%   ORIGINAL: h4/DeepSyntax/deep__form__11_c3
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/bool/TRUTH: T
% Assm: h4/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_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/DeepSyntax/UnrelatedBool0: h4/DeepSyntax/UnrelatedBool = h4/DeepSyntax/_20_40ind__typeDeepSyntax3
% Assm: h4/bool/DATATYPE__TAG__THM: !x. h4/bool/DATATYPE x <=> T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/DeepSyntax/datatype__deep__form: !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)
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/OR__DEF: $or = (\t1 t2. !t. (t1 ==> t) ==> (t2 ==> t) ==> t)
% Assm: h4/bool/F__DEF: F <=> (!t. t)
% Assm: h4/bool/T__DEF: T <=> (\x. x) = (\x. x)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/AND1__THM: !t2 t1. t1 /\ t2 ==> t1
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% 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/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/DeepSyntax/Negn0: h4/DeepSyntax/Negn = h4/DeepSyntax/_20_40ind__typeDeepSyntax2
% 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____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____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____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/LTx0: h4/DeepSyntax/LTx = h4/DeepSyntax/_20_40ind__typeDeepSyntax5
% Assm: h4/DeepSyntax/Disjn0: h4/DeepSyntax/Disjn = h4/DeepSyntax/_20_40ind__typeDeepSyntax1
% 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____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/xDivided0: h4/DeepSyntax/xDivided = h4/DeepSyntax/_20_40ind__typeDeepSyntax7
% 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/xEQ0: h4/DeepSyntax/xEQ = h4/DeepSyntax/_20_40ind__typeDeepSyntax6
% Assm: h4/DeepSyntax/Conjn0: h4/DeepSyntax/Conjn = h4/DeepSyntax/_20_40ind__typeDeepSyntax0
% Assm: h4/DeepSyntax/xLT0: h4/DeepSyntax/xLT = h4/DeepSyntax/_20_40ind__typeDeepSyntax4
% 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/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/bool/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/ind__type/FCONS0_c1: !n f a. h4/ind__type/FCONS a f (h4/num/SUC n) = f n
% Assm: h4/ind__type/FCONS0_c0: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm: h4/ind__type/CONSTR__REC: !Fn. ?f. !c i r. f (h4/ind__type/CONSTR c i r) = Fn c i r (\n. f (r n))
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/bool/NOT__DEF: $not = (\t. t ==> F)
% Assm: h4/bool/EXCLUDED__MIDDLE0: !t. t \/ ~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/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/LEFT__OR__CONG: !Q_27 Q P_27 P. (P <=> P_27) /\ (~P_27 ==> (Q <=> Q_27)) ==> (P \/ Q <=> P_27 \/ Q_27)
% Assm: h4/quotient/EQ__IMPLIES: !Q P. (P <=> Q) ==> P ==> Q
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/bool/OR__CONG: !Q_27 Q P_27 P. (~Q ==> (P <=> P_27)) /\ (~P_27 ==> (Q <=> Q_27)) ==> (P \/ Q <=> P_27 \/ Q_27)
% Assm: h4/bool/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm: h4/bool/LEFT__AND__CONG: !Q_27 Q P_27 P. (P <=> P_27) /\ (P_27 ==> (Q <=> Q_27)) ==> (P /\ Q <=> P_27 /\ Q_27)
% Assm: h4/ind__type/NUMSUM__INJ: !x2 x1 b2 b1. h4/ind__type/NUMSUM b1 x1 = h4/ind__type/NUMSUM b2 x2 <=> (b1 <=> b2) /\ x1 = x2
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/bool/bool__case__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/bool/EQ__EXPAND: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% Assm: h4/bool/AND__CONG: !Q_27 Q P_27 P. (Q ==> (P <=> P_27)) /\ (P_27 ==> (Q <=> Q_27)) ==> (P /\ Q <=> P_27 /\ Q_27)
% Assm: h4/quotient/COND__RSP: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a1 a2 b1 b2 c1 c2. (a1 <=> a2) /\ R b1 b2 /\ R c1 c2 ==> R (h4/bool/COND a1 b1 c1) (h4/bool/COND a2 b2 c2))
% Assm: h4/bool/ABS__REP__THM: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. abs (rep a) = a) /\ (!r. P r <=> rep (abs r) = r))
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/COND__DEF: h4/bool/COND = (\t t1 t2. h4/min/_40 (\x. ((t <=> T) ==> x = t1) /\ ((t <=> F) ==> x = t2)))
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% 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_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_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__11_c2: !a_27 a. h4/DeepSyntax/Negn a = h4/DeepSyntax/Negn a_27 <=> a = a_27
% 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_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/bool/AND__CLAUSES_c4: !t. t /\ t <=> t
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/OR__CLAUSES_c4: !t. t \/ t <=> t
