%   ORIGINAL: h4/patricia/PEEK__def_c2
% 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/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/PEEK__ind: !P. (!k. P h4/patricia/Empty k) /\ (!j d k. P (h4/patricia/Leaf j d) k) /\ (!p m l r k. P (h4/bool/COND (h4/bit/BIT m k) l r) k ==> P (h4/patricia/Branch p m l r) k) ==> (!v v1. P v v1)
% Assm: h4/bool/TRUTH: T
% Assm: h4/patricia/Branch0: h4/patricia/Branch = h4/patricia/_20_40ind__typepatricia2
% 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/hidden____20__40ind____typepatricia2____def: h4/patricia/_20_40ind__typepatricia2 = (\a0 a1 a2 a3. h4/patricia/_20_40ind__typepatricia3 ((\a00 a10 a20 a30. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C a00 (h4/pair/_2C h4/bool/ARB a10)) (h4/ind__type/FCONS a20 (h4/ind__type/FCONS a30 (\n. h4/ind__type/BOTTOM)))) a0 a1 (h4/patricia/_20_40ind__typepatricia4 a2) (h4/patricia/_20_40ind__typepatricia4 a3)))
% Assm: h4/patricia/ptree__repfns_c0: !a. h4/patricia/_20_40ind__typepatricia3 (h4/patricia/_20_40ind__typepatricia4 a) = a
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% 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/PEEK__def_c1: !k j d. h4/patricia/PEEK (h4/patricia/Leaf j d) k = h4/bool/COND (k = j) (h4/option/SOME d) h4/option/NONE
% Assm: h4/patricia/PEEK__def_c0: !k. h4/patricia/PEEK h4/patricia/Empty k = h4/option/NONE
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/patricia/hidden____20__40ind____typepatricia1____def: h4/patricia/_20_40ind__typepatricia1 = (\a0 a1. h4/patricia/_20_40ind__typepatricia3 ((\a00 a10. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C a00 (h4/pair/_2C a10 h4/bool/ARB)) (\n. h4/ind__type/BOTTOM)) a0 a1))
% Assm: h4/patricia/ptree__repfns_c1: !r. (\a0_27. !_27ptree_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C h4/bool/ARB (h4/pair/_2C h4/bool/ARB h4/bool/ARB)) (\n. h4/ind__type/BOTTOM) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C a00 (h4/pair/_2C a10 h4/bool/ARB)) (\n. h4/ind__type/BOTTOM)) a0 a1) \/ (?a0 a1 a2 a3. a0_270 = (\a00 a10 a20 a30. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C a00 (h4/pair/_2C h4/bool/ARB a10)) (h4/ind__type/FCONS a20 (h4/ind__type/FCONS a30 (\n. h4/ind__type/BOTTOM)))) a0 a1 a2 a3 /\ _27ptree_27 a2 /\ _27ptree_27 a3) ==> _27ptree_27 a0_270) ==> _27ptree_27 a0_27) r <=> h4/patricia/_20_40ind__typepatricia4 (h4/patricia/_20_40ind__typepatricia3 r) = r
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/patricia/Empty0: h4/patricia/Empty = h4/patricia/_20_40ind__typepatricia0
% Assm: h4/patricia/Leaf0: h4/patricia/Leaf = h4/patricia/_20_40ind__typepatricia1
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/patricia/hidden____20__40ind____typepatricia0____def: h4/patricia/_20_40ind__typepatricia0 = h4/patricia/_20_40ind__typepatricia3 (h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C h4/bool/ARB (h4/pair/_2C h4/bool/ARB h4/bool/ARB)) (\n. h4/ind__type/BOTTOM))
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/patricia/ptree__case__cong: !v_27 v f1_27 f1 f_27 f M_27 M. M = M_27 /\ (M_27 = h4/patricia/Empty ==> v = v_27) /\ (!a0 a1. M_27 = h4/patricia/Leaf a0 a1 ==> f a0 a1 = f_27 a0 a1) /\ (!a0 a1 a2 a3. M_27 = h4/patricia/Branch a0 a1 a2 a3 ==> f1 a0 a1 a2 a3 = f1_27 a0 a1 a2 a3) ==> h4/patricia/ptree__CASE M v f f1 = h4/patricia/ptree__CASE M_27 v_27 f_27 f1_27
% Assm: h4/patricia/ptree__11_c1: !a3_27 a3 a2_27 a2 a1_27 a1 a0_27 a0. h4/patricia/Branch a0 a1 a2 a3 = h4/patricia/Branch a0_27 a1_27 a2_27 a3_27 <=> a0 = a0_27 /\ a1 = a1_27 /\ a2 = a2_27 /\ a3 = a3_27
% Assm: h4/patricia/ptree__distinct_c2: !a3 a2 a1_27 a1 a0_27 a0. ~(h4/patricia/Leaf a0 a1 = h4/patricia/Branch a0_27 a1_27 a2 a3)
% Assm: h4/patricia/ptree__distinct_c1: !a3 a2 a1 a0. ~(h4/patricia/Empty = h4/patricia/Branch a0 a1 a2 a3)
% Assm: h4/patricia/ptree__nchotomy: !pp. pp = h4/patricia/Empty \/ (?n a. pp = h4/patricia/Leaf n a) \/ (?n0 n p p0. pp = h4/patricia/Branch n0 n p p0)
% Assm: h4/patricia/ptree__induction: !P. P h4/patricia/Empty /\ (!n a. P (h4/patricia/Leaf n a)) /\ (!p p0. P p /\ P p0 ==> (!n n0. P (h4/patricia/Branch n0 n p p0))) ==> (!p. P p)
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/DATATYPE__TAG__THM: !x. h4/bool/DATATYPE x <=> T
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% 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/datatype__ptree: !ptree. h4/bool/DATATYPE (ptree h4/patricia/Empty h4/patricia/Leaf h4/patricia/Branch)
% Assm: h4/patricia/ptree__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION (\a0_27. !_27ptree_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C h4/bool/ARB (h4/pair/_2C h4/bool/ARB h4/bool/ARB)) (\n. h4/ind__type/BOTTOM) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C a00 (h4/pair/_2C a10 h4/bool/ARB)) (\n. h4/ind__type/BOTTOM)) a0 a1) \/ (?a0 a1 a2 a3. a0_270 = (\a00 a10 a20 a30. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C a00 (h4/pair/_2C h4/bool/ARB a10)) (h4/ind__type/FCONS a20 (h4/ind__type/FCONS a30 (\n. h4/ind__type/BOTTOM)))) a0 a1 a2 a3 /\ _27ptree_27 a2 /\ _27ptree_27 a3) ==> _27ptree_27 a0_270) ==> _27ptree_27 a0_27) rep
% Assm: h4/bool/ABS__REP__THM: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. abs (rep a) = a) /\ (!r. P r <=> rep (abs r) = r))
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/BETA__THM: !y f. (\x. f x) y = f y
% Assm: h4/ind__type/CONSTR__REC: !Fn. ?f. !c i r. f (h4/ind__type/CONSTR c i r) = Fn c i r (\n. f (r n))
% Assm: h4/ind__type/FCONS0_c1: !n f a. h4/ind__type/FCONS a f (h4/num/SUC n) = f n
% Assm: h4/ind__type/FCONS0_c0: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm: h4/prim__rec/num__Axiom: !f e. ?fn. fn h4/num/0 = e /\ (!n. fn (h4/num/SUC n) = f n (fn n))
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/patricia/ptree__Axiom: !f2 f1 f0. ?fn. fn h4/patricia/Empty = f0 /\ (!a0 a1. fn (h4/patricia/Leaf a0 a1) = f1 a0 a1) /\ (!a0 a1 a2 a3. fn (h4/patricia/Branch a0 a1 a2 a3) = f2 a0 a1 a2 a3 (fn a2) (fn a3))
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> 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/numeral/numeral__distrib_c1: !n. h4/arithmetic/_2B n h4/num/0 = n
