%   ORIGINAL: h4/rat/rat__of__num__def_c0
% 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__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__0__def: h4/rat/rat__0 = h4/rat/abs__rat h4/frac/frac__0
% 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__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__bijections_c0: !a. h4/rat/abs__rat__CLASS (h4/rat/rep__rat__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/rat/rep__rat__def: !a. h4/rat/rep__rat a = h4/min/_40 (h4/rat/rep__rat__CLASS a)
% 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/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/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__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__1__def: h4/rat/rat__1 = h4/rat/abs__rat h4/frac/frac__1
% 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__geq__def: !r2 r1. h4/rat/rat__geq r1 r2 <=> h4/rat/rat__leq r2 r1
% 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__sgn__def: !r. h4/rat/rat__sgn r = h4/frac/frac__sgn (h4/rat/rep__rat r)
% 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/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__dnm__def: !r. h4/rat/rat__dnm r = h4/frac/frac__dnm (h4/rat/rep__rat r)
% Assm: h4/rat/rat__gre__def: !r2 r1. h4/rat/rat__gre r1 r2 <=> h4/rat/rat__les r2 r1
% 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__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__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__QUOTIENT: h4/quotient/QUOTIENT h4/rat/rat__equiv h4/rat/abs__rat h4/rat/rep__rat
% Assm: h4/rat/rat__def: 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/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/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/EQUIV__IMP__PARTIAL__EQUIV: !R. h4/quotient/EQUIV R ==> h4/quotient/PARTIAL__EQUIV R
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% 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/INT__MOD__P: !x c P. ~(c = h4/integer/int__of__num h4/num/0) ==> (P (h4/integer/int__mod x c) <=> (?k r. x = h4/integer/int__add (h4/integer/int__mul k c) r /\ (h4/integer/int__lt c (h4/integer/int__of__num h4/num/0) /\ h4/integer/int__lt c r /\ h4/integer/int__le r (h4/integer/int__of__num h4/num/0) \/ ~h4/integer/int__lt c (h4/integer/int__of__num h4/num/0) /\ h4/integer/int__le (h4/integer/int__of__num h4/num/0) r /\ h4/integer/int__lt r c) /\ P r))
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% 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/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% 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/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/integer/INT__DIVISION: !q. ~(q = h4/integer/int__of__num h4/num/0) ==> (!p. p = h4/integer/int__add (h4/integer/int__mul (h4/integer/int__div p q) q) (h4/integer/int__mod p q) /\ h4/bool/COND (h4/integer/int__lt q (h4/integer/int__of__num h4/num/0)) (h4/integer/int__lt q (h4/integer/int__mod p q) /\ h4/integer/int__le (h4/integer/int__mod p q) (h4/integer/int__of__num h4/num/0)) (h4/integer/int__le (h4/integer/int__of__num h4/num/0) (h4/integer/int__mod p q) /\ h4/integer/int__lt (h4/integer/int__mod p q) q))
% Assm: h4/integer/INT__MOD__UNIQUE: !m j i. (?q. i = h4/integer/int__add (h4/integer/int__mul q j) m /\ h4/bool/COND (h4/integer/int__lt j (h4/integer/int__of__num h4/num/0)) (h4/integer/int__lt j m /\ h4/integer/int__le m (h4/integer/int__of__num h4/num/0)) (h4/integer/int__le (h4/integer/int__of__num h4/num/0) m /\ h4/integer/int__lt m j)) ==> h4/integer/int__mod i j = m
% Assm: h4/bool/LEFT__FORALL__OR__THM: !Q P. (!x. P x \/ Q) <=> (!x. P x) \/ Q
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/dc__cond: !s r q p. (p <=> h4/bool/COND q r s) <=> (p \/ q \/ ~s) /\ (p \/ ~r \/ ~q) /\ (p \/ ~r \/ ~s) /\ (~q \/ r \/ ~p) /\ (q \/ s \/ ~p)
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/bool/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/bool/RIGHT__OR__OVER__AND: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm: h4/Omega/sumc__ADD: !vs ds cs. h4/integer/int__add (h4/Omega/sumc cs vs) (h4/Omega/sumc ds vs) = h4/Omega/sumc (h4/Omega/MAP2 (h4/integer/int__of__num h4/num/0) h4/integer/int__add cs ds) vs
% Assm: h4/bool/EQ__EXPAND: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/Omega/sumc__thm_c2: !vs v cs c. h4/Omega/sumc (h4/list/CONS c cs) (h4/list/CONS v vs) = h4/integer/int__add (h4/integer/int__mul c v) (h4/Omega/sumc cs vs)
% Assm: h4/Omega/sumc__thm_c1: !cs. h4/Omega/sumc cs h4/list/NIL = h4/integer/int__of__num h4/num/0
% Assm: h4/Omega/sumc__ind: !P. (!v0. P v0 h4/list/NIL) /\ (!v4 v5. P h4/list/NIL (h4/list/CONS v4 v5)) /\ (!c cs v vs. P cs vs ==> P (h4/list/CONS c cs) (h4/list/CONS v vs)) ==> (!v v1. P v v1)
% Assm: h4/Omega/MAP2__zero__ADD_c1: !xs. h4/Omega/MAP2 (h4/integer/int__of__num h4/num/0) h4/integer/int__add xs h4/list/NIL = xs
% Assm: h4/Omega/MAP2__zero__ADD_c0: !xs. h4/Omega/MAP2 (h4/integer/int__of__num h4/num/0) h4/integer/int__add h4/list/NIL xs = xs
% Assm: h4/Omega/MAP2__tupled__primitive__def: h4/Omega/MAP2__tupled = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!y ys f pad. R (h4/pair/_2C pad (h4/pair/_2C f (h4/pair/_2C h4/list/NIL ys))) (h4/pair/_2C pad (h4/pair/_2C f (h4/pair/_2C h4/list/NIL (h4/list/CONS y ys))))) /\ (!x xs f pad. R (h4/pair/_2C pad (h4/pair/_2C f (h4/pair/_2C xs h4/list/NIL))) (h4/pair/_2C pad (h4/pair/_2C f (h4/pair/_2C (h4/list/CONS x xs) h4/list/NIL)))) /\ (!y x ys xs f pad. R (h4/pair/_2C pad (h4/pair/_2C f (h4/pair/_2C xs ys))) (h4/pair/_2C pad (h4/pair/_2C f (h4/pair/_2C (h4/list/CONS x xs) (h4/list/CONS y ys))))))) (\MAP2__tupled a. h4/pair/pair__CASE a (\pad v1. h4/pair/pair__CASE v1 (\f v3. h4/pair/pair__CASE v3 (\v4 v5. h4/list/list__CASE v4 (h4/list/list__CASE v5 (h4/combin/I h4/list/NIL) (\y ys. h4/combin/I (h4/list/CONS (f pad y) (MAP2__tupled (h4/pair/_2C pad (h4/pair/_2C f (h4/pair/_2C h4/list/NIL ys))))))) (\x xs. h4/list/list__CASE v5 (h4/combin/I (h4/list/CONS (f x pad) (MAP2__tupled (h4/pair/_2C pad (h4/pair/_2C f (h4/pair/_2C xs h4/list/NIL)))))) (\y_27 ys_27. h4/combin/I (h4/list/CONS (f x y_27) (MAP2__tupled (h4/pair/_2C pad (h4/pair/_2C f (h4/pair/_2C xs ys_27)))))))))))
