%   ORIGINAL: h4/alist/ALOOKUP__NONE
% 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/finite__map/FUPDATE__FUPDATE__LIST__MEM: !v kvl k fm. h4/bool/IN k (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST kvl)) ==> h4/finite__map/FUPDATE__LIST (h4/finite__map/FUPDATE fm (h4/pair/_2C k v)) kvl = h4/finite__map/FUPDATE__LIST fm kvl
% Assm: h4/finite__map/FUPDATE__FUPDATE__LIST__COMMUTES: !v kvl k fm. ~h4/bool/IN k (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST kvl)) ==> h4/finite__map/FUPDATE__LIST (h4/finite__map/FUPDATE fm (h4/pair/_2C k v)) kvl = h4/finite__map/FUPDATE (h4/finite__map/FUPDATE__LIST fm kvl) (h4/pair/_2C k v)
% Assm: h4/finite__map/FUPDATE__LIST__APPLY__NOT__MEM__matchable: !v kvl k f. ~h4/bool/IN k (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST kvl)) /\ v = h4/finite__map/FAPPLY f k ==> h4/finite__map/FAPPLY (h4/finite__map/FUPDATE__LIST f kvl) k = v
% Assm: h4/finite__map/FUPDATE__LIST__CANCEL: !ls2 ls1 fm. (!k. h4/bool/IN k (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST ls1)) ==> h4/bool/IN k (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST ls2))) ==> h4/finite__map/FUPDATE__LIST (h4/finite__map/FUPDATE__LIST fm ls1) ls2 = h4/finite__map/FUPDATE__LIST fm ls2
% Assm: h4/alist/ALOOKUP__def_c0: !q. h4/alist/ALOOKUP h4/list/NIL q = h4/option/NONE
% Assm: h4/alist/ALOOKUP__curried__def: !x1 x. h4/alist/ALOOKUP x x1 = h4/alist/ALOOKUP__tupled (h4/pair/_2C x x1)
% Assm: h4/alist/ALOOKUP__def_c1: !y x t q. h4/alist/ALOOKUP (h4/list/CONS (h4/pair/_2C x y) t) q = h4/bool/COND (x = q) (h4/option/SOME y) (h4/alist/ALOOKUP t q)
% Assm: h4/alist/ALOOKUP__FAILS: !x l. h4/alist/ALOOKUP l x = h4/option/NONE <=> (!k v. h4/bool/IN (h4/pair/_2C k v) (h4/list/LIST__TO__SET l) ==> ~(k = x))
% Assm: h4/finite__map/FUPDATE__LIST__SAME__UPDATE: !kvl f2 f1. h4/finite__map/FUPDATE__LIST f1 kvl = h4/finite__map/FUPDATE__LIST f2 kvl <=> h4/finite__map/DRESTRICT f1 (h4/pred__set/COMPL (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST kvl))) = h4/finite__map/DRESTRICT f2 (h4/pred__set/COMPL (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST kvl)))
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/finite__map/FUPDATE__LIST__THM_c1: !t h f. h4/finite__map/FUPDATE__LIST f (h4/list/CONS h t) = h4/finite__map/FUPDATE__LIST (h4/finite__map/FUPDATE f h) t
% Assm: h4/list/LIST__TO__SET0_c1: !t h. h4/list/LIST__TO__SET (h4/list/CONS h t) = h4/pred__set/INSERT h (h4/list/LIST__TO__SET t)
% Assm: h4/pred__set/NOT__IN__EMPTY: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm: h4/pred__set/IN__INSERT: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm: h4/list/MAP0_c0: !f. h4/list/MAP f h4/list/NIL = h4/list/NIL
% Assm: h4/list/LIST__TO__SET0_c0: h4/list/LIST__TO__SET h4/list/NIL = h4/pred__set/EMPTY
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> 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/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% 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/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/finite__map/FUPDATE__LIST__THM_c0: !f. h4/finite__map/FUPDATE__LIST f h4/list/NIL = f
% Assm: h4/list/list__induction: !P. P h4/list/NIL /\ (!t. P t ==> (!h. P (h4/list/CONS h t))) ==> (!l. P l)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/list/MAP0_c1: !t h f. h4/list/MAP f (h4/list/CONS h t) = h4/list/CONS (f h) (h4/list/MAP f t)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/finite__map/FUPDATE__COMMUTES: !f d c b a. ~(a = c) ==> h4/finite__map/FUPDATE (h4/finite__map/FUPDATE f (h4/pair/_2C a b)) (h4/pair/_2C c d) = h4/finite__map/FUPDATE (h4/finite__map/FUPDATE f (h4/pair/_2C c d)) (h4/pair/_2C a b)
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/list/MAP__SNOC: !x l f. h4/list/MAP f (h4/list/SNOC x l) = h4/list/SNOC (f x) (h4/list/MAP f l)
% Assm: h4/list/SNOC__INDUCT: !P. P h4/list/NIL /\ (!l. P l ==> (!x. P (h4/list/SNOC x l))) ==> (!l. P l)
% Assm: h4/list/SNOC__APPEND: !x l. h4/list/SNOC x l = h4/list/APPEND l (h4/list/CONS x h4/list/NIL)
% Assm: h4/list/LIST__TO__SET__APPEND: !l2 l1. h4/list/LIST__TO__SET (h4/list/APPEND l1 l2) = h4/pred__set/UNION (h4/list/LIST__TO__SET l1) (h4/list/LIST__TO__SET l2)
