%   ORIGINAL: h4/EncodeVar/fixed__width__bnum
% 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/Coder/bnum__coder__def: !p m. h4/Coder/bnum__coder m p = h4/pair/_2C p (h4/pair/_2C (h4/Encode/encode__bnum m) (h4/Decode/decode__bnum m p))
% Assm: h4/Coder/wf__coder__bnum: !p m. h4/Encode/wf__pred__bnum m p ==> h4/Coder/wf__coder (h4/Coder/bnum__coder m p)
% 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/TRUTH: T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/EncodeVar/fixed__width__def: !n c. h4/EncodeVar/fixed__width n c <=> (!x. h4/Coder/domain c x ==> h4/list/LENGTH (h4/Coder/encoder c x) = n)
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/Decode/decode__bnum__def: !p m. h4/Decode/decode__bnum m p = h4/Decode/enc2dec p (h4/Encode/encode__bnum m)
% Assm: h4/Encode/wf__pred__bnum__def: !p m. h4/Encode/wf__pred__bnum m p <=> h4/Encode/wf__pred p /\ (!x. p x ==> h4/prim__rec/_3C x (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) m))
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/Encode/wf__encode__bnum: !p m. h4/Encode/wf__pred__bnum m p ==> h4/Encode/wf__encoder p (h4/Encode/encode__bnum m)
% Assm: h4/Coder/wf__coder__def: !p e d. h4/Coder/wf__coder (h4/pair/_2C p (h4/pair/_2C e d)) <=> h4/Encode/wf__pred p /\ h4/Encode/wf__encoder p e /\ d = h4/Decode/enc2dec p e
% Assm: h4/EncodeVar/fixed__width__prod: !n2 n1 c2 c1. h4/EncodeVar/fixed__width n1 c1 /\ h4/EncodeVar/fixed__width n2 c2 ==> h4/EncodeVar/fixed__width (h4/arithmetic/_2B n1 n2) (h4/Coder/prod__coder c1 c2)
% Assm: h4/EncodeVar/fixed__width__exists: !phi n c. h4/Coder/wf__coder c /\ h4/EncodeVar/fixed__width n c ==> ((?x. h4/Coder/domain c x /\ phi x) <=> h4/bool/RES__EXISTS (h4/EncodeVar/of__length n) (\w. phi (h4/Coder/decoder c w)))
% Assm: h4/EncodeVar/fixed__width__univ: !phi n c. h4/Coder/wf__coder c /\ h4/EncodeVar/fixed__width n c ==> ((!x. h4/Coder/domain c x ==> phi x) <=> h4/bool/RES__FORALL (h4/EncodeVar/of__length n) (\w. phi (h4/Coder/decoder c w)))
% Assm: h4/EncodeVar/fixed__width__sum: !n c2 c1. h4/EncodeVar/fixed__width n c1 /\ h4/EncodeVar/fixed__width n c2 ==> h4/EncodeVar/fixed__width (h4/num/SUC n) (h4/Coder/sum__coder c1 c2)
% Assm: h4/EncodeVar/fixed__width__bool: !p. h4/EncodeVar/fixed__width (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/Coder/bool__coder p)
% Assm: h4/EncodeVar/fixed__width__unit: !p. h4/EncodeVar/fixed__width h4/num/0 (h4/Coder/unit__coder p)
% Assm: h4/Coder/encoder__def: !p e d. h4/Coder/encoder (h4/pair/_2C p (h4/pair/_2C e d)) = e
% Assm: h4/Coder/domain__def: !p e d. h4/Coder/domain (h4/pair/_2C p (h4/pair/_2C e d)) = p
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/list/LENGTH0_c0: h4/list/LENGTH h4/list/NIL = h4/num/0
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/list/LENGTH0_c1: !t h. h4/list/LENGTH (h4/list/CONS h t) = h4/num/SUC (h4/list/LENGTH t)
% Assm: h4/Encode/encode__unit__def: !v0. h4/Encode/encode__unit v0 = h4/list/NIL
% Assm: h4/Coder/unit__coder__def: !p. h4/Coder/unit__coder p = h4/pair/_2C p (h4/pair/_2C h4/Encode/encode__unit (h4/Decode/decode__unit p))
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/Encode/encode__bool__def: !x. h4/Encode/encode__bool x = h4/list/CONS x h4/list/NIL
% Assm: h4/numeral/numeral__distrib_c13: h4/num/SUC h4/num/0 = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm: h4/Coder/bool__coder__def: !p. h4/Coder/bool__coder p = h4/pair/_2C p (h4/pair/_2C h4/Encode/encode__bool (h4/Decode/decode__bool p))
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/Coder/wf__coder0: !c. h4/Coder/wf__coder c ==> (!x. h4/Coder/domain c x ==> h4/Coder/decoder c (h4/Coder/encoder c x) = x)
% Assm: h4/Coder/wf__coder__closed: !c. h4/Coder/wf__coder c ==> (!l. h4/Coder/domain c (h4/Coder/decoder c l))
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/EncodeVar/of__length__def: !n l. h4/bool/IN l (h4/EncodeVar/of__length n) <=> h4/list/LENGTH l = n
% Assm: h4/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/bool/SKOLEM__THM: !P. (!x. ?y. P x y) <=> (?f. !x. P x (f x))
% Assm: h4/Decode/decode__num0: !p l. h4/Decode/decode__num p l = h4/list/list__CASE l h4/option/NONE (\v2 t_27. h4/bool/COND v2 (h4/list/list__CASE t_27 h4/option/NONE (\v6 t. h4/bool/COND v6 (h4/bool/COND (p h4/num/0) (h4/option/SOME (h4/pair/_2C h4/num/0 t)) h4/option/NONE) (h4/option/option__CASE (h4/Decode/decode__num (h4/combin/K T) t) h4/option/NONE (\v1. h4/pair/pair__CASE v1 (\v t_270. h4/bool/COND (p (h4/arithmetic/_2B (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) v) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/option/SOME (h4/pair/_2C (h4/arithmetic/_2B (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) v) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) t_270)) h4/option/NONE))))) (h4/option/option__CASE (h4/Decode/decode__num (h4/combin/K T) t_27) h4/option/NONE (\v1. h4/pair/pair__CASE v1 (\v t_270. h4/bool/COND (p (h4/arithmetic/_2B (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) v) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) (h4/option/SOME (h4/pair/_2C (h4/arithmetic/_2B (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) v) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) t_270)) h4/option/NONE))))
% Assm: h4/bool/RES__FORALL__THM: !f P. h4/bool/RES__FORALL P f <=> (!x. h4/bool/IN x P ==> f x)
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/Encode/lift__prod__def: !x p2 p1. h4/Encode/lift__prod p1 p2 x <=> p1 (h4/pair/FST x) /\ p2 (h4/pair/SND x)
% Assm: h4/list/LENGTH__APPEND: !l2 l1. h4/list/LENGTH (h4/list/APPEND l1 l2) = h4/arithmetic/_2B (h4/list/LENGTH l1) (h4/list/LENGTH l2)
% Assm: h4/arithmetic/EQ__MONO__ADD__EQ: !p n m. h4/arithmetic/_2B m p = h4/arithmetic/_2B n p <=> m = n
% Assm: h4/Encode/encode__prod__def: !yb y xb x. h4/Encode/encode__prod xb yb (h4/pair/_2C x y) = h4/list/APPEND (xb x) (yb y)
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/Coder/prod__coder__def: !p2 p1 e2 e1 d2 d1. h4/Coder/prod__coder (h4/pair/_2C p1 (h4/pair/_2C e1 d1)) (h4/pair/_2C p2 (h4/pair/_2C e2 d2)) = h4/pair/_2C (h4/Encode/lift__prod p1 p2) (h4/pair/_2C (h4/Encode/encode__prod e1 e2) (h4/Decode/decode__prod (h4/Encode/lift__prod p1 p2) d1 d2))
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/RES__EXISTS__THM: !f P. h4/bool/RES__EXISTS P f <=> (?x. h4/bool/IN x P /\ f x)
% Assm: h4/Encode/encode__sum__def_c1: !yb y xb. h4/Encode/encode__sum xb yb (h4/sum/INR y) = h4/list/CONS F (yb y)
% Assm: h4/Encode/lift__sum__def: !x p2 p1. h4/Encode/lift__sum p1 p2 x <=> h4/sum/sum__CASE x (\x1. p1 x1) (\x2. p2 x2)
% Assm: h4/Encode/encode__sum__def_c0: !yb xb x. h4/Encode/encode__sum xb yb (h4/sum/INL x) = h4/list/CONS T (xb x)
