%   ORIGINAL: h4/patricia__casts/word__ptree__Axiom
% 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/patricia__casts/word__ptree__case__def: !f a1 a0. h4/patricia__casts/word__ptree__CASE (h4/patricia__casts/Word__ptree a0 a1) f = f a0 a1
% Assm: h4/patricia__casts/Word__ptree0: h4/patricia__casts/Word__ptree = h4/patricia__casts/_20_40ind__typepatricia__casts0
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/patricia__casts/word__ptree__repfns_c1: !r. (\a0_27. !_27word__ptree_27. (!a0_270. (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a00 a10) (\n. h4/ind__type/BOTTOM)) a0 a1) ==> _27word__ptree_27 a0_270) ==> _27word__ptree_27 a0_27) r <=> h4/patricia__casts/_20_40ind__typepatricia__casts2 (h4/patricia__casts/_20_40ind__typepatricia__casts1 r) = r
% Assm: h4/patricia__casts/hidden____20__40ind____typepatricia____casts0____def: h4/patricia__casts/_20_40ind__typepatricia__casts0 = (\a0 a1. h4/patricia__casts/_20_40ind__typepatricia__casts1 ((\a00 a10. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a00 a10) (\n. h4/ind__type/BOTTOM)) a0 a1))
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/ind__type/CONSTR__REC: !Fn. ?f. !c i r. f (h4/ind__type/CONSTR c i r) = Fn c i r (\n. f (r n))
% Assm: h4/ind__type/FCONS0_c0: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/DATATYPE__TAG__THM: !x. h4/bool/DATATYPE x <=> T
% Assm: h4/patricia__casts/word__ptree__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION (\a0_27. !_27word__ptree_27. (!a0_270. (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a00 a10) (\n. h4/ind__type/BOTTOM)) a0 a1) ==> _27word__ptree_27 a0_270) ==> _27word__ptree_27 a0_27) rep
% Assm: h4/patricia__casts/datatype__word__ptree: !word__ptree. h4/bool/DATATYPE (word__ptree h4/patricia__casts/Word__ptree)
% Assm: h4/patricia__casts/word__ptree__repfns_c0: !a. h4/patricia__casts/_20_40ind__typepatricia__casts1 (h4/patricia__casts/_20_40ind__typepatricia__casts2 a) = a
% Assm: h4/bool/ABS__REP__THM: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. abs (rep a) = a) /\ (!r. P r <=> rep (abs r) = r))
% Assm: h4/patricia__casts/word__ptree__nchotomy: !ww. ?f p. ww = h4/patricia__casts/Word__ptree f p
% Assm: h4/patricia__casts/word__ptree__size__def: !f1 f a1 a0. h4/patricia__casts/word__ptree__size f f1 (h4/patricia__casts/Word__ptree a0 a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/patricia/ptree__size f1 a1)
% Assm: h4/patricia__casts/word__ptree__11: !a1_27 a1 a0_27 a0. h4/patricia__casts/Word__ptree a0 a1 = h4/patricia__casts/Word__ptree a0_27 a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm: h4/patricia__casts/word__ptree__case__cong: !f_27 f M_27 M. M = M_27 /\ (!a0 a1. M_27 = h4/patricia__casts/Word__ptree a0 a1 ==> f a0 a1 = f_27 a0 a1) ==> h4/patricia__casts/word__ptree__CASE M f = h4/patricia__casts/word__ptree__CASE M_27 f_27
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/patricia/PTREE__OF__NUMSET__def: !t s. h4/patricia/PTREE__OF__NUMSET t s = h4/list/FOLDL (h4/combin/C h4/patricia/INSERT__PTREE) t (h4/list/SET__TO__LIST s)
% Assm: h4/patricia/PTREE__OF__NUMSET__EMPTY: !t. h4/patricia/PTREE__OF__NUMSET t h4/pred__set/EMPTY = t
% Assm: h4/patricia/SIZE__PTREE__OF__NUMSET: !t s. h4/pred__set/FINITE s ==> h4/patricia/IS__PTREE t /\ h4/list/ALL__DISTINCT (h4/list/APPEND (h4/patricia/TRAVERSE t) (h4/list/SET__TO__LIST s)) ==> h4/patricia/SIZE (h4/patricia/PTREE__OF__NUMSET t s) = h4/arithmetic/_2B (h4/patricia/SIZE t) (h4/pred__set/CARD s)
% Assm: h4/patricia/NUMSET__OF__PTREE__def: !t. h4/patricia/NUMSET__OF__PTREE t = h4/list/LIST__TO__SET (h4/patricia/TRAVERSE t)
% Assm: h4/patricia/SIZE__PTREE__OF__NUMSET__EMPTY: !s. h4/pred__set/FINITE s ==> h4/patricia/SIZE (h4/patricia/PTREE__OF__NUMSET h4/patricia/Empty s) = h4/pred__set/CARD s
% Assm: h4/patricia/UNION__PTREE__EMPTY_c0: !t. h4/patricia/UNION__PTREE t h4/patricia/Empty = t
% Assm: h4/patricia/PERM__DELETE__PTREE: !t k. h4/patricia/IS__PTREE t /\ h4/bool/IN k (h4/list/LIST__TO__SET (h4/patricia/TRAVERSE t)) ==> h4/sorting/PERM (h4/patricia/TRAVERSE (h4/patricia/REMOVE t k)) (h4/list/FILTER (\x. ~(x = k)) (h4/patricia/TRAVERSE t))
% Assm: h4/patricia/PEEK__INSERT__PTREE: !t k j. h4/patricia/IS__PTREE t ==> h4/patricia/PEEK (h4/patricia/INSERT__PTREE k t) j = h4/bool/COND (k = j) (h4/option/SOME h4/one/one0) (h4/patricia/PEEK t j)
% Assm: h4/patricia__casts/PTREE__OF__STRINGSET__def: !t s. h4/patricia__casts/PTREE__OF__STRINGSET t s = h4/patricia/PTREE__OF__NUMSET t (h4/pred__set/IMAGE h4/patricia__casts/string__to__num s)
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/patricia__casts/IN__PTREEs__def: !w t. h4/patricia__casts/IN__PTREEs w t <=> h4/patricia/IN__PTREE (h4/patricia__casts/string__to__num w) t
% Assm: h4/patricia/IN__NUMSET__OF__PTREE: !t n. h4/patricia/IS__PTREE t ==> (h4/bool/IN n (h4/patricia/NUMSET__OF__PTREE t) <=> h4/patricia/IN__PTREE n t)
% Assm: h4/list/FOLDL0_c0: !f e. h4/list/FOLDL f e h4/list/NIL = e
% Assm: h4/patricia/IN__PTREE__INSERT__PTREE: !t n m. h4/patricia/IS__PTREE t ==> (h4/patricia/IN__PTREE n (h4/patricia/INSERT__PTREE m t) <=> m = n \/ h4/patricia/IN__PTREE n t)
% Assm: h4/patricia/ADD__INSERT: !v t n. h4/patricia/ADD t (h4/pair/_2C n v) = h4/patricia/INSERT__PTREE n t
% Assm: h4/patricia/PEEK__ADD: !t k j d. h4/patricia/IS__PTREE t ==> h4/patricia/PEEK (h4/patricia/ADD t (h4/pair/_2C k d)) j = h4/bool/COND (k = j) (h4/option/SOME d) (h4/patricia/PEEK t j)
% Assm: h4/one/one1: !v. v = h4/one/one0
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/patricia/NUMSET__OF__PTREE__EMPTY: h4/patricia/NUMSET__OF__PTREE h4/patricia/Empty = h4/pred__set/EMPTY
% Assm: h4/patricia/UNION__PTREE__def: !t2 t1. h4/patricia/UNION__PTREE t1 t2 = h4/patricia/PTREE__OF__NUMSET t1 (h4/patricia/NUMSET__OF__PTREE t2)
% Assm: h4/list/SET__TO__LIST__EMPTY: h4/list/SET__TO__LIST h4/pred__set/EMPTY = h4/list/NIL
