%   ORIGINAL: h4/hreal/HREAL__ADD__SYM
% 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/hreal/hreal__add0: !Y X. h4/hreal/hreal__add X Y = h4/hreal/hreal0 (\w. ?x y. w = h4/hrat/hrat__add x y /\ h4/hreal/cut X x /\ h4/hreal/cut Y y)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/hreal/CUT__ISACUT: !X. h4/hreal/isacut (h4/hreal/cut X)
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/hreal/isacut0: !C. h4/hreal/isacut C <=> (?x. C x) /\ (?x. ~C x) /\ (!x y. C x /\ h4/hreal/hrat__lt y x ==> C y) /\ (!x. C x ==> (?y. C y /\ h4/hreal/hrat__lt x y))
% Assm: h4/hreal/hreal__tybij_c0: !a. h4/hreal/hreal0 (h4/hreal/cut a) = a
% Assm: h4/hreal/hreal__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION h4/hreal/isacut rep
% Assm: h4/bool/ABS__REP__THM: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. abs (rep a) = a) /\ (!r. P r <=> rep (abs r) = r))
% Assm: h4/hreal/CUT__DOWN: !y x X. h4/hreal/cut X x /\ h4/hreal/hrat__lt y x ==> h4/hreal/cut X y
% Assm: h4/hreal/CUT__NONEMPTY: !X. ?x. h4/hreal/cut X x
% Assm: h4/hreal/CUT__STRADDLE: !y x X. h4/hreal/cut X x /\ ~h4/hreal/cut X y ==> h4/hreal/hrat__lt x y
% Assm: h4/hreal/CUT__BOUNDED: !X. ?x. ~h4/hreal/cut X x
% Assm: h4/hreal/CUT__UP: !x X. h4/hreal/cut X x ==> (?y. h4/hreal/cut X y /\ h4/hreal/hrat__lt x y)
% Assm: h4/hreal/CUT__UBOUND: !y x X. ~h4/hreal/cut X x /\ h4/hreal/hrat__lt x y ==> ~h4/hreal/cut X y
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/hreal/CUT__NEARTOP__ADD: !e X. ?x. h4/hreal/cut X x /\ ~h4/hreal/cut X (h4/hrat/hrat__add x e)
% Assm: h4/hreal/hreal__sup0: !P. h4/hreal/hreal__sup P = h4/hreal/hreal0 (\w. ?X. P X /\ h4/hreal/cut X w)
% Assm: h4/hreal/hreal__lt0: !Y X. h4/hreal/hreal__lt X Y <=> ~(X = Y) /\ (!x. h4/hreal/cut X x ==> h4/hreal/cut Y x)
% Assm: h4/hreal/hreal__mul0: !Y X. h4/hreal/hreal__mul X Y = h4/hreal/hreal0 (\w. ?x y. w = h4/hrat/hrat__mul x y /\ h4/hreal/cut X x /\ h4/hreal/cut Y y)
% Assm: h4/hreal/hreal__inv0: !X. h4/hreal/hreal__inv X = h4/hreal/hreal0 (\w. ?d. h4/hreal/hrat__lt d h4/hrat/hrat__1 /\ (!x. h4/hreal/cut X x ==> h4/hreal/hrat__lt (h4/hrat/hrat__mul w x) d))
% Assm: h4/hreal/HREAL__ADD__ISACUT: !Y X. h4/hreal/isacut (\w. ?x y. w = h4/hrat/hrat__add x y /\ h4/hreal/cut X x /\ h4/hreal/cut Y y)
% Assm: h4/hreal/HREAL__MUL__ISACUT: !Y X. h4/hreal/isacut (\w. ?x y. w = h4/hrat/hrat__mul x y /\ h4/hreal/cut X x /\ h4/hreal/cut Y y)
% Assm: h4/hreal/EQUAL__CUTS: !Y X. h4/hreal/cut X = h4/hreal/cut Y ==> X = Y
% Assm: h4/hreal/CUT__NEARTOP__MUL: !u X. h4/hreal/hrat__lt h4/hrat/hrat__1 u ==> (?x. h4/hreal/cut X x /\ ~h4/hreal/cut X (h4/hrat/hrat__mul u x))
% Assm: h4/hreal/hreal__tybij_c1: !r. h4/hreal/isacut r <=> h4/hreal/cut (h4/hreal/hreal0 r) = r
% Assm: h4/hreal/HREAL__INV__ISACUT: !X. h4/hreal/isacut (\w. ?d. h4/hreal/hrat__lt d h4/hrat/hrat__1 /\ (!x. h4/hreal/cut X x ==> h4/hreal/hrat__lt (h4/hrat/hrat__mul w x) d))
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/hrat/HRAT__MUL__ASSOC: !j i h. h4/hrat/hrat__mul h (h4/hrat/hrat__mul i j) = h4/hrat/hrat__mul (h4/hrat/hrat__mul h i) j
% Assm: h4/hreal/hreal__10: h4/hreal/hreal__1 = h4/hreal/hreal0 (h4/hreal/cut__of__hrat h4/hrat/hrat__1)
% Assm: h4/hrat/HRAT__MUL__LID: !h. h4/hrat/hrat__mul h4/hrat/hrat__1 h = h
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/hrat/HRAT__MUL__LINV: !h. h4/hrat/hrat__mul (h4/hrat/hrat__inv h) h = h4/hrat/hrat__1
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/hreal/HRAT__LT__RADD: !z y x. h4/hreal/hrat__lt (h4/hrat/hrat__add x z) (h4/hrat/hrat__add y z) <=> h4/hreal/hrat__lt x y
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/hreal/HRAT__LT__R1: !y x. h4/hreal/hrat__lt (h4/hrat/hrat__mul x (h4/hrat/hrat__inv y)) h4/hrat/hrat__1 <=> h4/hreal/hrat__lt x y
% Assm: h4/hreal/HRAT__LT__LMUL1: !y x. h4/hreal/hrat__lt (h4/hrat/hrat__mul x y) y <=> h4/hreal/hrat__lt x h4/hrat/hrat__1
% Assm: h4/hreal/HRAT__LT__NE: !y x. h4/hreal/hrat__lt x y ==> ~(x = y)
% Assm: h4/hreal/hrat__lt0: !y x. h4/hreal/hrat__lt x y <=> (?d. y = h4/hrat/hrat__add x d)
% Assm: h4/hreal/HRAT__LT__RMUL: !z y x. h4/hreal/hrat__lt (h4/hrat/hrat__mul x z) (h4/hrat/hrat__mul y z) <=> h4/hreal/hrat__lt x y
% Assm: h4/hrat/HRAT__MUL__SYM: !i h. h4/hrat/hrat__mul h i = h4/hrat/hrat__mul i h
% Assm: h4/hreal/HRAT__MUL__RID: !x. h4/hrat/hrat__mul x h4/hrat/hrat__1 = x
% Assm: h4/hrat/HRAT__LDISTRIB: !j i h. h4/hrat/hrat__mul h (h4/hrat/hrat__add i j) = h4/hrat/hrat__add (h4/hrat/hrat__mul h i) (h4/hrat/hrat__mul h j)
% Assm: h4/hreal/HRAT__LT__TOTAL: !y x. x = y \/ h4/hreal/hrat__lt x y \/ h4/hreal/hrat__lt y x
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/hreal/HRAT__DOWN: !x. ?y. h4/hreal/hrat__lt y x
