%   ORIGINAL: h4/quotient__pair/PAIR__REL__THM
% 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/quotient__pair/PAIR__REL: !R2 R1. h4/quotient__pair/_23_23_23 R1 R2 = h4/pair/UNCURRY (\a b. h4/pair/UNCURRY (\c d. R1 a c /\ R2 b d))
% Assm: h4/pair/PAIR__MAP__THM: !y x g f. h4/pair/_23_23 f g (h4/pair/_2C x y) = h4/pair/_2C (f x) (g y)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/relation/O__ASSOC: !R3 R2 R1. h4/relation/O R1 (h4/relation/O R2 R3) = h4/relation/O (h4/relation/O R1 R2) R3
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/pair/PAIR__MAP: !p g f. h4/pair/_23_23 f g p = h4/pair/_2C (f (h4/pair/FST p)) (g (h4/pair/SND p))
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/relation/O__DEF: !z x R2 R1. h4/relation/O R1 R2 x z <=> (?y. R2 x y /\ R1 y z)
% 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/LET__DEF: h4/bool/LET = (\f x. f x)
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/quotient/FUN__MAP: !g f. h4/quotient/_2D_2D_3E f g = (\h x. g (h (f x)))
% Assm: h4/quotient/FUN__REL: !g f R2 R1. h4/quotient/_3D_3D_3D_3E R1 R2 f g <=> (!x y. R1 x y ==> R2 (f x) (g y))
% Assm: h4/relation/RDOM__DEF: !x R. h4/relation/RDOM R x <=> (?y. R x y)
% Assm: h4/relation/RRANGE0: !y R. h4/relation/RRANGE R y <=> (?x. R x y)
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> 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__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/dc__imp: !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/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/pair/LET2__RATOR: !b N M. h4/bool/LET (h4/pair/UNCURRY (\x y. N x y)) M b = h4/bool/LET (h4/pair/UNCURRY (\x y. N x y b)) M
% Assm: h4/bool/LEFT__EXISTS__AND__THM: !Q P. (?x. P x /\ Q) <=> (?x. P x) /\ Q
% Assm: h4/bool/LEFT__OR__EXISTS__THM: !Q P. (?x. P x) \/ Q <=> (?x. P x \/ Q)
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/bool/RIGHT__OR__EXISTS__THM: !Q P. P \/ (?x. Q x) <=> (?x. P \/ Q x)
% Assm: h4/bool/RIGHT__EXISTS__AND__THM: !Q P. (?x. P /\ Q x) <=> P /\ (?x. Q x)
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/bool/EXISTS__OR__THM: !Q P. (?x. P x \/ Q x) <=> (?x. P x) \/ (?x. Q x)
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/bool/LEFT__FORALL__OR__THM: !Q P. (!x. P x \/ Q) <=> (!x. P x) \/ Q
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/pair/LET2__RAND: !P N M. P (h4/bool/LET (h4/pair/UNCURRY (\x y. N x y)) M) = h4/bool/LET (h4/pair/UNCURRY (\x y. P (N x y))) M
% Assm: h4/list/LIST__REL__def_c3: !bs b as a R. h4/list/LIST__REL R (h4/list/CONS a as) (h4/list/CONS b bs) <=> R a b /\ h4/list/LIST__REL R as bs
% Assm: h4/quotient/LAMBDA__PRS: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f. (\x. f x) = h4/quotient/_2D_2D_3E rep1 abs2 (\x. rep2 (f (abs1 x)))))
% Assm: h4/quotient/K__RSP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!x1 x2 y1 y2. R1 x1 x2 /\ R2 y1 y2 ==> R1 (h4/combin/K x1 y1) (h4/combin/K x2 y2)))
% Assm: h4/ind__type/ISO__FUN: !g_27 g f_27 f. h4/ind__type/ISO f f_27 /\ h4/ind__type/ISO g g_27 ==> h4/ind__type/ISO (\h a_27. g (h (f_27 a_27))) (\h a. g_27 (h (f a)))
% Assm: h4/quotient/LAMBDA__REP__ABS__RSP: !rep2 rep1 f2 f1 abs2 abs1 REL2 REL1. ((!r r_27. REL1 r r_27 ==> REL1 r (rep1 (abs1 r_27))) /\ (!r r_27. REL2 r r_27 ==> REL2 r (rep2 (abs2 r_27)))) /\ h4/quotient/_3D_3D_3D_3E REL1 REL2 f1 f2 ==> h4/quotient/_3D_3D_3D_3E REL1 REL2 f1 (h4/quotient/_2D_2D_3E abs1 rep2 (h4/quotient/_2D_2D_3E rep1 abs2 f2))
% Assm: h4/bool/IN__DEF: h4/bool/IN = (\x f. f x)
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/combin/K__THM: !y x. h4/combin/K x y = x
% Assm: h4/pair/C__UNCURRY__L: !x f. h4/combin/C (h4/pair/UNCURRY f) x = h4/pair/UNCURRY (h4/combin/C (h4/combin/o h4/combin/C f) x)
% Assm: h4/relation/RSUBSET0: !R2 R1. h4/relation/RSUBSET R1 R2 <=> (!x y. R1 x y ==> R2 x y)
% Assm: h4/list/EVERY2__THM_c3: !ys y xxs P. h4/list/LIST__REL P xxs (h4/list/CONS y ys) <=> (?x xs. xxs = h4/list/CONS x xs /\ P x y /\ h4/list/LIST__REL P xs ys)
% Assm: h4/list/EVERY2__THM_c1: !yys xs x P. h4/list/LIST__REL P (h4/list/CONS x xs) yys <=> (?y ys. yys = h4/list/CONS y ys /\ P x y /\ h4/list/LIST__REL P xs ys)
% Assm: h4/relation/nf__def: !x R. h4/relation/nf R x <=> (!y. ~R x y)
% Assm: h4/relation/IN__RDOM: !x R. h4/bool/IN x (h4/relation/RDOM R) <=> (?y. R x y)
% Assm: h4/relation/RUNION0: !y x R2 R1. h4/relation/RUNION R1 R2 x y <=> R1 x y \/ R2 x y
% Assm: h4/list/EVERY2__cong: !l2_27 l2 l1_27 l1 P_27 P. l1 = l1_27 /\ l2 = l2_27 /\ (!x y. h4/bool/IN x (h4/list/LIST__TO__SET l1_27) /\ h4/bool/IN y (h4/list/LIST__TO__SET l2_27) ==> (P x y <=> P_27 x y)) ==> (h4/list/LIST__REL P l1 l2 <=> h4/list/LIST__REL P_27 l1_27 l2_27)
% Assm: h4/list/LIST__REL__CONS1: !xs t h R. h4/list/LIST__REL R (h4/list/CONS h t) xs <=> (?h_27 t_27. xs = h4/list/CONS h_27 t_27 /\ R h h_27 /\ h4/list/LIST__REL R t t_27)
% Assm: h4/option/OPTREL__def: !y x R. h4/option/OPTREL R x y <=> x = h4/option/NONE /\ y = h4/option/NONE \/ (?x0 y0. x = h4/option/SOME x0 /\ y = h4/option/SOME y0 /\ R x0 y0)
% Assm: h4/list/LIST__REL__CONS2: !xs t h R. h4/list/LIST__REL R xs (h4/list/CONS h t) <=> (?h_27 t_27. xs = h4/list/CONS h_27 t_27 /\ R h_27 h /\ h4/list/LIST__REL R t_27 t)
% Assm: h4/relation/RINTER0: !y x R2 R1. h4/relation/RINTER R1 R2 x y <=> R1 x y /\ R2 x y
% Assm: h4/relation/RCOMPL0: !y x R. h4/relation/RCOMPL R x y <=> ~R x y
% Assm: h4/list/LIST__REL__cases: !a1 a0 R. h4/list/LIST__REL R a0 a1 <=> a0 = h4/list/NIL /\ a1 = h4/list/NIL \/ (?h1 h2 t1 t2. a0 = h4/list/CONS h1 t1 /\ a1 = h4/list/CONS h2 t2 /\ R h1 h2 /\ h4/list/LIST__REL R t1 t2)
% Assm: h4/relation/IN__RRANGE: !y R. h4/bool/IN y (h4/relation/RRANGE R) <=> (?x. R x y)
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/pair/UNCURRY__VAR: !v f. h4/pair/UNCURRY f v = f (h4/pair/FST v) (h4/pair/SND v)
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% 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/quotient/QUOTIENT__ABS__REP: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a. abs (rep a) = a)
% Assm: h4/quotient/FUN__MAP__THM: !x h g f. h4/quotient/_2D_2D_3E f g h x = g (h (f x))
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/list/hidden____20__40ind____typelist1____def: h4/list/_20_40ind__typelist1 = (\a0 a1. h4/list/_20_40ind__typelist2 ((\a00 a10. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM))) a0 (h4/list/_20_40ind__typelist3 a1)))
