%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : LCL888+1 : TPTP v9.3.1. Released v5.5.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n004.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 29 11:50:44 AM UTC 2026
% Result : Theorem 59.33s 8.05s
% Output : Proof 61.67s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : LCL888+1 : TPTP v9.3.1. Released v5.5.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.09/0.37 % Computer : n004.cluster.edu
% 0.09/0.37 % Model : x86_64 x86_64
% 0.09/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.37 % Memory : 8046.5625MB
% 0.09/0.37 % OS : Linux 6.8.0-71-generic
% 0.09/0.37 % CPULimit : 300
% 0.09/0.37 % WCLimit : 300
% 0.09/0.37 % DateTime : Sun Sep 27 17:04:21 UTC 2026
% 0.09/0.37 % CPUTime :
% 0.09/0.37 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 59.33/8.05 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 59.33/8.05
% 59.33/8.05 % SZS status Theorem
% 59.33/8.05
% 61.67/8.22 % SZS output start Proof
% 61.67/8.22 Axiom 1 (sos_08): X >= 0 = true.
% 61.67/8.22 Axiom 2 (sos_02): X + Y = Y + X.
% 61.67/8.22 Axiom 3 (sos_03): X + 0 = X.
% 61.67/8.22 Axiom 4 (goals_13): x19 = x19 ==> x18.
% 61.67/8.22 Axiom 5 (goals_13_1): x17 = x17 ==> x18.
% 61.67/8.22 Axiom 6 (sos_12): X + (X ==> Y) = Y + (Y ==> X).
% 61.67/8.22 Axiom 7 (sos_01): (X + Y) + Z = X + (Y + Z).
% 61.67/8.22 Axiom 8 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 61.67/8.22 Axiom 9 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 61.67/8.22 Axiom 10 (sos_09): ifeq(X >= Y, true, (X + Z) >= (Y + Z), true) = true.
% 61.67/8.22 Axiom 11 (sos_11): ifeq(X >= Y, true, (Z ==> X) >= (Z ==> Y), true) = true.
% 61.67/8.22 Axiom 12 (sos_07_1): ifeq((X + Y) >= Z, true, Y >= (X ==> Z), true) = true.
% 61.67/8.22 Axiom 13 (sos_06): ifeq2(X >= Y, true, ifeq2(Y >= X, true, Y, X), X) = X.
% 61.67/8.22
% 61.67/8.22 Lemma 14: 0 + X = X.
% 61.67/8.22 Proof:
% 61.67/8.22 0 + X
% 61.67/8.22 = { by axiom 2 (sos_02) R->L }
% 61.67/8.22 X + 0
% 61.67/8.22 = { by axiom 3 (sos_03) }
% 61.67/8.22 X
% 61.67/8.22
% 61.67/8.22 Lemma 15: Z + ((Z ==> X) + Y) = X + (Y + (X ==> Z)).
% 61.67/8.22 Proof:
% 61.67/8.22 Z + ((Z ==> X) + Y)
% 61.67/8.22 = { by axiom 7 (sos_01) R->L }
% 61.67/8.22 (Z + (Z ==> X)) + Y
% 61.67/8.22 = { by axiom 6 (sos_12) R->L }
% 61.67/8.22 (X + (X ==> Z)) + Y
% 61.67/8.22 = { by axiom 7 (sos_01) }
% 61.67/8.22 X + ((X ==> Z) + Y)
% 61.67/8.22 = { by axiom 2 (sos_02) }
% 61.67/8.22 X + (Y + (X ==> Z))
% 61.67/8.22
% 61.67/8.22 Lemma 16: Z + ((Y ==> X) + (Z ==> Y)) = X + ((X ==> Y) + (Y ==> Z)).
% 61.67/8.22 Proof:
% 61.67/8.22 Z + ((Y ==> X) + (Z ==> Y))
% 61.67/8.22 = { by lemma 15 R->L }
% 61.67/8.22 Y + ((Y ==> Z) + (Y ==> X))
% 61.67/8.22 = { by lemma 15 R->L }
% 61.67/8.22 X + ((X ==> Y) + (Y ==> Z))
% 61.67/8.22
% 61.67/8.22 Lemma 17: ifeq2(0 >= X, true, 0, X) = X.
% 61.67/8.22 Proof:
% 61.67/8.22 ifeq2(0 >= X, true, 0, X)
% 61.67/8.22 = { by axiom 9 (ifeq_axiom) R->L }
% 61.67/8.22 ifeq2(true, true, ifeq2(0 >= X, true, 0, X), X)
% 61.67/8.22 = { by axiom 1 (sos_08) R->L }
% 61.67/8.22 ifeq2(X >= 0, true, ifeq2(0 >= X, true, 0, X), X)
% 61.67/8.22 = { by axiom 13 (sos_06) }
% 61.67/8.22 X
% 61.67/8.22
% 61.67/8.22 Lemma 18: Y + (X + Z) = X + (Y + Z).
% 61.67/8.22 Proof:
% 61.67/8.22 Y + (X + Z)
% 61.67/8.22 = { by axiom 2 (sos_02) R->L }
% 61.67/8.22 (X + Z) + Y
% 61.67/8.22 = { by axiom 7 (sos_01) }
% 61.67/8.22 X + (Z + Y)
% 61.67/8.22 = { by axiom 2 (sos_02) }
% 61.67/8.22 X + (Y + Z)
% 61.67/8.22
% 61.67/8.22 Lemma 19: x18 ==> x19 = 0.
