%   ORIGINAL: h4/option/OPTION__GUARD__COND
% 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/bool/boolAxiom: !t2 t1. ?fn. fn T = t1 /\ fn F = t2
% Assm: h4/option/OPTION__GUARD__def_c0: h4/option/OPTION__GUARD T = h4/option/SOME h4/one/one0
% Assm: h4/option/OPTION__GUARD__def_c1: h4/option/OPTION__GUARD F = h4/option/NONE
% Assm: h4/one/one__axiom: !g f. f = g
% Assm: h4/bool/TRUTH: T
% Assm: h4/one/one1: !v. v = h4/one/one0
% Assm: h4/one/one__Axiom: !e. h4/bool/_3F_21 (\fn. fn h4/one/one0 = e)
% Assm: h4/bool/EXISTS__UNIQUE__DEF: h4/bool/_3F_21 = (\P. $exists P /\ (!x y. P x /\ P y ==> x = y))
% Assm: h4/one/one__induction: !P. P h4/one/one0 ==> (!x. P x)
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/option/NONE__DEF: h4/option/NONE = h4/option/option__ABS (h4/sum/INR h4/one/one0)
% Assm: h4/one/one__prim__rec: !e. ?fn. fn h4/one/one0 = e
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/one/FORALL__ONE: !P. (!x. P x) <=> P h4/one/one0
% Assm: h4/one/one__DEF: h4/one/one0 = h4/min/_40 (\x. T)
% Assm: h4/one/one__case__thm: !x. h4/one/one__CASE h4/one/one0 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__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION (\x. T) rep
% Assm: h4/bool/T__DEF: T <=> (\x. x) = (\x. x)
% Assm: h4/one/one__case__def: !x u. h4/one/one__CASE u x = x
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% 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/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm: h4/one/one__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION (\b. b) rep
% Assm: h4/option/SOME__DEF: !x. h4/option/SOME x = h4/option/option__ABS (h4/sum/INL x)
% Assm: h4/option/option__REP__ABS__DEF_c1: !r. (\x. T) r <=> h4/option/option__REP (h4/option/option__ABS r) = r
% 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/option/option__REP__ABS__DEF_c0: !a. h4/option/option__ABS (h4/option/option__REP a) = a
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/TYPE__DEFINITION__THM: !rep P. h4/bool/TYPE__DEFINITION P rep <=> (!x_27 x_27_27. rep x_27 = rep x_27_27 ==> x_27 = x_27_27) /\ (!x. P x <=> (?x_27. x = rep x_27))
% Assm: h4/option/NOT__NONE__SOME: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/option/IF__EQUALS__OPTION_c3: !y x P. h4/bool/COND P h4/option/NONE (h4/option/SOME x) = h4/option/SOME y <=> ~P /\ x = y
% Assm: h4/option/IF__EQUALS__OPTION_c1: !x P. h4/bool/COND P h4/option/NONE (h4/option/SOME x) = h4/option/NONE <=> P
% Assm: h4/option/IF__EQUALS__OPTION_c2: !y x P. h4/bool/COND P (h4/option/SOME x) h4/option/NONE = h4/option/SOME y <=> P /\ x = y
% Assm: h4/option/IF__EQUALS__OPTION_c0: !x P. h4/bool/COND P (h4/option/SOME x) h4/option/NONE = h4/option/NONE <=> ~P
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/option/option__nchotomy: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm: h4/option/IF__NONE__EQUALS__OPTION_c2: !x X P. h4/bool/COND P X h4/option/NONE = h4/option/SOME x <=> P /\ X = h4/option/SOME x
% Assm: h4/bool/COND__ID: !t b. h4/bool/COND b t t = t
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/option/IS__SOME__DEF_c1: h4/option/IS__SOME h4/option/NONE <=> F
% Assm: h4/option/IS__SOME__DEF_c0: !x. h4/option/IS__SOME (h4/option/SOME x) <=> T
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/option/IF__NONE__EQUALS__OPTION_c3: !x X P. h4/bool/COND P h4/option/NONE X = h4/option/SOME x <=> ~P /\ X = h4/option/SOME x
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> 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/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/option/IS__NONE__DEF_c0: !x. h4/option/IS__NONE (h4/option/SOME x) <=> F
% Assm: h4/option/IS__NONE__DEF_c1: h4/option/IS__NONE h4/option/NONE <=> T
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> 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/NOT__DEF: $not = (\t. t ==> F)
% 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/IF__NONE__EQUALS__OPTION_c0: !X P. h4/bool/COND P X h4/option/NONE = h4/option/NONE <=> P ==> h4/option/IS__NONE X
