%   ORIGINAL: h4/bag/BAG__EVERY__THM_c0
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/bag/BAG__EVERY0: !b P. h4/bag/BAG__EVERY P b <=> (!e. h4/bag/BAG__IN e b ==> P e)
% Assm: h4/bool/TRUTH: T
% Assm: h4/combin/K__THM: !y x. h4/combin/K x y = x
% Assm: h4/bag/EMPTY__BAG0: h4/bag/EMPTY__BAG = h4/combin/K h4/num/0
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bag/BAG__FILTER__EMPTY: !P. h4/bag/BAG__FILTER P h4/bag/EMPTY__BAG = h4/bag/EMPTY__BAG
% Assm: h4/bag/BAG__INSERT__NOT__EMPTY: !x b. ~(h4/bag/BAG__INSERT x b = h4/bag/EMPTY__BAG)
% Assm: h4/bag/BAG__DIFF__EMPTY_c0: !b. h4/bag/BAG__DIFF b b = h4/bag/EMPTY__BAG
% Assm: h4/bag/FINITE__BAG0: !b. h4/bag/FINITE__BAG b <=> (!P. P h4/bag/EMPTY__BAG /\ (!b0. P b0 ==> (!e. P (h4/bag/BAG__INSERT e b0))) ==> P b)
% Assm: h4/bag/EL__BAG__SUB__BAG: !x b. h4/bag/SUB__BAG (h4/bag/BAG__INSERT x h4/bag/EMPTY__BAG) b <=> h4/bag/BAG__IN x b
% Assm: h4/bag/ITBAG__THM: !f b acc. h4/bag/FINITE__BAG b ==> h4/bag/ITBAG f b acc = h4/bool/COND (b = h4/bag/EMPTY__BAG) acc (h4/bag/ITBAG f (h4/bag/BAG__REST b) (f (h4/bag/BAG__CHOICE b) acc))
% Assm: h4/bag/SUB__BAG__EMPTY_c1: !b. h4/bag/SUB__BAG b h4/bag/EMPTY__BAG <=> b = h4/bag/EMPTY__BAG
% Assm: h4/bag/BAG__REST__SING: !x. h4/bag/BAG__REST (h4/bag/BAG__INSERT x h4/bag/EMPTY__BAG) = h4/bag/EMPTY__BAG
% Assm: h4/bag/BAG__DIFF__EMPTY_c2: !b. h4/bag/BAG__DIFF h4/bag/EMPTY__BAG b = h4/bag/EMPTY__BAG
% Assm: h4/bag/NOT__IN__EMPTY__BAG: !x. ~h4/bag/BAG__IN x h4/bag/EMPTY__BAG
% Assm: h4/bag/BAG__UNION__EMPTY_c2: !b2 b1. h4/bag/BAG__UNION b1 b2 = h4/bag/EMPTY__BAG <=> b1 = h4/bag/EMPTY__BAG /\ b2 = h4/bag/EMPTY__BAG
% Assm: h4/bag/ITBAG__IND: !f P. (!b acc. (h4/bag/FINITE__BAG b /\ ~(b = h4/bag/EMPTY__BAG) ==> P (h4/bag/BAG__REST b) (f (h4/bag/BAG__CHOICE b) acc)) ==> P b acc) ==> (!v v1. P v v1)
% Assm: h4/bag/BAG__DIFF0: !b2 b1. h4/bag/BAG__DIFF b1 b2 = (\x. h4/arithmetic/_2D (b1 x) (b2 x))
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/numeral/numeral__distrib_c17: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__eq_c1: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Assm: h4/arithmetic/LESS__EQ__0: !n. h4/arithmetic/_3C_3D n h4/num/0 <=> n = h4/num/0
% Assm: h4/arithmetic/SUB__EQUAL__0: !c. h4/arithmetic/_2D c c = h4/num/0
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bag/BAG__FILTER__DEF: !b P. h4/bag/BAG__FILTER P b = (\e. h4/bool/COND (P e) (b e) h4/num/0)
% Assm: h4/bool/COND__ID: !t b. h4/bool/COND b t t = t
% Assm: h4/bag/SUB__BAG__LEQ: !b2 b1. h4/bag/SUB__BAG b1 b2 <=> (!x. h4/arithmetic/_3C_3D (b1 x) (b2 x))
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bag/BAG__IN0: !e b. h4/bag/BAG__IN e b <=> h4/bag/BAG__INN e (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) b
% Assm: h4/bag/BAG__INN0: !n e b. h4/bag/BAG__INN e n b <=> h4/arithmetic/_3E_3D (b e) n
% Assm: h4/arithmetic/ADD__EQ__0: !n m. h4/arithmetic/_2B m n = h4/num/0 <=> m = h4/num/0 /\ n = h4/num/0
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/bag/BAG__CHOICE__SING: !x. h4/bag/BAG__CHOICE (h4/bag/BAG__INSERT x h4/bag/EMPTY__BAG) = x
% Assm: h4/bag/BAG__DIFF__INSERT__same: !x b2 b1. h4/bag/BAG__DIFF (h4/bag/BAG__INSERT x b1) (h4/bag/BAG__INSERT x b2) = h4/bag/BAG__DIFF b1 b2
% Assm: h4/bag/BAG__REST__DEF: !b. h4/bag/BAG__REST b = h4/bag/BAG__DIFF b (h4/bag/EL__BAG (h4/bag/BAG__CHOICE b))
% Assm: h4/bag/BAG__DIFF__EMPTY__simple_c2: !b. h4/bag/BAG__DIFF h4/bag/EMPTY__BAG b = h4/bag/EMPTY__BAG
% Assm: h4/bag/EL__BAG0: !e. h4/bag/EL__BAG e = h4/bag/BAG__INSERT e h4/bag/EMPTY__BAG
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% 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/bag/BAG__INSERT0: !e b. h4/bag/BAG__INSERT e b = (\x. h4/bool/COND (x = e) (h4/arithmetic/_2B (b e) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (b x))
% Assm: h4/bool/COND__EXPAND: !t2 t1 b. h4/bool/COND b t1 t2 <=> (~b \/ t1) /\ (b \/ t2)
