TSTP Solution File: NUM925+5 by Prover9---1109a
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- Process Solution
%------------------------------------------------------------------------------
% File : Prover9---1109a
% Problem : NUM925+5 : TPTP v8.1.0. Released v5.3.0.
% Transfm : none
% Format : tptp:raw
% Command : tptp2X_and_run_prover9 %d %s
% Computer : n013.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 13:33:22 EDT 2022
% Result : Theorem 1.41s 1.75s
% Output : Refutation 1.41s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.13 % Problem : NUM925+5 : TPTP v8.1.0. Released v5.3.0.
% 0.13/0.13 % Command : tptp2X_and_run_prover9 %d %s
% 0.13/0.35 % Computer : n013.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 600
% 0.13/0.35 % DateTime : Wed Jul 6 17:12:59 EDT 2022
% 0.13/0.35 % CPUTime :
% 0.82/1.09 ============================== Prover9 ===============================
% 0.82/1.09 Prover9 (32) version 2009-11A, November 2009.
% 0.82/1.09 Process 16909 was started by sandbox2 on n013.cluster.edu,
% 0.82/1.09 Wed Jul 6 17:13:00 2022
% 0.82/1.09 The command was "/export/starexec/sandbox2/solver/bin/prover9 -t 300 -f /tmp/Prover9_16755_n013.cluster.edu".
% 0.82/1.09 ============================== end of head ===========================
% 0.82/1.09
% 0.82/1.09 ============================== INPUT =================================
% 0.82/1.09
% 0.82/1.09 % Reading from file /tmp/Prover9_16755_n013.cluster.edu
% 0.82/1.09
% 0.82/1.09 set(prolog_style_variables).
% 0.82/1.09 set(auto2).
% 0.82/1.09 % set(auto2) -> set(auto).
% 0.82/1.09 % set(auto) -> set(auto_inference).
% 0.82/1.09 % set(auto) -> set(auto_setup).
% 0.82/1.09 % set(auto_setup) -> set(predicate_elim).
% 0.82/1.09 % set(auto_setup) -> assign(eq_defs, unfold).
% 0.82/1.09 % set(auto) -> set(auto_limits).
% 0.82/1.09 % set(auto_limits) -> assign(max_weight, "100.000").
% 0.82/1.09 % set(auto_limits) -> assign(sos_limit, 20000).
% 0.82/1.09 % set(auto) -> set(auto_denials).
% 0.82/1.09 % set(auto) -> set(auto_process).
% 0.82/1.09 % set(auto2) -> assign(new_constants, 1).
% 0.82/1.09 % set(auto2) -> assign(fold_denial_max, 3).
% 0.82/1.09 % set(auto2) -> assign(max_weight, "200.000").
% 0.82/1.09 % set(auto2) -> assign(max_hours, 1).
% 0.82/1.09 % assign(max_hours, 1) -> assign(max_seconds, 3600).
% 0.82/1.09 % set(auto2) -> assign(max_seconds, 0).
% 0.82/1.09 % set(auto2) -> assign(max_minutes, 5).
% 0.82/1.09 % assign(max_minutes, 5) -> assign(max_seconds, 300).
% 0.82/1.09 % set(auto2) -> set(sort_initial_sos).
% 0.82/1.09 % set(auto2) -> assign(sos_limit, -1).
% 0.82/1.09 % set(auto2) -> assign(lrs_ticks, 3000).
% 0.82/1.09 % set(auto2) -> assign(max_megs, 400).
% 0.82/1.09 % set(auto2) -> assign(stats, some).
% 0.82/1.09 % set(auto2) -> clear(echo_input).
% 0.82/1.09 % set(auto2) -> set(quiet).
% 0.82/1.09 % set(auto2) -> clear(print_initial_clauses).
% 0.82/1.09 % set(auto2) -> clear(print_given).
% 0.82/1.09 assign(lrs_ticks,-1).
% 0.82/1.09 assign(sos_limit,10000).
% 0.82/1.09 assign(order,kbo).
% 0.82/1.09 set(lex_order_vars).
% 0.82/1.09 clear(print_given).
% 0.82/1.09
% 0.82/1.09 % formulas(sos). % not echoed (153 formulas)
% 0.82/1.09
% 0.82/1.09 ============================== end of input ==========================
% 0.82/1.09
% 0.82/1.09 % From the command line: assign(max_seconds, 300).
% 0.82/1.09
% 0.82/1.09 ============================== PROCESS NON-CLAUSAL FORMULAS ==========
% 0.82/1.09
% 0.82/1.09 % Formulas that are not ordinary clauses:
% 0.82/1.09 1 (all X_a (power(X_a) & semiring_0(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].
% 0.82/1.09 2 (all B_1 all B_2 all X_a (linord219039673up_add(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_0_arg1) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.09 3 (all B_1 all B_2 all X_a (linord219039673up_add(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_0_arg2) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.09 4 (all B_1 all B_2 all X_a (linord219039673up_add(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_0_res) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.09 5 (all B_1 all B_2 all X_a (number_semiring(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].
% 0.82/1.09 6 (all B_1 all B_2 all X_a (number_semiring(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].
% 0.82/1.09 7 (all B_1 all B_2 all X_a (number_semiring(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].
% 0.82/1.09 8 (all X_a (power(X_a) & mult_zero(X_a) & no_zero_divisors(X_a) & zero_neq_one(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].
% 0.82/1.09 9 (all X_a (linord219039673up_add(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].
% 0.82/1.09 10 (all X_a (power(X_a) & semiring_0(X_a) -> ti(X_a,zero_zero(X_a)) = zero_zero(X_a))) # label(tsy_c_Groups_Ozero__class_Ozero_2_res) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.09 11 (all X_a ti(X_a,undefined(X_a)) = undefined(X_a)) # label(tsy_c_HOL_Oundefined_res) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.09 12 (all B_1 bit0(ti(int,B_1)) = bit0(B_1)) # label(tsy_c_Int_OBit0_arg1) # label(hypothesis) # label(non_clause). [assumption].
% 0.82/1.09 13 (all B_1 ti(int,bit0(B_1)) = bit0(B_1)) # label(tsy_c_Int_OBit0_res) # label(hypothesis) # label(non_clause). [assumption].
% 0.82/1.09 14 (all B_1 bit1(ti(int,B_1)) = bit1(B_1)) # label(tsy_c_Int_OBit1_arg1) # label(hypothesis) # label(non_clause). [assumption].
% 0.82/1.09 15 (all B_1 ti(int,bit1(B_1)) = bit1(B_1)) # label(tsy_c_Int_OBit1_res) # label(hypothesis) # label(non_clause). [assumption].
% 0.82/1.09 16 (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].
% 0.82/1.09 17 (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].
