0.08/0.13 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.08/0.14 % Command : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn 0.15/0.36 % Computer : n017.cluster.edu 0.15/0.36 % Model : x86_64 x86_64 0.15/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.15/0.36 % Memory : 8042.1875MB 0.15/0.36 % OS : Linux 3.10.0-693.el7.x86_64 0.15/0.36 % CPULimit : 180 0.15/0.36 % DateTime : Thu Aug 29 12:56:04 EDT 2019 0.15/0.36 % CPUTime : 2.95/3.11 % SZS status Unsatisfiable 2.95/3.11 2.95/3.11 % SZS output start Proof 2.95/3.11 Take the following subset of the input axioms: 2.95/3.14 fof(composition_associativity_5, axiom, ![A, B, C]: composition(composition(A, B), C)=composition(A, composition(B, C))). 2.95/3.14 fof(composition_distributivity_7, axiom, ![A, B, C]: composition(join(A, B), C)=join(composition(A, C), composition(B, C))). 2.95/3.14 fof(composition_identity_6, axiom, ![A]: A=composition(A, one)). 2.95/3.14 fof(converse_additivity_9, axiom, ![A, B]: join(converse(A), converse(B))=converse(join(A, B))). 2.95/3.14 fof(converse_cancellativity_11, axiom, ![A, B]: join(composition(converse(A), complement(composition(A, B))), complement(B))=complement(B)). 2.95/3.14 fof(converse_idempotence_8, axiom, ![A]: converse(converse(A))=A). 2.95/3.14 fof(converse_multiplicativity_10, axiom, ![A, B]: converse(composition(A, B))=composition(converse(B), converse(A))). 2.95/3.14 fof(def_top_12, axiom, ![A]: join(A, complement(A))=top). 2.95/3.14 fof(def_zero_13, axiom, ![A]: zero=meet(A, complement(A))). 2.95/3.14 fof(goals_14, negated_conjecture, one=join(sk1, one)). 2.95/3.14 fof(goals_15, negated_conjecture, meet(complement(composition(sk1, top)), one)!=meet(complement(sk1), one)). 2.95/3.14 fof(maddux1_join_commutativity_1, axiom, ![A, B]: join(B, A)=join(A, B)). 2.95/3.14 fof(maddux2_join_associativity_2, axiom, ![A, B, C]: join(join(A, B), C)=join(A, join(B, C))). 2.95/3.14 fof(maddux3_a_kind_of_de_Morgan_3, axiom, ![A, B]: join(complement(join(complement(A), complement(B))), complement(join(complement(A), B)))=A). 2.95/3.14 fof(maddux4_definiton_of_meet_4, axiom, ![A, B]: complement(join(complement(A), complement(B)))=meet(A, B)). 2.95/3.14 2.95/3.14 Now clausify the problem and encode Horn clauses using encoding 3 of 2.95/3.14 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 2.95/3.14 We repeatedly replace C & s=t => u=v by the two clauses: 2.95/3.14 fresh(y, y, x1...xn) = u 2.95/3.14 C => fresh(s, t, x1...xn) = v 2.95/3.14 where fresh is a fresh function symbol and x1..xn are the free 2.95/3.14 variables of u and v. 2.95/3.14 A predicate p(X) is encoded as p(X)=true (this is sound, because the 2.95/3.14 input problem has no model of domain size 1). 2.95/3.14 2.95/3.14 The encoding turns the above axioms into the following unit equations and goals: 2.95/3.14 2.95/3.14 Axiom 1 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X). 2.95/3.14 Axiom 2 (maddux4_definiton_of_meet_4): complement(join(complement(X), complement(Y))) = meet(X, Y). 2.95/3.14 Axiom 3 (composition_identity_6): X = composition(X, one). 