TSTP Solution File: LAT394-1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LAT394-1 : TPTP v8.1.2. Released v5.4.0.
% 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 : n008.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 : Thu Aug 31 06:29:03 EDT 2023
% Result : Unsatisfiable 6.57s 1.38s
% Output : Proof 8.70s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LAT394-1 : TPTP v8.1.2. Released v5.4.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.17/0.34 % Computer : n008.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.35 % CPULimit : 300
% 0.17/0.35 % WCLimit : 300
% 0.17/0.35 % DateTime : Thu Aug 24 04:28:47 EDT 2023
% 0.17/0.35 % CPUTime :
% 6.57/1.38 Command-line arguments: --no-flatten-goal
% 6.57/1.38
% 6.57/1.38 % SZS status Unsatisfiable
% 6.57/1.38
% 8.70/1.51 % SZS output start Proof
% 8.70/1.51 Axiom 1 (sos_003): neg(X) = f(X, X).
% 8.70/1.51 Axiom 2 (sos_001): or(X, Y) = f(f(X, X), f(Y, Y)).
% 8.70/1.51 Axiom 3 (sos_002): and(X, Y) = f(f(X, Y), f(X, Y)).
% 8.70/1.51 Axiom 4 (sos): f(f(f(f(X, Y), f(Y, Z)), W), f(Y, f(f(Y, f(f(X, X), X)), Z))) = Y.
% 8.70/1.51
% 8.70/1.51 Lemma 5: f(neg(X), neg(Y)) = or(X, Y).
% 8.70/1.51 Proof:
% 8.70/1.51 f(neg(X), neg(Y))
% 8.70/1.51 = { by axiom 1 (sos_003) }
% 8.70/1.51 f(neg(X), f(Y, Y))
% 8.70/1.51 = { by axiom 1 (sos_003) }
% 8.70/1.51 f(f(X, X), f(Y, Y))
% 8.70/1.51 = { by axiom 2 (sos_001) R->L }
% 8.70/1.51 or(X, Y)
% 8.70/1.51
% 8.70/1.51 Lemma 6: neg(neg(X)) = or(X, X).
% 8.70/1.51 Proof:
% 8.70/1.51 neg(neg(X))
% 8.70/1.51 = { by axiom 1 (sos_003) }
% 8.70/1.51 f(neg(X), neg(X))
% 8.70/1.51 = { by lemma 5 }
% 8.70/1.51 or(X, X)
% 8.70/1.51
% 8.70/1.51 Lemma 7: f(or(X, X), neg(Y)) = or(neg(X), Y).
% 8.70/1.51 Proof:
% 8.70/1.51 f(or(X, X), neg(Y))
% 8.70/1.51 = { by lemma 6 R->L }
% 8.70/1.51 f(neg(neg(X)), neg(Y))
% 8.70/1.51 = { by lemma 5 }
% 8.70/1.51 or(neg(X), Y)
% 8.70/1.51
% 8.70/1.51 Lemma 8: f(f(f(f(X, Y), f(Y, Z)), W), f(Y, f(f(Y, f(neg(X), X)), Z))) = Y.
% 8.70/1.51 Proof:
% 8.70/1.51 f(f(f(f(X, Y), f(Y, Z)), W), f(Y, f(f(Y, f(neg(X), X)), Z)))
% 8.70/1.51 = { by axiom 1 (sos_003) }
% 8.70/1.51 f(f(f(f(X, Y), f(Y, Z)), W), f(Y, f(f(Y, f(f(X, X), X)), Z)))
% 8.70/1.51 = { by axiom 4 (sos) }
% 8.70/1.51 Y
% 8.70/1.51
% 8.70/1.51 Lemma 9: neg(f(X, Y)) = and(X, Y).
% 8.70/1.51 Proof:
% 8.70/1.51 neg(f(X, Y))
% 8.70/1.51 = { by axiom 1 (sos_003) }
% 8.70/1.51 f(f(X, Y), f(X, Y))
% 8.70/1.51 = { by axiom 3 (sos_002) R->L }
% 8.70/1.51 and(X, Y)
% 8.70/1.51
% 8.70/1.51 Lemma 10: f(X, f(f(X, Y), f(f(f(X, Y), f(and(Z, X), f(Z, X))), W))) = f(X, Y).
