%   ORIGINAL: h4/patricia/IN__PTREE__UNION__PTREE
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/patricia/UNION__PTREE__def: !t2 t1. h4/patricia/UNION__PTREE t1 t2 = h4/patricia/PTREE__OF__NUMSET t1 (h4/patricia/NUMSET__OF__PTREE t2)
% Assm: h4/patricia/IN__PTREE__def: !t n. h4/patricia/IN__PTREE n t <=> h4/option/IS__SOME (h4/patricia/PEEK t n)
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% 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/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/TRUTH: T
% Assm: h4/patricia/IN__NUMSET__OF__PTREE: !t n. h4/patricia/IS__PTREE t ==> (h4/bool/IN n (h4/patricia/NUMSET__OF__PTREE t) <=> h4/patricia/IN__PTREE n t)
% Assm: h4/patricia/IN__PTREE__OF__NUMSET: !t s n. h4/patricia/IS__PTREE t /\ h4/pred__set/FINITE s ==> (h4/patricia/IN__PTREE n (h4/patricia/PTREE__OF__NUMSET t s) <=> h4/patricia/IN__PTREE n t \/ h4/bool/IN n s)
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/patricia/PTREE__EXTENSION: !t2 t1. h4/patricia/IS__PTREE t1 /\ h4/patricia/IS__PTREE t2 ==> (t1 = t2 <=> (!x. h4/patricia/IN__PTREE x t1 <=> h4/patricia/IN__PTREE x t2))
% Assm: h4/patricia/IN__PTREE__REMOVE: !t n m. h4/patricia/IS__PTREE t ==> (h4/patricia/IN__PTREE n (h4/patricia/REMOVE t m) <=> ~(n = m) /\ h4/patricia/IN__PTREE n t)
% Assm: h4/patricia/IN__PTREE__INSERT__PTREE: !t n m. h4/patricia/IS__PTREE t ==> (h4/patricia/IN__PTREE n (h4/patricia/INSERT__PTREE m t) <=> m = n \/ h4/patricia/IN__PTREE n t)
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/patricia/PTREE__OF__NUMSET__IS__PTREE: !t s. h4/patricia/IS__PTREE t ==> h4/patricia/IS__PTREE (h4/patricia/PTREE__OF__NUMSET t s)
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/patricia/MEM__TRAVERSE__PEEK: !t k. h4/patricia/IS__PTREE t ==> (h4/bool/IN k (h4/list/LIST__TO__SET (h4/patricia/TRAVERSE t)) <=> h4/option/IS__SOME (h4/patricia/PEEK t k))
% Assm: h4/patricia/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/MEM__TRAVERSE: !t k. h4/patricia/IS__PTREE t ==> (h4/bool/IN k (h4/list/LIST__TO__SET (h4/patricia/TRAVERSE t)) <=> h4/bool/IN k (h4/patricia/NUMSET__OF__PTREE t))
% Assm: h4/patricia/PERM__INSERT__PTREE: !t s. h4/pred__set/FINITE s ==> h4/patricia/IS__PTREE t ==> h4/sorting/PERM (h4/patricia/TRAVERSE (h4/list/FOLDL (h4/combin/C h4/patricia/INSERT__PTREE) t (h4/list/SET__TO__LIST s))) (h4/list/SET__TO__LIST (h4/pred__set/UNION (h4/patricia/NUMSET__OF__PTREE t) s))
% Assm: h4/one/one1: !v. v = h4/one/one0
% Assm: h4/patricia/IN__PTREE__EMPTY: !n. ~h4/patricia/IN__PTREE n h4/patricia/Empty
% Assm: h4/patricia/INSERT__PTREE__IS__PTREE: !x t. h4/patricia/IS__PTREE t ==> h4/patricia/IS__PTREE (h4/patricia/INSERT__PTREE x t)
% Assm: h4/patricia/MEM__TRAVERSE__INSERT__PTREE: !x t h. h4/patricia/IS__PTREE t ==> (h4/bool/IN x (h4/list/LIST__TO__SET (h4/patricia/TRAVERSE (h4/patricia/INSERT__PTREE h t))) <=> x = h \/ ~(x = h) /\ h4/bool/IN x (h4/list/LIST__TO__SET (h4/patricia/TRAVERSE t)))
% Assm: h4/patricia/PTREE__OF__NUMSET__NUMSET__OF__PTREE: !t s. h4/patricia/IS__PTREE t /\ h4/pred__set/FINITE s ==> h4/patricia/PTREE__OF__NUMSET h4/patricia/Empty (h4/pred__set/UNION (h4/patricia/NUMSET__OF__PTREE t) s) = h4/patricia/PTREE__OF__NUMSET t s
% Assm: h4/patricia/NUMSET__OF__PTREE__PTREE__OF__NUMSET: !t s. h4/patricia/IS__PTREE t /\ h4/pred__set/FINITE s ==> h4/patricia/NUMSET__OF__PTREE (h4/patricia/PTREE__OF__NUMSET t s) = h4/pred__set/UNION (h4/patricia/NUMSET__OF__PTREE t) s
% 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/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/patricia/PEEK__ADD: !t k j d. h4/patricia/IS__PTREE t ==> h4/patricia/PEEK (h4/patricia/ADD t (h4/pair/_2C k d)) j = h4/bool/COND (k = j) (h4/option/SOME d) (h4/patricia/PEEK t j)
% Assm: h4/patricia/ADD__INSERT: !v t n. h4/patricia/ADD t (h4/pair/_2C n v) = h4/patricia/INSERT__PTREE n t
% Assm: h4/option/IS__SOME__DEF_c0: !x. h4/option/IS__SOME (h4/option/SOME x) <=> T
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/patricia/IN__PTREE__OF__NUMSET__EMPTY: !s n. h4/pred__set/FINITE s ==> (h4/bool/IN n s <=> h4/patricia/IN__PTREE n (h4/patricia/PTREE__OF__NUMSET h4/patricia/Empty s))
% Assm: h4/pred__set/IN__UNION: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x t
% Assm: h4/patricia/NUMSET__OF__PTREE__def: !t. h4/patricia/NUMSET__OF__PTREE t = h4/list/LIST__TO__SET (h4/patricia/TRAVERSE t)
% Assm: h4/option/IS__SOME__DEF_c1: h4/option/IS__SOME h4/option/NONE <=> F
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/pred__set/FINITE__UNION: !t s. h4/pred__set/FINITE (h4/pred__set/UNION s t) <=> h4/pred__set/FINITE s /\ h4/pred__set/FINITE t
% Assm: h4/patricia/FINITE__NUMSET__OF__PTREE: !t. h4/pred__set/FINITE (h4/patricia/NUMSET__OF__PTREE t)
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> 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/patricia/ADD__IS__PTREE: !x t. h4/patricia/IS__PTREE t ==> h4/patricia/IS__PTREE (h4/patricia/ADD t x)
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/sorting/PERM__MEM__EQ: !l2 l1. h4/sorting/PERM l1 l2 ==> (!x. h4/bool/IN x (h4/list/LIST__TO__SET l1) <=> h4/bool/IN x (h4/list/LIST__TO__SET l2))
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/patricia/PTREE__OF__NUMSET__def: !t s. h4/patricia/PTREE__OF__NUMSET t s = h4/list/FOLDL (h4/combin/C h4/patricia/INSERT__PTREE) t (h4/list/SET__TO__LIST s)
% Assm: h4/list/SET__TO__LIST__IN__MEM: !s. h4/pred__set/FINITE s ==> (!x. h4/bool/IN x s <=> h4/bool/IN x (h4/list/LIST__TO__SET (h4/list/SET__TO__LIST s)))
% Assm: h4/patricia/PEEK__REMOVE: !t k j. h4/patricia/IS__PTREE t ==> h4/patricia/PEEK (h4/patricia/REMOVE t k) j = h4/bool/COND (k = j) h4/option/NONE (h4/patricia/PEEK t j)
% Assm: h4/patricia/PTREE__EQ: !t2 t1. h4/patricia/IS__PTREE t1 /\ h4/patricia/IS__PTREE t2 ==> ((!k. h4/patricia/PEEK t1 k = h4/patricia/PEEK t2 k) <=> t1 = t2)
% Assm: h4/bool/BOOL__EQ__DISTINCT_c0: ~(T <=> F)
% 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/option/option__nchotomy: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm: h4/patricia/EMPTY__IS__PTREE: h4/patricia/IS__PTREE h4/patricia/Empty
% 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/combin/C__THM: !y x f. h4/combin/C f x y = f y x
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/pred__set/EXTENSION: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x 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/list/FOLDL0_c0: !f e. h4/list/FOLDL f e h4/list/NIL = e
% Assm: h4/patricia/PTREE__OF__NUMSET__IS__PTREE__EMPTY: !s. h4/patricia/IS__PTREE (h4/patricia/PTREE__OF__NUMSET h4/patricia/Empty s)
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/relation/WFREC__COROLLARY: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. f x = M (h4/relation/RESTRICT f R x) x)
