0.11/0.11	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.11/0.12	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.13/0.31	% Computer   : n003.cluster.edu
0.13/0.31	% Model      : x86_64 x86_64
0.13/0.31	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.13/0.31	% Memory     : 8042.1875MB
0.13/0.31	% OS         : Linux 3.10.0-693.el7.x86_64
0.13/0.31	% CPULimit   : 180
0.13/0.31	% DateTime   : Thu Aug 29 10:04:32 EDT 2019
0.13/0.32	% CPUTime    : 
125.58/125.86	% SZS status Theorem
125.58/125.86	
126.18/126.46	% SZS output start Proof
126.18/126.46	Take the following subset of the input axioms:
126.37/126.69	  fof(arity_Int_Oint___Groups_Oab__semigroup__mult, axiom, ab_semigroup_mult(int)).
126.37/126.69	  fof(arity_Int_Oint___Groups_Ogroup__add, axiom, group_add(int)).
126.37/126.69	  fof(arity_Int_Oint___Int_Onumber__ring, axiom, number_ring(int)).
126.37/126.69	  fof(arity_Int_Oint___Rings_Ocomm__semiring__1, axiom, comm_semiring_1(int)).
126.37/126.69	  fof(arity_Int_Oint___Rings_Oring__1__no__zero__divisors, axiom, ring_11004092258visors(int)).
126.37/126.69	  fof(arity_Nat_Onat___Int_Onumber__semiring, axiom, number_semiring(nat)).
126.37/126.69	  fof(arity_Nat_Onat___Rings_Olinordered__semidom, axiom, linordered_semidom(nat)).
126.37/126.69	  fof(conj_0, conjecture, hAPP(nat, int, power_power(int, plus_plus(int, one_one(int), hAPP(nat, int, semiring_1_of_nat(int), n))), number_number_of(nat, bit0(bit1(pls))))!=zero_zero(int)).
126.37/126.69	  fof(fact_107_of__nat__0__less__iff, axiom, ![X_a]: (![Na]: (hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), zero_zero(nat)), Na)) <=> hBOOL(hAPP(X_a, bool, hAPP(X_a, fun(X_a, bool), ord_less(X_a), zero_zero(X_a)), hAPP(nat, X_a, semiring_1_of_nat(X_a), Na)))) <= linordered_semidom(X_a))).
126.37/126.69	  fof(fact_115_less__zeroE, axiom, ![N]: ~hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), N), zero_zero(nat)))).
126.37/126.69	  fof(fact_117_less__not__refl, axiom, ![N]: ~hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), N), N))).
126.37/126.69	  fof(fact_118_not__add__less1, axiom, ![J_1, I_2]: ~hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), plus_plus(nat, I_2, J_1)), I_2))).
126.37/126.69	  fof(fact_119_not__add__less2, axiom, ![J_1, I_2]: ~hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), plus_plus(nat, J_1, I_2)), I_2))).
126.37/126.69	  fof(fact_120_number__of__is__id, axiom, ![K_1]: ti(int, K_1)=number_number_of(int, K_1)).
126.37/126.69	  fof(fact_121_nat__neq__iff, axiom, ![Ma, Na]: (Na!=Ma <=> (hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), Ma), Na)) | hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), Na), Ma))))).
126.37/126.69	  fof(fact_129_less__irrefl__nat, axiom, ![N]: ~hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), N), N))).
126.37/126.69	  fof(fact_130_less__not__refl2, axiom, ![N, M]: (M!=N <= hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), N), M)))).
126.37/126.69	  fof(fact_131_less__not__refl3, axiom, ![S_1, T_2]: (T_2!=S_1 <= hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), S_1), T_2)))).
126.37/126.69	  fof(fact_138_nat__less__cases, axiom, ![Ma, Na, P_1]: (((hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), P_1, Na), Ma)) <= Ma=Na) => (hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), P_1, Na), Ma)) <= (hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), Na), Ma)) => hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), P_1, Na), Ma))))) <= (hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), P_1, Na), Ma)) <= hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), Ma), Na))))).
126.37/126.69	  fof(fact_140_gr__implies__not0, axiom, ![N, M]: (hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), M), N)) => N!=zero_zero(nat))).
126.37/126.69	  fof(fact_142_less__nat__zero__code, axiom, ![N]: ~hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), N), zero_zero(nat)))).
126.37/126.69	  fof(fact_145_neq0__conv, axiom, ![Na]: (hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), zero_zero(nat)), Na)) <=> Na!=zero_zero(nat))).
126.37/126.69	  fof(fact_146_not__less0, axiom, ![N]: ~hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), N), zero_zero(nat)))).
126.37/126.69	  fof(fact_14_n0, axiom, hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), zero_zero(nat)), n))).
126.37/126.69	  fof(fact_163_one__neq__zero, axiom, ![X_a]: (zero_neq_one(X_a) => zero_zero(X_a)!=one_one(X_a))).
126.37/126.69	  fof(fact_164_zero__neq__one, axiom, ![X_a]: (zero_neq_one(X_a) => zero_zero(X_a)!=one_one(X_a))).
126.37/126.69	  fof(fact_168_less__imp__of__nat__less, axiom, ![X_a]: (linordered_semidom(X_a) => ![N, M]: (hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), M), N)) => hBOOL(hAPP(X_a, bool, hAPP(X_a, fun(X_a, bool), ord_less(X_a), hAPP(nat, X_a, semiring_1_of_nat(X_a), M)), hAPP(nat, X_a, semiring_1_of_nat(X_a), N)))))).
126.37/126.69	  fof(fact_177_not__one__less__zero, axiom, ![X_a]: (~hBOOL(hAPP(X_a, bool, hAPP(X_a, fun(X_a, bool), ord_less(X_a), one_one(X_a)), zero_zero(X_a))) <= linordered_semidom(X_a))).
126.37/126.69	  fof(fact_185_power__eq__0__iff, axiom, ![X_a]: (![Na, A_2]: (hAPP(nat, X_a, power_power(X_a, A_2), Na)=zero_zero(X_a) <=> (zero_zero(X_a)=ti(X_a, A_2) & Na!=zero_zero(nat))) <= (mult_zero(X_a) & (zero_neq_one(X_a) & (no_zero_divisors(X_a) & power(X_a)))))).
126.37/126.69	  fof(fact_186_of__nat__less__0__iff, axiom, ![X_a]: (linordered_semidom(X_a) => ![M]: ~hBOOL(hAPP(X_a, bool, hAPP(X_a, fun(X_a, bool), ord_less(X_a), hAPP(nat, X_a, semiring_1_of_nat(X_a), M)), zero_zero(X_a))))).
126.37/126.69	  fof(fact_228_nat__power__eq__0__iff, axiom, ![Ma, Na]: (zero_zero(nat)=hAPP(nat, nat, power_power(nat, Ma), Na) <=> (Ma=zero_zero(nat) & Na!=zero_zero(nat)))).
126.37/126.69	  fof(fact_27_int__eq__0__conv, axiom, ![Na]: (zero_zero(int)=hAPP(nat, int, semiring_1_of_nat(int), Na) <=> zero_zero(nat)=Na)).
126.37/126.69	  fof(fact_314_nat__le__0, axiom, ![Z]: (hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less_eq(int), Z), zero_zero(int))) => nat_1(Z)=zero_zero(nat))).
126.37/126.69	  fof(fact_326_add__diff__cancel, axiom, ![X_a]: (![A_1, B_1]: minus_minus(X_a, plus_plus(X_a, A_1, B_1), B_1)=ti(X_a, A_1) <= group_add(X_a))).
126.37/126.69	  fof(fact_346_nat__int, axiom, ![N]: N=nat_1(hAPP(nat, int, semiring_1_of_nat(int), N))).
126.37/126.69	  fof(fact_37_one__is__num__one, axiom, one_one(int)=number_number_of(int, bit1(pls))).
126.37/126.69	  fof(fact_380_zless__le, axiom, ![Z_1, Wa]: ((hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less_eq(int), Z_1), Wa)) & ti(int, Wa)!=ti(int, Z_1)) <=> hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less(int), Z_1), Wa)))).
126.37/126.69	  fof(fact_384_times__numeral__code_I5_J, axiom, ![V_1, W]: number_number_of(int, times_times(int, V_1, W))=times_times(int, number_number_of(int, V_1), number_number_of(int, W))).
126.37/126.69	  fof(fact_409_not__square__less__zero, axiom, ![X_a]: (![A_1]: ~hBOOL(hAPP(X_a, bool, hAPP(X_a, fun(X_a, bool), ord_less(X_a), times_times(X_a, A_1, A_1)), zero_zero(X_a))) <= linordered_ring(X_a))).
126.37/126.69	  fof(fact_427_nat__number__of__def, axiom, ![V_1]: nat_1(number_number_of(int, V_1))=number_number_of(nat, V_1)).
126.37/126.69	  fof(fact_441_not__one__le__zero, axiom, ![X_a]: (linordered_semidom(X_a) => ~hBOOL(hAPP(X_a, bool, hAPP(X_a, fun(X_a, bool), ord_less_eq(X_a), one_one(X_a)), zero_zero(X_a))))).
126.37/126.69	  fof(fact_448_le__number__of__eq__not__less, axiom, ![X_a]: ((linorder(X_a) & number(X_a)) => ![Wa, Va]: (~hBOOL(hAPP(X_a, bool, hAPP(X_a, fun(X_a, bool), ord_less(X_a), number_number_of(X_a, Wa)), number_number_of(X_a, Va))) <=> hBOOL(hAPP(X_a, bool, hAPP(X_a, fun(X_a, bool), ord_less_eq(X_a), number_number_of(X_a, Va)), number_number_of(X_a, Wa)))))).
126.37/126.69	  fof(fact_45_zadd__assoc, axiom, ![Z1, Z2, Z3]: plus_plus(int, Z1, plus_plus(int, Z2, Z3))=plus_plus(int, plus_plus(int, Z1, Z2), Z3)).
126.37/126.69	  fof(fact_465_rel__simps_I27_J, axiom, ![K]: (hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less_eq(int), bit0(K)), pls)) <=> hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less_eq(int), K), pls)))).
126.37/126.69	  fof(fact_47_zadd__commute, axiom, ![Z, W]: plus_plus(int, Z, W)=plus_plus(int, W, Z)).