% Assm: h4/Omega/sumc__thm_c0: !vs. h4/Omega/sumc h4/list/NIL vs = h4/integer/int__of__num h4/num/0
% Assm: h4/Omega/MAP2__curried__def: !x3 x2 x1 x. h4/Omega/MAP2 x x1 x2 x3 = h4/Omega/MAP2__tupled (h4/pair/_2C x (h4/pair/_2C x1 (h4/pair/_2C x2 x3)))
% Assm: h4/integer/INT__ADD__LID: !x. h4/integer/int__add (h4/integer/int__of__num h4/num/0) x = x
% Assm: h4/integer/INT__ADD__COMM: !y x. h4/integer/int__add x y = h4/integer/int__add y x
% 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__ADD__RID: !x. h4/integer/int__add x (h4/integer/int__of__num h4/num/0) = x
% Assm: h4/list/list__size__def_c0: !f. h4/list/list__size f h4/list/NIL = h4/num/0
% Assm: h4/integer/INT__RDISTRIB: !z y x. h4/integer/int__mul (h4/integer/int__add x y) z = h4/integer/int__add (h4/integer/int__mul x z) (h4/integer/int__mul y z)
% Assm: h4/list/list__case__def_c0: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm: h4/list/list__case__def_c1: !v f a1 a0. h4/list/list__CASE (h4/list/CONS a0 a1) v f = f a0 a1
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/list/list__size__def_c1: !f a1 a0. h4/list/list__size f (h4/list/CONS a0 a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (f a0) (h4/list/list__size f a1))
% Assm: h4/list/list__nchotomy: !l. l = h4/list/NIL \/ (?h t. l = h4/list/CONS h t)
% Assm: h4/numeral/numeral__distrib_c21: !n. h4/prim__rec/_3C h4/num/0 (h4/arithmetic/NUMERAL n) <=> h4/prim__rec/_3C h4/arithmetic/ZERO n
% Assm: h4/numeral/numeral__distrib_c27: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__distrib_c2: !n m. h4/arithmetic/_2B (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm: h4/arithmetic/SUC__ONE__ADD: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm: h4/numeral/numeral__add_c1: !n. h4/numeral/iZ (h4/arithmetic/_2B n h4/arithmetic/ZERO) = n
% Assm: h4/numeral/numeral__distrib_c1: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm: h4/arithmetic/ADD__MONO__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm: h4/arithmetic/LESS__MONO__ADD__EQ: !p n m. h4/prim__rec/_3C (h4/arithmetic/_2B m p) (h4/arithmetic/_2B n p) <=> h4/prim__rec/_3C m n
% Assm: h4/arithmetic/LESS__EQ__TRANS: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/numeral/numeral__add_c2: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT2 (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/arithmetic/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% 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__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/ADD__COMM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/numeral/numeral__lt_c0: !n. h4/prim__rec/_3C h4/arithmetic/ZERO (h4/arithmetic/BIT1 n) <=> T
% Assm: h4/arithmetic/ADD__ASSOC: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm: h4/arithmetic/ADD__0: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND 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/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/relation/RESTRICT__LEMMA: !z y f R. R y z ==> h4/relation/RESTRICT f R z y = f y
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% 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/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/numeral/numeral__lt_c1: !n. h4/prim__rec/_3C h4/arithmetic/ZERO (h4/arithmetic/BIT2 n) <=> T
% Assm: h4/prim__rec/WF__LESS: h4/relation/WF h4/prim__rec/_3C
% Assm: h4/bool/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Goal: h4/rat/rat__of__num h4/num/0 = h4/rat/rat__0
%   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_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_0u_u_def]: h4/rat/rat__0 = happ h4/rat/abs__rat h4/frac/frac__0
% 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_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_bijectionsu_c0]: !a. h4/rat/abs__rat__CLASS (h4/rat/rep__rat__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_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_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_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_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_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_1u_u_def]: h4/rat/rat__1 = happ h4/rat/abs__rat h4/frac/frac__1
% 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_gequ_u_def]: !r2 r1. h4/rat/rat__geq r1 r2 <=> h4/rat/rat__leq r2 r1
% 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_sgnu_u_def]: !r. h4/rat/rat__sgn r = h4/frac/frac__sgn (happ h4/rat/rep__rat r)
% 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_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_dnmu_u_def]: !r. h4/rat/rat__dnm r = h4/frac/frac__dnm (happ h4/rat/rep__rat r)
% 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_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_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_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_QUOTIENT]: h4/quotient/QUOTIENT h4/rat/rat__equiv h4/rat/abs__rat h4/rat/rep__rat
