%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : LCL274-3 : TPTP v9.3.1. Released v2.3.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n002.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 11:49:32 AM UTC 2026
% Result : Unsatisfiable 14.94s 2.47s
% Output : Proof 15.65s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.04 % Problem : LCL274-3 : TPTP v9.3.1. Released v2.3.0.
% 0.00/0.06 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.17/0.44 % Computer : n002.cluster.edu
% 0.17/0.44 % Model : x86_64 x86_64
% 0.17/0.44 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.44 % Memory : 8046.5625MB
% 0.17/0.44 % OS : Linux 6.8.0-71-generic
% 0.17/0.44 % CPULimit : 300
% 0.17/0.44 % WCLimit : 300
% 0.17/0.44 % DateTime : Sun Sep 27 15:35:06 UTC 2026
% 0.17/0.44 % CPUTime :
% 0.17/0.44 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 14.94/2.47 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
% 14.94/2.47
% 14.94/2.47 % SZS status Unsatisfiable
% 14.94/2.47
% 15.65/2.51 % SZS output start Proof
% 15.65/2.51 Axiom 1 (implies_definition): implies(X, Y) = or(not(X), Y).
% 15.65/2.51 Axiom 2 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 15.65/2.51 Axiom 3 (and_defn): and(X, Y) = not(or(not(X), not(Y))).
% 15.65/2.51 Axiom 4 (rule_1): ifeq(axiom(X), true, theorem(X), true) = true.
% 15.65/2.51 Axiom 5 (equivalent_defn): equivalent(X, Y) = and(implies(X, Y), implies(Y, X)).
% 15.65/2.51 Axiom 6 (axiom_1_4): axiom(implies(or(X, Y), or(Y, X))) = true.
% 15.65/2.51 Axiom 7 (rule_2): ifeq(theorem(implies(X, Y)), true, ifeq(theorem(X), true, theorem(Y), true), true) = true.
% 15.65/2.51 Axiom 8 (axiom_1_5): axiom(implies(or(X, or(Y, Z)), or(Y, or(X, Z)))) = true.
% 15.65/2.51
% 15.65/2.51 Goal 1 (prove_this): theorem(equivalent(or(p, q), or(q, p))) = true.
% 15.65/2.51 Proof:
% 15.65/2.51 theorem(equivalent(or(p, q), or(q, p)))
% 15.65/2.51 = { by axiom 5 (equivalent_defn) }
% 15.65/2.51 theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q))))
% 15.65/2.51 = { by axiom 2 (ifeq_axiom) R->L }
% 15.65/2.51 ifeq(true, true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.51 = { by axiom 7 (rule_2) R->L }
% 15.65/2.51 ifeq(ifeq(theorem(implies(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p)))), or(not(implies(or(p, q), or(q, p))), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))))))), true, ifeq(theorem(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))), true, theorem(or(not(implies(or(p, q), or(q, p))), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))))), true), true), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.51 = { by axiom 1 (implies_definition) }
% 15.65/2.51 ifeq(ifeq(theorem(implies(or(not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))), not(implies(or(p, q), or(q, p)))), or(not(implies(or(p, q), or(q, p))), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))))))), true, ifeq(theorem(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))), true, theorem(or(not(implies(or(p, q), or(q, p))), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))))), true), true), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.51 = { by axiom 2 (ifeq_axiom) R->L }
% 15.65/2.51 ifeq(ifeq(ifeq(true, true, theorem(implies(or(not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))), not(implies(or(p, q), or(q, p)))), or(not(implies(or(p, q), or(q, p))), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))))))), true), true, ifeq(theorem(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))), true, theorem(or(not(implies(or(p, q), or(q, p))), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))))), true), true), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.51 = { by axiom 6 (axiom_1_4) R->L }
% 15.65/2.51 ifeq(ifeq(ifeq(axiom(implies(or(not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))), not(implies(or(p, q), or(q, p)))), or(not(implies(or(p, q), or(q, p))), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))))))), true, theorem(implies(or(not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))), not(implies(or(p, q), or(q, p)))), or(not(implies(or(p, q), or(q, p))), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))))))), true), true, ifeq(theorem(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))), true, theorem(or(not(implies(or(p, q), or(q, p))), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))))), true), true), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.52 = { by axiom 4 (rule_1) }
% 15.65/2.52 ifeq(ifeq(true, true, ifeq(theorem(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))), true, theorem(or(not(implies(or(p, q), or(q, p))), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))))), true), true), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.52 = { by axiom 2 (ifeq_axiom) }
