%   ORIGINAL: h4/string/STRCAT__EQNS_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/string/STRCAT__def_c1: !l2 l1 h. h4/list/APPEND (h4/list/CONS h l1) l2 = h4/list/CONS h (h4/list/APPEND l1 l2)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/string/STRCAT: !s2 s1. h4/list/APPEND s1 s2 = h4/list/APPEND s1 s2
% Assm: h4/list/APPEND__11_c0: !l3 l2 l1. h4/list/APPEND l1 l2 = h4/list/APPEND l1 l3 <=> l2 = l3
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/list/NIL0: h4/list/NIL = h4/list/_20_40ind__typelist0
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/list/list__repfns_c1: !r. (\a0_27. !_27list_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM))) a0 a1 /\ _27list_27 a1) ==> _27list_27 a0_270) ==> _27list_27 a0_27) r <=> h4/list/_20_40ind__typelist3 (h4/list/_20_40ind__typelist2 r) = r
% Assm: h4/bool/BETA__THM: !y f. (\x. f x) y = f y
% 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/CONSTR__REC: !Fn. ?f. !c i r. f (h4/ind__type/CONSTR c i r) = Fn c i r (\n. f (r n))
% Assm: h4/ind__type/FCONS0_c0: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/list/hidden____20__40ind____typelist1____def: h4/list/_20_40ind__typelist1 = (\a0 a1. h4/list/_20_40ind__typelist2 ((\a00 a10. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM))) a0 (h4/list/_20_40ind__typelist3 a1)))
% Assm: h4/list/CONS0: h4/list/CONS = h4/list/_20_40ind__typelist1
% Assm: h4/list/hidden____20__40ind____typelist0____def: h4/list/_20_40ind__typelist0 = h4/list/_20_40ind__typelist2 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM))
% Assm: h4/list/list__repfns_c0: !a. h4/list/_20_40ind__typelist2 (h4/list/_20_40ind__typelist3 a) = a
% Assm: h4/string/EXPLODE__EQ__THM_c1: !t s h. h4/string/EXPLODE s = h4/list/CONS h t <=> s = h4/list/CONS h (h4/string/IMPLODE t)
% Assm: h4/string/STRING__ACYCLIC_c0: !s c. ~(h4/list/CONS c s = s)
% Assm: h4/string/DEST__STRING__LEMS_c1: !t s c. h4/string/DEST__STRING s = h4/option/SOME (h4/pair/_2C c t) <=> s = h4/list/CONS c t
% Assm: h4/string/IMPLODE__EQ__THM_c0: !s l c. h4/list/CONS c s = h4/string/IMPLODE l <=> l = h4/list/CONS c (h4/string/EXPLODE s)
% Assm: h4/string/FIELDS__ind: !P_27. (!P. P_27 P h4/list/NIL) /\ (!P h t. (!l r. h4/pair/_2C l r = h4/rich__list/SPLITP P (h4/list/CONS h t) /\ h4/list/NULL l ==> P_27 P (h4/list/TL r)) /\ (!l r. h4/pair/_2C l r = h4/rich__list/SPLITP P (h4/list/CONS h t) /\ ~h4/list/NULL l /\ ~h4/list/NULL r ==> P_27 P (h4/list/TL r)) ==> P_27 P (h4/list/CONS h t)) ==> (!v v1. P_27 v v1)
% Assm: h4/string/TOKENS__def_c1: !t h P. h4/string/TOKENS P (h4/list/CONS h t) = h4/bool/LET (h4/pair/UNCURRY (\l r. h4/bool/COND (h4/list/NULL l) (h4/string/TOKENS P (h4/list/TL r)) (h4/list/CONS l (h4/string/TOKENS P r)))) (h4/rich__list/SPLITP P (h4/list/CONS h t))
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/list/NOT__NIL__CONS: !a1 a0. ~(h4/list/NIL = h4/list/CONS a0 a1)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/list/CONS__11: !a1_27 a1 a0_27 a0. h4/list/CONS a0 a1 = h4/list/CONS a0_27 a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm: h4/list/list__nchotomy: !l. l = h4/list/NIL \/ (?h t. l = h4/list/CONS h t)
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/list/list__induction: !P. P h4/list/NIL /\ (!t. P t ==> (!h. P (h4/list/CONS h t))) ==> (!l. P l)
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/string/IMPLODE__EXPLODE: !s. h4/string/IMPLODE (h4/string/EXPLODE s) = s
% Assm: h4/string/EXPLODE__IMPLODE: !cs. h4/string/EXPLODE (h4/string/IMPLODE cs) = cs
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/pair/CLOSED__PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/option/NOT__NONE__SOME: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm: h4/string/DEST__STRING__def_c1: !rst c. h4/string/DEST__STRING (h4/list/CONS c rst) = h4/option/SOME (h4/pair/_2C c rst)
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/string/DEST__STRING__def_c0: h4/string/DEST__STRING h4/list/NIL = h4/option/NONE
% Assm: h4/string/EXPLODE__def_c1: !s c. h4/string/EXPLODE (h4/list/CONS c s) = h4/list/CONS c (h4/string/EXPLODE s)
% Assm: h4/string/EXPLODE__def_c0: h4/string/EXPLODE h4/list/NIL = h4/list/NIL
% Assm: h4/string/IMPLODE__def_c1: !cs c. h4/string/IMPLODE (h4/list/CONS c cs) = h4/list/CONS c (h4/string/IMPLODE cs)
% Assm: h4/string/IMPLODE__def_c0: h4/string/IMPLODE h4/list/NIL = h4/list/NIL
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/rich__list/SPLITP0_c0: !P. h4/rich__list/SPLITP P h4/list/NIL = h4/pair/_2C h4/list/NIL h4/list/NIL
% Assm: h4/rich__list/SPLITP0_c1: !x l P. h4/rich__list/SPLITP P (h4/list/CONS x l) = h4/bool/COND (P x) (h4/pair/_2C h4/list/NIL (h4/list/CONS x l)) (h4/pair/_2C (h4/list/CONS x (h4/pair/FST (h4/rich__list/SPLITP P l))) (h4/pair/SND (h4/rich__list/SPLITP P l)))
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/list/NULL__EQ: !l. h4/list/NULL l <=> l = h4/list/NIL
% Assm: h4/list/list__case__def_c1: !v f a1 a0. h4/list/list__CASE (h4/list/CONS a0 a1) v f = f a0 a1
% Assm: h4/list/list__case__def_c0: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/list/LENGTH0_c0: h4/list/LENGTH h4/list/NIL = h4/num/0
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/numeral/numeral__add_c2: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT2 (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm: h4/numeral/numeral__add_c0: !n. h4/numeral/iZ (h4/arithmetic/_2B h4/arithmetic/ZERO n) = n
% Assm: h4/numeral/numeral__distrib_c2: !n m. h4/arithmetic/_2B (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm: h4/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/list/LENGTH0_c1: !t h. h4/list/LENGTH (h4/list/CONS h t) = h4/num/SUC (h4/list/LENGTH t)
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/numeral/numeral__distrib_c1: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm: h4/arithmetic/LESS__EQ__MONO: !n m. h4/arithmetic/_3C_3D (h4/num/SUC n) (h4/num/SUC m) <=> h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/LESS__EQ__REFL: !m. h4/arithmetic/_3C_3D m m