% Assm: h4/list/NIL0: h4/list/NIL = h4/list/_20_40ind__typelist0
% Assm: h4/list/hidden____20__40ind____typelist0____def: h4/list/_20_40ind__typelist0 = h4/list/_20_40ind__typelist2 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM))
% Assm: h4/list/CONS0: h4/list/CONS = h4/list/_20_40ind__typelist1
% Assm: h4/list/list__repfns_c1: !r. (\a0_27. !_27list_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM))) a0 a1 /\ _27list_27 a1) ==> _27list_27 a0_270) ==> _27list_27 a0_27) r <=> h4/list/_20_40ind__typelist3 (h4/list/_20_40ind__typelist2 r) = r
% Assm: h4/list/list__repfns_c0: !a. h4/list/_20_40ind__typelist2 (h4/list/_20_40ind__typelist3 a) = a
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/pair/UNCURRY0: !v f. h4/pair/UNCURRY f v = f (h4/pair/FST v) (h4/pair/SND v)
% Assm: h4/combin/C__THM: !y x f. h4/combin/C f x y = f y x
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/ind__type/ISO0: !g f. h4/ind__type/ISO f g <=> (!x. f (g x) = x) /\ (!y. g (f y) = y)
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/BETA__THM: !y f. (\x. f x) y = f y
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~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/FCONS0_c0: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% 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/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/list/LIST__REL__def0: h4/list/LIST__REL = (\R a0 a1. !LIST__REL_27. (!a00 a10. a00 = h4/list/NIL /\ a10 = h4/list/NIL \/ (?h1 h2 t1 t2. a00 = h4/list/CONS h1 t1 /\ a10 = h4/list/CONS h2 t2 /\ R h1 h2 /\ LIST__REL_27 t1 t2) ==> LIST__REL_27 a00 a10) ==> LIST__REL_27 a0 a1)
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> t
% Assm: h4/list/LIST__REL__def_c1: !as a R. h4/list/LIST__REL R (h4/list/CONS a as) h4/list/NIL <=> F
% Assm: h4/list/LIST__REL__def_c2: !bs b R. h4/list/LIST__REL R h4/list/NIL (h4/list/CONS b bs) <=> F
% Assm: h4/list/EVERY2__EVERY: !l2 l1 f. h4/list/LIST__REL f l1 l2 <=> h4/list/LENGTH l1 = h4/list/LENGTH l2 /\ h4/list/EVERY (h4/pair/UNCURRY f) (h4/list/ZIP (h4/pair/_2C l1 l2))
% Assm: h4/marker/Cong__def: !x. h4/marker/Cong x <=> x
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/BOUNDED__THM: !v. h4/bool/BOUNDED v <=> T
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/list/LIST__REL__NIL_c0: !x R. h4/list/LIST__REL R h4/list/NIL x <=> x = h4/list/NIL
% Assm: h4/list/MEM_c1: !x t h. h4/bool/IN x (h4/list/LIST__TO__SET (h4/list/CONS h t)) <=> x = h \/ h4/bool/IN x (h4/list/LIST__TO__SET t)
% Assm: h4/list/MEM_c0: !x. h4/bool/IN x (h4/list/LIST__TO__SET h4/list/NIL) <=> F
% Assm: h4/list/list__INDUCT: !P. P h4/list/NIL /\ (!t. P t ==> (!h. P (h4/list/CONS h t))) ==> (!l. P l)
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% 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/UNWIND__FORALL__THM2: !v f. (!x. x = v ==> f x) <=> f v
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Goal: !d c b a R2 R1. h4/quotient__pair/_23_23_23 R1 R2 (h4/pair/_2C a b) (h4/pair/_2C c d) <=> R1 a c /\ R2 b d
%   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_quotientu_u_pairs_PAIRu_u_REL]: !_3. (!R1 a c R2 b d. happ (happ (happ (happ (happ (happ _3 R1) a) c) R2) b) d <=> happ (happ R1 a) c /\ happ (happ R2 b) d) ==> (!_2. (!R1 a R2 b c. happ (happ (happ (happ (happ _2 R1) a) R2) b) c = happ (happ (happ (happ (happ _3 R1) a) c) R2) b) ==> (!_1. (!R1 a R2 b. happ (happ (happ (happ _1 R1) a) R2) b = h4/pair/UNCURRY (happ (happ (happ (happ _2 R1) a) R2) b)) ==> (!_0. (!R1 R2 a. happ (happ (happ _0 R1) R2) a = happ (happ (happ _1 R1) a) R2) ==> (!R2 R1. h4/quotient__pair/_23_23_23 R1 R2 = h4/pair/UNCURRY (happ (happ _0 R1) R2)))))
% Assm [h4s_pairs_PAIRu_u_MAPu_u_THM]: !y x g f. h4/pair/_23_23 f g (h4/pair/_2C x y) = h4/pair/_2C (happ f x) (happ g y)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_relations_Ou_u_ASSOC]: !R3 R2 R1. h4/relation/O R1 (h4/relation/O R2 R3) = h4/relation/O (h4/relation/O R1 R2) R3
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_pairs_PAIRu_u_MAP]: !p g f. h4/pair/_23_23 f g p = h4/pair/_2C (happ f (h4/pair/FST p)) (happ g (h4/pair/SND p))
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_relations_Ou_u_DEF]: !z x R2 R1. happ (happ (h4/relation/O R1 R2) x) z <=> (?y. happ (happ R2 x) y /\ happ (happ R1 y) z)
% 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_LETu_u_DEF]: !x x. h4/bool/LET x x = happ x x
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_quotients_FUNu_u_MAP]: !g f x x. happ (h4/quotient/_2D_2D_3E f g x) x = happ g (happ x (happ f x))
% Assm [h4s_quotients_FUNu_u_REL]: !g f R2 R1. h4/quotient/_3D_3D_3D_3E R1 R2 f g <=> (!x y. happ (happ R1 x) y ==> happ (happ R2 (happ f x)) (happ g y))
% Assm [h4s_relations_RDOMu_u_DEF]: !x R. happ (h4/relation/RDOM R) x <=> (?y. happ (happ R x) y)
% Assm [h4s_relations_RRANGE0]: !y R. happ (h4/relation/RRANGE R) y <=> (?x. happ (happ R x) y)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> 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_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_dcu_u_imp]: !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_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_pairs_LET2u_u_RATOR]: !_3. (!N x b y. happ (happ (happ (happ _3 N) x) b) y = happ (happ (happ N x) y) b) ==> (!_2. (!N b x. happ (happ (happ _2 N) b) x = happ (happ (happ _3 N) x) b) ==> (!_1. (!N x y. happ (happ (happ _1 N) x) y = happ (happ N x) y) ==> (!_0. (!N x. happ (happ _0 N) x = happ (happ _1 N) x) ==> (!b N M. happ (h4/bool/LET (h4/pair/UNCURRY (happ _0 N)) M) b = h4/bool/LET (h4/pair/UNCURRY (happ (happ _2 N) b)) M))))
% 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_LEFTu_u_ORu_u_EXISTSu_u_THM]: !Q P. (?x. happ P x) \/ Q <=> (?x. happ P x \/ Q)
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM]: !Q P. P \/ (?x. happ Q x) <=> (?x. P \/ happ Q x)
% Assm [h4s_bools_RIGHTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. P /\ happ Q x) <=> P /\ (?x. 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_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_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. happ P x \/ Q) <=> (!x. happ P x) \/ Q
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_pairs_LET2u_u_RAND]: !_3. (!P N x y. happ (happ (happ (happ _3 P) N) x) y = happ P (happ (happ N x) y)) ==> (!_2. (!P N x. happ (happ (happ _2 P) N) x = happ (happ (happ _3 P) N) x) ==> (!_1. (!N x y. happ (happ (happ _1 N) x) y = happ (happ N x) y) ==> (!_0. (!N x. happ (happ _0 N) x = happ (happ _1 N) x) ==> (!P N M. happ P (h4/bool/LET (h4/pair/UNCURRY (happ _0 N)) M) = h4/bool/LET (h4/pair/UNCURRY (happ (happ _2 P) N)) M))))
