%   ORIGINAL: h4/rat/rat__of__num__def__compute_c1
% 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/rat/rat__of__num__def_c1: h4/rat/rat__of__num (h4/num/SUC h4/num/0) = h4/rat/rat__1
% Assm: h4/rat/rat__of__num__primitive__def: h4/rat/rat__of__num = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!n. R (h4/num/SUC n) (h4/num/SUC (h4/num/SUC n))))) (\rat__of__num a. h4/arithmetic/num__CASE a (h4/combin/I h4/rat/rat__0) (\v. h4/arithmetic/num__CASE v (h4/combin/I h4/rat/rat__1) (\n. h4/combin/I (h4/rat/rat__add (rat__of__num (h4/num/SUC n)) h4/rat/rat__1))))
% Assm: h4/rat/rat__of__num__def_c2: !n. h4/rat/rat__of__num (h4/num/SUC (h4/num/SUC n)) = h4/rat/rat__add (h4/rat/rat__of__num (h4/num/SUC n)) h4/rat/rat__1
% Assm: h4/rat/rat__1__def: h4/rat/rat__1 = h4/rat/abs__rat h4/frac/frac__1
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/relation/RESTRICT__LEMMA: !z y f R. R y z ==> h4/relation/RESTRICT f R z y = f y
% Assm: h4/arithmetic/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/num__case__def_c0: !v f. h4/arithmetic/num__CASE h4/num/0 v f = v
% Assm: h4/relation/WFREC__COROLLARY: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. f x = M (h4/relation/RESTRICT f R x) x)
% Assm: h4/relation/inv__image__def: !f R. h4/relation/inv__image R f = (\x y. R (f x) (f y))
% Assm: h4/relation/WF__inv__image: !f R. h4/relation/WF R ==> h4/relation/WF (h4/relation/inv__image R f)
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/num__case__def_c1: !v n f. h4/arithmetic/num__CASE (h4/num/SUC n) v f = f n
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/arithmetic/SUC__ONE__ADD: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm: h4/prim__rec/WF__LESS: h4/relation/WF h4/prim__rec/_3C
% Assm: h4/numeral/numeral__distrib_c27: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/LESS__MONO__EQ: !n m. h4/prim__rec/_3C (h4/num/SUC m) (h4/num/SUC n) <=> h4/prim__rec/_3C m n
% Assm: h4/bool/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Assm: h4/arithmetic/ADD__MONO__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/rat/rat__of__num__def_c0: h4/rat/rat__of__num h4/num/0 = h4/rat/rat__0
% Assm: h4/rat/rat__of__num__def__compute_c0: h4/rat/rat__of__num h4/num/0 = h4/rat/rat__0
% 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/rat/rat__bijections_c0: !a. h4/rat/abs__rat__CLASS (h4/rat/rep__rat__CLASS a) = a
% Assm: h4/rat/rat__bijections_c1: !r. (\c. ?r0. h4/rat/rat__equiv r0 r0 /\ c = h4/rat/rat__equiv r0) r <=> h4/rat/rep__rat__CLASS (h4/rat/abs__rat__CLASS r) = r
% Assm: h4/rat/rat__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION (\c. ?r. h4/rat/rat__equiv r r /\ c = h4/rat/rat__equiv r) rep
% Assm: h4/rat/rat__ABS__REP__CLASS_c1: !c. (?r. h4/rat/rat__equiv r r /\ c = h4/rat/rat__equiv r) <=> h4/rat/rep__rat__CLASS (h4/rat/abs__rat__CLASS c) = c
% Assm: h4/rat/abs__rat__def: !r. h4/rat/abs__rat r = h4/rat/abs__rat__CLASS (h4/rat/rat__equiv r)
% Assm: h4/rat/rep__rat__def: !a. h4/rat/rep__rat a = h4/min/_40 (h4/rat/rep__rat__CLASS a)
% Assm: h4/rat/rat__ABS__REP__CLASS_c0: !a. h4/rat/abs__rat__CLASS (h4/rat/rep__rat__CLASS a) = a
% Assm: h4/rat/rat__les__def: !r2 r1. h4/rat/rat__les r1 r2 <=> h4/rat/rat__sgn (h4/rat/rat__sub r2 r1) = h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/rat/rat__mul__def: !r2 r1. h4/rat/rat__mul r1 r2 = h4/rat/abs__rat (h4/frac/frac__mul (h4/rat/rep__rat r1) (h4/rat/rep__rat r2))
% Assm: h4/ringNorm/polynom__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION (\a0_27. !_27polynom_27. (!a0_270. (?a. a0_270 = (\a0. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a0 h4/bool/ARB) (\n. h4/ind__type/BOTTOM)) a) \/ (?a. a0_270 = (\a0. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C h4/bool/ARB a0) (\n. h4/ind__type/BOTTOM)) a) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) a0 a1 /\ _27polynom_27 a0 /\ _27polynom_27 a1) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) a0 a1 /\ _27polynom_27 a0 /\ _27polynom_27 a1) \/ (?a. a0_270 = (\a0. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0)))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a0 (\n. h4/ind__type/BOTTOM))) a /\ _27polynom_27 a) ==> _27polynom_27 a0_270) ==> _27polynom_27 a0_27) rep
% Assm: h4/rat/rat__dnm__def: !r. h4/rat/rat__dnm r = h4/frac/frac__dnm (h4/rat/rep__rat r)
% Assm: h4/rat/rat__minv__def: !r1. h4/rat/rat__minv r1 = h4/rat/abs__rat (h4/frac/frac__minv (h4/rat/rep__rat r1))
% Assm: h4/rat/rat__cons__def: !nmr dnm. h4/rat/rat__cons nmr dnm = h4/rat/abs__rat (h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__mul (h4/integer/int__mul (h4/intExtension/SGN nmr) (h4/intExtension/SGN dnm)) (h4/integer/ABS nmr)) (h4/integer/ABS dnm)))
% Assm: h4/rat/rat__leq__def: !r2 r1. h4/rat/rat__leq r1 r2 <=> h4/rat/rat__les r1 r2 \/ r1 = r2
% Assm: h4/rat/rat__sub__def: !r2 r1. h4/rat/rat__sub r1 r2 = h4/rat/abs__rat (h4/frac/frac__sub (h4/rat/rep__rat r1) (h4/rat/rep__rat r2))
% Assm: h4/rat/rat__sgn__def: !r. h4/rat/rat__sgn r = h4/frac/frac__sgn (h4/rat/rep__rat r)
% Assm: h4/rat/rat__ainv__def: !r1. h4/rat/rat__ainv r1 = h4/rat/abs__rat (h4/frac/frac__ainv (h4/rat/rep__rat r1))
% Assm: h4/rat/rat__nmr__def: !r. h4/rat/rat__nmr r = h4/frac/frac__nmr (h4/rat/rep__rat r)
% Assm: h4/rat/rat__geq__def: !r2 r1. h4/rat/rat__geq r1 r2 <=> h4/rat/rat__leq r2 r1