% 61.67/8.22 Proof:
% 61.67/8.22 x18 ==> x19
% 61.67/8.22 = { by lemma 17 R->L }
% 61.67/8.22 ifeq2(0 >= (x18 ==> x19), true, 0, x18 ==> x19)
% 61.67/8.22 = { by axiom 8 (ifeq_axiom) R->L }
% 61.67/8.22 ifeq2(ifeq(true, true, 0 >= (x18 ==> x19), true), true, 0, x18 ==> x19)
% 61.67/8.22 = { by axiom 12 (sos_07_1) R->L }
% 61.67/8.22 ifeq2(ifeq(ifeq((x19 + (x18 + 0)) >= x18, true, (x18 + 0) >= (x19 ==> x18), true), true, 0 >= (x18 ==> x19), true), true, 0, x18 ==> x19)
% 61.67/8.22 = { by lemma 18 }
% 61.67/8.22 ifeq2(ifeq(ifeq((x18 + (x19 + 0)) >= x18, true, (x18 + 0) >= (x19 ==> x18), true), true, 0 >= (x18 ==> x19), true), true, 0, x18 ==> x19)
% 61.67/8.22 = { by axiom 2 (sos_02) R->L }
% 61.67/8.22 ifeq2(ifeq(ifeq(((x19 + 0) + x18) >= x18, true, (x18 + 0) >= (x19 ==> x18), true), true, 0 >= (x18 ==> x19), true), true, 0, x18 ==> x19)
% 61.67/8.22 = { by axiom 8 (ifeq_axiom) R->L }
% 61.67/8.22 ifeq2(ifeq(ifeq(ifeq(true, true, ((x19 + 0) + x18) >= x18, true), true, (x18 + 0) >= (x19 ==> x18), true), true, 0 >= (x18 ==> x19), true), true, 0, x18 ==> x19)
% 61.67/8.22 = { by axiom 1 (sos_08) R->L }
% 61.67/8.22 ifeq2(ifeq(ifeq(ifeq((x19 + 0) >= 0, true, ((x19 + 0) + x18) >= x18, true), true, (x18 + 0) >= (x19 ==> x18), true), true, 0 >= (x18 ==> x19), true), true, 0, x18 ==> x19)
% 61.67/8.22 = { by lemma 14 R->L }
% 61.67/8.22 ifeq2(ifeq(ifeq(ifeq((x19 + 0) >= 0, true, ((x19 + 0) + x18) >= (0 + x18), true), true, (x18 + 0) >= (x19 ==> x18), true), true, 0 >= (x18 ==> x19), true), true, 0, x18 ==> x19)
% 61.67/8.22 = { by axiom 10 (sos_09) }
% 61.67/8.22 ifeq2(ifeq(ifeq(true, true, (x18 + 0) >= (x19 ==> x18), true), true, 0 >= (x18 ==> x19), true), true, 0, x18 ==> x19)
% 61.67/8.22 = { by axiom 8 (ifeq_axiom) }
% 61.67/8.22 ifeq2(ifeq((x18 + 0) >= (x19 ==> x18), true, 0 >= (x18 ==> x19), true), true, 0, x18 ==> x19)
% 61.67/8.22 = { by axiom 4 (goals_13) R->L }
% 61.67/8.22 ifeq2(ifeq((x18 + 0) >= x19, true, 0 >= (x18 ==> x19), true), true, 0, x18 ==> x19)
% 61.67/8.22 = { by axiom 12 (sos_07_1) }
% 61.67/8.22 ifeq2(true, true, 0, x18 ==> x19)
% 61.67/8.22 = { by axiom 9 (ifeq_axiom) }
% 61.67/8.22 0
% 61.67/8.22
% 61.67/8.22 Lemma 20: x17 ==> x19 = x19 ==> x17.
% 61.67/8.22 Proof:
% 61.67/8.22 x17 ==> x19
% 61.67/8.22 = { by axiom 13 (sos_06) R->L }
% 61.67/8.22 ifeq2((x17 ==> x19) >= (x19 ==> x17), true, ifeq2((x19 ==> x17) >= (x17 ==> x19), true, x19 ==> x17, x17 ==> x19), x17 ==> x19)
% 61.67/8.22 = { by axiom 8 (ifeq_axiom) R->L }
% 61.67/8.22 ifeq2(ifeq(true, true, (x17 ==> x19) >= (x19 ==> x17), true), true, ifeq2((x19 ==> x17) >= (x17 ==> x19), true, x19 ==> x17, x17 ==> x19), x17 ==> x19)
% 61.67/8.22 = { by axiom 12 (sos_07_1) R->L }
% 61.67/8.22 ifeq2(ifeq(ifeq((x17 + (x19 + (x17 ==> x19))) >= x18, true, (x19 + (x17 ==> x19)) >= (x17 ==> x18), true), true, (x17 ==> x19) >= (x19 ==> x17), true), true, ifeq2((x19 ==> x17) >= (x17 ==> x19), true, x19 ==> x17, x17 ==> x19), x17 ==> x19)
% 61.67/8.22 = { by axiom 5 (goals_13_1) R->L }
% 61.67/8.22 ifeq2(ifeq(ifeq((x17 + (x19 + (x17 ==> x19))) >= x18, true, (x19 + (x17 ==> x19)) >= x17, true), true, (x17 ==> x19) >= (x19 ==> x17), true), true, ifeq2((x19 ==> x17) >= (x17 ==> x19), true, x19 ==> x17, x17 ==> x19), x17 ==> x19)
% 61.67/8.22 = { by axiom 4 (goals_13) }
