%   ORIGINAL: h4/finite__map/fupdate__list__foldl
% 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/bool/TRUTH: T
% Assm: h4/finite__map/FUPDATE__LIST__SNOC: !xs x fm. h4/finite__map/FUPDATE__LIST fm (h4/list/SNOC x xs) = h4/finite__map/FUPDATE (h4/finite__map/FUPDATE__LIST fm xs) x
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/finite__map/fupdate__list__foldr: !m l. h4/list/FOLDR (h4/pair/UNCURRY (\k v env. h4/finite__map/FUPDATE env (h4/pair/_2C k v))) m l = h4/finite__map/FUPDATE__LIST m (h4/list/REVERSE l)
% Assm: h4/finite__map/FUPDATE__LIST__APPLY__MEM: !v n kvl k f. h4/prim__rec/_3C n (h4/list/LENGTH kvl) /\ k = h4/list/EL n (h4/list/MAP h4/pair/FST kvl) /\ v = h4/list/EL n (h4/list/MAP h4/pair/SND kvl) /\ (!m. h4/prim__rec/_3C n m /\ h4/prim__rec/_3C m (h4/list/LENGTH kvl) ==> ~(h4/list/EL m (h4/list/MAP h4/pair/FST kvl) = k)) ==> h4/finite__map/FAPPLY (h4/finite__map/FUPDATE__LIST f kvl) 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/IN__FRANGE__FUPDATE__LIST__suff: !ls fm P. (!v. h4/bool/IN v (h4/finite__map/FRANGE fm) ==> P v) /\ (!v. h4/bool/IN v (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/SND ls)) ==> P v) ==> (!v. h4/bool/IN v (h4/finite__map/FRANGE (h4/finite__map/FUPDATE__LIST fm ls)) ==> P 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/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/ETA__AX: !t. (\x. t x) = t
% 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/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/finite__map/FUPDATE__LIST__THM_c0: !f. h4/finite__map/FUPDATE__LIST f h4/list/NIL = f
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/list/MAP0_c0: !f. h4/list/MAP f h4/list/NIL = h4/list/NIL
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> 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/list/LIST__TO__SET0_c0: h4/list/LIST__TO__SET h4/list/NIL = 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/SNOC0_c0: !x. h4/list/SNOC x h4/list/NIL = h4/list/CONS x h4/list/NIL
% Assm: h4/list/SNOC0_c1: !x_27 x l. h4/list/SNOC x (h4/list/CONS x_27 l) = h4/list/CONS x_27 (h4/list/SNOC x l)
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% 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__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> 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/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% 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__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/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/rich__list/FOLDL__APPEND: !l2 l1 f e. h4/list/FOLDL f e (h4/list/APPEND l1 l2) = h4/list/FOLDL f (h4/list/FOLDL f e l1) l2
% Assm: h4/list/REVERSE__DEF_c1: !t h. h4/list/REVERSE (h4/list/CONS h t) = h4/list/APPEND (h4/list/REVERSE t) (h4/list/CONS h h4/list/NIL)
% Assm: h4/list/REVERSE__DEF_c0: h4/list/REVERSE h4/list/NIL = h4/list/NIL
% Assm: h4/list/FOLDL0_c1: !x l f e. h4/list/FOLDL f e (h4/list/CONS x l) = h4/list/FOLDL f (f e x) l
% Assm: h4/list/FOLDR0_c0: !f e. h4/list/FOLDR f e h4/list/NIL = e
% Assm: h4/list/FOLDR0_c1: !x l f e. h4/list/FOLDR f e (h4/list/CONS x l) = f x (h4/list/FOLDR f e l)
% Assm: h4/list/FOLDL0_c0: !f e. h4/list/FOLDL f e h4/list/NIL = e
% Assm: h4/pair/pair__CASES: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/finite__map/FUPDATE__LIST0: h4/finite__map/FUPDATE__LIST = h4/list/FOLDL h4/finite__map/FUPDATE
% 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/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/list/SNOC__INDUCT: !P. P h4/list/NIL /\ (!l. P l ==> (!x. P (h4/list/SNOC x l))) ==> (!l. P l)
