%   ORIGINAL: h4/lbtree/lbtree__ue__Axiom
% 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/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/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/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/AND__CLAUSES_c4: !t. t /\ t <=> t
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/OR__CLAUSES_c1: !t. t \/ T <=> T
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% 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/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/bool/RIGHT__OR__OVER__AND: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm: h4/bool/EXISTS__UNIQUE__THM: !P. h4/bool/_3F_21 (\x. P x) <=> (?x. P x) /\ (!x y. P x /\ P y ==> x = y)
% 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/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% 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/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% 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/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/option/option__nchotomy: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm: h4/option/option__case__def_c0: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm: h4/option/option__case__def_c1: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = f x
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/option/OPTION__MAP__DEF_c0: !x f. h4/option/OPTION__MAP f (h4/option/SOME x) = h4/option/SOME (f x)
% Assm: h4/option/OPTION__MAP__DEF_c1: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm: h4/pair/pair__CASES: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% Assm: h4/list/list__case__def_c0: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm: h4/list/list__case__def_c1: !v f a1 a0. h4/list/list__CASE (h4/list/CONS a0 a1) v f = f a0 a1
% 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/list__nchotomy: !l. l = h4/list/NIL \/ (?h t. l = h4/list/CONS h t)
% Assm: h4/lbtree/Lfrep__def: h4/lbtree/Lfrep = (\l. h4/option/NONE)
% Assm: h4/lbtree/Ndrep__def: !t2 t1 a. h4/lbtree/Ndrep a t1 t2 = (\l. h4/list/list__CASE l (h4/option/SOME a) (\v xs. h4/bool/COND v (t1 xs) (t2 xs)))
% Assm: h4/lbtree/is__lbtree__def: !t. h4/lbtree/is__lbtree t <=> (?P. (!t0. P t0 ==> t0 = h4/lbtree/Lfrep \/ (?a t1 t2. P t1 /\ P t2 /\ t0 = h4/lbtree/Ndrep a t1 t2)) /\ P t)
% Assm: h4/lbtree/lbtree__absrep_c0: !a. h4/lbtree/lbtree__abs (h4/lbtree/lbtree__rep a) = a
% Assm: h4/lbtree/lbtree__absrep_c1: !r. h4/lbtree/is__lbtree r <=> h4/lbtree/lbtree__rep (h4/lbtree/lbtree__abs r) = r
% Assm: h4/lbtree/path__follow__def_c0: !x g. h4/lbtree/path__follow g x h4/list/NIL = h4/option/OPTION__MAP h4/pair/FST (g x)
% Assm: h4/lbtree/path__follow__def_c1: !x t h g. h4/lbtree/path__follow g x (h4/list/CONS h t) = h4/option/option__CASE (g x) h4/option/NONE (\v. h4/pair/pair__CASE v (\a v2. h4/pair/pair__CASE v2 (\y z. h4/lbtree/path__follow g (h4/bool/COND h y z) t)))
% Assm: h4/lbtree/Lf__def: h4/lbtree/Lf = h4/lbtree/lbtree__abs h4/lbtree/Lfrep
% Assm: h4/lbtree/Nd__def: !t2 t1 a. h4/lbtree/Nd a t1 t2 = h4/lbtree/lbtree__abs (h4/lbtree/Ndrep a (h4/lbtree/lbtree__rep t1) (h4/lbtree/lbtree__rep t2))
% Goal: !f. h4/bool/_3F_21 (\g. !x. g x = h4/option/option__CASE (f x) h4/lbtree/Lf (\v. h4/pair/pair__CASE v (\b v2. h4/pair/pair__CASE v2 (\y z. h4/lbtree/Nd b (g y) (g z)))))
