%   ORIGINAL: h4/fmapal/optry__list__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/fmapal/optry__list__tupled__primitive: h4/fmapal/optry__list__tupled = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!l f. R (h4/pair/_2C f l) (h4/pair/_2C f (h4/list/CONS h4/option/NONE l))) /\ (!z l f. R (h4/pair/_2C f l) (h4/pair/_2C f (h4/list/CONS (h4/option/SOME z) l))))) (\optry__list__tupled a. h4/pair/pair__CASE a (\f v1. h4/list/list__CASE v1 (h4/combin/I h4/option/NONE) (\v2 l. h4/option/option__CASE v2 (h4/combin/I (optry__list__tupled (h4/pair/_2C f l))) (\z. h4/combin/I (h4/fmapal/optry (f z) (optry__list__tupled (h4/pair/_2C f l)))))))
% Assm: h4/option/option__Axiom: !f e. ?fn. fn h4/option/NONE = e /\ (!x. fn (h4/option/SOME x) = f x)
% Assm: h4/fmapal/optry__list__curried: !x1 x. h4/fmapal/optry__list x x1 = h4/fmapal/optry__list__tupled (h4/pair/_2C x x1)
% Assm: h4/option/OPTION__BIND__def_c1: !x f. h4/option/OPTION__BIND (h4/option/SOME x) f = f x
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/option/OPTION__MAP__DEF_c0: !x f. h4/option/OPTION__MAP f (h4/option/SOME x) = h4/option/SOME (f x)
% Assm: h4/option/option__case__def_c1: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = f x
% Assm: h4/finite__map/FLOOKUP__DEF: !x f. h4/finite__map/FLOOKUP f x = h4/bool/COND (h4/bool/IN x (h4/finite__map/FDOM f)) (h4/option/SOME (h4/finite__map/FAPPLY f x)) h4/option/NONE
% 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/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/option/OPTION__MAP2__THM_c2: !y f. h4/option/OPTION__MAP2 f h4/option/NONE (h4/option/SOME y) = h4/option/NONE
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/option/option__case__ID: !x. h4/option/option__CASE x h4/option/NONE h4/option/SOME = x
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/option/OPTION__MAP2__THM_c0: !y x f. h4/option/OPTION__MAP2 f (h4/option/SOME x) (h4/option/SOME y) = h4/option/SOME (f x y)
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/option/option__nchotomy: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm: h4/finite__map/SUBMAP__FUPDATE__FLOOKUP: !y x f. h4/finite__map/SUBMAP f (h4/finite__map/FUPDATE f (h4/pair/_2C x y)) <=> h4/finite__map/FLOOKUP f x = h4/option/NONE \/ h4/finite__map/FLOOKUP f x = h4/option/SOME y
% 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/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/option/OPTION__MAP2__cong: !y2 y1 x2 x1 f2 f1. x1 = x2 /\ y1 = y2 /\ (!x y. x2 = h4/option/SOME x /\ y2 = h4/option/SOME y ==> f1 x y = f2 x y) ==> h4/option/OPTION__MAP2 f1 x1 y1 = h4/option/OPTION__MAP2 f2 x2 y2
% Assm: h4/finite__map/FRANGE__FLOOKUP: !v f. h4/bool/IN v (h4/finite__map/FRANGE f) <=> (?k. h4/finite__map/FLOOKUP f k = h4/option/SOME v)
% Assm: h4/option/option__CLAUSES_c15: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = f x
% Assm: h4/finite__map/FEVERY__FLOOKUP: !v k f P. h4/finite__map/FEVERY P f /\ h4/finite__map/FLOOKUP f k = h4/option/SOME v ==> P (h4/pair/_2C k v)
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/finite__map/FLOOKUP__SUBMAP: !v k g f. h4/finite__map/SUBMAP f g /\ h4/finite__map/FLOOKUP f k = h4/option/SOME v ==> h4/finite__map/FLOOKUP g k = h4/option/SOME v
% Assm: h4/option/option__CLAUSES_c16: !x f. h4/option/OPTION__MAP f (h4/option/SOME x) = h4/option/SOME (f x)
% Assm: h4/option/SOME__SOME__APPLY: !x f. h4/option/OPTION__APPLY (h4/option/SOME f) (h4/option/SOME x) = h4/option/SOME (f x)
% Assm: h4/option/OPTION__MAP__EQ__SOME: !y x f. h4/option/OPTION__MAP f x = h4/option/SOME y <=> (?z. x = h4/option/SOME z /\ y = f z)
% Assm: h4/finite__map/flookup__thm_c1: !x v f. h4/finite__map/FLOOKUP f x = h4/option/SOME v <=> h4/bool/IN x (h4/finite__map/FDOM f) /\ h4/finite__map/FAPPLY f x = v
% Assm: h4/finite__map/IN__FRANGE__FLOOKUP: !v f. h4/bool/IN v (h4/finite__map/FRANGE f) <=> (?k. h4/finite__map/FLOOKUP f k = h4/option/SOME v)
% 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/finite__map/FLOOKUP__FUNION: !k f2 f1. h4/finite__map/FLOOKUP (h4/finite__map/FUNION f1 f2) k = h4/option/option__CASE (h4/finite__map/FLOOKUP f1 k) (h4/finite__map/FLOOKUP f2 k) (\v. h4/option/SOME v)
% Assm: h4/finite__map/FEVERY__ALL__FLOOKUP: !f P. h4/finite__map/FEVERY P f <=> (!k v. h4/finite__map/FLOOKUP f k = h4/option/SOME v ==> P (h4/pair/_2C k v))
% Assm: h4/option/OPTION__MAP__CONG: !opt2 opt1 f2 f1. opt1 = opt2 /\ (!x. opt2 = h4/option/SOME x ==> f1 x = f2 x) ==> h4/option/OPTION__MAP f1 opt1 = h4/option/OPTION__MAP f2 opt2
% Assm: h4/option/OPTION__MAP2__SOME: !v o2 o1 f. h4/option/OPTION__MAP2 f o1 o2 = h4/option/SOME v <=> (?x1 x2. o1 = h4/option/SOME x1 /\ o2 = h4/option/SOME x2 /\ v = f x1 x2)
% Assm: h4/finite__map/fmap__inverse__def: !m2 m1. h4/finite__map/fmap__inverse m1 m2 <=> (!k. h4/bool/IN k (h4/finite__map/FDOM m1) ==> (?v. h4/finite__map/FLOOKUP m1 k = h4/option/SOME v /\ h4/finite__map/FLOOKUP m2 v = h4/option/SOME k))
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% 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/finite__map/FEVERY__DEF: !f P. h4/finite__map/FEVERY P f <=> (!x. h4/bool/IN x (h4/finite__map/FDOM f) ==> P (h4/pair/_2C x (h4/finite__map/FAPPLY 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/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/THE__DEF: !x. h4/option/THE (h4/option/SOME x) = x
% Assm: h4/option/OPTION__APPLY__def_c1: !x f. h4/option/OPTION__APPLY (h4/option/SOME f) x = h4/option/OPTION__MAP f x
