%   ORIGINAL: h4/Coder/wf__coder__tree
% 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/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/Coder/tree__coder__def: !p e d. h4/Coder/tree__coder (h4/pair/_2C p (h4/pair/_2C e d)) = h4/pair/_2C (h4/Encode/lift__tree p) (h4/pair/_2C (h4/Encode/encode__tree e) (h4/Decode/decode__tree (h4/Encode/lift__tree p) d))
% Assm: h4/Coder/wf__coder__blist: !m c. h4/Coder/wf__coder c ==> h4/Coder/wf__coder (h4/Coder/blist__coder m c)
% Assm: h4/Coder/wf__coder__sum: !c2 c1. h4/Coder/wf__coder c1 /\ h4/Coder/wf__coder c2 ==> h4/Coder/wf__coder (h4/Coder/sum__coder c1 c2)
% Assm: h4/Coder/wf__coder__list: !c. h4/Coder/wf__coder c ==> h4/Coder/wf__coder (h4/Coder/list__coder c)
% 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__option: !c. h4/Coder/wf__coder c ==> h4/Coder/wf__coder (h4/Coder/option__coder c)
% Assm: h4/Coder/wf__coder__prod: !c2 c1. h4/Coder/wf__coder c1 /\ h4/Coder/wf__coder c2 ==> h4/Coder/wf__coder (h4/Coder/prod__coder c1 c2)
% Assm: h4/Coder/wf__coder__closed: !c. h4/Coder/wf__coder c ==> (!l. h4/Coder/domain c (h4/Coder/decoder c l))
% 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/pair/ABS__PAIR__THM: !x. ?q r. x = h4/pair/_2C q r
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/Coder/wf__coder__op: !p f e. (?x. p x) /\ h4/Encode/wf__encoder p e /\ (!x. p x ==> e x = f x) ==> h4/Coder/wf__coder (h4/pair/_2C p (h4/pair/_2C e (h4/Decode/enc2dec p f)))
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/Encode/wf__pred__def: !p. h4/Encode/wf__pred p <=> (?x. p x)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% 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/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/SKOLEM__THM: !P. (!x. ?y. P x y) <=> (?f. !x. P x (f x))
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% 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/Coder/decoder__def: !p e d. h4/Coder/decoder (h4/pair/_2C p (h4/pair/_2C e d)) = h4/Coder/decode p d
% Assm: h4/Coder/domain__def: !p e d. h4/Coder/domain (h4/pair/_2C p (h4/pair/_2C e d)) = p
% Assm: h4/Coder/decode__encode: !x p e. h4/Encode/wf__encoder p e /\ p x ==> h4/Coder/decode p (h4/Decode/enc2dec p e) (e x) = x
% Assm: h4/Coder/encoder__def: !p e d. h4/Coder/encoder (h4/pair/_2C p (h4/pair/_2C e d)) = e
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% 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/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/option/option__case__def_c0: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm: h4/option/option__case__def_c1: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = f x
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/list/EVERY__DEF_c0: !P. h4/list/EVERY P h4/list/NIL <=> T
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/option/option__nchotomy: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% 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/Coder/decode__def: !p l d. h4/Coder/decode p d l = h4/option/option__CASE (d l) (h4/min/_40 (\x. p x)) (\v. h4/pair/pair__CASE v (\x v2. x))
% 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/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/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/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/Decode/decode__prod__def: !p d2 d1. h4/Decode/decode__prod p d1 d2 = h4/Decode/enc2dec p (h4/Encode/encode__prod (h4/Decode/dec2enc d1) (h4/Decode/dec2enc d2))
% Assm: h4/Encode/wf__encode__prod: !p2 p1 e2 e1. h4/Encode/wf__encoder p1 e1 /\ h4/Encode/wf__encoder p2 e2 ==> h4/Encode/wf__encoder (h4/Encode/lift__prod p1 p2) (h4/Encode/encode__prod e1 e2)
% 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/Decode/decode__sum__def: !p d2 d1. h4/Decode/decode__sum p d1 d2 = h4/Decode/enc2dec p (h4/Encode/encode__sum (h4/Decode/dec2enc d1) (h4/Decode/dec2enc d2))
% 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/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/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/wf__encode__sum: !p2 p1 e2 e1. h4/Encode/wf__encoder p1 e1 /\ h4/Encode/wf__encoder p2 e2 ==> h4/Encode/wf__encoder (h4/Encode/lift__sum p1 p2) (h4/Encode/encode__sum e1 e2)
% 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/sum/sum__case__def_c1: !y f1 f. h4/sum/sum__CASE (h4/sum/INR y) f f1 = f1 y
% 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__CASES: !ss. (?x. ss = h4/sum/INL x) \/ (?y. ss = h4/sum/INR y)
% Assm: h4/Decode/decode__list__def: !p d. h4/Decode/decode__list p d = h4/Decode/enc2dec p (h4/Encode/encode__list (h4/Decode/dec2enc d))
% Assm: h4/Encode/wf__encode__list: !p e. h4/Encode/wf__encoder p e ==> h4/Encode/wf__encoder (h4/list/EVERY p) (h4/Encode/encode__list e)
% Assm: h4/Encode/encode__list__def_c0: !xb. h4/Encode/encode__list xb h4/list/NIL = h4/list/CONS F h4/list/NIL
% Assm: h4/Encode/encode__list__def_c1: !xs xb x. h4/Encode/encode__list xb (h4/list/CONS x xs) = h4/list/CONS T (h4/list/APPEND (xb x) (h4/Encode/encode__list xb xs))
