%   ORIGINAL: h4/finite__map/FAPPLY__FUPDATE__THM
% 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/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/finite__map/NOT__EQ__FAPPLY: !y x f a. ~(a = x) ==> h4/finite__map/FAPPLY (h4/finite__map/FUPDATE f (h4/pair/_2C x y)) a = h4/finite__map/FAPPLY f a
% Assm: h4/finite__map/FUPDATE__DEF: !y x f. h4/finite__map/FUPDATE f (h4/pair/_2C x y) = h4/finite__map/fmap__ABS (\a. h4/bool/COND (a = x) (h4/sum/INL y) (h4/finite__map/fmap__REP f a))
% Assm: h4/finite__map/FAPPLY__DEF: !x f. h4/finite__map/FAPPLY f x = h4/sum/OUTL (h4/finite__map/fmap__REP f x)
% Assm: h4/finite__map/fmap__ISO__DEF_c1: !r. h4/finite__map/is__fmap r <=> h4/finite__map/fmap__REP (h4/finite__map/fmap__ABS r) = r
% Assm: h4/finite__map/fmap__ISO__DEF_c0: !a. h4/finite__map/fmap__ABS (h4/finite__map/fmap__REP a) = a
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/finite__map/is__fmap__def: h4/finite__map/is__fmap = (\a0. !is__fmap_27. (!a00. a00 = (\a. h4/sum/INR h4/one/one0) \/ (?f a b. a00 = (\x. h4/bool/COND (x = a) (h4/sum/INL b) (f x)) /\ is__fmap_27 f) ==> is__fmap_27 a00) ==> is__fmap_27 a0)
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/finite__map/FAPPLY__FUPDATE: !y x f. h4/finite__map/FAPPLY (h4/finite__map/FUPDATE f (h4/pair/_2C x y)) x = y
% Assm: h4/pair/PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm: h4/finite__map/FDOM__DEF: !x f. h4/finite__map/FDOM f x <=> h4/sum/ISL (h4/finite__map/fmap__REP f x)
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/finite__map/fmap__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION h4/finite__map/is__fmap rep
% Assm: h4/finite__map/FDOM__FUPDATE: !f b a. h4/finite__map/FDOM (h4/finite__map/FUPDATE f (h4/pair/_2C a b)) = h4/pred__set/INSERT a (h4/finite__map/FDOM f)
% Assm: h4/finite__map/FUPDATE__EQ: !f c b a. h4/finite__map/FUPDATE (h4/finite__map/FUPDATE f (h4/pair/_2C a b)) (h4/pair/_2C a c) = h4/finite__map/FUPDATE f (h4/pair/_2C a c)
% Assm: h4/finite__map/FUPDATE__COMMUTES: !f d c b a. ~(a = c) ==> h4/finite__map/FUPDATE (h4/finite__map/FUPDATE f (h4/pair/_2C a b)) (h4/pair/_2C c d) = h4/finite__map/FUPDATE (h4/finite__map/FUPDATE f (h4/pair/_2C c d)) (h4/pair/_2C a b)
% 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/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/sum/OUTL0: !x. h4/sum/OUTL (h4/sum/INL x) = x
% Assm: h4/finite__map/FEMPTY__DEF: h4/finite__map/FEMPTY = h4/finite__map/fmap__ABS (\a. h4/sum/INR h4/one/one0)
% Assm: h4/finite__map/FDOM__FEMPTY: h4/finite__map/FDOM h4/finite__map/FEMPTY = h4/pred__set/EMPTY
% Assm: h4/bool/EQ__EXT: !g f. (!x. f x = g x) ==> f = g
% Assm: h4/pred__set/SPECIFICATION: !x P. h4/bool/IN x P <=> P x
% Assm: h4/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm: h4/list/list__Axiom: !f1 f0. ?fn. fn h4/list/NIL = f0 /\ (!a0 a1. fn (h4/list/CONS a0 a1) = f1 a0 a1 (fn a1))
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/sum/ISL0_c0: !x. h4/sum/ISL (h4/sum/INL x)
% Assm: h4/pred__set/IN__INSERT: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/ABS__SIMP: !t2 t1. (\x. t1) t2 = t1
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/pred__set/NOT__IN__EMPTY: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm: h4/sum/ISL0_c1: !y. ~h4/sum/ISL (h4/sum/INR y)
% Assm: h4/pred__set/ITSET__tupled__primitive__def: !f. h4/pred__set/ITSET__tupled f = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!b s. h4/pred__set/FINITE s /\ ~(s = h4/pred__set/EMPTY) ==> R (h4/pair/_2C (h4/pred__set/REST s) (f (h4/pred__set/CHOICE s) b)) (h4/pair/_2C s b)))) (\ITSET__tupled a. h4/pair/pair__CASE a (\s b. h4/combin/I (h4/bool/COND (h4/pred__set/FINITE s) (h4/bool/COND (s = h4/pred__set/EMPTY) b (ITSET__tupled (h4/pair/_2C (h4/pred__set/REST s) (f (h4/pred__set/CHOICE s) b)))) h4/bool/ARB)))
% Assm: h4/rich__list/SPLITP__AUX__def_c1: !t h acc P. h4/rich__list/SPLITP__AUX acc P (h4/list/CONS h t) = h4/bool/COND (P h) (h4/pair/_2C acc (h4/list/CONS h t)) (h4/rich__list/SPLITP__AUX (h4/list/APPEND acc (h4/list/CONS h h4/list/NIL)) P t)
% Assm: h4/numpair/nlistrec__tupled__primitive__def: h4/numpair/nlistrec__tupled = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!f n l. ~(l = h4/num/0) ==> R (h4/pair/_2C n (h4/pair/_2C f (h4/numpair/nsnd (h4/arithmetic/_2D l (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))))) (h4/pair/_2C n (h4/pair/_2C f l))))) (\nlistrec__tupled a. h4/pair/pair__CASE a (\n v1. h4/pair/pair__CASE v1 (\f l. h4/combin/I (h4/bool/COND (l = h4/num/0) n (f (h4/numpair/nfst (h4/arithmetic/_2D l (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/numpair/nsnd (h4/arithmetic/_2D l (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (nlistrec__tupled (h4/pair/_2C n (h4/pair/_2C f (h4/numpair/nsnd (h4/arithmetic/_2D l (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))))))))))
% Assm: h4/rich__list/SPLITP0_c1: !x l P. h4/rich__list/SPLITP P (h4/list/CONS x l) = h4/bool/COND (P x) (h4/pair/_2C h4/list/NIL (h4/list/CONS x l)) (h4/pair/_2C (h4/list/CONS x (h4/pair/FST (h4/rich__list/SPLITP P l))) (h4/pair/SND (h4/rich__list/SPLITP P l)))
% Assm: h4/arithmetic/DIVMOD__DEF: h4/arithmetic/DIVMOD = h4/relation/WFREC (h4/prim__rec/measure (h4/combin/o h4/pair/FST h4/pair/SND)) (\f. h4/pair/UNCURRY (\a. h4/pair/UNCURRY (\m n. h4/bool/COND (n = h4/num/0) (h4/pair/_2C h4/num/0 h4/num/0) (h4/bool/COND (h4/prim__rec/_3C m n) (h4/pair/_2C a m) (h4/bool/LET (\q. f (h4/pair/_2C (h4/arithmetic/_2B a q) (h4/pair/_2C (h4/arithmetic/_2D m (h4/arithmetic/_2A n q)) n))) (h4/arithmetic/findq (h4/pair/_2C (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/pair/_2C m n))))))))
% Assm: h4/arithmetic/findq__def: h4/arithmetic/findq = h4/relation/WFREC (h4/prim__rec/measure (h4/pair/UNCURRY (\a. h4/pair/UNCURRY (\m n. h4/arithmetic/_2D m n)))) (\f. h4/pair/UNCURRY (\a. h4/pair/UNCURRY (\m n. h4/bool/COND (n = h4/num/0) a (h4/bool/LET (\d. h4/bool/COND (h4/prim__rec/_3C m d) a (f (h4/pair/_2C (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) a) (h4/pair/_2C m d)))) (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n)))))
