%   ORIGINAL: h4/path/every__el
% 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/path/firstP__at__def: !p i P. h4/path/firstP__at P p i <=> h4/bool/IN i (h4/path/PL p) /\ P (h4/path/el i p) /\ (!j. h4/prim__rec/_3C j i ==> ~P (h4/path/el j p))
% Assm: h4/path/exists__el: !p P. h4/path/exists P p <=> (?i. h4/bool/IN i (h4/path/PL p) /\ P (h4/path/el i p))
% Assm: h4/path/every__def: !p P. h4/path/every P p <=> ~h4/path/exists (h4/combin/o $not P) p
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/path/PL__pcons: !x r q. h4/path/PL (h4/path/pcons x r q) = h4/pred__set/INSERT h4/num/0 (h4/pred__set/IMAGE h4/num/SUC (h4/path/PL q))
% 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/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/path/PL__downward__closed: !p i. h4/bool/IN i (h4/path/PL p) ==> (!j. h4/prim__rec/_3C j i ==> h4/bool/IN j (h4/path/PL p))
% Assm: h4/path/not__every: !p P. ~h4/path/every P p <=> h4/path/exists (h4/combin/o $not P) p
% Assm: h4/path/every__thm_c1: !x r p P. h4/path/every P (h4/path/pcons x r p) <=> P x /\ h4/path/every P p
% Assm: h4/path/infinite__PL: !p. ~h4/path/finite p ==> (!i. h4/bool/IN i (h4/path/PL p))
% Assm: h4/path/el__def_c1: !p n. h4/path/el (h4/num/SUC n) p = h4/path/el n (h4/path/tail p)
% Assm: h4/path/el__def_c0: !p. h4/path/el h4/num/0 p = h4/path/first p
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/path/PL__def: !p. h4/path/PL p = h4/pred__set/GSPEC (\i. h4/pair/_2C i (h4/path/finite p ==> h4/prim__rec/_3C i (h4/option/THE (h4/path/length p))))
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/arithmetic/SUC__ELIM__NUMERALS: !g f. (!n. g (h4/num/SUC n) = f n (h4/num/SUC n)) <=> (!n. g (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) = f (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) /\ (!n. g (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) = f (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)))
% Assm: h4/path/PL__thm_c1: !x r q. h4/path/PL (h4/path/pcons x r q) = h4/pred__set/INSERT h4/num/0 (h4/pred__set/IMAGE h4/num/SUC (h4/path/PL q))
% Assm: h4/path/PL__0: !p. h4/bool/IN h4/num/0 (h4/path/PL p)
% Assm: h4/path/not__exists: !p P. ~h4/path/exists P p <=> h4/path/every (h4/combin/o $not P) p
% Assm: h4/path/el__pmap: !p i g f. h4/bool/IN i (h4/path/PL p) ==> h4/path/el i (h4/path/pmap f g p) = f (h4/path/el i p)
% Assm: h4/sat/NOT__NOT: !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__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/path/exists__thm_c1: !x r p P. h4/path/exists P (h4/path/pcons x r p) <=> P x \/ h4/path/exists P p
% Assm: h4/path/every__thm_c0: !x P. h4/path/every P (h4/path/stopped__at x) <=> P x
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/path/el__def__compute_c2: !p n. h4/path/el (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) p = h4/path/el (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/path/tail p)
% Assm: h4/path/el__def__compute_c1: !p n. h4/path/el (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) p = h4/path/el (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/path/tail p)
% Assm: h4/path/el__def__compute_c0: !p. h4/path/el h4/num/0 p = h4/path/first p
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/path/exists__def: !p P. h4/path/exists P p <=> (?i. h4/path/firstP__at P p i)
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/bool/RIGHT__OR__EXISTS__THM: !Q P. P \/ (?x. Q x) <=> (?x. P \/ Q x)
% Assm: h4/bool/LEFT__EXISTS__AND__THM: !Q P. (?x. P x /\ Q) <=> (?x. P x) /\ Q
% Assm: h4/arithmetic/WOP: !P. (?n. P n) ==> (?n. P n /\ (!m. h4/prim__rec/_3C m n ==> ~P m))
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/bool/SKOLEM__THM: !P. (!x. ?y. P x y) <=> (?f. !x. P x (f x))
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/pred__set/IN__UNIV: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm: h4/pred__set/GSPEC__T: h4/pred__set/GSPEC (\x. h4/pair/_2C x T) = h4/pred__set/UNIV
% Assm: h4/path/exists__thm_c0: !x P. h4/path/exists P (h4/path/stopped__at x) <=> P x
% Assm: h4/path/nth__label__pmap: !p i g f. h4/bool/IN (h4/num/SUC i) (h4/path/PL p) ==> h4/path/nth__label i (h4/path/pmap f g p) = g (h4/path/nth__label i p)
% Assm: h4/num/NOT__SUC: !n. ~(h4/num/SUC n = h4/num/0)
