TSTP Solution File: NUM924+5 by Prover9---1109a
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- Process Solution
%------------------------------------------------------------------------------
% File : Prover9---1109a
% Problem : NUM924+5 : TPTP v8.1.0. Released v5.3.0.
% Transfm : none
% Format : tptp:raw
% Command : tptp2X_and_run_prover9 %d %s
% Computer : n019.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 : 600s
% DateTime : Mon Jul 18 13:33:16 EDT 2022
% Result : Theorem 0.74s 1.10s
% Output : Refutation 0.74s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : NUM924+5 : TPTP v8.1.0. Released v5.3.0.
% 0.07/0.13 % Command : tptp2X_and_run_prover9 %d %s
% 0.13/0.34 % Computer : n019.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 600
% 0.13/0.34 % DateTime : Wed Jul 6 21:34:53 EDT 2022
% 0.13/0.34 % CPUTime :
% 0.42/1.00 ============================== Prover9 ===============================
% 0.42/1.00 Prover9 (32) version 2009-11A, November 2009.
% 0.42/1.00 Process 1387 was started by sandbox on n019.cluster.edu,
% 0.42/1.00 Wed Jul 6 21:34:54 2022
% 0.42/1.00 The command was "/export/starexec/sandbox/solver/bin/prover9 -t 300 -f /tmp/Prover9_1233_n019.cluster.edu".
% 0.42/1.00 ============================== end of head ===========================
% 0.42/1.00
% 0.42/1.00 ============================== INPUT =================================
% 0.42/1.00
% 0.42/1.00 % Reading from file /tmp/Prover9_1233_n019.cluster.edu
% 0.42/1.00
% 0.42/1.00 set(prolog_style_variables).
% 0.42/1.00 set(auto2).
% 0.42/1.00 % set(auto2) -> set(auto).
% 0.42/1.00 % set(auto) -> set(auto_inference).
% 0.42/1.00 % set(auto) -> set(auto_setup).
% 0.42/1.00 % set(auto_setup) -> set(predicate_elim).
% 0.42/1.00 % set(auto_setup) -> assign(eq_defs, unfold).
% 0.42/1.00 % set(auto) -> set(auto_limits).
% 0.42/1.00 % set(auto_limits) -> assign(max_weight, "100.000").
% 0.42/1.00 % set(auto_limits) -> assign(sos_limit, 20000).
% 0.42/1.00 % set(auto) -> set(auto_denials).
% 0.42/1.00 % set(auto) -> set(auto_process).
% 0.42/1.00 % set(auto2) -> assign(new_constants, 1).
% 0.42/1.00 % set(auto2) -> assign(fold_denial_max, 3).
% 0.42/1.00 % set(auto2) -> assign(max_weight, "200.000").
% 0.42/1.00 % set(auto2) -> assign(max_hours, 1).
% 0.42/1.00 % assign(max_hours, 1) -> assign(max_seconds, 3600).
% 0.42/1.00 % set(auto2) -> assign(max_seconds, 0).
% 0.42/1.00 % set(auto2) -> assign(max_minutes, 5).
% 0.42/1.00 % assign(max_minutes, 5) -> assign(max_seconds, 300).
% 0.42/1.00 % set(auto2) -> set(sort_initial_sos).
% 0.42/1.00 % set(auto2) -> assign(sos_limit, -1).
% 0.42/1.00 % set(auto2) -> assign(lrs_ticks, 3000).
% 0.42/1.00 % set(auto2) -> assign(max_megs, 400).
% 0.42/1.00 % set(auto2) -> assign(stats, some).
% 0.42/1.00 % set(auto2) -> clear(echo_input).
% 0.42/1.00 % set(auto2) -> set(quiet).
% 0.42/1.00 % set(auto2) -> clear(print_initial_clauses).
% 0.42/1.00 % set(auto2) -> clear(print_given).
% 0.42/1.00 assign(lrs_ticks,-1).
% 0.42/1.00 assign(sos_limit,10000).
% 0.42/1.00 assign(order,kbo).
% 0.42/1.00 set(lex_order_vars).
% 0.42/1.00 clear(print_given).
% 0.42/1.00
% 0.42/1.00 % formulas(sos). % not echoed (155 formulas)
% 0.42/1.00
% 0.42/1.00 ============================== end of input ==========================
% 0.42/1.00
% 0.42/1.00 % From the command line: assign(max_seconds, 300).
% 0.42/1.00
% 0.42/1.00 ============================== PROCESS NON-CLAUSAL FORMULAS ==========
% 0.42/1.00
% 0.42/1.00 % Formulas that are not ordinary clauses:
% 0.42/1.00 1 (all X_a (semiring_1(X_a) -> ti(X_a,one_one(X_a)) = one_one(X_a))) # label(tsy_c_Groups_Oone__class_Oone_res) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 2 (all B_1 all B_2 all X_b (number(X_b) & semiring(X_b) -> plus_plus(X_b,ti(X_b,B_1),B_2) = plus_plus(X_b,B_1,B_2))) # label(tsy_c_Groups_Oplus__class_Oplus_0_arg1) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 3 (all B_1 all B_2 all X_b (number(X_b) & semiring(X_b) -> plus_plus(X_b,B_1,ti(X_b,B_2)) = plus_plus(X_b,B_1,B_2))) # label(tsy_c_Groups_Oplus__class_Oplus_0_arg2) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 4 (all B_1 all B_2 all X_b (number(X_b) & semiring(X_b) -> ti(X_b,plus_plus(X_b,B_1,B_2)) = plus_plus(X_b,B_1,B_2))) # label(tsy_c_Groups_Oplus__class_Oplus_0_res) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 5 (all B_1 all B_2 all X_a (linordered_ring(X_a) -> plus_plus(X_a,ti(X_a,B_1),B_2) = plus_plus(X_a,B_1,B_2))) # label(tsy_c_Groups_Oplus__class_Oplus_1_arg1) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 6 (all B_1 all B_2 all X_a (linordered_ring(X_a) -> plus_plus(X_a,B_1,ti(X_a,B_2)) = plus_plus(X_a,B_1,B_2))) # label(tsy_c_Groups_Oplus__class_Oplus_1_arg2) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 7 (all B_1 all B_2 all X_a (linordered_ring(X_a) -> ti(X_a,plus_plus(X_a,B_1,B_2)) = plus_plus(X_a,B_1,B_2))) # label(tsy_c_Groups_Oplus__class_Oplus_1_res) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 8 (all B_1 all B_2 all X_b (number(X_b) & semiring(X_b) -> times_times(X_b,ti(X_b,B_1),B_2) = times_times(X_b,B_1,B_2))) # label(tsy_c_Groups_Otimes__class_Otimes_0_arg1) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 9 (all B_1 all B_2 all X_b (number(X_b) & semiring(X_b) -> times_times(X_b,B_1,ti(X_b,B_2)) = times_times(X_b,B_1,B_2))) # label(tsy_c_Groups_Otimes__class_Otimes_0_arg2) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 10 (all B_1 all B_2 all X_b (number(X_b) & semiring(X_b) -> ti(X_b,times_times(X_b,B_1,B_2)) = times_times(X_b,B_1,B_2))) # label(tsy_c_Groups_Otimes__class_Otimes_0_res) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 11 (all B_1 all B_2 all X_a (linordered_ring(X_a) -> times_times(X_a,ti(X_a,B_1),B_2) = times_times(X_a,B_1,B_2))) # label(tsy_c_Groups_Otimes__class_Otimes_1_arg1) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 12 (all B_1 all B_2 all X_a (linordered_ring(X_a) -> times_times(X_a,B_1,ti(X_a,B_2)) = times_times(X_a,B_1,B_2))) # label(tsy_c_Groups_Otimes__class_Otimes_1_arg2) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 13 (all B_1 all B_2 all X_a (linordered_ring(X_a) -> ti(X_a,times_times(X_a,B_1,B_2)) = times_times(X_a,B_1,B_2))) # label(tsy_c_Groups_Otimes__class_Otimes_1_res) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 14 (all X_a (linordered_ring(X_a) -> ti(X_a,zero_zero(X_a)) = zero_zero(X_a))) # label(tsy_c_Groups_Ozero__class_Ozero_0_res) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 15 (all X_a (semiring_1(X_a) -> ti(X_a,zero_zero(X_a)) = zero_zero(X_a))) # label(tsy_c_Groups_Ozero__class_Ozero_1_res) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 16 (all X_a ti(X_a,undefined(X_a)) = undefined(X_a)) # label(tsy_c_HOL_Oundefined_res) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 17 (all B_1 (zprime(ti(int,B_1)) <-> zprime(B_1))) # label(tsy_c_IntPrimes_Ozprime_arg1) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 18 (all B_1 bit0(ti(int,B_1)) = bit0(B_1)) # label(tsy_c_Int_OBit0_arg1) # label(hypothesis) # label(non_clause). [assumption].
% 0.42/1.00 19 (all B_1 ti(int,bit0(B_1)) = bit0(B_1)) # label(tsy_c_Int_OBit0_res) # label(hypothesis) # label(non_clause). [assumption].
% 0.42/1.00 20 (all B_1 bit1(ti(int,B_1)) = bit1(B_1)) # label(tsy_c_Int_OBit1_arg1) # label(hypothesis) # label(non_clause). [assumption].
% 0.42/1.00 21 (all B_1 ti(int,bit1(B_1)) = bit1(B_1)) # label(tsy_c_Int_OBit1_res) # label(hypothesis) # label(non_clause). [assumption].
