0.00/0.09 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.00/0.10 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof 0.09/0.30 % Computer : n003.cluster.edu 0.09/0.30 % Model : x86_64 x86_64 0.09/0.30 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.09/0.30 % Memory : 8042.1875MB 0.09/0.30 % OS : Linux 3.10.0-693.el7.x86_64 0.09/0.30 % CPULimit : 1200 0.09/0.30 % WCLimit : 120 0.09/0.30 % DateTime : Tue Jul 13 16:45:39 EDT 2021 0.09/0.30 % CPUTime : 172.30/22.04 % SZS status Theorem 172.30/22.04 172.30/22.08 % SZS output start Proof 172.30/22.08 Take the following subset of the input axioms: 172.30/22.08 fof(conj_0, conjecture, hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less(int), hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int))), zero_zero(int)))). 172.30/22.08 fof(fact_111_one__is__num__one, axiom, number_number_of(int, bit1(pls))=one_one(int)). 172.30/22.08 fof(fact_122_Pls__def, axiom, zero_zero(int)=pls). 172.30/22.09 fof(fact_22_number__of__is__id, axiom, ![K_1]: number_number_of(int, K_1)=ti(int, K_1)). 172.30/22.09 fof(fact_23_zmult__commute, axiom, ![Z_1, W]: hAPP(int, int, times_times(int, Z_1), W)=hAPP(int, int, times_times(int, W), Z_1)). 172.30/22.09 fof(fact_2__096_I4_A_K_Am_A_L_A1_J_A_K_At_A_060_A_I4_A_K_Am_A_L_A1_J_A_K_A0_096, axiom, hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less(int), hAPP(int, int, times_times(int, hAPP(int, int, plus_plus(int, hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m)), one_one(int))), t)), hAPP(int, int, times_times(int, hAPP(int, int, plus_plus(int, hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m)), one_one(int))), zero_zero(int))))). 172.30/22.09 fof(fact_3_t, axiom, hAPP(int, int, times_times(int, hAPP(int, int, plus_plus(int, hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m)), one_one(int))), t)=hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int))). 172.30/22.09 fof(fact_62_mult__Pls, axiom, ![W]: hAPP(int, int, times_times(int, pls), W)=pls). 172.30/22.09 fof(fact_63_mult__Bit0, axiom, ![K_1, L_1]: hAPP(int, int, times_times(int, bit0(K_1)), L_1)=bit0(hAPP(int, int, times_times(int, K_1), L_1))). 172.30/22.09 fof(fact_66_zmult__1, axiom, ![Z_1]: ti(int, Z_1)=hAPP(int, int, times_times(int, one_one(int)), Z_1)). 172.30/22.09 fof(tsy_c_Int_OBit0_arg1, hypothesis, ![B_1_1]: bit0(ti(int, B_1_1))=bit0(B_1_1)). 172.30/22.09 fof(tsy_c_Int_OBit0_res, hypothesis, ![B_1_1]: bit0(B_1_1)=ti(int, bit0(B_1_1))). 172.30/22.09 fof(tsy_c_Int_OBit1_res, hypothesis, ![B_1_1]: bit1(B_1_1)=ti(int, bit1(B_1_1))). 172.30/22.09 172.30/22.09 Now clausify the problem and encode Horn clauses using encoding 3 of 172.30/22.09 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 172.30/22.09 We repeatedly replace C & s=t => u=v by the two clauses: 172.30/22.09 fresh(y, y, x1...xn) = u 172.30/22.09 C => fresh(s, t, x1...xn) = v 172.30/22.09 where fresh is a fresh function symbol and x1..xn are the free 172.30/22.09 variables of u and v. 172.30/22.09 A predicate p(X) is encoded as p(X)=true (this is sound, because the 172.30/22.09 input problem has no model of domain size 1). 172.30/22.09 172.30/22.09 The encoding turns the above axioms into the following unit equations and goals: 172.30/22.09 172.30/22.09 Axiom 1 (fact_122_Pls__def): zero_zero(int) = pls. 172.30/22.09 Axiom 2 (fact_22_number__of__is__id): number_number_of(int, X) = ti(int, X). 172.30/22.09 Axiom 3 (fact_111_one__is__num__one): number_number_of(int, bit1(pls)) = one_one(int). 172.30/22.09 Axiom 4 (tsy_c_Int_OBit1_res): bit1(X) = ti(int, bit1(X)). 