%   ORIGINAL: h4/rich__list/COUNT__LIST__SNOC_c1
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/prim__rec/num__Axiom: !f e. ?fn. fn h4/num/0 = e /\ (!n. fn (h4/num/SUC n) = f n (fn n))
% Assm: h4/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__GENLIST: !n. h4/rich__list/COUNT__LIST n = h4/list/GENLIST h4/combin/I n
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/rich__list/EL__COUNT__LIST: !n m. h4/prim__rec/_3C m n ==> h4/list/EL m (h4/rich__list/COUNT__LIST n) = m
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/rich__list/LENGTH__COUNT__LIST: !n. h4/list/LENGTH (h4/rich__list/COUNT__LIST n) = n
% Assm: h4/rich__list/MEM__COUNT__LIST: !n m. h4/bool/IN m (h4/list/LIST__TO__SET (h4/rich__list/COUNT__LIST n)) <=> h4/prim__rec/_3C m n
% Assm: h4/list/LENGTH__GENLIST: !n f. h4/list/LENGTH (h4/list/GENLIST f n) = n
% Assm: h4/rich__list/COUNT__LIST__def_c0: h4/rich__list/COUNT__LIST h4/num/0 = h4/list/NIL
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/rich__list/COUNT__LIST__SNOC_c0: h4/rich__list/COUNT__LIST h4/num/0 = h4/list/NIL
% Assm: h4/list/GENLIST0_c0: !f. h4/list/GENLIST f h4/num/0 = h4/list/NIL
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/IMP__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/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/list/EL__GENLIST: !x n f. h4/prim__rec/_3C x n ==> h4/list/EL x (h4/list/GENLIST f n) = f x
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/combin/I__o__ID_c1: !f. h4/combin/o f h4/combin/I = f
% Assm: h4/list/GENLIST__CONS: !n f. h4/list/GENLIST f (h4/num/SUC n) = h4/list/CONS (f h4/num/0) (h4/list/GENLIST (h4/combin/o f h4/num/SUC) n)
% Assm: h4/combin/I__o__ID_c0: !f. h4/combin/o h4/combin/I f = f
% Assm: h4/list/MAP__GENLIST: !n g f. h4/list/MAP f (h4/list/GENLIST g n) = h4/list/GENLIST (h4/combin/o f g) n
% Assm: h4/list/MEM__EL: !x l. h4/bool/IN x (h4/list/LIST__TO__SET l) <=> (?n. h4/prim__rec/_3C n (h4/list/LENGTH l) /\ x = h4/list/EL n l)
% Assm: h4/bool/AND__CONG: !Q_27 Q P_27 P. (Q ==> (P <=> P_27)) /\ (P_27 ==> (Q <=> Q_27)) ==> (P /\ Q <=> P_27 /\ Q_27)
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/list/GENLIST0_c1: !n f. h4/list/GENLIST f (h4/num/SUC n) = h4/list/SNOC (f n) (h4/list/GENLIST f n)
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/rich__list/ELL__SNOC: !n. h4/prim__rec/_3C h4/num/0 n ==> (!x l. h4/rich__list/ELL n (h4/list/SNOC x l) = h4/rich__list/ELL (h4/prim__rec/PRE n) l)
% Assm: h4/rich__list/REVERSE__FOLDR: !l. h4/list/REVERSE l = h4/list/FOLDR h4/list/SNOC h4/list/NIL l
% Assm: h4/prim__rec/LESS__0: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm: h4/prim__rec/PRE0_c1: !m. h4/prim__rec/PRE (h4/num/SUC m) = m
% Assm: h4/rich__list/BUTLASTN0_c1: !x n l. h4/rich__list/BUTLASTN (h4/num/SUC n) (h4/list/SNOC x l) = h4/rich__list/BUTLASTN n l
% Assm: h4/rich__list/ELL__SUC__SNOC: !x n l. h4/rich__list/ELL (h4/num/SUC n) (h4/list/SNOC x l) = h4/rich__list/ELL n l
% Assm: h4/rich__list/LASTN0_c1: !x n l. h4/rich__list/LASTN (h4/num/SUC n) (h4/list/SNOC x l) = h4/list/SNOC x (h4/rich__list/LASTN n l)
% 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/list/SNOC0_c1: !x_27 x l. h4/list/SNOC x (h4/list/CONS x_27 l) = h4/list/CONS x_27 (h4/list/SNOC x l)
% Assm: h4/list/LENGTH__SNOC: !x l. h4/list/LENGTH (h4/list/SNOC x l) = h4/num/SUC (h4/list/LENGTH l)
% Assm: h4/rich__list/SEG__0__SNOC: !x m l. h4/arithmetic/_3C_3D m (h4/list/LENGTH l) ==> h4/rich__list/SEG m h4/num/0 (h4/list/SNOC x l) = h4/rich__list/SEG m h4/num/0 l
% Assm: h4/list/SUM__SNOC: !x l. h4/list/SUM (h4/list/SNOC x l) = h4/arithmetic/_2B (h4/list/SUM l) x
% 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/rich__list/TAKE__SNOC: !n l. h4/arithmetic/_3C_3D n (h4/list/LENGTH l) ==> (!x. h4/list/TAKE n (h4/list/SNOC x l) = h4/list/TAKE n l)
