%   ORIGINAL: h4/defCNF/BIGSTEP
% 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/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/defCNF/UNIQUE__def_c2: !v n j conn a. h4/defCNF/UNIQUE v n (h4/pair/_2C conn (h4/pair/_2C (h4/sum/INR a) (h4/sum/INL j))) <=> v n <=> conn a (v j)
% Assm: h4/defCNF/UNIQUE__def_c1: !v n i conn b. h4/defCNF/UNIQUE v n (h4/pair/_2C conn (h4/pair/_2C (h4/sum/INL i) (h4/sum/INR b))) <=> v n <=> conn (v i) b
% Assm: h4/defCNF/UNIQUE__def_c3: !v n conn b a. h4/defCNF/UNIQUE v n (h4/pair/_2C conn (h4/pair/_2C (h4/sum/INR a) (h4/sum/INR b))) <=> v n <=> conn a b
% Assm: h4/defCNF/UNIQUE__ind: !P. (!v n conn i j. P v n (h4/pair/_2C conn (h4/pair/_2C (h4/sum/INL i) (h4/sum/INL j)))) /\ (!v n conn i b. P v n (h4/pair/_2C conn (h4/pair/_2C (h4/sum/INL i) (h4/sum/INR b)))) /\ (!v n conn a j. P v n (h4/pair/_2C conn (h4/pair/_2C (h4/sum/INR a) (h4/sum/INL j)))) /\ (!v n conn a b. P v n (h4/pair/_2C conn (h4/pair/_2C (h4/sum/INR a) (h4/sum/INR b)))) ==> (!v v1 v2 v3 v4. P v v1 (h4/pair/_2C v2 (h4/pair/_2C v3 v4)))
% Assm: h4/pred__set/MIN__SET__LEM: !s. ~(s = h4/pred__set/EMPTY) ==> h4/bool/IN (h4/pred__set/MIN__SET s) s /\ (!x. h4/bool/IN x s ==> h4/arithmetic/_3C_3D (h4/pred__set/MIN__SET s) x)
% Assm: h4/relation/WF__EMPTY__REL: h4/relation/WF h4/relation/EMPTY__REL
% Assm: h4/while/LEAST__DEF: !P. h4/while/LEAST P = h4/while/WHILE (h4/combin/o $not P) h4/num/SUC h4/num/0
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/TRUTH: T
% Assm: h4/pred__set/MAX__SET__UNION: !B A. h4/pred__set/FINITE A /\ h4/pred__set/FINITE B ==> h4/pred__set/MAX__SET (h4/pred__set/UNION A B) = h4/arithmetic/MAX (h4/pred__set/MAX__SET A) (h4/pred__set/MAX__SET B)
% Assm: h4/pred__set/PROD__SET__THM_c1: !x s. h4/pred__set/FINITE s ==> h4/pred__set/PROD__SET (h4/pred__set/INSERT x s) = h4/arithmetic/_2A x (h4/pred__set/PROD__SET (h4/pred__set/DELETE s x))
% Assm: h4/pred__set/SUM__SET__UNION: !t s. h4/pred__set/FINITE s /\ h4/pred__set/FINITE t ==> h4/pred__set/SUM__SET (h4/pred__set/UNION s t) = h4/arithmetic/_2D (h4/arithmetic/_2B (h4/pred__set/SUM__SET s) (h4/pred__set/SUM__SET t)) (h4/pred__set/SUM__SET (h4/pred__set/INTER s t))
% Assm: h4/rich__list/every__count__list: !n P. h4/list/EVERY P (h4/rich__list/COUNT__LIST n) <=> (!m. h4/prim__rec/_3C m n ==> P m)
% Assm: h4/pred__set/SUBSET__MIN__SET: !J I. ~(I = h4/pred__set/EMPTY) /\ ~(J = h4/pred__set/EMPTY) /\ h4/pred__set/SUBSET I J ==> h4/arithmetic/_3C_3D (h4/pred__set/MIN__SET J) (h4/pred__set/MIN__SET I)
% Assm: h4/defCNF/UNIQUE__curried__def: !x2 x1 x. h4/defCNF/UNIQUE x x1 x2 <=> h4/defCNF/UNIQUE__tupled (h4/pair/_2C x (h4/pair/_2C x1 x2))
% Assm: h4/defCNF/UNIQUE__tupled__primitive__def: h4/defCNF/UNIQUE__tupled = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R)) (\UNIQUE__tupled a_27. h4/pair/pair__CASE a_27 (\v v2. h4/pair/pair__CASE v2 (\n v4. h4/pair/pair__CASE v4 (\conn v6. h4/pair/pair__CASE v6 (\v7 v8. h4/sum/sum__CASE v7 (\i. h4/sum/sum__CASE v8 (\j. h4/combin/I (v n <=> conn (v i) (v j))) (\b. h4/combin/I (v n <=> conn (v i) b))) (\a. h4/sum/sum__CASE v8 (\j_27. h4/combin/I (v n <=> conn a (v j_27))) (\b_27. h4/combin/I (v n <=> conn a b_27))))))))
% Assm: h4/sum/sum__case__def_c1: !y f1 f. h4/sum/sum__CASE (h4/sum/INR y) f f1 = f1 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/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% Assm: h4/sum/sum__case__def_c0: !x f1 f. h4/sum/sum__CASE (h4/sum/INL x) f f1 = f x
% Assm: h4/pred__set/MIN__SET__THM_c1: !s e2 e1. h4/pred__set/MIN__SET (h4/pred__set/INSERT e1 (h4/pred__set/INSERT e2 s)) = h4/arithmetic/MIN e1 (h4/pred__set/MIN__SET (h4/pred__set/INSERT e2 s))
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> 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__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/dc__disj: !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/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% 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/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/pred__set/PROD__SET__DEF: h4/pred__set/PROD__SET = h4/pred__set/PROD__IMAGE h4/combin/I
% Assm: h4/pred__set/PROD__IMAGE__THM_c1: !s f e. h4/pred__set/FINITE s ==> h4/pred__set/PROD__IMAGE f (h4/pred__set/INSERT e s) = h4/arithmetic/_2A (f e) (h4/pred__set/PROD__IMAGE f (h4/pred__set/DELETE s e))
% Assm: h4/pred__set/SUM__IMAGE__UNION: !t s f. h4/pred__set/FINITE s /\ h4/pred__set/FINITE t ==> h4/pred__set/SUM__IMAGE f (h4/pred__set/UNION s t) = h4/arithmetic/_2D (h4/arithmetic/_2B (h4/pred__set/SUM__IMAGE f s) (h4/pred__set/SUM__IMAGE f t)) (h4/pred__set/SUM__IMAGE f (h4/pred__set/INTER s t))
% Assm: h4/pred__set/SUM__SET__DEF: h4/pred__set/SUM__SET = h4/pred__set/SUM__IMAGE h4/combin/I
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/bool/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/bool/LEFT__FORALL__OR__THM: !Q P. (!x. P x \/ Q) <=> (!x. P x) \/ Q
% 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/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/RIGHT__OR__EXISTS__THM: !Q P. P \/ (?x. Q x) <=> (?x. P \/ Q x)
% Assm: h4/bool/EXISTS__OR__THM: !Q P. (?x. P x \/ Q x) <=> (?x. P x) \/ (?x. Q x)
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/bool/LEFT__OR__EXISTS__THM: !Q P. (?x. P x) \/ Q <=> (?x. P x \/ Q)
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/LEFT__EXISTS__AND__THM: !Q P. (?x. P x /\ Q) <=> (?x. P x) /\ Q
% Assm: h4/sum/sum__CASES: !ss. (?x. ss = h4/sum/INL x) \/ (?y. ss = h4/sum/INR y)
