%   ORIGINAL: h4/toto/num__dt__distinct_c2
% 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/TRUTH: T
% Assm: h4/toto/bit10: h4/toto/bit1 = h4/toto/_20_40ind__typetoto1
% Assm: h4/toto/bit20: h4/toto/bit2 = h4/toto/_20_40ind__typetoto2
% Assm: h4/bool/DATATYPE__TAG__THM: !x. h4/bool/DATATYPE x <=> T
% Assm: h4/toto/hidden____20__40ind____typetoto2____def: h4/toto/_20_40ind__typetoto2 = (\a. h4/toto/_20_40ind__typetoto3 ((\a0. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB (h4/ind__type/FCONS a0 (\n. h4/ind__type/BOTTOM))) (h4/toto/_20_40ind__typetoto4 a)))
% Assm: h4/toto/num__dt__repfns_c0: !a. h4/toto/_20_40ind__typetoto3 (h4/toto/_20_40ind__typetoto4 a) = a
% Assm: h4/toto/hidden____20__40ind____typetoto1____def: h4/toto/_20_40ind__typetoto1 = (\a. h4/toto/_20_40ind__typetoto3 ((\a0. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB (h4/ind__type/FCONS a0 (\n. h4/ind__type/BOTTOM))) (h4/toto/_20_40ind__typetoto4 a)))
% Assm: h4/toto/datatype__num__dt: !num__dt. h4/bool/DATATYPE (num__dt h4/toto/zer h4/toto/bit1 h4/toto/bit2)
% Assm: h4/toto/num__dt__repfns_c1: !r. (\a0. !_27num__dt_27. (!a00. a00 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM) \/ (?a. a00 = (\a1. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB (h4/ind__type/FCONS a1 (\n. h4/ind__type/BOTTOM))) a /\ _27num__dt_27 a) \/ (?a. a00 = (\a1. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB (h4/ind__type/FCONS a1 (\n. h4/ind__type/BOTTOM))) a /\ _27num__dt_27 a) ==> _27num__dt_27 a00) ==> _27num__dt_27 a0) r <=> h4/toto/_20_40ind__typetoto4 (h4/toto/_20_40ind__typetoto3 r) = r
% Assm: h4/toto/hidden____20__40ind____typetoto0____def: h4/toto/_20_40ind__typetoto0 = h4/toto/_20_40ind__typetoto3 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM))
% Assm: h4/toto/zer0: h4/toto/zer = h4/toto/_20_40ind__typetoto0
% Assm: h4/toto/num__dt__distinct_c0: !a. ~(h4/toto/zer = h4/toto/bit1 a)
% Assm: h4/toto/num__dt__distinct_c1: !a. ~(h4/toto/zer = h4/toto/bit2 a)
% Assm: h4/toto/num__dt__11_c1: !a_27 a. h4/toto/bit2 a = h4/toto/bit2 a_27 <=> a = a_27
% Assm: h4/toto/num__dt__11_c0: !a_27 a. h4/toto/bit1 a = h4/toto/bit1 a_27 <=> a = a_27
% Assm: h4/toto/num__dt__case__def_c2: !v f1 f a. h4/toto/num__dt__CASE (h4/toto/bit2 a) v f f1 = f1 a
% Assm: h4/toto/num__dt__size__def_c2: !a. h4/toto/num__dt__size (h4/toto/bit2 a) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/toto/num__dt__size a)
% Assm: h4/toto/num__dt__size__def_c1: !a. h4/toto/num__dt__size (h4/toto/bit1 a) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/toto/num__dt__size a)
% Assm: h4/toto/num__dt__case__def_c1: !v f1 f a. h4/toto/num__dt__CASE (h4/toto/bit1 a) v f f1 = f a
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/BETA__THM: !y f. (\x. f x) y = f y
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/ind__type/FCONS0_c0: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm: h4/ind__type/FCONS0_c1: !n f a. h4/ind__type/FCONS a f (h4/num/SUC n) = f n
% Assm: h4/ind__type/CONSTR__REC: !Fn. ?f. !c i r. f (h4/ind__type/CONSTR c i r) = Fn c i r (\n. f (r n))
% Assm: h4/toto/num__dt__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION (\a0. !_27num__dt_27. (!a00. a00 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM) \/ (?a. a00 = (\a1. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB (h4/ind__type/FCONS a1 (\n. h4/ind__type/BOTTOM))) a /\ _27num__dt_27 a) \/ (?a. a00 = (\a1. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB (h4/ind__type/FCONS a1 (\n. h4/ind__type/BOTTOM))) a /\ _27num__dt_27 a) ==> _27num__dt_27 a00) ==> _27num__dt_27 a0) 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/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/toto/num__dt__case__def_c0: !v f1 f. h4/toto/num__dt__CASE h4/toto/zer v f f1 = v
% Assm: h4/toto/num__dt__size__def_c0: h4/toto/num__dt__size h4/toto/zer = h4/num/0
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/EXCLUDED__MIDDLE0: !t. t \/ ~t
% Assm: h4/list/MAP__TL: !l f. ~h4/list/NULL l ==> h4/list/MAP f (h4/list/TL l) = h4/list/TL (h4/list/MAP f l)
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/relation/WF__noloops: !y x R. h4/relation/WF R ==> h4/relation/TC R x y ==> ~(x = y)
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/list/CONS0: h4/list/CONS = h4/list/_20_40ind__typelist1
% Assm: h4/list/NIL0: h4/list/NIL = h4/list/_20_40ind__typelist0
% 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____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/list/hidden____20__40ind____typelist0____def: h4/list/_20_40ind__typelist0 = h4/list/_20_40ind__typelist2 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM))
% Assm: h4/list/NULL__DEF_c1: !t h. h4/list/NULL (h4/list/CONS h t) <=> F