% Assm [h4s_lists_LISTu_u_RELu_u_defu_c3]: !bs b as a R. h4/list/LIST__REL R (happ (happ h4/list/CONS a) as) (happ (happ h4/list/CONS b) bs) <=> happ (happ R a) b /\ h4/list/LIST__REL R as bs
% Assm [h4s_quotients_LAMBDAu_u_PRS]: !_0. (!rep2 f abs1 x. happ (happ (happ (happ _0 rep2) f) abs1) x = happ rep2 (happ f (happ abs1 x))) ==> (!rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f x. happ f x = happ (h4/quotient/_2D_2D_3E rep1 abs2 (happ (happ (happ _0 rep2) f) abs1)) x)))
% Assm [h4s_quotients_Ku_u_RSP]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!x1 x2 y1 y2. happ (happ R1 x1) x2 /\ happ (happ R2 y1) y2 ==> happ (happ R1 (h4/combin/K x1 y1)) (h4/combin/K x2 y2)))
% Assm [h4s_indu_u_types_ISOu_u_FUN]: !_3. (!g_27 h f a. happ (happ (happ (happ _3 g_27) h) f) a = happ g_27 (happ h (happ f a))) ==> (!_2. (!g_27 f h. happ (happ (happ _2 g_27) f) h = happ (happ (happ _3 g_27) h) f) ==> (!_1. (!g h f_27 a_27. happ (happ (happ (happ _1 g) h) f_27) a_27 = happ g (happ h (happ f_27 a_27))) ==> (!_0. (!g f_27 h. happ (happ (happ _0 g) f_27) h = happ (happ (happ _1 g) h) f_27) ==> (!g_27 g f_27 f. h4/ind__type/ISO f f_27 /\ h4/ind__type/ISO g g_27 ==> h4/ind__type/ISO (happ (happ _0 g) f_27) (happ (happ _2 g_27) f)))))
% Assm [h4s_quotients_LAMBDAu_u_REPu_u_ABSu_u_RSP]: !rep2 rep1 f2 f1 abs2 abs1 REL2 REL1. ((!r r_27. happ (happ REL1 r) r_27 ==> happ (happ REL1 r) (happ rep1 (happ abs1 r_27))) /\ (!r r_27. happ (happ REL2 r) r_27 ==> happ (happ REL2 r) (happ rep2 (happ abs2 r_27)))) /\ h4/quotient/_3D_3D_3D_3E REL1 REL2 f1 f2 ==> h4/quotient/_3D_3D_3D_3E REL1 REL2 f1 (h4/quotient/_2D_2D_3E abs1 rep2 (h4/quotient/_2D_2D_3E rep1 abs2 f2))
% Assm [h4s_bools_INu_u_DEF]: !x x. h4/bool/IN x x <=> happ x x
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. happ (h4/pair/UNCURRY f) (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_combins_Ku_u_THM]: !y x. h4/combin/K x y = x
% Assm [h4s_pairs_Cu_u_UNCURRYu_u_L]: !x f. happ (happ h4/combin/C (h4/pair/UNCURRY f)) x = h4/pair/UNCURRY (happ (happ h4/combin/C (h4/combin/o h4/combin/C f)) x)
% Assm [h4s_relations_RSUBSET0]: !R2 R1. h4/relation/RSUBSET R1 R2 <=> (!x y. happ (happ R1 x) y ==> happ (happ R2 x) y)
% Assm [h4s_lists_EVERY2u_u_THMu_c3]: !ys y xxs P. h4/list/LIST__REL P xxs (happ (happ h4/list/CONS y) ys) <=> (?x xs. xxs = happ (happ h4/list/CONS x) xs /\ happ (happ P x) y /\ h4/list/LIST__REL P xs ys)
% Assm [h4s_lists_EVERY2u_u_THMu_c1]: !yys xs x P. h4/list/LIST__REL P (happ (happ h4/list/CONS x) xs) yys <=> (?y ys. yys = happ (happ h4/list/CONS y) ys /\ happ (happ P x) y /\ h4/list/LIST__REL P xs ys)
% Assm [h4s_relations_nfu_u_def]: !x R. h4/relation/nf R x <=> (!y. ~happ (happ R x) y)
% Assm [h4s_relations_INu_u_RDOM]: !x R. h4/bool/IN x (h4/relation/RDOM R) <=> (?y. happ (happ R x) y)
% Assm [h4s_relations_RUNION0]: !y x R2 R1. h4/relation/RUNION R1 R2 x y <=> happ (happ R1 x) y \/ happ (happ R2 x) y
% Assm [h4s_lists_EVERY2u_u_cong]: !l2_27 l2 l1_27 l1 P_27 P. l1 = l1_27 /\ l2 = l2_27 /\ (!x y. h4/bool/IN x (h4/list/LIST__TO__SET l1_27) /\ h4/bool/IN y (h4/list/LIST__TO__SET l2_27) ==> (happ (happ P x) y <=> happ (happ P_27 x) y)) ==> (h4/list/LIST__REL P l1 l2 <=> h4/list/LIST__REL P_27 l1_27 l2_27)
% Assm [h4s_lists_LISTu_u_RELu_u_CONS1]: !xs t h R. h4/list/LIST__REL R (happ (happ h4/list/CONS h) t) xs <=> (?h_27 t_27. xs = happ (happ h4/list/CONS h_27) t_27 /\ happ (happ R h) h_27 /\ h4/list/LIST__REL R t t_27)
% Assm [h4s_options_OPTRELu_u_def]: !y x R. h4/option/OPTREL R x y <=> x = h4/option/NONE /\ y = h4/option/NONE \/ (?x0 y0. x = h4/option/SOME x0 /\ y = h4/option/SOME y0 /\ happ (happ R x0) y0)
% Assm [h4s_lists_LISTu_u_RELu_u_CONS2]: !xs t h R. h4/list/LIST__REL R xs (happ (happ h4/list/CONS h) t) <=> (?h_27 t_27. xs = happ (happ h4/list/CONS h_27) t_27 /\ happ (happ R h_27) h /\ h4/list/LIST__REL R t_27 t)
% Assm [h4s_relations_RINTER0]: !y x R2 R1. h4/relation/RINTER R1 R2 x y <=> happ (happ R1 x) y /\ happ (happ R2 x) y
% Assm [h4s_relations_RCOMPL0]: !y x R. h4/relation/RCOMPL R x y <=> ~happ (happ R x) y
% Assm [h4s_lists_LISTu_u_RELu_u_cases]: !a1 a0 R. h4/list/LIST__REL R a0 a1 <=> a0 = h4/list/NIL /\ a1 = h4/list/NIL \/ (?h1 h2 t1 t2. a0 = happ (happ h4/list/CONS h1) t1 /\ a1 = happ (happ h4/list/CONS h2) t2 /\ happ (happ R h1) h2 /\ h4/list/LIST__REL R t1 t2)
% Assm [h4s_relations_INu_u_RRANGE]: !y R. h4/bool/IN y (h4/relation/RRANGE R) <=> (?x. happ (happ R x) y)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_pairs_UNCURRYu_u_VAR]: !v f. happ (h4/pair/UNCURRY f) v = happ (happ f (h4/pair/FST v)) (h4/pair/SND v)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% 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_quotients_QUOTIENTu_u_ABSu_u_REP]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a. happ abs (happ rep a) = a)
% Assm [h4s_quotients_FUNu_u_MAPu_u_THM]: !x h g f. happ (h4/quotient/_2D_2D_3E f g h) x = happ g (happ h (happ f x))
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist1u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x. happ (happ h4/list/_20_40ind__typelist1 x) x = h4/list/_20_40ind__typelist2 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) x (h4/ind__type/FCONS (h4/list/_20_40ind__typelist3 x) _0)))
% Assm [h4s_lists_NIL0]: h4/list/NIL = h4/list/_20_40ind__typelist0
% Assm [h4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> h4/list/_20_40ind__typelist0 = h4/list/_20_40ind__typelist2 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0)
% Assm [h4s_lists_CONS0]: h4/list/CONS = h4/list/_20_40ind__typelist1
% Assm [h4s_lists_listu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27list_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0 \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a0 (h4/ind__type/FCONS a1 _0) /\ happ _27list_27 a1) ==> happ _27list_27 a0_270) ==> happ _27list_27 r) <=> h4/list/_20_40ind__typelist3 (h4/list/_20_40ind__typelist2 r) = r)
% Assm [h4s_lists_listu_u_repfnsu_c0]: !a. h4/list/_20_40ind__typelist2 (h4/list/_20_40ind__typelist3 a) = a
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_pairs_UNCURRY0]: !v f. happ (h4/pair/UNCURRY f) v = happ (happ f (h4/pair/FST v)) (h4/pair/SND v)
