%   ORIGINAL: h4/DeepSyntax/datatype__deep__form
% 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/DeepSyntax/xDivided0: h4/DeepSyntax/xDivided = h4/DeepSyntax/_20_40ind__typeDeepSyntax7
% Assm: h4/DeepSyntax/xEQ0: h4/DeepSyntax/xEQ = h4/DeepSyntax/_20_40ind__typeDeepSyntax6
% Assm: h4/DeepSyntax/xLT0: h4/DeepSyntax/xLT = h4/DeepSyntax/_20_40ind__typeDeepSyntax4
% Assm: h4/DeepSyntax/LTx0: h4/DeepSyntax/LTx = h4/DeepSyntax/_20_40ind__typeDeepSyntax5
% Assm: h4/DeepSyntax/UnrelatedBool0: h4/DeepSyntax/UnrelatedBool = h4/DeepSyntax/_20_40ind__typeDeepSyntax3
% Assm: h4/DeepSyntax/Negn0: h4/DeepSyntax/Negn = h4/DeepSyntax/_20_40ind__typeDeepSyntax2
% Assm: h4/DeepSyntax/Conjn0: h4/DeepSyntax/Conjn = h4/DeepSyntax/_20_40ind__typeDeepSyntax0
% Assm: h4/DeepSyntax/Disjn0: h4/DeepSyntax/Disjn = h4/DeepSyntax/_20_40ind__typeDeepSyntax1
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/DATATYPE__TAG__THM: !x. h4/bool/DATATYPE x <=> T
% Assm: h4/bool/DATATYPE__TAG__DEF: h4/bool/DATATYPE = (\x. T)
% Assm: h4/DeepSyntax/hidden____20__40ind____typeDeepSyntax0____def: h4/DeepSyntax/_20_40ind__typeDeepSyntax0 = (\a0 a1. h4/DeepSyntax/_20_40ind__typeDeepSyntax8 ((\a00 a10. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C h4/bool/ARB (h4/pair/_2C h4/bool/ARB h4/bool/ARB)) (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) (h4/DeepSyntax/_20_40ind__typeDeepSyntax9 a0) (h4/DeepSyntax/_20_40ind__typeDeepSyntax9 a1)))
% Assm: h4/DeepSyntax/hidden____20__40ind____typeDeepSyntax5____def: h4/DeepSyntax/_20_40ind__typeDeepSyntax5 = (\a. h4/DeepSyntax/_20_40ind__typeDeepSyntax8 ((\a0. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))))) (h4/pair/_2C h4/bool/ARB (h4/pair/_2C a0 h4/bool/ARB)) (\n. h4/ind__type/BOTTOM)) a))
% Assm: h4/DeepSyntax/hidden____20__40ind____typeDeepSyntax1____def: h4/DeepSyntax/_20_40ind__typeDeepSyntax1 = (\a0 a1. h4/DeepSyntax/_20_40ind__typeDeepSyntax8 ((\a00 a10. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C h4/bool/ARB (h4/pair/_2C h4/bool/ARB h4/bool/ARB)) (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) (h4/DeepSyntax/_20_40ind__typeDeepSyntax9 a0) (h4/DeepSyntax/_20_40ind__typeDeepSyntax9 a1)))
% Assm: h4/DeepSyntax/hidden____20__40ind____typeDeepSyntax6____def: h4/DeepSyntax/_20_40ind__typeDeepSyntax6 = (\a. h4/DeepSyntax/_20_40ind__typeDeepSyntax8 ((\a0. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0)))))) (h4/pair/_2C h4/bool/ARB (h4/pair/_2C a0 h4/bool/ARB)) (\n. h4/ind__type/BOTTOM)) a))
% Assm: h4/DeepSyntax/hidden____20__40ind____typeDeepSyntax3____def: h4/DeepSyntax/_20_40ind__typeDeepSyntax3 = (\a. h4/DeepSyntax/_20_40ind__typeDeepSyntax8 ((\a0. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) (h4/pair/_2C a0 (h4/pair/_2C h4/bool/ARB h4/bool/ARB)) (\n. h4/ind__type/BOTTOM)) a))
% Assm: h4/DeepSyntax/hidden____20__40ind____typeDeepSyntax4____def: h4/DeepSyntax/_20_40ind__typeDeepSyntax4 = (\a. h4/DeepSyntax/_20_40ind__typeDeepSyntax8 ((\a0. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0)))) (h4/pair/_2C h4/bool/ARB (h4/pair/_2C a0 h4/bool/ARB)) (\n. h4/ind__type/BOTTOM)) a))
% Assm: h4/DeepSyntax/hidden____20__40ind____typeDeepSyntax2____def: h4/DeepSyntax/_20_40ind__typeDeepSyntax2 = (\a. h4/DeepSyntax/_20_40ind__typeDeepSyntax8 ((\a0. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB (h4/pair/_2C h4/bool/ARB h4/bool/ARB)) (h4/ind__type/FCONS a0 (\n. h4/ind__type/BOTTOM))) (h4/DeepSyntax/_20_40ind__typeDeepSyntax9 a)))
% Assm: h4/DeepSyntax/deep__form__repfns_c0: !a. h4/DeepSyntax/_20_40ind__typeDeepSyntax8 (h4/DeepSyntax/_20_40ind__typeDeepSyntax9 a) = a
% Assm: h4/DeepSyntax/hidden____20__40ind____typeDeepSyntax7____def: h4/DeepSyntax/_20_40ind__typeDeepSyntax7 = (\a0 a1. h4/DeepSyntax/_20_40ind__typeDeepSyntax8 ((\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))))))) (h4/pair/_2C h4/bool/ARB (h4/pair/_2C a00 a10)) (\n. h4/ind__type/BOTTOM)) a0 a1))
% Assm: h4/DeepSyntax/deep__form__repfns_c1: !r. (\a0_27. !_27deep__form_27. (!a0_270. (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C h4/bool/ARB (h4/pair/_2C h4/bool/ARB h4/bool/ARB)) (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) a0 a1 /\ _27deep__form_27 a0 /\ _27deep__form_27 a1) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C h4/bool/ARB (h4/pair/_2C h4/bool/ARB h4/bool/ARB)) (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) a0 a1 /\ _27deep__form_27 a0 /\ _27deep__form_27 a1) \/ (?a. a0_270 = (\a0. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB (h4/pair/_2C h4/bool/ARB h4/bool/ARB)) (h4/ind__type/FCONS a0 (\n. h4/ind__type/BOTTOM))) a /\ _27deep__form_27 a) \/ (?a. a0_270 = (\a0. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) (h4/pair/_2C a0 (h4/pair/_2C h4/bool/ARB h4/bool/ARB)) (\n. h4/ind__type/BOTTOM)) a) \/ (?a. a0_270 = (\a0. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0)))) (h4/pair/_2C h4/bool/ARB (h4/pair/_2C a0 h4/bool/ARB)) (\n. h4/ind__type/BOTTOM)) a) \/ (?a. a0_270 = (\a0. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))))) (h4/pair/_2C h4/bool/ARB (h4/pair/_2C a0 h4/bool/ARB)) (\n. h4/ind__type/BOTTOM)) a) \/ (?a. a0_270 = (\a0. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0)))))) (h4/pair/_2C h4/bool/ARB (h4/pair/_2C a0 h4/bool/ARB)) (\n. h4/ind__type/BOTTOM)) a) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))))))) (h4/pair/_2C h4/bool/ARB (h4/pair/_2C a00 a10)) (\n. h4/ind__type/BOTTOM)) a0 a1) ==> _27deep__form_27 a0_270) ==> _27deep__form_27 a0_27) r <=> h4/DeepSyntax/_20_40ind__typeDeepSyntax9 (h4/DeepSyntax/_20_40ind__typeDeepSyntax8 r) = r
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/MONO__AND: !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/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/ind__type/FCONS0_c0: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm: h4/bool/BETA__THM: !y f. (\x. f x) y = f y
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/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/ind__type/FCONS0_c1: !n f a. h4/ind__type/FCONS a f (h4/num/SUC n) = f n
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/DeepSyntax/deep__form__case__def_c7: !f7 f6 f5 f4 f3 f2 f1 f a1 a0. h4/DeepSyntax/deep__form__CASE (h4/DeepSyntax/xDivided a0 a1) f f1 f2 f3 f4 f5 f6 f7 = f7 a0 a1
% Assm: h4/DeepSyntax/deep__form__case__def_c5: !f7 f6 f5 f4 f3 f2 f1 f a. h4/DeepSyntax/deep__form__CASE (h4/DeepSyntax/LTx a) f f1 f2 f3 f4 f5 f6 f7 = f5 a
% Assm: h4/DeepSyntax/deep__form__size__def_c6: !a. h4/DeepSyntax/deep__form__size (h4/DeepSyntax/xEQ a) = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm: h4/DeepSyntax/deep__form__case__def_c4: !f7 f6 f5 f4 f3 f2 f1 f a. h4/DeepSyntax/deep__form__CASE (h4/DeepSyntax/xLT a) f f1 f2 f3 f4 f5 f6 f7 = f4 a
% Assm: h4/DeepSyntax/deep__form__case__def_c6: !f7 f6 f5 f4 f3 f2 f1 f a. h4/DeepSyntax/deep__form__CASE (h4/DeepSyntax/xEQ a) f f1 f2 f3 f4 f5 f6 f7 = f6 a
% Assm: h4/DeepSyntax/deep__form__case__def_c0: !f7 f6 f5 f4 f3 f2 f1 f a1 a0. h4/DeepSyntax/deep__form__CASE (h4/DeepSyntax/Conjn a0 a1) f f1 f2 f3 f4 f5 f6 f7 = f a0 a1
% Assm: h4/DeepSyntax/deep__form__case__def_c3: !f7 f6 f5 f4 f3 f2 f1 f a. h4/DeepSyntax/deep__form__CASE (h4/DeepSyntax/UnrelatedBool a) f f1 f2 f3 f4 f5 f6 f7 = f3 a
% Assm: h4/DeepSyntax/deep__form__size__def_c4: !a. h4/DeepSyntax/deep__form__size (h4/DeepSyntax/xLT a) = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm: h4/DeepSyntax/deep__form__size__def_c7: !a1 a0. h4/DeepSyntax/deep__form__size (h4/DeepSyntax/xDivided a0 a1) = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm: h4/DeepSyntax/deep__form__case__def_c1: !f7 f6 f5 f4 f3 f2 f1 f a1 a0. h4/DeepSyntax/deep__form__CASE (h4/DeepSyntax/Disjn a0 a1) f f1 f2 f3 f4 f5 f6 f7 = f1 a0 a1
% Assm: h4/DeepSyntax/deep__form__case__def_c2: !f7 f6 f5 f4 f3 f2 f1 f a. h4/DeepSyntax/deep__form__CASE (h4/DeepSyntax/Negn a) f f1 f2 f3 f4 f5 f6 f7 = f2 a
% Assm: h4/DeepSyntax/deep__form__size__def_c5: !a. h4/DeepSyntax/deep__form__size (h4/DeepSyntax/LTx a) = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm: h4/DeepSyntax/deep__form__size__def_c0: !a1 a0. h4/DeepSyntax/deep__form__size (h4/DeepSyntax/Conjn a0 a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (h4/DeepSyntax/deep__form__size a0) (h4/DeepSyntax/deep__form__size a1))
% Assm: h4/DeepSyntax/deep__form__size__def_c2: !a. h4/DeepSyntax/deep__form__size (h4/DeepSyntax/Negn a) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/DeepSyntax/deep__form__size a)
% Assm: h4/DeepSyntax/deep__form__size__def_c1: !a1 a0. h4/DeepSyntax/deep__form__size (h4/DeepSyntax/Disjn a0 a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (h4/DeepSyntax/deep__form__size a0) (h4/DeepSyntax/deep__form__size a1))
% Assm: h4/DeepSyntax/deep__form__size__def_c3: !a. h4/DeepSyntax/deep__form__size (h4/DeepSyntax/UnrelatedBool a) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/basicSize/bool__size a)
% Assm: h4/arithmetic/datatype__num: !num. h4/bool/DATATYPE (num h4/num/0 h4/num/SUC)
% Assm: h4/sum/datatype__sum: !sum. h4/bool/DATATYPE (sum h4/sum/INL h4/sum/INR)
% Assm: h4/pair/datatype__pair: !pair. h4/bool/DATATYPE (pair h4/pair/_2C)
% Assm: h4/option/datatype__option: !option. h4/bool/DATATYPE (option h4/option/NONE h4/option/SOME)
% Assm: h4/list/datatype__list: !list. h4/bool/DATATYPE (list h4/list/NIL h4/list/CONS)
% Assm: h4/bool/DATATYPE__BOOL: !bool. h4/bool/DATATYPE (bool T F) <=> T
