TSTP Solution File: LCL274-3 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL274-3 : TPTP v8.1.2. Released v2.3.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n003.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 : 300s
% DateTime : Thu Aug 31 08:18:22 EDT 2023
% Result : Unsatisfiable 52.97s 7.13s
% Output : Proof 52.97s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LCL274-3 : TPTP v8.1.2. Released v2.3.0.
% 0.12/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.34 % Computer : n003.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Fri Aug 25 05:41:38 EDT 2023
% 0.12/0.34 % CPUTime :
% 52.97/7.13 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 52.97/7.13
% 52.97/7.13 % SZS status Unsatisfiable
% 52.97/7.13
% 52.97/7.14 % SZS output start Proof
% 52.97/7.14 Take the following subset of the input axioms:
% 52.97/7.14 fof(and_defn, axiom, ![P, Q]: and(P, Q)=not(or(not(P), not(Q)))).
% 52.97/7.14 fof(axiom_1_4, axiom, ![A, B]: axiom(implies(or(A, B), or(B, A)))).
% 52.97/7.14 fof(axiom_1_5, axiom, ![C, A2, B2]: axiom(implies(or(A2, or(B2, C)), or(B2, or(A2, C))))).
% 52.97/7.14 fof(equivalent_defn, axiom, ![P2, Q2]: equivalent(P2, Q2)=and(implies(P2, Q2), implies(Q2, P2))).
% 52.97/7.14 fof(implies_definition, axiom, ![X, Y]: implies(X, Y)=or(not(X), Y)).
% 52.97/7.14 fof(prove_this, negated_conjecture, ~theorem(equivalent(or(p, q), or(q, p)))).
% 52.97/7.14 fof(rule_1, axiom, ![X2]: (theorem(X2) | ~axiom(X2))).
% 52.97/7.14 fof(rule_2, axiom, ![X2, Y2]: (theorem(X2) | (~theorem(implies(Y2, X2)) | ~theorem(Y2)))).
% 52.97/7.15
% 52.97/7.15 Now clausify the problem and encode Horn clauses using encoding 3 of
% 52.97/7.15 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 52.97/7.15 We repeatedly replace C & s=t => u=v by the two clauses:
% 52.97/7.15 fresh(y, y, x1...xn) = u
% 52.97/7.15 C => fresh(s, t, x1...xn) = v
% 52.97/7.15 where fresh is a fresh function symbol and x1..xn are the free
% 52.97/7.15 variables of u and v.
% 52.97/7.15 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 52.97/7.15 input problem has no model of domain size 1).
% 52.97/7.15
% 52.97/7.15 The encoding turns the above axioms into the following unit equations and goals:
% 52.97/7.15
% 52.97/7.15 Axiom 1 (implies_definition): implies(X, Y) = or(not(X), Y).
% 52.97/7.15 Axiom 2 (rule_2): fresh(X, X, Y) = true.
% 52.97/7.15 Axiom 3 (rule_1): fresh2(X, X, Y) = true.
% 52.97/7.15 Axiom 4 (rule_2): fresh3(X, X, Y, Z) = theorem(Y).
% 52.97/7.15 Axiom 5 (rule_1): fresh2(axiom(X), true, X) = theorem(X).
% 52.97/7.15 Axiom 6 (and_defn): and(X, Y) = not(or(not(X), not(Y))).
% 52.97/7.15 Axiom 7 (equivalent_defn): equivalent(X, Y) = and(implies(X, Y), implies(Y, X)).
% 52.97/7.15 Axiom 8 (axiom_1_4): axiom(implies(or(X, Y), or(Y, X))) = true.
% 52.97/7.15 Axiom 9 (rule_2): fresh3(theorem(implies(X, Y)), true, Y, X) = fresh(theorem(X), true, Y).
% 52.97/7.15 Axiom 10 (axiom_1_5): axiom(implies(or(X, or(Y, Z)), or(Y, or(X, Z)))) = true.
% 52.97/7.15
% 52.97/7.15 Lemma 11: theorem(implies(or(X, Y), or(Y, X))) = true.