% 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__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__11_c0: !a1_27 a1 a0_27 a0. h4/DeepSyntax/Conjn a0 a1 = h4/DeepSyntax/Conjn a0_27 a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% 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_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_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__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__11_c1: !a1_27 a1 a0_27 a0. h4/DeepSyntax/Disjn a0 a1 = h4/DeepSyntax/Disjn a0_27 a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% 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__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/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/ind__type/NUMSUM0: !x b. h4/ind__type/NUMSUM b x = h4/bool/COND b (h4/num/SUC (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) x)) (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) x)
% Assm: h4/arithmetic/EVEN0_c1: !n. h4/arithmetic/EVEN (h4/num/SUC n) <=> ~h4/arithmetic/EVEN n
% Assm: h4/arithmetic/EQ__MULT__LCANCEL: !p n m. h4/arithmetic/_2A m n = h4/arithmetic/_2A m p <=> m = h4/num/0 \/ n = p
% Assm: h4/prim__rec/INV__SUC__EQ: !n m. h4/num/SUC m = h4/num/SUC n <=> m = n
% Assm: h4/arithmetic/EVEN__DOUBLE: !n. h4/arithmetic/EVEN (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n)
% Assm: h4/numeral/numeral__distrib_c17: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__eq_c3: !n. h4/arithmetic/BIT2 n = h4/arithmetic/ZERO <=> F
% Assm: h4/marker/Abbrev__def: !x. h4/marker/Abbrev x <=> x
% Assm: h4/marker/label__def: !lab argument. h4/marker/_3A_2D lab argument <=> argument
% Assm: h4/option/IF__NONE__EQUALS__OPTION_c3: !x X P. h4/bool/COND P h4/option/NONE X = h4/option/SOME x <=> ~P /\ X = h4/option/SOME x
% Assm: h4/bool/COND__ID: !t b. h4/bool/COND b t t = t
% Assm: h4/option/IF__EQUALS__OPTION_c3: !y x P. h4/bool/COND P h4/option/NONE (h4/option/SOME x) = h4/option/SOME y <=> ~P /\ x = y
% Assm: h4/option/IF__EQUALS__OPTION_c0: !x P. h4/bool/COND P (h4/option/SOME x) h4/option/NONE = h4/option/NONE <=> ~P
% Assm: h4/option/IF__EQUALS__OPTION_c1: !x P. h4/bool/COND P h4/option/NONE (h4/option/SOME x) = h4/option/NONE <=> P
% Assm: h4/option/IS__NONE__DEF_c1: h4/option/IS__NONE h4/option/NONE <=> T
% Assm: h4/option/IS__SOME__DEF_c1: h4/option/IS__SOME h4/option/NONE <=> F
% Assm: h4/option/IF__EQUALS__OPTION_c2: !y x P. h4/bool/COND P (h4/option/SOME x) h4/option/NONE = h4/option/SOME y <=> P /\ x = y
% Assm: h4/option/IS__NONE__DEF_c0: !x. h4/option/IS__NONE (h4/option/SOME x) <=> F
% Assm: h4/option/IS__SOME__DEF_c0: !x. h4/option/IS__SOME (h4/option/SOME x) <=> T
% Assm: h4/option/option__nchotomy: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/option/NOT__NONE__SOME: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Goal: !a_27 a. h4/DeepSyntax/UnrelatedBool a = h4/DeepSyntax/UnrelatedBool a_27 <=> a <=> a_27
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_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_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_DeepSyntaxs_UnrelatedBool0]: h4/DeepSyntax/UnrelatedBool = h4/DeepSyntax/_20_40ind__typeDeepSyntax3
% Assm [h4s_bools_DATATYPEu_u_TAGu_u_THM]: !x. h4/bool/DATATYPE x <=> T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_DeepSyntaxs_datatypeu_u_deepu_u_form]: !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)
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_ORu_u_DEF]: !x x'. $or x x' <=> (!t. (x ==> t) ==> (x' ==> t) ==> t)
% Assm [h4s_bools_Fu_u_DEF]: F <=> (!t. t)
% Assm [h4s_bools_Tu_u_DEF]: T <=> (!x. x <=> x)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_AND1u_u_THM]: !t2 t1. t1 /\ t2 ==> t1
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% 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 (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) (h4/pair/_2C x (h4/pair/_2C h4/bool/ARB h4/bool/ARB)) _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 (h4/pair/_2C h4/bool/ARB (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 (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 (h4/ind__type/FCONS a1 _0)) /\ happ _27deep__form_27 a0 /\ happ _27deep__form_27 a1) \/ (?a. a0_270 = 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 a _0) /\ happ _27deep__form_27 a) \/ (?a. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) (h4/pair/_2C a (h4/pair/_2C h4/bool/ARB h4/bool/ARB)) _0) \/ (?a. a0_270 = 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 a h4/bool/ARB)) _0) \/ (?a. a0_270 = 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 a h4/bool/ARB)) _0) \/ (?a. a0_270 = 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 a h4/bool/ARB)) _0) \/ (?a0 a1. a0_270 = 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 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_DeepSyntaxs_Negn0]: h4/DeepSyntax/Negn = h4/DeepSyntax/_20_40ind__typeDeepSyntax2