% Assm: h4/list/EVERY__DEF_c1: !t h P. h4/list/EVERY P (h4/list/CONS h t) <=> P h /\ h4/list/EVERY P t
% Assm: h4/list/list__induction: !P. P h4/list/NIL /\ (!t. P t ==> (!h. P (h4/list/CONS h t))) ==> (!l. P l)
% Assm: h4/Coder/list__coder__def: !p e d. h4/Coder/list__coder (h4/pair/_2C p (h4/pair/_2C e d)) = h4/pair/_2C (h4/list/EVERY p) (h4/pair/_2C (h4/Encode/encode__list e) (h4/Decode/decode__list (h4/list/EVERY p) d))
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/Encode/wf__encode__blist: !p m e. h4/Encode/wf__encoder p e ==> h4/Encode/wf__encoder (h4/Encode/lift__blist m p) (h4/Encode/encode__blist m e)
% Assm: h4/Encode/encode__blist__def_c1: !m l e. h4/Encode/encode__blist (h4/num/SUC m) e l = h4/list/APPEND (e (h4/list/HD l)) (h4/Encode/encode__blist m e (h4/list/TL l))
% Assm: h4/Encode/encode__blist__def_c0: !l e. h4/Encode/encode__blist h4/num/0 e l = h4/list/NIL
% Assm: h4/Encode/lift__blist__suc: !t p n h. h4/Encode/lift__blist (h4/num/SUC n) p (h4/list/CONS h t) <=> p h /\ h4/Encode/lift__blist n p t
% Assm: h4/Encode/lift__blist__def: !x p m. h4/Encode/lift__blist m p x <=> h4/list/EVERY p x /\ h4/list/LENGTH x = m
% Assm: h4/Decode/decode__blist__def: !p m d. h4/Decode/decode__blist p m d = h4/Decode/enc2dec p (h4/Encode/encode__blist m (h4/Decode/dec2enc d))
% Assm: h4/list/list__nchotomy: !l. l = h4/list/NIL \/ (?h t. l = h4/list/CONS h t)
% Assm: h4/list/LENGTH0_c0: h4/list/LENGTH h4/list/NIL = h4/num/0
% Assm: h4/list/HD0: !t h. h4/list/HD (h4/list/CONS h t) = h
% Assm: h4/list/TL0: !t h. h4/list/TL (h4/list/CONS h t) = t
% Assm: h4/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Assm: h4/Coder/blist__coder__def: !p m e d. h4/Coder/blist__coder m (h4/pair/_2C p (h4/pair/_2C e d)) = h4/pair/_2C (h4/Encode/lift__blist m p) (h4/pair/_2C (h4/Encode/encode__blist m e) (h4/Decode/decode__blist (h4/Encode/lift__blist m p) m d))
% Assm: h4/num/NOT__SUC: !n. ~(h4/num/SUC n = h4/num/0)
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/Decode/decode__option__def: !p d. h4/Decode/decode__option p d = h4/Decode/enc2dec p (h4/Encode/encode__option (h4/Decode/dec2enc d))
% Assm: h4/Encode/wf__encode__option: !p e. h4/Encode/wf__encoder p e ==> h4/Encode/wf__encoder (h4/Encode/lift__option p) (h4/Encode/encode__option e)
% Assm: h4/Encode/lift__option__def: !x p. h4/Encode/lift__option p x <=> h4/option/option__CASE x T (\y. p y)
% Assm: h4/Encode/encode__option__def_c1: !xb x. h4/Encode/encode__option xb (h4/option/SOME x) = h4/list/CONS T (xb x)
% Assm: h4/Encode/encode__option__def_c0: !xb. h4/Encode/encode__option xb h4/option/NONE = h4/list/CONS F h4/list/NIL
% Assm: h4/option/option__induction: !P. P h4/option/NONE /\ (!a. P (h4/option/SOME a)) ==> (!x. P x)
% Assm: h4/Coder/option__coder__def: !p e d. h4/Coder/option__coder (h4/pair/_2C p (h4/pair/_2C e d)) = h4/pair/_2C (h4/Encode/lift__option p) (h4/pair/_2C (h4/Encode/encode__option e) (h4/Decode/decode__option (h4/Encode/lift__option p) d))
% Assm: h4/Decode/enc2dec__some__alt: !x p l e. h4/Encode/wf__encoder p e ==> (h4/Decode/enc2dec p e l = h4/option/SOME x <=> p (h4/pair/FST x) /\ l = h4/list/APPEND (e (h4/pair/FST x)) (h4/pair/SND x))
% 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/bool/EQ__EXT: !g f. (!x. f x = g x) ==> f = g
% Assm: h4/Encode/wf__encoder__eq: !p f e. h4/Encode/wf__encoder p e /\ (!x. p x ==> e x = f x) ==> h4/Encode/wf__encoder p f
% Assm: h4/Encode/wf__encoder__alt: !p e. h4/Encode/wf__encoder p e <=> (!x y. p x /\ p y /\ h4/Encode/biprefix (e x) (e y) ==> x = y)
% Assm: h4/Encode/biprefix__refl: !x. h4/Encode/biprefix x x
% Assm: h4/Encode/biprefix__append: !d c b a. h4/Encode/biprefix (h4/list/APPEND a b) (h4/list/APPEND c d) ==> h4/Encode/biprefix a c
% Assm: h4/list/APPEND__11_c1: !l3 l2 l1. h4/list/APPEND l2 l1 = h4/list/APPEND l3 l1 <=> l2 = l3
% Assm: h4/bool/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/list/APPEND__11_c0: !l3 l2 l1. h4/list/APPEND l1 l2 = h4/list/APPEND l1 l3 <=> l2 = l3
% Assm: h4/bool/OR__DEF: $or = (\t1 t2. !t. (t1 ==> t) ==> (t2 ==> t) ==> t)
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/pair/CLOSED__PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/bool/RIGHT__OR__EXISTS__THM: !Q P. P \/ (?x. Q x) <=> (?x. P \/ Q x)
% Goal: !c. h4/Coder/wf__coder c ==> h4/Coder/wf__coder (h4/Coder/tree__coder c)
%   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_pairs_UNCURRYu_u_DEF]: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_Coders_treeu_u_coderu_u_def]: !p e d. h4/Coder/tree__coder (h4/pair/_2C p (h4/pair/_2C e d)) = h4/pair/_2C (h4/Encode/lift__tree p) (h4/pair/_2C (h4/Encode/encode__tree e) (h4/Decode/decode__tree (h4/Encode/lift__tree p) d))
% Assm [h4s_Coders_wfu_u_coderu_u_blist]: !m c. h4/Coder/wf__coder c ==> h4/Coder/wf__coder (h4/Coder/blist__coder m c)
% Assm [h4s_Coders_wfu_u_coderu_u_sum]: !c2 c1. h4/Coder/wf__coder c1 /\ h4/Coder/wf__coder c2 ==> h4/Coder/wf__coder (h4/Coder/sum__coder c1 c2)
% Assm [h4s_Coders_wfu_u_coderu_u_list]: !c. h4/Coder/wf__coder c ==> h4/Coder/wf__coder (h4/Coder/list__coder c)