% Assm [h4s_combins_Cu_u_THM]: !y x f. happ (happ (happ h4/combin/C f) x) y = happ (happ f y) x
% Assm [h4s_combins_ou_u_THM]: !x g f. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_indu_u_types_ISO0]: !g f. h4/ind__type/ISO f g <=> (!x. happ f (happ g x) = x) /\ (!y. happ g (happ f y) = y)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_BETAu_u_THM]: !y f. happ f y = happ f y
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~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_FCONS0u_c0]: !f a. happ (h4/ind__type/FCONS a f) h4/num/0 = a
% 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_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_lists_LISTu_u_RELu_u_def0]: !x x x. h4/list/LIST__REL x x x <=> (!LIST__REL_27. (!a00 a10. a00 = h4/list/NIL /\ a10 = h4/list/NIL \/ (?h1 h2 t1 t2. a00 = happ (happ h4/list/CONS h1) t1 /\ a10 = happ (happ h4/list/CONS h2) t2 /\ happ (happ x h1) h2 /\ happ (happ LIST__REL_27 t1) t2) ==> happ (happ LIST__REL_27 a00) a10) ==> happ (happ LIST__REL_27 x) x)
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> t
% Assm [h4s_lists_LISTu_u_RELu_u_defu_c1]: !as a R. h4/list/LIST__REL R (happ (happ h4/list/CONS a) as) h4/list/NIL <=> F
% Assm [h4s_lists_LISTu_u_RELu_u_defu_c2]: !bs b R. h4/list/LIST__REL R h4/list/NIL (happ (happ h4/list/CONS b) bs) <=> F
% Assm [h4s_lists_EVERY2u_u_EVERY]: !l2 l1 f. h4/list/LIST__REL f l1 l2 <=> h4/list/LENGTH l1 = h4/list/LENGTH l2 /\ h4/list/EVERY (h4/pair/UNCURRY f) (h4/list/ZIP (h4/pair/_2C l1 l2))
% Assm [h4s_markers_Congu_u_def]: !x. h4/marker/Cong x <=> x
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_BOUNDEDu_u_THM]: !v. h4/bool/BOUNDED v <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_lists_LISTu_u_RELu_u_NILu_c0]: !x R. h4/list/LIST__REL R h4/list/NIL x <=> x = h4/list/NIL
% Assm [h4s_lists_MEMu_c1]: !x t h. h4/bool/IN x (h4/list/LIST__TO__SET (happ (happ h4/list/CONS h) t)) <=> x = h \/ h4/bool/IN x (h4/list/LIST__TO__SET t)
% Assm [h4s_lists_MEMu_c0]: !x. h4/bool/IN x (h4/list/LIST__TO__SET h4/list/NIL) <=> F
% Assm [h4s_lists_listu_u_INDUCT]: !P. happ P h4/list/NIL /\ (!t. happ P t ==> (!h. happ P (happ (happ h4/list/CONS h) t))) ==> (!l. happ P l)
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% 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_UNWINDu_u_FORALLu_u_THM2]: !v f. (!x. x = v ==> happ f x) <=> happ f v
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Goal: !d c b a R2 R1. happ (happ (h4/quotient__pair/_23_23_23 R1 R2) (h4/pair/_2C a b)) (h4/pair/_2C c d) <=> happ (happ R1 a) c /\ happ (happ R2 b) d
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q1329545,TV_Q1329541]: ![V_f, V_g]: (![V_x]: s(TV_Q1329541,happ(s(t_fun(TV_Q1329545,TV_Q1329541),V_f),s(TV_Q1329545,V_x))) = s(TV_Q1329541,happ(s(t_fun(TV_Q1329545,TV_Q1329541),V_g),s(TV_Q1329545,V_x))) => s(t_fun(TV_Q1329545,TV_Q1329541),V_f) = s(t_fun(TV_Q1329545,TV_Q1329541),V_g))).
fof(ah4s_quotientu_u_pairs_PAIRu_u_REL, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b,TV_u_27d]: ![V_uu_3]: (![V_R1, V_a, V_c, V_R2, V_b, V_d]: (p(s(t_bool,happ(s(t_fun(TV_u_27d,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool))),happ(s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool))))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)))))),V_uu_3),s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1))),s(TV_u_27a,V_a))),s(TV_u_27c,V_c))),s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2))),s(TV_u_27b,V_b))),s(TV_u_27d,V_d)))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1),s(TV_u_27a,V_a))),s(TV_u_27c,V_c)))) & p(s(t_bool,happ(s(t_fun(TV_u_27d,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2),s(TV_u_27b,V_b))),s(TV_u_27d,V_d)))))) => ![V_uu_2]: (![V_R1, V_a, V_R2, V_b, V_c]: s(t_fun(TV_u_27d,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27d,t_bool)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,t_fun(TV_u_27d,t_bool))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27c,t_fun(TV_u_27d,t_bool)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27c,t_fun(TV_u_27d,t_bool))))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27c,t_fun(TV_u_27d,t_bool)))))),V_uu_2),s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1))),s(TV_u_27a,V_a))),s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2))),s(TV_u_27b,V_b))),s(TV_u_27c,V_c))) = s(t_fun(TV_u_27d,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool))),happ(s(t_fun(TV_u_27c,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool))))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27c,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)))))),V_uu_3),s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1))),s(TV_u_27a,V_a))),s(TV_u_27c,V_c))),s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2))),s(TV_u_27b,V_b))) => ![V_uu_1]: (![V_R1, V_a, V_R2, V_b]: s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool))))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1))),s(TV_u_27a,V_a))),s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2))),s(TV_u_27b,V_b))) = s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool),h4s_pairs_uncurry(s(t_fun(TV_u_27c,t_fun(TV_u_27d,t_bool)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,t_fun(TV_u_27d,t_bool))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27c,t_fun(TV_u_27d,t_bool)))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27c,t_fun(TV_u_27d,t_bool))))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27c,t_fun(TV_u_27d,t_bool)))))),V_uu_2),s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1))),s(TV_u_27a,V_a))),s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2))),s(TV_u_27b,V_b))))) => ![V_uu_0]: (![V_R1, V_R2, V_a]: s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool))))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1))),s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2))),s(TV_u_27a,V_a))) = s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool))))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1))),s(TV_u_27a,V_a))),s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2))) => ![V_R2, V_R1]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool)),h4s_quotientu_u_pairs_u_23u_23u_23(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool)),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool))))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1))),s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2)))))))))).
