/real__neg y) <=> h4/real/real__lte (h4/realax/real__add x y) (h4/real/real__of__num h4/num/0)
% Assm [h4s_reals_realu_u_lt]: !y x. h4/realax/real__lt x y <=> ~h4/real/real__lte y x
% Assm [h4s_reals_REALu_u_LEu_u_LNEG]: !y x. h4/real/real__lte (h4/realax/real__neg x) y <=> h4/real/real__lte (h4/real/real__of__num h4/num/0) (h4/realax/real__add x y)
% Assm [h4s_reals_REALu_u_LEu_u_ANTISYM]: !y x. h4/real/real__lte x y /\ h4/real/real__lte y x <=> x = y
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_reals_REALu_u_MULu_u_LZERO]: !x. h4/realax/real__mul (h4/real/real__of__num h4/num/0) x = h4/real/real__of__num h4/num/0
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_lists_EVERYu_u_DEFu_c0]: !P. h4/list/EVERY P h4/list/NIL <=> T
% Assm [h4s_lists_EVERYu_u_DEFu_c1]: !t h P. h4/list/EVERY P (h4/list/CONS h t) <=> happ P h /\ h4/list/EVERY P t
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_arithmetics_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_reals_REALu_u_ADD]: !n m. h4/realax/real__add (h4/real/real__of__num m) (h4/real/real__of__num n) = h4/real/real__of__num (h4/arithmetic/_2B m n)
% Assm [h4s_reals_REALu_u_MULu_u_RID]: !x. h4/realax/real__mul x (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) = x
% Assm [h4s_reals_REALu_u_ADDu_u_LID]: !x. h4/realax/real__add (h4/real/real__of__num h4/num/0) x = x
% Assm [h4s_reals_REALu_u_OFu_u_NUMu_u_LE]: !n m. h4/real/real__lte (h4/real/real__of__num m) (h4/real/real__of__num n) <=> h4/arithmetic/_3C_3D m n
% Assm [h4s_numerals_numeralu_u_distribu_c20]: !n. h4/prim__rec/_3C n h4/num/0 <=> F
% Assm [h4s_polys_POLYu_u_ZEROu_u_LEMMA]: !t h. h4/poly/poly (h4/list/CONS h t) = h4/poly/poly h4/list/NIL ==> h = h4/real/real__of__num h4/num/0 /\ h4/poly/poly t = h4/poly/poly h4/list/NIL
% Assm [h4s_arithmetics_LEu_c0]: !n. h4/arithmetic/_3C_3D n h4/num/0 <=> n = h4/num/0
% Assm [h4s_arithmetics_NOTu_u_LESSu_u_EQUAL]: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/prim__rec/_3C n m
% Assm [h4s_arithmetics_ADDu_u_EQu_u_0]: !n m. h4/arithmetic/_2B m n = h4/num/0 <=> m = h4/num/0 /\ n = h4/num/0
% Assm [h4s_primu_u_recs_numu_u_Axiom]: !f e. ?fn. happ fn h4/num/0 = e /\ (!n. happ fn (happ h4/num/SUC n) = happ (happ f n) (happ fn n))
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_polys_polyu_u_cmulu_u_defu_c1]: !t h c. happ (h4/poly/_23_23 c) (h4/list/CONS h t) = h4/list/CONS (h4/realax/real__mul c h) (happ (h4/poly/_23_23 c) t)
% Assm [h4s_polys_polyu_u_cmulu_u_defu_c0]: !c. happ (h4/poly/_23_23 c) h4/list/NIL = h4/list/NIL
% Assm [h4s_reals_equ_u_intsu_c0]: !n m. h4/real/real__of__num n = h4/real/real__of__num m <=> 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_polys_polyu_u_expu_u_defu_c1]: !p n. h4/poly/poly__exp p (happ h4/num/SUC n) = h4/poly/poly__mul p (h4/poly/poly__exp p n)
% Assm [h4s_polys_POLYu_u_DIFFu_u_EXP]: !p n. h4/poly/poly (h4/poly/diff (h4/poly/poly__exp p (happ h4/num/SUC n))) = h4/poly/poly (h4/poly/poly__mul (happ (h4/poly/_23_23 (h4/real/real__of__num (happ h4/num/SUC n))) (h4/poly/poly__exp p n)) (h4/poly/diff p))
% Assm [h4s_richu_u_lists_EVERYu_u_REPLICATE]: !x n f. h4/list/EVERY f (h4/rich__list/REPLICATE n x) <=> n = h4/num/0 \/ happ f x
% Assm [h4s_richu_u_lists_everyu_u_countu_u_list]: !n P. h4/list/EVERY P (h4/rich__list/COUNT__LIST n) <=> (!m. h4/prim__rec/_3C m n ==> happ P m)
% Assm [h4s_reals_REALu_u_ADDu_u_LDISTRIB]: !z y x. h4/realax/real__mul x (h4/realax/real__add y z) = h4/realax/real__add (h4/realax/real__mul x y) (h4/realax/real__mul x z)
% Assm [h4s_reals_REALu_u_NEGu_u_NEG]: !x. h4/realax/real__neg (h4/realax/real__neg x) = x
% Assm [h4s_reals_REALu_u_LEu_u_REFL]: !x. h4/real/real__lte x x
% Assm [h4s_reals_REALu_u_NEGu_u_ADD]: !y x. h4/realax/real__neg (h4/realax/real__add x y) = h4/realax/real__add (h4/realax/real__neg x) (h4/realax/real__neg y)
