0.07/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.07/0.12 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof 0.12/0.33 % Computer : n021.cluster.edu 0.12/0.33 % Model : x86_64 x86_64 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.12/0.33 % Memory : 8042.1875MB 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64 0.12/0.33 % CPULimit : 1200 0.12/0.33 % WCLimit : 120 0.12/0.33 % DateTime : Tue Jul 13 14:31:10 EDT 2021 0.12/0.33 % CPUTime : 14.27/2.15 % SZS status Theorem 14.27/2.15 14.27/2.19 % SZS output start Proof 14.27/2.19 Take the following subset of the input axioms: 14.27/2.19 fof(exemplo6GDDFULL012015, conjecture, ![A, B, C, H, D, E, F, G, K, I]: (cyclic(H, K, I, G) <= (coll(D, B, C) & (perp(E, B, A, C) & (perp(F, C, A, B) & (coll(F, A, B) & (coll(G, B, C) & (perp(H, F, A, C) & (coll(H, A, C) & (perp(K, E, A, B) & (coll(I, A, B) & (perp(I, D, A, B) & (coll(K, A, B) & (perp(G, F, B, C) & (coll(E, A, C) & perp(D, A, B, C)))))))))))))))). 14.27/2.19 fof(ruleD1, axiom, ![A, B, C]: (coll(A, B, C) => coll(A, C, B))). 14.27/2.19 fof(ruleD14, axiom, ![A, B, C, D]: (cyclic(A, B, D, C) <= cyclic(A, B, C, D))). 14.27/2.19 fof(ruleD15, axiom, ![A, B, C, D]: (cyclic(A, B, C, D) => cyclic(A, C, B, D))). 14.27/2.19 fof(ruleD16, axiom, ![A, B, C, D]: (cyclic(B, A, C, D) <= cyclic(A, B, C, D))). 14.27/2.19 fof(ruleD17, axiom, ![A, B, C, D, E]: ((cyclic(A, B, C, D) & cyclic(A, B, C, E)) => cyclic(B, C, D, E))). 14.27/2.19 fof(ruleD19, axiom, ![A, B, C, P, Q, D, U, V]: (eqangle(C, D, A, B, U, V, P, Q) <= eqangle(A, B, C, D, P, Q, U, V))). 14.27/2.19 fof(ruleD2, axiom, ![A, B, C]: (coll(B, A, C) <= coll(A, B, C))). 14.27/2.19 fof(ruleD3, axiom, ![A, B, C, D]: (coll(C, D, A) <= (coll(A, B, C) & coll(A, B, D)))). 14.27/2.19 fof(ruleD40, axiom, ![A, B, C, P, Q, D]: (eqangle(A, B, P, Q, C, D, P, Q) <= para(A, B, C, D))). 14.27/2.19 fof(ruleD42b, axiom, ![A, B, P, Q]: ((eqangle(P, A, P, B, Q, A, Q, B) & coll(P, Q, B)) => cyclic(A, B, P, Q))). 14.27/2.19 fof(ruleD8, axiom, ![A, B, C, D]: (perp(C, D, A, B) <= perp(A, B, C, D))). 14.27/2.19 fof(ruleD9, axiom, ![A, B, C, D, E, F]: (para(A, B, E, F) <= (perp(C, D, E, F) & perp(A, B, C, D)))). 14.27/2.19 14.27/2.19 Now clausify the problem and encode Horn clauses using encoding 3 of 14.27/2.19 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 14.27/2.19 We repeatedly replace C & s=t => u=v by the two clauses: 14.27/2.19 fresh(y, y, x1...xn) = u 14.27/2.19 C => fresh(s, t, x1...xn) = v 14.27/2.19 where fresh is a fresh function symbol and x1..xn are the free 14.27/2.19 variables of u and v. 14.27/2.19 A predicate p(X) is encoded as p(X)=true (this is sound, because the 14.27/2.19 input problem has no model of domain size 1). 14.27/2.19 14.27/2.19 The encoding turns the above axioms into the following unit equations and goals: 14.27/2.19 14.27/2.19 Axiom 1 (exemplo6GDDFULL012015_8): coll(f, a, b) = true. 14.27/2.19 Axiom 2 (exemplo6GDDFULL012015_1): perp(f, c, a, b) = true. 14.27/2.19 Axiom 3 (ruleD1): fresh147(X, X, Y, Z, W) = true. 14.27/2.19 Axiom 4 (ruleD2): fresh134(X, X, Y, Z, W) = true. 14.27/2.19 Axiom 5 (ruleD3): fresh120(X, X, Y, Z, W) = true. 14.27/2.19 Axiom 6 (ruleD14): fresh141(X, X, Y, Z, W, V) = true. 14.27/2.19 Axiom 7 (ruleD15): fresh140(X, X, Y, Z, W, V) = true. 14.27/2.19 Axiom 8 (ruleD16): fresh139(X, X, Y, Z, W, V) = true. 14.27/2.19 Axiom 9 (ruleD17): fresh137(X, X, Y, Z, W, V) = true. 14.27/2.19 Axiom 10 (ruleD3): fresh121(X, X, Y, Z, W, V) = coll(W, V, Y). 