TSTP Solution File: LCL272-3 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL272-3 : 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 : n012.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 15.01s 2.38s
% Output : Proof 15.01s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : LCL272-3 : TPTP v8.1.2. Released v2.3.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.15/0.36 % Computer : n012.cluster.edu
% 0.15/0.36 % Model : x86_64 x86_64
% 0.15/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.36 % Memory : 8042.1875MB
% 0.15/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.36 % CPULimit : 300
% 0.15/0.36 % WCLimit : 300
% 0.15/0.36 % DateTime : Fri Aug 25 06:17:56 EDT 2023
% 0.15/0.36 % CPUTime :
% 15.01/2.38 Command-line arguments: --no-flatten-goal
% 15.01/2.38
% 15.01/2.38 % SZS status Unsatisfiable
% 15.01/2.38
% 15.01/2.39 % SZS output start Proof
% 15.01/2.39 Take the following subset of the input axioms:
% 15.01/2.39 fof(and_defn, axiom, ![P, Q]: and(P, Q)=not(or(not(P), not(Q)))).
% 15.01/2.39 fof(axiom_1_2, axiom, ![A]: axiom(implies(or(A, A), A))).
% 15.01/2.39 fof(axiom_1_3, axiom, ![B, A2]: axiom(implies(A2, or(B, A2)))).
% 15.01/2.39 fof(axiom_1_4, axiom, ![A2, B2]: axiom(implies(or(A2, B2), or(B2, A2)))).
% 15.01/2.39 fof(axiom_1_5, axiom, ![C, A2, B2]: axiom(implies(or(A2, or(B2, C)), or(B2, or(A2, C))))).
% 15.01/2.39 fof(equivalent_defn, axiom, ![P2, Q2]: equivalent(P2, Q2)=and(implies(P2, Q2), implies(Q2, P2))).
% 15.01/2.39 fof(implies_definition, axiom, ![X, Y]: implies(X, Y)=or(not(X), Y)).
% 15.01/2.39 fof(prove_this, negated_conjecture, ~theorem(equivalent(p, or(p, p)))).
% 15.01/2.39 fof(rule_1, axiom, ![X2]: (theorem(X2) | ~axiom(X2))).
% 15.01/2.39 fof(rule_2, axiom, ![X2, Y2]: (theorem(X2) | (~theorem(implies(Y2, X2)) | ~theorem(Y2)))).
% 15.01/2.39
% 15.01/2.39 Now clausify the problem and encode Horn clauses using encoding 3 of
% 15.01/2.39 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 15.01/2.39 We repeatedly replace C & s=t => u=v by the two clauses:
% 15.01/2.39 fresh(y, y, x1...xn) = u
% 15.01/2.39 C => fresh(s, t, x1...xn) = v
% 15.01/2.39 where fresh is a fresh function symbol and x1..xn are the free
% 15.01/2.39 variables of u and v.
% 15.01/2.39 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 15.01/2.39 input problem has no model of domain size 1).
% 15.01/2.39
% 15.01/2.39 The encoding turns the above axioms into the following unit equations and goals:
% 15.01/2.39
% 15.01/2.39 Axiom 1 (rule_2): fresh(X, X, Y) = true.
% 15.01/2.39 Axiom 2 (rule_1): fresh2(X, X, Y) = true.
% 15.01/2.39 Axiom 3 (implies_definition): implies(X, Y) = or(not(X), Y).
% 15.01/2.39 Axiom 4 (rule_2): fresh3(X, X, Y, Z) = theorem(Y).
% 15.01/2.39 Axiom 5 (rule_1): fresh2(axiom(X), true, X) = theorem(X).
% 15.01/2.39 Axiom 6 (and_defn): and(X, Y) = not(or(not(X), not(Y))).
% 15.01/2.39 Axiom 7 (axiom_1_3): axiom(implies(X, or(Y, X))) = true.
% 15.01/2.39 Axiom 8 (axiom_1_2): axiom(implies(or(X, X), X)) = true.
% 15.01/2.39 Axiom 9 (equivalent_defn): equivalent(X, Y) = and(implies(X, Y), implies(Y, X)).
% 15.01/2.39 Axiom 10 (rule_2): fresh3(theorem(implies(X, Y)), true, Y, X) = fresh(theorem(X), true, Y).