% Assm: h4/list/TL0: !t h. h4/list/TL (h4/list/CONS h t) = t
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/numeral/numeral__lte_c2: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT2 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/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% 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/SUC__ONE__ADD: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% 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/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/prim__rec/WF__measure: !m. h4/relation/WF (h4/prim__rec/measure m)
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% Assm: h4/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm: h4/prim__rec/measure__def: h4/prim__rec/measure = h4/relation/inv__image h4/prim__rec/_3C
% 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__INDUCTION__THM: !R. h4/relation/WF R ==> (!P. (!x. (!y. R y x ==> P y) ==> P x) ==> (!x. P 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/o__DEF: !g f. h4/combin/o f g = (\x. f (g x))
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% 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/combin/I__THM: !x. h4/combin/I x = x
% 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/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% 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/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% 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__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/bool/LET__CONG: !g f N M. M = N /\ (!x. x = N ==> f x = g x) ==> h4/bool/LET f M = h4/bool/LET g N
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Assm: h4/string/FIELDS__curried__def: !x1 x. h4/string/FIELDS x x1 = h4/string/FIELDS__tupled (h4/pair/_2C x x1)
% Assm: h4/string/FIELDS__tupled__primitive__def: h4/string/FIELDS__tupled = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!t h P l r. h4/pair/_2C l r = h4/rich__list/SPLITP P (h4/list/CONS h t) /\ h4/list/NULL l ==> R (h4/pair/_2C P (h4/list/TL r)) (h4/pair/_2C P (h4/list/CONS h t))) /\ (!t h P l r. h4/pair/_2C l r = h4/rich__list/SPLITP P (h4/list/CONS h t) /\ ~h4/list/NULL l /\ ~h4/list/NULL r ==> R (h4/pair/_2C P (h4/list/TL r)) (h4/pair/_2C P (h4/list/CONS h t))))) (\FIELDS__tupled a. h4/pair/pair__CASE a (\P v1. h4/list/list__CASE v1 (h4/combin/I (h4/list/CONS h4/list/NIL h4/list/NIL)) (\h t. h4/combin/I (h4/bool/LET (h4/pair/UNCURRY (\l r. h4/bool/COND (h4/list/NULL l) (h4/list/CONS h4/list/NIL (FIELDS__tupled (h4/pair/_2C P (h4/list/TL r)))) (h4/bool/COND (h4/list/NULL r) (h4/list/CONS l h4/list/NIL) (h4/list/CONS l (FIELDS__tupled (h4/pair/_2C P (h4/list/TL r))))))) (h4/rich__list/SPLITP P (h4/list/CONS h t))))))
% Assm: h4/arithmetic/LESS__TRANS: !p n m. h4/prim__rec/_3C m n /\ h4/prim__rec/_3C n p ==> h4/prim__rec/_3C m p
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/string/TOKENS__curried__def: !x1 x. h4/string/TOKENS x x1 = h4/string/TOKENS__tupled (h4/pair/_2C x x1)
% Goal: !s2 s1 c. h4/list/APPEND (h4/list/CONS c s1) s2 = h4/list/CONS c (h4/list/APPEND s1 s2)
%   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_strings_STRCATu_u_defu_c1]: !l2 l1 h. h4/list/APPEND (happ (happ h4/list/CONS h) l1) l2 = happ (happ h4/list/CONS h) (h4/list/APPEND l1 l2)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_strings_STRCAT]: !s2 s1. h4/list/APPEND s1 s2 = h4/list/APPEND s1 s2
% Assm [h4s_lists_APPENDu_u_11u_c0]: !l3 l2 l1. h4/list/APPEND l1 l2 = h4/list/APPEND l1 l3 <=> l2 = l3
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% 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_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_lists_NIL0]: h4/list/NIL = h4/list/_20_40ind__typelist0
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_lists_listu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27list_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0 \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a0 (h4/ind__type/FCONS a1 _0) /\ happ _27list_27 a1) ==> happ _27list_27 a0_270) ==> happ _27list_27 r) <=> h4/list/_20_40ind__typelist3 (h4/list/_20_40ind__typelist2 r) = r)
% Assm [h4s_bools_BETAu_u_THM]: !y f. happ f y = happ f y
% 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_CONSTRu_u_REC]: !_0. (!f r n. happ (happ (happ _0 f) r) n = happ f (happ r n)) ==> (!Fn. ?f. !c i r. happ f (h4/ind__type/CONSTR c i r) = happ (happ (happ (happ Fn c) i) r) (happ (happ _0 f) r))
% Assm [h4s_indu_u_types_FCONS0u_c0]: !f a. happ (h4/ind__type/FCONS a f) h4/num/0 = a
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist1u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x. happ (happ h4/list/_20_40ind__typelist1 x) x = h4/list/_20_40ind__typelist2 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) x (h4/ind__type/FCONS (h4/list/_20_40ind__typelist3 x) _0)))
% Assm [h4s_lists_CONS0]: h4/list/CONS = h4/list/_20_40ind__typelist1
% Assm [h4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> h4/list/_20_40ind__typelist0 = h4/list/_20_40ind__typelist2 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0)
% Assm [h4s_lists_listu_u_repfnsu_c0]: !a. h4/list/_20_40ind__typelist2 (h4/list/_20_40ind__typelist3 a) = a
% Assm [h4s_strings_EXPLODEu_u_EQu_u_THMu_c1]: !t s h. h4/string/EXPLODE s = happ (happ h4/list/CONS h) t <=> s = happ (happ h4/list/CONS h) (h4/string/IMPLODE t)
% Assm [h4s_strings_STRINGu_u_ACYCLICu_c0]: !s c. ~(happ (happ h4/list/CONS c) s = s)
% Assm [h4s_strings_DESTu_u_STRINGu_u_LEMSu_c1]: !t s c. h4/string/DEST__STRING s = h4/option/SOME (h4/pair/_2C c t) <=> s = happ (happ h4/list/CONS c) t
% Assm [h4s_strings_IMPLODEu_u_EQu_u_THMu_c0]: !s l c. happ (happ h4/list/CONS c) s = h4/string/IMPLODE l <=> l = happ (happ h4/list/CONS c) (h4/string/EXPLODE s)
