%   ORIGINAL: h4/update/OVERRIDE__ind
% 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/update/OVERRIDE__primitive__def: h4/update/OVERRIDE = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!t x. R (h4/list/FILTER (\y. ~(h4/pair/FST x = h4/pair/FST y)) t) (h4/list/CONS x t)))) (\OVERRIDE a. h4/list/list__CASE a (h4/combin/I h4/list/NIL) (\x t. h4/combin/I (h4/list/CONS x (OVERRIDE (h4/list/FILTER (\y. ~(h4/pair/FST x = h4/pair/FST y)) t)))))
% Assm: h4/list/UNZIP0_c1: !x l. h4/list/UNZIP (h4/list/CONS x l) = h4/pair/_2C (h4/list/CONS (h4/pair/FST x) (h4/pair/FST (h4/list/UNZIP l))) (h4/list/CONS (h4/pair/SND x) (h4/pair/SND (h4/list/UNZIP l)))
% Assm: h4/bool/TRUTH: T
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/pred__set/IN__CROSS: !x Q P. h4/bool/IN x (h4/pred__set/CROSS P Q) <=> h4/bool/IN (h4/pair/FST x) P /\ h4/bool/IN (h4/pair/SND x) Q
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/rich__list/UNZIP__SNOC: !x l. h4/list/UNZIP (h4/list/SNOC x l) = h4/pair/_2C (h4/list/SNOC (h4/pair/FST x) (h4/pair/FST (h4/list/UNZIP l))) (h4/list/SNOC (h4/pair/SND x) (h4/pair/SND (h4/list/UNZIP l)))
% Assm: h4/list/mem__exists__set: !y x l. h4/bool/IN (h4/pair/_2C x y) (h4/list/LIST__TO__SET l) ==> (?z. x = h4/pair/FST z /\ h4/bool/IN z (h4/list/LIST__TO__SET l))
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/pred__set/CROSS__DEF: !Q P. h4/pred__set/CROSS P Q = h4/pred__set/GSPEC (\p. h4/pair/_2C p (h4/bool/IN (h4/pair/FST p) P /\ h4/bool/IN (h4/pair/SND p) Q))
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/rich__list/ALL__DISTINCT__MEM__ZIP__MAP: !x ls f. h4/list/ALL__DISTINCT ls ==> (h4/bool/IN x (h4/list/LIST__TO__SET (h4/list/ZIP (h4/pair/_2C ls (h4/list/MAP f ls)))) <=> h4/bool/IN (h4/pair/FST x) (h4/list/LIST__TO__SET ls) /\ h4/pair/SND x = f (h4/pair/FST x))
% Assm: h4/list/MEM__ZIP__MEM__MAP: !ps p. h4/list/LENGTH (h4/pair/FST ps) = h4/list/LENGTH (h4/pair/SND ps) /\ h4/bool/IN p (h4/list/LIST__TO__SET (h4/list/ZIP ps)) ==> h4/bool/IN (h4/pair/FST p) (h4/list/LIST__TO__SET (h4/pair/FST ps)) /\ h4/bool/IN (h4/pair/SND p) (h4/list/LIST__TO__SET (h4/pair/SND ps))
% Assm: h4/pair/ELIM__PFORALL: !P. (!p. P (h4/pair/FST p) (h4/pair/SND p)) <=> (!p1 p2. P p1 p2)
% Assm: h4/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm: h4/pair/PAIR__FST__SND__EQ: !q p. p = q <=> h4/pair/FST p = h4/pair/FST q /\ h4/pair/SND p = h4/pair/SND q
% Assm: h4/pair/ELIM__PEXISTS: !P. (?p. P (h4/pair/FST p) (h4/pair/SND p)) <=> (?p1 p2. P p1 p2)
% Assm: h4/pair/SWAP__def: !a. h4/pair/SWAP a = h4/pair/_2C (h4/pair/SND a) (h4/pair/FST a)
% Assm: h4/pred__set/CROSS__applied: !x Q P. h4/pred__set/CROSS P Q x <=> h4/bool/IN (h4/pair/FST x) P /\ h4/bool/IN (h4/pair/SND x) Q
% Assm: h4/marker/Cong__def: !x. h4/marker/Cong x <=> x
% Assm: h4/bool/SKOLEM__THM: !P. (!x. ?y. P x y) <=> (?f. !x. P x (f x))
% Assm: h4/pair/PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/bool/IN__DEF: h4/bool/IN = (\x f. f x)
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% 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/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/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/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/LIST__EQ__MAP__PAIR: !l2 l1. h4/list/MAP h4/pair/FST l1 = h4/list/MAP h4/pair/FST l2 /\ h4/list/MAP h4/pair/SND l1 = h4/list/MAP h4/pair/SND l2 ==> l1 = l2
% Assm: h4/list/UNZIP__MAP: !L. h4/list/UNZIP L = h4/pair/_2C (h4/list/MAP h4/pair/FST L) (h4/list/MAP h4/pair/SND L)
% Assm: h4/bool/BETA__THM: !y f. (\x. f x) y = f y
% 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/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% 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/list/UNZIP0_c0: h4/list/UNZIP h4/list/NIL = h4/pair/_2C h4/list/NIL h4/list/NIL
% Assm: h4/rich__list/UNZIP__FST__DEF: !l. h4/rich__list/UNZIP__FST l = h4/pair/FST (h4/list/UNZIP l)
% Assm: h4/list/LENGTH__UNZIP_c0: !pl. h4/list/LENGTH (h4/pair/FST (h4/list/UNZIP pl)) = h4/list/LENGTH pl
% Assm: h4/rich__list/MAP__SND__FILTER__NEQ: !z ls. h4/list/MAP h4/pair/SND (h4/list/FILTER (h4/pair/UNCURRY (\x y. ~(y = z))) ls) = h4/list/FILTER (\y. ~(z = y)) (h4/list/MAP h4/pair/SND ls)
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/list/UNZIP__THM_c1: !y x t. h4/list/UNZIP (h4/list/CONS (h4/pair/_2C x y) t) = h4/bool/LET (h4/pair/UNCURRY (\L1 L2. h4/pair/_2C (h4/list/CONS x L1) (h4/list/CONS y L2))) (h4/list/UNZIP t)
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/list/SNOC0_c1: !x_27 x l. h4/list/SNOC x (h4/list/CONS x_27 l) = h4/list/CONS x_27 (h4/list/SNOC x l)
% Assm: h4/list/SNOC0_c0: !x. h4/list/SNOC x h4/list/NIL = h4/list/CONS x h4/list/NIL
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/pred__set/GSPECIFICATION: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = f x)
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/pair/CLOSED__PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/list/MEM__ZIP: !p l2 l1. h4/list/LENGTH l1 = h4/list/LENGTH l2 ==> (h4/bool/IN p (h4/list/LIST__TO__SET (h4/list/ZIP (h4/pair/_2C l1 l2))) <=> (?n. h4/prim__rec/_3C n (h4/list/LENGTH l1) /\ p = h4/pair/_2C (h4/list/EL n l1) (h4/list/EL n l2)))