% Assm [h4s_rats_ratu_u_def]: 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_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_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_EQUIVu_u_IMPu_u_PARTIALu_u_EQUIV]: !R. h4/quotient/EQUIV R ==> h4/quotient/PARTIAL__EQUIV R
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% 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_INTu_u_MODu_u_P]: !x c P. ~(c = h4/integer/int__of__num h4/num/0) ==> (happ P (h4/integer/int__mod x c) <=> (?k r. x = happ (happ h4/integer/int__add (h4/integer/int__mul k c)) r /\ (h4/integer/int__lt c (h4/integer/int__of__num h4/num/0) /\ h4/integer/int__lt c r /\ h4/integer/int__le r (h4/integer/int__of__num h4/num/0) \/ ~h4/integer/int__lt c (h4/integer/int__of__num h4/num/0) /\ h4/integer/int__le (h4/integer/int__of__num h4/num/0) r /\ h4/integer/int__lt r c) /\ happ P r))
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% 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_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% 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_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_integers_INTu_u_DIVISION]: !q. ~(q = h4/integer/int__of__num h4/num/0) ==> (!p. p = happ (happ h4/integer/int__add (h4/integer/int__mul (h4/integer/int__div p q) q)) (h4/integer/int__mod p q) /\ (?v. (v <=> h4/integer/int__lt q (h4/integer/int__mod p q) /\ h4/integer/int__le (h4/integer/int__mod p q) (h4/integer/int__of__num h4/num/0)) /\ (?v'. (v' <=> h4/integer/int__le (h4/integer/int__of__num h4/num/0) (h4/integer/int__mod p q) /\ h4/integer/int__lt (h4/integer/int__mod p q) q) /\ h4/bool/COND (h4/integer/int__lt q (h4/integer/int__of__num h4/num/0)) v v')))
% Assm [h4s_integers_INTu_u_MODu_u_UNIQUE]: !m j i. (?q. i = happ (happ h4/integer/int__add (h4/integer/int__mul q j)) m /\ (?v. (v <=> h4/integer/int__lt j m /\ h4/integer/int__le m (h4/integer/int__of__num h4/num/0)) /\ (?v'. (v' <=> h4/integer/int__le (h4/integer/int__of__num h4/num/0) m /\ h4/integer/int__lt m j) /\ h4/bool/COND (h4/integer/int__lt j (h4/integer/int__of__num h4/num/0)) v v'))) ==> h4/integer/int__mod i j = m
% Assm [h4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. happ P x \/ Q) <=> (!x. happ P x) \/ Q
% Assm [h4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. P \/ happ Q x) <=> P \/ (!x. happ Q x)
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_dcu_u_cond]: !s r q p. (p <=> h4/bool/COND q r s) <=> (p \/ q \/ ~s) /\ (p \/ ~r \/ ~q) /\ (p \/ ~r \/ ~s) /\ (~q \/ r \/ ~p) /\ (q \/ s \/ ~p)
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_bools_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND]: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm [h4s_Omegas_sumcu_u_ADD]: !vs ds cs. happ (happ h4/integer/int__add (h4/Omega/sumc cs vs)) (h4/Omega/sumc ds vs) = h4/Omega/sumc (h4/Omega/MAP2 (h4/integer/int__of__num h4/num/0) h4/integer/int__add cs ds) vs
% Assm [h4s_bools_EQu_u_EXPAND]: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_Omegas_sumcu_u_thmu_c2]: !vs v cs c. h4/Omega/sumc (h4/list/CONS c cs) (h4/list/CONS v vs) = happ (happ h4/integer/int__add (h4/integer/int__mul c v)) (h4/Omega/sumc cs vs)
% Assm [h4s_Omegas_sumcu_u_thmu_c1]: !cs. h4/Omega/sumc cs h4/list/NIL = h4/integer/int__of__num h4/num/0
% Assm [h4s_Omegas_sumcu_u_ind]: !P. (!v0. happ (happ P v0) h4/list/NIL) /\ (!v4 v5. happ (happ P h4/list/NIL) (h4/list/CONS v4 v5)) /\ (!c cs v vs. happ (happ P cs) vs ==> happ (happ P (h4/list/CONS c cs)) (h4/list/CONS v vs)) ==> (!v v1. happ (happ P v) v1)
% Assm [h4s_Omegas_MAP2u_u_zerou_u_ADDu_c1]: !xs. h4/Omega/MAP2 (h4/integer/int__of__num h4/num/0) h4/integer/int__add xs h4/list/NIL = xs
% Assm [h4s_Omegas_MAP2u_u_zerou_u_ADDu_c0]: !xs. h4/Omega/MAP2 (h4/integer/int__of__num h4/num/0) h4/integer/int__add h4/list/NIL xs = xs
% Assm [h4s_Omegas_MAP2u_u_tupledu_u_primitiveu_u_def]: !_14. (!x y_27 MAP2__tupled pad f xs ys_27. happ (happ (happ (happ (happ (happ (happ _14 x) y_27) MAP2__tupled) pad) f) xs) ys_27 = h4/combin/I (h4/list/CONS (happ (happ f x) y_27) (happ MAP2__tupled (h4/pair/_2C pad (h4/pair/_2C f (h4/pair/_2C xs ys_27)))))) ==> (!_13. (!x MAP2__tupled pad f xs y_27. happ (happ (happ (happ (happ (happ _13 x) MAP2__tupled) pad) f) xs) y_27 = happ (happ (happ (happ (happ (happ _14 x) y_27) MAP2__tupled) pad) f) xs) ==> (!_12. (!v5 x MAP2__tupled pad f xs. happ (happ (happ (happ (happ (happ _12 v5) x) MAP2__tupled) pad) f) xs = h4/list/list__CASE v5 (h4/combin/I (h4/list/CONS (happ (happ f x) pad) (happ MAP2__tupled (h4/pair/_2C pad (h4/pair/_2C f (h4/pair/_2C xs h4/list/NIL)))))) (happ (happ (happ (happ (happ _13 x) MAP2__tupled) pad) f) xs)) ==> (!_11. (!v5 MAP2__tupled pad f x. happ (happ (happ (happ (happ _11 v5) MAP2__tupled) pad) f) x = happ (happ (happ (happ (happ _12 v5) x) MAP2__tupled) pad) f) ==> (!_10. (!y MAP2__tupled pad f ys. happ (happ (happ (happ (happ _10 y) MAP2__tupled) pad) f) ys = h4/combin/I (h4/list/CONS (happ (happ f pad) y) (happ MAP2__tupled (h4/pair/_2C pad (h4/pair/_2C f (h4/pair/_2C h4/list/NIL ys)))))) ==> (!_9. (!MAP2__tupled pad f y. happ (happ (happ (happ _9 MAP2__tupled) pad) f) y = happ (happ (happ (happ _10 y) MAP2__tupled) pad) f) ==> (!_8. (!v4 MAP2__tupled pad f v5. happ (happ (happ (happ (happ _8 v4) MAP2__tupled) pad) f) v5 = h4/list/list__CASE v4 (h4/list/list__CASE v5 (h4/combin/I h4/list/NIL) (happ (happ (happ _9 MAP2__tupled) pad) f)) (happ (happ (happ (happ _11 v5) MAP2__tupled) pad) f)) ==> (!_7. (!MAP2__tupled pad f v4. happ (happ (happ (happ _7 MAP2__tupled) pad) f) v4 = happ (happ (happ (happ _8 v4) MAP2__tupled) pad) f) ==> (!_6. (!MAP2__tupled pad f v3. happ (happ (happ (happ _6 MAP2__tupled) pad) f) v3 = h4/pair/pair__CASE v3 (happ (happ (happ _7 MAP2__tupled) pad) f)) ==> (!