0.00/0.04 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.00/0.04 % Command : twee %s --tstp --casc --quiet --conditional-encoding if --smaller --drop-non-horn 0.03/0.23 % Computer : n012.star.cs.uiowa.edu 0.03/0.23 % Model : x86_64 x86_64 0.03/0.23 % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz 0.03/0.23 % Memory : 32218.625MB 0.03/0.23 % OS : Linux 3.10.0-693.2.2.el7.x86_64 0.03/0.23 % CPULimit : 300 0.03/0.23 % DateTime : Sat Jul 14 05:48:25 CDT 2018 0.03/0.23 % CPUTime : 203.92/204.16 % SZS status Theorem 203.92/204.16 203.92/204.17 % SZS output start Proof 203.92/204.17 Take the following subset of the input axioms: 203.92/204.18 fof(arity_Complex__Ocomplex__Groups_Omonoid__mult, axiom, 203.92/204.18 class_Groups_Omonoid__mult(tc_Complex_Ocomplex)). 203.92/204.18 fof(arity_Complex__Ocomplex__Power_Opower, axiom, 203.92/204.18 class_Power_Opower(tc_Complex_Ocomplex)). 203.92/204.18 fof(arity_Complex__Ocomplex__Rings_Ocomm__semiring__1, axiom, 203.92/204.18 class_Rings_Ocomm__semiring__1(tc_Complex_Ocomplex)). 203.92/204.18 fof(arity_Complex__Ocomplex__Rings_Osemiring__0, axiom, 203.92/204.18 class_Rings_Osemiring__0(tc_Complex_Ocomplex)). 203.92/204.18 fof(arity_Complex__Ocomplex__Rings_Ozero__neq__one, axiom, 203.92/204.18 class_Rings_Ozero__neq__one(tc_Complex_Ocomplex)). 203.92/204.18 fof(conj_0, conjecture, 203.92/204.18 ?[B_x, B_y]: 203.92/204.18 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), B_y), 203.92/204.18 hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex), B_y), 203.92/204.18 c_Groups_Ominus__class_Ominus(tc_Nat_Onat, v_k____, 203.92/204.18 c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))))!=hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), 203.92/204.18 B_x), 203.92/204.18 hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex), 203.92/204.18 B_x), 203.92/204.18 c_Groups_Ominus__class_Ominus(tc_Nat_Onat, 203.92/204.18 v_k____, 203.92/204.18 c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))))). 203.92/204.18 fof(fact_One__nat__def, axiom, 203.92/204.18 c_Groups_Oone__class_Oone(tc_Nat_Onat)=c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). 203.92/204.18 fof(fact_Suc__n__not__le__n, axiom, 203.92/204.18 ![V_n]: 203.92/204.18 ~c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, c_Nat_OSuc(V_n), 203.92/204.18 V_n)). 203.92/204.18 fof(fact_Suc__n__not__n, axiom, ![V_n]: V_n!=c_Nat_OSuc(V_n)). 203.92/204.18 fof(fact_Suc__neq__Zero, axiom, 203.92/204.18 ![V_m]: c_Groups_Ozero__class_Ozero(tc_Nat_Onat)!=c_Nat_OSuc(V_m)). 203.92/204.18 fof(fact_Suc__not__Zero, axiom, 203.92/204.18 ![V_m]: c_Groups_Ozero__class_Ozero(tc_Nat_Onat)!=c_Nat_OSuc(V_m)). 203.92/204.18 fof(fact_Zero__neq__Suc, axiom, 203.92/204.18 ![V_m]: c_Groups_Ozero__class_Ozero(tc_Nat_Onat)!=c_Nat_OSuc(V_m)). 203.92/204.18 fof(fact_Zero__not__Suc, axiom, 203.92/204.18 ![V_m]: c_Nat_OSuc(V_m)!=c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). 203.92/204.18 fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J, 203.92/204.18 axiom, 203.92/204.18 ![T_a, V_x, V_q]: 203.92/204.18 (class_Rings_Ocomm__semiring__1(T_a) 203.92/204.18 => hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a), V_x), 203.92/204.18 hAPP(hAPP(c_Power_Opower__class_Opower(T_a), V_x), 203.92/204.18 V_q))=hAPP(hAPP(c_Power_Opower__class_Opower(T_a), V_x), 203.92/204.18 c_Nat_OSuc(V_q)))). 203.92/204.18 fof(fact_complex__i__not__number__of, axiom, 203.92/204.18 ![V_w]: 203.92/204.18 c_Complex_Oii!=c_Int_Onumber__class_Onumber__of(tc_Complex_Ocomplex, 203.92/204.18 V_w)). 203.92/204.18 fof(fact_gr__implies__not0, axiom, 203.92/204.18 ![V_m, V_n]: 203.92/204.18 (c_Orderings_Oord__class_Oless(tc_Nat_Onat, V_m, V_n) 203.92/204.18 => V_n!=c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). 203.92/204.18 fof(fact_leD, axiom, 203.92/204.18 ![T_a, V_x, V_y]: 203.92/204.18 ((c_Orderings_Oord__class_Oless__eq(T_a, V_y, V_x) 203.92/204.18 => ~c_Orderings_Oord__class_Oless(T_a, V_x, V_y)) 203.92/204.18 <= class_Orderings_Olinorder(T_a))). 