% 61.67/8.22 ifeq2(ifeq(ifeq((x17 + ((x19 ==> x18) + (x17 ==> x19))) >= x18, true, (x19 + (x17 ==> x19)) >= x17, true), true, (x17 ==> x19) >= (x19 ==> x17), true), true, ifeq2((x19 ==> x17) >= (x17 ==> x19), true, x19 ==> x17, x17 ==> x19), x17 ==> x19)
% 61.67/8.22 = { by lemma 16 }
% 61.67/8.23 ifeq2(ifeq(ifeq((x18 + ((x18 ==> x19) + (x19 ==> x17))) >= x18, true, (x19 + (x17 ==> x19)) >= x17, true), true, (x17 ==> x19) >= (x19 ==> x17), true), true, ifeq2((x19 ==> x17) >= (x17 ==> x19), true, x19 ==> x17, x17 ==> x19), x17 ==> x19)
% 61.67/8.23 = { by lemma 19 }
% 61.67/8.23 ifeq2(ifeq(ifeq((x18 + (0 + (x19 ==> x17))) >= x18, true, (x19 + (x17 ==> x19)) >= x17, true), true, (x17 ==> x19) >= (x19 ==> x17), true), true, ifeq2((x19 ==> x17) >= (x17 ==> x19), true, x19 ==> x17, x17 ==> x19), x17 ==> x19)
% 61.67/8.23 = { by lemma 14 }
% 61.67/8.23 ifeq2(ifeq(ifeq((x18 + (x19 ==> x17)) >= x18, true, (x19 + (x17 ==> x19)) >= x17, true), true, (x17 ==> x19) >= (x19 ==> x17), true), true, ifeq2((x19 ==> x17) >= (x17 ==> x19), true, x19 ==> x17, x17 ==> x19), x17 ==> x19)
% 61.67/8.23 = { by axiom 2 (sos_02) R->L }
% 61.67/8.23 ifeq2(ifeq(ifeq(((x19 ==> x17) + x18) >= x18, true, (x19 + (x17 ==> x19)) >= x17, true), true, (x17 ==> x19) >= (x19 ==> x17), true), true, ifeq2((x19 ==> x17) >= (x17 ==> x19), true, x19 ==> x17, x17 ==> x19), x17 ==> x19)
% 61.67/8.23 = { by axiom 8 (ifeq_axiom) R->L }
% 61.67/8.23 ifeq2(ifeq(ifeq(ifeq(true, true, ((x19 ==> x17) + x18) >= x18, true), true, (x19 + (x17 ==> x19)) >= x17, true), true, (x17 ==> x19) >= (x19 ==> x17), true), true, ifeq2((x19 ==> x17) >= (x17 ==> x19), true, x19 ==> x17, x17 ==> x19), x17 ==> x19)
% 61.67/8.23 = { by axiom 1 (sos_08) R->L }
% 61.67/8.23 ifeq2(ifeq(ifeq(ifeq((x19 ==> x17) >= 0, true, ((x19 ==> x17) + x18) >= x18, true), true, (x19 + (x17 ==> x19)) >= x17, true), true, (x17 ==> x19) >= (x19 ==> x17), true), true, ifeq2((x19 ==> x17) >= (x17 ==> x19), true, x19 ==> x17, x17 ==> x19), x17 ==> x19)
% 61.67/8.23 = { by lemma 14 R->L }
% 61.67/8.23 ifeq2(ifeq(ifeq(ifeq((x19 ==> x17) >= 0, true, ((x19 ==> x17) + x18) >= (0 + x18), true), true, (x19 + (x17 ==> x19)) >= x17, true), true, (x17 ==> x19) >= (x19 ==> x17), true), true, ifeq2((x19 ==> x17) >= (x17 ==> x19), true, x19 ==> x17, x17 ==> x19), x17 ==> x19)
% 61.67/8.23 = { by axiom 10 (sos_09) }
% 61.67/8.23 ifeq2(ifeq(ifeq(true, true, (x19 + (x17 ==> x19)) >= x17, true), true, (x17 ==> x19) >= (x19 ==> x17), true), true, ifeq2((x19 ==> x17) >= (x17 ==> x19), true, x19 ==> x17, x17 ==> x19), x17 ==> x19)
% 61.67/8.23 = { by axiom 8 (ifeq_axiom) }
% 61.67/8.23 ifeq2(ifeq((x19 + (x17 ==> x19)) >= x17, true, (x17 ==> x19) >= (x19 ==> x17), true), true, ifeq2((x19 ==> x17) >= (x17 ==> x19), true, x19 ==> x17, x17 ==> x19), x17 ==> x19)
% 61.67/8.23 = { by axiom 12 (sos_07_1) }
% 61.67/8.23 ifeq2(true, true, ifeq2((x19 ==> x17) >= (x17 ==> x19), true, x19 ==> x17, x17 ==> x19), x17 ==> x19)
% 61.67/8.23 = { by axiom 9 (ifeq_axiom) }
% 61.67/8.23 ifeq2((x19 ==> x17) >= (x17 ==> x19), true, x19 ==> x17, x17 ==> x19)
% 61.67/8.23 = { by axiom 8 (ifeq_axiom) R->L }
% 61.67/8.23 ifeq2(ifeq(true, true, (x19 ==> x17) >= (x17 ==> x19), true), true, x19 ==> x17, x17 ==> x19)
% 61.67/8.23 = { by axiom 12 (sos_07_1) R->L }
% 61.67/8.23 ifeq2(ifeq(ifeq((x19 + (x17 + (x19 ==> x17))) >= x18, true, (x17 + (x19 ==> x17)) >= (x19 ==> x18), true), true, (x19 ==> x17) >= (x17 ==> x19), true), true, x19 ==> x17, x17 ==> x19)
% 61.67/8.23 = { by axiom 4 (goals_13) R->L }
