%   ORIGINAL: h4/toto/qk__numeralOrd_c2
% 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/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% 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/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/bool/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/EQ__EXPAND: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% 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/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/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/relation/WF__INDUCTION__THM: !R. h4/relation/WF R ==> (!P. (!x. (!y. R y x ==> P y) ==> P x) ==> (!x. P x))
% Assm: h4/relation/inv__image__def: !f R. h4/relation/inv__image R f = (\x y. R (f x) (f y))
% Assm: h4/relation/WF__inv__image: !f R. h4/relation/WF R ==> h4/relation/WF (h4/relation/inv__image R f)
% Assm: h4/relation/RESTRICT__LEMMA: !z y f R. R y z ==> h4/relation/RESTRICT f R z y = f y
% 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/prim__rec/WF__LESS: h4/relation/WF h4/prim__rec/_3C
% Assm: h4/prim__rec/measure__def: h4/prim__rec/measure = h4/relation/inv__image h4/prim__rec/_3C
% Assm: h4/prim__rec/WF__measure: !m. h4/relation/WF (h4/prim__rec/measure m)
% Assm: h4/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% Assm: h4/arithmetic/ALT__ZERO: h4/arithmetic/ZERO = h4/num/0
% Assm: h4/arithmetic/BIT10: !n. h4/arithmetic/BIT1 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (h4/num/SUC h4/num/0))
% Assm: h4/arithmetic/BIT20: !n. h4/arithmetic/BIT2 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (h4/num/SUC (h4/num/SUC h4/num/0)))
% Assm: h4/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/ADD__ASSOC: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm: h4/arithmetic/num__CASES: !m. m = h4/num/0 \/ (?n. m = h4/num/SUC n)
% Assm: h4/arithmetic/LESS__EQ: !n m. h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm: h4/arithmetic/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n 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/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/MULT__COMM: !n m. h4/arithmetic/_2A m n = h4/arithmetic/_2A n m
% Assm: h4/arithmetic/RIGHT__ADD__DISTRIB: !p n m. h4/arithmetic/_2A (h4/arithmetic/_2B m n) p = h4/arithmetic/_2B (h4/arithmetic/_2A m p) (h4/arithmetic/_2A n p)
% Assm: h4/arithmetic/ADD__EQ__0: !n m. h4/arithmetic/_2B m n = h4/num/0 <=> m = h4/num/0 /\ n = h4/num/0
% 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/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/arithmetic/ADD__SUB: !c a. h4/arithmetic/_2D (h4/arithmetic/_2B a c) c = a
% 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/EQ__LESS__EQ: !n m. m = n <=> h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/ADD__MONO__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm: h4/arithmetic/NOT__NUM__EQ: !n m. ~(m = n) <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n \/ h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm: h4/arithmetic/SUC__ONE__ADD: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm: h4/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/DIV2__def: !n. h4/arithmetic/DIV2 n = h4/arithmetic/DIV n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))
% Assm: h4/arithmetic/DIV__LESS__EQ: !n. h4/prim__rec/_3C h4/num/0 n ==> (!k. h4/arithmetic/_3C_3D (h4/arithmetic/DIV k n) k)
% Assm: h4/arithmetic/MULT__DIV: !q n. h4/prim__rec/_3C h4/num/0 n ==> h4/arithmetic/DIV (h4/arithmetic/_2A q n) n = q
% Assm: h4/numeral/numeral__suc_c1: !n. h4/num/SUC (h4/arithmetic/BIT1 n) = h4/arithmetic/BIT2 n
% Assm: h4/numeral/numeral__distrib_c1: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm: h4/numeral/numeral__distrib_c2: !n m. h4/arithmetic/_2B (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm: h4/numeral/numeral__distrib_c13: h4/num/SUC h4/num/0 = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm: h4/numeral/numeral__distrib_c14: !n. h4/num/SUC (h4/arithmetic/NUMERAL n) = h4/arithmetic/NUMERAL (h4/num/SUC n)
% Assm: h4/numeral/numeral__distrib_c17: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__distrib_c21: !n. h4/prim__rec/_3C h4/num/0 (h4/arithmetic/NUMERAL n) <=> h4/prim__rec/_3C h4/arithmetic/ZERO n
% Assm: h4/numeral/numeral__distrib_c27: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__add_c1: !n. h4/numeral/iZ (h4/arithmetic/_2B n h4/arithmetic/ZERO) = n
% Assm: h4/numeral/numeral__add_c2: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT2 (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm: h4/numeral/numeral__eq_c1: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Assm: h4/numeral/numeral__eq_c3: !n. h4/arithmetic/BIT2 n = h4/arithmetic/ZERO <=> F
