%   ORIGINAL: h4/prim__rec/WF__PRED
% 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/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/num/INV__SUC: !n m. h4/num/SUC m = h4/num/SUC n ==> m = n
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/prim__rec/LESS__THM: !n m. h4/prim__rec/_3C m (h4/num/SUC n) <=> m = n \/ h4/prim__rec/_3C m n
% Assm: h4/prim__rec/SIMP__REC__THM_c1: !x m f. h4/prim__rec/SIMP__REC x f (h4/num/SUC m) = f (h4/prim__rec/SIMP__REC x f m)
% Assm: h4/prim__rec/LESS__DEF: !n m. h4/prim__rec/_3C m n <=> (?P. (!n0. P (h4/num/SUC n0) ==> P n0) /\ P m /\ ~P n)
% Assm: h4/num/NOT__SUC: !n. ~(h4/num/SUC n = h4/num/0)
% Assm: h4/prim__rec/LESS__LEMMA1: !n m. h4/prim__rec/_3C m (h4/num/SUC n) ==> m = n \/ h4/prim__rec/_3C m n
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/num/SUC__DEF: !m. h4/num/SUC m = h4/num/ABS__num (h4/num/SUC__REP (h4/num/REP__num m))
% Assm: h4/prim__rec/INV__SUC__EQ: !n m. h4/num/SUC m = h4/num/SUC n <=> m = n
% Assm: h4/prim__rec/LESS__LEMMA2: !n m. m = n \/ h4/prim__rec/_3C m n ==> h4/prim__rec/_3C m (h4/num/SUC n)
% Assm: h4/prim__rec/PRE0_c1: !m. h4/prim__rec/PRE (h4/num/SUC m) = m
% Assm: h4/prim__rec/LESS__SUC__REFL: !n. h4/prim__rec/_3C n (h4/num/SUC n)
% Assm: h4/prim__rec/PRE__DEF: !m. h4/prim__rec/PRE m = h4/bool/COND (m = h4/num/0) h4/num/0 (h4/min/_40 (\n. m = h4/num/SUC n))
% Assm: h4/prim__rec/SIMP__REC__REL0: !x n fun f. h4/prim__rec/SIMP__REC__REL fun x f n <=> fun h4/num/0 = x /\ (!m. h4/prim__rec/_3C m n ==> fun (h4/num/SUC m) = f (fun m))
% Assm: h4/prim__rec/DC: !a R P. P a /\ (!x. P x ==> (?y. P y /\ R x y)) ==> (?f. f h4/num/0 = a /\ (!n. P (f n) /\ R (f n) (f (h4/num/SUC n))))
% Assm: h4/prim__rec/SUC__ID: !n. ~(h4/num/SUC n = n)
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/prim__rec/PRIM__REC__THM_c1: !x m f. h4/prim__rec/PRIM__REC x f (h4/num/SUC m) = f (h4/prim__rec/PRIM__REC x f m) m
% Assm: h4/prim__rec/EQ__LESS: !n m. h4/num/SUC m = n ==> h4/prim__rec/_3C m n
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/prim__rec/LESS__SUC: !n m. h4/prim__rec/_3C m n ==> h4/prim__rec/_3C m (h4/num/SUC n)
% Assm: h4/prim__rec/PRIM__REC__FUN0: !x f. h4/prim__rec/PRIM__REC__FUN x f = h4/prim__rec/SIMP__REC (\n. x) (\fun n. f (fun (h4/prim__rec/PRE n)) n)
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/prim__rec/num__Axiom__old: !f e. h4/bool/_3F_21 (\fn1. fn1 h4/num/0 = e /\ (!n. fn1 (h4/num/SUC n) = f (fn1 n) n))
% Assm: h4/num/num__ISO__DEF_c1: !r. h4/num/IS__NUM__REP r <=> h4/num/REP__num (h4/num/ABS__num r) = r
% Assm: h4/num/num__ISO__DEF_c0: !a. h4/num/ABS__num (h4/num/REP__num a) = a
% Assm: h4/num/IS__NUM__REP0: !m. h4/num/IS__NUM__REP m <=> (!P. P h4/num/ZERO__REP /\ (!n. P n ==> P (h4/num/SUC__REP n)) ==> P m)
% Assm: h4/prim__rec/SIMP__REC0: !x n f_27. ?g. h4/prim__rec/SIMP__REC__REL g x f_27 (h4/num/SUC n) /\ h4/prim__rec/SIMP__REC x f_27 n = g n
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/prim__rec/LESS__SUC__IMP: !n m. h4/prim__rec/_3C m (h4/num/SUC n) ==> ~(m = n) ==> h4/prim__rec/_3C m n
% Assm: h4/prim__rec/PRIM__REC__EQN_c1: !x n m f. h4/prim__rec/PRIM__REC__FUN x f (h4/num/SUC m) n = f (h4/prim__rec/PRIM__REC__FUN x f m (h4/prim__rec/PRE n)) n
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/prim__rec/SIMP__REC__REL__UNIQUE__RESULT: !x n f. h4/bool/_3F_21 (\y. ?g. h4/prim__rec/SIMP__REC__REL g x f (h4/num/SUC n) /\ y = g n)
% Assm: h4/prim__rec/LESS__SUC__SUC_c1: !m. h4/prim__rec/_3C m (h4/num/SUC (h4/num/SUC m))
% Assm: h4/prim__rec/LESS__SUC__SUC_c0: !m. h4/prim__rec/_3C m (h4/num/SUC m)
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/prim__rec/wellfounded__def: !R. h4/prim__rec/wellfounded R <=> ~(?f. !n. R (f (h4/num/SUC n)) (f n))
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/num/ZERO__DEF: h4/num/0 = h4/num/ABS__num h4/num/ZERO__REP
% Assm: h4/prim__rec/SIMP__REC__THM_c0: !x f. h4/prim__rec/SIMP__REC x f h4/num/0 = x
% Assm: h4/num/SUC__REP__DEF_c0: h4/bool/ONE__ONE h4/num/SUC__REP
% Assm: h4/num/SUC__REP__DEF_c1: ~h4/bool/ONTO h4/num/SUC__REP
% Assm: h4/bool/ONE__ONE__THM: !f. h4/bool/ONE__ONE f <=> (!x1 x2. f x1 = f x2 ==> x1 = x2)
% Assm: h4/prim__rec/num__Axiom: !f e. ?fn. fn h4/num/0 = e /\ (!n. fn (h4/num/SUC n) = f n (fn n))
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/prim__rec/LESS__MONO: !n m. h4/prim__rec/_3C m n ==> h4/prim__rec/_3C (h4/num/SUC m) (h4/num/SUC n)
% Assm: h4/bool/OR__CLAUSES_c1: !t. t \/ T <=> T