% Assm [h4s_Coders_wfu_u_coder0]: !c. h4/Coder/wf__coder c ==> (!x. happ (h4/Coder/domain c) x ==> happ (h4/Coder/decoder c) (happ (h4/Coder/encoder c) x) = x)
% Assm [h4s_Coders_wfu_u_coderu_u_option]: !c. h4/Coder/wf__coder c ==> h4/Coder/wf__coder (h4/Coder/option__coder c)
% Assm [h4s_Coders_wfu_u_coderu_u_prod]: !c2 c1. h4/Coder/wf__coder c1 /\ h4/Coder/wf__coder c2 ==> h4/Coder/wf__coder (h4/Coder/prod__coder c1 c2)
% Assm [h4s_Coders_wfu_u_coderu_u_closed]: !c. h4/Coder/wf__coder c ==> (!l. happ (h4/Coder/domain c) (happ (h4/Coder/decoder c) l))
% 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_pairs_ABSu_u_PAIRu_u_THM]: !x. ?q r. x = h4/pair/_2C q r
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_Coders_wfu_u_coderu_u_op]: !p f e. (?x. happ p x) /\ h4/Encode/wf__encoder p e /\ (!x. happ p x ==> happ e x = happ f x) ==> h4/Coder/wf__coder (h4/pair/_2C p (h4/pair/_2C e (h4/Decode/enc2dec p f)))
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_Encodes_wfu_u_predu_u_def]: !p. h4/Encode/wf__pred p <=> (?x. happ p x)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% 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_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% 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_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_Decodes_dec2encu_u_enc2dec]: !x p e. h4/Encode/wf__encoder p e /\ happ p x ==> happ (h4/Decode/dec2enc (h4/Decode/enc2dec p e)) x = happ e x
% Assm [h4s_Coders_decoderu_u_def]: !p e d. h4/Coder/decoder (h4/pair/_2C p (h4/pair/_2C e d)) = h4/Coder/decode p d
% Assm [h4s_Coders_domainu_u_def]: !p e d. h4/Coder/domain (h4/pair/_2C p (h4/pair/_2C e d)) = p
% Assm [h4s_Coders_decodeu_u_encode]: !x p e. h4/Encode/wf__encoder p e /\ happ p x ==> happ (h4/Coder/decode p (h4/Decode/enc2dec p e)) (happ e x) = x
% Assm [h4s_Coders_encoderu_u_def]: !p e d. h4/Coder/encoder (h4/pair/_2C p (h4/pair/_2C e d)) = e
% 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_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_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_options_optionu_u_caseu_u_defu_c0]: !v f. h4/option/option__CASE h4/option/NONE v f = v
% 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
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_lists_EVERYu_u_DEFu_c0]: !P. happ (h4/list/EVERY P) h4/list/NIL <=> T
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_options_optionu_u_nchotomy]: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% 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_Coders_decodeu_u_def]: !_3. (!x v2. happ (happ _3 x) v2 = x) ==> (!_2. (!x. happ _2 x = happ _3 x) ==> (!_1. (!v. happ _1 v = h4/pair/pair__CASE v _2) ==> (!_0. (!p x. happ (happ _0 p) x <=> happ p x) ==> (!p l d. happ (h4/Coder/decode p d) l = h4/option/option__CASE (happ d l) (h4/min/_40 (happ _0 p)) _1))))
% 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_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_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_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_Decodes_decodeu_u_produ_u_def]: !p d2 d1. h4/Decode/decode__prod p d1 d2 = h4/Decode/enc2dec p (h4/Encode/encode__prod (h4/Decode/dec2enc d1) (h4/Decode/dec2enc d2))
% Assm [h4s_Encodes_wfu_u_encodeu_u_prod]: !p2 p1 e2 e1. h4/Encode/wf__encoder p1 e1 /\ h4/Encode/wf__encoder p2 e2 ==> h4/Encode/wf__encoder (h4/Encode/lift__prod p1 p2) (h4/Encode/encode__prod e1 e2)
% 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_Decodes_decodeu_u_sumu_u_def]: !p d2 d1. h4/Decode/decode__sum p d1 d2 = h4/Decode/enc2dec p (h4/Encode/encode__sum (h4/Decode/dec2enc d1) (h4/Decode/dec2enc d2))
% 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_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_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_wfu_u_encodeu_u_sum]: !p2 p1 e2 e1. h4/Encode/wf__encoder p1 e1 /\ h4/Encode/wf__encoder p2 e2 ==> h4/Encode/wf__encoder (h4/Encode/lift__sum p1 p2) (h4/Encode/encode__sum e1 e2)
% 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_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_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_CASES]: !ss. (?x. ss = h4/sum/INL x) \/ (?y. ss = h4/sum/INR y)
% Assm [h4s_Decodes_decodeu_u_listu_u_def]: !p d. h4/Decode/decode__list p d = h4/Decode/enc2dec p (h4/Encode/encode__list (h4/Decode/dec2enc d))
% Assm [h4s_Encodes_wfu_u_encodeu_u_list]: !p e. h4/Encode/wf__encoder p e ==> h4/Encode/wf__encoder (h4/list/EVERY p) (h4/Encode/encode__list e)
% Assm [h4s_Encodes_encodeu_u_listu_u_defu_c0]: !xb. happ (h4/Encode/encode__list xb) h4/list/NIL = h4/list/CONS F h4/list/NIL
% Assm [h4s_Encodes_encodeu_u_listu_u_defu_c1]: !xs xb x. happ (h4/Encode/encode__list xb) (h4/list/CONS x xs) = h4/list/CONS T (h4/list/APPEND (happ xb x) (happ (h4/Encode/encode__list xb) xs))
% Assm [h4s_lists_EVERYu_u_DEFu_c1]: !t h P. happ (h4/list/EVERY P) (h4/list/CONS h t) <=> happ P h /\ happ (h4/list/EVERY P) t