126.37/126.69	  fof(fact_485_not__sum__squares__lt__zero, axiom, ![X_a]: (linordered_ring(X_a) => ![X, Y]: ~hBOOL(hAPP(X_a, bool, hAPP(X_a, fun(X_a, bool), ord_less(X_a), plus_plus(X_a, times_times(X_a, X, X), times_times(X_a, Y, Y))), zero_zero(X_a))))).
126.37/126.69	  fof(fact_486_sum__squares__gt__zero__iff, axiom, ![X_a]: (![Xa, Ya]: ((zero_zero(X_a)!=ti(X_a, Ya) | ti(X_a, Xa)!=zero_zero(X_a)) <=> hBOOL(hAPP(X_a, bool, hAPP(X_a, fun(X_a, bool), ord_less(X_a), zero_zero(X_a)), plus_plus(X_a, times_times(X_a, Xa, Xa), times_times(X_a, Ya, Ya))))) <= linord581940658strict(X_a))).
126.37/126.69	  fof(fact_513_succ__Pls, axiom, bit1(pls)=succ(pls)).
126.37/126.69	  fof(fact_514_succ__Bit0, axiom, ![K_1]: succ(bit0(K_1))=bit1(K_1)).
126.37/126.69	  fof(fact_515_succ__Bit1, axiom, ![K_1]: succ(bit1(K_1))=bit0(succ(K_1))).
126.37/126.69	  fof(fact_519_succ__def, axiom, ![K_1]: plus_plus(int, K_1, one_one(int))=succ(K_1)).
126.37/126.69	  fof(fact_534_transfer__nat__int__numerals_I3_J, axiom, number_number_of(nat, bit0(bit1(pls)))=nat_1(number_number_of(int, bit0(bit1(pls))))).
126.37/126.69	  fof(fact_537_mult__2, axiom, ![X_a]: (![Z]: times_times(X_a, number_number_of(X_a, bit0(bit1(pls))), Z)=plus_plus(X_a, Z, Z) <= number_ring(X_a))).
126.37/126.69	  fof(fact_538_semiring__mult__2, axiom, ![X_a]: (number_semiring(X_a) => ![Z]: plus_plus(X_a, Z, Z)=times_times(X_a, number_number_of(X_a, bit0(bit1(pls))), Z))).
126.47/126.69	  fof(fact_541_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J, axiom, ![X_a]: (comm_semiring_1(X_a) => ![X]: hAPP(nat, X_a, power_power(X_a, X), number_number_of(nat, bit0(bit1(pls))))=times_times(X_a, X, X))).
126.47/126.69	  fof(fact_542_Nat__Transfer_Otransfer__nat__int__function__closures_I7_J, axiom, hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less_eq(int), zero_zero(int)), number_number_of(int, bit0(bit1(pls)))))).
126.47/126.69	  fof(fact_5_zero__eq__power2, axiom, ![X_a]: (ring_11004092258visors(X_a) => ![A_2]: (ti(X_a, A_2)=zero_zero(X_a) <=> hAPP(nat, X_a, power_power(X_a, A_2), number_number_of(nat, bit0(bit1(pls))))=zero_zero(X_a)))).
126.47/126.69	  fof(fact_609_nat__less__le, axiom, ![Ma, Na]: (hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), Ma), Na)) <=> (Ma!=Na & hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less_eq(nat), Ma), Na))))).
126.47/126.69	  fof(fact_610_real__less__def, axiom, ![Xa, Ya]: (hBOOL(hAPP(real, bool, hAPP(real, fun(real, bool), ord_less(real), Xa), Ya)) <=> (hBOOL(hAPP(real, bool, hAPP(real, fun(real, bool), ord_less_eq(real), Xa), Ya)) & Xa!=Ya))).
126.47/126.69	  fof(fact_61_int__less__0__conv, axiom, ![K_1]: ~hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less(int), hAPP(nat, int, semiring_1_of_nat(int), K_1)), zero_zero(int)))).
126.47/126.69	  fof(fact_653_not__real__square__gt__zero, axiom, ![Xa]: (~hBOOL(hAPP(real, bool, hAPP(real, fun(real, bool), ord_less(real), zero_zero(real)), times_times(real, Xa, Xa))) <=> zero_zero(real)=Xa)).
126.47/126.69	  fof(fact_66_rel__simps_I46_J, axiom, ![K_1]: bit1(K_1)!=pls).
126.47/126.69	  fof(fact_67_rel__simps_I39_J, axiom, ![L]: bit1(L)!=pls).
126.47/126.69	  fof(fact_685_nat__mult__distrib, axiom, ![Z, Z_2]: (nat_1(times_times(int, Z, Z_2))=times_times(nat, nat_1(Z), nat_1(Z_2)) <= hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less_eq(int), zero_zero(int)), Z)))).
126.47/126.69	  fof(fact_68_rel__simps_I50_J, axiom, ![K_1, L]: bit1(K_1)!=bit0(L)).
126.47/126.69	  fof(fact_69_rel__simps_I49_J, axiom, ![K_1, L]: bit0(K_1)!=bit1(L)).
126.47/126.69	  fof(fact_73_Pls__def, axiom, pls=zero_zero(int)).
126.47/126.69	  fof(fact_745_zdvd__not__zless, axiom, ![N, M]: (hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less(int), zero_zero(int)), M)) => (hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less(int), M), N)) => ~hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), dvd_dvd(int), N), M))))).
126.47/126.69	  fof(fact_758_rel__simps_I23_J, axiom, hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less_eq(int), min), pls))).
126.47/126.69	  fof(fact_765_rel__simps_I42_J, axiom, ![L]: bit0(L)!=min).
126.47/126.69	  fof(fact_766_rel__simps_I45_J, axiom, ![K_1]: min!=bit0(K_1)).
126.47/126.69	  fof(fact_776_succ__Min, axiom, succ(min)=pls).
126.47/126.69	  fof(fact_78_Bit0__def, axiom, ![K_1]: bit0(K_1)=plus_plus(int, K_1, K_1)).
126.47/126.69	  fof(fact_7_add__special_I3_J, axiom, ![X_a]: (![V_1]: number_number_of(X_a, plus_plus(int, V_1, bit1(pls)))=plus_plus(X_a, number_number_of(X_a, V_1), one_one(X_a)) <= number_ring(X_a))).
126.47/126.69	  fof(fact_803_nat__dvd__not__less, axiom, ![N, M]: (hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), zero_zero(nat)), M)) => (~hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), dvd_dvd(nat), N), M)) <= hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), M), N))))).
126.47/126.69	  fof(fact_821_zcong__not__zero, axiom, ![X, M]: (hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less(int), zero_zero(int)), X)) => (hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less(int), X), M)) => ~hBOOL(hAPP(int, bool, zcong(X, zero_zero(int)), M))))).
126.47/126.69	  fof(fact_830_divides__div__not, axiom, ![N, X, R_1, Q]: (X=plus_plus(nat, times_times(nat, Q, N), R_1) => (hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), zero_zero(nat)), R_1)) => (~hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), dvd_dvd(nat), N), X)) <= hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), R_1), N)))))).
126.47/126.69	  fof(fact_836_zcong__neg__1__impl__ne__1, axiom, ![X, P_2]: ((~hBOOL(hAPP(int, bool, zcong(X, one_one(int)), P_2)) <= hBOOL(hAPP(int, bool, zcong(X, number_number_of(int, min)), P_2))) <= hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less(int), number_number_of(int, bit0(bit1(pls)))), P_2)))).
126.47/126.69	  fof(fact_86_power__eq__0__iff__number__of, axiom, ![X_a]: (![Wa, A_2]: (zero_zero(X_a)=hAPP(nat, X_a, power_power(X_a, A_2), number_number_of(nat, Wa)) <=> (number_number_of(nat, Wa)!=zero_zero(nat) & ti(X_a, A_2)=zero_zero(X_a))) <= (power(X_a) & (mult_zero(X_a) & (zero_neq_one(X_a) & no_zero_divisors(X_a)))))).
126.47/126.69	  fof(fact_875_one__not__neg__one__mod__m, axiom, ![M]: (hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less(int), number_number_of(int, bit0(bit1(pls)))), M)) => ~hBOOL(hAPP(int, bool, zcong(one_one(int), number_number_of(int, min)), M)))).
126.47/126.69	  fof(fact_893_zcong__not, axiom, ![A_1, B_1, M]: (hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less(int), zero_zero(int)), A_1)) => (((~hBOOL(hAPP(int, bool, zcong(A_1, B_1), M)) <= hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less(int), B_1), A_1))) <= hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less(int), zero_zero(int)), B_1))) <= hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less(int), A_1), M))))).
126.47/126.69	  fof(fact_8_one__add__one__is__two, axiom, ![X_a]: (number_number_of(X_a, bit0(bit1(pls)))=plus_plus(X_a, one_one(X_a), one_one(X_a)) <= number_ring(X_a))).
126.47/126.69	  fof(fact_93_odd__nonzero, axiom, ![Z]: zero_zero(int)!=plus_plus(int, plus_plus(int, one_one(int), Z), Z)).
126.47/126.69	  fof(fact_945_zero__less__abs__iff, axiom, ![X_a]: (ordere142940540dd_abs(X_a) => ![A_2]: (ti(X_a, A_2)!=zero_zero(X_a) <=> hBOOL(hAPP(X_a, bool, hAPP(X_a, fun(X_a, bool), ord_less(X_a), zero_zero(X_a)), abs_abs(X_a, A_2)))))).
126.47/126.69	  fof(fact_946_abs__not__less__zero, axiom, ![X_a]: (ordere142940540dd_abs(X_a) => ![A_1]: ~hBOOL(hAPP(X_a, bool, hAPP(X_a, fun(X_a, bool), ord_less(X_a), abs_abs(X_a, A_1)), zero_zero(X_a))))).
126.47/126.69	  fof(fact_94_number__of__int, axiom, ![X_a]: (![N]: hAPP(nat, X_a, semiring_1_of_nat(X_a), N)=number_number_of(X_a, hAPP(nat, int, semiring_1_of_nat(int), N)) <= number_semiring(X_a))).