% Assm: h4/prim__rec/INV__SUC__EQ: !n m. h4/num/SUC m = h4/num/SUC n <=> m = n
% Assm: h4/sum/sum__case__def_c0: !x f1 f. h4/sum/sum__CASE (h4/sum/INL x) f f1 = f x
% Assm: h4/sum/sum__case__def_c1: !y f1 f. h4/sum/sum__CASE (h4/sum/INR y) f f1 = f1 y
% Assm: h4/sum/sum__CASES: !ss. (?x. ss = h4/sum/INL x) \/ (?y. ss = h4/sum/INR y)
% Assm: h4/Coder/sum__coder__def: !p2 p1 e2 e1 d2 d1. h4/Coder/sum__coder (h4/pair/_2C p1 (h4/pair/_2C e1 d1)) (h4/pair/_2C p2 (h4/pair/_2C e2 d2)) = h4/pair/_2C (h4/Encode/lift__sum p1 p2) (h4/pair/_2C (h4/Encode/encode__sum e1 e2) (h4/Decode/decode__sum (h4/Encode/lift__sum p1 p2) d1 d2))
% Assm: h4/Decode/decode__num__total: !l. h4/Decode/decode__num (h4/combin/K T) l = h4/list/list__CASE l h4/option/NONE (\v2 t_27. h4/bool/COND v2 (h4/list/list__CASE t_27 h4/option/NONE (\v6 t. h4/bool/COND v6 (h4/option/SOME (h4/pair/_2C h4/num/0 t)) (h4/option/option__CASE (h4/Decode/decode__num (h4/combin/K T) t) h4/option/NONE (\v1. h4/pair/pair__CASE v1 (\v t_270. h4/option/SOME (h4/pair/_2C (h4/arithmetic/_2B (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) v) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) t_270)))))) (h4/option/option__CASE (h4/Decode/decode__num (h4/combin/K T) t_27) h4/option/NONE (\v1. h4/pair/pair__CASE v1 (\v t_270. h4/option/SOME (h4/pair/_2C (h4/arithmetic/_2B (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) v) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) t_270)))))
% Assm: h4/Decode/decode__num__def: !p. h4/Decode/decode__num p = h4/Decode/enc2dec p h4/Encode/encode__num
% Assm: h4/Decode/wf__decode__num: !p. h4/Decode/wf__decoder p (h4/Decode/decode__num p)
% Assm: h4/Decode/dec2enc__enc2dec: !x p e. h4/Encode/wf__encoder p e /\ p x ==> h4/Decode/dec2enc (h4/Decode/enc2dec p e) x = e x
% Assm: h4/Decode/decode__dec2enc__append: !x t p d. h4/Decode/wf__decoder p d /\ p x ==> d (h4/list/APPEND (h4/Decode/dec2enc d x) t) = h4/option/SOME (h4/pair/_2C x t)
% Assm: h4/Decode/enc2dec__none: !p l e. h4/Decode/enc2dec p e l = h4/option/NONE <=> (!x t. p x ==> ~(l = h4/list/APPEND (e x) t))
% Assm: h4/Encode/wf__encode__num: !p. h4/Encode/wf__encoder p h4/Encode/encode__num
% Assm: h4/bool/EQ__EXT: !g f. (!x. f x = g x) ==> f = g
% Assm: h4/bool/BOOL__EQ__DISTINCT_c0: ~(T <=> F)
% Assm: h4/Decode/enc2dec__some: !x t p l e. h4/Encode/wf__encoder p e ==> (h4/Decode/enc2dec p e l = h4/option/SOME (h4/pair/_2C x t) <=> p x /\ l = h4/list/APPEND (e x) t)
% Assm: h4/Encode/encode__num__def: !n. h4/Encode/encode__num n = h4/bool/COND (n = h4/num/0) (h4/list/CONS T (h4/list/CONS T h4/list/NIL)) (h4/bool/COND (h4/arithmetic/EVEN n) (h4/list/CONS F (h4/Encode/encode__num (h4/arithmetic/DIV (h4/arithmetic/_2D n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))))) (h4/list/CONS T (h4/list/CONS F (h4/Encode/encode__num (h4/arithmetic/DIV (h4/arithmetic/_2D n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))))))
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/list/list__nchotomy: !l. l = h4/list/NIL \/ (?h t. l = h4/list/CONS h t)
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/list/APPEND0_c1: !l2 l1 h. h4/list/APPEND (h4/list/CONS h l1) l2 = h4/list/CONS h (h4/list/APPEND l1 l2)
% Assm: h4/list/list__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/APPEND0_c0: !l. h4/list/APPEND h4/list/NIL l = l
% 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/numeral/numeral__add_c1: !n. h4/numeral/iZ (h4/arithmetic/_2B n h4/arithmetic/ZERO) = n
% 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__distrib_c21: !n. h4/prim__rec/_3C h4/num/0 (h4/arithmetic/NUMERAL n) <=> h4/prim__rec/_3C h4/arithmetic/ZERO n
% Assm: h4/numeral/numeral__lt_c1: !n. h4/prim__rec/_3C h4/arithmetic/ZERO (h4/arithmetic/BIT2 n) <=> T
% 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/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/arithmetic/ODD__EXISTS: !n. h4/arithmetic/ODD n <=> (?m. n = h4/num/SUC (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) m))
% Assm: h4/arithmetic/EVEN__ODD: !n. h4/arithmetic/EVEN n <=> ~h4/arithmetic/ODD n
% Assm: h4/list/list__case__def_c0: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm: h4/arithmetic/MULT__DIV: !q n. h4/prim__rec/_3C h4/num/0 n ==> h4/arithmetic/DIV (h4/arithmetic/_2A q n) n = q
% Assm: h4/arithmetic/ADD__SUB: !c a. h4/arithmetic/_2D (h4/arithmetic/_2B a c) c = a
% Assm: h4/arithmetic/MULT__COMM: !n m. h4/arithmetic/_2A m n = h4/arithmetic/_2A n m
% Assm: h4/arithmetic/ADD__ASSOC: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm: h4/arithmetic/EVEN0_c1: !n. h4/arithmetic/EVEN (h4/num/SUC n) <=> ~h4/arithmetic/EVEN n
% 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/CLOSED__PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/arithmetic/num__CASES: !m. m = h4/num/0 \/ (?n. m = h4/num/SUC n)
% Assm: h4/option/option__case__def_c1: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = f x
% Goal: !p m. h4/EncodeVar/fixed__width m (h4/Coder/bnum__coder m p)
%   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_Coders_bnumu_u_coderu_u_def]: !p m. h4/Coder/bnum__coder m p = h4/pair/_2C p (h4/pair/_2C (h4/Encode/encode__bnum m) (h4/Decode/decode__bnum m p))
% Assm [h4s_Coders_wfu_u_coderu_u_bnum]: !p m. h4/Encode/wf__pred__bnum m p ==> h4/Coder/wf__coder (h4/Coder/bnum__coder m p)
% 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_TRUTH]: T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_EncodeVars_fixedu_u_widthu_u_def]: !n c. h4/EncodeVar/fixed__width n c <=> (!x. happ (h4/Coder/domain c) x ==> h4/list/LENGTH (happ (h4/Coder/encoder c) x) = n)
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_Decodes_decodeu_u_bnumu_u_def]: !p m. h4/Decode/decode__bnum m p = h4/Decode/enc2dec p (h4/Encode/encode__bnum m)
% Assm [h4s_Encodes_wfu_u_predu_u_bnumu_u_def]: !p m. h4/Encode/wf__pred__bnum m p <=> h4/Encode/wf__pred p /\ (!x. happ p x ==> h4/prim__rec/_3C x (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) m))
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_Encodes_wfu_u_encodeu_u_bnum]: !p m. h4/Encode/wf__pred__bnum m p ==> h4/Encode/wf__encoder p (h4/Encode/encode__bnum m)
% Assm [h4s_Coders_wfu_u_coderu_u_def]: !p e d. h4/Coder/wf__coder (h4/pair/_2C p (h4/pair/_2C e d)) <=> h4/Encode/wf__pred p /\ h4/Encode/wf__encoder p e /\ d = h4/Decode/enc2dec p e
% Assm [h4s_EncodeVars_fixedu_u_widthu_u_prod]: !n2 n1 c2 c1. h4/EncodeVar/fixed__width n1 c1 /\ h4/EncodeVar/fixed__width n2 c2 ==> h4/EncodeVar/fixed__width (h4/arithmetic/_2B n1 n2) (h4/Coder/prod__coder c1 c2)
