%   ORIGINAL: h4/quotient__option/OPTION__MAP__I
% 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/option/option__Axiom: !f e. ?fn. fn h4/option/NONE = e /\ (!x. fn (h4/option/SOME x) = f x)
% Assm: h4/option/OPTION__MAP__DEF_c1: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm: h4/option/OPTION__MAP__DEF_c0: !x f. h4/option/OPTION__MAP f (h4/option/SOME x) = h4/option/SOME (f x)
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/option/OPTION__MAP__EQ__NONE: !x f. h4/option/OPTION__MAP f x = h4/option/NONE <=> x = h4/option/NONE
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/option/option__nchotomy: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm: h4/option/option__CLAUSES_c17: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm: h4/option/option__CLAUSES_c16: !x f. h4/option/OPTION__MAP f (h4/option/SOME x) = h4/option/SOME (f x)
% Assm: h4/option/OPTION__APPLY__def_c1: !x f. h4/option/OPTION__APPLY (h4/option/SOME f) x = h4/option/OPTION__MAP f x
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/option/OPTION__MAP__EQ__NONE__both__ways_c0: !x f. h4/option/OPTION__MAP f x = h4/option/NONE <=> x = h4/option/NONE
% Assm: h4/option/OPTION__APPLY__MAP2: !y x f. h4/option/OPTION__APPLY (h4/option/OPTION__MAP f x) y = h4/option/OPTION__MAP2 f x y
% Assm: h4/option/OPTION__MAP__CONG: !opt2 opt1 f2 f1. opt1 = opt2 /\ (!x. opt2 = h4/option/SOME x ==> f1 x = f2 x) ==> h4/option/OPTION__MAP f1 opt1 = h4/option/OPTION__MAP f2 opt2
% Assm: h4/list/MAP__ID_c1: !l. h4/list/MAP h4/combin/I l = l
% Assm: h4/option/OPTION__MAP__EQ__SOME: !y x f. h4/option/OPTION__MAP f x = h4/option/SOME y <=> (?z. x = h4/option/SOME z /\ y = f z)
% Assm: h4/option/OPTION__MAP__COMPOSE: !x g f. h4/option/OPTION__MAP f (h4/option/OPTION__MAP g x) = h4/option/OPTION__MAP (h4/combin/o f g) x
% Assm: h4/option/OPTION__MAP__EQ__NONE__both__ways_c1: !x f. h4/option/NONE = h4/option/OPTION__MAP f x <=> x = h4/option/NONE
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/quotient/EXISTS__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f. $exists f <=> h4/bool/RES__EXISTS (h4/quotient/respects R) (h4/quotient/_2D_2D_3E abs h4/combin/I f))
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/quotient/QUOTIENT__ABS__REP: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a. abs (rep a) = a)
% Assm: h4/pred__set/SPECIFICATION: !x P. h4/bool/IN x P <=> P x
% Assm: h4/quotient/FUN__MAP__THM: !x h g f. h4/quotient/_2D_2D_3E f g h x = g (h (f x))
% Assm: h4/combin/I__o__ID_c1: !f. h4/combin/o f h4/combin/I = f
% Assm: h4/combin/I__o__ID_c0: !f. h4/combin/o h4/combin/I f = f
% Assm: h4/quotient/FUN__MAP__I: h4/quotient/_2D_2D_3E h4/combin/I h4/combin/I = h4/combin/I
% Assm: h4/quotient/FORALL__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f. $forall f <=> h4/bool/RES__FORALL (h4/quotient/respects R) (h4/quotient/_2D_2D_3E abs h4/combin/I f))
% Assm: h4/quotient/RES__EXISTS__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!P f. h4/bool/RES__EXISTS P f <=> h4/bool/RES__EXISTS (h4/quotient/_2D_2D_3E abs h4/combin/I P) (h4/quotient/_2D_2D_3E abs h4/combin/I f))
% Assm: h4/quotient/EXISTS__UNIQUE__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f. h4/bool/_3F_21 f <=> h4/quotient/RES__EXISTS__EQUIV R (h4/quotient/_2D_2D_3E abs h4/combin/I f))
% Assm: h4/pred__set/SUM__SET__DEF: h4/pred__set/SUM__SET = h4/pred__set/SUM__IMAGE h4/combin/I
% Assm: h4/list/dropWhile__splitAtPki: !P. h4/list/dropWhile P = h4/list/splitAtPki (h4/combin/C (h4/combin/o h4/combin/K (h4/combin/o $not P))) (h4/combin/K h4/combin/I)
% Assm: h4/quotient/RES__FORALL__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!P f. h4/bool/RES__FORALL P f <=> h4/bool/RES__FORALL (h4/quotient/_2D_2D_3E abs h4/combin/I P) (h4/quotient/_2D_2D_3E abs h4/combin/I f))
% Assm: h4/sum/SUM__MAP__I: h4/sum/_2B_2B h4/combin/I h4/combin/I = h4/combin/I
% Assm: h4/combin/I__DEF: h4/combin/I = h4/combin/S h4/combin/K h4/combin/K
% Assm: h4/list/LIST__BIND__ID_c1: !l. h4/list/LIST__BIND l h4/combin/I = h4/list/FLAT l
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/res__quan/RES__FORALL: !f P. h4/bool/RES__FORALL P f <=> (!x. h4/bool/IN x P ==> f x)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/res__quan/RES__EXISTS: !f P. h4/bool/RES__EXISTS P f <=> (?x. h4/bool/IN x P /\ f x)
% Assm: h4/quotient/respects__def: h4/quotient/respects = h4/combin/W
% Assm: h4/quotient/QUOTIENT__REP__REFL: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a. R (rep a) (rep a))
% Assm: h4/combin/W__THM: !x f. h4/combin/W f x = f x x
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/quotient/FUN__MAP: !g f. h4/quotient/_2D_2D_3E f g = (\h x. g (h (f x)))
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/option/option__CLAUSES_c3: !x. ~(h4/option/SOME x = h4/option/NONE)
% Assm: h4/list/LIST__BIND__DEF: !l f. h4/list/LIST__BIND l f = h4/list/FLAT (h4/list/MAP f l)
% Assm: h4/list/MAP__ID_c0: !l. h4/list/MAP (\x. x) l = l
% Assm: h4/option/OPTION__MAP2__NONE: !o2 o1 f. h4/option/OPTION__MAP2 f o1 o2 = h4/option/NONE <=> o1 = h4/option/NONE \/ o2 = h4/option/NONE
% Assm: h4/option/OPTION__APPLY__def_c0: !x. h4/option/OPTION__APPLY h4/option/NONE x = h4/option/NONE
% Assm: h4/option/OPTION__MAP2__THM_c1: !x f. h4/option/OPTION__MAP2 f (h4/option/SOME x) h4/option/NONE = h4/option/NONE