% 15.01/2.39 Axiom 11 (axiom_1_4): axiom(implies(or(X, Y), or(Y, X))) = true.
% 15.01/2.39 Axiom 12 (axiom_1_5): axiom(implies(or(X, or(Y, Z)), or(Y, or(X, Z)))) = true.
% 15.01/2.39
% 15.01/2.39 Lemma 13: theorem(implies(or(X, Y), or(Y, X))) = true.
% 15.01/2.39 Proof:
% 15.01/2.39 theorem(implies(or(X, Y), or(Y, X)))
% 15.01/2.39 = { by axiom 5 (rule_1) R->L }
% 15.01/2.39 fresh2(axiom(implies(or(X, Y), or(Y, X))), true, implies(or(X, Y), or(Y, X)))
% 15.01/2.39 = { by axiom 11 (axiom_1_4) }
% 15.01/2.39 fresh2(true, true, implies(or(X, Y), or(Y, X)))
% 15.01/2.39 = { by axiom 2 (rule_1) }
% 15.01/2.39 true
% 15.01/2.39
% 15.01/2.39 Goal 1 (prove_this): theorem(equivalent(p, or(p, p))) = true.
% 15.01/2.39 Proof:
% 15.01/2.39 theorem(equivalent(p, or(p, p)))
% 15.01/2.39 = { by axiom 4 (rule_2) R->L }
% 15.01/2.39 fresh3(true, true, equivalent(p, or(p, p)), implies(p, or(p, p)))
% 15.01/2.39 = { by axiom 1 (rule_2) R->L }
% 15.01/2.39 fresh3(fresh(true, true, implies(implies(p, or(p, p)), not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))))), true, equivalent(p, or(p, p)), implies(p, or(p, p)))
% 15.01/2.39 = { by axiom 1 (rule_2) R->L }
% 15.01/2.39 fresh3(fresh(fresh(true, true, implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p))))), true, implies(implies(p, or(p, p)), not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))))), true, equivalent(p, or(p, p)), implies(p, or(p, p)))
% 15.01/2.39 = { by axiom 2 (rule_1) R->L }
% 15.01/2.39 fresh3(fresh(fresh(fresh2(true, true, implies(or(p, p), p)), true, implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p))))), true, implies(implies(p, or(p, p)), not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))))), true, equivalent(p, or(p, p)), implies(p, or(p, p)))
% 15.01/2.39 = { by axiom 8 (axiom_1_2) R->L }
% 15.01/2.39 fresh3(fresh(fresh(fresh2(axiom(implies(or(p, p), p)), true, implies(or(p, p), p)), true, implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p))))), true, implies(implies(p, or(p, p)), not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))))), true, equivalent(p, or(p, p)), implies(p, or(p, p)))
% 15.01/2.39 = { by axiom 5 (rule_1) }
% 15.01/2.39 fresh3(fresh(fresh(theorem(implies(or(p, p), p)), true, implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p))))), true, implies(implies(p, or(p, p)), not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))))), true, equivalent(p, or(p, p)), implies(p, or(p, p)))
% 15.01/2.39 = { by axiom 10 (rule_2) R->L }
% 15.01/2.39 fresh3(fresh(fresh3(theorem(implies(implies(or(p, p), p), implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p)))))), true, implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p)))), implies(or(p, p), p)), true, implies(implies(p, or(p, p)), not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))))), true, equivalent(p, or(p, p)), implies(p, or(p, p)))
% 15.01/2.39 = { by axiom 3 (implies_definition) }
% 15.01/2.40 fresh3(fresh(fresh3(theorem(or(not(implies(or(p, p), p)), implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p)))))), true, implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p)))), implies(or(p, p), p)), true, implies(implies(p, or(p, p)), not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))))), true, equivalent(p, or(p, p)), implies(p, or(p, p)))
% 15.01/2.40 = { by axiom 4 (rule_2) R->L }
% 15.01/2.40 fresh3(fresh(fresh3(fresh3(true, true, or(not(implies(or(p, p), p)), implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p))))), implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), or(not(implies(or(p, p), p)), not(implies(p, or(p, p)))))), true, implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p)))), implies(or(p, p), p)), true, implies(implies(p, or(p, p)), not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))))), true, equivalent(p, or(p, p)), implies(p, or(p, p)))
% 15.01/2.40 = { by axiom 2 (rule_1) R->L }
% 15.01/2.40 fresh3(fresh(fresh3(fresh3(fresh2(true, true, implies(implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), or(not(implies(or(p, p), p)), not(implies(p, or(p, p))))), or(not(implies(or(p, p), p)), implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p))))))), true, or(not(implies(or(p, p), p)), implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p))))), implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), or(not(implies(or(p, p), p)), not(implies(p, or(p, p)))))), true, implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p)))), implies(or(p, p), p)), true, implies(implies(p, or(p, p)), not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))))), true, equivalent(p, or(p, p)), implies(p, or(p, p)))