% Assm [h4s_lists_listu_u_induction]: !P. happ P h4/list/NIL /\ (!t. happ P t ==> (!h. happ P (h4/list/CONS h t))) ==> (!l. happ P l)
% Assm [h4s_Coders_listu_u_coderu_u_def]: !p e d. h4/Coder/list__coder (h4/pair/_2C p (h4/pair/_2C e d)) = h4/pair/_2C (h4/list/EVERY p) (h4/pair/_2C (h4/Encode/encode__list e) (h4/Decode/decode__list (h4/list/EVERY p) d))
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_Encodes_wfu_u_encodeu_u_blist]: !p m e. h4/Encode/wf__encoder p e ==> h4/Encode/wf__encoder (h4/Encode/lift__blist m p) (h4/Encode/encode__blist m e)
% Assm [h4s_Encodes_encodeu_u_blistu_u_defu_c1]: !m l e. happ (h4/Encode/encode__blist (h4/num/SUC m) e) l = h4/list/APPEND (happ e (h4/list/HD l)) (happ (h4/Encode/encode__blist m e) (h4/list/TL l))
% Assm [h4s_Encodes_encodeu_u_blistu_u_defu_c0]: !l e. happ (h4/Encode/encode__blist h4/num/0 e) l = h4/list/NIL
% Assm [h4s_Encodes_liftu_u_blistu_u_suc]: !t p n h. happ (h4/Encode/lift__blist (h4/num/SUC n) p) (h4/list/CONS h t) <=> happ p h /\ happ (h4/Encode/lift__blist n p) t
% Assm [h4s_Encodes_liftu_u_blistu_u_def]: !x p m. happ (h4/Encode/lift__blist m p) x <=> happ (h4/list/EVERY p) x /\ h4/list/LENGTH x = m
% Assm [h4s_Decodes_decodeu_u_blistu_u_def]: !p m d. h4/Decode/decode__blist p m d = h4/Decode/enc2dec p (h4/Encode/encode__blist m (h4/Decode/dec2enc d))
% Assm [h4s_lists_listu_u_nchotomy]: !l. l = h4/list/NIL \/ (?h t. l = h4/list/CONS h t)
% Assm [h4s_lists_LENGTH0u_c0]: h4/list/LENGTH h4/list/NIL = h4/num/0
% Assm [h4s_lists_HD0]: !t h. h4/list/HD (h4/list/CONS h t) = h
% Assm [h4s_lists_TL0]: !t h. h4/list/TL (h4/list/CONS h t) = t
% Assm [h4s_nums_INDUCTION]: !P. happ P h4/num/0 /\ (!n. happ P n ==> happ P (h4/num/SUC n)) ==> (!n. happ P n)
% Assm [h4s_Coders_blistu_u_coderu_u_def]: !p m e d. h4/Coder/blist__coder m (h4/pair/_2C p (h4/pair/_2C e d)) = h4/pair/_2C (h4/Encode/lift__blist m p) (h4/pair/_2C (h4/Encode/encode__blist m e) (h4/Decode/decode__blist (h4/Encode/lift__blist m p) m d))
% Assm [h4s_nums_NOTu_u_SUC]: !n. ~(h4/num/SUC n = h4/num/0)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_Decodes_decodeu_u_optionu_u_def]: !p d. h4/Decode/decode__option p d = h4/Decode/enc2dec p (h4/Encode/encode__option (h4/Decode/dec2enc d))
% Assm [h4s_Encodes_wfu_u_encodeu_u_option]: !p e. h4/Encode/wf__encoder p e ==> h4/Encode/wf__encoder (h4/Encode/lift__option p) (h4/Encode/encode__option e)
% Assm [h4s_Encodes_liftu_u_optionu_u_def]: !_0. (!p y. happ (happ _0 p) y <=> happ p y) ==> (!x p. happ (h4/Encode/lift__option p) x <=> h4/option/option__CASE x T (happ _0 p))
% Assm [h4s_Encodes_encodeu_u_optionu_u_defu_c1]: !xb x. happ (h4/Encode/encode__option xb) (h4/option/SOME x) = h4/list/CONS T (happ xb x)
% Assm [h4s_Encodes_encodeu_u_optionu_u_defu_c0]: !xb. happ (h4/Encode/encode__option xb) h4/option/NONE = h4/list/CONS F h4/list/NIL
% Assm [h4s_options_optionu_u_induction]: !P. happ P h4/option/NONE /\ (!a. happ P (h4/option/SOME a)) ==> (!x. happ P x)
% Assm [h4s_Coders_optionu_u_coderu_u_def]: !p e d. h4/Coder/option__coder (h4/pair/_2C p (h4/pair/_2C e d)) = h4/pair/_2C (h4/Encode/lift__option p) (h4/pair/_2C (h4/Encode/encode__option e) (h4/Decode/decode__option (h4/Encode/lift__option p) d))
% Assm [h4s_Decodes_enc2decu_u_someu_u_alt]: !x p l e. h4/Encode/wf__encoder p e ==> (happ (h4/Decode/enc2dec p e) l = h4/option/SOME x <=> happ p (h4/pair/FST x) /\ l = h4/list/APPEND (happ e (h4/pair/FST x)) (h4/pair/SND x))
% 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_bools_EQu_u_EXT]: !g f. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_Encodes_wfu_u_encoderu_u_eq]: !p f e. h4/Encode/wf__encoder p e /\ (!x. happ p x ==> happ e x = happ f x) ==> h4/Encode/wf__encoder p f
% Assm [h4s_Encodes_wfu_u_encoderu_u_alt]: !p e. h4/Encode/wf__encoder p e <=> (!x y. happ p x /\ happ p y /\ h4/Encode/biprefix (happ e x) (happ e y) ==> x = y)
% Assm [h4s_Encodes_biprefixu_u_refl]: !x. h4/Encode/biprefix x x
% Assm [h4s_Encodes_biprefixu_u_append]: !d c b a. h4/Encode/biprefix (h4/list/APPEND a b) (h4/list/APPEND c d) ==> h4/Encode/biprefix a c
% Assm [h4s_lists_APPENDu_u_11u_c1]: !l3 l2 l1. h4/list/APPEND l2 l1 = h4/list/APPEND l3 l1 <=> l2 = l3
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_lists_APPENDu_u_11u_c0]: !l3 l2 l1. h4/list/APPEND l1 l2 = h4/list/APPEND l1 l3 <=> l2 = l3
% Assm [h4s_bools_ORu_u_DEF]: !x x'. $or x x' <=> (!t. (x ==> t) ==> (x' ==> t) ==> t)
% Assm [h4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% Assm [h4s_pairs_CLOSEDu_u_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM]: !Q P. P \/ (?x. happ Q x) <=> (?x. P \/ happ Q x)
% Goal: !c. h4/Coder/wf__coder c ==> h4/Coder/wf__coder (h4/Coder/tree__coder c)
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_Q1092583,TV_Q1092579]: ![V_f, V_g]: (![V_x]: s(TV_Q1092579,happ(s(t_fun(TV_Q1092583,TV_Q1092579),V_f),s(TV_Q1092583,V_x))) = s(TV_Q1092579,happ(s(t_fun(TV_Q1092583,TV_Q1092579),V_g),s(TV_Q1092583,V_x))) => s(t_fun(TV_Q1092583,TV_Q1092579),V_f) = s(t_fun(TV_Q1092583,TV_Q1092579),V_g))).
