0.00/0.03 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.00/0.04 % Command : twee %s --tstp --casc --quiet --conditional-encoding if --smaller --drop-non-horn 0.02/0.23 % Computer : n157.star.cs.uiowa.edu 0.02/0.23 % Model : x86_64 x86_64 0.02/0.23 % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz 0.02/0.23 % Memory : 32218.625MB 0.02/0.23 % OS : Linux 3.10.0-693.2.2.el7.x86_64 0.02/0.23 % CPULimit : 300 0.02/0.23 % DateTime : Sat Jul 14 05:04:40 CDT 2018 0.02/0.23 % CPUTime : 113.77/114.03 % SZS status Theorem 113.77/114.03 113.77/114.04 % SZS output start Proof 113.77/114.04 Take the following subset of the input axioms: 113.86/114.07 fof(modus_ponens_for_nand, axiom, 113.86/114.07 ![Q, P, R]: (p(R) <= (p(P) & p(nand(P, nand(Q, R)))))). 113.86/114.07 fof(tautology, conjecture, 113.86/114.07 ![A, B]: 113.86/114.07 p(nand(nand(A, nand(B, B)), 113.86/114.07 nand(nand(nand(B, B), nand(nand(A, A), nand(A, A))), 113.86/114.07 nand(nand(B, B), nand(nand(A, A), nand(A, A))))))). 113.86/114.07 fof(wajsbergs_axiom, axiom, 113.86/114.07 ![A, B, C, D]: 113.86/114.07 p(nand(nand(A, nand(B, C)), 113.86/114.07 nand(nand(nand(D, C), nand(nand(A, D), nand(A, D))), 113.86/114.07 nand(A, nand(A, B)))))). 113.86/114.07 113.86/114.07 Now clausify the problem and encode Horn clauses using encoding 3 of 113.86/114.07 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 113.86/114.07 We repeatedly replace C & s=t => u=v by the two clauses: 113.86/114.07 $$fresh(y, y, x1...xn) = u 113.86/114.07 C => $$fresh(s, t, x1...xn) = v 113.86/114.07 where $$fresh is a fresh function symbol and x1..xn are the free 113.86/114.07 variables of u and v. 113.86/114.07 A predicate p(X) is encoded as p(X)=$$true (this is sound, because the 113.86/114.07 input problem has no model of domain size 1). 113.86/114.07 113.86/114.07 The encoding turns the above axioms into the following unit equations and goals: 113.86/114.07 113.86/114.07 Axiom 1 (modus_ponens_for_nand): $$fresh2(X, X, Y, Z) = p(Z). 113.86/114.07 Axiom 2 (modus_ponens_for_nand): $$fresh(X, X, Y) = $$true. 113.86/114.07 Axiom 3 (modus_ponens_for_nand): $$fresh2(p(nand(X, nand(Y, Z))), $$true, X, Z) = $$fresh(p(X), $$true, Z). 113.86/114.07 Axiom 4 (wajsbergs_axiom): p(nand(nand(X, nand(Y, Z)), nand(nand(nand(W, Z), nand(nand(X, W), nand(X, W))), nand(X, nand(X, Y))))) = $$true. 113.86/114.07 113.86/114.07 Lemma 5: $$fresh(p(nand(X, nand(Y, Z))), $$true, nand(X, nand(X, Y))) = p(nand(X, nand(X, Y))). 113.86/114.07 Proof: 113.86/114.07 $$fresh(p(nand(X, nand(Y, Z))), $$true, nand(X, nand(X, Y))) 113.86/114.07 = { by axiom 3 (modus_ponens_for_nand) } 113.86/114.07 $$fresh2(p(nand(nand(X, nand(Y, Z)), nand(nand(nand(?, Z), nand(nand(X, ?), nand(X, ?))), nand(X, nand(X, Y))))), $$true, nand(X, nand(Y, Z)), nand(X, nand(X, Y))) 113.86/114.07 = { by axiom 4 (wajsbergs_axiom) } 113.86/114.07 $$fresh2($$true, $$true, nand(X, nand(Y, Z)), nand(X, nand(X, Y))) 113.86/114.07 = { by axiom 1 (modus_ponens_for_nand) } 113.86/114.07 p(nand(X, nand(X, Y))) 113.86/114.07 113.86/114.07 Lemma 6: p(nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(nand(W, Z), nand(nand(X, W), nand(X, W)))))) = $$true. 113.86/114.07 Proof: 113.86/114.07 p(nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(nand(W, Z), nand(nand(X, W), nand(X, W)))))) 113.86/114.07 = { by lemma 5 } 113.86/114.07 $$fresh(p(nand(nand(X, nand(Y, Z)), nand(nand(nand(W, Z), nand(nand(X, W), nand(X, W))), nand(X, nand(X, Y))))), $$true, nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(nand(W, Z), nand(nand(X, W), nand(X, W)))))) 113.86/114.07 = { by axiom 4 (wajsbergs_axiom) } 113.86/114.07 $$fresh($$true, $$true, nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(nand(W, Z), nand(nand(X, W), nand(X, W)))))) 113.86/114.07 = { by axiom 2 (modus_ponens_for_nand) } 113.86/114.07 $$true 113.86/114.07 113.86/114.07 Lemma 7: p(nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))))) = $$true. 113.86/114.07 Proof: 113.86/114.07 p(nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))))) 113.86/114.07 = { by lemma 5 } 113.86/114.07 $$fresh(p(nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(nand(?, Z), nand(nand(X, ?), nand(X, ?)))))), $$true, nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))))) 113.86/114.07 = { by lemma 6 } 113.86/114.07 $$fresh($$true, $$true, nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))))) 113.86/114.07 = { by axiom 2 (modus_ponens_for_nand) } 113.86/114.07 $$true 113.86/114.07 113.86/114.07 Lemma 8: $$fresh(p(nand(X, nand(Y, Z))), $$true, nand(nand(W, Z), nand(nand(X, W), nand(X, W)))) = p(nand(nand(W, Z), nand(nand(X, W), nand(X, W)))). 113.86/114.07 Proof: 113.86/114.07 $$fresh(p(nand(X, nand(Y, Z))), $$true, nand(nand(W, Z), nand(nand(X, W), nand(X, W)))) 113.86/114.07 = { by axiom 3 (modus_ponens_for_nand) } 113.86/114.07 $$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)))))), $$true, nand(X, nand(Y, Z)), nand(nand(W, Z), nand(nand(X, W), nand(X, W)))) 113.86/114.07 = { by lemma 6 } 113.86/114.07 $$fresh2($$true, $$true, nand(X, nand(Y, Z)), nand(nand(W, Z), nand(nand(X, W), nand(X, W)))) 113.86/114.07 = { by axiom 1 (modus_ponens_for_nand) } 113.86/114.07 p(nand(nand(W, Z), nand(nand(X, W), nand(X, W)))) 113.86/114.07 113.86/114.07 Lemma 9: $$fresh(p(nand(X, nand(Y, nand(Z, W)))), $$true, nand(nand(Y, nand(Z, W)), X)) = p(nand(nand(Y, nand(Z, W)), X)). 113.86/114.07 Proof: 113.86/114.07 $$fresh(p(nand(X, nand(Y, nand(Z, W)))), $$true, nand(nand(Y, nand(Z, W)), X)) 113.86/114.07 = { by axiom 3 (modus_ponens_for_nand) } 113.86/114.07 $$fresh2(p(nand(nand(X, nand(Y, nand(Z, W))), nand(nand(nand(Y, nand(Z, W)), X), nand(nand(Y, nand(Z, W)), X)))), $$true, nand(X, nand(Y, nand(Z, W))), nand(nand(Y, nand(Z, W)), X)) 113.86/114.07 = { by lemma 8 } 113.86/114.07 $$fresh2($$fresh(p(nand(nand(Y, nand(Z, W)), nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))))), $$true, nand(nand(X, nand(Y, nand(Z, W))), nand(nand(nand(Y, nand(Z, W)), X), nand(nand(Y, nand(Z, W)), X)))), $$true, nand(X, nand(Y, nand(Z, W))), nand(nand(Y, nand(Z, W)), X)) 113.86/114.07 = { by lemma 7 } 113.86/114.07 $$fresh2($$fresh($$true, $$true, nand(nand(X, nand(Y, nand(Z, W))), nand(nand(nand(Y, nand(Z, W)), X), nand(nand(Y, nand(Z, W)), X)))), $$true, nand(X, nand(Y, nand(Z, W))), nand(nand(Y, nand(Z, W)), X)) 113.86/114.07 = { by axiom 2 (modus_ponens_for_nand) } 113.86/114.07 $$fresh2($$true, $$true, nand(X, nand(Y, nand(Z, W))), nand(nand(Y, nand(Z, W)), X)) 113.86/114.07 = { by axiom 1 (modus_ponens_for_nand) } 113.86/114.07 p(nand(nand(Y, nand(Z, W)), X)) 113.86/114.07 113.86/114.07 Lemma 10: p(nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z)))) = $$true. 113.86/114.07 Proof: 113.86/114.07 p(nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z)))) 113.86/114.07 = { by lemma 9 } 113.86/114.07 $$fresh(p(nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))))), $$true, nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z)))) 113.86/114.07 = { by lemma 7 } 113.86/114.07 $$fresh($$true, $$true, nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z)))) 113.86/114.07 = { by axiom 2 (modus_ponens_for_nand) } 113.86/114.07 $$true 113.86/114.07 113.86/114.07 Lemma 11: $$fresh(p(nand(X, nand(Y, nand(Y, Z)))), $$true, nand(nand(Y, nand(Z, W)), X)) = p(nand(nand(Y, nand(Z, W)), X)). 