% Assm: h4/pred__set/IN__IMAGE: !y s f. h4/bool/IN y (h4/pred__set/IMAGE f s) <=> (?x. y = f x /\ h4/bool/IN x s)
% Assm: h4/pred__set/IN__INSERT: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm: h4/path/PL__thm_c0: !x. h4/path/PL (h4/path/stopped__at x) = h4/pred__set/INSERT h4/num/0 h4/pred__set/EMPTY
% Assm: h4/pred__set/NOT__IN__EMPTY: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm: h4/path/FORALL__path: !P. (!p. P p) <=> (!x. P (h4/path/stopped__at x)) /\ (!x r p. P (h4/path/pcons x r p))
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/pred__set/GSPECIFICATION: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = f x)
% Assm: h4/pair/PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/arithmetic/LESS__TRANS: !p n m. h4/prim__rec/_3C m n /\ h4/prim__rec/_3C n p ==> h4/prim__rec/_3C m p
% Assm: h4/combin/o__DEF: !g f. h4/combin/o f g = (\x. f (g x))
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/path/pmap__thm_c1: !x r p g f. h4/path/pmap f g (h4/path/pcons x r p) = h4/path/pcons (f x) (g r) (h4/path/pmap f g p)
% Assm: h4/path/tail__def: !x r p. h4/path/tail (h4/path/pcons x r p) = p
% Assm: h4/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Assm: h4/prim__rec/INV__SUC__EQ: !n m. h4/num/SUC m = h4/num/SUC n <=> m = n
% Assm: h4/path/first__pmap: !p g f. h4/path/first (h4/path/pmap f g p) = f (h4/path/first p)
% Assm: h4/path/first__thm_c1: !x r p. h4/path/first (h4/path/pcons x r p) = x
% Assm: h4/path/first__thm_c0: !x. h4/path/first (h4/path/stopped__at x) = x
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% 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__eq_c1: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Assm: h4/numeral/numeral__add_c0: !n. h4/numeral/iZ (h4/arithmetic/_2B h4/arithmetic/ZERO n) = n
% 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
% Assm: h4/numeral/numeral__distrib_c17: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% 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_c1: !n. h4/arithmetic/_2B n h4/num/0 = 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/LESS__EQ__TRANS: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% 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__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/bool/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Assm: h4/path/length__thm_c1: !x r p. h4/path/length (h4/path/pcons x r p) = h4/bool/COND (h4/path/finite p) (h4/option/SOME (h4/arithmetic/_2B (h4/option/THE (h4/path/length p)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) h4/option/NONE
% Assm: h4/path/finite__thm_c1: !x r p. h4/path/finite (h4/path/pcons x r p) <=> h4/path/finite p
% Assm: h4/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm: h4/arithmetic/EQ__MONO__ADD__EQ: !p n m. h4/arithmetic/_2B m p = h4/arithmetic/_2B n p <=> m = n
% Assm: h4/arithmetic/LESS__MONO__ADD__EQ: !p n m. h4/prim__rec/_3C (h4/arithmetic/_2B m p) (h4/arithmetic/_2B n p) <=> h4/prim__rec/_3C m n
% Assm: h4/arithmetic/ADD1: !m. h4/num/SUC m = h4/arithmetic/_2B m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% 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/arithmetic/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/LESS__EQ: !n m. h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/arithmetic/num__CASES: !m. m = h4/num/0 \/ (?n. m = h4/num/SUC n)
% Assm: h4/option/THE__DEF: !x. h4/option/THE (h4/option/SOME x) = x
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% 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/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm: h4/bool/EQ__EXPAND: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% Assm: h4/path/nth__label__def_c1: !p n. h4/path/nth__label (h4/num/SUC n) p = h4/path/nth__label n (h4/path/tail p)
% Assm: h4/path/nth__label__def_c0: !p. h4/path/nth__label h4/num/0 p = h4/path/first__label p
% Assm: h4/path/path__cases: !p. (?x. p = h4/path/stopped__at x) \/ (?x r q. p = h4/path/pcons x r q)
% Assm: h4/path/first__label__def: !x r p. h4/path/first__label (h4/path/pcons x r p) = r
% Goal: !p P. h4/path/every P p <=> (!i. h4/bool/IN i (h4/path/PL p) ==> P (h4/path/el i 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_paths_firstPu_u_atu_u_def]: !p i P. h4/path/firstP__at P p i <=> h4/bool/IN i (h4/path/PL p) /\ happ P (h4/path/el i p) /\ (!j. h4/prim__rec/_3C j i ==> ~happ P (h4/path/el j p))
