%   ORIGINAL: h4/patricia__casts/ADD__INSERT__WORD
% 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/one/one__axiom: !g f. f = g
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/one/one1: !v. v = h4/one/one0
% Assm: h4/patricia__casts/ADD__LISTw__def: h4/patricia__casts/ADD__LISTw = h4/list/FOLDL h4/patricia__casts/ADDw
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/one/one__induction: !P. P h4/one/one0 ==> (!x. P x)
% Assm: h4/bool/boolAxiom: !t2 t1. ?fn. fn T = t1 /\ fn F = t2
% Assm: h4/patricia__casts/ADDw__def: !w t d. h4/patricia__casts/ADDw t (h4/pair/_2C w d) = h4/patricia__casts/SOME__PTREE (h4/patricia/ADD (h4/patricia__casts/THE__PTREE t) (h4/pair/_2C (h4/words/w2n w) d))
% Assm: h4/one/one__Axiom: !e. h4/bool/_3F_21 (\fn. fn h4/one/one0 = e)
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/patricia__casts/ADD__INSERT__STRING: !w v t. h4/patricia__casts/ADDs t (h4/pair/_2C w v) = h4/patricia__casts/ADDs t (h4/pair/_2C w h4/one/one0)
% Assm: h4/one/FORALL__ONE: !P. (!x. P x) <=> P h4/one/one0
% Assm: h4/patricia__casts/INSERT__PTREEw__def: !w t. h4/patricia__casts/INSERT__PTREEw w t = h4/patricia__casts/SOME__PTREE (h4/patricia/INSERT__PTREE (h4/words/w2n w) (h4/patricia__casts/THE__PTREE t))
% Assm: h4/patricia__casts/IN__PTREEw__def: !w t. h4/patricia__casts/IN__PTREEw w t <=> h4/patricia/IN__PTREE (h4/words/w2n w) (h4/patricia__casts/THE__PTREE t)
% Assm: h4/patricia__casts/PTREE__OF__WORDSET__def: !t s. h4/patricia__casts/PTREE__OF__WORDSET t s = h4/patricia__casts/SOME__PTREE (h4/patricia/PTREE__OF__NUMSET (h4/patricia__casts/THE__PTREE t) (h4/pred__set/IMAGE h4/words/w2n s))
% Assm: h4/patricia__casts/WORDSET__OF__PTREE__def: !t. h4/patricia__casts/WORDSET__OF__PTREE t = h4/list/LIST__TO__SET (h4/patricia__casts/TRAVERSEw 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/INSERT__PTREE__def: !t n. h4/patricia/INSERT__PTREE n t = h4/patricia/ADD t (h4/pair/_2C n h4/one/one0)
% Assm: h4/bool/EXISTS__UNIQUE__DEF: h4/bool/_3F_21 = (\P. $exists P /\ (!x y. P x /\ P y ==> x = y))
% Assm: h4/one/one__case__def: !x u. h4/one/one__CASE u x = x
% Assm: h4/option/OPTION__GUARD__def_c0: h4/option/OPTION__GUARD T = h4/option/SOME h4/one/one0
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/patricia__casts/SOME__PTREE__def: !t. h4/patricia__casts/SOME__PTREE t = h4/patricia__casts/Word__ptree (h4/combin/K h4/one/one0) t
% Assm: h4/basicSize/one__size__def: !x. h4/basicSize/one__size x = h4/num/0
% Assm: h4/one/one__case__thm: !x. h4/one/one__CASE h4/one/one0 x = x
% Assm: h4/option/NONE__DEF: h4/option/NONE = h4/option/option__ABS (h4/sum/INR h4/one/one0)
% Assm: h4/one/one__prim__rec: !e. ?fn. fn h4/one/one0 = e
% Assm: h4/option/OPTION__GUARD__COND: !b. h4/option/OPTION__GUARD b = h4/bool/COND b (h4/option/SOME h4/one/one0) h4/option/NONE
% Assm: h4/one/one__DEF: h4/one/one0 = h4/min/_40 (\x. T)
% Assm: h4/option/OPTION__GUARD__EQ__THM_c0: !b. h4/option/OPTION__GUARD b = h4/option/SOME h4/one/one0 <=> b
% 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/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/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/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/patricia__casts/Word__ptree0: h4/patricia__casts/Word__ptree = h4/patricia__casts/_20_40ind__typepatricia__casts0
% 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/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% 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/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/option/OPTION__GUARD__def_c1: h4/option/OPTION__GUARD F = h4/option/NONE
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% 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/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/bool/DATATYPE__TAG__THM: !x. h4/bool/DATATYPE x <=> T
% 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/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/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% 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/UNION__PTREEw__def: !t2 t1. h4/patricia__casts/UNION__PTREEw t1 t2 = h4/patricia__casts/SOME__PTREE (h4/patricia/UNION__PTREE (h4/patricia__casts/THE__PTREE t1) (h4/patricia__casts/THE__PTREE t2))
% Assm: h4/patricia__casts/word__ptree__Axiom: !f. ?fn. !a0 a1. fn (h4/patricia__casts/Word__ptree a0 a1) = f a0 a1
% Assm: h4/patricia__casts/THE__PTREE__def: !t a. h4/patricia__casts/THE__PTREE (h4/patricia__casts/Word__ptree a t) = t
% 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__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/patricia__casts/datatype__word__ptree: !word__ptree. h4/bool/DATATYPE (word__ptree h4/patricia__casts/Word__ptree)
% Assm: h4/patricia__casts/word__ptree__induction: !P. (!f p. P (h4/patricia__casts/Word__ptree f p)) ==> (!w. P w)
% Assm: h4/patricia__casts/EVERY__LEAFw__def: !t P. h4/patricia__casts/EVERY__LEAFw P t <=> h4/patricia/EVERY__LEAF (\k d. P (h4/words/n2w k) d) (h4/patricia__casts/THE__PTREE t)
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/option/NOT__NONE__SOME: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/patricia__casts/KEYSw__def: !t. h4/patricia__casts/KEYSw t = h4/sorting/QSORT h4/words/word__lo (h4/patricia__casts/TRAVERSEw t)
% Assm: h4/patricia__casts/REMOVEw__def: !w t. h4/patricia__casts/REMOVEw t w = h4/patricia__casts/SOME__PTREE (h4/patricia/REMOVE (h4/patricia__casts/THE__PTREE t) (h4/words/w2n w))
% Assm: h4/patricia__casts/TRAVERSEw__def: !t. h4/patricia__casts/TRAVERSEw t = h4/list/MAP h4/words/n2w (h4/patricia/TRAVERSE (h4/patricia__casts/THE__PTREE t))
% Assm: h4/patricia__casts/PEEKw__def: !w t. h4/patricia__casts/PEEKw t w = h4/patricia/PEEK (h4/patricia__casts/THE__PTREE t) (h4/words/w2n w)
% Assm: h4/patricia__casts/FINDw__def: !w t. h4/patricia__casts/FINDw t w = h4/option/THE (h4/patricia__casts/PEEKw t w)
% Assm: h4/patricia__casts/EXISTS__LEAFw__def: !t P. h4/patricia__casts/EXISTS__LEAFw P t <=> h4/patricia/EXISTS__LEAF (\k d. P (h4/words/n2w k) d) (h4/patricia__casts/THE__PTREE t)