% 61.67/8.23 ifeq2(ifeq(ifeq((x19 + (x17 + (x19 ==> x17))) >= x18, true, (x17 + (x19 ==> x17)) >= x19, true), true, (x19 ==> x17) >= (x17 ==> x19), true), true, x19 ==> x17, x17 ==> x19)
% 61.67/8.23 = { by axiom 5 (goals_13_1) }
% 61.67/8.23 ifeq2(ifeq(ifeq((x19 + ((x17 ==> x18) + (x19 ==> x17))) >= x18, true, (x17 + (x19 ==> x17)) >= x19, true), true, (x19 ==> x17) >= (x17 ==> x19), true), true, x19 ==> x17, x17 ==> x19)
% 61.67/8.23 = { by lemma 16 }
% 61.67/8.23 ifeq2(ifeq(ifeq((x18 + ((x18 ==> x17) + (x17 ==> x19))) >= x18, true, (x17 + (x19 ==> x17)) >= x19, true), true, (x19 ==> x17) >= (x17 ==> x19), true), true, x19 ==> x17, x17 ==> x19)
% 61.67/8.23 = { by lemma 17 R->L }
% 61.67/8.23 ifeq2(ifeq(ifeq((x18 + (ifeq2(0 >= (x18 ==> x17), true, 0, x18 ==> x17) + (x17 ==> x19))) >= x18, true, (x17 + (x19 ==> x17)) >= x19, true), true, (x19 ==> x17) >= (x17 ==> x19), true), true, x19 ==> x17, x17 ==> x19)
% 61.67/8.23 = { by axiom 8 (ifeq_axiom) R->L }
% 61.67/8.23 ifeq2(ifeq(ifeq((x18 + (ifeq2(ifeq(true, true, 0 >= (x18 ==> x17), true), true, 0, x18 ==> x17) + (x17 ==> x19))) >= x18, true, (x17 + (x19 ==> x17)) >= x19, true), true, (x19 ==> x17) >= (x17 ==> x19), true), true, x19 ==> x17, x17 ==> x19)
% 61.67/8.23 = { by axiom 12 (sos_07_1) R->L }
% 61.67/8.23 ifeq2(ifeq(ifeq((x18 + (ifeq2(ifeq(ifeq((x17 + (x18 + 0)) >= x18, true, (x18 + 0) >= (x17 ==> x18), true), true, 0 >= (x18 ==> x17), true), true, 0, x18 ==> x17) + (x17 ==> x19))) >= x18, true, (x17 + (x19 ==> x17)) >= x19, true), true, (x19 ==> x17) >= (x17 ==> x19), true), true, x19 ==> x17, x17 ==> x19)
% 61.67/8.23 = { by lemma 18 }
% 61.67/8.23 ifeq2(ifeq(ifeq((x18 + (ifeq2(ifeq(ifeq((x18 + (x17 + 0)) >= x18, true, (x18 + 0) >= (x17 ==> x18), true), true, 0 >= (x18 ==> x17), true), true, 0, x18 ==> x17) + (x17 ==> x19))) >= x18, true, (x17 + (x19 ==> x17)) >= x19, true), true, (x19 ==> x17) >= (x17 ==> x19), true), true, x19 ==> x17, x17 ==> x19)
% 61.67/8.23 = { by axiom 2 (sos_02) R->L }
% 61.67/8.23 ifeq2(ifeq(ifeq((x18 + (ifeq2(ifeq(ifeq(((x17 + 0) + x18) >= x18, true, (x18 + 0) >= (x17 ==> x18), true), true, 0 >= (x18 ==> x17), true), true, 0, x18 ==> x17) + (x17 ==> x19))) >= x18, true, (x17 + (x19 ==> x17)) >= x19, true), true, (x19 ==> x17) >= (x17 ==> x19), true), true, x19 ==> x17, x17 ==> x19)
% 61.67/8.23 = { by axiom 8 (ifeq_axiom) R->L }
% 61.67/8.23 ifeq2(ifeq(ifeq((x18 + (ifeq2(ifeq(ifeq(ifeq(true, true, ((x17 + 0) + x18) >= x18, true), true, (x18 + 0) >= (x17 ==> x18), true), true, 0 >= (x18 ==> x17), true), true, 0, x18 ==> x17) + (x17 ==> x19))) >= x18, true, (x17 + (x19 ==> x17)) >= x19, true), true, (x19 ==> x17) >= (x17 ==> x19), true), true, x19 ==> x17, x17 ==> x19)
% 61.67/8.23 = { by axiom 1 (sos_08) R->L }
% 61.67/8.23 ifeq2(ifeq(ifeq((x18 + (ifeq2(ifeq(ifeq(ifeq((x17 + 0) >= 0, true, ((x17 + 0) + x18) >= x18, true), true, (x18 + 0) >= (x17 ==> x18), true), true, 0 >= (x18 ==> x17), true), true, 0, x18 ==> x17) + (x17 ==> x19))) >= x18, true, (x17 + (x19 ==> x17)) >= x19, true), true, (x19 ==> x17) >= (x17 ==> x19), true), true, x19 ==> x17, x17 ==> x19)
% 61.67/8.23 = { by lemma 14 R->L }
% 61.67/8.23 ifeq2(ifeq(ifeq((x18 + (ifeq2(ifeq(ifeq(ifeq((x17 + 0) >= 0, true, ((x17 + 0) + x18) >= (0 + x18), true), true, (x18 + 0) >= (x17 ==> x18), true), true, 0 >= (x18 ==> x17), true), true, 0, x18 ==> x17) + (x17 ==> x19))) >= x18, true, (x17 + (x19 ==> x17)) >= x19, true), true, (x19 ==> x17) >= (x17 ==> x19), true), true, x19 ==> x17, x17 ==> x19)