_5. (!MAP2__tupled pad f. happ (happ (happ _5 MAP2__tupled) pad) f = happ (happ (happ _6 MAP2__tupled) pad) f) ==> (!_4. (!MAP2__tupled pad v1. happ (happ (happ _4 MAP2__tupled) pad) v1 = h4/pair/pair__CASE v1 (happ (happ _5 MAP2__tupled) pad)) ==> (!_3. (!MAP2__tupled pad. happ (happ _3 MAP2__tupled) pad = happ (happ _4 MAP2__tupled) pad) ==> (!_2. (!MAP2__tupled a. happ (happ _2 MAP2__tupled) a = h4/pair/pair__CASE a (happ _3 MAP2__tupled)) ==> (!_1. (!MAP2__tupled. happ _1 MAP2__tupled = happ _2 MAP2__tupled) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R /\ (!y ys f pad. happ (happ R (h4/pair/_2C pad (h4/pair/_2C f (h4/pair/_2C h4/list/NIL ys)))) (h4/pair/_2C pad (h4/pair/_2C f (h4/pair/_2C h4/list/NIL (h4/list/CONS y ys))))) /\ (!x xs f pad. happ (happ R (h4/pair/_2C pad (h4/pair/_2C f (h4/pair/_2C xs h4/list/NIL)))) (h4/pair/_2C pad (h4/pair/_2C f (h4/pair/_2C (h4/list/CONS x xs) h4/list/NIL)))) /\ (!y x ys xs f pad. happ (happ R (h4/pair/_2C pad (h4/pair/_2C f (h4/pair/_2C xs ys)))) (h4/pair/_2C pad (h4/pair/_2C f (h4/pair/_2C (h4/list/CONS x xs) (h4/list/CONS y ys)))))) ==> h4/Omega/MAP2__tupled = h4/relation/WFREC (h4/min/_40 _0) _1))))))))))))))
% Assm [h4s_Omegas_sumcu_u_thmu_c0]: !vs. h4/Omega/sumc h4/list/NIL vs = h4/integer/int__of__num h4/num/0
% Assm [h4s_Omegas_MAP2u_u_curriedu_u_def]: !x3 x2 x1 x. h4/Omega/MAP2 x x1 x2 x3 = happ h4/Omega/MAP2__tupled (h4/pair/_2C x (h4/pair/_2C x1 (h4/pair/_2C x2 x3)))
% Assm [h4s_integers_INTu_u_ADDu_u_LID]: !x. happ (happ h4/integer/int__add (h4/integer/int__of__num h4/num/0)) x = x
% Assm [h4s_integers_INTu_u_ADDu_u_COMM]: !y x. happ (happ h4/integer/int__add x) y = happ (happ h4/integer/int__add y) x
% Assm [h4s_integers_INTu_u_ADDu_u_ASSOC]: !z y x. happ (happ h4/integer/int__add x) (happ (happ h4/integer/int__add y) z) = happ (happ h4/integer/int__add (happ (happ h4/integer/int__add x) y)) z
% Assm [h4s_integers_INTu_u_ADDu_u_RID]: !x. happ (happ h4/integer/int__add x) (h4/integer/int__of__num h4/num/0) = x
% Assm [h4s_lists_listu_u_sizeu_u_defu_c0]: !f. h4/list/list__size f h4/list/NIL = h4/num/0
% Assm [h4s_integers_INTu_u_RDISTRIB]: !z y x. h4/integer/int__mul (happ (happ h4/integer/int__add x) y) z = happ (happ h4/integer/int__add (h4/integer/int__mul x z)) (h4/integer/int__mul y z)
% Assm [h4s_lists_listu_u_caseu_u_defu_c0]: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm [h4s_lists_listu_u_caseu_u_defu_c1]: !v f a1 a0. h4/list/list__CASE (h4/list/CONS a0 a1) v f = happ (happ f a0) a1
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_lists_listu_u_sizeu_u_defu_c1]: !f a1 a0. h4/list/list__size f (h4/list/CONS a0 a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (happ f a0) (h4/list/list__size f a1))
% Assm [h4s_lists_listu_u_nchotomy]: !l. l = h4/list/NIL \/ (?h t. l = h4/list/CONS h t)
% Assm [h4s_numerals_numeralu_u_distribu_c21]: !n. happ (happ h4/prim__rec/_3C h4/num/0) (h4/arithmetic/NUMERAL n) <=> happ (happ h4/prim__rec/_3C h4/arithmetic/ZERO) n
% Assm [h4s_numerals_numeralu_u_distribu_c27]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_distribu_c2]: !n m. h4/arithmetic/_2B (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm [h4s_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_numerals_numeralu_u_addu_c1]: !n. h4/numeral/iZ (h4/arithmetic/_2B n h4/arithmetic/ZERO) = n
% Assm [h4s_numerals_numeralu_u_distribu_c1]: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm [h4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm [h4s_arithmetics_LESSu_u_MONOu_u_ADDu_u_EQ]: !p n m. happ (happ h4/prim__rec/_3C (h4/arithmetic/_2B m p)) (h4/arithmetic/_2B n p) <=> happ (happ h4/prim__rec/_3C m) n
% Assm [h4s_arithmetics_LESSu_u_EQu_u_TRANS]: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_numerals_numeralu_u_addu_c2]: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT2 (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm [h4s_arithmetics_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% 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_LESSu_u_EQ]: !n m. happ (happ h4/prim__rec/_3C m) n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_ADDu_u_COMM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_pairs_pairu_u_caseu_u_thm]: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = happ (happ f x) y
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_numerals_numeralu_u_ltu_c0]: !n. happ (happ h4/prim__rec/_3C h4/arithmetic/ZERO) (h4/arithmetic/BIT1 n) <=> T
% Assm [h4s_arithmetics_ADDu_u_ASSOC]: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm [h4s_arithmetics_ADDu_u_0]: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_pairs_PAIR]: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND 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_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% 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_combins_Iu_u_THM]: !x. h4/combin/I x = x
% 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_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_numerals_numeralu_u_ltu_c1]: !n. happ (happ h4/prim__rec/_3C h4/arithmetic/ZERO) (h4/arithmetic/BIT2 n) <=> T
% Assm [h4s_primu_u_recs_WFu_u_LESS]: h4/relation/WF h4/prim__rec/_3C
% Assm [h4s_bools_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% Goal: happ h4/rat/rat__of__num h4/num/0 = h4/rat/rat__0
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_Q1168776,TV_Q1168772]: ![V_f, V_g]: (![V_x]: s(TV_Q1168772,happ(s(t_fun(TV_Q1168776,TV_Q1168772),V_f),s(TV_Q1168776,V_x))) = s(TV_Q1168772,happ(s(t_fun(TV_Q1168776,TV_Q1168772),V_g),s(TV_Q1168776,V_x))) => s(t_fun(TV_Q1168776,TV_Q1168772),V_f) = s(t_fun(TV_Q1168776,TV_Q1168772),V_g))).