% 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/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/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% 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/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/patricia/ptree__case__def_c0: !v f1 f. h4/patricia/ptree__CASE h4/patricia/Empty v f f1 = v
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/arithmetic/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% 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__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/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/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/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% 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/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/words/word__bit__n2w: !n b. h4/words/word__bit b (h4/words/n2w n) <=> h4/arithmetic/_3C_3D b (h4/arithmetic/_2D (h4/fcp/dimindex h4/bool/the__value) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) /\ h4/bit/BIT b n
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 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/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/relation/RESTRICT__LEMMA: !z y f R. R y z ==> h4/relation/RESTRICT f R z y = f y
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/bit/EXISTS__BIT__IN__RANGE: !n b a. ~(n = h4/num/0) /\ h4/arithmetic/_3C_3D (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) a) n /\ h4/prim__rec/_3C n (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) b) ==> (?i. h4/arithmetic/_3C_3D a i /\ h4/prim__rec/_3C i b /\ h4/bit/BIT i n)
% Assm: h4/words/word__index__n2w: !n index_20too_20large i. h4/fcp/fcp__index (h4/words/n2w n) i <=> h4/bool/COND (h4/prim__rec/_3C i (h4/fcp/dimindex h4/bool/the__value)) (h4/bit/BIT i n) (h4/combin/FAIL h4/fcp/fcp__index index_20too_20large (h4/words/n2w n) i)
% Assm: h4/bit/LOWEST__SET__BIT__def: !n. h4/bit/LOWEST__SET__BIT n = h4/while/LEAST (\i. h4/bit/BIT i n)
% Assm: h4/bit/BIT__SLICE__THM3: !n b. ~h4/bit/BIT b n <=> h4/bit/SLICE b b n = h4/num/0
% Assm: h4/words/word__index: !n i. h4/prim__rec/_3C i (h4/fcp/dimindex h4/bool/the__value) ==> (h4/fcp/fcp__index (h4/words/n2w n) i <=> h4/bit/BIT i n)
% Assm: h4/bit/BIT__MODIFY__def_c1: !x n f. h4/bit/BIT__MODIFY (h4/num/SUC n) f x = h4/arithmetic/_2B (h4/bit/BIT__MODIFY n f x) (h4/bit/SBIT (f n (h4/bit/BIT n x)) n)
% Assm: h4/bit/EXISTS__BIT__LT: !n b. ~(n = h4/num/0) /\ h4/prim__rec/_3C n (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) b) ==> (?i. h4/prim__rec/_3C i b /\ h4/bit/BIT i n)
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/relation/WF__INDUCTION__THM: !R. h4/relation/WF R ==> (!P. (!x. (!y. R y x ==> P y) ==> P x) ==> (!x. P x))
% Assm: h4/arithmetic/SUC__ONE__ADD: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm: h4/arithmetic/LESS__EQ: !n m. h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/arithmetic/NOT__LEQ: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm: h4/arithmetic/SUB__LEFT__LESS__EQ: !p n m. h4/arithmetic/_3C_3D m (h4/arithmetic/_2D n p) <=> h4/arithmetic/_3C_3D (h4/arithmetic/_2B m p) n \/ h4/arithmetic/_3C_3D m h4/num/0
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bit/BIT__SLICE__THM: !x n. h4/bit/SBIT (h4/bit/BIT x n) x = h4/bit/SLICE x x n
% Assm: h4/bit/TWOEXP__NOT__ZERO: !n. ~(h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n = h4/num/0)
% Assm: h4/bit/SBIT__def: !n b. h4/bit/SBIT b n = h4/bool/COND b (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n) h4/num/0
% Assm: h4/words/word__bit0: !w b. h4/prim__rec/_3C b (h4/fcp/dimindex h4/bool/the__value) ==> (h4/fcp/fcp__index w b <=> h4/words/word__bit b w)
% Assm: h4/combin/FAIL__DEF: h4/combin/FAIL = (\x y. x)
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/arithmetic/NOT__NUM__EQ: !n m. ~(m = n) <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n \/ h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm: h4/numeral/TWO__EXP__THM_c0: h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) h4/num/0 = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm: h4/bool/AND__CONG: !Q_27 Q P_27 P. (Q ==> (P <=> P_27)) /\ (P_27 ==> (Q <=> Q_27)) ==> (P /\ Q <=> P_27 /\ Q_27)
% Assm: h4/words/DIMINDEX__GT__0: h4/prim__rec/_3C h4/num/0 (h4/fcp/dimindex h4/bool/the__value)
% Assm: h4/words/n2w__def: !n. h4/words/n2w n = h4/fcp/FCP (\i. h4/bit/BIT i n)
% Assm: h4/fcp/FCP__BETA: !i g. h4/prim__rec/_3C i (h4/fcp/dimindex h4/bool/the__value) ==> h4/fcp/fcp__index (h4/fcp/FCP g) i = g i
% Assm: h4/words/word__bit__def: !w b. h4/words/word__bit b w <=> h4/arithmetic/_3C_3D b (h4/arithmetic/_2D (h4/fcp/dimindex h4/bool/the__value) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) /\ h4/fcp/fcp__index w b
% Assm: h4/bool/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm: h4/bit/BIT__LOG2: !n. ~(n = h4/num/0) ==> h4/bit/BIT (h4/bit/LOG2 n) n
% Goal: !r p m l k. h4/patricia/PEEK (h4/patricia/Branch p m l r) k = h4/patricia/PEEK (h4/bool/COND (h4/bit/BIT m k) l r) k
%   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_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 (happ (happ (happ (happ h4/patricia/Branch p) m) l) r) k))) ==> h4/patricia/PEEK__tupled = h4/relation/WFREC (h4/min/_40 _0) _1))))))))))
% Assm [h4s_patricias_PEEKu_u_ind]: !P. (!k. happ (happ P h4/patricia/Empty) k) /\ (!j d k. happ (happ P (happ (happ h4/patricia/Leaf j) d)) k) /\ (!p m l r k. happ (happ P (h4/bool/COND (h4/bit/BIT m k) l r)) k ==> happ (happ P (happ (happ (happ (happ h4/patricia/Branch p) m) l) r)) k) ==> (!v v1. happ (happ P v) v1)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_patricias_Branch0]: h4/patricia/Branch = h4/patricia/_20_40ind__typepatricia2
% 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_hiddenu_u_u_u_20u_u_40indu_u_u_u_typepatricia2u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x' x x'. happ (happ (happ (happ h4/patricia/_20_40ind__typepatricia2 x) x') x) x' = h4/patricia/_20_40ind__typepatricia3 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C x (h4/pair/_2C h4/bool/ARB x')) (h4/ind__type/FCONS (h4/patricia/_20_40ind__typepatricia4 x) (h4/ind__type/FCONS (h4/patricia/_20_40ind__typepatricia4 x') _0))))
