%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : ALG071+1 : TPTP v9.3.1. Released v2.7.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n001.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 09:06:59 AM UTC 2026
% Result : Theorem 1.23s 0.39s
% Output : Proof 1.23s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : ALG071+1 : TPTP v9.3.1. Released v2.7.0.
% 0.00/0.03 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.09/0.17 % Computer : n001.cluster.edu
% 0.09/0.17 % Model : x86_64 x86_64
% 0.09/0.17 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.17 % Memory : 8046.5625MB
% 0.09/0.17 % OS : Linux 6.8.0-71-generic
% 0.09/0.17 % CPULimit : 300
% 0.09/0.17 % WCLimit : 300
% 0.09/0.17 % DateTime : Mon Sep 28 19:30:48 UTC 2026
% 0.09/0.17 % CPUTime :
% 0.09/0.17 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 1.23/0.39 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --complete-subsets --ground-joining-incomplete-limit 15 --flatten-every 2
% 1.23/0.39
% 1.23/0.39 % SZS status Theorem
% 1.23/0.39
% 1.23/0.42 % SZS output start Proof
% 1.23/0.42 Axiom 1 (ax4): sorti2(u) = true.
% 1.23/0.42 Axiom 2 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 1.23/0.42 Axiom 3 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 1.23/0.42 Axiom 4 (ifeq_axiom): ifeq3(X, X, Y, Z) = Y.
% 1.23/0.42 Axiom 5 (ax3_2): ifeq(sorti1(X), true, sorti1(v(X)), true) = true.
% 1.23/0.42 Axiom 6 (co1_2): ifeq2(sorti1(X), true, j(h(X)), X) = X.
% 1.23/0.42 Axiom 7 (co1): ifeq(sorti1(X), true, sorti2(h(X)), true) = true.
% 1.23/0.42 Axiom 8 (co1_3): ifeq(sorti2(X), true, sorti1(j(X)), true) = true.
% 1.23/0.42 Axiom 9 (co1_5): ifeq2(sorti2(X), true, h(j(X)), X) = X.
% 1.23/0.42 Axiom 10 (ax3_1): ifeq2(sorti1(X), true, op1(X, op1(v(X), X)), v(X)) = v(X).
% 1.23/0.42 Axiom 11 (ax1): ifeq(sorti1(X), true, ifeq(sorti1(Y), true, sorti1(op1(Y, X)), true), true) = true.
% 1.23/0.42 Axiom 12 (ax2): ifeq(sorti2(X), true, ifeq(sorti2(Y), true, sorti2(op2(Y, X)), true), true) = true.
% 1.23/0.42 Axiom 13 (ax4_1): ifeq2(sorti2(X), true, ifeq3(op2(u, op2(X, u)), X, op2(X, op2(X, u)), u), u) = u.
% 1.23/0.42 Axiom 14 (co1_1): ifeq2(sorti1(X), true, ifeq2(sorti1(Y), true, op2(h(Y), h(X)), h(op1(Y, X))), h(op1(Y, X))) = h(op1(Y, X)).
% 1.23/0.42 Axiom 15 (co1_4): ifeq2(sorti2(X), true, ifeq2(sorti2(Y), true, op1(j(Y), j(X)), j(op2(Y, X))), j(op2(Y, X))) = j(op2(Y, X)).
% 1.23/0.42
% 1.23/0.42 Lemma 16: h(j(u)) = u.
% 1.23/0.42 Proof:
% 1.23/0.42 h(j(u))
% 1.23/0.42 = { by axiom 2 (ifeq_axiom) R->L }
% 1.23/0.42 ifeq2(true, true, h(j(u)), u)
% 1.23/0.42 = { by axiom 1 (ax4) R->L }
% 1.23/0.42 ifeq2(sorti2(u), true, h(j(u)), u)
% 1.23/0.42 = { by axiom 9 (co1_5) }
% 1.23/0.42 u
% 1.23/0.42
% 1.23/0.42 Lemma 17: j(h(v(j(u)))) = v(j(u)).
