TSTP Solution File: SWW252+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SWW252+1 : TPTP v8.1.2. Released v5.2.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n012.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Fri Sep 1 00:54:31 EDT 2023
% Result : Theorem 252.14s 32.82s
% Output : Proof 252.49s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SWW252+1 : TPTP v8.1.2. Released v5.2.0.
% 0.13/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.34 % Computer : n012.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Sun Aug 27 20:46:25 EDT 2023
% 0.14/0.35 % CPUTime :
% 252.14/32.82 Command-line arguments: --kbo-weight0 --lhs-weight 5 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --goal-heuristic
% 252.14/32.82
% 252.14/32.82 % SZS status Theorem
% 252.14/32.82
% 252.14/32.82 % SZS output start Proof
% 252.14/32.82 Take the following subset of the input axioms:
% 252.49/32.83 fof(fact_Suc__n__not__le__n, axiom, ![V_n]: ~hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, hAPP(c_Nat_OSuc, V_n)), V_n))).
% 252.49/32.83 fof(fact_Suc__n__not__n, axiom, ![V_n2]: hAPP(c_Nat_OSuc, V_n2)!=V_n2).
% 252.49/32.83 fof(fact_Suc__neq__Zero, axiom, ![V_m]: hAPP(c_Nat_OSuc, V_m)!=c_Groups_Ozero__class_Ozero(tc_Nat_Onat)).
% 252.49/32.83 fof(fact_Suc__not__Zero, axiom, ![V_m2]: hAPP(c_Nat_OSuc, V_m2)!=c_Groups_Ozero__class_Ozero(tc_Nat_Onat)).
% 252.49/32.83 fof(fact_Zero__neq__Suc, axiom, ![V_m2]: c_Groups_Ozero__class_Ozero(tc_Nat_Onat)!=hAPP(c_Nat_OSuc, V_m2)).
% 252.49/32.83 fof(fact_Zero__not__Suc, axiom, ![V_m2]: c_Groups_Ozero__class_Ozero(tc_Nat_Onat)!=hAPP(c_Nat_OSuc, V_m2)).
% 252.49/32.83 fof(fact_gr__implies__not0, axiom, ![V_n2, V_m2]: (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat), V_m2), V_n2)) => V_n2!=c_Groups_Ozero__class_Ozero(tc_Nat_Onat))).
% 252.49/32.83 fof(fact_leD, axiom, ![T_a, V_x, V_y]: (class_Orderings_Olinorder(T_a) => (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a, V_y), V_x)) => ~hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a), V_x), V_y))))).
% 252.49/32.83 fof(fact_less__degree__imp, axiom, ![V_p, T_a2, V_n2]: (class_Groups_Ozero(T_a2) => (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat), V_n2), c_Polynomial_Odegree(T_a2, V_p))) => ?[B_i]: (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat), V_n2), B_i)) & hAPP(c_Polynomial_Ocoeff(T_a2, V_p), B_i)!=c_Groups_Ozero__class_Ozero(T_a2))))).
% 252.49/32.83 fof(fact_less__fun__def, axiom, ![V_g_2, V_f_2, T_b, T_a2]: (class_Orderings_Oord(T_b) => (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_a2, T_b)), V_f_2), V_g_2)) <=> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_a2, T_b), V_f_2), V_g_2)) & ~hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_a2, T_b), V_g_2), V_f_2)))))).
% 252.49/32.83 fof(fact_less__imp__neq, axiom, ![T_a2, V_x2, V_y2]: (class_Orderings_Oorder(T_a2) => (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), V_x2), V_y2)) => V_x2!=V_y2))).
% 252.49/32.83 fof(fact_less__irrefl__nat, axiom, ![V_n2]: ~hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat), V_n2), V_n2))).
% 252.49/32.83 fof(fact_less__le__not__le, axiom, ![V_y_2, V_x_2, T_a2]: (class_Orderings_Opreorder(T_a2) => (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), V_x_2), V_y_2)) <=> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a2, V_x_2), V_y_2)) & ~hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a2, V_y_2), V_x_2)))))).
% 252.49/32.83 fof(fact_less__nat__zero__code, axiom, ![V_n2]: ~hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat), V_n2), c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))).