% Assm: h4/numeral/numeral__lt_c0: !n. h4/prim__rec/_3C h4/arithmetic/ZERO (h4/arithmetic/BIT1 n) <=> T
% Assm: h4/numeral/numeral__lt_c1: !n. h4/prim__rec/_3C h4/arithmetic/ZERO (h4/arithmetic/BIT2 n) <=> T
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/numeral/numeral__evenodd_c4: !n. ~h4/arithmetic/ODD (h4/arithmetic/BIT2 n)
% Assm: h4/numeral/numeral__evenodd_c5: !n. h4/arithmetic/ODD (h4/arithmetic/BIT1 n)
% Assm: h4/toto/num__dt__case__def_c0: !v f1 f. h4/toto/num__dt__CASE h4/toto/zer v f f1 = v
% Assm: h4/toto/num__dt__case__def_c1: !v f1 f a. h4/toto/num__dt__CASE (h4/toto/bit1 a) v f f1 = f a
% Assm: h4/toto/num__dt__case__def_c2: !v f1 f a. h4/toto/num__dt__CASE (h4/toto/bit2 a) v f f1 = f1 a
% Assm: h4/toto/num__dt__size__def_c1: !a. h4/toto/num__dt__size (h4/toto/bit1 a) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/toto/num__dt__size a)
% Assm: h4/toto/num__dt__size__def_c2: !a. h4/toto/num__dt__size (h4/toto/bit2 a) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/toto/num__dt__size a)
% Assm: h4/toto/num__to__dt__primitive: h4/toto/num__to__dt = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!n. ~(n = h4/num/0) /\ h4/arithmetic/ODD n ==> R (h4/arithmetic/DIV2 (h4/arithmetic/_2D n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) n) /\ (!n. ~(n = h4/num/0) /\ ~h4/arithmetic/ODD n ==> R (h4/arithmetic/DIV2 (h4/arithmetic/_2D n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) n))) (\num__to__dt n. h4/combin/I (h4/bool/COND (n = h4/num/0) h4/toto/zer (h4/bool/COND (h4/arithmetic/ODD n) (h4/toto/bit1 (num__to__dt (h4/arithmetic/DIV2 (h4/arithmetic/_2D n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))))) (h4/toto/bit2 (num__to__dt (h4/arithmetic/DIV2 (h4/arithmetic/_2D n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))))))))
% Assm: h4/toto/num__dtOrd__tupled__primitive: h4/toto/num__dtOrd__tupled = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!y x. R (h4/pair/_2C x y) (h4/pair/_2C (h4/toto/bit1 x) (h4/toto/bit1 y))) /\ (!y x. R (h4/pair/_2C x y) (h4/pair/_2C (h4/toto/bit2 x) (h4/toto/bit2 y))))) (\num__dtOrd__tupled a. h4/pair/pair__CASE a (\v v1. h4/toto/num__dt__CASE v (h4/toto/num__dt__CASE v1 (h4/combin/I h4/toto/EQUAL) (\x. h4/combin/I h4/toto/LESS) (\x_27. h4/combin/I h4/toto/LESS)) (\x_27_27. h4/toto/num__dt__CASE v1 (h4/combin/I h4/toto/GREATER) (\y_27_27. h4/combin/I (num__dtOrd__tupled (h4/pair/_2C x_27_27 y_27_27))) (\y. h4/combin/I h4/toto/LESS)) (\x_27_27_27. h4/toto/num__dt__CASE v1 (h4/combin/I h4/toto/GREATER) (\y_27. h4/combin/I h4/toto/GREATER) (\y_27_27_27. h4/combin/I (num__dtOrd__tupled (h4/pair/_2C x_27_27_27 y_27_27_27))))))
% Assm: h4/toto/num__dtOrd__curried: !x1 x. h4/toto/num__dtOrd x x1 = h4/toto/num__dtOrd__tupled (h4/pair/_2C x x1)
% Assm: h4/toto/qk__numOrd__def: !n m. h4/toto/qk__numOrd m n = h4/toto/num__dtOrd (h4/toto/num__to__dt m) (h4/toto/num__to__dt n)
% Goal: !y. h4/toto/qk__numOrd h4/arithmetic/ZERO (h4/arithmetic/BIT2 y) = h4/toto/LESS
%   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_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% 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_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_bools_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_EQu_u_EXPAND]: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% 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_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_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_relations_WFu_u_INDUCTIONu_u_THM]: !R. h4/relation/WF R ==> (!P. (!x. (!y. happ (happ R y) x ==> happ P y) ==> happ P x) ==> (!x. happ P x))
% Assm [h4s_relations_invu_u_imageu_u_def]: !f R x x'. happ (happ (happ (h4/relation/inv__image R) f) x) x' <=> happ (happ R (happ f x)) (happ f x')
% Assm [h4s_relations_WFu_u_invu_u_image]: !f R. h4/relation/WF R ==> h4/relation/WF (happ (h4/relation/inv__image R) f)
% Assm [h4s_relations_RESTRICTu_u_LEMMA]: !z y f R. happ (happ R y) z ==> happ (h4/relation/RESTRICT f R z) y = happ f y
% Assm [h4s_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_primu_u_recs_WFu_u_LESS]: h4/relation/WF h4/prim__rec/_3C
% Assm [h4s_primu_u_recs_measureu_u_def]: h4/prim__rec/measure = h4/relation/inv__image h4/prim__rec/_3C
% Assm [h4s_primu_u_recs_WFu_u_measure]: !m. h4/relation/WF (happ h4/prim__rec/measure m)
% Assm [h4s_pairs_PAIR]: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_pairs_pairu_u_caseu_u_thm]: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = happ (happ f x) y
