%   ORIGINAL: h4/binary__ieee/float__sub__plus__infinity__finite
% 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/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/binary__ieee/float__value__case__def_c0: !v1 v f a. h4/binary__ieee/float__value__CASE (h4/binary__ieee/Float a) f v v1 = f a
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/TRUTH: T
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% 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/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/binary__ieee/float__sub__def: !y x mode. h4/binary__ieee/float__sub mode x y = h4/pair/pair__CASE (h4/pair/_2C (h4/binary__ieee/float__value0 x) (h4/binary__ieee/float__value0 y)) (\v v1. h4/binary__ieee/float__value__CASE v (\r1. h4/binary__ieee/float__value__CASE v1 (\r2. h4/binary__ieee/float__round mode (h4/bool/COND (r1 = h4/real/real__of__num h4/num/0 /\ r2 = h4/real/real__of__num h4/num/0 /\ ~(h4/binary__ieee/float__Sign x = h4/binary__ieee/float__Sign y)) (h4/binary__ieee/float__Sign x = h4/words/n2w (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (mode = h4/binary__ieee/roundTowardNegative)) (h4/real/real__sub r1 r2)) (h4/binary__ieee/float__negate y) (h4/binary__ieee/float__some__nan h4/bool/the__value)) (h4/binary__ieee/float__value__CASE v1 (\v7. x) (h4/bool/COND (h4/binary__ieee/float__Sign x = h4/binary__ieee/float__Sign y) (h4/binary__ieee/float__some__nan h4/bool/the__value) x) (h4/binary__ieee/float__some__nan h4/bool/the__value)) (h4/binary__ieee/float__some__nan h4/bool/the__value))
% Assm: h4/binary__ieee/float__sub__finite__plus__infinity: !x r mode. h4/binary__ieee/float__value0 x = h4/binary__ieee/Float r ==> h4/binary__ieee/float__sub mode x (h4/binary__ieee/float__plus__infinity h4/bool/the__value) = h4/binary__ieee/float__minus__infinity h4/bool/the__value
% Assm: h4/binary__ieee/float__sub__finite: !y x r2 r1 mode. h4/binary__ieee/float__value0 x = h4/binary__ieee/Float r1 /\ h4/binary__ieee/float__value0 y = h4/binary__ieee/Float r2 ==> h4/binary__ieee/float__sub mode x y = h4/binary__ieee/float__round mode (h4/bool/COND (r1 = h4/real/real__of__num h4/num/0 /\ r2 = h4/real/real__of__num h4/num/0 /\ ~(h4/binary__ieee/float__Sign x = h4/binary__ieee/float__Sign y)) (h4/binary__ieee/float__Sign x = h4/words/n2w (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (mode = h4/binary__ieee/roundTowardNegative)) (h4/real/real__sub r1 r2)
% Assm: h4/binary__ieee/float__value__case__def_c1: !v1 v f. h4/binary__ieee/float__value__CASE h4/binary__ieee/Infinity f v v1 = v
% Assm: h4/binary__ieee/float__add__finite__plus__infinity: !x r mode. h4/binary__ieee/float__value0 x = h4/binary__ieee/Float r ==> h4/binary__ieee/float__add mode x (h4/binary__ieee/float__plus__infinity h4/bool/the__value) = h4/binary__ieee/float__plus__infinity h4/bool/the__value
% Assm: h4/binary__ieee/float__add__plus__infinity__finite: !x r mode. h4/binary__ieee/float__value0 x = h4/binary__ieee/Float r ==> h4/binary__ieee/float__add mode (h4/binary__ieee/float__plus__infinity h4/bool/the__value) x = h4/binary__ieee/float__plus__infinity h4/bool/the__value
% 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/binary__ieee/float__add__def: !y x mode. h4/binary__ieee/float__add mode x y = h4/pair/pair__CASE (h4/pair/_2C (h4/binary__ieee/float__value0 x) (h4/binary__ieee/float__value0 y)) (\v v1. h4/binary__ieee/float__value__CASE v (\r1. h4/binary__ieee/float__value__CASE v1 (\r2. h4/binary__ieee/float__round mode (h4/bool/COND (r1 = h4/real/real__of__num h4/num/0 /\ r2 = h4/real/real__of__num h4/num/0 /\ h4/binary__ieee/float__Sign x = h4/binary__ieee/float__Sign y) (h4/binary__ieee/float__Sign x = h4/words/n2w (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (mode = h4/binary__ieee/roundTowardNegative)) (h4/realax/real__add r1 r2)) y (h4/binary__ieee/float__some__nan h4/bool/the__value)) (h4/binary__ieee/float__value__CASE v1 (\v7. x) (h4/bool/COND (h4/binary__ieee/float__Sign x = h4/binary__ieee/float__Sign y) x (h4/binary__ieee/float__some__nan h4/bool/the__value)) (h4/binary__ieee/float__some__nan h4/bool/the__value)) (h4/binary__ieee/float__some__nan h4/bool/the__value))
% Assm: h4/binary__ieee/float__sub__nan: !y x mode. h4/binary__ieee/float__value0 x = h4/binary__ieee/NaN \/ h4/binary__ieee/float__value0 y = h4/binary__ieee/NaN ==> h4/binary__ieee/float__sub mode x y = h4/binary__ieee/float__some__nan h4/bool/the__value
% Assm: h4/binary__ieee/float__add__minus__infinity__finite: !x r mode. h4/binary__ieee/float__value0 x = h4/binary__ieee/Float r ==> h4/binary__ieee/float__add mode (h4/binary__ieee/float__minus__infinity h4/bool/the__value) x = h4/binary__ieee/float__minus__infinity h4/bool/the__value
% Assm: h4/binary__ieee/float__add__finite__minus__infinity: !x r mode. h4/binary__ieee/float__value0 x = h4/binary__ieee/Float r ==> h4/binary__ieee/float__add mode x (h4/binary__ieee/float__minus__infinity h4/bool/the__value) = h4/binary__ieee/float__minus__infinity h4/bool/the__value
% Assm: h4/binary__ieee/float__add__finite: !y x r2 r1 mode. h4/binary__ieee/float__value0 x = h4/binary__ieee/Float r1 /\ h4/binary__ieee/float__value0 y = h4/binary__ieee/Float r2 ==> h4/binary__ieee/float__add mode x y = h4/binary__ieee/float__round mode (h4/bool/COND (r1 = h4/real/real__of__num h4/num/0 /\ r2 = h4/real/real__of__num h4/num/0 /\ h4/binary__ieee/float__Sign x = h4/binary__ieee/float__Sign y) (h4/binary__ieee/float__Sign x = h4/words/n2w (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (mode = h4/binary__ieee/roundTowardNegative)) (h4/realax/real__add r1 r2)
% Assm: h4/binary__ieee/round__roundTiesToEven0: !y x r. h4/binary__ieee/float__value0 y = h4/binary__ieee/Float r /\ (h4/binary__ieee/float__Significand y = h4/words/n2w h4/num/0 /\ ~(h4/binary__ieee/float__Exponent y = h4/words/n2w (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) /\ ~h4/real/real__lte (h4/real/abs r) (h4/real/abs x)) /\ h4/real/real__lte (h4/realax/real__mul (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/real/abs (h4/real/real__sub r x))) (h4/binary__ieee/ULP (h4/pair/_2C (h4/binary__ieee/float__Exponent y) h4/bool/the__value)) /\ h4/realax/real__lt (h4/binary__ieee/ulp h4/bool/the__value) (h4/realax/real__mul (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/real/abs x)) /\ h4/realax/real__lt (h4/real/abs x) (h4/binary__ieee/threshold h4/bool/the__value) ==> h4/binary__ieee/round h4/binary__ieee/roundTiesToEven x = y
% Assm: h4/binary__ieee/float__values_c0: h4/binary__ieee/float__value0 (h4/binary__ieee/float__plus__infinity h4/bool/the__value) = h4/binary__ieee/Infinity
% Assm: h4/binary__ieee/float__values_c1: h4/binary__ieee/float__value0 (h4/binary__ieee/float__minus__infinity h4/bool/the__value) = h4/binary__ieee/Infinity
% Assm: h4/binary__ieee/float__minus__infinity__def: h4/binary__ieee/float__minus__infinity h4/bool/the__value = h4/binary__ieee/float__negate (h4/binary__ieee/float__plus__infinity h4/bool/the__value)
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/binary__ieee/NaN0: h4/binary__ieee/NaN = h4/binary__ieee/_20_40ind__typebinary__ieee5
% Assm: h4/ind__type/FCONS0_c0: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm: h4/binary__ieee/hidden____20__40ind____typebinary____ieee3____def: h4/binary__ieee/_20_40ind__typebinary__ieee3 = (\a. h4/binary__ieee/_20_40ind__typebinary__ieee6 ((\a0. h4/ind__type/CONSTR h4/num/0 a0 (\n. h4/ind__type/BOTTOM)) a))
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/binary__ieee/hidden____20__40ind____typebinary____ieee4____def: h4/binary__ieee/_20_40ind__typebinary__ieee4 = h4/binary__ieee/_20_40ind__typebinary__ieee6 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB (\n. h4/ind__type/BOTTOM))
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/binary__ieee/float__value__repfns_c1: !r. (\a0. !_27float__value_27. (!a00. (?a. a00 = (\a1. h4/ind__type/CONSTR h4/num/0 a1 (\n. h4/ind__type/BOTTOM)) a) \/ a00 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB (\n. h4/ind__type/BOTTOM) \/ a00 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB (\n. h4/ind__type/BOTTOM) ==> _27float__value_27 a00) ==> _27float__value_27 a0) r <=> h4/binary__ieee/_20_40ind__typebinary__ieee7 (h4/binary__ieee/_20_40ind__typebinary__ieee6 r) = r
