%   ORIGINAL: h4/integer/INT__INJ0
% 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/integer/int__of__num0: !T1. h4/integer/int__of__num T1 = h4/integer/int__ABS (h4/integer/tint__of__num T1)
% Assm: h4/integer/int__of__num1_c1: !n. h4/integer/int__of__num (h4/num/SUC n) = h4/integer/int__add (h4/integer/int__of__num n) h4/integer/int__1
% Assm: h4/integer/int__of__num1_c0: h4/integer/int__of__num h4/num/0 = h4/integer/int__0
% Assm: h4/integer/int__QUOTIENT: h4/quotient/QUOTIENT h4/integer/tint__eq h4/integer/int__ABS h4/integer/int__REP
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/integer/TINT__EQ__EQUIV: !q p. h4/integer/tint__eq p q <=> h4/integer/tint__eq p = h4/integer/tint__eq q
% Assm: h4/quotient/EQUIV__def: !E. h4/quotient/EQUIV E <=> (!x y. E x y <=> E x = E y)
% Assm: h4/quotient/REP__ABS__RSP: !rep abs REL. h4/quotient/QUOTIENT REL abs rep ==> (!x1 x2. REL x1 x2 ==> REL x1 (rep (abs x2)))
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/integer/int__add0: !T2 T1. h4/integer/int__add T1 T2 = h4/integer/int__ABS (h4/integer/tint__add (h4/integer/int__REP T1) (h4/integer/int__REP T2))
% Assm: h4/quotient/APPLY__RSP: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g x y. h4/quotient/_3D_3D_3D_3E R1 R2 f g /\ R1 x y ==> R2 (f x) (g y)))
% Assm: h4/quotient/FUN__REL: !g f R2 R1. h4/quotient/_3D_3D_3D_3E R1 R2 f g <=> (!x y. R1 x y ==> R2 (f x) (g y))
% Assm: h4/quotient/FUN__QUOTIENT: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> h4/quotient/QUOTIENT (h4/quotient/_3D_3D_3D_3E R1 R2) (h4/quotient/_2D_2D_3E rep1 abs2) (h4/quotient/_2D_2D_3E abs1 rep2))
% Assm: h4/quotient/IDENTITY__QUOTIENT: h4/quotient/QUOTIENT $equals h4/combin/I h4/combin/I
% Assm: h4/integer/int__00: h4/integer/int__0 = h4/integer/int__ABS h4/integer/tint__0
% Assm: h4/integer/int__10: h4/integer/int__1 = h4/integer/int__ABS h4/integer/tint__1
% Assm: h4/quotient/EQUALS__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x y. x = y <=> R (rep x) (rep y))
% Assm: h4/quotient/EQUALS__RSP: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x1 x2 y1 y2. R x1 x2 /\ R y1 y2 ==> (R x1 y1 <=> R x2 y2))
% Assm: h4/integer/TINT__EQ__AP: !q p. p = q ==> h4/integer/tint__eq p q
% Assm: h4/integer/TINT__ADD__WELLDEF: !y2 y1 x2 x1. h4/integer/tint__eq x1 x2 /\ h4/integer/tint__eq y1 y2 ==> h4/integer/tint__eq (h4/integer/tint__add x1 y1) (h4/integer/tint__add x2 y2)
% Assm: h4/quotient/EQUIV__REFL__SYM__TRANS: !R. (!x y. R x y <=> R x = R y) <=> (!x. R x x) /\ (!x y. R x y ==> R y x) /\ (!x y z. R x y /\ R y z ==> R x z)
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/integer/tint__of__num0_c0: h4/integer/tint__of__num h4/num/0 = h4/integer/tint__0
% Assm: h4/integer/tint__of__num0_c1: !n. h4/integer/tint__of__num (h4/num/SUC n) = h4/integer/tint__add (h4/integer/tint__of__num n) h4/integer/tint__1
% Assm: h4/integer/int__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION (\c. ?r. h4/integer/tint__eq r r /\ c = h4/integer/tint__eq r) rep
% Assm: h4/integer/int__bijections_c1: !r. (\c. ?r0. h4/integer/tint__eq r0 r0 /\ c = h4/integer/tint__eq r0) r <=> h4/integer/int__REP__CLASS (h4/integer/int__ABS__CLASS r) = r
% Assm: h4/integer/int__bijections_c0: !a. h4/integer/int__ABS__CLASS (h4/integer/int__REP__CLASS a) = a
% Assm: h4/bool/ABS__REP__THM: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. abs (rep a) = a) /\ (!r. P r <=> rep (abs r) = r))
% Assm: h4/integer/int__lt0: !T2 T1. h4/integer/int__lt T1 T2 <=> h4/integer/tint__lt (h4/integer/int__REP T1) (h4/integer/int__REP T2)
% Assm: h4/integer/int__mul0: !T2 T1. h4/integer/int__mul T1 T2 = h4/integer/int__ABS (h4/integer/tint__mul (h4/integer/int__REP T1) (h4/integer/int__REP T2))
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/quotient/RES__FORALL__RSP: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f g. h4/quotient/_3D_3D_3D_3E R $equals f g ==> (h4/bool/RES__FORALL (h4/quotient/respects R) f <=> h4/bool/RES__FORALL (h4/quotient/respects R) g))
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/quotient/FORALL__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f. $forall f <=> h4/bool/RES__FORALL (h4/quotient/respects R) (h4/quotient/_2D_2D_3E abs h4/combin/I f))
% Assm: h4/quotient/LAMBDA__PRS: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f. (\x. f x) = h4/quotient/_2D_2D_3E rep1 abs2 (\x. rep2 (f (abs1 x)))))
% Assm: h4/quotient/EQUIV__RES__FORALL: !P E. h4/quotient/EQUIV E ==> (h4/bool/RES__FORALL (h4/quotient/respects E) P <=> $forall P)
% Assm: h4/integer/int__ABS__def: !r. h4/integer/int__ABS r = h4/integer/int__ABS__CLASS (h4/integer/tint__eq r)
% Assm: h4/integer/int__REP__def: !a. h4/integer/int__REP a = h4/min/_40 (h4/integer/int__REP__CLASS a)
% Assm: h4/integer/int__ABS__REP__CLASS_c0: !a. h4/integer/int__ABS__CLASS (h4/integer/int__REP__CLASS a) = a
% Assm: h4/integer/int__ABS__REP__CLASS_c1: !c. (?r. h4/integer/tint__eq r r /\ c = h4/integer/tint__eq r) <=> h4/integer/int__REP__CLASS (h4/integer/int__ABS__CLASS c) = c
% Assm: h4/integer/int__neg0: !T1. h4/integer/int__neg T1 = h4/integer/int__ABS (h4/integer/tint__neg (h4/integer/int__REP T1))
% Assm: h4/integer/INT__ADD__COMM: !y x. h4/integer/int__add x y = h4/integer/int__add y x