% Assm: h4/hreal/HRAT__MEAN: !y x. h4/hreal/hrat__lt x y ==> (?z. h4/hreal/hrat__lt x z /\ h4/hreal/hrat__lt z y)
% Assm: h4/hreal/HRAT__LT__TRANS: !z y x. h4/hreal/hrat__lt x y /\ h4/hreal/hrat__lt y z ==> h4/hreal/hrat__lt x z
% Assm: h4/hreal/HRAT__RDISTRIB: !z y x. h4/hrat/hrat__mul (h4/hrat/hrat__add x y) z = h4/hrat/hrat__add (h4/hrat/hrat__mul x z) (h4/hrat/hrat__mul y z)
% Assm: h4/hreal/HRAT__LT__LMUL: !z y x. h4/hreal/hrat__lt (h4/hrat/hrat__mul z x) (h4/hrat/hrat__mul z y) <=> h4/hreal/hrat__lt x y
% Assm: h4/hreal/HRAT__LT__LADD: !z y x. h4/hreal/hrat__lt (h4/hrat/hrat__add z x) (h4/hrat/hrat__add z y) <=> h4/hreal/hrat__lt x y
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/hreal/HRAT__LT__ADD2: !y x v u. h4/hreal/hrat__lt u x /\ h4/hreal/hrat__lt v y ==> h4/hreal/hrat__lt (h4/hrat/hrat__add u v) (h4/hrat/hrat__add x y)
% Assm: h4/hreal/HRAT__LT__MUL2: !y x v u. h4/hreal/hrat__lt u x /\ h4/hreal/hrat__lt v y ==> h4/hreal/hrat__lt (h4/hrat/hrat__mul u v) (h4/hrat/hrat__mul x y)
% Assm: h4/hreal/HRAT__GT__L1: !y x. h4/hreal/hrat__lt h4/hrat/hrat__1 (h4/hrat/hrat__mul (h4/hrat/hrat__inv x) y) <=> h4/hreal/hrat__lt x y
% Assm: h4/hreal/HRAT__LT__L1: !y x. h4/hreal/hrat__lt (h4/hrat/hrat__mul (h4/hrat/hrat__inv x) y) h4/hrat/hrat__1 <=> h4/hreal/hrat__lt y x
% Assm: h4/hreal/HRAT__GT__LMUL1: !y x. h4/hreal/hrat__lt y (h4/hrat/hrat__mul x y) <=> h4/hreal/hrat__lt h4/hrat/hrat__1 x
% Assm: h4/hreal/HRAT__LT__RMUL1: !y x. h4/hreal/hrat__lt (h4/hrat/hrat__mul x y) x <=> h4/hreal/hrat__lt y h4/hrat/hrat__1
% Assm: h4/hreal/HRAT__LT__GT: !y x. h4/hreal/hrat__lt x y ==> ~h4/hreal/hrat__lt y x
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/prim__rec/NOT__LESS__0: !n. ~h4/prim__rec/_3C n h4/num/0
% Assm: h4/prim__rec/LESS__SUC__REFL: !n. h4/prim__rec/_3C n (h4/num/SUC n)
% Assm: h4/hreal/HRAT__LT__ADDR: !y x. h4/hreal/hrat__lt y (h4/hrat/hrat__add x y)
% Assm: h4/hreal/HRAT__MUL__RINV: !x. h4/hrat/hrat__mul x (h4/hrat/hrat__inv x) = h4/hrat/hrat__1
% Assm: h4/hrat/HRAT__SUCINT_c1: !n. h4/hrat/hrat__sucint (h4/num/SUC n) = h4/hrat/hrat__add (h4/hrat/hrat__sucint n) h4/hrat/hrat__1
% Assm: h4/hrat/HRAT__SUCINT_c0: h4/hrat/hrat__sucint h4/num/0 = h4/hrat/hrat__1
% Assm: h4/hrat/HRAT__ARCH: !h. ?n d. h4/hrat/hrat__sucint n = h4/hrat/hrat__add h d
% Assm: h4/arithmetic/WOP: !P. (?n. P n) ==> (?n. P n /\ (!m. h4/prim__rec/_3C m n ==> ~P m))
% Assm: h4/arithmetic/num__CASES: !m. m = h4/num/0 \/ (?n. m = h4/num/SUC n)
% Assm: h4/hreal/cut__of__hrat0: !x. h4/hreal/cut__of__hrat x = (\y. h4/hreal/hrat__lt y x)
% Assm: h4/hreal/HRAT__UP: !x. ?y. h4/hreal/hrat__lt x y
% Assm: h4/bool/CONJ__SYM: !t2 t1. t1 /\ t2 <=> t2 /\ t1
% Assm: h4/hreal/HRAT__LT__ANTISYM: !y x. ~(h4/hreal/hrat__lt x y /\ h4/hreal/hrat__lt y x)
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/list/LENGTH__CONS: !n l. h4/list/LENGTH l = h4/num/SUC n <=> (?h l_27. h4/list/LENGTH l_27 = n /\ l = h4/list/CONS h l_27)
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/list/NOT__NIL__CONS: !a1 a0. ~(h4/list/NIL = h4/list/CONS a0 a1)
% Assm: h4/list/LENGTH0_c1: !t h. h4/list/LENGTH (h4/list/CONS h t) = h4/num/SUC (h4/list/LENGTH t)
% Assm: h4/list/list__INDUCT: !P. P h4/list/NIL /\ (!t. P t ==> (!h. P (h4/list/CONS h t))) ==> (!l. P l)
% Assm: h4/list/LENGTH0_c0: h4/list/LENGTH h4/list/NIL = h4/num/0
% Assm: h4/list/CONS__11: !a1_27 a1 a0_27 a0. h4/list/CONS a0 a1 = h4/list/CONS a0_27 a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm: h4/prim__rec/INV__SUC__EQ: !n m. h4/num/SUC m = h4/num/SUC n <=> m = n
% Assm: h4/num/NOT__SUC: !n. ~(h4/num/SUC n = h4/num/0)
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/arithmetic/MAX__DEF: !n m. h4/arithmetic/MAX m n = h4/bool/COND (h4/prim__rec/_3C m n) n m
% Assm: h4/list/LAST__APPEND__CONS: !l2 l1 h. h4/list/LAST (h4/list/APPEND l1 (h4/list/CONS h l2)) = h4/list/LAST (h4/list/CONS h l2)
% Assm: h4/bool/BETA__THM: !y f. (\x. f x) y = f y
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/list/LAST__CONS__cond: !t h. h4/list/LAST (h4/list/CONS h t) = h4/bool/COND (t = h4/list/NIL) h (h4/list/LAST t)
% Assm: h4/list/APPEND__eq__NIL_c0: !l2 l1. h4/list/NIL = h4/list/APPEND l1 l2 <=> l1 = h4/list/NIL /\ l2 = h4/list/NIL
% Assm: h4/list/APPEND0_c0: !l. h4/list/APPEND h4/list/NIL l = l
% Assm: h4/list/APPEND0_c1: !l2 l1 h. h4/list/APPEND (h4/list/CONS h l1) l2 = h4/list/CONS h (h4/list/APPEND l1 l2)
% Assm: h4/list/CONS0: h4/list/CONS = h4/list/_20_40ind__typelist1
% Assm: h4/list/hidden____20__40ind____typelist0____def: h4/list/_20_40ind__typelist0 = h4/list/_20_40ind__typelist2 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM))
% Assm: h4/list/list__repfns_c1: !r. (\a0_27. !_27list_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM))) a0 a1 /\ _27list_27 a1) ==> _27list_27 a0_270) ==> _27list_27 a0_27) r <=> h4/list/_20_40ind__typelist3 (h4/list/_20_40ind__typelist2 r) = r