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/numeral/numeral__distrib_c1: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm: h4/patricia/PEEK__curried__def: !x1 x. h4/patricia/PEEK x x1 = h4/patricia/PEEK__tupled (h4/pair/_2C x x1)
% Assm: h4/patricia/PEEK__tupled__primitive__def: h4/patricia/PEEK__tupled = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!p r l k m. R (h4/pair/_2C (h4/bool/COND (h4/bit/BIT m k) l r) k) (h4/pair/_2C (h4/patricia/Branch p m l r) k)))) (\PEEK__tupled a. h4/pair/pair__CASE a (\v k. h4/patricia/ptree__CASE v (h4/combin/I h4/option/NONE) (\j d. h4/combin/I (h4/bool/COND (k = j) (h4/option/SOME d) h4/option/NONE)) (\p m l r. h4/combin/I (PEEK__tupled (h4/pair/_2C (h4/bool/COND (h4/bit/BIT m k) l r) k)))))
% Assm: h4/patricia/ptree__size__def_c2: !f a3 a2 a1 a0. h4/patricia/ptree__size f (h4/patricia/Branch a0 a1 a2 a3) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B a0 (h4/arithmetic/_2B a1 (h4/arithmetic/_2B (h4/patricia/ptree__size f a2) (h4/patricia/ptree__size f a3))))
% Assm: h4/patricia/ptree__case__def_c2: !v f1 f a3 a2 a1 a0. h4/patricia/ptree__CASE (h4/patricia/Branch a0 a1 a2 a3) v f f1 = f1 a0 a1 a2 a3
% Assm: h4/patricia/ptree__case__def_c0: !v f1 f. h4/patricia/ptree__CASE h4/patricia/Empty v f f1 = v
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm: h4/patricia/ptree__case__def_c1: !v f1 f a1 a0. h4/patricia/ptree__CASE (h4/patricia/Leaf a0 a1) v f f1 = f a0 a1
% Assm: h4/arithmetic/LESS__MONO__ADD__EQ: !p n m. h4/prim__rec/_3C (h4/arithmetic/_2B m p) (h4/arithmetic/_2B n p) <=> h4/prim__rec/_3C m n
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/ADD__ASSOC: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% Assm: h4/arithmetic/ADD__COMM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/relation/RESTRICT__LEMMA: !z y f R. R y z ==> h4/relation/RESTRICT f R z y = f y
% Assm: h4/prim__rec/WF__LESS: h4/relation/WF h4/prim__rec/_3C
% Assm: h4/relation/WF__inv__image: !f R. h4/relation/WF R ==> h4/relation/WF (h4/relation/inv__image R f)
% Assm: h4/numeral/numeral__distrib_c27: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm: h4/relation/inv__image__def: !f R. h4/relation/inv__image R f = (\x y. R (f x) (f y))
% 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/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/arithmetic/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/ADD__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/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/bool/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Assm: h4/patricia/ALL__DISTINCT__TRAVERSE: !t. h4/patricia/IS__PTREE t ==> h4/list/ALL__DISTINCT (h4/patricia/TRAVERSE t)
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/patricia/MEM__ALL__DISTINCT__IMP__PERM: !l2 l1. h4/list/ALL__DISTINCT l1 /\ h4/list/ALL__DISTINCT l2 /\ (!x. h4/bool/IN x (h4/list/LIST__TO__SET l1) <=> h4/bool/IN x (h4/list/LIST__TO__SET l2)) ==> h4/sorting/PERM l1 l2
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/rich__list/FOLDL__FOLDR__REVERSE: !l f e. h4/list/FOLDL f e l = h4/list/FOLDR (\x y. f y x) e (h4/list/REVERSE l)
% Assm: h4/rich__list/FOLDR__APPEND: !l2 l1 f e. h4/list/FOLDR f e (h4/list/APPEND l1 l2) = h4/list/FOLDR f (h4/list/FOLDR f e l2) l1
% Assm: h4/list/ALL__DISTINCT__SET__TO__LIST: !s. h4/pred__set/FINITE s ==> h4/list/ALL__DISTINCT (h4/list/SET__TO__LIST s)
% Assm: h4/list/SET__TO__LIST__INV: !s. h4/pred__set/FINITE s ==> h4/list/LIST__TO__SET (h4/list/SET__TO__LIST s) = s
% Assm: h4/list/ALL__DISTINCT0_c1: !t h. h4/list/ALL__DISTINCT (h4/list/CONS h t) <=> ~h4/bool/IN h (h4/list/LIST__TO__SET t) /\ h4/list/ALL__DISTINCT t
% Assm: h4/list/MEM__SET__TO__LIST: !s. h4/pred__set/FINITE s ==> (!x. h4/bool/IN x (h4/list/LIST__TO__SET (h4/list/SET__TO__LIST s)) <=> h4/bool/IN x s)
% Assm: h4/list/FINITE__LIST__TO__SET: !l. h4/pred__set/FINITE (h4/list/LIST__TO__SET l)
% Assm: h4/list/ALL__DISTINCT0_c0: h4/list/ALL__DISTINCT h4/list/NIL <=> T
% Assm: h4/list/FOLDR0_c0: !f e. h4/list/FOLDR f e h4/list/NIL = e
% Assm: h4/list/REVERSE__DEF_c0: h4/list/REVERSE h4/list/NIL = h4/list/NIL
% Assm: h4/list/FOLDR0_c1: !x l f e. h4/list/FOLDR f e (h4/list/CONS x l) = f x (h4/list/FOLDR f e l)
% Assm: h4/list/LIST__TO__SET0_c1: !t h. h4/list/LIST__TO__SET (h4/list/CONS h t) = h4/pred__set/INSERT h (h4/list/LIST__TO__SET t)
% Assm: h4/pred__set/FINITE__INSERT: !x s. h4/pred__set/FINITE (h4/pred__set/INSERT x s) <=> h4/pred__set/FINITE s
% Assm: h4/pred__set/IN__INSERT: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm: h4/list/REVERSE__DEF_c1: !t h. h4/list/REVERSE (h4/list/CONS h t) = h4/list/APPEND (h4/list/REVERSE t) (h4/list/CONS h h4/list/NIL)
% Assm: h4/list/LIST__TO__SET0_c0: h4/list/LIST__TO__SET h4/list/NIL = h4/pred__set/EMPTY
% Assm: h4/pred__set/UNION__EMPTY_c1: !s. h4/pred__set/UNION s h4/pred__set/EMPTY = s
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/combin/o__DEF: !g f. h4/combin/o f g = (\x. f (g x))
% Assm: h4/combin/C__DEF: h4/combin/C = (\f x y. f y x)
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Goal: !t2 t1 n. h4/patricia/IS__PTREE t1 /\ h4/patricia/IS__PTREE t2 ==> (h4/patricia/IN__PTREE n (h4/patricia/UNION__PTREE t1 t2) <=> h4/patricia/IN__PTREE n t1 \/ h4/patricia/IN__PTREE n t2)
%   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_patricias_UNIONu_u_PTREEu_u_def]: !t2 t1. h4/patricia/UNION__PTREE t1 t2 = h4/patricia/PTREE__OF__NUMSET t1 (h4/patricia/NUMSET__OF__PTREE t2)
% Assm [h4s_patricias_INu_u_PTREEu_u_def]: !t n. h4/patricia/IN__PTREE n t <=> h4/option/IS__SOME (h4/patricia/PEEK t n)
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% 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_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_patricias_INu_u_NUMSETu_u_OFu_u_PTREE]: !t n. h4/patricia/IS__PTREE t ==> (h4/bool/IN n (h4/patricia/NUMSET__OF__PTREE t) <=> h4/patricia/IN__PTREE n t)
% Assm [h4s_patricias_INu_u_PTREEu_u_OFu_u_NUMSET]: !t s n. h4/patricia/IS__PTREE t /\ h4/pred__set/FINITE s ==> (h4/patricia/IN__PTREE n (h4/patricia/PTREE__OF__NUMSET t s) <=> h4/patricia/IN__PTREE n t \/ h4/bool/IN n s)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_patricias_PTREEu_u_EXTENSION]: !t2 t1. h4/patricia/IS__PTREE t1 /\ h4/patricia/IS__PTREE t2 ==> (t1 = t2 <=> (!x. h4/patricia/IN__PTREE x t1 <=> h4/patricia/IN__PTREE x t2))
% Assm [h4s_patricias_INu_u_PTREEu_u_REMOVE]: !t n m. h4/patricia/IS__PTREE t ==> (h4/patricia/IN__PTREE n (h4/patricia/REMOVE t m) <=> ~(n = m) /\ h4/patricia/IN__PTREE n t)
% Assm [h4s_patricias_INu_u_PTREEu_u_INSERTu_u_PTREE]: !t n m. h4/patricia/IS__PTREE t ==> (h4/patricia/IN__PTREE n (happ (happ h4/patricia/INSERT__PTREE m) t) <=> m = n \/ h4/patricia/IN__PTREE n t)