% Assm: h4/patricia__casts/DEPTHw__def: !t. h4/patricia__casts/DEPTHw t = h4/patricia/DEPTH (h4/patricia__casts/THE__PTREE t)
% Assm: h4/patricia__casts/SIZEw__def: !t. h4/patricia__casts/SIZEw t = h4/patricia/SIZE (h4/patricia__casts/THE__PTREE t)
% Assm: h4/patricia__casts/WordEmpty__def: h4/patricia__casts/WordEmpty = h4/patricia__casts/SOME__PTREE h4/patricia/Empty
% Assm: h4/patricia__casts/TRANSFORMw__def: !t f. h4/patricia__casts/TRANSFORMw f t = h4/patricia__casts/SOME__PTREE (h4/patricia/TRANSFORM f (h4/patricia__casts/THE__PTREE t))
% Assm: h4/pair/S__UNCURRY__R: !g f. h4/combin/S f (h4/pair/UNCURRY g) = h4/pair/UNCURRY (h4/combin/S (h4/combin/o h4/combin/S (h4/combin/o (h4/combin/o f) h4/pair/_2C)) g)
% Assm: h4/pair/datatype__pair: !pair. h4/bool/DATATYPE (pair h4/pair/_2C)
% Assm: h4/words/word__compare__def: !b a. h4/words/word__compare a b = h4/bool/COND (a = b) (h4/words/n2w (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/words/n2w h4/num/0)
% Assm: h4/words/foldl__reduce__or: !w. h4/words/reduce__or w = h4/bool/LET (\l. h4/list/FOLDL h4/words/word__or (h4/list/HD l) (h4/list/TL l)) (h4/list/GENLIST (\i. h4/bool/LET (\n. h4/words/word__extract n n w) (h4/arithmetic/_2D (h4/arithmetic/_2D (h4/fcp/dimindex h4/bool/the__value) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) i)) (h4/fcp/dimindex h4/bool/the__value))
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/pair/UNCURRY0: !v f. h4/pair/UNCURRY f v = f (h4/pair/FST v) (h4/pair/SND v)
% Assm: h4/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm: h4/combin/S__THM: !x g f. h4/combin/S f g x = f x (g x)
% Assm: h4/bool/EQ__EXPAND: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% Assm: h4/list/FOLDL0_c0: !f e. h4/list/FOLDL f e h4/list/NIL = e
% Assm: h4/list/MAP0_c0: !f. h4/list/MAP f h4/list/NIL = h4/list/NIL
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% 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/bool/LET__THM: !x f. h4/bool/LET f x = f x
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> 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/bool/EQ__REFL: !x. x = x
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% 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_c2: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT2 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_c1: !n. h4/arithmetic/_2B n h4/num/0 = 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/arithmetic/MIN__LE_c1: !p n m. h4/arithmetic/_3C_3D p (h4/arithmetic/MIN m n) <=> h4/arithmetic/_3C_3D p m /\ h4/arithmetic/_3C_3D p n
% Assm: h4/list/MAP__TL: !l f. ~h4/list/NULL l ==> h4/list/MAP f (h4/list/TL l) = h4/list/TL (h4/list/MAP f l)
% 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/arithmetic/SUB__RIGHT__EQ: !p n m. h4/arithmetic/_2D m n = p <=> m = h4/arithmetic/_2B n p \/ h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D p h4/num/0
% Assm: h4/arithmetic/SUB__LEFT__EQ: !p n m. m = h4/arithmetic/_2D n p <=> h4/arithmetic/_2B m p = n \/ h4/arithmetic/_3C_3D m h4/num/0 /\ h4/arithmetic/_3C_3D n p
% 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/SUB__LEFT__ADD: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2D n p) = h4/bool/COND (h4/arithmetic/_3C_3D n p) m (h4/arithmetic/_2D (h4/arithmetic/_2B m 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/NOT__LEQ: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm: h4/arithmetic/EQ__LESS__EQ: !n m. m = n <=> h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% 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/arithmetic/LESS__EQ__TRANS: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm: h4/arithmetic/SUB__LEFT__LESS__EQ: !p n m. h4/arithmetic/_3C_3D m (h4/arithmetic/_2D n p) <=> h4/arithmetic/_3C_3D (h4/arithmetic/_2B m p) n \/ h4/arithmetic/_3C_3D m h4/num/0
% Assm: h4/arithmetic/ADD__MONO__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm: h4/arithmetic/LESS__EQ__REFL: !m. h4/arithmetic/_3C_3D m m
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Goal: !w v t. h4/patricia__casts/ADDw t (h4/pair/_2C w v) = h4/patricia__casts/ADDw t (h4/pair/_2C w h4/one/one0)
%   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_ones_oneu_u_axiom]: !g f. f = g
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. happ (h4/pair/UNCURRY f) (happ (happ h4/pair/_2C x) y) = happ (happ f x) y
% Assm [h4s_ones_one1]: !v. v = h4/one/one0
% Assm [h4s_patriciau_u_castss_ADDu_u_LISTwu_u_def]: h4/patricia__casts/ADD__LISTw = h4/list/FOLDL h4/patricia__casts/ADDw
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_ones_oneu_u_induction]: !P. happ P h4/one/one0 ==> (!x. happ P x)
% Assm [h4s_bools_boolAxiom]: !t2 t1. ?fn. happ fn T = t1 /\ happ fn F = t2
% Assm [h4s_patriciau_u_castss_ADDwu_u_def]: !w t d. happ (happ h4/patricia__casts/ADDw t) (happ (happ h4/pair/_2C w) d) = h4/patricia__casts/SOME__PTREE (h4/patricia/ADD (h4/patricia__casts/THE__PTREE t) (happ (happ h4/pair/_2C (happ h4/words/w2n w)) d))
% Assm [h4s_ones_oneu_u_Axiom]: !_0. (!e fn. happ (happ _0 e) fn <=> happ fn h4/one/one0 = e) ==> (!e. h4/bool/_3F_21 (happ _0 e))
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_patriciau_u_castss_ADDu_u_INSERTu_u_STRING]: !w v t. h4/patricia__casts/ADDs t (happ (happ h4/pair/_2C w) v) = h4/patricia__casts/ADDs t (happ (happ h4/pair/_2C w) h4/one/one0)
% Assm [h4s_ones_FORALLu_u_ONE]: !P. (!x. happ P x) <=> happ P h4/one/one0
% Assm [h4s_patriciau_u_castss_INSERTu_u_PTREEwu_u_def]: !w t. h4/patricia__casts/INSERT__PTREEw w t = h4/patricia__casts/SOME__PTREE (h4/patricia/INSERT__PTREE (happ h4/words/w2n w) (h4/patricia__casts/THE__PTREE t))
% Assm [h4s_patriciau_u_castss_INu_u_PTREEwu_u_def]: !w t. h4/patricia__casts/IN__PTREEw w t <=> h4/patricia/IN__PTREE (happ h4/words/w2n w) (h4/patricia__casts/THE__PTREE t)
% Assm [h4s_patriciau_u_castss_PTREEu_u_OFu_u_WORDSETu_u_def]: !t s. h4/patricia__casts/PTREE__OF__WORDSET t s = h4/patricia__casts/SOME__PTREE (h4/patricia/PTREE__OF__NUMSET (h4/patricia__casts/THE__PTREE t) (h4/pred__set/IMAGE h4/words/w2n s))