126.47/126.69	  fof(fact_95_zero__less__power2, axiom, ![X_a]: (![A_2]: (ti(X_a, A_2)!=zero_zero(X_a) <=> hBOOL(hAPP(X_a, bool, hAPP(X_a, fun(X_a, bool), ord_less(X_a), zero_zero(X_a)), hAPP(nat, X_a, power_power(X_a, A_2), number_number_of(nat, bit0(bit1(pls))))))) <= linordered_idom(X_a))).
126.47/126.69	  fof(fact_96_power2__less__0, axiom, ![X_a]: (![A_1]: ~hBOOL(hAPP(X_a, bool, hAPP(X_a, fun(X_a, bool), ord_less(X_a), hAPP(nat, X_a, power_power(X_a, A_1), number_number_of(nat, bit0(bit1(pls))))), zero_zero(X_a))) <= linordered_idom(X_a))).
126.47/126.69	  fof(fact_973_abs__add__one__not__less__self, axiom, ![X]: ~hBOOL(hAPP(real, bool, hAPP(real, fun(real, bool), ord_less(real), plus_plus(real, abs_abs(real, X), one_one(real))), X))).
126.47/126.69	  fof(fact_97_sum__power2__gt__zero__iff, axiom, ![X_a]: (![Xa, Ya]: ((zero_zero(X_a)!=ti(X_a, Ya) | ti(X_a, Xa)!=zero_zero(X_a)) <=> hBOOL(hAPP(X_a, bool, hAPP(X_a, fun(X_a, bool), ord_less(X_a), zero_zero(X_a)), plus_plus(X_a, hAPP(nat, X_a, power_power(X_a, Xa), number_number_of(nat, bit0(bit1(pls)))), hAPP(nat, X_a, power_power(X_a, Ya), number_number_of(nat, bit0(bit1(pls)))))))) <= linordered_idom(X_a))).
126.47/126.69	  fof(fact_98_not__sum__power2__lt__zero, axiom, ![X_a]: (![X, Y]: ~hBOOL(hAPP(X_a, bool, hAPP(X_a, fun(X_a, bool), ord_less(X_a), plus_plus(X_a, hAPP(nat, X_a, power_power(X_a, X), number_number_of(nat, bit0(bit1(pls)))), hAPP(nat, X_a, power_power(X_a, Y), number_number_of(nat, bit0(bit1(pls)))))), zero_zero(X_a))) <= linordered_idom(X_a))).
126.47/126.69	  fof(fact_9_semiring__one__add__one__is__two, axiom, ![X_a]: (plus_plus(X_a, one_one(X_a), one_one(X_a))=number_number_of(X_a, bit0(bit1(pls))) <= number_semiring(X_a))).
126.47/126.69	  fof(tsy_c_Groups_Otimes__class_Otimes_5_res, axiom, ![X_a, B_1_1, B_2_1]: (ab_semigroup_mult(X_a) => times_times(X_a, B_1_1, B_2_1)=ti(X_a, times_times(X_a, B_1_1, B_2_1)))).
126.47/126.69	  fof(tsy_c_Int_OBit1_res, hypothesis, ![B_1_1]: ti(int, bit1(B_1_1))=bit1(B_1_1)).
126.47/126.69	  fof(tsy_c_Int_Onat_arg1, axiom, ![B_1_1]: nat_1(B_1_1)=nat_1(ti(int, B_1_1))).
126.47/126.69	  fof(tsy_c_Int_Osucc_arg1, axiom, ![B_1_1]: succ(B_1_1)=succ(ti(int, B_1_1))).
126.47/126.69	  fof(tsy_c_Int_Osucc_res, axiom, ![B_1_1]: succ(B_1_1)=ti(int, succ(B_1_1))).
126.47/126.69	
126.47/126.69	Now clausify the problem and encode Horn clauses using encoding 3 of
126.47/126.69	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
126.47/126.69	We repeatedly replace C & s=t => u=v by the two clauses:
126.47/126.69	  fresh(y, y, x1...xn) = u
126.47/126.69	  C => fresh(s, t, x1...xn) = v
126.47/126.69	where fresh is a fresh function symbol and x1..xn are the free
126.47/126.69	variables of u and v.
126.47/126.69	A predicate p(X) is encoded as p(X)=true (this is sound, because the
126.47/126.69	input problem has no model of domain size 1).
126.47/126.69	
126.47/126.69	The encoding turns the above axioms into the following unit equations and goals:
126.47/126.69	
126.47/126.69	Axiom 1 (fact_107_of__nat__0__less__iff_1): fresh1195(X, X, Y, Z) = true2.
126.47/126.69	Axiom 2 (fact_107_of__nat__0__less__iff_1): fresh1194(X, X, Y, Z) = fresh1195(hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), zero_zero(nat)), Z)), true2, Y, Z).
126.47/126.69	Axiom 3 (fact_141_nat__power__less__imp__less): fresh1160(X, X, Y, Z, W) = hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), Y), Z)).
126.47/126.69	Axiom 4 (fact_168_less__imp__of__nat__less): fresh1457(X, X, Y, Z, W) = fresh1458(hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), Z), W)), true2, Y, Z, W).
126.47/126.69	Axiom 5 (fact_194_zero__less__double__add__iff__zero__less__single__add_1): fresh1103(X, X, Y, Z) = hBOOL(hAPP(Y, bool, hAPP(Y, fun(Y, bool), ord_less(Y), zero_zero(Y)), Z)).
126.47/126.69	Axiom 6 (fact_216_double__zero__sym): fresh1066(X, X, Y, Z) = plus_plus(Y, Z, Z).
126.47/126.69	Axiom 7 (fact_220_add__less__imp__less__left): fresh1060(X, X, Y, Z, W, V) = hBOOL(hAPP(Y, bool, hAPP(Y, fun(Y, bool), ord_less(Y), W), V)).
126.47/126.69	Axiom 8 (fact_250_semiring__mult__number__of): fresh1017(X, X, Y, Z, W) = times_times(Y, number_number_of(Y, W), number_number_of(Y, Z)).
126.47/126.69	Axiom 9 (fact_300_mult__less__imp__less__left): fresh1478(X, X, Y, Z, W, V) = hBOOL(hAPP(Y, bool, hAPP(Y, fun(Y, bool), ord_less(Y), W), V)).
126.47/126.69	Axiom 10 (fact_314_nat__le__0): fresh922(X, X, Y) = zero_zero(nat).
126.47/126.69	Axiom 11 (fact_318_int__eq__iff): fresh917(X, X, Y, Z) = hAPP(nat, int, semiring_1_of_nat(int), Y).
126.47/126.69	Axiom 12 (fact_326_add__diff__cancel): fresh899(X, X, Y, Z, W) = ti(Y, Z).
126.47/126.69	Axiom 13 (fact_405_realpow__minus__mult): fresh791(X, X, Y, Z, W) = hAPP(nat, Y, power_power(Y, Z), W).
126.47/126.69	Axiom 14 (fact_406_less__iff__diff__less__0_1): fresh787(X, X, Y, Z, W) = hBOOL(hAPP(Y, bool, hAPP(Y, fun(Y, bool), ord_less(Y), Z), W)).
126.47/126.69	Axiom 15 (fact_465_rel__simps_I27_J): fresh714(X, X, Y) = true2.
126.47/126.69	Axiom 16 (fact_474_transfer__int__nat__quantifiers_I1_J_1): fresh702(X, X, Y, Z) = hBOOL(hAPP(int, bool, Y, Z)).
126.47/126.69	Axiom 17 (fact_480_semiring__number__of__add__1): fresh694(X, X, Y, Z) = plus_plus(Y, number_number_of(Y, Z), one_one(Y)).
126.47/126.69	Axiom 18 (fact_537_mult__2): fresh605(X, X, Y, Z) = plus_plus(Y, Z, Z).
126.47/126.69	Axiom 19 (fact_538_semiring__mult__2): fresh604(X, X, Y, Z) = plus_plus(Y, Z, Z).
126.47/126.69	Axiom 20 (fact_540_power2__eq__square): fresh601(X, X, Y, Z) = times_times(Y, Z, Z).
126.47/126.69	Axiom 21 (fact_541_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J): fresh600(X, X, Y, Z) = times_times(Y, Z, Z).
126.47/126.69	Axiom 22 (fact_545_power2__eq__square__number__of): fresh1227(X, X, Y, Z) = times_times(Y, number_number_of(Y, Z), number_number_of(Y, Z)).
126.47/126.69	Axiom 23 (fact_550_power2__sum): fresh592(X, X, Y, Z, W) = hAPP(nat, Y, power_power(Y, plus_plus(Y, Z, W)), number_number_of(nat, bit0(bit1(pls)))).
126.47/126.69	Axiom 24 (fact_551_power2__less__imp__less): fresh1294(X, X, Y, Z, W) = hBOOL(hAPP(Y, bool, hAPP(Y, fun(Y, bool), ord_less(Y), Z), W)).
126.47/126.69	Axiom 25 (fact_560_q__pos__lemma): fresh578(X, X, Y, Z, W) = hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less_eq(int), zero_zero(int)), Z)).
126.47/126.69	Axiom 26 (fact_5_zero__eq__power2_1): fresh543(X, X, Y, Z) = ti(Y, Z).
126.47/126.69	Axiom 27 (fact_5_zero__eq__power2_1): fresh542(X, X, Y, Z) = zero_zero(Y).
126.47/126.69	Axiom 28 (fact_662_less__diff__iff_1): fresh1389(X, X, Y, Z, W) = hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), W), Y)).
126.47/126.69	Axiom 29 (fact_683_transfer__nat__int__relations_I3_J): fresh1510(X, X, Y, Z) = hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less_eq(int), Z), Y)).
126.47/126.69	Axiom 30 (fact_685_nat__mult__distrib): fresh449(X, X, Y, Z) = nat_1(times_times(int, Z, Y)).
126.47/126.69	Axiom 31 (fact_709_Int2_Oaux__1): fresh429(X, X, Y) = nat_1(minus_minus(int, Y, number_number_of(int, bit0(bit1(pls))))).
126.47/126.69	Axiom 32 (fact_7_add__special_I3_J): fresh353(X, X, Y, Z) = plus_plus(Y, number_number_of(Y, Z), one_one(Y)).