% Assm: h4/integer/INT__10: ~(h4/integer/int__1 = h4/integer/int__0)
% Assm: h4/integer/INT__ADD__ASSOC: !z y x. h4/integer/int__add x (h4/integer/int__add y z) = h4/integer/int__add (h4/integer/int__add x y) z
% Assm: h4/integer/INT__LT__TOTAL: !y x. x = y \/ h4/integer/int__lt x y \/ h4/integer/int__lt y x
% Assm: h4/integer/INT__ADD__SYM: !y x. h4/integer/int__add x y = h4/integer/int__add y x
% Assm: h4/integer/INT__LDISTRIB: !z y x. h4/integer/int__mul x (h4/integer/int__add y z) = h4/integer/int__add (h4/integer/int__mul x y) (h4/integer/int__mul x z)
% Assm: h4/integer/INT__MUL__SYM: !y x. h4/integer/int__mul x y = h4/integer/int__mul y x
% Assm: h4/integer/INT__MUL__ASSOC: !z y x. h4/integer/int__mul x (h4/integer/int__mul y z) = h4/integer/int__mul (h4/integer/int__mul x y) z
% Assm: h4/integer/INT__MUL__COMM: !y x. h4/integer/int__mul x y = h4/integer/int__mul y x
% Assm: h4/integer/INT__LT__MUL: !y x. h4/integer/int__lt h4/integer/int__0 x /\ h4/integer/int__lt h4/integer/int__0 y ==> h4/integer/int__lt h4/integer/int__0 (h4/integer/int__mul x y)
% Assm: h4/integer/INT__LT__REFL: !x. ~h4/integer/int__lt x x
% Assm: h4/integer/INT__LT__LADD__IMP: !z y x. h4/integer/int__lt y z ==> h4/integer/int__lt (h4/integer/int__add x y) (h4/integer/int__add x z)
% Assm: h4/integer/INT__LT__TRANS: !z y x. h4/integer/int__lt x y /\ h4/integer/int__lt y z ==> h4/integer/int__lt x z
% Assm: h4/quotient/EQUIV__IMP__PARTIAL__EQUIV: !R. h4/quotient/EQUIV R ==> h4/quotient/PARTIAL__EQUIV R
% Assm: h4/quotient/PARTIAL__EQUIV__def: !R. h4/quotient/PARTIAL__EQUIV R <=> (?x. R x x) /\ (!x y. R x y <=> R x x /\ R y y /\ R x = R y)
% Assm: h4/integer/TINT__LT__WELLDEF: !y2 y1 x2 x1. h4/integer/tint__eq x1 x2 /\ h4/integer/tint__eq y1 y2 ==> (h4/integer/tint__lt x1 y1 <=> h4/integer/tint__lt x2 y2)
% Assm: h4/integer/TINT__MUL__WELLDEF: !y2 y1 x2 x1. h4/integer/tint__eq x1 x2 /\ h4/integer/tint__eq y1 y2 ==> h4/integer/tint__eq (h4/integer/tint__mul x1 y1) (h4/integer/tint__mul x2 y2)
% Assm: h4/integer/TINT__10: ~h4/integer/tint__eq h4/integer/tint__1 h4/integer/tint__0
% Assm: h4/integer/TINT__ADD__SYM: !y x. h4/integer/tint__add x y = h4/integer/tint__add y x
% Assm: h4/integer/TINT__MUL__SYM: !y x. h4/integer/tint__mul x y = h4/integer/tint__mul y x
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/quotient/QUOTIENT__def: !rep abs R. h4/quotient/QUOTIENT R abs rep <=> (!a. abs (rep a) = a) /\ (!a. R (rep a) (rep a)) /\ (!r s. R r s <=> R r r /\ R s s /\ abs r = abs s)
% Assm: h4/integer/TINT__ADD__ASSOC: !z y x. h4/integer/tint__add x (h4/integer/tint__add y z) = h4/integer/tint__add (h4/integer/tint__add x y) z
% Assm: h4/integer/TINT__LT__TOTAL: !y x. h4/integer/tint__eq x y \/ h4/integer/tint__lt x y \/ h4/integer/tint__lt y x
% Assm: h4/integer/TINT__LDISTRIB: !z y x. h4/integer/tint__mul x (h4/integer/tint__add y z) = h4/integer/tint__add (h4/integer/tint__mul x y) (h4/integer/tint__mul x z)
% Assm: h4/integer/TINT__MUL__ASSOC: !z y x. h4/integer/tint__mul x (h4/integer/tint__mul y z) = h4/integer/tint__mul (h4/integer/tint__mul x y) z
% Assm: h4/integer/TINT__LT__REFL: !x. ~h4/integer/tint__lt x x
% Assm: h4/integer/TINT__LT__MUL: !y x. h4/integer/tint__lt h4/integer/tint__0 x /\ h4/integer/tint__lt h4/integer/tint__0 y ==> h4/integer/tint__lt h4/integer/tint__0 (h4/integer/tint__mul x y)
% Assm: h4/integer/TINT__LT__TRANS: !z y x. h4/integer/tint__lt x y /\ h4/integer/tint__lt y z ==> h4/integer/tint__lt x z
% Assm: h4/integer/TINT__LT__ADD: !z y x. h4/integer/tint__lt y z ==> h4/integer/tint__lt (h4/integer/tint__add x y) (h4/integer/tint__add x z)
% Assm: h4/arithmetic/SUB__RIGHT__GREATER__EQ: !p n m. h4/arithmetic/_3E_3D (h4/arithmetic/_2D m n) p <=> h4/arithmetic/_3E_3D m (h4/arithmetic/_2B n p) \/ h4/arithmetic/_3E_3D h4/num/0 p
% Assm: h4/arithmetic/SUB__LEFT__LESS__EQ: !p n m. h4/arithmetic/_3C_3D m (h4/arithmetic/_2D n p) <=> h4/arithmetic/_3C_3D (h4/arithmetic/_2B m p) n \/ h4/arithmetic/_3C_3D m h4/num/0
% Assm: h4/arithmetic/LESS__OR__EQ: !n m. h4/arithmetic/_3C_3D m n <=> h4/prim__rec/_3C m n \/ m = n
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/GREATER__OR__EQ: !n m. h4/arithmetic/_3E_3D m n <=> h4/arithmetic/_3E m n \/ m = n
% Assm: h4/arithmetic/GREATER__DEF: !n m. h4/arithmetic/_3E m n <=> h4/prim__rec/_3C n m
% Assm: h4/arithmetic/SUB__LEFT__LESS: !p n m. h4/prim__rec/_3C m (h4/arithmetic/_2D n p) <=> h4/prim__rec/_3C (h4/arithmetic/_2B m p) n
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/arithmetic/LESS__EQ: !n m. h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm: h4/arithmetic/ADD_c1: !n m. h4/arithmetic/_2B (h4/num/SUC m) n = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm: h4/prim__rec/NOT__LESS__0: !n. ~h4/prim__rec/_3C n h4/num/0
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/ind__type/CONSTR__BOT: !r i c. ~(h4/ind__type/CONSTR c i r = h4/ind__type/BOTTOM)
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/ind__type/MK__REC__INJ: !y x. h4/ind__type/mk__rec x = h4/ind__type/mk__rec y ==> h4/ind__type/ZRECSPACE x /\ h4/ind__type/ZRECSPACE y ==> x = y
% Assm: h4/ind__type/CONSTR0: !r i c. h4/ind__type/CONSTR c i r = h4/ind__type/mk__rec (h4/ind__type/ZCONSTR c i (\n. h4/ind__type/dest__rec (r n)))