% Assm: h4/list/NULL__DEF_c0: h4/list/NULL h4/list/NIL <=> T
% Assm: h4/list/MAP0_c0: !f. h4/list/MAP f h4/list/NIL = h4/list/NIL
% Assm: h4/list/MAP0_c1: !t h f. h4/list/MAP f (h4/list/CONS h t) = h4/list/CONS (f h) (h4/list/MAP f t)
% Assm: h4/list/TL0: !t h. h4/list/TL (h4/list/CONS h t) = t
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% 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/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/relation/WF__TC__EQN: !R. h4/relation/WF (h4/relation/TC R) <=> h4/relation/WF R
% Assm: h4/relation/WF__NOT__REFL: !y x R. h4/relation/WF R ==> R x y ==> ~(x = y)
% Assm: h4/rich__list/EL__PRE__LENGTH: !l. ~(l = h4/list/NIL) ==> h4/list/EL (h4/prim__rec/PRE (h4/list/LENGTH l)) l = h4/list/LAST l
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/list/list__to__set__diff: !l2 l1. h4/pred__set/DIFF (h4/list/LIST__TO__SET l2) (h4/list/LIST__TO__SET l1) = h4/list/LIST__TO__SET (h4/list/FILTER (\x. ~h4/bool/IN x (h4/list/LIST__TO__SET l1)) l2)
% Assm: h4/list/SNOC__INDUCT: !P. P h4/list/NIL /\ (!l. P l ==> (!x. P (h4/list/SNOC x l))) ==> (!l. P l)
% Assm: h4/list/EL__LENGTH__SNOC: !x l. h4/list/EL (h4/list/LENGTH l) (h4/list/SNOC x l) = x
% Assm: h4/list/LAST__SNOC: !x l. h4/list/LAST (h4/list/SNOC x l) = x
% Assm: h4/list/LENGTH__SNOC: !x l. h4/list/LENGTH (h4/list/SNOC x l) = h4/num/SUC (h4/list/LENGTH l)
% Assm: h4/prim__rec/PRE0_c1: !m. h4/prim__rec/PRE (h4/num/SUC m) = m
% Assm: h4/pred__set/PROD__SET__IMAGE__REDUCTION: !x s f. h4/pred__set/FINITE (h4/pred__set/IMAGE f s) /\ ~h4/bool/IN (f x) (h4/pred__set/IMAGE f s) ==> h4/pred__set/PROD__SET (h4/pred__set/IMAGE f (h4/pred__set/INSERT x s)) = h4/arithmetic/_2A (f x) (h4/pred__set/PROD__SET (h4/pred__set/IMAGE f s))
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/bool/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/pred__set/INSERT__DIFF: !x t s. h4/pred__set/DIFF (h4/pred__set/INSERT x s) t = h4/bool/COND (h4/bool/IN x t) (h4/pred__set/DIFF s t) (h4/pred__set/INSERT x (h4/pred__set/DIFF s t))
% Assm: h4/pred__set/EMPTY__DIFF: !s. h4/pred__set/DIFF h4/pred__set/EMPTY s = h4/pred__set/EMPTY
% Assm: h4/list/FILTER0_c1: !t h P. h4/list/FILTER P (h4/list/CONS h t) = h4/bool/COND (P h) (h4/list/CONS h (h4/list/FILTER P t)) (h4/list/FILTER P t)
% Assm: h4/list/LIST__TO__SET0_c1: !t h. h4/list/LIST__TO__SET (h4/list/CONS h t) = h4/pred__set/INSERT h (h4/list/LIST__TO__SET t)
% Assm: h4/list/LIST__TO__SET0_c0: h4/list/LIST__TO__SET h4/list/NIL = h4/pred__set/EMPTY
% Assm: h4/list/FILTER0_c0: !P. h4/list/FILTER P h4/list/NIL = h4/list/NIL
% Assm: h4/pred__set/IN__DELETE: !y x s. h4/bool/IN x (h4/pred__set/DELETE s y) <=> h4/bool/IN x s /\ ~(x = y)
% Assm: h4/pred__set/IMAGE__INSERT: !x s f. h4/pred__set/IMAGE f (h4/pred__set/INSERT x s) = h4/pred__set/INSERT (f x) (h4/pred__set/IMAGE f s)
% Assm: h4/pred__set/DELETE__NON__ELEMENT: !x s. ~h4/bool/IN x s <=> h4/pred__set/DELETE s x = s
% Assm: h4/pred__set/PROD__SET__THM_c0: h4/pred__set/PROD__SET h4/pred__set/EMPTY = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm: h4/pred__set/PROD__SET__THM_c1: !x s. h4/pred__set/FINITE s ==> h4/pred__set/PROD__SET (h4/pred__set/INSERT x s) = h4/arithmetic/_2A x (h4/pred__set/PROD__SET (h4/pred__set/DELETE s x))
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/rich__list/SPLITP__EVERY: !l P. h4/list/EVERY (\x. ~P x) l ==> h4/rich__list/SPLITP P l = h4/pair/_2C l h4/list/NIL
% Assm: h4/pred__set/IN__INSERT: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm: h4/pred__set/NOT__IN__EMPTY: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/pred__set/DELETE__DEF: !x s. h4/pred__set/DELETE s x = h4/pred__set/DIFF s (h4/pred__set/INSERT x h4/pred__set/EMPTY)
% Assm: h4/pred__set/IN__DIFF: !x t s. h4/bool/IN x (h4/pred__set/DIFF s t) <=> h4/bool/IN x s /\ ~h4/bool/IN x t
% Assm: h4/pred__set/INFINITE__UNIV: ~h4/pred__set/FINITE h4/pred__set/UNIV <=> (?f. (!x y. f x = f y ==> x = y) /\ (?y. !x. ~(f x = y)))
% Assm: h4/rich__list/SPLITP0_c0: !P. h4/rich__list/SPLITP P h4/list/NIL = h4/pair/_2C h4/list/NIL h4/list/NIL
% Assm: h4/list/EVERY__DEF_c1: !t h P. h4/list/EVERY P (h4/list/CONS h t) <=> P h /\ h4/list/EVERY P t
% Assm: h4/rich__list/SPLITP0_c1: !x l P. h4/rich__list/SPLITP P (h4/list/CONS x l) = h4/bool/COND (P x) (h4/pair/_2C h4/list/NIL (h4/list/CONS x l)) (h4/pair/_2C (h4/list/CONS x (h4/pair/FST (h4/rich__list/SPLITP P l))) (h4/pair/SND (h4/rich__list/SPLITP P l)))
% Assm: h4/list/CONS__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__induction: !P. P h4/list/NIL /\ (!t. P t ==> (!h. P (h4/list/CONS h t))) ==> (!l. P l)
% Assm: h4/list/EVERY__DEF_c0: !P. h4/list/EVERY P h4/list/NIL <=> T
% 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/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/bool/AND__CLAUSES_c4: !t. t /\ t <=> t
