%   ORIGINAL: h4/enumeral/LESS__ALL__OU__UO__LEM
% 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/enumeral/LESS__ALL__UO__LEM: !s cmp a. h4/enumeral/LESS__ALL cmp a s ==> h4/enumeral/UO cmp (h4/pred__set/INSERT a h4/pred__set/EMPTY) s = h4/pred__set/INSERT a s
% Assm: h4/enumeral/LESS__ALL__OU: !x v u cmp. h4/enumeral/LESS__ALL cmp x (h4/enumeral/OU cmp u v) <=> h4/enumeral/LESS__ALL cmp x u /\ h4/enumeral/LESS__ALL cmp x v
% Assm: h4/enumeral/LESS__ALL0: !x s cmp. h4/enumeral/LESS__ALL cmp x s <=> (!y. h4/bool/IN y s ==> h4/toto/apto cmp x y = h4/toto/LESS)
% Assm: h4/enumeral/LESS__UO__LEM: !y x s cmp. (!z. h4/bool/IN z (h4/enumeral/UO cmp (h4/pred__set/INSERT x h4/pred__set/EMPTY) s) ==> h4/toto/apto cmp y z = h4/toto/LESS) <=> h4/toto/apto cmp y x = h4/toto/LESS
% Assm: h4/enumeral/OU0: !u t cmp. h4/enumeral/OU cmp t u = h4/pred__set/UNION (h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/bool/IN x t /\ (!z. h4/bool/IN z u ==> h4/toto/apto cmp x z = h4/toto/LESS)))) u
% Assm: h4/enumeral/UO0: !t s cmp. h4/enumeral/UO cmp s t = h4/pred__set/UNION s (h4/pred__set/GSPEC (\y. h4/pair/_2C y (h4/bool/IN y t /\ (!z. h4/bool/IN z s ==> h4/toto/apto cmp z y = h4/toto/LESS))))
% Assm: h4/enumeral/OU__ASSOC: !cmp c b a. h4/enumeral/OU cmp a (h4/enumeral/OU cmp b c) = h4/enumeral/OU cmp (h4/enumeral/OU cmp a b) c
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/pair/PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% 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/pred__set/IN__UNION: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x t
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = 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/enumeral/bl__to__set0_c2: !x t cmp b. h4/enumeral/bl__to__set cmp (h4/enumeral/onebl x t b) = h4/enumeral/OU cmp (h4/pred__set/UNION (h4/pred__set/INSERT x h4/pred__set/EMPTY) (h4/pred__set/GSPEC (\y. h4/pair/_2C y (h4/bool/IN y (h4/enumeral/ENUMERAL cmp t) /\ h4/toto/apto cmp x y = h4/toto/LESS)))) (h4/enumeral/bl__to__set cmp b)
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/bool/UNWIND__FORALL__THM2: !v f. (!x. x = v ==> f x) <=> f v
% Assm: h4/pred__set/NOT__IN__EMPTY: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% 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/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/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/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/enumeral/OU__EMPTY: !t cmp. h4/enumeral/OU cmp t h4/pred__set/EMPTY = t
% Assm: h4/enumeral/EMPTY__OU: !sl cmp. h4/enumeral/OU cmp h4/pred__set/EMPTY sl = sl
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/toto/toto__trans__less_c0: !z y x c. h4/toto/apto c x y = h4/toto/LESS /\ h4/toto/apto c y z = h4/toto/LESS ==> h4/toto/apto c x z = h4/toto/LESS
% Assm: h4/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% 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__SING: !y x. h4/bool/IN x (h4/pred__set/INSERT y h4/pred__set/EMPTY) <=> x = y
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ 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/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/toto/toto__refl: !x c. h4/toto/apto c x x = h4/toto/EQUAL
% 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/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/toto/all__cpn__distinct_c3: ~(h4/toto/EQUAL = h4/toto/LESS)
% Assm: h4/sat/dc__cond: !s r q p. (p <=> h4/bool/COND q r s) <=> (p \/ q \/ ~s) /\ (p \/ ~r \/ ~q) /\ (p \/ ~r \/ ~s) /\ (~q \/ r \/ ~p) /\ (q \/ s \/ ~p)
% 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/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/toto/toto__trans__less_c4: !z y x c. h4/toto/apto c x y = h4/toto/LESS /\ h4/toto/apto c y z = h4/toto/EQUAL ==> h4/toto/apto c x z = h4/toto/LESS
% Assm: h4/toto/toto__trans__less_c3: !z y x c. h4/toto/apto c y x = h4/toto/GREATER /\ h4/toto/apto c y z = h4/toto/LESS ==> h4/toto/apto c x z = h4/toto/LESS
% Assm: h4/toto/toto__equal__sym: !y x c. h4/toto/apto c x y = h4/toto/EQUAL <=> h4/toto/apto c y x = h4/toto/EQUAL
% Assm: h4/toto/cpn__nchotomy: !a. a = h4/toto/LESS \/ a = h4/toto/EQUAL \/ a = h4/toto/GREATER
% Assm: h4/toto/toto__trans__less_c1: !z y x c. h4/toto/apto c x y = h4/toto/LESS /\ h4/toto/apto c z y = h4/toto/GREATER ==> h4/toto/apto c x z = h4/toto/LESS
% Assm: h4/pred__set/SUBSET__DEF: !t s. h4/pred__set/SUBSET s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN x t)
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/bool/OR__CLAUSES_c1: !t. t \/ T <=> T
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/pred__set/GSPEC__F: h4/pred__set/GSPEC (\x. h4/pair/_2C x F) = h4/pred__set/EMPTY
% Assm: h4/pred__set/UNION__EMPTY_c0: !s. h4/pred__set/UNION h4/pred__set/EMPTY s = s
% Assm: h4/pred__set/GSPEC__ID: !y. h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/bool/IN x y)) = y
% Assm: h4/pred__set/UNION__EMPTY_c1: !s. h4/pred__set/UNION s h4/pred__set/EMPTY = s
% Assm: h4/enumeral/bt__to__set_c1: !x r l cmp. h4/enumeral/ENUMERAL cmp (h4/enumeral/node l x r) = h4/pred__set/UNION (h4/pred__set/UNION (h4/pred__set/GSPEC (\y. h4/pair/_2C y (h4/bool/IN y (h4/enumeral/ENUMERAL cmp l) /\ h4/toto/apto cmp y x = h4/toto/LESS))) (h4/pred__set/INSERT x h4/pred__set/EMPTY)) (h4/pred__set/GSPEC (\z. h4/pair/_2C z (h4/bool/IN z (h4/enumeral/ENUMERAL cmp r) /\ h4/toto/apto cmp x z = h4/toto/LESS)))
% Assm: h4/res__quan/RES__EXISTS__UNIQUE__NULL: !p m. h4/bool/RES__EXISTS__UNIQUE p (\x. m) <=> (?x. p = h4/pred__set/INSERT x h4/pred__set/EMPTY) /\ m
% Assm: h4/bool/RIGHT__OR__OVER__AND: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm: h4/bool/BETA__THM: !y f. (\x. f x) y = f y
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/ind__type/FCONS0_c1: !n f a. h4/ind__type/FCONS a f (h4/num/SUC n) = f n
% Assm: h4/ind__type/CONSTR__REC: !Fn. ?f. !c i r. f (h4/ind__type/CONSTR c i r) = Fn c i r (\n. f (r n))
% Assm: h4/ind__type/FCONS0_c0: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/enumeral/onebl0: h4/enumeral/onebl = h4/enumeral/_20_40ind__typeenumeral6
% Assm: h4/enumeral/zerbl0: h4/enumeral/zerbl = h4/enumeral/_20_40ind__typeenumeral5
