0.05/0.11 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.05/0.11 % Command : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn 0.10/0.32 % Computer : n028.cluster.edu 0.10/0.32 % Model : x86_64 x86_64 0.10/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.10/0.32 % Memory : 8042.1875MB 0.10/0.32 % OS : Linux 3.10.0-693.el7.x86_64 0.10/0.32 % CPULimit : 960 0.10/0.32 % WCLimit : 120 0.10/0.32 % DateTime : Thu Jul 2 07:45:37 EDT 2020 0.10/0.32 % CPUTime : 300.83/38.17 % SZS status Theorem 300.83/38.17 300.83/38.17 % SZS output start Proof 300.83/38.17 Take the following subset of the input axioms: 300.83/38.18 fof(modus_ponens_for_nand, axiom, ![Q, P, R]: (p(R) <= (p(nand(P, nand(Q, R))) & p(P)))). 300.83/38.18 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))))))). 300.83/38.18 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)))))). 300.83/38.18 300.83/38.18 Now clausify the problem and encode Horn clauses using encoding 3 of 300.83/38.18 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 300.83/38.18 We repeatedly replace C & s=t => u=v by the two clauses: 300.83/38.18 fresh(y, y, x1...xn) = u 300.83/38.18 C => fresh(s, t, x1...xn) = v 300.83/38.18 where fresh is a fresh function symbol and x1..xn are the free 300.83/38.18 variables of u and v. 300.83/38.18 A predicate p(X) is encoded as p(X)=true (this is sound, because the 300.83/38.18 input problem has no model of domain size 1). 300.83/38.18 300.83/38.18 The encoding turns the above axioms into the following unit equations and goals: 300.83/38.18 300.83/38.18 Axiom 1 (modus_ponens_for_nand): fresh(X, X, Y, Z) = p(Z). 300.83/38.18 Axiom 2 (modus_ponens_for_nand): fresh2(X, X, Y) = true. 300.83/38.18 Axiom 3 (modus_ponens_for_nand): fresh(p(nand(X, nand(Y, Z))), true, X, Z) = fresh2(p(X), true, Z). 300.83/38.18 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. 300.83/38.18 300.83/38.18 Lemma 5: fresh2(p(nand(W, nand(Z, Y))), true, nand(W, nand(W, Z))) = p(nand(W, nand(W, Z))). 300.83/38.18 Proof: 300.83/38.18 fresh2(p(nand(W, nand(Z, Y))), true, nand(W, nand(W, Z))) 300.83/38.18 = { by axiom 3 (modus_ponens_for_nand) } 300.83/38.18 fresh(p(nand(nand(W, nand(Z, Y)), nand(nand(nand(?, Y), nand(nand(W, ?), nand(W, ?))), nand(W, nand(W, Z))))), true, nand(W, nand(Z, Y)), nand(W, nand(W, Z))) 300.83/38.18 = { by axiom 4 (wajsbergs_axiom) } 300.83/38.18 fresh(true, true, nand(W, nand(Z, Y)), nand(W, nand(W, Z))) 300.83/38.18 = { by axiom 1 (modus_ponens_for_nand) } 300.83/38.19 p(nand(W, nand(W, Z))) 300.83/38.19 300.83/38.19 Lemma 6: p(nand(nand(W, nand(Z, Y)), nand(nand(W, nand(Z, Y)), nand(nand(X, Y), nand(nand(W, X), nand(W, X)))))) = true. 300.83/38.19 Proof: 300.83/38.19 p(nand(nand(W, nand(Z, Y)), nand(nand(W, nand(Z, Y)), nand(nand(X, Y), nand(nand(W, X), nand(W, X)))))) 300.83/38.19 = { by lemma 5 } 300.83/38.19 fresh2(p(nand(nand(W, nand(Z, Y)), nand(nand(nand(X, Y), nand(nand(W, X), nand(W, X))), nand(W, nand(W, Z))))), true, nand(nand(W, nand(Z, Y)), nand(nand(W, nand(Z, Y)), nand(nand(X, Y), nand(nand(W, X), nand(W, X)))))) 300.83/38.19 = { by axiom 4 (wajsbergs_axiom) } 300.83/38.19 fresh2(true, true, nand(nand(W, nand(Z, Y)), nand(nand(W, nand(Z, Y)), nand(nand(X, Y), nand(nand(W, X), nand(W, X)))))) 300.83/38.19 = { by axiom 2 (modus_ponens_for_nand) } 300.83/38.19 true 300.83/38.19 300.83/38.19 Lemma 7: fresh2(p(nand(W, nand(Z, Y))), true, nand(nand(X, Y), nand(nand(W, X), nand(W, X)))) = p(nand(nand(X, Y), nand(nand(W, X), nand(W, X)))). 300.83/38.19 Proof: 300.83/38.19 fresh2(p(nand(W, nand(Z, Y))), true, nand(nand(X, Y), nand(nand(W, X), nand(W, X)))) 300.83/38.19 = { by axiom 3 (modus_ponens_for_nand) } 300.83/38.19 fresh(p(nand(nand(W, nand(Z, Y)), nand(nand(W, nand(Z, Y)), nand(nand(X, Y), nand(nand(W, X), nand(W, X)))))), true, nand(W, nand(Z, Y)), nand(nand(X, Y), nand(nand(W, X), nand(W, X)))) 300.83/38.19 = { by lemma 6 } 300.83/38.19 fresh(true, true, nand(W, nand(Z, Y)), nand(nand(X, Y), nand(nand(W, X), nand(W, X)))) 300.83/38.19 = { by axiom 1 (modus_ponens_for_nand) } 300.83/38.19 p(nand(nand(X, Y), nand(nand(W, X), nand(W, X)))) 300.83/38.19 300.83/38.19 Lemma 8: p(nand(nand(W, nand(Z, Y)), nand(nand(W, nand(Z, Y)), nand(W, nand(Z, Y))))) = true. 300.83/38.19 Proof: 300.83/38.19 p(nand(nand(W, nand(Z, Y)), nand(nand(W, nand(Z, Y)), nand(W, nand(Z, Y))))) 300.83/38.19 = { by lemma 5 } 300.83/38.19 fresh2(p(nand(nand(W, nand(Z, Y)), nand(nand(W, nand(Z, Y)), nand(nand(?, Y), nand(nand(W, ?), nand(W, ?)))))), true, nand(nand(W, nand(Z, Y)), nand(nand(W, nand(Z, Y)), nand(W, nand(Z, Y))))) 300.83/38.19 = { by lemma 6 } 300.83/38.19 fresh2(true, true, nand(nand(W, nand(Z, Y)), nand(nand(W, nand(Z, Y)), nand(W, nand(Z, Y))))) 300.83/38.19 = { by axiom 2 (modus_ponens_for_nand) } 300.83/38.19 true 300.83/38.19 300.83/38.19 Lemma 9: fresh2(p(nand(W, nand(Z, nand(Y, X)))), true, nand(nand(Z, nand(Y, X)), W)) = p(nand(nand(Z, nand(Y, X)), W)). 