% 1.23/0.42 Proof:
% 1.23/0.42 j(h(v(j(u))))
% 1.23/0.42 = { by axiom 2 (ifeq_axiom) R->L }
% 1.23/0.42 ifeq2(true, true, j(h(v(j(u)))), v(j(u)))
% 1.23/0.42 = { by axiom 5 (ax3_2) R->L }
% 1.23/0.42 ifeq2(ifeq(sorti1(j(u)), true, sorti1(v(j(u))), true), true, j(h(v(j(u)))), v(j(u)))
% 1.23/0.42 = { by axiom 3 (ifeq_axiom) R->L }
% 1.23/0.42 ifeq2(ifeq(ifeq(true, true, sorti1(j(u)), true), true, sorti1(v(j(u))), true), true, j(h(v(j(u)))), v(j(u)))
% 1.23/0.42 = { by axiom 1 (ax4) R->L }
% 1.23/0.42 ifeq2(ifeq(ifeq(sorti2(u), true, sorti1(j(u)), true), true, sorti1(v(j(u))), true), true, j(h(v(j(u)))), v(j(u)))
% 1.23/0.42 = { by axiom 8 (co1_3) }
% 1.23/0.42 ifeq2(ifeq(true, true, sorti1(v(j(u))), true), true, j(h(v(j(u)))), v(j(u)))
% 1.23/0.42 = { by axiom 3 (ifeq_axiom) }
% 1.23/0.42 ifeq2(sorti1(v(j(u))), true, j(h(v(j(u)))), v(j(u)))
% 1.23/0.42 = { by axiom 6 (co1_2) }
% 1.23/0.42 v(j(u))
% 1.23/0.43
% 1.23/0.43 Lemma 18: op2(h(v(j(u))), op2(h(v(j(u))), u)) = u.
% 1.23/0.43 Proof:
% 1.23/0.43 op2(h(v(j(u))), op2(h(v(j(u))), u))
% 1.23/0.43 = { by axiom 4 (ifeq_axiom) R->L }
% 1.23/0.43 ifeq3(h(v(j(u))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u)
% 1.23/0.43 = { by axiom 2 (ifeq_axiom) R->L }
% 1.23/0.43 ifeq2(true, true, ifeq3(h(v(j(u))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 7 (co1) R->L }
% 1.23/0.43 ifeq2(ifeq(sorti1(v(j(u))), true, sorti2(h(v(j(u)))), true), true, ifeq3(h(v(j(u))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 3 (ifeq_axiom) R->L }
% 1.23/0.43 ifeq2(ifeq(ifeq(true, true, sorti1(v(j(u))), true), true, sorti2(h(v(j(u)))), true), true, ifeq3(h(v(j(u))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 8 (co1_3) R->L }
% 1.23/0.43 ifeq2(ifeq(ifeq(ifeq(sorti2(u), true, sorti1(j(u)), true), true, sorti1(v(j(u))), true), true, sorti2(h(v(j(u)))), true), true, ifeq3(h(v(j(u))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 1 (ax4) }
% 1.23/0.43 ifeq2(ifeq(ifeq(ifeq(true, true, sorti1(j(u)), true), true, sorti1(v(j(u))), true), true, sorti2(h(v(j(u)))), true), true, ifeq3(h(v(j(u))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 3 (ifeq_axiom) }
% 1.23/0.43 ifeq2(ifeq(ifeq(sorti1(j(u)), true, sorti1(v(j(u))), true), true, sorti2(h(v(j(u)))), true), true, ifeq3(h(v(j(u))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 5 (ax3_2) }
% 1.23/0.43 ifeq2(ifeq(true, true, sorti2(h(v(j(u)))), true), true, ifeq3(h(v(j(u))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 3 (ifeq_axiom) }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(h(v(j(u))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 10 (ax3_1) R->L }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(h(ifeq2(sorti1(j(u)), true, op1(j(u), op1(v(j(u)), j(u))), v(j(u)))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 3 (ifeq_axiom) R->L }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(h(ifeq2(ifeq(true, true, sorti1(j(u)), true), true, op1(j(u), op1(v(j(u)), j(u))), v(j(u)))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 1 (ax4) R->L }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(h(ifeq2(ifeq(sorti2(u), true, sorti1(j(u)), true), true, op1(j(u), op1(v(j(u)), j(u))), v(j(u)))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 8 (co1_3) }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(h(ifeq2(true, true, op1(j(u), op1(v(j(u)), j(u))), v(j(u)))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 2 (ifeq_axiom) }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(h(op1(j(u), op1(v(j(u)), j(u)))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 14 (co1_1) R->L }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(sorti1(op1(v(j(u)), j(u))), true, ifeq2(sorti1(j(u)), true, op2(h(j(u)), h(op1(v(j(u)), j(u)))), h(op1(j(u), op1(v(j(u)), j(u))))), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by