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% 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/finite__map/FUPDATE__LIST__APPEND: !kvl2 kvl1 fm. h4/finite__map/FUPDATE__LIST fm (h4/list/APPEND kvl1 kvl2) = h4/finite__map/FUPDATE__LIST (h4/finite__map/FUPDATE__LIST fm kvl1) kvl2
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/finite__map/FUPDATE__LIST__APPLY__NOT__MEM: !kvl k f. ~h4/bool/IN k (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST kvl)) ==> h4/finite__map/FAPPLY (h4/finite__map/FUPDATE__LIST f kvl) k = h4/finite__map/FAPPLY f k
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% 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/finite__map/FUPDATE__EQ: !f c b a. h4/finite__map/FUPDATE (h4/finite__map/FUPDATE f (h4/pair/_2C a b)) (h4/pair/_2C a c) = h4/finite__map/FUPDATE f (h4/pair/_2C a c)
% Assm: h4/bool/EQ__EXPAND: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/list/list__size__def_c1: !f a1 a0. h4/list/list__size f (h4/list/CONS a0 a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (f a0) (h4/list/list__size f a1))
% Assm: h4/list/list__case__def_c1: !v f a1 a0. h4/list/list__CASE (h4/list/CONS a0 a1) v f = f a0 a1
% Assm: h4/list/list__case__def_c0: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/numeral/numeral__distrib_c27: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__lt_c0: !n. h4/prim__rec/_3C h4/arithmetic/ZERO (h4/arithmetic/BIT1 n) <=> T
% Assm: h4/numeral/numeral__distrib_c21: !n. h4/prim__rec/_3C h4/num/0 (h4/arithmetic/NUMERAL n) <=> h4/prim__rec/_3C h4/arithmetic/ZERO n
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/arithmetic/SUC__ONE__ADD: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm: h4/arithmetic/ADD__MONO__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm: h4/arithmetic/LESS__EQ__TRANS: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/LESS__EQ: !n m. h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm: h4/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% Assm: h4/numeral/numeral__distrib_c1: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm: h4/arithmetic/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/ADD__0: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/alist/ALOOKUP__tupled__primitive__def: h4/alist/ALOOKUP__tupled = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!y t q x. ~(x = q) ==> R (h4/pair/_2C t q) (h4/pair/_2C (h4/list/CONS (h4/pair/_2C x y) t) q)))) (\ALOOKUP__tupled a. h4/pair/pair__CASE a (\v q. h4/list/list__CASE v (h4/combin/I h4/option/NONE) (\v2 t. h4/pair/pair__CASE v2 (\x y. h4/combin/I (h4/bool/COND (x = q) (h4/option/SOME y) (ALOOKUP__tupled (h4/pair/_2C t q)))))))
% Assm: h4/prim__rec/WF__LESS: h4/relation/WF h4/prim__rec/_3C
% Assm: h4/relation/WFREC__COROLLARY: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. f x = M (h4/relation/RESTRICT f R x) x)
% Assm: h4/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm: h4/relation/RESTRICT__LEMMA: !z y f R. R y z ==> h4/relation/RESTRICT f R z y = f y
% Assm: h4/relation/inv__image__def: !f R. h4/relation/inv__image R f = (\x y. R (f x) (f y))
% Assm: h4/relation/WF__inv__image: !f R. h4/relation/WF R ==> h4/relation/WF (h4/relation/inv__image R f)
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/bool/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% 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/list/MEM_c1: !x t h. h4/bool/IN x (h4/list/LIST__TO__SET (h4/list/CONS h t)) <=> x = h \/ h4/bool/IN x (h4/list/LIST__TO__SET t)
% Assm: h4/list/MEM_c0: !x. h4/bool/IN x (h4/list/LIST__TO__SET h4/list/NIL) <=> F
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/pair/CLOSED__PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/option/NOT__NONE__SOME: !x. ~(h4/option/NONE = h4/option/SOME x)
% 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/UNWIND__FORALL__THM2: !v f. (!x. x = v ==> f x) <=> f v
% Assm: h4/finite__map/FAPPLY__FUPDATE__THM: !x f b a. h4/finite__map/FAPPLY (h4/finite__map/FUPDATE f (h4/pair/_2C a b)) x = h4/bool/COND (x = a) b (h4/finite__map/FAPPLY f x)
% Assm: h4/finite__map/FDOM__FUPDATE: !f b a. h4/finite__map/FDOM (h4/finite__map/FUPDATE f (h4/pair/_2C a b)) = h4/pred__set/INSERT a (h4/finite__map/FDOM f)
% Assm: h4/finite__map/fmap__EQ__THM: !g f. h4/finite__map/FDOM f = h4/finite__map/FDOM g /\ (!x. h4/bool/IN x (h4/finite__map/FDOM f) ==> h4/finite__map/FAPPLY f x = h4/finite__map/FAPPLY g x) <=> f = g
