%   ORIGINAL: h4/sorting/FOLDR__PERM
% 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/operator/ASSOC__DEF: !f. h4/operator/ASSOC f <=> (!x y z. f x (f y z) = f (f x y) z)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/operator/COMM__DEF: !f. h4/operator/COMM f <=> (!x y. f x y = f y x)
% Assm: h4/rich__list/COMM__ASSOC__FOLDR__REVERSE: !f. h4/operator/COMM f ==> h4/operator/ASSOC f ==> (!e l. h4/list/FOLDR f e (h4/list/REVERSE l) = h4/list/FOLDR f e l)
% Assm: h4/list/FOLDL__EQ__FOLDR: !l f e. h4/operator/ASSOC f /\ h4/operator/COMM f ==> h4/list/FOLDL f e l = h4/list/FOLDR f e l
% Assm: h4/list/list__INDUCT: !P. P h4/list/NIL /\ (!t. P t ==> (!h. P (h4/list/CONS h t))) ==> (!l. P l)
% Assm: h4/rich__list/COMM__MONOID__FOLDR: !f. h4/operator/COMM f ==> (!e_27. h4/operator/MONOID f e_27 ==> (!e l. h4/list/FOLDR f e l = f e (h4/list/FOLDR f e_27 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/rich__list/COMM__ASSOC__FOLDL__REVERSE: !f. h4/operator/COMM f ==> h4/operator/ASSOC f ==> (!e l. h4/list/FOLDL f e (h4/list/REVERSE l) = h4/list/FOLDL f e l)
% Assm: h4/operator/MONOID__DEF: !f e. h4/operator/MONOID f e <=> h4/operator/ASSOC f /\ h4/operator/RIGHT__ID f e /\ h4/operator/LEFT__ID f e
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/list/FOLDL0_c0: !f e. h4/list/FOLDL f e h4/list/NIL = e
% 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/rich__list/COMM__MONOID__FOLDL: !f. h4/operator/COMM f ==> (!e_27. h4/operator/MONOID f e_27 ==> (!e l. h4/list/FOLDL f e l = f e (h4/list/FOLDL f e_27 l)))
% Assm: h4/operator/FCOMM__ASSOC: !f. h4/operator/FCOMM f f <=> h4/operator/ASSOC f
% Assm: h4/list/REVERSE__SNOC__DEF_c0: h4/list/REVERSE h4/list/NIL = h4/list/NIL
% Assm: h4/operator/LEFT__ID__DEF: !f e. h4/operator/LEFT__ID f e <=> (!x. f e x = x)
% Assm: h4/rich__list/FOLDR__FOLDL: !f e. h4/operator/MONOID f e ==> (!l. h4/list/FOLDR f e l = h4/list/FOLDL f e l)
% Assm: h4/rich__list/ASSOC__FOLDR__FLAT: !f. h4/operator/ASSOC f ==> (!e. h4/operator/LEFT__ID f e ==> (!l. h4/list/FOLDR f e (h4/list/FLAT l) = h4/list/FOLDR f e (h4/list/MAP (h4/list/FOLDR f e) l)))
% Assm: h4/rich__list/ASSOC__FOLDL__FLAT: !f. h4/operator/ASSOC f ==> (!e. h4/operator/RIGHT__ID f e ==> (!l. h4/list/FOLDL f e (h4/list/FLAT l) = h4/list/FOLDL f e (h4/list/MAP (h4/list/FOLDL f e) l)))
% Assm: h4/operator/ASSOC__SYM: !f. h4/operator/ASSOC f <=> (!x y z. f (f x y) z = f x (f y z))
% 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/FOLDL__SNOC: !x l f e. h4/list/FOLDL f e (h4/list/SNOC x l) = f (h4/list/FOLDL f e l) x
% Assm: h4/list/REVERSE__SNOC__DEF_c1: !x l. h4/list/REVERSE (h4/list/CONS x l) = h4/list/SNOC x (h4/list/REVERSE l)
% Assm: h4/rich__list/FOLDR__SNOC: !x l f e. h4/list/FOLDR f e (h4/list/SNOC x l) = h4/list/FOLDR f (f x e) l
% 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/RIGHT__FORALL__IMP__THM: !Q P. (!x. P ==> Q x) <=> P ==> (!x. Q x)
% Assm: h4/operator/FCOMM__DEF: !g f. h4/operator/FCOMM f g <=> (!x y z. g x (f y z) = f (g x y) z)
% Assm: h4/rich__list/FCOMM__FOLDR__APPEND: !g f. h4/operator/FCOMM g f ==> (!e. h4/operator/LEFT__ID g e ==> (!l1 l2. h4/list/FOLDR f e (h4/list/APPEND l1 l2) = g (h4/list/FOLDR f e l1) (h4/list/FOLDR f e l2)))
% Assm: h4/list/REVERSE__SNOC: !x l. h4/list/REVERSE (h4/list/SNOC x l) = h4/list/CONS x (h4/list/REVERSE l)
% Assm: h4/list/MAP0_c0: !f. h4/list/MAP f h4/list/NIL = h4/list/NIL
% Assm: h4/list/FLAT0_c0: h4/list/FLAT h4/list/NIL = h4/list/NIL
% Assm: h4/rich__list/ASSOC__APPEND: h4/operator/ASSOC h4/list/APPEND
% Assm: h4/operator/ASSOC__CONJ: h4/operator/ASSOC $and
% Assm: h4/operator/ASSOC__DISJ: h4/operator/ASSOC $or
% Assm: h4/list/MAP0_c1: !t h f. h4/list/MAP f (h4/list/CONS h t) = h4/list/CONS (f h) (h4/list/MAP f t)
% Assm: h4/rich__list/FOLDR__FILTER__REVERSE: !f. (!a b c. f a (f b c) = f b (f a c)) ==> (!e P l. h4/list/FOLDR f e (h4/list/FILTER P (h4/list/REVERSE l)) = h4/list/FOLDR f e (h4/list/FILTER P l))
% Assm: h4/rich__list/FCOMM__FOLDR__FLAT: !g f. h4/operator/FCOMM g f ==> (!e. h4/operator/LEFT__ID g e ==> (!l. h4/list/FOLDR f e (h4/list/FLAT l) = h4/list/FOLDR g e (h4/list/MAP (h4/list/FOLDR f e) l)))
% Assm: h4/rich__list/FOLDR__MAP__REVERSE: !f. (!a b c. f a (f b c) = f b (f a c)) ==> (!e g l. h4/list/FOLDR f e (h4/list/MAP g (h4/list/REVERSE l)) = h4/list/FOLDR f e (h4/list/MAP g l))
% Assm: h4/list/list__induction: !P. P h4/list/NIL /\ (!t. P t ==> (!h. P (h4/list/CONS h t))) ==> (!l. P l)
% Assm: h4/list/MAP__SNOC: !x l f. h4/list/MAP f (h4/list/SNOC x l) = h4/list/SNOC (f x) (h4/list/MAP f l)
% Assm: h4/list/FLAT0_c1: !t h. h4/list/FLAT (h4/list/CONS h t) = h4/list/APPEND h (h4/list/FLAT t)
% Assm: h4/operator/RIGHT__ID__DEF: !f e. h4/operator/RIGHT__ID f e <=> (!x. f x e = x)
% Assm: h4/sorting/PERM__DEF: !L2 L1. h4/sorting/PERM L1 L2 <=> (!x. h4/list/FILTER ($equals x) L1 = h4/list/FILTER ($equals x) L2)