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_Coders_treeu_u_coderu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_p, V_e, V_d]: s(t_h4s_pairs_prod(t_fun(t_h4s_Encodes_tree(TV_u_27a),t_bool),t_h4s_pairs_prod(t_fun(t_h4s_Encodes_tree(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_Encodes_tree(TV_u_27a),t_h4s_lists_list(t_bool)))))),h4s_coders_treeu_u_coder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,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_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_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_27a,t_h4s_lists_list(t_bool))))),h4s_pairs_u_2c(s(t_fun(TV_u_27b,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_h4s_pairs_prod(t_fun(t_h4s_Encodes_tree(TV_u_27a),t_bool),t_h4s_pairs_prod(t_fun(t_h4s_Encodes_tree(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_Encodes_tree(TV_u_27a),t_h4s_lists_list(t_bool)))))),h4s_pairs_u_2c(s(t_fun(t_h4s_Encodes_tree(TV_u_27a),t_bool),h4s_encodes_liftu_u_tree(s(t_fun(TV_u_27a,t_bool),V_p))),s(t_h4s_pairs_prod(t_fun(t_h4s_Encodes_tree(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_Encodes_tree(TV_u_27a),t_h4s_lists_list(t_bool))))),h4s_pairs_u_2c(s(t_fun(t_h4s_Encodes_tree(TV_u_27b),t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_tree(s(t_fun(TV_u_27b,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(t_h4s_Encodes_tree(TV_u_27a),t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_tree(s(t_fun(t_h4s_Encodes_tree(TV_u_27a),t_bool),h4s_encodes_liftu_u_tree(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)))))))).
fof(ah4s_Coders_wfu_u_coderu_u_blist, axiom, ![TV_u_27a]: ![V_m, 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_coders_wfu_u_coder(s(t_h4s_pairs_prod(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),t_h4s_pairs_prod(t_fun(t_h4s_lists_list(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(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(t_bool)))))),h4s_coders_blistu_u_coder(s(t_h4s_nums_num,V_m),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_Coders_wfu_u_coderu_u_sum, axiom, ![TV_u_27a,TV_u_27b]: ![V_c2, V_c1]: ((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_c1)))) & p(s(t_bool,h4s_coders_wfu_u_coder(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_coders_wfu_u_coder(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_Coders_wfu_u_coderu_u_list, 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)))) => p(s(t_bool,h4s_coders_wfu_u_coder(s(t_h4s_pairs_prod(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),t_h4s_pairs_prod(t_fun(t_h4s_lists_list(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(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(t_bool)))))),h4s_coders_listu_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)))))))).
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,happ(s(t_fun(t_h4s_lists_list(t_bool),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_option, 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)))) => p(s(t_bool,h4s_coders_wfu_u_coder(s(t_h4s_pairs_prod(t_fun(t_h4s_options_option(TV_u_27a),t_bool),t_h4s_pairs_prod(t_fun(t_h4s_options_option(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(t_h4s_options_option(TV_u_27a),t_h4s_lists_list(t_bool)))))),h4s_coders_optionu_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)))))))).
fof(ah4s_Coders_wfu_u_coderu_u_prod, axiom, ![TV_u_27a,TV_u_27b]: ![V_c2, V_c1]: ((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_c1)))) & p(s(t_bool,h4s_coders_wfu_u_coder(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_coders_wfu_u_coder(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_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,happ(s(t_fun(t_h4s_lists_list(t_bool),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_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_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_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_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_Coders_wfu_u_coderu_u_op, axiom, ![TV_u_27a]: ![V_p, V_f, V_e]: ((?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p),s(TV_u_27a,V_x)))) & (p(s(t_bool,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)))) & ![V_x]: (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),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),happ(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_f),s(TV_u_27a,V_x)))))) => 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)))),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_f)))))))))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_Encodes_wfu_u_predu_u_def, axiom, ![TV_u_27a]: ![V_p]: (p(s(t_bool,h4s_encodes_wfu_u_pred(s(t_fun(TV_u_27a,t_bool),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)))))).
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_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_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_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_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_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_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_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
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_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_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_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_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_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_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_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_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_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_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),happ(s(t_fun(TV_u_27a,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_Coders_decoderu_u_def, axiom, ![TV_u_27a]: ![V_p, V_e, V_d]: s(t_fun(t_h4s_lists_list(t_bool),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)))))),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(t_h4s_lists_list(t_bool),TV_u_27a),h4s_coders_decode(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)))).
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_Coders_decodeu_u_encode, 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(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_bool),TV_u_27a),h4s_coders_decode(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)))),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),happ(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e),s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x))).
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_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_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_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_options_optionu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,V_v)).