% Assm: h4/num/ZERO__REP__DEF: !y. ~(h4/num/ZERO__REP = h4/num/SUC__REP y)
% Assm: h4/prim__rec/PRIM__REC0: !x m f. h4/prim__rec/PRIM__REC x f m = h4/prim__rec/PRIM__REC__FUN x f m (h4/prim__rec/PRE m)
% Assm: h4/bool/SELECT__REFL__2: !x. h4/min/_40 (\y. x = y) = x
% Assm: h4/bool/EXISTS__UNIQUE__DEF: h4/bool/_3F_21 = (\P. $exists P /\ (!x y. P x /\ P y ==> x = y))
% Assm: h4/prim__rec/SIMP__REC__REL__UNIQUE: !x m2 m1 g2 g1 f. h4/prim__rec/SIMP__REC__REL g1 x f m1 /\ h4/prim__rec/SIMP__REC__REL g2 x f m2 ==> (!n. h4/prim__rec/_3C n m1 /\ h4/prim__rec/_3C n m2 ==> g1 n = g2 n)
% Assm: h4/bool/EXISTS__UNIQUE__THM: !P. h4/bool/_3F_21 (\x. P x) <=> (?x. P x) /\ (!x y. P x /\ P y ==> x = y)
% Assm: h4/prim__rec/LESS__0: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm: h4/prim__rec/SUC__LESS: !n m. h4/prim__rec/_3C (h4/num/SUC m) n ==> h4/prim__rec/_3C m n
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/prim__rec/PRIM__REC__THM_c0: !x f. h4/prim__rec/PRIM__REC x f h4/num/0 = x
% Assm: h4/bool/SKOLEM__THM: !P. (!x. ?y. P x y) <=> (?f. !x. P x (f x))
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/prim__rec/SIMP__REC__EXISTS: !x n f. ?fun. h4/prim__rec/SIMP__REC__REL fun x f n
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/bool/BETA__THM: !y f. (\x. f x) y = f y
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/bool/FORALL__AND__THM: !Q P. (!x. P x /\ Q x) <=> (!x. P x) /\ (!x. Q x)
% Assm: h4/relation/WF__EQ__INDUCTION__THM: !R. h4/relation/WF R <=> (!P. (!x. (!y. R y x ==> P y) ==> P x) ==> (!x. P x))
% Assm: h4/relation/WF__EQ__WFP: !R. h4/relation/WF R <=> (!x. h4/relation/WFP R x)
% Assm: h4/relation/INDUCTION__WF__THM: !R. (!P. (!x. (!y. R y x ==> P y) ==> P x) ==> (!x. P x)) ==> h4/relation/WF R
% Assm: h4/relation/WFREC__THM: !R M. h4/relation/WF R ==> (!x. h4/relation/WFREC R M x = M (h4/relation/RESTRICT (h4/relation/WFREC R M) R x) x)
% Assm: h4/relation/WF__INDUCTION__THM: !R. h4/relation/WF R ==> (!P. (!x. (!y. R y x ==> P y) ==> P x) ==> (!x. P x))
% Assm: h4/relation/WF__TC__EQN: !R. h4/relation/WF (h4/relation/TC R) <=> h4/relation/WF R
% Assm: h4/prim__rec/LESS__0__0: h4/prim__rec/_3C h4/num/0 (h4/num/SUC h4/num/0)
% Assm: h4/relation/WF__DEF: !R. h4/relation/WF R <=> (!B. (?w. B w) ==> (?min. B min /\ (!b. R b min ==> ~B b)))
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/relation/WF__RECURSION__THM: !R. h4/relation/WF R ==> (!M. h4/bool/_3F_21 (\f. !x. f x = M (h4/relation/RESTRICT f R x) x))
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/relation/WF__TC: !R. h4/relation/WF R ==> h4/relation/WF (h4/relation/TC R)
% Assm: h4/relation/TC__SUBSET: !y x R. R x y ==> h4/relation/TC R x y
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/bool/LEFT__AND__FORALL__THM: !Q P. (!x. P x) /\ Q <=> (!x. P x /\ Q)
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/LEFT__OR__EXISTS__THM: !Q P. (?x. P x) \/ Q <=> (?x. P x \/ Q)
% Assm: h4/bool/LEFT__EXISTS__AND__THM: !Q P. (?x. P x /\ Q) <=> (?x. P x) /\ Q
% Assm: h4/relation/WFP__RULES: !x R. (!y. R y x ==> h4/relation/WFP R y) ==> h4/relation/WFP R x
% Assm: h4/bool/RIGHT__EXISTS__AND__THM: !Q P. (?x. P /\ Q x) <=> P /\ (?x. Q x)
% Assm: h4/bool/LEFT__FORALL__OR__THM: !Q P. (!x. P x \/ Q) <=> (!x. P x) \/ Q
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/relation/WFP__STRONG__INDUCT: !R P. (!x. h4/relation/WFP R x /\ (!y. R y x ==> P y) ==> P x) ==> (!x. h4/relation/WFP R x ==> P x)
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/relation/WF__SUBSET: !R P. h4/relation/WF R /\ (!x y. P x y ==> R x y) ==> h4/relation/WF P
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/bool/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/relation/RESTRICT__DEF: !x f R. h4/relation/RESTRICT f R x = (\y. h4/bool/COND (R y x) (f y) h4/bool/ARB)
% Assm: h4/relation/WFREC__DEF: !R M. h4/relation/WFREC R M = (\x. M (h4/relation/RESTRICT (h4/relation/the__fun (h4/relation/TC R) (\f v. M (h4/relation/RESTRICT f R v) v) x) R x) x)
% Assm: h4/relation/the__fun__def: !x R M. h4/relation/the__fun R M x = h4/min/_40 (\f. h4/relation/approx R M x f)
% Assm: h4/relation/approx__def: !x f R M. h4/relation/approx R M x f <=> f = h4/relation/RESTRICT (\y. M (h4/relation/RESTRICT f R y) y) R x
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/relation/TC__TRANSITIVE: !R. h4/relation/transitive (h4/relation/TC R)
% Assm: h4/relation/transitive__def: !R. h4/relation/transitive R <=> (!x y z. R x y /\ R y z ==> R x z)