% Assm: h4/option/IF__NONE__EQUALS__OPTION_c1: !X P. h4/bool/COND P h4/option/NONE X = h4/option/NONE <=> h4/option/IS__SOME X ==> P
% Assm: h4/option/option__case__def_c1: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = f x
% Assm: h4/option/option__case__def_c0: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm: h4/option/option__case__compute: !x f e. h4/option/option__CASE x e f = h4/bool/COND (h4/option/IS__SOME x) (f (h4/option/THE x)) e
% Assm: h4/bool/bool__case__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/bool/AND1__THM: !t2 t1. t1 /\ t2 ==> t1
% Assm: h4/bool/EXCLUDED__MIDDLE0: !t. t \/ ~t
% Assm: h4/option/THE__DEF: !x. h4/option/THE (h4/option/SOME x) = x
% Assm: h4/sat/dc__cond: !s r q p. (p <=> h4/bool/COND q r s) <=> (p \/ q \/ ~s) /\ (p \/ ~r \/ ~q) /\ (p \/ ~r \/ ~s) /\ (~q \/ r \/ ~p) /\ (q \/ s \/ ~p)
% Assm: h4/bool/COND__EXPAND__IMP: !t2 t1 b. h4/bool/COND b t1 t2 <=> (b ==> t1) /\ (~b ==> t2)
% Assm: h4/bool/OR__CLAUSES_c4: !t. t \/ t <=> t
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/COND__ABS: !g f b. (\x. h4/bool/COND b (f x) (g x)) = h4/bool/COND b f g
% Assm: h4/bool/bool__case__thm_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/bool__case__thm_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/COND__EXPAND: !t2 t1 b. h4/bool/COND b t1 t2 <=> (~b \/ t1) /\ (b \/ t2)
% Assm: h4/sum/cond__sum__expand_c0: !z y x P. h4/bool/COND P (h4/sum/INR x) (h4/sum/INL y) = h4/sum/INR z <=> P /\ z = x
% 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/bool/COND__RAND: !y x f b. f (h4/bool/COND b x y) = h4/bool/COND b (f x) (f y)
% Assm: h4/sum/sum__distinct: !y x. ~(h4/sum/INL x = h4/sum/INR y)
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/COND__EXPAND__OR: !t2 t1 b. h4/bool/COND b t1 t2 <=> b /\ t1 \/ ~b /\ t2
% Assm: h4/bool/bool__case__ID: !t b. h4/bool/COND b t t = t
% Assm: h4/sum/cond__sum__expand_c3: !z y x P. h4/bool/COND P (h4/sum/INL x) (h4/sum/INR y) = h4/sum/INR z <=> ~P /\ z = y
% Assm: h4/sum/cond__sum__expand_c2: !z y x P. h4/bool/COND P (h4/sum/INL x) (h4/sum/INR y) = h4/sum/INL z <=> P /\ z = x
% Assm: h4/bool/COND__RATOR: !x g f b. h4/bool/COND b f g x = h4/bool/COND b (f x) (g x)
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/sum/cond__sum__expand_c1: !z y x P. h4/bool/COND P (h4/sum/INR x) (h4/sum/INL y) = h4/sum/INL z <=> ~P /\ z = y
% 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/MONO__COND: !z y x w b. (x ==> y) ==> (z ==> w) ==> h4/bool/COND b x z ==> h4/bool/COND b y w
% Assm: h4/combin/APPLY__UPDATE__THM: !f c b a. h4/combin/UPDATE a b f c = h4/bool/COND (a = c) b (f c)
% Assm: h4/option/option__case__ID: !x. h4/option/option__CASE x h4/option/NONE h4/option/SOME = x
% Assm: h4/option/option__case__SOME__ID: !x. h4/option/option__CASE x x h4/option/SOME = x
% Assm: h4/bool/literal__case__id: !u t a. h4/bool/literal__case (\x. h4/bool/COND (x = a) t u) a = t
% Assm: h4/sum/SUM__MAP: !z g f. h4/sum/_2B_2B f g z = h4/bool/COND (h4/sum/ISL z) (h4/sum/INL (f (h4/sum/OUTL z))) (h4/sum/INR (g (h4/sum/OUTR z)))
% Assm: h4/bool/literal__case__DEF: h4/bool/literal__case = (\f x. f x)
% Assm: h4/option/option__CLAUSES_c9: !x. h4/option/option__CASE x h4/option/NONE h4/option/SOME = x
% 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/SUM__MAP__def_c0: !g f a. h4/sum/_2B_2B f g (h4/sum/INL a) = h4/sum/INL (f a)
% 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/sum/FORALL__SUM: !P. (!s. P s) <=> (!x. P (h4/sum/INL x)) /\ (!y. P (h4/sum/INR y))
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/sum/ISL0_c0: !x. h4/sum/ISL (h4/sum/INL x)
% Assm: h4/sum/ISL0_c1: !y. ~h4/sum/ISL (h4/sum/INR y)
% Assm: h4/option/option__CLAUSES_c10: !x. h4/option/option__CASE x x h4/option/SOME = x
% Assm: h4/option/NOT__SOME__NONE: !x. ~(h4/option/SOME x = h4/option/NONE)
% Assm: h4/option/option__CLAUSES_c13: !x e. h4/option/IS__SOME x ==> h4/option/option__CASE x e h4/option/SOME = x
% Assm: h4/option/option__CLAUSES_c2: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm: h4/option/option__CLAUSES_c3: !x. ~(h4/option/SOME x = h4/option/NONE)
