%   ORIGINAL: h4/option/option__CLAUSES_c1
% 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/THE__DEF: !x. h4/option/THE (h4/option/SOME x) = x
% Assm: h4/option/OPTION__MAP2__DEF: !y x f. h4/option/OPTION__MAP2 f x y = h4/bool/COND (h4/option/IS__SOME x /\ h4/option/IS__SOME y) (h4/option/SOME (f (h4/option/THE x) (h4/option/THE y))) h4/option/NONE
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/bool/TRUTH: T
% Assm: h4/option/option__induction: !P. P h4/option/NONE /\ (!a. P (h4/option/SOME a)) ==> (!x. P x)
% Assm: h4/option/SOME__DEF: !x. h4/option/SOME x = h4/option/option__ABS (h4/sum/INL x)
% Assm: h4/option/option__nchotomy: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm: h4/option/IS__SOME__DEF_c0: !x. h4/option/IS__SOME (h4/option/SOME x) <=> T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/option/NOT__SOME__NONE: !x. ~(h4/option/SOME x = h4/option/NONE)
% Assm: h4/option/option__REP__ABS__DEF_c1: !r. (\x. T) r <=> h4/option/option__REP (h4/option/option__ABS r) = r
% Assm: h4/option/NONE__DEF: h4/option/NONE = h4/option/option__ABS (h4/sum/INR h4/one/one0)
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/option/EXISTS__OPTION: !P. (?opt. P opt) <=> P h4/option/NONE \/ (?x. P (h4/option/SOME x))
% Assm: h4/option/FORALL__OPTION: !P. (!opt. P opt) <=> P h4/option/NONE /\ (!x. P (h4/option/SOME x))
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% 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/IS__NONE__DEF_c0: !x. h4/option/IS__NONE (h4/option/SOME x) <=> F
% Assm: h4/option/NOT__NONE__SOME: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/option/option__CLAUSES_c0: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/sum/INR__neq__INL: !v2 v1. ~(h4/sum/INR v2 = h4/sum/INL v1)
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/option/IS__SOME__DEF_c1: h4/option/IS__SOME h4/option/NONE <=> F
% 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/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sum/INR__INL__11_c0: !y x. h4/sum/INL x = h4/sum/INL y <=> x = y
% Assm: h4/option/option__case__def_c0: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/option/option__case__ID: !x. h4/option/option__CASE x h4/option/NONE h4/option/SOME = x
% Assm: h4/option/option__REP__ABS__DEF_c0: !a. h4/option/option__ABS (h4/option/option__REP a) = a
% Assm: h4/bool/LEFT__FORALL__OR__THM: !Q P. (!x. P x \/ Q) <=> (!x. P x) \/ Q
% Assm: h4/option/option__case__SOME__ID: !x. h4/option/option__CASE x x h4/option/SOME = x
% Assm: h4/one/one1: !v. v = h4/one/one0
% Assm: h4/sum/sum__INDUCT: !P. (!x. P (h4/sum/INL x)) /\ (!y. P (h4/sum/INR y)) ==> (!s. P s)
% 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/option/OPTION__MAP__DEF_c0: !x f. h4/option/OPTION__MAP f (h4/option/SOME x) = h4/option/SOME (f x)
% Assm: h4/option/OPTION__JOIN__DEF_c1: !x. h4/option/OPTION__JOIN (h4/option/SOME x) = x
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/option/OPTION__MAP2__THM_c2: !y f. h4/option/OPTION__MAP2 f h4/option/NONE (h4/option/SOME y) = h4/option/NONE
% Assm: h4/sum/sum__Axiom: !g f. ?h. (!x. h (h4/sum/INL x) = f x) /\ (!y. h (h4/sum/INR y) = g y)
% Assm: h4/option/option__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION (\x. T) rep
% Assm: h4/bool/ABS__REP__THM: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. abs (rep a) = a) /\ (!r. P r <=> rep (abs r) = r))
% Assm: h4/option/NOT__IS__SOME__EQ__NONE: !x. ~h4/option/IS__SOME x <=> x = h4/option/NONE
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/option/IS__NONE__DEF_c1: h4/option/IS__NONE h4/option/NONE <=> T
% Assm: h4/option/IS__NONE__EQ__NONE: !x. h4/option/IS__NONE x <=> x = h4/option/NONE
% Assm: h4/option/OPTION__MAP2__THM_c3: !f. h4/option/OPTION__MAP2 f h4/option/NONE h4/option/NONE = h4/option/NONE
% Assm: h4/option/OPTION__MAP__DEF_c1: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm: h4/bool/T__DEF: T <=> (\x. x) = (\x. x)
% Assm: h4/option/OPTION__JOIN__DEF_c0: h4/option/OPTION__JOIN h4/option/NONE = h4/option/NONE
% Assm: h4/combin/FAIL__DEF: h4/combin/FAIL = (\x y. x)
% Assm: h4/combin/FAIL__THM: !y x. h4/combin/FAIL x y = x
% Assm: h4/one/one__case__def: !x u. h4/one/one__CASE u x = x
% Assm: h4/one/one__case__thm: !x. h4/one/one__CASE h4/one/one0 x = x
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/OR__DEF: $or = (\t1 t2. !t. (t1 ==> t) ==> (t2 ==> t) ==> t)
% Assm: h4/bool/F__DEF: F <=> (!t. t)
% Assm: h4/bool/COND__DEF: h4/bool/COND = (\t t1 t2. h4/min/_40 (\x. ((t <=> T) ==> x = t1) /\ ((t <=> F) ==> x = t2)))
% Assm: h4/bool/RES__ABSTRACT__DEF_c1: !p m2 m1. (!x. h4/bool/IN x p ==> m1 x = m2 x) ==> h4/bool/RES__ABSTRACT p m1 = h4/bool/RES__ABSTRACT p m2
% Assm: h4/bool/NOT__DEF: $not = (\t. t ==> F)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/EXCLUDED__MIDDLE0: !t. t \/ ~t
% Assm: h4/bool/EXISTS__UNIQUE__DEF: h4/bool/_3F_21 = (\P. $exists P /\ (!x y. P x /\ P y ==> x = y))
% Assm: h4/bool/EXISTS__UNIQUE__THM: !P. h4/bool/_3F_21 (\x. P x) <=> (?x. P x) /\ (!x y. P x /\ P y ==> x = y)