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Goal: h4/relation/WF (\x y. y = h4/num/SUC x)
%   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_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_nums_INDUCTION]: !P. happ P h4/num/0 /\ (!n. happ P n ==> happ P (h4/num/SUC n)) ==> (!n. happ P n)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_nums_INVu_u_SUC]: !n m. h4/num/SUC m = h4/num/SUC n ==> m = n
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_primu_u_recs_LESSu_u_THM]: !n m. h4/prim__rec/_3C m (h4/num/SUC n) <=> m = n \/ h4/prim__rec/_3C m n
% Assm [h4s_primu_u_recs_SIMPu_u_RECu_u_THMu_c1]: !x m f. happ (h4/prim__rec/SIMP__REC x f) (h4/num/SUC m) = happ f (happ (h4/prim__rec/SIMP__REC x f) m)
% Assm [h4s_primu_u_recs_LESSu_u_DEF]: !n m. h4/prim__rec/_3C m n <=> (?P. (!n0. happ P (h4/num/SUC n0) ==> happ P n0) /\ happ P m /\ ~happ P n)
% Assm [h4s_nums_NOTu_u_SUC]: !n. ~(h4/num/SUC n = h4/num/0)
% Assm [h4s_primu_u_recs_LESSu_u_LEMMA1]: !n m. h4/prim__rec/_3C m (h4/num/SUC n) ==> m = n \/ h4/prim__rec/_3C m n
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_nums_SUCu_u_DEF]: !m. h4/num/SUC m = h4/num/ABS__num (happ h4/num/SUC__REP (h4/num/REP__num m))
% Assm [h4s_primu_u_recs_INVu_u_SUCu_u_EQ]: !n m. h4/num/SUC m = h4/num/SUC n <=> m = n
% Assm [h4s_primu_u_recs_LESSu_u_LEMMA2]: !n m. m = n \/ h4/prim__rec/_3C m n ==> h4/prim__rec/_3C m (h4/num/SUC n)
% Assm [h4s_primu_u_recs_PRE0u_c1]: !m. h4/prim__rec/PRE (h4/num/SUC m) = m
% Assm [h4s_primu_u_recs_LESSu_u_SUCu_u_REFL]: !n. h4/prim__rec/_3C n (h4/num/SUC n)
% Assm [h4s_primu_u_recs_PREu_u_DEF]: !_0. (!m n. happ (happ _0 m) n <=> m = h4/num/SUC n) ==> (!m. ?v. (v <=> m = h4/num/0) /\ h4/prim__rec/PRE m = h4/bool/COND v h4/num/0 (h4/min/_40 (happ _0 m)))
% Assm [h4s_primu_u_recs_SIMPu_u_RECu_u_REL0]: !x n fun f. h4/prim__rec/SIMP__REC__REL fun x f n <=> happ fun h4/num/0 = x /\ (!m. h4/prim__rec/_3C m n ==> happ fun (h4/num/SUC m) = happ f (happ fun m))
% Assm [h4s_primu_u_recs_DC]: !a R P. happ P a /\ (!x. happ P x ==> (?y. happ P y /\ happ (happ R x) y)) ==> (?f. happ f h4/num/0 = a /\ (!n. happ P (happ f n) /\ happ (happ R (happ f n)) (happ f (h4/num/SUC n))))
% Assm [h4s_primu_u_recs_SUCu_u_ID]: !n. ~(h4/num/SUC n = n)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_primu_u_recs_PRIMu_u_RECu_u_THMu_c1]: !x m f. h4/prim__rec/PRIM__REC x f (h4/num/SUC m) = happ (happ f (h4/prim__rec/PRIM__REC x f m)) m
% Assm [h4s_primu_u_recs_EQu_u_LESS]: !n m. h4/num/SUC m = n ==> h4/prim__rec/_3C m n
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_primu_u_recs_LESSu_u_SUC]: !n m. h4/prim__rec/_3C m n ==> h4/prim__rec/_3C m (h4/num/SUC n)
% Assm [h4s_primu_u_recs_PRIMu_u_RECu_u_FUN0]: !_2. (!f fun n. happ (happ (happ _2 f) fun) n = happ (happ f (happ fun (h4/prim__rec/PRE n))) n) ==> (!_1. (!f fun. happ (happ _1 f) fun = happ (happ _2 f) fun) ==> (!_0. (!x n. happ (happ _0 x) n = x) ==> (!x f. h4/prim__rec/PRIM__REC__FUN x f = h4/prim__rec/SIMP__REC (happ _0 x) (happ _1 f))))
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_primu_u_recs_numu_u_Axiomu_u_old]: !_0. (!e f fn1. happ (happ (happ _0 e) f) fn1 <=> happ fn1 h4/num/0 = e /\ (!n. happ fn1 (h4/num/SUC n) = happ (happ f (happ fn1 n)) n)) ==> (!f e. h4/bool/_3F_21 (happ (happ _0 e) f))
% Assm [h4s_nums_numu_u_ISOu_u_DEFu_c1]: !r. h4/num/IS__NUM__REP r <=> h4/num/REP__num (h4/num/ABS__num r) = r
% Assm [h4s_nums_numu_u_ISOu_u_DEFu_c0]: !a. h4/num/ABS__num (h4/num/REP__num a) = a
% Assm [h4s_nums_ISu_u_NUMu_u_REP0]: !m. h4/num/IS__NUM__REP m <=> (!P. happ P h4/num/ZERO__REP /\ (!n. happ P n ==> happ P (happ h4/num/SUC__REP n)) ==> happ P m)
% Assm [h4s_primu_u_recs_SIMPu_u_REC0]: !x n f_27. ?g. h4/prim__rec/SIMP__REC__REL g x f_27 (h4/num/SUC n) /\ happ (h4/prim__rec/SIMP__REC x f_27) n = happ g n
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_primu_u_recs_LESSu_u_SUCu_u_IMP]: !n m. h4/prim__rec/_3C m (h4/num/SUC n) ==> ~(m = n) ==> h4/prim__rec/_3C m n
% Assm [h4s_primu_u_recs_PRIMu_u_RECu_u_EQNu_c1]: !x n m f. happ (happ (h4/prim__rec/PRIM__REC__FUN x f) (h4/num/SUC m)) n = happ (happ f (happ (happ (h4/prim__rec/PRIM__REC__FUN x f) m) (h4/prim__rec/PRE n))) n
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_primu_u_recs_SIMPu_u_RECu_u_RELu_u_UNIQUEu_u_RESULT]: !_0. (!x f n y. happ (happ (happ (happ _0 x) f) n) y <=> (?g. h4/prim__rec/SIMP__REC__REL g x f (h4/num/SUC n) /\ y = happ g n)) ==> (!x n f. h4/bool/_3F_21 (happ (happ (happ _0 x) f) n))