% Assm [h4s_EncodeVars_fixedu_u_widthu_u_exists]: !_0. (!phi c w. happ (happ (happ _0 phi) c) w <=> happ phi (h4/Coder/decoder c w)) ==> (!phi n c. h4/Coder/wf__coder c /\ h4/EncodeVar/fixed__width n c ==> ((?x. happ (h4/Coder/domain c) x /\ happ phi x) <=> h4/bool/RES__EXISTS (h4/EncodeVar/of__length n) (happ (happ _0 phi) c)))
% Assm [h4s_EncodeVars_fixedu_u_widthu_u_univ]: !_0. (!phi c w. happ (happ (happ _0 phi) c) w <=> happ phi (h4/Coder/decoder c w)) ==> (!phi n c. h4/Coder/wf__coder c /\ h4/EncodeVar/fixed__width n c ==> ((!x. happ (h4/Coder/domain c) x ==> happ phi x) <=> h4/bool/RES__FORALL (h4/EncodeVar/of__length n) (happ (happ _0 phi) c)))
% Assm [h4s_EncodeVars_fixedu_u_widthu_u_sum]: !n c2 c1. h4/EncodeVar/fixed__width n c1 /\ h4/EncodeVar/fixed__width n c2 ==> h4/EncodeVar/fixed__width (h4/num/SUC n) (h4/Coder/sum__coder c1 c2)
% Assm [h4s_EncodeVars_fixedu_u_widthu_u_bool]: !p. h4/EncodeVar/fixed__width (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/Coder/bool__coder p)
% Assm [h4s_EncodeVars_fixedu_u_widthu_u_unit]: !p. h4/EncodeVar/fixed__width h4/num/0 (h4/Coder/unit__coder p)
% Assm [h4s_Coders_encoderu_u_def]: !p e d. h4/Coder/encoder (h4/pair/_2C p (h4/pair/_2C e d)) = e
% Assm [h4s_Coders_domainu_u_def]: !p e d. h4/Coder/domain (h4/pair/_2C p (h4/pair/_2C e d)) = p
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_lists_LENGTH0u_c0]: h4/list/LENGTH h4/list/NIL = h4/num/0
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_lists_LENGTH0u_c1]: !t h. h4/list/LENGTH (h4/list/CONS h t) = h4/num/SUC (h4/list/LENGTH t)
% Assm [h4s_Encodes_encodeu_u_unitu_u_def]: !v0. happ h4/Encode/encode__unit v0 = h4/list/NIL
% Assm [h4s_Coders_unitu_u_coderu_u_def]: !p. h4/Coder/unit__coder p = h4/pair/_2C p (h4/pair/_2C h4/Encode/encode__unit (h4/Decode/decode__unit p))
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_Encodes_encodeu_u_boolu_u_def]: !x. happ h4/Encode/encode__bool x = h4/list/CONS x h4/list/NIL
% Assm [h4s_numerals_numeralu_u_distribu_c13]: h4/num/SUC h4/num/0 = h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)
% Assm [h4s_Coders_boolu_u_coderu_u_def]: !p. h4/Coder/bool__coder p = h4/pair/_2C p (h4/pair/_2C h4/Encode/encode__bool (h4/Decode/decode__bool p))
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_Coders_wfu_u_coder0]: !c. h4/Coder/wf__coder c ==> (!x. happ (h4/Coder/domain c) x ==> h4/Coder/decoder c (happ (h4/Coder/encoder c) x) = x)
% Assm [h4s_Coders_wfu_u_coderu_u_closed]: !c. h4/Coder/wf__coder c ==> (!l. happ (h4/Coder/domain c) (h4/Coder/decoder c l))
% Assm [h4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. P \/ happ Q x) <=> P \/ (!x. happ Q x)
% Assm [h4s_EncodeVars_ofu_u_lengthu_u_def]: !n l. h4/bool/IN l (h4/EncodeVar/of__length n) <=> h4/list/LENGTH l = n
% Assm [h4s_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_bools_SKOLEMu_u_THM]: !P. (!x. ?y. happ (happ P x) y) <=> (?f. !x. happ (happ P x) (happ f x))
% Assm [h4s_Decodes_decodeu_u_num0]: !_9. (!p v t_270. happ (happ (happ _9 p) v) t_270 = h4/bool/COND (happ p (h4/arithmetic/_2B (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) v) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))) (h4/option/SOME (h4/pair/_2C (h4/arithmetic/_2B (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) v) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) t_270)) h4/option/NONE) ==> (!_8. (!p v. happ (happ _8 p) v = happ (happ _9 p) v) ==> (!_7. (!p v1. happ (happ _7 p) v1 = h4/pair/pair__CASE v1 (happ _8 p)) ==> (!_6. (!p v t_270. happ (happ (happ _6 p) v) t_270 = h4/bool/COND (happ p (h4/arithmetic/_2B (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) v) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/option/SOME (h4/pair/_2C (h4/arithmetic/_2B (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) v) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) t_270)) h4/option/NONE) ==> (!_5. (!p v. happ (happ _5 p) v = happ (happ _6 p) v) ==> (!_4. (!p v1. happ (happ _4 p) v1 = h4/pair/pair__CASE v1 (happ _5 p)) ==> (!_3. (!v6 p t. happ (happ (happ _3 v6) p) t = h4/bool/COND v6 (h4/bool/COND (happ p h4/num/0) (h4/option/SOME (h4/pair/_2C h4/num/0 t)) h4/option/NONE) (h4/option/option__CASE (happ (h4/Decode/decode__num (h4/combin/K T)) t) h4/option/NONE (happ _4 p))) ==> (!_2. (!p v6. happ (happ _2 p) v6 = happ (happ _3 v6) p) ==> (!_1. (!v2 p t_27. happ (happ (happ _1 v2) p) t_27 = h4/bool/COND v2 (h4/list/list__CASE t_27 h4/option/NONE (happ _2 p)) (h4/option/option__CASE (happ (h4/Decode/decode__num (h4/combin/K T)) t_27) h4/option/NONE (happ _7 p))) ==> (!_0. (!p v2. happ (happ _0 p) v2 = happ (happ _1 v2) p) ==> (!p l. happ (h4/Decode/decode__num p) l = h4/list/list__CASE l h4/option/NONE (happ _0 p)))))))))))
% Assm [h4s_bools_RESu_u_FORALLu_u_THM]: !f P. h4/bool/RES__FORALL P f <=> (!x. h4/bool/IN x P ==> happ f x)
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_Encodes_liftu_u_produ_u_def]: !x p2 p1. happ (h4/Encode/lift__prod p1 p2) x <=> happ p1 (h4/pair/FST x) /\ happ p2 (h4/pair/SND x)
% Assm [h4s_lists_LENGTHu_u_APPEND]: !l2 l1. h4/list/LENGTH (h4/list/APPEND l1 l2) = h4/arithmetic/_2B (h4/list/LENGTH l1) (h4/list/LENGTH l2)
% Assm [h4s_arithmetics_EQu_u_MONOu_u_ADDu_u_EQ]: !p n m. h4/arithmetic/_2B m p = h4/arithmetic/_2B n p <=> m = n
% Assm [h4s_Encodes_encodeu_u_produ_u_def]: !yb y xb x. happ (h4/Encode/encode__prod xb yb) (h4/pair/_2C x y) = h4/list/APPEND (happ xb x) (happ yb y)
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_Coders_produ_u_coderu_u_def]: !p2 p1 e2 e1 d2 d1. h4/Coder/prod__coder (h4/pair/_2C p1 (h4/pair/_2C e1 d1)) (h4/pair/_2C p2 (h4/pair/_2C e2 d2)) = h4/pair/_2C (h4/Encode/lift__prod p1 p2) (h4/pair/_2C (h4/Encode/encode__prod e1 e2) (h4/Decode/decode__prod (h4/Encode/lift__prod p1 p2) d1 d2))
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_RESu_u_EXISTSu_u_THM]: !f P. h4/bool/RES__EXISTS P f <=> (?x. h4/bool/IN x P /\ happ f x)
% Assm [h4s_Encodes_encodeu_u_sumu_u_defu_c1]: !yb y xb. happ (h4/Encode/encode__sum xb yb) (h4/sum/INR y) = h4/list/CONS F (happ yb y)
% Assm [h4s_Encodes_liftu_u_sumu_u_def]: !_1. (!p2 x2. happ (happ _1 p2) x2 <=> happ p2 x2) ==> (!_0. (!p1 x1. happ (happ _0 p1) x1 <=> happ p1 x1) ==> (!x p2 p1. happ (h4/Encode/lift__sum p1 p2) x <=> h4/sum/sum__CASE x (happ _0 p1) (happ _1 p2)))
% Assm [h4s_Encodes_encodeu_u_sumu_u_defu_c0]: !yb xb x. happ (h4/Encode/encode__sum xb yb) (h4/sum/INL x) = h4/list/CONS T (happ xb x)
% Assm [h4s_primu_u_recs_INVu_u_SUCu_u_EQ]: !n m. h4/num/SUC m = h4/num/SUC n <=> m = n
% Assm [h4s_sums_sumu_u_caseu_u_defu_c0]: !x f1 f. h4/sum/sum__CASE (h4/sum/INL x) f f1 = happ f x
% Assm [h4s_sums_sumu_u_caseu_u_defu_c1]: !y f1 f. h4/sum/sum__CASE (h4/sum/INR y) f f1 = happ f1 y
% Assm [h4s_sums_sumu_u_CASES]: !ss. (?x. ss = h4/sum/INL x) \/ (?y. ss = h4/sum/INR y)