2.95/3.14 Axiom 4 (maddux3_a_kind_of_de_Morgan_3): join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))) = X. 2.95/3.14 Axiom 5 (converse_idempotence_8): converse(converse(X)) = X. 2.95/3.14 Axiom 6 (maddux2_join_associativity_2): join(join(X, Y), Z) = join(X, join(Y, Z)). 2.95/3.14 Axiom 7 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y). 2.95/3.14 Axiom 8 (def_zero_13): zero = meet(X, complement(X)). 2.95/3.14 Axiom 9 (def_top_12): join(X, complement(X)) = top. 2.95/3.14 Axiom 10 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)). 2.95/3.14 Axiom 11 (converse_additivity_9): join(converse(X), converse(Y)) = converse(join(X, Y)). 2.95/3.14 Axiom 12 (composition_associativity_5): composition(composition(X, Y), Z) = composition(X, composition(Y, Z)). 2.95/3.14 Axiom 13 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)). 2.95/3.14 Axiom 14 (goals_14): one = join(sk1, one). 2.95/3.14 2.95/3.14 Lemma 15: meet(X, Y) = meet(Y, X). 2.95/3.14 Proof: 2.95/3.14 meet(X, Y) 2.95/3.14 = { by axiom 2 (maddux4_definiton_of_meet_4) } 2.95/3.14 complement(join(complement(X), complement(Y))) 2.95/3.14 = { by axiom 1 (maddux1_join_commutativity_1) } 2.95/3.14 complement(join(complement(Y), complement(X))) 2.95/3.14 = { by axiom 2 (maddux4_definiton_of_meet_4) } 2.95/3.14 meet(Y, X) 2.95/3.14 2.95/3.14 Lemma 16: complement(top) = zero. 2.95/3.14 Proof: 2.95/3.14 complement(top) 2.95/3.14 = { by axiom 9 (def_top_12) } 2.95/3.14 complement(join(complement(?), complement(complement(?)))) 2.95/3.14 = { by axiom 2 (maddux4_definiton_of_meet_4) } 2.95/3.14 meet(?, complement(?)) 2.95/3.14 = { by axiom 8 (def_zero_13) } 2.95/3.14 zero 2.95/3.14 2.95/3.14 Lemma 17: complement(join(zero, complement(X))) = meet(X, top). 2.95/3.14 Proof: 2.95/3.14 complement(join(zero, complement(X))) 2.95/3.14 = { by lemma 16 } 2.95/3.14 complement(join(complement(top), complement(X))) 2.95/3.14 = { by axiom 2 (maddux4_definiton_of_meet_4) } 2.95/3.14 meet(top, X) 2.95/3.14 = { by lemma 15 } 2.95/3.14 meet(X, top) 2.95/3.14 2.95/3.14 Lemma 18: composition(converse(one), X) = X. 2.95/3.14 Proof: 2.95/3.14 composition(converse(one), X) 2.95/3.14 = { by axiom 5 (converse_idempotence_8) } 2.95/3.14 composition(converse(one), converse(converse(X))) 2.95/3.14 = { by axiom 10 (converse_multiplicativity_10) } 2.95/3.14 converse(composition(converse(X), one)) 2.95/3.14 = { by axiom 3 (composition_identity_6) } 2.95/3.14 converse(converse(X)) 2.95/3.14 = { by axiom 5 (converse_idempotence_8) } 2.95/3.14 X 2.95/3.14 2.95/3.14 Lemma 19: join(complement(Y), composition(converse(X), complement(composition(X, Y)))) = complement(Y). 2.95/3.14 Proof: 2.95/3.14 join(complement(Y), composition(converse(X), complement(composition(X, Y)))) 2.95/3.14 = { by axiom 1 (maddux1_join_commutativity_1) } 2.95/3.14 join(composition(converse(X), complement(composition(X, Y))), complement(Y)) 2.95/3.14 = { by axiom 7 (converse_cancellativity_11) } 2.95/3.14 complement(Y) 2.95/3.14 2.95/3.14 Lemma 20: join(complement(X), complement(X)) = complement(X). 