% 0.42/1.00 22 (all B_1 all X_a (number(X_a) -> number_number_of(X_a,ti(int,B_1)) = number_number_of(X_a,B_1))) # label(tsy_c_Int_Onumber__class_Onumber__of_arg1) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 23 (all B_1 all X_a (number(X_a) -> ti(X_a,number_number_of(X_a,B_1)) = number_number_of(X_a,B_1))) # label(tsy_c_Int_Onumber__class_Onumber__of_res) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 24 (all B_1 all B_2 all X_a (number(X_a) & linorder(X_a) -> (ord_less(X_a,ti(X_a,B_1),B_2) <-> ord_less(X_a,B_1,B_2)))) # label(tsy_c_Orderings_Oord__class_Oless_0_arg1) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 25 (all B_1 all B_2 all X_a (number(X_a) & linorder(X_a) -> (ord_less(X_a,B_1,ti(X_a,B_2)) <-> ord_less(X_a,B_1,B_2)))) # label(tsy_c_Orderings_Oord__class_Oless_0_arg2) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 26 (all B_1 all B_2 all X_a (linordered_idom(X_a) -> (ord_less(X_a,ti(X_a,B_1),B_2) <-> ord_less(X_a,B_1,B_2)))) # label(tsy_c_Orderings_Oord__class_Oless_1_arg1) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 27 (all B_1 all B_2 all X_a (linordered_idom(X_a) -> (ord_less(X_a,B_1,ti(X_a,B_2)) <-> ord_less(X_a,B_1,B_2)))) # label(tsy_c_Orderings_Oord__class_Oless_1_arg2) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 28 (all B_1 all B_2 all X_a (number(X_a) & linorder(X_a) -> (ord_less_eq(X_a,ti(X_a,B_1),B_2) <-> ord_less_eq(X_a,B_1,B_2)))) # label(tsy_c_Orderings_Oord__class_Oless__eq_0_arg1) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 29 (all B_1 all B_2 all X_a (number(X_a) & linorder(X_a) -> (ord_less_eq(X_a,B_1,ti(X_a,B_2)) <-> ord_less_eq(X_a,B_1,B_2)))) # label(tsy_c_Orderings_Oord__class_Oless__eq_0_arg2) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 30 (all B_1 all B_2 all X_a (linordered_ring(X_a) -> (ord_less_eq(X_a,ti(X_a,B_1),B_2) <-> ord_less_eq(X_a,B_1,B_2)))) # label(tsy_c_Orderings_Oord__class_Oless__eq_1_arg1) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 31 (all B_1 all B_2 all X_a (linordered_ring(X_a) -> (ord_less_eq(X_a,B_1,ti(X_a,B_2)) <-> ord_less_eq(X_a,B_1,B_2)))) # label(tsy_c_Orderings_Oord__class_Oless__eq_1_arg2) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 32 (all B_1 all B_2 all X_a (semiring_1(X_a) -> power_power(X_a,ti(X_a,B_1),B_2) = power_power(X_a,B_1,B_2))) # label(tsy_c_Power_Opower__class_Opower_arg1) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 33 (all B_1 all B_2 all X_a (semiring_1(X_a) -> power_power(X_a,B_1,ti(nat,B_2)) = power_power(X_a,B_1,B_2))) # label(tsy_c_Power_Opower__class_Opower_arg2) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 34 (all B_1 all B_2 all X_a (semiring_1(X_a) -> ti(X_a,power_power(X_a,B_1,B_2)) = power_power(X_a,B_1,B_2))) # label(tsy_c_Power_Opower__class_Opower_res) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 35 (all B_1 (twoSqu33214720sum2sq(ti(int,B_1)) <-> twoSqu33214720sum2sq(B_1))) # label(tsy_c_TwoSquares__Mirabelle__vsgmegnqdl_Ois__sum2sq_arg1) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 36 t = zero_zero(int) -> plus_plus(int,power_power(int,s,number_number_of(nat,bit0(bit1(pls)))),one_one(int)) = zero_zero(int) # label(fact_4_calculation_I2_J) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 37 (all X_a (linordered_idom(X_a) -> (all X all Y -ord_less(X_a,plus_plus(X_a,power_power(X_a,X,number_number_of(nat,bit0(bit1(pls)))),power_power(X_a,Y,number_number_of(nat,bit0(bit1(pls))))),zero_zero(X_a))))) # label(fact_7_not__sum__power2__lt__zero) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 38 (all X_a (linordered_idom(X_a) -> (all X_1 all Y_1 (ord_less(X_a,zero_zero(X_a),plus_plus(X_a,power_power(X_a,X_1,number_number_of(nat,bit0(bit1(pls)))),power_power(X_a,Y_1,number_number_of(nat,bit0(bit1(pls)))))) <-> ti(X_a,X_1) != zero_zero(X_a) | ti(X_a,Y_1) != zero_zero(X_a))))) # label(fact_8_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 39 (all X_a (linordered_idom(X_a) -> (all X_1 all Y_1 (plus_plus(X_a,power_power(X_a,X_1,number_number_of(nat,bit0(bit1(pls)))),power_power(X_a,Y_1,number_number_of(nat,bit0(bit1(pls))))) = zero_zero(X_a) <-> ti(X_a,X_1) = zero_zero(X_a) & ti(X_a,Y_1) = zero_zero(X_a))))) # label(fact_9_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 40 (all X_a (linordered_idom(X_a) -> (all A_1 -ord_less(X_a,power_power(X_a,A_1,number_number_of(nat,bit0(bit1(pls)))),zero_zero(X_a))))) # label(fact_10_power2__less__0) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 41 (all X_a (linordered_idom(X_a) -> (all A_2 (ord_less(X_a,zero_zero(X_a),power_power(X_a,A_2,number_number_of(nat,bit0(bit1(pls))))) <-> ti(X_a,A_2) != zero_zero(X_a))))) # label(fact_11_zero__less__power2) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 42 (all X_a (semiring_1(X_a) -> power_power(X_a,one_one(X_a),number_number_of(nat,bit0(bit1(pls)))) = one_one(X_a))) # label(fact_12_one__power2) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 43 (all X_a (semiring_1(X_a) -> power_power(X_a,zero_zero(X_a),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(X_a))) # label(fact_13_zero__power2) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 44 (all X_a (ring_11004092258visors(X_a) -> (all A_2 (power_power(X_a,A_2,number_number_of(nat,bit0(bit1(pls)))) = zero_zero(X_a) <-> ti(X_a,A_2) = zero_zero(X_a))))) # label(fact_14_zero__eq__power2) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 45 (all X_a (number_ring(X_a) -> (all W plus_plus(X_a,one_one(X_a),number_number_of(X_a,W)) = number_number_of(X_a,plus_plus(int,bit1(pls),W))))) # label(fact_15_add__special_I2_J) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 46 (all X_a (number_ring(X_a) -> (all V plus_plus(X_a,number_number_of(X_a,V),one_one(X_a)) = number_number_of(X_a,plus_plus(int,V,bit1(pls)))))) # label(fact_16_add__special_I3_J) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 47 -(all T_1 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_1)) # label(fact_18__096_B_Bthesis_O_A_I_B_Bt_O_As_A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A_L_A1_) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 48 (all W ord_less_eq(int,W,W)) # label(fact_21_zle__refl) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 49 (all K_1 number_number_of(int,K_1) = K_1) # label(fact_22_number__of__is__id) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 50 (all Z all W times_times(int,Z,W) = times_times(int,W,Z)) # label(fact_23_zmult__commute) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 51 (all Z all W (ord_less_eq(int,Z,W) | ord_less_eq(int,W,Z))) # label(fact_24_zle__linear) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 52 (all V all W times_times(int,number_number_of(int,V),number_number_of(int,W)) = number_number_of(int,times_times(int,V,W))) # label(fact_25_times__numeral__code_I5_J) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 53 (all K all L (ord_less_eq(int,number_number_of(int,K),number_number_of(int,L)) <-> ord_less_eq(int,K,L))) # label(fact_26_less__eq__number__of__int__code) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 54 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all X_1 all Y_1 (ord_less_eq(X_a,number_number_of(X_a,X_1),number_number_of(X_a,Y_1)) <-> ord_less_eq(int,X_1,Y_1))))) # label(fact_27_le__number__of) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 55 (all Z1 all Z2 all Z3 times_times(int,times_times(int,Z1,Z2),Z3) = times_times(int,Z1,times_times(int,Z2,Z3))) # label(fact_28_zmult__assoc) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 56 (all K_1 all I all J (ord_less_eq(int,I,J) -> (ord_less_eq(int,J,K_1) -> ord_less_eq(int,I,K_1)))) # label(fact_29_zle__trans) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 57 (all Z all W (ord_less_eq(int,Z,W) -> (ord_less_eq(int,W,Z) -> Z = W))) # label(fact_30_zle__antisym) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 58 (all X all Y all Z power_power(int,X,plus_plus(nat,Y,Z)) = times_times(int,power_power(int,X,Y),power_power(int,X,Z))) # label(fact_31_zpower__zadd__distrib) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 59 (all K1 all K2 (ord_less_eq(int,bit1(K1),bit1(K2)) <-> ord_less_eq(int,K1,K2))) # label(fact_32_less__eq__int__code_I16_J) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 60 (all K all L (ord_less_eq(int,bit1(K),bit1(L)) <-> ord_less_eq(int,K,L))) # label(fact_33_rel__simps_I34_J) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 61 (all K1 all K2 (ord_less_eq(int,bit0(K1),bit0(K2)) <-> ord_less_eq(int,K1,K2))) # label(fact_35_less__eq__int__code_I13_J) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 62 (all K all L (ord_less_eq(int,bit0(K),bit0(L)) <-> ord_less_eq(int,K,L))) # label(fact_36_rel__simps_I31_J) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 63 (all Z_1 all W_1 (ord_less(int,Z_1,W_1) <-> ord_less_eq(int,Z_1,W_1) & Z_1 != W_1)) # label(fact_37_zless__le) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 64 (all K_1 all I all J (ord_less_eq(int,I,J) -> ord_less_eq(int,plus_plus(int,K_1,I),plus_plus(int,K_1,J)))) # label(fact_38_zadd__left__mono) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 65 (all V_3 (number_number_of(nat,V_3) = zero_zero(nat) <-> ord_less_eq(int,V_3,pls))) # label(fact_39_eq__number__of__0) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 66 (all V_3 (zero_zero(nat) = number_number_of(nat,V_3) <-> ord_less_eq(int,V_3,pls))) # label(fact_40_eq__0__number__of) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 67 (all X_a (number_semiring(X_a) -> (all V_1 all V (ord_less_eq(int,pls,V) -> (ord_less_eq(int,pls,V_1) -> times_times(X_a,number_number_of(X_a,V),number_number_of(X_a,V_1)) = number_number_of(X_a,times_times(int,V,V_1))))))) # label(fact_41_semiring__mult__number__of) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 68 (all X_a (number_ring(X_a) -> (all V all W all Z times_times(X_a,number_number_of(X_a,V),times_times(X_a,number_number_of(X_a,W),Z)) = times_times(X_a,number_number_of(X_a,times_times(int,V,W)),Z)))) # label(fact_42_mult__number__of__left) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 69 (all X_a (number_ring(X_a) -> (all V all W times_times(X_a,number_number_of(X_a,V),number_number_of(X_a,W)) = number_number_of(X_a,times_times(int,V,W))))) # label(fact_43_arith__simps_I32_J) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 70 (all X_a (number_ring(X_a) -> (all V all W number_number_of(X_a,times_times(int,V,W)) = times_times(X_a,number_number_of(X_a,V),number_number_of(X_a,W))))) # label(fact_44_number__of__mult) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 71 (all X_a (linord581940658strict(X_a) -> (all X_1 all Y_1 (ord_less_eq(X_a,plus_plus(X_a,times_times(X_a,X_1,X_1),times_times(X_a,Y_1,Y_1)),zero_zero(X_a)) <-> ti(X_a,X_1) = zero_zero(X_a) & ti(X_a,Y_1) = zero_zero(X_a))))) # label(fact_45_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 72 (all X_a (linordered_ring(X_a) -> (all X all Y ord_less_eq(X_a,zero_zero(X_a),plus_plus(X_a,times_times(X_a,X,X),times_times(X_a,Y,Y)))))) # label(fact_46_sum__squares__ge__zero) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 73 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all X_1 (ord_less_eq(X_a,number_number_of(X_a,X_1),zero_zero(X_a)) <-> ord_less_eq(int,X_1,pls))))) # label(fact_47_le__special_I3_J) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 74 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all Y_1 (ord_less_eq(X_a,zero_zero(X_a),number_number_of(X_a,Y_1)) <-> ord_less_eq(int,pls,Y_1))))) # label(fact_48_le__special_I1_J) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 75 (all V_3 (ord_less(nat,zero_zero(nat),number_number_of(nat,V_3)) <-> ord_less(int,pls,V_3))) # label(fact_49_less__0__number__of) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 76 (all X_a (number(X_a) & linorder(X_a) -> (all V_3 all W_1 (ord_less_eq(X_a,number_number_of(X_a,V_3),number_number_of(X_a,W_1)) <-> -ord_less(X_a,number_number_of(X_a,W_1),number_number_of(X_a,V_3)))))) # label(fact_50_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 77 (all K (ord_less_eq(int,pls,bit1(K)) <-> ord_less_eq(int,pls,K))) # label(fact_51_rel__simps_I22_J) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 78 (all K1 all K2 (ord_less_eq(int,bit0(K1),bit1(K2)) <-> ord_less_eq(int,K1,K2))) # label(fact_52_less__eq__int__code_I14_J) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 79 (all K all L (ord_less_eq(int,bit0(K),bit1(L)) <-> ord_less_eq(int,K,L))) # label(fact_53_rel__simps_I32_J) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 80 (all K (ord_less_eq(int,bit0(K),pls) <-> ord_less_eq(int,K,pls))) # label(fact_54_rel__simps_I27_J) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 81 (all K (ord_less_eq(int,pls,bit0(K)) <-> ord_less_eq(int,pls,K))) # label(fact_55_rel__simps_I21_J) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 82 (all Z_2 all Z all W_2 all W (ord_less(int,W_2,W) -> (ord_less_eq(int,Z_2,Z) -> ord_less(int,plus_plus(int,W_2,Z_2),plus_plus(int,W,Z))))) # label(fact_56_zadd__zless__mono) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 83 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all X_1 (ord_less_eq(X_a,number_number_of(X_a,X_1),one_one(X_a)) <-> ord_less_eq(int,X_1,bit1(pls)))))) # label(fact_59_le__special_I4_J) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 84 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all Y_1 (ord_less_eq(X_a,one_one(X_a),number_number_of(X_a,Y_1)) <-> ord_less_eq(int,bit1(pls),Y_1))))) # label(fact_60_le__special_I2_J) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 85 (all W times_times(int,pls,W) = pls) # label(fact_62_mult__Pls) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 86 (all K_1 all L_1 times_times(int,bit0(K_1),L_1) = bit0(times_times(int,K_1,L_1))) # label(fact_63_mult__Bit0) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 87 (all K all L (ord_less(int,number_number_of(int,K),number_number_of(int,L)) <-> ord_less(int,K,L))) # label(fact_64_less__number__of__int__code) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 88 (all Z times_times(int,Z,one_one(int)) = Z) # label(fact_65_zmult__1__right) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 89 (all Z times_times(int,one_one(int),Z) = Z) # label(fact_66_zmult__1) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.00 90 (all V all W plus_plus(int,number_number_of(int,V),number_number_of(int,W)) = number_number_of(int,plus_plus(int,V,W))) # label(fact_67_plus__numeral__code_I9_J) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 91 (all Z1 all Z2 all W times_times(int,plus_plus(int,Z1,Z2),W) = plus_plus(int,times_times(int,Z1,W),times_times(int,Z2,W))) # label(fact_68_zadd__zmult__distrib) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 92 (all W all Z1 all Z2 times_times(int,W,plus_plus(int,Z1,Z2)) = plus_plus(int,times_times(int,W,Z1),times_times(int,W,Z2))) # label(fact_69_zadd__zmult__distrib2) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 93 (all K (ord_less_eq(int,bit1(K),pls) <-> ord_less(int,K,pls))) # label(fact_70_rel__simps_I29_J) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 94 (all K (ord_less(int,pls,bit1(K)) <-> ord_less_eq(int,pls,K))) # label(fact_71_rel__simps_I5_J) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 95 (all K1 all K2 (ord_less_eq(int,bit1(K1),bit0(K2)) <-> ord_less(int,K1,K2))) # label(fact_72_less__eq__int__code_I15_J) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 96 (all K all L (ord_less_eq(int,bit1(K),bit0(L)) <-> ord_less(int,K,L))) # label(fact_73_rel__simps_I33_J) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 97 (all K1 all K2 (ord_less(int,bit0(K1),bit1(K2)) <-> ord_less_eq(int,K1,K2))) # label(fact_74_less__int__code_I14_J) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 98 (all K all L (ord_less(int,bit0(K),bit1(L)) <-> ord_less_eq(int,K,L))) # label(fact_75_rel__simps_I15_J) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 99 (all V_3 all V_2 (ord_less(nat,number_number_of(nat,V_3),number_number_of(nat,V_2)) <-> (ord_less(int,V_3,V_2) -> ord_less(int,pls,V_2)) & ord_less(int,V_3,V_2))) # label(fact_76_less__nat__number__of) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 100 (all Z_1 (ord_less_eq(int,one_one(int),Z_1) <-> ord_less(int,zero_zero(int),Z_1))) # label(fact_77_int__one__le__iff__zero__less) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 101 (all W all Z (ord_less(int,W,Z) -> ord_less_eq(int,plus_plus(int,W,one_one(int)),Z))) # label(fact_80_zless__imp__add1__zle) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 102 (all W_1 all Z_1 (ord_less_eq(int,plus_plus(int,W_1,one_one(int)),Z_1) <-> ord_less(int,W_1,Z_1))) # label(fact_81_add1__zle__eq) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 103 (all W_1 all Z_1 (ord_less(int,W_1,plus_plus(int,Z_1,one_one(int))) <-> ord_less_eq(int,W_1,Z_1))) # label(fact_82_zle__add1__eq__le) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 104 (all X_a (number_semiring(X_a) -> (all V_1 all V (ord_less_eq(int,pls,V) -> (ord_less_eq(int,pls,V_1) -> plus_plus(X_a,number_number_of(X_a,V),number_number_of(X_a,V_1)) = number_number_of(X_a,plus_plus(int,V,V_1))))))) # label(fact_83_semiring__add__number__of) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 105 (all V_1 all V ((ord_less(int,V,pls) -> plus_plus(nat,number_number_of(nat,V),number_number_of(nat,V_1)) = number_number_of(nat,V_1)) & (-ord_less(int,V,pls) -> (ord_less(int,V_1,pls) -> plus_plus(nat,number_number_of(nat,V),number_number_of(nat,V_1)) = number_number_of(nat,V)) & (-ord_less(int,V_1,pls) -> plus_plus(nat,number_number_of(nat,V),number_number_of(nat,V_1)) = number_number_of(nat,plus_plus(int,V,V_1)))))) # label(fact_84_add__nat__number__of) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 106 (all Z (ord_less_eq(int,zero_zero(int),Z) -> ord_less(int,zero_zero(int),plus_plus(int,one_one(int),Z)))) # label(fact_85_le__imp__0__less) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 107 (all X_a (number_ring(X_a) & ring_char_0(X_a) -> (all X_1 all Y_1 (number_number_of(X_a,X_1) = number_number_of(X_a,Y_1) <-> X_1 = Y_1)))) # label(fact_86_eq__number__of) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 108 (all X_a (number(X_a) -> (all W_1 all X_1 (number_number_of(X_a,W_1) = ti(X_a,X_1) <-> ti(X_a,X_1) = number_number_of(X_a,W_1))))) # label(fact_87_number__of__reorient) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 109 (all K all L (bit1(K) = bit1(L) <-> K = L)) # label(fact_88_rel__simps_I51_J) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 110 (all K all L (bit0(K) = bit0(L) <-> K = L)) # label(fact_89_rel__simps_I48_J) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 111 (all X all Y (ord_less(int,X,Y) | X = Y | ord_less(int,Y,X))) # label(fact_90_zless__linear) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 112 (all X_a (linord581940658strict(X_a) -> (all X_1 all Y_1 (plus_plus(X_a,times_times(X_a,X_1,X_1),times_times(X_a,Y_1,Y_1)) = zero_zero(X_a) <-> ti(X_a,X_1) = zero_zero(X_a) & ti(X_a,Y_1) = zero_zero(X_a))))) # label(fact_91_sum__squares__eq__zero__iff) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 113 (all X_b (number(X_b) & semiring(X_b) -> (all A_1 all B all V times_times(X_b,plus_plus(X_b,A_1,B),number_number_of(X_b,V)) = plus_plus(X_b,times_times(X_b,A_1,number_number_of(X_b,V)),times_times(X_b,B,number_number_of(X_b,V)))))) # label(fact_92_left__distrib__number__of) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 114 (all X_b (number(X_b) & semiring(X_b) -> (all V all B all C times_times(X_b,number_number_of(X_b,V),plus_plus(X_b,B,C)) = plus_plus(X_b,times_times(X_b,number_number_of(X_b,V),B),times_times(X_b,number_number_of(X_b,V),C))))) # label(fact_93_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 115 (all Z1 all Z2 all Z3 plus_plus(int,plus_plus(int,Z1,Z2),Z3) = plus_plus(int,Z1,plus_plus(int,Z2,Z3))) # label(fact_94_zadd__assoc) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 116 (all X all Y all Z plus_plus(int,X,plus_plus(int,Y,Z)) = plus_plus(int,Y,plus_plus(int,X,Z))) # label(fact_95_zadd__left__commute) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 117 (all Z all W plus_plus(int,Z,W) = plus_plus(int,W,Z)) # label(fact_96_zadd__commute) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01 118 (all T all A ti(T,ti(T,A)) = ti(T,A)) # label(help_ti_idem) # label(axiom) # label(non_clause). [assumption].
% 0.42/1.01
% 0.42/1.01 ============================== end of process non-clausal formulas ===
% 0.42/1.01
% 0.42/1.01 ============================== PROCESS INITIAL CLAUSES ===============
% 0.42/1.01
% 0.42/1.01 ============================== PREDICATE ELIMINATION =================
% 0.42/1.01 119 semiring_1(int) # label(arity_Int_Oint___Rings_Osemiring__1) # label(axiom). [assumption].
% 0.42/1.01 120 -semiring_1(A) | ti(A,one_one(A)) = one_one(A) # label(tsy_c_Groups_Oone__class_Oone_res) # label(axiom). [clausify(1)].
% 0.42/1.01 121 -semiring_1(A) | zero_zero(A) = ti(A,zero_zero(A)) # label(tsy_c_Groups_Ozero__class_Ozero_1_res) # label(axiom). [clausify(15)].
% 0.42/1.01 122 -semiring_1(A) | power_power(A,ti(A,B),C) = power_power(A,B,C) # label(tsy_c_Power_Opower__class_Opower_arg1) # label(axiom). [clausify(32)].
% 0.42/1.01 123 -semiring_1(A) | power_power(A,B,ti(nat,C)) = power_power(A,B,C) # label(tsy_c_Power_Opower__class_Opower_arg2) # label(axiom). [clausify(33)].
