TSTP Solution File: LCL027-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL027-1 : 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 : n005.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 08:17:05 EDT 2023
% Result : Unsatisfiable 0.24s 0.43s
% Output : Proof 0.24s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.12 % Problem : LCL027-1 : TPTP v8.1.2. Released v1.0.0.
% 0.10/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.37 % Computer : n005.cluster.edu
% 0.14/0.37 % Model : x86_64 x86_64
% 0.14/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.37 % Memory : 8042.1875MB
% 0.14/0.37 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.37 % CPULimit : 300
% 0.14/0.37 % WCLimit : 300
% 0.14/0.37 % DateTime : Fri Aug 25 06:54:38 EDT 2023
% 0.14/0.37 % CPUTime :
% 0.24/0.43 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.24/0.43
% 0.24/0.43 % SZS status Unsatisfiable
% 0.24/0.43
% 0.24/0.44 % SZS output start Proof
% 0.24/0.44 Take the following subset of the input axioms:
% 0.24/0.44 fof(c0_2, axiom, ![X, Y]: is_a_theorem(implies(X, implies(Y, X)))).
% 0.24/0.44 fof(c0_5, axiom, ![X2]: is_a_theorem(implies(implies(implies(X2, falsehood), falsehood), X2))).
% 0.24/0.44 fof(c0_6, axiom, ![Z, X2, Y2]: is_a_theorem(implies(implies(X2, implies(Y2, Z)), implies(implies(X2, Y2), implies(X2, Z))))).
% 0.24/0.44 fof(condensed_detachment, axiom, ![X2, Y2]: (~is_a_theorem(implies(X2, Y2)) | (~is_a_theorem(X2) | is_a_theorem(Y2)))).
% 0.24/0.44 fof(prove_c0_4, negated_conjecture, ~is_a_theorem(implies(falsehood, a))).
% 0.24/0.44
% 0.24/0.44 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.24/0.44 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.24/0.44 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.24/0.44 fresh(y, y, x1...xn) = u
% 0.24/0.44 C => fresh(s, t, x1...xn) = v
% 0.24/0.44 where fresh is a fresh function symbol and x1..xn are the free
% 0.24/0.44 variables of u and v.
% 0.24/0.44 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.24/0.44 input problem has no model of domain size 1).
% 0.24/0.44
% 0.24/0.44 The encoding turns the above axioms into the following unit equations and goals:
% 0.24/0.44
% 0.24/0.44 Axiom 1 (condensed_detachment): fresh2(X, X, Y) = true.
% 0.24/0.44 Axiom 2 (condensed_detachment): fresh(X, X, Y, Z) = is_a_theorem(Z).
% 0.24/0.44 Axiom 3 (c0_2): is_a_theorem(implies(X, implies(Y, X))) = true.
% 0.24/0.44 Axiom 4 (c0_5): is_a_theorem(implies(implies(implies(X, falsehood), falsehood), X)) = true.
% 0.24/0.44 Axiom 5 (condensed_detachment): fresh(is_a_theorem(implies(X, Y)), true, X, Y) = fresh2(is_a_theorem(X), true, Y).
% 0.24/0.44 Axiom 6 (c0_6): is_a_theorem(implies(implies(X, implies(Y, Z)), implies(implies(X, Y), implies(X, Z)))) = true.
% 0.24/0.44
% 0.24/0.44 Goal 1 (prove_c0_4): is_a_theorem(implies(falsehood, a)) = true.