% Assm [h4s_patriciau_u_castss_WORDSETu_u_OFu_u_PTREEu_u_def]: !t. h4/patricia__casts/WORDSET__OF__PTREE t = h4/list/LIST__TO__SET (h4/patricia__casts/TRAVERSEw t)
% Assm [h4s_patricias_ADDu_u_INSERT]: !v t n. h4/patricia/ADD t (happ (happ h4/pair/_2C n) v) = h4/patricia/INSERT__PTREE n t
% Assm [h4s_patricias_INSERTu_u_PTREEu_u_def]: !t n. h4/patricia/INSERT__PTREE n t = h4/patricia/ADD t (happ (happ h4/pair/_2C n) h4/one/one0)
% Assm [h4s_bools_EXISTSu_u_UNIQUEu_u_DEF]: !x. h4/bool/_3F_21 x <=> $exists x /\ (!x y. happ x x /\ happ x y ==> x = y)
% Assm [h4s_ones_oneu_u_caseu_u_def]: !x u. h4/one/one__CASE u x = x
% Assm [h4s_options_OPTIONu_u_GUARDu_u_defu_c0]: h4/option/OPTION__GUARD T = h4/option/SOME h4/one/one0
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_patriciau_u_castss_SOMEu_u_PTREEu_u_def]: !t. h4/patricia__casts/SOME__PTREE t = happ (happ h4/patricia__casts/Word__ptree (h4/combin/K h4/one/one0)) t
% Assm [h4s_basicSizes_oneu_u_sizeu_u_def]: !x. h4/basicSize/one__size x = h4/num/0
% Assm [h4s_ones_oneu_u_caseu_u_thm]: !x. h4/one/one__CASE h4/one/one0 x = x
% Assm [h4s_options_NONEu_u_DEF]: h4/option/NONE = h4/option/option__ABS (h4/sum/INR h4/one/one0)
% Assm [h4s_ones_oneu_u_primu_u_rec]: !e. ?fn. happ fn h4/one/one0 = e
% Assm [h4s_options_OPTIONu_u_GUARDu_u_COND]: !b. h4/option/OPTION__GUARD b = h4/bool/COND b (h4/option/SOME h4/one/one0) h4/option/NONE
% Assm [h4s_ones_oneu_u_DEF]: !_0. (!x. happ _0 x <=> T) ==> h4/one/one0 = h4/min/_40 _0
% Assm [h4s_options_OPTIONu_u_GUARDu_u_EQu_u_THMu_c0]: !b. h4/option/OPTION__GUARD b = h4/option/SOME h4/one/one0 <=> b
% Assm [h4s_patricias_PEEKu_u_INSERTu_u_PTREE]: !t k j. h4/patricia/IS__PTREE t ==> (?v. (v <=> k = j) /\ h4/patricia/PEEK (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_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 (happ (happ 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_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 (happ (happ 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 (happ (happ h4/pair/_2C x) x) _0))
% 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 (happ (happ h4/pair/_2C k) d)) j = h4/bool/COND v (h4/option/SOME d) (h4/patricia/PEEK t j))
% Assm [h4s_patriciau_u_castss_Wordu_u_ptree0]: h4/patricia__casts/Word__ptree = h4/patricia__casts/_20_40ind__typepatricia__casts0
% 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_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% 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_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_options_OPTIONu_u_GUARDu_u_defu_c1]: h4/option/OPTION__GUARD F = h4/option/NONE
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (happ (happ 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_pairs_SND0]: !y x. h4/pair/SND (happ (happ h4/pair/_2C x) y) = y
% Assm [h4s_bools_DATATYPEu_u_TAGu_u_THM]: !x. h4/bool/DATATYPE x <=> T
% 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_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_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% 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_UNIONu_u_PTREEwu_u_def]: !t2 t1. h4/patricia__casts/UNION__PTREEw t1 t2 = h4/patricia__casts/SOME__PTREE (h4/patricia/UNION__PTREE (h4/patricia__casts/THE__PTREE t1) (h4/patricia__casts/THE__PTREE t2))
% Assm [h4s_patriciau_u_castss_wordu_u_ptreeu_u_Axiom]: !f. ?fn. !a0 a1. happ fn (happ (happ h4/patricia__casts/Word__ptree a0) a1) = happ (happ f a0) a1
% Assm [h4s_patriciau_u_castss_THEu_u_PTREEu_u_def]: !t a. h4/patricia__casts/THE__PTREE (happ (happ h4/patricia__casts/Word__ptree a) t) = t
% 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_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_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_induction]: !P. (!f p. happ P (happ (happ h4/patricia__casts/Word__ptree f) p)) ==> (!w. happ P w)
% Assm [h4s_patriciau_u_castss_EVERYu_u_LEAFwu_u_def]: !_1. (!P k d. happ (happ (happ _1 P) k) d <=> happ (happ P (happ h4/words/n2w k)) d) ==> (!_0. (!P k. happ (happ _0 P) k = happ (happ _1 P) k) ==> (!t P. h4/patricia__casts/EVERY__LEAFw P t <=> h4/patricia/EVERY__LEAF (happ _0 P) (h4/patricia__casts/THE__PTREE t)))
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_options_NOTu_u_NONEu_u_SOME]: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm [h4s_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_patriciau_u_castss_KEYSwu_u_def]: !t. h4/patricia__casts/KEYSw t = h4/sorting/QSORT h4/words/word__lo (h4/patricia__casts/TRAVERSEw t)
% Assm [h4s_patriciau_u_castss_REMOVEwu_u_def]: !w t. h4/patricia__casts/REMOVEw t w = h4/patricia__casts/SOME__PTREE (h4/patricia/REMOVE (h4/patricia__casts/THE__PTREE t) (happ h4/words/w2n w))
% Assm [h4s_patriciau_u_castss_TRAVERSEwu_u_def]: !t. h4/patricia__casts/TRAVERSEw t = h4/list/MAP h4/words/n2w (h4/patricia/TRAVERSE (h4/patricia__casts/THE__PTREE t))
% Assm [h4s_patriciau_u_castss_PEEKwu_u_def]: !w t. h4/patricia__casts/PEEKw t w = h4/patricia/PEEK (h4/patricia__casts/THE__PTREE t) (happ h4/words/w2n w)
% Assm [h4s_patriciau_u_castss_FINDwu_u_def]: !w t. h4/patricia__casts/FINDw t w = h4/option/THE (h4/patricia__casts/PEEKw t w)
% Assm [h4s_patriciau_u_castss_EXISTSu_u_LEAFwu_u_def]: !_1. (!P k d. happ (happ (happ _1 P) k) d <=> happ (happ P (happ h4/words/n2w k)) d) ==> (!_0. (!P k. happ (happ _0 P) k = happ (happ _1 P) k) ==> (!t P. h4/patricia__casts/EXISTS__LEAFw P t <=> h4/patricia/EXISTS__LEAF (happ _0 P) (h4/patricia__casts/THE__PTREE t)))
% Assm [h4s_patriciau_u_castss_DEPTHwu_u_def]: !t. h4/patricia__casts/DEPTHw t = h4/patricia/DEPTH (h4/patricia__casts/THE__PTREE t)
% Assm [h4s_patriciau_u_castss_SIZEwu_u_def]: !t. h4/patricia__casts/SIZEw t = h4/patricia/SIZE (h4/patricia__casts/THE__PTREE t)
% Assm [h4s_patriciau_u_castss_WordEmptyu_u_def]: h4/patricia__casts/WordEmpty = h4/patricia__casts/SOME__PTREE h4/patricia/Empty