% Assm: h4/binary__ieee/Float0: h4/binary__ieee/Float = h4/binary__ieee/_20_40ind__typebinary__ieee3
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/ind__type/FCONS0_c1: !n f a. h4/ind__type/FCONS a f (h4/num/SUC n) = f n
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/binary__ieee/Infinity0: h4/binary__ieee/Infinity = h4/binary__ieee/_20_40ind__typebinary__ieee4
% Assm: h4/binary__ieee/hidden____20__40ind____typebinary____ieee5____def: h4/binary__ieee/_20_40ind__typebinary__ieee5 = h4/binary__ieee/_20_40ind__typebinary__ieee6 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB (\n. h4/ind__type/BOTTOM))
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% 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/binary__ieee/float__value__case__def_c2: !v1 v f. h4/binary__ieee/float__value__CASE h4/binary__ieee/NaN f v v1 = v1
% Assm: h4/bool/OR__CLAUSES_c4: !t. t \/ t <=> t
% Assm: h4/binary__ieee/float__value__repfns_c0: !a. h4/binary__ieee/_20_40ind__typebinary__ieee6 (h4/binary__ieee/_20_40ind__typebinary__ieee7 a) = a
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/numeral/numeral__add_c4: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT2 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT1 (h4/num/SUC (h4/arithmetic/_2B n m))
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/numeral/numeral__add_c5: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT2 n) (h4/arithmetic/BIT2 m)) = h4/arithmetic/BIT2 (h4/num/SUC (h4/arithmetic/_2B n m))
% Assm: h4/bool/LET__THM: !x f. h4/bool/LET f x = f x
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/numeral/numeral__add_c2: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT2 (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm: h4/numeral/numeral__add_c6: !n. h4/num/SUC (h4/arithmetic/_2B h4/arithmetic/ZERO n) = h4/num/SUC n
% Assm: h4/numeral/numeral__add_c3: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT2 m)) = h4/arithmetic/BIT1 (h4/num/SUC (h4/arithmetic/_2B n m))
% Assm: h4/numeral/numeral__add_c0: !n. h4/numeral/iZ (h4/arithmetic/_2B h4/arithmetic/ZERO n) = n
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/numeral/numeral__distrib_c27: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__distrib_c26: !n. h4/arithmetic/_3C_3D h4/num/0 n <=> T
% Assm: h4/numeral/numeral__distrib_c32: !n. h4/arithmetic/ODD (h4/arithmetic/NUMERAL n) <=> h4/arithmetic/ODD n
% Assm: h4/numeral/numeral__distrib_c21: !n. h4/prim__rec/_3C h4/num/0 (h4/arithmetic/NUMERAL n) <=> h4/prim__rec/_3C h4/arithmetic/ZERO n
% Assm: h4/numeral/numeral__distrib_c20: !n. h4/prim__rec/_3C n h4/num/0 <=> F
% Assm: h4/numeral/numeral__distrib_c8: !n m. h4/arithmetic/_2D (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/arithmetic/_2D n m)
% Assm: h4/numeral/numeral__distrib_c17: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__distrib_c5: !n m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/arithmetic/_2A n m)
% Assm: h4/numeral/numeral__add_c7: !n. h4/num/SUC (h4/arithmetic/_2B n h4/arithmetic/ZERO) = h4/num/SUC n
% Assm: h4/numeral/numeral__distrib_c34: ~h4/arithmetic/ODD h4/num/0
% Assm: h4/numeral/numeral__distrib_c18: !n. h4/num/0 = h4/arithmetic/NUMERAL n <=> n = h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__distrib_c2: !n m. h4/arithmetic/_2B (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm: h4/bool/LET__DEF: h4/bool/LET = (\f x. f x)
% Assm: h4/arithmetic/EXP__1_c1: !n. h4/arithmetic/EXP n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = n
% Assm: h4/numeral/numeral__suc_c0: h4/num/SUC h4/arithmetic/ZERO = h4/arithmetic/BIT1 h4/arithmetic/ZERO
% Assm: h4/arithmetic/EXP__1_c0: !n. h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% 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/arithmetic/LESS__MOD: !n k. h4/prim__rec/_3C k n ==> h4/arithmetic/MOD k n = k
% Assm: h4/arithmetic/LE_c0: !n. h4/arithmetic/_3C_3D n h4/num/0 <=> n = h4/num/0
% Assm: h4/arithmetic/SUB__LEFT__ADD: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2D n p) = h4/bool/COND (h4/arithmetic/_3C_3D n p) m (h4/arithmetic/_2D (h4/arithmetic/_2B m n) p)
% Assm: h4/numeral/numeral__suc_c1: !n. h4/num/SUC (h4/arithmetic/BIT1 n) = h4/arithmetic/BIT2 n
% Assm: h4/arithmetic/ZERO__MOD: !n. h4/prim__rec/_3C h4/num/0 n ==> h4/arithmetic/MOD h4/num/0 n = h4/num/0
% Assm: h4/arithmetic/SUB__RIGHT__EQ: !p n m. h4/arithmetic/_2D m n = p <=> m = h4/arithmetic/_2B n p \/ h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D p h4/num/0
% Assm: h4/arithmetic/MULT__LESS__EQ__SUC: !p n m. h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/arithmetic/_2A (h4/num/SUC p) m) (h4/arithmetic/_2A (h4/num/SUC p) n)
% Assm: h4/arithmetic/ADD__MONO__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm: h4/arithmetic/NOT__LEQ: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm: h4/bool/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm: h4/bool/SELECT__ELIM__THM: !Q P. (?x. P x) /\ (!x. P x ==> Q x) ==> Q (h4/min/_40 P)
% Assm: h4/arithmetic/NOT__NUM__EQ: !n m. ~(m = n) <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n \/ h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm: h4/arithmetic/NOT__LESS__EQUAL: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/prim__rec/_3C n m
% Assm: h4/arithmetic/TIMES2: !n. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n = h4/arithmetic/_2B n n
% Assm: h4/arithmetic/ADD__EQ__0: !n m. h4/arithmetic/_2B m n = h4/num/0 <=> m = h4/num/0 /\ n = h4/num/0
% Assm: h4/arithmetic/LESS__EQ__TRANS: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm: h4/arithmetic/MULT__ASSOC: !p n m. h4/arithmetic/_2A m (h4/arithmetic/_2A n p) = h4/arithmetic/_2A (h4/arithmetic/_2A m n) p
% Assm: h4/arithmetic/MULT__CLAUSES_c3: !m. h4/arithmetic/_2A m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = m
% Assm: h4/numeral/numeral__add_c8: !n m. h4/num/SUC (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT1 (h4/num/SUC (h4/arithmetic/_2B n m))
% Assm: h4/numeral/numeral__distrib_c1: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm: h4/arithmetic/SUC__ONE__ADD: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/arithmetic/RIGHT__ADD__DISTRIB: !p n m. h4/arithmetic/_2A (h4/arithmetic/_2B m n) p = h4/arithmetic/_2B (h4/arithmetic/_2A m p) (h4/arithmetic/_2A n p)
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/ADD__0: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/arithmetic/ADD__ASSOC: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm: h4/arithmetic/TWO: h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO) = h4/num/SUC (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/relation/WFREC__COROLLARY: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. f x = M (h4/relation/RESTRICT f R x) x)
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/arithmetic/LESS__EQ: !n m. h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm: h4/relation/WF__EMPTY__REL: h4/relation/WF h4/relation/EMPTY__REL
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/bool/LEFT__FORALL__IMP__THM: !Q P. (!x. P x ==> Q) <=> (?x. P x) ==> Q
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Goal: !x r mode. h4/binary__ieee/float__value0 x = h4/binary__ieee/Float r ==> h4/binary__ieee/float__sub mode (h4/binary__ieee/float__plus__infinity h4/bool/the__value) x = h4/binary__ieee/float__plus__infinity h4/bool/the__value
%   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_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_binaryu_u_ieees_floatu_u_valueu_u_caseu_u_defu_c0]: !v1 v f a. h4/binary__ieee/float__value__CASE (happ h4/binary__ieee/Float a) f v v1 = happ f a
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_pairs_pairu_u_caseu_u_thm]: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = happ (happ f x) y