% Assm: h4/arithmetic/SUB__0_c0: !m. h4/arithmetic/_2D h4/num/0 m = h4/num/0
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% 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/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__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/bool/EXISTS__OR__THM: !Q P. (?x. P x \/ Q x) <=> (?x. P x) \/ (?x. Q x)
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/bool/EXISTS__REFL: !a. ?x. x = a
% 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/NOT__CLAUSES_c1: ~T <=> F
% 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/FALSITY: !t. F ==> t
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/numeral/numeral__distrib_c30: !n. h4/arithmetic/_3E_3D h4/num/0 n <=> n = h4/num/0
% Assm: h4/bag/PSUB__BAG__CARD: !b2 b1. h4/bag/FINITE__BAG b2 /\ h4/bag/PSUB__BAG b1 b2 ==> h4/prim__rec/_3C (h4/bag/BAG__CARD b1) (h4/bag/BAG__CARD b2)
% Assm: h4/bag/ITBAG__tupled__primitive__def: !f. h4/bag/ITBAG__tupled f = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!acc b. h4/bag/FINITE__BAG b /\ ~(b = h4/bag/EMPTY__BAG) ==> R (h4/pair/_2C (h4/bag/BAG__REST b) (f (h4/bag/BAG__CHOICE b) acc)) (h4/pair/_2C b acc)))) (\ITBAG__tupled a. h4/pair/pair__CASE a (\b acc. h4/combin/I (h4/bool/COND (h4/bag/FINITE__BAG b) (h4/bool/COND (b = h4/bag/EMPTY__BAG) acc (ITBAG__tupled (h4/pair/_2C (h4/bag/BAG__REST b) (f (h4/bag/BAG__CHOICE b) acc)))) h4/bool/ARB)))
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bag/ITBAG__curried__def: !x1 x f. h4/bag/ITBAG f x x1 = h4/bag/ITBAG__tupled f (h4/pair/_2C x x1)
% Assm: h4/bag/PSUB__BAG__REST: !b. ~(b = h4/bag/EMPTY__BAG) ==> h4/bag/PSUB__BAG (h4/bag/BAG__REST b) b
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% 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/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/relation/WF__INDUCTION__THM: !R. h4/relation/WF R ==> (!P. (!x. (!y. R y x ==> P y) ==> P x) ==> (!x. P x))
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/relation/RESTRICT__LEMMA: !z y f R. R y z ==> h4/relation/RESTRICT f R z y = f y
% Assm: h4/prim__rec/WF__measure: !m. h4/relation/WF (h4/prim__rec/measure m)
% Assm: h4/relation/inv__image__def: !f R. h4/relation/inv__image R f = (\x y. R (f x) (f y))
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/combin/o__DEF: !g f. h4/combin/o f g = (\x. f (g x))
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% 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/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bag/SUB__BAG0: !b2 b1. h4/bag/SUB__BAG b1 b2 <=> (!x n. h4/bag/BAG__INN x n b1 ==> h4/bag/BAG__INN x n b2)
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% 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/arithmetic/NOT__LEQ: !n m. ~h4/arithmetic/_3C_3D 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/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/arithmetic/GREATER__EQ: !n m. h4/arithmetic/_3E_3D n m <=> h4/arithmetic/_3C_3D m n
% Assm: h4/bool/UNWIND__FORALL__THM2: !v f. (!x. x = v ==> f x) <=> f v
% Assm: h4/arithmetic/LESS__EQ__REFL: !m. h4/arithmetic/_3C_3D m m
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/bool/FORALL__AND__THM: !Q P. (!x. P x /\ Q x) <=> (!x. P x) /\ (!x. Q x)
% Assm: h4/arithmetic/LESS__EQ__TRANS: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/ADD__0: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/RIGHT__FORALL__IMP__THM: !Q P. (!x. P ==> Q x) <=> P ==> (!x. Q x)
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/bool/DISJ__COMM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/LEFT__AND__OVER__OR: !C B A. A /\ (B \/ C) <=> A /\ B \/ A /\ C
% Assm: h4/bool/RIGHT__AND__OVER__OR: !C B A. (B \/ C) /\ A <=> B /\ A \/ C /\ A
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/bool/DISJ__IMP__THM: !R Q P. P \/ Q ==> R <=> (P ==> R) /\ (Q ==> R)
