%   ORIGINAL: h4/pair/pair__induction
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/pair/FORALL__PROD: !P. (!p. P p) <=> (!p__1 p__2. P (h4/pair/_2C p__1 p__2))
% Assm: h4/pair/EXISTS__PROD: !P. (?p. P p) <=> (?p__1 p__2. P (h4/pair/_2C p__1 p__2))
% Assm: h4/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm: h4/pair/COMMA__DEF: !y x. h4/pair/_2C x y = h4/pair/ABS__prod (\a b. a = x /\ b = y)
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% 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/pair/ABS__REP__prod_c0: !a. h4/pair/ABS__prod (h4/pair/REP__prod a) = a
% Assm: h4/pair/ABS__REP__prod_c1: !r. (\p. ?x y. p = (\a b. a = x /\ b = y)) r <=> h4/pair/REP__prod (h4/pair/ABS__prod r) = r
% Assm: h4/pair/pair__CASES: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/pair/prod__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION (\p. ?x y. p = (\a b. a = x /\ b = y)) rep
% Assm: h4/bool/ABS__REP__THM: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. abs (rep a) = a) /\ (!r. P r <=> rep (abs r) = r))
% 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/pair/SWAP__def: !a. h4/pair/SWAP a = h4/pair/_2C (h4/pair/SND a) (h4/pair/FST a)
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/bool/SKOLEM__THM: !P. (!x. ?y. P x y) <=> (?f. !x. P x (f 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/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/pair/UNCURRY0: !v f. h4/pair/UNCURRY f v = f (h4/pair/FST v) (h4/pair/SND v)
% Assm: h4/pair/CURRY__DEF: !y x f. h4/pair/CURRY f x y = f (h4/pair/_2C x y)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/pair/pair__Axiom: !f. ?fn. !x y. fn (h4/pair/_2C x y) = f x y
% Assm: h4/pair/LAMBDA__PROD: !P. (\p. P p) = h4/pair/UNCURRY (\p1 p2. P (h4/pair/_2C p1 p2))
% Assm: h4/pair/UNCURRY__CONG: !f_27 f M_27 M. M = M_27 /\ (!x y. M_27 = h4/pair/_2C x y ==> f x y = f_27 x y) ==> h4/pair/UNCURRY f M = h4/pair/UNCURRY f_27 M_27
% Assm: h4/bool/SWAP__EXISTS__THM: !P. (?x y. P x y) <=> (?y x. P x y)
% Assm: h4/bool/UNWIND__FORALL__THM2: !v f. (!x. x = v ==> f x) <=> f v
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/DISJ__COMM: !B A. A \/ B <=> B \/ A
% 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/pair/CURRY__ONE__ONE__THM: !g f. h4/pair/CURRY f = h4/pair/CURRY g <=> f = g
% Assm: h4/pair/ELIM__UNCURRY: !f. h4/pair/UNCURRY f = (\x. f (h4/pair/FST x) (h4/pair/SND x))
% Assm: h4/pair/UNCURRY__VAR: !v f. h4/pair/UNCURRY f v = f (h4/pair/FST v) (h4/pair/SND v)
% Assm: h4/pair/CURRY__UNCURRY__THM: !f. h4/pair/CURRY (h4/pair/UNCURRY f) = f
% Assm: h4/pair/UNCURRY__CURRY__THM: !f. h4/pair/UNCURRY (h4/pair/CURRY f) = f
% Assm: h4/pair/UNCURRY__ONE__ONE__THM: !g f. h4/pair/UNCURRY f = h4/pair/UNCURRY g <=> f = g
% Assm: h4/combin/UPDATE__def: !b a. h4/combin/UPDATE a b = (\f c. h4/bool/COND (a = c) b (f c))
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/combin/UPDATE__COMMUTES: !f d c b a. ~(a = b) ==> h4/combin/UPDATE a c (h4/combin/UPDATE b d f) = h4/combin/UPDATE b d (h4/combin/UPDATE a c f)
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/relation/RRANGE0: !y R. h4/relation/RRANGE R y <=> (?x. R x y)
% Assm: h4/combin/UPDATE__APPLY__ID: !f b a. f a = b <=> h4/combin/UPDATE a b f = f
% Assm: h4/bool/TYPE__DEFINITION0: h4/bool/TYPE__DEFINITION = (\P rep. (!x_27 x_27_27. rep x_27 = rep x_27_27 ==> x_27 = x_27_27) /\ (!x. P x <=> (?x_27. x = rep x_27)))
% Assm: h4/combin/APPLY__UPDATE__THM: !f c b a. h4/combin/UPDATE a b f c = h4/bool/COND (a = c) b (f c)
% Assm: h4/combin/UPD11__SAME__BASE: !f d c b a. h4/combin/UPDATE a c f = h4/combin/UPDATE b d f <=> a = b /\ c = d \/ ~(a = b) /\ h4/combin/UPDATE a c f = f /\ h4/combin/UPDATE b d f = f
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/AND__CLAUSES_c4: !t. t /\ t <=> t
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/combin/FAIL__DEF: h4/combin/FAIL = (\x y. x)
% Assm: h4/combin/FAIL__THM: !y x. h4/combin/FAIL x y = x
% Assm: h4/bool/IN__DEF: h4/bool/IN = (\x f. f x)
% Assm: h4/relation/IN__RRANGE: !y R. h4/bool/IN y (h4/relation/RRANGE R) <=> (?x. R x y)
% Assm: h4/relation/RUNION0: !y x R2 R1. h4/relation/RUNION R1 R2 x y <=> R1 x y \/ R2 x y
% Assm: h4/relation/RUNIV0: !y x. h4/relation/RUNIV x y <=> T
% Assm: h4/bool/TYPE__DEFINITION__THM: !rep P. h4/bool/TYPE__DEFINITION P rep <=> (!x_27 x_27_27. rep x_27 = rep x_27_27 ==> x_27 = x_27_27) /\ (!x. P x <=> (?x_27. x = rep x_27))
% Assm: h4/combin/UPDATE__EQ: !f c b a. h4/combin/UPDATE a c (h4/combin/UPDATE a b f) = h4/combin/UPDATE a c f
% Assm: h4/relation/O__DEF: !z x R2 R1. h4/relation/O R1 R2 x z <=> (?y. R2 x y /\ R1 y z)
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/ONTO__DEF: h4/bool/ONTO = (\f. !y. ?x. y = f x)
% Assm: h4/bool/ONTO__THM: !f. h4/bool/ONTO f <=> (!y. ?x. y = f x)