% Assm [h4s_patricias_ptreeu_u_repfnsu_c0]: !a. h4/patricia/_20_40ind__typepatricia3 (h4/patricia/_20_40ind__typepatricia4 a) = a
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_patricias_ptreeu_u_sizeu_u_defu_c2]: !f a3 a2 a1 a0. h4/patricia/ptree__size f (happ (happ (happ (happ 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_PEEKu_u_defu_c1]: !k j d. ?v. (v <=> k = j) /\ h4/patricia/PEEK (happ (happ h4/patricia/Leaf j) d) k = h4/bool/COND v (h4/option/SOME d) h4/option/NONE
% Assm [h4s_patricias_PEEKu_u_defu_c0]: !k. h4/patricia/PEEK h4/patricia/Empty k = h4/option/NONE
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_patricias_hiddenu_u_u_u_20u_u_40indu_u_u_u_typepatricia1u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x. happ (happ h4/patricia/_20_40ind__typepatricia1 x) x = h4/patricia/_20_40ind__typepatricia3 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C x (h4/pair/_2C x h4/bool/ARB)) _0))
% Assm [h4s_patricias_ptreeu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27ptree_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C h4/bool/ARB (h4/pair/_2C h4/bool/ARB h4/bool/ARB)) _0 \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C a0 (h4/pair/_2C a1 h4/bool/ARB)) _0) \/ (?a0 a1 a2 a3. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C a0 (h4/pair/_2C h4/bool/ARB a1)) (h4/ind__type/FCONS a2 (h4/ind__type/FCONS a3 _0)) /\ happ _27ptree_27 a2 /\ happ _27ptree_27 a3) ==> happ _27ptree_27 a0_270) ==> happ _27ptree_27 r) <=> h4/patricia/_20_40ind__typepatricia4 (h4/patricia/_20_40ind__typepatricia3 r) = r)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_patricias_Empty0]: h4/patricia/Empty = h4/patricia/_20_40ind__typepatricia0
% Assm [h4s_patricias_Leaf0]: h4/patricia/Leaf = h4/patricia/_20_40ind__typepatricia1
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_patricias_hiddenu_u_u_u_20u_u_40indu_u_u_u_typepatricia0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> h4/patricia/_20_40ind__typepatricia0 = h4/patricia/_20_40ind__typepatricia3 (h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C h4/bool/ARB (h4/pair/_2C h4/bool/ARB h4/bool/ARB)) _0)
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_patricias_ptreeu_u_caseu_u_cong]: !v_27 v f1_27 f1 f_27 f M_27 M. M = M_27 /\ (M_27 = h4/patricia/Empty ==> v = v_27) /\ (!a0 a1. M_27 = happ (happ h4/patricia/Leaf a0) a1 ==> happ (happ f a0) a1 = happ (happ f_27 a0) a1) /\ (!a0 a1 a2 a3. M_27 = happ (happ (happ (happ h4/patricia/Branch a0) a1) a2) a3 ==> happ (happ (happ (happ f1 a0) a1) a2) a3 = happ (happ (happ (happ f1_27 a0) a1) a2) a3) ==> h4/patricia/ptree__CASE M v f f1 = h4/patricia/ptree__CASE M_27 v_27 f_27 f1_27
% Assm [h4s_patricias_ptreeu_u_11u_c1]: !a3_27 a3 a2_27 a2 a1_27 a1 a0_27 a0. happ (happ (happ (happ h4/patricia/Branch a0) a1) a2) a3 = happ (happ (happ (happ h4/patricia/Branch a0_27) a1_27) a2_27) a3_27 <=> a0 = a0_27 /\ a1 = a1_27 /\ a2 = a2_27 /\ a3 = a3_27
% Assm [h4s_patricias_ptreeu_u_distinctu_c2]: !a3 a2 a1_27 a1 a0_27 a0. ~(happ (happ h4/patricia/Leaf a0) a1 = happ (happ (happ (happ h4/patricia/Branch a0_27) a1_27) a2) a3)
% Assm [h4s_patricias_ptreeu_u_distinctu_c1]: !a3 a2 a1 a0. ~(h4/patricia/Empty = happ (happ (happ (happ h4/patricia/Branch a0) a1) a2) a3)
% Assm [h4s_patricias_ptreeu_u_nchotomy]: !pp. pp = h4/patricia/Empty \/ (?n a. pp = happ (happ h4/patricia/Leaf n) a) \/ (?n0 n p p0. pp = happ (happ (happ (happ h4/patricia/Branch n0) n) p) p0)
% Assm [h4s_patricias_ptreeu_u_induction]: !P. happ P h4/patricia/Empty /\ (!n a. happ P (happ (happ h4/patricia/Leaf n) a)) /\ (!p p0. happ P p /\ happ P p0 ==> (!n n0. happ P (happ (happ (happ (happ h4/patricia/Branch n0) n) p) p0))) ==> (!p. happ P p)
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_DATATYPEu_u_TAGu_u_THM]: !x. h4/bool/DATATYPE x <=> T
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_patricias_ptreeu_u_caseu_u_defu_c2]: !v f1 f a3 a2 a1 a0. h4/patricia/ptree__CASE (happ (happ (happ (happ h4/patricia/Branch a0) a1) a2) a3) v f f1 = happ (happ (happ (happ f1 a0) a1) a2) a3
% Assm [h4s_patricias_datatypeu_u_ptree]: !ptree. h4/bool/DATATYPE (happ (happ (happ ptree h4/patricia/Empty) h4/patricia/Leaf) h4/patricia/Branch)
% Assm [h4s_patricias_ptreeu_u_TYu_u_DEF]: !_1. (!n. happ _1 n = h4/ind__type/BOTTOM) ==> (!_0. (!a0_27. happ _0 a0_27 <=> (!_27ptree_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C h4/bool/ARB (h4/pair/_2C h4/bool/ARB h4/bool/ARB)) _1 \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C a0 (h4/pair/_2C a1 h4/bool/ARB)) _1) \/ (?a0 a1 a2 a3. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C a0 (h4/pair/_2C h4/bool/ARB a1)) (h4/ind__type/FCONS a2 (h4/ind__type/FCONS a3 _1)) /\ happ _27ptree_27 a2 /\ happ _27ptree_27 a3) ==> happ _27ptree_27 a0_270) ==> happ _27ptree_27 a0_27)) ==> (?rep. h4/bool/TYPE__DEFINITION _0 rep))
% Assm [h4s_bools_ABSu_u_REPu_u_THM]: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. happ abs (happ rep a) = a) /\ (!r. happ P r <=> happ rep (happ abs r) = r))
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_BETAu_u_THM]: !y f. happ f y = happ f y
% Assm [h4s_indu_u_types_CONSTRu_u_REC]: !_0. (!f r n. happ (happ (happ _0 f) r) n = happ f (happ r n)) ==> (!Fn. ?f. !c i r. happ f (h4/ind__type/CONSTR c i r) = happ (happ (happ (happ Fn c) i) r) (happ (happ _0 f) r))
% Assm [h4s_indu_u_types_FCONS0u_c1]: !n f a. happ (h4/ind__type/FCONS a f) (h4/num/SUC n) = happ f n
% Assm [h4s_indu_u_types_FCONS0u_c0]: !f a. happ (h4/ind__type/FCONS a f) h4/num/0 = a
% Assm [h4s_primu_u_recs_numu_u_Axiom]: !f e. ?fn. happ fn h4/num/0 = e /\ (!n. happ fn (h4/num/SUC n) = happ (happ f n) (happ fn n))
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_patricias_ptreeu_u_Axiom]: !f2 f1 f0. ?fn. happ fn h4/patricia/Empty = f0 /\ (!a0 a1. happ fn (happ (happ h4/patricia/Leaf a0) a1) = happ (happ f1 a0) a1) /\ (!a0 a1 a2 a3. happ fn (happ (happ (happ (happ h4/patricia/Branch a0) a1) a2) a3) = happ (happ (happ (happ (happ (happ f2 a0) a1) a2) a3) (happ fn a2)) (happ fn a3))