% 61.67/8.23 = { by axiom 10 (sos_09) }
% 61.67/8.23 ifeq2(ifeq(ifeq((x18 + (ifeq2(ifeq(ifeq(true, true, (x18 + 0) >= (x17 ==> x18), true), true, 0 >= (x18 ==> x17), true), true, 0, x18 ==> x17) + (x17 ==> x19))) >= x18, true, (x17 + (x19 ==> x17)) >= x19, true), true, (x19 ==> x17) >= (x17 ==> x19), true), true, x19 ==> x17, x17 ==> x19)
% 61.67/8.23 = { by axiom 8 (ifeq_axiom) }
% 61.67/8.23 ifeq2(ifeq(ifeq((x18 + (ifeq2(ifeq((x18 + 0) >= (x17 ==> x18), true, 0 >= (x18 ==> x17), true), true, 0, x18 ==> x17) + (x17 ==> x19))) >= x18, true, (x17 + (x19 ==> x17)) >= x19, true), true, (x19 ==> x17) >= (x17 ==> x19), true), true, x19 ==> x17, x17 ==> x19)
% 61.67/8.23 = { by axiom 5 (goals_13_1) R->L }
% 61.67/8.23 ifeq2(ifeq(ifeq((x18 + (ifeq2(ifeq((x18 + 0) >= x17, true, 0 >= (x18 ==> x17), true), true, 0, x18 ==> x17) + (x17 ==> x19))) >= x18, true, (x17 + (x19 ==> x17)) >= x19, true), true, (x19 ==> x17) >= (x17 ==> x19), true), true, x19 ==> x17, x17 ==> x19)
% 61.67/8.24 = { by axiom 12 (sos_07_1) }
% 61.67/8.24 ifeq2(ifeq(ifeq((x18 + (ifeq2(true, true, 0, x18 ==> x17) + (x17 ==> x19))) >= x18, true, (x17 + (x19 ==> x17)) >= x19, true), true, (x19 ==> x17) >= (x17 ==> x19), true), true, x19 ==> x17, x17 ==> x19)
% 61.67/8.24 = { by axiom 9 (ifeq_axiom) }
% 61.67/8.24 ifeq2(ifeq(ifeq((x18 + (0 + (x17 ==> x19))) >= x18, true, (x17 + (x19 ==> x17)) >= x19, true), true, (x19 ==> x17) >= (x17 ==> x19), true), true, x19 ==> x17, x17 ==> x19)
% 61.67/8.24 = { by lemma 14 }
% 61.67/8.24 ifeq2(ifeq(ifeq((x18 + (x17 ==> x19)) >= x18, true, (x17 + (x19 ==> x17)) >= x19, true), true, (x19 ==> x17) >= (x17 ==> x19), true), true, x19 ==> x17, x17 ==> x19)
% 61.67/8.24 = { by axiom 2 (sos_02) R->L }
% 61.67/8.24 ifeq2(ifeq(ifeq(((x17 ==> x19) + x18) >= x18, true, (x17 + (x19 ==> x17)) >= x19, true), true, (x19 ==> x17) >= (x17 ==> x19), true), true, x19 ==> x17, x17 ==> x19)
% 61.67/8.24 = { by axiom 8 (ifeq_axiom) R->L }
% 61.67/8.24 ifeq2(ifeq(ifeq(ifeq(true, true, ((x17 ==> x19) + x18) >= x18, true), true, (x17 + (x19 ==> x17)) >= x19, true), true, (x19 ==> x17) >= (x17 ==> x19), true), true, x19 ==> x17, x17 ==> x19)
% 61.67/8.24 = { by axiom 1 (sos_08) R->L }
% 61.67/8.24 ifeq2(ifeq(ifeq(ifeq((x17 ==> x19) >= 0, true, ((x17 ==> x19) + x18) >= x18, true), true, (x17 + (x19 ==> x17)) >= x19, true), true, (x19 ==> x17) >= (x17 ==> x19), true), true, x19 ==> x17, x17 ==> x19)
% 61.67/8.24 = { by lemma 14 R->L }
% 61.67/8.24 ifeq2(ifeq(ifeq(ifeq((x17 ==> x19) >= 0, true, ((x17 ==> x19) + x18) >= (0 + x18), true), true, (x17 + (x19 ==> x17)) >= x19, true), true, (x19 ==> x17) >= (x17 ==> x19), true), true, x19 ==> x17, x17 ==> x19)
% 61.67/8.24 = { by axiom 10 (sos_09) }
% 61.67/8.24 ifeq2(ifeq(ifeq(true, true, (x17 + (x19 ==> x17)) >= x19, true), true, (x19 ==> x17) >= (x17 ==> x19), true), true, x19 ==> x17, x17 ==> x19)
% 61.67/8.24 = { by axiom 8 (ifeq_axiom) }
% 61.67/8.24 ifeq2(ifeq((x17 + (x19 ==> x17)) >= x19, true, (x19 ==> x17) >= (x17 ==> x19), true), true, x19 ==> x17, x17 ==> x19)
% 61.67/8.24 = { by axiom 12 (sos_07_1) }
% 61.67/8.24 ifeq2(true, true, x19 ==> x17, x17 ==> x19)
% 61.67/8.24 = { by axiom 9 (ifeq_axiom) }
% 61.67/8.24 x19 ==> x17
% 61.67/8.24
% 61.67/8.24 Lemma 21: (X ==> x17) + ((X ==> x17) ==> (X ==> x18)) = X ==> x18.