% Assm: h4/bool/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/list/MEM__EL: !x l. h4/bool/IN x (h4/list/LIST__TO__SET l) <=> (?n. h4/prim__rec/_3C n (h4/list/LENGTH l) /\ x = h4/list/EL n l)
% Assm: h4/list/MAP0_c0: !f. h4/list/MAP f h4/list/NIL = h4/list/NIL
% Assm: h4/list/MAP0_c1: !t h f. h4/list/MAP f (h4/list/CONS h t) = h4/list/CONS (f h) (h4/list/MAP f t)
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/pair/FORALL__PROD: !P. (!p. P p) <=> (!p__1 p__2. P (h4/pair/_2C p__1 p__2))
% Assm: h4/list/LENGTH0_c1: !t h. h4/list/LENGTH (h4/list/CONS h t) = h4/num/SUC (h4/list/LENGTH t)
% Assm: h4/list/LENGTH0_c0: h4/list/LENGTH h4/list/NIL = h4/num/0
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% 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/bool/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/list/EL__ALL__DISTINCT__EL__EQ: !l. h4/list/ALL__DISTINCT l <=> (!n1 n2. h4/prim__rec/_3C n1 (h4/list/LENGTH l) /\ h4/prim__rec/_3C n2 (h4/list/LENGTH l) ==> (h4/list/EL n1 l = h4/list/EL n2 l <=> n1 = n2))
% Assm: h4/list/EL__MAP: !n l. h4/prim__rec/_3C n (h4/list/LENGTH l) ==> (!f. h4/list/EL n (h4/list/MAP f l) = f (h4/list/EL n l))
% Assm: h4/list/LENGTH__MAP: !l f. h4/list/LENGTH (h4/list/MAP f l) = h4/list/LENGTH l
% Assm: h4/bool/LEFT__FORALL__OR__THM: !Q P. (!x. P x \/ Q) <=> (!x. P x) \/ Q
% Assm: h4/rich__list/FILTER__MAP: !l f2 f1. h4/list/FILTER f1 (h4/list/MAP f2 l) = h4/list/MAP f2 (h4/list/FILTER (h4/combin/o f1 f2) l)
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% 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/list/MAP__EQ__EVERY2: !l2 l1 f2 f1. h4/list/MAP f1 l1 = h4/list/MAP f2 l2 <=> h4/list/LENGTH l1 = h4/list/LENGTH l2 /\ h4/list/LIST__REL (\x y. f1 x = f2 y) l1 l2
% Assm: h4/list/LIST__EQ__REWRITE: !l2 l1. l1 = l2 <=> h4/list/LENGTH l1 = h4/list/LENGTH l2 /\ (!x. h4/prim__rec/_3C x (h4/list/LENGTH l1) ==> h4/list/EL x l1 = h4/list/EL x l2)
% Assm: h4/list/EVERY__MEM: !l P. h4/list/EVERY P l <=> (!e. h4/bool/IN e (h4/list/LIST__TO__SET l) ==> P e)
% Assm: h4/pair/pair__CASES: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/bool/LEFT__FORALL__IMP__THM: !Q P. (!x. P x ==> Q) <=> (?x. P x) ==> Q
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/pair/S__UNCURRY__R: !g f. h4/combin/S f (h4/pair/UNCURRY g) = h4/pair/UNCURRY (h4/combin/S (h4/combin/o h4/combin/S (h4/combin/o (h4/combin/o f) h4/pair/_2C)) g)
% Assm: h4/pair/FORALL__UNCURRY: !f. $forall (h4/pair/UNCURRY f) <=> $forall (h4/combin/o $forall f)
% Assm: h4/pair/o__UNCURRY__R: !g f. h4/combin/o f (h4/pair/UNCURRY g) = h4/pair/UNCURRY (h4/combin/o (h4/combin/o f) g)
% Assm: h4/combin/LET__FORALL__ELIM: !v f. h4/bool/LET f v <=> $forall (h4/combin/S (h4/combin/o $imply (h4/combin/o h4/marker/Abbrev (h4/combin/C $equals v))) f)
% Assm: h4/combin/C__THM: !y x f. h4/combin/C f x y = f y x
% Assm: h4/combin/S__ABS__R: !g f. h4/combin/S f (\x. g x) = (\x. f x (g x))
% Assm: h4/combin/o__ABS__R: !g f. h4/combin/o f (\x. g x) = (\x. f (g x))
% Assm: h4/combin/GEN__LET__RAND: !v f P. P (h4/bool/LET f v) = h4/bool/LET (h4/combin/o P f) v
% Goal: !P. P h4/list/NIL /\ (!x t. P (h4/list/FILTER (\y. ~(h4/pair/FST x = h4/pair/FST y)) t) ==> P (h4/list/CONS x t)) ==> (!v. P v)
%   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_updates_OVERRIDEu_u_primitiveu_u_def]: !_5. (!OVERRIDE _1 x t. happ (happ (happ (happ _5 OVERRIDE) _1) x) t = h4/combin/I (happ (happ h4/list/CONS x) (happ OVERRIDE (h4/list/FILTER (happ _1 x) t)))) ==> (!_4. (!OVERRIDE _1 x. happ (happ (happ _4 OVERRIDE) _1) x = happ (happ (happ _5 OVERRIDE) _1) x) ==> (!_3. (!OVERRIDE _1 a. happ (happ (happ _3 OVERRIDE) _1) a = h4/list/list__CASE a (h4/combin/I h4/list/NIL) (happ (happ _4 OVERRIDE) _1)) ==> (!_2. (!_1 OVERRIDE. happ (happ _2 _1) OVERRIDE = happ (happ _3 OVERRIDE) _1) ==> (!_1. (!x y. happ (happ _1 x) y <=> ~(happ h4/pair/FST x = happ h4/pair/FST y)) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R /\ (!t x. happ (happ R (h4/list/FILTER (happ _1 x) t)) (happ (happ h4/list/CONS x) t))) ==> h4/update/OVERRIDE = h4/relation/WFREC (h4/min/_40 _0) (happ _2 _1))))))
% Assm [h4s_lists_UNZIP0u_c1]: !x l. h4/list/UNZIP (happ (happ h4/list/CONS x) l) = happ (happ h4/pair/_2C (happ (happ h4/list/CONS (happ h4/pair/FST x)) (happ h4/pair/FST (h4/list/UNZIP l)))) (happ (happ h4/list/CONS (happ h4/pair/SND x)) (happ h4/pair/SND (h4/list/UNZIP l)))
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_pairs_SND0]: !y x. happ h4/pair/SND (happ (happ h4/pair/_2C x) y) = y
% Assm [h4s_pairs_FST0]: !y x. happ h4/pair/FST (happ (happ h4/pair/_2C x) y) = x
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_predu_u_sets_INu_u_CROSS]: !x Q P. h4/bool/IN x (h4/pred__set/CROSS P Q) <=> h4/bool/IN (happ h4/pair/FST x) P /\ h4/bool/IN (happ h4/pair/SND x) Q
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = happ (happ h4/pair/_2C q) r