% Assm: h4/pred__set/COMPL__EMPTY: h4/pred__set/COMPL h4/pred__set/EMPTY = h4/pred__set/UNIV
% Assm: h4/pred__set/INTER__UNIV_c1: !s. h4/pred__set/INTER s h4/pred__set/UNIV = s
% Assm: h4/pred__set/IN__INTER: !x t s. h4/bool/IN x (h4/pred__set/INTER s t) <=> h4/bool/IN x s /\ h4/bool/IN x t
% Assm: h4/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm: h4/finite__map/DRESTRICT__DEF_c0: !r f. h4/finite__map/FDOM (h4/finite__map/DRESTRICT f r) = h4/pred__set/INTER (h4/finite__map/FDOM f) r
% Assm: h4/pred__set/IN__COMPL: !x s. h4/bool/IN x (h4/pred__set/COMPL s) <=> ~h4/bool/IN x s
% Goal: !x l. h4/alist/ALOOKUP l x = h4/option/NONE <=> ~h4/bool/IN x (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST l))
%   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_finiteu_u_maps_FUPDATEu_u_FUPDATEu_u_LISTu_u_MEM]: !v kvl k fm. h4/bool/IN k (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST kvl)) ==> h4/finite__map/FUPDATE__LIST (h4/finite__map/FUPDATE fm (h4/pair/_2C k v)) kvl = h4/finite__map/FUPDATE__LIST fm kvl
% Assm [h4s_finiteu_u_maps_FUPDATEu_u_FUPDATEu_u_LISTu_u_COMMUTES]: !v kvl k fm. ~h4/bool/IN k (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST kvl)) ==> h4/finite__map/FUPDATE__LIST (h4/finite__map/FUPDATE fm (h4/pair/_2C k v)) kvl = h4/finite__map/FUPDATE (h4/finite__map/FUPDATE__LIST fm kvl) (h4/pair/_2C k v)
% Assm [h4s_finiteu_u_maps_FUPDATEu_u_LISTu_u_APPLYu_u_NOTu_u_MEMu_u_matchable]: !v kvl k f. ~h4/bool/IN k (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST kvl)) /\ v = h4/finite__map/FAPPLY f k ==> h4/finite__map/FAPPLY (h4/finite__map/FUPDATE__LIST f kvl) k = v
% Assm [h4s_finiteu_u_maps_FUPDATEu_u_LISTu_u_CANCEL]: !ls2 ls1 fm. (!k. h4/bool/IN k (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST ls1)) ==> h4/bool/IN k (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST ls2))) ==> h4/finite__map/FUPDATE__LIST (h4/finite__map/FUPDATE__LIST fm ls1) ls2 = h4/finite__map/FUPDATE__LIST fm ls2
% Assm [h4s_alists_ALOOKUPu_u_defu_c0]: !q. h4/alist/ALOOKUP h4/list/NIL q = h4/option/NONE
% Assm [h4s_alists_ALOOKUPu_u_curriedu_u_def]: !x1 x. h4/alist/ALOOKUP x x1 = happ h4/alist/ALOOKUP__tupled (h4/pair/_2C x x1)
% Assm [h4s_alists_ALOOKUPu_u_defu_c1]: !y x t q. ?v. (v <=> x = q) /\ h4/alist/ALOOKUP (h4/list/CONS (h4/pair/_2C x y) t) q = h4/bool/COND v (h4/option/SOME y) (h4/alist/ALOOKUP t q)
% Assm [h4s_alists_ALOOKUPu_u_FAILS]: !x l. h4/alist/ALOOKUP l x = h4/option/NONE <=> (!k v. h4/bool/IN (h4/pair/_2C k v) (h4/list/LIST__TO__SET l) ==> ~(k = x))
% Assm [h4s_finiteu_u_maps_FUPDATEu_u_LISTu_u_SAMEu_u_UPDATE]: !kvl f2 f1. h4/finite__map/FUPDATE__LIST f1 kvl = h4/finite__map/FUPDATE__LIST f2 kvl <=> h4/finite__map/DRESTRICT f1 (h4/pred__set/COMPL (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST kvl))) = h4/finite__map/DRESTRICT f2 (h4/pred__set/COMPL (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST kvl)))
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_pairs_FST0]: !y x. happ h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_finiteu_u_maps_FUPDATEu_u_LISTu_u_THMu_c1]: !t h f. h4/finite__map/FUPDATE__LIST f (h4/list/CONS h t) = h4/finite__map/FUPDATE__LIST (h4/finite__map/FUPDATE f h) t
% Assm [h4s_lists_LISTu_u_TOu_u_SET0u_c1]: !t h. h4/list/LIST__TO__SET (h4/list/CONS h t) = h4/pred__set/INSERT h (h4/list/LIST__TO__SET t)
% Assm [h4s_predu_u_sets_NOTu_u_INu_u_EMPTY]: !x. ~h4/bool/IN x h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_INu_u_INSERT]: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm [h4s_lists_MAP0u_c0]: !f. h4/list/MAP f h4/list/NIL = h4/list/NIL
% Assm [h4s_lists_LISTu_u_TOu_u_SET0u_c0]: h4/list/LIST__TO__SET h4/list/NIL = h4/pred__set/EMPTY
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> 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_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% 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_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_finiteu_u_maps_FUPDATEu_u_LISTu_u_THMu_c0]: !f. h4/finite__map/FUPDATE__LIST f h4/list/NIL = f