203.92/204.18 fof(fact_le__number__of__eq__not__less, axiom, 203.92/204.18 ![V_w_2, T_a, V_v_2]: 203.92/204.18 ((class_Orderings_Olinorder(T_a) & class_Int_Onumber(T_a)) 203.92/204.18 => (c_Orderings_Oord__class_Oless__eq(T_a, 203.92/204.18 c_Int_Onumber__class_Onumber__of(T_a, V_v_2), 203.92/204.18 c_Int_Onumber__class_Onumber__of(T_a, V_w_2)) 203.92/204.18 <=> ~c_Orderings_Oord__class_Oless(T_a, 203.92/204.18 c_Int_Onumber__class_Onumber__of(T_a, V_w_2), 203.92/204.18 c_Int_Onumber__class_Onumber__of(T_a, V_v_2))))). 203.92/204.18 fof(fact_less__fun__def, axiom, 203.92/204.18 ![T_a, V_g_2, V_f_2, T_b]: 203.92/204.18 (class_Orderings_Oord(T_b) 203.92/204.18 => ((~c_Orderings_Oord__class_Oless__eq(tc_fun(T_a, T_b), V_g_2, 203.92/204.18 V_f_2) 203.92/204.18 & c_Orderings_Oord__class_Oless__eq(tc_fun(T_a, T_b), V_f_2, 203.92/204.18 V_g_2)) 203.92/204.18 <=> c_Orderings_Oord__class_Oless(tc_fun(T_a, T_b), V_f_2, 203.92/204.18 V_g_2)))). 203.92/204.18 fof(fact_less__imp__neq, axiom, 203.92/204.18 ![T_a, V_x, V_y]: 203.92/204.18 (class_Orderings_Oorder(T_a) 203.92/204.18 => (V_y!=V_x <= c_Orderings_Oord__class_Oless(T_a, V_x, V_y)))). 203.92/204.18 fof(fact_less__irrefl__nat, axiom, 203.92/204.18 ![V_n]: ~c_Orderings_Oord__class_Oless(tc_Nat_Onat, V_n, V_n)). 203.92/204.18 fof(fact_less__le__not__le, axiom, 203.92/204.18 ![T_a, V_x_2, V_y_2]: 203.92/204.18 (class_Orderings_Opreorder(T_a) 203.92/204.18 => (c_Orderings_Oord__class_Oless(T_a, V_x_2, V_y_2) 203.92/204.18 <=> (c_Orderings_Oord__class_Oless__eq(T_a, V_x_2, V_y_2) 203.92/204.18 & ~c_Orderings_Oord__class_Oless__eq(T_a, V_y_2, V_x_2))))). 203.92/204.18 fof(fact_less__nat__zero__code, axiom, 203.92/204.18 ![V_n]: 203.92/204.18 ~c_Orderings_Oord__class_Oless(tc_Nat_Onat, V_n, 203.92/204.18 c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). 203.92/204.18 fof(fact_less__not__refl, axiom, 203.92/204.18 ![V_n]: ~c_Orderings_Oord__class_Oless(tc_Nat_Onat, V_n, V_n)). 203.92/204.18 fof(fact_less__not__refl2, axiom, 203.92/204.18 ![V_m, V_n]: 203.92/204.18 (c_Orderings_Oord__class_Oless(tc_Nat_Onat, V_n, V_m) 203.92/204.18 => V_m!=V_n)). 203.92/204.18 fof(fact_less__not__refl3, axiom, 203.92/204.18 ![V_t, V_s]: 203.92/204.18 (V_s!=V_t 203.92/204.18 <= c_Orderings_Oord__class_Oless(tc_Nat_Onat, V_s, V_t))). 203.92/204.18 fof(fact_less__zeroE, axiom, 203.92/204.18 ![V_n]: 203.92/204.18 ~c_Orderings_Oord__class_Oless(tc_Nat_Onat, V_n, 203.92/204.18 c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). 203.92/204.18 fof(fact_linorder__antisym__conv2, axiom, 203.92/204.18 ![T_a, V_x_2, V_y_2]: 203.92/204.18 ((c_Orderings_Oord__class_Oless__eq(T_a, V_x_2, V_y_2) 203.92/204.18 => (~c_Orderings_Oord__class_Oless(T_a, V_x_2, V_y_2) 203.92/204.18 <=> V_x_2=V_y_2)) 203.92/204.18 <= class_Orderings_Olinorder(T_a))). 203.92/204.18 fof(fact_linorder__neq__iff, axiom, 203.92/204.18 ![T_a, V_x_2, V_y_2]: 203.92/204.18 ((V_x_2!=V_y_2 203.92/204.18 <=> (c_Orderings_Oord__class_Oless(T_a, V_x_2, V_y_2) 203.92/204.18 | c_Orderings_Oord__class_Oless(T_a, V_y_2, V_x_2))) 203.92/204.18 <= class_Orderings_Olinorder(T_a))). 203.92/204.18 fof(fact_linorder__not__le, axiom, 203.92/204.18 ![T_a, V_x_2, V_y_2]: 203.92/204.18 ((c_Orderings_Oord__class_Oless(T_a, V_y_2, V_x_2) 203.92/204.18 <=> ~c_Orderings_Oord__class_Oless__eq(T_a, V_x_2, V_y_2)) 203.92/204.18 <= class_Orderings_Olinorder(T_a))). 203.92/204.18 fof(fact_linorder__not__less, axiom, 203.92/204.18 ![T_a, V_x_2, V_y_2]: 203.92/204.18 (class_Orderings_Olinorder(T_a) 203.92/204.18 => (~c_Orderings_Oord__class_Oless(T_a, V_x_2, V_y_2) 203.92/204.18 <=> c_Orderings_Oord__class_Oless__eq(T_a, V_y_2, V_x_2)))). 203.92/204.18 fof(fact_n__not__Suc__n, axiom, ![V_n]: V_n!=c_Nat_OSuc(V_n)). 203.92/204.18 fof(fact_nat_Osimps_I2_J, axiom, 203.92/204.18 ![V_nat_H]: 203.92/204.18 c_Nat_OSuc(V_nat_H)!=c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). 203.92/204.18 fof(fact_nat_Osimps_I3_J, axiom, 203.92/204.18 ![V_nat_H_1]: 203.92/204.18 c_Nat_OSuc(V_nat_H_1)!