% Assm: h4/ind__type/ZCONSTR__ZBOT: !r i c. ~(h4/ind__type/ZCONSTR c i r = h4/ind__type/ZBOT)
% Assm: h4/ind__type/ZRECSPACE__def: h4/ind__type/ZRECSPACE = (\a0. !ZRECSPACE_27. (!a00. a00 = h4/ind__type/ZBOT \/ (?c i r. a00 = h4/ind__type/ZCONSTR c i r /\ (!n. ZRECSPACE_27 (r n))) ==> ZRECSPACE_27 a00) ==> ZRECSPACE_27 a0)
% Assm: h4/ind__type/recspace__repfns_c0: !a. h4/ind__type/mk__rec (h4/ind__type/dest__rec a) = a
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/ind__type/recspace__repfns_c1: !r. h4/ind__type/ZRECSPACE r <=> h4/ind__type/dest__rec (h4/ind__type/mk__rec r) = r
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/bool/MONO__ALL: !Q P. (!x. P x ==> Q x) ==> (!x. P x) ==> (!x. Q x)
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/ind__type/BOTTOM0: h4/ind__type/BOTTOM = h4/ind__type/mk__rec h4/ind__type/ZBOT
% Assm: h4/numpair/nlistrec__curried__def: !x2 x1 x. h4/numpair/nlistrec x x1 x2 = h4/numpair/nlistrec__tupled (h4/pair/_2C x (h4/pair/_2C x1 x2))
% Assm: h4/arithmetic/MIN__MAX__LT: !n m. h4/prim__rec/_3C (h4/arithmetic/MIN m n) (h4/arithmetic/MAX m n) <=> ~(m = n)
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/arithmetic/MIN__DEF: !n m. h4/arithmetic/MIN m n = h4/bool/COND (h4/prim__rec/_3C m n) m n
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% 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/arithmetic/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/prim__rec/LESS__REFL: !n. ~h4/prim__rec/_3C n n
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Goal: !n m. h4/integer/int__of__num m = h4/integer/int__of__num n <=> m = n
%   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_integers_intu_u_ofu_u_num0]: !T1. h4/integer/int__of__num T1 = happ h4/integer/int__ABS (h4/integer/tint__of__num T1)
% Assm [h4s_integers_intu_u_ofu_u_num1u_c1]: !n. h4/integer/int__of__num (h4/num/SUC n) = h4/integer/int__add (h4/integer/int__of__num n) h4/integer/int__1
% Assm [h4s_integers_intu_u_ofu_u_num1u_c0]: h4/integer/int__of__num h4/num/0 = h4/integer/int__0
% Assm [h4s_integers_intu_u_QUOTIENT]: h4/quotient/QUOTIENT h4/integer/tint__eq h4/integer/int__ABS h4/integer/int__REP
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_integers_TINTu_u_EQu_u_EQUIV]: !q p. happ (happ h4/integer/tint__eq p) q <=> happ h4/integer/tint__eq p = happ h4/integer/tint__eq q
% Assm [h4s_quotients_EQUIVu_u_def]: !E. h4/quotient/EQUIV E <=> (!x y. happ (happ E x) y <=> happ E x = happ E y)
% Assm [h4s_quotients_REPu_u_ABSu_u_RSP]: !rep abs REL. h4/quotient/QUOTIENT REL abs rep ==> (!x1 x2. happ (happ REL x1) x2 ==> happ (happ REL x1) (happ rep (happ abs x2)))
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_integers_intu_u_add0]: !T2 T1. h4/integer/int__add T1 T2 = happ h4/integer/int__ABS (h4/integer/tint__add (happ h4/integer/int__REP T1) (happ h4/integer/int__REP T2))
% Assm [h4s_quotients_APPLYu_u_RSP]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f g x y. happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) f) g /\ happ (happ R1 x) y ==> happ (happ R2 (happ f x)) (happ g y)))
% Assm [h4s_quotients_FUNu_u_REL]: !g f R2 R1. happ (happ (h4/quotient/_3D_3D_3D_3E R1 R2) f) g <=> (!x y. happ (happ R1 x) y ==> happ (happ R2 (happ f x)) (happ g y))
% Assm [h4s_quotients_FUNu_u_QUOTIENT]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> h4/quotient/QUOTIENT (h4/quotient/_3D_3D_3D_3E R1 R2) (h4/quotient/_2D_2D_3E rep1 abs2) (h4/quotient/_2D_2D_3E abs1 rep2))
% Assm [h4s_quotients_IDENTITYu_u_QUOTIENT]: h4/quotient/QUOTIENT $equals h4/combin/I h4/combin/I
% Assm [h4s_integers_intu_u_00]: h4/integer/int__0 = happ h4/integer/int__ABS h4/integer/tint__0
% Assm [h4s_integers_intu_u_10]: h4/integer/int__1 = happ h4/integer/int__ABS h4/integer/tint__1
% Assm [h4s_quotients_EQUALSu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x y. x = y <=> happ (happ R (happ rep x)) (happ rep y))
% Assm [h4s_quotients_EQUALSu_u_RSP]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x1 x2 y1 y2. happ (happ R x1) x2 /\ happ (happ R y1) y2 ==> (happ (happ R x1) y1 <=> happ (happ R x2) y2))
% Assm [h4s_integers_TINTu_u_EQu_u_AP]: !q p. p = q ==> happ (happ h4/integer/tint__eq p) q
% Assm [h4s_integers_TINTu_u_ADDu_u_WELLDEF]: !y2 y1 x2 x1. happ (happ h4/integer/tint__eq x1) x2 /\ happ (happ h4/integer/tint__eq y1) y2 ==> happ (happ h4/integer/tint__eq (h4/integer/tint__add x1 y1)) (h4/integer/tint__add x2 y2)
% Assm [h4s_quotients_EQUIVu_u_REFLu_u_SYMu_u_TRANS]: !R. (!x y. happ (happ R x) y <=> happ R x = happ R y) <=> (!x. happ (happ R x) x) /\ (!x y. happ (happ R x) y ==> happ (happ R y) x) /\ (!x y z. happ (happ R x) y /\ happ (happ R y) z ==> happ (happ R x) z)
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_integers_tintu_u_ofu_u_num0u_c0]: h4/integer/tint__of__num h4/num/0 = h4/integer/tint__0
% Assm [h4s_integers_tintu_u_ofu_u_num0u_c1]: !n. h4/integer/tint__of__num (h4/num/SUC n) = h4/integer/tint__add (h4/integer/tint__of__num n) h4/integer/tint__1
% Assm [h4s_integers_intu_u_TYu_u_DEF]: !_0. (!c. happ _0 c <=> (?r. happ (happ h4/integer/tint__eq r) r /\ c = happ h4/integer/tint__eq r)) ==> (?rep. h4/bool/TYPE__DEFINITION _0 rep)