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/ADD__0: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/patricia/MEM__TRAVERSE__PEEK: !t k. h4/patricia/IS__PTREE t ==> (h4/bool/IN k (h4/list/LIST__TO__SET (h4/patricia/TRAVERSE t)) <=> h4/option/IS__SOME (h4/patricia/PEEK t k))
% Assm: h4/patricia/SIZE0_c0: h4/patricia/SIZE h4/patricia/Empty = h4/num/0
% Assm: h4/patricia/EMPTY__IS__PTREE: h4/patricia/IS__PTREE h4/patricia/Empty
% Assm: h4/list/ALL__DISTINCT__SET__TO__LIST: !s. h4/pred__set/FINITE s ==> h4/list/ALL__DISTINCT (h4/list/SET__TO__LIST s)
% Assm: h4/patricia/TRAVERSE__def_c0: h4/patricia/TRAVERSE h4/patricia/Empty = h4/list/NIL
% Assm: h4/list/APPEND0_c0: !l. h4/list/APPEND h4/list/NIL l = l
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/patricia/ALL__DISTINCT__TRAVERSE: !t. h4/patricia/IS__PTREE t ==> h4/list/ALL__DISTINCT (h4/patricia/TRAVERSE t)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% 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__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/patricia/SIZE__ADD: !t k d. h4/patricia/IS__PTREE t ==> h4/patricia/SIZE (h4/patricia/ADD t (h4/pair/_2C k d)) = h4/bool/COND (h4/bool/IN k (h4/list/LIST__TO__SET (h4/patricia/TRAVERSE t))) (h4/patricia/SIZE t) (h4/arithmetic/_2B (h4/patricia/SIZE t) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% Assm: h4/patricia/MEM__TRAVERSE__INSERT__PTREE: !x t h. h4/patricia/IS__PTREE t ==> (h4/bool/IN x (h4/list/LIST__TO__SET (h4/patricia/TRAVERSE (h4/patricia/INSERT__PTREE h t))) <=> x = h \/ ~(x = h) /\ h4/bool/IN x (h4/list/LIST__TO__SET (h4/patricia/TRAVERSE t)))
% Assm: h4/patricia/INSERT__PTREE__IS__PTREE: !x t. h4/patricia/IS__PTREE t ==> h4/patricia/IS__PTREE (h4/patricia/INSERT__PTREE x t)
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/patricia/INSERT__PTREE__def: !t n. h4/patricia/INSERT__PTREE n t = h4/patricia/ADD t (h4/pair/_2C n h4/one/one0)
% Assm: h4/list/SET__TO__LIST__CARD: !s. h4/pred__set/FINITE s ==> h4/list/LENGTH (h4/list/SET__TO__LIST s) = h4/pred__set/CARD s
% Assm: h4/list/ALL__DISTINCT__APPEND: !l2 l1. h4/list/ALL__DISTINCT (h4/list/APPEND l1 l2) <=> h4/list/ALL__DISTINCT l1 /\ h4/list/ALL__DISTINCT l2 /\ (!e. h4/bool/IN e (h4/list/LIST__TO__SET l1) ==> ~h4/bool/IN e (h4/list/LIST__TO__SET l2))
% Assm: h4/list/ALL__DISTINCT0_c1: !t h. h4/list/ALL__DISTINCT (h4/list/CONS h t) <=> ~h4/bool/IN h (h4/list/LIST__TO__SET t) /\ h4/list/ALL__DISTINCT t
% Assm: h4/list/APPEND__NIL: !l. h4/list/APPEND l h4/list/NIL = l
% 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/list__induction: !P. P h4/list/NIL /\ (!t. P t ==> (!h. P (h4/list/CONS h t))) ==> (!l. P l)
% 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/LENGTH0_c1: !t h. h4/list/LENGTH (h4/list/CONS h t) = h4/num/SUC (h4/list/LENGTH t)
% Assm: h4/numeral/numeral__add_c2: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT2 (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/numeral/numeral__add_c0: !n. h4/numeral/iZ (h4/arithmetic/_2B h4/arithmetic/ZERO n) = n
% Assm: h4/numeral/numeral__distrib_c2: !n m. h4/arithmetic/_2B (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm: h4/numeral/numeral__distrib_c27: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__distrib_c1: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm: h4/arithmetic/ADD__MONO__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm: h4/arithmetic/NOT__NUM__EQ: !n m. ~(m = n) <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n \/ h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm: h4/arithmetic/EQ__MONO__ADD__EQ: !p n m. h4/arithmetic/_2B m p = h4/arithmetic/_2B n p <=> m = n
% Assm: h4/arithmetic/SUC__ONE__ADD: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/ADD__ASSOC: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm: h4/combin/C__THM: !y x f. h4/combin/C f x y = f y x
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/list/LENGTH0_c0: h4/list/LENGTH h4/list/NIL = h4/num/0
% Assm: h4/bool/RIGHT__OR__OVER__AND: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% 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/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/patricia/IN__PTREE__def: !t n. h4/patricia/IN__PTREE n t <=> h4/option/IS__SOME (h4/patricia/PEEK t n)
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/patricia/MEM__ALL__DISTINCT__IMP__PERM: !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/patricia/PEEK__REMOVE: !t k j. h4/patricia/IS__PTREE t ==> h4/patricia/PEEK (h4/patricia/REMOVE t k) j = h4/bool/COND (k = j) h4/option/NONE (h4/patricia/PEEK t j)
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/patricia/REMOVE__IS__PTREE: !t k. h4/patricia/IS__PTREE t ==> h4/patricia/IS__PTREE (h4/patricia/REMOVE t k)
% Assm: h4/rich__list/FILTER__FILTER: !l Q P. h4/list/FILTER P (h4/list/FILTER Q l) = h4/list/FILTER (\x. P x /\ Q x) l
% 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__FILTER: !x P L. h4/bool/IN x (h4/list/LIST__TO__SET (h4/list/FILTER P L)) <=> P x /\ h4/bool/IN x (h4/list/LIST__TO__SET L)
% Assm: h4/option/IS__SOME__DEF_c1: h4/option/IS__SOME h4/option/NONE <=> F
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/OR__CLAUSES_c3: !t. t \/ F <=> t
% Assm: h4/bool/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Goal: !f. ?fn. !a0 a1. fn (h4/patricia__casts/Word__ptree a0 a1) = f a0 a1
%   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_patriciau_u_castss_wordu_u_ptreeu_u_caseu_u_def]: !f a1 a0. h4/patricia__casts/word__ptree__CASE (happ (happ h4/patricia__casts/Word__ptree a0) a1) f = happ (happ f a0) a1
% Assm [h4s_patriciau_u_castss_Wordu_u_ptree0]: h4/patricia__casts/Word__ptree = h4/patricia__casts/_20_40ind__typepatricia__casts0