% 15.01/2.40 = { by axiom 12 (axiom_1_5) R->L }
% 15.01/2.40 fresh3(fresh(fresh3(fresh3(fresh2(axiom(implies(or(not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))), or(not(implies(or(p, p), p)), not(implies(p, or(p, p))))), or(not(implies(or(p, p), p)), or(not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))), not(implies(p, or(p, p))))))), true, implies(implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), or(not(implies(or(p, p), p)), not(implies(p, or(p, p))))), or(not(implies(or(p, p), p)), implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p))))))), true, or(not(implies(or(p, p), p)), implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p))))), implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), or(not(implies(or(p, p), p)), not(implies(p, or(p, p)))))), true, implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p)))), implies(or(p, p), p)), true, implies(implies(p, or(p, p)), not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))))), true, equivalent(p, or(p, p)), implies(p, or(p, p)))
% 15.01/2.40 = { by axiom 3 (implies_definition) R->L }
% 15.01/2.40 fresh3(fresh(fresh3(fresh3(fresh2(axiom(implies(implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), or(not(implies(or(p, p), p)), not(implies(p, or(p, p))))), or(not(implies(or(p, p), p)), or(not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))), not(implies(p, or(p, p))))))), true, implies(implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), or(not(implies(or(p, p), p)), not(implies(p, or(p, p))))), or(not(implies(or(p, p), p)), implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p))))))), true, or(not(implies(or(p, p), p)), implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p))))), implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), or(not(implies(or(p, p), p)), not(implies(p, or(p, p)))))), true, implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p)))), implies(or(p, p), p)), true, implies(implies(p, or(p, p)), not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))))), true, equivalent(p, or(p, p)), implies(p, or(p, p)))
% 15.01/2.40 = { by axiom 3 (implies_definition) R->L }
% 15.01/2.40 fresh3(fresh(fresh3(fresh3(fresh2(axiom(implies(implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), or(not(implies(or(p, p), p)), not(implies(p, or(p, p))))), or(not(implies(or(p, p), p)), implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p))))))), true, implies(implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), or(not(implies(or(p, p), p)), not(implies(p, or(p, p))))), or(not(implies(or(p, p), p)), implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p))))))), true, or(not(implies(or(p, p), p)), implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p))))), implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), or(not(implies(or(p, p), p)), not(implies(p, or(p, p)))))), true, implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p)))), implies(or(p, p), p)), true, implies(implies(p, or(p, p)), not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))))), true, equivalent(p, or(p, p)), implies(p, or(p, p)))
% 15.01/2.40 = { by axiom 5 (rule_1) }
% 15.01/2.40 fresh3(fresh(fresh3(fresh3(theorem(implies(implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), or(not(implies(or(p, p), p)), not(implies(p, or(p, p))))), or(not(implies(or(p, p), p)), implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p))))))), true, or(not(implies(or(p, p), p)), implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p))))), implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), or(not(implies(or(p, p), p)), not(implies(p, or(p, p)))))), true, implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p)))), implies(or(p, p), p)), true, implies(implies(p, or(p, p)), not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))))), true, equivalent(p, or(p, p)), implies(p, or(p, p)))
% 15.01/2.40 = { by axiom 10 (rule_2) }