% Assm: h4/combin/UPD__SAME__KEY__UNWIND: !f2 f1 c b a. h4/combin/UPDATE a b f1 = h4/combin/UPDATE a c f2 ==> b = c /\ (!v. h4/combin/UPDATE a v f1 = h4/combin/UPDATE a v f2)
% Assm: h4/combin/SAME__KEY__UPDATE__DIFFER: !f c b a. ~(b = c) ==> ~(h4/combin/UPDATE a b f = h4/combin/UPDATE a c f)
% Assm: h4/bool/SWAP__FORALL__THM: !P. (!x y. P x y) <=> (!y x. P x y)
% Assm: h4/relation/RRESTRICT__DEF: !y x s R. h4/relation/RRESTRICT R s x y <=> R x y /\ h4/bool/IN x s
% Assm: h4/bool/ABS__SIMP: !t2 t1. (\x. t1) t2 = t1
% Assm: h4/combin/K__DEF: h4/combin/K = (\x y. x)
% Assm: h4/combin/K__THM: !y x. h4/combin/K x y = x
% Assm: h4/relation/nf__def: !x R. h4/relation/nf R x <=> (!y. ~R x y)
% Assm: h4/combin/UPDATE__APPLY_c0: !x f a. h4/combin/UPDATE a x f a = x
% Assm: h4/relation/RCOMPL0: !y x R. h4/relation/RCOMPL R x y <=> ~R x y
% Assm: h4/relation/RINTER0: !y x R2 R1. h4/relation/RINTER R1 R2 x y <=> R1 x y /\ R2 x y
% Assm: h4/bool/literal__case__id: !u t a. h4/bool/literal__case (\x. h4/bool/COND (x = a) t u) a = t
% Assm: h4/bool/OR__DEF: $or = (\t1 t2. !t. (t1 ==> t) ==> (t2 ==> t) ==> t)
% Assm: h4/bool/F__DEF: F <=> (!t. t)
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/T__DEF: T <=> (\x. x) = (\x. x)
% Assm: h4/bool/literal__case__DEF: h4/bool/literal__case = (\f x. f x)
% Assm: h4/bool/COND__DEF: h4/bool/COND = (\t t1 t2. h4/min/_40 (\x. ((t <=> T) ==> x = t1) /\ ((t <=> F) ==> x = t2)))
% Assm: h4/bool/NOT__DEF: $not = (\t. t ==> F)
% Assm: h4/relation/RSUBSET0: !R2 R1. h4/relation/RSUBSET R1 R2 <=> (!x y. R1 x y ==> R2 x y)
% Assm: h4/relation/RDOM__DELETE__DEF: !x v u R. h4/relation/RDOM__DELETE R x u v <=> R u v /\ ~(u = x)
% Assm: h4/relation/RDOM__DEF: !x R. h4/relation/RDOM R x <=> (?y. R x y)
% Assm: h4/relation/IN__RDOM: !x R. h4/bool/IN x (h4/relation/RDOM R) <=> (?y. R x y)
% Assm: h4/combin/UPDATE__APPLY__IMP__ID: !f b a. f a = b ==> h4/combin/UPDATE a b f = f
% Assm: h4/combin/UPDATE__APPLY_c1: !x f b a. ~(a = b) ==> h4/combin/UPDATE a x f b = f b
% Assm: h4/bool/LET__THM: !x f. h4/bool/LET f x = f x
% Assm: h4/combin/UPD11__SAME__KEY__AND__BASE: !f c b a. h4/combin/UPDATE a b f = h4/combin/UPDATE a c f <=> b = c
% Assm: h4/combin/C__THM: !y x f. h4/combin/C f x y = f y x
% Assm: h4/combin/GEN__LET__RATOR: !x v f. h4/bool/LET f v x = h4/bool/LET (h4/combin/C f x) v
% Assm: h4/combin/C__DEF: h4/combin/C = (\f x y. f y x)
% Assm: h4/combin/GEN__literal__case__RATOR: !x v f. h4/bool/literal__case f v x = h4/bool/literal__case (h4/combin/C f x) v
% Assm: h4/bool/literal__case__THM: !x f. h4/bool/literal__case f x = f x
% Assm: h4/bool/BETA__THM: !y f. (\x. f x) y = f y
% Assm: h4/bool/LET__RAND: !P N M. P (h4/bool/LET (\x. N x) M) <=> h4/bool/LET (\x. P (N x)) M
% Assm: h4/bool/LET__RATOR: !b N M. h4/bool/LET (\x. N x) M b = h4/bool/LET (\x. N x b) M
% Assm: h4/bool/LET__CONG: !g f N M. M = N /\ (!x. x = N ==> f x = g x) ==> h4/bool/LET f M = h4/bool/LET g N
% Assm: h4/bool/LET__DEF: h4/bool/LET = (\f x. f x)
% Assm: h4/bool/AND1__THM: !t2 t1. t1 /\ t2 ==> t1
% Assm: h4/relation/IN__RDOM__RRESTRICT: !x s R. h4/bool/IN x (h4/relation/RDOM (h4/relation/RRESTRICT R s)) <=> h4/bool/IN x (h4/relation/RDOM R) /\ h4/bool/IN x s
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% 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/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Goal: !P. (!p__1 p__2. P (h4/pair/_2C p__1 p__2)) ==> (!p. P p)
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_pairs_FORALLu_u_PROD]: !P. (!p. happ P p) <=> (!p__1 p__2. happ P (h4/pair/_2C p__1 p__2))
% Assm [h4s_pairs_EXISTSu_u_PROD]: !P. (?p. happ P p) <=> (?p__1 p__2. happ P (h4/pair/_2C p__1 p__2))
% Assm [h4s_pairs_PAIR]: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm [h4s_pairs_COMMAu_u_DEF]: !_1. (!a x y b. happ (happ (happ (happ _1 a) x) y) b <=> a = x /\ b = y) ==> (!_0. (!x y a. happ (happ (happ _0 x) y) a = happ (happ (happ _1 a) x) y) ==> (!y x. h4/pair/_2C x y = h4/pair/ABS__prod (happ (happ _0 x) y)))
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_pairs_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_pairs_ABSu_u_REPu_u_produ_c0]: !a. h4/pair/ABS__prod (h4/pair/REP__prod a) = a
% Assm [h4s_pairs_ABSu_u_REPu_u_produ_c1]: !r. (?x y. !x' x. happ (happ r x') x <=> x' = x /\ x = y) <=> h4/pair/REP__prod (h4/pair/ABS__prod r) = r
% Assm [h4s_pairs_pairu_u_CASES]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_pairs_produ_u_TYu_u_DEF]: !_0. (!p. happ _0 p <=> (?x y. !x' x. happ (happ p x') x <=> x' = x /\ x = y)) ==> (?rep. h4/bool/TYPE__DEFINITION _0 rep)
% Assm [h4s_bools_ABSu_u_REPu_u_THM]: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. happ abs (happ rep a) = a) /\ (!r. happ P r <=> happ rep (happ abs r) = r))
