0.03/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.03/0.13 % 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 : n026.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 : 1200 0.12/0.34 % WCLimit : 120 0.12/0.34 % DateTime : Tue Jul 13 10:48:59 EDT 2021 0.12/0.35 % CPUTime : 0.18/0.42 % SZS status Theorem 0.18/0.42 0.18/0.43 % SZS output start Proof 0.18/0.43 Take the following subset of the input axioms: 0.18/0.43 fof(ax2, axiom, ![U]: (![V]: (sorti2(op2(U, V)) <= sorti2(V)) <= sorti2(U))). 0.18/0.43 fof(ax3, axiom, ?[U]: (sorti1(U) & ![V]: (sorti1(V) => op1(V, V)=U))). 0.18/0.43 fof(ax4, axiom, ~?[U]: (![V]: (sorti2(V) => op2(V, V)=U) & sorti2(U))). 0.18/0.43 fof(co1, conjecture, (![U]: (sorti1(U) => sorti2(h(U))) & ![V]: (sorti2(V) => sorti1(j(V)))) => ~(![W]: (![X]: (sorti1(X) => h(op1(W, X))=op2(h(W), h(X))) <= sorti1(W)) & (![X1]: (sorti2(X1) => X1=h(j(X1))) & (![X2]: (sorti1(X2) => j(h(X2))=X2) & ![Y]: (sorti2(Y) => ![Z]: (sorti2(Z) => j(op2(Y, Z))=op1(j(Y), j(Z)))))))). 0.18/0.43 0.18/0.43 Now clausify the problem and encode Horn clauses using encoding 3 of 0.18/0.43 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 0.18/0.43 We repeatedly replace C & s=t => u=v by the two clauses: 0.18/0.43 fresh(y, y, x1...xn) = u 0.18/0.43 C => fresh(s, t, x1...xn) = v 0.18/0.43 where fresh is a fresh function symbol and x1..xn are the free 0.18/0.43 variables of u and v. 0.18/0.43 A predicate p(X) is encoded as p(X)=true (this is sound, because the 0.18/0.43 input problem has no model of domain size 1). 0.18/0.43 0.18/0.43 The encoding turns the above axioms into the following unit equations and goals: 0.18/0.43 0.18/0.43 Axiom 1 (ax3): sorti1(u) = true2. 0.18/0.43 Axiom 2 (co1_4): fresh(X, X, Y) = Y. 0.18/0.43 Axiom 3 (ax3_1): fresh10(X, X, Y) = u. 0.18/0.43 Axiom 4 (ax4_1): fresh9(X, X, Y) = true2. 0.18/0.43 Axiom 5 (co1): fresh8(X, X, Y) = true2. 0.18/0.43 Axiom 6 (co1_3): fresh5(X, X, Y) = true2. 0.18/0.43 Axiom 7 (co1_4): fresh(sorti2(X), true2, X) = h(j(X)). 0.18/0.43 Axiom 8 (ax2): fresh12(X, X, Y, Z) = sorti2(op2(Y, Z)). 0.18/0.43 Axiom 9 (ax2): fresh11(X, X, Y, Z) = true2. 0.18/0.43 Axiom 10 (ax3_1): fresh10(sorti1(X), true2, X) = op1(X, X). 0.18/0.43 Axiom 11 (ax4_1): fresh9(sorti2(X), true2, X) = sorti2(v(X)). 0.18/0.43 Axiom 12 (co1): fresh8(sorti1(X), true2, X) = sorti2(h(X)). 0.18/0.43 Axiom 13 (co1_3): fresh5(sorti2(X), true2, X) = sorti1(j(X)). 0.18/0.43 Axiom 14 (co1_5): fresh4(X, X, Y, Z) = op1(j(Y), j(Z)). 0.18/0.43 Axiom 15 (co1_5): fresh3(X, X, Y, Z) = j(op2(Y, Z)). 0.18/0.43 Axiom 16 (ax2): fresh12(sorti2(X), true2, Y, X) = fresh11(sorti2(Y), true2, Y, X). 0.18/0.43 Axiom 17 (co1_5): fresh4(sorti2(X), true2, Y, X) = fresh3(sorti2(Y), true2, Y, X). 0.18/0.43 0.18/0.43 Lemma 18: sorti2(h(u)) = true2. 