% 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/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/bool/RIGHT__OR__OVER__AND: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm: h4/pred__set/IN__INSERT: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% 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/SUC__ONE__ADD: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% 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: !n m. h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm: h4/arithmetic/ADD__MONO__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm: h4/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% 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/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/bool/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Assm: h4/pred__set/MAX__SET__REWRITES_c0: h4/pred__set/MAX__SET h4/pred__set/EMPTY = h4/num/0
% Assm: h4/pred__set/MAX__SET__THM_c1: !s e. h4/pred__set/FINITE s ==> h4/pred__set/MAX__SET (h4/pred__set/INSERT e s) = h4/arithmetic/MAX e (h4/pred__set/MAX__SET s)
% Assm: h4/pred__set/FINITE__UNION: !t s. h4/pred__set/FINITE (h4/pred__set/UNION s t) <=> h4/pred__set/FINITE s /\ h4/pred__set/FINITE t
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/pred__set/FINITE__INDUCT: !P. P h4/pred__set/EMPTY /\ (!s. h4/pred__set/FINITE s /\ P s ==> (!e. ~h4/bool/IN e s ==> P (h4/pred__set/INSERT e s))) ==> (!s. h4/pred__set/FINITE s ==> P s)
% Assm: h4/pred__set/SET__CASES: !s. s = h4/pred__set/EMPTY \/ (?x t. s = h4/pred__set/INSERT x t /\ ~h4/bool/IN x t)
% Assm: h4/pred__set/FINITE__INSERT: !x s. h4/pred__set/FINITE (h4/pred__set/INSERT x s) <=> h4/pred__set/FINITE s
% Assm: h4/pred__set/UNION__EMPTY_c1: !s. h4/pred__set/UNION s h4/pred__set/EMPTY = s
% Assm: h4/pred__set/IN__UNION: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x t
% Assm: h4/pred__set/UNION__EMPTY_c0: !s. h4/pred__set/UNION h4/pred__set/EMPTY s = s
% Assm: h4/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm: h4/arithmetic/MAX__0_c1: !n. h4/arithmetic/MAX h4/num/0 n = n
% Assm: h4/arithmetic/MAX__ASSOC: !p n m. h4/arithmetic/MAX m (h4/arithmetic/MAX n p) = h4/arithmetic/MAX (h4/arithmetic/MAX m n) p
% Assm: h4/arithmetic/MAX__COMM: !n m. h4/arithmetic/MAX m n = h4/arithmetic/MAX n m
% Assm: h4/arithmetic/MAX__0_c0: !n. h4/arithmetic/MAX n h4/num/0 = n
% Assm: h4/marker/AC__DEF: !b2 b1. h4/marker/AC b1 b2 <=> b1 /\ b2
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/DISJ__COMM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/LCOMM__THM: !f. (!x y z. f x (f y z) = f (f x y) z) ==> (!x y. f x y = f y x) ==> (!x y z. f x (f y z) = f y (f x z))
% Assm: h4/pred__set/SUBSET__DEF: !t s. h4/pred__set/SUBSET s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN x t)
% Assm: h4/pred__set/MEMBER__NOT__EMPTY: !s. (?x. h4/bool/IN x s) <=> ~(s = h4/pred__set/EMPTY)
% Assm: h4/pred__set/MIN__SET__DEF: h4/pred__set/MIN__SET = h4/while/LEAST
% Assm: h4/bool/IN__DEF: h4/bool/IN = (\x f. f x)
% Assm: h4/while/FULL__LEAST__INTRO: !x P. P x ==> P (h4/while/LEAST P) /\ h4/arithmetic/_3C_3D (h4/while/LEAST P) x
% Assm: h4/bool/LEFT__AND__FORALL__THM: !Q P. (!x. P x) /\ Q <=> (!x. P x /\ Q)
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/numeral/numeral__add_c0: !n. h4/numeral/iZ (h4/arithmetic/_2B h4/arithmetic/ZERO n) = n
% Assm: h4/numeral/numeral__distrib_c2: !n m. h4/arithmetic/_2B (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm: h4/numeral/numeral__add_c2: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT2 (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm: h4/rich__list/COUNT__LIST__def_c0: h4/rich__list/COUNT__LIST h4/num/0 = h4/list/NIL
% Assm: h4/rich__list/COUNT__LIST__def_c1: !n. h4/rich__list/COUNT__LIST (h4/num/SUC n) = h4/list/CONS h4/num/0 (h4/list/MAP h4/num/SUC (h4/rich__list/COUNT__LIST n))
% Assm: h4/arithmetic/LESS__MONO__EQ: !n m. h4/prim__rec/_3C (h4/num/SUC m) (h4/num/SUC n) <=> h4/prim__rec/_3C m n
% Assm: h4/arithmetic/num__CASES: !m. m = h4/num/0 \/ (?n. m = h4/num/SUC n)
% 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/prim__rec/LESS__0: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm: h4/prim__rec/NOT__LESS__0: !n. ~h4/prim__rec/_3C n h4/num/0
% Assm: h4/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Assm: h4/list/EVERY__MAP: !l f P. h4/list/EVERY P (h4/list/MAP f l) <=> h4/list/EVERY (\x. P (f x)) l
% 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/EVERY__DEF_c0: !P. h4/list/EVERY P h4/list/NIL <=> T
% Goal: !R Q P. (!v. P v ==> (Q <=> R v)) ==> ((?v. P v) /\ Q <=> (?v. P v /\ R v))
%   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_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_combins_Iu_u_THM]: !x. happ h4/combin/I x = x
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_defCNFs_UNIQUEu_u_defu_c2]: !v n j conn a. h4/defCNF/UNIQUE v n (h4/pair/_2C conn (h4/pair/_2C (h4/sum/INR a) (h4/sum/INL j))) <=> happ v n <=> happ (happ conn a) (happ v j)
% Assm [h4s_defCNFs_UNIQUEu_u_defu_c1]: !v n i conn b. h4/defCNF/UNIQUE v n (h4/pair/_2C conn (h4/pair/_2C (h4/sum/INL i) (h4/sum/INR b))) <=> happ v n <=> happ (happ conn (happ v i)) b
% Assm [h4s_defCNFs_UNIQUEu_u_defu_c3]: !v n conn b a. h4/defCNF/UNIQUE v n (h4/pair/_2C conn (h4/pair/_2C (h4/sum/INR a) (h4/sum/INR b))) <=> happ v n <=> happ (happ conn a) b
% Assm [h4s_defCNFs_UNIQUEu_u_ind]: !P. (!v n conn i j. happ (happ (happ P v) n) (h4/pair/_2C conn (h4/pair/_2C (h4/sum/INL i) (h4/sum/INL j)))) /\ (!v n conn i b. happ (happ (happ P v) n) (h4/pair/_2C conn (h4/pair/_2C (h4/sum/INL i) (h4/sum/INR b)))) /\ (!v n conn a j. happ (happ (happ P v) n) (h4/pair/_2C conn (h4/pair/_2C (h4/sum/INR a) (h4/sum/INL j)))) /\ (!v n conn a b. happ (happ (happ P v) n) (h4/pair/_2C conn (h4/pair/_2C (h4/sum/INR a) (h4/sum/INR b)))) ==> (!v v1 v2 v3 v4. happ (happ (happ P v) v1) (h4/pair/_2C v2 (h4/pair/_2C v3 v4)))
% Assm [h4s_predu_u_sets_MINu_u_SETu_u_LEM]: !s. ~(s = h4/pred__set/EMPTY) ==> h4/bool/IN (happ h4/pred__set/MIN__SET s) s /\ (!x. h4/bool/IN x s ==> h4/arithmetic/_3C_3D (happ h4/pred__set/MIN__SET s) x)
% Assm [h4s_relations_WFu_u_EMPTYu_u_REL]: h4/relation/WF h4/relation/EMPTY__REL
% Assm [h4s_whiles_LEASTu_u_DEF]: !P. happ h4/while/LEAST P = h4/while/WHILE (h4/combin/o $not P) h4/num/SUC h4/num/0