% Assm [h4s_lists_listu_u_induction]: !P. happ P h4/list/NIL /\ (!t. happ P t ==> (!h. happ P (h4/list/CONS h t))) ==> (!l. happ P l)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_lists_MAP0u_c1]: !t h f. h4/list/MAP f (h4/list/CONS h t) = h4/list/CONS (happ f h) (h4/list/MAP f t)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_finiteu_u_maps_FUPDATEu_u_COMMUTES]: !f d c b a. ~(a = c) ==> h4/finite__map/FUPDATE (h4/finite__map/FUPDATE f (h4/pair/_2C a b)) (h4/pair/_2C c d) = h4/finite__map/FUPDATE (h4/finite__map/FUPDATE f (h4/pair/_2C c d)) (h4/pair/_2C a b)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% 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_lists_MAPu_u_SNOC]: !x l f. h4/list/MAP f (h4/list/SNOC x l) = h4/list/SNOC (happ f x) (h4/list/MAP f l)
% Assm [h4s_lists_SNOCu_u_INDUCT]: !P. happ P h4/list/NIL /\ (!l. happ P l ==> (!x. happ P (h4/list/SNOC x l))) ==> (!l. happ P l)
% Assm [h4s_lists_SNOCu_u_APPEND]: !x l. h4/list/SNOC x l = h4/list/APPEND l (h4/list/CONS x h4/list/NIL)
% Assm [h4s_lists_LISTu_u_TOu_u_SETu_u_APPEND]: !l2 l1. h4/list/LIST__TO__SET (h4/list/APPEND l1 l2) = h4/pred__set/UNION (h4/list/LIST__TO__SET l1) (h4/list/LIST__TO__SET l2)
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% 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_finiteu_u_maps_FUPDATEu_u_LISTu_u_APPEND]: !kvl2 kvl1 fm. h4/finite__map/FUPDATE__LIST fm (h4/list/APPEND kvl1 kvl2) = h4/finite__map/FUPDATE__LIST (h4/finite__map/FUPDATE__LIST fm kvl1) kvl2
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_finiteu_u_maps_FUPDATEu_u_LISTu_u_APPLYu_u_NOTu_u_MEM]: !kvl k f. ~h4/bool/IN k (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST kvl)) ==> h4/finite__map/FAPPLY (h4/finite__map/FUPDATE__LIST f kvl) k = h4/finite__map/FAPPLY f k
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% 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_nn]: !p. ~ ~p ==> p
% 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_finiteu_u_maps_FUPDATEu_u_EQ]: !f c b a. h4/finite__map/FUPDATE (h4/finite__map/FUPDATE f (h4/pair/_2C a b)) (h4/pair/_2C a c) = h4/finite__map/FUPDATE f (h4/pair/_2C a c)
% Assm [h4s_bools_EQu_u_EXPAND]: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_lists_listu_u_sizeu_u_defu_c1]: !f a1 a0. h4/list/list__size f (h4/list/CONS a0 a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (happ f a0) (h4/list/list__size f a1))
% Assm [h4s_lists_listu_u_caseu_u_defu_c1]: !v f a1 a0. h4/list/list__CASE (h4/list/CONS a0 a1) v f = happ (happ f a0) a1
% Assm [h4s_lists_listu_u_caseu_u_defu_c0]: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_numerals_numeralu_u_distribu_c27]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_ltu_c0]: !n. happ (happ h4/prim__rec/_3C h4/arithmetic/ZERO) (h4/arithmetic/BIT1 n) <=> T
% Assm [h4s_numerals_numeralu_u_distribu_c21]: !n. happ (happ h4/prim__rec/_3C h4/num/0) (h4/arithmetic/NUMERAL n) <=> happ (happ h4/prim__rec/_3C h4/arithmetic/ZERO) n
% Assm [h4s_arithmetics_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_arithmetics_SUCu_u_ONEu_u_ADD]: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm [h4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm [h4s_arithmetics_LESSu_u_EQu_u_TRANS]: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_LESSu_u_EQ]: !n m. happ (happ h4/prim__rec/_3C m) n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm [h4s_pairs_pairu_u_caseu_u_thm]: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = happ (happ f x) y
% Assm [h4s_numerals_numeralu_u_distribu_c1]: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm [h4s_arithmetics_NOTu_u_LESS]: !n m. ~happ (happ h4/prim__rec/_3C m) n <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_ADDu_u_0]: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm [h4s_alists_ALOOKUPu_u_tupledu_u_primitiveu_u_def]: !_8. (!x ALOOKUP__tupled t q y. ?v. (v <=> x = q) /\ happ (happ (happ (happ (happ _8 x) ALOOKUP__tupled) t) q) y = h4/combin/I (h4/bool/COND v (h4/option/SOME y) (happ ALOOKUP__tupled (h4/pair/_2C t q)))) ==> (!_7. (!ALOOKUP__tupled t q x. happ (happ (happ (happ _7 ALOOKUP__tupled) t) q) x = happ (happ (happ (happ _8 x) ALOOKUP__tupled) t) q) ==> (!_6. (!v2 ALOOKUP__tupled q t. happ (happ (happ (happ _6 v2) ALOOKUP__tupled) q) t = h4/pair/pair__CASE v2 (happ (happ (happ _7 ALOOKUP__tupled) t) q)) ==> (!_5. (!ALOOKUP__tupled q v2. happ (happ (happ _5 ALOOKUP__tupled) q) v2 = happ (happ (happ _6 v2) ALOOKUP__tupled) q) ==> (!_4. (!v ALOOKUP__tupled q. happ (happ (happ _4 v) ALOOKUP__tupled) q = h4/list/list__CASE v (h4/combin/I h4/option/NONE) (happ (happ _5 ALOOKUP__tupled) q)) ==> (!_3. (!ALOOKUP__tupled v. happ (happ _3 ALOOKUP__tupled) v = happ (happ _4 v) ALOOKUP__tupled) ==> (!_2. (!ALOOKUP__tupled a. happ (happ _2 ALOOKUP__tupled) a = h4/pair/pair__CASE a (happ _3 ALOOKUP__tupled)) ==> (!_1. (!ALOOKUP__tupled. happ _1 ALOOKUP__tupled = happ _2 ALOOKUP__tupled) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R /\ (!y t q x. ~(x = q) ==> happ (happ R (h4/pair/_2C t q)) (h4/pair/_2C (h4/list/CONS (h4/pair/_2C x y) t) q))) ==> h4/alist/ALOOKUP__tupled = h4/relation/WFREC (h4/min/_40 _0) _1))))))))