fof(ah4s_pairs_PAIRu_u_MAPu_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27d,TV_u_27c]: ![V_y, V_x, V_g, V_f]: s(t_h4s_pairs_prod(TV_u_27b,TV_u_27d),h4s_pairs_u_23u_23(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27d),V_g),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27c,V_y))))) = s(t_h4s_pairs_prod(TV_u_27b,TV_u_27d),h4s_pairs_u_2c(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_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_g),s(TV_u_27c,V_y)))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_relations_Ou_u_ASSOC, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a,TV_u_27d]: ![V_R3, V_R2, V_R1]: s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),h4s_relations_o(s(t_fun(TV_u_27c,t_fun(TV_u_27b,t_bool)),V_R1),s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),h4s_relations_o(s(t_fun(TV_u_27d,t_fun(TV_u_27c,t_bool)),V_R2),s(t_fun(TV_u_27a,t_fun(TV_u_27d,t_bool)),V_R3))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),h4s_relations_o(s(t_fun(TV_u_27d,t_fun(TV_u_27b,t_bool)),h4s_relations_o(s(t_fun(TV_u_27c,t_fun(TV_u_27b,t_bool)),V_R1),s(t_fun(TV_u_27d,t_fun(TV_u_27c,t_bool)),V_R2))),s(t_fun(TV_u_27a,t_fun(TV_u_27d,t_bool)),V_R3)))).
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_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_PAIRu_u_MAP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27a,TV_u_27b]: ![V_p, V_g, V_f]: s(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),h4s_pairs_u_23u_23(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))) = s(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),h4s_pairs_u_2c(s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))))),s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p)))))))).
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_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_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_relations_Ou_u_DEF, axiom, ![TV_u_27g,TV_u_27h,TV_u_27k]: ![V_z, V_x, V_R2, V_R1]: (p(s(t_bool,happ(s(t_fun(TV_u_27k,t_bool),happ(s(t_fun(TV_u_27g,t_fun(TV_u_27k,t_bool)),h4s_relations_o(s(t_fun(TV_u_27h,t_fun(TV_u_27k,t_bool)),V_R1),s(t_fun(TV_u_27g,t_fun(TV_u_27h,t_bool)),V_R2))),s(TV_u_27g,V_x))),s(TV_u_27k,V_z)))) <=> ?[V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27h,t_bool),happ(s(t_fun(TV_u_27g,t_fun(TV_u_27h,t_bool)),V_R2),s(TV_u_27g,V_x))),s(TV_u_27h,V_y)))) & p(s(t_bool,happ(s(t_fun(TV_u_27k,t_bool),happ(s(t_fun(TV_u_27h,t_fun(TV_u_27k,t_bool)),V_R1),s(TV_u_27h,V_y))),s(TV_u_27k,V_z))))))).
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_LETu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_x0]: s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(TV_u_27a,V_x0))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(TV_u_27a,V_x0)))).
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_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_quotients_FUNu_u_MAP, axiom, ![TV_u_27d,TV_u_27b,TV_u_27c,TV_u_27a]: ![V_g, V_f, V_x, V_x0]: s(TV_u_27d,happ(s(t_fun(TV_u_27a,TV_u_27d),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g),s(t_fun(TV_u_27c,TV_u_27b),V_x))),s(TV_u_27a,V_x0))) = s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_x),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x0)))))))).
fof(ah4s_quotients_FUNu_u_REL, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f, V_R2, V_R1]: (p(s(t_bool,h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_g)))) <=> ![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)),V_R1),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),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_y))))))))).
fof(ah4s_relations_RDOMu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_R]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),h4s_relations_rdom(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R))),s(TV_u_27a,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_R),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))))).
fof(ah4s_relations_RRANGE0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_R]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),h4s_relations_rrange(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R))),s(TV_u_27b,V_y)))) <=> ?[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_R),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f))))).
fof(ah4s_sats_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_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_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_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_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,f))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_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_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_pairs_LET2u_u_RATOR, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a1,TV_u_27a2]: ![V_uu_3]: (![V_N, V_x, V_b, V_y]: s(TV_u_27c,happ(s(t_fun(TV_u_27a2,TV_u_27c),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a2,TV_u_27c)),happ(s(t_fun(TV_u_27a1,t_fun(TV_u_27b,t_fun(TV_u_27a2,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),t_fun(TV_u_27a1,t_fun(TV_u_27b,t_fun(TV_u_27a2,TV_u_27c)))),V_uu_3),s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),V_N))),s(TV_u_27a1,V_x))),s(TV_u_27b,V_b))),s(TV_u_27a2,V_y))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),V_N),s(TV_u_27a1,V_x))),s(TV_u_27a2,V_y))),s(TV_u_27b,V_b))) => ![V_uu_2]: (![V_N, V_b, V_x]: s(t_fun(TV_u_27a2,TV_u_27c),happ(s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,TV_u_27c)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a1,t_fun(TV_u_27a2,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),t_fun(TV_u_27b,t_fun(TV_u_27a1,t_fun(TV_u_27a2,TV_u_27c)))),V_uu_2),s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),V_N))),s(TV_u_27b,V_b))),s(TV_u_27a1,V_x))) = s(t_fun(TV_u_27a2,TV_u_27c),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a2,TV_u_27c)),happ(s(t_fun(TV_u_27a1,t_fun(TV_u_27b,t_fun(TV_u_27a2,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),t_fun(TV_u_27a1,t_fun(TV_u_27b,t_fun(TV_u_27a2,TV_u_27c)))),V_uu_3),s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),V_N))),s(TV_u_27a1,V_x))),s(TV_u_27b,V_b))) => ![V_uu_1]: (![V_N, V_x, V_y]: s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c)))),V_uu_1),s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),V_N))),s(TV_u_27a1,V_x))),s(TV_u_27a2,V_y))) = s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),V_N),s(TV_u_27a1,V_x))),s(TV_u_27a2,V_y))) => ![V_uu_0]: (![V_N, V_x]: s(t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c)))),V_uu_0),s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),V_N))),s(TV_u_27a1,V_x))) = s(t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c)))),V_uu_1),s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),V_N))),s(TV_u_27a1,V_x))) => ![V_b, V_N, V_M]: s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),h4s_bools_let(s(t_fun(t_h4s_pairs_prod(TV_u_27a1,TV_u_27a2),t_fun(TV_u_27b,TV_u_27c)),h4s_pairs_uncurry(s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c)))),V_uu_0),s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),V_N))))),s(t_h4s_pairs_prod(TV_u_27a1,TV_u_27a2),V_M))),s(TV_u_27b,V_b))) = s(TV_u_27c,h4s_bools_let(s(t_fun(t_h4s_pairs_prod(TV_u_27a1,TV_u_27a2),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,TV_u_27c)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a1,t_fun(TV_u_27a2,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),t_fun(TV_u_27b,t_fun(TV_u_27a1,t_fun(TV_u_27a2,TV_u_27c)))),V_uu_2),s(t_fun(TV_u_27a1,t_fun(TV_u_27a2,t_fun(TV_u_27b,TV_u_27c))),V_N))),s(TV_u_27b,V_b))))),s(t_h4s_pairs_prod(TV_u_27a1,TV_u_27a2),V_M)))))))).