% Assm [h4s_reals_REALu_u_ADDu_u_RINV]: !x. h4/realax/real__add x (h4/realax/real__neg x) = h4/real/real__of__num h4/num/0
% Assm [h4s_reals_REALu_u_ADDu_u_RID]: !x. h4/realax/real__add x (h4/real/real__of__num h4/num/0) = x
% Assm [h4s_reals_REALu_u_ADDu_u_LINV]: !x. h4/realax/real__add (h4/realax/real__neg x) x = h4/real/real__of__num h4/num/0
% Assm [h4s_reals_REALu_u_ADDu_u_ASSOC]: !z y x. h4/realax/real__add x (h4/realax/real__add y z) = h4/realax/real__add (h4/realax/real__add x y) z
% Assm [h4s_reals_REALu_u_ADDu_u_SYM]: !y x. h4/realax/real__add x y = h4/realax/real__add y x
% Assm [h4s_reals_REALu_u_MULu_u_LNEG]: !y x. h4/realax/real__mul (h4/realax/real__neg x) y = h4/realax/real__neg (h4/realax/real__mul x y)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_nums_INDUCTION]: !P. happ P h4/num/0 /\ (!n. happ P n ==> happ P (happ h4/num/SUC n)) ==> (!n. happ P n)
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_numerals_numeralu_u_addu_c0]: !n. h4/numeral/iZ (h4/arithmetic/_2B h4/arithmetic/ZERO n) = n
% Assm [h4s_numerals_numeralu_u_addu_c2]: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT2 (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm [h4s_numerals_numeralu_u_distribu_c2]: !n m. h4/arithmetic/_2B (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm [h4s_richu_u_lists_REPLICATE0u_c1]: !x n. h4/rich__list/REPLICATE (happ h4/num/SUC n) x = h4/list/CONS x (h4/rich__list/REPLICATE n x)
% Assm [h4s_richu_u_lists_REPLICATE0u_c0]: !x. h4/rich__list/REPLICATE h4/num/0 x = h4/list/NIL
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_nums_NOTu_u_SUC]: !n. ~(happ h4/num/SUC n = h4/num/0)
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_reals_REALu_u_OFu_u_NUMu_u_ADD]: !n m. h4/realax/real__add (h4/real/real__of__num m) (h4/real/real__of__num n) = h4/real/real__of__num (h4/arithmetic/_2B m n)
% Assm [h4s_polys_POLYu_u_DIFFu_u_MUL]: !p2 p1. h4/poly/poly (h4/poly/diff (h4/poly/poly__mul p1 p2)) = h4/poly/poly (h4/poly/poly__add (h4/poly/poly__mul p1 (h4/poly/diff p2)) (h4/poly/poly__mul (h4/poly/diff p1) p2))
% Assm [h4s_polys_POLYu_u_MUL]: !x p2 p1. happ (h4/poly/poly (h4/poly/poly__mul p1 p2)) x = h4/realax/real__mul (happ (h4/poly/poly p1) x) (happ (h4/poly/poly p2) x)
% Assm [h4s_polys_POLYu_u_CMUL]: !x p c. happ (h4/poly/poly (happ (h4/poly/_23_23 c) p)) x = h4/realax/real__mul c (happ (h4/poly/poly p) x)
% Assm [h4s_arithmetics_ADD1]: !m. happ h4/num/SUC m = h4/arithmetic/_2B m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_polys_POLYu_u_DIFFu_u_CLAUSESu_c1]: !c. h4/poly/diff (h4/list/CONS c h4/list/NIL) = h4/list/NIL
% Assm [h4s_polys_POLYu_u_ADD]: !x p2 p1. happ (h4/poly/poly (h4/poly/poly__add p1 p2)) x = h4/realax/real__add (happ (h4/poly/poly p1) x) (happ (h4/poly/poly p2) x)
% Assm [h4s_polys_polyu_u_expu_u_defu_c0]: !p. h4/poly/poly__exp p h4/num/0 = h4/list/CONS (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/list/NIL
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_richu_u_lists_COUNTu_u_LISTu_u_defu_c1]: !n. h4/rich__list/COUNT__LIST (happ h4/num/SUC n) = h4/list/CONS h4/num/0 (h4/list/MAP h4/num/SUC (h4/rich__list/COUNT__LIST n))
% Assm [h4s_richu_u_lists_COUNTu_u_LISTu_u_defu_c0]: h4/rich__list/COUNT__LIST h4/num/0 = h4/list/NIL
% Assm [h4s_lists_EVERYu_u_MAP]: !_0. (!P f x. happ (happ (happ _0 P) f) x <=> happ P (happ f x)) ==> (!l f P. h4/list/EVERY P (h4/list/MAP f l) <=> h4/list/EVERY (happ (happ _0 P) f) l)
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_numerals_numeralu_u_distribu_c27]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Goal: !_0. (!c. happ _0 c <=> c = h4/real/real__of__num h4/num/0) ==> (!p n. h4/list/EVERY _0 (h4/poly/poly__diff__aux (happ h4/num/SUC n) p) <=> h4/list/EVERY _0 p)