% Assm: h4/rat/rat__add__def: !r2 r1. h4/rat/rat__add r1 r2 = h4/rat/abs__rat (h4/frac/frac__add (h4/rat/rep__rat r1) (h4/rat/rep__rat r2))
% Assm: h4/rat/rat__div__def: !r2 r1. h4/rat/rat__div r1 r2 = h4/rat/abs__rat (h4/frac/frac__div (h4/rat/rep__rat r1) (h4/rat/rep__rat r2))
% Assm: h4/rat/rat__gre__def: !r2 r1. h4/rat/rat__gre r1 r2 <=> h4/rat/rat__les r2 r1
% Assm: h4/rat/rat__0__def: h4/rat/rat__0 = h4/rat/abs__rat h4/frac/frac__0
% Assm: h4/rat/rat__def: h4/quotient/QUOTIENT h4/rat/rat__equiv h4/rat/abs__rat h4/rat/rep__rat
% Assm: h4/rat/rat__QUOTIENT: h4/quotient/QUOTIENT h4/rat/rat__equiv h4/rat/abs__rat h4/rat/rep__rat
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/rat/RAT__EQUIV: !f2 f1. h4/rat/rat__equiv f1 f2 <=> h4/rat/rat__equiv f1 = h4/rat/rat__equiv f2
% Assm: h4/quotient/EQUIV__def: !E. h4/quotient/EQUIV E <=> (!x y. E x y <=> E x = E y)
% 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/quotient/EQUIV__IMP__PARTIAL__EQUIV: !R. h4/quotient/EQUIV R ==> h4/quotient/PARTIAL__EQUIV R
% Assm: h4/ringNorm/polynom__repfns_c1: !r. (\a0_27. !_27polynom_27. (!a0_270. (?a. a0_270 = (\a0. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a0 h4/bool/ARB) (\n. h4/ind__type/BOTTOM)) a) \/ (?a. a0_270 = (\a0. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C h4/bool/ARB a0) (\n. h4/ind__type/BOTTOM)) a) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) a0 a1 /\ _27polynom_27 a0 /\ _27polynom_27 a1) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) a0 a1 /\ _27polynom_27 a0 /\ _27polynom_27 a1) \/ (?a. a0_270 = (\a0. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0)))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a0 (\n. h4/ind__type/BOTTOM))) a /\ _27polynom_27 a) ==> _27polynom_27 a0_270) ==> _27polynom_27 a0_27) r <=> h4/ringNorm/_20_40ind__typeringNorm6 (h4/ringNorm/_20_40ind__typeringNorm5 r) = r
% Assm: h4/numeral/numeral__distrib_c13: h4/num/SUC h4/num/0 = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm: h4/ringNorm/hidden____20__40ind____typeringNorm3____def: h4/ringNorm/_20_40ind__typeringNorm3 = (\a0 a1. h4/ringNorm/_20_40ind__typeringNorm5 ((\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) (h4/ringNorm/_20_40ind__typeringNorm6 a0) (h4/ringNorm/_20_40ind__typeringNorm6 a1)))
% Assm: h4/list/list__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION (\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) rep
% Assm: h4/arithmetic/SUC__ELIM__NUMERALS: !g f. (!n. g (h4/num/SUC n) = f n (h4/num/SUC n)) <=> (!n. g (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) = f (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) /\ (!n. g (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) = f (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)))
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% 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/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__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/numeral/iZ0: !x. h4/numeral/iZ x = x
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/arithmetic/MULT__CLAUSES_c0: !m. h4/arithmetic/_2A h4/num/0 m = h4/num/0
% Assm: h4/arithmetic/MULT__CLAUSES_c1: !m. h4/arithmetic/_2A m h4/num/0 = h4/num/0
% Assm: h4/arithmetic/SUB__0_c1: !m. h4/arithmetic/_2D m h4/num/0 = m
% Assm: h4/arithmetic/SUB__0_c0: !m. h4/arithmetic/_2D h4/num/0 m = h4/num/0
% Assm: h4/arithmetic/LESS__0__CASES: !m. h4/num/0 = m \/ h4/prim__rec/_3C h4/num/0 m
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/EXISTS__OR__THM: !Q P. (?x. P x \/ Q x) <=> (?x. P x) \/ (?x. Q x)
% Assm: h4/arithmetic/ADD__CLAUSES_c1: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% 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/GREATER__OR__EQ: !n m. h4/arithmetic/_3E_3D m n <=> h4/arithmetic/_3E m n \/ m = n
% Assm: h4/arithmetic/EVEN0_c0: h4/arithmetic/EVEN h4/num/0 <=> T
% Assm: h4/arithmetic/ADD__CLAUSES_c3: !n m. h4/arithmetic/_2B m (h4/num/SUC n) = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm: h4/arithmetic/EXP0_c1: !n m. h4/arithmetic/EXP m (h4/num/SUC n) = h4/arithmetic/_2A m (h4/arithmetic/EXP m n)
% Assm: h4/arithmetic/ODD0_c0: h4/arithmetic/ODD h4/num/0 <=> F
% Assm: h4/arithmetic/GREATER__DEF: !n m. h4/arithmetic/_3E m n <=> h4/prim__rec/_3C n m
% Assm: h4/arithmetic/EXP0_c0: !m. h4/arithmetic/EXP m h4/num/0 = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm: h4/arithmetic/ALT__ZERO: h4/arithmetic/ZERO = h4/num/0
% Assm: h4/arithmetic/NUMERAL__DEF: !x. h4/arithmetic/NUMERAL x = x
% Assm: h4/arithmetic/BIT10: !n. h4/arithmetic/BIT1 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (h4/num/SUC h4/num/0))
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/arithmetic/BIT20: !n. h4/arithmetic/BIT2 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (h4/num/SUC (h4/num/SUC h4/num/0)))
% Assm: h4/prim__rec/PRE0_c0: h4/prim__rec/PRE h4/num/0 = h4/num/0
% Assm: h4/prim__rec/NOT__LESS__0: !n. ~h4/prim__rec/_3C n h4/num/0
% 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/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/OR__CLAUSES_c1: !t. t \/ T <=> T
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~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/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/DISJ__IMP__THM: !R Q P. P \/ Q ==> R <=> (P ==> R) /\ (Q ==> R)