14.27/2.19 Axiom 11 (ruleD42b): fresh103(X, X, Y, Z, W, V) = cyclic(Y, Z, W, V). 14.27/2.19 Axiom 12 (ruleD42b): fresh102(X, X, Y, Z, W, V) = true. 14.27/2.19 Axiom 13 (ruleD8): fresh52(X, X, Y, Z, W, V) = true. 14.27/2.19 Axiom 14 (ruleD9): fresh50(X, X, Y, Z, W, V) = true. 14.27/2.19 Axiom 15 (ruleD1): fresh147(coll(X, Y, Z), true, X, Y, Z) = coll(X, Z, Y). 14.27/2.19 Axiom 16 (ruleD17): fresh138(X, X, Y, Z, W, V, U) = cyclic(Z, W, V, U). 14.27/2.19 Axiom 17 (ruleD2): fresh134(coll(X, Y, Z), true, X, Y, Z) = coll(Y, X, Z). 14.27/2.19 Axiom 18 (ruleD3): fresh121(coll(X, Y, Z), true, X, Y, W, Z) = fresh120(coll(X, Y, W), true, X, W, Z). 14.27/2.19 Axiom 19 (ruleD40): fresh105(X, X, Y, Z, W, V, U, T) = true. 14.27/2.19 Axiom 20 (ruleD9): fresh51(X, X, Y, Z, W, V, U, T) = para(Y, Z, U, T). 14.27/2.19 Axiom 21 (ruleD14): fresh141(cyclic(X, Y, Z, W), true, X, Y, Z, W) = cyclic(X, Y, W, Z). 14.27/2.19 Axiom 22 (ruleD15): fresh140(cyclic(X, Y, Z, W), true, X, Y, Z, W) = cyclic(X, Z, Y, W). 14.27/2.19 Axiom 23 (ruleD16): fresh139(cyclic(X, Y, Z, W), true, X, Y, Z, W) = cyclic(Y, X, Z, W). 14.27/2.19 Axiom 24 (ruleD8): fresh52(perp(X, Y, Z, W), true, X, Y, Z, W) = perp(Z, W, X, Y). 14.27/2.19 Axiom 25 (ruleD17): fresh138(cyclic(X, Y, Z, W), true, X, Y, Z, V, W) = fresh137(cyclic(X, Y, Z, V), true, Y, Z, V, W). 14.27/2.19 Axiom 26 (ruleD19): fresh135(X, X, Y, Z, W, V, U, T, S, X2) = true. 14.27/2.19 Axiom 27 (ruleD40): fresh105(para(X, Y, Z, W), true, X, Y, Z, W, V, U) = eqangle(X, Y, V, U, Z, W, V, U). 14.27/2.19 Axiom 28 (ruleD9): fresh51(perp(X, Y, Z, W), true, V, U, X, Y, Z, W) = fresh50(perp(V, U, X, Y), true, V, U, Z, W). 14.27/2.19 Axiom 29 (ruleD42b): fresh103(eqangle(X, Y, X, Z, W, Y, W, Z), true, Y, Z, X, W) = fresh102(coll(X, W, Z), true, Y, Z, X, W). 14.27/2.19 Axiom 30 (ruleD19): fresh135(eqangle(X, Y, Z, W, V, U, T, S), true, X, Y, Z, W, V, U, T, S) = eqangle(Z, W, X, Y, T, S, V, U). 14.27/2.19 14.27/2.19 Lemma 31: coll(f, b, a) = true. 14.27/2.19 Proof: 14.27/2.19 coll(f, b, a) 14.27/2.19 = { by axiom 15 (ruleD1) R->L } 14.27/2.19 fresh147(coll(f, a, b), true, f, a, b) 14.27/2.19 = { by axiom 1 (exemplo6GDDFULL012015_8) } 14.27/2.19 fresh147(true, true, f, a, b) 14.27/2.19 = { by axiom 3 (ruleD1) } 14.27/2.19 true 14.27/2.19 14.27/2.19 Lemma 32: cyclic(b, a, a, X) = true. 14.27/2.19 Proof: 14.27/2.19 cyclic(b, a, a, X) 14.27/2.19 = { by axiom 21 (ruleD14) R->L } 14.27/2.19 fresh141(cyclic(b, a, X, a), true, b, a, X, a) 14.27/2.19 = { by axiom 22 (ruleD15) R->L } 14.27/2.19 fresh141(fresh140(cyclic(b, X, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.19 = { by axiom 23 (ruleD16) R->L } 14.27/2.19 fresh141(fresh140(fresh139(cyclic(X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.19 = { by axiom 11 (ruleD42b) R->L } 14.27/2.19 fresh141(fresh140(fresh139(fresh103(true, true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 26 (ruleD19) R->L } 14.27/2.20 fresh141(fresh140(fresh139(fresh103(fresh135(true, true, a, b, a, X, a, b, a, X), true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 19 (ruleD40) R->L } 14.27/2.20 fresh141(fresh140(fresh139(fresh103(fresh135(fresh105(true, true, a, b, a, b, a, X), true, a, b, a, X, a, b, a, X), true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 14 (ruleD9) R->L } 14.27/2.20 fresh141(fresh140(fresh139(fresh103(fresh135(fresh105(fresh50(true, true, a, b, a, b), true, a, b, a, b, a, X), true, a, b, a, X, a, b, a, X), true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 