% Assm [h4s_arithmetics_ALTu_u_ZERO]: h4/arithmetic/ZERO = h4/num/0
% Assm [h4s_arithmetics_BIT10]: !n. h4/arithmetic/BIT1 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (h4/num/SUC h4/num/0))
% Assm [h4s_arithmetics_BIT20]: !n. h4/arithmetic/BIT2 n = h4/arithmetic/_2B n (h4/arithmetic/_2B n (h4/num/SUC (h4/num/SUC h4/num/0)))
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_ADDu_u_ASSOC]: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm [h4s_arithmetics_numu_u_CASES]: !m. m = h4/num/0 \/ (?n. m = h4/num/SUC n)
% Assm [h4s_arithmetics_LESSu_u_EQ]: !n m. happ (happ h4/prim__rec/_3C m) n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm [h4s_arithmetics_NOTu_u_LESS]: !n m. ~happ (happ h4/prim__rec/_3C m) n <=> h4/arithmetic/_3C_3D n 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_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_MULTu_u_COMM]: !n m. h4/arithmetic/_2A m n = h4/arithmetic/_2A n m
% Assm [h4s_arithmetics_RIGHTu_u_ADDu_u_DISTRIB]: !p n m. h4/arithmetic/_2A (h4/arithmetic/_2B m n) p = h4/arithmetic/_2B (h4/arithmetic/_2A m p) (h4/arithmetic/_2A n p)
% Assm [h4s_arithmetics_ADDu_u_EQu_u_0]: !n m. h4/arithmetic/_2B m n = h4/num/0 <=> m = h4/num/0 /\ n = h4/num/0
% 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_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_arithmetics_ADDu_u_SUB]: !c a. h4/arithmetic/_2D (h4/arithmetic/_2B a c) c = a
% Assm [h4s_arithmetics_LESSu_u_EQu_u_LESSu_u_TRANS]: !p n m. h4/arithmetic/_3C_3D m n /\ happ (happ h4/prim__rec/_3C n) p ==> happ (happ h4/prim__rec/_3C m) p
% Assm [h4s_arithmetics_EQu_u_LESSu_u_EQ]: !n m. m = n <=> h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm [h4s_arithmetics_NOTu_u_NUMu_u_EQ]: !n m. ~(m = n) <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n \/ h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm [h4s_arithmetics_SUCu_u_ONEu_u_ADD]: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm [h4s_arithmetics_SUBu_u_RIGHTu_u_LESS]: !p n m. happ (happ h4/prim__rec/_3C (h4/arithmetic/_2D m n)) p <=> happ (happ h4/prim__rec/_3C m) (h4/arithmetic/_2B n p) /\ happ (happ h4/prim__rec/_3C h4/num/0) p
% Assm [h4s_arithmetics_DIV2u_u_def]: !n. h4/arithmetic/DIV2 n = h4/arithmetic/DIV n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))
% Assm [h4s_arithmetics_DIVu_u_LESSu_u_EQ]: !n. happ (happ h4/prim__rec/_3C h4/num/0) n ==> (!k. h4/arithmetic/_3C_3D (h4/arithmetic/DIV k n) k)
% Assm [h4s_arithmetics_MULTu_u_DIV]: !q n. happ (happ h4/prim__rec/_3C h4/num/0) n ==> h4/arithmetic/DIV (h4/arithmetic/_2A q n) n = q
% Assm [h4s_numerals_numeralu_u_sucu_c1]: !n. h4/num/SUC (h4/arithmetic/BIT1 n) = h4/arithmetic/BIT2 n
% Assm [h4s_numerals_numeralu_u_distribu_c1]: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm [h4s_numerals_numeralu_u_distribu_c2]: !n m. h4/arithmetic/_2B (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm [h4s_numerals_numeralu_u_distribu_c13]: h4/num/SUC h4/num/0 = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm [h4s_numerals_numeralu_u_distribu_c14]: !n. h4/num/SUC (h4/arithmetic/NUMERAL n) = h4/arithmetic/NUMERAL (h4/num/SUC n)
% Assm [h4s_numerals_numeralu_u_distribu_c17]: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_distribu_c21]: !n. happ (happ h4/prim__rec/_3C h4/num/0) (h4/arithmetic/NUMERAL n) <=> happ (happ h4/prim__rec/_3C h4/arithmetic/ZERO) n
% Assm [h4s_numerals_numeralu_u_distribu_c27]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_addu_c1]: !n. h4/numeral/iZ (h4/arithmetic/_2B n h4/arithmetic/ZERO) = n
% Assm [h4s_numerals_numeralu_u_addu_c2]: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT2 (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm [h4s_numerals_numeralu_u_equ_c1]: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Assm [h4s_numerals_numeralu_u_equ_c3]: !n. h4/arithmetic/BIT2 n = h4/arithmetic/ZERO <=> F
% Assm [h4s_numerals_numeralu_u_ltu_c0]: !n. happ (happ h4/prim__rec/_3C h4/arithmetic/ZERO) (h4/arithmetic/BIT1 n) <=> T
% Assm [h4s_numerals_numeralu_u_ltu_c1]: !n. happ (happ h4/prim__rec/_3C h4/arithmetic/ZERO) (h4/arithmetic/BIT2 n) <=> T
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_numerals_numeralu_u_evenoddu_c4]: !n. ~h4/arithmetic/ODD (h4/arithmetic/BIT2 n)
% Assm [h4s_numerals_numeralu_u_evenoddu_c5]: !n. h4/arithmetic/ODD (h4/arithmetic/BIT1 n)
% Assm [h4s_totos_numu_u_dtu_u_caseu_u_defu_c0]: !v f1 f. h4/toto/num__dt__CASE h4/toto/zer v f f1 = v
% Assm [h4s_totos_numu_u_dtu_u_caseu_u_defu_c1]: !v f1 f a. h4/toto/num__dt__CASE (h4/toto/bit1 a) v f f1 = happ f a
