%   ORIGINAL: h4/path/length__take
% 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/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% 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/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/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/option/THE__DEF: !x. h4/option/THE (h4/option/SOME x) = x
% Assm: h4/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Assm: h4/arithmetic/ADD__0: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% 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/EQ__MONO__ADD__EQ: !p n m. h4/arithmetic/_2B m p = h4/arithmetic/_2B n p <=> m = n
% Assm: h4/numeral/numeral__distrib_c17: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__eq_c1: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Assm: h4/pred__set/NOT__IN__EMPTY: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% 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/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/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/path/first__thm_c1: !x r p. h4/path/first (h4/path/pcons x r p) = x
% Assm: h4/path/tail__def: !x r p. h4/path/tail (h4/path/pcons x r p) = p
% Assm: h4/path/first__label__def: !x r p. h4/path/first__label (h4/path/pcons x r p) = r
% Assm: h4/path/length__thm_c0: !x. h4/path/length (h4/path/stopped__at x) = h4/option/SOME (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% 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/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/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/take__def_c0: !p. h4/path/take h4/num/0 p = h4/path/stopped__at (h4/path/first p)
% Assm: h4/path/take__def_c1: !p n. h4/path/take (h4/num/SUC n) p = h4/path/pcons (h4/path/first p) (h4/path/first__label p) (h4/path/take n (h4/path/tail p))
% Assm: h4/path/finite__take: !p i. h4/bool/IN i (h4/path/PL p) ==> h4/path/finite (h4/path/take i p)
% Goal: !p i. h4/bool/IN i (h4/path/PL p) ==> h4/path/length (h4/path/take i p) = h4/option/SOME (h4/arithmetic/_2B i (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
%   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_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% 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_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% 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_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_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_options_THEu_u_DEF]: !x. h4/option/THE (h4/option/SOME x) = x
% 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_arithmetics_ADDu_u_0]: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% 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_EQu_u_MONOu_u_ADDu_u_EQ]: !p n m. h4/arithmetic/_2B m p = h4/arithmetic/_2B n p <=> m = n
% Assm [h4s_numerals_numeralu_u_distribu_c17]: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_equ_c1]: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Assm [h4s_predu_u_sets_NOTu_u_INu_u_EMPTY]: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% 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_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_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_paths_firstu_u_thmu_c1]: !x r p. h4/path/first (h4/path/pcons x r p) = x
% Assm [h4s_paths_tailu_u_def]: !x r p. h4/path/tail (h4/path/pcons x r p) = p
% Assm [h4s_paths_firstu_u_labelu_u_def]: !x r p. h4/path/first__label (h4/path/pcons x r p) = r
% Assm [h4s_paths_lengthu_u_thmu_c0]: !x. h4/path/length (h4/path/stopped__at x) = h4/option/SOME (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% 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_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_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_takeu_u_defu_c0]: !p. h4/path/take h4/num/0 p = h4/path/stopped__at (h4/path/first p)
% Assm [h4s_paths_takeu_u_defu_c1]: !p n. h4/path/take (happ h4/num/SUC n) p = h4/path/pcons (h4/path/first p) (h4/path/first__label p) (h4/path/take n (h4/path/tail p))
% Assm [h4s_paths_finiteu_u_take]: !p i. h4/bool/IN i (h4/path/PL p) ==> h4/path/finite (h4/path/take i p)
% Goal: !p i. h4/bool/IN i (h4/path/PL p) ==> h4/path/length (h4/path/take i p) = h4/option/SOME (h4/arithmetic/_2B i (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
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_Q142631,TV_Q142627]: ![V_f, V_g]: (![V_x]: s(TV_Q142627,happ(s(t_fun(TV_Q142631,TV_Q142627),V_f),s(TV_Q142631,V_x))) = s(TV_Q142627,happ(s(t_fun(TV_Q142631,TV_Q142627),V_g),s(TV_Q142631,V_x))) => s(t_fun(TV_Q142631,TV_Q142627),V_f) = s(t_fun(TV_Q142631,TV_Q142627),V_g))).
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_TRUTH, axiom, 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_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
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_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_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_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_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_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_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_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_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_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_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_options_SOMEu_u_11, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
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_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_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_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_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_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_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_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_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_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_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_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_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_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_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(ah4s_paths_lengthu_u_thmu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: s(t_h4s_options_option(t_h4s_nums_num),h4s_paths_length(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_stoppedu_u_at(s(TV_u_27a,V_x))))) = s(t_h4s_options_option(t_h4s_nums_num),h4s_options_some(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_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_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_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_takeu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_p]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_take(s(t_h4s_nums_num,h4s_nums_0),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,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))).
fof(ah4s_paths_takeu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_n]: s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_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_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,h4s_paths_first(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p))),s(TV_u_27b,h4s_paths_firstu_u_label(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_take(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_finiteu_u_take, 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)))))) => p(s(t_bool,h4s_paths_finite(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_take(s(t_h4s_nums_num,V_i),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),V_p)))))))).
fof(ch4s_paths_lengthu_u_take, conjecture, ![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)))))) => s(t_h4s_options_option(t_h4s_nums_num),h4s_paths_length(s(t_h4s_paths_path(TV_u_27a,TV_u_27b),h4s_paths_take(s(t_h4s_nums_num,V_i),s(t_h4s_paths_path(TV_u_27a,TV_u_27b),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,V_i),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))))).
