0.06/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.06/0.12 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof 0.13/0.34 % Computer : n002.cluster.edu 0.13/0.34 % Model : x86_64 x86_64 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.13/0.34 % Memory : 8042.1875MB 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64 0.13/0.34 % CPULimit : 1200 0.13/0.34 % WCLimit : 120 0.13/0.34 % DateTime : Tue Jul 13 12:37:33 EDT 2021 0.13/0.34 % CPUTime : 1.78/0.64 % SZS status Theorem 1.78/0.64 2.40/0.66 % SZS output start Proof 2.40/0.66 Take the following subset of the input axioms: 2.40/0.66 fof(modus_ponens_for_nand, axiom, ![P, Q, R]: (p(R) <= (p(nand(P, nand(Q, R))) & p(P)))). 2.40/0.66 fof(tautology, conjecture, ![A, B]: p(nand(nand(A, nand(B, B)), nand(nand(nand(B, B), nand(nand(A, A), nand(A, A))), nand(nand(B, B), nand(nand(A, A), nand(A, A))))))). 2.40/0.66 fof(wajsbergs_axiom, axiom, ![A, B, C, D]: p(nand(nand(A, nand(B, C)), nand(nand(nand(D, C), nand(nand(A, D), nand(A, D))), nand(A, nand(A, B)))))). 2.40/0.66 2.40/0.66 Now clausify the problem and encode Horn clauses using encoding 3 of 2.40/0.66 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 2.40/0.66 We repeatedly replace C & s=t => u=v by the two clauses: 2.40/0.66 fresh(y, y, x1...xn) = u 2.40/0.66 C => fresh(s, t, x1...xn) = v 2.40/0.66 where fresh is a fresh function symbol and x1..xn are the free 2.40/0.66 variables of u and v. 2.40/0.66 A predicate p(X) is encoded as p(X)=true (this is sound, because the 2.40/0.66 input problem has no model of domain size 1). 2.40/0.66 2.40/0.66 The encoding turns the above axioms into the following unit equations and goals: 2.40/0.66 2.40/0.66 Axiom 1 (modus_ponens_for_nand): fresh2(X, X, Y) = true. 2.40/0.66 Axiom 2 (modus_ponens_for_nand): fresh(X, X, Y, Z) = p(Z). 2.40/0.66 Axiom 3 (modus_ponens_for_nand): fresh(p(nand(X, nand(Y, Z))), true, X, Z) = fresh2(p(X), true, Z). 2.40/0.66 Axiom 4 (wajsbergs_axiom): p(nand(nand(X, nand(Y, Z)), nand(nand(nand(W, Z), nand(nand(X, W), nand(X, W))), nand(X, nand(X, Y))))) = true. 2.40/0.66 2.40/0.66 Lemma 5: fresh2(p(nand(X, nand(Y, Z))), true, nand(X, nand(X, Y))) = p(nand(X, nand(X, Y))). 2.40/0.66 Proof: 2.40/0.66 fresh2(p(nand(X, nand(Y, Z))), true, nand(X, nand(X, Y))) 2.40/0.66 = { by axiom 3 (modus_ponens_for_nand) R->L } 2.40/0.66 fresh(p(nand(nand(X, nand(Y, Z)), nand(nand(nand(W, Z), nand(nand(X, W), nand(X, W))), nand(X, nand(X, Y))))), true, nand(X, nand(Y, Z)), nand(X, nand(X, Y))) 2.40/0.66 = { by axiom 4 (wajsbergs_axiom) } 2.40/0.66 fresh(true, true, nand(X, nand(Y, Z)), nand(X, nand(X, Y))) 2.40/0.66 = { by axiom 2 (modus_ponens_for_nand) } 2.40/0.66 p(nand(X, nand(X, Y))) 2.40/0.66 2.40/0.67 Lemma 6: p(nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(nand(W, Z), nand(nand(X, W), nand(X, W)))))) = true. 2.40/0.67 Proof: 2.40/0.67 p(nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(nand(W, Z), nand(nand(X, W), nand(X, W)))))) 2.40/0.67 = { by lemma 5 R->L } 2.40/0.67 fresh2(p(nand(nand(X, nand(Y, Z)), nand(nand(nand(W, Z), nand(nand(X, W), nand(X, W))), nand(X, nand(X, Y))))), true, nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(nand(W, Z), nand(nand(X, W), nand(X, W)))))) 2.40/0.67 = { by axiom 4 (wajsbergs_axiom) } 2.40/0.67 fresh2(true, true, nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(nand(W, Z), nand(nand(X, W), nand(X, W)))))) 2.40/0.67 = { by axiom 1 (modus_ponens_for_nand) } 2.40/0.67 true 2.40/0.67 2.40/0.67 Lemma 7: p(nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))))) = true. 2.40/0.67 Proof: 2.40/0.67 p(nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))))) 2.40/0.67 = { by lemma 5 R->L } 2.40/0.67 fresh2(p(nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(nand(W, Z), nand(nand(X, W), nand(X, W)))))), true, nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))))) 2.40/0.67 = { by lemma 6 } 2.40/0.67 fresh2(true, true, nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))))) 2.40/0.67 = { by axiom 1 (modus_ponens_for_nand) } 2.40/0.67 true 2.40/0.67 2.40/0.67 Lemma 8: fresh2(p(nand(X, nand(Y, Z))), true, nand(nand(W, Z), nand(nand(X, W), nand(X, W)))) = p(nand(nand(W, Z), nand(nand(X, W), nand(X, W)))). 