% Assm [h4s_totos_numu_u_dtu_u_caseu_u_defu_c2]: !v f1 f a. h4/toto/num__dt__CASE (h4/toto/bit2 a) v f f1 = happ f1 a
% Assm [h4s_totos_numu_u_dtu_u_sizeu_u_defu_c1]: !a. h4/toto/num__dt__size (h4/toto/bit1 a) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/toto/num__dt__size a)
% Assm [h4s_totos_numu_u_dtu_u_sizeu_u_defu_c2]: !a. h4/toto/num__dt__size (h4/toto/bit2 a) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/toto/num__dt__size a)
% Assm [h4s_totos_numu_u_tou_u_dtu_u_primitive]: !_2. (!num__to__dt n. ?v. (v <=> n = h4/num/0) /\ happ (happ _2 num__to__dt) n = h4/combin/I (h4/bool/COND v h4/toto/zer (h4/bool/COND (h4/arithmetic/ODD n) (h4/toto/bit1 (happ num__to__dt (h4/arithmetic/DIV2 (h4/arithmetic/_2D n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))))) (h4/toto/bit2 (happ num__to__dt (h4/arithmetic/DIV2 (h4/arithmetic/_2D n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))))))))) ==> (!_1. (!num__to__dt. happ _1 num__to__dt = happ _2 num__to__dt) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R /\ (!n. ~(n = h4/num/0) /\ h4/arithmetic/ODD n ==> happ (happ R (h4/arithmetic/DIV2 (h4/arithmetic/_2D n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))))) n) /\ (!n. ~(n = h4/num/0) /\ ~h4/arithmetic/ODD n ==> happ (happ R (h4/arithmetic/DIV2 (h4/arithmetic/_2D n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))))) n)) ==> h4/toto/num__to__dt = h4/relation/WFREC (h4/min/_40 _0) _1))
% Assm [h4s_totos_numu_u_dtOrdu_u_tupledu_u_primitive]: !_9. (!y_27. happ _9 y_27 = h4/combin/I h4/toto/GREATER) ==> (!_8. (!v1 _7 num__dtOrd__tupled x_27_27_27. happ (happ (happ (happ _8 v1) _7) num__dtOrd__tupled) x_27_27_27 = h4/toto/num__dt__CASE v1 (h4/combin/I h4/toto/GREATER) _9 (happ (happ _7 num__dtOrd__tupled) x_27_27_27)) ==> (!_7. (!num__dtOrd__tupled x_27_27 y_27_27. happ (happ (happ _7 num__dtOrd__tupled) x_27_27) y_27_27 = h4/combin/I (happ num__dtOrd__tupled (h4/pair/_2C x_27_27 y_27_27))) ==> (!_6. (!v1 num__dtOrd__tupled _5 x_27_27. happ (happ (happ (happ _6 v1) num__dtOrd__tupled) _5) x_27_27 = h4/toto/num__dt__CASE v1 (h4/combin/I h4/toto/GREATER) (happ (happ _7 num__dtOrd__tupled) x_27_27) _5) ==> (!_5. (!x. happ _5 x = h4/combin/I h4/toto/LESS) ==> (!_4. (!v num__dtOrd__tupled v1. happ (happ (happ _4 v) num__dtOrd__tupled) v1 = h4/toto/num__dt__CASE v (h4/toto/num__dt__CASE v1 (h4/combin/I h4/toto/EQUAL) _5 _5) (happ (happ (happ _6 v1) num__dtOrd__tupled) _5) (happ (happ (happ _8 v1) _7) num__dtOrd__tupled)) ==> (!_3. (!num__dtOrd__tupled v. happ (happ _3 num__dtOrd__tupled) v = happ (happ _4 v) num__dtOrd__tupled) ==> (!_2. (!num__dtOrd__tupled a. happ (happ _2 num__dtOrd__tupled) a = h4/pair/pair__CASE a (happ _3 num__dtOrd__tupled)) ==> (!_1. (!num__dtOrd__tupled. happ _1 num__dtOrd__tupled = happ _2 num__dtOrd__tupled) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R /\ (!y x. happ (happ R (h4/pair/_2C x y)) (h4/pair/_2C (h4/toto/bit1 x) (h4/toto/bit1 y))) /\ (!y x. happ (happ R (h4/pair/_2C x y)) (h4/pair/_2C (h4/toto/bit2 x) (h4/toto/bit2 y)))) ==> h4/toto/num__dtOrd__tupled = h4/relation/WFREC (h4/min/_40 _0) _1)))))))))
% Assm [h4s_totos_numu_u_dtOrdu_u_curried]: !x1 x. h4/toto/num__dtOrd x x1 = happ h4/toto/num__dtOrd__tupled (h4/pair/_2C x x1)
% Assm [h4s_totos_qku_u_numOrdu_u_def]: !n m. h4/toto/qk__numOrd m n = h4/toto/num__dtOrd (happ h4/toto/num__to__dt m) (happ h4/toto/num__to__dt n)
% Goal: !y. h4/toto/qk__numOrd h4/arithmetic/ZERO (h4/arithmetic/BIT2 y) = h4/toto/LESS
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q252485,TV_Q252481]: ![V_f, V_g]: (![V_x]: s(TV_Q252481,happ(s(t_fun(TV_Q252485,TV_Q252481),V_f),s(TV_Q252485,V_x))) = s(TV_Q252481,happ(s(t_fun(TV_Q252485,TV_Q252481),V_g),s(TV_Q252485,V_x))) => s(t_fun(TV_Q252485,TV_Q252481),V_f) = s(t_fun(TV_Q252485,TV_Q252481),V_g))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_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_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_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_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c0, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) & p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) | ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_IMPu_u_DISJu_u_THM, 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_IMPu_u_Fu_u_EQu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_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_EQu_u_EXPAND, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) | (~ (p(s(t_bool,V_t1))) & ~ (p(s(t_bool,V_t2))))))).