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(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_lists_EVERYu_u_DEFu_c0, axiom, ![TV_u_27a]: ![V_P]: s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_bool,t)).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_options_optionu_u_nchotomy, axiom, ![TV_u_27a]: ![V_opt]: (s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) | ?[V_x]: s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_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_Coders_decodeu_u_def, axiom, ![TV_u_27a]: ![V_uu_3]: (![V_x, V_v2]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_bool),TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_bool),TV_u_27a)),V_uu_3),s(TV_u_27a,V_x))),s(t_h4s_lists_list(t_bool),V_v2))) = s(TV_u_27a,V_x) => ![V_uu_2]: (![V_x]: s(t_fun(t_h4s_lists_list(t_bool),TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_bool),TV_u_27a)),V_uu_2),s(TV_u_27a,V_x))) = s(t_fun(t_h4s_lists_list(t_bool),TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_bool),TV_u_27a)),V_uu_3),s(TV_u_27a,V_x))) => ![V_uu_1]: (![V_v]: s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)),TV_u_27a),V_uu_1),s(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)),V_v))) = s(TV_u_27a,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)),V_v),s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_bool),TV_u_27a)),V_uu_2))) => ![V_uu_0]: (![V_p, V_x]: 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_p))),s(TV_u_27a,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p),s(TV_u_27a,V_x))) => ![V_p, V_l, V_d]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_bool),TV_u_27a),h4s_coders_decode(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))),s(t_h4s_lists_list(t_bool),V_l))) = s(TV_u_27a,h4s_options_optionu_u_case(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),V_l))),s(TV_u_27a,h4s_mins_u_40(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_p))))),s(t_fun(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)),TV_u_27a),V_uu_1)))))))).
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_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27b,V_y)).
fof(ah4s_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_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_Decodes_decodeu_u_produ_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_d2, V_d1]: 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),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_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))) = 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_enc2dec(s(t_fun(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),t_bool),V_p),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)),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_d1))),s(t_fun(TV_u_27b,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_27b,t_h4s_lists_list(t_bool)))),V_d2)))))))).
fof(ah4s_Encodes_wfu_u_encodeu_u_prod, axiom, ![TV_u_27a,TV_u_27b]: ![V_p2, V_p1, V_e2, V_e1]: ((p(s(t_bool,h4s_encodes_wfu_u_encoder(s(t_fun(TV_u_27a,t_bool),V_p1),s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e1)))) & p(s(t_bool,h4s_encodes_wfu_u_encoder(s(t_fun(TV_u_27b,t_bool),V_p2),s(t_fun(TV_u_27b,t_h4s_lists_list(t_bool)),V_e2))))) => p(s(t_bool,h4s_encodes_wfu_u_encoder(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_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_e1),s(t_fun(TV_u_27b,t_h4s_lists_list(t_bool)),V_e2)))))))).
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_Decodes_decodeu_u_sumu_u_def, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_d2, V_d1]: 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),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_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))) = 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_enc2dec(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),V_p),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)),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_d1))),s(t_fun(TV_u_27b,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_27b,t_h4s_lists_list(t_bool)))),V_d2)))))))).
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_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_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_wfu_u_encodeu_u_sum, axiom, ![TV_u_27a,TV_u_27b]: ![V_p2, V_p1, V_e2, V_e1]: ((p(s(t_bool,h4s_encodes_wfu_u_encoder(s(t_fun(TV_u_27a,t_bool),V_p1),s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e1)))) & p(s(t_bool,h4s_encodes_wfu_u_encoder(s(t_fun(TV_u_27b,t_bool),V_p2),s(t_fun(TV_u_27b,t_h4s_lists_list(t_bool)),V_e2))))) => p(s(t_bool,h4s_encodes_wfu_u_encoder(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_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_e1),s(t_fun(TV_u_27b,t_h4s_lists_list(t_bool)),V_e2)))))))).
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_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_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_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_Decodes_decodeu_u_listu_u_def, axiom, ![TV_u_27a]: ![V_p, V_d]: s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_list(s(t_fun(t_h4s_lists_list(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))) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(t_bool)))),h4s_decodes_enc2dec(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_p),s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_list(s(t_fun(TV_u_27a,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)))))))).
fof(ah4s_Encodes_wfu_u_encodeu_u_list, axiom, ![TV_u_27a]: ![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,h4s_encodes_wfu_u_encoder(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_p))),s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_list(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e)))))))).
fof(ah4s_Encodes_encodeu_u_listu_u_defu_c0, axiom, ![TV_u_27a]: ![V_xb]: s(t_h4s_lists_list(t_bool),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_list(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_xb))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_lists_list(t_bool),h4s_lists_cons(s(t_bool,f),s(t_h4s_lists_list(t_bool),h4s_lists_nil)))).
fof(ah4s_Encodes_encodeu_u_listu_u_defu_c1, axiom, ![TV_u_27a]: ![V_xs, V_xb, V_x]: s(t_h4s_lists_list(t_bool),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_list(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_xb))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_x),s(t_h4s_lists_list(TV_u_27a),V_xs))))) = s(t_h4s_lists_list(t_bool),h4s_lists_cons(s(t_bool,t),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(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_list(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_xb))),s(t_h4s_lists_list(TV_u_27a),V_xs)))))))).
fof(ah4s_lists_EVERYu_u_DEFu_c1, axiom, ![TV_u_27a]: ![V_t, V_h, V_P]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t)))))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_h)))) & p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P))),s(t_h4s_lists_list(TV_u_27a),V_t))))))).