% Assm: h4/list/EL__SNOC: !n l. h4/prim__rec/_3C n (h4/list/LENGTH l) ==> (!x. h4/list/EL n (h4/list/SNOC x l) = h4/list/EL n l)
% Assm: h4/rich__list/SEG__SNOC: !n m l. h4/arithmetic/_3C_3D (h4/arithmetic/_2B n m) (h4/list/LENGTH l) ==> (!x. h4/rich__list/SEG n m (h4/list/SNOC x l) = h4/rich__list/SEG n m l)
% Assm: h4/list/LUPDATE__SNOC: !ys y x k. h4/list/LUPDATE x k (h4/list/SNOC y ys) = h4/bool/COND (k = h4/list/LENGTH ys) (h4/list/SNOC x ys) (h4/list/SNOC y (h4/list/LUPDATE x k ys))
% Assm: h4/list/SNOC0_c0: !x. h4/list/SNOC x h4/list/NIL = h4/list/CONS x h4/list/NIL
% Assm: h4/list/REVERSE__SNOC__DEF_c0: h4/list/REVERSE h4/list/NIL = h4/list/NIL
% Assm: h4/list/REVERSE__SNOC__DEF_c1: !x l. h4/list/REVERSE (h4/list/CONS x l) = h4/list/SNOC x (h4/list/REVERSE l)
% 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/FOLDR0_c1: !x l f e. h4/list/FOLDR f e (h4/list/CONS x l) = f x (h4/list/FOLDR f e l)
% Assm: h4/list/FOLDR0_c0: !f e. h4/list/FOLDR f e h4/list/NIL = e
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/prim__rec/NOT__LESS__0: !n. ~h4/prim__rec/_3C n h4/num/0
% Assm: h4/list/FRONT__SNOC: !x l. h4/list/FRONT (h4/list/SNOC x l) = l
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/bool/LEFT__AND__FORALL__THM: !Q P. (!x. P x) /\ Q <=> (!x. P x /\ Q)
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/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/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/num/NOT__SUC: !n. ~(h4/num/SUC n = h4/num/0)
% Assm: h4/arithmetic/LESS__EQ__MONO: !n m. h4/arithmetic/_3C_3D (h4/num/SUC n) (h4/num/SUC m) <=> h4/arithmetic/_3C_3D n m
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% 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/ADD__0: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/arithmetic/ADD_c0: !n. h4/arithmetic/_2B h4/num/0 n = n
% Assm: h4/arithmetic/LESS__OR__EQ: !n m. h4/arithmetic/_3C_3D m n <=> h4/prim__rec/_3C m n \/ m = n
% Assm: h4/list/LAST__SNOC: !x l. h4/list/LAST (h4/list/SNOC x l) = x
% Assm: h4/list/SUM0_c1: !t h. h4/list/SUM (h4/list/CONS h t) = h4/arithmetic/_2B h (h4/list/SUM t)
% Assm: h4/list/SUM0_c0: h4/list/SUM h4/list/NIL = h4/num/0
% Assm: h4/rich__list/SEG0_c0: !l k. h4/rich__list/SEG h4/num/0 k l = h4/list/NIL
% Assm: h4/rich__list/SEG0_c1: !x m l. h4/rich__list/SEG (h4/num/SUC m) h4/num/0 (h4/list/CONS x l) = h4/list/CONS x (h4/rich__list/SEG m h4/num/0 l)
% Assm: h4/rich__list/ELL0_c1: !n l. h4/rich__list/ELL (h4/num/SUC n) l = h4/rich__list/ELL n (h4/list/FRONT l)
% Assm: h4/rich__list/ELL0_c0: !l. h4/rich__list/ELL h4/num/0 l = h4/list/LAST l
% Assm: h4/prim__rec/PRE0_c0: h4/prim__rec/PRE h4/num/0 = h4/num/0
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/rich__list/TAKE_c1: !x n l. h4/list/TAKE (h4/num/SUC n) (h4/list/CONS x l) = h4/list/CONS x (h4/list/TAKE n l)
% Assm: h4/rich__list/TAKE_c0: !l. h4/list/TAKE h4/num/0 l = h4/list/NIL
% Assm: h4/list/EL0_c1: !n l. h4/list/EL (h4/num/SUC n) l = h4/list/EL n (h4/list/TL l)
% Assm: h4/list/EL0_c0: !l. h4/list/EL h4/num/0 l = h4/list/HD l
% Assm: h4/list/TL0: !t h. h4/list/TL (h4/list/CONS h t) = t
% Assm: h4/arithmetic/LESS__MONO__EQ: !n m. h4/prim__rec/_3C (h4/num/SUC m) (h4/num/SUC n) <=> h4/prim__rec/_3C m n
% Assm: h4/list/HD0: !t h. h4/list/HD (h4/list/CONS h t) = h
% Assm: h4/list/LUPDATE__def_c2: !x n l e. h4/list/LUPDATE e (h4/num/SUC n) (h4/list/CONS x l) = h4/list/CONS x (h4/list/LUPDATE e n l)
% Assm: h4/list/LUPDATE__def_c1: !x l e. h4/list/LUPDATE e h4/num/0 (h4/list/CONS x l) = h4/list/CONS e l
% Assm: h4/list/LUPDATE__def_c0: !n e. h4/list/LUPDATE e n h4/list/NIL = h4/list/NIL
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/arithmetic/num__CASES: !m. m = h4/num/0 \/ (?n. m = h4/num/SUC n)
% Assm: h4/prim__rec/INV__SUC__EQ: !n m. h4/num/SUC m = h4/num/SUC n <=> m = n