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Goal: h4/rat/rat__of__num (h4/num/SUC h4/num/0) = h4/rat/rat__1
%   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_rats_ratu_u_ofu_u_numu_u_defu_c1]: happ h4/rat/rat__of__num (h4/num/SUC h4/num/0) = h4/rat/rat__1
% Assm [h4s_rats_ratu_u_ofu_u_numu_u_primitiveu_u_def]: !_4. (!rat__of__num n. happ (happ _4 rat__of__num) n = h4/combin/I (h4/rat/rat__add (happ rat__of__num (h4/num/SUC n)) h4/rat/rat__1)) ==> (!_3. (!rat__of__num v. happ (happ _3 rat__of__num) v = h4/arithmetic/num__CASE v (h4/combin/I h4/rat/rat__1) (happ _4 rat__of__num)) ==> (!_2. (!rat__of__num a. happ (happ _2 rat__of__num) a = h4/arithmetic/num__CASE a (h4/combin/I h4/rat/rat__0) (happ _3 rat__of__num)) ==> (!_1. (!rat__of__num. happ _1 rat__of__num = happ _2 rat__of__num) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R /\ (!n. happ (happ R (h4/num/SUC n)) (h4/num/SUC (h4/num/SUC n)))) ==> h4/rat/rat__of__num = h4/relation/WFREC (h4/min/_40 _0) _1))))
% Assm [h4s_rats_ratu_u_ofu_u_numu_u_defu_c2]: !n. happ h4/rat/rat__of__num (h4/num/SUC (h4/num/SUC n)) = h4/rat/rat__add (happ h4/rat/rat__of__num (h4/num/SUC n)) h4/rat/rat__1
% Assm [h4s_rats_ratu_u_1u_u_def]: h4/rat/rat__1 = happ h4/rat/abs__rat h4/frac/frac__1
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm [h4s_relations_RESTRICTu_u_LEMMA]: !z y f R. happ (happ R y) z ==> happ (h4/relation/RESTRICT f R z) y = happ f y
% Assm [h4s_arithmetics_NOTu_u_LESS]: !n m. ~happ (happ h4/prim__rec/_3C m) n <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_numu_u_caseu_u_defu_c0]: !v f. h4/arithmetic/num__CASE h4/num/0 v f = v
% Assm [h4s_relations_WFRECu_u_COROLLARY]: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. happ f x = happ (happ M (h4/relation/RESTRICT f R x)) x)
% Assm [h4s_relations_invu_u_imageu_u_def]: !f R x x'. happ (happ (h4/relation/inv__image R f) x) x' <=> happ (happ R (happ f x)) (happ f x')
% Assm [h4s_relations_WFu_u_invu_u_image]: !f R. h4/relation/WF R ==> h4/relation/WF (h4/relation/inv__image R f)
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_numu_u_caseu_u_defu_c1]: !v n f. h4/arithmetic/num__CASE (h4/num/SUC n) v f = happ f n
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_arithmetics_SUCu_u_ONEu_u_ADD]: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm [h4s_primu_u_recs_WFu_u_LESS]: h4/relation/WF h4/prim__rec/_3C
% Assm [h4s_numerals_numeralu_u_distribu_c27]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_LESSu_u_MONOu_u_EQ]: !n m. happ (happ h4/prim__rec/_3C (h4/num/SUC m)) (h4/num/SUC n) <=> happ (happ h4/prim__rec/_3C m) n
% Assm [h4s_bools_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% Assm [h4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_rats_ratu_u_ofu_u_numu_u_defu_c0]: happ h4/rat/rat__of__num h4/num/0 = h4/rat/rat__0
% Assm [h4s_rats_ratu_u_ofu_u_numu_u_defu_u_computeu_c0]: happ h4/rat/rat__of__num h4/num/0 = h4/rat/rat__0
% 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_rats_ratu_u_bijectionsu_c0]: !a. h4/rat/abs__rat__CLASS (h4/rat/rep__rat__CLASS a) = a
% Assm [h4s_rats_ratu_u_bijectionsu_c1]: !r. (?r0. happ (happ h4/rat/rat__equiv r0) r0 /\ r = happ h4/rat/rat__equiv r0) <=> h4/rat/rep__rat__CLASS (h4/rat/abs__rat__CLASS r) = r
% Assm [h4s_rats_ratu_u_TYu_u_DEF]: !_0. (!c. happ _0 c <=> (?r. happ (happ h4/rat/rat__equiv r) r /\ c = happ h4/rat/rat__equiv r)) ==> (?rep. h4/bool/TYPE__DEFINITION _0 rep)
% Assm [h4s_rats_ratu_u_ABSu_u_REPu_u_CLASSu_c1]: !c. (?r. happ (happ h4/rat/rat__equiv r) r /\ c = happ h4/rat/rat__equiv r) <=> h4/rat/rep__rat__CLASS (h4/rat/abs__rat__CLASS c) = c
% Assm [h4s_rats_absu_u_ratu_u_def]: !r. happ h4/rat/abs__rat r = h4/rat/abs__rat__CLASS (happ h4/rat/rat__equiv r)
% Assm [h4s_rats_repu_u_ratu_u_def]: !a. happ h4/rat/rep__rat a = h4/min/_40 (h4/rat/rep__rat__CLASS a)
% Assm [h4s_rats_ratu_u_ABSu_u_REPu_u_CLASSu_c0]: !a. h4/rat/abs__rat__CLASS (h4/rat/rep__rat__CLASS a) = a
% Assm [h4s_rats_ratu_u_lesu_u_def]: !r2 r1. h4/rat/rat__les r1 r2 <=> h4/rat/rat__sgn (h4/rat/rat__sub r2 r1) = h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_rats_ratu_u_mulu_u_def]: !r2 r1. h4/rat/rat__mul r1 r2 = happ h4/rat/abs__rat (h4/frac/frac__mul (happ h4/rat/rep__rat r1) (happ h4/rat/rep__rat r2))
% Assm [h4s_ringNorms_polynomu_u_TYu_u_DEF]: !_1. (!n. happ _1 n = h4/ind__type/BOTTOM) ==> (!_0. (!a0_27. happ _0 a0_27 <=> (!_27polynom_27. (!a0_270. (?a. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a h4/bool/ARB) _1) \/ (?a. a0_270 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C h4/bool/ARB a) _1) \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a0 (h4/ind__type/FCONS a1 _1)) /\ happ _27polynom_27 a0 /\ happ _27polynom_27 a1) \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a0 (h4/ind__type/FCONS a1 _1)) /\ happ _27polynom_27 a0 /\ happ _27polynom_27 a1) \/ (?a. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0)))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a _1) /\ happ _27polynom_27 a) ==> happ _27polynom_27 a0_270) ==> happ _27polynom_27 a0_27)) ==> (?rep. h4/bool/TYPE__DEFINITION _0 rep))
% Assm [h4s_rats_ratu_u_dnmu_u_def]: !r. h4/rat/rat__dnm r = h4/frac/frac__dnm (happ h4/rat/rep__rat r)
% Assm [h4s_rats_ratu_u_minvu_u_def]: !r1. h4/rat/rat__minv r1 = happ h4/rat/abs__rat (h4/frac/frac__minv (happ h4/rat/rep__rat r1))