% 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_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 (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0)))) (h4/pair/_2C h4/bool/ARB (h4/pair/_2C x h4/bool/ARB)) _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 (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 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 (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 (h4/DeepSyntax/_20_40ind__typeDeepSyntax9 x) _0)))
% Assm [h4s_DeepSyntaxs_LTx0]: h4/DeepSyntax/LTx = h4/DeepSyntax/_20_40ind__typeDeepSyntax5
% Assm [h4s_DeepSyntaxs_Disjn0]: h4/DeepSyntax/Disjn = h4/DeepSyntax/_20_40ind__typeDeepSyntax1
% 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 (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 x h4/bool/ARB)) _0))
% 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 (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 x x')) _0))
% Assm [h4s_DeepSyntaxs_xDivided0]: h4/DeepSyntax/xDivided = h4/DeepSyntax/_20_40ind__typeDeepSyntax7
% 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 (h4/pair/_2C h4/bool/ARB (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_xEQ0]: h4/DeepSyntax/xEQ = h4/DeepSyntax/_20_40ind__typeDeepSyntax6
% Assm [h4s_DeepSyntaxs_Conjn0]: h4/DeepSyntax/Conjn = h4/DeepSyntax/_20_40ind__typeDeepSyntax0
% Assm [h4s_DeepSyntaxs_xLT0]: h4/DeepSyntax/xLT = h4/DeepSyntax/_20_40ind__typeDeepSyntax4
% 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 (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 (h4/DeepSyntax/_20_40ind__typeDeepSyntax9 x) (h4/ind__type/FCONS (h4/DeepSyntax/_20_40ind__typeDeepSyntax9 x') _0))))
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_bools_BETAu_u_THM]: !y f. happ f y = happ f y
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_indu_u_types_FCONS0u_c1]: !n f a. happ (h4/ind__type/FCONS a f) (h4/num/SUC n) = happ f n
% Assm [h4s_indu_u_types_FCONS0u_c0]: !f a. happ (h4/ind__type/FCONS a f) h4/num/0 = a
% Assm [h4s_indu_u_types_CONSTRu_u_REC]: !_0. (!f r n. happ (happ (happ _0 f) r) n = happ f (happ r n)) ==> (!Fn. ?f. !c i r. happ f (h4/ind__type/CONSTR c i r) = happ (happ (happ (happ Fn c) i) r) (happ (happ _0 f) r))
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_bools_NOTu_u_DEF]: !x. $not x <=> x ==> F
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE0]: !t. t \/ ~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 (h4/pair/_2C h4/bool/ARB (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 (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 (h4/ind__type/FCONS a1 _1)) /\ happ _27deep__form_27 a0 /\ happ _27deep__form_27 a1) \/ (?a. a0_270 = 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 a _1) /\ happ _27deep__form_27 a) \/ (?a. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) (h4/pair/_2C a (h4/pair/_2C h4/bool/ARB h4/bool/ARB)) _1) \/ (?a. a0_270 = 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 a h4/bool/ARB)) _1) \/ (?a. a0_270 = 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 a h4/bool/ARB)) _1) \/ (?a. a0_270 = 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 a h4/bool/ARB)) _1) \/ (?a0 a1. a0_270 = 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 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_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_LEFTu_u_ORu_u_CONG]: !Q_27 Q P_27 P. (P <=> P_27) /\ (~P_27 ==> (Q <=> Q_27)) ==> (P \/ Q <=> P_27 \/ Q_27)
% Assm [h4s_quotients_EQu_u_IMPLIES]: !Q P. (P <=> Q) ==> P ==> Q
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_bools_ORu_u_CONG]: !Q_27 Q P_27 P. (~Q ==> (P <=> P_27)) /\ (~P_27 ==> (Q <=> Q_27)) ==> (P \/ Q <=> P_27 \/ Q_27)
% Assm [h4s_bools_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm [h4s_bools_LEFTu_u_ANDu_u_CONG]: !Q_27 Q P_27 P. (P <=> P_27) /\ (P_27 ==> (Q <=> Q_27)) ==> (P /\ Q <=> P_27 /\ Q_27)
% Assm [h4s_indu_u_types_NUMSUMu_u_INJ]: !x2 x1 b2 b1. h4/ind__type/NUMSUM b1 x1 = h4/ind__type/NUMSUM b2 x2 <=> (b1 <=> b2) /\ x1 = x2
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_bools_boolu_u_caseu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_bools_EQu_u_EXPAND]: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% Assm [h4s_bools_ANDu_u_CONG]: !Q_27 Q P_27 P. (Q ==> (P <=> P_27)) /\ (P_27 ==> (Q <=> Q_27)) ==> (P /\ Q <=> P_27 /\ Q_27)
% Assm [h4s_quotients_CONDu_u_RSP]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a1 a2 b1 b2 c1 c2. (a1 <=> a2) /\ happ (happ R b1) b2 /\ happ (happ R c1) c2 ==> happ (happ R (h4/bool/COND a1 b1 c1)) (h4/bool/COND a2 b2 c2))