% Assm: h4/DeepSyntax/deep__form__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION (\a0_27. !_27deep__form_27. (!a0_270. (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C h4/bool/ARB (h4/pair/_2C h4/bool/ARB h4/bool/ARB)) (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) a0 a1 /\ _27deep__form_27 a0 /\ _27deep__form_27 a1) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C h4/bool/ARB (h4/pair/_2C h4/bool/ARB h4/bool/ARB)) (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) a0 a1 /\ _27deep__form_27 a0 /\ _27deep__form_27 a1) \/ (?a. a0_270 = (\a0. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB (h4/pair/_2C h4/bool/ARB h4/bool/ARB)) (h4/ind__type/FCONS a0 (\n. h4/ind__type/BOTTOM))) a /\ _27deep__form_27 a) \/ (?a. a0_270 = (\a0. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) (h4/pair/_2C a0 (h4/pair/_2C h4/bool/ARB h4/bool/ARB)) (\n. h4/ind__type/BOTTOM)) a) \/ (?a. a0_270 = (\a0. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0)))) (h4/pair/_2C h4/bool/ARB (h4/pair/_2C a0 h4/bool/ARB)) (\n. h4/ind__type/BOTTOM)) a) \/ (?a. a0_270 = (\a0. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))))) (h4/pair/_2C h4/bool/ARB (h4/pair/_2C a0 h4/bool/ARB)) (\n. h4/ind__type/BOTTOM)) a) \/ (?a. a0_270 = (\a0. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0)))))) (h4/pair/_2C h4/bool/ARB (h4/pair/_2C a0 h4/bool/ARB)) (\n. h4/ind__type/BOTTOM)) a) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))))))) (h4/pair/_2C h4/bool/ARB (h4/pair/_2C a00 a10)) (\n. h4/ind__type/BOTTOM)) a0 a1) ==> _27deep__form_27 a0_270) ==> _27deep__form_27 a0_27) 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/integer/INT__LDISTRIB: !z y x. h4/integer/int__mul x (h4/integer/int__add y z) = h4/integer/int__add (h4/integer/int__mul x y) (h4/integer/int__mul x z)
% 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/integer/TINT__EQ__EQUIV: !q p. h4/integer/tint__eq p q <=> h4/integer/tint__eq p = h4/integer/tint__eq q
% Assm: h4/integer/int__mul0: !T2 T1. h4/integer/int__mul T1 T2 = h4/integer/int__ABS (h4/integer/tint__mul (h4/integer/int__REP T1) (h4/integer/int__REP T2))
% Assm: h4/integer/int__QUOTIENT: h4/quotient/QUOTIENT h4/integer/tint__eq h4/integer/int__ABS h4/integer/int__REP
% Assm: h4/integer/int__add0: !T2 T1. h4/integer/int__add T1 T2 = h4/integer/int__ABS (h4/integer/tint__add (h4/integer/int__REP T1) (h4/integer/int__REP T2))
% Assm: h4/integer/TINT__MUL__WELLDEF: !y2 y1 x2 x1. h4/integer/tint__eq x1 x2 /\ h4/integer/tint__eq y1 y2 ==> h4/integer/tint__eq (h4/integer/tint__mul x1 y1) (h4/integer/tint__mul x2 y2)
% Assm: h4/integer/TINT__EQ__AP: !q p. p = q ==> h4/integer/tint__eq p q
% Assm: h4/quotient/EQUIV__RES__FORALL: !P E. h4/quotient/EQUIV E ==> (h4/bool/RES__FORALL (h4/quotient/respects E) P <=> $forall P)
% Assm: h4/quotient/RES__FORALL__RSP: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f g. h4/quotient/_3D_3D_3D_3E R $equals f g ==> (h4/bool/RES__FORALL (h4/quotient/respects R) f <=> h4/bool/RES__FORALL (h4/quotient/respects R) g))
% Assm: h4/quotient/EQUALS__RSP: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x1 x2 y1 y2. R x1 x2 /\ R y1 y2 ==> (R x1 y1 <=> R x2 y2))
% Assm: h4/quotient/FUN__QUOTIENT: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> h4/quotient/QUOTIENT (h4/quotient/_3D_3D_3D_3E R1 R2) (h4/quotient/_2D_2D_3E rep1 abs2) (h4/quotient/_2D_2D_3E abs1 rep2))
% Assm: h4/quotient/FORALL__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f. $forall f <=> h4/bool/RES__FORALL (h4/quotient/respects R) (h4/quotient/_2D_2D_3E abs h4/combin/I f))
% Assm: h4/quotient/EQUALS__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x y. x = y <=> R (rep x) (rep y))
% Assm: h4/quotient/FUN__REL: !g f R2 R1. h4/quotient/_3D_3D_3D_3E R1 R2 f g <=> (!x y. R1 x y ==> R2 (f x) (g y))
% Assm: h4/quotient/LAMBDA__PRS: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f. (\x. f x) = h4/quotient/_2D_2D_3E rep1 abs2 (\x. rep2 (f (abs1 x)))))
% Assm: h4/quotient/APPLY__RSP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g x y. h4/quotient/_3D_3D_3D_3E R1 R2 f g /\ R1 x y ==> R2 (f x) (g y)))
% Assm: h4/integer/TINT__ADD__WELLDEF: !y2 y1 x2 x1. h4/integer/tint__eq x1 x2 /\ h4/integer/tint__eq y1 y2 ==> h4/integer/tint__eq (h4/integer/tint__add x1 y1) (h4/integer/tint__add x2 y2)
% Assm: h4/quotient/EQUIV__def: !E. h4/quotient/EQUIV E <=> (!x y. E x y <=> E x = E y)
% Assm: h4/integer/TINT__LDISTRIB: !z y x. h4/integer/tint__mul x (h4/integer/tint__add y z) = h4/integer/tint__add (h4/integer/tint__mul x y) (h4/integer/tint__mul x z)
% Assm: h4/quotient/REP__ABS__RSP: !rep abs REL. h4/quotient/QUOTIENT REL abs rep ==> (!x1 x2. REL x1 x2 ==> REL x1 (rep (abs x2)))
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/quotient/IDENTITY__QUOTIENT: h4/quotient/QUOTIENT $equals h4/combin/I h4/combin/I
% Assm: h4/integer/INT__EQ__REDUCE_c2: !n. h4/integer/int__of__num h4/num/0 = h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) <=> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/BIT10: !n. h4/arithmetic/BIT1 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (h4/num/SUC h4/num/0))
% Assm: h4/arithmetic/BIT20: !n. h4/arithmetic/BIT2 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (h4/num/SUC (h4/num/SUC h4/num/0)))
% Assm: h4/arithmetic/NUMERAL__DEF: !x. h4/arithmetic/NUMERAL x = x
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/integer/INT__EQ__CALCULATE_c2: !n m. h4/integer/int__of__num n = h4/integer/int__neg (h4/integer/int__of__num m) <=> n = h4/num/0 /\ m = h4/num/0
% Assm: h4/integer/INT__EQ__CALCULATE_c1: !y x. h4/integer/int__neg x = h4/integer/int__neg y <=> x = y
% Assm: h4/integer/INT__EQ__CALCULATE_c0: !n m. h4/integer/int__of__num m = h4/integer/int__of__num n <=> m = n
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/bool/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm: h4/numeral/numeral__lte_c2: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT2 n) h4/arithmetic/ZERO <=> F
% Assm: h4/numeral/numeral__distrib_c27: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/numeral/numeral__distrib_c1: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm: h4/arithmetic/MULT__LESS__EQ__SUC: !p n m. h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/arithmetic/_2A (h4/num/SUC p) m) (h4/arithmetic/_2A (h4/num/SUC p) n)
% Assm: h4/arithmetic/ADD__MONO__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm: h4/arithmetic/EQ__LESS__EQ: !n m. m = n <=> h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/arithmetic/LESS__EQ__TRANS: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm: h4/arithmetic/TIMES2: !n. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n = h4/arithmetic/_2B n n
% Assm: h4/arithmetic/MULT__ASSOC: !p n m. h4/arithmetic/_2A m (h4/arithmetic/_2A n p) = h4/arithmetic/_2A (h4/arithmetic/_2A m n) p
% Assm: h4/arithmetic/MULT__CLAUSES_c1: !m. h4/arithmetic/_2A m h4/num/0 = h4/num/0
% Assm: h4/arithmetic/MULT__CLAUSES_c3: !m. h4/arithmetic/_2A m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = m
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/ADD__ASSOC: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm: h4/arithmetic/TWO: h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO) = h4/num/SUC (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/arithmetic/ONE: h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) = h4/num/SUC h4/num/0
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Assm: h4/integer/INT__DIV__REDUCE_c6: !n m. h4/integer/int__div (h4/integer/int__of__num (h4/arithmetic/NUMERAL m)) (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)))) = h4/integer/int__add (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/DIV (h4/arithmetic/NUMERAL m) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))))) (h4/bool/COND (h4/arithmetic/MOD (h4/arithmetic/NUMERAL m) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) = h4/num/0) (h4/integer/int__of__num h4/num/0) (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))))
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/integer/INT__DIV__NEG: !q p. ~(q = h4/integer/int__of__num h4/num/0) ==> h4/integer/int__div p (h4/integer/int__neg q) = h4/integer/int__div (h4/integer/int__neg p) q
% Assm: h4/integer/INT__DIV: !n m. ~(m = h4/num/0) ==> h4/integer/int__div (h4/integer/int__of__num n) (h4/integer/int__of__num m) = h4/integer/int__of__num (h4/arithmetic/DIV n m)
% Assm: h4/integer/int__div0: !j i. ~(j = h4/integer/int__of__num h4/num/0) ==> h4/integer/int__div i j = h4/bool/COND (h4/integer/int__lt (h4/integer/int__of__num h4/num/0) j) (h4/bool/COND (h4/integer/int__le (h4/integer/int__of__num h4/num/0) i) (h4/integer/int__of__num (h4/arithmetic/DIV (h4/integer/Num i) (h4/integer/Num j))) (h4/integer/int__add (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/DIV (h4/integer/Num (h4/integer/int__neg i)) (h4/integer/Num j)))) (h4/bool/COND (h4/arithmetic/MOD (h4/integer/Num (h4/integer/int__neg i)) (h4/integer/Num j) = h4/num/0) (h4/integer/int__of__num h4/num/0) (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))))) (h4/bool/COND (h4/integer/int__le (h4/integer/int__of__num h4/num/0) i) (h4/integer/int__add (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/DIV (h4/integer/Num i) (h4/integer/Num (h4/integer/int__neg j))))) (h4/bool/COND (h4/arithmetic/MOD (h4/integer/Num i) (h4/integer/Num (h4/integer/int__neg j)) = h4/num/0) (h4/integer/int__of__num h4/num/0) (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))))) (h4/integer/int__of__num (h4/arithmetic/DIV (h4/integer/Num (h4/integer/int__neg i)) (h4/integer/Num (h4/integer/int__neg j)))))
% Assm: h4/integer/NUM__OF__INT: !n. h4/integer/Num (h4/integer/int__of__num n) = n
% Assm: h4/integer/INT__LE: !n m. h4/integer/int__le (h4/integer/int__of__num m) (h4/integer/int__of__num n) <=> h4/arithmetic/_3C_3D m n
% Goal: !deep__form. h4/bool/DATATYPE (deep__form h4/DeepSyntax/Conjn h4/DeepSyntax/Disjn h4/DeepSyntax/Negn h4/DeepSyntax/UnrelatedBool h4/DeepSyntax/xLT h4/DeepSyntax/LTx h4/DeepSyntax/xEQ h4/DeepSyntax/xDivided)
%   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_DeepSyntaxs_xDivided0]: h4/DeepSyntax/xDivided = h4/DeepSyntax/_20_40ind__typeDeepSyntax7
% Assm [h4s_DeepSyntaxs_xEQ0]: h4/DeepSyntax/xEQ = h4/DeepSyntax/_20_40ind__typeDeepSyntax6
% Assm [h4s_DeepSyntaxs_xLT0]: h4/DeepSyntax/xLT = h4/DeepSyntax/_20_40ind__typeDeepSyntax4