2.40/0.67 Proof: 2.40/0.67 fresh2(p(nand(X, nand(Y, Z))), true, nand(nand(W, Z), nand(nand(X, W), nand(X, W)))) 2.40/0.67 = { by axiom 3 (modus_ponens_for_nand) R->L } 2.40/0.67 fresh(p(nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(nand(W, Z), nand(nand(X, W), nand(X, W)))))), true, nand(X, nand(Y, Z)), nand(nand(W, Z), nand(nand(X, W), nand(X, W)))) 2.40/0.67 = { by lemma 6 } 2.40/0.67 fresh(true, true, nand(X, nand(Y, Z)), nand(nand(W, Z), nand(nand(X, W), nand(X, W)))) 2.40/0.67 = { by axiom 2 (modus_ponens_for_nand) } 2.40/0.67 p(nand(nand(W, Z), nand(nand(X, W), nand(X, W)))) 2.40/0.67 2.40/0.67 Lemma 9: p(nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z)))) = true. 2.40/0.67 Proof: 2.40/0.67 p(nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z)))) 2.40/0.67 = { by axiom 2 (modus_ponens_for_nand) R->L } 2.40/0.67 fresh(true, true, nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z)))), nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z)))) 2.40/0.67 = { by axiom 1 (modus_ponens_for_nand) R->L } 2.40/0.67 fresh(fresh2(true, true, nand(nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z)))), nand(nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z))), nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z)))))), true, nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z)))), nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z)))) 2.40/0.67 = { by lemma 7 R->L } 2.40/0.67 fresh(fresh2(p(nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z)))))), true, nand(nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z)))), nand(nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z))), nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z)))))), true, nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z)))), nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z)))) 2.40/0.67 = { by lemma 8 } 2.40/0.67 fresh(p(nand(nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z)))), nand(nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z))), nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z)))))), true, nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z)))), nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z)))) 2.40/0.67 = { by axiom 3 (modus_ponens_for_nand) } 2.40/0.67 fresh2(p(nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))))), true, nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z)))) 2.40/0.67 = { by lemma 7 } 2.40/0.67 fresh2(true, true, nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z)))) 2.40/0.67 = { by axiom 1 (modus_ponens_for_nand) } 2.40/0.67 true 2.40/0.67 2.40/0.67 Lemma 10: p(nand(nand(X, Y), nand(nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), X), nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), X)))) = true. 2.40/0.67 Proof: 2.40/0.67 p(nand(nand(X, Y), nand(nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), X), nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), X)))) 2.40/0.67 = { by lemma 8 R->L } 2.40/0.67 fresh2(p(nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), Y))), true, nand(nand(X, Y), nand(nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), X), nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), X)))) 2.40/0.67 = { by lemma 5 R->L } 2.40/0.67 fresh2(fresh2(p(nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), nand(Y, nand(Z, W)))), true, nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), Y))), true, nand(nand(X, Y), nand(nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), X), nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), X)))) 2.40/0.67 = { by lemma 9 } 2.40/0.67 fresh2(fresh2(true, true, nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), Y))), true, nand(nand(X, Y), nand(nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), X), nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), X)))) 2.40/0.67 = { by axiom 1 (modus_ponens_for_nand) } 2.40/0.67 fresh2(true, true, nand(nand(X, Y), nand(nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), X), nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), X)))) 2.40/0.67 = { by axiom 1 (modus_ponens_for_nand) } 2.40/0.67 true 2.40/0.67 2.40/0.67 Goal 1 (tautology): p(nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))))) = true. 2.40/0.67 Proof: 2.40/0.67 p(nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))))) 2.40/0.67 = { by axiom 2 (modus_ponens_for_nand) R->L } 2.40/0.67 fresh(true, true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a)), nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))))) 2.40/0.67 = { by axiom 1 (modus_ponens_for_nand) R->L } 2.40/0.67 fresh(fresh2(true, true, nand(nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a)), nand(nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a))))), nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))))))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a)), nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))))) 2.40/0.67 = { by lemma 10 R->L } 2.40/0.67 fresh(fresh2(p(nand(nand(a, nand(b, b)), nand(nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a)))), true, nand(nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a)), nand(nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a))))), nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))))))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a)), nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))))) 