% Assm [h4s_primu_u_recs_LESSu_u_SUCu_u_SUCu_c1]: !m. h4/prim__rec/_3C m (h4/num/SUC (h4/num/SUC m))
% Assm [h4s_primu_u_recs_LESSu_u_SUCu_u_SUCu_c0]: !m. h4/prim__rec/_3C m (h4/num/SUC m)
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_primu_u_recs_wellfoundedu_u_def]: !R. h4/prim__rec/wellfounded R <=> ~(?f. !n. happ (happ R (happ f (h4/num/SUC n))) (happ f n))
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_nums_ZEROu_u_DEF]: h4/num/0 = h4/num/ABS__num h4/num/ZERO__REP
% Assm [h4s_primu_u_recs_SIMPu_u_RECu_u_THMu_c0]: !x f. happ (h4/prim__rec/SIMP__REC x f) h4/num/0 = x
% Assm [h4s_nums_SUCu_u_REPu_u_DEFu_c0]: h4/bool/ONE__ONE h4/num/SUC__REP
% Assm [h4s_nums_SUCu_u_REPu_u_DEFu_c1]: ~h4/bool/ONTO h4/num/SUC__REP
% Assm [h4s_bools_ONEu_u_ONEu_u_THM]: !f. h4/bool/ONE__ONE f <=> (!x1 x2. happ f x1 = happ f x2 ==> x1 = x2)
% Assm [h4s_primu_u_recs_numu_u_Axiom]: !f e. ?fn. happ fn h4/num/0 = e /\ (!n. happ fn (h4/num/SUC n) = happ (happ f n) (happ fn n))
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_primu_u_recs_LESSu_u_MONO]: !n m. h4/prim__rec/_3C m n ==> h4/prim__rec/_3C (h4/num/SUC m) (h4/num/SUC n)
% Assm [h4s_bools_ORu_u_CLAUSESu_c1]: !t. t \/ T <=> T
% Assm [h4s_nums_ZEROu_u_REPu_u_DEF]: !y. ~(h4/num/ZERO__REP = happ h4/num/SUC__REP y)
% Assm [h4s_primu_u_recs_PRIMu_u_REC0]: !x m f. h4/prim__rec/PRIM__REC x f m = happ (happ (h4/prim__rec/PRIM__REC__FUN x f) m) (h4/prim__rec/PRE m)
% Assm [h4s_bools_SELECTu_u_REFLu_u_2]: !_0. (!x y. happ (happ _0 x) y <=> x = y) ==> (!x. h4/min/_40 (happ _0 x) = x)
% Assm [h4s_bools_EXISTSu_u_UNIQUEu_u_DEF]: !x. h4/bool/_3F_21 x <=> $exists x /\ (!x y. happ x x /\ happ x y ==> x = y)
% Assm [h4s_primu_u_recs_SIMPu_u_RECu_u_RELu_u_UNIQUE]: !x m2 m1 g2 g1 f. h4/prim__rec/SIMP__REC__REL g1 x f m1 /\ h4/prim__rec/SIMP__REC__REL g2 x f m2 ==> (!n. h4/prim__rec/_3C n m1 /\ h4/prim__rec/_3C n m2 ==> happ g1 n = happ g2 n)
% Assm [h4s_bools_EXISTSu_u_UNIQUEu_u_THM]: !_0. (!P x. happ (happ _0 P) x <=> happ P x) ==> (!P. h4/bool/_3F_21 (happ _0 P) <=> (?x. happ P x) /\ (!x y. happ P x /\ happ P y ==> x = y))
% Assm [h4s_primu_u_recs_LESSu_u_0]: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm [h4s_primu_u_recs_SUCu_u_LESS]: !n m. h4/prim__rec/_3C (h4/num/SUC m) n ==> h4/prim__rec/_3C m n
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_primu_u_recs_PRIMu_u_RECu_u_THMu_c0]: !x f. h4/prim__rec/PRIM__REC x f h4/num/0 = x
% Assm [h4s_bools_SKOLEMu_u_THM]: !P. (!x. ?y. happ (happ P x) y) <=> (?f. !x. happ (happ P x) (happ f x))
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_primu_u_recs_SIMPu_u_RECu_u_EXISTS]: !x n f. ?fun. h4/prim__rec/SIMP__REC__REL fun x f n
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_bools_BETAu_u_THM]: !y f. happ f y = happ f y
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_bools_FORALLu_u_ANDu_u_THM]: !Q P. (!x. happ P x /\ happ Q x) <=> (!x. happ P x) /\ (!x. happ Q x)
% Assm [h4s_relations_WFu_u_EQu_u_INDUCTIONu_u_THM]: !R. h4/relation/WF R <=> (!P. (!x. (!y. happ (happ R y) x ==> happ P y) ==> happ P x) ==> (!x. happ P x))
% Assm [h4s_relations_WFu_u_EQu_u_WFP]: !R. h4/relation/WF R <=> (!x. h4/relation/WFP R x)
% Assm [h4s_relations_INDUCTIONu_u_WFu_u_THM]: !R. (!P. (!x. (!y. happ (happ R y) x ==> happ P y) ==> happ P x) ==> (!x. happ P x)) ==> h4/relation/WF R
% Assm [h4s_relations_WFRECu_u_THM]: !R M. h4/relation/WF R ==> (!x. happ (h4/relation/WFREC R M) x = happ (happ M (h4/relation/RESTRICT (h4/relation/WFREC R M) R x)) x)
% Assm [h4s_relations_WFu_u_INDUCTIONu_u_THM]: !R. h4/relation/WF R ==> (!P. (!x. (!y. happ (happ R y) x ==> happ P y) ==> happ P x) ==> (!x. happ P x))
% Assm [h4s_relations_WFu_u_TCu_u_EQN]: !R. h4/relation/WF (h4/relation/TC R) <=> h4/relation/WF R
% Assm [h4s_primu_u_recs_LESSu_u_0u_u_0]: h4/prim__rec/_3C h4/num/0 (h4/num/SUC h4/num/0)
% Assm [h4s_relations_WFu_u_DEF]: !R. h4/relation/WF R <=> (!B. (?w. happ B w) ==> (?min. happ B min /\ (!b. happ (happ R b) min ==> ~happ B b)))
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_relations_WFu_u_RECURSIONu_u_THM]: !_0. (!M R f. happ (happ (happ _0 M) R) f <=> (!x. happ f x = happ (happ M (h4/relation/RESTRICT f R x)) x)) ==> (!R. h4/relation/WF R ==> (!M. h4/bool/_3F_21 (happ (happ _0 M) R)))
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_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_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_relations_WFu_u_TC]: !R. h4/relation/WF R ==> h4/relation/WF (h4/relation/TC R)