2.95/3.14 Proof: 2.95/3.14 join(complement(X), complement(X)) 2.95/3.14 = { by lemma 18 } 2.95/3.14 join(complement(X), composition(converse(one), complement(X))) 2.95/3.14 = { by lemma 18 } 2.95/3.14 join(complement(X), composition(converse(one), complement(composition(converse(one), X)))) 2.95/3.14 = { by axiom 3 (composition_identity_6) } 2.95/3.14 join(complement(X), composition(converse(one), complement(composition(composition(converse(one), one), X)))) 2.95/3.14 = { by axiom 12 (composition_associativity_5) } 2.95/3.14 join(complement(X), composition(converse(one), complement(composition(converse(one), composition(one, X))))) 2.95/3.14 = { by lemma 18 } 2.95/3.14 join(complement(X), composition(converse(one), complement(composition(one, X)))) 2.95/3.14 = { by lemma 19 } 2.95/3.14 complement(X) 2.95/3.14 2.95/3.14 Lemma 21: join(meet(X, Y), complement(join(complement(X), Y))) = X. 2.95/3.14 Proof: 2.95/3.14 join(meet(X, Y), complement(join(complement(X), Y))) 2.95/3.14 = { by axiom 2 (maddux4_definiton_of_meet_4) } 2.95/3.14 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))) 2.95/3.14 = { by axiom 4 (maddux3_a_kind_of_de_Morgan_3) } 2.95/3.14 X 2.95/3.14 2.95/3.14 Lemma 22: join(X, join(Y, Z)) = join(Z, join(X, Y)). 2.95/3.14 Proof: 2.95/3.14 join(X, join(Y, Z)) 2.95/3.14 = { by axiom 6 (maddux2_join_associativity_2) } 2.95/3.14 join(join(X, Y), Z) 2.95/3.14 = { by axiom 1 (maddux1_join_commutativity_1) } 2.95/3.14 join(Z, join(X, Y)) 2.95/3.14 2.95/3.14 Lemma 23: join(X, join(complement(X), Y)) = join(Y, top). 2.95/3.14 Proof: 2.95/3.14 join(X, join(complement(X), Y)) 2.95/3.14 = { by lemma 22 } 2.95/3.14 join(complement(X), join(Y, X)) 2.95/3.14 = { by lemma 22 } 2.95/3.14 join(Y, join(X, complement(X))) 2.95/3.14 = { by axiom 9 (def_top_12) } 2.95/3.14 join(Y, top) 2.95/3.14 2.95/3.14 Lemma 24: join(X, top) = top. 2.95/3.14 Proof: 2.95/3.14 join(X, top) 2.95/3.14 = { by axiom 9 (def_top_12) } 2.95/3.14 join(X, join(complement(X), complement(complement(X)))) 2.95/3.14 = { by lemma 23 } 2.95/3.14 join(complement(complement(X)), top) 2.95/3.14 = { by axiom 1 (maddux1_join_commutativity_1) } 2.95/3.14 join(top, complement(complement(X))) 2.95/3.14 = { by axiom 9 (def_top_12) } 2.95/3.14 join(join(complement(X), complement(complement(X))), complement(complement(X))) 2.95/3.14 = { by axiom 6 (maddux2_join_associativity_2) } 2.95/3.14 join(complement(X), join(complement(complement(X)), complement(complement(X)))) 2.95/3.14 = { by lemma 20 } 2.95/3.14 join(complement(X), complement(complement(X))) 2.95/3.14 = { by axiom 9 (def_top_12) } 2.95/3.14 top 2.95/3.14 2.95/3.14 Lemma 25: join(zero, meet(X, top)) = X. 2.95/3.14 Proof: 2.95/3.14 join(zero, meet(X, top)) 2.95/3.14 = { by axiom 1 (maddux1_join_commutativity_1) } 2.95/3.14 join(meet(X, top), zero) 2.95/3.14 = { by lemma 16 } 2.95/3.14 join(meet(X, top), complement(top)) 2.95/3.14 = { by lemma 24 } 2.95/3.14 join(meet(X, top), complement(join(complement(X), top))) 2.95/3.14 = { by lemma 21 } 2.95/3.14 X 2.95/3.14 2.95/3.14 Lemma 26: join(zero, complement(X)) = complement(X). 