% 0.42/1.01 124 -semiring_1(A) | power_power(A,B,C) = ti(A,power_power(A,B,C)) # label(tsy_c_Power_Opower__class_Opower_res) # label(axiom). [clausify(34)].
% 0.42/1.01 125 -semiring_1(A) | power_power(A,one_one(A),number_number_of(nat,bit0(bit1(pls)))) = one_one(A) # label(fact_12_one__power2) # label(axiom). [clausify(42)].
% 0.42/1.01 126 -semiring_1(A) | power_power(A,zero_zero(A),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(A) # label(fact_13_zero__power2) # label(axiom). [clausify(43)].
% 0.42/1.01 Derived: ti(int,one_one(int)) = one_one(int). [resolve(119,a,120,a)].
% 0.42/1.01 Derived: zero_zero(int) = ti(int,zero_zero(int)). [resolve(119,a,121,a)].
% 0.42/1.01 Derived: power_power(int,ti(int,A),B) = power_power(int,A,B). [resolve(119,a,122,a)].
% 0.42/1.01 Derived: power_power(int,A,ti(nat,B)) = power_power(int,A,B). [resolve(119,a,123,a)].
% 0.42/1.01 Derived: power_power(int,A,B) = ti(int,power_power(int,A,B)). [resolve(119,a,124,a)].
% 0.42/1.01 Derived: power_power(int,one_one(int),number_number_of(nat,bit0(bit1(pls)))) = one_one(int). [resolve(119,a,125,a)].
% 0.42/1.01 Derived: power_power(int,zero_zero(int),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(int). [resolve(119,a,126,a)].
% 0.42/1.01 127 semiring_1(nat) # label(arity_Nat_Onat___Rings_Osemiring__1) # label(axiom). [assumption].
% 0.42/1.01 Derived: ti(nat,one_one(nat)) = one_one(nat). [resolve(127,a,120,a)].
% 0.42/1.01 Derived: zero_zero(nat) = ti(nat,zero_zero(nat)). [resolve(127,a,121,a)].
% 0.42/1.01 Derived: power_power(nat,ti(nat,A),B) = power_power(nat,A,B). [resolve(127,a,122,a)].
% 0.42/1.01 Derived: power_power(nat,A,ti(nat,B)) = power_power(nat,A,B). [resolve(127,a,123,a)].
% 0.42/1.01 Derived: power_power(nat,A,B) = ti(nat,power_power(nat,A,B)). [resolve(127,a,124,a)].
% 0.42/1.01 Derived: power_power(nat,one_one(nat),number_number_of(nat,bit0(bit1(pls)))) = one_one(nat). [resolve(127,a,125,a)].
% 0.42/1.01 Derived: power_power(nat,zero_zero(nat),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(nat). [resolve(127,a,126,a)].
% 0.42/1.01 128 number(int) # label(arity_Int_Oint___Int_Onumber) # label(axiom). [assumption].
% 0.42/1.01 129 -number(A) | -semiring(A) | plus_plus(A,ti(A,B),C) = plus_plus(A,B,C) # label(tsy_c_Groups_Oplus__class_Oplus_0_arg1) # label(axiom). [clausify(2)].
% 0.42/1.01 130 -number(A) | -semiring(A) | plus_plus(A,B,ti(A,C)) = plus_plus(A,B,C) # label(tsy_c_Groups_Oplus__class_Oplus_0_arg2) # label(axiom). [clausify(3)].
% 0.42/1.01 131 -number(A) | -semiring(A) | plus_plus(A,B,C) = ti(A,plus_plus(A,B,C)) # label(tsy_c_Groups_Oplus__class_Oplus_0_res) # label(axiom). [clausify(4)].
% 0.42/1.01 132 -number(A) | -semiring(A) | times_times(A,ti(A,B),C) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_0_arg1) # label(axiom). [clausify(8)].
% 0.42/1.01 133 -number(A) | -semiring(A) | times_times(A,B,ti(A,C)) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_0_arg2) # label(axiom). [clausify(9)].
% 0.42/1.01 134 -number(A) | -semiring(A) | times_times(A,B,C) = ti(A,times_times(A,B,C)) # label(tsy_c_Groups_Otimes__class_Otimes_0_res) # label(axiom). [clausify(10)].
% 0.42/1.01 135 -number(A) | number_number_of(A,ti(int,B)) = number_number_of(A,B) # label(tsy_c_Int_Onumber__class_Onumber__of_arg1) # label(axiom). [clausify(22)].
% 0.42/1.01 136 -number(A) | number_number_of(A,B) = ti(A,number_number_of(A,B)) # label(tsy_c_Int_Onumber__class_Onumber__of_res) # label(axiom). [clausify(23)].
% 0.42/1.01 137 -number(A) | -linorder(A) | -ord_less(A,ti(A,B),C) | ord_less(A,B,C) # label(tsy_c_Orderings_Oord__class_Oless_0_arg1) # label(axiom). [clausify(24)].
% 0.42/1.01 138 -number(A) | -linorder(A) | ord_less(A,ti(A,B),C) | -ord_less(A,B,C) # label(tsy_c_Orderings_Oord__class_Oless_0_arg1) # label(axiom). [clausify(24)].
% 0.42/1.01 139 -number(A) | -linorder(A) | -ord_less(A,B,ti(A,C)) | ord_less(A,B,C) # label(tsy_c_Orderings_Oord__class_Oless_0_arg2) # label(axiom). [clausify(25)].
% 0.42/1.01 140 -number(A) | -linorder(A) | ord_less(A,B,ti(A,C)) | -ord_less(A,B,C) # label(tsy_c_Orderings_Oord__class_Oless_0_arg2) # label(axiom). [clausify(25)].
% 0.42/1.01 141 -number(A) | -linorder(A) | -ord_less_eq(A,ti(A,B),C) | ord_less_eq(A,B,C) # label(tsy_c_Orderings_Oord__class_Oless__eq_0_arg1) # label(axiom). [clausify(28)].
% 0.42/1.01 142 -number(A) | -linorder(A) | ord_less_eq(A,ti(A,B),C) | -ord_less_eq(A,B,C) # label(tsy_c_Orderings_Oord__class_Oless__eq_0_arg1) # label(axiom). [clausify(28)].
% 0.42/1.01 143 -number(A) | -linorder(A) | -ord_less_eq(A,B,ti(A,C)) | ord_less_eq(A,B,C) # label(tsy_c_Orderings_Oord__class_Oless__eq_0_arg2) # label(axiom). [clausify(29)].
% 0.42/1.01 144 -number(A) | -linorder(A) | ord_less_eq(A,B,ti(A,C)) | -ord_less_eq(A,B,C) # label(tsy_c_Orderings_Oord__class_Oless__eq_0_arg2) # label(axiom). [clausify(29)].
% 0.42/1.01 145 -number(A) | -linorder(A) | -ord_less_eq(A,number_number_of(A,B),number_number_of(A,C)) | -ord_less(A,number_number_of(A,C),number_number_of(A,B)) # label(fact_50_le__number__of__eq__not__less) # label(axiom). [clausify(76)].
% 0.42/1.01 146 -number(A) | -linorder(A) | ord_less_eq(A,number_number_of(A,B),number_number_of(A,C)) | ord_less(A,number_number_of(A,C),number_number_of(A,B)) # label(fact_50_le__number__of__eq__not__less) # label(axiom). [clausify(76)].
% 0.42/1.01 147 -number(A) | -semiring(A) | times_times(A,plus_plus(A,B,C),number_number_of(A,D)) = plus_plus(A,times_times(A,B,number_number_of(A,D)),times_times(A,C,number_number_of(A,D))) # label(fact_92_left__distrib__number__of) # label(axiom). [clausify(113)].
% 0.42/1.01 148 -number(A) | -semiring(A) | times_times(A,number_number_of(A,B),plus_plus(A,C,D)) = plus_plus(A,times_times(A,number_number_of(A,B),C),times_times(A,number_number_of(A,B),D)) # label(fact_93_right__distrib__number__of) # label(axiom). [clausify(114)].
% 0.42/1.01 Derived: -semiring(int) | plus_plus(int,ti(int,A),B) = plus_plus(int,A,B). [resolve(128,a,129,a)].
% 0.42/1.01 Derived: -semiring(int) | plus_plus(int,A,ti(int,B)) = plus_plus(int,A,B). [resolve(128,a,130,a)].
% 0.42/1.01 Derived: -semiring(int) | plus_plus(int,A,B) = ti(int,plus_plus(int,A,B)). [resolve(128,a,131,a)].
% 0.42/1.01 Derived: -semiring(int) | times_times(int,ti(int,A),B) = times_times(int,A,B). [resolve(128,a,132,a)].
% 0.42/1.01 Derived: -semiring(int) | times_times(int,A,ti(int,B)) = times_times(int,A,B). [resolve(128,a,133,a)].
% 0.42/1.01 Derived: -semiring(int) | times_times(int,A,B) = ti(int,times_times(int,A,B)). [resolve(128,a,134,a)].
% 0.42/1.01 Derived: number_number_of(int,ti(int,A)) = number_number_of(int,A). [resolve(128,a,135,a)].
% 0.42/1.01 Derived: number_number_of(int,A) = ti(int,number_number_of(int,A)). [resolve(128,a,136,a)].
% 0.42/1.01 Derived: -linorder(int) | -ord_less(int,ti(int,A),B) | ord_less(int,A,B). [resolve(128,a,137,a)].
% 0.42/1.01 Derived: -linorder(int) | ord_less(int,ti(int,A),B) | -ord_less(int,A,B). [resolve(128,a,138,a)].
% 0.42/1.01 Derived: -linorder(int) | -ord_less(int,A,ti(int,B)) | ord_less(int,A,B). [resolve(128,a,139,a)].
% 0.42/1.01 Derived: -linorder(int) | ord_less(int,A,ti(int,B)) | -ord_less(int,A,B). [resolve(128,a,140,a)].
% 0.42/1.01 Derived: -linorder(int) | -ord_less_eq(int,ti(int,A),B) | ord_less_eq(int,A,B). [resolve(128,a,141,a)].
% 0.42/1.01 Derived: -linorder(int) | ord_less_eq(int,ti(int,A),B) | -ord_less_eq(int,A,B). [resolve(128,a,142,a)].
% 0.42/1.01 Derived: -linorder(int) | -ord_less_eq(int,A,ti(int,B)) | ord_less_eq(int,A,B). [resolve(128,a,143,a)].
% 0.42/1.01 Derived: -linorder(int) | ord_less_eq(int,A,ti(int,B)) | -ord_less_eq(int,A,B). [resolve(128,a,144,a)].
% 0.42/1.01 Derived: -linorder(int) | -ord_less_eq(int,number_number_of(int,A),number_number_of(int,B)) | -ord_less(int,number_number_of(int,B),number_number_of(int,A)). [resolve(128,a,145,a)].