% Assm: h4/enumeral/nbl0: h4/enumeral/nbl = h4/enumeral/_20_40ind__typeenumeral4
% Assm: h4/enumeral/hidden____20__40ind____typeenumeral6____def: h4/enumeral/_20_40ind__typeenumeral6 = (\a0 a1 a2. h4/enumeral/_20_40ind__typeenumeral7 ((\a00 a10 a20. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C a00 a10) (h4/ind__type/FCONS a20 (\n. h4/ind__type/BOTTOM))) a0 a1 (h4/enumeral/_20_40ind__typeenumeral8 a2)))
% Assm: h4/enumeral/bl__repfns_c1: !r. (\a0_27. !_27bl_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (\n. h4/ind__type/BOTTOM) \/ (?a. a0_270 = (\a0. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a0 (\n. h4/ind__type/BOTTOM))) a /\ _27bl_27 a) \/ (?a0 a1 a2. a0_270 = (\a00 a10 a20. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C a00 a10) (h4/ind__type/FCONS a20 (\n. h4/ind__type/BOTTOM))) a0 a1 a2 /\ _27bl_27 a2) ==> _27bl_27 a0_270) ==> _27bl_27 a0_27) r <=> h4/enumeral/_20_40ind__typeenumeral8 (h4/enumeral/_20_40ind__typeenumeral7 r) = r
% Assm: h4/enumeral/bl__repfns_c0: !a. h4/enumeral/_20_40ind__typeenumeral7 (h4/enumeral/_20_40ind__typeenumeral8 a) = a
% Assm: h4/enumeral/hidden____20__40ind____typeenumeral4____def: h4/enumeral/_20_40ind__typeenumeral4 = h4/enumeral/_20_40ind__typeenumeral7 (h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (\n. h4/ind__type/BOTTOM))
% Assm: h4/enumeral/hidden____20__40ind____typeenumeral5____def: h4/enumeral/_20_40ind__typeenumeral5 = (\a. h4/enumeral/_20_40ind__typeenumeral7 ((\a0. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a0 (\n. h4/ind__type/BOTTOM))) (h4/enumeral/_20_40ind__typeenumeral8 a)))
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/enumeral/node0: h4/enumeral/node = h4/enumeral/_20_40ind__typeenumeral1
% Assm: h4/enumeral/nt0: h4/enumeral/nt = h4/enumeral/_20_40ind__typeenumeral0
% Assm: h4/enumeral/bt__repfns_c1: !r. (\a0_27. !_27bt_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM) \/ (?a0 a1 a2. a0_270 = (\a00 a10 a20. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a10 (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a20 (\n. h4/ind__type/BOTTOM)))) a0 a1 a2 /\ _27bt_27 a0 /\ _27bt_27 a2) ==> _27bt_27 a0_270) ==> _27bt_27 a0_27) r <=> h4/enumeral/_20_40ind__typeenumeral3 (h4/enumeral/_20_40ind__typeenumeral2 r) = r
% Assm: h4/enumeral/hidden____20__40ind____typeenumeral1____def: h4/enumeral/_20_40ind__typeenumeral1 = (\a0 a1 a2. h4/enumeral/_20_40ind__typeenumeral2 ((\a00 a10 a20. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a10 (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a20 (\n. h4/ind__type/BOTTOM)))) (h4/enumeral/_20_40ind__typeenumeral3 a0) a1 (h4/enumeral/_20_40ind__typeenumeral3 a2)))
% Assm: h4/enumeral/hidden____20__40ind____typeenumeral0____def: h4/enumeral/_20_40ind__typeenumeral0 = h4/enumeral/_20_40ind__typeenumeral2 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM))
% Assm: h4/enumeral/bt__repfns_c0: !a. h4/enumeral/_20_40ind__typeenumeral2 (h4/enumeral/_20_40ind__typeenumeral3 a) = a
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/res__quan/RES__EXISTS__NULL: !p m. h4/bool/RES__EXISTS p (\x. m) <=> ~(p = h4/pred__set/EMPTY) /\ m
% Assm: h4/res__quan/RES__EXISTS__UNIQUE: !f P. h4/bool/RES__EXISTS__UNIQUE P f <=> h4/bool/RES__EXISTS P (\x. f x) /\ h4/bool/RES__FORALL P (\x. h4/bool/RES__FORALL P (\y. f x /\ f y ==> x = y))
% Assm: h4/res__quan/RES__FORALL: !f P. h4/bool/RES__FORALL P f <=> (!x. h4/bool/IN x P ==> f x)
% 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/EXISTS__OR__THM: !Q P. (?x. P x \/ Q x) <=> (?x. P x) \/ (?x. Q x)
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/LEFT__OR__EXISTS__THM: !Q P. (?x. P x) \/ Q <=> (?x. P x \/ Q)
% 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/bool/RIGHT__EXISTS__AND__THM: !Q P. (?x. P /\ Q x) <=> P /\ (?x. Q x)
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__CLAUSES_c4: !t. t /\ t <=> t
% Assm: h4/bool/SKOLEM__THM: !P. (!x. ?y. P x y) <=> (?f. !x. P x (f x))
% Goal: !t s cmp a. h4/enumeral/LESS__ALL cmp a s /\ h4/enumeral/LESS__ALL cmp a t ==> h4/enumeral/OU cmp (h4/enumeral/UO cmp (h4/pred__set/INSERT a h4/pred__set/EMPTY) s) t = h4/pred__set/INSERT a (h4/enumeral/OU cmp s t)
%   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_enumerals_LESSu_u_ALLu_u_UOu_u_LEM]: !s cmp a. h4/enumeral/LESS__ALL cmp a s ==> h4/enumeral/UO cmp (h4/pred__set/INSERT a h4/pred__set/EMPTY) s = h4/pred__set/INSERT a s
% Assm [h4s_enumerals_LESSu_u_ALLu_u_OU]: !x v u cmp. h4/enumeral/LESS__ALL cmp x (h4/enumeral/OU cmp u v) <=> h4/enumeral/LESS__ALL cmp x u /\ h4/enumeral/LESS__ALL cmp x v
% Assm [h4s_enumerals_LESSu_u_ALL0]: !x s cmp. h4/enumeral/LESS__ALL cmp x s <=> (!y. h4/bool/IN y s ==> h4/toto/apto cmp x y = h4/toto/LESS)
% Assm [h4s_enumerals_LESSu_u_UOu_u_LEM]: !y x s cmp. (!z. h4/bool/IN z (h4/enumeral/UO cmp (h4/pred__set/INSERT x h4/pred__set/EMPTY) s) ==> h4/toto/apto cmp y z = h4/toto/LESS) <=> h4/toto/apto cmp y x = h4/toto/LESS
% Assm [h4s_enumerals_OU0]: !_0. (!t u cmp x. ?v. (v <=> h4/bool/IN x t /\ (!z. h4/bool/IN z u ==> h4/toto/apto cmp x z = h4/toto/LESS)) /\ happ (happ (happ (happ _0 t) u) cmp) x = h4/pair/_2C x v) ==> (!u t cmp. h4/enumeral/OU cmp t u = h4/pred__set/UNION (h4/pred__set/GSPEC (happ (happ (happ _0 t) u) cmp)) u)
% Assm [h4s_enumerals_UO0]: !_0. (!t s cmp y. ?v. (v <=> h4/bool/IN y t /\ (!z. h4/bool/IN z s ==> h4/toto/apto cmp z y = h4/toto/LESS)) /\ happ (happ (happ (happ _0 t) s) cmp) y = h4/pair/_2C y v) ==> (!t s cmp. h4/enumeral/UO cmp s t = h4/pred__set/UNION s (h4/pred__set/GSPEC (happ (happ (happ _0 t) s) cmp)))
% Assm [h4s_enumerals_OUu_u_ASSOC]: !cmp c b a. h4/enumeral/OU cmp a (h4/enumeral/OU cmp b c) = h4/enumeral/OU cmp (h4/enumeral/OU cmp a b) c
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_pairs_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% 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_predu_u_sets_INu_u_UNION]: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x t