% 15.65/2.52 ifeq(ifeq(theorem(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))), true, theorem(or(not(implies(or(p, q), or(q, p))), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))))), true), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.52 = { by axiom 1 (implies_definition) R->L }
% 15.65/2.52 ifeq(ifeq(theorem(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))), true, theorem(implies(implies(or(p, q), or(q, p)), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))))), true), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.52 = { by axiom 2 (ifeq_axiom) R->L }
% 15.65/2.52 ifeq(ifeq(ifeq(true, true, theorem(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))), true), true, theorem(implies(implies(or(p, q), or(q, p)), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))))), true), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.52 = { by axiom 7 (rule_2) R->L }
% 15.65/2.52 ifeq(ifeq(ifeq(ifeq(theorem(implies(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), or(not(implies(or(q, p), or(p, q))), not(implies(or(p, q), or(q, p))))), or(not(implies(or(q, p), or(p, q))), implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))))), true, ifeq(theorem(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), or(not(implies(or(q, p), or(p, q))), not(implies(or(p, q), or(q, p)))))), true, theorem(or(not(implies(or(q, p), or(p, q))), implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p)))))), true), true), true, theorem(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))), true), true, theorem(implies(implies(or(p, q), or(q, p)), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))))), true), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.53 = { by axiom 2 (ifeq_axiom) R->L }
% 15.65/2.53 ifeq(ifeq(ifeq(ifeq(theorem(implies(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), or(not(implies(or(q, p), or(p, q))), not(implies(or(p, q), or(q, p))))), or(not(implies(or(q, p), or(p, q))), implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))))), true, ifeq(ifeq(true, true, theorem(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), or(not(implies(or(q, p), or(p, q))), not(implies(or(p, q), or(q, p)))))), true), true, theorem(or(not(implies(or(q, p), or(p, q))), implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p)))))), true), true), true, theorem(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))), true), true, theorem(implies(implies(or(p, q), or(q, p)), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))))), true), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.53 = { by axiom 6 (axiom_1_4) R->L }
% 15.65/2.53 ifeq(ifeq(ifeq(ifeq(theorem(implies(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), or(not(implies(or(q, p), or(p, q))), not(implies(or(p, q), or(q, p))))), or(not(implies(or(q, p), or(p, q))), implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))))), true, ifeq(ifeq(axiom(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), or(not(implies(or(q, p), or(p, q))), not(implies(or(p, q), or(q, p)))))), true, theorem(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), or(not(implies(or(q, p), or(p, q))), not(implies(or(p, q), or(q, p)))))), true), true, theorem(or(not(implies(or(q, p), or(p, q))), implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p)))))), true), true), true, theorem(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))), true), true, theorem(implies(implies(or(p, q), or(q, p)), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))))), true), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.53 = { by axiom 4 (rule_1) }
% 15.65/2.53 ifeq(ifeq(ifeq(ifeq(theorem(implies(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), or(not(implies(or(q, p), or(p, q))), not(implies(or(p, q), or(q, p))))), or(not(implies(or(q, p), or(p, q))), implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))))), true, ifeq(true, true, theorem(or(not(implies(or(q, p), or(p, q))), implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p)))))), true), true), true, theorem(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))), true), true, theorem(implies(implies(or(p, q), or(q, p)), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))))), true), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.53 = { by axiom 2 (ifeq_axiom) }