% 252.49/32.83 fof(fact_less__not__refl, axiom, ![V_n2]: ~hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat), V_n2), V_n2))).
% 252.49/32.83 fof(fact_less__not__refl2, axiom, ![V_n2, V_m2]: (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat), V_n2), V_m2)) => V_m2!=V_n2)).
% 252.49/32.84 fof(fact_less__not__refl3, axiom, ![V_t, V_s]: (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat), V_s), V_t)) => V_s!=V_t)).
% 252.49/32.84 fof(fact_less__zeroE, axiom, ![V_n2]: ~hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat), V_n2), c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))).
% 252.49/32.84 fof(fact_linorder__antisym__conv2, axiom, ![T_a2, V_x_2_2, V_y_2_2]: (class_Orderings_Olinorder(T_a2) => (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a2, V_x_2_2), V_y_2_2)) => (~hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), V_x_2_2), V_y_2_2)) <=> V_x_2_2=V_y_2_2)))).
% 252.49/32.84 fof(fact_linorder__neq__iff, axiom, ![T_a2, V_x_2_2, V_y_2_2]: (class_Orderings_Olinorder(T_a2) => (V_x_2_2!=V_y_2_2 <=> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), V_x_2_2), V_y_2_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), V_y_2_2), V_x_2_2)))))).
% 252.49/32.84 fof(fact_linorder__not__le, axiom, ![T_a2, V_x_2_2, V_y_2_2]: (class_Orderings_Olinorder(T_a2) => (~hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a2, V_x_2_2), V_y_2_2)) <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), V_y_2_2), V_x_2_2))))).
% 252.49/32.84 fof(fact_linorder__not__less, axiom, ![T_a2, V_x_2_2, V_y_2_2]: (class_Orderings_Olinorder(T_a2) => (~hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), V_x_2_2), V_y_2_2)) <=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a2, V_y_2_2), V_x_2_2))))).
% 252.49/32.84 fof(fact_n__not__Suc__n, axiom, ![V_n2]: V_n2!=hAPP(c_Nat_OSuc, V_n2)).
% 252.49/32.84 fof(fact_nat_Osimps_I2_J, axiom, ![V_nat_H]: c_Groups_Ozero__class_Ozero(tc_Nat_Onat)!=hAPP(c_Nat_OSuc, V_nat_H)).
% 252.49/32.84 fof(fact_nat_Osimps_I3_J, axiom, ![V_nat_H_1]: hAPP(c_Nat_OSuc, V_nat_H_1)!=c_Groups_Ozero__class_Ozero(tc_Nat_Onat)).
% 252.49/32.84 fof(fact_nat__less__cases, axiom, ![V_n_2, V_P_2, V_m_2]: ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat), V_m_2), V_n_2)) => hBOOL(hAPP(hAPP(V_P_2, V_n_2), V_m_2))) => ((V_m_2=V_n_2 => hBOOL(hAPP(hAPP(V_P_2, V_n_2), V_m_2))) => ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat), V_n_2), V_m_2)) => hBOOL(hAPP(hAPP(V_P_2, V_n_2), V_m_2))) => hBOOL(hAPP(hAPP(V_P_2, V_n_2), V_m_2)))))).
% 252.49/32.84 fof(fact_nat__less__le, axiom, ![V_n_2_2, V_m_2_2]: (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat), V_m_2_2), V_n_2_2)) <=> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, V_m_2_2), V_n_2_2)) & V_m_2_2!=V_n_2_2))).
% 252.49/32.84 fof(fact_nat__neq__iff, axiom, ![V_n_2_2, V_m_2_2]: (V_m_2_2!=V_n_2_2 <=> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat), V_m_2_2), V_n_2_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat), V_n_2_2), V_m_2_2))))).
% 252.49/32.84 fof(fact_neq0__conv, axiom, ![V_n_2_2]: (V_n_2_2!=c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat), c_Groups_Ozero__class_Ozero(tc_Nat_Onat)), V_n_2_2)))).
% 252.49/32.84 fof(fact_norm__not__less__zero, axiom, ![T_a2, V_x2]: (class_RealVector_Oreal__normed__vector(T_a2) => ~hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal), c_RealVector_Onorm__class_Onorm(T_a2, V_x2)), c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))))).