113.86/114.07 Proof: 113.86/114.07 $$fresh(p(nand(X, nand(Y, nand(Y, Z)))), $$true, nand(nand(Y, nand(Z, W)), X)) 113.86/114.07 = { by axiom 3 (modus_ponens_for_nand) } 113.86/114.07 $$fresh2(p(nand(nand(X, nand(Y, nand(Y, Z))), nand(nand(nand(Y, nand(Z, W)), X), nand(nand(Y, nand(Z, W)), X)))), $$true, nand(X, nand(Y, nand(Y, Z))), nand(nand(Y, nand(Z, W)), X)) 113.86/114.07 = { by lemma 8 } 113.86/114.07 $$fresh2($$fresh(p(nand(nand(Y, nand(Z, W)), nand(nand(nand(?, W), nand(nand(Y, ?), nand(Y, ?))), nand(Y, nand(Y, Z))))), $$true, nand(nand(X, nand(Y, nand(Y, Z))), nand(nand(nand(Y, nand(Z, W)), X), nand(nand(Y, nand(Z, W)), X)))), $$true, nand(X, nand(Y, nand(Y, Z))), nand(nand(Y, nand(Z, W)), X)) 113.86/114.07 = { by axiom 4 (wajsbergs_axiom) } 113.86/114.07 $$fresh2($$fresh($$true, $$true, nand(nand(X, nand(Y, nand(Y, Z))), nand(nand(nand(Y, nand(Z, W)), X), nand(nand(Y, nand(Z, W)), X)))), $$true, nand(X, nand(Y, nand(Y, Z))), nand(nand(Y, nand(Z, W)), X)) 113.86/114.07 = { by axiom 2 (modus_ponens_for_nand) } 113.86/114.07 $$fresh2($$true, $$true, nand(X, nand(Y, nand(Y, Z))), nand(nand(Y, nand(Z, W)), X)) 113.86/114.07 = { by axiom 1 (modus_ponens_for_nand) } 113.86/114.07 p(nand(nand(Y, nand(Z, W)), X)) 113.86/114.07 113.86/114.07 Lemma 12: $$fresh(p(nand(X, nand(Y, Z))), $$true, nand(X, nand(X, X))) = p(nand(X, nand(X, X))). 113.86/114.07 Proof: 113.86/114.07 $$fresh(p(nand(X, nand(Y, Z))), $$true, nand(X, nand(X, X))) 113.86/114.07 = { by axiom 3 (modus_ponens_for_nand) } 113.86/114.07 $$fresh2(p(nand(nand(X, nand(Y, Z)), nand(nand(X, nand(X, X)), nand(X, nand(X, X))))), $$true, nand(X, nand(Y, Z)), nand(X, nand(X, X))) 113.86/114.07 = { by lemma 11 } 113.86/114.07 $$fresh2($$fresh(p(nand(nand(nand(X, nand(X, X)), nand(X, nand(X, X))), nand(X, nand(X, Y)))), $$true, nand(nand(X, nand(Y, Z)), nand(nand(X, nand(X, X)), nand(X, nand(X, X))))), $$true, nand(X, nand(Y, Z)), nand(X, nand(X, X))) 113.86/114.07 = { by lemma 9 } 113.86/114.07 $$fresh2($$fresh($$fresh(p(nand(nand(X, nand(X, Y)), nand(nand(X, nand(X, X)), nand(X, nand(X, X))))), $$true, nand(nand(nand(X, nand(X, X)), nand(X, nand(X, X))), nand(X, nand(X, Y)))), $$true, nand(nand(X, nand(Y, Z)), nand(nand(X, nand(X, X)), nand(X, nand(X, X))))), $$true, nand(X, nand(Y, Z)), nand(X, nand(X, X))) 113.86/114.07 = { by lemma 11 } 113.86/114.07 $$fresh2($$fresh($$fresh($$fresh(p(nand(nand(nand(X, nand(X, X)), nand(X, nand(X, X))), nand(X, nand(X, X)))), $$true, nand(nand(X, nand(X, Y)), nand(nand(X, nand(X, X)), nand(X, nand(X, X))))), $$true, nand(nand(nand(X, nand(X, X)), nand(X, nand(X, X))), nand(X, nand(X, Y)))), $$true, nand(nand(X, nand(Y, Z)), nand(nand(X, nand(X, X)), nand(X, nand(X, X))))), $$true, nand(X, nand(Y, Z)), nand(X, nand(X, X))) 113.86/114.07 = { by lemma 10 } 113.86/114.07 $$fresh2($$fresh($$fresh($$fresh($$true, $$true, nand(nand(X, nand(X, Y)), nand(nand(X, nand(X, X)), nand(X, nand(X, X))))), $$true, nand(nand(nand(X, nand(X, X)), nand(X, nand(X, X))), nand(X, nand(X, Y)))), $$true, nand(nand(X, nand(Y, Z)), nand(nand(X, nand(X, X)), nand(X, nand(X, X))))), $$true, nand(X, nand(Y, Z)), nand(X, nand(X, X))) 113.86/114.07 = { by axiom 2 (modus_ponens_for_nand) } 113.86/114.07 $$fresh2($$fresh($$fresh($$true, $$true, nand(nand(nand(X, nand(X, X)), nand(X, nand(X, X))), nand(X, nand(X, Y)))), $$true, nand(nand(X, nand(Y, Z)), nand(nand(X, nand(X, X)), nand(X, nand(X, X))))), $$true, nand(X, nand(Y, Z)), nand(X, nand(X, X))) 113.86/114.07 = { by axiom 2 (modus_ponens_for_nand) } 113.86/114.07 $$fresh2($$fresh($$true, $$true, nand(nand(X, nand(Y, Z)), nand(nand(X, nand(X, X)), nand(X, nand(X, X))))), $$true, nand(X, nand(Y, Z)), nand(X, nand(X, X))) 113.86/114.07 = { by axiom 2 (modus_ponens_for_nand) } 113.86/114.07 $$fresh2($$true, $$true, nand(X, nand(Y, Z)), nand(X, nand(X, X))) 113.86/114.07 = { by axiom 1 (modus_ponens_for_nand) } 113.86/114.07 p(nand(X, nand(X, X))) 113.86/114.07 113.86/114.07 Lemma 13: p(nand(nand(X, Y), nand(nand(X, Y), nand(X, Y)))) = $$true. 113.86/114.07 Proof: 113.86/114.07 p(nand(nand(X, Y), nand(nand(X, Y), nand(X, Y)))) 113.86/114.07 = { by lemma 12 } 113.86/114.07 $$fresh(p(nand(nand(X, Y), nand(nand(nand(nand(Y, nand(?, ?)), nand(Y, nand(?, ?))), X), nand(nand(nand(Y, nand(?, ?)), nand(Y, nand(?, ?))), X)))), $$true, nand(nand(X, Y), nand(nand(X, Y), nand(X, Y)))) 113.86/114.07 = { by lemma 8 } 113.86/114.07 $$fresh($$fresh(p(nand(nand(nand(Y, nand(?, ?)), nand(Y, nand(?, ?))), nand(nand(nand(Y, nand(?, ?)), nand(Y, nand(?, ?))), Y))), $$true, nand(nand(X, Y), nand(nand(nand(nand(Y, nand(?, ?)), nand(Y, nand(?, ?))), X), nand(nand(nand(Y, nand(?, ?)), nand(Y, nand(?, ?))), X)))), $$true, nand(nand(X, Y), nand(nand(X, Y), nand(X, Y)))) 113.86/114.07 = { by lemma 5 } 113.86/114.07 $$fresh($$fresh($$fresh(p(nand(nand(nand(Y, nand(?, ?)), nand(Y, nand(?, ?))), nand(Y, nand(?, ?)))), $$true, nand(nand(nand(Y, nand(?, ?)), nand(Y, nand(?, ?))), nand(nand(nand(Y, nand(?, ?)), nand(Y, nand(?, ?))), Y))), $$true, nand(nand(X, Y), nand(nand(nand(nand(Y, nand(?, ?)), nand(Y, nand(?, ?))), X), nand(nand(nand(Y, nand(?, ?)), nand(Y, nand(?, ?))), X)))), $$true, nand(nand(X, Y), nand(nand(X, Y), nand(X, Y)))) 113.86/114.07 = { by lemma 10 } 113.86/114.07 $$fresh($$fresh($$fresh($$true, $$true, nand(nand(nand(Y, nand(?, ?)), nand(Y, nand(?, ?))), nand(nand(nand(Y, nand(?, ?)), nand(Y, nand(?, ?))), Y))), $$true, nand(nand(X, Y), nand(nand(nand(nand(Y, nand(?, ?)), nand(Y, nand(?, ?))), X), nand(nand(nand(Y, nand(?, ?)), nand(Y, nand(?, ?))), X)))), $$true, nand(nand(X, Y), nand(nand(X, Y), nand(X, Y)))) 113.86/114.07 = { by axiom 2 (modus_ponens_for_nand) } 113.86/114.07 $$fresh($$fresh($$true, $$true, nand(nand(X, Y), nand(nand(nand(nand(Y, nand(?, ?)), nand(Y, nand(?, ?))), X), nand(nand(nand(Y, nand(?, ?)), nand(Y, nand(?, ?))), X)))), $$true, nand(nand(X, Y), nand(nand(X, Y), nand(X, Y)))) 113.86/114.07 = { by axiom 2 (modus_ponens_for_nand) } 113.86/114.07 $$fresh($$true, $$true, nand(nand(X, Y), nand(nand(X, Y), nand(X, Y)))) 113.86/114.07 = { by axiom 2 (modus_ponens_for_nand) } 113.88/114.10 $$true 113.88/114.10 113.88/114.10 Lemma 14: p(nand(nand(X, Y), nand(nand(Y, X), nand(Y, X)))) = $$true. 113.88/114.10 Proof: 113.88/114.10 p(nand(nand(X, Y), nand(nand(Y, X), nand(Y, X)))) 113.88/114.10 = { by lemma 8 } 113.88/114.10 $$fresh(p(nand(Y, nand(Y, Y))), $$true, nand(nand(X, Y), nand(nand(Y, X), nand(Y, X)))) 113.88/114.10 = { by lemma 12 } 113.88/114.10 $$fresh($$fresh(p(nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), $$true, nand(Y, nand(Y, Y))), $$true, nand(nand(X, Y), nand(nand(Y, X), nand(Y, X)))) 113.88/114.10 = { by axiom 1 (modus_ponens_for_nand) } 113.88/114.10 $$fresh($$fresh($$fresh2($$true, $$true, nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), $$true, nand(Y, nand(Y, Y))), $$true, nand(nand(X, Y), nand(nand(Y, X), nand(Y, X)))) 113.88/114.10 = { by axiom 2 (modus_ponens_for_nand) } 113.88/114.10 $$fresh($$fresh($$fresh2($$fresh($$true, $$true, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))))), $$true, nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), $$true, nand(Y, nand(Y, Y))), $$true, nand(nand(X, Y), nand(nand(Y, X), nand(Y, X)))) 113.88/114.10 = { by axiom 2 (modus_ponens_for_nand) } 113.88/114.10 $$fresh($$fresh($$fresh2($$fresh($$fresh($$true, $$true, nand(nand(nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), $$true, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))))), $$true, nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), $$true, nand(Y, nand(Y, Y))), $$true, nand(nand(X, Y), nand(nand(Y, X), nand(Y, X)))) 113.88/114.10 = { by lemma 7 } 113.88/114.10 $$fresh($$fresh($$fresh2($$fresh($$fresh(p(nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), $$true, nand(nand(nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), $$true, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))))), $$true, nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), $$true, nand(Y, nand(Y, Y))), $$true, nand(nand(X, Y), nand(nand(Y, X), nand(Y, X)))) 