% Assm: h4/rich__list/FCOMM__FOLDL__APPEND: !g f. h4/operator/FCOMM f g ==> (!e. h4/operator/RIGHT__ID g e ==> (!l1 l2. h4/list/FOLDL f e (h4/list/APPEND l1 l2) = g (h4/list/FOLDL f e l1) (h4/list/FOLDL f e l2)))
% Assm: h4/rich__list/FLAT__SNOC: !x l. h4/list/FLAT (h4/list/SNOC x l) = h4/list/APPEND (h4/list/FLAT l) x
% Assm: h4/sorting/PERM__CONS__IFF: !x l2 l1. h4/sorting/PERM (h4/list/CONS x l1) (h4/list/CONS x l2) <=> h4/sorting/PERM l1 l2
% Assm: h4/sorting/PERM__ALL__DISTINCT: !l2 l1. h4/list/ALL__DISTINCT l1 /\ h4/list/ALL__DISTINCT l2 /\ (!x. h4/bool/IN x (h4/list/LIST__TO__SET l1) <=> h4/bool/IN x (h4/list/LIST__TO__SET l2)) ==> h4/sorting/PERM l1 l2
% Assm: h4/sorting/PERM__SWAP__AT__FRONT: !y x l2 l1. h4/sorting/PERM (h4/list/CONS x (h4/list/CONS y l1)) (h4/list/CONS y (h4/list/CONS x l2)) <=> h4/sorting/PERM l1 l2
% Assm: h4/list/APPEND__ASSOC: !l3 l2 l1. h4/list/APPEND l1 (h4/list/APPEND l2 l3) = h4/list/APPEND (h4/list/APPEND l1 l2) l3
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/list/FILTER0_c1: !t h P. h4/list/FILTER P (h4/list/CONS h t) = h4/bool/COND (P h) (h4/list/CONS h (h4/list/FILTER P t)) (h4/list/FILTER P t)
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/list/APPEND0_c1: !l2 l1 h. h4/list/APPEND (h4/list/CONS h l1) l2 = h4/list/CONS h (h4/list/APPEND l1 l2)
% Assm: h4/list/APPEND0_c0: !l. h4/list/APPEND h4/list/NIL l = l
% Assm: h4/list/FILTER0_c0: !P. h4/list/FILTER P h4/list/NIL = h4/list/NIL
% Assm: h4/rich__list/FILTER__SNOC: !x l P. h4/list/FILTER P (h4/list/SNOC x l) = h4/bool/COND (P x) (h4/list/SNOC x (h4/list/FILTER P l)) (h4/list/FILTER P l)
% 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/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/sorting/PERM__MONO: !x l2 l1. h4/sorting/PERM l1 l2 ==> h4/sorting/PERM (h4/list/CONS x l1) (h4/list/CONS x l2)
% Assm: h4/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/list/list__11: !a1_27 a1 a0_27 a0. h4/list/CONS a0 a1 = h4/list/CONS a0_27 a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/list/NOT__NIL__CONS: !a1 a0. ~(h4/list/NIL = h4/list/CONS a0 a1)
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% 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/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> 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/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/list/ALL__DISTINCT__FILTER: !l. h4/list/ALL__DISTINCT l <=> (!x. h4/bool/IN x (h4/list/LIST__TO__SET l) ==> h4/list/FILTER ($equals x) l = h4/list/CONS x h4/list/NIL)
% 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/sorting/PERM__NIL_c1: !L. h4/sorting/PERM h4/list/NIL L <=> L = h4/list/NIL
% Assm: h4/sorting/PERM__TRANS: !z y x. h4/sorting/PERM x y /\ h4/sorting/PERM y z ==> h4/sorting/PERM x z
% Assm: h4/sorting/PERM__REFL: !L. h4/sorting/PERM L L
% Assm: h4/sorting/PERM__CONS__EQ__APPEND: !t h L. h4/sorting/PERM (h4/list/CONS h t) L <=> (?M N. L = h4/list/APPEND M (h4/list/CONS h N) /\ h4/sorting/PERM t (h4/list/APPEND M N))
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> t
% Assm: h4/list/APPEND__eq__NIL_c0: !l2 l1. h4/list/NIL = h4/list/APPEND l1 l2 <=> l1 = h4/list/NIL /\ l2 = h4/list/NIL
% Assm: h4/list/CONS__11: !a1_27 a1 a0_27 a0. h4/list/CONS a0 a1 = h4/list/CONS a0_27 a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm: h4/list/list__nchotomy: !l. l = h4/list/NIL \/ (?h t. l = h4/list/CONS h t)
% Assm: h4/bool/FORALL__AND__THM: !Q P. (!x. P x /\ Q x) <=> (!x. P x) /\ (!x. Q x)
% Assm: h4/list/APPEND__11__LENGTH_c0: !l2_27 l2 l1_27 l1. h4/list/LENGTH l1 = h4/list/LENGTH l1_27 ==> (h4/list/APPEND l1 l2 = h4/list/APPEND l1_27 l2_27 <=> l1 = l1_27 /\ l2 = l2_27)
% Assm: h4/bool/LEFT__AND__FORALL__THM: !Q P. (!x. P x) /\ Q <=> (!x. P x /\ Q)
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/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/bool/LEFT__FORALL__OR__THM: !Q P. (!x. P x \/ Q) <=> (!x. P x) \/ Q
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/prim__rec/INV__SUC__EQ: !n m. h4/num/SUC m = h4/num/SUC n <=> m = n
% Assm: h4/marker/label__def: !lab argument. h4/marker/_3A_2D lab argument <=> argument
% Assm: h4/bool/DISJ__COMM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/UNWIND__FORALL__THM2: !v f. (!x. x = v ==> f x) <=> f v
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/bool/EXISTS__REFL: !a. ?x. x = a
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Goal: !l2 l1 f e. h4/operator/ASSOC f /\ h4/operator/COMM f ==> h4/sorting/PERM l1 l2 ==> h4/list/FOLDR f e l1 = h4/list/FOLDR f e l2
%   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_operators_ASSOCu_u_DEF]: !f. h4/operator/ASSOC f <=> (!x y z. happ (happ f x) (happ (happ f y) z) = happ (happ f (happ (happ f x) y)) z)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_operators_COMMu_u_DEF]: !f. h4/operator/COMM f <=> (!x y. happ (happ f x) y = happ (happ f y) x)
