%   ORIGINAL: h4/set__relation/nth__min__ind
% 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/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/set__relation/nth__min__tupled__primitive__def: h4/set__relation/nth__min__tupled = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!n r s r_27 min. min = h4/set__relation/get__min r_27 (h4/pair/_2C s r) /\ ~(min = h4/option/NONE) ==> R (h4/pair/_2C r_27 (h4/pair/_2C (h4/pair/_2C (h4/pred__set/DELETE s (h4/option/THE min)) r) n)) (h4/pair/_2C r_27 (h4/pair/_2C (h4/pair/_2C s r) (h4/num/SUC n)))))) (\nth__min__tupled a. h4/pair/pair__CASE a (\r_27 v1. h4/pair/pair__CASE v1 (\v2 v3. h4/pair/pair__CASE v2 (\s r. h4/arithmetic/num__CASE v3 (h4/combin/I (h4/set__relation/get__min r_27 (h4/pair/_2C s r))) (\n. h4/combin/I (h4/bool/LET (\min. h4/bool/COND (min = h4/option/NONE) h4/option/NONE (nth__min__tupled (h4/pair/_2C r_27 (h4/pair/_2C (h4/pair/_2C (h4/pred__set/DELETE s (h4/option/THE min)) r) n)))) (h4/set__relation/get__min r_27 (h4/pair/_2C s r))))))))
% Assm: h4/set__relation/get__min__def: !s r_27 r. h4/set__relation/get__min r_27 (h4/pair/_2C s r) = h4/bool/LET (\mins. h4/bool/COND (h4/pred__set/SING mins) (h4/option/SOME (h4/pred__set/CHOICE mins)) h4/option/NONE) (h4/set__relation/minimal__elements (h4/set__relation/minimal__elements s r) r_27)
% Assm: h4/option/THE__DEF: !x. h4/option/THE (h4/option/SOME x) = x
% Assm: h4/option/option__Axiom: !f e. ?fn. fn h4/option/NONE = e /\ (!x. fn (h4/option/SOME x) = f x)
% Assm: h4/option/OPTION__MAP2__DEF: !y x f. h4/option/OPTION__MAP2 f x y = h4/bool/COND (h4/option/IS__SOME x /\ h4/option/IS__SOME y) (h4/option/SOME (f (h4/option/THE x) (h4/option/THE y))) h4/option/NONE
% Assm: h4/option/option__case__compute: !x f e. h4/option/option__CASE x e f = h4/bool/COND (h4/option/IS__SOME x) (f (h4/option/THE x)) e
% Assm: h4/option/option__CLAUSES_c8: !x. h4/option/IS__SOME x ==> h4/option/SOME (h4/option/THE x) = x
% Assm: h4/option/option__nchotomy: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/option/IS__SOME__DEF_c0: !x. h4/option/IS__SOME (h4/option/SOME x) <=> T
% Assm: h4/option/IS__SOME__DEF_c1: h4/option/IS__SOME h4/option/NONE <=> F
% Assm: h4/option/option__case__def_c0: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm: h4/option/option__case__def_c1: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = f x
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/option/option__CLAUSES_c1: !x. h4/option/THE (h4/option/SOME x) = x
% Assm: h4/option/option__CLAUSES_c12: !x f e. h4/option/IS__SOME x ==> h4/option/option__CASE x e f = f (h4/option/THE x)
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/option/NOT__SOME__NONE: !x. ~(h4/option/SOME x = h4/option/NONE)
% Assm: h4/option/NOT__IS__SOME__EQ__NONE: !x. ~h4/option/IS__SOME x <=> x = h4/option/NONE
% Assm: h4/option/option__case__ID: !x. h4/option/option__CASE x h4/option/NONE h4/option/SOME = x
% Assm: h4/option/IS__NONE__EQ__NONE: !x. h4/option/IS__NONE x <=> x = h4/option/NONE
% Assm: h4/option/OPTION__JOIN__DEF_c1: !x. h4/option/OPTION__JOIN (h4/option/SOME x) = x
% Assm: h4/option/option__case__SOME__ID: !x. h4/option/option__CASE x x h4/option/SOME = x
% Assm: h4/option/option__induction: !P. P h4/option/NONE /\ (!a. P (h4/option/SOME a)) ==> (!x. P x)
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/option/OPTION__CHOICE__def_c0: !m2. h4/option/OPTION__CHOICE h4/option/NONE m2 = m2
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/option/NOT__NONE__SOME: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm: h4/option/option__CLAUSES_c7: !x. ~h4/option/IS__SOME x <=> x = h4/option/NONE
% Assm: h4/option/IF__NONE__EQUALS__OPTION_c3: !x X P. h4/bool/COND P h4/option/NONE X = h4/option/SOME x <=> ~P /\ X = h4/option/SOME x
% Assm: h4/option/OPTREL__def: !y x R. h4/option/OPTREL R x y <=> x = h4/option/NONE /\ y = h4/option/NONE \/ (?x0 y0. x = h4/option/SOME x0 /\ y = h4/option/SOME y0 /\ R x0 y0)
% Assm: h4/option/OPTION__CHOICE__EQ__NONE: !m2 m1. h4/option/OPTION__CHOICE m1 m2 = h4/option/NONE <=> m1 = h4/option/NONE /\ m2 = h4/option/NONE
% Assm: h4/option/OPTION__BIND__EQUALS__OPTION_c0: !p f. h4/option/OPTION__BIND p f = h4/option/NONE <=> p = h4/option/NONE \/ (?x. p = h4/option/SOME x /\ f x = h4/option/NONE)
% Assm: h4/option/IF__NONE__EQUALS__OPTION_c2: !x X P. h4/bool/COND P X h4/option/NONE = h4/option/SOME x <=> P /\ X = h4/option/SOME x
% Assm: h4/option/FORALL__OPTION: !P. (!opt. P opt) <=> P h4/option/NONE /\ (!x. P (h4/option/SOME x))
% Assm: h4/option/OPTION__APPLY__def_c0: !x. h4/option/OPTION__APPLY h4/option/NONE x = h4/option/NONE
% Assm: h4/option/IF__NONE__EQUALS__OPTION_c1: !X P. h4/bool/COND P h4/option/NONE X = h4/option/NONE <=> h4/option/IS__SOME X ==> P
% Assm: h4/option/OPTION__CHOICE__def_c1: !x m2. h4/option/OPTION__CHOICE (h4/option/SOME x) m2 = h4/option/SOME x
% Assm: h4/option/EXISTS__OPTION: !P. (?opt. P opt) <=> P h4/option/NONE \/ (?x. P (h4/option/SOME x))
% Assm: h4/option/option__case__cong: !v_27 v f_27 f M_27 M. M = M_27 /\ (M_27 = h4/option/NONE ==> v = v_27) /\ (!x. M_27 = h4/option/SOME x ==> f x = f_27 x) ==> h4/option/option__CASE M v f = h4/option/option__CASE M_27 v_27 f_27
% Assm: h4/option/option__CLAUSES_c9: !x. h4/option/option__CASE x h4/option/NONE h4/option/SOME = x
% Assm: h4/option/option__CLAUSES_c6: !x. h4/option/IS__NONE x <=> x = h4/option/NONE
% Assm: h4/option/IS__NONE__DEF_c1: h4/option/IS__NONE h4/option/NONE <=> T
% Assm: h4/option/IS__NONE__DEF_c0: !x. h4/option/IS__NONE (h4/option/SOME x) <=> F