% Assm [h4s_rats_ratu_u_consu_u_def]: !nmr dnm. h4/rat/rat__cons nmr dnm = happ h4/rat/abs__rat (h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__mul (h4/integer/int__mul (h4/intExtension/SGN nmr) (h4/intExtension/SGN dnm)) (h4/integer/ABS nmr)) (h4/integer/ABS dnm)))
% Assm [h4s_rats_ratu_u_lequ_u_def]: !r2 r1. h4/rat/rat__leq r1 r2 <=> h4/rat/rat__les r1 r2 \/ r1 = r2
% Assm [h4s_rats_ratu_u_subu_u_def]: !r2 r1. h4/rat/rat__sub r1 r2 = happ h4/rat/abs__rat (h4/frac/frac__sub (happ h4/rat/rep__rat r1) (happ h4/rat/rep__rat r2))
% Assm [h4s_rats_ratu_u_sgnu_u_def]: !r. h4/rat/rat__sgn r = h4/frac/frac__sgn (happ h4/rat/rep__rat r)
% Assm [h4s_rats_ratu_u_ainvu_u_def]: !r1. h4/rat/rat__ainv r1 = happ h4/rat/abs__rat (h4/frac/frac__ainv (happ h4/rat/rep__rat r1))
% Assm [h4s_rats_ratu_u_nmru_u_def]: !r. h4/rat/rat__nmr r = h4/frac/frac__nmr (happ h4/rat/rep__rat r)
% Assm [h4s_rats_ratu_u_gequ_u_def]: !r2 r1. h4/rat/rat__geq r1 r2 <=> h4/rat/rat__leq r2 r1
% Assm [h4s_rats_ratu_u_addu_u_def]: !r2 r1. h4/rat/rat__add r1 r2 = happ h4/rat/abs__rat (h4/frac/frac__add (happ h4/rat/rep__rat r1) (happ h4/rat/rep__rat r2))
% Assm [h4s_rats_ratu_u_divu_u_def]: !r2 r1. h4/rat/rat__div r1 r2 = happ h4/rat/abs__rat (h4/frac/frac__div (happ h4/rat/rep__rat r1) (happ h4/rat/rep__rat r2))
% Assm [h4s_rats_ratu_u_greu_u_def]: !r2 r1. h4/rat/rat__gre r1 r2 <=> h4/rat/rat__les r2 r1
% Assm [h4s_rats_ratu_u_0u_u_def]: h4/rat/rat__0 = happ h4/rat/abs__rat h4/frac/frac__0
% Assm [h4s_rats_ratu_u_def]: h4/quotient/QUOTIENT h4/rat/rat__equiv h4/rat/abs__rat h4/rat/rep__rat
% Assm [h4s_rats_ratu_u_QUOTIENT]: h4/quotient/QUOTIENT h4/rat/rat__equiv h4/rat/abs__rat h4/rat/rep__rat
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_rats_RATu_u_EQUIV]: !f2 f1. happ (happ h4/rat/rat__equiv f1) f2 <=> happ h4/rat/rat__equiv f1 = happ h4/rat/rat__equiv f2
% 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_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_quotients_EQUIVu_u_IMPu_u_PARTIALu_u_EQUIV]: !R. h4/quotient/EQUIV R ==> h4/quotient/PARTIAL__EQUIV R
% Assm [h4s_ringNorms_polynomu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27polynom_27. (!a0_270. (?a. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a h4/bool/ARB) _0) \/ (?a. a0_270 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C h4/bool/ARB a) _0) \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a0 (h4/ind__type/FCONS a1 _0)) /\ happ _27polynom_27 a0 /\ happ _27polynom_27 a1) \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a0 (h4/ind__type/FCONS a1 _0)) /\ happ _27polynom_27 a0 /\ happ _27polynom_27 a1) \/ (?a. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0)))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a _0) /\ happ _27polynom_27 a) ==> happ _27polynom_27 a0_270) ==> happ _27polynom_27 r) <=> h4/ringNorm/_20_40ind__typeringNorm6 (h4/ringNorm/_20_40ind__typeringNorm5 r) = r)
% Assm [h4s_numerals_numeralu_u_distribu_c13]: h4/num/SUC h4/num/0 = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm [h4s_ringNorms_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeringNorm3u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x'. h4/ringNorm/_20_40ind__typeringNorm3 x x' = h4/ringNorm/_20_40ind__typeringNorm5 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS (h4/ringNorm/_20_40ind__typeringNorm6 x) (h4/ind__type/FCONS (h4/ringNorm/_20_40ind__typeringNorm6 x') _0))))
% Assm [h4s_lists_listu_u_TYu_u_DEF]: !_1. (!n. happ _1 n = h4/ind__type/BOTTOM) ==> (!_0. (!a0_27. happ _0 a0_27 <=> (!_27list_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _1 \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a0 (h4/ind__type/FCONS a1 _1) /\ happ _27list_27 a1) ==> happ _27list_27 a0_270) ==> happ _27list_27 a0_27)) ==> (?rep. h4/bool/TYPE__DEFINITION _0 rep))
% Assm [h4s_arithmetics_SUCu_u_ELIMu_u_NUMERALS]: !g f. (!n. happ g (h4/num/SUC n) = happ (happ f n) (h4/num/SUC n)) <=> (!n. happ g (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) = happ (happ f (h4/arithmetic/_2D (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n)) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) /\ (!n. happ g (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)) = happ (happ f (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 n))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 n)))
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% 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_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_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% 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_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_numerals_iZ0]: !x. h4/numeral/iZ x = x
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2A h4/num/0 m = h4/num/0
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c1]: !m. h4/arithmetic/_2A m h4/num/0 = h4/num/0
% Assm [h4s_arithmetics_SUBu_u_0u_c1]: !m. h4/arithmetic/_2D m h4/num/0 = m
% Assm [h4s_arithmetics_SUBu_u_0u_c0]: !m. h4/arithmetic/_2D h4/num/0 m = h4/num/0
% Assm [h4s_arithmetics_LESSu_u_0u_u_CASES]: !m. h4/num/0 = m \/ happ (happ h4/prim__rec/_3C h4/num/0) m
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_EXISTSu_u_ORu_u_THM]: !Q P. (?x. happ P x \/ happ Q x) <=> (?x. happ P x) \/ (?x. happ Q x)