% 15.01/2.40 fresh3(fresh(fresh3(fresh(theorem(implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), or(not(implies(or(p, p), p)), not(implies(p, or(p, p)))))), true, or(not(implies(or(p, p), p)), implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p)))))), true, implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p)))), implies(or(p, p), p)), true, implies(implies(p, or(p, p)), not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))))), true, equivalent(p, or(p, p)), implies(p, or(p, p)))
% 15.01/2.40 = { by lemma 13 }
% 15.01/2.40 fresh3(fresh(fresh3(fresh(true, true, or(not(implies(or(p, p), p)), implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p)))))), true, implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p)))), implies(or(p, p), p)), true, implies(implies(p, or(p, p)), not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))))), true, equivalent(p, or(p, p)), implies(p, or(p, p)))
% 15.01/2.40 = { by axiom 1 (rule_2) }
% 15.01/2.40 fresh3(fresh(fresh3(true, true, implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p)))), implies(or(p, p), p)), true, implies(implies(p, or(p, p)), not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))))), true, equivalent(p, or(p, p)), implies(p, or(p, p)))
% 15.01/2.40 = { by axiom 4 (rule_2) }
% 15.01/2.40 fresh3(fresh(theorem(implies(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))), not(implies(p, or(p, p))))), true, implies(implies(p, or(p, p)), not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))))), true, equivalent(p, or(p, p)), implies(p, or(p, p)))
% 15.01/2.40 = { by axiom 3 (implies_definition) }
% 15.01/2.40 fresh3(fresh(theorem(or(not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))), not(implies(p, or(p, p))))), true, implies(implies(p, or(p, p)), not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))))), true, equivalent(p, or(p, p)), implies(p, or(p, p)))
% 15.01/2.40 = { by axiom 10 (rule_2) R->L }
% 15.01/2.40 fresh3(fresh3(theorem(implies(or(not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))), not(implies(p, or(p, p)))), implies(implies(p, or(p, p)), not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))))))), true, implies(implies(p, or(p, p)), not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))))), or(not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))), not(implies(p, or(p, p))))), true, equivalent(p, or(p, p)), implies(p, or(p, p)))
% 15.01/2.40 = { by axiom 3 (implies_definition) }
% 15.01/2.40 fresh3(fresh3(theorem(implies(or(not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))), not(implies(p, or(p, p)))), or(not(implies(p, or(p, p))), not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))))))), true, implies(implies(p, or(p, p)), not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))))), or(not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))), not(implies(p, or(p, p))))), true, equivalent(p, or(p, p)), implies(p, or(p, p)))
% 15.01/2.40 = { by lemma 13 }
% 15.01/2.40 fresh3(fresh3(true, true, implies(implies(p, or(p, p)), not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p))))), or(not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))), not(implies(p, or(p, p))))), true, equivalent(p, or(p, p)), implies(p, or(p, p)))
% 15.01/2.40 = { by axiom 4 (rule_2) }
% 15.01/2.40 fresh3(theorem(implies(implies(p, or(p, p)), not(or(not(implies(p, or(p, p))), not(implies(or(p, p), p)))))), true, equivalent(p, or(p, p)), implies(p, or(p, p)))
% 15.01/2.40 = { by axiom 6 (and_defn) R->L }
% 15.01/2.40 fresh3(theorem(implies(implies(p, or(p, p)), and(implies(p, or(p, p)), implies(or(p, p), p)))), true, equivalent(p, or(p, p)), implies(p, or(p, p)))
% 15.01/2.40 = { by axiom 9 (equivalent_defn) R->L }
% 15.01/2.40 fresh3(theorem(implies(implies(p, or(p, p)), equivalent(p, or(p, p)))), true, equivalent(p, or(p, p)), implies(p, or(p, p)))
% 15.01/2.40 = { by axiom 10 (rule_2) }
% 15.01/2.40 fresh(theorem(implies(p, or(p, p))), true, equivalent(p, or(p, p)))
% 15.01/2.40 = { by axiom 5 (rule_1) R->L }
% 15.01/2.40 fresh(fresh2(axiom(implies(p, or(p, p))), true, implies(p, or(p, p))), true, equivalent(p, or(p, p)))
% 15.01/2.40 = { by axiom 7 (axiom_1_3) }
% 15.01/2.40 fresh(fresh2(true, true, implies(p, or(p, p))), true, equivalent(p, or(p, p)))
% 15.01/2.40 = { by axiom 2 (rule_1) }
% 15.01/2.40 fresh(true, true, equivalent(p, or(p, p)))
% 15.01/2.40 = { by axiom 1 (rule_2) }
% 15.01/2.40 true
% 15.01/2.40 % SZS output end Proof
% 15.01/2.40
% 15.01/2.40 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------