% 0.82/1.09 18 (all B_1 all X_a (number_semiring(X_a) -> semiring_1_of_nat(X_a,ti(nat,B_1)) = semiring_1_of_nat(X_a,B_1))) # label(tsy_c_Nat_Osemiring__1__class_Oof__nat_arg1) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.09 19 (all B_1 all X_a (number_semiring(X_a) -> ti(X_a,semiring_1_of_nat(X_a,B_1)) = semiring_1_of_nat(X_a,B_1))) # label(tsy_c_Nat_Osemiring__1__class_Oof__nat_res) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.09 20 (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_0_arg1) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.09 21 (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_0_arg2) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.09 22 (all B_1 all B_2 (ord_less(nat,ti(nat,B_1),B_2) <-> ord_less(nat,B_1,B_2))) # label(tsy_c_Orderings_Oord__class_Oless_1_arg1) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.09 23 (all B_1 all B_2 (ord_less(nat,B_1,ti(nat,B_2)) <-> ord_less(nat,B_1,B_2))) # label(tsy_c_Orderings_Oord__class_Oless_1_arg2) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.09 24 (all B_1 all B_2 all X_a (power(X_a) & mult_zero(X_a) & no_zero_divisors(X_a) & zero_neq_one(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_0_arg1) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.09 25 (all B_1 all B_2 all X_a (power(X_a) & mult_zero(X_a) & no_zero_divisors(X_a) & zero_neq_one(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_0_arg2) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.09 26 (all B_1 all B_2 all X_a (power(X_a) & mult_zero(X_a) & no_zero_divisors(X_a) & zero_neq_one(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_0_res) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.09 27 (all B_1 all B_2 all X_a (power(X_a) & semiring_0(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_1_arg1) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.09 28 (all B_1 all B_2 all X_a (power(X_a) & semiring_0(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_1_arg2) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.09 29 (all B_1 all B_2 all X_a (power(X_a) & semiring_0(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_1_res) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.09 30 (all X_a (linordered_idom(X_a) -> (all Xa all Ya (plus_plus(X_a,power_power(X_a,Xa,number_number_of(nat,bit0(bit1(pls)))),power_power(X_a,Ya,number_number_of(nat,bit0(bit1(pls))))) = zero_zero(X_a) <-> ti(X_a,Xa) = zero_zero(X_a) & ti(X_a,Ya) = zero_zero(X_a))))) # label(fact_2_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 31 (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_3_one__power2) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 32 (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_4_zero__power2) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 33 (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_5_zero__eq__power2) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 34 (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_6_add__special_I2_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 35 (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_7_add__special_I3_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 36 (all X_a (number_ring(X_a) -> plus_plus(X_a,one_one(X_a),one_one(X_a)) = number_number_of(X_a,bit0(bit1(pls))))) # label(fact_8_one__add__one__is__two) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 37 (all X_a (number_semiring(X_a) -> plus_plus(X_a,one_one(X_a),one_one(X_a)) = number_number_of(X_a,bit0(bit1(pls))))) # label(fact_9_semiring__one__add__one__is__two) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 38 (all X power_power(int,power_power(int,X,number_number_of(nat,bit0(bit1(pls)))),number_number_of(nat,bit0(bit1(pls)))) = power_power(int,X,number_number_of(nat,bit0(bit0(bit1(pls)))))) # label(fact_10_quartic__square__square) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 39 (all X_a (power(X_a) & semiring_0(X_a) -> (all W ((number_number_of(nat,W) = zero_zero(nat) -> power_power(X_a,zero_zero(X_a),number_number_of(nat,W)) = one_one(X_a)) & (number_number_of(nat,W) != zero_zero(nat) -> power_power(X_a,zero_zero(X_a),number_number_of(nat,W)) = zero_zero(X_a)))))) # label(fact_11_power__0__left__number__of) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 40 (all X_a (number_ring(X_a) -> one_one(X_a) = number_number_of(X_a,bit1(pls)))) # label(fact_12_semiring__norm_I110_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 41 (all X_a (number_ring(X_a) -> number_number_of(X_a,bit1(pls)) = one_one(X_a))) # label(fact_13_numeral__1__eq__1) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 42 (all X all Y (ord_less(int,X,Y) | X = Y | ord_less(int,Y,X))) # label(fact_15_zless__linear) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 43 (all K_1 all L_1 (ord_less(int,number_number_of(int,K_1),number_number_of(int,L_1)) <-> ord_less(int,K_1,L_1))) # label(fact_16_less__number__of__int__code) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 44 (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_17_plus__numeral__code_I9_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 45 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all Xa all Ya (ord_less(X_a,number_number_of(X_a,Xa),number_number_of(X_a,Ya)) <-> ord_less(int,Xa,Ya))))) # label(fact_18_less__number__of) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 46 (all M all N power_power(int,semiring_1_of_nat(int,M),N) = semiring_1_of_nat(int,power_power(nat,M,N))) # label(fact_20_zpower__int) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 47 (all M all N semiring_1_of_nat(int,power_power(nat,M,N)) = power_power(int,semiring_1_of_nat(int,M),N)) # label(fact_21_int__power) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 48 (all M all N all Z plus_plus(int,semiring_1_of_nat(int,M),plus_plus(int,semiring_1_of_nat(int,N),Z)) = plus_plus(int,semiring_1_of_nat(int,plus_plus(nat,M,N)),Z)) # label(fact_22_zadd__int__left) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 49 (all M all N plus_plus(int,semiring_1_of_nat(int,M),semiring_1_of_nat(int,N)) = semiring_1_of_nat(int,plus_plus(nat,M,N))) # label(fact_23_zadd__int) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 50 (all Na (semiring_1_of_nat(int,Na) = zero_zero(int) <-> Na = zero_zero(nat))) # label(fact_27_int__eq__0__conv) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 51 (all K1 all K2 (ord_less(int,bit1(K1),bit1(K2)) <-> ord_less(int,K1,K2))) # label(fact_30_less__int__code_I16_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 52 (all K_1 all L_1 (ord_less(int,bit1(K_1),bit1(L_1)) <-> ord_less(int,K_1,L_1))) # label(fact_31_rel__simps_I17_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 53 (all K1 all K2 (ord_less(int,bit0(K1),bit0(K2)) <-> ord_less(int,K1,K2))) # label(fact_33_less__int__code_I13_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 54 (all K_1 all L_1 (ord_less(int,bit0(K_1),bit0(L_1)) <-> ord_less(int,K_1,L_1))) # label(fact_34_rel__simps_I14_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 55 (all K all I all J (ord_less(int,I,J) -> ord_less(int,plus_plus(int,I,K),plus_plus(int,J,K)))) # label(fact_35_zadd__strict__right__mono) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 56 (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_36_add__nat__number__of) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 57 (all X_a (number_ring(X_a) & ring_char_0(X_a) -> (all Xa all Ya (number_number_of(X_a,Xa) = number_number_of(X_a,Ya) <-> Xa = Ya)))) # label(fact_40_eq__number__of) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 58 (all X_a (number(X_a) -> (all Wa all Xa (number_number_of(X_a,Wa) = ti(X_a,Xa) <-> ti(X_a,Xa) = number_number_of(X_a,Wa))))) # label(fact_41_number__of__reorient) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 59 (all K_1 all L_1 (bit1(K_1) = bit1(L_1) <-> K_1 = L_1)) # label(fact_42_rel__simps_I51_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 60 (all K_1 all L_1 (bit0(K_1) = bit0(L_1) <-> K_1 = L_1)) # label(fact_43_rel__simps_I48_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 61 (all X_a (linordered_idom(X_a) -> (all A_2 (ord_less(X_a,plus_plus(X_a,A_2,A_2),zero_zero(X_a)) <-> ord_less(X_a,A_2,zero_zero(X_a)))))) # label(fact_44_even__less__0__iff) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 62 (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_45_zadd__assoc) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 63 (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_46_zadd__left__commute) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 64 (all Z all W plus_plus(int,Z,W) = plus_plus(int,W,Z)) # label(fact_47_zadd__commute) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 65 (all Ma all Na (semiring_1_of_nat(int,Ma) = semiring_1_of_nat(int,Na) <-> Ma = Na)) # label(fact_48_int__int__eq) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 66 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all Xa (ord_less(X_a,number_number_of(X_a,Xa),zero_zero(X_a)) <-> ord_less(int,Xa,pls))))) # label(fact_49_less__special_I3_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 67 