fof(ah4s_bools_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_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_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_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_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_relations_WFu_u_INDUCTIONu_u_THM, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_P]: (![V_x]: (![V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) => ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_relations_invu_u_imageu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_x, V_xi_]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_invu_u_image(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_x))),s(TV_u_27a,V_xi_))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_xi_)))))).
fof(ah4s_relations_WFu_u_invu_u_image, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R)))) => p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_invu_u_image(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R))),s(t_fun(TV_u_27a,TV_u_27b),V_f)))))))).
fof(ah4s_relations_RESTRICTu_u_LEMMA, axiom, ![TV_u_27b,TV_u_27a]: ![V_z, V_y, V_f, V_R]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_z)))) => s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_z))),s(TV_u_27a,V_y))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))))).
fof(ah4s_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_primu_u_recs_WFu_u_LESS, axiom, p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c))))).
fof(ah4s_primu_u_recs_measureu_u_def, axiom, ![TV_u_27a]: s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_primu_u_recs_measure) = s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_invu_u_image(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c)))).
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)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),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_pairs_PAIR, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x)).
fof(ah4s_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_pairs_pairu_u_caseu_u_thm, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_y, V_x, V_f]: s(TV_u_27a,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27b,V_x),s(TV_u_27c,V_y))),s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f))) = s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27b,V_x))),s(TV_u_27c,V_y)))).
fof(ah4s_arithmetics_ALTu_u_ZERO, axiom, s(t_h4s_nums_num,h4s_arithmetics_zero) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_arithmetics_BIT10, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0)))))))).
fof(ah4s_arithmetics_BIT20, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0)))))))))).
fof(ah4s_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_ADDu_u_SYM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_ADDu_u_ASSOC, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_numu_u_CASES, axiom, ![V_m]: (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0) | ?[V_n]: s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_LESSu_u_EQ, axiom, ![V_n, V_m]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_NOTu_u_LESS, axiom, ![V_n, V_m]: (~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_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_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_COMM, 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_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_RIGHTu_u_ADDu_u_DISTRIB, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(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))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_ADDu_u_EQu_u_0, 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_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_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_ZEROu_u_LESSu_u_EQ, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_ADDu_u_SUB, axiom, ![V_c, V_a]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_a),s(t_h4s_nums_num,V_c))),s(t_h4s_nums_num,V_c))) = s(t_h4s_nums_num,V_a)).
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,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_EQu_u_LESSu_u_EQ, axiom, ![V_n, V_m]: (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_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))))))).
fof(ah4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_NOTu_u_NUMu_u_EQ, axiom, ![V_n, V_m]: (~ (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,h4s_nums_suc(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,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m))))))).
fof(ah4s_arithmetics_SUCu_u_ONEu_u_ADD, axiom, ![V_n]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_SUBu_u_RIGHTu_u_LESS, axiom, ![V_p, V_n, V_m]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))))) & p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_nums_num,V_p))))))).
fof(ah4s_arithmetics_DIV2u_u_def, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_div2(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))).
fof(ah4s_arithmetics_DIVu_u_LESSu_u_EQ, axiom, ![V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_nums_num,V_n)))) => ![V_k]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,V_k),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_k)))))).
fof(ah4s_arithmetics_MULTu_u_DIV, axiom, ![V_q, V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_nums_num,V_n)))) => s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_q),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_q))).
fof(ah4s_numerals_numeralu_u_sucu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n)))).
fof(ah4s_numerals_numeralu_u_distribu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_numerals_numeralu_u_distribu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_numerals_numeralu_u_distribu_c13, axiom, s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))).
fof(ah4s_numerals_numeralu_u_distribu_c14, axiom, ![V_n]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))).
fof(ah4s_numerals_numeralu_u_distribu_c17, axiom, ![V_n]: (s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_arithmetics_zero))).
fof(ah4s_numerals_numeralu_u_distribu_c21, axiom, ![V_n]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))))) = s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_arithmetics_zero))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_numerals_numeralu_u_distribu_c27, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ah4s_numerals_numeralu_u_addu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero))))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_numerals_numeralu_u_addu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_m))))))) = s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_numerals_numeralu_u_equ_c1, axiom, ![V_n]: (s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_zero) <=> p(s(t_bool,f)))).
fof(ah4s_numerals_numeralu_u_equ_c3, axiom, ![V_n]: (s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_zero) <=> p(s(t_bool,f)))).
fof(ah4s_numerals_numeralu_u_ltu_c0, axiom, ![V_n]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_arithmetics_zero))),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))) = s(t_bool,t)).
fof(ah4s_numerals_numeralu_u_ltu_c1, axiom, ![V_n]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_arithmetics_zero))),s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))))) = s(t_bool,t)).
fof(ah4s_numerals_numeralu_u_lteu_c1, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
fof(ah4s_numerals_numeralu_u_evenoddu_c4, axiom, ![V_n]: ~ (p(s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n)))))))).
fof(ah4s_numerals_numeralu_u_evenoddu_c5, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))))).