% 52.97/7.15 Proof:
% 52.97/7.15 theorem(implies(or(X, Y), or(Y, X)))
% 52.97/7.15 = { by axiom 5 (rule_1) R->L }
% 52.97/7.15 fresh2(axiom(implies(or(X, Y), or(Y, X))), true, implies(or(X, Y), or(Y, X)))
% 52.97/7.15 = { by axiom 8 (axiom_1_4) }
% 52.97/7.15 fresh2(true, true, implies(or(X, Y), or(Y, X)))
% 52.97/7.15 = { by axiom 3 (rule_1) }
% 52.97/7.15 true
% 52.97/7.15
% 52.97/7.15 Goal 1 (prove_this): theorem(equivalent(or(p, q), or(q, p))) = true.
% 52.97/7.15 Proof:
% 52.97/7.15 theorem(equivalent(or(p, q), or(q, p)))
% 52.97/7.15 = { by axiom 4 (rule_2) R->L }
% 52.97/7.15 fresh3(true, true, equivalent(or(p, q), or(q, p)), implies(or(p, q), or(q, p)))
% 52.97/7.15 = { by axiom 2 (rule_2) R->L }
% 52.97/7.15 fresh3(fresh(true, true, or(not(implies(or(p, q), or(q, p))), equivalent(or(p, q), or(q, p)))), true, equivalent(or(p, q), or(q, p)), implies(or(p, q), or(q, p)))
% 52.97/7.15 = { by axiom 2 (rule_2) R->L }
% 52.97/7.15 fresh3(fresh(fresh(true, true, or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p))))), true, or(not(implies(or(p, q), or(q, p))), equivalent(or(p, q), or(q, p)))), true, equivalent(or(p, q), or(q, p)), implies(or(p, q), or(q, p)))
% 52.97/7.15 = { by lemma 11 R->L }
% 52.97/7.15 fresh3(fresh(fresh(theorem(implies(or(q, p), or(p, q))), true, or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p))))), true, or(not(implies(or(p, q), or(q, p))), equivalent(or(p, q), or(q, p)))), true, equivalent(or(p, q), or(q, p)), implies(or(p, q), or(q, p)))
% 52.97/7.15 = { by axiom 9 (rule_2) R->L }
% 52.97/7.15 fresh3(fresh(fresh3(theorem(implies(implies(or(q, p), or(p, q)), or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))))), true, or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))), implies(or(q, p), or(p, q))), true, or(not(implies(or(p, q), or(q, p))), equivalent(or(p, q), or(q, p)))), true, equivalent(or(p, q), or(q, p)), implies(or(p, q), or(q, p)))
% 52.97/7.15 = { by axiom 4 (rule_2) R->L }
% 52.97/7.15 fresh3(fresh(fresh3(fresh3(true, true, implies(implies(or(q, p), or(p, q)), or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p))))), or(equivalent(or(p, q), or(q, p)), implies(implies(or(q, p), or(p, q)), not(implies(or(p, q), or(q, p)))))), true, or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))), implies(or(q, p), or(p, q))), true, or(not(implies(or(p, q), or(q, p))), equivalent(or(p, q), or(q, p)))), true, equivalent(or(p, q), or(q, p)), implies(or(p, q), or(q, p)))
% 52.97/7.15 = { by axiom 3 (rule_1) R->L }
% 52.97/7.15 fresh3(fresh(fresh3(fresh3(fresh2(true, true, implies(or(equivalent(or(p, q), or(q, p)), implies(implies(or(q, p), or(p, q)), not(implies(or(p, q), or(q, p))))), implies(implies(or(q, p), or(p, q)), or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p))))))), true, implies(implies(or(q, p), or(p, q)), or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p))))), or(equivalent(or(p, q), or(q, p)), implies(implies(or(q, p), or(p, q)), not(implies(or(p, q), or(q, p)))))), true, or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))), implies(or(q, p), or(p, q))), true, or(not(implies(or(p, q), or(q, p))), equivalent(or(p, q), or(q, p)))), true, equivalent(or(p, q), or(q, p)), implies(or(p, q), or(q, p)))
% 52.97/7.15 = { by axiom 10 (axiom_1_5) R->L }
% 52.97/7.15 fresh3(fresh(fresh3(fresh3(fresh2(axiom(implies(or(equivalent(or(p, q), or(q, p)), 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(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p))))))), true, implies(or(equivalent(or(p, q), or(q, p)), implies(implies(or(q, p), or(p, q)), not(implies(or(p, q), or(q, p))))), implies(implies(or(q, p), or(p, q)), or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p))))))), true, implies(implies(or(q, p), or(p, q)), or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p))))), or(equivalent(or(p, q), or(q, p)), implies(implies(or(q, p), or(p, q)), not(implies(or(p, q), or(q, p)))))), true, or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))), implies(or(q, p), or(p, q))), true, or(not(implies(or(p, q), or(q, p))), equivalent(or(p, q), or(q, p)))), true, equivalent(or(p, q), or(q, p)), implies(or(p, q), or(q, p)))