% Assm [h4s_integers_intu_u_bijectionsu_c1]: !r. (?r0. happ (happ h4/integer/tint__eq r0) r0 /\ r = happ h4/integer/tint__eq r0) <=> h4/integer/int__REP__CLASS (h4/integer/int__ABS__CLASS r) = r
% Assm [h4s_integers_intu_u_bijectionsu_c0]: !a. h4/integer/int__ABS__CLASS (h4/integer/int__REP__CLASS a) = a
% Assm [h4s_bools_ABSu_u_REPu_u_THM]: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. happ abs (happ rep a) = a) /\ (!r. happ P r <=> happ rep (happ abs r) = r))
% Assm [h4s_integers_intu_u_lt0]: !T2 T1. h4/integer/int__lt T1 T2 <=> h4/integer/tint__lt (happ h4/integer/int__REP T1) (happ h4/integer/int__REP T2)
% Assm [h4s_integers_intu_u_mul0]: !T2 T1. h4/integer/int__mul T1 T2 = happ h4/integer/int__ABS (h4/integer/tint__mul (happ h4/integer/int__REP T1) (happ h4/integer/int__REP T2))
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_quotients_RESu_u_FORALLu_u_RSP]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f g. happ (happ (h4/quotient/_3D_3D_3D_3E R $equals) f) g ==> (h4/bool/RES__FORALL (h4/quotient/respects R) f <=> h4/bool/RES__FORALL (h4/quotient/respects R) g))
% Assm [h4s_combins_Iu_u_THM]: !x. happ h4/combin/I x = x
% Assm [h4s_quotients_FORALLu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f. $forall f <=> h4/bool/RES__FORALL (h4/quotient/respects R) (happ (h4/quotient/_2D_2D_3E abs h4/combin/I) f))
% Assm [h4s_quotients_LAMBDAu_u_PRS]: !_0. (!rep2 f abs1 x. happ (happ (happ (happ _0 rep2) f) abs1) x = happ rep2 (happ f (happ abs1 x))) ==> (!rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f x. happ f x = happ (happ (h4/quotient/_2D_2D_3E rep1 abs2) (happ (happ (happ _0 rep2) f) abs1)) x)))
% Assm [h4s_quotients_EQUIVu_u_RESu_u_FORALL]: !P E. h4/quotient/EQUIV E ==> (h4/bool/RES__FORALL (h4/quotient/respects E) P <=> $forall P)
% Assm [h4s_integers_intu_u_ABSu_u_def]: !r. happ h4/integer/int__ABS r = h4/integer/int__ABS__CLASS (happ h4/integer/tint__eq r)
% Assm [h4s_integers_intu_u_REPu_u_def]: !a. happ h4/integer/int__REP a = h4/min/_40 (h4/integer/int__REP__CLASS a)
% Assm [h4s_integers_intu_u_ABSu_u_REPu_u_CLASSu_c0]: !a. h4/integer/int__ABS__CLASS (h4/integer/int__REP__CLASS a) = a
% Assm [h4s_integers_intu_u_ABSu_u_REPu_u_CLASSu_c1]: !c. (?r. happ (happ h4/integer/tint__eq r) r /\ c = happ h4/integer/tint__eq r) <=> h4/integer/int__REP__CLASS (h4/integer/int__ABS__CLASS c) = c
% Assm [h4s_integers_intu_u_neg0]: !T1. h4/integer/int__neg T1 = happ h4/integer/int__ABS (h4/integer/tint__neg (happ h4/integer/int__REP T1))
% Assm [h4s_integers_INTu_u_ADDu_u_COMM]: !y x. h4/integer/int__add x y = h4/integer/int__add y x
% Assm [h4s_integers_INTu_u_10]: ~(h4/integer/int__1 = h4/integer/int__0)
% Assm [h4s_integers_INTu_u_ADDu_u_ASSOC]: !z y x. h4/integer/int__add x (h4/integer/int__add y z) = h4/integer/int__add (h4/integer/int__add x y) z
% Assm [h4s_integers_INTu_u_LTu_u_TOTAL]: !y x. x = y \/ h4/integer/int__lt x y \/ h4/integer/int__lt y x
% Assm [h4s_integers_INTu_u_ADDu_u_SYM]: !y x. h4/integer/int__add x y = h4/integer/int__add y x
% Assm [h4s_integers_INTu_u_LDISTRIB]: !z y x. h4/integer/int__mul x (h4/integer/int__add y z) = h4/integer/int__add (h4/integer/int__mul x y) (h4/integer/int__mul x z)
% Assm [h4s_integers_INTu_u_MULu_u_SYM]: !y x. h4/integer/int__mul x y = h4/integer/int__mul y x
% Assm [h4s_integers_INTu_u_MULu_u_ASSOC]: !z y x. h4/integer/int__mul x (h4/integer/int__mul y z) = h4/integer/int__mul (h4/integer/int__mul x y) z
% Assm [h4s_integers_INTu_u_MULu_u_COMM]: !y x. h4/integer/int__mul x y = h4/integer/int__mul y x
% Assm [h4s_integers_INTu_u_LTu_u_MUL]: !y x. h4/integer/int__lt h4/integer/int__0 x /\ h4/integer/int__lt h4/integer/int__0 y ==> h4/integer/int__lt h4/integer/int__0 (h4/integer/int__mul x y)
% Assm [h4s_integers_INTu_u_LTu_u_REFL]: !x. ~h4/integer/int__lt x x
% Assm [h4s_integers_INTu_u_LTu_u_LADDu_u_IMP]: !z y x. h4/integer/int__lt y z ==> h4/integer/int__lt (h4/integer/int__add x y) (h4/integer/int__add x z)
% Assm [h4s_integers_INTu_u_LTu_u_TRANS]: !z y x. h4/integer/int__lt x y /\ h4/integer/int__lt y z ==> h4/integer/int__lt x z
% Assm [h4s_quotients_EQUIVu_u_IMPu_u_PARTIALu_u_EQUIV]: !R. h4/quotient/EQUIV R ==> h4/quotient/PARTIAL__EQUIV R
% Assm [h4s_quotients_PARTIALu_u_EQUIVu_u_def]: !R. h4/quotient/PARTIAL__EQUIV R <=> (?x. happ (happ R x) x) /\ (!x y. happ (happ R x) y <=> happ (happ R x) x /\ happ (happ R y) y /\ happ R x = happ R y)
% Assm [h4s_integers_TINTu_u_LTu_u_WELLDEF]: !y2 y1 x2 x1. happ (happ h4/integer/tint__eq x1) x2 /\ happ (happ h4/integer/tint__eq y1) y2 ==> (h4/integer/tint__lt x1 y1 <=> h4/integer/tint__lt x2 y2)
% Assm [h4s_integers_TINTu_u_MULu_u_WELLDEF]: !y2 y1 x2 x1. happ (happ h4/integer/tint__eq x1) x2 /\ happ (happ h4/integer/tint__eq y1) y2 ==> happ (happ h4/integer/tint__eq (h4/integer/tint__mul x1 y1)) (h4/integer/tint__mul x2 y2)
% Assm [h4s_integers_TINTu_u_10]: ~happ (happ h4/integer/tint__eq h4/integer/tint__1) h4/integer/tint__0
% Assm [h4s_integers_TINTu_u_ADDu_u_SYM]: !y x. h4/integer/tint__add x y = h4/integer/tint__add y x
% Assm [h4s_integers_TINTu_u_MULu_u_SYM]: !y x. h4/integer/tint__mul x y = h4/integer/tint__mul y x
