%   ORIGINAL: h4/list/list__case__cong
% 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/list/list__case__def_c0: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm: h4/bool/TRUTH: T
% Assm: h4/list/NIL0: h4/list/NIL = h4/list/_20_40ind__typelist0
% Assm: h4/list/list__distinct: !a1 a0. ~(h4/list/NIL = h4/list/CONS a0 a1)
% Assm: h4/list/CONS0: h4/list/CONS = h4/list/_20_40ind__typelist1
% 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/list__repfns_c0: !a. h4/list/_20_40ind__typelist2 (h4/list/_20_40ind__typelist3 a) = a
% Assm: h4/bool/DATATYPE__TAG__THM: !x. h4/bool/DATATYPE x <=> T
% 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/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/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/list/list__11: !a1_27 a1 a0_27 a0. h4/list/CONS a0 a1 = h4/list/CONS a0_27 a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm: h4/list/list__size__def_c1: !f a1 a0. h4/list/list__size f (h4/list/CONS a0 a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (f a0) (h4/list/list__size f a1))
% 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/datatype__list: !list. h4/bool/DATATYPE (list h4/list/NIL h4/list/CONS)
% Assm: h4/list/list__case__def_c1: !v f a1 a0. h4/list/list__CASE (h4/list/CONS a0 a1) v f = f a0 a1
% Assm: h4/list/list__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION (\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) rep
% Assm: h4/list/list__size__def_c0: !f. h4/list/list__size f h4/list/NIL = h4/num/0
% 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/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/relation/RSUBSET0: !R2 R1. h4/relation/RSUBSET R1 R2 <=> (!x y. R1 x y ==> R2 x y)
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% 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/COMMA__DEF: !y x. h4/pair/_2C x y = h4/pair/ABS__prod (\a b. a = x /\ b = y)
% Assm: h4/pred__set/IN__IMAGE: !y s f. h4/bool/IN y (h4/pred__set/IMAGE f s) <=> (?x. y = f x /\ h4/bool/IN x s)
% Assm: h4/relation/RDOM__DEF: !x R. h4/relation/RDOM R x <=> (?y. R x y)
% Assm: h4/pred__set/IMAGE__DEF: !s f. h4/pred__set/IMAGE f s = h4/pred__set/GSPEC (\x. h4/pair/_2C (f x) (h4/bool/IN x s))
% 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/pair/PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/relation/IN__RDOM: !x R. h4/bool/IN x (h4/relation/RDOM R) <=> (?y. R x y)
% Assm: h4/bool/IN__DEF: h4/bool/IN = (\x f. f x)
% Assm: h4/sum/sum__case__cong: !f1_27 f1 f_27 f M_27 M. M = M_27 /\ (!x. M_27 = h4/sum/INL x ==> f x = f_27 x) /\ (!y. M_27 = h4/sum/INR y ==> f1 y = f1_27 y) ==> h4/sum/sum__CASE M f f1 = h4/sum/sum__CASE M_27 f_27 f1_27
% Assm: h4/sum/sum__case__def_c0: !x f1 f. h4/sum/sum__CASE (h4/sum/INL x) f f1 = f x
% Assm: h4/sum/sum__CASES: !ss. (?x. ss = h4/sum/INL x) \/ (?y. ss = h4/sum/INR y)
% Assm: h4/sum/sum__case__def_c1: !y f1 f. h4/sum/sum__CASE (h4/sum/INR y) f f1 = f1 y
% 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/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/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/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/relation/RUNIV0: !y x. h4/relation/RUNIV x y <=> 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/pair/LEX__DEF: !R2 R1. h4/pair/LEX R1 R2 = h4/pair/UNCURRY (\s t. h4/pair/UNCURRY (\u v. R1 s u \/ s = u /\ R2 t v))
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/pair/LEX__DEF__THM: !d c b a R2 R1. h4/pair/LEX R1 R2 (h4/pair/_2C a b) (h4/pair/_2C c d) <=> R1 a c \/ a = c /\ R2 b d
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% 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/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/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/AND__CLAUSES_c4: !t. t /\ t <=> t