13 (ruleD8) R->L } 14.27/2.20 fresh141(fresh140(fresh139(fresh103(fresh135(fresh105(fresh50(fresh52(true, true, f, c, a, b), true, a, b, a, b), true, a, b, a, b, a, X), true, a, b, a, X, a, b, a, X), true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 2 (exemplo6GDDFULL012015_1) R->L } 14.27/2.20 fresh141(fresh140(fresh139(fresh103(fresh135(fresh105(fresh50(fresh52(perp(f, c, a, b), true, f, c, a, b), true, a, b, a, b), true, a, b, a, b, a, X), true, a, b, a, X, a, b, a, X), true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 24 (ruleD8) } 14.27/2.20 fresh141(fresh140(fresh139(fresh103(fresh135(fresh105(fresh50(perp(a, b, f, c), true, a, b, a, b), true, a, b, a, b, a, X), true, a, b, a, X, a, b, a, X), true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 28 (ruleD9) R->L } 14.27/2.20 fresh141(fresh140(fresh139(fresh103(fresh135(fresh105(fresh51(perp(f, c, a, b), true, a, b, f, c, a, b), true, a, b, a, b, a, X), true, a, b, a, X, a, b, a, X), true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 2 (exemplo6GDDFULL012015_1) } 14.27/2.20 fresh141(fresh140(fresh139(fresh103(fresh135(fresh105(fresh51(true, true, a, b, f, c, a, b), true, a, b, a, b, a, X), true, a, b, a, X, a, b, a, X), true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 20 (ruleD9) } 14.27/2.20 fresh141(fresh140(fresh139(fresh103(fresh135(fresh105(para(a, b, a, b), true, a, b, a, b, a, X), true, a, b, a, X, a, b, a, X), true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 27 (ruleD40) } 14.27/2.20 fresh141(fresh140(fresh139(fresh103(fresh135(eqangle(a, b, a, X, a, b, a, X), true, a, b, a, X, a, b, a, X), true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 30 (ruleD19) } 14.27/2.20 fresh141(fresh140(fresh139(fresh103(eqangle(a, X, a, b, a, X, a, b), true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 29 (ruleD42b) } 14.27/2.20 fresh141(fresh140(fresh139(fresh102(coll(a, a, b), true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 15 (ruleD1) R->L } 14.27/2.20 fresh141(fresh140(fresh139(fresh102(fresh147(coll(a, b, a), true, a, b, a), true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 10 (ruleD3) R->L } 14.27/2.20 fresh141(fresh140(fresh139(fresh102(fresh147(fresh121(true, true, a, f, a, b), true, a, b, a), true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 4 (ruleD2) R->L } 14.27/2.20 fresh141(fresh140(fresh139(fresh102(fresh147(fresh121(fresh134(true, true, f, a, b), true, a, f, a, b), true, a, b, a), true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 1 (exemplo6GDDFULL012015_8) R->L } 14.27/2.20 fresh141(fresh140(fresh139(fresh102(fresh147(fresh121(fresh134(coll(f, a, b), true, f, a, b), true, a, f, a, b), true, a, b, a), true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 17 (ruleD2) } 14.27/2.20 fresh141(fresh140(fresh139(fresh102(fresh147(fresh121(coll(a, f, b), true, a, f, a, b), true, a, b, a), true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 18 (ruleD3) } 14.27/2.20 fresh141(fresh140(fresh139(fresh102(fresh147(fresh120(coll(a, f, a), true, a, a, b), true, a, b, a), true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 15 (ruleD1) R->L } 14.27/2.20 fresh141(fresh140(fresh139(fresh102(fresh147(fresh120(fresh147(coll(a, a, f), true, a, a, f), true, a, a, b), true, a, b, a), true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 10 (ruleD3) R->L } 14.27/2.20 fresh141(fresh140(fresh139(fresh102(fresh147(fresh120(fresh147(fresh121(true, true, f, b, a, a), true, a, a, f), true, a, a, b), true, a, b, a), true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by