TSTP Solution File: LCL110-1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL110-1 : TPTP v8.1.2. Released v1.0.0.
% 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 08:17:29 EDT 2023
% Result : Unsatisfiable 0.19s 0.53s
% 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 : LCL110-1 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n025.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Aug 25 01:21:06 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.19/0.53 Command-line arguments: --no-flatten-goal
% 0.19/0.53
% 0.19/0.53 % SZS status Unsatisfiable
% 0.19/0.53
% 0.19/0.54 % SZS output start Proof
% 0.19/0.54 Take the following subset of the input axioms:
% 0.19/0.54 fof(condensed_detachment, axiom, ![X, Y]: (~is_a_theorem(implies(X, Y)) | (~is_a_theorem(X) | is_a_theorem(Y)))).
% 0.19/0.54 fof(mv_1, axiom, ![X2, Y2]: is_a_theorem(implies(X2, implies(Y2, X2)))).
% 0.19/0.54 fof(mv_2, axiom, ![Z, X2, Y2]: is_a_theorem(implies(implies(X2, Y2), implies(implies(Y2, Z), implies(X2, Z))))).
% 0.19/0.54 fof(mv_3, axiom, ![X2, Y2]: is_a_theorem(implies(implies(implies(X2, Y2), Y2), implies(implies(Y2, X2), X2)))).
% 0.19/0.54 fof(mv_5, axiom, ![X2, Y2]: is_a_theorem(implies(implies(not(X2), not(Y2)), implies(Y2, X2)))).
% 0.19/0.54 fof(prove_mv_24, negated_conjecture, ~is_a_theorem(implies(not(not(a)), a))).
% 0.19/0.54
% 0.19/0.54 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.54 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.54 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.54 fresh(y, y, x1...xn) = u
% 0.19/0.54 C => fresh(s, t, x1...xn) = v
% 0.19/0.54 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.54 variables of u and v.
% 0.19/0.54 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.54 input problem has no model of domain size 1).
% 0.19/0.54
% 0.19/0.54 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.54
% 0.19/0.54 Axiom 1 (condensed_detachment): fresh2(X, X, Y) = true.
% 0.19/0.54 Axiom 2 (condensed_detachment): fresh(X, X, Y, Z) = is_a_theorem(Z).
% 0.19/0.54 Axiom 3 (mv_1): is_a_theorem(implies(X, implies(Y, X))) = true.
% 0.19/0.54 Axiom 4 (condensed_detachment): fresh(is_a_theorem(implies(X, Y)), true, X, Y) = fresh2(is_a_theorem(X), true, Y).
% 0.19/0.54 Axiom 5 (mv_5): is_a_theorem(implies(implies(not(X), not(Y)), implies(Y, X))) = true.
% 0.19/0.54 Axiom 6 (mv_2): is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))) = true.
% 0.19/0.54 Axiom 7 (mv_3): is_a_theorem(implies(implies(implies(X, Y), Y), implies(implies(Y, X), X))) = true.
% 0.19/0.54
% 0.19/0.54 Lemma 8: is_a_theorem(implies(implies(implies(X, implies(Y, X)), Z), Z)) = true.