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c1]: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_arithmetics_LESSu_u_ORu_u_EQ]: !n m. h4/arithmetic/_3C_3D m n <=> happ (happ h4/prim__rec/_3C m) n \/ m = n
% 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_EVEN0u_c0]: h4/arithmetic/EVEN h4/num/0 <=> T
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c3]: !n m. h4/arithmetic/_2B m (h4/num/SUC n) = h4/num/SUC (h4/arithmetic/_2B m n)
% Assm [h4s_arithmetics_EXP0u_c1]: !n m. h4/arithmetic/EXP m (h4/num/SUC n) = h4/arithmetic/_2A m (h4/arithmetic/EXP m n)
% Assm [h4s_arithmetics_ODD0u_c0]: h4/arithmetic/ODD h4/num/0 <=> F
% Assm [h4s_arithmetics_GREATERu_u_DEF]: !n m. h4/arithmetic/_3E m n <=> happ (happ h4/prim__rec/_3C n) m
% Assm [h4s_arithmetics_EXP0u_c0]: !m. h4/arithmetic/EXP m h4/num/0 = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm [h4s_arithmetics_ALTu_u_ZERO]: h4/arithmetic/ZERO = h4/num/0
% Assm [h4s_arithmetics_NUMERALu_u_DEF]: !x. h4/arithmetic/NUMERAL x = x
% Assm [h4s_arithmetics_BIT10]: !n. h4/arithmetic/BIT1 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (h4/num/SUC h4/num/0))
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_arithmetics_BIT20]: !n. h4/arithmetic/BIT2 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (h4/num/SUC (h4/num/SUC h4/num/0)))
% Assm [h4s_primu_u_recs_PRE0u_c0]: h4/prim__rec/PRE h4/num/0 = h4/num/0
% Assm [h4s_primu_u_recs_NOTu_u_LESSu_u_0]: !n. ~happ (happ h4/prim__rec/_3C n) h4/num/0
% 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_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c1]: !t. t \/ T <=> T
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~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_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_DISJu_u_IMPu_u_THM]: !R Q P. P \/ Q ==> R <=> (P ==> R) /\ (Q ==> R)
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Goal: happ h4/rat/rat__of__num (h4/num/SUC h4/num/0) = h4/rat/rat__1
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_Q1168880,TV_Q1168876]: ![V_f, V_g]: (![V_x]: s(TV_Q1168876,happ(s(t_fun(TV_Q1168880,TV_Q1168876),V_f),s(TV_Q1168880,V_x))) = s(TV_Q1168876,happ(s(t_fun(TV_Q1168880,TV_Q1168876),V_g),s(TV_Q1168880,V_x))) => s(t_fun(TV_Q1168880,TV_Q1168876),V_f) = s(t_fun(TV_Q1168880,TV_Q1168876),V_g))).
fof(ah4s_rats_ratu_u_ofu_u_numu_u_defu_c1, axiom, s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_1)).
fof(ah4s_rats_ratu_u_ofu_u_numu_u_primitiveu_u_def, axiom, ![V_uu_4]: (![V_ratu_u_ofu_u_num, V_n]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_rats_rat),t_fun(t_h4s_nums_num,t_h4s_rats_rat)),V_uu_4),s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),V_ratu_u_ofu_u_num))),s(t_h4s_nums_num,V_n))) = s(t_h4s_rats_rat,h4s_combins_i(s(t_h4s_rats_rat,h4s_rats_ratu_u_add(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),V_ratu_u_ofu_u_num),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))),s(t_h4s_rats_rat,h4s_rats_ratu_u_1))))) => ![V_uu_3]: (![V_ratu_u_ofu_u_num, V_v]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_rats_rat),t_fun(t_h4s_nums_num,t_h4s_rats_rat)),V_uu_3),s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),V_ratu_u_ofu_u_num))),s(t_h4s_nums_num,V_v))) = s(t_h4s_rats_rat,h4s_arithmetics_numu_u_case(s(t_h4s_nums_num,V_v),s(t_h4s_rats_rat,h4s_combins_i(s(t_h4s_rats_rat,h4s_rats_ratu_u_1))),s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_rats_rat),t_fun(t_h4s_nums_num,t_h4s_rats_rat)),V_uu_4),s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),V_ratu_u_ofu_u_num))))) => ![V_uu_2]: (![V_ratu_u_ofu_u_num, V_a]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_rats_rat),t_fun(t_h4s_nums_num,t_h4s_rats_rat)),V_uu_2),s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),V_ratu_u_ofu_u_num))),s(t_h4s_nums_num,V_a))) = s(t_h4s_rats_rat,h4s_arithmetics_numu_u_case(s(t_h4s_nums_num,V_a),s(t_h4s_rats_rat,h4s_combins_i(s(t_h4s_rats_rat,h4s_rats_ratu_u_0))),s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_rats_rat),t_fun(t_h4s_nums_num,t_h4s_rats_rat)),V_uu_3),s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),V_ratu_u_ofu_u_num))))) => ![V_uu_1]: (![V_ratu_u_ofu_u_num]: s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_rats_rat),t_fun(t_h4s_nums_num,t_h4s_rats_rat)),V_uu_1),s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),V_ratu_u_ofu_u_num))) = s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_rats_rat),t_fun(t_h4s_nums_num,t_h4s_rats_rat)),V_uu_2),s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),V_ratu_u_ofu_u_num))) => ![V_uu_0]: (![V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_R)))) & ![V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_R),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))))) => s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num) = s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_relations_wfrec(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_nums_num,t_h4s_rats_rat),t_fun(t_h4s_nums_num,t_h4s_rats_rat)),V_uu_1))))))))).
fof(ah4s_rats_ratu_u_ofu_u_numu_u_defu_c2, axiom, ![V_n]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_add(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))),s(t_h4s_rats_rat,h4s_rats_ratu_u_1)))).
fof(ah4s_rats_ratu_u_1u_u_def, axiom, s(t_h4s_rats_rat,h4s_rats_ratu_u_1) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,h4s_fracs_fracu_u_1)))).