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% 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_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_numerals_numeralu_u_distribu_c27]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_distribu_c1]: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm [h4s_arithmetics_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_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_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% 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_patricias_ptreeu_u_caseu_u_defu_c1]: !v f1 f a1 a0. h4/patricia/ptree__CASE (happ (happ h4/patricia/Leaf a0) a1) v f f1 = happ (happ f a0) a1
% 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_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_arithmetics_NOTu_u_LESS]: !n m. ~happ (happ h4/prim__rec/_3C m) n <=> h4/arithmetic/_3C_3D n m
% 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_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_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_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_PAIR]: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% 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_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_wordss_wordu_u_bitu_u_n2w]: !n b. h4/words/word__bit b (h4/words/n2w n) <=> h4/arithmetic/_3C_3D b (h4/arithmetic/_2D (h4/fcp/dimindex h4/bool/the__value) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) /\ h4/bit/BIT b n
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 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_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_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_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_bits_EXISTSu_u_BITu_u_INu_u_RANGE]: !n b a. ~(n = h4/num/0) /\ h4/arithmetic/_3C_3D (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) a) n /\ happ (happ h4/prim__rec/_3C n) (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) b) ==> (?i. h4/arithmetic/_3C_3D a i /\ happ (happ h4/prim__rec/_3C i) b /\ h4/bit/BIT i n)
% Assm [h4s_wordss_wordu_u_indexu_u_n2w]: !n index_20too_20large i. happ (happ h4/fcp/fcp__index (h4/words/n2w n)) i <=> h4/bool/COND (happ (happ h4/prim__rec/_3C i) (h4/fcp/dimindex h4/bool/the__value)) (h4/bit/BIT i n) (happ (happ (h4/combin/FAIL h4/fcp/fcp__index index_20too_20large) (h4/words/n2w n)) i)
% Assm [h4s_bits_LOWESTu_u_SETu_u_BITu_u_def]: !_0. (!n i. happ (happ _0 n) i <=> h4/bit/BIT i n) ==> (!n. h4/bit/LOWEST__SET__BIT n = h4/while/LEAST (happ _0 n))
% Assm [h4s_bits_BITu_u_SLICEu_u_THM3]: !n b. ~h4/bit/BIT b n <=> h4/bit/SLICE b b n = h4/num/0
% Assm [h4s_wordss_wordu_u_index]: !n i. happ (happ h4/prim__rec/_3C i) (h4/fcp/dimindex h4/bool/the__value) ==> (happ (happ h4/fcp/fcp__index (h4/words/n2w n)) i <=> h4/bit/BIT i n)
% Assm [h4s_bits_BITu_u_MODIFYu_u_defu_c1]: !x n f. h4/bit/BIT__MODIFY (h4/num/SUC n) f x = h4/arithmetic/_2B (h4/bit/BIT__MODIFY n f x) (h4/bit/SBIT (happ (happ f n) (h4/bit/BIT n x)) n)
% Assm [h4s_bits_EXISTSu_u_BITu_u_LT]: !n b. ~(n = h4/num/0) /\ happ (happ h4/prim__rec/_3C n) (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) b) ==> (?i. happ (happ h4/prim__rec/_3C i) b /\ h4/bit/BIT i n)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_relations_WFu_u_INDUCTIONu_u_THM]: !R. h4/relation/WF R ==> (!P. (!x. (!y. happ (happ R y) x ==> happ P y) ==> happ P x) ==> (!x. happ P x))
% Assm [h4s_arithmetics_SUCu_u_ONEu_u_ADD]: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm [h4s_arithmetics_LESSu_u_EQ]: !n m. happ (happ h4/prim__rec/_3C m) n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_arithmetics_NOTu_u_LEQ]: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm [h4s_arithmetics_SUBu_u_LEFTu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D m (h4/arithmetic/_2D n p) <=> h4/arithmetic/_3C_3D (h4/arithmetic/_2B m p) n \/ h4/arithmetic/_3C_3D m h4/num/0
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bits_BITu_u_SLICEu_u_THM]: !x n. h4/bit/SBIT (h4/bit/BIT x n) x = h4/bit/SLICE x x n
% Assm [h4s_bits_TWOEXPu_u_NOTu_u_ZERO]: !n. ~(h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n = h4/num/0)
% Assm [h4s_bits_SBITu_u_def]: !n b. h4/bit/SBIT b n = h4/bool/COND b (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n) h4/num/0
% Assm [h4s_wordss_wordu_u_bit0]: !w b. happ (happ h4/prim__rec/_3C b) (h4/fcp/dimindex h4/bool/the__value) ==> (happ (happ h4/fcp/fcp__index w) b <=> h4/words/word__bit b w)
% Assm [h4s_combins_FAILu_u_DEF]: !x x. h4/combin/FAIL x x = x
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_arithmetics_NOTu_u_NUMu_u_EQ]: !n m. ~(m = n) <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n \/ h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm [h4s_numerals_TWOu_u_EXPu_u_THMu_c0]: h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) h4/num/0 = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm [h4s_bools_ANDu_u_CONG]: !Q_27 Q P_27 P. (Q ==> (P <=> P_27)) /\ (P_27 ==> (Q <=> Q_27)) ==> (P /\ Q <=> P_27 /\ Q_27)
% Assm [h4s_wordss_DIMINDEXu_u_GTu_u_0]: happ (happ h4/prim__rec/_3C h4/num/0) (h4/fcp/dimindex h4/bool/the__value)
% Assm [h4s_wordss_n2wu_u_def]: !_0. (!n i. happ (happ _0 n) i <=> h4/bit/BIT i n) ==> (!n. h4/words/n2w n = h4/fcp/FCP (happ _0 n))
% Assm [h4s_fcps_FCPu_u_BETA]: !i g. happ (happ h4/prim__rec/_3C i) (h4/fcp/dimindex h4/bool/the__value) ==> happ (happ h4/fcp/fcp__index (h4/fcp/FCP g)) i = happ g i
% Assm [h4s_wordss_wordu_u_bitu_u_def]: !w b. h4/words/word__bit b w <=> h4/arithmetic/_3C_3D b (h4/arithmetic/_2D (h4/fcp/dimindex h4/bool/the__value) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) /\ happ (happ h4/fcp/fcp__index w) b
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm [h4s_bits_BITu_u_LOG2]: !n. ~(n = h4/num/0) ==> h4/bit/BIT (h4/bit/LOG2 n) n
% Goal: !r p m l k. h4/patricia/PEEK (happ (happ (happ (happ h4/patricia/Branch p) m) l) r) k = h4/patricia/PEEK (h4/bool/COND (h4/bit/BIT m k) l r) k
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_Q1379366,TV_Q1379362]: ![V_f, V_g]: (![V_x]: s(TV_Q1379362,happ(s(t_fun(TV_Q1379366,TV_Q1379362),V_f),s(TV_Q1379366,V_x))) = s(TV_Q1379362,happ(s(t_fun(TV_Q1379366,TV_Q1379362),V_g),s(TV_Q1379366,V_x))) => s(t_fun(TV_Q1379366,TV_Q1379362),V_f) = s(t_fun(TV_Q1379366,TV_Q1379362),V_g))).
