0.04/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.04/0.13 % Command : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn 0.12/0.34 % Computer : n020.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 : 960 0.12/0.34 % WCLimit : 120 0.12/0.34 % DateTime : Thu Jul 2 07:03:17 EDT 2020 0.12/0.34 % CPUTime : 0.19/0.48 % SZS status Theorem 0.19/0.48 0.19/0.48 % SZS output start Proof 0.19/0.48 Take the following subset of the input axioms: 0.19/0.48 fof(ax56, axiom, ![U, V]: (contains_pq(U, V) <=> pi_sharp_remove(U, V))). 0.19/0.48 fof(ax57, axiom, ![U, V]: (pi_sharp_remove(i(U), V) <=> pi_remove(U, V))). 0.19/0.48 fof(co3, conjecture, ![U, V, W, X]: ((phi(remove_cpq(triple(U, V, W), X)) => (pi_sharp_remove(i(triple(U, V, W)), X) & i(remove_cpq(triple(U, V, W), X))=remove_pq(i(triple(U, V, W)), X))) <= pi_remove(triple(U, V, W), X))). 0.19/0.48 fof(main3_li12, lemma, ![U, V, W, X, Y]: i(triple(U, W, X))=i(triple(V, W, Y))). 0.19/0.48 fof(main3_li34, lemma, ![U, V, W, X]: (remove_pq(i(triple(U, V, W)), X)=i(remove_cpq(triple(U, V, W), X)) <= contains_pq(i(triple(U, V, W)), X))). 0.19/0.48 0.19/0.48 Now clausify the problem and encode Horn clauses using encoding 3 of 0.19/0.48 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 0.19/0.48 We repeatedly replace C & s=t => u=v by the two clauses: 0.19/0.48 fresh(y, y, x1...xn) = u 0.19/0.48 C => fresh(s, t, x1...xn) = v 0.19/0.48 where fresh is a fresh function symbol and x1..xn are the free 0.19/0.48 variables of u and v. 0.19/0.48 A predicate p(X) is encoded as p(X)=true (this is sound, because the 0.19/0.48 input problem has no model of domain size 1). 0.19/0.48 0.19/0.48 The encoding turns the above axioms into the following unit equations and goals: 0.19/0.48 0.19/0.48 Axiom 1 (ax56_1): fresh27(X, X, Y, Z) = true2. 0.19/0.48 Axiom 2 (ax57_1): fresh25(X, X, Y, Z) = true2. 0.19/0.48 Axiom 3 (main3_li34): fresh7(X, X, Y, Z, W, V) = i(remove_cpq(triple(Y, Z, W), V)). 0.19/0.48 Axiom 4 (ax57_1): fresh25(pi_remove(X, Y), true2, X, Y) = pi_sharp_remove(i(X), Y). 0.19/0.48 Axiom 5 (ax56_1): fresh27(pi_sharp_remove(X, Y), true2, X, Y) = contains_pq(X, Y). 0.19/0.48 Axiom 6 (main3_li12): i(triple(X, Y, Z)) = i(triple(W, Y, V)). 0.19/0.48 Axiom 7 (main3_li34): fresh7(contains_pq(i(triple(X, Y, Z)), W), true2, X, Y, Z, W) = remove_pq(i(triple(X, Y, Z)), W). 0.19/0.48 Axiom 8 (co3): pi_remove(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W), sK1_co3_X) = true2. 0.19/0.48 0.19/0.48 Lemma 9: fresh7(X, X, Y, Z, W, V) = fresh7(?, ?, Y, Z, W, V). 0.19/0.48 Proof: 0.19/0.48 fresh7(X, X, Y, Z, W, V) 0.19/0.48 = { by axiom 3 (main3_li34) } 0.19/0.48 i(remove_cpq(triple(Y, Z, W), V)) 0.19/0.48 = { by axiom 3 (main3_li34) } 0.19/0.48 fresh7(?, ?, Y, Z, W, V) 0.19/0.48 0.19/0.48 Lemma 10: i(triple(X, Y, Z)) = i(triple(?, Y, ?)). 0.19/0.48 Proof: 0.19/0.48 i(triple(X, Y, Z)) 0.19/0.48 = { by axiom 6 (main3_li12) } 0.19/0.48 i(triple(W, Y, V)) 0.19/0.48 = { by axiom 6 (main3_li12) } 0.19/0.48 i(triple(?, Y, ?)) 0.19/0.48 0.19/0.48 Lemma 11: pi_sharp_remove(i(triple(?, sK3_co3_V, ?)), sK1_co3_X) = true2. 