% Assm: h4/option/OPTION__MAP2__THM_c0: !y x f. h4/option/OPTION__MAP2 f (h4/option/SOME x) (h4/option/SOME y) = h4/option/SOME (f x y)
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/sum/FORALL__SUM: !P. (!s. P s) <=> (!x. P (h4/sum/INL x)) /\ (!y. P (h4/sum/INR y))
% Assm: h4/sum/SUM__MAP__def_c1: !g f b. h4/sum/_2B_2B f g (h4/sum/INR b) = h4/sum/INR (g b)
% Assm: h4/sum/INR__INL__11_c1: !y x. h4/sum/INR x = h4/sum/INR y <=> x = y
% Assm: h4/sum/INR__INL__11_c0: !y x. h4/sum/INL x = h4/sum/INL y <=> x = y
% Assm: h4/sum/SUM__MAP__def_c0: !g f a. h4/sum/_2B_2B f g (h4/sum/INL a) = h4/sum/INL (f a)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/bool/FORALL__DEF: $forall = (\P. P = (\x. T))
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/BETA__THM: !y f. (\x. f x) y = f y
% Assm: h4/list/CONS0: h4/list/CONS = h4/list/_20_40ind__typelist1
% Assm: h4/list/NIL0: h4/list/NIL = h4/list/_20_40ind__typelist0
% Assm: h4/list/hidden____20__40ind____typelist0____def: h4/list/_20_40ind__typelist0 = h4/list/_20_40ind__typelist2 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM))
% Assm: h4/list/hidden____20__40ind____typelist1____def: h4/list/_20_40ind__typelist1 = (\a0 a1. h4/list/_20_40ind__typelist2 ((\a00 a10. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM))) a0 (h4/list/_20_40ind__typelist3 a1)))
% Assm: h4/list/list__repfns_c1: !r. (\a0_27. !_27list_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB (\n. h4/ind__type/BOTTOM) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM))) a0 a1 /\ _27list_27 a1) ==> _27list_27 a0_270) ==> _27list_27 a0_27) r <=> h4/list/_20_40ind__typelist3 (h4/list/_20_40ind__typelist2 r) = r
% Assm: h4/list/list__repfns_c0: !a. h4/list/_20_40ind__typelist2 (h4/list/_20_40ind__typelist3 a) = a
% Assm: h4/ind__type/FCONS0_c1: !n f a. h4/ind__type/FCONS a f (h4/num/SUC n) = f n
% Assm: h4/ind__type/FCONS0_c0: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm: h4/ind__type/CONSTR__REC: !Fn. ?f. !c i r. f (h4/ind__type/CONSTR c i r) = Fn c i r (\n. f (r n))
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/option/NOT__NONE__SOME: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% 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/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/quotient/QUOTIENT__REL__ABS__EQ: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!r s. R r r ==> R s s ==> (R r s <=> abs r = abs s))
% Assm: h4/quotient/RES__EXISTS__EQUIV0: !m R. h4/quotient/RES__EXISTS__EQUIV R m <=> h4/bool/RES__EXISTS (h4/quotient/respects R) (\x. m x) /\ h4/bool/RES__FORALL (h4/quotient/respects R) (\x. h4/bool/RES__FORALL (h4/quotient/respects R) (\y. m x /\ m y ==> R x y))
% Assm: h4/quotient/QUOTIENT__REL: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!r s. R r s <=> R r r /\ R s s /\ abs r = abs s)
% Assm: h4/bool/EXISTS__UNIQUE__DEF: h4/bool/_3F_21 = (\P. $exists P /\ (!x y. P x /\ P y ==> x = y))
% Assm: h4/list/MAP0_c1: !t h f. h4/list/MAP f (h4/list/CONS h t) = h4/list/CONS (f h) (h4/list/MAP f t)
% Assm: h4/list/MAP0_c0: !f. h4/list/MAP f h4/list/NIL = h4/list/NIL
% Assm: h4/list/dropWhile__DEF_c0: !P. h4/list/dropWhile P h4/list/NIL = h4/list/NIL
% Assm: h4/list/dropWhile__DEF_c1: !t h P. h4/list/dropWhile P (h4/list/CONS h t) = h4/bool/COND (P h) (h4/list/dropWhile P t) (h4/list/CONS h t)
% Assm: h4/list/splitAtPki__DEF_c1: !t k h P. h4/list/splitAtPki P k (h4/list/CONS h t) = h4/bool/COND (P h4/num/0 h) (k h4/list/NIL (h4/list/CONS h t)) (h4/list/splitAtPki (h4/combin/o P h4/num/SUC) (\p s. k (h4/list/CONS h p) s) t)
% Assm: h4/list/splitAtPki__DEF_c0: !k P. h4/list/splitAtPki P k h4/list/NIL = k h4/list/NIL h4/list/NIL
% Assm: h4/list/list__INDUCT: !P. P h4/list/NIL /\ (!t. P t ==> (!h. P (h4/list/CONS h t))) ==> (!l. P l)
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/combin/C__THM: !y x f. h4/combin/C f x y = f y x
% Assm: h4/combin/K__THM: !y x. h4/combin/K x y = x
% Assm: h4/marker/Abbrev__def: !x. h4/marker/Abbrev x <=> x
% Goal: h4/option/OPTION__MAP h4/combin/I = h4/combin/I
%   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_options_optionu_u_Axiom]: !f e. ?fn. happ fn h4/option/NONE = e /\ (!x. happ fn (h4/option/SOME x) = happ f x)
% Assm [h4s_options_OPTIONu_u_MAPu_u_DEFu_c1]: !f. happ (h4/option/OPTION__MAP f) h4/option/NONE = h4/option/NONE
% Assm [h4s_options_OPTIONu_u_MAPu_u_DEFu_c0]: !x f. happ (h4/option/OPTION__MAP f) (h4/option/SOME x) = h4/option/SOME (happ f x)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_options_OPTIONu_u_MAPu_u_EQu_u_NONE]: !x f. happ (h4/option/OPTION__MAP f) x = h4/option/NONE <=> x = h4/option/NONE
% Assm [h4s_combins_Iu_u_THM]: !x. happ h4/combin/I x = x
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_options_optionu_u_nchotomy]: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm [h4s_options_optionu_u_CLAUSESu_c17]: !f. happ (h4/option/OPTION__MAP f) h4/option/NONE = h4/option/NONE
% Assm [h4s_options_optionu_u_CLAUSESu_c16]: !x f. happ (h4/option/OPTION__MAP f) (h4/option/SOME x) = h4/option/SOME (happ f x)
% Assm [h4s_options_OPTIONu_u_APPLYu_u_defu_c1]: !x f. h4/option/OPTION__APPLY (h4/option/SOME f) x = happ (h4/option/OPTION__MAP f) x