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),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(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_patricias_ptree(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_patricias_ptree(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_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_PEEKu_u_ind, axiom, ![TV_u_27a]: ![V_P]: ((![V_k]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),V_P),s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_empty))),s(t_h4s_nums_num,V_k)))) & (![V_j, V_d, V_k]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),V_P),s(t_h4s_patricias_ptree(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a))),h4s_patricias_leaf),s(t_h4s_nums_num,V_j))),s(TV_u_27a,V_d))))),s(t_h4s_nums_num,V_k)))) & ![V_p, V_m, V_l, V_r, V_k]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),V_P),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)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),V_P),s(t_h4s_patricias_ptree(TV_u_27a),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(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_patricias_ptree(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_patricias_ptree(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_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))))))) => ![V_v, V_v1]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),V_P),s(t_h4s_patricias_ptree(TV_u_27a),V_v))),s(t_h4s_nums_num,V_v1)))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_patricias_Branch0, axiom, ![TV_u_27a]: 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_patricias_ptree(TV_u_27a))))),h4s_patricias_branch) = 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_patricias_ptree(TV_u_27a))))),h4s_patricias_u_20u_40indu_u_typepatricia2)).
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_hiddenu_u_u_u_20u_u_40indu_u_u_u_typepatricia2u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_indu_u_types_bottom) => ![V_x, V_xi_, V_x0, V_xi_0]: s(t_h4s_patricias_ptree(TV_u_27a),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(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_patricias_ptree(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_patricias_ptree(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_patricias_ptree(TV_u_27a))))),h4s_patricias_u_20u_40indu_u_typepatricia2),s(t_h4s_nums_num,V_x))),s(t_h4s_nums_num,V_xi_))),s(t_h4s_patricias_ptree(TV_u_27a),V_x0))),s(t_h4s_patricias_ptree(TV_u_27a),V_xi_0))) = s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_u_20u_40indu_u_typepatricia3(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,V_x),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num),h4s_pairs_u_2c(s(TV_u_27a,h4s_bools_arb),s(t_h4s_nums_num,V_xi_))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_patricias_u_20u_40indu_u_typepatricia4(s(t_h4s_patricias_ptree(TV_u_27a),V_x0))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_patricias_u_20u_40indu_u_typepatricia4(s(t_h4s_patricias_ptree(TV_u_27a),V_xi_0))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),V_uu_0))))))))))).
fof(ah4s_patricias_ptreeu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_u_20u_40indu_u_typepatricia3(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_patricias_u_20u_40indu_u_typepatricia4(s(t_h4s_patricias_ptree(TV_u_27a),V_a))))) = s(t_h4s_patricias_ptree(TV_u_27a),V_a)).
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_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),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(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_patricias_ptree(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_patricias_ptree(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_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_PEEKu_u_defu_c1, axiom, ![TV_u_27a]: ![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),h4s_patricias_peek(s(t_h4s_patricias_ptree(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a))),h4s_patricias_leaf),s(t_h4s_nums_num,V_j))),s(TV_u_27a,V_d))),s(t_h4s_nums_num,V_k))) = 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))))).
fof(ah4s_patricias_PEEKu_u_defu_c0, axiom, ![TV_u_27a]: ![V_k]: s(t_h4s_options_option(TV_u_27a),h4s_patricias_peek(s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_empty),s(t_h4s_nums_num,V_k))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_patricias_hiddenu_u_u_u_20u_u_40indu_u_u_u_typepatricia1u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_indu_u_types_bottom) => ![V_x, V_x0]: s(t_h4s_patricias_ptree(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a))),h4s_patricias_u_20u_40indu_u_typepatricia1),s(t_h4s_nums_num,V_x))),s(TV_u_27a,V_x0))) = s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_u_20u_40indu_u_typepatricia3(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,V_x),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num),h4s_pairs_u_2c(s(TV_u_27a,V_x0),s(t_h4s_nums_num,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),V_uu_0))))))).
fof(ah4s_patricias_ptreeu_u_repfnsu_c1, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27ptreeu_27]: (![V_a0u_270]: ((s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_bools_arb),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num),h4s_pairs_u_2c(s(TV_u_27a,h4s_bools_arb),s(t_h4s_nums_num,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),V_uu_0))) | (?[V_a0, V_a1]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,V_a0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num),h4s_pairs_u_2c(s(TV_u_27a,V_a1),s(t_h4s_nums_num,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),V_uu_0))) | ?[V_a0, V_a1, V_a2, V_a3]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,V_a0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num),h4s_pairs_u_2c(s(TV_u_27a,h4s_bools_arb),s(t_h4s_nums_num,V_a1))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_a2),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_a3),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),V_uu_0))))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),t_bool),V_uu_27ptreeu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_a2)))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),t_bool),V_uu_27ptreeu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_a3)))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),t_bool),V_uu_27ptreeu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),t_bool),V_uu_27ptreeu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_r))))) <=> s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_patricias_u_20u_40indu_u_typepatricia4(s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_u_20u_40indu_u_typepatricia3(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_r))))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_r)))).
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_bools_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_patricias_Empty0, axiom, ![TV_u_27a]: s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_empty) = s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_u_20u_40indu_u_typepatricia0)).
fof(ah4s_patricias_Leaf0, axiom, ![TV_u_27a]: s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a))),h4s_patricias_leaf) = s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a))),h4s_patricias_u_20u_40indu_u_typepatricia1)).
fof(ah4s_bools_MONOu_u_AND, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) & p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) & p(s(t_bool,V_w)))))).
fof(ah4s_patricias_hiddenu_u_u_u_20u_u_40indu_u_u_u_typepatricia0u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_indu_u_types_bottom) => s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_u_20u_40indu_u_typepatricia0) = s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_u_20u_40indu_u_typepatricia3(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_bools_arb),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num),h4s_pairs_u_2c(s(TV_u_27a,h4s_bools_arb),s(t_h4s_nums_num,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),V_uu_0))))))).
fof(ah4s_bools_MONOu_u_OR, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) | p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) | p(s(t_bool,V_w)))))).