% 8.70/1.51 Proof:
% 8.70/1.51 f(X, f(f(X, Y), f(f(f(X, Y), f(and(Z, X), f(Z, X))), W)))
% 8.70/1.51 = { by lemma 9 R->L }
% 8.70/1.51 f(X, f(f(X, Y), f(f(f(X, Y), f(neg(f(Z, X)), f(Z, X))), W)))
% 8.70/1.51 = { by lemma 8 R->L }
% 8.70/1.51 f(f(f(f(f(Z, X), f(X, Y)), f(f(X, Y), W)), f(X, f(f(X, f(neg(Z), Z)), Y))), f(f(X, Y), f(f(f(X, Y), f(neg(f(Z, X)), f(Z, X))), W)))
% 8.70/1.51 = { by lemma 8 }
% 8.70/1.51 f(X, Y)
% 8.70/1.51
% 8.70/1.51 Lemma 11: f(f(X, Y), f(f(f(X, Y), f(Y, Z)), Y)) = f(f(X, Y), f(Y, Z)).
% 8.70/1.51 Proof:
% 8.70/1.51 f(f(X, Y), f(f(f(X, Y), f(Y, Z)), Y))
% 8.70/1.51 = { by lemma 8 R->L }
% 8.70/1.51 f(f(X, Y), f(f(f(X, Y), f(Y, Z)), f(f(f(f(X, Y), f(Y, Z)), f(and(W, f(X, Y)), f(W, f(X, Y)))), f(Y, f(f(Y, f(neg(X), X)), Z)))))
% 8.70/1.51 = { by lemma 10 }
% 8.70/1.51 f(f(X, Y), f(Y, Z))
% 8.70/1.51
% 8.70/1.51 Lemma 12: f(f(f(f(X, Y), f(Y, Z)), Y), neg(Y)) = Y.
% 8.70/1.51 Proof:
% 8.70/1.51 f(f(f(f(X, Y), f(Y, Z)), Y), neg(Y))
% 8.70/1.51 = { by axiom 1 (sos_003) }
% 8.70/1.51 f(f(f(f(X, Y), f(Y, Z)), Y), f(Y, Y))
% 8.70/1.51 = { by lemma 8 R->L }
% 8.70/1.51 f(f(f(f(X, Y), f(Y, Z)), Y), f(f(f(f(f(X, Y), f(Y, Z)), Y), f(Y, f(f(Y, f(neg(X), X)), Z))), Y))
% 8.70/1.51 = { by lemma 11 }
% 8.70/1.51 f(f(f(f(X, Y), f(Y, Z)), Y), f(Y, f(f(Y, f(neg(X), X)), Z)))
% 8.70/1.51 = { by lemma 8 }
% 8.70/1.51 Y
% 8.70/1.51
% 8.70/1.51 Lemma 13: or(X, X) = X.
% 8.70/1.51 Proof:
% 8.70/1.51 or(X, X)
% 8.70/1.51 = { by lemma 5 R->L }
% 8.70/1.51 f(neg(X), neg(X))
% 8.70/1.51 = { by axiom 1 (sos_003) }
% 8.70/1.51 f(f(X, X), neg(X))
% 8.70/1.51 = { by lemma 8 R->L }
% 8.70/1.51 f(f(f(f(f(f(Y, X), f(X, Z)), X), f(X, f(f(X, f(neg(Y), Y)), Z))), X), neg(X))
% 8.70/1.51 = { by lemma 12 }
% 8.70/1.51 X
% 8.70/1.51
% 8.70/1.51 Lemma 14: or(neg(X), neg(Y)) = f(X, Y).
% 8.70/1.51 Proof:
% 8.70/1.51 or(neg(X), neg(Y))
% 8.70/1.51 = { by lemma 7 R->L }
% 8.70/1.51 f(or(X, X), neg(neg(Y)))
% 8.70/1.51 = { by lemma 6 }
% 8.70/1.52 f(or(X, X), or(Y, Y))
% 8.70/1.52 = { by lemma 13 }
% 8.70/1.52 f(X, or(Y, Y))
% 8.70/1.52 = { by lemma 13 }
% 8.70/1.52 f(X, Y)
% 8.70/1.52
% 8.70/1.52 Lemma 15: and(neg(X), neg(Y)) = neg(or(X, Y)).
% 8.70/1.52 Proof:
% 8.70/1.52 and(neg(X), neg(Y))
% 8.70/1.52 = { by lemma 9 R->L }
% 8.70/1.52 neg(f(neg(X), neg(Y)))
% 8.70/1.52 = { by lemma 5 }
% 8.70/1.52 neg(or(X, Y))
% 8.70/1.52
% 8.70/1.52 Lemma 16: neg(neg(X)) = X.
% 8.70/1.52 Proof:
% 8.70/1.52 neg(neg(X))
% 8.70/1.52 = { by lemma 6 }
% 8.70/1.52 or(X, X)
% 8.70/1.52 = { by lemma 13 }
% 8.70/1.52 X
% 8.70/1.52
% 8.70/1.52 Lemma 17: neg(and(X, Y)) = f(X, Y).