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_quotients_QUOTIENTu_u_def]: !rep abs R. h4/quotient/QUOTIENT R abs rep <=> (!a. happ abs (happ rep a) = a) /\ (!a. happ (happ R (happ rep a)) (happ rep a)) /\ (!r s. happ (happ R r) s <=> happ (happ R r) r /\ happ (happ R s) s /\ happ abs r = happ abs s)
% Assm [h4s_integers_TINTu_u_ADDu_u_ASSOC]: !z y x. h4/integer/tint__add x (h4/integer/tint__add y z) = h4/integer/tint__add (h4/integer/tint__add x y) z
% Assm [h4s_integers_TINTu_u_LTu_u_TOTAL]: !y x. happ (happ h4/integer/tint__eq x) y \/ h4/integer/tint__lt x y \/ h4/integer/tint__lt y x
% Assm [h4s_integers_TINTu_u_LDISTRIB]: !z y x. h4/integer/tint__mul x (h4/integer/tint__add y z) = h4/integer/tint__add (h4/integer/tint__mul x y) (h4/integer/tint__mul x z)
% Assm [h4s_integers_TINTu_u_MULu_u_ASSOC]: !z y x. h4/integer/tint__mul x (h4/integer/tint__mul y z) = h4/integer/tint__mul (h4/integer/tint__mul x y) z
% Assm [h4s_integers_TINTu_u_LTu_u_REFL]: !x. ~h4/integer/tint__lt x x
% Assm [h4s_integers_TINTu_u_LTu_u_MUL]: !y x. h4/integer/tint__lt h4/integer/tint__0 x /\ h4/integer/tint__lt h4/integer/tint__0 y ==> h4/integer/tint__lt h4/integer/tint__0 (h4/integer/tint__mul x y)
% Assm [h4s_integers_TINTu_u_LTu_u_TRANS]: !z y x. h4/integer/tint__lt x y /\ h4/integer/tint__lt y z ==> h4/integer/tint__lt x z
% Assm [h4s_integers_TINTu_u_LTu_u_ADD]: !z y x. h4/integer/tint__lt y z ==> h4/integer/tint__lt (h4/integer/tint__add x y) (h4/integer/tint__add x z)
% Assm [h4s_arithmetics_SUBu_u_RIGHTu_u_GREATERu_u_EQ]: !p n m. h4/arithmetic/_3E_3D (h4/arithmetic/_2D m n) p <=> h4/arithmetic/_3E_3D m (h4/arithmetic/_2B n p) \/ h4/arithmetic/_3E_3D h4/num/0 p
% Assm [h4s_arithmetics_SUBu_u_LEFTu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D m (h4/arithmetic/_2D n p) <=> h4/arithmetic/_3C_3D (h4/arithmetic/_2B m p) n \/ h4/arithmetic/_3C_3D m h4/num/0
% Assm [h4s_arithmetics_LESSu_u_ORu_u_EQ]: !n m. h4/arithmetic/_3C_3D m n <=> h4/prim__rec/_3C m n \/ m = n
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_GREATERu_u_ORu_u_EQ]: !n m. h4/arithmetic/_3E_3D m n <=> h4/arithmetic/_3E m n \/ m = n
% Assm [h4s_arithmetics_GREATERu_u_DEF]: !n m. h4/arithmetic/_3E m n <=> h4/prim__rec/_3C n m
% Assm [h4s_arithmetics_SUBu_u_LEFTu_u_LESS]: !p n m. h4/prim__rec/_3C m (h4/arithmetic/_2D n p) <=> h4/prim__rec/_3C (h4/arithmetic/_2B m p) n
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% 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_arithmetics_ADDu_c1]: !n m. h4/arithmetic/_2B (h4/num/SUC m) n = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm [h4s_primu_u_recs_NOTu_u_LESSu_u_0]: !n. ~h4/prim__rec/_3C n h4/num/0
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% Assm [h4s_indu_u_types_CONSTRu_u_BOT]: !r i c. ~(h4/ind__type/CONSTR c i r = h4/ind__type/BOTTOM)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_indu_u_types_MKu_u_RECu_u_INJ]: !y x. h4/ind__type/mk__rec x = h4/ind__type/mk__rec y ==> h4/ind__type/ZRECSPACE x /\ h4/ind__type/ZRECSPACE y ==> x = y
% Assm [h4s_indu_u_types_CONSTR0]: !_0. (!r n. happ (happ _0 r) n = h4/ind__type/dest__rec (happ r n)) ==> (!r i c. h4/ind__type/CONSTR c i r = h4/ind__type/mk__rec (h4/ind__type/ZCONSTR c i (happ _0 r)))
% Assm [h4s_indu_u_types_ZCONSTRu_u_ZBOT]: !r i c. ~(h4/ind__type/ZCONSTR c i r = h4/ind__type/ZBOT)
% Assm [h4s_indu_u_types_ZRECSPACEu_u_def]: !x. h4/ind__type/ZRECSPACE x <=> (!ZRECSPACE_27. (!a00. a00 = h4/ind__type/ZBOT \/ (?c i r. a00 = h4/ind__type/ZCONSTR c i r /\ (!n. happ ZRECSPACE_27 (happ r n))) ==> happ ZRECSPACE_27 a00) ==> happ ZRECSPACE_27 x)
% Assm [h4s_indu_u_types_recspaceu_u_repfnsu_c0]: !a. h4/ind__type/mk__rec (h4/ind__type/dest__rec a) = a
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_indu_u_types_recspaceu_u_repfnsu_c1]: !r. h4/ind__type/ZRECSPACE r <=> h4/ind__type/dest__rec (h4/ind__type/mk__rec r) = r
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_bools_MONOu_u_ALL]: !Q P. (!x. happ P x ==> happ Q x) ==> (!x. happ P x) ==> (!x. happ Q x)
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_indu_u_types_BOTTOM0]: h4/ind__type/BOTTOM = h4/ind__type/mk__rec h4/ind__type/ZBOT
% Assm [h4s_numpairs_nlistrecu_u_curriedu_u_def]: !x2 x1 x. h4/numpair/nlistrec x x1 x2 = h4/numpair/nlistrec__tupled (h4/pair/_2C x (h4/pair/_2C x1 x2))
% Assm [h4s_arithmetics_MINu_u_MAXu_u_LT]: !n m. h4/prim__rec/_3C (h4/arithmetic/MIN m n) (h4/arithmetic/MAX m n) <=> ~(m = n)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_arithmetics_MINu_u_DEF]: !n m. h4/arithmetic/MIN m n = h4/bool/COND (h4/prim__rec/_3C m n) m n
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% 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_arithmetics_NOTu_u_LESS]: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_primu_u_recs_LESSu_u_REFL]: !n. ~h4/prim__rec/_3C n n
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Goal: !n m. h4/integer/int__of__num m = h4/integer/int__of__num n <=> m = n
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_Q1075983,TV_Q1075979]: ![V_f, V_g]: (![V_x]: s(TV_Q1075979,happ(s(t_fun(TV_Q1075983,TV_Q1075979),V_f),s(TV_Q1075983,V_x))) = s(TV_Q1075979,happ(s(t_fun(TV_Q1075983,TV_Q1075979),V_g),s(TV_Q1075983,V_x))) => s(t_fun(TV_Q1075983,TV_Q1075979),V_f) = s(t_fun(TV_Q1075983,TV_Q1075979),V_g))).
