TSTP Solution File: NUM924+7 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : NUM924+7 : TPTP v8.1.2. Released v5.3.0.
% 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 : n014.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 : Thu Aug 31 11:58:57 EDT 2023
% Result : Theorem 141.95s 18.86s
% Output : Proof 144.48s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : NUM924+7 : TPTP v8.1.2. Released v5.3.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.16/0.35 % Computer : n014.cluster.edu
% 0.16/0.35 % Model : x86_64 x86_64
% 0.16/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.35 % Memory : 8042.1875MB
% 0.16/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.35 % CPULimit : 300
% 0.16/0.35 % WCLimit : 300
% 0.16/0.35 % DateTime : Fri Aug 25 09:27:18 EDT 2023
% 0.16/0.35 % CPUTime :
% 141.95/18.86 Command-line arguments: --flip-ordering --lhs-weight 1 --depth-weight 60 --distributivity-heuristic
% 141.95/18.86
% 141.95/18.86 % SZS status Theorem
% 141.95/18.86
% 141.95/18.93 % SZS output start Proof
% 141.95/18.93 Take the following subset of the input axioms:
% 141.95/18.93 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)))).
% 141.95/18.93 fof(fact_111_one__is__num__one, axiom, one_one(int)=number_number_of(int, bit1(pls))).
% 141.95/18.93 fof(fact_122_Pls__def, axiom, pls=zero_zero(int)).
% 141.95/18.93 fof(fact_22_number__of__is__id, axiom, ![K_1]: number_number_of(int, K_1)=ti(int, K_1)).
% 141.95/18.93 fof(fact_23_zmult__commute, axiom, ![W, Z_1]: hAPP(int, int, times_times(int, Z_1), W)=hAPP(int, int, times_times(int, W), Z_1)).
% 141.95/18.93 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))))).
% 141.95/18.93 fof(fact_3_t, axiom, 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, 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)).
% 141.95/18.93 fof(fact_62_mult__Pls, axiom, ![W2]: hAPP(int, int, times_times(int, pls), W2)=pls).
% 141.95/18.93 fof(fact_63_mult__Bit0, axiom, ![L_1, K_1_2]: hAPP(int, int, times_times(int, bit0(K_1_2)), L_1)=bit0(hAPP(int, int, times_times(int, K_1_2), L_1))).
% 141.95/18.93 fof(fact_66_zmult__1, axiom, ![Z_1_2]: hAPP(int, int, times_times(int, one_one(int)), Z_1_2)=ti(int, Z_1_2)).
% 141.95/18.93 fof(tsy_c_Int_OBit0_arg1, hypothesis, ![B_1_1]: bit0(ti(int, B_1_1))=bit0(B_1_1)).
% 141.95/18.93 fof(tsy_c_Int_OBit0_res, hypothesis, ![B_1_1_2]: ti(int, bit0(B_1_1_2))=bit0(B_1_1_2)).
% 141.95/18.93 fof(tsy_c_Int_OBit1_res, hypothesis, ![B_1_1_2]: ti(int, bit1(B_1_1_2))=bit1(B_1_1_2)).
% 141.95/18.93
% 141.95/18.93 Now clausify the problem and encode Horn clauses using encoding 3 of
% 141.95/18.93 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 141.95/18.93 We repeatedly replace C & s=t => u=v by the two clauses:
% 141.95/18.93 fresh(y, y, x1...xn) = u
% 141.95/18.93 C => fresh(s, t, x1...xn) = v
% 141.95/18.93 where fresh is a fresh function symbol and x1..xn are the free
% 141.95/18.93 variables of u and v.
% 141.95/18.93 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 141.95/18.93 input problem has no model of domain size 1).
% 141.95/18.93
% 141.95/18.93 The encoding turns the above axioms into the following unit equations and goals:
% 141.95/18.93
% 141.95/18.93 Axiom 1 (fact_122_Pls__def): pls = zero_zero(int).
% 141.95/18.93 Axiom 2 (fact_22_number__of__is__id): number_number_of(int, X) = ti(int, X).
% 141.95/18.93 Axiom 3 (fact_111_one__is__num__one): one_one(int) = number_number_of(int, bit1(pls)).
% 141.95/18.93 Axiom 4 (tsy_c_Int_OBit1_res): ti(int, bit1(X)) = bit1(X).