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_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_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_arithmetics_ADDu_u_CLAUSESu_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_relations_RESTRICTu_u_LEMMA, axiom, ![TV_u_27b,TV_u_27a]: ![V_z, V_y, V_f, 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_y))),s(TV_u_27a,V_z)))) => s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_z))),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_arithmetics_NOTu_u_LESS, axiom, ![V_n, V_m]: (~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,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_arithmetics_numu_u_caseu_u_defu_c0, axiom, ![TV_u_27a]: ![V_v, V_f]: s(TV_u_27a,h4s_arithmetics_numu_u_case(s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,V_v),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))) = s(TV_u_27a,V_v)).
fof(ah4s_relations_WFRECu_u_COROLLARY, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_M]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_wfrec(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M))) => (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M),s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))))),s(TV_u_27a,V_x)))))).
fof(ah4s_relations_invu_u_imageu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_x, V_xi_]: 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)),h4s_relations_invu_u_image(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_x))),s(TV_u_27a,V_xi_))) = 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_R),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_f),s(TV_u_27a,V_xi_)))))).
fof(ah4s_relations_WFu_u_invu_u_image, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R)))) => p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_invu_u_image(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_f)))))))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c2, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_numu_u_caseu_u_defu_c1, axiom, ![TV_u_27a]: ![V_v, V_n, V_f]: s(TV_u_27a,h4s_arithmetics_numu_u_case(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_v),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))) = 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_numerals_numeralu_u_lteu_c1, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
fof(ah4s_arithmetics_SUCu_u_ONEu_u_ADD, axiom, ![V_n]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_primu_u_recs_WFu_u_LESS, axiom, p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c))))).
fof(ah4s_numerals_numeralu_u_distribu_c27, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ah4s_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_LESSu_u_MONOu_u_EQ, axiom, ![V_n, V_m]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),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_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_bools_IMPu_u_Fu_u_EQu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_rats_ratu_u_ofu_u_numu_u_defu_c0, axiom, s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_0)).
fof(ah4s_rats_ratu_u_ofu_u_numu_u_defu_u_computeu_c0, axiom, s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_0)).
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_rats_ratu_u_bijectionsu_c0, axiom, ![V_a]: s(t_h4s_rats_rat,h4s_rats_absu_u_ratu_u_class(s(t_fun(t_h4s_fracs_frac,t_bool),h4s_rats_repu_u_ratu_u_class(s(t_h4s_rats_rat,V_a))))) = s(t_h4s_rats_rat,V_a)).
fof(ah4s_rats_ratu_u_bijectionsu_c1, axiom, ![V_r]: (?[V_r0]: (p(s(t_bool,happ(s(t_fun(t_h4s_fracs_frac,t_bool),happ(s(t_fun(t_h4s_fracs_frac,t_fun(t_h4s_fracs_frac,t_bool)),h4s_rats_ratu_u_equiv),s(t_h4s_fracs_frac,V_r0))),s(t_h4s_fracs_frac,V_r0)))) & s(t_fun(t_h4s_fracs_frac,t_bool),V_r) = s(t_fun(t_h4s_fracs_frac,t_bool),happ(s(t_fun(t_h4s_fracs_frac,t_fun(t_h4s_fracs_frac,t_bool)),h4s_rats_ratu_u_equiv),s(t_h4s_fracs_frac,V_r0)))) <=> s(t_fun(t_h4s_fracs_frac,t_bool),h4s_rats_repu_u_ratu_u_class(s(t_h4s_rats_rat,h4s_rats_absu_u_ratu_u_class(s(t_fun(t_h4s_fracs_frac,t_bool),V_r))))) = s(t_fun(t_h4s_fracs_frac,t_bool),V_r))).
fof(ah4s_rats_ratu_u_TYu_u_DEF, axiom, ![V_uu_0]: (![V_c]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_fracs_frac,t_bool),t_bool),V_uu_0),s(t_fun(t_h4s_fracs_frac,t_bool),V_c)))) <=> ?[V_r]: (p(s(t_bool,happ(s(t_fun(t_h4s_fracs_frac,t_bool),happ(s(t_fun(t_h4s_fracs_frac,t_fun(t_h4s_fracs_frac,t_bool)),h4s_rats_ratu_u_equiv),s(t_h4s_fracs_frac,V_r))),s(t_h4s_fracs_frac,V_r)))) & s(t_fun(t_h4s_fracs_frac,t_bool),V_c) = s(t_fun(t_h4s_fracs_frac,t_bool),happ(s(t_fun(t_h4s_fracs_frac,t_fun(t_h4s_fracs_frac,t_bool)),h4s_rats_ratu_u_equiv),s(t_h4s_fracs_frac,V_r))))) => ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_fun(t_h4s_fracs_frac,t_bool),t_bool),V_uu_0),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_fracs_frac,t_bool)),V_rep)))))).
fof(ah4s_rats_ratu_u_ABSu_u_REPu_u_CLASSu_c1, axiom, ![V_c]: (?[V_r]: (p(s(t_bool,happ(s(t_fun(t_h4s_fracs_frac,t_bool),happ(s(t_fun(t_h4s_fracs_frac,t_fun(t_h4s_fracs_frac,t_bool)),h4s_rats_ratu_u_equiv),s(t_h4s_fracs_frac,V_r))),s(t_h4s_fracs_frac,V_r)))) & s(t_fun(t_h4s_fracs_frac,t_bool),V_c) = s(t_fun(t_h4s_fracs_frac,t_bool),happ(s(t_fun(t_h4s_fracs_frac,t_fun(t_h4s_fracs_frac,t_bool)),h4s_rats_ratu_u_equiv),s(t_h4s_fracs_frac,V_r)))) <=> s(t_fun(t_h4s_fracs_frac,t_bool),h4s_rats_repu_u_ratu_u_class(s(t_h4s_rats_rat,h4s_rats_absu_u_ratu_u_class(s(t_fun(t_h4s_fracs_frac,t_bool),V_c))))) = s(t_fun(t_h4s_fracs_frac,t_bool),V_c))).
fof(ah4s_rats_absu_u_ratu_u_def, axiom, ![V_r]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,V_r))) = s(t_h4s_rats_rat,h4s_rats_absu_u_ratu_u_class(s(t_fun(t_h4s_fracs_frac,t_bool),happ(s(t_fun(t_h4s_fracs_frac,t_fun(t_h4s_fracs_frac,t_bool)),h4s_rats_ratu_u_equiv),s(t_h4s_fracs_frac,V_r)))))).
fof(ah4s_rats_repu_u_ratu_u_def, axiom, ![V_a]: s(t_h4s_fracs_frac,happ(s(t_fun(t_h4s_rats_rat,t_h4s_fracs_frac),h4s_rats_repu_u_rat),s(t_h4s_rats_rat,V_a))) = s(t_h4s_fracs_frac,h4s_mins_u_40(s(t_fun(t_h4s_fracs_frac,t_bool),h4s_rats_repu_u_ratu_u_class(s(t_h4s_rats_rat,V_a)))))).
fof(ah4s_rats_ratu_u_ABSu_u_REPu_u_CLASSu_c0, axiom, ![V_a]: s(t_h4s_rats_rat,h4s_rats_absu_u_ratu_u_class(s(t_fun(t_h4s_fracs_frac,t_bool),h4s_rats_repu_u_ratu_u_class(s(t_h4s_rats_rat,V_a))))) = s(t_h4s_rats_rat,V_a)).
fof(ah4s_rats_ratu_u_lesu_u_def, axiom, ![V_r2, V_r1]: (p(s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2)))) <=> s(t_h4s_integers_int,h4s_rats_ratu_u_sgn(s(t_h4s_rats_rat,h4s_rats_ratu_u_sub(s(t_h4s_rats_rat,V_r2),s(t_h4s_rats_rat,V_r1))))) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))).