% Assm [h4s_bools_ABSu_u_REPu_u_THM]: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. happ abs (happ rep a) = a) /\ (!r. happ P r <=> happ rep (happ abs r) = r))
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_CONDu_u_DEF]: !_0. (!x x x' x''. happ (happ (happ (happ _0 x) x) x') x'' <=> ((x <=> T) ==> x'' = x) /\ ((x <=> F) ==> x'' = x')) ==> (!x x x'. h4/bool/COND x x x' = h4/min/_40 (happ (happ (happ _0 x) x) x'))
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% 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_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_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_11u_c2]: !a_27 a. happ h4/DeepSyntax/Negn a = happ h4/DeepSyntax/Negn a_27 <=> a = a_27
% 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_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_bools_ANDu_u_CLAUSESu_c4]: !t. t /\ t <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c4]: !t. t \/ t <=> t
% 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_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_11u_c0]: !a1_27 a1 a0_27 a0. happ (happ h4/DeepSyntax/Conjn a0) a1 = happ (happ h4/DeepSyntax/Conjn a0_27) a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% 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_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_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_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_11u_c1]: !a1_27 a1 a0_27 a0. happ (happ h4/DeepSyntax/Disjn a0) a1 = happ (happ h4/DeepSyntax/Disjn a0_27) a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% 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_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_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_indu_u_types_NUMSUM0]: !x b. h4/ind__type/NUMSUM b x = h4/bool/COND b (h4/num/SUC (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) x)) (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) x)
% Assm [h4s_arithmetics_EVEN0u_c1]: !n. h4/arithmetic/EVEN (h4/num/SUC n) <=> ~h4/arithmetic/EVEN n
% Assm [h4s_arithmetics_EQu_u_MULTu_u_LCANCEL]: !p n m. h4/arithmetic/_2A m n = h4/arithmetic/_2A m p <=> m = h4/num/0 \/ n = p
% Assm [h4s_primu_u_recs_INVu_u_SUCu_u_EQ]: !n m. h4/num/SUC m = h4/num/SUC n <=> m = n
% Assm [h4s_arithmetics_EVENu_u_DOUBLE]: !n. h4/arithmetic/EVEN (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n)
% Assm [h4s_numerals_numeralu_u_distribu_c17]: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_equ_c3]: !n. h4/arithmetic/BIT2 n = h4/arithmetic/ZERO <=> F
% Assm [h4s_markers_Abbrevu_u_def]: !x. h4/marker/Abbrev x <=> x
% Assm [h4s_markers_labelu_u_def]: !lab argument. h4/marker/_3A_2D lab argument <=> argument
% Assm [h4s_options_IFu_u_NONEu_u_EQUALSu_u_OPTIONu_c3]: !x X P. h4/bool/COND P h4/option/NONE X = h4/option/SOME x <=> ~P /\ X = h4/option/SOME x
% Assm [h4s_bools_CONDu_u_ID]: !t b. h4/bool/COND b t t = t
% Assm [h4s_options_IFu_u_EQUALSu_u_OPTIONu_c3]: !y x P. h4/bool/COND P h4/option/NONE (h4/option/SOME x) = h4/option/SOME y <=> ~P /\ x = y
% Assm [h4s_options_IFu_u_EQUALSu_u_OPTIONu_c0]: !x P. h4/bool/COND P (h4/option/SOME x) h4/option/NONE = h4/option/NONE <=> ~P
% Assm [h4s_options_IFu_u_EQUALSu_u_OPTIONu_c1]: !x P. h4/bool/COND P h4/option/NONE (h4/option/SOME x) = h4/option/NONE <=> P
% Assm [h4s_options_ISu_u_NONEu_u_DEFu_c1]: h4/option/IS__NONE h4/option/NONE <=> T
% Assm [h4s_options_ISu_u_SOMEu_u_DEFu_c1]: h4/option/IS__SOME h4/option/NONE <=> F
% Assm [h4s_options_IFu_u_EQUALSu_u_OPTIONu_c2]: !y x P. h4/bool/COND P (h4/option/SOME x) h4/option/NONE = h4/option/SOME y <=> P /\ x = y
% Assm [h4s_options_ISu_u_NONEu_u_DEFu_c0]: !x. h4/option/IS__NONE (h4/option/SOME x) <=> F
% Assm [h4s_options_ISu_u_SOMEu_u_DEFu_c0]: !x. h4/option/IS__SOME (h4/option/SOME x) <=> T
% Assm [h4s_options_optionu_u_nchotomy]: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm [h4s_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_options_NOTu_u_NONEu_u_SOME]: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Goal: !a_27 a. happ h4/DeepSyntax/UnrelatedBool a = happ h4/DeepSyntax/UnrelatedBool a_27 <=> a <=> a_27
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q1445762,TV_Q1445758]: ![V_f, V_g]: (![V_x]: s(TV_Q1445758,happ(s(t_fun(TV_Q1445762,TV_Q1445758),V_f),s(TV_Q1445762,V_x))) = s(TV_Q1445758,happ(s(t_fun(TV_Q1445762,TV_Q1445758),V_g),s(TV_Q1445762,V_x))) => s(t_fun(TV_Q1445762,TV_Q1445758),V_f) = s(t_fun(TV_Q1445762,TV_Q1445758),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
