%   ORIGINAL: h4/relation/STRORD__RC
% 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/relation/StrongOrder0: !Z. h4/relation/StrongOrder Z <=> h4/relation/irreflexive Z /\ h4/relation/transitive Z
% Assm: h4/relation/StrongOrd__Ord: !R. h4/relation/StrongOrder R ==> h4/relation/Order R
% Assm: h4/relation/STRORD__Strong: !R. h4/relation/Order R <=> h4/relation/StrongOrder (h4/relation/STRORD R)
% Assm: h4/bool/TRUTH: T
% Assm: h4/relation/STRORD0: !b a R. h4/relation/STRORD R a b <=> R a b /\ ~(a = b)
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/relation/STRORD__AND__NOT__Id: !R. h4/relation/STRORD R = h4/relation/RINTER R (h4/relation/RCOMPL $equals)
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% 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/relation/Order0: !Z. h4/relation/Order Z <=> h4/relation/antisymmetric Z /\ h4/relation/transitive Z
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/relation/irrefl__trans__implies__antisym: !R. h4/relation/irreflexive R /\ h4/relation/transitive R ==> h4/relation/antisymmetric R
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/FUN__EQ__THM: !g f. f = g <=> (!x. f x = g x)
% Assm: h4/relation/RINTER0: !y x R2 R1. h4/relation/RINTER R1 R2 x y <=> R1 x y /\ R2 x y
% Assm: h4/relation/RCOMPL0: !y x R. h4/relation/RCOMPL R x y <=> ~R x y
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/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/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/relation/RC__DEF: !y x R. h4/relation/RC R x y <=> x = y \/ R x y
% Assm: h4/relation/TC__RC__EQNS_c1: !R. h4/relation/TC (h4/relation/RC R) = h4/relation/RTC R
% Assm: h4/relation/RC__MOVES__OUT_c2: !R. h4/relation/TC (h4/relation/RC R) = h4/relation/RC (h4/relation/TC R)
% Assm: h4/relation/RC__MOVES__OUT_c0: !R. h4/relation/SC (h4/relation/RC R) = h4/relation/RC (h4/relation/SC R)
% Assm: h4/relation/RC__MOVES__OUT_c1: !R. h4/relation/RC (h4/relation/RC R) = h4/relation/RC R
% Assm: h4/relation/reflexive__RC__identity: !R. h4/relation/reflexive R ==> h4/relation/RC R = R
% Assm: h4/relation/RC__REFLEXIVE: !R. h4/relation/reflexive (h4/relation/RC R)
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/relation/inv__RC: !R. h4/relation/inv (h4/relation/RC R) = h4/relation/RC (h4/relation/inv R)
% Assm: h4/bool/LEFT__EXISTS__AND__THM: !Q P. (?x. P x /\ Q) <=> (?x. P x) /\ Q
% Assm: h4/bool/RIGHT__OR__EXISTS__THM: !Q P. P \/ (?x. Q x) <=> (?x. P \/ Q x)
% Assm: h4/relation/RC__IDEM: !R. h4/relation/RC (h4/relation/RC R) = h4/relation/RC R
% Assm: h4/relation/transitive__def: !R. h4/relation/transitive R <=> (!x y z. R x y /\ R y z ==> R x z)
% Assm: h4/bool/RIGHT__EXISTS__AND__THM: !Q P. (?x. P /\ Q x) <=> P /\ (?x. Q x)
% Assm: h4/bool/EXISTS__OR__THM: !Q P. (?x. P x \/ Q x) <=> (?x. P x) \/ (?x. Q x)
% Assm: h4/relation/antisymmetric__def: !R. h4/relation/antisymmetric R <=> (!x y. R x y /\ R y x ==> x = y)
% Assm: h4/relation/EQC__MOVES__IN_c0: !R. h4/relation/EQC (h4/relation/RC R) = h4/relation/EQC R
% Assm: h4/relation/irreflexive__def: !R. h4/relation/irreflexive R <=> (!x. ~R x x)
% Assm: h4/bool/AND__CLAUSES_c3: !t. t /\ F <=> F
% Assm: h4/relation/EQC__DEF: !R. h4/relation/EQC R = h4/relation/RC (h4/relation/TC (h4/relation/SC R))
% Assm: h4/relation/RC__OR__Id: !R. h4/relation/RC R = h4/relation/RUNION R $equals
% Assm: h4/relation/symmetric__RC: !R. h4/relation/symmetric (h4/relation/RC R) <=> h4/relation/symmetric R
% Assm: h4/relation/antisymmetric__RC: !R. h4/relation/antisymmetric (h4/relation/RC R) <=> h4/relation/antisymmetric R
% Assm: h4/relation/transitive__RC: !R. h4/relation/transitive R ==> h4/relation/transitive (h4/relation/RC R)
% Assm: h4/relation/reflexive__RC: !R. h4/relation/reflexive (h4/relation/RC R)
% Assm: h4/relation/RC__Weak: !R. h4/relation/Order R <=> h4/relation/WeakOrder (h4/relation/RC R)
% Assm: h4/relation/TC__RC__EQNS_c0: !R. h4/relation/RC (h4/relation/TC R) = h4/relation/RTC R
% Assm: h4/relation/inv__MOVES__OUT_c2: !R. h4/relation/RC (h4/relation/inv R) = h4/relation/inv (h4/relation/RC R)