% 61.67/8.24 Proof:
% 61.67/8.24 (X ==> x17) + ((X ==> x17) ==> (X ==> x18))
% 61.67/8.24 = { by axiom 6 (sos_12) R->L }
% 61.67/8.24 (X ==> x18) + ((X ==> x18) ==> (X ==> x17))
% 61.67/8.24 = { by axiom 9 (ifeq_axiom) R->L }
% 61.67/8.24 (X ==> x18) + ifeq2(true, true, (X ==> x18) ==> (X ==> x17), 0)
% 61.67/8.24 = { by axiom 1 (sos_08) R->L }
% 61.67/8.24 (X ==> x18) + ifeq2(((X ==> x18) ==> (X ==> x17)) >= 0, true, (X ==> x18) ==> (X ==> x17), 0)
% 61.67/8.24 = { by axiom 9 (ifeq_axiom) R->L }
% 61.67/8.24 (X ==> x18) + ifeq2(true, true, ifeq2(((X ==> x18) ==> (X ==> x17)) >= 0, true, (X ==> x18) ==> (X ==> x17), 0), 0)
% 61.67/8.24 = { by axiom 12 (sos_07_1) R->L }
% 61.67/8.24 (X ==> x18) + ifeq2(ifeq(((X ==> x18) + 0) >= (X ==> x17), true, 0 >= ((X ==> x18) ==> (X ==> x17)), true), true, ifeq2(((X ==> x18) ==> (X ==> x17)) >= 0, true, (X ==> x18) ==> (X ==> x17), 0), 0)
% 61.67/8.24 = { by axiom 3 (sos_03) }
% 61.67/8.24 (X ==> x18) + ifeq2(ifeq((X ==> x18) >= (X ==> x17), true, 0 >= ((X ==> x18) ==> (X ==> x17)), true), true, ifeq2(((X ==> x18) ==> (X ==> x17)) >= 0, true, (X ==> x18) ==> (X ==> x17), 0), 0)
% 61.67/8.24 = { by axiom 8 (ifeq_axiom) R->L }
% 61.67/8.24 (X ==> x18) + ifeq2(ifeq(ifeq(true, true, (X ==> x18) >= (X ==> x17), true), true, 0 >= ((X ==> x18) ==> (X ==> x17)), true), true, ifeq2(((X ==> x18) ==> (X ==> x17)) >= 0, true, (X ==> x18) ==> (X ==> x17), 0), 0)
% 61.67/8.24 = { by axiom 12 (sos_07_1) R->L }
% 61.67/8.24 (X ==> x18) + ifeq2(ifeq(ifeq(ifeq((x17 + x18) >= x18, true, x18 >= (x17 ==> x18), true), true, (X ==> x18) >= (X ==> x17), true), true, 0 >= ((X ==> x18) ==> (X ==> x17)), true), true, ifeq2(((X ==> x18) ==> (X ==> x17)) >= 0, true, (X ==> x18) ==> (X ==> x17), 0), 0)
% 61.67/8.24 = { by axiom 8 (ifeq_axiom) R->L }
% 61.67/8.24 (X ==> x18) + ifeq2(ifeq(ifeq(ifeq(ifeq(true, true, (x17 + x18) >= x18, true), true, x18 >= (x17 ==> x18), true), true, (X ==> x18) >= (X ==> x17), true), true, 0 >= ((X ==> x18) ==> (X ==> x17)), true), true, ifeq2(((X ==> x18) ==> (X ==> x17)) >= 0, true, (X ==> x18) ==> (X ==> x17), 0), 0)
% 61.67/8.24 = { by axiom 1 (sos_08) R->L }
% 61.67/8.24 (X ==> x18) + ifeq2(ifeq(ifeq(ifeq(ifeq(x17 >= 0, true, (x17 + x18) >= x18, true), true, x18 >= (x17 ==> x18), true), true, (X ==> x18) >= (X ==> x17), true), true, 0 >= ((X ==> x18) ==> (X ==> x17)), true), true, ifeq2(((X ==> x18) ==> (X ==> x17)) >= 0, true, (X ==> x18) ==> (X ==> x17), 0), 0)
% 61.67/8.24 = { by lemma 14 R->L }
% 61.67/8.24 (X ==> x18) + ifeq2(ifeq(ifeq(ifeq(ifeq(x17 >= 0, true, (x17 + x18) >= (0 + x18), true), true, x18 >= (x17 ==> x18), true), true, (X ==> x18) >= (X ==> x17), true), true, 0 >= ((X ==> x18) ==> (X ==> x17)), true), true, ifeq2(((X ==> x18) ==> (X ==> x17)) >= 0, true, (X ==> x18) ==> (X ==> x17), 0), 0)
% 61.67/8.24 = { by axiom 10 (sos_09) }
% 61.67/8.24 (X ==> x18) + ifeq2(ifeq(ifeq(ifeq(true, true, x18 >= (x17 ==> x18), true), true, (X ==> x18) >= (X ==> x17), true), true, 0 >= ((X ==> x18) ==> (X ==> x17)), true), true, ifeq2(((X ==> x18) ==> (X ==> x17)) >= 0, true, (X ==> x18) ==> (X ==> x17), 0), 0)
% 61.67/8.24 = { by axiom 8 (ifeq_axiom) }
% 61.67/8.24 (X ==> x18) + ifeq2(ifeq(ifeq(x18 >= (x17 ==> x18), true, (X ==> x18) >= (X ==> x17), true), true, 0 >= ((X ==> x18) ==> (X ==> x17)), true), true, ifeq2(((X ==> x18) ==> (X ==> x17)) >= 0, true, (X ==> x18) ==> (X ==> x17), 0), 0)
% 61.67/8.24 = { by axiom 5 (goals_13_1) R->L }
% 61.67/8.24 (X ==> x18) + ifeq2(ifeq(ifeq(x18 >= x17, true, (X ==> x18) >= (X ==> x17), true), true, 0 >= ((X ==> x18) ==> (X ==> x17)), true), true, ifeq2(((X ==> x18) ==> (X ==> x17)) >= 0, true, (X ==> x18) ==> (X ==> x17), 0), 0)
% 61.67/8.24 = { by axiom 11 (sos_11) }
% 61.67/8.24 (X ==> x18) + ifeq2(ifeq(true, true, 0 >= ((X ==> x18) ==> (X ==> x17)), true), true, ifeq2(((X ==> x18) ==> (X ==> x17)) >= 0, true, (X ==> x18) ==> (X ==> x17), 0), 0)
% 61.67/8.24 = { by axiom 8 (ifeq_axiom) }
% 61.67/8.24 (X ==> x18) + ifeq2(0 >= ((X ==> x18) ==> (X ==> x17)), true, ifeq2(((X ==> x18) ==> (X ==> x17)) >= 0, true, (X ==> x18) ==> (X ==> x17), 0), 0)
% 61.67/8.24 = { by axiom 13 (sos_06) }
% 61.67/8.24 (X ==> x18) + 0
% 61.67/8.24 = { by axiom 3 (sos_03) }
% 61.67/8.24 X ==> x18
% 61.67/8.24
% 61.67/8.24 Lemma 22: X + (W + (Y + Z)) = X + (Y + (Z + W)).