% Assm [h4s_richu_u_lists_COMMu_u_ASSOCu_u_FOLDRu_u_REVERSE]: !f. h4/operator/COMM f ==> h4/operator/ASSOC f ==> (!e l. happ (h4/list/FOLDR f e) (h4/list/REVERSE l) = happ (h4/list/FOLDR f e) l)
% Assm [h4s_lists_FOLDLu_u_EQu_u_FOLDR]: !l f e. h4/operator/ASSOC f /\ h4/operator/COMM f ==> happ (h4/list/FOLDL f e) l = happ (h4/list/FOLDR f e) l
% Assm [h4s_lists_listu_u_INDUCT]: !P. happ P h4/list/NIL /\ (!t. happ P t ==> (!h. happ P (h4/list/CONS h t))) ==> (!l. happ P l)
% Assm [h4s_richu_u_lists_COMMu_u_MONOIDu_u_FOLDR]: !f. h4/operator/COMM f ==> (!e_27. h4/operator/MONOID f e_27 ==> (!e l. happ (h4/list/FOLDR f e) l = happ (happ f e) (happ (h4/list/FOLDR f e_27) l)))
% Assm [h4s_lists_FOLDR0u_c0]: !f e. happ (h4/list/FOLDR f e) h4/list/NIL = e
% Assm [h4s_lists_FOLDR0u_c1]: !x l f e. happ (h4/list/FOLDR f e) (h4/list/CONS x l) = happ (happ f x) (happ (h4/list/FOLDR f e) l)
% Assm [h4s_richu_u_lists_COMMu_u_ASSOCu_u_FOLDLu_u_REVERSE]: !f. h4/operator/COMM f ==> h4/operator/ASSOC f ==> (!e l. happ (h4/list/FOLDL f e) (h4/list/REVERSE l) = happ (h4/list/FOLDL f e) l)
% Assm [h4s_operators_MONOIDu_u_DEF]: !f e. h4/operator/MONOID f e <=> h4/operator/ASSOC f /\ h4/operator/RIGHT__ID f e /\ h4/operator/LEFT__ID f e
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_lists_FOLDL0u_c0]: !f e. happ (h4/list/FOLDL f e) h4/list/NIL = e
% Assm [h4s_lists_FOLDL0u_c1]: !x l f e. happ (h4/list/FOLDL f e) (h4/list/CONS x l) = happ (h4/list/FOLDL f (happ (happ f e) x)) l
% Assm [h4s_richu_u_lists_COMMu_u_MONOIDu_u_FOLDL]: !f. h4/operator/COMM f ==> (!e_27. h4/operator/MONOID f e_27 ==> (!e l. happ (h4/list/FOLDL f e) l = happ (happ f e) (happ (h4/list/FOLDL f e_27) l)))
% Assm [h4s_operators_FCOMMu_u_ASSOC]: !f. h4/operator/FCOMM f f <=> h4/operator/ASSOC f
% Assm [h4s_lists_REVERSEu_u_SNOCu_u_DEFu_c0]: h4/list/REVERSE h4/list/NIL = h4/list/NIL
% Assm [h4s_operators_LEFTu_u_IDu_u_DEF]: !f e. h4/operator/LEFT__ID f e <=> (!x. happ (happ f e) x = x)
% Assm [h4s_richu_u_lists_FOLDRu_u_FOLDL]: !f e. h4/operator/MONOID f e ==> (!l. happ (h4/list/FOLDR f e) l = happ (h4/list/FOLDL f e) l)
% Assm [h4s_richu_u_lists_ASSOCu_u_FOLDRu_u_FLAT]: !f. h4/operator/ASSOC f ==> (!e. h4/operator/LEFT__ID f e ==> (!l. happ (h4/list/FOLDR f e) (h4/list/FLAT l) = happ (h4/list/FOLDR f e) (h4/list/MAP (h4/list/FOLDR f e) l)))
% Assm [h4s_richu_u_lists_ASSOCu_u_FOLDLu_u_FLAT]: !f. h4/operator/ASSOC f ==> (!e. h4/operator/RIGHT__ID f e ==> (!l. happ (h4/list/FOLDL f e) (h4/list/FLAT l) = happ (h4/list/FOLDL f e) (h4/list/MAP (h4/list/FOLDL f e) l)))
% Assm [h4s_operators_ASSOCu_u_SYM]: !f. h4/operator/ASSOC f <=> (!x y z. happ (happ f (happ (happ f x) y)) z = happ (happ f x) (happ (happ f y) z))
% 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_FOLDLu_u_SNOC]: !x l f e. happ (h4/list/FOLDL f e) (h4/list/SNOC x l) = happ (happ f (happ (h4/list/FOLDL f e) l)) x
% Assm [h4s_lists_REVERSEu_u_SNOCu_u_DEFu_c1]: !x l. h4/list/REVERSE (h4/list/CONS x l) = h4/list/SNOC x (h4/list/REVERSE l)
% Assm [h4s_richu_u_lists_FOLDRu_u_SNOC]: !x l f e. happ (h4/list/FOLDR f e) (h4/list/SNOC x l) = happ (h4/list/FOLDR f (happ (happ f x) e)) l
% 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_RIGHTu_u_FORALLu_u_IMPu_u_THM]: !Q P. (!x. P ==> happ Q x) <=> P ==> (!x. happ Q x)
% Assm [h4s_operators_FCOMMu_u_DEF]: !g f. h4/operator/FCOMM f g <=> (!x y z. happ (happ g x) (happ (happ f y) z) = happ (happ f (happ (happ g x) y)) z)
% Assm [h4s_richu_u_lists_FCOMMu_u_FOLDRu_u_APPEND]: !g f. h4/operator/FCOMM g f ==> (!e. h4/operator/LEFT__ID g e ==> (!l1 l2. happ (h4/list/FOLDR f e) (happ (happ h4/list/APPEND l1) l2) = happ (happ g (happ (h4/list/FOLDR f e) l1)) (happ (h4/list/FOLDR f e) l2)))
% Assm [h4s_lists_REVERSEu_u_SNOC]: !x l. h4/list/REVERSE (h4/list/SNOC x l) = h4/list/CONS x (h4/list/REVERSE l)
% Assm [h4s_lists_MAP0u_c0]: !f. h4/list/MAP f h4/list/NIL = h4/list/NIL
% Assm [h4s_lists_FLAT0u_c0]: h4/list/FLAT h4/list/NIL = h4/list/NIL
% Assm [h4s_richu_u_lists_ASSOCu_u_APPEND]: h4/operator/ASSOC h4/list/APPEND
% Assm [h4s_operators_ASSOCu_u_CONJ]: h4/operator/ASSOC $and
% Assm [h4s_operators_ASSOCu_u_DISJ]: h4/operator/ASSOC $or
% Assm [h4s_lists_MAP0u_c1]: !t h f. h4/list/MAP f (h4/list/CONS h t) = h4/list/CONS (happ f h) (h4/list/MAP f t)
% Assm [h4s_richu_u_lists_FOLDRu_u_FILTERu_u_REVERSE]: !f. (!a b c. happ (happ f a) (happ (happ f b) c) = happ (happ f b) (happ (happ f a) c)) ==> (!e P l. happ (h4/list/FOLDR f e) (h4/list/FILTER P (h4/list/REVERSE l)) = happ (h4/list/FOLDR f e) (h4/list/FILTER P l))
% Assm [h4s_richu_u_lists_FCOMMu_u_FOLDRu_u_FLAT]: !g f. h4/operator/FCOMM g f ==> (!e. h4/operator/LEFT__ID g e ==> (!l. happ (h4/list/FOLDR f e) (h4/list/FLAT l) = happ (h4/list/FOLDR g e) (h4/list/MAP (h4/list/FOLDR f e) l)))
% Assm [h4s_richu_u_lists_FOLDRu_u_MAPu_u_REVERSE]: !f. (!a b c. happ (happ f a) (happ (happ f b) c) = happ (happ f b) (happ (happ f a) c)) ==> (!e g l. happ (h4/list/FOLDR f e) (h4/list/MAP g (h4/list/REVERSE l)) = happ (h4/list/FOLDR f e) (h4/list/MAP g l))
