TSTP Solution File: LCL041-1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : LCL041-1 : 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 : n011.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 08:17:08 EDT 2023
% Result : Unsatisfiable 0.20s 0.47s
% 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
% 0.07/0.12 % Problem : LCL041-1 : TPTP v8.1.2. Released v1.0.0.
% 0.07/0.13 % 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 : n011.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 : Thu Aug 24 18:12:22 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.20/0.47 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
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% 0.20/0.47 % SZS status Unsatisfiable
% 0.20/0.47
% 0.20/0.48 % SZS output start Proof
% 0.20/0.48 Take the following subset of the input axioms:
% 0.20/0.48 fof(cn_21, axiom, ![X, Y, Z]: is_a_theorem(implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z))))).
% 0.20/0.48 fof(cn_3, axiom, ![X2, Y2]: is_a_theorem(implies(X2, implies(not(X2), Y2)))).
% 0.20/0.48 fof(cn_54, axiom, ![X2, Y2]: is_a_theorem(implies(implies(X2, Y2), implies(implies(not(X2), Y2), Y2)))).
% 0.20/0.48 fof(condensed_detachment, axiom, ![X2, Y2]: (~is_a_theorem(implies(X2, Y2)) | (~is_a_theorem(X2) | is_a_theorem(Y2)))).
% 0.20/0.48 fof(prove_cn_30, negated_conjecture, ~is_a_theorem(implies(implies(a, implies(a, b)), implies(a, b)))).
% 0.20/0.48
% 0.20/0.48 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.48 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.48 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.48 fresh(y, y, x1...xn) = u
% 0.20/0.48 C => fresh(s, t, x1...xn) = v
% 0.20/0.48 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.48 variables of u and v.
% 0.20/0.48 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.48 input problem has no model of domain size 1).
% 0.20/0.48
% 0.20/0.48 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.48
% 0.20/0.48 Axiom 1 (condensed_detachment): fresh2(X, X, Y) = true.
% 0.20/0.48 Axiom 2 (condensed_detachment): fresh(X, X, Y, Z) = is_a_theorem(Z).
% 0.20/0.48 Axiom 3 (cn_3): is_a_theorem(implies(X, implies(not(X), Y))) = true.
% 0.20/0.48 Axiom 4 (condensed_detachment): fresh(is_a_theorem(implies(X, Y)), true, X, Y) = fresh2(is_a_theorem(X), true, Y).
% 0.20/0.48 Axiom 5 (cn_54): is_a_theorem(implies(implies(X, Y), implies(implies(not(X), Y), Y))) = true.
% 0.20/0.48 Axiom 6 (cn_21): is_a_theorem(implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))) = true.
% 0.20/0.48
% 0.20/0.48 Lemma 7: fresh2(is_a_theorem(implies(X, implies(Y, Z))), true, implies(Y, implies(X, Z))) = is_a_theorem(implies(Y, implies(X, Z))).
% 0.20/0.48 Proof:
% 0.20/0.48 fresh2(is_a_theorem(implies(X, implies(Y, Z))), true, implies(Y, implies(X, Z)))
% 0.20/0.48 = { by axiom 4 (condensed_detachment) R->L }
% 0.20/0.48 fresh(is_a_theorem(implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))), true, implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))
% 0.20/0.48 = { by axiom 6 (cn_21) }
% 0.20/0.48 fresh(true, true, implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))
% 0.20/0.48 = { by axiom 2 (condensed_detachment) }
% 0.20/0.48 is_a_theorem(implies(Y, implies(X, Z)))
% 0.20/0.48
% 0.20/0.48 Goal 1 (prove_cn_30): is_a_theorem(implies(implies(a, implies(a, b)), implies(a, b))) = true.
% 0.20/0.48 Proof:
% 0.20/0.48 is_a_theorem(implies(implies(a, implies(a, b)), implies(a, b)))
% 0.20/0.48 = { by axiom 2 (condensed_detachment) R->L }
% 0.20/0.48 fresh(true, true, implies(not(a), implies(a, b)), implies(implies(a, implies(a, b)), implies(a, b)))
% 0.20/0.48 = { by axiom 1 (condensed_detachment) R->L }
% 0.20/0.48 fresh(fresh2(true, true, implies(implies(not(a), implies(a, b)), implies(implies(a, implies(a, b)), implies(a, b)))), true, implies(not(a), implies(a, b)), implies(implies(a, implies(a, b)), implies(a, b)))
% 0.20/0.48 = { by axiom 5 (cn_54) R->L }
% 0.20/0.48 fresh(fresh2(is_a_theorem(implies(implies(a, implies(a, b)), implies(implies(not(a), implies(a, b)), implies(a, b)))), true, implies(implies(not(a), implies(a, b)), implies(implies(a, implies(a, b)), implies(a, b)))), true, implies(not(a), implies(a, b)), implies(implies(a, implies(a, b)), implies(a, b)))
% 0.20/0.48 = { by lemma 7 }
% 0.20/0.48 fresh(is_a_theorem(implies(implies(not(a), implies(a, b)), implies(implies(a, implies(a, b)), implies(a, b)))), true, implies(not(a), implies(a, b)), implies(implies(a, implies(a, b)), implies(a, b)))
% 0.20/0.48 = { by axiom 4 (condensed_detachment) }
% 0.20/0.48 fresh2(is_a_theorem(implies(not(a), implies(a, b))), true, implies(implies(a, implies(a, b)), implies(a, b)))
% 0.20/0.48 = { by lemma 7 R->L }
% 0.20/0.48 fresh2(fresh2(is_a_theorem(implies(a, implies(not(a), b))), true, implies(not(a), implies(a, b))), true, implies(implies(a, implies(a, b)), implies(a, b)))
% 0.20/0.48 = { by axiom 3 (cn_3) }
% 0.20/0.48 fresh2(fresh2(true, true, implies(not(a), implies(a, b))), true, implies(implies(a, implies(a, b)), implies(a, b)))
% 0.20/0.48 = { by axiom 1 (condensed_detachment) }
% 0.20/0.48 fresh2(true, true, implies(implies(a, implies(a, b)), implies(a, b)))
% 0.20/0.48 = { by axiom 1 (condensed_detachment) }
% 0.20/0.48 true
% 0.20/0.48 % SZS output end Proof
% 0.20/0.48
% 0.20/0.48 RESULT: Unsatisfiable (the axioms are contradictory).
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