300.83/38.19 Proof: 300.83/38.19 fresh2(p(nand(W, nand(Z, nand(Y, X)))), true, nand(nand(Z, nand(Y, X)), W)) 300.83/38.19 = { by axiom 3 (modus_ponens_for_nand) } 300.83/38.19 fresh(p(nand(nand(W, nand(Z, nand(Y, X))), nand(nand(nand(Z, nand(Y, X)), W), nand(nand(Z, nand(Y, X)), W)))), true, nand(W, nand(Z, nand(Y, X))), nand(nand(Z, nand(Y, X)), W)) 300.83/38.19 = { by lemma 7 } 300.83/38.19 fresh(fresh2(p(nand(nand(Z, nand(Y, X)), nand(nand(Z, nand(Y, X)), nand(Z, nand(Y, X))))), true, nand(nand(W, nand(Z, nand(Y, X))), nand(nand(nand(Z, nand(Y, X)), W), nand(nand(Z, nand(Y, X)), W)))), true, nand(W, nand(Z, nand(Y, X))), nand(nand(Z, nand(Y, X)), W)) 300.83/38.19 = { by lemma 8 } 300.83/38.19 fresh(fresh2(true, true, nand(nand(W, nand(Z, nand(Y, X))), nand(nand(nand(Z, nand(Y, X)), W), nand(nand(Z, nand(Y, X)), W)))), true, nand(W, nand(Z, nand(Y, X))), nand(nand(Z, nand(Y, X)), W)) 300.83/38.19 = { by axiom 2 (modus_ponens_for_nand) } 300.83/38.19 fresh(true, true, nand(W, nand(Z, nand(Y, X))), nand(nand(Z, nand(Y, X)), W)) 300.83/38.19 = { by axiom 1 (modus_ponens_for_nand) } 300.83/38.19 p(nand(nand(Z, nand(Y, X)), W)) 300.83/38.19 300.83/38.19 Lemma 10: p(nand(nand(W, Z), nand(nand(W, Z), nand(W, Z)))) = true. 300.83/38.19 Proof: 300.83/38.20 p(nand(nand(W, Z), nand(nand(W, Z), nand(W, Z)))) 300.83/38.20 = { by lemma 5 } 300.83/38.20 fresh2(p(nand(nand(W, Z), nand(nand(W, Z), nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), W)))), true, nand(nand(W, Z), nand(nand(W, Z), nand(W, Z)))) 300.83/38.20 = { by lemma 5 } 300.83/38.20 fresh2(fresh2(p(nand(nand(W, Z), nand(nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), W), nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), W)))), true, nand(nand(W, Z), nand(nand(W, Z), nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), W)))), true, nand(nand(W, Z), nand(nand(W, Z), nand(W, Z)))) 300.83/38.20 = { by lemma 7 } 300.83/38.20 fresh2(fresh2(fresh2(p(nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), Z))), true, nand(nand(W, Z), nand(nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), W), nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), W)))), true, nand(nand(W, Z), nand(nand(W, Z), nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), W)))), true, nand(nand(W, Z), nand(nand(W, Z), nand(W, Z)))) 300.83/38.20 = { by lemma 5 } 300.83/38.20 fresh2(fresh2(fresh2(fresh2(p(nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), nand(Z, nand(?, ?)))), true, nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), Z))), true, nand(nand(W, Z), nand(nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), W), nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), W)))), true, nand(nand(W, Z), nand(nand(W, Z), nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), W)))), true, nand(nand(W, Z), nand(nand(W, Z), nand(W, Z)))) 300.83/38.20 = { by lemma 9 } 300.83/38.20 fresh2(fresh2(fresh2(fresh2(fresh2(p(nand(nand(Z, nand(?, ?)), nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))))), true, nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), nand(Z, nand(?, ?)))), true, nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), Z))), true, nand(nand(W, Z), nand(nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), W), nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), W)))), true, nand(nand(W, Z), nand(nand(W, Z), nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), W)))), true, nand(nand(W, Z), nand(nand(W, Z), nand(W, Z)))) 300.83/38.20 = { by lemma 8 } 300.83/38.20 fresh2(fresh2(fresh2(fresh2(fresh2(true, true, nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), nand(Z, nand(?, ?)))), true, nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), Z))), true, nand(nand(W, Z), nand(nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), W), nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), W)))), true, nand(nand(W, Z), nand(nand(W, Z), nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), W)))), true, nand(nand(W, Z), nand(nand(W, Z), nand(W, Z)))) 300.83/38.20 = { by axiom 2 (modus_ponens_for_nand) } 300.83/38.20 