%   ORIGINAL: h4/Omega/dark__implies__real
% 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/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/TRUTH: 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_c1: !t. t /\ T <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% 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/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/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/relation/inv__image__def: !f R. h4/relation/inv__image R f = (\x y. R (f x) (f y))
% Assm: h4/relation/WF__inv__image: !f R. h4/relation/WF R ==> h4/relation/WF (h4/relation/inv__image R f)
% Assm: h4/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/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/FORALL__PROD: !P. (!p. P p) <=> (!p__1 p__2. P (h4/pair/_2C p__1 p__2))
% 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/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__COMM: !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/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/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__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/numeral/numeral__distrib_c1: !n. h4/arithmetic/_2B n h4/num/0 = 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__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/list/list__case__def_c0: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm: h4/list/list__case__def_c1: !v f a1 a0. h4/list/list__CASE (h4/list/CONS a0 a1) v f = f a0 a1
% Assm: h4/list/list__size__def_c1: !f a1 a0. h4/list/list__size f (h4/list/CONS a0 a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (f a0) (h4/list/list__size f a1))
% Assm: h4/list/EVERY__DEF_c0: !P. h4/list/EVERY P h4/list/NIL <=> T
% Assm: h4/list/EVERY__DEF_c1: !t h P. h4/list/EVERY P (h4/list/CONS h t) <=> P h /\ h4/list/EVERY P t
% Assm: h4/list/list__induction: !P. P h4/list/NIL /\ (!t. P t ==> (!h. P (h4/list/CONS h t))) ==> (!l. P l)
% Assm: h4/Omega/fst__nzero__def: !x. h4/Omega/fst__nzero x <=> h4/prim__rec/_3C h4/num/0 (h4/pair/FST x)
% Assm: h4/Omega/real__shadow__def_c0: !lowers. h4/Omega/real__shadow h4/list/NIL lowers <=> T
% Assm: h4/Omega/real__shadow__def_c1: !upper ls lowers. h4/Omega/real__shadow (h4/list/CONS upper ls) lowers <=> h4/Omega/rshadow__row upper lowers /\ h4/Omega/real__shadow ls lowers
% Assm: h4/Omega/dark__shadow__tupled__primitive__def: h4/Omega/dark__shadow__tupled = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!L c lowers uppers. R (h4/pair/_2C uppers lowers) (h4/pair/_2C (h4/list/CONS (h4/pair/_2C c L) uppers) lowers)))) (\dark__shadow__tupled a. h4/pair/pair__CASE a (\v lowers. h4/list/list__CASE v (h4/combin/I T) (\v2 uppers. h4/pair/pair__CASE v2 (\c L. h4/combin/I (h4/Omega/dark__shadow__row c L lowers /\ dark__shadow__tupled (h4/pair/_2C uppers lowers))))))
% Assm: h4/Omega/dark__shadow__curried__def: !x1 x. h4/Omega/dark__shadow x x1 <=> h4/Omega/dark__shadow__tupled (h4/pair/_2C x x1)
% Assm: h4/Omega/darkrow__implies__realrow: !lowers c L. h4/prim__rec/_3C h4/num/0 c /\ h4/list/EVERY h4/Omega/fst__nzero lowers /\ h4/Omega/dark__shadow__row c L lowers ==> h4/Omega/rshadow__row (h4/pair/_2C c L) lowers
% Goal: !uppers lowers. h4/list/EVERY h4/Omega/fst__nzero uppers /\ h4/list/EVERY h4/Omega/fst__nzero lowers /\ h4/Omega/dark__shadow uppers lowers ==> h4/Omega/real__shadow uppers lowers
%   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_ETAu_u_AX]: !t x. happ t x = happ t x
% 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_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% 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_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_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_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_relations_invu_u_imageu_u_def]: !f R x x'. happ (happ (h4/relation/inv__image R f) x) x' <=> happ (happ R (happ f x)) (happ f x')
% Assm [h4s_relations_WFu_u_invu_u_image]: !f R. h4/relation/WF R ==> h4/relation/WF (h4/relation/inv__image R f)
% Assm [h4s_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_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_FORALLu_u_PROD]: !P. (!p. happ P p) <=> (!p__1 p__2. happ P (h4/pair/_2C p__1 p__2))