% Assm: h4/option/IF__NONE__EQUALS__OPTION_c0: !X P. h4/bool/COND P X h4/option/NONE = h4/option/NONE <=> P ==> h4/option/IS__NONE X
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/option/option__REP__ABS__DEF_c0: !a. h4/option/option__ABS (h4/option/option__REP a) = a
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/option/option__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION (\x. T) rep
% Assm: h4/bool/ABS__REP__THM: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. abs (rep a) = a) /\ (!r. P r <=> rep (abs r) = r))
% 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/option/option__CLAUSES_c13: !x e. h4/option/IS__SOME x ==> h4/option/option__CASE x e h4/option/SOME = x
% Assm: h4/option/OPTION__BIND__EQUALS__OPTION_c1: !y p f. h4/option/OPTION__BIND p f = h4/option/SOME y <=> (?x. p = h4/option/SOME x /\ f x = h4/option/SOME y)
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> t
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/option/option__CLAUSES_c11: !x f e. h4/option/IS__NONE x ==> h4/option/option__CASE x e f = e
% Assm: h4/option/OPTION__IGNORE__BIND__def: !m2 m1. h4/option/OPTION__IGNORE__BIND m1 m2 = h4/option/OPTION__BIND m1 (h4/combin/K m2)
% Assm: h4/option/option__CLAUSES_c10: !x. h4/option/option__CASE x x h4/option/SOME = x
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/option/OPTION__BIND__cong: !o2 o1 f2 f1. o1 = o2 /\ (!x. o2 = h4/option/SOME x ==> f1 x = f2 x) ==> h4/option/OPTION__BIND o1 f1 = h4/option/OPTION__BIND o2 f2
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/option/OPTION__BIND__def_c0: !f. h4/option/OPTION__BIND h4/option/NONE f = h4/option/NONE
% Assm: h4/option/OPTION__BIND__def_c1: !x f. h4/option/OPTION__BIND (h4/option/SOME x) f = f x
% Assm: h4/option/option__CLAUSES_c19: !x. h4/option/OPTION__JOIN (h4/option/SOME x) = x
% Assm: h4/option/IF__EQUALS__OPTION_c3: !y x P. h4/bool/COND P h4/option/NONE (h4/option/SOME x) = h4/option/SOME y <=> ~P /\ x = y
% Assm: h4/option/IF__EQUALS__OPTION_c2: !y x P. h4/bool/COND P (h4/option/SOME x) h4/option/NONE = h4/option/SOME y <=> P /\ x = y
% Assm: h4/option/IF__EQUALS__OPTION_c1: !x P. h4/bool/COND P h4/option/NONE (h4/option/SOME x) = h4/option/NONE <=> P
% Assm: h4/option/IF__EQUALS__OPTION_c0: !x P. h4/bool/COND P (h4/option/SOME x) h4/option/NONE = h4/option/NONE <=> ~P
% Assm: h4/bool/COND__ID: !t b. h4/bool/COND b t t = t
% Assm: h4/option/OPTREL__MONO: !y x Q P. (!x0 y0. P x0 y0 ==> Q x0 y0) ==> h4/option/OPTREL P x y ==> h4/option/OPTREL Q x y
% Assm: h4/option/OPTREL__refl: !R. (!x. R x x) ==> (!x. h4/option/OPTREL R x x)
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/set__relation/per__delete: !xss xs e. h4/set__relation/per xs xss ==> h4/set__relation/per (h4/pred__set/DELETE xs e) (h4/pred__set/GSPEC (\es. h4/pair/_2C es (h4/bool/IN es (h4/pred__set/IMAGE (\es0. h4/pred__set/DELETE es0 e) xss) /\ ~(es = h4/pred__set/EMPTY))))
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/option/NONE__DEF: h4/option/NONE = h4/option/option__ABS (h4/sum/INR h4/one/one0)
% Assm: h4/option/SOME__DEF: !x. h4/option/SOME x = h4/option/option__ABS (h4/sum/INL x)
% Assm: h4/one/one1: !v. v = h4/one/one0
% Assm: h4/sum/sum__INDUCT: !P. (!x. P (h4/sum/INL x)) /\ (!y. P (h4/sum/INR y)) ==> (!s. P s)
% Assm: h4/bool/LEFT__FORALL__OR__THM: !Q P. (!x. P x \/ Q) <=> (!x. P x) \/ Q
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/bool/UNWIND__FORALL__THM2: !v f. (!x. x = v ==> f x) <=> f v
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/RIGHT__EXISTS__AND__THM: !Q P. (?x. P /\ Q x) <=> P /\ (?x. Q x)
% Assm: h4/bool/AND__CLAUSES_c4: !t. t /\ t <=> t
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Goal: !P. (!r_27 s r. P r_27 (h4/pair/_2C s r) h4/num/0) /\ (!r_27 s r n. (!min. min = h4/set__relation/get__min r_27 (h4/pair/_2C s r) /\ ~(min = h4/option/NONE) ==> P r_27 (h4/pair/_2C (h4/pred__set/DELETE s (h4/option/THE min)) r) n) ==> P r_27 (h4/pair/_2C s r) (h4/num/SUC n)) ==> (!v v1 v2 v3. P v (h4/pair/_2C v1 v2) v3)
%   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_pairs_UNCURRYu_u_DEF]: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_setu_u_relations_nthu_u_minu_u_tupledu_u_primitiveu_u_def]: !_10. (!nth__min__tupled r_27 s r n min. ?v. (v <=> min = h4/option/NONE) /\ happ (happ (happ (happ (happ (happ _10 nth__min__tupled) r_27) s) r) n) min = h4/bool/COND v h4/option/NONE (happ nth__min__tupled (h4/pair/_2C r_27 (h4/pair/_2C (h4/pair/_2C (h4/pred__set/DELETE s (h4/option/THE min)) r) n)))) ==> (!_9. (!nth__min__tupled r_27 s r n. happ (happ (happ (happ (happ _9 nth__min__tupled) r_27) s) r) n = h4/combin/I (h4/bool/LET (happ (happ (happ (happ (happ _10 nth__min__tupled) r_27) s) r) n) (h4/set__relation/get__min r_27 (h4/pair/_2C s r)))) ==> (!_8. (!v3 nth__min__tupled r_27 s r. happ (happ (happ (happ (happ _8 v3) nth__min__tupled) r_27) s) r = h4/arithmetic/num__CASE v3 (h4/combin/I (h4/set__relation/get__min r_27 (h4/pair/_2C s r))) (happ (happ (happ (happ _9 nth__min__tupled) r_27) s) r)) ==> (!_7. (!v3 nth__min__tupled r_27 s. happ (happ (happ (happ _7 v3) nth__min__tupled) r_27) s = happ (happ (happ (happ _8 v3) nth__min__tupled) r_27) s) ==> (!_6. (!v2 nth__min__tupled r_27 v3. happ (happ (happ (happ _6 v2) nth__min__tupled) r_27) v3 = h4/pair/pair__CASE v2 (happ (happ (happ _7 v3) nth__min__tupled) r_27)) ==> (!_5. (!nth__min__tupled r_27 v2. happ (happ (happ _5 nth__min__tupled) r_27) v2 = happ (happ (happ _6 v2) nth__min__tupled) r_27) ==> (!