=c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). 203.92/204.18 fof(fact_nat__less__cases, axiom, 203.92/204.18 ![V_n_2, V_m_2, V_P_2]: 203.92/204.18 (((hBOOL(hAPP(hAPP(V_P_2, V_n_2), V_m_2)) 203.92/204.18 <= (hBOOL(hAPP(hAPP(V_P_2, V_n_2), V_m_2)) 203.92/204.18 <= c_Orderings_Oord__class_Oless(tc_Nat_Onat, V_n_2, V_m_2))) 203.92/204.18 <= (hBOOL(hAPP(hAPP(V_P_2, V_n_2), V_m_2)) <= V_m_2=V_n_2)) 203.92/204.18 <= (c_Orderings_Oord__class_Oless(tc_Nat_Onat, V_m_2, V_n_2) 203.92/204.18 => hBOOL(hAPP(hAPP(V_P_2, V_n_2), V_m_2))))). 203.92/204.18 fof(fact_nat__less__le, axiom, 203.92/204.18 ![V_n_2, V_m_2]: 203.92/204.18 ((c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, V_m_2, V_n_2) 203.92/204.18 & V_m_2!=V_n_2) 203.92/204.18 <=> c_Orderings_Oord__class_Oless(tc_Nat_Onat, V_m_2, V_n_2))). 203.92/204.18 fof(fact_nat__neq__iff, axiom, 203.92/204.18 ![V_n_2, V_m_2]: 203.92/204.18 ((c_Orderings_Oord__class_Oless(tc_Nat_Onat, V_n_2, V_m_2) 203.92/204.18 | c_Orderings_Oord__class_Oless(tc_Nat_Onat, V_m_2, V_n_2)) 203.92/204.18 <=> V_n_2!=V_m_2)). 203.92/204.18 fof(fact_neq0__conv, axiom, 203.92/204.18 ![V_n_2]: 203.92/204.18 (V_n_2!=c_Groups_Ozero__class_Ozero(tc_Nat_Onat) 203.92/204.18 <=> c_Orderings_Oord__class_Oless(tc_Nat_Onat, 203.92/204.18 c_Groups_Ozero__class_Ozero(tc_Nat_Onat), V_n_2))). 203.92/204.18 fof(fact_norm__not__less__zero, axiom, 203.92/204.18 ![T_a, V_x]: 203.92/204.18 (class_RealVector_Oreal__normed__vector(T_a) 203.92/204.18 => ~c_Orderings_Oord__class_Oless(tc_RealDef_Oreal, 203.92/204.18 c_RealVector_Onorm__class_Onorm(T_a, V_x), 203.92/204.18 c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)))). 203.92/204.18 fof(fact_not__add__less1, axiom, 203.92/204.18 ![V_i, V_j]: 203.92/204.18 ~c_Orderings_Oord__class_Oless(tc_Nat_Onat, 203.92/204.18 c_Groups_Oplus__class_Oplus(tc_Nat_Onat, V_i, V_j), V_i)). 203.92/204.18 fof(fact_not__add__less2, axiom, 203.92/204.18 ![V_i, V_j]: 203.92/204.18 ~c_Orderings_Oord__class_Oless(tc_Nat_Onat, 203.92/204.18 c_Groups_Oplus__class_Oplus(tc_Nat_Onat, V_j, V_i), V_i)). 203.92/204.18 fof(fact_not__less0, axiom, 203.92/204.18 ![V_n]: 203.92/204.18 ~c_Orderings_Oord__class_Oless(tc_Nat_Onat, V_n, 203.92/204.18 c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). 203.92/204.18 fof(fact_not__less__eq, axiom, 203.92/204.18 ![V_n_2, V_m_2]: 203.92/204.18 (~c_Orderings_Oord__class_Oless(tc_Nat_Onat, V_m_2, V_n_2) 203.92/204.18 <=> c_Orderings_Oord__class_Oless(tc_Nat_Onat, V_n_2, 203.92/204.18 c_Nat_OSuc(V_m_2)))). 203.92/204.18 fof(fact_not__less__eq__eq, axiom, 203.92/204.18 ![V_n_2, V_m_2]: 203.92/204.18 (~c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, V_m_2, V_n_2) 203.92/204.18 <=> c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, 203.92/204.18 c_Nat_OSuc(V_n_2), V_m_2))). 203.92/204.18 fof(fact_not__less__iff__gr__or__eq, axiom, 203.92/204.18 ![T_a, V_x_2, V_y_2]: 203.92/204.18 (((c_Orderings_Oord__class_Oless(T_a, V_y_2, V_x_2) | V_y_2=V_x_2) 203.92/204.18 <=> ~c_Orderings_Oord__class_Oless(T_a, V_x_2, V_y_2)) 203.92/204.18 <= class_Orderings_Olinorder(T_a))). 203.92/204.18 fof(fact_not__one__le__zero, axiom, 203.92/204.18 ![T_a]: 203.92/204.18 (class_Rings_Olinordered__semidom(T_a) 203.92/204.18 => ~c_Orderings_Oord__class_Oless__eq(T_a, 203.92/204.19 c_Groups_Oone__class_Oone(T_a), 203.92/204.19 c_Groups_Ozero__class_Ozero(T_a)))). 203.92/204.19 fof(fact_not__one__less__zero, axiom, 203.92/204.19 ![T_a]: 203.92/204.19 (class_Rings_Olinordered__semidom(T_a) 203.92/204.19 => ~c_Orderings_Oord__class_Oless(T_a, 203.92/204.19 c_Groups_Oone__class_Oone(T_a), 203.92/204.19 c_Groups_Ozero__class_Ozero(T_a)))). 203.92/204.19 fof(fact_not__real__square__gt__zero, axiom, 203.92/204.19 ![V_x_2]: 203.92/204.19 (V_x_2=c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal) 203.92/204.19 <=> ~c_Orderings_Oord__class_Oless(tc_RealDef_Oreal, 203.92/204.19 c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal), 203.92/204.19 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), 203.92/204.19 V_x_2), 203.92/204.19 V_x_2)))). 