TSTP Solution File: GRP012+5 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP012+5 : 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 01:16:34 EDT 2023
% Result : Theorem 0.20s 0.46s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : GRP012+5 : 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
% 0.16/0.35 % Computer : n008.cluster.edu
% 0.16/0.35 % Model : x86_64 x86_64
% 0.16/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.35 % Memory : 8042.1875MB
% 0.16/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.35 % CPULimit : 300
% 0.16/0.35 % WCLimit : 300
% 0.16/0.35 % DateTime : Tue Aug 29 02:42:02 EDT 2023
% 0.16/0.35 % CPUTime :
% 0.20/0.46 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.20/0.46
% 0.20/0.46 % SZS status Theorem
% 0.20/0.46
% 0.20/0.50 % SZS output start Proof
% 0.20/0.50 Take the following subset of the input axioms:
% 0.20/0.51 fof(prove_distribution, 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)) & (![X2, U2, V2, W2, 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, U2, V2, W2]: ((product(inverse(U2), inverse(V2), W2) & product(V2, U2, X2)) => product(inverse(W2), inverse(X2), E)))).
% 0.20/0.51
% 0.20/0.51 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.51 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.51 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.51 fresh(y, y, x1...xn) = u
% 0.20/0.51 C => fresh(s, t, x1...xn) = v
% 0.20/0.51 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.51 variables of u and v.
% 0.20/0.51 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.51 input problem has no model of domain size 1).
% 0.20/0.51
% 0.20/0.51 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.51
% 0.20/0.51 Axiom 1 (prove_distribution_1): product(X, e, X) = true.
% 0.20/0.51 Axiom 2 (prove_distribution_6): product(v, u, x) = true.
% 0.20/0.51 Axiom 3 (prove_distribution_5): product(e, X, X) = true.
% 0.20/0.51 Axiom 4 (prove_distribution_2): product(X, inverse(X), e) = true.
% 0.20/0.51 Axiom 5 (prove_distribution_3): product(inverse(X), X, e) = true.
% 0.20/0.51 Axiom 6 (prove_distribution_7): fresh6(X, X, Y, Z, W) = true.
% 0.20/0.51 Axiom 7 (prove_distribution_8): fresh4(X, X, Y, Z, W) = true.
% 0.20/0.51 Axiom 8 (prove_distribution_4): product(inverse(u), inverse(v), w) = true.
% 0.20/0.51 Axiom 9 (prove_distribution_8): fresh(X, X, Y, Z, W, V, U) = product(V, W, U).
% 0.20/0.51 Axiom 10 (prove_distribution_7): fresh2(X, X, Y, Z, W, V, U) = product(Y, V, U).
% 0.20/0.51 Axiom 11 (prove_distribution_7): fresh5(X, X, Y, Z, W, V, U, T) = fresh6(product(Y, Z, V), true, Y, U, T).
% 0.20/0.51 Axiom 12 (prove_distribution_8): fresh3(X, X, Y, Z, W, V, U, T) = fresh4(product(Y, Z, V), true, W, V, T).
% 0.20/0.51 Axiom 13 (prove_distribution_7): fresh5(product(X, Y, Z), true, W, V, Y, X, U, Z) = fresh2(product(V, Y, U), true, W, V, X, U, Z).
% 0.20/0.51 Axiom 14 (prove_distribution_8): fresh3(product(X, Y, Z), true, W, X, Y, V, Z, U) = fresh(product(W, Z, U), true, W, X, Y, V, U).
% 0.20/0.51
% 0.20/0.51 Lemma 15: fresh3(product(X, Y, e), true, Z, X, Y, W, e, Z) = product(W, Y, Z).
% 0.20/0.51 Proof:
% 0.20/0.51 fresh3(product(X, Y, e), true, Z, X, Y, W, e, Z)
% 0.20/0.51 = { by axiom 14 (prove_distribution_8) }
% 0.20/0.51 fresh(product(Z, e, Z), true, Z, X, Y, W, Z)
% 0.20/0.51 = { by axiom 1 (prove_distribution_1) }
% 0.20/0.51 fresh(true, true, Z, X, Y, W, Z)
% 0.20/0.51 = { by axiom 9 (prove_distribution_8) }
% 0.20/0.51 product(W, Y, Z)
% 0.20/0.51
% 0.20/0.51 Lemma 16: fresh4(product(X, e, Y), true, e, Y, X) = product(Y, e, X).