% Assm [h4s_primu_u_recs_WFu_u_LESS]: h4/relation/WF h4/prim__rec/_3C
% Assm [h4s_relations_WFRECu_u_COROLLARY]: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. happ f x = happ (happ M (h4/relation/RESTRICT f R x)) x)
% Assm [h4s_pairs_PAIR]: !x. h4/pair/_2C (happ h4/pair/FST x) (h4/pair/SND x) = x
% Assm [h4s_relations_RESTRICTu_u_LEMMA]: !z y f R. happ (happ R y) z ==> happ (h4/relation/RESTRICT f R z) y = happ f y
% Assm [h4s_relations_invu_u_imageu_u_def]: !f R x x'. happ (happ (h4/relation/inv__image R f) x) x' <=> happ (happ R (happ f x)) (happ f x')
% Assm [h4s_relations_WFu_u_invu_u_image]: !f R. h4/relation/WF R ==> h4/relation/WF (h4/relation/inv__image R f)
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_bools_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% 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_lists_MEMu_c1]: !x t h. h4/bool/IN x (h4/list/LIST__TO__SET (h4/list/CONS h t)) <=> x = h \/ h4/bool/IN x (h4/list/LIST__TO__SET t)
% Assm [h4s_lists_MEMu_c0]: !x. h4/bool/IN x (h4/list/LIST__TO__SET h4/list/NIL) <=> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_pairs_CLOSEDu_u_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_options_NOTu_u_NONEu_u_SOME]: !x. ~(h4/option/NONE = h4/option/SOME x)
% 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_UNWINDu_u_FORALLu_u_THM2]: !v f. (!x. x = v ==> happ f x) <=> happ f v
% Assm [h4s_finiteu_u_maps_FAPPLYu_u_FUPDATEu_u_THM]: !x f b a. ?v. (v <=> x = a) /\ h4/finite__map/FAPPLY (h4/finite__map/FUPDATE f (h4/pair/_2C a b)) x = h4/bool/COND v b (h4/finite__map/FAPPLY f x)
% Assm [h4s_finiteu_u_maps_FDOMu_u_FUPDATE]: !f b a. h4/finite__map/FDOM (h4/finite__map/FUPDATE f (h4/pair/_2C a b)) = h4/pred__set/INSERT a (h4/finite__map/FDOM f)
% Assm [h4s_finiteu_u_maps_fmapu_u_EQu_u_THM]: !g f. h4/finite__map/FDOM f = h4/finite__map/FDOM g /\ (!x. h4/bool/IN x (h4/finite__map/FDOM f) ==> h4/finite__map/FAPPLY f x = h4/finite__map/FAPPLY g x) <=> f = g
% Assm [h4s_predu_u_sets_COMPLu_u_EMPTY]: h4/pred__set/COMPL h4/pred__set/EMPTY = h4/pred__set/UNIV
% Assm [h4s_predu_u_sets_INTERu_u_UNIVu_c1]: !s. h4/pred__set/INTER s h4/pred__set/UNIV = s
% Assm [h4s_predu_u_sets_INu_u_INTER]: !x t s. h4/bool/IN x (h4/pred__set/INTER s t) <=> h4/bool/IN x s /\ h4/bool/IN x t
% Assm [h4s_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm [h4s_finiteu_u_maps_DRESTRICTu_u_DEFu_c0]: !r f. h4/finite__map/FDOM (h4/finite__map/DRESTRICT f r) = h4/pred__set/INTER (h4/finite__map/FDOM f) r
% Assm [h4s_predu_u_sets_INu_u_COMPL]: !x s. h4/bool/IN x (h4/pred__set/COMPL s) <=> ~h4/bool/IN x s
% Goal: !x l. h4/alist/ALOOKUP l x = h4/option/NONE <=> ~h4/bool/IN x (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/FST l))
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q1363693,TV_Q1363689]: ![V_f, V_g]: (![V_x]: s(TV_Q1363689,happ(s(t_fun(TV_Q1363693,TV_Q1363689),V_f),s(TV_Q1363693,V_x))) = s(TV_Q1363689,happ(s(t_fun(TV_Q1363693,TV_Q1363689),V_g),s(TV_Q1363693,V_x))) => s(t_fun(TV_Q1363693,TV_Q1363689),V_f) = s(t_fun(TV_Q1363693,TV_Q1363689),V_g))).
fof(ah4s_finiteu_u_maps_FUPDATEu_u_FUPDATEu_u_LISTu_u_MEM, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_kvl, V_k, V_fm]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_k),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_kvl)))))))) => s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdateu_u_list(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_fm),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))))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_kvl))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdateu_u_list(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_kvl))))).