% Assm [h4s_Coders_sumu_u_coderu_u_def]: !p2 p1 e2 e1 d2 d1. h4/Coder/sum__coder (h4/pair/_2C p1 (h4/pair/_2C e1 d1)) (h4/pair/_2C p2 (h4/pair/_2C e2 d2)) = h4/pair/_2C (h4/Encode/lift__sum p1 p2) (h4/pair/_2C (h4/Encode/encode__sum e1 e2) (h4/Decode/decode__sum (h4/Encode/lift__sum p1 p2) d1 d2))
% Assm [h4s_Decodes_decodeu_u_numu_u_total]: !_9. (!v t_270. happ (happ _9 v) t_270 = h4/option/SOME (h4/pair/_2C (h4/arithmetic/_2B (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) v) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) t_270)) ==> (!_8. (!v. happ _8 v = happ _9 v) ==> (!_7. (!v1. happ _7 v1 = h4/pair/pair__CASE v1 _8) ==> (!_6. (!v t_270. happ (happ _6 v) t_270 = h4/option/SOME (h4/pair/_2C (h4/arithmetic/_2B (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) v) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) t_270)) ==> (!_5. (!v. happ _5 v = happ _6 v) ==> (!_4. (!v1. happ _4 v1 = h4/pair/pair__CASE v1 _5) ==> (!_3. (!v6 t. happ (happ _3 v6) t = h4/bool/COND v6 (h4/option/SOME (h4/pair/_2C h4/num/0 t)) (h4/option/option__CASE (happ (h4/Decode/decode__num (h4/combin/K T)) t) h4/option/NONE _4)) ==> (!_2. (!v6. happ _2 v6 = happ _3 v6) ==> (!_1. (!v2 t_27. happ (happ _1 v2) t_27 = h4/bool/COND v2 (h4/list/list__CASE t_27 h4/option/NONE _2) (h4/option/option__CASE (happ (h4/Decode/decode__num (h4/combin/K T)) t_27) h4/option/NONE _7)) ==> (!_0. (!v2. happ _0 v2 = happ _1 v2) ==> (!l. happ (h4/Decode/decode__num (h4/combin/K T)) l = h4/list/list__CASE l h4/option/NONE _0))))))))))
% Assm [h4s_Decodes_decodeu_u_numu_u_def]: !p. h4/Decode/decode__num p = h4/Decode/enc2dec p h4/Encode/encode__num
% Assm [h4s_Decodes_wfu_u_decodeu_u_num]: !p. h4/Decode/wf__decoder p (h4/Decode/decode__num p)
% Assm [h4s_Decodes_dec2encu_u_enc2dec]: !x p e. h4/Encode/wf__encoder p e /\ happ p x ==> h4/Decode/dec2enc (h4/Decode/enc2dec p e) x = happ e x
% Assm [h4s_Decodes_decodeu_u_dec2encu_u_append]: !x t p d. h4/Decode/wf__decoder p d /\ happ p x ==> happ d (h4/list/APPEND (h4/Decode/dec2enc d x) t) = h4/option/SOME (h4/pair/_2C x t)
% Assm [h4s_Decodes_enc2decu_u_none]: !p l e. happ (h4/Decode/enc2dec p e) l = h4/option/NONE <=> (!x t. happ p x ==> ~(l = h4/list/APPEND (happ e x) t))
% Assm [h4s_Encodes_wfu_u_encodeu_u_num]: !p. h4/Encode/wf__encoder p h4/Encode/encode__num
% Assm [h4s_bools_EQu_u_EXT]: !g f. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_BOOLu_u_EQu_u_DISTINCTu_c0]: ~(T <=> F)
% Assm [h4s_Decodes_enc2decu_u_some]: !x t p l e. h4/Encode/wf__encoder p e ==> (happ (h4/Decode/enc2dec p e) l = h4/option/SOME (h4/pair/_2C x t) <=> happ p x /\ l = h4/list/APPEND (happ e x) t)
% Assm [h4s_Encodes_encodeu_u_numu_u_def]: !n. ?v. (v <=> n = h4/num/0) /\ happ h4/Encode/encode__num n = h4/bool/COND v (h4/list/CONS T (h4/list/CONS T h4/list/NIL)) (h4/bool/COND (h4/arithmetic/EVEN n) (h4/list/CONS F (happ h4/Encode/encode__num (h4/arithmetic/DIV (h4/arithmetic/_2D n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))))) (h4/list/CONS T (h4/list/CONS F (happ h4/Encode/encode__num (h4/arithmetic/DIV (h4/arithmetic/_2D n (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)))))))
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_lists_listu_u_nchotomy]: !l. l = h4/list/NIL \/ (?h t. l = h4/list/CONS h t)
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_lists_APPEND0u_c1]: !l2 l1 h. h4/list/APPEND (h4/list/CONS h l1) l2 = h4/list/CONS h (h4/list/APPEND l1 l2)
% Assm [h4s_lists_listu_u_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 [h4s_lists_APPEND0u_c0]: !l. h4/list/APPEND h4/list/NIL l = l
% Assm [h4s_lists_listu_u_caseu_u_defu_c1]: !v f a1 a0. h4/list/list__CASE (h4/list/CONS a0 a1) v f = happ (happ f a0) a1
% Assm [h4s_numerals_numeralu_u_addu_c1]: !n. h4/numeral/iZ (h4/arithmetic/_2B n h4/arithmetic/ZERO) = n
% 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_distribu_c21]: !n. h4/prim__rec/_3C h4/num/0 (h4/arithmetic/NUMERAL n) <=> h4/prim__rec/_3C h4/arithmetic/ZERO n
% Assm [h4s_numerals_numeralu_u_ltu_c1]: !n. h4/prim__rec/_3C h4/arithmetic/ZERO (h4/arithmetic/BIT2 n) <=> T
% 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_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_arithmetics_ODDu_u_EXISTS]: !n. h4/arithmetic/ODD n <=> (?m. n = h4/num/SUC (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) m))
% Assm [h4s_arithmetics_EVENu_u_ODD]: !n. h4/arithmetic/EVEN n <=> ~h4/arithmetic/ODD n
% Assm [h4s_lists_listu_u_caseu_u_defu_c0]: !v f. h4/list/list__CASE h4/list/NIL v f = v
% Assm [h4s_arithmetics_MULTu_u_DIV]: !q n. h4/prim__rec/_3C h4/num/0 n ==> h4/arithmetic/DIV (h4/arithmetic/_2A q n) n = q
% Assm [h4s_arithmetics_ADDu_u_SUB]: !c a. h4/arithmetic/_2D (h4/arithmetic/_2B a c) c = a
% Assm [h4s_arithmetics_MULTu_u_COMM]: !n m. h4/arithmetic/_2A m n = h4/arithmetic/_2A n m
% Assm [h4s_arithmetics_ADDu_u_ASSOC]: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2B n p) = h4/arithmetic/_2B (h4/arithmetic/_2B m n) p
% Assm [h4s_arithmetics_EVEN0u_c1]: !n. h4/arithmetic/EVEN (h4/num/SUC n) <=> ~h4/arithmetic/EVEN n
% 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_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_arithmetics_numu_u_CASES]: !m. m = h4/num/0 \/ (?n. m = h4/num/SUC n)
% Assm [h4s_options_optionu_u_caseu_u_defu_c1]: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = happ f x
% Goal: !p m. h4/EncodeVar/fixed__width m (h4/Coder/bnum__coder m p)
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_Q1148020,TV_Q1148016]: ![V_f, V_g]: (![V_x]: s(TV_Q1148016,happ(s(t_fun(TV_Q1148020,TV_Q1148016),V_f),s(TV_Q1148020,V_x))) = s(TV_Q1148016,happ(s(t_fun(TV_Q1148020,TV_Q1148016),V_g),s(TV_Q1148020,V_x))) => s(t_fun(TV_Q1148020,TV_Q1148016),V_f) = s(t_fun(TV_Q1148020,TV_Q1148016),V_g))).
fof(ah4s_Coders_bnumu_u_coderu_u_def, axiom, ![V_p, V_m]: s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),h4s_coders_bnumu_u_coder(s(t_h4s_nums_num,V_m),s(t_fun(t_h4s_nums_num,t_bool),V_p))) = s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),h4s_pairs_u_2c(s(t_fun(t_h4s_nums_num,t_bool),V_p),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),h4s_pairs_u_2c(s(t_fun(t_h4s_nums_num,t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_bnum(s(t_h4s_nums_num,V_m))),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_bnum(s(t_h4s_nums_num,V_m),s(t_fun(t_h4s_nums_num,t_bool),V_p)))))))).
fof(ah4s_Coders_wfu_u_coderu_u_bnum, axiom, ![V_p, V_m]: (p(s(t_bool,h4s_encodes_wfu_u_predu_u_bnum(s(t_h4s_nums_num,V_m),s(t_fun(t_h4s_nums_num,t_bool),V_p)))) => p(s(t_bool,h4s_coders_wfu_u_coder(s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),h4s_coders_bnumu_u_coder(s(t_h4s_nums_num,V_m),s(t_fun(t_h4s_nums_num,t_bool),V_p)))))))).
