%   ORIGINAL: h4/quotient__option/IS__NONE__PRS
% 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/bool/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/option/IS__NONE__EQ__NONE: !x. h4/option/IS__NONE x <=> x = h4/option/NONE
% Assm: h4/option/OPTION__MAP__DEF_c0: !x f. h4/option/OPTION__MAP f (h4/option/SOME x) = h4/option/SOME (f x)
% Assm: h4/option/OPTION__MAP__DEF_c1: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm: h4/option/OPTION__MAP__EQ__NONE: !x f. h4/option/OPTION__MAP f x = h4/option/NONE <=> x = h4/option/NONE
% Assm: h4/option/option__nchotomy: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm: h4/quotient/QUOTIENT__ABS__REP: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a. abs (rep a) = a)
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/quotient__option/IS__SOME__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x. h4/option/IS__SOME x <=> h4/option/IS__SOME (h4/option/OPTION__MAP rep x))
% 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/option/OPTION__APPLY__def_c1: !x f. h4/option/OPTION__APPLY (h4/option/SOME f) x = h4/option/OPTION__MAP f x
% Assm: h4/option/OPTION__MAP__EQ__SOME: !y x f. h4/option/OPTION__MAP f x = h4/option/SOME y <=> (?z. x = h4/option/SOME z /\ y = f z)
% Assm: h4/option/OPTION__MAP__EQ__NONE__both__ways_c1: !x f. h4/option/NONE = h4/option/OPTION__MAP f x <=> x = h4/option/NONE
% Assm: h4/option/OPTION__MAP__CONG: !opt2 opt1 f2 f1. opt1 = opt2 /\ (!x. opt2 = h4/option/SOME x ==> f1 x = f2 x) ==> h4/option/OPTION__MAP f1 opt1 = h4/option/OPTION__MAP f2 opt2
% Assm: h4/option/OPTION__MAP__EQ__NONE__both__ways_c0: !x f. h4/option/OPTION__MAP f x = h4/option/NONE <=> x = h4/option/NONE
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/option/option__CLAUSES_c11: !x f e. h4/option/IS__NONE x ==> h4/option/option__CASE x e f = e
% 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/option__CLAUSES_c6: !x. h4/option/IS__NONE x <=> x = h4/option/NONE
% Assm: h4/quotient/FUN__MAP__THM: !x h g f. h4/quotient/_2D_2D_3E f g h x = g (h (f x))
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/option/option__CLAUSES_c17: !f. h4/option/OPTION__MAP f h4/option/NONE = h4/option/NONE
% Assm: h4/option/option__CLAUSES_c16: !x f. h4/option/OPTION__MAP f (h4/option/SOME x) = h4/option/SOME (f x)
% Assm: h4/option/IS__SOME__DEF_c0: !x. h4/option/IS__SOME (h4/option/SOME x) <=> T
% Assm: h4/option/IS__SOME__DEF_c1: h4/option/IS__SOME h4/option/NONE <=> F
% Assm: h4/quotient__option/SOME__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x. h4/option/SOME x = h4/option/OPTION__MAP abs (h4/option/SOME (rep x)))
% Assm: h4/option/OPTION__APPLY__MAP2: !y x f. h4/option/OPTION__APPLY (h4/option/OPTION__MAP f x) y = h4/option/OPTION__MAP2 f x y
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/option/option__CLAUSES_c3: !x. ~(h4/option/SOME x = h4/option/NONE)
% Assm: h4/quotient__option/NONE__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> h4/option/NONE = h4/option/OPTION__MAP abs h4/option/NONE
% Assm: h4/quotient__option/OPTION__QUOTIENT: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> h4/quotient/QUOTIENT (h4/option/OPTREL R) (h4/option/OPTION__MAP abs) (h4/option/OPTION__MAP rep)
% Assm: h4/quotient__option/IS__SOME__RSP: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x y. h4/option/OPTREL R x y ==> (h4/option/IS__SOME x <=> h4/option/IS__SOME y))
% Assm: h4/option/OPTION__MAP__COMPOSE: !x g f. h4/option/OPTION__MAP f (h4/option/OPTION__MAP g x) = h4/option/OPTION__MAP (h4/combin/o f g) x
% Assm: h4/quotient/FUN__MAP: !g f. h4/quotient/_2D_2D_3E f g = (\h x. g (h (f x)))
% Assm: h4/quotient__option/OPTION__MAP__I: h4/option/OPTION__MAP h4/combin/I = h4/combin/I
% 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/quotient/QUOTIENT__REL__REP: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a b. R (rep a) (rep b) <=> a = b)
% Assm: h4/quotient/FORALL__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f. $forall f <=> h4/bool/RES__FORALL (h4/quotient/respects R) (h4/quotient/_2D_2D_3E abs h4/combin/I f))
% Assm: h4/quotient/K__PRS: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!x y. h4/combin/K x y = abs1 (h4/combin/K (rep1 x) (rep2 y))))
% Assm: h4/quotient/literal__case__PRS: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f x. h4/bool/literal__case f x = abs2 (h4/bool/literal__case (h4/quotient/_2D_2D_3E abs1 rep2 f) (rep1 x))))
% Assm: h4/quotient/LAMBDA__PRS: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f. (\x. f x) = h4/quotient/_2D_2D_3E rep1 abs2 (\x. rep2 (f (abs1 x)))))
% Assm: h4/quotient/LET__PRS: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f x. h4/bool/LET f x = abs2 (h4/bool/LET (h4/quotient/_2D_2D_3E abs1 rep2 f) (rep1 x))))
% Assm: h4/quotient/RES__EXISTS__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!P f. h4/bool/RES__EXISTS P f <=> h4/bool/RES__EXISTS (h4/quotient/_2D_2D_3E abs h4/combin/I P) (h4/quotient/_2D_2D_3E abs h4/combin/I f))
% Assm: h4/quotient/o__PRS: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!R3 abs3 rep3. h4/quotient/QUOTIENT R3 abs3 rep3 ==> (!f g. h4/combin/o f g = h4/quotient/_2D_2D_3E rep1 abs3 (h4/combin/o (h4/quotient/_2D_2D_3E abs2 rep3 f) (h4/quotient/_2D_2D_3E abs1 rep2 g)))))
% Assm: h4/quotient/W__PRS: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f x. h4/combin/W f x = abs2 (h4/combin/W (h4/quotient/_2D_2D_3E abs1 (h4/quotient/_2D_2D_3E abs1 rep2) f) (rep1 x))))
% Assm: h4/quotient/APPLY__PRS: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f x. f x = abs2 (h4/quotient/_2D_2D_3E abs1 rep2 f (rep1 x))))
% Assm: h4/quotient/I__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!e. h4/combin/I e = abs (h4/combin/I (rep e)))