% Assm [h4s_lists_listu_u_induction]: !P. happ P h4/list/NIL /\ (!t. happ P t ==> (!h. happ P (h4/list/CONS h t))) ==> (!l. happ P l)
% Assm [h4s_lists_MAPu_u_SNOC]: !x l f. h4/list/MAP f (h4/list/SNOC x l) = h4/list/SNOC (happ f x) (h4/list/MAP f l)
% Assm [h4s_lists_FLAT0u_c1]: !t h. h4/list/FLAT (h4/list/CONS h t) = happ (happ h4/list/APPEND h) (h4/list/FLAT t)
% Assm [h4s_operators_RIGHTu_u_IDu_u_DEF]: !f e. h4/operator/RIGHT__ID f e <=> (!x. happ (happ f x) e = x)
% Assm [h4s_sortings_PERMu_u_DEF]: !L2 L1. h4/sorting/PERM L1 L2 <=> (!x. h4/list/FILTER ($equals x) L1 = h4/list/FILTER ($equals x) L2)
% Assm [h4s_richu_u_lists_FCOMMu_u_FOLDLu_u_APPEND]: !g f. h4/operator/FCOMM f g ==> (!e. h4/operator/RIGHT__ID g e ==> (!l1 l2. happ (h4/list/FOLDL f e) (happ (happ h4/list/APPEND l1) l2) = happ (happ g (happ (h4/list/FOLDL f e) l1)) (happ (h4/list/FOLDL f e) l2)))
% Assm [h4s_richu_u_lists_FLATu_u_SNOC]: !x l. h4/list/FLAT (h4/list/SNOC x l) = happ (happ h4/list/APPEND (h4/list/FLAT l)) x
% Assm [h4s_sortings_PERMu_u_CONSu_u_IFF]: !x l2 l1. h4/sorting/PERM (h4/list/CONS x l1) (h4/list/CONS x l2) <=> h4/sorting/PERM l1 l2
% Assm [h4s_sortings_PERMu_u_ALLu_u_DISTINCT]: !l2 l1. h4/list/ALL__DISTINCT l1 /\ h4/list/ALL__DISTINCT l2 /\ (!x. h4/bool/IN x (h4/list/LIST__TO__SET l1) <=> h4/bool/IN x (h4/list/LIST__TO__SET l2)) ==> h4/sorting/PERM l1 l2
% Assm [h4s_sortings_PERMu_u_SWAPu_u_ATu_u_FRONT]: !y x l2 l1. h4/sorting/PERM (h4/list/CONS x (h4/list/CONS y l1)) (h4/list/CONS y (h4/list/CONS x l2)) <=> h4/sorting/PERM l1 l2
% Assm [h4s_lists_APPENDu_u_ASSOC]: !l3 l2 l1. happ (happ h4/list/APPEND l1) (happ (happ h4/list/APPEND l2) l3) = happ (happ h4/list/APPEND (happ (happ h4/list/APPEND l1) l2)) l3
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_lists_FILTER0u_c1]: !t h P. h4/list/FILTER P (h4/list/CONS h t) = h4/bool/COND (happ P h) (h4/list/CONS h (h4/list/FILTER P t)) (h4/list/FILTER P t)
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_lists_APPEND0u_c1]: !l2 l1 h. happ (happ h4/list/APPEND (h4/list/CONS h l1)) l2 = h4/list/CONS h (happ (happ h4/list/APPEND l1) l2)
% Assm [h4s_lists_APPEND0u_c0]: !l. happ (happ h4/list/APPEND h4/list/NIL) l = l
% Assm [h4s_lists_FILTER0u_c0]: !P. h4/list/FILTER P h4/list/NIL = h4/list/NIL
% Assm [h4s_richu_u_lists_FILTERu_u_SNOC]: !x l P. h4/list/FILTER P (h4/list/SNOC x l) = h4/bool/COND (happ P x) (h4/list/SNOC x (h4/list/FILTER P l)) (h4/list/FILTER P l)
% 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_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_sortings_PERMu_u_MONO]: !x l2 l1. h4/sorting/PERM l1 l2 ==> h4/sorting/PERM (h4/list/CONS x l1) (h4/list/CONS x l2)
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_lists_listu_u_11]: !a1_27 a1 a0_27 a0. h4/list/CONS a0 a1 = h4/list/CONS a0_27 a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_lists_NOTu_u_NILu_u_CONS]: !a1 a0. ~(h4/list/NIL = h4/list/CONS a0 a1)
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% 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_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> 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_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ 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_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_lists_ALLu_u_DISTINCTu_u_FILTER]: !l. h4/list/ALL__DISTINCT l <=> (!x. h4/bool/IN x (h4/list/LIST__TO__SET l) ==> h4/list/FILTER ($equals x) l = h4/list/CONS x h4/list/NIL)
% 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_sortings_PERMu_u_NILu_c1]: !L. h4/sorting/PERM h4/list/NIL L <=> L = h4/list/NIL
% Assm [h4s_sortings_PERMu_u_TRANS]: !z y x. h4/sorting/PERM x y /\ h4/sorting/PERM y z ==> h4/sorting/PERM x z
% Assm [h4s_sortings_PERMu_u_REFL]: !L. h4/sorting/PERM L L
% Assm [h4s_sortings_PERMu_u_CONSu_u_EQu_u_APPEND]: !t h L. h4/sorting/PERM (h4/list/CONS h t) L <=> (?M N. L = happ (happ h4/list/APPEND M) (h4/list/CONS h N) /\ h4/sorting/PERM t (happ (happ h4/list/APPEND M) N))
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> t
% Assm [h4s_lists_APPENDu_u_equ_u_NILu_c0]: !l2 l1. h4/list/NIL = happ (happ h4/list/APPEND l1) l2 <=> l1 = h4/list/NIL /\ l2 = h4/list/NIL
% Assm [h4s_lists_CONSu_u_11]: !a1_27 a1 a0_27 a0. h4/list/CONS a0 a1 = h4/list/CONS a0_27 a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm [h4s_lists_listu_u_nchotomy]: !l. l = h4/list/NIL \/ (?h t. l = h4/list/CONS h t)
% Assm [h4s_bools_FORALLu_u_ANDu_u_THM]: !Q P. (!x. happ P x /\ happ Q x) <=> (!x. happ P x) /\ (!x. happ Q x)
% Assm [h4s_lists_APPENDu_u_11u_u_LENGTHu_c0]: !l2_27 l2 l1_27 l1. h4/list/LENGTH l1 = h4/list/LENGTH l1_27 ==> (happ (happ h4/list/APPEND l1) l2 = happ (happ h4/list/APPEND l1_27) l2_27 <=> l1 = l1_27 /\ l2 = l2_27)