% Assm [h4s_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_options_OPTIONu_u_MAPu_u_EQu_u_NONEu_u_bothu_u_waysu_c0]: !x f. happ (h4/option/OPTION__MAP f) x = h4/option/NONE <=> x = h4/option/NONE
% Assm [h4s_options_OPTIONu_u_APPLYu_u_MAP2]: !y x f. h4/option/OPTION__APPLY (happ (h4/option/OPTION__MAP f) x) y = h4/option/OPTION__MAP2 f x y
% Assm [h4s_options_OPTIONu_u_MAPu_u_CONG]: !opt2 opt1 f2 f1. opt1 = opt2 /\ (!x. opt2 = h4/option/SOME x ==> happ f1 x = happ f2 x) ==> happ (h4/option/OPTION__MAP f1) opt1 = happ (h4/option/OPTION__MAP f2) opt2
% Assm [h4s_lists_MAPu_u_IDu_c1]: !l. h4/list/MAP h4/combin/I l = l
% Assm [h4s_options_OPTIONu_u_MAPu_u_EQu_u_SOME]: !y x f. happ (h4/option/OPTION__MAP f) x = h4/option/SOME y <=> (?z. x = h4/option/SOME z /\ y = happ f z)
% Assm [h4s_options_OPTIONu_u_MAPu_u_COMPOSE]: !x g f. happ (h4/option/OPTION__MAP f) (happ (h4/option/OPTION__MAP g) x) = happ (h4/option/OPTION__MAP (h4/combin/o f g)) x
% Assm [h4s_options_OPTIONu_u_MAPu_u_EQu_u_NONEu_u_bothu_u_waysu_c1]: !x f. h4/option/NONE = happ (h4/option/OPTION__MAP f) x <=> x = h4/option/NONE
% Assm [h4s_combins_ou_u_THM]: !x g f. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_quotients_EXISTSu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f. $exists f <=> h4/bool/RES__EXISTS (happ h4/quotient/respects R) (happ (h4/quotient/_2D_2D_3E abs h4/combin/I) f))
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_quotients_QUOTIENTu_u_ABSu_u_REP]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a. happ abs (happ rep a) = a)
% Assm [h4s_predu_u_sets_SPECIFICATION]: !x P. h4/bool/IN x P <=> happ P x
% Assm [h4s_quotients_FUNu_u_MAPu_u_THM]: !x h g f. happ (happ (h4/quotient/_2D_2D_3E f g) h) x = happ g (happ h (happ f x))
% Assm [h4s_combins_Iu_u_ou_u_IDu_c1]: !f. h4/combin/o f h4/combin/I = f
% Assm [h4s_combins_Iu_u_ou_u_IDu_c0]: !f. h4/combin/o h4/combin/I f = f
% Assm [h4s_quotients_FUNu_u_MAPu_u_I]: h4/quotient/_2D_2D_3E h4/combin/I h4/combin/I = h4/combin/I
% Assm [h4s_quotients_FORALLu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f. $forall f <=> h4/bool/RES__FORALL (happ h4/quotient/respects R) (happ (h4/quotient/_2D_2D_3E abs h4/combin/I) f))
% Assm [h4s_quotients_RESu_u_EXISTSu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!P f. h4/bool/RES__EXISTS P f <=> h4/bool/RES__EXISTS (happ (h4/quotient/_2D_2D_3E abs h4/combin/I) P) (happ (h4/quotient/_2D_2D_3E abs h4/combin/I) f))
% Assm [h4s_quotients_EXISTSu_u_UNIQUEu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f. h4/bool/_3F_21 f <=> h4/quotient/RES__EXISTS__EQUIV R (happ (h4/quotient/_2D_2D_3E abs h4/combin/I) f))
% Assm [h4s_predu_u_sets_SUMu_u_SETu_u_DEF]: h4/pred__set/SUM__SET = h4/pred__set/SUM__IMAGE h4/combin/I
% Assm [h4s_lists_dropWhileu_u_splitAtPki]: !P. h4/list/dropWhile P = h4/list/splitAtPki (h4/combin/C (h4/combin/o h4/combin/K (h4/combin/o $not P))) (happ h4/combin/K h4/combin/I)
% Assm [h4s_quotients_RESu_u_FORALLu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!P f. h4/bool/RES__FORALL P f <=> h4/bool/RES__FORALL (happ (h4/quotient/_2D_2D_3E abs h4/combin/I) P) (happ (h4/quotient/_2D_2D_3E abs h4/combin/I) f))
% Assm [h4s_sums_SUMu_u_MAPu_u_I]: h4/sum/_2B_2B h4/combin/I h4/combin/I = h4/combin/I
% Assm [h4s_combins_Iu_u_DEF]: h4/combin/I = h4/combin/S h4/combin/K h4/combin/K
% Assm [h4s_lists_LISTu_u_BINDu_u_IDu_c1]: !l. h4/list/LIST__BIND l h4/combin/I = h4/list/FLAT l
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_resu_u_quans_RESu_u_FORALL]: !f P. h4/bool/RES__FORALL P f <=> (!x. h4/bool/IN x P ==> happ f x)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_resu_u_quans_RESu_u_EXISTS]: !f P. h4/bool/RES__EXISTS P f <=> (?x. h4/bool/IN x P /\ happ f x)
% Assm [h4s_quotients_respectsu_u_def]: h4/quotient/respects = h4/combin/W
% Assm [h4s_quotients_QUOTIENTu_u_REPu_u_REFL]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a. happ (happ R (happ rep a)) (happ rep a))
% Assm [h4s_combins_Wu_u_THM]: !x f. happ (happ h4/combin/W f) x = happ (happ f x) x
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_quotients_FUNu_u_MAP]: !g f x x. happ (happ (h4/quotient/_2D_2D_3E f g) x) x = happ g (happ x (happ f x))
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_options_optionu_u_CLAUSESu_c3]: !x. ~(h4/option/SOME x = h4/option/NONE)
% Assm [h4s_lists_LISTu_u_BINDu_u_DEF]: !l f. h4/list/LIST__BIND l f = h4/list/FLAT (h4/list/MAP f l)
% Assm [h4s_lists_MAPu_u_IDu_c0]: !_0. (!x. happ _0 x = x) ==> (!l. h4/list/MAP _0 l = l)
% Assm [h4s_options_OPTIONu_u_MAP2u_u_NONE]: !o2 o1 f. h4/option/OPTION__MAP2 f o1 o2 = h4/option/NONE <=> o1 = h4/option/NONE \/ o2 = h4/option/NONE
% Assm [h4s_options_OPTIONu_u_APPLYu_u_defu_c0]: !x. h4/option/OPTION__APPLY h4/option/NONE x = h4/option/NONE
% Assm [h4s_options_OPTIONu_u_MAP2u_u_THMu_c1]: !x f. h4/option/OPTION__MAP2 f (h4/option/SOME x) h4/option/NONE = h4/option/NONE
% Assm [h4s_options_OPTIONu_u_MAP2u_u_THMu_c0]: !y x f. h4/option/OPTION__MAP2 f (h4/option/SOME x) (h4/option/SOME y) = h4/option/SOME (happ (happ f x) y)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_sums_FORALLu_u_SUM]: !P. (!s. happ P s) <=> (!x. happ P (h4/sum/INL x)) /\ (!y. happ P (h4/sum/INR y))
% Assm [h4s_sums_SUMu_u_MAPu_u_defu_c1]: !g f b. happ (h4/sum/_2B_2B f g) (h4/sum/INR b) = h4/sum/INR (happ g b)