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_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_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_EncodeVars_fixedu_u_widthu_u_def, axiom, ![TV_u_27a]: ![V_n, V_c]: (p(s(t_bool,h4s_encodevars_fixedu_u_width(s(t_h4s_nums_num,V_n),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c)))) <=> ![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),h4s_coders_domain(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c))),s(TV_u_27a,V_x)))) => s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(t_bool),happ(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),h4s_coders_encoder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c))),s(TV_u_27a,V_x))))) = s(t_h4s_nums_num,V_n)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_Decodes_decodeu_u_bnumu_u_def, axiom, ![V_p, V_m]: s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_bnum(s(t_h4s_nums_num,V_m),s(t_fun(t_h4s_nums_num,t_bool),V_p))) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),h4s_decodes_enc2dec(s(t_fun(t_h4s_nums_num,t_bool),V_p),s(t_fun(t_h4s_nums_num,t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_bnum(s(t_h4s_nums_num,V_m)))))).
fof(ah4s_Encodes_wfu_u_predu_u_bnumu_u_def, axiom, ![V_p, V_m]: (p(s(t_bool,h4s_encodes_wfu_u_predu_u_bnum(s(t_h4s_nums_num,V_m),s(t_fun(t_h4s_nums_num,t_bool),V_p)))) <=> (p(s(t_bool,h4s_encodes_wfu_u_pred(s(t_fun(t_h4s_nums_num,t_bool),V_p)))) & ![V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_p),s(t_h4s_nums_num,V_x)))) => p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_x),s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m)))))))))).
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_Encodes_wfu_u_encodeu_u_bnum, axiom, ![V_p, V_m]: (p(s(t_bool,h4s_encodes_wfu_u_predu_u_bnum(s(t_h4s_nums_num,V_m),s(t_fun(t_h4s_nums_num,t_bool),V_p)))) => p(s(t_bool,h4s_encodes_wfu_u_encoder(s(t_fun(t_h4s_nums_num,t_bool),V_p),s(t_fun(t_h4s_nums_num,t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_bnum(s(t_h4s_nums_num,V_m)))))))).
fof(ah4s_Coders_wfu_u_coderu_u_def, axiom, ![TV_u_27a]: ![V_p, V_e, V_d]: (p(s(t_bool,h4s_coders_wfu_u_coder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_p),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool))))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),V_d)))))))) <=> (p(s(t_bool,h4s_encodes_wfu_u_pred(s(t_fun(TV_u_27a,t_bool),V_p)))) & (p(s(t_bool,h4s_encodes_wfu_u_encoder(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e)))) & s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),V_d) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),h4s_decodes_enc2dec(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e))))))).
fof(ah4s_EncodeVars_fixedu_u_widthu_u_prod, axiom, ![TV_u_27a,TV_u_27b]: ![V_n2, V_n1, V_c2, V_c1]: ((p(s(t_bool,h4s_encodevars_fixedu_u_width(s(t_h4s_nums_num,V_n1),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c1)))) & p(s(t_bool,h4s_encodevars_fixedu_u_width(s(t_h4s_nums_num,V_n2),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27b,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27b,t_h4s_lists_list(t_bool)))))),V_c2))))) => p(s(t_bool,h4s_encodevars_fixedu_u_width(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n1),s(t_h4s_nums_num,V_n2))),s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_lists_list(t_bool)))))),h4s_coders_produ_u_coder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c1),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27b,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27b,t_h4s_lists_list(t_bool)))))),V_c2)))))))).
fof(ah4s_EncodeVars_fixedu_u_widthu_u_exists, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_phi, V_c, V_w]: s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),t_fun(t_h4s_lists_list(t_bool),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),t_fun(t_h4s_lists_list(t_bool),t_bool))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_phi))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c))),s(t_h4s_lists_list(t_bool),V_w))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_phi),s(TV_u_27a,h4s_coders_decoder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c),s(t_h4s_lists_list(t_bool),V_w))))) => ![V_phi, V_n, V_c]: ((p(s(t_bool,h4s_coders_wfu_u_coder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c)))) & p(s(t_bool,h4s_encodevars_fixedu_u_width(s(t_h4s_nums_num,V_n),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c))))) => (?[V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),h4s_coders_domain(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c))),s(TV_u_27a,V_x)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_phi),s(TV_u_27a,V_x))))) <=> p(s(t_bool,h4s_bools_resu_u_exists(s(t_fun(t_h4s_lists_list(t_bool),t_bool),h4s_encodevars_ofu_u_length(s(t_h4s_nums_num,V_n))),s(t_fun(t_h4s_lists_list(t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),t_fun(t_h4s_lists_list(t_bool),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),t_fun(t_h4s_lists_list(t_bool),t_bool))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_phi))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c)))))))))).
fof(ah4s_EncodeVars_fixedu_u_widthu_u_univ, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_phi, V_c, V_w]: s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),t_fun(t_h4s_lists_list(t_bool),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),t_fun(t_h4s_lists_list(t_bool),t_bool))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_phi))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c))),s(t_h4s_lists_list(t_bool),V_w))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_phi),s(TV_u_27a,h4s_coders_decoder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c),s(t_h4s_lists_list(t_bool),V_w))))) => ![V_phi, V_n, V_c]: ((p(s(t_bool,h4s_coders_wfu_u_coder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c)))) & p(s(t_bool,h4s_encodevars_fixedu_u_width(s(t_h4s_nums_num,V_n),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c))))) => (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),h4s_coders_domain(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c))),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_phi),s(TV_u_27a,V_x))))) <=> p(s(t_bool,h4s_bools_resu_u_forall(s(t_fun(t_h4s_lists_list(t_bool),t_bool),h4s_encodevars_ofu_u_length(s(t_h4s_nums_num,V_n))),s(t_fun(t_h4s_lists_list(t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),t_fun(t_h4s_lists_list(t_bool),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),t_fun(t_h4s_lists_list(t_bool),t_bool))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_phi))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c)))))))))).
fof(ah4s_EncodeVars_fixedu_u_widthu_u_sum, axiom, ![TV_u_27a,TV_u_27b]: ![V_n, V_c2, V_c1]: ((p(s(t_bool,h4s_encodevars_fixedu_u_width(s(t_h4s_nums_num,V_n),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c1)))) & p(s(t_bool,h4s_encodevars_fixedu_u_width(s(t_h4s_nums_num,V_n),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27b,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27b,t_h4s_lists_list(t_bool)))))),V_c2))))) => p(s(t_bool,h4s_encodevars_fixedu_u_width(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_pairs_prod(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),t_h4s_pairs_prod(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_lists_list(t_bool)))))),h4s_coders_sumu_u_coder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c1),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27b,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27b,t_h4s_lists_list(t_bool)))))),V_c2)))))))).
fof(ah4s_EncodeVars_fixedu_u_widthu_u_bool, axiom, ![V_p]: p(s(t_bool,h4s_encodevars_fixedu_u_width(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_pairs_prod(t_fun(t_bool,t_bool),t_h4s_pairs_prod(t_fun(t_bool,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_bool,t_h4s_lists_list(t_bool)))))),h4s_coders_boolu_u_coder(s(t_fun(t_bool,t_bool),V_p))))))).
fof(ah4s_EncodeVars_fixedu_u_widthu_u_unit, axiom, ![V_p]: p(s(t_bool,h4s_encodevars_fixedu_u_width(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(t_fun(t_h4s_ones_one,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_ones_one,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_ones_one,t_h4s_lists_list(t_bool)))))),h4s_coders_unitu_u_coder(s(t_fun(t_h4s_ones_one,t_bool),V_p))))))).
fof(ah4s_Coders_encoderu_u_def, axiom, ![TV_u_27a]: ![V_p, V_e, V_d]: s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),h4s_coders_encoder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_p),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool))))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),V_d))))))) = s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e)).
fof(ah4s_Coders_domainu_u_def, axiom, ![TV_u_27a]: ![V_p, V_e, V_d]: s(t_fun(TV_u_27a,t_bool),h4s_coders_domain(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_p),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool))))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),V_d))))))) = s(t_fun(TV_u_27a,t_bool),V_p)).