% Assm [h4s_bools_LEFTu_u_ANDu_u_FORALLu_u_THM]: !Q P. (!x. happ P x) /\ Q <=> (!x. happ P x /\ Q)
% Assm [h4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% 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_bools_LEFTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. happ P x \/ Q) <=> (!x. happ P x) \/ Q
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_primu_u_recs_INVu_u_SUCu_u_EQ]: !n m. h4/num/SUC m = h4/num/SUC n <=> m = n
% Assm [h4s_markers_labelu_u_def]: !lab argument. h4/marker/_3A_2D lab argument <=> argument
% Assm [h4s_bools_DISJu_u_COMM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_UNWINDu_u_FORALLu_u_THM2]: !v f. (!x. x = v ==> happ f x) <=> happ f v
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_bools_EXISTSu_u_REFL]: !a. ?x. x = a
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Goal: !l2 l1 f e. h4/operator/ASSOC f /\ h4/operator/COMM f ==> h4/sorting/PERM l1 l2 ==> happ (h4/list/FOLDR f e) l1 = happ (h4/list/FOLDR f e) l2
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f0)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f0))).
fof(aHLu_EXT, axiom, ![TV_Q1370090,TV_Q1370086]: ![V_f, V_g]: (![V_x]: s(TV_Q1370086,happ(s(t_fun(TV_Q1370090,TV_Q1370086),V_f),s(TV_Q1370090,V_x))) = s(TV_Q1370086,happ(s(t_fun(TV_Q1370090,TV_Q1370086),V_g),s(TV_Q1370090,V_x))) => s(t_fun(TV_Q1370090,TV_Q1370086),V_f) = s(t_fun(TV_Q1370090,TV_Q1370086),V_g))).
fof(ah4s_operators_ASSOCu_u_DEF, axiom, ![TV_u_27a]: ![V_f]: (p(s(t_bool,h4s_operators_assoc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)))) <=> ![V_x, V_y, V_z]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_x))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_y))),s(TV_u_27a,V_z))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))),s(TV_u_27a,V_z))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,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_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_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_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_operators_COMMu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_f]: (p(s(t_bool,h4s_operators_comm(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),V_f)))) <=> ![V_x, V_y]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27a,V_y))),s(TV_u_27a,V_x))))).
fof(ah4s_richu_u_lists_COMMu_u_ASSOCu_u_FOLDRu_u_REVERSE, axiom, ![TV_u_27a]: ![V_f]: (p(s(t_bool,h4s_operators_comm(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)))) => (p(s(t_bool,h4s_operators_assoc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)))) => ![V_e, V_l]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_reverse(s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_lists_FOLDLu_u_EQu_u_FOLDR, axiom, ![TV_u_27a]: ![V_l, V_f, V_e]: ((p(s(t_bool,h4s_operators_assoc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)))) & p(s(t_bool,h4s_operators_comm(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))))) => s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_foldl(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27a),V_l))))).
fof(ah4s_lists_listu_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_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_richu_u_lists_COMMu_u_MONOIDu_u_FOLDR, axiom, ![TV_u_27a]: ![V_f]: (p(s(t_bool,h4s_operators_comm(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)))) => ![V_eu_27]: (p(s(t_bool,h4s_operators_monoid(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_eu_27)))) => ![V_e, V_l]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_eu_27))),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,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),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,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),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,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),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_richu_u_lists_COMMu_u_ASSOCu_u_FOLDLu_u_REVERSE, axiom, ![TV_u_27a]: ![V_f]: (p(s(t_bool,h4s_operators_comm(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)))) => (p(s(t_bool,h4s_operators_assoc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)))) => ![V_e, V_l]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_foldl(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_reverse(s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_foldl(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_operators_MONOIDu_u_DEF, axiom, ![TV_u_27a]: ![V_f, V_e]: (p(s(t_bool,h4s_operators_monoid(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e)))) <=> (p(s(t_bool,h4s_operators_assoc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)))) & (p(s(t_bool,h4s_operators_rightu_u_id(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e)))) & p(s(t_bool,h4s_operators_leftu_u_id(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e)))))))).