% Assm [h4s_DeepSyntaxs_LTx0]: h4/DeepSyntax/LTx = h4/DeepSyntax/_20_40ind__typeDeepSyntax5
% Assm [h4s_DeepSyntaxs_UnrelatedBool0]: h4/DeepSyntax/UnrelatedBool = h4/DeepSyntax/_20_40ind__typeDeepSyntax3
% Assm [h4s_DeepSyntaxs_Negn0]: h4/DeepSyntax/Negn = h4/DeepSyntax/_20_40ind__typeDeepSyntax2
% Assm [h4s_DeepSyntaxs_Conjn0]: h4/DeepSyntax/Conjn = h4/DeepSyntax/_20_40ind__typeDeepSyntax0
% Assm [h4s_DeepSyntaxs_Disjn0]: h4/DeepSyntax/Disjn = h4/DeepSyntax/_20_40ind__typeDeepSyntax1
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_DATATYPEu_u_TAGu_u_THM]: !x. h4/bool/DATATYPE x <=> T
% Assm [h4s_bools_DATATYPEu_u_TAGu_u_DEF]: !x. h4/bool/DATATYPE x <=> T
% Assm [h4s_DeepSyntaxs_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeDeepSyntax0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x'. happ (happ h4/DeepSyntax/_20_40ind__typeDeepSyntax0 x) x' = h4/DeepSyntax/_20_40ind__typeDeepSyntax8 (h4/ind__type/CONSTR h4/num/0 (happ (happ h4/pair/_2C h4/bool/ARB) (happ (happ h4/pair/_2C h4/bool/ARB) h4/bool/ARB)) (h4/ind__type/FCONS (h4/DeepSyntax/_20_40ind__typeDeepSyntax9 x) (h4/ind__type/FCONS (h4/DeepSyntax/_20_40ind__typeDeepSyntax9 x') _0))))
% Assm [h4s_DeepSyntaxs_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeDeepSyntax5u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x. happ h4/DeepSyntax/_20_40ind__typeDeepSyntax5 x = h4/DeepSyntax/_20_40ind__typeDeepSyntax8 (h4/ind__type/CONSTR (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC h4/num/0))))) (happ (happ h4/pair/_2C h4/bool/ARB) (happ (happ h4/pair/_2C x) h4/bool/ARB)) _0))
% Assm [h4s_DeepSyntaxs_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeDeepSyntax1u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x'. happ (happ h4/DeepSyntax/_20_40ind__typeDeepSyntax1 x) x' = h4/DeepSyntax/_20_40ind__typeDeepSyntax8 (h4/ind__type/CONSTR (happ h4/num/SUC h4/num/0) (happ (happ h4/pair/_2C h4/bool/ARB) (happ (happ h4/pair/_2C h4/bool/ARB) h4/bool/ARB)) (h4/ind__type/FCONS (h4/DeepSyntax/_20_40ind__typeDeepSyntax9 x) (h4/ind__type/FCONS (h4/DeepSyntax/_20_40ind__typeDeepSyntax9 x') _0))))
% Assm [h4s_DeepSyntaxs_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeDeepSyntax6u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x. happ h4/DeepSyntax/_20_40ind__typeDeepSyntax6 x = h4/DeepSyntax/_20_40ind__typeDeepSyntax8 (h4/ind__type/CONSTR (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC h4/num/0)))))) (happ (happ h4/pair/_2C h4/bool/ARB) (happ (happ h4/pair/_2C x) h4/bool/ARB)) _0))
% Assm [h4s_DeepSyntaxs_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeDeepSyntax3u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x. happ h4/DeepSyntax/_20_40ind__typeDeepSyntax3 x = h4/DeepSyntax/_20_40ind__typeDeepSyntax8 (h4/ind__type/CONSTR (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC h4/num/0))) (happ (happ h4/pair/_2C x) (happ (happ h4/pair/_2C h4/bool/ARB) h4/bool/ARB)) _0))
% Assm [h4s_DeepSyntaxs_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeDeepSyntax4u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x. happ h4/DeepSyntax/_20_40ind__typeDeepSyntax4 x = h4/DeepSyntax/_20_40ind__typeDeepSyntax8 (h4/ind__type/CONSTR (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC h4/num/0)))) (happ (happ h4/pair/_2C h4/bool/ARB) (happ (happ h4/pair/_2C x) h4/bool/ARB)) _0))
% Assm [h4s_DeepSyntaxs_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeDeepSyntax2u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x. happ h4/DeepSyntax/_20_40ind__typeDeepSyntax2 x = h4/DeepSyntax/_20_40ind__typeDeepSyntax8 (h4/ind__type/CONSTR (happ h4/num/SUC (happ h4/num/SUC h4/num/0)) (happ (happ h4/pair/_2C h4/bool/ARB) (happ (happ h4/pair/_2C h4/bool/ARB) h4/bool/ARB)) (h4/ind__type/FCONS (h4/DeepSyntax/_20_40ind__typeDeepSyntax9 x) _0)))
% Assm [h4s_DeepSyntaxs_deepu_u_formu_u_repfnsu_c0]: !a. h4/DeepSyntax/_20_40ind__typeDeepSyntax8 (h4/DeepSyntax/_20_40ind__typeDeepSyntax9 a) = a
% Assm [h4s_DeepSyntaxs_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeDeepSyntax7u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x'. happ (happ h4/DeepSyntax/_20_40ind__typeDeepSyntax7 x) x' = h4/DeepSyntax/_20_40ind__typeDeepSyntax8 (h4/ind__type/CONSTR (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC h4/num/0))))))) (happ (happ h4/pair/_2C h4/bool/ARB) (happ (happ h4/pair/_2C x) x')) _0))
% Assm [h4s_DeepSyntaxs_deepu_u_formu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27deep__form_27. (!a0_270. (?a0 a1. a0_270 = h4/ind__type/CONSTR h4/num/0 (happ (happ h4/pair/_2C h4/bool/ARB) (happ (happ h4/pair/_2C h4/bool/ARB) h4/bool/ARB)) (h4/ind__type/FCONS a0 (h4/ind__type/FCONS a1 _0)) /\ happ _27deep__form_27 a0 /\ happ _27deep__form_27 a1) \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (happ h4/num/SUC h4/num/0) (happ (happ h4/pair/_2C h4/bool/ARB) (happ (happ h4/pair/_2C h4/bool/ARB) h4/bool/ARB)) (h4/ind__type/FCONS a0 (h4/ind__type/FCONS a1 _0)) /\ happ _27deep__form_27 a0 /\ happ _27deep__form_27 a1) \/ (?a. a0_270 = h4/ind__type/CONSTR (happ h4/num/SUC (happ h4/num/SUC h4/num/0)) (happ (happ h4/pair/_2C h4/bool/ARB) (happ (happ h4/pair/_2C h4/bool/ARB) h4/bool/ARB)) (h4/ind__type/FCONS a _0) /\ happ _27deep__form_27 a) \/ (?a. a0_270 = h4/ind__type/CONSTR (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC h4/num/0))) (happ (happ h4/pair/_2C a) (happ (happ h4/pair/_2C h4/bool/ARB) h4/bool/ARB)) _0) \/ (?a. a0_270 = h4/ind__type/CONSTR (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC h4/num/0)))) (happ (happ h4/pair/_2C h4/bool/ARB) (happ (happ h4/pair/_2C a) h4/bool/ARB)) _0) \/ (?a. a0_270 = h4/ind__type/CONSTR (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC h4/num/0))))) (happ (happ h4/pair/_2C h4/bool/ARB) (happ (happ h4/pair/_2C a) h4/bool/ARB)) _0) \/ (?a. a0_270 = h4/ind__type/CONSTR (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC h4/num/0)))))) (happ (happ h4/pair/_2C h4/bool/ARB) (happ (happ h4/pair/_2C a) h4/bool/ARB)) _0) \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC h4/num/0))))))) (happ (happ h4/pair/_2C h4/bool/ARB) (happ (happ h4/pair/_2C a0) a1)) _0) ==> happ _27deep__form_27 a0_270) ==> happ _27deep__form_27 r) <=> h4/DeepSyntax/_20_40ind__typeDeepSyntax9 (h4/DeepSyntax/_20_40ind__typeDeepSyntax8 r) = r)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_MONOu_u_AND]: !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_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_indu_u_types_FCONS0u_c0]: !f a. happ (h4/ind__type/FCONS a f) h4/num/0 = a
% Assm [h4s_bools_BETAu_u_THM]: !y f. happ f y = happ f y
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (happ (happ h4/pair/_2C x) y) = y
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (happ (happ h4/pair/_2C x) y) = x
% 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_indu_u_types_FCONS0u_c1]: !n f a. happ (h4/ind__type/FCONS a f) (happ h4/num/SUC n) = happ f n
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_DeepSyntaxs_deepu_u_formu_u_caseu_u_defu_c7]: !f7 f6 f5 f4 f3 f2 f1 f a1 a0. h4/DeepSyntax/deep__form__CASE (happ (happ h4/DeepSyntax/xDivided a0) a1) f f1 f2 f3 f4 f5 f6 f7 = happ (happ f7 a0) a1
% Assm [h4s_DeepSyntaxs_deepu_u_formu_u_caseu_u_defu_c5]: !f7 f6 f5 f4 f3 f2 f1 f a. h4/DeepSyntax/deep__form__CASE (happ h4/DeepSyntax/LTx a) f f1 f2 f3 f4 f5 f6 f7 = happ f5 a
% Assm [h4s_DeepSyntaxs_deepu_u_formu_u_sizeu_u_defu_c6]: !a. h4/DeepSyntax/deep__form__size (happ h4/DeepSyntax/xEQ a) = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm [h4s_DeepSyntaxs_deepu_u_formu_u_caseu_u_defu_c4]: !f7 f6 f5 f4 f3 f2 f1 f a. h4/DeepSyntax/deep__form__CASE (happ h4/DeepSyntax/xLT a) f f1 f2 f3 f4 f5 f6 f7 = happ f4 a
% Assm [h4s_DeepSyntaxs_deepu_u_formu_u_caseu_u_defu_c6]: !f7 f6 f5 f4 f3 f2 f1 f a. h4/DeepSyntax/deep__form__CASE (happ h4/DeepSyntax/xEQ a) f f1 f2 f3 f4 f5 f6 f7 = happ f6 a
% Assm [h4s_DeepSyntaxs_deepu_u_formu_u_caseu_u_defu_c0]: !f7 f6 f5 f4 f3 f2 f1 f a1 a0. h4/DeepSyntax/deep__form__CASE (happ (happ h4/DeepSyntax/Conjn a0) a1) f f1 f2 f3 f4 f5 f6 f7 = happ (happ f a0) a1
% Assm [h4s_DeepSyntaxs_deepu_u_formu_u_caseu_u_defu_c3]: !f7 f6 f5 f4 f3 f2 f1 f a. h4/DeepSyntax/deep__form__CASE (happ h4/DeepSyntax/UnrelatedBool a) f f1 f2 f3 f4 f5 f6 f7 = happ f3 a
% Assm [h4s_DeepSyntaxs_deepu_u_formu_u_sizeu_u_defu_c4]: !a. h4/DeepSyntax/deep__form__size (happ h4/DeepSyntax/xLT a) = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm [h4s_DeepSyntaxs_deepu_u_formu_u_sizeu_u_defu_c7]: !a1 a0. h4/DeepSyntax/deep__form__size (happ (happ h4/DeepSyntax/xDivided a0) a1) = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm [h4s_DeepSyntaxs_deepu_u_formu_u_caseu_u_defu_c1]: !f7 f6 f5 f4 f3 f2 f1 f a1 a0. h4/DeepSyntax/deep__form__CASE (happ (happ h4/DeepSyntax/Disjn a0) a1) f f1 f2 f3 f4 f5 f6 f7 = happ (happ f1 a0) a1
% Assm [h4s_DeepSyntaxs_deepu_u_formu_u_caseu_u_defu_c2]: !f7 f6 f5 f4 f3 f2 f1 f a. h4/DeepSyntax/deep__form__CASE (happ h4/DeepSyntax/Negn a) f f1 f2 f3 f4 f5 f6 f7 = happ f2 a
% Assm [h4s_DeepSyntaxs_deepu_u_formu_u_sizeu_u_defu_c5]: !a. h4/DeepSyntax/deep__form__size (happ h4/DeepSyntax/LTx a) = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm [h4s_DeepSyntaxs_deepu_u_formu_u_sizeu_u_defu_c0]: !a1 a0. h4/DeepSyntax/deep__form__size (happ (happ h4/DeepSyntax/Conjn a0) a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (h4/DeepSyntax/deep__form__size a0) (h4/DeepSyntax/deep__form__size a1))
% Assm [h4s_DeepSyntaxs_deepu_u_formu_u_sizeu_u_defu_c2]: !a. h4/DeepSyntax/deep__form__size (happ h4/DeepSyntax/Negn a) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/DeepSyntax/deep__form__size a)
% Assm [h4s_DeepSyntaxs_deepu_u_formu_u_sizeu_u_defu_c1]: !a1 a0. h4/DeepSyntax/deep__form__size (happ (happ h4/DeepSyntax/Disjn a0) a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (h4/DeepSyntax/deep__form__size a0) (h4/DeepSyntax/deep__form__size a1))