_4. (!nth__min__tupled r_27 v1. happ (happ (happ _4 nth__min__tupled) r_27) v1 = h4/pair/pair__CASE v1 (happ (happ _5 nth__min__tupled) r_27)) ==> (!_3. (!nth__min__tupled r_27. happ (happ _3 nth__min__tupled) r_27 = happ (happ _4 nth__min__tupled) r_27) ==> (!_2. (!nth__min__tupled a. happ (happ _2 nth__min__tupled) a = h4/pair/pair__CASE a (happ _3 nth__min__tupled)) ==> (!_1. (!nth__min__tupled. happ _1 nth__min__tupled = happ _2 nth__min__tupled) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R /\ (!n r s r_27 min. min = h4/set__relation/get__min r_27 (h4/pair/_2C s r) /\ ~(min = h4/option/NONE) ==> happ (happ R (h4/pair/_2C r_27 (h4/pair/_2C (h4/pair/_2C (h4/pred__set/DELETE s (h4/option/THE min)) r) n))) (h4/pair/_2C r_27 (h4/pair/_2C (h4/pair/_2C s r) (h4/num/SUC n))))) ==> h4/set__relation/nth__min__tupled = h4/relation/WFREC (h4/min/_40 _0) _1))))))))))
% Assm [h4s_setu_u_relations_getu_u_minu_u_def]: !_0. (!mins. happ _0 mins = h4/bool/COND (h4/pred__set/SING mins) (happ h4/option/SOME (h4/pred__set/CHOICE mins)) h4/option/NONE) ==> (!s r_27 r. h4/set__relation/get__min r_27 (h4/pair/_2C s r) = h4/bool/LET _0 (h4/set__relation/minimal__elements (h4/set__relation/minimal__elements s r) r_27))
% Assm [h4s_options_THEu_u_DEF]: !x. h4/option/THE (happ h4/option/SOME x) = x
% Assm [h4s_options_optionu_u_Axiom]: !f e. ?fn. happ fn h4/option/NONE = e /\ (!x. happ fn (happ h4/option/SOME x) = happ f x)
% Assm [h4s_options_OPTIONu_u_MAP2u_u_DEF]: !y x f. ?v. (v <=> h4/option/IS__SOME x /\ h4/option/IS__SOME y) /\ h4/option/OPTION__MAP2 f x y = h4/bool/COND v (happ h4/option/SOME (happ (happ f (h4/option/THE x)) (h4/option/THE y))) h4/option/NONE
% Assm [h4s_options_optionu_u_caseu_u_compute]: !x f e. h4/option/option__CASE x e f = h4/bool/COND (h4/option/IS__SOME x) (happ f (h4/option/THE x)) e
% Assm [h4s_options_optionu_u_CLAUSESu_c8]: !x. h4/option/IS__SOME x ==> happ h4/option/SOME (h4/option/THE x) = x
% Assm [h4s_options_optionu_u_nchotomy]: !opt. opt = h4/option/NONE \/ (?x. opt = happ h4/option/SOME x)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_options_ISu_u_SOMEu_u_DEFu_c0]: !x. h4/option/IS__SOME (happ h4/option/SOME x) <=> T
% Assm [h4s_options_ISu_u_SOMEu_u_DEFu_c1]: h4/option/IS__SOME h4/option/NONE <=> F
% Assm [h4s_options_optionu_u_caseu_u_defu_c0]: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm [h4s_options_optionu_u_caseu_u_defu_c1]: !x v f. h4/option/option__CASE (happ h4/option/SOME x) v f = happ f x
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_options_optionu_u_CLAUSESu_c1]: !x. h4/option/THE (happ h4/option/SOME x) = x
% Assm [h4s_options_optionu_u_CLAUSESu_c12]: !x f e. h4/option/IS__SOME x ==> h4/option/option__CASE x e f = happ f (h4/option/THE x)
% Assm [h4s_options_SOMEu_u_11]: !y x. happ h4/option/SOME x = happ h4/option/SOME y <=> x = y
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_options_NOTu_u_SOMEu_u_NONE]: !x. ~(happ h4/option/SOME x = h4/option/NONE)
% Assm [h4s_options_NOTu_u_ISu_u_SOMEu_u_EQu_u_NONE]: !x. ~h4/option/IS__SOME x <=> x = h4/option/NONE
% Assm [h4s_options_optionu_u_caseu_u_ID]: !x. h4/option/option__CASE x h4/option/NONE h4/option/SOME = x
% Assm [h4s_options_ISu_u_NONEu_u_EQu_u_NONE]: !x. h4/option/IS__NONE x <=> x = h4/option/NONE
% Assm [h4s_options_OPTIONu_u_JOINu_u_DEFu_c1]: !x. h4/option/OPTION__JOIN (happ h4/option/SOME x) = x
% Assm [h4s_options_optionu_u_caseu_u_SOMEu_u_ID]: !x. h4/option/option__CASE x x h4/option/SOME = x
% Assm [h4s_options_optionu_u_induction]: !P. happ P h4/option/NONE /\ (!a. happ P (happ h4/option/SOME a)) ==> (!x. happ P x)
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_options_OPTIONu_u_CHOICEu_u_defu_c0]: !m2. h4/option/OPTION__CHOICE h4/option/NONE m2 = m2
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_options_NOTu_u_NONEu_u_SOME]: !x. ~(h4/option/NONE = happ h4/option/SOME x)
% Assm [h4s_options_optionu_u_CLAUSESu_c7]: !x. ~h4/option/IS__SOME x <=> x = h4/option/NONE
% Assm [h4s_options_IFu_u_NONEu_u_EQUALSu_u_OPTIONu_c3]: !x X P. h4/bool/COND P h4/option/NONE X = happ h4/option/SOME x <=> ~P /\ X = happ h4/option/SOME x
% Assm [h4s_options_OPTRELu_u_def]: !y x R. h4/option/OPTREL R x y <=> x = h4/option/NONE /\ y = h4/option/NONE \/ (?x0 y0. x = happ h4/option/SOME x0 /\ y = happ h4/option/SOME y0 /\ happ (happ R x0) y0)
% Assm [h4s_options_OPTIONu_u_CHOICEu_u_EQu_u_NONE]: !m2 m1. h4/option/OPTION__CHOICE m1 m2 = h4/option/NONE <=> m1 = h4/option/NONE /\ m2 = h4/option/NONE
% Assm [h4s_options_OPTIONu_u_BINDu_u_EQUALSu_u_OPTIONu_c0]: !p f. h4/option/OPTION__BIND p f = h4/option/NONE <=> p = h4/option/NONE \/ (?x. p = happ h4/option/SOME x /\ happ f x = h4/option/NONE)
% Assm [h4s_options_IFu_u_NONEu_u_EQUALSu_u_OPTIONu_c2]: !x X P. h4/bool/COND P X h4/option/NONE = happ h4/option/SOME x <=> P /\ X = happ h4/option/SOME x
% Assm [h4s_options_FORALLu_u_OPTION]: !P. (!opt. happ P opt) <=> happ P h4/option/NONE /\ (!x. happ P (happ h4/option/SOME x))
% Assm [h4s_options_OPTIONu_u_APPLYu_u_defu_c0]: !x. h4/option/OPTION__APPLY h4/option/NONE x = h4/option/NONE
% Assm [h4s_options_IFu_u_NONEu_u_EQUALSu_u_OPTIONu_c1]: !X P. h4/bool/COND P h4/option/NONE X = h4/option/NONE <=> h4/option/IS__SOME X ==> P
% Assm [h4s_options_OPTIONu_u_CHOICEu_u_defu_c1]: !x m2. h4/option/OPTION__CHOICE (happ h4/option/SOME x) m2 = happ h4/option/SOME x
% Assm [h4s_options_EXISTSu_u_OPTION]: !P. (?opt. happ P opt) <=> happ P h4/option/NONE \/ (?x. happ P (happ h4/option/SOME x))