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_Q1306606,TV_Q1306602]: ![V_f, V_g]: (![V_x]: s(TV_Q1306602,happ(s(t_fun(TV_Q1306606,TV_Q1306602),V_f),s(TV_Q1306606,V_x))) = s(TV_Q1306602,happ(s(t_fun(TV_Q1306606,TV_Q1306602),V_g),s(TV_Q1306606,V_x))) => s(t_fun(TV_Q1306606,TV_Q1306602),V_f) = s(t_fun(TV_Q1306606,TV_Q1306602),V_g))).
fof(ah4s_lists_listu_u_Axiom, axiom, ![TV_u_27b,TV_u_27a]: ![V_f1, V_f0]: ?[V_fn]: (s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(TV_u_27b,V_f0) & ![V_a0, V_a1]: s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a0),s(t_h4s_lists_list(TV_u_27a),V_a1))))) = s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,TV_u_27b))),V_f1),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1))),s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_lists_list(TV_u_27a),V_a1))))))).
fof(ah4s_polys_POLYu_u_ZERO, axiom, ![V_uu_0]: (![V_c]: (p(s(t_bool,happ(s(t_fun(t_h4s_realaxs_real,t_bool),V_uu_0),s(t_h4s_realaxs_real,V_c)))) <=> s(t_h4s_realaxs_real,V_c) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) => ![V_p]: (s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),V_p))) = s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_nil))) <=> p(s(t_bool,h4s_lists_every(s(t_fun(t_h4s_realaxs_real,t_bool),V_uu_0),s(t_h4s_lists_list(t_h4s_realaxs_real),V_p))))))).
fof(ah4s_polys_polyu_u_diffu_u_auxu_u_defu_c1, axiom, ![V_t, V_n, V_h]: s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_diffu_u_aux(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_cons(s(t_h4s_realaxs_real,V_h),s(t_h4s_lists_list(t_h4s_realaxs_real),V_t))))) = s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_cons(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_n))),s(t_h4s_realaxs_real,V_h))),s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_diffu_u_aux(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))),s(t_h4s_lists_list(t_h4s_realaxs_real),V_t)))))).
fof(ah4s_polys_POLYu_u_DIFFu_u_LEMMA, axiom, ![V_uu_0]: (![V_n, V_l, V_x0]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_lists_list(t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_h4s_nums_num,V_n))),s(t_h4s_lists_list(t_h4s_realaxs_real),V_l))),s(t_h4s_realaxs_real,V_x0))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_reals_pow(s(t_h4s_realaxs_real,V_x0),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),V_l))),s(t_h4s_realaxs_real,V_x0))))) => ![V_x, V_n, V_l]: p(s(t_bool,h4s_lims_diffl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_lists_list(t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_h4s_nums_num,V_n))),s(t_h4s_lists_list(t_h4s_realaxs_real),V_l))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_reals_pow(s(t_h4s_realaxs_real,V_x),s(t_h4s_nums_num,V_n))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_diffu_u_aux(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))),s(t_h4s_lists_list(t_h4s_realaxs_real),V_l))))),s(t_h4s_realaxs_real,V_x))))),s(t_h4s_realaxs_real,V_x)))))).
fof(ah4s_polys_POLYu_u_DIFFu_u_AUXu_u_MULu_u_LEMMA, axiom, ![V_p, V_n]: s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_diffu_u_aux(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))),s(t_h4s_lists_list(t_h4s_realaxs_real),V_p))))) = s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_add(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_diffu_u_aux(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(t_h4s_realaxs_real),V_p))),s(t_h4s_lists_list(t_h4s_realaxs_real),V_p)))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_polys_polyu_u_diffu_u_auxu_u_defu_c0, axiom, ![V_n]: s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_diffu_u_aux(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_nil))) = s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_nil)).