% Assm [h4s_richu_u_lists_UNZIPu_u_SNOC]: !x l. h4/list/UNZIP (h4/list/SNOC x l) = happ (happ h4/pair/_2C (h4/list/SNOC (happ h4/pair/FST x) (happ h4/pair/FST (h4/list/UNZIP l)))) (h4/list/SNOC (happ h4/pair/SND x) (happ h4/pair/SND (h4/list/UNZIP l)))
% Assm [h4s_lists_memu_u_existsu_u_set]: !y x l. h4/bool/IN (happ (happ h4/pair/_2C x) y) (h4/list/LIST__TO__SET l) ==> (?z. x = happ h4/pair/FST z /\ h4/bool/IN z (h4/list/LIST__TO__SET l))
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_predu_u_sets_CROSSu_u_DEF]: !_0. (!P Q p. ?v. (v <=> h4/bool/IN (happ h4/pair/FST p) P /\ h4/bool/IN (happ h4/pair/SND p) Q) /\ happ (happ (happ _0 P) Q) p = happ (happ h4/pair/_2C p) v) ==> (!Q P. h4/pred__set/CROSS P Q = h4/pred__set/GSPEC (happ (happ _0 P) Q))
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_richu_u_lists_ALLu_u_DISTINCTu_u_MEMu_u_ZIPu_u_MAP]: !x ls f. h4/list/ALL__DISTINCT ls ==> (h4/bool/IN x (h4/list/LIST__TO__SET (h4/list/ZIP (happ (happ h4/pair/_2C ls) (h4/list/MAP f ls)))) <=> h4/bool/IN (happ h4/pair/FST x) (h4/list/LIST__TO__SET ls) /\ happ h4/pair/SND x = happ f (happ h4/pair/FST x))
% Assm [h4s_lists_MEMu_u_ZIPu_u_MEMu_u_MAP]: !ps p. h4/list/LENGTH (happ h4/pair/FST ps) = h4/list/LENGTH (happ h4/pair/SND ps) /\ h4/bool/IN p (h4/list/LIST__TO__SET (h4/list/ZIP ps)) ==> h4/bool/IN (happ h4/pair/FST p) (h4/list/LIST__TO__SET (happ h4/pair/FST ps)) /\ h4/bool/IN (happ h4/pair/SND p) (h4/list/LIST__TO__SET (happ h4/pair/SND ps))
% Assm [h4s_pairs_ELIMu_u_PFORALL]: !P. (!p. happ (happ P (happ h4/pair/FST p)) (happ h4/pair/SND p)) <=> (!p1 p2. happ (happ P p1) p2)
% Assm [h4s_pairs_PAIR]: !x. happ (happ h4/pair/_2C (happ h4/pair/FST x)) (happ h4/pair/SND x) = x
% Assm [h4s_pairs_PAIRu_u_FSTu_u_SNDu_u_EQ]: !q p. p = q <=> happ h4/pair/FST p = happ h4/pair/FST q /\ happ h4/pair/SND p = happ h4/pair/SND q
% Assm [h4s_pairs_ELIMu_u_PEXISTS]: !P. (?p. happ (happ P (happ h4/pair/FST p)) (happ h4/pair/SND p)) <=> (?p1 p2. happ (happ P p1) p2)
% Assm [h4s_pairs_SWAPu_u_def]: !a. h4/pair/SWAP a = happ (happ h4/pair/_2C (happ h4/pair/SND a)) (happ h4/pair/FST a)
% Assm [h4s_predu_u_sets_CROSSu_u_applied]: !x Q P. happ (h4/pred__set/CROSS P Q) x <=> h4/bool/IN (happ h4/pair/FST x) P /\ h4/bool/IN (happ h4/pair/SND x) Q
% Assm [h4s_markers_Congu_u_def]: !x. h4/marker/Cong x <=> x
% Assm [h4s_bools_SKOLEMu_u_THM]: !P. (!x. ?y. happ (happ P x) y) <=> (?f. !x. happ (happ P x) (happ f x))
% Assm [h4s_pairs_PAIRu_u_EQ]: !y x b a. happ (happ h4/pair/_2C x) y = happ (happ h4/pair/_2C a) b <=> x = a /\ y = b
% Assm [h4s_bools_INu_u_DEF]: !x x. h4/bool/IN x x <=> happ x x
% 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_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_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_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_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_LISTu_u_EQu_u_MAPu_u_PAIR]: !l2 l1. h4/list/MAP h4/pair/FST l1 = h4/list/MAP h4/pair/FST l2 /\ h4/list/MAP h4/pair/SND l1 = h4/list/MAP h4/pair/SND l2 ==> l1 = l2
% Assm [h4s_lists_UNZIPu_u_MAP]: !L. h4/list/UNZIP L = happ (happ h4/pair/_2C (h4/list/MAP h4/pair/FST L)) (h4/list/MAP h4/pair/SND L)
% Assm [h4s_bools_BETAu_u_THM]: !y f. happ f y = happ f y
% 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_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% 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_lists_UNZIP0u_c0]: h4/list/UNZIP h4/list/NIL = happ (happ h4/pair/_2C h4/list/NIL) h4/list/NIL
% Assm [h4s_richu_u_lists_UNZIPu_u_FSTu_u_DEF]: !l. h4/rich__list/UNZIP__FST l = happ h4/pair/FST (h4/list/UNZIP l)
% Assm [h4s_lists_LENGTHu_u_UNZIPu_c0]: !pl. h4/list/LENGTH (happ h4/pair/FST (h4/list/UNZIP pl)) = h4/list/LENGTH pl
% Assm [h4s_richu_u_lists_MAPu_u_SNDu_u_FILTERu_u_NEQ]: !_2. (!z y. happ (happ _2 z) y <=> ~(z = y)) ==> (!_1. (!z y. happ (happ _1 z) y <=> ~(y = z)) ==> (!_0. (!z x. happ (happ _0 z) x = happ _1 z) ==> (!z ls. h4/list/MAP h4/pair/SND (h4/list/FILTER (h4/pair/UNCURRY (happ _0 z)) ls) = h4/list/FILTER (happ _2 z) (h4/list/MAP h4/pair/SND ls))))
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_lists_UNZIPu_u_THMu_c1]: !_1. (!x L1 y L2. happ (happ (happ (happ _1 x) L1) y) L2 = happ (happ h4/pair/_2C (happ (happ h4/list/CONS x) L1)) (happ (happ h4/list/CONS y) L2)) ==> (!_0. (!x y L1. happ (happ (happ _0 x) y) L1 = happ (happ (happ _1 x) L1) y) ==> (!y x t. h4/list/UNZIP (happ (happ h4/list/CONS (happ (happ h4/pair/_2C x) y)) t) = h4/bool/LET (h4/pair/UNCURRY (happ (happ _0 x) y)) (h4/list/UNZIP t)))
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_lists_SNOC0u_c1]: !x_27 x l. h4/list/SNOC x (happ (happ h4/list/CONS x_27) l) = happ (happ h4/list/CONS x_27) (h4/list/SNOC x l)
% Assm [h4s_lists_SNOC0u_c0]: !x. h4/list/SNOC x h4/list/NIL = happ (happ h4/list/CONS x) h4/list/NIL
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_predu_u_sets_GSPECIFICATION]: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. happ (happ h4/pair/_2C v) T = happ f x)
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_pairs_CLOSEDu_u_PAIRu_u_EQ]: !y x b a. happ (happ h4/pair/_2C x) y = happ (happ h4/pair/_2C a) b <=> x = a /\ y = b
