TSTP Solution File: BOO109+1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : BOO109+1 : TPTP v8.1.2. Released v3.5.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n026.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Aug 30 18:11:42 EDT 2023
% Result : Theorem 0.21s 0.56s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : BOO109+1 : TPTP v8.1.2. Released v3.5.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.18/0.34 % Computer : n026.cluster.edu
% 0.18/0.34 % Model : x86_64 x86_64
% 0.18/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.34 % Memory : 8042.1875MB
% 0.18/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.18/0.34 % CPULimit : 300
% 0.18/0.34 % WCLimit : 300
% 0.18/0.34 % DateTime : Sun Aug 27 08:23:04 EDT 2023
% 0.18/0.34 % CPUTime :
% 0.21/0.56 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.21/0.56
% 0.21/0.56 % SZS status Theorem
% 0.21/0.56
% 0.21/0.58 % SZS output start Proof
% 0.21/0.58 Take the following subset of the input axioms:
% 0.21/0.58 fof(modus_ponens_for_nand, axiom, ![P, Q, R]: ((p(nand(P, nand(Q, R))) & p(P)) => p(R))).
% 0.21/0.58 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))))))).
% 0.21/0.58 fof(wajsbergs_axiom, axiom, ![C, D, A2, B2]: p(nand(nand(A2, nand(B2, C)), nand(nand(nand(D, C), nand(nand(A2, D), nand(A2, D))), nand(A2, nand(A2, B2)))))).
% 0.21/0.58
% 0.21/0.58 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.58 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.58 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.58 fresh(y, y, x1...xn) = u
% 0.21/0.58 C => fresh(s, t, x1...xn) = v
% 0.21/0.58 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.58 variables of u and v.
% 0.21/0.58 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.58 input problem has no model of domain size 1).
% 0.21/0.58
% 0.21/0.58 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.58
% 0.21/0.58 Axiom 1 (modus_ponens_for_nand): fresh2(X, X, Y) = true.
% 0.21/0.58 Axiom 2 (modus_ponens_for_nand): fresh(X, X, Y, Z) = p(Z).
% 0.21/0.58 Axiom 3 (modus_ponens_for_nand): fresh(p(nand(X, nand(Y, Z))), true, X, Z) = fresh2(p(X), true, Z).
% 0.21/0.58 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.
% 0.21/0.58
% 0.21/0.58 Lemma 5: fresh2(p(nand(X, nand(Y, Z))), true, nand(X, nand(X, Y))) = p(nand(X, nand(X, Y))).
% 0.21/0.58 Proof:
% 0.21/0.58 fresh2(p(nand(X, nand(Y, Z))), true, nand(X, nand(X, Y)))
% 0.21/0.58 = { by axiom 3 (modus_ponens_for_nand) R->L }
% 0.21/0.58 fresh(p(nand(nand(X, nand(Y, Z)), nand(nand(nand(W, Z), nand(nand(X, W), nand(X, W))), nand(X, nand(X, Y))))), true, nand(X, nand(Y, Z)), nand(X, nand(X, Y)))
% 0.21/0.58 = { by axiom 4 (wajsbergs_axiom) }
% 0.21/0.58 fresh(true, true, nand(X, nand(Y, Z)), nand(X, nand(X, Y)))
% 0.21/0.58 = { by axiom 2 (modus_ponens_for_nand) }
% 0.21/0.58 p(nand(X, nand(X, Y)))
% 0.21/0.58
% 0.21/0.58 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.
% 0.21/0.58 Proof:
% 0.21/0.58 p(nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(nand(W, Z), nand(nand(X, W), nand(X, W))))))
% 0.21/0.58 = { by lemma 5 R->L }
% 0.21/0.58 fresh2(p(nand(nand(X, nand(Y, Z)), nand(nand(nand(W, Z), nand(nand(X, W), nand(X, W))), nand(X, nand(X, Y))))), true, nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(nand(W, Z), nand(nand(X, W), nand(X, W))))))
% 0.21/0.58 = { by axiom 4 (wajsbergs_axiom) }
% 0.21/0.58 fresh2(true, true, nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(nand(W, Z), nand(nand(X, W), nand(X, W))))))
% 0.21/0.58 = { by axiom 1 (modus_ponens_for_nand) }
% 0.21/0.58 true
% 0.21/0.58
% 0.21/0.58 Lemma 7: p(nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))))) = true.
% 0.21/0.58 Proof:
% 0.21/0.58 p(nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z)))))
% 0.21/0.58 = { by lemma 5 R->L }
% 0.21/0.58 fresh2(p(nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(nand(W, Z), nand(nand(X, W), nand(X, W)))))), true, nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z)))))
% 0.21/0.58 = { by lemma 6 }
% 0.21/0.58 fresh2(true, true, nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z)))))
% 0.21/0.58 = { by axiom 1 (modus_ponens_for_nand) }
% 0.21/0.58 true
% 0.21/0.58
% 0.21/0.58 Lemma 8: fresh2(p(nand(X, nand(Y, Z))), true, nand(nand(W, Z), nand(nand(X, W), nand(X, W)))) = p(nand(nand(W, Z), nand(nand(X, W), nand(X, W)))).