%   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_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_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_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c4]: !t. t /\ t <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c1]: !t. t \/ T <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% 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_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% 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_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% 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_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND]: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm [h4s_bools_EXISTSu_u_UNIQUEu_u_THM]: !_0. (!P x. happ (happ _0 P) x <=> happ P x) ==> (!P. h4/bool/_3F_21 (happ _0 P) <=> (?x. happ P x) /\ (!x y. happ P x /\ happ P y ==> x = y))
% 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_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% 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_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% 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_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_options_optionu_u_nchotomy]: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm [h4s_options_optionu_u_caseu_u_defu_c0]: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm [h4s_options_optionu_u_caseu_u_defu_c1]: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = happ f x
% Assm [h4s_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_options_OPTIONu_u_MAPu_u_DEFu_c0]: !x f. h4/option/OPTION__MAP f (h4/option/SOME x) = h4/option/SOME (happ f x)
% Assm [h4s_options_OPTIONu_u_MAPu_u_DEFu_c1]: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm [h4s_pairs_pairu_u_CASES]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_pairs_FST0]: !y x. happ h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_pairs_pairu_u_caseu_u_thm]: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = happ (happ f x) y
% Assm [h4s_lists_listu_u_caseu_u_defu_c0]: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm [h4s_lists_listu_u_caseu_u_defu_c1]: !v f a1 a0. h4/list/list__CASE (h4/list/CONS a0 a1) v f = happ (happ f a0) a1
% 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_listu_u_nchotomy]: !l. l = h4/list/NIL \/ (?h t. l = h4/list/CONS h t)
% Assm [h4s_lbtrees_Lfrepu_u_def]: !x. happ h4/lbtree/Lfrep x = h4/option/NONE
% Assm [h4s_lbtrees_Ndrepu_u_def]: !_1. (!v t1 t2 xs. happ (happ (happ (happ _1 v) t1) t2) xs = h4/bool/COND v (happ t1 xs) (happ t2 xs)) ==> (!_0. (!t1 t2 v. happ (happ (happ _0 t1) t2) v = happ (happ (happ _1 v) t1) t2) ==> (!t2 t1 a x. happ (h4/lbtree/Ndrep a t1 t2) x = h4/list/list__CASE x (h4/option/SOME a) (happ (happ _0 t1) t2)))
% Assm [h4s_lbtrees_isu_u_lbtreeu_u_def]: !t. h4/lbtree/is__lbtree t <=> (?P. (!t0. happ P t0 ==> t0 = h4/lbtree/Lfrep \/ (?a t1 t2. happ P t1 /\ happ P t2 /\ t0 = h4/lbtree/Ndrep a t1 t2)) /\ happ P t)
% Assm [h4s_lbtrees_lbtreeu_u_absrepu_c0]: !a. h4/lbtree/lbtree__abs (h4/lbtree/lbtree__rep a) = a
% Assm [h4s_lbtrees_lbtreeu_u_absrepu_c1]: !r. h4/lbtree/is__lbtree r <=> h4/lbtree/lbtree__rep (h4/lbtree/lbtree__abs r) = r
% Assm [h4s_lbtrees_pathu_u_followu_u_defu_c0]: !x g. h4/lbtree/path__follow g x h4/list/NIL = h4/option/OPTION__MAP h4/pair/FST (happ g x)
% Assm [h4s_lbtrees_pathu_u_followu_u_defu_c1]: !_4. (!g h y t z. happ (happ (happ (happ (happ _4 g) h) y) t) z = h4/lbtree/path__follow g (h4/bool/COND h y z) t) ==> (!_3. (!g h t y. happ (happ (happ (happ _3 g) h) t) y = happ (happ (happ (happ _4 g) h) y) t) ==> (!_2. (!g h t v2. happ (happ (happ (happ _2 g) h) t) v2 = h4/pair/pair__CASE v2 (happ (happ (happ _3 g) h) t)) ==> (!_1. (!g h t a. happ (happ (happ (happ _1 g) h) t) a = happ (happ (happ _2 g) h) t) ==> (!_0. (!g h t v. happ (happ (happ (happ _0 g) h) t) v = h4/pair/pair__CASE v (happ (happ (happ _1 g) h) t)) ==> (!x t h g. h4/lbtree/path__follow g x (h4/list/CONS h t) = h4/option/option__CASE (happ g x) h4/option/NONE (happ (happ (happ _0 g) h) t))))))