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_predu_u_sets_MAXu_u_SETu_u_UNION]: !B A. h4/pred__set/FINITE A /\ h4/pred__set/FINITE B ==> h4/pred__set/MAX__SET (h4/pred__set/UNION A B) = h4/arithmetic/MAX (h4/pred__set/MAX__SET A) (h4/pred__set/MAX__SET B)
% Assm [h4s_predu_u_sets_PRODu_u_SETu_u_THMu_c1]: !x s. h4/pred__set/FINITE s ==> happ h4/pred__set/PROD__SET (h4/pred__set/INSERT x s) = h4/arithmetic/_2A x (happ h4/pred__set/PROD__SET (h4/pred__set/DELETE s x))
% Assm [h4s_predu_u_sets_SUMu_u_SETu_u_UNION]: !t s. h4/pred__set/FINITE s /\ h4/pred__set/FINITE t ==> happ h4/pred__set/SUM__SET (h4/pred__set/UNION s t) = h4/arithmetic/_2D (h4/arithmetic/_2B (happ h4/pred__set/SUM__SET s) (happ h4/pred__set/SUM__SET t)) (happ h4/pred__set/SUM__SET (h4/pred__set/INTER s t))
% Assm [h4s_richu_u_lists_everyu_u_countu_u_list]: !n P. h4/list/EVERY P (h4/rich__list/COUNT__LIST n) <=> (!m. h4/prim__rec/_3C m n ==> happ P m)
% Assm [h4s_predu_u_sets_SUBSETu_u_MINu_u_SET]: !J I. ~(I = h4/pred__set/EMPTY) /\ ~(J = h4/pred__set/EMPTY) /\ h4/pred__set/SUBSET I J ==> h4/arithmetic/_3C_3D (happ h4/pred__set/MIN__SET J) (happ h4/pred__set/MIN__SET I)
% Assm [h4s_defCNFs_UNIQUEu_u_curriedu_u_def]: !x2 x1 x. h4/defCNF/UNIQUE x x1 x2 <=> happ h4/defCNF/UNIQUE__tupled (h4/pair/_2C x (h4/pair/_2C x1 x2))
% Assm [h4s_defCNFs_UNIQUEu_u_tupledu_u_primitiveu_u_def]: !_16. (!v n conn a b_27. ?v. (v <=> happ v n <=> happ (happ conn a) b_27) /\ (happ (happ (happ (happ (happ _16 v) n) conn) a) b_27 <=> happ h4/combin/I v)) ==> (!_15. (!n conn a v j_27. ?v. (v <=> happ v n <=> happ (happ conn a) (happ v j_27)) /\ (happ (happ (happ (happ (happ _15 n) conn) a) v) j_27 <=> happ h4/combin/I v)) ==> (!_14. (!v8 v n conn a. happ (happ (happ (happ (happ _14 v8) v) n) conn) a <=> h4/sum/sum__CASE v8 (happ (happ (happ (happ _15 n) conn) a) v) (happ (happ (happ (happ _16 v) n) conn) a)) ==> (!_13. (!n conn v i b. ?v. (v <=> happ v n <=> happ (happ conn (happ v i)) b) /\ (happ (happ (happ (happ (happ _13 n) conn) v) i) b <=> happ h4/combin/I v)) ==> (!_12. (!n conn i v j. ?v. (v <=> happ v n <=> happ (happ conn (happ v i)) (happ v j)) /\ (happ (happ (happ (happ (happ _12 n) conn) i) v) j <=> happ h4/combin/I v)) ==> (!_11. (!v8 n conn v i. happ (happ (happ (happ (happ _11 v8) n) conn) v) i <=> h4/sum/sum__CASE v8 (happ (happ (happ (happ _12 n) conn) i) v) (happ (happ (happ (happ _13 n) conn) v) i)) ==> (!_10. (!v7 v n conn v8. happ (happ (happ (happ (happ _10 v7) v) n) conn) v8 <=> h4/sum/sum__CASE v7 (happ (happ (happ (happ _11 v8) n) conn) v) (happ (happ (happ (happ _14 v8) v) n) conn)) ==> (!_9. (!v n conn v7. happ (happ (happ (happ _9 v) n) conn) v7 = happ (happ (happ (happ _10 v7) v) n) conn) ==> (!_8. (!v n conn v6. happ (happ (happ (happ _8 v) n) conn) v6 <=> h4/pair/pair__CASE v6 (happ (happ (happ _9 v) n) conn)) ==> (!_7. (!v n conn. happ (happ (happ _7 v) n) conn = happ (happ (happ _8 v) n) conn) ==> (!_6. (!v n v4. happ (happ (happ _6 v) n) v4 <=> h4/pair/pair__CASE v4 (happ (happ _7 v) n)) ==> (!_5. (!v n. happ (happ _5 v) n = happ (happ _6 v) n) ==> (!_4. (!v v2. happ (happ _4 v) v2 <=> h4/pair/pair__CASE v2 (happ _5 v)) ==> (!_3. (!v. happ _3 v = happ _4 v) ==> (!_2. (!a_27. happ _2 a_27 <=> h4/pair/pair__CASE a_27 _3) ==> (!_1. (!UNIQUE__tupled. happ _1 UNIQUE__tupled = _2) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R) ==> h4/defCNF/UNIQUE__tupled = h4/relation/WFREC (h4/min/_40 _0) _1))))))))))))))))
% Assm [h4s_sums_sumu_u_caseu_u_defu_c1]: !y f1 f. h4/sum/sum__CASE (h4/sum/INR y) f f1 = happ f1 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_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_sums_sumu_u_caseu_u_defu_c0]: !x f1 f. h4/sum/sum__CASE (h4/sum/INL x) f f1 = happ f x
% Assm [h4s_predu_u_sets_MINu_u_SETu_u_THMu_c1]: !s e2 e1. happ h4/pred__set/MIN__SET (h4/pred__set/INSERT e1 (h4/pred__set/INSERT e2 s)) = h4/arithmetic/MIN e1 (happ h4/pred__set/MIN__SET (h4/pred__set/INSERT e2 s))
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> 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_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_dcu_u_disj]: !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_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% 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_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_predu_u_sets_PRODu_u_SETu_u_DEF]: h4/pred__set/PROD__SET = h4/pred__set/PROD__IMAGE h4/combin/I
% Assm [h4s_predu_u_sets_PRODu_u_IMAGEu_u_THMu_c1]: !s f e. h4/pred__set/FINITE s ==> happ (h4/pred__set/PROD__IMAGE f) (h4/pred__set/INSERT e s) = h4/arithmetic/_2A (happ f e) (happ (h4/pred__set/PROD__IMAGE f) (h4/pred__set/DELETE s e))
% Assm [h4s_predu_u_sets_SUMu_u_IMAGEu_u_UNION]: !t s f. h4/pred__set/FINITE s /\ h4/pred__set/FINITE t ==> happ (h4/pred__set/SUM__IMAGE f) (h4/pred__set/UNION s t) = h4/arithmetic/_2D (h4/arithmetic/_2B (happ (h4/pred__set/SUM__IMAGE f) s) (happ (h4/pred__set/SUM__IMAGE f) t)) (happ (h4/pred__set/SUM__IMAGE f) (h4/pred__set/INTER s t))
% Assm [h4s_predu_u_sets_SUMu_u_SETu_u_DEF]: h4/pred__set/SUM__SET = h4/pred__set/SUM__IMAGE h4/combin/I
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_bools_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% Assm [h4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. happ P x \/ Q) <=> (!x. happ P x) \/ Q
% 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_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM]: !Q P. P \/ (?x. happ Q x) <=> (?x. P \/ happ Q x)
% 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_pairs_UNCURRYu_u_DEF]: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_bools_LEFTu_u_ORu_u_EXISTSu_u_THM]: !Q P. (?x. happ P x) \/ Q <=> (?x. happ P x \/ Q)
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_bools_LEFTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. happ P x /\ Q) <=> (?x. happ P x) /\ Q