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% 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_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_binaryu_u_ieees_floatu_u_subu_u_def]: !_4. (!x v7. happ (happ _4 x) v7 = x) ==> (!_3. (!y x mode r1 r2. ?v. (v <=> mode = h4/binary__ieee/roundTowardNegative) /\ (?v'. (v' <=> h4/binary__ieee/float__Sign x = h4/words/n2w (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) /\ (?v''. (v'' <=> r1 = h4/real/real__of__num h4/num/0 /\ r2 = h4/real/real__of__num h4/num/0 /\ ~(h4/binary__ieee/float__Sign x = h4/binary__ieee/float__Sign y)) /\ happ (happ (happ (happ (happ _3 y) x) mode) r1) r2 = h4/binary__ieee/float__round mode (h4/bool/COND v'' v' v) (h4/real/real__sub r1 r2)))) ==> (!_2. (!v1 x mode y r1. happ (happ (happ (happ (happ _2 v1) x) mode) y) r1 = h4/binary__ieee/float__value__CASE v1 (happ (happ (happ (happ _3 y) x) mode) r1) (h4/binary__ieee/float__negate y) (h4/binary__ieee/float__some__nan h4/bool/the__value)) ==> (!_1. (!v mode y x v1. ?v. (v <=> h4/binary__ieee/float__Sign x = h4/binary__ieee/float__Sign y) /\ happ (happ (happ (happ (happ _1 v) mode) y) x) v1 = h4/binary__ieee/float__value__CASE v (happ (happ (happ (happ _2 v1) x) mode) y) (h4/binary__ieee/float__value__CASE v1 (happ _4 x) (h4/bool/COND v (h4/binary__ieee/float__some__nan h4/bool/the__value) x) (h4/binary__ieee/float__some__nan h4/bool/the__value)) (h4/binary__ieee/float__some__nan h4/bool/the__value)) ==> (!_0. (!mode y x v. happ (happ (happ (happ _0 mode) y) x) v = happ (happ (happ (happ _1 v) mode) y) x) ==> (!y x mode. h4/binary__ieee/float__sub mode x y = h4/pair/pair__CASE (h4/pair/_2C (h4/binary__ieee/float__value0 x) (h4/binary__ieee/float__value0 y)) (happ (happ (happ _0 mode) y) x))))))
% Assm [h4s_binaryu_u_ieees_floatu_u_subu_u_finiteu_u_plusu_u_infinity]: !x r mode. h4/binary__ieee/float__value0 x = happ h4/binary__ieee/Float r ==> h4/binary__ieee/float__sub mode x (h4/binary__ieee/float__plus__infinity h4/bool/the__value) = h4/binary__ieee/float__minus__infinity h4/bool/the__value
% Assm [h4s_binaryu_u_ieees_floatu_u_subu_u_finite]: !y x r2 r1 mode. h4/binary__ieee/float__value0 x = happ h4/binary__ieee/Float r1 /\ h4/binary__ieee/float__value0 y = happ h4/binary__ieee/Float r2 ==> (?v. (v <=> mode = h4/binary__ieee/roundTowardNegative) /\ (?v'. (v' <=> h4/binary__ieee/float__Sign x = h4/words/n2w (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) /\ (?v''. (v'' <=> r1 = h4/real/real__of__num h4/num/0 /\ r2 = h4/real/real__of__num h4/num/0 /\ ~(h4/binary__ieee/float__Sign x = h4/binary__ieee/float__Sign y)) /\ h4/binary__ieee/float__sub mode x y = h4/binary__ieee/float__round mode (h4/bool/COND v'' v' v) (h4/real/real__sub r1 r2))))
% Assm [h4s_binaryu_u_ieees_floatu_u_valueu_u_caseu_u_defu_c1]: !v1 v f. h4/binary__ieee/float__value__CASE h4/binary__ieee/Infinity f v v1 = v
% Assm [h4s_binaryu_u_ieees_floatu_u_addu_u_finiteu_u_plusu_u_infinity]: !x r mode. h4/binary__ieee/float__value0 x = happ h4/binary__ieee/Float r ==> h4/binary__ieee/float__add mode x (h4/binary__ieee/float__plus__infinity h4/bool/the__value) = h4/binary__ieee/float__plus__infinity h4/bool/the__value
% Assm [h4s_binaryu_u_ieees_floatu_u_addu_u_plusu_u_infinityu_u_finite]: !x r mode. h4/binary__ieee/float__value0 x = happ h4/binary__ieee/Float r ==> h4/binary__ieee/float__add mode (h4/binary__ieee/float__plus__infinity h4/bool/the__value) x = h4/binary__ieee/float__plus__infinity h4/bool/the__value
% 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_binaryu_u_ieees_floatu_u_addu_u_def]: !_4. (!x v7. happ (happ _4 x) v7 = x) ==> (!_3. (!y x mode r1 r2. ?v. (v <=> mode = h4/binary__ieee/roundTowardNegative) /\ (?v'. (v' <=> h4/binary__ieee/float__Sign x = h4/words/n2w (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) /\ (?v''. (v'' <=> r1 = h4/real/real__of__num h4/num/0 /\ r2 = h4/real/real__of__num h4/num/0 /\ h4/binary__ieee/float__Sign x = h4/binary__ieee/float__Sign y) /\ happ (happ (happ (happ (happ _3 y) x) mode) r1) r2 = h4/binary__ieee/float__round mode (h4/bool/COND v'' v' v) (h4/realax/real__add r1 r2)))) ==> (!_2. (!v1 x mode y r1. happ (happ (happ (happ (happ _2 v1) x) mode) y) r1 = h4/binary__ieee/float__value__CASE v1 (happ (happ (happ (happ _3 y) x) mode) r1) y (h4/binary__ieee/float__some__nan h4/bool/the__value)) ==> (!_1. (!v mode y x v1. ?v. (v <=> h4/binary__ieee/float__Sign x = h4/binary__ieee/float__Sign y) /\ happ (happ (happ (happ (happ _1 v) mode) y) x) v1 = h4/binary__ieee/float__value__CASE v (happ (happ (happ (happ _2 v1) x) mode) y) (h4/binary__ieee/float__value__CASE v1 (happ _4 x) (h4/bool/COND v x (h4/binary__ieee/float__some__nan h4/bool/the__value)) (h4/binary__ieee/float__some__nan h4/bool/the__value)) (h4/binary__ieee/float__some__nan h4/bool/the__value)) ==> (!_0. (!mode y x v. happ (happ (happ (happ _0 mode) y) x) v = happ (happ (happ (happ _1 v) mode) y) x) ==> (!y x mode. h4/binary__ieee/float__add mode x y = h4/pair/pair__CASE (h4/pair/_2C (h4/binary__ieee/float__value0 x) (h4/binary__ieee/float__value0 y)) (happ (happ (happ _0 mode) y) x))))))
% Assm [h4s_binaryu_u_ieees_floatu_u_subu_u_nan]: !y x mode. h4/binary__ieee/float__value0 x = h4/binary__ieee/NaN \/ h4/binary__ieee/float__value0 y = h4/binary__ieee/NaN ==> h4/binary__ieee/float__sub mode x y = h4/binary__ieee/float__some__nan h4/bool/the__value
% Assm [h4s_binaryu_u_ieees_floatu_u_addu_u_minusu_u_infinityu_u_finite]: !x r mode. h4/binary__ieee/float__value0 x = happ h4/binary__ieee/Float r ==> h4/binary__ieee/float__add mode (h4/binary__ieee/float__minus__infinity h4/bool/the__value) x = h4/binary__ieee/float__minus__infinity h4/bool/the__value
% Assm [h4s_binaryu_u_ieees_floatu_u_addu_u_finiteu_u_minusu_u_infinity]: !x r mode. h4/binary__ieee/float__value0 x = happ h4/binary__ieee/Float r ==> h4/binary__ieee/float__add mode x (h4/binary__ieee/float__minus__infinity h4/bool/the__value) = h4/binary__ieee/float__minus__infinity h4/bool/the__value
% Assm [h4s_binaryu_u_ieees_floatu_u_addu_u_finite]: !y x r2 r1 mode. h4/binary__ieee/float__value0 x = happ h4/binary__ieee/Float r1 /\ h4/binary__ieee/float__value0 y = happ h4/binary__ieee/Float r2 ==> (?v. (v <=> mode = h4/binary__ieee/roundTowardNegative) /\ (?v'. (v' <=> h4/binary__ieee/float__Sign x = h4/words/n2w (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) /\ (?v''. (v'' <=> r1 = h4/real/real__of__num h4/num/0 /\ r2 = h4/real/real__of__num h4/num/0 /\ h4/binary__ieee/float__Sign x = h4/binary__ieee/float__Sign y) /\ h4/binary__ieee/float__add mode x y = h4/binary__ieee/float__round mode (h4/bool/COND v'' v' v) (h4/realax/real__add r1 r2))))
% Assm [h4s_binaryu_u_ieees_roundu_u_roundTiesToEven0]: !y x r. h4/binary__ieee/float__value0 y = happ h4/binary__ieee/Float r /\ (h4/binary__ieee/float__Significand y = h4/words/n2w h4/num/0 /\ ~(h4/binary__ieee/float__Exponent y = h4/words/n2w (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) /\ ~h4/real/real__lte (h4/real/abs r) (h4/real/abs x)) /\ h4/real/real__lte (h4/realax/real__mul (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/real/abs (h4/real/real__sub r x))) (h4/binary__ieee/ULP (h4/pair/_2C (h4/binary__ieee/float__Exponent y) h4/bool/the__value)) /\ h4/realax/real__lt (h4/binary__ieee/ulp h4/bool/the__value) (h4/realax/real__mul (h4/real/real__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/real/abs x)) /\ h4/realax/real__lt (h4/real/abs x) (h4/binary__ieee/threshold h4/bool/the__value) ==> h4/binary__ieee/round h4/binary__ieee/roundTiesToEven x = y
% Assm [h4s_binaryu_u_ieees_floatu_u_valuesu_c0]: h4/binary__ieee/float__value0 (h4/binary__ieee/float__plus__infinity h4/bool/the__value) = h4/binary__ieee/Infinity
% Assm [h4s_binaryu_u_ieees_floatu_u_valuesu_c1]: h4/binary__ieee/float__value0 (h4/binary__ieee/float__minus__infinity h4/bool/the__value) = h4/binary__ieee/Infinity
% Assm [h4s_binaryu_u_ieees_floatu_u_minusu_u_infinityu_u_def]: h4/binary__ieee/float__minus__infinity h4/bool/the__value = h4/binary__ieee/float__negate (h4/binary__ieee/float__plus__infinity h4/bool/the__value)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_binaryu_u_ieees_NaN0]: h4/binary__ieee/NaN = h4/binary__ieee/_20_40ind__typebinary__ieee5