% Assm [h4s_relations_TCu_u_SUBSET]: !y x R. happ (happ R x) y ==> happ (happ (h4/relation/TC R) x) y
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ 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_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_LEFTu_u_ORu_u_EXISTSu_u_THM]: !Q P. (?x. happ P x) \/ Q <=> (?x. happ P x \/ Q)
% Assm [h4s_bools_LEFTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. happ P x /\ Q) <=> (?x. happ P x) /\ Q
% Assm [h4s_relations_WFPu_u_RULES]: !x R. (!y. happ (happ R y) x ==> h4/relation/WFP R y) ==> h4/relation/WFP R x
% Assm [h4s_bools_RIGHTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. P /\ happ Q x) <=> P /\ (?x. 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_bools_RIGHTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. P \/ happ Q x) <=> P \/ (!x. happ Q x)
% Assm [h4s_relations_WFPu_u_STRONGu_u_INDUCT]: !R P. (!x. h4/relation/WFP R x /\ (!y. happ (happ R y) x ==> happ P y) ==> happ P x) ==> (!x. h4/relation/WFP R x ==> happ P x)
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_relations_WFu_u_SUBSET]: !R P. h4/relation/WF R /\ (!x y. happ (happ P x) y ==> happ (happ R x) y) ==> h4/relation/WF P
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_bools_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_relations_RESTRICTu_u_DEF]: !x f R x'. happ (h4/relation/RESTRICT f R x) x' = h4/bool/COND (happ (happ R x') x) (happ f x') h4/bool/ARB
% Assm [h4s_relations_WFRECu_u_DEF]: !_1. (!M f R v. happ (happ (happ (happ _1 M) f) R) v = happ (happ M (h4/relation/RESTRICT f R v)) v) ==> (!_0. (!M R f. happ (happ (happ _0 M) R) f = happ (happ (happ _1 M) f) R) ==> (!R M x. happ (h4/relation/WFREC R M) x = happ (happ M (h4/relation/RESTRICT (h4/relation/the__fun (h4/relation/TC R) (happ (happ _0 M) R) x) R x)) x))
% Assm [h4s_relations_theu_u_funu_u_def]: !_0. (!R M x f. happ (happ (happ (happ _0 R) M) x) f <=> h4/relation/approx R M x f) ==> (!x R M. h4/relation/the__fun R M x = h4/min/_40 (happ (happ (happ _0 R) M) x))
% Assm [h4s_relations_approxu_u_def]: !_0. (!M f R y. happ (happ (happ (happ _0 M) f) R) y = happ (happ M (h4/relation/RESTRICT f R y)) y) ==> (!x f R M. h4/relation/approx R M x f <=> f = h4/relation/RESTRICT (happ (happ (happ _0 M) f) R) R x)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_relations_TCu_u_TRANSITIVE]: !R. h4/relation/transitive (h4/relation/TC R)
% Assm [h4s_relations_transitiveu_u_def]: !R. h4/relation/transitive R <=> (!x y z. happ (happ R x) y /\ happ (happ R y) z ==> happ (happ R x) z)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Goal: !_1. (!x y. happ (happ _1 x) y <=> y = h4/num/SUC x) ==> (!_0. (!x. happ _0 x = happ _1 x) ==> h4/relation/WF _0)
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_Q1277984,TV_Q1277980]: ![V_f, V_g]: (![V_x]: s(TV_Q1277980,happ(s(t_fun(TV_Q1277984,TV_Q1277980),V_f),s(TV_Q1277984,V_x))) = s(TV_Q1277980,happ(s(t_fun(TV_Q1277984,TV_Q1277980),V_g),s(TV_Q1277984,V_x))) => s(t_fun(TV_Q1277984,TV_Q1277980),V_f) = s(t_fun(TV_Q1277984,TV_Q1277980),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_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_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_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,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_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_nums_INVu_u_SUC, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) => s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
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_primu_u_recs_LESSu_u_THM, 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,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))) <=> (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n) | p(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_primu_u_recs_SIMPu_u_RECu_u_THMu_c1, axiom, ![TV_u_27a]: ![V_x, V_m, V_f]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_primu_u_recs_simpu_u_rec(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_primu_u_recs_simpu_u_rec(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_primu_u_recs_LESSu_u_DEF, 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)))) <=> ?[V_P]: (![V_n0]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n0)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n0))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_m)))) & ~ (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_nums_NOTu_u_SUC, axiom, ![V_n]: ~ (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0))).
