%   ORIGINAL: h4/state__transformer/FOR__def
% 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/FALSITY: !t. F ==> t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% 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/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% 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/COND__RATOR: !x g f b. h4/bool/COND b f g x = h4/bool/COND b (f x) (g x)
% Assm: h4/bool/COND__RAND: !y x f b. f (h4/bool/COND b x y) = h4/bool/COND b (f x) (f y)
% Assm: h4/bool/COND__EXPAND: !t2 t1 b. h4/bool/COND b t1 t2 <=> (~b \/ t1) /\ (b \/ 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/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/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/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% 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/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% 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/TWO: h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO) = h4/num/SUC (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% 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/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__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/MULT__CLAUSES_c3: !m. h4/arithmetic/_2A m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = m
% Assm: h4/arithmetic/LEFT__ADD__DISTRIB: !p n m. h4/arithmetic/_2A p (h4/arithmetic/_2B m n) = h4/arithmetic/_2B (h4/arithmetic/_2A p m) (h4/arithmetic/_2A p n)
% Assm: h4/arithmetic/MULT__ASSOC: !p n m. h4/arithmetic/_2A m (h4/arithmetic/_2A n p) = h4/arithmetic/_2A (h4/arithmetic/_2A m n) p
% 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/TIMES2: !n. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n = h4/arithmetic/_2B n n
% Assm: h4/arithmetic/ADD__MONO__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm: h4/arithmetic/NOT__LEQ: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC n) m
% 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/MULT__LESS__EQ__SUC: !p n m. h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/arithmetic/_2A (h4/num/SUC p) m) (h4/arithmetic/_2A (h4/num/SUC p) n)
% Assm: h4/arithmetic/SUB__RIGHT__ADD: !p n m. h4/arithmetic/_2B (h4/arithmetic/_2D m n) p = h4/bool/COND (h4/arithmetic/_3C_3D m n) p (h4/arithmetic/_2D (h4/arithmetic/_2B m p) n)
% Assm: h4/arithmetic/SUB__LEFT__SUB: !p n m. h4/arithmetic/_2D m (h4/arithmetic/_2D n p) = h4/bool/COND (h4/arithmetic/_3C_3D n p) m (h4/arithmetic/_2D (h4/arithmetic/_2B m p) n)
% Assm: h4/arithmetic/SUB__RIGHT__SUB: !p n m. h4/arithmetic/_2D (h4/arithmetic/_2D m n) p = h4/arithmetic/_2D m (h4/arithmetic/_2B n p)
% 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__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/numeral/numeral__suc_c0: h4/num/SUC h4/arithmetic/ZERO = h4/arithmetic/BIT1 h4/arithmetic/ZERO
% 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_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_c0: !n. h4/numeral/iZ (h4/arithmetic/_2B h4/arithmetic/ZERO n) = 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__add_c3: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT2 m)) = h4/arithmetic/BIT1 (h4/num/SUC (h4/arithmetic/_2B n m))
% Assm: h4/numeral/numeral__add_c4: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT2 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT1 (h4/num/SUC (h4/arithmetic/_2B n m))
% Assm: h4/numeral/numeral__add_c6: !n. h4/num/SUC (h4/arithmetic/_2B h4/arithmetic/ZERO n) = h4/num/SUC n
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/numeral/numeral__lte_c2: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT2 n) h4/arithmetic/ZERO <=> F
% Assm: h4/state__transformer/FOR__primitive__def: h4/state__transformer/FOR = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!a j i. ~(i = j) ==> R (h4/pair/_2C (h4/bool/COND (h4/prim__rec/_3C i j) (h4/arithmetic/_2B i (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/_2D i (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/pair/_2C j a)) (h4/pair/_2C i (h4/pair/_2C j a))))) (\FOR a_27. h4/pair/pair__CASE a_27 (\i v1. h4/pair/pair__CASE v1 (\j a. h4/combin/I (h4/bool/COND (i = j) (a i) (h4/state__transformer/BIND (a i) (\u. FOR (h4/pair/_2C (h4/bool/COND (h4/prim__rec/_3C i j) (h4/arithmetic/_2B i (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/_2D i (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/pair/_2C j a))))))))
% Goal: !j i a. h4/state__transformer/FOR (h4/pair/_2C i (h4/pair/_2C j a)) = h4/bool/COND (i = j) (a i) (h4/state__transformer/BIND (a i) (\u. h4/state__transformer/FOR (h4/pair/_2C (h4/bool/COND (h4/prim__rec/_3C i j) (h4/arithmetic/_2B i (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/_2D i (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/pair/_2C j a))))
%   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_FALSITY]: !t. F ==> t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% 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_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_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_CONDu_u_RATOR]: !x g f b. happ (h4/bool/COND b f g) x = h4/bool/COND b (happ f x) (happ g x)