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% 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)
% Goal: !a_27 a. ~(h4/toto/bit1 a = h4/toto/bit2 a_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_bools_TRUTH]: T
% Assm [h4s_totos_bit10]: h4/toto/bit1 = h4/toto/_20_40ind__typetoto1
% Assm [h4s_totos_bit20]: h4/toto/bit2 = h4/toto/_20_40ind__typetoto2
% Assm [h4s_bools_DATATYPEu_u_TAGu_u_THM]: !x. h4/bool/DATATYPE x <=> T
% Assm [h4s_totos_hiddenu_u_u_u_20u_u_40indu_u_u_u_typetoto2u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x. happ h4/toto/_20_40ind__typetoto2 x = h4/toto/_20_40ind__typetoto3 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB (h4/ind__type/FCONS (h4/toto/_20_40ind__typetoto4 x) _0)))
% Assm [h4s_totos_numu_u_dtu_u_repfnsu_c0]: !a. h4/toto/_20_40ind__typetoto3 (h4/toto/_20_40ind__typetoto4 a) = a
% Assm [h4s_totos_hiddenu_u_u_u_20u_u_40indu_u_u_u_typetoto1u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x. happ h4/toto/_20_40ind__typetoto1 x = h4/toto/_20_40ind__typetoto3 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB (h4/ind__type/FCONS (h4/toto/_20_40ind__typetoto4 x) _0)))
% Assm [h4s_totos_datatypeu_u_numu_u_dt]: !num__dt. h4/bool/DATATYPE (happ (happ (happ num__dt h4/toto/zer) h4/toto/bit1) h4/toto/bit2)
% Assm [h4s_totos_numu_u_dtu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27num__dt_27. (!a00. a00 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0 \/ (?a. a00 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB (h4/ind__type/FCONS a _0) /\ happ _27num__dt_27 a) \/ (?a. a00 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB (h4/ind__type/FCONS a _0) /\ happ _27num__dt_27 a) ==> happ _27num__dt_27 a00) ==> happ _27num__dt_27 r) <=> h4/toto/_20_40ind__typetoto4 (h4/toto/_20_40ind__typetoto3 r) = r)
% Assm [h4s_totos_hiddenu_u_u_u_20u_u_40indu_u_u_u_typetoto0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> h4/toto/_20_40ind__typetoto0 = h4/toto/_20_40ind__typetoto3 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0)
% Assm [h4s_totos_zer0]: h4/toto/zer = h4/toto/_20_40ind__typetoto0
% Assm [h4s_totos_numu_u_dtu_u_distinctu_c0]: !a. ~(h4/toto/zer = happ h4/toto/bit1 a)
% Assm [h4s_totos_numu_u_dtu_u_distinctu_c1]: !a. ~(h4/toto/zer = happ h4/toto/bit2 a)
% Assm [h4s_totos_numu_u_dtu_u_11u_c1]: !a_27 a. happ h4/toto/bit2 a = happ h4/toto/bit2 a_27 <=> a = a_27
% Assm [h4s_totos_numu_u_dtu_u_11u_c0]: !a_27 a. happ h4/toto/bit1 a = happ h4/toto/bit1 a_27 <=> a = a_27
% Assm [h4s_totos_numu_u_dtu_u_caseu_u_defu_c2]: !v f1 f a. h4/toto/num__dt__CASE (happ h4/toto/bit2 a) v f f1 = happ f1 a
% Assm [h4s_totos_numu_u_dtu_u_sizeu_u_defu_c2]: !a. h4/toto/num__dt__size (happ h4/toto/bit2 a) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/toto/num__dt__size a)
% Assm [h4s_totos_numu_u_dtu_u_sizeu_u_defu_c1]: !a. h4/toto/num__dt__size (happ h4/toto/bit1 a) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/toto/num__dt__size a)
% Assm [h4s_totos_numu_u_dtu_u_caseu_u_defu_c1]: !v f1 f a. h4/toto/num__dt__CASE (happ h4/toto/bit1 a) v f f1 = happ f a
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% 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_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_BETAu_u_THM]: !y f. happ f y = happ f y
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% 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_FCONS0u_c1]: !n f a. happ (h4/ind__type/FCONS a f) (h4/num/SUC n) = happ f n
% Assm [h4s_indu_u_types_CONSTRu_u_REC]: !_0. (!f r n. happ (happ (happ _0 f) r) n = happ f (happ r n)) ==> (!Fn. ?f. !c i r. happ f (h4/ind__type/CONSTR c i r) = happ (happ (happ (happ Fn c) i) r) (happ (happ _0 f) r))
% Assm [h4s_totos_numu_u_dtu_u_TYu_u_DEF]: !_1. (!n. happ _1 n = h4/ind__type/BOTTOM) ==> (!_0. (!a0. happ _0 a0 <=> (!_27num__dt_27. (!a00. a00 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _1 \/ (?a. a00 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB (h4/ind__type/FCONS a _1) /\ happ _27num__dt_27 a) \/ (?a. a00 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB (h4/ind__type/FCONS a _1) /\ happ _27num__dt_27 a) ==> happ _27num__dt_27 a00) ==> happ _27num__dt_27 a0)) ==> (?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_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_totos_numu_u_dtu_u_caseu_u_defu_c0]: !v f1 f. h4/toto/num__dt__CASE h4/toto/zer v f f1 = v
% Assm [h4s_totos_numu_u_dtu_u_sizeu_u_defu_c0]: h4/toto/num__dt__size h4/toto/zer = h4/num/0
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE0]: !t. t \/ ~t
% Assm [h4s_lists_MAPu_u_TL]: !l f. ~h4/list/NULL l ==> h4/list/MAP f (h4/list/TL l) = h4/list/TL (h4/list/MAP f l)