% Assm: h4/finite__map/IN__FRANGE: !v f. h4/bool/IN v (h4/finite__map/FRANGE f) <=> (?k. h4/bool/IN k (h4/finite__map/FDOM f) /\ h4/finite__map/FAPPLY f k = v)
% Assm: h4/option/OPTION__MAP__DEF_c1: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/DATATYPE__TAG__THM: !x. h4/bool/DATATYPE x <=> T
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/option/NOT__NONE__SOME: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm: h4/finite__map/SUBMAP__FUPDATE__EQN: !y x f. h4/finite__map/SUBMAP f (h4/finite__map/FUPDATE f (h4/pair/_2C x y)) <=> ~h4/bool/IN x (h4/finite__map/FDOM f) \/ h4/finite__map/FAPPLY f x = y /\ h4/bool/IN x (h4/finite__map/FDOM f)
% Assm: h4/bool/CONJ__COMM: !t2 t1. t1 /\ t2 <=> t2 /\ t1
% Assm: h4/bool/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% 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/marker/AC__DEF: !b2 b1. h4/marker/AC b1 b2 <=> b1 /\ b2
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/option/OPTION__MAP2__THM_c3: !f. h4/option/OPTION__MAP2 f h4/option/NONE h4/option/NONE = h4/option/NONE
% Assm: h4/option/OPTION__MAP2__THM_c1: !x f. h4/option/OPTION__MAP2 f (h4/option/SOME x) h4/option/NONE = h4/option/NONE
% Assm: h4/option/SOME__DEF: !x. h4/option/SOME x = h4/option/option__ABS (h4/sum/INL x)
% Assm: h4/option/NONE__DEF: h4/option/NONE = h4/option/option__ABS (h4/sum/INR h4/one/one0)
% Assm: h4/option/option__REP__ABS__DEF_c1: !r. (\x. T) r <=> h4/option/option__REP (h4/option/option__ABS r) = r
% Assm: h4/option/option__REP__ABS__DEF_c0: !a. h4/option/option__ABS (h4/option/option__REP a) = a
% Assm: h4/sum/sum__Axiom: !g f. ?h. (!x. h (h4/sum/INL x) = f x) /\ (!y. h (h4/sum/INR y) = g y)
% Assm: h4/option/datatype__option: !option. h4/bool/DATATYPE (option h4/option/NONE h4/option/SOME)
% Assm: h4/option/option__CLAUSES_c9: !x. h4/option/option__CASE x h4/option/NONE h4/option/SOME = x
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> t
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/finite__map/FRANGE__DEF: !f. h4/finite__map/FRANGE f = h4/pred__set/GSPEC (\y. h4/pair/_2C y (?x. h4/bool/IN x (h4/finite__map/FDOM f) /\ h4/finite__map/FAPPLY f x = y))
% Assm: h4/pair/PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/pred__set/GSPECIFICATION: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = f x)
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/option/option__case__def_c0: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm: h4/enumeral/OL__sublists__ind: !P. (!cmp. P cmp h4/list/NIL) /\ (!cmp lol. P cmp lol ==> P cmp (h4/list/CONS h4/option/NONE lol)) /\ (!cmp m lol. P cmp lol ==> P cmp (h4/list/CONS (h4/option/SOME m) lol)) ==> (!v v1. P v v1)
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/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/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/bool/UNWIND__FORALL__THM2: !v f. (!x. x = v ==> f x) <=> f v
% Assm: h4/bool/DISJ__COMM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/option/OPTION__BIND__def_c0: !f. h4/option/OPTION__BIND h4/option/NONE f = h4/option/NONE
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/finite__map/SUBMAP__DEF: !g f. h4/finite__map/SUBMAP f g <=> (!x. h4/bool/IN x (h4/finite__map/FDOM f) ==> h4/bool/IN x (h4/finite__map/FDOM g) /\ h4/finite__map/FAPPLY f x = h4/finite__map/FAPPLY g x)
% Assm: h4/bool/RIGHT__OR__OVER__AND: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm: h4/finite__map/FUNION__DEF_c1: !x g f. h4/finite__map/FAPPLY (h4/finite__map/FUNION f g) x = h4/bool/COND (h4/bool/IN x (h4/finite__map/FDOM f)) (h4/finite__map/FAPPLY f x) (h4/finite__map/FAPPLY g x)
% Assm: h4/finite__map/FUNION__DEF_c0: !g f. h4/finite__map/FDOM (h4/finite__map/FUNION f g) = h4/pred__set/UNION (h4/finite__map/FDOM f) (h4/finite__map/FDOM g)
% Assm: h4/pred__set/IN__UNION: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x t
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~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/EQ__EXPAND: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/arithmetic/ADD__MONO__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm: h4/arithmetic/LESS__EQ__TRANS: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm: h4/arithmetic/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Goal: !P. (!f. P f h4/list/NIL) /\ (!f l. P f l ==> P f (h4/list/CONS h4/option/NONE l)) /\ (!f z l. P f l ==> P f (h4/list/CONS (h4/option/SOME z) l)) ==> (!v v1. P v v1)
%   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_fmapals_optryu_u_listu_u_tupledu_u_primitive]: !_7. (!optry__list__tupled f l z. happ (happ (happ (happ _7 optry__list__tupled) f) l) z = h4/combin/I (h4/fmapal/optry (happ f z) (happ optry__list__tupled (h4/pair/_2C f l)))) ==> (!_6. (!v2 optry__list__tupled f l. happ (happ (happ (happ _6 v2) optry__list__tupled) f) l = h4/option/option__CASE v2 (h4/combin/I (happ optry__list__tupled (h4/pair/_2C f l))) (happ (happ (happ _7 optry__list__tupled) f) l)) ==> (!_5. (!optry__list__tupled f v2. happ (happ (happ _5 optry__list__tupled) f) v2 = happ (happ (happ _6 v2) optry__list__tupled) f) ==> (!_4. (!optry__list__tupled f v1. happ (happ (happ _4 optry__list__tupled) f) v1 = h4/list/list__CASE v1 (h4/combin/I h4/option/NONE) (happ (happ _5 optry__list__tupled) f)) ==> (!_3. (!optry__list__tupled f. happ (happ _3 optry__list__tupled) f = happ (happ _4 optry__list__tupled) f) ==> (!_2. (!optry__list__tupled a. happ (happ _2 optry__list__tupled) a = h4/pair/pair__CASE a (happ _3 optry__list__tupled)) ==> (!_1. (!optry__list__tupled. happ _1 optry__list__tupled = happ _2 optry__list__tupled) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R /\ (!l f. happ (happ R (h4/pair/_2C f l)) (h4/pair/_2C f (h4/list/CONS h4/option/NONE l))) /\ (!z l f. happ (happ R (h4/pair/_2C f l)) (h4/pair/_2C f (h4/list/CONS (happ h4/option/SOME z) l)))) ==> h4/fmapal/optry__list__tupled = h4/relation/WFREC (h4/min/_40 _0) _1)))))))