% Assm [h4s_sums_sumu_u_CASES]: !ss. (?x. ss = h4/sum/INL x) \/ (?y. ss = h4/sum/INR y)
% 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_RIGHTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. P \/ happ Q x) <=> P \/ (!x. happ Q x)
% Assm [h4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND]: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm [h4s_predu_u_sets_INu_u_INSERT]: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% 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_SUCu_u_ONEu_u_ADD]: !n. happ h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm [h4s_arithmetics_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_arithmetics_NOTu_u_LESS]: !n m. ~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_EQ]: !n m. h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D (happ h4/num/SUC m) n
% Assm [h4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% 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_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_bools_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% Assm [h4s_predu_u_sets_MAXu_u_SETu_u_REWRITESu_c0]: h4/pred__set/MAX__SET h4/pred__set/EMPTY = h4/num/0
% Assm [h4s_predu_u_sets_MAXu_u_SETu_u_THMu_c1]: !s e. h4/pred__set/FINITE s ==> h4/pred__set/MAX__SET (h4/pred__set/INSERT e s) = h4/arithmetic/MAX e (h4/pred__set/MAX__SET s)
% Assm [h4s_predu_u_sets_FINITEu_u_UNION]: !t s. h4/pred__set/FINITE (h4/pred__set/UNION s t) <=> h4/pred__set/FINITE s /\ h4/pred__set/FINITE t
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_predu_u_sets_FINITEu_u_INDUCT]: !P. happ P h4/pred__set/EMPTY /\ (!s. h4/pred__set/FINITE s /\ happ P s ==> (!e. ~h4/bool/IN e s ==> happ P (h4/pred__set/INSERT e s))) ==> (!s. h4/pred__set/FINITE s ==> happ P s)
% Assm [h4s_predu_u_sets_SETu_u_CASES]: !s. s = h4/pred__set/EMPTY \/ (?x t. s = h4/pred__set/INSERT x t /\ ~h4/bool/IN x t)
% Assm [h4s_predu_u_sets_FINITEu_u_INSERT]: !x s. h4/pred__set/FINITE (h4/pred__set/INSERT x s) <=> h4/pred__set/FINITE s
% Assm [h4s_predu_u_sets_UNIONu_u_EMPTYu_c1]: !s. h4/pred__set/UNION s h4/pred__set/EMPTY = s
% Assm [h4s_predu_u_sets_INu_u_UNION]: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x t
% Assm [h4s_predu_u_sets_UNIONu_u_EMPTYu_c0]: !s. h4/pred__set/UNION h4/pred__set/EMPTY s = s
% Assm [h4s_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm [h4s_arithmetics_MAXu_u_0u_c1]: !n. h4/arithmetic/MAX h4/num/0 n = n
% Assm [h4s_arithmetics_MAXu_u_ASSOC]: !p n m. h4/arithmetic/MAX m (h4/arithmetic/MAX n p) = h4/arithmetic/MAX (h4/arithmetic/MAX m n) p
% Assm [h4s_arithmetics_MAXu_u_COMM]: !n m. h4/arithmetic/MAX m n = h4/arithmetic/MAX n m
% Assm [h4s_arithmetics_MAXu_u_0u_c0]: !n. h4/arithmetic/MAX n h4/num/0 = n
% Assm [h4s_markers_ACu_u_DEF]: !b2 b1. h4/marker/AC b1 b2 <=> b1 /\ b2
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_DISJu_u_COMM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_LCOMMu_u_THM]: !f. (!x y z. happ (happ f x) (happ (happ f y) z) = happ (happ f (happ (happ f x) y)) z) ==> (!x y. happ (happ f x) y = happ (happ f y) x) ==> (!x y z. happ (happ f x) (happ (happ f y) z) = happ (happ f y) (happ (happ f x) z))
% Assm [h4s_predu_u_sets_SUBSETu_u_DEF]: !t s. h4/pred__set/SUBSET s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN x t)
% Assm [h4s_predu_u_sets_MEMBERu_u_NOTu_u_EMPTY]: !s. (?x. h4/bool/IN x s) <=> ~(s = h4/pred__set/EMPTY)
% Assm [h4s_predu_u_sets_MINu_u_SETu_u_DEF]: h4/pred__set/MIN__SET = h4/while/LEAST
% Assm [h4s_bools_INu_u_DEF]: !x x. h4/bool/IN x x <=> happ x x
% Assm [h4s_whiles_FULLu_u_LEASTu_u_INTRO]: !x P. happ P x ==> happ P (happ h4/while/LEAST P) /\ h4/arithmetic/_3C_3D (happ h4/while/LEAST P) x
% Assm [h4s_bools_LEFTu_u_ANDu_u_FORALLu_u_THM]: !Q P. (!x. happ P x) /\ Q <=> (!x. happ P x /\ Q)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_numerals_numeralu_u_addu_c0]: !n. h4/numeral/iZ (h4/arithmetic/_2B h4/arithmetic/ZERO n) = n
% Assm [h4s_numerals_numeralu_u_distribu_c2]: !n m. h4/arithmetic/_2B (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm [h4s_numerals_numeralu_u_addu_c2]: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT2 (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm [h4s_richu_u_lists_COUNTu_u_LISTu_u_defu_c0]: h4/rich__list/COUNT__LIST h4/num/0 = h4/list/NIL
% Assm [h4s_richu_u_lists_COUNTu_u_LISTu_u_defu_c1]: !n. h4/rich__list/COUNT__LIST (happ h4/num/SUC n) = h4/list/CONS h4/num/0 (h4/list/MAP h4/num/SUC (h4/rich__list/COUNT__LIST n))
% Assm [h4s_arithmetics_LESSu_u_MONOu_u_EQ]: !n m. h4/prim__rec/_3C (happ h4/num/SUC m) (happ h4/num/SUC n) <=> h4/prim__rec/_3C m n
% Assm [h4s_arithmetics_numu_u_CASES]: !m. m = h4/num/0 \/ (?n. m = happ h4/num/SUC n)
% 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_primu_u_recs_LESSu_u_0]: !n. h4/prim__rec/_3C h4/num/0 (happ h4/num/SUC n)
% Assm [h4s_primu_u_recs_NOTu_u_LESSu_u_0]: !n. ~h4/prim__rec/_3C n h4/num/0
% Assm [h4s_nums_INDUCTION]: !P. happ P h4/num/0 /\ (!n. happ P n ==> happ P (happ h4/num/SUC n)) ==> (!n. happ P n)
% Assm [h4s_lists_EVERYu_u_MAP]: !_0. (!P f x. happ (happ (happ _0 P) f) x <=> happ P (happ f x)) ==> (!l f P. h4/list/EVERY P (h4/list/MAP f l) <=> h4/list/EVERY (happ (happ _0 P) f) l)
% 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_EVERYu_u_DEFu_c0]: !P. h4/list/EVERY P h4/list/NIL <=> T
% Goal: !R Q P. (!v. happ P v ==> (Q <=> happ R v)) ==> ((?v. happ P v) /\ Q <=> (?v. happ P v /\ happ R v))
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_Q1260867,TV_Q1260863]: ![V_f, V_g]: (![V_x]: s(TV_Q1260863,happ(s(t_fun(TV_Q1260867,TV_Q1260863),V_f),s(TV_Q1260867,V_x))) = s(TV_Q1260863,happ(s(t_fun(TV_Q1260867,TV_Q1260863),V_g),s(TV_Q1260867,V_x))) => s(t_fun(TV_Q1260867,TV_Q1260863),V_f) = s(t_fun(TV_Q1260867,TV_Q1260863),V_g))).