% 8.70/1.52 Proof:
% 8.70/1.52 neg(and(X, Y))
% 8.70/1.52 = { by lemma 9 R->L }
% 8.70/1.52 neg(neg(f(X, Y)))
% 8.70/1.52 = { by lemma 16 }
% 8.70/1.52 f(X, Y)
% 8.70/1.52
% 8.70/1.52 Lemma 18: and(or(X, X), neg(Y)) = neg(or(neg(X), Y)).
% 8.70/1.52 Proof:
% 8.70/1.52 and(or(X, X), neg(Y))
% 8.70/1.52 = { by lemma 6 R->L }
% 8.70/1.52 and(neg(neg(X)), neg(Y))
% 8.70/1.52 = { by lemma 15 }
% 8.70/1.52 neg(or(neg(X), Y))
% 8.70/1.52
% 8.70/1.52 Lemma 19: neg(or(neg(X), Y)) = and(X, neg(Y)).
% 8.70/1.52 Proof:
% 8.70/1.52 neg(or(neg(X), Y))
% 8.70/1.52 = { by lemma 18 R->L }
% 8.70/1.52 and(or(X, X), neg(Y))
% 8.70/1.52 = { by lemma 13 }
% 8.70/1.52 and(X, neg(Y))
% 8.70/1.52
% 8.70/1.52 Lemma 20: f(neg(X), Y) = or(X, neg(Y)).
% 8.70/1.52 Proof:
% 8.70/1.52 f(neg(X), Y)
% 8.70/1.52 = { by lemma 13 R->L }
% 8.70/1.52 f(neg(X), or(Y, Y))
% 8.70/1.52 = { by lemma 6 R->L }
% 8.70/1.52 f(neg(X), neg(neg(Y)))
% 8.70/1.52 = { by lemma 5 }
% 8.70/1.52 or(X, neg(Y))
% 8.70/1.52
% 8.70/1.52 Lemma 21: or(neg(X), and(Y, Z)) = f(X, f(Y, Z)).
% 8.70/1.52 Proof:
% 8.70/1.52 or(neg(X), and(Y, Z))
% 8.70/1.52 = { by lemma 9 R->L }
% 8.70/1.52 or(neg(X), neg(f(Y, Z)))
% 8.70/1.52 = { by lemma 14 }
% 8.70/1.52 f(X, f(Y, Z))
% 8.70/1.52
% 8.70/1.52 Lemma 22: f(X, neg(Y)) = or(neg(X), Y).
% 8.70/1.52 Proof:
% 8.70/1.52 f(X, neg(Y))
% 8.70/1.52 = { by lemma 13 R->L }
% 8.70/1.52 f(or(X, X), neg(Y))
% 8.70/1.52 = { by lemma 7 }
% 8.70/1.52 or(neg(X), Y)
% 8.70/1.52
% 8.70/1.52 Lemma 23: or(and(X, Y), neg(Z)) = f(f(X, Y), Z).
% 8.70/1.52 Proof:
% 8.70/1.52 or(and(X, Y), neg(Z))
% 8.70/1.52 = { by lemma 9 R->L }
% 8.70/1.52 or(neg(f(X, Y)), neg(Z))
% 8.70/1.52 = { by lemma 14 }
% 8.70/1.52 f(f(X, Y), Z)
% 8.70/1.52
% 8.70/1.52 Lemma 24: f(f(X, Y), or(neg(Z), W)) = or(and(X, Y), and(Z, neg(W))).
% 8.70/1.52 Proof:
% 8.70/1.52 f(f(X, Y), or(neg(Z), W))
% 8.70/1.52 = { by lemma 23 R->L }
% 8.70/1.52 or(and(X, Y), neg(or(neg(Z), W)))
% 8.70/1.52 = { by lemma 19 }
% 8.70/1.52 or(and(X, Y), and(Z, neg(W)))
% 8.70/1.52
% 8.70/1.52 Lemma 25: and(X, X) = or(X, X).
% 8.70/1.52 Proof:
% 8.70/1.52 and(X, X)
% 8.70/1.52 = { by lemma 9 R->L }
% 8.70/1.52 neg(f(X, X))
% 8.70/1.52 = { by axiom 1 (sos_003) R->L }
% 8.70/1.52 neg(neg(X))
% 8.70/1.52 = { by lemma 6 }
% 8.70/1.52 or(X, X)
% 8.70/1.52
% 8.70/1.52 Lemma 26: f(f(or(X, X), Y), f(X, f(f(X, f(neg(X), X)), X))) = X.