% 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_fmapals_optryu_u_listu_u_curried]: !x1 x. h4/fmapal/optry__list x x1 = happ h4/fmapal/optry__list__tupled (h4/pair/_2C x x1)
% 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_bools_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_options_OPTIONu_u_MAPu_u_DEFu_c0]: !x f. h4/option/OPTION__MAP f (happ h4/option/SOME x) = happ h4/option/SOME (happ f x)
% 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_finiteu_u_maps_FLOOKUPu_u_DEF]: !x f. h4/finite__map/FLOOKUP f x = h4/bool/COND (h4/bool/IN x (h4/finite__map/FDOM f)) (happ h4/option/SOME (h4/finite__map/FAPPLY f x)) h4/option/NONE
% 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_SOMEu_u_11]: !y x. happ h4/option/SOME x = happ h4/option/SOME y <=> x = y
% Assm [h4s_options_OPTIONu_u_MAP2u_u_THMu_c2]: !y f. h4/option/OPTION__MAP2 f h4/option/NONE (happ h4/option/SOME y) = h4/option/NONE
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_options_optionu_u_caseu_u_ID]: !x. h4/option/option__CASE x h4/option/NONE h4/option/SOME = x
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_options_OPTIONu_u_MAP2u_u_THMu_c0]: !y x f. h4/option/OPTION__MAP2 f (happ h4/option/SOME x) (happ h4/option/SOME y) = happ h4/option/SOME (happ (happ f x) y)
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_options_optionu_u_nchotomy]: !opt. opt = h4/option/NONE \/ (?x. opt = happ h4/option/SOME x)
% Assm [h4s_finiteu_u_maps_SUBMAPu_u_FUPDATEu_u_FLOOKUP]: !y x f. h4/finite__map/SUBMAP f (h4/finite__map/FUPDATE f (h4/pair/_2C x y)) <=> h4/finite__map/FLOOKUP f x = h4/option/NONE \/ h4/finite__map/FLOOKUP f x = happ h4/option/SOME y
% 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_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_options_OPTIONu_u_MAP2u_u_cong]: !y2 y1 x2 x1 f2 f1. x1 = x2 /\ y1 = y2 /\ (!x y. x2 = happ h4/option/SOME x /\ y2 = happ h4/option/SOME y ==> happ (happ f1 x) y = happ (happ f2 x) y) ==> h4/option/OPTION__MAP2 f1 x1 y1 = h4/option/OPTION__MAP2 f2 x2 y2
% Assm [h4s_finiteu_u_maps_FRANGEu_u_FLOOKUP]: !v f. h4/bool/IN v (h4/finite__map/FRANGE f) <=> (?k. h4/finite__map/FLOOKUP f k = happ h4/option/SOME v)
% Assm [h4s_options_optionu_u_CLAUSESu_c15]: !x v f. h4/option/option__CASE (happ h4/option/SOME x) v f = happ f x
% Assm [h4s_finiteu_u_maps_FEVERYu_u_FLOOKUP]: !v k f P. h4/finite__map/FEVERY P f /\ h4/finite__map/FLOOKUP f k = happ h4/option/SOME v ==> happ P (h4/pair/_2C k v)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_finiteu_u_maps_FLOOKUPu_u_SUBMAP]: !v k g f. h4/finite__map/SUBMAP f g /\ h4/finite__map/FLOOKUP f k = happ h4/option/SOME v ==> h4/finite__map/FLOOKUP g k = happ h4/option/SOME v
% Assm [h4s_options_optionu_u_CLAUSESu_c16]: !x f. h4/option/OPTION__MAP f (happ h4/option/SOME x) = happ h4/option/SOME (happ f x)
% Assm [h4s_options_SOMEu_u_SOMEu_u_APPLY]: !x f. h4/option/OPTION__APPLY (happ h4/option/SOME f) (happ h4/option/SOME x) = happ h4/option/SOME (happ f x)
% Assm [h4s_options_OPTIONu_u_MAPu_u_EQu_u_SOME]: !y x f. h4/option/OPTION__MAP f x = happ h4/option/SOME y <=> (?z. x = happ h4/option/SOME z /\ y = happ f z)
% Assm [h4s_finiteu_u_maps_flookupu_u_thmu_c1]: !x v f. h4/finite__map/FLOOKUP f x = happ h4/option/SOME v <=> h4/bool/IN x (h4/finite__map/FDOM f) /\ h4/finite__map/FAPPLY f x = v
% Assm [h4s_finiteu_u_maps_INu_u_FRANGEu_u_FLOOKUP]: !v f. h4/bool/IN v (h4/finite__map/FRANGE f) <=> (?k. h4/finite__map/FLOOKUP f k = happ h4/option/SOME v)
% 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_finiteu_u_maps_FLOOKUPu_u_FUNION]: !_0. (!v. happ _0 v = happ h4/option/SOME v) ==> (!k f2 f1. h4/finite__map/FLOOKUP (h4/finite__map/FUNION f1 f2) k = h4/option/option__CASE (h4/finite__map/FLOOKUP f1 k) (h4/finite__map/FLOOKUP f2 k) _0)
% Assm [h4s_finiteu_u_maps_FEVERYu_u_ALLu_u_FLOOKUP]: !f P. h4/finite__map/FEVERY P f <=> (!k v. h4/finite__map/FLOOKUP f k = happ h4/option/SOME v ==> happ P (h4/pair/_2C k v))
% Assm [h4s_options_OPTIONu_u_MAPu_u_CONG]: !opt2 opt1 f2 f1. opt1 = opt2 /\ (!x. opt2 = happ h4/option/SOME x ==> happ f1 x = happ f2 x) ==> h4/option/OPTION__MAP f1 opt1 = h4/option/OPTION__MAP f2 opt2
% Assm [h4s_options_OPTIONu_u_MAP2u_u_SOME]: !v o2 o1 f. h4/option/OPTION__MAP2 f o1 o2 = happ h4/option/SOME v <=> (?x1 x2. o1 = happ h4/option/SOME x1 /\ o2 = happ h4/option/SOME x2 /\ v = happ (happ f x1) x2)
% Assm [h4s_finiteu_u_maps_fmapu_u_inverseu_u_def]: !m2 m1. h4/finite__map/fmap__inverse m1 m2 <=> (!k. h4/bool/IN k (h4/finite__map/FDOM m1) ==> (?v. h4/finite__map/FLOOKUP m1 k = happ h4/option/SOME v /\ h4/finite__map/FLOOKUP m2 v = happ h4/option/SOME k))
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% 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_finiteu_u_maps_FEVERYu_u_DEF]: !f P. h4/finite__map/FEVERY P f <=> (!x. h4/bool/IN x (h4/finite__map/FDOM f) ==> happ P (h4/pair/_2C x (h4/finite__map/FAPPLY 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_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_THEu_u_DEF]: !x. h4/option/THE (happ h4/option/SOME x) = x