fof(ah4s_totos_numu_u_dtu_u_caseu_u_defu_c0, axiom, ![TV_u_27a]: ![V_v, V_f1, V_f]: s(TV_u_27a,h4s_totos_numu_u_dtu_u_case(s(t_h4s_totos_numu_u_dt,h4s_totos_zer),s(TV_u_27a,V_v),s(t_fun(t_h4s_totos_numu_u_dt,TV_u_27a),V_f),s(t_fun(t_h4s_totos_numu_u_dt,TV_u_27a),V_f1))) = s(TV_u_27a,V_v)).
fof(ah4s_totos_numu_u_dtu_u_caseu_u_defu_c1, axiom, ![TV_u_27a]: ![V_v, V_f1, V_f, V_a]: s(TV_u_27a,h4s_totos_numu_u_dtu_u_case(s(t_h4s_totos_numu_u_dt,h4s_totos_bit1(s(t_h4s_totos_numu_u_dt,V_a))),s(TV_u_27a,V_v),s(t_fun(t_h4s_totos_numu_u_dt,TV_u_27a),V_f),s(t_fun(t_h4s_totos_numu_u_dt,TV_u_27a),V_f1))) = s(TV_u_27a,happ(s(t_fun(t_h4s_totos_numu_u_dt,TV_u_27a),V_f),s(t_h4s_totos_numu_u_dt,V_a)))).
fof(ah4s_totos_numu_u_dtu_u_caseu_u_defu_c2, axiom, ![TV_u_27a]: ![V_v, V_f1, V_f, V_a]: s(TV_u_27a,h4s_totos_numu_u_dtu_u_case(s(t_h4s_totos_numu_u_dt,h4s_totos_bit2(s(t_h4s_totos_numu_u_dt,V_a))),s(TV_u_27a,V_v),s(t_fun(t_h4s_totos_numu_u_dt,TV_u_27a),V_f),s(t_fun(t_h4s_totos_numu_u_dt,TV_u_27a),V_f1))) = s(TV_u_27a,happ(s(t_fun(t_h4s_totos_numu_u_dt,TV_u_27a),V_f1),s(t_h4s_totos_numu_u_dt,V_a)))).
fof(ah4s_totos_numu_u_dtu_u_sizeu_u_defu_c1, axiom, ![V_a]: s(t_h4s_nums_num,h4s_totos_numu_u_dtu_u_size(s(t_h4s_totos_numu_u_dt,h4s_totos_bit1(s(t_h4s_totos_numu_u_dt,V_a))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,h4s_totos_numu_u_dtu_u_size(s(t_h4s_totos_numu_u_dt,V_a)))))).
fof(ah4s_totos_numu_u_dtu_u_sizeu_u_defu_c2, axiom, ![V_a]: s(t_h4s_nums_num,h4s_totos_numu_u_dtu_u_size(s(t_h4s_totos_numu_u_dt,h4s_totos_bit2(s(t_h4s_totos_numu_u_dt,V_a))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,h4s_totos_numu_u_dtu_u_size(s(t_h4s_totos_numu_u_dt,V_a)))))).
fof(ah4s_totos_numu_u_tou_u_dtu_u_primitive, axiom, ![V_uu_2]: (![V_numu_u_tou_u_dt, 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_totos_numu_u_dt,happ(s(t_fun(t_h4s_nums_num,t_h4s_totos_numu_u_dt),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_totos_numu_u_dt),t_fun(t_h4s_nums_num,t_h4s_totos_numu_u_dt)),V_uu_2),s(t_fun(t_h4s_nums_num,t_h4s_totos_numu_u_dt),V_numu_u_tou_u_dt))),s(t_h4s_nums_num,V_n))) = s(t_h4s_totos_numu_u_dt,h4s_combins_i(s(t_h4s_totos_numu_u_dt,h4s_bools_cond(s(t_bool,V_v),s(t_h4s_totos_numu_u_dt,h4s_totos_zer),s(t_h4s_totos_numu_u_dt,h4s_bools_cond(s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,V_n))),s(t_h4s_totos_numu_u_dt,h4s_totos_bit1(s(t_h4s_totos_numu_u_dt,happ(s(t_fun(t_h4s_nums_num,t_h4s_totos_numu_u_dt),V_numu_u_tou_u_dt),s(t_h4s_nums_num,h4s_arithmetics_div2(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))))))),s(t_h4s_totos_numu_u_dt,h4s_totos_bit2(s(t_h4s_totos_numu_u_dt,happ(s(t_fun(t_h4s_nums_num,t_h4s_totos_numu_u_dt),V_numu_u_tou_u_dt),s(t_h4s_nums_num,h4s_arithmetics_div2(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))))))))))))) => ![V_uu_1]: (![V_numu_u_tou_u_dt]: s(t_fun(t_h4s_nums_num,t_h4s_totos_numu_u_dt),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_totos_numu_u_dt),t_fun(t_h4s_nums_num,t_h4s_totos_numu_u_dt)),V_uu_1),s(t_fun(t_h4s_nums_num,t_h4s_totos_numu_u_dt),V_numu_u_tou_u_dt))) = s(t_fun(t_h4s_nums_num,t_h4s_totos_numu_u_dt),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_totos_numu_u_dt),t_fun(t_h4s_nums_num,t_h4s_totos_numu_u_dt)),V_uu_2),s(t_fun(t_h4s_nums_num,t_h4s_totos_numu_u_dt),V_numu_u_tou_u_dt))) => ![V_uu_0]: (![V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_R)))) & (![V_n]: ((~ (s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) & p(s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,V_n))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_R),s(t_h4s_nums_num,h4s_arithmetics_div2(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))))),s(t_h4s_nums_num,V_n))))) & ![V_n]: ((~ (s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) & ~ (p(s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,V_n)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_R),s(t_h4s_nums_num,h4s_arithmetics_div2(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))))),s(t_h4s_nums_num,V_n)))))))) => s(t_fun(t_h4s_nums_num,t_h4s_totos_numu_u_dt),h4s_totos_numu_u_tou_u_dt) = s(t_fun(t_h4s_nums_num,t_h4s_totos_numu_u_dt),h4s_relations_wfrec(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_nums_num,t_h4s_totos_numu_u_dt),t_fun(t_h4s_nums_num,t_h4s_totos_numu_u_dt)),V_uu_1))))))).