% Assm: h4/arithmetic/DIVMOD__THM: !n m a. h4/arithmetic/DIVMOD (h4/pair/_2C a (h4/pair/_2C m n)) = h4/bool/COND (n = h4/num/0) (h4/pair/_2C h4/num/0 h4/num/0) (h4/bool/COND (h4/prim__rec/_3C m n) (h4/pair/_2C a m) (h4/bool/LET (\q. h4/arithmetic/DIVMOD (h4/pair/_2C (h4/arithmetic/_2B a q) (h4/pair/_2C (h4/arithmetic/_2D m (h4/arithmetic/_2A n q)) n))) (h4/arithmetic/findq (h4/pair/_2C (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/pair/_2C m n)))))
% Assm: h4/arithmetic/findq__thm: !n m a. h4/arithmetic/findq (h4/pair/_2C a (h4/pair/_2C m n)) = h4/bool/COND (n = h4/num/0) a (h4/bool/LET (\d. h4/bool/COND (h4/prim__rec/_3C m d) a (h4/arithmetic/findq (h4/pair/_2C (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) a) (h4/pair/_2C m d)))) (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n))
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/arithmetic/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/arithmetic/NOT__ZERO__LT__ZERO: !n. ~(n = h4/num/0) <=> h4/prim__rec/_3C h4/num/0 n
% Assm: h4/arithmetic/ONE: h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) = h4/num/SUC h4/num/0
% Assm: h4/prim__rec/measure__thm: !y x f. h4/prim__rec/measure f x y <=> h4/prim__rec/_3C (f x) (f y)
% Assm: h4/prim__rec/WF__measure: !m. h4/relation/WF (h4/prim__rec/measure m)
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/relation/RESTRICT__DEF: !x f R. h4/relation/RESTRICT f R x = (\y. h4/bool/COND (R y x) (f y) h4/bool/ARB)
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% 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/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/arithmetic/ADD__COMM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> 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/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/LET__THM: !x f. h4/bool/LET f x = f x
% Assm: h4/arithmetic/findq__eq__0: !n m a. h4/arithmetic/findq (h4/pair/_2C a (h4/pair/_2C m n)) = h4/num/0 <=> a = h4/num/0
% Assm: h4/arithmetic/SUB__LEFT__LESS__EQ: !p n m. h4/arithmetic/_3C_3D m (h4/arithmetic/_2D n p) <=> h4/arithmetic/_3C_3D (h4/arithmetic/_2B m p) n \/ h4/arithmetic/_3C_3D m h4/num/0
% Assm: h4/arithmetic/MULT__EQ__0: !n m. h4/arithmetic/_2A m n = h4/num/0 <=> m = h4/num/0 \/ n = h4/num/0
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/arithmetic/LESS__LESS__EQ__TRANS: !p n m. h4/prim__rec/_3C m n /\ h4/arithmetic/_3C_3D n p ==> h4/prim__rec/_3C m p
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/LESS__ADD__NONZERO: !n m. ~(n = h4/num/0) ==> h4/prim__rec/_3C m (h4/arithmetic/_2B m n)
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/num/NOT__SUC: !n. ~(h4/num/SUC n = h4/num/0)
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/arithmetic/SUB__LEFT__LESS: !p n m. h4/prim__rec/_3C m (h4/arithmetic/_2D n p) <=> h4/prim__rec/_3C (h4/arithmetic/_2B m p) n
% Assm: h4/arithmetic/SUB__LEFT__ADD: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2D n p) = h4/bool/COND (h4/arithmetic/_3C_3D n p) m (h4/arithmetic/_2D (h4/arithmetic/_2B m n) p)
% Assm: h4/arithmetic/SUB__RIGHT__LESS: !p n m. h4/prim__rec/_3C (h4/arithmetic/_2D m n) p <=> h4/prim__rec/_3C m (h4/arithmetic/_2B n p) /\ h4/prim__rec/_3C h4/num/0 p
% Assm: h4/arithmetic/LT__MULT__RCANCEL: !p n m. h4/prim__rec/_3C (h4/arithmetic/_2A m n) (h4/arithmetic/_2A p n) <=> h4/prim__rec/_3C h4/num/0 n /\ h4/prim__rec/_3C m p
% Assm: h4/arithmetic/LE__MULT__RCANCEL: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2A m n) (h4/arithmetic/_2A p n) <=> n = h4/num/0 \/ h4/arithmetic/_3C_3D m p
% Assm: h4/arithmetic/LESS__EQ__LESS__TRANS: !p n m. h4/arithmetic/_3C_3D m n /\ h4/prim__rec/_3C n p ==> h4/prim__rec/_3C m p
% Assm: h4/arithmetic/NOT__LESS__EQUAL: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/prim__rec/_3C n m
% 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/LT__ADD__LCANCEL: !p n m. h4/prim__rec/_3C (h4/arithmetic/_2B p m) (h4/arithmetic/_2B p n) <=> h4/prim__rec/_3C m n
% Assm: h4/arithmetic/LESS__EQ__MONO: !n m. h4/arithmetic/_3C_3D (h4/num/SUC n) (h4/num/SUC m) <=> h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/MULT__LEFT__1: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/NOT__SUC__LESS__EQ__0: !n. ~h4/arithmetic/_3C_3D (h4/num/SUC n) h4/num/0
% Assm: h4/arithmetic/LESS__MONO__EQ: !n m. h4/prim__rec/_3C (h4/num/SUC m) (h4/num/SUC n) <=> h4/prim__rec/_3C m n
% Assm: h4/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/arithmetic/TWO: h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO) = h4/num/SUC (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/combin/LET__FORALL__ELIM: !v f. h4/bool/LET f v <=> $forall (h4/combin/S (h4/combin/o $imply (h4/combin/o h4/marker/Abbrev (h4/combin/C $equals v))) f)
% Assm: h4/combin/C__ABS__L: !y f. h4/combin/C (\x. f x) y = (\x. f x y)
% Assm: h4/combin/GEN__LET__RAND: !v f P. P (h4/bool/LET f v) = h4/bool/LET (h4/combin/o P f) v
% Assm: h4/combin/C__THM: !y x f. h4/combin/C f x y = f y x
% Assm: h4/combin/GEN__LET__RATOR: !x v f. h4/bool/LET f v x = h4/bool/LET (h4/combin/C f x) v
% Assm: h4/combin/S__ABS__R: !g f. h4/combin/S f (\x. g x) = (\x. f x (g x))
% Assm: h4/combin/o__ABS__R: !g f. h4/combin/o f (\x. g x) = (\x. f (g x))
% Goal: !x f b a. h4/finite__map/FAPPLY (h4/finite__map/FUPDATE f (h4/pair/_2C a b)) x = h4/bool/COND (x = a) b (h4/finite__map/FAPPLY f x)
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. happ (h4/pair/UNCURRY f) (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_finiteu_u_maps_NOTu_u_EQu_u_FAPPLY]: !y x f a. ~(a = x) ==> h4/finite__map/FAPPLY (h4/finite__map/FUPDATE f (h4/pair/_2C x y)) a = h4/finite__map/FAPPLY f a