% Assm [h4s_options_OPTIONu_u_APPLYu_u_defu_c1]: !x f. h4/option/OPTION__APPLY (happ h4/option/SOME f) x = h4/option/OPTION__MAP f x
% Assm [h4s_finiteu_u_maps_INu_u_FRANGE]: !v f. h4/bool/IN v (h4/finite__map/FRANGE f) <=> (?k. h4/bool/IN k (h4/finite__map/FDOM f) /\ h4/finite__map/FAPPLY f k = v)
% Assm [h4s_options_OPTIONu_u_MAPu_u_DEFu_c1]: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_DATATYPEu_u_TAGu_u_THM]: !x. h4/bool/DATATYPE x <=> T
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_options_NOTu_u_NONEu_u_SOME]: !x. ~(h4/option/NONE = happ h4/option/SOME x)
% Assm [h4s_finiteu_u_maps_SUBMAPu_u_FUPDATEu_u_EQN]: !y x f. h4/finite__map/SUBMAP f (h4/finite__map/FUPDATE f (h4/pair/_2C x y)) <=> ~h4/bool/IN x (h4/finite__map/FDOM f) \/ h4/finite__map/FAPPLY f x = y /\ h4/bool/IN x (h4/finite__map/FDOM f)
% Assm [h4s_bools_CONJu_u_COMM]: !t2 t1. t1 /\ t2 <=> t2 /\ t1
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% 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_markers_ACu_u_DEF]: !b2 b1. h4/marker/AC b1 b2 <=> b1 /\ b2
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_options_OPTIONu_u_MAP2u_u_THMu_c3]: !f. h4/option/OPTION__MAP2 f h4/option/NONE h4/option/NONE = h4/option/NONE
% Assm [h4s_options_OPTIONu_u_MAP2u_u_THMu_c1]: !x f. h4/option/OPTION__MAP2 f (happ h4/option/SOME x) h4/option/NONE = h4/option/NONE
% Assm [h4s_options_SOMEu_u_DEF]: !x. happ h4/option/SOME x = h4/option/option__ABS (h4/sum/INL x)
% Assm [h4s_options_NONEu_u_DEF]: h4/option/NONE = h4/option/option__ABS (h4/sum/INR h4/one/one0)
% Assm [h4s_options_optionu_u_REPu_u_ABSu_u_DEFu_c1]: !r. T <=> h4/option/option__REP (h4/option/option__ABS r) = r
% Assm [h4s_options_optionu_u_REPu_u_ABSu_u_DEFu_c0]: !a. h4/option/option__ABS (h4/option/option__REP a) = a
% Assm [h4s_sums_sumu_u_Axiom]: !g f. ?h. (!x. happ h (h4/sum/INL x) = happ f x) /\ (!y. happ h (h4/sum/INR y) = happ g y)
% Assm [h4s_options_datatypeu_u_option]: !option. h4/bool/DATATYPE (happ (happ option h4/option/NONE) h4/option/SOME)
% Assm [h4s_options_optionu_u_CLAUSESu_c9]: !x. h4/option/option__CASE x h4/option/NONE h4/option/SOME = x
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> t
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_finiteu_u_maps_FRANGEu_u_DEF]: !_0. (!f y. ?v. (v <=> (?x. h4/bool/IN x (h4/finite__map/FDOM f) /\ h4/finite__map/FAPPLY f x = y)) /\ happ (happ _0 f) y = h4/pair/_2C y v) ==> (!f. h4/finite__map/FRANGE f = h4/pred__set/GSPEC (happ _0 f))
% Assm [h4s_pairs_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_predu_u_sets_GSPECIFICATION]: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = happ f x)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_options_optionu_u_caseu_u_defu_c0]: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm [h4s_enumerals_OLu_u_sublistsu_u_ind]: !P. (!cmp. happ (happ P cmp) h4/list/NIL) /\ (!cmp lol. happ (happ P cmp) lol ==> happ (happ P cmp) (h4/list/CONS h4/option/NONE lol)) /\ (!cmp m lol. happ (happ P cmp) lol ==> happ (happ P cmp) (h4/list/CONS (happ h4/option/SOME m) lol)) ==> (!v v1. happ (happ P v) v1)
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_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_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_bools_UNWINDu_u_FORALLu_u_THM2]: !v f. (!x. x = v ==> happ f x) <=> happ f v
% Assm [h4s_bools_DISJu_u_COMM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_options_OPTIONu_u_BINDu_u_defu_c0]: !f. h4/option/OPTION__BIND h4/option/NONE f = h4/option/NONE
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_finiteu_u_maps_SUBMAPu_u_DEF]: !g f. h4/finite__map/SUBMAP f g <=> (!x. h4/bool/IN x (h4/finite__map/FDOM f) ==> h4/bool/IN x (h4/finite__map/FDOM g) /\ h4/finite__map/FAPPLY f x = h4/finite__map/FAPPLY g x)
% Assm [h4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND]: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm [h4s_finiteu_u_maps_FUNIONu_u_DEFu_c1]: !x g f. h4/finite__map/FAPPLY (h4/finite__map/FUNION f g) x = h4/bool/COND (h4/bool/IN x (h4/finite__map/FDOM f)) (h4/finite__map/FAPPLY f x) (h4/finite__map/FAPPLY g x)
% Assm [h4s_finiteu_u_maps_FUNIONu_u_DEFu_c0]: !g f. h4/finite__map/FDOM (h4/finite__map/FUNION f g) = h4/pred__set/UNION (h4/finite__map/FDOM f) (h4/finite__map/FDOM g)
% Assm [h4s_predu_u_sets_INu_u_UNION]: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x t
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~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_EQu_u_EXPAND]: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% Assm [h4s_arithmetics_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm [h4s_arithmetics_LESSu_u_EQu_u_TRANS]: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm [h4s_arithmetics_NOTu_u_LESS]: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Goal: !P. (!f. happ (happ P f) h4/list/NIL) /\ (!f l. happ (happ P f) l ==> happ (happ P f) (h4/list/CONS h4/option/NONE l)) /\ (!f z l. happ (happ P f) l ==> happ (happ P f) (h4/list/CONS (happ h4/option/SOME z) l)) ==> (!v v1. happ (happ P v) v1)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f0)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(aHLu_EXT, axiom, ![TV_Q1310057,TV_Q1310053]: ![V_f, V_g]: (![V_x]: s(TV_Q1310053,happ(s(t_fun(TV_Q1310057,TV_Q1310053),V_f),s(TV_Q1310057,V_x))) = s(TV_Q1310053,happ(s(t_fun(TV_Q1310057,TV_Q1310053),V_g),s(TV_Q1310057,V_x))) => s(t_fun(TV_Q1310057,TV_Q1310053),V_f) = s(t_fun(TV_Q1310057,TV_Q1310053),V_g))).