% 0.42/1.01 Derived: -linorder(int) | ord_less_eq(int,number_number_of(int,A),number_number_of(int,B)) | ord_less(int,number_number_of(int,B),number_number_of(int,A)). [resolve(128,a,146,a)].
% 0.42/1.01 Derived: -semiring(int) | times_times(int,plus_plus(int,A,B),number_number_of(int,C)) = plus_plus(int,times_times(int,A,number_number_of(int,C)),times_times(int,B,number_number_of(int,C))). [resolve(128,a,147,a)].
% 0.42/1.01 Derived: -semiring(int) | times_times(int,number_number_of(int,A),plus_plus(int,B,C)) = plus_plus(int,times_times(int,number_number_of(int,A),B),times_times(int,number_number_of(int,A),C)). [resolve(128,a,148,a)].
% 0.42/1.01 149 number(nat) # label(arity_Nat_Onat___Int_Onumber) # label(axiom). [assumption].
% 0.42/1.01 Derived: -semiring(nat) | plus_plus(nat,ti(nat,A),B) = plus_plus(nat,A,B). [resolve(149,a,129,a)].
% 0.42/1.01 Derived: -semiring(nat) | plus_plus(nat,A,ti(nat,B)) = plus_plus(nat,A,B). [resolve(149,a,130,a)].
% 0.42/1.01 Derived: -semiring(nat) | plus_plus(nat,A,B) = ti(nat,plus_plus(nat,A,B)). [resolve(149,a,131,a)].
% 0.42/1.01 Derived: -semiring(nat) | times_times(nat,ti(nat,A),B) = times_times(nat,A,B). [resolve(149,a,132,a)].
% 0.42/1.01 Derived: -semiring(nat) | times_times(nat,A,ti(nat,B)) = times_times(nat,A,B). [resolve(149,a,133,a)].
% 0.42/1.01 Derived: -semiring(nat) | times_times(nat,A,B) = ti(nat,times_times(nat,A,B)). [resolve(149,a,134,a)].
% 0.42/1.01 Derived: number_number_of(nat,ti(int,A)) = number_number_of(nat,A). [resolve(149,a,135,a)].
% 0.42/1.01 Derived: number_number_of(nat,A) = ti(nat,number_number_of(nat,A)). [resolve(149,a,136,a)].
% 0.42/1.01 Derived: -linorder(nat) | -ord_less(nat,ti(nat,A),B) | ord_less(nat,A,B). [resolve(149,a,137,a)].
% 0.42/1.01 Derived: -linorder(nat) | ord_less(nat,ti(nat,A),B) | -ord_less(nat,A,B). [resolve(149,a,138,a)].
% 0.42/1.01 Derived: -linorder(nat) | -ord_less(nat,A,ti(nat,B)) | ord_less(nat,A,B). [resolve(149,a,139,a)].
% 0.42/1.01 Derived: -linorder(nat) | ord_less(nat,A,ti(nat,B)) | -ord_less(nat,A,B). [resolve(149,a,140,a)].
% 0.42/1.01 Derived: -linorder(nat) | -ord_less_eq(nat,ti(nat,A),B) | ord_less_eq(nat,A,B). [resolve(149,a,141,a)].
% 0.42/1.01 Derived: -linorder(nat) | ord_less_eq(nat,ti(nat,A),B) | -ord_less_eq(nat,A,B). [resolve(149,a,142,a)].
% 0.42/1.01 Derived: -linorder(nat) | -ord_less_eq(nat,A,ti(nat,B)) | ord_less_eq(nat,A,B). [resolve(149,a,143,a)].
% 0.42/1.02 Derived: -linorder(nat) | ord_less_eq(nat,A,ti(nat,B)) | -ord_less_eq(nat,A,B). [resolve(149,a,144,a)].
% 0.42/1.02 Derived: -linorder(nat) | -ord_less_eq(nat,number_number_of(nat,A),number_number_of(nat,B)) | -ord_less(nat,number_number_of(nat,B),number_number_of(nat,A)). [resolve(149,a,145,a)].
% 0.42/1.02 Derived: -linorder(nat) | ord_less_eq(nat,number_number_of(nat,A),number_number_of(nat,B)) | ord_less(nat,number_number_of(nat,B),number_number_of(nat,A)). [resolve(149,a,146,a)].
% 0.42/1.02 Derived: -semiring(nat) | times_times(nat,plus_plus(nat,A,B),number_number_of(nat,C)) = plus_plus(nat,times_times(nat,A,number_number_of(nat,C)),times_times(nat,B,number_number_of(nat,C))). [resolve(149,a,147,a)].
% 0.42/1.02 Derived: -semiring(nat) | times_times(nat,number_number_of(nat,A),plus_plus(nat,B,C)) = plus_plus(nat,times_times(nat,number_number_of(nat,A),B),times_times(nat,number_number_of(nat,A),C)). [resolve(149,a,148,a)].
% 0.42/1.02 150 linordered_ring(int) # label(arity_Int_Oint___Rings_Olinordered__ring) # label(axiom). [assumption].
% 0.42/1.02 151 -linordered_ring(A) | plus_plus(A,ti(A,B),C) = plus_plus(A,B,C) # label(tsy_c_Groups_Oplus__class_Oplus_1_arg1) # label(axiom). [clausify(5)].
% 0.42/1.02 152 -linordered_ring(A) | plus_plus(A,B,ti(A,C)) = plus_plus(A,B,C) # label(tsy_c_Groups_Oplus__class_Oplus_1_arg2) # label(axiom). [clausify(6)].
% 0.42/1.02 153 -linordered_ring(A) | plus_plus(A,B,C) = ti(A,plus_plus(A,B,C)) # label(tsy_c_Groups_Oplus__class_Oplus_1_res) # label(axiom). [clausify(7)].
% 0.42/1.02 154 -linordered_ring(A) | times_times(A,ti(A,B),C) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_1_arg1) # label(axiom). [clausify(11)].
% 0.42/1.02 155 -linordered_ring(A) | times_times(A,B,ti(A,C)) = times_times(A,B,C) # label(tsy_c_Groups_Otimes__class_Otimes_1_arg2) # label(axiom). [clausify(12)].
% 0.42/1.02 156 -linordered_ring(A) | times_times(A,B,C) = ti(A,times_times(A,B,C)) # label(tsy_c_Groups_Otimes__class_Otimes_1_res) # label(axiom). [clausify(13)].
% 0.42/1.02 157 -linordered_ring(A) | zero_zero(A) = ti(A,zero_zero(A)) # label(tsy_c_Groups_Ozero__class_Ozero_0_res) # label(axiom). [clausify(14)].
% 0.42/1.02 158 -linordered_ring(A) | -ord_less_eq(A,ti(A,B),C) | ord_less_eq(A,B,C) # label(tsy_c_Orderings_Oord__class_Oless__eq_1_arg1) # label(axiom). [clausify(30)].
% 0.42/1.02 159 -linordered_ring(A) | ord_less_eq(A,ti(A,B),C) | -ord_less_eq(A,B,C) # label(tsy_c_Orderings_Oord__class_Oless__eq_1_arg1) # label(axiom). [clausify(30)].
% 0.42/1.02 160 -linordered_ring(A) | -ord_less_eq(A,B,ti(A,C)) | ord_less_eq(A,B,C) # label(tsy_c_Orderings_Oord__class_Oless__eq_1_arg2) # label(axiom). [clausify(31)].
% 0.42/1.02 161 -linordered_ring(A) | ord_less_eq(A,B,ti(A,C)) | -ord_less_eq(A,B,C) # label(tsy_c_Orderings_Oord__class_Oless__eq_1_arg2) # label(axiom). [clausify(31)].
% 0.42/1.02 162 -linordered_ring(A) | ord_less_eq(A,zero_zero(A),plus_plus(A,times_times(A,B,B),times_times(A,C,C))) # label(fact_46_sum__squares__ge__zero) # label(axiom). [clausify(72)].
% 0.42/1.02 Derived: plus_plus(int,ti(int,A),B) = plus_plus(int,A,B). [resolve(150,a,151,a)].
% 0.42/1.02 Derived: plus_plus(int,A,ti(int,B)) = plus_plus(int,A,B). [resolve(150,a,152,a)].
% 0.42/1.02 Derived: plus_plus(int,A,B) = ti(int,plus_plus(int,A,B)). [resolve(150,a,153,a)].
% 0.42/1.02 Derived: times_times(int,ti(int,A),B) = times_times(int,A,B). [resolve(150,a,154,a)].
% 0.42/1.02 Derived: times_times(int,A,ti(int,B)) = times_times(int,A,B). [resolve(150,a,155,a)].
% 0.42/1.02 Derived: times_times(int,A,B) = ti(int,times_times(int,A,B)). [resolve(150,a,156,a)].
% 0.42/1.02 Derived: -ord_less_eq(int,ti(int,A),B) | ord_less_eq(int,A,B). [resolve(150,a,158,a)].
% 0.42/1.02 Derived: ord_less_eq(int,ti(int,A),B) | -ord_less_eq(int,A,B). [resolve(150,a,159,a)].
% 0.42/1.02 Derived: -ord_less_eq(int,A,ti(int,B)) | ord_less_eq(int,A,B). [resolve(150,a,160,a)].
% 0.42/1.02 Derived: ord_less_eq(int,A,ti(int,B)) | -ord_less_eq(int,A,B). [resolve(150,a,161,a)].
% 0.42/1.02 Derived: ord_less_eq(int,zero_zero(int),plus_plus(int,times_times(int,A,A),times_times(int,B,B))). [resolve(150,a,162,a)].
% 0.42/1.02 163 linordered_idom(int) # label(arity_Int_Oint___Rings_Olinordered__idom) # label(axiom). [assumption].
% 0.42/1.02 164 -linordered_idom(A) | -ord_less(A,ti(A,B),C) | ord_less(A,B,C) # label(tsy_c_Orderings_Oord__class_Oless_1_arg1) # label(axiom). [clausify(26)].
% 0.42/1.02 165 -linordered_idom(A) | ord_less(A,ti(A,B),C) | -ord_less(A,B,C) # label(tsy_c_Orderings_Oord__class_Oless_1_arg1) # label(axiom). [clausify(26)].
% 0.42/1.02 166 -linordered_idom(A) | -ord_less(A,B,ti(A,C)) | ord_less(A,B,C) # label(tsy_c_Orderings_Oord__class_Oless_1_arg2) # label(axiom). [clausify(27)].
% 0.42/1.02 167 -linordered_idom(A) | ord_less(A,B,ti(A,C)) | -ord_less(A,B,C) # label(tsy_c_Orderings_Oord__class_Oless_1_arg2) # label(axiom). [clausify(27)].
% 0.42/1.02 168 -linordered_idom(A) | -ord_less(A,plus_plus(A,power_power(A,B,number_number_of(nat,bit0(bit1(pls)))),power_power(A,C,number_number_of(nat,bit0(bit1(pls))))),zero_zero(A)) # label(fact_7_not__sum__power2__lt__zero) # label(axiom). [clausify(37)].