% Assm [h4s_patriciau_u_castss_TRANSFORMwu_u_def]: !t f. h4/patricia__casts/TRANSFORMw f t = h4/patricia__casts/SOME__PTREE (h4/patricia/TRANSFORM f (h4/patricia__casts/THE__PTREE t))
% Assm [h4s_pairs_Su_u_UNCURRYu_u_R]: !g f. happ (happ h4/combin/S f) (h4/pair/UNCURRY g) = h4/pair/UNCURRY (happ (happ h4/combin/S (happ (h4/combin/o h4/combin/S) (happ (h4/combin/o (h4/combin/o f)) h4/pair/_2C))) g)
% Assm [h4s_pairs_datatypeu_u_pair]: !pair. h4/bool/DATATYPE (happ pair h4/pair/_2C)
% Assm [h4s_wordss_wordu_u_compareu_u_def]: !b a. ?v. (v <=> a = b) /\ h4/words/word__compare a b = h4/bool/COND v (happ h4/words/n2w (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (happ h4/words/n2w h4/num/0)
% Assm [h4s_wordss_foldlu_u_reduceu_u_or]: !_2. (!w n. happ (happ _2 w) n = h4/words/word__extract n n w) ==> (!_1. (!w i. happ (happ _1 w) i = h4/bool/LET (happ _2 w) (h4/arithmetic/_2D (h4/arithmetic/_2D (h4/fcp/dimindex h4/bool/the__value) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) i)) ==> (!_0. (!l. happ _0 l = happ (happ (h4/list/FOLDL h4/words/word__or) (h4/list/HD l)) (h4/list/TL l)) ==> (!w. h4/words/reduce__or w = h4/bool/LET _0 (h4/list/GENLIST (happ _1 w) (h4/fcp/dimindex h4/bool/the__value)))))
% Assm [h4s_combins_ou_u_THM]: !x g f. happ (happ (h4/combin/o f) g) x = happ f (happ g x)
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_pairs_UNCURRY0]: !v f. happ (h4/pair/UNCURRY f) v = happ (happ f (h4/pair/FST v)) (h4/pair/SND v)
% Assm [h4s_pairs_PAIR]: !x. happ (happ h4/pair/_2C (h4/pair/FST x)) (h4/pair/SND x) = x
% Assm [h4s_combins_Su_u_THM]: !x g f. happ (happ (happ h4/combin/S f) g) x = happ (happ f x) (happ g x)
% Assm [h4s_bools_EQu_u_EXPAND]: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% Assm [h4s_lists_FOLDL0u_c0]: !f e. happ (happ (h4/list/FOLDL f) e) h4/list/NIL = e
% Assm [h4s_lists_MAP0u_c0]: !f. h4/list/MAP f h4/list/NIL = h4/list/NIL
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_lists_FOLDL0u_c1]: !x l f e. happ (happ (h4/list/FOLDL f) e) (h4/list/CONS x l) = happ (happ (h4/list/FOLDL f) (happ (happ f e) x)) l
% Assm [h4s_bools_LETu_u_THM]: !x f. h4/bool/LET f x = happ f x
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> 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_bools_EQu_u_REFL]: !x. x = x
% 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_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_c2]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT2 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_c1]: !n. h4/arithmetic/_2B n h4/num/0 = 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_arithmetics_MINu_u_LEu_c1]: !p n m. h4/arithmetic/_3C_3D p (h4/arithmetic/MIN m n) <=> h4/arithmetic/_3C_3D p m /\ h4/arithmetic/_3C_3D p n
% Assm [h4s_lists_MAPu_u_TL]: !l f. ~h4/list/NULL l ==> h4/list/MAP f (h4/list/TL l) = h4/list/TL (h4/list/MAP f l)
% 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_arithmetics_SUBu_u_RIGHTu_u_EQ]: !p n m. h4/arithmetic/_2D m n = p <=> m = h4/arithmetic/_2B n p \/ h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D p h4/num/0
% Assm [h4s_arithmetics_SUBu_u_LEFTu_u_EQ]: !p n m. m = h4/arithmetic/_2D n p <=> h4/arithmetic/_2B m p = n \/ h4/arithmetic/_3C_3D m h4/num/0 /\ h4/arithmetic/_3C_3D n p
% 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_SUBu_u_LEFTu_u_ADD]: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2D n p) = h4/bool/COND (h4/arithmetic/_3C_3D n p) m (h4/arithmetic/_2D (h4/arithmetic/_2B m 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_NOTu_u_LEQ]: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm [h4s_arithmetics_EQu_u_LESSu_u_EQ]: !n m. m = n <=> h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% 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_arithmetics_LESSu_u_EQu_u_TRANS]: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm [h4s_arithmetics_SUBu_u_LEFTu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D m (h4/arithmetic/_2D n p) <=> h4/arithmetic/_3C_3D (h4/arithmetic/_2B m p) n \/ h4/arithmetic/_3C_3D m h4/num/0
% Assm [h4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm [h4s_arithmetics_LESSu_u_EQu_u_REFL]: !m. h4/arithmetic/_3C_3D m m
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Goal: !w v t. happ (happ h4/patricia__casts/ADDw t) (happ (happ h4/pair/_2C w) v) = happ (happ h4/patricia__casts/ADDw t) (happ (happ h4/pair/_2C w) h4/one/one0)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t0))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q1295504,TV_Q1295500]: ![V_f, V_g]: (![V_x]: s(TV_Q1295500,happ(s(t_fun(TV_Q1295504,TV_Q1295500),V_f),s(TV_Q1295504,V_x))) = s(TV_Q1295500,happ(s(t_fun(TV_Q1295504,TV_Q1295500),V_g),s(TV_Q1295504,V_x))) => s(t_fun(TV_Q1295504,TV_Q1295500),V_f) = s(t_fun(TV_Q1295504,TV_Q1295500),V_g))).
fof(ah4s_ones_oneu_u_axiom, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(TV_u_27a,t_h4s_ones_one),V_f) = s(t_fun(TV_u_27a,t_h4s_ones_one),V_g)).
fof(ah4s_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),h4s_pairs_u_2c),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_ones_one1, axiom, ![V_v]: s(t_h4s_ones_one,V_v) = s(t_h4s_ones_one,h4s_ones_one0)).
fof(ah4s_patriciau_u_castss_ADDu_u_LISTwu_u_def, axiom, ![TV_u_27a,TV_u_27b]: s(t_fun(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),TV_u_27b)),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b))),h4s_patriciau_u_castss_addu_u_listw) = s(t_fun(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),t_fun(t_h4s_lists_list(t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),TV_u_27b)),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b))),h4s_lists_foldl(s(t_fun(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b))),h4s_patriciau_u_castss_addw)))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t0))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) <=> p(s(t_bool,V_t)))).
fof(ah4s_ones_oneu_u_induction, axiom, ![V_P]: (p(s(t_bool,happ(s(t_fun(t_h4s_ones_one,t_bool),V_P),s(t_h4s_ones_one,h4s_ones_one0)))) => ![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_ones_one,t_bool),V_P),s(t_h4s_ones_one,V_x)))))).
