TSTP Solution File: GRP001+6 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : GRP001+6 : TPTP v8.1.2. Released v3.1.0.
% 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 : n008.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 01:16:29 EDT 2023
% Result : Theorem 0.20s 0.45s
% 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
% 0.00/0.12 % Problem : GRP001+6 : TPTP v8.1.2. Released v3.1.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.34 % Computer : n008.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Mon Aug 28 21:47:02 EDT 2023
% 0.12/0.35 % CPUTime :
% 0.20/0.45 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.20/0.45
% 0.20/0.45 % SZS status Theorem
% 0.20/0.45
% 0.20/0.46 % SZS output start Proof
% 0.20/0.46 Take the following subset of the input axioms:
% 0.20/0.47 fof(commutativity, conjecture, ![E]: ((![X, Y]: ?[Z]: product(X, Y, Z) & (![U, V, W, X2, Y2, Z2]: ((product(X2, Y2, U) & (product(Y2, Z2, V) & product(U, Z2, W))) => product(X2, V, W)) & (![U2, V2, W2, X2, Y2, Z2]: ((product(X2, Y2, U2) & (product(Y2, Z2, V2) & product(X2, V2, W2))) => product(U2, Z2, W2)) & (![X2]: product(X2, E, X2) & (![X2]: product(E, X2, X2) & (![X2]: product(X2, inverse(X2), E) & ![X2]: product(inverse(X2), X2, E))))))) => (![X2]: product(X2, X2, E) => ![U2, V2, W2]: (product(U2, V2, W2) => product(V2, U2, W2))))).
% 0.20/0.47
% 0.20/0.47 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.47 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.47 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.47 fresh(y, y, x1...xn) = u
% 0.20/0.47 C => fresh(s, t, x1...xn) = v
% 0.20/0.47 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.47 variables of u and v.
% 0.20/0.47 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.47 input problem has no model of domain size 1).
% 0.20/0.47
% 0.20/0.47 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.47
% 0.20/0.47 Axiom 1 (commutativity_3): product(X, X, e) = true.
% 0.20/0.47 Axiom 2 (commutativity_1): product(X, e, X) = true.
% 0.20/0.47 Axiom 3 (commutativity_6): product(u, v, w) = true.
% 0.20/0.47 Axiom 4 (commutativity_5): product(e, X, X) = true.
% 0.20/0.47 Axiom 5 (commutativity_7): fresh6(X, X, Y, Z, W) = true.
% 0.20/0.47 Axiom 6 (commutativity_8): fresh4(X, X, Y, Z, W) = true.
% 0.20/0.47 Axiom 7 (commutativity_8): fresh(X, X, Y, Z, W, V, U) = product(V, W, U).
% 0.20/0.47 Axiom 8 (commutativity_7): fresh2(X, X, Y, Z, W, V, U) = product(Y, V, U).
% 0.20/0.47 Axiom 9 (commutativity_7): fresh5(X, X, Y, Z, W, V, U, T) = fresh6(product(Y, Z, V), true, Y, U, T).
% 0.20/0.47 Axiom 10 (commutativity_8): fresh3(X, X, Y, Z, W, V, U, T) = fresh4(product(Y, Z, V), true, W, V, T).
% 0.20/0.47 Axiom 11 (commutativity_7): fresh5(product(X, Y, Z), true, W, V, Y, X, U, Z) = fresh2(product(V, Y, U), true, W, V, X, U, Z).
% 0.20/0.47 Axiom 12 (commutativity_8): fresh3(product(X, Y, Z), true, W, X, Y, V, Z, U) = fresh(product(W, Z, U), true, W, X, Y, V, U).
% 0.20/0.47
% 0.20/0.47 Lemma 13: fresh4(product(X, Y, Z), true, Y, Z, X) = product(Z, Y, X).
% 0.20/0.47 Proof:
% 0.20/0.47 fresh4(product(X, Y, Z), true, Y, Z, X)
% 0.20/0.48 = { by axiom 10 (commutativity_8) R->L }
% 0.20/0.48 fresh3(true, true, X, Y, Y, Z, e, X)
% 0.20/0.48 = { by axiom 1 (commutativity_3) R->L }
% 0.20/0.48 fresh3(product(Y, Y, e), true, X, Y, Y, Z, e, X)
% 0.20/0.48 = { by axiom 12 (commutativity_8) }
% 0.20/0.48 fresh(product(X, e, X), true, X, Y, Y, Z, X)
% 0.20/0.48 = { by axiom 2 (commutativity_1) }
% 0.20/0.48 fresh(true, true, X, Y, Y, Z, X)
% 0.20/0.48 = { by axiom 7 (commutativity_8) }
% 0.20/0.48 product(Z, Y, X)
% 0.20/0.48
% 0.20/0.48 Goal 1 (commutativity_9): product(v, u, w) = true.
% 0.20/0.48 Proof:
% 0.20/0.48 product(v, u, w)
% 0.20/0.48 = { by lemma 13 R->L }
% 0.20/0.48 fresh4(product(w, u, v), true, u, v, w)
% 0.20/0.48 = { by axiom 8 (commutativity_7) R->L }
% 0.20/0.48 fresh4(fresh2(true, true, w, w, e, u, v), true, u, v, w)
% 0.20/0.48 = { by axiom 6 (commutativity_8) R->L }
% 0.20/0.48 fresh4(fresh2(fresh4(true, true, v, w, u), true, w, w, e, u, v), true, u, v, w)
% 0.20/0.48 = { by axiom 3 (commutativity_6) R->L }
% 0.20/0.48 fresh4(fresh2(fresh4(product(u, v, w), true, v, w, u), true, w, w, e, u, v), true, u, v, w)
% 0.20/0.48 = { by lemma 13 }
% 0.20/0.48 fresh4(fresh2(product(w, v, u), true, w, w, e, u, v), true, u, v, w)
% 0.20/0.48 = { by axiom 11 (commutativity_7) R->L }
% 0.20/0.48 fresh4(fresh5(product(e, v, v), true, w, w, v, e, u, v), true, u, v, w)
% 0.20/0.48 = { by axiom 4 (commutativity_5) }
% 0.20/0.48 fresh4(fresh5(true, true, w, w, v, e, u, v), true, u, v, w)
% 0.20/0.48 = { by axiom 9 (commutativity_7) }
% 0.20/0.48 fresh4(fresh6(product(w, w, e), true, w, u, v), true, u, v, w)
% 0.20/0.48 = { by axiom 1 (commutativity_3) }
% 0.20/0.48 fresh4(fresh6(true, true, w, u, v), true, u, v, w)
% 0.20/0.48 = { by axiom 5 (commutativity_7) }
% 0.20/0.48 fresh4(true, true, u, v, w)
% 0.20/0.48 = { by axiom 6 (commutativity_8) }
% 0.20/0.48 true
% 0.20/0.48 % SZS output end Proof
% 0.20/0.48
% 0.20/0.48 RESULT: Theorem (the conjecture is true).
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