TSTP Solution File: LCL189-3 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : LCL189-3 : 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 : n025.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 0.20s 0.66s
% Output : Proof 0.20s
% 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 : LCL189-3 : 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 : n025.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
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Fri Aug 25 01:50:36 EDT 2023
% 0.20/0.34 % CPUTime :
% 0.20/0.66 Command-line arguments: --flip-ordering --lhs-weight 1 --depth-weight 60 --distributivity-heuristic
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% 0.20/0.66 % SZS status Unsatisfiable
% 0.20/0.66
% 0.20/0.67 % SZS output start Proof
% 0.20/0.67 Take the following subset of the input axioms:
% 0.20/0.67 fof(axiom_1_2, axiom, ![A]: axiom(implies(or(A, A), A))).
% 0.20/0.67 fof(axiom_1_3, axiom, ![B, A2]: axiom(implies(A2, or(B, A2)))).
% 0.20/0.67 fof(axiom_1_5, axiom, ![C, A2, B2]: axiom(implies(or(A2, or(B2, C)), or(B2, or(A2, C))))).
% 0.20/0.67 fof(implies_definition, axiom, ![X, Y]: implies(X, Y)=or(not(X), Y)).
% 0.20/0.67 fof(prove_this, negated_conjecture, ~theorem(or(not(p), implies(implies(p, q), q)))).
% 0.20/0.67 fof(rule_1, axiom, ![X2]: (theorem(X2) | ~axiom(X2))).
% 0.20/0.67 fof(rule_2, axiom, ![X2, Y2]: (theorem(X2) | (~theorem(implies(Y2, X2)) | ~theorem(Y2)))).
% 0.20/0.67
% 0.20/0.67 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.67 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.67 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.67 fresh(y, y, x1...xn) = u
% 0.20/0.67 C => fresh(s, t, x1...xn) = v
% 0.20/0.67 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.67 variables of u and v.
% 0.20/0.67 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.67 input problem has no model of domain size 1).
% 0.20/0.67
% 0.20/0.67 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.67
% 0.20/0.67 Axiom 1 (implies_definition): implies(X, Y) = or(not(X), Y).
% 0.20/0.67 Axiom 2 (rule_2): fresh(X, X, Y) = true.
% 0.20/0.67 Axiom 3 (rule_1): fresh2(X, X, Y) = true.
% 0.20/0.67 Axiom 4 (rule_2): fresh3(X, X, Y, Z) = theorem(Y).
% 0.20/0.67 Axiom 5 (rule_1): fresh2(axiom(X), true, X) = theorem(X).
% 0.20/0.67 Axiom 6 (axiom_1_3): axiom(implies(X, or(Y, X))) = true.
% 0.20/0.67 Axiom 7 (axiom_1_2): axiom(implies(or(X, X), X)) = true.
% 0.20/0.67 Axiom 8 (rule_2): fresh3(theorem(implies(X, Y)), true, Y, X) = fresh(theorem(X), true, Y).
% 0.20/0.67 Axiom 9 (axiom_1_5): axiom(implies(or(X, or(Y, Z)), or(Y, or(X, Z)))) = true.
% 0.20/0.67
% 0.20/0.67 Lemma 10: fresh(theorem(implies(X, or(Y, Z))), true, or(Y, implies(X, Z))) = theorem(or(Y, implies(X, Z))).
% 0.20/0.67 Proof:
% 0.20/0.67 fresh(theorem(implies(X, or(Y, Z))), true, or(Y, implies(X, Z)))
% 0.20/0.67 = { by axiom 8 (rule_2) R->L }
% 0.20/0.67 fresh3(theorem(implies(implies(X, or(Y, Z)), or(Y, implies(X, Z)))), true, or(Y, implies(X, Z)), implies(X, or(Y, Z)))
% 0.20/0.67 = { by axiom 5 (rule_1) R->L }
% 0.20/0.67 fresh3(fresh2(axiom(implies(implies(X, or(Y, Z)), or(Y, implies(X, Z)))), true, implies(implies(X, or(Y, Z)), or(Y, implies(X, Z)))), true, or(Y, implies(X, Z)), implies(X, or(Y, Z)))
% 0.20/0.67 = { by axiom 1 (implies_definition) }
% 0.20/0.67 fresh3(fresh2(axiom(implies(implies(X, or(Y, Z)), or(Y, or(not(X), Z)))), true, implies(implies(X, or(Y, Z)), or(Y, implies(X, Z)))), true, or(Y, implies(X, Z)), implies(X, or(Y, Z)))
% 0.20/0.67 = { by axiom 1 (implies_definition) }
% 0.20/0.67 fresh3(fresh2(axiom(implies(or(not(X), or(Y, Z)), or(Y, or(not(X), Z)))), true, implies(implies(X, or(Y, Z)), or(Y, implies(X, Z)))), true, or(Y, implies(X, Z)), implies(X, or(Y, Z)))
% 0.20/0.67 = { by axiom 9 (axiom_1_5) }
% 0.20/0.67 fresh3(fresh2(true, true, implies(implies(X, or(Y, Z)), or(Y, implies(X, Z)))), true, or(Y, implies(X, Z)), implies(X, or(Y, Z)))
% 0.20/0.67 = { by axiom 3 (rule_1) }
% 0.20/0.67 fresh3(true, true, or(Y, implies(X, Z)), implies(X, or(Y, Z)))
% 0.20/0.67 = { by axiom 4 (rule_2) }
% 0.20/0.67 theorem(or(Y, implies(X, Z)))
% 0.20/0.67
% 0.20/0.67 Goal 1 (prove_this): theorem(or(not(p), implies(implies(p, q), q))) = true.