% Assm [h4s_lbtrees_Lfu_u_def]: h4/lbtree/Lf = h4/lbtree/lbtree__abs h4/lbtree/Lfrep
% Assm [h4s_lbtrees_Ndu_u_def]: !t2 t1 a. h4/lbtree/Nd a t1 t2 = h4/lbtree/lbtree__abs (h4/lbtree/Ndrep a (h4/lbtree/lbtree__rep t1) (h4/lbtree/lbtree__rep t2))
% Goal: !_5. (!b y g z. happ (happ (happ (happ _5 b) y) g) z = h4/lbtree/Nd b (happ g y) (happ g z)) ==> (!_4. (!b g y. happ (happ (happ _4 b) g) y = happ (happ (happ _5 b) y) g) ==> (!_3. (!b g v2. happ (happ (happ _3 b) g) v2 = h4/pair/pair__CASE v2 (happ (happ _4 b) g)) ==> (!_2. (!g b. happ (happ _2 g) b = happ (happ _3 b) g) ==> (!_1. (!g v. happ (happ _1 g) v = h4/pair/pair__CASE v (happ _2 g)) ==> (!_0. (!f g. happ (happ _0 f) g <=> (!x. happ g x = h4/option/option__CASE (happ f x) h4/lbtree/Lf (happ _1 g))) ==> (!f. h4/bool/_3F_21 (happ _0 f)))))))
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f0)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(aHLu_EXT, axiom, ![TV_Q197392,TV_Q197388]: ![V_f, V_g]: (![V_x]: s(TV_Q197388,happ(s(t_fun(TV_Q197392,TV_Q197388),V_f),s(TV_Q197392,V_x))) = s(TV_Q197388,happ(s(t_fun(TV_Q197392,TV_Q197388),V_g),s(TV_Q197392,V_x))) => s(t_fun(TV_Q197392,TV_Q197388),V_f) = s(t_fun(TV_Q197392,TV_Q197388),V_g))).
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_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_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,f0)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_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_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_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_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_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_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
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_FUNu_u_EQu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g) <=> ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))).
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_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,f0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
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_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_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_bools_LEFTu_u_ORu_u_OVERu_u_AND, 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_A)) | p(s(t_bool,V_C)))))).
fof(ah4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND, axiom, ![V_C, V_B, V_A]: (((p(s(t_bool,V_B)) & p(s(t_bool,V_C))) | p(s(t_bool,V_A))) <=> ((p(s(t_bool,V_B)) | p(s(t_bool,V_A))) & (p(s(t_bool,V_C)) | p(s(t_bool,V_A)))))).
fof(ah4s_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_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_EXISTSu_u_UNIQUEu_u_THM, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_P, V_x]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),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))) => ![V_P]: (p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),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, V_y]: ((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_y))))) => s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))))).
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_bools_boolu_u_caseu_u_thmu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
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_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f0))))).
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,f0))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
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,f0))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f0))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => p(s(t_bool,f0))))).