% Assm: h4/quotient/EXISTS__UNIQUE__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f. h4/bool/_3F_21 f <=> h4/quotient/RES__EXISTS__EQUIV R (h4/quotient/_2D_2D_3E abs h4/combin/I f))
% Assm: h4/quotient/C__PRS: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!R3 abs3 rep3. h4/quotient/QUOTIENT R3 abs3 rep3 ==> (!f x y. h4/combin/C f x y = abs3 (h4/combin/C (h4/quotient/_2D_2D_3E abs1 (h4/quotient/_2D_2D_3E abs2 rep3) f) (rep2 x) (rep1 y)))))
% Assm: h4/option/NOT__SOME__NONE: !x. ~(h4/option/SOME x = h4/option/NONE)
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/combin/W__THM: !x f. h4/combin/W f x = f x x
% Assm: h4/combin/o__THM: !x g f. h4/combin/o f g x = f (g x)
% Assm: h4/quotient/QUOTIENT__def: !rep abs R. h4/quotient/QUOTIENT R abs rep <=> (!a. abs (rep a) = a) /\ (!a. R (rep a) (rep a)) /\ (!r s. R r s <=> R r r /\ R s s /\ abs r = abs s)
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/option/NOT__NONE__SOME: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/combin/K__THM: !y x. h4/combin/K x y = x
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/pred__set/SPECIFICATION: !x P. h4/bool/IN x P <=> P x
% Assm: h4/bool/literal__case__DEF: h4/bool/literal__case = (\f x. f x)
% Assm: h4/bool/LET__DEF: h4/bool/LET = (\f x. f x)
% Assm: h4/quotient/QUOTIENT__REP__REFL: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a. R (rep a) (rep a))
% Assm: h4/combin/C__THM: !y x f. h4/combin/C f x y = f y x
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/option/OPTION__APPLY__def_c0: !x. h4/option/OPTION__APPLY h4/option/NONE x = h4/option/NONE
% Assm: h4/option/OPTION__MAP2__NONE: !o2 o1 f. h4/option/OPTION__MAP2 f o1 o2 = h4/option/NONE <=> o1 = h4/option/NONE \/ o2 = h4/option/NONE
% Assm: h4/option/OPTION__MAP2__THM_c1: !x f. h4/option/OPTION__MAP2 f (h4/option/SOME x) h4/option/NONE = h4/option/NONE
% Assm: h4/option/OPTION__MAP2__THM_c0: !y x f. h4/option/OPTION__MAP2 f (h4/option/SOME x) (h4/option/SOME y) = h4/option/SOME (f x y)
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/EXISTS__SIMP: !t. (?x. t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/bool/UNWIND__THM2: !a P. (?x. x = a /\ P x) <=> P a
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/quotient__option/OPTION__REL__def_c1: !x R. h4/option/OPTREL R (h4/option/SOME x) h4/option/NONE <=> F
% Assm: h4/quotient__option/OPTION__REL__def_c3: !y x R. h4/option/OPTREL R (h4/option/SOME x) (h4/option/SOME y) <=> R x y
% Assm: h4/quotient__option/OPTION__REL__def_c2: !y R. h4/option/OPTREL R h4/option/NONE (h4/option/SOME y) <=> F
% Assm: h4/quotient__option/OPTION__REL__def_c0: !R. h4/option/OPTREL R h4/option/NONE h4/option/NONE <=> T
% Assm: h4/bool/COND__ID: !t b. h4/bool/COND b t t = t
% 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/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_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_c1: !x P. h4/bool/COND P h4/option/NONE (h4/option/SOME x) = h4/option/NONE <=> P
% 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/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/quotient/QUOTIENT__REL: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!r s. R r s <=> R r r /\ R s s /\ abs r = abs s)
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/quotient/respects__def: h4/quotient/respects = h4/combin/W
% Assm: h4/res__quan/RES__FORALL: !f P. h4/bool/RES__FORALL P f <=> (!x. h4/bool/IN x P ==> f x)
% Assm: h4/res__quan/RES__EXISTS: !f P. h4/bool/RES__EXISTS P f <=> (?x. h4/bool/IN x P /\ f x)
% Assm: h4/option/option__CLAUSES_c2: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm: h4/option/option__CLAUSES_c0: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/bool/FORALL__DEF: $forall = (\P. P = (\x. T))
% Assm: h4/bool/EXISTS__UNIQUE__DEF: h4/bool/_3F_21 = (\P. $exists P /\ (!x y. P x /\ P y ==> x = y))
% Assm: h4/quotient/EXISTS__PRS: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f. $exists f <=> h4/bool/RES__EXISTS (h4/quotient/respects R) (h4/quotient/_2D_2D_3E abs h4/combin/I f))
% Assm: h4/quotient/RES__EXISTS__EQUIV0: !m R. h4/quotient/RES__EXISTS__EQUIV R m <=> h4/bool/RES__EXISTS (h4/quotient/respects R) (\x. m x) /\ h4/bool/RES__FORALL (h4/quotient/respects R) (\x. h4/bool/RES__FORALL (h4/quotient/respects R) (\y. m x /\ m y ==> R x y))
% Goal: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x. h4/option/IS__NONE x <=> h4/option/IS__NONE (h4/option/OPTION__MAP rep 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 (h4/option/SOME x) = happ f x)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> 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_MAPu_u_DEFu_c0]: !x f. happ (h4/option/OPTION__MAP f) (h4/option/SOME x) = h4/option/SOME (happ f x)
% Assm [h4s_options_OPTIONu_u_MAPu_u_DEFu_c1]: !f. happ (h4/option/OPTION__MAP f) h4/option/NONE = h4/option/NONE
% Assm [h4s_options_OPTIONu_u_MAPu_u_EQu_u_NONE]: !x f. happ (h4/option/OPTION__MAP f) x = h4/option/NONE <=> x = h4/option/NONE
% Assm [h4s_options_optionu_u_nchotomy]: !opt. opt = h4/option/NONE \/ (?x. opt = h4/option/SOME x)
% Assm [h4s_quotients_QUOTIENTu_u_ABSu_u_REP]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a. happ abs (happ rep a) = a)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_quotientu_u_options_ISu_u_SOMEu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x. h4/option/IS__SOME x <=> h4/option/IS__SOME (happ (h4/option/OPTION__MAP rep) x))
% Assm [h4s_options_ISu_u_NONEu_u_DEFu_c0]: !x. h4/option/IS__NONE (h4/option/SOME x) <=> F
% Assm [h4s_options_ISu_u_NONEu_u_DEFu_c1]: h4/option/IS__NONE h4/option/NONE <=> T
% Assm [h4s_options_OPTIONu_u_APPLYu_u_defu_c1]: !x f. h4/option/OPTION__APPLY (h4/option/SOME f) x = happ (h4/option/OPTION__MAP f) x