% 8.70/1.52 Proof:
% 8.70/1.52 f(f(or(X, X), Y), f(X, f(f(X, f(neg(X), X)), X)))
% 8.70/1.52 = { by lemma 25 R->L }
% 8.70/1.52 f(f(and(X, X), Y), f(X, f(f(X, f(neg(X), X)), X)))
% 8.70/1.52 = { by lemma 9 R->L }
% 8.70/1.52 f(f(neg(f(X, X)), Y), f(X, f(f(X, f(neg(X), X)), X)))
% 8.70/1.52 = { by axiom 1 (sos_003) }
% 8.70/1.52 f(f(f(f(X, X), f(X, X)), Y), f(X, f(f(X, f(neg(X), X)), X)))
% 8.70/1.52 = { by lemma 8 }
% 8.70/1.52 X
% 8.70/1.52
% 8.70/1.52 Lemma 27: f(f(neg(neg(X)), Y), f(X, f(f(X, f(neg(X), X)), X))) = X.
% 8.70/1.52 Proof:
% 8.70/1.52 f(f(neg(neg(X)), Y), f(X, f(f(X, f(neg(X), X)), X)))
% 8.70/1.52 = { by lemma 6 }
% 8.70/1.52 f(f(or(X, X), Y), f(X, f(f(X, f(neg(X), X)), X)))
% 8.70/1.52 = { by lemma 26 }
% 8.70/1.52 X
% 8.70/1.52
% 8.70/1.52 Lemma 28: and(f(neg(neg(X)), Y), f(X, f(f(X, f(neg(X), X)), X))) = neg(X).
% 8.70/1.52 Proof:
% 8.70/1.52 and(f(neg(neg(X)), Y), f(X, f(f(X, f(neg(X), X)), X)))
% 8.70/1.52 = { by lemma 9 R->L }
% 8.70/1.52 neg(f(f(neg(neg(X)), Y), f(X, f(f(X, f(neg(X), X)), X))))
% 8.70/1.52 = { by lemma 27 }
% 8.70/1.52 neg(X)
% 8.70/1.52
% 8.70/1.52 Lemma 29: f(f(X, Y), f(Z, W)) = or(and(X, Y), and(Z, W)).
% 8.70/1.52 Proof:
% 8.70/1.52 f(f(X, Y), f(Z, W))
% 8.70/1.52 = { by lemma 23 R->L }
% 8.70/1.52 or(and(X, Y), neg(f(Z, W)))
% 8.70/1.52 = { by lemma 9 }
% 8.70/1.52 or(and(X, Y), and(Z, W))
% 8.70/1.52
% 8.70/1.52 Lemma 30: f(X, or(and(X, Y), and(and(Z, X), f(Z, X)))) = f(X, Y).
% 8.70/1.52 Proof:
% 8.70/1.52 f(X, or(and(X, Y), and(and(Z, X), f(Z, X))))
% 8.70/1.52 = { by lemma 29 R->L }
% 8.70/1.52 f(X, f(f(X, Y), f(and(Z, X), f(Z, X))))
% 8.70/1.52 = { by lemma 10 R->L }
% 8.70/1.52 f(X, f(f(X, Y), f(f(f(X, Y), f(and(Z, X), f(Z, X))), f(f(f(f(X, Y), f(and(Z, X), f(Z, X))), f(and(W, f(X, Y)), f(W, f(X, Y)))), V))))
% 8.70/1.52 = { by lemma 10 }
% 8.70/1.52 f(X, Y)
% 8.70/1.52
% 8.70/1.52 Lemma 31: f(X, or(and(X, Y), and(X, neg(X)))) = f(X, Y).
% 8.70/1.52 Proof:
% 8.70/1.52 f(X, or(and(X, Y), and(X, neg(X))))
% 8.70/1.52 = { by lemma 24 R->L }
% 8.70/1.52 f(X, f(f(X, Y), or(neg(X), X)))
% 8.70/1.52 = { by lemma 23 R->L }
% 8.70/1.52 f(X, or(and(X, Y), neg(or(neg(X), X))))
% 8.70/1.52 = { by lemma 18 R->L }
% 8.70/1.52 f(X, or(and(X, Y), and(or(X, X), neg(X))))
% 8.70/1.52 = { by lemma 25 R->L }
% 8.70/1.52 f(X, or(and(X, Y), and(and(X, X), neg(X))))
% 8.70/1.52 = { by axiom 1 (sos_003) }
% 8.70/1.52 f(X, or(and(X, Y), and(and(X, X), f(X, X))))
% 8.70/1.52 = { by lemma 30 }
% 8.70/1.52 f(X, Y)
% 8.70/1.52
% 8.70/1.52 Lemma 32: and(or(X, Y), or(X, and(or(X, Y), neg(or(X, Y))))) = X.
