TSTP Solution File: NUM924+5 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : NUM924+5 : 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 : n026.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 11:58:56 EDT 2023
% Result : Theorem 3.41s 0.84s
% Output : Proof 3.41s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : NUM924+5 : TPTP v8.1.2. Released v5.3.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.36 % Computer : n026.cluster.edu
% 0.14/0.36 % Model : x86_64 x86_64
% 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.36 % Memory : 8042.1875MB
% 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.36 % CPULimit : 300
% 0.14/0.36 % WCLimit : 300
% 0.14/0.36 % DateTime : Fri Aug 25 14:08:32 EDT 2023
% 0.14/0.36 % CPUTime :
% 3.41/0.84 Command-line arguments: --no-flatten-goal
% 3.41/0.84
% 3.41/0.84 % SZS status Theorem
% 3.41/0.84
% 3.41/0.85 % SZS output start Proof
% 3.41/0.85 Take the following subset of the input axioms:
% 3.41/0.85 fof(conj_0, conjecture, ord_less(int, plus_plus(int, power_power(int, s, number_number_of(nat, bit0(bit1(pls)))), one_one(int)), zero_zero(int))).
% 3.41/0.85 fof(fact_22_number__of__is__id, axiom, ![K_1]: number_number_of(int, K_1)=K_1).
% 3.41/0.85 fof(fact_23_zmult__commute, axiom, ![W, Z]: times_times(int, Z, W)=times_times(int, W, Z)).
% 3.41/0.85 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, ord_less(int, times_times(int, plus_plus(int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls)))), m), one_one(int)), t), times_times(int, plus_plus(int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls)))), m), one_one(int)), zero_zero(int)))).
% 3.41/0.85 fof(fact_3_t, axiom, plus_plus(int, power_power(int, s, number_number_of(nat, bit0(bit1(pls)))), one_one(int))=times_times(int, plus_plus(int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls)))), m), one_one(int)), t)).
% 3.41/0.85 fof(fact_4_calculation_I2_J, axiom, t=zero_zero(int) => plus_plus(int, power_power(int, s, number_number_of(nat, bit0(bit1(pls)))), one_one(int))=zero_zero(int)).
% 3.41/0.85 fof(fact_62_mult__Pls, axiom, ![W2]: times_times(int, pls, W2)=pls).
% 3.41/0.85 fof(fact_63_mult__Bit0, axiom, ![L_1, K_1_2]: times_times(int, bit0(K_1_2), L_1)=bit0(times_times(int, K_1_2, L_1))).
% 3.41/0.85 fof(fact_96_zadd__commute, axiom, ![Z2, W2]: plus_plus(int, Z2, W2)=plus_plus(int, W2, Z2)).
% 3.41/0.85 fof(fact_97_zero__is__num__zero, axiom, zero_zero(int)=number_number_of(int, pls)).
% 3.41/0.85
% 3.41/0.85 Now clausify the problem and encode Horn clauses using encoding 3 of
% 3.41/0.85 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 3.41/0.85 We repeatedly replace C & s=t => u=v by the two clauses:
% 3.41/0.85 fresh(y, y, x1...xn) = u
% 3.41/0.85 C => fresh(s, t, x1...xn) = v
% 3.41/0.85 where fresh is a fresh function symbol and x1..xn are the free
% 3.41/0.85 variables of u and v.
% 3.41/0.85 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 3.41/0.85 input problem has no model of domain size 1).
% 3.41/0.85
% 3.41/0.85 The encoding turns the above axioms into the following unit equations and goals:
% 3.41/0.85
% 3.41/0.85 Axiom 1 (fact_22_number__of__is__id): number_number_of(int, X) = X.
% 3.41/0.85 Axiom 2 (fact_97_zero__is__num__zero): zero_zero(int) = number_number_of(int, pls).
% 3.41/0.85 Axiom 3 (fact_96_zadd__commute): plus_plus(int, X, Y) = plus_plus(int, Y, X).
% 3.41/0.85 Axiom 4 (fact_23_zmult__commute): times_times(int, X, Y) = times_times(int, Y, X).