% Assm: h4/combin/APPLY__UPDATE__ID: !f a. h4/combin/UPDATE a (f a) f = f
% Assm: h4/combin/UPDATE__APPLY__ID: !f b a. f a = b <=> h4/combin/UPDATE a b f = f
% Assm: h4/bool/LEFT__EXISTS__AND__THM: !Q P. (?x. P x /\ Q) <=> (?x. P x) /\ Q
% Assm: h4/bool/COND__ID: !t b. h4/bool/COND b t t = t
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> t
% Assm: h4/sum/INL__DEF: !e. h4/sum/INL e = h4/sum/ABS__sum (\b x y. x = e /\ b)
% Assm: h4/sum/INR__DEF: !e. h4/sum/INR e = h4/sum/ABS__sum (\b x y. y = e /\ ~b)
% Assm: h4/sum/sum__ISO__DEF_c1: !r. h4/sum/IS__SUM__REP r <=> h4/sum/REP__sum (h4/sum/ABS__sum r) = r
% Assm: h4/sum/sum__ISO__DEF_c0: !a. h4/sum/ABS__sum (h4/sum/REP__sum a) = a
% Assm: h4/sum/IS__SUM__REP0: !f. h4/sum/IS__SUM__REP f <=> (?v1 v2. f = (\b x y. x = v1 /\ b) \/ f = (\b x y. y = v2 /\ ~b))
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/sum/sum__axiom: !g f. h4/bool/_3F_21 (\h. h4/combin/o h h4/sum/INL = f /\ h4/combin/o h h4/sum/INR = g)
% Assm: h4/combin/UPDATE__def: !b a. h4/combin/UPDATE a b = (\f c. h4/bool/COND (a = c) b (f c))
% Assm: h4/bool/PULL__EXISTS_c1: !Q P. (?x. P x) /\ Q <=> (?x. P x /\ Q)
% Assm: h4/bool/IN__DEF: h4/bool/IN = (\x f. f x)
% Assm: h4/bool/UNWIND__THM1: !a P. (?x. a = x /\ P x) <=> P a
% Assm: h4/bool/ABS__SIMP: !t2 t1. (\x. t1) t2 = t1
% Assm: h4/bool/SKOLEM__THM: !P. (!x. ?y. P x y) <=> (?f. !x. P x (f x))
% Assm: h4/bool/boolAxiom: !t2 t1. ?fn. fn T = t1 /\ fn F = t2
% Assm: h4/combin/LET__FORALL__ELIM: !v f. h4/bool/LET f v <=> $forall (h4/combin/S (h4/combin/o $imply (h4/combin/o h4/marker/Abbrev (h4/combin/C $equals v))) f)
% Assm: h4/bool/LET__THM: !x f. h4/bool/LET f x = f x
% Assm: h4/combin/C__DEF: h4/combin/C = (\f x y. f y x)
% Assm: h4/combin/S__DEF: h4/combin/S = (\f g x. f x (g x))
% Assm: h4/marker/Abbrev__def: !x. h4/marker/Abbrev x <=> x
% Assm: h4/combin/UPD11__SAME__KEY__AND__BASE: !f c b a. h4/combin/UPDATE a b f = h4/combin/UPDATE a c f <=> b = c
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/bool/RES__FORALL__DEF: h4/bool/RES__FORALL = (\p m. !x. h4/bool/IN x p ==> m x)
% Assm: h4/bool/RES__FORALL__THM: !f P. h4/bool/RES__FORALL P f <=> (!x. h4/bool/IN x P ==> f x)
% Assm: h4/bool/RES__EXISTS__UNIQUE__DEF: h4/bool/RES__EXISTS__UNIQUE = (\p m. h4/bool/RES__EXISTS p (\x. m x) /\ h4/bool/RES__FORALL p (\x. h4/bool/RES__FORALL p (\y. m x /\ m y ==> x = y)))
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/RES__FORALL__CONG: !g f Q P. P = Q ==> (!x. h4/bool/IN x Q ==> (f x <=> g x)) ==> (h4/bool/RES__FORALL P f <=> h4/bool/RES__FORALL Q g)
% Assm: h4/bool/SELECT__REFL: !x. h4/min/_40 (\y. y = x) = x
% Goal: !x. h4/option/THE (h4/option/SOME x) = x
%   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 (happ h4/option/SOME x) = happ f x)
% Assm [h4s_options_THEu_u_DEF]: !x. h4/option/THE (happ h4/option/SOME x) = x
% Assm [h4s_options_OPTIONu_u_MAP2u_u_DEF]: !y x f. ?v. (v <=> h4/option/IS__SOME x /\ h4/option/IS__SOME y) /\ h4/option/OPTION__MAP2 f x y = h4/bool/COND v (happ h4/option/SOME (happ (happ f (h4/option/THE x)) (h4/option/THE y))) h4/option/NONE
% Assm [h4s_options_SOMEu_u_11]: !y x. happ h4/option/SOME x = happ h4/option/SOME y <=> x = y
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_options_optionu_u_induction]: !P. happ P h4/option/NONE /\ (!a. happ P (happ h4/option/SOME a)) ==> (!x. happ P x)
% Assm [h4s_options_SOMEu_u_DEF]: !x. happ h4/option/SOME x = h4/option/option__ABS (happ h4/sum/INL x)
% Assm [h4s_options_optionu_u_nchotomy]: !opt. opt = h4/option/NONE \/ (?x. opt = happ h4/option/SOME x)
% Assm [h4s_options_ISu_u_SOMEu_u_DEFu_c0]: !x. h4/option/IS__SOME (happ h4/option/SOME x) <=> T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_options_NOTu_u_SOMEu_u_NONE]: !x. ~(happ h4/option/SOME x = h4/option/NONE)
% Assm [h4s_options_optionu_u_REPu_u_ABSu_u_DEFu_c1]: !r. T <=> h4/option/option__REP (h4/option/option__ABS r) = r
% Assm [h4s_options_NONEu_u_DEF]: h4/option/NONE = h4/option/option__ABS (happ h4/sum/INR h4/one/one0)
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_options_EXISTSu_u_OPTION]: !P. (?opt. happ P opt) <=> happ P h4/option/NONE \/ (?x. happ P (happ h4/option/SOME x))
% Assm [h4s_options_FORALLu_u_OPTION]: !P. (!opt. happ P opt) <=> happ P h4/option/NONE /\ (!x. happ P (happ h4/option/SOME x))
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_options_OPTIONu_u_MAP2u_u_THMu_c1]: !x f. h4/option/OPTION__MAP2 f (happ h4/option/SOME x) h4/option/NONE = h4/option/NONE
% Assm [h4s_options_ISu_u_NONEu_u_DEFu_c0]: !x. h4/option/IS__NONE (happ h4/option/SOME x) <=> F
% Assm [h4s_options_NOTu_u_NONEu_u_SOME]: !x. ~(h4/option/NONE = happ h4/option/SOME x)
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_options_optionu_u_CLAUSESu_c0]: !y x. happ h4/option/SOME x = happ h4/option/SOME y <=> x = y
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_sums_INRu_u_nequ_u_INL]: !v2 v1. ~(happ h4/sum/INR v2 = happ h4/sum/INL v1)