% Assm [h4s_options_OPTIONu_u_MAPu_u_EQu_u_SOME]: !y x f. happ (h4/option/OPTION__MAP f) x = h4/option/SOME y <=> (?z. x = h4/option/SOME z /\ y = happ f z)
% Assm [h4s_options_OPTIONu_u_MAPu_u_EQu_u_NONEu_u_bothu_u_waysu_c1]: !x f. h4/option/NONE = happ (h4/option/OPTION__MAP f) x <=> x = h4/option/NONE
% Assm [h4s_options_OPTIONu_u_MAPu_u_CONG]: !opt2 opt1 f2 f1. opt1 = opt2 /\ (!x. opt2 = h4/option/SOME x ==> happ f1 x = happ f2 x) ==> happ (h4/option/OPTION__MAP f1) opt1 = happ (h4/option/OPTION__MAP f2) opt2
% Assm [h4s_options_OPTIONu_u_MAPu_u_EQu_u_NONEu_u_bothu_u_waysu_c0]: !x f. happ (h4/option/OPTION__MAP f) x = h4/option/NONE <=> x = h4/option/NONE
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_options_optionu_u_CLAUSESu_c11]: !x f e. h4/option/IS__NONE x ==> h4/option/option__CASE x e f = e
% 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_optionu_u_CLAUSESu_c6]: !x. h4/option/IS__NONE x <=> x = h4/option/NONE
% Assm [h4s_quotients_FUNu_u_MAPu_u_THM]: !x h g f. happ (happ (h4/quotient/_2D_2D_3E f g) h) x = happ g (happ h (happ f x))
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_options_optionu_u_CLAUSESu_c17]: !f. happ (h4/option/OPTION__MAP f) h4/option/NONE = h4/option/NONE
% Assm [h4s_options_optionu_u_CLAUSESu_c16]: !x f. happ (h4/option/OPTION__MAP f) (h4/option/SOME x) = h4/option/SOME (happ f x)
% Assm [h4s_options_ISu_u_SOMEu_u_DEFu_c0]: !x. h4/option/IS__SOME (h4/option/SOME x) <=> T
% Assm [h4s_options_ISu_u_SOMEu_u_DEFu_c1]: h4/option/IS__SOME h4/option/NONE <=> F
% Assm [h4s_quotientu_u_options_SOMEu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x. h4/option/SOME x = happ (h4/option/OPTION__MAP abs) (h4/option/SOME (happ rep x)))
% Assm [h4s_options_OPTIONu_u_APPLYu_u_MAP2]: !y x f. h4/option/OPTION__APPLY (happ (h4/option/OPTION__MAP f) x) y = h4/option/OPTION__MAP2 f x y
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_combins_Iu_u_THM]: !x. happ h4/combin/I x = x
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_options_optionu_u_CLAUSESu_c3]: !x. ~(h4/option/SOME x = h4/option/NONE)
% Assm [h4s_quotientu_u_options_NONEu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> h4/option/NONE = happ (h4/option/OPTION__MAP abs) h4/option/NONE
% Assm [h4s_quotientu_u_options_OPTIONu_u_QUOTIENT]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> h4/quotient/QUOTIENT (h4/option/OPTREL R) (h4/option/OPTION__MAP abs) (h4/option/OPTION__MAP rep)
% Assm [h4s_quotientu_u_options_ISu_u_SOMEu_u_RSP]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x y. happ (happ (h4/option/OPTREL R) x) y ==> (h4/option/IS__SOME x <=> h4/option/IS__SOME y))
% Assm [h4s_options_OPTIONu_u_MAPu_u_COMPOSE]: !x g f. happ (h4/option/OPTION__MAP f) (happ (h4/option/OPTION__MAP g) x) = happ (h4/option/OPTION__MAP (h4/combin/o f g)) x
% Assm [h4s_quotients_FUNu_u_MAP]: !g f x x. happ (happ (h4/quotient/_2D_2D_3E f g) x) x = happ g (happ x (happ f x))
% Assm [h4s_quotientu_u_options_OPTIONu_u_MAPu_u_I]: h4/option/OPTION__MAP h4/combin/I = h4/combin/I
% Assm [h4s_options_optionu_u_caseu_u_defu_c1]: !x v f. h4/option/option__CASE (h4/option/SOME x) v f = happ f x
% Assm [h4s_options_optionu_u_caseu_u_defu_c0]: !v f. h4/option/option__CASE h4/option/NONE v f = v
% Assm [h4s_quotients_QUOTIENTu_u_RELu_u_REP]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a b. happ (happ R (happ rep a)) (happ rep b) <=> a = b)
% Assm [h4s_quotients_FORALLu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f. $forall f <=> h4/bool/RES__FORALL (happ h4/quotient/respects R) (happ (h4/quotient/_2D_2D_3E abs h4/combin/I) f))
% Assm [h4s_quotients_Ku_u_PRS]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!x y. h4/combin/K x y = happ abs1 (h4/combin/K (happ rep1 x) (happ rep2 y))))
% Assm [h4s_quotients_literalu_u_caseu_u_PRS]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f x. h4/bool/literal__case f x = happ abs2 (h4/bool/literal__case (happ (h4/quotient/_2D_2D_3E abs1 rep2) f) (happ rep1 x))))
% Assm [h4s_quotients_LAMBDAu_u_PRS]: !_0. (!rep2 f abs1 x. happ (happ (happ (happ _0 rep2) f) abs1) x = happ rep2 (happ f (happ abs1 x))) ==> (!rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f x. happ f x = happ (happ (h4/quotient/_2D_2D_3E rep1 abs2) (happ (happ (happ _0 rep2) f) abs1)) x)))
% Assm [h4s_quotients_LETu_u_PRS]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f x. h4/bool/LET f x = happ abs2 (h4/bool/LET (happ (h4/quotient/_2D_2D_3E abs1 rep2) f) (happ rep1 x))))
% Assm [h4s_quotients_RESu_u_EXISTSu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!P f. h4/bool/RES__EXISTS P f <=> h4/bool/RES__EXISTS (happ (h4/quotient/_2D_2D_3E abs h4/combin/I) P) (happ (h4/quotient/_2D_2D_3E abs h4/combin/I) f))
% Assm [h4s_quotients_ou_u_PRS]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!R3 abs3 rep3. h4/quotient/QUOTIENT R3 abs3 rep3 ==> (!f g. h4/combin/o f g = happ (h4/quotient/_2D_2D_3E rep1 abs3) (h4/combin/o (happ (h4/quotient/_2D_2D_3E abs2 rep3) f) (happ (h4/quotient/_2D_2D_3E abs1 rep2) g)))))
% Assm [h4s_quotients_Wu_u_PRS]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f x. happ (happ h4/combin/W f) x = happ abs2 (happ (happ h4/combin/W (happ (h4/quotient/_2D_2D_3E abs1 (h4/quotient/_2D_2D_3E abs1 rep2)) f)) (happ rep1 x))))
% Assm [h4s_quotients_APPLYu_u_PRS]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!f x. happ f x = happ abs2 (happ (happ (h4/quotient/_2D_2D_3E abs1 rep2) f) (happ rep1 x))))
% Assm [h4s_quotients_Iu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!e. happ h4/combin/I e = happ abs (happ h4/combin/I (happ rep e)))
% Assm [h4s_quotients_EXISTSu_u_UNIQUEu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f. h4/bool/_3F_21 f <=> h4/quotient/RES__EXISTS__EQUIV R (happ (h4/quotient/_2D_2D_3E abs h4/combin/I) f))