% Assm: h4/list/hidden____20__40ind____typelist1____def: h4/list/_20_40ind__typelist1 = (\a0 a1. h4/list/_20_40ind__typelist2 ((\a00 a10. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM))) a0 (h4/list/_20_40ind__typelist3 a1)))
% Assm: h4/list/list__repfns_c0: !a. h4/list/_20_40ind__typelist2 (h4/list/_20_40ind__typelist3 a) = a
% Assm: h4/list/NIL0: h4/list/NIL = h4/list/_20_40ind__typelist0
% Assm: h4/ind__type/FCONS0_c1: !n f a. h4/ind__type/FCONS a f (h4/num/SUC n) = f n
% Assm: h4/ind__type/FCONS0_c0: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm: h4/ind__type/CONSTR__REC: !Fn. ?f. !c i r. f (h4/ind__type/CONSTR c i r) = Fn c i r (\n. f (r n))
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Goal: !Y X. h4/hreal/hreal__add X Y = h4/hreal/hreal__add Y X
%   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_hreals_hrealu_u_add0]: !_0. (!X Y w. happ (happ (happ _0 X) Y) w <=> (?x y. w = h4/hrat/hrat__add x y /\ happ (h4/hreal/cut X) x /\ happ (h4/hreal/cut Y) y)) ==> (!Y X. h4/hreal/hreal__add X Y = h4/hreal/hreal0 (happ (happ _0 X) Y))
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_hreals_CUTu_u_ISACUT]: !X. happ h4/hreal/isacut (h4/hreal/cut X)
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_hreals_isacut0]: !C. happ h4/hreal/isacut C <=> (?x. happ C x) /\ (?x. ~happ C x) /\ (!x y. happ C x /\ h4/hreal/hrat__lt y x ==> happ C y) /\ (!x. happ C x ==> (?y. happ C y /\ h4/hreal/hrat__lt x y))
% Assm [h4s_hreals_hrealu_u_tybiju_c0]: !a. h4/hreal/hreal0 (h4/hreal/cut a) = a
% Assm [h4s_hreals_hrealu_u_TYu_u_DEF]: ?rep. h4/bool/TYPE__DEFINITION h4/hreal/isacut rep
% Assm [h4s_bools_ABSu_u_REPu_u_THM]: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. happ abs (happ rep a) = a) /\ (!r. happ P r <=> happ rep (happ abs r) = r))
% Assm [h4s_hreals_CUTu_u_DOWN]: !y x X. happ (h4/hreal/cut X) x /\ h4/hreal/hrat__lt y x ==> happ (h4/hreal/cut X) y
% Assm [h4s_hreals_CUTu_u_NONEMPTY]: !X. ?x. happ (h4/hreal/cut X) x
% Assm [h4s_hreals_CUTu_u_STRADDLE]: !y x X. happ (h4/hreal/cut X) x /\ ~happ (h4/hreal/cut X) y ==> h4/hreal/hrat__lt x y
% Assm [h4s_hreals_CUTu_u_BOUNDED]: !X. ?x. ~happ (h4/hreal/cut X) x
% Assm [h4s_hreals_CUTu_u_UP]: !x X. happ (h4/hreal/cut X) x ==> (?y. happ (h4/hreal/cut X) y /\ h4/hreal/hrat__lt x y)
% Assm [h4s_hreals_CUTu_u_UBOUND]: !y x X. ~happ (h4/hreal/cut X) x /\ h4/hreal/hrat__lt x y ==> ~happ (h4/hreal/cut X) y
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_hreals_CUTu_u_NEARTOPu_u_ADD]: !e X. ?x. happ (h4/hreal/cut X) x /\ ~happ (h4/hreal/cut X) (h4/hrat/hrat__add x e)
% Assm [h4s_hreals_hrealu_u_sup0]: !_0. (!P w. happ (happ _0 P) w <=> (?X. happ P X /\ happ (h4/hreal/cut X) w)) ==> (!P. h4/hreal/hreal__sup P = h4/hreal/hreal0 (happ _0 P))
% Assm [h4s_hreals_hrealu_u_lt0]: !Y X. h4/hreal/hreal__lt X Y <=> ~(X = Y) /\ (!x. happ (h4/hreal/cut X) x ==> happ (h4/hreal/cut Y) x)
% Assm [h4s_hreals_hrealu_u_mul0]: !_0. (!X Y w. happ (happ (happ _0 X) Y) w <=> (?x y. w = h4/hrat/hrat__mul x y /\ happ (h4/hreal/cut X) x /\ happ (h4/hreal/cut Y) y)) ==> (!Y X. h4/hreal/hreal__mul X Y = h4/hreal/hreal0 (happ (happ _0 X) Y))
% Assm [h4s_hreals_hrealu_u_inv0]: !_0. (!X w. happ (happ _0 X) w <=> (?d. h4/hreal/hrat__lt d h4/hrat/hrat__1 /\ (!x. happ (h4/hreal/cut X) x ==> h4/hreal/hrat__lt (h4/hrat/hrat__mul w x) d))) ==> (!X. h4/hreal/hreal__inv X = h4/hreal/hreal0 (happ _0 X))
% Assm [h4s_hreals_HREALu_u_ADDu_u_ISACUT]: !_0. (!X Y w. happ (happ (happ _0 X) Y) w <=> (?x y. w = h4/hrat/hrat__add x y /\ happ (h4/hreal/cut X) x /\ happ (h4/hreal/cut Y) y)) ==> (!Y X. happ h4/hreal/isacut (happ (happ _0 X) Y))
% Assm [h4s_hreals_HREALu_u_MULu_u_ISACUT]: !_0. (!X Y w. happ (happ (happ _0 X) Y) w <=> (?x y. w = h4/hrat/hrat__mul x y /\ happ (h4/hreal/cut X) x /\ happ (h4/hreal/cut Y) y)) ==> (!Y X. happ h4/hreal/isacut (happ (happ _0 X) Y))
% Assm [h4s_hreals_EQUALu_u_CUTS]: !Y X. h4/hreal/cut X = h4/hreal/cut Y ==> X = Y
% Assm [h4s_hreals_CUTu_u_NEARTOPu_u_MUL]: !u X. h4/hreal/hrat__lt h4/hrat/hrat__1 u ==> (?x. happ (h4/hreal/cut X) x /\ ~happ (h4/hreal/cut X) (h4/hrat/hrat__mul u x))
% Assm [h4s_hreals_hrealu_u_tybiju_c1]: !r. happ h4/hreal/isacut r <=> h4/hreal/cut (h4/hreal/hreal0 r) = r
% Assm [h4s_hreals_HREALu_u_INVu_u_ISACUT]: !_0. (!X w. happ (happ _0 X) w <=> (?d. h4/hreal/hrat__lt d h4/hrat/hrat__1 /\ (!x. happ (h4/hreal/cut X) x ==> h4/hreal/hrat__lt (h4/hrat/hrat__mul w x) d))) ==> (!X. happ h4/hreal/isacut (happ _0 X))
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_hrats_HRATu_u_MULu_u_ASSOC]: !j i h. h4/hrat/hrat__mul h (h4/hrat/hrat__mul i j) = h4/hrat/hrat__mul (h4/hrat/hrat__mul h i) j