% 141.95/18.93 Axiom 5 (tsy_c_Int_OBit0_res): ti(int, bit0(X)) = bit0(X).
% 141.95/18.93 Axiom 6 (tsy_c_Int_OBit0_arg1): bit0(ti(int, X)) = bit0(X).
% 141.95/18.93 Axiom 7 (fact_317_mult__eq__0__iff): fresh735(X, X, Y, Z, W) = hAPP(Y, Y, times_times(Y, Z), W).
% 141.95/18.93 Axiom 8 (fact_219_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J): fresh826(X, X, Y, Z, W) = hAPP(Y, Y, times_times(Y, W), Z).
% 141.95/18.93 Axiom 9 (fact_238_add__0__iff_1): fresh804(X, X, Y, Z, W) = hAPP(Y, Y, plus_plus(Y, Z), W).
% 141.95/18.93 Axiom 10 (fact_801_add__nonneg__eq__0__iff): fresh1086(X, X, Y, Z, W) = hAPP(Y, Y, plus_plus(Y, W), Z).
% 141.95/18.93 Axiom 11 (fact_23_zmult__commute): hAPP(int, int, times_times(int, X), Y) = hAPP(int, int, times_times(int, Y), X).
% 141.95/18.93 Axiom 12 (fact_304_zmult__eq__1__iff): fresh748(X, X, Y, Z) = hAPP(int, int, times_times(int, Y), Z).
% 141.95/18.93 Axiom 13 (fact_113_pos__zmult__eq__1__iff): fresh929(X, X, Y, Z) = hAPP(int, int, times_times(int, Z), Y).
% 141.95/18.93 Axiom 14 (fact_62_mult__Pls): hAPP(int, int, times_times(int, pls), X) = pls.
% 141.95/18.93 Axiom 15 (fact_963_mod__pos__neg__trivial): fresh135(X, X, Y, Z) = hAPP(int, int, plus_plus(int, Z), Y).
% 141.95/18.94 Axiom 16 (fact_715_realpow__minus__mult): fresh1198(X, X, Y, Z, W) = hAPP(nat, Y, power_power(Y, Z), W).
% 141.95/18.94 Axiom 17 (fact_66_zmult__1): hAPP(int, int, times_times(int, one_one(int)), X) = ti(int, X).
% 141.95/18.94 Axiom 18 (fact_63_mult__Bit0): hAPP(int, int, times_times(int, bit0(X)), Y) = bit0(hAPP(int, int, times_times(int, X), Y)).
% 141.95/18.94 Axiom 19 (fact_149_less__number__of): fresh1666(X, X, Y, Z, W) = hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less(int), Z), W)).
% 141.95/18.94 Axiom 20 (fact_175_less__special_I3_J): fresh1660(X, X, Y, Z) = hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less(int), Z), pls)).
% 141.95/18.94 Axiom 21 (fact_3_t): 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, 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).
% 141.95/18.94 Axiom 22 (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.
% 141.95/18.94
% 141.95/18.94 Lemma 23: number_number_of(int, bit0(X)) = bit0(X).
% 141.95/18.94 Proof:
% 141.95/18.94 number_number_of(int, bit0(X))
% 141.95/18.94 = { by axiom 2 (fact_22_number__of__is__id) }
% 141.95/18.94 ti(int, bit0(X))
% 141.95/18.94 = { by axiom 5 (tsy_c_Int_OBit0_res) }
% 141.95/18.94 bit0(X)
% 141.95/18.94
% 141.95/18.94 Lemma 24: fresh735(X, X, Y, Z, W) = fresh826(V, V, Y, W, Z).
% 141.95/18.94 Proof:
% 141.95/18.94 fresh735(X, X, Y, Z, W)
% 141.95/18.94 = { by axiom 7 (fact_317_mult__eq__0__iff) }
% 141.95/18.94 hAPP(Y, Y, times_times(Y, Z), W)
% 141.95/18.94 = { by axiom 8 (fact_219_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J) R->L }
% 141.95/18.94 fresh826(V, V, Y, W, Z)
% 141.95/18.94
% 141.95/18.94 Lemma 25: fresh826(X, X, int, Y, Z) = fresh748(W, W, Z, Y).