fof(ah4s_fmapals_optryu_u_listu_u_tupledu_u_primitive, axiom, ![TV_u_27z,TV_u_27g]: ![V_uu_7]: (![V_optryu_u_listu_u_tupled, V_f, V_l, V_z]: s(t_h4s_options_option(TV_u_27g),happ(s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),happ(s(t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g))),happ(s(t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g))))),V_uu_7),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),V_optryu_u_listu_u_tupled))),s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),V_f))),s(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),V_l))),s(TV_u_27z,V_z))) = s(t_h4s_options_option(TV_u_27g),h4s_combins_i(s(t_h4s_options_option(TV_u_27g),h4s_fmapals_optry(s(t_h4s_options_option(TV_u_27g),happ(s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),V_f),s(TV_u_27z,V_z))),s(t_h4s_options_option(TV_u_27g),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),V_optryu_u_listu_u_tupled),s(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),h4s_pairs_u_2c(s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),V_f),s(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),V_l))))))))) => ![V_uu_6]: (![V_v2, V_optryu_u_listu_u_tupled, V_f, V_l]: s(t_h4s_options_option(TV_u_27g),happ(s(t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_h4s_options_option(TV_u_27g)),happ(s(t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_h4s_options_option(TV_u_27g))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_h4s_options_option(TV_u_27g)))),happ(s(t_fun(t_h4s_options_option(TV_u_27z),t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_h4s_options_option(TV_u_27g))))),V_uu_6),s(t_h4s_options_option(TV_u_27z),V_v2))),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),V_optryu_u_listu_u_tupled))),s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),V_f))),s(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),V_l))) = s(t_h4s_options_option(TV_u_27g),h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27z),V_v2),s(t_h4s_options_option(TV_u_27g),h4s_combins_i(s(t_h4s_options_option(TV_u_27g),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),V_optryu_u_listu_u_tupled),s(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),h4s_pairs_u_2c(s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),V_f),s(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),V_l))))))),s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),happ(s(t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g))),happ(s(t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g))))),V_uu_7),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),V_optryu_u_listu_u_tupled))),s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),V_f))),s(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),V_l))))) => ![V_uu_5]: (![V_optryu_u_listu_u_tupled, V_f, V_v2]: s(t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_h4s_options_option(TV_u_27g)),happ(s(t_fun(t_h4s_options_option(TV_u_27z),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_h4s_options_option(TV_u_27g))),happ(s(t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_options_option(TV_u_27z),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_h4s_options_option(TV_u_27g)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_options_option(TV_u_27z),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_h4s_options_option(TV_u_27g))))),V_uu_5),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),V_optryu_u_listu_u_tupled))),s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),V_f))),s(t_h4s_options_option(TV_u_27z),V_v2))) = s(t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_h4s_options_option(TV_u_27g)),happ(s(t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_h4s_options_option(TV_u_27g))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_h4s_options_option(TV_u_27g)))),happ(s(t_fun(t_h4s_options_option(TV_u_27z),t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_h4s_options_option(TV_u_27g))))),V_uu_6),s(t_h4s_options_option(TV_u_27z),V_v2))),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),V_optryu_u_listu_u_tupled))),s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),V_f))) => ![V_uu_4]: (![V_optryu_u_listu_u_tupled, V_f, V_v1]: s(t_h4s_options_option(TV_u_27g),happ(s(t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_h4s_options_option(TV_u_27g)),happ(s(t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_h4s_options_option(TV_u_27g))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_h4s_options_option(TV_u_27g)))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),V_optryu_u_listu_u_tupled))),s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),V_f))),s(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),V_v1))) = s(t_h4s_options_option(TV_u_27g),h4s_lists_listu_u_case(s(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),V_v1),s(t_h4s_options_option(TV_u_27g),h4s_combins_i(s(t_h4s_options_option(TV_u_27g),h4s_options_none))),s(t_fun(t_h4s_options_option(TV_u_27z),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_h4s_options_option(TV_u_27g))),happ(s(t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_options_option(TV_u_27z),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_h4s_options_option(TV_u_27g)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_options_option(TV_u_27z),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_h4s_options_option(TV_u_27g))))),V_uu_5),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),V_optryu_u_listu_u_tupled))),s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),V_f))))) => ![V_uu_3]: (![V_optryu_u_listu_u_tupled, V_f]: s(t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_h4s_options_option(TV_u_27g)),happ(s(t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_h4s_options_option(TV_u_27g))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_h4s_options_option(TV_u_27g)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),V_optryu_u_listu_u_tupled))),s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),V_f))) = s(t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_h4s_options_option(TV_u_27g)),happ(s(t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_h4s_options_option(TV_u_27g))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_h4s_options_option(TV_u_27g)))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),V_optryu_u_listu_u_tupled))),s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),V_f))) => ![V_uu_2]: (![V_optryu_u_listu_u_tupled, V_a]: s(t_h4s_options_option(TV_u_27g),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),V_optryu_u_listu_u_tupled))),s(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),V_a))) = s(t_h4s_options_option(TV_u_27g),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),V_a),s(t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_h4s_options_option(TV_u_27g))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_h4s_options_option(TV_u_27g)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),V_optryu_u_listu_u_tupled))))) => ![V_uu_1]: (![V_optryu_u_listu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g))),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),V_optryu_u_listu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),V_optryu_u_listu_u_tupled))) => ![V_uu_0]: (![V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_bool)),V_R)))) & (![V_l, V_f]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_bool)),V_R),s(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),h4s_pairs_u_2c(s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),V_f),s(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),V_l))))),s(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),h4s_pairs_u_2c(s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),V_f),s(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),h4s_lists_cons(s(t_h4s_options_option(TV_u_27z),h4s_options_none),s(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),V_l)))))))) & ![V_z, V_l, V_f]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_bool)),V_R),s(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),h4s_pairs_u_2c(s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),V_f),s(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),V_l))))),s(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),h4s_pairs_u_2c(s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),V_f),s(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),h4s_lists_cons(s(t_h4s_options_option(TV_u_27z),happ(s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27z)),h4s_options_some),s(TV_u_27z,V_z))),s(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),V_l))))))))))) => s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),h4s_fmapals_optryu_u_listu_u_tupled) = s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g))),V_uu_1)))))))))))).