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_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_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_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_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_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_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_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_options_optionu_u_nchotomy, axiom, ![TV_u_27a]: ![V_opt]: (s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) | ?[V_x]: s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_options_optionu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,V_v)).
fof(ah4s_options_optionu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
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_OPTIONu_u_MAPu_u_DEFu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))))).
fof(ah4s_options_OPTIONu_u_MAPu_u_DEFu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_h4s_options_option(TV_u_27b),h4s_options_none)).
fof(ah4s_pairs_pairu_u_CASES, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: ?[V_q, V_r]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_q),s(TV_u_27b,V_r)))).
fof(ah4s_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_pairs_pairu_u_caseu_u_thm, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_y, V_x, V_f]: s(TV_u_27a,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27b,V_x),s(TV_u_27c,V_y))),s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f))) = s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27b,V_x))),s(TV_u_27c,V_y)))).
fof(ah4s_lists_listu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27b,h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f))) = s(TV_u_27b,V_v)).
fof(ah4s_lists_listu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_f, V_a1, V_a0]: s(TV_u_27b,h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a0),s(t_h4s_lists_list(TV_u_27a),V_a1))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f))) = s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1)))).
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_listu_u_nchotomy, axiom, ![TV_u_27a]: ![V_l]: (s(t_h4s_lists_list(TV_u_27a),V_l) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) | ?[V_h, V_t]: 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_h),s(t_h4s_lists_list(TV_u_27a),V_t))))).
fof(ah4s_lbtrees_Lfrepu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27b)),h4s_lbtrees_lfrep),s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27b),h4s_options_none)).
fof(ah4s_lbtrees_Ndrepu_u_def, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_v, V_t1, V_t2, V_xs]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_bool,t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a))))),V_uu_1),s(t_bool,V_v))),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t1))),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t2))),s(t_h4s_lists_list(t_bool),V_xs))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t1),s(t_h4s_lists_list(t_bool),V_xs))),s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t2),s(t_h4s_lists_list(t_bool),V_xs))))) => ![V_uu_0]: (![V_t1, V_t2, V_v]: s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a))))),V_uu_0),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t1))),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t2))),s(t_bool,V_v))) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_bool,t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a))))),V_uu_1),s(t_bool,V_v))),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t1))),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t2))) => ![V_t2, V_t1, V_a, V_x]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),h4s_lbtrees_ndrep(s(TV_u_27a,V_a),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t1),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t2))),s(t_h4s_lists_list(t_bool),V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_lists_listu_u_case(s(t_h4s_lists_list(t_bool),V_x),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_a))),s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a))))),V_uu_0),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t1))),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t2)))))))).
fof(ah4s_lbtrees_isu_u_lbtreeu_u_def, axiom, ![TV_u_27a]: ![V_t]: (p(s(t_bool,h4s_lbtrees_isu_u_lbtree(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t)))) <=> ?[V_P]: (![V_t0]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_bool),V_P),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t0)))) => (s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t0) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),h4s_lbtrees_lfrep) | ?[V_a, V_t1, V_t2]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_bool),V_P),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t1)))) & (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_bool),V_P),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t2)))) & s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t0) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),h4s_lbtrees_ndrep(s(TV_u_27a,V_a),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t1),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t2))))))) & p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),t_bool),V_P),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_t))))))).
fof(ah4s_lbtrees_lbtreeu_u_absrepu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_lbtreeu_u_abs(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),h4s_lbtrees_lbtreeu_u_rep(s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a))))) = s(t_h4s_lbtrees_lbtree(TV_u_27a),V_a)).
fof(ah4s_lbtrees_lbtreeu_u_absrepu_c1, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_lbtrees_isu_u_lbtree(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_r)))) <=> s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),h4s_lbtrees_lbtreeu_u_rep(s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_lbtreeu_u_abs(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_r))))) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),V_r))).
fof(ah4s_lbtrees_pathu_u_followu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_g]: s(t_h4s_options_option(TV_u_27a),h4s_lbtrees_pathu_u_follow(s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g),s(TV_u_27b,V_x),s(t_h4s_lists_list(t_bool),h4s_lists_nil))) = s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)),TV_u_27a),h4s_pairs_fst),s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b))),happ(s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g),s(TV_u_27b,V_x)))))).