% Assm [h4s_strings_FIELDSu_u_ind]: !P_27. (!P. happ (happ P_27 P) h4/list/NIL) /\ (!P h t. (!l r. h4/pair/_2C l r = h4/rich__list/SPLITP P (happ (happ h4/list/CONS h) t) /\ h4/list/NULL l ==> happ (happ P_27 P) (h4/list/TL r)) /\ (!l r. h4/pair/_2C l r = h4/rich__list/SPLITP P (happ (happ h4/list/CONS h) t) /\ ~h4/list/NULL l /\ ~h4/list/NULL r ==> happ (happ P_27 P) (h4/list/TL r)) ==> happ (happ P_27 P) (happ (happ h4/list/CONS h) t)) ==> (!v v1. happ (happ P_27 v) v1)
% Assm [h4s_strings_TOKENSu_u_defu_c1]: !_1. (!l P r. happ (happ (happ _1 l) P) r = h4/bool/COND (h4/list/NULL l) (h4/string/TOKENS P (h4/list/TL r)) (happ (happ h4/list/CONS l) (h4/string/TOKENS P r))) ==> (!_0. (!P l. happ (happ _0 P) l = happ (happ _1 l) P) ==> (!t h P. h4/string/TOKENS P (happ (happ h4/list/CONS h) t) = h4/bool/LET (h4/pair/UNCURRY (happ _0 P)) (h4/rich__list/SPLITP P (happ (happ h4/list/CONS h) t))))
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_lists_NOTu_u_NILu_u_CONS]: !a1 a0. ~(h4/list/NIL = happ (happ h4/list/CONS a0) a1)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_lists_CONSu_u_11]: !a1_27 a1 a0_27 a0. happ (happ h4/list/CONS a0) a1 = happ (happ h4/list/CONS a0_27) a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm [h4s_lists_listu_u_nchotomy]: !l. l = h4/list/NIL \/ (?h t. l = happ (happ h4/list/CONS h) t)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_lists_listu_u_induction]: !P. happ P h4/list/NIL /\ (!t. happ P t ==> (!h. happ P (happ (happ h4/list/CONS h) t))) ==> (!l. happ P l)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_strings_IMPLODEu_u_EXPLODE]: !s. h4/string/IMPLODE (h4/string/EXPLODE s) = s
% Assm [h4s_strings_EXPLODEu_u_IMPLODE]: !cs. h4/string/EXPLODE (h4/string/IMPLODE cs) = cs
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_pairs_CLOSEDu_u_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_options_NOTu_u_NONEu_u_SOME]: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm [h4s_strings_DESTu_u_STRINGu_u_defu_c1]: !rst c. h4/string/DEST__STRING (happ (happ h4/list/CONS c) rst) = h4/option/SOME (h4/pair/_2C c rst)
% Assm [h4s_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_strings_DESTu_u_STRINGu_u_defu_c0]: h4/string/DEST__STRING h4/list/NIL = h4/option/NONE
% Assm [h4s_strings_EXPLODEu_u_defu_c1]: !s c. h4/string/EXPLODE (happ (happ h4/list/CONS c) s) = happ (happ h4/list/CONS c) (h4/string/EXPLODE s)
% Assm [h4s_strings_EXPLODEu_u_defu_c0]: h4/string/EXPLODE h4/list/NIL = h4/list/NIL
% Assm [h4s_strings_IMPLODEu_u_defu_c1]: !cs c. h4/string/IMPLODE (happ (happ h4/list/CONS c) cs) = happ (happ h4/list/CONS c) (h4/string/IMPLODE cs)
% Assm [h4s_strings_IMPLODEu_u_defu_c0]: h4/string/IMPLODE h4/list/NIL = h4/list/NIL
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_richu_u_lists_SPLITP0u_c0]: !P. h4/rich__list/SPLITP P h4/list/NIL = h4/pair/_2C h4/list/NIL h4/list/NIL
% Assm [h4s_richu_u_lists_SPLITP0u_c1]: !x l P. h4/rich__list/SPLITP P (happ (happ h4/list/CONS x) l) = h4/bool/COND (happ P x) (h4/pair/_2C h4/list/NIL (happ (happ h4/list/CONS x) l)) (h4/pair/_2C (happ (happ h4/list/CONS x) (h4/pair/FST (h4/rich__list/SPLITP P l))) (h4/pair/SND (h4/rich__list/SPLITP P l)))
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_lists_NULLu_u_EQ]: !l. h4/list/NULL l <=> l = h4/list/NIL
% Assm [h4s_lists_listu_u_caseu_u_defu_c1]: !v f a1 a0. h4/list/list__CASE (happ (happ h4/list/CONS a0) a1) v f = happ (happ f a0) a1
% Assm [h4s_lists_listu_u_caseu_u_defu_c0]: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_lists_LENGTH0u_c0]: h4/list/LENGTH h4/list/NIL = h4/num/0
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_numerals_numeralu_u_addu_c2]: !n m. h4/numeral/iZ (h4/arithmetic/_2B (h4/arithmetic/BIT1 n) (h4/arithmetic/BIT1 m)) = h4/arithmetic/BIT2 (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm [h4s_numerals_numeralu_u_addu_c0]: !n. h4/numeral/iZ (h4/arithmetic/_2B h4/arithmetic/ZERO n) = n
% Assm [h4s_numerals_numeralu_u_distribu_c2]: !n m. h4/arithmetic/_2B (h4/arithmetic/NUMERAL n) (h4/arithmetic/NUMERAL m) = h4/arithmetic/NUMERAL (h4/numeral/iZ (h4/arithmetic/_2B n m))
% Assm [h4s_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_lists_LENGTH0u_c1]: !t h. h4/list/LENGTH (happ (happ h4/list/CONS h) t) = h4/num/SUC (h4/list/LENGTH t)
% Assm [h4s_arithmetics_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_numerals_numeralu_u_distribu_c1]: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm [h4s_arithmetics_LESSu_u_EQu_u_MONO]: !n m. h4/arithmetic/_3C_3D (h4/num/SUC n) (h4/num/SUC m) <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_LESSu_u_EQu_u_REFL]: !m. h4/arithmetic/_3C_3D m m
% Assm [h4s_lists_TL0]: !t h. h4/list/TL (happ (happ h4/list/CONS h) t) = t
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_numerals_numeralu_u_lteu_c2]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT2 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_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. happ (h4/pair/UNCURRY f) (h4/pair/_2C x y) = happ (happ f x) y
% 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_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_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% 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_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_primu_u_recs_WFu_u_measure]: !m. h4/relation/WF (happ h4/prim__rec/measure m)
% 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_pairs_PAIR]: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm [h4s_primu_u_recs_measureu_u_def]: h4/prim__rec/measure = h4/relation/inv__image h4/prim__rec/_3C
% Assm [h4s_relations_invu_u_imageu_u_def]: !f R x x'. happ (happ (happ (h4/relation/inv__image R) f) x) x' <=> happ (happ R (happ f x)) (happ f x')
% 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_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_ou_u_DEF]: !g f x. h4/combin/o f g x = happ f (happ g x)