0.19/0.48 Proof: 0.19/0.48 pi_sharp_remove(i(triple(?, sK3_co3_V, ?)), sK1_co3_X) 0.19/0.48 = { by lemma 10 } 0.19/0.48 pi_sharp_remove(i(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W)), sK1_co3_X) 0.19/0.48 = { by axiom 4 (ax57_1) } 0.19/0.48 fresh25(pi_remove(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W), sK1_co3_X), true2, triple(sK4_co3_U, sK3_co3_V, sK2_co3_W), sK1_co3_X) 0.19/0.48 = { by axiom 8 (co3) } 0.19/0.48 fresh25(true2, true2, triple(sK4_co3_U, sK3_co3_V, sK2_co3_W), sK1_co3_X) 0.19/0.48 = { by axiom 2 (ax57_1) } 0.19/0.48 true2 0.19/0.48 0.19/0.48 Lemma 12: contains_pq(i(triple(?, sK3_co3_V, ?)), sK1_co3_X) = true2. 0.19/0.48 Proof: 0.19/0.48 contains_pq(i(triple(?, sK3_co3_V, ?)), sK1_co3_X) 0.19/0.48 = { by axiom 5 (ax56_1) } 0.19/0.48 fresh27(pi_sharp_remove(i(triple(?, sK3_co3_V, ?)), sK1_co3_X), true2, i(triple(?, sK3_co3_V, ?)), sK1_co3_X) 0.19/0.48 = { by lemma 11 } 0.19/0.48 fresh27(true2, true2, i(triple(?, sK3_co3_V, ?)), sK1_co3_X) 0.19/0.48 = { by axiom 1 (ax56_1) } 0.19/0.48 true2 0.19/0.48 0.19/0.48 Lemma 13: fresh7(contains_pq(i(triple(?, Y, ?)), W), true2, X, Y, Z, W) = remove_pq(i(triple(?, Y, ?)), W). 0.19/0.48 Proof: 0.19/0.48 fresh7(contains_pq(i(triple(?, Y, ?)), W), true2, X, Y, Z, W) 0.19/0.48 = { by lemma 10 } 0.19/0.48 fresh7(contains_pq(i(triple(X, Y, Z)), W), true2, X, Y, Z, W) 0.19/0.48 = { by axiom 7 (main3_li34) } 0.19/0.48 remove_pq(i(triple(X, Y, Z)), W) 0.19/0.48 = { by lemma 10 } 0.19/0.48 remove_pq(i(triple(?, Y, ?)), W) 0.19/0.48 0.19/0.48 Goal 1 (co3_2): tuple2(i(remove_cpq(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W), sK1_co3_X)), pi_sharp_remove(i(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W)), sK1_co3_X)) = tuple2(remove_pq(i(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W)), sK1_co3_X), true2). 0.19/0.48 Proof: 0.19/0.48 tuple2(i(remove_cpq(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W), sK1_co3_X)), pi_sharp_remove(i(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W)), sK1_co3_X)) 0.19/0.48 = { by axiom 3 (main3_li34) } 0.19/0.48 tuple2(fresh7(?, ?, sK4_co3_U, sK3_co3_V, sK2_co3_W, sK1_co3_X), pi_sharp_remove(i(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W)), sK1_co3_X)) 0.19/0.48 = { by lemma 9 } 0.19/0.48 tuple2(fresh7(true2, true2, sK4_co3_U, sK3_co3_V, sK2_co3_W, sK1_co3_X), pi_sharp_remove(i(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W)), sK1_co3_X)) 0.19/0.48 = { by lemma 12 } 0.19/0.48 tuple2(fresh7(contains_pq(i(triple(?, sK3_co3_V, ?)), sK1_co3_X), true2, sK4_co3_U, sK3_co3_V, sK2_co3_W, sK1_co3_X), pi_sharp_remove(i(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W)), sK1_co3_X)) 0.19/0.48 = { by lemma 13 } 0.19/0.48 tuple2(remove_pq(i(triple(?, sK3_co3_V, ?)), sK1_co3_X), pi_sharp_remove(i(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W)), sK1_co3_X)) 0.19/0.48 = { by lemma 10 } 0.19/0.48 tuple2(remove_pq(i(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W)), sK1_co3_X), pi_sharp_remove(i(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W)), sK1_co3_X)) 0.19/0.48 = { by lemma 10 } 0.19/0.48 tuple2(remove_pq(i(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W)), sK1_co3_X), pi_sharp_remove(i(triple(?, sK3_co3_V, ?)), sK1_co3_X)) 0.19/0.48 = { by lemma 11 } 0.19/0.48 tuple2(remove_pq(i(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W)), sK1_co3_X), true2) 0.19/0.48 % SZS output end Proof 0.19/0.48 0.19/0.48 RESULT: Theorem (the conjecture is true). 0.19/0.48 EOF