fof(ah4s_totos_numu_u_dtOrdu_u_tupledu_u_primitive, axiom, ![V_uu_9]: (![V_yu_27]: s(t_h4s_totos_cpn,happ(s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn),V_uu_9),s(t_h4s_totos_numu_u_dt,V_yu_27))) = s(t_h4s_totos_cpn,h4s_combins_i(s(t_h4s_totos_cpn,h4s_totos_greater))) => ![V_uu_8]: (![V_v1, V_uu_7, V_numu_u_dtOrdu_u_tupled, V_xu_27u_27u_27]: s(t_h4s_totos_cpn,happ(s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn)),happ(s(t_fun(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_h4s_totos_numu_u_dt,t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn))),t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn))),happ(s(t_fun(t_h4s_totos_numu_u_dt,t_fun(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_h4s_totos_numu_u_dt,t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn))),t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn)))),V_uu_8),s(t_h4s_totos_numu_u_dt,V_v1))),s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_h4s_totos_numu_u_dt,t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn))),V_uu_7))),s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),V_numu_u_dtOrdu_u_tupled))),s(t_h4s_totos_numu_u_dt,V_xu_27u_27u_27))) = s(t_h4s_totos_cpn,h4s_totos_numu_u_dtu_u_case(s(t_h4s_totos_numu_u_dt,V_v1),s(t_h4s_totos_cpn,h4s_combins_i(s(t_h4s_totos_cpn,h4s_totos_greater))),s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn),V_uu_9),s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn),happ(s(t_fun(t_h4s_totos_numu_u_dt,t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_h4s_totos_numu_u_dt,t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn))),V_uu_7),s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),V_numu_u_dtOrdu_u_tupled))),s(t_h4s_totos_numu_u_dt,V_xu_27u_27u_27))))) => ![V_uu_7]: (![V_numu_u_dtOrdu_u_tupled, V_xu_27u_27, V_yu_27u_27]: s(t_h4s_totos_cpn,happ(s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn),happ(s(t_fun(t_h4s_totos_numu_u_dt,t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_h4s_totos_numu_u_dt,t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn))),V_uu_7),s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),V_numu_u_dtOrdu_u_tupled))),s(t_h4s_totos_numu_u_dt,V_xu_27u_27))),s(t_h4s_totos_numu_u_dt,V_yu_27u_27))) = s(t_h4s_totos_cpn,h4s_combins_i(s(t_h4s_totos_cpn,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),V_numu_u_dtOrdu_u_tupled),s(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),h4s_pairs_u_2c(s(t_h4s_totos_numu_u_dt,V_xu_27u_27),s(t_h4s_totos_numu_u_dt,V_yu_27u_27))))))) => ![V_uu_6]: (![V_v1, V_numu_u_dtOrdu_u_tupled, V_uu_5, V_xu_27u_27]: s(t_h4s_totos_cpn,happ(s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn),happ(s(t_fun(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn),t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn),t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn))),happ(s(t_fun(t_h4s_totos_numu_u_dt,t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn),t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn)))),V_uu_6),s(t_h4s_totos_numu_u_dt,V_v1))),s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),V_numu_u_dtOrdu_u_tupled))),s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn),V_uu_5))),s(t_h4s_totos_numu_u_dt,V_xu_27u_27))) = s(t_h4s_totos_cpn,h4s_totos_numu_u_dtu_u_case(s(t_h4s_totos_numu_u_dt,V_v1),s(t_h4s_totos_cpn,h4s_combins_i(s(t_h4s_totos_cpn,h4s_totos_greater))),s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn),happ(s(t_fun(t_h4s_totos_numu_u_dt,t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_h4s_totos_numu_u_dt,t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn))),V_uu_7),s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),V_numu_u_dtOrdu_u_tupled))),s(t_h4s_totos_numu_u_dt,V_xu_27u_27))),s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn),V_uu_5))) => ![V_uu_5]: (![V_x]: s(t_h4s_totos_cpn,happ(s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn),V_uu_5),s(t_h4s_totos_numu_u_dt,V_x))) = s(t_h4s_totos_cpn,h4s_combins_i(s(t_h4s_totos_cpn,h4s_totos_less))) => ![V_uu_4]: (![V_v, V_numu_u_dtOrdu_u_tupled, V_v1]: s(t_h4s_totos_cpn,happ(s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn)),happ(s(t_fun(t_h4s_totos_numu_u_dt,t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn))),V_uu_4),s(t_h4s_totos_numu_u_dt,V_v))),s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),V_numu_u_dtOrdu_u_tupled))),s(t_h4s_totos_numu_u_dt,V_v1))) = s(t_h4s_totos_cpn,h4s_totos_numu_u_dtu_u_case(s(t_h4s_totos_numu_u_dt,V_v),s(t_h4s_totos_cpn,h4s_totos_numu_u_dtu_u_case(s(t_h4s_totos_numu_u_dt,V_v1),s(t_h4s_totos_cpn,h4s_combins_i(s(t_h4s_totos_cpn,h4s_totos_equal))),s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn),V_uu_5),s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn),V_uu_5))),s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn),happ(s(t_fun(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn),t