113.88/114.10 = { by axiom 3 (modus_ponens_for_nand) } 113.88/114.10 $$fresh($$fresh($$fresh2($$fresh($$fresh2(p(nand(nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(nand(nand(nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))), nand(nand(nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))))), $$true, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(nand(nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), $$true, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))))), $$true, nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), $$true, nand(Y, nand(Y, Y))), $$true, nand(nand(X, Y), nand(nand(Y, X), nand(Y, X)))) 113.88/114.10 = { by lemma 8 } 113.88/114.10 $$fresh($$fresh($$fresh2($$fresh($$fresh2($$fresh(p(nand(nand(nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))))), $$true, nand(nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(nand(nand(nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))), nand(nand(nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))))), $$true, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(nand(nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), $$true, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))))), $$true, nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), $$true, nand(Y, nand(Y, Y))), $$true, nand(nand(X, Y), nand(nand(Y, X), nand(Y, X)))) 113.88/114.10 = { by lemma 10 } 113.88/114.10 $$fresh($$fresh($$fresh2($$fresh($$fresh2($$fresh($$true, $$true, nand(nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(nand(nand(nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))), nand(nand(nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))))), $$true, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(nand(nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), $$true, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))))), $$true, nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), $$true, nand(Y, nand(Y, Y))), $$true, nand(nand(X, Y), nand(nand(Y, X), nand(Y, X)))) 113.88/114.10 = { by axiom 2 (modus_ponens_for_nand) } 113.88/114.10 $$fresh($$fresh($$fresh2($$fresh($$fresh2($$true, $$true, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(nand(nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), $$true, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))))), $$true, nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), $$true, nand(Y, nand(Y, Y))), $$true, nand(nand(X, Y), nand(nand(Y, X), nand(Y, X)))) 113.88/114.10 = { by axiom 1 (modus_ponens_for_nand) } 113.88/114.10 $$fresh($$fresh($$fresh2($$fresh(p(nand(nand(nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), $$true, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))))), $$true, nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), $$true, nand(Y, nand(Y, Y))), $$true, nand(nand(X, Y), nand(nand(Y, X), nand(Y, X)))) 113.88/114.10 = { by lemma 9 } 113.88/114.10 $$fresh($$fresh($$fresh2(p(nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))))), $$true, nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), $$true, nand(Y, nand(Y, Y))), $$true, nand(nand(X, Y), nand(nand(Y, X), nand(Y, X)))) 113.88/114.10 = { by axiom 3 (modus_ponens_for_nand) } 113.88/114.10 $$fresh($$fresh($$fresh(p(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))), $$true, nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), $$true, nand(Y, nand(Y, Y))), $$true, nand(nand(X, Y), nand(nand(Y, X), nand(Y, X)))) 113.88/114.10 = { by lemma 13 } 113.88/114.10 $$fresh($$fresh($$fresh($$true, $$true, nand(Y, nand(nand(nand(?, ?), nand(nand(?, ?), nand(?, ?))), nand(nand(?, ?), nand(nand(?, ?), nand(?, ?)))))), $$true, nand(Y, nand(Y, Y))), $$true, nand(nand(X, Y), nand(nand(Y, X), nand(Y, X)))) 