% Assm [h4s_DeepSyntaxs_deepu_u_formu_u_sizeu_u_defu_c3]: !a. h4/DeepSyntax/deep__form__size (happ h4/DeepSyntax/UnrelatedBool a) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/basicSize/bool__size a)
% Assm [h4s_arithmetics_datatypeu_u_num]: !num. h4/bool/DATATYPE (happ (happ num h4/num/0) h4/num/SUC)
% Assm [h4s_sums_datatypeu_u_sum]: !sum. h4/bool/DATATYPE (happ (happ sum h4/sum/INL) h4/sum/INR)
% Assm [h4s_pairs_datatypeu_u_pair]: !pair. h4/bool/DATATYPE (happ pair h4/pair/_2C)
% Assm [h4s_options_datatypeu_u_option]: !option. h4/bool/DATATYPE (happ (happ option h4/option/NONE) h4/option/SOME)
% Assm [h4s_lists_datatypeu_u_list]: !list. h4/bool/DATATYPE (happ (happ list h4/list/NIL) h4/list/CONS)
% Assm [h4s_bools_DATATYPEu_u_BOOL]: !bool. h4/bool/DATATYPE (happ (happ bool T) F) <=> T
% Assm [h4s_DeepSyntaxs_deepu_u_formu_u_TYu_u_DEF]: !_1. (!n. happ _1 n = h4/ind__type/BOTTOM) ==> (!_0. (!a0_27. happ _0 a0_27 <=> (!_27deep__form_27. (!a0_270. (?a0 a1. a0_270 = h4/ind__type/CONSTR h4/num/0 (happ (happ h4/pair/_2C h4/bool/ARB) (happ (happ h4/pair/_2C h4/bool/ARB) h4/bool/ARB)) (h4/ind__type/FCONS a0 (h4/ind__type/FCONS a1 _1)) /\ happ _27deep__form_27 a0 /\ happ _27deep__form_27 a1) \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (happ h4/num/SUC h4/num/0) (happ (happ h4/pair/_2C h4/bool/ARB) (happ (happ h4/pair/_2C h4/bool/ARB) h4/bool/ARB)) (h4/ind__type/FCONS a0 (h4/ind__type/FCONS a1 _1)) /\ happ _27deep__form_27 a0 /\ happ _27deep__form_27 a1) \/ (?a. a0_270 = h4/ind__type/CONSTR (happ h4/num/SUC (happ h4/num/SUC h4/num/0)) (happ (happ h4/pair/_2C h4/bool/ARB) (happ (happ h4/pair/_2C h4/bool/ARB) h4/bool/ARB)) (h4/ind__type/FCONS a _1) /\ happ _27deep__form_27 a) \/ (?a. a0_270 = h4/ind__type/CONSTR (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC h4/num/0))) (happ (happ h4/pair/_2C a) (happ (happ h4/pair/_2C h4/bool/ARB) h4/bool/ARB)) _1) \/ (?a. a0_270 = h4/ind__type/CONSTR (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC h4/num/0)))) (happ (happ h4/pair/_2C h4/bool/ARB) (happ (happ h4/pair/_2C a) h4/bool/ARB)) _1) \/ (?a. a0_270 = h4/ind__type/CONSTR (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC h4/num/0))))) (happ (happ h4/pair/_2C h4/bool/ARB) (happ (happ h4/pair/_2C a) h4/bool/ARB)) _1) \/ (?a. a0_270 = h4/ind__type/CONSTR (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC h4/num/0)))))) (happ (happ h4/pair/_2C h4/bool/ARB) (happ (happ h4/pair/_2C a) h4/bool/ARB)) _1) \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC (happ h4/num/SUC h4/num/0))))))) (happ (happ h4/pair/_2C h4/bool/ARB) (happ (happ h4/pair/_2C a0) a1)) _1) ==> happ _27deep__form_27 a0_270) ==> happ _27deep__form_27 a0_27)) ==> (?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_integers_INTu_u_LDISTRIB]: !z y x. h4/integer/int__mul x (h4/integer/int__add y z) = h4/integer/int__add (h4/integer/int__mul x y) (h4/integer/int__mul x z)
% 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_integers_TINTu_u_EQu_u_EQUIV]: !q p. happ (happ h4/integer/tint__eq p) q <=> happ h4/integer/tint__eq p = happ h4/integer/tint__eq q
% Assm [h4s_integers_intu_u_mul0]: !T2 T1. h4/integer/int__mul T1 T2 = happ h4/integer/int__ABS (h4/integer/tint__mul (happ h4/integer/int__REP T1) (happ h4/integer/int__REP T2))
% Assm [h4s_integers_intu_u_QUOTIENT]: h4/quotient/QUOTIENT h4/integer/tint__eq h4/integer/int__ABS h4/integer/int__REP
% Assm [h4s_integers_intu_u_add0]: !T2 T1. h4/integer/int__add T1 T2 = happ h4/integer/int__ABS (h4/integer/tint__add (happ h4/integer/int__REP T1) (happ h4/integer/int__REP T2))
% Assm [h4s_integers_TINTu_u_MULu_u_WELLDEF]: !y2 y1 x2 x1. happ (happ h4/integer/tint__eq x1) x2 /\ happ (happ h4/integer/tint__eq y1) y2 ==> happ (happ h4/integer/tint__eq (h4/integer/tint__mul x1 y1)) (h4/integer/tint__mul x2 y2)
% Assm [h4s_integers_TINTu_u_EQu_u_AP]: !q p. p = q ==> happ (happ h4/integer/tint__eq p) q
% Assm [h4s_quotients_EQUIVu_u_RESu_u_FORALL]: !P E. h4/quotient/EQUIV E ==> (h4/bool/RES__FORALL (h4/quotient/respects E) P <=> $forall P)
% Assm [h4s_quotients_RESu_u_FORALLu_u_RSP]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f g. happ (happ (h4/quotient/_3D_3D_3D_3E R $equals) f) g ==> (h4/bool/RES__FORALL (h4/quotient/respects R) f <=> h4/bool/RES__FORALL (h4/quotient/respects R) g))
% Assm [h4s_quotients_EQUALSu_u_RSP]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x1 x2 y1 y2. happ (happ R x1) x2 /\ happ (happ R y1) y2 ==> (happ (happ R x1) y1 <=> happ (happ R x2) y2))
% Assm [h4s_quotients_FUNu_u_QUOTIENT]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> h4/quotient/QUOTIENT (h4/quotient/_3D_3D_3D_3E R1 R2) (h4/quotient/_2D_2D_3E rep1 abs2) (h4/quotient/_2D_2D_3E abs1 rep2))
% Assm [h4s_quotients_FORALLu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f. $forall f <=> h4/bool/RES__FORALL (h4/quotient/respects R) (happ (h4/quotient/_2D_2D_3E abs h4/combin/I) f))
% Assm [h4s_quotients_EQUALSu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x y. x = y <=> happ (happ R (happ rep x)) (happ rep y))
% Assm [h4s_quotients_FUNu_u_REL]: !g f R2 R1. happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) f) g <=> (!x y. happ (happ R1 x) y ==> happ (happ R2 (happ f x)) (happ g y))
% Assm [h4s_quotients_LAMBDAu_u_PRS]: !_0. (!rep2 f abs1 x. happ (happ (happ (happ _0 rep2) f) abs1) x = happ rep2 (happ f (happ abs1 x))) ==> (!rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f x. happ f x = happ (happ (h4/quotient/_2D_2D_3E rep1 abs2) (happ (happ (happ _0 rep2) f) abs1)) x)))
% Assm [h4s_quotients_APPLYu_u_RSP]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g x y. happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) f) g /\ happ (happ R1 x) y ==> happ (happ R2 (happ f x)) (happ g y)))
% Assm [h4s_integers_TINTu_u_ADDu_u_WELLDEF]: !y2 y1 x2 x1. happ (happ h4/integer/tint__eq x1) x2 /\ happ (happ h4/integer/tint__eq y1) y2 ==> happ (happ h4/integer/tint__eq (h4/integer/tint__add x1 y1)) (h4/integer/tint__add x2 y2)
% Assm [h4s_quotients_EQUIVu_u_def]: !E. h4/quotient/EQUIV E <=> (!x y. happ (happ E x) y <=> happ E x = happ E y)
% Assm [h4s_integers_TINTu_u_LDISTRIB]: !z y x. h4/integer/tint__mul x (h4/integer/tint__add y z) = h4/integer/tint__add (h4/integer/tint__mul x y) (h4/integer/tint__mul x z)
% Assm [h4s_quotients_REPu_u_ABSu_u_RSP]: !rep abs REL. h4/quotient/QUOTIENT REL abs rep ==> (!x1 x2. happ (happ REL x1) x2 ==> happ (happ REL x1) (happ rep (happ abs x2)))
% Assm [h4s_combins_Iu_u_THM]: !x. happ h4/combin/I x = x
% Assm [h4s_quotients_IDENTITYu_u_QUOTIENT]: h4/quotient/QUOTIENT $equals h4/combin/I h4/combin/I
% Assm [h4s_integers_INTu_u_EQu_u_REDUCEu_c2]: !n. h4/integer/int__of__num h4/num/0 = h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_BIT10]: !n. h4/arithmetic/BIT1 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (happ h4/num/SUC h4/num/0))
% Assm [h4s_arithmetics_BIT20]: !n. h4/arithmetic/BIT2 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (happ h4/num/SUC (happ h4/num/SUC h4/num/0)))
% Assm [h4s_arithmetics_NUMERALu_u_DEF]: !x. h4/arithmetic/NUMERAL x = x
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_integers_INTu_u_EQu_u_CALCULATEu_c2]: !n m. h4/integer/int__of__num n = h4/integer/int__neg (h4/integer/int__of__num m) <=> n = h4/num/0 /\ m = h4/num/0
% Assm [h4s_integers_INTu_u_EQu_u_CALCULATEu_c1]: !y x. h4/integer/int__neg x = h4/integer/int__neg y <=> x = y
% Assm [h4s_integers_INTu_u_EQu_u_CALCULATEu_c0]: !n m. h4/integer/int__of__num m = h4/integer/int__of__num n <=> m = n
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm [h4s_numerals_numeralu_u_lteu_c2]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT2 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_numerals_numeralu_u_distribu_c27]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_numerals_numeralu_u_distribu_c1]: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm [h4s_arithmetics_MULTu_u_LESSu_u_EQu_u_SUC]: !p n m. h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/arithmetic/_2A (happ h4/num/SUC p) m) (h4/arithmetic/_2A (happ h4/num/SUC p) n)
% Assm [h4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm [h4s_arithmetics_EQu_u_LESSu_u_EQ]: !n m. m = n <=> h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_arithmetics_LESSu_u_EQu_u_TRANS]: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm [h4s_arithmetics_TIMES2]: !n. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n = h4/arithmetic/_2B n n
% Assm [h4s_arithmetics_MULTu_u_ASSOC]: !p n m. h4/arithmetic/_2A m (h4/arithmetic/_2A n p) = h4/arithmetic/_2A (h4/arithmetic/_2A m n) p
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c1]: !m. h4/arithmetic/_2A m h4/num/0 = h4/num/0
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c3]: !m. h4/arithmetic/_2A m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = m
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_ADDu_u_ASSOC]: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm [h4s_arithmetics_TWO]: h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO) = happ h4/num/SUC (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_arithmetics_ONE]: h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) = happ h4/num/SUC h4/num/0
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% Assm [h4s_integers_INTu_u_DIVu_u_REDUCEu_c6]: !n m. ?v. (v <=> h4/arithmetic/MOD (h4/arithmetic/NUMERAL m) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) = h4/num/0) /\ h4/integer/int__div (h4/integer/int__of__num (h4/arithmetic/NUMERAL m)) (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)))) = h4/integer/int__add (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/DIV (h4/arithmetic/NUMERAL m) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))))) (h4/bool/COND v (h4/integer/int__of__num h4/num/0) (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))))
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_integers_INTu_u_DIVu_u_NEG]: !q p. ~(q = h4/integer/int__of__num h4/num/0) ==> h4/integer/int__div p (h4/integer/int__neg q) = h4/integer/int__div (h4/integer/int__neg p) q
