TSTP Solution File: LCL191-3 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL191-3 : 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 : n016.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 08:17:53 EDT 2023
% Result : Unsatisfiable 190.21s 25.07s
% Output : Proof 190.41s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : LCL191-3 : 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
% 0.15/0.35 % Computer : n016.cluster.edu
% 0.15/0.35 % Model : x86_64 x86_64
% 0.15/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.35 % Memory : 8042.1875MB
% 0.15/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.35 % CPULimit : 300
% 0.15/0.35 % WCLimit : 300
% 0.15/0.35 % DateTime : Fri Aug 25 05:52:40 EDT 2023
% 0.15/0.35 % CPUTime :
% 190.21/25.07 Command-line arguments: --flip-ordering --lhs-weight 1 --depth-weight 60 --distributivity-heuristic
% 190.21/25.07
% 190.21/25.07 % SZS status Unsatisfiable
% 190.21/25.07
% 190.41/25.08 % SZS output start Proof
% 190.41/25.08 Take the following subset of the input axioms:
% 190.41/25.08 fof(axiom_1_4, axiom, ![A, B]: axiom(implies(or(A, B), or(B, A)))).
% 190.41/25.08 fof(axiom_1_5, axiom, ![C, A2, B2]: axiom(implies(or(A2, or(B2, C)), or(B2, or(A2, C))))).
% 190.41/25.08 fof(axiom_1_6, axiom, ![A2, B2, C2]: axiom(implies(implies(A2, B2), implies(or(C2, A2), or(C2, B2))))).
% 190.41/25.08 fof(implies_definition, axiom, ![X, Y]: implies(X, Y)=or(not(X), Y)).
% 190.41/25.08 fof(prove_this, negated_conjecture, ~theorem(implies(or(p, or(q, r)), or(or(p, q), r)))).
% 190.41/25.08 fof(rule_1, axiom, ![X2]: (theorem(X2) | ~axiom(X2))).
% 190.41/25.08 fof(rule_2, axiom, ![X2, Y2]: (theorem(X2) | (~theorem(implies(Y2, X2)) | ~theorem(Y2)))).
% 190.41/25.08
% 190.41/25.08 Now clausify the problem and encode Horn clauses using encoding 3 of
% 190.41/25.08 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 190.41/25.08 We repeatedly replace C & s=t => u=v by the two clauses:
% 190.41/25.08 fresh(y, y, x1...xn) = u
% 190.41/25.08 C => fresh(s, t, x1...xn) = v
% 190.41/25.08 where fresh is a fresh function symbol and x1..xn are the free
% 190.41/25.08 variables of u and v.
% 190.41/25.08 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 190.41/25.08 input problem has no model of domain size 1).
% 190.41/25.08
% 190.41/25.08 The encoding turns the above axioms into the following unit equations and goals:
% 190.41/25.08
% 190.41/25.08 Axiom 1 (implies_definition): implies(X, Y) = or(not(X), Y).
% 190.41/25.08 Axiom 2 (rule_2): fresh(X, X, Y) = true.
% 190.41/25.08 Axiom 3 (rule_1): fresh2(X, X, Y) = true.
% 190.41/25.08 Axiom 4 (rule_2): fresh3(X, X, Y, Z) = theorem(Y).
% 190.41/25.08 Axiom 5 (rule_1): fresh2(axiom(X), true, X) = theorem(X).
% 190.41/25.08 Axiom 6 (axiom_1_4): axiom(implies(or(X, Y), or(Y, X))) = true.
% 190.41/25.08 Axiom 7 (rule_2): fresh3(theorem(implies(X, Y)), true, Y, X) = fresh(theorem(X), true, Y).
% 190.41/25.08 Axiom 8 (axiom_1_6): axiom(implies(implies(X, Y), implies(or(Z, X), or(Z, Y)))) = true.
% 190.41/25.08 Axiom 9 (axiom_1_5): axiom(implies(or(X, or(Y, Z)), or(Y, or(X, Z)))) = true.
% 190.41/25.08
% 190.41/25.08 Lemma 10: theorem(implies(or(X, Y), or(Y, X))) = true.