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_options_ISu_u_SOMEu_u_DEFu_c1]: h4/option/IS__SOME h4/option/NONE <=> F
% Assm [h4s_options_optionu_u_caseu_u_defu_c1]: !x v f. h4/option/option__CASE (happ h4/option/SOME x) v f = happ f x
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sums_INRu_u_INLu_u_11u_c0]: !y x. happ h4/sum/INL x = happ h4/sum/INL y <=> x = y
% Assm [h4s_options_optionu_u_caseu_u_defu_c0]: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% Assm [h4s_options_optionu_u_caseu_u_ID]: !x. h4/option/option__CASE x h4/option/NONE h4/option/SOME = x
% Assm [h4s_options_optionu_u_REPu_u_ABSu_u_DEFu_c0]: !a. h4/option/option__ABS (h4/option/option__REP a) = a
% Assm [h4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. happ P x \/ Q) <=> (!x. happ P x) \/ Q
% Assm [h4s_options_optionu_u_caseu_u_SOMEu_u_ID]: !x. h4/option/option__CASE x x h4/option/SOME = x
% Assm [h4s_ones_one1]: !v. v = h4/one/one0
% Assm [h4s_sums_sumu_u_INDUCT]: !P. (!x. happ P (happ h4/sum/INL x)) /\ (!y. happ P (happ h4/sum/INR y)) ==> (!s. happ P s)
% Assm [h4s_options_OPTIONu_u_MAP2u_u_THMu_c0]: !y x f. h4/option/OPTION__MAP2 f (happ h4/option/SOME x) (happ h4/option/SOME y) = happ h4/option/SOME (happ (happ f x) y)
% Assm [h4s_options_OPTIONu_u_MAPu_u_DEFu_c0]: !x f. h4/option/OPTION__MAP f (happ h4/option/SOME x) = happ h4/option/SOME (happ f x)
% Assm [h4s_options_OPTIONu_u_JOINu_u_DEFu_c1]: !x. h4/option/OPTION__JOIN (happ h4/option/SOME x) = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_options_OPTIONu_u_MAP2u_u_THMu_c2]: !y f. h4/option/OPTION__MAP2 f h4/option/NONE (happ h4/option/SOME y) = h4/option/NONE
% Assm [h4s_sums_sumu_u_Axiom]: !g f. ?h. (!x. happ h (happ h4/sum/INL x) = happ f x) /\ (!y. happ h (happ h4/sum/INR y) = happ g y)
% Assm [h4s_options_optionu_u_TYu_u_DEF]: !_0. (!x. happ _0 x <=> T) ==> (?rep. h4/bool/TYPE__DEFINITION _0 rep)
% Assm [h4s_bools_ABSu_u_REPu_u_THM]: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. happ abs (happ rep a) = a) /\ (!r. happ P r <=> happ rep (happ abs r) = r))
% Assm [h4s_options_NOTu_u_ISu_u_SOMEu_u_EQu_u_NONE]: !x. ~h4/option/IS__SOME x <=> x = h4/option/NONE
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_options_ISu_u_NONEu_u_DEFu_c1]: h4/option/IS__NONE h4/option/NONE <=> T
% Assm [h4s_options_ISu_u_NONEu_u_EQu_u_NONE]: !x. h4/option/IS__NONE x <=> x = h4/option/NONE
% Assm [h4s_options_OPTIONu_u_MAP2u_u_THMu_c3]: !f. h4/option/OPTION__MAP2 f h4/option/NONE h4/option/NONE = h4/option/NONE
% Assm [h4s_options_OPTIONu_u_MAPu_u_DEFu_c1]: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm [h4s_bools_Tu_u_DEF]: T <=> (!x. x <=> x)
% Assm [h4s_options_OPTIONu_u_JOINu_u_DEFu_c0]: h4/option/OPTION__JOIN h4/option/NONE = h4/option/NONE
% Assm [h4s_combins_FAILu_u_DEF]: !x x. h4/combin/FAIL x x = x
% Assm [h4s_combins_FAILu_u_THM]: !y x. h4/combin/FAIL x y = x
% Assm [h4s_ones_oneu_u_caseu_u_def]: !x u. h4/one/one__CASE u x = x
% Assm [h4s_ones_oneu_u_caseu_u_thm]: !x. h4/one/one__CASE h4/one/one0 x = x
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_ORu_u_DEF]: !x x'. $or x x' <=> (!t. (x ==> t) ==> (x' ==> t) ==> t)
% Assm [h4s_bools_Fu_u_DEF]: F <=> (!t. t)
% Assm [h4s_bools_CONDu_u_DEF]: !_0. (!x x x' x''. happ (happ (happ (happ _0 x) x) x') x'' <=> ((x <=> T) ==> x'' = x) /\ ((x <=> F) ==> x'' = x')) ==> (!x x x'. h4/bool/COND x x x' = h4/min/_40 (happ (happ (happ _0 x) x) x'))
% Assm [h4s_bools_RESu_u_ABSTRACTu_u_DEFu_c1]: !p m2 m1. (!x. h4/bool/IN x p ==> happ m1 x = happ m2 x) ==> h4/bool/RES__ABSTRACT p m1 = h4/bool/RES__ABSTRACT p m2
% Assm [h4s_bools_NOTu_u_DEF]: !x. $not x <=> x ==> F
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE0]: !t. t \/ ~t
% 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_bools_EXISTSu_u_UNIQUEu_u_THM]: !_0. (!P x. happ (happ _0 P) x <=> happ P x) ==> (!P. h4/bool/_3F_21 (happ _0 P) <=> (?x. happ P x) /\ (!x y. happ P x /\ happ P y ==> x = y))
% Assm [h4s_combins_APPLYu_u_UPDATEu_u_ID]: !f a. h4/combin/UPDATE a (happ f a) f = f
% Assm [h4s_combins_UPDATEu_u_APPLYu_u_ID]: !f b a. happ f a = b <=> h4/combin/UPDATE a b f = f
% Assm [h4s_bools_LEFTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. happ P x /\ Q) <=> (?x. happ P x) /\ Q
% Assm [h4s_bools_CONDu_u_ID]: !t b. h4/bool/COND b t t = t
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> t
% Assm [h4s_sums_INLu_u_DEF]: !_2. (!x e b y. happ (happ (happ (happ _2 x) e) b) y <=> x = e /\ b) ==> (!_1. (!e b x. happ (happ (happ _1 e) b) x = happ (happ (happ _2 x) e) b) ==> (!_0. (!e b. happ (happ _0 e) b = happ (happ _1 e) b) ==> (!e. happ h4/sum/INL e = h4/sum/ABS__sum (happ _0 e))))
% Assm [h4s_sums_INRu_u_DEF]: !_2. (!e b y. happ (happ (happ _2 e) b) y <=> y = e /\ ~b) ==> (!_1. (!e b x. happ (happ (happ _1 e) b) x = happ (happ _2 e) b) ==> (!_0. (!e b. happ (happ _0 e) b = happ (happ _1 e) b) ==> (!e. happ h4/sum/INR e = h4/sum/ABS__sum (happ _0 e))))