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn),t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn))),happ(s(t_fun(t_h4s_totos_numu_u_dt,t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn),t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn)))),V_uu_6),s(t_h4s_totos_numu_u_dt,V_v1))),s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),V_numu_u_dtOrdu_u_tupled))),s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn),V_uu_5))),s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn)),happ(s(t_fun(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_h4s_totos_numu_u_dt,t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn))),t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn))),happ(s(t_fun(t_h4s_totos_numu_u_dt,t_fun(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_h4s_totos_numu_u_dt,t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn))),t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn)))),V_uu_8),s(t_h4s_totos_numu_u_dt,V_v1))),s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_h4s_totos_numu_u_dt,t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn))),V_uu_7))),s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),V_numu_u_dtOrdu_u_tupled))))) => ![V_uu_3]: (![V_numu_u_dtOrdu_u_tupled, V_v]: s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn),happ(s(t_fun(t_h4s_totos_numu_u_dt,t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_h4s_totos_numu_u_dt,t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),V_numu_u_dtOrdu_u_tupled))),s(t_h4s_totos_numu_u_dt,V_v))) = s(t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn)),happ(s(t_fun(t_h4s_totos_numu_u_dt,t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn))),V_uu_4),s(t_h4s_totos_numu_u_dt,V_v))),s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),V_numu_u_dtOrdu_u_tupled))) => ![V_uu_2]: (![V_numu_u_dtOrdu_u_tupled, V_a]: s(t_h4s_totos_cpn,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn)),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),V_numu_u_dtOrdu_u_tupled))),s(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),V_a))) = s(t_h4s_totos_cpn,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),V_a),s(t_fun(t_h4s_totos_numu_u_dt,t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_h4s_totos_numu_u_dt,t_fun(t_h4s_totos_numu_u_dt,t_h4s_totos_cpn))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),V_numu_u_dtOrdu_u_tupled))))) => ![V_uu_1]: (![V_numu_u_dtOrdu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn)),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),V_numu_u_dtOrdu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn)),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),V_numu_u_dtOrdu_u_tupled))) => ![V_uu_0]: (![V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_bool)),V_R)))) & (![V_y, V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_bool)),V_R),s(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),h4s_pairs_u_2c(s(t_h4s_totos_numu_u_dt,V_x),s(t_h4s_totos_numu_u_dt,V_y))))),s(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),h4s_pairs_u_2c(s(t_h4s_totos_numu_u_dt,h4s_totos_bit1(s(t_h4s_totos_numu_u_dt,V_x))),s(t_h4s_totos_numu_u_dt,h4s_totos_bit1(s(t_h4s_totos_numu_u_dt,V_y)))))))) & ![V_y, V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_bool)),V_R),s(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),h4s_pairs_u_2c(s(t_h4s_totos_numu_u_dt,V_x),s(t_h4s_totos_numu_u_dt,V_y))))),s(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),h4s_pairs_u_2c(s(t_h4s_totos_numu_u_dt,h4s_totos_bit2(s(t_h4s_totos_numu_u_dt,V_x))),s(t_h4s_totos_numu_u_dt,h4s_totos_bit2(s(t_h4s_totos_numu_u_dt,V_y))))))))))) => s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),h4s_totos_numu_u_dtordu_u_tupled) = s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn)),V_uu_1)))))))))))))).
fof(ah4s_totos_numu_u_dtOrdu_u_curried, axiom, ![V_x1, V_x]: s(t_h4s_totos_cpn,h4s_totos_numu_u_dtord(s(t_h4s_totos_numu_u_dt,V_x),s(t_h4s_totos_numu_u_dt,V_x1))) = s(t_h4s_totos_cpn,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),t_h4s_totos_cpn),h4s_totos_numu_u_dtordu_u_tupled),s(t_h4s_pairs_prod(t_h4s_totos_numu_u_dt,t_h4s_totos_numu_u_dt),h4s_pairs_u_2c(s(t_h4s_totos_numu_u_dt,V_x),s(t_h4s_totos_numu_u_dt,V_x1)))))).
fof(ah4s_totos_qku_u_numOrdu_u_def, axiom, ![V_n, V_m]: s(t_h4s_totos_cpn,h4s_totos_qku_u_numord(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_totos_cpn,h4s_totos_numu_u_dtord(s(t_h4s_totos_numu_u_dt,happ(s(t_fun(t_h4s_nums_num,t_h4s_totos_numu_u_dt),h4s_totos_numu_u_tou_u_dt),s(t_h4s_nums_num,V_m))),s(t_h4s_totos_numu_u_dt,happ(s(t_fun(t_h4s_nums_num,t_h4s_totos_numu_u_dt),h4s_totos_numu_u_tou_u_dt),s(t_h4s_nums_num,V_n)))))).
fof(ch4s_totos_qku_u_numeralOrdu_c2, conjecture, ![V_y]: s(t_h4s_totos_cpn,h4s_totos_qku_u_numord(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_y))))) = s(t_h4s_totos_cpn,h4s_totos_less)).