% Assm [h4s_integers_INTu_u_DIV]: !n m. ~(m = h4/num/0) ==> h4/integer/int__div (h4/integer/int__of__num n) (h4/integer/int__of__num m) = h4/integer/int__of__num (h4/arithmetic/DIV n m)
% Assm [h4s_integers_intu_u_div0]: !j i. ~(j = h4/integer/int__of__num h4/num/0) ==> (?v. (v <=> h4/arithmetic/MOD (h4/integer/Num i) (h4/integer/Num (h4/integer/int__neg j)) = h4/num/0) /\ (?v'. (v' <=> h4/arithmetic/MOD (h4/integer/Num (h4/integer/int__neg i)) (h4/integer/Num j) = h4/num/0) /\ h4/integer/int__div i j = h4/bool/COND (h4/integer/int__lt (h4/integer/int__of__num h4/num/0) j) (h4/bool/COND (h4/integer/int__le (h4/integer/int__of__num h4/num/0) i) (h4/integer/int__of__num (h4/arithmetic/DIV (h4/integer/Num i) (h4/integer/Num j))) (h4/integer/int__add (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/DIV (h4/integer/Num (h4/integer/int__neg i)) (h4/integer/Num j)))) (h4/bool/COND v' (h4/integer/int__of__num h4/num/0) (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))))) (h4/bool/COND (h4/integer/int__le (h4/integer/int__of__num h4/num/0) i) (h4/integer/int__add (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/DIV (h4/integer/Num i) (h4/integer/Num (h4/integer/int__neg j))))) (h4/bool/COND v (h4/integer/int__of__num h4/num/0) (h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))))) (h4/integer/int__of__num (h4/arithmetic/DIV (h4/integer/Num (h4/integer/int__neg i)) (h4/integer/Num (h4/integer/int__neg j)))))))
% Assm [h4s_integers_NUMu_u_OFu_u_INT]: !n. h4/integer/Num (h4/integer/int__of__num n) = n
% Assm [h4s_integers_INTu_u_LE]: !n m. h4/integer/int__le (h4/integer/int__of__num m) (h4/integer/int__of__num n) <=> h4/arithmetic/_3C_3D m n
% Goal: !deep__form. h4/bool/DATATYPE (happ (happ (happ (happ (happ (happ (happ (happ deep__form h4/DeepSyntax/Conjn) h4/DeepSyntax/Disjn) h4/DeepSyntax/Negn) h4/DeepSyntax/UnrelatedBool) h4/DeepSyntax/xLT) h4/DeepSyntax/LTx) h4/DeepSyntax/xEQ) h4/DeepSyntax/xDivided)
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_Q1445659,TV_Q1445655]: ![V_f, V_g]: (![V_x]: s(TV_Q1445655,happ(s(t_fun(TV_Q1445659,TV_Q1445655),V_f),s(TV_Q1445659,V_x))) = s(TV_Q1445655,happ(s(t_fun(TV_Q1445659,TV_Q1445655),V_g),s(TV_Q1445659,V_x))) => s(t_fun(TV_Q1445659,TV_Q1445655),V_f) = s(t_fun(TV_Q1445659,TV_Q1445655),V_g))).
fof(ah4s_DeepSyntaxs_xDivided0, axiom, s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form)),h4s_deepsyntaxs_xdivided) = s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form)),h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax7)).
fof(ah4s_DeepSyntaxs_xEQ0, axiom, s(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_xeq) = s(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax6)).
fof(ah4s_DeepSyntaxs_xLT0, axiom, s(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_xlt) = s(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax4)).
fof(ah4s_DeepSyntaxs_LTx0, axiom, s(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_ltx) = s(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax5)).
fof(ah4s_DeepSyntaxs_UnrelatedBool0, axiom, s(t_fun(t_bool,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_unrelatedbool) = s(t_fun(t_bool,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax3)).
fof(ah4s_DeepSyntaxs_Negn0, axiom, s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_negn) = s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax2)).
fof(ah4s_DeepSyntaxs_Conjn0, axiom, s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form)),h4s_deepsyntaxs_conjn) = s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form)),h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax0)).
fof(ah4s_DeepSyntaxs_Disjn0, axiom, s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form)),h4s_deepsyntaxs_disjn) = s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form)),h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax1)).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
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_bools_DATATYPEu_u_TAGu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,h4s_bools_datatype(s(TV_u_27a,V_x))) = s(t_bool,t)).
fof(ah4s_DeepSyntaxs_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeDeepSyntax0u_u_u_u_def, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_bottom) => ![V_x, V_xi_]: s(t_h4s_DeepSyntaxs_deepu_u_form,happ(s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form),happ(s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form)),h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax0),s(t_h4s_DeepSyntaxs_deepu_u_form,V_x))),s(t_h4s_DeepSyntaxs_deepu_u_form,V_xi_))) = s(t_h4s_DeepSyntaxs_deepu_u_form,h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax8(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_bool,t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_pairs_u_2c),s(t_bool,h4s_bools_arb))),s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_pairs_u_2c),s(t_h4s_integers_int,h4s_bools_arb))),s(t_h4s_integers_int,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax9(s(t_h4s_DeepSyntaxs_deepu_u_form,V_x))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax9(s(t_h4s_DeepSyntaxs_deepu_u_form,V_xi_))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_0))))))))))).
fof(ah4s_DeepSyntaxs_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeDeepSyntax5u_u_u_u_def, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_bottom) => ![V_x]: s(t_h4s_DeepSyntaxs_deepu_u_form,happ(s(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax5),s(t_h4s_integers_int,V_x))) = s(t_h4s_DeepSyntaxs_deepu_u_form,h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax8(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_constr(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_nums_0))))))))))),s(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_bool,t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_pairs_u_2c),s(t_bool,h4s_bools_arb))),s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_pairs_u_2c),s(t_h4s_integers_int,V_x))),s(t_h4s_integers_int,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_0))))))).
fof(ah4s_DeepSyntaxs_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeDeepSyntax1u_u_u_u_def, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_bottom) => ![V_x, V_xi_]: s(t_h4s_DeepSyntaxs_deepu_u_form,happ(s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form),happ(s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form)),h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax1),s(t_h4s_DeepSyntaxs_deepu_u_form,V_x))),s(t_h4s_DeepSyntaxs_deepu_u_form,V_xi_))) = s(t_h4s_DeepSyntaxs_deepu_u_form,h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax8(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_constr(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_bool,t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_pairs_u_2c),s(t_bool,h4s_bools_arb))),s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_pairs_u_2c),s(t_h4s_integers_int,h4s_bools_arb))),s(t_h4s_integers_int,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax9(s(t_h4s_DeepSyntaxs_deepu_u_form,V_x))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax9(s(t_h4s_DeepSyntaxs_deepu_u_form,V_xi_))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_0))))))))))).
fof(ah4s_DeepSyntaxs_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeDeepSyntax6u_u_u_u_def, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_bottom) => ![V_x]: s(t_h4s_DeepSyntaxs_deepu_u_form,happ(s(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax6),s(t_h4s_integers_int,V_x))) = s(t_h4s_DeepSyntaxs_deepu_u_form,h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax8(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_constr(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_nums_0))))))))))))),s(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_bool,t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_pairs_u_2c),s(t_bool,h4s_bools_arb))),s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_pairs_u_2c),s(t_h4s_integers_int,V_x))),s(t_h4s_integers_int,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_0))))))).
fof(ah4s_DeepSyntaxs_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeDeepSyntax3u_u_u_u_def, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_bottom) => ![V_x]: s(t_h4s_DeepSyntaxs_deepu_u_form,happ(s(t_fun(t_bool,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax3),s(t_bool,V_x))) = s(t_h4s_DeepSyntaxs_deepu_u_form,h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax8(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_constr(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_nums_0))))))),s(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_bool,t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_pairs_u_2c),s(t_bool,V_x))),s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_pairs_u_2c),s(t_h4s_integers_int,h4s_bools_arb))),s(t_h4s_integers_int,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_0))))))).
fof(ah4s_DeepSyntaxs_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeDeepSyntax4u_u_u_u_def, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_bottom) => ![V_x]: s(t_h4s_DeepSyntaxs_deepu_u_form,happ(s(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax4),s(t_h4s_integers_int,V_x))) = s(t_h4s_DeepSyntaxs_deepu_u_form,h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax8(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_constr(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_nums_0))))))))),s(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_bool,t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_pairs_u_2c),s(t_bool,h4s_bools_arb))),s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_pairs_u_2c),s(t_h4s_integers_int,V_x))),s(t_h4s_integers_int,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_0))))))).