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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_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_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_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_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_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_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_INTu_u_MODu_u_P, axiom, ![V_x, V_c, V_P]: (~ (s(t_h4s_integers_int,V_c) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) => (p(s(t_bool,happ(s(t_fun(t_h4s_integers_int,t_bool),V_P),s(t_h4s_integers_int,h4s_integers_intu_u_mod(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_c)))))) <=> ?[V_k, V_r]: (s(t_h4s_integers_int,V_x) = s(t_h4s_integers_int,happ(s(t_fun(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,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_k),s(t_h4s_integers_int,V_c))))),s(t_h4s_integers_int,V_r))) & (((p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_c),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))) & (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_c),s(t_h4s_integers_int,V_r)))) & p(s(t_bool,h4s_integers_intu_u_le(s(t_h4s_integers_int,V_r),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))))) | (~ (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_c),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))))) & (p(s(t_bool,h4s_integers_intu_u_le(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,V_r)))) & p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_r),s(t_h4s_integers_int,V_c))))))) & p(s(t_bool,happ(s(t_fun(t_h4s_integers_int,t_bool),V_P),s(t_h4s_integers_int,V_r))))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_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_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_integers_INTu_u_DIVISION, axiom, ![V_q]: (~ (s(t_h4s_integers_int,V_q) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) => ![V_p]: (s(t_h4s_integers_int,V_p) = s(t_h4s_integers_int,happ(s(t_fun(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_add),s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_integers_intu_u_div(s(t_h4s_integers_int,V_p),s(t_h4s_integers_int,V_q))),s(t_h4s_integers_int,V_q))))),s(t_h4s_integers_int,h4s_integers_intu_u_mod(s(t_h4s_integers_int,V_p),s(t_h4s_integers_int,V_q))))) & ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_q),s(t_h4s_integers_int,h4s_integers_intu_u_mod(s(t_h4s_integers_int,V_p),s(t_h4s_integers_int,V_q)))))) & p(s(t_bool,h4s_integers_intu_u_le(s(t_h4s_integers_int,h4s_integers_intu_u_mod(s(t_h4s_integers_int,V_p),s(t_h4s_integers_int,V_q))),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))))) & ?[V_vi_]: ((p(s(t_bool,V_vi_)) <=> (p(s(t_bool,h4s_integers_intu_u_le(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_mod(s(t_h4s_integers_int,V_p),s(t_h4s_integers_int,V_q)))))) & p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_mod(s(t_h4s_integers_int,V_p),s(t_h4s_integers_int,V_q))),s(t_h4s_integers_int,V_q)))))) & p(s(t_bool,h4s_bools_cond(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_q),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))),s(t_bool,V_v),s(t_bool,V_vi_))))))))).
fof(ah4s_integers_INTu_u_MODu_u_UNIQUE, axiom, ![V_m, V_j, V_i]: (?[V_q]: (s(t_h4s_integers_int,V_i) = s(t_h4s_integers_int,happ(s(t_fun(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,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_q),s(t_h4s_integers_int,V_j))))),s(t_h4s_integers_int,V_m))) & ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_j),s(t_h4s_integers_int,V_m)))) & p(s(t_bool,h4s_integers_intu_u_le(s(t_h4s_integers_int,V_m),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))))) & ?[V_vi_]: ((p(s(t_bool,V_vi_)) <=> (p(s(t_bool,h4s_integers_intu_u_le(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,V_m)))) & p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_m),s(t_h4s_integers_int,V_j)))))) & p(s(t_bool,h4s_bools_cond(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,V_j),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))),s(t_bool,V_v),s(t_bool,V_vi_))))))) => s(t_h4s_integers_int,h4s_integers_intu_u_mod(s(t_h4s_integers_int,V_i),s(t_h4s_integers_int,V_j))) = s(t_h4s_integers_int,V_m))).
fof(ah4s_bools_LEFTu_u_FORALLu_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,V_Q))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))))).
fof(ah4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,V_P)) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (p(s(t_bool,V_P)) | ![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_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_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_cond, axiom, ![V_s, V_r, V_q, V_p]: (s(t_bool,V_p) = s(t_bool,h4s_bools_cond(s(t_bool,V_q),s(t_bool,V_r),s(t_bool,V_s))) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_s))))) & ((p(s(t_bool,V_p)) | (~ (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_s))))) & ((~ (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_s)) | ~ (p(s(t_bool,V_p))))))))))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_bools_LEFTu_u_ORu_u_OVERu_u_AND, 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_A)) | p(s(t_bool,V_C)))))).
fof(ah4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND, axiom, ![V_C, V_B, 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_A))) & (p(s(t_bool,V_C)) | p(s(t_bool,V_A)))))).
fof(ah4s_Omegas_sumcu_u_ADD, axiom, ![V_vs, V_ds, V_cs]: s(t_h4s_integers_int,happ(s(t_fun(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_add),s(t_h4s_integers_int,h4s_omegas_sumc(s(t_h4s_lists_list(t_h4s_integers_int),V_cs),s(t_h4s_lists_list(t_h4s_integers_int),V_vs))))),s(t_h4s_integers_int,h4s_omegas_sumc(s(t_h4s_lists_list(t_h4s_integers_int),V_ds),s(t_h4s_lists_list(t_h4s_integers_int),V_vs))))) = s(t_h4s_integers_int,h4s_omegas_sumc(s(t_h4s_lists_list(t_h4s_integers_int),h4s_omegas_map2(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_add),s(t_h4s_lists_list(t_h4s_integers_int),V_cs),s(t_h4s_lists_list(t_h4s_integers_int),V_ds))),s(t_h4s_lists_list(t_h4s_integers_int),V_vs)))).