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% 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_enumerals_blu_u_tou_u_set0u_c2]: !_0. (!t cmp x y. ?v. (v <=> h4/bool/IN y (h4/enumeral/ENUMERAL cmp t) /\ h4/toto/apto cmp x y = h4/toto/LESS) /\ happ (happ (happ (happ _0 t) cmp) x) y = h4/pair/_2C y v) ==> (!x t cmp b. h4/enumeral/bl__to__set cmp (happ (happ (happ h4/enumeral/onebl x) t) b) = h4/enumeral/OU cmp (h4/pred__set/UNION (h4/pred__set/INSERT x h4/pred__set/EMPTY) (h4/pred__set/GSPEC (happ (happ (happ _0 t) cmp) x))) (h4/enumeral/bl__to__set cmp b))
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_bools_UNWINDu_u_FORALLu_u_THM2]: !v f. (!x. x = v ==> happ f x) <=> happ f v
% Assm [h4s_predu_u_sets_NOTu_u_INu_u_EMPTY]: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% 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_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_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_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_enumerals_OUu_u_EMPTY]: !t cmp. h4/enumeral/OU cmp t h4/pred__set/EMPTY = t
% Assm [h4s_enumerals_EMPTYu_u_OU]: !sl cmp. h4/enumeral/OU cmp h4/pred__set/EMPTY sl = sl
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_totos_totou_u_transu_u_lessu_c0]: !z y x c. h4/toto/apto c x y = h4/toto/LESS /\ h4/toto/apto c y z = h4/toto/LESS ==> h4/toto/apto c x z = h4/toto/LESS
% Assm [h4s_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% 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_SING]: !y x. h4/bool/IN x (h4/pred__set/INSERT y h4/pred__set/EMPTY) <=> x = y
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ 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_NOTu_u_CLAUSESu_c2]: ~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_totos_totou_u_refl]: !x c. h4/toto/apto c x x = h4/toto/EQUAL
% 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_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_totos_allu_u_cpnu_u_distinctu_c3]: ~(h4/toto/EQUAL = h4/toto/LESS)
% Assm [h4s_sats_dcu_u_cond]: !s r q p. (p <=> h4/bool/COND q r s) <=> (p \/ q \/ ~s) /\ (p \/ ~r \/ ~q) /\ (p \/ ~r \/ ~s) /\ (~q \/ r \/ ~p) /\ (q \/ s \/ ~p)
% 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_Fu_u_IMP]: !t. ~t ==> t ==> F
% 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_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_totos_totou_u_transu_u_lessu_c4]: !z y x c. h4/toto/apto c x y = h4/toto/LESS /\ h4/toto/apto c y z = h4/toto/EQUAL ==> h4/toto/apto c x z = h4/toto/LESS
% Assm [h4s_totos_totou_u_transu_u_lessu_c3]: !z y x c. h4/toto/apto c y x = h4/toto/GREATER /\ h4/toto/apto c y z = h4/toto/LESS ==> h4/toto/apto c x z = h4/toto/LESS
% Assm [h4s_totos_totou_u_equalu_u_sym]: !y x c. h4/toto/apto c x y = h4/toto/EQUAL <=> h4/toto/apto c y x = h4/toto/EQUAL
% Assm [h4s_totos_cpnu_u_nchotomy]: !a. a = h4/toto/LESS \/ a = h4/toto/EQUAL \/ a = h4/toto/GREATER
% Assm [h4s_totos_totou_u_transu_u_lessu_c1]: !z y x c. h4/toto/apto c x y = h4/toto/LESS /\ h4/toto/apto c z y = h4/toto/GREATER ==> h4/toto/apto c x z = h4/toto/LESS
% Assm [h4s_predu_u_sets_SUBSETu_u_DEF]: !t s. h4/pred__set/SUBSET s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN x t)
% 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_ORu_u_CLAUSESu_c1]: !t. t \/ T <=> T
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_predu_u_sets_GSPECu_u_F]: !_0. (!x. happ _0 x = h4/pair/_2C x F) ==> h4/pred__set/GSPEC _0 = h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_UNIONu_u_EMPTYu_c0]: !s. h4/pred__set/UNION h4/pred__set/EMPTY s = s
% Assm [h4s_predu_u_sets_GSPECu_u_ID]: !_0. (!y x. happ (happ _0 y) x = h4/pair/_2C x (h4/bool/IN x y)) ==> (!y. h4/pred__set/GSPEC (happ _0 y) = y)
% Assm [h4s_predu_u_sets_UNIONu_u_EMPTYu_c1]: !s. h4/pred__set/UNION s h4/pred__set/EMPTY = s
% Assm [h4s_enumerals_btu_u_tou_u_setu_c1]: !_1. (!r cmp x z. ?v. (v <=> h4/bool/IN z (h4/enumeral/ENUMERAL cmp r) /\ h4/toto/apto cmp x z = h4/toto/LESS) /\ happ (happ (happ (happ _1 r) cmp) x) z = h4/pair/_2C z v) ==> (!_0. (!l cmp x y. ?v. (v <=> h4/bool/IN y (h4/enumeral/ENUMERAL cmp l) /\ h4/toto/apto cmp y x = h4/toto/LESS) /\ happ (happ (happ (happ _0 l) cmp) x) y = h4/pair/_2C y v) ==> (!x r l cmp. h4/enumeral/ENUMERAL cmp (happ (happ (happ h4/enumeral/node l) x) r) = h4/pred__set/UNION (h4/pred__set/UNION (h4/pred__set/GSPEC (happ (happ (happ _0 l) cmp) x)) (h4/pred__set/INSERT x h4/pred__set/EMPTY)) (h4/pred__set/GSPEC (happ (happ (happ _1 r) cmp) x))))
% Assm [h4s_resu_u_quans_RESu_u_EXISTSu_u_UNIQUEu_u_NULL]: !_0. (!m x. happ (happ _0 m) x <=> m) ==> (!p m. h4/bool/RES__EXISTS__UNIQUE p (happ _0 m) <=> (?x. p = h4/pred__set/INSERT x h4/pred__set/EMPTY) /\ m)
% Assm [h4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND]: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm [h4s_bools_BETAu_u_THM]: !y f. happ f y = happ f y
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_indu_u_types_FCONS0u_c1]: !n f a. happ (h4/ind__type/FCONS a f) (h4/num/SUC n) = happ f n
% Assm [h4s_indu_u_types_CONSTRu_u_REC]: !_0. (!f r n. happ (happ (happ _0 f) r) n = happ f (happ r n)) ==> (!Fn. ?f. !c i r. happ f (h4/ind__type/CONSTR c i r) = happ (happ (happ (happ Fn c) i) r) (happ (happ _0 f) r))
% Assm [h4s_indu_u_types_FCONS0u_c0]: !f a. happ (h4/ind__type/FCONS a f) h4/num/0 = a
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_enumerals_onebl0]: h4/enumeral/onebl = h4/enumeral/_20_40ind__typeenumeral6
% Assm [h4s_enumerals_zerbl0]: h4/enumeral/zerbl = h4/enumeral/_20_40ind__typeenumeral5
% Assm [h4s_enumerals_nbl0]: h4/enumeral/nbl = h4/enumeral/_20_40ind__typeenumeral4
% Assm [h4s_enumerals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeenumeral6u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x x. happ (happ (happ h4/enumeral/_20_40ind__typeenumeral6 x) x) x = h4/enumeral/_20_40ind__typeenumeral7 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C x x) (h4/ind__type/FCONS (h4/enumeral/_20_40ind__typeenumeral8 x) _0)))
% Assm [h4s_enumerals_blu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27bl_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C h4/bool/ARB h4/bool/ARB) _0 \/ (?a. a0_270 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a _0) /\ happ _27bl_27 a) \/ (?a0 a1 a2. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C a0 a1) (h4/ind__type/FCONS a2 _0) /\ happ _27bl_27 a2) ==> happ _27bl_27 a0_270) ==> happ _27bl_27 r) <=> h4/enumeral/_20_40ind__typeenumeral8 (h4/enumeral/_20_40ind__typeenumeral7 r) = r)