% 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__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/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/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% 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/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% 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__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/finite__map/FRANGE__FUPDATE__LIST__SUBSET: !ls fm. h4/pred__set/SUBSET (h4/finite__map/FRANGE (h4/finite__map/FUPDATE__LIST fm ls)) (h4/pred__set/UNION (h4/finite__map/FRANGE fm) (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/SND ls)))
% Assm: h4/pred__set/UNION__DEF: !t s. h4/pred__set/UNION s t = h4/pred__set/GSPEC (\x. h4/pair/_2C x (h4/bool/IN x s \/ h4/bool/IN x t))
% Assm: h4/pred__set/SUBSET__DEF: !t s. h4/pred__set/SUBSET s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN x t)
% Assm: h4/pred__set/GSPECIFICATION: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = f x)
% 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/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/rich__list/CONS__APPEND: !x l. h4/list/CONS x l = h4/list/APPEND (h4/list/CONS x h4/list/NIL) l
% Assm: h4/finite__map/fmap__ISO__DEF_c1: !r. h4/finite__map/is__fmap r <=> h4/finite__map/fmap__REP (h4/finite__map/fmap__ABS r) = r
% Assm: h4/finite__map/fmap__ISO__DEF_c0: !a. h4/finite__map/fmap__ABS (h4/finite__map/fmap__REP a) = a
% Assm: h4/finite__map/is__fmap__def: h4/finite__map/is__fmap = (\a0. !is__fmap_27. (!a00. a00 = (\a. h4/sum/INR h4/one/one0) \/ (?f a b. a00 = (\x. h4/bool/COND (x = a) (h4/sum/INL b) (f x)) /\ is__fmap_27 f) ==> is__fmap_27 a00) ==> is__fmap_27 a0)
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/list/MEM__EL: !x l. h4/bool/IN x (h4/list/LIST__TO__SET l) <=> (?n. h4/prim__rec/_3C n (h4/list/LENGTH l) /\ x = h4/list/EL n l)
% Assm: h4/finite__map/FUPDATE__DEF: !y x f. h4/finite__map/FUPDATE f (h4/pair/_2C x y) = h4/finite__map/fmap__ABS (\a. h4/bool/COND (a = x) (h4/sum/INL y) (h4/finite__map/fmap__REP f a))
% Assm: h4/list/MEM__MAP: !x l f. h4/bool/IN x (h4/list/LIST__TO__SET (h4/list/MAP f l)) <=> (?y. x = f y /\ h4/bool/IN y (h4/list/LIST__TO__SET l))
% Assm: h4/list/EL__restricted_c1: !n ls l. h4/list/EL (h4/num/SUC n) (h4/list/CONS l ls) = h4/list/EL n ls
% Assm: h4/list/EL__restricted_c0: h4/list/EL h4/num/0 = h4/list/HD
% Assm: h4/list/EL__MAP: !n l. h4/prim__rec/_3C n (h4/list/LENGTH l) ==> (!f. h4/list/EL n (h4/list/MAP f l) = f (h4/list/EL n l))
% Assm: h4/list/LENGTH0_c1: !t h. h4/list/LENGTH (h4/list/CONS h t) = h4/num/SUC (h4/list/LENGTH t)
% Assm: h4/list/LENGTH0_c0: h4/list/LENGTH h4/list/NIL = h4/num/0
% Assm: h4/list/HD0: !t h. h4/list/HD (h4/list/CONS h t) = h
% Assm: h4/finite__map/FAPPLY__DEF: !x f. h4/finite__map/FAPPLY f x = h4/sum/OUTL (h4/finite__map/fmap__REP f x)
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/arithmetic/num__CASES: !m. m = h4/num/0 \/ (?n. m = h4/num/SUC n)
% Assm: h4/arithmetic/LESS__MONO__EQ: !n m. h4/prim__rec/_3C (h4/num/SUC m) (h4/num/SUC n) <=> h4/prim__rec/_3C m n
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/prim__rec/LESS__0: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm: h4/prim__rec/NOT__LESS__0: !n. ~h4/prim__rec/_3C n h4/num/0
% Assm: h4/sum/OUTL0: !x. h4/sum/OUTL (h4/sum/INL x) = x
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/pair/EXISTS__PROD: !P. (?p. P p) <=> (?p__1 p__2. P (h4/pair/_2C p__1 p__2))
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/pred__set/COMPL__EMPTY: h4/pred__set/COMPL h4/pred__set/EMPTY = h4/pred__set/UNIV
% Assm: h4/pred__set/IN__COMPL: !x s. h4/bool/IN x (h4/pred__set/COMPL s) <=> ~h4/bool/IN x s