% 141.95/18.94 Proof:
% 141.95/18.94 fresh826(X, X, int, Y, Z)
% 141.95/18.94 = { by lemma 24 R->L }
% 141.95/18.94 fresh735(V, V, int, Z, Y)
% 141.95/18.94 = { by axiom 7 (fact_317_mult__eq__0__iff) }
% 141.95/18.94 hAPP(int, int, times_times(int, Z), Y)
% 141.95/18.94 = { by axiom 12 (fact_304_zmult__eq__1__iff) R->L }
% 141.95/18.94 fresh748(W, W, Z, Y)
% 141.95/18.94
% 141.95/18.94 Lemma 26: fresh748(X, X, Y, Z) = fresh929(W, W, Z, Y).
% 141.95/18.94 Proof:
% 141.95/18.94 fresh748(X, X, Y, Z)
% 141.95/18.94 = { by lemma 25 R->L }
% 141.95/18.94 fresh826(V, V, int, Z, Y)
% 141.95/18.94 = { by lemma 24 R->L }
% 141.95/18.94 fresh735(U, U, int, Y, Z)
% 141.95/18.94 = { by axiom 7 (fact_317_mult__eq__0__iff) }
% 141.95/18.94 hAPP(int, int, times_times(int, Y), Z)
% 141.95/18.94 = { by axiom 13 (fact_113_pos__zmult__eq__1__iff) R->L }
% 141.95/18.94 fresh929(W, W, Z, Y)
% 141.95/18.94
% 141.95/18.94 Lemma 27: fresh804(X, X, Y, Z, W) = fresh1086(V, V, Y, W, Z).
% 141.95/18.94 Proof:
% 141.95/18.94 fresh804(X, X, Y, Z, W)
% 141.95/18.94 = { by axiom 9 (fact_238_add__0__iff_1) }
% 141.95/18.94 hAPP(Y, Y, plus_plus(Y, Z), W)
% 141.95/18.94 = { by axiom 10 (fact_801_add__nonneg__eq__0__iff) R->L }
% 141.95/18.94 fresh1086(V, V, Y, W, Z)
% 141.95/18.94
% 141.95/18.94 Lemma 28: fresh929(X, X, Y, bit0(Z)) = bit0(fresh929(W, W, Y, Z)).
% 141.95/18.94 Proof:
% 141.95/18.94 fresh929(X, X, Y, bit0(Z))
% 141.95/18.94 = { by lemma 26 R->L }
% 141.95/18.94 fresh748(V, V, bit0(Z), Y)
% 141.95/18.94 = { by lemma 25 R->L }
% 141.95/18.94 fresh826(U, U, int, Y, bit0(Z))
% 141.95/18.94 = { by lemma 24 R->L }
% 141.95/18.94 fresh735(T, T, int, bit0(Z), Y)
% 141.95/18.94 = { by axiom 7 (fact_317_mult__eq__0__iff) }
% 141.95/18.94 hAPP(int, int, times_times(int, bit0(Z)), Y)
% 141.95/18.94 = { by axiom 18 (fact_63_mult__Bit0) }
% 141.95/18.94 bit0(hAPP(int, int, times_times(int, Z), Y))
% 141.95/18.94 = { by axiom 7 (fact_317_mult__eq__0__iff) R->L }
% 141.95/18.94 bit0(fresh735(S, S, int, Z, Y))
% 141.95/18.94 = { by lemma 24 }
% 141.95/18.94 bit0(fresh826(X2, X2, int, Y, Z))
% 141.95/18.94 = { by lemma 25 }
% 141.95/18.94 bit0(fresh748(Y2, Y2, Z, Y))
% 141.95/18.94 = { by lemma 26 }
% 141.95/18.94 bit0(fresh929(W, W, Y, Z))
% 141.95/18.94
% 141.95/18.94 Lemma 29: fresh1666(X, X, Y, Z, pls) = fresh1660(W, W, V, Z).
% 141.95/18.94 Proof:
% 141.95/18.94 fresh1666(X, X, Y, Z, pls)
% 141.95/18.94 = { by axiom 19 (fact_149_less__number__of) }
% 141.95/18.94 hBOOL(hAPP(int, bool, hAPP(int, fun(int, bool), ord_less(int), Z), pls))
% 141.95/18.94 = { by axiom 20 (fact_175_less__special_I3_J) R->L }
% 141.95/18.94 fresh1660(W, W, V, Z)
% 141.95/18.94
% 141.95/18.94 Lemma 30: fresh1086(X, X, int, Y, Z) = fresh135(W, W, Y, Z).