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_fmapals_optryu_u_listu_u_curried, axiom, ![TV_u_27g,TV_u_27z]: ![V_x1, V_x]: s(t_h4s_options_option(TV_u_27g),h4s_fmapals_optryu_u_list(s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),V_x),s(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),V_x1))) = s(t_h4s_options_option(TV_u_27g),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),t_h4s_options_option(TV_u_27g)),h4s_fmapals_optryu_u_listu_u_tupled),s(t_h4s_pairs_prod(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_h4s_lists_list(t_h4s_options_option(TV_u_27z))),h4s_pairs_u_2c(s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),V_x),s(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),V_x1)))))).
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_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_OPTIONu_u_MAPu_u_DEFu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),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_27b,t_h4s_options_option(TV_u_27b)),h4s_options_some),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_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_finiteu_u_maps_FLOOKUPu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f]: s(t_h4s_options_option(TV_u_27b),h4s_finiteu_u_maps_flookup(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27b),h4s_bools_cond(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,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,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,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_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_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_options_OPTIONu_u_MAP2u_u_THMu_c2, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_y, V_f]: 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),h4s_options_none),s(t_h4s_options_option(TV_u_27c),happ(s(t_fun(TV_u_27c,t_h4s_options_option(TV_u_27c)),h4s_options_some),s(TV_u_27c,V_y))))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)).
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_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_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_MAP2u_u_THMu_c0, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_y, V_x, V_f]: 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),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_h4s_options_option(TV_u_27c),happ(s(t_fun(TV_u_27c,t_h4s_options_option(TV_u_27c)),h4s_options_some),s(TV_u_27c,V_y))))) = 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,V_x))),s(TV_u_27c,V_y)))))).
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_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_finiteu_u_maps_SUBMAPu_u_FUPDATEu_u_FLOOKUP, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: (p(s(t_bool,h4s_finiteu_u_maps_submap(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),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(t_h4s_options_option(TV_u_27b),h4s_finiteu_u_maps_flookup(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27b),h4s_options_none) | s(t_h4s_options_option(TV_u_27b),h4s_finiteu_u_maps_flookup(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,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_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_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_options_OPTIONu_u_MAP2u_u_cong, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y2, V_y1, V_x2, V_x1, V_f2, V_f1]: ((s(t_h4s_options_option(TV_u_27a),V_x1) = s(t_h4s_options_option(TV_u_27a),V_x2) & (s(t_h4s_options_option(TV_u_27b),V_y1) = s(t_h4s_options_option(TV_u_27b),V_y2) & ![V_x, V_y]: ((s(t_h4s_options_option(TV_u_27a),V_x2) = 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),V_y2) = 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)))) => 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_f1),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_f2),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))))) => s(t_h4s_options_option(TV_u_27c),h4s_options_optionu_u_map2(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f1),s(t_h4s_options_option(TV_u_27a),V_x1),s(t_h4s_options_option(TV_u_27b),V_y1))) = s(t_h4s_options_option(TV_u_27c),h4s_options_optionu_u_map2(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f2),s(t_h4s_options_option(TV_u_27a),V_x2),s(t_h4s_options_option(TV_u_27b),V_y2))))).
fof(ah4s_finiteu_u_maps_FRANGEu_u_FLOOKUP, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_v),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_frange(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_f)))))) <=> ?[V_k]: s(t_h4s_options_option(TV_u_27a),h4s_finiteu_u_maps_flookup(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_f),s(TV_u_27b,V_k))) = 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_v))))).
fof(ah4s_options_optionu_u_CLAUSESu_c15, 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_finiteu_u_maps_FEVERYu_u_FLOOKUP, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_k, V_f, V_P]: ((p(s(t_bool,h4s_finiteu_u_maps_fevery(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f)))) & s(t_h4s_options_option(TV_u_27b),h4s_finiteu_u_maps_flookup(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_k))) = 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_v)))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_k),s(TV_u_27b,V_v)))))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f0)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f0)))).
fof(ah4s_finiteu_u_maps_FLOOKUPu_u_SUBMAP, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_k, V_g, V_f]: ((p(s(t_bool,h4s_finiteu_u_maps_submap(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g)))) & s(t_h4s_options_option(TV_u_27b),h4s_finiteu_u_maps_flookup(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_k))) = 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_v)))) => s(t_h4s_options_option(TV_u_27b),h4s_finiteu_u_maps_flookup(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_k))) = 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_v))))).
fof(ah4s_options_optionu_u_CLAUSESu_c16, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),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_27b,t_h4s_options_option(TV_u_27b)),h4s_options_some),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_SOMEu_u_SOMEu_u_APPLY, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f]: 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)),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27a),t_h4s_options_option(t_fun(TV_u_27b,TV_u_27a))),h4s_options_some),s(t_fun(TV_u_27b,TV_u_27a),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_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,happ(s(t_fun(TV_u_27b,TV_u_27a),V_f),s(TV_u_27b,V_x)))))).
fof(ah4s_options_OPTIONu_u_MAPu_u_EQu_u_SOME, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x, V_f]: (s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_options_option(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))) <=> ?[V_z]: (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_z))) & s(TV_u_27b,V_y) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_z)))))).
fof(ah4s_finiteu_u_maps_flookupu_u_thmu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_v, V_f]: (s(t_h4s_options_option(TV_u_27b),h4s_finiteu_u_maps_flookup(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,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_v))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f)))))) & s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,V_v)))).
fof(ah4s_finiteu_u_maps_INu_u_FRANGEu_u_FLOOKUP, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_v),s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_frange(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f)))))) <=> ?[V_k]: s(t_h4s_options_option(TV_u_27b),h4s_finiteu_u_maps_flookup(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_k))) = 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_v))))).