203.92/204.19 fof(fact_not__square__less__zero, axiom, 203.92/204.19 ![T_a, V_a]: 203.92/204.19 (class_Rings_Olinordered__ring(T_a) 203.92/204.19 => ~c_Orderings_Oord__class_Oless(T_a, 203.92/204.19 hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a), V_a), V_a), 203.92/204.19 c_Groups_Ozero__class_Ozero(T_a)))). 203.92/204.19 fof(fact_not__sum__squares__lt__zero, axiom, 203.92/204.19 ![T_a, V_x, V_y]: 203.92/204.19 (class_Rings_Olinordered__ring(T_a) 203.92/204.19 => ~c_Orderings_Oord__class_Oless(T_a, 203.92/204.19 c_Groups_Oplus__class_Oplus(T_a, 203.92/204.19 hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a), 203.92/204.19 V_x), 203.92/204.19 V_x), 203.92/204.19 hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a), 203.92/204.19 V_y), 203.92/204.19 V_y)), 203.92/204.19 c_Groups_Ozero__class_Ozero(T_a)))). 203.92/204.19 fof(fact_odd__nonzero, axiom, 203.92/204.19 ![V_z]: 203.92/204.19 c_Groups_Oplus__class_Oplus(tc_Int_Oint, 203.92/204.19 c_Groups_Oplus__class_Oplus(tc_Int_Oint, 203.92/204.19 c_Groups_Oone__class_Oone(tc_Int_Oint), 203.92/204.19 V_z), 203.92/204.19 V_z)!=c_Groups_Ozero__class_Ozero(tc_Int_Oint)). 203.92/204.19 fof(fact_one__neq__zero, axiom, 203.92/204.19 ![T_a]: 203.92/204.19 (c_Groups_Ozero__class_Ozero(T_a)!=c_Groups_Oone__class_Oone(T_a) 203.92/204.19 <= class_Rings_Ozero__neq__one(T_a))). 203.92/204.19 fof(fact_order__less__asym, axiom, 203.92/204.19 ![T_a, V_x, V_y]: 203.92/204.19 ((~c_Orderings_Oord__class_Oless(T_a, V_y, V_x) 203.92/204.19 <= c_Orderings_Oord__class_Oless(T_a, V_x, V_y)) 203.92/204.19 <= class_Orderings_Opreorder(T_a))). 203.92/204.19 fof(fact_order__less__asym_H, axiom, 203.92/204.19 ![T_a, V_b, V_a]: 203.92/204.19 ((c_Orderings_Oord__class_Oless(T_a, V_a, V_b) 203.92/204.19 => ~c_Orderings_Oord__class_Oless(T_a, V_b, V_a)) 203.92/204.19 <= class_Orderings_Opreorder(T_a))). 203.92/204.19 fof(fact_order__less__imp__not__eq, axiom, 203.92/204.19 ![T_a, V_x, V_y]: 203.92/204.19 ((c_Orderings_Oord__class_Oless(T_a, V_x, V_y) => V_y!=V_x) 203.92/204.19 <= class_Orderings_Oorder(T_a))). 203.92/204.19 fof(fact_order__less__imp__not__eq2, axiom, 203.92/204.19 ![T_a, V_x, V_y]: 203.92/204.19 ((V_x!=V_y <= c_Orderings_Oord__class_Oless(T_a, V_x, V_y)) 203.92/204.19 <= class_Orderings_Oorder(T_a))). 203.92/204.19 fof(fact_order__less__imp__not__less, axiom, 203.92/204.19 ![T_a, V_x, V_y]: 203.92/204.19 (class_Orderings_Opreorder(T_a) 203.92/204.19 => (~c_Orderings_Oord__class_Oless(T_a, V_y, V_x) 203.92/204.19 <= c_Orderings_Oord__class_Oless(T_a, V_x, V_y)))). 203.92/204.19 fof(fact_order__less__irrefl, axiom, 203.92/204.19 ![T_a, V_x]: 203.92/204.19 (class_Orderings_Opreorder(T_a) 203.92/204.19 => ~c_Orderings_Oord__class_Oless(T_a, V_x, V_x))). 203.92/204.19 fof(fact_order__less__le, axiom, 203.92/204.19 ![T_a, V_x_2, V_y_2]: 203.92/204.19 (class_Orderings_Oorder(T_a) 203.92/204.19 => (c_Orderings_Oord__class_Oless(T_a, V_x_2, V_y_2) 203.92/204.19 <=> (c_Orderings_Oord__class_Oless__eq(T_a, V_x_2, V_y_2) 203.92/204.19 & V_x_2!=V_y_2)))). 203.92/204.19 fof(fact_order__less__not__sym, axiom, 203.92/204.19 ![T_a, V_x, V_y]: 203.92/204.19 (class_Orderings_Opreorder(T_a) 203.92/204.19 => (c_Orderings_Oord__class_Oless(T_a, V_x, V_y) 203.92/204.19 => ~c_Orderings_Oord__class_Oless(T_a, V_y, V_x)))). 203.92/204.19 fof(fact_power__0__Suc, axiom, 203.92/204.19 ![T_a, V_n]: 203.92/204.19 (hAPP(hAPP(c_Power_Opower__class_Opower(T_a), 203.92/204.19 c_Groups_Ozero__class_Ozero(T_a)), 203.92/204.19 c_Nat_OSuc(V_n))=c_Groups_Ozero__class_Ozero(T_a) 203.92/204.19 <= (class_Power_Opower(T_a) & class_Rings_Osemiring__0(T_a)))). 203.92/204.19 fof(fact_power__eq__0__iff, axiom, 203.92/204.19 ![V_n_2, V_aa_2, T_a]: 203.92/204.19 (((c_Groups_Ozero__class_Ozero(T_a)=V_aa_2 203.92/204.19 & V_n_2!