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/rich__list/SEG0_c2: !x m l k. h4/rich__list/SEG (h4/num/SUC m) (h4/num/SUC k) (h4/list/CONS x l) = h4/rich__list/SEG (h4/num/SUC m) k l
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/list/CONS__11: !a1_27 a1 a0_27 a0. h4/list/CONS a0 a1 = h4/list/CONS a0_27 a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% 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/numeral/numeral__distrib_c27: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__distrib_c1: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm: h4/arithmetic/NOT__NUM__EQ: !n m. ~(m = n) <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n \/ h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm: h4/numeral/numeral__add_c0: !n. h4/numeral/iZ (h4/arithmetic/_2B h4/arithmetic/ZERO n) = n
% Assm: h4/arithmetic/SUC__ONE__ADD: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm: h4/arithmetic/NOT__SUC__LESS__EQ__0: !n. ~h4/arithmetic/_3C_3D (h4/num/SUC n) h4/num/0
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% 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/arithmetic/ADD__MONO__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Goal: !n. h4/rich__list/COUNT__LIST (h4/num/SUC n) = h4/list/SNOC n (h4/rich__list/COUNT__LIST n)
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_primu_u_recs_numu_u_Axiom]: !f e. ?fn. happ fn h4/num/0 = e /\ (!n. happ fn (happ h4/num/SUC n) = happ (happ f n) (happ fn n))
% 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_GENLIST]: !n. h4/rich__list/COUNT__LIST n = h4/list/GENLIST h4/combin/I n
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_richu_u_lists_ELu_u_COUNTu_u_LIST]: !n m. h4/prim__rec/_3C m n ==> h4/list/EL m (h4/rich__list/COUNT__LIST n) = m
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_richu_u_lists_LENGTHu_u_COUNTu_u_LIST]: !n. h4/list/LENGTH (h4/rich__list/COUNT__LIST n) = n
% Assm [h4s_richu_u_lists_MEMu_u_COUNTu_u_LIST]: !n m. h4/bool/IN m (h4/list/LIST__TO__SET (h4/rich__list/COUNT__LIST n)) <=> h4/prim__rec/_3C m n
% Assm [h4s_lists_LENGTHu_u_GENLIST]: !n f. h4/list/LENGTH (h4/list/GENLIST f n) = 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_combins_Iu_u_THM]: !x. happ h4/combin/I x = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_richu_u_lists_COUNTu_u_LISTu_u_SNOCu_c0]: h4/rich__list/COUNT__LIST h4/num/0 = h4/list/NIL
% Assm [h4s_lists_GENLIST0u_c0]: !f. h4/list/GENLIST f h4/num/0 = h4/list/NIL
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_IMPu_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_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_lists_ELu_u_GENLIST]: !x n f. h4/prim__rec/_3C x n ==> h4/list/EL x (h4/list/GENLIST f n) = happ f x
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_combins_Iu_u_ou_u_IDu_c1]: !f. h4/combin/o f h4/combin/I = f
% Assm [h4s_lists_GENLISTu_u_CONS]: !n f. h4/list/GENLIST f (happ h4/num/SUC n) = h4/list/CONS (happ f h4/num/0) (h4/list/GENLIST (h4/combin/o f h4/num/SUC) n)
% Assm [h4s_combins_Iu_u_ou_u_IDu_c0]: !f. h4/combin/o h4/combin/I f = f
% Assm [h4s_lists_MAPu_u_GENLIST]: !n g f. h4/list/MAP f (h4/list/GENLIST g n) = h4/list/GENLIST (h4/combin/o f g) n
% Assm [h4s_lists_MEMu_u_EL]: !x l. h4/bool/IN x (h4/list/LIST__TO__SET l) <=> (?n. h4/prim__rec/_3C n (h4/list/LENGTH l) /\ x = h4/list/EL n l)
% Assm [h4s_bools_ANDu_u_CONG]: !Q_27 Q P_27 P. (Q ==> (P <=> P_27)) /\ (P_27 ==> (Q <=> Q_27)) ==> (P /\ Q <=> P_27 /\ Q_27)
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_lists_GENLIST0u_c1]: !n f. h4/list/GENLIST f (happ h4/num/SUC n) = happ (happ h4/list/SNOC (happ f n)) (h4/list/GENLIST f n)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_richu_u_lists_ELLu_u_SNOC]: !n. h4/prim__rec/_3C h4/num/0 n ==> (!x l. h4/rich__list/ELL n (happ (happ h4/list/SNOC x) l) = h4/rich__list/ELL (h4/prim__rec/PRE n) l)