2.40/0.67 = { by lemma 8 } 2.40/0.67 fresh(p(nand(nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a)), nand(nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a))))), nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))))))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a)), nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))))) 2.40/0.67 = { by axiom 3 (modus_ponens_for_nand) } 2.40/0.67 fresh2(p(nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a))), true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))))) 2.40/0.67 = { by lemma 5 R->L } 2.40/0.67 fresh2(fresh2(p(nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a))), true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))))) 2.40/0.67 = { by axiom 2 (modus_ponens_for_nand) R->L } 2.40/0.68 fresh2(fresh2(fresh(true, true, nand(nand(a, a), nand(nand(a, a), nand(a, a))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a))), true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))))) 2.40/0.68 = { by axiom 1 (modus_ponens_for_nand) R->L } 2.40/0.68 fresh2(fresh2(fresh(fresh2(true, true, nand(nand(nand(a, a), nand(nand(a, a), nand(a, a))), nand(nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a)), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))))), true, nand(nand(a, a), nand(nand(a, a), nand(a, a))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a))), true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))))) 2.40/0.68 = { by lemma 9 R->L } 2.40/0.68 fresh2(fresh2(fresh(fresh2(p(nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(b, b), nand(nand(a, a), nand(a, a))))), true, nand(nand(nand(a, a), nand(nand(a, a), nand(a, a))), nand(nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a)), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))))), true, nand(nand(a, a), nand(nand(a, a), nand(a, a))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a))), true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))))) 2.40/0.68 = { by lemma 8 } 2.40/0.68 fresh2(fresh2(fresh(p(nand(nand(nand(a, a), nand(nand(a, a), nand(a, a))), nand(nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a)), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))))), true, nand(nand(a, a), nand(nand(a, a), nand(a, a))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a))), true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))))) 2.54/0.68 = { by axiom 3 (modus_ponens_for_nand) } 2.54/0.68 fresh2(fresh2(fresh2(p(nand(nand(a, a), nand(nand(a, a), nand(a, a)))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a))), true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))))) 2.54/0.68 = { by lemma 5 R->L } 2.54/0.68 fresh2(fresh2(fresh2(fresh2(p(nand(nand(a, a), nand(nand(a, a), nand(nand(nand(a, nand(X, Y)), nand(a, nand(X, Y))), a)))), true, nand(nand(a, a), nand(nand(a, a), nand(a, a)))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a))), true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))))) 2.54/0.68 = { by lemma 5 R->L } 2.54/0.68 fresh2(fresh2(fresh2(fresh2(fresh2(p(nand(nand(a, a), nand(nand(nand(nand(a, nand(X, Y)), nand(a, nand(X, Y))), a), nand(nand(nand(a, nand(X, Y)), nand(a, nand(X, Y))), a)))), true, nand(nand(a, a), nand(nand(a, a), nand(nand(nand(a, nand(X, Y)), nand(a, nand(X, Y))), a)))), true, nand(nand(a, a), nand(nand(a, a), nand(a, a)))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a))), true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))))) 2.54/0.68 = { by lemma 10 } 2.54/0.68 fresh2(fresh2(fresh2(fresh2(fresh2(true, true, nand(nand(a, a), nand(nand(a, a), nand(nand(nand(a, nand(X, Y)), nand(a, nand(X, Y))), a)))), true, nand(nand(a, a), nand(nand(a, a), nand(a, a)))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a))), true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))))) 2.54/0.68 = { by axiom 1 (modus_ponens_for_nand) } 2.54/0.68 fresh2(fresh2(fresh2(fresh2(true, true, nand(nand(a, a), nand(nand(a, a), nand(a, a)))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a))), true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))))) 2.54/0.68 = { by axiom 1 (modus_ponens_for_nand) } 2.54/0.68 fresh2(fresh2(fresh2(true, true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a))), true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))))) 2.54/0.68 = { by axiom 1 (modus_ponens_for_nand) } 2.54/0.68 fresh2(fresh2(true, true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a))), true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))))) 2.54/0.68 = { by axiom 1 (modus_ponens_for_nand) } 2.54/0.68 fresh2(true, true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))))) 2.54/0.68 = { by axiom 1 (modus_ponens_for_nand) } 2.54/0.68 true 2.54/0.68 % SZS output end Proof 2.54/0.68 2.54/0.68 RESULT: Theorem (the conjecture is true). 2.54/0.68 EOF