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_relations_WFu_u_noloops]: !y x R. h4/relation/WF R ==> happ (happ (h4/relation/TC R) x) y ==> ~(x = y)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_lists_CONS0]: h4/list/CONS = h4/list/_20_40ind__typelist1
% Assm [h4s_lists_NIL0]: h4/list/NIL = h4/list/_20_40ind__typelist0
% 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_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_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> h4/list/_20_40ind__typelist0 = h4/list/_20_40ind__typelist2 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0)
% Assm [h4s_lists_NULLu_u_DEFu_c1]: !t h. h4/list/NULL (happ (happ h4/list/CONS h) t) <=> F
% Assm [h4s_lists_NULLu_u_DEFu_c0]: h4/list/NULL h4/list/NIL <=> T
% Assm [h4s_lists_MAP0u_c0]: !f. h4/list/MAP f h4/list/NIL = h4/list/NIL
% Assm [h4s_lists_MAP0u_c1]: !t h f. h4/list/MAP f (happ (happ h4/list/CONS h) t) = happ (happ h4/list/CONS (happ f h)) (h4/list/MAP f t)
% Assm [h4s_lists_TL0]: !t h. h4/list/TL (happ (happ h4/list/CONS h) t) = t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% 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_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_relations_WFu_u_TCu_u_EQN]: !R. h4/relation/WF (h4/relation/TC R) <=> h4/relation/WF R
% Assm [h4s_relations_WFu_u_NOTu_u_REFL]: !y x R. h4/relation/WF R ==> happ (happ R x) y ==> ~(x = y)
% Assm [h4s_richu_u_lists_ELu_u_PREu_u_LENGTH]: !l. ~(l = h4/list/NIL) ==> h4/list/EL (h4/prim__rec/PRE (h4/list/LENGTH l)) l = h4/list/LAST l
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_lists_listu_u_tou_u_setu_u_diff]: !_0. (!l1 x. happ (happ _0 l1) x <=> ~h4/bool/IN x (h4/list/LIST__TO__SET l1)) ==> (!l2 l1. h4/pred__set/DIFF (h4/list/LIST__TO__SET l2) (h4/list/LIST__TO__SET l1) = h4/list/LIST__TO__SET (h4/list/FILTER (happ _0 l1) l2))
% Assm [h4s_lists_SNOCu_u_INDUCT]: !P. happ P h4/list/NIL /\ (!l. happ P l ==> (!x. happ P (h4/list/SNOC x l))) ==> (!l. happ P l)
% Assm [h4s_lists_ELu_u_LENGTHu_u_SNOC]: !x l. h4/list/EL (h4/list/LENGTH l) (h4/list/SNOC x l) = x
% Assm [h4s_lists_LASTu_u_SNOC]: !x l. h4/list/LAST (h4/list/SNOC x l) = x
% Assm [h4s_lists_LENGTHu_u_SNOC]: !x l. h4/list/LENGTH (h4/list/SNOC x l) = h4/num/SUC (h4/list/LENGTH l)
% Assm [h4s_primu_u_recs_PRE0u_c1]: !m. h4/prim__rec/PRE (h4/num/SUC m) = m
% Assm [h4s_predu_u_sets_PRODu_u_SETu_u_IMAGEu_u_REDUCTION]: !x s f. h4/pred__set/FINITE (h4/pred__set/IMAGE f s) /\ ~h4/bool/IN (happ f x) (h4/pred__set/IMAGE f s) ==> h4/pred__set/PROD__SET (h4/pred__set/IMAGE f (h4/pred__set/INSERT x s)) = h4/arithmetic/_2A (happ f x) (h4/pred__set/PROD__SET (h4/pred__set/IMAGE f s))
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_bools_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_predu_u_sets_INSERTu_u_DIFF]: !x t s. h4/pred__set/DIFF (h4/pred__set/INSERT x s) t = h4/bool/COND (h4/bool/IN x t) (h4/pred__set/DIFF s t) (h4/pred__set/INSERT x (h4/pred__set/DIFF s t))
% Assm [h4s_predu_u_sets_EMPTYu_u_DIFF]: !s. h4/pred__set/DIFF h4/pred__set/EMPTY s = h4/pred__set/EMPTY
% Assm [h4s_lists_FILTER0u_c1]: !t h P. h4/list/FILTER P (happ (happ h4/list/CONS h) t) = h4/bool/COND (happ P h) (happ (happ h4/list/CONS h) (h4/list/FILTER P t)) (h4/list/FILTER P t)
% Assm [h4s_lists_LISTu_u_TOu_u_SET0u_c1]: !t h. h4/list/LIST__TO__SET (happ (happ h4/list/CONS h) t) = h4/pred__set/INSERT h (h4/list/LIST__TO__SET t)
% Assm [h4s_lists_LISTu_u_TOu_u_SET0u_c0]: h4/list/LIST__TO__SET h4/list/NIL = h4/pred__set/EMPTY
% Assm [h4s_lists_FILTER0u_c0]: !P. h4/list/FILTER P h4/list/NIL = h4/list/NIL
% Assm [h4s_predu_u_sets_INu_u_DELETE]: !y x s. h4/bool/IN x (h4/pred__set/DELETE s y) <=> h4/bool/IN x s /\ ~(x = y)
% Assm [h4s_predu_u_sets_IMAGEu_u_INSERT]: !x s f. h4/pred__set/IMAGE f (h4/pred__set/INSERT x s) = h4/pred__set/INSERT (happ f x) (h4/pred__set/IMAGE f s)
% Assm [h4s_predu_u_sets_DELETEu_u_NONu_u_ELEMENT]: !x s. ~h4/bool/IN x s <=> h4/pred__set/DELETE s x = s
% Assm [h4s_predu_u_sets_PRODu_u_SETu_u_THMu_c0]: h4/pred__set/PROD__SET h4/pred__set/EMPTY = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm [h4s_predu_u_sets_PRODu_u_SETu_u_THMu_c1]: !x s. h4/pred__set/FINITE s ==> h4/pred__set/PROD__SET (h4/pred__set/INSERT x s) = h4/arithmetic/_2A x (h4/pred__set/PROD__SET (h4/pred__set/DELETE s x))
% Assm [h4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% Assm [h4s_richu_u_lists_SPLITPu_u_EVERY]: !_0. (!P x. happ (happ _0 P) x <=> ~happ P x) ==> (!l P. h4/list/EVERY (happ _0 P) l ==> h4/rich__list/SPLITP P l = h4/pair/_2C l h4/list/NIL)
% Assm [h4s_predu_u_sets_INu_u_INSERT]: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm [h4s_predu_u_sets_NOTu_u_INu_u_EMPTY]: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% Assm [h4s_predu_u_sets_DELETEu_u_DEF]: !x s. h4/pred__set/DELETE s x = h4/pred__set/DIFF s (h4/pred__set/INSERT x h4/pred__set/EMPTY)