% Assm [h4s_hreals_hrealu_u_10]: h4/hreal/hreal__1 = h4/hreal/hreal0 (h4/hreal/cut__of__hrat h4/hrat/hrat__1)
% Assm [h4s_hrats_HRATu_u_MULu_u_LID]: !h. h4/hrat/hrat__mul h4/hrat/hrat__1 h = h
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_hrats_HRATu_u_MULu_u_LINV]: !h. h4/hrat/hrat__mul (h4/hrat/hrat__inv h) h = h4/hrat/hrat__1
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_hreals_HRATu_u_LTu_u_RADD]: !z y x. h4/hreal/hrat__lt (h4/hrat/hrat__add x z) (h4/hrat/hrat__add y z) <=> h4/hreal/hrat__lt x y
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_hreals_HRATu_u_LTu_u_R1]: !y x. h4/hreal/hrat__lt (h4/hrat/hrat__mul x (h4/hrat/hrat__inv y)) h4/hrat/hrat__1 <=> h4/hreal/hrat__lt x y
% Assm [h4s_hreals_HRATu_u_LTu_u_LMUL1]: !y x. h4/hreal/hrat__lt (h4/hrat/hrat__mul x y) y <=> h4/hreal/hrat__lt x h4/hrat/hrat__1
% Assm [h4s_hreals_HRATu_u_LTu_u_NE]: !y x. h4/hreal/hrat__lt x y ==> ~(x = y)
% Assm [h4s_hreals_hratu_u_lt0]: !y x. h4/hreal/hrat__lt x y <=> (?d. y = h4/hrat/hrat__add x d)
% Assm [h4s_hreals_HRATu_u_LTu_u_RMUL]: !z y x. h4/hreal/hrat__lt (h4/hrat/hrat__mul x z) (h4/hrat/hrat__mul y z) <=> h4/hreal/hrat__lt x y
% Assm [h4s_hrats_HRATu_u_MULu_u_SYM]: !i h. h4/hrat/hrat__mul h i = h4/hrat/hrat__mul i h
% Assm [h4s_hreals_HRATu_u_MULu_u_RID]: !x. h4/hrat/hrat__mul x h4/hrat/hrat__1 = x
% Assm [h4s_hrats_HRATu_u_LDISTRIB]: !j i h. h4/hrat/hrat__mul h (h4/hrat/hrat__add i j) = h4/hrat/hrat__add (h4/hrat/hrat__mul h i) (h4/hrat/hrat__mul h j)
% Assm [h4s_hreals_HRATu_u_LTu_u_TOTAL]: !y x. x = y \/ h4/hreal/hrat__lt x y \/ h4/hreal/hrat__lt y x
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_hreals_HRATu_u_DOWN]: !x. ?y. h4/hreal/hrat__lt y x
% Assm [h4s_hreals_HRATu_u_MEAN]: !y x. h4/hreal/hrat__lt x y ==> (?z. h4/hreal/hrat__lt x z /\ h4/hreal/hrat__lt z y)
% Assm [h4s_hreals_HRATu_u_LTu_u_TRANS]: !z y x. h4/hreal/hrat__lt x y /\ h4/hreal/hrat__lt y z ==> h4/hreal/hrat__lt x z
% Assm [h4s_hreals_HRATu_u_RDISTRIB]: !z y x. h4/hrat/hrat__mul (h4/hrat/hrat__add x y) z = h4/hrat/hrat__add (h4/hrat/hrat__mul x z) (h4/hrat/hrat__mul y z)
% Assm [h4s_hreals_HRATu_u_LTu_u_LMUL]: !z y x. h4/hreal/hrat__lt (h4/hrat/hrat__mul z x) (h4/hrat/hrat__mul z y) <=> h4/hreal/hrat__lt x y
% Assm [h4s_hreals_HRATu_u_LTu_u_LADD]: !z y x. h4/hreal/hrat__lt (h4/hrat/hrat__add z x) (h4/hrat/hrat__add z y) <=> h4/hreal/hrat__lt x y
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_hreals_HRATu_u_LTu_u_ADD2]: !y x v u. h4/hreal/hrat__lt u x /\ h4/hreal/hrat__lt v y ==> h4/hreal/hrat__lt (h4/hrat/hrat__add u v) (h4/hrat/hrat__add x y)
% Assm [h4s_hreals_HRATu_u_LTu_u_MUL2]: !y x v u. h4/hreal/hrat__lt u x /\ h4/hreal/hrat__lt v y ==> h4/hreal/hrat__lt (h4/hrat/hrat__mul u v) (h4/hrat/hrat__mul x y)
% Assm [h4s_hreals_HRATu_u_GTu_u_L1]: !y x. h4/hreal/hrat__lt h4/hrat/hrat__1 (h4/hrat/hrat__mul (h4/hrat/hrat__inv x) y) <=> h4/hreal/hrat__lt x y
% Assm [h4s_hreals_HRATu_u_LTu_u_L1]: !y x. h4/hreal/hrat__lt (h4/hrat/hrat__mul (h4/hrat/hrat__inv x) y) h4/hrat/hrat__1 <=> h4/hreal/hrat__lt y x
% Assm [h4s_hreals_HRATu_u_GTu_u_LMUL1]: !y x. h4/hreal/hrat__lt y (h4/hrat/hrat__mul x y) <=> h4/hreal/hrat__lt h4/hrat/hrat__1 x
% Assm [h4s_hreals_HRATu_u_LTu_u_RMUL1]: !y x. h4/hreal/hrat__lt (h4/hrat/hrat__mul x y) x <=> h4/hreal/hrat__lt y h4/hrat/hrat__1
% Assm [h4s_hreals_HRATu_u_LTu_u_GT]: !y x. h4/hreal/hrat__lt x y ==> ~h4/hreal/hrat__lt y x
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_primu_u_recs_NOTu_u_LESSu_u_0]: !n. ~h4/prim__rec/_3C n h4/num/0
% Assm [h4s_primu_u_recs_LESSu_u_SUCu_u_REFL]: !n. h4/prim__rec/_3C n (h4/num/SUC n)
% Assm [h4s_hreals_HRATu_u_LTu_u_ADDR]: !y x. h4/hreal/hrat__lt y (h4/hrat/hrat__add x y)
% Assm [h4s_hreals_HRATu_u_MULu_u_RINV]: !x. h4/hrat/hrat__mul x (h4/hrat/hrat__inv x) = h4/hrat/hrat__1
% Assm [h4s_hrats_HRATu_u_SUCINTu_c1]: !n. h4/hrat/hrat__sucint (h4/num/SUC n) = h4/hrat/hrat__add (h4/hrat/hrat__sucint n) h4/hrat/hrat__1
% Assm [h4s_hrats_HRATu_u_SUCINTu_c0]: h4/hrat/hrat__sucint h4/num/0 = h4/hrat/hrat__1
% Assm [h4s_hrats_HRATu_u_ARCH]: !h. ?n d. h4/hrat/hrat__sucint n = h4/hrat/hrat__add h d
% Assm [h4s_arithmetics_WOP]: !P. (?n. happ P n) ==> (?n. happ P n /\ (!m. h4/prim__rec/_3C m n ==> ~happ P m))
% Assm [h4s_arithmetics_numu_u_CASES]: !m. m = h4/num/0 \/ (?n. m = h4/num/SUC n)
% Assm [h4s_hreals_cutu_u_ofu_u_hrat0]: !x x'. happ (h4/hreal/cut__of__hrat x) x' <=> h4/hreal/hrat__lt x' x
% Assm [h4s_hreals_HRATu_u_UP]: !x. ?y. h4/hreal/hrat__lt x y
% Assm [h4s_bools_CONJu_u_SYM]: !t2 t1. t1 /\ t2 <=> t2 /\ t1
% Assm [h4s_hreals_HRATu_u_LTu_u_ANTISYM]: !y x. ~(h4/hreal/hrat__lt x y /\ h4/hreal/hrat__lt y x)