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all Ya (ord_less(X_a,zero_zero(X_a),number_number_of(X_a,Ya)) <-> ord_less(int,pls,Ya))))) # label(fact_50_less__special_I1_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 68 (all K_1 (ord_less(int,bit1(K_1),pls) <-> ord_less(int,K_1,pls))) # label(fact_51_rel__simps_I12_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 69 (all K1 all K2 (ord_less(int,bit1(K1),bit0(K2)) <-> ord_less(int,K1,K2))) # label(fact_52_less__int__code_I15_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 70 (all K_1 all L_1 (ord_less(int,bit1(K_1),bit0(L_1)) <-> ord_less(int,K_1,L_1))) # label(fact_53_rel__simps_I16_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 71 (all K_1 (ord_less(int,bit0(K_1),pls) <-> ord_less(int,K_1,pls))) # label(fact_54_rel__simps_I10_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 72 (all K_1 (ord_less(int,pls,bit0(K_1)) <-> ord_less(int,pls,K_1))) # label(fact_55_rel__simps_I4_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 73 (all Wa (ord_less(int,bit1(Wa),zero_zero(int)) <-> ord_less(int,Wa,zero_zero(int)))) # label(fact_56_bin__less__0__simps_I4_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 74 (all Wa (ord_less(int,bit0(Wa),zero_zero(int)) <-> ord_less(int,Wa,zero_zero(int)))) # label(fact_58_bin__less__0__simps_I3_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 75 (all Wa all Z_1 (ord_less(int,Wa,plus_plus(int,Z_1,one_one(int))) <-> ord_less(int,Wa,Z_1) | Wa = Z_1)) # label(fact_60_zless__add1__eq) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 76 (all K -ord_less(int,semiring_1_of_nat(int,K),zero_zero(int))) # label(fact_61_int__less__0__conv) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 77 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all Xa (ord_less(X_a,number_number_of(X_a,Xa),one_one(X_a)) <-> ord_less(int,Xa,bit1(pls)))))) # label(fact_62_less__special_I4_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 78 (all X_a (number_ring(X_a) & linordered_idom(X_a) -> (all Ya (ord_less(X_a,one_one(X_a),number_number_of(X_a,Ya)) <-> ord_less(int,bit1(pls),Ya))))) # label(fact_63_less__special_I2_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 79 (all Z_1 (ord_less(int,plus_plus(int,plus_plus(int,one_one(int),Z_1),Z_1),zero_zero(int)) <-> ord_less(int,Z_1,zero_zero(int)))) # label(fact_64_odd__less__0) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 80 (all X_a (linord219039673up_add(X_a) -> (all A_2 (plus_plus(X_a,A_2,A_2) = zero_zero(X_a) <-> ti(X_a,A_2) = zero_zero(X_a))))) # label(fact_65_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 81 (all K bit1(K) != pls) # label(fact_66_rel__simps_I46_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 82 (all L pls != bit1(L)) # label(fact_67_rel__simps_I39_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 83 (all K all L bit1(K) != bit0(L)) # label(fact_68_rel__simps_I50_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 84 (all K all L bit0(K) != bit1(L)) # label(fact_69_rel__simps_I49_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 85 (all K_1 (bit0(K_1) = pls <-> K_1 = pls)) # label(fact_70_rel__simps_I44_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 86 (all L_1 (pls = bit0(L_1) <-> pls = L_1)) # label(fact_71_rel__simps_I38_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 87 (all K plus_plus(int,K,pls) = K) # label(fact_75_add__Pls__right) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 88 (all K plus_plus(int,pls,K) = K) # label(fact_76_add__Pls) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 89 (all K all L plus_plus(int,bit0(K),bit0(L)) = bit0(plus_plus(int,K,L))) # label(fact_77_add__Bit0__Bit0) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 90 (all K bit0(K) = plus_plus(int,K,K)) # label(fact_78_Bit0__def) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 91 (all Z plus_plus(int,Z,zero_zero(int)) = Z) # label(fact_79_zadd__0__right) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 92 (all Z plus_plus(int,zero_zero(int),Z) = Z) # label(fact_80_zadd__0) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 93 (all X_a (number_semiring(X_a) -> number_number_of(X_a,pls) = zero_zero(X_a))) # label(fact_81_semiring__numeral__0__eq__0) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 94 (all X_a (number_ring(X_a) -> number_number_of(X_a,pls) = zero_zero(X_a))) # label(fact_82_number__of__Pls) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 95 (all X_a (number_ring(X_a) -> zero_zero(X_a) = number_number_of(X_a,pls))) # label(fact_83_semiring__norm_I112_J) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 96 (all X_a (number_ring(X_a) -> (all A_1 plus_plus(X_a,number_number_of(X_a,pls),A_1) = ti(X_a,A_1)))) # label(fact_84_add__numeral__0) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 97 (all X_a (number_ring(X_a) -> (all A_1 plus_plus(X_a,A_1,number_number_of(X_a,pls)) = ti(X_a,A_1)))) # label(fact_85_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 98 (all X_a (power(X_a) & mult_zero(X_a) & no_zero_divisors(X_a) & zero_neq_one(X_a) -> (all A_2 all Wa (power_power(X_a,A_2,number_number_of(nat,Wa)) = zero_zero(X_a) <-> ti(X_a,A_2) = zero_zero(X_a) & number_number_of(nat,Wa) != zero_zero(nat))))) # label(fact_86_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 99 (all X_a (number_ring(X_a) -> (all V all W all Z plus_plus(X_a,number_number_of(X_a,V),plus_plus(X_a,number_number_of(X_a,W),Z)) = plus_plus(X_a,number_number_of(X_a,plus_plus(int,V,W)),Z)))) # label(fact_87_add__number__of__left) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 100 (all X_a (number_ring(X_a) -> (all V all W plus_plus(X_a,number_number_of(X_a,V),number_number_of(X_a,W)) = number_number_of(X_a,plus_plus(int,V,W))))) # label(fact_88_add__number__of__eq) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 101 (all X_a (number_ring(X_a) -> (all V all W number_number_of(X_a,plus_plus(int,V,W)) = plus_plus(X_a,number_number_of(X_a,V),number_number_of(X_a,W))))) # label(fact_89_number__of__add) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 102 (all K all L plus_plus(int,bit1(K),bit0(L)) = bit1(plus_plus(int,K,L))) # label(fact_90_add__Bit1__Bit0) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 103 (all K all L plus_plus(int,bit0(K),bit1(L)) = bit1(plus_plus(int,K,L))) # label(fact_91_add__Bit0__Bit1) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 104 (all K bit1(K) = plus_plus(int,plus_plus(int,one_one(int),K),K)) # label(fact_92_Bit1__def) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 105 (all Z plus_plus(int,plus_plus(int,one_one(int),Z),Z) != zero_zero(int)) # label(fact_93_odd__nonzero) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 106 (all X_a (number_semiring(X_a) -> (all N number_number_of(X_a,semiring_1_of_nat(int,N)) = semiring_1_of_nat(X_a,N)))) # label(fact_94_number__of__int) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 107 (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_95_zero__less__power2) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 108 (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_96_power2__less__0) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 109 (all X_a (linordered_idom(X_a) -> (all Xa all Ya (ord_less(X_a,zero_zero(X_a),plus_plus(X_a,power_power(X_a,Xa,number_number_of(nat,bit0(bit1(pls)))),power_power(X_a,Ya,number_number_of(nat,bit0(bit1(pls)))))) <-> ti(X_a,Xa) != zero_zero(X_a) | ti(X_a,Ya) != zero_zero(X_a))))) # label(fact_97_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 110 (all T all A ti(T,ti(T,A)) = ti(T,A)) # label(help_ti_idem) # label(axiom) # label(non_clause). [assumption].
% 0.82/1.10 111 -(power_power(int,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)),number_number_of(nat,bit0(bit1(pls)))) != zero_zero(int)) # label(conj_0) # label(negated_conjecture) # label(non_clause). [assumption].
% 0.82/1.10
% 0.82/1.10 ============================== end of process non-clausal formulas ===
% 0.82/1.10
% 0.82/1.10 ============================== PROCESS INITIAL CLAUSES ===============
% 0.82/1.10
% 0.82/1.10 ============================== PREDICATE ELIMINATION =================
% 0.82/1.10 112 power(int) # label(arity_Int_Oint___Power_Opower) # label(axiom). [assumption].
% 0.82/1.10 113 -power(A) | -semiring_0(A) | ti(A,one_one(A)) = one_one(A) # label(tsy_c_Groups_Oone__class_Oone_res) # label(axiom). [clausify(1)].
% 0.82/1.10 114 -power(A) | -mult_zero(A) | -no_zero_divisors(A) | -zero_neq_one(A) | zero_zero(A) = ti(A,zero_zero(A)) # label(tsy_c_Groups_Ozero__class_Ozero_0_res) # label(axiom). [clausify(8)].
% 0.82/1.10 115 -power(A) | -semiring_0(A) | zero_zero(A) = ti(A,zero_zero(A)) # label(tsy_c_Groups_Ozero__class_Ozero_2_res) # label(axiom). [clausify(10)].
% 0.82/1.10 116 -power(A) | -mult_zero(A) | -no_zero_divisors(A) | -zero_neq_one(A) | power_power(A,ti(A,B),C) = power_power(A,B,C) # label(tsy_c_Power_Opower__class_Opower_0_arg1) # label(axiom). [clausify(24)].
% 0.82/1.10 117 -power(A) | -mult_zero(A) | -no_zero_divisors(A) | -zero_neq_one(A) | power_power(A,B,ti(nat,C)) = power_power(A,B,C) # label(tsy_c_Power_Opower__class_Opower_0_arg2) # label(axiom). [clausify(25)].