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/ind__type/ISO0: !g f. h4/ind__type/ISO f g <=> (!x. f (g x) = x) /\ (!y. g (f y) = y)
% Assm: h4/ind__type/ISO__USAGE: !g f. h4/ind__type/ISO f g ==> (!P. (!x. P x) <=> (!x. P (g x))) /\ (!P. (?x. P x) <=> (?x. P (g x))) /\ (!a b. a = g b <=> f a = b)
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/pair/ELIM__PEXISTS: !P. (?p. P (h4/pair/FST p) (h4/pair/SND p)) <=> (?p1 p2. P p1 p2)
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/bool/LEFT__OR__EXISTS__THM: !Q P. (?x. P x) \/ Q <=> (?x. P x \/ Q)
% Assm: h4/bool/RIGHT__OR__EXISTS__THM: !Q P. P \/ (?x. Q x) <=> (?x. P \/ Q x)
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/bool/LEFT__EXISTS__AND__THM: !Q P. (?x. P x /\ Q) <=> (?x. P x) /\ Q
% Assm: h4/bool/RIGHT__EXISTS__AND__THM: !Q P. (?x. P /\ Q x) <=> P /\ (?x. Q x)
% Assm: h4/bool/EXISTS__OR__THM: !Q P. (?x. P x \/ Q x) <=> (?x. P x) \/ (?x. Q x)
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% 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/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/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% 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/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/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/option/OPTREL__def: !y x R. h4/option/OPTREL R x y <=> x = h4/option/NONE /\ y = h4/option/NONE \/ (?x0 y0. x = h4/option/SOME x0 /\ y = h4/option/SOME y0 /\ R x0 y0)
% Assm: h4/relation/nf__def: !x R. h4/relation/nf R x <=> (!y. ~R x y)
% 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/bool/ONTO__DEF: h4/bool/ONTO = (\f. !y. ?x. y = f x)
% 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__APPLY_c1: !x f b a. ~(a = b) ==> h4/combin/UPDATE a x f b = f b
% 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/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% 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/OR__DEF: $or = (\t1 t2. !t. (t1 ==> t) ==> (t2 ==> t) ==> t)
% Assm: h4/bool/F__DEF: F <=> (!t. t)
% Assm: h4/bool/T__DEF: T <=> (\x. x) = (\x. x)
% Assm: h4/bool/NOT__DEF: $not = (\t. t ==> F)
% Assm: h4/pair/PFORALL__THM: !P. (!x y. P x y) <=> $forall (h4/pair/UNCURRY (\x y. P x y))
% Assm: h4/pair/ELIM__PFORALL: !P. (!p. P (h4/pair/FST p) (h4/pair/SND p)) <=> (!p1 p2. P p1 p2)
% Goal: !v_27 v f_27 f M_27 M. M = M_27 /\ (M_27 = h4/list/NIL ==> v = v_27) /\ (!a0 a1. M_27 = h4/list/CONS a0 a1 ==> f a0 a1 = f_27 a0 a1) ==> h4/list/list__CASE M v f = h4/list/list__CASE M_27 v_27 f_27
%   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_lists_listu_u_caseu_u_defu_c0]: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_lists_NIL0]: h4/list/NIL = h4/list/_20_40ind__typelist0
% Assm [h4s_lists_listu_u_distinct]: !a1 a0. ~(h4/list/NIL = happ (happ h4/list/CONS a0) a1)
% Assm [h4s_lists_CONS0]: h4/list/CONS = h4/list/_20_40ind__typelist1
% 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_listu_u_repfnsu_c0]: !a. h4/list/_20_40ind__typelist2 (h4/list/_20_40ind__typelist3 a) = a
% Assm [h4s_bools_DATATYPEu_u_TAGu_u_THM]: !x. h4/bool/DATATYPE x <=> T
% 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_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_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_lists_listu_u_11]: !a1_27 a1 a0_27 a0. happ (happ h4/list/CONS a0) a1 = happ (happ h4/list/CONS a0_27) a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm [h4s_lists_listu_u_sizeu_u_defu_c1]: !f a1 a0. h4/list/list__size f (happ (happ h4/list/CONS a0) a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (happ f a0) (h4/list/list__size f a1))