% 15.65/2.53 ifeq(ifeq(ifeq(ifeq(theorem(implies(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), or(not(implies(or(q, p), or(p, q))), not(implies(or(p, q), or(q, p))))), or(not(implies(or(q, p), or(p, q))), implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))))), true, theorem(or(not(implies(or(q, p), or(p, q))), implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p)))))), true), true, theorem(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))), true), true, theorem(implies(implies(or(p, q), or(q, p)), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))))), true), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.53 = { by axiom 2 (ifeq_axiom) R->L }
% 15.65/2.53 ifeq(ifeq(ifeq(ifeq(ifeq(true, true, theorem(implies(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), or(not(implies(or(q, p), or(p, q))), not(implies(or(p, q), or(q, p))))), or(not(implies(or(q, p), or(p, q))), implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))))), true), true, theorem(or(not(implies(or(q, p), or(p, q))), implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p)))))), true), true, theorem(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))), true), true, theorem(implies(implies(or(p, q), or(q, p)), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))))), true), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.53 = { by axiom 8 (axiom_1_5) R->L }
% 15.65/2.53 ifeq(ifeq(ifeq(ifeq(ifeq(axiom(implies(or(not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))), or(not(implies(or(q, p), or(p, q))), not(implies(or(p, q), or(q, p))))), or(not(implies(or(q, p), or(p, q))), or(not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))), not(implies(or(p, q), or(q, p))))))), true, theorem(implies(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), or(not(implies(or(q, p), or(p, q))), not(implies(or(p, q), or(q, p))))), or(not(implies(or(q, p), or(p, q))), implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))))), true), true, theorem(or(not(implies(or(q, p), or(p, q))), implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p)))))), true), true, theorem(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))), true), true, theorem(implies(implies(or(p, q), or(q, p)), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))))), true), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.53 = { by axiom 1 (implies_definition) R->L }
% 15.65/2.53 ifeq(ifeq(ifeq(ifeq(ifeq(axiom(implies(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), or(not(implies(or(q, p), or(p, q))), not(implies(or(p, q), or(q, p))))), or(not(implies(or(q, p), or(p, q))), or(not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))), not(implies(or(p, q), or(q, p))))))), true, theorem(implies(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), or(not(implies(or(q, p), or(p, q))), not(implies(or(p, q), or(q, p))))), or(not(implies(or(q, p), or(p, q))), implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))))), true), true, theorem(or(not(implies(or(q, p), or(p, q))), implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p)))))), true), true, theorem(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))), true), true, theorem(implies(implies(or(p, q), or(q, p)), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))))), true), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.54 = { by axiom 1 (implies_definition) R->L }
% 15.65/2.54 ifeq(ifeq(ifeq(ifeq(ifeq(axiom(implies(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), or(not(implies(or(q, p), or(p, q))), not(implies(or(p, q), or(q, p))))), or(not(implies(or(q, p), or(p, q))), implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))))), true, theorem(implies(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), or(not(implies(or(q, p), or(p, q))), not(implies(or(p, q), or(q, p))))), or(not(implies(or(q, p), or(p, q))), implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))))), true), true, theorem(or(not(implies(or(q, p), or(p, q))), implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p)))))), true), true, theorem(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))), true), true, theorem(implies(implies(or(p, q), or(q, p)), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))))), true), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.54 = { by axiom 4 (rule_1) }
% 15.65/2.54 ifeq(ifeq(ifeq(ifeq(true, true, theorem(or(not(implies(or(q, p), or(p, q))), implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p)))))), true), true, theorem(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))), true), true, theorem(implies(implies(or(p, q), or(q, p)), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))))), true), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.54 = { by axiom 2 (ifeq_axiom) }