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_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_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_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_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_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_dcu_u_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_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_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_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_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_ANDu_u_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_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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),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_Encodes_encodeu_u_unitu_u_def, axiom, ![V_v0]: s(t_h4s_lists_list(t_bool),happ(s(t_fun(t_h4s_ones_one,t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_unit),s(t_h4s_ones_one,V_v0))) = s(t_h4s_lists_list(t_bool),h4s_lists_nil)).
fof(ah4s_Coders_unitu_u_coderu_u_def, axiom, ![V_p]: s(t_h4s_pairs_prod(t_fun(t_h4s_ones_one,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_ones_one,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_ones_one,t_h4s_lists_list(t_bool)))))),h4s_coders_unitu_u_coder(s(t_fun(t_h4s_ones_one,t_bool),V_p))) = s(t_h4s_pairs_prod(t_fun(t_h4s_ones_one,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_ones_one,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_ones_one,t_h4s_lists_list(t_bool)))))),h4s_pairs_u_2c(s(t_fun(t_h4s_ones_one,t_bool),V_p),s(t_h4s_pairs_prod(t_fun(t_h4s_ones_one,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_ones_one,t_h4s_lists_list(t_bool))))),h4s_pairs_u_2c(s(t_fun(t_h4s_ones_one,t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_unit),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_ones_one,t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_unit(s(t_fun(t_h4s_ones_one,t_bool),V_p)))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![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))))) <=> ![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_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_Encodes_encodeu_u_boolu_u_def, axiom, ![V_x]: s(t_h4s_lists_list(t_bool),happ(s(t_fun(t_bool,t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_bool),s(t_bool,V_x))) = s(t_h4s_lists_list(t_bool),h4s_lists_cons(s(t_bool,V_x),s(t_h4s_lists_list(t_bool),h4s_lists_nil)))).
fof(ah4s_numerals_numeralu_u_distribu_c13, axiom, s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))).
fof(ah4s_Coders_boolu_u_coderu_u_def, axiom, ![V_p]: s(t_h4s_pairs_prod(t_fun(t_bool,t_bool),t_h4s_pairs_prod(t_fun(t_bool,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_bool,t_h4s_lists_list(t_bool)))))),h4s_coders_boolu_u_coder(s(t_fun(t_bool,t_bool),V_p))) = s(t_h4s_pairs_prod(t_fun(t_bool,t_bool),t_h4s_pairs_prod(t_fun(t_bool,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_bool,t_h4s_lists_list(t_bool)))))),h4s_pairs_u_2c(s(t_fun(t_bool,t_bool),V_p),s(t_h4s_pairs_prod(t_fun(t_bool,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_bool,t_h4s_lists_list(t_bool))))),h4s_pairs_u_2c(s(t_fun(t_bool,t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_bool),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_bool,t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_bool(s(t_fun(t_bool,t_bool),V_p)))))))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(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_Coders_wfu_u_coder0, axiom, ![TV_u_27a]: ![V_c]: (p(s(t_bool,h4s_coders_wfu_u_coder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c)))) => ![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),h4s_coders_domain(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c))),s(TV_u_27a,V_x)))) => s(TV_u_27a,h4s_coders_decoder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c),s(t_h4s_lists_list(t_bool),happ(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),h4s_coders_encoder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c))),s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x)))).
fof(ah4s_Coders_wfu_u_coderu_u_closed, axiom, ![TV_u_27a]: ![V_c]: (p(s(t_bool,h4s_coders_wfu_u_coder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c)))) => ![V_l]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),h4s_coders_domain(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c))),s(TV_u_27a,h4s_coders_decoder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c),s(t_h4s_lists_list(t_bool),V_l)))))))).
fof(ah4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,V_P)) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (p(s(t_bool,V_P)) | ![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_EncodeVars_ofu_u_lengthu_u_def, axiom, ![TV_u_27a]: ![V_n, V_l]: (p(s(t_bool,h4s_bools_in(s(t_h4s_lists_list(TV_u_27a),V_l),s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),h4s_encodevars_ofu_u_length(s(t_h4s_nums_num,V_n)))))) <=> s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l))) = 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_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_bools_SKOLEMu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (![V_x]: ?[V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) <=> ?[V_f]: ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))))))).
fof(ah4s_Decodes_decodeu_u_num0, axiom, ![V_uu_9]: (![V_p, V_v, V_tu_270]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),V_uu_9),s(t_fun(t_h4s_nums_num,t_bool),V_p))),s(t_h4s_nums_num,V_v))),s(t_h4s_lists_list(t_bool),V_tu_270))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_bools_cond(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_p),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_v))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_some(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_v))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_lists_list(t_bool),V_tu_270))))),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_none))) => ![V_uu_8]: (![V_p, V_v]: s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),V_uu_8),s(t_fun(t_h4s_nums_num,t_bool),V_p))),s(t_h4s_nums_num,V_v))) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),V_uu_9),s(t_fun(t_h4s_nums_num,t_bool),V_p))),s(t_h4s_nums_num,V_v))) => ![V_uu_7]: (![V_p, V_v1]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_7),s(t_fun(t_h4s_nums_num,t_bool),V_p))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),V_v1))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),V_v1),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),V_uu_8),s(t_fun(t_h4s_nums_num,t_bool),V_p))))) => ![V_uu_6]: (![V_p, V_v, V_tu_270]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),V_uu_6),s(t_fun(t_h4s_nums_num,t_bool),V_p))),s(t_h4s_nums_num,V_v))),s(t_h4s_lists_list(t_bool),V_tu_270))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_bools_cond(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_p),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_v))),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_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_some(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_v))),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_lists_list(t_bool),V_tu_270))))),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_none))) => ![V_uu_5]: (![V_p, V_v]: s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),V_uu_5),s(t_fun(t_h4s_nums_num,t_bool),V_p))),s(t_h4s_nums_num,V_v))) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),V_uu_6),s(t_fun(t_h4s_nums_num,t_bool),V_p))),s(t_h4s_nums_num,V_v))) => ![V_uu_4]: (![V_p, V_v1]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_4),s(t_fun(t_h4s_nums_num,t_bool),V_p))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),V_v1))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),V_v1),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),V_uu_5),s(t_fun(t_h4s_nums_num,t_bool),V_p))))) => ![V_uu_3]: (![V_v6, V_p, V_t]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),happ(s(t_fun(t_bool,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),V_uu_3),s(t_bool,V_v6))),s(t_fun(t_h4s_nums_num,t_bool),V_p))),s(t_h4s_lists_list(t_bool),V_t))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_bools_cond(s(t_bool,V_v6),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_bools_cond(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_p),s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_some(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_lists_list(t_bool),V_t))))),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_none))),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_optionu_u_case(s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_num(s(t_fun(t_h4s_nums_num,t_bool),h4s_combins_k(s(t_bool,t))))),s(t_h4s_lists_list(t_bool),V_t))),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_none),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_4),s(t_fun(t_h4s_nums_num,t_bool),V_p))))))) => ![V_uu_2]: (![V_p, V_v6]: s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),V_uu_2),s(t_fun(t_h4s_nums_num,t_bool),V_p))),s(t_bool,V_v6))) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),happ(s(t_fun(t_bool,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),V_uu_3),s(t_bool,V_v6))),s(t_fun(t_h4s_nums_num,t_bool),V_p))) => ![V_uu_1]: (![V_v2, V_p, V_tu_27]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),happ(s(t_fun(t_bool,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),V_uu_1),s(t_bool,V_v2))),s(t_fun(t_h4s_nums_num,t_bool),V_p))),s(t_h4s_lists_list(t_bool),V_tu_27))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_bools_cond(s(t_bool,V_v2),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_lists_listu_u_case(s(t_h4s_lists_list(t_bool),V_tu_27),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_none),s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),V_uu_2),s(t_fun(t_h4s_nums_num,t_bool),V_p))))),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_optionu_u_case(s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_num(s(t_fun(t_h4s_nums_num,t_bool),h4s_combins_k(s(t_bool,t))))),s(t_h4s_lists_list(t_bool),V_tu_27))),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_none),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_7),s(t_fun(t_h4s_nums_num,t_bool),V_p))))))) => ![V_uu_0]: (![V_p, V_v2]: s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),V_uu_0),s(t_fun(t_h4s_nums_num,t_bool),V_p))),s(t_bool,V_v2))) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),happ(s(t_fun(t_bool,t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),V_uu_1),s(t_bool,V_v2))),s(t_fun(t_h4s_nums_num,t_bool),V_p))) => ![V_p, V_l]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_num(s(t_fun(t_h4s_nums_num,t_bool),V_p))),s(t_h4s_lists_list(t_bool),V_l))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_lists_listu_u_case(s(t_h4s_lists_list(t_bool),V_l),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_none),s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),happ(s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),V_uu_0),s(t_fun(t_h4s_nums_num,t_bool),V_p)))))))))))))))).
fof(ah4s_bools_RESu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_f, V_P]: (p(s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_f)))) <=> ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c0, 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_Encodes_liftu_u_produ_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_p2, V_p1]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_encodes_liftu_u_prod(s(t_fun(TV_u_27a,t_bool),V_p1),s(t_fun(TV_u_27b,t_bool),V_p2))),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x)))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p1),s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x)))))) & p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_p2),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))))))))).
fof(ah4s_lists_LENGTHu_u_APPEND, axiom, ![TV_u_27a]: ![V_l2, V_l1]: s(t_h4s_nums_num,h4s_lists_length(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_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l1))),s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l2)))))).