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_patricias_PTREEu_u_OFu_u_NUMSETu_u_ISu_u_PTREE]: !t s. h4/patricia/IS__PTREE t ==> h4/patricia/IS__PTREE (h4/patricia/PTREE__OF__NUMSET t s)
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_patricias_MEMu_u_TRAVERSEu_u_PEEK]: !t k. h4/patricia/IS__PTREE t ==> (h4/bool/IN k (h4/list/LIST__TO__SET (h4/patricia/TRAVERSE t)) <=> h4/option/IS__SOME (h4/patricia/PEEK t k))
% Assm [h4s_patricias_PEEKu_u_INSERTu_u_PTREE]: !t k j. h4/patricia/IS__PTREE t ==> (?v. (v <=> k = j) /\ h4/patricia/PEEK (happ (happ h4/patricia/INSERT__PTREE k) t) j = h4/bool/COND v (h4/option/SOME h4/one/one0) (h4/patricia/PEEK t j))
% Assm [h4s_patricias_MEMu_u_TRAVERSE]: !t k. h4/patricia/IS__PTREE t ==> (h4/bool/IN k (h4/list/LIST__TO__SET (h4/patricia/TRAVERSE t)) <=> h4/bool/IN k (h4/patricia/NUMSET__OF__PTREE t))
% Assm [h4s_patricias_PERMu_u_INSERTu_u_PTREE]: !t s. h4/pred__set/FINITE s ==> h4/patricia/IS__PTREE t ==> h4/sorting/PERM (h4/patricia/TRAVERSE (h4/list/FOLDL (h4/combin/C h4/patricia/INSERT__PTREE) t (h4/list/SET__TO__LIST s))) (h4/list/SET__TO__LIST (h4/pred__set/UNION (h4/patricia/NUMSET__OF__PTREE t) s))
% Assm [h4s_ones_one1]: !v. v = h4/one/one0
% Assm [h4s_patricias_INu_u_PTREEu_u_EMPTY]: !n. ~h4/patricia/IN__PTREE n h4/patricia/Empty
% Assm [h4s_patricias_INSERTu_u_PTREEu_u_ISu_u_PTREE]: !x t. h4/patricia/IS__PTREE t ==> h4/patricia/IS__PTREE (happ (happ h4/patricia/INSERT__PTREE x) t)
% Assm [h4s_patricias_MEMu_u_TRAVERSEu_u_INSERTu_u_PTREE]: !x t h. h4/patricia/IS__PTREE t ==> (h4/bool/IN x (h4/list/LIST__TO__SET (h4/patricia/TRAVERSE (happ (happ h4/patricia/INSERT__PTREE h) t))) <=> x = h \/ ~(x = h) /\ h4/bool/IN x (h4/list/LIST__TO__SET (h4/patricia/TRAVERSE t)))
% Assm [h4s_patricias_PTREEu_u_OFu_u_NUMSETu_u_NUMSETu_u_OFu_u_PTREE]: !t s. h4/patricia/IS__PTREE t /\ h4/pred__set/FINITE s ==> h4/patricia/PTREE__OF__NUMSET h4/patricia/Empty (h4/pred__set/UNION (h4/patricia/NUMSET__OF__PTREE t) s) = h4/patricia/PTREE__OF__NUMSET t s
% Assm [h4s_patricias_NUMSETu_u_OFu_u_PTREEu_u_PTREEu_u_OFu_u_NUMSET]: !t s. h4/patricia/IS__PTREE t /\ h4/pred__set/FINITE s ==> h4/patricia/NUMSET__OF__PTREE (h4/patricia/PTREE__OF__NUMSET t s) = h4/pred__set/UNION (h4/patricia/NUMSET__OF__PTREE t) s
% 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_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_patricias_PEEKu_u_ADD]: !t k j d. h4/patricia/IS__PTREE t ==> (?v. (v <=> k = j) /\ h4/patricia/PEEK (h4/patricia/ADD t (h4/pair/_2C k d)) j = h4/bool/COND v (h4/option/SOME d) (h4/patricia/PEEK t j))
% Assm [h4s_patricias_ADDu_u_INSERT]: !v t n. h4/patricia/ADD t (h4/pair/_2C n v) = happ (happ h4/patricia/INSERT__PTREE n) t
% Assm [h4s_options_ISu_u_SOMEu_u_DEFu_c0]: !x. h4/option/IS__SOME (h4/option/SOME x) <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_patricias_INu_u_PTREEu_u_OFu_u_NUMSETu_u_EMPTY]: !s n. h4/pred__set/FINITE s ==> (h4/bool/IN n s <=> h4/patricia/IN__PTREE n (h4/patricia/PTREE__OF__NUMSET h4/patricia/Empty s))
% Assm [h4s_predu_u_sets_INu_u_UNION]: !x t s. h4/bool/IN x (h4/pred__set/UNION s t) <=> h4/bool/IN x s \/ h4/bool/IN x t
% Assm [h4s_patricias_NUMSETu_u_OFu_u_PTREEu_u_def]: !t. h4/patricia/NUMSET__OF__PTREE t = h4/list/LIST__TO__SET (h4/patricia/TRAVERSE t)
% Assm [h4s_options_ISu_u_SOMEu_u_DEFu_c1]: h4/option/IS__SOME h4/option/NONE <=> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_predu_u_sets_FINITEu_u_UNION]: !t s. h4/pred__set/FINITE (h4/pred__set/UNION s t) <=> h4/pred__set/FINITE s /\ h4/pred__set/FINITE t
% Assm [h4s_patricias_FINITEu_u_NUMSETu_u_OFu_u_PTREE]: !t. h4/pred__set/FINITE (h4/patricia/NUMSET__OF__PTREE t)
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_patricias_INSERTu_u_PTREEu_u_def]: !t n. happ (happ h4/patricia/INSERT__PTREE n) t = h4/patricia/ADD t (h4/pair/_2C n h4/one/one0)
% Assm [h4s_patricias_ADDu_u_ISu_u_PTREE]: !x t. h4/patricia/IS__PTREE t ==> h4/patricia/IS__PTREE (h4/patricia/ADD t x)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_sortings_PERMu_u_MEMu_u_EQ]: !l2 l1. h4/sorting/PERM l1 l2 ==> (!x. h4/bool/IN x (h4/list/LIST__TO__SET l1) <=> h4/bool/IN x (h4/list/LIST__TO__SET l2))
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_patricias_PTREEu_u_OFu_u_NUMSETu_u_def]: !t s. h4/patricia/PTREE__OF__NUMSET t s = h4/list/FOLDL (h4/combin/C h4/patricia/INSERT__PTREE) t (h4/list/SET__TO__LIST s)
% Assm [h4s_lists_SETu_u_TOu_u_LISTu_u_INu_u_MEM]: !s. h4/pred__set/FINITE s ==> (!x. h4/bool/IN x s <=> h4/bool/IN x (h4/list/LIST__TO__SET (h4/list/SET__TO__LIST s)))
% Assm [h4s_patricias_PEEKu_u_REMOVE]: !t k j. h4/patricia/IS__PTREE t ==> (?v. (v <=> k = j) /\ h4/patricia/PEEK (h4/patricia/REMOVE t k) j = h4/bool/COND v h4/option/NONE (h4/patricia/PEEK t j))
% Assm [h4s_patricias_PTREEu_u_EQ]: !t2 t1. h4/patricia/IS__PTREE t1 /\ h4/patricia/IS__PTREE t2 ==> ((!k. h4/patricia/PEEK t1 k = h4/patricia/PEEK t2 k) <=> t1 = t2)
% Assm [h4s_bools_BOOLu_u_EQu_u_DISTINCTu_c0]: ~(T <=> F)
% 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_options_optionu_u_nchotomy]: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm [h4s_patricias_EMPTYu_u_ISu_u_PTREE]: h4/patricia/IS__PTREE h4/patricia/Empty
% 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_combins_Cu_u_THM]: !y x f. happ (happ (h4/combin/C f) x) y = happ (happ f y) x
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_predu_u_sets_EXTENSION]: !t s. s = t <=> (!x. h4/bool/IN x s <=> h4/bool/IN x t)
% Assm [h4s_lists_FOLDL0u_c1]: !x l f e. h4/list/FOLDL f e (h4/list/CONS x l) = h4/list/FOLDL f (happ (happ f e) x) l
% Assm [h4s_lists_FOLDL0u_c0]: !f e. h4/list/FOLDL f e h4/list/NIL = e
% Assm [h4s_patricias_PTREEu_u_OFu_u_NUMSETu_u_ISu_u_PTREEu_u_EMPTY]: !s. h4/patricia/IS__PTREE (h4/patricia/PTREE__OF__NUMSET h4/patricia/Empty s)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_relations_WFRECu_u_COROLLARY]: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. happ f x = happ (happ M (h4/relation/RESTRICT f R x)) x)
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_numerals_numeralu_u_distribu_c1]: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm [h4s_patricias_PEEKu_u_curriedu_u_def]: !x1 x. h4/patricia/PEEK x x1 = happ h4/patricia/PEEK__tupled (h4/pair/_2C x x1)