fof(ah4s_primu_u_recs_LESSu_u_LEMMA1, 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,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))) => (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n) | p(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_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_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_nums_SUCu_u_DEF, axiom, ![V_m]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_nums_absu_u_num(s(t_h4s_mins_ind,happ(s(t_fun(t_h4s_mins_ind,t_h4s_mins_ind),h4s_nums_sucu_u_rep),s(t_h4s_mins_ind,h4s_nums_repu_u_num(s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_primu_u_recs_INVu_u_SUCu_u_EQ, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_primu_u_recs_LESSu_u_LEMMA2, axiom, ![V_n, V_m]: ((s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n) | 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_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))).
fof(ah4s_primu_u_recs_PRE0u_c1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_primu_u_recs_LESSu_u_SUCu_u_REFL, 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_suc(s(t_h4s_nums_num,V_n))))))).
fof(ah4s_primu_u_recs_PREu_u_DEF, axiom, ![V_uu_0]: (![V_m, V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))) => ![V_m]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0)) & s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,V_v),s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_mins_u_40(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_h4s_nums_num,V_m)))))))))).
fof(ah4s_primu_u_recs_SIMPu_u_RECu_u_REL0, axiom, ![TV_u_27a]: ![V_x, V_n, V_fun, V_f]: (p(s(t_bool,h4s_primu_u_recs_simpu_u_recu_u_rel(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fun),s(TV_u_27a,V_x),s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,V_n)))) <=> (s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fun),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_x) & ![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)))) => s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fun),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fun),s(t_h4s_nums_num,V_m))))))))).
fof(ah4s_primu_u_recs_DC, axiom, ![TV_u_27a]: ![V_a, V_R, V_P]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_a)))) & ![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_y]: (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),happ(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_y))))))) => ?[V_f]: (s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_a) & ![V_n]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n)))))) & 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,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n))))),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))))))).
fof(ah4s_primu_u_recs_SUCu_u_ID, axiom, ![V_n]: ~ (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_n))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_primu_u_recs_PRIMu_u_RECu_u_THMu_c1, axiom, ![TV_u_27a]: ![V_x, V_m, V_f]: s(TV_u_27a,h4s_primu_u_recs_primu_u_rec(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(TV_u_27a,h4s_primu_u_recs_primu_u_rec(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(t_h4s_nums_num,V_m))))),s(t_h4s_nums_num,V_m)))).
fof(ah4s_primu_u_recs_EQu_u_LESS, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_n) => p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_bools_IMPu_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_primu_u_recs_LESSu_u_SUC, 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_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))).
fof(ah4s_primu_u_recs_PRIMu_u_RECu_u_FUN0, axiom, ![TV_u_27a]: ![V_uu_2]: (![V_f, V_fun, V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27a))),V_uu_2),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f))),s(t_fun(t_h4s_nums_num,TV_u_27a),V_fun))),s(t_h4s_nums_num,V_n))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fun),s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,V_n))))))),s(t_h4s_nums_num,V_n))) => ![V_uu_1]: (![V_f, V_fun]: s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27a))),V_uu_1),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f))),s(t_fun(t_h4s_nums_num,TV_u_27a),V_fun))) = s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27a))),V_uu_2),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f))),s(t_fun(t_h4s_nums_num,TV_u_27a),V_fun))) => ![V_uu_0]: (![V_x, V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_uu_0),s(TV_u_27a,V_x))),s(t_h4s_nums_num,V_n))) = s(TV_u_27a,V_x) => ![V_x, V_f]: s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),h4s_primu_u_recs_primu_u_recu_u_fun(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f))) = s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),h4s_primu_u_recs_simpu_u_rec(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_uu_0),s(TV_u_27a,V_x))),s(t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_fun(t_h4s_nums_num,TV_u_27a))),V_uu_1),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f))))))))).
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_primu_u_recs_numu_u_Axiomu_u_old, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_e, V_f, V_fn1]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_bool))),V_uu_0),s(TV_u_27a,V_e))),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f))),s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn1)))) <=> (s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn1),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_e) & ![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn1),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn1),s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,V_n))))) => ![V_f, V_e]: p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27a),t_bool))),V_uu_0),s(TV_u_27a,V_e))),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f)))))))).
fof(ah4s_nums_numu_u_ISOu_u_DEFu_c1, axiom, ![V_r]: (p(s(t_bool,h4s_nums_isu_u_numu_u_rep(s(t_h4s_mins_ind,V_r)))) <=> s(t_h4s_mins_ind,h4s_nums_repu_u_num(s(t_h4s_nums_num,h4s_nums_absu_u_num(s(t_h4s_mins_ind,V_r))))) = s(t_h4s_mins_ind,V_r))).
fof(ah4s_nums_numu_u_ISOu_u_DEFu_c0, axiom, ![V_a]: s(t_h4s_nums_num,h4s_nums_absu_u_num(s(t_h4s_mins_ind,h4s_nums_repu_u_num(s(t_h4s_nums_num,V_a))))) = s(t_h4s_nums_num,V_a)).
fof(ah4s_nums_ISu_u_NUMu_u_REP0, axiom, ![V_m]: (p(s(t_bool,h4s_nums_isu_u_numu_u_rep(s(t_h4s_mins_ind,V_m)))) <=> ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_mins_ind,t_bool),V_P),s(t_h4s_mins_ind,h4s_nums_zerou_u_rep)))) & ![V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_mins_ind,t_bool),V_P),s(t_h4s_mins_ind,V_n)))) => p(s(t_bool,happ(s(t_fun(t_h4s_mins_ind,t_bool),V_P),s(t_h4s_mins_ind,happ(s(t_fun(t_h4s_mins_ind,t_h4s_mins_ind),h4s_nums_sucu_u_rep),s(t_h4s_mins_ind,V_n)))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_mins_ind,t_bool),V_P),s(t_h4s_mins_ind,V_m))))))).
