%   ORIGINAL: h4/quantHeuristics/DISJ__IMP__INTRO
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/quantHeuristics/GUESS__FORALL__GAP__def: !i P. h4/quantHeuristics/GUESS__FORALL__GAP i P <=> (!v. ~P v ==> (?fv. v = i fv))
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/quantHeuristics/GUESS__EXISTS__FORALL__REWRITES_c1: !i P. h4/quantHeuristics/GUESS__FORALL i P <=> (!v. ~P v ==> (?fv. ~P (i fv)))
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/list/EVERY__NOT__EXISTS: !l P. h4/list/EVERY P l <=> ~h4/list/EXISTS (\x. ~P x) l
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/list/FILTER__EQ__NIL: !l P. h4/list/FILTER P l = h4/list/NIL <=> h4/list/EVERY (\x. ~P x) l
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/quantHeuristics/GUESS__EXISTS__FORALL__REWRITES_c0: !i P. h4/quantHeuristics/GUESS__EXISTS i P <=> (!v. P v ==> (?fv. P (i fv)))
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/list/FILTER__COND__REWRITE_c2: !l h P. ~P h ==> h4/list/FILTER P (h4/list/CONS h l) = h4/list/FILTER P l
% Assm: h4/bool/FALSITY: !t. F ==> t
% 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/list/FILTER__NEQ__ID: !l P. ~(h4/list/FILTER P l = l) <=> (?x. h4/bool/IN x (h4/list/LIST__TO__SET l) /\ ~P x)
% Assm: h4/quantHeuristics/GUESS__REWRITES_c1: !i P. h4/quantHeuristics/GUESS__FORALL i P <=> (!v. ~P v ==> (?fv. ~P (i fv)))
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/while/WHILE__RULE: !R P C B. h4/relation/WF R /\ (!s. B s ==> R (C s) s) ==> h4/while/HOARE__SPEC (\s. P s /\ B s) C P ==> h4/while/HOARE__SPEC P (h4/while/WHILE B C) (\s. P s /\ ~B s)
% Assm: h4/quantHeuristics/UNWIND__EXISTS__THM: !a P. (?x. P x) <=> (!x. ~(x = a) ==> ~P x) ==> P a
% 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__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% 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/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/list/list__INDUCT: !P. P h4/list/NIL /\ (!t. P t ==> (!h. P (h4/list/CONS h t))) ==> (!l. P l)
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/list/FILTER0_c1: !t h P. h4/list/FILTER P (h4/list/CONS h t) = h4/bool/COND (P h) (h4/list/CONS h (h4/list/FILTER P t)) (h4/list/FILTER P t)
% Assm: h4/list/FILTER0_c0: !P. h4/list/FILTER P h4/list/NIL = h4/list/NIL
% Assm: h4/list/EVERY__DEF_c0: !P. h4/list/EVERY P h4/list/NIL <=> T
% Assm: h4/list/EVERY__DEF_c1: !t h P. h4/list/EVERY P (h4/list/CONS h t) <=> P h /\ h4/list/EVERY P t
% Assm: h4/bool/COND__CONG: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm: h4/list/EXISTS__MEM: !l P. h4/list/EXISTS P l <=> (?e. h4/bool/IN e (h4/list/LIST__TO__SET l) /\ P e)
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/list/EXISTS__NOT__EVERY: !l P. h4/list/EXISTS P l <=> ~h4/list/EVERY (\x. ~P x) l
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/COND__RAND: !y x f b. f (h4/bool/COND b x y) = h4/bool/COND b (f x) (f y)
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/list/NOT__CONS__NIL: !a1 a0. ~(h4/list/CONS a0 a1 = h4/list/NIL)
% Assm: h4/bool/COND__RATOR: !x g f b. h4/bool/COND b f g x = h4/bool/COND b (f x) (g x)
% Assm: h4/bool/F__DEF: F <=> (!t. t)
% Assm: h4/while/OWHILE__IND: !f P G. (!s. ~G s ==> P s s) /\ (!s1 s2. G s1 /\ P (f s1) s2 ==> P s1 s2) ==> (!s1 s2. h4/while/OWHILE G f s1 = h4/option/SOME s2 ==> P s1 s2)
% Assm: h4/list/FILTER__EQ__ID: !l P. h4/list/FILTER P l = l <=> h4/list/EVERY P l
% Assm: h4/quantHeuristics/IMP__NEG__CONTRA: !x i P. ~P i ==> P x ==> ~(x = i)
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/EXISTS__OR__THM: !Q P. (?x. P x \/ Q x) <=> (?x. P x) \/ (?x. Q x)
% Assm: h4/bool/LEFT__EXISTS__AND__THM: !Q P. (?x. P x /\ Q) <=> (?x. P x) /\ Q
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/option/some__elim: !Q P. Q (h4/option/some P) ==> (?x. P x /\ Q (h4/option/SOME x)) \/ (!x. ~P x) /\ Q h4/option/NONE
% Assm: h4/bool/RIGHT__OR__OVER__AND: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm: h4/while/HOARE__SPEC__DEF: !Q P C. h4/while/HOARE__SPEC P C Q <=> (!s. P s ==> Q (C s))
% Assm: h4/while/WHILE__INDUCTION: !R C B. h4/relation/WF R /\ (!s. B s ==> R (C s) s) ==> (!P. (!s. (B s ==> P (C s)) ==> P s) ==> (!v. P v))
% Assm: h4/while/WHILE0: !x g P. h4/while/WHILE P g x = h4/bool/COND (P x) (h4/while/WHILE P g (g x)) x
% Assm: h4/sat/dc__cond: !s r q p. (p <=> h4/bool/COND q r s) <=> (p \/ q \/ ~s) /\ (p \/ ~r \/ ~q) /\ (p \/ ~r \/ ~s) /\ (~q \/ r \/ ~p) /\ (q \/ s \/ ~p)