fof(ah4s_finiteu_u_maps_FUPDATEu_u_FUPDATEu_u_LISTu_u_COMMUTES, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_kvl, V_k, V_fm]: (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_k),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_kvl))))))))) => s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdateu_u_list(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_fm),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))))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_kvl))) = 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),h4s_finiteu_u_maps_fupdateu_u_list(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_kvl))),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_finiteu_u_maps_FUPDATEu_u_LISTu_u_APPLYu_u_NOTu_u_MEMu_u_matchable, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_kvl, V_k, V_f]: ((~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_k),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_kvl))))))))) & s(TV_u_27b,V_v) = 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,h4s_finiteu_u_maps_fapply(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdateu_u_list(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_kvl))),s(TV_u_27a,V_k))) = s(TV_u_27b,V_v))).
fof(ah4s_finiteu_u_maps_FUPDATEu_u_LISTu_u_CANCEL, axiom, ![TV_u_27a,TV_u_27b]: ![V_ls2, V_ls1, V_fm]: (![V_k]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_k),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_ls1)))))))) => p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_k),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_ls2))))))))) => s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdateu_u_list(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdateu_u_list(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_ls1))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_ls2))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdateu_u_list(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_ls2))))).
fof(ah4s_alists_ALOOKUPu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_q]: s(t_h4s_options_option(TV_u_27a),h4s_alists_alookup(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),h4s_lists_nil),s(TV_u_27b,V_q))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)).
fof(ah4s_alists_ALOOKUPu_u_curriedu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_x1, V_x]: s(t_h4s_options_option(TV_u_27a),h4s_alists_alookup(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_x),s(TV_u_27b,V_x1))) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),h4s_alists_alookupu_u_tupled),s(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),h4s_pairs_u_2c(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_x),s(TV_u_27b,V_x1)))))).
fof(ah4s_alists_ALOOKUPu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_t, V_q]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(TV_u_27b,V_x) = s(TV_u_27b,V_q)) & s(t_h4s_options_option(TV_u_27a),h4s_alists_alookup(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),h4s_lists_cons(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27b,V_x),s(TV_u_27a,V_y))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_t))),s(TV_u_27b,V_q))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))),s(t_h4s_options_option(TV_u_27a),h4s_alists_alookup(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_t),s(TV_u_27b,V_q))))))).
fof(ah4s_alists_ALOOKUPu_u_FAILS, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_l]: (s(t_h4s_options_option(TV_u_27a),h4s_alists_alookup(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_l),s(TV_u_27b,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) <=> ![V_k, V_v]: (p(s(t_bool,h4s_bools_in(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27b,V_k),s(TV_u_27a,V_v))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_l)))))) => ~ (s(TV_u_27b,V_k) = s(TV_u_27b,V_x))))).
fof(ah4s_finiteu_u_maps_FUPDATEu_u_LISTu_u_SAMEu_u_UPDATE, axiom, ![TV_u_27a,TV_u_27b]: ![V_kvl, V_f2, V_f1]: (s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdateu_u_list(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f1),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_kvl))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdateu_u_list(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f2),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_kvl))) <=> s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_drestrict(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f1),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_compl(s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_kvl))))))))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_drestrict(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f2),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_compl(s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_kvl))))))))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_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_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_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_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_pairs_ABSu_u_PAIRu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: ?[V_q, V_r]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_q),s(TV_u_27b,V_r)))).
fof(ah4s_finiteu_u_maps_FUPDATEu_u_LISTu_u_THMu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_t, V_h, V_f]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdateu_u_list(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_cons(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_h),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_t))))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdateu_u_list(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),V_h))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_t)))).
fof(ah4s_lists_LISTu_u_TOu_u_SET0u_c1, axiom, ![TV_u_27b]: ![V_t, V_h]: s(t_fun(TV_u_27b,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,V_h),s(t_h4s_lists_list(TV_u_27b),V_t))))) = s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_insert(s(TV_u_27b,V_h),s(t_fun(TV_u_27b,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27b),V_t)))))).
fof(ah4s_predu_u_sets_NOTu_u_INu_u_EMPTY, axiom, ![TV_u_27a]: ![V_x]: ~ (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_empty)))))).
fof(ah4s_predu_u_sets_INu_u_INSERT, axiom, ![TV_u_27a]: ![V_y, V_x, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_lists_MAP0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil)).
fof(ah4s_lists_LISTu_u_TOu_u_SET0u_c0, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_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,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_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_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_finiteu_u_maps_FUPDATEu_u_LISTu_u_THMu_c0, axiom, ![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_fupdateu_u_list(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_nil))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f)).
fof(ah4s_lists_listu_u_induction, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))) & ![V_t]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t)))) => ![V_h]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t)))))))) => ![V_l]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
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_lists_MAP0u_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_h, V_f]: s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t))))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),V_t)))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_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_finiteu_u_maps_FUPDATEu_u_COMMUTES, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_d, V_c, V_b, V_a]: (~ (s(TV_u_27a,V_a) = s(TV_u_27a,V_c)) => 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),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_a),s(TV_u_27b,V_b))))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_c),s(TV_u_27b,V_d))))) = 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),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_c),s(TV_u_27b,V_d))))),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))))))).