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_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_NOTu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ![V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) | ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ?[V_x]: (p(s(t_bool,V_P)) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_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_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_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_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c0, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) & p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) | ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_LEFTu_u_FORALLu_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,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_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_pairs_LET2u_u_RAND, axiom, ![TV_u_27d,TV_u_27c,TV_u_27a,TV_u_27b]: ![V_uu_3]: (![V_P, V_N, V_x, V_y]: s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d)))),V_uu_3),s(t_fun(TV_u_27c,TV_u_27d),V_P))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_N))),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) = s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_P),s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_N),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))) => ![V_uu_2]: (![V_P, V_N, V_x]: s(t_fun(TV_u_27b,TV_u_27d),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d)))),V_uu_2),s(t_fun(TV_u_27c,TV_u_27d),V_P))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_N))),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27b,TV_u_27d),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d)))),V_uu_3),s(t_fun(TV_u_27c,TV_u_27d),V_P))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_N))),s(TV_u_27a,V_x))) => ![V_uu_1]: (![V_N, V_x, V_y]: s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_N))),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_N),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) => ![V_uu_0]: (![V_N, V_x]: s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_N))),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_N))),s(TV_u_27a,V_x))) => ![V_P, V_N, V_M]: s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_P),s(TV_u_27c,h4s_bools_let(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_N))))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_M))))) = s(TV_u_27d,h4s_bools_let(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27d),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d))),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d)))),V_uu_2),s(t_fun(TV_u_27c,TV_u_27d),V_P))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_N))))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_M)))))))).
fof(ah4s_lists_LISTu_u_RELu_u_defu_c3, axiom, ![TV_u_27a,TV_u_27b]: ![V_bs, V_b, V_as, V_a, V_R]: (p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_a))),s(t_h4s_lists_list(TV_u_27a),V_as))),s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),h4s_lists_cons),s(TV_u_27b,V_b))),s(t_h4s_lists_list(TV_u_27b),V_bs)))))) <=> (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_R),s(TV_u_27a,V_a))),s(TV_u_27b,V_b)))) & p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),V_as),s(t_h4s_lists_list(TV_u_27b),V_bs))))))).
fof(ah4s_quotients_LAMBDAu_u_PRS, axiom, ![TV_u_27b,TV_u_27d,TV_u_27a,TV_u_27c]: ![V_uu_0]: (![V_rep2, V_f, V_abs1, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))),s(t_fun(TV_u_27c,TV_u_27d),V_f))),s(t_fun(TV_u_27a,TV_u_27c),V_abs1))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27d,TV_u_27b),V_rep2),s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(TV_u_27a,V_x))))))) => ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_x]: s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_f),s(TV_u_27c,V_x))) = s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))),s(t_fun(TV_u_27c,TV_u_27d),V_f))),s(t_fun(TV_u_27a,TV_u_27c),V_abs1))))),s(TV_u_27c,V_x))))))).
fof(ah4s_quotients_Ku_u_RSP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27a,TV_u_27b]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_x1, V_x2, V_y1, V_y2]: ((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)),V_R1),s(TV_u_27a,V_x1))),s(TV_u_27a,V_x2)))) & p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,V_y1))),s(TV_u_27b,V_y2))))) => 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)),V_R1),s(TV_u_27a,h4s_combins_k(s(TV_u_27a,V_x1),s(TV_u_27b,V_y1))))),s(TV_u_27a,h4s_combins_k(s(TV_u_27a,V_x2),s(TV_u_27b,V_y2)))))))))).
fof(ah4s_indu_u_types_ISOu_u_FUN, axiom, ![TV_u_27d,TV_u_27b,TV_u_27a,TV_u_27c]: ![V_uu_3]: (![V_gu_27, V_h, V_f, V_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)))),V_uu_3),s(t_fun(TV_u_27d,TV_u_27b),V_gu_27))),s(t_fun(TV_u_27c,TV_u_27d),V_h))),s(t_fun(TV_u_27a,TV_u_27c),V_f))),s(TV_u_27a,V_a))) = s(TV_u_27b,happ(s(t_fun(TV_u_27d,TV_u_27b),V_gu_27),s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_h),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_a))))))) => ![V_uu_2]: (![V_gu_27, V_f, V_h]: s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)))),V_uu_2),s(t_fun(TV_u_27d,TV_u_27b),V_gu_27))),s(t_fun(TV_u_27a,TV_u_27c),V_f))),s(t_fun(TV_u_27c,TV_u_27d),V_h))) = s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)))),V_uu_3),s(t_fun(TV_u_27d,TV_u_27b),V_gu_27))),s(t_fun(TV_u_27c,TV_u_27d),V_h))),s(t_fun(TV_u_27a,TV_u_27c),V_f))) => ![V_uu_1]: (![V_g, V_h, V_fu_27, V_au_27]: s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27d)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27d))),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27d)))),V_uu_1),s(t_fun(TV_u_27b,TV_u_27d),V_g))),s(t_fun(TV_u_27a,TV_u_27b),V_h))),s(t_fun(TV_u_27c,TV_u_27a),V_fu_27))),s(TV_u_27c,V_au_27))) = s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_h),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_fu_27),s(TV_u_27c,V_au_27))))))) => ![V_uu_0]: (![V_g, V_fu_27, V_h]: s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d))),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)))),V_uu_0),s(t_fun(TV_u_27b,TV_u_27d),V_g))),s(t_fun(TV_u_27c,TV_u_27a),V_fu_27))),s(t_fun(TV_u_27a,TV_u_27b),V_h))) = s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27d)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27d))),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27d)))),V_uu_1),s(t_fun(TV_u_27b,TV_u_27d),V_g))),s(t_fun(TV_u_27a,TV_u_27b),V_h))),s(t_fun(TV_u_27c,TV_u_27a),V_fu_27))) => ![V_gu_27, V_g, V_fu_27, V_f]: ((p(s(t_bool,h4s_indu_u_types_iso(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27c,TV_u_27a),V_fu_27)))) & p(s(t_bool,h4s_indu_u_types_iso(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(t_fun(TV_u_27d,TV_u_27b),V_gu_27))))) => p(s(t_bool,h4s_indu_u_types_iso(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d))),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)))),V_uu_0),s(t_fun(TV_u_27b,TV_u_27d),V_g))),s(t_fun(TV_u_27c,TV_u_27a),V_fu_27))),s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)))),V_uu_2),s(t_fun(TV_u_27d,TV_u_27b),V_gu_27))),s(t_fun(TV_u_27a,TV_u_27c),V_f)))))))))))).
fof(ah4s_quotients_LAMBDAu_u_REPu_u_ABSu_u_RSP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27a,TV_u_27b]: ![V_rep2, V_rep1, V_f2, V_f1, V_abs2, V_abs1, V_REL2, V_REL1]: (((![V_r, V_ru_27]: (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)),V_REL1),s(TV_u_27a,V_r))),s(TV_u_27a,V_ru_27)))) => 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)),V_REL1),s(TV_u_27a,V_r))),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(TV_u_27a,V_ru_27))))))))) & ![V_r, V_ru_27]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_REL2),s(TV_u_27b,V_r))),s(TV_u_27b,V_ru_27)))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_REL2),s(TV_u_27b,V_r))),s(TV_u_27b,happ(s(t_fun(TV_u_27d,TV_u_27b),V_rep2),s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(TV_u_27b,V_ru_27)))))))))) & p(s(t_bool,h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_REL1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_REL2),s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(t_fun(TV_u_27a,TV_u_27b),V_f2))))) => p(s(t_bool,h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_REL1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_REL2),s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(t_fun(TV_u_27a,TV_u_27b),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27d,TV_u_27b),V_rep2),s(t_fun(TV_u_27c,TV_u_27d),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27a,TV_u_27b),V_f2)))))))))).
fof(ah4s_bools_INu_u_DEF, axiom, ![TV_u_27a]: ![V_x, V_x0]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_x0))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x0),s(TV_u_27a,V_x)))).
fof(ah4s_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),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_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_pairs_ABSu_u_PAIRu_u_THM, 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_combins_Ku_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_combins_k(s(TV_u_27a,V_x),s(TV_u_27b,V_y))) = s(TV_u_27a,V_x)).