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_lists_LENGTHu_u_CONS]: !n l. h4/list/LENGTH l = h4/num/SUC n <=> (?h l_27. h4/list/LENGTH l_27 = n /\ l = happ (happ h4/list/CONS h) l_27)
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_lists_NOTu_u_NILu_u_CONS]: !a1 a0. ~(h4/list/NIL = happ (happ h4/list/CONS a0) a1)
% Assm [h4s_lists_LENGTH0u_c1]: !t h. h4/list/LENGTH (happ (happ h4/list/CONS h) t) = h4/num/SUC (h4/list/LENGTH t)
% Assm [h4s_lists_listu_u_INDUCT]: !P. happ P h4/list/NIL /\ (!t. happ P t ==> (!h. happ P (happ (happ h4/list/CONS h) t))) ==> (!l. happ P l)
% Assm [h4s_lists_LENGTH0u_c0]: h4/list/LENGTH h4/list/NIL = h4/num/0
% Assm [h4s_lists_CONSu_u_11]: !a1_27 a1 a0_27 a0. happ (happ h4/list/CONS a0) a1 = happ (happ h4/list/CONS a0_27) a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm [h4s_primu_u_recs_INVu_u_SUCu_u_EQ]: !n m. h4/num/SUC m = h4/num/SUC n <=> m = n
% Assm [h4s_nums_NOTu_u_SUC]: !n. ~(h4/num/SUC n = h4/num/0)
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_arithmetics_MAXu_u_DEF]: !n m. h4/arithmetic/MAX m n = h4/bool/COND (h4/prim__rec/_3C m n) n m
% Assm [h4s_lists_LASTu_u_APPENDu_u_CONS]: !l2 l1 h. h4/list/LAST (h4/list/APPEND l1 (happ (happ h4/list/CONS h) l2)) = h4/list/LAST (happ (happ h4/list/CONS h) l2)
% Assm [h4s_bools_BETAu_u_THM]: !y f. happ f y = happ f y
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_lists_LASTu_u_CONSu_u_cond]: !t h. ?v. (v <=> t = h4/list/NIL) /\ h4/list/LAST (happ (happ h4/list/CONS h) t) = h4/bool/COND v h (h4/list/LAST t)
% Assm [h4s_lists_APPENDu_u_equ_u_NILu_c0]: !l2 l1. h4/list/NIL = h4/list/APPEND l1 l2 <=> l1 = h4/list/NIL /\ l2 = h4/list/NIL
% Assm [h4s_lists_APPEND0u_c0]: !l. h4/list/APPEND h4/list/NIL l = l
% Assm [h4s_lists_APPEND0u_c1]: !l2 l1 h. h4/list/APPEND (happ (happ h4/list/CONS h) l1) l2 = happ (happ h4/list/CONS h) (h4/list/APPEND l1 l2)
% Assm [h4s_lists_CONS0]: h4/list/CONS = h4/list/_20_40ind__typelist1
% Assm [h4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> h4/list/_20_40ind__typelist0 = h4/list/_20_40ind__typelist2 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0)
% Assm [h4s_lists_listu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27list_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0 \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a0 (h4/ind__type/FCONS a1 _0) /\ happ _27list_27 a1) ==> happ _27list_27 a0_270) ==> happ _27list_27 r) <=> h4/list/_20_40ind__typelist3 (h4/list/_20_40ind__typelist2 r) = r)
% Assm [h4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist1u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x. happ (happ h4/list/_20_40ind__typelist1 x) x = h4/list/_20_40ind__typelist2 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) x (h4/ind__type/FCONS (h4/list/_20_40ind__typelist3 x) _0)))
% Assm [h4s_lists_listu_u_repfnsu_c0]: !a. h4/list/_20_40ind__typelist2 (h4/list/_20_40ind__typelist3 a) = a
% Assm [h4s_lists_NIL0]: h4/list/NIL = h4/list/_20_40ind__typelist0
% Assm [h4s_indu_u_types_FCONS0u_c1]: !n f a. happ (h4/ind__type/FCONS a f) (h4/num/SUC n) = happ f n
% Assm [h4s_indu_u_types_FCONS0u_c0]: !f a. happ (h4/ind__type/FCONS a f) h4/num/0 = a
% Assm [h4s_indu_u_types_CONSTRu_u_REC]: !_0. (!f r n. happ (happ (happ _0 f) r) n = happ f (happ r n)) ==> (!Fn. ?f. !c i r. happ f (h4/ind__type/CONSTR c i r) = happ (happ (happ (happ Fn c) i) r) (happ (happ _0 f) r))
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Goal: !Y X. h4/hreal/hreal__add X Y = h4/hreal/hreal__add Y X
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_Q1347796,TV_Q1347792]: ![V_f, V_g]: (![V_x]: s(TV_Q1347792,happ(s(t_fun(TV_Q1347796,TV_Q1347792),V_f),s(TV_Q1347796,V_x))) = s(TV_Q1347792,happ(s(t_fun(TV_Q1347796,TV_Q1347792),V_g),s(TV_Q1347796,V_x))) => s(t_fun(TV_Q1347796,TV_Q1347792),V_f) = s(t_fun(TV_Q1347796,TV_Q1347792),V_g))).
fof(ah4s_hreals_hrealu_u_add0, axiom, ![V_uu_0]: (![V_X, V_Y, V_w]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool))),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,V_w)))) <=> ?[V_x, V_y]: (s(t_h4s_hrats_hrat,V_w) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y))) & (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))) & p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,V_y))))))) => ![V_Y, V_X]: s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y))) = s(t_h4s_hreals_hreal,h4s_hreals_hreal0(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool))),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hreals_hreal,V_Y))))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_hreals_CUTu_u_ISACUT, axiom, ![V_X]: p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_hrats_hrat,t_bool),t_bool),h4s_hreals_isacut),s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))))))).