% 252.49/32.84 fof(fact_not__add__less1, axiom, ![V_j, V_i]: ~hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat), hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat), V_i), V_j)), V_i))).
% 252.49/32.84 fof(fact_not__add__less2, axiom, ![V_j2, V_i2]: ~hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat), hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat), V_j2), V_i2)), V_i2))).
% 252.49/32.84 fof(fact_not__less0, axiom, ![V_n2]: ~hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat), V_n2), c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))).
% 252.49/32.84 fof(fact_not__less__Least, axiom, ![V_k_2, V_P_2_2, T_a2]: (class_Orderings_Owellorder(T_a2) => (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), V_k_2), c_Orderings_Oord__class_OLeast(T_a2, V_P_2_2))) => ~hBOOL(hAPP(V_P_2_2, V_k_2))))).
% 252.49/32.84 fof(fact_not__less__eq, axiom, ![V_n_2_2, V_m_2_2]: (~hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat), V_m_2_2), V_n_2_2)) <=> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat), V_n_2_2), hAPP(c_Nat_OSuc, V_m_2_2))))).
% 252.49/32.84 fof(fact_not__less__eq__eq, axiom, ![V_n_2_2, V_m_2_2]: (~hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, V_m_2_2), V_n_2_2)) <=> hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat, hAPP(c_Nat_OSuc, V_n_2_2)), V_m_2_2)))).
% 252.49/32.84 fof(fact_not__less__iff__gr__or__eq, axiom, ![T_a2, V_x_2_2, V_y_2_2]: (class_Orderings_Olinorder(T_a2) => (~hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), V_x_2_2), V_y_2_2)) <=> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), V_y_2_2), V_x_2_2)) | V_x_2_2=V_y_2_2)))).
% 252.49/32.84 fof(fact_not__one__le__zero, axiom, ![T_a2]: (class_Rings_Olinordered__semidom(T_a2) => ~hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a2, c_Groups_Oone__class_Oone(T_a2)), c_Groups_Ozero__class_Ozero(T_a2))))).
% 252.49/32.84 fof(fact_not__one__less__zero, axiom, ![T_a2]: (class_Rings_Olinordered__semidom(T_a2) => ~hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), c_Groups_Oone__class_Oone(T_a2)), c_Groups_Ozero__class_Ozero(T_a2))))).
% 252.49/32.84 fof(fact_not__pos__poly__0, axiom, ![T_a2]: (class_Rings_Olinordered__idom(T_a2) => ~c_Polynomial_Opos__poly(T_a2, c_Groups_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a2))))).
% 252.49/32.84 fof(fact_not__real__of__nat__less__zero, axiom, ![V_n2]: ~hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal), c_RealDef_Oreal(tc_Nat_Onat, V_n2)), c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)))).
% 252.49/32.84 fof(fact_not__real__square__gt__zero, axiom, ![V_x_2_2]: (~hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal), c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)), hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), V_x_2_2), V_x_2_2))) <=> V_x_2_2=c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal))).
% 252.49/32.84 fof(fact_not__square__less__zero, axiom, ![V_a, T_a2]: (class_Rings_Olinordered__ring(T_a2) => ~hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a2), V_a), V_a)), c_Groups_Ozero__class_Ozero(T_a2))))).
% 252.49/32.84 fof(fact_not__sum__squares__lt__zero, axiom, ![T_a2, V_x2, V_y2]: (class_Rings_Olinordered__ring(T_a2) => ~hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a2), hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a2), V_x2), V_x2)), hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a2), V_y2), V_y2))), c_Groups_Ozero__class_Ozero(T_a2))))).
% 252.49/32.84 fof(fact_odd__nonzero, axiom, ![V_z]: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint), hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint), c_Groups_Oone__class_Oone(tc_Int_Oint)), V_z)), V_z)!=c_Groups_Ozero__class_Ozero(tc_Int_Oint)).
% 252.49/32.84 fof(fact_one__neq__zero, axiom, ![T_a2]: (class_Rings_Ozero__neq__one(T_a2) => c_Groups_Oone__class_Oone(T_a2)!=c_Groups_Ozero__class_Ozero(T_a2))).