% Assm [h4s_paths_existsu_u_el]: !p P. h4/path/exists P p <=> (?i. h4/bool/IN i (h4/path/PL p) /\ happ P (h4/path/el i p))
% Assm [h4s_paths_everyu_u_def]: !p P. h4/path/every P p <=> ~h4/path/exists (h4/combin/o $not P) p
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_paths_PLu_u_pcons]: !x r q. h4/path/PL (h4/path/pcons x r q) = h4/pred__set/INSERT h4/num/0 (h4/pred__set/IMAGE h4/num/SUC (h4/path/PL q))
% 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_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_paths_PLu_u_downwardu_u_closed]: !p i. h4/bool/IN i (h4/path/PL p) ==> (!j. h4/prim__rec/_3C j i ==> h4/bool/IN j (h4/path/PL p))
% Assm [h4s_paths_notu_u_every]: !p P. ~h4/path/every P p <=> h4/path/exists (h4/combin/o $not P) p
% Assm [h4s_paths_everyu_u_thmu_c1]: !x r p P. h4/path/every P (h4/path/pcons x r p) <=> happ P x /\ h4/path/every P p
% Assm [h4s_paths_infiniteu_u_PL]: !p. ~h4/path/finite p ==> (!i. h4/bool/IN i (h4/path/PL p))
% Assm [h4s_paths_elu_u_defu_c1]: !p n. h4/path/el (happ h4/num/SUC n) p = h4/path/el n (h4/path/tail p)
% Assm [h4s_paths_elu_u_defu_c0]: !p. h4/path/el h4/num/0 p = h4/path/first p
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_paths_PLu_u_def]: !_0. (!p i. ?v. (v <=> h4/path/finite p ==> h4/prim__rec/_3C i (h4/option/THE (h4/path/length p))) /\ happ (happ _0 p) i = h4/pair/_2C i v) ==> (!p. h4/path/PL p = h4/pred__set/GSPEC (happ _0 p))
% Assm [h4s_combins_ou_u_THM]: !x g f. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_arithmetics_SUCu_u_ELIMu_u_NUMERALS]: !g f. (!n. happ g (happ h4/num/SUC n) = happ (happ f n) (happ h4/num/SUC n)) <=> (!n. happ g (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) = happ (happ f (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) /\ (!n. happ g (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) = happ (happ f (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)))
% Assm [h4s_paths_PLu_u_thmu_c1]: !x r q. h4/path/PL (h4/path/pcons x r q) = h4/pred__set/INSERT h4/num/0 (h4/pred__set/IMAGE h4/num/SUC (h4/path/PL q))
% Assm [h4s_paths_PLu_u_0]: !p. h4/bool/IN h4/num/0 (h4/path/PL p)
% Assm [h4s_paths_notu_u_exists]: !p P. ~h4/path/exists P p <=> h4/path/every (h4/combin/o $not P) p
% Assm [h4s_paths_elu_u_pmap]: !p i g f. h4/bool/IN i (h4/path/PL p) ==> h4/path/el i (h4/path/pmap f g p) = happ f (h4/path/el i p)
% Assm [h4s_sats_NOTu_u_NOT]: !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_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_paths_existsu_u_thmu_c1]: !x r p P. h4/path/exists P (h4/path/pcons x r p) <=> happ P x \/ h4/path/exists P p
% Assm [h4s_paths_everyu_u_thmu_c0]: !x P. h4/path/every P (h4/path/stopped__at x) <=> happ P x
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_paths_elu_u_defu_u_computeu_c2]: !p n. h4/path/el (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) p = h4/path/el (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/path/tail p)
% Assm [h4s_paths_elu_u_defu_u_computeu_c1]: !p n. h4/path/el (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) p = h4/path/el (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/path/tail p)
% Assm [h4s_paths_elu_u_defu_u_computeu_c0]: !p. h4/path/el h4/num/0 p = h4/path/first p
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_paths_existsu_u_def]: !p P. h4/path/exists P p <=> (?i. h4/path/firstP__at P p i)
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM]: !Q P. P \/ (?x. happ Q x) <=> (?x. P \/ happ Q x)
% Assm [h4s_bools_LEFTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. happ P x /\ Q) <=> (?x. happ P x) /\ Q
% Assm [h4s_arithmetics_WOP]: !P. (?n. happ P n) ==> (?n. happ P n /\ (!m. h4/prim__rec/_3C m n ==> ~happ P m))
% Assm [h4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. P \/ happ Q x) <=> P \/ (!x. happ Q x)
% Assm [h4s_bools_SKOLEMu_u_THM]: !P. (!x. ?y. happ (happ P x) y) <=> (?f. !x. happ (happ P x) (happ f x))
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_predu_u_sets_INu_u_UNIV]: !x. h4/bool/IN x h4/pred__set/UNIV
% Assm [h4s_predu_u_sets_GSPECu_u_T]: !_0. (!x. happ _0 x = h4/pair/_2C x T) ==> h4/pred__set/GSPEC _0 = h4/pred__set/UNIV