% Goal: !m l. h4/list/FOLDL (\env. h4/pair/UNCURRY (\k v. h4/finite__map/FUPDATE env (h4/pair/_2C k v))) m l = h4/finite__map/FUPDATE__LIST m 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)) ==> happ (happ h4/finite__map/FUPDATE__LIST (happ (happ h4/finite__map/FUPDATE fm) (h4/pair/_2C k v))) kvl = happ (happ 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)) ==> happ (happ h4/finite__map/FUPDATE__LIST (happ (happ h4/finite__map/FUPDATE fm) (h4/pair/_2C k v))) kvl = happ (happ h4/finite__map/FUPDATE (happ (happ h4/finite__map/FUPDATE__LIST fm) kvl)) (h4/pair/_2C k v)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_finiteu_u_maps_FUPDATEu_u_LISTu_u_SNOC]: !xs x fm. happ (happ h4/finite__map/FUPDATE__LIST fm) (h4/list/SNOC x xs) = happ (happ h4/finite__map/FUPDATE (happ (happ h4/finite__map/FUPDATE__LIST fm) xs)) x
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_finiteu_u_maps_fupdateu_u_listu_u_foldr]: !_2. (!k v env. happ (happ (happ _2 k) v) env = happ (happ h4/finite__map/FUPDATE env) (h4/pair/_2C k v)) ==> (!_1. (!k v. happ (happ _1 k) v = happ (happ _2 k) v) ==> (!_0. (!k. happ _0 k = happ _1 k) ==> (!m l. h4/list/FOLDR (h4/pair/UNCURRY _0) m l = happ (happ h4/finite__map/FUPDATE__LIST m) (h4/list/REVERSE l))))
% Assm [h4s_finiteu_u_maps_FUPDATEu_u_LISTu_u_APPLYu_u_MEM]: !v n kvl k f. h4/prim__rec/_3C n (h4/list/LENGTH kvl) /\ k = happ (h4/list/EL n) (h4/list/MAP h4/pair/FST kvl) /\ v = happ (h4/list/EL n) (h4/list/MAP h4/pair/SND kvl) /\ (!m. h4/prim__rec/_3C n m /\ h4/prim__rec/_3C m (h4/list/LENGTH kvl) ==> ~(happ (h4/list/EL m) (h4/list/MAP h4/pair/FST kvl) = k)) ==> h4/finite__map/FAPPLY (happ (happ h4/finite__map/FUPDATE__LIST f) kvl) 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 (happ (happ h4/finite__map/FUPDATE__LIST f) kvl) k = v
% Assm [h4s_finiteu_u_maps_INu_u_FRANGEu_u_FUPDATEu_u_LISTu_u_suff]: !ls fm P. (!v. h4/bool/IN v (h4/finite__map/FRANGE fm) ==> happ P v) /\ (!v. h4/bool/IN v (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/SND ls)) ==> happ P v) ==> (!v. h4/bool/IN v (h4/finite__map/FRANGE (happ (happ h4/finite__map/FUPDATE__LIST fm) ls)) ==> happ P 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))) ==> happ (happ h4/finite__map/FUPDATE__LIST (happ (happ h4/finite__map/FUPDATE__LIST fm) ls1)) ls2 = happ (happ h4/finite__map/FUPDATE__LIST fm) ls2
% Assm [h4s_finiteu_u_maps_FUPDATEu_u_LISTu_u_SAMEu_u_UPDATE]: !kvl f2 f1. happ (happ h4/finite__map/FUPDATE__LIST f1) kvl = happ (happ 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_ETAu_u_AX]: !t x. happ t x = happ t x
% 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_finiteu_u_maps_FUPDATEu_u_LISTu_u_THMu_c1]: !t h f. happ (happ h4/finite__map/FUPDATE__LIST f) (h4/list/CONS h t) = happ (happ h4/finite__map/FUPDATE__LIST (happ (happ h4/finite__map/FUPDATE f) h)) t
% Assm [h4s_finiteu_u_maps_FUPDATEu_u_LISTu_u_THMu_c0]: !f. happ (happ h4/finite__map/FUPDATE__LIST f) h4/list/NIL = f
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_pairs_FST0]: !y x. happ h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_lists_MAP0u_c0]: !f. h4/list/MAP f h4/list/NIL = h4/list/NIL
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> 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_lists_LISTu_u_TOu_u_SET0u_c0]: h4/list/LIST__TO__SET h4/list/NIL = 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_SNOC0u_c0]: !x. h4/list/SNOC x h4/list/NIL = h4/list/CONS x h4/list/NIL
% Assm [h4s_lists_SNOC0u_c1]: !x_27 x l. h4/list/SNOC x (h4/list/CONS x_27 l) = h4/list/CONS x_27 (h4/list/SNOC x l)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_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_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> 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_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_finiteu_u_maps_FUPDATEu_u_COMMUTES]: !f d c b a. ~(a = c) ==> happ (happ h4/finite__map/FUPDATE (happ (happ h4/finite__map/FUPDATE f) (h4/pair/_2C a b))) (h4/pair/_2C c d) = happ (happ h4/finite__map/FUPDATE (happ (happ h4/finite__map/FUPDATE f) (h4/pair/_2C c d))) (h4/pair/_2C a b)