% Assm [h4s_indu_u_types_FCONS0u_c0]: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm [h4s_binaryu_u_ieees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typebinaryu_u_u_u_ieee3u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x. happ h4/binary__ieee/_20_40ind__typebinary__ieee3 x = h4/binary__ieee/_20_40ind__typebinary__ieee6 (h4/ind__type/CONSTR h4/num/0 x _0))
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_binaryu_u_ieees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typebinaryu_u_u_u_ieee4u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> h4/binary__ieee/_20_40ind__typebinary__ieee4 = h4/binary__ieee/_20_40ind__typebinary__ieee6 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB _0)
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_binaryu_u_ieees_floatu_u_valueu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27float__value_27. (!a00. (?a. a00 = h4/ind__type/CONSTR h4/num/0 a _0) \/ a00 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB _0 \/ a00 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB _0 ==> happ _27float__value_27 a00) ==> happ _27float__value_27 r) <=> h4/binary__ieee/_20_40ind__typebinary__ieee7 (h4/binary__ieee/_20_40ind__typebinary__ieee6 r) = r)
% Assm [h4s_binaryu_u_ieees_Float0]: h4/binary__ieee/Float = h4/binary__ieee/_20_40ind__typebinary__ieee3
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_indu_u_types_FCONS0u_c1]: !n f a. h4/ind__type/FCONS a f (h4/num/SUC n) = happ f n
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_binaryu_u_ieees_Infinity0]: h4/binary__ieee/Infinity = h4/binary__ieee/_20_40ind__typebinary__ieee4
% Assm [h4s_binaryu_u_ieees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typebinaryu_u_u_u_ieee5u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> h4/binary__ieee/_20_40ind__typebinary__ieee5 = h4/binary__ieee/_20_40ind__typebinary__ieee6 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB _0)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% 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_binaryu_u_ieees_floatu_u_valueu_u_caseu_u_defu_c2]: !v1 v f. h4/binary__ieee/float__value__CASE h4/binary__ieee/NaN f v v1 = v1
% Assm [h4s_bools_ORu_u_CLAUSESu_c4]: !t. t \/ t <=> t
% Assm [h4s_binaryu_u_ieees_floatu_u_valueu_u_repfnsu_c0]: !a. h4/binary__ieee/_20_40ind__typebinary__ieee6 (h4/binary__ieee/_20_40ind__typebinary__ieee7 a) = a
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_numerals_numeralu_u_addu_c4]: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT2 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT1 (h4/num/SUC (h4/arithmetic/_2B n m))
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_numerals_numeralu_u_addu_c5]: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT2 n) (h4/arithmetic/BIT2 m)) = h4/arithmetic/BIT2 (h4/num/SUC (h4/arithmetic/_2B n m))
% Assm [h4s_bools_LETu_u_THM]: !x f. h4/bool/LET f x = happ f x
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_numerals_numeralu_u_addu_c2]: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT2 (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm [h4s_numerals_numeralu_u_addu_c6]: !n. h4/num/SUC (h4/arithmetic/_2B h4/arithmetic/ZERO n) = h4/num/SUC n
% Assm [h4s_numerals_numeralu_u_addu_c3]: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT2 m)) = h4/arithmetic/BIT1 (h4/num/SUC (h4/arithmetic/_2B n m))
% Assm [h4s_numerals_numeralu_u_addu_c0]: !n. h4/numeral/iZ (h4/arithmetic/_2B h4/arithmetic/ZERO n) = n
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_numerals_numeralu_u_distribu_c27]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_distribu_c26]: !n. h4/arithmetic/_3C_3D h4/num/0 n <=> T
% Assm [h4s_numerals_numeralu_u_distribu_c32]: !n. h4/arithmetic/ODD (h4/arithmetic/NUMERAL n) <=> h4/arithmetic/ODD n
% Assm [h4s_numerals_numeralu_u_distribu_c21]: !n. h4/prim__rec/_3C h4/num/0 (h4/arithmetic/NUMERAL n) <=> h4/prim__rec/_3C h4/arithmetic/ZERO n
% Assm [h4s_numerals_numeralu_u_distribu_c20]: !n. h4/prim__rec/_3C n h4/num/0 <=> F
% Assm [h4s_numerals_numeralu_u_distribu_c8]: !n m. h4/arithmetic/_2D (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/arithmetic/_2D n m)
% Assm [h4s_numerals_numeralu_u_distribu_c17]: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_distribu_c5]: !n m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/arithmetic/_2A n m)
% Assm [h4s_numerals_numeralu_u_addu_c7]: !n. h4/num/SUC (h4/arithmetic/_2B n h4/arithmetic/ZERO) = h4/num/SUC n
% Assm [h4s_numerals_numeralu_u_distribu_c34]: ~h4/arithmetic/ODD h4/num/0
% Assm [h4s_numerals_numeralu_u_distribu_c18]: !n. h4/num/0 = h4/arithmetic/NUMERAL n <=> n = h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_distribu_c2]: !n m. h4/arithmetic/_2B (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm [h4s_bools_LETu_u_DEF]: !x x. h4/bool/LET x x = happ x x
% Assm [h4s_arithmetics_EXPu_u_1u_c1]: !n. h4/arithmetic/EXP n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = n
% Assm [h4s_numerals_numeralu_u_sucu_c0]: h4/num/SUC h4/arithmetic/ZERO = h4/arithmetic/BIT1 h4/arithmetic/ZERO
% Assm [h4s_arithmetics_EXPu_u_1u_c0]: !n. h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% 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_arithmetics_LESSu_u_MOD]: !n k. h4/prim__rec/_3C k n ==> h4/arithmetic/MOD k n = k
% Assm [h4s_arithmetics_LEu_c0]: !n. h4/arithmetic/_3C_3D n h4/num/0 <=> n = h4/num/0
% Assm [h4s_arithmetics_SUBu_u_LEFTu_u_ADD]: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2D n p) = h4/bool/COND (h4/arithmetic/_3C_3D n p) m (h4/arithmetic/_2D (h4/arithmetic/_2B m n) p)
% Assm [h4s_numerals_numeralu_u_sucu_c1]: !n. h4/num/SUC (h4/arithmetic/BIT1 n) = h4/arithmetic/BIT2 n
% Assm [h4s_arithmetics_ZEROu_u_MOD]: !n. h4/prim__rec/_3C h4/num/0 n ==> h4/arithmetic/MOD h4/num/0 n = h4/num/0
% Assm [h4s_arithmetics_SUBu_u_RIGHTu_u_EQ]: !p n m. h4/arithmetic/_2D m n = p <=> m = h4/arithmetic/_2B n p \/ h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D p h4/num/0
% Assm [h4s_arithmetics_MULTu_u_LESSu_u_EQu_u_SUC]: !p n m. h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/arithmetic/_2A (h4/num/SUC p) m) (h4/arithmetic/_2A (h4/num/SUC p) n)
% Assm [h4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm [h4s_arithmetics_NOTu_u_LEQ]: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm [h4s_bools_SELECTu_u_ELIMu_u_THM]: !Q P. (?x. happ P x) /\ (!x. happ P x ==> happ Q x) ==> happ Q (h4/min/_40 P)
% Assm [h4s_arithmetics_NOTu_u_NUMu_u_EQ]: !n m. ~(m = n) <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n \/ h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm [h4s_arithmetics_NOTu_u_LESSu_u_EQUAL]: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/prim__rec/_3C n m
% Assm [h4s_arithmetics_TIMES2]: !n. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n = h4/arithmetic/_2B n n
% Assm [h4s_arithmetics_ADDu_u_EQu_u_0]: !n m. h4/arithmetic/_2B m n = h4/num/0 <=> m = h4/num/0 /\ n = h4/num/0
% Assm [h4s_arithmetics_LESSu_u_EQu_u_TRANS]: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm [h4s_arithmetics_MULTu_u_ASSOC]: !p n m. h4/arithmetic/_2A m (h4/arithmetic/_2A n p) = h4/arithmetic/_2A (h4/arithmetic/_2A m n) p
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c3]: !m. h4/arithmetic/_2A m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = m
% Assm [h4s_numerals_numeralu_u_addu_c8]: !n m. h4/num/SUC (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT1 (h4/num/SUC (h4/arithmetic/_2B n m))
% Assm [h4s_numerals_numeralu_u_distribu_c1]: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm [h4s_arithmetics_SUCu_u_ONEu_u_ADD]: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm [h4s_arithmetics_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_arithmetics_RIGHTu_u_ADDu_u_DISTRIB]: !p n m. h4/arithmetic/_2A (h4/arithmetic/_2B m n) p = h4/arithmetic/_2B (h4/arithmetic/_2A m p) (h4/arithmetic/_2A n p)
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_ADDu_u_0]: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm [h4s_arithmetics_ADDu_u_ASSOC]: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm [h4s_arithmetics_TWO]: h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO) = h4/num/SUC (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_relations_WFRECu_u_COROLLARY]: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. happ f x = happ (happ M (h4/relation/RESTRICT f R x)) x)