% Assm [h4s_pairs_CLOSEDu_u_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_pairs_SWAPu_u_def]: !a. h4/pair/SWAP a = h4/pair/_2C (h4/pair/SND a) (h4/pair/FST a)
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = 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_FSTu_u_SNDu_u_EQ]: !q p. p = q <=> h4/pair/FST p = h4/pair/FST q /\ h4/pair/SND p = h4/pair/SND q
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. happ (h4/pair/UNCURRY f) (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_pairs_UNCURRY0]: !v f. happ (h4/pair/UNCURRY f) v = happ (happ f (h4/pair/FST v)) (h4/pair/SND v)
% Assm [h4s_pairs_CURRYu_u_DEF]: !y x f. happ (happ (h4/pair/CURRY f) x) y = happ f (h4/pair/_2C x y)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_pairs_pairu_u_Axiom]: !f. ?fn. !x y. happ fn (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_pairs_LAMBDAu_u_PROD]: !_1. (!P p1 p2. happ (happ (happ _1 P) p1) p2 = happ P (h4/pair/_2C p1 p2)) ==> (!_0. (!P p1. happ (happ _0 P) p1 = happ (happ _1 P) p1) ==> (!P x. happ P x = happ (h4/pair/UNCURRY (happ _0 P)) x))
% Assm [h4s_pairs_UNCURRYu_u_CONG]: !f_27 f M_27 M. M = M_27 /\ (!x y. M_27 = h4/pair/_2C x y ==> happ (happ f x) y = happ (happ f_27 x) y) ==> happ (h4/pair/UNCURRY f) M = happ (h4/pair/UNCURRY f_27) M_27
% Assm [h4s_bools_SWAPu_u_EXISTSu_u_THM]: !P. (?x y. happ (happ P x) y) <=> (?y x. happ (happ P x) y)
% Assm [h4s_bools_UNWINDu_u_FORALLu_u_THM2]: !v f. (!x. x = v ==> happ f x) <=> happ f v
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_DISJu_u_COMM]: !B A. A \/ B <=> B \/ A
% 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_pairs_CURRYu_u_ONEu_u_ONEu_u_THM]: !g f. h4/pair/CURRY f = h4/pair/CURRY g <=> f = g
% Assm [h4s_pairs_ELIMu_u_UNCURRY]: !f x. happ (h4/pair/UNCURRY f) x = happ (happ f (h4/pair/FST x)) (h4/pair/SND x)
% Assm [h4s_pairs_UNCURRYu_u_VAR]: !v f. happ (h4/pair/UNCURRY f) v = happ (happ f (h4/pair/FST v)) (h4/pair/SND v)
% Assm [h4s_pairs_CURRYu_u_UNCURRYu_u_THM]: !f. h4/pair/CURRY (h4/pair/UNCURRY f) = f
% Assm [h4s_pairs_UNCURRYu_u_CURRYu_u_THM]: !f. h4/pair/UNCURRY (h4/pair/CURRY f) = f
% Assm [h4s_pairs_UNCURRYu_u_ONEu_u_ONEu_u_THM]: !g f. h4/pair/UNCURRY f = h4/pair/UNCURRY g <=> f = g
% Assm [h4s_combins_UPDATEu_u_def]: !b a x x. ?v. (v <=> a = x) /\ happ (h4/combin/UPDATE a b x) x = h4/bool/COND v b (happ x x)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_combins_UPDATEu_u_COMMUTES]: !f d c b a. ~(a = b) ==> h4/combin/UPDATE a c (h4/combin/UPDATE b d f) = h4/combin/UPDATE b d (h4/combin/UPDATE a c f)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_relations_RRANGE0]: !y R. happ (h4/relation/RRANGE R) y <=> (?x. happ (happ R x) y)
% Assm [h4s_combins_UPDATEu_u_APPLYu_u_ID]: !f b a. happ f a = b <=> h4/combin/UPDATE a b f = f
% Assm [h4s_bools_TYPEu_u_DEFINITION0]: !x x. h4/bool/TYPE__DEFINITION x x <=> (!x_27 x_27_27. happ x x_27 = happ x x_27_27 ==> x_27 = x_27_27) /\ (!x. happ x x <=> (?x_27. x = happ x x_27))
% Assm [h4s_combins_APPLYu_u_UPDATEu_u_THM]: !f c b a. ?v. (v <=> a = c) /\ happ (h4/combin/UPDATE a b f) c = h4/bool/COND v b (happ f c)
% Assm [h4s_combins_UPD11u_u_SAMEu_u_BASE]: !f d c b a. h4/combin/UPDATE a c f = h4/combin/UPDATE b d f <=> a = b /\ c = d \/ ~(a = b) /\ h4/combin/UPDATE a c f = f /\ h4/combin/UPDATE b d f = f
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_ANDu_u_CLAUSESu_c4]: !t. t /\ t <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% Assm [h4s_combins_FAILu_u_DEF]: !x x. h4/combin/FAIL x x = x
% Assm [h4s_combins_FAILu_u_THM]: !y x. h4/combin/FAIL x y = x
% Assm [h4s_bools_INu_u_DEF]: !x x. h4/bool/IN x x <=> happ x x
% Assm [h4s_relations_INu_u_RRANGE]: !y R. h4/bool/IN y (h4/relation/RRANGE R) <=> (?x. happ (happ R x) y)
% Assm [h4s_relations_RUNION0]: !y x R2 R1. h4/relation/RUNION R1 R2 x y <=> happ (happ R1 x) y \/ happ (happ R2 x) y
% Assm [h4s_relations_RUNIV0]: !y x. h4/relation/RUNIV x y <=> T
% Assm [h4s_bools_TYPEu_u_DEFINITIONu_u_THM]: !rep P. h4/bool/TYPE__DEFINITION P rep <=> (!x_27 x_27_27. happ rep x_27 = happ rep x_27_27 ==> x_27 = x_27_27) /\ (!x. happ P x <=> (?x_27. x = happ rep x_27))
% Assm [h4s_combins_UPDATEu_u_EQ]: !f c b a. h4/combin/UPDATE a c (h4/combin/UPDATE a b f) = h4/combin/UPDATE a c f
% Assm [h4s_relations_Ou_u_DEF]: !z x R2 R1. h4/relation/O R1 R2 x z <=> (?y. happ (happ R2 x) y /\ happ (happ R1 y) z)
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_ONTOu_u_DEF]: !x. h4/bool/ONTO x <=> (!y. ?x. y = happ x x)
% Assm [h4s_bools_ONTOu_u_THM]: !f. h4/bool/ONTO f <=> (!y. ?x. y = happ f x)
% Assm [h4s_combins_UPDu_u_SAMEu_u_KEYu_u_UNWIND]: !f2 f1 c b a. h4/combin/UPDATE a b f1 = h4/combin/UPDATE a c f2 ==> b = c /\ (!v. h4/combin/UPDATE a v f1 = h4/combin/UPDATE a v f2)
% Assm [h4s_combins_SAMEu_u_KEYu_u_UPDATEu_u_DIFFER]: !f c b a. ~(b = c) ==> ~(h4/combin/UPDATE a b f = h4/combin/UPDATE a c f)