% 52.97/7.15 = { by axiom 1 (implies_definition) R->L }
% 52.97/7.15 fresh3(fresh(fresh3(fresh3(fresh2(axiom(implies(or(equivalent(or(p, q), or(q, p)), implies(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(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p))))))), true, implies(or(equivalent(or(p, q), or(q, p)), implies(implies(or(q, p), or(p, q)), not(implies(or(p, q), or(q, p))))), implies(implies(or(q, p), or(p, q)), or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p))))))), true, implies(implies(or(q, p), or(p, q)), or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p))))), or(equivalent(or(p, q), or(q, p)), implies(implies(or(q, p), or(p, q)), not(implies(or(p, q), or(q, p)))))), true, or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))), implies(or(q, p), or(p, q))), true, or(not(implies(or(p, q), or(q, p))), equivalent(or(p, q), or(q, p)))), true, equivalent(or(p, q), or(q, p)), implies(or(p, q), or(q, p)))
% 52.97/7.15 = { by axiom 1 (implies_definition) R->L }
% 52.97/7.15 fresh3(fresh(fresh3(fresh3(fresh2(axiom(implies(or(equivalent(or(p, q), or(q, p)), implies(implies(or(q, p), or(p, q)), not(implies(or(p, q), or(q, p))))), implies(implies(or(q, p), or(p, q)), or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p))))))), true, implies(or(equivalent(or(p, q), or(q, p)), implies(implies(or(q, p), or(p, q)), not(implies(or(p, q), or(q, p))))), implies(implies(or(q, p), or(p, q)), or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p))))))), true, implies(implies(or(q, p), or(p, q)), or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p))))), or(equivalent(or(p, q), or(q, p)), implies(implies(or(q, p), or(p, q)), not(implies(or(p, q), or(q, p)))))), true, or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))), implies(or(q, p), or(p, q))), true, or(not(implies(or(p, q), or(q, p))), equivalent(or(p, q), or(q, p)))), true, equivalent(or(p, q), or(q, p)), implies(or(p, q), or(q, p)))
% 52.97/7.15 = { by axiom 5 (rule_1) }
% 52.97/7.15 fresh3(fresh(fresh3(fresh3(theorem(implies(or(equivalent(or(p, q), or(q, p)), implies(implies(or(q, p), or(p, q)), not(implies(or(p, q), or(q, p))))), implies(implies(or(q, p), or(p, q)), or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p))))))), true, implies(implies(or(q, p), or(p, q)), or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p))))), or(equivalent(or(p, q), or(q, p)), implies(implies(or(q, p), or(p, q)), not(implies(or(p, q), or(q, p)))))), true, or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))), implies(or(q, p), or(p, q))), true, or(not(implies(or(p, q), or(q, p))), equivalent(or(p, q), or(q, p)))), true, equivalent(or(p, q), or(q, p)), implies(or(p, q), or(q, p)))
% 52.97/7.15 = { by axiom 9 (rule_2) }
% 52.97/7.15 fresh3(fresh(fresh3(fresh(theorem(or(equivalent(or(p, q), or(q, p)), implies(implies(or(q, p), or(p, q)), not(implies(or(p, q), or(q, p)))))), true, implies(implies(or(q, p), or(p, q)), or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))))), true, or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))), implies(or(q, p), or(p, q))), true, or(not(implies(or(p, q), or(q, p))), equivalent(or(p, q), or(q, p)))), true, equivalent(or(p, q), or(q, p)), implies(or(p, q), or(q, p)))
% 52.97/7.15 = { by axiom 5 (rule_1) R->L }