126.47/126.69	Axiom 33 (fact_8_one__add__one__is__two): fresh264(X, X, Y) = number_number_of(Y, bit0(bit1(pls))).
126.47/126.69	Axiom 34 (fact_935_abs__of__nat): fresh209(X, X, Y, Z) = hAPP(nat, Y, semiring_1_of_nat(Y), Z).
126.47/126.69	Axiom 35 (fact_94_number__of__int): fresh195(X, X, Y, Z) = hAPP(nat, Y, semiring_1_of_nat(Y), Z).
126.47/126.69	Axiom 36 (fact_964_abs__power2): fresh178(X, X, Y, Z) = hAPP(nat, Y, power_power(Y, Z), number_number_of(nat, bit0(bit1(pls)))).
126.47/126.69	Axiom 37 (tsy_c_Groups_Otimes__class_Otimes_5_res): fresh101(X, X, Y, Z, W) = times_times(W, Y, Z).
126.47/126.69	Axiom 38 (fact_73_Pls__def): pls = zero_zero(int).
126.47/126.69	Axiom 39 (fact_384_times__numeral__code_I5_J): number_number_of(int, times_times(int, X, Y)) = times_times(int, number_number_of(int, X), number_number_of(int, Y)).
126.47/126.69	Axiom 40 (fact_47_zadd__commute): plus_plus(int, X, Y) = plus_plus(int, Y, X).
126.47/126.69	Axiom 41 (tsy_c_Int_Osucc_arg1): succ(X) = succ(ti(int, X)).
126.47/126.69	Axiom 42 (fact_314_nat__le__0): fresh922(hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less_eq(int), X), zero_zero(int))), true2, X) = nat_1(X).
126.47/126.69	Axiom 43 (fact_541_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J): fresh600(comm_semiring_1(X), true2, X, Y) = hAPP(nat, X, power_power(X, Y), number_number_of(nat, bit0(bit1(pls)))).
126.47/126.69	Axiom 44 (fact_37_one__is__num__one): one_one(int) = number_number_of(int, bit1(pls)).
126.47/126.69	Axiom 45 (fact_776_succ__Min): succ(min) = pls.
126.47/126.69	Axiom 46 (tsy_c_Int_Osucc_res): succ(X) = ti(int, succ(X)).
126.47/126.69	Axiom 47 (fact_534_transfer__nat__int__numerals_I3_J): number_number_of(nat, bit0(bit1(pls))) = nat_1(number_number_of(int, bit0(bit1(pls)))).
126.47/126.69	Axiom 48 (fact_758_rel__simps_I23_J): hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less_eq(int), min), pls)) = true2.
126.47/126.69	Axiom 49 (fact_326_add__diff__cancel): fresh899(group_add(X), true2, X, Y, Z) = minus_minus(X, plus_plus(X, Y, Z), Z).
126.47/126.69	Axiom 50 (fact_465_rel__simps_I27_J): fresh714(hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less_eq(int), X), pls)), true2, X) = hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less_eq(int), bit0(X)), pls)).
126.47/126.69	Axiom 51 (fact_685_nat__mult__distrib): fresh449(hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less_eq(int), zero_zero(int)), X)), true2, Y, X) = times_times(nat, nat_1(X), nat_1(Y)).
126.47/126.69	Axiom 52 (fact_427_nat__number__of__def): nat_1(number_number_of(int, X)) = number_number_of(nat, X).
126.47/126.69	Axiom 53 (arity_Nat_Onat___Int_Onumber__semiring): number_semiring(nat) = true2.
126.47/126.69	Axiom 54 (arity_Nat_Onat___Rings_Olinordered__semidom): linordered_semidom(nat) = true2.
126.47/126.69	Axiom 55 (fact_9_semiring__one__add__one__is__two): fresh134(number_semiring(X), true2, X) = plus_plus(X, one_one(X), one_one(X)).
126.47/126.69	Axiom 56 (tsy_c_Groups_Otimes__class_Otimes_5_res): fresh101(ab_semigroup_mult(X), true2, Y, Z, X) = ti(X, times_times(X, Y, Z)).
126.47/126.69	Axiom 57 (arity_Int_Oint___Groups_Oab__semigroup__mult): ab_semigroup_mult(int) = true2.
126.47/126.69	Axiom 58 (fact_514_succ__Bit0): succ(bit0(X)) = bit1(X).
126.47/126.69	Axiom 59 (fact_537_mult__2): fresh605(number_ring(X), true2, X, Y) = times_times(X, number_number_of(X, bit0(bit1(pls))), Y).
126.47/126.69	Axiom 60 (arity_Int_Oint___Int_Onumber__ring): number_ring(int) = true2.
126.47/126.69	Axiom 61 (fact_538_semiring__mult__2): fresh604(number_semiring(X), true2, X, Y) = times_times(X, number_number_of(X, bit0(bit1(pls))), Y).
126.47/126.69	Axiom 62 (arity_Int_Oint___Rings_Ocomm__semiring__1): comm_semiring_1(int) = true2.
126.47/126.69	Axiom 63 (arity_Int_Oint___Groups_Ogroup__add): group_add(int) = true2.
126.47/126.69	Axiom 64 (fact_7_add__special_I3_J): fresh353(number_ring(X), true2, X, Y) = number_number_of(X, plus_plus(int, Y, bit1(pls))).
126.47/126.69	Axiom 65 (fact_542_Nat__Transfer_Otransfer__nat__int__function__closures_I7_J): hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less_eq(int), zero_zero(int)), number_number_of(int, bit0(bit1(pls))))) = true2.
126.47/126.69	Axiom 66 (fact_519_succ__def): plus_plus(int, X, one_one(int)) = succ(X).
126.47/126.69	Axiom 67 (fact_168_less__imp__of__nat__less): fresh1457(linordered_semidom(X), true2, X, Y, Z) = hBOOL(hAPP(X, bool, hAPP(X, fun(X, bool), ord_less(X), hAPP(nat, X, semiring_1_of_nat(X), Y)), hAPP(nat, X, semiring_1_of_nat(X), Z))).
126.47/126.69	Axiom 68 (fact_120_number__of__is__id): ti(int, X) = number_number_of(int, X).
126.47/126.69	Axiom 69 (tsy_c_Int_OBit1_res): ti(int, bit1(X)) = bit1(X).
126.47/126.69	Axiom 70 (fact_515_succ__Bit1): succ(bit1(X)) = bit0(succ(X)).
126.47/126.69	Axiom 71 (arity_Int_Oint___Rings_Oring__1__no__zero__divisors): ring_11004092258visors(int) = true2.
126.47/126.69	Axiom 72 (fact_346_nat__int): X = nat_1(hAPP(nat, int, semiring_1_of_nat(int), X)).
126.47/126.69	Axiom 73 (fact_8_one__add__one__is__two): fresh264(number_ring(X), true2, X) = plus_plus(X, one_one(X), one_one(X)).
126.47/126.69	Axiom 74 (fact_45_zadd__assoc): plus_plus(int, X, plus_plus(int, Y, Z)) = plus_plus(int, plus_plus(int, X, Y), Z).
126.47/126.69	Axiom 75 (fact_513_succ__Pls): bit1(pls) = succ(pls).
126.47/126.69	Axiom 76 (tsy_c_Int_Onat_arg1): nat_1(X) = nat_1(ti(int, X)).
126.47/126.69	Axiom 77 (fact_78_Bit0__def): bit0(X) = plus_plus(int, X, X).
126.47/126.69	Axiom 78 (fact_27_int__eq__0__conv): fresh989(zero_zero(nat), X, X) = hAPP(nat, int, semiring_1_of_nat(int), X).
126.47/126.69	Axiom 79 (fact_5_zero__eq__power2_1): fresh543(ring_11004092258visors(X), true2, X, Y) = fresh542(hAPP(nat, X, power_power(X, Y), number_number_of(nat, bit0(bit1(pls)))), zero_zero(X), X, Y).
126.47/126.69	Axiom 80 (fact_94_number__of__int): fresh195(number_semiring(X), true2, X, Y) = number_number_of(X, hAPP(nat, int, semiring_1_of_nat(int), Y)).
126.47/126.69	Axiom 81 (fact_14_n0): hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), zero_zero(nat)), n)) = true2.
126.47/126.69	Axiom 82 (fact_107_of__nat__0__less__iff_1): fresh1194(linordered_semidom(X), true2, X, Y) = hBOOL(hAPP(X, bool, hAPP(X, fun(X, bool), ord_less(X), zero_zero(X)), hAPP(nat, X, semiring_1_of_nat(X), Y))).
126.47/126.70	Axiom 83 (conj_0): hAPP(nat, int, power_power(int, plus_plus(int, one_one(int), hAPP(nat, int, semiring_1_of_nat(int), n))), number_number_of(nat, bit0(bit1(pls)))) = zero_zero(int).
126.47/126.70	
126.47/126.70	Lemma 84: number_number_of(nat, X) = nat_1(X).
126.47/126.70	Proof:
126.47/126.70	  number_number_of(nat, X)
126.47/126.70	= { by axiom 52 (fact_427_nat__number__of__def) }
126.47/126.70	  nat_1(number_number_of(int, X))
126.47/126.70	= { by axiom 68 (fact_120_number__of__is__id) }
126.47/126.70	  nat_1(ti(int, X))
126.47/126.70	= { by axiom 76 (tsy_c_Int_Onat_arg1) }
126.47/126.70	  nat_1(X)
126.47/126.70	
126.47/126.70	Lemma 85: nat_1(fresh917(?, ?, X, ?)) = X.
126.47/126.70	Proof:
126.47/126.70	  nat_1(fresh917(?, ?, X, ?))
126.47/126.70	= { by axiom 11 (fact_318_int__eq__iff) }
126.47/126.70	  nat_1(hAPP(nat, int, semiring_1_of_nat(int), X))
126.47/126.70	= { by axiom 72 (fact_346_nat__int) }
126.47/126.70	  X
126.47/126.70	
126.47/126.70	Lemma 86: fresh209(?, ?, nat, X) = X.