fof(ah4s_patricias_ptreeu_u_caseu_u_cong, axiom, ![TV_u_27a,TV_u_27b]: ![V_vu_27, V_v, V_f1u_27, V_f1, V_fu_27, V_f, V_Mu_27, V_M]: ((s(t_h4s_patricias_ptree(TV_u_27a),V_M) = s(t_h4s_patricias_ptree(TV_u_27a),V_Mu_27) & ((s(t_h4s_patricias_ptree(TV_u_27a),V_Mu_27) = s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_empty) => s(TV_u_27b,V_v) = s(TV_u_27b,V_vu_27)) & (![V_a0, V_a1]: (s(t_h4s_patricias_ptree(TV_u_27a),V_Mu_27) = s(t_h4s_patricias_ptree(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,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,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))) = 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_fu_27),s(t_h4s_nums_num,V_a0))),s(TV_u_27a,V_a1)))) & ![V_a0, V_a1, V_a2, V_a3]: (s(t_h4s_patricias_ptree(TV_u_27a),V_Mu_27) = s(t_h4s_patricias_ptree(TV_u_27a),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(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_patricias_ptree(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_patricias_ptree(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_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,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))) = 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_f1u_27),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,h4s_patricias_ptreeu_u_case(s(t_h4s_patricias_ptree(TV_u_27a),V_M),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,h4s_patricias_ptreeu_u_case(s(t_h4s_patricias_ptree(TV_u_27a),V_Mu_27),s(TV_u_27b,V_vu_27),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27b)),V_fu_27),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_f1u_27))))).
fof(ah4s_patricias_ptreeu_u_11u_c1, axiom, ![TV_u_27a]: ![V_a3u_27, V_a3, V_a2u_27, V_a2, V_a1u_27, V_a1, V_a0u_27, V_a0]: (s(t_h4s_patricias_ptree(TV_u_27a),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(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_patricias_ptree(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_patricias_ptree(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_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_patricias_ptree(TV_u_27a),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(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_patricias_ptree(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_patricias_ptree(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_patricias_ptree(TV_u_27a))))),h4s_patricias_branch),s(t_h4s_nums_num,V_a0u_27))),s(t_h4s_nums_num,V_a1u_27))),s(t_h4s_patricias_ptree(TV_u_27a),V_a2u_27))),s(t_h4s_patricias_ptree(TV_u_27a),V_a3u_27))) <=> (s(t_h4s_nums_num,V_a0) = s(t_h4s_nums_num,V_a0u_27) & (s(t_h4s_nums_num,V_a1) = s(t_h4s_nums_num,V_a1u_27) & (s(t_h4s_patricias_ptree(TV_u_27a),V_a2) = s(t_h4s_patricias_ptree(TV_u_27a),V_a2u_27) & s(t_h4s_patricias_ptree(TV_u_27a),V_a3) = s(t_h4s_patricias_ptree(TV_u_27a),V_a3u_27)))))).
fof(ah4s_patricias_ptreeu_u_distinctu_c2, axiom, ![TV_u_27a]: ![V_a3, V_a2, V_a1u_27, V_a1, V_a0u_27, V_a0]: ~ (s(t_h4s_patricias_ptree(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a))),h4s_patricias_leaf),s(t_h4s_nums_num,V_a0))),s(TV_u_27a,V_a1))) = s(t_h4s_patricias_ptree(TV_u_27a),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(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_patricias_ptree(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_patricias_ptree(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_patricias_ptree(TV_u_27a))))),h4s_patricias_branch),s(t_h4s_nums_num,V_a0u_27))),s(t_h4s_nums_num,V_a1u_27))),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_distinctu_c1, axiom, ![TV_u_27a]: ![V_a3, V_a2, V_a1, V_a0]: ~ (s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_empty) = s(t_h4s_patricias_ptree(TV_u_27a),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(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_patricias_ptree(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_patricias_ptree(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_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))))).
fof(ah4s_patricias_ptreeu_u_nchotomy, axiom, ![TV_u_27a]: ![V_pp]: (s(t_h4s_patricias_ptree(TV_u_27a),V_pp) = s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_empty) | (?[V_n, V_a]: s(t_h4s_patricias_ptree(TV_u_27a),V_pp) = s(t_h4s_patricias_ptree(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a))),h4s_patricias_leaf),s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_a))) | ?[V_n0, V_n, V_p, V_p0]: s(t_h4s_patricias_ptree(TV_u_27a),V_pp) = s(t_h4s_patricias_ptree(TV_u_27a),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(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_patricias_ptree(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_patricias_ptree(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_patricias_ptree(TV_u_27a))))),h4s_patricias_branch),s(t_h4s_nums_num,V_n0))),s(t_h4s_nums_num,V_n))),s(t_h4s_patricias_ptree(TV_u_27a),V_p))),s(t_h4s_patricias_ptree(TV_u_27a),V_p0)))))).
fof(ah4s_patricias_ptreeu_u_induction, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool),V_P),s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_empty)))) & (![V_n, V_a]: p(s(t_bool,happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool),V_P),s(t_h4s_patricias_ptree(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a))),h4s_patricias_leaf),s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_a)))))) & ![V_p, V_p0]: ((p(s(t_bool,happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool),V_P),s(t_h4s_patricias_ptree(TV_u_27a),V_p)))) & p(s(t_bool,happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool),V_P),s(t_h4s_patricias_ptree(TV_u_27a),V_p0))))) => ![V_n, V_n0]: p(s(t_bool,happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool),V_P),s(t_h4s_patricias_ptree(TV_u_27a),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(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_patricias_ptree(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_patricias_ptree(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_patricias_ptree(TV_u_27a))))),h4s_patricias_branch),s(t_h4s_nums_num,V_n0))),s(t_h4s_nums_num,V_n))),s(t_h4s_patricias_ptree(TV_u_27a),V_p))),s(t_h4s_patricias_ptree(TV_u_27a),V_p0))))))))) => ![V_p]: p(s(t_bool,happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_bool),V_P),s(t_h4s_patricias_ptree(TV_u_27a),V_p)))))).
fof(ah4s_bools_JRHu_u_INDUCTu_u_UTIL, axiom, ![TV_u_27a]: ![V_t, V_P]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_t) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) => p(s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_P)))))).
fof(ah4s_bools_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_DATATYPEu_u_TAGu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,h4s_bools_datatype(s(TV_u_27a,V_x))) = s(t_bool,t)).
fof(ah4s_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_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_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),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(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_patricias_ptree(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_patricias_ptree(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_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_datatypeu_u_ptree, axiom, ![TV_u_27a]: ![V_ptree]: p(s(t_bool,h4s_bools_datatype(s(t_bool,happ(s(t_fun(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_patricias_ptree(TV_u_27a))))),t_bool),happ(s(t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a))),t_fun(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_patricias_ptree(TV_u_27a))))),t_bool)),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a))),t_fun(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_patricias_ptree(TV_u_27a))))),t_bool))),V_ptree),s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_empty))),s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a))),h4s_patricias_leaf))),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_patricias_ptree(TV_u_27a))))),h4s_patricias_branch))))))).