fof(ah4s_bools_EQu_u_EXPAND, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) | (~ (p(s(t_bool,V_t1))) & ~ (p(s(t_bool,V_t2))))))).
fof(ah4s_bools_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_Omegas_sumcu_u_thmu_c2, axiom, ![V_vs, V_v, V_cs, V_c]: s(t_h4s_integers_int,h4s_omegas_sumc(s(t_h4s_lists_list(t_h4s_integers_int),h4s_lists_cons(s(t_h4s_integers_int,V_c),s(t_h4s_lists_list(t_h4s_integers_int),V_cs))),s(t_h4s_lists_list(t_h4s_integers_int),h4s_lists_cons(s(t_h4s_integers_int,V_v),s(t_h4s_lists_list(t_h4s_integers_int),V_vs))))) = s(t_h4s_integers_int,happ(s(t_fun(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,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_c),s(t_h4s_integers_int,V_v))))),s(t_h4s_integers_int,h4s_omegas_sumc(s(t_h4s_lists_list(t_h4s_integers_int),V_cs),s(t_h4s_lists_list(t_h4s_integers_int),V_vs)))))).
fof(ah4s_Omegas_sumcu_u_thmu_c1, axiom, ![V_cs]: s(t_h4s_integers_int,h4s_omegas_sumc(s(t_h4s_lists_list(t_h4s_integers_int),V_cs),s(t_h4s_lists_list(t_h4s_integers_int),h4s_lists_nil))) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_Omegas_sumcu_u_ind, axiom, ![V_P]: ((![V_v0]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_integers_int),t_bool),happ(s(t_fun(t_h4s_lists_list(t_h4s_integers_int),t_fun(t_h4s_lists_list(t_h4s_integers_int),t_bool)),V_P),s(t_h4s_lists_list(t_h4s_integers_int),V_v0))),s(t_h4s_lists_list(t_h4s_integers_int),h4s_lists_nil)))) & (![V_v4, V_v5]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_integers_int),t_bool),happ(s(t_fun(t_h4s_lists_list(t_h4s_integers_int),t_fun(t_h4s_lists_list(t_h4s_integers_int),t_bool)),V_P),s(t_h4s_lists_list(t_h4s_integers_int),h4s_lists_nil))),s(t_h4s_lists_list(t_h4s_integers_int),h4s_lists_cons(s(t_h4s_integers_int,V_v4),s(t_h4s_lists_list(t_h4s_integers_int),V_v5)))))) & ![V_c, V_cs, V_v, V_vs]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_integers_int),t_bool),happ(s(t_fun(t_h4s_lists_list(t_h4s_integers_int),t_fun(t_h4s_lists_list(t_h4s_integers_int),t_bool)),V_P),s(t_h4s_lists_list(t_h4s_integers_int),V_cs))),s(t_h4s_lists_list(t_h4s_integers_int),V_vs)))) => p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_integers_int),t_bool),happ(s(t_fun(t_h4s_lists_list(t_h4s_integers_int),t_fun(t_h4s_lists_list(t_h4s_integers_int),t_bool)),V_P),s(t_h4s_lists_list(t_h4s_integers_int),h4s_lists_cons(s(t_h4s_integers_int,V_c),s(t_h4s_lists_list(t_h4s_integers_int),V_cs))))),s(t_h4s_lists_list(t_h4s_integers_int),h4s_lists_cons(s(t_h4s_integers_int,V_v),s(t_h4s_lists_list(t_h4s_integers_int),V_vs))))))))) => ![V_v, V_v1]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_integers_int),t_bool),happ(s(t_fun(t_h4s_lists_list(t_h4s_integers_int),t_fun(t_h4s_lists_list(t_h4s_integers_int),t_bool)),V_P),s(t_h4s_lists_list(t_h4s_integers_int),V_v))),s(t_h4s_lists_list(t_h4s_integers_int),V_v1)))))).
fof(ah4s_Omegas_MAP2u_u_zerou_u_ADDu_c1, axiom, ![V_xs]: s(t_h4s_lists_list(t_h4s_integers_int),h4s_omegas_map2(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_add),s(t_h4s_lists_list(t_h4s_integers_int),V_xs),s(t_h4s_lists_list(t_h4s_integers_int),h4s_lists_nil))) = s(t_h4s_lists_list(t_h4s_integers_int),V_xs)).
fof(ah4s_Omegas_MAP2u_u_zerou_u_ADDu_c0, axiom, ![V_xs]: s(t_h4s_lists_list(t_h4s_integers_int),h4s_omegas_map2(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_add),s(t_h4s_lists_list(t_h4s_integers_int),h4s_lists_nil),s(t_h4s_lists_list(t_h4s_integers_int),V_xs))) = s(t_h4s_lists_list(t_h4s_integers_int),V_xs)).