% 3.41/0.85 Axiom 5 (fact_62_mult__Pls): times_times(int, pls, X) = pls.
% 3.41/0.85 Axiom 6 (fact_63_mult__Bit0): times_times(int, bit0(X), Y) = bit0(times_times(int, X, Y)).
% 3.41/0.85 Axiom 7 (fact_14_zero__eq__power2): fresh153(X, X, Y, Z) = power_power(Y, Z, number_number_of(nat, bit0(bit1(pls)))).
% 3.41/0.85 Axiom 8 (fact_4_calculation_I2_J): fresh113(t, zero_zero(int)) = plus_plus(int, power_power(int, s, number_number_of(nat, bit0(bit1(pls)))), one_one(int)).
% 3.41/0.85 Axiom 9 (fact_3_t): plus_plus(int, power_power(int, s, number_number_of(nat, bit0(bit1(pls)))), one_one(int)) = times_times(int, plus_plus(int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls)))), m), one_one(int)), t).
% 3.41/0.85 Axiom 10 (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): ord_less(int, times_times(int, plus_plus(int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls)))), m), one_one(int)), t), times_times(int, plus_plus(int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls)))), m), one_one(int)), zero_zero(int))) = true2.
% 3.41/0.85
% 3.41/0.85 Lemma 11: zero_zero(int) = pls.
% 3.41/0.85 Proof:
% 3.41/0.85 zero_zero(int)
% 3.41/0.85 = { by axiom 2 (fact_97_zero__is__num__zero) }
% 3.41/0.85 number_number_of(int, pls)
% 3.41/0.85 = { by axiom 1 (fact_22_number__of__is__id) }
% 3.41/0.85 pls
% 3.41/0.85
% 3.41/0.85 Lemma 12: plus_plus(int, one_one(int), fresh153(X, X, int, s)) = fresh113(t, pls).
% 3.41/0.85 Proof:
% 3.41/0.85 plus_plus(int, one_one(int), fresh153(X, X, int, s))
% 3.41/0.85 = { by axiom 3 (fact_96_zadd__commute) R->L }
% 3.41/0.85 plus_plus(int, fresh153(X, X, int, s), one_one(int))
% 3.41/0.85 = { by axiom 7 (fact_14_zero__eq__power2) }
% 3.41/0.85 plus_plus(int, power_power(int, s, number_number_of(nat, bit0(bit1(pls)))), one_one(int))
% 3.41/0.85 = { by axiom 8 (fact_4_calculation_I2_J) R->L }
% 3.41/0.85 fresh113(t, zero_zero(int))
% 3.41/0.85 = { by lemma 11 }
% 3.41/0.85 fresh113(t, pls)
% 3.41/0.85
% 3.41/0.85 Goal 1 (conj_0): ord_less(int, plus_plus(int, power_power(int, s, number_number_of(nat, bit0(bit1(pls)))), one_one(int)), zero_zero(int)) = true2.