% Assm [h4s_sums_sumu_u_ISOu_u_DEFu_c1]: !r. h4/sum/IS__SUM__REP r <=> h4/sum/REP__sum (h4/sum/ABS__sum r) = r
% Assm [h4s_sums_sumu_u_ISOu_u_DEFu_c0]: !a. h4/sum/ABS__sum (h4/sum/REP__sum a) = a
% Assm [h4s_sums_ISu_u_SUMu_u_REP0]: !f. h4/sum/IS__SUM__REP f <=> (?v1 v2. (!x x x. happ (happ (happ f x) x) x <=> x = v1 /\ x) \/ (!x x x. happ (happ (happ f x) x) x <=> x = v2 /\ ~x))
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_combins_ou_u_THM]: !x g f. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_sums_sumu_u_axiom]: !_0. (!f g h. happ (happ (happ _0 f) g) h <=> h4/combin/o h h4/sum/INL = f /\ h4/combin/o h h4/sum/INR = g) ==> (!g f. h4/bool/_3F_21 (happ (happ _0 f) g))
% Assm [h4s_combins_UPDATEu_u_def]: !b a x x. ?v. (v <=> a = x) /\ happ (h4/combin/UPDATE a b x) x = h4/bool/COND v b (happ x x)
% Assm [h4s_bools_PULLu_u_EXISTSu_c1]: !Q P. (?x. happ P x) /\ Q <=> (?x. happ P x /\ Q)
% Assm [h4s_bools_INu_u_DEF]: !x x. h4/bool/IN x x <=> happ x x
% Assm [h4s_bools_UNWINDu_u_THM1]: !a P. (?x. a = x /\ happ P x) <=> happ P a
% Assm [h4s_bools_ABSu_u_SIMP]: !t2 t1. t1 = t1
% 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_boolAxiom]: !t2 t1. ?fn. happ fn T = t1 /\ happ fn F = t2
% Assm [h4s_combins_LETu_u_FORALLu_u_ELIM]: !v f. h4/bool/LET f v <=> $forall (h4/combin/S (h4/combin/o $imply (h4/combin/o h4/marker/Abbrev (h4/combin/C $equals v))) f)
% Assm [h4s_bools_LETu_u_THM]: !x f. h4/bool/LET f x = happ f x
% Assm [h4s_combins_Cu_u_DEF]: !x x x. happ (h4/combin/C x x) x = happ (happ x x) x
% Assm [h4s_combins_Su_u_DEF]: !x x x. happ (h4/combin/S x x) x = happ (happ x x) (happ x x)
% Assm [h4s_markers_Abbrevu_u_def]: !x. happ h4/marker/Abbrev x <=> x
% Assm [h4s_combins_UPD11u_u_SAMEu_u_KEYu_u_ANDu_u_BASE]: !f c b a. h4/combin/UPDATE a b f = h4/combin/UPDATE a c f <=> b = c
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_bools_RESu_u_FORALLu_u_DEF]: !x x'. h4/bool/RES__FORALL x x' <=> (!x. h4/bool/IN x x ==> happ x' x)
% Assm [h4s_bools_RESu_u_FORALLu_u_THM]: !f P. h4/bool/RES__FORALL P f <=> (!x. h4/bool/IN x P ==> happ f x)
% Assm [h4s_bools_RESu_u_EXISTSu_u_UNIQUEu_u_DEF]: !_2. (!x' x y. happ (happ (happ _2 x') x) y <=> happ x' x /\ happ x' y ==> x = y) ==> (!_1. (!x x' x. happ (happ (happ _1 x) x') x <=> h4/bool/RES__FORALL x (happ (happ _2 x') x)) ==> (!_0. (!x' x. happ (happ _0 x') x <=> happ x' x) ==> (!x x'. h4/bool/RES__EXISTS__UNIQUE x x' <=> h4/bool/RES__EXISTS x (happ _0 x') /\ h4/bool/RES__FORALL x (happ _0 (happ (happ _1 x) x')))))
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_RESu_u_FORALLu_u_CONG]: !g f Q P. P = Q ==> (!x. h4/bool/IN x Q ==> (happ f x <=> happ g x)) ==> (h4/bool/RES__FORALL P f <=> h4/bool/RES__FORALL Q g)
% Assm [h4s_bools_SELECTu_u_REFL]: !_0. (!x y. happ (happ _0 x) y <=> y = x) ==> (!x. h4/min/_40 (happ _0 x) = x)
% Goal: !x. h4/option/THE (happ h4/option/SOME x) = x
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_Q1134094,TV_Q1134090]: ![V_f, V_g]: (![V_x]: s(TV_Q1134090,happ(s(t_fun(TV_Q1134094,TV_Q1134090),V_f),s(TV_Q1134094,V_x))) = s(TV_Q1134090,happ(s(t_fun(TV_Q1134094,TV_Q1134090),V_g),s(TV_Q1134094,V_x))) => s(t_fun(TV_Q1134094,TV_Q1134090),V_f) = s(t_fun(TV_Q1134094,TV_Q1134090),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),happ(s(t_fun(TV_u_27a,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_THEu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_options_the(s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x)).
fof(ah4s_options_OPTIONu_u_MAP2u_u_DEF, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_y, V_x, V_f]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27b),V_x)))) & p(s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27c),V_y)))))) & 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),V_x),s(t_h4s_options_option(TV_u_27c),V_y))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,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,h4s_options_the(s(t_h4s_options_option(TV_u_27b),V_x))))),s(TV_u_27c,h4s_options_the(s(t_h4s_options_option(TV_u_27c),V_y))))))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))))).
fof(ah4s_options_SOMEu_u_11, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,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_TRUTH, axiom, p(s(t_bool,t))).
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),happ(s(t_fun(TV_u_27a,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_options_SOMEu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,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_optionu_u_abs(s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one)),h4s_sums_inl),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),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))))).