lemma 16 }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(sorti1(op1(v(j(u)), j(u))), true, ifeq2(sorti1(j(u)), true, op2(u, h(op1(v(j(u)), j(u)))), h(op1(j(u), op1(v(j(u)), j(u))))), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 3 (ifeq_axiom) R->L }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(sorti1(op1(v(j(u)), j(u))), true, ifeq2(ifeq(true, true, sorti1(j(u)), true), true, op2(u, h(op1(v(j(u)), j(u)))), h(op1(j(u), op1(v(j(u)), j(u))))), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 1 (ax4) R->L }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(sorti1(op1(v(j(u)), j(u))), true, ifeq2(ifeq(sorti2(u), true, sorti1(j(u)), true), true, op2(u, h(op1(v(j(u)), j(u)))), h(op1(j(u), op1(v(j(u)), j(u))))), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 8 (co1_3) }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(sorti1(op1(v(j(u)), j(u))), true, ifeq2(true, true, op2(u, h(op1(v(j(u)), j(u)))), h(op1(j(u), op1(v(j(u)), j(u))))), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 2 (ifeq_axiom) }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(sorti1(op1(v(j(u)), j(u))), true, op2(u, h(op1(v(j(u)), j(u)))), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 14 (co1_1) R->L }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(sorti1(op1(v(j(u)), j(u))), true, op2(u, ifeq2(sorti1(j(u)), true, ifeq2(sorti1(v(j(u))), true, op2(h(v(j(u))), h(j(u))), h(op1(v(j(u)), j(u)))), h(op1(v(j(u)), j(u))))), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by lemma 16 }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(sorti1(op1(v(j(u)), j(u))), true, op2(u, ifeq2(sorti1(j(u)), true, ifeq2(sorti1(v(j(u))), true, op2(h(v(j(u))), u), h(op1(v(j(u)), j(u)))), h(op1(v(j(u)), j(u))))), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 3 (ifeq_axiom) R->L }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(sorti1(op1(v(j(u)), j(u))), true, op2(u, ifeq2(ifeq(true, true, sorti1(j(u)), true), true, ifeq2(sorti1(v(j(u))), true, op2(h(v(j(u))), u), h(op1(v(j(u)), j(u)))), h(op1(v(j(u)), j(u))))), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 1 (ax4) R->L }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(sorti1(op1(v(j(u)), j(u))), true, op2(u, ifeq2(ifeq(sorti2(u), true, sorti1(j(u)), true), true, ifeq2(sorti1(v(j(u))), true, op2(h(v(j(u))), u), h(op1(v(j(u)), j(u)))), h(op1(v(j(u)), j(u))))), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 8 (co1_3) }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(sorti1(op1(v(j(u)), j(u))), true, op2(u, ifeq2(true, true, ifeq2(sorti1(v(j(u))), true, op2(h(v(j(u))), u), h(op1(v(j(u)), j(u)))), h(op1(v(j(u)), j(u))))), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 2 (ifeq_axiom) }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(sorti1(op1(v(j(u)), j(u))), true, op2(u, ifeq2(sorti1(v(j(u))), true, op2(h(v(j(u))), u), h(op1(v(j(u)), j(u))))), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 3 (ifeq_axiom) R->L }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(sorti1(op1(v(j(u)), j(u))), true, op2(u, ifeq2(ifeq(true, true, sorti1(v(j(u))), true), true, op2(h(v(j(u))), u), h(op1(v(j(u)), j(u))))), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 8 (co1_3) R->L }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(sorti1(op1(v(j(u)), j(u))), true, op2(u, ifeq2(ifeq(ifeq(sorti2(u), true, sorti1(j(u)), true), true, sorti1(v(j(u))), true), true, op2(h(v(j(u))), u), h(op1(v(j(u)), j(u))))), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 1 (ax4) }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(sorti1(op1(v(j(u)), j(u))), true, op2(u, ifeq2(ifeq(ifeq(true, true, sorti1(j(u)), true), true, sorti1(v(j(u))), true), true, op2(h(v(j(u))), u), h(op1(v(j(u)), j(u))))), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 