% Assm [h4s_paths_existsu_u_thmu_c0]: !x P. h4/path/exists P (h4/path/stopped__at x) <=> happ P x
% Assm [h4s_paths_nthu_u_labelu_u_pmap]: !p i g f. h4/bool/IN (happ h4/num/SUC i) (h4/path/PL p) ==> h4/path/nth__label i (h4/path/pmap f g p) = happ g (h4/path/nth__label i p)
% Assm [h4s_nums_NOTu_u_SUC]: !n. ~(happ h4/num/SUC n = h4/num/0)
% Assm [h4s_predu_u_sets_INu_u_IMAGE]: !y s f. h4/bool/IN y (h4/pred__set/IMAGE f s) <=> (?x. y = happ f x /\ h4/bool/IN x s)
% Assm [h4s_predu_u_sets_INu_u_INSERT]: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm [h4s_paths_PLu_u_thmu_c0]: !x. h4/path/PL (h4/path/stopped__at x) = h4/pred__set/INSERT h4/num/0 h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_NOTu_u_INu_u_EMPTY]: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm [h4s_paths_FORALLu_u_path]: !P. (!p. happ P p) <=> (!x. happ P (h4/path/stopped__at x)) /\ (!x r p. happ P (h4/path/pcons x r p))
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_predu_u_sets_GSPECIFICATION]: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = happ f x)
% Assm [h4s_pairs_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_arithmetics_LESSu_u_TRANS]: !p n m. h4/prim__rec/_3C m n /\ h4/prim__rec/_3C n p ==> h4/prim__rec/_3C m p
% Assm [h4s_combins_ou_u_DEF]: !g f x. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_paths_pmapu_u_thmu_c1]: !x r p g f. h4/path/pmap f g (h4/path/pcons x r p) = h4/path/pcons (happ f x) (happ g r) (h4/path/pmap f g p)
% Assm [h4s_paths_tailu_u_def]: !x r p. h4/path/tail (h4/path/pcons x r p) = p
% 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_primu_u_recs_INVu_u_SUCu_u_EQ]: !n m. happ h4/num/SUC m = happ h4/num/SUC n <=> m = n
% Assm [h4s_paths_firstu_u_pmap]: !p g f. h4/path/first (h4/path/pmap f g p) = happ f (h4/path/first p)
% Assm [h4s_paths_firstu_u_thmu_c1]: !x r p. h4/path/first (h4/path/pcons x r p) = x
% Assm [h4s_paths_firstu_u_thmu_c0]: !x. h4/path/first (h4/path/stopped__at x) = x
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% 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_equ_c1]: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Assm [h4s_numerals_numeralu_u_addu_c0]: !n. h4/numeral/iZ (h4/arithmetic/_2B h4/arithmetic/ZERO n) = n
% 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
% Assm [h4s_numerals_numeralu_u_distribu_c17]: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% 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_c1]: !n. h4/arithmetic/_2B n h4/num/0 = 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_LESSu_u_EQu_u_TRANS]: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm [h4s_arithmetics_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% 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_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_bools_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% Assm [h4s_paths_lengthu_u_thmu_c1]: !x r p. h4/path/length (h4/path/pcons x r p) = h4/bool/COND (h4/path/finite p) (h4/option/SOME (h4/arithmetic/_2B (h4/option/THE (h4/path/length p)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) h4/option/NONE
% Assm [h4s_paths_finiteu_u_thmu_c1]: !x r p. h4/path/finite (h4/path/pcons x r p) <=> h4/path/finite p
% Assm [h4s_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm [h4s_arithmetics_EQu_u_MONOu_u_ADDu_u_EQ]: !p n m. h4/arithmetic/_2B m p = h4/arithmetic/_2B n p <=> m = n
% Assm [h4s_arithmetics_LESSu_u_MONOu_u_ADDu_u_EQ]: !p n m. h4/prim__rec/_3C (h4/arithmetic/_2B m p) (h4/arithmetic/_2B n p) <=> h4/prim__rec/_3C m n
% 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_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_arithmetics_NOTu_u_LESS]: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_LESSu_u_EQ]: !n m. h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D (happ h4/num/SUC m) n
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm [h4s_arithmetics_numu_u_CASES]: !m. m = h4/num/0 \/ (?n. m = happ h4/num/SUC n)
% Assm [h4s_options_THEu_u_DEF]: !x. h4/option/THE (h4/option/SOME x) = x
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% 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_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm [h4s_bools_EQu_u_EXPAND]: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% Assm [h4s_paths_nthu_u_labelu_u_defu_c1]: !p n. h4/path/nth__label (happ h4/num/SUC n) p = h4/path/nth__label n (h4/path/tail p)