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_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_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_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_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_DeepSyntaxs_datatypeu_u_deepu_u_form, axiom, ![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))))))).
fof(ah4s_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_bools_ORu_u_DEF, axiom, ![V_x, V_xi_]: (p(s(t_bool,d_or(s(t_bool,V_x),s(t_bool,V_xi_)))) <=> ![V_t]: ((p(s(t_bool,V_x)) => p(s(t_bool,V_t))) => ((p(s(t_bool,V_xi_)) => p(s(t_bool,V_t))) => p(s(t_bool,V_t)))))).
fof(ah4s_bools_Fu_u_DEF, axiom, (p(s(t_bool,f)) <=> ![V_t]: p(s(t_bool,V_t)))).
fof(ah4s_bools_Tu_u_DEF, axiom, (p(s(t_bool,t)) <=> ![V_x]: s(t_bool,V_x) = s(t_bool,V_x))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_AND1u_u_THM, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t1)))).
fof(ah4s_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_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,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))),s(t_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),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_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)),h4s_pairs_u_2c(s(t_bool,h4s_bools_arb),s(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,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)),h4s_pairs_u_2c(s(t_bool,h4s_bools_arb),s(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,h4s_nums_suc(s(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)),h4s_pairs_u_2c(s(t_bool,h4s_bools_arb),s(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,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))),s(t_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),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,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))))),s(t_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),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,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))))))),s(t_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),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,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))))))))),s(t_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),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,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))))))))))),s(t_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),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_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_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_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,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))))),s(t_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),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_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,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))))))),s(t_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),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,h4s_nums_suc(s(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)),h4s_pairs_u_2c(s(t_bool,h4s_bools_arb),s(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_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_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_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,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))))))))),s(t_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),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_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,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))))))))))),s(t_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),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_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_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)),h4s_pairs_u_2c(s(t_bool,h4s_bools_arb),s(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_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_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_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_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,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)),h4s_pairs_u_2c(s(t_bool,h4s_bools_arb),s(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_bools_MONOu_u_OR, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) | p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) | p(s(t_bool,V_w)))))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_bools_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_MONOu_u_AND, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) & p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) & p(s(t_bool,V_w)))))).