% 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_datatypeu_u_list]: !list. h4/bool/DATATYPE (happ (happ list h4/list/NIL) h4/list/CONS)
% Assm [h4s_lists_listu_u_caseu_u_defu_c1]: !v f a1 a0. h4/list/list__CASE (happ (happ h4/list/CONS a0) a1) v f = happ (happ f a0) a1
% Assm [h4s_lists_listu_u_TYu_u_DEF]: !_1. (!n. happ _1 n = h4/ind__type/BOTTOM) ==> (!_0. (!a0_27. happ _0 a0_27 <=> (!_27list_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _1 \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a0 (h4/ind__type/FCONS a1 _1) /\ happ _27list_27 a1) ==> happ _27list_27 a0_270) ==> happ _27list_27 a0_27)) ==> (?rep. h4/bool/TYPE__DEFINITION _0 rep))
% Assm [h4s_lists_listu_u_sizeu_u_defu_c0]: !f. h4/list/list__size f h4/list/NIL = h4/num/0
% 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_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% 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_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% 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_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_predu_u_sets_INu_u_IMAGE]: !y s f. h4/bool/IN y (h4/pred__set/IMAGE f s) <=> (?x. y = happ f x /\ h4/bool/IN x s)
% Assm [h4s_relations_RDOMu_u_DEF]: !x R. happ (h4/relation/RDOM R) x <=> (?y. happ (happ R x) y)
% Assm [h4s_predu_u_sets_IMAGEu_u_DEF]: !_0. (!f s x. happ (happ (happ _0 f) s) x = h4/pair/_2C (happ f x) (h4/bool/IN x s)) ==> (!s f. h4/pred__set/IMAGE f s = h4/pred__set/GSPEC (happ (happ _0 f) s))
% Assm [h4s_predu_u_sets_GSPECIFICATION]: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = happ f x)
% Assm [h4s_pairs_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_relations_INu_u_RDOM]: !x R. h4/bool/IN x (h4/relation/RDOM R) <=> (?y. happ (happ R x) y)
% Assm [h4s_bools_INu_u_DEF]: !x x. h4/bool/IN x x <=> happ x x
% Assm [h4s_sums_sumu_u_caseu_u_cong]: !f1_27 f1 f_27 f M_27 M. M = M_27 /\ (!x. M_27 = h4/sum/INL x ==> happ f x = happ f_27 x) /\ (!y. M_27 = h4/sum/INR y ==> happ f1 y = happ f1_27 y) ==> h4/sum/sum__CASE M f f1 = h4/sum/sum__CASE M_27 f_27 f1_27
% Assm [h4s_sums_sumu_u_caseu_u_defu_c0]: !x f1 f. h4/sum/sum__CASE (h4/sum/INL x) f f1 = happ f x
% Assm [h4s_sums_sumu_u_CASES]: !ss. (?x. ss = h4/sum/INL x) \/ (?y. ss = h4/sum/INR y)
% Assm [h4s_sums_sumu_u_caseu_u_defu_c1]: !y f1 f. h4/sum/sum__CASE (h4/sum/INR y) f f1 = happ f1 y
% 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_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_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_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_relations_RUNIV0]: !y x. h4/relation/RUNIV x y <=> 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_pairs_LEXu_u_DEF]: !_3. (!R1 s u R2 t v. happ (happ (happ (happ (happ (happ _3 R1) s) u) R2) t) v <=> happ (happ R1 s) u \/ s = u /\ happ (happ R2 t) v) ==> (!_2. (!R1 s R2 t u. happ (happ (happ (happ (happ _2 R1) s) R2) t) u = happ (happ (happ (happ (happ _3 R1) s) u) R2) t) ==> (!_1. (!R1 s R2 t. happ (happ (happ (happ _1 R1) s) R2) t = h4/pair/UNCURRY (happ (happ (happ (happ _2 R1) s) R2) t)) ==> (!_0. (!R1 R2 s. happ (happ (happ _0 R1) R2) s = happ (happ (happ _1 R1) s) R2) ==> (!R2 R1. h4/pair/LEX R1 R2 = h4/pair/UNCURRY (happ (happ _0 R1) R2)))))
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. happ (h4/pair/UNCURRY f) (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_pairs_LEXu_u_DEFu_u_THM]: !d c b a R2 R1. happ (happ (h4/pair/LEX R1 R2) (h4/pair/_2C a b)) (h4/pair/_2C c d) <=> happ (happ R1 a) c \/ a = c /\ happ (happ R2 b) d