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_arithmetics_LESSu_u_EQ]: !n m. h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm [h4s_relations_WFu_u_EMPTYu_u_REL]: h4/relation/WF h4/relation/EMPTY__REL
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_bools_LEFTu_u_FORALLu_u_IMPu_u_THM]: !Q P. (!x. happ P x ==> Q) <=> (?x. happ P x) ==> Q
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Goal: !x r mode. h4/binary__ieee/float__value0 x = happ h4/binary__ieee/Float r ==> h4/binary__ieee/float__sub mode (h4/binary__ieee/float__plus__infinity h4/bool/the__value) x = h4/binary__ieee/float__plus__infinity h4/bool/the__value
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_Q1191770,TV_Q1191766]: ![V_f, V_g]: (![V_x]: s(TV_Q1191766,happ(s(t_fun(TV_Q1191770,TV_Q1191766),V_f),s(TV_Q1191770,V_x))) = s(TV_Q1191766,happ(s(t_fun(TV_Q1191770,TV_Q1191766),V_g),s(TV_Q1191770,V_x))) => s(t_fun(TV_Q1191770,TV_Q1191766),V_f) = s(t_fun(TV_Q1191770,TV_Q1191766),V_g))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_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_binaryu_u_ieees_floatu_u_valueu_u_caseu_u_defu_c0, axiom, ![TV_u_27a]: ![V_v1, V_v, V_f, V_a]: s(TV_u_27a,h4s_binaryu_u_ieees_floatu_u_valueu_u_case(s(t_h4s_binaryu_u_ieees_floatu_u_value,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_floatu_u_value),h4s_binaryu_u_ieees_float),s(t_h4s_realaxs_real,V_a))),s(t_fun(t_h4s_realaxs_real,TV_u_27a),V_f),s(TV_u_27a,V_v),s(TV_u_27a,V_v1))) = s(TV_u_27a,happ(s(t_fun(t_h4s_realaxs_real,TV_u_27a),V_f),s(t_h4s_realaxs_real,V_a)))).
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_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_pairs_pairu_u_caseu_u_thm, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_y, V_x, V_f]: s(TV_u_27a,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27b,V_x),s(TV_u_27c,V_y))),s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f))) = s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27b,V_x))),s(TV_u_27c,V_y)))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_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_binaryu_u_ieees_floatu_u_subu_u_def, axiom, ![TV_u_27t,TV_u_27w]: ![V_uu_4]: (![V_x, V_v7]: s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),V_uu_4),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_realaxs_real,V_v7))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x) => ![V_uu_3]: (![V_y, V_x, V_mode, V_r1, V_r2]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_binaryu_u_ieees_rounding,V_mode) = s(t_h4s_binaryu_u_ieees_rounding,h4s_binaryu_u_ieees_roundtowardnegative)) & ?[V_vi_]: ((p(s(t_bool,V_vi_)) <=> s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_binaryu_u_ieees_floatu_u_sign(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))) = s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_wordss_n2w(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))) & ?[V_vi_i_]: ((p(s(t_bool,V_vi_i_)) <=> (s(t_h4s_realaxs_real,V_r1) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) & (s(t_h4s_realaxs_real,V_r2) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) & ~ (s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_binaryu_u_ieees_floatu_u_sign(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))) = s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_binaryu_u_ieees_floatu_u_sign(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))))))) & s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),happ(s(t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))))),V_uu_3),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_rounding,V_mode))),s(t_h4s_realaxs_real,V_r1))),s(t_h4s_realaxs_real,V_r2))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_round(s(t_h4s_binaryu_u_ieees_rounding,V_mode),s(t_bool,h4s_bools_cond(s(t_bool,V_vi_i_),s(t_bool,V_vi_),s(t_bool,V_v))),s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_r1),s(t_h4s_realaxs_real,V_r2)))))))) => ![V_uu_2]: (![V_v1, V_x, V_mode, V_y, V_r1]: s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),happ(s(t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))))),happ(s(t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))))),V_uu_2),s(t_h4s_binaryu_u_ieees_floatu_u_value,V_v1))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_rounding,V_mode))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))),s(t_h4s_realaxs_real,V_r1))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_valueu_u_case(s(t_h4s_binaryu_u_ieees_floatu_u_value,V_v1),s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),happ(s(t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))))),V_uu_3),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_rounding,V_mode))),s(t_h4s_realaxs_real,V_r1))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_negate(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_someu_u_nan(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))))) => ![V_uu_1]: (![V_v, V_mode, V_y, V_x, V_v1]: ?[V_v0]: ((p(s(t_bool,V_v0)) <=> s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_binaryu_u_ieees_floatu_u_sign(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))) = s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_binaryu_u_ieees_floatu_u_sign(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y)))) & s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),happ(s(t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))),happ(s(t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))))),happ(s(t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))))),V_uu_1),s(t_h4s_binaryu_u_ieees_floatu_u_value,V_v))),s(t_h4s_binaryu_u_ieees_rounding,V_mode))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_floatu_u_value,V_v1))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_valueu_u_case(s(t_h4s_binaryu_u_ieees_floatu_u_value,V_v),s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),happ(s(t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))))),happ(s(t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))))),V_uu_2),s(t_h4s_binaryu_u_ieees_floatu_u_value,V_v1))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_rounding,V_mode))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_valueu_u_case(s(t_h4s_binaryu_u_ieees_floatu_u_value,V_v1),s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),V_uu_4),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_bools_cond(s(t_bool,V_v0),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_someu_u_nan(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_someu_u_nan(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_someu_u_nan(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value)))))) => ![V_uu_0]: (![V_mode, V_y, V_x, V_v]: s(t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))))),happ(s(t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))))),V_uu_0),s(t_h4s_binaryu_u_ieees_rounding,V_mode))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_floatu_u_value,V_v))) = s(t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))),happ(s(t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))))),happ(s(t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))))),V_uu_1),s(t_h4s_binaryu_u_ieees_floatu_u_value,V_v))),s(t_h4s_binaryu_u_ieees_rounding,V_mode))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))) => ![V_y, V_x, V_mode]: s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_sub(s(t_h4s_binaryu_u_ieees_rounding,V_mode),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_floatu_u_value),h4s_pairs_u_2c(s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_floatu_u_value0(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_floatu_u_value0(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))))),s(t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))))),happ(s(t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))))),V_uu_0),s(t_h4s_binaryu_u_ieees_rounding,V_mode))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))))))))))).