% Assm: h4/bool/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/bag/BAG__UNION0: !c b. h4/bag/BAG__UNION b c = (\x. h4/arithmetic/_2B (b x) (c x))
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Goal: !P. h4/bag/BAG__EVERY P h4/bag/EMPTY__BAG
%   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_bags_BAGu_u_EVERY0]: !b P. h4/bag/BAG__EVERY P b <=> (!e. h4/bag/BAG__IN e b ==> happ P e)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_combins_Ku_u_THM]: !y x. happ (h4/combin/K x) y = x
% Assm [h4s_bags_EMPTYu_u_BAG0]: h4/bag/EMPTY__BAG = h4/combin/K h4/num/0
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bags_BAGu_u_FILTERu_u_EMPTY]: !P. h4/bag/BAG__FILTER P h4/bag/EMPTY__BAG = h4/bag/EMPTY__BAG
% Assm [h4s_bags_BAGu_u_INSERTu_u_NOTu_u_EMPTY]: !x b. ~(h4/bag/BAG__INSERT x b = h4/bag/EMPTY__BAG)
% Assm [h4s_bags_BAGu_u_DIFFu_u_EMPTYu_c0]: !b. h4/bag/BAG__DIFF b b = h4/bag/EMPTY__BAG
% Assm [h4s_bags_FINITEu_u_BAG0]: !b. h4/bag/FINITE__BAG b <=> (!P. happ P h4/bag/EMPTY__BAG /\ (!b0. happ P b0 ==> (!e. happ P (h4/bag/BAG__INSERT e b0))) ==> happ P b)
% Assm [h4s_bags_ELu_u_BAGu_u_SUBu_u_BAG]: !x b. h4/bag/SUB__BAG (h4/bag/BAG__INSERT x h4/bag/EMPTY__BAG) b <=> h4/bag/BAG__IN x b
% Assm [h4s_bags_ITBAGu_u_THM]: !f b acc. h4/bag/FINITE__BAG b ==> (?v. (v <=> b = h4/bag/EMPTY__BAG) /\ h4/bag/ITBAG f b acc = h4/bool/COND v acc (h4/bag/ITBAG f (h4/bag/BAG__REST b) (happ (happ f (h4/bag/BAG__CHOICE b)) acc)))
% Assm [h4s_bags_SUBu_u_BAGu_u_EMPTYu_c1]: !b. h4/bag/SUB__BAG b h4/bag/EMPTY__BAG <=> b = h4/bag/EMPTY__BAG
% Assm [h4s_bags_BAGu_u_RESTu_u_SING]: !x. h4/bag/BAG__REST (h4/bag/BAG__INSERT x h4/bag/EMPTY__BAG) = h4/bag/EMPTY__BAG
% Assm [h4s_bags_BAGu_u_DIFFu_u_EMPTYu_c2]: !b. h4/bag/BAG__DIFF h4/bag/EMPTY__BAG b = h4/bag/EMPTY__BAG
% Assm [h4s_bags_NOTu_u_INu_u_EMPTYu_u_BAG]: !x. ~h4/bag/BAG__IN x h4/bag/EMPTY__BAG
% Assm [h4s_bags_BAGu_u_UNIONu_u_EMPTYu_c2]: !b2 b1. h4/bag/BAG__UNION b1 b2 = h4/bag/EMPTY__BAG <=> b1 = h4/bag/EMPTY__BAG /\ b2 = h4/bag/EMPTY__BAG
% Assm [h4s_bags_ITBAGu_u_IND]: !f P. (!b acc. (h4/bag/FINITE__BAG b /\ ~(b = h4/bag/EMPTY__BAG) ==> happ (happ P (h4/bag/BAG__REST b)) (happ (happ f (h4/bag/BAG__CHOICE b)) acc)) ==> happ (happ P b) acc) ==> (!v v1. happ (happ P v) v1)
% Assm [h4s_bags_BAGu_u_DIFF0]: !b2 b1 x. happ (h4/bag/BAG__DIFF b1 b2) x = h4/arithmetic/_2D (happ b1 x) (happ b2 x)
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_numerals_numeralu_u_distribu_c17]: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_equ_c1]: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Assm [h4s_arithmetics_LESSu_u_EQu_u_0]: !n. h4/arithmetic/_3C_3D n h4/num/0 <=> n = h4/num/0
% Assm [h4s_arithmetics_SUBu_u_EQUALu_u_0]: !c. h4/arithmetic/_2D c c = h4/num/0
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bags_BAGu_u_FILTERu_u_DEF]: !b P x. happ (h4/bag/BAG__FILTER P b) x = h4/bool/COND (happ P x) (happ b x) h4/num/0
% Assm [h4s_bools_CONDu_u_ID]: !t b. h4/bool/COND b t t = t
% Assm [h4s_bags_SUBu_u_BAGu_u_LEQ]: !b2 b1. h4/bag/SUB__BAG b1 b2 <=> (!x. h4/arithmetic/_3C_3D (happ b1 x) (happ b2 x))
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bags_BAGu_u_IN0]: !e b. h4/bag/BAG__IN e b <=> h4/bag/BAG__INN e (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) b
% Assm [h4s_bags_BAGu_u_INN0]: !n e b. h4/bag/BAG__INN e n b <=> h4/arithmetic/_3E_3D (happ b e) n
% Assm [h4s_arithmetics_ADDu_u_EQu_u_0]: !n m. h4/arithmetic/_2B m n = h4/num/0 <=> m = h4/num/0 /\ n = h4/num/0
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_bags_BAGu_u_CHOICEu_u_SING]: !x. h4/bag/BAG__CHOICE (h4/bag/BAG__INSERT x h4/bag/EMPTY__BAG) = x
% Assm [h4s_bags_BAGu_u_DIFFu_u_INSERTu_u_same]: !x b2 b1. h4/bag/BAG__DIFF (h4/bag/BAG__INSERT x b1) (h4/bag/BAG__INSERT x b2) = h4/bag/BAG__DIFF b1 b2
% Assm [h4s_bags_BAGu_u_RESTu_u_DEF]: !b. h4/bag/BAG__REST b = h4/bag/BAG__DIFF b (h4/bag/EL__BAG (h4/bag/BAG__CHOICE b))