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_lists_MEMu_u_ZIP]: !p l2 l1. h4/list/LENGTH l1 = h4/list/LENGTH l2 ==> (h4/bool/IN p (h4/list/LIST__TO__SET (h4/list/ZIP (happ (happ h4/pair/_2C l1) l2))) <=> (?n. h4/prim__rec/_3C n (h4/list/LENGTH l1) /\ p = happ (happ h4/pair/_2C (h4/list/EL n l1)) (h4/list/EL n l2)))
% Assm [h4s_bools_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_lists_MEMu_u_EL]: !x l. h4/bool/IN x (h4/list/LIST__TO__SET l) <=> (?n. h4/prim__rec/_3C n (h4/list/LENGTH l) /\ x = h4/list/EL n l)
% Assm [h4s_lists_MAP0u_c0]: !f. h4/list/MAP f h4/list/NIL = h4/list/NIL
% Assm [h4s_lists_MAP0u_c1]: !t h f. h4/list/MAP f (happ (happ h4/list/CONS h) t) = happ (happ h4/list/CONS (happ f h)) (h4/list/MAP f t)
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. happ (h4/pair/UNCURRY f) (happ (happ h4/pair/_2C x) y) = happ (happ f x) y
% Assm [h4s_pairs_FORALLu_u_PROD]: !P. (!p. happ P p) <=> (!p__1 p__2. happ P (happ (happ h4/pair/_2C p__1) p__2))
% Assm [h4s_lists_LENGTH0u_c1]: !t h. h4/list/LENGTH (happ (happ h4/list/CONS h) t) = h4/num/SUC (h4/list/LENGTH t)
% Assm [h4s_lists_LENGTH0u_c0]: h4/list/LENGTH h4/list/NIL = h4/num/0
% Assm [h4s_combins_ou_u_THM]: !x g f. happ (happ (h4/combin/o f) g) x = happ f (happ g x)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% 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_bools_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_lists_ELu_u_ALLu_u_DISTINCTu_u_ELu_u_EQ]: !l. h4/list/ALL__DISTINCT l <=> (!n1 n2. h4/prim__rec/_3C n1 (h4/list/LENGTH l) /\ h4/prim__rec/_3C n2 (h4/list/LENGTH l) ==> (h4/list/EL n1 l = h4/list/EL n2 l <=> n1 = n2))
% Assm [h4s_lists_ELu_u_MAP]: !n l. h4/prim__rec/_3C n (h4/list/LENGTH l) ==> (!f. h4/list/EL n (h4/list/MAP f l) = happ f (h4/list/EL n l))
% Assm [h4s_lists_LENGTHu_u_MAP]: !l f. h4/list/LENGTH (h4/list/MAP f l) = h4/list/LENGTH l
% Assm [h4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. happ P x \/ Q) <=> (!x. happ P x) \/ Q
% Assm [h4s_richu_u_lists_FILTERu_u_MAP]: !l f2 f1. h4/list/FILTER f1 (h4/list/MAP f2 l) = h4/list/MAP f2 (h4/list/FILTER (happ (h4/combin/o f1) f2) l)
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% 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 (happ (happ h4/pair/_2C l1) l2))
% Assm [h4s_lists_MAPu_u_EQu_u_EVERY2]: !_1. (!f1 x f2 y. happ (happ (happ (happ _1 f1) x) f2) y <=> happ f1 x = happ f2 y) ==> (!_0. (!f1 f2 x. happ (happ (happ _0 f1) f2) x = happ (happ (happ _1 f1) x) f2) ==> (!l2 l1 f2 f1. h4/list/MAP f1 l1 = h4/list/MAP f2 l2 <=> h4/list/LENGTH l1 = h4/list/LENGTH l2 /\ h4/list/LIST__REL (happ (happ _0 f1) f2) l1 l2))
% Assm [h4s_lists_LISTu_u_EQu_u_REWRITE]: !l2 l1. l1 = l2 <=> h4/list/LENGTH l1 = h4/list/LENGTH l2 /\ (!x. h4/prim__rec/_3C x (h4/list/LENGTH l1) ==> h4/list/EL x l1 = h4/list/EL x l2)
% Assm [h4s_lists_EVERYu_u_MEM]: !l P. h4/list/EVERY P l <=> (!e. h4/bool/IN e (h4/list/LIST__TO__SET l) ==> happ P e)
% Assm [h4s_pairs_pairu_u_CASES]: !x. ?q r. x = happ (happ h4/pair/_2C q) r
% Assm [h4s_bools_LEFTu_u_FORALLu_u_IMPu_u_THM]: !Q P. (!x. happ P x ==> Q) <=> (?x. happ P x) ==> Q
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_pairs_Su_u_UNCURRYu_u_R]: !g f. happ (happ h4/combin/S f) (h4/pair/UNCURRY g) = h4/pair/UNCURRY (happ (happ h4/combin/S (happ (h4/combin/o h4/combin/S) (happ (h4/combin/o (h4/combin/o f)) h4/pair/_2C))) g)
% Assm [h4s_pairs_FORALLu_u_UNCURRY]: !f. happ $forall (h4/pair/UNCURRY f) <=> happ $forall (happ (h4/combin/o $forall) f)
% Assm [h4s_pairs_ou_u_UNCURRYu_u_R]: !g f. happ (h4/combin/o f) (h4/pair/UNCURRY g) = h4/pair/UNCURRY (happ (h4/combin/o (h4/combin/o f)) g)
% Assm [h4s_combins_LETu_u_FORALLu_u_ELIM]: !v f. h4/bool/LET f v <=> happ $forall (happ (happ h4/combin/S (happ (h4/combin/o $imply) (happ (h4/combin/o h4/marker/Abbrev) (h4/combin/C $equals v)))) f)
% Assm [h4s_combins_Cu_u_THM]: !y x f. happ (h4/combin/C f x) y = happ (happ f y) x
% Assm [h4s_combins_Su_u_ABSu_u_R]: !_0. (!g x. happ (happ _0 g) x = happ g x) ==> (!g f x. happ (happ (happ h4/combin/S f) (happ _0 g)) x = happ (happ f x) (happ g x))
% Assm [h4s_combins_ou_u_ABSu_u_R]: !_0. (!g x. happ (happ _0 g) x = happ g x) ==> (!g f x. happ (happ (h4/combin/o f) (happ _0 g)) x = happ f (happ g x))
% Assm [h4s_combins_GENu_u_LETu_u_RAND]: !v f P. happ P (h4/bool/LET f v) = h4/bool/LET (happ (h4/combin/o P) f) v
% Goal: !_0. (!x y. happ (happ _0 x) y <=> ~(happ h4/pair/FST x = happ h4/pair/FST y)) ==> (!P. happ P h4/list/NIL /\ (!x t. happ P (h4/list/FILTER (happ _0 x) t) ==> happ P (happ (happ h4/list/CONS x) t)) ==> (!v. happ P v))
fof(aHLu_TRUTH, axiom, p(s(t_bool,t0))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q1304469,TV_Q1304465]: ![V_f, V_g]: (![V_x]: s(TV_Q1304465,happ(s(t_fun(TV_Q1304469,TV_Q1304465),V_f),s(TV_Q1304469,V_x))) = s(TV_Q1304465,happ(s(t_fun(TV_Q1304469,TV_Q1304465),V_g),s(TV_Q1304469,V_x))) => s(t_fun(TV_Q1304469,TV_Q1304465),V_f) = s(t_fun(TV_Q1304469,TV_Q1304465),V_g))).