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_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),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_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),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),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_richu_u_lists_COMMu_u_MONOIDu_u_FOLDL, axiom, ![TV_u_27a]: ![V_f]: (p(s(t_bool,h4s_operators_comm(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)))) => ![V_eu_27]: (p(s(t_bool,h4s_operators_monoid(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_eu_27)))) => ![V_e, V_l]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_foldl(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_foldl(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_eu_27))),s(t_h4s_lists_list(TV_u_27a),V_l)))))))).
fof(ah4s_operators_FCOMMu_u_ASSOC, axiom, ![TV_u_27a]: ![V_f]: s(t_bool,h4s_operators_fcomm(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))) = s(t_bool,h4s_operators_assoc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)))).
fof(ah4s_lists_REVERSEu_u_SNOCu_u_DEFu_c0, axiom, ![TV_u_27b]: s(t_h4s_lists_list(TV_u_27b),h4s_lists_reverse(s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil)).
fof(ah4s_operators_LEFTu_u_IDu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_e]: (p(s(t_bool,h4s_operators_leftu_u_id(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27a,V_e)))) <=> ![V_x]: 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_e))),s(TV_u_27b,V_x))) = s(TV_u_27b,V_x))).
fof(ah4s_richu_u_lists_FOLDRu_u_FOLDL, axiom, ![TV_u_27a]: ![V_f, V_e]: (p(s(t_bool,h4s_operators_monoid(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e)))) => ![V_l]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_foldl(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27a),V_l))))).
fof(ah4s_richu_u_lists_ASSOCu_u_FOLDRu_u_FLAT, axiom, ![TV_u_27a]: ![V_f]: (p(s(t_bool,h4s_operators_assoc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)))) => ![V_e]: (p(s(t_bool,h4s_operators_leftu_u_id(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e)))) => ![V_l]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_flat(s(t_h4s_lists_list(t_h4s_lists_list(TV_u_27a)),V_l))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(t_h4s_lists_list(t_h4s_lists_list(TV_u_27a)),V_l)))))))).
fof(ah4s_richu_u_lists_ASSOCu_u_FOLDLu_u_FLAT, axiom, ![TV_u_27a]: ![V_f]: (p(s(t_bool,h4s_operators_assoc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)))) => ![V_e]: (p(s(t_bool,h4s_operators_rightu_u_id(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e)))) => ![V_l]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_foldl(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_flat(s(t_h4s_lists_list(t_h4s_lists_list(TV_u_27a)),V_l))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_foldl(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_foldl(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(t_h4s_lists_list(t_h4s_lists_list(TV_u_27a)),V_l)))))))).
fof(ah4s_operators_ASSOCu_u_SYM, axiom, ![TV_u_27a]: ![V_f]: (p(s(t_bool,h4s_operators_assoc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)))) <=> ![V_x, V_y, V_z]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))),s(TV_u_27a,V_z))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_x))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_y))),s(TV_u_27a,V_z))))))).
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_FOLDLu_u_SNOC, 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),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_snoc(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_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,happ(s(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),V_l))))),s(TV_u_27a,V_x)))).
fof(ah4s_lists_REVERSEu_u_SNOCu_u_DEFu_c1, axiom, ![TV_u_27a]: ![V_x, V_l]: 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_x),s(t_h4s_lists_list(TV_u_27a),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_reverse(s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_richu_u_lists_FOLDRu_u_SNOC, 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),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_snoc(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),h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),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,V_e))))),s(t_h4s_lists_list(TV_u_27a),V_l)))).
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_RIGHTu_u_FORALLu_u_IMPu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,V_P)) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (p(s(t_bool,V_P)) => ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_operators_FCOMMu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_g, V_f]: (p(s(t_bool,h4s_operators_fcomm(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_f),s(t_fun(TV_u_27c,t_fun(TV_u_27a,TV_u_27a)),V_g)))) <=> ![V_x, V_y, V_z]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,TV_u_27a)),V_g),s(TV_u_27c,V_x))),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_f),s(TV_u_27a,V_y))),s(TV_u_27b,V_z))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27c,t_fun(TV_u_27a,TV_u_27a)),V_g),s(TV_u_27c,V_x))),s(TV_u_27a,V_y))))),s(TV_u_27b,V_z))))).
fof(ah4s_richu_u_lists_FCOMMu_u_FOLDRu_u_APPEND, axiom, ![TV_u_27a,TV_u_27b]: ![V_g, V_f]: (p(s(t_bool,h4s_operators_fcomm(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_g),s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27a)),V_f)))) => ![V_e]: (p(s(t_bool,h4s_operators_leftu_u_id(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_g),s(TV_u_27a,V_e)))) => ![V_l1, V_l2]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a),h4s_lists_foldr(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),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(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_g),s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a),h4s_lists_foldr(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27b),V_l1))))),s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a),h4s_lists_foldr(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27b),V_l2)))))))).
fof(ah4s_lists_REVERSEu_u_SNOC, axiom, ![TV_u_27a]: ![V_x, V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_reverse(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_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),h4s_lists_reverse(s(t_h4s_lists_list(TV_u_27a),V_l)))))).
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_FLAT0u_c0, axiom, ![TV_u_27a]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_flat(s(t_h4s_lists_list(t_h4s_lists_list(TV_u_27a)),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)).
fof(ah4s_richu_u_lists_ASSOCu_u_APPEND, axiom, ![TV_u_27a]: p(s(t_bool,h4s_operators_assoc(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_append))))).
fof(ah4s_operators_ASSOCu_u_CONJ, axiom, p(s(t_bool,h4s_operators_assoc(s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_and))))).
fof(ah4s_operators_ASSOCu_u_DISJ, axiom, p(s(t_bool,h4s_operators_assoc(s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_or))))).