% Assm [h4s_enumerals_blu_u_repfnsu_c0]: !a. h4/enumeral/_20_40ind__typeenumeral7 (h4/enumeral/_20_40ind__typeenumeral8 a) = a
% Assm [h4s_enumerals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeenumeral4u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> h4/enumeral/_20_40ind__typeenumeral4 = h4/enumeral/_20_40ind__typeenumeral7 (h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C h4/bool/ARB h4/bool/ARB) _0)
% Assm [h4s_enumerals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeenumeral5u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x. happ h4/enumeral/_20_40ind__typeenumeral5 x = h4/enumeral/_20_40ind__typeenumeral7 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS (h4/enumeral/_20_40ind__typeenumeral8 x) _0)))
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_enumerals_node0]: h4/enumeral/node = h4/enumeral/_20_40ind__typeenumeral1
% Assm [h4s_enumerals_nt0]: h4/enumeral/nt = h4/enumeral/_20_40ind__typeenumeral0
% Assm [h4s_enumerals_btu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27bt_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0 \/ (?a0 a1 a2. a0_270 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a1 (h4/ind__type/FCONS a0 (h4/ind__type/FCONS a2 _0)) /\ happ _27bt_27 a0 /\ happ _27bt_27 a2) ==> happ _27bt_27 a0_270) ==> happ _27bt_27 r) <=> h4/enumeral/_20_40ind__typeenumeral3 (h4/enumeral/_20_40ind__typeenumeral2 r) = r)
% Assm [h4s_enumerals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeenumeral1u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x x'. happ (happ (happ h4/enumeral/_20_40ind__typeenumeral1 x) x) x' = h4/enumeral/_20_40ind__typeenumeral2 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) x (h4/ind__type/FCONS (h4/enumeral/_20_40ind__typeenumeral3 x) (h4/ind__type/FCONS (h4/enumeral/_20_40ind__typeenumeral3 x') _0))))
% Assm [h4s_enumerals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeenumeral0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> h4/enumeral/_20_40ind__typeenumeral0 = h4/enumeral/_20_40ind__typeenumeral2 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0)
% Assm [h4s_enumerals_btu_u_repfnsu_c0]: !a. h4/enumeral/_20_40ind__typeenumeral2 (h4/enumeral/_20_40ind__typeenumeral3 a) = a
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_resu_u_quans_RESu_u_EXISTSu_u_NULL]: !_0. (!m x. happ (happ _0 m) x <=> m) ==> (!p m. h4/bool/RES__EXISTS p (happ _0 m) <=> ~(p = h4/pred__set/EMPTY) /\ m)
% Assm [h4s_resu_u_quans_RESu_u_EXISTSu_u_UNIQUE]: !_2. (!f x y. happ (happ (happ _2 f) x) y <=> happ f x /\ happ f y ==> x = y) ==> (!_1. (!P f x. happ (happ (happ _1 P) f) x <=> h4/bool/RES__FORALL P (happ (happ _2 f) x)) ==> (!_0. (!f x. happ (happ _0 f) x <=> happ f x) ==> (!f P. h4/bool/RES__EXISTS__UNIQUE P f <=> h4/bool/RES__EXISTS P (happ _0 f) /\ h4/bool/RES__FORALL P (happ _0 (happ (happ _1 P) f)))))
% Assm [h4s_resu_u_quans_RESu_u_FORALL]: !f P. h4/bool/RES__FORALL P f <=> (!x. h4/bool/IN x P ==> happ f x)
% 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_EXISTSu_u_ORu_u_THM]: !Q P. (?x. happ P x \/ happ Q x) <=> (?x. happ P x) \/ (?x. happ Q x)
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_LEFTu_u_ORu_u_EXISTSu_u_THM]: !Q P. (?x. happ P x) \/ Q <=> (?x. happ P x \/ Q)
% 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_bools_RIGHTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. P /\ happ Q x) <=> P /\ (?x. happ Q x)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c4]: !t. t /\ t <=> t
% Assm [h4s_bools_SKOLEMu_u_THM]: !P. (!x. ?y. happ (happ P x) y) <=> (?f. !x. happ (happ P x) (happ f x))
% Goal: !t s cmp a. h4/enumeral/LESS__ALL cmp a s /\ h4/enumeral/LESS__ALL cmp a t ==> h4/enumeral/OU cmp (h4/enumeral/UO cmp (h4/pred__set/INSERT a h4/pred__set/EMPTY) s) t = h4/pred__set/INSERT a (h4/enumeral/OU cmp s t)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t0))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q1429169,TV_Q1429165]: ![V_f, V_g]: (![V_x]: s(TV_Q1429165,happ(s(t_fun(TV_Q1429169,TV_Q1429165),V_f),s(TV_Q1429169,V_x))) = s(TV_Q1429165,happ(s(t_fun(TV_Q1429169,TV_Q1429165),V_g),s(TV_Q1429169,V_x))) => s(t_fun(TV_Q1429169,TV_Q1429165),V_f) = s(t_fun(TV_Q1429169,TV_Q1429165),V_g))).
fof(ah4s_enumerals_LESSu_u_ALLu_u_UOu_u_LEM, axiom, ![TV_u_27a]: ![V_s, V_cmp, V_a]: (p(s(t_bool,h4s_enumerals_lessu_u_all(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(TV_u_27a,V_a),s(t_fun(TV_u_27a,t_bool),V_s)))) => s(t_fun(TV_u_27a,t_bool),h4s_enumerals_uo(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_a),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_a),s(t_fun(TV_u_27a,t_bool),V_s))))).
fof(ah4s_enumerals_LESSu_u_ALLu_u_OU, axiom, ![TV_u_27a]: ![V_x, V_v, V_u, V_cmp]: (p(s(t_bool,h4s_enumerals_lessu_u_all(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_enumerals_ou(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_fun(TV_u_27a,t_bool),V_u),s(t_fun(TV_u_27a,t_bool),V_v)))))) <=> (p(s(t_bool,h4s_enumerals_lessu_u_all(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_u)))) & p(s(t_bool,h4s_enumerals_lessu_u_all(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_v))))))).
fof(ah4s_enumerals_LESSu_u_ALL0, axiom, ![TV_u_27a]: ![V_x, V_s, V_cmp]: (p(s(t_bool,h4s_enumerals_lessu_u_all(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) <=> ![V_y]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s)))) => s(t_h4s_totos_cpn,h4s_totos_apto(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(t_h4s_totos_cpn,h4s_totos_less)))).
fof(ah4s_enumerals_LESSu_u_UOu_u_LEM, axiom, ![TV_u_27a]: ![V_y, V_x, V_s, V_cmp]: (![V_z]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_z),s(t_fun(TV_u_27a,t_bool),h4s_enumerals_uo(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))),s(t_fun(TV_u_27a,t_bool),V_s)))))) => s(t_h4s_totos_cpn,h4s_totos_apto(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(TV_u_27a,V_y),s(TV_u_27a,V_z))) = s(t_h4s_totos_cpn,h4s_totos_less)) <=> s(t_h4s_totos_cpn,h4s_totos_apto(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(TV_u_27a,V_y),s(TV_u_27a,V_x))) = s(t_h4s_totos_cpn,h4s_totos_less))).