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% 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_combins_Iu_u_THM]: !x. h4/combin/I x = x
% 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_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% 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_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% 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_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_bools_LETu_u_CONG]: !g f N M. M = N /\ (!x. x = N ==> happ f x = happ g x) ==> h4/bool/LET f M = h4/bool/LET g N
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% Assm [h4s_strings_FIELDSu_u_curriedu_u_def]: !x1 x. h4/string/FIELDS x x1 = happ h4/string/FIELDS__tupled (h4/pair/_2C x x1)
% Assm [h4s_strings_FIELDSu_u_tupledu_u_primitiveu_u_def]: !_8. (!l FIELDS__tupled P r. happ (happ (happ (happ _8 l) FIELDS__tupled) P) r = h4/bool/COND (h4/list/NULL l) (happ (happ h4/list/CONS h4/list/NIL) (happ FIELDS__tupled (h4/pair/_2C P (h4/list/TL r)))) (h4/bool/COND (h4/list/NULL r) (happ (happ h4/list/CONS l) h4/list/NIL) (happ (happ h4/list/CONS l) (happ FIELDS__tupled (h4/pair/_2C P (h4/list/TL r)))))) ==> (!_7. (!FIELDS__tupled P l. happ (happ (happ _7 FIELDS__tupled) P) l = happ (happ (happ _8 l) FIELDS__tupled) P) ==> (!_6. (!FIELDS__tupled P h t. happ (happ (happ (happ _6 FIELDS__tupled) P) h) t = h4/combin/I (h4/bool/LET (h4/pair/UNCURRY (happ (happ _7 FIELDS__tupled) P)) (h4/rich__list/SPLITP P (happ (happ h4/list/CONS h) t)))) ==> (!_5. (!FIELDS__tupled P h. happ (happ (happ _5 FIELDS__tupled) P) h = happ (happ (happ _6 FIELDS__tupled) P) h) ==> (!_4. (!FIELDS__tupled P v1. happ (happ (happ _4 FIELDS__tupled) P) v1 = h4/list/list__CASE v1 (h4/combin/I (happ (happ h4/list/CONS h4/list/NIL) h4/list/NIL)) (happ (happ _5 FIELDS__tupled) P)) ==> (!_3. (!FIELDS__tupled P. happ (happ _3 FIELDS__tupled) P = happ (happ _4 FIELDS__tupled) P) ==> (!_2. (!FIELDS__tupled a. happ (happ _2 FIELDS__tupled) a = h4/pair/pair__CASE a (happ _3 FIELDS__tupled)) ==> (!_1. (!FIELDS__tupled. happ _1 FIELDS__tupled = happ _2 FIELDS__tupled) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R /\ (!t h P l r. h4/pair/_2C l r = h4/rich__list/SPLITP P (happ (happ h4/list/CONS h) t) /\ h4/list/NULL l ==> happ (happ R (h4/pair/_2C P (h4/list/TL r))) (h4/pair/_2C P (happ (happ h4/list/CONS h) t))) /\ (!t h P l r. h4/pair/_2C l r = h4/rich__list/SPLITP P (happ (happ h4/list/CONS h) t) /\ ~h4/list/NULL l /\ ~h4/list/NULL r ==> happ (happ R (h4/pair/_2C P (h4/list/TL r))) (h4/pair/_2C P (happ (happ h4/list/CONS h) t)))) ==> h4/string/FIELDS__tupled = h4/relation/WFREC (h4/min/_40 _0) _1))))))))
% Assm [h4s_arithmetics_LESSu_u_TRANS]: !p n m. happ (happ h4/prim__rec/_3C m) n /\ happ (happ h4/prim__rec/_3C n) p ==> happ (happ h4/prim__rec/_3C m) p
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_strings_TOKENSu_u_curriedu_u_def]: !x1 x. h4/string/TOKENS x x1 = h4/string/TOKENS__tupled (h4/pair/_2C x x1)
% Goal: !s2 s1 c. h4/list/APPEND (happ (happ h4/list/CONS c) s1) s2 = happ (happ h4/list/CONS c) (h4/list/APPEND s1 s2)
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_Q1303957,TV_Q1303953]: ![V_f, V_g]: (![V_x]: s(TV_Q1303953,happ(s(t_fun(TV_Q1303957,TV_Q1303953),V_f),s(TV_Q1303957,V_x))) = s(TV_Q1303953,happ(s(t_fun(TV_Q1303957,TV_Q1303953),V_g),s(TV_Q1303957,V_x))) => s(t_fun(TV_Q1303957,TV_Q1303953),V_f) = s(t_fun(TV_Q1303957,TV_Q1303953),V_g))).
fof(ah4s_strings_STRCATu_u_defu_c1, axiom, ![V_l2, V_l1, V_h]: s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_append(s(t_h4s_lists_list(t_h4s_strings_char),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),h4s_lists_cons),s(t_h4s_strings_char,V_h))),s(t_h4s_lists_list(t_h4s_strings_char),V_l1))),s(t_h4s_lists_list(t_h4s_strings_char),V_l2))) = s(t_h4s_lists_list(t_h4s_strings_char),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),h4s_lists_cons),s(t_h4s_strings_char,V_h))),s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_append(s(t_h4s_lists_list(t_h4s_strings_char),V_l1),s(t_h4s_lists_list(t_h4s_strings_char),V_l2)))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
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_strings_STRCAT, axiom, ![V_s2, V_s1]: s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_append(s(t_h4s_lists_list(t_h4s_strings_char),V_s1),s(t_h4s_lists_list(t_h4s_strings_char),V_s2))) = s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_append(s(t_h4s_lists_list(t_h4s_strings_char),V_s1),s(t_h4s_lists_list(t_h4s_strings_char),V_s2)))).
fof(ah4s_lists_APPENDu_u_11u_c0, axiom, ![TV_u_27a]: ![V_l3, V_l2, V_l1]: (s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l3))) <=> s(t_h4s_lists_list(TV_u_27a),V_l2) = s(t_h4s_lists_list(TV_u_27a),V_l3))).
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_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_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_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_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_lists_NIL0, axiom, ![TV_u_27a]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist0)).
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_lists_listu_u_repfnsu_c1, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: 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_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27listu_27]: (![V_a0u_270]: ((s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))) | ?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(TV_u_27a,V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27listu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27listu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27listu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r))))) <=> s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r))))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r)))).