% Assm [h4s_finiteu_u_maps_FUPDATEu_u_DEF]: !_0. (!x y f a. ?v. (v <=> a = x) /\ happ (happ (happ (happ _0 x) y) f) a = h4/bool/COND v (h4/sum/INL y) (happ (h4/finite__map/fmap__REP f) a)) ==> (!y x f. h4/finite__map/FUPDATE f (h4/pair/_2C x y) = h4/finite__map/fmap__ABS (happ (happ (happ _0 x) y) f))
% Assm [h4s_finiteu_u_maps_FAPPLYu_u_DEF]: !x f. h4/finite__map/FAPPLY f x = h4/sum/OUTL (happ (h4/finite__map/fmap__REP f) x)
% Assm [h4s_finiteu_u_maps_fmapu_u_ISOu_u_DEFu_c1]: !r. happ h4/finite__map/is__fmap r <=> h4/finite__map/fmap__REP (h4/finite__map/fmap__ABS r) = r
% Assm [h4s_finiteu_u_maps_fmapu_u_ISOu_u_DEFu_c0]: !a. h4/finite__map/fmap__ABS (h4/finite__map/fmap__REP a) = a
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_finiteu_u_maps_isu_u_fmapu_u_def]: !x. happ h4/finite__map/is__fmap x <=> (!is__fmap_27. (!a00. (!x. happ a00 x = h4/sum/INR h4/one/one0) \/ (?f a b. (!x. ?v. (v <=> x = a) /\ happ a00 x = h4/bool/COND v (h4/sum/INL b) (happ f x)) /\ happ is__fmap_27 f) ==> happ is__fmap_27 a00) ==> happ is__fmap_27 x)
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_finiteu_u_maps_FAPPLYu_u_FUPDATE]: !y x f. h4/finite__map/FAPPLY (h4/finite__map/FUPDATE f (h4/pair/_2C x y)) x = y
% Assm [h4s_pairs_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_pairs_PAIR]: !x. h4/pair/_2C (happ h4/pair/FST x) (happ h4/pair/SND x) = x
% Assm [h4s_finiteu_u_maps_FDOMu_u_DEF]: !x f. happ (h4/finite__map/FDOM f) x <=> h4/sum/ISL (happ (h4/finite__map/fmap__REP f) x)
% Assm [h4s_pairs_FST0]: !y x. happ h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_finiteu_u_maps_fmapu_u_TYu_u_DEF]: ?rep. h4/bool/TYPE__DEFINITION h4/finite__map/is__fmap rep
% Assm [h4s_finiteu_u_maps_FDOMu_u_FUPDATE]: !f b a. h4/finite__map/FDOM (h4/finite__map/FUPDATE f (h4/pair/_2C a b)) = h4/pred__set/INSERT a (h4/finite__map/FDOM f)
% Assm [h4s_finiteu_u_maps_FUPDATEu_u_EQ]: !f c b a. h4/finite__map/FUPDATE (h4/finite__map/FUPDATE f (h4/pair/_2C a b)) (h4/pair/_2C a c) = h4/finite__map/FUPDATE f (h4/pair/_2C a c)
% Assm [h4s_finiteu_u_maps_FUPDATEu_u_COMMUTES]: !f d c b a. ~(a = c) ==> h4/finite__map/FUPDATE (h4/finite__map/FUPDATE f (h4/pair/_2C a b)) (h4/pair/_2C c d) = h4/finite__map/FUPDATE (h4/finite__map/FUPDATE f (h4/pair/_2C c d)) (h4/pair/_2C a b)
% 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_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_sums_OUTL0]: !x. h4/sum/OUTL (h4/sum/INL x) = x
% Assm [h4s_finiteu_u_maps_FEMPTYu_u_DEF]: !_0. (!a. happ _0 a = h4/sum/INR h4/one/one0) ==> h4/finite__map/FEMPTY = h4/finite__map/fmap__ABS _0
% Assm [h4s_finiteu_u_maps_FDOMu_u_FEMPTY]: h4/finite__map/FDOM h4/finite__map/FEMPTY = h4/pred__set/EMPTY
% Assm [h4s_bools_EQu_u_EXT]: !g f. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_predu_u_sets_SPECIFICATION]: !x P. h4/bool/IN x P <=> happ P x
% Assm [h4s_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm [h4s_lists_listu_u_Axiom]: !f1 f0. ?fn. happ fn h4/list/NIL = f0 /\ (!a0 a1. happ fn (h4/list/CONS a0 a1) = happ (happ (happ f1 a0) a1) (happ fn a1))
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_sums_ISL0u_c0]: !x. h4/sum/ISL (h4/sum/INL x)
% Assm [h4s_predu_u_sets_INu_u_INSERT]: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_ABSu_u_SIMP]: !t2 t1. t1 = t1
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_predu_u_sets_NOTu_u_INu_u_EMPTY]: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm [h4s_sums_ISL0u_c1]: !y. ~h4/sum/ISL (h4/sum/INR y)
% Assm [h4s_predu_u_sets_ITSETu_u_tupledu_u_primitiveu_u_def]: !_4. (!ITSET__tupled f s b. ?v. (v <=> s = h4/pred__set/EMPTY) /\ happ (happ (happ (happ _4 ITSET__tupled) f) s) b = h4/combin/I (h4/bool/COND (h4/pred__set/FINITE s) (h4/bool/COND v b (happ ITSET__tupled (h4/pair/_2C (h4/pred__set/REST s) (happ (happ f (h4/pred__set/CHOICE s)) b)))) h4/bool/ARB)) ==> (!_3. (!ITSET__tupled f s. happ (happ (happ _3 ITSET__tupled) f) s = happ (happ (happ _4 ITSET__tupled) f) s) ==> (!_2. (!ITSET__tupled f a. happ (happ (happ _2 ITSET__tupled) f) a = h4/pair/pair__CASE a (happ (happ _3 ITSET__tupled) f)) ==> (!_1. (!f ITSET__tupled. happ (happ _1 f) ITSET__tupled = happ (happ _2 ITSET__tupled) f) ==> (!_0. (!f R. happ (happ _0 f) R <=> h4/relation/WF R /\ (!b s. h4/pred__set/FINITE s /\ ~(s = h4/pred__set/EMPTY) ==> happ (happ R (h4/pair/_2C (h4/pred__set/REST s) (happ (happ f (h4/pred__set/CHOICE s)) b))) (h4/pair/_2C s b))) ==> (!f. h4/pred__set/ITSET__tupled f = h4/relation/WFREC (h4/min/_40 (happ _0 f)) (happ _1 f))))))
% Assm [h4s_richu_u_lists_SPLITPu_u_AUXu_u_defu_c1]: !t h acc P. h4/rich__list/SPLITP__AUX acc P (h4/list/CONS h t) = h4/bool/COND (happ P h) (h4/pair/_2C acc (h4/list/CONS h t)) (h4/rich__list/SPLITP__AUX (h4/list/APPEND acc (h4/list/CONS h h4/list/NIL)) P t)
% Assm [h4s_numpairs_nlistrecu_u_tupledu_u_primitiveu_u_def]: !_6. (!nlistrec__tupled n f l. ?v. (v <=> l = h4/num/0) /\ happ (happ (happ (happ _6 nlistrec__tupled) n) f) l = h4/combin/I (h4/bool/COND v n (happ (happ (happ f (h4/numpair/nfst (h4/arithmetic/_2D l (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))) (h4/numpair/nsnd (h4/arithmetic/_2D l (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))) (happ nlistrec__tupled (h4/pair/_2C n (h4/pair/_2C f (h4/numpair/nsnd (h4/arithmetic/_2D l (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))))))))) ==> (!_5. (!nlistrec__tupled n f. happ (happ (happ _5 nlistrec__tupled) n) f = happ (happ (happ _6 nlistrec__tupled) n) f) ==> (!_4. (!nlistrec__tupled n v1. happ (happ (happ _4 nlistrec__tupled) n) v1 = h4/pair/pair__CASE v1 (happ (happ _5 nlistrec__tupled) n)) ==> (!_3. (!nlistrec__tupled n. happ (happ _3 nlistrec__tupled) n = happ (happ _4 nlistrec__tupled) n) ==> (!_2. (!nlistrec__tupled a. happ (happ _2 nlistrec__tupled) a = h4/pair/pair__CASE a (happ _3 nlistrec__tupled)) ==> (!_1. (!nlistrec__tupled. happ _1 nlistrec__tupled = happ _2 nlistrec__tupled) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R /\ (!f n l. ~(l = h4/num/0) ==> happ (happ R (h4/pair/_2C n (h4/pair/_2C f (h4/numpair/nsnd (h4/arithmetic/_2D l (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))))) (h4/pair/_2C n (h4/pair/_2C f l)))) ==> h4/numpair/nlistrec__tupled = h4/relation/WFREC (h4/min/_40 _0) _1))))))