% 252.49/32.84 fof(fact_order__less__asym, axiom, ![T_a2, V_x2, V_y2]: (class_Orderings_Opreorder(T_a2) => (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), V_x2), V_y2)) => ~hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), V_y2), V_x2))))).
% 252.49/32.84 fof(fact_order__less__asym_H, axiom, ![V_b, T_a2, V_a2]: (class_Orderings_Opreorder(T_a2) => (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), V_a2), V_b)) => ~hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), V_b), V_a2))))).
% 252.49/32.84 fof(fact_order__less__imp__not__eq, axiom, ![T_a2, V_x2, V_y2]: (class_Orderings_Oorder(T_a2) => (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), V_x2), V_y2)) => V_x2!=V_y2))).
% 252.49/32.84 fof(fact_order__less__imp__not__eq2, axiom, ![T_a2, V_x2, V_y2]: (class_Orderings_Oorder(T_a2) => (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), V_x2), V_y2)) => V_y2!=V_x2))).
% 252.49/32.84 fof(fact_order__less__imp__not__less, axiom, ![T_a2, V_x2, V_y2]: (class_Orderings_Opreorder(T_a2) => (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), V_x2), V_y2)) => ~hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), V_y2), V_x2))))).
% 252.49/32.84 fof(fact_order__less__irrefl, axiom, ![T_a2, V_x2]: (class_Orderings_Opreorder(T_a2) => ~hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), V_x2), V_x2)))).
% 252.49/32.84 fof(fact_order__less__le, axiom, ![T_a2, V_x_2_2, V_y_2_2]: (class_Orderings_Oorder(T_a2) => (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), V_x_2_2), V_y_2_2)) <=> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a2, V_x_2_2), V_y_2_2)) & V_x_2_2!=V_y_2_2)))).
% 252.49/32.84 fof(fact_order__less__not__sym, axiom, ![T_a2, V_x2, V_y2]: (class_Orderings_Opreorder(T_a2) => (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), V_x2), V_y2)) => ~hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), V_y2), V_x2))))).
% 252.49/32.84 fof(fact_power__eq__0__iff, axiom, ![V_a_2, T_a2, V_n_2_2]: ((class_Power_Opower(T_a2) & (class_Rings_Omult__zero(T_a2) & (class_Rings_Ono__zero__divisors(T_a2) & class_Rings_Ozero__neq__one(T_a2)))) => (hAPP(hAPP(c_Power_Opower__class_Opower(T_a2), V_a_2), V_n_2_2)=c_Groups_Ozero__class_Ozero(T_a2) <=> (V_a_2=c_Groups_Ozero__class_Ozero(T_a2) & V_n_2_2!=c_Groups_Ozero__class_Ozero(tc_Nat_Onat))))).
% 252.49/32.84 fof(fact_real__less__def, axiom, ![V_x_2_2, V_y_2_2]: (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal), V_x_2_2), V_y_2_2)) <=> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal, V_x_2_2), V_y_2_2)) & V_x_2_2!=V_y_2_2))).
% 252.49/32.84 fof(fact_real__mult__1, axiom, ![V_z2]: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)), V_z2)=V_z2).
% 252.49/32.84 fof(fact_split__zdiv, axiom, ![V_P_2_2, V_n_2_2, V_k_2_2]: (hBOOL(hAPP(V_P_2_2, c_Divides_Odiv__class_Odiv(tc_Int_Oint, V_n_2_2, V_k_2_2))) <=> ((V_k_2_2=c_Groups_Ozero__class_Ozero(tc_Int_Oint) => hBOOL(hAPP(V_P_2_2, c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) & ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint), c_Groups_Ozero__class_Ozero(tc_Int_Oint)), V_k_2_2)) => ![B_i2]: (?[B_j]: (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint, c_Groups_Ozero__class_Ozero(tc_Int_Oint)), B_j)) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint), B_j), V_k_2_2)) & V_n_2_2=hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint), hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint), V_k_2_2), B_i2)), B_j))) => hBOOL(hAPP(V_P_2_2, B_i2)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint), V_k_2_2), c_Groups_Ozero__class_Ozero(tc_Int_Oint))) => ![B_i2]: (?[B_j2]: (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint), V_k_2_2), B_j2)) & (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint, B_j2), c_Groups_Ozero__class_Ozero(tc_Int_Oint))) & V_n_2_2=hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint), hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint), V_k_2_2), B_i2)), B_j2))) => hBOOL(hAPP(V_P_2_2, B_i2)))))))).