2.95/3.14 Proof: 2.95/3.14 join(zero, complement(X)) 2.95/3.14 = { by lemma 21 } 2.95/3.14 join(zero, complement(join(meet(X, complement(X)), complement(join(complement(X), complement(X)))))) 2.95/3.14 = { by axiom 2 (maddux4_definiton_of_meet_4) } 2.95/3.14 join(zero, complement(join(meet(X, complement(X)), meet(X, X)))) 2.95/3.14 = { by axiom 1 (maddux1_join_commutativity_1) } 2.95/3.14 join(zero, complement(join(meet(X, X), meet(X, complement(X))))) 2.95/3.14 = { by axiom 2 (maddux4_definiton_of_meet_4) } 2.95/3.14 join(zero, complement(join(complement(join(complement(X), complement(X))), meet(X, complement(X))))) 2.95/3.14 = { by lemma 20 } 2.95/3.14 join(zero, complement(join(complement(complement(X)), meet(X, complement(X))))) 2.95/3.14 = { by axiom 8 (def_zero_13) } 2.95/3.14 join(zero, complement(join(complement(complement(X)), zero))) 2.95/3.14 = { by axiom 1 (maddux1_join_commutativity_1) } 2.95/3.14 join(zero, complement(join(zero, complement(complement(X))))) 2.95/3.14 = { by lemma 17 } 2.95/3.14 join(zero, meet(complement(X), top)) 2.95/3.14 = { by lemma 25 } 2.95/3.14 complement(X) 2.95/3.14 2.95/3.14 Lemma 27: complement(join(X, complement(Y))) = meet(Y, complement(X)). 2.95/3.14 Proof: 2.95/3.14 complement(join(X, complement(Y))) 2.95/3.14 = { by axiom 1 (maddux1_join_commutativity_1) } 2.95/3.14 complement(join(complement(Y), X)) 2.95/3.14 = { by lemma 25 } 2.95/3.14 complement(join(complement(Y), join(zero, meet(X, top)))) 2.95/3.14 = { by lemma 17 } 2.95/3.14 complement(join(complement(Y), join(zero, complement(join(zero, complement(X)))))) 2.95/3.14 = { by lemma 26 } 2.95/3.14 complement(join(complement(Y), complement(join(zero, complement(X))))) 2.95/3.14 = { by lemma 17 } 2.95/3.14 complement(join(complement(Y), meet(X, top))) 2.95/3.14 = { by axiom 1 (maddux1_join_commutativity_1) } 2.95/3.14 complement(join(meet(X, top), complement(Y))) 2.95/3.14 = { by axiom 2 (maddux4_definiton_of_meet_4) } 2.95/3.14 complement(join(complement(join(complement(X), complement(top))), complement(Y))) 2.95/3.14 = { by axiom 2 (maddux4_definiton_of_meet_4) } 2.95/3.14 meet(join(complement(X), complement(top)), Y) 2.95/3.14 = { by lemma 15 } 2.95/3.14 meet(Y, join(complement(X), complement(top))) 2.95/3.14 = { by axiom 1 (maddux1_join_commutativity_1) } 2.95/3.14 meet(Y, join(complement(top), complement(X))) 2.95/3.14 = { by lemma 16 } 2.95/3.14 meet(Y, join(zero, complement(X))) 2.95/3.14 = { by lemma 26 } 2.95/3.14 meet(Y, complement(X)) 2.95/3.14 2.95/3.14 Lemma 28: converse(join(X, converse(Y))) = join(Y, converse(X)). 2.95/3.14 Proof: 2.95/3.14 converse(join(X, converse(Y))) 2.95/3.14 = { by axiom 1 (maddux1_join_commutativity_1) } 2.95/3.14 converse(join(converse(Y), X)) 2.95/3.14 = { by axiom 11 (converse_additivity_9) } 2.95/3.14 join(converse(converse(Y)), converse(X)) 2.95/3.14 = { by axiom 5 (converse_idempotence_8) } 2.95/3.14 join(Y, converse(X)) 2.95/3.14 2.95/3.14 Lemma 29: converse(join(converse(X), Y)) = join(X, converse(Y)). 2.95/3.14 Proof: 2.95/3.14 converse(join(converse(X), Y)) 2.95/3.14 = { by axiom 1 (maddux1_join_commutativity_1) } 2.95/3.14 converse(join(Y, converse(X))) 2.95/3.14 = { by lemma 28 } 2.95/3.14 join(X, converse(Y)) 2.95/3.14 2.95/3.14 Lemma 30: join(X, converse(top)) = converse(top). 2.95/3.14 Proof: 2.95/3.14 join(X, converse(top)) 2.95/3.14 = { by lemma 29 } 2.95/3.14 converse(join(converse(X), top)) 2.95/3.14 = { by lemma 24 } 2.95/3.14 converse(top) 2.95/3.14 2.95/3.14 Lemma 31: converse(top) = top. 2.95/3.14 Proof: 2.95/3.14 converse(top) 2.95/3.14 = { by lemma 30 } 2.95/3.14 join(?, converse(top)) 2.95/3.14 = { by lemma 30 } 2.95/3.14 join(?, join(complement(?), converse(top))) 2.95/3.14 = { by lemma 23 } 2.95/3.14 join(converse(top), top) 2.95/3.14 = { by lemma 24 } 2.95/3.16 top 2.95/3.16 2.95/3.16 Goal 1 (goals_15): meet(complement(composition(sk1, top)), one) = meet(complement(sk1), one). 