fof(ah4s_DeepSyntaxs_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeDeepSyntax2u_u_u_u_def, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_bottom) => ![V_x]: s(t_h4s_DeepSyntaxs_deepu_u_form,happ(s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax2),s(t_h4s_DeepSyntaxs_deepu_u_form,V_x))) = s(t_h4s_DeepSyntaxs_deepu_u_form,h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax8(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_constr(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_bool,t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_pairs_u_2c),s(t_bool,h4s_bools_arb))),s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_pairs_u_2c),s(t_h4s_integers_int,h4s_bools_arb))),s(t_h4s_integers_int,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax9(s(t_h4s_DeepSyntaxs_deepu_u_form,V_x))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_0))))))))).
fof(ah4s_DeepSyntaxs_deepu_u_formu_u_repfnsu_c0, axiom, ![V_a]: s(t_h4s_DeepSyntaxs_deepu_u_form,h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax8(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax9(s(t_h4s_DeepSyntaxs_deepu_u_form,V_a))))) = s(t_h4s_DeepSyntaxs_deepu_u_form,V_a)).
fof(ah4s_DeepSyntaxs_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeDeepSyntax7u_u_u_u_def, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_bottom) => ![V_x, V_xi_]: s(t_h4s_DeepSyntaxs_deepu_u_form,happ(s(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form)),h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax7),s(t_h4s_integers_int,V_x))),s(t_h4s_integers_int,V_xi_))) = s(t_h4s_DeepSyntaxs_deepu_u_form,h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax8(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_constr(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_nums_0))))))))))))))),s(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_bool,t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_pairs_u_2c),s(t_bool,h4s_bools_arb))),s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_pairs_u_2c),s(t_h4s_integers_int,V_x))),s(t_h4s_integers_int,V_xi_))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_0))))))).
fof(ah4s_DeepSyntaxs_deepu_u_formu_u_repfnsu_c1, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27deepu_u_formu_27]: (![V_a0u_270]: ((?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_bool,t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_pairs_u_2c),s(t_bool,h4s_bools_arb))),s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_pairs_u_2c),s(t_h4s_integers_int,h4s_bools_arb))),s(t_h4s_integers_int,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_0))))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),t_bool),V_uu_27deepu_u_formu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0)))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),t_bool),V_uu_27deepu_u_formu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a1)))))) | (?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_constr(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_bool,t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_pairs_u_2c),s(t_bool,h4s_bools_arb))),s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_pairs_u_2c),s(t_h4s_integers_int,h4s_bools_arb))),s(t_h4s_integers_int,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_0))))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),t_bool),V_uu_27deepu_u_formu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0)))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),t_bool),V_uu_27deepu_u_formu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a1)))))) | (?[V_a]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_constr(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_bool,t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_pairs_u_2c),s(t_bool,h4s_bools_arb))),s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_pairs_u_2c),s(t_h4s_integers_int,h4s_bools_arb))),s(t_h4s_integers_int,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_0))))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),t_bool),V_uu_27deepu_u_formu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a))))) | (?[V_a]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_constr(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_nums_0))))))),s(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_bool,t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_pairs_u_2c),s(t_bool,V_a))),s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_pairs_u_2c),s(t_h4s_integers_int,h4s_bools_arb))),s(t_h4s_integers_int,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_0))) | (?[V_a]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_constr(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_nums_0))))))))),s(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_bool,t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_pairs_u_2c),s(t_bool,h4s_bools_arb))),s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_pairs_u_2c),s(t_h4s_integers_int,V_a))),s(t_h4s_integers_int,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_0))) | (?[V_a]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_constr(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_nums_0))))))))))),s(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_bool,t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_pairs_u_2c),s(t_bool,h4s_bools_arb))),s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_pairs_u_2c),s(t_h4s_integers_int,V_a))),s(t_h4s_integers_int,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_0))) | (?[V_a]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_constr(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_nums_0))))))))))))),s(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_bool,t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_pairs_u_2c),s(t_bool,h4s_bools_arb))),s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_pairs_u_2c),s(t_h4s_integers_int,V_a))),s(t_h4s_integers_int,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_0))) | ?[V_a0, V_a1]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_constr(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_nums_0))))))))))))))),s(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_bool,t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_pairs_u_2c),s(t_bool,h4s_bools_arb))),s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_pairs_u_2c),s(t_h4s_integers_int,V_a0))),s(t_h4s_integers_int,V_a1))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_0)))))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),t_bool),V_uu_27deepu_u_formu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),t_bool),V_uu_27deepu_u_formu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_r))))) <=> s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax9(s(t_h4s_DeepSyntaxs_deepu_u_form,h4s_deepsyntaxs_u_20u_40indu_u_typedeepsyntax8(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_r))))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_r)))).
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_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_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_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_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_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_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
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_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_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),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),h4s_pairs_u_2c),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))) = s(TV_u_27b,V_y)).
fof(ah4s_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),h4s_pairs_u_2c),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_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_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,happ(s(t_fun(t_h4s_nums_num,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_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_DeepSyntaxs_deepu_u_formu_u_caseu_u_defu_c7, axiom, ![TV_u_27a]: ![V_f7, V_f6, V_f5, V_f4, V_f3, V_f2, V_f1, V_f, V_a1, V_a0]: s(TV_u_27a,h4s_deepsyntaxs_deepu_u_formu_u_case(s(t_h4s_DeepSyntaxs_deepu_u_form,happ(s(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form)),h4s_deepsyntaxs_xdivided),s(t_h4s_integers_int,V_a0))),s(t_h4s_integers_int,V_a1))),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a)),V_f),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a)),V_f1),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a),V_f2),s(t_fun(t_bool,TV_u_27a),V_f3),s(t_fun(t_h4s_integers_int,TV_u_27a),V_f4),s(t_fun(t_h4s_integers_int,TV_u_27a),V_f5),s(t_fun(t_h4s_integers_int,TV_u_27a),V_f6),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,TV_u_27a)),V_f7))) = s(TV_u_27a,happ(s(t_fun(t_h4s_integers_int,TV_u_27a),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,TV_u_27a)),V_f7),s(t_h4s_integers_int,V_a0))),s(t_h4s_integers_int,V_a1)))).
fof(ah4s_DeepSyntaxs_deepu_u_formu_u_caseu_u_defu_c5, axiom, ![TV_u_27a]: ![V_f7, V_f6, V_f5, V_f4, V_f3, V_f2, V_f1, V_f, V_a]: s(TV_u_27a,h4s_deepsyntaxs_deepu_u_formu_u_case(s(t_h4s_DeepSyntaxs_deepu_u_form,happ(s(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_ltx),s(t_h4s_integers_int,V_a))),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a)),V_f),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a)),V_f1),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a),V_f2),s(t_fun(t_bool,TV_u_27a),V_f3),s(t_fun(t_h4s_integers_int,TV_u_27a),V_f4),s(t_fun(t_h4s_integers_int,TV_u_27a),V_f5),s(t_fun(t_h4s_integers_int,TV_u_27a),V_f6),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,TV_u_27a)),V_f7))) = s(TV_u_27a,happ(s(t_fun(t_h4s_integers_int,TV_u_27a),V_f5),s(t_h4s_integers_int,V_a)))).
fof(ah4s_DeepSyntaxs_deepu_u_formu_u_sizeu_u_defu_c6, axiom, ![V_a]: s(t_h4s_nums_num,h4s_deepsyntaxs_deepu_u_formu_u_size(s(t_h4s_DeepSyntaxs_deepu_u_form,happ(s(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_xeq),s(t_h4s_integers_int,V_a))))) = 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_DeepSyntaxs_deepu_u_formu_u_caseu_u_defu_c4, axiom, ![TV_u_27a]: ![V_f7, V_f6, V_f5, V_f4, V_f3, V_f2, V_f1, V_f, V_a]: s(TV_u_27a,h4s_deepsyntaxs_deepu_u_formu_u_case(s(t_h4s_DeepSyntaxs_deepu_u_form,happ(s(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_xlt),s(t_h4s_integers_int,V_a))),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a)),V_f),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a)),V_f1),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a),V_f2),s(t_fun(t_bool,TV_u_27a),V_f3),s(t_fun(t_h4s_integers_int,TV_u_27a),V_f4),s(t_fun(t_h4s_integers_int,TV_u_27a),V_f5),s(t_fun(t_h4s_integers_int,TV_u_27a),V_f6),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,TV_u_27a)),V_f7))) = s(TV_u_27a,happ(s(t_fun(t_h4s_integers_int,TV_u_27a),V_f4),s(t_h4s_integers_int,V_a)))).
fof(ah4s_DeepSyntaxs_deepu_u_formu_u_caseu_u_defu_c6, axiom, ![TV_u_27a]: ![V_f7, V_f6, V_f5, V_f4, V_f3, V_f2, V_f1, V_f, V_a]: s(TV_u_27a,h4s_deepsyntaxs_deepu_u_formu_u_case(s(t_h4s_DeepSyntaxs_deepu_u_form,happ(s(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_xeq),s(t_h4s_integers_int,V_a))),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a)),V_f),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a)),V_f1),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a),V_f2),s(t_fun(t_bool,TV_u_27a),V_f3),s(t_fun(t_h4s_integers_int,TV_u_27a),V_f4),s(t_fun(t_h4s_integers_int,TV_u_27a),V_f5),s(t_fun(t_h4s_integers_int,TV_u_27a),V_f6),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,TV_u_27a)),V_f7))) = s(TV_u_27a,happ(s(t_fun(t_h4s_integers_int,TV_u_27a),V_f6),s(t_h4s_integers_int,V_a)))).
fof(ah4s_DeepSyntaxs_deepu_u_formu_u_caseu_u_defu_c0, axiom, ![TV_u_27a]: ![V_f7, V_f6, V_f5, V_f4, V_f3, V_f2, V_f1, V_f, V_a1, V_a0]: s(TV_u_27a,h4s_deepsyntaxs_deepu_u_formu_u_case(s(t_h4s_DeepSyntaxs_deepu_u_form,happ(s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form),happ(s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form)),h4s_deepsyntaxs_conjn),s(t_h4s_DeepSyntaxs_deepu_u_form,V_a0))),s(t_h4s_DeepSyntaxs_deepu_u_form,V_a1))),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a)),V_f),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a)),V_f1),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a),V_f2),s(t_fun(t_bool,TV_u_27a),V_f3),s(t_fun(t_h4s_integers_int,TV_u_27a),V_f4),s(t_fun(t_h4s_integers_int,TV_u_27a),V_f5),s(t_fun(t_h4s_integers_int,TV_u_27a),V_f6),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,TV_u_27a)),V_f7))) = s(TV_u_27a,happ(s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a),happ(s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a)),V_f),s(t_h4s_DeepSyntaxs_deepu_u_form,V_a0))),s(t_h4s_DeepSyntaxs_deepu_u_form,V_a1)))).
fof(ah4s_DeepSyntaxs_deepu_u_formu_u_caseu_u_defu_c3, axiom, ![TV_u_27a]: ![V_f7, V_f6, V_f5, V_f4, V_f3, V_f2, V_f1, V_f, V_a]: s(TV_u_27a,h4s_deepsyntaxs_deepu_u_formu_u_case(s(t_h4s_DeepSyntaxs_deepu_u_form,happ(s(t_fun(t_bool,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_unrelatedbool),s(t_bool,V_a))),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a)),V_f),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a)),V_f1),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a),V_f2),s(t_fun(t_bool,TV_u_27a),V_f3),s(t_fun(t_h4s_integers_int,TV_u_27a),V_f4),s(t_fun(t_h4s_integers_int,TV_u_27a),V_f5),s(t_fun(t_h4s_integers_int,TV_u_27a),V_f6),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,TV_u_27a)),V_f7))) = s(TV_u_27a,happ(s(t_fun(t_bool,TV_u_27a),V_f3),s(t_bool,V_a)))).