fof(ah4s_bools_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),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27b,V_y)).
fof(ah4s_indu_u_types_FCONS0u_c1, axiom, ![TV_u_27a]: ![V_n, V_f, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n)))).
fof(ah4s_indu_u_types_FCONS0u_c0, axiom, ![TV_u_27a]: ![V_f, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_a)).
fof(ah4s_indu_u_types_CONSTRu_u_REC, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_f, V_r, V_n]: s(TV_u_27b,happ(s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_h4s_nums_num,V_n))) = s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r),s(t_h4s_nums_num,V_n))))) => ![V_Fn]: ?[V_f]: ![V_c, V_i, V_r]: s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,V_c),s(TV_u_27a,V_i),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))) = s(TV_u_27b,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)))),V_Fn),s(t_h4s_nums_num,V_c))),s(TV_u_27a,V_i))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))))).
fof(ah4s_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_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_bools_NOTu_u_DEF, axiom, ![V_x]: (p(s(t_bool,d_not(s(t_bool,V_x)))) <=> (p(s(t_bool,V_x)) => p(s(t_bool,f))))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE0, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_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)),h4s_pairs_u_2c(s(t_bool,h4s_bools_arb),s(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,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)),h4s_pairs_u_2c(s(t_bool,h4s_bools_arb),s(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,h4s_nums_suc(s(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)),h4s_pairs_u_2c(s(t_bool,h4s_bools_arb),s(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,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))),s(t_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),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,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))))),s(t_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),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,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))))))),s(t_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),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,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))))))))),s(t_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),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,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))))))))))),s(t_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),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_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_LEFTu_u_ORu_u_CONG, axiom, ![V_Qu_27, V_Q, V_Pu_27, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Pu_27) & (~ (p(s(t_bool,V_Pu_27))) => s(t_bool,V_Q) = s(t_bool,V_Qu_27))) => ((p(s(t_bool,V_P)) | p(s(t_bool,V_Q))) <=> (p(s(t_bool,V_Pu_27)) | p(s(t_bool,V_Qu_27)))))).
fof(ah4s_quotients_EQu_u_IMPLIES, axiom, ![V_Q, V_P]: (s(t_bool,V_P) = s(t_bool,V_Q) => (p(s(t_bool,V_P)) => p(s(t_bool,V_Q))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_bools_ORu_u_CONG, axiom, ![V_Qu_27, V_Q, V_Pu_27, V_P]: (((~ (p(s(t_bool,V_Q))) => s(t_bool,V_P) = s(t_bool,V_Pu_27)) & (~ (p(s(t_bool,V_Pu_27))) => s(t_bool,V_Q) = s(t_bool,V_Qu_27))) => ((p(s(t_bool,V_P)) | p(s(t_bool,V_Q))) <=> (p(s(t_bool,V_Pu_27)) | p(s(t_bool,V_Qu_27)))))).
fof(ah4s_bools_EQu_u_IMPu_u_THM, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) & (p(s(t_bool,V_t2)) => p(s(t_bool,V_t1)))))).
fof(ah4s_bools_LEFTu_u_ANDu_u_CONG, axiom, ![V_Qu_27, V_Q, V_Pu_27, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Pu_27) & (p(s(t_bool,V_Pu_27)) => s(t_bool,V_Q) = s(t_bool,V_Qu_27))) => ((p(s(t_bool,V_P)) & p(s(t_bool,V_Q))) <=> (p(s(t_bool,V_Pu_27)) & p(s(t_bool,V_Qu_27)))))).
fof(ah4s_indu_u_types_NUMSUMu_u_INJ, axiom, ![V_x2, V_x1, V_b2, V_b1]: (s(t_h4s_nums_num,h4s_indu_u_types_numsum(s(t_bool,V_b1),s(t_h4s_nums_num,V_x1))) = s(t_h4s_nums_num,h4s_indu_u_types_numsum(s(t_bool,V_b2),s(t_h4s_nums_num,V_x2))) <=> (s(t_bool,V_b1) = s(t_bool,V_b2) & s(t_h4s_nums_num,V_x1) = s(t_h4s_nums_num,V_x2)))).