fresh2(fresh2(fresh2(fresh2(true, true, nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), Z))), true, nand(nand(W, Z), nand(nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), W), nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), W)))), true, nand(nand(W, Z), nand(nand(W, Z), nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), W)))), true, nand(nand(W, Z), nand(nand(W, Z), nand(W, Z)))) 300.83/38.20 = { by axiom 2 (modus_ponens_for_nand) } 300.83/38.20 fresh2(fresh2(fresh2(true, true, nand(nand(W, Z), nand(nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), W), nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), W)))), true, nand(nand(W, Z), nand(nand(W, Z), nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), W)))), true, nand(nand(W, Z), nand(nand(W, Z), nand(W, Z)))) 300.83/38.20 = { by axiom 2 (modus_ponens_for_nand) } 300.83/38.20 fresh2(fresh2(true, true, nand(nand(W, Z), nand(nand(W, Z), nand(nand(nand(Z, nand(?, ?)), nand(Z, nand(?, ?))), W)))), true, nand(nand(W, Z), nand(nand(W, Z), nand(W, Z)))) 300.83/38.20 = { by axiom 2 (modus_ponens_for_nand) } 300.83/38.20 fresh2(true, true, nand(nand(W, Z), nand(nand(W, Z), nand(W, Z)))) 300.83/38.20 = { by axiom 2 (modus_ponens_for_nand) } 300.83/38.20 true 300.83/38.20 300.83/38.20 Lemma 11: p(nand(nand(nand(Y, X), nand(Y, X)), nand(Y, X))) = true. 300.83/38.20 Proof: 300.83/38.20 p(nand(nand(nand(Y, X), nand(Y, X)), nand(Y, X))) 300.83/38.20 = { by lemma 9 } 300.83/38.20 fresh2(p(nand(nand(Y, X), nand(nand(Y, X), nand(Y, X)))), true, nand(nand(nand(Y, X), nand(Y, X)), nand(Y, X))) 300.83/38.20 = { by lemma 10 } 300.83/38.20 fresh2(true, true, nand(nand(nand(Y, X), nand(Y, X)), nand(Y, X))) 300.83/38.20 = { by axiom 2 (modus_ponens_for_nand) } 301.56/38.22 true 301.56/38.22 301.56/38.22 Goal 1 (tautology): p(nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) = true. 301.56/38.22 Proof: 301.56/38.22 p(nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 301.56/38.22 = { by axiom 1 (modus_ponens_for_nand) } 301.56/38.22 fresh(true, true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A)), nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 301.56/38.22 = { by axiom 2 (modus_ponens_for_nand) } 301.56/38.22 fresh(fresh2(true, true, nand(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A)), nand(nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))))), nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))))), true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A)), nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 301.56/38.22 = { by axiom 2 (modus_ponens_for_nand) } 301.56/38.22 fresh(fresh2(fresh2(true, true, nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A)))), true, nand(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A)), nand(nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))))), nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))))), true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A)), nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 301.56/38.22 = { by axiom 2 (modus_ponens_for_nand) } 301.56/38.22 fresh(fresh2(fresh2(fresh2(true, true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(sK1_tautology_B, sK1_tautology_B)))), true, nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A)))), true, nand(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A)), nand(nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))))), nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))))), true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A)), nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 301.56/38.22 = { by lemma 11 } 301.56/38.23 fresh(fresh2(fresh2(fresh2(p(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))))), true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(sK1_tautology_B, sK1_tautology_B)))), true, nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A)))), true, nand(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A)), nand(nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))))), nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))))), true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A)), nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 301.56/38.23 = { by lemma 5 } 301.56/38.23 fresh(fresh2(fresh2(p(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(sK1_tautology_B, sK1_tautology_B)))), true, nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A)))), true, nand(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A)), nand(nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))))), nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))))), true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A)), nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 