% Assm [h4s_options_optionu_u_caseu_u_cong]: !v_27 v f_27 f M_27 M. M = M_27 /\ (M_27 = h4/option/NONE ==> v = v_27) /\ (!x. M_27 = happ h4/option/SOME x ==> happ f x = happ f_27 x) ==> h4/option/option__CASE M v f = h4/option/option__CASE M_27 v_27 f_27
% Assm [h4s_options_optionu_u_CLAUSESu_c9]: !x. h4/option/option__CASE x h4/option/NONE h4/option/SOME = x
% Assm [h4s_options_optionu_u_CLAUSESu_c6]: !x. h4/option/IS__NONE x <=> x = h4/option/NONE
% Assm [h4s_options_ISu_u_NONEu_u_DEFu_c1]: h4/option/IS__NONE h4/option/NONE <=> T
% Assm [h4s_options_ISu_u_NONEu_u_DEFu_c0]: !x. h4/option/IS__NONE (happ h4/option/SOME x) <=> F
% Assm [h4s_options_IFu_u_NONEu_u_EQUALSu_u_OPTIONu_c0]: !X P. h4/bool/COND P X h4/option/NONE = h4/option/NONE <=> P ==> h4/option/IS__NONE X
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_options_optionu_u_REPu_u_ABSu_u_DEFu_c0]: !a. h4/option/option__ABS (h4/option/option__REP a) = a
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_options_optionu_u_TYu_u_DEF]: !_0. (!x. happ _0 x <=> T) ==> (?rep. h4/bool/TYPE__DEFINITION _0 rep)
% Assm [h4s_bools_ABSu_u_REPu_u_THM]: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. happ abs (happ rep a) = a) /\ (!r. happ P r <=> happ rep (happ abs r) = r))
% 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_options_optionu_u_CLAUSESu_c13]: !x e. h4/option/IS__SOME x ==> h4/option/option__CASE x e h4/option/SOME = x
% Assm [h4s_options_OPTIONu_u_BINDu_u_EQUALSu_u_OPTIONu_c1]: !y p f. h4/option/OPTION__BIND p f = happ h4/option/SOME y <=> (?x. p = happ h4/option/SOME x /\ happ f x = happ h4/option/SOME y)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_options_optionu_u_CLAUSESu_c11]: !x f e. h4/option/IS__NONE x ==> h4/option/option__CASE x e f = e
% Assm [h4s_options_OPTIONu_u_IGNOREu_u_BINDu_u_def]: !m2 m1. h4/option/OPTION__IGNORE__BIND m1 m2 = h4/option/OPTION__BIND m1 (h4/combin/K m2)
% Assm [h4s_options_optionu_u_CLAUSESu_c10]: !x. h4/option/option__CASE x x h4/option/SOME = x
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_options_OPTIONu_u_BINDu_u_cong]: !o2 o1 f2 f1. o1 = o2 /\ (!x. o2 = happ h4/option/SOME x ==> happ f1 x = happ f2 x) ==> h4/option/OPTION__BIND o1 f1 = h4/option/OPTION__BIND o2 f2
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_options_OPTIONu_u_BINDu_u_defu_c0]: !f. h4/option/OPTION__BIND h4/option/NONE f = h4/option/NONE
% Assm [h4s_options_OPTIONu_u_BINDu_u_defu_c1]: !x f. h4/option/OPTION__BIND (happ h4/option/SOME x) f = happ f x
% Assm [h4s_options_optionu_u_CLAUSESu_c19]: !x. h4/option/OPTION__JOIN (happ h4/option/SOME x) = x
% Assm [h4s_options_IFu_u_EQUALSu_u_OPTIONu_c3]: !y x P. h4/bool/COND P h4/option/NONE (happ h4/option/SOME x) = happ h4/option/SOME y <=> ~P /\ x = y
% Assm [h4s_options_IFu_u_EQUALSu_u_OPTIONu_c2]: !y x P. h4/bool/COND P (happ h4/option/SOME x) h4/option/NONE = happ h4/option/SOME y <=> P /\ x = y
% Assm [h4s_options_IFu_u_EQUALSu_u_OPTIONu_c1]: !x P. h4/bool/COND P h4/option/NONE (happ h4/option/SOME x) = h4/option/NONE <=> P
% Assm [h4s_options_IFu_u_EQUALSu_u_OPTIONu_c0]: !x P. h4/bool/COND P (happ h4/option/SOME x) h4/option/NONE = h4/option/NONE <=> ~P
% Assm [h4s_bools_CONDu_u_ID]: !t b. h4/bool/COND b t t = t
% Assm [h4s_options_OPTRELu_u_MONO]: !y x Q P. (!x0 y0. happ (happ P x0) y0 ==> happ (happ Q x0) y0) ==> h4/option/OPTREL P x y ==> h4/option/OPTREL Q x y
% Assm [h4s_options_OPTRELu_u_refl]: !R. (!x. happ (happ R x) x) ==> (!x. h4/option/OPTREL R x x)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_setu_u_relations_peru_u_delete]: !_1. (!e es0. happ (happ _1 e) es0 = h4/pred__set/DELETE es0 e) ==> (!_0. (!e xss es. ?v. (v <=> h4/bool/IN es (h4/pred__set/IMAGE (happ _1 e) xss) /\ ~(es = h4/pred__set/EMPTY)) /\ happ (happ (happ _0 e) xss) es = h4/pair/_2C es v) ==> (!xss xs e. h4/set__relation/per xs xss ==> h4/set__relation/per (h4/pred__set/DELETE xs e) (h4/pred__set/GSPEC (happ (happ _0 e) xss))))
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_options_NONEu_u_DEF]: h4/option/NONE = h4/option/option__ABS (h4/sum/INR h4/one/one0)
% Assm [h4s_options_SOMEu_u_DEF]: !x. happ h4/option/SOME x = h4/option/option__ABS (h4/sum/INL x)
% Assm [h4s_ones_one1]: !v. v = h4/one/one0
% Assm [h4s_sums_sumu_u_INDUCT]: !P. (!x. happ P (h4/sum/INL x)) /\ (!y. happ P (h4/sum/INR y)) ==> (!s. happ P s)
% 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_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_bools_UNWINDu_u_FORALLu_u_THM2]: !v f. (!x. x = v ==> happ f x) <=> happ f v
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% 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_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% 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_ANDu_u_CLAUSESu_c4]: !t. t /\ t <=> t
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Goal: !P. (!r_27 s r. happ (happ (happ P r_27) (h4/pair/_2C s r)) h4/num/0) /\ (!r_27 s r n. (!min. min = h4/set__relation/get__min r_27 (h4/pair/_2C s r) /\ ~(min = h4/option/NONE) ==> happ (happ (happ P r_27) (h4/pair/_2C (h4/pred__set/DELETE s (h4/option/THE min)) r)) n) ==> happ (happ (happ P r_27) (h4/pair/_2C s r)) (h4/num/SUC n)) ==> (!v v1 v2 v3. happ (happ (happ P v) (h4/pair/_2C v1 v2)) v3)
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_Q1225416,TV_Q1225412]: ![V_f, V_g]: (![V_x]: s(TV_Q1225412,happ(s(t_fun(TV_Q1225416,TV_Q1225412),V_f),s(TV_Q1225416,V_x))) = s(TV_Q1225412,happ(s(t_fun(TV_Q1225416,TV_Q1225412),V_g),s(TV_Q1225416,V_x))) => s(t_fun(TV_Q1225416,TV_Q1225412),V_f) = s(t_fun(TV_Q1225416,TV_Q1225412),V_g))).