% Assm [h4s_patricias_PEEKu_u_tupledu_u_primitiveu_u_def]: !_10. (!PEEK__tupled m l k r. happ (happ (happ (happ (happ _10 PEEK__tupled) m) l) k) r = h4/combin/I (happ PEEK__tupled (h4/pair/_2C (h4/bool/COND (h4/bit/BIT m k) l r) k))) ==> (!_9. (!PEEK__tupled m k l. happ (happ (happ (happ _9 PEEK__tupled) m) k) l = happ (happ (happ (happ _10 PEEK__tupled) m) l) k) ==> (!_8. (!PEEK__tupled k m. happ (happ (happ _8 PEEK__tupled) k) m = happ (happ (happ _9 PEEK__tupled) m) k) ==> (!_7. (!PEEK__tupled k p. happ (happ (happ _7 PEEK__tupled) k) p = happ (happ _8 PEEK__tupled) k) ==> (!_6. (!k j d. ?v. (v <=> k = j) /\ happ (happ (happ _6 k) j) d = h4/combin/I (h4/bool/COND v (h4/option/SOME d) h4/option/NONE)) ==> (!_5. (!k j. happ (happ _5 k) j = happ (happ _6 k) j) ==> (!_4. (!v PEEK__tupled k. happ (happ (happ _4 v) PEEK__tupled) k = h4/patricia/ptree__CASE v (h4/combin/I h4/option/NONE) (happ _5 k) (happ (happ _7 PEEK__tupled) k)) ==> (!_3. (!PEEK__tupled v. happ (happ _3 PEEK__tupled) v = happ (happ _4 v) PEEK__tupled) ==> (!_2. (!PEEK__tupled a. happ (happ _2 PEEK__tupled) a = h4/pair/pair__CASE a (happ _3 PEEK__tupled)) ==> (!_1. (!PEEK__tupled. happ _1 PEEK__tupled = happ _2 PEEK__tupled) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R /\ (!p r l k m. happ (happ R (h4/pair/_2C (h4/bool/COND (h4/bit/BIT m k) l r) k)) (h4/pair/_2C (h4/patricia/Branch p m l r) k))) ==> h4/patricia/PEEK__tupled = h4/relation/WFREC (h4/min/_40 _0) _1))))))))))
% Assm [h4s_patricias_ptreeu_u_sizeu_u_defu_c2]: !f a3 a2 a1 a0. h4/patricia/ptree__size f (h4/patricia/Branch a0 a1 a2 a3) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B a0 (h4/arithmetic/_2B a1 (h4/arithmetic/_2B (h4/patricia/ptree__size f a2) (h4/patricia/ptree__size f a3))))
% Assm [h4s_patricias_ptreeu_u_caseu_u_defu_c2]: !v f1 f a3 a2 a1 a0. h4/patricia/ptree__CASE (h4/patricia/Branch a0 a1 a2 a3) v f f1 = happ (happ (happ (happ f1 a0) a1) a2) a3
% Assm [h4s_patricias_ptreeu_u_caseu_u_defu_c0]: !v f1 f. h4/patricia/ptree__CASE h4/patricia/Empty v f f1 = v
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_pairs_PAIR]: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm [h4s_patricias_ptreeu_u_caseu_u_defu_c1]: !v f1 f a1 a0. h4/patricia/ptree__CASE (h4/patricia/Leaf a0 a1) v f f1 = happ (happ f a0) a1
% Assm [h4s_arithmetics_LESSu_u_MONOu_u_ADDu_u_EQ]: !p n m. happ (happ h4/prim__rec/_3C (h4/arithmetic/_2B m p)) (h4/arithmetic/_2B n p) <=> happ (happ h4/prim__rec/_3C m) n
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_ADDu_u_ASSOC]: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm [h4s_pairs_pairu_u_caseu_u_thm]: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = happ (happ f x) y
% Assm [h4s_arithmetics_ADDu_u_COMM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_relations_RESTRICTu_u_LEMMA]: !z y f R. happ (happ R y) z ==> happ (h4/relation/RESTRICT f R z) y = happ f y
% Assm [h4s_primu_u_recs_WFu_u_LESS]: h4/relation/WF h4/prim__rec/_3C
% Assm [h4s_relations_WFu_u_invu_u_image]: !f R. h4/relation/WF R ==> h4/relation/WF (h4/relation/inv__image R f)
% Assm [h4s_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_relations_invu_u_imageu_u_def]: !f R x x'. happ (happ (h4/relation/inv__image R f) x) x' <=> happ (happ R (happ f x)) (happ f x')
% Assm [h4s_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_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_arithmetics_NOTu_u_LESS]: !n m. ~happ (happ h4/prim__rec/_3C m) n <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_ADDu_u_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_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_bools_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% Assm [h4s_patricias_ALLu_u_DISTINCTu_u_TRAVERSE]: !t. h4/patricia/IS__PTREE t ==> h4/list/ALL__DISTINCT (h4/patricia/TRAVERSE t)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_patricias_MEMu_u_ALLu_u_DISTINCTu_u_IMPu_u_PERM]: !l2 l1. h4/list/ALL__DISTINCT l1 /\ h4/list/ALL__DISTINCT l2 /\ (!x. h4/bool/IN x (h4/list/LIST__TO__SET l1) <=> h4/bool/IN x (h4/list/LIST__TO__SET l2)) ==> h4/sorting/PERM l1 l2
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_richu_u_lists_FOLDLu_u_FOLDRu_u_REVERSE]: !_1. (!f x y. happ (happ (happ _1 f) x) y = happ (happ f y) x) ==> (!_0. (!f x. happ (happ _0 f) x = happ (happ _1 f) x) ==> (!l f e. h4/list/FOLDL f e l = h4/list/FOLDR (happ _0 f) e (h4/list/REVERSE l)))
% Assm [h4s_richu_u_lists_FOLDRu_u_APPEND]: !l2 l1 f e. h4/list/FOLDR f e (h4/list/APPEND l1 l2) = h4/list/FOLDR f (h4/list/FOLDR f e l2) l1
% Assm [h4s_lists_ALLu_u_DISTINCTu_u_SETu_u_TOu_u_LIST]: !s. h4/pred__set/FINITE s ==> h4/list/ALL__DISTINCT (h4/list/SET__TO__LIST s)
% Assm [h4s_lists_SETu_u_TOu_u_LISTu_u_INV]: !s. h4/pred__set/FINITE s ==> h4/list/LIST__TO__SET (h4/list/SET__TO__LIST s) = s
% Assm [h4s_lists_ALLu_u_DISTINCT0u_c1]: !t h. h4/list/ALL__DISTINCT (h4/list/CONS h t) <=> ~h4/bool/IN h (h4/list/LIST__TO__SET t) /\ h4/list/ALL__DISTINCT t
% Assm [h4s_lists_MEMu_u_SETu_u_TOu_u_LIST]: !s. h4/pred__set/FINITE s ==> (!x. h4/bool/IN x (h4/list/LIST__TO__SET (h4/list/SET__TO__LIST s)) <=> h4/bool/IN x s)
% Assm [h4s_lists_FINITEu_u_LISTu_u_TOu_u_SET]: !l. h4/pred__set/FINITE (h4/list/LIST__TO__SET l)
% Assm [h4s_lists_ALLu_u_DISTINCT0u_c0]: h4/list/ALL__DISTINCT h4/list/NIL <=> T
% Assm [h4s_lists_FOLDR0u_c0]: !f e. h4/list/FOLDR f e h4/list/NIL = e
% Assm [h4s_lists_REVERSEu_u_DEFu_c0]: h4/list/REVERSE h4/list/NIL = h4/list/NIL
% Assm [h4s_lists_FOLDR0u_c1]: !x l f e. h4/list/FOLDR f e (h4/list/CONS x l) = happ (happ f x) (h4/list/FOLDR f e l)
% Assm [h4s_lists_LISTu_u_TOu_u_SET0u_c1]: !t h. h4/list/LIST__TO__SET (h4/list/CONS h t) = h4/pred__set/INSERT h (h4/list/LIST__TO__SET t)
% Assm [h4s_predu_u_sets_FINITEu_u_INSERT]: !x s. h4/pred__set/FINITE (h4/pred__set/INSERT x s) <=> h4/pred__set/FINITE s
% Assm [h4s_predu_u_sets_INu_u_INSERT]: !y x s. h4/bool/IN x (h4/pred__set/INSERT y s) <=> x = y \/ h4/bool/IN x s
% Assm [h4s_lists_REVERSEu_u_DEFu_c1]: !t h. h4/list/REVERSE (h4/list/CONS h t) = h4/list/APPEND (h4/list/REVERSE t) (h4/list/CONS h h4/list/NIL)
% Assm [h4s_lists_LISTu_u_TOu_u_SET0u_c0]: h4/list/LIST__TO__SET h4/list/NIL = h4/pred__set/EMPTY
% Assm [h4s_predu_u_sets_UNIONu_u_EMPTYu_c1]: !s. h4/pred__set/UNION s h4/pred__set/EMPTY = s
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_combins_ou_u_DEF]: !g f x. h4/combin/o f g x = happ f (happ g x)
% Assm [h4s_combins_Cu_u_DEF]: !x x x. happ (happ (h4/combin/C x) x) x = happ (happ x x) x
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Goal: !t2 t1 n. h4/patricia/IS__PTREE t1 /\ h4/patricia/IS__PTREE t2 ==> (h4/patricia/IN__PTREE n (h4/patricia/UNION__PTREE t1 t2) <=> h4/patricia/IN__PTREE n t1 \/ h4/patricia/IN__PTREE n t2)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q1381546,TV_Q1381542]: ![V_f, V_g]: (![V_x]: s(TV_Q1381542,happ(s(t_fun(TV_Q1381546,TV_Q1381542),V_f),s(TV_Q1381546,V_x))) = s(TV_Q1381542,happ(s(t_fun(TV_Q1381546,TV_Q1381542),V_g),s(TV_Q1381546,V_x))) => s(t_fun(TV_Q1381546,TV_Q1381542),V_f) = s(t_fun(TV_Q1381546,TV_Q1381542),V_g))).