=c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) 203.92/204.19 <=> hAPP(hAPP(c_Power_Opower__class_Opower(T_a), V_aa_2), 203.92/204.19 V_n_2)=c_Groups_Ozero__class_Ozero(T_a)) 203.92/204.19 <= (class_Power_Opower(T_a) 203.92/204.19 & (class_Rings_Omult__zero(T_a) 203.92/204.19 & (class_Rings_Ozero__neq__one(T_a) 203.92/204.19 & class_Rings_Ono__zero__divisors(T_a)))))). 203.92/204.19 fof(fact_power__eq__0__iff__number__of, axiom, 203.92/204.19 ![V_w_2, V_aa_2, T_a]: 203.92/204.19 ((hAPP(hAPP(c_Power_Opower__class_Opower(T_a), V_aa_2), 203.92/204.19 c_Int_Onumber__class_Onumber__of(tc_Nat_Onat, 203.92/204.19 V_w_2))=c_Groups_Ozero__class_Ozero(T_a) 203.92/204.19 <=> (V_aa_2=c_Groups_Ozero__class_Ozero(T_a) 203.92/204.19 & c_Int_Onumber__class_Onumber__of(tc_Nat_Onat, 203.92/204.19 V_w_2)!=c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) 203.92/204.19 <= (class_Power_Opower(T_a) 203.92/204.19 & (class_Rings_Omult__zero(T_a) 203.92/204.19 & (class_Rings_Ozero__neq__one(T_a) 203.92/204.19 & class_Rings_Ono__zero__divisors(T_a)))))). 203.92/204.19 fof(fact_power__one, axiom, 203.92/204.19 ![T_a, V_n]: 203.92/204.19 (c_Groups_Oone__class_Oone(T_a)=hAPP(hAPP(c_Power_Opower__class_Opower(T_a), 203.92/204.19 c_Groups_Oone__class_Oone(T_a)), 203.92/204.19 V_n) 203.92/204.19 <= class_Groups_Omonoid__mult(T_a))). 203.92/204.19 fof(fact_real__less__def, axiom, 203.92/204.19 ![V_x_2, V_y_2]: 203.92/204.19 ((V_y_2!=V_x_2 203.92/204.19 & c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal, V_x_2, 203.92/204.19 V_y_2)) 203.92/204.19 <=> c_Orderings_Oord__class_Oless(tc_RealDef_Oreal, V_x_2, 203.92/204.19 V_y_2))). 203.92/204.19 fof(fact_split__zdiv, axiom, 203.92/204.19 ![V_n_2, V_P_2, V_ka_2]: 203.92/204.19 (hBOOL(hAPP(V_P_2, 203.92/204.19 c_Divides_Odiv__class_Odiv(tc_Int_Oint, V_n_2, V_ka_2))) 203.92/204.19 <=> ((![B_i]: 203.92/204.19 (hBOOL(hAPP(V_P_2, B_i)) 203.92/204.19 <= ?[B_j]: 203.92/204.19 (c_Groups_Oplus__class_Oplus(tc_Int_Oint, 203.92/204.19 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint), 203.92/204.19 V_ka_2), 203.92/204.19 B_i), 203.92/204.19 B_j)=V_n_2 203.92/204.19 & (c_Orderings_Oord__class_Oless__eq(tc_Int_Oint, B_j, 203.92/204.19 c_Groups_Ozero__class_Ozero(tc_Int_Oint)) 203.92/204.19 & c_Orderings_Oord__class_Oless(tc_Int_Oint, V_ka_2, B_j)))) 203.92/204.19 <= c_Orderings_Oord__class_Oless(tc_Int_Oint, V_ka_2, 203.92/204.19 c_Groups_Ozero__class_Ozero(tc_Int_Oint))) 203.92/204.19 & ((c_Orderings_Oord__class_Oless(tc_Int_Oint, 203.92/204.19 c_Groups_Ozero__class_Ozero(tc_Int_Oint), V_ka_2) 203.92/204.19 => ![B_i]: 203.92/204.19 (?[B_j]: 203.92/204.19 (c_Orderings_Oord__class_Oless(tc_Int_Oint, B_j, V_ka_2) 203.92/204.19 & (c_Groups_Oplus__class_Oplus(tc_Int_Oint, 203.92/204.19 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint), 203.92/204.19 V_ka_2), 203.92/204.19 B_i), 203.92/204.19 B_j)=V_n_2 203.92/204.19 & c_Orderings_Oord__class_Oless__eq(tc_Int_Oint, 203.92/204.19 c_Groups_Ozero__class_Ozero(tc_Int_Oint), 203.92/204.19 B_j))) 203.92/204.19 => hBOOL(hAPP(V_P_2, B_i)))) 203.92/204.19 & (c_Groups_Ozero__class_Ozero(tc_Int_Oint)=V_ka_2 203.92/204.19 => hBOOL(hAPP(V_P_2, 203.92/204.19 c_Groups_Ozero__class_Ozero(tc_Int_Oint)))))))). 203.92/204.19 fof(fact_sum__squares__gt__zero__iff, axiom, 203.92/204.19 ![T_a, V_x_2, V_y_2]: 203.92/204.19 ((c_Orderings_Oord__class_Oless(T_a, 203.92/204.19 c_Groups_Ozero__class_Ozero(T_a), 203.92/204.19 c_Groups_Oplus__class_Oplus(T_a, 203.92/204.19 hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a), 203.92/204.19 V_x_2), 203.92/204.19 V_x_2), 203.92/204.19 hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a), 203.92/204.19 V_y_2), 203.92/204.19 V_y_2))) 203.92/204.19 <=> (c_Groups_Ozero__class_Ozero(T_a)!=V_x_2 203.92/204.19 | c_Groups_Ozero__class_Ozero(T_a)!=V_y_2)) 203.92/204.19 <= class_Rings_Olinordered__ring__strict(T_a))). 203.92/204.19 fof(fact_xt1_I9_J, axiom, 203.92/204.19 ![T_a, V_b, V_a]: 203.92/204.19 (class_Orderings_Oorder(T_a) 203.92/204.19 => (~c_Orderings_Oord__class_Oless(T_a, V_a, V_b) 203.92/204.19 <= c_Orderings_Oord__class_Oless(T_a, V_b, V_a)))). 