% 0.82/1.10 118 -power(A) | -mult_zero(A) | -no_zero_divisors(A) | -zero_neq_one(A) | power_power(A,B,C) = ti(A,power_power(A,B,C)) # label(tsy_c_Power_Opower__class_Opower_0_res) # label(axiom). [clausify(26)].
% 0.82/1.10 119 -power(A) | -semiring_0(A) | power_power(A,ti(A,B),C) = power_power(A,B,C) # label(tsy_c_Power_Opower__class_Opower_1_arg1) # label(axiom). [clausify(27)].
% 0.82/1.10 120 -power(A) | -semiring_0(A) | power_power(A,B,ti(nat,C)) = power_power(A,B,C) # label(tsy_c_Power_Opower__class_Opower_1_arg2) # label(axiom). [clausify(28)].
% 0.82/1.10 121 -power(A) | -semiring_0(A) | power_power(A,B,C) = ti(A,power_power(A,B,C)) # label(tsy_c_Power_Opower__class_Opower_1_res) # label(axiom). [clausify(29)].
% 0.82/1.10 122 -power(A) | -semiring_0(A) | number_number_of(nat,B) != zero_zero(nat) | power_power(A,zero_zero(A),number_number_of(nat,B)) = one_one(A) # label(fact_11_power__0__left__number__of) # label(axiom). [clausify(39)].
% 0.82/1.10 123 -power(A) | -semiring_0(A) | number_number_of(nat,B) = zero_zero(nat) | power_power(A,zero_zero(A),number_number_of(nat,B)) = zero_zero(A) # label(fact_11_power__0__left__number__of) # label(axiom). [clausify(39)].
% 0.82/1.10 124 -power(A) | -mult_zero(A) | -no_zero_divisors(A) | -zero_neq_one(A) | power_power(A,B,number_number_of(nat,C)) != zero_zero(A) | zero_zero(A) = ti(A,B) # label(fact_86_power__eq__0__iff__number__of) # label(axiom). [clausify(98)].
% 0.82/1.10 125 -power(A) | -mult_zero(A) | -no_zero_divisors(A) | -zero_neq_one(A) | power_power(A,B,number_number_of(nat,C)) != zero_zero(A) | number_number_of(nat,C) != zero_zero(nat) # label(fact_86_power__eq__0__iff__number__of) # label(axiom). [clausify(98)].
% 0.82/1.10 126 -power(A) | -mult_zero(A) | -no_zero_divisors(A) | -zero_neq_one(A) | power_power(A,B,number_number_of(nat,C)) = zero_zero(A) | zero_zero(A) != ti(A,B) | number_number_of(nat,C) = zero_zero(nat) # label(fact_86_power__eq__0__iff__number__of) # label(axiom). [clausify(98)].
% 0.82/1.10 Derived: -semiring_0(int) | ti(int,one_one(int)) = one_one(int). [resolve(112,a,113,a)].
% 0.82/1.10 Derived: -mult_zero(int) | -no_zero_divisors(int) | -zero_neq_one(int) | zero_zero(int) = ti(int,zero_zero(int)). [resolve(112,a,114,a)].
% 0.82/1.10 Derived: -semiring_0(int) | zero_zero(int) = ti(int,zero_zero(int)). [resolve(112,a,115,a)].
% 0.82/1.10 Derived: -mult_zero(int) | -no_zero_divisors(int) | -zero_neq_one(int) | power_power(int,ti(int,A),B) = power_power(int,A,B). [resolve(112,a,116,a)].
% 0.82/1.10 Derived: -mult_zero(int) | -no_zero_divisors(int) | -zero_neq_one(int) | power_power(int,A,ti(nat,B)) = power_power(int,A,B). [resolve(112,a,117,a)].
% 0.82/1.10 Derived: -mult_zero(int) | -no_zero_divisors(int) | -zero_neq_one(int) | power_power(int,A,B) = ti(int,power_power(int,A,B)). [resolve(112,a,118,a)].
% 0.82/1.10 Derived: -semiring_0(int) | power_power(int,ti(int,A),B) = power_power(int,A,B). [resolve(112,a,119,a)].
% 0.82/1.10 Derived: -semiring_0(int) | power_power(int,A,ti(nat,B)) = power_power(int,A,B). [resolve(112,a,120,a)].
% 0.82/1.10 Derived: -semiring_0(int) | power_power(int,A,B) = ti(int,power_power(int,A,B)). [resolve(112,a,121,a)].
% 0.82/1.10 Derived: -semiring_0(int) | number_number_of(nat,A) != zero_zero(nat) | power_power(int,zero_zero(int),number_number_of(nat,A)) = one_one(int). [resolve(112,a,122,a)].
% 0.82/1.10 Derived: -semiring_0(int) | number_number_of(nat,A) = zero_zero(nat) | power_power(int,zero_zero(int),number_number_of(nat,A)) = zero_zero(int). [resolve(112,a,123,a)].
% 0.82/1.10 Derived: -mult_zero(int) | -no_zero_divisors(int) | -zero_neq_one(int) | power_power(int,A,number_number_of(nat,B)) != zero_zero(int) | zero_zero(int) = ti(int,A). [resolve(112,a,124,a)].
% 0.82/1.10 Derived: -mult_zero(int) | -no_zero_divisors(int) | -zero_neq_one(int) | power_power(int,A,number_number_of(nat,B)) != zero_zero(int) | number_number_of(nat,B) != zero_zero(nat). [resolve(112,a,125,a)].
% 0.82/1.10 Derived: -mult_zero(int) | -no_zero_divisors(int) | -zero_neq_one(int) | power_power(int,A,number_number_of(nat,B)) = zero_zero(int) | zero_zero(int) != ti(int,A) | number_number_of(nat,B) = zero_zero(nat). [resolve(112,a,126,a)].
% 0.82/1.10 127 power(nat) # label(arity_Nat_Onat___Power_Opower) # label(axiom). [assumption].
% 0.82/1.10 Derived: -semiring_0(nat) | ti(nat,one_one(nat)) = one_one(nat). [resolve(127,a,113,a)].
% 0.82/1.10 Derived: -mult_zero(nat) | -no_zero_divisors(nat) | -zero_neq_one(nat) | zero_zero(nat) = ti(nat,zero_zero(nat)). [resolve(127,a,114,a)].
% 0.82/1.10 Derived: -semiring_0(nat) | zero_zero(nat) = ti(nat,zero_zero(nat)). [resolve(127,a,115,a)].
% 0.82/1.10 Derived: -mult_zero(nat) | -no_zero_divisors(nat) | -zero_neq_one(nat) | power_power(nat,ti(nat,A),B) = power_power(nat,A,B). [resolve(127,a,116,a)].
% 0.82/1.10 Derived: -mult_zero(nat) | -no_zero_divisors(nat) | -zero_neq_one(nat) | power_power(nat,A,ti(nat,B)) = power_power(nat,A,B). [resolve(127,a,117,a)].
% 0.82/1.10 Derived: -mult_zero(nat) | -no_zero_divisors(nat) | -zero_neq_one(nat) | power_power(nat,A,B) = ti(nat,power_power(nat,A,B)). [resolve(127,a,118,a)].
% 0.82/1.10 Derived: -semiring_0(nat) | power_power(nat,ti(nat,A),B) = power_power(nat,A,B). [resolve(127,a,119,a)].
% 0.82/1.10 Derived: -semiring_0(nat) | power_power(nat,A,ti(nat,B)) = power_power(nat,A,B). [resolve(127,a,120,a)].
% 0.82/1.10 Derived: -semiring_0(nat) | power_power(nat,A,B) = ti(nat,power_power(nat,A,B)). [resolve(127,a,121,a)].
% 0.82/1.10 Derived: -semiring_0(nat) | number_number_of(nat,A) != zero_zero(nat) | power_power(nat,zero_zero(nat),number_number_of(nat,A)) = one_one(nat). [resolve(127,a,122,a)].
% 0.82/1.10 Derived: -semiring_0(nat) | number_number_of(nat,A) = zero_zero(nat) | power_power(nat,zero_zero(nat),number_number_of(nat,A)) = zero_zero(nat). [resolve(127,a,123,a)].