% Assm: h4/bool/CONJ__ASSOC: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm: h4/relation/RC__RTC: !y x R. h4/relation/RC R x y ==> h4/relation/RTC R x y
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/bool/FORALL__AND__THM: !Q P. (!x. P x /\ Q x) <=> (!x. P x) /\ (!x. Q x)
% Assm: h4/relation/TC__RULES_c1: !z y x R. h4/relation/TC R x y /\ h4/relation/TC R y z ==> h4/relation/TC R x z
% Assm: h4/relation/TC__RULES_c0: !y x R. R x y ==> h4/relation/TC R x y
% Assm: h4/relation/SC__DEF: !y x R. h4/relation/SC R x y <=> R x y \/ R y x
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/relation/TC__INDUCT: !R P. (!x y. R x y ==> P x y) /\ (!x y z. P x y /\ P y z ==> P x z) ==> (!u v. h4/relation/TC R u v ==> P u v)
% Assm: h4/relation/RTC__RTC: !y x R. h4/relation/RTC R x y ==> (!z. h4/relation/RTC R y z ==> h4/relation/RTC R x z)
% Assm: h4/relation/RTC__INDUCT: !R P. (!x. P x x) /\ (!x y z. R x y /\ P y z ==> P x z) ==> (!x y. h4/relation/RTC R x y ==> P x y)
% Assm: h4/relation/inv__DEF: !y x R. h4/relation/inv R x y <=> R y x
% Assm: h4/relation/RUNION0: !y x R2 R1. h4/relation/RUNION R1 R2 x y <=> R1 x y \/ R2 x y
% Assm: h4/relation/symmetric__def: !R. h4/relation/symmetric R <=> (!x y. R x y <=> R y x)
% Assm: h4/relation/reflexive__def: !R. h4/relation/reflexive R <=> (!x. R x x)
% Assm: h4/relation/inv__SC_c1: !R. h4/relation/SC (h4/relation/inv R) = h4/relation/SC R
% Assm: h4/relation/inv__TC: !R. h4/relation/inv (h4/relation/TC R) = h4/relation/TC (h4/relation/inv R)
% Assm: h4/relation/inv__inv: !R. h4/relation/inv (h4/relation/inv R) = R
% Assm: h4/relation/WeakOrder0: !Z. h4/relation/WeakOrder Z <=> h4/relation/reflexive Z /\ h4/relation/antisymmetric Z /\ h4/relation/transitive Z
% Assm: h4/bool/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/bool/SKOLEM__THM: !P. (!x. ?y. P x y) <=> (?f. !x. P x (f x))
% Assm: h4/relation/RC__lifts__invariants: !R P. (!x y. P x /\ R x y ==> P y) ==> (!x y. P x /\ h4/relation/RC R x y ==> P y)
% Assm: h4/relation/RC__SUBSET: !y x R. R x y ==> h4/relation/RC R x y
% Assm: h4/relation/RTC__TC__RC: !y x R. h4/relation/RTC R x y ==> h4/relation/RC R x y \/ h4/relation/TC R x y
% Assm: h4/relation/RTC__STRONG__INDUCT: !R P. (!x. P x x) /\ (!x y z. R x y /\ h4/relation/RTC R y z /\ P y z ==> P x z) ==> (!x y. h4/relation/RTC R x y ==> P x y)
% Assm: h4/relation/RTC__RULES_c1: !z y x R. R x y /\ h4/relation/RTC R y z ==> h4/relation/RTC R x z
% Assm: h4/relation/RTC__RULES_c0: !x R. h4/relation/RTC R x x
% Assm: h4/relation/TC__RTC: !y x R. h4/relation/TC R x y ==> h4/relation/RTC R x y
% Assm: h4/relation/RC__MONOTONE: !y x R Q. (!x0 y0. R x0 y0 ==> Q x0 y0) ==> h4/relation/RC R x y ==> h4/relation/RC Q x y
% Assm: h4/relation/RC__lifts__monotonicities: !f R. (!x y. R x y ==> R (f x) (f y)) ==> (!x y. h4/relation/RC R x y ==> h4/relation/RC R (f x) (f y))
% Assm: h4/bool/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/relation/RC__lifts__equalities: !f R. (!x y. R x y ==> f x = f y) ==> (!x y. h4/relation/RC R x y ==> f x = f y)
% Assm: h4/bool/OR__CLAUSES_c1: !t. t \/ T <=> T
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/relation/equivalence__def: !R. h4/relation/equivalence R <=> h4/relation/reflexive R /\ h4/relation/symmetric R /\ h4/relation/transitive R
% Assm: h4/relation/EXTEND__RTC__TC__EQN: !z x R. h4/relation/TC R x z <=> (?y. R x y /\ h4/relation/RTC R y z)
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/relation/EXTEND__RTC__TC: !z y x R. R x y /\ h4/relation/RTC R y z ==> h4/relation/TC R x z
% Assm: h4/relation/RTC__RULES__RIGHT1_c1: !z y x R. h4/relation/RTC R x y /\ R y z ==> h4/relation/RTC R x z
% Assm: h4/relation/RTC__RULES__RIGHT1_c0: !x R. h4/relation/RTC R x x
% Assm: h4/relation/RTC__TRANSITIVE: !R. h4/relation/transitive (h4/relation/RTC R)
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/relation/reflexive__inv: !R. h4/relation/reflexive (h4/relation/inv R) <=> h4/relation/reflexive R
% Assm: h4/relation/WF__DEF: !R. h4/relation/WF R <=> (!B. (?w. B w) ==> (?min. B min /\ (!b. R b min ==> ~B b)))
% Assm: h4/relation/TC__implies__one__step: !y x R. h4/relation/TC R x y /\ ~(x = y) ==> (?z. R x z /\ ~(x = z))