fof(ah4s_options_ISu_u_SOMEu_u_DEFu_c0, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))))) = s(t_bool,t)).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_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_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_options_NOTu_u_SOMEu_u_NONE, axiom, ![TV_u_27a]: ![V_x]: ~ (s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,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_options_optionu_u_REPu_u_ABSu_u_DEFu_c1, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,t)) <=> s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),h4s_options_optionu_u_rep(s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_abs(s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),V_r))))) = s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),V_r))).
fof(ah4s_options_NONEu_u_DEF, axiom, ![TV_u_27a]: s(t_h4s_options_option(TV_u_27a),h4s_options_none) = s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_abs(s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),happ(s(t_fun(t_h4s_ones_one,t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one)),h4s_sums_inr),s(t_h4s_ones_one,h4s_ones_one0)))))).
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_options_EXISTSu_u_OPTION, axiom, ![TV_u_27a]: ![V_P]: (?[V_opt]: 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_opt)))) <=> (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_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),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))))))))).
fof(ah4s_options_FORALLu_u_OPTION, axiom, ![TV_u_27a]: ![V_P]: (![V_opt]: 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_opt)))) <=> (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_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),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),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_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),happ(s(t_fun(TV_u_27b,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_ISu_u_NONEu_u_DEFu_c0, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,h4s_options_isu_u_none(s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))))) = 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),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))))).
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_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,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_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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_sums_INRu_u_nequ_u_INL, axiom, ![TV_u_27b,TV_u_27a]: ![V_v2, V_v1]: ~ (s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_v2))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl),s(TV_u_27a,V_v1))))).
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_options_ISu_u_SOMEu_u_DEFu_c1, axiom, ![TV_u_27a]: s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_bool,f)).
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),happ(s(t_fun(TV_u_27a,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_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_bools_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_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_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_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_dcu_u_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_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ANDu_u_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_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_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),happ(s(t_fun(TV_u_27a,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),happ(s(t_fun(TV_u_27a,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_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_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_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_options_optionu_u_caseu_u_ID, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),V_x),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some))) = s(t_h4s_options_option(TV_u_27a),V_x)).
fof(ah4s_options_optionu_u_REPu_u_ABSu_u_DEFu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_abs(s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),h4s_options_optionu_u_rep(s(t_h4s_options_option(TV_u_27a),V_a))))) = s(t_h4s_options_option(TV_u_27a),V_a)).
fof(ah4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM, 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,V_Q))) <=> (![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,V_Q))))).
fof(ah4s_options_optionu_u_caseu_u_SOMEu_u_ID, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),V_x),s(t_h4s_options_option(TV_u_27a),V_x),s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some))) = s(t_h4s_options_option(TV_u_27a),V_x)).
fof(ah4s_ones_one1, axiom, ![V_v]: s(t_h4s_ones_one,V_v) = s(t_h4s_ones_one,h4s_ones_one0)).
fof(ah4s_sums_sumu_u_INDUCT, axiom, ![TV_u_27a,TV_u_27b]: ![V_P]: ((![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),happ(s(t_fun(TV_u_27a,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),happ(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_y))))))) => ![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)))))).