fof(ah4s_patricias_ptreeu_u_TYu_u_DEF, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),V_uu_1),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_indu_u_types_bottom) => ![V_uu_0]: (![V_a0u_27]: (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),t_bool),V_uu_0),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_a0u_27)))) <=> ![V_uu_27ptreeu_27]: (![V_a0u_270]: ((s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_bools_arb),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num),h4s_pairs_u_2c(s(TV_u_27a,h4s_bools_arb),s(t_h4s_nums_num,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),V_uu_1))) | (?[V_a0, V_a1]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,V_a0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num),h4s_pairs_u_2c(s(TV_u_27a,V_a1),s(t_h4s_nums_num,h4s_bools_arb))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),V_uu_1))) | ?[V_a0, V_a1, V_a2, V_a3]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)),h4s_pairs_u_2c(s(t_h4s_nums_num,V_a0),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num),h4s_pairs_u_2c(s(TV_u_27a,h4s_bools_arb),s(t_h4s_nums_num,V_a1))))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_a2),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_a3),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),V_uu_1))))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),t_bool),V_uu_27ptreeu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_a2)))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),t_bool),V_uu_27ptreeu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_a3)))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),t_bool),V_uu_27ptreeu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),t_bool),V_uu_27ptreeu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),V_a0u_27)))))) => ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num))),t_bool),V_uu_0),s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(TV_u_27a,t_h4s_nums_num)))),V_rep))))))).
fof(ah4s_bools_ABSu_u_REPu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ?[V_rep, V_abs]: (![V_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))) = s(TV_u_27b,V_a) & ![V_r]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_r)))) <=> s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))))) = s(TV_u_27a,V_r))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_BETAu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),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_indu_u_types_CONSTRu_u_REC, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_f, V_r, V_n]: s(TV_u_27b,happ(s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_h4s_nums_num,V_n))) = s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r),s(t_h4s_nums_num,V_n))))) => ![V_Fn]: ?[V_f]: ![V_c, V_i, V_r]: s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,V_c),s(TV_u_27a,V_i),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))) = s(TV_u_27b,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)))),V_Fn),s(t_h4s_nums_num,V_c))),s(TV_u_27a,V_i))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))))).
fof(ah4s_indu_u_types_FCONS0u_c1, axiom, ![TV_u_27a]: ![V_n, V_f, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n)))).
fof(ah4s_indu_u_types_FCONS0u_c0, axiom, ![TV_u_27a]: ![V_f, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_a)).
fof(ah4s_primu_u_recs_numu_u_Axiom, axiom, ![TV_u_27a]: ![V_f, V_e]: ?[V_fn]: (s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_e) & ![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_h4s_nums_num,V_n))),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_patricias_ptreeu_u_Axiom, axiom, ![TV_u_27b,TV_u_27a]: ![V_f2, V_f1, V_f0]: ?[V_fn]: (s(TV_u_27b,happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_patricias_ptree(TV_u_27a),h4s_patricias_empty))) = s(TV_u_27b,V_f0) & (![V_a0, V_a1]: s(TV_u_27b,happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_patricias_ptree(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,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,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_f1),s(t_h4s_nums_num,V_a0))),s(TV_u_27a,V_a1))) & ![V_a0, V_a1, V_a2, V_a3]: s(TV_u_27b,happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_patricias_ptree(TV_u_27a),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(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_patricias_ptree(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_patricias_ptree(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_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,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27b))),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(t_h4s_patricias_ptree(TV_u_27a),t_fun(TV_u_27b,t_fun(TV_u_27b,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),t_fun(TV_u_27b,t_fun(TV_u_27b,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),t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27b)))))),V_f2),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,happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_patricias_ptree(TV_u_27a),V_a2))))),s(TV_u_27b,happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_patricias_ptree(TV_u_27a),V_a3)))))))).
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_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
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_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_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_numerals_numeralu_u_lteu_c1, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
fof(ah4s_numerals_numeralu_u_distribu_c27, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ah4s_numerals_numeralu_u_distribu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_arithmetics_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_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_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
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_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_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_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),happ(s(t_fun(TV_u_27a,t_h4s_patricias_ptree(TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,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_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_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_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_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_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_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_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_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_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_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_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_wordss_wordu_u_bitu_u_n2w, axiom, ![TV_u_27a]: ![V_n, V_b]: (p(s(t_bool,h4s_wordss_wordu_u_bit(s(t_h4s_nums_num,V_b),s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_n2w(s(t_h4s_nums_num,V_n)))))) <=> (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_b),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_fcps_dimindex(s(t_h4s_bools_itself(TV_u_27a),h4s_bools_theu_u_value))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))) & p(s(t_bool,h4s_bits_bit(s(t_h4s_nums_num,V_b),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_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_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_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_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_bits_EXISTSu_u_BITu_u_INu_u_RANGE, axiom, ![V_n, V_b, V_a]: ((~ (s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) & (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_a))),s(t_h4s_nums_num,V_n)))) & 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_n))),s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_b)))))))) => ?[V_i]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_a),s(t_h4s_nums_num,V_i)))) & (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_i))),s(t_h4s_nums_num,V_b)))) & p(s(t_bool,h4s_bits_bit(s(t_h4s_nums_num,V_i),s(t_h4s_nums_num,V_n)))))))).
fof(ah4s_wordss_wordu_u_indexu_u_n2w, axiom, ![TV_u_27a]: ![V_n, V_indexu_20toou_20large, V_i]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),h4s_fcps_fcpu_u_index),s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_n2w(s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,V_i))) = s(t_bool,h4s_bools_cond(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_i))),s(t_h4s_nums_num,h4s_fcps_dimindex(s(t_h4s_bools_itself(TV_u_27a),h4s_bools_theu_u_value))))),s(t_bool,h4s_bits_bit(s(t_h4s_nums_num,V_i),s(t_h4s_nums_num,V_n))),s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),h4s_combins_fail(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),h4s_fcps_fcpu_u_index),s(t_bool,V_indexu_20toou_20large))),s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_n2w(s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,V_i)))))).
fof(ah4s_bits_LOWESTu_u_SETu_u_BITu_u_def, axiom, ![V_uu_0]: (![V_n, V_i]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_i))) = s(t_bool,h4s_bits_bit(s(t_h4s_nums_num,V_i),s(t_h4s_nums_num,V_n))) => ![V_n]: s(t_h4s_nums_num,h4s_bits_lowestu_u_setu_u_bit(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_whiles_least(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_bits_BITu_u_SLICEu_u_THM3, axiom, ![V_n, V_b]: (~ (p(s(t_bool,h4s_bits_bit(s(t_h4s_nums_num,V_b),s(t_h4s_nums_num,V_n))))) <=> s(t_h4s_nums_num,h4s_bits_slice(s(t_h4s_nums_num,V_b),s(t_h4s_nums_num,V_b),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0))).
fof(ah4s_wordss_wordu_u_index, axiom, ![TV_u_27a]: ![V_n, V_i]: (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_i))),s(t_h4s_nums_num,h4s_fcps_dimindex(s(t_h4s_bools_itself(TV_u_27a),h4s_bools_theu_u_value)))))) => s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),h4s_fcps_fcpu_u_index),s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_n2w(s(t_h4s_nums_num,V_n))))),s(t_h4s_nums_num,V_i))) = s(t_bool,h4s_bits_bit(s(t_h4s_nums_num,V_i),s(t_h4s_nums_num,V_n))))).