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_patriciau_u_castss_wordu_u_ptreeu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27word__ptree_27. (!a0_270. (?a0 a1. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a0 a1) _0) ==> happ _27word__ptree_27 a0_270) ==> happ _27word__ptree_27 r) <=> h4/patricia__casts/_20_40ind__typepatricia__casts2 (h4/patricia__casts/_20_40ind__typepatricia__casts1 r) = r)
% Assm [h4s_patriciau_u_castss_hiddenu_u_u_u_20u_u_40indu_u_u_u_typepatriciau_u_u_u_casts0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x. happ (happ h4/patricia__casts/_20_40ind__typepatricia__casts0 x) x = h4/patricia__casts/_20_40ind__typepatricia__casts1 (h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C x x) _0))
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_indu_u_types_CONSTRu_u_REC]: !_0. (!f r n. happ (happ (happ _0 f) r) n = happ f (happ r n)) ==> (!Fn. ?f. !c i r. happ f (h4/ind__type/CONSTR c i r) = happ (happ (happ (happ Fn c) i) r) (happ (happ _0 f) r))
% Assm [h4s_indu_u_types_FCONS0u_c0]: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_DATATYPEu_u_TAGu_u_THM]: !x. h4/bool/DATATYPE x <=> T
% Assm [h4s_patriciau_u_castss_wordu_u_ptreeu_u_TYu_u_DEF]: !_1. (!n. happ _1 n = h4/ind__type/BOTTOM) ==> (!_0. (!a0_27. happ _0 a0_27 <=> (!_27word__ptree_27. (!a0_270. (?a0 a1. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a0 a1) _1) ==> happ _27word__ptree_27 a0_270) ==> happ _27word__ptree_27 a0_27)) ==> (?rep. h4/bool/TYPE__DEFINITION _0 rep))
% Assm [h4s_patriciau_u_castss_datatypeu_u_wordu_u_ptree]: !word__ptree. h4/bool/DATATYPE (happ word__ptree h4/patricia__casts/Word__ptree)
% Assm [h4s_patriciau_u_castss_wordu_u_ptreeu_u_repfnsu_c0]: !a. h4/patricia__casts/_20_40ind__typepatricia__casts1 (h4/patricia__casts/_20_40ind__typepatricia__casts2 a) = a
% Assm [h4s_bools_ABSu_u_REPu_u_THM]: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. happ abs (happ rep a) = a) /\ (!r. happ P r <=> happ rep (happ abs r) = r))
% Assm [h4s_patriciau_u_castss_wordu_u_ptreeu_u_nchotomy]: !ww. ?f p. ww = happ (happ h4/patricia__casts/Word__ptree f) p
% Assm [h4s_patriciau_u_castss_wordu_u_ptreeu_u_sizeu_u_def]: !f1 f a1 a0. h4/patricia__casts/word__ptree__size f f1 (happ (happ h4/patricia__casts/Word__ptree a0) a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/patricia/ptree__size f1 a1)
% Assm [h4s_patriciau_u_castss_wordu_u_ptreeu_u_11]: !a1_27 a1 a0_27 a0. happ (happ h4/patricia__casts/Word__ptree a0) a1 = happ (happ h4/patricia__casts/Word__ptree a0_27) a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm [h4s_patriciau_u_castss_wordu_u_ptreeu_u_caseu_u_cong]: !f_27 f M_27 M. M = M_27 /\ (!a0 a1. M_27 = happ (happ h4/patricia__casts/Word__ptree a0) a1 ==> happ (happ f a0) a1 = happ (happ f_27 a0) a1) ==> h4/patricia__casts/word__ptree__CASE M f = h4/patricia__casts/word__ptree__CASE M_27 f_27
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_patricias_PTREEu_u_OFu_u_NUMSETu_u_def]: !t s. h4/patricia/PTREE__OF__NUMSET t s = h4/list/FOLDL (h4/combin/C h4/patricia/INSERT__PTREE) t (h4/list/SET__TO__LIST s)
% Assm [h4s_patricias_PTREEu_u_OFu_u_NUMSETu_u_EMPTY]: !t. h4/patricia/PTREE__OF__NUMSET t h4/pred__set/EMPTY = t
% Assm [h4s_patricias_SIZEu_u_PTREEu_u_OFu_u_NUMSET]: !t s. h4/pred__set/FINITE s ==> h4/patricia/IS__PTREE t /\ h4/list/ALL__DISTINCT (h4/list/APPEND (h4/patricia/TRAVERSE t) (h4/list/SET__TO__LIST s)) ==> h4/patricia/SIZE (h4/patricia/PTREE__OF__NUMSET t s) = h4/arithmetic/_2B (h4/patricia/SIZE t) (h4/pred__set/CARD s)
% Assm [h4s_patricias_NUMSETu_u_OFu_u_PTREEu_u_def]: !t. h4/patricia/NUMSET__OF__PTREE t = h4/list/LIST__TO__SET (h4/patricia/TRAVERSE t)
% Assm [h4s_patricias_SIZEu_u_PTREEu_u_OFu_u_NUMSETu_u_EMPTY]: !s. h4/pred__set/FINITE s ==> h4/patricia/SIZE (h4/patricia/PTREE__OF__NUMSET h4/patricia/Empty s) = h4/pred__set/CARD s
% Assm [h4s_patricias_UNIONu_u_PTREEu_u_EMPTYu_c0]: !t. h4/patricia/UNION__PTREE t h4/patricia/Empty = t
% Assm [h4s_patricias_PERMu_u_DELETEu_u_PTREE]: !_0. (!k x. happ (happ _0 k) x <=> ~(x = k)) ==> (!t k. h4/patricia/IS__PTREE t /\ h4/bool/IN k (h4/list/LIST__TO__SET (h4/patricia/TRAVERSE t)) ==> h4/sorting/PERM (h4/patricia/TRAVERSE (h4/patricia/REMOVE t k)) (h4/list/FILTER (happ _0 k) (h4/patricia/TRAVERSE t)))
% Assm [h4s_patricias_PEEKu_u_INSERTu_u_PTREE]: !t k j. h4/patricia/IS__PTREE t ==> (?v. (v <=> k = j) /\ h4/patricia/PEEK (happ (happ h4/patricia/INSERT__PTREE k) t) j = h4/bool/COND v (h4/option/SOME h4/one/one0) (h4/patricia/PEEK t j))
% Assm [h4s_patriciau_u_castss_PTREEu_u_OFu_u_STRINGSETu_u_def]: !t s. h4/patricia__casts/PTREE__OF__STRINGSET t s = h4/patricia/PTREE__OF__NUMSET t (h4/pred__set/IMAGE h4/patricia__casts/string__to__num s)
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_patriciau_u_castss_INu_u_PTREEsu_u_def]: !w t. h4/patricia__casts/IN__PTREEs w t <=> h4/patricia/IN__PTREE (happ h4/patricia__casts/string__to__num w) t
% Assm [h4s_patricias_INu_u_NUMSETu_u_OFu_u_PTREE]: !t n. h4/patricia/IS__PTREE t ==> (h4/bool/IN n (h4/patricia/NUMSET__OF__PTREE t) <=> h4/patricia/IN__PTREE n t)
% Assm [h4s_lists_FOLDL0u_c0]: !f e. h4/list/FOLDL f e h4/list/NIL = e
% Assm [h4s_patricias_INu_u_PTREEu_u_INSERTu_u_PTREE]: !t n m. h4/patricia/IS__PTREE t ==> (h4/patricia/IN__PTREE n (happ (happ h4/patricia/INSERT__PTREE m) t) <=> m = n \/ h4/patricia/IN__PTREE n t)
% Assm [h4s_patricias_ADDu_u_INSERT]: !v t n. h4/patricia/ADD t (h4/pair/_2C n v) = happ (happ h4/patricia/INSERT__PTREE n) t
% Assm [h4s_patricias_PEEKu_u_ADD]: !t k j d. h4/patricia/IS__PTREE t ==> (?v. (v <=> k = j) /\ h4/patricia/PEEK (h4/patricia/ADD t (h4/pair/_2C k d)) j = h4/bool/COND v (h4/option/SOME d) (h4/patricia/PEEK t j))
% Assm [h4s_ones_one1]: !v. v = h4/one/one0
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_patricias_NUMSETu_u_OFu_u_PTREEu_u_EMPTY]: h4/patricia/NUMSET__OF__PTREE h4/patricia/Empty = h4/pred__set/EMPTY
% Assm [h4s_patricias_UNIONu_u_PTREEu_u_def]: !t2 t1. h4/patricia/UNION__PTREE t1 t2 = h4/patricia/PTREE__OF__NUMSET t1 (h4/patricia/NUMSET__OF__PTREE t2)