fof(ah4s_lbtrees_pathu_u_followu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_4]: (![V_g, V_h, V_y, V_t, V_z]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_bool,t_fun(TV_u_27b,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),t_fun(t_bool,t_fun(TV_u_27b,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)))))),V_uu_4),s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g))),s(t_bool,V_h))),s(TV_u_27b,V_y))),s(t_h4s_lists_list(t_bool),V_t))),s(TV_u_27b,V_z))) = s(t_h4s_options_option(TV_u_27a),h4s_lbtrees_pathu_u_follow(s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g),s(TV_u_27b,h4s_bools_cond(s(t_bool,V_h),s(TV_u_27b,V_y),s(TV_u_27b,V_z))),s(t_h4s_lists_list(t_bool),V_t))) => ![V_uu_3]: (![V_g, V_h, V_t, V_y]: s(t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)))))),V_uu_3),s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g))),s(t_bool,V_h))),s(t_h4s_lists_list(t_bool),V_t))),s(TV_u_27b,V_y))) = s(t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_bool,t_fun(TV_u_27b,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),t_fun(t_bool,t_fun(TV_u_27b,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)))))),V_uu_4),s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g))),s(t_bool,V_h))),s(TV_u_27b,V_y))),s(t_h4s_lists_list(t_bool),V_t))) => ![V_uu_2]: (![V_g, V_h, V_t, V_v2]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_bool),t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a))))),V_uu_2),s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g))),s(t_bool,V_h))),s(t_h4s_lists_list(t_bool),V_t))),s(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),V_v2))) = s(t_h4s_options_option(TV_u_27a),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),V_v2),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27b,t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)))))),V_uu_3),s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g))),s(t_bool,V_h))),s(t_h4s_lists_list(t_bool),V_t))))) => ![V_uu_1]: (![V_g, V_h, V_t, V_a]: s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a)))))),V_uu_1),s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g))),s(t_bool,V_h))),s(t_h4s_lists_list(t_bool),V_t))),s(TV_u_27a,V_a))) = s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_bool),t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a))))),V_uu_2),s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g))),s(t_bool,V_h))),s(t_h4s_lists_list(t_bool),V_t))) => ![V_uu_0]: (![V_g, V_h, V_t, V_v]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)),t_h4s_options_option(TV_u_27a))))),V_uu_0),s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g))),s(t_bool,V_h))),s(t_h4s_lists_list(t_bool),V_t))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)),V_v))) = s(t_h4s_options_option(TV_u_27a),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)),V_v),s(t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27b),t_h4s_options_option(TV_u_27a)))))),V_uu_1),s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g))),s(t_bool,V_h))),s(t_h4s_lists_list(t_bool),V_t))))) => ![V_x, V_t, V_h, V_g]: s(t_h4s_options_option(TV_u_27a),h4s_lbtrees_pathu_u_follow(s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g),s(TV_u_27b,V_x),s(t_h4s_lists_list(t_bool),h4s_lists_cons(s(t_bool,V_h),s(t_h4s_lists_list(t_bool),V_t))))) = s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_case(s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b))),happ(s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g),s(TV_u_27b,V_x))),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)),t_h4s_options_option(TV_u_27a))))),V_uu_0),s(t_fun(TV_u_27b,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,TV_u_27b)))),V_g))),s(t_bool,V_h))),s(t_h4s_lists_list(t_bool),V_t))))))))))).
fof(ah4s_lbtrees_Lfu_u_def, axiom, ![TV_u_27a]: s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_lf) = s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_lbtreeu_u_abs(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),h4s_lbtrees_lfrep)))).
fof(ah4s_lbtrees_Ndu_u_def, axiom, ![TV_u_27a]: ![V_t2, V_t1, V_a]: s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_nd(s(TV_u_27a,V_a),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1),s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2))) = s(t_h4s_lbtrees_lbtree(TV_u_27a),h4s_lbtrees_lbtreeu_u_abs(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),h4s_lbtrees_ndrep(s(TV_u_27a,V_a),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),h4s_lbtrees_lbtreeu_u_rep(s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t1))),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(TV_u_27a)),h4s_lbtrees_lbtreeu_u_rep(s(t_h4s_lbtrees_lbtree(TV_u_27a),V_t2)))))))).