fof(ah4s_bools_boolAxiom, axiom, ![TV_u_27a]: ![V_t2, V_t1]: ?[V_fn]: (s(TV_u_27a,happ(s(t_fun(t_bool,TV_u_27a),V_fn),s(t_bool,t0))) = s(TV_u_27a,V_t1) & s(TV_u_27a,happ(s(t_fun(t_bool,TV_u_27a),V_fn),s(t_bool,f))) = s(TV_u_27a,V_t2))).
fof(ah4s_patriciau_u_castss_ADDwu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_w, V_t, V_d]: s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),t_fun(t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),TV_u_27b),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b))),h4s_patriciau_u_castss_addw),s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),V_t))),s(t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),TV_u_27b)),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(TV_u_27b,t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),TV_u_27b))),h4s_pairs_u_2c),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))),s(TV_u_27b,V_d))))) = s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),h4s_patriciau_u_castss_someu_u_ptree(s(t_h4s_patricias_ptree(TV_u_27b),h4s_patricias_add(s(t_h4s_patricias_ptree(TV_u_27b),h4s_patriciau_u_castss_theu_u_ptree(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),V_t))),s(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27b))),h4s_pairs_u_2c),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_nums_num),h4s_wordss_w2n),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))))),s(TV_u_27b,V_d)))))))).
fof(ah4s_ones_oneu_u_Axiom, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_e, V_fn]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_ones_one,TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_ones_one,TV_u_27a),t_bool)),V_uu_0),s(TV_u_27a,V_e))),s(t_fun(t_h4s_ones_one,TV_u_27a),V_fn)))) <=> s(TV_u_27a,happ(s(t_fun(t_h4s_ones_one,TV_u_27a),V_fn),s(t_h4s_ones_one,h4s_ones_one0))) = s(TV_u_27a,V_e)) => ![V_e]: p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(t_fun(t_h4s_ones_one,TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_ones_one,TV_u_27a),t_bool)),V_uu_0),s(TV_u_27a,V_e)))))))).
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_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_ADDu_u_INSERTu_u_STRING, axiom, ![V_w, V_v, V_t]: s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patriciau_u_castss_adds(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_strings_char),t_h4s_ones_one),happ(s(t_fun(t_h4s_ones_one,t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_strings_char),t_h4s_ones_one)),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_fun(t_h4s_ones_one,t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_strings_char),t_h4s_ones_one))),h4s_pairs_u_2c),s(t_h4s_lists_list(t_h4s_strings_char),V_w))),s(t_h4s_ones_one,V_v))))) = s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patriciau_u_castss_adds(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_strings_char),t_h4s_ones_one),happ(s(t_fun(t_h4s_ones_one,t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_strings_char),t_h4s_ones_one)),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_fun(t_h4s_ones_one,t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_strings_char),t_h4s_ones_one))),h4s_pairs_u_2c),s(t_h4s_lists_list(t_h4s_strings_char),V_w))),s(t_h4s_ones_one,h4s_ones_one0)))))).
fof(ah4s_ones_FORALLu_u_ONE, axiom, ![V_P]: (![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_ones_one,t_bool),V_P),s(t_h4s_ones_one,V_x)))) <=> p(s(t_bool,happ(s(t_fun(t_h4s_ones_one,t_bool),V_P),s(t_h4s_ones_one,h4s_ones_one0)))))).
fof(ah4s_patriciau_u_castss_INSERTu_u_PTREEwu_u_def, axiom, ![TV_u_27a]: ![V_w, V_t]: s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,t_h4s_ones_one),h4s_patriciau_u_castss_insertu_u_ptreew(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w),s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,t_h4s_ones_one),V_t))) = s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,t_h4s_ones_one),h4s_patriciau_u_castss_someu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_insertu_u_ptree(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_nums_num),h4s_wordss_w2n),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))),s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patriciau_u_castss_theu_u_ptree(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,t_h4s_ones_one),V_t)))))))).
fof(ah4s_patriciau_u_castss_INu_u_PTREEwu_u_def, axiom, ![TV_u_27a]: ![V_w, V_t]: s(t_bool,h4s_patriciau_u_castss_inu_u_ptreew(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w),s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,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_fcps_cart(t_bool,TV_u_27a),t_h4s_nums_num),h4s_wordss_w2n),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))),s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patriciau_u_castss_theu_u_ptree(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,t_h4s_ones_one),V_t)))))).
fof(ah4s_patriciau_u_castss_PTREEu_u_OFu_u_WORDSETu_u_def, axiom, ![TV_u_27a]: ![V_t, V_s]: s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,t_h4s_ones_one),h4s_patriciau_u_castss_ptreeu_u_ofu_u_wordset(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,t_h4s_ones_one),V_t),s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_bool),V_s))) = s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,t_h4s_ones_one),h4s_patriciau_u_castss_someu_u_ptree(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_patriciau_u_castss_theu_u_ptree(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,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_fcps_cart(t_bool,TV_u_27a),t_h4s_nums_num),h4s_wordss_w2n),s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_bool),V_s)))))))).
fof(ah4s_patriciau_u_castss_WORDSETu_u_OFu_u_PTREEu_u_def, axiom, ![TV_u_27a]: ![V_t]: s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_bool),h4s_patriciau_u_castss_wordsetu_u_ofu_u_ptree(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,t_h4s_ones_one),V_t))) = s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_fcps_cart(t_bool,TV_u_27a)),h4s_patriciau_u_castss_traversew(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,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),happ(s(t_fun(t_h4s_ones_one,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_ones_one)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_ones_one,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),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_INSERTu_u_PTREEu_u_def, axiom, ![V_t, V_n]: s(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),happ(s(t_fun(t_h4s_ones_one,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_ones_one)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_ones_one,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_bools_EXISTSu_u_UNIQUEu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: (p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(TV_u_27a,t_bool),V_x)))) <=> (p(s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x)))) & ![V_x0, V_y]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_x0)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_y))))) => s(TV_u_27a,V_x0) = s(TV_u_27a,V_y))))).
fof(ah4s_ones_oneu_u_caseu_u_def, axiom, ![TV_u_27a]: ![V_x, V_u]: s(TV_u_27a,h4s_ones_oneu_u_case(s(t_h4s_ones_one,V_u),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_options_OPTIONu_u_GUARDu_u_defu_c0, axiom, s(t_h4s_options_option(t_h4s_ones_one),h4s_options_optionu_u_guard(s(t_bool,t0))) = s(t_h4s_options_option(t_h4s_ones_one),h4s_options_some(s(t_h4s_ones_one,h4s_ones_one0)))).