% Assm [h4s_richu_u_lists_SPLITP0u_c1]: !x l P. h4/rich__list/SPLITP P (h4/list/CONS x l) = h4/bool/COND (happ P x) (h4/pair/_2C h4/list/NIL (h4/list/CONS x l)) (h4/pair/_2C (h4/list/CONS x (happ h4/pair/FST (h4/rich__list/SPLITP P l))) (happ h4/pair/SND (h4/rich__list/SPLITP P l)))
% Assm [h4s_arithmetics_DIVMODu_u_DEF]: !_4. (!f a m n q. happ (happ (happ (happ (happ _4 f) a) m) n) q = happ f (h4/pair/_2C (h4/arithmetic/_2B a q) (h4/pair/_2C (h4/arithmetic/_2D m (h4/arithmetic/_2A n q)) n))) ==> (!_3. (!f a m n. ?v. (v <=> n = h4/num/0) /\ happ (happ (happ (happ _3 f) a) m) n = h4/bool/COND v (h4/pair/_2C h4/num/0 h4/num/0) (h4/bool/COND (h4/prim__rec/_3C m n) (h4/pair/_2C a m) (h4/bool/LET (happ (happ (happ (happ _4 f) a) m) n) (happ h4/arithmetic/findq (h4/pair/_2C (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/pair/_2C m n)))))) ==> (!_2. (!f a m. happ (happ (happ _2 f) a) m = happ (happ (happ _3 f) a) m) ==> (!_1. (!f a. happ (happ _1 f) a = h4/pair/UNCURRY (happ (happ _2 f) a)) ==> (!_0. (!f. happ _0 f = h4/pair/UNCURRY (happ _1 f)) ==> h4/arithmetic/DIVMOD = h4/relation/WFREC (h4/prim__rec/measure (h4/combin/o h4/pair/FST h4/pair/SND)) _0))))
% Assm [h4s_arithmetics_findqu_u_def]: !_7. (!f a m d. happ (happ (happ (happ _7 f) a) m) d = h4/bool/COND (h4/prim__rec/_3C m d) a (happ f (h4/pair/_2C (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) a) (h4/pair/_2C m d)))) ==> (!_6. (!f a m n. ?v. (v <=> n = h4/num/0) /\ happ (happ (happ (happ _6 f) a) m) n = h4/bool/COND v a (h4/bool/LET (happ (happ (happ _7 f) a) m) (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n))) ==> (!_5. (!f a m. happ (happ (happ _5 f) a) m = happ (happ (happ _6 f) a) m) ==> (!_4. (!f a. happ (happ _4 f) a = h4/pair/UNCURRY (happ (happ _5 f) a)) ==> (!_3. (!f. happ _3 f = h4/pair/UNCURRY (happ _4 f)) ==> (!_2. (!m n. happ (happ _2 m) n = h4/arithmetic/_2D m n) ==> (!_1. (!m. happ _1 m = happ _2 m) ==> (!_0. (!a. happ _0 a = h4/pair/UNCURRY _1) ==> h4/arithmetic/findq = h4/relation/WFREC (h4/prim__rec/measure (h4/pair/UNCURRY _0)) _3)))))))
% Assm [h4s_arithmetics_DIVMODu_u_THM]: !_0. (!a m n q. happ (happ (happ (happ _0 a) m) n) q = happ h4/arithmetic/DIVMOD (h4/pair/_2C (h4/arithmetic/_2B a q) (h4/pair/_2C (h4/arithmetic/_2D m (h4/arithmetic/_2A n q)) n))) ==> (!n m a. ?v. (v <=> n = h4/num/0) /\ happ h4/arithmetic/DIVMOD (h4/pair/_2C a (h4/pair/_2C m n)) = h4/bool/COND v (h4/pair/_2C h4/num/0 h4/num/0) (h4/bool/COND (h4/prim__rec/_3C m n) (h4/pair/_2C a m) (h4/bool/LET (happ (happ (happ _0 a) m) n) (happ h4/arithmetic/findq (h4/pair/_2C (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/pair/_2C m n))))))
% Assm [h4s_arithmetics_findqu_u_thm]: !_0. (!a m d. happ (happ (happ _0 a) m) d = h4/bool/COND (h4/prim__rec/_3C m d) a (happ h4/arithmetic/findq (h4/pair/_2C (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) a) (h4/pair/_2C m d)))) ==> (!n m a. ?v. (v <=> n = h4/num/0) /\ happ h4/arithmetic/findq (h4/pair/_2C a (h4/pair/_2C m n)) = h4/bool/COND v a (h4/bool/LET (happ (happ _0 a) m) (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n)))
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% 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_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_arithmetics_NOTu_u_LESS]: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_arithmetics_NOTu_u_ZEROu_u_LTu_u_ZERO]: !n. ~(n = h4/num/0) <=> h4/prim__rec/_3C h4/num/0 n
% Assm [h4s_arithmetics_ONE]: h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO) = h4/num/SUC h4/num/0
% Assm [h4s_primu_u_recs_measureu_u_thm]: !y x f. happ (happ (h4/prim__rec/measure f) x) y <=> h4/prim__rec/_3C (happ f x) (happ f y)
% Assm [h4s_primu_u_recs_WFu_u_measure]: !m. h4/relation/WF (h4/prim__rec/measure m)
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_relations_RESTRICTu_u_DEF]: !x f R x'. happ (h4/relation/RESTRICT f R x) x' = h4/bool/COND (happ (happ R x') x) (happ f x') h4/bool/ARB
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% 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_combins_ou_u_THM]: !x g f. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_arithmetics_ADDu_u_COMM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> 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_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_LETu_u_THM]: !x f. h4/bool/LET f x = happ f x
% Assm [h4s_arithmetics_findqu_u_equ_u_0]: !n m a. happ h4/arithmetic/findq (h4/pair/_2C a (h4/pair/_2C m n)) = h4/num/0 <=> a = h4/num/0
% Assm [h4s_arithmetics_SUBu_u_LEFTu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D m (h4/arithmetic/_2D n p) <=> h4/arithmetic/_3C_3D (h4/arithmetic/_2B m p) n \/ h4/arithmetic/_3C_3D m h4/num/0
% Assm [h4s_arithmetics_MULTu_u_EQu_u_0]: !n m. h4/arithmetic/_2A m n = h4/num/0 <=> m = h4/num/0 \/ n = h4/num/0
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_arithmetics_LESSu_u_LESSu_u_EQu_u_TRANS]: !p n m. h4/prim__rec/_3C m n /\ h4/arithmetic/_3C_3D n p ==> h4/prim__rec/_3C m p
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_LESSu_u_ADDu_u_NONZERO]: !n m. ~(n = h4/num/0) ==> h4/prim__rec/_3C m (h4/arithmetic/_2B m n)
% Assm [h4s_pairs_SND0]: !y x. happ h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_nums_NOTu_u_SUC]: !n. ~(h4/num/SUC n = h4/num/0)
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_arithmetics_SUBu_u_LEFTu_u_LESS]: !p n m. h4/prim__rec/_3C m (h4/arithmetic/_2D n p) <=> h4/prim__rec/_3C (h4/arithmetic/_2B m p) n