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/LEFT__OR__EXISTS__THM: !Q P. (?x. P x) \/ Q <=> (?x. P x \/ Q)
% Assm: h4/list/FILTER__NEQ__NIL: !l P. ~(h4/list/FILTER P l = h4/list/NIL) <=> (?x. h4/bool/IN x (h4/list/LIST__TO__SET l) /\ P x)
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/while/OWHILE__ENDCOND: !s_27 s f G. h4/while/OWHILE G f s = h4/option/SOME s_27 ==> ~G s_27
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/bool/EXISTS__DEF: $exists = (\P. P (h4/min/_40 P))
% Assm: h4/bool/LEFT__AND__FORALL__THM: !Q P. (!x. P x) /\ Q <=> (!x. P x /\ Q)
% Assm: h4/quantHeuristics/GUESS__REWRITES_c5: !i P. h4/quantHeuristics/GUESS__FORALL__GAP i P <=> (!v. ~P v ==> (?fv. v = i fv))
% Assm: h4/bool/LEFT__FORALL__OR__THM: !Q P. (!x. P x \/ Q) <=> (!x. P x) \/ Q
% Assm: h4/list/NULL__FILTER: !ls P. h4/list/NULL (h4/list/FILTER P ls) <=> (!x. h4/bool/IN x (h4/list/LIST__TO__SET ls) ==> ~P x)
% Assm: h4/while/OWHILE__def: !s f G. h4/while/OWHILE G f s = h4/bool/COND (?n. ~G (h4/arithmetic/FUNPOW f n s)) (h4/option/SOME (h4/arithmetic/FUNPOW f (h4/while/LEAST (\n. ~G (h4/arithmetic/FUNPOW f n s))) s)) h4/option/NONE
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/quantHeuristics/GUESS__FORALL__def: !i P. h4/quantHeuristics/GUESS__FORALL i P <=> (!v. P v) <=> (!fv. P (i fv))
% Assm: h4/quantHeuristics/GUESS__EXISTS__def: !i P. h4/quantHeuristics/GUESS__EXISTS i P <=> (?v. P v) <=> (?fv. P (i fv))
% Assm: h4/bool/RIGHT__AND__FORALL__THM: !Q P. P /\ (!x. Q x) <=> (!x. P /\ Q x)
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/while/LEAST__EXISTS__IMP: !p. (?n. p n) ==> p (h4/while/LEAST p) /\ (!n. h4/prim__rec/_3C n (h4/while/LEAST p) ==> ~p n)
% Assm: h4/arithmetic/FUNPOW0_c1: !x n f. h4/arithmetic/FUNPOW f (h4/num/SUC n) x = h4/arithmetic/FUNPOW f n (f x)
% Assm: h4/arithmetic/FUNPOW0_c0: !x f. h4/arithmetic/FUNPOW f h4/num/0 x = x
% Assm: h4/prim__rec/LESS__0: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm: h4/num/INDUCTION: !P. P h4/num/0 /\ (!n. P n ==> P (h4/num/SUC n)) ==> (!n. P n)
% Assm: h4/prim__rec/LESS__MONO: !n m. h4/prim__rec/_3C m n ==> h4/prim__rec/_3C (h4/num/SUC m) (h4/num/SUC n)
% Assm: h4/option/NOT__NONE__SOME: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/DISJ__COMM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/UNWIND__FORALL__THM2: !v f. (!x. x = v ==> f x) <=> f v
% Assm: h4/bool/PULL__EXISTS_c0: !Q P. (?x. P x) ==> Q <=> (!x. P x ==> Q)
% Assm: h4/bool/IMP__DISJ__THM: !B A. A ==> B <=> ~A \/ B
% Assm: h4/quantHeuristics/GUESS__RULES__TRIVIAL__FORALL__POINT: !i P. ~P i ==> h4/quantHeuristics/GUESS__FORALL__POINT (\xxx. i) P
% Assm: h4/quantHeuristics/EXISTS__NOT__FORALL__THM: !P. (?x. P x) <=> ~(!x. ~P x)
% Assm: h4/option/some__def: !P. h4/option/some P = h4/bool/COND (?x. P x) (h4/option/SOME (h4/min/_40 (\x. P x))) h4/option/NONE
% Assm: h4/option/some__intro: !Q P. (!x. P x ==> Q (h4/option/SOME x)) /\ ((!x. ~P x) ==> Q h4/option/NONE) ==> Q (h4/option/some P)
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/quantHeuristics/MOVE__EXISTS__IMP__THM: !R Q P. (?x. (!y. ~P x y ==> R y) ==> Q x) <=> (!y. ~(!x. P x y) ==> R y) ==> (?x. Q x)
% Assm: h4/while/LEAST__ELIM: !Q P. (?n. P n) /\ (!n. (!m. h4/prim__rec/_3C m n ==> ~P m) /\ P n ==> Q n) ==> Q (h4/while/LEAST P)
% Assm: h4/option/IF__EQUALS__OPTION_c2: !y x P. h4/bool/COND P (h4/option/SOME x) h4/option/NONE = h4/option/SOME y <=> P /\ x = y
% Assm: h4/relation/WF__DEF: !R. h4/relation/WF R <=> (!B. (?w. B w) ==> (?min. B min /\ (!b. R b min ==> ~B b)))
% Assm: h4/list/EXISTS__DEF_c1: !t h P. h4/list/EXISTS P (h4/list/CONS h t) <=> P h \/ h4/list/EXISTS P t
% Assm: h4/list/EXISTS__DEF_c0: !P. h4/list/EXISTS P h4/list/NIL <=> F
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/EQ__REFL: !x. x = x
% Assm: h4/list/MEM_c1: !x t h. h4/bool/IN x (h4/list/LIST__TO__SET (h4/list/CONS h t)) <=> x = h \/ h4/bool/IN x (h4/list/LIST__TO__SET t)
% Assm: h4/list/MEM_c0: !x. h4/bool/IN x (h4/list/LIST__TO__SET h4/list/NIL) <=> F
% Assm: h4/list/NULL0_c1: !t h. ~h4/list/NULL (h4/list/CONS h t)
% Assm: h4/list/NULL0_c0: h4/list/NULL h4/list/NIL
% Assm: h4/bool/NOT__IMP: !B A. ~(A ==> B) <=> A /\ ~B
% Assm: h4/bool/LEFT__OR__OVER__AND: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm: h4/quantHeuristics/GUESS__REWRITES_c3: !i P. h4/quantHeuristics/GUESS__FORALL__POINT i P <=> (!fv. ~P (i fv))
% Assm: h4/bool/RIGHT__OR__EXISTS__THM: !Q P. P \/ (?x. Q x) <=> (?x. P \/ Q x)
% Assm: h4/bool/RIGHT__EXISTS__AND__THM: !Q P. (?x. P /\ Q x) <=> P /\ (?x. Q x)