fof(ah4s_pairs_Cu_u_UNCURRYu_u_L, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c,TV_u_27d]: ![V_x, V_f]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),happ(s(t_fun(TV_u_27d,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(TV_u_27d,TV_u_27c)),t_fun(TV_u_27d,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c))),h4s_combins_c),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(TV_u_27d,TV_u_27c)),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27d,TV_u_27c))),V_f))))),s(TV_u_27d,V_x))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(TV_u_27d,t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27d,t_fun(TV_u_27b,TV_u_27c))),t_fun(TV_u_27d,t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)))),h4s_combins_c),s(t_fun(TV_u_27a,t_fun(TV_u_27d,t_fun(TV_u_27b,TV_u_27c))),h4s_combins_o(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,TV_u_27c)),t_fun(TV_u_27d,t_fun(TV_u_27b,TV_u_27c))),h4s_combins_c),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27d,TV_u_27c))),V_f))))),s(TV_u_27d,V_x)))))).
fof(ah4s_relations_RSUBSET0, axiom, ![TV_u_27a,TV_u_27b]: ![V_R2, V_R1]: (p(s(t_bool,h4s_relations_rsubset(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R2)))) <=> ![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_R1),s(TV_u_27a,V_x))),s(TV_u_27b,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_R2),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))))).
fof(ah4s_lists_EVERY2u_u_THMu_c3, axiom, ![TV_u_27g,TV_u_27h]: ![V_ys, V_y, V_xxs, V_P]: (p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27g,t_fun(TV_u_27h,t_bool)),V_P),s(t_h4s_lists_list(TV_u_27g),V_xxs),s(t_h4s_lists_list(TV_u_27h),happ(s(t_fun(t_h4s_lists_list(TV_u_27h),t_h4s_lists_list(TV_u_27h)),happ(s(t_fun(TV_u_27h,t_fun(t_h4s_lists_list(TV_u_27h),t_h4s_lists_list(TV_u_27h))),h4s_lists_cons),s(TV_u_27h,V_y))),s(t_h4s_lists_list(TV_u_27h),V_ys)))))) <=> ?[V_x, V_xs]: (s(t_h4s_lists_list(TV_u_27g),V_xxs) = s(t_h4s_lists_list(TV_u_27g),happ(s(t_fun(t_h4s_lists_list(TV_u_27g),t_h4s_lists_list(TV_u_27g)),happ(s(t_fun(TV_u_27g,t_fun(t_h4s_lists_list(TV_u_27g),t_h4s_lists_list(TV_u_27g))),h4s_lists_cons),s(TV_u_27g,V_x))),s(t_h4s_lists_list(TV_u_27g),V_xs))) & (p(s(t_bool,happ(s(t_fun(TV_u_27h,t_bool),happ(s(t_fun(TV_u_27g,t_fun(TV_u_27h,t_bool)),V_P),s(TV_u_27g,V_x))),s(TV_u_27h,V_y)))) & p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27g,t_fun(TV_u_27h,t_bool)),V_P),s(t_h4s_lists_list(TV_u_27g),V_xs),s(t_h4s_lists_list(TV_u_27h),V_ys)))))))).
fof(ah4s_lists_EVERY2u_u_THMu_c1, axiom, ![TV_u_27c,TV_u_27d]: ![V_yys, V_xs, V_x, V_P]: (p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27c,t_fun(TV_u_27d,t_bool)),V_P),s(t_h4s_lists_list(TV_u_27c),happ(s(t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_lists_list(TV_u_27c)),happ(s(t_fun(TV_u_27c,t_fun(t_h4s_lists_list(TV_u_27c),t_h4s_lists_list(TV_u_27c))),h4s_lists_cons),s(TV_u_27c,V_x))),s(t_h4s_lists_list(TV_u_27c),V_xs))),s(t_h4s_lists_list(TV_u_27d),V_yys)))) <=> ?[V_y, V_ys]: (s(t_h4s_lists_list(TV_u_27d),V_yys) = s(t_h4s_lists_list(TV_u_27d),happ(s(t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_lists_list(TV_u_27d)),happ(s(t_fun(TV_u_27d,t_fun(t_h4s_lists_list(TV_u_27d),t_h4s_lists_list(TV_u_27d))),h4s_lists_cons),s(TV_u_27d,V_y))),s(t_h4s_lists_list(TV_u_27d),V_ys))) & (p(s(t_bool,happ(s(t_fun(TV_u_27d,t_bool),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27d,t_bool)),V_P),s(TV_u_27c,V_x))),s(TV_u_27d,V_y)))) & p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27c,t_fun(TV_u_27d,t_bool)),V_P),s(t_h4s_lists_list(TV_u_27c),V_xs),s(t_h4s_lists_list(TV_u_27d),V_ys)))))))).
fof(ah4s_relations_nfu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_R]: (p(s(t_bool,h4s_relations_nf(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27a,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_R),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))))).
fof(ah4s_relations_INu_u_RDOM, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_R]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_relations_rdom(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R)))))) <=> ?[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_R),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))))).
fof(ah4s_relations_RUNION0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_R2, V_R1]: (p(s(t_bool,h4s_relations_runion(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27a,V_x),s(TV_u_27b,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_R1),s(TV_u_27a,V_x))),s(TV_u_27b,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_R2),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))))).
fof(ah4s_lists_EVERY2u_u_cong, axiom, ![TV_u_27a,TV_u_27b]: ![V_l2u_27, V_l2, V_l1u_27, V_l1, V_Pu_27, V_P]: ((s(t_h4s_lists_list(TV_u_27a),V_l1) = s(t_h4s_lists_list(TV_u_27a),V_l1u_27) & (s(t_h4s_lists_list(TV_u_27b),V_l2) = s(t_h4s_lists_list(TV_u_27b),V_l2u_27) & ![V_x, V_y]: ((p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l1u_27)))))) & p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_y),s(t_fun(TV_u_27b,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27b),V_l2u_27))))))) => 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))) = 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_Pu_27),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))))) => s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27b),V_l2))) = s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_Pu_27),s(t_h4s_lists_list(TV_u_27a),V_l1u_27),s(t_h4s_lists_list(TV_u_27b),V_l2u_27))))).
fof(ah4s_lists_LISTu_u_RELu_u_CONS1, axiom, ![TV_u_27a,TV_u_27b]: ![V_xs, V_t, V_h, V_R]: (p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t))),s(t_h4s_lists_list(TV_u_27b),V_xs)))) <=> ?[V_hu_27, V_tu_27]: (s(t_h4s_lists_list(TV_u_27b),V_xs) = s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),h4s_lists_cons),s(TV_u_27b,V_hu_27))),s(t_h4s_lists_list(TV_u_27b),V_tu_27))) & (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_R),s(TV_u_27a,V_h))),s(TV_u_27b,V_hu_27)))) & p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),V_t),s(t_h4s_lists_list(TV_u_27b),V_tu_27)))))))).
fof(ah4s_options_OPTRELu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_R]: (p(s(t_bool,h4s_options_optrel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_options_option(TV_u_27a),V_x),s(t_h4s_options_option(TV_u_27b),V_y)))) <=> ((s(t_h4s_options_option(TV_u_27a),V_x) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) & s(t_h4s_options_option(TV_u_27b),V_y) = s(t_h4s_options_option(TV_u_27b),h4s_options_none)) | ?[V_x0, V_y0]: (s(t_h4s_options_option(TV_u_27a),V_x) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x0))) & (s(t_h4s_options_option(TV_u_27b),V_y) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,V_y0))) & 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_R),s(TV_u_27a,V_x0))),s(TV_u_27b,V_y0))))))))).
fof(ah4s_lists_LISTu_u_RELu_u_CONS2, axiom, ![TV_u_27a,TV_u_27b]: ![V_xs, V_t, V_h, V_R]: (p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),V_xs),s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),h4s_lists_cons),s(TV_u_27b,V_h))),s(t_h4s_lists_list(TV_u_27b),V_t)))))) <=> ?[V_hu_27, V_tu_27]: (s(t_h4s_lists_list(TV_u_27a),V_xs) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_hu_27))),s(t_h4s_lists_list(TV_u_27a),V_tu_27))) & (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_R),s(TV_u_27a,V_hu_27))),s(TV_u_27b,V_h)))) & p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),V_tu_27),s(t_h4s_lists_list(TV_u_27b),V_t)))))))).