% Assm [h4s_lists_SETu_u_TOu_u_LISTu_u_EMPTY]: h4/list/SET__TO__LIST h4/pred__set/EMPTY = h4/list/NIL
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_ADDu_u_0]: !m. h4/arithmetic/_2B m h4/num/0 = m
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_patricias_MEMu_u_TRAVERSEu_u_PEEK]: !t k. h4/patricia/IS__PTREE t ==> (h4/bool/IN k (h4/list/LIST__TO__SET (h4/patricia/TRAVERSE t)) <=> h4/option/IS__SOME (h4/patricia/PEEK t k))
% Assm [h4s_patricias_SIZE0u_c0]: h4/patricia/SIZE h4/patricia/Empty = h4/num/0
% Assm [h4s_patricias_EMPTYu_u_ISu_u_PTREE]: h4/patricia/IS__PTREE h4/patricia/Empty
% Assm [h4s_lists_ALLu_u_DISTINCTu_u_SETu_u_TOu_u_LIST]: !s. h4/pred__set/FINITE s ==> h4/list/ALL__DISTINCT (h4/list/SET__TO__LIST s)
% Assm [h4s_patricias_TRAVERSEu_u_defu_c0]: h4/patricia/TRAVERSE h4/patricia/Empty = h4/list/NIL
% Assm [h4s_lists_APPEND0u_c0]: !l. h4/list/APPEND h4/list/NIL l = l
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_patricias_ALLu_u_DISTINCTu_u_TRAVERSE]: !t. h4/patricia/IS__PTREE t ==> h4/list/ALL__DISTINCT (h4/patricia/TRAVERSE t)
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% 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_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_patricias_SIZEu_u_ADD]: !t k d. h4/patricia/IS__PTREE t ==> h4/patricia/SIZE (h4/patricia/ADD t (h4/pair/_2C k d)) = h4/bool/COND (h4/bool/IN k (h4/list/LIST__TO__SET (h4/patricia/TRAVERSE t))) (h4/patricia/SIZE t) (h4/arithmetic/_2B (h4/patricia/SIZE t) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% Assm [h4s_patricias_MEMu_u_TRAVERSEu_u_INSERTu_u_PTREE]: !x t h. h4/patricia/IS__PTREE t ==> (h4/bool/IN x (h4/list/LIST__TO__SET (h4/patricia/TRAVERSE (happ (happ h4/patricia/INSERT__PTREE h) t))) <=> x = h \/ ~(x = h) /\ h4/bool/IN x (h4/list/LIST__TO__SET (h4/patricia/TRAVERSE t)))
% Assm [h4s_patricias_INSERTu_u_PTREEu_u_ISu_u_PTREE]: !x t. h4/patricia/IS__PTREE t ==> h4/patricia/IS__PTREE (happ (happ h4/patricia/INSERT__PTREE x) t)
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_patricias_INSERTu_u_PTREEu_u_def]: !t n. happ (happ h4/patricia/INSERT__PTREE n) t = h4/patricia/ADD t (h4/pair/_2C n h4/one/one0)
% Assm [h4s_lists_SETu_u_TOu_u_LISTu_u_CARD]: !s. h4/pred__set/FINITE s ==> h4/list/LENGTH (h4/list/SET__TO__LIST s) = h4/pred__set/CARD s
% Assm [h4s_lists_ALLu_u_DISTINCTu_u_APPEND]: !l2 l1. h4/list/ALL__DISTINCT (h4/list/APPEND l1 l2) <=> h4/list/ALL__DISTINCT l1 /\ h4/list/ALL__DISTINCT l2 /\ (!e. h4/bool/IN e (h4/list/LIST__TO__SET l1) ==> ~h4/bool/IN e (h4/list/LIST__TO__SET l2))
% Assm [h4s_lists_ALLu_u_DISTINCT0u_c1]: !t h. h4/list/ALL__DISTINCT (h4/list/CONS h t) <=> ~h4/bool/IN h (h4/list/LIST__TO__SET t) /\ h4/list/ALL__DISTINCT t
% Assm [h4s_lists_APPENDu_u_NIL]: !l. h4/list/APPEND l h4/list/NIL = l
% 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_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_FOLDL0u_c1]: !x l f e. h4/list/FOLDL f e (h4/list/CONS x l) = h4/list/FOLDL f (happ (happ f e) x) 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_numerals_numeralu_u_addu_c2]: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT2 (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_numerals_numeralu_u_addu_c0]: !n. h4/numeral/iZ (h4/arithmetic/_2B h4/arithmetic/ZERO n) = n
% Assm [h4s_numerals_numeralu_u_distribu_c2]: !n m. h4/arithmetic/_2B (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm [h4s_numerals_numeralu_u_distribu_c27]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_distribu_c1]: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm [h4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm [h4s_arithmetics_NOTu_u_NUMu_u_EQ]: !n m. ~(m = n) <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n \/ h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm [h4s_arithmetics_EQu_u_MONOu_u_ADDu_u_EQ]: !p n m. h4/arithmetic/_2B m p = h4/arithmetic/_2B n p <=> m = n
% Assm [h4s_arithmetics_SUCu_u_ONEu_u_ADD]: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_ADDu_u_ASSOC]: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm [h4s_combins_Cu_u_THM]: !y x f. happ (happ (h4/combin/C f) x) y = happ (happ f y) x
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_lists_LENGTH0u_c0]: h4/list/LENGTH h4/list/NIL = h4/num/0
% Assm [h4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND]: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% 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_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_patricias_INu_u_PTREEu_u_def]: !t n. h4/patricia/IN__PTREE n t <=> h4/option/IS__SOME (h4/patricia/PEEK t n)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_patricias_MEMu_u_ALLu_u_DISTINCTu_u_IMPu_u_PERM]: !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_patricias_PEEKu_u_REMOVE]: !t k j. h4/patricia/IS__PTREE t ==> (?v. (v <=> k = j) /\ h4/patricia/PEEK (h4/patricia/REMOVE t k) j = h4/bool/COND v h4/option/NONE (h4/patricia/PEEK t j))
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_patricias_REMOVEu_u_ISu_u_PTREE]: !t k. h4/patricia/IS__PTREE t ==> h4/patricia/IS__PTREE (h4/patricia/REMOVE t k)
% Assm [h4s_richu_u_lists_FILTERu_u_FILTER]: !_0. (!P Q x. happ (happ (happ _0 P) Q) x <=> happ P x /\ happ Q x) ==> (!l Q P. h4/list/FILTER P (h4/list/FILTER Q l) = h4/list/FILTER (happ (happ _0 P) Q) l)
% 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_u_FILTER]: !x P L. h4/bool/IN x (h4/list/LIST__TO__SET (h4/list/FILTER P L)) <=> happ P x /\ h4/bool/IN x (h4/list/LIST__TO__SET L)
% Assm [h4s_options_ISu_u_SOMEu_u_DEFu_c1]: h4/option/IS__SOME h4/option/NONE <=> F
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_ORu_u_CLAUSESu_c3]: !t. t \/ F <=> t
% Assm [h4s_bools_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Goal: !f. ?fn. !a0 a1. happ fn (happ (happ h4/patricia__casts/Word__ptree a0) a1) = happ (happ f a0) a1