fof(ah4s_bools_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_bools_boolu_u_caseu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_bools_EQu_u_EXPAND, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) | (~ (p(s(t_bool,V_t1))) & ~ (p(s(t_bool,V_t2))))))).
fof(ah4s_bools_ANDu_u_CONG, axiom, ![V_Qu_27, V_Q, V_Pu_27, V_P]: (((p(s(t_bool,V_Q)) => s(t_bool,V_P) = s(t_bool,V_Pu_27)) & (p(s(t_bool,V_Pu_27)) => s(t_bool,V_Q) = s(t_bool,V_Qu_27))) => ((p(s(t_bool,V_P)) & p(s(t_bool,V_Q))) <=> (p(s(t_bool,V_Pu_27)) & p(s(t_bool,V_Qu_27)))))).
fof(ah4s_quotients_CONDu_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_a1, V_a2, V_b1, V_b2, V_c1, V_c2]: ((s(t_bool,V_a1) = s(t_bool,V_a2) & (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_b1))),s(TV_u_27a,V_b2)))) & 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_c1))),s(TV_u_27a,V_c2)))))) => 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,h4s_bools_cond(s(t_bool,V_a1),s(TV_u_27a,V_b1),s(TV_u_27a,V_c1))))),s(TV_u_27a,h4s_bools_cond(s(t_bool,V_a2),s(TV_u_27a,V_b2),s(TV_u_27a,V_c2))))))))).
fof(ah4s_bools_ABSu_u_REPu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ?[V_rep, V_abs]: (![V_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))) = s(TV_u_27b,V_a) & ![V_r]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_r)))) <=> s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))))) = s(TV_u_27a,V_r))))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_bools_CONDu_u_DEF, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x, V_x0, V_xi_, V_xi_i_]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)))),V_uu_0),s(TV_u_27a,V_x))),s(t_bool,V_x0))),s(TV_u_27a,V_xi_))),s(TV_u_27a,V_xi_i_)))) <=> ((s(t_bool,V_x0) = s(t_bool,t) => s(TV_u_27a,V_xi_i_) = s(TV_u_27a,V_x)) & (s(t_bool,V_x0) = s(t_bool,f) => s(TV_u_27a,V_xi_i_) = s(TV_u_27a,V_xi_)))) => ![V_x, V_x0, V_xi_]: s(TV_u_27a,h4s_bools_cond(s(t_bool,V_x),s(TV_u_27a,V_x0),s(TV_u_27a,V_xi_))) = s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)))),V_uu_0),s(TV_u_27a,V_x0))),s(t_bool,V_x))),s(TV_u_27a,V_xi_))))))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_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_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_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,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_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_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_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_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_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_11u_c2, axiom, ![V_au_27, V_a]: (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_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_au_27))) <=> s(t_h4s_DeepSyntaxs_deepu_u_form,V_a) = s(t_h4s_DeepSyntaxs_deepu_u_form,V_au_27))).
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_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_bools_ANDu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_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_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_11u_c0, axiom, ![V_a1u_27, V_a1, V_a0u_27, V_a0]: (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_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_a0u_27))),s(t_h4s_DeepSyntaxs_deepu_u_form,V_a1u_27))) <=> (s(t_h4s_DeepSyntaxs_deepu_u_form,V_a0) = s(t_h4s_DeepSyntaxs_deepu_u_form,V_a0u_27) & s(t_h4s_DeepSyntaxs_deepu_u_form,V_a1) = s(t_h4s_DeepSyntaxs_deepu_u_form,V_a1u_27)))).
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_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_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_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_11u_c1, axiom, ![V_a1u_27, V_a1, V_a0u_27, V_a0]: (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_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_a0u_27))),s(t_h4s_DeepSyntaxs_deepu_u_form,V_a1u_27))) <=> (s(t_h4s_DeepSyntaxs_deepu_u_form,V_a0) = s(t_h4s_DeepSyntaxs_deepu_u_form,V_a0u_27) & s(t_h4s_DeepSyntaxs_deepu_u_form,V_a1) = s(t_h4s_DeepSyntaxs_deepu_u_form,V_a1u_27)))).