fof(ah4s_enumerals_OU0, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_t, V_u, V_cmp, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t)))) & ![V_z]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_z),s(t_fun(TV_u_27a,t_bool),V_u)))) => s(t_h4s_totos_cpn,h4s_totos_apto(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(TV_u_27a,V_x),s(TV_u_27a,V_z))) = s(t_h4s_totos_cpn,h4s_totos_less)))) & 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)),happ(s(t_fun(t_h4s_totos_toto(TV_u_27a),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_totos_toto(TV_u_27a),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_totos_toto(TV_u_27a),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),V_t))),s(t_fun(TV_u_27a,t_bool),V_u))),s(t_h4s_totos_toto(TV_u_27a),V_cmp))),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,V_v)))) => ![V_u, V_t, V_cmp]: s(t_fun(TV_u_27a,t_bool),h4s_enumerals_ou(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),V_u))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(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)),happ(s(t_fun(t_h4s_totos_toto(TV_u_27a),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_totos_toto(TV_u_27a),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_totos_toto(TV_u_27a),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),V_t))),s(t_fun(TV_u_27a,t_bool),V_u))),s(t_h4s_totos_toto(TV_u_27a),V_cmp))))),s(t_fun(TV_u_27a,t_bool),V_u))))).
fof(ah4s_enumerals_UO0, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_t, V_s, V_cmp, V_y]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_t)))) & ![V_z]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_z),s(t_fun(TV_u_27a,t_bool),V_s)))) => s(t_h4s_totos_cpn,h4s_totos_apto(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(TV_u_27a,V_z),s(TV_u_27a,V_y))) = s(t_h4s_totos_cpn,h4s_totos_less)))) & 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)),happ(s(t_fun(t_h4s_totos_toto(TV_u_27a),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_totos_toto(TV_u_27a),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_totos_toto(TV_u_27a),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),V_t))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_totos_toto(TV_u_27a),V_cmp))),s(TV_u_27a,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_y),s(t_bool,V_v)))) => ![V_t, V_s, V_cmp]: s(t_fun(TV_u_27a,t_bool),h4s_enumerals_uo(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),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)),happ(s(t_fun(t_h4s_totos_toto(TV_u_27a),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_totos_toto(TV_u_27a),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_totos_toto(TV_u_27a),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),V_t))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_totos_toto(TV_u_27a),V_cmp))))))))).
fof(ah4s_enumerals_OUu_u_ASSOC, axiom, ![TV_u_27a]: ![V_cmp, V_c, V_b, V_a]: s(t_fun(TV_u_27a,t_bool),h4s_enumerals_ou(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_fun(TV_u_27a,t_bool),V_a),s(t_fun(TV_u_27a,t_bool),h4s_enumerals_ou(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_fun(TV_u_27a,t_bool),V_b),s(t_fun(TV_u_27a,t_bool),V_c))))) = s(t_fun(TV_u_27a,t_bool),h4s_enumerals_ou(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_fun(TV_u_27a,t_bool),h4s_enumerals_ou(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_fun(TV_u_27a,t_bool),V_a),s(t_fun(TV_u_27a,t_bool),V_b))),s(t_fun(TV_u_27a,t_bool),V_c)))).
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_TRUTH, axiom, p(s(t_bool,t0))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_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_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,t0))) = 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_predu_u_sets_INu_u_UNION, axiom, ![TV_u_27a]: ![V_x, V_t, 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_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t0) = s(t_bool,V_t) <=> p(s(t_bool,V_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,t0)))).
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_enumerals_blu_u_tou_u_set0u_c2, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_t, V_cmp, V_x, V_y]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),h4s_enumerals_enumeral(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_h4s_enumerals_bt(TV_u_27a),V_t)))))) & s(t_h4s_totos_cpn,h4s_totos_apto(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(t_h4s_totos_cpn,h4s_totos_less))) & 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)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_h4s_totos_toto(TV_u_27a),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_h4s_enumerals_bt(TV_u_27a),t_fun(t_h4s_totos_toto(TV_u_27a),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))))),V_uu_0),s(t_h4s_enumerals_bt(TV_u_27a),V_t))),s(t_h4s_totos_toto(TV_u_27a),V_cmp))),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_y),s(t_bool,V_v)))) => ![V_x, V_t, V_cmp, V_b]: s(t_fun(TV_u_27a,t_bool),h4s_enumerals_blu_u_tou_u_set(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_h4s_enumerals_bl(TV_u_27a),happ(s(t_fun(t_h4s_enumerals_bl(TV_u_27a),t_h4s_enumerals_bl(TV_u_27a)),happ(s(t_fun(t_h4s_enumerals_bt(TV_u_27a),t_fun(t_h4s_enumerals_bl(TV_u_27a),t_h4s_enumerals_bl(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_enumerals_bt(TV_u_27a),t_fun(t_h4s_enumerals_bl(TV_u_27a),t_h4s_enumerals_bl(TV_u_27a)))),h4s_enumerals_onebl),s(TV_u_27a,V_x))),s(t_h4s_enumerals_bt(TV_u_27a),V_t))),s(t_h4s_enumerals_bl(TV_u_27a),V_b))))) = s(t_fun(TV_u_27a,t_bool),h4s_enumerals_ou(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))),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)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_h4s_totos_toto(TV_u_27a),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_h4s_enumerals_bt(TV_u_27a),t_fun(t_h4s_totos_toto(TV_u_27a),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))))),V_uu_0),s(t_h4s_enumerals_bt(TV_u_27a),V_t))),s(t_h4s_totos_toto(TV_u_27a),V_cmp))),s(TV_u_27a,V_x))))))),s(t_fun(TV_u_27a,t_bool),h4s_enumerals_blu_u_tou_u_set(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_h4s_enumerals_bl(TV_u_27a),V_b))))))).
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_UNWINDu_u_FORALLu_u_THM2, axiom, ![TV_u_27a]: ![V_v, V_f]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_v) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_v)))))).
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_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_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_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_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_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_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_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_bools_ORu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,f))) <=> 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_enumerals_OUu_u_EMPTY, axiom, ![TV_u_27a]: ![V_t, V_cmp]: s(t_fun(TV_u_27a,t_bool),h4s_enumerals_ou(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_fun(TV_u_27a,t_bool),V_t),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) = s(t_fun(TV_u_27a,t_bool),V_t)).
fof(ah4s_enumerals_EMPTYu_u_OU, axiom, ![TV_u_27a]: ![V_sl, V_cmp]: s(t_fun(TV_u_27a,t_bool),h4s_enumerals_ou(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty),s(t_fun(TV_u_27a,t_bool),V_sl))) = s(t_fun(TV_u_27a,t_bool),V_sl)).
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_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_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_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
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_totos_totou_u_transu_u_lessu_c0, axiom, ![TV_u_27a]: ![V_z, V_y, V_x, V_c]: ((s(t_h4s_totos_cpn,h4s_totos_apto(s(t_h4s_totos_toto(TV_u_27a),V_c),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(t_h4s_totos_cpn,h4s_totos_less) & s(t_h4s_totos_cpn,h4s_totos_apto(s(t_h4s_totos_toto(TV_u_27a),V_c),s(TV_u_27a,V_y),s(TV_u_27a,V_z))) = s(t_h4s_totos_cpn,h4s_totos_less)) => s(t_h4s_totos_cpn,h4s_totos_apto(s(t_h4s_totos_toto(TV_u_27a),V_c),s(TV_u_27a,V_x),s(TV_u_27a,V_z))) = s(t_h4s_totos_cpn,h4s_totos_less))).