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),happ(s(t_fun(TV_u_27b,t_h4s_options_option(TV_u_27b)),h4s_options_some),s(TV_u_27b,V_x))),s(t_h4s_options_option(TV_u_27c),happ(s(t_fun(TV_u_27c,t_h4s_options_option(TV_u_27c)),h4s_options_some),s(TV_u_27c,V_y))))) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,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_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),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))))) = s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(TV_u_27b,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_JOINu_u_DEFu_c1, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_join(s(t_h4s_options_option(t_h4s_options_option(TV_u_27a)),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(t_h4s_options_option(TV_u_27a))),h4s_options_some),s(t_h4s_options_option(TV_u_27a),V_x))))) = s(t_h4s_options_option(TV_u_27a),V_x)).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_options_OPTIONu_u_MAP2u_u_THMu_c2, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_y, 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_none),s(t_h4s_options_option(TV_u_27c),happ(s(t_fun(TV_u_27c,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_none)).
fof(ah4s_sums_sumu_u_Axiom, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b]: ![V_g, V_f]: ?[V_h]: (![V_x]: s(TV_u_27c,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),V_h),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl),s(TV_u_27a,V_x))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x))) & ![V_y]: s(TV_u_27c,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),V_h),s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),V_g),s(TV_u_27b,V_y))))).
fof(ah4s_options_optionu_u_TYu_u_DEF, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x]: s(t_bool,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),t_bool),V_uu_0),s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),V_x))) = s(t_bool,t) => ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),t_bool),V_uu_0),s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one)),V_rep)))))).
fof(ah4s_bools_ABSu_u_REPu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ?[V_rep, V_abs]: (![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) & ![V_r]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_r)))) <=> s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),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_27a,V_r))))).
fof(ah4s_options_NOTu_u_ISu_u_SOMEu_u_EQu_u_NONE, axiom, ![TV_u_27a]: ![V_x]: (~ (p(s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),V_x))))) <=> s(t_h4s_options_option(TV_u_27a),V_x) = s(t_h4s_options_option(TV_u_27a),h4s_options_none))).
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_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_options_ISu_u_NONEu_u_DEFu_c1, axiom, ![TV_u_27a]: s(t_bool,h4s_options_isu_u_none(s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_bool,t)).
fof(ah4s_options_ISu_u_NONEu_u_EQu_u_NONE, axiom, ![TV_u_27a]: ![V_x]: (p(s(t_bool,h4s_options_isu_u_none(s(t_h4s_options_option(TV_u_27a),V_x)))) <=> s(t_h4s_options_option(TV_u_27a),V_x) = s(t_h4s_options_option(TV_u_27a),h4s_options_none))).
fof(ah4s_options_OPTIONu_u_MAP2u_u_THMu_c3, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![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_none),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_MAPu_u_DEFu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(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_bools_Tu_u_DEF, axiom, (p(s(t_bool,t)) <=> ![V_x]: s(t_bool,V_x) = s(t_bool,V_x))).
fof(ah4s_options_OPTIONu_u_JOINu_u_DEFu_c0, axiom, ![TV_u_27a]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_join(s(t_h4s_options_option(t_h4s_options_option(TV_u_27a)),h4s_options_none))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)).
fof(ah4s_combins_FAILu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_x0]: s(TV_u_27a,h4s_combins_fail(s(TV_u_27a,V_x),s(TV_u_27b,V_x0))) = s(TV_u_27a,V_x)).
fof(ah4s_combins_FAILu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_combins_fail(s(TV_u_27a,V_x),s(TV_u_27b,V_y))) = s(TV_u_27a,V_x)).
fof(ah4s_ones_oneu_u_caseu_u_def, axiom, ![TV_u_27a]: ![V_x, V_u]: s(TV_u_27a,h4s_ones_oneu_u_case(s(t_h4s_ones_one,V_u),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_ones_oneu_u_caseu_u_thm, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_ones_oneu_u_case(s(t_h4s_ones_one,h4s_ones_one0),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
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_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_Fu_u_DEF, axiom, (p(s(t_bool,f)) <=> ![V_t]: p(s(t_bool,V_t)))).
fof(ah4s_bools_CONDu_u_DEF, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x, V_x0, V_xi_, V_xi_i_]: (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_bool,t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)))),V_uu_0),s(TV_u_27a,V_x))),s(t_bool,V_x0))),s(TV_u_27a,V_xi_))),s(TV_u_27a,V_xi_i_)))) <=> ((s(t_bool,V_x0) = s(t_bool,t) => s(TV_u_27a,V_xi_i_) = s(TV_u_27a,V_x)) & (s(t_bool,V_x0) = s(t_bool,f) => s(TV_u_27a,V_xi_i_) = s(TV_u_27a,V_xi_)))) => ![V_x, V_x0, V_xi_]: s(TV_u_27a,h4s_bools_cond(s(t_bool,V_x),s(TV_u_27a,V_x0),s(TV_u_27a,V_xi_))) = s(TV_u_27a,h4s_mins_u_40(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_bool,t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)))),V_uu_0),s(TV_u_27a,V_x0))),s(t_bool,V_x))),s(TV_u_27a,V_xi_))))))).
fof(ah4s_bools_RESu_u_ABSTRACTu_u_DEFu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_p, V_m2, V_m1]: (![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)))) => s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_m1),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_m2),s(TV_u_27a,V_x)))) => s(t_fun(TV_u_27a,TV_u_27b),h4s_bools_resu_u_abstract(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,TV_u_27b),V_m1))) = s(t_fun(TV_u_27a,TV_u_27b),h4s_bools_resu_u_abstract(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,TV_u_27b),V_m2))))).
fof(ah4s_bools_NOTu_u_DEF, axiom, ![V_x]: (p(s(t_bool,d_not(s(t_bool,V_x)))) <=> (p(s(t_bool,V_x)) => p(s(t_bool,f))))).