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_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_polys_POLYu_u_DIFFu_u_AUXu_u_CMUL, axiom, ![V_p, V_n, V_c]: s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_diffu_u_aux(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(t_h4s_realaxs_real),happ(s(t_fun(t_h4s_lists_list(t_h4s_realaxs_real),t_h4s_lists_list(t_h4s_realaxs_real)),h4s_polys_u_23u_23(s(t_h4s_realaxs_real,V_c))),s(t_h4s_lists_list(t_h4s_realaxs_real),V_p))))))) = s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),happ(s(t_fun(t_h4s_lists_list(t_h4s_realaxs_real),t_h4s_lists_list(t_h4s_realaxs_real)),h4s_polys_u_23u_23(s(t_h4s_realaxs_real,V_c))),s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_diffu_u_aux(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(t_h4s_realaxs_real),V_p)))))))).
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_polys_polyu_u_defu_c1, axiom, ![V_x, V_t, V_h]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_cons(s(t_h4s_realaxs_real,V_h),s(t_h4s_lists_list(t_h4s_realaxs_real),V_t))))),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_h),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),V_t))),s(t_h4s_realaxs_real,V_x)))))))).
fof(ah4s_lists_listu_u_INDUCT, 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_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_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_polys_polyu_u_diffu_u_def, axiom, ![V_l]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_lists_list(t_h4s_realaxs_real),V_l) = s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_nil)) & s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_diff(s(t_h4s_lists_list(t_h4s_realaxs_real),V_l))) = s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_nil),s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_diffu_u_aux(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_lists_list(t_h4s_realaxs_real),h4s_lists_tl(s(t_h4s_lists_list(t_h4s_realaxs_real),V_l))))))))).
fof(ah4s_polys_POLYu_u_DIFFu_u_CLAUSESu_c2, axiom, ![V_t, V_h]: s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_diff(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_cons(s(t_h4s_realaxs_real,V_h),s(t_h4s_lists_list(t_h4s_realaxs_real),V_t))))) = s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_diffu_u_aux(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_lists_list(t_h4s_realaxs_real),V_t)))).
fof(ah4s_polys_POLYu_u_DIFFu_u_AUXu_u_NEG, axiom, ![V_p, V_n]: s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_diffu_u_aux(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(t_h4s_realaxs_real),happ(s(t_fun(t_h4s_lists_list(t_h4s_realaxs_real),t_h4s_lists_list(t_h4s_realaxs_real)),h4s_polys_polyu_u_neg),s(t_h4s_lists_list(t_h4s_realaxs_real),V_p))))))) = s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),happ(s(t_fun(t_h4s_lists_list(t_h4s_realaxs_real),t_h4s_lists_list(t_h4s_realaxs_real)),h4s_polys_polyu_u_neg),s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_diffu_u_aux(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(t_h4s_realaxs_real),V_p)))))))).
fof(ah4s_bools_EQu_u_EXT, axiom, ![TV_u_27a,TV_u_27b]: ![V_g, V_f]: (![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))) => s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g))).
fof(ah4s_polys_POLYu_u_DIFFu_u_AUXu_u_ADD, axiom, ![V_p2, V_p1, V_n]: s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_diffu_u_aux(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_add(s(t_h4s_lists_list(t_h4s_realaxs_real),V_p1),s(t_h4s_lists_list(t_h4s_realaxs_real),V_p2))))))) = s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_add(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_diffu_u_aux(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(t_h4s_realaxs_real),V_p1))),s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_diffu_u_aux(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(t_h4s_realaxs_real),V_p2)))))))).
fof(ah4s_polys_polyu_u_defu_c0, axiom, ![V_x]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_nil))),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
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_reals_REALu_u_MULu_u_RZERO, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
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_lists_TL0, axiom, ![TV_u_27a]: ![V_t, V_h]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_tl(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_lists_list(TV_u_27a),V_t)).
fof(ah4s_lists_NOTu_u_CONSu_u_NIL, axiom, ![TV_u_27a]: ![V_a1, V_a0]: ~ (s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a0),s(t_h4s_lists_list(TV_u_27a),V_a1))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))).
fof(ah4s_reals_REALu_u_MULu_u_SYM, axiom, ![V_y, V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_x)))).