fof(ah4s_integers_intu_u_ofu_u_num0, axiom, ![V_T1]: s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_T1))) = s(t_h4s_integers_int,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_integers_int),h4s_integers_intu_u_abs),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_ofu_u_num(s(t_h4s_nums_num,V_T1)))))).
fof(ah4s_integers_intu_u_ofu_u_num1u_c1, axiom, ![V_n]: s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_n))),s(t_h4s_integers_int,h4s_integers_intu_u_1)))).
fof(ah4s_integers_intu_u_ofu_u_num1u_c0, axiom, s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_integers_int,h4s_integers_intu_u_0)).
fof(ah4s_integers_intu_u_QUOTIENT, axiom, p(s(t_bool,h4s_quotients_quotient(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_integers_int),h4s_integers_intu_u_abs),s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_integers_intu_u_rep))))).
fof(ah4s_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_integers_TINTu_u_EQu_u_EQUIV, axiom, ![V_q, V_p]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q)))) <=> s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p))) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q))))).
fof(ah4s_quotients_EQUIVu_u_def, axiom, ![TV_u_27a]: ![V_E]: (p(s(t_bool,h4s_quotients_equiv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_E)))) <=> ![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_E),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) <=> s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_E),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_E),s(TV_u_27a,V_y)))))).
fof(ah4s_quotients_REPu_u_ABSu_u_RSP, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_REL]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_REL),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_x1, V_x2]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_REL),s(TV_u_27a,V_x1))),s(TV_u_27a,V_x2)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_REL),s(TV_u_27a,V_x1))),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_x2))))))))))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_integers_intu_u_add0, axiom, ![V_T2, V_T1]: s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,V_T1),s(t_h4s_integers_int,V_T2))) = s(t_h4s_integers_int,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_integers_int),h4s_integers_intu_u_abs),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_integers_intu_u_rep),s(t_h4s_integers_int,V_T1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_integers_intu_u_rep),s(t_h4s_integers_int,V_T2)))))))).
fof(ah4s_quotients_APPLYu_u_RSP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27b,TV_u_27a]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_g, V_x, V_y]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,TV_u_27b),V_g)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_y)))))))))).
fof(ah4s_quotients_FUNu_u_REL, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f, V_R2, V_R1]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,TV_u_27b),V_g)))) <=> ![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_y))))))))).
fof(ah4s_quotients_FUNu_u_QUOTIENT, axiom, ![TV_u_27a,TV_u_27c,TV_u_27d,TV_u_27b]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => p(s(t_bool,h4s_quotients_quotient(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2))),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))))))))).
fof(ah4s_quotients_IDENTITYu_u_QUOTIENT, axiom, ![TV_u_27a]: p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),d_equals),s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i))))).
fof(ah4s_integers_intu_u_00, axiom, s(t_h4s_integers_int,h4s_integers_intu_u_0) = s(t_h4s_integers_int,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_integers_int),h4s_integers_intu_u_abs),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_0)))).
fof(ah4s_integers_intu_u_10, axiom, s(t_h4s_integers_int,h4s_integers_intu_u_1) = s(t_h4s_integers_int,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_integers_int),h4s_integers_intu_u_abs),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_1)))).
fof(ah4s_quotients_EQUALSu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_x, V_y]: (s(TV_u_27b,V_x) = s(TV_u_27b,V_y) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_x))))),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_y))))))))).
fof(ah4s_quotients_EQUALSu_u_RSP, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_x1, V_x2, V_y1, V_y2]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x1))),s(TV_u_27a,V_x2)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y1))),s(TV_u_27a,V_y2))))) => s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x1))),s(TV_u_27a,V_y1))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x2))),s(TV_u_27a,V_y2)))))).
fof(ah4s_integers_TINTu_u_EQu_u_AP, axiom, ![V_q, V_p]: (s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_p))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_q)))))).
fof(ah4s_integers_TINTu_u_ADDu_u_WELLDEF, axiom, ![V_y2, V_y1, V_x2, V_x1]: ((p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x2)))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y2))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x1),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y1))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x2),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y2)))))))).
fof(ah4s_quotients_EQUIVu_u_REFLu_u_SYMu_u_TRANS, axiom, ![TV_u_27a]: ![V_R]: (![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) <=> s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y)))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_x)))) & (![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_x))))) & ![V_x, V_y, V_z]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_z))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_z))))))))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_integers_tintu_u_ofu_u_num0u_c0, axiom, s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_0)).
fof(ah4s_integers_tintu_u_ofu_u_num0u_c1, axiom, ![V_n]: s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_ofu_u_num(s(t_h4s_nums_num,V_n))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_1)))).
fof(ah4s_integers_intu_u_TYu_u_DEF, axiom, ![V_uu_0]: (![V_c]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),V_c)))) <=> ?[V_r]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r)))) & s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),V_c) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r))))) => ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),t_bool),V_uu_0),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),V_rep)))))).
fof(ah4s_integers_intu_u_bijectionsu_c1, axiom, ![V_r]: (?[V_r0]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r0))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r0)))) & s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),V_r) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r0)))) <=> s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_integers_intu_u_repu_u_class(s(t_h4s_integers_int,h4s_integers_intu_u_absu_u_class(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),V_r))))) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),V_r))).
fof(ah4s_integers_intu_u_bijectionsu_c0, axiom, ![V_a]: s(t_h4s_integers_int,h4s_integers_intu_u_absu_u_class(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_integers_intu_u_repu_u_class(s(t_h4s_integers_int,V_a))))) = s(t_h4s_integers_int,V_a)).
fof(ah4s_bools_ABSu_u_REPu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ?[V_rep, V_abs]: (![V_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))) = s(TV_u_27b,V_a) & ![V_r]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_r)))) <=> s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))))) = s(TV_u_27a,V_r))))).
fof(ah4s_integers_intu_u_lt0, axiom, ![V_T2, V_T1]: s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_T1),s(t_h4s_integers_int,V_T2))) = s(t_bool,h4s_integers_tintu_u_lt(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_integers_intu_u_rep),s(t_h4s_integers_int,V_T1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_integers_intu_u_rep),s(t_h4s_integers_int,V_T2)))))).
fof(ah4s_integers_intu_u_mul0, axiom, ![V_T2, V_T1]: s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_T1),s(t_h4s_integers_int,V_T2))) = s(t_h4s_integers_int,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_integers_int),h4s_integers_intu_u_abs),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_integers_intu_u_rep),s(t_h4s_integers_int,V_T1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_integers_intu_u_rep),s(t_h4s_integers_int,V_T2)))))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_quotients_RESu_u_FORALLu_u_RSP, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_f, V_g]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_bool)),h4s_quotients_u_3du_3du_3du_3e(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_equals))),s(t_fun(TV_u_27a,t_bool),V_f))),s(t_fun(TV_u_27a,t_bool),V_g)))) => s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),h4s_quotients_respects(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),V_f))) = s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),h4s_quotients_respects(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),V_g)))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_quotients_FORALLu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_f]: s(t_bool,d_forall(s(t_fun(TV_u_27b,t_bool),V_f))) = s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),h4s_quotients_respects(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_bool)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(t_bool,t_bool),h4s_combins_i))),s(t_fun(TV_u_27b,t_bool),V_f))))))).