3 (ifeq_axiom) }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(sorti1(op1(v(j(u)), j(u))), true, op2(u, ifeq2(ifeq(sorti1(j(u)), true, sorti1(v(j(u))), true), true, op2(h(v(j(u))), u), h(op1(v(j(u)), j(u))))), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 5 (ax3_2) }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(sorti1(op1(v(j(u)), j(u))), true, op2(u, ifeq2(true, true, op2(h(v(j(u))), u), h(op1(v(j(u)), j(u))))), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 2 (ifeq_axiom) }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(sorti1(op1(v(j(u)), j(u))), true, op2(u, op2(h(v(j(u))), u)), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 3 (ifeq_axiom) R->L }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(ifeq(true, true, sorti1(op1(v(j(u)), j(u))), true), true, op2(u, op2(h(v(j(u))), u)), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 5 (ax3_2) R->L }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(ifeq(ifeq(sorti1(j(u)), true, sorti1(v(j(u))), true), true, sorti1(op1(v(j(u)), j(u))), true), true, op2(u, op2(h(v(j(u))), u)), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 3 (ifeq_axiom) R->L }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(ifeq(ifeq(ifeq(true, true, sorti1(j(u)), true), true, sorti1(v(j(u))), true), true, sorti1(op1(v(j(u)), j(u))), true), true, op2(u, op2(h(v(j(u))), u)), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 1 (ax4) R->L }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(ifeq(ifeq(ifeq(sorti2(u), true, sorti1(j(u)), true), true, sorti1(v(j(u))), true), true, sorti1(op1(v(j(u)), j(u))), true), true, op2(u, op2(h(v(j(u))), u)), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 8 (co1_3) }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(ifeq(ifeq(true, true, sorti1(v(j(u))), true), true, sorti1(op1(v(j(u)), j(u))), true), true, op2(u, op2(h(v(j(u))), u)), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 3 (ifeq_axiom) }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(ifeq(sorti1(v(j(u))), true, sorti1(op1(v(j(u)), j(u))), true), true, op2(u, op2(h(v(j(u))), u)), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 3 (ifeq_axiom) R->L }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(ifeq(true, true, ifeq(sorti1(v(j(u))), true, sorti1(op1(v(j(u)), j(u))), true), true), true, op2(u, op2(h(v(j(u))), u)), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 8 (co1_3) R->L }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(ifeq(ifeq(sorti2(u), true, sorti1(j(u)), true), true, ifeq(sorti1(v(j(u))), true, sorti1(op1(v(j(u)), j(u))), true), true), true, op2(u, op2(h(v(j(u))), u)), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 1 (ax4) }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(ifeq(ifeq(true, true, sorti1(j(u)), true), true, ifeq(sorti1(v(j(u))), true, sorti1(op1(v(j(u)), j(u))), true), true), true, op2(u, op2(h(v(j(u))), u)), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 3 (ifeq_axiom) }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(ifeq(sorti1(j(u)), true, ifeq(sorti1(v(j(u))), true, sorti1(op1(v(j(u)), j(u))), true), true), true, op2(u, op2(h(v(j(u))), u)), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 11 (ax1) }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(ifeq2(true, true, op2(u, op2(h(v(j(u))), u)), h(op1(j(u), op1(v(j(u)), j(u))))), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 2 (ifeq_axiom) }
% 1.23/0.43 ifeq2(sorti2(h(v(j(u)))), true, ifeq3(op2(u, op2(h(v(j(u))), u)), h(v(j(u))), op2(h(v(j(u))), op2(h(v(j(u))), u)), u), u)
% 1.23/0.43 = { by axiom 13 (ax4_1) }
% 1.23/0.44 u
% 1.23/0.44
% 1.23/0.44 Goal 1 (ax3): tuple(op1(v(X), op1(v(X), X)), sorti1(X)) = tuple(X, true).