% 0.20/0.51 Proof:
% 0.20/0.51 fresh4(product(X, e, Y), true, e, Y, X)
% 0.20/0.51 = { by axiom 12 (prove_distribution_8) R->L }
% 0.20/0.51 fresh3(true, true, X, e, e, Y, e, X)
% 0.20/0.51 = { by axiom 1 (prove_distribution_1) R->L }
% 0.20/0.51 fresh3(product(e, e, e), true, X, e, e, Y, e, X)
% 0.20/0.51 = { by lemma 15 }
% 0.20/0.51 product(Y, e, X)
% 0.20/0.51
% 0.20/0.51 Lemma 17: fresh3(X, X, Y, inverse(Z), Z, W, V, Y) = product(W, Z, Y).
% 0.20/0.51 Proof:
% 0.20/0.51 fresh3(X, X, Y, inverse(Z), Z, W, V, Y)
% 0.20/0.51 = { by axiom 12 (prove_distribution_8) }
% 0.20/0.51 fresh4(product(Y, inverse(Z), W), true, Z, W, Y)
% 0.20/0.51 = { by axiom 12 (prove_distribution_8) R->L }
% 0.20/0.51 fresh3(true, true, Y, inverse(Z), Z, W, e, Y)
% 0.20/0.51 = { by axiom 5 (prove_distribution_3) R->L }
% 0.20/0.51 fresh3(product(inverse(Z), Z, e), true, Y, inverse(Z), Z, W, e, Y)
% 0.20/0.51 = { by lemma 15 }
% 0.20/0.51 product(W, Z, Y)
% 0.20/0.51
% 0.20/0.51 Lemma 18: fresh5(product(X, e, Y), true, Z, W, e, X, W, Y) = product(Z, W, Y).
% 0.20/0.51 Proof:
% 0.20/0.51 fresh5(product(X, e, Y), true, Z, W, e, X, W, Y)
% 0.20/0.51 = { by axiom 13 (prove_distribution_7) }
% 0.20/0.51 fresh2(product(W, e, W), true, Z, W, X, W, Y)
% 0.20/0.51 = { by axiom 1 (prove_distribution_1) }
% 0.20/0.51 fresh2(true, true, Z, W, X, W, Y)
% 0.20/0.51 = { by axiom 10 (prove_distribution_7) }
% 0.20/0.51 product(Z, W, Y)
% 0.20/0.51
% 0.20/0.51 Goal 1 (prove_distribution_9): product(inverse(w), inverse(x), e) = true.
% 0.20/0.51 Proof:
% 0.20/0.51 product(inverse(w), inverse(x), e)
% 0.20/0.51 = { by lemma 18 R->L }
% 0.20/0.51 fresh5(product(inverse(e), e, e), true, inverse(w), inverse(x), e, inverse(e), inverse(x), e)
% 0.20/0.51 = { by axiom 5 (prove_distribution_3) }
% 0.20/0.51 fresh5(true, true, inverse(w), inverse(x), e, inverse(e), inverse(x), e)
% 0.20/0.51 = { by axiom 11 (prove_distribution_7) }
% 0.20/0.51 fresh6(product(inverse(w), inverse(x), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.51 = { by lemma 15 R->L }
% 0.20/0.51 fresh6(fresh3(product(x, inverse(x), e), true, inverse(e), x, inverse(x), inverse(w), e, inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.51 = { by axiom 4 (prove_distribution_2) }
% 0.20/0.51 fresh6(fresh3(true, true, inverse(e), x, inverse(x), inverse(w), e, inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.51 = { by axiom 12 (prove_distribution_8) }
% 0.20/0.51 fresh6(fresh4(product(inverse(e), x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.51 = { by lemma 18 R->L }
% 0.20/0.51 fresh6(fresh4(fresh5(product(x, e, inverse(w)), true, inverse(e), x, e, x, x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.51 = { by lemma 16 R->L }
% 0.20/0.51 fresh6(fresh4(fresh5(fresh4(product(inverse(w), e, x), true, e, x, inverse(w)), true, inverse(e), x, e, x, x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.51 = { by axiom 10 (prove_distribution_7) R->L }
% 0.20/0.51 fresh6(fresh4(fresh5(fresh4(fresh2(true, true, inverse(w), inverse(u), v, e, x), true, e, x, inverse(w)), true, inverse(e), x, e, x, x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.51 = { by axiom 5 (prove_distribution_3) R->L }
% 0.20/0.51 fresh6(fresh4(fresh5(fresh4(fresh2(product(inverse(u), u, e), true, inverse(w), inverse(u), v, e, x), true, e, x, inverse(w)), true, inverse(e), x, e, x, x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.51 = { by axiom 13 (prove_distribution_7) R->L }