fof(ah4s_binaryu_u_ieees_floatu_u_subu_u_finiteu_u_plusu_u_infinity, axiom, ![TV_u_27t,TV_u_27w]: ![V_x, V_r, V_mode]: (s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_floatu_u_value0(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))) = s(t_h4s_binaryu_u_ieees_floatu_u_value,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_floatu_u_value),h4s_binaryu_u_ieees_float),s(t_h4s_realaxs_real,V_r))) => s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_sub(s(t_h4s_binaryu_u_ieees_rounding,V_mode),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_plusu_u_infinity(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_minusu_u_infinity(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))))).
fof(ah4s_binaryu_u_ieees_floatu_u_subu_u_finite, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_r2, V_r1, V_mode]: ((s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_floatu_u_value0(s(t_h4s_binaryu_u_ieees_float(TV_u_27a,TV_u_27b),V_x))) = s(t_h4s_binaryu_u_ieees_floatu_u_value,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_floatu_u_value),h4s_binaryu_u_ieees_float),s(t_h4s_realaxs_real,V_r1))) & s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_floatu_u_value0(s(t_h4s_binaryu_u_ieees_float(TV_u_27a,TV_u_27b),V_y))) = s(t_h4s_binaryu_u_ieees_floatu_u_value,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_floatu_u_value),h4s_binaryu_u_ieees_float),s(t_h4s_realaxs_real,V_r2)))) => ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_binaryu_u_ieees_rounding,V_mode) = s(t_h4s_binaryu_u_ieees_rounding,h4s_binaryu_u_ieees_roundtowardnegative)) & ?[V_vi_]: ((p(s(t_bool,V_vi_)) <=> s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_binaryu_u_ieees_floatu_u_sign(s(t_h4s_binaryu_u_ieees_float(TV_u_27a,TV_u_27b),V_x))) = s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_wordss_n2w(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))) & ?[V_vi_i_]: ((p(s(t_bool,V_vi_i_)) <=> (s(t_h4s_realaxs_real,V_r1) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) & (s(t_h4s_realaxs_real,V_r2) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) & ~ (s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_binaryu_u_ieees_floatu_u_sign(s(t_h4s_binaryu_u_ieees_float(TV_u_27a,TV_u_27b),V_x))) = s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_binaryu_u_ieees_floatu_u_sign(s(t_h4s_binaryu_u_ieees_float(TV_u_27a,TV_u_27b),V_y))))))) & s(t_h4s_binaryu_u_ieees_float(TV_u_27a,TV_u_27b),h4s_binaryu_u_ieees_floatu_u_sub(s(t_h4s_binaryu_u_ieees_rounding,V_mode),s(t_h4s_binaryu_u_ieees_float(TV_u_27a,TV_u_27b),V_x),s(t_h4s_binaryu_u_ieees_float(TV_u_27a,TV_u_27b),V_y))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27a,TV_u_27b),h4s_binaryu_u_ieees_floatu_u_round(s(t_h4s_binaryu_u_ieees_rounding,V_mode),s(t_bool,h4s_bools_cond(s(t_bool,V_vi_i_),s(t_bool,V_vi_),s(t_bool,V_v))),s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_r1),s(t_h4s_realaxs_real,V_r2)))))))))).
fof(ah4s_binaryu_u_ieees_floatu_u_valueu_u_caseu_u_defu_c1, axiom, ![TV_u_27a]: ![V_v1, V_v, V_f]: s(TV_u_27a,h4s_binaryu_u_ieees_floatu_u_valueu_u_case(s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_infinity),s(t_fun(t_h4s_realaxs_real,TV_u_27a),V_f),s(TV_u_27a,V_v),s(TV_u_27a,V_v1))) = s(TV_u_27a,V_v)).
fof(ah4s_binaryu_u_ieees_floatu_u_addu_u_finiteu_u_plusu_u_infinity, axiom, ![TV_u_27t,TV_u_27w]: ![V_x, V_r, V_mode]: (s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_floatu_u_value0(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))) = s(t_h4s_binaryu_u_ieees_floatu_u_value,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_floatu_u_value),h4s_binaryu_u_ieees_float),s(t_h4s_realaxs_real,V_r))) => s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_add(s(t_h4s_binaryu_u_ieees_rounding,V_mode),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_plusu_u_infinity(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_plusu_u_infinity(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))))).
fof(ah4s_binaryu_u_ieees_floatu_u_addu_u_plusu_u_infinityu_u_finite, axiom, ![TV_u_27t,TV_u_27w]: ![V_x, V_r, V_mode]: (s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_floatu_u_value0(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))) = s(t_h4s_binaryu_u_ieees_floatu_u_value,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_floatu_u_value),h4s_binaryu_u_ieees_float),s(t_h4s_realaxs_real,V_r))) => s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_add(s(t_h4s_binaryu_u_ieees_rounding,V_mode),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_plusu_u_infinity(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_plusu_u_infinity(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))))).
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_binaryu_u_ieees_floatu_u_addu_u_def, axiom, ![TV_u_27t,TV_u_27w]: ![V_uu_4]: (![V_x, V_v7]: s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),V_uu_4),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_realaxs_real,V_v7))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x) => ![V_uu_3]: (![V_y, V_x, V_mode, V_r1, V_r2]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_binaryu_u_ieees_rounding,V_mode) = s(t_h4s_binaryu_u_ieees_rounding,h4s_binaryu_u_ieees_roundtowardnegative)) & ?[V_vi_]: ((p(s(t_bool,V_vi_)) <=> s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_binaryu_u_ieees_floatu_u_sign(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))) = s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_wordss_n2w(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))) & ?[V_vi_i_]: ((p(s(t_bool,V_vi_i_)) <=> (s(t_h4s_realaxs_real,V_r1) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) & (s(t_h4s_realaxs_real,V_r2) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) & s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_binaryu_u_ieees_floatu_u_sign(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))) = s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_binaryu_u_ieees_floatu_u_sign(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y)))))) & s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),happ(s(t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))))),V_uu_3),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_rounding,V_mode))),s(t_h4s_realaxs_real,V_r1))),s(t_h4s_realaxs_real,V_r2))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_round(s(t_h4s_binaryu_u_ieees_rounding,V_mode),s(t_bool,h4s_bools_cond(s(t_bool,V_vi_i_),s(t_bool,V_vi_),s(t_bool,V_v))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_r1),s(t_h4s_realaxs_real,V_r2)))))))) => ![V_uu_2]: (![V_v1, V_x, V_mode, V_y, V_r1]: s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),happ(s(t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))))),happ(s(t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))))),V_uu_2),s(t_h4s_binaryu_u_ieees_floatu_u_value,V_v1))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_rounding,V_mode))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))),s(t_h4s_realaxs_real,V_r1))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_valueu_u_case(s(t_h4s_binaryu_u_ieees_floatu_u_value,V_v1),s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),happ(s(t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))))),V_uu_3),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_rounding,V_mode))),s(t_h4s_realaxs_real,V_r1))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_someu_u_nan(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))))) => ![V_uu_1]: (![V_v, V_mode, V_y, V_x, V_v1]: ?[V_v0]: ((p(s(t_bool,V_v0)) <=> s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_binaryu_u_ieees_floatu_u_sign(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))) = s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_binaryu_u_ieees_floatu_u_sign(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y)))) & s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),happ(s(t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))),happ(s(t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))))),happ(s(t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))))),V_uu_1),s(t_h4s_binaryu_u_ieees_floatu_u_value,V_v))),s(t_h4s_binaryu_u_ieees_rounding,V_mode))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_floatu_u_value,V_v1))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_valueu_u_case(s(t_h4s_binaryu_u_ieees_floatu_u_value,V_v),s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),happ(s(t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))))),happ(s(t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))))),V_uu_2),s(t_h4s_binaryu_u_ieees_floatu_u_value,V_v1))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_rounding,V_mode))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_valueu_u_case(s(t_h4s_binaryu_u_ieees_floatu_u_value,V_v1),s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),V_uu_4),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_bools_cond(s(t_bool,V_v0),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_someu_u_nan(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_someu_u_nan(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_someu_u_nan(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value)))))) => ![V_uu_0]: (![V_mode, V_y, V_x, V_v]: s(t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))))),happ(s(t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))))),V_uu_0),s(t_h4s_binaryu_u_ieees_rounding,V_mode))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_floatu_u_value,V_v))) = s(t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))),happ(s(t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))))),happ(s(t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))))),V_uu_1),s(t_h4s_binaryu_u_ieees_floatu_u_value,V_v))),s(t_h4s_binaryu_u_ieees_rounding,V_mode))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))) => ![V_y, V_x, V_mode]: s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_add(s(t_h4s_binaryu_u_ieees_rounding,V_mode),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_floatu_u_value),h4s_pairs_u_2c(s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_floatu_u_value0(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_floatu_u_value0(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))))),s(t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))))),happ(s(t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))))),V_uu_0),s(t_h4s_binaryu_u_ieees_rounding,V_mode))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))))))))))).