% 61.67/8.24 Proof:
% 61.67/8.24 X + (W + (Y + Z))
% 61.67/8.24 = { by axiom 2 (sos_02) }
% 61.67/8.24 X + ((Y + Z) + W)
% 61.67/8.24 = { by axiom 7 (sos_01) }
% 61.67/8.24 X + (Y + (Z + W))
% 61.67/8.24
% 61.67/8.24 Lemma 23: x19 ==> x17 = 0.
% 61.67/8.24 Proof:
% 61.67/8.24 x19 ==> x17
% 61.67/8.24 = { by lemma 20 R->L }
% 61.67/8.24 x17 ==> x19
% 61.67/8.24 = { by axiom 4 (goals_13) }
% 61.67/8.24 x17 ==> (x19 ==> x18)
% 61.67/8.24 = { by lemma 14 R->L }
% 61.67/8.24 x17 ==> (x19 ==> (0 + x18))
% 61.67/8.24 = { by axiom 2 (sos_02) R->L }
% 61.67/8.24 x17 ==> (x19 ==> (x18 + 0))
% 61.67/8.24 = { by axiom 3 (sos_03) R->L }
% 61.67/8.24 x17 ==> (x19 ==> ((x18 + 0) + 0))
% 61.67/8.24 = { by lemma 19 R->L }
% 61.67/8.24 x17 ==> (x19 ==> ((x18 + (x18 ==> x19)) + 0))
% 61.67/8.24 = { by axiom 6 (sos_12) }
% 61.67/8.24 x17 ==> (x19 ==> ((x19 + (x19 ==> x18)) + 0))
% 61.67/8.24 = { by axiom 4 (goals_13) R->L }
% 61.67/8.24 x17 ==> (x19 ==> ((x19 + x19) + 0))
% 61.67/8.24 = { by axiom 7 (sos_01) }
% 61.67/8.24 x17 ==> (x19 ==> (x19 + (x19 + 0)))
% 61.67/8.24 = { by axiom 2 (sos_02) }
% 61.67/8.24 x17 ==> (x19 ==> (x19 + (0 + x19)))
% 61.67/8.24 = { by axiom 4 (goals_13) }
% 61.67/8.24 x17 ==> (x19 ==> (x19 + (0 + (x19 ==> x18))))
% 61.67/8.24 = { by lemma 21 R->L }
% 61.67/8.24 x17 ==> (x19 ==> (x19 + (0 + ((x19 ==> x17) + ((x19 ==> x17) ==> (x19 ==> x18))))))
% 61.67/8.24 = { by lemma 22 R->L }
% 61.67/8.24 x17 ==> (x19 ==> (x19 + (((x19 ==> x17) ==> (x19 ==> x18)) + (0 + (x19 ==> x17)))))
% 61.67/8.24 = { by axiom 7 (sos_01) R->L }
% 61.67/8.24 x17 ==> (x19 ==> (x19 + ((((x19 ==> x17) ==> (x19 ==> x18)) + 0) + (x19 ==> x17))))
% 61.67/8.24 = { by lemma 15 R->L }
% 61.67/8.24 x17 ==> (x19 ==> (x17 + ((x17 ==> x19) + (((x19 ==> x17) ==> (x19 ==> x18)) + 0))))
% 61.67/8.24 = { by axiom 2 (sos_02) }
% 61.67/8.24 x17 ==> (x19 ==> (x17 + ((((x19 ==> x17) ==> (x19 ==> x18)) + 0) + (x17 ==> x19))))
% 61.67/8.24 = { by axiom 7 (sos_01) }
% 61.67/8.24 x17 ==> (x19 ==> (x17 + (((x19 ==> x17) ==> (x19 ==> x18)) + (0 + (x17 ==> x19)))))
% 61.67/8.24 = { by lemma 22 }
% 61.67/8.24 x17 ==> (x19 ==> (x17 + (0 + ((x17 ==> x19) + ((x19 ==> x17) ==> (x19 ==> x18))))))
% 61.67/8.24 = { by lemma 20 }
% 61.67/8.24 x17 ==> (x19 ==> (x17 + (0 + ((x19 ==> x17) + ((x19 ==> x17) ==> (x19 ==> x18))))))
% 61.67/8.24 = { by lemma 21 }
% 61.67/8.24 x17 ==> (x19 ==> (x17 + (0 + (x19 ==> x18))))
% 61.67/8.24 = { by axiom 4 (goals_13) R->L }
% 61.67/8.24 x17 ==> (x19 ==> (x17 + (0 + x19)))
% 61.67/8.24 = { by lemma 18 R->L }
% 61.67/8.24 x17 ==> (x19 ==> (0 + (x17 + x19)))
% 61.67/8.24 = { by axiom 2 (sos_02) }
% 61.67/8.24 x17 ==> (x19 ==> (0 + (x19 + x17)))
% 61.67/8.24 = { by lemma 14 }
% 61.67/8.24 x17 ==> (x19 ==> (x19 + x17))
% 61.67/8.24 = { by axiom 2 (sos_02) R->L }
% 61.67/8.24 x17 ==> (x19 ==> (x17 + x19))
% 61.67/8.24 = { by lemma 17 R->L }
% 61.67/8.24 ifeq2(0 >= (x17 ==> (x19 ==> (x17 + x19))), true, 0, x17 ==> (x19 ==> (x17 + x19)))
% 61.67/8.24 = { by axiom 8 (ifeq_axiom) R->L }