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_setu_u_relations_nthu_u_minu_u_tupledu_u_primitiveu_u_def, axiom, ![TV_u_27a]: ![V_uu_10]: (![V_nthu_u_minu_u_tupled, V_ru_27, V_s, V_r, V_n, V_min]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_options_option(TV_u_27a),V_min) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)) & s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))))))),V_uu_10),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,V_n))),s(t_h4s_options_option(TV_u_27a),V_min))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled),s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),h4s_pairs_u_2c(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(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,h4s_options_the(s(t_h4s_options_option(TV_u_27a),V_min))))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,V_n)))))))))) => ![V_uu_9]: (![V_nthu_u_minu_u_tupled, V_ru_27, V_s, V_r, V_n]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))))),V_uu_9),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,V_n))) = s(t_h4s_options_option(TV_u_27a),h4s_combins_i(s(t_h4s_options_option(TV_u_27a),h4s_bools_let(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a))))))),V_uu_10),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,V_n))),s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_getu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))))))) => ![V_uu_8]: (![V_v3, V_nthu_u_minu_u_tupled, V_ru_27, V_s, V_r]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)))))),V_uu_8),s(t_h4s_nums_num,V_v3))),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))) = s(t_h4s_options_option(TV_u_27a),h4s_arithmetics_numu_u_case(s(t_h4s_nums_num,V_v3),s(t_h4s_options_option(TV_u_27a),h4s_combins_i(s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_getu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))))),s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))))),V_uu_9),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))) => ![V_uu_7]: (![V_v3, V_nthu_u_minu_u_tupled, V_ru_27, V_s]: s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)))))),V_uu_7),s(t_h4s_nums_num,V_v3))),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)))))),V_uu_8),s(t_h4s_nums_num,V_v3))),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_fun(TV_u_27a,t_bool),V_s))) => ![V_uu_6]: (![V_v2, V_nthu_u_minu_u_tupled, V_ru_27, V_v3]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))))),V_uu_6),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),V_v2))),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_h4s_nums_num,V_v3))) = s(t_h4s_options_option(TV_u_27a),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),V_v2),s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_options_option(TV_u_27a)))))),V_uu_7),s(t_h4s_nums_num,V_v3))),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))))) => ![V_uu_5]: (![V_nthu_u_minu_u_tupled, V_ru_27, V_v2]: s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))))),V_uu_5),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),V_v2))) = s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))))),V_uu_6),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),V_v2))),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))) => ![V_uu_4]: (![V_nthu_u_minu_u_tupled, V_ru_27, V_v1]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),V_v1))) = s(t_h4s_options_option(TV_u_27a),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),V_v1),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))))),V_uu_5),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))))) => ![V_uu_3]: (![V_nthu_u_minu_u_tupled, V_ru_27]: s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))) = s(t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))) => ![V_uu_2]: (![V_nthu_u_minu_u_tupled, V_a]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))),s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),V_a))) = s(t_h4s_options_option(TV_u_27a),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),V_a),s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))))) => ![V_uu_1]: (![V_nthu_u_minu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a))),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),V_nthu_u_minu_u_tupled))) => ![V_uu_0]: (![V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_bool)),V_R)))) & ![V_n, V_r, V_s, V_ru_27, V_min]: ((s(t_h4s_options_option(TV_u_27a),V_min) = s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_getu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))) & ~ (s(t_h4s_options_option(TV_u_27a),V_min) = s(t_h4s_options_option(TV_u_27a),h4s_options_none))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_bool)),V_R),s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),h4s_pairs_u_2c(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(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,h4s_options_the(s(t_h4s_options_option(TV_u_27a),V_min))))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,V_n))))))),s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),h4s_pairs_u_2c(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))))))))))) => s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),h4s_setu_u_relations_nthu_u_minu_u_tupled) = s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_h4s_pairs_prod(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_h4s_nums_num)),t_h4s_options_option(TV_u_27a))),V_uu_1))))))))))))))).
fof(ah4s_setu_u_relations_getu_u_minu_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_mins]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_options_option(TV_u_27a)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_mins))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,h4s_predu_u_sets_sing(s(t_fun(TV_u_27a,t_bool),V_mins))),s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,h4s_predu_u_sets_choice(s(t_fun(TV_u_27a,t_bool),V_mins))))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) => ![V_s, V_ru_27, V_r]: s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_getu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_let(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_options_option(TV_u_27a)),V_uu_0),s(t_fun(TV_u_27a,t_bool),h4s_setu_u_relations_minimalu_u_elements(s(t_fun(TV_u_27a,t_bool),h4s_setu_u_relations_minimalu_u_elements(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))))))).
fof(ah4s_options_THEu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_options_the(s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x)).