% 52.97/7.15 fresh3(fresh(fresh3(fresh(fresh2(axiom(or(equivalent(or(p, q), or(q, p)), implies(implies(or(q, p), or(p, q)), not(implies(or(p, q), or(q, p)))))), true, or(equivalent(or(p, q), or(q, p)), implies(implies(or(q, p), or(p, q)), not(implies(or(p, q), or(q, p)))))), true, implies(implies(or(q, p), or(p, q)), or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))))), true, or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))), implies(or(q, p), or(p, q))), true, or(not(implies(or(p, q), or(q, p))), equivalent(or(p, q), or(q, p)))), true, equivalent(or(p, q), or(q, p)), implies(or(p, q), or(q, p)))
% 52.97/7.15 = { by axiom 7 (equivalent_defn) }
% 52.97/7.15 fresh3(fresh(fresh3(fresh(fresh2(axiom(or(and(implies(or(p, q), or(q, p)), implies(or(q, p), or(p, q))), implies(implies(or(q, p), or(p, q)), not(implies(or(p, q), or(q, p)))))), true, or(equivalent(or(p, q), or(q, p)), implies(implies(or(q, p), or(p, q)), not(implies(or(p, q), or(q, p)))))), true, implies(implies(or(q, p), or(p, q)), or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))))), true, or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))), implies(or(q, p), or(p, q))), true, or(not(implies(or(p, q), or(q, p))), equivalent(or(p, q), or(q, p)))), true, equivalent(or(p, q), or(q, p)), implies(or(p, q), or(q, p)))
% 52.97/7.15 = { by axiom 1 (implies_definition) }
% 52.97/7.16 fresh3(fresh(fresh3(fresh(fresh2(axiom(or(and(implies(or(p, q), or(q, p)), 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, or(equivalent(or(p, q), or(q, p)), implies(implies(or(q, p), or(p, q)), not(implies(or(p, q), or(q, p)))))), true, implies(implies(or(q, p), or(p, q)), or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))))), true, or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))), implies(or(q, p), or(p, q))), true, or(not(implies(or(p, q), or(q, p))), equivalent(or(p, q), or(q, p)))), true, equivalent(or(p, q), or(q, p)), implies(or(p, q), or(q, p)))
% 52.97/7.16 = { by axiom 6 (and_defn) }
% 52.97/7.16 fresh3(fresh(fresh3(fresh(fresh2(axiom(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)))))), true, or(equivalent(or(p, q), or(q, p)), implies(implies(or(q, p), or(p, q)), not(implies(or(p, q), or(q, p)))))), true, implies(implies(or(q, p), or(p, q)), or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))))), true, or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))), implies(or(q, p), or(p, q))), true, or(not(implies(or(p, q), or(q, p))), equivalent(or(p, q), or(q, p)))), true, equivalent(or(p, q), or(q, p)), implies(or(p, q), or(q, p)))
% 52.97/7.16 = { by axiom 1 (implies_definition) R->L }
% 52.97/7.16 fresh3(fresh(fresh3(fresh(fresh2(axiom(or(not(implies(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, or(equivalent(or(p, q), or(q, p)), implies(implies(or(q, p), or(p, q)), not(implies(or(p, q), or(q, p)))))), true, implies(implies(or(q, p), or(p, q)), or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))))), true, or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))), implies(or(q, p), or(p, q))), true, or(not(implies(or(p, q), or(q, p))), equivalent(or(p, q), or(q, p)))), true, equivalent(or(p, q), or(q, p)), implies(or(p, q), or(q, p)))
% 52.97/7.16 = { by axiom 1 (implies_definition) R->L }
% 52.97/7.16 fresh3(fresh(fresh3(fresh(fresh2(axiom(implies(implies(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, or(equivalent(or(p, q), or(q, p)), implies(implies(or(q, p), or(p, q)), not(implies(or(p, q), or(q, p)))))), true, implies(implies(or(q, p), or(p, q)), or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))))), true, or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))), implies(or(q, p), or(p, q))), true, or(not(implies(or(p, q), or(q, p))), equivalent(or(p, q), or(q, p)))), true, equivalent(or(p, q), or(q, p)), implies(or(p, q), or(q, p)))