% Assm: h4/option/FORALL__OPTION: !P. (!opt. P opt) <=> P h4/option/NONE /\ (!x. P (h4/option/SOME x))
% Goal: !b. h4/option/OPTION__GUARD b = h4/bool/COND b (h4/option/SOME h4/one/one0) h4/option/NONE
%   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_bools_boolAxiom]: !t2 t1. ?fn. happ fn T = t1 /\ happ fn F = t2
% Assm [h4s_options_OPTIONu_u_GUARDu_u_defu_c0]: h4/option/OPTION__GUARD T = happ h4/option/SOME h4/one/one0
% Assm [h4s_options_OPTIONu_u_GUARDu_u_defu_c1]: h4/option/OPTION__GUARD F = h4/option/NONE
% Assm [h4s_ones_oneu_u_axiom]: !g f. f = g
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_ones_one1]: !v. v = h4/one/one0
% Assm [h4s_ones_oneu_u_Axiom]: !_0. (!e fn. happ (happ _0 e) fn <=> happ fn h4/one/one0 = e) ==> (!e. h4/bool/_3F_21 (happ _0 e))
% 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_ones_oneu_u_induction]: !P. happ P h4/one/one0 ==> (!x. happ P x)
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_options_NONEu_u_DEF]: h4/option/NONE = h4/option/option__ABS (happ h4/sum/INR h4/one/one0)
% Assm [h4s_ones_oneu_u_primu_u_rec]: !e. ?fn. happ fn h4/one/one0 = e
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_ones_FORALLu_u_ONE]: !P. (!x. happ P x) <=> happ P h4/one/one0
% Assm [h4s_ones_oneu_u_DEF]: !_0. (!x. happ _0 x <=> T) ==> h4/one/one0 = h4/min/_40 _0
% Assm [h4s_ones_oneu_u_caseu_u_thm]: !x. h4/one/one__CASE h4/one/one0 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_TYu_u_DEF]: !_0. (!x. happ _0 x <=> T) ==> (?rep. h4/bool/TYPE__DEFINITION _0 rep)
% Assm [h4s_bools_Tu_u_DEF]: T <=> (!x. x <=> x)
% Assm [h4s_ones_oneu_u_caseu_u_def]: !x u. h4/one/one__CASE u x = x
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% 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_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm [h4s_ones_oneu_u_TYu_u_DEF]: !_0. (!b. happ _0 b <=> b) ==> (?rep. h4/bool/TYPE__DEFINITION _0 rep)
% 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_REPu_u_ABSu_u_DEFu_c1]: !r. T <=> h4/option/option__REP (h4/option/option__ABS r) = r
% 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_options_optionu_u_REPu_u_ABSu_u_DEFu_c0]: !a. h4/option/option__ABS (h4/option/option__REP a) = a
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_TYPEu_u_DEFINITIONu_u_THM]: !rep P. h4/bool/TYPE__DEFINITION P rep <=> (!x_27 x_27_27. happ rep x_27 = happ rep x_27_27 ==> x_27 = x_27_27) /\ (!x. happ P x <=> (?x_27. x = happ rep x_27))
% Assm [h4s_options_NOTu_u_NONEu_u_SOME]: !x. ~(h4/option/NONE = happ h4/option/SOME x)
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_options_IFu_u_EQUALSu_u_OPTIONu_c3]: !y x P. h4/bool/COND P h4/option/NONE (happ h4/option/SOME x) = happ h4/option/SOME y <=> ~P /\ x = y
% Assm [h4s_options_IFu_u_EQUALSu_u_OPTIONu_c1]: !x P. h4/bool/COND P h4/option/NONE (happ h4/option/SOME x) = h4/option/NONE <=> P
% Assm [h4s_options_IFu_u_EQUALSu_u_OPTIONu_c2]: !y x P. h4/bool/COND P (happ h4/option/SOME x) h4/option/NONE = happ h4/option/SOME y <=> P /\ x = y
% Assm [h4s_options_IFu_u_EQUALSu_u_OPTIONu_c0]: !x P. h4/bool/COND P (happ h4/option/SOME x) h4/option/NONE = h4/option/NONE <=> ~P
% Assm [h4s_options_SOMEu_u_11]: !y x. happ h4/option/SOME x = happ h4/option/SOME y <=> x = y
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_options_optionu_u_nchotomy]: !opt. opt = h4/option/NONE \/ (?x. opt = happ h4/option/SOME x)
% Assm [h4s_options_IFu_u_NONEu_u_EQUALSu_u_OPTIONu_c2]: !x X P. h4/bool/COND P X h4/option/NONE = happ h4/option/SOME x <=> P /\ X = happ h4/option/SOME x
% Assm [h4s_bools_CONDu_u_ID]: !t b. h4/bool/COND b t t = t
% Assm [h4s_bools_EQu_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_options_ISu_u_SOMEu_u_DEFu_c1]: h4/option/IS__SOME h4/option/NONE <=> F
% Assm [h4s_options_ISu_u_SOMEu_u_DEFu_c0]: !x. h4/option/IS__SOME (happ h4/option/SOME x) <=> T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_options_IFu_u_NONEu_u_EQUALSu_u_OPTIONu_c3]: !x X P. h4/bool/COND P h4/option/NONE X = happ h4/option/SOME x <=> ~P /\ X = happ h4/option/SOME x