% Assm [h4s_quotients_Cu_u_PRS]: !rep1 abs1 R1. h4/quotient/QUOTIENT R1 abs1 rep1 ==> (!R2 abs2 rep2. h4/quotient/QUOTIENT R2 abs2 rep2 ==> (!R3 abs3 rep3. h4/quotient/QUOTIENT R3 abs3 rep3 ==> (!f x y. h4/combin/C f x y = happ abs3 (h4/combin/C (happ (h4/quotient/_2D_2D_3E abs1 (h4/quotient/_2D_2D_3E abs2 rep3)) f) (happ rep2 x) (happ rep1 y)))))
% Assm [h4s_options_NOTu_u_SOMEu_u_NONE]: !x. ~(h4/option/SOME x = h4/option/NONE)
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_combins_Wu_u_THM]: !x f. happ (happ h4/combin/W f) x = happ (happ f x) x
% Assm [h4s_combins_ou_u_THM]: !x g f. happ (h4/combin/o f g) x = happ f (happ g x)
% Assm [h4s_quotients_QUOTIENTu_u_def]: !rep abs R. h4/quotient/QUOTIENT R abs rep <=> (!a. happ abs (happ rep a) = a) /\ (!a. happ (happ R (happ rep a)) (happ rep a)) /\ (!r s. happ (happ R r) s <=> happ (happ R r) r /\ happ (happ R s) s /\ happ abs r = happ abs s)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_options_NOTu_u_NONEu_u_SOME]: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_combins_Ku_u_THM]: !y x. h4/combin/K x y = x
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_predu_u_sets_SPECIFICATION]: !x P. h4/bool/IN x P <=> happ P x
% Assm [h4s_bools_literalu_u_caseu_u_DEF]: !x x. h4/bool/literal__case x x = happ x x
% Assm [h4s_bools_LETu_u_DEF]: !x x. h4/bool/LET x x = happ x x
% Assm [h4s_quotients_QUOTIENTu_u_REPu_u_REFL]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!a. happ (happ R (happ rep a)) (happ rep a))
% Assm [h4s_combins_Cu_u_THM]: !y x f. h4/combin/C f x y = happ (happ f y) x
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_options_OPTIONu_u_APPLYu_u_defu_c0]: !x. h4/option/OPTION__APPLY h4/option/NONE x = h4/option/NONE
% Assm [h4s_options_OPTIONu_u_MAP2u_u_NONE]: !o2 o1 f. h4/option/OPTION__MAP2 f o1 o2 = h4/option/NONE <=> o1 = h4/option/NONE \/ o2 = h4/option/NONE
% Assm [h4s_options_OPTIONu_u_MAP2u_u_THMu_c1]: !x f. h4/option/OPTION__MAP2 f (h4/option/SOME x) h4/option/NONE = h4/option/NONE
% Assm [h4s_options_OPTIONu_u_MAP2u_u_THMu_c0]: !y x f. h4/option/OPTION__MAP2 f (h4/option/SOME x) (h4/option/SOME y) = h4/option/SOME (happ (happ f x) y)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_EXISTSu_u_SIMP]: !t. (?x. t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_bools_UNWINDu_u_THM2]: !a P. (?x. x = a /\ happ P x) <=> happ P a
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_quotientu_u_options_OPTIONu_u_RELu_u_defu_c1]: !x R. happ (happ (h4/option/OPTREL R) (h4/option/SOME x)) h4/option/NONE <=> F
% Assm [h4s_quotientu_u_options_OPTIONu_u_RELu_u_defu_c3]: !y x R. happ (happ (h4/option/OPTREL R) (h4/option/SOME x)) (h4/option/SOME y) <=> happ (happ R x) y
% Assm [h4s_quotientu_u_options_OPTIONu_u_RELu_u_defu_c2]: !y R. happ (happ (h4/option/OPTREL R) h4/option/NONE) (h4/option/SOME y) <=> F
% Assm [h4s_quotientu_u_options_OPTIONu_u_RELu_u_defu_c0]: !R. happ (happ (h4/option/OPTREL R) h4/option/NONE) h4/option/NONE <=> T
% Assm [h4s_bools_CONDu_u_ID]: !t b. h4/bool/COND b t t = t
% Assm [h4s_options_IFu_u_EQUALSu_u_OPTIONu_c0]: !x P. h4/bool/COND P (h4/option/SOME x) h4/option/NONE = h4/option/NONE <=> ~P
% Assm [h4s_options_IFu_u_EQUALSu_u_OPTIONu_c3]: !y x P. h4/bool/COND P h4/option/NONE (h4/option/SOME x) = h4/option/SOME y <=> ~P /\ x = y
% Assm [h4s_options_IFu_u_EQUALSu_u_OPTIONu_c2]: !y x P. h4/bool/COND P (h4/option/SOME x) h4/option/NONE = 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 (h4/option/SOME x) = h4/option/NONE <=> P
% 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_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_quotients_QUOTIENTu_u_REL]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!r s. happ (happ R r) s <=> happ (happ R r) r /\ happ (happ R s) s /\ happ abs r = happ abs s)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_quotients_respectsu_u_def]: h4/quotient/respects = h4/combin/W
% Assm [h4s_resu_u_quans_RESu_u_FORALL]: !f P. h4/bool/RES__FORALL P f <=> (!x. h4/bool/IN x P ==> happ f x)
% Assm [h4s_resu_u_quans_RESu_u_EXISTS]: !f P. h4/bool/RES__EXISTS P f <=> (?x. h4/bool/IN x P /\ happ f x)
% Assm [h4s_options_optionu_u_CLAUSESu_c2]: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm [h4s_options_optionu_u_CLAUSESu_c0]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_bools_FORALLu_u_DEF]: !x. $forall x <=> (!x. happ x x <=> 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_quotients_EXISTSu_u_PRS]: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!f. $exists f <=> h4/bool/RES__EXISTS (happ h4/quotient/respects R) (happ (h4/quotient/_2D_2D_3E abs h4/combin/I) f))
% Assm [h4s_quotients_RESu_u_EXISTSu_u_EQUIV0]: !_2. (!m R x y. happ (happ (happ (happ _2 m) R) x) y <=> happ m x /\ happ m y ==> happ (happ R x) y) ==> (!_1. (!m R x. happ (happ (happ _1 m) R) x <=> h4/bool/RES__FORALL (happ h4/quotient/respects R) (happ (happ (happ _2 m) R) x)) ==> (!_0. (!m x. happ (happ _0 m) x <=> happ m x) ==> (!m R. h4/quotient/RES__EXISTS__EQUIV R m <=> h4/bool/RES__EXISTS (happ h4/quotient/respects R) (happ _0 m) /\ h4/bool/RES__FORALL (happ h4/quotient/respects R) (happ _0 (happ (happ _1 m) R)))))
% Goal: !rep abs R. h4/quotient/QUOTIENT R abs rep ==> (!x. h4/option/IS__NONE x <=> h4/option/IS__NONE (happ (h4/option/OPTION__MAP rep) 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_Q1295928,TV_Q1295924]: ![V_f, V_g]: (![V_x]: s(TV_Q1295924,happ(s(t_fun(TV_Q1295928,TV_Q1295924),V_f),s(TV_Q1295928,V_x))) = s(TV_Q1295924,happ(s(t_fun(TV_Q1295928,TV_Q1295924),V_g),s(TV_Q1295928,V_x))) => s(t_fun(TV_Q1295928,TV_Q1295924),V_f) = s(t_fun(TV_Q1295928,TV_Q1295924),V_g))).