126.47/126.70	Proof:
126.47/126.70	  fresh209(?, ?, nat, X)
126.47/126.70	= { by axiom 34 (fact_935_abs__of__nat) }
126.47/126.70	  hAPP(nat, nat, semiring_1_of_nat(nat), X)
126.47/126.70	= { by axiom 35 (fact_94_number__of__int) }
126.47/126.70	  fresh195(true2, true2, nat, X)
126.47/126.70	= { by axiom 53 (arity_Nat_Onat___Int_Onumber__semiring) }
126.47/126.70	  fresh195(number_semiring(nat), true2, nat, X)
126.47/126.70	= { by axiom 80 (fact_94_number__of__int) }
126.47/126.70	  number_number_of(nat, hAPP(nat, int, semiring_1_of_nat(int), X))
126.47/126.70	= { by axiom 11 (fact_318_int__eq__iff) }
126.47/126.70	  number_number_of(nat, fresh917(?, ?, X, ?))
126.47/126.70	= { by lemma 84 }
126.47/126.70	  nat_1(fresh917(?, ?, X, ?))
126.47/126.70	= { by lemma 85 }
126.47/126.70	  X
126.47/126.70	
126.47/126.70	Lemma 87: fresh1060(?, ?, Y, ?, W, V) = fresh1478(?, ?, Y, ?, W, V).
126.47/126.70	Proof:
126.47/126.70	  fresh1060(?, ?, Y, ?, W, V)
126.47/126.70	= { by axiom 7 (fact_220_add__less__imp__less__left) }
126.47/126.70	  hBOOL(hAPP(Y, bool, hAPP(Y, fun(Y, bool), ord_less(Y), W), V))
126.47/126.70	= { by axiom 9 (fact_300_mult__less__imp__less__left) }
126.47/126.70	  fresh1478(?, ?, Y, ?, W, V)
126.47/126.70	
126.47/126.70	Lemma 88: fresh1478(?, ?, Y, ?, Z, W) = fresh787(?, ?, Y, Z, W).
126.47/126.70	Proof:
126.47/126.70	  fresh1478(?, ?, Y, ?, Z, W)
126.47/126.70	= { by lemma 87 }
126.47/126.70	  fresh1060(?, ?, Y, ?, Z, W)
126.47/126.70	= { by axiom 7 (fact_220_add__less__imp__less__left) }
126.47/126.70	  hBOOL(hAPP(Y, bool, hAPP(Y, fun(Y, bool), ord_less(Y), Z), W))
126.47/126.70	= { by axiom 14 (fact_406_less__iff__diff__less__0_1) }
126.47/126.70	  fresh787(?, ?, Y, Z, W)
126.47/126.70	
126.47/126.70	Lemma 89: fresh787(?, ?, Y, Z, W) = fresh1294(?, ?, Y, Z, W).
126.47/126.70	Proof:
126.47/126.70	  fresh787(?, ?, Y, Z, W)
126.47/126.70	= { by lemma 88 }
126.47/126.70	  fresh1478(?, ?, Y, ?, Z, W)
126.47/126.70	= { by lemma 87 }
126.47/126.70	  fresh1060(?, ?, Y, ?, Z, W)
126.47/126.70	= { by axiom 7 (fact_220_add__less__imp__less__left) }
126.47/126.70	  hBOOL(hAPP(Y, bool, hAPP(Y, fun(Y, bool), ord_less(Y), Z), W))
126.47/126.70	= { by axiom 24 (fact_551_power2__less__imp__less) }
126.47/126.70	  fresh1294(?, ?, Y, Z, W)
126.47/126.70	
126.47/126.70	Lemma 90: fresh1160(?, ?, W, Y, ?) = fresh1389(?, ?, Y, ?, W).
126.47/126.70	Proof:
126.47/126.70	  fresh1160(?, ?, W, Y, ?)
126.47/126.70	= { by axiom 3 (fact_141_nat__power__less__imp__less) }
126.47/126.70	  hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), W), Y))
126.47/126.70	= { by axiom 28 (fact_662_less__diff__iff_1) }
126.47/126.70	  fresh1389(?, ?, Y, ?, W)
126.47/126.70	
126.47/126.70	Lemma 91: fresh1294(?, ?, nat, Y, Z) = fresh1389(?, ?, Z, ?, Y).
126.47/126.70	Proof:
126.47/126.70	  fresh1294(?, ?, nat, Y, Z)
126.47/126.70	= { by lemma 89 }
126.47/126.70	  fresh787(?, ?, nat, Y, Z)
126.47/126.70	= { by lemma 88 }
126.47/126.70	  fresh1478(?, ?, nat, ?, Y, Z)
126.47/126.70	= { by lemma 87 }
126.47/126.70	  fresh1060(?, ?, nat, ?, Y, Z)
126.47/126.70	= { by axiom 7 (fact_220_add__less__imp__less__left) }
126.47/126.70	  hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), Y), Z))
126.47/126.70	= { by axiom 3 (fact_141_nat__power__less__imp__less) }
126.47/126.70	  fresh1160(?, ?, Y, Z, ?)
126.47/126.70	= { by lemma 90 }
126.47/126.70	  fresh1389(?, ?, Z, ?, Y)
126.47/126.70	
126.47/126.70	Lemma 92: fresh1389(?, ?, X, ?, Y) = fresh1457(?, ?, nat, Y, X).
126.47/126.70	Proof:
126.47/126.70	  fresh1389(?, ?, X, ?, Y)
126.47/126.70	= { by lemma 86 }
126.47/126.70	  fresh1389(?, ?, fresh209(?, ?, nat, X), ?, Y)
126.47/126.70	= { by lemma 86 }
126.47/126.70	  fresh1389(?, ?, fresh209(?, ?, nat, X), ?, fresh209(?, ?, nat, Y))
126.47/126.70	= { by lemma 91 }
126.47/126.70	  fresh1294(?, ?, nat, fresh209(?, ?, nat, Y), fresh209(?, ?, nat, X))
126.47/126.70	= { by axiom 34 (fact_935_abs__of__nat) }
126.47/126.70	  fresh1294(?, ?, nat, hAPP(nat, nat, semiring_1_of_nat(nat), Y), fresh209(?, ?, nat, X))
126.47/126.70	= { by axiom 34 (fact_935_abs__of__nat) }
126.47/126.70	  fresh1294(?, ?, nat, hAPP(nat, nat, semiring_1_of_nat(nat), Y), hAPP(nat, nat, semiring_1_of_nat(nat), X))
126.47/126.70	= { by lemma 89 }
126.47/126.70	  fresh787(?, ?, nat, hAPP(nat, nat, semiring_1_of_nat(nat), Y), hAPP(nat, nat, semiring_1_of_nat(nat), X))
126.47/126.70	= { by lemma 88 }
126.47/126.70	  fresh1478(?, ?, nat, ?, hAPP(nat, nat, semiring_1_of_nat(nat), Y), hAPP(nat, nat, semiring_1_of_nat(nat), X))
126.47/126.70	= { by lemma 87 }
126.47/126.70	  fresh1060(?, ?, nat, ?, hAPP(nat, nat, semiring_1_of_nat(nat), Y), hAPP(nat, nat, semiring_1_of_nat(nat), X))
126.47/126.70	= { by axiom 7 (fact_220_add__less__imp__less__left) }
126.47/126.70	  hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), hAPP(nat, nat, semiring_1_of_nat(nat), Y)), hAPP(nat, nat, semiring_1_of_nat(nat), X)))
126.47/126.70	= { by axiom 67 (fact_168_less__imp__of__nat__less) }
126.47/126.70	  fresh1457(linordered_semidom(nat), true2, nat, Y, X)
126.47/126.70	= { by axiom 54 (arity_Nat_Onat___Rings_Olinordered__semidom) }
126.47/126.70	  fresh1457(true2, true2, nat, Y, X)
126.47/126.70	= { by axiom 4 (fact_168_less__imp__of__nat__less) }
126.47/126.70	  fresh1458(hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), Y), X)), true2, nat, Y, X)
126.47/126.70	= { by axiom 4 (fact_168_less__imp__of__nat__less) }
126.47/126.70	  fresh1457(?, ?, nat, Y, X)
126.47/126.70	
126.47/126.70	Lemma 93: one_one(int) = bit1(pls).
126.47/126.70	Proof:
126.47/126.70	  one_one(int)
126.47/126.70	= { by axiom 44 (fact_37_one__is__num__one) }
126.47/126.70	  number_number_of(int, bit1(pls))
126.47/126.70	= { by axiom 68 (fact_120_number__of__is__id) }
126.47/126.70	  ti(int, bit1(pls))
126.47/126.70	= { by axiom 69 (tsy_c_Int_OBit1_res) }
126.47/126.70	  bit1(pls)
126.47/126.70	
126.47/126.70	Lemma 94: fresh264(?, ?, int) = bit0(bit1(pls)).
126.47/126.70	Proof:
126.47/126.70	  fresh264(?, ?, int)
126.47/126.70	= { by axiom 33 (fact_8_one__add__one__is__two) }
126.47/126.70	  number_number_of(int, bit0(bit1(pls)))
126.47/126.70	= { by axiom 33 (fact_8_one__add__one__is__two) }
126.47/126.70	  fresh264(true2, true2, int)
126.47/126.70	= { by axiom 60 (arity_Int_Oint___Int_Onumber__ring) }
126.47/126.70	  fresh264(number_ring(int), true2, int)
126.47/126.70	= { by axiom 73 (fact_8_one__add__one__is__two) }
126.47/126.70	  plus_plus(int, one_one(int), one_one(int))
126.47/126.70	= { by axiom 77 (fact_78_Bit0__def) }
126.47/126.70	  bit0(one_one(int))
126.47/126.70	= { by lemma 93 }
126.47/126.70	  bit0(bit1(pls))
126.47/126.70	
126.47/126.70	Lemma 95: fresh702(?, ?, hAPP(int, fun(int, bool), ord_less_eq(int), Z), Y) = fresh1510(?, ?, Y, Z).
126.47/126.70	Proof:
126.47/126.70	  fresh702(?, ?, hAPP(int, fun(int, bool), ord_less_eq(int), Z), Y)
126.47/126.70	= { by axiom 16 (fact_474_transfer__int__nat__quantifiers_I1_J_1) }
126.47/126.70	  hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less_eq(int), Z), Y))
126.47/126.70	= { by axiom 29 (fact_683_transfer__nat__int__relations_I3_J) }
126.47/126.70	  fresh1510(?, ?, Y, Z)
126.47/126.70	
126.47/126.70	Lemma 96: fresh578(?, ?, ?, Z, ?) = fresh1510(?, ?, Z, pls).