% 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_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_COMM]: !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_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_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_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_numerals_numeralu_u_distribu_c1]: !n. h4/arithmetic/_2B n h4/num/0 = 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_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_lists_listu_u_caseu_u_defu_c0]: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm [h4s_lists_listu_u_caseu_u_defu_c1]: !v f a1 a0. h4/list/list__CASE (h4/list/CONS a0 a1) v f = happ (happ f a0) a1
% Assm [h4s_lists_listu_u_sizeu_u_defu_c1]: !f a1 a0. h4/list/list__size f (h4/list/CONS a0 a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (happ f a0) (h4/list/list__size f a1))
% Assm [h4s_lists_EVERYu_u_DEFu_c0]: !P. h4/list/EVERY P h4/list/NIL <=> T
% Assm [h4s_lists_EVERYu_u_DEFu_c1]: !t h P. h4/list/EVERY P (h4/list/CONS h t) <=> happ P h /\ h4/list/EVERY P t
% Assm [h4s_lists_listu_u_induction]: !P. happ P h4/list/NIL /\ (!t. happ P t ==> (!h. happ P (h4/list/CONS h t))) ==> (!l. happ P l)
% Assm [h4s_Omegas_fstu_u_nzerou_u_def]: !x. happ h4/Omega/fst__nzero x <=> happ (happ h4/prim__rec/_3C h4/num/0) (h4/pair/FST x)
% Assm [h4s_Omegas_realu_u_shadowu_u_defu_c0]: !lowers. h4/Omega/real__shadow h4/list/NIL lowers <=> T
% Assm [h4s_Omegas_realu_u_shadowu_u_defu_c1]: !upper ls lowers. h4/Omega/real__shadow (h4/list/CONS upper ls) lowers <=> h4/Omega/rshadow__row upper lowers /\ h4/Omega/real__shadow ls lowers
% Assm [h4s_Omegas_darku_u_shadowu_u_tupledu_u_primitiveu_u_def]: !_8. (!c dark__shadow__tupled uppers lowers L. ?v. (v <=> h4/Omega/dark__shadow__row c L lowers /\ happ dark__shadow__tupled (h4/pair/_2C uppers lowers)) /\ (happ (happ (happ (happ (happ _8 c) dark__shadow__tupled) uppers) lowers) L <=> h4/combin/I v)) ==> (!_7. (!dark__shadow__tupled uppers lowers c. happ (happ (happ (happ _7 dark__shadow__tupled) uppers) lowers) c = happ (happ (happ (happ _8 c) dark__shadow__tupled) uppers) lowers) ==> (!_6. (!v2 dark__shadow__tupled lowers uppers. happ (happ (happ (happ _6 v2) dark__shadow__tupled) lowers) uppers <=> h4/pair/pair__CASE v2 (happ (happ (happ _7 dark__shadow__tupled) uppers) lowers)) ==> (!_5. (!dark__shadow__tupled lowers v2. happ (happ (happ _5 dark__shadow__tupled) lowers) v2 = happ (happ (happ _6 v2) dark__shadow__tupled) lowers) ==> (!_4. (!v dark__shadow__tupled lowers. happ (happ (happ _4 v) dark__shadow__tupled) lowers <=> h4/list/list__CASE v (h4/combin/I T) (happ (happ _5 dark__shadow__tupled) lowers)) ==> (!_3. (!dark__shadow__tupled v. happ (happ _3 dark__shadow__tupled) v = happ (happ _4 v) dark__shadow__tupled) ==> (!_2. (!dark__shadow__tupled a. happ (happ _2 dark__shadow__tupled) a <=> h4/pair/pair__CASE a (happ _3 dark__shadow__tupled)) ==> (!_1. (!dark__shadow__tupled. happ _1 dark__shadow__tupled = happ _2 dark__shadow__tupled) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R /\ (!L c lowers uppers. happ (happ R (h4/pair/_2C uppers lowers)) (h4/pair/_2C (h4/list/CONS (h4/pair/_2C c L) uppers) lowers))) ==> h4/Omega/dark__shadow__tupled = h4/relation/WFREC (h4/min/_40 _0) _1))))))))
% Assm [h4s_Omegas_darku_u_shadowu_u_curriedu_u_def]: !x1 x. h4/Omega/dark__shadow x x1 <=> happ h4/Omega/dark__shadow__tupled (h4/pair/_2C x x1)
% Assm [h4s_Omegas_darkrowu_u_impliesu_u_realrow]: !lowers c L. happ (happ h4/prim__rec/_3C h4/num/0) c /\ h4/list/EVERY h4/Omega/fst__nzero lowers /\ h4/Omega/dark__shadow__row c L lowers ==> h4/Omega/rshadow__row (h4/pair/_2C c L) lowers
% Goal: !uppers lowers. h4/list/EVERY h4/Omega/fst__nzero uppers /\ h4/list/EVERY h4/Omega/fst__nzero lowers /\ h4/Omega/dark__shadow uppers lowers ==> h4/Omega/real__shadow uppers lowers
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_Q19234,TV_Q19230]: ![V_f, V_g]: (![V_x]: s(TV_Q19230,happ(s(t_fun(TV_Q19234,TV_Q19230),V_f),s(TV_Q19234,V_x))) = s(TV_Q19230,happ(s(t_fun(TV_Q19234,TV_Q19230),V_g),s(TV_Q19234,V_x))) => s(t_fun(TV_Q19234,TV_Q19230),V_f) = s(t_fun(TV_Q19234,TV_Q19230),V_g))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_bools_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_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_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_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_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_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_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_invu_u_imageu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_x, V_xi_]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_invu_u_image(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_x))),s(TV_u_27a,V_xi_))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_xi_)))))).