% 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_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_richu_u_lists_FOLDLu_u_APPEND]: !l2 l1 f e. happ (happ (h4/list/FOLDL f) e) (h4/list/APPEND l1 l2) = happ (happ (h4/list/FOLDL f) (happ (happ (h4/list/FOLDL f) e) l1)) l2
% Assm [h4s_lists_REVERSEu_u_DEFu_c1]: !t h. h4/list/REVERSE (h4/list/CONS h t) = h4/list/APPEND (h4/list/REVERSE t) (h4/list/CONS h h4/list/NIL)
% Assm [h4s_lists_REVERSEu_u_DEFu_c0]: h4/list/REVERSE h4/list/NIL = h4/list/NIL
% Assm [h4s_lists_FOLDL0u_c1]: !x l f e. happ (happ (h4/list/FOLDL f) e) (h4/list/CONS x l) = happ (happ (h4/list/FOLDL f) (happ (happ f e) x)) l
% Assm [h4s_lists_FOLDR0u_c0]: !f e. h4/list/FOLDR f e h4/list/NIL = e
% Assm [h4s_lists_FOLDR0u_c1]: !x l f e. h4/list/FOLDR f e (h4/list/CONS x l) = happ (happ f x) (h4/list/FOLDR f e l)
% Assm [h4s_lists_FOLDL0u_c0]: !f e. happ (happ (h4/list/FOLDL f) e) h4/list/NIL = e
% Assm [h4s_pairs_pairu_u_CASES]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. happ (h4/pair/UNCURRY f) (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_finiteu_u_maps_FUPDATEu_u_LIST0]: h4/finite__map/FUPDATE__LIST = h4/list/FOLDL h4/finite__map/FUPDATE
% Assm [h4s_finiteu_u_maps_FUPDATEu_u_LISTu_u_APPEND]: !kvl2 kvl1 fm. happ (happ h4/finite__map/FUPDATE__LIST fm) (h4/list/APPEND kvl1 kvl2) = happ (happ h4/finite__map/FUPDATE__LIST (happ (happ h4/finite__map/FUPDATE__LIST fm) kvl1)) kvl2
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_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_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_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_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_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% 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 (happ (happ h4/finite__map/FUPDATE__LIST f) kvl) k = h4/finite__map/FAPPLY f k
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_finiteu_u_maps_FUPDATEu_u_EQ]: !f c b a. happ (happ h4/finite__map/FUPDATE (happ (happ h4/finite__map/FUPDATE f) (h4/pair/_2C a b))) (h4/pair/_2C a c) = happ (happ h4/finite__map/FUPDATE f) (h4/pair/_2C a c)
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_finiteu_u_maps_FRANGEu_u_FUPDATEu_u_LISTu_u_SUBSET]: !ls fm. h4/pred__set/SUBSET (h4/finite__map/FRANGE (happ (happ h4/finite__map/FUPDATE__LIST fm) ls)) (h4/pred__set/UNION (h4/finite__map/FRANGE fm) (h4/list/LIST__TO__SET (h4/list/MAP h4/pair/SND ls)))
% Assm [h4s_predu_u_sets_UNIONu_u_DEF]: !_0. (!s t x. ?v. (v <=> h4/bool/IN x s \/ h4/bool/IN x t) /\ happ (happ (happ _0 s) t) x = h4/pair/_2C x v) ==> (!t s. h4/pred__set/UNION s t = h4/pred__set/GSPEC (happ (happ _0 s) t))
% Assm [h4s_predu_u_sets_SUBSETu_u_DEF]: !t s. h4/pred__set/SUBSET s t <=> (!x. h4/bool/IN x s ==> h4/bool/IN x t)
% Assm [h4s_predu_u_sets_GSPECIFICATION]: !v f. h4/bool/IN v (h4/pred__set/GSPEC f) <=> (?x. h4/pair/_2C v T = happ f x)
% 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_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_richu_u_lists_CONSu_u_APPEND]: !x l. h4/list/CONS x l = h4/list/APPEND (h4/list/CONS x h4/list/NIL) l
% Assm [h4s_finiteu_u_maps_fmapu_u_ISOu_u_DEFu_c1]: !r. h4/finite__map/is__fmap r <=> h4/finite__map/fmap__REP (h4/finite__map/fmap__ABS r) = r
% Assm [h4s_finiteu_u_maps_fmapu_u_ISOu_u_DEFu_c0]: !a. h4/finite__map/fmap__ABS (h4/finite__map/fmap__REP a) = a
% Assm [h4s_finiteu_u_maps_isu_u_fmapu_u_def]: !x. h4/finite__map/is__fmap x <=> (!is__fmap_27. (!a00. (!x. happ a00 x = h4/sum/INR h4/one/one0) \/ (?f a b. (!x. ?v. (v <=> x = a) /\ happ a00 x = h4/bool/COND v (h4/sum/INL b) (happ f x)) /\ happ is__fmap_27 f) ==> happ is__fmap_27 a00) ==> happ is__fmap_27 x)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_lists_MEMu_u_EL]: !x l. h4/bool/IN x (h4/list/LIST__TO__SET l) <=> (?n. h4/prim__rec/_3C n (h4/list/LENGTH l) /\ x = happ (h4/list/EL n) l)