% Assm [h4s_paths_nthu_u_labelu_u_defu_c0]: !p. h4/path/nth__label h4/num/0 p = h4/path/first__label p
% Assm [h4s_paths_pathu_u_cases]: !p. (?x. p = h4/path/stopped__at x) \/ (?x r q. p = h4/path/pcons x r q)
% Assm [h4s_paths_firstu_u_labelu_u_def]: !x r p. h4/path/first__label (h4/path/pcons x r p) = r
% Goal: !p P. h4/path/every P p <=> (!i. h4/bool/IN i (h4/path/PL p) ==> happ P (h4/path/el i 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_Q1228893,TV_Q1228889]: ![V_f, V_g]: (![V_x]: s(TV_Q1228889,happ(s(t_fun(TV_Q1228893,TV_Q1228889),V_f),s(TV_Q1228893,V_x))) = s(TV_Q1228889,happ(s(t_fun(TV_Q1228893,TV_Q1228889),V_g),s(TV_Q1228893,V_x))) => s(t_fun(TV_Q1228893,TV_Q1228889),V_f) = s(t_fun(TV_Q1228893,TV_Q1228889),V_g))).
fof(ah4s_paths_firstPu_u_atu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_i, V_P]: (p(s(t_bool,h4s_paths_firstpu_u_at(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p),s(t_h4s_nums_num,V_i)))) <=> (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_i),s(t_fun(t_h4s_nums_num,t_bool),h4s_paths_pl(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))) & (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_paths_el(s(t_h4s_nums_num,V_i),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))) & ![V_j]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_j),s(t_h4s_nums_num,V_i)))) => ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_paths_el(s(t_h4s_nums_num,V_j),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))))))))).
fof(ah4s_paths_existsu_u_el, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_P]: (p(s(t_bool,h4s_paths_exists(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))) <=> ?[V_i]: (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_i),s(t_fun(t_h4s_nums_num,t_bool),h4s_paths_pl(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_paths_el(s(t_h4s_nums_num,V_i),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))))))).
fof(ah4s_paths_everyu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_P]: (p(s(t_bool,h4s_paths_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))) <=> ~ (p(s(t_bool,h4s_paths_exists(s(t_fun(TV_u_27a,t_bool),h4s_combins_o(s(t_fun(t_bool,t_bool),d_not),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_paths_PLu_u_pcons, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_r, V_q]: s(t_fun(t_h4s_nums_num,t_bool),h4s_paths_pl(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_q))))) = s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_insert(s(t_h4s_nums_num,h4s_nums_0),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_fun(t_h4s_nums_num,t_bool),h4s_paths_pl(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_q)))))))).
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_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_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_paths_PLu_u_downwardu_u_closed, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_i]: (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_i),s(t_fun(t_h4s_nums_num,t_bool),h4s_paths_pl(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))) => ![V_j]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_j),s(t_h4s_nums_num,V_i)))) => p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_j),s(t_fun(t_h4s_nums_num,t_bool),h4s_paths_pl(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))))))).
fof(ah4s_paths_notu_u_every, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_P]: (~ (p(s(t_bool,h4s_paths_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))) <=> p(s(t_bool,h4s_paths_exists(s(t_fun(TV_u_27a,t_bool),h4s_combins_o(s(t_fun(t_bool,t_bool),d_not),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))).
fof(ah4s_paths_everyu_u_thmu_c1, axiom, ![TV_u_27a,TV_u_27c]: ![V_x, V_r, V_p, V_P]: (p(s(t_bool,h4s_paths_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27c),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27c,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27c),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,h4s_paths_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27c),V_p))))))).