% Assm [h4s_sums_INRu_u_INLu_u_11u_c1]: !y x. h4/sum/INR x = h4/sum/INR y <=> x = y
% Assm [h4s_sums_INRu_u_INLu_u_11u_c0]: !y x. h4/sum/INL x = h4/sum/INL y <=> x = y
% Assm [h4s_sums_SUMu_u_MAPu_u_defu_c0]: !g f a. happ (h4/sum/_2B_2B f g) (h4/sum/INL a) = h4/sum/INL (happ f a)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_bools_FORALLu_u_DEF]: !x. $forall x <=> (!x. happ x x <=> T)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_BETAu_u_THM]: !y f. happ f y = happ f y
% Assm [h4s_lists_CONS0]: h4/list/CONS = h4/list/_20_40ind__typelist1
% Assm [h4s_lists_NIL0]: h4/list/NIL = h4/list/_20_40ind__typelist0
% Assm [h4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> h4/list/_20_40ind__typelist0 = h4/list/_20_40ind__typelist2 (h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0)
% Assm [h4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist1u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x. happ (happ h4/list/_20_40ind__typelist1 x) x = h4/list/_20_40ind__typelist2 (h4/ind__type/CONSTR (happ h4/num/SUC h4/num/0) x (h4/ind__type/FCONS (h4/list/_20_40ind__typelist3 x) _0)))
% Assm [h4s_lists_listu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27list_27. (!a0_270. a0_270 = h4/ind__type/CONSTR h4/num/0 h4/bool/ARB _0 \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (happ h4/num/SUC h4/num/0) a0 (h4/ind__type/FCONS a1 _0) /\ happ _27list_27 a1) ==> happ _27list_27 a0_270) ==> happ _27list_27 r) <=> h4/list/_20_40ind__typelist3 (h4/list/_20_40ind__typelist2 r) = r)
% Assm [h4s_lists_listu_u_repfnsu_c0]: !a. h4/list/_20_40ind__typelist2 (h4/list/_20_40ind__typelist3 a) = a
% Assm [h4s_indu_u_types_FCONS0u_c1]: !n f a. happ (h4/ind__type/FCONS a f) (happ h4/num/SUC n) = happ f n
% Assm [h4s_indu_u_types_FCONS0u_c0]: !f a. happ (h4/ind__type/FCONS a f) h4/num/0 = a
% Assm [h4s_indu_u_types_CONSTRu_u_REC]: !_0. (!f r n. happ (happ (happ _0 f) r) n = happ f (happ r n)) ==> (!Fn. ?f. !c i r. happ f (h4/ind__type/CONSTR c i r) = happ (happ (happ (happ Fn c) i) r) (happ (happ _0 f) r))
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_options_NOTu_u_NONEu_u_SOME]: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% 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_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_quotients_QUOTIENTu_u_RELu_u_ABSu_u_EQ]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!r s. happ (happ R r) r ==> happ (happ R s) s ==> (happ (happ R r) s <=> happ abs r = happ abs s))
% Assm [h4s_quotients_RESu_u_EXISTSu_u_EQUIV0]: !_2. (!m R x y. happ (happ (happ (happ _2 m) R) x) y <=> happ m x /\ happ m y ==> happ (happ R x) y) ==> (!_1. (!m R x. happ (happ (happ _1 m) R) x <=> h4/bool/RES__FORALL (happ h4/quotient/respects R) (happ (happ (happ _2 m) R) x)) ==> (!_0. (!m x. happ (happ _0 m) x <=> happ m x) ==> (!m R. h4/quotient/RES__EXISTS__EQUIV R m <=> h4/bool/RES__EXISTS (happ h4/quotient/respects R) (happ _0 m) /\ h4/bool/RES__FORALL (happ h4/quotient/respects R) (happ _0 (happ (happ _1 m) R)))))
% Assm [h4s_quotients_QUOTIENTu_u_REL]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!r s. happ (happ R r) s <=> happ (happ R r) r /\ happ (happ R s) s /\ happ abs r = happ abs s)
% Assm [h4s_bools_EXISTSu_u_UNIQUEu_u_DEF]: !x. h4/bool/_3F_21 x <=> $exists x /\ (!x y. happ x x /\ happ x y ==> x = y)
% Assm [h4s_lists_MAP0u_c1]: !t h f. h4/list/MAP f (happ (happ h4/list/CONS h) t) = happ (happ h4/list/CONS (happ f h)) (h4/list/MAP f t)
% Assm [h4s_lists_MAP0u_c0]: !f. h4/list/MAP f h4/list/NIL = h4/list/NIL
% Assm [h4s_lists_dropWhileu_u_DEFu_c0]: !P. happ (h4/list/dropWhile P) h4/list/NIL = h4/list/NIL
% Assm [h4s_lists_dropWhileu_u_DEFu_c1]: !t h P. happ (h4/list/dropWhile P) (happ (happ h4/list/CONS h) t) = h4/bool/COND (happ P h) (happ (h4/list/dropWhile P) t) (happ (happ h4/list/CONS h) t)
% Assm [h4s_lists_splitAtPkiu_u_DEFu_c1]: !_1. (!k h p s. happ (happ (happ (happ _1 k) h) p) s = happ (happ k (happ (happ h4/list/CONS h) p)) s) ==> (!_0. (!k h p. happ (happ (happ _0 k) h) p = happ (happ (happ _1 k) h) p) ==> (!t k h P. happ (h4/list/splitAtPki P k) (happ (happ h4/list/CONS h) t) = h4/bool/COND (happ (happ P h4/num/0) h) (happ (happ k h4/list/NIL) (happ (happ h4/list/CONS h) t)) (happ (h4/list/splitAtPki (h4/combin/o P h4/num/SUC) (happ (happ _0 k) h)) t)))
% Assm [h4s_lists_splitAtPkiu_u_DEFu_c0]: !k P. happ (h4/list/splitAtPki P k) h4/list/NIL = happ (happ k h4/list/NIL) h4/list/NIL
% Assm [h4s_lists_listu_u_INDUCT]: !P. happ P h4/list/NIL /\ (!t. happ P t ==> (!h. happ P (happ (happ h4/list/CONS h) t))) ==> (!l. happ P l)
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_combins_Cu_u_THM]: !y x f. happ (happ (h4/combin/C f) x) y = happ (happ f y) x
% Assm [h4s_combins_Ku_u_THM]: !y x. happ (happ h4/combin/K x) y = x
% Assm [h4s_markers_Abbrevu_u_def]: !x. h4/marker/Abbrev x <=> x
% Goal: h4/option/OPTION__MAP h4/combin/I = h4/combin/I
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_Q1295559,TV_Q1295555]: ![V_f, V_g]: (![V_x]: s(TV_Q1295555,happ(s(t_fun(TV_Q1295559,TV_Q1295555),V_f),s(TV_Q1295559,V_x))) = s(TV_Q1295555,happ(s(t_fun(TV_Q1295559,TV_Q1295555),V_g),s(TV_Q1295559,V_x))) => s(t_fun(TV_Q1295559,TV_Q1295555),V_f) = s(t_fun(TV_Q1295559,TV_Q1295555),V_g))).