126.47/126.70	Proof:
126.47/126.70	  fresh578(?, ?, ?, Z, ?)
126.47/126.70	= { by axiom 25 (fact_560_q__pos__lemma) }
126.47/126.70	  hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less_eq(int), zero_zero(int)), Z))
126.47/126.70	= { by axiom 16 (fact_474_transfer__int__nat__quantifiers_I1_J_1) }
126.47/126.70	  fresh702(?, ?, hAPP(int, fun(int, bool), ord_less_eq(int), zero_zero(int)), Z)
126.47/126.70	= { by lemma 95 }
126.47/126.70	  fresh1510(?, ?, Z, zero_zero(int))
126.47/126.70	= { by axiom 38 (fact_73_Pls__def) }
126.47/126.70	  fresh1510(?, ?, Z, pls)
126.47/126.70	
126.47/126.70	Lemma 97: fresh702(?, ?, hAPP(int, fun(int, bool), ord_less_eq(int), zero_zero(int)), Z) = fresh578(?, ?, ?, Z, ?).
126.47/126.70	Proof:
126.47/126.70	  fresh702(?, ?, hAPP(int, fun(int, bool), ord_less_eq(int), zero_zero(int)), Z)
126.47/126.70	= { by axiom 16 (fact_474_transfer__int__nat__quantifiers_I1_J_1) }
126.47/126.70	  hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less_eq(int), zero_zero(int)), Z))
126.47/126.70	= { by axiom 25 (fact_560_q__pos__lemma) }
126.47/126.70	  fresh578(?, ?, ?, Z, ?)
126.47/126.70	
126.47/126.70	Lemma 98: fresh917(?, ?, X, ?) = fresh989(zero_zero(nat), X, X).
126.47/126.70	Proof:
126.47/126.70	  fresh917(?, ?, X, ?)
126.47/126.70	= { by axiom 11 (fact_318_int__eq__iff) }
126.47/126.70	  hAPP(nat, int, semiring_1_of_nat(int), X)
126.47/126.70	= { by axiom 78 (fact_27_int__eq__0__conv) }
126.47/126.70	  fresh989(zero_zero(nat), X, X)
126.47/126.70	
126.47/126.70	Lemma 99: plus_plus(int, X, bit1(pls)) = succ(X).
126.47/126.70	Proof:
126.47/126.70	  plus_plus(int, X, bit1(pls))
126.47/126.70	= { by lemma 93 }
126.47/126.70	  plus_plus(int, X, one_one(int))
126.47/126.70	= { by axiom 66 (fact_519_succ__def) }
126.47/126.70	  succ(X)
126.47/126.70	
126.47/126.70	Lemma 100: succ(number_number_of(int, X)) = succ(X).
126.47/126.70	Proof:
126.47/126.70	  succ(number_number_of(int, X))
126.47/126.70	= { by axiom 68 (fact_120_number__of__is__id) }
126.47/126.70	  succ(ti(int, X))
126.47/126.70	= { by axiom 41 (tsy_c_Int_Osucc_arg1) }
126.48/126.70	  succ(X)
126.48/126.70	
126.48/126.70	Lemma 101: fresh429(?, ?, bit0(succ(X))) = nat_1(bit0(X)).
126.48/126.70	Proof:
126.48/126.70	  fresh429(?, ?, bit0(succ(X)))
126.48/126.70	= { by axiom 70 (fact_515_succ__Bit1) }
126.48/126.70	  fresh429(?, ?, succ(bit1(X)))
126.48/126.70	= { by axiom 58 (fact_514_succ__Bit0) }
126.48/126.70	  fresh429(?, ?, succ(succ(bit0(X))))
126.48/126.70	= { by lemma 100 }
126.48/126.70	  fresh429(?, ?, succ(succ(number_number_of(int, bit0(X)))))
126.48/126.70	= { by lemma 99 }
126.48/126.70	  fresh429(?, ?, succ(plus_plus(int, number_number_of(int, bit0(X)), bit1(pls))))
126.48/126.70	= { by lemma 93 }
126.48/126.70	  fresh429(?, ?, succ(plus_plus(int, number_number_of(int, bit0(X)), one_one(int))))
126.48/126.70	= { by axiom 32 (fact_7_add__special_I3_J) }
126.48/126.70	  fresh429(?, ?, succ(fresh353(true2, true2, int, bit0(X))))
126.48/126.70	= { by axiom 60 (arity_Int_Oint___Int_Onumber__ring) }
126.48/126.70	  fresh429(?, ?, succ(fresh353(number_ring(int), true2, int, bit0(X))))
126.48/126.70	= { by axiom 64 (fact_7_add__special_I3_J) }
126.48/126.70	  fresh429(?, ?, succ(number_number_of(int, plus_plus(int, bit0(X), bit1(pls)))))
126.48/126.70	= { by lemma 100 }
126.48/126.70	  fresh429(?, ?, succ(plus_plus(int, bit0(X), bit1(pls))))
126.48/126.70	= { by lemma 99 }
126.48/126.70	  fresh429(?, ?, plus_plus(int, plus_plus(int, bit0(X), bit1(pls)), bit1(pls)))
126.48/126.70	= { by axiom 74 (fact_45_zadd__assoc) }
126.48/126.70	  fresh429(?, ?, plus_plus(int, bit0(X), plus_plus(int, bit1(pls), bit1(pls))))
126.48/126.70	= { by lemma 99 }
126.48/126.70	  fresh429(?, ?, plus_plus(int, bit0(X), succ(bit1(pls))))
126.48/126.70	= { by axiom 70 (fact_515_succ__Bit1) }
126.48/126.70	  fresh429(?, ?, plus_plus(int, bit0(X), bit0(succ(pls))))
126.48/126.70	= { by axiom 75 (fact_513_succ__Pls) }
126.48/126.70	  fresh429(?, ?, plus_plus(int, bit0(X), bit0(bit1(pls))))
126.48/126.70	= { by lemma 94 }
126.48/126.70	  fresh429(?, ?, plus_plus(int, bit0(X), fresh264(?, ?, int)))
126.48/126.70	= { by axiom 31 (fact_709_Int2_Oaux__1) }
126.48/126.70	  nat_1(minus_minus(int, plus_plus(int, bit0(X), fresh264(?, ?, int)), number_number_of(int, bit0(bit1(pls)))))
126.48/126.70	= { by axiom 33 (fact_8_one__add__one__is__two) }
126.48/126.70	  nat_1(minus_minus(int, plus_plus(int, bit0(X), fresh264(?, ?, int)), fresh264(?, ?, int)))
126.48/126.70	= { by axiom 49 (fact_326_add__diff__cancel) }
126.48/126.70	  nat_1(fresh899(group_add(int), true2, int, bit0(X), fresh264(?, ?, int)))
126.48/126.70	= { by axiom 63 (arity_Int_Oint___Groups_Ogroup__add) }
126.48/126.70	  nat_1(fresh899(true2, true2, int, bit0(X), fresh264(?, ?, int)))
126.48/126.70	= { by lemma 94 }
126.48/126.70	  nat_1(fresh899(true2, true2, int, bit0(X), bit0(bit1(pls))))
126.48/126.70	= { by axiom 12 (fact_326_add__diff__cancel) }
126.48/126.70	  nat_1(ti(int, bit0(X)))
126.48/126.70	= { by axiom 68 (fact_120_number__of__is__id) }
126.48/126.70	  nat_1(number_number_of(int, bit0(X)))
126.48/126.70	= { by axiom 52 (fact_427_nat__number__of__def) }
126.48/126.70	  number_number_of(nat, bit0(X))
126.48/126.70	= { by lemma 84 }
126.48/126.70	  nat_1(bit0(X))
126.48/126.70	
126.48/126.70	Lemma 102: fresh178(?, ?, int, X) = fresh1227(?, ?, int, X).
126.48/126.70	Proof:
126.48/126.70	  fresh178(?, ?, int, X)
126.48/126.70	= { by axiom 36 (fact_964_abs__power2) }
126.48/126.70	  hAPP(nat, int, power_power(int, X), number_number_of(nat, bit0(bit1(pls))))
126.48/126.70	= { by axiom 43 (fact_541_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) }
126.48/126.70	  fresh600(comm_semiring_1(int), true2, int, X)
126.48/126.70	= { by axiom 62 (arity_Int_Oint___Rings_Ocomm__semiring__1) }
126.48/126.70	  fresh600(true2, true2, int, X)
126.48/126.70	= { by axiom 21 (fact_541_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) }
126.48/126.70	  times_times(int, X, X)
126.48/126.70	= { by axiom 37 (tsy_c_Groups_Otimes__class_Otimes_5_res) }
126.48/126.70	  fresh101(true2, true2, X, X, int)
126.48/126.70	= { by axiom 57 (arity_Int_Oint___Groups_Oab__semigroup__mult) }
126.48/126.70	  fresh101(ab_semigroup_mult(int), true2, X, X, int)
126.48/126.70	= { by axiom 56 (tsy_c_Groups_Otimes__class_Otimes_5_res) }
126.48/126.70	  ti(int, times_times(int, X, X))
126.48/126.70	= { by axiom 68 (fact_120_number__of__is__id) }
126.48/126.70	  number_number_of(int, times_times(int, X, X))
126.48/126.70	= { by axiom 39 (fact_384_times__numeral__code_I5_J) }
126.48/126.70	  times_times(int, number_number_of(int, X), number_number_of(int, X))
126.48/126.70	= { by axiom 22 (fact_545_power2__eq__square__number__of) }
126.48/126.70	  fresh1227(?, ?, int, X)
126.48/126.70	
126.48/126.70	Lemma 103: fresh791(?, ?, Y, Z, nat_1(bit0(bit1(pls)))) = fresh178(?, ?, Y, Z).