% Assm [h4s_predu_u_sets_INu_u_DIFF]: !x t s. h4/bool/IN x (h4/pred__set/DIFF s t) <=> h4/bool/IN x s /\ ~h4/bool/IN x t
% Assm [h4s_predu_u_sets_INFINITEu_u_UNIV]: ~h4/pred__set/FINITE h4/pred__set/UNIV <=> (?f. (!x y. happ f x = happ f y ==> x = y) /\ (?y. !x. ~(happ f x = y)))
% Assm [h4s_richu_u_lists_SPLITP0u_c0]: !P. h4/rich__list/SPLITP P h4/list/NIL = h4/pair/_2C h4/list/NIL h4/list/NIL
% Assm [h4s_lists_EVERYu_u_DEFu_c1]: !t h P. h4/list/EVERY P (happ (happ h4/list/CONS h) t) <=> happ P h /\ h4/list/EVERY P t
% Assm [h4s_richu_u_lists_SPLITP0u_c1]: !x l P. h4/rich__list/SPLITP P (happ (happ h4/list/CONS x) l) = h4/bool/COND (happ P x) (h4/pair/_2C h4/list/NIL (happ (happ h4/list/CONS x) l)) (h4/pair/_2C (happ (happ h4/list/CONS x) (h4/pair/FST (h4/rich__list/SPLITP P l))) (h4/pair/SND (h4/rich__list/SPLITP P l)))
% Assm [h4s_lists_CONSu_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_induction]: !P. happ P h4/list/NIL /\ (!t. happ P t ==> (!h. happ P (happ (happ h4/list/CONS h) t))) ==> (!l. happ P l)
% Assm [h4s_lists_EVERYu_u_DEFu_c0]: !P. h4/list/EVERY P h4/list/NIL <=> T
% 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_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_bools_ANDu_u_CLAUSESu_c4]: !t. t /\ t <=> t
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% 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)
% Goal: !a_27 a. ~(happ h4/toto/bit1 a = happ h4/toto/bit2 a_27)
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_Q1345066,TV_Q1345062]: ![V_f, V_g]: (![V_x]: s(TV_Q1345062,happ(s(t_fun(TV_Q1345066,TV_Q1345062),V_f),s(TV_Q1345066,V_x))) = s(TV_Q1345062,happ(s(t_fun(TV_Q1345066,TV_Q1345062),V_g),s(TV_Q1345066,V_x))) => s(t_fun(TV_Q1345066,TV_Q1345062),V_f) = s(t_fun(TV_Q1345066,TV_Q1345062),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_totos_bit10, axiom, s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),h4s_totos_bit1) = s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),h4s_totos_u_20u_40indu_u_typetoto1)).
fof(ah4s_totos_bit20, axiom, s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),h4s_totos_bit2) = s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),h4s_totos_u_20u_40indu_u_typetoto2)).
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_totos_hiddenu_u_u_u_20u_u_40indu_u_u_u_typetoto2u_u_u_u_def, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_bool),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_bool),h4s_indu_u_types_bottom) => ![V_x]: s(t_h4s_totos_numu_u_dt,happ(s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),h4s_totos_u_20u_40indu_u_typetoto2),s(t_h4s_totos_numu_u_dt,V_x))) = s(t_h4s_totos_numu_u_dt,h4s_totos_u_20u_40indu_u_typetoto3(s(t_h4s_indu_u_types_recspace(t_bool),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(t_bool,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_bool),h4s_totos_u_20u_40indu_u_typetoto4(s(t_h4s_totos_numu_u_dt,V_x))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),V_uu_0))))))))).
fof(ah4s_totos_numu_u_dtu_u_repfnsu_c0, axiom, ![V_a]: s(t_h4s_totos_numu_u_dt,h4s_totos_u_20u_40indu_u_typetoto3(s(t_h4s_indu_u_types_recspace(t_bool),h4s_totos_u_20u_40indu_u_typetoto4(s(t_h4s_totos_numu_u_dt,V_a))))) = s(t_h4s_totos_numu_u_dt,V_a)).
fof(ah4s_totos_hiddenu_u_u_u_20u_u_40indu_u_u_u_typetoto1u_u_u_u_def, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_bool),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_bool),h4s_indu_u_types_bottom) => ![V_x]: s(t_h4s_totos_numu_u_dt,happ(s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),h4s_totos_u_20u_40indu_u_typetoto1),s(t_h4s_totos_numu_u_dt,V_x))) = s(t_h4s_totos_numu_u_dt,h4s_totos_u_20u_40indu_u_typetoto3(s(t_h4s_indu_u_types_recspace(t_bool),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_bool,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_bool),h4s_totos_u_20u_40indu_u_typetoto4(s(t_h4s_totos_numu_u_dt,V_x))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),V_uu_0))))))))).
fof(ah4s_totos_datatypeu_u_numu_u_dt, axiom, ![V_numu_u_dt]: p(s(t_bool,h4s_bools_datatype(s(t_bool,happ(s(t_fun(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_bool),happ(s(t_fun(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_fun(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_bool)),happ(s(t_fun(t_h4s_totos_numu_u_dt,t_fun(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_fun(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_bool))),V_numu_u_dt),s(t_h4s_totos_numu_u_dt,h4s_totos_zer))),s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),h4s_totos_bit1))),s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),h4s_totos_bit2))))))).