fof(ah4s_paths_infiniteu_u_PL, axiom, ![TV_u_27a,TV_u_27b]: ![V_p]: (~ (p(s(t_bool,h4s_paths_finite(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))) => ![V_i]: p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_i),s(t_fun(t_h4s_nums_num,t_bool),h4s_paths_pl(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))))).
fof(ah4s_paths_elu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_n]: s(TV_u_27a,h4s_paths_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_paths_path(TV_u_27a,TV_u_27b),V_p))) = s(TV_u_27a,h4s_paths_el(s(t_h4s_nums_num,V_n),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_tail(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))).
fof(ah4s_paths_elu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_p]: s(TV_u_27a,h4s_paths_el(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))) = s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))).
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_paths_PLu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_p, V_i]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_paths_finite(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))) => p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_i),s(t_h4s_nums_num,h4s_options_the(s(t_h4s_options_option(t_h4s_nums_num),h4s_paths_length(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))))))) & s(t_h4s_pairs_prod(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_bool))),V_uu_0),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))),s(t_h4s_nums_num,V_i))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_bool),h4s_pairs_u_2c(s(t_h4s_nums_num,V_i),s(t_bool,V_v)))) => ![V_p]: s(t_fun(t_h4s_nums_num,t_bool),h4s_paths_pl(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))) = s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_bool))),V_uu_0),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))))).
fof(ah4s_combins_ou_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_x, V_g, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27c,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,V_x)))))).
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_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_arithmetics_SUCu_u_ELIMu_u_NUMERALS, axiom, ![TV_u_27a]: ![V_g, V_f]: (![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g),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(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),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_n))))) <=> (![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))) & ![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n)))))))))).
fof(ah4s_paths_PLu_u_thmu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_r, V_q]: s(t_fun(t_h4s_nums_num,t_bool),h4s_paths_pl(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_q))))) = s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_insert(s(t_h4s_nums_num,h4s_nums_0),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_fun(t_h4s_nums_num,t_bool),h4s_paths_pl(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_q)))))))).
fof(ah4s_paths_PLu_u_0, axiom, ![TV_u_27a,TV_u_27b]: ![V_p]: p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,h4s_nums_0),s(t_fun(t_h4s_nums_num,t_bool),h4s_paths_pl(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))))).
fof(ah4s_paths_notu_u_exists, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_P]: (~ (p(s(t_bool,h4s_paths_exists(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))) <=> p(s(t_bool,h4s_paths_every(s(t_fun(TV_u_27a,t_bool),h4s_combins_o(s(t_fun(t_bool,t_bool),d_not),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))).
fof(ah4s_paths_elu_u_pmap, axiom, ![TV_u_27d,TV_u_27c,TV_u_27a,TV_u_27b]: ![V_p, V_i, V_g, V_f]: (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_i),s(t_fun(t_h4s_nums_num,t_bool),h4s_paths_pl(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))) => s(TV_u_27c,h4s_paths_el(s(t_h4s_nums_num,V_i),s(t_h4s_paths_path(TV_u_27c,TV_u_27d),h4s_paths_pmap(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,h4s_paths_el(s(t_h4s_nums_num,V_i),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),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_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_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_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_paths_existsu_u_thmu_c1, axiom, ![TV_u_27a,TV_u_27c]: ![V_x, V_r, V_p, V_P]: (p(s(t_bool,h4s_paths_exists(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27c),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27c,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27c),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,h4s_paths_exists(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27c),V_p))))))).
fof(ah4s_paths_everyu_u_thmu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_P]: s(t_bool,h4s_paths_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))).
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_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
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_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_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_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_bools_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_paths_elu_u_defu_u_computeu_c2, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_n]: s(TV_u_27a,h4s_paths_el(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))))),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))) = s(TV_u_27a,h4s_paths_el(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_tail(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))).
fof(ah4s_paths_elu_u_defu_u_computeu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_n]: s(TV_u_27a,h4s_paths_el(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))) = s(TV_u_27a,h4s_paths_el(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(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_paths_path(TV_u_27a,TV_u_27b),h4s_paths_tail(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))).
fof(ah4s_paths_elu_u_defu_u_computeu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_p]: s(TV_u_27a,h4s_paths_el(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))) = s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))).