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_reals_REALu_u_MULu_u_ASSOC, axiom, ![V_z, V_y, V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_z))))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))),s(t_h4s_realaxs_real,V_z)))).
fof(ah4s_polys_polyu_u_negu_u_def, axiom, s(t_fun(t_h4s_lists_list(t_h4s_realaxs_real),t_h4s_lists_list(t_h4s_realaxs_real)),h4s_polys_polyu_u_neg) = s(t_fun(t_h4s_lists_list(t_h4s_realaxs_real),t_h4s_lists_list(t_h4s_realaxs_real)),h4s_polys_u_23u_23(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(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_reals_REALu_u_ADDu_u_RDISTRIB, axiom, ![V_z, V_y, V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))),s(t_h4s_realaxs_real,V_z))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_z))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_z)))))).
fof(ah4s_reals_REALu_u_MULu_u_LID, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(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_realaxs_real,V_x))) = s(t_h4s_realaxs_real,V_x)).
fof(ah4s_polys_polyu_u_addu_u_defu_c0, axiom, ![V_l2]: s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_add(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_nil),s(t_h4s_lists_list(t_h4s_realaxs_real),V_l2))) = s(t_h4s_lists_list(t_h4s_realaxs_real),V_l2)).
fof(ah4s_polys_polyu_u_addu_u_defu_c1, axiom, ![V_t, V_l2, V_h]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_lists_list(t_h4s_realaxs_real),V_l2) = s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_nil)) & s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_add(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_cons(s(t_h4s_realaxs_real,V_h),s(t_h4s_lists_list(t_h4s_realaxs_real),V_t))),s(t_h4s_lists_list(t_h4s_realaxs_real),V_l2))) = s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_cons(s(t_h4s_realaxs_real,V_h),s(t_h4s_lists_list(t_h4s_realaxs_real),V_t))),s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_cons(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_h),s(t_h4s_realaxs_real,h4s_lists_hd(s(t_h4s_lists_list(t_h4s_realaxs_real),V_l2))))),s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_add(s(t_h4s_lists_list(t_h4s_realaxs_real),V_t),s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_tl(s(t_h4s_lists_list(t_h4s_realaxs_real),V_l2))))))))))).
fof(ah4s_lists_HD0, axiom, ![TV_u_27a]: ![V_t, V_h]: s(TV_u_27a,h4s_lists_hd(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(TV_u_27a,V_h)).
fof(ah4s_lims_DIFFu_u_POW, axiom, ![V_uu_0]: (![V_n, V_x0]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),V_uu_0),s(t_h4s_nums_num,V_n))),s(t_h4s_realaxs_real,V_x0))) = s(t_h4s_realaxs_real,h4s_reals_pow(s(t_h4s_realaxs_real,V_x0),s(t_h4s_nums_num,V_n))) => ![V_x, V_n]: p(s(t_bool,h4s_lims_diffl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),V_uu_0),s(t_h4s_nums_num,V_n))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_n))),s(t_h4s_realaxs_real,h4s_reals_pow(s(t_h4s_realaxs_real,V_x),s(t_h4s_nums_num,h4s_arithmetics_u_2d(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_realaxs_real,V_x)))))).
fof(ah4s_lims_DIFFu_u_CMUL, axiom, ![V_uu_0]: (![V_c, V_f, V_x0]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_h4s_realaxs_real,V_c))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_h4s_realaxs_real,V_x0))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_c),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x0))))) => ![V_x, V_l, V_f, V_c]: (p(s(t_bool,h4s_lims_diffl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,V_x)))) => p(s(t_bool,h4s_lims_diffl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_h4s_realaxs_real,V_c))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_c),s(t_h4s_realaxs_real,V_l))),s(t_h4s_realaxs_real,V_x))))))).
fof(ah4s_lims_DIFFu_u_CONST, axiom, ![V_uu_0]: (![V_k, V_x0]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),V_uu_0),s(t_h4s_realaxs_real,V_k))),s(t_h4s_realaxs_real,V_x0))) = s(t_h4s_realaxs_real,V_k) => ![V_x, V_k]: p(s(t_bool,h4s_lims_diffl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),V_uu_0),s(t_h4s_realaxs_real,V_k))),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_x)))))).
fof(ah4s_reals_pow0u_c1, axiom, ![V_x, V_n]: s(t_h4s_realaxs_real,h4s_reals_pow(s(t_h4s_realaxs_real,V_x),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_reals_pow(s(t_h4s_realaxs_real,V_x),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_reals_REALu_u_LDISTRIB, axiom, ![V_z, V_y, V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_z))))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_z)))))).