fof(ah4s_options_optionu_u_Axiom, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_e]: ?[V_fn]: (s(TV_u_27b,happ(s(t_fun(t_h4s_options_option(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(TV_u_27b,V_e) & ![V_x]: s(TV_u_27b,happ(s(t_fun(t_h4s_options_option(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_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_MAPu_u_DEFu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27b)),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))))).
fof(ah4s_options_OPTIONu_u_MAPu_u_DEFu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27b)),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_h4s_options_option(TV_u_27b),h4s_options_none)).
fof(ah4s_options_OPTIONu_u_MAPu_u_EQu_u_NONE, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: (s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27b)),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_options_option(TV_u_27a),V_x))) = s(t_h4s_options_option(TV_u_27b),h4s_options_none) <=> s(t_h4s_options_option(TV_u_27a),V_x) = s(t_h4s_options_option(TV_u_27a),h4s_options_none))).
fof(ah4s_options_optionu_u_nchotomy, axiom, ![TV_u_27a]: ![V_opt]: (s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) | ?[V_x]: s(t_h4s_options_option(TV_u_27a),V_opt) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_quotients_QUOTIENTu_u_ABSu_u_REP, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))) = s(TV_u_27b,V_a))).
fof(ah4s_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_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_quotientu_u_options_ISu_u_SOMEu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_x]: s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27b),V_x))) = s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_options_option(TV_u_27b),t_h4s_options_option(TV_u_27a)),h4s_options_optionu_u_map(s(t_fun(TV_u_27b,TV_u_27a),V_rep))),s(t_h4s_options_option(TV_u_27b),V_x))))))).
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),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_options_OPTIONu_u_APPLYu_u_defu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(TV_u_27b,TV_u_27a)),h4s_options_some(s(t_fun(TV_u_27b,TV_u_27a),V_f))),s(t_h4s_options_option(TV_u_27b),V_x))) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_options_option(TV_u_27b),t_h4s_options_option(TV_u_27a)),h4s_options_optionu_u_map(s(t_fun(TV_u_27b,TV_u_27a),V_f))),s(t_h4s_options_option(TV_u_27b),V_x)))).
fof(ah4s_options_OPTIONu_u_MAPu_u_EQu_u_SOME, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x, V_f]: (s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27b)),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_options_option(TV_u_27a),V_x))) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,V_y))) <=> ?[V_z]: (s(t_h4s_options_option(TV_u_27a),V_x) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_z))) & s(TV_u_27b,V_y) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_z)))))).
fof(ah4s_options_OPTIONu_u_MAPu_u_EQu_u_NONEu_u_bothu_u_waysu_c1, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f]: (s(t_h4s_options_option(TV_u_27a),h4s_options_none) = s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_options_option(TV_u_27b),t_h4s_options_option(TV_u_27a)),h4s_options_optionu_u_map(s(t_fun(TV_u_27b,TV_u_27a),V_f))),s(t_h4s_options_option(TV_u_27b),V_x))) <=> s(t_h4s_options_option(TV_u_27b),V_x) = s(t_h4s_options_option(TV_u_27b),h4s_options_none))).
fof(ah4s_options_OPTIONu_u_MAPu_u_CONG, axiom, ![TV_u_27b,TV_u_27a]: ![V_opt2, V_opt1, V_f2, V_f1]: ((s(t_h4s_options_option(TV_u_27a),V_opt1) = s(t_h4s_options_option(TV_u_27a),V_opt2) & ![V_x]: (s(t_h4s_options_option(TV_u_27a),V_opt2) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) => s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f2),s(TV_u_27a,V_x))))) => s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27b)),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f1))),s(t_h4s_options_option(TV_u_27a),V_opt1))) = s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27b)),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f2))),s(t_h4s_options_option(TV_u_27a),V_opt2))))).
fof(ah4s_options_OPTIONu_u_MAPu_u_EQu_u_NONEu_u_bothu_u_waysu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f]: (s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_options_option(TV_u_27b),t_h4s_options_option(TV_u_27a)),h4s_options_optionu_u_map(s(t_fun(TV_u_27b,TV_u_27a),V_f))),s(t_h4s_options_option(TV_u_27b),V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) <=> s(t_h4s_options_option(TV_u_27b),V_x) = s(t_h4s_options_option(TV_u_27b),h4s_options_none))).
fof(ah4s_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_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_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_options_SOMEu_u_11, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_options_optionu_u_CLAUSESu_c11, axiom, ![TV_u_27a,TV_u_27b]: ![V_x, V_f, V_e]: (p(s(t_bool,h4s_options_isu_u_none(s(t_h4s_options_option(TV_u_27a),V_x)))) => 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,V_e))).
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_optionu_u_CLAUSESu_c6, 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_quotients_FUNu_u_MAPu_u_THM, axiom, ![TV_u_27d,TV_u_27b,TV_u_27c,TV_u_27a]: ![V_x, V_h, V_g, V_f]: s(TV_u_27d,happ(s(t_fun(TV_u_27a,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27a,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g))),s(t_fun(TV_u_27c,TV_u_27b),V_h))),s(TV_u_27a,V_x))) = s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_h),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x)))))))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_options_optionu_u_CLAUSESu_c17, axiom, ![TV_u_27a,TV_u_27b]: ![V_f]: s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27b)),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_h4s_options_option(TV_u_27b),h4s_options_none)).
fof(ah4s_options_optionu_u_CLAUSESu_c16, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27b)),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))) = s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))))).
fof(ah4s_options_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),h4s_options_some(s(TV_u_27a,V_x))))) = s(t_bool,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_quotientu_u_options_SOMEu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_x]: s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,V_x))) = s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27b)),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_abs))),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_x))))))))).
fof(ah4s_options_OPTIONu_u_APPLYu_u_MAP2, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(t_h4s_options_option(TV_u_27c),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(t_fun(TV_u_27b,TV_u_27c))),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f))),s(t_h4s_options_option(TV_u_27a),V_x))),s(t_h4s_options_option(TV_u_27b),V_y))) = s(t_h4s_options_option(TV_u_27c),h4s_options_optionu_u_map2(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(t_h4s_options_option(TV_u_27a),V_x),s(t_h4s_options_option(TV_u_27b),V_y)))).
fof(ah4s_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_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_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_options_optionu_u_CLAUSESu_c3, axiom, ![TV_u_27a]: ![V_x]: ~ (s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none))).