% Assm [h4s_bools_SWAPu_u_FORALLu_u_THM]: !P. (!x y. happ (happ P x) y) <=> (!y x. happ (happ P x) y)
% Assm [h4s_relations_RRESTRICTu_u_DEF]: !y x s R. happ (happ (h4/relation/RRESTRICT R s) x) y <=> happ (happ R x) y /\ h4/bool/IN x s
% Assm [h4s_bools_ABSu_u_SIMP]: !t2 t1. t1 = t1
% Assm [h4s_combins_Ku_u_DEF]: !x x. h4/combin/K x x = x
% Assm [h4s_combins_Ku_u_THM]: !y x. h4/combin/K x y = x
% Assm [h4s_relations_nfu_u_def]: !x R. h4/relation/nf R x <=> (!y. ~happ (happ R x) y)
% Assm [h4s_combins_UPDATEu_u_APPLYu_c0]: !x f a. happ (h4/combin/UPDATE a x f) a = x
% Assm [h4s_relations_RCOMPL0]: !y x R. h4/relation/RCOMPL R x y <=> ~happ (happ R x) y
% Assm [h4s_relations_RINTER0]: !y x R2 R1. h4/relation/RINTER R1 R2 x y <=> happ (happ R1 x) y /\ happ (happ R2 x) y
% Assm [h4s_bools_literalu_u_caseu_u_id]: !_0. (!a t u x. ?v. (v <=> x = a) /\ happ (happ (happ (happ _0 a) t) u) x = h4/bool/COND v t u) ==> (!u t a. h4/bool/literal__case (happ (happ (happ _0 a) t) u) a = t)
% Assm [h4s_bools_ORu_u_DEF]: !x x'. $or x x' <=> (!t. (x ==> t) ==> (x' ==> t) ==> t)
% Assm [h4s_bools_Fu_u_DEF]: F <=> (!t. t)
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_Tu_u_DEF]: T <=> (!x. x <=> x)
% Assm [h4s_bools_literalu_u_caseu_u_DEF]: !x x. h4/bool/literal__case x x = happ x x
% Assm [h4s_bools_CONDu_u_DEF]: !_0. (!x x x' x''. happ (happ (happ (happ _0 x) x) x') x'' <=> ((x <=> T) ==> x'' = x) /\ ((x <=> F) ==> x'' = x')) ==> (!x x x'. h4/bool/COND x x x' = h4/min/_40 (happ (happ (happ _0 x) x) x'))
% Assm [h4s_bools_NOTu_u_DEF]: !x. $not x <=> x ==> F
% Assm [h4s_relations_RSUBSET0]: !R2 R1. h4/relation/RSUBSET R1 R2 <=> (!x y. happ (happ R1 x) y ==> happ (happ R2 x) y)
% Assm [h4s_relations_RDOMu_u_DELETEu_u_DEF]: !x v u R. h4/relation/RDOM__DELETE R x u v <=> happ (happ R u) v /\ ~(u = x)
% Assm [h4s_relations_RDOMu_u_DEF]: !x R. happ (h4/relation/RDOM R) x <=> (?y. happ (happ R x) y)
% Assm [h4s_relations_INu_u_RDOM]: !x R. h4/bool/IN x (h4/relation/RDOM R) <=> (?y. happ (happ R x) y)
% Assm [h4s_combins_UPDATEu_u_APPLYu_u_IMPu_u_ID]: !f b a. happ f a = b ==> h4/combin/UPDATE a b f = f
% Assm [h4s_combins_UPDATEu_u_APPLYu_c1]: !x f b a. ~(a = b) ==> happ (h4/combin/UPDATE a x f) b = happ f b
% Assm [h4s_bools_LETu_u_THM]: !x f. h4/bool/LET f x = happ f x
% Assm [h4s_combins_UPD11u_u_SAMEu_u_KEYu_u_ANDu_u_BASE]: !f c b a. h4/combin/UPDATE a b f = h4/combin/UPDATE a c f <=> b = c
% Assm [h4s_combins_Cu_u_THM]: !y x f. happ (h4/combin/C f x) y = happ (happ f y) x
% Assm [h4s_combins_GENu_u_LETu_u_RATOR]: !x v f. happ (h4/bool/LET f v) x = h4/bool/LET (h4/combin/C f x) v
% Assm [h4s_combins_Cu_u_DEF]: !x x x. happ (h4/combin/C x x) x = happ (happ x x) x
% Assm [h4s_combins_GENu_u_literalu_u_caseu_u_RATOR]: !x v f. happ (h4/bool/literal__case f v) x = h4/bool/literal__case (h4/combin/C f x) v
% Assm [h4s_bools_literalu_u_caseu_u_THM]: !x f. h4/bool/literal__case f x = happ f x
% Assm [h4s_bools_BETAu_u_THM]: !y f. happ f y = happ f y
% Assm [h4s_bools_LETu_u_RAND]: !_1. (!P N x. happ (happ (happ _1 P) N) x <=> happ P (happ N x)) ==> (!_0. (!N x. happ (happ _0 N) x = happ N x) ==> (!P N M. happ P (h4/bool/LET (happ _0 N) M) <=> h4/bool/LET (happ (happ _1 P) N) M))
% Assm [h4s_bools_LETu_u_RATOR]: !_1. (!N b x. happ (happ (happ _1 N) b) x = happ (happ N x) b) ==> (!_0. (!N x. happ (happ _0 N) x = happ N x) ==> (!b N M. happ (h4/bool/LET (happ _0 N) M) b = h4/bool/LET (happ (happ _1 N) b) M))
% Assm [h4s_bools_LETu_u_CONG]: !g f N M. M = N /\ (!x. x = N ==> happ f x = happ g x) ==> h4/bool/LET f M = h4/bool/LET g N
% Assm [h4s_bools_LETu_u_DEF]: !x x. h4/bool/LET x x = happ x x
% Assm [h4s_bools_AND1u_u_THM]: !t2 t1. t1 /\ t2 ==> t1
% Assm [h4s_relations_INu_u_RDOMu_u_RRESTRICT]: !x s R. h4/bool/IN x (h4/relation/RDOM (h4/relation/RRESTRICT R s)) <=> h4/bool/IN x (h4/relation/RDOM R) /\ h4/bool/IN x s
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% 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_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Goal: !P. (!p__1 p__2. happ P (h4/pair/_2C p__1 p__2)) ==> (!p. happ P p)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q1378234,TV_Q1378230]: ![V_f, V_g]: (![V_x]: s(TV_Q1378230,happ(s(t_fun(TV_Q1378234,TV_Q1378230),V_f),s(TV_Q1378234,V_x))) = s(TV_Q1378230,happ(s(t_fun(TV_Q1378234,TV_Q1378230),V_g),s(TV_Q1378234,V_x))) => s(t_fun(TV_Q1378234,TV_Q1378230),V_f) = s(t_fun(TV_Q1378234,TV_Q1378230),V_g))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_pairs_ABSu_u_PAIRu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: ?[V_q, V_r]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_q),s(TV_u_27b,V_r)))).
fof(ah4s_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),h4s_pairs_u_2c(s(TV_u_27a,V_pu_u_1),s(TV_u_27b,V_pu_u_2)))))))).