% Assm [h4s_bags_BAGu_u_DIFFu_u_EMPTYu_u_simpleu_c2]: !b. h4/bag/BAG__DIFF h4/bag/EMPTY__BAG b = h4/bag/EMPTY__BAG
% Assm [h4s_bags_ELu_u_BAG0]: !e. h4/bag/EL__BAG e = h4/bag/BAG__INSERT e h4/bag/EMPTY__BAG
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% 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_bags_BAGu_u_INSERT0]: !e b x. ?v. (v <=> x = e) /\ happ (h4/bag/BAG__INSERT e b) x = h4/bool/COND v (h4/arithmetic/_2B (happ b e) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (happ b x)
% Assm [h4s_bools_CONDu_u_EXPAND]: !t2 t1 b. h4/bool/COND b t1 t2 <=> (~b \/ t1) /\ (b \/ t2)
% Assm [h4s_arithmetics_SUBu_u_0u_c0]: !m. h4/arithmetic/_2D h4/num/0 m = h4/num/0
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% 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_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_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_bools_EXISTSu_u_ORu_u_THM]: !Q P. (?x. happ P x \/ happ Q x) <=> (?x. happ P x) \/ (?x. happ Q x)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% Assm [h4s_bools_EXISTSu_u_REFL]: !a. ?x. x = a
% 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_NOTu_u_CLAUSESu_c1]: ~T <=> F
% 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_FALSITY]: !t. F ==> t
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_numerals_numeralu_u_distribu_c30]: !n. h4/arithmetic/_3E_3D h4/num/0 n <=> n = h4/num/0
% Assm [h4s_bags_PSUBu_u_BAGu_u_CARD]: !b2 b1. h4/bag/FINITE__BAG b2 /\ h4/bag/PSUB__BAG b1 b2 ==> happ (happ h4/prim__rec/_3C (h4/bag/BAG__CARD b1)) (h4/bag/BAG__CARD b2)
% Assm [h4s_bags_ITBAGu_u_tupledu_u_primitiveu_u_def]: !_4. (!ITBAG__tupled f b acc. ?v. (v <=> b = h4/bag/EMPTY__BAG) /\ happ (happ (happ (happ _4 ITBAG__tupled) f) b) acc = h4/combin/I (h4/bool/COND (h4/bag/FINITE__BAG b) (h4/bool/COND v acc (happ ITBAG__tupled (h4/pair/_2C (h4/bag/BAG__REST b) (happ (happ f (h4/bag/BAG__CHOICE b)) acc)))) h4/bool/ARB)) ==> (!_3. (!ITBAG__tupled f b. happ (happ (happ _3 ITBAG__tupled) f) b = happ (happ (happ _4 ITBAG__tupled) f) b) ==> (!_2. (!ITBAG__tupled f a. happ (happ (happ _2 ITBAG__tupled) f) a = h4/pair/pair__CASE a (happ (happ _3 ITBAG__tupled) f)) ==> (!_1. (!f ITBAG__tupled. happ (happ _1 f) ITBAG__tupled = happ (happ _2 ITBAG__tupled) f) ==> (!_0. (!f R. happ (happ _0 f) R <=> h4/relation/WF R /\ (!acc b. h4/bag/FINITE__BAG b /\ ~(b = h4/bag/EMPTY__BAG) ==> happ (happ R (h4/pair/_2C (h4/bag/BAG__REST b) (happ (happ f (h4/bag/BAG__CHOICE b)) acc))) (h4/pair/_2C b acc))) ==> (!f. h4/bag/ITBAG__tupled f = h4/relation/WFREC (h4/min/_40 (happ _0 f)) (happ _1 f))))))
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bags_ITBAGu_u_curriedu_u_def]: !x1 x f. h4/bag/ITBAG f x x1 = happ (h4/bag/ITBAG__tupled f) (h4/pair/_2C x x1)
% Assm [h4s_bags_PSUBu_u_BAGu_u_REST]: !b. ~(b = h4/bag/EMPTY__BAG) ==> h4/bag/PSUB__BAG (h4/bag/BAG__REST b) b
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% 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_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_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_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> 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_primu_u_recs_WFu_u_measure]: !m. h4/relation/WF (happ h4/prim__rec/measure m)
% 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_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_combins_ou_u_DEF]: !g f x. h4/combin/o f g x = happ f (happ g x)
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% 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_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bags_SUBu_u_BAG0]: !b2 b1. h4/bag/SUB__BAG b1 b2 <=> (!x n. h4/bag/BAG__INN x n b1 ==> h4/bag/BAG__INN x n b2)