% 252.49/32.84 fof(fact_split__zmod, axiom, ![V_P_2_2, V_n_2_2, V_k_2_2]: (hBOOL(hAPP(V_P_2_2, c_Divides_Odiv__class_Omod(tc_Int_Oint, V_n_2_2, V_k_2_2))) <=> ((V_k_2_2=c_Groups_Ozero__class_Ozero(tc_Int_Oint) => hBOOL(hAPP(V_P_2_2, V_n_2_2))) & ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint), c_Groups_Ozero__class_Ozero(tc_Int_Oint)), V_k_2_2)) => ![B_j2, B_i2]: ((hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint, c_Groups_Ozero__class_Ozero(tc_Int_Oint)), B_j2)) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint), B_j2), V_k_2_2)) & V_n_2_2=hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint), hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint), V_k_2_2), B_i2)), B_j2))) => hBOOL(hAPP(V_P_2_2, B_j2)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint), V_k_2_2), c_Groups_Ozero__class_Ozero(tc_Int_Oint))) => ![B_j2, B_i2]: ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint), V_k_2_2), B_j2)) & (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint, B_j2), c_Groups_Ozero__class_Ozero(tc_Int_Oint))) & V_n_2_2=hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint), hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint), V_k_2_2), B_i2)), B_j2))) => hBOOL(hAPP(V_P_2_2, B_j2)))))))).
% 252.49/32.84 fof(fact_sum__squares__gt__zero__iff, axiom, ![T_a2, V_x_2_2, V_y_2_2]: (class_Rings_Olinordered__ring__strict(T_a2) => (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), c_Groups_Ozero__class_Ozero(T_a2)), hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a2), hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a2), V_x_2_2), V_x_2_2)), hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a2), V_y_2_2), V_y_2_2)))) <=> (V_x_2_2!=c_Groups_Ozero__class_Ozero(T_a2) | V_y_2_2!=c_Groups_Ozero__class_Ozero(T_a2))))).
% 252.49/32.84 fof(fact_xt1_I9_J, axiom, ![T_a2, V_b2, V_a2]: (class_Orderings_Oorder(T_a2) => (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), V_b2), V_a2)) => ~hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a2), V_a2), V_b2))))).
% 252.49/32.84 fof(fact_zero__less__norm__iff, axiom, ![T_a2, V_x_2_2]: (class_RealVector_Oreal__normed__vector(T_a2) => (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal), c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)), c_RealVector_Onorm__class_Onorm(T_a2, V_x_2_2))) <=> V_x_2_2!=c_Groups_Ozero__class_Ozero(T_a2)))).
% 252.49/32.84 fof(fact_zero__neq__one, axiom, ![T_a2]: (class_Rings_Ozero__neq__one(T_a2) => c_Groups_Ozero__class_Ozero(T_a2)!=c_Groups_Oone__class_Oone(T_a2))).
% 252.49/32.84 fof(fact_zless__le, axiom, ![V_z_2, V_w_2]: (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint), V_z_2), V_w_2)) <=> (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint, V_z_2), V_w_2)) & V_z_2!=V_w_2))).
% 252.49/32.84 fof(help_c__COMBB__1, axiom, ![V_Q_2, V_R_2, T_c, V_P_2_2, T_a2, T_b2]: hAPP(hAPP(c_COMBB(T_b2, T_a2, T_c, V_P_2_2), V_Q_2), V_R_2)=hAPP(V_P_2_2, hAPP(V_Q_2, V_R_2))).
% 252.49/32.84 fof(help_c__COMBS__1, axiom, ![V_P_2_2, T_a2, T_b2, V_Q_2_2, V_R_2_2, T_c2]: hAPP(hAPP(hAPP(c_COMBS(T_b2, T_c2, T_a2), V_P_2_2), V_Q_2_2), V_R_2_2)=hAPP(hAPP(V_P_2_2, V_R_2_2), hAPP(V_Q_2_2, V_R_2_2))).