% 0.42/1.02 169 -linordered_idom(A) | -ord_less(A,zero_zero(A),plus_plus(A,power_power(A,B,number_number_of(nat,bit0(bit1(pls)))),power_power(A,C,number_number_of(nat,bit0(bit1(pls)))))) | zero_zero(A) != ti(A,B) | zero_zero(A) != ti(A,C) # label(fact_8_sum__power2__gt__zero__iff) # label(axiom). [clausify(38)].
% 0.42/1.02 170 -linordered_idom(A) | ord_less(A,zero_zero(A),plus_plus(A,power_power(A,B,number_number_of(nat,bit0(bit1(pls)))),power_power(A,C,number_number_of(nat,bit0(bit1(pls)))))) | zero_zero(A) = ti(A,B) # label(fact_8_sum__power2__gt__zero__iff) # label(axiom). [clausify(38)].
% 0.42/1.02 171 -linordered_idom(A) | ord_less(A,zero_zero(A),plus_plus(A,power_power(A,B,number_number_of(nat,bit0(bit1(pls)))),power_power(A,C,number_number_of(nat,bit0(bit1(pls)))))) | zero_zero(A) = ti(A,C) # label(fact_8_sum__power2__gt__zero__iff) # label(axiom). [clausify(38)].
% 0.42/1.02 172 -linordered_idom(A) | zero_zero(A) != plus_plus(A,power_power(A,B,number_number_of(nat,bit0(bit1(pls)))),power_power(A,C,number_number_of(nat,bit0(bit1(pls))))) | zero_zero(A) = ti(A,B) # label(fact_9_sum__power2__eq__zero__iff) # label(axiom). [clausify(39)].
% 0.42/1.02 173 -linordered_idom(A) | zero_zero(A) != plus_plus(A,power_power(A,B,number_number_of(nat,bit0(bit1(pls)))),power_power(A,C,number_number_of(nat,bit0(bit1(pls))))) | zero_zero(A) = ti(A,C) # label(fact_9_sum__power2__eq__zero__iff) # label(axiom). [clausify(39)].
% 0.42/1.02 174 -linordered_idom(A) | zero_zero(A) = plus_plus(A,power_power(A,B,number_number_of(nat,bit0(bit1(pls)))),power_power(A,C,number_number_of(nat,bit0(bit1(pls))))) | zero_zero(A) != ti(A,B) | zero_zero(A) != ti(A,C) # label(fact_9_sum__power2__eq__zero__iff) # label(axiom). [clausify(39)].
% 0.42/1.02 175 -linordered_idom(A) | -ord_less(A,power_power(A,B,number_number_of(nat,bit0(bit1(pls)))),zero_zero(A)) # label(fact_10_power2__less__0) # label(axiom). [clausify(40)].
% 0.42/1.02 176 -linordered_idom(A) | -ord_less(A,zero_zero(A),power_power(A,B,number_number_of(nat,bit0(bit1(pls))))) | zero_zero(A) != ti(A,B) # label(fact_11_zero__less__power2) # label(axiom). [clausify(41)].
% 0.42/1.02 177 -linordered_idom(A) | ord_less(A,zero_zero(A),power_power(A,B,number_number_of(nat,bit0(bit1(pls))))) | zero_zero(A) = ti(A,B) # label(fact_11_zero__less__power2) # label(axiom). [clausify(41)].
% 0.42/1.02 178 -number_ring(A) | -linordered_idom(A) | -ord_less_eq(A,number_number_of(A,B),number_number_of(A,C)) | ord_less_eq(int,B,C) # label(fact_27_le__number__of) # label(axiom). [clausify(54)].
% 0.42/1.02 179 -number_ring(A) | -linordered_idom(A) | ord_less_eq(A,number_number_of(A,B),number_number_of(A,C)) | -ord_less_eq(int,B,C) # label(fact_27_le__number__of) # label(axiom). [clausify(54)].
% 0.42/1.02 180 -number_ring(A) | -linordered_idom(A) | -ord_less_eq(A,number_number_of(A,B),zero_zero(A)) | ord_less_eq(int,B,pls) # label(fact_47_le__special_I3_J) # label(axiom). [clausify(73)].
% 0.42/1.02 181 -number_ring(A) | -linordered_idom(A) | ord_less_eq(A,number_number_of(A,B),zero_zero(A)) | -ord_less_eq(int,B,pls) # label(fact_47_le__special_I3_J) # label(axiom). [clausify(73)].
% 0.42/1.02 182 -number_ring(A) | -linordered_idom(A) | -ord_less_eq(A,zero_zero(A),number_number_of(A,B)) | ord_less_eq(int,pls,B) # label(fact_48_le__special_I1_J) # label(axiom). [clausify(74)].
% 0.42/1.02 183 -number_ring(A) | -linordered_idom(A) | ord_less_eq(A,zero_zero(A),number_number_of(A,B)) | -ord_less_eq(int,pls,B) # label(fact_48_le__special_I1_J) # label(axiom). [clausify(74)].
% 0.42/1.02 184 -number_ring(A) | -linordered_idom(A) | -ord_less_eq(A,number_number_of(A,B),one_one(A)) | ord_less_eq(int,B,bit1(pls)) # label(fact_59_le__special_I4_J) # label(axiom). [clausify(83)].
% 0.42/1.02 185 -number_ring(A) | -linordered_idom(A) | ord_less_eq(A,number_number_of(A,B),one_one(A)) | -ord_less_eq(int,B,bit1(pls)) # label(fact_59_le__special_I4_J) # label(axiom). [clausify(83)].
% 0.42/1.02 186 -number_ring(A) | -linordered_idom(A) | -ord_less_eq(A,one_one(A),number_number_of(A,B)) | ord_less_eq(int,bit1(pls),B) # label(fact_60_le__special_I2_J) # label(axiom). [clausify(84)].
% 0.42/1.02 187 -number_ring(A) | -linordered_idom(A) | ord_less_eq(A,one_one(A),number_number_of(A,B)) | -ord_less_eq(int,bit1(pls),B) # label(fact_60_le__special_I2_J) # label(axiom). [clausify(84)].
% 0.42/1.02 Derived: -ord_less(int,ti(int,A),B) | ord_less(int,A,B). [resolve(163,a,164,a)].
% 0.42/1.02 Derived: ord_less(int,ti(int,A),B) | -ord_less(int,A,B). [resolve(163,a,165,a)].
% 0.42/1.02 Derived: -ord_less(int,A,ti(int,B)) | ord_less(int,A,B). [resolve(163,a,166,a)].
% 0.42/1.02 Derived: ord_less(int,A,ti(int,B)) | -ord_less(int,A,B). [resolve(163,a,167,a)].
% 0.42/1.02 Derived: -ord_less(int,plus_plus(int,power_power(int,A,number_number_of(nat,bit0(bit1(pls)))),power_power(int,B,number_number_of(nat,bit0(bit1(pls))))),zero_zero(int)). [resolve(163,a,168,a)].
% 0.42/1.02 Derived: -ord_less(int,zero_zero(int),plus_plus(int,power_power(int,A,number_number_of(nat,bit0(bit1(pls)))),power_power(int,B,number_number_of(nat,bit0(bit1(pls)))))) | zero_zero(int) != ti(int,A) | zero_zero(int) != ti(int,B). [resolve(163,a,169,a)].
% 0.42/1.02 Derived: ord_less(int,zero_zero(int),plus_plus(int,power_power(int,A,number_number_of(nat,bit0(bit1(pls)))),power_power(int,B,number_number_of(nat,bit0(bit1(pls)))))) | zero_zero(int) = ti(int,A). [resolve(163,a,170,a)].
% 0.42/1.02 Derived: ord_less(int,zero_zero(int),plus_plus(int,power_power(int,A,number_number_of(nat,bit0(bit1(pls)))),power_power(int,B,number_number_of(nat,bit0(bit1(pls)))))) | zero_zero(int) = ti(int,B). [resolve(163,a,171,a)].
% 0.42/1.02 Derived: zero_zero(int) != plus_plus(int,power_power(int,A,number_number_of(nat,bit0(bit1(pls)))),power_power(int,B,number_number_of(nat,bit0(bit1(pls))))) | zero_zero(int) = ti(int,A). [resolve(163,a,172,a)].
% 0.42/1.02 Derived: zero_zero(int) != plus_plus(int,power_power(int,A,number_number_of(nat,bit0(bit1(pls)))),power_power(int,B,number_number_of(nat,bit0(bit1(pls))))) | zero_zero(int) = ti(int,B). [resolve(163,a,173,a)].
% 0.42/1.02 Derived: zero_zero(int) = plus_plus(int,power_power(int,A,number_number_of(nat,bit0(bit1(pls)))),power_power(int,B,number_number_of(nat,bit0(bit1(pls))))) | zero_zero(int) != ti(int,A) | zero_zero(int) != ti(int,B). [resolve(163,a,174,a)].
% 0.42/1.02 Derived: -ord_less(int,power_power(int,A,number_number_of(nat,bit0(bit1(pls)))),zero_zero(int)). [resolve(163,a,175,a)].
% 0.42/1.02 Derived: -ord_less(int,zero_zero(int),power_power(int,A,number_number_of(nat,bit0(bit1(pls))))) | zero_zero(int) != ti(int,A). [resolve(163,a,176,a)].
% 0.42/1.02 Derived: ord_less(int,zero_zero(int),power_power(int,A,number_number_of(nat,bit0(bit1(pls))))) | zero_zero(int) = ti(int,A). [resolve(163,a,177,a)].
% 0.42/1.02 Derived: -number_ring(int) | -ord_less_eq(int,number_number_of(int,A),number_number_of(int,B)) | ord_less_eq(int,A,B). [resolve(163,a,178,b)].
% 0.42/1.02 Derived: -number_ring(int) | ord_less_eq(int,number_number_of(int,A),number_number_of(int,B)) | -ord_less_eq(int,A,B). [resolve(163,a,179,b)].
% 0.42/1.02 Derived: -number_ring(int) | -ord_less_eq(int,number_number_of(int,A),zero_zero(int)) | ord_less_eq(int,A,pls). [resolve(163,a,180,b)].
% 0.42/1.02 Derived: -number_ring(int) | ord_less_eq(int,number_number_of(int,A),zero_zero(int)) | -ord_less_eq(int,A,pls). [resolve(163,a,181,b)].
% 0.42/1.02 Derived: -number_ring(int) | -ord_less_eq(int,zero_zero(int),number_number_of(int,A)) | ord_less_eq(int,pls,A). [resolve(163,a,182,b)].
% 0.42/1.02 Derived: -number_ring(int) | ord_less_eq(int,zero_zero(int),number_number_of(int,A)) | -ord_less_eq(int,pls,A). [resolve(163,a,183,b)].
% 0.42/1.02 Derived: -number_ring(int) | -ord_less_eq(int,number_number_of(int,A),one_one(int)) | ord_less_eq(int,A,bit1(pls)). [resolve(163,a,184,b)].
% 0.42/1.02 Derived: -number_ring(int) | ord_less_eq(int,number_number_of(int,A),one_one(int)) | -ord_less_eq(int,A,bit1(pls)). [resolve(163,a,185,b)].