% Assm [h4s_richu_u_lists_REVERSEu_u_FOLDR]: !l. h4/list/REVERSE l = h4/list/FOLDR h4/list/SNOC h4/list/NIL l
% Assm [h4s_primu_u_recs_LESSu_u_0]: !n. h4/prim__rec/_3C h4/num/0 (happ h4/num/SUC n)
% Assm [h4s_primu_u_recs_PRE0u_c1]: !m. h4/prim__rec/PRE (happ h4/num/SUC m) = m
% Assm [h4s_richu_u_lists_BUTLASTN0u_c1]: !x n l. h4/rich__list/BUTLASTN (happ h4/num/SUC n) (happ (happ h4/list/SNOC x) l) = h4/rich__list/BUTLASTN n l
% Assm [h4s_richu_u_lists_ELLu_u_SUCu_u_SNOC]: !x n l. h4/rich__list/ELL (happ h4/num/SUC n) (happ (happ h4/list/SNOC x) l) = h4/rich__list/ELL n l
% Assm [h4s_richu_u_lists_LASTN0u_c1]: !x n l. h4/rich__list/LASTN (happ h4/num/SUC n) (happ (happ h4/list/SNOC x) l) = happ (happ h4/list/SNOC x) (h4/rich__list/LASTN n l)
% 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_lists_SNOC0u_c1]: !x_27 x l. happ (happ h4/list/SNOC x) (h4/list/CONS x_27 l) = h4/list/CONS x_27 (happ (happ h4/list/SNOC x) l)
% Assm [h4s_lists_LENGTHu_u_SNOC]: !x l. h4/list/LENGTH (happ (happ h4/list/SNOC x) l) = happ h4/num/SUC (h4/list/LENGTH l)
% Assm [h4s_richu_u_lists_SEGu_u_0u_u_SNOC]: !x m l. h4/arithmetic/_3C_3D m (h4/list/LENGTH l) ==> h4/rich__list/SEG m h4/num/0 (happ (happ h4/list/SNOC x) l) = h4/rich__list/SEG m h4/num/0 l
% Assm [h4s_lists_SUMu_u_SNOC]: !x l. h4/list/SUM (happ (happ h4/list/SNOC x) l) = h4/arithmetic/_2B (h4/list/SUM l) x
% Assm [h4s_lists_LENGTH0u_c1]: !t h. h4/list/LENGTH (h4/list/CONS h t) = happ h4/num/SUC (h4/list/LENGTH t)
% Assm [h4s_lists_LENGTH0u_c0]: h4/list/LENGTH h4/list/NIL = h4/num/0
% Assm [h4s_richu_u_lists_TAKEu_u_SNOC]: !n l. h4/arithmetic/_3C_3D n (h4/list/LENGTH l) ==> (!x. h4/list/TAKE n (happ (happ h4/list/SNOC x) l) = h4/list/TAKE n l)
% Assm [h4s_lists_ELu_u_SNOC]: !n l. h4/prim__rec/_3C n (h4/list/LENGTH l) ==> (!x. h4/list/EL n (happ (happ h4/list/SNOC x) l) = h4/list/EL n l)
% Assm [h4s_richu_u_lists_SEGu_u_SNOC]: !n m l. h4/arithmetic/_3C_3D (h4/arithmetic/_2B n m) (h4/list/LENGTH l) ==> (!x. h4/rich__list/SEG n m (happ (happ h4/list/SNOC x) l) = h4/rich__list/SEG n m l)
% Assm [h4s_lists_LUPDATEu_u_SNOC]: !ys y x k. ?v. (v <=> k = h4/list/LENGTH ys) /\ h4/list/LUPDATE x k (happ (happ h4/list/SNOC y) ys) = h4/bool/COND v (happ (happ h4/list/SNOC x) ys) (happ (happ h4/list/SNOC y) (h4/list/LUPDATE x k ys))
% Assm [h4s_lists_SNOC0u_c0]: !x. happ (happ h4/list/SNOC x) h4/list/NIL = h4/list/CONS x h4/list/NIL
% Assm [h4s_lists_REVERSEu_u_SNOCu_u_DEFu_c0]: h4/list/REVERSE h4/list/NIL = h4/list/NIL
% Assm [h4s_lists_REVERSEu_u_SNOCu_u_DEFu_c1]: !x l. h4/list/REVERSE (h4/list/CONS x l) = happ (happ h4/list/SNOC x) (h4/list/REVERSE l)
% 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_FOLDR0u_c1]: !x l f e. h4/list/FOLDR f e (h4/list/CONS x l) = happ (happ f x) (h4/list/FOLDR f e l)
% Assm [h4s_lists_FOLDR0u_c0]: !f e. h4/list/FOLDR f e h4/list/NIL = e
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_IMPu_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_lists_FRONTu_u_SNOC]: !x l. h4/list/FRONT (happ (happ h4/list/SNOC x) l) = l
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_bools_LEFTu_u_ANDu_u_FORALLu_u_THM]: !Q P. (!x. happ P x) /\ Q <=> (!x. happ P x /\ Q)
% Assm [h4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_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_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_nums_NOTu_u_SUC]: !n. ~(happ h4/num/SUC n = h4/num/0)
% Assm [h4s_arithmetics_LESSu_u_EQu_u_MONO]: !n m. h4/arithmetic/_3C_3D (happ h4/num/SUC n) (happ h4/num/SUC m) <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% 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_ADDu_u_0]: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm [h4s_arithmetics_ADDu_c0]: !n. h4/arithmetic/_2B h4/num/0 n = n