fof(ah4s_rats_ratu_u_mulu_u_def, axiom, ![V_r2, V_r1]: s(t_h4s_rats_rat,h4s_rats_ratu_u_mul(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,h4s_fracs_fracu_u_mul(s(t_h4s_fracs_frac,happ(s(t_fun(t_h4s_rats_rat,t_h4s_fracs_frac),h4s_rats_repu_u_rat),s(t_h4s_rats_rat,V_r1))),s(t_h4s_fracs_frac,happ(s(t_fun(t_h4s_rats_rat,t_h4s_fracs_frac),h4s_rats_repu_u_rat),s(t_h4s_rats_rat,V_r2)))))))).
fof(ah4s_ringNorms_polynomu_u_TYu_u_DEF, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_1),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_bottom) => ![V_uu_0]: (![V_a0u_27]: (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_0),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_27)))) <=> ![V_uu_27polynomu_27]: (![V_a0u_270]: ((?[V_a]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,V_a),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_1))) | (?[V_a]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,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(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,V_a))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_1))) | (?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_1))))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0)))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a1)))))) | (?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_1))))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0)))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a1)))))) | ?[V_a]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))))),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_1))))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a))))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_27)))))) => ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_0),s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_rep))))))).
fof(ah4s_rats_ratu_u_dnmu_u_def, axiom, ![V_r]: s(t_h4s_integers_int,h4s_rats_ratu_u_dnm(s(t_h4s_rats_rat,V_r))) = s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,happ(s(t_fun(t_h4s_rats_rat,t_h4s_fracs_frac),h4s_rats_repu_u_rat),s(t_h4s_rats_rat,V_r)))))).
fof(ah4s_rats_ratu_u_minvu_u_def, axiom, ![V_r1]: s(t_h4s_rats_rat,h4s_rats_ratu_u_minv(s(t_h4s_rats_rat,V_r1))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,h4s_fracs_fracu_u_minv(s(t_h4s_fracs_frac,happ(s(t_fun(t_h4s_rats_rat,t_h4s_fracs_frac),h4s_rats_repu_u_rat),s(t_h4s_rats_rat,V_r1)))))))).
fof(ah4s_rats_ratu_u_consu_u_def, axiom, ![V_nmr, V_dnm]: s(t_h4s_rats_rat,h4s_rats_ratu_u_cons(s(t_h4s_integers_int,V_nmr),s(t_h4s_integers_int,V_dnm))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,h4s_fracs_absu_u_frac(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),h4s_pairs_u_2c(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,h4s_intextensions_sgn(s(t_h4s_integers_int,V_nmr))),s(t_h4s_integers_int,h4s_intextensions_sgn(s(t_h4s_integers_int,V_dnm))))),s(t_h4s_integers_int,h4s_integers_abs(s(t_h4s_integers_int,V_nmr))))),s(t_h4s_integers_int,h4s_integers_abs(s(t_h4s_integers_int,V_dnm)))))))))).
fof(ah4s_rats_ratu_u_lequ_u_def, axiom, ![V_r2, V_r1]: (p(s(t_bool,h4s_rats_ratu_u_leq(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2)))) <=> (p(s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2)))) | s(t_h4s_rats_rat,V_r1) = s(t_h4s_rats_rat,V_r2)))).
fof(ah4s_rats_ratu_u_subu_u_def, axiom, ![V_r2, V_r1]: s(t_h4s_rats_rat,h4s_rats_ratu_u_sub(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,h4s_fracs_fracu_u_sub(s(t_h4s_fracs_frac,happ(s(t_fun(t_h4s_rats_rat,t_h4s_fracs_frac),h4s_rats_repu_u_rat),s(t_h4s_rats_rat,V_r1))),s(t_h4s_fracs_frac,happ(s(t_fun(t_h4s_rats_rat,t_h4s_fracs_frac),h4s_rats_repu_u_rat),s(t_h4s_rats_rat,V_r2)))))))).
fof(ah4s_rats_ratu_u_sgnu_u_def, axiom, ![V_r]: s(t_h4s_integers_int,h4s_rats_ratu_u_sgn(s(t_h4s_rats_rat,V_r))) = s(t_h4s_integers_int,h4s_fracs_fracu_u_sgn(s(t_h4s_fracs_frac,happ(s(t_fun(t_h4s_rats_rat,t_h4s_fracs_frac),h4s_rats_repu_u_rat),s(t_h4s_rats_rat,V_r)))))).
fof(ah4s_rats_ratu_u_ainvu_u_def, axiom, ![V_r1]: s(t_h4s_rats_rat,h4s_rats_ratu_u_ainv(s(t_h4s_rats_rat,V_r1))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,h4s_fracs_fracu_u_ainv(s(t_h4s_fracs_frac,happ(s(t_fun(t_h4s_rats_rat,t_h4s_fracs_frac),h4s_rats_repu_u_rat),s(t_h4s_rats_rat,V_r1)))))))).
fof(ah4s_rats_ratu_u_nmru_u_def, axiom, ![V_r]: s(t_h4s_integers_int,h4s_rats_ratu_u_nmr(s(t_h4s_rats_rat,V_r))) = s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,happ(s(t_fun(t_h4s_rats_rat,t_h4s_fracs_frac),h4s_rats_repu_u_rat),s(t_h4s_rats_rat,V_r)))))).
fof(ah4s_rats_ratu_u_gequ_u_def, axiom, ![V_r2, V_r1]: s(t_bool,h4s_rats_ratu_u_geq(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2))) = s(t_bool,h4s_rats_ratu_u_leq(s(t_h4s_rats_rat,V_r2),s(t_h4s_rats_rat,V_r1)))).
fof(ah4s_rats_ratu_u_addu_u_def, axiom, ![V_r2, V_r1]: s(t_h4s_rats_rat,h4s_rats_ratu_u_add(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,h4s_fracs_fracu_u_add(s(t_h4s_fracs_frac,happ(s(t_fun(t_h4s_rats_rat,t_h4s_fracs_frac),h4s_rats_repu_u_rat),s(t_h4s_rats_rat,V_r1))),s(t_h4s_fracs_frac,happ(s(t_fun(t_h4s_rats_rat,t_h4s_fracs_frac),h4s_rats_repu_u_rat),s(t_h4s_rats_rat,V_r2)))))))).