fof(ah4s_totos_numu_u_dtu_u_repfnsu_c1, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_bool),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_bool),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27numu_u_dtu_27]: (![V_a00]: ((s(t_h4s_indu_u_types_recspace(t_bool),V_a00) = s(t_h4s_indu_u_types_recspace(t_bool),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_bool,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),V_uu_0))) | (?[V_a]: (s(t_h4s_indu_u_types_recspace(t_bool),V_a00) = s(t_h4s_indu_u_types_recspace(t_bool),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_bool,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_bool),V_a),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),V_uu_0))))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_bool),t_bool),V_uu_27numu_u_dtu_27),s(t_h4s_indu_u_types_recspace(t_bool),V_a))))) | ?[V_a]: (s(t_h4s_indu_u_types_recspace(t_bool),V_a00) = s(t_h4s_indu_u_types_recspace(t_bool),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(t_bool,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_bool),V_a),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),V_uu_0))))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_bool),t_bool),V_uu_27numu_u_dtu_27),s(t_h4s_indu_u_types_recspace(t_bool),V_a))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_bool),t_bool),V_uu_27numu_u_dtu_27),s(t_h4s_indu_u_types_recspace(t_bool),V_a00))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_bool),t_bool),V_uu_27numu_u_dtu_27),s(t_h4s_indu_u_types_recspace(t_bool),V_r))))) <=> s(t_h4s_indu_u_types_recspace(t_bool),h4s_totos_u_20u_40indu_u_typetoto4(s(t_h4s_totos_numu_u_dt,h4s_totos_u_20u_40indu_u_typetoto3(s(t_h4s_indu_u_types_recspace(t_bool),V_r))))) = s(t_h4s_indu_u_types_recspace(t_bool),V_r)))).
fof(ah4s_totos_hiddenu_u_u_u_20u_u_40indu_u_u_u_typetoto0u_u_u_u_def, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_bool),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_bool),h4s_indu_u_types_bottom) => s(t_h4s_totos_numu_u_dt,h4s_totos_u_20u_40indu_u_typetoto0) = s(t_h4s_totos_numu_u_dt,h4s_totos_u_20u_40indu_u_typetoto3(s(t_h4s_indu_u_types_recspace(t_bool),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_bool,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),V_uu_0))))))).
fof(ah4s_totos_zer0, axiom, s(t_h4s_totos_numu_u_dt,h4s_totos_zer) = s(t_h4s_totos_numu_u_dt,h4s_totos_u_20u_40indu_u_typetoto0)).
fof(ah4s_totos_numu_u_dtu_u_distinctu_c0, axiom, ![V_a]: ~ (s(t_h4s_totos_numu_u_dt,h4s_totos_zer) = s(t_h4s_totos_numu_u_dt,happ(s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),h4s_totos_bit1),s(t_h4s_totos_numu_u_dt,V_a))))).
fof(ah4s_totos_numu_u_dtu_u_distinctu_c1, axiom, ![V_a]: ~ (s(t_h4s_totos_numu_u_dt,h4s_totos_zer) = s(t_h4s_totos_numu_u_dt,happ(s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),h4s_totos_bit2),s(t_h4s_totos_numu_u_dt,V_a))))).
fof(ah4s_totos_numu_u_dtu_u_11u_c1, axiom, ![V_au_27, V_a]: (s(t_h4s_totos_numu_u_dt,happ(s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),h4s_totos_bit2),s(t_h4s_totos_numu_u_dt,V_a))) = s(t_h4s_totos_numu_u_dt,happ(s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),h4s_totos_bit2),s(t_h4s_totos_numu_u_dt,V_au_27))) <=> s(t_h4s_totos_numu_u_dt,V_a) = s(t_h4s_totos_numu_u_dt,V_au_27))).
fof(ah4s_totos_numu_u_dtu_u_11u_c0, axiom, ![V_au_27, V_a]: (s(t_h4s_totos_numu_u_dt,happ(s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),h4s_totos_bit1),s(t_h4s_totos_numu_u_dt,V_a))) = s(t_h4s_totos_numu_u_dt,happ(s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),h4s_totos_bit1),s(t_h4s_totos_numu_u_dt,V_au_27))) <=> s(t_h4s_totos_numu_u_dt,V_a) = s(t_h4s_totos_numu_u_dt,V_au_27))).
fof(ah4s_totos_numu_u_dtu_u_caseu_u_defu_c2, axiom, ![TV_u_27a]: ![V_v, V_f1, V_f, V_a]: s(TV_u_27a,h4s_totos_numu_u_dtu_u_case(s(t_h4s_totos_numu_u_dt,happ(s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),h4s_totos_bit2),s(t_h4s_totos_numu_u_dt,V_a))),s(TV_u_27a,V_v),s(t_fun(t_h4s_totos_numu_u_dt,TV_u_27a),V_f),s(t_fun(t_h4s_totos_numu_u_dt,TV_u_27a),V_f1))) = s(TV_u_27a,happ(s(t_fun(t_h4s_totos_numu_u_dt,TV_u_27a),V_f1),s(t_h4s_totos_numu_u_dt,V_a)))).
fof(ah4s_totos_numu_u_dtu_u_sizeu_u_defu_c2, axiom, ![V_a]: s(t_h4s_nums_num,h4s_totos_numu_u_dtu_u_size(s(t_h4s_totos_numu_u_dt,happ(s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),h4s_totos_bit2),s(t_h4s_totos_numu_u_dt,V_a))))) = 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_totos_numu_u_dtu_u_size(s(t_h4s_totos_numu_u_dt,V_a)))))).
fof(ah4s_totos_numu_u_dtu_u_sizeu_u_defu_c1, axiom, ![V_a]: s(t_h4s_nums_num,h4s_totos_numu_u_dtu_u_size(s(t_h4s_totos_numu_u_dt,happ(s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),h4s_totos_bit1),s(t_h4s_totos_numu_u_dt,V_a))))) = 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_totos_numu_u_dtu_u_size(s(t_h4s_totos_numu_u_dt,V_a)))))).
fof(ah4s_totos_numu_u_dtu_u_caseu_u_defu_c1, axiom, ![TV_u_27a]: ![V_v, V_f1, V_f, V_a]: s(TV_u_27a,h4s_totos_numu_u_dtu_u_case(s(t_h4s_totos_numu_u_dt,happ(s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),h4s_totos_bit1),s(t_h4s_totos_numu_u_dt,V_a))),s(TV_u_27a,V_v),s(t_fun(t_h4s_totos_numu_u_dt,TV_u_27a),V_f),s(t_fun(t_h4s_totos_numu_u_dt,TV_u_27a),V_f1))) = s(TV_u_27a,happ(s(t_fun(t_h4s_totos_numu_u_dt,TV_u_27a),V_f),s(t_h4s_totos_numu_u_dt,V_a)))).