301.56/38.23 = { by lemma 7 } 301.56/38.23 fresh(fresh2(p(nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A)))), true, nand(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A)), nand(nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))))), nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))))), true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A)), nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 301.56/38.23 = { by lemma 7 } 301.56/38.23 fresh(p(nand(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A)), nand(nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))))), nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))))), true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A)), nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 301.56/38.23 = { by axiom 3 (modus_ponens_for_nand) } 301.56/38.23 fresh2(p(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A))), true, nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 301.56/38.23 = { by lemma 5 } 301.56/38.23 fresh2(fresh2(p(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(sK2_tautology_A, sK2_tautology_A))), true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A))), true, nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 301.56/38.23 = { by axiom 1 (modus_ponens_for_nand) } 301.56/38.23 fresh2(fresh2(fresh(true, true, nand(nand(sK2_tautology_A, sK2_tautology_A), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(sK2_tautology_A, sK2_tautology_A))), true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A))), true, nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 301.56/38.23 = { by axiom 2 (modus_ponens_for_nand) } 301.56/38.23 fresh2(fresh2(fresh(fresh2(true, true, nand(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(sK2_tautology_A, sK2_tautology_A)), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(sK2_tautology_A, sK2_tautology_A))))), true, nand(nand(sK2_tautology_A, sK2_tautology_A), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(sK2_tautology_A, sK2_tautology_A))), true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A))), true, nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 301.56/38.23 = { by lemma 11 } 301.56/38.24 fresh2(fresh2(fresh(fresh2(p(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))))), true, nand(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(sK2_tautology_A, sK2_tautology_A)), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(sK2_tautology_A, sK2_tautology_A))))), true, nand(nand(sK2_tautology_A, sK2_tautology_A), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(sK2_tautology_A, sK2_tautology_A))), true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A))), true, nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 301.56/38.24 = { by lemma 7 } 301.56/38.24 fresh2(fresh2(fresh(p(nand(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(sK2_tautology_A, sK2_tautology_A)), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(sK2_tautology_A, sK2_tautology_A))))), true, nand(nand(sK2_tautology_A, sK2_tautology_A), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(sK2_tautology_A, sK2_tautology_A))), true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A))), true, nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 301.56/38.24 = { by axiom 3 (modus_ponens_for_nand) } 301.56/38.24 fresh2(fresh2(fresh2(p(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(sK2_tautology_A, sK2_tautology_A))), true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A))), true, nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 301.56/38.24 = { by lemma 10 } 301.56/38.24 fresh2(fresh2(fresh2(true, true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(sK2_tautology_A, sK2_tautology_A))), true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A))), true, nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 301.56/38.24 = { by axiom 2 (modus_ponens_for_nand) } 301.56/38.24 fresh2(fresh2(true, true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), sK2_tautology_A))), true, nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 301.56/38.24 = { by axiom 2 (modus_ponens_for_nand) } 301.56/38.24 fresh2(true, true, nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 301.56/38.24 = { by axiom 2 (modus_ponens_for_nand) } 301.56/38.24 true 301.56/38.24 % SZS output end Proof 301.56/38.24 301.56/38.24 RESULT: Theorem (the conjecture is true). 301.56/38.27 EOF