% 8.70/1.52 Proof:
% 8.70/1.52 and(or(X, Y), or(X, and(or(X, Y), neg(or(X, Y)))))
% 8.70/1.52 = { by lemma 9 R->L }
% 8.70/1.52 neg(f(or(X, Y), or(X, and(or(X, Y), neg(or(X, Y))))))
% 8.70/1.52 = { by lemma 19 R->L }
% 8.70/1.52 neg(f(or(X, Y), or(X, neg(or(neg(or(X, Y)), or(X, Y))))))
% 8.70/1.52 = { by lemma 20 R->L }
% 8.70/1.52 neg(f(or(X, Y), f(neg(X), or(neg(or(X, Y)), or(X, Y)))))
% 8.70/1.52 = { by lemma 21 R->L }
% 8.70/1.52 neg(or(neg(or(X, Y)), and(neg(X), or(neg(or(X, Y)), or(X, Y)))))
% 8.70/1.52 = { by lemma 15 R->L }
% 8.70/1.52 neg(or(and(neg(X), neg(Y)), and(neg(X), or(neg(or(X, Y)), or(X, Y)))))
% 8.70/1.52 = { by lemma 15 R->L }
% 8.70/1.52 neg(or(and(neg(X), neg(Y)), and(neg(X), or(and(neg(X), neg(Y)), or(X, Y)))))
% 8.70/1.52 = { by lemma 5 R->L }
% 8.70/1.52 neg(or(and(neg(X), neg(Y)), and(neg(X), or(and(neg(X), neg(Y)), f(neg(X), neg(Y))))))
% 8.70/1.52 = { by lemma 9 R->L }
% 8.70/1.52 neg(or(and(neg(X), neg(Y)), and(neg(X), or(neg(f(neg(X), neg(Y))), f(neg(X), neg(Y))))))
% 8.70/1.52 = { by lemma 22 R->L }
% 8.70/1.52 neg(or(and(neg(X), neg(Y)), and(neg(X), f(f(neg(X), neg(Y)), neg(f(neg(X), neg(Y)))))))
% 8.70/1.52 = { by lemma 17 R->L }
% 8.70/1.52 neg(or(and(neg(X), neg(Y)), and(neg(X), neg(and(f(neg(X), neg(Y)), neg(f(neg(X), neg(Y))))))))
% 8.70/1.52 = { by lemma 16 R->L }
% 8.70/1.52 neg(or(and(neg(neg(neg(X))), neg(Y)), and(neg(X), neg(and(f(neg(X), neg(Y)), neg(f(neg(X), neg(Y))))))))
% 8.70/1.52 = { by lemma 16 R->L }
% 8.70/1.52 neg(or(and(neg(neg(neg(X))), neg(Y)), and(neg(X), neg(and(f(neg(neg(neg(X))), neg(Y)), neg(f(neg(X), neg(Y))))))))
% 8.70/1.52 = { by lemma 16 R->L }
% 8.70/1.52 neg(or(and(neg(neg(neg(X))), neg(Y)), and(neg(X), neg(and(f(neg(neg(neg(X))), neg(Y)), neg(f(neg(neg(neg(X))), neg(Y))))))))
% 8.70/1.52 = { by lemma 24 R->L }
% 8.70/1.52 neg(f(f(neg(neg(neg(X))), neg(Y)), or(neg(neg(X)), and(f(neg(neg(neg(X))), neg(Y)), neg(f(neg(neg(neg(X))), neg(Y)))))))
% 8.70/1.52 = { by lemma 28 R->L }
% 8.70/1.52 neg(f(f(neg(neg(neg(X))), neg(Y)), or(and(f(neg(neg(neg(X))), neg(Y)), f(neg(X), f(f(neg(X), f(neg(neg(X)), neg(X))), neg(X)))), and(f(neg(neg(neg(X))), neg(Y)), neg(f(neg(neg(neg(X))), neg(Y)))))))
% 8.70/1.52 = { by lemma 31 }
% 8.70/1.52 neg(f(f(neg(neg(neg(X))), neg(Y)), f(neg(X), f(f(neg(X), f(neg(neg(X)), neg(X))), neg(X)))))
% 8.70/1.52 = { by lemma 27 }
% 8.70/1.52 neg(neg(X))
% 8.70/1.52 = { by lemma 16 }
% 8.70/1.52 X
% 8.70/1.52
% 8.70/1.52 Lemma 33: f(and(X, Y), neg(Z)) = or(f(X, Y), Z).
% 8.70/1.52 Proof:
% 8.70/1.52 f(and(X, Y), neg(Z))
% 8.70/1.52 = { by lemma 9 R->L }
% 8.70/1.52 f(neg(f(X, Y)), neg(Z))
% 8.70/1.52 = { by lemma 5 }
% 8.70/1.52 or(f(X, Y), Z)
% 8.70/1.52
% 8.70/1.52 Lemma 34: f(and(X, Y), and(Z, W)) = or(f(X, Y), f(Z, W)).