% 1.23/0.44 The goal is true when:
% 1.23/0.44 X = j(u)
% 1.23/0.44
% 1.23/0.44 Proof:
% 1.23/0.44 tuple(op1(v(j(u)), op1(v(j(u)), j(u))), sorti1(j(u)))
% 1.23/0.44 = { by lemma 17 R->L }
% 1.23/0.44 tuple(op1(v(j(u)), op1(j(h(v(j(u)))), j(u))), sorti1(j(u)))
% 1.23/0.44 = { by axiom 2 (ifeq_axiom) R->L }
% 1.23/0.44 tuple(op1(v(j(u)), ifeq2(true, true, op1(j(h(v(j(u)))), j(u)), j(op2(h(v(j(u))), u)))), sorti1(j(u)))
% 1.23/0.44 = { by axiom 7 (co1) R->L }
% 1.23/0.44 tuple(op1(v(j(u)), ifeq2(ifeq(sorti1(v(j(u))), true, sorti2(h(v(j(u)))), true), true, op1(j(h(v(j(u)))), j(u)), j(op2(h(v(j(u))), u)))), sorti1(j(u)))
% 1.23/0.44 = { by axiom 3 (ifeq_axiom) R->L }
% 1.23/0.44 tuple(op1(v(j(u)), ifeq2(ifeq(ifeq(true, true, sorti1(v(j(u))), true), true, sorti2(h(v(j(u)))), true), true, op1(j(h(v(j(u)))), j(u)), j(op2(h(v(j(u))), u)))), sorti1(j(u)))
% 1.23/0.44 = { by axiom 8 (co1_3) R->L }
% 1.23/0.44 tuple(op1(v(j(u)), ifeq2(ifeq(ifeq(ifeq(sorti2(u), true, sorti1(j(u)), true), true, sorti1(v(j(u))), true), true, sorti2(h(v(j(u)))), true), true, op1(j(h(v(j(u)))), j(u)), j(op2(h(v(j(u))), u)))), sorti1(j(u)))
% 1.23/0.44 = { by axiom 1 (ax4) }
% 1.23/0.44 tuple(op1(v(j(u)), ifeq2(ifeq(ifeq(ifeq(true, true, sorti1(j(u)), true), true, sorti1(v(j(u))), true), true, sorti2(h(v(j(u)))), true), true, op1(j(h(v(j(u)))), j(u)), j(op2(h(v(j(u))), u)))), sorti1(j(u)))
% 1.23/0.44 = { by axiom 3 (ifeq_axiom) }
% 1.23/0.44 tuple(op1(v(j(u)), ifeq2(ifeq(ifeq(sorti1(j(u)), true, sorti1(v(j(u))), true), true, sorti2(h(v(j(u)))), true), true, op1(j(h(v(j(u)))), j(u)), j(op2(h(v(j(u))), u)))), sorti1(j(u)))
% 1.23/0.44 = { by axiom 5 (ax3_2) }
% 1.23/0.44 tuple(op1(v(j(u)), ifeq2(ifeq(true, true, sorti2(h(v(j(u)))), true), true, op1(j(h(v(j(u)))), j(u)), j(op2(h(v(j(u))), u)))), sorti1(j(u)))
% 1.23/0.44 = { by axiom 3 (ifeq_axiom) }
% 1.23/0.44 tuple(op1(v(j(u)), ifeq2(sorti2(h(v(j(u)))), true, op1(j(h(v(j(u)))), j(u)), j(op2(h(v(j(u))), u)))), sorti1(j(u)))
% 1.23/0.44 = { by axiom 2 (ifeq_axiom) R->L }
% 1.23/0.44 tuple(op1(v(j(u)), ifeq2(true, true, ifeq2(sorti2(h(v(j(u)))), true, op1(j(h(v(j(u)))), j(u)), j(op2(h(v(j(u))), u))), j(op2(h(v(j(u))), u)))), sorti1(j(u)))