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_richu_u_lists_FOLDRu_u_FILTERu_u_REVERSE, axiom, ![TV_u_27a]: ![V_f]: (![V_a, V_b, V_c]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_a))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_b))),s(TV_u_27a,V_c))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_b))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_a))),s(TV_u_27a,V_c))))) => ![V_e, V_P, V_l]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_reverse(s(t_h4s_lists_list(TV_u_27a),V_l))))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l))))))).
fof(ah4s_richu_u_lists_FCOMMu_u_FOLDRu_u_FLAT, axiom, ![TV_u_27a,TV_u_27b]: ![V_g, V_f]: (p(s(t_bool,h4s_operators_fcomm(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_g),s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27a)),V_f)))) => ![V_e]: (p(s(t_bool,h4s_operators_leftu_u_id(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_g),s(TV_u_27a,V_e)))) => ![V_l]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a),h4s_lists_foldr(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_flat(s(t_h4s_lists_list(t_h4s_lists_list(TV_u_27b)),V_l))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_g),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a),h4s_lists_foldr(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(t_h4s_lists_list(t_h4s_lists_list(TV_u_27b)),V_l)))))))).
fof(ah4s_richu_u_lists_FOLDRu_u_MAPu_u_REVERSE, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: (![V_a, V_b, V_c]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_a))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_b))),s(TV_u_27a,V_c))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_b))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_a))),s(TV_u_27a,V_c))))) => ![V_e, V_g, V_l]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(TV_u_27b,TV_u_27a),V_g),s(t_h4s_lists_list(TV_u_27b),h4s_lists_reverse(s(t_h4s_lists_list(TV_u_27b),V_l))))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(TV_u_27b,TV_u_27a),V_g),s(t_h4s_lists_list(TV_u_27b),V_l))))))).
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_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_FLAT0u_c1, axiom, ![TV_u_27a]: ![V_t, V_h]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_flat(s(t_h4s_lists_list(t_h4s_lists_list(TV_u_27a)),h4s_lists_cons(s(t_h4s_lists_list(TV_u_27a),V_h),s(t_h4s_lists_list(t_h4s_lists_list(TV_u_27a)),V_t))))) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_append),s(t_h4s_lists_list(TV_u_27a),V_h))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_flat(s(t_h4s_lists_list(t_h4s_lists_list(TV_u_27a)),V_t)))))).
fof(ah4s_operators_RIGHTu_u_IDu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_e]: (p(s(t_bool,h4s_operators_rightu_u_id(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_f),s(TV_u_27b,V_e)))) <=> ![V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_e))) = s(TV_u_27a,V_x))).
fof(ah4s_sortings_PERMu_u_DEF, axiom, ![TV_u_27a]: ![V_L2, V_L1]: (p(s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),V_L1),s(t_h4s_lists_list(TV_u_27a),V_L2)))) <=> ![V_x]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),d_equals(s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_L1))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),d_equals(s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_L2))))).
fof(ah4s_richu_u_lists_FCOMMu_u_FOLDLu_u_APPEND, axiom, ![TV_u_27a,TV_u_27b]: ![V_g, V_f]: (p(s(t_bool,h4s_operators_fcomm(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_g)))) => ![V_e]: (p(s(t_bool,h4s_operators_rightu_u_id(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_g),s(TV_u_27a,V_e)))) => ![V_l1, V_l2]: s(TV_u_27a,happ(s(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),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),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(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_g),s(TV_u_27a,happ(s(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(TV_u_27a,happ(s(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_l2)))))))).
fof(ah4s_richu_u_lists_FLATu_u_SNOC, axiom, ![TV_u_27a]: ![V_x, V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_flat(s(t_h4s_lists_list(t_h4s_lists_list(TV_u_27a)),h4s_lists_snoc(s(t_h4s_lists_list(TV_u_27a),V_x),s(t_h4s_lists_list(t_h4s_lists_list(TV_u_27a)),V_l))))) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_append),s(t_h4s_lists_list(TV_u_27a),h4s_lists_flat(s(t_h4s_lists_list(t_h4s_lists_list(TV_u_27a)),V_l))))),s(t_h4s_lists_list(TV_u_27a),V_x)))).
fof(ah4s_sortings_PERMu_u_CONSu_u_IFF, axiom, ![TV_u_27a]: ![V_x, V_l2, V_l1]: s(t_bool,h4s_sortings_perm(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_l1))),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_l2))))) = s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2)))).
fof(ah4s_sortings_PERMu_u_ALLu_u_DISTINCT, axiom, ![TV_u_27a]: ![V_l2, V_l1]: ((p(s(t_bool,h4s_lists_allu_u_distinct(s(t_h4s_lists_list(TV_u_27a),V_l1)))) & (p(s(t_bool,h4s_lists_allu_u_distinct(s(t_h4s_lists_list(TV_u_27a),V_l2)))) & ![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),V_l1))))) = 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_l2))))))) => p(s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2)))))).
fof(ah4s_sortings_PERMu_u_SWAPu_u_ATu_u_FRONT, axiom, ![TV_u_27a]: ![V_y, V_x, V_l2, V_l1]: s(t_bool,h4s_sortings_perm(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_cons(s(TV_u_27a,V_y),s(t_h4s_lists_list(TV_u_27a),V_l1))))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_y),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_l2))))))) = s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2)))).
fof(ah4s_lists_APPENDu_u_ASSOC, axiom, ![TV_u_27a]: ![V_l3, V_l2, V_l1]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),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),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_append),s(t_h4s_lists_list(TV_u_27a),V_l2))),s(t_h4s_lists_list(TV_u_27a),V_l3))))) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_append),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),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_h4s_lists_list(TV_u_27a),V_l3)))).
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_CONJu_u_ASSOC, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) & (p(s(t_bool,V_t2)) & p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) & p(s(t_bool,V_t3))))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f0)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_lists_FILTER0u_c1, axiom, ![TV_u_27a]: ![V_t, V_h, V_P]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(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))))) = s(t_h4s_lists_list(TV_u_27a),h4s_bools_cond(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_h))),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_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t))))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),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_lists_APPEND0u_c1, axiom, ![TV_u_27a]: ![V_l2, V_l1, V_h]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),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_h),s(t_h4s_lists_list(TV_u_27a),V_l1))))),s(t_h4s_lists_list(TV_u_27a),V_l2))) = 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),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),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)))))).
fof(ah4s_lists_APPEND0u_c0, axiom, ![TV_u_27a]: ![V_l]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_append),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),V_l)).
fof(ah4s_lists_FILTER0u_c0, axiom, ![TV_u_27a]: ![V_P]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)).