fof(ah4s_updates_OVERRIDEu_u_primitiveu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_5]: (![V_OVERRIDE, V_uu_1, V_x, V_t]: s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)))))),V_uu_5),s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_OVERRIDE))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),V_uu_1))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_t))) = s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_combins_i(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)))),h4s_lists_cons),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_OVERRIDE),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_filter(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),V_uu_1),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_t))))))))) => ![V_uu_4]: (![V_OVERRIDE, V_uu_1, V_x]: s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)))))),V_uu_4),s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_OVERRIDE))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),V_uu_1))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))) = s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)))))),V_uu_5),s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_OVERRIDE))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),V_uu_1))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))) => ![V_uu_3]: (![V_OVERRIDE, V_uu_1, V_a]: s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))))),V_uu_3),s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_OVERRIDE))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),V_uu_1))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_a))) = s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_listu_u_case(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_a),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_combins_i(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_nil))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)))))),V_uu_4),s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_OVERRIDE))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),V_uu_1))))) => ![V_uu_2]: (![V_uu_1, V_OVERRIDE]: s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))))),V_uu_2),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),V_uu_1))),s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_OVERRIDE))) = s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)))),happ(s(t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))))),V_uu_3),s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),V_OVERRIDE))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),V_uu_1))) => ![V_uu_1]: (![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),V_uu_1),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_y)))) <=> ~ (s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))) = s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_y))))) => ![V_uu_0]: (![V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_bool)),V_R)))) & ![V_t, V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_bool),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_bool)),V_R),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_filter(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),V_uu_1),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_t))))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)))),h4s_lists_cons),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_t)))))))) => s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),h4s_updates_override) = s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),h4s_relations_wfrec(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),t_fun(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))))),V_uu_2),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),V_uu_1)))))))))))).
fof(ah4s_lists_UNZIP0u_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_l]: s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_lists_unzip(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)))),h4s_lists_cons),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l))))) = s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),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),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,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))))),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),t_h4s_lists_list(TV_u_27a)),h4s_pairs_fst),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_lists_unzip(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l))))))))),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,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))))),s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),t_h4s_lists_list(TV_u_27b)),h4s_pairs_snd),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_lists_unzip(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l)))))))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t0))).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,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_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,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_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) <=> p(s(t_bool,V_t)))).