fof(ah4s_lims_DIFFu_u_ADD, axiom, ![V_uu_0]: (![V_f, V_g, V_x0]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g))),s(t_h4s_realaxs_real,V_x0))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_x0))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g),s(t_h4s_realaxs_real,V_x0))))) => ![V_x, V_m, V_l, V_g, V_f]: ((p(s(t_bool,h4s_lims_diffl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f),s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,V_x)))) & p(s(t_bool,h4s_lims_diffl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g),s(t_h4s_realaxs_real,V_m),s(t_h4s_realaxs_real,V_x))))) => p(s(t_bool,h4s_lims_diffl(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real)),happ(s(t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real))),V_uu_0),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_f))),s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),V_g))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_l),s(t_h4s_realaxs_real,V_m))),s(t_h4s_realaxs_real,V_x))))))).
fof(ah4s_arithmetics_SUCu_u_SUB1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),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_reals_REALu_u_OFu_u_NUMu_u_SUC, axiom, ![V_n]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_n))),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(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_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_dcu_u_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_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_reals_REALu_u_LEu_u_NEG2, axiom, ![V_y, V_x]: s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_y))))) = s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_x)))).
fof(ah4s_reals_REALu_u_LEu_u_RNEG, axiom, ![V_y, V_x]: s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_y))))) = s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))).
fof(ah4s_reals_realu_u_lt, axiom, ![V_y, V_x]: (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))) <=> ~ (p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_x))))))).
fof(ah4s_reals_REALu_u_LEu_u_LNEG, axiom, ![V_y, V_x]: s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,V_y))) = s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))))).
fof(ah4s_reals_REALu_u_LEu_u_ANTISYM, axiom, ![V_y, V_x]: ((p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))) & p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_x))))) <=> s(t_h4s_realaxs_real,V_x) = s(t_h4s_realaxs_real,V_y))).
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_reals_REALu_u_MULu_u_LZERO, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c0, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) & p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) | ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_lists_EVERYu_u_DEFu_c0, axiom, ![TV_u_27a]: ![V_P]: s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_bool,t)).
fof(ah4s_lists_EVERYu_u_DEFu_c1, axiom, ![TV_u_27a]: ![V_t, V_h, V_P]: (p(s(t_bool,h4s_lists_every(s(t_fun(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)))))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_h)))) & p(s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t))))))).
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_arithmetics_ZEROu_u_LESSu_u_EQ, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))))).
fof(ah4s_reals_REALu_u_ADD, axiom, ![V_n, V_m]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_m))),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_n))))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(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_reals_REALu_u_MULu_u_RID, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(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_realaxs_real,V_x)).
fof(ah4s_reals_REALu_u_ADDu_u_LID, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,V_x)).
fof(ah4s_reals_REALu_u_OFu_u_NUMu_u_LE, axiom, ![V_n, V_m]: s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_m))),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_n))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))).
fof(ah4s_numerals_numeralu_u_distribu_c20, axiom, ![V_n]: s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,f)).
fof(ah4s_polys_POLYu_u_ZEROu_u_LEMMA, axiom, ![V_t, V_h]: (s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_cons(s(t_h4s_realaxs_real,V_h),s(t_h4s_lists_list(t_h4s_realaxs_real),V_t))))) = s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_nil))) => (s(t_h4s_realaxs_real,V_h) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) & s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),V_t))) = s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_nil)))))).
fof(ah4s_arithmetics_LEu_c0, axiom, ![V_n]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0)))) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0))).
fof(ah4s_arithmetics_NOTu_u_LESSu_u_EQUAL, 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_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_ADDu_u_EQu_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,h4s_nums_0) <=> (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0) & s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)))).
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,happ(s(t_fun(t_h4s_nums_num,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_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_polys_polyu_u_cmulu_u_defu_c1, axiom, ![V_t, V_h, V_c]: s(t_h4s_lists_list(t_h4s_realaxs_real),happ(s(t_fun(t_h4s_lists_list(t_h4s_realaxs_real),t_h4s_lists_list(t_h4s_realaxs_real)),h4s_polys_u_23u_23(s(t_h4s_realaxs_real,V_c))),s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_cons(s(t_h4s_realaxs_real,V_h),s(t_h4s_lists_list(t_h4s_realaxs_real),V_t))))) = s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_cons(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_c),s(t_h4s_realaxs_real,V_h))),s(t_h4s_lists_list(t_h4s_realaxs_real),happ(s(t_fun(t_h4s_lists_list(t_h4s_realaxs_real),t_h4s_lists_list(t_h4s_realaxs_real)),h4s_polys_u_23u_23(s(t_h4s_realaxs_real,V_c))),s(t_h4s_lists_list(t_h4s_realaxs_real),V_t)))))).