fof(ah4s_options_optionu_u_Axiom, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_e]: ?[V_fn]: (s(TV_u_27b,happ(s(t_fun(t_h4s_options_option(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(TV_u_27b,V_e) & ![V_x]: s(TV_u_27b,happ(s(t_fun(t_h4s_options_option(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_options_option(TV_u_27a),h4s_options_some(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_options_OPTIONu_u_MAPu_u_DEFu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27b)),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_h4s_options_option(TV_u_27b),h4s_options_none)).
fof(ah4s_options_OPTIONu_u_MAPu_u_DEFu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27b)),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(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_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_options_OPTIONu_u_MAPu_u_EQu_u_NONE, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: (s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27b)),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_options_option(TV_u_27a),V_x))) = s(t_h4s_options_option(TV_u_27b),h4s_options_none) <=> s(t_h4s_options_option(TV_u_27a),V_x) = s(t_h4s_options_option(TV_u_27a),h4s_options_none))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_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_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_options_optionu_u_CLAUSESu_c17, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27b)),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_h4s_options_option(TV_u_27b),h4s_options_none)).
fof(ah4s_options_optionu_u_CLAUSESu_c16, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27b)),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))))).
fof(ah4s_options_OPTIONu_u_APPLYu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(TV_u_27b,TV_u_27a)),h4s_options_some(s(t_fun(TV_u_27b,TV_u_27a),V_f))),s(t_h4s_options_option(TV_u_27b),V_x))) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_options_option(TV_u_27b),t_h4s_options_option(TV_u_27a)),h4s_options_optionu_u_map(s(t_fun(TV_u_27b,TV_u_27a),V_f))),s(t_h4s_options_option(TV_u_27b),V_x)))).
fof(ah4s_options_SOMEu_u_11, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_bools_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_options_OPTIONu_u_MAPu_u_EQu_u_NONEu_u_bothu_u_waysu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f]: (s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_options_option(TV_u_27b),t_h4s_options_option(TV_u_27a)),h4s_options_optionu_u_map(s(t_fun(TV_u_27b,TV_u_27a),V_f))),s(t_h4s_options_option(TV_u_27b),V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) <=> s(t_h4s_options_option(TV_u_27b),V_x) = s(t_h4s_options_option(TV_u_27b),h4s_options_none))).
fof(ah4s_options_OPTIONu_u_APPLYu_u_MAP2, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(t_h4s_options_option(TV_u_27c),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(t_fun(TV_u_27b,TV_u_27c))),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(t_h4s_options_option(TV_u_27a),V_x))),s(t_h4s_options_option(TV_u_27b),V_y))) = s(t_h4s_options_option(TV_u_27c),h4s_options_optionu_u_map2(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(t_h4s_options_option(TV_u_27a),V_x),s(t_h4s_options_option(TV_u_27b),V_y)))).
fof(ah4s_options_OPTIONu_u_MAPu_u_CONG, axiom, ![TV_u_27b,TV_u_27a]: ![V_opt2, V_opt1, V_f2, V_f1]: ((s(t_h4s_options_option(TV_u_27a),V_opt1) = s(t_h4s_options_option(TV_u_27a),V_opt2) & ![V_x]: (s(t_h4s_options_option(TV_u_27a),V_opt2) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) => s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f2),s(TV_u_27a,V_x))))) => s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27b)),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f1))),s(t_h4s_options_option(TV_u_27a),V_opt1))) = s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27b)),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f2))),s(t_h4s_options_option(TV_u_27a),V_opt2))))).
fof(ah4s_lists_MAPu_u_IDu_c1, axiom, ![TV_u_27a]: ![V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),V_l)).
fof(ah4s_options_OPTIONu_u_MAPu_u_EQu_u_SOME, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x, V_f]: (s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27b)),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_options_option(TV_u_27a),V_x))) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,V_y))) <=> ?[V_z]: (s(t_h4s_options_option(TV_u_27a),V_x) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_z))) & s(TV_u_27b,V_y) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_z)))))).
fof(ah4s_options_OPTIONu_u_MAPu_u_COMPOSE, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_x, V_g, V_f]: s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(t_h4s_options_option(TV_u_27c),t_h4s_options_option(TV_u_27b)),h4s_options_optionu_u_map(s(t_fun(TV_u_27c,TV_u_27b),V_f))),s(t_h4s_options_option(TV_u_27c),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27c)),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27c),V_g))),s(t_h4s_options_option(TV_u_27a),V_x))))) = s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27b)),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27c),V_g))))),s(t_h4s_options_option(TV_u_27a),V_x)))).
fof(ah4s_options_OPTIONu_u_MAPu_u_EQu_u_NONEu_u_bothu_u_waysu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f]: (s(t_h4s_options_option(TV_u_27a),h4s_options_none) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_options_option(TV_u_27b),t_h4s_options_option(TV_u_27a)),h4s_options_optionu_u_map(s(t_fun(TV_u_27b,TV_u_27a),V_f))),s(t_h4s_options_option(TV_u_27b),V_x))) <=> s(t_h4s_options_option(TV_u_27b),V_x) = s(t_h4s_options_option(TV_u_27b),h4s_options_none))).