% Assm [h4s_arithmetics_SUBu_u_LEFTu_u_ADD]: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2D n p) = h4/bool/COND (h4/arithmetic/_3C_3D n p) m (h4/arithmetic/_2D (h4/arithmetic/_2B m n) p)
% Assm [h4s_arithmetics_SUBu_u_RIGHTu_u_LESS]: !p n m. h4/prim__rec/_3C (h4/arithmetic/_2D m n) p <=> h4/prim__rec/_3C m (h4/arithmetic/_2B n p) /\ h4/prim__rec/_3C h4/num/0 p
% Assm [h4s_arithmetics_LTu_u_MULTu_u_RCANCEL]: !p n m. h4/prim__rec/_3C (h4/arithmetic/_2A m n) (h4/arithmetic/_2A p n) <=> h4/prim__rec/_3C h4/num/0 n /\ h4/prim__rec/_3C m p
% Assm [h4s_arithmetics_LEu_u_MULTu_u_RCANCEL]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2A m n) (h4/arithmetic/_2A p n) <=> n = h4/num/0 \/ h4/arithmetic/_3C_3D m p
% Assm [h4s_arithmetics_LESSu_u_EQu_u_LESSu_u_TRANS]: !p n m. h4/arithmetic/_3C_3D m n /\ h4/prim__rec/_3C n p ==> h4/prim__rec/_3C m p
% Assm [h4s_arithmetics_NOTu_u_LESSu_u_EQUAL]: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/prim__rec/_3C n m
% Assm [h4s_arithmetics_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_LTu_u_ADDu_u_LCANCEL]: !p n m. h4/prim__rec/_3C (h4/arithmetic/_2B p m) (h4/arithmetic/_2B p n) <=> h4/prim__rec/_3C m n
% Assm [h4s_arithmetics_LESSu_u_EQu_u_MONO]: !n m. h4/arithmetic/_3C_3D (h4/num/SUC n) (h4/num/SUC m) <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_MULTu_u_LEFTu_u_1]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_NOTu_u_SUCu_u_LESSu_u_EQu_u_0]: !n. ~h4/arithmetic/_3C_3D (h4/num/SUC n) h4/num/0
% Assm [h4s_arithmetics_LESSu_u_MONOu_u_EQ]: !n m. h4/prim__rec/_3C (h4/num/SUC m) (h4/num/SUC n) <=> h4/prim__rec/_3C m n
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm [h4s_arithmetics_TWO]: h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO) = h4/num/SUC (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_combins_LETu_u_FORALLu_u_ELIM]: !v f. h4/bool/LET f v <=> $forall (h4/combin/S (h4/combin/o $imply (h4/combin/o h4/marker/Abbrev (h4/combin/C $equals v))) f)
% Assm [h4s_combins_Cu_u_ABSu_u_L]: !_0. (!f x. happ (happ _0 f) x = happ f x) ==> (!y f x. happ (h4/combin/C (happ _0 f) y) x = happ (happ f x) y)
% Assm [h4s_combins_GENu_u_LETu_u_RAND]: !v f P. happ P (h4/bool/LET f v) = h4/bool/LET (h4/combin/o P f) v
% Assm [h4s_combins_Cu_u_THM]: !y x f. happ (h4/combin/C f x) y = happ (happ f y) x
% Assm [h4s_combins_GENu_u_LETu_u_RATOR]: !x v f. happ (h4/bool/LET f v) x = h4/bool/LET (h4/combin/C f x) v
% Assm [h4s_combins_Su_u_ABSu_u_R]: !_0. (!g x. happ (happ _0 g) x = happ g x) ==> (!g f x. happ (h4/combin/S f (happ _0 g)) x = happ (happ f x) (happ g x))
% Assm [h4s_combins_ou_u_ABSu_u_R]: !_0. (!g x. happ (happ _0 g) x = happ g x) ==> (!g f x. happ (h4/combin/o f (happ _0 g)) x = happ f (happ g x))
% Goal: !x f b a. ?v. (v <=> x = a) /\ h4/finite__map/FAPPLY (h4/finite__map/FUPDATE f (h4/pair/_2C a b)) x = h4/bool/COND v b (h4/finite__map/FAPPLY f x)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f0)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(aHLu_EXT, axiom, ![TV_Q1148636,TV_Q1148632]: ![V_f, V_g]: (![V_x]: s(TV_Q1148632,happ(s(t_fun(TV_Q1148636,TV_Q1148632),V_f),s(TV_Q1148636,V_x))) = s(TV_Q1148632,happ(s(t_fun(TV_Q1148636,TV_Q1148632),V_g),s(TV_Q1148636,V_x))) => s(t_fun(TV_Q1148636,TV_Q1148632),V_f) = s(t_fun(TV_Q1148636,TV_Q1148632),V_g))).
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,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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_finiteu_u_maps_NOTu_u_EQu_u_FAPPLY, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x, V_f, V_a]: (~ (s(TV_u_27a,V_a) = s(TV_u_27a,V_x)) => s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),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_27a,V_a))) = s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_a))))).
fof(ah4s_finiteu_u_maps_FUPDATEu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_x, V_y, V_f, V_a]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(TV_u_27a,V_a) = s(TV_u_27a,V_x)) & s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one))),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one))))),V_uu_0),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_a))) = s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),h4s_sums_inl(s(TV_u_27b,V_y))),s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),h4s_finiteu_u_maps_fmapu_u_rep(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_a)))))) => ![V_y, V_x, V_f]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),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(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fmapu_u_abs(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one))),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one))))),V_uu_0),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))))))).
fof(ah4s_finiteu_u_maps_FAPPLYu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,h4s_sums_outl(s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),h4s_finiteu_u_maps_fmapu_u_rep(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_x)))))).
fof(ah4s_finiteu_u_maps_fmapu_u_ISOu_u_DEFu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_r]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),t_bool),h4s_finiteu_u_maps_isu_u_fmap),s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_r)))) <=> s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),h4s_finiteu_u_maps_fmapu_u_rep(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fmapu_u_abs(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_r))))) = s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_r))).
fof(ah4s_finiteu_u_maps_fmapu_u_ISOu_u_DEFu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_a]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fmapu_u_abs(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),h4s_finiteu_u_maps_fmapu_u_rep(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_a))))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_a)).