% Assm [h4s_finiteu_u_maps_FUPDATEu_u_DEF]: !_0. (!x y f a. ?v. (v <=> a = x) /\ happ (happ (happ (happ _0 x) y) f) a = h4/bool/COND v (h4/sum/INL y) (happ (h4/finite__map/fmap__REP f) a)) ==> (!y x f. happ (happ h4/finite__map/FUPDATE f) (h4/pair/_2C x y) = h4/finite__map/fmap__ABS (happ (happ (happ _0 x) y) f))
% Assm [h4s_lists_MEMu_u_MAP]: !x l f. h4/bool/IN x (h4/list/LIST__TO__SET (h4/list/MAP f l)) <=> (?y. x = happ f y /\ h4/bool/IN y (h4/list/LIST__TO__SET l))
% Assm [h4s_lists_ELu_u_restrictedu_c1]: !n ls l. happ (h4/list/EL (h4/num/SUC n)) (h4/list/CONS l ls) = happ (h4/list/EL n) ls
% Assm [h4s_lists_ELu_u_restrictedu_c0]: h4/list/EL h4/num/0 = h4/list/HD
% Assm [h4s_lists_ELu_u_MAP]: !n l. h4/prim__rec/_3C n (h4/list/LENGTH l) ==> (!f. happ (h4/list/EL n) (h4/list/MAP f l) = happ f (happ (h4/list/EL n) l))
% Assm [h4s_lists_LENGTH0u_c1]: !t h. h4/list/LENGTH (h4/list/CONS h t) = h4/num/SUC (h4/list/LENGTH t)
% Assm [h4s_lists_LENGTH0u_c0]: h4/list/LENGTH h4/list/NIL = h4/num/0
% Assm [h4s_lists_HD0]: !t h. happ h4/list/HD (h4/list/CONS h t) = h
% Assm [h4s_finiteu_u_maps_FAPPLYu_u_DEF]: !x f. h4/finite__map/FAPPLY f x = h4/sum/OUTL (happ (h4/finite__map/fmap__REP f) x)
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_arithmetics_numu_u_CASES]: !m. m = h4/num/0 \/ (?n. m = h4/num/SUC n)
% Assm [h4s_arithmetics_LESSu_u_MONOu_u_EQ]: !n m. h4/prim__rec/_3C (h4/num/SUC m) (h4/num/SUC n) <=> h4/prim__rec/_3C m n
% Assm [h4s_pairs_SND0]: !y x. happ h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_primu_u_recs_LESSu_u_0]: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm [h4s_primu_u_recs_NOTu_u_LESSu_u_0]: !n. ~h4/prim__rec/_3C n h4/num/0
% Assm [h4s_sums_OUTL0]: !x. h4/sum/OUTL (h4/sum/INL x) = x
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_pairs_EXISTSu_u_PROD]: !P. (?p. happ P p) <=> (?p__1 p__2. happ P (h4/pair/_2C p__1 p__2))
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% 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_INu_u_COMPL]: !x s. h4/bool/IN x (h4/pred__set/COMPL s) <=> ~h4/bool/IN x s
% Goal: !_2. (!env k v. happ (happ (happ _2 env) k) v = happ (happ h4/finite__map/FUPDATE env) (h4/pair/_2C k v)) ==> (!_1. (!env k. happ (happ _1 env) k = happ (happ _2 env) k) ==> (!_0. (!env. happ _0 env = h4/pair/UNCURRY (happ _1 env)) ==> (!m l. happ (happ (h4/list/FOLDL _0) m) l = happ (happ h4/finite__map/FUPDATE__LIST m) 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_Q1155406,TV_Q1155402]: ![V_f, V_g]: (![V_x]: s(TV_Q1155402,happ(s(t_fun(TV_Q1155406,TV_Q1155402),V_f),s(TV_Q1155406,V_x))) = s(TV_Q1155402,happ(s(t_fun(TV_Q1155406,TV_Q1155402),V_g),s(TV_Q1155406,V_x))) => s(t_fun(TV_Q1155406,TV_Q1155402),V_f) = s(t_fun(TV_Q1155406,TV_Q1155402),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),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),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),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),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),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),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),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),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_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_finiteu_u_maps_FUPDATEu_u_LISTu_u_SNOC, axiom, ![TV_u_27a,TV_u_27b]: ![V_xs, V_x, V_fm]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),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_snoc(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_xs))))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),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_xs))))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x)))).