fof(ah4s_polys_polyu_u_cmulu_u_defu_c0, axiom, ![V_c]: s(t_h4s_lists_list(t_h4s_realaxs_real),happ(s(t_fun(t_h4s_lists_list(t_h4s_realaxs_real),t_h4s_lists_list(t_h4s_realaxs_real)),h4s_polys_u_23u_23(s(t_h4s_realaxs_real,V_c))),s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_nil))) = s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_nil)).
fof(ah4s_reals_equ_u_intsu_c0, axiom, ![V_n, V_m]: (s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_n))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_m))) <=> 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_polys_polyu_u_expu_u_defu_c1, axiom, ![V_p, V_n]: s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_exp(s(t_h4s_lists_list(t_h4s_realaxs_real),V_p),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))))) = s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_mul(s(t_h4s_lists_list(t_h4s_realaxs_real),V_p),s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_exp(s(t_h4s_lists_list(t_h4s_realaxs_real),V_p),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_polys_POLYu_u_DIFFu_u_EXP, axiom, ![V_p, V_n]: s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_diff(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_exp(s(t_h4s_lists_list(t_h4s_realaxs_real),V_p),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))))))))) = s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_mul(s(t_h4s_lists_list(t_h4s_realaxs_real),happ(s(t_fun(t_h4s_lists_list(t_h4s_realaxs_real),t_h4s_lists_list(t_h4s_realaxs_real)),h4s_polys_u_23u_23(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))))))),s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_exp(s(t_h4s_lists_list(t_h4s_realaxs_real),V_p),s(t_h4s_nums_num,V_n))))),s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_diff(s(t_h4s_lists_list(t_h4s_realaxs_real),V_p)))))))).
fof(ah4s_richu_u_lists_EVERYu_u_REPLICATE, axiom, ![TV_u_27a]: ![V_x, V_n, V_f]: (p(s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_f),s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_replicate(s(t_h4s_nums_num,V_n),s(TV_u_27a,V_x)))))) <=> (s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))))).
fof(ah4s_richu_u_lists_everyu_u_countu_u_list, axiom, ![V_n, V_P]: (p(s(t_bool,h4s_lists_every(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_lists_list(t_h4s_nums_num),h4s_richu_u_lists_countu_u_list(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_reals_REALu_u_ADDu_u_LDISTRIB, axiom, ![V_z, V_y, V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_z))))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_z)))))).
fof(ah4s_reals_REALu_u_NEGu_u_NEG, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_x))))) = s(t_h4s_realaxs_real,V_x)).
fof(ah4s_reals_REALu_u_LEu_u_REFL, axiom, ![V_x]: p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_x))))).
fof(ah4s_reals_REALu_u_NEGu_u_ADD, axiom, ![V_y, V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_y)))))).
fof(ah4s_reals_REALu_u_ADDu_u_RINV, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_x))))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_reals_REALu_u_ADDu_u_RID, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))) = s(t_h4s_realaxs_real,V_x)).
fof(ah4s_reals_REALu_u_ADDu_u_LINV, axiom, ![V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_reals_REALu_u_ADDu_u_ASSOC, axiom, ![V_z, V_y, V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_z))))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))),s(t_h4s_realaxs_real,V_z)))).
fof(ah4s_reals_REALu_u_ADDu_u_SYM, axiom, ![V_y, V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_y),s(t_h4s_realaxs_real,V_x)))).
fof(ah4s_reals_REALu_u_MULu_u_LNEG, axiom, ![V_y, V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,V_y))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y)))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_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,happ(s(t_fun(t_h4s_nums_num,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_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_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_numerals_numeralu_u_addu_c0, axiom, ![V_n]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_numerals_numeralu_u_addu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_m))))))) = s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_numerals_iz(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_numerals_numeralu_u_distribu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_numerals_iz(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_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_richu_u_lists_REPLICATE0u_c1, axiom, ![TV_u_27a]: ![V_x, V_n]: s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_replicate(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_x))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_replicate(s(t_h4s_nums_num,V_n),s(TV_u_27a,V_x)))))).
fof(ah4s_richu_u_lists_REPLICATE0u_c0, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_replicate(s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,V_x))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)).