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_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_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_MONOu_u_OR, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) | p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) | p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_finiteu_u_maps_isu_u_fmapu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),t_bool),h4s_finiteu_u_maps_isu_u_fmap),s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_x)))) <=> ![V_isu_u_fmapu_27]: (![V_a00]: ((![V_x0]: s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_a00),s(TV_u_27a,V_x0))) = s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),h4s_sums_inr(s(t_h4s_ones_one,h4s_ones_one0))) | ?[V_f, V_a, V_b]: (![V_x0]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(TV_u_27a,V_x0) = s(TV_u_27a,V_a)) & s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_a00),s(TV_u_27a,V_x0))) = s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),h4s_sums_inl(s(TV_u_27b,V_b))),s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_f),s(TV_u_27a,V_x0)))))) & p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),t_bool),V_isu_u_fmapu_27),s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_f)))))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),t_bool),V_isu_u_fmapu_27),s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_a00))))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),t_bool),V_isu_u_fmapu_27),s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_x))))))).
fof(ah4s_bools_MONOu_u_AND, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) & p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) & p(s(t_bool,V_w)))))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f0)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_finiteu_u_maps_FAPPLYu_u_FUPDATE, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),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_27a,V_x))) = s(TV_u_27b,V_y)).
fof(ah4s_pairs_PAIRu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_b, V_a]: (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(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & s(TV_u_27b,V_y) = s(TV_u_27b,V_b)))).
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,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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_finiteu_u_maps_FDOMu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_x))) = s(t_bool,h4s_sums_isl(s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),h4s_finiteu_u_maps_fmapu_u_rep(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_x)))))).
fof(ah4s_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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_finiteu_u_maps_fmapu_u_TYu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),t_bool),h4s_finiteu_u_maps_isu_u_fmap),s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one))),V_rep))))).
fof(ah4s_finiteu_u_maps_FDOMu_u_FUPDATE, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_b, V_a]: s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))))))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_a),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f)))))).
fof(ah4s_finiteu_u_maps_FUPDATEu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_c, V_b, V_a]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_c))))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_c)))))).
fof(ah4s_finiteu_u_maps_FUPDATEu_u_COMMUTES, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_d, V_c, V_b, V_a]: (~ (s(TV_u_27a,V_a) = s(TV_u_27a,V_c)) => s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_c),s(TV_u_27b,V_d))))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_c),s(TV_u_27b,V_d))))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))))))).
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_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_sums_OUTL0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_sums_outl(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x)).
fof(ah4s_finiteu_u_maps_FEMPTYu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_a]: s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_uu_0),s(TV_u_27a,V_a))) = s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),h4s_sums_inr(s(t_h4s_ones_one,h4s_ones_one0))) => s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fempty) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fmapu_u_abs(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_uu_0))))).
fof(ah4s_finiteu_u_maps_FDOMu_u_FEMPTY, axiom, ![TV_u_27b,TV_u_27a]: s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fempty))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_bools_EQu_u_EXT, axiom, ![TV_u_27a,TV_u_27b]: ![V_g, V_f]: (![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),V_g),s(TV_u_27a,V_x))) => s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g))).
fof(ah4s_predu_u_sets_SPECIFICATION, axiom, ![TV_u_27a]: ![V_x, V_P]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))).
fof(ah4s_predu_u_sets_EXTENSION, axiom, ![TV_u_27a]: ![V_t, V_s]: (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),V_t) <=> ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))).
fof(ah4s_lists_listu_u_Axiom, axiom, ![TV_u_27b,TV_u_27a]: ![V_f1, V_f0]: ?[V_fn]: (s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(TV_u_27b,V_f0) & ![V_a0, V_a1]: s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn),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,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,TV_u_27b))),V_f1),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1))),s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_lists_list(TV_u_27a),V_a1))))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f0)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sums_ISL0u_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: p(s(t_bool,h4s_sums_isl(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_x))))))).
fof(ah4s_predu_u_sets_INu_u_INSERT, axiom, ![TV_u_27a]: ![V_y, V_x, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f0))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f0) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_ABSu_u_SIMP, axiom, ![TV_u_27b,TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,V_t1) = s(TV_u_27a,V_t1)).
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_predu_u_sets_NOTu_u_INu_u_EMPTY, axiom, ![TV_u_27a]: ![V_x]: ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))))).
fof(ah4s_sums_ISL0u_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_y]: ~ (p(s(t_bool,h4s_sums_isl(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_y)))))))).
fof(ah4s_predu_u_sets_ITSETu_u_tupledu_u_primitiveu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_4]: (![V_ITSETu_u_tupled, V_f, V_s, V_b]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)) & s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b)))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),V_ITSETu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27b,V_b))) = s(TV_u_27b,h4s_combins_i(s(TV_u_27b,h4s_bools_cond(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27b,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27b,V_b),s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),V_ITSETu_u_tupled),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27a,h4s_predu_u_sets_choice(s(t_fun(TV_u_27a,t_bool),V_s))))),s(TV_u_27b,V_b))))))))),s(TV_u_27b,h4s_bools_arb)))))) => ![V_uu_3]: (![V_ITSETu_u_tupled, V_f, V_s]: s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),V_ITSETu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b)))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),V_ITSETu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_fun(TV_u_27a,t_bool),V_s))) => ![V_uu_2]: (![V_ITSETu_u_tupled, V_f, V_a]: s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),V_ITSETu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),V_a))) = s(TV_u_27b,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),V_a),s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27b,TV_u_27b)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),V_ITSETu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))))) => ![V_uu_1]: (![V_f, V_ITSETu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),V_ITSETu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),V_ITSETu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))) => ![V_uu_0]: (![V_f, V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_bool)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_bool)),t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_bool)),V_R)))) & ![V_b, V_s]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & ~ (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_bool)),V_R),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_rest(s(t_fun(TV_u_27a,t_bool),V_s))),s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27a,h4s_predu_u_sets_choice(s(t_fun(TV_u_27a,t_bool),V_s))))),s(TV_u_27b,V_b))))))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27b,V_b))))))))) => ![V_f]: s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),h4s_predu_u_sets_itsetu_u_tupled(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))) = s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_bool)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),t_bool)),t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))))),s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),TV_u_27b),TV_u_27b))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))))))))))).
fof(ah4s_richu_u_lists_SPLITPu_u_AUXu_u_defu_c1, axiom, ![TV_u_27a]: ![V_t, V_h, V_acc, V_P]: s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_richu_u_lists_splitpu_u_aux(s(t_h4s_lists_list(TV_u_27a),V_acc),s(t_fun(TV_u_27a,t_bool),V_P),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))))) = s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_bools_cond(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_h))),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),V_acc),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))))),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_richu_u_lists_splitpu_u_aux(s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_acc),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),h4s_lists_nil))))),s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t)))))).