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_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_CONSTRu_u_REC, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_f, V_r, V_n]: s(TV_u_27b,happ(s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_h4s_nums_num,V_n))) = s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r),s(t_h4s_nums_num,V_n))))) => ![V_Fn]: ?[V_f]: ![V_c, V_i, V_r]: s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,V_c),s(TV_u_27a,V_i),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))) = s(TV_u_27b,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)))),V_Fn),s(t_h4s_nums_num,V_c))),s(TV_u_27a,V_i))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))))).
fof(ah4s_indu_u_types_FCONS0u_c0, axiom, ![TV_u_27a]: ![V_f, V_a]: s(TV_u_27a,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_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_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist1u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: 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_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => ![V_x, V_x0]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_u_20u_40indu_u_typelist1),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_x0))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(TV_u_27a,V_x),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(TV_u_27a),V_x0))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))))).
fof(ah4s_lists_CONS0, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons) = s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_u_20u_40indu_u_typelist1)).
fof(ah4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist0u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: 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_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist0) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))).
fof(ah4s_lists_listu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(TV_u_27a),V_a))))) = s(t_h4s_lists_list(TV_u_27a),V_a)).
fof(ah4s_strings_EXPLODEu_u_EQu_u_THMu_c1, axiom, ![V_t, V_s, V_h]: (s(t_h4s_lists_list(t_h4s_strings_char),h4s_strings_explode(s(t_h4s_lists_list(t_h4s_strings_char),V_s))) = s(t_h4s_lists_list(t_h4s_strings_char),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),h4s_lists_cons),s(t_h4s_strings_char,V_h))),s(t_h4s_lists_list(t_h4s_strings_char),V_t))) <=> s(t_h4s_lists_list(t_h4s_strings_char),V_s) = s(t_h4s_lists_list(t_h4s_strings_char),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),h4s_lists_cons),s(t_h4s_strings_char,V_h))),s(t_h4s_lists_list(t_h4s_strings_char),h4s_strings_implode(s(t_h4s_lists_list(t_h4s_strings_char),V_t))))))).
fof(ah4s_strings_STRINGu_u_ACYCLICu_c0, axiom, ![V_s, V_c]: ~ (s(t_h4s_lists_list(t_h4s_strings_char),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),h4s_lists_cons),s(t_h4s_strings_char,V_c))),s(t_h4s_lists_list(t_h4s_strings_char),V_s))) = s(t_h4s_lists_list(t_h4s_strings_char),V_s))).
fof(ah4s_strings_DESTu_u_STRINGu_u_LEMSu_c1, axiom, ![V_t, V_s, V_c]: (s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_strings_char,t_h4s_lists_list(t_h4s_strings_char))),h4s_strings_destu_u_string(s(t_h4s_lists_list(t_h4s_strings_char),V_s))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_strings_char,t_h4s_lists_list(t_h4s_strings_char))),h4s_options_some(s(t_h4s_pairs_prod(t_h4s_strings_char,t_h4s_lists_list(t_h4s_strings_char)),h4s_pairs_u_2c(s(t_h4s_strings_char,V_c),s(t_h4s_lists_list(t_h4s_strings_char),V_t))))) <=> s(t_h4s_lists_list(t_h4s_strings_char),V_s) = s(t_h4s_lists_list(t_h4s_strings_char),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),h4s_lists_cons),s(t_h4s_strings_char,V_c))),s(t_h4s_lists_list(t_h4s_strings_char),V_t))))).
fof(ah4s_strings_IMPLODEu_u_EQu_u_THMu_c0, axiom, ![V_s, V_l, V_c]: (s(t_h4s_lists_list(t_h4s_strings_char),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),h4s_lists_cons),s(t_h4s_strings_char,V_c))),s(t_h4s_lists_list(t_h4s_strings_char),V_s))) = s(t_h4s_lists_list(t_h4s_strings_char),h4s_strings_implode(s(t_h4s_lists_list(t_h4s_strings_char),V_l))) <=> s(t_h4s_lists_list(t_h4s_strings_char),V_l) = s(t_h4s_lists_list(t_h4s_strings_char),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),h4s_lists_cons),s(t_h4s_strings_char,V_c))),s(t_h4s_lists_list(t_h4s_strings_char),h4s_strings_explode(s(t_h4s_lists_list(t_h4s_strings_char),V_s))))))).
fof(ah4s_strings_FIELDSu_u_ind, axiom, ![V_Pu_27]: ((![V_P]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_bool),happ(s(t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_bool)),V_Pu_27),s(t_fun(t_h4s_strings_char,t_bool),V_P))),s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_nil)))) & ![V_P, V_h, V_t]: ((![V_l, V_r]: ((s(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),h4s_pairs_u_2c(s(t_h4s_lists_list(t_h4s_strings_char),V_l),s(t_h4s_lists_list(t_h4s_strings_char),V_r))) = s(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),h4s_richu_u_lists_splitp(s(t_fun(t_h4s_strings_char,t_bool),V_P),s(t_h4s_lists_list(t_h4s_strings_char),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),h4s_lists_cons),s(t_h4s_strings_char,V_h))),s(t_h4s_lists_list(t_h4s_strings_char),V_t))))) & p(s(t_bool,h4s_lists_null(s(t_h4s_lists_list(t_h4s_strings_char),V_l))))) => p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_bool),happ(s(t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_bool)),V_Pu_27),s(t_fun(t_h4s_strings_char,t_bool),V_P))),s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_tl(s(t_h4s_lists_list(t_h4s_strings_char),V_r))))))) & ![V_l, V_r]: ((s(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),h4s_pairs_u_2c(s(t_h4s_lists_list(t_h4s_strings_char),V_l),s(t_h4s_lists_list(t_h4s_strings_char),V_r))) = s(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),h4s_richu_u_lists_splitp(s(t_fun(t_h4s_strings_char,t_bool),V_P),s(t_h4s_lists_list(t_h4s_strings_char),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),h4s_lists_cons),s(t_h4s_strings_char,V_h))),s(t_h4s_lists_list(t_h4s_strings_char),V_t))))) & (~ (p(s(t_bool,h4s_lists_null(s(t_h4s_lists_list(t_h4s_strings_char),V_l))))) & ~ (p(s(t_bool,h4s_lists_null(s(t_h4s_lists_list(t_h4s_strings_char),V_r))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_bool),happ(s(t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_bool)),V_Pu_27),s(t_fun(t_h4s_strings_char,t_bool),V_P))),s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_tl(s(t_h4s_lists_list(t_h4s_strings_char),V_r)))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_bool),happ(s(t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_bool)),V_Pu_27),s(t_fun(t_h4s_strings_char,t_bool),V_P))),s(t_h4s_lists_list(t_h4s_strings_char),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),h4s_lists_cons),s(t_h4s_strings_char,V_h))),s(t_h4s_lists_list(t_h4s_strings_char),V_t)))))))) => ![V_v, V_v1]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_bool),happ(s(t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_bool)),V_Pu_27),s(t_fun(t_h4s_strings_char,t_bool),V_v))),s(t_h4s_lists_list(t_h4s_strings_char),V_v1)))))).