fof(ah4s_pairs_EXISTSu_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),h4s_pairs_u_2c(s(TV_u_27a,V_pu_u_1),s(TV_u_27b,V_pu_u_2)))))))).
fof(ah4s_pairs_PAIR, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(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_COMMAu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_a, V_x, V_y, V_b]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)))),V_uu_1),s(TV_u_27a,V_a))),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))),s(TV_u_27b,V_b)))) <=> (s(TV_u_27a,V_a) = s(TV_u_27a,V_x) & s(TV_u_27b,V_b) = s(TV_u_27b,V_y))) => ![V_uu_0]: (![V_x, V_y, V_a]: s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_0),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))),s(TV_u_27a,V_a))) = s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)))),V_uu_1),s(TV_u_27a,V_a))),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) => ![V_y, V_x]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_absu_u_prod(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_0),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))))))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_pairs_PAIRu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_b, V_a]: (s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & s(TV_u_27b,V_y) = s(TV_u_27b,V_b)))).
fof(ah4s_pairs_ABSu_u_REPu_u_produ_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_a]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_absu_u_prod(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),h4s_pairs_repu_u_prod(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_a))))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_a)).
fof(ah4s_pairs_ABSu_u_REPu_u_produ_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_r]: (?[V_x, V_y]: ![V_xi_, V_x0]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_r),s(TV_u_27a,V_xi_))),s(TV_u_27b,V_x0)))) <=> (s(TV_u_27a,V_xi_) = s(TV_u_27a,V_x) & s(TV_u_27b,V_x0) = s(TV_u_27b,V_y))) <=> s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),h4s_pairs_repu_u_prod(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_absu_u_prod(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_r))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_r))).
fof(ah4s_pairs_pairu_u_CASES, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: ?[V_q, V_r]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_q),s(TV_u_27b,V_r)))).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27b,V_y)).
fof(ah4s_pairs_produ_u_TYu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_p]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_bool),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_p)))) <=> ?[V_x, V_y]: ![V_xi_, V_x0]: (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_xi_))),s(TV_u_27b,V_x0)))) <=> (s(TV_u_27a,V_xi_) = s(TV_u_27a,V_x) & s(TV_u_27b,V_x0) = s(TV_u_27b,V_y)))) => ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_rep)))))).
fof(ah4s_bools_ABSu_u_REPu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ?[V_rep, V_abs]: (![V_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))) = s(TV_u_27b,V_a) & ![V_r]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_r)))) <=> s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))))) = s(TV_u_27a,V_r))))).
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),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & s(TV_u_27b,V_y) = s(TV_u_27b,V_b)))).
fof(ah4s_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),h4s_pairs_u_2c(s(TV_u_27a,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),V_a))),s(TV_u_27b,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),V_a)))))).
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_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_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_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,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))) = s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_q))) & s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p))) = s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_q)))))).
fof(ah4s_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_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_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_pairs_UNCURRY0, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_v))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_v))))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_v)))))).
fof(ah4s_pairs_CURRYu_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(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),h4s_pairs_curry(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),V_f))),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) = s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),V_f),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y)))))).