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> 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_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_options_ISu_u_NONEu_u_DEFu_c0]: !x. h4/option/IS__NONE (happ h4/option/SOME x) <=> F
% Assm [h4s_options_ISu_u_NONEu_u_DEFu_c1]: h4/option/IS__NONE h4/option/NONE <=> T
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> 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_NOTu_u_DEF]: !x. $not x <=> x ==> F
% 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_IFu_u_NONEu_u_EQUALSu_u_OPTIONu_c0]: !X P. h4/bool/COND P X h4/option/NONE = h4/option/NONE <=> P ==> h4/option/IS__NONE X
% Assm [h4s_options_IFu_u_NONEu_u_EQUALSu_u_OPTIONu_c1]: !X P. h4/bool/COND P h4/option/NONE X = h4/option/NONE <=> h4/option/IS__SOME X ==> P
% 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_options_optionu_u_caseu_u_defu_c0]: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm [h4s_options_optionu_u_caseu_u_compute]: !x f e. h4/option/option__CASE x e f = h4/bool/COND (h4/option/IS__SOME x) (happ f (h4/option/THE x)) e
% Assm [h4s_bools_boolu_u_caseu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_bools_AND1u_u_THM]: !t2 t1. t1 /\ t2 ==> t1
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE0]: !t. t \/ ~t
% Assm [h4s_options_THEu_u_DEF]: !x. h4/option/THE (happ h4/option/SOME x) = x
% Assm [h4s_sats_dcu_u_cond]: !s r q p. (p <=> h4/bool/COND q r s) <=> (p \/ q \/ ~s) /\ (p \/ ~r \/ ~q) /\ (p \/ ~r \/ ~s) /\ (~q \/ r \/ ~p) /\ (q \/ s \/ ~p)
% Assm [h4s_bools_CONDu_u_EXPANDu_u_IMP]: !t2 t1 b. h4/bool/COND b t1 t2 <=> (b ==> t1) /\ (~b ==> t2)
% Assm [h4s_bools_ORu_u_CLAUSESu_c4]: !t. t \/ t <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_CONDu_u_ABS]: !g f b x. h4/bool/COND b (happ f x) (happ g x) = happ (h4/bool/COND b f g) x
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_boolu_u_caseu_u_thmu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_CONDu_u_EXPAND]: !t2 t1 b. h4/bool/COND b t1 t2 <=> (~b \/ t1) /\ (b \/ t2)
% Assm [h4s_sums_condu_u_sumu_u_expandu_c0]: !z y x P. h4/bool/COND P (happ h4/sum/INR x) (happ h4/sum/INL y) = happ h4/sum/INR z <=> P /\ z = x
% Assm [h4s_sums_INRu_u_INLu_u_11u_c1]: !y x. happ h4/sum/INR x = happ h4/sum/INR y <=> x = y
% 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_bools_CONDu_u_RAND]: !y x f b. happ f (h4/bool/COND b x y) = h4/bool/COND b (happ f x) (happ f y)
% Assm [h4s_sums_sumu_u_distinct]: !y x. ~(happ h4/sum/INL x = happ h4/sum/INR y)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_CONDu_u_EXPANDu_u_OR]: !t2 t1 b. h4/bool/COND b t1 t2 <=> b /\ t1 \/ ~b /\ t2
% Assm [h4s_bools_boolu_u_caseu_u_ID]: !t b. h4/bool/COND b t t = t
% Assm [h4s_sums_condu_u_sumu_u_expandu_c3]: !z y x P. h4/bool/COND P (happ h4/sum/INL x) (happ h4/sum/INR y) = happ h4/sum/INR z <=> ~P /\ z = y
% Assm [h4s_sums_condu_u_sumu_u_expandu_c2]: !z y x P. h4/bool/COND P (happ h4/sum/INL x) (happ h4/sum/INR y) = happ h4/sum/INL z <=> P /\ z = x
% Assm [h4s_bools_CONDu_u_RATOR]: !x g f b. happ (h4/bool/COND b f g) x = h4/bool/COND b (happ f x) (happ g x)
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_sums_condu_u_sumu_u_expandu_c1]: !z y x P. h4/bool/COND P (happ h4/sum/INR x) (happ h4/sum/INL y) = happ h4/sum/INL z <=> ~P /\ z = y
% Assm [h4s_combins_UPDATEu_u_def]: !b a x x. ?v. (v <=> a = x) /\ h4/combin/UPDATE a b x x = h4/bool/COND v b (happ x x)
% Assm [h4s_bools_MONOu_u_COND]: !z y x w b. (x ==> y) ==> (z ==> w) ==> h4/bool/COND b x z ==> h4/bool/COND b y w
% Assm [h4s_combins_APPLYu_u_UPDATEu_u_THM]: !f c b a. ?v. (v <=> a = c) /\ h4/combin/UPDATE a b f c = h4/bool/COND v b (happ f c)
% 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_caseu_u_SOMEu_u_ID]: !x. h4/option/option__CASE x x h4/option/SOME = x
% Assm [h4s_bools_literalu_u_caseu_u_id]: !_0. (!a t u x. ?v. (v <=> x = a) /\ happ (happ (happ (happ _0 a) t) u) x = h4/bool/COND v t u) ==> (!u t a. h4/bool/literal__case (happ (happ (happ _0 a) t) u) a = t)