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_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t0))) <=> p(s(t_bool,V_t)))).
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_SING, axiom, ![TV_u_27a]: ![V_y, 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_insert(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t0)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t0)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t0)) => 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,t0))) <=> p(s(t_bool,t0)))).
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,t0)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t0)))).
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_totos_totou_u_refl, axiom, ![TV_u_27a]: ![V_x, V_c]: s(t_h4s_totos_cpn,h4s_totos_apto(s(t_h4s_totos_toto(TV_u_27a),V_c),s(TV_u_27a,V_x),s(TV_u_27a,V_x))) = s(t_h4s_totos_cpn,h4s_totos_equal)).
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,t0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_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_totos_allu_u_cpnu_u_distinctu_c3, axiom, ~ (s(t_h4s_totos_cpn,h4s_totos_equal) = s(t_h4s_totos_cpn,h4s_totos_less))).
fof(ah4s_sats_dcu_u_cond, axiom, ![V_s, V_r, V_q, V_p]: (s(t_bool,V_p) = s(t_bool,h4s_bools_cond(s(t_bool,V_q),s(t_bool,V_r),s(t_bool,V_s))) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_s))))) & ((p(s(t_bool,V_p)) | (~ (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_s))))) & ((~ (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_s)) | ~ (p(s(t_bool,V_p))))))))))).
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_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_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_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_totos_totou_u_transu_u_lessu_c4, axiom, ![TV_u_27a]: ![V_z, V_y, V_x, V_c]: ((s(t_h4s_totos_cpn,h4s_totos_apto(s(t_h4s_totos_toto(TV_u_27a),V_c),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(t_h4s_totos_cpn,h4s_totos_less) & s(t_h4s_totos_cpn,h4s_totos_apto(s(t_h4s_totos_toto(TV_u_27a),V_c),s(TV_u_27a,V_y),s(TV_u_27a,V_z))) = s(t_h4s_totos_cpn,h4s_totos_equal)) => s(t_h4s_totos_cpn,h4s_totos_apto(s(t_h4s_totos_toto(TV_u_27a),V_c),s(TV_u_27a,V_x),s(TV_u_27a,V_z))) = s(t_h4s_totos_cpn,h4s_totos_less))).
fof(ah4s_totos_totou_u_transu_u_lessu_c3, axiom, ![TV_u_27a]: ![V_z, V_y, V_x, V_c]: ((s(t_h4s_totos_cpn,h4s_totos_apto(s(t_h4s_totos_toto(TV_u_27a),V_c),s(TV_u_27a,V_y),s(TV_u_27a,V_x))) = s(t_h4s_totos_cpn,h4s_totos_greater) & s(t_h4s_totos_cpn,h4s_totos_apto(s(t_h4s_totos_toto(TV_u_27a),V_c),s(TV_u_27a,V_y),s(TV_u_27a,V_z))) = s(t_h4s_totos_cpn,h4s_totos_less)) => s(t_h4s_totos_cpn,h4s_totos_apto(s(t_h4s_totos_toto(TV_u_27a),V_c),s(TV_u_27a,V_x),s(TV_u_27a,V_z))) = s(t_h4s_totos_cpn,h4s_totos_less))).
fof(ah4s_totos_totou_u_equalu_u_sym, axiom, ![TV_u_27a]: ![V_y, V_x, V_c]: (s(t_h4s_totos_cpn,h4s_totos_apto(s(t_h4s_totos_toto(TV_u_27a),V_c),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(t_h4s_totos_cpn,h4s_totos_equal) <=> s(t_h4s_totos_cpn,h4s_totos_apto(s(t_h4s_totos_toto(TV_u_27a),V_c),s(TV_u_27a,V_y),s(TV_u_27a,V_x))) = s(t_h4s_totos_cpn,h4s_totos_equal))).
fof(ah4s_totos_cpnu_u_nchotomy, axiom, ![V_a]: (s(t_h4s_totos_cpn,V_a) = s(t_h4s_totos_cpn,h4s_totos_less) | (s(t_h4s_totos_cpn,V_a) = s(t_h4s_totos_cpn,h4s_totos_equal) | s(t_h4s_totos_cpn,V_a) = s(t_h4s_totos_cpn,h4s_totos_greater)))).
fof(ah4s_totos_totou_u_transu_u_lessu_c1, axiom, ![TV_u_27a]: ![V_z, V_y, V_x, V_c]: ((s(t_h4s_totos_cpn,h4s_totos_apto(s(t_h4s_totos_toto(TV_u_27a),V_c),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(t_h4s_totos_cpn,h4s_totos_less) & s(t_h4s_totos_cpn,h4s_totos_apto(s(t_h4s_totos_toto(TV_u_27a),V_c),s(TV_u_27a,V_z),s(TV_u_27a,V_y))) = s(t_h4s_totos_cpn,h4s_totos_greater)) => s(t_h4s_totos_cpn,h4s_totos_apto(s(t_h4s_totos_toto(TV_u_27a),V_c),s(TV_u_27a,V_x),s(TV_u_27a,V_z))) = s(t_h4s_totos_cpn,h4s_totos_less))).
fof(ah4s_predu_u_sets_SUBSETu_u_DEF, axiom, ![TV_u_27a]: ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) <=> ![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)))) => p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))))).
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_ORu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,t0))) <=> p(s(t_bool,t0)))).
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_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_predu_u_sets_GSPECu_u_F, 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,f))) => 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_empty))).
fof(ah4s_predu_u_sets_UNIONu_u_EMPTYu_c0, axiom, ![TV_u_27a]: ![V_s]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(TV_u_27a,t_bool),V_s)).
fof(ah4s_predu_u_sets_GSPECu_u_ID, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_y, 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)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),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),V_y))),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,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_y))))) => ![V_y]: 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)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),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),V_y))))) = s(t_fun(TV_u_27a,t_bool),V_y))).
fof(ah4s_predu_u_sets_UNIONu_u_EMPTYu_c1, axiom, ![TV_u_27a]: ![V_s]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) = s(t_fun(TV_u_27a,t_bool),V_s)).
fof(ah4s_enumerals_btu_u_tou_u_setu_c1, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_r, V_cmp, V_x, V_z]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_z),s(t_fun(TV_u_27a,t_bool),h4s_enumerals_enumeral(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_h4s_enumerals_bt(TV_u_27a),V_r)))))) & s(t_h4s_totos_cpn,h4s_totos_apto(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(TV_u_27a,V_x),s(TV_u_27a,V_z))) = s(t_h4s_totos_cpn,h4s_totos_less))) & 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)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_h4s_totos_toto(TV_u_27a),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_h4s_enumerals_bt(TV_u_27a),t_fun(t_h4s_totos_toto(TV_u_27a),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))))),V_uu_1),s(t_h4s_enumerals_bt(TV_u_27a),V_r))),s(t_h4s_totos_toto(TV_u_27a),V_cmp))),s(TV_u_27a,V_x))),s(TV_u_27a,V_z))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_z),s(t_bool,V_v)))) => ![V_uu_0]: (![V_l, V_cmp, V_x, V_y]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),h4s_enumerals_enumeral(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_h4s_enumerals_bt(TV_u_27a),V_l)))))) & s(t_h4s_totos_cpn,h4s_totos_apto(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(TV_u_27a,V_y),s(TV_u_27a,V_x))) = s(t_h4s_totos_cpn,h4s_totos_less))) & 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)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_h4s_totos_toto(TV_u_27a),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_h4s_enumerals_bt(TV_u_27a),t_fun(t_h4s_totos_toto(TV_u_27a),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))))),V_uu_0),s(t_h4s_enumerals_bt(TV_u_27a),V_l))),s(t_h4s_totos_toto(TV_u_27a),V_cmp))),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_y),s(t_bool,V_v)))) => ![V_x, V_r, V_l, V_cmp]: s(t_fun(TV_u_27a,t_bool),h4s_enumerals_enumeral(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_h4s_enumerals_bt(TV_u_27a),happ(s(t_fun(t_h4s_enumerals_bt(TV_u_27a),t_h4s_enumerals_bt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_enumerals_bt(TV_u_27a),t_h4s_enumerals_bt(TV_u_27a))),happ(s(t_fun(t_h4s_enumerals_bt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_enumerals_bt(TV_u_27a),t_h4s_enumerals_bt(TV_u_27a)))),h4s_enumerals_node),s(t_h4s_enumerals_bt(TV_u_27a),V_l))),s(TV_u_27a,V_x))),s(t_h4s_enumerals_bt(TV_u_27a),V_r))))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(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)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_h4s_totos_toto(TV_u_27a),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_h4s_enumerals_bt(TV_u_27a),t_fun(t_h4s_totos_toto(TV_u_27a),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))))),V_uu_0),s(t_h4s_enumerals_bt(TV_u_27a),V_l))),s(t_h4s_totos_toto(TV_u_27a),V_cmp))),s(TV_u_27a,V_x))))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))))),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)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_h4s_totos_toto(TV_u_27a),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),happ(s(t_fun(t_h4s_enumerals_bt(TV_u_27a),t_fun(t_h4s_totos_toto(TV_u_27a),t_fun(TV_u_27a,t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))))),V_uu_1),s(t_h4s_enumerals_bt(TV_u_27a),V_r))),s(t_h4s_totos_toto(TV_u_27a),V_cmp))),s(TV_u_27a,V_x)))))))))).