fof(ah4s_primu_u_recs_SIMPu_u_REC0, axiom, ![TV_u_27a]: ![V_x, V_n, V_fu_27]: ?[V_g]: (p(s(t_bool,h4s_primu_u_recs_simpu_u_recu_u_rel(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g),s(TV_u_27a,V_x),s(t_fun(TV_u_27a,TV_u_27a),V_fu_27),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))) & s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_primu_u_recs_simpu_u_rec(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,TV_u_27a),V_fu_27))),s(t_h4s_nums_num,V_n))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g),s(t_h4s_nums_num,V_n))))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_primu_u_recs_LESSu_u_SUCu_u_IMP, 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,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))) => (~ (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n)) => p(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_primu_u_recs_PRIMu_u_RECu_u_EQNu_c1, axiom, ![TV_u_27a]: ![V_x, V_n, V_m, V_f]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),h4s_primu_u_recs_primu_u_recu_u_fun(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))),s(t_h4s_nums_num,V_n))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),h4s_primu_u_recs_primu_u_recu_u_fun(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,V_n))))))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_primu_u_recs_SIMPu_u_RECu_u_RELu_u_UNIQUEu_u_RESULT, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x, V_f, V_n, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)))),V_uu_0),s(TV_u_27a,V_x))),s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_y)))) <=> ?[V_g]: (p(s(t_bool,h4s_primu_u_recs_simpu_u_recu_u_rel(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g),s(TV_u_27a,V_x),s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))) & s(TV_u_27a,V_y) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g),s(t_h4s_nums_num,V_n))))) => ![V_x, V_n, V_f]: p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)))),V_uu_0),s(TV_u_27a,V_x))),s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_nums_num,V_n)))))))).
fof(ah4s_primu_u_recs_LESSu_u_SUCu_u_SUCu_c1, axiom, ![V_m]: p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))))))).
fof(ah4s_primu_u_recs_LESSu_u_SUCu_u_SUCu_c0, axiom, ![V_m]: p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))))))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_bools_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_primu_u_recs_wellfoundedu_u_def, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_primu_u_recs_wellfounded(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> ~ (?[V_f]: ![V_n]: 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,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))))),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n))))))))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_bools_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_nums_ZEROu_u_DEF, axiom, s(t_h4s_nums_num,h4s_nums_0) = s(t_h4s_nums_num,h4s_nums_absu_u_num(s(t_h4s_mins_ind,h4s_nums_zerou_u_rep)))).
fof(ah4s_primu_u_recs_SIMPu_u_RECu_u_THMu_c0, axiom, ![TV_u_27a]: ![V_x, V_f]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_primu_u_recs_simpu_u_rec(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_x)).
fof(ah4s_nums_SUCu_u_REPu_u_DEFu_c0, axiom, p(s(t_bool,h4s_bools_oneu_u_one(s(t_fun(t_h4s_mins_ind,t_h4s_mins_ind),h4s_nums_sucu_u_rep))))).
fof(ah4s_nums_SUCu_u_REPu_u_DEFu_c1, axiom, ~ (p(s(t_bool,h4s_bools_onto(s(t_fun(t_h4s_mins_ind,t_h4s_mins_ind),h4s_nums_sucu_u_rep)))))).
fof(ah4s_bools_ONEu_u_ONEu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_f]: (p(s(t_bool,h4s_bools_oneu_u_one(s(t_fun(TV_u_27a,TV_u_27b),V_f)))) <=> ![V_x1, V_x2]: (s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x1))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x2))) => s(TV_u_27a,V_x1) = s(TV_u_27a,V_x2)))).
fof(ah4s_primu_u_recs_numu_u_Axiom, axiom, ![TV_u_27a]: ![V_f, V_e]: ?[V_fn]: (s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_e) & ![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_h4s_nums_num,V_n))),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_primu_u_recs_LESSu_u_MONO, 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_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_nums_ZEROu_u_REPu_u_DEF, axiom, ![V_y]: ~ (s(t_h4s_mins_ind,h4s_nums_zerou_u_rep) = s(t_h4s_mins_ind,happ(s(t_fun(t_h4s_mins_ind,t_h4s_mins_ind),h4s_nums_sucu_u_rep),s(t_h4s_mins_ind,V_y))))).
fof(ah4s_primu_u_recs_PRIMu_u_REC0, axiom, ![TV_u_27a]: ![V_x, V_m, V_f]: s(TV_u_27a,h4s_primu_u_recs_primu_u_rec(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(t_h4s_nums_num,V_m))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,TV_u_27a)),h4s_primu_u_recs_primu_u_recu_u_fun(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_primu_u_recs_pre(s(t_h4s_nums_num,V_m)))))).
fof(ah4s_bools_SELECTu_u_REFLu_u_2, axiom, ![TV_u_27a]: ![V_uu_0]: (![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_uu_0),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y)) => ![V_x]: s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_uu_0),s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x))).
fof(ah4s_bools_EXISTSu_u_UNIQUEu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: (p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(TV_u_27a,t_bool),V_x)))) <=> (p(s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x)))) & ![V_x0, V_y]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_x0)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_y))))) => s(TV_u_27a,V_x0) = s(TV_u_27a,V_y))))).
fof(ah4s_primu_u_recs_SIMPu_u_RECu_u_RELu_u_UNIQUE, axiom, ![TV_u_27a]: ![V_x, V_m2, V_m1, V_g2, V_g1, V_f]: ((p(s(t_bool,h4s_primu_u_recs_simpu_u_recu_u_rel(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g1),s(TV_u_27a,V_x),s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,V_m1)))) & p(s(t_bool,h4s_primu_u_recs_simpu_u_recu_u_rel(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g2),s(TV_u_27a,V_x),s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,V_m2))))) => ![V_n]: ((p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m1)))) & p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m2))))) => s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g1),s(t_h4s_nums_num,V_n))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g2),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_bools_EXISTSu_u_UNIQUEu_u_THM, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_P, V_x]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(TV_u_27a,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))) => ![V_P]: (p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),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, V_y]: ((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,V_y))))) => s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))))).
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,h4s_nums_suc(s(t_h4s_nums_num,V_n))))))).
fof(ah4s_primu_u_recs_SUCu_u_LESS, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_primu_u_recs_PRIMu_u_RECu_u_THMu_c0, axiom, ![TV_u_27a]: ![V_x, V_f]: s(TV_u_27a,h4s_primu_u_recs_primu_u_rec(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,TV_u_27a)),V_f),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_SKOLEMu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (![V_x]: ?[V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) <=> ?[V_f]: ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))))))).