fof(ah4s_strings_TOKENSu_u_defu_c1, axiom, ![V_uu_1]: (![V_l, V_P, V_r]: s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),happ(s(t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))))),V_uu_1),s(t_h4s_lists_list(t_h4s_strings_char),V_l))),s(t_fun(t_h4s_strings_char,t_bool),V_P))),s(t_h4s_lists_list(t_h4s_strings_char),V_r))) = s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),h4s_bools_cond(s(t_bool,h4s_lists_null(s(t_h4s_lists_list(t_h4s_strings_char),V_l))),s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),h4s_strings_tokens(s(t_fun(t_h4s_strings_char,t_bool),V_P),s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_tl(s(t_h4s_lists_list(t_h4s_strings_char),V_r))))),s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_fun(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))),h4s_lists_cons),s(t_h4s_lists_list(t_h4s_strings_char),V_l))),s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),h4s_strings_tokens(s(t_fun(t_h4s_strings_char,t_bool),V_P),s(t_h4s_lists_list(t_h4s_strings_char),V_r))))))) => ![V_uu_0]: (![V_P, V_l]: s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))),happ(s(t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))))),V_uu_0),s(t_fun(t_h4s_strings_char,t_bool),V_P))),s(t_h4s_lists_list(t_h4s_strings_char),V_l))) = s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),happ(s(t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))))),V_uu_1),s(t_h4s_lists_list(t_h4s_strings_char),V_l))),s(t_fun(t_h4s_strings_char,t_bool),V_P))) => ![V_t, V_h, V_P]: s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),h4s_strings_tokens(s(t_fun(t_h4s_strings_char,t_bool),V_P),s(t_h4s_lists_list(t_h4s_strings_char),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),h4s_lists_cons),s(t_h4s_strings_char,V_h))),s(t_h4s_lists_list(t_h4s_strings_char),V_t))))) = s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),h4s_bools_let(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),h4s_pairs_uncurry(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))),happ(s(t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))))),V_uu_0),s(t_fun(t_h4s_strings_char,t_bool),V_P))))),s(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),h4s_richu_u_lists_splitp(s(t_fun(t_h4s_strings_char,t_bool),V_P),s(t_h4s_lists_list(t_h4s_strings_char),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),h4s_lists_cons),s(t_h4s_strings_char,V_h))),s(t_h4s_lists_list(t_h4s_strings_char),V_t)))))))))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_lists_NOTu_u_NILu_u_CONS, axiom, ![TV_u_27a]: ![V_a1, V_a0]: ~ (s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1))))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_lists_CONSu_u_11, axiom, ![TV_u_27a]: ![V_a1u_27, V_a1, V_a0u_27, V_a0]: (s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1))) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_a0u_27))),s(t_h4s_lists_list(TV_u_27a),V_a1u_27))) <=> (s(TV_u_27a,V_a0) = s(TV_u_27a,V_a0u_27) & s(t_h4s_lists_list(TV_u_27a),V_a1) = s(t_h4s_lists_list(TV_u_27a),V_a1u_27)))).
fof(ah4s_lists_listu_u_nchotomy, axiom, ![TV_u_27a]: ![V_l]: (s(t_h4s_lists_list(TV_u_27a),V_l) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) | ?[V_h, V_t]: s(t_h4s_lists_list(TV_u_27a),V_l) = s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_lists_listu_u_induction, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))) & ![V_t]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t)))) => ![V_h]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t)))))))) => ![V_l]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_strings_IMPLODEu_u_EXPLODE, axiom, ![V_s]: s(t_h4s_lists_list(t_h4s_strings_char),h4s_strings_implode(s(t_h4s_lists_list(t_h4s_strings_char),h4s_strings_explode(s(t_h4s_lists_list(t_h4s_strings_char),V_s))))) = s(t_h4s_lists_list(t_h4s_strings_char),V_s)).
fof(ah4s_strings_EXPLODEu_u_IMPLODE, axiom, ![V_cs]: s(t_h4s_lists_list(t_h4s_strings_char),h4s_strings_explode(s(t_h4s_lists_list(t_h4s_strings_char),h4s_strings_implode(s(t_h4s_lists_list(t_h4s_strings_char),V_cs))))) = s(t_h4s_lists_list(t_h4s_strings_char),V_cs)).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_IMPu_u_THM, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) & (p(s(t_bool,V_t2)) => p(s(t_bool,V_t1)))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,f))) <=> p(s(t_bool,f)))).
fof(ah4s_pairs_CLOSEDu_u_PAIRu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_b, V_a]: (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(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & s(TV_u_27b,V_y) = s(TV_u_27b,V_b)))).
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_options_NOTu_u_NONEu_u_SOME, axiom, ![TV_u_27a]: ![V_x]: ~ (s(t_h4s_options_option(TV_u_27a),h4s_options_none) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_strings_DESTu_u_STRINGu_u_defu_c1, axiom, ![V_rst, V_c]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_strings_char,t_h4s_lists_list(t_h4s_strings_char))),h4s_strings_destu_u_string(s(t_h4s_lists_list(t_h4s_strings_char),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),h4s_lists_cons),s(t_h4s_strings_char,V_c))),s(t_h4s_lists_list(t_h4s_strings_char),V_rst))))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_strings_char,t_h4s_lists_list(t_h4s_strings_char))),h4s_options_some(s(t_h4s_pairs_prod(t_h4s_strings_char,t_h4s_lists_list(t_h4s_strings_char)),h4s_pairs_u_2c(s(t_h4s_strings_char,V_c),s(t_h4s_lists_list(t_h4s_strings_char),V_rst)))))).
fof(ah4s_options_SOMEu_u_11, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_strings_DESTu_u_STRINGu_u_defu_c0, axiom, s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_strings_char,t_h4s_lists_list(t_h4s_strings_char))),h4s_strings_destu_u_string(s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_nil))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_strings_char,t_h4s_lists_list(t_h4s_strings_char))),h4s_options_none)).
fof(ah4s_strings_EXPLODEu_u_defu_c1, axiom, ![V_s, V_c]: s(t_h4s_lists_list(t_h4s_strings_char),h4s_strings_explode(s(t_h4s_lists_list(t_h4s_strings_char),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),h4s_lists_cons),s(t_h4s_strings_char,V_c))),s(t_h4s_lists_list(t_h4s_strings_char),V_s))))) = s(t_h4s_lists_list(t_h4s_strings_char),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),h4s_lists_cons),s(t_h4s_strings_char,V_c))),s(t_h4s_lists_list(t_h4s_strings_char),h4s_strings_explode(s(t_h4s_lists_list(t_h4s_strings_char),V_s)))))).
fof(ah4s_strings_EXPLODEu_u_defu_c0, axiom, s(t_h4s_lists_list(t_h4s_strings_char),h4s_strings_explode(s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_nil))) = s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_nil)).
fof(ah4s_strings_IMPLODEu_u_defu_c1, axiom, ![V_cs, V_c]: s(t_h4s_lists_list(t_h4s_strings_char),h4s_strings_implode(s(t_h4s_lists_list(t_h4s_strings_char),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),h4s_lists_cons),s(t_h4s_strings_char,V_c))),s(t_h4s_lists_list(t_h4s_strings_char),V_cs))))) = s(t_h4s_lists_list(t_h4s_strings_char),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),h4s_lists_cons),s(t_h4s_strings_char,V_c))),s(t_h4s_lists_list(t_h4s_strings_char),h4s_strings_implode(s(t_h4s_lists_list(t_h4s_strings_char),V_cs)))))).
fof(ah4s_strings_IMPLODEu_u_defu_c0, axiom, s(t_h4s_lists_list(t_h4s_strings_char),h4s_strings_implode(s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_nil))) = s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_nil)).