fof(ah4s_resu_u_quans_RESu_u_EXISTSu_u_UNIQUEu_u_NULL, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_m, V_x]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_bool,V_m))),s(TV_u_27a,V_x))) = s(t_bool,V_m) => ![V_p, V_m]: (p(s(t_bool,h4s_bools_resu_u_existsu_u_unique(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_bool,V_m)))))) <=> (?[V_x]: s(t_fun(TV_u_27a,t_bool),V_p) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) & p(s(t_bool,V_m)))))).
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_BETAu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t0)))).
fof(ah4s_indu_u_types_FCONS0u_c1, axiom, ![TV_u_27a]: ![V_n, V_f, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))),s(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),V_f),s(t_h4s_nums_num,V_n)))).
fof(ah4s_indu_u_types_CONSTRu_u_REC, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_f, V_r, V_n]: s(TV_u_27b,happ(s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_h4s_nums_num,V_n))) = s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r),s(t_h4s_nums_num,V_n))))) => ![V_Fn]: ?[V_f]: ![V_c, V_i, V_r]: s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,V_c),s(TV_u_27a,V_i),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))) = s(TV_u_27b,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)))),V_Fn),s(t_h4s_nums_num,V_c))),s(TV_u_27a,V_i))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))))).
fof(ah4s_indu_u_types_FCONS0u_c0, axiom, ![TV_u_27a]: ![V_f, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_a)).
fof(ah4s_bools_JRHu_u_INDUCTu_u_UTIL, axiom, ![TV_u_27a]: ![V_t, V_P]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_t) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) => p(s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_P)))))).
fof(ah4s_bools_MONOu_u_EXISTS, 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,happ(s(t_fun(TV_u_27a,t_bool),V_Q),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)))) => ?[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_MONOu_u_OR, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) | p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) | p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_AND, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) & p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) & p(s(t_bool,V_w)))))).
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_enumerals_onebl0, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_fun(t_h4s_enumerals_bt(TV_u_27a),t_fun(t_h4s_enumerals_bl(TV_u_27a),t_h4s_enumerals_bl(TV_u_27a)))),h4s_enumerals_onebl) = s(t_fun(TV_u_27a,t_fun(t_h4s_enumerals_bt(TV_u_27a),t_fun(t_h4s_enumerals_bl(TV_u_27a),t_h4s_enumerals_bl(TV_u_27a)))),h4s_enumerals_u_20u_40indu_u_typeenumeral6)).
fof(ah4s_enumerals_zerbl0, axiom, ![TV_u_27a]: s(t_fun(t_h4s_enumerals_bl(TV_u_27a),t_h4s_enumerals_bl(TV_u_27a)),h4s_enumerals_zerbl) = s(t_fun(t_h4s_enumerals_bl(TV_u_27a),t_h4s_enumerals_bl(TV_u_27a)),h4s_enumerals_u_20u_40indu_u_typeenumeral5)).
fof(ah4s_enumerals_nbl0, axiom, ![TV_u_27a]: s(t_h4s_enumerals_bl(TV_u_27a),h4s_enumerals_nbl) = s(t_h4s_enumerals_bl(TV_u_27a),h4s_enumerals_u_20u_40indu_u_typeenumeral4)).
fof(ah4s_enumerals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeenumeral6u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),h4s_indu_u_types_bottom) => ![V_x, V_x0, V_x1]: s(t_h4s_enumerals_bl(TV_u_27a),happ(s(t_fun(t_h4s_enumerals_bl(TV_u_27a),t_h4s_enumerals_bl(TV_u_27a)),happ(s(t_fun(t_h4s_enumerals_bt(TV_u_27a),t_fun(t_h4s_enumerals_bl(TV_u_27a),t_h4s_enumerals_bl(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_enumerals_bt(TV_u_27a),t_fun(t_h4s_enumerals_bl(TV_u_27a),t_h4s_enumerals_bl(TV_u_27a)))),h4s_enumerals_u_20u_40indu_u_typeenumeral6),s(TV_u_27a,V_x))),s(t_h4s_enumerals_bt(TV_u_27a),V_x0))),s(t_h4s_enumerals_bl(TV_u_27a),V_x1))) = s(t_h4s_enumerals_bl(TV_u_27a),h4s_enumerals_u_20u_40indu_u_typeenumeral7(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_h4s_enumerals_bt(TV_u_27a),V_x0))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),h4s_enumerals_u_20u_40indu_u_typeenumeral8(s(t_h4s_enumerals_bl(TV_u_27a),V_x1))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a)))),V_uu_0))))))))).
fof(ah4s_enumerals_blu_u_repfnsu_c1, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27blu_27]: (![V_a0u_270]: ((s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,h4s_bools_arb),s(t_h4s_enumerals_bt(TV_u_27a),h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a)))),V_uu_0))) | (?[V_a]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,h4s_bools_arb),s(t_h4s_enumerals_bt(TV_u_27a),h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),V_a),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a)))),V_uu_0))))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),t_bool),V_uu_27blu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),V_a))))) | ?[V_a0, V_a1, V_a2]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,V_a0),s(t_h4s_enumerals_bt(TV_u_27a),V_a1))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),V_a2),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a)))),V_uu_0))))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),t_bool),V_uu_27blu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),V_a2))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),t_bool),V_uu_27blu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),t_bool),V_uu_27blu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),V_r))))) <=> s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),h4s_enumerals_u_20u_40indu_u_typeenumeral8(s(t_h4s_enumerals_bl(TV_u_27a),h4s_enumerals_u_20u_40indu_u_typeenumeral7(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),V_r))))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),V_r)))).
fof(ah4s_enumerals_blu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_enumerals_bl(TV_u_27a),h4s_enumerals_u_20u_40indu_u_typeenumeral7(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),h4s_enumerals_u_20u_40indu_u_typeenumeral8(s(t_h4s_enumerals_bl(TV_u_27a),V_a))))) = s(t_h4s_enumerals_bl(TV_u_27a),V_a)).