% 61.67/8.24 ifeq2(ifeq(true, true, 0 >= (x17 ==> (x19 ==> (x17 + x19))), true), true, 0, x17 ==> (x19 ==> (x17 + x19)))
% 61.67/8.24 = { by axiom 12 (sos_07_1) R->L }
% 61.67/8.24 ifeq2(ifeq(ifeq((x19 + (x17 + 0)) >= (x19 + x17), true, (x17 + 0) >= (x19 ==> (x19 + x17)), true), true, 0 >= (x17 ==> (x19 ==> (x17 + x19))), true), true, 0, x17 ==> (x19 ==> (x17 + x19)))
% 61.67/8.24 = { by axiom 7 (sos_01) R->L }
% 61.67/8.24 ifeq2(ifeq(ifeq(((x19 + x17) + 0) >= (x19 + x17), true, (x17 + 0) >= (x19 ==> (x19 + x17)), true), true, 0 >= (x17 ==> (x19 ==> (x17 + x19))), true), true, 0, x17 ==> (x19 ==> (x17 + x19)))
% 61.67/8.24 = { by axiom 2 (sos_02) R->L }
% 61.67/8.24 ifeq2(ifeq(ifeq((0 + (x19 + x17)) >= (x19 + x17), true, (x17 + 0) >= (x19 ==> (x19 + x17)), true), true, 0 >= (x17 ==> (x19 ==> (x17 + x19))), true), true, 0, x17 ==> (x19 ==> (x17 + x19)))
% 61.67/8.24 = { by axiom 8 (ifeq_axiom) R->L }
% 61.67/8.24 ifeq2(ifeq(ifeq(ifeq(true, true, (0 + (x19 + x17)) >= (x19 + x17), true), true, (x17 + 0) >= (x19 ==> (x19 + x17)), true), true, 0 >= (x17 ==> (x19 ==> (x17 + x19))), true), true, 0, x17 ==> (x19 ==> (x17 + x19)))
% 61.67/8.24 = { by axiom 1 (sos_08) R->L }
% 61.67/8.24 ifeq2(ifeq(ifeq(ifeq(0 >= 0, true, (0 + (x19 + x17)) >= (x19 + x17), true), true, (x17 + 0) >= (x19 ==> (x19 + x17)), true), true, 0 >= (x17 ==> (x19 ==> (x17 + x19))), true), true, 0, x17 ==> (x19 ==> (x17 + x19)))
% 61.67/8.24 = { by lemma 14 R->L }
% 61.67/8.24 ifeq2(ifeq(ifeq(ifeq(0 >= 0, true, (0 + (x19 + x17)) >= (0 + (x19 + x17)), true), true, (x17 + 0) >= (x19 ==> (x19 + x17)), true), true, 0 >= (x17 ==> (x19 ==> (x17 + x19))), true), true, 0, x17 ==> (x19 ==> (x17 + x19)))
% 61.67/8.24 = { by axiom 10 (sos_09) }
% 61.67/8.24 ifeq2(ifeq(ifeq(true, true, (x17 + 0) >= (x19 ==> (x19 + x17)), true), true, 0 >= (x17 ==> (x19 ==> (x17 + x19))), true), true, 0, x17 ==> (x19 ==> (x17 + x19)))
% 61.67/8.24 = { by axiom 8 (ifeq_axiom) }
% 61.67/8.24 ifeq2(ifeq((x17 + 0) >= (x19 ==> (x19 + x17)), true, 0 >= (x17 ==> (x19 ==> (x17 + x19))), true), true, 0, x17 ==> (x19 ==> (x17 + x19)))
% 61.67/8.24 = { by axiom 2 (sos_02) }
% 61.67/8.24 ifeq2(ifeq((x17 + 0) >= (x19 ==> (x17 + x19)), true, 0 >= (x17 ==> (x19 ==> (x17 + x19))), true), true, 0, x17 ==> (x19 ==> (x17 + x19)))
% 61.67/8.24 = { by axiom 12 (sos_07_1) }
% 61.67/8.24 ifeq2(true, true, 0, x17 ==> (x19 ==> (x17 + x19)))
% 61.67/8.24 = { by axiom 9 (ifeq_axiom) }
% 61.67/8.24 0
% 61.67/8.24
% 61.67/8.24 Goal 1 (goals_13_2): x17 = x19.
% 61.67/8.24 Proof:
% 61.67/8.24 x17
% 61.67/8.24 = { by axiom 3 (sos_03) R->L }
% 61.67/8.24 x17 + 0
% 61.67/8.24 = { by lemma 23 R->L }
% 61.67/8.24 x17 + (x19 ==> x17)
% 61.67/8.24 = { by lemma 20 R->L }
% 61.67/8.24 x17 + (x17 ==> x19)
% 61.67/8.24 = { by axiom 6 (sos_12) }
% 61.67/8.24 x19 + (x19 ==> x17)
% 61.67/8.24 = { by lemma 23 }
% 61.67/8.24 x19 + 0
% 61.67/8.24 = { by axiom 3 (sos_03) }
% 61.67/8.24 x19
% 61.67/8.24 % SZS output end Proof
% 61.67/8.24
% 61.67/8.24 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------