fof(ah4s_combins_ou_u_THM, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_x, V_g, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27c,TV_u_27a),V_g))),s(TV_u_27c,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_g),s(TV_u_27c,V_x)))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_quotients_EXISTSu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_f]: s(t_bool,d_exists(s(t_fun(TV_u_27b,t_bool),V_f))) = s(t_bool,h4s_bools_resu_u_exists(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),h4s_quotients_respects),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_bool)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(t_bool,t_bool),h4s_combins_i))),s(t_fun(TV_u_27b,t_bool),V_f))))))).
fof(ah4s_bools_FUNu_u_EQu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g) <=> ![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))))).
fof(ah4s_quotients_QUOTIENTu_u_ABSu_u_REP, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))) = s(TV_u_27b,V_a))).
fof(ah4s_predu_u_sets_SPECIFICATION, axiom, ![TV_u_27a]: ![V_x, V_P]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))).
fof(ah4s_quotients_FUNu_u_MAPu_u_THM, axiom, ![TV_u_27d,TV_u_27b,TV_u_27c,TV_u_27a]: ![V_x, V_h, V_g, V_f]: s(TV_u_27d,happ(s(t_fun(TV_u_27a,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27a,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g))),s(t_fun(TV_u_27c,TV_u_27b),V_h))),s(TV_u_27a,V_x))) = s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_h),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x)))))))).
fof(ah4s_combins_Iu_u_ou_u_IDu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i))) = s(t_fun(TV_u_27a,TV_u_27b),V_f)).
fof(ah4s_combins_Iu_u_ou_u_IDu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27b,TV_u_27b),h4s_combins_i),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27a,TV_u_27b),V_f)).
fof(ah4s_quotients_FUNu_u_MAPu_u_I, axiom, ![TV_u_27a,TV_u_27b]: s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(t_fun(TV_u_27b,TV_u_27b),h4s_combins_i))) = s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),h4s_combins_i)).
fof(ah4s_quotients_FORALLu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_f]: s(t_bool,d_forall(s(t_fun(TV_u_27b,t_bool),V_f))) = s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),h4s_quotients_respects),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_bool)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(t_bool,t_bool),h4s_combins_i))),s(t_fun(TV_u_27b,t_bool),V_f))))))).
fof(ah4s_quotients_RESu_u_EXISTSu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_P, V_f]: s(t_bool,h4s_bools_resu_u_exists(s(t_fun(TV_u_27b,t_bool),V_P),s(t_fun(TV_u_27b,t_bool),V_f))) = s(t_bool,h4s_bools_resu_u_exists(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_bool)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(t_bool,t_bool),h4s_combins_i))),s(t_fun(TV_u_27b,t_bool),V_P))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_bool)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(t_bool,t_bool),h4s_combins_i))),s(t_fun(TV_u_27b,t_bool),V_f))))))).
fof(ah4s_quotients_EXISTSu_u_UNIQUEu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_f]: s(t_bool,h4s_bools_u_3fu_21(s(t_fun(TV_u_27b,t_bool),V_f))) = s(t_bool,h4s_quotients_resu_u_existsu_u_equiv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_bool)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(t_bool,t_bool),h4s_combins_i))),s(t_fun(TV_u_27b,t_bool),V_f))))))).
fof(ah4s_predu_u_sets_SUMu_u_SETu_u_DEF, axiom, s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_set) = s(t_fun(t_fun(t_h4s_nums_num,t_bool),t_h4s_nums_num),h4s_predu_u_sets_sumu_u_image(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_combins_i)))).
fof(ah4s_lists_dropWhileu_u_splitAtPki, axiom, ![TV_u_27a]: ![V_P]: s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_lists_dropwhile(s(t_fun(TV_u_27a,t_bool),V_P))) = s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_lists_splitatpki(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_bool)),h4s_combins_c(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_bool)),h4s_combins_o(s(t_fun(t_bool,t_fun(t_h4s_nums_num,t_bool)),h4s_combins_k),s(t_fun(TV_u_27a,t_bool),h4s_combins_o(s(t_fun(t_bool,t_bool),d_not),s(t_fun(TV_u_27a,t_bool),V_P))))))),s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),t_fun(t_h4s_lists_list(TV_u_27a),t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)))),h4s_combins_k),s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_combins_i)))))).
fof(ah4s_quotients_RESu_u_FORALLu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_P, V_f]: s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27b,t_bool),V_P),s(t_fun(TV_u_27b,t_bool),V_f))) = s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_bool)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(t_bool,t_bool),h4s_combins_i))),s(t_fun(TV_u_27b,t_bool),V_P))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_bool)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(t_bool,t_bool),h4s_combins_i))),s(t_fun(TV_u_27b,t_bool),V_f))))))).
fof(ah4s_sums_SUMu_u_MAPu_u_I, axiom, ![TV_u_27a,TV_u_27b]: s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_u_2bu_2b(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(t_fun(TV_u_27b,TV_u_27b),h4s_combins_i))) = s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_combins_i)).
fof(ah4s_combins_Iu_u_DEF, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i) = s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_s(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27a,TV_u_27a),TV_u_27a)),h4s_combins_k),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_combins_k)))).
fof(ah4s_lists_LISTu_u_BINDu_u_IDu_c1, axiom, ![TV_u_27a]: ![V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(s(t_h4s_lists_list(t_h4s_lists_list(TV_u_27a)),V_l),s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_combins_i))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_flat(s(t_h4s_lists_list(t_h4s_lists_list(TV_u_27a)),V_l)))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_resu_u_quans_RESu_u_FORALL, axiom, ![TV_u_27a]: ![V_f, V_P]: (p(s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_f)))) <=> ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_resu_u_quans_RESu_u_EXISTS, axiom, ![TV_u_27a]: ![V_f, V_P]: (p(s(t_bool,h4s_bools_resu_u_exists(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_f)))) <=> ?[V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))))).
fof(ah4s_quotients_respectsu_u_def, axiom, ![TV_u_27a,TV_u_27b]: s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),t_fun(TV_u_27a,TV_u_27b)),h4s_quotients_respects) = s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),t_fun(TV_u_27a,TV_u_27b)),h4s_combins_w)).
fof(ah4s_quotients_QUOTIENTu_u_REPu_u_REFL, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_a]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a)))))))).
fof(ah4s_combins_Wu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),t_fun(TV_u_27a,TV_u_27b)),h4s_combins_w),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),V_f))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27a,V_x))),s(TV_u_27a,V_x)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,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_quotients_FUNu_u_MAP, axiom, ![TV_u_27d,TV_u_27b,TV_u_27c,TV_u_27a]: ![V_g, V_f, V_x, V_x0]: s(TV_u_27d,happ(s(t_fun(TV_u_27a,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27a,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g))),s(t_fun(TV_u_27c,TV_u_27b),V_x))),s(TV_u_27a,V_x0))) = s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_x),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x0)))))))).