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_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_pairs_pairu_u_Axiom, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_f]: ?[V_fn]: ![V_x, V_y]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),V_fn),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_pairs_LAMBDAu_u_PROD, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_P, V_p1, V_p2]: s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),V_uu_1),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),V_P))),s(TV_u_27a,V_p1))),s(TV_u_27b,V_p2))) = s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),V_P),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_p1),s(TV_u_27b,V_p2))))) => ![V_uu_0]: (![V_P, V_p1]: s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),V_P))),s(TV_u_27a,V_p1))) = s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),V_uu_1),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),V_P))),s(TV_u_27a,V_p1))) => ![V_P, V_x]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),V_P),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))) = 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)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),V_P))))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x)))))).
fof(ah4s_pairs_UNCURRYu_u_CONG, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_fu_27, V_f, V_Mu_27, V_M]: ((s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_M) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_Mu_27) & ![V_x, V_y]: (s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_Mu_27) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))) => s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) = 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_fu_27),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))) => s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_M))) = 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_fu_27))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_Mu_27))))).
fof(ah4s_bools_SWAPu_u_EXISTSu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![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_y, V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))))).
fof(ah4s_bools_UNWINDu_u_FORALLu_u_THM2, axiom, ![TV_u_27a]: ![V_v, V_f]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_v) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_v)))))).
fof(ah4s_bools_IMPu_u_DISJu_u_THM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) => p(s(t_bool,V_B))) <=> (~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B))))).
fof(ah4s_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_bools_DISJu_u_COMM, 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_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_pairs_CURRYu_u_ONEu_u_ONEu_u_THM, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_g, V_f]: (s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),h4s_pairs_curry(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),V_f))) = s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),h4s_pairs_curry(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),V_g))) <=> s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),V_f) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),V_g))).
fof(ah4s_pairs_ELIMu_u_UNCURRY, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_f, V_x]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x)))))).
fof(ah4s_pairs_UNCURRYu_u_VAR, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_v))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_v))))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_v)))))).
fof(ah4s_pairs_CURRYu_u_UNCURRYu_u_THM, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_f]: s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),h4s_pairs_curry(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_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f)).
fof(ah4s_pairs_UNCURRYu_u_CURRYu_u_THM, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_f]: 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)),h4s_pairs_curry(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),V_f))))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),V_f)).
fof(ah4s_pairs_UNCURRYu_u_ONEu_u_ONEu_u_THM, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_g, V_f]: (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_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_g))) <=> s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f) = s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_g))).
fof(ah4s_combins_UPDATEu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_b, V_a, V_x, V_x0]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(TV_u_27a,V_a) = s(TV_u_27a,V_x0)) & s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_b),s(t_fun(TV_u_27a,TV_u_27b),V_x))),s(TV_u_27a,V_x0))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27b,V_b),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(TV_u_27a,V_x0))))))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_combins_UPDATEu_u_COMMUTES, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_d, V_c, V_b, V_a]: (~ (s(TV_u_27a,V_a) = s(TV_u_27a,V_b)) => s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_c),s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_b),s(TV_u_27b,V_d),s(t_fun(TV_u_27a,TV_u_27b),V_f))))) = s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_b),s(TV_u_27b,V_d),s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_c),s(t_fun(TV_u_27a,TV_u_27b),V_f))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_relations_RRANGE0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_R]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),h4s_relations_rrange(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R))),s(TV_u_27b,V_y)))) <=> ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))))).
fof(ah4s_combins_UPDATEu_u_APPLYu_u_ID, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_b, V_a]: (s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_a))) = s(TV_u_27b,V_b) <=> s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_b),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27a,TV_u_27b),V_f))).
fof(ah4s_bools_TYPEu_u_DEFINITION0, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_x0]: (p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(TV_u_27a,t_bool),V_x),s(t_fun(TV_u_27b,TV_u_27a),V_x0)))) <=> (![V_xu_27, V_xu_27u_27]: (s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_x0),s(TV_u_27b,V_xu_27))) = s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_x0),s(TV_u_27b,V_xu_27u_27))) => s(TV_u_27b,V_xu_27) = s(TV_u_27b,V_xu_27u_27)) & ![V_x1]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_x1)))) <=> ?[V_xu_27]: s(TV_u_27a,V_x1) = s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_x0),s(TV_u_27b,V_xu_27))))))).
fof(ah4s_combins_APPLYu_u_UPDATEu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_c, V_b, V_a]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(TV_u_27a,V_a) = s(TV_u_27a,V_c)) & s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_b),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_c))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27b,V_b),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_c))))))).
fof(ah4s_combins_UPD11u_u_SAMEu_u_BASE, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_d, V_c, V_b, V_a]: (s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_c),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_b),s(TV_u_27b,V_d),s(t_fun(TV_u_27a,TV_u_27b),V_f))) <=> ((s(TV_u_27a,V_a) = s(TV_u_27a,V_b) & s(TV_u_27b,V_c) = s(TV_u_27b,V_d)) | (~ (s(TV_u_27a,V_a) = s(TV_u_27a,V_b)) & (s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_c),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27a,TV_u_27b),V_f) & s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_b),s(TV_u_27b,V_d),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27a,TV_u_27b),V_f)))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,f))) <=> p(s(t_bool,V_t)))).
fof(ah4s_combins_FAILu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_x0]: s(TV_u_27a,h4s_combins_fail(s(TV_u_27a,V_x),s(TV_u_27b,V_x0))) = s(TV_u_27a,V_x)).
fof(ah4s_combins_FAILu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_combins_fail(s(TV_u_27a,V_x),s(TV_u_27b,V_y))) = s(TV_u_27a,V_x)).
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_relations_INu_u_RRANGE, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_R]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),h4s_relations_rrange(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool)),V_R)))))) <=> ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27b,V_x))),s(TV_u_27a,V_y)))))).
fof(ah4s_relations_RUNION0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_R2, V_R1]: (p(s(t_bool,h4s_relations_runion(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27a,V_x),s(TV_u_27b,V_y)))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R1),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) | p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))))).
fof(ah4s_relations_RUNIV0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(t_bool,h4s_relations_runiv(s(TV_u_27a,V_x),s(TV_u_27b,V_y))) = s(t_bool,t)).
fof(ah4s_bools_TYPEu_u_DEFINITIONu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_P]: (p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) <=> (![V_xu_27, V_xu_27u_27]: (s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_xu_27))) = s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_xu_27u_27))) => s(TV_u_27b,V_xu_27) = s(TV_u_27b,V_xu_27u_27)) & ![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) <=> ?[V_xu_27]: s(TV_u_27a,V_x) = s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_xu_27))))))).
fof(ah4s_combins_UPDATEu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_c, V_b, V_a]: s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_c),s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_b),s(t_fun(TV_u_27a,TV_u_27b),V_f))))) = s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_c),s(t_fun(TV_u_27a,TV_u_27b),V_f)))).