fof(ah4s_numpairs_nlistrecu_u_tupledu_u_primitiveu_u_def, axiom, ![TV_u_27a]: ![V_uu_6]: (![V_nlistrecu_u_tupled, V_n, V_f, V_l]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_l) = s(t_h4s_nums_num,h4s_nums_0)) & s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_fun(t_h4s_nums_num,TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_fun(t_h4s_nums_num,TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_fun(t_h4s_nums_num,TV_u_27a)))),V_uu_6),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),V_nlistrecu_u_tupled))),s(TV_u_27a,V_n))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),V_f))),s(t_h4s_nums_num,V_l))) = s(TV_u_27a,h4s_combins_i(s(TV_u_27a,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27a,V_n),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),V_f),s(t_h4s_nums_num,h4s_numpairs_nfst(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_l),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_numpairs_nsnd(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_l),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(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),V_nlistrecu_u_tupled),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),h4s_pairs_u_2c(s(TV_u_27a,V_n),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num),h4s_pairs_u_2c(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),V_f),s(t_h4s_nums_num,h4s_numpairs_nsnd(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_l),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))))))))))))))) => ![V_uu_5]: (![V_nlistrecu_u_tupled, V_n, V_f]: s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_fun(t_h4s_nums_num,TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_fun(t_h4s_nums_num,TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_fun(t_h4s_nums_num,TV_u_27a)))),V_uu_5),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),V_nlistrecu_u_tupled))),s(TV_u_27a,V_n))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),V_f))) = s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_fun(t_h4s_nums_num,TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_fun(t_h4s_nums_num,TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_fun(t_h4s_nums_num,TV_u_27a)))),V_uu_6),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),V_nlistrecu_u_tupled))),s(TV_u_27a,V_n))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),V_f))) => ![V_uu_4]: (![V_nlistrecu_u_tupled, V_n, V_v1]: s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num),TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num),TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num),TV_u_27a))),V_uu_4),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),V_nlistrecu_u_tupled))),s(TV_u_27a,V_n))),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num),V_v1))) = s(TV_u_27a,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num),V_v1),s(t_fun(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_fun(t_h4s_nums_num,TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_fun(t_h4s_nums_num,TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_fun(t_h4s_nums_num,TV_u_27a)))),V_uu_5),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),V_nlistrecu_u_tupled))),s(TV_u_27a,V_n))))) => ![V_uu_3]: (![V_nlistrecu_u_tupled, V_n]: s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num),TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num),TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num),TV_u_27a))),V_uu_3),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),V_nlistrecu_u_tupled))),s(TV_u_27a,V_n))) = s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num),TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num),TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num),TV_u_27a))),V_uu_4),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),V_nlistrecu_u_tupled))),s(TV_u_27a,V_n))) => ![V_uu_2]: (![V_nlistrecu_u_tupled, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a)),V_uu_2),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),V_nlistrecu_u_tupled))),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),V_a))) = s(TV_u_27a,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),V_a),s(t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num),TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num),TV_u_27a))),V_uu_3),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),V_nlistrecu_u_tupled))))) => ![V_uu_1]: (![V_nlistrecu_u_tupled]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a)),V_uu_1),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),V_nlistrecu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a)),V_uu_2),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),V_nlistrecu_u_tupled))) => ![V_uu_0]: (![V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),t_bool)),V_R)))) & ![V_f, V_n, V_l]: (~ (s(t_h4s_nums_num,V_l) = s(t_h4s_nums_num,h4s_nums_0)) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),t_bool),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),t_bool)),V_R),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),h4s_pairs_u_2c(s(TV_u_27a,V_n),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num),h4s_pairs_u_2c(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),V_f),s(t_h4s_nums_num,h4s_numpairs_nsnd(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_l),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_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),h4s_pairs_u_2c(s(TV_u_27a,V_n),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num),h4s_pairs_u_2c(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),V_f),s(t_h4s_nums_num,V_l))))))))))) => s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),h4s_numpairs_nlistrecu_u_tupled) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a),t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a))),t_h4s_nums_num)),TV_u_27a)),V_uu_1))))))))))).
fof(ah4s_richu_u_lists_SPLITP0u_c1, axiom, ![TV_u_27a]: ![V_x, V_l, V_P]: s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_richu_u_lists_splitp(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_bools_cond(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l))))),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),t_h4s_lists_list(TV_u_27a)),h4s_pairs_fst),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_richu_u_lists_splitp(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l))))))),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),t_h4s_lists_list(TV_u_27a)),h4s_pairs_snd),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_richu_u_lists_splitp(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))))))))).
fof(ah4s_arithmetics_DIVMODu_u_DEF, axiom, ![V_uu_4]: (![V_f, V_a, V_m, V_n, V_q]: s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f))),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_q))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_a),s(t_h4s_nums_num,V_q))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_q))))),s(t_h4s_nums_num,V_n))))))) => ![V_uu_3]: (![V_f, V_a, V_m, V_n]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) & s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f))),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_bools_cond(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_a),s(t_h4s_nums_num,V_m))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_bools_let(s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f))),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),h4s_arithmetics_findq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_u_2c(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_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))))))))))) => ![V_uu_2]: (![V_f, V_a, V_m]: s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f))),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_m))) = s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f))),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_m))) => ![V_uu_1]: (![V_f, V_a]: s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f))),s(t_h4s_nums_num,V_a))) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_uncurry(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f))),s(t_h4s_nums_num,V_a))))) => ![V_uu_0]: (![V_f]: s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f))) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_uncurry(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_f))))) => s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_divmod) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool)),h4s_primu_u_recs_measure(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),h4s_combins_o(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_nums_num),h4s_pairs_fst),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_snd))))),s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),V_uu_0))))))))).
fof(ah4s_arithmetics_findqu_u_def, axiom, ![V_uu_7]: (![V_f, V_a, V_m, V_d]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)))),V_uu_7),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),V_f))),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_d))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_d))),s(t_h4s_nums_num,V_a),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),V_f),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_a))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_d))))))))) => ![V_uu_6]: (![V_f, V_a, V_m, V_n]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) & s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)))),V_uu_6),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),V_f))),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,V_v),s(t_h4s_nums_num,V_a),s(t_h4s_nums_num,h4s_bools_let(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)))),V_uu_7),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),V_f))),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n)))))))) => ![V_uu_5]: (![V_f, V_a, V_m]: s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)))),V_uu_5),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),V_f))),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_m))) = s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)))),V_uu_6),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),V_f))),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_m))) => ![V_uu_4]: (![V_f, V_a]: s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_nums_num)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_nums_num))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),V_f))),s(t_h4s_nums_num,V_a))) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_nums_num),h4s_pairs_uncurry(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)))),V_uu_5),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),V_f))),s(t_h4s_nums_num,V_a))))) => ![V_uu_3]: (![V_f]: s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num)),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),V_f))) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),h4s_pairs_uncurry(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_nums_num)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_nums_num))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),V_f))))) => ![V_uu_2]: (![V_m, V_n]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),V_uu_2),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) => ![V_uu_1]: (![V_m]: s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),V_uu_1),s(t_h4s_nums_num,V_m))) = s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),V_uu_2),s(t_h4s_nums_num,V_m))) => ![V_uu_0]: (![V_a]: s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_nums_num)),V_uu_0),s(t_h4s_nums_num,V_a))) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_nums_num),h4s_pairs_uncurry(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),V_uu_1))) => s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),h4s_arithmetics_findq) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool)),h4s_primu_u_recs_measure(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),h4s_pairs_uncurry(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_h4s_nums_num)),V_uu_0))))),s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num)),V_uu_3)))))))))))).
fof(ah4s_arithmetics_DIVMODu_u_THM, axiom, ![V_uu_0]: (![V_a, V_m, V_n, V_q]: s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))))),V_uu_0),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_q))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_divmod),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_a),s(t_h4s_nums_num,V_q))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_q))))),s(t_h4s_nums_num,V_n))))))) => ![V_n, V_m, V_a]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) & s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_divmod),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,V_a),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))))) = s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_bools_cond(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_a),s(t_h4s_nums_num,V_m))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_bools_let(s(t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num))))),V_uu_0),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),h4s_arithmetics_findq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_u_2c(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_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))))))))))))).
fof(ah4s_arithmetics_findqu_u_thm, axiom, ![V_uu_0]: (![V_a, V_m, V_d]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num))),V_uu_0),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_d))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_d))),s(t_h4s_nums_num,V_a),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),h4s_arithmetics_findq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_a))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_d))))))))) => ![V_n, V_m, V_a]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) & s(t_h4s_nums_num,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),h4s_arithmetics_findq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,V_a),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,V_v),s(t_h4s_nums_num,V_a),s(t_h4s_nums_num,h4s_bools_let(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num))),V_uu_0),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,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_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n)))))))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f0)))).