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_Q1295160,TV_Q1295156]: ![V_f, V_g]: (![V_x]: s(TV_Q1295156,happ(s(t_fun(TV_Q1295160,TV_Q1295156),V_f),s(TV_Q1295160,V_x))) = s(TV_Q1295156,happ(s(t_fun(TV_Q1295160,TV_Q1295156),V_g),s(TV_Q1295160,V_x))) => s(t_fun(TV_Q1295160,TV_Q1295156),V_f) = s(t_fun(TV_Q1295160,TV_Q1295156),V_g))).
fof(ah4s_patriciau_u_castss_wordu_u_ptreeu_u_caseu_u_def, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_f, V_a1, V_a0]: s(TV_u_27c,h4s_patriciau_u_castss_wordu_u_ptreeu_u_case(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b))),h4s_patriciau_u_castss_wordu_u_ptree),s(t_fun(TV_u_27a,t_h4s_ones_one),V_a0))),s(t_h4s_patricias_ptree(TV_u_27b),V_a1))),s(t_fun(t_fun(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27b),TV_u_27c)),V_f))) = s(TV_u_27c,happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27b),TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27b),TV_u_27c)),V_f),s(t_fun(TV_u_27a,t_h4s_ones_one),V_a0))),s(t_h4s_patricias_ptree(TV_u_27b),V_a1)))).
fof(ah4s_patriciau_u_castss_Wordu_u_ptree0, axiom, ![TV_u_27a,TV_u_27b]: s(t_fun(t_fun(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b))),h4s_patriciau_u_castss_wordu_u_ptree) = s(t_fun(t_fun(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b))),h4s_patriciau_u_castss_u_20u_40indu_u_typepatriciau_u_casts0)).
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_patriciau_u_castss_wordu_u_ptreeu_u_repfnsu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27wordu_u_ptreeu_27]: (![V_a0u_270]: (?[V_a0, V_a1]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_h4s_ones_one),V_a0),s(t_h4s_patricias_ptree(TV_u_27b),V_a1))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b)))),V_uu_0))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),t_bool),V_uu_27wordu_u_ptreeu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),t_bool),V_uu_27wordu_u_ptreeu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),V_r))))) <=> s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),h4s_patriciau_u_castss_u_20u_40indu_u_typepatriciau_u_casts2(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),h4s_patriciau_u_castss_u_20u_40indu_u_typepatriciau_u_casts1(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),V_r))))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),V_r)))).
fof(ah4s_patriciau_u_castss_hiddenu_u_u_u_20u_u_40indu_u_u_u_typepatriciau_u_u_u_casts0u_u_u_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),h4s_indu_u_types_bottom) => ![V_x, V_x0]: s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b))),h4s_patriciau_u_castss_u_20u_40indu_u_typepatriciau_u_casts0),s(t_fun(TV_u_27a,t_h4s_ones_one),V_x))),s(t_h4s_patricias_ptree(TV_u_27b),V_x0))) = s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),h4s_patriciau_u_castss_u_20u_40indu_u_typepatriciau_u_casts1(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_h4s_ones_one),V_x),s(t_h4s_patricias_ptree(TV_u_27b),V_x0))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b)))),V_uu_0))))))).
fof(ah4s_bools_JRHu_u_INDUCTu_u_UTIL, axiom, ![TV_u_27a]: ![V_t, V_P]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_t) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) => p(s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_P)))))).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(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_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(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_indu_u_types_CONSTRu_u_REC, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_f, V_r, V_n]: s(TV_u_27b,happ(s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_h4s_nums_num,V_n))) = s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r),s(t_h4s_nums_num,V_n))))) => ![V_Fn]: ?[V_f]: ![V_c, V_i, V_r]: s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,V_c),s(TV_u_27a,V_i),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))) = s(TV_u_27b,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)))),V_Fn),s(t_h4s_nums_num,V_c))),s(TV_u_27a,V_i))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))))).
fof(ah4s_indu_u_types_FCONS0u_c0, axiom, ![TV_u_27a]: ![V_f, V_a]: s(TV_u_27a,h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_a)).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_DATATYPEu_u_TAGu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,h4s_bools_datatype(s(TV_u_27a,V_x))) = s(t_bool,t)).
fof(ah4s_patriciau_u_castss_wordu_u_ptreeu_u_TYu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b)))),V_uu_1),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),h4s_indu_u_types_bottom) => ![V_uu_0]: (![V_a0u_27]: (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),t_bool),V_uu_0),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),V_a0u_27)))) <=> ![V_uu_27wordu_u_ptreeu_27]: (![V_a0u_270]: (?[V_a0, V_a1]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b)),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_h4s_ones_one),V_a0),s(t_h4s_patricias_ptree(TV_u_27b),V_a1))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b)))),V_uu_1))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),t_bool),V_uu_27wordu_u_ptreeu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),t_bool),V_uu_27wordu_u_ptreeu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),V_a0u_27)))))) => ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),t_bool),V_uu_0),s(t_fun(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b)))),V_rep))))))).
fof(ah4s_patriciau_u_castss_datatypeu_u_wordu_u_ptree, axiom, ![TV_u_27a,TV_u_27b]: ![V_wordu_u_ptree]: p(s(t_bool,h4s_bools_datatype(s(t_bool,happ(s(t_fun(t_fun(t_fun(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b))),t_bool),V_wordu_u_ptree),s(t_fun(t_fun(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b))),h4s_patriciau_u_castss_wordu_u_ptree))))))).