203.92/204.19 fof(fact_zero__less__norm__iff, axiom, 203.92/204.19 ![T_a, V_x_2]: 203.92/204.19 (class_RealVector_Oreal__normed__vector(T_a) 203.92/204.19 => (c_Orderings_Oord__class_Oless(tc_RealDef_Oreal, 203.92/204.19 c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal), 203.92/204.19 c_RealVector_Onorm__class_Onorm(T_a, V_x_2)) 203.92/204.19 <=> c_Groups_Ozero__class_Ozero(T_a)!=V_x_2))). 204.02/204.19 fof(fact_zero__neq__one, axiom, 204.02/204.19 ![T_a]: 204.02/204.19 (c_Groups_Oone__class_Oone(T_a)!=c_Groups_Ozero__class_Ozero(T_a) 204.02/204.19 <= class_Rings_Ozero__neq__one(T_a))). 204.02/204.19 fof(fact_zless__le, axiom, 204.02/204.19 ![V_w_2, V_z_2]: 204.02/204.19 ((V_z_2!=V_w_2 204.02/204.19 & c_Orderings_Oord__class_Oless__eq(tc_Int_Oint, V_z_2, V_w_2)) 204.02/204.19 <=> c_Orderings_Oord__class_Oless(tc_Int_Oint, V_z_2, V_w_2))). 204.02/204.19 204.02/204.19 Now clausify the problem and encode Horn clauses using encoding 3 of 204.02/204.19 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 204.02/204.19 We repeatedly replace C & s=t => u=v by the two clauses: 204.02/204.19 $$fresh(y, y, x1...xn) = u 204.02/204.19 C => $$fresh(s, t, x1...xn) = v 204.02/204.19 where $$fresh is a fresh function symbol and x1..xn are the free 204.02/204.19 variables of u and v. 204.02/204.19 A predicate p(X) is encoded as p(X)=$$true (this is sound, because the 204.02/204.19 input problem has no model of domain size 1). 204.02/204.19 204.02/204.19 The encoding turns the above axioms into the following unit equations and goals: 204.02/204.19 204.02/204.19 Axiom 291 (fact_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J): $$fresh974(X, X, Y, Z, W) = hAPP(hAPP(c_Power_Opower__class_Opower(W), Z), c_Nat_OSuc(Y)). 204.02/204.19 Axiom 1346 (fact_power__0__Suc): $$fresh1213(X, X, Y, Z) = c_Groups_Ozero__class_Ozero(Z). 204.02/204.19 Axiom 1347 (fact_power__0__Suc): $$fresh1214(X, X, Y, Z) = $$fresh1213(class_Power_Opower(Z), $$true2, Y, Z). 204.02/204.19 Axiom 1428 (fact_power__one): $$fresh330(X, X, Y, Z) = c_Groups_Oone__class_Oone(Z). 204.02/204.19 Axiom 1773 (fact_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J): $$fresh974(class_Rings_Ocomm__semiring__1(X), $$true2, Y, Z, X) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(X), Z), hAPP(hAPP(c_Power_Opower__class_Opower(X), Z), Y)). 204.02/204.19 Axiom 2050 (arity_Complex__Ocomplex__Rings_Ocomm__semiring__1): class_Rings_Ocomm__semiring__1(tc_Complex_Ocomplex) = $$true2. 204.02/204.19 Axiom 2326 (arity_Complex__Ocomplex__Power_Opower): class_Power_Opower(tc_Complex_Ocomplex) = $$true2. 204.02/204.19 Axiom 2416 (arity_Complex__Ocomplex__Groups_Omonoid__mult): class_Groups_Omonoid__mult(tc_Complex_Ocomplex) = $$true2. 204.02/204.19 Axiom 2471 (fact_power__one): $$fresh330(class_Groups_Omonoid__mult(X), $$true2, Y, X) = hAPP(hAPP(c_Power_Opower__class_Opower(X), c_Groups_Oone__class_Oone(X)), Y). 204.02/204.19 Axiom 2635 (arity_Complex__Ocomplex__Rings_Osemiring__0): class_Rings_Osemiring__0(tc_Complex_Ocomplex) = $$true2. 204.02/204.19 Axiom 2736 (fact_One__nat__def): c_Groups_Oone__class_Oone(tc_Nat_Onat) = c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). 204.02/204.19 Axiom 2792 (arity_Complex__Ocomplex__Rings_Ozero__neq__one): class_Rings_Ozero__neq__one(tc_Complex_Ocomplex) = $$true2. 204.02/204.19 Axiom 2980 (fact_power__0__Suc): $$fresh1214(class_Rings_Osemiring__0(X), $$true2, Y, X) = hAPP(hAPP(c_Power_Opower__class_Opower(X), c_Groups_Ozero__class_Ozero(X)), c_Nat_OSuc(Y)). 204.02/204.19 Axiom 3089 (conj_0): hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), X), hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex), X), c_Groups_Ominus__class_Ominus(tc_Nat_Onat, v_k____, c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), Y), hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex), Y), c_Groups_Ominus__class_Ominus(tc_Nat_Onat, v_k____, c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))))). 