% 1.23/0.44 = { by axiom 1 (ax4) R->L }
% 1.23/0.44 tuple(op1(v(j(u)), ifeq2(sorti2(u), true, ifeq2(sorti2(h(v(j(u)))), true, op1(j(h(v(j(u)))), j(u)), j(op2(h(v(j(u))), u))), j(op2(h(v(j(u))), u)))), sorti1(j(u)))
% 1.23/0.44 = { by axiom 15 (co1_4) }
% 1.23/0.44 tuple(op1(v(j(u)), j(op2(h(v(j(u))), u))), sorti1(j(u)))
% 1.23/0.44 = { by lemma 17 R->L }
% 1.23/0.44 tuple(op1(j(h(v(j(u)))), j(op2(h(v(j(u))), u))), sorti1(j(u)))
% 1.23/0.44 = { by axiom 2 (ifeq_axiom) R->L }
% 1.23/0.44 tuple(ifeq2(true, true, op1(j(h(v(j(u)))), j(op2(h(v(j(u))), u))), j(u)), sorti1(j(u)))
% 1.23/0.44 = { by axiom 7 (co1) R->L }
% 1.23/0.44 tuple(ifeq2(ifeq(sorti1(v(j(u))), true, sorti2(h(v(j(u)))), true), true, op1(j(h(v(j(u)))), j(op2(h(v(j(u))), u))), j(u)), sorti1(j(u)))
% 1.23/0.44 = { by axiom 3 (ifeq_axiom) R->L }
% 1.23/0.44 tuple(ifeq2(ifeq(ifeq(true, true, sorti1(v(j(u))), true), true, sorti2(h(v(j(u)))), true), true, op1(j(h(v(j(u)))), j(op2(h(v(j(u))), u))), j(u)), sorti1(j(u)))
% 1.23/0.45 = { by axiom 8 (co1_3) R->L }
% 1.23/0.45 tuple(ifeq2(ifeq(ifeq(ifeq(sorti2(u), true, sorti1(j(u)), true), true, sorti1(v(j(u))), true), true, sorti2(h(v(j(u)))), true), true, op1(j(h(v(j(u)))), j(op2(h(v(j(u))), u))), j(u)), sorti1(j(u)))
% 1.23/0.45 = { by axiom 1 (ax4) }
% 1.23/0.45 tuple(ifeq2(ifeq(ifeq(ifeq(true, true, sorti1(j(u)), true), true, sorti1(v(j(u))), true), true, sorti2(h(v(j(u)))), true), true, op1(j(h(v(j(u)))), j(op2(h(v(j(u))), u))), j(u)), sorti1(j(u)))
% 1.23/0.45 = { by axiom 3 (ifeq_axiom) }
% 1.23/0.45 tuple(ifeq2(ifeq(ifeq(sorti1(j(u)), true, sorti1(v(j(u))), true), true, sorti2(h(v(j(u)))), true), true, op1(j(h(v(j(u)))), j(op2(h(v(j(u))), u))), j(u)), sorti1(j(u)))
% 1.23/0.45 = { by axiom 5 (ax3_2) }
% 1.23/0.45 tuple(ifeq2(ifeq(true, true, sorti2(h(v(j(u)))), true), true, op1(j(h(v(j(u)))), j(op2(h(v(j(u))), u))), j(u)), sorti1(j(u)))
% 1.23/0.45 = { by axiom 3 (ifeq_axiom) }
% 1.23/0.45 tuple(ifeq2(sorti2(h(v(j(u)))), true, op1(j(h(v(j(u)))), j(op2(h(v(j(u))), u))), j(u)), sorti1(j(u)))
% 1.23/0.45 = { by axiom 2 (ifeq_axiom) R->L }