2.95/3.16 Proof: 2.95/3.16 meet(complement(composition(sk1, top)), one) 2.95/3.16 = { by lemma 15 } 2.95/3.17 meet(one, complement(composition(sk1, top))) 2.95/3.17 = { by lemma 27 } 2.95/3.17 complement(join(composition(sk1, top), complement(one))) 2.95/3.17 = { by axiom 1 (maddux1_join_commutativity_1) } 2.95/3.17 complement(join(complement(one), composition(sk1, top))) 2.95/3.17 = { by axiom 5 (converse_idempotence_8) } 2.95/3.17 complement(join(complement(one), composition(converse(converse(sk1)), top))) 2.95/3.17 = { by lemma 31 } 2.95/3.17 complement(join(complement(one), composition(converse(converse(sk1)), converse(top)))) 2.95/3.17 = { by lemma 24 } 2.95/3.17 complement(join(complement(one), composition(converse(converse(sk1)), converse(join(one, top))))) 2.95/3.17 = { by lemma 31 } 2.95/3.17 complement(join(complement(one), composition(converse(converse(sk1)), converse(join(one, converse(top)))))) 2.95/3.17 = { by axiom 9 (def_top_12) } 2.95/3.17 complement(join(complement(one), composition(converse(converse(sk1)), converse(join(one, converse(join(converse(sk1), complement(converse(sk1))))))))) 2.95/3.17 = { by lemma 29 } 2.95/3.17 complement(join(complement(one), composition(converse(converse(sk1)), converse(join(one, join(sk1, converse(complement(converse(sk1))))))))) 2.95/3.17 = { by axiom 6 (maddux2_join_associativity_2) } 2.95/3.17 complement(join(complement(one), composition(converse(converse(sk1)), converse(join(join(one, sk1), converse(complement(converse(sk1)))))))) 2.95/3.17 = { by axiom 1 (maddux1_join_commutativity_1) } 2.95/3.17 complement(join(complement(one), composition(converse(converse(sk1)), converse(join(join(sk1, one), converse(complement(converse(sk1)))))))) 2.95/3.17 = { by axiom 14 (goals_14) } 2.95/3.17 complement(join(complement(one), composition(converse(converse(sk1)), converse(join(one, converse(complement(converse(sk1)))))))) 2.95/3.17 = { by lemma 28 } 2.95/3.17 complement(join(complement(one), composition(converse(converse(sk1)), join(complement(converse(sk1)), converse(one))))) 2.95/3.17 = { by axiom 3 (composition_identity_6) } 2.95/3.17 complement(join(complement(one), composition(converse(converse(sk1)), join(complement(converse(sk1)), composition(converse(one), one))))) 2.95/3.17 = { by lemma 18 } 2.95/3.17 complement(join(complement(one), composition(converse(converse(sk1)), join(complement(converse(sk1)), one)))) 2.95/3.17 = { by axiom 5 (converse_idempotence_8) } 2.95/3.17 complement(join(complement(one), converse(converse(composition(converse(converse(sk1)), join(complement(converse(sk1)), one)))))) 2.95/3.17 = { by axiom 1 (maddux1_join_commutativity_1) } 2.95/3.17 complement(join(complement(one), converse(converse(composition(converse(converse(sk1)), join(one, complement(converse(sk1)))))))) 2.95/3.17 = { by axiom 10 (converse_multiplicativity_10) } 2.95/3.17 