% 0.19/0.54 Proof:
% 0.19/0.54 is_a_theorem(implies(implies(implies(X, implies(Y, X)), Z), Z))
% 0.19/0.54 = { by axiom 2 (condensed_detachment) R->L }
% 0.19/0.54 fresh(true, true, implies(implies(Z, implies(X, implies(Y, X))), implies(X, implies(Y, X))), implies(implies(implies(X, implies(Y, X)), Z), Z))
% 0.19/0.54 = { by axiom 7 (mv_3) R->L }
% 0.19/0.54 fresh(is_a_theorem(implies(implies(implies(Z, implies(X, implies(Y, X))), implies(X, implies(Y, X))), implies(implies(implies(X, implies(Y, X)), Z), Z))), true, implies(implies(Z, implies(X, implies(Y, X))), implies(X, implies(Y, X))), implies(implies(implies(X, implies(Y, X)), Z), Z))
% 0.19/0.54 = { by axiom 4 (condensed_detachment) }
% 0.19/0.55 fresh2(is_a_theorem(implies(implies(Z, implies(X, implies(Y, X))), implies(X, implies(Y, X)))), true, implies(implies(implies(X, implies(Y, X)), Z), Z))
% 0.19/0.55 = { by axiom 2 (condensed_detachment) R->L }
% 0.19/0.55 fresh2(fresh(true, true, implies(X, implies(Y, X)), implies(implies(Z, implies(X, implies(Y, X))), implies(X, implies(Y, X)))), true, implies(implies(implies(X, implies(Y, X)), Z), Z))
% 0.19/0.55 = { by axiom 3 (mv_1) R->L }
% 0.19/0.55 fresh2(fresh(is_a_theorem(implies(implies(X, implies(Y, X)), implies(implies(Z, implies(X, implies(Y, X))), implies(X, implies(Y, X))))), true, implies(X, implies(Y, X)), implies(implies(Z, implies(X, implies(Y, X))), implies(X, implies(Y, X)))), true, implies(implies(implies(X, implies(Y, X)), Z), Z))
% 0.19/0.55 = { by axiom 4 (condensed_detachment) }
% 0.19/0.55 fresh2(fresh2(is_a_theorem(implies(X, implies(Y, X))), true, implies(implies(Z, implies(X, implies(Y, X))), implies(X, implies(Y, X)))), true, implies(implies(implies(X, implies(Y, X)), Z), Z))
% 0.19/0.55 = { by axiom 3 (mv_1) }
% 0.19/0.55 fresh2(fresh2(true, true, implies(implies(Z, implies(X, implies(Y, X))), implies(X, implies(Y, X)))), true, implies(implies(implies(X, implies(Y, X)), Z), Z))
% 0.19/0.55 = { by axiom 1 (condensed_detachment) }
% 0.19/0.55 fresh2(true, true, implies(implies(implies(X, implies(Y, X)), Z), Z))
% 0.19/0.55 = { by axiom 1 (condensed_detachment) }
% 0.19/0.55 true
% 0.19/0.55
% 0.19/0.55 Lemma 9: fresh2(is_a_theorem(implies(X, Y)), true, implies(implies(Y, Z), implies(X, Z))) = is_a_theorem(implies(implies(Y, Z), implies(X, Z))).
% 0.19/0.55 Proof:
% 0.19/0.55 fresh2(is_a_theorem(implies(X, Y)), true, implies(implies(Y, Z), implies(X, Z)))
% 0.19/0.55 = { by axiom 4 (condensed_detachment) R->L }
% 0.19/0.55 fresh(is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))), true, implies(X, Y), implies(implies(Y, Z), implies(X, Z)))
% 0.19/0.55 = { by axiom 6 (mv_2) }
% 0.19/0.55 fresh(true, true, implies(X, Y), implies(implies(Y, Z), implies(X, Z)))
% 0.19/0.55 = { by axiom 2 (condensed_detachment) }
% 0.19/0.55 is_a_theorem(implies(implies(Y, Z), implies(X, Z)))
% 0.19/0.55
% 0.19/0.55 Goal 1 (prove_mv_24): is_a_theorem(implies(not(not(a)), a)) = true.