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_options_optionu_u_CLAUSESu_c3, axiom, ![TV_u_27a]: ![V_x]: ~ (s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none))).
fof(ah4s_lists_LISTu_u_BINDu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_l, V_f]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_listu_u_bind(s(t_h4s_lists_list(TV_u_27b),V_l),s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_f))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_flat(s(t_h4s_lists_list(t_h4s_lists_list(TV_u_27a)),h4s_lists_map(s(t_fun(TV_u_27b,t_h4s_lists_list(TV_u_27a)),V_f),s(t_h4s_lists_list(TV_u_27b),V_l)))))).
fof(ah4s_lists_MAPu_u_IDu_c0, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_uu_0),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x) => ![V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27a),V_uu_0),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),V_l))).
fof(ah4s_options_OPTIONu_u_MAP2u_u_NONE, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_o2, V_o1, V_f]: (s(t_h4s_options_option(TV_u_27c),h4s_options_optionu_u_map2(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(t_h4s_options_option(TV_u_27a),V_o1),s(t_h4s_options_option(TV_u_27b),V_o2))) = s(t_h4s_options_option(TV_u_27c),h4s_options_none) <=> (s(t_h4s_options_option(TV_u_27a),V_o1) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) | s(t_h4s_options_option(TV_u_27b),V_o2) = s(t_h4s_options_option(TV_u_27b),h4s_options_none)))).
fof(ah4s_options_OPTIONu_u_APPLYu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(TV_u_27b,TV_u_27a)),h4s_options_none),s(t_h4s_options_option(TV_u_27b),V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)).
fof(ah4s_options_OPTIONu_u_MAP2u_u_THMu_c1, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_x, V_f]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_map2(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,V_x))),s(t_h4s_options_option(TV_u_27c),h4s_options_none))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)).
fof(ah4s_options_OPTIONu_u_MAP2u_u_THMu_c0, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_y, V_x, V_f]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_map2(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,V_x))),s(t_h4s_options_option(TV_u_27c),h4s_options_some(s(TV_u_27c,V_y))))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(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_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_bools_EXISTSu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_x)))))).
fof(ah4s_sums_FORALLu_u_SUM, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: (![V_s]: p(s(t_bool,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_s)))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_x)))))) & ![V_y]: p(s(t_bool,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_bool),V_P),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_y))))))))).
fof(ah4s_sums_SUMu_u_MAPu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27c,TV_u_27d,TV_u_27b]: ![V_g, V_f, V_b]: s(t_h4s_sums_sum(TV_u_27c,TV_u_27d),happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27c,TV_u_27d)),h4s_sums_u_2bu_2b(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g))),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_b))))) = s(t_h4s_sums_sum(TV_u_27c,TV_u_27d),h4s_sums_inr(s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(TV_u_27b,V_b)))))).
fof(ah4s_sums_INRu_u_INLu_u_11u_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: (s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_x))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inr(s(TV_u_27b,V_y))) <=> s(TV_u_27b,V_x) = s(TV_u_27b,V_y))).
fof(ah4s_sums_INRu_u_INLu_u_11u_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: (s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_x))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_sums_SUMu_u_MAPu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27d,TV_u_27c,TV_u_27a]: ![V_g, V_f, V_a]: s(t_h4s_sums_sum(TV_u_27c,TV_u_27d),happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),t_h4s_sums_sum(TV_u_27c,TV_u_27d)),h4s_sums_u_2bu_2b(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g))),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_inl(s(TV_u_27a,V_a))))) = s(t_h4s_sums_sum(TV_u_27c,TV_u_27d),h4s_sums_inl(s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_a)))))).
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_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
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_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_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_bools_FORALLu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: (p(s(t_bool,d_forall(s(t_fun(TV_u_27a,t_bool),V_x)))) <=> ![V_x0]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_x0))) = s(t_bool,t))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_BETAu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y)))).
fof(ah4s_lists_CONS0, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons) = s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_u_20u_40indu_u_typelist1)).
fof(ah4s_lists_NIL0, axiom, ![TV_u_27a]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist0)).
fof(ah4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist0u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist0) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))).
fof(ah4s_lists_hiddenu_u_u_u_20u_u_40indu_u_u_u_typelist1u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => ![V_x, V_x0]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_u_20u_40indu_u_typelist1),s(TV_u_27a,V_x))),s(t_h4s_lists_list(TV_u_27a),V_x0))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_nums_0))),s(TV_u_27a,V_x),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(TV_u_27a),V_x0))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))))))).
fof(ah4s_lists_listu_u_repfnsu_c1, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27listu_27]: (![V_a0u_270]: ((s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))) | ?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270) = s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,h4s_nums_0))),s(TV_u_27a,V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_uu_0))))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27listu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a1)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27listu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),t_bool),V_uu_27listu_27),s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r))))) <=> s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r))))) = s(t_h4s_indu_u_types_recspace(TV_u_27a),V_r)))).
fof(ah4s_lists_listu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist2(s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_lists_u_20u_40indu_u_typelist3(s(t_h4s_lists_list(TV_u_27a),V_a))))) = s(t_h4s_lists_list(TV_u_27a),V_a)).
fof(ah4s_indu_u_types_FCONS0u_c1, axiom, ![TV_u_27a]: ![V_n, V_f, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc),s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n)))).
fof(ah4s_indu_u_types_FCONS0u_c0, axiom, ![TV_u_27a]: ![V_f, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f))),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_a)).
fof(ah4s_indu_u_types_CONSTRu_u_REC, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_f, V_r, V_n]: s(TV_u_27b,happ(s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_h4s_nums_num,V_n))) = s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r),s(t_h4s_nums_num,V_n))))) => ![V_Fn]: ?[V_f]: ![V_c, V_i, V_r]: s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,V_c),s(TV_u_27a,V_i),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))) = s(TV_u_27b,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)))),V_Fn),s(t_h4s_nums_num,V_c))),s(TV_u_27a,V_i))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))))).
fof(ah4s_bools_JRHu_u_INDUCTu_u_UTIL, axiom, ![TV_u_27a]: ![V_t, V_P]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_t) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) => p(s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_P)))))).
fof(ah4s_bools_MONOu_u_OR, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) | p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) | p(s(t_bool,V_w)))))).
fof(ah4s_bools_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_MONOu_u_AND, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) & p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) & p(s(t_bool,V_w)))))).