complement(join(complement(one), converse(composition(converse(join(one, complement(converse(sk1)))), converse(converse(converse(sk1))))))) 2.95/3.17 = { by axiom 11 (converse_additivity_9) } 2.95/3.17 complement(join(complement(one), converse(composition(join(converse(one), converse(complement(converse(sk1)))), converse(converse(converse(sk1))))))) 2.95/3.17 = { by axiom 1 (maddux1_join_commutativity_1) } 2.95/3.17 complement(join(complement(one), converse(composition(join(converse(complement(converse(sk1))), converse(one)), converse(converse(converse(sk1))))))) 2.95/3.17 = { by axiom 13 (composition_distributivity_7) } 2.95/3.17 complement(join(complement(one), converse(join(composition(converse(complement(converse(sk1))), converse(converse(converse(sk1)))), composition(converse(one), converse(converse(converse(sk1)))))))) 2.95/3.17 = { by axiom 10 (converse_multiplicativity_10) } 2.95/3.17 complement(join(complement(one), converse(join(converse(composition(converse(converse(sk1)), complement(converse(sk1)))), composition(converse(one), converse(converse(converse(sk1)))))))) 2.95/3.17 = { by axiom 1 (maddux1_join_commutativity_1) } 2.95/3.17 complement(join(complement(one), converse(join(composition(converse(one), converse(converse(converse(sk1)))), converse(composition(converse(converse(sk1)), complement(converse(sk1)))))))) 2.95/3.17 = { by lemma 18 } 2.95/3.17 complement(join(complement(one), converse(join(converse(converse(converse(sk1))), converse(composition(converse(converse(sk1)), complement(converse(sk1)))))))) 2.95/3.17 = { by axiom 11 (converse_additivity_9) } 2.95/3.17 complement(join(complement(one), converse(converse(join(converse(converse(sk1)), composition(converse(converse(sk1)), complement(converse(sk1)))))))) 2.95/3.17 = { by axiom 5 (converse_idempotence_8) } 2.95/3.17 complement(join(complement(one), join(converse(converse(sk1)), composition(converse(converse(sk1)), complement(converse(sk1)))))) 2.95/3.17 = { by axiom 6 (maddux2_join_associativity_2) } 2.95/3.17 complement(join(join(complement(one), converse(converse(sk1))), composition(converse(converse(sk1)), complement(converse(sk1))))) 2.95/3.17 = { by axiom 1 (maddux1_join_commutativity_1) } 2.95/3.17 complement(join(join(converse(converse(sk1)), complement(one)), composition(converse(converse(sk1)), complement(converse(sk1))))) 2.95/3.17 = { by axiom 6 (maddux2_join_associativity_2) } 2.95/3.17 complement(join(converse(converse(sk1)), join(complement(one), composition(converse(converse(sk1)), complement(converse(sk1)))))) 2.95/3.17 = { by axiom 5 (converse_idempotence_8) } 2.95/3.17 complement(join(sk1, join(complement(one), composition(converse(converse(sk1)), complement(converse(sk1)))))) 2.95/3.17 = { by axiom 3 (composition_identity_6) } 2.95/3.17 complement(join(sk1, join(complement(one), composition(converse(converse(sk1)), complement(composition(converse(sk1), one)))))) 2.95/3.17 = { by lemma 19 } 2.95/3.17 complement(join(sk1, complement(one))) 2.95/3.17 = { by lemma 27 } 2.95/3.17 meet(one, complement(sk1)) 2.95/3.17 = { by lemma 15 } 2.95/3.17 meet(complement(sk1), one) 2.95/3.17 % SZS output end Proof 2.95/3.17 2.95/3.17 RESULT: Unsatisfiable (the axioms are contradictory). 2.95/3.17 EOF