% 190.41/25.08 Proof:
% 190.41/25.08 theorem(implies(or(X, Y), or(Y, X)))
% 190.41/25.08 = { by axiom 5 (rule_1) R->L }
% 190.41/25.08 fresh2(axiom(implies(or(X, Y), or(Y, X))), true, implies(or(X, Y), or(Y, X)))
% 190.41/25.08 = { by axiom 6 (axiom_1_4) }
% 190.41/25.08 fresh2(true, true, implies(or(X, Y), or(Y, X)))
% 190.41/25.08 = { by axiom 3 (rule_1) }
% 190.41/25.08 true
% 190.41/25.08
% 190.41/25.08 Lemma 11: theorem(implies(implies(X, Y), implies(or(Z, X), or(Z, Y)))) = true.
% 190.41/25.08 Proof:
% 190.41/25.08 theorem(implies(implies(X, Y), implies(or(Z, X), or(Z, Y))))
% 190.41/25.08 = { by axiom 5 (rule_1) R->L }
% 190.41/25.08 fresh2(axiom(implies(implies(X, Y), implies(or(Z, X), or(Z, Y)))), true, implies(implies(X, Y), implies(or(Z, X), or(Z, Y))))
% 190.41/25.08 = { by axiom 8 (axiom_1_6) }
% 190.41/25.08 fresh2(true, true, implies(implies(X, Y), implies(or(Z, X), or(Z, Y))))
% 190.41/25.08 = { by axiom 3 (rule_1) }
% 190.41/25.08 true
% 190.41/25.08
% 190.41/25.08 Lemma 12: fresh(theorem(implies(X, Y)), true, implies(or(Z, X), or(Z, Y))) = theorem(implies(or(Z, X), or(Z, Y))).
% 190.41/25.08 Proof:
% 190.41/25.08 fresh(theorem(implies(X, Y)), true, implies(or(Z, X), or(Z, Y)))
% 190.41/25.08 = { by axiom 7 (rule_2) R->L }
% 190.41/25.08 fresh3(theorem(implies(implies(X, Y), implies(or(Z, X), or(Z, Y)))), true, implies(or(Z, X), or(Z, Y)), implies(X, Y))
% 190.41/25.08 = { by lemma 11 }
% 190.41/25.08 fresh3(true, true, implies(or(Z, X), or(Z, Y)), implies(X, Y))
% 190.41/25.08 = { by axiom 4 (rule_2) }
% 190.41/25.08 theorem(implies(or(Z, X), or(Z, Y)))
% 190.41/25.08
% 190.41/25.08 Goal 1 (prove_this): theorem(implies(or(p, or(q, r)), or(or(p, q), r))) = true.
% 190.41/25.08 Proof:
% 190.41/25.08 theorem(implies(or(p, or(q, r)), or(or(p, q), r)))
% 190.41/25.08 = { by axiom 4 (rule_2) R->L }
% 190.41/25.08 fresh3(true, true, implies(or(p, or(q, r)), or(or(p, q), r)), implies(or(p, or(q, r)), or(r, or(p, q))))
% 190.41/25.08 = { by axiom 2 (rule_2) R->L }
% 190.41/25.08 fresh3(fresh(true, true, implies(or(not(or(p, or(q, r))), or(r, or(p, q))), or(not(or(p, or(q, r))), or(or(p, q), r)))), true, implies(or(p, or(q, r)), or(or(p, q), r)), implies(or(p, or(q, r)), or(r, or(p, q))))
% 190.41/25.08 = { by lemma 10 R->L }
% 190.41/25.08 fresh3(fresh(theorem(implies(or(r, or(p, q)), or(or(p, q), r))), true, implies(or(not(or(p, or(q, r))), or(r, or(p, q))), or(not(or(p, or(q, r))), or(or(p, q), r)))), true, implies(or(p, or(q, r)), or(or(p, q), r)), implies(or(p, or(q, r)), or(r, or(p, q))))
% 190.41/25.08 = { by lemma 12 }
% 190.41/25.08 fresh3(theorem(implies(or(not(or(p, or(q, r))), or(r, or(p, q))), or(not(or(p, or(q, r))), or(or(p, q), r)))), true, implies(or(p, or(q, r)), or(or(p, q), r)), implies(or(p, or(q, r)), or(r, or(p, q))))