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_EXCLUDEDu_u_MIDDLE0, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
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_bools_EXISTSu_u_UNIQUEu_u_THM, axiom, ![TV_u_27a]: ![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]: (p(s(t_bool,h4s_bools_u_3fu_21(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)))))) <=> (?[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, 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))))) => s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))))).
fof(ah4s_combins_APPLYu_u_UPDATEu_u_ID, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_a]: s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_a))),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27a,TV_u_27b),V_f)).
fof(ah4s_combins_UPDATEu_u_APPLYu_u_ID, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_b, V_a]: (s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_a))) = s(TV_u_27b,V_b) <=> s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_b),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27a,TV_u_27b),V_f))).
fof(ah4s_bools_LEFTu_u_EXISTSu_u_ANDu_u_THM, 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,V_Q))) <=> (?[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,V_Q))))).
fof(ah4s_bools_CONDu_u_ID, axiom, ![TV_u_27a]: ![V_t, V_b]: s(TV_u_27a,h4s_bools_cond(s(t_bool,V_b),s(TV_u_27a,V_t),s(TV_u_27a,V_t))) = s(TV_u_27a,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_sums_INLu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_2]: (![V_x, V_e, V_b, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_bool,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27b,t_bool)))),V_uu_2),s(TV_u_27a,V_x))),s(TV_u_27a,V_e))),s(t_bool,V_b))),s(TV_u_27b,V_y)))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_e) & p(s(t_bool,V_b)))) => ![V_uu_1]: (![V_e, V_b, V_x]: s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_1),s(TV_u_27a,V_e))),s(t_bool,V_b))),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_bool,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27b,t_bool)))),V_uu_2),s(TV_u_27a,V_x))),s(TV_u_27a,V_e))),s(t_bool,V_b))) => ![V_uu_0]: (![V_e, V_b]: s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_0),s(TV_u_27a,V_e))),s(t_bool,V_b))) = s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_1),s(TV_u_27a,V_e))),s(t_bool,V_b))) => ![V_e]: s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl),s(TV_u_27a,V_e))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_absu_u_sum(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27a,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_0),s(TV_u_27a,V_e))))))))).
fof(ah4s_sums_INRu_u_DEF, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_2]: (![V_e, V_b, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_bool,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(TV_u_27b,t_fun(t_bool,t_fun(TV_u_27b,t_bool))),V_uu_2),s(TV_u_27b,V_e))),s(t_bool,V_b))),s(TV_u_27b,V_y)))) <=> (s(TV_u_27b,V_y) = s(TV_u_27b,V_e) & ~ (p(s(t_bool,V_b))))) => ![V_uu_1]: (![V_e, V_b, V_x]: s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27b,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_1),s(TV_u_27b,V_e))),s(t_bool,V_b))),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(t_bool,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(TV_u_27b,t_fun(t_bool,t_fun(TV_u_27b,t_bool))),V_uu_2),s(TV_u_27b,V_e))),s(t_bool,V_b))) => ![V_uu_0]: (![V_e, V_b]: s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27b,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_0),s(TV_u_27b,V_e))),s(t_bool,V_b))) = s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27b,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_1),s(TV_u_27b,V_e))),s(t_bool,V_b))) => ![V_e]: s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_e))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_absu_u_sum(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),happ(s(t_fun(TV_u_27b,t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)))),V_uu_0),s(TV_u_27b,V_e))))))))).
fof(ah4s_sums_sumu_u_ISOu_u_DEFu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_r]: (p(s(t_bool,h4s_sums_isu_u_sumu_u_rep(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_r)))) <=> s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),h4s_sums_repu_u_sum(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_absu_u_sum(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_r))))) = s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_r))).
fof(ah4s_sums_sumu_u_ISOu_u_DEFu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_a]: s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),h4s_sums_absu_u_sum(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),h4s_sums_repu_u_sum(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_a))))) = s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_a)).
fof(ah4s_sums_ISu_u_SUMu_u_REP0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: (p(s(t_bool,h4s_sums_isu_u_sumu_u_rep(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_f)))) <=> ?[V_v1, V_v2]: (![V_x, V_x0, V_x1]: (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)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_f),s(t_bool,V_x))),s(TV_u_27a,V_x0))),s(TV_u_27b,V_x1)))) <=> (s(TV_u_27a,V_x0) = s(TV_u_27a,V_v1) & p(s(t_bool,V_x)))) | ![V_x, V_x0, V_x1]: (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)),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool))),V_f),s(t_bool,V_x))),s(TV_u_27a,V_x0))),s(TV_u_27b,V_x1)))) <=> (s(TV_u_27b,V_x1) = s(TV_u_27b,V_v2) & ~ (p(s(t_bool,V_x)))))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_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_sums_sumu_u_axiom, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_uu_0]: (![V_f, V_g, V_h]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),t_bool),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),t_bool))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_f))),s(t_fun(TV_u_27b,TV_u_27c),V_g))),s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),V_h)))) <=> (s(t_fun(TV_u_27a,TV_u_27c),h4s_combins_o(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),V_h),s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inl))) = s(t_fun(TV_u_27a,TV_u_27c),V_f) & s(t_fun(TV_u_27b,TV_u_27c),h4s_combins_o(s(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),V_h),s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr))) = s(t_fun(TV_u_27b,TV_u_27c),V_g))) => ![V_g, V_f]: p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),t_bool),happ(s(t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(t_fun(TV_u_27b,TV_u_27c),t_fun(t_fun(t_h4s_sums_sum(TV_u_27a,TV_u_27b),TV_u_27c),t_bool))),V_uu_0),s(t_fun(TV_u_27a,TV_u_27c),V_f))),s(t_fun(TV_u_27b,TV_u_27c),V_g)))))))).
fof(ah4s_combins_UPDATEu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_b, V_a, V_x, V_x0]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(TV_u_27a,V_a) = s(TV_u_27a,V_x0)) & s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_b),s(t_fun(TV_u_27a,TV_u_27b),V_x))),s(TV_u_27a,V_x0))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27b,V_b),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(TV_u_27a,V_x0))))))).
fof(ah4s_bools_PULLu_u_EXISTSu_c1, 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,V_Q))) <=> ?[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,V_Q))))).
fof(ah4s_bools_INu_u_DEF, axiom, ![TV_u_27a]: ![V_x, V_x0]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_x0))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x0),s(TV_u_27a,V_x)))).
fof(ah4s_bools_UNWINDu_u_THM1, axiom, ![TV_u_27a]: ![V_a, V_P]: (?[V_x]: (s(TV_u_27a,V_a) = 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_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_a)))))).
fof(ah4s_bools_ABSu_u_SIMP, axiom, ![TV_u_27b,TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,V_t1) = s(TV_u_27a,V_t1)).