fof(ah4s_DeepSyntaxs_deepu_u_formu_u_sizeu_u_defu_c4, axiom, ![V_a]: s(t_h4s_nums_num,h4s_deepsyntaxs_deepu_u_formu_u_size(s(t_h4s_DeepSyntaxs_deepu_u_form,happ(s(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_xlt),s(t_h4s_integers_int,V_a))))) = 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_DeepSyntaxs_deepu_u_formu_u_sizeu_u_defu_c7, axiom, ![V_a1, V_a0]: s(t_h4s_nums_num,h4s_deepsyntaxs_deepu_u_formu_u_size(s(t_h4s_DeepSyntaxs_deepu_u_form,happ(s(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form)),h4s_deepsyntaxs_xdivided),s(t_h4s_integers_int,V_a0))),s(t_h4s_integers_int,V_a1))))) = 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_DeepSyntaxs_deepu_u_formu_u_caseu_u_defu_c1, axiom, ![TV_u_27a]: ![V_f7, V_f6, V_f5, V_f4, V_f3, V_f2, V_f1, V_f, V_a1, V_a0]: s(TV_u_27a,h4s_deepsyntaxs_deepu_u_formu_u_case(s(t_h4s_DeepSyntaxs_deepu_u_form,happ(s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form),happ(s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form)),h4s_deepsyntaxs_disjn),s(t_h4s_DeepSyntaxs_deepu_u_form,V_a0))),s(t_h4s_DeepSyntaxs_deepu_u_form,V_a1))),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a)),V_f),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a)),V_f1),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a),V_f2),s(t_fun(t_bool,TV_u_27a),V_f3),s(t_fun(t_h4s_integers_int,TV_u_27a),V_f4),s(t_fun(t_h4s_integers_int,TV_u_27a),V_f5),s(t_fun(t_h4s_integers_int,TV_u_27a),V_f6),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,TV_u_27a)),V_f7))) = s(TV_u_27a,happ(s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a),happ(s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a)),V_f1),s(t_h4s_DeepSyntaxs_deepu_u_form,V_a0))),s(t_h4s_DeepSyntaxs_deepu_u_form,V_a1)))).
fof(ah4s_DeepSyntaxs_deepu_u_formu_u_caseu_u_defu_c2, axiom, ![TV_u_27a]: ![V_f7, V_f6, V_f5, V_f4, V_f3, V_f2, V_f1, V_f, V_a]: s(TV_u_27a,h4s_deepsyntaxs_deepu_u_formu_u_case(s(t_h4s_DeepSyntaxs_deepu_u_form,happ(s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_negn),s(t_h4s_DeepSyntaxs_deepu_u_form,V_a))),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a)),V_f),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a)),V_f1),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a),V_f2),s(t_fun(t_bool,TV_u_27a),V_f3),s(t_fun(t_h4s_integers_int,TV_u_27a),V_f4),s(t_fun(t_h4s_integers_int,TV_u_27a),V_f5),s(t_fun(t_h4s_integers_int,TV_u_27a),V_f6),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,TV_u_27a)),V_f7))) = s(TV_u_27a,happ(s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,TV_u_27a),V_f2),s(t_h4s_DeepSyntaxs_deepu_u_form,V_a)))).
fof(ah4s_DeepSyntaxs_deepu_u_formu_u_sizeu_u_defu_c5, axiom, ![V_a]: s(t_h4s_nums_num,h4s_deepsyntaxs_deepu_u_formu_u_size(s(t_h4s_DeepSyntaxs_deepu_u_form,happ(s(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_ltx),s(t_h4s_integers_int,V_a))))) = 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_DeepSyntaxs_deepu_u_formu_u_sizeu_u_defu_c0, axiom, ![V_a1, V_a0]: s(t_h4s_nums_num,h4s_deepsyntaxs_deepu_u_formu_u_size(s(t_h4s_DeepSyntaxs_deepu_u_form,happ(s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form),happ(s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form)),h4s_deepsyntaxs_conjn),s(t_h4s_DeepSyntaxs_deepu_u_form,V_a0))),s(t_h4s_DeepSyntaxs_deepu_u_form,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,h4s_deepsyntaxs_deepu_u_formu_u_size(s(t_h4s_DeepSyntaxs_deepu_u_form,V_a0))),s(t_h4s_nums_num,h4s_deepsyntaxs_deepu_u_formu_u_size(s(t_h4s_DeepSyntaxs_deepu_u_form,V_a1)))))))).
fof(ah4s_DeepSyntaxs_deepu_u_formu_u_sizeu_u_defu_c2, axiom, ![V_a]: s(t_h4s_nums_num,h4s_deepsyntaxs_deepu_u_formu_u_size(s(t_h4s_DeepSyntaxs_deepu_u_form,happ(s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_negn),s(t_h4s_DeepSyntaxs_deepu_u_form,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_deepsyntaxs_deepu_u_formu_u_size(s(t_h4s_DeepSyntaxs_deepu_u_form,V_a)))))).
fof(ah4s_DeepSyntaxs_deepu_u_formu_u_sizeu_u_defu_c1, axiom, ![V_a1, V_a0]: s(t_h4s_nums_num,h4s_deepsyntaxs_deepu_u_formu_u_size(s(t_h4s_DeepSyntaxs_deepu_u_form,happ(s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form),happ(s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form)),h4s_deepsyntaxs_disjn),s(t_h4s_DeepSyntaxs_deepu_u_form,V_a0))),s(t_h4s_DeepSyntaxs_deepu_u_form,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,h4s_deepsyntaxs_deepu_u_formu_u_size(s(t_h4s_DeepSyntaxs_deepu_u_form,V_a0))),s(t_h4s_nums_num,h4s_deepsyntaxs_deepu_u_formu_u_size(s(t_h4s_DeepSyntaxs_deepu_u_form,V_a1)))))))).
fof(ah4s_DeepSyntaxs_deepu_u_formu_u_sizeu_u_defu_c3, axiom, ![V_a]: s(t_h4s_nums_num,h4s_deepsyntaxs_deepu_u_formu_u_size(s(t_h4s_DeepSyntaxs_deepu_u_form,happ(s(t_fun(t_bool,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_unrelatedbool),s(t_bool,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_basicsizes_boolu_u_size(s(t_bool,V_a)))))).
fof(ah4s_arithmetics_datatypeu_u_num, axiom, ![TV_u_27a]: ![V_num]: p(s(t_bool,h4s_bools_datatype(s(TV_u_27a,happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_nums_num),TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_h4s_nums_num),TV_u_27a)),V_num),s(t_h4s_nums_num,h4s_nums_0))),s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc))))))).
fof(ah4s_sums_datatypeu_u_sum, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_sum]: p(s(t_bool,h4s_bools_datatype(s(TV_u_27c,happ(s(t_fun(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),t_fun(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),TV_u_27c)),V_sum),s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl))),s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr))))))).
fof(ah4s_pairs_datatypeu_u_pair, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_pair]: p(s(t_bool,h4s_bools_datatype(s(TV_u_27c,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),TV_u_27c),V_pair),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),h4s_pairs_u_2c))))))).
fof(ah4s_options_datatypeu_u_option, axiom, ![TV_u_27b,TV_u_27a]: ![V_option]: p(s(t_bool,h4s_bools_datatype(s(TV_u_27b,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),TV_u_27b),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_fun(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),TV_u_27b)),V_option),s(t_h4s_options_option(TV_u_27a),h4s_options_none))),s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some))))))).
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_bools_DATATYPEu_u_BOOL, axiom, ![V_bool]: s(t_bool,h4s_bools_datatype(s(t_bool,happ(s(t_fun(t_bool,t_bool),happ(s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_bool),s(t_bool,t))),s(t_bool,f))))) = s(t_bool,t)).
fof(ah4s_DeepSyntaxs_deepu_u_formu_u_TYu_u_DEF, axiom, ![V_uu_1]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_1),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),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(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),t_bool),V_uu_0),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0u_27)))) <=> ![V_uu_27deepu_u_formu_27]: (![V_a0u_270]: ((?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_bool,t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_pairs_u_2c),s(t_bool,h4s_bools_arb))),s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_pairs_u_2c),s(t_h4s_integers_int,h4s_bools_arb))),s(t_h4s_integers_int,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_1))))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),t_bool),V_uu_27deepu_u_formu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0)))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),t_bool),V_uu_27deepu_u_formu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a1)))))) | (?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_constr(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_bool,t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_pairs_u_2c),s(t_bool,h4s_bools_arb))),s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_pairs_u_2c),s(t_h4s_integers_int,h4s_bools_arb))),s(t_h4s_integers_int,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_1))))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),t_bool),V_uu_27deepu_u_formu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0)))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),t_bool),V_uu_27deepu_u_formu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a1)))))) | (?[V_a]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_constr(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_bool,t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_pairs_u_2c),s(t_bool,h4s_bools_arb))),s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_pairs_u_2c),s(t_h4s_integers_int,h4s_bools_arb))),s(t_h4s_integers_int,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_1))))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),t_bool),V_uu_27deepu_u_formu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a))))) | (?[V_a]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_constr(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_nums_0))))))),s(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_bool,t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_pairs_u_2c),s(t_bool,V_a))),s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_pairs_u_2c),s(t_h4s_integers_int,h4s_bools_arb))),s(t_h4s_integers_int,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_1))) | (?[V_a]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_constr(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_nums_0))))))))),s(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_bool,t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_pairs_u_2c),s(t_bool,h4s_bools_arb))),s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_pairs_u_2c),s(t_h4s_integers_int,V_a))),s(t_h4s_integers_int,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_1))) | (?[V_a]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_constr(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_nums_0))))))))))),s(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_bool,t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_pairs_u_2c),s(t_bool,h4s_bools_arb))),s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_pairs_u_2c),s(t_h4s_integers_int,V_a))),s(t_h4s_integers_int,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_1))) | (?[V_a]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_constr(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_nums_0))))))))))))),s(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_bool,t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_pairs_u_2c),s(t_bool,h4s_bools_arb))),s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_pairs_u_2c),s(t_h4s_integers_int,V_a))),s(t_h4s_integers_int,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_1))) | ?[V_a0, V_a1]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_indu_u_types_constr(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_nums_0))))))))))))))),s(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),happ(s(t_fun(t_bool,t_fun(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),h4s_pairs_u_2c),s(t_bool,h4s_bools_arb))),s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),h4s_pairs_u_2c),s(t_h4s_integers_int,V_a0))),s(t_h4s_integers_int,V_a1))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),V_uu_1)))))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),t_bool),V_uu_27deepu_u_formu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),t_bool),V_uu_27deepu_u_formu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),V_a0u_27)))))) => ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int))),t_bool),V_uu_0),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_bool,t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int)))),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_integers_INTu_u_LDISTRIB, axiom, ![V_z, V_y, V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,V_y),s(t_h4s_integers_int,V_z))))) = s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y))),s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_z)))))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_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_integers_TINTu_u_EQu_u_EQUIV, axiom, ![V_q, V_p]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q)))) <=> s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p))) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q))))).
fof(ah4s_integers_intu_u_mul0, axiom, ![V_T2, V_T1]: s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_T1),s(t_h4s_integers_int,V_T2))) = s(t_h4s_integers_int,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_integers_int),h4s_integers_intu_u_abs),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_integers_intu_u_rep),s(t_h4s_integers_int,V_T1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_integers_intu_u_rep),s(t_h4s_integers_int,V_T2)))))))).
fof(ah4s_integers_intu_u_QUOTIENT, axiom, p(s(t_bool,h4s_quotients_quotient(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_integers_int),h4s_integers_intu_u_abs),s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_integers_intu_u_rep))))).