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_finiteu_u_maps_FLOOKUPu_u_FUNION, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_v]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),V_uu_0),s(TV_u_27a,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,V_v))) => ![V_k, V_f2, V_f1]: s(t_h4s_options_option(TV_u_27a),h4s_finiteu_u_maps_flookup(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),h4s_finiteu_u_maps_funion(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_f1),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_f2))),s(TV_u_27b,V_k))) = s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_finiteu_u_maps_flookup(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_f1),s(TV_u_27b,V_k))),s(t_h4s_options_option(TV_u_27a),h4s_finiteu_u_maps_flookup(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_f2),s(TV_u_27b,V_k))),s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),V_uu_0))))).
fof(ah4s_finiteu_u_maps_FEVERYu_u_ALLu_u_FLOOKUP, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_P]: (p(s(t_bool,h4s_finiteu_u_maps_fevery(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f)))) <=> ![V_k, V_v]: (s(t_h4s_options_option(TV_u_27b),h4s_finiteu_u_maps_flookup(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_k))) = 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_v))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_k),s(TV_u_27b,V_v))))))))).
fof(ah4s_options_OPTIONu_u_MAPu_u_CONG, axiom, ![TV_u_27b,TV_u_27a]: ![V_opt2, V_opt1, V_f2, V_f1]: ((s(t_h4s_options_option(TV_u_27a),V_opt1) = s(t_h4s_options_option(TV_u_27a),V_opt2) & ![V_x]: (s(t_h4s_options_option(TV_u_27a),V_opt2) = 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_f1),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f2),s(TV_u_27a,V_x))))) => s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(t_h4s_options_option(TV_u_27a),V_opt1))) = s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f2),s(t_h4s_options_option(TV_u_27a),V_opt2))))).
fof(ah4s_options_OPTIONu_u_MAP2u_u_SOME, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_v, V_o2, V_o1, V_f]: (s(t_h4s_options_option(TV_u_27c),h4s_options_optionu_u_map2(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(t_h4s_options_option(TV_u_27a),V_o1),s(t_h4s_options_option(TV_u_27b),V_o2))) = s(t_h4s_options_option(TV_u_27c),happ(s(t_fun(TV_u_27c,t_h4s_options_option(TV_u_27c)),h4s_options_some),s(TV_u_27c,V_v))) <=> ?[V_x1, V_x2]: (s(t_h4s_options_option(TV_u_27a),V_o1) = 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_x1))) & (s(t_h4s_options_option(TV_u_27b),V_o2) = 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_x2))) & s(TV_u_27c,V_v) = 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_x1))),s(TV_u_27b,V_x2))))))).
fof(ah4s_finiteu_u_maps_fmapu_u_inverseu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_m2, V_m1]: (p(s(t_bool,h4s_finiteu_u_maps_fmapu_u_inverse(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_m1),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_m2)))) <=> ![V_k]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_k),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_m1)))))) => ?[V_v]: (s(t_h4s_options_option(TV_u_27b),h4s_finiteu_u_maps_flookup(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_m1),s(TV_u_27a,V_k))) = 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_v))) & s(t_h4s_options_option(TV_u_27a),h4s_finiteu_u_maps_flookup(s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_m2),s(TV_u_27b,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,V_k))))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_CONDu_u_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,f0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_finiteu_u_maps_FEVERYu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_P]: (p(s(t_bool,h4s_finiteu_u_maps_fevery(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f)))) <=> ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(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_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,f0)).
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_APPLYu_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_apply(s(t_h4s_options_option(t_fun(TV_u_27b,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27a),t_h4s_options_option(t_fun(TV_u_27b,TV_u_27a))),h4s_options_some),s(t_fun(TV_u_27b,TV_u_27a),V_f))),s(t_h4s_options_option(TV_u_27b),V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_map(s(t_fun(TV_u_27b,TV_u_27a),V_f),s(t_h4s_options_option(TV_u_27b),V_x)))).
fof(ah4s_finiteu_u_maps_INu_u_FRANGE, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_v),s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_frange(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f)))))) <=> ?[V_k]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_k),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f)))))) & s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_k))) = s(TV_u_27b,V_v)))).
fof(ah4s_options_OPTIONu_u_MAPu_u_DEFu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_h4s_options_option(TV_u_27b),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_h4s_options_option(TV_u_27b),h4s_options_none)).
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_DATATYPEu_u_TAGu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,h4s_bools_datatype(s(TV_u_27a,V_x))) = s(t_bool,t)).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f0)) => 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_finiteu_u_maps_SUBMAPu_u_FUPDATEu_u_EQN, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: (p(s(t_bool,h4s_finiteu_u_maps_submap(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdate(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),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)))))))) <=> (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))))))) | (s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,V_y) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f)))))))))).
fof(ah4s_bools_CONJu_u_COMM, 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))))).
fof(ah4s_bools_CONJu_u_ASSOC, 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_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_markers_ACu_u_DEF, axiom, ![V_b2, V_b1]: (p(s(t_bool,h4s_markers_ac(s(t_bool,V_b1),s(t_bool,V_b2)))) <=> (p(s(t_bool,V_b1)) & p(s(t_bool,V_b2))))).
fof(ah4s_bools_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_options_OPTIONu_u_MAP2u_u_THMu_c3, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_f]: 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),h4s_options_none),s(t_h4s_options_option(TV_u_27c),h4s_options_none))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)).
fof(ah4s_options_OPTIONu_u_MAP2u_u_THMu_c1, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_x, V_f]: 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),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_h4s_options_option(TV_u_27c),h4s_options_none))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)).
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_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_optionu_u_REPu_u_ABSu_u_DEFu_c1, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,t)) <=> 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),h4s_options_optionu_u_abs(s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),V_r))))) = s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),V_r))).
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_sums_sumu_u_Axiom, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_g, V_f]: ?[V_h]: (![V_x]: s(TV_u_27c,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),V_h),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_x))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x))) & ![V_y]: s(TV_u_27c,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),V_h),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),V_g),s(TV_u_27b,V_y))))).
fof(ah4s_options_datatypeu_u_option, axiom, ![TV_u_27b,TV_u_27a]: ![V_option]: p(s(t_bool,h4s_bools_datatype(s(TV_u_27b,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),TV_u_27b),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_fun(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),TV_u_27b)),V_option),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))))))).