fof(ah4s_bits_BITu_u_MODIFYu_u_defu_c1, axiom, ![V_x, V_n, V_f]: s(t_h4s_nums_num,h4s_bits_bitu_u_modify(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_fun(t_h4s_nums_num,t_fun(t_bool,t_bool)),V_f),s(t_h4s_nums_num,V_x))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_bits_bitu_u_modify(s(t_h4s_nums_num,V_n),s(t_fun(t_h4s_nums_num,t_fun(t_bool,t_bool)),V_f),s(t_h4s_nums_num,V_x))),s(t_h4s_nums_num,h4s_bits_sbit(s(t_bool,happ(s(t_fun(t_bool,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_bool,t_bool)),V_f),s(t_h4s_nums_num,V_n))),s(t_bool,h4s_bits_bit(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_x))))),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_bits_EXISTSu_u_BITu_u_LT, axiom, ![V_n, V_b]: ((~ (s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) & 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_n))),s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_b))))))) => ?[V_i]: (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_i))),s(t_h4s_nums_num,V_b)))) & p(s(t_bool,h4s_bits_bit(s(t_h4s_nums_num,V_i),s(t_h4s_nums_num,V_n))))))).
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_pairs_ABSu_u_PAIRu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: ?[V_q, V_r]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_q),s(TV_u_27b,V_r)))).
fof(ah4s_relations_WFu_u_INDUCTIONu_u_THM, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_P]: (![V_x]: (![V_y]: (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_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) => ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_arithmetics_SUCu_u_ONEu_u_ADD, axiom, ![V_n]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_LESSu_u_EQ, axiom, ![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,V_m))),s(t_h4s_nums_num,V_n))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_IMPu_u_DISJu_u_THM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) => p(s(t_bool,V_B))) <=> (~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B))))).
fof(ah4s_bools_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_arithmetics_NOTu_u_LEQ, axiom, ![V_n, V_m]: (~ (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_SUBu_u_LEFTu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))))) <=> (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,V_n)))) | p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0))))))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_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_bits_BITu_u_SLICEu_u_THM, axiom, ![V_x, V_n]: s(t_h4s_nums_num,h4s_bits_sbit(s(t_bool,h4s_bits_bit(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_x))) = s(t_h4s_nums_num,h4s_bits_slice(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,V_n)))).
fof(ah4s_bits_TWOEXPu_u_NOTu_u_ZERO, axiom, ![V_n]: ~ (s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0))).
fof(ah4s_bits_SBITu_u_def, axiom, ![V_n, V_b]: s(t_h4s_nums_num,h4s_bits_sbit(s(t_bool,V_b),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,V_b),s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_wordss_wordu_u_bit0, axiom, ![TV_u_27a]: ![V_w, V_b]: (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_b))),s(t_h4s_nums_num,h4s_fcps_dimindex(s(t_h4s_bools_itself(TV_u_27a),h4s_bools_theu_u_value)))))) => s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),h4s_fcps_fcpu_u_index),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))),s(t_h4s_nums_num,V_b))) = s(t_bool,h4s_wordss_wordu_u_bit(s(t_h4s_nums_num,V_b),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))))).
fof(ah4s_combins_FAILu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_x0]: s(TV_u_27a,h4s_combins_fail(s(TV_u_27a,V_x),s(TV_u_27b,V_x0))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_bools_IMPu_u_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_arithmetics_NOTu_u_NUMu_u_EQ, axiom, ![V_n, V_m]: (~ (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n)) <=> (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))) | p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m))))))).
fof(ah4s_numerals_TWOu_u_EXPu_u_THMu_c0, axiom, s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))).
fof(ah4s_bools_ANDu_u_CONG, axiom, ![V_Qu_27, V_Q, V_Pu_27, V_P]: (((p(s(t_bool,V_Q)) => s(t_bool,V_P) = s(t_bool,V_Pu_27)) & (p(s(t_bool,V_Pu_27)) => s(t_bool,V_Q) = s(t_bool,V_Qu_27))) => ((p(s(t_bool,V_P)) & p(s(t_bool,V_Q))) <=> (p(s(t_bool,V_Pu_27)) & p(s(t_bool,V_Qu_27)))))).
fof(ah4s_wordss_DIMINDEXu_u_GTu_u_0, axiom, ![TV_u_27a]: 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,h4s_nums_0))),s(t_h4s_nums_num,h4s_fcps_dimindex(s(t_h4s_bools_itself(TV_u_27a),h4s_bools_theu_u_value))))))).
fof(ah4s_wordss_n2wu_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n, V_i]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_i))) = s(t_bool,h4s_bits_bit(s(t_h4s_nums_num,V_i),s(t_h4s_nums_num,V_n))) => ![V_n]: s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_wordss_n2w(s(t_h4s_nums_num,V_n))) = s(t_h4s_fcps_cart(t_bool,TV_u_27a),h4s_fcps_fcp(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),V_uu_0),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_fcps_FCPu_u_BETA, axiom, ![TV_u_27b,TV_u_27a]: ![V_i, V_g]: (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_i))),s(t_h4s_nums_num,h4s_fcps_dimindex(s(t_h4s_bools_itself(TV_u_27b),h4s_bools_theu_u_value)))))) => s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),happ(s(t_fun(t_h4s_fcps_cart(TV_u_27a,TV_u_27b),t_fun(t_h4s_nums_num,TV_u_27a)),h4s_fcps_fcpu_u_index),s(t_h4s_fcps_cart(TV_u_27a,TV_u_27b),h4s_fcps_fcp(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g))))),s(t_h4s_nums_num,V_i))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_g),s(t_h4s_nums_num,V_i))))).
fof(ah4s_wordss_wordu_u_bitu_u_def, axiom, ![TV_u_27a]: ![V_w, V_b]: (p(s(t_bool,h4s_wordss_wordu_u_bit(s(t_h4s_nums_num,V_b),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w)))) <=> (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_b),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_fcps_dimindex(s(t_h4s_bools_itself(TV_u_27a),h4s_bools_theu_u_value))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))) & p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_fcps_cart(t_bool,TV_u_27a),t_fun(t_h4s_nums_num,t_bool)),h4s_fcps_fcpu_u_index),s(t_h4s_fcps_cart(t_bool,TV_u_27a),V_w))),s(t_h4s_nums_num,V_b))))))).
fof(ah4s_bools_CONJu_u_ASSOC, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) & (p(s(t_bool,V_t2)) & p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) & p(s(t_bool,V_t3))))).
fof(ah4s_bits_BITu_u_LOG2, axiom, ![V_n]: (~ (s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) => p(s(t_bool,h4s_bits_bit(s(t_h4s_nums_num,h4s_bits_log2(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_n)))))).
fof(ch4s_patricias_PEEKu_u_defu_c2, conjecture, ![TV_u_27a]: ![V_r, V_p, V_m, V_l, V_k]: s(t_h4s_options_option(TV_u_27a),h4s_patricias_peek(s(t_h4s_patricias_ptree(TV_u_27a),happ(s(t_fun(t_h4s_patricias_ptree(TV_u_27a),t_h4s_patricias_ptree(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_patricias_ptree(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_patricias_ptree(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_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_h4s_options_option(TV_u_27a),h4s_patricias_peek(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)))).