% Assm [h4s_bools_CONDu_u_RAND]: !y x f b. happ f (h4/bool/COND b x y) = h4/bool/COND b (happ f x) (happ f y)
% Assm [h4s_bools_CONDu_u_EXPAND]: !t2 t1 b. h4/bool/COND b t1 t2 <=> (~b \/ t1) /\ (b \/ 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_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_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_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% 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_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% 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_TWO]: h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO) = h4/num/SUC (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% 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_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_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c3]: !m. h4/arithmetic/_2A m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) = m
% Assm [h4s_arithmetics_LEFTu_u_ADDu_u_DISTRIB]: !p n m. h4/arithmetic/_2A p (h4/arithmetic/_2B m n) = h4/arithmetic/_2B (h4/arithmetic/_2A p m) (h4/arithmetic/_2A p n)
% Assm [h4s_arithmetics_MULTu_u_ASSOC]: !p n m. h4/arithmetic/_2A m (h4/arithmetic/_2A n p) = h4/arithmetic/_2A (h4/arithmetic/_2A m n) p
% 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_TIMES2]: !n. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n = h4/arithmetic/_2B n n
% Assm [h4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm [h4s_arithmetics_NOTu_u_LEQ]: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC n) m
% 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_MULTu_u_LESSu_u_EQu_u_SUC]: !p n m. h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/arithmetic/_2A (h4/num/SUC p) m) (h4/arithmetic/_2A (h4/num/SUC p) n)
% Assm [h4s_arithmetics_SUBu_u_RIGHTu_u_ADD]: !p n m. h4/arithmetic/_2B (h4/arithmetic/_2D m n) p = h4/bool/COND (h4/arithmetic/_3C_3D m n) p (h4/arithmetic/_2D (h4/arithmetic/_2B m p) n)
% Assm [h4s_arithmetics_SUBu_u_LEFTu_u_SUB]: !p n m. h4/arithmetic/_2D m (h4/arithmetic/_2D n p) = h4/bool/COND (h4/arithmetic/_3C_3D n p) m (h4/arithmetic/_2D (h4/arithmetic/_2B m p) n)
% Assm [h4s_arithmetics_SUBu_u_RIGHTu_u_SUB]: !p n m. h4/arithmetic/_2D (h4/arithmetic/_2D m n) p = h4/arithmetic/_2D m (h4/arithmetic/_2B n p)
% Assm [h4s_arithmetics_SUBu_u_LEFTu_u_LESS]: !p n m. happ (happ h4/prim__rec/_3C m) (h4/arithmetic/_2D n p) <=> happ (happ h4/prim__rec/_3C (h4/arithmetic/_2B m p)) 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_numerals_numeralu_u_sucu_c0]: h4/num/SUC h4/arithmetic/ZERO = h4/arithmetic/BIT1 h4/arithmetic/ZERO
% 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_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_c0]: !n. h4/numeral/iZ (h4/arithmetic/_2B h4/arithmetic/ZERO n) = 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_addu_c3]: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT2 m)) = h4/arithmetic/BIT1 (h4/num/SUC (h4/arithmetic/_2B n m))
% Assm [h4s_numerals_numeralu_u_addu_c4]: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT2 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT1 (h4/num/SUC (h4/arithmetic/_2B n m))
% Assm [h4s_numerals_numeralu_u_addu_c6]: !n. h4/num/SUC (h4/arithmetic/_2B h4/arithmetic/ZERO n) = h4/num/SUC n
% 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_lteu_c2]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT2 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_stateu_u_transformers_FORu_u_primitiveu_u_def]: !_7. (!FOR i j a u. happ (happ (happ (happ (happ _7 FOR) i) j) a) u = happ FOR (h4/pair/_2C (h4/bool/COND (happ (happ h4/prim__rec/_3C i) j) (h4/arithmetic/_2B i (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/_2D i (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/pair/_2C j a))) ==> (!_6. (!FOR i j a. ?v. (v <=> i = j) /\ happ (happ (happ (happ _6 FOR) i) j) a = h4/combin/I (h4/bool/COND v (happ a i) (h4/state__transformer/BIND (happ a i) (happ (happ (happ (happ _7 FOR) i) j) a)))) ==> (!_5. (!FOR i j. happ (happ (happ _5 FOR) i) j = happ (happ (happ _6 FOR) i) j) ==> (!_4. (!FOR i v1. happ (happ (happ _4 FOR) i) v1 = h4/pair/pair__CASE v1 (happ (happ _5 FOR) i)) ==> (!_3. (!FOR i. happ (happ _3 FOR) i = happ (happ _4 FOR) i) ==> (!_2. (!FOR a_27. happ (happ _2 FOR) a_27 = h4/pair/pair__CASE a_27 (happ _3 FOR)) ==> (!_1. (!FOR. happ _1 FOR = happ _2 FOR) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R /\ (!a j i. ~(i = j) ==> happ (happ R (h4/pair/_2C (h4/bool/COND (happ (happ h4/prim__rec/_3C i) j) (h4/arithmetic/_2B i (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/_2D i (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/pair/_2C j a))) (h4/pair/_2C i (h4/pair/_2C j a)))) ==> h4/state__transformer/FOR = h4/relation/WFREC (h4/min/_40 _0) _1)))))))
% Goal: !_0. (!i j a u. happ (happ (happ (happ _0 i) j) a) u = happ h4/state__transformer/FOR (h4/pair/_2C (h4/bool/COND (happ (happ h4/prim__rec/_3C i) j) (h4/arithmetic/_2B i (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/_2D i (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/pair/_2C j a))) ==> (!j i a. ?v. (v <=> i = j) /\ happ h4/state__transformer/FOR (h4/pair/_2C i (h4/pair/_2C j a)) = h4/bool/COND v (happ a i) (h4/state__transformer/BIND (happ a i) (happ (happ (happ _0 i) j) a)))