172.30/22.09 Axiom 5 (tsy_c_Int_OBit0_res): bit0(X) = ti(int, bit0(X)). 172.30/22.09 Axiom 6 (tsy_c_Int_OBit0_arg1): bit0(ti(int, X)) = bit0(X). 172.30/22.09 Axiom 7 (fact_113_pos__zmult__eq__1__iff): fresh927(X, X, Y, Z) = hAPP(int, int, times_times(int, Z), Y). 172.30/22.09 Axiom 8 (fact_23_zmult__commute): hAPP(int, int, times_times(int, X), Y) = hAPP(int, int, times_times(int, Y), X). 172.30/22.09 Axiom 9 (fact_62_mult__Pls): hAPP(int, int, times_times(int, pls), X) = pls. 172.30/22.09 Axiom 10 (fact_963_mod__pos__neg__trivial): fresh137(X, X, Y, Z) = hAPP(int, int, plus_plus(int, Z), Y). 172.30/22.09 Axiom 11 (fact_66_zmult__1): ti(int, X) = hAPP(int, int, times_times(int, one_one(int)), X). 172.30/22.09 Axiom 12 (fact_63_mult__Bit0): hAPP(int, int, times_times(int, bit0(X)), Y) = bit0(hAPP(int, int, times_times(int, X), Y)). 172.30/22.09 Axiom 13 (fact_104_power2__less__imp__less): fresh935(X, X, Y, Z, W) = hBOOL(hAPP(Y, bool, hAPP(Y, fun(Y, bool), ord_less(Y), Z), W)). 172.30/22.09 Axiom 14 (fact_149_less__number__of): fresh1395(X, X, Y, Z, W) = hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less(int), Z), W)). 172.30/22.09 Axiom 15 (fact_175_less__special_I3_J): fresh1231(X, X, Y, Z) = hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less(int), Z), pls)). 172.30/22.09 Axiom 16 (fact_3_t): hAPP(int, int, times_times(int, hAPP(int, int, plus_plus(int, hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m)), one_one(int))), t) = hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)). 172.30/22.09 Axiom 17 (fact_2__096_I4_A_K_Am_A_L_A1_J_A_K_At_A_060_A_I4_A_K_Am_A_L_A1_J_A_K_A0_096): hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less(int), hAPP(int, int, times_times(int, hAPP(int, int, plus_plus(int, hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m)), one_one(int))), t)), hAPP(int, int, times_times(int, hAPP(int, int, plus_plus(int, hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m)), one_one(int))), zero_zero(int)))) = true2. 172.30/22.09 172.30/22.09 Lemma 18: number_number_of(int, bit0(X)) = bit0(X). 172.30/22.09 Proof: 172.30/22.09 number_number_of(int, bit0(X)) 172.30/22.09 = { by axiom 2 (fact_22_number__of__is__id) } 172.30/22.09 ti(int, bit0(X)) 172.30/22.09 = { by axiom 5 (tsy_c_Int_OBit0_res) R->L } 172.30/22.09 bit0(X) 172.30/22.09 172.30/22.09 Lemma 19: fresh935(X, X, int, Y, Z) = fresh1395(W, W, V, Y, Z). 172.30/22.09 Proof: 172.30/22.09 fresh935(X, X, int, Y, Z) 172.30/22.09 = { by axiom 13 (fact_104_power2__less__imp__less) } 172.30/22.09 hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less(int), Y), Z)) 172.30/22.09 = { by axiom 14 (fact_149_less__number__of) R->L } 172.30/22.09 fresh1395(W, W, V, Y, Z) 172.30/22.09 172.30/22.09 Lemma 20: fresh927(X, X, Y, bit0(Z)) = bit0(fresh927(W, W, Y, Z)). 172.30/22.09 Proof: 172.30/22.09 fresh927(X, X, Y, bit0(Z)) 172.30/22.09 = { by axiom 7 (fact_113_pos__zmult__eq__1__iff) } 172.30/22.09 hAPP(int, int, times_times(int, bit0(Z)), Y) 172.30/22.09 = { by axiom 12 (fact_63_mult__Bit0) } 172.30/22.09 bit0(hAPP(int, int, times_times(int, Z), Y)) 172.30/22.09 = { by axiom 7 (fact_113_pos__zmult__eq__1__iff) R->L } 172.30/22.09 bit0(fresh927(W, W, Y, Z)) 172.30/22.09 172.30/22.09 Lemma 21: fresh1395(X, X, Y, Z, pls) = fresh1231(W, W, V, Z). 