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_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_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f0))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => p(s(t_bool,f0))))).
fof(ah4s_arithmetics_NOTu_u_LESS, axiom, ![V_n, V_m]: (~ (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f0))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
fof(ah4s_arithmetics_NOTu_u_ZEROu_u_LTu_u_ZERO, axiom, ![V_n]: (~ (s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) <=> p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_ONE, axiom, 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_nums_suc(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_primu_u_recs_measureu_u_thm, axiom, ![TV_u_27a]: ![V_y, V_x, V_f]: 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_primu_u_recs_measure(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f))),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))) = s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(TV_u_27a,V_x))),s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(TV_u_27a,V_y)))))).
fof(ah4s_primu_u_recs_WFu_u_measure, axiom, ![TV_u_27a]: ![V_m]: p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_primu_u_recs_measure(s(t_fun(TV_u_27a,t_h4s_nums_num),V_m))))))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_relations_RESTRICTu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f, V_R, V_xi_]: 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_x))),s(TV_u_27a,V_xi_))) = s(TV_u_27b,h4s_bools_cond(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_xi_))),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_))),s(TV_u_27b,h4s_bools_arb)))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f0))))).
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_combins_ou_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_x, V_g, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27c,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,V_x)))))).
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_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f0))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
fof(ah4s_bools_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_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_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_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,f0))))).
fof(ah4s_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
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_bools_boolu_u_caseu_u_thmu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_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_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_LETu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_arithmetics_findqu_u_equ_u_0, axiom, ![V_n, V_m, V_a]: (s(t_h4s_nums_num,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_nums_num),h4s_arithmetics_findq),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,V_a),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))))) = s(t_h4s_nums_num,h4s_nums_0) <=> s(t_h4s_nums_num,V_a) = s(t_h4s_nums_num,h4s_nums_0))).
fof(ah4s_arithmetics_SUBu_u_LEFTu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))))) <=> (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,V_n)))) | p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))))))).
fof(ah4s_arithmetics_MULTu_u_EQu_u_0, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0) <=> (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0) | s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f0) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_arithmetics_LESSu_u_LESSu_u_EQu_u_TRANS, axiom, ![V_p, V_n, V_m]: ((p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_ADDu_u_SYM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_LESSu_u_ADDu_u_NONZERO, axiom, ![V_n, V_m]: (~ (s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) => p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,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)))))))).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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_nums_NOTu_u_SUC, axiom, ![V_n]: ~ (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0))).
fof(ah4s_bools_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_ORu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,f0))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f0)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_arithmetics_SUBu_u_LEFTu_u_LESS, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_SUBu_u_LEFTu_u_ADD, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_SUBu_u_RIGHTu_u_LESS, axiom, ![V_p, V_n, V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p)))) <=> (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))))) & p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_p))))))).
fof(ah4s_arithmetics_LTu_u_MULTu_u_RCANCEL, axiom, ![V_p, V_n, V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_n)))))) <=> (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))))).
fof(ah4s_arithmetics_LEu_u_MULTu_u_RCANCEL, axiom, ![V_p, V_n, V_m]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_n)))))) <=> (s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0) | 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_LESSu_u_EQu_u_LESSu_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_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_NOTu_u_LESSu_u_EQUAL, axiom, ![V_n, V_m]: (~ (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_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_LTu_u_ADDu_u_LCANCEL, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_n))))) = s(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_MONO, axiom, ![V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))) = 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_MULTu_u_CLAUSESu_c2, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_MULTu_u_LEFTu_u_1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_NOTu_u_SUCu_u_LESSu_u_EQu_u_0, axiom, ![V_n]: ~ (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0)))))).
fof(ah4s_arithmetics_LESSu_u_MONOu_u_EQ, axiom, ![V_n, V_m]: s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))).
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_arithmetics_TWO, axiom, s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))).
fof(ah4s_combins_LETu_u_FORALLu_u_ELIM, axiom, ![TV_u_27a]: ![V_v, V_f]: s(t_bool,h4s_bools_let(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_v))) = s(t_bool,d_forall(s(t_fun(TV_u_27a,t_bool),h4s_combins_s(s(t_fun(TV_u_27a,t_fun(t_bool,t_bool)),h4s_combins_o(s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_imply),s(t_fun(TV_u_27a,t_bool),h4s_combins_o(s(t_fun(t_bool,t_bool),h4s_markers_abbrev),s(t_fun(TV_u_27a,t_bool),h4s_combins_c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),d_equals),s(TV_u_27a,V_v))))))),s(t_fun(TV_u_27a,t_bool),V_f)))))).
fof(ah4s_combins_Cu_u_ABSu_u_L, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_uu_0]: (![V_f, V_x]: s(t_fun(TV_u_27c,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27b)),t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27b))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27b)),V_f))),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27c,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27b)),V_f),s(TV_u_27a,V_x))) => ![V_y, V_f, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_c(s(t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27b)),t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27b))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27b)),V_f))),s(TV_u_27c,V_y))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27b)),V_f),s(TV_u_27a,V_x))),s(TV_u_27c,V_y))))).
fof(ah4s_combins_GENu_u_LETu_u_RAND, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_v, V_f, V_P]: s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_P),s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,V_v))))) = s(TV_u_27a,h4s_bools_let(s(t_fun(TV_u_27c,TV_u_27a),h4s_combins_o(s(t_fun(TV_u_27b,TV_u_27a),V_P),s(t_fun(TV_u_27c,TV_u_27b),V_f))),s(TV_u_27c,V_v)))).
fof(ah4s_combins_Cu_u_THM, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),h4s_combins_c(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27b,V_x))),s(TV_u_27a,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_y))),s(TV_u_27b,V_x)))).
fof(ah4s_combins_GENu_u_LETu_u_RATOR, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_x, V_v, V_f]: s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),h4s_bools_let(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27b,V_v))),s(TV_u_27c,V_x))) = s(TV_u_27a,h4s_bools_let(s(t_fun(TV_u_27b,TV_u_27a),h4s_combins_c(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27c,V_x))),s(TV_u_27b,V_v)))).
fof(ah4s_combins_Su_u_ABSu_u_R, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_uu_0]: (![V_g, V_x]: s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27c)),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x))) => ![V_g, V_f, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_s(s(t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27b)),V_f),s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27c)),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_g))))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27b)),V_f),s(TV_u_27a,V_x))),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x))))))).
fof(ah4s_combins_ou_u_ABSu_u_R, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_uu_0]: (![V_g, V_x]: s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27c)),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x))) => ![V_g, V_f, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27c)),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_g))))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x))))))).
fof(ch4s_finiteu_u_maps_FAPPLYu_u_FUPDATEu_u_THM, conjecture, ![TV_u_27b,TV_u_27a]: ![V_x, V_f, V_b, V_a]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_a)) & s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))))),s(TV_u_27a,V_x))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27b,V_b),s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))))).