% 8.70/1.52 Proof:
% 8.70/1.52 f(and(X, Y), and(Z, W))
% 8.70/1.52 = { by lemma 9 R->L }
% 8.70/1.52 f(and(X, Y), neg(f(Z, W)))
% 8.70/1.52 = { by lemma 33 }
% 8.70/1.52 or(f(X, Y), f(Z, W))
% 8.70/1.52
% 8.70/1.52 Lemma 35: f(or(X, X), and(Y, Z)) = or(neg(X), f(Y, Z)).
% 8.70/1.52 Proof:
% 8.70/1.52 f(or(X, X), and(Y, Z))
% 8.70/1.52 = { by lemma 6 R->L }
% 8.70/1.52 f(neg(neg(X)), and(Y, Z))
% 8.70/1.52 = { by lemma 9 R->L }
% 8.70/1.52 f(neg(neg(X)), neg(f(Y, Z)))
% 8.70/1.52 = { by lemma 5 }
% 8.70/1.52 or(neg(X), f(Y, Z))
% 8.70/1.52
% 8.70/1.52 Lemma 36: or(neg(X), f(Y, Z)) = f(X, and(Y, Z)).
% 8.70/1.52 Proof:
% 8.70/1.52 or(neg(X), f(Y, Z))
% 8.70/1.52 = { by lemma 35 R->L }
% 8.70/1.52 f(or(X, X), and(Y, Z))
% 8.70/1.52 = { by lemma 13 }
% 8.70/1.52 f(X, and(Y, Z))
% 8.70/1.52
% 8.70/1.52 Lemma 37: f(neg(and(X, Y)), or(neg(and(X, Y)), f(X, Y))) = neg(neg(and(X, Y))).
% 8.70/1.52 Proof:
% 8.70/1.52 f(neg(and(X, Y)), or(neg(and(X, Y)), f(X, Y)))
% 8.70/1.52 = { by lemma 35 R->L }
% 8.70/1.52 f(neg(and(X, Y)), f(or(and(X, Y), and(X, Y)), and(X, Y)))
% 8.70/1.52 = { by lemma 6 R->L }
% 8.70/1.52 f(neg(and(X, Y)), f(neg(neg(and(X, Y))), and(X, Y)))
% 8.70/1.52 = { by axiom 1 (sos_003) }
% 8.70/1.52 f(f(and(X, Y), and(X, Y)), f(neg(neg(and(X, Y))), and(X, Y)))
% 8.70/1.52 = { by axiom 1 (sos_003) }
% 8.70/1.52 f(f(and(X, Y), and(X, Y)), f(neg(f(and(X, Y), and(X, Y))), and(X, Y)))
% 8.70/1.52 = { by axiom 1 (sos_003) }
% 8.70/1.52 f(f(and(X, Y), and(X, Y)), f(f(f(and(X, Y), and(X, Y)), f(and(X, Y), and(X, Y))), and(X, Y)))
% 8.70/1.52 = { by lemma 11 }
% 8.70/1.52 f(f(and(X, Y), and(X, Y)), f(and(X, Y), and(X, Y)))
% 8.70/1.52 = { by axiom 1 (sos_003) R->L }
% 8.70/1.52 neg(f(and(X, Y), and(X, Y)))
% 8.70/1.52 = { by lemma 9 }
% 8.70/1.52 and(and(X, Y), and(X, Y))
% 8.70/1.52 = { by lemma 25 }
% 8.70/1.52 or(and(X, Y), and(X, Y))
% 8.70/1.52 = { by lemma 6 R->L }
% 8.70/1.52 neg(neg(and(X, Y)))
% 8.70/1.52
% 8.70/1.52 Lemma 38: f(f(or(X, X), X), neg(X)) = X.
% 8.70/1.52 Proof:
% 8.70/1.52 f(f(or(X, X), X), neg(X))
% 8.70/1.52 = { by lemma 6 R->L }
% 8.70/1.52 f(f(neg(neg(X)), X), neg(X))
% 8.70/1.52 = { by axiom 1 (sos_003) }
% 8.70/1.52 f(f(neg(neg(X)), X), f(X, X))
% 8.70/1.52 = { by lemma 27 R->L }
% 8.70/1.52 f(f(neg(neg(X)), X), f(f(f(neg(neg(X)), X), f(X, f(f(X, f(neg(X), X)), X))), X))
% 8.70/1.52 = { by lemma 11 }
% 8.70/1.52 f(f(neg(neg(X)), X), f(X, f(f(X, f(neg(X), X)), X)))
% 8.70/1.52 = { by lemma 6 }
% 8.70/1.52 f(f(or(X, X), X), f(X, f(f(X, f(neg(X), X)), X)))
% 8.70/1.52 = { by lemma 26 }
% 8.70/1.52 X
% 8.70/1.52
% 8.70/1.52 Lemma 39: f(X, or(and(X, or(X, neg(Y))), X)) = f(X, or(X, neg(Y))).