fof(ah4s_patriciau_u_castss_wordu_u_ptreeu_u_repfnsu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_a]: s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),h4s_patriciau_u_castss_u_20u_40indu_u_typepatriciau_u_casts1(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_ones_one),t_h4s_patricias_ptree(TV_u_27b))),h4s_patriciau_u_castss_u_20u_40indu_u_typepatriciau_u_casts2(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),V_a))))) = s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),V_a)).
fof(ah4s_bools_ABSu_u_REPu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ?[V_rep, V_abs]: (![V_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))) = s(TV_u_27b,V_a) & ![V_r]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_r)))) <=> s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))))) = s(TV_u_27a,V_r))))).
fof(ah4s_patriciau_u_castss_wordu_u_ptreeu_u_nchotomy, axiom, ![TV_u_27a,TV_u_27b]: ![V_ww]: ?[V_f, V_p]: s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),V_ww) = s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b))),h4s_patriciau_u_castss_wordu_u_ptree),s(t_fun(TV_u_27a,t_h4s_ones_one),V_f))),s(t_h4s_patricias_ptree(TV_u_27b),V_p)))).
fof(ah4s_patriciau_u_castss_wordu_u_ptreeu_u_sizeu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_f1, V_f, V_a1, V_a0]: s(t_h4s_nums_num,h4s_patriciau_u_castss_wordu_u_ptreeu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_fun(TV_u_27b,t_h4s_nums_num),V_f1),s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b))),h4s_patriciau_u_castss_wordu_u_ptree),s(t_fun(TV_u_27a,t_h4s_ones_one),V_a0))),s(t_h4s_patricias_ptree(TV_u_27b),V_a1))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,h4s_patricias_ptreeu_u_size(s(t_fun(TV_u_27b,t_h4s_nums_num),V_f1),s(t_h4s_patricias_ptree(TV_u_27b),V_a1)))))).
fof(ah4s_patriciau_u_castss_wordu_u_ptreeu_u_11, axiom, ![TV_u_27a,TV_u_27b]: ![V_a1u_27, V_a1, V_a0u_27, V_a0]: (s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b))),h4s_patriciau_u_castss_wordu_u_ptree),s(t_fun(TV_u_27a,t_h4s_ones_one),V_a0))),s(t_h4s_patricias_ptree(TV_u_27b),V_a1))) = s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b))),h4s_patriciau_u_castss_wordu_u_ptree),s(t_fun(TV_u_27a,t_h4s_ones_one),V_a0u_27))),s(t_h4s_patricias_ptree(TV_u_27b),V_a1u_27))) <=> (s(t_fun(TV_u_27a,t_h4s_ones_one),V_a0) = s(t_fun(TV_u_27a,t_h4s_ones_one),V_a0u_27) & s(t_h4s_patricias_ptree(TV_u_27b),V_a1) = s(t_h4s_patricias_ptree(TV_u_27b),V_a1u_27)))).
fof(ah4s_patriciau_u_castss_wordu_u_ptreeu_u_caseu_u_cong, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_fu_27, V_f, V_Mu_27, V_M]: ((s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),V_M) = s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),V_Mu_27) & ![V_a0, V_a1]: (s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),V_Mu_27) = s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b))),h4s_patriciau_u_castss_wordu_u_ptree),s(t_fun(TV_u_27a,t_h4s_ones_one),V_a0))),s(t_h4s_patricias_ptree(TV_u_27b),V_a1))) => s(TV_u_27c,happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27b),TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27b),TV_u_27c)),V_f),s(t_fun(TV_u_27a,t_h4s_ones_one),V_a0))),s(t_h4s_patricias_ptree(TV_u_27b),V_a1))) = s(TV_u_27c,happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27b),TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27b),TV_u_27c)),V_fu_27),s(t_fun(TV_u_27a,t_h4s_ones_one),V_a0))),s(t_h4s_patricias_ptree(TV_u_27b),V_a1))))) => s(TV_u_27c,h4s_patriciau_u_castss_wordu_u_ptreeu_u_case(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),V_M),s(t_fun(t_fun(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27b),TV_u_27c)),V_f))) = s(TV_u_27c,h4s_patriciau_u_castss_wordu_u_ptreeu_u_case(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),V_Mu_27),s(t_fun(t_fun(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27b),TV_u_27c)),V_fu_27))))).
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_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_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_patricias_PTREEu_u_OFu_u_NUMSETu_u_def, axiom, ![V_t, V_s]: s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_ptreeu_u_ofu_u_numset(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_fun(t_h4s_nums_num,t_bool),V_s))) = s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_lists_foldl(s(t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_fun(t_h4s_nums_num,t_h4s_patricias_ptree(t_h4s_ones_one))),h4s_combins_c(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one))),h4s_patricias_insertu_u_ptree))),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_setu_u_tou_u_list(s(t_fun(t_h4s_nums_num,t_bool),V_s)))))).
fof(ah4s_patricias_PTREEu_u_OFu_u_NUMSETu_u_EMPTY, axiom, ![V_t]: s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_ptreeu_u_ofu_u_numset(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_empty))) = s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)).
fof(ah4s_patricias_SIZEu_u_PTREEu_u_OFu_u_NUMSET, axiom, ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),V_s)))) => ((p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))) & p(s(t_bool,h4s_lists_allu_u_distinct(s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_append(s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))),s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_setu_u_tou_u_list(s(t_fun(t_h4s_nums_num,t_bool),V_s))))))))) => s(t_h4s_nums_num,h4s_patricias_size(s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_ptreeu_u_ofu_u_numset(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_fun(t_h4s_nums_num,t_bool),V_s))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_patricias_size(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))),s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(t_h4s_nums_num,t_bool),V_s)))))))).
fof(ah4s_patricias_NUMSETu_u_OFu_u_PTREEu_u_def, axiom, ![V_t]: s(t_fun(t_h4s_nums_num,t_bool),h4s_patricias_numsetu_u_ofu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))) = s(t_fun(t_h4s_nums_num,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))))).
fof(ah4s_patricias_SIZEu_u_PTREEu_u_OFu_u_NUMSETu_u_EMPTY, axiom, ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),V_s)))) => s(t_h4s_nums_num,h4s_patricias_size(s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_ptreeu_u_ofu_u_numset(s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_empty),s(t_fun(t_h4s_nums_num,t_bool),V_s))))) = s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(t_h4s_nums_num,t_bool),V_s))))).
fof(ah4s_patricias_UNIONu_u_PTREEu_u_EMPTYu_c0, axiom, ![V_t]: s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_unionu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_empty))) = s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)).
fof(ah4s_patricias_PERMu_u_DELETEu_u_PTREE, axiom, ![V_uu_0]: (![V_k, V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_h4s_nums_num,V_k))),s(t_h4s_nums_num,V_x)))) <=> ~ (s(t_h4s_nums_num,V_x) = s(t_h4s_nums_num,V_k))) => ![V_t, V_k]: ((p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))) & p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_k),s(t_fun(t_h4s_nums_num,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))))))))) => p(s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_remove(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_h4s_nums_num,V_k))))),s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_filter(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_h4s_nums_num,V_k))),s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))))))))))).
