TSTP Solution File: LCL186-3 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : LCL186-3 : 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 : n028.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 08:17:51 EDT 2023
% Result : Unsatisfiable 0.19s 0.80s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LCL186-3 : TPTP v8.1.2. Released v2.3.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.33 % Computer : n028.cluster.edu
% 0.14/0.33 % Model : x86_64 x86_64
% 0.14/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.33 % Memory : 8042.1875MB
% 0.14/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.33 % CPULimit : 300
% 0.14/0.33 % WCLimit : 300
% 0.14/0.33 % DateTime : Thu Aug 24 17:20:35 EDT 2023
% 0.14/0.33 % CPUTime :
% 0.19/0.80 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.19/0.80
% 0.19/0.80 % SZS status Unsatisfiable
% 0.19/0.81
% 0.19/0.82 % SZS output start Proof
% 0.19/0.82 Take the following subset of the input axioms:
% 0.19/0.82 fof(axiom_1_3, axiom, ![A, B]: axiom(implies(A, or(B, A)))).
% 0.19/0.82 fof(axiom_1_4, axiom, ![A2, B2]: axiom(implies(or(A2, B2), or(B2, A2)))).
% 0.19/0.82 fof(axiom_1_6, axiom, ![C, A2, B2]: axiom(implies(implies(A2, B2), implies(or(C, A2), or(C, B2))))).
% 0.19/0.82 fof(implies_definition, axiom, ![X, Y]: implies(X, Y)=or(not(X), Y)).
% 0.19/0.82 fof(prove_this, negated_conjecture, ~theorem(implies(not(p), implies(p, q)))).
% 0.19/0.82 fof(rule_1, axiom, ![X2]: (theorem(X2) | ~axiom(X2))).
% 0.19/0.82 fof(rule_2, axiom, ![X2, Y2]: (theorem(X2) | (~theorem(implies(Y2, X2)) | ~theorem(Y2)))).
% 0.19/0.82
% 0.19/0.82 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.82 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.82 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.82 fresh(y, y, x1...xn) = u
% 0.19/0.82 C => fresh(s, t, x1...xn) = v
% 0.19/0.82 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.82 variables of u and v.
% 0.19/0.82 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.82 input problem has no model of domain size 1).
% 0.19/0.82
% 0.19/0.82 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.82
% 0.19/0.82 Axiom 1 (implies_definition): implies(X, Y) = or(not(X), Y).
% 0.19/0.82 Axiom 2 (rule_2): fresh(X, X, Y) = true.
% 0.19/0.82 Axiom 3 (rule_1): fresh2(X, X, Y) = true.
% 0.19/0.82 Axiom 4 (rule_2): fresh3(X, X, Y, Z) = theorem(Y).
% 0.19/0.82 Axiom 5 (rule_1): fresh2(axiom(X), true, X) = theorem(X).
% 0.19/0.82 Axiom 6 (axiom_1_3): axiom(implies(X, or(Y, X))) = true.
% 0.19/0.82 Axiom 7 (axiom_1_4): axiom(implies(or(X, Y), or(Y, X))) = true.
% 0.19/0.82 Axiom 8 (rule_2): fresh3(theorem(implies(X, Y)), true, Y, X) = fresh(theorem(X), true, Y).
% 0.19/0.82 Axiom 9 (axiom_1_6): axiom(implies(implies(X, Y), implies(or(Z, X), or(Z, Y)))) = true.
% 0.19/0.82
% 0.19/0.82 Goal 1 (prove_this): theorem(implies(not(p), implies(p, q))) = true.