% Goal: !y Q P. (!x. P x \/ Q x) ==> (~P y ==> Q y) /\ (~Q y ==> P y)
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_quantHeuristicss_GUESSu_u_FORALLu_u_GAPu_u_def]: !i P. h4/quantHeuristics/GUESS__FORALL__GAP i P <=> (!v. ~happ P v ==> (?fv. v = happ i fv))
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_quantHeuristicss_GUESSu_u_EXISTSu_u_FORALLu_u_REWRITESu_c1]: !i P. h4/quantHeuristics/GUESS__FORALL i P <=> (!v. ~happ P v ==> (?fv. ~happ P (happ i fv)))
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_lists_EVERYu_u_NOTu_u_EXISTS]: !_0. (!P x. happ (happ _0 P) x <=> ~happ P x) ==> (!l P. h4/list/EVERY P l <=> ~h4/list/EXISTS (happ _0 P) l)
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_lists_FILTERu_u_EQu_u_NIL]: !_0. (!P x. happ (happ _0 P) x <=> ~happ P x) ==> (!l P. h4/list/FILTER P l = h4/list/NIL <=> h4/list/EVERY (happ _0 P) l)
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_quantHeuristicss_GUESSu_u_EXISTSu_u_FORALLu_u_REWRITESu_c0]: !i P. h4/quantHeuristics/GUESS__EXISTS i P <=> (!v. happ P v ==> (?fv. happ P (happ i fv)))
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_lists_FILTERu_u_CONDu_u_REWRITEu_c2]: !l h P. ~happ P h ==> h4/list/FILTER P (h4/list/CONS h l) = h4/list/FILTER P l
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% 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_lists_FILTERu_u_NEQu_u_ID]: !l P. ~(h4/list/FILTER P l = l) <=> (?x. h4/bool/IN x (h4/list/LIST__TO__SET l) /\ ~happ P x)
% Assm [h4s_quantHeuristicss_GUESSu_u_REWRITESu_c1]: !i P. h4/quantHeuristics/GUESS__FORALL i P <=> (!v. ~happ P v ==> (?fv. ~happ P (happ i fv)))
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_whiles_WHILEu_u_RULE]: !_1. (!B s. happ (happ _1 B) s <=> ~happ B s) ==> (!_0. (!P B s. happ (happ (happ _0 P) B) s <=> happ P s /\ happ B s) ==> (!R P C B. h4/relation/WF R /\ (!s. happ B s ==> happ (happ R (happ C s)) s) ==> h4/while/HOARE__SPEC (happ (happ _0 P) B) C P ==> h4/while/HOARE__SPEC P (h4/while/WHILE B C) (happ (happ _0 P) (happ _1 B))))
% Assm [h4s_quantHeuristicss_UNWINDu_u_EXISTSu_u_THM]: !a P. (?x. happ P x) <=> (!x. ~(x = a) ==> ~happ P x) ==> happ P a
% 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_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% 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_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_lists_listu_u_INDUCT]: !P. happ P h4/list/NIL /\ (!t. happ P t ==> (!h. happ P (h4/list/CONS h t))) ==> (!l. happ P l)
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_lists_FILTER0u_c1]: !t h P. h4/list/FILTER P (h4/list/CONS h t) = h4/bool/COND (happ P h) (h4/list/CONS h (h4/list/FILTER P t)) (h4/list/FILTER P t)
% Assm [h4s_lists_FILTER0u_c0]: !P. h4/list/FILTER P h4/list/NIL = h4/list/NIL
% Assm [h4s_lists_EVERYu_u_DEFu_c0]: !P. h4/list/EVERY P h4/list/NIL <=> T
% Assm [h4s_lists_EVERYu_u_DEFu_c1]: !t h P. h4/list/EVERY P (h4/list/CONS h t) <=> happ P h /\ h4/list/EVERY P t
% Assm [h4s_bools_CONDu_u_CONG]: !y_27 y x_27 x Q P. (P <=> Q) /\ (Q ==> x = x_27) /\ (~Q ==> y = y_27) ==> h4/bool/COND P x y = h4/bool/COND Q x_27 y_27
% Assm [h4s_lists_EXISTSu_u_MEM]: !l P. h4/list/EXISTS P l <=> (?e. h4/bool/IN e (h4/list/LIST__TO__SET l) /\ happ P e)
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_lists_EXISTSu_u_NOTu_u_EVERY]: !_0. (!P x. happ (happ _0 P) x <=> ~happ P x) ==> (!l P. h4/list/EXISTS P l <=> ~h4/list/EVERY (happ _0 P) l)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_CONDu_u_RAND]: !y x f b. happ f (h4/bool/COND b x y) = h4/bool/COND b (happ f x) (happ f y)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_lists_NOTu_u_CONSu_u_NIL]: !a1 a0. ~(h4/list/CONS a0 a1 = h4/list/NIL)
% Assm [h4s_bools_CONDu_u_RATOR]: !x g f b. happ (h4/bool/COND b f g) x = h4/bool/COND b (happ f x) (happ g x)
% Assm [h4s_bools_Fu_u_DEF]: F <=> (!t. t)
% Assm [h4s_whiles_OWHILEu_u_IND]: !f P G. (!s. ~happ G s ==> happ (happ P s) s) /\ (!s1 s2. happ G s1 /\ happ (happ P (happ f s1)) s2 ==> happ (happ P s1) s2) ==> (!s1 s2. h4/while/OWHILE G f s1 = h4/option/SOME s2 ==> happ (happ P s1) s2)
% Assm [h4s_lists_FILTERu_u_EQu_u_ID]: !l P. h4/list/FILTER P l = l <=> h4/list/EVERY P l
% Assm [h4s_quantHeuristicss_IMPu_u_NEGu_u_CONTRA]: !x i P. ~happ P i ==> happ P x ==> ~(x = i)
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% 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_bools_LEFTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. happ P x /\ Q) <=> (?x. happ P x) /\ Q