% 8.70/1.52 Proof:
% 8.70/1.52 f(X, or(and(X, or(X, neg(Y))), X))
% 8.70/1.52 = { by lemma 20 R->L }
% 8.70/1.52 f(X, or(and(X, f(neg(X), Y)), X))
% 8.70/1.52 = { by lemma 38 R->L }
% 8.70/1.52 f(X, or(and(f(f(or(X, X), X), neg(X)), f(neg(X), Y)), X))
% 8.70/1.52 = { by lemma 9 R->L }
% 8.70/1.52 f(X, or(neg(f(f(f(or(X, X), X), neg(X)), f(neg(X), Y))), X))
% 8.70/1.52 = { by lemma 22 R->L }
% 8.70/1.52 f(X, f(f(f(f(or(X, X), X), neg(X)), f(neg(X), Y)), neg(X)))
% 8.70/1.52 = { by lemma 38 R->L }
% 8.70/1.52 f(f(f(or(X, X), X), neg(X)), f(f(f(f(or(X, X), X), neg(X)), f(neg(X), Y)), neg(X)))
% 8.70/1.52 = { by lemma 11 }
% 8.70/1.52 f(f(f(or(X, X), X), neg(X)), f(neg(X), Y))
% 8.70/1.52 = { by lemma 29 }
% 8.70/1.52 or(and(f(or(X, X), X), neg(X)), and(neg(X), Y))
% 8.70/1.52 = { by lemma 9 R->L }
% 8.70/1.52 or(neg(f(f(or(X, X), X), neg(X))), and(neg(X), Y))
% 8.70/1.52 = { by lemma 38 }
% 8.70/1.52 or(neg(X), and(neg(X), Y))
% 8.70/1.52 = { by lemma 21 }
% 8.70/1.52 f(X, f(neg(X), Y))
% 8.70/1.52 = { by lemma 20 }
% 8.70/1.52 f(X, or(X, neg(Y)))
% 8.70/1.52
% 8.70/1.52 Lemma 40: or(and(or(and(X, Y), Y), Y), Y) = Y.
% 8.70/1.52 Proof:
% 8.70/1.52 or(and(or(and(X, Y), Y), Y), Y)
% 8.70/1.52 = { by lemma 9 R->L }
% 8.70/1.52 or(and(or(neg(f(X, Y)), Y), Y), Y)
% 8.70/1.52 = { by lemma 22 R->L }
% 8.70/1.52 or(and(f(f(X, Y), neg(Y)), Y), Y)
% 8.70/1.53 = { by lemma 9 R->L }
% 8.70/1.53 or(neg(f(f(f(X, Y), neg(Y)), Y)), Y)
% 8.70/1.53 = { by lemma 22 R->L }
% 8.70/1.53 f(f(f(f(X, Y), neg(Y)), Y), neg(Y))
% 8.70/1.53 = { by axiom 1 (sos_003) }
% 8.70/1.53 f(f(f(f(X, Y), f(Y, Y)), Y), neg(Y))
% 8.70/1.53 = { by lemma 12 }
% 8.70/1.53 Y
% 8.70/1.53
% 8.70/1.53 Lemma 41: f(or(neg(X), Y), f(X, f(f(X, f(neg(X), X)), X))) = X.
% 8.70/1.53 Proof:
% 8.70/1.53 f(or(neg(X), Y), f(X, f(f(X, f(neg(X), X)), X)))
% 8.70/1.53 = { by lemma 7 R->L }
% 8.70/1.53 f(f(or(X, X), neg(Y)), f(X, f(f(X, f(neg(X), X)), X)))
% 8.70/1.53 = { by lemma 26 }
% 8.70/1.53 X
% 8.70/1.53
% 8.70/1.53 Lemma 42: and(or(neg(X), Y), f(X, f(f(X, f(neg(X), X)), X))) = neg(X).
% 8.70/1.53 Proof:
% 8.70/1.53 and(or(neg(X), Y), f(X, f(f(X, f(neg(X), X)), X)))
% 8.70/1.53 = { by lemma 9 R->L }
% 8.70/1.53 neg(f(or(neg(X), Y), f(X, f(f(X, f(neg(X), X)), X))))
% 8.70/1.53 = { by lemma 41 }
% 8.70/1.53 neg(X)
% 8.70/1.53
% 8.70/1.53 Lemma 43: and(X, f(f(X, or(X, neg(X))), X)) = X.