% 190.41/25.08 = { by axiom 1 (implies_definition) R->L }
% 190.41/25.08 fresh3(theorem(implies(implies(or(p, or(q, r)), or(r, or(p, q))), or(not(or(p, or(q, r))), or(or(p, q), r)))), true, implies(or(p, or(q, r)), or(or(p, q), r)), implies(or(p, or(q, r)), or(r, or(p, q))))
% 190.41/25.08 = { by axiom 1 (implies_definition) R->L }
% 190.41/25.08 fresh3(theorem(implies(implies(or(p, or(q, r)), or(r, or(p, q))), implies(or(p, or(q, r)), or(or(p, q), r)))), true, implies(or(p, or(q, r)), or(or(p, q), r)), implies(or(p, or(q, r)), or(r, or(p, q))))
% 190.41/25.08 = { by axiom 7 (rule_2) }
% 190.41/25.08 fresh(theorem(implies(or(p, or(q, r)), or(r, or(p, q)))), true, implies(or(p, or(q, r)), or(or(p, q), r)))
% 190.41/25.08 = { by axiom 4 (rule_2) R->L }
% 190.41/25.08 fresh(fresh3(true, true, implies(or(p, or(q, r)), or(r, or(p, q))), or(r, implies(or(p, or(q, r)), or(p, q)))), true, implies(or(p, or(q, r)), or(or(p, q), r)))
% 190.41/25.08 = { by axiom 3 (rule_1) R->L }
% 190.41/25.08 fresh(fresh3(fresh2(true, true, implies(or(r, implies(or(p, or(q, r)), or(p, q))), implies(or(p, or(q, r)), or(r, or(p, q))))), true, implies(or(p, or(q, r)), or(r, or(p, q))), or(r, implies(or(p, or(q, r)), or(p, q)))), true, implies(or(p, or(q, r)), or(or(p, q), r)))
% 190.41/25.08 = { by axiom 9 (axiom_1_5) R->L }
% 190.41/25.08 fresh(fresh3(fresh2(axiom(implies(or(r, or(not(or(p, or(q, r))), or(p, q))), or(not(or(p, or(q, r))), or(r, or(p, q))))), true, implies(or(r, implies(or(p, or(q, r)), or(p, q))), implies(or(p, or(q, r)), or(r, or(p, q))))), true, implies(or(p, or(q, r)), or(r, or(p, q))), or(r, implies(or(p, or(q, r)), or(p, q)))), true, implies(or(p, or(q, r)), or(or(p, q), r)))
% 190.41/25.08 = { by axiom 1 (implies_definition) R->L }
% 190.41/25.08 fresh(fresh3(fresh2(axiom(implies(or(r, implies(or(p, or(q, r)), or(p, q))), or(not(or(p, or(q, r))), or(r, or(p, q))))), true, implies(or(r, implies(or(p, or(q, r)), or(p, q))), implies(or(p, or(q, r)), or(r, or(p, q))))), true, implies(or(p, or(q, r)), or(r, or(p, q))), or(r, implies(or(p, or(q, r)), or(p, q)))), true, implies(or(p, or(q, r)), or(or(p, q), r)))
% 190.41/25.08 = { by axiom 1 (implies_definition) R->L }
% 190.41/25.08 fresh(fresh3(fresh2(axiom(implies(or(r, implies(or(p, or(q, r)), or(p, q))), implies(or(p, or(q, r)), or(r, or(p, q))))), true, implies(or(r, implies(or(p, or(q, r)), or(p, q))), implies(or(p, or(q, r)), or(r, or(p, q))))), true, implies(or(p, or(q, r)), or(r, or(p, q))), or(r, implies(or(p, or(q, r)), or(p, q)))), true, implies(or(p, or(q, r)), or(or(p, q), r)))
% 190.41/25.08 = { by axiom 5 (rule_1) }
% 190.41/25.08 fresh(fresh3(theorem(implies(or(r, implies(or(p, or(q, r)), or(p, q))), implies(or(p, or(q, r)), or(r, or(p, q))))), true, implies(or(p, or(q, r)), or(r, or(p, q))), or(r, implies(or(p, or(q, r)), or(p, q)))), true, implies(or(p, or(q, r)), or(or(p, q), r)))