fof(ah4s_quotientu_u_options_NONEu_u_PRS, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => s(t_h4s_options_option(TV_u_27b),h4s_options_none) = s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27b)),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_abs))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))))).
fof(ah4s_quotientu_u_options_OPTIONu_u_QUOTIENT, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => p(s(t_bool,h4s_quotients_quotient(s(t_fun(t_h4s_options_option(TV_u_27a),t_fun(t_h4s_options_option(TV_u_27a),t_bool)),h4s_options_optrel(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27b)),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),V_abs))),s(t_fun(t_h4s_options_option(TV_u_27b),t_h4s_options_option(TV_u_27a)),h4s_options_optionu_u_map(s(t_fun(TV_u_27b,TV_u_27a),V_rep)))))))).
fof(ah4s_quotientu_u_options_ISu_u_SOMEu_u_RSP, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_fun(t_h4s_options_option(TV_u_27a),t_bool)),h4s_options_optrel(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_h4s_options_option(TV_u_27a),V_x))),s(t_h4s_options_option(TV_u_27a),V_y)))) => s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),V_x))) = s(t_bool,h4s_options_isu_u_some(s(t_h4s_options_option(TV_u_27a),V_y)))))).
fof(ah4s_options_OPTIONu_u_MAPu_u_COMPOSE, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_x, V_g, V_f]: s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(t_h4s_options_option(TV_u_27c),t_h4s_options_option(TV_u_27b)),h4s_options_optionu_u_map(s(t_fun(TV_u_27c,TV_u_27b),V_f))),s(t_h4s_options_option(TV_u_27c),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27c)),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27c),V_g))),s(t_h4s_options_option(TV_u_27a),V_x))))) = s(t_h4s_options_option(TV_u_27b),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27b)),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27b),h4s_combins_o(s(t_fun(TV_u_27c,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27c),V_g))))),s(t_h4s_options_option(TV_u_27a),V_x)))).
fof(ah4s_quotients_FUNu_u_MAP, axiom, ![TV_u_27d,TV_u_27b,TV_u_27c,TV_u_27a]: ![V_g, V_f, V_x, V_x0]: s(TV_u_27d,happ(s(t_fun(TV_u_27a,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27a,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(t_fun(TV_u_27b,TV_u_27d),V_g))),s(t_fun(TV_u_27c,TV_u_27b),V_x))),s(TV_u_27a,V_x0))) = s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_g),s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_x),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_f),s(TV_u_27a,V_x0)))))))).
fof(ah4s_quotientu_u_options_OPTIONu_u_MAPu_u_I, axiom, ![TV_u_27a]: s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)),h4s_options_optionu_u_map(s(t_fun(TV_u_27a,TV_u_27a),h4s_combins_i))) = s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_options_option(TV_u_27a)),h4s_combins_i)).
fof(ah4s_options_optionu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_v, V_f]: s(TV_u_27b,h4s_options_optionu_u_case(s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,TV_u_27b),V_f))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))).
fof(ah4s_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_quotients_QUOTIENTu_u_RELu_u_REP, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_a, V_b]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_b)))))) <=> s(TV_u_27b,V_a) = s(TV_u_27b,V_b)))).
fof(ah4s_quotients_FORALLu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_f]: s(t_bool,d_forall(s(t_fun(TV_u_27b,t_bool),V_f))) = s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),h4s_quotients_respects),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_bool)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(t_bool,t_bool),h4s_combins_i))),s(t_fun(TV_u_27b,t_bool),V_f))))))).
fof(ah4s_quotients_Ku_u_PRS, axiom, ![TV_u_27a,TV_u_27c,TV_u_27b,TV_u_27d]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_x, V_y]: s(TV_u_27c,h4s_combins_k(s(TV_u_27c,V_x),s(TV_u_27d,V_y))) = s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(TV_u_27a,h4s_combins_k(s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(TV_u_27c,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27d,TV_u_27b),V_rep2),s(TV_u_27d,V_y)))))))))).
fof(ah4s_quotients_literalu_u_caseu_u_PRS, axiom, ![TV_u_27b,TV_u_27d,TV_u_27a,TV_u_27c]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_x]: s(TV_u_27d,h4s_bools_literalu_u_case(s(t_fun(TV_u_27c,TV_u_27d),V_f),s(TV_u_27c,V_x))) = s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(TV_u_27b,h4s_bools_literalu_u_case(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))),s(t_fun(TV_u_27c,TV_u_27d),V_f))),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(TV_u_27c,V_x)))))))))).
fof(ah4s_quotients_LAMBDAu_u_PRS, axiom, ![TV_u_27b,TV_u_27d,TV_u_27a,TV_u_27c]: ![V_uu_0]: (![V_rep2, V_f, V_abs1, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))),s(t_fun(TV_u_27c,TV_u_27d),V_f))),s(t_fun(TV_u_27a,TV_u_27c),V_abs1))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27d,TV_u_27b),V_rep2),s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_f),s(TV_u_27c,happ(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(TV_u_27a,V_x))))))) => ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_x]: s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_f),s(TV_u_27c,V_x))) = s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27c,TV_u_27d)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(t_fun(TV_u_27b,TV_u_27d),V_abs2))),s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b))),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27b),t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27a,TV_u_27b)))),V_uu_0),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))),s(t_fun(TV_u_27c,TV_u_27d),V_f))),s(t_fun(TV_u_27a,TV_u_27c),V_abs1))))),s(TV_u_27c,V_x))))))).
fof(ah4s_quotients_LETu_u_PRS, axiom, ![TV_u_27b,TV_u_27d,TV_u_27a,TV_u_27c]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_x]: s(TV_u_27d,h4s_bools_let(s(t_fun(TV_u_27c,TV_u_27d),V_f),s(TV_u_27c,V_x))) = s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(TV_u_27b,h4s_bools_let(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))),s(t_fun(TV_u_27c,TV_u_27d),V_f))),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(TV_u_27c,V_x)))))))))).
fof(ah4s_quotients_RESu_u_EXISTSu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_P, V_f]: s(t_bool,h4s_bools_resu_u_exists(s(t_fun(TV_u_27b,t_bool),V_P),s(t_fun(TV_u_27b,t_bool),V_f))) = s(t_bool,h4s_bools_resu_u_exists(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_bool)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(t_bool,t_bool),h4s_combins_i))),s(t_fun(TV_u_27b,t_bool),V_P))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_bool)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(t_bool,t_bool),h4s_combins_i))),s(t_fun(TV_u_27b,t_bool),V_f))))))).