% 0.24/0.44 Proof:
% 0.24/0.44 is_a_theorem(implies(falsehood, a))
% 0.24/0.44 = { by axiom 2 (condensed_detachment) R->L }
% 0.24/0.44 fresh(true, true, implies(falsehood, implies(implies(a, falsehood), falsehood)), implies(falsehood, a))
% 0.24/0.44 = { by axiom 1 (condensed_detachment) R->L }
% 0.24/0.44 fresh(fresh2(true, true, implies(implies(falsehood, implies(implies(a, falsehood), falsehood)), implies(falsehood, a))), true, implies(falsehood, implies(implies(a, falsehood), falsehood)), implies(falsehood, a))
% 0.24/0.44 = { by axiom 1 (condensed_detachment) R->L }
% 0.24/0.44 fresh(fresh2(fresh2(true, true, implies(falsehood, implies(implies(implies(a, falsehood), falsehood), a))), true, implies(implies(falsehood, implies(implies(a, falsehood), falsehood)), implies(falsehood, a))), true, implies(falsehood, implies(implies(a, falsehood), falsehood)), implies(falsehood, a))
% 0.24/0.44 = { by axiom 4 (c0_5) R->L }
% 0.24/0.44 fresh(fresh2(fresh2(is_a_theorem(implies(implies(implies(a, falsehood), falsehood), a)), true, implies(falsehood, implies(implies(implies(a, falsehood), falsehood), a))), true, implies(implies(falsehood, implies(implies(a, falsehood), falsehood)), implies(falsehood, a))), true, implies(falsehood, implies(implies(a, falsehood), falsehood)), implies(falsehood, a))
% 0.24/0.44 = { by axiom 5 (condensed_detachment) R->L }
% 0.24/0.44 fresh(fresh2(fresh(is_a_theorem(implies(implies(implies(implies(a, falsehood), falsehood), a), implies(falsehood, implies(implies(implies(a, falsehood), falsehood), a)))), true, implies(implies(implies(a, falsehood), falsehood), a), implies(falsehood, implies(implies(implies(a, falsehood), falsehood), a))), true, implies(implies(falsehood, implies(implies(a, falsehood), falsehood)), implies(falsehood, a))), true, implies(falsehood, implies(implies(a, falsehood), falsehood)), implies(falsehood, a))
% 0.24/0.44 = { by axiom 3 (c0_2) }
% 0.24/0.44 fresh(fresh2(fresh(true, true, implies(implies(implies(a, falsehood), falsehood), a), implies(falsehood, implies(implies(implies(a, falsehood), falsehood), a))), true, implies(implies(falsehood, implies(implies(a, falsehood), falsehood)), implies(falsehood, a))), true, implies(falsehood, implies(implies(a, falsehood), falsehood)), implies(falsehood, a))
% 0.24/0.44 = { by axiom 2 (condensed_detachment) }
% 0.24/0.44 fresh(fresh2(is_a_theorem(implies(falsehood, implies(implies(implies(a, falsehood), falsehood), a))), true, implies(implies(falsehood, implies(implies(a, falsehood), falsehood)), implies(falsehood, a))), true, implies(falsehood, implies(implies(a, falsehood), falsehood)), implies(falsehood, a))
% 0.24/0.44 = { by axiom 5 (condensed_detachment) R->L }
% 0.24/0.44 fresh(fresh(is_a_theorem(implies(implies(falsehood, implies(implies(implies(a, falsehood), falsehood), a)), implies(implies(falsehood, implies(implies(a, falsehood), falsehood)), implies(falsehood, a)))), true, implies(falsehood, implies(implies(implies(a, falsehood), falsehood), a)), implies(implies(falsehood, implies(implies(a, falsehood), falsehood)), implies(falsehood, a))), true, implies(falsehood, implies(implies(a, falsehood), falsehood)), implies(falsehood, a))
% 0.24/0.44 = { by axiom 6 (c0_6) }
% 0.24/0.44 fresh(fresh(true, true, implies(falsehood, implies(implies(implies(a, falsehood), falsehood), a)), implies(implies(falsehood, implies(implies(a, falsehood), falsehood)), implies(falsehood, a))), true, implies(falsehood, implies(implies(a, falsehood), falsehood)), implies(falsehood, a))
% 0.24/0.44 = { by axiom 2 (condensed_detachment) }
% 0.24/0.44 fresh(is_a_theorem(implies(implies(falsehood, implies(implies(a, falsehood), falsehood)), implies(falsehood, a))), true, implies(falsehood, implies(implies(a, falsehood), falsehood)), implies(falsehood, a))
% 0.24/0.44 = { by axiom 5 (condensed_detachment) }
% 0.24/0.44 fresh2(is_a_theorem(implies(falsehood, implies(implies(a, falsehood), falsehood))), true, implies(falsehood, a))
% 0.24/0.44 = { by axiom 3 (c0_2) }
% 0.24/0.44 fresh2(true, true, implies(falsehood, a))
% 0.24/0.44 = { by axiom 1 (condensed_detachment) }
% 0.24/0.44 true
% 0.24/0.44 % SZS output end Proof
% 0.24/0.44
% 0.24/0.44 RESULT: Unsatisfiable (the axioms are contradictory).
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