fof(ah4s_predu_u_sets_INu_u_CROSS, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_Q, V_P]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,t_bool),V_Q)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(t_fun(TV_u_27a,t_bool),V_P)))) & p(s(t_bool,h4s_bools_in(s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(t_fun(TV_u_27b,t_bool),V_Q))))))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t0)))).
fof(ah4s_bools_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_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),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,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_richu_u_lists_UNZIPu_u_SNOC, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_l]: s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_lists_unzip(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_snoc(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l))))) = s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),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),h4s_lists_snoc(s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),t_h4s_lists_list(TV_u_27a)),h4s_pairs_fst),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_lists_unzip(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l))))))))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_snoc(s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),t_h4s_lists_list(TV_u_27b)),h4s_pairs_snd),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_lists_unzip(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l)))))))))).
fof(ah4s_lists_memu_u_existsu_u_set, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_l]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),h4s_pairs_u_2c),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l)))))) => ?[V_z]: (s(TV_u_27a,V_x) = s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_z))) & p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_z),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l))))))))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t0)))).
fof(ah4s_predu_u_sets_CROSSu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_P, V_Q, V_p]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))),s(t_fun(TV_u_27a,t_bool),V_P)))) & p(s(t_bool,h4s_bools_in(s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))),s(t_fun(TV_u_27b,t_bool),V_Q)))))) & s(t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_fun(TV_u_27b,t_bool),V_Q))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))) = s(t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_bool,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_bool,t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool))),h4s_pairs_u_2c),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))),s(t_bool,V_v)))) => ![V_Q, V_P]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,t_bool),V_Q))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_fun(TV_u_27b,t_bool),V_Q))))))).
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_richu_u_lists_ALLu_u_DISTINCTu_u_MEMu_u_ZIPu_u_MAP, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_ls, V_f]: (p(s(t_bool,h4s_lists_allu_u_distinct(s(t_h4s_lists_list(TV_u_27a),V_ls)))) => (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_lists_listu_u_tou_u_set(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)),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),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_ls))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),V_ls)))))))))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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_ls)))))) & s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))))))))).
fof(ah4s_lists_MEMu_u_ZIPu_u_MEMu_u_MAP, axiom, ![TV_u_27a,TV_u_27b]: ![V_ps, V_p]: ((s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),t_h4s_lists_list(TV_u_27a)),h4s_pairs_fst),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),V_ps))))) = s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),t_h4s_lists_list(TV_u_27b)),h4s_pairs_snd),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),V_ps))))) & p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_lists_listu_u_tou_u_set(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)),V_ps))))))))) => (p(s(t_bool,h4s_bools_in(s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))),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_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),t_h4s_lists_list(TV_u_27a)),h4s_pairs_fst),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),V_ps)))))))) & p(s(t_bool,h4s_bools_in(s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))),s(t_fun(TV_u_27b,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),t_h4s_lists_list(TV_u_27b)),h4s_pairs_snd),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),V_ps))))))))))).
fof(ah4s_pairs_ELIMu_u_PFORALL, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: (![V_p]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))))),s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p)))))) <=> ![V_p1, V_p2]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_p1))),s(TV_u_27b,V_p2)))))).
fof(ah4s_pairs_PAIR, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),h4s_pairs_u_2c),s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))))),s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x)).
fof(ah4s_pairs_PAIRu_u_FSTu_u_SNDu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_q, V_p]: (s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_q) <=> (s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))) = s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_q))) & s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))) = s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_q)))))).
fof(ah4s_pairs_ELIMu_u_PEXISTS, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: (?[V_p]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))))),s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p)))))) <=> ?[V_p1, V_p2]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_p1))),s(TV_u_27b,V_p2)))))).
fof(ah4s_pairs_SWAPu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_a]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_swap(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),V_a))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),h4s_pairs_u_2c),s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),TV_u_27a),h4s_pairs_snd),s(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),V_a))))),s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),TV_u_27b),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),V_a)))))).
fof(ah4s_predu_u_sets_CROSSu_u_applied, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_Q, V_P]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_predu_u_sets_cross(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,t_bool),V_Q))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x)))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(t_fun(TV_u_27a,t_bool),V_P)))) & p(s(t_bool,h4s_bools_in(s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(t_fun(TV_u_27b,t_bool),V_Q))))))).
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_SKOLEMu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (![V_x]: ?[V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) <=> ?[V_f]: ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))))))).
fof(ah4s_pairs_PAIRu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_b, V_a]: (s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),h4s_pairs_u_2c),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),h4s_pairs_u_2c),s(TV_u_27a,V_a))),s(TV_u_27b,V_b))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & s(TV_u_27b,V_y) = s(TV_u_27b,V_b)))).
fof(ah4s_bools_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_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_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_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_MONOu_u_OR, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) | p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) | p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_AND, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) & p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) & p(s(t_bool,V_w)))))).
fof(ah4s_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_LISTu_u_EQu_u_MAPu_u_PAIR, axiom, ![TV_u_27a,TV_u_27b]: ![V_l2, V_l1]: ((s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l1))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l2))) & s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l1))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l2)))) => s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l1) = s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l2))).
fof(ah4s_lists_UNZIPu_u_MAP, axiom, ![TV_u_27a,TV_u_27b]: ![V_L]: s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_lists_unzip(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_L))) = s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),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),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_L))))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_L)))))).
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_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_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_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_lists_UNZIP0u_c0, axiom, ![TV_u_27a,TV_u_27b]: s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_lists_unzip(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_nil))) = s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),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),h4s_lists_nil))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil)))).
fof(ah4s_richu_u_lists_UNZIPu_u_FSTu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_richu_u_lists_unzipu_u_fst(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l))) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),t_h4s_lists_list(TV_u_27a)),h4s_pairs_fst),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_lists_unzip(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l)))))).
fof(ah4s_lists_LENGTHu_u_UNZIPu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_pl]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),t_h4s_lists_list(TV_u_27a)),h4s_pairs_fst),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_lists_unzip(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_pl))))))) = s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_pl)))).