% 0.21/0.58 Proof:
% 0.21/0.58 fresh2(p(nand(X, nand(Y, Z))), true, nand(nand(W, Z), nand(nand(X, W), nand(X, W))))
% 0.21/0.58 = { by axiom 3 (modus_ponens_for_nand) R->L }
% 0.21/0.58 fresh(p(nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(nand(W, Z), nand(nand(X, W), nand(X, W)))))), true, nand(X, nand(Y, Z)), nand(nand(W, Z), nand(nand(X, W), nand(X, W))))
% 0.21/0.58 = { by lemma 6 }
% 0.21/0.58 fresh(true, true, nand(X, nand(Y, Z)), nand(nand(W, Z), nand(nand(X, W), nand(X, W))))
% 0.21/0.58 = { by axiom 2 (modus_ponens_for_nand) }
% 0.21/0.58 p(nand(nand(W, Z), nand(nand(X, W), nand(X, W))))
% 0.21/0.58
% 0.21/0.58 Lemma 9: p(nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z)))) = true.
% 0.21/0.58 Proof:
% 0.21/0.58 p(nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z))))
% 0.21/0.58 = { by axiom 2 (modus_ponens_for_nand) R->L }
% 0.21/0.58 fresh(true, true, nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z)))), nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z))))
% 0.21/0.58 = { by axiom 1 (modus_ponens_for_nand) R->L }
% 0.21/0.58 fresh(fresh2(true, true, nand(nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z)))), nand(nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z))), nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z)))))), true, nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z)))), nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z))))
% 0.21/0.58 = { by lemma 7 R->L }
% 0.21/0.58 fresh(fresh2(p(nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z)))))), true, nand(nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z)))), nand(nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z))), nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z)))))), true, nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z)))), nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z))))
% 0.21/0.58 = { by lemma 8 }
% 0.21/0.58 fresh(p(nand(nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z)))), nand(nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z))), nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z)))))), true, nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z)))), nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z))))
% 0.21/0.58 = { by axiom 3 (modus_ponens_for_nand) }
% 0.21/0.58 fresh2(p(nand(nand(X, nand(Y, Z)), nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))))), true, nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z))))
% 0.21/0.58 = { by lemma 7 }
% 0.21/0.58 fresh2(true, true, nand(nand(nand(X, nand(Y, Z)), nand(X, nand(Y, Z))), nand(X, nand(Y, Z))))
% 0.21/0.58 = { by axiom 1 (modus_ponens_for_nand) }
% 0.21/0.58 true
% 0.21/0.58
% 0.21/0.58 Lemma 10: p(nand(nand(X, Y), nand(nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), X), nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), X)))) = true.
% 0.21/0.58 Proof:
% 0.21/0.58 p(nand(nand(X, Y), nand(nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), X), nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), X))))
% 0.21/0.58 = { by lemma 8 R->L }
% 0.21/0.58 fresh2(p(nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), Y))), true, nand(nand(X, Y), nand(nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), X), nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), X))))
% 0.21/0.58 = { by lemma 5 R->L }
% 0.21/0.58 fresh2(fresh2(p(nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), nand(Y, nand(Z, W)))), true, nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), Y))), true, nand(nand(X, Y), nand(nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), X), nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), X))))
% 0.21/0.58 = { by lemma 9 }
% 0.21/0.58 fresh2(fresh2(true, true, nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), Y))), true, nand(nand(X, Y), nand(nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), X), nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), X))))
% 0.21/0.58 = { by axiom 1 (modus_ponens_for_nand) }
% 0.21/0.58 fresh2(true, true, nand(nand(X, Y), nand(nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), X), nand(nand(nand(Y, nand(Z, W)), nand(Y, nand(Z, W))), X))))
% 0.21/0.58 = { by axiom 1 (modus_ponens_for_nand) }
% 0.21/0.58 true
% 0.21/0.58
% 0.21/0.58 Goal 1 (tautology): p(nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))))) = true.