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_options_someu_u_elim]: !Q P. happ Q (h4/option/some P) ==> (?x. happ P x /\ happ Q (h4/option/SOME x)) \/ (!x. ~happ P x) /\ happ Q h4/option/NONE
% Assm [h4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND]: !C B A. B /\ C \/ A <=> (B \/ A) /\ (C \/ A)
% Assm [h4s_whiles_HOAREu_u_SPECu_u_DEF]: !Q P C. h4/while/HOARE__SPEC P C Q <=> (!s. happ P s ==> happ Q (happ C s))
% Assm [h4s_whiles_WHILEu_u_INDUCTION]: !R C B. h4/relation/WF R /\ (!s. happ B s ==> happ (happ R (happ C s)) s) ==> (!P. (!s. (happ B s ==> happ P (happ C s)) ==> happ P s) ==> (!v. happ P v))
% Assm [h4s_whiles_WHILE0]: !x g P. happ (h4/while/WHILE P g) x = h4/bool/COND (happ P x) (happ (h4/while/WHILE P g) (happ g x)) x
% Assm [h4s_sats_dcu_u_cond]: !s r q p. (p <=> h4/bool/COND q r s) <=> (p \/ q \/ ~s) /\ (p \/ ~r \/ ~q) /\ (p \/ ~r \/ ~s) /\ (~q \/ r \/ ~p) /\ (q \/ s \/ ~p)
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_LEFTu_u_ORu_u_EXISTSu_u_THM]: !Q P. (?x. happ P x) \/ Q <=> (?x. happ P x \/ Q)
% Assm [h4s_lists_FILTERu_u_NEQu_u_NIL]: !l P. ~(h4/list/FILTER P l = h4/list/NIL) <=> (?x. h4/bool/IN x (h4/list/LIST__TO__SET l) /\ happ P x)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_whiles_OWHILEu_u_ENDCOND]: !s_27 s f G. h4/while/OWHILE G f s = h4/option/SOME s_27 ==> ~happ G s_27
% 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_EXISTSu_u_DEF]: !x. $exists x <=> happ x (h4/min/_40 x)
% Assm [h4s_bools_LEFTu_u_ANDu_u_FORALLu_u_THM]: !Q P. (!x. happ P x) /\ Q <=> (!x. happ P x /\ Q)
% Assm [h4s_quantHeuristicss_GUESSu_u_REWRITESu_c5]: !i P. h4/quantHeuristics/GUESS__FORALL__GAP i P <=> (!v. ~happ P v ==> (?fv. v = happ i fv))
% Assm [h4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. happ P x \/ Q) <=> (!x. happ P x) \/ Q
% Assm [h4s_lists_NULLu_u_FILTER]: !ls P. h4/list/NULL (h4/list/FILTER P ls) <=> (!x. h4/bool/IN x (h4/list/LIST__TO__SET ls) ==> ~happ P x)
% Assm [h4s_whiles_OWHILEu_u_def]: !_0. (!G f s n. happ (happ (happ (happ _0 G) f) s) n <=> ~happ G (h4/arithmetic/FUNPOW f n s)) ==> (!s f G. ?v. (v <=> (?n. ~happ G (h4/arithmetic/FUNPOW f n s))) /\ h4/while/OWHILE G f s = h4/bool/COND v (h4/option/SOME (h4/arithmetic/FUNPOW f (h4/while/LEAST (happ (happ (happ _0 G) f) s)) s)) h4/option/NONE)
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_quantHeuristicss_GUESSu_u_FORALLu_u_def]: !i P. h4/quantHeuristics/GUESS__FORALL i P <=> (!v. happ P v) <=> (!fv. happ P (happ i fv))
% Assm [h4s_quantHeuristicss_GUESSu_u_EXISTSu_u_def]: !i P. h4/quantHeuristics/GUESS__EXISTS i P <=> (?v. happ P v) <=> (?fv. happ P (happ i fv))
% 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_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_whiles_LEASTu_u_EXISTSu_u_IMP]: !p. (?n. happ p n) ==> happ p (h4/while/LEAST p) /\ (!n. h4/prim__rec/_3C n (h4/while/LEAST p) ==> ~happ p n)
% Assm [h4s_arithmetics_FUNPOW0u_c1]: !x n f. h4/arithmetic/FUNPOW f (h4/num/SUC n) x = h4/arithmetic/FUNPOW f n (happ f x)
% Assm [h4s_arithmetics_FUNPOW0u_c0]: !x f. h4/arithmetic/FUNPOW f h4/num/0 x = x
% Assm [h4s_primu_u_recs_LESSu_u_0]: !n. h4/prim__rec/_3C h4/num/0 (h4/num/SUC n)
% Assm [h4s_nums_INDUCTION]: !P. happ P h4/num/0 /\ (!n. happ P n ==> happ P (h4/num/SUC n)) ==> (!n. happ P n)
% Assm [h4s_primu_u_recs_LESSu_u_MONO]: !n m. h4/prim__rec/_3C m n ==> h4/prim__rec/_3C (h4/num/SUC m) (h4/num/SUC n)
% Assm [h4s_options_NOTu_u_NONEu_u_SOME]: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm [h4s_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_DISJu_u_COMM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_UNWINDu_u_FORALLu_u_THM2]: !v f. (!x. x = v ==> happ f x) <=> happ f v
% Assm [h4s_bools_PULLu_u_EXISTSu_c0]: !Q P. (?x. happ P x) ==> Q <=> (!x. happ P x ==> Q)
% Assm [h4s_bools_IMPu_u_DISJu_u_THM]: !B A. A ==> B <=> ~A \/ B
% Assm [h4s_quantHeuristicss_GUESSu_u_RULESu_u_TRIVIALu_u_FORALLu_u_POINT]: !_0. (!i xxx. happ (happ _0 i) xxx = i) ==> (!i P. ~happ P i ==> h4/quantHeuristics/GUESS__FORALL__POINT (happ _0 i) P)
% Assm [h4s_quantHeuristicss_EXISTSu_u_NOTu_u_FORALLu_u_THM]: !P. (?x. happ P x) <=> ~(!x. ~happ P x)
% Assm [h4s_options_someu_u_def]: !_0. (!P x. happ (happ _0 P) x <=> happ P x) ==> (!P. ?v. (v <=> (?x. happ P x)) /\ h4/option/some P = h4/bool/COND v (h4/option/SOME (h4/min/_40 (happ _0 P))) h4/option/NONE)