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_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_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_MONOu_u_OR, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) | p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) | p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_AND, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) & p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) & p(s(t_bool,V_w)))))).
fof(ah4s_bools_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_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_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_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_FCONS0u_c1, axiom, ![TV_u_27a]: ![V_n, V_f, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n)))).
fof(ah4s_indu_u_types_CONSTRu_u_REC, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_f, V_r, V_n]: s(TV_u_27b,happ(s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_h4s_nums_num,V_n))) = s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r),s(t_h4s_nums_num,V_n))))) => ![V_Fn]: ?[V_f]: ![V_c, V_i, V_r]: s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,V_c),s(TV_u_27a,V_i),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))) = s(TV_u_27b,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)))),V_Fn),s(t_h4s_nums_num,V_c))),s(TV_u_27a,V_i))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))))).
fof(ah4s_totos_numu_u_dtu_u_TYu_u_DEF, axiom, ![V_uu_1]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_bool),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),V_uu_1),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_bool),h4s_indu_u_types_bottom) => ![V_uu_0]: (![V_a0]: (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_bool),t_bool),V_uu_0),s(t_h4s_indu_u_types_recspace(t_bool),V_a0)))) <=> ![V_uu_27numu_u_dtu_27]: (![V_a00]: ((s(t_h4s_indu_u_types_recspace(t_bool),V_a00) = s(t_h4s_indu_u_types_recspace(t_bool),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_bool,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),V_uu_1))) | (?[V_a]: (s(t_h4s_indu_u_types_recspace(t_bool),V_a00) = s(t_h4s_indu_u_types_recspace(t_bool),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_bool,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_bool),V_a),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),V_uu_1))))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_bool),t_bool),V_uu_27numu_u_dtu_27),s(t_h4s_indu_u_types_recspace(t_bool),V_a))))) | ?[V_a]: (s(t_h4s_indu_u_types_recspace(t_bool),V_a00) = s(t_h4s_indu_u_types_recspace(t_bool),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(t_bool,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_bool),V_a),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_bool)),V_uu_1))))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_bool),t_bool),V_uu_27numu_u_dtu_27),s(t_h4s_indu_u_types_recspace(t_bool),V_a))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_bool),t_bool),V_uu_27numu_u_dtu_27),s(t_h4s_indu_u_types_recspace(t_bool),V_a00))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_bool),t_bool),V_uu_27numu_u_dtu_27),s(t_h4s_indu_u_types_recspace(t_bool),V_a0)))))) => ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_h4s_indu_u_types_recspace(t_bool),t_bool),V_uu_0),s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_indu_u_types_recspace(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_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_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_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_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_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_c2, axiom, ![V_t]: (s(t_bool,f) = 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,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_totos_numu_u_dtu_u_caseu_u_defu_c0, axiom, ![TV_u_27a]: ![V_v, V_f1, V_f]: s(TV_u_27a,h4s_totos_numu_u_dtu_u_case(s(t_h4s_totos_numu_u_dt,h4s_totos_zer),s(TV_u_27a,V_v),s(t_fun(t_h4s_totos_numu_u_dt,TV_u_27a),V_f),s(t_fun(t_h4s_totos_numu_u_dt,TV_u_27a),V_f1))) = s(TV_u_27a,V_v)).
fof(ah4s_totos_numu_u_dtu_u_sizeu_u_defu_c0, axiom, s(t_h4s_nums_num,h4s_totos_numu_u_dtu_u_size(s(t_h4s_totos_numu_u_dt,h4s_totos_zer))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE0, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_lists_MAPu_u_TL, axiom, ![TV_u_27b,TV_u_27a]: ![V_l, V_f]: (~ (p(s(t_bool,h4s_lists_null(s(t_h4s_lists_list(TV_u_27a),V_l))))) => s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),h4s_lists_tl(s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_tl(s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),V_l))))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_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_relations_WFu_u_noloops, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => (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)),h4s_relations_tc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) => ~ (s(TV_u_27a,V_x) = s(TV_u_27a,V_y))))).
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_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_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_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_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_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist0u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist0) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))).
fof(ah4s_lists_NULLu_u_DEFu_c1, axiom, ![TV_u_27a]: ![V_t, V_h]: s(t_bool,h4s_lists_null(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t))))) = s(t_bool,f)).
fof(ah4s_lists_NULLu_u_DEFu_c0, axiom, ![TV_u_27a]: s(t_bool,h4s_lists_null(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_bool,t)).
fof(ah4s_lists_MAP0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil)).
fof(ah4s_lists_MAP0u_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_h, V_f]: s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t))))) = s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),h4s_lists_cons),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_h))))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),V_t)))))).
fof(ah4s_lists_TL0, axiom, ![TV_u_27a]: ![V_t, V_h]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_tl(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t))))) = s(t_h4s_lists_list(TV_u_27a),V_t)).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_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_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_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_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_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_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_relations_WFu_u_TCu_u_EQN, axiom, ![TV_u_27a]: ![V_R]: s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_tc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) = s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))).
fof(ah4s_relations_WFu_u_NOTu_u_REFL, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => (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_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) => ~ (s(TV_u_27a,V_x) = s(TV_u_27a,V_y))))).
fof(ah4s_richu_u_lists_ELu_u_PREu_u_LENGTH, axiom, ![TV_u_27a]: ![V_l]: (~ (s(t_h4s_lists_list(TV_u_27a),V_l) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)) => s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l))))),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(TV_u_27a,h4s_lists_last(s(t_h4s_lists_list(TV_u_27a),V_l))))).
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_lists_listu_u_tou_u_setu_u_diff, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_l1, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_h4s_lists_list(TV_u_27a),V_l1))),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),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l1)))))))) => ![V_l2, V_l1]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_diff(s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l2))),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l1))))) = s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_h4s_lists_list(TV_u_27a),V_l1))),s(t_h4s_lists_list(TV_u_27a),V_l2))))))).
fof(ah4s_lists_SNOCu_u_INDUCT, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))) & ![V_l]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))) => ![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_snoc(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l)))))))) => ![V_l]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_lists_ELu_u_LENGTHu_u_SNOC, axiom, ![TV_u_27a]: ![V_x, V_l]: s(TV_u_27a,h4s_lists_el(s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_snoc(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(TV_u_27a,V_x)).
fof(ah4s_lists_LASTu_u_SNOC, axiom, ![TV_u_27a]: ![V_x, V_l]: s(TV_u_27a,h4s_lists_last(s(t_h4s_lists_list(TV_u_27a),h4s_lists_snoc(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(TV_u_27a,V_x)).