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_finiteu_u_maps_fupdateu_u_listu_u_foldr, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_2]: (![V_k, V_v, V_env]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)))),V_uu_2),s(TV_u_27a,V_k))),s(TV_u_27b,V_v))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_env))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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_env))),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))))) => ![V_uu_1]: (![V_k, V_v]: s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)))),V_uu_1),s(TV_u_27a,V_k))),s(TV_u_27b,V_v))) = s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)))),V_uu_2),s(TV_u_27a,V_k))),s(TV_u_27b,V_v))) => ![V_uu_0]: (![V_k]: s(t_fun(TV_u_27b,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)))),V_uu_0),s(TV_u_27a,V_k))) = s(t_fun(TV_u_27b,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)))),V_uu_1),s(TV_u_27a,V_k))) => ![V_m, V_l]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_lists_foldr(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b))),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)))),V_uu_0))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_m),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),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_m))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),h4s_lists_reverse(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l))))))))).
fof(ah4s_finiteu_u_maps_FUPDATEu_u_LISTu_u_APPLYu_u_MEM, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_n, V_kvl, V_k, V_f]: ((p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_lists_length(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_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_el(s(t_h4s_nums_num,V_n))),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,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27b),h4s_lists_el(s(t_h4s_nums_num,V_n))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_kvl))))) & ![V_m]: ((p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))) & p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_kvl))))))) => ~ (s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_el(s(t_h4s_nums_num,V_m))),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_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),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),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_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),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),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_INu_u_FRANGEu_u_FUPDATEu_u_LISTu_u_suff, axiom, ![TV_u_27b,TV_u_27a]: ![V_ls, V_fm, V_P]: ((![V_v]: (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_fm)))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_v))))) & ![V_v]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_v),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_27b,TV_u_27a),TV_u_27a),h4s_pairs_snd),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_ls)))))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_v)))))) => ![V_v]: (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),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a))),h4s_finiteu_u_maps_fupdateu_u_list),s(t_h4s_finiteu_u_maps_fmap(TV_u_27b,TV_u_27a),V_fm))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27b,TV_u_27a)),V_ls)))))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,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),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),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),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),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),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),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_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),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),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),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),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_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_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_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),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),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),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),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),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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_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),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),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_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_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
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_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_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_pairs_ABSu_u_PAIRu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: ?[V_q, V_r]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_q),s(TV_u_27b,V_r)))).
fof(ah4s_bools_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_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_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_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_SNOC0u_c0, axiom, ![TV_u_27a]: ![V_x]: 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),h4s_lists_nil))) = 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_SNOC0u_c1, axiom, ![TV_u_27a]: ![V_xu_27, 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),h4s_lists_cons(s(TV_u_27a,V_xu_27),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_xu_27),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)))))).
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_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_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_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,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_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_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
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),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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_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_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_richu_u_lists_FOLDLu_u_APPEND, axiom, ![TV_u_27a,TV_u_27b]: ![V_l2, V_l1, V_f, V_e]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),h4s_lists_foldl(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_f))),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_append(s(t_h4s_lists_list(TV_u_27b),V_l1),s(t_h4s_lists_list(TV_u_27b),V_l2))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),h4s_lists_foldl(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_f))),s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),h4s_lists_foldl(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_f))),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27b),V_l1))))),s(t_h4s_lists_list(TV_u_27b),V_l2)))).
fof(ah4s_lists_REVERSEu_u_DEFu_c1, axiom, ![TV_u_27a]: ![V_t, V_h]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_reverse(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_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),h4s_lists_reverse(s(t_h4s_lists_list(TV_u_27a),V_t))),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),h4s_lists_nil)))))).