% 141.95/18.94 Proof:
% 141.95/18.94 fresh1086(X, X, int, Y, Z)
% 141.95/18.94 = { by lemma 27 R->L }
% 141.95/18.94 fresh804(V, V, int, Z, Y)
% 141.95/18.94 = { by axiom 9 (fact_238_add__0__iff_1) }
% 141.95/18.94 hAPP(int, int, plus_plus(int, Z), Y)
% 141.95/18.94 = { by axiom 15 (fact_963_mod__pos__neg__trivial) R->L }
% 144.48/18.94 fresh135(W, W, Y, Z)
% 144.48/18.94
% 144.48/18.94 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.
% 144.48/18.94 Proof:
% 144.48/18.94 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)))
% 144.48/18.94 = { by axiom 19 (fact_149_less__number__of) R->L }
% 144.48/18.94 fresh1666(X, X, Y, 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))
% 144.48/18.94 = { by axiom 9 (fact_238_add__0__iff_1) R->L }
% 144.48/18.94 fresh1666(X, X, Y, fresh804(Z, Z, int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls)))), one_one(int)), zero_zero(int))
% 144.48/18.94 = { by lemma 27 }
% 144.48/18.94 fresh1666(X, X, Y, fresh1086(W, W, int, one_one(int), hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), zero_zero(int))
% 144.48/18.94 = { by lemma 30 }
% 144.48/18.94 fresh1666(X, X, Y, fresh135(V, V, one_one(int), hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), zero_zero(int))
% 144.48/18.94 = { by axiom 16 (fact_715_realpow__minus__mult) R->L }
% 144.48/18.94 fresh1666(X, X, Y, fresh135(V, V, one_one(int), fresh1198(U, U, int, s, number_number_of(nat, bit0(bit1(pls))))), zero_zero(int))
% 144.48/18.94 = { by axiom 1 (fact_122_Pls__def) R->L }
% 144.48/18.94 fresh1666(X, X, Y, fresh135(V, V, one_one(int), fresh1198(U, U, int, s, number_number_of(nat, bit0(bit1(pls))))), pls)
% 144.48/18.94 = { by lemma 29 }
% 144.48/18.94 fresh1660(T, T, S, fresh135(V, V, one_one(int), fresh1198(U, U, int, s, number_number_of(nat, bit0(bit1(pls))))))
% 144.48/18.94 = { by axiom 16 (fact_715_realpow__minus__mult) }
% 144.48/18.94 fresh1660(T, T, S, fresh135(V, V, one_one(int), hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))))
% 144.48/18.94 = { by lemma 30 R->L }
% 144.48/18.94 fresh1660(T, T, S, fresh1086(X2, X2, int, one_one(int), hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))))
% 144.48/18.94 = { by lemma 27 R->L }
% 144.48/18.94 fresh1660(T, T, S, fresh804(Y2, Y2, int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls)))), one_one(int)))
% 144.48/18.94 = { by axiom 9 (fact_238_add__0__iff_1) }
% 144.48/18.94 fresh1660(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)))
% 144.48/18.94 = { by lemma 29 R->L }
% 144.48/18.94 fresh1666(Z2, Z2, W2, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), pls)
% 144.48/18.94 = { by axiom 14 (fact_62_mult__Pls) R->L }
% 144.48/18.94 fresh1666(Z2, Z2, W2, 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), fresh135(V2, V2, one_one(int), bit0(bit0(m)))))
% 144.48/18.94 = { by axiom 11 (fact_23_zmult__commute) R->L }
% 144.48/18.94 fresh1666(Z2, Z2, W2, 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, fresh135(V2, V2, one_one(int), bit0(bit0(m)))), pls))
% 144.48/18.94 = { by axiom 7 (fact_317_mult__eq__0__iff) R->L }
% 144.48/18.94 fresh1666(Z2, Z2, W2, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), fresh735(U2, U2, int, fresh135(V2, V2, one_one(int), bit0(bit0(m))), pls))
% 144.48/18.94 = { by lemma 24 }
% 144.48/18.94 fresh1666(Z2, Z2, W2, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), fresh826(T2, T2, int, pls, fresh135(V2, V2, one_one(int), bit0(bit0(m)))))
% 144.48/18.94 = { by lemma 25 }
% 144.48/18.94 fresh1666(Z2, Z2, W2, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), fresh748(S2, S2, fresh135(V2, V2, one_one(int), bit0(bit0(m))), pls))
% 144.48/18.94 = { by lemma 26 }