% 0.82/1.10 Derived: -mult_zero(nat) | -no_zero_divisors(nat) | -zero_neq_one(nat) | power_power(nat,A,number_number_of(nat,B)) != zero_zero(nat) | zero_zero(nat) = ti(nat,A). [resolve(127,a,124,a)].
% 0.82/1.10 Derived: -mult_zero(nat) | -no_zero_divisors(nat) | -zero_neq_one(nat) | power_power(nat,A,number_number_of(nat,B)) != zero_zero(nat) | number_number_of(nat,B) != zero_zero(nat). [resolve(127,a,125,a)].
% 0.82/1.10 Derived: -mult_zero(nat) | -no_zero_divisors(nat) | -zero_neq_one(nat) | power_power(nat,A,number_number_of(nat,B)) = zero_zero(nat) | zero_zero(nat) != ti(nat,A) | number_number_of(nat,B) = zero_zero(nat). [resolve(127,a,126,a)].
% 0.82/1.10 128 linord219039673up_add(int) # label(arity_Int_Oint___Groups_Olinordered__ab__group__add) # label(axiom). [assumption].
% 0.82/1.10 129 -linord219039673up_add(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)].
% 0.82/1.10 130 -linord219039673up_add(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)].
% 0.82/1.10 131 -linord219039673up_add(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)].
% 0.82/1.10 132 -linord219039673up_add(A) | zero_zero(A) = ti(A,zero_zero(A)) # label(tsy_c_Groups_Ozero__class_Ozero_1_res) # label(axiom). [clausify(9)].
% 0.82/1.10 133 -linord219039673up_add(A) | zero_zero(A) != plus_plus(A,B,B) | zero_zero(A) = ti(A,B) # label(fact_65_double__eq__0__iff) # label(axiom). [clausify(80)].
% 0.82/1.10 134 -linord219039673up_add(A) | zero_zero(A) = plus_plus(A,B,B) | zero_zero(A) != ti(A,B) # label(fact_65_double__eq__0__iff) # label(axiom). [clausify(80)].
% 0.82/1.10 Derived: plus_plus(int,ti(int,A),B) = plus_plus(int,A,B). [resolve(128,a,129,a)].
% 0.82/1.10 Derived: plus_plus(int,A,ti(int,B)) = plus_plus(int,A,B). [resolve(128,a,130,a)].
% 0.82/1.10 Derived: plus_plus(int,A,B) = ti(int,plus_plus(int,A,B)). [resolve(128,a,131,a)].
% 0.82/1.10 Derived: zero_zero(int) = ti(int,zero_zero(int)). [resolve(128,a,132,a)].
% 0.82/1.10 Derived: zero_zero(int) != plus_plus(int,A,A) | zero_zero(int) = ti(int,A). [resolve(128,a,133,a)].
% 0.82/1.11 Derived: zero_zero(int) = plus_plus(int,A,A) | zero_zero(int) != ti(int,A). [resolve(128,a,134,a)].
% 0.82/1.11 135 number_semiring(int) # label(arity_Int_Oint___Int_Onumber__semiring) # label(axiom). [assumption].
% 0.82/1.11 136 -number_semiring(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)].
% 0.82/1.11 137 -number_semiring(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)].
% 0.82/1.11 138 -number_semiring(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)].
% 0.82/1.11 139 -number_semiring(A) | semiring_1_of_nat(A,ti(nat,B)) = semiring_1_of_nat(A,B) # label(tsy_c_Nat_Osemiring__1__class_Oof__nat_arg1) # label(axiom). [clausify(18)].
% 0.82/1.11 140 -number_semiring(A) | semiring_1_of_nat(A,B) = ti(A,semiring_1_of_nat(A,B)) # label(tsy_c_Nat_Osemiring__1__class_Oof__nat_res) # label(axiom). [clausify(19)].
% 0.82/1.11 141 -number_semiring(A) | number_number_of(A,bit0(bit1(pls))) = plus_plus(A,one_one(A),one_one(A)) # label(fact_9_semiring__one__add__one__is__two) # label(axiom). [clausify(37)].
% 0.82/1.11 142 -number_semiring(A) | number_number_of(A,pls) = zero_zero(A) # label(fact_81_semiring__numeral__0__eq__0) # label(axiom). [clausify(93)].
% 0.82/1.11 143 -number_semiring(A) | semiring_1_of_nat(A,B) = number_number_of(A,semiring_1_of_nat(int,B)) # label(fact_94_number__of__int) # label(axiom). [clausify(106)].
% 0.82/1.11 Derived: semiring_1_of_nat(int,ti(nat,A)) = semiring_1_of_nat(int,A). [resolve(135,a,139,a)].
% 0.82/1.11 Derived: semiring_1_of_nat(int,A) = ti(int,semiring_1_of_nat(int,A)). [resolve(135,a,140,a)].
% 0.82/1.11 Derived: number_number_of(int,bit0(bit1(pls))) = plus_plus(int,one_one(int),one_one(int)). [resolve(135,a,141,a)].
% 0.82/1.11 Derived: number_number_of(int,pls) = zero_zero(int). [resolve(135,a,142,a)].
% 0.82/1.11 Derived: semiring_1_of_nat(int,A) = number_number_of(int,semiring_1_of_nat(int,A)). [resolve(135,a,143,a)].
% 0.82/1.11 144 number_semiring(nat) # label(arity_Nat_Onat___Int_Onumber__semiring) # label(axiom). [assumption].
% 0.82/1.11 Derived: plus_plus(nat,ti(nat,A),B) = plus_plus(nat,A,B). [resolve(144,a,136,a)].
% 0.82/1.11 Derived: plus_plus(nat,A,ti(nat,B)) = plus_plus(nat,A,B). [resolve(144,a,137,a)].
% 0.82/1.11 Derived: plus_plus(nat,A,B) = ti(nat,plus_plus(nat,A,B)). [resolve(144,a,138,a)].
% 0.82/1.11 Derived: semiring_1_of_nat(nat,ti(nat,A)) = semiring_1_of_nat(nat,A). [resolve(144,a,139,a)].
% 0.82/1.11 Derived: semiring_1_of_nat(nat,A) = ti(nat,semiring_1_of_nat(nat,A)). [resolve(144,a,140,a)].
% 0.82/1.11 Derived: number_number_of(nat,bit0(bit1(pls))) = plus_plus(nat,one_one(nat),one_one(nat)). [resolve(144,a,141,a)].
% 0.82/1.11 Derived: number_number_of(nat,pls) = zero_zero(nat). [resolve(144,a,142,a)].
% 0.82/1.11 Derived: semiring_1_of_nat(nat,A) = number_number_of(nat,semiring_1_of_nat(int,A)). [resolve(144,a,143,a)].
% 0.82/1.11 145 number(int) # label(arity_Int_Oint___Int_Onumber) # label(axiom). [assumption].
% 0.82/1.11 146 -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(16)].
% 0.82/1.11 147 -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(17)].
% 0.82/1.11 Derived: number_number_of(int,ti(int,A)) = number_number_of(int,A). [resolve(145,a,146,a)].
% 0.82/1.11 Derived: number_number_of(int,A) = ti(int,number_number_of(int,A)). [resolve(145,a,147,a)].
% 0.82/1.11 148 number(nat) # label(arity_Nat_Onat___Int_Onumber) # label(axiom). [assumption].
% 0.82/1.11 Derived: number_number_of(nat,ti(int,A)) = number_number_of(nat,A). [resolve(148,a,146,a)].
% 0.82/1.11 Derived: number_number_of(nat,A) = ti(nat,number_number_of(nat,A)). [resolve(148,a,147,a)].
% 0.82/1.11 149 linordered_idom(int) # label(arity_Int_Oint___Rings_Olinordered__idom) # label(axiom). [assumption].
% 0.82/1.11 150 -linordered_idom(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(20)].
% 0.82/1.11 151 -linordered_idom(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(20)].
% 0.82/1.11 152 -linordered_idom(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(21)].
% 0.82/1.11 153 -linordered_idom(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(21)].
% 0.82/1.11 154 -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_2_sum__power2__eq__zero__iff) # label(axiom). [clausify(30)].
% 0.82/1.11 155 -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_2_sum__power2__eq__zero__iff) # label(axiom). [clausify(30)].