fof(ah4s_quotients_ou_u_PRS, axiom, ![TV_u_27c,TV_u_27f,TV_u_27a,TV_u_27b,TV_u_27d,TV_u_27e]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27d),V_abs1),s(t_fun(TV_u_27d,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27e),V_abs2),s(t_fun(TV_u_27e,TV_u_27b),V_rep2)))) => ![V_R3, V_abs3, V_rep3]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27c,t_fun(TV_u_27c,t_bool)),V_R3),s(t_fun(TV_u_27c,TV_u_27f),V_abs3),s(t_fun(TV_u_27f,TV_u_27c),V_rep3)))) => ![V_f, V_g]: s(t_fun(TV_u_27d,TV_u_27f),h4s_combins_o(s(t_fun(TV_u_27e,TV_u_27f),V_f),s(t_fun(TV_u_27d,TV_u_27e),V_g))) = s(t_fun(TV_u_27d,TV_u_27f),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27c),t_fun(TV_u_27d,TV_u_27f)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27d,TV_u_27a),V_rep1),s(t_fun(TV_u_27c,TV_u_27f),V_abs3))),s(t_fun(TV_u_27a,TV_u_27c),h4s_combins_o(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(t_fun(TV_u_27e,TV_u_27f),t_fun(TV_u_27b,TV_u_27c)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27b,TV_u_27e),V_abs2),s(t_fun(TV_u_27f,TV_u_27c),V_rep3))),s(t_fun(TV_u_27e,TV_u_27f),V_f))),s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27d,TV_u_27e),t_fun(TV_u_27a,TV_u_27b)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27d),V_abs1),s(t_fun(TV_u_27e,TV_u_27b),V_rep2))),s(t_fun(TV_u_27d,TV_u_27e),V_g))))))))))).
fof(ah4s_quotients_Wu_u_PRS, axiom, ![TV_u_27b,TV_u_27d,TV_u_27a,TV_u_27c]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_x]: s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27c,TV_u_27d)),t_fun(TV_u_27c,TV_u_27d)),h4s_combins_w),s(t_fun(TV_u_27c,t_fun(TV_u_27c,TV_u_27d)),V_f))),s(TV_u_27c,V_x))) = s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),t_fun(TV_u_27a,TV_u_27b)),h4s_combins_w),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),happ(s(t_fun(t_fun(TV_u_27c,t_fun(TV_u_27c,TV_u_27d)),t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b))),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))))),s(t_fun(TV_u_27c,t_fun(TV_u_27c,TV_u_27d)),V_f))))),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(TV_u_27c,V_x)))))))))).
fof(ah4s_quotients_APPLYu_u_PRS, axiom, ![TV_u_27b,TV_u_27d,TV_u_27a,TV_u_27c]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27c,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(t_fun(TV_u_27d,TV_u_27b),V_rep2)))) => ![V_f, V_x]: s(TV_u_27d,happ(s(t_fun(TV_u_27c,TV_u_27d),V_f),s(TV_u_27c,V_x))) = s(TV_u_27d,happ(s(t_fun(TV_u_27b,TV_u_27d),V_abs2),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27d),t_fun(TV_u_27a,TV_u_27b)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27c),V_abs1),s(t_fun(TV_u_27d,TV_u_27b),V_rep2))),s(t_fun(TV_u_27c,TV_u_27d),V_f))),s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),V_rep1),s(TV_u_27c,V_x)))))))))).
fof(ah4s_quotients_Iu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_e]: s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),h4s_combins_i),s(TV_u_27b,V_e))) = 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_27a,TV_u_27a),h4s_combins_i),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_e))))))))).
fof(ah4s_quotients_EXISTSu_u_UNIQUEu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_f]: s(t_bool,h4s_bools_u_3fu_21(s(t_fun(TV_u_27b,t_bool),V_f))) = s(t_bool,h4s_quotients_resu_u_existsu_u_equiv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_bool)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(t_bool,t_bool),h4s_combins_i))),s(t_fun(TV_u_27b,t_bool),V_f))))))).
fof(ah4s_quotients_Cu_u_PRS, axiom, ![TV_u_27c,TV_u_27f,TV_u_27b,TV_u_27e,TV_u_27a,TV_u_27d]: ![V_rep1, V_abs1, V_R1]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R1),s(t_fun(TV_u_27a,TV_u_27d),V_abs1),s(t_fun(TV_u_27d,TV_u_27a),V_rep1)))) => ![V_R2, V_abs2, V_rep2]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R2),s(t_fun(TV_u_27b,TV_u_27e),V_abs2),s(t_fun(TV_u_27e,TV_u_27b),V_rep2)))) => ![V_R3, V_abs3, V_rep3]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27c,t_fun(TV_u_27c,t_bool)),V_R3),s(t_fun(TV_u_27c,TV_u_27f),V_abs3),s(t_fun(TV_u_27f,TV_u_27c),V_rep3)))) => ![V_f, V_x, V_y]: s(TV_u_27f,h4s_combins_c(s(t_fun(TV_u_27d,t_fun(TV_u_27e,TV_u_27f)),V_f),s(TV_u_27e,V_x),s(TV_u_27d,V_y))) = s(TV_u_27f,happ(s(t_fun(TV_u_27c,TV_u_27f),V_abs3),s(TV_u_27c,h4s_combins_c(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),happ(s(t_fun(t_fun(TV_u_27d,t_fun(TV_u_27e,TV_u_27f)),t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c))),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27d),V_abs1),s(t_fun(t_fun(TV_u_27e,TV_u_27f),t_fun(TV_u_27b,TV_u_27c)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27b,TV_u_27e),V_abs2),s(t_fun(TV_u_27f,TV_u_27c),V_rep3))))),s(t_fun(TV_u_27d,t_fun(TV_u_27e,TV_u_27f)),V_f))),s(TV_u_27b,happ(s(t_fun(TV_u_27e,TV_u_27b),V_rep2),s(TV_u_27e,V_x))),s(TV_u_27a,happ(s(t_fun(TV_u_27d,TV_u_27a),V_rep1),s(TV_u_27d,V_y))))))))))).
fof(ah4s_options_NOTu_u_SOMEu_u_NONE, axiom, ![TV_u_27a]: ![V_x]: ~ (s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none))).
fof(ah4s_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_Wu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),t_fun(TV_u_27a,TV_u_27b)),h4s_combins_w),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),V_f))),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),V_f),s(TV_u_27a,V_x))),s(TV_u_27a,V_x)))).
fof(ah4s_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_quotients_QUOTIENTu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) <=> (![V_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_a]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a)))))) & ![V_r, V_s]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_r))),s(TV_u_27a,V_s)))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_r))),s(TV_u_27a,V_r)))) & (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_s))),s(TV_u_27a,V_s)))) & s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_s)))))))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_bools_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_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_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_options_NOTu_u_NONEu_u_SOME, axiom, ![TV_u_27a]: ![V_x]: ~ (s(t_h4s_options_option(TV_u_27a),h4s_options_none) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_combins_Ku_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_combins_k(s(TV_u_27a,V_x),s(TV_u_27b,V_y))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_predu_u_sets_SPECIFICATION, axiom, ![TV_u_27a]: ![V_x, V_P]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))).
fof(ah4s_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_bools_LETu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_x0]: s(TV_u_27b,h4s_bools_let(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_quotients_QUOTIENTu_u_REPu_u_REFL, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_a]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a)))))))).
fof(ah4s_combins_Cu_u_THM, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,h4s_combins_c(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27b,V_x),s(TV_u_27a,V_y))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_y))),s(TV_u_27b,V_x)))).