126.48/126.70	Proof:
126.48/126.70	  fresh791(?, ?, Y, Z, nat_1(bit0(bit1(pls))))
126.48/126.70	= { by lemma 84 }
126.48/126.70	  fresh791(?, ?, Y, Z, number_number_of(nat, bit0(bit1(pls))))
126.48/126.70	= { by axiom 13 (fact_405_realpow__minus__mult) }
126.48/126.70	  hAPP(nat, Y, power_power(Y, Z), number_number_of(nat, bit0(bit1(pls))))
126.48/126.70	= { by axiom 36 (fact_964_abs__power2) }
126.67/126.93	  fresh178(?, ?, Y, Z)
126.67/126.93	
126.67/126.93	Goal 1 (fact_118_not__add__less1): hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), plus_plus(nat, X, Y)), X)) = true2.
126.67/126.93	The goal is true when:
126.67/126.93	  X = n
126.67/126.93	  Y = n
126.67/126.93	
126.67/126.93	Proof:
126.67/126.93	  hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), plus_plus(nat, n, n)), n))
126.67/126.93	= { by axiom 3 (fact_141_nat__power__less__imp__less) }
126.67/126.93	  fresh1160(?, ?, plus_plus(nat, n, n), n, ?)
126.67/126.93	= { by lemma 90 }
126.67/126.93	  fresh1389(?, ?, n, ?, plus_plus(nat, n, n))
126.67/126.93	= { by lemma 92 }
126.67/126.93	  fresh1457(?, ?, nat, plus_plus(nat, n, n), n)
126.67/126.93	= { by axiom 6 (fact_216_double__zero__sym) }
126.67/126.93	  fresh1457(?, ?, nat, fresh1066(?, ?, nat, n), n)
126.67/126.93	= { by lemma 85 }
126.67/126.93	  fresh1457(?, ?, nat, fresh1066(?, ?, nat, nat_1(fresh917(?, ?, n, ?))), n)
126.67/126.93	= { by lemma 98 }
126.67/126.93	  fresh1457(?, ?, nat, fresh1066(?, ?, nat, nat_1(fresh989(zero_zero(nat), n, n))), n)
126.67/126.93	= { by axiom 6 (fact_216_double__zero__sym) }
126.67/126.93	  fresh1457(?, ?, nat, plus_plus(nat, nat_1(fresh989(zero_zero(nat), n, n)), nat_1(fresh989(zero_zero(nat), n, n))), n)
126.67/126.93	= { by axiom 19 (fact_538_semiring__mult__2) }
126.67/126.93	  fresh1457(?, ?, nat, fresh604(true2, true2, nat, nat_1(fresh989(zero_zero(nat), n, n))), n)
126.67/126.93	= { by axiom 53 (arity_Nat_Onat___Int_Onumber__semiring) }
126.67/126.93	  fresh1457(?, ?, nat, fresh604(number_semiring(nat), true2, nat, nat_1(fresh989(zero_zero(nat), n, n))), n)
126.67/126.93	= { by axiom 61 (fact_538_semiring__mult__2) }
126.67/126.93	  fresh1457(?, ?, nat, times_times(nat, number_number_of(nat, bit0(bit1(pls))), nat_1(fresh989(zero_zero(nat), n, n))), n)
126.67/126.93	= { by axiom 47 (fact_534_transfer__nat__int__numerals_I3_J) }
126.67/126.93	  fresh1457(?, ?, nat, times_times(nat, nat_1(number_number_of(int, bit0(bit1(pls)))), nat_1(fresh989(zero_zero(nat), n, n))), n)
126.67/126.93	= { by axiom 33 (fact_8_one__add__one__is__two) }
126.67/126.93	  fresh1457(?, ?, nat, times_times(nat, nat_1(fresh264(?, ?, int)), nat_1(fresh989(zero_zero(nat), n, n))), n)
126.67/126.93	= { by axiom 51 (fact_685_nat__mult__distrib) }
126.67/126.93	  fresh1457(?, ?, nat, fresh449(hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less_eq(int), zero_zero(int)), fresh264(?, ?, int))), true2, fresh989(zero_zero(nat), n, n), fresh264(?, ?, int)), n)
126.67/126.93	= { by axiom 16 (fact_474_transfer__int__nat__quantifiers_I1_J_1) }
126.67/126.93	  fresh1457(?, ?, nat, fresh449(fresh702(?, ?, hAPP(int, fun(int, bool), ord_less_eq(int), zero_zero(int)), fresh264(?, ?, int)), true2, fresh989(zero_zero(nat), n, n), fresh264(?, ?, int)), n)
126.67/126.93	= { by lemma 97 }
126.67/126.93	  fresh1457(?, ?, nat, fresh449(fresh578(?, ?, ?, fresh264(?, ?, int), ?), true2, fresh989(zero_zero(nat), n, n), fresh264(?, ?, int)), n)
126.67/126.93	= { by axiom 33 (fact_8_one__add__one__is__two) }
126.67/126.93	  fresh1457(?, ?, nat, fresh449(fresh578(?, ?, ?, number_number_of(int, bit0(bit1(pls))), ?), true2, fresh989(zero_zero(nat), n, n), fresh264(?, ?, int)), n)
126.67/126.93	= { by lemma 97 }
126.67/126.93	  fresh1457(?, ?, nat, fresh449(fresh702(?, ?, hAPP(int, fun(int, bool), ord_less_eq(int), zero_zero(int)), number_number_of(int, bit0(bit1(pls)))), true2, fresh989(zero_zero(nat), n, n), fresh264(?, ?, int)), n)
126.67/126.93	= { by axiom 16 (fact_474_transfer__int__nat__quantifiers_I1_J_1) }
126.67/126.93	  fresh1457(?, ?, nat, fresh449(hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less_eq(int), zero_zero(int)), number_number_of(int, bit0(bit1(pls))))), true2, fresh989(zero_zero(nat), n, n), fresh264(?, ?, int)), n)
126.67/126.93	= { by axiom 65 (fact_542_Nat__Transfer_Otransfer__nat__int__function__closures_I7_J) }
126.67/126.93	  fresh1457(?, ?, nat, fresh449(true2, true2, fresh989(zero_zero(nat), n, n), fresh264(?, ?, int)), n)
126.67/126.93	= { by axiom 30 (fact_685_nat__mult__distrib) }
126.67/126.93	  fresh1457(?, ?, nat, nat_1(times_times(int, fresh264(?, ?, int), fresh989(zero_zero(nat), n, n))), n)
126.67/126.93	= { by axiom 33 (fact_8_one__add__one__is__two) }
126.67/126.93	  fresh1457(?, ?, nat, nat_1(times_times(int, number_number_of(int, bit0(bit1(pls))), fresh989(zero_zero(nat), n, n))), n)
126.67/126.93	= { by axiom 59 (fact_537_mult__2) }
126.67/126.93	  fresh1457(?, ?, nat, nat_1(fresh605(number_ring(int), true2, int, fresh989(zero_zero(nat), n, n))), n)
126.67/126.93	= { by axiom 60 (arity_Int_Oint___Int_Onumber__ring) }
126.67/126.93	  fresh1457(?, ?, nat, nat_1(fresh605(true2, true2, int, fresh989(zero_zero(nat), n, n))), n)
126.67/126.93	= { by axiom 18 (fact_537_mult__2) }
126.67/126.93	  fresh1457(?, ?, nat, nat_1(plus_plus(int, fresh989(zero_zero(nat), n, n), fresh989(zero_zero(nat), n, n))), n)
126.67/126.93	= { by axiom 77 (fact_78_Bit0__def) }
126.67/126.93	  fresh1457(?, ?, nat, nat_1(bit0(fresh989(zero_zero(nat), n, n))), n)
126.67/126.93	= { by lemma 101 }
126.67/126.93	  fresh1457(?, ?, nat, fresh429(?, ?, bit0(succ(fresh989(zero_zero(nat), n, n)))), n)
126.67/126.93	= { by axiom 46 (tsy_c_Int_Osucc_res) }
126.67/126.93	  fresh1457(?, ?, nat, fresh429(?, ?, bit0(ti(int, succ(fresh989(zero_zero(nat), n, n))))), n)
126.67/126.93	= { by axiom 26 (fact_5_zero__eq__power2_1) }
126.67/126.93	  fresh1457(?, ?, nat, fresh429(?, ?, bit0(fresh543(true2, true2, int, succ(fresh989(zero_zero(nat), n, n))))), n)
126.67/126.93	= { by axiom 71 (arity_Int_Oint___Rings_Oring__1__no__zero__divisors) }
126.67/126.93	  fresh1457(?, ?, nat, fresh429(?, ?, bit0(fresh543(ring_11004092258visors(int), true2, int, succ(fresh989(zero_zero(nat), n, n))))), n)
126.67/126.93	= { by axiom 79 (fact_5_zero__eq__power2_1) }
126.67/126.93	  fresh1457(?, ?, nat, fresh429(?, ?, bit0(fresh542(hAPP(nat, int, power_power(int, succ(fresh989(zero_zero(nat), n, n))), number_number_of(nat, bit0(bit1(pls)))), zero_zero(int), int, succ(fresh989(zero_zero(nat), n, n))))), n)
126.67/126.93	= { by axiom 13 (fact_405_realpow__minus__mult) }
126.67/126.93	  fresh1457(?, ?, nat, fresh429(?, ?, bit0(fresh542(fresh791(?, ?, int, succ(fresh989(zero_zero(nat), n, n)), number_number_of(nat, bit0(bit1(pls)))), zero_zero(int), int, succ(fresh989(zero_zero(nat), n, n))))), n)
126.67/126.93	= { by lemma 84 }
126.67/126.93	  fresh1457(?, ?, nat, fresh429(?, ?, bit0(fresh542(fresh791(?, ?, int, succ(fresh989(zero_zero(nat), n, n)), nat_1(bit0(bit1(pls)))), zero_zero(int), int, succ(fresh989(zero_zero(nat), n, n))))), n)
126.67/126.93	= { by lemma 103 }
126.67/126.93	  fresh1457(?, ?, nat, fresh429(?, ?, bit0(fresh542(fresh178(?, ?, int, succ(fresh989(zero_zero(nat), n, n))), zero_zero(int), int, succ(fresh989(zero_zero(nat), n, n))))), n)
126.67/126.93	= { by lemma 99 }