113.88/114.10 = { by axiom 2 (modus_ponens_for_nand) } 113.88/114.10 $$fresh($$fresh($$true, $$true, nand(Y, nand(Y, Y))), $$true, nand(nand(X, Y), nand(nand(Y, X), nand(Y, X)))) 113.88/114.10 = { by axiom 2 (modus_ponens_for_nand) } 113.88/114.10 $$fresh($$true, $$true, nand(nand(X, Y), nand(nand(Y, X), nand(Y, X)))) 113.88/114.10 = { by axiom 2 (modus_ponens_for_nand) } 113.93/114.13 $$true 113.93/114.13 113.93/114.13 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)))))) = $$true. 113.93/114.13 Proof: 113.93/114.13 p(nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 113.93/114.13 = { by axiom 1 (modus_ponens_for_nand) } 113.93/114.13 $$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(sK1_tautology_B, sK1_tautology_B), 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)))))) 113.93/114.13 = { by axiom 2 (modus_ponens_for_nand) } 113.93/114.14 $$fresh2($$fresh($$true, $$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(sK1_tautology_B, sK1_tautology_B), 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)))))))), $$true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A)), nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 113.93/114.14 = { by lemma 14 } 113.93/114.14 $$fresh2($$fresh(p(nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A)))), $$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(sK1_tautology_B, sK1_tautology_B), 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)))))))), $$true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A)), nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 113.93/114.14 = { by lemma 8 } 113.93/114.14 $$fresh2(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(sK1_tautology_B, sK1_tautology_B), 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)))))))), $$true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A)), nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 113.93/114.14 = { by axiom 3 (modus_ponens_for_nand) } 113.93/114.14 $$fresh(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), sK2_tautology_A))), $$true, nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 113.93/114.14 = { by axiom 1 (modus_ponens_for_nand) } 113.93/114.14 $$fresh($$fresh2($$true, $$true, nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A))), $$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)))))) 113.93/114.14 = { by lemma 14 } 113.93/114.14 $$fresh($$fresh2(p(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), 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(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), 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(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A))))), $$true, nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A))), $$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)))))) 113.93/114.14 = { by axiom 3 (modus_ponens_for_nand) } 113.93/114.14 $$fresh($$fresh(p(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), 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)))))), $$true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A))), $$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)))))) 113.93/114.14 = { by axiom 1 (modus_ponens_for_nand) } 113.93/114.14 $$fresh($$fresh($$fresh2($$true, $$true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B))), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))), $$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(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A))), $$true, nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 113.93/114.14 = { by axiom 2 (modus_ponens_for_nand) } 113.93/114.14 $$fresh($$fresh($$fresh2($$fresh($$true, $$true, nand(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))))), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))))), $$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(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B))), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), 