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_Q351499,TV_Q351495]: ![V_f, V_g]: (![V_x]: s(TV_Q351495,happ(s(t_fun(TV_Q351499,TV_Q351495),V_f),s(TV_Q351499,V_x))) = s(TV_Q351495,happ(s(t_fun(TV_Q351499,TV_Q351495),V_g),s(TV_Q351499,V_x))) => s(t_fun(TV_Q351499,TV_Q351495),V_f) = s(t_fun(TV_Q351499,TV_Q351495),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_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_CONJu_u_ASSOC, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) & (p(s(t_bool,V_t2)) & p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) & p(s(t_bool,V_t3))))).
fof(ah4s_bools_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_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_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_CONDu_u_RATOR, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_g, V_f, V_b]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_bools_cond(s(t_bool,V_b),s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_b),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)))))).
fof(ah4s_bools_CONDu_u_RAND, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x, V_f, V_b]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,h4s_bools_cond(s(t_bool,V_b),s(TV_u_27a,V_x),s(TV_u_27a,V_y))))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_b),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_y)))))).
fof(ah4s_bools_CONDu_u_EXPAND, axiom, ![V_t2, V_t1, V_b]: (p(s(t_bool,h4s_bools_cond(s(t_bool,V_b),s(t_bool,V_t1),s(t_bool,V_t2)))) <=> ((~ (p(s(t_bool,V_b))) | p(s(t_bool,V_t1))) & (p(s(t_bool,V_b)) | 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_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_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_ABSu_u_PAIRu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: ?[V_q, V_r]: s(t_h4s_pairs_prod(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,V_q),s(TV_u_27b,V_r)))).
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_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27b,V_y)).
fof(ah4s_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_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_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_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_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_CLAUSESu_c3, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_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)).
fof(ah4s_arithmetics_LEFTu_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,V_p),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,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_MULTu_u_ASSOC, axiom, ![V_p, 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,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(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,V_p)))).
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_TIMES2, axiom, ![V_n]: 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))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_NOTu_u_LEQ, 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_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_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_MULTu_u_LESSu_u_EQu_u_SUC, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(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_arithmetics_u_2a(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_SUBu_u_RIGHTu_u_ADD, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(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))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,h4s_arithmetics_u_3cu_3d(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_2d(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_SUB, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(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_p))),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_arithmetics_SUBu_u_RIGHTu_u_SUB, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(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))) = s(t_h4s_nums_num,h4s_arithmetics_u_2d(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)))))).
fof(ah4s_arithmetics_SUBu_u_LEFTu_u_LESS, axiom, ![V_p, V_n, V_m]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))),s(t_h4s_nums_num,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_numerals_numeralu_u_sucu_c0, axiom, s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))).
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_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_c0, 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,h4s_arithmetics_zero),s(t_h4s_nums_num,V_n))))) = 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_addu_c3, 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_bit2(s(t_h4s_nums_num,V_m))))))) = s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_nums_suc(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_addu_c4, 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_bit2(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_bit1(s(t_h4s_nums_num,h4s_nums_suc(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_addu_c6, axiom, ![V_n]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))).