% 3.41/0.85 Proof:
% 3.41/0.85 ord_less(int, plus_plus(int, power_power(int, s, number_number_of(nat, bit0(bit1(pls)))), one_one(int)), zero_zero(int))
% 3.41/0.85 = { by lemma 11 }
% 3.41/0.85 ord_less(int, plus_plus(int, power_power(int, s, number_number_of(nat, bit0(bit1(pls)))), one_one(int)), pls)
% 3.41/0.85 = { by axiom 3 (fact_96_zadd__commute) }
% 3.41/0.85 ord_less(int, plus_plus(int, one_one(int), power_power(int, s, number_number_of(nat, bit0(bit1(pls))))), pls)
% 3.41/0.85 = { by axiom 7 (fact_14_zero__eq__power2) R->L }
% 3.41/0.85 ord_less(int, plus_plus(int, one_one(int), fresh153(X, X, int, s)), pls)
% 3.41/0.85 = { by lemma 12 }
% 3.41/0.85 ord_less(int, fresh113(t, pls), pls)
% 3.41/0.85 = { by axiom 5 (fact_62_mult__Pls) R->L }
% 3.41/0.85 ord_less(int, fresh113(t, pls), times_times(int, pls, plus_plus(int, one_one(int), bit0(bit0(times_times(int, m, bit1(pls)))))))
% 3.41/0.85 = { by axiom 4 (fact_23_zmult__commute) R->L }
% 3.41/0.85 ord_less(int, fresh113(t, pls), times_times(int, plus_plus(int, one_one(int), bit0(bit0(times_times(int, m, bit1(pls))))), pls))
% 3.41/0.85 = { by lemma 12 R->L }
% 3.41/0.85 ord_less(int, plus_plus(int, one_one(int), fresh153(Y, Y, int, s)), times_times(int, plus_plus(int, one_one(int), bit0(bit0(times_times(int, m, bit1(pls))))), pls))
% 3.41/0.85 = { by axiom 3 (fact_96_zadd__commute) R->L }
% 3.41/0.85 ord_less(int, plus_plus(int, fresh153(Y, Y, int, s), one_one(int)), times_times(int, plus_plus(int, one_one(int), bit0(bit0(times_times(int, m, bit1(pls))))), pls))
% 3.41/0.85 = { by axiom 7 (fact_14_zero__eq__power2) }
% 3.41/0.85 ord_less(int, plus_plus(int, power_power(int, s, number_number_of(nat, bit0(bit1(pls)))), one_one(int)), times_times(int, plus_plus(int, one_one(int), bit0(bit0(times_times(int, m, bit1(pls))))), pls))
% 3.41/0.85 = { by axiom 9 (fact_3_t) }
% 3.41/0.85 ord_less(int, times_times(int, plus_plus(int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls)))), m), one_one(int)), t), times_times(int, plus_plus(int, one_one(int), bit0(bit0(times_times(int, m, bit1(pls))))), pls))
% 3.41/0.85 = { by axiom 1 (fact_22_number__of__is__id) }
% 3.41/0.85 ord_less(int, times_times(int, plus_plus(int, times_times(int, bit0(bit0(bit1(pls))), m), one_one(int)), t), times_times(int, plus_plus(int, one_one(int), bit0(bit0(times_times(int, m, bit1(pls))))), pls))
% 3.41/0.85 = { by axiom 6 (fact_63_mult__Bit0) }
% 3.41/0.85 ord_less(int, times_times(int, plus_plus(int, bit0(times_times(int, bit0(bit1(pls)), m)), one_one(int)), t), times_times(int, plus_plus(int, one_one(int), bit0(bit0(times_times(int, m, bit1(pls))))), pls))
% 3.41/0.85 = { by axiom 6 (fact_63_mult__Bit0) }
% 3.41/0.85 ord_less(int, times_times(int, plus_plus(int, bit0(bit0(times_times(int, bit1(pls), m))), one_one(int)), t), times_times(int, plus_plus(int, one_one(int), bit0(bit0(times_times(int, m, bit1(pls))))), pls))
% 3.41/0.85 = { by axiom 4 (fact_23_zmult__commute) }
% 3.41/0.85 ord_less(int, times_times(int, t, plus_plus(int, bit0(bit0(times_times(int, bit1(pls), m))), one_one(int))), times_times(int, plus_plus(int, one_one(int), bit0(bit0(times_times(int, m, bit1(pls))))), pls))
% 3.41/0.85 = { by axiom 3 (fact_96_zadd__commute) }
% 3.41/0.85 ord_less(int, times_times(int, t, plus_plus(int, one_one(int), bit0(bit0(times_times(int, bit1(pls), m))))), times_times(int, plus_plus(int, one_one(int), bit0(bit0(times_times(int, m, bit1(pls))))), pls))