% 52.97/7.16 = { by axiom 1 (implies_definition) }
% 52.97/7.16 fresh3(fresh(fresh3(fresh(fresh2(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, or(equivalent(or(p, q), or(q, p)), implies(implies(or(q, p), or(p, q)), not(implies(or(p, q), or(q, p)))))), true, implies(implies(or(q, p), or(p, q)), or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))))), true, or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))), implies(or(q, p), or(p, q))), true, or(not(implies(or(p, q), or(q, p))), equivalent(or(p, q), or(q, p)))), true, equivalent(or(p, q), or(q, p)), implies(or(p, q), or(q, p)))
% 52.97/7.16 = { by axiom 8 (axiom_1_4) }
% 52.97/7.16 fresh3(fresh(fresh3(fresh(fresh2(true, true, or(equivalent(or(p, q), or(q, p)), implies(implies(or(q, p), or(p, q)), not(implies(or(p, q), or(q, p)))))), true, implies(implies(or(q, p), or(p, q)), or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))))), true, or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))), implies(or(q, p), or(p, q))), true, or(not(implies(or(p, q), or(q, p))), equivalent(or(p, q), or(q, p)))), true, equivalent(or(p, q), or(q, p)), implies(or(p, q), or(q, p)))
% 52.97/7.16 = { by axiom 3 (rule_1) }
% 52.97/7.16 fresh3(fresh(fresh3(fresh(true, true, implies(implies(or(q, p), or(p, q)), or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))))), true, or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))), implies(or(q, p), or(p, q))), true, or(not(implies(or(p, q), or(q, p))), equivalent(or(p, q), or(q, p)))), true, equivalent(or(p, q), or(q, p)), implies(or(p, q), or(q, p)))
% 52.97/7.16 = { by axiom 2 (rule_2) }
% 52.97/7.16 fresh3(fresh(fresh3(true, true, or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))), implies(or(q, p), or(p, q))), true, or(not(implies(or(p, q), or(q, p))), equivalent(or(p, q), or(q, p)))), true, equivalent(or(p, q), or(q, p)), implies(or(p, q), or(q, p)))
% 52.97/7.16 = { by axiom 4 (rule_2) }
% 52.97/7.16 fresh3(fresh(theorem(or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p))))), true, or(not(implies(or(p, q), or(q, p))), equivalent(or(p, q), or(q, p)))), true, equivalent(or(p, q), or(q, p)), implies(or(p, q), or(q, p)))
% 52.97/7.16 = { by axiom 9 (rule_2) R->L }
% 52.97/7.16 fresh3(fresh3(theorem(implies(or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p)))), or(not(implies(or(p, q), or(q, p))), equivalent(or(p, q), or(q, p))))), true, or(not(implies(or(p, q), or(q, p))), equivalent(or(p, q), or(q, p))), or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p))))), true, equivalent(or(p, q), or(q, p)), implies(or(p, q), or(q, p)))
% 52.97/7.16 = { by lemma 11 }
% 52.97/7.16 fresh3(fresh3(true, true, or(not(implies(or(p, q), or(q, p))), equivalent(or(p, q), or(q, p))), or(equivalent(or(p, q), or(q, p)), not(implies(or(p, q), or(q, p))))), true, equivalent(or(p, q), or(q, p)), implies(or(p, q), or(q, p)))
% 52.97/7.16 = { by axiom 4 (rule_2) }
% 52.97/7.16 fresh3(theorem(or(not(implies(or(p, q), or(q, p))), equivalent(or(p, q), or(q, p)))), true, equivalent(or(p, q), or(q, p)), implies(or(p, q), or(q, p)))
% 52.97/7.16 = { by axiom 1 (implies_definition) R->L }
% 52.97/7.16 fresh3(theorem(implies(implies(or(p, q), or(q, p)), equivalent(or(p, q), or(q, p)))), true, equivalent(or(p, q), or(q, p)), implies(or(p, q), or(q, p)))
% 52.97/7.16 = { by axiom 9 (rule_2) }
% 52.97/7.16 fresh(theorem(implies(or(p, q), or(q, p))), true, equivalent(or(p, q), or(q, p)))
% 52.97/7.16 = { by lemma 11 }
% 52.97/7.16 fresh(true, true, equivalent(or(p, q), or(q, p)))
% 52.97/7.16 = { by axiom 2 (rule_2) }
% 52.97/7.16 true
% 52.97/7.16 % SZS output end Proof
% 52.97/7.16
% 52.97/7.16 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------