% 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_distribu_c27]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% 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_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_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_arithmetics_GREATERu_u_EQ]: !n m. h4/arithmetic/_3E_3D n m <=> h4/arithmetic/_3C_3D m n
% Assm [h4s_bools_UNWINDu_u_FORALLu_u_THM2]: !v f. (!x. x = v ==> happ f x) <=> happ f v
% Assm [h4s_arithmetics_LESSu_u_EQu_u_REFL]: !m. h4/arithmetic/_3C_3D m m
% Assm [h4s_arithmetics_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_bools_FORALLu_u_ANDu_u_THM]: !Q P. (!x. happ P x /\ happ Q x) <=> (!x. happ P x) /\ (!x. happ Q x)
% Assm [h4s_arithmetics_LESSu_u_EQu_u_TRANS]: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_ADDu_u_0]: !m. h4/arithmetic/_2B m h4/num/0 = m
% 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_RIGHTu_u_FORALLu_u_IMPu_u_THM]: !Q P. (!x. P ==> happ Q x) <=> P ==> (!x. happ Q x)
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_bools_DISJu_u_COMM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_LEFTu_u_ANDu_u_OVERu_u_OR]: !C B A. A /\ (B \/ C) <=> A /\ B \/ A /\ C
% Assm [h4s_bools_RIGHTu_u_ANDu_u_OVERu_u_OR]: !C B A. (B \/ C) /\ A <=> B /\ A \/ C /\ A
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_bools_DISJu_u_IMPu_u_THM]: !R Q P. P \/ Q ==> R <=> (P ==> R) /\ (Q ==> R)
% Assm [h4s_bools_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_bags_BAGu_u_UNION0]: !c b x. happ (h4/bag/BAG__UNION b c) x = h4/arithmetic/_2B (happ b x) (happ c x)
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Goal: !P. h4/bag/BAG__EVERY P h4/bag/EMPTY__BAG
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_Q1324510,TV_Q1324506]: ![V_f, V_g]: (![V_x]: s(TV_Q1324506,happ(s(t_fun(TV_Q1324510,TV_Q1324506),V_f),s(TV_Q1324510,V_x))) = s(TV_Q1324506,happ(s(t_fun(TV_Q1324510,TV_Q1324506),V_g),s(TV_Q1324510,V_x))) => s(t_fun(TV_Q1324510,TV_Q1324506),V_f) = s(t_fun(TV_Q1324510,TV_Q1324506),V_g))).
fof(ah4s_bags_BAGu_u_EVERY0, axiom, ![TV_u_27a]: ![V_b, V_P]: (p(s(t_bool,h4s_bags_bagu_u_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)))) <=> ![V_e]: (p(s(t_bool,h4s_bags_bagu_u_in(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_e))))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_combins_Ku_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),h4s_combins_k(s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) = s(TV_u_27a,V_x)).
fof(ah4s_bags_EMPTYu_u_BAG0, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_combins_k(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_FUNu_u_EQu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g) <=> ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))).
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_bags_BAGu_u_FILTERu_u_EMPTY, axiom, ![TV_u_27a]: ![V_P]: s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag))) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag)).
fof(ah4s_bags_BAGu_u_INSERTu_u_NOTu_u_EMPTY, axiom, ![TV_u_27a]: ![V_x, V_b]: ~ (s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag))).
fof(ah4s_bags_BAGu_u_DIFFu_u_EMPTYu_c0, axiom, ![TV_u_27a]: ![V_b]: s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_diff(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag)).
fof(ah4s_bags_FINITEu_u_BAG0, axiom, ![TV_u_27a]: ![V_b]: (p(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)))) <=> ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool),V_P),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag)))) & ![V_b0]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool),V_P),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b0)))) => ![V_e]: p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool),V_P),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b0)))))))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_bool),V_P),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))))))).