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)))))), $$true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A))), $$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)))))) 113.93/114.14 = { by axiom 2 (modus_ponens_for_nand) } 113.93/114.14 $$fresh($$fresh($$fresh2($$fresh($$fresh($$true, $$true, nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), nand(nand(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B))))), $$true, nand(nand(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(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), 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(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), 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)))))))), $$true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B))), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))), $$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(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A))), $$true, nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 113.93/114.14 = { by axiom 2 (modus_ponens_for_nand) } 113.93/114.14 $$fresh($$fresh($$fresh2($$fresh($$fresh($$fresh($$true, $$true, nand(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK2_tautology_A, sK2_tautology_A))), $$true, nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), nand(nand(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B))))), $$true, nand(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))))), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))))), $$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(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B))), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), 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)))))), $$true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A))), $$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)))))) 113.93/114.14 = { by lemma 13 } 113.93/114.14 $$fresh($$fresh($$fresh2($$fresh($$fresh($$fresh(p(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), $$true, nand(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK2_tautology_A, sK2_tautology_A))), $$true, nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), nand(nand(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B))))), $$true, nand(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))))), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))))), $$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(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B))), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), 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)))))), $$true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A))), $$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)))))) 113.93/114.14 = { by lemma 9 } 113.93/114.14 $$fresh($$fresh($$fresh2($$fresh($$fresh(p(nand(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK2_tautology_A, sK2_tautology_A))), $$true, nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), nand(nand(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B))))), $$true, nand(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))))), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))))), $$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(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B))), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), 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)))))), $$true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A))), $$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)))))) 113.93/114.14 = { by lemma 8 } 113.93/114.14 $$fresh($$fresh($$fresh2($$fresh(p(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), nand(nand(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B))))), $$true, nand(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))))), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))))), $$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(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B))), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), 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)))))), $$true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A))), $$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)))))) 113.93/114.14 = { by lemma 8 } 113.93/114.14 $$fresh($$fresh($$fresh2(p(nand(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B))), nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))))), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))))), $$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(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B))), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), 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)))))), $$true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A))), $$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)))))) 113.93/114.14 = { by axiom 3 (modus_ponens_for_nand) } 113.93/114.14 $$fresh($$fresh($$fresh(p(nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B)))), $$true, nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), 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)))))), $$true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A))), $$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)))))) 113.93/114.14 = { by lemma 9 } 113.93/114.14 $$fresh($$fresh($$fresh($$fresh(p(nand(nand(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, 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)))))), $$true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B)))), $$true, nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), 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)))))), $$true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A))), $$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)))))) 113.93/114.14 = { by lemma 14 } 113.93/114.14 $$fresh($$fresh($$fresh($$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(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)), nand(sK1_tautology_B, sK1_tautology_B)))), $$true, nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))), $$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(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A))), $$true, nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 113.93/114.14 = { by axiom 2 (modus_ponens_for_nand) } 113.93/114.15 $$fresh($$fresh($$fresh($$true, $$true, nand(nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A), 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)))))), $$true, nand(nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))), nand(nand(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A))), $$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)))))) 113.93/114.15 = { by axiom 2 (modus_ponens_for_nand) } 113.93/114.15 $$fresh($$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(sK1_tautology_B, sK1_tautology_B), sK2_tautology_A))), $$true, nand(nand(sK2_tautology_A, nand(sK1_tautology_B, sK1_tautology_B)), nand(nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A))), nand(nand(sK1_tautology_B, sK1_tautology_B), nand(nand(sK2_tautology_A, sK2_tautology_A), nand(sK2_tautology_A, sK2_tautology_A)))))) 113.93/114.15 = { by axiom 2 (modus_ponens_for_nand) } 113.93/114.15 $$fresh($$true, $$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)))))) 113.93/114.15 = { by axiom 2 (modus_ponens_for_nand) } 113.93/114.15 $$true 113.93/114.15 % SZS output end Proof 113.93/114.15 113.93/114.15 RESULT: Theorem (the conjecture is true). 113.93/114.16 EOF