fof(ah4s_relations_Ou_u_DEF, axiom, ![TV_u_27g,TV_u_27h,TV_u_27k]: ![V_z, V_x, V_R2, V_R1]: (p(s(t_bool,h4s_relations_o(s(t_fun(TV_u_27h,t_fun(TV_u_27k,t_bool)),V_R1),s(t_fun(TV_u_27g,t_fun(TV_u_27h,t_bool)),V_R2),s(TV_u_27g,V_x),s(TV_u_27k,V_z)))) <=> ?[V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27h,t_bool),happ(s(t_fun(TV_u_27g,t_fun(TV_u_27h,t_bool)),V_R2),s(TV_u_27g,V_x))),s(TV_u_27h,V_y)))) & p(s(t_bool,happ(s(t_fun(TV_u_27k,t_bool),happ(s(t_fun(TV_u_27h,t_fun(TV_u_27k,t_bool)),V_R1),s(TV_u_27h,V_y))),s(TV_u_27k,V_z))))))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_bools_ONTOu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: (p(s(t_bool,h4s_bools_onto(s(t_fun(TV_u_27a,TV_u_27b),V_x)))) <=> ![V_y]: ?[V_x0]: s(TV_u_27b,V_y) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(TV_u_27a,V_x0))))).
fof(ah4s_bools_ONTOu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_f]: (p(s(t_bool,h4s_bools_onto(s(t_fun(TV_u_27a,TV_u_27b),V_f)))) <=> ![V_y]: ?[V_x]: s(TV_u_27b,V_y) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))).
fof(ah4s_combins_UPDu_u_SAMEu_u_KEYu_u_UNWIND, axiom, ![TV_u_27a,TV_u_27b]: ![V_f2, V_f1, V_c, V_b, V_a]: (s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_b),s(t_fun(TV_u_27a,TV_u_27b),V_f1))) = s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_c),s(t_fun(TV_u_27a,TV_u_27b),V_f2))) => (s(TV_u_27b,V_b) = s(TV_u_27b,V_c) & ![V_v]: s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f1))) = s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f2)))))).
fof(ah4s_combins_SAMEu_u_KEYu_u_UPDATEu_u_DIFFER, axiom, ![TV_u_27c,TV_u_27d]: ![V_f, V_c, V_b, V_a]: (~ (s(TV_u_27d,V_b) = s(TV_u_27d,V_c)) => ~ (s(t_fun(TV_u_27c,TV_u_27d),h4s_combins_update(s(TV_u_27c,V_a),s(TV_u_27d,V_b),s(t_fun(TV_u_27c,TV_u_27d),V_f))) = s(t_fun(TV_u_27c,TV_u_27d),h4s_combins_update(s(TV_u_27c,V_a),s(TV_u_27d,V_c),s(t_fun(TV_u_27c,TV_u_27d),V_f)))))).
fof(ah4s_bools_SWAPu_u_FORALLu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![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_y, V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))))).
fof(ah4s_relations_RRESTRICTu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x, V_s, V_R]: (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)),h4s_relations_rrestrict(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_bools_ABSu_u_SIMP, axiom, ![TV_u_27b,TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,V_t1) = s(TV_u_27a,V_t1)).
fof(ah4s_combins_Ku_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_x0]: s(TV_u_27a,h4s_combins_k(s(TV_u_27a,V_x),s(TV_u_27b,V_x0))) = s(TV_u_27a,V_x)).
fof(ah4s_combins_Ku_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_combins_k(s(TV_u_27a,V_x),s(TV_u_27b,V_y))) = s(TV_u_27a,V_x)).
fof(ah4s_relations_nfu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_R]: (p(s(t_bool,h4s_relations_nf(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27a,V_x)))) <=> ![V_y]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))))).
fof(ah4s_combins_UPDATEu_u_APPLYu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f, V_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_x),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_a))) = s(TV_u_27b,V_x)).
fof(ah4s_relations_RCOMPL0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_R]: (p(s(t_bool,h4s_relations_rcompl(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27a,V_x),s(TV_u_27b,V_y)))) <=> ~ (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))))).
fof(ah4s_relations_RINTER0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_R2, V_R1]: (p(s(t_bool,h4s_relations_rinter(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27a,V_x),s(TV_u_27b,V_y)))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R1),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) & p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))))).
fof(ah4s_bools_literalu_u_caseu_u_id, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_a, V_t, V_u, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_a)) & s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)))),V_uu_0),s(TV_u_27a,V_a))),s(TV_u_27b,V_t))),s(TV_u_27b,V_u))),s(TV_u_27a,V_x))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27b,V_t),s(TV_u_27b,V_u)))) => ![V_u, V_t, V_a]: s(TV_u_27b,h4s_bools_literalu_u_case(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)))),V_uu_0),s(TV_u_27a,V_a))),s(TV_u_27b,V_t))),s(TV_u_27b,V_u))),s(TV_u_27a,V_a))) = s(TV_u_27b,V_t))).
fof(ah4s_bools_ORu_u_DEF, axiom, ![V_x, V_xi_]: (p(s(t_bool,d_or(s(t_bool,V_x),s(t_bool,V_xi_)))) <=> ![V_t]: ((p(s(t_bool,V_x)) => p(s(t_bool,V_t))) => ((p(s(t_bool,V_xi_)) => p(s(t_bool,V_t))) => p(s(t_bool,V_t)))))).
fof(ah4s_bools_Fu_u_DEF, axiom, (p(s(t_bool,f)) <=> ![V_t]: p(s(t_bool,V_t)))).
fof(ah4s_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_bools_Tu_u_DEF, axiom, (p(s(t_bool,t)) <=> ![V_x]: s(t_bool,V_x) = s(t_bool,V_x))).
fof(ah4s_bools_literalu_u_caseu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_x0]: s(TV_u_27b,h4s_bools_literalu_u_case(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(TV_u_27a,V_x0))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(TV_u_27a,V_x0)))).
fof(ah4s_bools_CONDu_u_DEF, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x, V_x0, V_xi_, V_xi_i_]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)))),V_uu_0),s(TV_u_27a,V_x))),s(t_bool,V_x0))),s(TV_u_27a,V_xi_))),s(TV_u_27a,V_xi_i_)))) <=> ((s(t_bool,V_x0) = s(t_bool,t) => s(TV_u_27a,V_xi_i_) = s(TV_u_27a,V_x)) & (s(t_bool,V_x0) = s(t_bool,f) => s(TV_u_27a,V_xi_i_) = s(TV_u_27a,V_xi_)))) => ![V_x, V_x0, V_xi_]: s(TV_u_27a,h4s_bools_cond(s(t_bool,V_x),s(TV_u_27a,V_x0),s(TV_u_27a,V_xi_))) = s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)))),V_uu_0),s(TV_u_27a,V_x0))),s(t_bool,V_x))),s(TV_u_27a,V_xi_))))))).