fof(ah4s_richu_u_lists_MAPu_u_SNDu_u_FILTERu_u_NEQ, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_2]: (![V_z, 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_uu_2),s(TV_u_27a,V_z))),s(TV_u_27a,V_y)))) <=> ~ (s(TV_u_27a,V_z) = s(TV_u_27a,V_y))) => ![V_uu_1]: (![V_z, 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_uu_1),s(TV_u_27a,V_z))),s(TV_u_27a,V_y)))) <=> ~ (s(TV_u_27a,V_y) = s(TV_u_27a,V_z))) => ![V_uu_0]: (![V_z, V_x]: s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),V_uu_0),s(TV_u_27a,V_z))),s(TV_u_27b,V_x))) = s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_uu_1),s(TV_u_27a,V_z))) => ![V_z, V_ls]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),TV_u_27a),h4s_pairs_snd),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),h4s_lists_filter(s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_bool),h4s_pairs_uncurry(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool))),V_uu_0),s(TV_u_27a,V_z))))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_ls))))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_uu_2),s(TV_u_27a,V_z))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),TV_u_27a),h4s_pairs_snd),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_ls))))))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t0)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_lists_UNZIPu_u_THMu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_x, V_L1, V_y, V_L2]: s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),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_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)))),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)))))),V_uu_1),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_L1))),s(TV_u_27b,V_y))),s(t_h4s_lists_list(TV_u_27b),V_L2))) = s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),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),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_x))),s(t_h4s_lists_list(TV_u_27a),V_L1))))),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_y))),s(t_h4s_lists_list(TV_u_27b),V_L2))))) => ![V_uu_0]: (![V_x, V_y, V_L1]: s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)))),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)))))),V_uu_0),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))),s(t_h4s_lists_list(TV_u_27a),V_L1))) = s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),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_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)))),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)))))),V_uu_1),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_L1))),s(TV_u_27b,V_y))) => ![V_y, V_x, V_t]: s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_lists_unzip(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)))),h4s_lists_cons),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),h4s_pairs_u_2c),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_t))))) = s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_bools_let(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),h4s_pairs_uncurry(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)))),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)))))),V_uu_0),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b)),h4s_lists_unzip(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_t)))))))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_lists_SNOC0u_c1, axiom, ![TV_u_27a]: ![V_xu_27, V_x, V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_snoc(s(TV_u_27a,V_x),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_xu_27))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = 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_xu_27))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_snoc(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_lists_SNOC0u_c0, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_snoc(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = 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_x))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t0) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_dcu_u_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_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_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_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_predu_u_sets_GSPECIFICATION, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_v),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f)))))) <=> ?[V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(t_bool,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_h4s_pairs_prod(TV_u_27a,t_bool))),h4s_pairs_u_2c),s(TV_u_27a,V_v))),s(t_bool,t0))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f),s(TV_u_27b,V_x))))).
fof(ah4s_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_pairs_CLOSEDu_u_PAIRu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_b, V_a]: (s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),h4s_pairs_u_2c),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),h4s_pairs_u_2c),s(TV_u_27a,V_a))),s(TV_u_27b,V_b))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & s(TV_u_27b,V_y) = s(TV_u_27b,V_b)))).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_pthu_u_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_lists_MEMu_u_ZIP, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_l2, V_l1]: (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_bools_in(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_lists_listu_u_tou_u_set(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)),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),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)))))))))) <=> ?[V_n]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l1)))))) & s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),h4s_pairs_u_2c),s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27a),V_l1))))),s(TV_u_27b,h4s_lists_el(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27b),V_l2))))))))).
fof(ah4s_bools_EQu_u_IMPu_u_THM, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) & (p(s(t_bool,V_t2)) => p(s(t_bool,V_t1)))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t0)))).
fof(ah4s_lists_MEMu_u_EL, axiom, ![TV_u_27a]: ![V_x, V_l]: (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_l)))))) <=> ?[V_n]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l)))))) & s(TV_u_27a,V_x) = s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_lists_MAP0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil)).
fof(ah4s_lists_MAP0u_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_h, V_f]: s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),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),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,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_h))))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),V_t)))))).
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),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,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_FORALLu_u_PROD, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: (![V_p]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p)))) <=> ![V_pu_u_1, V_pu_u_2]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),h4s_pairs_u_2c),s(TV_u_27a,V_pu_u_1))),s(TV_u_27b,V_pu_u_2)))))))).
fof(ah4s_lists_LENGTH0u_c1, axiom, ![TV_u_27a]: ![V_t, V_h]: s(t_h4s_nums_num,h4s_lists_length(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_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_t)))))).
fof(ah4s_lists_LENGTH0u_c0, axiom, ![TV_u_27a]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_nums_num,h4s_nums_0)).
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),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27b)),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27c,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,V_x)))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t0)))).
fof(ah4s_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_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_lists_ELu_u_ALLu_u_DISTINCTu_u_ELu_u_EQ, axiom, ![TV_u_27a]: ![V_l]: (p(s(t_bool,h4s_lists_allu_u_distinct(s(t_h4s_lists_list(TV_u_27a),V_l)))) <=> ![V_n1, V_n2]: ((p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n1),s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l)))))) & p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n2),s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l))))))) => (s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,V_n1),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,V_n2),s(t_h4s_lists_list(TV_u_27a),V_l))) <=> s(t_h4s_nums_num,V_n1) = s(t_h4s_nums_num,V_n2))))).
fof(ah4s_lists_ELu_u_MAP, axiom, ![TV_u_27b,TV_u_27a]: ![V_n, V_l]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l)))))) => ![V_f]: s(TV_u_27b,h4s_lists_el(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,V_n),s(t_h4s_lists_list(TV_u_27a),V_l))))))).
fof(ah4s_lists_LENGTHu_u_MAP, axiom, ![TV_u_27b,TV_u_27a]: ![V_l, V_f]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l)))).
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_richu_u_lists_FILTERu_u_MAP, axiom, ![TV_u_27a,TV_u_27b]: ![V_l, V_f2, V_f1]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_f1),s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(TV_u_27b,TV_u_27a),V_f2),s(t_h4s_lists_list(TV_u_27b),V_l))))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(TV_u_27b,TV_u_27a),V_f2),s(t_h4s_lists_list(TV_u_27b),h4s_lists_filter(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27a),t_fun(TV_u_27b,t_bool)),h4s_combins_o(s(t_fun(TV_u_27a,t_bool),V_f1))),s(t_fun(TV_u_27b,TV_u_27a),V_f2))),s(t_h4s_lists_list(TV_u_27b),V_l)))))).