fof(ah4s_binaryu_u_ieees_floatu_u_subu_u_nan, axiom, ![TV_u_27t,TV_u_27w]: ![V_y, V_x, V_mode]: ((s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_floatu_u_value0(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))) = s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_nan) | s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_floatu_u_value0(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))) = s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_nan)) => s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_sub(s(t_h4s_binaryu_u_ieees_rounding,V_mode),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_someu_u_nan(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))))).
fof(ah4s_binaryu_u_ieees_floatu_u_addu_u_minusu_u_infinityu_u_finite, axiom, ![TV_u_27t,TV_u_27w]: ![V_x, V_r, V_mode]: (s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_floatu_u_value0(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))) = s(t_h4s_binaryu_u_ieees_floatu_u_value,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_floatu_u_value),h4s_binaryu_u_ieees_float),s(t_h4s_realaxs_real,V_r))) => s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_add(s(t_h4s_binaryu_u_ieees_rounding,V_mode),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_minusu_u_infinity(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_minusu_u_infinity(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))))).
fof(ah4s_binaryu_u_ieees_floatu_u_addu_u_finiteu_u_minusu_u_infinity, axiom, ![TV_u_27t,TV_u_27w]: ![V_x, V_r, V_mode]: (s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_floatu_u_value0(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))) = s(t_h4s_binaryu_u_ieees_floatu_u_value,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_floatu_u_value),h4s_binaryu_u_ieees_float),s(t_h4s_realaxs_real,V_r))) => s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_add(s(t_h4s_binaryu_u_ieees_rounding,V_mode),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_minusu_u_infinity(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_minusu_u_infinity(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))))).
fof(ah4s_binaryu_u_ieees_floatu_u_addu_u_finite, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_r2, V_r1, V_mode]: ((s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_floatu_u_value0(s(t_h4s_binaryu_u_ieees_float(TV_u_27a,TV_u_27b),V_x))) = s(t_h4s_binaryu_u_ieees_floatu_u_value,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_floatu_u_value),h4s_binaryu_u_ieees_float),s(t_h4s_realaxs_real,V_r1))) & s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_floatu_u_value0(s(t_h4s_binaryu_u_ieees_float(TV_u_27a,TV_u_27b),V_y))) = s(t_h4s_binaryu_u_ieees_floatu_u_value,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_floatu_u_value),h4s_binaryu_u_ieees_float),s(t_h4s_realaxs_real,V_r2)))) => ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_binaryu_u_ieees_rounding,V_mode) = s(t_h4s_binaryu_u_ieees_rounding,h4s_binaryu_u_ieees_roundtowardnegative)) & ?[V_vi_]: ((p(s(t_bool,V_vi_)) <=> s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_binaryu_u_ieees_floatu_u_sign(s(t_h4s_binaryu_u_ieees_float(TV_u_27a,TV_u_27b),V_x))) = s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_wordss_n2w(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))) & ?[V_vi_i_]: ((p(s(t_bool,V_vi_i_)) <=> (s(t_h4s_realaxs_real,V_r1) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) & (s(t_h4s_realaxs_real,V_r2) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) & s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_binaryu_u_ieees_floatu_u_sign(s(t_h4s_binaryu_u_ieees_float(TV_u_27a,TV_u_27b),V_x))) = s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_binaryu_u_ieees_floatu_u_sign(s(t_h4s_binaryu_u_ieees_float(TV_u_27a,TV_u_27b),V_y)))))) & s(t_h4s_binaryu_u_ieees_float(TV_u_27a,TV_u_27b),h4s_binaryu_u_ieees_floatu_u_add(s(t_h4s_binaryu_u_ieees_rounding,V_mode),s(t_h4s_binaryu_u_ieees_float(TV_u_27a,TV_u_27b),V_x),s(t_h4s_binaryu_u_ieees_float(TV_u_27a,TV_u_27b),V_y))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27a,TV_u_27b),h4s_binaryu_u_ieees_floatu_u_round(s(t_h4s_binaryu_u_ieees_rounding,V_mode),s(t_bool,h4s_bools_cond(s(t_bool,V_vi_i_),s(t_bool,V_vi_),s(t_bool,V_v))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_add(s(t_h4s_realaxs_real,V_r1),s(t_h4s_realaxs_real,V_r2)))))))))).
fof(ah4s_binaryu_u_ieees_roundu_u_roundTiesToEven0, axiom, ![TV_u_27t,TV_u_27w]: ![V_y, V_x, V_r]: ((s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_floatu_u_value0(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))) = s(t_h4s_binaryu_u_ieees_floatu_u_value,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_floatu_u_value),h4s_binaryu_u_ieees_float),s(t_h4s_realaxs_real,V_r))) & ((s(t_h4s_fcps_cart(t_bool,TV_u_27t),h4s_binaryu_u_ieees_floatu_u_significand(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))) = s(t_h4s_fcps_cart(t_bool,TV_u_27t),h4s_wordss_n2w(s(t_h4s_nums_num,h4s_nums_0))) & (~ (s(t_h4s_fcps_cart(t_bool,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_exponent(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))) = s(t_h4s_fcps_cart(t_bool,TV_u_27w),h4s_wordss_n2w(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))) & ~ (p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,V_r))),s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,V_x))))))))) & (p(s(t_bool,h4s_reals_realu_u_lte(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,h4s_reals_realu_u_sub(s(t_h4s_realaxs_real,V_r),s(t_h4s_realaxs_real,V_x))))))),s(t_h4s_realaxs_real,h4s_binaryu_u_ieees_ulp(s(t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27w),t_h4s_bools_itself(TV_u_27t)),h4s_pairs_u_2c(s(t_h4s_fcps_cart(t_bool,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_exponent(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))),s(t_h4s_bools_itself(TV_u_27t),h4s_bools_theu_u_value)))))))) & (p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_binaryu_u_ieees_ulp0(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,V_x)))))))) & p(s(t_bool,h4s_realaxs_realu_u_lt(s(t_h4s_realaxs_real,h4s_reals_abs(s(t_h4s_realaxs_real,V_x))),s(t_h4s_realaxs_real,h4s_binaryu_u_ieees_threshold(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value)))))))))) => s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_round(s(t_h4s_binaryu_u_ieees_rounding,h4s_binaryu_u_ieees_roundtiestoeven),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))).
fof(ah4s_binaryu_u_ieees_floatu_u_valuesu_c0, axiom, ![TV_u_27t,TV_u_27w]: s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_floatu_u_value0(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_plusu_u_infinity(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))))) = s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_infinity)).
fof(ah4s_binaryu_u_ieees_floatu_u_valuesu_c1, axiom, ![TV_u_27t,TV_u_27w]: s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_floatu_u_value0(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_minusu_u_infinity(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))))) = s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_infinity)).
fof(ah4s_binaryu_u_ieees_floatu_u_minusu_u_infinityu_u_def, axiom, ![TV_u_27t,TV_u_27w]: s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_minusu_u_infinity(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_negate(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_plusu_u_infinity(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value)))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_binaryu_u_ieees_NaN0, axiom, s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_nan) = s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_u_20u_40indu_u_typebinaryu_u_ieee5)).