% Assm: h4/relation/SC__IDEM: !R. h4/relation/SC (h4/relation/SC R) = h4/relation/SC R
% Assm: h4/relation/symmetric__SC__identity: !R. h4/relation/symmetric R ==> h4/relation/SC R = R
% Goal: !R. h4/relation/StrongOrder R ==> h4/relation/STRORD (h4/relation/RC R) = R
%   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_relations_StrongOrder0]: !Z. h4/relation/StrongOrder Z <=> h4/relation/irreflexive Z /\ h4/relation/transitive Z
% Assm [h4s_relations_StrongOrdu_u_Ord]: !R. h4/relation/StrongOrder R ==> h4/relation/Order R
% Assm [h4s_relations_STRORDu_u_Strong]: !R. h4/relation/Order R <=> h4/relation/StrongOrder (h4/relation/STRORD R)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_relations_STRORD0]: !b a R. happ (happ (h4/relation/STRORD R) a) b <=> happ (happ R a) b /\ ~(a = b)
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_relations_STRORDu_u_ANDu_u_NOTu_u_Id]: !R. h4/relation/STRORD R = h4/relation/RINTER R (h4/relation/RCOMPL $equals)
% 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_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_relations_Order0]: !Z. h4/relation/Order Z <=> h4/relation/antisymmetric Z /\ h4/relation/transitive Z
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_relations_irreflu_u_transu_u_impliesu_u_antisym]: !R. h4/relation/irreflexive R /\ h4/relation/transitive R ==> h4/relation/antisymmetric R
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_FUNu_u_EQu_u_THM]: !g f. f = g <=> (!x. happ f x = happ g x)
% Assm [h4s_relations_RINTER0]: !y x R2 R1. happ (happ (h4/relation/RINTER R1 R2) x) y <=> happ (happ R1 x) y /\ happ (happ R2 x) y
% Assm [h4s_relations_RCOMPL0]: !y x R. happ (happ (h4/relation/RCOMPL R) x) y <=> ~happ (happ R x) y
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_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_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_relations_RCu_u_DEF]: !y x R. happ (happ (h4/relation/RC R) x) y <=> x = y \/ happ (happ R x) y
% Assm [h4s_relations_TCu_u_RCu_u_EQNSu_c1]: !R. h4/relation/TC (h4/relation/RC R) = h4/relation/RTC R
% Assm [h4s_relations_RCu_u_MOVESu_u_OUTu_c2]: !R. h4/relation/TC (h4/relation/RC R) = h4/relation/RC (h4/relation/TC R)
% Assm [h4s_relations_RCu_u_MOVESu_u_OUTu_c0]: !R. h4/relation/SC (h4/relation/RC R) = h4/relation/RC (h4/relation/SC R)
% Assm [h4s_relations_RCu_u_MOVESu_u_OUTu_c1]: !R. h4/relation/RC (h4/relation/RC R) = h4/relation/RC R
% Assm [h4s_relations_reflexiveu_u_RCu_u_identity]: !R. h4/relation/reflexive R ==> h4/relation/RC R = R
% Assm [h4s_relations_RCu_u_REFLEXIVE]: !R. h4/relation/reflexive (h4/relation/RC R)
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_relations_invu_u_RC]: !R. h4/relation/inv (h4/relation/RC R) = h4/relation/RC (h4/relation/inv R)
% Assm [h4s_bools_LEFTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. happ P x /\ Q) <=> (?x. happ P x) /\ Q
% Assm [h4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM]: !Q P. P \/ (?x. happ Q x) <=> (?x. P \/ happ Q x)
% Assm [h4s_relations_RCu_u_IDEM]: !R. h4/relation/RC (h4/relation/RC R) = h4/relation/RC R
% Assm [h4s_relations_transitiveu_u_def]: !R. h4/relation/transitive R <=> (!x y z. happ (happ R x) y /\ happ (happ R y) z ==> happ (happ R x) z)
% Assm [h4s_bools_RIGHTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. P /\ happ Q x) <=> P /\ (?x. happ Q x)
% Assm [h4s_bools_EXISTSu_u_ORu_u_THM]: !Q P. (?x. happ P x \/ happ Q x) <=> (?x. happ P x) \/ (?x. happ Q x)
% Assm [h4s_relations_antisymmetricu_u_def]: !R. h4/relation/antisymmetric R <=> (!x y. happ (happ R x) y /\ happ (happ R y) x ==> x = y)
% Assm [h4s_relations_EQCu_u_MOVESu_u_INu_c0]: !R. h4/relation/EQC (h4/relation/RC R) = h4/relation/EQC R
% Assm [h4s_relations_irreflexiveu_u_def]: !R. h4/relation/irreflexive R <=> (!x. ~happ (happ R x) x)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c3]: !t. t /\ F <=> F
% Assm [h4s_relations_EQCu_u_DEF]: !R. h4/relation/EQC R = h4/relation/RC (h4/relation/TC (h4/relation/SC R))
% Assm [h4s_relations_RCu_u_ORu_u_Id]: !R. h4/relation/RC R = h4/relation/RUNION R $equals