% 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_lists_LASTu_u_SNOC]: !x l. h4/list/LAST (happ (happ h4/list/SNOC x) l) = x
% Assm [h4s_lists_SUM0u_c1]: !t h. h4/list/SUM (h4/list/CONS h t) = h4/arithmetic/_2B h (h4/list/SUM t)
% Assm [h4s_lists_SUM0u_c0]: h4/list/SUM h4/list/NIL = h4/num/0
% Assm [h4s_richu_u_lists_SEG0u_c0]: !l k. h4/rich__list/SEG h4/num/0 k l = h4/list/NIL
% Assm [h4s_richu_u_lists_SEG0u_c1]: !x m l. h4/rich__list/SEG (happ h4/num/SUC m) h4/num/0 (h4/list/CONS x l) = h4/list/CONS x (h4/rich__list/SEG m h4/num/0 l)
% Assm [h4s_richu_u_lists_ELL0u_c1]: !n l. h4/rich__list/ELL (happ h4/num/SUC n) l = h4/rich__list/ELL n (h4/list/FRONT l)
% Assm [h4s_richu_u_lists_ELL0u_c0]: !l. h4/rich__list/ELL h4/num/0 l = h4/list/LAST l
% Assm [h4s_primu_u_recs_PRE0u_c0]: h4/prim__rec/PRE h4/num/0 = h4/num/0
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_richu_u_lists_TAKEu_c1]: !x n l. h4/list/TAKE (happ h4/num/SUC n) (h4/list/CONS x l) = h4/list/CONS x (h4/list/TAKE n l)
% Assm [h4s_richu_u_lists_TAKEu_c0]: !l. h4/list/TAKE h4/num/0 l = h4/list/NIL
% Assm [h4s_lists_EL0u_c1]: !n l. h4/list/EL (happ h4/num/SUC n) l = h4/list/EL n (h4/list/TL l)
% Assm [h4s_lists_EL0u_c0]: !l. h4/list/EL h4/num/0 l = h4/list/HD l
% Assm [h4s_lists_TL0]: !t h. h4/list/TL (h4/list/CONS h t) = t
% Assm [h4s_arithmetics_LESSu_u_MONOu_u_EQ]: !n m. h4/prim__rec/_3C (happ h4/num/SUC m) (happ h4/num/SUC n) <=> h4/prim__rec/_3C m n
% Assm [h4s_lists_HD0]: !t h. h4/list/HD (h4/list/CONS h t) = h
% Assm [h4s_lists_LUPDATEu_u_defu_c2]: !x n l e. h4/list/LUPDATE e (happ h4/num/SUC n) (h4/list/CONS x l) = h4/list/CONS x (h4/list/LUPDATE e n l)
% Assm [h4s_lists_LUPDATEu_u_defu_c1]: !x l e. h4/list/LUPDATE e h4/num/0 (h4/list/CONS x l) = h4/list/CONS e l
% Assm [h4s_lists_LUPDATEu_u_defu_c0]: !n e. h4/list/LUPDATE e n h4/list/NIL = h4/list/NIL
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_arithmetics_numu_u_CASES]: !m. m = h4/num/0 \/ (?n. m = happ h4/num/SUC n)
% Assm [h4s_primu_u_recs_INVu_u_SUCu_u_EQ]: !n m. happ h4/num/SUC m = happ h4/num/SUC n <=> m = n
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_richu_u_lists_SEG0u_c2]: !x m l k. h4/rich__list/SEG (happ h4/num/SUC m) (happ h4/num/SUC k) (h4/list/CONS x l) = h4/rich__list/SEG (happ h4/num/SUC m) k l
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_lists_CONSu_u_11]: !a1_27 a1 a0_27 a0. h4/list/CONS a0 a1 = h4/list/CONS a0_27 a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% 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_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
% Assm [h4s_numerals_numeralu_u_distribu_c1]: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm [h4s_arithmetics_NOTu_u_NUMu_u_EQ]: !n m. ~(m = n) <=> h4/arithmetic/_3C_3D (happ h4/num/SUC m) n \/ h4/arithmetic/_3C_3D (happ h4/num/SUC n) m
% Assm [h4s_numerals_numeralu_u_addu_c0]: !n. h4/numeral/iZ (h4/arithmetic/_2B h4/arithmetic/ZERO n) = n
% Assm [h4s_arithmetics_SUCu_u_ONEu_u_ADD]: !n. happ h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm [h4s_arithmetics_NOTu_u_SUCu_u_LESSu_u_EQu_u_0]: !n. ~h4/arithmetic/_3C_3D (happ h4/num/SUC n) h4/num/0
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% 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_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Goal: !n. h4/rich__list/COUNT__LIST (happ h4/num/SUC n) = happ (happ h4/list/SNOC n) (h4/rich__list/COUNT__LIST n)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q1252266,TV_Q1252262]: ![V_f, V_g]: (![V_x]: s(TV_Q1252262,happ(s(t_fun(TV_Q1252266,TV_Q1252262),V_f),s(TV_Q1252266,V_x))) = s(TV_Q1252262,happ(s(t_fun(TV_Q1252266,TV_Q1252262),V_g),s(TV_Q1252266,V_x))) => s(t_fun(TV_Q1252266,TV_Q1252262),V_f) = s(t_fun(TV_Q1252266,TV_Q1252262),V_g))).