fof(ah4s_quotients_LAMBDAu_u_PRS, axiom, ![TV_u_27b,TV_u_27d,TV_u_27a,TV_u_27c]: ![V_uu_0]: (![V_rep2, V_f, V_abs1, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))),s(t_fun(TV_u_27c,TV_u_27d),V_f))),s(t_fun(TV_u_27a,TV_u_27c),V_abs1))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27d,TV_u_27b),V_rep2),s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(TV_u_27a,V_x))))))) => ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_x]: s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_f),s(TV_u_27c,V_x))) = s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))),s(t_fun(TV_u_27c,TV_u_27d),V_f))),s(t_fun(TV_u_27a,TV_u_27c),V_abs1))))),s(TV_u_27c,V_x))))))).
fof(ah4s_quotients_EQUIVu_u_RESu_u_FORALL, axiom, ![TV_u_27a]: ![V_P, V_E]: (p(s(t_bool,h4s_quotients_equiv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_E)))) => s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),h4s_quotients_respects(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_E))),s(t_fun(TV_u_27a,t_bool),V_P))) = s(t_bool,d_forall(s(t_fun(TV_u_27a,t_bool),V_P))))).
fof(ah4s_integers_intu_u_ABSu_u_def, axiom, ![V_r]: s(t_h4s_integers_int,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_integers_int),h4s_integers_intu_u_abs),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r))) = s(t_h4s_integers_int,h4s_integers_intu_u_absu_u_class(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r)))))).
fof(ah4s_integers_intu_u_REPu_u_def, axiom, ![V_a]: s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_integers_intu_u_rep),s(t_h4s_integers_int,V_a))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_mins_u_40(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_integers_intu_u_repu_u_class(s(t_h4s_integers_int,V_a)))))).
fof(ah4s_integers_intu_u_ABSu_u_REPu_u_CLASSu_c0, axiom, ![V_a]: s(t_h4s_integers_int,h4s_integers_intu_u_absu_u_class(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_integers_intu_u_repu_u_class(s(t_h4s_integers_int,V_a))))) = s(t_h4s_integers_int,V_a)).
fof(ah4s_integers_intu_u_ABSu_u_REPu_u_CLASSu_c1, axiom, ![V_c]: (?[V_r]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r)))) & s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),V_c) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_r)))) <=> s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),h4s_integers_intu_u_repu_u_class(s(t_h4s_integers_int,h4s_integers_intu_u_absu_u_class(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),V_c))))) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),V_c))).
fof(ah4s_integers_intu_u_neg0, axiom, ![V_T1]: s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_T1))) = s(t_h4s_integers_int,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_integers_int),h4s_integers_intu_u_abs),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_neg(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_integers_int,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_integers_intu_u_rep),s(t_h4s_integers_int,V_T1)))))))).
fof(ah4s_integers_INTu_u_ADDu_u_COMM, axiom, ![V_y, V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y))) = s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,V_y),s(t_h4s_integers_int,V_x)))).
fof(ah4s_integers_INTu_u_10, axiom, ~ (s(t_h4s_integers_int,h4s_integers_intu_u_1) = s(t_h4s_integers_int,h4s_integers_intu_u_0))).
fof(ah4s_integers_INTu_u_ADDu_u_ASSOC, axiom, ![V_z, V_y, V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,V_y),s(t_h4s_integers_int,V_z))))) = s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y))),s(t_h4s_integers_int,V_z)))).
fof(ah4s_integers_INTu_u_LTu_u_TOTAL, axiom, ![V_y, V_x]: (s(t_h4s_integers_int,V_x) = s(t_h4s_integers_int,V_y) | (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y)))) | p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_y),s(t_h4s_integers_int,V_x))))))).
fof(ah4s_integers_INTu_u_ADDu_u_SYM, axiom, ![V_y, V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y))) = s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,V_y),s(t_h4s_integers_int,V_x)))).
fof(ah4s_integers_INTu_u_LDISTRIB, axiom, ![V_z, V_y, V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,V_y),s(t_h4s_integers_int,V_z))))) = s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y))),s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_z)))))).
fof(ah4s_integers_INTu_u_MULu_u_SYM, axiom, ![V_y, V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y))) = s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_y),s(t_h4s_integers_int,V_x)))).
fof(ah4s_integers_INTu_u_MULu_u_ASSOC, axiom, ![V_z, V_y, V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_y),s(t_h4s_integers_int,V_z))))) = s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y))),s(t_h4s_integers_int,V_z)))).
fof(ah4s_integers_INTu_u_MULu_u_COMM, axiom, ![V_y, V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y))) = s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_y),s(t_h4s_integers_int,V_x)))).
fof(ah4s_integers_INTu_u_LTu_u_MUL, axiom, ![V_y, V_x]: ((p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_0),s(t_h4s_integers_int,V_x)))) & p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_0),s(t_h4s_integers_int,V_y))))) => p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_0),s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y)))))))).
fof(ah4s_integers_INTu_u_LTu_u_REFL, axiom, ![V_x]: ~ (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_x)))))).
fof(ah4s_integers_INTu_u_LTu_u_LADDu_u_IMP, axiom, ![V_z, V_y, V_x]: (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_y),s(t_h4s_integers_int,V_z)))) => p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y))),s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_z)))))))).
fof(ah4s_integers_INTu_u_LTu_u_TRANS, axiom, ![V_z, V_y, V_x]: ((p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y)))) & p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_y),s(t_h4s_integers_int,V_z))))) => p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_z)))))).
fof(ah4s_quotients_EQUIVu_u_IMPu_u_PARTIALu_u_EQUIV, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_quotients_equiv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => p(s(t_bool,h4s_quotients_partialu_u_equiv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))).
fof(ah4s_quotients_PARTIALu_u_EQUIVu_u_def, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_quotients_partialu_u_equiv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_x)))) & ![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_x)))) & (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_y)))) & s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))))))))).
fof(ah4s_integers_TINTu_u_LTu_u_WELLDEF, axiom, ![V_y2, V_y1, V_x2, V_x1]: ((p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x2)))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y2))))) => s(t_bool,h4s_integers_tintu_u_lt(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x1),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y1))) = s(t_bool,h4s_integers_tintu_u_lt(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x2),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y2))))).
fof(ah4s_integers_TINTu_u_MULu_u_WELLDEF, axiom, ![V_y2, V_y1, V_x2, V_x1]: ((p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x2)))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y2))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x1),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y1))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x2),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y2)))))))).
fof(ah4s_integers_TINTu_u_10, axiom, ~ (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_1))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_0)))))).
fof(ah4s_integers_TINTu_u_ADDu_u_SYM, axiom, ![V_y, V_x]: s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x)))).
fof(ah4s_integers_TINTu_u_MULu_u_SYM, axiom, ![V_y, V_x]: s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x)))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_quotients_QUOTIENTu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) <=> (![V_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_a]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a)))))) & ![V_r, V_s]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_r))),s(TV_u_27a,V_s)))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_r))),s(TV_u_27a,V_r)))) & (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_s))),s(TV_u_27a,V_s)))) & 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_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_s)))))))))).
fof(ah4s_integers_TINTu_u_ADDu_u_ASSOC, axiom, ![V_z, V_y, V_x]: s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_z))))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_z)))).