lemma 31 R->L } 14.27/2.20 fresh141(fresh140(fresh139(fresh102(fresh147(fresh120(fresh147(fresh121(coll(f, b, a), true, f, b, a, a), true, a, a, f), true, a, a, b), true, a, b, a), true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 18 (ruleD3) } 14.27/2.20 fresh141(fresh140(fresh139(fresh102(fresh147(fresh120(fresh147(fresh120(coll(f, b, a), true, f, a, a), true, a, a, f), true, a, a, b), true, a, b, a), true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by lemma 31 } 14.27/2.20 fresh141(fresh140(fresh139(fresh102(fresh147(fresh120(fresh147(fresh120(true, true, f, a, a), true, a, a, f), true, a, a, b), true, a, b, a), true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 5 (ruleD3) } 14.27/2.20 fresh141(fresh140(fresh139(fresh102(fresh147(fresh120(fresh147(true, true, a, a, f), true, a, a, b), true, a, b, a), true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 3 (ruleD1) } 14.27/2.20 fresh141(fresh140(fresh139(fresh102(fresh147(fresh120(true, true, a, a, b), true, a, b, a), true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 5 (ruleD3) } 14.27/2.20 fresh141(fresh140(fresh139(fresh102(fresh147(true, true, a, b, a), true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 3 (ruleD1) } 14.27/2.20 fresh141(fresh140(fresh139(fresh102(true, true, X, b, a, a), true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 12 (ruleD42b) } 14.27/2.20 fresh141(fresh140(fresh139(true, true, X, b, a, a), true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 8 (ruleD16) } 14.27/2.20 fresh141(fresh140(true, true, b, X, a, a), true, b, a, X, a) 14.27/2.20 = { by axiom 7 (ruleD15) } 14.27/2.20 fresh141(true, true, b, a, X, a) 14.27/2.20 = { by axiom 6 (ruleD14) } 14.27/2.20 true 14.27/2.20 14.27/2.20 Lemma 33: cyclic(a, a, X, Y) = true. 14.27/2.20 Proof: 14.27/2.20 cyclic(a, a, X, Y) 14.27/2.20 = { by axiom 16 (ruleD17) R->L } 14.27/2.20 fresh138(true, true, b, a, a, X, Y) 14.27/2.20 = { by lemma 32 R->L } 14.27/2.20 fresh138(cyclic(b, a, a, Y), true, b, a, a, X, Y) 14.27/2.20 = { by axiom 25 (ruleD17) } 14.27/2.20 fresh137(cyclic(b, a, a, X), true, a, a, X, Y) 14.27/2.20 = { by lemma 32 } 14.27/2.20 fresh137(true, true, a, a, X, Y) 14.27/2.20 = { by axiom 9 (ruleD17) } 14.27/2.20 true 14.27/2.20 14.27/2.20 Lemma 34: cyclic(a, X, Y, Z) = true. 14.27/2.20 Proof: 14.27/2.20 cyclic(a, X, Y, Z) 14.27/2.20 = { by axiom 16 (ruleD17) R->L } 14.27/2.20 fresh138(true, true, a, a, X, Y, Z) 14.27/2.20 = { by lemma 33 R->L } 14.27/2.20 fresh138(cyclic(a, a, X, Z), true, a, a, X, Y, Z) 14.27/2.20 = { by axiom 25 (ruleD17) } 14.27/2.20 fresh137(cyclic(a, a, X, Y), true, a, X, Y, Z) 14.27/2.20 = { by lemma 33 } 14.27/2.20 fresh137(true, true, a, X, Y, Z) 14.27/2.20 = { by axiom 9 (ruleD17) } 14.27/2.20 true 14.27/2.20 14.27/2.20 Goal 1 (exemplo6GDDFULL012015_14): cyclic(h, k, i, g) = true. 14.27/2.20 Proof: 14.27/2.20 cyclic(h, k, i, g) 14.27/2.20 = { by axiom 16 (ruleD17) R->L } 14.27/2.20 fresh138(true, true, a, h, k, i, g) 14.27/2.20 = { by lemma 34 R->L } 14.27/2.20 fresh138(cyclic(a, h, k, g), true, a, h, k, i, g) 14.27/2.20 = { by axiom 25 (ruleD17) } 14.27/2.20 fresh137(cyclic(a, h, k, i), true, h, k, i, g) 14.27/2.20 = { by lemma 34 } 14.27/2.20 fresh137(true, true, h, k, i, g) 14.27/2.20 = { by axiom 9 (ruleD17) } 14.27/2.20 true 14.27/2.20 % SZS output end Proof 14.27/2.20 14.27/2.20 RESULT: Theorem (the conjecture is true). 14.27/2.21 EOF