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_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,t0)))).
fof(ah4s_patriciau_u_castss_SOMEu_u_PTREEu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_t]: s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),h4s_patriciau_u_castss_someu_u_ptree(s(t_h4s_patricias_ptree(TV_u_27b),V_t))) = 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),h4s_combins_k(s(t_h4s_ones_one,h4s_ones_one0))))),s(t_h4s_patricias_ptree(TV_u_27b),V_t)))).
fof(ah4s_basicSizes_oneu_u_sizeu_u_def, axiom, ![V_x]: s(t_h4s_nums_num,h4s_basicsizes_oneu_u_size(s(t_h4s_ones_one,V_x))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_ones_oneu_u_caseu_u_thm, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_ones_oneu_u_case(s(t_h4s_ones_one,h4s_ones_one0),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_options_NONEu_u_DEF, axiom, ![TV_u_27a]: s(t_h4s_options_option(TV_u_27a),h4s_options_none) = s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_abs(s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),h4s_sums_inr(s(t_h4s_ones_one,h4s_ones_one0)))))).
fof(ah4s_ones_oneu_u_primu_u_rec, axiom, ![TV_u_27a]: ![V_e]: ?[V_fn]: s(TV_u_27a,happ(s(t_fun(t_h4s_ones_one,TV_u_27a),V_fn),s(t_h4s_ones_one,h4s_ones_one0))) = s(TV_u_27a,V_e)).
fof(ah4s_options_OPTIONu_u_GUARDu_u_COND, axiom, ![V_b]: s(t_h4s_options_option(t_h4s_ones_one),h4s_options_optionu_u_guard(s(t_bool,V_b))) = s(t_h4s_options_option(t_h4s_ones_one),h4s_bools_cond(s(t_bool,V_b),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_options_none)))).
fof(ah4s_ones_oneu_u_DEF, axiom, ![V_uu_0]: (![V_x]: s(t_bool,happ(s(t_fun(t_h4s_ones_one,t_bool),V_uu_0),s(t_h4s_ones_one,V_x))) = s(t_bool,t0) => s(t_h4s_ones_one,h4s_ones_one0) = s(t_h4s_ones_one,h4s_mins_u_40(s(t_fun(t_h4s_ones_one,t_bool),V_uu_0))))).
fof(ah4s_options_OPTIONu_u_GUARDu_u_EQu_u_THMu_c0, axiom, ![V_b]: (s(t_h4s_options_option(t_h4s_ones_one),h4s_options_optionu_u_guard(s(t_bool,V_b))) = s(t_h4s_options_option(t_h4s_ones_one),h4s_options_some(s(t_h4s_ones_one,h4s_ones_one0))) <=> p(s(t_bool,V_b)))).
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),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_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)),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27b),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_fun(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27b),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_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)),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27b),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_fun(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27b),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)),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27b),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_fun(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27b),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_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),happ(s(t_fun(TV_u_27a,t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,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_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_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_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t0)) & 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,t0))) <=> p(s(t_bool,t0)))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_EQu_u_IMPu_u_THM, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) & (p(s(t_bool,V_t2)) => p(s(t_bool,V_t1)))))).
fof(ah4s_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_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_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_options_OPTIONu_u_GUARDu_u_defu_c1, axiom, s(t_h4s_options_option(t_h4s_ones_one),h4s_options_optionu_u_guard(s(t_bool,f))) = s(t_h4s_options_option(t_h4s_ones_one),h4s_options_none)).
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_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),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,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_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),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,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_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,t0)).
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_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t0),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_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,t0) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
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_UNIONu_u_PTREEwu_u_def, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_t2, V_t1]: s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,t_h4s_ones_one),h4s_patriciau_u_castss_unionu_u_ptreew(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27b,t_h4s_ones_one),V_t1),s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27c,t_h4s_ones_one),V_t2))) = s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,t_h4s_ones_one),h4s_patriciau_u_castss_someu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_unionu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patriciau_u_castss_theu_u_ptree(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27b,t_h4s_ones_one),V_t1))),s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patriciau_u_castss_theu_u_ptree(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27c,t_h4s_ones_one),V_t2)))))))).
fof(ah4s_patriciau_u_castss_wordu_u_ptreeu_u_Axiom, axiom, ![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)))).
fof(ah4s_patriciau_u_castss_THEu_u_PTREEu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_a]: s(t_h4s_patricias_ptree(TV_u_27a),h4s_patriciau_u_castss_theu_u_ptree(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27b,TV_u_27a),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27b,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27b,t_h4s_ones_one),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27b,TV_u_27a))),h4s_patriciau_u_castss_wordu_u_ptree),s(t_fun(TV_u_27b,t_h4s_ones_one),V_a))),s(t_h4s_patricias_ptree(TV_u_27a),V_t))))) = s(t_h4s_patricias_ptree(TV_u_27a),V_t)).
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_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_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_induction, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: (![V_f, V_p]: p(s(t_bool,happ(s(t_fun(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),t_bool),V_P),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)))))) => ![V_w]: p(s(t_bool,happ(s(t_fun(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),V_w)))))).
fof(ah4s_patriciau_u_castss_EVERYu_u_LEAFwu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_P, V_k, V_d]: s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool))),V_uu_1),s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(TV_u_27b,t_bool)),V_P))),s(t_h4s_nums_num,V_k))),s(TV_u_27b,V_d))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(TV_u_27b,t_bool)),V_P),s(t_h4s_fcps_cart(t_bool,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_fcps_cart(t_bool,TV_u_27a)),h4s_wordss_n2w),s(t_h4s_nums_num,V_k))))),s(TV_u_27b,V_d))) => ![V_uu_0]: (![V_P, V_k]: s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool))),V_uu_0),s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(TV_u_27b,t_bool)),V_P))),s(t_h4s_nums_num,V_k))) = s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool))),V_uu_1),s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(TV_u_27b,t_bool)),V_P))),s(t_h4s_nums_num,V_k))) => ![V_t, V_P]: s(t_bool,h4s_patriciau_u_castss_everyu_u_leafw(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(TV_u_27b,t_bool)),V_P),s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),V_t))) = s(t_bool,h4s_patricias_everyu_u_leaf(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool))),V_uu_0),s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(TV_u_27b,t_bool)),V_P))),s(t_h4s_patricias_ptree(TV_u_27b),h4s_patriciau_u_castss_theu_u_ptree(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),V_t)))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t0)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t0))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_options_NOTu_u_NONEu_u_SOME, axiom, ![TV_u_27a]: ![V_x]: ~ (s(t_h4s_options_option(TV_u_27a),h4s_options_none) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_options_SOMEu_u_11, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) | s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_patriciau_u_castss_KEYSwu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_t]: s(t_h4s_lists_list(t_h4s_fcps_cart(t_bool,TV_u_27a)),h4s_patriciau_u_castss_keysw(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),V_t))) = s(t_h4s_lists_list(t_h4s_fcps_cart(t_bool,TV_u_27a)),h4s_sortings_qsort(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_bool)),h4s_wordss_wordu_u_lo),s(t_h4s_lists_list(t_h4s_fcps_cart(t_bool,TV_u_27a)),h4s_patriciau_u_castss_traversew(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),V_t)))))).
fof(ah4s_patriciau_u_castss_REMOVEwu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_w, V_t]: s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),h4s_patriciau_u_castss_removew(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),V_t),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))) = s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),h4s_patriciau_u_castss_someu_u_ptree(s(t_h4s_patricias_ptree(TV_u_27b),h4s_patricias_remove(s(t_h4s_patricias_ptree(TV_u_27b),h4s_patriciau_u_castss_theu_u_ptree(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),V_t))),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_nums_num),h4s_wordss_w2n),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w)))))))).