fof(ah4s_patricias_UNIONu_u_PTREEu_u_def, axiom, ![V_t2, V_t1]: s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_unionu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t1),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t2))) = s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_ptreeu_u_ofu_u_numset(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t1),s(t_fun(t_h4s_nums_num,t_bool),h4s_patricias_numsetu_u_ofu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t2)))))).
fof(ah4s_patricias_INu_u_PTREEu_u_def, axiom, ![V_t, V_n]: s(t_bool,h4s_patricias_inu_u_ptree(s(t_h4s_nums_num,V_n),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))) = s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(t_h4s_ones_one),h4s_patricias_peek(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_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_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_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_patricias_INu_u_NUMSETu_u_OFu_u_PTREE, axiom, ![V_t, V_n]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))) => s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_n),s(t_fun(t_h4s_nums_num,t_bool),h4s_patricias_numsetu_u_ofu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))))) = s(t_bool,h4s_patricias_inu_u_ptree(s(t_h4s_nums_num,V_n),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))))).
fof(ah4s_patricias_INu_u_PTREEu_u_OFu_u_NUMSET, axiom, ![V_t, V_s, V_n]: ((p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))) & p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),V_s))))) => (p(s(t_bool,h4s_patricias_inu_u_ptree(s(t_h4s_nums_num,V_n),s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_ptreeu_u_ofu_u_numset(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_fun(t_h4s_nums_num,t_bool),V_s)))))) <=> (p(s(t_bool,h4s_patricias_inu_u_ptree(s(t_h4s_nums_num,V_n),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))) | p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_n),s(t_fun(t_h4s_nums_num,t_bool),V_s)))))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_patricias_PTREEu_u_EXTENSION, axiom, ![V_t2, V_t1]: ((p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t1)))) & p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t2))))) => (s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t1) = s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t2) <=> ![V_x]: s(t_bool,h4s_patricias_inu_u_ptree(s(t_h4s_nums_num,V_x),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t1))) = s(t_bool,h4s_patricias_inu_u_ptree(s(t_h4s_nums_num,V_x),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t2)))))).
fof(ah4s_patricias_INu_u_PTREEu_u_REMOVE, axiom, ![V_t, V_n, V_m]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))) => (p(s(t_bool,h4s_patricias_inu_u_ptree(s(t_h4s_nums_num,V_n),s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_remove(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_h4s_nums_num,V_m)))))) <=> (~ (s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,V_m)) & p(s(t_bool,h4s_patricias_inu_u_ptree(s(t_h4s_nums_num,V_n),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))))))).
fof(ah4s_patricias_INu_u_PTREEu_u_INSERTu_u_PTREE, axiom, ![V_t, V_n, V_m]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))) => (p(s(t_bool,h4s_patricias_inu_u_ptree(s(t_h4s_nums_num,V_n),s(t_h4s_patricias_ptree(t_h4s_ones_one),happ(s(t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one))),h4s_patricias_insertu_u_ptree),s(t_h4s_nums_num,V_m))),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))))) <=> (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n) | p(s(t_bool,h4s_patricias_inu_u_ptree(s(t_h4s_nums_num,V_n),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))))))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_patricias_PTREEu_u_OFu_u_NUMSETu_u_ISu_u_PTREE, axiom, ![V_t, V_s]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))) => p(s(t_bool,h4s_patricias_isu_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),V_t),s(t_fun(t_h4s_nums_num,t_bool),V_s)))))))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_patricias_MEMu_u_TRAVERSEu_u_PEEK, axiom, ![TV_u_27a]: ![V_t, V_k]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(TV_u_27a),V_t)))) => s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_k),s(t_fun(t_h4s_nums_num,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(TV_u_27a),V_t))))))) = s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),h4s_patricias_peek(s(t_h4s_patricias_ptree(TV_u_27a),V_t),s(t_h4s_nums_num,V_k))))))).
fof(ah4s_patricias_PEEKu_u_INSERTu_u_PTREE, axiom, ![V_t, V_k, V_j]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))) => ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,V_j)) & s(t_h4s_options_option(t_h4s_ones_one),h4s_patricias_peek(s(t_h4s_patricias_ptree(t_h4s_ones_one),happ(s(t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one))),h4s_patricias_insertu_u_ptree),s(t_h4s_nums_num,V_k))),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))),s(t_h4s_nums_num,V_j))) = s(t_h4s_options_option(t_h4s_ones_one),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(t_h4s_ones_one),h4s_options_some(s(t_h4s_ones_one,h4s_ones_one0))),s(t_h4s_options_option(t_h4s_ones_one),h4s_patricias_peek(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_h4s_nums_num,V_j)))))))).
fof(ah4s_patricias_MEMu_u_TRAVERSE, axiom, ![V_t, V_k]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))) => s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_k),s(t_fun(t_h4s_nums_num,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))))))) = s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_k),s(t_fun(t_h4s_nums_num,t_bool),h4s_patricias_numsetu_u_ofu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))))))).
fof(ah4s_patricias_PERMu_u_INSERTu_u_PTREE, axiom, ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),V_s)))) => (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))) => p(s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_lists_foldl(s(t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_fun(t_h4s_nums_num,t_h4s_patricias_ptree(t_h4s_ones_one))),h4s_combins_c(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one))),h4s_patricias_insertu_u_ptree))),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_setu_u_tou_u_list(s(t_fun(t_h4s_nums_num,t_bool),V_s))))))),s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_setu_u_tou_u_list(s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_union(s(t_fun(t_h4s_nums_num,t_bool),h4s_patricias_numsetu_u_ofu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))),s(t_fun(t_h4s_nums_num,t_bool),V_s))))))))))).
fof(ah4s_ones_one1, axiom, ![V_v]: s(t_h4s_ones_one,V_v) = s(t_h4s_ones_one,h4s_ones_one0)).
fof(ah4s_patricias_INu_u_PTREEu_u_EMPTY, axiom, ![V_n]: ~ (p(s(t_bool,h4s_patricias_inu_u_ptree(s(t_h4s_nums_num,V_n),s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_empty)))))).
fof(ah4s_patricias_INSERTu_u_PTREEu_u_ISu_u_PTREE, axiom, ![V_x, V_t]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))) => p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),happ(s(t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one))),h4s_patricias_insertu_u_ptree),s(t_h4s_nums_num,V_x))),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))))))).
fof(ah4s_patricias_MEMu_u_TRAVERSEu_u_INSERTu_u_PTREE, axiom, ![V_x, V_t, V_h]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))) => (p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_x),s(t_fun(t_h4s_nums_num,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(t_h4s_ones_one),happ(s(t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one))),h4s_patricias_insertu_u_ptree),s(t_h4s_nums_num,V_h))),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))))))))) <=> (s(t_h4s_nums_num,V_x) = s(t_h4s_nums_num,V_h) | (~ (s(t_h4s_nums_num,V_x) = s(t_h4s_nums_num,V_h)) & p(s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_x),s(t_fun(t_h4s_nums_num,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))))))))))))).