fof(ah4s_enumerals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeenumeral4u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),h4s_indu_u_types_bottom) => s(t_h4s_enumerals_bl(TV_u_27a),h4s_enumerals_u_20u_40indu_u_typeenumeral4) = s(t_h4s_enumerals_bl(TV_u_27a),h4s_enumerals_u_20u_40indu_u_typeenumeral7(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,h4s_bools_arb),s(t_h4s_enumerals_bt(TV_u_27a),h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a)))),V_uu_0))))))).
fof(ah4s_enumerals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeenumeral5u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),h4s_indu_u_types_bottom) => ![V_x]: s(t_h4s_enumerals_bl(TV_u_27a),happ(s(t_fun(t_h4s_enumerals_bl(TV_u_27a),t_h4s_enumerals_bl(TV_u_27a)),h4s_enumerals_u_20u_40indu_u_typeenumeral5),s(t_h4s_enumerals_bl(TV_u_27a),V_x))) = s(t_h4s_enumerals_bl(TV_u_27a),h4s_enumerals_u_20u_40indu_u_typeenumeral7(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a)),h4s_pairs_u_2c(s(TV_u_27a,h4s_bools_arb),s(t_h4s_enumerals_bt(TV_u_27a),h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a))),h4s_enumerals_u_20u_40indu_u_typeenumeral8(s(t_h4s_enumerals_bl(TV_u_27a),V_x))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(TV_u_27a,t_h4s_enumerals_bt(TV_u_27a)))),V_uu_0))))))))).
fof(ah4s_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(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_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(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_27b,V_y)).
fof(ah4s_enumerals_node0, axiom, ![TV_u_27a]: s(t_fun(t_h4s_enumerals_bt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_enumerals_bt(TV_u_27a),t_h4s_enumerals_bt(TV_u_27a)))),h4s_enumerals_node) = s(t_fun(t_h4s_enumerals_bt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_enumerals_bt(TV_u_27a),t_h4s_enumerals_bt(TV_u_27a)))),h4s_enumerals_u_20u_40indu_u_typeenumeral1)).
fof(ah4s_enumerals_nt0, axiom, ![TV_u_27a]: s(t_h4s_enumerals_bt(TV_u_27a),h4s_enumerals_nt) = s(t_h4s_enumerals_bt(TV_u_27a),h4s_enumerals_u_20u_40indu_u_typeenumeral0)).
fof(ah4s_enumerals_btu_u_repfnsu_c1, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27btu_27]: (![V_a0u_270]: ((s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))) | ?[V_a0, V_a1, V_a2]: (s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(TV_u_27a,V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a2),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27btu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0)))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27btu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a2))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27btu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27btu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r))))) <=> s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_enumerals_u_20u_40indu_u_typeenumeral3(s(t_h4s_enumerals_bt(TV_u_27a),h4s_enumerals_u_20u_40indu_u_typeenumeral2(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r))))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r)))).
fof(ah4s_enumerals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeenumeral1u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => ![V_x, V_x0, V_xi_]: s(t_h4s_enumerals_bt(TV_u_27a),happ(s(t_fun(t_h4s_enumerals_bt(TV_u_27a),t_h4s_enumerals_bt(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_enumerals_bt(TV_u_27a),t_h4s_enumerals_bt(TV_u_27a))),happ(s(t_fun(t_h4s_enumerals_bt(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_enumerals_bt(TV_u_27a),t_h4s_enumerals_bt(TV_u_27a)))),h4s_enumerals_u_20u_40indu_u_typeenumeral1),s(t_h4s_enumerals_bt(TV_u_27a),V_x))),s(TV_u_27a,V_x0))),s(t_h4s_enumerals_bt(TV_u_27a),V_xi_))) = s(t_h4s_enumerals_bt(TV_u_27a),h4s_enumerals_u_20u_40indu_u_typeenumeral2(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(TV_u_27a,V_x0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_enumerals_u_20u_40indu_u_typeenumeral3(s(t_h4s_enumerals_bt(TV_u_27a),V_x))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_enumerals_u_20u_40indu_u_typeenumeral3(s(t_h4s_enumerals_bt(TV_u_27a),V_xi_))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))))))).
fof(ah4s_enumerals_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeenumeral0u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => s(t_h4s_enumerals_bt(TV_u_27a),h4s_enumerals_u_20u_40indu_u_typeenumeral0) = s(t_h4s_enumerals_bt(TV_u_27a),h4s_enumerals_u_20u_40indu_u_typeenumeral2(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))).
fof(ah4s_enumerals_btu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_enumerals_bt(TV_u_27a),h4s_enumerals_u_20u_40indu_u_typeenumeral2(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_enumerals_u_20u_40indu_u_typeenumeral3(s(t_h4s_enumerals_bt(TV_u_27a),V_a))))) = s(t_h4s_enumerals_bt(TV_u_27a),V_a)).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t0))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_resu_u_quans_RESu_u_EXISTSu_u_NULL, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_m, V_x]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_bool,V_m))),s(TV_u_27a,V_x))) = s(t_bool,V_m) => ![V_p, V_m]: (p(s(t_bool,h4s_bools_resu_u_exists(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_bool,V_m)))))) <=> (~ (s(t_fun(TV_u_27a,t_bool),V_p) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)) & p(s(t_bool,V_m)))))).
fof(ah4s_resu_u_quans_RESu_u_EXISTSu_u_UNIQUE, axiom, ![TV_u_27a]: ![V_uu_2]: (![V_f, V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),V_uu_2),s(t_fun(TV_u_27a,t_bool),V_f))),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) <=> ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_y))))) => s(TV_u_27a,V_x) = s(TV_u_27a,V_y))) => ![V_uu_1]: (![V_P, V_f, V_x]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool))),V_uu_1),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_fun(TV_u_27a,t_bool),V_f))),s(TV_u_27a,V_x))) = s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),V_uu_2),s(t_fun(TV_u_27a,t_bool),V_f))),s(TV_u_27a,V_x))))) => ![V_uu_0]: (![V_f, V_x]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_f))),s(TV_u_27a,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))) => ![V_f, V_P]: (p(s(t_bool,h4s_bools_resu_u_existsu_u_unique(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_f)))) <=> (p(s(t_bool,h4s_bools_resu_u_exists(s(t_fun(TV_u_27a,t_bool),V_P),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_f)))))) & p(s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),V_P),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),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool))),V_uu_1),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_fun(TV_u_27a,t_bool),V_f)))))))))))))).
fof(ah4s_resu_u_quans_RESu_u_FORALL, axiom, ![TV_u_27a]: ![V_f, V_P]: (p(s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_f)))) <=> ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))))).
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_EXISTSu_u_ORu_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,happ(s(t_fun(TV_u_27a,t_bool),V_Q),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)))) | ?[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_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_LEFTu_u_ORu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))) <=> ?[V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))))).
fof(ah4s_bools_RIGHTu_u_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_bools_RIGHTu_u_EXISTSu_u_ANDu_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_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t0)) & p(s(t_bool,V_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_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(ch4s_enumerals_LESSu_u_ALLu_u_OUu_u_UOu_u_LEM, conjecture, ![TV_u_27a]: ![V_t, V_s, V_cmp, V_a]: ((p(s(t_bool,h4s_enumerals_lessu_u_all(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(TV_u_27a,V_a),s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_enumerals_lessu_u_all(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(TV_u_27a,V_a),s(t_fun(TV_u_27a,t_bool),V_t))))) => s(t_fun(TV_u_27a,t_bool),h4s_enumerals_ou(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_fun(TV_u_27a,t_bool),h4s_enumerals_uo(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_a),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(TV_u_27a,t_bool),V_t))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_a),s(t_fun(TV_u_27a,t_bool),h4s_enumerals_ou(s(t_h4s_totos_toto(TV_u_27a),V_cmp),s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))))))).