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_paths_existsu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_P]: (p(s(t_bool,h4s_paths_exists(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))) <=> ?[V_i]: p(s(t_bool,h4s_paths_firstpu_u_at(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p),s(t_h4s_nums_num,V_i)))))).
fof(ah4s_bools_NOTu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ?[V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_NOTu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ![V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) | ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ?[V_x]: (p(s(t_bool,V_P)) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_LEFTu_u_EXISTSu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (?[V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))) <=> (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))))).
fof(ah4s_arithmetics_WOP, axiom, ![V_P]: (?[V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) => ?[V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) & ![V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) => ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_m))))))))).
fof(ah4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![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))))) <=> (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))))))).
fof(ah4s_bools_SKOLEMu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (![V_x]: ?[V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) <=> ?[V_f]: ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_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_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_predu_u_sets_INu_u_UNIV, axiom, ![TV_u_27a]: ![V_x]: p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ))))).
fof(ah4s_predu_u_sets_GSPECu_u_T, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_uu_0),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,t))) => s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_uu_0))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ))).
fof(ah4s_paths_existsu_u_thmu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_P]: s(t_bool,h4s_paths_exists(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))).
fof(ah4s_paths_nthu_u_labelu_u_pmap, axiom, ![TV_u_27d,TV_u_27c,TV_u_27a,TV_u_27b]: ![V_p, V_i, V_g, V_f]: (p(s(t_bool,h4s_bools_in(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_i))),s(t_fun(t_h4s_nums_num,t_bool),h4s_paths_pl(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))) => s(TV_u_27c,h4s_paths_nthu_u_label(s(t_h4s_nums_num,V_i),s(t_h4s_paths_path(TV_u_27d,TV_u_27c),h4s_paths_pmap(s(t_fun(TV_u_27a,TV_u_27d),V_f),s(t_fun(TV_u_27b,TV_u_27c),V_g),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),V_g),s(TV_u_27b,h4s_paths_nthu_u_label(s(t_h4s_nums_num,V_i),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))))).
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_predu_u_sets_INu_u_IMAGE, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_s, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_y),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> ?[V_x]: (s(TV_u_27b,V_y) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_predu_u_sets_INu_u_INSERT, axiom, ![TV_u_27a]: ![V_y, V_x, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_paths_PLu_u_thmu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: s(t_fun(t_h4s_nums_num,t_bool),h4s_paths_pl(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))))) = s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_insert(s(t_h4s_nums_num,h4s_nums_0),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_empty)))).
fof(ah4s_predu_u_sets_NOTu_u_INu_u_EMPTY, axiom, ![TV_u_27a]: ![V_x]: ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))))).
fof(ah4s_paths_FORALLu_u_path, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: (![V_p]: p(s(t_bool,happ(s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x)))))) & ![V_x, V_r, V_p]: p(s(t_bool,happ(s(t_fun(t_h4s_paths_path(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))))))).
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_predu_u_sets_GSPECIFICATION, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_v),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f)))))) <=> ?[V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_v),s(t_bool,t))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f),s(TV_u_27b,V_x))))).
fof(ah4s_pairs_PAIRu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_b, V_a]: (s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & s(TV_u_27b,V_y) = s(TV_u_27b,V_b)))).
fof(ah4s_bools_EXISTSu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (?[V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_arithmetics_LESSu_u_TRANS, axiom, ![V_p, V_n, V_m]: ((p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_combins_ou_u_DEF, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_g, V_f, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27c),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x)))))).
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_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_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_paths_pmapu_u_thmu_c1, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a,TV_u_27d]: ![V_x, V_r, V_p, V_g, V_f]: s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_paths_pmap(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27d,TV_u_27c),V_g),s(t_h4s_paths_path(TV_u_27a,TV_u_27d),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27d,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27d),V_p))))) = s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_paths_pcons(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_27c,happ(s(t_fun(TV_u_27d,TV_u_27c),V_g),s(TV_u_27d,V_r))),s(t_h4s_paths_path(TV_u_27b,TV_u_27c),h4s_paths_pmap(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27d,TV_u_27c),V_g),s(t_h4s_paths_path(TV_u_27a,TV_u_27d),V_p)))))).
fof(ah4s_paths_tailu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_r, V_p]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_tail(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)).
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_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_paths_firstu_u_pmap, axiom, ![TV_u_27d,TV_u_27c,TV_u_27a,TV_u_27b]: ![V_p, V_g, V_f]: s(TV_u_27c,h4s_paths_first(s(t_h4s_paths_path(TV_u_27c,TV_u_27d),h4s_paths_pmap(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))).
fof(ah4s_paths_firstu_u_thmu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_r, V_p]: s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))) = s(TV_u_27a,V_x)).
fof(ah4s_paths_firstu_u_thmu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
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_equ_c1, axiom, ![V_n]: (s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_zero) <=> p(s(t_bool,f)))).