fof(ah4s_patricias_PTREEu_u_OFu_u_NUMSETu_u_NUMSETu_u_OFu_u_PTREE, axiom, ![V_t, V_s]: ((p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))) & p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),V_s))))) => s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_ptreeu_u_ofu_u_numset(s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_empty),s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_union(s(t_fun(t_h4s_nums_num,t_bool),h4s_patricias_numsetu_u_ofu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))),s(t_fun(t_h4s_nums_num,t_bool),V_s))))) = s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_ptreeu_u_ofu_u_numset(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_fun(t_h4s_nums_num,t_bool),V_s))))).
fof(ah4s_patricias_NUMSETu_u_OFu_u_PTREEu_u_PTREEu_u_OFu_u_NUMSET, axiom, ![V_t, V_s]: ((p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))) & p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),V_s))))) => s(t_fun(t_h4s_nums_num,t_bool),h4s_patricias_numsetu_u_ofu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_ptreeu_u_ofu_u_numset(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_fun(t_h4s_nums_num,t_bool),V_s))))) = s(t_fun(t_h4s_nums_num,t_bool),h4s_predu_u_sets_union(s(t_fun(t_h4s_nums_num,t_bool),h4s_patricias_numsetu_u_ofu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))),s(t_fun(t_h4s_nums_num,t_bool),V_s))))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_patricias_PEEKu_u_ADD, axiom, ![TV_u_27a]: ![V_t, V_k, V_j, V_d]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(TV_u_27a),V_t)))) => ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,V_j)) & s(t_h4s_options_option(TV_u_27a),h4s_patricias_peek(s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_add(s(t_h4s_patricias_ptree(TV_u_27a),V_t),s(t_h4s_pairs_prod(t_h4s_nums_num,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_nums_num,V_k),s(TV_u_27a,V_d))))),s(t_h4s_nums_num,V_j))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_d))),s(t_h4s_options_option(TV_u_27a),h4s_patricias_peek(s(t_h4s_patricias_ptree(TV_u_27a),V_t),s(t_h4s_nums_num,V_j)))))))).
fof(ah4s_patricias_ADDu_u_INSERT, axiom, ![V_v, V_t, V_n]: s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_add(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_ones_one),h4s_pairs_u_2c(s(t_h4s_nums_num,V_n),s(t_h4s_ones_one,V_v))))) = s(t_h4s_patricias_ptree(t_h4s_ones_one),happ(s(t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one))),h4s_patricias_insertu_u_ptree),s(t_h4s_nums_num,V_n))),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))).
fof(ah4s_options_ISu_u_SOMEu_u_DEFu_c0, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))) = s(t_bool,t)).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_patricias_INu_u_PTREEu_u_OFu_u_NUMSETu_u_EMPTY, axiom, ![V_s, V_n]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),V_s)))) => s(t_bool,h4s_bools_in(s(t_h4s_nums_num,V_n),s(t_fun(t_h4s_nums_num,t_bool),V_s))) = s(t_bool,h4s_patricias_inu_u_ptree(s(t_h4s_nums_num,V_n),s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_ptreeu_u_ofu_u_numset(s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_empty),s(t_fun(t_h4s_nums_num,t_bool),V_s))))))).
fof(ah4s_predu_u_sets_INu_u_UNION, axiom, ![TV_u_27a]: ![V_x, V_t, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) <=> (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s)))) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_patricias_NUMSETu_u_OFu_u_PTREEu_u_def, axiom, ![V_t]: s(t_fun(t_h4s_nums_num,t_bool),h4s_patricias_numsetu_u_ofu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))) = s(t_fun(t_h4s_nums_num,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t)))))).
fof(ah4s_options_ISu_u_SOMEu_u_DEFu_c1, axiom, ![TV_u_27a]: s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_bool,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_predu_u_sets_FINITEu_u_UNION, axiom, ![TV_u_27a]: ![V_t, V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),V_t)))))) <=> (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) & p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_t))))))).
fof(ah4s_patricias_FINITEu_u_NUMSETu_u_OFu_u_PTREE, axiom, ![V_t]: p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_nums_num,t_bool),h4s_patricias_numsetu_u_ofu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_patricias_INSERTu_u_PTREEu_u_def, axiom, ![V_t, V_n]: s(t_h4s_patricias_ptree(t_h4s_ones_one),happ(s(t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one))),h4s_patricias_insertu_u_ptree),s(t_h4s_nums_num,V_n))),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t))) = s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_add(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_ones_one),h4s_pairs_u_2c(s(t_h4s_nums_num,V_n),s(t_h4s_ones_one,h4s_ones_one0)))))).
fof(ah4s_patricias_ADDu_u_ISu_u_PTREE, axiom, ![TV_u_27a]: ![V_x, V_t]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(TV_u_27a),V_t)))) => p(s(t_bool,h4s_patricias_isu_u_ptree(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),V_x)))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
fof(ah4s_sortings_PERMu_u_MEMu_u_EQ, axiom, ![TV_u_27a]: ![V_l2, V_l1]: (p(s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2)))) => ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l1))))) = s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l2))))))).
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_patricias_PTREEu_u_OFu_u_NUMSETu_u_def, axiom, ![V_t, V_s]: s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_ptreeu_u_ofu_u_numset(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_fun(t_h4s_nums_num,t_bool),V_s))) = s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_lists_foldl(s(t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_fun(t_h4s_nums_num,t_h4s_patricias_ptree(t_h4s_ones_one))),h4s_combins_c(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(t_h4s_ones_one),t_h4s_patricias_ptree(t_h4s_ones_one))),h4s_patricias_insertu_u_ptree))),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t),s(t_h4s_lists_list(t_h4s_nums_num),h4s_lists_setu_u_tou_u_list(s(t_fun(t_h4s_nums_num,t_bool),V_s)))))).
fof(ah4s_lists_SETu_u_TOu_u_LISTu_u_INu_u_MEM, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_setu_u_tou_u_list(s(t_fun(TV_u_27a,t_bool),V_s))))))))).
fof(ah4s_patricias_PEEKu_u_REMOVE, axiom, ![TV_u_27a]: ![V_t, V_k, V_j]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(TV_u_27a),V_t)))) => ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,V_j)) & s(t_h4s_options_option(TV_u_27a),h4s_patricias_peek(s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_remove(s(t_h4s_patricias_ptree(TV_u_27a),V_t),s(t_h4s_nums_num,V_k))),s(t_h4s_nums_num,V_j))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_h4s_options_option(TV_u_27a),h4s_patricias_peek(s(t_h4s_patricias_ptree(TV_u_27a),V_t),s(t_h4s_nums_num,V_j)))))))).
fof(ah4s_patricias_PTREEu_u_EQ, axiom, ![TV_u_27a]: ![V_t2, V_t1]: ((p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(TV_u_27a),V_t1)))) & p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(TV_u_27a),V_t2))))) => (![V_k]: s(t_h4s_options_option(TV_u_27a),h4s_patricias_peek(s(t_h4s_patricias_ptree(TV_u_27a),V_t1),s(t_h4s_nums_num,V_k))) = s(t_h4s_options_option(TV_u_27a),h4s_patricias_peek(s(t_h4s_patricias_ptree(TV_u_27a),V_t2),s(t_h4s_nums_num,V_k))) <=> s(t_h4s_patricias_ptree(TV_u_27a),V_t1) = s(t_h4s_patricias_ptree(TV_u_27a),V_t2)))).
fof(ah4s_bools_BOOLu_u_EQu_u_DISTINCTu_c0, axiom, ~ (s(t_bool,t) = s(t_bool,f))).