fof(ah4s_integers_TINTu_u_LTu_u_TOTAL, axiom, ![V_y, V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_bool)),h4s_integers_tintu_u_eq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y)))) | (p(s(t_bool,h4s_integers_tintu_u_lt(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y)))) | p(s(t_bool,h4s_integers_tintu_u_lt(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x))))))).
fof(ah4s_integers_TINTu_u_LDISTRIB, axiom, ![V_z, V_y, V_x]: s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_z))))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_z)))))).
fof(ah4s_integers_TINTu_u_MULu_u_ASSOC, axiom, ![V_z, V_y, V_x]: s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_z))))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_z)))).
fof(ah4s_integers_TINTu_u_LTu_u_REFL, axiom, ![V_x]: ~ (p(s(t_bool,h4s_integers_tintu_u_lt(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x)))))).
fof(ah4s_integers_TINTu_u_LTu_u_MUL, axiom, ![V_y, V_x]: ((p(s(t_bool,h4s_integers_tintu_u_lt(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_0),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x)))) & p(s(t_bool,h4s_integers_tintu_u_lt(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_0),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y))))) => p(s(t_bool,h4s_integers_tintu_u_lt(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_0),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_mul(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y)))))))).
fof(ah4s_integers_TINTu_u_LTu_u_TRANS, axiom, ![V_z, V_y, V_x]: ((p(s(t_bool,h4s_integers_tintu_u_lt(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y)))) & p(s(t_bool,h4s_integers_tintu_u_lt(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_z))))) => p(s(t_bool,h4s_integers_tintu_u_lt(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_z)))))).
fof(ah4s_integers_TINTu_u_LTu_u_ADD, axiom, ![V_z, V_y, V_x]: (p(s(t_bool,h4s_integers_tintu_u_lt(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_z)))) => p(s(t_bool,h4s_integers_tintu_u_lt(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_y))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_integers_tintu_u_add(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_x),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_z)))))))).
fof(ah4s_arithmetics_SUBu_u_RIGHTu_u_GREATERu_u_EQ, axiom, ![V_p, V_n, V_m]: (p(s(t_bool,h4s_arithmetics_u_3eu_3d(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)))) <=> (p(s(t_bool,h4s_arithmetics_u_3eu_3d(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_3eu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_p))))))).
fof(ah4s_arithmetics_SUBu_u_LEFTu_u_LESSu_u_EQ, 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,h4s_arithmetics_u_2d(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,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,V_n)))) | p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))))))).
fof(ah4s_arithmetics_LESSu_u_ORu_u_EQ, 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_m),s(t_h4s_nums_num,V_n)))) | s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n)))).
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_GREATERu_u_ORu_u_EQ, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_arithmetics_u_3eu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) <=> (p(s(t_bool,h4s_arithmetics_u_3e(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) | s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_GREATERu_u_DEF, axiom, ![V_n, V_m]: s(t_bool,h4s_arithmetics_u_3e(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = 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_SUBu_u_LEFTu_u_LESS, axiom, ![V_p, 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,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_primu_u_recs_u_3c(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_h4s_nums_num,V_n)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
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_arithmetics_ADDu_c1, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,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_m),s(t_h4s_nums_num,V_n)))))).
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_bools_ORu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,f))) <=> p(s(t_bool,V_t)))).
fof(ah4s_indu_u_types_CONSTRu_u_BOT, axiom, ![TV_u_27a]: ![V_r, V_i, V_c]: ~ (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(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_indu_u_types_MKu_u_RECu_u_INJ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_mku_u_rec(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_x))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_mku_u_rec(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_y))) => ((p(s(t_bool,h4s_indu_u_types_zrecspace(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_x)))) & p(s(t_bool,h4s_indu_u_types_zrecspace(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_y))))) => s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_x) = s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_y)))).
fof(ah4s_indu_u_types_CONSTR0, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_r, V_n]: s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool))),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,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)))),V_uu_0),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(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),h4s_indu_u_types_destu_u_rec(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_r, V_i, V_c]: 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(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_mku_u_rec(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),h4s_indu_u_types_zconstr(s(t_h4s_nums_num,V_c),s(TV_u_27a,V_i),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool))),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,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))))))).
fof(ah4s_indu_u_types_ZCONSTRu_u_ZBOT, axiom, ![TV_u_27a]: ![V_r, V_i, V_c]: ~ (s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),h4s_indu_u_types_zconstr(s(t_h4s_nums_num,V_c),s(TV_u_27a,V_i),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool))),V_r))) = s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),h4s_indu_u_types_zbot))).
fof(ah4s_indu_u_types_ZRECSPACEu_u_def, axiom, ![TV_u_27a]: ![V_x]: (p(s(t_bool,h4s_indu_u_types_zrecspace(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_x)))) <=> ![V_ZRECSPACEu_27]: (![V_a00]: ((s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a00) = s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),h4s_indu_u_types_zbot) | ?[V_c, V_i, V_r]: (s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a00) = s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),h4s_indu_u_types_zconstr(s(t_h4s_nums_num,V_c),s(TV_u_27a,V_i),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool))),V_r))) & ![V_n]: p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_bool),V_ZRECSPACEu_27),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool))),V_r),s(t_h4s_nums_num,V_n)))))))) => p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_bool),V_ZRECSPACEu_27),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_a00))))) => p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),t_bool),V_ZRECSPACEu_27),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_x))))))).
fof(ah4s_indu_u_types_recspaceu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_mku_u_rec(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),h4s_indu_u_types_destu_u_rec(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a))))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a)).
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_indu_u_types_recspaceu_u_repfnsu_c1, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_indu_u_types_zrecspace(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_r)))) <=> s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),h4s_indu_u_types_destu_u_rec(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_mku_u_rec(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_r))))) = s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),V_r))).
fof(ah4s_bools_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_MONOu_u_OR, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) | p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) | p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_ALL, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_MONOu_u_AND, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) & p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) & p(s(t_bool,V_w)))))).
fof(ah4s_indu_u_types_BOTTOM0, axiom, ![TV_u_27a]: s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_mku_u_rec(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),h4s_indu_u_types_zbot)))).
fof(ah4s_numpairs_nlistrecu_u_curriedu_u_def, axiom, ![TV_u_27a]: ![V_x2, V_x1, V_x]: s(TV_u_27a,h4s_numpairs_nlistrec(s(TV_u_27a,V_x),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),V_x1),s(t_h4s_nums_num,V_x2))) = s(TV_u_27a,h4s_numpairs_nlistrecu_u_tupled(s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num),h4s_pairs_u_2c(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),V_x1),s(t_h4s_nums_num,V_x2)))))))).
fof(ah4s_arithmetics_MINu_u_MAXu_u_LT, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_arithmetics_min(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),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,V_m) = s(t_h4s_nums_num,V_n)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_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_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_NOTu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![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))))))).
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_arithmetics_MINu_u_DEF, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_min(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_m),s(t_h4s_nums_num,V_n)))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f))))).
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_arithmetics_NOTu_u_LESS, axiom, ![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,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
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_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_primu_u_recs_LESSu_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,V_n)))))).
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_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_dcu_u_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_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_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
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_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
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_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(ch4s_integers_INTu_u_INJ0, conjecture, ![V_n, V_m]: (s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_m))) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_n))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