% Assm [h4s_sums_SUMu_u_MAP]: !z g f. h4/sum/_2B_2B f g z = h4/bool/COND (h4/sum/ISL z) (happ h4/sum/INL (happ f (h4/sum/OUTL z))) (happ h4/sum/INR (happ g (h4/sum/OUTR z)))
% Assm [h4s_bools_literalu_u_caseu_u_DEF]: !x x. h4/bool/literal__case x x = happ x x
% Assm [h4s_options_optionu_u_CLAUSESu_c9]: !x. h4/option/option__CASE x h4/option/NONE h4/option/SOME = x
% Assm [h4s_sums_SUMu_u_MAPu_u_defu_c1]: !g f b. h4/sum/_2B_2B f g (happ h4/sum/INR b) = happ h4/sum/INR (happ g b)
% Assm [h4s_sums_SUMu_u_MAPu_u_defu_c0]: !g f a. h4/sum/_2B_2B f g (happ h4/sum/INL a) = happ h4/sum/INL (happ f a)
% 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_sums_FORALLu_u_SUM]: !P. (!s. happ P s) <=> (!x. happ P (happ h4/sum/INL x)) /\ (!y. happ P (happ h4/sum/INR y))
% Assm [h4s_combins_ou_u_THM]: !x g f. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_sums_ISL0u_c0]: !x. h4/sum/ISL (happ h4/sum/INL x)
% Assm [h4s_sums_ISL0u_c1]: !y. ~h4/sum/ISL (happ h4/sum/INR y)
% Assm [h4s_options_optionu_u_CLAUSESu_c10]: !x. h4/option/option__CASE x x h4/option/SOME = x
% Assm [h4s_options_NOTu_u_SOMEu_u_NONE]: !x. ~(happ h4/option/SOME x = h4/option/NONE)
% Assm [h4s_options_optionu_u_CLAUSESu_c13]: !x e. h4/option/IS__SOME x ==> h4/option/option__CASE x e h4/option/SOME = x
% Assm [h4s_options_optionu_u_CLAUSESu_c2]: !x. ~(h4/option/NONE = happ h4/option/SOME x)
% Assm [h4s_options_optionu_u_CLAUSESu_c3]: !x. ~(happ h4/option/SOME x = h4/option/NONE)
% Assm [h4s_options_FORALLu_u_OPTION]: !P. (!opt. happ P opt) <=> happ P h4/option/NONE /\ (!x. happ P (happ h4/option/SOME x))
% Goal: !b. h4/option/OPTION__GUARD b = h4/bool/COND b (happ h4/option/SOME h4/one/one0) h4/option/NONE
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_Q1135264,TV_Q1135260]: ![V_f, V_g]: (![V_x]: s(TV_Q1135260,happ(s(t_fun(TV_Q1135264,TV_Q1135260),V_f),s(TV_Q1135264,V_x))) = s(TV_Q1135260,happ(s(t_fun(TV_Q1135264,TV_Q1135260),V_g),s(TV_Q1135264,V_x))) => s(t_fun(TV_Q1135264,TV_Q1135260),V_f) = s(t_fun(TV_Q1135264,TV_Q1135260),V_g))).
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_options_OPTIONu_u_GUARDu_u_defu_c0, axiom, s(t_h4s_options_option(t_h4s_ones_one),h4s_options_optionu_u_guard(s(t_bool,t))) = s(t_h4s_options_option(t_h4s_ones_one),happ(s(t_fun(t_h4s_ones_one,t_h4s_options_option(t_h4s_ones_one)),h4s_options_some),s(t_h4s_ones_one,h4s_ones_one0)))).
fof(ah4s_options_OPTIONu_u_GUARDu_u_defu_c1, axiom, s(t_h4s_options_option(t_h4s_ones_one),h4s_options_optionu_u_guard(s(t_bool,f))) = s(t_h4s_options_option(t_h4s_ones_one),h4s_options_none)).
fof(ah4s_ones_oneu_u_axiom, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(TV_u_27a,t_h4s_ones_one),V_f) = s(t_fun(TV_u_27a,t_h4s_ones_one),V_g)).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_ones_one1, axiom, ![V_v]: s(t_h4s_ones_one,V_v) = s(t_h4s_ones_one,h4s_ones_one0)).
fof(ah4s_ones_oneu_u_Axiom, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_e, V_fn]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_ones_one,TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_ones_one,TV_u_27a),t_bool)),V_uu_0),s(TV_u_27a,V_e))),s(t_fun(t_h4s_ones_one,TV_u_27a),V_fn)))) <=> s(TV_u_27a,happ(s(t_fun(t_h4s_ones_one,TV_u_27a),V_fn),s(t_h4s_ones_one,h4s_ones_one0))) = s(TV_u_27a,V_e)) => ![V_e]: p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(t_fun(t_h4s_ones_one,TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_ones_one,TV_u_27a),t_bool)),V_uu_0),s(TV_u_27a,V_e)))))))).
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_ones_oneu_u_induction, axiom, ![V_P]: (p(s(t_bool,happ(s(t_fun(t_h4s_ones_one,t_bool),V_P),s(t_h4s_ones_one,h4s_ones_one0)))) => ![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_ones_one,t_bool),V_P),s(t_h4s_ones_one,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_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_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_ones_oneu_u_primu_u_rec, axiom, ![TV_u_27a]: ![V_e]: ?[V_fn]: s(TV_u_27a,happ(s(t_fun(t_h4s_ones_one,TV_u_27a),V_fn),s(t_h4s_ones_one,h4s_ones_one0))) = s(TV_u_27a,V_e)).