% 0.19/0.55 Proof:
% 0.19/0.55 is_a_theorem(implies(not(not(a)), a))
% 0.19/0.55 = { by axiom 2 (condensed_detachment) R->L }
% 0.19/0.55 fresh(true, true, implies(implies(not(a), not(implies(X, implies(Y, X)))), a), implies(not(not(a)), a))
% 0.19/0.55 = { by axiom 1 (condensed_detachment) R->L }
% 0.19/0.55 fresh(fresh2(true, true, implies(implies(implies(not(a), not(implies(X, implies(Y, X)))), a), implies(not(not(a)), a))), true, implies(implies(not(a), not(implies(X, implies(Y, X)))), a), implies(not(not(a)), a))
% 0.19/0.55 = { by axiom 1 (condensed_detachment) R->L }
% 0.19/0.55 fresh(fresh2(fresh2(true, true, implies(not(not(a)), implies(not(a), not(implies(X, implies(Y, X)))))), true, implies(implies(implies(not(a), not(implies(X, implies(Y, X)))), a), implies(not(not(a)), a))), true, implies(implies(not(a), not(implies(X, implies(Y, X)))), a), implies(not(not(a)), a))
% 0.19/0.55 = { by axiom 5 (mv_5) R->L }
% 0.19/0.55 fresh(fresh2(fresh2(is_a_theorem(implies(implies(not(not(implies(X, implies(Y, X)))), not(not(a))), implies(not(a), not(implies(X, implies(Y, X)))))), true, implies(not(not(a)), implies(not(a), not(implies(X, implies(Y, X)))))), true, implies(implies(implies(not(a), not(implies(X, implies(Y, X)))), a), implies(not(not(a)), a))), true, implies(implies(not(a), not(implies(X, implies(Y, X)))), a), implies(not(not(a)), a))
% 0.19/0.55 = { by axiom 4 (condensed_detachment) R->L }
% 0.19/0.55 fresh(fresh2(fresh(is_a_theorem(implies(implies(implies(not(not(implies(X, implies(Y, X)))), not(not(a))), implies(not(a), not(implies(X, implies(Y, X))))), implies(not(not(a)), implies(not(a), not(implies(X, implies(Y, X))))))), true, implies(implies(not(not(implies(X, implies(Y, X)))), not(not(a))), implies(not(a), not(implies(X, implies(Y, X))))), implies(not(not(a)), implies(not(a), not(implies(X, implies(Y, X)))))), true, implies(implies(implies(not(a), not(implies(X, implies(Y, X)))), a), implies(not(not(a)), a))), true, implies(implies(not(a), not(implies(X, implies(Y, X)))), a), implies(not(not(a)), a))
% 0.19/0.55 = { by axiom 2 (condensed_detachment) R->L }
% 0.19/0.55 fresh(fresh2(fresh(fresh(true, true, implies(implies(not(not(a)), implies(not(not(implies(X, implies(Y, X)))), not(not(a)))), implies(implies(implies(not(not(implies(X, implies(Y, X)))), not(not(a))), implies(not(a), not(implies(X, implies(Y, X))))), implies(not(not(a)), implies(not(a), not(implies(X, implies(Y, X))))))), implies(implies(implies(not(not(implies(X, implies(Y, X)))), not(not(a))), implies(not(a), not(implies(X, implies(Y, X))))), implies(not(not(a)), implies(not(a), not(implies(X, implies(Y, X))))))), true, implies(implies(not(not(implies(X, implies(Y, X)))), not(not(a))), implies(not(a), not(implies(X, implies(Y, X))))), implies(not(not(a)), implies(not(a), not(implies(X, implies(Y, X)))))), true, implies(implies(implies(not(a), not(implies(X, implies(Y, X)))), a), implies(not(not(a)), a))), true, implies(implies(not(a), not(implies(X, implies(Y, X)))), a), implies(not(not(a)), a))
% 0.19/0.55 = { by lemma 8 R->L }