204.02/204.19 204.02/204.19 Lemma 3090: $$fresh974(X, X, Y, Z, W) = $$fresh974(?, ?, Y, Z, W). 204.02/204.19 Proof: 204.02/204.19 $$fresh974(X, X, Y, Z, W) 204.02/204.19 = { by axiom 291 (fact_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) } 204.02/204.19 hAPP(hAPP(c_Power_Opower__class_Opower(W), Z), c_Nat_OSuc(Y)) 204.02/204.19 = { by axiom 291 (fact_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) } 204.02/204.19 $$fresh974(?, ?, Y, Z, W) 204.02/204.19 204.02/204.19 Goal 1 (fact_zero__neq__one): tuple5(c_Groups_Oone__class_Oone(X), class_Rings_Ozero__neq__one(X)) = tuple5(c_Groups_Ozero__class_Ozero(X), $$true2). 204.02/204.19 The goal is true when: 204.02/204.19 X = tc_Complex_Ocomplex 204.02/204.19 204.02/204.19 Proof: 204.02/204.19 tuple5(c_Groups_Oone__class_Oone(tc_Complex_Ocomplex), class_Rings_Ozero__neq__one(tc_Complex_Ocomplex)) 204.02/204.19 = { by axiom 1428 (fact_power__one) } 204.02/204.19 tuple5($$fresh330($$true2, $$true2, c_Nat_OSuc(c_Groups_Ominus__class_Ominus(tc_Nat_Onat, v_k____, c_Groups_Oone__class_Oone(tc_Nat_Onat))), tc_Complex_Ocomplex), class_Rings_Ozero__neq__one(tc_Complex_Ocomplex)) 204.02/204.19 = { by axiom 2416 (arity_Complex__Ocomplex__Groups_Omonoid__mult) } 204.02/204.19 tuple5($$fresh330(class_Groups_Omonoid__mult(tc_Complex_Ocomplex), $$true2, c_Nat_OSuc(c_Groups_Ominus__class_Ominus(tc_Nat_Onat, v_k____, c_Groups_Oone__class_Oone(tc_Nat_Onat))), tc_Complex_Ocomplex), class_Rings_Ozero__neq__one(tc_Complex_Ocomplex)) 204.02/204.19 = { by axiom 2471 (fact_power__one) } 204.02/204.19 tuple5(hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex), c_Groups_Oone__class_Oone(tc_Complex_Ocomplex)), c_Nat_OSuc(c_Groups_Ominus__class_Ominus(tc_Nat_Onat, v_k____, c_Groups_Oone__class_Oone(tc_Nat_Onat)))), class_Rings_Ozero__neq__one(tc_Complex_Ocomplex)) 204.02/204.19 = { by axiom 291 (fact_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) } 204.02/204.19 tuple5($$fresh974(?, ?, c_Groups_Ominus__class_Ominus(tc_Nat_Onat, v_k____, c_Groups_Oone__class_Oone(tc_Nat_Onat)), c_Groups_Oone__class_Oone(tc_Complex_Ocomplex), tc_Complex_Ocomplex), class_Rings_Ozero__neq__one(tc_Complex_Ocomplex)) 204.02/204.19 = { by lemma 3090 } 204.02/204.19 tuple5($$fresh974($$true2, $$true2, c_Groups_Ominus__class_Ominus(tc_Nat_Onat, v_k____, c_Groups_Oone__class_Oone(tc_Nat_Onat)), c_Groups_Oone__class_Oone(tc_Complex_Ocomplex), tc_Complex_Ocomplex), class_Rings_Ozero__neq__one(tc_Complex_Ocomplex)) 204.02/204.19 = { by axiom 2050 (arity_Complex__Ocomplex__Rings_Ocomm__semiring__1) } 204.02/204.19 tuple5($$fresh974(class_Rings_Ocomm__semiring__1(tc_Complex_Ocomplex), $$true2, c_Groups_Ominus__class_Ominus(tc_Nat_Onat, v_k____, c_Groups_Oone__class_Oone(tc_Nat_Onat)), c_Groups_Oone__class_Oone(tc_Complex_Ocomplex), tc_Complex_Ocomplex), class_Rings_Ozero__neq__one(tc_Complex_Ocomplex)) 204.02/204.19 = { by axiom 2736 (fact_One__nat__def) } 204.02/204.19 tuple5($$fresh974(class_Rings_Ocomm__semiring__1(tc_Complex_Ocomplex), $$true2, c_Groups_Ominus__class_Ominus(tc_Nat_Onat, v_k____, c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))), c_Groups_Oone__class_Oone(tc_Complex_Ocomplex), tc_Complex_Ocomplex), class_Rings_Ozero__neq__one(tc_Complex_Ocomplex)) 204.02/204.19 = { by axiom 1773 (fact_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) } 204.02/204.19 tuple5(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Groups_Oone__class_Oone(tc_Complex_Ocomplex)), hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex), c_Groups_Oone__class_Oone(tc_Complex_Ocomplex)), c_Groups_Ominus__class_Ominus(tc_Nat_Onat, v_k____, c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))))), class_Rings_Ozero__neq__one(tc_Complex_Ocomplex)) 204.02/204.19 = { by axiom 3089 (conj_0) } 204.02/204.19 tuple5(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), ?), hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex), ?), c_Groups_Ominus__class_Ominus(tc_Nat_Onat, v_k____, c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))))), class_Rings_Ozero__neq__one(tc_Complex_Ocomplex)) 