172.30/22.09 Proof: 172.30/22.09 fresh1395(X, X, Y, Z, pls) 172.30/22.09 = { by lemma 19 R->L } 172.30/22.09 fresh935(U, U, int, Z, pls) 172.30/22.09 = { by axiom 13 (fact_104_power2__less__imp__less) } 172.30/22.09 hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less(int), Z), pls)) 172.30/22.09 = { by axiom 15 (fact_175_less__special_I3_J) R->L } 172.30/22.09 fresh1231(W, W, V, Z) 172.30/22.09 172.30/22.09 Goal 1 (conj_0): hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less(int), hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int))), zero_zero(int))) = true2. 172.30/22.09 Proof: 172.30/22.09 hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less(int), hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int))), zero_zero(int))) 172.30/22.09 = { by axiom 13 (fact_104_power2__less__imp__less) R->L } 172.30/22.09 fresh935(X, X, int, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), zero_zero(int)) 172.30/22.09 = { by lemma 19 } 172.30/22.09 fresh1395(Y, Y, Z, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), zero_zero(int)) 172.30/22.09 = { by axiom 10 (fact_963_mod__pos__neg__trivial) R->L } 172.30/22.09 fresh1395(Y, Y, Z, fresh137(W, W, one_one(int), hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), zero_zero(int)) 172.30/22.09 = { by axiom 1 (fact_122_Pls__def) } 172.30/22.09 fresh1395(Y, Y, Z, fresh137(W, W, one_one(int), hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), pls) 172.30/22.09 = { by lemma 21 } 172.30/22.09 fresh1231(V, V, U, fresh137(W, W, one_one(int), hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls)))))) 172.30/22.09 = { by axiom 10 (fact_963_mod__pos__neg__trivial) } 172.30/22.09 fresh1231(V, V, U, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int))) 172.30/22.09 = { by lemma 21 R->L } 172.30/22.09 fresh1395(T, T, S, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), pls) 172.30/22.09 = { by axiom 9 (fact_62_mult__Pls) R->L } 172.30/22.09 fresh1395(T, T, S, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), hAPP(int, int, times_times(int, pls), fresh137(X2, X2, one_one(int), bit0(bit0(m))))) 172.30/22.09 = { by axiom 8 (fact_23_zmult__commute) R->L } 172.30/22.09 fresh1395(T, T, S, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), hAPP(int, int, times_times(int, fresh137(X2, X2, one_one(int), bit0(bit0(m)))), pls)) 172.30/22.09 = { by axiom 7 (fact_113_pos__zmult__eq__1__iff) R->L } 172.30/22.09 fresh1395(T, T, S, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), fresh927(Y2, Y2, pls, fresh137(X2, X2, one_one(int), bit0(bit0(m))))) 172.30/22.09 = { by axiom 6 (tsy_c_Int_OBit0_arg1) R->L } 172.30/22.09 fresh1395(T, T, S, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), fresh927(Y2, Y2, pls, fresh137(X2, X2, one_one(int), bit0(bit0(ti(int, m)))))) 172.30/22.09 = { by axiom 11 (fact_66_zmult__1) } 172.30/22.09 fresh1395(T, T, S, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), fresh927(Y2, Y2, pls, fresh137(X2, X2, one_one(int), bit0(bit0(hAPP(int, int, times_times(int, one_one(int)), m)))))) 172.30/22.09 = { by axiom 7 (fact_113_pos__zmult__eq__1__iff) R->L } 172.30/22.09 fresh1395(T, T, S, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), fresh927(Y2, Y2, pls, fresh137(X2, X2, one_one(int), bit0(bit0(fresh927(Z2, Z2, m, one_one(int))))))) 172.30/22.09 = { by axiom 3 (fact_111_one__is__num__one) R->L } 172.30/22.09 fresh1395(T, T, S, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), fresh927(Y2, Y2, pls, fresh137(X2, X2, one_one(int), bit0(bit0(fresh927(Z2, Z2, m, number_number_of(int, bit1(pls)))))))) 172.30/22.09 = { by axiom 2 (fact_22_number__of__is__id) } 172.30/22.09 fresh1395(T, T, S, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), fresh927(Y2, Y2, pls, fresh137(X2, X2, one_one(int), bit0(bit0(fresh927(Z2, Z2, m, ti(int, bit1(pls)))))))) 