% Assm [h4s_options_someu_u_intro]: !Q P. (!x. happ P x ==> happ Q (h4/option/SOME x)) /\ ((!x. ~happ P x) ==> happ Q h4/option/NONE) ==> happ Q (h4/option/some P)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_quantHeuristicss_MOVEu_u_EXISTSu_u_IMPu_u_THM]: !R Q P. (?x. (!y. ~happ (happ P x) y ==> happ R y) ==> happ Q x) <=> (!y. ~(!x. happ (happ P x) y) ==> happ R y) ==> (?x. happ Q x)
% Assm [h4s_whiles_LEASTu_u_ELIM]: !Q P. (?n. happ P n) /\ (!n. (!m. h4/prim__rec/_3C m n ==> ~happ P m) /\ happ P n ==> happ Q n) ==> happ Q (h4/while/LEAST P)
% 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_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_lists_EXISTSu_u_DEFu_c1]: !t h P. h4/list/EXISTS P (h4/list/CONS h t) <=> happ P h \/ h4/list/EXISTS P t
% Assm [h4s_lists_EXISTSu_u_DEFu_c0]: !P. h4/list/EXISTS P h4/list/NIL <=> F
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_EQu_u_REFL]: !x. x = x
% Assm [h4s_lists_MEMu_c1]: !x t h. h4/bool/IN x (h4/list/LIST__TO__SET (h4/list/CONS h t)) <=> x = h \/ h4/bool/IN x (h4/list/LIST__TO__SET t)
% Assm [h4s_lists_MEMu_c0]: !x. h4/bool/IN x (h4/list/LIST__TO__SET h4/list/NIL) <=> F
% Assm [h4s_lists_NULL0u_c1]: !t h. ~h4/list/NULL (h4/list/CONS h t)
% Assm [h4s_lists_NULL0u_c0]: h4/list/NULL h4/list/NIL
% Assm [h4s_bools_NOTu_u_IMP]: !B A. ~(A ==> B) <=> A /\ ~B
% Assm [h4s_bools_LEFTu_u_ORu_u_OVERu_u_AND]: !C B A. A \/ B /\ C <=> (A \/ B) /\ (A \/ C)
% Assm [h4s_quantHeuristicss_GUESSu_u_REWRITESu_c3]: !i P. h4/quantHeuristics/GUESS__FORALL__POINT i P <=> (!fv. ~happ P (happ i fv))
% Assm [h4s_bools_RIGHTu_u_ORu_u_EXISTSu_u_THM]: !Q P. P \/ (?x. happ Q x) <=> (?x. P \/ happ Q x)
% Assm [h4s_bools_RIGHTu_u_EXISTSu_u_ANDu_u_THM]: !Q P. (?x. P /\ happ Q x) <=> P /\ (?x. happ Q x)
% Goal: !y Q P. (!x. happ P x \/ happ Q x) ==> (~happ P y ==> happ Q y) /\ (~happ Q y ==> happ P y)
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_Q1098773,TV_Q1098769]: ![V_f, V_g]: (![V_x]: s(TV_Q1098769,happ(s(t_fun(TV_Q1098773,TV_Q1098769),V_f),s(TV_Q1098773,V_x))) = s(TV_Q1098769,happ(s(t_fun(TV_Q1098773,TV_Q1098769),V_g),s(TV_Q1098773,V_x))) => s(t_fun(TV_Q1098773,TV_Q1098769),V_f) = s(t_fun(TV_Q1098773,TV_Q1098769),V_g))).
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_quantHeuristicss_GUESSu_u_FORALLu_u_GAPu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_i, V_P]: (p(s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))) <=> ![V_v]: (~ (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_v))))) => ?[V_fv]: s(TV_u_27b,V_v) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(TV_u_27a,V_fv)))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,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_quantHeuristicss_GUESSu_u_EXISTSu_u_FORALLu_u_REWRITESu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_i, V_P]: (p(s(t_bool,h4s_quantheuristicss_guessu_u_forall(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))) <=> ![V_v]: (~ (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_v))))) => ?[V_fv]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(TV_u_27a,V_fv)))))))))).
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_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_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_lists_EVERYu_u_NOTu_u_EXISTS, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(TV_u_27a,V_x)))) <=> ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))))) => ![V_l, V_P]: (p(s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))) <=> ~ (p(s(t_bool,h4s_lists_exists(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_h4s_lists_list(TV_u_27a),V_l)))))))).
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_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_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_lists_FILTERu_u_EQu_u_NIL, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(TV_u_27a,V_x)))) <=> ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))))) => ![V_l, V_P]: (s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil) <=> p(s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_h4s_lists_list(TV_u_27a),V_l))))))).
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_quantHeuristicss_GUESSu_u_EXISTSu_u_FORALLu_u_REWRITESu_c0, axiom, ![TV_u_27b,TV_u_27a]: ![V_i, V_P]: (p(s(t_bool,h4s_quantheuristicss_guessu_u_exists(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))) <=> ![V_v]: (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_v)))) => ?[V_fv]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(TV_u_27a,V_fv))))))))).