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_nums_NOTu_u_SUC, axiom, ![V_n]: ~ (s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_reals_REALu_u_OFu_u_NUMu_u_ADD, axiom, ![V_n, V_m]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_m))),s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,V_n))))) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(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_polys_POLYu_u_DIFFu_u_MUL, axiom, ![V_p2, V_p1]: s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_diff(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_mul(s(t_h4s_lists_list(t_h4s_realaxs_real),V_p1),s(t_h4s_lists_list(t_h4s_realaxs_real),V_p2))))))) = s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_add(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_mul(s(t_h4s_lists_list(t_h4s_realaxs_real),V_p1),s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_diff(s(t_h4s_lists_list(t_h4s_realaxs_real),V_p2))))),s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_mul(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_diff(s(t_h4s_lists_list(t_h4s_realaxs_real),V_p1))),s(t_h4s_lists_list(t_h4s_realaxs_real),V_p2)))))))).
fof(ah4s_polys_POLYu_u_MUL, axiom, ![V_x, V_p2, V_p1]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_mul(s(t_h4s_lists_list(t_h4s_realaxs_real),V_p1),s(t_h4s_lists_list(t_h4s_realaxs_real),V_p2))))),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),V_p1))),s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),V_p2))),s(t_h4s_realaxs_real,V_x)))))).
fof(ah4s_polys_POLYu_u_CMUL, axiom, ![V_x, V_p, V_c]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),happ(s(t_fun(t_h4s_lists_list(t_h4s_realaxs_real),t_h4s_lists_list(t_h4s_realaxs_real)),h4s_polys_u_23u_23(s(t_h4s_realaxs_real,V_c))),s(t_h4s_lists_list(t_h4s_realaxs_real),V_p))))),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_c),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),V_p))),s(t_h4s_realaxs_real,V_x)))))).
fof(ah4s_arithmetics_ADD1, axiom, ![V_m]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))).
fof(ah4s_polys_POLYu_u_DIFFu_u_CLAUSESu_c1, axiom, ![V_c]: s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_diff(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_cons(s(t_h4s_realaxs_real,V_c),s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_nil))))) = s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_nil)).
fof(ah4s_polys_POLYu_u_ADD, axiom, ![V_x, V_p2, V_p1]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_add(s(t_h4s_lists_list(t_h4s_realaxs_real),V_p1),s(t_h4s_lists_list(t_h4s_realaxs_real),V_p2))))),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),V_p1))),s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_realaxs_real),h4s_polys_poly(s(t_h4s_lists_list(t_h4s_realaxs_real),V_p2))),s(t_h4s_realaxs_real,V_x)))))).
fof(ah4s_polys_polyu_u_expu_u_defu_c0, axiom, ![V_p]: s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_exp(s(t_h4s_lists_list(t_h4s_realaxs_real),V_p),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_lists_cons(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(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_lists_list(t_h4s_realaxs_real),h4s_lists_nil)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_richu_u_lists_COUNTu_u_LISTu_u_defu_c1, axiom, ![V_n]: s(t_h4s_lists_list(t_h4s_nums_num),h4s_richu_u_lists_countu_u_list(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))))) = s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_cons(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_map(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_lists_list(t_h4s_nums_num),h4s_richu_u_lists_countu_u_list(s(t_h4s_nums_num,V_n)))))))).
fof(ah4s_richu_u_lists_COUNTu_u_LISTu_u_defu_c0, axiom, s(t_h4s_lists_list(t_h4s_nums_num),h4s_richu_u_lists_countu_u_list(s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_nil)).
fof(ah4s_lists_EVERYu_u_MAP, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_P, V_f, V_x]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_bool))),V_uu_0),s(t_fun(TV_u_27b,t_bool),V_P))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))) => ![V_l, V_f, V_P]: s(t_bool,h4s_lists_every(s(t_fun(TV_u_27b,t_bool),V_P),s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_bool))),V_uu_0),s(t_fun(TV_u_27b,t_bool),V_P))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_lists_list(TV_u_27a),V_l))))).
fof(ah4s_numerals_numeralu_u_lteu_c1, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
fof(ah4s_numerals_numeralu_u_distribu_c27, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ch4s_polys_POLYu_u_DIFFu_u_AUXu_u_ISZERO, conjecture, ![V_uu_0]: (![V_c]: (p(s(t_bool,happ(s(t_fun(t_h4s_realaxs_real,t_bool),V_uu_0),s(t_h4s_realaxs_real,V_c)))) <=> s(t_h4s_realaxs_real,V_c) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) => ![V_p, V_n]: s(t_bool,h4s_lists_every(s(t_fun(t_h4s_realaxs_real,t_bool),V_uu_0),s(t_h4s_lists_list(t_h4s_realaxs_real),h4s_polys_polyu_u_diffu_u_aux(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))),s(t_h4s_lists_list(t_h4s_realaxs_real),V_p))))) = s(t_bool,h4s_lists_every(s(t_fun(t_h4s_realaxs_real,t_bool),V_uu_0),s(t_h4s_lists_list(t_h4s_realaxs_real),V_p))))).