fof(ah4s_patriciau_u_castss_TRAVERSEwu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_t]: s(t_h4s_lists_list(t_h4s_fcps_cart(t_bool,TV_u_27a)),h4s_patriciau_u_castss_traversew(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),V_t))) = s(t_h4s_lists_list(t_h4s_fcps_cart(t_bool,TV_u_27a)),h4s_lists_map(s(t_fun(t_h4s_nums_num,t_h4s_fcps_cart(t_bool,TV_u_27a)),h4s_wordss_n2w),s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(TV_u_27b),h4s_patriciau_u_castss_theu_u_ptree(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),V_t)))))))).
fof(ah4s_patriciau_u_castss_PEEKwu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_w, V_t]: s(t_h4s_options_option(TV_u_27b),h4s_patriciau_u_castss_peekw(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),V_t),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))) = s(t_h4s_options_option(TV_u_27b),h4s_patricias_peek(s(t_h4s_patricias_ptree(TV_u_27b),h4s_patriciau_u_castss_theu_u_ptree(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),V_t))),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_nums_num),h4s_wordss_w2n),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w)))))).
fof(ah4s_patriciau_u_castss_FINDwu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_w, V_t]: s(TV_u_27a,h4s_patriciau_u_castss_findw(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27b,TV_u_27a),V_t),s(t_h4s_fcps_cart(t_bool,TV_u_27b),V_w))) = s(TV_u_27a,h4s_options_the(s(t_h4s_options_option(TV_u_27a),h4s_patriciau_u_castss_peekw(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27b,TV_u_27a),V_t),s(t_h4s_fcps_cart(t_bool,TV_u_27b),V_w)))))).
fof(ah4s_patriciau_u_castss_EXISTSu_u_LEAFwu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_P, V_k, V_d]: s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool))),V_uu_1),s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(TV_u_27b,t_bool)),V_P))),s(t_h4s_nums_num,V_k))),s(TV_u_27b,V_d))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(TV_u_27b,t_bool)),V_P),s(t_h4s_fcps_cart(t_bool,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_fcps_cart(t_bool,TV_u_27a)),h4s_wordss_n2w),s(t_h4s_nums_num,V_k))))),s(TV_u_27b,V_d))) => ![V_uu_0]: (![V_P, V_k]: s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool))),V_uu_0),s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(TV_u_27b,t_bool)),V_P))),s(t_h4s_nums_num,V_k))) = s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool))),V_uu_1),s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(TV_u_27b,t_bool)),V_P))),s(t_h4s_nums_num,V_k))) => ![V_t, V_P]: s(t_bool,h4s_patriciau_u_castss_existsu_u_leafw(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(TV_u_27b,t_bool)),V_P),s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),V_t))) = s(t_bool,h4s_patricias_existsu_u_leaf(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(TV_u_27b,t_bool)),t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool))),V_uu_0),s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(TV_u_27b,t_bool)),V_P))),s(t_h4s_patricias_ptree(TV_u_27b),h4s_patriciau_u_castss_theu_u_ptree(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),V_t)))))))).
fof(ah4s_patriciau_u_castss_DEPTHwu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_t]: s(t_h4s_nums_num,h4s_patriciau_u_castss_depthw(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),V_t))) = s(t_h4s_nums_num,h4s_patricias_depth(s(t_h4s_patricias_ptree(TV_u_27b),h4s_patriciau_u_castss_theu_u_ptree(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),V_t)))))).
fof(ah4s_patriciau_u_castss_SIZEwu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_t]: s(t_h4s_nums_num,h4s_patriciau_u_castss_sizew(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),V_t))) = s(t_h4s_nums_num,h4s_patricias_size(s(t_h4s_patricias_ptree(TV_u_27b),h4s_patriciau_u_castss_theu_u_ptree(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),V_t)))))).
fof(ah4s_patriciau_u_castss_WordEmptyu_u_def, axiom, ![TV_u_27a,TV_u_27b]: s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),h4s_patriciau_u_castss_wordempty) = s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,TV_u_27b),h4s_patriciau_u_castss_someu_u_ptree(s(t_h4s_patricias_ptree(TV_u_27b),h4s_patricias_empty)))).
fof(ah4s_patriciau_u_castss_TRANSFORMwu_u_def, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_t, V_f]: s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27c,TV_u_27b),h4s_patriciau_u_castss_transformw(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27c,TV_u_27a),V_t))) = s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27c,TV_u_27b),h4s_patriciau_u_castss_someu_u_ptree(s(t_h4s_patricias_ptree(TV_u_27b),h4s_patricias_transform(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_patricias_ptree(TV_u_27a),h4s_patriciau_u_castss_theu_u_ptree(s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27c,TV_u_27a),V_t)))))))).
fof(ah4s_pairs_Su_u_UNCURRYu_u_R, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b,TV_u_27d]: ![V_g, V_f]: s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27d),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(TV_u_27d,TV_u_27c)),t_fun(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27d),t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c))),h4s_combins_s),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(TV_u_27d,TV_u_27c)),V_f))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27d),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d)),V_g))))) = s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(TV_u_27b,TV_u_27c))),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)))),h4s_combins_s),s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(TV_u_27b,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27d,TV_u_27c))),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(TV_u_27b,TV_u_27c)))),h4s_combins_o(s(t_fun(t_fun(TV_u_27b,t_fun(TV_u_27d,TV_u_27c)),t_fun(t_fun(TV_u_27b,TV_u_27d),t_fun(TV_u_27b,TV_u_27c))),h4s_combins_s))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27d,TV_u_27c))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27d,TV_u_27c)))),h4s_combins_o(s(t_fun(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),t_fun(TV_u_27b,t_fun(TV_u_27d,TV_u_27c))),h4s_combins_o(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_fun(TV_u_27d,TV_u_27c)),V_f))))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),h4s_pairs_u_2c))))))),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27d)),V_g)))))).
fof(ah4s_pairs_datatypeu_u_pair, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_pair]: p(s(t_bool,h4s_bools_datatype(s(TV_u_27c,happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),TV_u_27c),V_pair),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),h4s_pairs_u_2c))))))).
fof(ah4s_wordss_wordu_u_compareu_u_def, axiom, ![TV_u_27a]: ![V_b, V_a]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_a) = s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_b)) & s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_wordss_wordu_u_compare(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_a),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_b))) = s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),happ(s(t_fun(t_h4s_nums_num,t_h4s_fcps_cart(t_bool,t_h4s_ones_one)),h4s_wordss_n2w),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_fcps_cart(t_bool,t_h4s_ones_one),happ(s(t_fun(t_h4s_nums_num,t_h4s_fcps_cart(t_bool,t_h4s_ones_one)),h4s_wordss_n2w),s(t_h4s_nums_num,h4s_nums_0))))))).