% 0.20/0.51 fresh6(fresh4(fresh5(fresh4(fresh5(product(v, u, x), true, inverse(w), inverse(u), u, v, e, x), true, e, x, inverse(w)), true, inverse(e), x, e, x, x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.51 = { by axiom 2 (prove_distribution_6) }
% 0.20/0.51 fresh6(fresh4(fresh5(fresh4(fresh5(true, true, inverse(w), inverse(u), u, v, e, x), true, e, x, inverse(w)), true, inverse(e), x, e, x, x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.51 = { by axiom 11 (prove_distribution_7) }
% 0.20/0.51 fresh6(fresh4(fresh5(fresh4(fresh6(product(inverse(w), inverse(u), v), true, inverse(w), e, x), true, e, x, inverse(w)), true, inverse(e), x, e, x, x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.51 = { by axiom 10 (prove_distribution_7) R->L }
% 0.20/0.51 fresh6(fresh4(fresh5(fresh4(fresh6(fresh2(true, true, inverse(w), w, e, inverse(u), v), true, inverse(w), e, x), true, e, x, inverse(w)), true, inverse(e), x, e, x, x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.51 = { by axiom 7 (prove_distribution_8) R->L }
% 0.20/0.51 fresh6(fresh4(fresh5(fresh4(fresh6(fresh2(fresh4(true, true, v, w, inverse(u)), true, inverse(w), w, e, inverse(u), v), true, inverse(w), e, x), true, e, x, inverse(w)), true, inverse(e), x, e, x, x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.51 = { by axiom 8 (prove_distribution_4) R->L }
% 0.20/0.51 fresh6(fresh4(fresh5(fresh4(fresh6(fresh2(fresh4(product(inverse(u), inverse(v), w), true, v, w, inverse(u)), true, inverse(w), w, e, inverse(u), v), true, inverse(w), e, x), true, e, x, inverse(w)), true, inverse(e), x, e, x, x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.51 = { by axiom 12 (prove_distribution_8) R->L }
% 0.20/0.51 fresh6(fresh4(fresh5(fresh4(fresh6(fresh2(fresh3(X, X, inverse(u), inverse(v), v, w, Y, inverse(u)), true, inverse(w), w, e, inverse(u), v), true, inverse(w), e, x), true, e, x, inverse(w)), true, inverse(e), x, e, x, x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.51 = { by lemma 17 }
% 0.20/0.51 fresh6(fresh4(fresh5(fresh4(fresh6(fresh2(product(w, v, inverse(u)), true, inverse(w), w, e, inverse(u), v), true, inverse(w), e, x), true, e, x, inverse(w)), true, inverse(e), x, e, x, x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.51 = { by axiom 13 (prove_distribution_7) R->L }
% 0.20/0.51 fresh6(fresh4(fresh5(fresh4(fresh6(fresh5(product(e, v, v), true, inverse(w), w, v, e, inverse(u), v), true, inverse(w), e, x), true, e, x, inverse(w)), true, inverse(e), x, e, x, x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.51 = { by axiom 3 (prove_distribution_5) }
% 0.20/0.52 fresh6(fresh4(fresh5(fresh4(fresh6(fresh5(true, true, inverse(w), w, v, e, inverse(u), v), true, inverse(w), e, x), true, e, x, inverse(w)), true, inverse(e), x, e, x, x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.52 = { by axiom 11 (prove_distribution_7) }
% 0.20/0.52 fresh6(fresh4(fresh5(fresh4(fresh6(fresh6(product(inverse(w), w, e), true, inverse(w), inverse(u), v), true, inverse(w), e, x), true, e, x, inverse(w)), true, inverse(e), x, e, x, x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.52 = { by axiom 5 (prove_distribution_3) }
% 0.20/0.52 fresh6(fresh4(fresh5(fresh4(fresh6(fresh6(true, true, inverse(w), inverse(u), v), true, inverse(w), e, x), true, e, x, inverse(w)), true, inverse(e), x, e, x, x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.52 = { by axiom 6 (prove_distribution_7) }