fof(ah4s_relations_RINTER0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_R2, V_R1]: (p(s(t_bool,h4s_relations_rinter(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27a,V_x),s(TV_u_27b,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_R1),s(TV_u_27a,V_x))),s(TV_u_27b,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_R2),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))))).
fof(ah4s_relations_RCOMPL0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_R]: (p(s(t_bool,h4s_relations_rcompl(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27a,V_x),s(TV_u_27b,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_R),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))))).
fof(ah4s_lists_LISTu_u_RELu_u_cases, axiom, ![TV_u_27a,TV_u_27b]: ![V_a1, V_a0, V_R]: (p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),V_a0),s(t_h4s_lists_list(TV_u_27b),V_a1)))) <=> ((s(t_h4s_lists_list(TV_u_27a),V_a0) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) & s(t_h4s_lists_list(TV_u_27b),V_a1) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil)) | ?[V_h1, V_h2, V_t1, V_t2]: (s(t_h4s_lists_list(TV_u_27a),V_a0) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h1))),s(t_h4s_lists_list(TV_u_27a),V_t1))) & (s(t_h4s_lists_list(TV_u_27b),V_a1) = s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),h4s_lists_cons),s(TV_u_27b,V_h2))),s(t_h4s_lists_list(TV_u_27b),V_t2))) & (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_R),s(TV_u_27a,V_h1))),s(TV_u_27b,V_h2)))) & p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),V_t1),s(t_h4s_lists_list(TV_u_27b),V_t2)))))))))).
fof(ah4s_relations_INu_u_RRANGE, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_R]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),h4s_relations_rrange(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool)),V_R)))))) <=> ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27b,V_x))),s(TV_u_27a,V_y)))))).
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_pairs_UNCURRYu_u_VAR, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_v))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_v))))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_v)))))).
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,t)))).
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_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_quotients_QUOTIENTu_u_ABSu_u_REP, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))) = s(TV_u_27b,V_a))).
fof(ah4s_quotients_FUNu_u_MAPu_u_THM, axiom, ![TV_u_27d,TV_u_27b,TV_u_27c,TV_u_27a]: ![V_x, V_h, V_g, V_f]: s(TV_u_27d,happ(s(t_fun(TV_u_27a,TV_u_27d),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g),s(t_fun(TV_u_27c,TV_u_27b),V_h))),s(TV_u_27a,V_x))) = s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_h),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),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_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_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist1u_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]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_u_20u_40indu_u_typelist1),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_x0))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(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_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_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(TV_u_27a),V_x0))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))))).
fof(ah4s_lists_NIL0, axiom, ![TV_u_27a]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist0)).
fof(ah4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist0u_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_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist0) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(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_lists_CONS0, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons) = s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_u_20u_40indu_u_typelist1)).
fof(ah4s_lists_listu_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_27listu_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]: (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_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_a1),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_27listu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27listu_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_27listu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r))))) <=> s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(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_lists_listu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(TV_u_27a),V_a))))) = s(t_h4s_lists_list(TV_u_27a),V_a)).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
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_pairs_UNCURRY0, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_v))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_v))))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_v)))))).
fof(ah4s_combins_Cu_u_THM, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27c))),h4s_combins_c),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(TV_u_27b,V_x))),s(TV_u_27a,V_y))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_y))),s(TV_u_27b,V_x)))).
fof(ah4s_combins_ou_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_x, V_g, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27c,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,V_x)))))).
fof(ah4s_indu_u_types_ISO0, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (p(s(t_bool,h4s_indu_u_types_iso(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27b,TV_u_27a),V_g)))) <=> (![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_g),s(TV_u_27b,V_x))))) = s(TV_u_27b,V_x) & ![V_y]: s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_g),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_27a,V_y)))).
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_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_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_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_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_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_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_bools_UNWINDu_u_THM2, axiom, ![TV_u_27a]: ![V_a, V_P]: (?[V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_a)))))).
fof(ah4s_lists_LISTu_u_RELu_u_def0, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_x0, V_x1]: (p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_x),s(t_h4s_lists_list(TV_u_27a),V_x0),s(t_h4s_lists_list(TV_u_27b),V_x1)))) <=> ![V_LISTu_u_RELu_27]: (![V_a00, V_a10]: (((s(t_h4s_lists_list(TV_u_27a),V_a00) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) & s(t_h4s_lists_list(TV_u_27b),V_a10) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil)) | ?[V_h1, V_h2, V_t1, V_t2]: (s(t_h4s_lists_list(TV_u_27a),V_a00) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h1))),s(t_h4s_lists_list(TV_u_27a),V_t1))) & (s(t_h4s_lists_list(TV_u_27b),V_a10) = s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),h4s_lists_cons),s(TV_u_27b,V_h2))),s(t_h4s_lists_list(TV_u_27b),V_t2))) & (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_x),s(TV_u_27a,V_h1))),s(TV_u_27b,V_h2)))) & p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_bool),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_bool)),V_LISTu_u_RELu_27),s(t_h4s_lists_list(TV_u_27a),V_t1))),s(t_h4s_lists_list(TV_u_27b),V_t2)))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_bool),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_bool)),V_LISTu_u_RELu_27),s(t_h4s_lists_list(TV_u_27a),V_a00))),s(t_h4s_lists_list(TV_u_27b),V_a10))))) => p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_bool),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_bool)),V_LISTu_u_RELu_27),s(t_h4s_lists_list(TV_u_27a),V_x0))),s(t_h4s_lists_list(TV_u_27b),V_x1))))))).
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_EXISTSu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (?[V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_lists_LISTu_u_RELu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_as, V_a, V_R]: s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_a))),s(t_h4s_lists_list(TV_u_27a),V_as))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil))) = s(t_bool,f)).
fof(ah4s_lists_LISTu_u_RELu_u_defu_c2, axiom, ![TV_u_27a,TV_u_27b]: ![V_bs, V_b, V_R]: s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),h4s_lists_cons),s(TV_u_27b,V_b))),s(t_h4s_lists_list(TV_u_27b),V_bs))))) = s(t_bool,f)).
fof(ah4s_lists_EVERY2u_u_EVERY, axiom, ![TV_u_27a,TV_u_27b]: ![V_l2, V_l1, V_f]: (p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_f),s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27b),V_l2)))) <=> (s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l1))) = s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27b),V_l2))) & p(s(t_bool,h4s_lists_every(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_f))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_zip(s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27b),V_l2))))))))))).
fof(ah4s_markers_Congu_u_def, axiom, ![V_x]: s(t_bool,h4s_markers_cong(s(t_bool,V_x))) = s(t_bool,V_x)).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_BOUNDEDu_u_THM, axiom, ![V_v]: s(t_bool,h4s_bools_bounded(s(t_bool,V_v))) = s(t_bool,t)).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_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_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_lists_LISTu_u_RELu_u_NILu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_R]: (p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27b),V_x)))) <=> s(t_h4s_lists_list(TV_u_27b),V_x) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil))).
fof(ah4s_lists_MEMu_c1, axiom, ![TV_u_27a]: ![V_x, V_t, V_h]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t)))))))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_h) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_t))))))))).
fof(ah4s_lists_MEMu_c0, axiom, ![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),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))))) = s(t_bool,f)).
fof(ah4s_lists_listu_u_INDUCT, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))) & ![V_t]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t)))) => ![V_h]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_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_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_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_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_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_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_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_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_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ch4s_quotientu_u_pairs_PAIRu_u_RELu_u_THM, conjecture, ![TV_u_27a,TV_u_27c,TV_u_27b,TV_u_27d]: ![V_d, V_c, V_b, V_a, V_R2, V_R1]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_bool)),h4s_quotientu_u_pairs_u_23u_23u_23(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2))),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(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),h4s_pairs_u_2c(s(TV_u_27c,V_c),s(TV_u_27d,V_d)))))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),V_R1),s(TV_u_27a,V_a))),s(TV_u_27c,V_c)))) & p(s(t_bool,happ(s(t_fun(TV_u_27d,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27d,t_bool)),V_R2),s(TV_u_27b,V_b))),s(TV_u_27d,V_d))))))).