% 144.48/18.94 fresh1666(Z2, Z2, W2, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), fresh929(X3, X3, pls, fresh135(V2, V2, one_one(int), bit0(bit0(m)))))
% 144.48/18.94 = { by axiom 6 (tsy_c_Int_OBit0_arg1) R->L }
% 144.48/18.94 fresh1666(Z2, Z2, W2, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), fresh929(X3, X3, pls, fresh135(V2, V2, one_one(int), bit0(bit0(ti(int, m))))))
% 144.48/18.94 = { by axiom 17 (fact_66_zmult__1) R->L }
% 144.48/18.94 fresh1666(Z2, Z2, W2, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), fresh929(X3, X3, pls, fresh135(V2, V2, one_one(int), bit0(bit0(hAPP(int, int, times_times(int, one_one(int)), m))))))
% 144.48/18.94 = { by axiom 7 (fact_317_mult__eq__0__iff) R->L }
% 144.48/18.94 fresh1666(Z2, Z2, W2, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), fresh929(X3, X3, pls, fresh135(V2, V2, one_one(int), bit0(bit0(fresh735(Y3, Y3, int, one_one(int), m))))))
% 144.48/18.94 = { by lemma 24 }
% 144.48/18.94 fresh1666(Z2, Z2, W2, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), fresh929(X3, X3, pls, fresh135(V2, V2, one_one(int), bit0(bit0(fresh826(Z3, Z3, int, m, one_one(int)))))))
% 144.48/18.94 = { by lemma 25 }
% 144.48/18.94 fresh1666(Z2, Z2, W2, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), fresh929(X3, X3, pls, fresh135(V2, V2, one_one(int), bit0(bit0(fresh748(W3, W3, one_one(int), m))))))
% 144.48/18.95 = { by lemma 26 }
% 144.48/18.95 fresh1666(Z2, Z2, W2, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), fresh929(X3, X3, pls, fresh135(V2, V2, one_one(int), bit0(bit0(fresh929(V3, V3, m, one_one(int)))))))
% 144.48/18.95 = { by axiom 3 (fact_111_one__is__num__one) }
% 144.48/18.95 fresh1666(Z2, Z2, W2, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), fresh929(X3, X3, pls, fresh135(V2, V2, one_one(int), bit0(bit0(fresh929(V3, V3, m, number_number_of(int, bit1(pls))))))))
% 144.48/18.95 = { by axiom 2 (fact_22_number__of__is__id) }
% 144.48/18.95 fresh1666(Z2, Z2, W2, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), fresh929(X3, X3, pls, fresh135(V2, V2, one_one(int), bit0(bit0(fresh929(V3, V3, m, ti(int, bit1(pls))))))))
% 144.48/18.95 = { by axiom 4 (tsy_c_Int_OBit1_res) }
% 144.48/18.95 fresh1666(Z2, Z2, W2, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), fresh929(X3, X3, pls, fresh135(V2, V2, one_one(int), bit0(bit0(fresh929(V3, V3, m, bit1(pls)))))))
% 144.48/18.95 = { by axiom 1 (fact_122_Pls__def) }
% 144.48/18.95 fresh1666(Z2, Z2, W2, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), fresh929(X3, X3, zero_zero(int), fresh135(V2, V2, one_one(int), bit0(bit0(fresh929(V3, V3, m, bit1(pls)))))))
% 144.48/18.95 = { by lemma 26 R->L }
% 144.48/18.95 fresh1666(Z2, Z2, W2, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), fresh748(U3, U3, fresh135(V2, V2, one_one(int), bit0(bit0(fresh929(V3, V3, m, bit1(pls))))), zero_zero(int)))
% 144.48/18.95 = { by lemma 25 R->L }
% 144.48/18.95 fresh1666(Z2, Z2, W2, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), fresh826(T3, T3, int, zero_zero(int), fresh135(V2, V2, one_one(int), bit0(bit0(fresh929(V3, V3, m, bit1(pls)))))))
% 144.48/18.95 = { by lemma 24 R->L }
% 144.48/18.95 fresh1666(Z2, Z2, W2, hAPP(int, int, plus_plus(int, hAPP(nat, int, power_power(int, s), number_number_of(nat, bit0(bit1(pls))))), one_one(int)), fresh735(S3, S3, int, fresh135(V2, V2, one_one(int), bit0(bit0(fresh929(V3, V3, m, bit1(pls))))), zero_zero(int)))
% 144.48/18.95 = { by axiom 7 (fact_317_mult__eq__0__iff) }
% 144.48/18.95 fresh1666(Z2, Z2, W2, 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, fresh135(V2, V2, one_one(int), bit0(bit0(fresh929(V3, V3, m, bit1(pls)))))), zero_zero(int)))
% 144.48/18.95 = { by axiom 21 (fact_3_t) }