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_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_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_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
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_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_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_indu_u_types_NUMSUM0, axiom, ![V_x, V_b]: s(t_h4s_nums_num,h4s_indu_u_types_numsum(s(t_bool,V_b),s(t_h4s_nums_num,V_x))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,V_b),s(t_h4s_nums_num,h4s_nums_suc(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_x))))),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_x)))))).
fof(ah4s_arithmetics_EVEN0u_c1, axiom, ![V_n]: (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))) <=> ~ (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_n))))))).
fof(ah4s_arithmetics_EQu_u_MULTu_u_LCANCEL, 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,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))) <=> (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0) | s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,V_p)))).
fof(ah4s_primu_u_recs_INVu_u_SUCu_u_EQ, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_arithmetics_EVENu_u_DOUBLE, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_even(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))))))).
fof(ah4s_numerals_numeralu_u_distribu_c17, axiom, ![V_n]: (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_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_arithmetics_zero))).
fof(ah4s_numerals_numeralu_u_equ_c3, 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_zero) <=> p(s(t_bool,f)))).
fof(ah4s_markers_Abbrevu_u_def, axiom, ![V_x]: s(t_bool,h4s_markers_abbrev(s(t_bool,V_x))) = s(t_bool,V_x)).
fof(ah4s_markers_labelu_u_def, axiom, ![V_lab, V_argument]: s(t_bool,h4s_markers_u_3au_2d(s(t_h4s_markers_label,V_lab),s(t_bool,V_argument))) = s(t_bool,V_argument)).
fof(ah4s_options_IFu_u_NONEu_u_EQUALSu_u_OPTIONu_c3, axiom, ![TV_u_27a]: ![V_x, V_X, V_P]: (s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_h4s_options_option(TV_u_27a),V_X))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) <=> (~ (p(s(t_bool,V_P))) & s(t_h4s_options_option(TV_u_27a),V_X) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x)))))).
fof(ah4s_bools_CONDu_u_ID, axiom, ![TV_u_27a]: ![V_t, V_b]: s(TV_u_27a,h4s_bools_cond(s(t_bool,V_b),s(TV_u_27a,V_t),s(TV_u_27a,V_t))) = s(TV_u_27a,V_t)).
fof(ah4s_options_IFu_u_EQUALSu_u_OPTIONu_c3, axiom, ![TV_u_27a]: ![V_y, V_x, V_P]: (s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> (~ (p(s(t_bool,V_P))) & s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))).
fof(ah4s_options_IFu_u_EQUALSu_u_OPTIONu_c0, axiom, ![TV_u_27a]: ![V_x, V_P]: (s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) <=> ~ (p(s(t_bool,V_P))))).
fof(ah4s_options_IFu_u_EQUALSu_u_OPTIONu_c1, axiom, ![TV_u_27a]: ![V_x, V_P]: (s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) <=> p(s(t_bool,V_P)))).
fof(ah4s_options_ISu_u_NONEu_u_DEFu_c1, axiom, ![TV_u_27a]: s(t_bool,h4s_options_isu_u_none(s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_bool,t)).
fof(ah4s_options_ISu_u_SOMEu_u_DEFu_c1, axiom, ![TV_u_27a]: s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_bool,f)).
fof(ah4s_options_IFu_u_EQUALSu_u_OPTIONu_c2, axiom, ![TV_u_27a]: ![V_y, V_x, V_P]: (s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> (p(s(t_bool,V_P)) & s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))).
fof(ah4s_options_ISu_u_NONEu_u_DEFu_c0, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,h4s_options_isu_u_none(s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))) = s(t_bool,f)).
fof(ah4s_options_ISu_u_SOMEu_u_DEFu_c0, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))) = s(t_bool,t)).
fof(ah4s_options_optionu_u_nchotomy, axiom, ![TV_u_27a]: ![V_opt]: (s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) | ?[V_x]: s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_options_SOMEu_u_11, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_options_NOTu_u_NONEu_u_SOME, axiom, ![TV_u_27a]: ![V_x]: ~ (s(t_h4s_options_option(TV_u_27a),h4s_options_none) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,f))) <=> p(s(t_bool,f)))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ch4s_DeepSyntaxs_deepu_u_formu_u_11u_c3, conjecture, ![V_au_27, V_a]: (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_DeepSyntaxs_deepu_u_form,happ(s(t_fun(t_bool,t_h4s_DeepSyntaxs_deepu_u_form),h4s_deepsyntaxs_unrelatedbool),s(t_bool,V_au_27))) <=> s(t_bool,V_a) = s(t_bool,V_au_27))).