fof(ah4s_options_optionu_u_Axiom, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_e]: ?[V_fn]: (s(TV_u_27b,happ(s(t_fun(t_h4s_options_option(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(TV_u_27b,V_e) & ![V_x]: s(TV_u_27b,happ(s(t_fun(t_h4s_options_option(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),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_options_OPTIONu_u_MAP2u_u_DEF, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_y, V_x, V_f]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27b),V_x)))) & p(s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27c),V_y)))))) & s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_map2(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(t_h4s_options_option(TV_u_27b),V_x),s(t_h4s_options_option(TV_u_27c),V_y))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),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,h4s_options_the(s(t_h4s_options_option(TV_u_27b),V_x))))),s(TV_u_27c,h4s_options_the(s(t_h4s_options_option(TV_u_27c),V_y))))))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))))).
fof(ah4s_options_optionu_u_caseu_u_compute, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f, V_e]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),V_x),s(TV_u_27b,V_e),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(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,h4s_options_the(s(t_h4s_options_option(TV_u_27a),V_x))))),s(TV_u_27b,V_e)))).
fof(ah4s_options_optionu_u_CLAUSESu_c8, axiom, ![TV_u_27a]: ![V_x]: (p(s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),V_x)))) => s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,h4s_options_the(s(t_h4s_options_option(TV_u_27a),V_x))))) = s(t_h4s_options_option(TV_u_27a),V_x))).
fof(ah4s_options_optionu_u_nchotomy, axiom, ![TV_u_27a]: ![V_opt]: (s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) | ?[V_x]: s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,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_options_ISu_u_SOMEu_u_DEFu_c0, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))))) = s(t_bool,t)).
fof(ah4s_options_ISu_u_SOMEu_u_DEFu_c1, axiom, ![TV_u_27a]: s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_bool,f)).
fof(ah4s_options_optionu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,V_v)).
fof(ah4s_options_optionu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = 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_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
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_options_optionu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_options_the(s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x)).
fof(ah4s_options_optionu_u_CLAUSESu_c12, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f, V_e]: (p(s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),V_x)))) => s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),V_x),s(TV_u_27b,V_e),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,h4s_options_the(s(t_h4s_options_option(TV_u_27a),V_x))))))).
fof(ah4s_options_SOMEu_u_11, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_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_options_NOTu_u_SOMEu_u_NONE, axiom, ![TV_u_27a]: ![V_x]: ~ (s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none))).
fof(ah4s_options_NOTu_u_ISu_u_SOMEu_u_EQu_u_NONE, axiom, ![TV_u_27a]: ![V_x]: (~ (p(s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),V_x))))) <=> s(t_h4s_options_option(TV_u_27a),V_x) = s(t_h4s_options_option(TV_u_27a),h4s_options_none))).
fof(ah4s_options_optionu_u_caseu_u_ID, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),V_x),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some))) = s(t_h4s_options_option(TV_u_27a),V_x)).
fof(ah4s_options_ISu_u_NONEu_u_EQu_u_NONE, axiom, ![TV_u_27a]: ![V_x]: (p(s(t_bool,h4s_options_isu_u_none(s(t_h4s_options_option(TV_u_27a),V_x)))) <=> s(t_h4s_options_option(TV_u_27a),V_x) = s(t_h4s_options_option(TV_u_27a),h4s_options_none))).
fof(ah4s_options_OPTIONu_u_JOINu_u_DEFu_c1, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_join(s(t_h4s_options_option(t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(t_h4s_options_option(TV_u_27a))),h4s_options_some),s(t_h4s_options_option(TV_u_27a),V_x))))) = s(t_h4s_options_option(TV_u_27a),V_x)).
fof(ah4s_options_optionu_u_caseu_u_SOMEu_u_ID, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),V_x),s(t_h4s_options_option(TV_u_27a),V_x),s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some))) = s(t_h4s_options_option(TV_u_27a),V_x)).
fof(ah4s_options_optionu_u_induction, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),V_P),s(t_h4s_options_option(TV_u_27a),h4s_options_none)))) & ![V_a]: p(s(t_bool,happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),V_P),s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_a))))))) => ![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),V_P),s(t_h4s_options_option(TV_u_27a),V_x)))))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_JRHu_u_INDUCTu_u_UTIL, axiom, ![TV_u_27a]: ![V_t, V_P]: (![V_x]: (s(TV_u_27a,V_x) = s(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_x))))) => p(s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_P)))))).
fof(ah4s_options_OPTIONu_u_CHOICEu_u_defu_c0, axiom, ![TV_u_27a]: ![V_m2]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_choice(s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_h4s_options_option(TV_u_27a),V_m2))) = s(t_h4s_options_option(TV_u_27a),V_m2)).
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_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_options_NOTu_u_NONEu_u_SOME, axiom, ![TV_u_27a]: ![V_x]: ~ (s(t_h4s_options_option(TV_u_27a),h4s_options_none) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))))).
fof(ah4s_options_optionu_u_CLAUSESu_c7, axiom, ![TV_u_27a]: ![V_x]: (~ (p(s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),V_x))))) <=> s(t_h4s_options_option(TV_u_27a),V_x) = s(t_h4s_options_option(TV_u_27a),h4s_options_none))).
fof(ah4s_options_IFu_u_NONEu_u_EQUALSu_u_OPTIONu_c3, axiom, ![TV_u_27a]: ![V_x, V_X, V_P]: (s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_h4s_options_option(TV_u_27a),V_X))) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))) <=> (~ (p(s(t_bool,V_P))) & s(t_h4s_options_option(TV_u_27a),V_X) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x)))))).
fof(ah4s_options_OPTRELu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_R]: (p(s(t_bool,h4s_options_optrel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(t_h4s_options_option(TV_u_27a),V_x),s(t_h4s_options_option(TV_u_27b),V_y)))) <=> ((s(t_h4s_options_option(TV_u_27a),V_x) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) & s(t_h4s_options_option(TV_u_27b),V_y) = s(t_h4s_options_option(TV_u_27b),h4s_options_none)) | ?[V_x0, V_y0]: (s(t_h4s_options_option(TV_u_27a),V_x) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x0))) & (s(t_h4s_options_option(TV_u_27b),V_y) = s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_options_option(TV_u_27b)),h4s_options_some),s(TV_u_27b,V_y0))) & 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_R),s(TV_u_27a,V_x0))),s(TV_u_27b,V_y0))))))))).
fof(ah4s_options_OPTIONu_u_CHOICEu_u_EQu_u_NONE, axiom, ![TV_u_27a]: ![V_m2, V_m1]: (s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_choice(s(t_h4s_options_option(TV_u_27a),V_m1),s(t_h4s_options_option(TV_u_27a),V_m2))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) <=> (s(t_h4s_options_option(TV_u_27a),V_m1) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) & s(t_h4s_options_option(TV_u_27a),V_m2) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)))).
fof(ah4s_options_OPTIONu_u_BINDu_u_EQUALSu_u_OPTIONu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_f]: (s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_bind(s(t_h4s_options_option(TV_u_27a),V_p),s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27b)),V_f))) = s(t_h4s_options_option(TV_u_27b),h4s_options_none) <=> (s(t_h4s_options_option(TV_u_27a),V_p) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) | ?[V_x]: (s(t_h4s_options_option(TV_u_27a),V_p) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))) & s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27b)),V_f),s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27b),h4s_options_none))))).