fof(ah4s_rats_ratu_u_divu_u_def, axiom, ![V_r2, V_r1]: s(t_h4s_rats_rat,h4s_rats_ratu_u_div(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,h4s_fracs_fracu_u_div(s(t_h4s_fracs_frac,happ(s(t_fun(t_h4s_rats_rat,t_h4s_fracs_frac),h4s_rats_repu_u_rat),s(t_h4s_rats_rat,V_r1))),s(t_h4s_fracs_frac,happ(s(t_fun(t_h4s_rats_rat,t_h4s_fracs_frac),h4s_rats_repu_u_rat),s(t_h4s_rats_rat,V_r2)))))))).
fof(ah4s_rats_ratu_u_greu_u_def, axiom, ![V_r2, V_r1]: s(t_bool,h4s_rats_ratu_u_gre(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2))) = s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,V_r2),s(t_h4s_rats_rat,V_r1)))).
fof(ah4s_rats_ratu_u_0u_u_def, axiom, s(t_h4s_rats_rat,h4s_rats_ratu_u_0) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_h4s_fracs_frac,h4s_fracs_fracu_u_0)))).
fof(ah4s_rats_ratu_u_def, axiom, p(s(t_bool,h4s_quotients_quotient(s(t_fun(t_h4s_fracs_frac,t_fun(t_h4s_fracs_frac,t_bool)),h4s_rats_ratu_u_equiv),s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_fun(t_h4s_rats_rat,t_h4s_fracs_frac),h4s_rats_repu_u_rat))))).
fof(ah4s_rats_ratu_u_QUOTIENT, axiom, p(s(t_bool,h4s_quotients_quotient(s(t_fun(t_h4s_fracs_frac,t_fun(t_h4s_fracs_frac,t_bool)),h4s_rats_ratu_u_equiv),s(t_fun(t_h4s_fracs_frac,t_h4s_rats_rat),h4s_rats_absu_u_rat),s(t_fun(t_h4s_rats_rat,t_h4s_fracs_frac),h4s_rats_repu_u_rat))))).
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_rats_RATu_u_EQUIV, axiom, ![V_f2, V_f1]: (p(s(t_bool,happ(s(t_fun(t_h4s_fracs_frac,t_bool),happ(s(t_fun(t_h4s_fracs_frac,t_fun(t_h4s_fracs_frac,t_bool)),h4s_rats_ratu_u_equiv),s(t_h4s_fracs_frac,V_f1))),s(t_h4s_fracs_frac,V_f2)))) <=> s(t_fun(t_h4s_fracs_frac,t_bool),happ(s(t_fun(t_h4s_fracs_frac,t_fun(t_h4s_fracs_frac,t_bool)),h4s_rats_ratu_u_equiv),s(t_h4s_fracs_frac,V_f1))) = s(t_fun(t_h4s_fracs_frac,t_bool),happ(s(t_fun(t_h4s_fracs_frac,t_fun(t_h4s_fracs_frac,t_bool)),h4s_rats_ratu_u_equiv),s(t_h4s_fracs_frac,V_f2))))).
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_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_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_ringNorms_polynomu_u_repfnsu_c1, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27polynomu_27]: (![V_a0u_270]: ((?[V_a]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,V_a),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))) | (?[V_a]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,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(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,V_a))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))) | (?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0)))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a1)))))) | (?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0)))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a1)))))) | ?[V_a]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))))),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a))))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_r))))) <=> s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm6(s(t_h4s_ringNorms_polynom(TV_u_27a),h4s_ringnorms_u_20u_40indu_u_typeringnorm5(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_r))))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_r)))).
fof(ah4s_numerals_numeralu_u_distribu_c13, axiom, s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))).
fof(ah4s_ringNorms_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeringNorm3u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_bottom) => ![V_x, V_xi_]: s(t_h4s_ringNorms_polynom(TV_u_27a),h4s_ringnorms_u_20u_40indu_u_typeringnorm3(s(t_h4s_ringNorms_polynom(TV_u_27a),V_x),s(t_h4s_ringNorms_polynom(TV_u_27a),V_xi_))) = s(t_h4s_ringNorms_polynom(TV_u_27a),h4s_ringnorms_u_20u_40indu_u_typeringnorm5(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm6(s(t_h4s_ringNorms_polynom(TV_u_27a),V_x))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm6(s(t_h4s_ringNorms_polynom(TV_u_27a),V_xi_))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))))))))))).
fof(ah4s_lists_listu_u_TYu_u_DEF, axiom, ![TV_u_27a]: ![V_uu_1]: (![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_1),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => ![V_uu_0]: (![V_a0u_27]: (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_0),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_27)))) <=> ![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_1))) | ?[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_1))))) & 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_a0u_27)))))) => ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_0),s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_indu_u_types_recspace(TV_u_27a)),V_rep))))))).
fof(ah4s_arithmetics_SUCu_u_ELIMu_u_NUMERALS, axiom, ![TV_u_27a]: ![V_g, V_f]: (![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g),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),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) <=> (![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))) & ![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n)))))))))).
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_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_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_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_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_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_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_AND, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) & p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) & p(s(t_bool,V_w)))))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_bools_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_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_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_numerals_iZ0, axiom, ![V_x]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,V_x))) = s(t_h4s_nums_num,V_x)).
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_arithmetics_MULTu_u_CLAUSESu_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_arithmetics_SUBu_u_0u_c1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_SUBu_u_0u_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_arithmetics_LESSu_u_0u_u_CASES, axiom, ![V_m]: (s(t_h4s_nums_num,h4s_nums_0) = s(t_h4s_nums_num,V_m) | p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_nums_num,V_m)))))).
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_EXISTSu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (?[V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,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_arithmetics_ADDu_u_CLAUSESu_c1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_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_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,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,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_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_EVEN0u_c0, axiom, s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,t)).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c3, 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,h4s_nums_suc(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_arithmetics_EXP0u_c1, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_exp(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,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_ODD0u_c0, axiom, s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,f)).
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,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,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_EXP0u_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))).
fof(ah4s_arithmetics_ALTu_u_ZERO, axiom, s(t_h4s_nums_num,h4s_arithmetics_zero) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_arithmetics_NUMERALu_u_DEF, axiom, ![V_x]: s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_x))) = s(t_h4s_nums_num,V_x)).
fof(ah4s_arithmetics_BIT10, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0)))))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_arithmetics_BIT20, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0)))))))))).
fof(ah4s_primu_u_recs_PRE0u_c0, axiom, s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_primu_u_recs_NOTu_u_LESSu_u_0, axiom, ![V_n]: ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,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_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_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_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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,t))) <=> p(s(t_bool,t)))).
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_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_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_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_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_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_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_DISJu_u_IMPu_u_THM, 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_R))) & (p(s(t_bool,V_Q)) => p(s(t_bool,V_R)))))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
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(ch4s_rats_ratu_u_ofu_u_numu_u_defu_u_computeu_c1, conjecture, s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_nums_num,t_h4s_rats_rat),h4s_rats_ratu_u_ofu_u_num),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_1)).