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% 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_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_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c4]: !t. t /\ t <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% Assm [h4s_indu_u_types_ISO0]: !g f. h4/ind__type/ISO f g <=> (!x. happ f (happ g x) = x) /\ (!y. happ g (happ f y) = y)
% Assm [h4s_indu_u_types_ISOu_u_USAGE]: !g f. h4/ind__type/ISO f g ==> (!P. (!x. happ P x) <=> (!x. happ P (happ g x))) /\ (!P. (?x. happ P x) <=> (?x. happ P (happ g x))) /\ (!a b. a = happ g b <=> happ f a = b)
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_pairs_ELIMu_u_PEXISTS]: !P. (?p. happ (happ P (h4/pair/FST p)) (h4/pair/SND p)) <=> (?p1 p2. happ (happ P p1) p2)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_bools_LEFTu_u_ORu_u_EXISTSu_u_THM]: !Q P. (?x. happ P x) \/ Q <=> (?x. happ P x \/ Q)
% Assm [h4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM]: !Q P. P \/ (?x. happ Q x) <=> (?x. P \/ happ Q x)
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_bools_LEFTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. happ P x /\ Q) <=> (?x. happ P x) /\ Q
% Assm [h4s_bools_RIGHTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. P /\ happ Q x) <=> P /\ (?x. happ Q x)
% Assm [h4s_bools_EXISTSu_u_ORu_u_THM]: !Q P. (?x. happ P x \/ happ Q x) <=> (?x. happ P x) \/ (?x. happ Q x)
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% 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_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_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% 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_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% 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_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_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_options_OPTRELu_u_def]: !y x R. h4/option/OPTREL R x y <=> x = h4/option/NONE /\ y = h4/option/NONE \/ (?x0 y0. x = h4/option/SOME x0 /\ y = h4/option/SOME y0 /\ happ (happ R x0) y0)
% Assm [h4s_relations_nfu_u_def]: !x R. h4/relation/nf R x <=> (!y. ~happ (happ R x) y)
% 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_bools_ONTOu_u_DEF]: !x. h4/bool/ONTO x <=> (!y. ?x. y = happ x x)
% 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_APPLYu_c1]: !x f b a. ~(a = b) ==> happ (h4/combin/UPDATE a x f) b = happ f b
% 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_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% 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_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_Tu_u_DEF]: T <=> (!x. x <=> x)
% Assm [h4s_bools_NOTu_u_DEF]: !x. $not x <=> x ==> F
% Assm [h4s_pairs_PFORALLu_u_THM]: !_1. (!P x y. happ (happ (happ _1 P) x) y <=> happ (happ P x) y) ==> (!_0. (!P x. happ (happ _0 P) x = happ (happ _1 P) x) ==> (!P. (!x y. happ (happ P x) y) <=> $forall (h4/pair/UNCURRY (happ _0 P))))
% Assm [h4s_pairs_ELIMu_u_PFORALL]: !P. (!p. happ (happ P (h4/pair/FST p)) (h4/pair/SND p)) <=> (!p1 p2. happ (happ P p1) p2)
% Goal: !v_27 v f_27 f M_27 M. M = M_27 /\ (M_27 = h4/list/NIL ==> v = v_27) /\ (!a0 a1. M_27 = happ (happ h4/list/CONS a0) a1 ==> happ (happ f a0) a1 = happ (happ f_27 a0) a1) ==> h4/list/list__CASE M v f = h4/list/list__CASE M_27 v_27 f_27
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f0)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(aHLu_EXT, axiom, ![TV_Q1331566,TV_Q1331562]: ![V_f, V_g]: (![V_x]: s(TV_Q1331562,happ(s(t_fun(TV_Q1331566,TV_Q1331562),V_f),s(TV_Q1331566,V_x))) = s(TV_Q1331562,happ(s(t_fun(TV_Q1331566,TV_Q1331562),V_g),s(TV_Q1331566,V_x))) => s(t_fun(TV_Q1331566,TV_Q1331562),V_f) = s(t_fun(TV_Q1331566,TV_Q1331562),V_g))).