fof(ah4s_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_options_OPTIONu_u_APPLYu_u_defu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_x]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_apply(s(t_h4s_options_option(t_fun(TV_u_27b,TV_u_27a)),h4s_options_none),s(t_h4s_options_option(TV_u_27b),V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)).
fof(ah4s_options_OPTIONu_u_MAP2u_u_NONE, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_o2, V_o1, V_f]: (s(t_h4s_options_option(TV_u_27c),h4s_options_optionu_u_map2(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(t_h4s_options_option(TV_u_27a),V_o1),s(t_h4s_options_option(TV_u_27b),V_o2))) = s(t_h4s_options_option(TV_u_27c),h4s_options_none) <=> (s(t_h4s_options_option(TV_u_27a),V_o1) = s(t_h4s_options_option(TV_u_27a),h4s_options_none) | s(t_h4s_options_option(TV_u_27b),V_o2) = s(t_h4s_options_option(TV_u_27b),h4s_options_none)))).
fof(ah4s_options_OPTIONu_u_MAP2u_u_THMu_c1, axiom, ![TV_u_27b,TV_u_27c,TV_u_27a]: ![V_x, V_f]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_map2(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,V_x))),s(t_h4s_options_option(TV_u_27c),h4s_options_none))) = s(t_h4s_options_option(TV_u_27a),h4s_options_none)).
fof(ah4s_options_OPTIONu_u_MAP2u_u_THMu_c0, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_y, V_x, V_f]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_map2(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(t_h4s_options_option(TV_u_27b),h4s_options_some(s(TV_u_27b,V_x))),s(t_h4s_options_option(TV_u_27c),h4s_options_some(s(TV_u_27c,V_y))))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27b,V_x))),s(TV_u_27c,V_y)))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EXISTSu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (?[V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_bools_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_bools_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_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_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_UNWINDu_u_THM2, axiom, ![TV_u_27a]: ![V_a, V_P]: (?[V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_a)))))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_quotientu_u_options_OPTIONu_u_RELu_u_defu_c1, axiom, ![TV_u_27a]: ![V_x, V_R]: s(t_bool,happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_fun(t_h4s_options_option(TV_u_27a),t_bool)),h4s_options_optrel(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_bool,f)).
fof(ah4s_quotientu_u_options_OPTIONu_u_RELu_u_defu_c3, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: s(t_bool,happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_fun(t_h4s_options_option(TV_u_27a),t_bool)),h4s_options_optrel(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))).
fof(ah4s_quotientu_u_options_OPTIONu_u_RELu_u_defu_c2, axiom, ![TV_u_27a]: ![V_y, V_R]: s(t_bool,happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_fun(t_h4s_options_option(TV_u_27a),t_bool)),h4s_options_optrel(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))))) = s(t_bool,f)).
fof(ah4s_quotientu_u_options_OPTIONu_u_RELu_u_defu_c0, axiom, ![TV_u_27a]: ![V_R]: s(t_bool,happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_fun(t_h4s_options_option(TV_u_27a),t_bool)),h4s_options_optrel(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))),s(t_h4s_options_option(TV_u_27a),h4s_options_none))) = s(t_bool,t)).
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_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),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_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),h4s_options_some(s(TV_u_27a,V_x))))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> (~ (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_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),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_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),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_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_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_quotients_QUOTIENTu_u_REL, axiom, ![TV_u_27b,TV_u_27a]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_r, V_s]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_r))),s(TV_u_27a,V_s)))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_r))),s(TV_u_27a,V_r)))) & (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_s))),s(TV_u_27a,V_s)))) & s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_s)))))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_quotients_respectsu_u_def, axiom, ![TV_u_27a,TV_u_27b]: s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),t_fun(TV_u_27a,TV_u_27b)),h4s_quotients_respects) = s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27b)),t_fun(TV_u_27a,TV_u_27b)),h4s_combins_w)).
fof(ah4s_resu_u_quans_RESu_u_FORALL, axiom, ![TV_u_27a]: ![V_f, V_P]: (p(s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_f)))) <=> ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))))).
fof(ah4s_resu_u_quans_RESu_u_EXISTS, axiom, ![TV_u_27a]: ![V_f, V_P]: (p(s(t_bool,h4s_bools_resu_u_exists(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_f)))) <=> ?[V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))))).
fof(ah4s_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),h4s_options_some(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),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_bools_FORALLu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: (p(s(t_bool,d_forall(s(t_fun(TV_u_27a,t_bool),V_x)))) <=> ![V_x0]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_x0))) = s(t_bool,t))).
fof(ah4s_bools_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_quotients_EXISTSu_u_PRS, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_f]: s(t_bool,d_exists(s(t_fun(TV_u_27b,t_bool),V_f))) = s(t_bool,h4s_bools_resu_u_exists(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),h4s_quotients_respects),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27b,t_bool),t_fun(TV_u_27a,t_bool)),h4s_quotients_u_2du_2du_3e(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(t_bool,t_bool),h4s_combins_i))),s(t_fun(TV_u_27b,t_bool),V_f))))))).
fof(ah4s_quotients_RESu_u_EXISTSu_u_EQUIV0, axiom, ![TV_u_27a]: ![V_uu_2]: (![V_m, V_R, V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)))),V_uu_2),s(t_fun(TV_u_27a,t_bool),V_m))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) <=> ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_m),s(TV_u_27a,V_x)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_m),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))))) => ![V_uu_1]: (![V_m, V_R, V_x]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool))),V_uu_1),s(t_fun(TV_u_27a,t_bool),V_m))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_x))) = s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),h4s_quotients_respects),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)))),V_uu_2),s(t_fun(TV_u_27a,t_bool),V_m))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_x))))) => ![V_uu_0]: (![V_m, V_x]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_m))),s(TV_u_27a,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_m),s(TV_u_27a,V_x))) => ![V_m, V_R]: (p(s(t_bool,h4s_quotients_resu_u_existsu_u_equiv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_bool),V_m)))) <=> (p(s(t_bool,h4s_bools_resu_u_exists(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),h4s_quotients_respects),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_m)))))) & p(s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),h4s_quotients_respects),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),t_fun(TV_u_27a,t_bool))),V_uu_1),s(t_fun(TV_u_27a,t_bool),V_m))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))))))))))).
fof(ch4s_quotientu_u_options_ISu_u_NONEu_u_PRS, conjecture, ![TV_u_27a,TV_u_27b]: ![V_rep, V_abs, V_R]: (p(s(t_bool,h4s_quotients_quotient(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ![V_x]: s(t_bool,h4s_options_isu_u_none(s(t_h4s_options_option(TV_u_27b),V_x))) = s(t_bool,h4s_options_isu_u_none(s(t_h4s_options_option(TV_u_27a),happ(s(t_fun(t_h4s_options_option(TV_u_27b),t_h4s_options_option(TV_u_27a)),h4s_options_optionu_u_map(s(t_fun(TV_u_27b,TV_u_27a),V_rep))),s(t_h4s_options_option(TV_u_27b),V_x))))))).