fof(ah4s_lists_REVERSEu_u_DEFu_c0, axiom, ![TV_u_27a]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_reverse(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)).
fof(ah4s_lists_FOLDL0u_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_l, V_f, V_e]: s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),h4s_lists_foldl(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),V_f))),s(TV_u_27b,V_e))),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),V_l))))) = s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),h4s_lists_foldl(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),V_f))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27b,V_e))),s(TV_u_27a,V_x))))),s(t_h4s_lists_list(TV_u_27a),V_l)))).
fof(ah4s_lists_FOLDR0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_e]: s(TV_u_27b,h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(TV_u_27b,V_e)).
fof(ah4s_lists_FOLDR0u_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_l, V_f, V_e]: s(TV_u_27b,h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27b,V_e),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),V_l))))) = s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_lists_FOLDL0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_e]: s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),h4s_lists_foldl(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),V_f))),s(TV_u_27b,V_e))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(TV_u_27b,V_e)).
fof(ah4s_pairs_pairu_u_CASES, 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_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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_finiteu_u_maps_FUPDATEu_u_LIST0, axiom, ![TV_u_27a,TV_u_27b]: s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b))),h4s_finiteu_u_maps_fupdateu_u_list) = s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b))),h4s_lists_foldl(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b))),h4s_finiteu_u_maps_fupdate)))).
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),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),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),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),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),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),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_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_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_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_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_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_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_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_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_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_ORu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,f))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_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),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),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_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_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_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_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_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_bools_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_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),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_finiteu_u_maps_FRANGEu_u_FUPDATEu_u_LISTu_u_SUBSET, axiom, ![TV_u_27a,TV_u_27b]: ![V_ls, V_fm]: p(s(t_bool,h4s_predu_u_sets_subset(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),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),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_ls))))),s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_union(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_fm))),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_map(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_ls))))))))))).
fof(ah4s_predu_u_sets_UNIONu_u_DEF, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_s, V_t, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (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)))))) & s(t_h4s_pairs_prod(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(TV_u_27a,t_bool),V_t))),s(TV_u_27a,V_x))) = s(t_h4s_pairs_prod(TV_u_27a,t_bool),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_bool,V_v)))) => ![V_t, V_s]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_gspec(s(t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_h4s_pairs_prod(TV_u_27a,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_s))),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_predu_u_sets_SUBSETu_u_DEF, axiom, ![TV_u_27a]: ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_subset(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))) <=> ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_predu_u_sets_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_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_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_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_richu_u_lists_CONSu_u_APPEND, axiom, ![TV_u_27a]: ![V_x, 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),V_l))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(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))),s(t_h4s_lists_list(TV_u_27a),V_l)))).
fof(ah4s_finiteu_u_maps_fmapu_u_ISOu_u_DEFu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_r]: (p(s(t_bool,h4s_finiteu_u_maps_isu_u_fmap(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_r)))) <=> s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),h4s_finiteu_u_maps_fmapu_u_rep(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fmapu_u_abs(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_r))))) = s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_r))).
fof(ah4s_finiteu_u_maps_fmapu_u_ISOu_u_DEFu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_a]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fmapu_u_abs(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),h4s_finiteu_u_maps_fmapu_u_rep(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_a))))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_a)).
fof(ah4s_finiteu_u_maps_isu_u_fmapu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: (p(s(t_bool,h4s_finiteu_u_maps_isu_u_fmap(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_x)))) <=> ![V_isu_u_fmapu_27]: (![V_a00]: ((![V_x0]: s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_a00),s(TV_u_27a,V_x0))) = s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),h4s_sums_inr(s(t_h4s_ones_one,h4s_ones_one0))) | ?[V_f, V_a, V_b]: (![V_x0]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(TV_u_27a,V_x0) = s(TV_u_27a,V_a)) & s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_a00),s(TV_u_27a,V_x0))) = s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),h4s_sums_inl(s(TV_u_27b,V_b))),s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_f),s(TV_u_27a,V_x0)))))) & p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),t_bool),V_isu_u_fmapu_27),s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_f)))))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),t_bool),V_isu_u_fmapu_27),s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_a00))))) => p(s(t_bool,happ(s(t_fun(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),t_bool),V_isu_u_fmapu_27),s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),V_x))))))).