% 0.82/1.11 156 -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_2_sum__power2__eq__zero__iff) # label(axiom). [clausify(30)].
% 0.82/1.11 157 -number_ring(A) | -linordered_idom(A) | -ord_less(A,number_number_of(A,B),number_number_of(A,C)) | ord_less(int,B,C) # label(fact_18_less__number__of) # label(axiom). [clausify(45)].
% 0.82/1.11 158 -number_ring(A) | -linordered_idom(A) | ord_less(A,number_number_of(A,B),number_number_of(A,C)) | -ord_less(int,B,C) # label(fact_18_less__number__of) # label(axiom). [clausify(45)].
% 0.82/1.11 159 -linordered_idom(A) | -ord_less(A,plus_plus(A,B,B),zero_zero(A)) | ord_less(A,B,zero_zero(A)) # label(fact_44_even__less__0__iff) # label(axiom). [clausify(61)].
% 0.82/1.11 160 -linordered_idom(A) | ord_less(A,plus_plus(A,B,B),zero_zero(A)) | -ord_less(A,B,zero_zero(A)) # label(fact_44_even__less__0__iff) # label(axiom). [clausify(61)].
% 0.82/1.11 161 -number_ring(A) | -linordered_idom(A) | -ord_less(A,number_number_of(A,B),zero_zero(A)) | ord_less(int,B,pls) # label(fact_49_less__special_I3_J) # label(axiom). [clausify(66)].
% 0.82/1.11 162 -number_ring(A) | -linordered_idom(A) | ord_less(A,number_number_of(A,B),zero_zero(A)) | -ord_less(int,B,pls) # label(fact_49_less__special_I3_J) # label(axiom). [clausify(66)].
% 0.82/1.11 163 -number_ring(A) | -linordered_idom(A) | -ord_less(A,zero_zero(A),number_number_of(A,B)) | ord_less(int,pls,B) # label(fact_50_less__special_I1_J) # label(axiom). [clausify(67)].
% 0.82/1.11 164 -number_ring(A) | -linordered_idom(A) | ord_less(A,zero_zero(A),number_number_of(A,B)) | -ord_less(int,pls,B) # label(fact_50_less__special_I1_J) # label(axiom). [clausify(67)].
% 0.82/1.11 165 -number_ring(A) | -linordered_idom(A) | -ord_less(A,number_number_of(A,B),one_one(A)) | ord_less(int,B,bit1(pls)) # label(fact_62_less__special_I4_J) # label(axiom). [clausify(77)].
% 0.82/1.11 166 -number_ring(A) | -linordered_idom(A) | ord_less(A,number_number_of(A,B),one_one(A)) | -ord_less(int,B,bit1(pls)) # label(fact_62_less__special_I4_J) # label(axiom). [clausify(77)].
% 0.82/1.11 167 -number_ring(A) | -linordered_idom(A) | -ord_less(A,one_one(A),number_number_of(A,B)) | ord_less(int,bit1(pls),B) # label(fact_63_less__special_I2_J) # label(axiom). [clausify(78)].
% 0.82/1.11 168 -number_ring(A) | -linordered_idom(A) | ord_less(A,one_one(A),number_number_of(A,B)) | -ord_less(int,bit1(pls),B) # label(fact_63_less__special_I2_J) # label(axiom). [clausify(78)].
% 0.82/1.11 169 -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_95_zero__less__power2) # label(axiom). [clausify(107)].
% 0.82/1.11 170 -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_95_zero__less__power2) # label(axiom). [clausify(107)].
% 0.82/1.11 171 -linordered_idom(A) | -ord_less(A,power_power(A,B,number_number_of(nat,bit0(bit1(pls)))),zero_zero(A)) # label(fact_96_power2__less__0) # label(axiom). [clausify(108)].
% 0.82/1.11 172 -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_97_sum__power2__gt__zero__iff) # label(axiom). [clausify(109)].
% 0.82/1.11 173 -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_97_sum__power2__gt__zero__iff) # label(axiom). [clausify(109)].
% 0.82/1.11 174 -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_97_sum__power2__gt__zero__iff) # label(axiom). [clausify(109)].
% 0.82/1.11 Derived: -ord_less(int,ti(int,A),B) | ord_less(int,A,B). [resolve(149,a,150,a)].
% 0.82/1.11 Derived: ord_less(int,ti(int,A),B) | -ord_less(int,A,B). [resolve(149,a,151,a)].
% 0.82/1.11 Derived: -ord_less(int,A,ti(int,B)) | ord_less(int,A,B). [resolve(149,a,152,a)].
% 0.82/1.11 Derived: ord_less(int,A,ti(int,B)) | -ord_less(int,A,B). [resolve(149,a,153,a)].
% 0.82/1.11 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(149,a,154,a)].
% 0.82/1.11 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(149,a,155,a)].
% 0.82/1.11 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(149,a,156,a)].
% 0.82/1.11 Derived: -number_ring(int) | -ord_less(int,number_number_of(int,A),number_number_of(int,B)) | ord_less(int,A,B). [resolve(149,a,157,b)].
% 0.82/1.11 Derived: -number_ring(int) | ord_less(int,number_number_of(int,A),number_number_of(int,B)) | -ord_less(int,A,B). [resolve(149,a,158,b)].
% 0.82/1.11 Derived: -ord_less(int,plus_plus(int,A,A),zero_zero(int)) | ord_less(int,A,zero_zero(int)). [resolve(149,a,159,a)].
% 0.82/1.11 Derived: ord_less(int,plus_plus(int,A,A),zero_zero(int)) | -ord_less(int,A,zero_zero(int)). [resolve(149,a,160,a)].
% 0.82/1.11 Derived: -number_ring(int) | -ord_less(int,number_number_of(int,A),zero_zero(int)) | ord_less(int,A,pls). [resolve(149,a,161,b)].
% 0.82/1.11 Derived: -number_ring(int) | ord_less(int,number_number_of(int,A),zero_zero(int)) | -ord_less(int,A,pls). [resolve(149,a,162,b)].
% 0.82/1.11 Derived: -number_ring(int) | -ord_less(int,zero_zero(int),number_number_of(int,A)) | ord_less(int,pls,A). [resolve(149,a,163,b)].
% 0.82/1.11 Derived: -number_ring(int) | ord_less(int,zero_zero(int),number_number_of(int,A)) | -ord_less(int,pls,A). [resolve(149,a,164,b)].
% 0.82/1.11 Derived: -number_ring(int) | -ord_less(int,number_number_of(int,A),one_one(int)) | ord_less(int,A,bit1(pls)). [resolve(149,a,165,b)].
% 0.82/1.11 Derived: -number_ring(int) | ord_less(int,number_number_of(int,A),one_one(int)) | -ord_less(int,A,bit1(pls)). [resolve(149,a,166,b)].
% 0.82/1.11 Derived: -number_ring(int) | -ord_less(int,one_one(int),number_number_of(int,A)) | ord_less(int,bit1(pls),A). [resolve(149,a,167,b)].
% 0.82/1.11 Derived: -number_ring(int) | ord_less(int,one_one(int),number_number_of(int,A)) | -ord_less(int,bit1(pls),A). [resolve(149,a,168,b)].
% 0.82/1.11 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(149,a,169,a)].
% 0.82/1.11 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(149,a,170,a)].
% 0.82/1.11 Derived: -ord_less(int,power_power(int,A,number_number_of(nat,bit0(bit1(pls)))),zero_zero(int)). [resolve(149,a,171,a)].
% 0.82/1.11 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(149,a,172,a)].
% 0.82/1.11 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(149,a,173,a)].
% 1.41/1.75 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(149,a,174,a)].
% 1.41/1.75 175 semiring_1(int) # label(arity_Int_Oint___Rings_Osemiring__1) # label(axiom). [assumption].
% 1.41/1.75 176 -semiring_1(A) | power_power(A,one_one(A),number_number_of(nat,bit0(bit1(pls)))) = one_one(A) # label(fact_3_one__power2) # label(axiom). [clausify(31)].
% 1.41/1.75 177 -semiring_1(A) | power_power(A,zero_zero(A),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(A) # label(fact_4_zero__power2) # label(axiom). [clausify(32)].