204.02/204.19 = { by axiom 3089 (conj_0) } 204.02/204.19 tuple5(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Complex_Ocomplex), c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex)), hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex), c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex)), c_Groups_Ominus__class_Ominus(tc_Nat_Onat, v_k____, c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))))), class_Rings_Ozero__neq__one(tc_Complex_Ocomplex)) 204.02/204.19 = { by axiom 1773 (fact_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) } 204.02/204.19 tuple5($$fresh974(class_Rings_Ocomm__semiring__1(tc_Complex_Ocomplex), $$true2, c_Groups_Ominus__class_Ominus(tc_Nat_Onat, v_k____, c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))), c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex), tc_Complex_Ocomplex), class_Rings_Ozero__neq__one(tc_Complex_Ocomplex)) 204.02/204.19 = { by axiom 2050 (arity_Complex__Ocomplex__Rings_Ocomm__semiring__1) } 204.02/204.19 tuple5($$fresh974($$true2, $$true2, c_Groups_Ominus__class_Ominus(tc_Nat_Onat, v_k____, c_Nat_OSuc(c_Groups_Ozero__class_Ozero(tc_Nat_Onat))), c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex), tc_Complex_Ocomplex), class_Rings_Ozero__neq__one(tc_Complex_Ocomplex)) 204.02/204.19 = { by axiom 2736 (fact_One__nat__def) } 204.02/204.19 tuple5($$fresh974($$true2, $$true2, c_Groups_Ominus__class_Ominus(tc_Nat_Onat, v_k____, c_Groups_Oone__class_Oone(tc_Nat_Onat)), c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex), tc_Complex_Ocomplex), class_Rings_Ozero__neq__one(tc_Complex_Ocomplex)) 204.02/204.19 = { by lemma 3090 } 204.02/204.19 tuple5($$fresh974(?, ?, c_Groups_Ominus__class_Ominus(tc_Nat_Onat, v_k____, c_Groups_Oone__class_Oone(tc_Nat_Onat)), c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex), tc_Complex_Ocomplex), class_Rings_Ozero__neq__one(tc_Complex_Ocomplex)) 204.02/204.19 = { by axiom 291 (fact_comm__semiring__1__class_Onormalizing__semiring__rules_I27_J) } 204.02/204.19 tuple5(hAPP(hAPP(c_Power_Opower__class_Opower(tc_Complex_Ocomplex), c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex)), c_Nat_OSuc(c_Groups_Ominus__class_Ominus(tc_Nat_Onat, v_k____, c_Groups_Oone__class_Oone(tc_Nat_Onat)))), class_Rings_Ozero__neq__one(tc_Complex_Ocomplex)) 204.02/204.19 = { by axiom 2980 (fact_power__0__Suc) } 204.02/204.19 tuple5($$fresh1214(class_Rings_Osemiring__0(tc_Complex_Ocomplex), $$true2, c_Groups_Ominus__class_Ominus(tc_Nat_Onat, v_k____, c_Groups_Oone__class_Oone(tc_Nat_Onat)), tc_Complex_Ocomplex), class_Rings_Ozero__neq__one(tc_Complex_Ocomplex)) 204.02/204.19 = { by axiom 2635 (arity_Complex__Ocomplex__Rings_Osemiring__0) } 204.02/204.19 tuple5($$fresh1214($$true2, $$true2, c_Groups_Ominus__class_Ominus(tc_Nat_Onat, v_k____, c_Groups_Oone__class_Oone(tc_Nat_Onat)), tc_Complex_Ocomplex), class_Rings_Ozero__neq__one(tc_Complex_Ocomplex)) 204.02/204.19 = { by axiom 1347 (fact_power__0__Suc) } 204.02/204.19 tuple5($$fresh1213(class_Power_Opower(tc_Complex_Ocomplex), $$true2, c_Groups_Ominus__class_Ominus(tc_Nat_Onat, v_k____, c_Groups_Oone__class_Oone(tc_Nat_Onat)), tc_Complex_Ocomplex), class_Rings_Ozero__neq__one(tc_Complex_Ocomplex)) 204.02/204.19 = { by axiom 2326 (arity_Complex__Ocomplex__Power_Opower) } 204.02/204.19 tuple5($$fresh1213($$true2, $$true2, c_Groups_Ominus__class_Ominus(tc_Nat_Onat, v_k____, c_Groups_Oone__class_Oone(tc_Nat_Onat)), tc_Complex_Ocomplex), class_Rings_Ozero__neq__one(tc_Complex_Ocomplex)) 204.02/204.19 = { by axiom 1346 (fact_power__0__Suc) } 204.02/204.19 tuple5(c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex), class_Rings_Ozero__neq__one(tc_Complex_Ocomplex)) 204.02/204.19 = { by axiom 2792 (arity_Complex__Ocomplex__Rings_Ozero__neq__one) } 204.02/204.19 tuple5(c_Groups_Ozero__class_Ozero(tc_Complex_Ocomplex), $$true2) 204.02/204.19 % SZS output end Proof 204.02/204.19 204.02/204.19 RESULT: Theorem (the conjecture is true). 204.03/204.26 EOF