fof(ah4s_patricias_PEEKu_u_INSERTu_u_PTREE, axiom, ![V_t, V_k, V_j]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))) => ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,V_j)) & s(t_h4s_options_option(t_h4s_ones_one),h4s_patricias_peek(s(t_h4s_patricias_ptree(t_h4s_ones_one),happ(s(t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one))),h4s_patricias_insertu_u_ptree),s(t_h4s_nums_num,V_k))),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))),s(t_h4s_nums_num,V_j))) = s(t_h4s_options_option(t_h4s_ones_one),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(t_h4s_ones_one),h4s_options_some(s(t_h4s_ones_one,h4s_ones_one0))),s(t_h4s_options_option(t_h4s_ones_one),h4s_patricias_peek(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_h4s_nums_num,V_j)))))))).
fof(ah4s_patriciau_u_castss_PTREEu_u_OFu_u_STRINGSETu_u_def, axiom, ![V_t, V_s]: s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patriciau_u_castss_ptreeu_u_ofu_u_stringset(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_bool),V_s))) = s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_ptreeu_u_ofu_u_numset(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_image(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_nums_num),h4s_patriciau_u_castss_stringu_u_tou_u_num),s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_bool),V_s)))))).
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_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_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_patriciau_u_castss_INu_u_PTREEsu_u_def, axiom, ![V_w, V_t]: s(t_bool,h4s_patriciau_u_castss_inu_u_ptrees(s(t_h4s_lists_list(t_h4s_strings_char),V_w),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))) = s(t_bool,h4s_patricias_inu_u_ptree(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_nums_num),h4s_patriciau_u_castss_stringu_u_tou_u_num),s(t_h4s_lists_list(t_h4s_strings_char),V_w))),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))).
fof(ah4s_patricias_INu_u_NUMSETu_u_OFu_u_PTREE, axiom, ![V_t, V_n]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))) => s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_n),s(t_fun(t_h4s_nums_num,t_bool),h4s_patricias_numsetu_u_ofu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))))) = s(t_bool,h4s_patricias_inu_u_ptree(s(t_h4s_nums_num,V_n),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))))).
fof(ah4s_lists_FOLDL0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_e]: s(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_patricias_INu_u_PTREEu_u_INSERTu_u_PTREE, axiom, ![V_t, V_n, V_m]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))) => (p(s(t_bool,h4s_patricias_inu_u_ptree(s(t_h4s_nums_num,V_n),s(t_h4s_patricias_ptree(t_h4s_ones_one),happ(s(t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one))),h4s_patricias_insertu_u_ptree),s(t_h4s_nums_num,V_m))),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))))) <=> (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n) | p(s(t_bool,h4s_patricias_inu_u_ptree(s(t_h4s_nums_num,V_n),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))))))).
fof(ah4s_patricias_ADDu_u_INSERT, axiom, ![V_v, V_t, V_n]: s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_add(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_ones_one),h4s_pairs_u_2c(s(t_h4s_nums_num,V_n),s(t_h4s_ones_one,V_v))))) = s(t_h4s_patricias_ptree(t_h4s_ones_one),happ(s(t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one))),h4s_patricias_insertu_u_ptree),s(t_h4s_nums_num,V_n))),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))).
fof(ah4s_patricias_PEEKu_u_ADD, axiom, ![TV_u_27a]: ![V_t, V_k, V_j, V_d]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(TV_u_27a),V_t)))) => ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,V_j)) & s(t_h4s_options_option(TV_u_27a),h4s_patricias_peek(s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_add(s(t_h4s_patricias_ptree(TV_u_27a),V_t),s(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_nums_num,V_k),s(TV_u_27a,V_d))))),s(t_h4s_nums_num,V_j))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_d))),s(t_h4s_options_option(TV_u_27a),h4s_patricias_peek(s(t_h4s_patricias_ptree(TV_u_27a),V_t),s(t_h4s_nums_num,V_j)))))))).
fof(ah4s_ones_one1, axiom, ![V_v]: s(t_h4s_ones_one,V_v) = s(t_h4s_ones_one,h4s_ones_one0)).
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_patricias_NUMSETu_u_OFu_u_PTREEu_u_EMPTY, axiom, s(t_fun(t_h4s_nums_num,t_bool),h4s_patricias_numsetu_u_ofu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_empty))) = s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_patricias_UNIONu_u_PTREEu_u_def, axiom, ![V_t2, V_t1]: s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_unionu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t1),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t2))) = s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_ptreeu_u_ofu_u_numset(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t1),s(t_fun(t_h4s_nums_num,t_bool),h4s_patricias_numsetu_u_ofu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t2)))))).
fof(ah4s_lists_SETu_u_TOu_u_LISTu_u_EMPTY, axiom, ![TV_u_27a]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_setu_u_tou_u_list(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f0)) => p(s(t_bool,V_t)))).
fof(ah4s_arithmetics_ADDu_u_SYM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_ADDu_u_0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_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_patricias_MEMu_u_TRAVERSEu_u_PEEK, axiom, ![TV_u_27a]: ![V_t, V_k]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(TV_u_27a),V_t)))) => s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_k),s(t_fun(t_h4s_nums_num,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(TV_u_27a),V_t))))))) = s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),h4s_patricias_peek(s(t_h4s_patricias_ptree(TV_u_27a),V_t),s(t_h4s_nums_num,V_k))))))).
fof(ah4s_patricias_SIZE0u_c0, axiom, ![TV_u_27a]: s(t_h4s_nums_num,h4s_patricias_size(s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_empty))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_patricias_EMPTYu_u_ISu_u_PTREE, axiom, ![TV_u_27a]: p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_empty))))).
fof(ah4s_lists_ALLu_u_DISTINCTu_u_SETu_u_TOu_u_LIST, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_lists_allu_u_distinct(s(t_h4s_lists_list(TV_u_27a),h4s_lists_setu_u_tou_u_list(s(t_fun(TV_u_27a,t_bool),V_s)))))))).
fof(ah4s_patricias_TRAVERSEu_u_defu_c0, axiom, ![TV_u_27a]: s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_empty))) = s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_nil)).
fof(ah4s_lists_APPEND0u_c0, axiom, ![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_nil),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),V_l)).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f0))) <=> 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_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f0))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f0)))))).
fof(ah4s_sats_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_patricias_ALLu_u_DISTINCTu_u_TRAVERSE, axiom, ![TV_u_27a]: ![V_t]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(TV_u_27a),V_t)))) => p(s(t_bool,h4s_lists_allu_u_distinct(s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(TV_u_27a),V_t)))))))).