% Assm [h4s_relations_symmetricu_u_RC]: !R. h4/relation/symmetric (h4/relation/RC R) <=> h4/relation/symmetric R
% Assm [h4s_relations_antisymmetricu_u_RC]: !R. h4/relation/antisymmetric (h4/relation/RC R) <=> h4/relation/antisymmetric R
% Assm [h4s_relations_transitiveu_u_RC]: !R. h4/relation/transitive R ==> h4/relation/transitive (h4/relation/RC R)
% Assm [h4s_relations_reflexiveu_u_RC]: !R. h4/relation/reflexive (h4/relation/RC R)
% Assm [h4s_relations_RCu_u_Weak]: !R. h4/relation/Order R <=> h4/relation/WeakOrder (h4/relation/RC R)
% Assm [h4s_relations_TCu_u_RCu_u_EQNSu_c0]: !R. h4/relation/RC (h4/relation/TC R) = h4/relation/RTC R
% Assm [h4s_relations_invu_u_MOVESu_u_OUTu_c2]: !R. h4/relation/RC (h4/relation/inv R) = h4/relation/inv (h4/relation/RC R)
% Assm [h4s_bools_CONJu_u_ASSOC]: !t3 t2 t1. t1 /\ t2 /\ t3 <=> (t1 /\ t2) /\ t3
% Assm [h4s_relations_RCu_u_RTC]: !y x R. happ (happ (h4/relation/RC R) x) y ==> happ (happ (h4/relation/RTC R) x) y
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. P \/ happ Q x) <=> P \/ (!x. happ Q x)
% Assm [h4s_bools_FORALLu_u_ANDu_u_THM]: !Q P. (!x. happ P x /\ happ Q x) <=> (!x. happ P x) /\ (!x. happ Q x)
% Assm [h4s_relations_TCu_u_RULESu_c1]: !z y x R. happ (happ (h4/relation/TC R) x) y /\ happ (happ (h4/relation/TC R) y) z ==> happ (happ (h4/relation/TC R) x) z
% Assm [h4s_relations_TCu_u_RULESu_c0]: !y x R. happ (happ R x) y ==> happ (happ (h4/relation/TC R) x) y
% Assm [h4s_relations_SCu_u_DEF]: !y x R. happ (happ (h4/relation/SC R) x) y <=> happ (happ R x) y \/ happ (happ R y) x
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_relations_TCu_u_INDUCT]: !R P. (!x y. happ (happ R x) y ==> happ (happ P x) y) /\ (!x y z. happ (happ P x) y /\ happ (happ P y) z ==> happ (happ P x) z) ==> (!u v. happ (happ (h4/relation/TC R) u) v ==> happ (happ P u) v)
% Assm [h4s_relations_RTCu_u_RTC]: !y x R. happ (happ (h4/relation/RTC R) x) y ==> (!z. happ (happ (h4/relation/RTC R) y) z ==> happ (happ (h4/relation/RTC R) x) z)
% Assm [h4s_relations_RTCu_u_INDUCT]: !R P. (!x. happ (happ P x) x) /\ (!x y z. happ (happ R x) y /\ happ (happ P y) z ==> happ (happ P x) z) ==> (!x y. happ (happ (h4/relation/RTC R) x) y ==> happ (happ P x) y)
% Assm [h4s_relations_invu_u_DEF]: !y x R. happ (happ (h4/relation/inv R) x) y <=> happ (happ R y) x
% Assm [h4s_relations_RUNION0]: !y x R2 R1. happ (happ (h4/relation/RUNION R1 R2) x) y <=> happ (happ R1 x) y \/ happ (happ R2 x) y
% Assm [h4s_relations_symmetricu_u_def]: !R. h4/relation/symmetric R <=> (!x y. happ (happ R x) y <=> happ (happ R y) x)
% Assm [h4s_relations_reflexiveu_u_def]: !R. h4/relation/reflexive R <=> (!x. happ (happ R x) x)
% Assm [h4s_relations_invu_u_SCu_c1]: !R. h4/relation/SC (h4/relation/inv R) = h4/relation/SC R
% Assm [h4s_relations_invu_u_TC]: !R. h4/relation/inv (h4/relation/TC R) = h4/relation/TC (h4/relation/inv R)
% Assm [h4s_relations_invu_u_inv]: !R. h4/relation/inv (h4/relation/inv R) = R
% Assm [h4s_relations_WeakOrder0]: !Z. h4/relation/WeakOrder Z <=> h4/relation/reflexive Z /\ h4/relation/antisymmetric Z /\ h4/relation/transitive Z
% Assm [h4s_bools_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm [h4s_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM]: !Q P. P /\ (!x. happ Q x) <=> (!x. P /\ happ Q x)
% Assm [h4s_bools_SKOLEMu_u_THM]: !P. (!x. ?y. happ (happ P x) y) <=> (?f. !x. happ (happ P x) (happ f x))
% Assm [h4s_relations_RCu_u_liftsu_u_invariants]: !R P. (!x y. happ P x /\ happ (happ R x) y ==> happ P y) ==> (!x y. happ P x /\ happ (happ (h4/relation/RC R) x) y ==> happ P y)
% Assm [h4s_relations_RCu_u_SUBSET]: !y x R. happ (happ R x) y ==> happ (happ (h4/relation/RC R) x) y
% Assm [h4s_relations_RTCu_u_TCu_u_RC]: !y x R. happ (happ (h4/relation/RTC R) x) y ==> happ (happ (h4/relation/RC R) x) y \/ happ (happ (h4/relation/TC R) x) y
% Assm [h4s_relations_RTCu_u_STRONGu_u_INDUCT]: !R P. (!x. happ (happ P x) x) /\ (!x y z. happ (happ R x) y /\ happ (happ (h4/relation/RTC R) y) z /\ happ (happ P y) z ==> happ (happ P x) z) ==> (!x y. happ (happ (h4/relation/RTC R) x) y ==> happ (happ P x) y)
% Assm [h4s_relations_RTCu_u_RULESu_c1]: !z y x R. happ (happ R x) y /\ happ (happ (h4/relation/RTC R) y) z ==> happ (happ (h4/relation/RTC R) x) z