fof(ah4s_arithmetics_EQu_u_MONOu_u_ADDu_u_EQ, 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,V_p))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_Encodes_encodeu_u_produ_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_yb, V_y, V_xb, V_x]: s(t_h4s_lists_list(t_bool),happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_prod(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_xb),s(t_fun(TV_u_27b,t_h4s_lists_list(t_bool)),V_yb))),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_lists_list(t_bool),h4s_lists_append(s(t_h4s_lists_list(t_bool),happ(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_xb),s(TV_u_27a,V_x))),s(t_h4s_lists_list(t_bool),happ(s(t_fun(TV_u_27b,t_h4s_lists_list(t_bool)),V_yb),s(TV_u_27b,V_y)))))).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27b,V_y)).
fof(ah4s_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_Coders_produ_u_coderu_u_def, axiom, ![TV_u_27c,TV_u_27d,TV_u_27a,TV_u_27b]: ![V_p2, V_p1, V_e2, V_e1, V_d2, V_d1]: s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_lists_list(t_bool)))))),h4s_coders_produ_u_coder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27c,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_p1),s(t_h4s_pairs_prod(t_fun(TV_u_27c,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool))))),h4s_pairs_u_2c(s(t_fun(TV_u_27c,t_h4s_lists_list(t_bool)),V_e1),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),V_d1))))),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27d,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27b,t_h4s_lists_list(t_bool)))))),h4s_pairs_u_2c(s(t_fun(TV_u_27b,t_bool),V_p2),s(t_h4s_pairs_prod(t_fun(TV_u_27d,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27b,t_h4s_lists_list(t_bool))))),h4s_pairs_u_2c(s(t_fun(TV_u_27d,t_h4s_lists_list(t_bool)),V_e2),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27b,t_h4s_lists_list(t_bool)))),V_d2))))))) = s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_lists_list(t_bool)))))),h4s_pairs_u_2c(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_encodes_liftu_u_prod(s(t_fun(TV_u_27a,t_bool),V_p1),s(t_fun(TV_u_27b,t_bool),V_p2))),s(t_h4s_pairs_prod(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_lists_list(t_bool))))),h4s_pairs_u_2c(s(t_fun(t_h4s_pairs_prod(TV_u_27c,TV_u_27d),t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_prod(s(t_fun(TV_u_27c,t_h4s_lists_list(t_bool)),V_e1),s(t_fun(TV_u_27d,t_h4s_lists_list(t_bool)),V_e2))),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_prod(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),h4s_encodes_liftu_u_prod(s(t_fun(TV_u_27a,t_bool),V_p1),s(t_fun(TV_u_27b,t_bool),V_p2))),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),V_d1),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27b,t_h4s_lists_list(t_bool)))),V_d2)))))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_bools_RESu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_f, V_P]: (p(s(t_bool,h4s_bools_resu_u_exists(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_f)))) <=> ?[V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))))).
fof(ah4s_Encodes_encodeu_u_sumu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_yb, V_y, V_xb]: s(t_h4s_lists_list(t_bool),happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_sum(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_xb),s(t_fun(TV_u_27b,t_h4s_lists_list(t_bool)),V_yb))),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_y))))) = s(t_h4s_lists_list(t_bool),h4s_lists_cons(s(t_bool,f),s(t_h4s_lists_list(t_bool),happ(s(t_fun(TV_u_27b,t_h4s_lists_list(t_bool)),V_yb),s(TV_u_27b,V_y)))))).
fof(ah4s_Encodes_liftu_u_sumu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_p2, V_x2]: s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_p2))),s(TV_u_27b,V_x2))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_p2),s(TV_u_27b,V_x2))) => ![V_uu_0]: (![V_p1, V_x1]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_p1))),s(TV_u_27a,V_x1))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p1),s(TV_u_27a,V_x1))) => ![V_x, V_p2, V_p1]: s(t_bool,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),h4s_encodes_liftu_u_sum(s(t_fun(TV_u_27a,t_bool),V_p1),s(t_fun(TV_u_27b,t_bool),V_p2))),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_x))) = s(t_bool,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_x),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_p1))),s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27b,t_bool)),V_uu_1),s(t_fun(TV_u_27b,t_bool),V_p2)))))))).
fof(ah4s_Encodes_encodeu_u_sumu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_yb, V_xb, V_x]: s(t_h4s_lists_list(t_bool),happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_sum(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_xb),s(t_fun(TV_u_27b,t_h4s_lists_list(t_bool)),V_yb))),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_x))))) = s(t_h4s_lists_list(t_bool),h4s_lists_cons(s(t_bool,t),s(t_h4s_lists_list(t_bool),happ(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_xb),s(TV_u_27a,V_x)))))).
fof(ah4s_primu_u_recs_INVu_u_SUCu_u_EQ, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_sums_sumu_u_caseu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_x, V_f1, V_f]: s(TV_u_27c,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_x))),s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27c),V_f1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_sums_sumu_u_caseu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_y, V_f1, V_f]: s(TV_u_27c,h4s_sums_sumu_u_case(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_y))),s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27c),V_f1))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),V_f1),s(TV_u_27b,V_y)))).
fof(ah4s_sums_sumu_u_CASES, axiom, ![TV_u_27a,TV_u_27b]: ![V_ss]: (?[V_x]: s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_ss) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_x))) | ?[V_y]: s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_ss) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_y))))).
fof(ah4s_Coders_sumu_u_coderu_u_def, axiom, ![TV_u_27c,TV_u_27d,TV_u_27a,TV_u_27b]: ![V_p2, V_p1, V_e2, V_e1, V_d2, V_d1]: s(t_h4s_pairs_prod(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),t_h4s_pairs_prod(t_fun(t_h4s_sums_sum(TV_u_27c,TV_u_27d),t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_lists_list(t_bool)))))),h4s_coders_sumu_u_coder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27c,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_p1),s(t_h4s_pairs_prod(t_fun(TV_u_27c,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool))))),h4s_pairs_u_2c(s(t_fun(TV_u_27c,t_h4s_lists_list(t_bool)),V_e1),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),V_d1))))),s(t_h4s_pairs_prod(t_fun(TV_u_27b,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27d,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27b,t_h4s_lists_list(t_bool)))))),h4s_pairs_u_2c(s(t_fun(TV_u_27b,t_bool),V_p2),s(t_h4s_pairs_prod(t_fun(TV_u_27d,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27b,t_h4s_lists_list(t_bool))))),h4s_pairs_u_2c(s(t_fun(TV_u_27d,t_h4s_lists_list(t_bool)),V_e2),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27b,t_h4s_lists_list(t_bool)))),V_d2))))))) = s(t_h4s_pairs_prod(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),t_h4s_pairs_prod(t_fun(t_h4s_sums_sum(TV_u_27c,TV_u_27d),t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_lists_list(t_bool)))))),h4s_pairs_u_2c(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),h4s_encodes_liftu_u_sum(s(t_fun(TV_u_27a,t_bool),V_p1),s(t_fun(TV_u_27b,t_bool),V_p2))),s(t_h4s_pairs_prod(t_fun(t_h4s_sums_sum(TV_u_27c,TV_u_27d),t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_lists_list(t_bool))))),h4s_pairs_u_2c(s(t_fun(t_h4s_sums_sum(TV_u_27c,TV_u_27d),t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_sum(s(t_fun(TV_u_27c,t_h4s_lists_list(t_bool)),V_e1),s(t_fun(TV_u_27d,t_h4s_lists_list(t_bool)),V_e2))),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_sum(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),h4s_encodes_liftu_u_sum(s(t_fun(TV_u_27a,t_bool),V_p1),s(t_fun(TV_u_27b,t_bool),V_p2))),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),V_d1),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27b,t_h4s_lists_list(t_bool)))),V_d2)))))))).