% 3.41/0.85 = { by axiom 4 (fact_23_zmult__commute) R->L }
% 3.41/0.85 ord_less(int, times_times(int, t, plus_plus(int, one_one(int), bit0(bit0(times_times(int, bit1(pls), m))))), times_times(int, plus_plus(int, one_one(int), bit0(bit0(times_times(int, bit1(pls), m)))), pls))
% 3.41/0.85 = { by axiom 3 (fact_96_zadd__commute) R->L }
% 3.41/0.85 ord_less(int, times_times(int, t, plus_plus(int, bit0(bit0(times_times(int, bit1(pls), m))), one_one(int))), times_times(int, plus_plus(int, one_one(int), bit0(bit0(times_times(int, bit1(pls), m)))), pls))
% 3.41/0.85 = { by axiom 3 (fact_96_zadd__commute) R->L }
% 3.41/0.85 ord_less(int, times_times(int, t, plus_plus(int, bit0(bit0(times_times(int, bit1(pls), m))), one_one(int))), times_times(int, plus_plus(int, bit0(bit0(times_times(int, bit1(pls), m))), one_one(int)), pls))
% 3.41/0.85 = { by axiom 4 (fact_23_zmult__commute) R->L }
% 3.41/0.85 ord_less(int, times_times(int, plus_plus(int, bit0(bit0(times_times(int, bit1(pls), m))), one_one(int)), t), times_times(int, plus_plus(int, bit0(bit0(times_times(int, bit1(pls), m))), one_one(int)), pls))
% 3.41/0.85 = { by lemma 11 R->L }
% 3.41/0.85 ord_less(int, times_times(int, plus_plus(int, bit0(bit0(times_times(int, bit1(pls), m))), one_one(int)), t), times_times(int, plus_plus(int, bit0(bit0(times_times(int, bit1(pls), m))), one_one(int)), zero_zero(int)))
% 3.41/0.85 = { by axiom 6 (fact_63_mult__Bit0) R->L }
% 3.41/0.85 ord_less(int, times_times(int, plus_plus(int, bit0(times_times(int, bit0(bit1(pls)), m)), one_one(int)), t), times_times(int, plus_plus(int, bit0(bit0(times_times(int, bit1(pls), m))), one_one(int)), zero_zero(int)))
% 3.41/0.85 = { by axiom 6 (fact_63_mult__Bit0) R->L }
% 3.41/0.85 ord_less(int, times_times(int, plus_plus(int, bit0(times_times(int, bit0(bit1(pls)), m)), one_one(int)), t), times_times(int, plus_plus(int, bit0(times_times(int, bit0(bit1(pls)), m)), one_one(int)), zero_zero(int)))
% 3.41/0.85 = { by axiom 6 (fact_63_mult__Bit0) R->L }
% 3.41/0.85 ord_less(int, times_times(int, plus_plus(int, times_times(int, bit0(bit0(bit1(pls))), m), one_one(int)), t), times_times(int, plus_plus(int, bit0(times_times(int, bit0(bit1(pls)), m)), one_one(int)), zero_zero(int)))
% 3.41/0.85 = { by axiom 6 (fact_63_mult__Bit0) R->L }
% 3.41/0.85 ord_less(int, times_times(int, plus_plus(int, times_times(int, bit0(bit0(bit1(pls))), m), one_one(int)), t), times_times(int, plus_plus(int, times_times(int, bit0(bit0(bit1(pls))), m), one_one(int)), zero_zero(int)))
% 3.41/0.85 = { by axiom 1 (fact_22_number__of__is__id) R->L }
% 3.41/0.85 ord_less(int, times_times(int, plus_plus(int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls)))), m), one_one(int)), t), times_times(int, plus_plus(int, times_times(int, bit0(bit0(bit1(pls))), m), one_one(int)), zero_zero(int)))
% 3.41/0.85 = { by axiom 1 (fact_22_number__of__is__id) R->L }
% 3.41/0.85 ord_less(int, times_times(int, plus_plus(int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls)))), m), one_one(int)), t), times_times(int, plus_plus(int, times_times(int, number_number_of(int, bit0(bit0(bit1(pls)))), m), one_one(int)), zero_zero(int)))
% 3.41/0.85 = { by axiom 10 (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) }
% 3.41/0.85 true2
% 3.41/0.85 % SZS output end Proof
% 3.41/0.85
% 3.41/0.85 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------