fof(ah4s_bools_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_EXISTSu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (?[V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_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_options_NOTu_u_NONEu_u_SOME, axiom, ![TV_u_27a]: ![V_x]: ~ (s(t_h4s_options_option(TV_u_27a),h4s_options_none) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_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_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_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_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_bools_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_UNWINDu_u_THM2, axiom, ![TV_u_27a]: ![V_a, V_P]: (?[V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & 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_a)))))).
fof(ah4s_quotients_QUOTIENTu_u_RELu_u_ABSu_u_EQ, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_r, V_s]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_r))),s(TV_u_27a,V_r)))) => (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_s))),s(TV_u_27a,V_s)))) => (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_r))),s(TV_u_27a,V_s)))) <=> s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_s)))))))).
fof(ah4s_quotients_RESu_u_EXISTSu_u_EQUIV0, axiom, ![TV_u_27a]: ![V_uu_2]: (![V_m, V_R, V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)))),V_uu_2),s(t_fun(TV_u_27a,t_bool),V_m))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) <=> ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_m),s(TV_u_27a,V_x)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_m),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))))) => ![V_uu_1]: (![V_m, V_R, V_x]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool))),V_uu_1),s(t_fun(TV_u_27a,t_bool),V_m))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_x))) = s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),h4s_quotients_respects),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)))),V_uu_2),s(t_fun(TV_u_27a,t_bool),V_m))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_x))))) => ![V_uu_0]: (![V_m, 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_m))),s(TV_u_27a,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_m),s(TV_u_27a,V_x))) => ![V_m, V_R]: (p(s(t_bool,h4s_quotients_resu_u_existsu_u_equiv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),V_m)))) <=> (p(s(t_bool,h4s_bools_resu_u_exists(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),h4s_quotients_respects),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),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_m)))))) & p(s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),h4s_quotients_respects),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),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),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool))),V_uu_1),s(t_fun(TV_u_27a,t_bool),V_m))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))))))))))).
fof(ah4s_quotients_QUOTIENTu_u_REL, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_r, V_s]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_r))),s(TV_u_27a,V_s)))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_r))),s(TV_u_27a,V_r)))) & (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_s))),s(TV_u_27a,V_s)))) & s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_s)))))))).
fof(ah4s_bools_EXISTSu_u_UNIQUEu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: (p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(TV_u_27a,t_bool),V_x)))) <=> (p(s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x)))) & ![V_x0, V_y]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_x0)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_y))))) => s(TV_u_27a,V_x0) = s(TV_u_27a,V_y))))).
fof(ah4s_lists_MAP0u_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_h, V_f]: s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t))))) = s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),h4s_lists_cons),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_h))))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),V_t)))))).
fof(ah4s_lists_MAP0u_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil)).
fof(ah4s_lists_dropWhileu_u_DEFu_c0, axiom, ![TV_u_27a]: ![V_P]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_lists_dropwhile(s(t_fun(TV_u_27a,t_bool),V_P))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)).
fof(ah4s_lists_dropWhileu_u_DEFu_c1, axiom, ![TV_u_27a]: ![V_t, V_h, V_P]: s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_lists_dropwhile(s(t_fun(TV_u_27a,t_bool),V_P))),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t))))) = s(t_h4s_lists_list(TV_u_27a),h4s_bools_cond(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),h4s_lists_dropwhile(s(t_fun(TV_u_27a,t_bool),V_P))),s(t_h4s_lists_list(TV_u_27a),V_t))),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t)))))).
fof(ah4s_lists_splitAtPkiu_u_DEFu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_1]: (![V_k, V_h, V_p, V_s]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)))),V_uu_1),s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),V_k))),s(TV_u_27b,V_h))),s(t_h4s_lists_list(TV_u_27b),V_p))),s(t_h4s_lists_list(TV_u_27b),V_s))) = s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),V_k),s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),h4s_lists_cons),s(TV_u_27b,V_h))),s(t_h4s_lists_list(TV_u_27b),V_p))))),s(t_h4s_lists_list(TV_u_27b),V_s))) => ![V_uu_0]: (![V_k, V_h, V_p]: s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)))),V_uu_0),s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),V_k))),s(TV_u_27b,V_h))),s(t_h4s_lists_list(TV_u_27b),V_p))) = s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)))),V_uu_1),s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),V_k))),s(TV_u_27b,V_h))),s(t_h4s_lists_list(TV_u_27b),V_p))) => ![V_t, V_k, V_h, V_P]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a),h4s_lists_splitatpki(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool)),V_P),s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),V_k))),s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),h4s_lists_cons),s(TV_u_27b,V_h))),s(t_h4s_lists_list(TV_u_27b),V_t))))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool)),V_P),s(t_h4s_nums_num,h4s_nums_0))),s(TV_u_27b,V_h))),s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),V_k),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil))),s(t_h4s_lists_list(TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_h4s_lists_list(TV_u_27b))),h4s_lists_cons),s(TV_u_27b,V_h))),s(t_h4s_lists_list(TV_u_27b),V_t))))),s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a),h4s_lists_splitatpki(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool)),h4s_combins_o(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool)),V_P),s(t_fun(t_h4s_nums_num,t_h4s_nums_num),h4s_nums_suc))),s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),happ(s(t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),t_fun(TV_u_27b,t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)))),V_uu_0),s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),V_k))),s(TV_u_27b,V_h))))),s(t_h4s_lists_list(TV_u_27b),V_t)))))))).
fof(ah4s_lists_splitAtPkiu_u_DEFu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_k, V_P]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a),h4s_lists_splitatpki(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27b,t_bool)),V_P),s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),V_k))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil))) = s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27b),t_fun(t_h4s_lists_list(TV_u_27b),TV_u_27a)),V_k),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil))),s(t_h4s_lists_list(TV_u_27b),h4s_lists_nil)))).
fof(ah4s_lists_listu_u_INDUCT, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))) & ![V_t]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t)))) => ![V_h]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_h4s_lists_list(TV_u_27a))),h4s_lists_cons),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),V_t)))))))) => ![V_l]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_combins_Cu_u_THM, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27c)),h4s_combins_c(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(TV_u_27b,V_x))),s(TV_u_27a,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_y))),s(TV_u_27b,V_x)))).
fof(ah4s_combins_Ku_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),h4s_combins_k),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))) = s(TV_u_27a,V_x)).
fof(ah4s_markers_Abbrevu_u_def, axiom, ![V_x]: s(t_bool,h4s_markers_abbrev(s(t_bool,V_x))) = s(t_bool,V_x)).
fof(ch4s_quotientu_u_options_OPTIONu_u_MAPu_u_I, conjecture, ![TV_u_27a]: s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i))) = s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)),h4s_combins_i)).