% 0.19/0.55 fresh(fresh2(fresh(fresh(is_a_theorem(implies(implies(implies(not(not(a)), implies(not(not(implies(X, implies(Y, X)))), not(not(a)))), implies(implies(implies(not(not(implies(X, implies(Y, X)))), not(not(a))), implies(not(a), not(implies(X, implies(Y, X))))), implies(not(not(a)), implies(not(a), not(implies(X, implies(Y, X))))))), implies(implies(implies(not(not(implies(X, implies(Y, X)))), not(not(a))), implies(not(a), not(implies(X, implies(Y, X))))), implies(not(not(a)), implies(not(a), not(implies(X, implies(Y, X)))))))), true, implies(implies(not(not(a)), implies(not(not(implies(X, implies(Y, X)))), not(not(a)))), implies(implies(implies(not(not(implies(X, implies(Y, X)))), not(not(a))), implies(not(a), not(implies(X, implies(Y, X))))), implies(not(not(a)), implies(not(a), not(implies(X, implies(Y, X))))))), implies(implies(implies(not(not(implies(X, implies(Y, X)))), not(not(a))), implies(not(a), not(implies(X, implies(Y, X))))), implies(not(not(a)), implies(not(a), not(implies(X, implies(Y, X))))))), true, implies(implies(not(not(implies(X, implies(Y, X)))), not(not(a))), implies(not(a), not(implies(X, implies(Y, X))))), implies(not(not(a)), implies(not(a), not(implies(X, implies(Y, X)))))), true, implies(implies(implies(not(a), not(implies(X, implies(Y, X)))), a), implies(not(not(a)), a))), true, implies(implies(not(a), not(implies(X, implies(Y, X)))), a), implies(not(not(a)), a))
% 0.19/0.55 = { by axiom 4 (condensed_detachment) }
% 0.19/0.55 fresh(fresh2(fresh(fresh2(is_a_theorem(implies(implies(not(not(a)), implies(not(not(implies(X, implies(Y, X)))), not(not(a)))), implies(implies(implies(not(not(implies(X, implies(Y, X)))), not(not(a))), implies(not(a), not(implies(X, implies(Y, X))))), implies(not(not(a)), implies(not(a), not(implies(X, implies(Y, X)))))))), true, implies(implies(implies(not(not(implies(X, implies(Y, X)))), not(not(a))), implies(not(a), not(implies(X, implies(Y, X))))), implies(not(not(a)), implies(not(a), not(implies(X, implies(Y, X))))))), true, implies(implies(not(not(implies(X, implies(Y, X)))), not(not(a))), implies(not(a), not(implies(X, implies(Y, X))))), implies(not(not(a)), implies(not(a), not(implies(X, implies(Y, X)))))), true, implies(implies(implies(not(a), not(implies(X, implies(Y, X)))), a), implies(not(not(a)), a))), true, implies(implies(not(a), not(implies(X, implies(Y, X)))), a), implies(not(not(a)), a))
% 0.19/0.55 = { by axiom 6 (mv_2) }
% 0.19/0.55 fresh(fresh2(fresh(fresh2(true, true, implies(implies(implies(not(not(implies(X, implies(Y, X)))), not(not(a))), implies(not(a), not(implies(X, implies(Y, X))))), implies(not(not(a)), implies(not(a), not(implies(X, implies(Y, X))))))), true, implies(implies(not(not(implies(X, implies(Y, X)))), not(not(a))), implies(not(a), not(implies(X, implies(Y, X))))), implies(not(not(a)), implies(not(a), not(implies(X, implies(Y, X)))))), true, implies(implies(implies(not(a), not(implies(X, implies(Y, X)))), a), implies(not(not(a)), a))), true, implies(implies(not(a), not(implies(X, implies(Y, X)))), a), implies(not(not(a)), a))
% 0.19/0.55 = { by axiom 1 (condensed_detachment) }
