TSTP Solution File: GRP043-2 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : GRP043-2 : 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 : n032.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 01:16:45 EDT 2023
% Result : Unsatisfiable 0.15s 0.36s
% Output : Proof 0.15s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.10 % Problem : GRP043-2 : TPTP v8.1.2. Released v1.0.0.
% 0.06/0.11 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.11/0.31 % Computer : n032.cluster.edu
% 0.11/0.31 % Model : x86_64 x86_64
% 0.11/0.31 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.31 % Memory : 8042.1875MB
% 0.11/0.31 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.31 % CPULimit : 300
% 0.11/0.31 % WCLimit : 300
% 0.11/0.31 % DateTime : Mon Aug 28 22:21:16 EDT 2023
% 0.11/0.31 % CPUTime :
% 0.15/0.36 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
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% 0.15/0.36 % SZS status Unsatisfiable
% 0.15/0.36
% 0.15/0.36 % SZS output start Proof
% 0.15/0.36 Take the following subset of the input axioms:
% 0.15/0.36 fof(a_equals_b, hypothesis, equalish(a, b)).
% 0.15/0.36 fof(b_equals_c, hypothesis, equalish(b, c)).
% 0.15/0.36 fof(left_identity, axiom, ![X]: product(identity, X, X)).
% 0.15/0.36 fof(product_substitution3, axiom, ![Y, Z, W, X2]: (~equalish(X2, Y) | (~product(W, Z, X2) | product(W, Z, Y)))).
% 0.15/0.36 fof(prove_transitivity, negated_conjecture, ~equalish(a, c)).
% 0.15/0.36 fof(total_function2, axiom, ![X2, Y2, Z2, W2]: (~product(X2, Y2, Z2) | (~product(X2, Y2, W2) | equalish(Z2, W2)))).
% 0.15/0.36
% 0.15/0.36 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.15/0.36 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.15/0.36 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.15/0.36 fresh(y, y, x1...xn) = u
% 0.15/0.36 C => fresh(s, t, x1...xn) = v
% 0.15/0.36 where fresh is a fresh function symbol and x1..xn are the free
% 0.15/0.36 variables of u and v.
% 0.15/0.36 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.15/0.36 input problem has no model of domain size 1).
% 0.15/0.36
% 0.15/0.36 The encoding turns the above axioms into the following unit equations and goals:
% 0.15/0.36
% 0.15/0.36 Axiom 1 (a_equals_b): equalish(a, b) = true.
% 0.15/0.36 Axiom 2 (b_equals_c): equalish(b, c) = true.
% 0.15/0.36 Axiom 3 (left_identity): product(identity, X, X) = true.
% 0.15/0.36 Axiom 4 (total_function2): fresh(X, X, Y, Z) = true.
% 0.15/0.36 Axiom 5 (product_substitution3): fresh3(X, X, Y, Z, W) = true.
% 0.15/0.36 Axiom 6 (product_substitution3): fresh4(X, X, Y, Z, W, V) = product(W, V, Z).
% 0.15/0.36 Axiom 7 (total_function2): fresh2(X, X, Y, Z, W, V) = equalish(W, V).
% 0.15/0.36 Axiom 8 (product_substitution3): fresh4(equalish(X, Y), true, X, Y, Z, W) = fresh3(product(Z, W, X), true, Y, Z, W).
% 0.15/0.36 Axiom 9 (total_function2): fresh2(product(X, Y, Z), true, X, Y, W, Z) = fresh(product(X, Y, W), true, W, Z).
% 0.15/0.36
% 0.15/0.36 Lemma 10: fresh4(equalish(X, Y), true, X, Y, identity, X) = true.
% 0.15/0.36 Proof:
% 0.15/0.36 fresh4(equalish(X, Y), true, X, Y, identity, X)
% 0.15/0.36 = { by axiom 8 (product_substitution3) }
% 0.15/0.36 fresh3(product(identity, X, X), true, Y, identity, X)
% 0.15/0.36 = { by axiom 3 (left_identity) }
% 0.15/0.36 fresh3(true, true, Y, identity, X)
% 0.15/0.36 = { by axiom 5 (product_substitution3) }
% 0.15/0.36 true
% 0.15/0.36
% 0.15/0.36 Goal 1 (prove_transitivity): equalish(a, c) = true.
% 0.15/0.36 Proof:
% 0.15/0.36 equalish(a, c)
% 0.15/0.36 = { by axiom 7 (total_function2) R->L }
% 0.15/0.36 fresh2(true, true, identity, b, a, c)
% 0.15/0.36 = { by lemma 10 R->L }
% 0.15/0.36 fresh2(fresh4(equalish(b, c), true, b, c, identity, b), true, identity, b, a, c)
% 0.15/0.36 = { by axiom 2 (b_equals_c) }
% 0.15/0.36 fresh2(fresh4(true, true, b, c, identity, b), true, identity, b, a, c)
% 0.15/0.36 = { by axiom 6 (product_substitution3) }
% 0.15/0.36 fresh2(product(identity, b, c), true, identity, b, a, c)
% 0.15/0.36 = { by axiom 9 (total_function2) }
% 0.15/0.36 fresh(product(identity, b, a), true, a, c)
% 0.15/0.36 = { by axiom 6 (product_substitution3) R->L }
% 0.15/0.36 fresh(fresh4(true, true, b, a, identity, b), true, a, c)
% 0.15/0.36 = { by axiom 4 (total_function2) R->L }
% 0.15/0.36 fresh(fresh4(fresh(true, true, b, a), true, b, a, identity, b), true, a, c)
% 0.15/0.36 = { by lemma 10 R->L }
% 0.15/0.36 fresh(fresh4(fresh(fresh4(equalish(a, b), true, a, b, identity, a), true, b, a), true, b, a, identity, b), true, a, c)
% 0.15/0.36 = { by axiom 1 (a_equals_b) }
% 0.15/0.36 fresh(fresh4(fresh(fresh4(true, true, a, b, identity, a), true, b, a), true, b, a, identity, b), true, a, c)
% 0.15/0.36 = { by axiom 6 (product_substitution3) }
% 0.15/0.36 fresh(fresh4(fresh(product(identity, a, b), true, b, a), true, b, a, identity, b), true, a, c)
% 0.15/0.36 = { by axiom 9 (total_function2) R->L }
% 0.15/0.36 fresh(fresh4(fresh2(product(identity, a, a), true, identity, a, b, a), true, b, a, identity, b), true, a, c)
% 0.15/0.36 = { by axiom 3 (left_identity) }
% 0.15/0.36 fresh(fresh4(fresh2(true, true, identity, a, b, a), true, b, a, identity, b), true, a, c)
% 0.15/0.36 = { by axiom 7 (total_function2) }
% 0.15/0.36 fresh(fresh4(equalish(b, a), true, b, a, identity, b), true, a, c)
% 0.15/0.36 = { by lemma 10 }
% 0.15/0.36 fresh(true, true, a, c)
% 0.15/0.36 = { by axiom 4 (total_function2) }
% 0.15/0.36 true
% 0.15/0.36 % SZS output end Proof
% 0.15/0.36
% 0.15/0.36 RESULT: Unsatisfiable (the axioms are contradictory).
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