% 15.65/2.54 ifeq(ifeq(ifeq(theorem(or(not(implies(or(q, p), or(p, q))), implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p)))))), true, theorem(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))), true), true, theorem(implies(implies(or(p, q), or(q, p)), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))))), true), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.54 = { by axiom 1 (implies_definition) R->L }
% 15.65/2.54 ifeq(ifeq(ifeq(theorem(implies(implies(or(q, p), or(p, q)), implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p)))))), true, theorem(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))), true), true, theorem(implies(implies(or(p, q), or(q, p)), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))))), true), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.54 = { by axiom 2 (ifeq_axiom) R->L }
% 15.65/2.54 ifeq(ifeq(ifeq(theorem(implies(implies(or(q, p), or(p, q)), implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p)))))), true, ifeq(true, true, theorem(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))), true), true), true, theorem(implies(implies(or(p, q), or(q, p)), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))))), true), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.54 = { by axiom 4 (rule_1) R->L }
% 15.65/2.54 ifeq(ifeq(ifeq(theorem(implies(implies(or(q, p), or(p, q)), implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p)))))), true, ifeq(ifeq(axiom(implies(or(q, p), or(p, q))), true, theorem(implies(or(q, p), or(p, q))), true), true, theorem(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))), true), true), true, theorem(implies(implies(or(p, q), or(q, p)), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))))), true), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.54 = { by axiom 6 (axiom_1_4) }
% 15.65/2.54 ifeq(ifeq(ifeq(theorem(implies(implies(or(q, p), or(p, q)), implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p)))))), true, ifeq(ifeq(true, true, theorem(implies(or(q, p), or(p, q))), true), true, theorem(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))), true), true), true, theorem(implies(implies(or(p, q), or(q, p)), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))))), true), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.54 = { by axiom 2 (ifeq_axiom) }
% 15.65/2.54 ifeq(ifeq(ifeq(theorem(implies(implies(or(q, p), or(p, q)), implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p)))))), true, ifeq(theorem(implies(or(q, p), or(p, q))), true, theorem(implies(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q)))), not(implies(or(p, q), or(q, p))))), true), true), true, theorem(implies(implies(or(p, q), or(q, p)), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))))), true), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.54 = { by axiom 7 (rule_2) }
% 15.65/2.54 ifeq(ifeq(true, true, theorem(implies(implies(or(p, q), or(q, p)), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))))), true), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.54 = { by axiom 2 (ifeq_axiom) }
% 15.65/2.54 ifeq(theorem(implies(implies(or(p, q), or(q, p)), not(or(not(implies(or(p, q), or(q, p))), not(implies(or(q, p), or(p, q))))))), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.54 = { by axiom 3 (and_defn) R->L }
% 15.65/2.54 ifeq(theorem(implies(implies(or(p, q), or(q, p)), and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q))))), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true)
% 15.65/2.54 = { by axiom 2 (ifeq_axiom) R->L }
% 15.65/2.54 ifeq(theorem(implies(implies(or(p, q), or(q, p)), and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q))))), true, ifeq(true, true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true), true)
% 15.65/2.54 = { by axiom 4 (rule_1) R->L }
% 15.65/2.54 ifeq(theorem(implies(implies(or(p, q), or(q, p)), and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q))))), true, ifeq(ifeq(axiom(implies(or(p, q), or(q, p))), true, theorem(implies(or(p, q), or(q, p))), true), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true), true)
% 15.65/2.54 = { by axiom 6 (axiom_1_4) }
% 15.65/2.54 ifeq(theorem(implies(implies(or(p, q), or(q, p)), and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q))))), true, ifeq(ifeq(true, true, theorem(implies(or(p, q), or(q, p))), true), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true), true)
% 15.65/2.54 = { by axiom 2 (ifeq_axiom) }
% 15.65/2.54 ifeq(theorem(implies(implies(or(p, q), or(q, p)), and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q))))), true, ifeq(theorem(implies(or(p, q), or(q, p))), true, theorem(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q)))), true), true)
% 15.65/2.54 = { by axiom 7 (rule_2) }
% 15.65/2.54 true
% 15.65/2.54 % SZS output end Proof
% 15.65/2.54
% 15.65/2.54 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------