% 190.41/25.08 = { by axiom 7 (rule_2) }
% 190.41/25.08 fresh(fresh(theorem(or(r, implies(or(p, or(q, r)), or(p, q)))), true, implies(or(p, or(q, r)), or(r, or(p, q)))), true, implies(or(p, or(q, r)), or(or(p, q), r)))
% 190.41/25.08 = { by axiom 4 (rule_2) R->L }
% 190.41/25.08 fresh(fresh(fresh3(true, true, or(r, implies(or(p, or(q, r)), or(p, q))), or(r, implies(or(q, r), q))), true, implies(or(p, or(q, r)), or(r, or(p, q)))), true, implies(or(p, or(q, r)), or(or(p, q), r)))
% 190.41/25.08 = { by axiom 2 (rule_2) R->L }
% 190.41/25.09 fresh(fresh(fresh3(fresh(true, true, implies(or(r, implies(or(q, r), q)), or(r, implies(or(p, or(q, r)), or(p, q))))), true, or(r, implies(or(p, or(q, r)), or(p, q))), or(r, implies(or(q, r), q))), true, implies(or(p, or(q, r)), or(r, or(p, q)))), true, implies(or(p, or(q, r)), or(or(p, q), r)))
% 190.41/25.09 = { by lemma 11 R->L }
% 190.41/25.09 fresh(fresh(fresh3(fresh(theorem(implies(implies(or(q, r), q), implies(or(p, or(q, r)), or(p, q)))), true, implies(or(r, implies(or(q, r), q)), or(r, implies(or(p, or(q, r)), or(p, q))))), true, or(r, implies(or(p, or(q, r)), or(p, q))), or(r, implies(or(q, r), q))), true, implies(or(p, or(q, r)), or(r, or(p, q)))), true, implies(or(p, or(q, r)), or(or(p, q), r)))
% 190.41/25.09 = { by lemma 12 }
% 190.41/25.09 fresh(fresh(fresh3(theorem(implies(or(r, implies(or(q, r), q)), or(r, implies(or(p, or(q, r)), or(p, q))))), true, or(r, implies(or(p, or(q, r)), or(p, q))), or(r, implies(or(q, r), q))), true, implies(or(p, or(q, r)), or(r, or(p, q)))), true, implies(or(p, or(q, r)), or(or(p, q), r)))
% 190.41/25.09 = { by axiom 7 (rule_2) }
% 190.41/25.09 fresh(fresh(fresh(theorem(or(r, implies(or(q, r), q))), true, or(r, implies(or(p, or(q, r)), or(p, q)))), true, implies(or(p, or(q, r)), or(r, or(p, q)))), true, implies(or(p, or(q, r)), or(or(p, q), r)))
% 190.41/25.09 = { by axiom 4 (rule_2) R->L }
% 190.41/25.09 fresh(fresh(fresh(fresh3(true, true, or(r, implies(or(q, r), q)), implies(or(q, r), or(r, q))), true, or(r, implies(or(p, or(q, r)), or(p, q)))), true, implies(or(p, or(q, r)), or(r, or(p, q)))), true, implies(or(p, or(q, r)), or(or(p, q), r)))
% 190.41/25.09 = { by axiom 3 (rule_1) R->L }
% 190.41/25.09 fresh(fresh(fresh(fresh3(fresh2(true, true, implies(implies(or(q, r), or(r, q)), or(r, implies(or(q, r), q)))), true, or(r, implies(or(q, r), q)), implies(or(q, r), or(r, q))), true, or(r, implies(or(p, or(q, r)), or(p, q)))), true, implies(or(p, or(q, r)), or(r, or(p, q)))), true, implies(or(p, or(q, r)), or(or(p, q), r)))
% 190.41/25.09 = { by axiom 9 (axiom_1_5) R->L }