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_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(ah4s_numerals_numeralu_u_distribu_c17, axiom, ![V_n]: (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_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_arithmetics_zero))).
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_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_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_LESSu_u_EQu_u_TRANS, axiom, ![V_p, V_n, V_m]: ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,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_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_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_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_bools_IMPu_u_Fu_u_EQu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_paths_lengthu_u_thmu_c1, axiom, ![TV_u_27c,TV_u_27d]: ![V_x, V_r, V_p]: s(t_h4s_options_option(t_h4s_nums_num),h4s_paths_length(s(t_h4s_paths_path(TV_u_27c,TV_u_27d),h4s_paths_pcons(s(TV_u_27c,V_x),s(TV_u_27d,V_r),s(t_h4s_paths_path(TV_u_27c,TV_u_27d),V_p))))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_bools_cond(s(t_bool,h4s_paths_finite(s(t_h4s_paths_path(TV_u_27c,TV_u_27d),V_p))),s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_options_the(s(t_h4s_options_option(t_h4s_nums_num),h4s_paths_length(s(t_h4s_paths_path(TV_u_27c,TV_u_27d),V_p))))),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_options_option(t_h4s_nums_num),h4s_options_none)))).
fof(ah4s_paths_finiteu_u_thmu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_r, V_p]: s(t_bool,h4s_paths_finite(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))) = s(t_bool,h4s_paths_finite(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))).
fof(ah4s_predu_u_sets_EXTENSION, axiom, ![TV_u_27a]: ![V_t, V_s]: (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),V_t) <=> ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))).
fof(ah4s_arithmetics_EQu_u_MONOu_u_ADDu_u_EQ, 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,V_p))) = 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,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_arithmetics_LESSu_u_MONOu_u_ADDu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_primu_u_recs_u_3c(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_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))).
fof(ah4s_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_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_arithmetics_NOTu_u_LESS, axiom, ![V_n, V_m]: (~ (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_LESSu_u_EQ, axiom, ![V_n, V_m]: 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_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)))).
fof(ah4s_arithmetics_ADDu_u_SYM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_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_options_THEu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_options_the(s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_ANDu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,f))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_boolu_u_caseu_u_thmu_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_IMPu_u_DISJu_u_THM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) => p(s(t_bool,V_B))) <=> (~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B))))).
fof(ah4s_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_EQu_u_IMPu_u_THM, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) & (p(s(t_bool,V_t2)) => p(s(t_bool,V_t1)))))).
fof(ah4s_bools_EQu_u_EXPAND, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) | (~ (p(s(t_bool,V_t1))) & ~ (p(s(t_bool,V_t2))))))).
fof(ah4s_paths_nthu_u_labelu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_p, V_n]: s(TV_u_27a,h4s_paths_nthu_u_label(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_paths_path(TV_u_27b,TV_u_27a),V_p))) = s(TV_u_27a,h4s_paths_nthu_u_label(s(t_h4s_nums_num,V_n),s(t_h4s_paths_path(TV_u_27b,TV_u_27a),h4s_paths_tail(s(t_h4s_paths_path(TV_u_27b,TV_u_27a),V_p)))))).
fof(ah4s_paths_nthu_u_labelu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_p]: s(TV_u_27a,h4s_paths_nthu_u_label(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_paths_path(TV_u_27b,TV_u_27a),V_p))) = s(TV_u_27a,h4s_paths_firstu_u_label(s(t_h4s_paths_path(TV_u_27b,TV_u_27a),V_p)))).
fof(ah4s_paths_pathu_u_cases, axiom, ![TV_u_27a,TV_u_27b]: ![V_p]: (?[V_x]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))) | ?[V_x, V_r, V_q]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p) = s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_q))))).
fof(ah4s_paths_firstu_u_labelu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_r, V_p]: s(TV_u_27b,h4s_paths_firstu_u_label(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_pcons(s(TV_u_27a,V_x),s(TV_u_27b,V_r),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))) = s(TV_u_27b,V_r)).
fof(ch4s_paths_everyu_u_el, conjecture, ![TV_u_27a,TV_u_27b]: ![V_p, V_P]: (p(s(t_bool,h4s_paths_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))) <=> ![V_i]: (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_i),s(t_fun(t_h4s_nums_num,t_bool),h4s_paths_pl(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_paths_el(s(t_h4s_nums_num,V_i),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))))))))).