% 144.48/18.95 fresh1666(Z2, Z2, W2, 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, fresh135(V2, V2, one_one(int), bit0(bit0(fresh929(V3, V3, m, bit1(pls)))))), zero_zero(int)))
% 144.48/18.95 = { by axiom 7 (fact_317_mult__eq__0__iff) R->L }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh735(X4, X4, 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, fresh135(V2, V2, one_one(int), bit0(bit0(fresh929(V3, V3, m, bit1(pls)))))), zero_zero(int)))
% 144.48/18.95 = { by lemma 24 }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh826(Y4, Y4, int, 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, fresh135(V2, V2, one_one(int), bit0(bit0(fresh929(V3, V3, m, bit1(pls)))))), zero_zero(int)))
% 144.48/18.95 = { by lemma 25 }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh748(Z4, Z4, 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, fresh135(V2, V2, one_one(int), bit0(bit0(fresh929(V3, V3, m, bit1(pls)))))), zero_zero(int)))
% 144.48/18.95 = { by lemma 26 }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, 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, fresh135(V2, V2, one_one(int), bit0(bit0(fresh929(V3, V3, m, bit1(pls)))))), zero_zero(int)))
% 144.48/18.95 = { by axiom 9 (fact_238_add__0__iff_1) R->L }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, t, fresh804(V4, V4, 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, fresh135(V2, V2, one_one(int), bit0(bit0(fresh929(V3, V3, m, bit1(pls)))))), zero_zero(int)))
% 144.48/18.95 = { by lemma 27 }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, t, fresh1086(U4, U4, int, one_one(int), hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m))), hAPP(int, int, times_times(int, fresh135(V2, V2, one_one(int), bit0(bit0(fresh929(V3, V3, m, bit1(pls)))))), zero_zero(int)))
% 144.48/18.95 = { by lemma 30 }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, t, fresh135(T4, T4, one_one(int), hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m))), hAPP(int, int, times_times(int, fresh135(V2, V2, one_one(int), bit0(bit0(fresh929(V3, V3, m, bit1(pls)))))), zero_zero(int)))
% 144.48/18.95 = { by axiom 7 (fact_317_mult__eq__0__iff) R->L }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, t, fresh135(T4, T4, one_one(int), fresh735(S4, S4, int, number_number_of(int, bit0(bit0(bit1(pls)))), m))), hAPP(int, int, times_times(int, fresh135(V2, V2, one_one(int), bit0(bit0(fresh929(V3, V3, m, bit1(pls)))))), zero_zero(int)))
% 144.48/18.95 = { by lemma 24 }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, t, fresh135(T4, T4, one_one(int), fresh826(X5, X5, int, m, number_number_of(int, bit0(bit0(bit1(pls))))))), hAPP(int, int, times_times(int, fresh135(V2, V2, one_one(int), bit0(bit0(fresh929(V3, V3, m, bit1(pls)))))), zero_zero(int)))
% 144.48/18.95 = { by lemma 25 }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, t, fresh135(T4, T4, one_one(int), fresh748(Y5, Y5, number_number_of(int, bit0(bit0(bit1(pls)))), m))), hAPP(int, int, times_times(int, fresh135(V2, V2, one_one(int), bit0(bit0(fresh929(V3, V3, m, bit1(pls)))))), zero_zero(int)))
% 144.48/18.95 = { by lemma 26 }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, t, fresh135(T4, T4, one_one(int), fresh929(Z5, Z5, m, number_number_of(int, bit0(bit0(bit1(pls))))))), hAPP(int, int, times_times(int, fresh135(V2, V2, one_one(int), bit0(bit0(fresh929(V3, V3, m, bit1(pls)))))), zero_zero(int)))
% 144.48/18.95 = { by lemma 23 }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, t, fresh135(T4, T4, one_one(int), fresh929(Z5, Z5, m, bit0(bit0(bit1(pls)))))), hAPP(int, int, times_times(int, fresh135(V2, V2, one_one(int), bit0(bit0(fresh929(V3, V3, m, bit1(pls)))))), zero_zero(int)))