% 252.49/32.84
% 252.49/32.84 Now clausify the problem and encode Horn clauses using encoding 3 of
% 252.49/32.84 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 252.49/32.84 We repeatedly replace C & s=t => u=v by the two clauses:
% 252.49/32.84 fresh(y, y, x1...xn) = u
% 252.49/32.84 C => fresh(s, t, x1...xn) = v
% 252.49/32.84 where fresh is a fresh function symbol and x1..xn are the free
% 252.49/32.84 variables of u and v.
% 252.49/32.84 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 252.49/32.84 input problem has no model of domain size 1).
% 252.49/32.84
% 252.49/32.84 The encoding turns the above axioms into the following unit equations and goals:
% 252.49/32.84
% 252.49/32.84 Axiom 1 (fact_real__mult__1): hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)), X) = X.
% 252.49/32.84 Axiom 2 (help_c__COMBS__1): hAPP(hAPP(hAPP(c_COMBS(X, Y, Z), W), V), U) = hAPP(hAPP(W, U), hAPP(V, U)).
% 252.49/32.84 Axiom 3 (help_c__COMBB__1): hAPP(hAPP(c_COMBB(X, Y, Z, W), V), U) = hAPP(W, hAPP(V, U)).
% 252.49/32.84
% 252.49/32.84 Goal 1 (fact_n__not__Suc__n): X = hAPP(c_Nat_OSuc, X).
% 252.49/32.84 The goal is true when:
% 252.49/32.84 X = hAPP(hAPP(hAPP(c_COMBS(X, Y, Z), c_COMBB(W, V, U, c_Nat_OSuc)), hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Oone__class_Oone(tc_RealDef_Oreal))), hAPP(hAPP(c_COMBS(X, Y, Z), c_COMBB(W, V, U, c_Nat_OSuc)), hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Oone__class_Oone(tc_RealDef_Oreal))))
% 252.49/32.84
% 252.49/32.84 Proof:
% 252.49/32.84 hAPP(hAPP(hAPP(c_COMBS(X, Y, Z), c_COMBB(W, V, U, c_Nat_OSuc)), hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Oone__class_Oone(tc_RealDef_Oreal))), hAPP(hAPP(c_COMBS(X, Y, Z), c_COMBB(W, V, U, c_Nat_OSuc)), hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Oone__class_Oone(tc_RealDef_Oreal))))
% 252.49/32.84 = { by axiom 2 (help_c__COMBS__1) }
% 252.49/32.84 hAPP(hAPP(c_COMBB(W, V, U, c_Nat_OSuc), hAPP(hAPP(c_COMBS(X, Y, Z), c_COMBB(W, V, U, c_Nat_OSuc)), hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)))), hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)), hAPP(hAPP(c_COMBS(X, Y, Z), c_COMBB(W, V, U, c_Nat_OSuc)), hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)))))
% 252.49/32.84 = { by axiom 3 (help_c__COMBB__1) }
% 252.49/32.84 hAPP(c_Nat_OSuc, hAPP(hAPP(hAPP(c_COMBS(X, Y, Z), c_COMBB(W, V, U, c_Nat_OSuc)), hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Oone__class_Oone(tc_RealDef_Oreal))), hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)), hAPP(hAPP(c_COMBS(X, Y, Z), c_COMBB(W, V, U, c_Nat_OSuc)), hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Oone__class_Oone(tc_RealDef_Oreal))))))
% 252.49/32.84 = { by axiom 1 (fact_real__mult__1) }
% 252.49/32.84 hAPP(c_Nat_OSuc, hAPP(hAPP(hAPP(c_COMBS(X, Y, Z), c_COMBB(W, V, U, c_Nat_OSuc)), hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Oone__class_Oone(tc_RealDef_Oreal))), hAPP(hAPP(c_COMBS(X, Y, Z), c_COMBB(W, V, U, c_Nat_OSuc)), hAPP(c_Groups_Otimes__class_Otimes(tc_RealDef_Oreal), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)))))
% 252.49/32.84 % SZS output end Proof
% 252.49/32.84
% 252.49/32.84 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------