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_lists_MEMu_u_EL, axiom, ![TV_u_27a]: ![V_x, V_l]: (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_l)))))) <=> ?[V_n]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l)))))) & s(TV_u_27a,V_x) = s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_el(s(t_h4s_nums_num,V_n))),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_finiteu_u_maps_FUPDATEu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_x, V_y, V_f, V_a]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(TV_u_27a,V_a) = s(TV_u_27a,V_x)) & s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one))),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one))))),V_uu_0),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_a))) = s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),h4s_sums_inl(s(TV_u_27b,V_y))),s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),h4s_finiteu_u_maps_fmapu_u_rep(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_a)))))) => ![V_y, V_x, V_f]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),h4s_finiteu_u_maps_fmapu_u_abs(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one))),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one))))),V_uu_0),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))))))).
fof(ah4s_lists_MEMu_u_MAP, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_l, V_f]: (p(s(t_bool,h4s_bools_in(s(TV_u_27b,V_x),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_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),V_l)))))))) <=> ?[V_y]: (s(TV_u_27b,V_x) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))) & p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l))))))))).
fof(ah4s_lists_ELu_u_restrictedu_c1, axiom, ![TV_u_27b]: ![V_n, V_ls, V_l]: s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27b),h4s_lists_el(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,V_l),s(t_h4s_lists_list(TV_u_27b),V_ls))))) = s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27b),h4s_lists_el(s(t_h4s_nums_num,V_n))),s(t_h4s_lists_list(TV_u_27b),V_ls)))).
fof(ah4s_lists_ELu_u_restrictedu_c0, axiom, ![TV_u_27a]: s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_el(s(t_h4s_nums_num,h4s_nums_0))) = s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_hd)).
fof(ah4s_lists_ELu_u_MAP, axiom, ![TV_u_27b,TV_u_27a]: ![V_n, V_l]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l)))))) => ![V_f]: s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27b),h4s_lists_el(s(t_h4s_nums_num,V_n))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_el(s(t_h4s_nums_num,V_n))),s(t_h4s_lists_list(TV_u_27a),V_l))))))).
fof(ah4s_lists_LENGTH0u_c1, axiom, ![TV_u_27a]: ![V_t, V_h]: s(t_h4s_nums_num,h4s_lists_length(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_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_t)))))).
fof(ah4s_lists_LENGTH0u_c0, axiom, ![TV_u_27a]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_lists_HD0, axiom, ![TV_u_27a]: ![V_t, V_h]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_hd),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_h)).
fof(ah4s_finiteu_u_maps_FAPPLYu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, 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_sums_outl(s(t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,t_h4s_ones_one)),h4s_finiteu_u_maps_fmapu_u_rep(s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_x)))))).
fof(ah4s_bools_NOTu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ![V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_arithmetics_numu_u_CASES, axiom, ![V_m]: (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0) | ?[V_n]: s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_LESSu_u_MONOu_u_EQ, axiom, ![V_n, V_m]: s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27b),h4s_pairs_snd),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_27b,V_y)).
fof(ah4s_primu_u_recs_LESSu_u_0, axiom, ![V_n]: p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))))).
fof(ah4s_primu_u_recs_NOTu_u_LESSu_u_0, axiom, ![V_n]: ~ (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0)))))).
fof(ah4s_sums_OUTL0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_sums_outl(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,f))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c0, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) & p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) | ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_MONOu_u_OR, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) | p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) | p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_AND, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) & p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) & p(s(t_bool,V_w)))))).
fof(ah4s_pairs_EXISTSu_u_PROD, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: (?[V_p]: 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),V_p)))) <=> ?[V_pu_u_1, V_pu_u_2]: 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_pu_u_1),s(TV_u_27b,V_pu_u_2)))))))).
fof(ah4s_bools_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_UNWINDu_u_THM2, axiom, ![TV_u_27a]: ![V_a, V_P]: (?[V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_a)))))).
fof(ah4s_bools_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_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_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_finiteu_u_maps_fupdateu_u_listu_u_foldl, conjecture, ![TV_u_27a,TV_u_27b]: ![V_uu_2]: (![V_env, V_k, V_v]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)))),V_uu_2),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_env))),s(TV_u_27a,V_k))),s(TV_u_27b,V_v))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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_env))),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))))) => ![V_uu_1]: (![V_env, V_k]: s(t_fun(TV_u_27b,t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)))),V_uu_1),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_env))),s(TV_u_27a,V_k))) = s(t_fun(TV_u_27b,t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)))),V_uu_2),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_env))),s(TV_u_27a,V_k))) => ![V_uu_0]: (![V_env]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b))),V_uu_0),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_env))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)))),V_uu_1),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_env))))) => ![V_m, V_l]: s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b))),h4s_lists_foldl(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b))),V_uu_0))),s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),V_m))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l))) = s(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_finiteu_u_maps_fmap(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),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_m))),s(t_h4s_lists_list(t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),V_l))))))).