fof(ah4s_lists_listu_u_induction, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))) & ![V_t]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t)))) => ![V_h]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t)))))))) => ![V_l]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_Coders_listu_u_coderu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_p, V_e, V_d]: s(t_h4s_pairs_prod(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),t_h4s_pairs_prod(t_fun(t_h4s_lists_list(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_lists_list(TV_u_27a),t_h4s_lists_list(t_bool)))))),h4s_coders_listu_u_coder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,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_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_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_27a,t_h4s_lists_list(t_bool))))),h4s_pairs_u_2c(s(t_fun(TV_u_27b,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_h4s_pairs_prod(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),t_h4s_pairs_prod(t_fun(t_h4s_lists_list(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_lists_list(TV_u_27a),t_h4s_lists_list(t_bool)))))),h4s_pairs_u_2c(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_p))),s(t_h4s_pairs_prod(t_fun(t_h4s_lists_list(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_lists_list(TV_u_27a),t_h4s_lists_list(t_bool))))),h4s_pairs_u_2c(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_list(s(t_fun(TV_u_27b,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(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_list(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),h4s_lists_every(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)))))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_Encodes_wfu_u_encodeu_u_blist, axiom, ![TV_u_27a]: ![V_p, V_m, 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,h4s_encodes_wfu_u_encoder(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),h4s_encodes_liftu_u_blist(s(t_h4s_nums_num,V_m),s(t_fun(TV_u_27a,t_bool),V_p))),s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_blist(s(t_h4s_nums_num,V_m),s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e)))))))).
fof(ah4s_Encodes_encodeu_u_blistu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_m, V_l, V_e]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)),h4s_encodes_encodeu_u_blist(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_e))),s(t_h4s_lists_list(TV_u_27b),V_l))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_e),s(TV_u_27b,h4s_lists_hd(s(t_h4s_lists_list(TV_u_27b),V_l))))),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)),h4s_encodes_encodeu_u_blist(s(t_h4s_nums_num,V_m),s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_e))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_tl(s(t_h4s_lists_list(TV_u_27b),V_l)))))))).
fof(ah4s_Encodes_encodeu_u_blistu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_l, V_e]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27a)),h4s_encodes_encodeu_u_blist(s(t_h4s_nums_num,h4s_nums_0),s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_e))),s(t_h4s_lists_list(TV_u_27b),V_l))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)).
fof(ah4s_Encodes_liftu_u_blistu_u_suc, axiom, ![TV_u_27a]: ![V_t, V_p, V_n, V_h]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),h4s_encodes_liftu_u_blist(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_fun(TV_u_27a,t_bool),V_p))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t)))))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p),s(TV_u_27a,V_h)))) & p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),h4s_encodes_liftu_u_blist(s(t_h4s_nums_num,V_n),s(t_fun(TV_u_27a,t_bool),V_p))),s(t_h4s_lists_list(TV_u_27a),V_t))))))).
fof(ah4s_Encodes_liftu_u_blistu_u_def, axiom, ![TV_u_27a]: ![V_x, V_p, V_m]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),h4s_encodes_liftu_u_blist(s(t_h4s_nums_num,V_m),s(t_fun(TV_u_27a,t_bool),V_p))),s(t_h4s_lists_list(TV_u_27a),V_x)))) <=> (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_p))),s(t_h4s_lists_list(TV_u_27a),V_x)))) & s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_x))) = s(t_h4s_nums_num,V_m)))).
fof(ah4s_Decodes_decodeu_u_blistu_u_def, axiom, ![TV_u_27a]: ![V_p, V_m, V_d]: s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_blist(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_p),s(t_h4s_nums_num,V_m),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(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(t_bool)))),h4s_decodes_enc2dec(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_p),s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_blist(s(t_h4s_nums_num,V_m),s(t_fun(TV_u_27a,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)))))))).
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_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_lists_HD0, axiom, ![TV_u_27a]: ![V_t, V_h]: s(TV_u_27a,h4s_lists_hd(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(TV_u_27a,V_h)).
fof(ah4s_lists_TL0, axiom, ![TV_u_27a]: ![V_t, V_h]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_tl(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t))))) = s(t_h4s_lists_list(TV_u_27a),V_t)).
fof(ah4s_nums_INDUCTION, axiom, ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_0)))) & ![V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))) => ![V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_Coders_blistu_u_coderu_u_def, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_p, V_m, V_e, V_d]: s(t_h4s_pairs_prod(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),t_h4s_pairs_prod(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27c)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(t_bool)))))),h4s_coders_blistu_u_coder(s(t_h4s_nums_num,V_m),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27c)),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_27b,t_h4s_lists_list(TV_u_27c)),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_27b,t_h4s_lists_list(TV_u_27c)),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_h4s_pairs_prod(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),t_h4s_pairs_prod(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27c)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(t_bool)))))),h4s_pairs_u_2c(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),h4s_encodes_liftu_u_blist(s(t_h4s_nums_num,V_m),s(t_fun(TV_u_27a,t_bool),V_p))),s(t_h4s_pairs_prod(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27c)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(t_bool))))),h4s_pairs_u_2c(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27c)),h4s_encodes_encodeu_u_blist(s(t_h4s_nums_num,V_m),s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27c)),V_e))),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_blist(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),h4s_encodes_liftu_u_blist(s(t_h4s_nums_num,V_m),s(t_fun(TV_u_27a,t_bool),V_p))),s(t_h4s_nums_num,V_m),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)))))))).
fof(ah4s_nums_NOTu_u_SUC, axiom, ![V_n]: ~ (s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,f))) <=> p(s(t_bool,f)))).
fof(ah4s_Decodes_decodeu_u_optionu_u_def, axiom, ![TV_u_27a]: ![V_p, V_d]: s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_options_option(TV_u_27a),t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_option(s(t_fun(t_h4s_options_option(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))) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_options_option(TV_u_27a),t_h4s_lists_list(t_bool)))),h4s_decodes_enc2dec(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),V_p),s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_option(s(t_fun(TV_u_27a,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)))))))).
fof(ah4s_Encodes_wfu_u_encodeu_u_option, axiom, ![TV_u_27a]: ![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,h4s_encodes_wfu_u_encoder(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),h4s_encodes_liftu_u_option(s(t_fun(TV_u_27a,t_bool),V_p))),s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_option(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e)))))))).
fof(ah4s_Encodes_liftu_u_optionu_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_p, V_y]: 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_p))),s(TV_u_27a,V_y))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p),s(TV_u_27a,V_y))) => ![V_x, V_p]: s(t_bool,happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),h4s_encodes_liftu_u_option(s(t_fun(TV_u_27a,t_bool),V_p))),s(t_h4s_options_option(TV_u_27a),V_x))) = s(t_bool,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),V_x),s(t_bool,t),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_p))))))).