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_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_NOTu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ?[V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f))))).
fof(ah4s_bools_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_lists_MAPu_u_SNOC, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_l, V_f]: s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),h4s_lists_snoc(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_snoc(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_lists_SNOCu_u_INDUCT, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))) & ![V_l]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))) => ![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_snoc(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l)))))))) => ![V_l]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_lists_SNOCu_u_APPEND, axiom, ![TV_u_27a]: ![V_x, V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_snoc(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))))).
fof(ah4s_lists_LISTu_u_TOu_u_SETu_u_APPEND, axiom, ![TV_u_27a]: ![V_l2, V_l1]: s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2))))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l1))),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l2)))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,f))) <=> p(s(t_bool,V_t)))).
fof(ah4s_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_finiteu_u_maps_FUPDATEu_u_LISTu_u_APPEND, axiom, ![TV_u_27a,TV_u_27b]: ![V_kvl2, V_kvl1, V_fm]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdateu_u_list(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_append(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_kvl1),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_kvl2))))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdateu_u_list(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fupdateu_u_list(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_fm),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_kvl1))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_kvl2)))).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_finiteu_u_maps_FUPDATEu_u_LISTu_u_APPLYu_u_NOTu_u_MEM, axiom, ![TV_u_27b,TV_u_27a]: ![V_kvl, V_k, V_f]: (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_k),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_kvl))))))))) => 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_fupdateu_u_list(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_kvl))),s(TV_u_27a,V_k))) = 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))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c0, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) & p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) | ~ (p(s(t_bool,V_B)))))).
fof(ah4s_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_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
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_finiteu_u_maps_FUPDATEu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_c, V_b, V_a]: 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),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_a),s(TV_u_27b,V_b))))),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_c))))) = 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_a),s(TV_u_27b,V_c)))))).
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_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_lists_listu_u_sizeu_u_defu_c1, axiom, ![TV_u_27a]: ![V_f, V_a1, V_a0]: s(t_h4s_nums_num,h4s_lists_listu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a0),s(t_h4s_lists_list(TV_u_27a),V_a1))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(TV_u_27a,V_a0))),s(t_h4s_nums_num,h4s_lists_listu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_lists_list(TV_u_27a),V_a1)))))))).
fof(ah4s_lists_listu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_f, V_a1, V_a0]: s(TV_u_27b,h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a0),s(t_h4s_lists_list(TV_u_27a),V_a1))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f))) = s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1)))).
fof(ah4s_lists_listu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27b,h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f))) = s(TV_u_27b,V_v)).
fof(ah4s_numerals_numeralu_u_lteu_c1, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
fof(ah4s_numerals_numeralu_u_distribu_c27, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ah4s_numerals_numeralu_u_ltu_c0, axiom, ![V_n]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_arithmetics_zero))),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))) = s(t_bool,t)).
fof(ah4s_numerals_numeralu_u_distribu_c21, axiom, ![V_n]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))))) = s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_arithmetics_zero))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_ZEROu_u_LESSu_u_EQ, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_SUCu_u_ONEu_u_ADD, axiom, ![V_n]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_LESSu_u_EQu_u_TRANS, axiom, ![V_p, V_n, V_m]: ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c2, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_SYM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_LESSu_u_EQ, axiom, ![V_n, V_m]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_pairs_pairu_u_caseu_u_thm, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_y, V_x, V_f]: s(TV_u_27a,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27b,V_x),s(TV_u_27c,V_y))),s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f))) = s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27b,V_x))),s(TV_u_27c,V_y)))).
fof(ah4s_numerals_numeralu_u_distribu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_arithmetics_NOTu_u_LESS, axiom, ![V_n, V_m]: (~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_ADDu_u_0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_alists_ALOOKUPu_u_tupledu_u_primitiveu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_8]: (![V_x, V_ALOOKUPu_u_tupled, V_t, V_q, V_y]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(TV_u_27b,V_x) = s(TV_u_27b,V_q)) & s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)))))),V_uu_8),s(TV_u_27b,V_x))),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),V_ALOOKUPu_u_tupled))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_t))),s(TV_u_27b,V_q))),s(TV_u_27a,V_y))) = s(t_h4s_options_option(TV_u_27a),h4s_combins_i(s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))),s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),V_ALOOKUPu_u_tupled),s(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),h4s_pairs_u_2c(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_t),s(TV_u_27b,V_q)))))))))) => ![V_uu_7]: (![V_ALOOKUPu_u_tupled, V_t, V_q, V_x]: s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)))))),V_uu_7),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),V_ALOOKUPu_u_tupled))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_t))),s(TV_u_27b,V_q))),s(TV_u_27b,V_x))) = s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(TV_u_27b,t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)))))),V_uu_8),s(TV_u_27b,V_x))),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),V_ALOOKUPu_u_tupled))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_t))),s(TV_u_27b,V_q))) => ![V_uu_6]: (![V_v2, V_ALOOKUPu_u_tupled, V_q, V_t]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_h4s_options_option(TV_u_27a))))),V_uu_6),s(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),V_v2))),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),V_ALOOKUPu_u_tupled))),s(TV_u_27b,V_q))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_t))) = s(t_h4s_options_option(TV_u_27a),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),V_v2),s(t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(TV_u_27b,t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)))))),V_uu_7),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),V_ALOOKUPu_u_tupled))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_t))),s(TV_u_27b,V_q))))) => ![V_uu_5]: (![V_ALOOKUPu_u_tupled, V_q, V_v2]: s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_h4s_options_option(TV_u_27a))))),V_uu_5),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),V_ALOOKUPu_u_tupled))),s(TV_u_27b,V_q))),s(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),V_v2))) = s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_h4s_options_option(TV_u_27a))))),V_uu_6),s(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),V_v2))),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),V_ALOOKUPu_u_tupled))),s(TV_u_27b,V_q))) => ![V_uu_4]: (![V_v, V_ALOOKUPu_u_tupled, V_q]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)))),V_uu_4),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_v))),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),V_ALOOKUPu_u_tupled))),s(TV_u_27b,V_q))) = s(t_h4s_options_option(TV_u_27a),h4s_lists_listu_u_case(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_v),s(t_h4s_options_option(TV_u_27a),h4s_combins_i(s(t_h4s_options_option(TV_u_27a),h4s_options_none))),s(t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_h4s_options_option(TV_u_27a))))),V_uu_5),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),V_ALOOKUPu_u_tupled))),s(TV_u_27b,V_q))))) => ![V_uu_3]: (![V_ALOOKUPu_u_tupled, V_v]: s(t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),V_ALOOKUPu_u_tupled))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_v))) = s(t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)))),V_uu_4),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_v))),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),V_ALOOKUPu_u_tupled))) => ![V_uu_2]: (![V_ALOOKUPu_u_tupled, V_a]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),V_ALOOKUPu_u_tupled))),s(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),V_a))) = s(t_h4s_options_option(TV_u_27a),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),V_a),s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_h4s_options_option(TV_u_27a)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),V_ALOOKUPu_u_tupled))))) => ![V_uu_1]: (![V_ALOOKUPu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a))),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),V_ALOOKUPu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),V_ALOOKUPu_u_tupled))) => ![V_uu_0]: (![V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_bool)),V_R)))) & ![V_y, V_t, V_q, V_x]: (~ (s(TV_u_27b,V_x) = s(TV_u_27b,V_q)) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_bool)),V_R),s(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),h4s_pairs_u_2c(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_t),s(TV_u_27b,V_q))))),s(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),h4s_pairs_u_2c(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),h4s_lists_cons(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27a),h4s_pairs_u_2c(s(TV_u_27b,V_x),s(TV_u_27a,V_y))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_t))),s(TV_u_27b,V_q))))))))) => s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),h4s_alists_alookupu_u_tupled) = s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),TV_u_27b),t_h4s_options_option(TV_u_27a))),V_uu_1))))))))))))).
fof(ah4s_primu_u_recs_WFu_u_LESS, axiom, p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c))))).
fof(ah4s_relations_WFRECu_u_COROLLARY, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_M]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_wfrec(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M))) => (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M),s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))))),s(TV_u_27a,V_x)))))).
fof(ah4s_pairs_PAIR, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x)).
fof(ah4s_relations_RESTRICTu_u_LEMMA, axiom, ![TV_u_27b,TV_u_27a]: ![V_z, V_y, V_f, V_R]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_z)))) => s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_z))),s(TV_u_27a,V_y))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))))).
fof(ah4s_relations_invu_u_imageu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_x, V_xi_]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_invu_u_image(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_x))),s(TV_u_27a,V_xi_))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_xi_)))))).
fof(ah4s_relations_WFu_u_invu_u_image, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R)))) => p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_invu_u_image(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_f)))))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_bools_IMPu_u_Fu_u_EQu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_lists_MEMu_c1, axiom, ![TV_u_27a]: ![V_x, V_t, V_h]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t)))))))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_h) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_t))))))))).
fof(ah4s_lists_MEMu_c0, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))))) = s(t_bool,f)).
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_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_pairs_CLOSEDu_u_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_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),h4s_options_some(s(TV_u_27a,V_x))))).
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_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_finiteu_u_maps_FAPPLYu_u_FUPDATEu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f, V_b, V_a]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_a)) & 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_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_a),s(TV_u_27b,V_b))))),s(TV_u_27a,V_x))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27b,V_b),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_finiteu_u_maps_FDOMu_u_FUPDATE, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_b, V_a]: 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_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_a),s(TV_u_27b,V_b))))))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_a),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_finiteu_u_maps_fmapu_u_EQu_u_THM, 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),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))) & ![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,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))))) <=> 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))).
fof(ah4s_predu_u_sets_COMPLu_u_EMPTY, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_compl(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ)).
fof(ah4s_predu_u_sets_INTERu_u_UNIVu_c1, axiom, ![TV_u_27a]: ![V_s]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_univ))) = s(t_fun(TV_u_27a,t_bool),V_s)).
fof(ah4s_predu_u_sets_INu_u_INTER, 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_inter(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_predu_u_sets_EXTENSION, axiom, ![TV_u_27a]: ![V_t, V_s]: (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),V_t) <=> ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))).
fof(ah4s_finiteu_u_maps_DRESTRICTu_u_DEFu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_r, 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_drestrict(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_bool),V_r))))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_inter(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),V_r)))).
fof(ah4s_predu_u_sets_INu_u_COMPL, axiom, ![TV_u_27a]: ![V_x, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_compl(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_s))))))).
fof(ch4s_alists_ALOOKUPu_u_NONE, conjecture, ![TV_u_27a,TV_u_27b]: ![V_x, V_l]: (s(t_h4s_options_option(TV_u_27b),h4s_alists_alookup(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l),s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27b),h4s_options_none) <=> ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27a),h4s_pairs_fst),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l))))))))))).