% Assm [h4s_relations_RTCu_u_RULESu_c0]: !x R. happ (happ (h4/relation/RTC R) x) x
% Assm [h4s_relations_TCu_u_RTC]: !y x R. happ (happ (h4/relation/TC R) x) y ==> happ (happ (h4/relation/RTC R) x) y
% Assm [h4s_relations_RCu_u_MONOTONE]: !y x R Q. (!x0 y0. happ (happ R x0) y0 ==> happ (happ Q x0) y0) ==> happ (happ (h4/relation/RC R) x) y ==> happ (happ (h4/relation/RC Q) x) y
% Assm [h4s_relations_RCu_u_liftsu_u_monotonicities]: !f R. (!x y. happ (happ R x) y ==> happ (happ R (happ f x)) (happ f y)) ==> (!x y. happ (happ (h4/relation/RC R) x) y ==> happ (happ (h4/relation/RC R) (happ f x)) (happ f y))
% Assm [h4s_bools_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_relations_RCu_u_liftsu_u_equalities]: !f R. (!x y. happ (happ R x) y ==> happ f x = happ f y) ==> (!x y. happ (happ (h4/relation/RC R) x) y ==> happ f x = happ f y)
% Assm [h4s_bools_ORu_u_CLAUSESu_c1]: !t. t \/ T <=> T
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_relations_equivalenceu_u_def]: !R. h4/relation/equivalence R <=> h4/relation/reflexive R /\ h4/relation/symmetric R /\ h4/relation/transitive R
% Assm [h4s_relations_EXTENDu_u_RTCu_u_TCu_u_EQN]: !z x R. happ (happ (h4/relation/TC R) x) z <=> (?y. happ (happ R x) y /\ happ (happ (h4/relation/RTC R) y) z)
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_relations_EXTENDu_u_RTCu_u_TC]: !z y x R. happ (happ R x) y /\ happ (happ (h4/relation/RTC R) y) z ==> happ (happ (h4/relation/TC R) x) z
% Assm [h4s_relations_RTCu_u_RULESu_u_RIGHT1u_c1]: !z y x R. happ (happ (h4/relation/RTC R) x) y /\ happ (happ R y) z ==> happ (happ (h4/relation/RTC R) x) z
% Assm [h4s_relations_RTCu_u_RULESu_u_RIGHT1u_c0]: !x R. happ (happ (h4/relation/RTC R) x) x
% Assm [h4s_relations_RTCu_u_TRANSITIVE]: !R. h4/relation/transitive (h4/relation/RTC R)
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_relations_reflexiveu_u_inv]: !R. h4/relation/reflexive (h4/relation/inv R) <=> h4/relation/reflexive R
% Assm [h4s_relations_WFu_u_DEF]: !R. h4/relation/WF R <=> (!B. (?w. happ B w) ==> (?min. happ B min /\ (!b. happ (happ R b) min ==> ~happ B b)))
% Assm [h4s_relations_TCu_u_impliesu_u_oneu_u_step]: !y x R. happ (happ (h4/relation/TC R) x) y /\ ~(x = y) ==> (?z. happ (happ R x) z /\ ~(x = z))
% Assm [h4s_relations_SCu_u_IDEM]: !R. h4/relation/SC (h4/relation/SC R) = h4/relation/SC R
% Assm [h4s_relations_symmetricu_u_SCu_u_identity]: !R. h4/relation/symmetric R ==> h4/relation/SC R = R
% Goal: !R. h4/relation/StrongOrder R ==> h4/relation/STRORD (h4/relation/RC R) = R
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_Q1358922,TV_Q1358918]: ![V_f, V_g]: (![V_x]: s(TV_Q1358918,happ(s(t_fun(TV_Q1358922,TV_Q1358918),V_f),s(TV_Q1358922,V_x))) = s(TV_Q1358918,happ(s(t_fun(TV_Q1358922,TV_Q1358918),V_g),s(TV_Q1358922,V_x))) => s(t_fun(TV_Q1358922,TV_Q1358918),V_f) = s(t_fun(TV_Q1358922,TV_Q1358918),V_g))).
fof(ah4s_relations_StrongOrder0, axiom, ![TV_u_27g]: ![V_Z]: (p(s(t_bool,h4s_relations_strongorder(s(t_fun(TV_u_27g,t_fun(TV_u_27g,t_bool)),V_Z)))) <=> (p(s(t_bool,h4s_relations_irreflexive(s(t_fun(TV_u_27g,t_fun(TV_u_27g,t_bool)),V_Z)))) & p(s(t_bool,h4s_relations_transitive(s(t_fun(TV_u_27g,t_fun(TV_u_27g,t_bool)),V_Z))))))).
fof(ah4s_relations_StrongOrdu_u_Ord, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_strongorder(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => p(s(t_bool,h4s_relations_order(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))).
fof(ah4s_relations_STRORDu_u_Strong, axiom, ![TV_u_27a]: ![V_R]: s(t_bool,h4s_relations_order(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))) = s(t_bool,h4s_relations_strongorder(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_strord(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_relations_STRORD0, axiom, ![TV_u_27a]: ![V_b, V_a, 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)),h4s_relations_strord(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_a))),s(TV_u_27a,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,V_a))),s(TV_u_27a,V_b)))) & ~ (s(TV_u_27a,V_a) = s(TV_u_27a,V_b))))).