fof(ah4s_primu_u_recs_numu_u_Axiom, axiom, ![TV_u_27a]: ![V_f, V_e]: ?[V_fn]: (s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_e) & ![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,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_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_GENLIST, axiom, ![V_n]: s(t_h4s_lists_list(t_h4s_nums_num),h4s_richu_u_lists_countu_u_list(s(t_h4s_nums_num,V_n))) = s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_combins_i),s(t_h4s_nums_num,V_n)))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_richu_u_lists_ELu_u_COUNTu_u_LIST, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) => s(t_h4s_nums_num,h4s_lists_el(s(t_h4s_nums_num,V_m),s(t_h4s_lists_list(t_h4s_nums_num),h4s_richu_u_lists_countu_u_list(s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,V_m))).
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_richu_u_lists_LENGTHu_u_COUNTu_u_LIST, axiom, ![V_n]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(t_h4s_nums_num),h4s_richu_u_lists_countu_u_list(s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_richu_u_lists_MEMu_u_COUNTu_u_LIST, axiom, ![V_n, V_m]: s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_m),s(t_fun(t_h4s_nums_num,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_nums_num),h4s_richu_u_lists_countu_u_list(s(t_h4s_nums_num,V_n))))))) = s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))).
fof(ah4s_lists_LENGTHu_u_GENLIST, axiom, ![TV_u_27a]: ![V_n, V_f]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n))))) = 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_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
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_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_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_richu_u_lists_COUNTu_u_LISTu_u_SNOCu_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_GENLIST0u_c0, axiom, ![TV_u_27a]: ![V_f]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)).
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_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_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_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_lists_ELu_u_GENLIST, axiom, ![TV_u_27a]: ![V_x, V_n, V_f]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,V_n)))) => s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,V_x),s(t_h4s_lists_list(TV_u_27a),h4s_lists_genlist(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,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_combins_Iu_u_ou_u_IDu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i))) = s(t_fun(TV_u_27a,TV_u_27b),V_f)).
fof(ah4s_lists_GENLISTu_u_CONS, axiom, ![TV_u_27a]: ![V_n, V_f]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),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(TV_u_27a),h4s_lists_cons(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(t_h4s_lists_list(TV_u_27a),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_combins_o(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc))),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_combins_Iu_u_ou_u_IDu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27b,TV_u_27b),h4s_combins_i),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27a,TV_u_27b),V_f)).
fof(ah4s_lists_MAPu_u_GENLIST, axiom, ![TV_u_27b,TV_u_27a]: ![V_n, V_g, V_f]: 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),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g),s(t_h4s_nums_num,V_n))))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(t_h4s_nums_num,TV_u_27a),V_g))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_lists_MEMu_u_EL, axiom, ![TV_u_27a]: ![V_x, V_l]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l)))))) <=> ?[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_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l)))))) & s(TV_u_27a,V_x) = s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_bools_ANDu_u_CONG, axiom, ![V_Qu_27, V_Q, V_Pu_27, V_P]: (((p(s(t_bool,V_Q)) => s(t_bool,V_P) = s(t_bool,V_Pu_27)) & (p(s(t_bool,V_Pu_27)) => s(t_bool,V_Q) = s(t_bool,V_Qu_27))) => ((p(s(t_bool,V_P)) & p(s(t_bool,V_Q))) <=> (p(s(t_bool,V_Pu_27)) & p(s(t_bool,V_Qu_27)))))).
fof(ah4s_bools_UNWINDu_u_THM2, axiom, ![TV_u_27a]: ![V_a, V_P]: (?[V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_a)))))).
fof(ah4s_lists_GENLIST0u_c1, axiom, ![TV_u_27a]: ![V_n, V_f]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),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(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_snoc),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(t_h4s_lists_list(TV_u_27a),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n)))))).
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_richu_u_lists_ELLu_u_SNOC, axiom, ![TV_u_27a]: ![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_l]: s(TV_u_27a,h4s_richu_u_lists_ell(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_snoc),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(TV_u_27a,h4s_richu_u_lists_ell(s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,V_n))),s(t_h4s_lists_list(TV_u_27a),V_l))))).
fof(ah4s_richu_u_lists_REVERSEu_u_FOLDR, axiom, ![TV_u_27a]: ![V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_reverse(s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_snoc),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27a),V_l)))).