fof(ah4s_bags_ELu_u_BAGu_u_SUBu_u_BAG, axiom, ![TV_u_27a]: ![V_x, V_b]: s(t_bool,h4s_bags_subu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag))),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))) = s(t_bool,h4s_bags_bagu_u_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)))).
fof(ah4s_bags_ITBAGu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_b, V_acc]: (p(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)))) => ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_fun(TV_u_27a,t_h4s_nums_num),V_b) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag)) & s(TV_u_27b,h4s_bags_itbag(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b),s(TV_u_27b,V_acc))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27b,V_acc),s(TV_u_27b,h4s_bags_itbag(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_rest(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))),s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27a,h4s_bags_bagu_u_choice(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))))),s(TV_u_27b,V_acc)))))))))).
fof(ah4s_bags_SUBu_u_BAGu_u_EMPTYu_c1, axiom, ![TV_u_27a]: ![V_b]: (p(s(t_bool,h4s_bags_subu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag)))) <=> s(t_fun(TV_u_27a,t_h4s_nums_num),V_b) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag))).
fof(ah4s_bags_BAGu_u_RESTu_u_SING, axiom, ![TV_u_27a]: ![V_x]: s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_rest(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag))))) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag)).
fof(ah4s_bags_BAGu_u_DIFFu_u_EMPTYu_c2, axiom, ![TV_u_27c]: ![V_b]: s(t_fun(TV_u_27c,t_h4s_nums_num),h4s_bags_bagu_u_diff(s(t_fun(TV_u_27c,t_h4s_nums_num),h4s_bags_emptyu_u_bag),s(t_fun(TV_u_27c,t_h4s_nums_num),V_b))) = s(t_fun(TV_u_27c,t_h4s_nums_num),h4s_bags_emptyu_u_bag)).
fof(ah4s_bags_NOTu_u_INu_u_EMPTYu_u_BAG, axiom, ![TV_u_27a]: ![V_x]: ~ (p(s(t_bool,h4s_bags_bagu_u_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag)))))).
fof(ah4s_bags_BAGu_u_UNIONu_u_EMPTYu_c2, axiom, ![TV_u_27c]: ![V_b2, V_b1]: (s(t_fun(TV_u_27c,t_h4s_nums_num),h4s_bags_bagu_u_union(s(t_fun(TV_u_27c,t_h4s_nums_num),V_b1),s(t_fun(TV_u_27c,t_h4s_nums_num),V_b2))) = s(t_fun(TV_u_27c,t_h4s_nums_num),h4s_bags_emptyu_u_bag) <=> (s(t_fun(TV_u_27c,t_h4s_nums_num),V_b1) = s(t_fun(TV_u_27c,t_h4s_nums_num),h4s_bags_emptyu_u_bag) & s(t_fun(TV_u_27c,t_h4s_nums_num),V_b2) = s(t_fun(TV_u_27c,t_h4s_nums_num),h4s_bags_emptyu_u_bag)))).
fof(ah4s_bags_ITBAGu_u_IND, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_P]: (![V_b, V_acc]: (((p(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)))) & ~ (s(t_fun(TV_u_27a,t_h4s_nums_num),V_b) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27b,t_bool)),V_P),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_rest(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))))),s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27a,h4s_bags_bagu_u_choice(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))))),s(TV_u_27b,V_acc))))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27b,t_bool)),V_P),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))),s(TV_u_27b,V_acc))))) => ![V_v, V_v1]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27b,t_bool)),V_P),s(t_fun(TV_u_27a,t_h4s_nums_num),V_v))),s(TV_u_27b,V_v1)))))).
fof(ah4s_bags_BAGu_u_DIFF0, axiom, ![TV_u_27a]: ![V_b2, V_b1, V_x]: s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_diff(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2))),s(TV_u_27a,V_x))) = s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1),s(TV_u_27a,V_x))),s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2),s(TV_u_27a,V_x)))))).
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_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_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_numerals_numeralu_u_distribu_c17, axiom, ![V_n]: (s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_arithmetics_zero))).
fof(ah4s_numerals_numeralu_u_equ_c1, axiom, ![V_n]: (s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_zero) <=> p(s(t_bool,f)))).
fof(ah4s_arithmetics_LESSu_u_EQu_u_0, axiom, ![V_n]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0)))) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0))).
fof(ah4s_arithmetics_SUBu_u_EQUALu_u_0, axiom, ![V_c]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_c),s(t_h4s_nums_num,V_c))) = s(t_h4s_nums_num,h4s_nums_0)).