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_options_optionu_u_nchotomy, axiom, ![TV_u_27a]: ![V_opt]: (s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) | ?[V_x]: s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_patricias_EMPTYu_u_ISu_u_PTREE, axiom, ![TV_u_27a]: p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_empty))))).
fof(ah4s_lists_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_combins_Cu_u_THM, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27c)),h4s_combins_c(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(TV_u_27b,V_x))),s(TV_u_27a,V_y))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_y))),s(TV_u_27b,V_x)))).
fof(ah4s_bools_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_predu_u_sets_EXTENSION, axiom, ![TV_u_27a]: ![V_t, V_s]: (s(t_fun(TV_u_27a,t_bool),V_s) = s(t_fun(TV_u_27a,t_bool),V_t) <=> ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))) = s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_t))))).
fof(ah4s_lists_FOLDL0u_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_l, V_f, V_e]: s(TV_u_27b,h4s_lists_foldl(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(TV_u_27b,h4s_lists_foldl(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27b,V_e))),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_l)))).
fof(ah4s_lists_FOLDL0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_e]: s(TV_u_27b,h4s_lists_foldl(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(TV_u_27b,V_e)).
fof(ah4s_patricias_PTREEu_u_OFu_u_NUMSETu_u_ISu_u_PTREEu_u_EMPTY, axiom, ![V_s]: p(s(t_bool,h4s_patricias_isu_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_patricias_empty),s(t_fun(t_h4s_nums_num,t_bool),V_s))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_relations_WFRECu_u_COROLLARY, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_M]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_wfrec(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M))) => (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M),s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))))),s(TV_u_27a,V_x)))))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_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_patricias_PEEKu_u_curriedu_u_def, axiom, ![TV_u_27a]: ![V_x1, V_x]: s(t_h4s_options_option(TV_u_27a),h4s_patricias_peek(s(t_h4s_patricias_ptree(TV_u_27a),V_x),s(t_h4s_nums_num,V_x1))) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),h4s_patricias_peeku_u_tupled),s(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_patricias_ptree(TV_u_27a),V_x),s(t_h4s_nums_num,V_x1)))))).
fof(ah4s_patricias_PEEKu_u_tupledu_u_primitiveu_u_def, axiom, ![TV_u_27a]: ![V_uu_10]: (![V_PEEKu_u_tupled, V_m, V_l, V_k, V_r]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a)))))),V_uu_10),s(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),V_PEEKu_u_tupled))),s(t_h4s_nums_num,V_m))),s(t_h4s_patricias_ptree(TV_u_27a),V_l))),s(t_h4s_nums_num,V_k))),s(t_h4s_patricias_ptree(TV_u_27a),V_r))) = s(t_h4s_options_option(TV_u_27a),h4s_combins_i(s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),V_PEEKu_u_tupled),s(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_patricias_ptree(TV_u_27a),h4s_bools_cond(s(t_bool,h4s_bits_bit(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_k))),s(t_h4s_patricias_ptree(TV_u_27a),V_l),s(t_h4s_patricias_ptree(TV_u_27a),V_r))),s(t_h4s_nums_num,V_k))))))) => ![V_uu_9]: (![V_PEEKu_u_tupled, V_m, V_k, V_l]: s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a)))))),V_uu_9),s(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),V_PEEKu_u_tupled))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_k))),s(t_h4s_patricias_ptree(TV_u_27a),V_l))) = s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a)))))),V_uu_10),s(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),V_PEEKu_u_tupled))),s(t_h4s_nums_num,V_m))),s(t_h4s_patricias_ptree(TV_u_27a),V_l))),s(t_h4s_nums_num,V_k))) => ![V_uu_8]: (![V_PEEKu_u_tupled, V_k, V_m]: s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a)))))),V_uu_8),s(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),V_PEEKu_u_tupled))),s(t_h4s_nums_num,V_k))),s(t_h4s_nums_num,V_m))) = s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a)))))),V_uu_9),s(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),V_PEEKu_u_tupled))),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_k))) => ![V_uu_7]: (![V_PEEKu_u_tupled, V_k, V_p]: s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a)))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a))))))),V_uu_7),s(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),V_PEEKu_u_tupled))),s(t_h4s_nums_num,V_k))),s(t_h4s_nums_num,V_p))) = s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a)))))),V_uu_8),s(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),V_PEEKu_u_tupled))),s(t_h4s_nums_num,V_k))) => ![V_uu_6]: (![V_k, V_j, V_d]: ?[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),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)))),V_uu_6),s(t_h4s_nums_num,V_k))),s(t_h4s_nums_num,V_j))),s(TV_u_27a,V_d))) = s(t_h4s_options_option(TV_u_27a),h4s_combins_i(s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_d))),s(t_h4s_options_option(TV_u_27a),h4s_options_none)))))) => ![V_uu_5]: (![V_k, V_j]: s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)))),V_uu_5),s(t_h4s_nums_num,V_k))),s(t_h4s_nums_num,V_j))) = s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)))),V_uu_6),s(t_h4s_nums_num,V_k))),s(t_h4s_nums_num,V_j))) => ![V_uu_4]: (![V_v, V_PEEKu_u_tupled, V_k]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))),V_uu_4),s(t_h4s_patricias_ptree(TV_u_27a),V_v))),s(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),V_PEEKu_u_tupled))),s(t_h4s_nums_num,V_k))) = s(t_h4s_options_option(TV_u_27a),h4s_patricias_ptreeu_u_case(s(t_h4s_patricias_ptree(TV_u_27a),V_v),s(t_h4s_options_option(TV_u_27a),h4s_combins_i(s(t_h4s_options_option(TV_u_27a),h4s_options_none))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)))),V_uu_5),s(t_h4s_nums_num,V_k))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a))))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a)))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_options_option(TV_u_27a))))))),V_uu_7),s(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),V_PEEKu_u_tupled))),s(t_h4s_nums_num,V_k))))) => ![V_uu_3]: (![V_PEEKu_u_tupled, V_v]: s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),V_PEEKu_u_tupled))),s(t_h4s_patricias_ptree(TV_u_27a),V_v))) = s(t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))),V_uu_4),s(t_h4s_patricias_ptree(TV_u_27a),V_v))),s(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),V_PEEKu_u_tupled))) => ![V_uu_2]: (![V_PEEKu_u_tupled, V_a]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),V_PEEKu_u_tupled))),s(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),V_a))) = s(t_h4s_options_option(TV_u_27a),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),V_a),s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_nums_num,t_h4s_options_option(TV_u_27a)))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),V_PEEKu_u_tupled))))) => ![V_uu_1]: (![V_PEEKu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a))),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),V_PEEKu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),V_PEEKu_u_tupled))) => ![V_uu_0]: (![V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_bool)),V_R)))) & ![V_p, V_r, V_l, V_k, V_m]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_bool)),V_R),s(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_patricias_ptree(TV_u_27a),h4s_bools_cond(s(t_bool,h4s_bits_bit(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_k))),s(t_h4s_patricias_ptree(TV_u_27a),V_l),s(t_h4s_patricias_ptree(TV_u_27a),V_r))),s(t_h4s_nums_num,V_k))))),s(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),h4s_pairs_u_2c(s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_branch(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,V_m),s(t_h4s_patricias_ptree(TV_u_27a),V_l),s(t_h4s_patricias_ptree(TV_u_27a),V_r))),s(t_h4s_nums_num,V_k)))))))) => s(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),h4s_patricias_peeku_u_tupled) = s(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_patricias_ptree(TV_u_27a),t_h4s_nums_num),t_h4s_options_option(TV_u_27a))),V_uu_1))))))))))))))).
fof(ah4s_patricias_ptreeu_u_sizeu_u_defu_c2, axiom, ![TV_u_27a]: ![V_f, V_a3, V_a2, V_a1, V_a0]: s(t_h4s_nums_num,h4s_patricias_ptreeu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_branch(s(t_h4s_nums_num,V_a0),s(t_h4s_nums_num,V_a1),s(t_h4s_patricias_ptree(TV_u_27a),V_a2),s(t_h4s_patricias_ptree(TV_u_27a),V_a3))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_a0),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_a1),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_patricias_ptreeu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_patricias_ptree(TV_u_27a),V_a2))),s(t_h4s_nums_num,h4s_patricias_ptreeu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_patricias_ptree(TV_u_27a),V_a3)))))))))))).
fof(ah4s_patricias_ptreeu_u_caseu_u_defu_c2, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_f1, V_f, V_a3, V_a2, V_a1, V_a0]: s(TV_u_27b,h4s_patricias_ptreeu_u_case(s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_branch(s(t_h4s_nums_num,V_a0),s(t_h4s_nums_num,V_a1),s(t_h4s_patricias_ptree(TV_u_27a),V_a2),s(t_h4s_patricias_ptree(TV_u_27a),V_a3))),s(TV_u_27b,V_v),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27b)),V_f),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b)))),V_f1))) = s(TV_u_27b,happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b)))),V_f1),s(t_h4s_nums_num,V_a0))),s(t_h4s_nums_num,V_a1))),s(t_h4s_patricias_ptree(TV_u_27a),V_a2))),s(t_h4s_patricias_ptree(TV_u_27a),V_a3)))).
fof(ah4s_patricias_ptreeu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f1, V_f]: s(TV_u_27b,h4s_patricias_ptreeu_u_case(s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_empty),s(TV_u_27b,V_v),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27b)),V_f),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b)))),V_f1))) = s(TV_u_27b,V_v)).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27b,V_y)).
fof(ah4s_pairs_PAIR, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,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_patricias_ptreeu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_f1, V_f, V_a1, V_a0]: s(TV_u_27b,h4s_patricias_ptreeu_u_case(s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_leaf(s(t_h4s_nums_num,V_a0),s(TV_u_27a,V_a1))),s(TV_u_27b,V_v),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27b)),V_f),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b)))),V_f1))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27b)),V_f),s(t_h4s_nums_num,V_a0))),s(TV_u_27a,V_a1)))).