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_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_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_finiteu_u_maps_FRANGEu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_f, V_y]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ?[V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f)))))) & s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,V_y))) & s(t_h4s_pairs_prod(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27b,t_bool)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27b,t_bool))),V_uu_0),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27b,V_y))) = s(t_h4s_pairs_prod(TV_u_27b,t_bool),h4s_pairs_u_2c(s(TV_u_27b,V_y),s(t_bool,V_v)))) => ![V_f]: s(t_fun(TV_u_27b,t_bool),h4s_finiteu_u_maps_frange(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27b,t_bool)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27b,t_bool))),V_uu_0),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))))))).
fof(ah4s_pairs_PAIRu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_b, V_a]: (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(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & s(TV_u_27b,V_y) = s(TV_u_27b,V_b)))).
fof(ah4s_predu_u_sets_GSPECIFICATION, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_v),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f)))))) <=> ?[V_x]: s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_v),s(t_bool,t))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,t_bool)),V_f),s(TV_u_27b,V_x))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f0))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_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_enumerals_OLu_u_sublistsu_u_ind, axiom, ![TV_u_27a]: ![V_P]: ((![V_cmp]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),t_bool),happ(s(t_fun(t_h4s_totos_toto(TV_u_27a),t_fun(t_h4s_lists_list(t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),t_bool)),V_P),s(t_h4s_totos_toto(TV_u_27a),V_cmp))),s(t_h4s_lists_list(t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),h4s_lists_nil)))) & (![V_cmp, V_lol]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),t_bool),happ(s(t_fun(t_h4s_totos_toto(TV_u_27a),t_fun(t_h4s_lists_list(t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),t_bool)),V_P),s(t_h4s_totos_toto(TV_u_27a),V_cmp))),s(t_h4s_lists_list(t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),V_lol)))) => p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),t_bool),happ(s(t_fun(t_h4s_totos_toto(TV_u_27a),t_fun(t_h4s_lists_list(t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),t_bool)),V_P),s(t_h4s_totos_toto(TV_u_27a),V_cmp))),s(t_h4s_lists_list(t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),h4s_lists_cons(s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),h4s_options_none),s(t_h4s_lists_list(t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),V_lol))))))) & ![V_cmp, V_m, V_lol]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),t_bool),happ(s(t_fun(t_h4s_totos_toto(TV_u_27a),t_fun(t_h4s_lists_list(t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),t_bool)),V_P),s(t_h4s_totos_toto(TV_u_27a),V_cmp))),s(t_h4s_lists_list(t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),V_lol)))) => p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),t_bool),happ(s(t_fun(t_h4s_totos_toto(TV_u_27a),t_fun(t_h4s_lists_list(t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),t_bool)),V_P),s(t_h4s_totos_toto(TV_u_27a),V_cmp))),s(t_h4s_lists_list(t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),h4s_lists_cons(s(t_h4s_options_option(t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),h4s_options_some),s(t_h4s_lists_list(TV_u_27a),V_m))),s(t_h4s_lists_list(t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),V_lol))))))))) => ![V_v, V_v1]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),t_bool),happ(s(t_fun(t_h4s_totos_toto(TV_u_27a),t_fun(t_h4s_lists_list(t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),t_bool)),V_P),s(t_h4s_totos_toto(TV_u_27a),V_v))),s(t_h4s_lists_list(t_h4s_options_option(t_h4s_lists_list(TV_u_27a))),V_v1)))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f0)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f0) = 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_boolu_u_caseu_u_thmu_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_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_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,f0))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
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,f0))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
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,f0))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => p(s(t_bool,f0))))).
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,f0))))).
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_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_DISJu_u_COMM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_bools_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_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f0)))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f0))))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
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_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_pthu_u_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_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_finiteu_u_maps_SUBMAPu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (p(s(t_bool,h4s_finiteu_u_maps_submap(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g)))) <=> ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f)))))) => (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g)))))) & s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND, axiom, ![V_C, V_B, V_A]: (((p(s(t_bool,V_B)) & p(s(t_bool,V_C))) | p(s(t_bool,V_A))) <=> ((p(s(t_bool,V_B)) | p(s(t_bool,V_A))) & (p(s(t_bool,V_C)) | p(s(t_bool,V_A)))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f0)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,f0))) <=> p(s(t_bool,f0)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_IMPu_u_THM, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) & (p(s(t_bool,V_t2)) => p(s(t_bool,V_t1)))))).
fof(ah4s_finiteu_u_maps_FUNIONu_u_DEFu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_g, V_f]: s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_funion(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))))),s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x)))))).
fof(ah4s_finiteu_u_maps_FUNIONu_u_DEFu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_g, V_f]: s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_funion(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g))))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27a,t_bool),h4s_finiteu_u_maps_fdom(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_g)))))).
fof(ah4s_predu_u_sets_INu_u_UNION, axiom, ![TV_u_27a]: ![V_x, V_t, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_bools_boolu_u_caseu_u_thmu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f0) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) => ~ (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,f0))).
fof(ah4s_bools_EQu_u_EXPAND, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) | (~ (p(s(t_bool,V_t1))) & ~ (p(s(t_bool,V_t2))))))).
fof(ah4s_arithmetics_ZEROu_u_LESSu_u_EQ, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_LESSu_u_EQu_u_TRANS, axiom, ![V_p, V_n, V_m]: ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_NOTu_u_LESS, axiom, ![V_n, V_m]: (~ (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ch4s_fmapals_optryu_u_listu_u_ind, conjecture, ![TV_u_27g,TV_u_27z]: ![V_P]: ((![V_f]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_bool),happ(s(t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_bool)),V_P),s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),V_f))),s(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),h4s_lists_nil)))) & (![V_f, V_l]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_bool),happ(s(t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_bool)),V_P),s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),V_f))),s(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),V_l)))) => p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_bool),happ(s(t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_bool)),V_P),s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),V_f))),s(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),h4s_lists_cons(s(t_h4s_options_option(TV_u_27z),h4s_options_none),s(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),V_l))))))) & ![V_f, V_z, V_l]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_bool),happ(s(t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_bool)),V_P),s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),V_f))),s(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),V_l)))) => p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_bool),happ(s(t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_bool)),V_P),s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),V_f))),s(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),h4s_lists_cons(s(t_h4s_options_option(TV_u_27z),happ(s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27z)),h4s_options_some),s(TV_u_27z,V_z))),s(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),V_l))))))))) => ![V_v, V_v1]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_bool),happ(s(t_fun(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),t_fun(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),t_bool)),V_P),s(t_fun(TV_u_27z,t_h4s_options_option(TV_u_27g)),V_v))),s(t_h4s_lists_list(t_h4s_options_option(TV_u_27z)),V_v1)))))).