% 1.41/1.75 Derived: power_power(int,one_one(int),number_number_of(nat,bit0(bit1(pls)))) = one_one(int). [resolve(175,a,176,a)].
% 1.41/1.75 Derived: power_power(int,zero_zero(int),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(int). [resolve(175,a,177,a)].
% 1.41/1.75 178 semiring_1(nat) # label(arity_Nat_Onat___Rings_Osemiring__1) # label(axiom). [assumption].
% 1.41/1.75 Derived: power_power(nat,one_one(nat),number_number_of(nat,bit0(bit1(pls)))) = one_one(nat). [resolve(178,a,176,a)].
% 1.41/1.75 Derived: power_power(nat,zero_zero(nat),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(nat). [resolve(178,a,177,a)].
% 1.41/1.75 179 ring_11004092258visors(int) # label(arity_Int_Oint___Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption].
% 1.41/1.75 180 -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_5_zero__eq__power2) # label(axiom). [clausify(33)].
% 1.41/1.75 181 -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_5_zero__eq__power2) # label(axiom). [clausify(33)].
% 1.41/1.75 Derived: power_power(int,A,number_number_of(nat,bit0(bit1(pls)))) != zero_zero(int) | zero_zero(int) = ti(int,A). [resolve(179,a,180,a)].
% 1.41/1.75 Derived: power_power(int,A,number_number_of(nat,bit0(bit1(pls)))) = zero_zero(int) | zero_zero(int) != ti(int,A). [resolve(179,a,181,a)].
% 1.41/1.75 182 ring_char_0(int) # label(arity_Int_Oint___Int_Oring__char__0) # label(axiom). [assumption].
% 1.41/1.75 183 -number_ring(A) | -ring_char_0(A) | number_number_of(A,B) != number_number_of(A,C) | B = C # label(fact_40_eq__number__of) # label(axiom). [clausify(57)].
% 1.41/1.75 184 -number_ring(A) | -ring_char_0(A) | number_number_of(A,B) = number_number_of(A,C) | B != C # label(fact_40_eq__number__of) # label(axiom). [clausify(57)].
% 1.41/1.75 Derived: -number_ring(int) | number_number_of(int,A) != number_number_of(int,B) | A = B. [resolve(182,a,183,b)].
% 1.41/1.75 Derived: -number_ring(int) | number_number_of(int,A) = number_number_of(int,B) | A != B. [resolve(182,a,184,b)].
% 1.41/1.75
% 1.41/1.75 ============================== end predicate elimination =============
% 1.41/1.75
% 1.41/1.75 Auto_denials: (non-Horn, no changes).
% 1.41/1.75
% 1.41/1.75 Term ordering decisions:
% 1.41/1.75 Function symbol KB weights: int=1. nat=1. pls=1. n=1. t=1. number_number_of=1. ti=1. semiring_1_of_nat=1. zero_zero=1. bit1=1. bit0=1. one_one=1. undefined=1. plus_plus=1. power_power=1.
% 1.41/1.75
% 1.41/1.75 ============================== end of process initial clauses ========
% 1.41/1.75
% 1.41/1.75 ============================== CLAUSES FOR SEARCH ====================
% 1.41/1.75
% 1.41/1.75 ============================== end of clauses for search =============
% 1.41/1.75
% 1.41/1.75 ============================== SEARCH ================================
% 1.41/1.75
% 1.41/1.75 % Starting search at 0.07 seconds.
% 1.41/1.75
% 1.41/1.75 NOTE: Back_subsumption disabled, ratio of kept to back_subsumed is 45 (0.00 of 0.49 sec).
% 1.41/1.75
% 1.41/1.75 ============================== PROOF =================================
% 1.41/1.75 % SZS status Theorem
% 1.41/1.75 % SZS output start Refutation
% 1.41/1.75
% 1.41/1.75 % Proof 1 at 0.66 (+ 0.02) seconds.
% 1.41/1.75 % Length of proof is 57.
% 1.41/1.75 % Level of proof is 8.
% 1.41/1.75 % Maximum clause weight is 36.000.
% 1.41/1.75 % Given clauses 621.
% 1.41/1.75
% 1.41/1.75 51 (all K1 all K2 (ord_less(int,bit1(K1),bit1(K2)) <-> ord_less(int,K1,K2))) # label(fact_30_less__int__code_I16_J) # label(axiom) # label(non_clause). [assumption].
% 1.41/1.75 55 (all K all I all J (ord_less(int,I,J) -> ord_less(int,plus_plus(int,I,K),plus_plus(int,J,K)))) # label(fact_35_zadd__strict__right__mono) # label(axiom) # label(non_clause). [assumption].
% 1.41/1.75 62 (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_45_zadd__assoc) # label(axiom) # label(non_clause). [assumption].
% 1.41/1.75 64 (all Z all W plus_plus(int,Z,W) = plus_plus(int,W,Z)) # label(fact_47_zadd__commute) # label(axiom) # label(non_clause). [assumption].
% 1.41/1.75 87 (all K plus_plus(int,K,pls) = K) # label(fact_75_add__Pls__right) # label(axiom) # label(non_clause). [assumption].
% 1.41/1.75 90 (all K bit0(K) = plus_plus(int,K,K)) # label(fact_78_Bit0__def) # label(axiom) # label(non_clause). [assumption].
% 1.41/1.75 93 (all X_a (number_semiring(X_a) -> number_number_of(X_a,pls) = zero_zero(X_a))) # label(fact_81_semiring__numeral__0__eq__0) # label(axiom) # label(non_clause). [assumption].
% 1.41/1.75 97 (all X_a (number_ring(X_a) -> (all A_1 plus_plus(X_a,A_1,number_number_of(X_a,pls)) = ti(X_a,A_1)))) # label(fact_85_add__numeral__0__right) # label(axiom) # label(non_clause). [assumption].
% 1.41/1.75 98 (all X_a (power(X_a) & mult_zero(X_a) & no_zero_divisors(X_a) & zero_neq_one(X_a) -> (all A_2 all Wa (power_power(X_a,A_2,number_number_of(nat,Wa)) = zero_zero(X_a) <-> ti(X_a,A_2) = zero_zero(X_a) & number_number_of(nat,Wa) != zero_zero(nat))))) # label(fact_86_power__eq__0__iff__number__of) # label(axiom) # label(non_clause). [assumption].
% 1.41/1.75 104 (all K bit1(K) = plus_plus(int,plus_plus(int,one_one(int),K),K)) # label(fact_92_Bit1__def) # label(axiom) # label(non_clause). [assumption].
% 1.41/1.75 111 -(power_power(int,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)),number_number_of(nat,bit0(bit1(pls)))) != zero_zero(int)) # label(conj_0) # label(negated_conjecture) # label(non_clause). [assumption].
% 1.41/1.75 112 power(int) # label(arity_Int_Oint___Power_Opower) # label(axiom). [assumption].
% 1.41/1.75 124 -power(A) | -mult_zero(A) | -no_zero_divisors(A) | -zero_neq_one(A) | power_power(A,B,number_number_of(nat,C)) != zero_zero(A) | zero_zero(A) = ti(A,B) # label(fact_86_power__eq__0__iff__number__of) # label(axiom). [clausify(98)].
% 1.41/1.75 135 number_semiring(int) # label(arity_Int_Oint___Int_Onumber__semiring) # label(axiom). [assumption].
% 1.41/1.75 142 -number_semiring(A) | number_number_of(A,pls) = zero_zero(A) # label(fact_81_semiring__numeral__0__eq__0) # label(axiom). [clausify(93)].
% 1.41/1.75 200 ord_less(int,zero_zero(int),plus_plus(int,one_one(int),semiring_1_of_nat(int,n))) # label(fact_0_n1pos) # label(axiom). [assumption].
% 1.41/1.75 229 -ord_less(int,bit1(A),bit1(B)) | ord_less(int,A,B) # label(fact_30_less__int__code_I16_J) # label(axiom). [clausify(51)].
% 1.41/1.75 234 -ord_less(int,A,B) | ord_less(int,plus_plus(int,A,C),plus_plus(int,B,C)) # label(fact_35_zadd__strict__right__mono) # label(axiom). [clausify(55)].