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_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_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_richu_u_lists_SPLITP0u_c0, axiom, ![TV_u_27a]: ![V_P]: s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_richu_u_lists_splitp(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))).
fof(ah4s_richu_u_lists_SPLITP0u_c1, axiom, ![TV_u_27a]: ![V_x, V_l, V_P]: s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_richu_u_lists_splitp(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_bools_cond(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_l))))),s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_pairs_u_2c(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),h4s_pairs_fst(s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_richu_u_lists_splitp(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l))))))),s(t_h4s_lists_list(TV_u_27a),h4s_pairs_snd(s(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_richu_u_lists_splitp(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))))))))).
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_lists_NULLu_u_EQ, axiom, ![TV_u_27a]: ![V_l]: (p(s(t_bool,h4s_lists_null(s(t_h4s_lists_list(TV_u_27a),V_l)))) <=> s(t_h4s_lists_list(TV_u_27a),V_l) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))).
fof(ah4s_lists_listu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_f, V_a1, V_a0]: s(TV_u_27b,h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f))) = s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1)))).
fof(ah4s_lists_listu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27b,h4s_lists_listu_u_case(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b)),V_f))) = s(TV_u_27b,V_v)).
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_lists_LENGTH0u_c0, axiom, ![TV_u_27a]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_bools_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_numerals_numeralu_u_addu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_m))))))) = s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_numerals_numeralu_u_addu_c0, axiom, ![V_n]: s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,V_n))))) = s(t_h4s_nums_num,V_n)).
fof(ah4s_numerals_numeralu_u_distribu_c2, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_m))))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_numerals_iz(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_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_lists_LENGTH0u_c1, axiom, ![TV_u_27a]: ![V_t, V_h]: s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t))))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_t)))))).
fof(ah4s_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_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_MONO, axiom, ![V_n, V_m]: 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,h4s_nums_suc(s(t_h4s_nums_num,V_m))))) = 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_LESSu_u_EQu_u_REFL, axiom, ![V_m]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_m))))).
fof(ah4s_lists_TL0, axiom, ![TV_u_27a]: ![V_t, V_h]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_tl(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t))))) = s(t_h4s_lists_list(TV_u_27a),V_t)).
fof(ah4s_arithmetics_ADDu_u_SYM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_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_lteu_c2, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
fof(ah4s_numerals_numeralu_u_distribu_c27, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ah4s_arithmetics_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_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),TV_u_27c),h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),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_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_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_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_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_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_primu_u_recs_WFu_u_measure, axiom, ![TV_u_27a]: ![V_m]: p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_primu_u_recs_measure),s(t_fun(TV_u_27a,t_h4s_nums_num),V_m))))))).
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_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_measureu_u_def, axiom, ![TV_u_27a]: s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_primu_u_recs_measure) = s(t_fun(t_fun(TV_u_27a,t_h4s_nums_num),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),h4s_relations_invu_u_image(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c)))).
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)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),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_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_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_ou_u_DEF, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_g, V_f, V_x]: s(TV_u_27b,h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_g),s(TV_u_27a,V_x)))))).
fof(ah4s_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_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_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_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_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_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_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_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_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_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_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_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_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_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_LETu_u_CONG, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f, V_N, V_M]: ((s(TV_u_27a,V_M) = s(TV_u_27a,V_N) & ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_N) => s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))) => s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_M))) = s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_N))))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_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_strings_FIELDSu_u_curriedu_u_def, axiom, ![V_x1, V_x]: s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),h4s_strings_fields(s(t_fun(t_h4s_strings_char,t_bool),V_x),s(t_h4s_lists_list(t_h4s_strings_char),V_x1))) = s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),h4s_strings_fieldsu_u_tupled),s(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),h4s_pairs_u_2c(s(t_fun(t_h4s_strings_char,t_bool),V_x),s(t_h4s_lists_list(t_h4s_strings_char),V_x1)))))).
fof(ah4s_strings_FIELDSu_u_tupledu_u_primitiveu_u_def, axiom, ![V_uu_8]: (![V_l, V_FIELDSu_u_tupled, V_P, V_r]: s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),happ(s(t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))))),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))))),V_uu_8),s(t_h4s_lists_list(t_h4s_strings_char),V_l))),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),V_FIELDSu_u_tupled))),s(t_fun(t_h4s_strings_char,t_bool),V_P))),s(t_h4s_lists_list(t_h4s_strings_char),V_r))) = s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),h4s_bools_cond(s(t_bool,h4s_lists_null(s(t_h4s_lists_list(t_h4s_strings_char),V_l))),s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_fun(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))),h4s_lists_cons),s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_nil))),s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),V_FIELDSu_u_tupled),s(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),h4s_pairs_u_2c(s(t_fun(t_h4s_strings_char,t_bool),V_P),s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_tl(s(t_h4s_lists_list(t_h4s_strings_char),V_r))))))))),s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),h4s_bools_cond(s(t_bool,h4s_lists_null(s(t_h4s_lists_list(t_h4s_strings_char),V_r))),s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_fun(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))),h4s_lists_cons),s(t_h4s_lists_list(t_h4s_strings_char),V_l))),s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),h4s_lists_nil))),s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_fun(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))),h4s_lists_cons),s(t_h4s_lists_list(t_h4s_strings_char),V_l))),s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),V_FIELDSu_u_tupled),s(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),h4s_pairs_u_2c(s(t_fun(t_h4s_strings_char,t_bool),V_P),s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_tl(s(t_h4s_lists_list(t_h4s_strings_char),V_r))))))))))))) => ![V_uu_7]: (![V_FIELDSu_u_tupled, V_P, V_l]: s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))),happ(s(t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))))),V_uu_7),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),V_FIELDSu_u_tupled))),s(t_fun(t_h4s_strings_char,t_bool),V_P))),s(t_h4s_lists_list(t_h4s_strings_char),V_l))) = s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),happ(s(t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))))),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))))),V_uu_8),s(t_h4s_lists_list(t_h4s_strings_char),V_l))),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),V_FIELDSu_u_tupled))),s(t_fun(t_h4s_strings_char,t_bool),V_P))) => ![V_uu_6]: (![V_FIELDSu_u_tupled, V_P, V_h, V_t]: s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))),happ(s(t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))))),V_uu_6),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),V_FIELDSu_u_tupled))),s(t_fun(t_h4s_strings_char,t_bool),V_P))),s(t_h4s_strings_char,V_h))),s(t_h4s_lists_list(t_h4s_strings_char),V_t))) = s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),h4s_combins_i(s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),h4s_bools_let(s(t_fun(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),h4s_pairs_uncurry(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))),happ(s(t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))))),V_uu_7),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),V_FIELDSu_u_tupled))),s(t_fun(t_h4s_strings_char,t_bool),V_P))))),s(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),h4s_richu_u_lists_splitp(s(t_fun(t_h4s_strings_char,t_bool),V_P),s(t_h4s_lists_list(t_h4s_strings_char),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),h4s_lists_cons),s(t_h4s_strings_char,V_h))),s(t_h4s_lists_list(t_h4s_strings_char),V_t))))))))) => ![V_uu_5]: (![V_FIELDSu_u_tupled, V_P, V_h]: s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))),happ(s(t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))))),V_uu_5),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),V_FIELDSu_u_tupled))),s(t_fun(t_h4s_strings_char,t_bool),V_P))),s(t_h4s_strings_char,V_h))) = s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))),happ(s(t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))))),V_uu_6),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),V_FIELDSu_u_tupled))),s(t_fun(t_h4s_strings_char,t_bool),V_P))),s(t_h4s_strings_char,V_h))) => ![V_uu_4]: (![V_FIELDSu_u_tupled, V_P, V_v1]: s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),happ(s(t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),V_FIELDSu_u_tupled))),s(t_fun(t_h4s_strings_char,t_bool),V_P))),s(t_h4s_lists_list(t_h4s_strings_char),V_v1))) = s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),h4s_lists_listu_u_case(s(t_h4s_lists_list(t_h4s_strings_char),V_v1),s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),h4s_combins_i(s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_fun(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))),h4s_lists_cons),s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_nil))),s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),h4s_lists_nil))))),s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))),happ(s(t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))))),V_uu_5),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),V_FIELDSu_u_tupled))),s(t_fun(t_h4s_strings_char,t_bool),V_P))))) => ![V_uu_3]: (![V_FIELDSu_u_tupled, V_P]: s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),happ(s(t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),V_FIELDSu_u_tupled))),s(t_fun(t_h4s_strings_char,t_bool),V_P))) = s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),happ(s(t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))))),V_uu_4),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),V_FIELDSu_u_tupled))),s(t_fun(t_h4s_strings_char,t_bool),V_P))) => ![V_uu_2]: (![V_FIELDSu_u_tupled, V_a]: s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),V_FIELDSu_u_tupled))),s(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),V_a))) = s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),V_a),s(t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),t_fun(t_fun(t_h4s_strings_char,t_bool),t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),V_FIELDSu_u_tupled))))) => ![V_uu_1]: (![V_FIELDSu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),V_FIELDSu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),V_FIELDSu_u_tupled))) => ![V_uu_0]: (![V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_bool)),V_R)))) & (![V_t, V_h, V_P, V_l, V_r]: ((s(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),h4s_pairs_u_2c(s(t_h4s_lists_list(t_h4s_strings_char),V_l),s(t_h4s_lists_list(t_h4s_strings_char),V_r))) = s(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),h4s_richu_u_lists_splitp(s(t_fun(t_h4s_strings_char,t_bool),V_P),s(t_h4s_lists_list(t_h4s_strings_char),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),h4s_lists_cons),s(t_h4s_strings_char,V_h))),s(t_h4s_lists_list(t_h4s_strings_char),V_t))))) & p(s(t_bool,h4s_lists_null(s(t_h4s_lists_list(t_h4s_strings_char),V_l))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_bool)),V_R),s(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),h4s_pairs_u_2c(s(t_fun(t_h4s_strings_char,t_bool),V_P),s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_tl(s(t_h4s_lists_list(t_h4s_strings_char),V_r))))))),s(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),h4s_pairs_u_2c(s(t_fun(t_h4s_strings_char,t_bool),V_P),s(t_h4s_lists_list(t_h4s_strings_char),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),h4s_lists_cons),s(t_h4s_strings_char,V_h))),s(t_h4s_lists_list(t_h4s_strings_char),V_t))))))))) & ![V_t, V_h, V_P, V_l, V_r]: ((s(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),h4s_pairs_u_2c(s(t_h4s_lists_list(t_h4s_strings_char),V_l),s(t_h4s_lists_list(t_h4s_strings_char),V_r))) = s(t_h4s_pairs_prod(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),h4s_richu_u_lists_splitp(s(t_fun(t_h4s_strings_char,t_bool),V_P),s(t_h4s_lists_list(t_h4s_strings_char),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),h4s_lists_cons),s(t_h4s_strings_char,V_h))),s(t_h4s_lists_list(t_h4s_strings_char),V_t))))) & (~ (p(s(t_bool,h4s_lists_null(s(t_h4s_lists_list(t_h4s_strings_char),V_l))))) & ~ (p(s(t_bool,h4s_lists_null(s(t_h4s_lists_list(t_h4s_strings_char),V_r))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_bool)),V_R),s(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),h4s_pairs_u_2c(s(t_fun(t_h4s_strings_char,t_bool),V_P),s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_tl(s(t_h4s_lists_list(t_h4s_strings_char),V_r))))))),s(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),h4s_pairs_u_2c(s(t_fun(t_h4s_strings_char,t_bool),V_P),s(t_h4s_lists_list(t_h4s_strings_char),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),h4s_lists_cons),s(t_h4s_strings_char,V_h))),s(t_h4s_lists_list(t_h4s_strings_char),V_t)))))))))))) => s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),h4s_strings_fieldsu_u_tupled) = s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char))),t_fun(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)))),V_uu_1))))))))))))).
fof(ah4s_arithmetics_LESSu_u_TRANS, axiom, ![V_p, 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,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,V_p))))) => 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_p)))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_strings_TOKENSu_u_curriedu_u_def, axiom, ![V_x1, V_x]: s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),h4s_strings_tokens(s(t_fun(t_h4s_strings_char,t_bool),V_x),s(t_h4s_lists_list(t_h4s_strings_char),V_x1))) = s(t_h4s_lists_list(t_h4s_lists_list(t_h4s_strings_char)),h4s_strings_tokensu_u_tupled(s(t_h4s_pairs_prod(t_fun(t_h4s_strings_char,t_bool),t_h4s_lists_list(t_h4s_strings_char)),h4s_pairs_u_2c(s(t_fun(t_h4s_strings_char,t_bool),V_x),s(t_h4s_lists_list(t_h4s_strings_char),V_x1)))))).
fof(ch4s_strings_STRCATu_u_EQNSu_c2, conjecture, ![V_s2, V_s1, V_c]: s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_append(s(t_h4s_lists_list(t_h4s_strings_char),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),h4s_lists_cons),s(t_h4s_strings_char,V_c))),s(t_h4s_lists_list(t_h4s_strings_char),V_s1))),s(t_h4s_lists_list(t_h4s_strings_char),V_s2))) = s(t_h4s_lists_list(t_h4s_strings_char),happ(s(t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char)),happ(s(t_fun(t_h4s_strings_char,t_fun(t_h4s_lists_list(t_h4s_strings_char),t_h4s_lists_list(t_h4s_strings_char))),h4s_lists_cons),s(t_h4s_strings_char,V_c))),s(t_h4s_lists_list(t_h4s_strings_char),h4s_lists_append(s(t_h4s_lists_list(t_h4s_strings_char),V_s1),s(t_h4s_lists_list(t_h4s_strings_char),V_s2)))))).