fof(ah4s_lists_LENGTHu_u_SNOC, axiom, ![TV_u_27a]: ![V_x, V_l]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),h4s_lists_snoc(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_primu_u_recs_PRE0u_c1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_predu_u_sets_PRODu_u_SETu_u_IMAGEu_u_REDUCTION, axiom, ![TV_u_27a]: ![V_x, V_s, V_f]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))) & ~ (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(TV_u_27a,V_x))),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_fun(TV_u_27a,t_bool),V_s)))))))) => s(t_h4s_nums_num,h4s_predu_u_sets_produ_u_set(s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(TV_u_27a,V_x))),s(t_h4s_nums_num,h4s_predu_u_sets_produ_u_set(s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_image(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_fun(TV_u_27a,t_bool),V_s))))))))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_bools_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_predu_u_sets_INSERTu_u_DIFF, axiom, ![TV_u_27a]: ![V_x, V_t, V_s]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_diff(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(TV_u_27a,t_bool),V_t))) = s(t_fun(TV_u_27a,t_bool),h4s_bools_cond(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_diff(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_diff(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))))).
fof(ah4s_predu_u_sets_EMPTYu_u_DIFF, axiom, ![TV_u_27a]: ![V_s]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_diff(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_lists_FILTER0u_c1, axiom, ![TV_u_27a]: ![V_t, V_h, V_P]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t))))) = s(t_h4s_lists_list(TV_u_27a),h4s_bools_cond(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t))))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t)))))).
fof(ah4s_lists_LISTu_u_TOu_u_SET0u_c1, axiom, ![TV_u_27b]: ![V_t, V_h]: s(t_fun(TV_u_27b,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),h4s_lists_cons),s(TV_u_27b,V_h))),s(t_h4s_lists_list(TV_u_27b),V_t))))) = s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_insert(s(TV_u_27b,V_h),s(t_fun(TV_u_27b,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27b),V_t)))))).
fof(ah4s_lists_LISTu_u_TOu_u_SET0u_c0, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_lists_FILTER0u_c0, axiom, ![TV_u_27a]: ![V_P]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)).
fof(ah4s_predu_u_sets_INu_u_DELETE, axiom, ![TV_u_27a]: ![V_y, V_x, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_y)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & ~ (s(TV_u_27a,V_x) = s(TV_u_27a,V_y))))).
fof(ah4s_predu_u_sets_IMAGEu_u_INSERT, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, 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),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))) = s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_insert(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),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)))))).
fof(ah4s_predu_u_sets_DELETEu_u_NONu_u_ELEMENT, axiom, ![TV_u_27a]: ![V_x, V_s]: (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))) <=> s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27a,t_bool),V_s))).
fof(ah4s_predu_u_sets_PRODu_u_SETu_u_THMu_c0, axiom, s(t_h4s_nums_num,h4s_predu_u_sets_produ_u_set(s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_empty))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))).
fof(ah4s_predu_u_sets_PRODu_u_SETu_u_THMu_c1, axiom, ![V_x, V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),V_s)))) => s(t_h4s_nums_num,h4s_predu_u_sets_produ_u_set(s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_insert(s(t_h4s_nums_num,V_x),s(t_fun(t_h4s_nums_num,t_bool),V_s))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,h4s_predu_u_sets_produ_u_set(s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_delete(s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_h4s_nums_num,V_x))))))))).
fof(ah4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ![V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_richu_u_lists_SPLITPu_u_EVERY, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(TV_u_27a,V_x)))) <=> ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))))) => ![V_l, V_P]: (p(s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_h4s_lists_list(TV_u_27a),V_l)))) => s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_richu_u_lists_splitp(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),V_l),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))))).
fof(ah4s_predu_u_sets_INu_u_INSERT, axiom, ![TV_u_27a]: ![V_y, V_x, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_predu_u_sets_NOTu_u_INu_u_EMPTY, axiom, ![TV_u_27a]: ![V_x]: ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))))).
fof(ah4s_bools_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_predu_u_sets_DELETEu_u_DEF, axiom, ![TV_u_27a]: ![V_x, V_s]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_diff(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))))).
fof(ah4s_predu_u_sets_INu_u_DIFF, axiom, ![TV_u_27a]: ![V_x, V_t, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_diff(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t)))))))).
fof(ah4s_predu_u_sets_INFINITEu_u_UNIV, axiom, ![TV_u_27a]: (~ (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ))))) <=> ?[V_f]: (![V_x, V_y]: (s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_x))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_y))) => s(TV_u_27a,V_x) = s(TV_u_27a,V_y)) & ?[V_y]: ![V_x]: ~ (s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_x))) = s(TV_u_27a,V_y))))).
fof(ah4s_richu_u_lists_SPLITP0u_c0, axiom, ![TV_u_27a]: ![V_P]: s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_richu_u_lists_splitp(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))).
fof(ah4s_lists_EVERYu_u_DEFu_c1, axiom, ![TV_u_27a]: ![V_t, V_h, V_P]: (p(s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t)))))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_h)))) & p(s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t))))))).
fof(ah4s_richu_u_lists_SPLITP0u_c1, axiom, ![TV_u_27a]: ![V_x, V_l, V_P]: s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_richu_u_lists_splitp(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_bools_cond(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_l))))),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),h4s_pairs_fst(s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_richu_u_lists_splitp(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l))))))),s(t_h4s_lists_list(TV_u_27a),h4s_pairs_snd(s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_richu_u_lists_splitp(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))))))))).
fof(ah4s_lists_CONSu_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_induction, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))) & ![V_t]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t)))) => ![V_h]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t)))))))) => ![V_l]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_lists_EVERYu_u_DEFu_c0, axiom, ![TV_u_27a]: ![V_P]: s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_bool,t)).
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_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_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_boolu_u_caseu_u_thmu_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_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(ch4s_totos_numu_u_dtu_u_distinctu_c2, conjecture, ![V_au_27, V_a]: ~ (s(t_h4s_totos_numu_u_dt,happ(s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),h4s_totos_bit1),s(t_h4s_totos_numu_u_dt,V_a))) = s(t_h4s_totos_numu_u_dt,happ(s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),h4s_totos_bit2),s(t_h4s_totos_numu_u_dt,V_au_27))))).