% 1.41/1.75 249 plus_plus(int,plus_plus(int,A,B),C) = plus_plus(int,A,plus_plus(int,B,C)) # label(fact_45_zadd__assoc) # label(axiom). [clausify(62)].
% 1.41/1.75 251 plus_plus(int,A,B) = plus_plus(int,B,A) # label(fact_47_zadd__commute) # label(axiom). [clausify(64)].
% 1.41/1.75 264 -ord_less(int,pls,zero_zero(int)) # label(fact_57_bin__less__0__simps_I1_J) # label(axiom). [assumption].
% 1.41/1.75 285 pls = zero_zero(int) # label(fact_73_Pls__def) # label(axiom). [assumption].
% 1.41/1.75 286 zero_zero(int) != one_one(int) # label(fact_74_int__0__neq__1) # label(axiom). [assumption].
% 1.41/1.75 287 one_one(int) != zero_zero(int). [copy(286),flip(a)].
% 1.41/1.75 288 plus_plus(int,A,pls) = A # label(fact_75_add__Pls__right) # label(axiom). [clausify(87)].
% 1.41/1.75 289 plus_plus(int,A,zero_zero(int)) = A. [copy(288),rewrite([285(2)])].
% 1.41/1.75 293 bit0(A) = plus_plus(int,A,A) # label(fact_78_Bit0__def) # label(axiom). [clausify(90)].
% 1.41/1.75 301 -number_ring(A) | plus_plus(A,B,number_number_of(A,pls)) = ti(A,B) # label(fact_85_add__numeral__0__right) # label(axiom). [clausify(97)].
% 1.41/1.75 302 -number_ring(A) | plus_plus(A,B,number_number_of(A,zero_zero(int))) = ti(A,B). [copy(301),rewrite([285(2)])].
% 1.41/1.75 312 bit1(A) = plus_plus(int,plus_plus(int,one_one(int),A),A) # label(fact_92_Bit1__def) # label(axiom). [clausify(104)].
% 1.41/1.75 313 bit1(A) = plus_plus(int,A,plus_plus(int,A,one_one(int))). [copy(312),rewrite([251(6),251(7)])].
% 1.41/1.75 316 no_zero_divisors(int) # label(arity_Int_Oint___Rings_Ono__zero__divisors) # label(axiom). [assumption].
% 1.41/1.75 317 zero_neq_one(int) # label(arity_Int_Oint___Rings_Ozero__neq__one) # label(axiom). [assumption].
% 1.41/1.75 319 mult_zero(int) # label(arity_Int_Oint___Rings_Omult__zero) # label(axiom). [assumption].
% 1.41/1.75 320 number_ring(int) # label(arity_Int_Oint___Int_Onumber__ring) # label(axiom). [assumption].
% 1.41/1.75 326 power_power(int,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(int) # label(conj_0) # label(negated_conjecture). [clausify(111)].
% 1.41/1.75 327 power_power(int,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)),number_number_of(nat,plus_plus(int,zero_zero(int),plus_plus(int,zero_zero(int),plus_plus(int,one_one(int),plus_plus(int,zero_zero(int),plus_plus(int,zero_zero(int),one_one(int)))))))) = zero_zero(int). [copy(326),rewrite([285(10),313(12),293(20),249(31),249(30)])].
% 1.41/1.75 346 -mult_zero(int) | -no_zero_divisors(int) | -zero_neq_one(int) | power_power(int,A,number_number_of(nat,B)) != zero_zero(int) | zero_zero(int) = ti(int,A). [resolve(112,a,124,a)].
% 1.41/1.75 347 power_power(int,A,number_number_of(nat,B)) != zero_zero(int) | ti(int,A) = zero_zero(int). [copy(346),flip(e),unit_del(a,319),unit_del(b,316),unit_del(c,317)].
% 1.41/1.75 391 number_number_of(int,pls) = zero_zero(int). [resolve(135,a,142,a)].
% 1.41/1.75 392 number_number_of(int,zero_zero(int)) = zero_zero(int). [copy(391),rewrite([285(2)])].
% 1.41/1.75 471 plus_plus(int,A,plus_plus(int,B,C)) = plus_plus(int,C,plus_plus(int,A,B)). [back_rewrite(249),rewrite([251(4)]),flip(a)].
% 1.41/1.75 476 -ord_less(int,zero_zero(int),zero_zero(int)). [back_rewrite(264),rewrite([285(2)])].
% 1.41/1.75 489 one_one(int) = c_0. [new_symbol(287)].
% 1.41/1.75 505 -ord_less(int,plus_plus(int,A,plus_plus(int,A,c_0)),plus_plus(int,B,plus_plus(int,B,c_0))) | ord_less(int,A,B). [back_rewrite(229),rewrite([313(2),489(5),313(7),489(10)])].
% 1.41/1.75 538 power_power(int,plus_plus(int,c_0,semiring_1_of_nat(int,n)),number_number_of(nat,plus_plus(int,zero_zero(int),plus_plus(int,zero_zero(int),plus_plus(int,c_0,c_0))))) = zero_zero(int). [back_rewrite(327),rewrite([489(4),489(17),489(24),251(24),289(24),251(21),289(21)])].
% 1.41/1.75 548 ord_less(int,zero_zero(int),plus_plus(int,c_0,semiring_1_of_nat(int,n))). [back_rewrite(200),rewrite([489(6)])].
% 1.41/1.75 570 ti(int,A) = A. [resolve(320,a,302,a),rewrite([392(5),289(4)]),flip(a)].
% 1.41/1.75 583 power_power(int,A,number_number_of(nat,B)) != zero_zero(int) | zero_zero(int) = A. [back_rewrite(347),rewrite([570(9)]),flip(b)].
% 1.41/1.75 632 plus_plus(int,zero_zero(int),plus_plus(int,A,B)) = plus_plus(int,A,B). [para(289(a,1),471(a,1,3)),flip(a)].
% 1.41/1.75 645 power_power(int,plus_plus(int,c_0,semiring_1_of_nat(int,n)),number_number_of(nat,plus_plus(int,c_0,c_0))) = zero_zero(int). [back_rewrite(538),rewrite([632(19),632(16)])].
% 1.41/1.75 846 ord_less(int,A,plus_plus(int,A,plus_plus(int,c_0,semiring_1_of_nat(int,n)))). [resolve(548,a,234,a),rewrite([251(5),289(5),251(9)])].
% 1.41/1.75 1704 -ord_less(int,c_0,plus_plus(int,A,plus_plus(int,A,c_0))) | ord_less(int,zero_zero(int),A). [para(632(a,1),505(a,2)),rewrite([251(6),289(6)])].
% 1.41/1.75 3781 -ord_less(int,c_0,c_0). [para(632(a,1),1704(a,3)),rewrite([251(7),289(7)]),unit_del(b,476)].
% 1.41/1.75 4281 plus_plus(int,c_0,semiring_1_of_nat(int,n)) = zero_zero(int). [resolve(645,a,583,a),flip(a)].
% 1.41/1.75 4284 ord_less(int,A,A). [back_rewrite(846),rewrite([4281(8),289(5)])].
% 1.41/1.75 4285 $F. [resolve(4284,a,3781,a)].
% 1.41/1.75
% 1.41/1.75 % SZS output end Refutation
% 1.41/1.75 ============================== end of proof ==========================
% 1.41/1.75
% 1.41/1.75 ============================== STATISTICS ============================
% 1.41/1.75
% 1.41/1.75 Given=621. Generated=15374. Kept=3989. proofs=1.
% 1.41/1.75 Usable=584. Sos=3009. Demods=191. Limbo=3, Disabled=672. Hints=0.
% 1.41/1.75 Megabytes=5.89.
% 1.41/1.75 User_CPU=0.66, System_CPU=0.02, Wall_clock=1.
% 1.41/1.75
% 1.41/1.75 ============================== end of statistics =====================
% 1.41/1.75
% 1.41/1.75 ============================== end of search =========================
% 1.41/1.75
% 1.41/1.75 THEOREM PROVED
% 1.41/1.75 % SZS status Theorem
% 1.41/1.75
% 1.41/1.75 Exiting with 1 proof.
% 1.41/1.75
% 1.41/1.75 Process 16909 exit (max_proofs) Wed Jul 6 17:13:01 2022
% 1.41/1.75 Prover9 interrupted
%------------------------------------------------------------------------------