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_relations_STRORDu_u_ANDu_u_NOTu_u_Id, axiom, ![TV_u_27a]: ![V_R]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_strord(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rinter(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rcompl(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),d_equals)))))).
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_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_relations_Order0, axiom, ![TV_u_27g]: ![V_Z]: (p(s(t_bool,h4s_relations_order(s(t_fun(TV_u_27g,t_fun(TV_u_27g,t_bool)),V_Z)))) <=> (p(s(t_bool,h4s_relations_antisymmetric(s(t_fun(TV_u_27g,t_fun(TV_u_27g,t_bool)),V_Z)))) & p(s(t_bool,h4s_relations_transitive(s(t_fun(TV_u_27g,t_fun(TV_u_27g,t_bool)),V_Z))))))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_relations_irreflu_u_transu_u_impliesu_u_antisym, axiom, ![TV_u_27a]: ![V_R]: ((p(s(t_bool,h4s_relations_irreflexive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) & p(s(t_bool,h4s_relations_transitive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) => p(s(t_bool,h4s_relations_antisymmetric(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_FUNu_u_EQu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_g, V_f]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),V_g) <=> ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x))))).
fof(ah4s_relations_RINTER0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_R2, V_R1]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),h4s_relations_rinter(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R2))),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R1),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) & p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))))).
fof(ah4s_relations_RCOMPL0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_R]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),h4s_relations_rcompl(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R))),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) <=> ~ (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_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_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_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_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_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_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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_bools_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_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_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_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_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_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_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_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_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_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_NOTu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ?[V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_relations_RCu_u_DEF, axiom, ![TV_u_27a]: ![V_y, V_x, 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)),h4s_relations_rc(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)))) <=> (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),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_relations_TCu_u_RCu_u_EQNSu_c1, axiom, ![TV_u_27a]: ![V_R]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_tc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rtc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))).
fof(ah4s_relations_RCu_u_MOVESu_u_OUTu_c2, axiom, ![TV_u_27a]: ![V_R]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_tc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_tc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))).
fof(ah4s_relations_RCu_u_MOVESu_u_OUTu_c0, axiom, ![TV_u_27a]: ![V_R]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_sc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_sc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))).
fof(ah4s_relations_RCu_u_MOVESu_u_OUTu_c1, axiom, ![TV_u_27a]: ![V_R]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))).
fof(ah4s_relations_reflexiveu_u_RCu_u_identity, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_reflexive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))).
fof(ah4s_relations_RCu_u_REFLEXIVE, axiom, ![TV_u_27a]: ![V_R]: p(s(t_bool,h4s_relations_reflexive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_relations_invu_u_RC, axiom, ![TV_u_27a]: ![V_R]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_inv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_inv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))).
fof(ah4s_bools_LEFTu_u_EXISTSu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (?[V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))) <=> (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,V_Q))))).
fof(ah4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) | ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ?[V_x]: (p(s(t_bool,V_P)) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_relations_RCu_u_IDEM, axiom, ![TV_u_27a]: ![V_R]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))).
fof(ah4s_relations_transitiveu_u_def, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_transitive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> ![V_x, V_y, V_z]: ((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)))) & 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_y))),s(TV_u_27a,V_z))))) => 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_z))))))).
fof(ah4s_bools_RIGHTu_u_EXISTSu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (?[V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (p(s(t_bool,V_P)) & ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_EXISTSu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (?[V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_relations_antisymmetricu_u_def, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_antisymmetric(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),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)),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),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_x))))) => s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))).
fof(ah4s_relations_EQCu_u_MOVESu_u_INu_c0, axiom, ![TV_u_27a]: ![V_R]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_eqc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_eqc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))).
fof(ah4s_relations_irreflexiveu_u_def, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_irreflexive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> ![V_x]: ~ (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_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_relations_EQCu_u_DEF, axiom, ![TV_u_27a]: ![V_R]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_eqc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_tc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_sc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))))).
fof(ah4s_relations_RCu_u_ORu_u_Id, axiom, ![TV_u_27a]: ![V_R]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_runion(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),d_equals)))).
fof(ah4s_relations_symmetricu_u_RC, axiom, ![TV_u_27a]: ![V_R]: s(t_bool,h4s_relations_symmetric(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) = s(t_bool,h4s_relations_symmetric(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))).
fof(ah4s_relations_antisymmetricu_u_RC, axiom, ![TV_u_27a]: ![V_R]: s(t_bool,h4s_relations_antisymmetric(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) = s(t_bool,h4s_relations_antisymmetric(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))).
fof(ah4s_relations_transitiveu_u_RC, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_transitive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => p(s(t_bool,h4s_relations_transitive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))))).
fof(ah4s_relations_reflexiveu_u_RC, axiom, ![TV_u_27a]: ![V_R]: p(s(t_bool,h4s_relations_reflexive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))))).
fof(ah4s_relations_RCu_u_Weak, axiom, ![TV_u_27a]: ![V_R]: s(t_bool,h4s_relations_order(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))) = s(t_bool,h4s_relations_weakorder(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))).