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_lists_FILTERu_u_CONDu_u_REWRITEu_c2, axiom, ![TV_u_27a]: ![V_l, V_h, V_P]: (~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_h))))) => s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_l))))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l))))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
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_lists_FILTERu_u_NEQu_u_ID, axiom, ![TV_u_27a]: ![V_l, V_P]: (~ (s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),V_l)) <=> ?[V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l)))))) & ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))))))).
fof(ah4s_quantHeuristicss_GUESSu_u_REWRITESu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_i, V_P]: (p(s(t_bool,h4s_quantheuristicss_guessu_u_forall(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))) <=> ![V_v]: (~ (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_v))))) => ?[V_fv]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(TV_u_27a,V_fv)))))))))).
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_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_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_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_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> 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_whiles_WHILEu_u_RULE, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_B, V_s]: (p(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_1),s(t_fun(TV_u_27a,t_bool),V_B))),s(TV_u_27a,V_s)))) <=> ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_B),s(TV_u_27a,V_s)))))) => ![V_uu_0]: (![V_P, V_B, V_s]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_fun(TV_u_27a,t_bool),V_B))),s(TV_u_27a,V_s)))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_s)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_B),s(TV_u_27a,V_s)))))) => ![V_R, V_P, V_C, V_B]: ((p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) & ![V_s]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_B),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,happ(s(t_fun(TV_u_27a,TV_u_27a),V_C),s(TV_u_27a,V_s))))),s(TV_u_27a,V_s)))))) => (p(s(t_bool,h4s_whiles_hoareu_u_spec(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_fun(TV_u_27a,t_bool),V_B))),s(t_fun(TV_u_27a,TV_u_27a),V_C),s(t_fun(TV_u_27a,t_bool),V_P)))) => p(s(t_bool,h4s_whiles_hoareu_u_spec(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,TV_u_27a),h4s_whiles_while(s(t_fun(TV_u_27a,t_bool),V_B),s(t_fun(TV_u_27a,TV_u_27a),V_C))),s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),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_1),s(t_fun(TV_u_27a,t_bool),V_B))))))))))))).
fof(ah4s_quantHeuristicss_UNWINDu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_a, 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]: (~ (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_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_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_sats_ORu_u_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_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_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_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_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_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_lists_listu_u_INDUCT, axiom, ![TV_u_27a]: ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)))) & ![V_t]: (p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t)))) => ![V_h]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t)))))))) => ![V_l]: p(s(t_bool,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (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_lists_FILTER0u_c1, axiom, ![TV_u_27a]: ![V_t, V_h, V_P]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t))))) = s(t_h4s_lists_list(TV_u_27a),h4s_bools_cond(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_h))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t))))),s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t)))))).
fof(ah4s_lists_FILTER0u_c0, axiom, ![TV_u_27a]: ![V_P]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)).
fof(ah4s_lists_EVERYu_u_DEFu_c0, axiom, ![TV_u_27a]: ![V_P]: s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_bool,t)).
fof(ah4s_lists_EVERYu_u_DEFu_c1, axiom, ![TV_u_27a]: ![V_t, V_h, V_P]: (p(s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t)))))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_h)))) & p(s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t))))))).
fof(ah4s_bools_CONDu_u_CONG, axiom, ![TV_u_27a]: ![V_yu_27, V_y, V_xu_27, V_x, V_Q, V_P]: ((s(t_bool,V_P) = s(t_bool,V_Q) & ((p(s(t_bool,V_Q)) => s(TV_u_27a,V_x) = s(TV_u_27a,V_xu_27)) & (~ (p(s(t_bool,V_Q))) => s(TV_u_27a,V_y) = s(TV_u_27a,V_yu_27)))) => s(TV_u_27a,h4s_bools_cond(s(t_bool,V_P),s(TV_u_27a,V_x),s(TV_u_27a,V_y))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,V_Q),s(TV_u_27a,V_xu_27),s(TV_u_27a,V_yu_27))))).
fof(ah4s_lists_EXISTSu_u_MEM, axiom, ![TV_u_27a]: ![V_l, V_P]: (p(s(t_bool,h4s_lists_exists(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))) <=> ?[V_e]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_e),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l)))))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_e))))))).
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_lists_EXISTSu_u_NOTu_u_EVERY, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(TV_u_27a,V_x)))) <=> ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))))) => ![V_l, V_P]: (p(s(t_bool,h4s_lists_exists(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))) <=> ~ (p(s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_h4s_lists_list(TV_u_27a),V_l)))))))).
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_CONDu_u_RAND, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x, V_f, V_b]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,h4s_bools_cond(s(t_bool,V_b),s(TV_u_27a,V_x),s(TV_u_27a,V_y))))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_b),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y)))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_lists_NOTu_u_CONSu_u_NIL, axiom, ![TV_u_27a]: ![V_a1, V_a0]: ~ (s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a0),s(t_h4s_lists_list(TV_u_27a),V_a1))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))).
fof(ah4s_bools_CONDu_u_RATOR, axiom, ![TV_u_27b,TV_u_27a]: ![V_x, V_g, V_f, V_b]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_bools_cond(s(t_bool,V_b),s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27b,h4s_bools_cond(s(t_bool,V_b),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_g),s(TV_u_27a,V_x)))))).
fof(ah4s_bools_Fu_u_DEF, axiom, (p(s(t_bool,f)) <=> ![V_t]: p(s(t_bool,V_t)))).
fof(ah4s_whiles_OWHILEu_u_IND, axiom, ![TV_u_27a]: ![V_f, V_P, V_G]: ((![V_s]: (~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_G),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_P),s(TV_u_27a,V_s))),s(TV_u_27a,V_s))))) & ![V_s1, V_s2]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_G),s(TV_u_27a,V_s1)))) & 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,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_s1))))),s(TV_u_27a,V_s2))))) => 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_s1))),s(TV_u_27a,V_s2)))))) => ![V_s1, V_s2]: (s(t_h4s_options_option(TV_u_27a),h4s_whiles_owhile(s(t_fun(TV_u_27a,t_bool),V_G),s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_s1))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_s2))) => 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_s1))),s(TV_u_27a,V_s2))))))).