fof(ah4s_lists_listu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27b,h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f))) = s(TV_u_27b,V_v)).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
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_listu_u_distinct, axiom, ![TV_u_27a]: ![V_a1, V_a0]: ~ (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_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1))))).
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_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_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_DATATYPEu_u_TAGu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,h4s_bools_datatype(s(TV_u_27a,V_x))) = s(t_bool,t)).
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_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_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_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_MONOu_u_AND, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) & p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) & p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_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_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,t)))).
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_lists_listu_u_11, axiom, ![TV_u_27a]: ![V_a1u_27, V_a1, V_a0u_27, V_a0]: (s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1))) = 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_a0u_27))),s(t_h4s_lists_list(TV_u_27a),V_a1u_27))) <=> (s(TV_u_27a,V_a0) = s(TV_u_27a,V_a0u_27) & s(t_h4s_lists_list(TV_u_27a),V_a1) = s(t_h4s_lists_list(TV_u_27a),V_a1u_27)))).
fof(ah4s_lists_listu_u_sizeu_u_defu_c1, axiom, ![TV_u_27a]: ![V_f, V_a1, V_a0]: s(t_h4s_nums_num,h4s_lists_listu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),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_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(TV_u_27a,V_a0))),s(t_h4s_nums_num,h4s_lists_listu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_lists_list(TV_u_27a),V_a1)))))))).
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_datatypeu_u_list, axiom, ![TV_u_27a]: ![V_list]: p(s(t_bool,h4s_bools_datatype(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),t_bool),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),t_bool)),V_list),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))),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))))))).
fof(ah4s_lists_listu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_f, V_a1, V_a0]: s(TV_u_27b,h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27a),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_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f))) = s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1)))).
fof(ah4s_lists_listu_u_TYu_u_DEF, axiom, ![TV_u_27a]: ![V_uu_1]: (![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_1),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => ![V_uu_0]: (![V_a0u_27]: (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_0),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_27)))) <=> ![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_1))) | ?[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_1))))) & 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_a0u_27)))))) => ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_0),s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_indu_u_types_recspace(TV_u_27a)),V_rep))))))).
fof(ah4s_lists_listu_u_sizeu_u_defu_c0, axiom, ![TV_u_27a]: ![V_f]: s(t_h4s_nums_num,h4s_lists_listu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_nums_num,h4s_nums_0)).