fof(ah4s_arithmetics_LESSu_u_MONOu_u_ADDu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c2, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_ASSOC, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p)))).
fof(ah4s_pairs_pairu_u_caseu_u_thm, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_y, V_x, V_f]: s(TV_u_27a,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27b,V_x),s(TV_u_27c,V_y))),s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f))) = s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27b,V_x))),s(TV_u_27c,V_y)))).
fof(ah4s_arithmetics_ADDu_u_COMM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_relations_RESTRICTu_u_LEMMA, axiom, ![TV_u_27b,TV_u_27a]: ![V_z, V_y, V_f, V_R]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_z)))) => s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_z))),s(TV_u_27a,V_y))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))))).
fof(ah4s_primu_u_recs_WFu_u_LESS, axiom, p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c))))).
fof(ah4s_relations_WFu_u_invu_u_image, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R)))) => p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_invu_u_image(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_f)))))))).
fof(ah4s_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_relations_invu_u_imageu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_x, V_xi_]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_invu_u_image(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_x))),s(TV_u_27a,V_xi_))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_xi_)))))).
fof(ah4s_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_ZEROu_u_LESSu_u_EQ, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_NOTu_u_LESS, axiom, ![V_n, V_m]: (~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_ADDu_u_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_ADDu_u_CLAUSESu_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_SYM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_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_bools_IMPu_u_Fu_u_EQu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_patricias_ALLu_u_DISTINCTu_u_TRAVERSE, axiom, ![TV_u_27a]: ![V_t]: (p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(TV_u_27a),V_t)))) => p(s(t_bool,h4s_lists_allu_u_distinct(s(t_h4s_lists_list(t_h4s_nums_num),h4s_patricias_traverse(s(t_h4s_patricias_ptree(TV_u_27a),V_t)))))))).
fof(ah4s_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_patricias_MEMu_u_ALLu_u_DISTINCTu_u_IMPu_u_PERM, axiom, ![TV_u_27a]: ![V_l2, V_l1]: ((p(s(t_bool,h4s_lists_allu_u_distinct(s(t_h4s_lists_list(TV_u_27a),V_l1)))) & (p(s(t_bool,h4s_lists_allu_u_distinct(s(t_h4s_lists_list(TV_u_27a),V_l2)))) & ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l1))))) = s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l2))))))) => p(s(t_bool,h4s_sortings_perm(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2)))))).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_richu_u_lists_FOLDLu_u_FOLDRu_u_REVERSE, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_f, V_x, V_y]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27a))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_f))),s(TV_u_27b,V_x))),s(TV_u_27a,V_y))) = s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_f),s(TV_u_27a,V_y))),s(TV_u_27b,V_x))) => ![V_uu_0]: (![V_f, V_x]: s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27a))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_f))),s(TV_u_27b,V_x))) = s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27a))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_f))),s(TV_u_27b,V_x))) => ![V_l, V_f, V_e]: s(TV_u_27a,h4s_lists_foldl(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_f),s(TV_u_27a,V_e),s(t_h4s_lists_list(TV_u_27b),V_l))) = s(TV_u_27a,h4s_lists_foldr(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27a))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_f))),s(TV_u_27a,V_e),s(t_h4s_lists_list(TV_u_27b),h4s_lists_reverse(s(t_h4s_lists_list(TV_u_27b),V_l)))))))).
fof(ah4s_richu_u_lists_FOLDRu_u_APPEND, axiom, ![TV_u_27b,TV_u_27a]: ![V_l2, V_l1, V_f, V_e]: s(TV_u_27b,h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2))))) = s(TV_u_27b,h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27b,h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27a),V_l2))),s(t_h4s_lists_list(TV_u_27a),V_l1)))).
fof(ah4s_lists_ALLu_u_DISTINCTu_u_SETu_u_TOu_u_LIST, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => p(s(t_bool,h4s_lists_allu_u_distinct(s(t_h4s_lists_list(TV_u_27a),h4s_lists_setu_u_tou_u_list(s(t_fun(TV_u_27a,t_bool),V_s)))))))).
fof(ah4s_lists_SETu_u_TOu_u_LISTu_u_INV, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_setu_u_tou_u_list(s(t_fun(TV_u_27a,t_bool),V_s))))) = s(t_fun(TV_u_27a,t_bool),V_s))).
fof(ah4s_lists_ALLu_u_DISTINCT0u_c1, axiom, ![TV_u_27a]: ![V_t, V_h]: (p(s(t_bool,h4s_lists_allu_u_distinct(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t)))))) <=> (~ (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_h),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_t))))))) & p(s(t_bool,h4s_lists_allu_u_distinct(s(t_h4s_lists_list(TV_u_27a),V_t))))))).
fof(ah4s_lists_MEMu_u_SETu_u_TOu_u_LIST, axiom, ![TV_u_27a]: ![V_s]: (p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))) => ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_setu_u_tou_u_list(s(t_fun(TV_u_27a,t_bool),V_s))))))) = s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))).
fof(ah4s_lists_FINITEu_u_LISTu_u_TOu_u_SET, axiom, ![TV_u_27a]: ![V_l]: p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l))))))).
fof(ah4s_lists_ALLu_u_DISTINCT0u_c0, axiom, ![TV_u_27a]: s(t_bool,h4s_lists_allu_u_distinct(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_bool,t)).
fof(ah4s_lists_FOLDR0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_e]: s(TV_u_27b,h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(TV_u_27b,V_e)).
fof(ah4s_lists_REVERSEu_u_DEFu_c0, axiom, ![TV_u_27a]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_reverse(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)).
fof(ah4s_lists_FOLDR0u_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_l, V_f, V_e]: s(TV_u_27b,h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,h4s_lists_foldr(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27b)),V_f),s(TV_u_27b,V_e),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_lists_LISTu_u_TOu_u_SET0u_c1, axiom, ![TV_u_27b]: ![V_t, V_h]: s(t_fun(TV_u_27b,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27b),h4s_lists_cons(s(TV_u_27b,V_h),s(t_h4s_lists_list(TV_u_27b),V_t))))) = s(t_fun(TV_u_27b,t_bool),h4s_predu_u_sets_insert(s(TV_u_27b,V_h),s(t_fun(TV_u_27b,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27b),V_t)))))).
fof(ah4s_predu_u_sets_FINITEu_u_INSERT, axiom, ![TV_u_27a]: ![V_x, V_s]: s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))) = s(t_bool,h4s_predu_u_sets_finite(s(t_fun(TV_u_27a,t_bool),V_s)))).
fof(ah4s_predu_u_sets_INu_u_INSERT, axiom, ![TV_u_27a]: ![V_y, V_x, V_s]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_insert(s(TV_u_27a,V_y),s(t_fun(TV_u_27a,t_bool),V_s)))))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_s))))))).
fof(ah4s_lists_REVERSEu_u_DEFu_c1, axiom, ![TV_u_27a]: ![V_t, V_h]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_reverse(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t))))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),h4s_lists_reverse(s(t_h4s_lists_list(TV_u_27a),V_t))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))))).
fof(ah4s_lists_LISTu_u_TOu_u_SET0u_c0, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty)).
fof(ah4s_predu_u_sets_UNIONu_u_EMPTYu_c1, axiom, ![TV_u_27a]: ![V_s]: s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_union(s(t_fun(TV_u_27a,t_bool),V_s),s(t_fun(TV_u_27a,t_bool),h4s_predu_u_sets_empty))) = s(t_fun(TV_u_27a,t_bool),V_s)).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_combins_ou_u_DEF, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_g, V_f, V_x]: s(TV_u_27b,h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x)))))).
fof(ah4s_combins_Cu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_x, V_x0, V_x1]: s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27c)),h4s_combins_c(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x))),s(TV_u_27b,V_x0))),s(TV_u_27a,V_x1))) = 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_x),s(TV_u_27a,V_x1))),s(TV_u_27b,V_x0)))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ch4s_patricias_INu_u_PTREEu_u_UNIONu_u_PTREE, conjecture, ![V_t2, V_t1, V_n]: ((p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t1)))) & p(s(t_bool,h4s_patricias_isu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t2))))) => (p(s(t_bool,h4s_patricias_inu_u_ptree(s(t_h4s_nums_num,V_n),s(t_h4s_patricias_ptree(t_h4s_ones_one),h4s_patricias_unionu_u_ptree(s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t1),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t2)))))) <=> (p(s(t_bool,h4s_patricias_inu_u_ptree(s(t_h4s_nums_num,V_n),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t1)))) | p(s(t_bool,h4s_patricias_inu_u_ptree(s(t_h4s_nums_num,V_n),s(t_h4s_patricias_ptree(t_h4s_ones_one),V_t2)))))))).