% 0.74/1.03 Derived: -number_ring(int) | -ord_less_eq(int,one_one(int),number_number_of(int,A)) | ord_less_eq(int,bit1(pls),A). [resolve(163,a,186,b)].
% 0.74/1.03 Derived: -number_ring(int) | ord_less_eq(int,one_one(int),number_number_of(int,A)) | -ord_less_eq(int,bit1(pls),A). [resolve(163,a,187,b)].
% 0.74/1.03 188 ring_11004092258visors(int) # label(arity_Int_Oint___Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption].
% 0.74/1.03 189 -ring_11004092258visors(A) | power_power(A,B,number_number_of(nat,bit0(bit1(pls)))) != zero_zero(A) | zero_zero(A) = ti(A,B) # label(fact_14_zero__eq__power2) # label(axiom). [clausify(44)].
% 0.74/1.03 190 -ring_11004092258visors(A) | power_power(A,B,number_number_of(nat,bit0(bit1(pls)))) = zero_zero(A) | zero_zero(A) != ti(A,B) # label(fact_14_zero__eq__power2) # label(axiom). [clausify(44)].
% 0.74/1.03 Derived: power_power(int,A,number_number_of(nat,bit0(bit1(pls)))) != zero_zero(int) | zero_zero(int) = ti(int,A). [resolve(188,a,189,a)].
% 0.74/1.03 Derived: power_power(int,A,number_number_of(nat,bit0(bit1(pls)))) = zero_zero(int) | zero_zero(int) != ti(int,A). [resolve(188,a,190,a)].
% 0.74/1.03 191 number_semiring(int) # label(arity_Int_Oint___Int_Onumber__semiring) # label(axiom). [assumption].
% 0.74/1.03 192 -number_semiring(A) | -ord_less_eq(int,pls,B) | -ord_less_eq(int,pls,C) | number_number_of(A,times_times(int,B,C)) = times_times(A,number_number_of(A,B),number_number_of(A,C)) # label(fact_41_semiring__mult__number__of) # label(axiom). [clausify(67)].
% 0.74/1.03 193 -number_semiring(A) | -ord_less_eq(int,pls,B) | -ord_less_eq(int,pls,C) | number_number_of(A,plus_plus(int,B,C)) = plus_plus(A,number_number_of(A,B),number_number_of(A,C)) # label(fact_83_semiring__add__number__of) # label(axiom). [clausify(104)].
% 0.74/1.03 Derived: -ord_less_eq(int,pls,A) | -ord_less_eq(int,pls,B) | number_number_of(int,times_times(int,A,B)) = times_times(int,number_number_of(int,A),number_number_of(int,B)). [resolve(191,a,192,a)].
% 0.74/1.03 Derived: -ord_less_eq(int,pls,A) | -ord_less_eq(int,pls,B) | number_number_of(int,plus_plus(int,A,B)) = plus_plus(int,number_number_of(int,A),number_number_of(int,B)). [resolve(191,a,193,a)].
% 0.74/1.03 194 number_semiring(nat) # label(arity_Nat_Onat___Int_Onumber__semiring) # label(axiom). [assumption].
% 0.74/1.03 Derived: -ord_less_eq(int,pls,A) | -ord_less_eq(int,pls,B) | number_number_of(nat,times_times(int,A,B)) = times_times(nat,number_number_of(nat,A),number_number_of(nat,B)). [resolve(194,a,192,a)].
% 0.74/1.03 Derived: -ord_less_eq(int,pls,A) | -ord_less_eq(int,pls,B) | number_number_of(nat,plus_plus(int,A,B)) = plus_plus(nat,number_number_of(nat,A),number_number_of(nat,B)). [resolve(194,a,193,a)].
% 0.74/1.03 195 linord581940658strict(int) # label(arity_Int_Oint___Rings_Olinordered__ring__strict) # label(axiom). [assumption].
% 0.74/1.03 196 -linord581940658strict(A) | -ord_less_eq(A,plus_plus(A,times_times(A,B,B),times_times(A,C,C)),zero_zero(A)) | zero_zero(A) = ti(A,B) # label(fact_45_sum__squares__le__zero__iff) # label(axiom). [clausify(71)].
% 0.74/1.03 197 -linord581940658strict(A) | -ord_less_eq(A,plus_plus(A,times_times(A,B,B),times_times(A,C,C)),zero_zero(A)) | zero_zero(A) = ti(A,C) # label(fact_45_sum__squares__le__zero__iff) # label(axiom). [clausify(71)].
% 0.74/1.03 198 -linord581940658strict(A) | ord_less_eq(A,plus_plus(A,times_times(A,B,B),times_times(A,C,C)),zero_zero(A)) | zero_zero(A) != ti(A,B) | zero_zero(A) != ti(A,C) # label(fact_45_sum__squares__le__zero__iff) # label(axiom). [clausify(71)].
% 0.74/1.03 199 -linord581940658strict(A) | zero_zero(A) != plus_plus(A,times_times(A,B,B),times_times(A,C,C)) | zero_zero(A) = ti(A,B) # label(fact_91_sum__squares__eq__zero__iff) # label(axiom). [clausify(112)].
% 0.74/1.03 200 -linord581940658strict(A) | zero_zero(A) != plus_plus(A,times_times(A,B,B),times_times(A,C,C)) | zero_zero(A) = ti(A,C) # label(fact_91_sum__squares__eq__zero__iff) # label(axiom). [clausify(112)].
% 0.74/1.03 201 -linord581940658strict(A) | zero_zero(A) = plus_plus(A,times_times(A,B,B),times_times(A,C,C)) | zero_zero(A) != ti(A,B) | zero_zero(A) != ti(A,C) # label(fact_91_sum__squares__eq__zero__iff) # label(axiom). [clausify(112)].
% 0.74/1.10 Derived: -ord_less_eq(int,plus_plus(int,times_times(int,A,A),times_times(int,B,B)),zero_zero(int)) | zero_zero(int) = ti(int,A). [resolve(195,a,196,a)].
% 0.74/1.10 Derived: -ord_less_eq(int,plus_plus(int,times_times(int,A,A),times_times(int,B,B)),zero_zero(int)) | zero_zero(int) = ti(int,B). [resolve(195,a,197,a)].
% 0.74/1.10 Derived: ord_less_eq(int,plus_plus(int,times_times(int,A,A),times_times(int,B,B)),zero_zero(int)) | zero_zero(int) != ti(int,A) | zero_zero(int) != ti(int,B). [resolve(195,a,198,a)].
% 0.74/1.10 Derived: zero_zero(int) != plus_plus(int,times_times(int,A,A),times_times(int,B,B)) | zero_zero(int) = ti(int,A). [resolve(195,a,199,a)].
% 0.74/1.10 Derived: zero_zero(int) != plus_plus(int,times_times(int,A,A),times_times(int,B,B)) | zero_zero(int) = ti(int,B). [resolve(195,a,200,a)].
% 0.74/1.10 Derived: zero_zero(int) = plus_plus(int,times_times(int,A,A),times_times(int,B,B)) | zero_zero(int) != ti(int,A) | zero_zero(int) != ti(int,B). [resolve(195,a,201,a)].
% 0.74/1.10 202 ring_char_0(int) # label(arity_Int_Oint___Int_Oring__char__0) # label(axiom). [assumption].
% 0.74/1.10 203 -number_ring(A) | -ring_char_0(A) | number_number_of(A,B) != number_number_of(A,C) | B = C # label(fact_86_eq__number__of) # label(axiom). [clausify(107)].
% 0.74/1.10 204 -number_ring(A) | -ring_char_0(A) | number_number_of(A,B) = number_number_of(A,C) | B != C # label(fact_86_eq__number__of) # label(axiom). [clausify(107)].
% 0.74/1.10 Derived: -number_ring(int) | number_number_of(int,A) != number_number_of(int,B) | A = B. [resolve(202,a,203,b)].
% 0.74/1.10 Derived: -number_ring(int) | number_number_of(int,A) = number_number_of(int,B) | A != B. [resolve(202,a,204,b)].
% 0.74/1.10
% 0.74/1.10 ============================== end predicate elimination =============
% 0.74/1.10
% 0.74/1.10 Auto_denials: (non-Horn, no changes).
% 0.74/1.10
% 0.74/1.10 Term ordering decisions:
% 0.74/1.10 Function symbol KB weights: int=1. nat=1. pls=1. m=1. t=1. s=1. c1=1. number_number_of=1. ti=1. bit0=1. bit1=1. zero_zero=1. one_one=1. undefined=1. plus_plus=1. times_times=1. power_power=1.
% 0.74/1.10
% 0.74/1.10 ============================== end of process initial clauses ========
% 0.74/1.10
% 0.74/1.10 ============================== CLAUSES FOR SEARCH ====================
% 0.74/1.10
% 0.74/1.10 ============================== end of clauses for search =============
% 0.74/1.10
% 0.74/1.10 ============================== SEARCH ================================
% 0.74/1.10
% 0.74/1.10 % Starting search at 0.07 seconds.
% 0.74/1.10
% 0.74/1.10 ============================== PROOF =================================
% 0.74/1.10 % SZS status Theorem
% 0.74/1.10 % SZS output start Refutation
% 0.74/1.10
% 0.74/1.10 % Proof 1 at 0.12 (+ 0.01) seconds.
% 0.74/1.10 % Length of proof is 51.
% 0.74/1.10 % Level of proof is 7.
% 0.74/1.10 % Maximum clause weight is 35.000.
% 0.74/1.10 % Given clauses 148.
% 0.74/1.10
% 0.74/1.10 22 (all B_1 all X_a (number(X_a) -> number_number_of(X_a,ti(int,B_1)) = number_number_of(X_a,B_1))) # label(tsy_c_Int_Onumber__class_Onumber__of_arg1) # label(axiom) # label(non_clause). [assumption].
% 0.74/1.10 37 (all X_a (linordered_idom(X_a) -> (all X all Y -ord_less(X_a,plus_plus(X_a,power_power(X_a,X,number_number_of(nat,bit0(bit1(pls)))),power_power(X_a,Y,number_number_of(nat,bit0(bit1(pls))))),zero_zero(X_a))))) # label(fact_7_not__sum__power2__lt__zero) # label(axiom) # label(non_clause). [assumption].
% 0.74/1.10 39 (all X_a (linordered_idom(X_a) -> (all X_1 all Y_1 (plus_plus(X_a,power_power(X_a,X_1,number_number_of(nat,bit0(bit1(pls)))),power_power(X_a,Y_1,number_number_of(nat,bit0(bit1(pls))))) = zero_zero(X_a) <-> ti(X_a,X_1) = zero_zero(X_a) & ti(X_a,Y_1) = zero_zero(X_a))))) # label(fact_9_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption].
% 0.74/1.10 42 (all X_a (semiring_1(X_a) -> power_power(X_a,one_one(X_a),number_number_of(nat,bit0(bit1(pls)))) = one_one(X_a))) # label(fact_12_one__power2) # label(axiom) # label(non_clause). [assumption].