fof(ah4s_options_IFu_u_NONEu_u_EQUALSu_u_OPTIONu_c2, axiom, ![TV_u_27a]: ![V_x, V_X, V_P]: (s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_options_option(TV_u_27a),V_X),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))) <=> (p(s(t_bool,V_P)) & s(t_h4s_options_option(TV_u_27a),V_X) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x)))))).
fof(ah4s_options_FORALLu_u_OPTION, axiom, ![TV_u_27a]: ![V_P]: (![V_opt]: p(s(t_bool,happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),V_P),s(t_h4s_options_option(TV_u_27a),V_opt)))) <=> (p(s(t_bool,happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),V_P),s(t_h4s_options_option(TV_u_27a),h4s_options_none)))) & ![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),V_P),s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))))))))).
fof(ah4s_options_OPTIONu_u_APPLYu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(TV_u_27b,TV_u_27a)),h4s_options_none),s(t_h4s_options_option(TV_u_27b),V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)).
fof(ah4s_options_IFu_u_NONEu_u_EQUALSu_u_OPTIONu_c1, axiom, ![TV_u_27a]: ![V_X, V_P]: (s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_h4s_options_option(TV_u_27a),V_X))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) <=> (p(s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),V_X)))) => p(s(t_bool,V_P))))).
fof(ah4s_options_OPTIONu_u_CHOICEu_u_defu_c1, axiom, ![TV_u_27a]: ![V_x, V_m2]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_choice(s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))),s(t_h4s_options_option(TV_u_27a),V_m2))) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x)))).
fof(ah4s_options_EXISTSu_u_OPTION, axiom, ![TV_u_27a]: ![V_P]: (?[V_opt]: p(s(t_bool,happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),V_P),s(t_h4s_options_option(TV_u_27a),V_opt)))) <=> (p(s(t_bool,happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),V_P),s(t_h4s_options_option(TV_u_27a),h4s_options_none)))) | ?[V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),V_P),s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))))))))).
fof(ah4s_options_optionu_u_caseu_u_cong, axiom, ![TV_u_27a,TV_u_27b]: ![V_vu_27, V_v, V_fu_27, V_f, V_Mu_27, V_M]: ((s(t_h4s_options_option(TV_u_27a),V_M) = s(t_h4s_options_option(TV_u_27a),V_Mu_27) & ((s(t_h4s_options_option(TV_u_27a),V_Mu_27) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) => s(TV_u_27b,V_v) = s(TV_u_27b,V_vu_27)) & ![V_x]: (s(t_h4s_options_option(TV_u_27a),V_Mu_27) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),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))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_fu_27),s(TV_u_27a,V_x)))))) => s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),V_M),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),V_Mu_27),s(TV_u_27b,V_vu_27),s(t_fun(TV_u_27a,TV_u_27b),V_fu_27))))).
fof(ah4s_options_optionu_u_CLAUSESu_c9, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),V_x),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some))) = s(t_h4s_options_option(TV_u_27a),V_x)).
fof(ah4s_options_optionu_u_CLAUSESu_c6, axiom, ![TV_u_27a]: ![V_x]: (p(s(t_bool,h4s_options_isu_u_none(s(t_h4s_options_option(TV_u_27a),V_x)))) <=> s(t_h4s_options_option(TV_u_27a),V_x) = s(t_h4s_options_option(TV_u_27a),h4s_options_none))).
fof(ah4s_options_ISu_u_NONEu_u_DEFu_c1, axiom, ![TV_u_27a]: s(t_bool,h4s_options_isu_u_none(s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_bool,t)).
fof(ah4s_options_ISu_u_NONEu_u_DEFu_c0, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,h4s_options_isu_u_none(s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))))) = s(t_bool,f)).
fof(ah4s_options_IFu_u_NONEu_u_EQUALSu_u_OPTIONu_c0, axiom, ![TV_u_27a]: ![V_X, V_P]: (s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_options_option(TV_u_27a),V_X),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) <=> (p(s(t_bool,V_P)) => p(s(t_bool,h4s_options_isu_u_none(s(t_h4s_options_option(TV_u_27a),V_X))))))).
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_options_optionu_u_REPu_u_ABSu_u_DEFu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_abs(s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),h4s_options_optionu_u_rep(s(t_h4s_options_option(TV_u_27a),V_a))))) = s(t_h4s_options_option(TV_u_27a),V_a)).
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_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_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_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_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_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_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_options_optionu_u_TYu_u_DEF, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x]: s(t_bool,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),t_bool),V_uu_0),s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),V_x))) = s(t_bool,t) => ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),t_bool),V_uu_0),s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one)),V_rep)))))).
fof(ah4s_bools_ABSu_u_REPu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ?[V_rep, V_abs]: (![V_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))) = s(TV_u_27b,V_a) & ![V_r]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_r)))) <=> s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))))) = s(TV_u_27a,V_r))))).
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_options_optionu_u_CLAUSESu_c13, axiom, ![TV_u_27a]: ![V_x, V_e]: (p(s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),V_x)))) => s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),V_x),s(t_h4s_options_option(TV_u_27a),V_e),s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some))) = s(t_h4s_options_option(TV_u_27a),V_x))).
fof(ah4s_options_OPTIONu_u_BINDu_u_EQUALSu_u_OPTIONu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_p, V_f]: (s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_bind(s(t_h4s_options_option(TV_u_27a),V_p),s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27b)),V_f))) = s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_options_option(TV_u_27b)),h4s_options_some),s(TV_u_27b,V_y))) <=> ?[V_x]: (s(t_h4s_options_option(TV_u_27a),V_p) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))) & s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27b)),V_f),s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_options_option(TV_u_27b)),h4s_options_some),s(TV_u_27b,V_y)))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EXISTSu_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_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_options_optionu_u_CLAUSESu_c11, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f, V_e]: (p(s(t_bool,h4s_options_isu_u_none(s(t_h4s_options_option(TV_u_27a),V_x)))) => s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),V_x),s(TV_u_27b,V_e),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,V_e))).
fof(ah4s_options_OPTIONu_u_IGNOREu_u_BINDu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_m2, V_m1]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_ignoreu_u_bind(s(t_h4s_options_option(TV_u_27b),V_m1),s(t_h4s_options_option(TV_u_27a),V_m2))) = s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_bind(s(t_h4s_options_option(TV_u_27b),V_m1),s(t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)),h4s_combins_k(s(t_h4s_options_option(TV_u_27a),V_m2)))))).
fof(ah4s_options_optionu_u_CLAUSESu_c10, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),V_x),s(t_h4s_options_option(TV_u_27a),V_x),s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some))) = s(t_h4s_options_option(TV_u_27a),V_x)).