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_hreals_isacut0, axiom, ![V_C]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_hrats_hrat,t_bool),t_bool),h4s_hreals_isacut),s(t_fun(t_h4s_hrats_hrat,t_bool),V_C)))) <=> (?[V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),V_C),s(t_h4s_hrats_hrat,V_x)))) & (?[V_x]: ~ (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),V_C),s(t_h4s_hrats_hrat,V_x))))) & (![V_x, V_y]: ((p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),V_C),s(t_h4s_hrats_hrat,V_x)))) & p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_y),s(t_h4s_hrats_hrat,V_x))))) => p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),V_C),s(t_h4s_hrats_hrat,V_y))))) & ![V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),V_C),s(t_h4s_hrats_hrat,V_x)))) => ?[V_y]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),V_C),s(t_h4s_hrats_hrat,V_y)))) & p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y))))))))))).
fof(ah4s_hreals_hrealu_u_tybiju_c0, axiom, ![V_a]: s(t_h4s_hreals_hreal,h4s_hreals_hreal0(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_a))))) = s(t_h4s_hreals_hreal,V_a)).
fof(ah4s_hreals_hrealu_u_TYu_u_DEF, axiom, ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_fun(t_h4s_hrats_hrat,t_bool),t_bool),h4s_hreals_isacut),s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),V_rep))))).
fof(ah4s_bools_ABSu_u_REPu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ?[V_rep, V_abs]: (![V_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))) = s(TV_u_27b,V_a) & ![V_r]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_r)))) <=> s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))))) = s(TV_u_27a,V_r))))).
fof(ah4s_hreals_CUTu_u_DOWN, axiom, ![V_y, V_x, V_X]: ((p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))) & p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_y),s(t_h4s_hrats_hrat,V_x))))) => p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_y)))))).
fof(ah4s_hreals_CUTu_u_NONEMPTY, axiom, ![V_X]: ?[V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x))))).
fof(ah4s_hreals_CUTu_u_STRADDLE, axiom, ![V_y, V_x, V_X]: ((p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))) & ~ (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_y)))))) => p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))))).
fof(ah4s_hreals_CUTu_u_BOUNDED, axiom, ![V_X]: ?[V_x]: ~ (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))))).
fof(ah4s_hreals_CUTu_u_UP, axiom, ![V_x, V_X]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))) => ?[V_y]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_y)))) & p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y))))))).
fof(ah4s_hreals_CUTu_u_UBOUND, axiom, ![V_y, V_x, V_X]: ((~ (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x))))) & p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y))))) => ~ (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_y))))))).
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_hreals_CUTu_u_NEARTOPu_u_ADD, axiom, ![V_e, V_X]: ?[V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))) & ~ (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_e))))))))).
fof(ah4s_hreals_hrealu_u_sup0, axiom, ![V_uu_0]: (![V_P, V_w]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_fun(t_h4s_hreals_hreal,t_bool),t_fun(t_h4s_hrats_hrat,t_bool)),V_uu_0),s(t_fun(t_h4s_hreals_hreal,t_bool),V_P))),s(t_h4s_hrats_hrat,V_w)))) <=> ?[V_X]: (p(s(t_bool,happ(s(t_fun(t_h4s_hreals_hreal,t_bool),V_P),s(t_h4s_hreals_hreal,V_X)))) & p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_w)))))) => ![V_P]: s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_sup(s(t_fun(t_h4s_hreals_hreal,t_bool),V_P))) = s(t_h4s_hreals_hreal,h4s_hreals_hreal0(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_fun(t_h4s_hreals_hreal,t_bool),t_fun(t_h4s_hrats_hrat,t_bool)),V_uu_0),s(t_fun(t_h4s_hreals_hreal,t_bool),V_P))))))).
fof(ah4s_hreals_hrealu_u_lt0, axiom, ![V_Y, V_X]: (p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y)))) <=> (~ (s(t_h4s_hreals_hreal,V_X) = s(t_h4s_hreals_hreal,V_Y)) & ![V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))) => p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,V_x)))))))).
fof(ah4s_hreals_hrealu_u_mul0, axiom, ![V_uu_0]: (![V_X, V_Y, V_w]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool))),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,V_w)))) <=> ?[V_x, V_y]: (s(t_h4s_hrats_hrat,V_w) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y))) & (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))) & p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,V_y))))))) => ![V_Y, V_X]: s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_mul(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y))) = s(t_h4s_hreals_hreal,h4s_hreals_hreal0(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool))),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hreals_hreal,V_Y))))))).
fof(ah4s_hreals_hrealu_u_inv0, axiom, ![V_uu_0]: (![V_X, V_w]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_w)))) <=> ?[V_d]: (p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_d),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_1)))) & ![V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))) => p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_w),s(t_h4s_hrats_hrat,V_x))),s(t_h4s_hrats_hrat,V_d))))))) => ![V_X]: s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_inv(s(t_h4s_hreals_hreal,V_X))) = s(t_h4s_hreals_hreal,h4s_hreals_hreal0(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),V_uu_0),s(t_h4s_hreals_hreal,V_X))))))).
fof(ah4s_hreals_HREALu_u_ADDu_u_ISACUT, axiom, ![V_uu_0]: (![V_X, V_Y, V_w]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool))),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,V_w)))) <=> ?[V_x, V_y]: (s(t_h4s_hrats_hrat,V_w) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y))) & (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))) & p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,V_y))))))) => ![V_Y, V_X]: p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_hrats_hrat,t_bool),t_bool),h4s_hreals_isacut),s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool))),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hreals_hreal,V_Y)))))))).
fof(ah4s_hreals_HREALu_u_MULu_u_ISACUT, axiom, ![V_uu_0]: (![V_X, V_Y, V_w]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool))),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,V_w)))) <=> ?[V_x, V_y]: (s(t_h4s_hrats_hrat,V_w) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y))) & (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))) & p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,V_y))))))) => ![V_Y, V_X]: p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_hrats_hrat,t_bool),t_bool),h4s_hreals_isacut),s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool))),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hreals_hreal,V_Y)))))))).
fof(ah4s_hreals_EQUALu_u_CUTS, axiom, ![V_Y, V_X]: (s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))) = s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_Y))) => s(t_h4s_hreals_hreal,V_X) = s(t_h4s_hreals_hreal,V_Y))).
fof(ah4s_hreals_CUTu_u_NEARTOPu_u_MUL, axiom, ![V_u, V_X]: (p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_1),s(t_h4s_hrats_hrat,V_u)))) => ?[V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))) & ~ (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_u),s(t_h4s_hrats_hrat,V_x)))))))))).
fof(ah4s_hreals_hrealu_u_tybiju_c1, axiom, ![V_r]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_hrats_hrat,t_bool),t_bool),h4s_hreals_isacut),s(t_fun(t_h4s_hrats_hrat,t_bool),V_r)))) <=> s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,h4s_hreals_hreal0(s(t_fun(t_h4s_hrats_hrat,t_bool),V_r))))) = s(t_fun(t_h4s_hrats_hrat,t_bool),V_r))).