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_primu_u_recs_SIMPu_u_RECu_u_EXISTS, axiom, ![TV_u_27a]: ![V_x, V_n, V_f]: ?[V_fun]: p(s(t_bool,h4s_primu_u_recs_simpu_u_recu_u_rel(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fun),s(TV_u_27a,V_x),s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,V_n))))).
fof(ah4s_bools_UNWINDu_u_THM2, axiom, ![TV_u_27a]: ![V_a, V_P]: (?[V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & 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,V_a)))))).
fof(ah4s_bools_BETAu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y)))).
fof(ah4s_bools_FUNu_u_EQu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g) <=> ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))).
fof(ah4s_bools_FORALLu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_relations_WFu_u_EQu_u_INDUCTIONu_u_THM, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> ![V_P]: (![V_x]: (![V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) => ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_relations_WFu_u_EQu_u_WFP, 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_x]: p(s(t_bool,h4s_relations_wfp(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x)))))).
fof(ah4s_relations_INDUCTIONu_u_WFu_u_THM, axiom, ![TV_u_27a]: ![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))))) => p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))).
fof(ah4s_relations_WFRECu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_R, 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),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))),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),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))),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_relations_WFu_u_INDUCTIONu_u_THM, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_P]: (![V_x]: (![V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) => ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_relations_WFu_u_TCu_u_EQN, axiom, ![TV_u_27a]: ![V_R]: s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_tc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) = s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))).
fof(ah4s_primu_u_recs_LESSu_u_0u_u_0, axiom, p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))).
fof(ah4s_relations_WFu_u_DEF, 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_B]: (?[V_w]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_B),s(TV_u_27a,V_w)))) => ?[V_min]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_B),s(TV_u_27a,V_min)))) & ![V_b]: (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_b))),s(TV_u_27a,V_min)))) => ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_B),s(TV_u_27a,V_b)))))))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,f))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_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_relations_WFu_u_RECURSIONu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_M, V_R, V_f]: (p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool))),V_uu_0),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,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,TV_u_27b),V_f)))) <=> ![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)))) => ![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_M]: p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool))),V_uu_0),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,t_fun(TV_u_27a,t_bool)),V_R))))))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_dcu_u_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_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_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_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_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_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_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_relations_WFu_u_TC, 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)))) => p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_tc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))))).
fof(ah4s_relations_TCu_u_SUBSET, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,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)),h4s_relations_tc(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_y)))))).
fof(ah4s_bools_NOTu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ?[V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_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_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
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_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_relations_WFPu_u_RULES, axiom, ![TV_u_27a]: ![V_x, V_R]: (![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,h4s_relations_wfp(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))))) => p(s(t_bool,h4s_relations_wfp(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x)))))).
fof(ah4s_bools_RIGHTu_u_EXISTSu_u_ANDu_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_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_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_relations_WFPu_u_STRONGu_u_INDUCT, axiom, ![TV_u_27a]: ![V_R, V_P]: (![V_x]: ((p(s(t_bool,h4s_relations_wfp(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,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,h4s_relations_wfp(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),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_x))))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_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_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_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_relations_WFu_u_SUBSET, axiom, ![TV_u_27a]: ![V_R, V_P]: ((p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) & ![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_P),s(TV_u_27a,V_x))),s(TV_u_27a,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_x))),s(TV_u_27a,V_y)))))) => p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_P)))))).
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_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_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_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_relations_RESTRICTu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f, V_R, V_xi_]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_xi_))) = s(TV_u_27b,h4s_bools_cond(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_xi_))),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_xi_))),s(TV_u_27b,h4s_bools_arb)))).
fof(ah4s_relations_WFRECu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_1]: (![V_M, V_f, V_R, V_v]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,TV_u_27b)))),V_uu_1),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),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_v))) = 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_v))))),s(TV_u_27a,V_v))) => ![V_uu_0]: (![V_M, V_R, V_f]: 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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)))),V_uu_0),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,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,TV_u_27b)))),V_uu_1),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),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))) => ![V_R, V_M, V_x]: s(TV_u_27b,happ(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))),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),h4s_relations_theu_u_fun(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_tc(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)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)))),V_uu_0),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,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_x))),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_relations_theu_u_funu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_R, V_M, V_x, V_f]: s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)))),V_uu_0),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))),s(TV_u_27a,V_x))),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_bool,h4s_relations_approx(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),s(TV_u_27a,V_x),s(t_fun(TV_u_27a,TV_u_27b),V_f))) => ![V_x, V_R, V_M]: s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_theu_u_fun(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),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27a,TV_u_27b),h4s_mins_u_40(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,TV_u_27b),t_bool)))),V_uu_0),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))),s(TV_u_27a,V_x))))))).
fof(ah4s_relations_approxu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_M, V_f, V_R, V_y]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,TV_u_27b)))),V_uu_0),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),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_y))) = 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_y))))),s(TV_u_27a,V_y))) => ![V_x, V_f, V_R, V_M]: (p(s(t_bool,h4s_relations_approx(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),s(TV_u_27a,V_x),s(t_fun(TV_u_27a,TV_u_27b),V_f)))) <=> s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,TV_u_27b)))),V_uu_0),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),V_f))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x)))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_relations_TCu_u_TRANSITIVE, axiom, ![TV_u_27a]: ![V_R]: p(s(t_bool,h4s_relations_transitive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_tc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))))).
fof(ah4s_relations_transitiveu_u_def, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_transitive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> ![V_x, V_y, V_z]: ((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_x))),s(TV_u_27a,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_z))))) => 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_x))),s(TV_u_27a,V_z))))))).
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(ch4s_primu_u_recs_WFu_u_PRED, conjecture, ![V_uu_1]: (![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_uu_1),s(t_h4s_nums_num,V_x))),s(t_h4s_nums_num,V_y)))) <=> s(t_h4s_nums_num,V_y) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_x)))) => ![V_uu_0]: (![V_x]: s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_h4s_nums_num,V_x))) = s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_uu_1),s(t_h4s_nums_num,V_x))) => p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_uu_0))))))).