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_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_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f0)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f0))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f0) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_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_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_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_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_predu_u_sets_INu_u_IMAGE, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_s, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_y),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> ?[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))) & 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_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_predu_u_sets_IMAGEu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_f, V_s, V_x]: s(t_h4s_pairs_prod(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27b,t_bool),h4s_pairs_u_2c(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))) => ![V_s, V_f]: s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27b,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_predu_u_sets_GSPECIFICATION, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_v),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f)))))) <=> ?[V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_v),s(t_bool,t))) = 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_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_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_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_sums_sumu_u_caseu_u_cong, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_f1u_27, V_f1, V_fu_27, V_f, V_Mu_27, V_M]: ((s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_M) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_Mu_27) & (![V_x]: (s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_Mu_27) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_x))) => s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_fu_27),s(TV_u_27a,V_x)))) & ![V_y]: (s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_Mu_27) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_y))) => s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),V_f1),s(TV_u_27b,V_y))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),V_f1u_27),s(TV_u_27b,V_y)))))) => s(TV_u_27c,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_M),s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27c),V_f1))) = s(TV_u_27c,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_Mu_27),s(t_fun(TV_u_27a,TV_u_27c),V_fu_27),s(t_fun(TV_u_27b,TV_u_27c),V_f1u_27))))).
fof(ah4s_sums_sumu_u_caseu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_x, V_f1, V_f]: s(TV_u_27c,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_x))),s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27c),V_f1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_sums_sumu_u_CASES, axiom, ![TV_u_27a,TV_u_27b]: ![V_ss]: (?[V_x]: s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_ss) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_x))) | ?[V_y]: s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_ss) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_y))))).
fof(ah4s_sums_sumu_u_caseu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_y, V_f1, V_f]: s(TV_u_27c,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_y))),s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27c),V_f1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),V_f1),s(TV_u_27b,V_y)))).
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,f0)) => 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_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
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_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_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f0)) => 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_pairs_LEXu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_3]: (![V_R1, V_s, V_u, V_R2, V_t, V_v]: (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(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),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(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool))))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)))))),V_uu_3),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(TV_u_27a,V_s))),s(TV_u_27a,V_u))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(TV_u_27b,V_t))),s(TV_u_27b,V_v)))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(TV_u_27a,V_s))),s(TV_u_27a,V_u)))) | (s(TV_u_27a,V_s) = s(TV_u_27a,V_u) & p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,V_t))),s(TV_u_27b,V_v))))))) => ![V_uu_2]: (![V_R1, V_s, V_R2, V_t, V_u]: 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(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),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(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))))),V_uu_2),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(TV_u_27a,V_s))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(TV_u_27b,V_t))),s(TV_u_27a,V_u))) = 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(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),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(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool))))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)))))),V_uu_3),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(TV_u_27a,V_s))),s(TV_u_27a,V_u))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(TV_u_27b,V_t))) => ![V_uu_1]: (![V_R1, V_s, V_R2, V_t]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool))))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(TV_u_27a,V_s))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(TV_u_27b,V_t))) = 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)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),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(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))))),V_uu_2),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(TV_u_27a,V_s))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(TV_u_27b,V_t))))) => ![V_uu_0]: (![V_R1, V_R2, V_s]: s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool))))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(TV_u_27a,V_s))) = s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool))))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(TV_u_27a,V_s))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))) => ![V_R2, V_R1]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)),h4s_pairs_lex(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool)))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool))))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2)))))))))).
fof(ah4s_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_pairs_LEXu_u_DEFu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_d, V_c, V_b, V_a, V_R2, V_R1]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_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)),h4s_pairs_lex(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_c),s(TV_u_27b,V_d)))))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(TV_u_27a,V_a))),s(TV_u_27a,V_c)))) | (s(TV_u_27a,V_a) = s(TV_u_27a,V_c) & p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,V_b))),s(TV_u_27b,V_d)))))))).