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_bags_BAGu_u_FILTERu_u_DEF, axiom, ![TV_u_27a]: ![V_b, V_P, V_x]: s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))),s(TV_u_27a,V_x))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))),s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b),s(TV_u_27a,V_x))),s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_bools_CONDu_u_ID, axiom, ![TV_u_27a]: ![V_t, V_b]: s(TV_u_27a,h4s_bools_cond(s(t_bool,V_b),s(TV_u_27a,V_t),s(TV_u_27a,V_t))) = s(TV_u_27a,V_t)).
fof(ah4s_bags_SUBu_u_BAGu_u_LEQ, axiom, ![TV_u_27a]: ![V_b2, V_b1]: (p(s(t_bool,h4s_bags_subu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2)))) <=> ![V_x]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1),s(TV_u_27a,V_x))),s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2),s(TV_u_27a,V_x)))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bags_BAGu_u_IN0, axiom, ![TV_u_27a]: ![V_e, V_b]: s(t_bool,h4s_bags_bagu_u_in(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))) = s(t_bool,h4s_bags_bagu_u_inn(s(TV_u_27a,V_e),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_fun(TV_u_27a,t_h4s_nums_num),V_b)))).
fof(ah4s_bags_BAGu_u_INN0, axiom, ![TV_u_27a]: ![V_n, V_e, V_b]: s(t_bool,h4s_bags_bagu_u_inn(s(TV_u_27a,V_e),s(t_h4s_nums_num,V_n),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))) = s(t_bool,h4s_arithmetics_u_3eu_3d(s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b),s(TV_u_27a,V_e))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_ADDu_u_EQu_u_0, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0) <=> (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0) & s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_bools_NOTu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ?[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_bags_BAGu_u_CHOICEu_u_SING, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_bags_bagu_u_choice(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag))))) = s(TV_u_27a,V_x)).
fof(ah4s_bags_BAGu_u_DIFFu_u_INSERTu_u_same, axiom, ![TV_u_27a]: ![V_x, V_b2, V_b1]: s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_diff(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1))),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2))))) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_diff(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2)))).
fof(ah4s_bags_BAGu_u_RESTu_u_DEF, axiom, ![TV_u_27a]: ![V_b]: s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_rest(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_diff(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_elu_u_bag(s(TV_u_27a,h4s_bags_bagu_u_choice(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)))))))).
fof(ah4s_bags_BAGu_u_DIFFu_u_EMPTYu_u_simpleu_c2, axiom, ![TV_u_27c]: ![V_b]: s(t_fun(TV_u_27c,t_h4s_nums_num),h4s_bags_bagu_u_diff(s(t_fun(TV_u_27c,t_h4s_nums_num),h4s_bags_emptyu_u_bag),s(t_fun(TV_u_27c,t_h4s_nums_num),V_b))) = s(t_fun(TV_u_27c,t_h4s_nums_num),h4s_bags_emptyu_u_bag)).
fof(ah4s_bags_ELu_u_BAG0, axiom, ![TV_u_27a]: ![V_e]: s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_elu_u_bag(s(TV_u_27a,V_e))) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag)))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_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_bags_BAGu_u_INSERT0, axiom, ![TV_u_27a]: ![V_e, V_b, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_e)) & s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))),s(TV_u_27a,V_x))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,V_v),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_b),s(TV_u_27a,V_e))),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,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b),s(TV_u_27a,V_x))))))).
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_arithmetics_SUBu_u_0u_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_nums_0)).
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_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_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_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_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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EXISTSu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (?[V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,f))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c0, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) & p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) | ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,f))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EXISTSu_u_REFL, axiom, ![TV_u_27a]: ![V_a]: ?[V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_a)).
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_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => 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_numerals_numeralu_u_distribu_c30, axiom, ![V_n]: (p(s(t_bool,h4s_arithmetics_u_3eu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n)))) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0))).
fof(ah4s_bags_PSUBu_u_BAGu_u_CARD, axiom, ![TV_u_27a]: ![V_b2, V_b1]: ((p(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2)))) & p(s(t_bool,h4s_bags_psubu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2))))) => 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_bags_bagu_u_card(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1))))),s(t_h4s_nums_num,h4s_bags_bagu_u_card(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2)))))))).
fof(ah4s_bags_ITBAGu_u_tupledu_u_primitiveu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_4]: (![V_ITBAGu_u_tupled, V_f, V_b, V_acc]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_fun(TV_u_27a,t_h4s_nums_num),V_b) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag)) & s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27b,TV_u_27b))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27b,TV_u_27b)))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b),V_ITBAGu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))),s(TV_u_27b,V_acc))) = s(TV_u_27b,h4s_combins_i(s(TV_u_27b,h4s_bools_cond(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))),s(TV_u_27b,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27b,V_acc),s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b),V_ITBAGu_u_tupled),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_rest(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))),s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27a,h4s_bags_bagu_u_choice(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))))),s(TV_u_27b,V_acc))))))))),s(TV_u_27b,h4s_bools_arb)))))) => ![V_uu_3]: (![V_ITBAGu_u_tupled, V_f, V_b]: s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27b,TV_u_27b))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27b,TV_u_27b)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b),V_ITBAGu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))) = s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27b,TV_u_27b))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27b,TV_u_27b)))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b),V_ITBAGu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))) => ![V_uu_2]: (![V_ITBAGu_u_tupled, V_f, V_a]: s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b),V_ITBAGu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),V_a))) = s(TV_u_27b,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),V_a),s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27b,TV_u_27b))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27b,TV_u_27b)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b),V_ITBAGu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))))) => ![V_uu_1]: (![V_f, V_ITBAGu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b),V_ITBAGu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b),V_ITBAGu_u_tupled))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))) => ![V_uu_0]: (![V_f, V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),t_bool)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),t_bool)),t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),t_bool)),V_R)))) & ![V_acc, V_b]: ((p(s(t_bool,h4s_bags_finiteu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)))) & ~ (s(t_fun(TV_u_27a,t_h4s_nums_num),V_b) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),t_bool)),V_R),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_rest(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))),s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27a,h4s_bags_bagu_u_choice(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))))),s(TV_u_27b,V_acc))))))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b),s(TV_u_27b,V_acc))))))))) => ![V_f]: s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b),h4s_bags_itbagu_u_tupled(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))) = s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),t_bool)),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),t_bool)),t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))))),s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))))))))))).