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_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(TV_u_27a,V_x))) = 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_defCNFs_UNIQUEu_u_defu_c2, axiom, ![V_v, V_n, V_j, V_conn, V_a]: (p(s(t_bool,h4s_defcnfs_unique(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_n),s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),h4s_pairs_u_2c(s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn),s(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),h4s_pairs_u_2c(s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),h4s_sums_inr(s(t_bool,V_a))),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),h4s_sums_inl(s(t_h4s_nums_num,V_j)))))))))) <=> s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_n))) = s(t_bool,happ(s(t_fun(t_bool,t_bool),happ(s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn),s(t_bool,V_a))),s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_j))))))).
fof(ah4s_defCNFs_UNIQUEu_u_defu_c1, axiom, ![V_v, V_n, V_i, V_conn, V_b]: (p(s(t_bool,h4s_defcnfs_unique(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_n),s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),h4s_pairs_u_2c(s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn),s(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),h4s_pairs_u_2c(s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),h4s_sums_inl(s(t_h4s_nums_num,V_i))),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),h4s_sums_inr(s(t_bool,V_b)))))))))) <=> s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_n))) = s(t_bool,happ(s(t_fun(t_bool,t_bool),happ(s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn),s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_i))))),s(t_bool,V_b))))).
fof(ah4s_defCNFs_UNIQUEu_u_defu_c3, axiom, ![V_v, V_n, V_conn, V_b, V_a]: (p(s(t_bool,h4s_defcnfs_unique(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_n),s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),h4s_pairs_u_2c(s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn),s(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),h4s_pairs_u_2c(s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),h4s_sums_inr(s(t_bool,V_a))),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),h4s_sums_inr(s(t_bool,V_b)))))))))) <=> s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_n))) = s(t_bool,happ(s(t_fun(t_bool,t_bool),happ(s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn),s(t_bool,V_a))),s(t_bool,V_b))))).
fof(ah4s_defCNFs_UNIQUEu_u_ind, axiom, ![V_P]: ((![V_v, V_n, V_conn, V_i, V_j]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool))),V_P),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),h4s_pairs_u_2c(s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn),s(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),h4s_pairs_u_2c(s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),h4s_sums_inl(s(t_h4s_nums_num,V_i))),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),h4s_sums_inl(s(t_h4s_nums_num,V_j)))))))))) & (![V_v, V_n, V_conn, V_i, V_b]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool))),V_P),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),h4s_pairs_u_2c(s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn),s(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),h4s_pairs_u_2c(s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),h4s_sums_inl(s(t_h4s_nums_num,V_i))),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),h4s_sums_inr(s(t_bool,V_b)))))))))) & (![V_v, V_n, V_conn, V_a, V_j]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool))),V_P),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),h4s_pairs_u_2c(s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn),s(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),h4s_pairs_u_2c(s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),h4s_sums_inr(s(t_bool,V_a))),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),h4s_sums_inl(s(t_h4s_nums_num,V_j)))))))))) & ![V_v, V_n, V_conn, V_a, V_b]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool))),V_P),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),h4s_pairs_u_2c(s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn),s(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),h4s_pairs_u_2c(s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),h4s_sums_inr(s(t_bool,V_a))),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),h4s_sums_inr(s(t_bool,V_b))))))))))))) => ![V_v, V_v1, V_v2, V_v3, V_v4]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool))),V_P),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_v1))),s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),h4s_pairs_u_2c(s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_v2),s(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),h4s_pairs_u_2c(s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v3),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v4)))))))))).
fof(ah4s_predu_u_sets_MINu_u_SETu_u_LEM, axiom, ![V_s]: (~ (s(t_fun(t_h4s_nums_num,t_bool),V_s) = s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_empty)) => (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_minu_u_set),s(t_fun(t_h4s_nums_num,t_bool),V_s))),s(t_fun(t_h4s_nums_num,t_bool),V_s)))) & ![V_x]: (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_x),s(t_fun(t_h4s_nums_num,t_bool),V_s)))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_minu_u_set),s(t_fun(t_h4s_nums_num,t_bool),V_s))),s(t_h4s_nums_num,V_x)))))))).
fof(ah4s_relations_WFu_u_EMPTYu_u_REL, axiom, ![TV_u_27a]: p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_emptyu_u_rel))))).
fof(ah4s_whiles_LEASTu_u_DEF, axiom, ![V_P]: s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_whiles_least),s(t_fun(t_h4s_nums_num,t_bool),V_P))) = s(t_h4s_nums_num,h4s_whiles_while(s(t_fun(t_h4s_nums_num,t_bool),h4s_combins_o(s(t_fun(t_bool,t_bool),d_not),s(t_fun(t_h4s_nums_num,t_bool),V_P))),s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_nums_0)))).