fof(ch4s_lbtrees_lbtreeu_u_ueu_u_Axiom, conjecture, ![TV_u_27b,TV_u_27a]: ![V_uu_5]: (![V_b, V_y, V_g, V_z]: s(t_h4s_lbtrees_lbtree(TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)))),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b))))),V_uu_5),s(TV_u_27b,V_b))),s(TV_u_27a,V_y))),s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g))),s(TV_u_27a,V_z))) = s(t_h4s_lbtrees_lbtree(TV_u_27b),h4s_lbtrees_nd(s(TV_u_27b,V_b),s(t_h4s_lbtrees_lbtree(TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g),s(TV_u_27a,V_y))),s(t_h4s_lbtrees_lbtree(TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g),s(TV_u_27a,V_z))))) => ![V_uu_4]: (![V_b, V_g, V_y]: s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b))))),V_uu_4),s(TV_u_27b,V_b))),s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g))),s(TV_u_27a,V_y))) = s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)))),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b))))),V_uu_5),s(TV_u_27b,V_b))),s(TV_u_27a,V_y))),s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g))) => ![V_uu_3]: (![V_b, V_g, V_v2]: s(t_h4s_lbtrees_lbtree(TV_u_27b),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_h4s_lbtrees_lbtree(TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_h4s_lbtrees_lbtree(TV_u_27b))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_h4s_lbtrees_lbtree(TV_u_27b)))),V_uu_3),s(TV_u_27b,V_b))),s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),V_v2))) = s(t_h4s_lbtrees_lbtree(TV_u_27b),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),V_v2),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b))))),V_uu_4),s(TV_u_27b,V_b))),s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g))))) => ![V_uu_2]: (![V_g, V_b]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_h4s_lbtrees_lbtree(TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_h4s_lbtrees_lbtree(TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_h4s_lbtrees_lbtree(TV_u_27b)))),V_uu_2),s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g))),s(TV_u_27b,V_b))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_h4s_lbtrees_lbtree(TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_h4s_lbtrees_lbtree(TV_u_27b))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_h4s_lbtrees_lbtree(TV_u_27b)))),V_uu_3),s(TV_u_27b,V_b))),s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g))) => ![V_uu_1]: (![V_g, V_v]: s(t_h4s_lbtrees_lbtree(TV_u_27b),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27a)),t_h4s_lbtrees_lbtree(TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27a)),t_h4s_lbtrees_lbtree(TV_u_27b))),V_uu_1),s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g))),s(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27a)),V_v))) = s(t_h4s_lbtrees_lbtree(TV_u_27b),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27a)),V_v),s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_h4s_lbtrees_lbtree(TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_h4s_lbtrees_lbtree(TV_u_27b)))),V_uu_2),s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g))))) => ![V_uu_0]: (![V_f, V_g]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27a)))),t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27a)))),V_f))),s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g)))) <=> ![V_x]: s(t_h4s_lbtrees_lbtree(TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g),s(TV_u_27a,V_x))) = s(t_h4s_lbtrees_lbtree(TV_u_27b),h4s_options_optionu_u_case(s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27a))),happ(s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27a)))),V_f),s(TV_u_27a,V_x))),s(t_h4s_lbtrees_lbtree(TV_u_27b),h4s_lbtrees_lf),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27a)),t_h4s_lbtrees_lbtree(TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27a)),t_h4s_lbtrees_lbtree(TV_u_27b))),V_uu_1),s(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),V_g)))))) => ![V_f]: p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27a)))),t_fun(t_fun(TV_u_27a,t_h4s_lbtrees_lbtree(TV_u_27b)),t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_h4s_options_option(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27a)))),V_f))))))))))))).