0.18/0.43 Proof: 0.18/0.43 sorti2(h(u)) 0.18/0.43 = { by axiom 12 (co1) R->L } 0.18/0.43 fresh8(sorti1(u), true2, u) 0.18/0.43 = { by axiom 1 (ax3) } 0.18/0.43 fresh8(true2, true2, u) 0.18/0.43 = { by axiom 5 (co1) } 0.18/0.43 true2 0.18/0.43 0.18/0.43 Lemma 19: sorti2(v(h(u))) = true2. 0.18/0.43 Proof: 0.18/0.43 sorti2(v(h(u))) 0.18/0.43 = { by axiom 11 (ax4_1) R->L } 0.18/0.43 fresh9(sorti2(h(u)), true2, h(u)) 0.18/0.43 = { by lemma 18 } 0.18/0.43 fresh9(true2, true2, h(u)) 0.18/0.43 = { by axiom 4 (ax4_1) } 0.18/0.44 true2 0.18/0.44 0.18/0.44 Goal 1 (ax4): tuple(op2(v(X), v(X)), sorti2(X)) = tuple(X, true2). 0.18/0.44 The goal is true when: 0.18/0.44 X = h(u) 0.18/0.44 0.18/0.44 Proof: 0.18/0.44 tuple(op2(v(h(u)), v(h(u))), sorti2(h(u))) 0.18/0.44 = { by axiom 2 (co1_4) R->L } 0.18/0.44 tuple(fresh(true2, true2, op2(v(h(u)), v(h(u)))), sorti2(h(u))) 0.18/0.44 = { by axiom 9 (ax2) R->L } 0.18/0.44 tuple(fresh(fresh11(true2, true2, v(h(u)), v(h(u))), true2, op2(v(h(u)), v(h(u)))), sorti2(h(u))) 0.18/0.44 = { by lemma 19 R->L } 0.18/0.44 tuple(fresh(fresh11(sorti2(v(h(u))), true2, v(h(u)), v(h(u))), true2, op2(v(h(u)), v(h(u)))), sorti2(h(u))) 0.18/0.44 = { by axiom 16 (ax2) R->L } 0.18/0.44 tuple(fresh(fresh12(sorti2(v(h(u))), true2, v(h(u)), v(h(u))), true2, op2(v(h(u)), v(h(u)))), sorti2(h(u))) 0.18/0.44 = { by lemma 19 } 0.18/0.44 tuple(fresh(fresh12(true2, true2, v(h(u)), v(h(u))), true2, op2(v(h(u)), v(h(u)))), sorti2(h(u))) 0.18/0.44 = { by axiom 8 (ax2) } 0.18/0.44 tuple(fresh(sorti2(op2(v(h(u)), v(h(u)))), true2, op2(v(h(u)), v(h(u)))), sorti2(h(u))) 0.18/0.44 = { by axiom 7 (co1_4) } 0.18/0.44 tuple(h(j(op2(v(h(u)), v(h(u))))), sorti2(h(u))) 0.18/0.44 = { by axiom 15 (co1_5) R->L } 0.18/0.44 tuple(h(fresh3(true2, true2, v(h(u)), v(h(u)))), sorti2(h(u))) 0.18/0.44 = { by lemma 19 R->L } 0.18/0.44 tuple(h(fresh3(sorti2(v(h(u))), true2, v(h(u)), v(h(u)))), sorti2(h(u))) 0.18/0.44 = { by axiom 17 (co1_5) R->L } 0.18/0.44 tuple(h(fresh4(sorti2(v(h(u))), true2, v(h(u)), v(h(u)))), sorti2(h(u))) 0.18/0.44 = { by lemma 19 } 0.18/0.44 tuple(h(fresh4(true2, true2, v(h(u)), v(h(u)))), sorti2(h(u))) 0.18/0.44 = { by axiom 14 (co1_5) } 0.18/0.44 tuple(h(op1(j(v(h(u))), j(v(h(u))))), sorti2(h(u))) 0.18/0.44 = { by axiom 10 (ax3_1) R->L } 0.18/0.44 tuple(h(fresh10(sorti1(j(v(h(u)))), true2, j(v(h(u))))), sorti2(h(u))) 0.18/0.44 = { by axiom 13 (co1_3) R->L } 0.18/0.44 tuple(h(fresh10(fresh5(sorti2(v(h(u))), true2, v(h(u))), true2, j(v(h(u))))), sorti2(h(u))) 0.18/0.44 = { by lemma 19 } 0.18/0.44 tuple(h(fresh10(fresh5(true2, true2, v(h(u))), true2, j(v(h(u))))), sorti2(h(u))) 0.18/0.44 = { by axiom 6 (co1_3) } 0.18/0.44 tuple(h(fresh10(true2, true2, j(v(h(u))))), sorti2(h(u))) 0.18/0.44 = { by axiom 3 (ax3_1) } 0.18/0.44 tuple(h(u), sorti2(h(u))) 0.18/0.44 = { by lemma 18 } 0.18/0.44 tuple(h(u), true2) 0.18/0.44 % SZS output end Proof 0.18/0.44 0.18/0.44 RESULT: Theorem (the conjecture is true). 0.18/0.44 EOF