fof(ah4s_integers_intu_u_add0, axiom, ![V_T2, V_T1]: s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,V_T1),s(t_h4s_integers_int,V_T2))) = s(t_h4s_integers_int,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_integers_int),h4s_integers_intu_u_abs),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_integers_intu_u_rep),s(t_h4s_integers_int,V_T1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_integers_intu_u_rep),s(t_h4s_integers_int,V_T2)))))))).
fof(ah4s_integers_TINTu_u_MULu_u_WELLDEF, axiom, ![V_y2, V_y1, V_x2, V_x1]: ((p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x2)))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y2))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x1),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y1))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x2),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y2)))))))).
fof(ah4s_integers_TINTu_u_EQu_u_AP, axiom, ![V_q, V_p]: (s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q)))))).
fof(ah4s_quotients_EQUIVu_u_RESu_u_FORALL, axiom, ![TV_u_27a]: ![V_P, V_E]: (p(s(t_bool,h4s_quotients_equiv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_E)))) => s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),h4s_quotients_respects(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_E))),s(t_fun(TV_u_27a,t_bool),V_P))) = s(t_bool,d_forall(s(t_fun(TV_u_27a,t_bool),V_P))))).
fof(ah4s_quotients_RESu_u_FORALLu_u_RSP, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_f, V_g]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_equals))),s(t_fun(TV_u_27a,t_bool),V_f))),s(t_fun(TV_u_27a,t_bool),V_g)))) => s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),h4s_quotients_respects(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),V_f))) = s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),h4s_quotients_respects(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),V_g)))))).
fof(ah4s_quotients_EQUALSu_u_RSP, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_x1, V_x2, V_y1, V_y2]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x1))),s(TV_u_27a,V_x2)))) & 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_y1))),s(TV_u_27a,V_y2))))) => 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_x1))),s(TV_u_27a,V_y1))) = 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_x2))),s(TV_u_27a,V_y2)))))).
fof(ah4s_quotients_FUNu_u_QUOTIENT, axiom, ![TV_u_27a,TV_u_27c,TV_u_27d,TV_u_27b]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => p(s(t_bool,h4s_quotients_quotient(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))))))))).
fof(ah4s_quotients_FORALLu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_f]: s(t_bool,d_forall(s(t_fun(TV_u_27b,t_bool),V_f))) = s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),h4s_quotients_respects(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_bool)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(t_bool,t_bool),h4s_combins_i))),s(t_fun(TV_u_27b,t_bool),V_f))))))).
fof(ah4s_quotients_EQUALSu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_x, V_y]: (s(TV_u_27b,V_x) = s(TV_u_27b,V_y) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_x))))),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_y))))))))).
fof(ah4s_quotients_FUNu_u_REL, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f, V_R2, V_R1]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,TV_u_27b),V_g)))) <=> ![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_y))))))))).
fof(ah4s_quotients_LAMBDAu_u_PRS, axiom, ![TV_u_27b,TV_u_27d,TV_u_27a,TV_u_27c]: ![V_uu_0]: (![V_rep2, V_f, V_abs1, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))),s(t_fun(TV_u_27c,TV_u_27d),V_f))),s(t_fun(TV_u_27a,TV_u_27c),V_abs1))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27d,TV_u_27b),V_rep2),s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(TV_u_27a,V_x))))))) => ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_x]: s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_f),s(TV_u_27c,V_x))) = s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))),s(t_fun(TV_u_27c,TV_u_27d),V_f))),s(t_fun(TV_u_27a,TV_u_27c),V_abs1))))),s(TV_u_27c,V_x))))))).
fof(ah4s_quotients_APPLYu_u_RSP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27b,TV_u_27a]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_g, V_x, V_y]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,TV_u_27b),V_g)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_y)))))))))).
fof(ah4s_integers_TINTu_u_ADDu_u_WELLDEF, axiom, ![V_y2, V_y1, V_x2, V_x1]: ((p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x2)))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y2))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x1),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y1))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x2),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y2)))))))).
fof(ah4s_quotients_EQUIVu_u_def, axiom, ![TV_u_27a]: ![V_E]: (p(s(t_bool,h4s_quotients_equiv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_E)))) <=> ![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_E),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) <=> s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_E),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_E),s(TV_u_27a,V_y)))))).
fof(ah4s_integers_TINTu_u_LDISTRIB, axiom, ![V_z, V_y, V_x]: s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_z))))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_z)))))).
fof(ah4s_quotients_REPu_u_ABSu_u_RSP, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_REL]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_REL),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_x1, V_x2]: (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_REL),s(TV_u_27a,V_x1))),s(TV_u_27a,V_x2)))) => 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_REL),s(TV_u_27a,V_x1))),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_x2))))))))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_quotients_IDENTITYu_u_QUOTIENT, axiom, ![TV_u_27a]: p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),d_equals),s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i))))).
fof(ah4s_integers_INTu_u_EQu_u_REDUCEu_c2, axiom, ![V_n]: (s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))))))) <=> 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_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_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_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_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_arithmetics_ADDu_u_SYM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_BIT10, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_nums_0)))))))).
fof(ah4s_arithmetics_BIT20, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_nums_0)))))))))).
fof(ah4s_arithmetics_NUMERALu_u_DEF, axiom, ![V_x]: s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_x))) = s(t_h4s_nums_num,V_x)).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_integers_INTu_u_EQu_u_CALCULATEu_c2, axiom, ![V_n, V_m]: (s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_n))) = s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_m))))) <=> (s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0) & s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_integers_INTu_u_EQu_u_CALCULATEu_c1, axiom, ![V_y, V_x]: (s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_x))) = s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_y))) <=> s(t_h4s_integers_int,V_x) = s(t_h4s_integers_int,V_y))).
fof(ah4s_integers_INTu_u_EQu_u_CALCULATEu_c0, axiom, ![V_n, V_m]: (s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_m))) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_n))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_CONJu_u_ASSOC, 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_numerals_numeralu_u_lteu_c2, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
fof(ah4s_numerals_numeralu_u_distribu_c27, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ah4s_numerals_numeralu_u_lteu_c1, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
fof(ah4s_numerals_numeralu_u_distribu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_arithmetics_MULTu_u_LESSu_u_EQu_u_SUC, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_EQu_u_LESSu_u_EQ, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n) <=> (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))))))).
fof(ah4s_arithmetics_ZEROu_u_LESSu_u_EQ, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_LESSu_u_EQu_u_TRANS, axiom, ![V_p, V_n, V_m]: ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_TIMES2, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_MULTu_u_ASSOC, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c3, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),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,V_m)).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c2, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(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,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_ASSOC, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_TWO, axiom, s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),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_arithmetics_ONE, axiom, 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,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c0, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) & p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) | ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_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_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_Fu_u_EQu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_integers_INTu_u_DIVu_u_REDUCEu_c6, axiom, ![V_n, V_m]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))) = s(t_h4s_nums_num,h4s_nums_0)) & s(t_h4s_integers_int,h4s_integers_intu_u_div(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_m))))),s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))))))) = s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))))))),s(t_h4s_integers_int,h4s_bools_cond(s(t_bool,V_v),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(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_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_integers_INTu_u_DIVu_u_NEG, axiom, ![V_q, V_p]: (~ (s(t_h4s_integers_int,V_q) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) => s(t_h4s_integers_int,h4s_integers_intu_u_div(s(t_h4s_integers_int,V_p),s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_q))))) = s(t_h4s_integers_int,h4s_integers_intu_u_div(s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_p))),s(t_h4s_integers_int,V_q))))).
fof(ah4s_integers_INTu_u_DIV, axiom, ![V_n, V_m]: (~ (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0)) => s(t_h4s_integers_int,h4s_integers_intu_u_div(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_n))),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_m))))) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))))))).
fof(ah4s_integers_intu_u_div0, axiom, ![V_j, V_i]: (~ (s(t_h4s_integers_int,V_j) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) => ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,h4s_integers_num(s(t_h4s_integers_int,V_i))),s(t_h4s_nums_num,h4s_integers_num(s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_j))))))) = s(t_h4s_nums_num,h4s_nums_0)) & ?[V_vi_]: ((p(s(t_bool,V_vi_)) <=> s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,h4s_integers_num(s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_i))))),s(t_h4s_nums_num,h4s_integers_num(s(t_h4s_integers_int,V_j))))) = s(t_h4s_nums_num,h4s_nums_0)) & s(t_h4s_integers_int,h4s_integers_intu_u_div(s(t_h4s_integers_int,V_i),s(t_h4s_integers_int,V_j))) = s(t_h4s_integers_int,h4s_bools_cond(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_j))),s(t_h4s_integers_int,h4s_bools_cond(s(t_bool,h4s_integers_intu_u_le(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_i))),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_integers_num(s(t_h4s_integers_int,V_i))),s(t_h4s_nums_num,h4s_integers_num(s(t_h4s_integers_int,V_j))))))),s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_integers_num(s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_i))))),s(t_h4s_nums_num,h4s_integers_num(s(t_h4s_integers_int,V_j))))))))),s(t_h4s_integers_int,h4s_bools_cond(s(t_bool,V_vi_),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(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_integers_int,h4s_bools_cond(s(t_bool,h4s_integers_intu_u_le(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_i))),s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_integers_num(s(t_h4s_integers_int,V_i))),s(t_h4s_nums_num,h4s_integers_num(s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_j))))))))))),s(t_h4s_integers_int,h4s_bools_cond(s(t_bool,V_v),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(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_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_integers_num(s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_i))))),s(t_h4s_nums_num,h4s_integers_num(s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_j))))))))))))))))).
fof(ah4s_integers_NUMu_u_OFu_u_INT, axiom, ![V_n]: s(t_h4s_nums_num,h4s_integers_num(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_integers_INTu_u_LE, axiom, ![V_n, V_m]: s(t_bool,h4s_integers_intu_u_le(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_m))),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_n))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))).
fof(ch4s_DeepSyntaxs_datatypeu_u_deepu_u_form, conjecture, ![V_deepu_u_form]: p(s(t_bool,h4s_bools_datatype(s(t_bool,happ(s(t_fun(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form)),t_bool),happ(s(t_fun(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),t_fun(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form)),t_bool)),happ(s(t_fun(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),t_fun(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),t_fun(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form)),t_bool))),happ(s(t_fun(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),t_fun(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),t_fun(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),t_fun(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form)),t_bool)))),happ(s(t_fun(t_fun(t_bool,t_h4s_DeepSyntaxs_deepu_u_form),t_fun(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),t_fun(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),t_fun(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),t_fun(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form)),t_bool))))),happ(s(t_fun(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form),t_fun(t_fun(t_bool,t_h4s_DeepSyntaxs_deepu_u_form),t_fun(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),t_fun(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),t_fun(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),t_fun(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form)),t_bool)))))),happ(s(t_fun(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form)),t_fun(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form),t_fun(t_fun(t_bool,t_h4s_DeepSyntaxs_deepu_u_form),t_fun(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),t_fun(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),t_fun(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),t_fun(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form)),t_bool))))))),happ(s(t_fun(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form)),t_fun(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form)),t_fun(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form),t_fun(t_fun(t_bool,t_h4s_DeepSyntaxs_deepu_u_form),t_fun(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),t_fun(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),t_fun(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),t_fun(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form)),t_bool)))))))),V_deepu_u_form),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form)),h4s_deepsyntaxs_conjn))),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form)),h4s_deepsyntaxs_disjn))),s(t_fun(t_h4s_DeepSyntaxs_deepu_u_form,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_negn))),s(t_fun(t_bool,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_unrelatedbool))),s(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_xlt))),s(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_ltx))),s(t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_xeq))),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_DeepSyntaxs_deepu_u_form)),h4s_deepsyntaxs_xdivided))))))).