172.30/22.10 = { by axiom 4 (tsy_c_Int_OBit1_res) R->L } 172.30/22.10 fresh1395(T, T, S, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), fresh927(Y2, Y2, pls, fresh137(X2, X2, one_one(int), bit0(bit0(fresh927(Z2, Z2, m, bit1(pls))))))) 172.30/22.10 = { by axiom 1 (fact_122_Pls__def) R->L } 172.30/22.10 fresh1395(T, T, S, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), fresh927(Y2, Y2, zero_zero(int), fresh137(X2, X2, one_one(int), bit0(bit0(fresh927(Z2, Z2, m, bit1(pls))))))) 172.30/22.10 = { by axiom 7 (fact_113_pos__zmult__eq__1__iff) } 172.30/22.10 fresh1395(T, T, S, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), hAPP(int, int, times_times(int, fresh137(X2, X2, one_one(int), bit0(bit0(fresh927(Z2, Z2, m, bit1(pls)))))), zero_zero(int))) 172.30/22.10 = { by axiom 16 (fact_3_t) R->L } 172.30/22.10 fresh1395(T, T, S, hAPP(int, int, times_times(int, hAPP(int, int, plus_plus(int, hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m)), one_one(int))), t), hAPP(int, int, times_times(int, fresh137(X2, X2, one_one(int), bit0(bit0(fresh927(Z2, Z2, m, bit1(pls)))))), zero_zero(int))) 172.30/22.10 = { by axiom 7 (fact_113_pos__zmult__eq__1__iff) R->L } 172.30/22.10 fresh1395(T, T, S, fresh927(W2, W2, t, hAPP(int, int, plus_plus(int, hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m)), one_one(int))), hAPP(int, int, times_times(int, fresh137(X2, X2, one_one(int), bit0(bit0(fresh927(Z2, Z2, m, bit1(pls)))))), zero_zero(int))) 172.30/22.10 = { by axiom 10 (fact_963_mod__pos__neg__trivial) R->L } 172.30/22.10 fresh1395(T, T, S, fresh927(W2, W2, t, fresh137(V2, V2, one_one(int), hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m))), hAPP(int, int, times_times(int, fresh137(X2, X2, one_one(int), bit0(bit0(fresh927(Z2, Z2, m, bit1(pls)))))), zero_zero(int))) 172.30/22.10 = { by axiom 7 (fact_113_pos__zmult__eq__1__iff) R->L } 172.30/22.10 fresh1395(T, T, S, fresh927(W2, W2, t, fresh137(V2, V2, one_one(int), fresh927(U2, U2, m, number_number_of(int, bit0(bit0(bit1(pls))))))), hAPP(int, int, times_times(int, fresh137(X2, X2, one_one(int), bit0(bit0(fresh927(Z2, Z2, m, bit1(pls)))))), zero_zero(int))) 172.30/22.10 = { by lemma 18 } 172.30/22.10 fresh1395(T, T, S, fresh927(W2, W2, t, fresh137(V2, V2, one_one(int), fresh927(U2, U2, m, bit0(bit0(bit1(pls)))))), hAPP(int, int, times_times(int, fresh137(X2, X2, one_one(int), bit0(bit0(fresh927(Z2, Z2, m, bit1(pls)))))), zero_zero(int))) 172.30/22.10 = { by lemma 20 } 172.30/22.10 fresh1395(T, T, S, fresh927(W2, W2, t, fresh137(V2, V2, one_one(int), bit0(fresh927(T2, T2, m, bit0(bit1(pls)))))), hAPP(int, int, times_times(int, fresh137(X2, X2, one_one(int), bit0(bit0(fresh927(Z2, Z2, m, bit1(pls)))))), zero_zero(int))) 172.30/22.10 = { by lemma 20 R->L } 172.30/22.10 fresh1395(T, T, S, fresh927(W2, W2, t, fresh137(V2, V2, one_one(int), bit0(fresh927(T2, T2, m, bit0(bit1(pls)))))), hAPP(int, int, times_times(int, fresh137(X2, X2, one_one(int), bit0(fresh927(S2, S2, m, bit0(bit1(pls)))))), zero_zero(int))) 172.30/22.10 = { by lemma 20 R->L } 172.30/22.10 fresh1395(T, T, S, fresh927(W2, W2, t, fresh137(V2, V2, one_one(int), fresh927(X3, X3, m, bit0(bit0(bit1(pls)))))), hAPP(int, int, times_times(int, fresh137(X2, X2, one_one(int), bit0(fresh927(S2, S2, m, bit0(bit1(pls)))))), zero_zero(int))) 172.30/22.10 = { by lemma 20 R->L } 172.30/22.10 fresh1395(T, T, S, fresh927(W2, W2, t, fresh137(V2, V2, one_one(int), fresh927(X3, X3, m, bit0(bit0(bit1(pls)))))), hAPP(int, int, times_times(int, fresh137(X2, X2, one_one(int), fresh927(Y3, Y3, m, bit0(bit0(bit1(pls)))))), zero_zero(int))) 