fof(ah4s_relations_TCu_u_RCu_u_EQNSu_c0, axiom, ![TV_u_27a]: ![V_R]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_tc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rtc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))).
fof(ah4s_relations_invu_u_MOVESu_u_OUTu_c2, axiom, ![TV_u_27a]: ![V_R]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_inv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_inv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))).
fof(ah4s_bools_CONJu_u_ASSOC, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) & (p(s(t_bool,V_t2)) & p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) & p(s(t_bool,V_t3))))).
fof(ah4s_relations_RCu_u_RTC, axiom, ![TV_u_27a]: ![V_y, V_x, 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)),h4s_relations_rc(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),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rtc(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_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,V_P)) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (p(s(t_bool,V_P)) | ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_FORALLu_u_ANDu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_relations_TCu_u_RULESu_c1, axiom, ![TV_u_27a]: ![V_z, V_y, V_x, 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)),h4s_relations_tc(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),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_tc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_y))),s(TV_u_27a,V_z))))) => 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)),h4s_relations_tc(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_z)))))).
fof(ah4s_relations_TCu_u_RULESu_c0, axiom, ![TV_u_27a]: ![V_y, V_x, 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_x))),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)),h4s_relations_tc(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_relations_SCu_u_DEF, axiom, ![TV_u_27a]: ![V_y, V_x, 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)),h4s_relations_sc(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),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)))) | 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_y))),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_relations_TCu_u_INDUCT, axiom, ![TV_u_27a]: ![V_R, V_P]: ((![V_x, V_y]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_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),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))) & ![V_x, V_y, V_z]: ((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_P),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),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27a,V_y))),s(TV_u_27a,V_z))))) => 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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_z)))))) => ![V_u, V_v]: (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)),h4s_relations_tc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_u))),s(TV_u_27a,V_v)))) => 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_P),s(TV_u_27a,V_u))),s(TV_u_27a,V_v))))))).
fof(ah4s_relations_RTCu_u_RTC, axiom, ![TV_u_27a]: ![V_y, V_x, 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)),h4s_relations_rtc(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_z]: (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)),h4s_relations_rtc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_y))),s(TV_u_27a,V_z)))) => 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)),h4s_relations_rtc(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_z))))))).
fof(ah4s_relations_RTCu_u_INDUCT, axiom, ![TV_u_27a]: ![V_R, V_P]: ((![V_x]: 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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_x)))) & ![V_x, V_y, V_z]: ((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)))) & 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_P),s(TV_u_27a,V_y))),s(TV_u_27a,V_z))))) => 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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_z)))))) => ![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)),h4s_relations_rtc(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),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))))).
fof(ah4s_relations_invu_u_DEF, axiom, ![TV_u_27a]: ![V_y, V_x, V_R]: 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)),h4s_relations_inv(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))) = 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_y))),s(TV_u_27a,V_x)))).
fof(ah4s_relations_RUNION0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_R2, V_R1]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),h4s_relations_runion(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R2))),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R1),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) | p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_R2),s(TV_u_27a,V_x))),s(TV_u_27b,V_y))))))).
fof(ah4s_relations_symmetricu_u_def, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_symmetric(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> ![V_x, 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))) = 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_y))),s(TV_u_27a,V_x))))).
fof(ah4s_relations_reflexiveu_u_def, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_reflexive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> ![V_x]: 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_x)))))).
fof(ah4s_relations_invu_u_SCu_c1, axiom, ![TV_u_27a]: ![V_R]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_sc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_inv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_sc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))).
fof(ah4s_relations_invu_u_TC, axiom, ![TV_u_27a]: ![V_R]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_inv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_tc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_tc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_inv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))).
fof(ah4s_relations_invu_u_inv, axiom, ![TV_u_27a]: ![V_R]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_inv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_inv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)).
fof(ah4s_relations_WeakOrder0, axiom, ![TV_u_27g]: ![V_Z]: (p(s(t_bool,h4s_relations_weakorder(s(t_fun(TV_u_27g,t_fun(TV_u_27g,t_bool)),V_Z)))) <=> (p(s(t_bool,h4s_relations_reflexive(s(t_fun(TV_u_27g,t_fun(TV_u_27g,t_bool)),V_Z)))) & (p(s(t_bool,h4s_relations_antisymmetric(s(t_fun(TV_u_27g,t_fun(TV_u_27g,t_bool)),V_Z)))) & p(s(t_bool,h4s_relations_transitive(s(t_fun(TV_u_27g,t_fun(TV_u_27g,t_bool)),V_Z)))))))).
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_bools_RIGHTu_u_ANDu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: ((p(s(t_bool,V_P)) & ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> ![V_x]: (p(s(t_bool,V_P)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_SKOLEMu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (![V_x]: ?[V_y]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))) <=> ?[V_f]: ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x)))))))).
fof(ah4s_relations_RCu_u_liftsu_u_invariants, axiom, ![TV_u_27a]: ![V_R, V_P]: (![V_x, V_y]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),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))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_y))))) => ![V_x, V_y]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(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_P),s(TV_u_27a,V_y))))))).