fof(ah4s_lists_FILTERu_u_EQu_u_ID, axiom, ![TV_u_27a]: ![V_l, V_P]: (s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),V_l) <=> p(s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ah4s_quantHeuristicss_IMPu_u_NEGu_u_CONTRA, axiom, ![TV_u_27a]: ![V_x, V_i, V_P]: (~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_i))))) => (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ~ (s(TV_u_27a,V_x) = s(TV_u_27a,V_i))))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_bools_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_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_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_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_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_options_someu_u_elim, axiom, ![TV_u_27a]: ![V_Q, V_P]: (p(s(t_bool,happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),V_Q),s(t_h4s_options_option(TV_u_27a),h4s_options_some0(s(t_fun(TV_u_27a,t_bool),V_P)))))) => (?[V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) & p(s(t_bool,happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),V_Q),s(t_h4s_options_option(TV_u_27a),h4s_options_some(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))))) & p(s(t_bool,happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),V_Q),s(t_h4s_options_option(TV_u_27a),h4s_options_none)))))))).
fof(ah4s_bools_RIGHTu_u_ORu_u_OVERu_u_AND, axiom, ![V_C, V_B, 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)) | p(s(t_bool,V_A)))))).
fof(ah4s_whiles_HOAREu_u_SPECu_u_DEF, axiom, ![TV_u_27b,TV_u_27a]: ![V_Q, V_P, V_C]: (p(s(t_bool,h4s_whiles_hoareu_u_spec(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,TV_u_27b),V_C),s(t_fun(TV_u_27b,t_bool),V_Q)))) <=> ![V_s]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_s)))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_Q),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_C),s(TV_u_27a,V_s))))))))).
fof(ah4s_whiles_WHILEu_u_INDUCTION, axiom, ![TV_u_27a]: ![V_R, V_C, V_B]: ((p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) & ![V_s]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_B),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,happ(s(t_fun(TV_u_27a,TV_u_27a),V_C),s(TV_u_27a,V_s))))),s(TV_u_27a,V_s)))))) => ![V_P]: (![V_s]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_B),s(TV_u_27a,V_s)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_C),s(TV_u_27a,V_s))))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_s))))) => ![V_v]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_v))))))).
fof(ah4s_whiles_WHILE0, axiom, ![TV_u_27a]: ![V_x, V_g, V_P]: s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_whiles_while(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(TV_u_27a,V_x))) = s(TV_u_27a,h4s_bools_cond(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_whiles_while(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,TV_u_27a),V_g))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_g),s(TV_u_27a,V_x))))),s(TV_u_27a,V_x)))).
fof(ah4s_sats_dcu_u_cond, axiom, ![V_s, V_r, V_q, V_p]: (s(t_bool,V_p) = s(t_bool,h4s_bools_cond(s(t_bool,V_q),s(t_bool,V_r),s(t_bool,V_s))) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_s))))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_s))))) & ((~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_s)) | ~ (p(s(t_bool,V_p))))))))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_LEFTu_u_ORu_u_EXISTSu_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_lists_FILTERu_u_NEQu_u_NIL, axiom, ![TV_u_27a]: ![V_l, V_P]: (~ (s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil)) <=> ?[V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_l)))))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_whiles_OWHILEu_u_ENDCOND, axiom, ![TV_u_27a]: ![V_su_27, V_s, V_f, V_G]: (s(t_h4s_options_option(TV_u_27a),h4s_whiles_owhile(s(t_fun(TV_u_27a,t_bool),V_G),s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_s))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_su_27))) => ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_G),s(TV_u_27a,V_su_27))))))).
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_EXISTSu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_x)))))).
fof(ah4s_bools_LEFTu_u_ANDu_u_FORALLu_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_quantHeuristicss_GUESSu_u_REWRITESu_c5, axiom, ![TV_u_27b,TV_u_27a]: ![V_i, V_P]: (p(s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_gap(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))) <=> ![V_v]: (~ (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_v))))) => ?[V_fv]: s(TV_u_27b,V_v) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(TV_u_27a,V_fv)))))).
fof(ah4s_bools_LEFTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))) <=> (![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) | p(s(t_bool,V_Q))))).
fof(ah4s_lists_NULLu_u_FILTER, axiom, ![TV_u_27a]: ![V_ls, V_P]: (p(s(t_bool,h4s_lists_null(s(t_h4s_lists_list(TV_u_27a),h4s_lists_filter(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_ls)))))) <=> ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_ls)))))) => ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))))))).
fof(ah4s_whiles_OWHILEu_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_G, V_f, V_s, V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_G))),s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(TV_u_27a,V_s))),s(t_h4s_nums_num,V_n)))) <=> ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_G),s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,V_n),s(TV_u_27a,V_s)))))))) => ![V_s, V_f, V_G]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ?[V_n]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_G),s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,V_n),s(TV_u_27a,V_s)))))))) & s(t_h4s_options_option(TV_u_27a),h4s_whiles_owhile(s(t_fun(TV_u_27a,t_bool),V_G),s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_s))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_whiles_least(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_fun(TV_u_27a,TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_nums_num,t_bool)))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_G))),s(t_fun(TV_u_27a,TV_u_27a),V_f))),s(TV_u_27a,V_s))))),s(TV_u_27a,V_s))))),s(t_h4s_options_option(TV_u_27a),h4s_options_none)))))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_quantHeuristicss_GUESSu_u_FORALLu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_i, V_P]: (p(s(t_bool,h4s_quantheuristicss_guessu_u_forall(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))) <=> (![V_v]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_v)))) <=> ![V_fv]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(TV_u_27a,V_fv))))))))).
fof(ah4s_quantHeuristicss_GUESSu_u_EXISTSu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_i, V_P]: (p(s(t_bool,h4s_quantheuristicss_guessu_u_exists(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))) <=> (?[V_v]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,V_v)))) <=> ?[V_fv]: p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(TV_u_27a,V_fv))))))))).