fof(ah4s_Decodes_decodeu_u_numu_u_total, axiom, ![V_uu_9]: (![V_v, V_tu_270]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_9),s(t_h4s_nums_num,V_v))),s(t_h4s_lists_list(t_bool),V_tu_270))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_some(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_v))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_lists_list(t_bool),V_tu_270))))) => ![V_uu_8]: (![V_v]: s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_8),s(t_h4s_nums_num,V_v))) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_9),s(t_h4s_nums_num,V_v))) => ![V_uu_7]: (![V_v1]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),V_uu_7),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),V_v1))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),V_v1),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_8))) => ![V_uu_6]: (![V_v, V_tu_270]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_6),s(t_h4s_nums_num,V_v))),s(t_h4s_lists_list(t_bool),V_tu_270))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_some(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_v))),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_lists_list(t_bool),V_tu_270))))) => ![V_uu_5]: (![V_v]: s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_5),s(t_h4s_nums_num,V_v))) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_6),s(t_h4s_nums_num,V_v))) => ![V_uu_4]: (![V_v1]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),V_uu_4),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),V_v1))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),V_v1),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_5))) => ![V_uu_3]: (![V_v6, V_t]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_3),s(t_bool,V_v6))),s(t_h4s_lists_list(t_bool),V_t))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_bools_cond(s(t_bool,V_v6),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_some(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_lists_list(t_bool),V_t))))),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_optionu_u_case(s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_num(s(t_fun(t_h4s_nums_num,t_bool),h4s_combins_k(s(t_bool,t))))),s(t_h4s_lists_list(t_bool),V_t))),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_none),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),V_uu_4))))) => ![V_uu_2]: (![V_v6]: s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_2),s(t_bool,V_v6))) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_3),s(t_bool,V_v6))) => ![V_uu_1]: (![V_v2, V_tu_27]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_1),s(t_bool,V_v2))),s(t_h4s_lists_list(t_bool),V_tu_27))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_bools_cond(s(t_bool,V_v2),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_lists_listu_u_case(s(t_h4s_lists_list(t_bool),V_tu_27),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_none),s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_2))),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_optionu_u_case(s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_num(s(t_fun(t_h4s_nums_num,t_bool),h4s_combins_k(s(t_bool,t))))),s(t_h4s_lists_list(t_bool),V_tu_27))),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_none),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),V_uu_7))))) => ![V_uu_0]: (![V_v2]: s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_0),s(t_bool,V_v2))) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_1),s(t_bool,V_v2))) => ![V_l]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_num(s(t_fun(t_h4s_nums_num,t_bool),h4s_combins_k(s(t_bool,t))))),s(t_h4s_lists_list(t_bool),V_l))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_lists_listu_u_case(s(t_h4s_lists_list(t_bool),V_l),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_none),s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_0)))))))))))))).
fof(ah4s_Decodes_decodeu_u_numu_u_def, axiom, ![V_p]: s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_num(s(t_fun(t_h4s_nums_num,t_bool),V_p))) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),h4s_decodes_enc2dec(s(t_fun(t_h4s_nums_num,t_bool),V_p),s(t_fun(t_h4s_nums_num,t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_num)))).
fof(ah4s_Decodes_wfu_u_decodeu_u_num, axiom, ![V_p]: p(s(t_bool,h4s_decodes_wfu_u_decoder(s(t_fun(t_h4s_nums_num,t_bool),V_p),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_num(s(t_fun(t_h4s_nums_num,t_bool),V_p))))))).
fof(ah4s_Decodes_dec2encu_u_enc2dec, axiom, ![TV_u_27a]: ![V_x, V_p, V_e]: ((p(s(t_bool,h4s_encodes_wfu_u_encoder(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p),s(TV_u_27a,V_x))))) => s(t_h4s_lists_list(t_bool),h4s_decodes_dec2enc(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),h4s_decodes_enc2dec(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e))),s(TV_u_27a,V_x))) = s(t_h4s_lists_list(t_bool),happ(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e),s(TV_u_27a,V_x))))).
fof(ah4s_Decodes_decodeu_u_dec2encu_u_append, axiom, ![TV_u_27a]: ![V_x, V_t, V_p, V_d]: ((p(s(t_bool,h4s_decodes_wfu_u_decoder(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),V_d)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p),s(TV_u_27a,V_x))))) => s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),V_d),s(t_h4s_lists_list(t_bool),h4s_lists_append(s(t_h4s_lists_list(t_bool),h4s_decodes_dec2enc(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),V_d),s(TV_u_27a,V_x))),s(t_h4s_lists_list(t_bool),V_t))))) = s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool))),h4s_options_some(s(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_h4s_lists_list(t_bool),V_t))))))).
fof(ah4s_Decodes_enc2decu_u_none, axiom, ![TV_u_27a]: ![V_p, V_l, V_e]: (s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),h4s_decodes_enc2dec(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e))),s(t_h4s_lists_list(t_bool),V_l))) = s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool))),h4s_options_none) <=> ![V_x, V_t]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p),s(TV_u_27a,V_x)))) => ~ (s(t_h4s_lists_list(t_bool),V_l) = s(t_h4s_lists_list(t_bool),h4s_lists_append(s(t_h4s_lists_list(t_bool),happ(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e),s(TV_u_27a,V_x))),s(t_h4s_lists_list(t_bool),V_t))))))).
fof(ah4s_Encodes_wfu_u_encodeu_u_num, axiom, ![V_p]: p(s(t_bool,h4s_encodes_wfu_u_encoder(s(t_fun(t_h4s_nums_num,t_bool),V_p),s(t_fun(t_h4s_nums_num,t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_num))))).
fof(ah4s_bools_EQu_u_EXT, axiom, ![TV_u_27a,TV_u_27b]: ![V_g, V_f]: (![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))) => s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g))).
fof(ah4s_bools_BOOLu_u_EQu_u_DISTINCTu_c0, axiom, ~ (s(t_bool,t) = s(t_bool,f))).
fof(ah4s_Decodes_enc2decu_u_some, axiom, ![TV_u_27a]: ![V_x, V_t, V_p, V_l, V_e]: (p(s(t_bool,h4s_encodes_wfu_u_encoder(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e)))) => (s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),h4s_decodes_enc2dec(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e))),s(t_h4s_lists_list(t_bool),V_l))) = s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool))),h4s_options_some(s(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_h4s_lists_list(t_bool),V_t))))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p),s(TV_u_27a,V_x)))) & s(t_h4s_lists_list(t_bool),V_l) = s(t_h4s_lists_list(t_bool),h4s_lists_append(s(t_h4s_lists_list(t_bool),happ(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e),s(TV_u_27a,V_x))),s(t_h4s_lists_list(t_bool),V_t))))))).
fof(ah4s_Encodes_encodeu_u_numu_u_def, axiom, ![V_n]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)) & s(t_h4s_lists_list(t_bool),happ(s(t_fun(t_h4s_nums_num,t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_num),s(t_h4s_nums_num,V_n))) = s(t_h4s_lists_list(t_bool),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_lists_list(t_bool),h4s_lists_cons(s(t_bool,t),s(t_h4s_lists_list(t_bool),h4s_lists_cons(s(t_bool,t),s(t_h4s_lists_list(t_bool),h4s_lists_nil))))),s(t_h4s_lists_list(t_bool),h4s_bools_cond(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_n))),s(t_h4s_lists_list(t_bool),h4s_lists_cons(s(t_bool,f),s(t_h4s_lists_list(t_bool),happ(s(t_fun(t_h4s_nums_num,t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_num),s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))))),s(t_h4s_lists_list(t_bool),h4s_lists_cons(s(t_bool,t),s(t_h4s_lists_list(t_bool),h4s_lists_cons(s(t_bool,f),s(t_h4s_lists_list(t_bool),happ(s(t_fun(t_h4s_nums_num,t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_num),s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))))))))))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
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),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t))))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_lists_APPEND0u_c1, axiom, ![TV_u_27a]: ![V_l2, V_l1, V_h]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_l1))),s(t_h4s_lists_list(TV_u_27a),V_l2))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2)))))).
fof(ah4s_lists_listu_u_11, axiom, ![TV_u_27a]: ![V_a1u_27, V_a1, V_a0u_27, V_a0]: (s(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),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_APPEND0u_c0, axiom, ![TV_u_27a]: ![V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),V_l)).
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),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_numerals_numeralu_u_addu_c1, 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,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero))))) = s(t_h4s_nums_num,V_n)).
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_distribu_c21, axiom, ![V_n]: s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))))) = s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,V_n)))).
fof(ah4s_numerals_numeralu_u_ltu_c1, axiom, ![V_n]: s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,V_n))))) = s(t_bool,t)).
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_bools_NOTu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![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))))) <=> ?[V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_arithmetics_ODDu_u_EXISTS, axiom, ![V_n]: (p(s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,V_n)))) <=> ?[V_m]: s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m))))))).
fof(ah4s_arithmetics_EVENu_u_ODD, axiom, ![V_n]: (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_n)))) <=> ~ (p(s(t_bool,h4s_arithmetics_odd(s(t_h4s_nums_num,V_n))))))).
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_arithmetics_MULTu_u_DIV, axiom, ![V_q, V_n]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n)))) => s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_q),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_q))).
fof(ah4s_arithmetics_ADDu_u_SUB, axiom, ![V_c, V_a]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_a),s(t_h4s_nums_num,V_c))),s(t_h4s_nums_num,V_c))) = s(t_h4s_nums_num,V_a)).
fof(ah4s_arithmetics_MULTu_u_COMM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_ADDu_u_ASSOC, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_EVEN0u_c1, axiom, ![V_n]: (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))) <=> ~ (p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,V_n))))))).
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_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_arithmetics_numu_u_CASES, axiom, ![V_m]: (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0) | ?[V_n]: s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))).
fof(ah4s_options_optionu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
fof(ch4s_EncodeVars_fixedu_u_widthu_u_bnum, conjecture, ![V_p, V_m]: p(s(t_bool,h4s_encodevars_fixedu_u_width(s(t_h4s_nums_num,V_m),s(t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_nums_num,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),h4s_coders_bnumu_u_coder(s(t_h4s_nums_num,V_m),s(t_fun(t_h4s_nums_num,t_bool),V_p))))))).