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_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_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_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f0)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f0)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f0)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,f0))) <=> p(s(t_bool,V_t)))).
fof(ah4s_indu_u_types_ISO0, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (p(s(t_bool,h4s_indu_u_types_iso(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27b,TV_u_27a),V_g)))) <=> (![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_g),s(TV_u_27b,V_x))))) = s(TV_u_27b,V_x) & ![V_y]: s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_g),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))))) = s(TV_u_27a,V_y)))).
fof(ah4s_indu_u_types_ISOu_u_USAGE, axiom, ![TV_u_27a,TV_u_27b]: ![V_g, V_f]: (p(s(t_bool,h4s_indu_u_types_iso(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27b,TV_u_27a),V_g)))) => (![V_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,happ(s(t_fun(TV_u_27b,TV_u_27a),V_g),s(TV_u_27b,V_x))))))) & (![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,happ(s(t_fun(TV_u_27b,TV_u_27a),V_g),s(TV_u_27b,V_x))))))) & ![V_a, V_b]: (s(TV_u_27a,V_a) = s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_g),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_a))) = s(TV_u_27b,V_b)))))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_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,f0))).
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,h4s_pairs_fst(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_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_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f0))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
fof(ah4s_bools_NOTu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ?[V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_NOTu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ![V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_LEFTu_u_ORu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))) <=> ?[V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))))).
fof(ah4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) | ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ?[V_x]: (p(s(t_bool,V_P)) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f0))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f0))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => p(s(t_bool,f0))))).
fof(ah4s_bools_LEFTu_u_EXISTSu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (?[V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))) <=> (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))))).
fof(ah4s_bools_RIGHTu_u_EXISTSu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (?[V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (p(s(t_bool,V_P)) & ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_EXISTSu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (?[V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_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_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_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => 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_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_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_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_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,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f0))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f0))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
fof(ah4s_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_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_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27b,V_y)).
fof(ah4s_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_options_OPTRELu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_R]: (p(s(t_bool,h4s_options_optrel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_options_option(TV_u_27a),V_x),s(t_h4s_options_option(TV_u_27b),V_y)))) <=> ((s(t_h4s_options_option(TV_u_27a),V_x) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) & s(t_h4s_options_option(TV_u_27b),V_y) = s(t_h4s_options_option(TV_u_27b),h4s_options_none)) | ?[V_x0, V_y0]: (s(t_h4s_options_option(TV_u_27a),V_x) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x0))) & (s(t_h4s_options_option(TV_u_27b),V_y) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,V_y0))) & p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27a,V_x0))),s(TV_u_27b,V_y0))))))))).
fof(ah4s_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_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_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_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_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_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_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_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,f0) => 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_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,f0)) <=> ![V_t]: p(s(t_bool,V_t)))).
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_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,f0))))).
fof(ah4s_pairs_PFORALLu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_P, V_x, V_y]: s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P))),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) => ![V_uu_0]: (![V_P, V_x]: 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(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P))),s(TV_u_27a,V_x))) = 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(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P))),s(TV_u_27a,V_x))) => ![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)))) <=> p(s(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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P)))))))))))).
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,h4s_pairs_fst(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_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(ch4s_lists_listu_u_caseu_u_cong, conjecture, ![TV_u_27a,TV_u_27b]: ![V_vu_27, V_v, V_fu_27, V_f, V_Mu_27, V_M]: ((s(t_h4s_lists_list(TV_u_27a),V_M) = s(t_h4s_lists_list(TV_u_27a),V_Mu_27) & ((s(t_h4s_lists_list(TV_u_27a),V_Mu_27) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) => s(TV_u_27b,V_v) = s(TV_u_27b,V_vu_27)) & ![V_a0, V_a1]: (s(t_h4s_lists_list(TV_u_27a),V_Mu_27) = 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_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1))) => s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1))) = s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_fu_27),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1)))))) => s(TV_u_27b,h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27a),V_M),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f))) = s(TV_u_27b,h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27a),V_Mu_27),s(TV_u_27b,V_vu_27),s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_fu_27))))).