126.67/126.93	  fresh1457(?, ?, nat, fresh429(?, ?, bit0(fresh542(fresh178(?, ?, int, plus_plus(int, fresh989(zero_zero(nat), n, n), bit1(pls))), zero_zero(int), int, succ(fresh989(zero_zero(nat), n, n))))), n)
126.67/126.93	= { by axiom 40 (fact_47_zadd__commute) }
126.67/126.93	  fresh1457(?, ?, nat, fresh429(?, ?, bit0(fresh542(fresh178(?, ?, int, plus_plus(int, bit1(pls), fresh989(zero_zero(nat), n, n))), zero_zero(int), int, succ(fresh989(zero_zero(nat), n, n))))), n)
126.67/126.93	= { by lemma 103 }
126.67/126.93	  fresh1457(?, ?, nat, fresh429(?, ?, bit0(fresh542(fresh791(?, ?, int, plus_plus(int, bit1(pls), fresh989(zero_zero(nat), n, n)), nat_1(bit0(bit1(pls)))), zero_zero(int), int, succ(fresh989(zero_zero(nat), n, n))))), n)
126.67/126.93	= { by lemma 84 }
126.67/126.93	  fresh1457(?, ?, nat, fresh429(?, ?, bit0(fresh542(fresh791(?, ?, int, plus_plus(int, bit1(pls), fresh989(zero_zero(nat), n, n)), number_number_of(nat, bit0(bit1(pls)))), zero_zero(int), int, succ(fresh989(zero_zero(nat), n, n))))), n)
126.67/126.93	= { by axiom 13 (fact_405_realpow__minus__mult) }
126.67/126.93	  fresh1457(?, ?, nat, fresh429(?, ?, bit0(fresh542(hAPP(nat, int, power_power(int, plus_plus(int, bit1(pls), fresh989(zero_zero(nat), n, n))), number_number_of(nat, bit0(bit1(pls)))), zero_zero(int), int, succ(fresh989(zero_zero(nat), n, n))))), n)
126.67/126.93	= { by axiom 23 (fact_550_power2__sum) }
126.67/126.93	  fresh1457(?, ?, nat, fresh429(?, ?, bit0(fresh542(fresh592(?, ?, int, bit1(pls), fresh989(zero_zero(nat), n, n)), zero_zero(int), int, succ(fresh989(zero_zero(nat), n, n))))), n)
126.67/126.93	= { by lemma 93 }
126.67/126.93	  fresh1457(?, ?, nat, fresh429(?, ?, bit0(fresh542(fresh592(?, ?, int, one_one(int), fresh989(zero_zero(nat), n, n)), zero_zero(int), int, succ(fresh989(zero_zero(nat), n, n))))), n)
126.67/126.93	= { by lemma 98 }
126.67/126.93	  fresh1457(?, ?, nat, fresh429(?, ?, bit0(fresh542(fresh592(?, ?, int, one_one(int), fresh917(?, ?, n, ?)), zero_zero(int), int, succ(fresh989(zero_zero(nat), n, n))))), n)
126.67/126.93	= { by axiom 11 (fact_318_int__eq__iff) }
126.67/126.93	  fresh1457(?, ?, nat, fresh429(?, ?, bit0(fresh542(fresh592(?, ?, int, one_one(int), hAPP(nat, int, semiring_1_of_nat(int), n)), zero_zero(int), int, succ(fresh989(zero_zero(nat), n, n))))), n)
126.67/126.93	= { by axiom 23 (fact_550_power2__sum) }
126.67/126.93	  fresh1457(?, ?, nat, fresh429(?, ?, bit0(fresh542(hAPP(nat, int, power_power(int, plus_plus(int, one_one(int), hAPP(nat, int, semiring_1_of_nat(int), n))), number_number_of(nat, bit0(bit1(pls)))), zero_zero(int), int, succ(fresh989(zero_zero(nat), n, n))))), n)
126.67/126.93	= { by axiom 83 (conj_0) }
126.67/126.93	  fresh1457(?, ?, nat, fresh429(?, ?, bit0(fresh542(zero_zero(int), zero_zero(int), int, succ(fresh989(zero_zero(nat), n, n))))), n)
126.67/126.93	= { by axiom 38 (fact_73_Pls__def) }
126.67/126.93	  fresh1457(?, ?, nat, fresh429(?, ?, bit0(fresh542(pls, zero_zero(int), int, succ(fresh989(zero_zero(nat), n, n))))), n)
126.67/126.93	= { by axiom 38 (fact_73_Pls__def) }
126.67/126.93	  fresh1457(?, ?, nat, fresh429(?, ?, bit0(fresh542(pls, pls, int, succ(fresh989(zero_zero(nat), n, n))))), n)
126.67/126.93	= { by axiom 27 (fact_5_zero__eq__power2_1) }
126.67/126.93	  fresh1457(?, ?, nat, fresh429(?, ?, bit0(zero_zero(int))), n)
126.67/126.93	= { by axiom 38 (fact_73_Pls__def) }
126.67/126.93	  fresh1457(?, ?, nat, fresh429(?, ?, bit0(pls)), n)
126.67/126.93	= { by axiom 45 (fact_776_succ__Min) }
126.67/126.93	  fresh1457(?, ?, nat, fresh429(?, ?, bit0(succ(min))), n)
126.67/126.93	= { by lemma 101 }
126.67/126.93	  fresh1457(?, ?, nat, nat_1(bit0(min)), n)
126.67/126.93	= { by axiom 42 (fact_314_nat__le__0) }
126.67/126.93	  fresh1457(?, ?, nat, fresh922(hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less_eq(int), bit0(min)), zero_zero(int))), true2, bit0(min)), n)
126.67/126.93	= { by axiom 16 (fact_474_transfer__int__nat__quantifiers_I1_J_1) }
126.67/126.93	  fresh1457(?, ?, nat, fresh922(fresh702(?, ?, hAPP(int, fun(int, bool), ord_less_eq(int), bit0(min)), zero_zero(int)), true2, bit0(min)), n)
126.67/126.93	= { by lemma 95 }
126.67/126.93	  fresh1457(?, ?, nat, fresh922(fresh1510(?, ?, zero_zero(int), bit0(min)), true2, bit0(min)), n)
126.67/126.93	= { by axiom 38 (fact_73_Pls__def) }
126.67/126.93	  fresh1457(?, ?, nat, fresh922(fresh1510(?, ?, pls, bit0(min)), true2, bit0(min)), n)
126.67/126.93	= { by lemma 95 }
126.67/126.93	  fresh1457(?, ?, nat, fresh922(fresh702(?, ?, hAPP(int, fun(int, bool), ord_less_eq(int), bit0(min)), pls), true2, bit0(min)), n)
126.67/126.93	= { by axiom 16 (fact_474_transfer__int__nat__quantifiers_I1_J_1) }
126.67/126.93	  fresh1457(?, ?, nat, fresh922(hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less_eq(int), bit0(min)), pls)), true2, bit0(min)), n)
126.67/126.93	= { by axiom 50 (fact_465_rel__simps_I27_J) }
126.67/126.93	  fresh1457(?, ?, nat, fresh922(fresh714(hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less_eq(int), min), pls)), true2, min), true2, bit0(min)), n)
126.67/126.93	= { by axiom 48 (fact_758_rel__simps_I23_J) }
126.67/126.93	  fresh1457(?, ?, nat, fresh922(fresh714(true2, true2, min), true2, bit0(min)), n)
126.67/126.93	= { by axiom 15 (fact_465_rel__simps_I27_J) }
126.67/126.93	  fresh1457(?, ?, nat, fresh922(true2, true2, bit0(min)), n)
126.67/126.93	= { by axiom 10 (fact_314_nat__le__0) }
126.67/126.93	  fresh1457(?, ?, nat, zero_zero(nat), n)
126.67/126.93	= { by lemma 92 }
126.67/126.93	  fresh1389(?, ?, n, ?, zero_zero(nat))
126.67/126.93	= { by lemma 91 }
126.67/126.93	  fresh1294(?, ?, nat, zero_zero(nat), n)
126.67/126.93	= { by lemma 89 }
126.67/126.93	  fresh787(?, ?, nat, zero_zero(nat), n)
126.67/126.93	= { by lemma 88 }
126.67/126.93	  fresh1478(?, ?, nat, ?, zero_zero(nat), n)
126.67/126.93	= { by lemma 87 }
126.67/126.93	  fresh1060(?, ?, nat, ?, zero_zero(nat), n)
126.67/126.93	= { by axiom 7 (fact_220_add__less__imp__less__left) }
126.67/126.93	  hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), zero_zero(nat)), n))
126.67/126.93	= { by axiom 5 (fact_194_zero__less__double__add__iff__zero__less__single__add_1) }
126.67/126.93	  fresh1103(?, ?, nat, n)
126.67/126.93	= { by lemma 86 }
126.67/126.93	  fresh1103(?, ?, nat, fresh209(?, ?, nat, n))
126.67/126.93	= { by axiom 34 (fact_935_abs__of__nat) }
126.67/126.93	  fresh1103(?, ?, nat, hAPP(nat, nat, semiring_1_of_nat(nat), n))
126.67/126.93	= { by axiom 5 (fact_194_zero__less__double__add__iff__zero__less__single__add_1) }
126.67/126.93	  hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), zero_zero(nat)), hAPP(nat, nat, semiring_1_of_nat(nat), n)))
126.67/126.93	= { by axiom 82 (fact_107_of__nat__0__less__iff_1) }
126.67/126.93	  fresh1194(linordered_semidom(nat), true2, nat, n)
126.67/126.93	= { by axiom 54 (arity_Nat_Onat___Rings_Olinordered__semidom) }
126.67/126.93	  fresh1194(true2, true2, nat, n)
126.67/126.93	= { by axiom 2 (fact_107_of__nat__0__less__iff_1) }
126.67/126.93	  fresh1195(hBOOL(hAPP(nat, bool, hAPP(nat, fun(nat, bool), ord_less(nat), zero_zero(nat)), n)), true2, nat, n)
126.67/126.93	= { by axiom 81 (fact_14_n0) }
126.67/126.93	  fresh1195(true2, true2, nat, n)
126.67/126.93	= { by axiom 1 (fact_107_of__nat__0__less__iff_1) }
126.67/126.93	  true2
126.67/126.93	% SZS output end Proof
126.67/126.93	
126.67/126.93	RESULT: Theorem (the conjecture is true).
126.67/126.97	EOF