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_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t0))) <=> p(s(t_bool,t0)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t0))) <=> p(s(t_bool,f)))).
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_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_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f))))).
fof(ah4s_bools_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t0)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t0)))).
fof(ah4s_bools_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_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) | s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_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_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)),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27b))),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27b),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_lists_MAPu_u_EQu_u_EVERY2, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_uu_1]: (![V_f1, V_x, V_f2, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,t_bool)))),V_uu_1),s(t_fun(TV_u_27a,TV_u_27b),V_f1))),s(TV_u_27a,V_x))),s(t_fun(TV_u_27c,TV_u_27b),V_f2))),s(TV_u_27c,V_y)))) <=> s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_f2),s(TV_u_27c,V_y)))) => ![V_uu_0]: (![V_f1, V_f2, V_x]: s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_f1))),s(t_fun(TV_u_27c,TV_u_27b),V_f2))),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27c,t_bool),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,t_bool)))),V_uu_1),s(t_fun(TV_u_27a,TV_u_27b),V_f1))),s(TV_u_27a,V_x))),s(t_fun(TV_u_27c,TV_u_27b),V_f2))) => ![V_l2, V_l1, V_f2, V_f1]: (s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(t_h4s_lists_list(TV_u_27a),V_l1))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27c,TV_u_27b),V_f2),s(t_h4s_lists_list(TV_u_27c),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_27c),V_l2))) & p(s(t_bool,h4s_lists_listu_u_rel(s(t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27a,t_fun(TV_u_27c,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_f1))),s(t_fun(TV_u_27c,TV_u_27b),V_f2))),s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27c),V_l2))))))))).
fof(ah4s_lists_LISTu_u_EQu_u_REWRITE, axiom, ![TV_u_27a]: ![V_l2, V_l1]: (s(t_h4s_lists_list(TV_u_27a),V_l1) = s(t_h4s_lists_list(TV_u_27a),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_27a),V_l2))) & ![V_x]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l1)))))) => s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,V_x),s(t_h4s_lists_list(TV_u_27a),V_l1))) = s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,V_x),s(t_h4s_lists_list(TV_u_27a),V_l2))))))).
fof(ah4s_lists_EVERYu_u_MEM, axiom, ![TV_u_27a]: ![V_l, V_P]: (p(s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))) <=> ![V_e]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l)))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_e))))))).
fof(ah4s_pairs_pairu_u_CASES, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: ?[V_q, V_r]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,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_bools_LEFTu_u_FORALLu_u_IMPu_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_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_pairs_Su_u_UNCURRYu_u_R, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b,TV_u_27d]: ![V_g, V_f]: s(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),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(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27d),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c))),h4s_combins_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(TV_u_27d,TV_u_27c)),V_f))),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)),V_g))))) = 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_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(t_fun(TV_u_27b,TV_u_27d),t_fun(TV_u_27b,TV_u_27c))),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)))),h4s_combins_s),s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27d,TV_u_27c))),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,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(t_fun(TV_u_27b,TV_u_27d),t_fun(TV_u_27b,TV_u_27c))),h4s_combins_s))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27d,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27d,TV_u_27c)))),h4s_combins_o(s(t_fun(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_fun(TV_u_27b,t_fun(TV_u_27d,TV_u_27c))),h4s_combins_o(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(TV_u_27d,TV_u_27c)),V_f))))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),h4s_pairs_u_2c))))))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d)),V_g)))))).
fof(ah4s_pairs_FORALLu_u_UNCURRY, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_bool),d_forall),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_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),d_forall),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_bool)),h4s_combins_o(s(t_fun(t_fun(TV_u_27b,t_bool),t_bool),d_forall))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_f)))))).
fof(ah4s_pairs_ou_u_UNCURRYu_u_R, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b,TV_u_27d]: ![V_g, V_f]: s(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),TV_u_27d),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c)),h4s_combins_o(s(t_fun(TV_u_27d,TV_u_27c),V_f))),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)),V_g))))) = 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_27d)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),h4s_combins_o(s(t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(TV_u_27b,TV_u_27c)),h4s_combins_o(s(t_fun(TV_u_27d,TV_u_27c),V_f))))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d)),V_g)))))).
fof(ah4s_combins_LETu_u_FORALLu_u_ELIM, axiom, ![TV_u_27a]: ![V_v, V_f]: s(t_bool,h4s_bools_let(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_v))) = s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),d_forall),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(t_bool,t_bool)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool))),h4s_combins_s),s(t_fun(TV_u_27a,t_fun(t_bool,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_fun(t_bool,t_bool))),h4s_combins_o(s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_imply))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),h4s_combins_o(s(t_fun(t_bool,t_bool),h4s_markers_abbrev))),s(t_fun(TV_u_27a,t_bool),h4s_combins_c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),d_equals),s(TV_u_27a,V_v))))))))),s(t_fun(TV_u_27a,t_bool),V_f)))))).
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),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_Su_u_ABSu_u_R, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_uu_0]: (![V_g, V_x]: s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27c)),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x))) => ![V_g, V_f, 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_27a,t_fun(TV_u_27c,TV_u_27b)),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b))),h4s_combins_s),s(t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27b)),V_f))),s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27c)),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_g))))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27b)),V_f),s(TV_u_27a,V_x))),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x))))))).
fof(ah4s_combins_ou_u_ABSu_u_R, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_uu_0]: (![V_g, V_x]: s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27c)),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x))) => ![V_g, V_f, 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)),h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27b),V_f))),s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27c)),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_g))))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x))))))).
fof(ah4s_combins_GENu_u_LETu_u_RAND, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_v, V_f, V_P]: s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_P),s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,V_v))))) = s(TV_u_27a,h4s_bools_let(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,TV_u_27a)),h4s_combins_o(s(t_fun(TV_u_27b,TV_u_27a),V_P))),s(t_fun(TV_u_27c,TV_u_27b),V_f))),s(TV_u_27c,V_v)))).
fof(ch4s_updates_OVERRIDEu_u_ind, conjecture, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),V_uu_0),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_y)))) <=> ~ (s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))) = s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_y))))) => ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_bool),V_P),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_nil)))) & ![V_x, V_t]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_bool),V_P),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_filter(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),V_uu_0),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_t)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_bool),V_P),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)))),h4s_lists_cons),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_t)))))))) => ![V_v]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_bool),V_P),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_v))))))).