fof(ah4s_Encodes_encodeu_u_optionu_u_defu_c1, axiom, ![TV_u_27a]: ![V_xb, V_x]: s(t_h4s_lists_list(t_bool),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_option(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_xb))),s(t_h4s_options_option(TV_u_27a),h4s_options_some(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_Encodes_encodeu_u_optionu_u_defu_c0, axiom, ![TV_u_27a]: ![V_xb]: s(t_h4s_lists_list(t_bool),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_option(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_xb))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_h4s_lists_list(t_bool),h4s_lists_cons(s(t_bool,f),s(t_h4s_lists_list(t_bool),h4s_lists_nil)))).
fof(ah4s_options_optionu_u_induction, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),V_P),s(t_h4s_options_option(TV_u_27a),h4s_options_none)))) & ![V_a]: p(s(t_bool,happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),V_P),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_a))))))) => ![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),V_P),s(t_h4s_options_option(TV_u_27a),V_x)))))).
fof(ah4s_Coders_optionu_u_coderu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_p, V_e, V_d]: s(t_h4s_pairs_prod(t_fun(t_h4s_options_option(TV_u_27a),t_bool),t_h4s_pairs_prod(t_fun(t_h4s_options_option(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_options_option(TV_u_27a),t_h4s_lists_list(t_bool)))))),h4s_coders_optionu_u_coder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,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_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_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_27a,t_h4s_lists_list(t_bool))))),h4s_pairs_u_2c(s(t_fun(TV_u_27b,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_h4s_pairs_prod(t_fun(t_h4s_options_option(TV_u_27a),t_bool),t_h4s_pairs_prod(t_fun(t_h4s_options_option(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_options_option(TV_u_27a),t_h4s_lists_list(t_bool)))))),h4s_pairs_u_2c(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),h4s_encodes_liftu_u_option(s(t_fun(TV_u_27a,t_bool),V_p))),s(t_h4s_pairs_prod(t_fun(t_h4s_options_option(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_options_option(TV_u_27a),t_h4s_lists_list(t_bool))))),h4s_pairs_u_2c(s(t_fun(t_h4s_options_option(TV_u_27b),t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_option(s(t_fun(TV_u_27b,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(t_h4s_options_option(TV_u_27a),t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_option(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),h4s_encodes_liftu_u_option(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)))))))).
fof(ah4s_Decodes_enc2decu_u_someu_u_alt, axiom, ![TV_u_27a]: ![V_x, 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)),V_x))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p),s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)),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,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)),V_x))))),s(t_h4s_lists_list(t_bool),h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)),V_x))))))))).
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_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_Encodes_wfu_u_encoderu_u_eq, axiom, ![TV_u_27a]: ![V_p, V_f, 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)))) & ![V_x]: (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),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),happ(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_f),s(TV_u_27a,V_x))))) => 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_f)))))).
fof(ah4s_Encodes_wfu_u_encoderu_u_alt, axiom, ![TV_u_27a]: ![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)))) <=> ![V_x, V_y]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p),s(TV_u_27a,V_x)))) & (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p),s(TV_u_27a,V_y)))) & p(s(t_bool,h4s_encodes_biprefix(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),happ(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e),s(TV_u_27a,V_y)))))))) => s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))).
fof(ah4s_Encodes_biprefixu_u_refl, axiom, ![TV_u_27a]: ![V_x]: p(s(t_bool,h4s_encodes_biprefix(s(t_h4s_lists_list(TV_u_27a),V_x),s(t_h4s_lists_list(TV_u_27a),V_x))))).
fof(ah4s_Encodes_biprefixu_u_append, axiom, ![TV_u_27a]: ![V_d, V_c, V_b, V_a]: (p(s(t_bool,h4s_encodes_biprefix(s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_a),s(t_h4s_lists_list(TV_u_27a),V_b))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_c),s(t_h4s_lists_list(TV_u_27a),V_d)))))) => p(s(t_bool,h4s_encodes_biprefix(s(t_h4s_lists_list(TV_u_27a),V_a),s(t_h4s_lists_list(TV_u_27a),V_c)))))).
fof(ah4s_lists_APPENDu_u_11u_c1, axiom, ![TV_u_27a]: ![V_l3, V_l2, V_l1]: (s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l2),s(t_h4s_lists_list(TV_u_27a),V_l1))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l3),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),V_l3))).
fof(ah4s_bools_CONJu_u_ASSOC, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) & (p(s(t_bool,V_t2)) & p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) & p(s(t_bool,V_t3))))).
fof(ah4s_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_lists_APPENDu_u_11u_c0, axiom, ![TV_u_27a]: ![V_l3, V_l2, V_l1]: (s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l2))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),V_l1),s(t_h4s_lists_list(TV_u_27a),V_l3))) <=> s(t_h4s_lists_list(TV_u_27a),V_l2) = s(t_h4s_lists_list(TV_u_27a),V_l3))).
fof(ah4s_bools_ORu_u_DEF, axiom, ![V_x, V_xi_]: (p(s(t_bool,d_or(s(t_bool,V_x),s(t_bool,V_xi_)))) <=> ![V_t]: ((p(s(t_bool,V_x)) => p(s(t_bool,V_t))) => ((p(s(t_bool,V_xi_)) => p(s(t_bool,V_t))) => p(s(t_bool,V_t)))))).
fof(ah4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((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))))) <=> ![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))))))).
fof(ah4s_pairs_CLOSEDu_u_PAIRu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_b, V_a]: (s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & s(TV_u_27b,V_y) = s(TV_u_27b,V_b)))).
fof(ah4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((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))))) <=> ?[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))))))).
fof(ch4s_Coders_wfu_u_coderu_u_tree, conjecture, ![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)))) => p(s(t_bool,h4s_coders_wfu_u_coder(s(t_h4s_pairs_prod(t_fun(t_h4s_Encodes_tree(TV_u_27a),t_bool),t_h4s_pairs_prod(t_fun(t_h4s_Encodes_tree(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(t_h4s_Encodes_tree(TV_u_27a),t_h4s_lists_list(t_bool)))))),h4s_coders_treeu_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)))))))).