% 1.23/0.45 tuple(ifeq2(true, true, ifeq2(sorti2(h(v(j(u)))), true, op1(j(h(v(j(u)))), j(op2(h(v(j(u))), u))), j(u)), j(op2(h(v(j(u))), op2(h(v(j(u))), u)))), sorti1(j(u)))
% 1.23/0.45 = { by axiom 12 (ax2) R->L }
% 1.23/0.45 tuple(ifeq2(ifeq(sorti2(u), true, ifeq(sorti2(h(v(j(u)))), true, sorti2(op2(h(v(j(u))), u)), true), true), true, ifeq2(sorti2(h(v(j(u)))), true, op1(j(h(v(j(u)))), j(op2(h(v(j(u))), u))), j(u)), j(op2(h(v(j(u))), op2(h(v(j(u))), u)))), sorti1(j(u)))
% 1.23/0.45 = { by axiom 1 (ax4) }
% 1.23/0.45 tuple(ifeq2(ifeq(true, true, ifeq(sorti2(h(v(j(u)))), true, sorti2(op2(h(v(j(u))), u)), true), true), true, ifeq2(sorti2(h(v(j(u)))), true, op1(j(h(v(j(u)))), j(op2(h(v(j(u))), u))), j(u)), j(op2(h(v(j(u))), op2(h(v(j(u))), u)))), sorti1(j(u)))
% 1.23/0.45 = { by axiom 3 (ifeq_axiom) }
% 1.23/0.45 tuple(ifeq2(ifeq(sorti2(h(v(j(u)))), true, sorti2(op2(h(v(j(u))), u)), true), true, ifeq2(sorti2(h(v(j(u)))), true, op1(j(h(v(j(u)))), j(op2(h(v(j(u))), u))), j(u)), j(op2(h(v(j(u))), op2(h(v(j(u))), u)))), sorti1(j(u)))
% 1.23/0.45 = { by axiom 3 (ifeq_axiom) R->L }
% 1.23/0.45 tuple(ifeq2(ifeq(ifeq(true, true, sorti2(h(v(j(u)))), true), true, sorti2(op2(h(v(j(u))), u)), true), true, ifeq2(sorti2(h(v(j(u)))), true, op1(j(h(v(j(u)))), j(op2(h(v(j(u))), u))), j(u)), j(op2(h(v(j(u))), op2(h(v(j(u))), u)))), sorti1(j(u)))
% 1.23/0.45 = { by axiom 5 (ax3_2) R->L }
% 1.23/0.45 tuple(ifeq2(ifeq(ifeq(ifeq(sorti1(j(u)), true, sorti1(v(j(u))), true), true, sorti2(h(v(j(u)))), true), true, sorti2(op2(h(v(j(u))), u)), true), true, ifeq2(sorti2(h(v(j(u)))), true, op1(j(h(v(j(u)))), j(op2(h(v(j(u))), u))), j(u)), j(op2(h(v(j(u))), op2(h(v(j(u))), u)))), sorti1(j(u)))
% 1.23/0.45 = { by axiom 3 (ifeq_axiom) R->L }
% 1.23/0.45 tuple(ifeq2(ifeq(ifeq(ifeq(ifeq(true, true, sorti1(j(u)), true), true, sorti1(v(j(u))), true), true, sorti2(h(v(j(u)))), true), true, sorti2(op2(h(v(j(u))), u)), true), true, ifeq2(sorti2(h(v(j(u)))), true, op1(j(h(v(j(u)))), j(op2(h(v(j(u))), u))), j(u)), j(op2(h(v(j(u))), op2(h(v(j(u))), u)))), sorti1(j(u)))
% 1.23/0.45 = { by axiom 1 (ax4) R->L }