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_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
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_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_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_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_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_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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_dcu_u_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_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_patricias_SIZEu_u_ADD, axiom, ![TV_u_27a]: ![V_t, V_k, V_d]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(TV_u_27a),V_t)))) => s(t_h4s_nums_num,h4s_patricias_size(s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_add(s(t_h4s_patricias_ptree(TV_u_27a),V_t),s(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_nums_num,V_k),s(TV_u_27a,V_d))))))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_k),s(t_fun(t_h4s_nums_num,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(TV_u_27a),V_t))))))),s(t_h4s_nums_num,h4s_patricias_size(s(t_h4s_patricias_ptree(TV_u_27a),V_t))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_patricias_size(s(t_h4s_patricias_ptree(TV_u_27a),V_t))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))))).
fof(ah4s_patricias_MEMu_u_TRAVERSEu_u_INSERTu_u_PTREE, axiom, ![V_x, V_t, V_h]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))) => (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_x),s(t_fun(t_h4s_nums_num,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(t_h4s_ones_one),happ(s(t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one))),h4s_patricias_insertu_u_ptree),s(t_h4s_nums_num,V_h))),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))))))))) <=> (s(t_h4s_nums_num,V_x) = s(t_h4s_nums_num,V_h) | (~ (s(t_h4s_nums_num,V_x) = s(t_h4s_nums_num,V_h)) & p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_x),s(t_fun(t_h4s_nums_num,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))))))))))))).
fof(ah4s_patricias_INSERTu_u_PTREEu_u_ISu_u_PTREE, axiom, ![V_x, V_t]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))) => p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),happ(s(t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one))),h4s_patricias_insertu_u_ptree),s(t_h4s_nums_num,V_x))),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))))))).
fof(ah4s_bools_boolu_u_caseu_u_thmu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_patricias_INSERTu_u_PTREEu_u_def, axiom, ![V_t, V_n]: s(t_h4s_patricias_ptree(t_h4s_ones_one),happ(s(t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one))),h4s_patricias_insertu_u_ptree),s(t_h4s_nums_num,V_n))),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))) = s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_add(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_ones_one),h4s_pairs_u_2c(s(t_h4s_nums_num,V_n),s(t_h4s_ones_one,h4s_ones_one0)))))).
fof(ah4s_lists_SETu_u_TOu_u_LISTu_u_CARD, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),h4s_lists_setu_u_tou_u_list(s(t_fun(TV_u_27a,t_bool),V_s))))) = s(t_h4s_nums_num,h4s_predu_u_sets_card(s(t_fun(TV_u_27a,t_bool),V_s))))).
fof(ah4s_lists_ALLu_u_DISTINCTu_u_APPEND, 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),h4s_lists_append(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_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_e]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_e),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)))))) => ~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_e),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_lists_ALLu_u_DISTINCT0u_c1, axiom, ![TV_u_27a]: ![V_t, V_h]: (p(s(t_bool,h4s_lists_allu_u_distinct(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)))))) <=> (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_h),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))))))) & p(s(t_bool,h4s_lists_allu_u_distinct(s(t_h4s_lists_list(TV_u_27a),V_t))))))).
fof(ah4s_lists_APPENDu_u_NIL, axiom, ![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_nil))) = s(t_h4s_lists_list(TV_u_27a),V_l)).
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_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_FOLDL0u_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_l, V_f, V_e]: s(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,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_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_numerals_numeralu_u_addu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_m))))))) = s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_numerals_numeralu_u_lteu_c1, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f0)).
fof(ah4s_numerals_numeralu_u_addu_c0, axiom, ![V_n]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_numerals_numeralu_u_distribu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_numerals_numeralu_u_distribu_c27, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ah4s_numerals_numeralu_u_distribu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_NOTu_u_NUMu_u_EQ, axiom, ![V_n, V_m]: (~ (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n)) <=> (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))) | p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m))))))).
fof(ah4s_arithmetics_EQu_u_MONOu_u_ADDu_u_EQ, axiom, ![V_p, V_n, V_m]: (s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_arithmetics_SUCu_u_ONEu_u_ADD, axiom, ![V_n]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c2, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_ASSOC, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p)))).
fof(ah4s_combins_Cu_u_THM, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27c)),h4s_combins_c(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(TV_u_27b,V_x))),s(TV_u_27a,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_y))),s(TV_u_27b,V_x)))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(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_RIGHTu_u_ORu_u_OVERu_u_AND, axiom, ![V_C, V_B, V_A]: (((p(s(t_bool,V_B)) & p(s(t_bool,V_C))) | p(s(t_bool,V_A))) <=> ((p(s(t_bool,V_B)) | p(s(t_bool,V_A))) & (p(s(t_bool,V_C)) | p(s(t_bool,V_A)))))).
fof(ah4s_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_IMPu_u_Fu_u_EQu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f0))) <=> s(t_bool,V_t) = s(t_bool,f0))).
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_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_patricias_INu_u_PTREEu_u_def, axiom, ![V_t, V_n]: s(t_bool,h4s_patricias_inu_u_ptree(s(t_h4s_nums_num,V_n),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))) = s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(t_h4s_ones_one),h4s_patricias_peek(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f0)))).
fof(ah4s_patricias_MEMu_u_ALLu_u_DISTINCTu_u_IMPu_u_PERM, 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_patricias_PEEKu_u_REMOVE, axiom, ![TV_u_27a]: ![V_t, V_k, V_j]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(TV_u_27a),V_t)))) => ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,V_j)) & s(t_h4s_options_option(TV_u_27a),h4s_patricias_peek(s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_remove(s(t_h4s_patricias_ptree(TV_u_27a),V_t),s(t_h4s_nums_num,V_k))),s(t_h4s_nums_num,V_j))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_h4s_options_option(TV_u_27a),h4s_patricias_peek(s(t_h4s_patricias_ptree(TV_u_27a),V_t),s(t_h4s_nums_num,V_j)))))))).
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_patricias_REMOVEu_u_ISu_u_PTREE, axiom, ![TV_u_27a]: ![V_t, V_k]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(TV_u_27a),V_t)))) => p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_remove(s(t_h4s_patricias_ptree(TV_u_27a),V_t),s(t_h4s_nums_num,V_k)))))))).
fof(ah4s_richu_u_lists_FILTERu_u_FILTER, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_P, V_Q, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_fun(TV_u_27a,t_bool),V_Q))),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_x)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x)))))) => ![V_l, V_Q, 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_filter(s(t_fun(TV_u_27a,t_bool),V_Q),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),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(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_bool))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_fun(TV_u_27a,t_bool),V_Q))),s(t_h4s_lists_list(TV_u_27a),V_l))))).
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_u_FILTER, axiom, ![TV_u_27a]: ![V_x, V_P, 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),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_L)))))))) <=> (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,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))))))))).
fof(ah4s_options_ISu_u_SOMEu_u_DEFu_c1, axiom, ![TV_u_27a]: s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_bool,f0)).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f0)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f0)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,f0))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_LEFTu_u_ORu_u_OVERu_u_AND, 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_A)) | p(s(t_bool,V_C)))))).
fof(ah4s_bools_EXISTSu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_x)))))).
fof(ch4s_patriciau_u_castss_wordu_u_ptreeu_u_Axiom, conjecture, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_f]: ?[V_fn]: ![V_a0, V_a1]: s(TV_u_27c,happ(s(t_fun(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),TV_u_27c),V_fn),s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b))),h4s_patriciau_u_castss_wordu_u_ptree),s(t_fun(TV_u_27a,t_h4s_ones_one),V_a0))),s(t_h4s_patricias_ptree(TV_u_27b),V_a1))))) = s(TV_u_27c,happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27b),TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27b),TV_u_27c)),V_f),s(t_fun(TV_u_27a,t_h4s_ones_one),V_a0))),s(t_h4s_patricias_ptree(TV_u_27b),V_a1)))).