172.30/22.10 = { by lemma 18 R->L } 172.30/22.10 fresh1395(T, T, S, fresh927(W2, W2, t, fresh137(V2, V2, one_one(int), fresh927(X3, X3, m, number_number_of(int, bit0(bit0(bit1(pls))))))), hAPP(int, int, times_times(int, fresh137(X2, X2, one_one(int), fresh927(Y3, Y3, m, bit0(bit0(bit1(pls)))))), zero_zero(int))) 172.30/22.10 = { by lemma 18 R->L } 172.30/22.10 fresh1395(T, T, S, fresh927(W2, W2, t, fresh137(V2, V2, one_one(int), fresh927(X3, X3, m, number_number_of(int, bit0(bit0(bit1(pls))))))), hAPP(int, int, times_times(int, fresh137(X2, X2, one_one(int), fresh927(Y3, Y3, m, number_number_of(int, bit0(bit0(bit1(pls))))))), zero_zero(int))) 172.30/22.10 = { by axiom 7 (fact_113_pos__zmult__eq__1__iff) } 172.30/22.10 fresh1395(T, T, S, fresh927(W2, W2, t, fresh137(V2, V2, one_one(int), hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m))), hAPP(int, int, times_times(int, fresh137(X2, X2, one_one(int), fresh927(Y3, Y3, m, number_number_of(int, bit0(bit0(bit1(pls))))))), zero_zero(int))) 172.30/22.10 = { by axiom 7 (fact_113_pos__zmult__eq__1__iff) } 172.30/22.10 fresh1395(T, T, S, fresh927(W2, W2, t, fresh137(V2, V2, one_one(int), hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m))), hAPP(int, int, times_times(int, fresh137(X2, X2, one_one(int), hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m))), zero_zero(int))) 172.30/22.10 = { by axiom 10 (fact_963_mod__pos__neg__trivial) } 172.30/22.10 fresh1395(T, T, S, fresh927(W2, W2, t, hAPP(int, int, plus_plus(int, hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m)), one_one(int))), hAPP(int, int, times_times(int, fresh137(X2, X2, one_one(int), hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m))), zero_zero(int))) 172.30/22.10 = { by axiom 10 (fact_963_mod__pos__neg__trivial) } 172.30/22.10 fresh1395(T, T, S, fresh927(W2, W2, t, hAPP(int, int, plus_plus(int, hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m)), one_one(int))), hAPP(int, int, times_times(int, hAPP(int, int, plus_plus(int, hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m)), one_one(int))), zero_zero(int))) 172.30/22.10 = { by axiom 7 (fact_113_pos__zmult__eq__1__iff) } 172.30/22.10 fresh1395(T, T, S, hAPP(int, int, times_times(int, hAPP(int, int, plus_plus(int, hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m)), one_one(int))), t), hAPP(int, int, times_times(int, hAPP(int, int, plus_plus(int, hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m)), one_one(int))), zero_zero(int))) 172.30/22.10 = { by lemma 19 R->L } 172.30/22.10 fresh935(Z3, Z3, int, hAPP(int, int, times_times(int, hAPP(int, int, plus_plus(int, hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m)), one_one(int))), t), hAPP(int, int, times_times(int, hAPP(int, int, plus_plus(int, hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m)), one_one(int))), zero_zero(int))) 172.30/22.10 = { by axiom 13 (fact_104_power2__less__imp__less) } 172.30/22.10 hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less(int), hAPP(int, int, times_times(int, hAPP(int, int, plus_plus(int, hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m)), one_one(int))), t)), hAPP(int, int, times_times(int, hAPP(int, int, plus_plus(int, hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m)), one_one(int))), zero_zero(int)))) 172.30/22.10 = { by axiom 17 (fact_2__096_I4_A_K_Am_A_L_A1_J_A_K_At_A_060_A_I4_A_K_Am_A_L_A1_J_A_K_A0_096) } 172.30/22.10 true2 172.30/22.10 % SZS output end Proof 172.30/22.10 172.30/22.10 RESULT: Theorem (the conjecture is true). 172.84/22.19 EOF