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_boolAxiom, axiom, ![TV_u_27a]: ![V_t2, V_t1]: ?[V_fn]: (s(TV_u_27a,happ(s(t_fun(t_bool,TV_u_27a),V_fn),s(t_bool,t))) = s(TV_u_27a,V_t1) & s(TV_u_27a,happ(s(t_fun(t_bool,TV_u_27a),V_fn),s(t_bool,f))) = s(TV_u_27a,V_t2))).
fof(ah4s_combins_LETu_u_FORALLu_u_ELIM, axiom, ![TV_u_27a]: ![V_v, V_f]: s(t_bool,h4s_bools_let(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_v))) = s(t_bool,d_forall(s(t_fun(TV_u_27a,t_bool),h4s_combins_s(s(t_fun(TV_u_27a,t_fun(t_bool,t_bool)),h4s_combins_o(s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_imply),s(t_fun(TV_u_27a,t_bool),h4s_combins_o(s(t_fun(t_bool,t_bool),h4s_markers_abbrev),s(t_fun(TV_u_27a,t_bool),h4s_combins_c(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),d_equals),s(TV_u_27a,V_v))))))),s(t_fun(TV_u_27a,t_bool),V_f)))))).
fof(ah4s_bools_LETu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_combins_Cu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_x, V_x0, V_x1]: s(TV_u_27c,happ(s(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_x),s(TV_u_27b,V_x0))),s(TV_u_27a,V_x1))) = 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_x),s(TV_u_27a,V_x1))),s(TV_u_27b,V_x0)))).
fof(ah4s_combins_Su_u_DEF, axiom, ![TV_u_27c,TV_u_27b,TV_u_27a]: ![V_x, V_x0, V_x1]: s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),h4s_combins_s(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_x),s(t_fun(TV_u_27a,TV_u_27b),V_x0))),s(TV_u_27a,V_x1))) = 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_x),s(TV_u_27a,V_x1))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_x0),s(TV_u_27a,V_x1)))))).
fof(ah4s_markers_Abbrevu_u_def, axiom, ![V_x]: s(t_bool,happ(s(t_fun(t_bool,t_bool),h4s_markers_abbrev),s(t_bool,V_x))) = s(t_bool,V_x)).
fof(ah4s_combins_UPD11u_u_SAMEu_u_KEYu_u_ANDu_u_BASE, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_c, V_b, V_a]: (s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_b),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_update(s(TV_u_27a,V_a),s(TV_u_27b,V_c),s(t_fun(TV_u_27a,TV_u_27b),V_f))) <=> s(TV_u_27b,V_b) = s(TV_u_27b,V_c))).
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_bools_RESu_u_FORALLu_u_DEF, axiom, ![TV_u_27a]: ![V_x, V_xi_]: (p(s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),V_x),s(t_fun(TV_u_27a,t_bool),V_xi_)))) <=> ![V_x0]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x0),s(t_fun(TV_u_27a,t_bool),V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_xi_),s(TV_u_27a,V_x0))))))).
fof(ah4s_bools_RESu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_f, V_P]: (p(s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_f)))) <=> ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_RESu_u_EXISTSu_u_UNIQUEu_u_DEF, axiom, ![TV_u_27a]: ![V_uu_2]: (![V_xi_, 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_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_xi_))),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_xi_),s(TV_u_27a,V_x)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_xi_),s(TV_u_27a,V_y))))) => s(TV_u_27a,V_x) = s(TV_u_27a,V_y))) => ![V_uu_1]: (![V_x, V_xi_, V_x0]: 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)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(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_x))),s(t_fun(TV_u_27a,t_bool),V_xi_))),s(TV_u_27a,V_x0))) = s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),V_x),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_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_xi_))),s(TV_u_27a,V_x0))))) => ![V_uu_0]: (![V_xi_, 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_xi_))),s(TV_u_27a,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_xi_),s(TV_u_27a,V_x))) => ![V_x, V_xi_]: (p(s(t_bool,h4s_bools_resu_u_existsu_u_unique(s(t_fun(TV_u_27a,t_bool),V_x),s(t_fun(TV_u_27a,t_bool),V_xi_)))) <=> (p(s(t_bool,h4s_bools_resu_u_exists(s(t_fun(TV_u_27a,t_bool),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_xi_)))))) & p(s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),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),happ(s(t_fun(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_bool),t_fun(TV_u_27a,t_bool))),V_uu_1),s(t_fun(TV_u_27a,t_bool),V_x))),s(t_fun(TV_u_27a,t_bool),V_xi_)))))))))))))).
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_bools_boolu_u_caseu_u_thmu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_bools_RESu_u_FORALLu_u_CONG, axiom, ![TV_u_27a]: ![V_g, V_f, V_Q, V_P]: (s(t_fun(TV_u_27a,t_bool),V_P) = s(t_fun(TV_u_27a,t_bool),V_Q) => (![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_Q)))) => s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_g),s(TV_u_27a,V_x)))) => 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))) = s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),V_Q),s(t_fun(TV_u_27a,t_bool),V_g)))))).
fof(ah4s_bools_SELECTu_u_REFL, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_uu_0),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x)) => ![V_x]: s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_uu_0),s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x))).
fof(ch4s_options_optionu_u_CLAUSESu_c1, conjecture, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_options_the(s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_options_option(TV_u_27a)),h4s_options_some),s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x)).