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_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_whiles_LEASTu_u_EXISTSu_u_IMP, axiom, ![V_p]: (?[V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_p),s(t_h4s_nums_num,V_n)))) => (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_p),s(t_h4s_nums_num,h4s_whiles_least(s(t_fun(t_h4s_nums_num,t_bool),V_p)))))) & ![V_n]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_whiles_least(s(t_fun(t_h4s_nums_num,t_bool),V_p)))))) => ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_p),s(t_h4s_nums_num,V_n))))))))).
fof(ah4s_arithmetics_FUNPOW0u_c1, axiom, ![TV_u_27a]: ![V_x, V_n, V_f]: s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(TV_u_27a,V_x))) = s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,V_n),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(TV_u_27a,V_x)))))).
fof(ah4s_arithmetics_FUNPOW0u_c0, axiom, ![TV_u_27a]: ![V_x, V_f]: s(TV_u_27a,h4s_arithmetics_funpow(s(t_fun(TV_u_27a,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_0),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_primu_u_recs_LESSu_u_0, axiom, ![V_n]: p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))))).
fof(ah4s_nums_INDUCTION, axiom, ![V_P]: ((p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_0)))) & ![V_n]: (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))) => ![V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))))).
fof(ah4s_primu_u_recs_LESSu_u_MONO, axiom, ![V_n, V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) => p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n)))))))).
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_options_SOMEu_u_11, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_bools_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_COMM, 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_UNWINDu_u_FORALLu_u_THM2, axiom, ![TV_u_27a]: ![V_v, V_f]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_v) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))) <=> p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_v)))))).
fof(ah4s_bools_PULLu_u_EXISTSu_c0, 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_IMPu_u_DISJu_u_THM, 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_quantHeuristicss_GUESSu_u_RULESu_u_TRIVIALu_u_FORALLu_u_POINT, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_i, V_xxx]: s(TV_u_27a,happ(s(t_fun(t_h4s_ones_one,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_ones_one,TV_u_27a)),V_uu_0),s(TV_u_27a,V_i))),s(t_h4s_ones_one,V_xxx))) = s(TV_u_27a,V_i) => ![V_i, V_P]: (~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_i))))) => p(s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_point(s(t_fun(t_h4s_ones_one,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_ones_one,TV_u_27a)),V_uu_0),s(TV_u_27a,V_i))),s(t_fun(TV_u_27a,t_bool),V_P))))))).
fof(ah4s_quantHeuristicss_EXISTSu_u_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_options_someu_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_P, V_x]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(TV_u_27a,V_x))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))) => ![V_P]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) & s(t_h4s_options_option(TV_u_27a),h4s_options_some0(s(t_fun(TV_u_27a,t_bool),V_P))) = s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))))))),s(t_h4s_options_option(TV_u_27a),h4s_options_none)))))).
fof(ah4s_options_someu_u_intro, 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(t_h4s_options_option(TV_u_27a),t_bool),V_Q),s(t_h4s_options_option(TV_u_27a),h4s_options_some(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))))) => p(s(t_bool,happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),V_Q),s(t_h4s_options_option(TV_u_27a),h4s_options_none)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_options_option(TV_u_27a),t_bool),V_Q),s(t_h4s_options_option(TV_u_27a),h4s_options_some0(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
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_quantHeuristicss_MOVEu_u_EXISTSu_u_IMPu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_R, V_Q, 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))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_R),s(TV_u_27b,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (![V_y]: (~ (![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,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_R),s(TV_u_27b,V_y))))) => ?[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_whiles_LEASTu_u_ELIM, axiom, ![V_Q, V_P]: ((?[V_n]: p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n)))) & ![V_n]: ((![V_m]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) => ~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_m)))))) & p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_P),s(t_h4s_nums_num,V_n))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_Q),s(t_h4s_nums_num,V_n)))))) => p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),V_Q),s(t_h4s_nums_num,h4s_whiles_least(s(t_fun(t_h4s_nums_num,t_bool),V_P)))))))).
fof(ah4s_options_IFu_u_EQUALSu_u_OPTIONu_c2, axiom, ![TV_u_27a]: ![V_y, V_x, V_P]: (s(t_h4s_options_option(TV_u_27a),h4s_bools_cond(s(t_bool,V_P),s(t_h4s_options_option(TV_u_27a),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_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_lists_EXISTSu_u_DEFu_c1, axiom, ![TV_u_27a]: ![V_t, V_h, V_P]: (p(s(t_bool,h4s_lists_exists(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t)))))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_h)))) | p(s(t_bool,h4s_lists_exists(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_t))))))).
fof(ah4s_lists_EXISTSu_u_DEFu_c0, axiom, ![TV_u_27a]: ![V_P]: s(t_bool,h4s_lists_exists(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(t_bool,f)).
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_EQu_u_REFL, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,V_x) = s(TV_u_27a,V_x)).
fof(ah4s_lists_MEMu_c1, axiom, ![TV_u_27a]: ![V_x, V_t, V_h]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t)))))))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_h) | p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),V_t))))))))).
fof(ah4s_lists_MEMu_c0, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),h4s_lists_listu_u_tou_u_set(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))))) = s(t_bool,f)).
fof(ah4s_lists_NULL0u_c1, axiom, ![TV_u_27a]: ![V_t, V_h]: ~ (p(s(t_bool,h4s_lists_null(s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_h),s(t_h4s_lists_list(TV_u_27a),V_t)))))))).
fof(ah4s_lists_NULL0u_c0, axiom, ![TV_u_27a]: p(s(t_bool,h4s_lists_null(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))))).
fof(ah4s_bools_NOTu_u_IMP, 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_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_quantHeuristicss_GUESSu_u_REWRITESu_c3, axiom, ![TV_u_27b,TV_u_27a]: ![V_i, V_P]: (p(s(t_bool,h4s_quantheuristicss_guessu_u_forallu_u_point(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(t_fun(TV_u_27b,t_bool),V_P)))) <=> ![V_fv]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),V_P),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_i),s(TV_u_27a,V_fv))))))))).
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_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(ch4s_quantHeuristicss_DISJu_u_IMPu_u_INTRO, conjecture, ![TV_u_27a]: ![V_y, 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))))) => ((~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_y))))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_y))))) & (~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),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)))))))).