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,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_primu_u_recs_PRE0u_c1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,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,V_m)).
fof(ah4s_richu_u_lists_BUTLASTN0u_c1, axiom, ![TV_u_27a]: ![V_x, V_n, V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_butlastn(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(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_snoc),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_butlastn(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27a),V_l)))).
fof(ah4s_richu_u_lists_ELLu_u_SUCu_u_SNOC, axiom, ![TV_u_27a]: ![V_x, V_n, V_l]: s(TV_u_27a,h4s_richu_u_lists_ell(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(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_snoc),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(TV_u_27a,h4s_richu_u_lists_ell(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27a),V_l)))).
fof(ah4s_richu_u_lists_LASTN0u_c1, axiom, ![TV_u_27a]: ![V_x, V_n, V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_lastn(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(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_snoc),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_snoc),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_lastn(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
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_lists_SNOC0u_c1, axiom, ![TV_u_27a]: ![V_xu_27, V_x, V_l]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_snoc),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_xu_27),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_xu_27),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_snoc),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_lists_LENGTHu_u_SNOC, axiom, ![TV_u_27a]: ![V_x, V_l]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_snoc),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = 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,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_richu_u_lists_SEGu_u_0u_u_SNOC, axiom, ![TV_u_27a]: ![V_x, V_m, V_l]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l)))))) => s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_seg(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_snoc),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_seg(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_lists_list(TV_u_27a),V_l))))).
fof(ah4s_lists_SUMu_u_SNOC, axiom, ![V_x, V_l]: s(t_h4s_nums_num,h4s_lists_sum(s(t_h4s_lists_list(t_h4s_nums_num),happ(s(t_fun(t_h4s_lists_list(t_h4s_nums_num),t_h4s_lists_list(t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_h4s_nums_num),t_h4s_lists_list(t_h4s_nums_num))),h4s_lists_snoc),s(t_h4s_nums_num,V_x))),s(t_h4s_lists_list(t_h4s_nums_num),V_l))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_lists_sum(s(t_h4s_lists_list(t_h4s_nums_num),V_l))),s(t_h4s_nums_num,V_x)))).
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,happ(s(t_fun(t_h4s_nums_num,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_richu_u_lists_TAKEu_u_SNOC, axiom, ![TV_u_27a]: ![V_n, V_l]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l)))))) => ![V_x]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_take(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_snoc),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_take(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27a),V_l))))).
fof(ah4s_lists_ELu_u_SNOC, axiom, ![TV_u_27a]: ![V_n, V_l]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l)))))) => ![V_x]: s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_snoc),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27a),V_l))))).
fof(ah4s_richu_u_lists_SEGu_u_SNOC, axiom, ![TV_u_27a]: ![V_n, V_m, V_l]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l)))))) => ![V_x]: s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_seg(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_snoc),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_seg(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m),s(t_h4s_lists_list(TV_u_27a),V_l))))).
fof(ah4s_lists_LUPDATEu_u_SNOC, axiom, ![TV_u_27a]: ![V_ys, V_y, V_x, V_k]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_ys)))) & s(t_h4s_lists_list(TV_u_27a),h4s_lists_lupdate(s(TV_u_27a,V_x),s(t_h4s_nums_num,V_k),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_snoc),s(TV_u_27a,V_y))),s(t_h4s_lists_list(TV_u_27a),V_ys))))) = s(t_h4s_lists_list(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_snoc),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_ys))),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_snoc),s(TV_u_27a,V_y))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_lupdate(s(TV_u_27a,V_x),s(t_h4s_nums_num,V_k),s(t_h4s_lists_list(TV_u_27a),V_ys))))))))).
fof(ah4s_lists_SNOC0u_c0, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_snoc),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = 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_lists_nil)))).
fof(ah4s_lists_REVERSEu_u_SNOCu_u_DEFu_c0, axiom, ![TV_u_27b]: s(t_h4s_lists_list(TV_u_27b),h4s_lists_reverse(s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil)).
fof(ah4s_lists_REVERSEu_u_SNOCu_u_DEFu_c1, axiom, ![TV_u_27a]: ![V_x, V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_reverse(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),V_l))))) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_snoc),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_reverse(s(t_h4s_lists_list(TV_u_27a),V_l)))))).
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_FOLDR0u_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_l, V_f, V_e]: s(TV_u_27b,h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27b,V_e),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),V_l))))) = s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_lists_FOLDR0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_e]: s(TV_u_27b,h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(TV_u_27b,V_e)).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
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_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_lists_FRONTu_u_SNOC, axiom, ![TV_u_27a]: ![V_x, V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_front(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_snoc),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27a),V_l)).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_sats_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_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
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_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_bools_LEFTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))) <=> ![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))))).
fof(ah4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ![V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_bools_EXISTSu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_x)))))).
fof(ah4s_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_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_arithmetics_LESSu_u_EQu_u_MONO, axiom, ![V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(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,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_m))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_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_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_ADDu_u_0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_c0, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_arithmetics_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_lists_LASTu_u_SNOC, axiom, ![TV_u_27a]: ![V_x, V_l]: s(TV_u_27a,h4s_lists_last(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_snoc),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(TV_u_27a,V_x)).
fof(ah4s_lists_SUM0u_c1, axiom, ![V_t, V_h]: s(t_h4s_nums_num,h4s_lists_sum(s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_cons(s(t_h4s_nums_num,V_h),s(t_h4s_lists_list(t_h4s_nums_num),V_t))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_h),s(t_h4s_nums_num,h4s_lists_sum(s(t_h4s_lists_list(t_h4s_nums_num),V_t)))))).
