TSTP Solution File: LCL046-1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : LCL046-1 : 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 : 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 08:17:09 EDT 2023
% Result : Unsatisfiable 0.18s 0.40s
% Output : Proof 0.18s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : LCL046-1 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.18/0.36 % Computer : n016.cluster.edu
% 0.18/0.36 % Model : x86_64 x86_64
% 0.18/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.36 % Memory : 8042.1875MB
% 0.18/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.18/0.36 % CPULimit : 300
% 0.18/0.36 % WCLimit : 300
% 0.18/0.36 % DateTime : Fri Aug 25 06:27:55 EDT 2023
% 0.18/0.36 % CPUTime :
% 0.18/0.40 Command-line arguments: --no-flatten-goal
% 0.18/0.40
% 0.18/0.40 % SZS status Unsatisfiable
% 0.18/0.40
% 0.18/0.41 % SZS output start Proof
% 0.18/0.41 Take the following subset of the input axioms:
% 0.18/0.41 fof(cn_1, axiom, ![X, Y, Z]: is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))))).
% 0.18/0.41 fof(cn_2, axiom, ![X2]: is_a_theorem(implies(implies(not(X2), X2), X2))).
% 0.18/0.41 fof(cn_3, axiom, ![X2, Y2]: is_a_theorem(implies(X2, implies(not(X2), Y2)))).
% 0.18/0.41 fof(condensed_detachment, axiom, ![X2, Y2]: (~is_a_theorem(implies(X2, Y2)) | (~is_a_theorem(X2) | is_a_theorem(Y2)))).
% 0.18/0.41 fof(prove_cn_16, negated_conjecture, ~is_a_theorem(implies(a, a))).
% 0.18/0.41
% 0.18/0.41 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.18/0.41 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.18/0.41 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.18/0.41 fresh(y, y, x1...xn) = u
% 0.18/0.41 C => fresh(s, t, x1...xn) = v
% 0.18/0.41 where fresh is a fresh function symbol and x1..xn are the free
% 0.18/0.41 variables of u and v.
% 0.18/0.41 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.18/0.41 input problem has no model of domain size 1).
% 0.18/0.41
% 0.18/0.41 The encoding turns the above axioms into the following unit equations and goals:
% 0.18/0.41
% 0.18/0.41 Axiom 1 (condensed_detachment): fresh2(X, X, Y) = true.
% 0.18/0.41 Axiom 2 (condensed_detachment): fresh(X, X, Y, Z) = is_a_theorem(Z).
% 0.18/0.41 Axiom 3 (cn_3): is_a_theorem(implies(X, implies(not(X), Y))) = true.
% 0.18/0.41 Axiom 4 (cn_2): is_a_theorem(implies(implies(not(X), X), X)) = true.
% 0.18/0.41 Axiom 5 (condensed_detachment): fresh(is_a_theorem(implies(X, Y)), true, X, Y) = fresh2(is_a_theorem(X), true, Y).
% 0.18/0.41 Axiom 6 (cn_1): is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))) = true.
% 0.18/0.41
% 0.18/0.41 Goal 1 (prove_cn_16): is_a_theorem(implies(a, a)) = true.
% 0.18/0.41 Proof:
% 0.18/0.41 is_a_theorem(implies(a, a))
% 0.18/0.41 = { by axiom 2 (condensed_detachment) R->L }
% 0.18/0.41 fresh(true, true, implies(implies(not(a), a), a), implies(a, a))
% 0.18/0.41 = { by axiom 1 (condensed_detachment) R->L }
% 0.18/0.41 fresh(fresh2(true, true, implies(implies(implies(not(a), a), a), implies(a, a))), true, implies(implies(not(a), a), a), implies(a, a))
% 0.18/0.41 = { by axiom 3 (cn_3) R->L }
% 0.18/0.41 fresh(fresh2(is_a_theorem(implies(a, implies(not(a), a))), true, implies(implies(implies(not(a), a), a), implies(a, a))), true, implies(implies(not(a), a), a), implies(a, a))
% 0.18/0.41 = { by axiom 5 (condensed_detachment) R->L }
% 0.18/0.41 fresh(fresh(is_a_theorem(implies(implies(a, implies(not(a), a)), implies(implies(implies(not(a), a), a), implies(a, a)))), true, implies(a, implies(not(a), a)), implies(implies(implies(not(a), a), a), implies(a, a))), true, implies(implies(not(a), a), a), implies(a, a))
% 0.18/0.41 = { by axiom 6 (cn_1) }
% 0.18/0.41 fresh(fresh(true, true, implies(a, implies(not(a), a)), implies(implies(implies(not(a), a), a), implies(a, a))), true, implies(implies(not(a), a), a), implies(a, a))
% 0.18/0.41 = { by axiom 2 (condensed_detachment) }
% 0.18/0.41 fresh(is_a_theorem(implies(implies(implies(not(a), a), a), implies(a, a))), true, implies(implies(not(a), a), a), implies(a, a))
% 0.18/0.41 = { by axiom 5 (condensed_detachment) }
% 0.18/0.41 fresh2(is_a_theorem(implies(implies(not(a), a), a)), true, implies(a, a))
% 0.18/0.41 = { by axiom 4 (cn_2) }
% 0.18/0.41 fresh2(true, true, implies(a, a))
% 0.18/0.41 = { by axiom 1 (condensed_detachment) }
% 0.18/0.41 true
% 0.18/0.41 % SZS output end Proof
% 0.18/0.41
% 0.18/0.41 RESULT: Unsatisfiable (the axioms are contradictory).
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