fof(ah4s_Omegas_MAP2u_u_tupledu_u_primitiveu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_14]: (![V_x, V_yu_27, V_MAP2u_u_tupled, V_pad, V_f, V_xs, V_ysu_27]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))))))),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))))))),V_uu_14),s(TV_u_27b,V_x))),s(TV_u_27b,V_yu_27))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled))),s(TV_u_27b,V_pad))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f))),s(t_h4s_lists_list(TV_u_27b),V_xs))),s(t_h4s_lists_list(TV_u_27b),V_ysu_27))) = s(t_h4s_lists_list(TV_u_27a),h4s_combins_i(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f),s(TV_u_27b,V_x))),s(TV_u_27b,V_yu_27))),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled),s(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),h4s_pairs_u_2c(s(TV_u_27b,V_pad),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),h4s_pairs_u_2c(s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27b),V_xs),s(t_h4s_lists_list(TV_u_27b),V_ysu_27))))))))))))) => ![V_uu_13]: (![V_x, V_MAP2u_u_tupled, V_pad, V_f, V_xs, V_yu_27]: s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))))))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))))))),V_uu_13),s(TV_u_27b,V_x))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled))),s(TV_u_27b,V_pad))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f))),s(t_h4s_lists_list(TV_u_27b),V_xs))),s(TV_u_27b,V_yu_27))) = s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))))))),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))))))),V_uu_14),s(TV_u_27b,V_x))),s(TV_u_27b,V_yu_27))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled))),s(TV_u_27b,V_pad))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f))),s(t_h4s_lists_list(TV_u_27b),V_xs))) => ![V_uu_12]: (![V_v5, V_x, V_MAP2u_u_tupled, V_pad, V_f, V_xs]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))))),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(TV_u_27b,t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))))))),V_uu_12),s(t_h4s_lists_list(TV_u_27b),V_v5))),s(TV_u_27b,V_x))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled))),s(TV_u_27b,V_pad))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f))),s(t_h4s_lists_list(TV_u_27b),V_xs))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27b),V_v5),s(t_h4s_lists_list(TV_u_27a),h4s_combins_i(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f),s(TV_u_27b,V_x))),s(TV_u_27b,V_pad))),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled),s(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),h4s_pairs_u_2c(s(TV_u_27b,V_pad),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),h4s_pairs_u_2c(s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27b),V_xs),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil))))))))))))),s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))))))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))))))),V_uu_13),s(TV_u_27b,V_x))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled))),s(TV_u_27b,V_pad))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f))),s(t_h4s_lists_list(TV_u_27b),V_xs))))) => ![V_uu_11]: (![V_v5, V_MAP2u_u_tupled, V_pad, V_f, V_x]: s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))))),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))))))),V_uu_11),s(t_h4s_lists_list(TV_u_27b),V_v5))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled))),s(TV_u_27b,V_pad))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f))),s(TV_u_27b,V_x))) = s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))))),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(TV_u_27b,t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))))))),V_uu_12),s(t_h4s_lists_list(TV_u_27b),V_v5))),s(TV_u_27b,V_x))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled))),s(TV_u_27b,V_pad))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f))) => ![V_uu_10]: (![V_y, V_MAP2u_u_tupled, V_pad, V_f, V_ys]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))))),V_uu_10),s(TV_u_27b,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled))),s(TV_u_27b,V_pad))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f))),s(t_h4s_lists_list(TV_u_27b),V_ys))) = s(t_h4s_lists_list(TV_u_27a),h4s_combins_i(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f),s(TV_u_27b,V_pad))),s(TV_u_27b,V_y))),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled),s(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),h4s_pairs_u_2c(s(TV_u_27b,V_pad),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),h4s_pairs_u_2c(s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27b),V_ys))))))))))))) => ![V_uu_9]: (![V_MAP2u_u_tupled, V_pad, V_f, V_y]: s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))))),V_uu_9),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled))),s(TV_u_27b,V_pad))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f))),s(TV_u_27b,V_y))) = s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))))),V_uu_10),s(TV_u_27b,V_y))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled))),s(TV_u_27b,V_pad))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f))) => ![V_uu_8]: (![V_v4, V_MAP2u_u_tupled, V_pad, V_f, V_v5]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))))),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))))),V_uu_8),s(t_h4s_lists_list(TV_u_27b),V_v4))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled))),s(TV_u_27b,V_pad))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f))),s(t_h4s_lists_list(TV_u_27b),V_v5))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27b),V_v4),s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27b),V_v5),s(t_h4s_lists_list(TV_u_27a),h4s_combins_i(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))),s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))))),V_uu_9),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled))),s(TV_u_27b,V_pad))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f))))),s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))))),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))))))),V_uu_11),s(t_h4s_lists_list(TV_u_27b),V_v5))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled))),s(TV_u_27b,V_pad))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f))))) => ![V_uu_7]: (![V_MAP2u_u_tupled, V_pad, V_f, V_v4]: s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))))),V_uu_7),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled))),s(TV_u_27b,V_pad))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f))),s(t_h4s_lists_list(TV_u_27b),V_v4))) = s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))))),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))))),V_uu_8),s(t_h4s_lists_list(TV_u_27b),V_v4))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled))),s(TV_u_27b,V_pad))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f))) => ![V_uu_6]: (![V_MAP2u_u_tupled, V_pad, V_f, V_v3]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),t_h4s_lists_list(TV_u_27a))))),V_uu_6),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled))),s(TV_u_27b,V_pad))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f))),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),V_v3))) = s(t_h4s_lists_list(TV_u_27a),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),V_v3),s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)))))),V_uu_7),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled))),s(TV_u_27b,V_pad))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f))))) => ![V_uu_5]: (![V_MAP2u_u_tupled, V_pad, V_f]: s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),t_h4s_lists_list(TV_u_27a))))),V_uu_5),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled))),s(TV_u_27b,V_pad))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f))) = s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),t_h4s_lists_list(TV_u_27a))))),V_uu_6),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled))),s(TV_u_27b,V_pad))),s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f))) => ![V_uu_4]: (![V_MAP2u_u_tupled, V_pad, V_v1]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),t_h4s_lists_list(TV_u_27a)))),V_uu_4),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled))),s(TV_u_27b,V_pad))),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),V_v1))) = s(t_h4s_lists_list(TV_u_27a),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),V_v1),s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),t_h4s_lists_list(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),t_h4s_lists_list(TV_u_27a))))),V_uu_5),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled))),s(TV_u_27b,V_pad))))) => ![V_uu_3]: (![V_MAP2u_u_tupled, V_pad]: s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),t_h4s_lists_list(TV_u_27a)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled))),s(TV_u_27b,V_pad))) = s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),t_h4s_lists_list(TV_u_27a)))),V_uu_4),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled))),s(TV_u_27b,V_pad))) => ![V_uu_2]: (![V_MAP2u_u_tupled, V_a]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a))),V_uu_2),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled))),s(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),V_a))) = s(t_h4s_lists_list(TV_u_27a),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),V_a),s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),t_h4s_lists_list(TV_u_27a)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled))))) => ![V_uu_1]: (![V_MAP2u_u_tupled]: s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a))),V_uu_1),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled))) = s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a))),V_uu_2),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),V_MAP2u_u_tupled))) => ![V_uu_0]: (![V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_bool)),V_R)))) & (![V_y, V_ys, V_f, V_pad]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_bool),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_bool)),V_R),s(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),h4s_pairs_u_2c(s(TV_u_27b,V_pad),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),h4s_pairs_u_2c(s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27b),V_ys))))))))),s(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),h4s_pairs_u_2c(s(TV_u_27b,V_pad),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),h4s_pairs_u_2c(s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,V_y),s(t_h4s_lists_list(TV_u_27b),V_ys)))))))))))) & (![V_x, V_xs, V_f, V_pad]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_bool),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_bool)),V_R),s(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),h4s_pairs_u_2c(s(TV_u_27b,V_pad),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),h4s_pairs_u_2c(s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27b),V_xs),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil))))))))),s(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),h4s_pairs_u_2c(s(TV_u_27b,V_pad),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),h4s_pairs_u_2c(s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,V_x),s(t_h4s_lists_list(TV_u_27b),V_xs))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil)))))))))) & ![V_y, V_x, V_ys, V_xs, V_f, V_pad]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_bool),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_bool)),V_R),s(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),h4s_pairs_u_2c(s(TV_u_27b,V_pad),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),h4s_pairs_u_2c(s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27b),V_xs),s(t_h4s_lists_list(TV_u_27b),V_ys))))))))),s(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),h4s_pairs_u_2c(s(TV_u_27b,V_pad),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),h4s_pairs_u_2c(s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_f),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,V_x),s(t_h4s_lists_list(TV_u_27b),V_xs))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,V_y),s(t_h4s_lists_list(TV_u_27b),V_ys)))))))))))))))) => s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),h4s_omegas_map2u_u_tupled) = s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a))),V_uu_1))))))))))))))))))).