fof(ah4s_lists_SUM0u_c0, axiom, s(t_h4s_nums_num,h4s_lists_sum(s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_nil))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_richu_u_lists_SEG0u_c0, axiom, ![TV_u_27a]: ![V_l, V_k]: s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_seg(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_k),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)).
fof(ah4s_richu_u_lists_SEG0u_c1, axiom, ![TV_u_27a]: ![V_x, V_m, V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_seg(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_nums_0),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),V_l))))) = 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_seg(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_richu_u_lists_ELL0u_c1, axiom, ![TV_u_27a]: ![V_n, V_l]: s(TV_u_27a,h4s_richu_u_lists_ell(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(TV_u_27a),V_l))) = s(TV_u_27a,h4s_richu_u_lists_ell(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27a),h4s_lists_front(s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_richu_u_lists_ELL0u_c0, axiom, ![TV_u_27a]: ![V_l]: s(TV_u_27a,h4s_richu_u_lists_ell(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(TV_u_27a,h4s_lists_last(s(t_h4s_lists_list(TV_u_27a),V_l)))).
fof(ah4s_primu_u_recs_PRE0u_c0, axiom, s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_richu_u_lists_TAKEu_c1, axiom, ![TV_u_27a]: ![V_x, V_n, V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_take(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(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l))))) = 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_lists_take(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_richu_u_lists_TAKEu_c0, axiom, ![TV_u_27a]: ![V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_take(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)).
fof(ah4s_lists_EL0u_c1, axiom, ![TV_u_27a]: ![V_n, V_l]: s(TV_u_27a,h4s_lists_el(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(TV_u_27a),V_l))) = s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27a),h4s_lists_tl(s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_lists_EL0u_c0, axiom, ![TV_u_27a]: ![V_l]: s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(TV_u_27a,h4s_lists_hd(s(t_h4s_lists_list(TV_u_27a),V_l)))).
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_arithmetics_LESSu_u_MONOu_u_EQ, axiom, ![V_n, V_m]: s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))))) = s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))).
fof(ah4s_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_lists_LUPDATEu_u_defu_c2, axiom, ![TV_u_27a]: ![V_x, V_n, V_l, V_e]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_lupdate(s(TV_u_27a,V_e),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(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l))))) = 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_lists_lupdate(s(TV_u_27a,V_e),s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_lists_LUPDATEu_u_defu_c1, axiom, ![TV_u_27a]: ![V_x, V_l, V_e]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_lupdate(s(TV_u_27a,V_e),s(t_h4s_nums_num,h4s_nums_0),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),V_l))))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_e),s(t_h4s_lists_list(TV_u_27a),V_l)))).
fof(ah4s_lists_LUPDATEu_u_defu_c0, axiom, ![TV_u_27a]: ![V_n, V_e]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_lupdate(s(TV_u_27a,V_e),s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)).
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_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_arithmetics_numu_u_CASES, axiom, ![V_m]: (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0) | ?[V_n]: s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))))).
fof(ah4s_primu_u_recs_INVu_u_SUCu_u_EQ, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,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,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,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_bools_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_richu_u_lists_SEG0u_c2, axiom, ![TV_u_27a]: ![V_x, V_m, V_l, V_k]: s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_seg(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,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_k))),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),V_l))))) = s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_seg(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,V_k),s(t_h4s_lists_list(TV_u_27a),V_l)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_lists_CONSu_u_11, axiom, ![TV_u_27a]: ![V_a1u_27, V_a1, V_a0u_27, 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_cons(s(TV_u_27a,V_a0u_27),s(t_h4s_lists_list(TV_u_27a),V_a1u_27))) <=> (s(TV_u_27a,V_a0) = s(TV_u_27a,V_a0u_27) & s(t_h4s_lists_list(TV_u_27a),V_a1) = s(t_h4s_lists_list(TV_u_27a),V_a1u_27)))).
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_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(ah4s_numerals_numeralu_u_distribu_c1, axiom, ![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_0))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_arithmetics_NOTu_u_NUMu_u_EQ, axiom, ![V_n, V_m]: (~ (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n)) <=> (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,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,V_n)))) | p(s(t_bool,h4s_arithmetics_u_3cu_3d(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,V_m))))))).
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_arithmetics_SUCu_u_ONEu_u_ADD, 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_arithmetics_u_2b(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_n)))).
fof(ah4s_arithmetics_NOTu_u_SUCu_u_LESSu_u_EQu_u_0, axiom, ![V_n]: ~ (p(s(t_bool,h4s_arithmetics_u_3cu_3d(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_arithmetics_MULTu_u_CLAUSESu_c2, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_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_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))).
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(ch4s_richu_u_lists_COUNTu_u_LISTu_u_SNOCu_c1, conjecture, ![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),happ(s(t_fun(t_h4s_lists_list(t_h4s_nums_num),t_h4s_lists_list(t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_h4s_nums_num),t_h4s_lists_list(t_h4s_nums_num))),h4s_lists_snoc),s(t_h4s_nums_num,V_n))),s(t_h4s_lists_list(t_h4s_nums_num),h4s_richu_u_lists_countu_u_list(s(t_h4s_nums_num,V_n)))))).