% 0.19/0.55 fresh(fresh2(fresh(true, true, implies(implies(not(not(implies(X, implies(Y, X)))), not(not(a))), implies(not(a), not(implies(X, implies(Y, X))))), implies(not(not(a)), implies(not(a), not(implies(X, implies(Y, X)))))), true, implies(implies(implies(not(a), not(implies(X, implies(Y, X)))), a), implies(not(not(a)), a))), true, implies(implies(not(a), not(implies(X, implies(Y, X)))), a), implies(not(not(a)), a))
% 0.19/0.55 = { by axiom 2 (condensed_detachment) }
% 0.19/0.55 fresh(fresh2(is_a_theorem(implies(not(not(a)), implies(not(a), not(implies(X, implies(Y, X)))))), true, implies(implies(implies(not(a), not(implies(X, implies(Y, X)))), a), implies(not(not(a)), a))), true, implies(implies(not(a), not(implies(X, implies(Y, X)))), a), implies(not(not(a)), a))
% 0.19/0.55 = { by lemma 9 }
% 0.19/0.55 fresh(is_a_theorem(implies(implies(implies(not(a), not(implies(X, implies(Y, X)))), a), implies(not(not(a)), a))), true, implies(implies(not(a), not(implies(X, implies(Y, X)))), a), implies(not(not(a)), a))
% 0.19/0.55 = { by axiom 4 (condensed_detachment) }
% 0.19/0.55 fresh2(is_a_theorem(implies(implies(not(a), not(implies(X, implies(Y, X)))), a)), true, implies(not(not(a)), a))
% 0.19/0.55 = { by axiom 2 (condensed_detachment) R->L }
% 0.19/0.55 fresh2(fresh(true, true, implies(implies(implies(X, implies(Y, X)), a), a), implies(implies(not(a), not(implies(X, implies(Y, X)))), a)), true, implies(not(not(a)), a))
% 0.19/0.55 = { by axiom 1 (condensed_detachment) R->L }
% 0.19/0.55 fresh2(fresh(fresh2(true, true, implies(implies(implies(implies(X, implies(Y, X)), a), a), implies(implies(not(a), not(implies(X, implies(Y, X)))), a))), true, implies(implies(implies(X, implies(Y, X)), a), a), implies(implies(not(a), not(implies(X, implies(Y, X)))), a)), true, implies(not(not(a)), a))
% 0.19/0.55 = { by axiom 5 (mv_5) R->L }
% 0.19/0.55 fresh2(fresh(fresh2(is_a_theorem(implies(implies(not(a), not(implies(X, implies(Y, X)))), implies(implies(X, implies(Y, X)), a))), true, implies(implies(implies(implies(X, implies(Y, X)), a), a), implies(implies(not(a), not(implies(X, implies(Y, X)))), a))), true, implies(implies(implies(X, implies(Y, X)), a), a), implies(implies(not(a), not(implies(X, implies(Y, X)))), a)), true, implies(not(not(a)), a))
% 0.19/0.55 = { by lemma 9 }
% 0.19/0.55 fresh2(fresh(is_a_theorem(implies(implies(implies(implies(X, implies(Y, X)), a), a), implies(implies(not(a), not(implies(X, implies(Y, X)))), a))), true, implies(implies(implies(X, implies(Y, X)), a), a), implies(implies(not(a), not(implies(X, implies(Y, X)))), a)), true, implies(not(not(a)), a))
% 0.19/0.55 = { by axiom 4 (condensed_detachment) }
% 0.19/0.55 fresh2(fresh2(is_a_theorem(implies(implies(implies(X, implies(Y, X)), a), a)), true, implies(implies(not(a), not(implies(X, implies(Y, X)))), a)), true, implies(not(not(a)), a))
% 0.19/0.55 = { by lemma 8 }
% 0.19/0.55 fresh2(fresh2(true, true, implies(implies(not(a), not(implies(X, implies(Y, X)))), a)), true, implies(not(not(a)), a))
% 0.19/0.55 = { by axiom 1 (condensed_detachment) }
% 0.19/0.55 fresh2(true, true, implies(not(not(a)), a))
% 0.19/0.55 = { by axiom 1 (condensed_detachment) }
% 0.19/0.55 true
% 0.19/0.55 % SZS output end Proof
% 0.19/0.55
% 0.19/0.55 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------