fof(ah4s_bools_NOTu_u_DEF, axiom, ![V_x]: (p(s(t_bool,d_not(s(t_bool,V_x)))) <=> (p(s(t_bool,V_x)) => p(s(t_bool,f))))).
fof(ah4s_relations_RSUBSET0, axiom, ![TV_u_27a,TV_u_27b]: ![V_R2, V_R1]: (p(s(t_bool,h4s_relations_rsubset(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R2)))) <=> ![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R1),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))))).
fof(ah4s_relations_RDOMu_u_DELETEu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_v, V_u, V_R]: (p(s(t_bool,h4s_relations_rdomu_u_delete(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27a,V_x),s(TV_u_27a,V_u),s(TV_u_27b,V_v)))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27a,V_u))),s(TV_u_27b,V_v)))) & ~ (s(TV_u_27a,V_u) = s(TV_u_27a,V_x))))).
fof(ah4s_relations_RDOMu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_R]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),h4s_relations_rdom(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R))),s(TV_u_27a,V_x)))) <=> ?[V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))))).
fof(ah4s_relations_INu_u_RDOM, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_R]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_relations_rdom(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R)))))) <=> ?[V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))))).
fof(ah4s_combins_UPDATEu_u_APPLYu_u_IMPu_u_ID, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_b, V_a]: (s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_a))) = s(TV_u_27b,V_b) => s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_b),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27a,TV_u_27b),V_f))).
fof(ah4s_combins_UPDATEu_u_APPLYu_c1, axiom, ![TV_u_27d,TV_u_27c]: ![V_x, V_f, V_b, V_a]: (~ (s(TV_u_27c,V_a) = s(TV_u_27c,V_b)) => s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),h4s_combins_update(s(TV_u_27c,V_a),s(TV_u_27d,V_x),s(t_fun(TV_u_27c,TV_u_27d),V_f))),s(TV_u_27c,V_b))) = s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_f),s(TV_u_27c,V_b))))).
fof(ah4s_bools_LETu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,h4s_bools_let(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_f),s(TV_u_27a,V_x)))).
fof(ah4s_combins_UPD11u_u_SAMEu_u_KEYu_u_ANDu_u_BASE, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_c, V_b, V_a]: (s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_b),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_c),s(t_fun(TV_u_27a,TV_u_27b),V_f))) <=> s(TV_u_27b,V_b) = s(TV_u_27b,V_c))).
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_GENu_u_LETu_u_RATOR, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_x, V_v, V_f]: s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),h4s_bools_let(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27b,V_v))),s(TV_u_27c,V_x))) = s(TV_u_27a,h4s_bools_let(s(t_fun(TV_u_27b,TV_u_27a),h4s_combins_c(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27c,V_x))),s(TV_u_27b,V_v)))).
fof(ah4s_combins_Cu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_x, V_x0, V_x1]: 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_x),s(TV_u_27b,V_x0))),s(TV_u_27a,V_x1))) = 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_x),s(TV_u_27a,V_x1))),s(TV_u_27b,V_x0)))).
fof(ah4s_combins_GENu_u_literalu_u_caseu_u_RATOR, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_x, V_v, V_f]: s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),h4s_bools_literalu_u_case(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27b,V_v))),s(TV_u_27c,V_x))) = s(TV_u_27a,h4s_bools_literalu_u_case(s(t_fun(TV_u_27b,TV_u_27a),h4s_combins_c(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27c,V_x))),s(TV_u_27b,V_v)))).
fof(ah4s_bools_literalu_u_caseu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,h4s_bools_literalu_u_case(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_f),s(TV_u_27a,V_x)))).
fof(ah4s_bools_BETAu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y)))).
fof(ah4s_bools_LETu_u_RAND, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_1]: (![V_P, V_N, V_x]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_bool))),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_P))),s(t_fun(TV_u_27a,TV_u_27b),V_N))),s(TV_u_27a,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_N),s(TV_u_27a,V_x))))) => ![V_uu_0]: (![V_N, 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_27b),t_fun(TV_u_27a,TV_u_27b)),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_N))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_N),s(TV_u_27a,V_x))) => ![V_P, V_N, V_M]: s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_N))),s(TV_u_27a,V_M))))) = s(t_bool,h4s_bools_let(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_bool))),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_P))),s(t_fun(TV_u_27a,TV_u_27b),V_N))),s(TV_u_27a,V_M)))))).
fof(ah4s_bools_LETu_u_RATOR, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_uu_1]: (![V_N, V_b, V_x]: s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27c))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_N))),s(TV_u_27b,V_b))),s(TV_u_27a,V_x))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_N),s(TV_u_27a,V_x))),s(TV_u_27b,V_b))) => ![V_uu_0]: (![V_N, V_x]: s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_N))),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_N),s(TV_u_27a,V_x))) => ![V_b, V_N, V_M]: s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),h4s_bools_let(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_N))),s(TV_u_27a,V_M))),s(TV_u_27b,V_b))) = s(TV_u_27c,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27c))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_N))),s(TV_u_27b,V_b))),s(TV_u_27a,V_M)))))).
fof(ah4s_bools_LETu_u_CONG, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f, V_N, V_M]: ((s(TV_u_27a,V_M) = s(TV_u_27a,V_N) & ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_N) => 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))))) => s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_M))) = s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_N))))).
fof(ah4s_bools_LETu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_x0]: s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(TV_u_27a,V_x0))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(TV_u_27a,V_x0)))).
fof(ah4s_bools_AND1u_u_THM, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t1)))).
fof(ah4s_relations_INu_u_RDOMu_u_RRESTRICT, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_s, V_R]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_relations_rdom(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),h4s_relations_rrestrict(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),V_s)))))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_relations_rdom(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R)))))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_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_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f))))).
fof(ah4s_sats_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_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_dcu_u_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_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(ch4s_pairs_pairu_u_induction, conjecture, ![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),h4s_pairs_u_2c(s(TV_u_27a,V_pu_u_1),s(TV_u_27b,V_pu_u_2)))))) => ![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)))))).