fof(ah4s_relations_RCu_u_SUBSET, axiom, ![TV_u_27a]: ![V_y, V_x, 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_x))),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)),h4s_relations_rc(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_relations_RTCu_u_TCu_u_RC, axiom, ![TV_u_27a]: ![V_y, V_x, 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)),h4s_relations_rtc(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),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(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),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_tc(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_relations_RTCu_u_STRONGu_u_INDUCT, axiom, ![TV_u_27a]: ![V_R, V_P]: ((![V_x]: 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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_x)))) & ![V_x, V_y, V_z]: ((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)))) & (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)),h4s_relations_rtc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_y))),s(TV_u_27a,V_z)))) & 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_P),s(TV_u_27a,V_y))),s(TV_u_27a,V_z)))))) => 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_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_z)))))) => ![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)),h4s_relations_rtc(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),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_P),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))))).
fof(ah4s_relations_RTCu_u_RULESu_c1, axiom, ![TV_u_27a]: ![V_z, V_y, V_x, 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_x))),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)),h4s_relations_rtc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_y))),s(TV_u_27a,V_z))))) => 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)),h4s_relations_rtc(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_z)))))).
fof(ah4s_relations_RTCu_u_RULESu_c0, axiom, ![TV_u_27a]: ![V_x, 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)),h4s_relations_rtc(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_x))))).
fof(ah4s_relations_TCu_u_RTC, axiom, ![TV_u_27a]: ![V_y, V_x, 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)),h4s_relations_tc(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),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rtc(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_relations_RCu_u_MONOTONE, axiom, ![TV_u_27a]: ![V_y, V_x, V_R, V_Q]: (![V_x0, V_y0]: (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_x0))),s(TV_u_27a,V_y0)))) => 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_Q),s(TV_u_27a,V_x0))),s(TV_u_27a,V_y0))))) => (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)),h4s_relations_rc(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),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_Q))),s(TV_u_27a,V_x))),s(TV_u_27a,V_y))))))).
fof(ah4s_relations_RCu_u_liftsu_u_monotonicities, axiom, ![TV_u_27a]: ![V_f, 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)),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),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_27a,TV_u_27a),V_f),s(TV_u_27a,V_x))))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_y))))))) => ![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)),h4s_relations_rc(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),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(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_27a,TV_u_27a),V_f),s(TV_u_27a,V_x))))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_y))))))))).
fof(ah4s_bools_LEFTu_u_ORu_u_OVERu_u_AND, 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_A)) | p(s(t_bool,V_C)))))).
fof(ah4s_relations_RCu_u_liftsu_u_equalities, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, 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)),V_R),s(TV_u_27a,V_x))),s(TV_u_27a,V_y)))) => s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y)))) => ![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)),h4s_relations_rc(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)))) => 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_bools_ORu_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_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_relations_equivalenceu_u_def, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_equivalence(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_reflexive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) & (p(s(t_bool,h4s_relations_symmetric(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) & p(s(t_bool,h4s_relations_transitive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))))))).
fof(ah4s_relations_EXTENDu_u_RTCu_u_TCu_u_EQN, axiom, ![TV_u_27a]: ![V_z, V_x, 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)),h4s_relations_tc(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_z)))) <=> ?[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)))) & 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)),h4s_relations_rtc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_y))),s(TV_u_27a,V_z))))))).
fof(ah4s_bools_NOTu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ![V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_relations_EXTENDu_u_RTCu_u_TC, axiom, ![TV_u_27a]: ![V_z, V_y, V_x, 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_x))),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)),h4s_relations_rtc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))),s(TV_u_27a,V_y))),s(TV_u_27a,V_z))))) => 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)),h4s_relations_tc(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_z)))))).
fof(ah4s_relations_RTCu_u_RULESu_u_RIGHT1u_c1, axiom, ![TV_u_27a]: ![V_z, V_y, V_x, 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)),h4s_relations_rtc(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),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_z))))) => 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)),h4s_relations_rtc(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_z)))))).
fof(ah4s_relations_RTCu_u_RULESu_u_RIGHT1u_c0, axiom, ![TV_u_27a]: ![V_x, 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)),h4s_relations_rtc(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_x))))).
fof(ah4s_relations_RTCu_u_TRANSITIVE, axiom, ![TV_u_27a]: ![V_R]: p(s(t_bool,h4s_relations_transitive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rtc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c0, 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_relations_reflexiveu_u_inv, axiom, ![TV_u_27a]: ![V_R]: s(t_bool,h4s_relations_reflexive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_inv(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) = s(t_bool,h4s_relations_reflexive(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))).
fof(ah4s_relations_WFu_u_DEF, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) <=> ![V_B]: (?[V_w]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_B),s(TV_u_27a,V_w)))) => ?[V_min]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_B),s(TV_u_27a,V_min)))) & ![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,V_b))),s(TV_u_27a,V_min)))) => ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_B),s(TV_u_27a,V_b)))))))))).
fof(ah4s_relations_TCu_u_impliesu_u_oneu_u_step, axiom, ![TV_u_27a]: ![V_y, V_x, 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)),h4s_relations_tc(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)))) & ~ (s(TV_u_27a,V_x) = s(TV_u_27a,V_y))) => ?[V_z]: (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_z)))) & ~ (s(TV_u_27a,V_x) = s(TV_u_27a,V_z))))).
fof(ah4s_relations_SCu_u_IDEM, axiom, ![TV_u_27a]: ![V_R]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_sc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_sc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_sc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))).
fof(ah4s_relations_symmetricu_u_SCu_u_identity, axiom, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_symmetric(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_sc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))).
fof(ch4s_relations_STRORDu_u_RC, conjecture, ![TV_u_27a]: ![V_R]: (p(s(t_bool,h4s_relations_strongorder(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_strord(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_rc(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R))).