% 0.20/0.67 Proof:
% 0.20/0.67 theorem(or(not(p), implies(implies(p, q), q)))
% 0.20/0.67 = { by lemma 10 R->L }
% 0.20/0.67 fresh(theorem(implies(implies(p, q), or(not(p), q))), true, or(not(p), implies(implies(p, q), q)))
% 0.20/0.67 = { by axiom 1 (implies_definition) R->L }
% 0.20/0.67 fresh(theorem(implies(implies(p, q), implies(p, q))), true, or(not(p), implies(implies(p, q), q)))
% 0.20/0.67 = { by axiom 4 (rule_2) R->L }
% 0.20/0.67 fresh(fresh3(true, true, implies(implies(p, q), implies(p, q)), or(implies(implies(p, q), implies(p, q)), implies(implies(p, q), implies(p, q)))), true, or(not(p), implies(implies(p, q), q)))
% 0.20/0.67 = { by axiom 3 (rule_1) R->L }
% 0.20/0.67 fresh(fresh3(fresh2(true, true, implies(or(implies(implies(p, q), implies(p, q)), implies(implies(p, q), implies(p, q))), implies(implies(p, q), implies(p, q)))), true, implies(implies(p, q), implies(p, q)), or(implies(implies(p, q), implies(p, q)), implies(implies(p, q), implies(p, q)))), true, or(not(p), implies(implies(p, q), q)))
% 0.20/0.67 = { by axiom 7 (axiom_1_2) R->L }
% 0.20/0.67 fresh(fresh3(fresh2(axiom(implies(or(implies(implies(p, q), implies(p, q)), implies(implies(p, q), implies(p, q))), implies(implies(p, q), implies(p, q)))), true, implies(or(implies(implies(p, q), implies(p, q)), implies(implies(p, q), implies(p, q))), implies(implies(p, q), implies(p, q)))), true, implies(implies(p, q), implies(p, q)), or(implies(implies(p, q), implies(p, q)), implies(implies(p, q), implies(p, q)))), true, or(not(p), implies(implies(p, q), q)))
% 0.20/0.67 = { by axiom 5 (rule_1) }
% 0.20/0.67 fresh(fresh3(theorem(implies(or(implies(implies(p, q), implies(p, q)), implies(implies(p, q), implies(p, q))), implies(implies(p, q), implies(p, q)))), true, implies(implies(p, q), implies(p, q)), or(implies(implies(p, q), implies(p, q)), implies(implies(p, q), implies(p, q)))), true, or(not(p), implies(implies(p, q), q)))
% 0.20/0.67 = { by axiom 8 (rule_2) }
% 0.20/0.67 fresh(fresh(theorem(or(implies(implies(p, q), implies(p, q)), implies(implies(p, q), implies(p, q)))), true, implies(implies(p, q), implies(p, q))), true, or(not(p), implies(implies(p, q), q)))
% 0.20/0.67 = { by lemma 10 R->L }
% 0.20/0.67 fresh(fresh(fresh(theorem(implies(implies(p, q), or(implies(implies(p, q), implies(p, q)), implies(p, q)))), true, or(implies(implies(p, q), implies(p, q)), implies(implies(p, q), implies(p, q)))), true, implies(implies(p, q), implies(p, q))), true, or(not(p), implies(implies(p, q), q)))
% 0.20/0.67 = { by axiom 5 (rule_1) R->L }
% 0.20/0.67 fresh(fresh(fresh(fresh2(axiom(implies(implies(p, q), or(implies(implies(p, q), implies(p, q)), implies(p, q)))), true, implies(implies(p, q), or(implies(implies(p, q), implies(p, q)), implies(p, q)))), true, or(implies(implies(p, q), implies(p, q)), implies(implies(p, q), implies(p, q)))), true, implies(implies(p, q), implies(p, q))), true, or(not(p), implies(implies(p, q), q)))
% 0.20/0.67 = { by axiom 6 (axiom_1_3) }
% 0.20/0.67 fresh(fresh(fresh(fresh2(true, true, implies(implies(p, q), or(implies(implies(p, q), implies(p, q)), implies(p, q)))), true, or(implies(implies(p, q), implies(p, q)), implies(implies(p, q), implies(p, q)))), true, implies(implies(p, q), implies(p, q))), true, or(not(p), implies(implies(p, q), q)))
% 0.20/0.67 = { by axiom 3 (rule_1) }
% 0.20/0.67 fresh(fresh(fresh(true, true, or(implies(implies(p, q), implies(p, q)), implies(implies(p, q), implies(p, q)))), true, implies(implies(p, q), implies(p, q))), true, or(not(p), implies(implies(p, q), q)))
% 0.20/0.67 = { by axiom 2 (rule_2) }
% 0.20/0.67 fresh(fresh(true, true, implies(implies(p, q), implies(p, q))), true, or(not(p), implies(implies(p, q), q)))
% 0.20/0.67 = { by axiom 2 (rule_2) }
% 0.20/0.67 fresh(true, true, or(not(p), implies(implies(p, q), q)))
% 0.20/0.67 = { by axiom 2 (rule_2) }
% 0.20/0.67 true
% 0.20/0.67 % SZS output end Proof
% 0.20/0.67
% 0.20/0.67 RESULT: Unsatisfiable (the axioms are contradictory).
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