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_ones_FORALLu_u_ONE, axiom, ![V_P]: (![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_ones_one,t_bool),V_P),s(t_h4s_ones_one,V_x)))) <=> p(s(t_bool,happ(s(t_fun(t_h4s_ones_one,t_bool),V_P),s(t_h4s_ones_one,h4s_ones_one0)))))).
fof(ah4s_ones_oneu_u_DEF, axiom, ![V_uu_0]: (![V_x]: s(t_bool,happ(s(t_fun(t_h4s_ones_one,t_bool),V_uu_0),s(t_h4s_ones_one,V_x))) = s(t_bool,t) => s(t_h4s_ones_one,h4s_ones_one0) = s(t_h4s_ones_one,h4s_mins_u_40(s(t_fun(t_h4s_ones_one,t_bool),V_uu_0))))).
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_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_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_Tu_u_DEF, axiom, (p(s(t_bool,t)) <=> ![V_x]: s(t_bool,V_x) = s(t_bool,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_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_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_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_IMPu_u_THM, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) & (p(s(t_bool,V_t2)) => p(s(t_bool,V_t1)))))).
fof(ah4s_ones_oneu_u_TYu_u_DEF, axiom, ![V_uu_0]: (![V_b]: s(t_bool,happ(s(t_fun(t_bool,t_bool),V_uu_0),s(t_bool,V_b))) = s(t_bool,V_b) => ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_bool,t_bool),V_uu_0),s(t_fun(t_h4s_ones_one,t_bool),V_rep)))))).
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_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_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_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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_TYPEu_u_DEFINITIONu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_P]: (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_xu_27, V_xu_27u_27]: (s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_xu_27))) = s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_xu_27u_27))) => s(TV_u_27b,V_xu_27) = s(TV_u_27b,V_xu_27u_27)) & ![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) <=> ?[V_xu_27]: s(TV_u_27a,V_x) = s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_xu_27))))))).
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_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_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_options_IFu_u_EQUALSu_u_OPTIONu_c3, axiom, ![TV_u_27a]: ![V_y, V_x, V_P]: (s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_P),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))))) = 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))) <=> (~ (p(s(t_bool,V_P))) & s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))).
fof(ah4s_options_IFu_u_EQUALSu_u_OPTIONu_c1, axiom, ![TV_u_27a]: ![V_x, V_P]: (s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_P),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))))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) <=> p(s(t_bool,V_P)))).
fof(ah4s_options_IFu_u_EQUALSu_u_OPTIONu_c2, axiom, ![TV_u_27a]: ![V_y, V_x, V_P]: (s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(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))),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_y))) <=> (p(s(t_bool,V_P)) & s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))).
fof(ah4s_options_IFu_u_EQUALSu_u_OPTIONu_c0, axiom, ![TV_u_27a]: ![V_x, V_P]: (s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(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))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) <=> ~ (p(s(t_bool,V_P))))).
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_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_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_IFu_u_NONEu_u_EQUALSu_u_OPTIONu_c2, axiom, ![TV_u_27a]: ![V_x, V_X, V_P]: (s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_options_option(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))) <=> (p(s(t_bool,V_P)) & s(t_h4s_options_option(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)))))).
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_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_options_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_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_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
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_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_options_IFu_u_NONEu_u_EQUALSu_u_OPTIONu_c3, axiom, ![TV_u_27a]: ![V_x, V_X, V_P]: (s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_h4s_options_option(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))) <=> (~ (p(s(t_bool,V_P))) & s(t_h4s_options_option(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)))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_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_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_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_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_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_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_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_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_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_bools_ANDu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,f))) <=> p(s(t_bool,f)))).
fof(ah4s_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_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_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_IFu_u_NONEu_u_EQUALSu_u_OPTIONu_c0, axiom, ![TV_u_27a]: ![V_X, V_P]: (s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_options_option(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_none) <=> (p(s(t_bool,V_P)) => p(s(t_bool,h4s_options_isu_u_none(s(t_h4s_options_option(TV_u_27a),V_X))))))).
fof(ah4s_options_IFu_u_NONEu_u_EQUALSu_u_OPTIONu_c1, axiom, ![TV_u_27a]: ![V_X, V_P]: (s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(t_h4s_options_option(TV_u_27a),V_X))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) <=> (p(s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),V_X)))) => p(s(t_bool,V_P))))).
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_options_optionu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_options_none),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,V_v)).
fof(ah4s_options_optionu_u_caseu_u_compute, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f, V_e]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),V_x),s(TV_u_27b,V_e),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(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,h4s_options_the(s(t_h4s_options_option(TV_u_27a),V_x))))),s(TV_u_27b,V_e)))).
fof(ah4s_bools_boolu_u_caseu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_bools_AND1u_u_THM, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t1)))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE0, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
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_sats_dcu_u_cond, axiom, ![V_s, V_r, V_q, V_p]: (s(t_bool,V_p) = s(t_bool,h4s_bools_cond(s(t_bool,V_q),s(t_bool,V_r),s(t_bool,V_s))) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_s))))) & ((p(s(t_bool,V_p)) | (~ (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_s))))) & ((~ (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_s)) | ~ (p(s(t_bool,V_p))))))))))).