fof(ah4s_indu_u_types_FCONS0u_c0, axiom, ![TV_u_27a]: ![V_f, V_a]: s(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_binaryu_u_ieees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typebinaryu_u_u_u_ieee3u_u_u_u_def, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_bottom) => ![V_x]: s(t_h4s_binaryu_u_ieees_floatu_u_value,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_floatu_u_value),h4s_binaryu_u_ieees_u_20u_40indu_u_typebinaryu_u_ieee3),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_u_20u_40indu_u_typebinaryu_u_ieee6(s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_realaxs_real,V_x),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_binaryu_u_ieees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typebinaryu_u_u_u_ieee4u_u_u_u_def, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_bottom) => s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_u_20u_40indu_u_typebinaryu_u_ieee4) = s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_u_20u_40indu_u_typebinaryu_u_ieee6(s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0))))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_binaryu_u_ieees_floatu_u_valueu_u_repfnsu_c1, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27floatu_u_valueu_27]: (![V_a00]: ((?[V_a]: s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_a00) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_realaxs_real,V_a),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0))) | (s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_a00) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0))) | s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_a00) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),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_realaxs_real,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),t_bool),V_uu_27floatu_u_valueu_27),s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_a00))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),t_bool),V_uu_27floatu_u_valueu_27),s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_r))))) <=> s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_binaryu_u_ieees_u_20u_40indu_u_typebinaryu_u_ieee7(s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_u_20u_40indu_u_typebinaryu_u_ieee6(s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_r))))) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_r)))).
fof(ah4s_binaryu_u_ieees_Float0, axiom, s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_floatu_u_value),h4s_binaryu_u_ieees_float) = s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_floatu_u_value),h4s_binaryu_u_ieees_u_20u_40indu_u_typebinaryu_u_ieee3)).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_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_indu_u_types_FCONS0u_c1, axiom, ![TV_u_27a]: ![V_n, V_f, V_a]: s(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_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_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
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_binaryu_u_ieees_Infinity0, axiom, s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_infinity) = s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_u_20u_40indu_u_typebinaryu_u_ieee4)).
fof(ah4s_binaryu_u_ieees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typebinaryu_u_u_u_ieee5u_u_u_u_def, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_bottom) => s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_u_20u_40indu_u_typebinaryu_u_ieee5) = s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_u_20u_40indu_u_typebinaryu_u_ieee6(s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),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_realaxs_real,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0))))))).
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_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_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
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_binaryu_u_ieees_floatu_u_valueu_u_caseu_u_defu_c2, axiom, ![TV_u_27a]: ![V_v1, V_v, V_f]: s(TV_u_27a,h4s_binaryu_u_ieees_floatu_u_valueu_u_case(s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_nan),s(t_fun(t_h4s_realaxs_real,TV_u_27a),V_f),s(TV_u_27a,V_v),s(TV_u_27a,V_v1))) = s(TV_u_27a,V_v1)).
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_binaryu_u_ieees_floatu_u_valueu_u_repfnsu_c0, axiom, ![V_a]: s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_u_20u_40indu_u_typebinaryu_u_ieee6(s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_binaryu_u_ieees_u_20u_40indu_u_typebinaryu_u_ieee7(s(t_h4s_binaryu_u_ieees_floatu_u_value,V_a))))) = s(t_h4s_binaryu_u_ieees_floatu_u_value,V_a)).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c0, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) & p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) | ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_numerals_numeralu_u_addu_c4, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_m))))))) = s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_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_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_numerals_numeralu_u_addu_c5, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_m))))))) = s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_bools_LETu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_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_numerals_numeralu_u_addu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_m))))))) = s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_numerals_numeralu_u_addu_c6, axiom, ![V_n]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))).
fof(ah4s_numerals_numeralu_u_addu_c3, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_m))))))) = s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_numerals_numeralu_u_addu_c0, axiom, ![V_n]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_numerals_numeralu_u_distribu_c27, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ah4s_numerals_numeralu_u_distribu_c26, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))) = s(t_bool,t)).
fof(ah4s_numerals_numeralu_u_distribu_c32, axiom, ![V_n]: s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))))) = s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,V_n)))).
fof(ah4s_numerals_numeralu_u_distribu_c21, axiom, ![V_n]: s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))))) = s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,V_n)))).
fof(ah4s_numerals_numeralu_u_distribu_c20, axiom, ![V_n]: s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,f)).
fof(ah4s_numerals_numeralu_u_distribu_c8, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
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_distribu_c5, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_numerals_numeralu_u_addu_c7, axiom, ![V_n]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))).
fof(ah4s_numerals_numeralu_u_distribu_c34, axiom, ~ (p(s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,h4s_nums_0)))))).
fof(ah4s_numerals_numeralu_u_distribu_c18, axiom, ![V_n]: (s(t_h4s_nums_num,h4s_nums_0) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_arithmetics_zero))).
fof(ah4s_numerals_numeralu_u_distribu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_bools_LETu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_x0]: s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(TV_u_27a,V_x0))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(TV_u_27a,V_x0)))).
fof(ah4s_arithmetics_EXPu_u_1u_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_numerals_numeralu_u_sucu_c0, axiom, s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ah4s_arithmetics_EXPu_u_1u_c0, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n))) = 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_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_arithmetics_LESSu_u_MOD, axiom, ![V_n, V_k]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_k),s(t_h4s_nums_num,V_n)))) => s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,V_k),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_k))).
fof(ah4s_arithmetics_LEu_c0, axiom, ![V_n]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(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_nums_0))).
fof(ah4s_arithmetics_SUBu_u_LEFTu_u_ADD, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_numerals_numeralu_u_sucu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_ZEROu_u_MOD, axiom, ![V_n]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n)))) => s(t_h4s_nums_num,h4s_arithmetics_mod(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0))).
fof(ah4s_arithmetics_SUBu_u_RIGHTu_u_EQ, axiom, ![V_p, V_n, V_m]: (s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_p) <=> (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))) | (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,h4s_nums_0)))))))).
fof(ah4s_arithmetics_MULTu_u_LESSu_u_EQu_u_SUC, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_NOTu_u_LEQ, axiom, ![V_n, V_m]: (~ (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_bools_CONJu_u_ASSOC, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) & (p(s(t_bool,V_t2)) & p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) & p(s(t_bool,V_t3))))).
fof(ah4s_bools_SELECTu_u_ELIMu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & ![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)))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_arithmetics_NOTu_u_NUMu_u_EQ, axiom, ![V_n, V_m]: (~ (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n)) <=> (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))) | p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m))))))).
fof(ah4s_arithmetics_NOTu_u_LESSu_u_EQUAL, axiom, ![V_n, V_m]: (~ (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_TIMES2, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_ADDu_u_EQu_u_0, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0) <=> (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,h4s_nums_0)))).
fof(ah4s_arithmetics_LESSu_u_EQu_u_TRANS, axiom, ![V_p, V_n, V_m]: ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_MULTu_u_ASSOC, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c3, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_numerals_numeralu_u_addu_c8, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_m))))))) = s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_numerals_numeralu_u_distribu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_arithmetics_SUCu_u_ONEu_u_ADD, axiom, ![V_n]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = 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,V_n)))).
fof(ah4s_arithmetics_ZEROu_u_LESSu_u_EQ, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_RIGHTu_u_ADDu_u_DISTRIB, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(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,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c2, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_SYM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_ADDu_u_0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_ASSOC, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_TWO, axiom, s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))).
fof(ah4s_relations_WFRECu_u_COROLLARY, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_M]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_wfrec(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M))) => (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M),s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))))),s(TV_u_27a,V_x)))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_arithmetics_LESSu_u_EQ, axiom, ![V_n, V_m]: s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_relations_WFu_u_EMPTYu_u_REL, axiom, ![TV_u_27a]: p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_emptyu_u_rel))))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_bools_LEFTu_u_FORALLu_u_IMPu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,V_Q))) <=> (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,V_Q))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ch4s_binaryu_u_ieees_floatu_u_subu_u_plusu_u_infinityu_u_finite, conjecture, ![TV_u_27t,TV_u_27w]: ![V_x, V_r, V_mode]: (s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_floatu_u_value0(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))) = s(t_h4s_binaryu_u_ieees_floatu_u_value,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_floatu_u_value),h4s_binaryu_u_ieees_float),s(t_h4s_realaxs_real,V_r))) => s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_sub(s(t_h4s_binaryu_u_ieees_rounding,V_mode),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_plusu_u_infinity(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_plusu_u_infinity(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))))).