fof(ah4s_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_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_bags_ITBAGu_u_curriedu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_x1, V_x, V_f]: s(TV_u_27b,h4s_bags_itbag(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(t_fun(TV_u_27a,t_h4s_nums_num),V_x),s(TV_u_27b,V_x1))) = s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),TV_u_27b),h4s_bags_itbagu_u_tupled(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_nums_num),TV_u_27b),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_h4s_nums_num),V_x),s(TV_u_27b,V_x1)))))).
fof(ah4s_bags_PSUBu_u_BAGu_u_REST, axiom, ![TV_u_27a]: ![V_b]: (~ (s(t_fun(TV_u_27a,t_h4s_nums_num),V_b) = s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag)) => p(s(t_bool,h4s_bags_psubu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_rest(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b))),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b)))))).
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_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_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_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_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
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_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,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_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_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_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_combins_ou_u_DEF, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_g, V_f, V_x]: s(TV_u_27b,h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x)))))).
fof(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_bools_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_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_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bags_SUBu_u_BAG0, axiom, ![TV_u_27a]: ![V_b2, V_b1]: (p(s(t_bool,h4s_bags_subu_u_bag(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2)))) <=> ![V_x, V_n]: (p(s(t_bool,h4s_bags_bagu_u_inn(s(TV_u_27a,V_x),s(t_h4s_nums_num,V_n),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b1)))) => p(s(t_bool,h4s_bags_bagu_u_inn(s(TV_u_27a,V_x),s(t_h4s_nums_num,V_n),s(t_fun(TV_u_27a,t_h4s_nums_num),V_b2))))))).
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_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_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_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_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_arithmetics_GREATERu_u_EQ, axiom, ![V_n, V_m]: s(t_bool,h4s_arithmetics_u_3eu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))).
fof(ah4s_bools_UNWINDu_u_FORALLu_u_THM2, axiom, ![TV_u_27a]: ![V_v, V_f]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_v) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_v)))))).
fof(ah4s_arithmetics_LESSu_u_EQu_u_REFL, axiom, ![V_m]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_m))))).
fof(ah4s_arithmetics_ZEROu_u_LESSu_u_EQ, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))))).
fof(ah4s_bools_FORALLu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_arithmetics_LESSu_u_EQu_u_TRANS, axiom, ![V_p, V_n, V_m]: ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c2, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_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_RIGHTu_u_FORALLu_u_IMPu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,V_P)) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (p(s(t_bool,V_P)) => ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_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_bools_DISJu_u_COMM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_bools_LEFTu_u_ANDu_u_OVERu_u_OR, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) & (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) & p(s(t_bool,V_B))) | (p(s(t_bool,V_A)) & p(s(t_bool,V_C)))))).
fof(ah4s_bools_RIGHTu_u_ANDu_u_OVERu_u_OR, axiom, ![V_C, V_B, V_A]: (((p(s(t_bool,V_B)) | p(s(t_bool,V_C))) & p(s(t_bool,V_A))) <=> ((p(s(t_bool,V_B)) & p(s(t_bool,V_A))) | (p(s(t_bool,V_C)) & p(s(t_bool,V_A)))))).
fof(ah4s_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_DISJu_u_IMPu_u_THM, 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_R))) & (p(s(t_bool,V_Q)) => p(s(t_bool,V_R)))))).
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_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_EQu_u_IMPu_u_THM, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) & (p(s(t_bool,V_t2)) => p(s(t_bool,V_t1)))))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_bags_BAGu_u_UNION0, axiom, ![TV_u_27a]: ![V_c, V_b, V_x]: s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_bagu_u_union(s(t_fun(TV_u_27a,t_h4s_nums_num),V_b),s(t_fun(TV_u_27a,t_h4s_nums_num),V_c))),s(TV_u_27a,V_x))) = 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_b),s(TV_u_27a,V_x))),s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_c),s(TV_u_27a,V_x)))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ch4s_bags_BAGu_u_EVERYu_u_THMu_c0, conjecture, ![TV_u_27a]: ![V_P]: p(s(t_bool,h4s_bags_bagu_u_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,t_h4s_nums_num),h4s_bags_emptyu_u_bag))))).