% 0.74/1.10 43 (all X_a (semiring_1(X_a) -> power_power(X_a,zero_zero(X_a),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(X_a))) # label(fact_13_zero__power2) # label(axiom) # label(non_clause). [assumption].
% 0.74/1.10 49 (all K_1 number_number_of(int,K_1) = K_1) # label(fact_22_number__of__is__id) # label(axiom) # label(non_clause). [assumption].
% 0.74/1.10 50 (all Z all W times_times(int,Z,W) = times_times(int,W,Z)) # label(fact_23_zmult__commute) # label(axiom) # label(non_clause). [assumption].
% 0.74/1.10 85 (all W times_times(int,pls,W) = pls) # label(fact_62_mult__Pls) # label(axiom) # label(non_clause). [assumption].
% 0.74/1.10 86 (all K_1 all L_1 times_times(int,bit0(K_1),L_1) = bit0(times_times(int,K_1,L_1))) # label(fact_63_mult__Bit0) # label(axiom) # label(non_clause). [assumption].
% 0.74/1.10 91 (all Z1 all Z2 all W times_times(int,plus_plus(int,Z1,Z2),W) = plus_plus(int,times_times(int,Z1,W),times_times(int,Z2,W))) # label(fact_68_zadd__zmult__distrib) # label(axiom) # label(non_clause). [assumption].
% 0.74/1.10 117 (all Z all W plus_plus(int,Z,W) = plus_plus(int,W,Z)) # label(fact_96_zadd__commute) # label(axiom) # label(non_clause). [assumption].
% 0.74/1.10 119 semiring_1(int) # label(arity_Int_Oint___Rings_Osemiring__1) # label(axiom). [assumption].
% 0.74/1.10 125 -semiring_1(A) | power_power(A,one_one(A),number_number_of(nat,bit0(bit1(pls)))) = one_one(A) # label(fact_12_one__power2) # label(axiom). [clausify(42)].
% 0.74/1.10 126 -semiring_1(A) | power_power(A,zero_zero(A),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(A) # label(fact_13_zero__power2) # label(axiom). [clausify(43)].
% 0.74/1.10 128 number(int) # label(arity_Int_Oint___Int_Onumber) # label(axiom). [assumption].
% 0.74/1.10 135 -number(A) | number_number_of(A,ti(int,B)) = number_number_of(A,B) # label(tsy_c_Int_Onumber__class_Onumber__of_arg1) # label(axiom). [clausify(22)].
% 0.74/1.10 163 linordered_idom(int) # label(arity_Int_Oint___Rings_Olinordered__idom) # label(axiom). [assumption].
% 0.74/1.10 168 -linordered_idom(A) | -ord_less(A,plus_plus(A,power_power(A,B,number_number_of(nat,bit0(bit1(pls)))),power_power(A,C,number_number_of(nat,bit0(bit1(pls))))),zero_zero(A)) # label(fact_7_not__sum__power2__lt__zero) # label(axiom). [clausify(37)].
% 0.74/1.10 174 -linordered_idom(A) | zero_zero(A) = plus_plus(A,power_power(A,B,number_number_of(nat,bit0(bit1(pls)))),power_power(A,C,number_number_of(nat,bit0(bit1(pls))))) | zero_zero(A) != ti(A,B) | zero_zero(A) != ti(A,C) # label(fact_9_sum__power2__eq__zero__iff) # label(axiom). [clausify(39)].
% 0.74/1.10 223 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))) # label(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) # label(axiom). [assumption].
% 0.74/1.10 224 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) # label(fact_3_t) # label(axiom). [assumption].
% 0.74/1.10 225 times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),t) = plus_plus(int,power_power(int,s,number_number_of(nat,bit0(bit1(pls)))),one_one(int)). [copy(224),flip(a)].
% 0.74/1.10 238 number_number_of(int,A) = A # label(fact_22_number__of__is__id) # label(axiom). [clausify(49)].
% 0.74/1.10 239 times_times(int,A,B) = times_times(int,B,A) # label(fact_23_zmult__commute) # label(axiom). [clausify(50)].
% 0.74/1.10 279 times_times(int,pls,A) = pls # label(fact_62_mult__Pls) # label(axiom). [clausify(85)].
% 0.74/1.10 280 times_times(int,A,pls) = pls. [copy(279),rewrite([239(3)])].
% 0.74/1.10 281 bit0(times_times(int,A,B)) = times_times(int,bit0(A),B) # label(fact_63_mult__Bit0) # label(axiom). [clausify(86)].
% 0.74/1.10 282 times_times(int,A,bit0(B)) = bit0(times_times(int,A,B)). [copy(281),rewrite([239(6)]),flip(a),rewrite([239(5)])].
% 0.74/1.10 288 times_times(int,plus_plus(int,A,B),C) = plus_plus(int,times_times(int,A,C),times_times(int,B,C)) # label(fact_68_zadd__zmult__distrib) # label(axiom). [clausify(91)].
% 0.74/1.10 289 plus_plus(int,times_times(int,A,B),times_times(int,B,C)) = times_times(int,B,plus_plus(int,A,C)). [copy(288),rewrite([239(4)]),flip(a),rewrite([239(5)])].
% 0.74/1.10 325 plus_plus(int,A,B) = plus_plus(int,B,A) # label(fact_96_zadd__commute) # label(axiom). [clausify(117)].
% 0.74/1.10 326 zero_zero(int) = number_number_of(int,pls) # label(fact_97_zero__is__num__zero) # label(axiom). [assumption].
% 0.74/1.10 327 zero_zero(int) = pls. [copy(326),rewrite([238(5)])].
% 0.74/1.10 342 power_power(int,one_one(int),number_number_of(nat,bit0(bit1(pls)))) = one_one(int). [resolve(119,a,125,a)].
% 0.74/1.10 343 power_power(int,zero_zero(int),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(int). [resolve(119,a,126,a)].
% 0.74/1.10 344 power_power(int,pls,number_number_of(nat,bit0(bit1(pls)))) = pls. [copy(343),rewrite([327(3),327(10)])].
% 0.74/1.10 364 number_number_of(int,ti(int,A)) = number_number_of(int,A). [resolve(128,a,135,a)].
% 0.74/1.10 365 ti(int,A) = A. [copy(364),rewrite([238(4),238(4)])].
% 0.74/1.10 436 -ord_less(int,plus_plus(int,power_power(int,A,number_number_of(nat,bit0(bit1(pls)))),power_power(int,B,number_number_of(nat,bit0(bit1(pls))))),zero_zero(int)). [resolve(163,a,168,a)].
% 0.74/1.10 437 -ord_less(int,plus_plus(int,power_power(int,A,number_number_of(nat,bit0(bit1(pls)))),power_power(int,B,number_number_of(nat,bit0(bit1(pls))))),pls). [copy(436),rewrite([327(19)])].
% 0.74/1.10 448 zero_zero(int) = plus_plus(int,power_power(int,A,number_number_of(nat,bit0(bit1(pls)))),power_power(int,B,number_number_of(nat,bit0(bit1(pls))))) | zero_zero(int) != ti(int,A) | zero_zero(int) != ti(int,B). [resolve(163,a,174,a)].
% 0.74/1.10 449 plus_plus(int,power_power(int,A,number_number_of(nat,bit0(bit1(pls)))),power_power(int,B,number_number_of(nat,bit0(bit1(pls))))) = pls | pls != A | pls != B. [copy(448),rewrite([327(2),327(20),365(21),327(22),365(23)]),flip(a)].
% 0.74/1.10 494 ord_less(int,times_times(int,t,plus_plus(int,one_one(int),bit0(bit0(times_times(int,m,bit1(pls)))))),times_times(int,pls,plus_plus(int,one_one(int),bit0(bit0(times_times(int,m,bit1(pls))))))). [back_rewrite(223),rewrite([238(10),239(10),282(10),282(9),325(13),239(15),238(24),239(24),282(24),282(23),325(27),327(29),239(29)])].
% 0.74/1.10 499 times_times(int,t,plus_plus(int,one_one(int),bit0(bit0(times_times(int,m,bit1(pls)))))) = plus_plus(int,one_one(int),power_power(int,s,number_number_of(nat,bit0(bit1(pls))))). [back_rewrite(225),rewrite([238(9),239(9),282(9),282(8),325(12),239(14),325(26)])].
% 0.74/1.10 510 plus_plus(int,power_power(int,A,number_number_of(nat,bit0(bit1(pls)))),power_power(int,A,number_number_of(nat,bit0(bit1(pls))))) = pls | pls != A. [factor(449,b,c)].
% 0.74/1.10 515 ord_less(int,plus_plus(int,one_one(int),power_power(int,s,number_number_of(nat,bit0(bit1(pls))))),times_times(int,pls,plus_plus(int,one_one(int),bit0(bit0(times_times(int,m,bit1(pls))))))). [back_rewrite(494),rewrite([499(15)])].
% 0.74/1.10 553 times_times(int,pls,plus_plus(int,A,B)) = plus_plus(int,pls,pls). [para(280(a,1),289(a,1,2)),rewrite([239(5),280(5),325(8)]),flip(a)].
% 0.74/1.10 559 ord_less(int,plus_plus(int,one_one(int),power_power(int,s,number_number_of(nat,bit0(bit1(pls))))),plus_plus(int,pls,pls)). [back_rewrite(515),rewrite([553(27)])].
% 0.74/1.10 709 -ord_less(int,plus_plus(int,one_one(int),power_power(int,A,number_number_of(nat,bit0(bit1(pls))))),pls). [para(342(a,1),437(a,2,2))].
% 0.74/1.10 865 plus_plus(int,pls,pls) = pls. [resolve(510,b,365,a(flip)),rewrite([365(5),344(9),365(6),344(10)])].
% 0.74/1.10 867 $F. [back_rewrite(559),rewrite([865(17)]),unit_del(a,709)].
% 0.74/1.10
% 0.74/1.10 % SZS output end Refutation
% 0.74/1.10 ============================== end of proof ==========================
% 0.74/1.10
% 0.74/1.10 ============================== STATISTICS ============================
% 0.74/1.10
% 0.74/1.10 Given=148. Generated=1399. Kept=539. proofs=1.
% 0.74/1.10 Usable=142. Sos=324. Demods=85. Limbo=2, Disabled=386. Hints=0.
% 0.74/1.10 Megabytes=1.06.
% 0.74/1.10 User_CPU=0.12, System_CPU=0.01, Wall_clock=0.
% 0.74/1.10
% 0.74/1.10 ============================== end of statistics =====================
% 0.74/1.10
% 0.74/1.10 ============================== end of search =========================
% 0.74/1.10
% 0.74/1.10 THEOREM PROVED
% 0.74/1.10 % SZS status Theorem
% 0.74/1.10
% 0.74/1.10 Exiting with 1 proof.
% 0.74/1.10
% 0.74/1.10 Process 1387 exit (max_proofs) Wed Jul 6 21:34:54 2022
% 0.74/1.10 Prover9 interrupted
%------------------------------------------------------------------------------