fof(ah4s_hreals_HREALu_u_INVu_u_ISACUT, axiom, ![V_uu_0]: (![V_X, V_w]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_w)))) <=> ?[V_d]: (p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_d),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_1)))) & ![V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))) => p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_w),s(t_h4s_hrats_hrat,V_x))),s(t_h4s_hrats_hrat,V_d))))))) => ![V_X]: p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_hrats_hrat,t_bool),t_bool),h4s_hreals_isacut),s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),V_uu_0),s(t_h4s_hreals_hreal,V_X)))))))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_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_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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_hrats_HRATu_u_MULu_u_ASSOC, axiom, ![V_j, V_i, V_h]: s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_h),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_i),s(t_h4s_hrats_hrat,V_j))))) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_h),s(t_h4s_hrats_hrat,V_i))),s(t_h4s_hrats_hrat,V_j)))).
fof(ah4s_hreals_hrealu_u_10, axiom, s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_1) = s(t_h4s_hreals_hreal,h4s_hreals_hreal0(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cutu_u_ofu_u_hrat(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_1)))))).
fof(ah4s_hrats_HRATu_u_MULu_u_LID, axiom, ![V_h]: s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_1),s(t_h4s_hrats_hrat,V_h))) = s(t_h4s_hrats_hrat,V_h)).
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_hrats_HRATu_u_MULu_u_LINV, axiom, ![V_h]: s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_inv(s(t_h4s_hrats_hrat,V_h))),s(t_h4s_hrats_hrat,V_h))) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_1)).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_hreals_HRATu_u_LTu_u_RADD, axiom, ![V_z, V_y, V_x]: s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_z))),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_y),s(t_h4s_hrats_hrat,V_z))))) = s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_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_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_hreals_HRATu_u_LTu_u_R1, axiom, ![V_y, V_x]: s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_inv(s(t_h4s_hrats_hrat,V_y))))),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_1))) = s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))).
fof(ah4s_hreals_HRATu_u_LTu_u_LMUL1, axiom, ![V_y, V_x]: s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y))),s(t_h4s_hrats_hrat,V_y))) = s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_1)))).
fof(ah4s_hreals_HRATu_u_LTu_u_NE, axiom, ![V_y, V_x]: (p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))) => ~ (s(t_h4s_hrats_hrat,V_x) = s(t_h4s_hrats_hrat,V_y)))).
fof(ah4s_hreals_hratu_u_lt0, axiom, ![V_y, V_x]: (p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))) <=> ?[V_d]: s(t_h4s_hrats_hrat,V_y) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_d))))).
fof(ah4s_hreals_HRATu_u_LTu_u_RMUL, axiom, ![V_z, V_y, V_x]: s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_z))),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_y),s(t_h4s_hrats_hrat,V_z))))) = s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))).
fof(ah4s_hrats_HRATu_u_MULu_u_SYM, axiom, ![V_i, V_h]: s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_h),s(t_h4s_hrats_hrat,V_i))) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_i),s(t_h4s_hrats_hrat,V_h)))).
fof(ah4s_hreals_HRATu_u_MULu_u_RID, axiom, ![V_x]: s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_1))) = s(t_h4s_hrats_hrat,V_x)).
fof(ah4s_hrats_HRATu_u_LDISTRIB, axiom, ![V_j, V_i, V_h]: s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_h),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_i),s(t_h4s_hrats_hrat,V_j))))) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_h),s(t_h4s_hrats_hrat,V_i))),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_h),s(t_h4s_hrats_hrat,V_j)))))).
fof(ah4s_hreals_HRATu_u_LTu_u_TOTAL, axiom, ![V_y, V_x]: (s(t_h4s_hrats_hrat,V_x) = s(t_h4s_hrats_hrat,V_y) | (p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))) | p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_y),s(t_h4s_hrats_hrat,V_x))))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> 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_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_hreals_HRATu_u_DOWN, axiom, ![V_x]: ?[V_y]: p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_y),s(t_h4s_hrats_hrat,V_x))))).
fof(ah4s_hreals_HRATu_u_MEAN, axiom, ![V_y, V_x]: (p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))) => ?[V_z]: (p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_z)))) & p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_z),s(t_h4s_hrats_hrat,V_y))))))).
fof(ah4s_hreals_HRATu_u_LTu_u_TRANS, axiom, ![V_z, V_y, V_x]: ((p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))) & p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_y),s(t_h4s_hrats_hrat,V_z))))) => p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_z)))))).
fof(ah4s_hreals_HRATu_u_RDISTRIB, axiom, ![V_z, V_y, V_x]: s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y))),s(t_h4s_hrats_hrat,V_z))) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_z))),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_y),s(t_h4s_hrats_hrat,V_z)))))).
fof(ah4s_hreals_HRATu_u_LTu_u_LMUL, axiom, ![V_z, V_y, V_x]: s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_z),s(t_h4s_hrats_hrat,V_x))),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_z),s(t_h4s_hrats_hrat,V_y))))) = s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))).
fof(ah4s_hreals_HRATu_u_LTu_u_LADD, axiom, ![V_z, V_y, V_x]: s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_z),s(t_h4s_hrats_hrat,V_x))),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_z),s(t_h4s_hrats_hrat,V_y))))) = s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_hreals_HRATu_u_LTu_u_ADD2, axiom, ![V_y, V_x, V_v, V_u]: ((p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_u),s(t_h4s_hrats_hrat,V_x)))) & p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_v),s(t_h4s_hrats_hrat,V_y))))) => p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_u),s(t_h4s_hrats_hrat,V_v))),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))))))).
fof(ah4s_hreals_HRATu_u_LTu_u_MUL2, axiom, ![V_y, V_x, V_v, V_u]: ((p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_u),s(t_h4s_hrats_hrat,V_x)))) & p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_v),s(t_h4s_hrats_hrat,V_y))))) => p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_u),s(t_h4s_hrats_hrat,V_v))),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))))))).
fof(ah4s_hreals_HRATu_u_GTu_u_L1, axiom, ![V_y, V_x]: s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_1),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_inv(s(t_h4s_hrats_hrat,V_x))),s(t_h4s_hrats_hrat,V_y))))) = s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))).
fof(ah4s_hreals_HRATu_u_LTu_u_L1, axiom, ![V_y, V_x]: s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_inv(s(t_h4s_hrats_hrat,V_x))),s(t_h4s_hrats_hrat,V_y))),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_1))) = s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_y),s(t_h4s_hrats_hrat,V_x)))).
fof(ah4s_hreals_HRATu_u_GTu_u_LMUL1, axiom, ![V_y, V_x]: s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_y),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y))))) = s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_1),s(t_h4s_hrats_hrat,V_x)))).
fof(ah4s_hreals_HRATu_u_LTu_u_RMUL1, axiom, ![V_y, V_x]: s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y))),s(t_h4s_hrats_hrat,V_x))) = s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_y),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_1)))).
fof(ah4s_hreals_HRATu_u_LTu_u_GT, axiom, ![V_y, V_x]: (p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))) => ~ (p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_y),s(t_h4s_hrats_hrat,V_x))))))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_primu_u_recs_NOTu_u_LESSu_u_0, axiom, ![V_n]: ~ (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0)))))).
fof(ah4s_primu_u_recs_LESSu_u_SUCu_u_REFL, axiom, ![V_n]: p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))))).