fof(ah4s_bools_CONDu_u_EXPANDu_u_IMP, axiom, ![V_t2, V_t1, V_b]: (p(s(t_bool,h4s_bools_cond(s(t_bool,V_b),s(t_bool,V_t1),s(t_bool,V_t2)))) <=> ((p(s(t_bool,V_b)) => p(s(t_bool,V_t1))) & (~ (p(s(t_bool,V_b))) => p(s(t_bool,V_t2)))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,V_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_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_CONDu_u_ABS, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f, V_b, V_x]: s(TV_u_27b,h4s_bools_cond(s(t_bool,V_b),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_bools_cond(s(t_bool,V_b),s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_g))),s(TV_u_27a,V_x)))).
fof(ah4s_bools_boolu_u_caseu_u_thmu_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_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_CONDu_u_EXPAND, axiom, ![V_t2, V_t1, V_b]: (p(s(t_bool,h4s_bools_cond(s(t_bool,V_b),s(t_bool,V_t1),s(t_bool,V_t2)))) <=> ((~ (p(s(t_bool,V_b))) | p(s(t_bool,V_t1))) & (p(s(t_bool,V_b)) | p(s(t_bool,V_t2)))))).
fof(ah4s_sums_condu_u_sumu_u_expandu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_z, V_y, V_x, V_P]: (s(t_h4s_sums_sum(TV_u_27b,TV_u_27a),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_sums_sum(TV_u_27b,TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,TV_u_27a)),h4s_sums_inr),s(TV_u_27a,V_x))),s(t_h4s_sums_sum(TV_u_27b,TV_u_27a),happ(s(t_fun(TV_u_27b,t_h4s_sums_sum(TV_u_27b,TV_u_27a)),h4s_sums_inl),s(TV_u_27b,V_y))))) = s(t_h4s_sums_sum(TV_u_27b,TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_sums_sum(TV_u_27b,TV_u_27a)),h4s_sums_inr),s(TV_u_27a,V_z))) <=> (p(s(t_bool,V_P)) & s(TV_u_27a,V_z) = s(TV_u_27a,V_x)))).
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),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_x))) = 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_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),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_bools_CONDu_u_RAND, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x, V_f, V_b]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,h4s_bools_cond(s(t_bool,V_b),s(TV_u_27a,V_x),s(TV_u_27a,V_y))))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_b),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y)))))).
fof(ah4s_sums_sumu_u_distinct, axiom, ![TV_u_27a,TV_u_27b]: ![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_27b,t_h4s_sums_sum(TV_u_27a,TV_u_27b)),h4s_sums_inr),s(TV_u_27b,V_y))))).
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_CONDu_u_EXPANDu_u_OR, axiom, ![V_t2, V_t1, V_b]: (p(s(t_bool,h4s_bools_cond(s(t_bool,V_b),s(t_bool,V_t1),s(t_bool,V_t2)))) <=> ((p(s(t_bool,V_b)) & p(s(t_bool,V_t1))) | (~ (p(s(t_bool,V_b))) & p(s(t_bool,V_t2)))))).
fof(ah4s_bools_boolu_u_caseu_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_sums_condu_u_sumu_u_expandu_c3, axiom, ![TV_u_27g,TV_u_27h]: ![V_z, V_y, V_x, V_P]: (s(t_h4s_sums_sum(TV_u_27g,TV_u_27h),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_sums_sum(TV_u_27g,TV_u_27h),happ(s(t_fun(TV_u_27g,t_h4s_sums_sum(TV_u_27g,TV_u_27h)),h4s_sums_inl),s(TV_u_27g,V_x))),s(t_h4s_sums_sum(TV_u_27g,TV_u_27h),happ(s(t_fun(TV_u_27h,t_h4s_sums_sum(TV_u_27g,TV_u_27h)),h4s_sums_inr),s(TV_u_27h,V_y))))) = s(t_h4s_sums_sum(TV_u_27g,TV_u_27h),happ(s(t_fun(TV_u_27h,t_h4s_sums_sum(TV_u_27g,TV_u_27h)),h4s_sums_inr),s(TV_u_27h,V_z))) <=> (~ (p(s(t_bool,V_P))) & s(TV_u_27h,V_z) = s(TV_u_27h,V_y)))).
fof(ah4s_sums_condu_u_sumu_u_expandu_c2, axiom, ![TV_u_27f,TV_u_27e]: ![V_z, V_y, V_x, V_P]: (s(t_h4s_sums_sum(TV_u_27e,TV_u_27f),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_sums_sum(TV_u_27e,TV_u_27f),happ(s(t_fun(TV_u_27e,t_h4s_sums_sum(TV_u_27e,TV_u_27f)),h4s_sums_inl),s(TV_u_27e,V_x))),s(t_h4s_sums_sum(TV_u_27e,TV_u_27f),happ(s(t_fun(TV_u_27f,t_h4s_sums_sum(TV_u_27e,TV_u_27f)),h4s_sums_inr),s(TV_u_27f,V_y))))) = s(t_h4s_sums_sum(TV_u_27e,TV_u_27f),happ(s(t_fun(TV_u_27e,t_h4s_sums_sum(TV_u_27e,TV_u_27f)),h4s_sums_inl),s(TV_u_27e,V_z))) <=> (p(s(t_bool,V_P)) & s(TV_u_27e,V_z) = s(TV_u_27e,V_x)))).