% 144.48/18.95 = { by lemma 28 }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, t, fresh135(T4, T4, one_one(int), bit0(fresh929(W5, W5, m, bit0(bit1(pls)))))), hAPP(int, int, times_times(int, fresh135(V2, V2, one_one(int), bit0(bit0(fresh929(V3, V3, m, bit1(pls)))))), zero_zero(int)))
% 144.48/18.95 = { by lemma 28 R->L }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, t, fresh135(T4, T4, one_one(int), bit0(fresh929(W5, W5, m, bit0(bit1(pls)))))), hAPP(int, int, times_times(int, fresh135(V2, V2, one_one(int), bit0(fresh929(V5, V5, m, bit0(bit1(pls)))))), zero_zero(int)))
% 144.48/18.95 = { by lemma 28 R->L }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, t, fresh135(T4, T4, one_one(int), fresh929(U5, U5, m, bit0(bit0(bit1(pls)))))), hAPP(int, int, times_times(int, fresh135(V2, V2, one_one(int), bit0(fresh929(V5, V5, m, bit0(bit1(pls)))))), zero_zero(int)))
% 144.48/18.95 = { by lemma 28 R->L }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, t, fresh135(T4, T4, one_one(int), fresh929(U5, U5, m, bit0(bit0(bit1(pls)))))), hAPP(int, int, times_times(int, fresh135(V2, V2, one_one(int), fresh929(T5, T5, m, bit0(bit0(bit1(pls)))))), zero_zero(int)))
% 144.48/18.95 = { by lemma 23 R->L }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, t, fresh135(T4, T4, one_one(int), fresh929(U5, U5, m, number_number_of(int, bit0(bit0(bit1(pls))))))), hAPP(int, int, times_times(int, fresh135(V2, V2, one_one(int), fresh929(T5, T5, m, bit0(bit0(bit1(pls)))))), zero_zero(int)))
% 144.48/18.95 = { by lemma 23 R->L }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, t, fresh135(T4, T4, one_one(int), fresh929(U5, U5, m, number_number_of(int, bit0(bit0(bit1(pls))))))), hAPP(int, int, times_times(int, fresh135(V2, V2, one_one(int), fresh929(T5, T5, m, number_number_of(int, bit0(bit0(bit1(pls))))))), zero_zero(int)))
% 144.48/18.95 = { by lemma 26 R->L }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, t, fresh135(T4, T4, one_one(int), fresh748(S5, S5, number_number_of(int, bit0(bit0(bit1(pls)))), m))), hAPP(int, int, times_times(int, fresh135(V2, V2, one_one(int), fresh929(T5, T5, m, number_number_of(int, bit0(bit0(bit1(pls))))))), zero_zero(int)))
% 144.48/18.95 = { by lemma 26 R->L }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, t, fresh135(T4, T4, one_one(int), fresh748(S5, S5, number_number_of(int, bit0(bit0(bit1(pls)))), m))), hAPP(int, int, times_times(int, fresh135(V2, V2, one_one(int), fresh748(X6, X6, number_number_of(int, bit0(bit0(bit1(pls)))), m))), zero_zero(int)))
% 144.48/18.95 = { by lemma 25 R->L }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, t, fresh135(T4, T4, one_one(int), fresh826(Y6, Y6, int, m, number_number_of(int, bit0(bit0(bit1(pls))))))), hAPP(int, int, times_times(int, fresh135(V2, V2, one_one(int), fresh748(X6, X6, number_number_of(int, bit0(bit0(bit1(pls)))), m))), zero_zero(int)))
% 144.48/18.95 = { by lemma 25 R->L }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, t, fresh135(T4, T4, one_one(int), fresh826(Y6, Y6, int, m, number_number_of(int, bit0(bit0(bit1(pls))))))), hAPP(int, int, times_times(int, fresh135(V2, V2, one_one(int), fresh826(Z6, Z6, int, m, number_number_of(int, bit0(bit0(bit1(pls))))))), zero_zero(int)))
% 144.48/18.95 = { by lemma 24 R->L }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, t, fresh135(T4, T4, one_one(int), fresh735(W6, W6, int, number_number_of(int, bit0(bit0(bit1(pls)))), m))), hAPP(int, int, times_times(int, fresh135(V2, V2, one_one(int), fresh826(Z6, Z6, int, m, number_number_of(int, bit0(bit0(bit1(pls))))))), zero_zero(int)))