fof(ah4s_hreals_HRATu_u_LTu_u_ADDR, axiom, ![V_y, V_x]: p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_y),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y))))))).
fof(ah4s_hreals_HRATu_u_MULu_u_RINV, axiom, ![V_x]: s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_inv(s(t_h4s_hrats_hrat,V_x))))) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_1)).
fof(ah4s_hrats_HRATu_u_SUCINTu_c1, axiom, ![V_n]: s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_sucint(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_sucint(s(t_h4s_nums_num,V_n))),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_1)))).
fof(ah4s_hrats_HRATu_u_SUCINTu_c0, axiom, s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_sucint(s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_1)).
fof(ah4s_hrats_HRATu_u_ARCH, axiom, ![V_h]: ?[V_n, V_d]: s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_sucint(s(t_h4s_nums_num,V_n))) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_h),s(t_h4s_hrats_hrat,V_d)))).
fof(ah4s_arithmetics_WOP, axiom, ![V_P]: (?[V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) => ?[V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) & ![V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) => ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_m))))))))).
fof(ah4s_arithmetics_numu_u_CASES, axiom, ![V_m]: (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0) | ?[V_n]: s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))).
fof(ah4s_hreals_cutu_u_ofu_u_hrat0, axiom, ![V_x, V_xi_]: s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cutu_u_ofu_u_hrat(s(t_h4s_hrats_hrat,V_x))),s(t_h4s_hrats_hrat,V_xi_))) = s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_xi_),s(t_h4s_hrats_hrat,V_x)))).
fof(ah4s_hreals_HRATu_u_UP, axiom, ![V_x]: ?[V_y]: p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y))))).
fof(ah4s_bools_CONJu_u_SYM, 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))))).
fof(ah4s_hreals_HRATu_u_LTu_u_ANTISYM, axiom, ![V_y, V_x]: ~ ((p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))) & p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_y),s(t_h4s_hrats_hrat,V_x))))))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => 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_lists_LENGTHu_u_CONS, axiom, ![TV_u_27a]: ![V_n, V_l]: (s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) <=> ?[V_h, V_lu_27]: (s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_lu_27))) = s(t_h4s_nums_num,V_n) & s(t_h4s_lists_list(TV_u_27a),V_l) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_lu_27)))))).
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_ANDu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,f))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_EXISTSu_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_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_lists_NOTu_u_NILu_u_CONS, axiom, ![TV_u_27a]: ![V_a1, V_a0]: ~ (s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1))))).
fof(ah4s_lists_LENGTH0u_c1, axiom, ![TV_u_27a]: ![V_t, V_h]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_t)))))).
fof(ah4s_lists_listu_u_INDUCT, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))) & ![V_t]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t)))) => ![V_h]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t)))))))) => ![V_l]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_lists_LENGTH0u_c0, axiom, ![TV_u_27a]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_lists_CONSu_u_11, axiom, ![TV_u_27a]: ![V_a1u_27, V_a1, V_a0u_27, V_a0]: (s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1))) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_a0u_27))),s(t_h4s_lists_list(TV_u_27a),V_a1u_27))) <=> (s(TV_u_27a,V_a0) = s(TV_u_27a,V_a0u_27) & s(t_h4s_lists_list(TV_u_27a),V_a1) = s(t_h4s_lists_list(TV_u_27a),V_a1u_27)))).
fof(ah4s_primu_u_recs_INVu_u_SUCu_u_EQ, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_nums_NOTu_u_SUC, axiom, ![V_n]: ~ (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0))).
fof(ah4s_bools_UNWINDu_u_THM2, axiom, ![TV_u_27a]: ![V_a, V_P]: (?[V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & 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,V_a)))))).
fof(ah4s_arithmetics_MAXu_u_DEF, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_max(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_bools_cond(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_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_lists_LASTu_u_APPENDu_u_CONS, axiom, ![TV_u_27a]: ![V_l2, V_l1, V_h]: s(TV_u_27a,h4s_lists_last(s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_l2))))))) = s(TV_u_27a,h4s_lists_last(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_l2)))))).
fof(ah4s_bools_BETAu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
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_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_lists_LASTu_u_CONSu_u_cond, axiom, ![TV_u_27a]: ![V_t, V_h]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_lists_list(TV_u_27a),V_t) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)) & s(TV_u_27a,h4s_lists_last(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t))))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27a,V_h),s(TV_u_27a,h4s_lists_last(s(t_h4s_lists_list(TV_u_27a),V_t))))))).
fof(ah4s_lists_APPENDu_u_equ_u_NILu_c0, axiom, ![TV_u_27a]: ![V_l2, V_l1]: (s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2))) <=> (s(t_h4s_lists_list(TV_u_27a),V_l1) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) & s(t_h4s_lists_list(TV_u_27a),V_l2) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))).
fof(ah4s_lists_APPEND0u_c0, axiom, ![TV_u_27a]: ![V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),V_l)).
fof(ah4s_lists_APPEND0u_c1, axiom, ![TV_u_27a]: ![V_l2, V_l1, V_h]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_l1))),s(t_h4s_lists_list(TV_u_27a),V_l2))) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2)))))).
fof(ah4s_lists_CONS0, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons) = s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_u_20u_40indu_u_typelist1)).
fof(ah4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist0u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist0) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))).
fof(ah4s_lists_listu_u_repfnsu_c1, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27listu_27]: (![V_a0u_270]: ((s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))) | ?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(TV_u_27a,V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27listu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27listu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27listu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r))))) <=> s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r))))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r)))).
fof(ah4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist1u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => ![V_x, V_x0]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_u_20u_40indu_u_typelist1),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_x0))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(TV_u_27a,V_x),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(TV_u_27a),V_x0))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))))).
fof(ah4s_lists_listu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(TV_u_27a),V_a))))) = s(t_h4s_lists_list(TV_u_27a),V_a)).
fof(ah4s_lists_NIL0, axiom, ![TV_u_27a]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist0)).
fof(ah4s_indu_u_types_FCONS0u_c1, axiom, ![TV_u_27a]: ![V_n, V_f, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n)))).
fof(ah4s_indu_u_types_FCONS0u_c0, axiom, ![TV_u_27a]: ![V_f, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_a)).
fof(ah4s_indu_u_types_CONSTRu_u_REC, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_f, V_r, V_n]: s(TV_u_27b,happ(s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_h4s_nums_num,V_n))) = s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r),s(t_h4s_nums_num,V_n))))) => ![V_Fn]: ?[V_f]: ![V_c, V_i, V_r]: s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,V_c),s(TV_u_27a,V_i),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))) = s(TV_u_27b,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)))),V_Fn),s(t_h4s_nums_num,V_c))),s(TV_u_27a,V_i))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))))).
fof(ah4s_bools_JRHu_u_INDUCTu_u_UTIL, axiom, ![TV_u_27a]: ![V_t, V_P]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_t) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) => p(s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_P)))))).
fof(ah4s_bools_boolu_u_caseu_u_thmu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_bools_MONOu_u_OR, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) | p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) | p(s(t_bool,V_w)))))).
fof(ch4s_hreals_HREALu_u_ADDu_u_SYM, conjecture, ![V_Y, V_X]: s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y))) = s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_Y),s(t_h4s_hreals_hreal,V_X)))).