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_ANDu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,f))) <=> p(s(t_bool,f)))).
fof(ah4s_options_OPTIONu_u_BINDu_u_cong, axiom, ![TV_u_27a,TV_u_27b]: ![V_o2, V_o1, V_f2, V_f1]: ((s(t_h4s_options_option(TV_u_27a),V_o1) = s(t_h4s_options_option(TV_u_27a),V_o2) & ![V_x]: (s(t_h4s_options_option(TV_u_27a),V_o2) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))) => s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27b)),V_f1),s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27b)),V_f2),s(TV_u_27a,V_x))))) => s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_bind(s(t_h4s_options_option(TV_u_27a),V_o1),s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27b)),V_f1))) = s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_bind(s(t_h4s_options_option(TV_u_27a),V_o2),s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27b)),V_f2))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_options_OPTIONu_u_BINDu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_f]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_bind(s(t_h4s_options_option(TV_u_27b),h4s_options_none),s(t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)),V_f))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)).
fof(ah4s_options_OPTIONu_u_BINDu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_bind(s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_options_option(TV_u_27b)),h4s_options_some),s(TV_u_27b,V_x))),s(t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)),V_f))) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)),V_f),s(TV_u_27b,V_x)))).
fof(ah4s_options_optionu_u_CLAUSESu_c19, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_join(s(t_h4s_options_option(t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(t_h4s_options_option(TV_u_27a))),h4s_options_some),s(t_h4s_options_option(TV_u_27a),V_x))))) = s(t_h4s_options_option(TV_u_27a),V_x)).
fof(ah4s_options_IFu_u_EQUALSu_u_OPTIONu_c3, axiom, ![TV_u_27a]: ![V_y, V_x, V_P]: (s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))))) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_y))) <=> (~ (p(s(t_bool,V_P))) & s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))).
fof(ah4s_options_IFu_u_EQUALSu_u_OPTIONu_c2, axiom, ![TV_u_27a]: ![V_y, V_x, V_P]: (s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_y))) <=> (p(s(t_bool,V_P)) & s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))).
fof(ah4s_options_IFu_u_EQUALSu_u_OPTIONu_c1, axiom, ![TV_u_27a]: ![V_x, V_P]: (s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) <=> p(s(t_bool,V_P)))).
fof(ah4s_options_IFu_u_EQUALSu_u_OPTIONu_c0, axiom, ![TV_u_27a]: ![V_x, V_P]: (s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) <=> ~ (p(s(t_bool,V_P))))).
fof(ah4s_bools_CONDu_u_ID, axiom, ![TV_u_27a]: ![V_t, V_b]: s(TV_u_27a,h4s_bools_cond(s(t_bool,V_b),s(TV_u_27a,V_t),s(TV_u_27a,V_t))) = s(TV_u_27a,V_t)).
fof(ah4s_options_OPTRELu_u_MONO, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_Q, V_P]: (![V_x0, V_y0]: (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_x0))),s(TV_u_27b,V_y0)))) => 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_Q),s(TV_u_27a,V_x0))),s(TV_u_27b,V_y0))))) => (p(s(t_bool,h4s_options_optrel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(t_h4s_options_option(TV_u_27a),V_x),s(t_h4s_options_option(TV_u_27b),V_y)))) => p(s(t_bool,h4s_options_optrel(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_Q),s(t_h4s_options_option(TV_u_27a),V_x),s(t_h4s_options_option(TV_u_27b),V_y))))))).
fof(ah4s_options_OPTRELu_u_refl, axiom, ![TV_u_27a]: ![V_R]: (![V_x]: 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_x)))) => ![V_x]: p(s(t_bool,h4s_options_optrel(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_h4s_options_option(TV_u_27a),V_x),s(t_h4s_options_option(TV_u_27a),V_x)))))).
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_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_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_setu_u_relations_peru_u_delete, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_e, V_es0]: 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)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool))),V_uu_1),s(TV_u_27a,V_e))),s(t_fun(TV_u_27a,t_bool),V_es0))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_es0),s(TV_u_27a,V_e))) => ![V_uu_0]: (![V_e, V_xss, V_es]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_bools_in(s(t_fun(TV_u_27a,t_bool),V_es),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_image(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool))),V_uu_1),s(TV_u_27a,V_e))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_xss)))))) & ~ (s(t_fun(TV_u_27a,t_bool),V_es) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)))) & s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_bool)),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_bool)))),V_uu_0),s(TV_u_27a,V_e))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_xss))),s(t_fun(TV_u_27a,t_bool),V_es))) = s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_bool),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_es),s(t_bool,V_v)))) => ![V_xss, V_xs, V_e]: (p(s(t_bool,h4s_setu_u_relations_per(s(t_fun(TV_u_27a,t_bool),V_xs),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_xss)))) => p(s(t_bool,h4s_setu_u_relations_per(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_delete(s(t_fun(TV_u_27a,t_bool),V_xs),s(TV_u_27a,V_e))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_bool)),happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_fun(TV_u_27a,t_bool),t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_bool)))),V_uu_0),s(TV_u_27a,V_e))),s(t_fun(t_fun(TV_u_27a,t_bool),t_bool),V_xss)))))))))))).
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_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_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_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_options_NONEu_u_DEF, axiom, ![TV_u_27a]: s(t_h4s_options_option(TV_u_27a),h4s_options_none) = s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_abs(s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),h4s_sums_inr(s(t_h4s_ones_one,h4s_ones_one0)))))).
fof(ah4s_options_SOMEu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_abs(s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),h4s_sums_inl(s(TV_u_27a,V_x)))))).
fof(ah4s_ones_one1, axiom, ![V_v]: s(t_h4s_ones_one,V_v) = s(t_h4s_ones_one,h4s_ones_one0)).
fof(ah4s_sums_sumu_u_INDUCT, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: ((![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_x)))))) & ![V_y]: p(s(t_bool,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_y))))))) => ![V_s]: p(s(t_bool,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_s)))))).
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_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_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_UNWINDu_u_FORALLu_u_THM2, axiom, ![TV_u_27a]: ![V_v, V_f]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_v) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_v)))))).
fof(ah4s_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_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_DEu_u_MORGANu_u_THMu_c0, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) & p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) | ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
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_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_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_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_ANDu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
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_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
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_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(ch4s_setu_u_relations_nthu_u_minu_u_ind, conjecture, ![TV_u_27a]: ![V_P]: ((![V_ru_27, V_s, V_r]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_bool))),V_P),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))),s(t_h4s_nums_num,h4s_nums_0)))) & ![V_ru_27, V_s, V_r, V_n]: (![V_min]: ((s(t_h4s_options_option(TV_u_27a),V_min) = s(t_h4s_options_option(TV_u_27a),h4s_setu_u_relations_getu_u_min(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))) & ~ (s(t_h4s_options_option(TV_u_27a),V_min) = s(t_h4s_options_option(TV_u_27a),h4s_options_none))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_bool))),V_P),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(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,h4s_options_the(s(t_h4s_options_option(TV_u_27a),V_min))))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))),s(t_h4s_nums_num,V_n))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_bool))),V_P),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_ru_27))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_r))))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))) => ![V_v, V_v1, V_v2, V_v3]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),t_fun(t_h4s_nums_num,t_bool))),V_P),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_v))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_v1),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27a),t_bool),V_v2))))),s(t_h4s_nums_num,V_v3)))))).