% 0.20/0.52 fresh6(fresh4(fresh5(fresh4(fresh6(true, true, inverse(w), e, x), true, e, x, inverse(w)), true, inverse(e), x, e, x, x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.52 = { by axiom 6 (prove_distribution_7) }
% 0.20/0.52 fresh6(fresh4(fresh5(fresh4(true, true, e, x, inverse(w)), true, inverse(e), x, e, x, x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.52 = { by axiom 7 (prove_distribution_8) }
% 0.20/0.52 fresh6(fresh4(fresh5(true, true, inverse(e), x, e, x, x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.52 = { by axiom 11 (prove_distribution_7) }
% 0.20/0.52 fresh6(fresh4(fresh6(product(inverse(e), x, x), true, inverse(e), x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.52 = { by lemma 17 R->L }
% 0.20/0.52 fresh6(fresh4(fresh6(fresh3(Z, Z, x, inverse(x), x, inverse(e), W, x), true, inverse(e), x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.52 = { by axiom 12 (prove_distribution_8) }
% 0.20/0.52 fresh6(fresh4(fresh6(fresh4(product(x, inverse(x), inverse(e)), true, x, inverse(e), x), true, inverse(e), x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.52 = { by lemma 18 R->L }
% 0.20/0.52 fresh6(fresh4(fresh6(fresh4(fresh5(product(e, e, inverse(e)), true, x, inverse(x), e, e, inverse(x), inverse(e)), true, x, inverse(e), x), true, inverse(e), x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.52 = { by lemma 16 R->L }
% 0.20/0.52 fresh6(fresh4(fresh6(fresh4(fresh5(fresh4(product(inverse(e), e, e), true, e, e, inverse(e)), true, x, inverse(x), e, e, inverse(x), inverse(e)), true, x, inverse(e), x), true, inverse(e), x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.52 = { by axiom 5 (prove_distribution_3) }
% 0.20/0.52 fresh6(fresh4(fresh6(fresh4(fresh5(fresh4(true, true, e, e, inverse(e)), true, x, inverse(x), e, e, inverse(x), inverse(e)), true, x, inverse(e), x), true, inverse(e), x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.52 = { by axiom 7 (prove_distribution_8) }
% 0.20/0.52 fresh6(fresh4(fresh6(fresh4(fresh5(true, true, x, inverse(x), e, e, inverse(x), inverse(e)), true, x, inverse(e), x), true, inverse(e), x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.52 = { by axiom 11 (prove_distribution_7) }
% 0.20/0.52 fresh6(fresh4(fresh6(fresh4(fresh6(product(x, inverse(x), e), true, x, inverse(x), inverse(e)), true, x, inverse(e), x), true, inverse(e), x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.52 = { by axiom 4 (prove_distribution_2) }
% 0.20/0.52 fresh6(fresh4(fresh6(fresh4(fresh6(true, true, x, inverse(x), inverse(e)), true, x, inverse(e), x), true, inverse(e), x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.52 = { by axiom 6 (prove_distribution_7) }
% 0.20/0.52 fresh6(fresh4(fresh6(fresh4(true, true, x, inverse(e), x), true, inverse(e), x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.52 = { by axiom 7 (prove_distribution_8) }
% 0.20/0.52 fresh6(fresh4(fresh6(true, true, inverse(e), x, inverse(w)), true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.52 = { by axiom 6 (prove_distribution_7) }
% 0.20/0.52 fresh6(fresh4(true, true, inverse(x), inverse(w), inverse(e)), true, inverse(w), inverse(x), e)
% 0.20/0.52 = { by axiom 7 (prove_distribution_8) }
% 0.20/0.52 fresh6(true, true, inverse(w), inverse(x), e)
% 0.20/0.52 = { by axiom 6 (prove_distribution_7) }
% 0.20/0.52 true
% 0.20/0.52 % SZS output end Proof
% 0.20/0.52
% 0.20/0.52 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------