fof(ah4s_wordss_foldlu_u_reduceu_u_or, axiom, ![TV_u_27a]: ![V_uu_2]: (![V_w, V_n]: s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),happ(s(t_fun(t_h4s_nums_num,t_h4s_fcps_cart(t_bool,t_h4s_ones_one)),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_nums_num,t_h4s_fcps_cart(t_bool,t_h4s_ones_one))),V_uu_2),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))),s(t_h4s_nums_num,V_n))) = s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_wordss_wordu_u_extract(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_n),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))) => ![V_uu_1]: (![V_w, V_i]: s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),happ(s(t_fun(t_h4s_nums_num,t_h4s_fcps_cart(t_bool,t_h4s_ones_one)),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_nums_num,t_h4s_fcps_cart(t_bool,t_h4s_ones_one))),V_uu_1),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))),s(t_h4s_nums_num,V_i))) = s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_bools_let(s(t_fun(t_h4s_nums_num,t_h4s_fcps_cart(t_bool,t_h4s_ones_one)),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_nums_num,t_h4s_fcps_cart(t_bool,t_h4s_ones_one))),V_uu_2),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_fcps_dimindex(s(t_h4s_bools_itself(TV_u_27a),h4s_bools_theu_u_value))),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_i))))) => ![V_uu_0]: (![V_l]: s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),happ(s(t_fun(t_h4s_lists_list(t_h4s_fcps_cart(t_bool,t_h4s_ones_one)),t_h4s_fcps_cart(t_bool,t_h4s_ones_one)),V_uu_0),s(t_h4s_lists_list(t_h4s_fcps_cart(t_bool,t_h4s_ones_one)),V_l))) = s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),happ(s(t_fun(t_h4s_lists_list(t_h4s_fcps_cart(t_bool,t_h4s_ones_one)),t_h4s_fcps_cart(t_bool,t_h4s_ones_one)),happ(s(t_fun(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),t_fun(t_h4s_lists_list(t_h4s_fcps_cart(t_bool,t_h4s_ones_one)),t_h4s_fcps_cart(t_bool,t_h4s_ones_one))),h4s_lists_foldl(s(t_fun(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),t_fun(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),t_h4s_fcps_cart(t_bool,t_h4s_ones_one))),h4s_wordss_wordu_u_or))),s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_lists_hd(s(t_h4s_lists_list(t_h4s_fcps_cart(t_bool,t_h4s_ones_one)),V_l))))),s(t_h4s_lists_list(t_h4s_fcps_cart(t_bool,t_h4s_ones_one)),h4s_lists_tl(s(t_h4s_lists_list(t_h4s_fcps_cart(t_bool,t_h4s_ones_one)),V_l))))) => ![V_w]: s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_wordss_reduceu_u_or(s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))) = s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_bools_let(s(t_fun(t_h4s_lists_list(t_h4s_fcps_cart(t_bool,t_h4s_ones_one)),t_h4s_fcps_cart(t_bool,t_h4s_ones_one)),V_uu_0),s(t_h4s_lists_list(t_h4s_fcps_cart(t_bool,t_h4s_ones_one)),h4s_lists_genlist(s(t_fun(t_h4s_nums_num,t_h4s_fcps_cart(t_bool,t_h4s_ones_one)),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_nums_num,t_h4s_fcps_cart(t_bool,t_h4s_ones_one))),V_uu_1),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))),s(t_h4s_nums_num,h4s_fcps_dimindex(s(t_h4s_bools_itself(TV_u_27a),h4s_bools_theu_u_value))))))))))).
fof(ah4s_combins_ou_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_x, V_g, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27a),t_fun(TV_u_27c,TV_u_27b)),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27c,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,V_x)))))).
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_pairs_UNCURRY0, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_v))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_v))))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_v)))))).
fof(ah4s_pairs_PAIR, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_h4s_pairs_prod(TV_u_27a,TV_u_27b))),h4s_pairs_u_2c),s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x)).
fof(ah4s_combins_Su_u_THM, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_x, V_g, V_f]: s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27c))),h4s_combins_s),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(t_fun(TV_u_27a,TV_u_27b),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x)))))).
fof(ah4s_bools_EQu_u_EXPAND, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) | (~ (p(s(t_bool,V_t1))) & ~ (p(s(t_bool,V_t2))))))).
fof(ah4s_lists_FOLDL0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_e]: s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),h4s_lists_foldl(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),V_f))),s(TV_u_27b,V_e))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(TV_u_27b,V_e)).
fof(ah4s_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_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_lists_FOLDL0u_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_l, V_f, V_e]: s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),h4s_lists_foldl(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),V_f))),s(TV_u_27b,V_e))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),h4s_lists_foldl(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),V_f))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27b,V_e))),s(TV_u_27a,V_x))))),s(t_h4s_lists_list(TV_u_27a),V_l)))).
fof(ah4s_bools_LETu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,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_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_numerals_numeralu_u_lteu_c1, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
fof(ah4s_numerals_numeralu_u_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_c2, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
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_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_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_arithmetics_MINu_u_LEu_c1, axiom, ![V_p, V_n, V_m]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,h4s_arithmetics_min(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))))) <=> (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_m)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_lists_MAPu_u_TL, axiom, ![TV_u_27b,TV_u_27a]: ![V_l, V_f]: (~ (p(s(t_bool,h4s_lists_null(s(t_h4s_lists_list(TV_u_27a),V_l))))) => 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_tl(s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_tl(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_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_arithmetics_SUBu_u_RIGHTu_u_EQ, axiom, ![V_p, V_n, V_m]: (s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),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,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))) | (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,h4s_nums_0)))))))).
fof(ah4s_arithmetics_SUBu_u_LEFTu_u_EQ, axiom, ![V_p, V_n, V_m]: (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_arithmetics_u_2d(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,V_m),s(t_h4s_nums_num,V_p))) = s(t_h4s_nums_num,V_n) | (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0)))) & 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_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_SUBu_u_LEFTu_u_ADD, 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_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,h4s_arithmetics_u_3cu_3d(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,h4s_arithmetics_u_2d(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_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_NOTu_u_LEQ, axiom, ![V_n, V_m]: (~ (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_EQu_u_LESSu_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,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m))))))).
fof(ah4s_arithmetics_ZEROu_u_LESSu_u_EQ, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))))).
fof(ah4s_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_arithmetics_LESSu_u_EQu_u_TRANS, axiom, ![V_p, V_n, V_m]: ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_SUBu_u_LEFTu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))))) <=> (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,V_n)))) | p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))))))).
fof(ah4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_LESSu_u_EQu_u_REFL, axiom, ![V_m]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_m))))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c2, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_SYM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ch4s_patriciau_u_castss_ADDu_u_INSERTu_u_WORD, conjecture, ![TV_u_27a]: ![V_w, V_v, V_t]: s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,t_h4s_ones_one),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_ones_one),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,t_h4s_ones_one)),happ(s(t_fun(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_ones_one),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,t_h4s_ones_one))),h4s_patriciau_u_castss_addw),s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,t_h4s_ones_one),V_t))),s(t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_ones_one),happ(s(t_fun(t_h4s_ones_one,t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_ones_one)),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_ones_one,t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_ones_one))),h4s_pairs_u_2c),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))),s(t_h4s_ones_one,V_v))))) = s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,t_h4s_ones_one),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_ones_one),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,t_h4s_ones_one)),happ(s(t_fun(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,t_h4s_ones_one),t_fun(t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_ones_one),t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,t_h4s_ones_one))),h4s_patriciau_u_castss_addw),s(t_h4s_patriciau_u_castss_wordu_u_ptree(TV_u_27a,t_h4s_ones_one),V_t))),s(t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_ones_one),happ(s(t_fun(t_h4s_ones_one,t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_ones_one)),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_ones_one,t_h4s_pairs_prod(t_h4s_fcps_cart(t_bool,TV_u_27a),t_h4s_ones_one))),h4s_pairs_u_2c),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))),s(t_h4s_ones_one,h4s_ones_one0)))))).