fof(ah4s_numerals_numeralu_u_lteu_c1, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
fof(ah4s_numerals_numeralu_u_lteu_c2, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
fof(ah4s_stateu_u_transformers_FORu_u_primitiveu_u_def, axiom, ![TV_u_27state]: ![V_uu_7]: (![V_FOR, V_i, V_j, V_a, V_u]: s(t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)),happ(s(t_fun(t_h4s_ones_one,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_ones_one,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_ones_one,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_ones_one,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_ones_one,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))))),V_uu_7),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_FOR))),s(t_h4s_nums_num,V_i))),s(t_h4s_nums_num,V_j))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_a))),s(t_h4s_ones_one,V_u))) = s(t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_FOR),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_bools_cond(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_i))),s(t_h4s_nums_num,V_j))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_i),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_i),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_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),h4s_pairs_u_2c(s(t_h4s_nums_num,V_j),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_a))))))) => ![V_uu_6]: (![V_FOR, V_i, V_j, V_a]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_i) = s(t_h4s_nums_num,V_j)) & s(t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))))),V_uu_6),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_FOR))),s(t_h4s_nums_num,V_i))),s(t_h4s_nums_num,V_j))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_a))) = s(t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)),h4s_combins_i(s(t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)),h4s_bools_cond(s(t_bool,V_v),s(t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_a),s(t_h4s_nums_num,V_i))),s(t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)),h4s_stateu_u_transformers_bind(s(t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_a),s(t_h4s_nums_num,V_i))),s(t_fun(t_h4s_ones_one,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_ones_one,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_ones_one,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_ones_one,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_ones_one,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))))),V_uu_7),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_FOR))),s(t_h4s_nums_num,V_i))),s(t_h4s_nums_num,V_j))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_a)))))))))) => ![V_uu_5]: (![V_FOR, V_i, V_j]: s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))))),V_uu_5),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_FOR))),s(t_h4s_nums_num,V_i))),s(t_h4s_nums_num,V_j))) = s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))))),V_uu_6),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_FOR))),s(t_h4s_nums_num,V_i))),s(t_h4s_nums_num,V_j))) => ![V_uu_4]: (![V_FOR, V_i, V_v1]: s(t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_FOR))),s(t_h4s_nums_num,V_i))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),V_v1))) = s(t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),V_v1),s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))))),V_uu_5),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_FOR))),s(t_h4s_nums_num,V_i))))) => ![V_uu_3]: (![V_FOR, V_i]: s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_FOR))),s(t_h4s_nums_num,V_i))) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_FOR))),s(t_h4s_nums_num,V_i))) => ![V_uu_2]: (![V_FOR, V_au_27]: s(t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_FOR))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),V_au_27))) = s(t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),V_au_27),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_FOR))))) => ![V_uu_1]: (![V_FOR]: s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_FOR))) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_FOR))) => ![V_uu_0]: (![V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_bool)),V_R)))) & ![V_a, V_j, V_i]: (~ (s(t_h4s_nums_num,V_i) = s(t_h4s_nums_num,V_j)) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_bool)),V_R),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_bools_cond(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_i))),s(t_h4s_nums_num,V_j))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_i),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_i),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_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),h4s_pairs_u_2c(s(t_h4s_nums_num,V_j),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_a))))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),h4s_pairs_u_2c(s(t_h4s_nums_num,V_i),s(t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),h4s_pairs_u_2c(s(t_h4s_nums_num,V_j),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_a))))))))))) => s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),h4s_stateu_u_transformers_for) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_bool)),t_bool),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_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),V_uu_1)))))))))))).
fof(ch4s_stateu_u_transformers_FORu_u_def, conjecture, ![TV_u_27state]: ![V_uu_0]: (![V_i, V_j, V_a, V_u]: s(t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)),happ(s(t_fun(t_h4s_ones_one,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_ones_one,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_ones_one,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_ones_one,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))))),V_uu_0),s(t_h4s_nums_num,V_i))),s(t_h4s_nums_num,V_j))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_a))),s(t_h4s_ones_one,V_u))) = s(t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),h4s_stateu_u_transformers_for),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_bools_cond(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_i))),s(t_h4s_nums_num,V_j))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_i),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_i),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_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),h4s_pairs_u_2c(s(t_h4s_nums_num,V_j),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_a))))))) => ![V_j, V_i, V_a]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_i) = s(t_h4s_nums_num,V_j)) & s(t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),h4s_stateu_u_transformers_for),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),h4s_pairs_u_2c(s(t_h4s_nums_num,V_i),s(t_h4s_pairs_prod(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),h4s_pairs_u_2c(s(t_h4s_nums_num,V_j),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_a))))))) = s(t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)),h4s_bools_cond(s(t_bool,V_v),s(t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_a),s(t_h4s_nums_num,V_i))),s(t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)),h4s_stateu_u_transformers_bind(s(t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_a),s(t_h4s_nums_num,V_i))),s(t_fun(t_h4s_ones_one,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_ones_one,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_ones_one,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),t_fun(t_h4s_ones_one,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state)))))),V_uu_0),s(t_h4s_nums_num,V_i))),s(t_h4s_nums_num,V_j))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27state,t_h4s_pairs_prod(t_h4s_ones_one,TV_u_27state))),V_a)))))))))).
