%   ORIGINAL: h4/patricia/PEEK__ind
% 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/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% 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/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> 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/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% 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/relation/inv__image__def: !f R. h4/relation/inv__image R f = (\x y. R (f x) (f y))
% Assm: h4/relation/WF__inv__image: !f R. h4/relation/WF R ==> h4/relation/WF (h4/relation/inv__image R f)
% Assm: h4/prim__rec/WF__LESS: h4/relation/WF h4/prim__rec/_3C
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% 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/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% 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/arithmetic/ADD__COMM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/ADD__ASSOC: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm: h4/arithmetic/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/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/LESS__EQ__TRANS: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/arithmetic/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/numeral/numeral__distrib_c1: !n. h4/arithmetic/_2B n h4/num/0 = n
% 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__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/patricia/ptree__repfns_c0: !a. h4/patricia/_20_40ind__typepatricia3 (h4/patricia/_20_40ind__typepatricia4 a) = a
% 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/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/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/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/Empty0: h4/patricia/Empty = h4/patricia/_20_40ind__typepatricia0
% Assm: h4/patricia/Leaf0: h4/patricia/Leaf = h4/patricia/_20_40ind__typepatricia1
% Assm: h4/patricia/Branch0: h4/patricia/Branch = h4/patricia/_20_40ind__typepatricia2
% 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))))
% Goal: !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)
%   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_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% 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_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% 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_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> 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_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% 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_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_relations_WFu_u_invu_u_image]: !f R. h4/relation/WF R ==> h4/relation/WF (h4/relation/inv__image R f)
% Assm [h4s_primu_u_recs_WFu_u_LESS]: h4/relation/WF h4/prim__rec/_3C
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_pairs_PAIR]: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% 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_pairs_UNCURRYu_u_DEF]: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = happ (happ f x) y
% 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_arithmetics_ADDu_u_COMM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_ADDu_u_ASSOC]: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm [h4s_arithmetics_NOTu_u_LESS]: !n m. ~happ (happ h4/prim__rec/_3C m) n <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_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_LESSu_u_EQu_u_TRANS]: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm [h4s_arithmetics_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_arithmetics_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_numerals_numeralu_u_distribu_c1]: !n. h4/arithmetic/_2B n h4/num/0 = n
% 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_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_patricias_ptreeu_u_repfnsu_c0]: !a. h4/patricia/_20_40ind__typepatricia3 (h4/patricia/_20_40ind__typepatricia4 a) = a
% 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_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_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_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_Empty0]: h4/patricia/Empty = h4/patricia/_20_40ind__typepatricia0
% Assm [h4s_patricias_Leaf0]: h4/patricia/Leaf = h4/patricia/_20_40ind__typepatricia1
% Assm [h4s_patricias_Branch0]: h4/patricia/Branch = h4/patricia/_20_40ind__typepatricia2
% 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))))
% Goal: !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)
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_Q285301,TV_Q285297]: ![V_f, V_g]: (![V_x]: s(TV_Q285297,happ(s(t_fun(TV_Q285301,TV_Q285297),V_f),s(TV_Q285301,V_x))) = s(TV_Q285297,happ(s(t_fun(TV_Q285301,TV_Q285297),V_g),s(TV_Q285301,V_x))) => s(t_fun(TV_Q285301,TV_Q285297),V_f) = s(t_fun(TV_Q285301,TV_Q285297),V_g))).
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_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_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_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_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_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_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_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_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_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_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_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_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_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
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_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_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_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_dcu_u_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_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_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_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_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_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_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_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_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_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_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_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_NOTu_u_LESS, axiom, ![V_n, V_m]: (~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_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_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_LESSu_u_EQu_u_TRANS, axiom, ![V_p, V_n, V_m]: ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_ZEROu_u_LESSu_u_EQ, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_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_numerals_numeralu_u_distribu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_numerals_numeralu_u_distribu_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_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_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_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_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_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_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_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_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_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(ch4s_patricias_PEEKu_u_ind, conjecture, ![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)))))).