fof(ah4s_relations_WFu_u_invu_u_image, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R)))) => p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_invu_u_image(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_f)))))))).
fof(ah4s_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_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_FORALLu_u_PROD, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: (![V_p]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_p)))) <=> ![V_pu_u_1, V_pu_u_2]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_pu_u_1),s(TV_u_27b,V_pu_u_2)))))))).
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_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_COMM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_ADDu_u_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_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_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_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_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_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_lteu_c1, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
fof(ah4s_lists_listu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27b,h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f))) = s(TV_u_27b,V_v)).
fof(ah4s_lists_listu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_f, V_a1, V_a0]: s(TV_u_27b,h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a0),s(t_h4s_lists_list(TV_u_27a),V_a1))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f))) = s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1)))).
fof(ah4s_lists_listu_u_sizeu_u_defu_c1, axiom, ![TV_u_27a]: ![V_f, V_a1, V_a0]: s(t_h4s_nums_num,h4s_lists_listu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a0),s(t_h4s_lists_list(TV_u_27a),V_a1))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(TV_u_27a,V_a0))),s(t_h4s_nums_num,h4s_lists_listu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_lists_list(TV_u_27a),V_a1)))))))).
fof(ah4s_lists_EVERYu_u_DEFu_c0, axiom, ![TV_u_27a]: ![V_P]: s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_bool,t)).
fof(ah4s_lists_EVERYu_u_DEFu_c1, axiom, ![TV_u_27a]: ![V_t, V_h, V_P]: (p(s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t)))))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_h)))) & p(s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t))))))).
fof(ah4s_lists_listu_u_induction, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))) & ![V_t]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t)))) => ![V_h]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t)))))))) => ![V_l]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_Omegas_fstu_u_nzerou_u_def, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a),t_bool),h4s_omegas_fstu_u_nzero),s(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a),V_x))) = 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_pairs_fst(s(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a),V_x)))))).
fof(ah4s_Omegas_realu_u_shadowu_u_defu_c0, axiom, ![V_lowers]: s(t_bool,h4s_omegas_realu_u_shadow(s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),h4s_lists_nil),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_lowers))) = s(t_bool,t)).
fof(ah4s_Omegas_realu_u_shadowu_u_defu_c1, axiom, ![V_upper, V_ls, V_lowers]: (p(s(t_bool,h4s_omegas_realu_u_shadow(s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),h4s_lists_cons(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int),V_upper),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_ls))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_lowers)))) <=> (p(s(t_bool,h4s_omegas_rshadowu_u_row(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int),V_upper),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_lowers)))) & p(s(t_bool,h4s_omegas_realu_u_shadow(s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_ls),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_lowers))))))).
fof(ah4s_Omegas_darku_u_shadowu_u_tupledu_u_primitiveu_u_def, axiom, ![V_uu_8]: (![V_c, V_darku_u_shadowu_u_tupled, V_uppers, V_lowers, V_L]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_omegas_darku_u_shadowu_u_row(s(t_h4s_nums_num,V_c),s(t_h4s_integers_int,V_L),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_lowers)))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),V_darku_u_shadowu_u_tupled),s(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),h4s_pairs_u_2c(s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_uppers),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_lowers)))))))) & s(t_bool,happ(s(t_fun(t_h4s_integers_int,t_bool),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_integers_int,t_bool)),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_integers_int,t_bool))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_integers_int,t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_integers_int,t_bool))))),V_uu_8),s(t_h4s_nums_num,V_c))),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),V_darku_u_shadowu_u_tupled))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_uppers))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_lowers))),s(t_h4s_integers_int,V_L))) = s(t_bool,h4s_combins_i(s(t_bool,V_v)))) => ![V_uu_7]: (![V_darku_u_shadowu_u_tupled, V_uppers, V_lowers, V_c]: s(t_fun(t_h4s_integers_int,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_integers_int,t_bool)),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_nums_num,t_fun(t_h4s_integers_int,t_bool))),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_nums_num,t_fun(t_h4s_integers_int,t_bool)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_nums_num,t_fun(t_h4s_integers_int,t_bool))))),V_uu_7),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),V_darku_u_shadowu_u_tupled))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_uppers))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_lowers))),s(t_h4s_nums_num,V_c))) = s(t_fun(t_h4s_integers_int,t_bool),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_integers_int,t_bool)),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_integers_int,t_bool))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_integers_int,t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_integers_int,t_bool))))),V_uu_8),s(t_h4s_nums_num,V_c))),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),V_darku_u_shadowu_u_tupled))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_uppers))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_lowers))) => ![V_uu_6]: (![V_v2, V_darku_u_shadowu_u_tupled, V_lowers, V_uppers]: s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_bool),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_bool))),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int),t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_bool)))),V_uu_6),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int),V_v2))),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),V_darku_u_shadowu_u_tupled))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_lowers))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_uppers))) = s(t_bool,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int),V_v2),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_integers_int,t_bool)),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_nums_num,t_fun(t_h4s_integers_int,t_bool))),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_nums_num,t_fun(t_h4s_integers_int,t_bool)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_nums_num,t_fun(t_h4s_integers_int,t_bool))))),V_uu_7),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),V_darku_u_shadowu_u_tupled))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_uppers))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_lowers))))) => ![V_uu_5]: (![V_darku_u_shadowu_u_tupled, V_lowers, V_v2]: s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_bool)),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_bool))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_bool)))),V_uu_5),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),V_darku_u_shadowu_u_tupled))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_lowers))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int),V_v2))) = s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_bool),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_bool))),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int),t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_bool)))),V_uu_6),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int),V_v2))),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),V_darku_u_shadowu_u_tupled))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_lowers))) => ![V_uu_4]: (![V_v, V_darku_u_shadowu_u_tupled, V_lowers]: s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_bool),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_bool)),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_bool))),V_uu_4),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_v))),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),V_darku_u_shadowu_u_tupled))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_lowers))) = s(t_bool,h4s_lists_listu_u_case(s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_v),s(t_bool,h4s_combins_i(s(t_bool,t))),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_bool)),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_bool))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_bool)))),V_uu_5),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),V_darku_u_shadowu_u_tupled))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_lowers))))) => ![V_uu_3]: (![V_darku_u_shadowu_u_tupled, V_v]: s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_bool),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_bool))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),V_darku_u_shadowu_u_tupled))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_v))) = s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_bool),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_bool)),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_bool))),V_uu_4),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_v))),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),V_darku_u_shadowu_u_tupled))) => ![V_uu_2]: (![V_darku_u_shadowu_u_tupled, V_a]: s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool)),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),V_darku_u_shadowu_u_tupled))),s(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),V_a))) = s(t_bool,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),V_a),s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_bool))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),V_darku_u_shadowu_u_tupled))))) => ![V_uu_1]: (![V_darku_u_shadowu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool)),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),V_darku_u_shadowu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool)),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),V_darku_u_shadowu_u_tupled))) => ![V_uu_0]: (![V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool)),V_R)))) & ![V_L, V_c, V_lowers, V_uppers]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool)),V_R),s(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),h4s_pairs_u_2c(s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_uppers),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_lowers))))),s(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),h4s_pairs_u_2c(s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),h4s_lists_cons(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_nums_num,V_c),s(t_h4s_integers_int,V_L))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_uppers))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_lowers)))))))) => s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),h4s_omegas_darku_u_shadowu_u_tupled) = s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool)),V_uu_1))))))))))))).
fof(ah4s_Omegas_darku_u_shadowu_u_curriedu_u_def, axiom, ![V_x1, V_x]: s(t_bool,h4s_omegas_darku_u_shadow(s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_x),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_x1))) = s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),t_bool),h4s_omegas_darku_u_shadowu_u_tupled),s(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int))),h4s_pairs_u_2c(s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_x),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_x1)))))).
fof(ah4s_Omegas_darkrowu_u_impliesu_u_realrow, axiom, ![V_lowers, V_c, V_L]: ((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_c)))) & (p(s(t_bool,h4s_lists_every(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int),t_bool),h4s_omegas_fstu_u_nzero),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_lowers)))) & p(s(t_bool,h4s_omegas_darku_u_shadowu_u_row(s(t_h4s_nums_num,V_c),s(t_h4s_integers_int,V_L),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_lowers)))))) => p(s(t_bool,h4s_omegas_rshadowu_u_row(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_nums_num,V_c),s(t_h4s_integers_int,V_L))),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_lowers)))))).
fof(ch4s_Omegas_darku_u_impliesu_u_real, conjecture, ![V_uppers, V_lowers]: ((p(s(t_bool,h4s_lists_every(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int),t_bool),h4s_omegas_fstu_u_nzero),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_uppers)))) & (p(s(t_bool,h4s_lists_every(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int),t_bool),h4s_omegas_fstu_u_nzero),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_lowers)))) & p(s(t_bool,h4s_omegas_darku_u_shadow(s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_uppers),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_lowers)))))) => p(s(t_bool,h4s_omegas_realu_u_shadow(s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_uppers),s(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_integers_int)),V_lowers)))))).