fof(ah4s_Omegas_sumcu_u_thmu_c0, axiom, ![V_vs]: s(t_h4s_integers_int,h4s_omegas_sumc(s(t_h4s_lists_list(t_h4s_integers_int),h4s_lists_nil),s(t_h4s_lists_list(t_h4s_integers_int),V_vs))) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_Omegas_MAP2u_u_curriedu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_x3, V_x2, V_x1, V_x]: s(t_h4s_lists_list(TV_u_27a),h4s_omegas_map2(s(TV_u_27b,V_x),s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_x1),s(t_h4s_lists_list(TV_u_27b),V_x2),s(t_h4s_lists_list(TV_u_27b),V_x3))) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),t_h4s_lists_list(TV_u_27a)),h4s_omegas_map2u_u_tupled),s(t_h4s_pairs_prod(TV_u_27b,t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)))),h4s_pairs_u_2c(s(TV_u_27b,V_x),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),h4s_pairs_u_2c(s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27a)),V_x1),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27b),V_x2),s(t_h4s_lists_list(TV_u_27b),V_x3)))))))))).
fof(ah4s_integers_INTu_u_ADDu_u_LID, axiom, ![V_x]: s(t_h4s_integers_int,happ(s(t_fun(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,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,h4s_nums_0))))),s(t_h4s_integers_int,V_x))) = s(t_h4s_integers_int,V_x)).
fof(ah4s_integers_INTu_u_ADDu_u_COMM, axiom, ![V_y, V_x]: s(t_h4s_integers_int,happ(s(t_fun(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,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,happ(s(t_fun(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,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_ADDu_u_ASSOC, axiom, ![V_z, V_y, V_x]: s(t_h4s_integers_int,happ(s(t_fun(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_add),s(t_h4s_integers_int,V_x))),s(t_h4s_integers_int,happ(s(t_fun(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,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,happ(s(t_fun(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_add),s(t_h4s_integers_int,happ(s(t_fun(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,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_ADDu_u_RID, axiom, ![V_x]: s(t_h4s_integers_int,happ(s(t_fun(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,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_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))) = s(t_h4s_integers_int,V_x)).
fof(ah4s_lists_listu_u_sizeu_u_defu_c0, axiom, ![TV_u_27a]: ![V_f]: s(t_h4s_nums_num,h4s_lists_listu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_integers_INTu_u_RDISTRIB, axiom, ![V_z, V_y, V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,happ(s(t_fun(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,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))) = s(t_h4s_integers_int,happ(s(t_fun(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,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_z))))),s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_y),s(t_h4s_integers_int,V_z)))))).
fof(ah4s_lists_listu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27b,h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f))) = s(TV_u_27b,V_v)).
fof(ah4s_lists_listu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_f, V_a1, V_a0]: s(TV_u_27b,h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a0),s(t_h4s_lists_list(TV_u_27a),V_a1))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f))) = s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1)))).
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_lists_listu_u_sizeu_u_defu_c1, axiom, ![TV_u_27a]: ![V_f, V_a1, V_a0]: s(t_h4s_nums_num,h4s_lists_listu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a0),s(t_h4s_lists_list(TV_u_27a),V_a1))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(TV_u_27a,V_a0))),s(t_h4s_nums_num,h4s_lists_listu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_lists_list(TV_u_27a),V_a1)))))))).
fof(ah4s_lists_listu_u_nchotomy, axiom, ![TV_u_27a]: ![V_l]: (s(t_h4s_lists_list(TV_u_27a),V_l) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) | ?[V_h, V_t]: s(t_h4s_lists_list(TV_u_27a),V_l) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t))))).
fof(ah4s_numerals_numeralu_u_distribu_c21, axiom, ![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,h4s_nums_0))),s(t_h4s_nums_num,h4s_arithmetics_numeral(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,h4s_arithmetics_zero))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_numerals_numeralu_u_distribu_c27, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ah4s_numerals_numeralu_u_distribu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_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_numerals_numeralu_u_addu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero))))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_numerals_numeralu_u_distribu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_LESSu_u_MONOu_u_ADDu_u_EQ, axiom, ![V_p, 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_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_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_arithmetics_LESSu_u_EQu_u_TRANS, axiom, ![V_p, V_n, V_m]: ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_MULTu_u_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_numerals_numeralu_u_addu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_m))))))) = s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_arithmetics_ZEROu_u_LESSu_u_EQ, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_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_LESSu_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,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_u_SYM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_ADDu_u_COMM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_ADDu_u_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_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_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_pairs_pairu_u_caseu_u_thm, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_y, V_x, V_f]: s(TV_u_27a,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27b,V_x),s(TV_u_27c,V_y))),s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f))) = s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27b,V_x))),s(TV_u_27c,V_y)))).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27b,V_y)).
fof(ah4s_numerals_numeralu_u_ltu_c0, axiom, ![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,h4s_arithmetics_zero))),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))) = s(t_bool,t)).
fof(ah4s_arithmetics_ADDu_u_ASSOC, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_ADDu_u_0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_pairs_PAIR, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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_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_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_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_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_bools_DEu_u_MORGANu_u_THMu_c0, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) & p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) | ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_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_numerals_numeralu_u_ltu_c1, axiom, ![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,h4s_arithmetics_zero))),s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))))) = s(t_bool,t)).
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_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(ch4s_rats_ratu_u_ofu_u_numu_u_defu_c0, 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_0))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_0)).