% 144.48/18.95 = { by lemma 24 R->L }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, t, fresh135(T4, T4, one_one(int), fresh735(W6, W6, int, number_number_of(int, bit0(bit0(bit1(pls)))), m))), hAPP(int, int, times_times(int, fresh135(V2, V2, one_one(int), fresh735(V6, V6, int, number_number_of(int, bit0(bit0(bit1(pls)))), m))), zero_zero(int)))
% 144.48/18.95 = { by axiom 7 (fact_317_mult__eq__0__iff) }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, t, fresh135(T4, T4, one_one(int), hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m))), hAPP(int, int, times_times(int, fresh135(V2, V2, one_one(int), fresh735(V6, V6, int, number_number_of(int, bit0(bit0(bit1(pls)))), m))), zero_zero(int)))
% 144.48/18.95 = { by axiom 7 (fact_317_mult__eq__0__iff) }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, t, fresh135(T4, T4, one_one(int), hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m))), hAPP(int, int, times_times(int, fresh135(V2, V2, one_one(int), hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m))), zero_zero(int)))
% 144.48/18.95 = { by lemma 30 R->L }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, t, fresh1086(U6, U6, int, one_one(int), hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m))), hAPP(int, int, times_times(int, fresh135(V2, V2, one_one(int), hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m))), zero_zero(int)))
% 144.48/18.95 = { by lemma 30 R->L }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, t, fresh1086(U6, U6, int, one_one(int), hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m))), hAPP(int, int, times_times(int, fresh1086(T6, T6, int, one_one(int), hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m))), zero_zero(int)))
% 144.48/18.95 = { by lemma 27 R->L }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, t, fresh804(S6, S6, 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, fresh1086(T6, T6, int, one_one(int), hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m))), zero_zero(int)))
% 144.48/18.95 = { by lemma 27 R->L }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, t, fresh804(S6, S6, 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, fresh804(X7, X7, int, hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m), one_one(int))), zero_zero(int)))
% 144.48/18.95 = { by axiom 9 (fact_238_add__0__iff_1) }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, 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, fresh804(X7, X7, int, hAPP(int, int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls))))), m), one_one(int))), zero_zero(int)))
% 144.48/18.95 = { by axiom 9 (fact_238_add__0__iff_1) }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh929(W4, W4, 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)))
% 144.48/18.95 = { by lemma 26 R->L }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh748(Y7, Y7, 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)))
% 144.48/18.95 = { by lemma 25 R->L }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh826(Z7, Z7, int, 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)))
% 144.48/18.95 = { by lemma 24 R->L }
% 144.48/18.95 fresh1666(Z2, Z2, W2, fresh735(W7, W7, 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)))
% 144.48/18.95 = { by axiom 7 (fact_317_mult__eq__0__iff) }
% 144.48/18.95 fresh1666(Z2, Z2, W2, 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)))
% 144.48/18.95 = { by axiom 19 (fact_149_less__number__of) }
% 144.48/18.96 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))))
% 144.48/18.96 = { by axiom 22 (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) }
% 144.48/18.96 true2
% 144.48/18.96 % SZS output end Proof
% 144.48/18.96
% 144.48/18.96 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------