fof(ah4s_richu_u_lists_FILTERu_u_SNOC, axiom, ![TV_u_27a]: ![V_x, V_l, V_P]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(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))))) = s(t_h4s_lists_list(TV_u_27a),h4s_bools_cond(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(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_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l))))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
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_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_sortings_PERMu_u_MONO, axiom, ![TV_u_27a]: ![V_x, V_l2, V_l1]: (p(s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2)))) => p(s(t_bool,h4s_sortings_perm(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_l1))),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_l2)))))))).
fof(ah4s_bools_boolu_u_caseu_u_thmu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_lists_listu_u_11, axiom, ![TV_u_27a]: ![V_a1u_27, V_a1, V_a0u_27, V_a0]: (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_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a0u_27),s(t_h4s_lists_list(TV_u_27a),V_a1u_27))) <=> (s(TV_u_27a,V_a0) = s(TV_u_27a,V_a0u_27) & s(t_h4s_lists_list(TV_u_27a),V_a1) = s(t_h4s_lists_list(TV_u_27a),V_a1u_27)))).
fof(ah4s_bools_boolu_u_caseu_u_thmu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
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_lists_NOTu_u_NILu_u_CONS, axiom, ![TV_u_27a]: ![V_a1, V_a0]: ~ (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_a0),s(t_h4s_lists_list(TV_u_27a),V_a1))))).
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_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_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_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_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_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_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f0))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => p(s(t_bool,f0))))).
fof(ah4s_sats_dcu_u_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,f0))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f0)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f0))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f0))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
fof(ah4s_sats_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_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,f0))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f0)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f0))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f0) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_lists_ALLu_u_DISTINCTu_u_FILTER, axiom, ![TV_u_27a]: ![V_l]: (p(s(t_bool,h4s_lists_allu_u_distinct(s(t_h4s_lists_list(TV_u_27a),V_l)))) <=> ![V_x]: (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)))))) => s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),d_equals(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_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))))).
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,f0)).
fof(ah4s_sortings_PERMu_u_NILu_c1, axiom, ![TV_u_27a]: ![V_L]: (p(s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27a),V_L)))) <=> s(t_h4s_lists_list(TV_u_27a),V_L) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))).
fof(ah4s_sortings_PERMu_u_TRANS, axiom, ![TV_u_27a]: ![V_z, V_y, V_x]: ((p(s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),V_x),s(t_h4s_lists_list(TV_u_27a),V_y)))) & p(s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),V_y),s(t_h4s_lists_list(TV_u_27a),V_z))))) => p(s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),V_x),s(t_h4s_lists_list(TV_u_27a),V_z)))))).
fof(ah4s_sortings_PERMu_u_REFL, axiom, ![TV_u_27a]: ![V_L]: p(s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),V_L),s(t_h4s_lists_list(TV_u_27a),V_L))))).
fof(ah4s_sortings_PERMu_u_CONSu_u_EQu_u_APPEND, axiom, ![TV_u_27a]: ![V_t, V_h, V_L]: (p(s(t_bool,h4s_sortings_perm(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),V_L)))) <=> ?[V_M, V_N]: (s(t_h4s_lists_list(TV_u_27a),V_L) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_append),s(t_h4s_lists_list(TV_u_27a),V_M))),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_N))))) & p(s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),V_t),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_append),s(t_h4s_lists_list(TV_u_27a),V_M))),s(t_h4s_lists_list(TV_u_27a),V_N))))))))).
fof(ah4s_bools_FUNu_u_EQu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g) <=> ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))).
fof(ah4s_bools_EXISTSu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (?[V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_lists_APPENDu_u_equ_u_NILu_c0, axiom, ![TV_u_27a]: ![V_l2, V_l1]: (s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),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_h4s_lists_list(TV_u_27a),V_l1) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) & s(t_h4s_lists_list(TV_u_27a),V_l2) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))).
fof(ah4s_lists_CONSu_u_11, axiom, ![TV_u_27a]: ![V_a1u_27, V_a1, V_a0u_27, V_a0]: (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_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a0u_27),s(t_h4s_lists_list(TV_u_27a),V_a1u_27))) <=> (s(TV_u_27a,V_a0) = s(TV_u_27a,V_a0u_27) & s(t_h4s_lists_list(TV_u_27a),V_a1) = s(t_h4s_lists_list(TV_u_27a),V_a1u_27)))).
fof(ah4s_lists_listu_u_nchotomy, axiom, ![TV_u_27a]: ![V_l]: (s(t_h4s_lists_list(TV_u_27a),V_l) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) | ?[V_h, V_t]: 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_h),s(t_h4s_lists_list(TV_u_27a),V_t))))).
fof(ah4s_bools_FORALLu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_lists_APPENDu_u_11u_u_LENGTHu_c0, axiom, ![TV_u_27a]: ![V_l2u_27, V_l2, V_l1u_27, V_l1]: (s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l1))) = s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l1u_27))) => (s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),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_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_append),s(t_h4s_lists_list(TV_u_27a),V_l1u_27))),s(t_h4s_lists_list(TV_u_27a),V_l2u_27))) <=> (s(t_h4s_lists_list(TV_u_27a),V_l1) = s(t_h4s_lists_list(TV_u_27a),V_l1u_27) & s(t_h4s_lists_list(TV_u_27a),V_l2) = s(t_h4s_lists_list(TV_u_27a),V_l2u_27))))).
fof(ah4s_bools_LEFTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))) <=> ![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))))).
fof(ah4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ![V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_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_bools_LEFTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))))).
fof(ah4s_bools_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_primu_u_recs_INVu_u_SUCu_u_EQ, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_markers_labelu_u_def, axiom, ![V_lab, V_argument]: s(t_bool,h4s_markers_u_3au_2d(s(t_h4s_markers_label,V_lab),s(t_bool,V_argument))) = s(t_bool,V_argument)).
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_UNWINDu_u_FORALLu_u_THM2, axiom, ![TV_u_27a]: ![V_v, V_f]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_v) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_v)))))).
fof(ah4s_bools_IMPu_u_DISJu_u_THM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) => p(s(t_bool,V_B))) <=> (~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B))))).
fof(ah4s_bools_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_EXISTSu_u_REFL, axiom, ![TV_u_27a]: ![V_a]: ?[V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_a)).
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(ch4s_sortings_FOLDRu_u_PERM, conjecture, ![TV_u_27a]: ![V_l2, V_l1, V_f, V_e]: ((p(s(t_bool,h4s_operators_assoc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)))) & p(s(t_bool,h4s_operators_comm(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f))))) => (p(s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2)))) => s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27a),V_l1))) = s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27a),h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(TV_u_27a,V_e))),s(t_h4s_lists_list(TV_u_27a),V_l2)))))).