% 190.41/25.09 fresh(fresh(fresh(fresh3(fresh2(axiom(implies(or(not(or(q, r)), or(r, q)), or(r, or(not(or(q, r)), q)))), true, implies(implies(or(q, r), or(r, q)), or(r, implies(or(q, r), q)))), true, or(r, implies(or(q, r), q)), implies(or(q, r), or(r, q))), true, or(r, implies(or(p, or(q, r)), or(p, q)))), true, implies(or(p, or(q, r)), or(r, or(p, q)))), true, implies(or(p, or(q, r)), or(or(p, q), r)))
% 190.41/25.09 = { by axiom 1 (implies_definition) R->L }
% 190.41/25.09 fresh(fresh(fresh(fresh3(fresh2(axiom(implies(implies(or(q, r), or(r, q)), or(r, or(not(or(q, r)), q)))), true, implies(implies(or(q, r), or(r, q)), or(r, implies(or(q, r), q)))), true, or(r, implies(or(q, r), q)), implies(or(q, r), or(r, q))), true, or(r, implies(or(p, or(q, r)), or(p, q)))), true, implies(or(p, or(q, r)), or(r, or(p, q)))), true, implies(or(p, or(q, r)), or(or(p, q), r)))
% 190.41/25.09 = { by axiom 1 (implies_definition) R->L }
% 190.41/25.09 fresh(fresh(fresh(fresh3(fresh2(axiom(implies(implies(or(q, r), or(r, q)), or(r, implies(or(q, r), q)))), true, implies(implies(or(q, r), or(r, q)), or(r, implies(or(q, r), q)))), true, or(r, implies(or(q, r), q)), implies(or(q, r), or(r, q))), true, or(r, implies(or(p, or(q, r)), or(p, q)))), true, implies(or(p, or(q, r)), or(r, or(p, q)))), true, implies(or(p, or(q, r)), or(or(p, q), r)))
% 190.41/25.09 = { by axiom 5 (rule_1) }
% 190.41/25.09 fresh(fresh(fresh(fresh3(theorem(implies(implies(or(q, r), or(r, q)), or(r, implies(or(q, r), q)))), true, or(r, implies(or(q, r), q)), implies(or(q, r), or(r, q))), true, or(r, implies(or(p, or(q, r)), or(p, q)))), true, implies(or(p, or(q, r)), or(r, or(p, q)))), true, implies(or(p, or(q, r)), or(or(p, q), r)))
% 190.41/25.09 = { by axiom 7 (rule_2) }
% 190.41/25.09 fresh(fresh(fresh(fresh(theorem(implies(or(q, r), or(r, q))), true, or(r, implies(or(q, r), q))), true, or(r, implies(or(p, or(q, r)), or(p, q)))), true, implies(or(p, or(q, r)), or(r, or(p, q)))), true, implies(or(p, or(q, r)), or(or(p, q), r)))
% 190.41/25.09 = { by lemma 10 }
% 190.41/25.09 fresh(fresh(fresh(fresh(true, true, or(r, implies(or(q, r), q))), true, or(r, implies(or(p, or(q, r)), or(p, q)))), true, implies(or(p, or(q, r)), or(r, or(p, q)))), true, implies(or(p, or(q, r)), or(or(p, q), r)))
% 190.41/25.09 = { by axiom 2 (rule_2) }
% 190.41/25.09 fresh(fresh(fresh(true, true, or(r, implies(or(p, or(q, r)), or(p, q)))), true, implies(or(p, or(q, r)), or(r, or(p, q)))), true, implies(or(p, or(q, r)), or(or(p, q), r)))
% 190.41/25.09 = { by axiom 2 (rule_2) }
% 190.41/25.09 fresh(fresh(true, true, implies(or(p, or(q, r)), or(r, or(p, q)))), true, implies(or(p, or(q, r)), or(or(p, q), r)))
% 190.41/25.09 = { by axiom 2 (rule_2) }
% 190.41/25.09 fresh(true, true, implies(or(p, or(q, r)), or(or(p, q), r)))
% 190.41/25.09 = { by axiom 2 (rule_2) }
% 190.41/25.09 true
% 190.41/25.09 % SZS output end Proof
% 190.41/25.09
% 190.41/25.09 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------