% 1.23/0.45 tuple(ifeq2(ifeq(ifeq(ifeq(ifeq(sorti2(u), true, sorti1(j(u)), true), true, sorti1(v(j(u))), true), true, sorti2(h(v(j(u)))), true), true, sorti2(op2(h(v(j(u))), u)), true), true, ifeq2(sorti2(h(v(j(u)))), true, op1(j(h(v(j(u)))), j(op2(h(v(j(u))), u))), j(u)), j(op2(h(v(j(u))), op2(h(v(j(u))), u)))), sorti1(j(u)))
% 1.23/0.45 = { by axiom 8 (co1_3) }
% 1.23/0.45 tuple(ifeq2(ifeq(ifeq(ifeq(true, true, sorti1(v(j(u))), true), true, sorti2(h(v(j(u)))), true), true, sorti2(op2(h(v(j(u))), u)), true), true, ifeq2(sorti2(h(v(j(u)))), true, op1(j(h(v(j(u)))), j(op2(h(v(j(u))), u))), j(u)), j(op2(h(v(j(u))), op2(h(v(j(u))), u)))), sorti1(j(u)))
% 1.23/0.45 = { by axiom 3 (ifeq_axiom) }
% 1.23/0.45 tuple(ifeq2(ifeq(ifeq(sorti1(v(j(u))), true, sorti2(h(v(j(u)))), true), true, sorti2(op2(h(v(j(u))), u)), true), true, ifeq2(sorti2(h(v(j(u)))), true, op1(j(h(v(j(u)))), j(op2(h(v(j(u))), u))), j(u)), j(op2(h(v(j(u))), op2(h(v(j(u))), u)))), sorti1(j(u)))
% 1.23/0.45 = { by axiom 7 (co1) }
% 1.23/0.45 tuple(ifeq2(ifeq(true, true, sorti2(op2(h(v(j(u))), u)), true), true, ifeq2(sorti2(h(v(j(u)))), true, op1(j(h(v(j(u)))), j(op2(h(v(j(u))), u))), j(u)), j(op2(h(v(j(u))), op2(h(v(j(u))), u)))), sorti1(j(u)))
% 1.23/0.45 = { by axiom 3 (ifeq_axiom) }
% 1.23/0.45 tuple(ifeq2(sorti2(op2(h(v(j(u))), u)), true, ifeq2(sorti2(h(v(j(u)))), true, op1(j(h(v(j(u)))), j(op2(h(v(j(u))), u))), j(u)), j(op2(h(v(j(u))), op2(h(v(j(u))), u)))), sorti1(j(u)))
% 1.23/0.45 = { by lemma 18 R->L }
% 1.23/0.45 tuple(ifeq2(sorti2(op2(h(v(j(u))), u)), true, ifeq2(sorti2(h(v(j(u)))), true, op1(j(h(v(j(u)))), j(op2(h(v(j(u))), u))), j(op2(h(v(j(u))), op2(h(v(j(u))), u)))), j(op2(h(v(j(u))), op2(h(v(j(u))), u)))), sorti1(j(u)))
% 1.23/0.45 = { by axiom 15 (co1_4) }
% 1.23/0.45 tuple(j(op2(h(v(j(u))), op2(h(v(j(u))), u))), sorti1(j(u)))
% 1.23/0.45 = { by lemma 18 }
% 1.23/0.45 tuple(j(u), sorti1(j(u)))
% 1.23/0.45 = { by axiom 3 (ifeq_axiom) R->L }
% 1.23/0.45 tuple(j(u), ifeq(true, true, sorti1(j(u)), true))
% 1.23/0.45 = { by axiom 1 (ax4) R->L }
% 1.23/0.45 tuple(j(u), ifeq(sorti2(u), true, sorti1(j(u)), true))
% 1.23/0.45 = { by axiom 8 (co1_3) }
% 1.23/0.45 tuple(j(u), true)
% 1.23/0.45 % SZS output end Proof
% 1.23/0.45
% 1.23/0.45 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------