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_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_predu_u_sets_MAXu_u_SETu_u_UNION, axiom, ![V_B, V_A]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),V_A)))) & p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),V_B))))) => s(t_h4s_nums_num,h4s_predu_u_sets_maxu_u_set(s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_union(s(t_fun(t_h4s_nums_num,t_bool),V_A),s(t_fun(t_h4s_nums_num,t_bool),V_B))))) = s(t_h4s_nums_num,h4s_arithmetics_max(s(t_h4s_nums_num,h4s_predu_u_sets_maxu_u_set(s(t_fun(t_h4s_nums_num,t_bool),V_A))),s(t_h4s_nums_num,h4s_predu_u_sets_maxu_u_set(s(t_fun(t_h4s_nums_num,t_bool),V_B))))))).
fof(ah4s_predu_u_sets_PRODu_u_SETu_u_THMu_c1, axiom, ![V_x, V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),V_s)))) => s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_produ_u_set),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_insert(s(t_h4s_nums_num,V_x),s(t_fun(t_h4s_nums_num,t_bool),V_s))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_produ_u_set),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_delete(s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_h4s_nums_num,V_x))))))))).
fof(ah4s_predu_u_sets_SUMu_u_SETu_u_UNION, axiom, ![V_t, V_s]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),V_s)))) & p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),V_t))))) => s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_set),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_union(s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_bool),V_t))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_set),s(t_fun(t_h4s_nums_num,t_bool),V_s))),s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_set),s(t_fun(t_h4s_nums_num,t_bool),V_t))))),s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_set),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_inter(s(t_fun(t_h4s_nums_num,t_bool),V_s),s(t_fun(t_h4s_nums_num,t_bool),V_t))))))))).
fof(ah4s_richu_u_lists_everyu_u_countu_u_list, axiom, ![V_n, V_P]: (p(s(t_bool,h4s_lists_every(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_lists_list(t_h4s_nums_num),h4s_richu_u_lists_countu_u_list(s(t_h4s_nums_num,V_n)))))) <=> ![V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_m))))))).
fof(ah4s_predu_u_sets_SUBSETu_u_MINu_u_SET, axiom, ![V_J, V_I]: ((~ (s(t_fun(t_h4s_nums_num,t_bool),V_I) = s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_empty)) & (~ (s(t_fun(t_h4s_nums_num,t_bool),V_J) = s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_empty)) & p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(t_h4s_nums_num,t_bool),V_I),s(t_fun(t_h4s_nums_num,t_bool),V_J)))))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_minu_u_set),s(t_fun(t_h4s_nums_num,t_bool),V_J))),s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_minu_u_set),s(t_fun(t_h4s_nums_num,t_bool),V_I)))))))).
fof(ah4s_defCNFs_UNIQUEu_u_curriedu_u_def, axiom, ![V_x2, V_x1, V_x]: s(t_bool,h4s_defcnfs_unique(s(t_fun(t_h4s_nums_num,t_bool),V_x),s(t_h4s_nums_num,V_x1),s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),V_x2))) = s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool),h4s_defcnfs_uniqueu_u_tupled),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),h4s_pairs_u_2c(s(t_fun(t_h4s_nums_num,t_bool),V_x),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),h4s_pairs_u_2c(s(t_h4s_nums_num,V_x1),s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),V_x2)))))))).
fof(ah4s_defCNFs_UNIQUEu_u_tupledu_u_primitiveu_u_def, axiom, ![V_uu_16]: (![V_v, V_n, V_conn, V_a, V_bu_27]: ?[V_v0]: ((p(s(t_bool,V_v0)) <=> s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_n))) = s(t_bool,happ(s(t_fun(t_bool,t_bool),happ(s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn),s(t_bool,V_a))),s(t_bool,V_bu_27)))) & s(t_bool,happ(s(t_fun(t_bool,t_bool),happ(s(t_fun(t_bool,t_fun(t_bool,t_bool)),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_fun(t_bool,t_bool))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_fun(t_bool,t_bool)))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_fun(t_bool,t_bool))))),V_uu_16),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_bool,V_a))),s(t_bool,V_bu_27))) = s(t_bool,happ(s(t_fun(t_bool,t_bool),h4s_combins_i),s(t_bool,V_v0)))) => ![V_uu_15]: (![V_n, V_conn, V_a, V_v, V_ju_27]: ?[V_v0]: ((p(s(t_bool,V_v0)) <=> s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_n))) = s(t_bool,happ(s(t_fun(t_bool,t_bool),happ(s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn),s(t_bool,V_a))),s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_ju_27)))))) & s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_bool,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))))),V_uu_15),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_bool,V_a))),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_ju_27))) = s(t_bool,happ(s(t_fun(t_bool,t_bool),h4s_combins_i),s(t_bool,V_v0)))) => ![V_uu_14]: (![V_v8, V_v, V_n, V_conn, V_a]: s(t_bool,happ(s(t_fun(t_bool,t_bool),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_bool))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_bool)))),happ(s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_bool))))),V_uu_14),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v8))),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_bool,V_a))) = s(t_bool,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v8),s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_bool,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))))),V_uu_15),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_bool,V_a))),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_fun(t_bool,t_bool),happ(s(t_fun(t_bool,t_fun(t_bool,t_bool)),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_fun(t_bool,t_bool))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_fun(t_bool,t_bool)))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_fun(t_bool,t_bool))))),V_uu_16),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_bool,V_a))))) => ![V_uu_13]: (![V_n, V_conn, V_v, V_i, V_b]: ?[V_v0]: ((p(s(t_bool,V_v0)) <=> s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_n))) = s(t_bool,happ(s(t_fun(t_bool,t_bool),happ(s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn),s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_i))))),s(t_bool,V_b)))) & s(t_bool,happ(s(t_fun(t_bool,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_bool,t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_bool,t_bool))),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_bool,t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_bool,t_bool))))),V_uu_13),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_i))),s(t_bool,V_b))) = s(t_bool,happ(s(t_fun(t_bool,t_bool),h4s_combins_i),s(t_bool,V_v0)))) => ![V_uu_12]: (![V_n, V_conn, V_i, V_v, V_j]: ?[V_v0]: ((p(s(t_bool,V_v0)) <=> s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_n))) = s(t_bool,happ(s(t_fun(t_bool,t_bool),happ(s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn),s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_i))))),s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_v),s(t_h4s_nums_num,V_j)))))) & s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))))),V_uu_12),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_h4s_nums_num,V_i))),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_j))) = s(t_bool,happ(s(t_fun(t_bool,t_bool),h4s_combins_i),s(t_bool,V_v0)))) => ![V_uu_11]: (![V_v8, V_n, V_conn, V_v, V_i]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)))),happ(s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))))),V_uu_11),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v8))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_i))) = s(t_bool,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v8),s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))))),V_uu_12),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_h4s_nums_num,V_i))),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_fun(t_bool,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_bool,t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_bool,t_bool))),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_bool,t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_bool,t_bool))))),V_uu_13),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_i))))) => ![V_uu_10]: (![V_v7, V_v, V_n, V_conn, V_v8]: s(t_bool,happ(s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool)))),happ(s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool))))),V_uu_10),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v7))),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v8))) = s(t_bool,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v7),s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool)))),happ(s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_bool))))),V_uu_11),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v8))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_fun(t_bool,t_bool),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_bool))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_bool)))),happ(s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_bool,t_bool))))),V_uu_14),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v8))),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))))) => ![V_uu_9]: (![V_v, V_n, V_conn, V_v7]: s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool),happ(s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool)),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool)))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool))))),V_uu_9),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v7))) = s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool)))),happ(s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool))))),V_uu_10),s(t_h4s_sums_sum(t_h4s_nums_num,t_bool),V_v7))),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))) => ![V_uu_8]: (![V_v, V_n, V_conn, V_v6]: s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool)))),V_uu_8),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))),s(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),V_v6))) = s(t_bool,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),V_v6),s(t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool)),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool)))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_fun(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_bool))))),V_uu_9),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))))) => ![V_uu_7]: (![V_v, V_n, V_conn]: s(t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool)))),V_uu_7),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))) = s(t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool),happ(s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool)))),V_uu_8),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_fun(t_bool,t_fun(t_bool,t_bool)),V_conn))) => ![V_uu_6]: (![V_v, V_n, V_v4]: s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool))),V_uu_6),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))),s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),V_v4))) = s(t_bool,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),V_v4),s(t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_fun(t_bool,t_fun(t_bool,t_bool)),t_fun(t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)),t_bool)))),V_uu_7),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))))) => ![V_uu_5]: (![V_v, V_n]: s(t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool))),V_uu_5),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))) = s(t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool))),V_uu_6),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_nums_num,V_n))) => ![V_uu_4]: (![V_v, V_v2]: s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool)),V_uu_4),s(t_fun(t_h4s_nums_num,t_bool),V_v))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),V_v2))) = s(t_bool,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),V_v2),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool)),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))),t_bool))),V_uu_5),s(t_fun(t_h4s_nums_num,t_bool),V_v))))) => ![V_uu_3]: (![V_v]: s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool)),V_uu_3),s(t_fun(t_h4s_nums_num,t_bool),V_v))) = s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool)),V_uu_4),s(t_fun(t_h4s_nums_num,t_bool),V_v))) => ![V_uu_2]: (![V_au_27]: s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool),V_uu_2),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),V_au_27))) = s(t_bool,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),V_au_27),s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool)))),t_bool)),V_uu_3))) => ![V_uu_1]: (![V_UNIQUEu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool)),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool),V_UNIQUEu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool),V_uu_2) => ![V_uu_0]: (![V_R]: s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool)),V_R))) = s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool)),V_R))) => s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool),h4s_defcnfs_uniqueu_u_tupled) = s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_fun(t_bool,t_fun(t_bool,t_bool)),t_h4s_pairs_prod(t_h4s_sums_sum(t_h4s_nums_num,t_bool),t_h4s_sums_sum(t_h4s_nums_num,t_bool))))),t_bool)),V_uu_1))))))))))))))))))))).
fof(ah4s_sums_sumu_u_caseu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_y, V_f1, V_f]: s(TV_u_27c,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_y))),s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27c),V_f1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),V_f1),s(TV_u_27b,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_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_sums_sumu_u_caseu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_x, V_f1, V_f]: s(TV_u_27c,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_x))),s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27c),V_f1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_predu_u_sets_MINu_u_SETu_u_THMu_c1, axiom, ![V_s, V_e2, V_e1]: s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_minu_u_set),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_insert(s(t_h4s_nums_num,V_e1),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_insert(s(t_h4s_nums_num,V_e2),s(t_fun(t_h4s_nums_num,t_bool),V_s))))))) = s(t_h4s_nums_num,h4s_arithmetics_min(s(t_h4s_nums_num,V_e1),s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_minu_u_set),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_insert(s(t_h4s_nums_num,V_e2),s(t_fun(t_h4s_nums_num,t_bool),V_s)))))))).
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_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_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_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_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_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_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_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_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ANDu_u_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_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_bools_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_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(ah4s_predu_u_sets_PRODu_u_SETu_u_DEF, axiom, s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_produ_u_set) = s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_produ_u_image(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_combins_i)))).
fof(ah4s_predu_u_sets_PRODu_u_IMAGEu_u_THMu_c1, axiom, ![TV_u_27a]: ![V_s, V_f, V_e]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => s(t_h4s_nums_num,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_nums_num),h4s_predu_u_sets_produ_u_image(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(TV_u_27a,V_e))),s(t_h4s_nums_num,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_nums_num),h4s_predu_u_sets_produ_u_image(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_s),s(TV_u_27a,V_e))))))))).
fof(ah4s_predu_u_sets_SUMu_u_IMAGEu_u_UNION, axiom, ![TV_u_27a]: ![V_t, V_s, V_f]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_t))))) => s(t_h4s_nums_num,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_h4s_nums_num,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f))),s(t_fun(TV_u_27a,t_bool),V_t))))),s(t_h4s_nums_num,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_image(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f))),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))))))))).
fof(ah4s_predu_u_sets_SUMu_u_SETu_u_DEF, axiom, s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_set) = s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_image(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_combins_i)))).
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_EXISTSu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_x)))))).
fof(ah4s_bools_LEFTu_u_ORu_u_OVERu_u_AND, 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_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((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))))) <=> ![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))))))).
fof(ah4s_bools_LEFTu_u_FORALLu_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,V_Q))) <=> (![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,V_Q))))).
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_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_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((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))))) <=> ?[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))))))).
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_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_bools_NOTu_u_EXISTSu_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_bools_LEFTu_u_ORu_u_EXISTSu_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,V_Q))) <=> ?[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,V_Q))))).
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_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_bools_LEFTu_u_EXISTSu_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,V_Q))) <=> (?[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,V_Q))))).
fof(ah4s_sums_sumu_u_CASES, axiom, ![TV_u_27a,TV_u_27b]: ![V_ss]: (?[V_x]: s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_ss) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_x))) | ?[V_y]: s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_ss) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_y))))).