fof(ah4s_bools_CONDu_u_RATOR, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_g, V_f, V_b]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_bools_cond(s(t_bool,V_b),s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_b),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_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_sums_condu_u_sumu_u_expandu_c1, axiom, ![TV_u_27c,TV_u_27d]: ![V_z, V_y, V_x, V_P]: (s(t_h4s_sums_sum(TV_u_27d,TV_u_27c),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_sums_sum(TV_u_27d,TV_u_27c),happ(s(t_fun(TV_u_27c,t_h4s_sums_sum(TV_u_27d,TV_u_27c)),h4s_sums_inr),s(TV_u_27c,V_x))),s(t_h4s_sums_sum(TV_u_27d,TV_u_27c),happ(s(t_fun(TV_u_27d,t_h4s_sums_sum(TV_u_27d,TV_u_27c)),h4s_sums_inl),s(TV_u_27d,V_y))))) = s(t_h4s_sums_sum(TV_u_27d,TV_u_27c),happ(s(t_fun(TV_u_27d,t_h4s_sums_sum(TV_u_27d,TV_u_27c)),h4s_sums_inl),s(TV_u_27d,V_z))) <=> (~ (p(s(t_bool,V_P))) & s(TV_u_27d,V_z) = s(TV_u_27d,V_y)))).
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,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_MONOu_u_COND, axiom, ![V_z, V_y, V_x, V_w, V_b]: ((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,h4s_bools_cond(s(t_bool,V_b),s(t_bool,V_x),s(t_bool,V_z)))) => p(s(t_bool,h4s_bools_cond(s(t_bool,V_b),s(t_bool,V_y),s(t_bool,V_w)))))))).
fof(ah4s_combins_APPLYu_u_UPDATEu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_c, V_b, V_a]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(TV_u_27a,V_a) = s(TV_u_27a,V_c)) & s(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(TV_u_27a,V_c))) = 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_f),s(TV_u_27a,V_c))))))).
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_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_bools_literalu_u_caseu_u_id, axiom, ![TV_u_27a,TV_u_27b]: ![V_uu_0]: (![V_a, V_t, V_u, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_a)) & s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)))),V_uu_0),s(TV_u_27a,V_a))),s(TV_u_27b,V_t))),s(TV_u_27b,V_u))),s(TV_u_27a,V_x))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_v),s(TV_u_27b,V_t),s(TV_u_27b,V_u)))) => ![V_u, V_t, V_a]: s(TV_u_27b,h4s_bools_literalu_u_case(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_fun(TV_u_27b,t_fun(TV_u_27a,TV_u_27b)))),V_uu_0),s(TV_u_27a,V_a))),s(TV_u_27b,V_t))),s(TV_u_27b,V_u))),s(TV_u_27a,V_a))) = s(TV_u_27b,V_t))).
fof(ah4s_sums_SUMu_u_MAP, axiom, ![TV_u_27c,TV_u_27d,TV_u_27a,TV_u_27b]: ![V_z, V_g, V_f]: s(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),V_z))) = s(t_h4s_sums_sum(TV_u_27c,TV_u_27d),h4s_bools_cond(s(t_bool,h4s_sums_isl(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_z))),s(t_h4s_sums_sum(TV_u_27c,TV_u_27d),happ(s(t_fun(TV_u_27c,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,h4s_sums_outl(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_z))))))),s(t_h4s_sums_sum(TV_u_27c,TV_u_27d),happ(s(t_fun(TV_u_27d,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,h4s_sums_outr(s(t_h4s_sums_sum(TV_u_27a,TV_u_27b),V_z)))))))))).
fof(ah4s_bools_literalu_u_caseu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_x0]: s(TV_u_27b,h4s_bools_literalu_u_case(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(TV_u_27a,V_x0))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_x),s(TV_u_27a,V_x0)))).
fof(ah4s_options_optionu_u_CLAUSESu_c9, 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_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),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),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_b))))) = s(t_h4s_sums_sum(TV_u_27c,TV_u_27d),happ(s(t_fun(TV_u_27d,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_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),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),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_a))))) = s(t_h4s_sums_sum(TV_u_27c,TV_u_27d),happ(s(t_fun(TV_u_27c,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_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_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),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))))))))).
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_ISL0u_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: p(s(t_bool,h4s_sums_isl(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))))))).
fof(ah4s_sums_ISL0u_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_y]: ~ (p(s(t_bool,h4s_sums_isl(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)))))))).
fof(ah4s_options_optionu_u_CLAUSESu_c10, 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_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_CLAUSESu_c13, axiom, ![TV_u_27a]: ![V_x, V_e]: (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),h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),V_x),s(t_h4s_options_option(TV_u_27a),V_e),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_CLAUSESu_c2, 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_options_optionu_u_CLAUSESu_c3, 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_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(ch4s_options_OPTIONu_u_GUARDu_u_COND, conjecture, ![V_b]: s(t_h4s_options_option(t_h4s_ones_one),h4s_options_optionu_u_guard(s(t_bool,V_b))) = s(t_h4s_options_option(t_h4s_ones_one),h4s_bools_cond(s(t_bool,V_b),s(t_h4s_options_option(t_h4s_ones_one),happ(s(t_fun(t_h4s_ones_one,t_h4s_options_option(t_h4s_ones_one)),h4s_options_some),s(t_h4s_ones_one,h4s_ones_one0))),s(t_h4s_options_option(t_h4s_ones_one),h4s_options_none)))).