% 0.19/0.82 Proof:
% 0.19/0.82 theorem(implies(not(p), implies(p, q)))
% 0.19/0.82 = { by axiom 4 (rule_2) R->L }
% 0.19/0.82 fresh3(true, true, implies(not(p), implies(p, q)), implies(not(p), or(q, not(p))))
% 0.19/0.82 = { by axiom 2 (rule_2) R->L }
% 0.19/0.82 fresh3(fresh(true, true, implies(or(not(not(p)), or(q, not(p))), or(not(not(p)), implies(p, q)))), true, implies(not(p), implies(p, q)), implies(not(p), or(q, not(p))))
% 0.19/0.82 = { by axiom 3 (rule_1) R->L }
% 0.19/0.82 fresh3(fresh(fresh2(true, true, implies(or(q, not(p)), implies(p, q))), true, implies(or(not(not(p)), or(q, not(p))), or(not(not(p)), implies(p, q)))), true, implies(not(p), implies(p, q)), implies(not(p), or(q, not(p))))
% 0.19/0.82 = { by axiom 7 (axiom_1_4) R->L }
% 0.19/0.82 fresh3(fresh(fresh2(axiom(implies(or(q, not(p)), or(not(p), q))), true, implies(or(q, not(p)), implies(p, q))), true, implies(or(not(not(p)), or(q, not(p))), or(not(not(p)), implies(p, q)))), true, implies(not(p), implies(p, q)), implies(not(p), or(q, not(p))))
% 0.19/0.82 = { by axiom 1 (implies_definition) R->L }
% 0.19/0.82 fresh3(fresh(fresh2(axiom(implies(or(q, not(p)), implies(p, q))), true, implies(or(q, not(p)), implies(p, q))), true, implies(or(not(not(p)), or(q, not(p))), or(not(not(p)), implies(p, q)))), true, implies(not(p), implies(p, q)), implies(not(p), or(q, not(p))))
% 0.19/0.82 = { by axiom 5 (rule_1) }
% 0.19/0.82 fresh3(fresh(theorem(implies(or(q, not(p)), implies(p, q))), true, implies(or(not(not(p)), or(q, not(p))), or(not(not(p)), implies(p, q)))), true, implies(not(p), implies(p, q)), implies(not(p), or(q, not(p))))
% 0.19/0.82 = { by axiom 8 (rule_2) R->L }
% 0.19/0.82 fresh3(fresh3(theorem(implies(implies(or(q, not(p)), implies(p, q)), implies(or(not(not(p)), or(q, not(p))), or(not(not(p)), implies(p, q))))), true, implies(or(not(not(p)), or(q, not(p))), or(not(not(p)), implies(p, q))), implies(or(q, not(p)), implies(p, q))), true, implies(not(p), implies(p, q)), implies(not(p), or(q, not(p))))
% 0.19/0.82 = { by axiom 5 (rule_1) R->L }
% 0.19/0.82 fresh3(fresh3(fresh2(axiom(implies(implies(or(q, not(p)), implies(p, q)), implies(or(not(not(p)), or(q, not(p))), or(not(not(p)), implies(p, q))))), true, implies(implies(or(q, not(p)), implies(p, q)), implies(or(not(not(p)), or(q, not(p))), or(not(not(p)), implies(p, q))))), true, implies(or(not(not(p)), or(q, not(p))), or(not(not(p)), implies(p, q))), implies(or(q, not(p)), implies(p, q))), true, implies(not(p), implies(p, q)), implies(not(p), or(q, not(p))))
% 0.19/0.82 = { by axiom 9 (axiom_1_6) }
% 0.19/0.82 fresh3(fresh3(fresh2(true, true, implies(implies(or(q, not(p)), implies(p, q)), implies(or(not(not(p)), or(q, not(p))), or(not(not(p)), implies(p, q))))), true, implies(or(not(not(p)), or(q, not(p))), or(not(not(p)), implies(p, q))), implies(or(q, not(p)), implies(p, q))), true, implies(not(p), implies(p, q)), implies(not(p), or(q, not(p))))
% 0.19/0.82 = { by axiom 3 (rule_1) }
% 0.19/0.82 fresh3(fresh3(true, true, implies(or(not(not(p)), or(q, not(p))), or(not(not(p)), implies(p, q))), implies(or(q, not(p)), implies(p, q))), true, implies(not(p), implies(p, q)), implies(not(p), or(q, not(p))))
% 0.19/0.82 = { by axiom 4 (rule_2) }
% 0.19/0.82 fresh3(theorem(implies(or(not(not(p)), or(q, not(p))), or(not(not(p)), implies(p, q)))), true, implies(not(p), implies(p, q)), implies(not(p), or(q, not(p))))
% 0.19/0.82 = { by axiom 1 (implies_definition) R->L }
% 0.19/0.82 fresh3(theorem(implies(implies(not(p), or(q, not(p))), or(not(not(p)), implies(p, q)))), true, implies(not(p), implies(p, q)), implies(not(p), or(q, not(p))))
% 0.19/0.82 = { by axiom 1 (implies_definition) R->L }
% 0.19/0.82 fresh3(theorem(implies(implies(not(p), or(q, not(p))), implies(not(p), implies(p, q)))), true, implies(not(p), implies(p, q)), implies(not(p), or(q, not(p))))
% 0.19/0.82 = { by axiom 8 (rule_2) }
% 0.19/0.82 fresh(theorem(implies(not(p), or(q, not(p)))), true, implies(not(p), implies(p, q)))
% 0.19/0.82 = { by axiom 5 (rule_1) R->L }
% 0.19/0.82 fresh(fresh2(axiom(implies(not(p), or(q, not(p)))), true, implies(not(p), or(q, not(p)))), true, implies(not(p), implies(p, q)))
% 0.19/0.82 = { by axiom 6 (axiom_1_3) }
% 0.19/0.82 fresh(fresh2(true, true, implies(not(p), or(q, not(p)))), true, implies(not(p), implies(p, q)))
% 0.19/0.82 = { by axiom 3 (rule_1) }
% 0.19/0.82 fresh(true, true, implies(not(p), implies(p, q)))
% 0.19/0.82 = { by axiom 2 (rule_2) }
% 0.19/0.82 true
% 0.19/0.82 % SZS output end Proof
% 0.19/0.82
% 0.19/0.82 RESULT: Unsatisfiable (the axioms are contradictory).
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