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_RIGHTu_u_FORALLu_u_ORu_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_bools_RIGHTu_u_ORu_u_OVERu_u_AND, 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_predu_u_sets_INu_u_INSERT, axiom, ![TV_u_27a]: ![V_y, V_x, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_bools_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_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
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_SUCu_u_ONEu_u_ADD, axiom, ![V_n]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,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_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_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_arithmetics_NOTu_u_LESS, axiom, ![V_n, V_m]: (~ (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_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_EQ, axiom, ![V_n, V_m]: s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_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_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
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_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_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_IMPu_u_DISJu_u_THM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) => p(s(t_bool,V_B))) <=> (~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B))))).
fof(ah4s_bools_IMPu_u_Fu_u_EQu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_predu_u_sets_MAXu_u_SETu_u_REWRITESu_c0, axiom, s(t_h4s_nums_num,h4s_predu_u_sets_maxu_u_set(s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_empty))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_predu_u_sets_MAXu_u_SETu_u_THMu_c1, axiom, ![V_s, V_e]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),V_s)))) => s(t_h4s_nums_num,h4s_predu_u_sets_maxu_u_set(s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_insert(s(t_h4s_nums_num,V_e),s(t_fun(t_h4s_nums_num,t_bool),V_s))))) = s(t_h4s_nums_num,h4s_arithmetics_max(s(t_h4s_nums_num,V_e),s(t_h4s_nums_num,h4s_predu_u_sets_maxu_u_set(s(t_fun(t_h4s_nums_num,t_bool),V_s))))))).
fof(ah4s_predu_u_sets_FINITEu_u_UNION, axiom, ![TV_u_27a]: ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) <=> (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_t))))))).
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_predu_u_sets_FINITEu_u_INDUCT, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))) & ![V_s]: ((p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_s))))) => ![V_e]: (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s))))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),V_s))))))))) => ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_predu_u_sets_SETu_u_CASES, axiom, ![TV_u_27a]: ![V_s]: (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty) | ?[V_x, V_t]: (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))) & ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t)))))))).
fof(ah4s_predu_u_sets_FINITEu_u_INSERT, axiom, ![TV_u_27a]: ![V_x, V_s]: s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))) = s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))).
fof(ah4s_predu_u_sets_UNIONu_u_EMPTYu_c1, axiom, ![TV_u_27a]: ![V_s]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) = s(t_fun(TV_u_27a,t_bool),V_s)).
fof(ah4s_predu_u_sets_INu_u_UNION, axiom, ![TV_u_27a]: ![V_x, V_t, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_predu_u_sets_UNIONu_u_EMPTYu_c0, axiom, ![TV_u_27a]: ![V_s]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(TV_u_27a,t_bool),V_s)).
fof(ah4s_predu_u_sets_EXTENSION, axiom, ![TV_u_27a]: ![V_t, V_s]: (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),V_t) <=> ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))).
fof(ah4s_arithmetics_MAXu_u_0u_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_max(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_arithmetics_MAXu_u_ASSOC, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_max(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_max(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_arithmetics_max(s(t_h4s_nums_num,h4s_arithmetics_max(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_MAXu_u_COMM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_max(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_max(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_MAXu_u_0u_c0, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_max(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_markers_ACu_u_DEF, axiom, ![V_b2, V_b1]: (p(s(t_bool,h4s_markers_ac(s(t_bool,V_b1),s(t_bool,V_b2)))) <=> (p(s(t_bool,V_b1)) & p(s(t_bool,V_b2))))).
fof(ah4s_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_bools_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_LCOMMu_u_THM, axiom, ![TV_u_27a]: ![V_f]: (![V_x, V_y, V_z]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_x))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_y))),s(TV_u_27a,V_z))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))),s(TV_u_27a,V_z))) => (![V_x, V_y]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_y))),s(TV_u_27a,V_x))) => ![V_x, V_y, V_z]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_x))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_y))),s(TV_u_27a,V_z))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_y))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_x))),s(TV_u_27a,V_z)))))))).
fof(ah4s_predu_u_sets_SUBSETu_u_DEF, axiom, ![TV_u_27a]: ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) <=> ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_predu_u_sets_MEMBERu_u_NOTu_u_EMPTY, axiom, ![TV_u_27a]: ![V_s]: (?[V_x]: p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) <=> ~ (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))).
fof(ah4s_predu_u_sets_MINu_u_SETu_u_DEF, axiom, s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_minu_u_set) = s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_whiles_least)).
fof(ah4s_bools_INu_u_DEF, axiom, ![TV_u_27a]: ![V_x, V_x0]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_x0))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x0),s(TV_u_27a,V_x)))).
fof(ah4s_whiles_FULLu_u_LEASTu_u_INTRO, axiom, ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_x)))) => (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_whiles_least),s(t_fun(t_h4s_nums_num,t_bool),V_P)))))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_whiles_least),s(t_fun(t_h4s_nums_num,t_bool),V_P))),s(t_h4s_nums_num,V_x))))))).
fof(ah4s_bools_LEFTu_u_ANDu_u_FORALLu_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,V_Q))) <=> ![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,V_Q))))).
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_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_numerals_numeralu_u_addu_c0, axiom, ![V_n]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_numerals_numeralu_u_distribu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_numerals_numeralu_u_addu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_m))))))) = s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_richu_u_lists_COUNTu_u_LISTu_u_defu_c0, axiom, s(t_h4s_lists_list(t_h4s_nums_num),h4s_richu_u_lists_countu_u_list(s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_nil)).
fof(ah4s_richu_u_lists_COUNTu_u_LISTu_u_defu_c1, axiom, ![V_n]: s(t_h4s_lists_list(t_h4s_nums_num),h4s_richu_u_lists_countu_u_list(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))))) = s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_cons(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_map(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_lists_list(t_h4s_nums_num),h4s_richu_u_lists_countu_u_list(s(t_h4s_nums_num,V_n)))))))).
fof(ah4s_arithmetics_LESSu_u_MONOu_u_EQ, axiom, ![V_n, V_m]: s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))))) = s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_numu_u_CASES, axiom, ![V_m]: (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0) | ?[V_n]: s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))))).
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_primu_u_recs_LESSu_u_0, axiom, ![V_n]: p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_primu_u_recs_NOTu_u_LESSu_u_0, axiom, ![V_n]: ~ (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0)))))).
fof(ah4s_nums_INDUCTION, axiom, ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_0)))) & ![V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n)))))))) => ![V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_lists_EVERYu_u_MAP, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_P, V_f, V_x]: s(t_bool,happ(s(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_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_bool))),V_uu_0),s(t_fun(TV_u_27b,t_bool),V_P))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))) => ![V_l, V_f, V_P]: s(t_bool,h4s_lists_every(s(t_fun(TV_u_27b,t_bool),V_P),s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_bool,h4s_lists_every(s(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_bool)),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_bool))),V_uu_0),s(t_fun(TV_u_27b,t_bool),V_P))),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_lists_list(TV_u_27a),V_l))))).
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_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(ch4s_defCNFs_BIGSTEP, conjecture, ![V_R, V_Q, V_P]: (![V_v]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_bool),V_P),s(t_fun(t_h4s_nums_num,t_bool),V_v)))) => s(t_bool,V_Q) = s(t_bool,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_bool),V_R),s(t_fun(t_h4s_nums_num,t_bool),V_v)))) => ((?[V_v]: p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_bool),V_P),s(t_fun(t_h4s_nums_num,t_bool),V_v)))) & p(s(t_bool,V_Q))) <=> ?[V_v]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_bool),V_P),s(t_fun(t_h4s_nums_num,t_bool),V_v)))) & p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_bool),V_R),s(t_fun(t_h4s_nums_num,t_bool),V_v)))))))).