% 0.21/0.58 Proof:
% 0.21/0.58 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))))))
% 0.21/0.58 = { by axiom 2 (modus_ponens_for_nand) R->L }
% 0.21/0.58 fresh(true, true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a)), nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a))))))
% 0.21/0.58 = { by axiom 1 (modus_ponens_for_nand) R->L }
% 0.21/0.58 fresh(fresh2(true, true, nand(nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a)), nand(nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a))))), nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))))))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a)), nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a))))))
% 0.21/0.58 = { by lemma 10 R->L }
% 0.21/0.59 fresh(fresh2(p(nand(nand(a, nand(b, b)), nand(nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a)))), true, nand(nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a)), nand(nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a))))), nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))))))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a)), nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a))))))
% 0.21/0.59 = { by lemma 8 }
% 0.21/0.59 fresh(p(nand(nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a)), nand(nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a))))), nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))))))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a)), nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a))))))
% 0.21/0.59 = { by axiom 3 (modus_ponens_for_nand) }
% 0.21/0.59 fresh2(p(nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a))), true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a))))))
% 0.21/0.59 = { by lemma 5 R->L }
% 0.21/0.59 fresh2(fresh2(p(nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a))), true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a))))))
% 0.21/0.59 = { by axiom 2 (modus_ponens_for_nand) R->L }
% 0.21/0.59 fresh2(fresh2(fresh(true, true, nand(nand(a, a), nand(nand(a, a), nand(a, a))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a))), true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a))))))
% 0.21/0.59 = { by axiom 1 (modus_ponens_for_nand) R->L }
% 0.21/0.59 fresh2(fresh2(fresh(fresh2(true, true, nand(nand(nand(a, a), nand(nand(a, a), nand(a, a))), nand(nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a)), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))))), true, nand(nand(a, a), nand(nand(a, a), nand(a, a))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a))), true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a))))))
% 0.21/0.59 = { by lemma 9 R->L }
% 0.21/0.59 fresh2(fresh2(fresh(fresh2(p(nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(b, b), nand(nand(a, a), nand(a, a))))), true, nand(nand(nand(a, a), nand(nand(a, a), nand(a, a))), nand(nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a)), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))))), true, nand(nand(a, a), nand(nand(a, a), nand(a, a))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a))), true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a))))))
% 0.21/0.59 = { by lemma 8 }
% 0.21/0.59 fresh2(fresh2(fresh(p(nand(nand(nand(a, a), nand(nand(a, a), nand(a, a))), nand(nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a)), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))))), true, nand(nand(a, a), nand(nand(a, a), nand(a, a))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a))), true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a))))))
% 0.21/0.59 = { by axiom 3 (modus_ponens_for_nand) }
% 0.21/0.59 fresh2(fresh2(fresh2(p(nand(nand(a, a), nand(nand(a, a), nand(a, a)))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a))), true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a))))))
% 0.21/0.59 = { by lemma 5 R->L }
% 0.21/0.59 fresh2(fresh2(fresh2(fresh2(p(nand(nand(a, a), nand(nand(a, a), nand(nand(nand(a, nand(X, Y)), nand(a, nand(X, Y))), a)))), true, nand(nand(a, a), nand(nand(a, a), nand(a, a)))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a))), true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a))))))
% 0.21/0.59 = { by lemma 5 R->L }
% 0.21/0.59 fresh2(fresh2(fresh2(fresh2(fresh2(p(nand(nand(a, a), nand(nand(nand(nand(a, nand(X, Y)), nand(a, nand(X, Y))), a), nand(nand(nand(a, nand(X, Y)), nand(a, nand(X, Y))), a)))), true, nand(nand(a, a), nand(nand(a, a), nand(nand(nand(a, nand(X, Y)), nand(a, nand(X, Y))), a)))), true, nand(nand(a, a), nand(nand(a, a), nand(a, a)))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a))), true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a))))))
% 0.21/0.59 = { by lemma 10 }
% 0.21/0.59 fresh2(fresh2(fresh2(fresh2(fresh2(true, true, nand(nand(a, a), nand(nand(a, a), nand(nand(nand(a, nand(X, Y)), nand(a, nand(X, Y))), a)))), true, nand(nand(a, a), nand(nand(a, a), nand(a, a)))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a))), true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a))))))
% 0.21/0.59 = { by axiom 1 (modus_ponens_for_nand) }
% 0.21/0.59 fresh2(fresh2(fresh2(fresh2(true, true, nand(nand(a, a), nand(nand(a, a), nand(a, a)))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a))), true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a))))))
% 0.21/0.59 = { by axiom 1 (modus_ponens_for_nand) }
% 0.21/0.59 fresh2(fresh2(fresh2(true, true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(a, a))), true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a))), true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a))))))
% 0.21/0.59 = { by axiom 1 (modus_ponens_for_nand) }
% 0.21/0.59 fresh2(fresh2(true, true, nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), nand(nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a)))), a))), true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a))))))
% 0.21/0.59 = { by axiom 1 (modus_ponens_for_nand) }
% 0.21/0.59 fresh2(true, true, nand(nand(a, nand(b, b)), nand(nand(nand(b, b), nand(nand(a, a), nand(a, a))), nand(nand(b, b), nand(nand(a, a), nand(a, a))))))
% 0.21/0.59 = { by axiom 1 (modus_ponens_for_nand) }
% 0.21/0.59 true
% 0.21/0.59 % SZS output end Proof
% 0.21/0.59
% 0.21/0.59 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------