% 8.70/1.53 Proof:
% 8.70/1.53 and(X, f(f(X, or(X, neg(X))), X))
% 8.70/1.53 = { by lemma 20 R->L }
% 8.70/1.53 and(X, f(f(X, f(neg(X), X)), X))
% 8.70/1.53 = { by lemma 9 R->L }
% 8.70/1.53 neg(f(X, f(f(X, f(neg(X), X)), X)))
% 8.70/1.53 = { by lemma 40 R->L }
% 8.70/1.53 or(and(or(and(neg(neg(X)), neg(f(X, f(f(X, f(neg(X), X)), X)))), neg(f(X, f(f(X, f(neg(X), X)), X)))), neg(f(X, f(f(X, f(neg(X), X)), X)))), neg(f(X, f(f(X, f(neg(X), X)), X))))
% 8.70/1.53 = { by lemma 15 }
% 8.70/1.53 or(and(or(neg(or(neg(X), f(X, f(f(X, f(neg(X), X)), X)))), neg(f(X, f(f(X, f(neg(X), X)), X)))), neg(f(X, f(f(X, f(neg(X), X)), X)))), neg(f(X, f(f(X, f(neg(X), X)), X))))
% 8.70/1.53 = { by lemma 23 }
% 8.70/1.53 f(f(or(neg(or(neg(X), f(X, f(f(X, f(neg(X), X)), X)))), neg(f(X, f(f(X, f(neg(X), X)), X)))), neg(f(X, f(f(X, f(neg(X), X)), X)))), f(X, f(f(X, f(neg(X), X)), X)))
% 8.70/1.53 = { by lemma 22 }
% 8.70/1.53 f(or(neg(or(neg(or(neg(X), f(X, f(f(X, f(neg(X), X)), X)))), neg(f(X, f(f(X, f(neg(X), X)), X))))), f(X, f(f(X, f(neg(X), X)), X))), f(X, f(f(X, f(neg(X), X)), X)))
% 8.70/1.53 = { by lemma 19 }
% 8.70/1.53 f(or(and(or(neg(X), f(X, f(f(X, f(neg(X), X)), X))), neg(neg(f(X, f(f(X, f(neg(X), X)), X))))), f(X, f(f(X, f(neg(X), X)), X))), f(X, f(f(X, f(neg(X), X)), X)))
% 8.70/1.53 = { by lemma 16 }
% 8.70/1.53 f(or(and(or(neg(X), f(X, f(f(X, f(neg(X), X)), X))), f(X, f(f(X, f(neg(X), X)), X))), f(X, f(f(X, f(neg(X), X)), X))), f(X, f(f(X, f(neg(X), X)), X)))
% 8.70/1.53 = { by lemma 42 }
% 8.70/1.53 f(or(neg(X), f(X, f(f(X, f(neg(X), X)), X))), f(X, f(f(X, f(neg(X), X)), X)))
% 8.70/1.53 = { by lemma 41 }
% 8.70/1.53 X
% 8.70/1.53
% 8.70/1.53 Lemma 44: or(and(X, neg(Y)), X) = X.
% 8.70/1.53 Proof:
% 8.70/1.53 or(and(X, neg(Y)), X)
% 8.70/1.53 = { by lemma 43 R->L }
% 8.70/1.53 or(and(X, neg(Y)), and(X, f(f(X, or(X, neg(X))), X)))
% 8.70/1.53 = { by lemma 20 R->L }
% 8.70/1.53 or(and(X, neg(Y)), and(X, f(f(X, f(neg(X), X)), X)))
% 8.70/1.53 = { by lemma 19 R->L }
% 8.70/1.53 or(neg(or(neg(X), Y)), and(X, f(f(X, f(neg(X), X)), X)))
% 8.70/1.53 = { by lemma 21 }
% 8.70/1.53 f(or(neg(X), Y), f(X, f(f(X, f(neg(X), X)), X)))
% 8.70/1.53 = { by lemma 41 }
% 8.70/1.53 X
% 8.70/1.53
% 8.70/1.53 Lemma 45: or(and(X, Y), X) = X.
% 8.70/1.53 Proof:
% 8.70/1.53 or(and(X, Y), X)
% 8.70/1.53 = { by lemma 13 R->L }
% 8.70/1.53 or(and(X, or(Y, Y)), X)
% 8.70/1.53 = { by lemma 6 R->L }
% 8.70/1.53 or(and(X, neg(neg(Y))), X)
% 8.70/1.53 = { by lemma 44 }
% 8.70/1.53 X
% 8.70/1.53
% 8.70/1.53 Lemma 46: and(X, or(X, Y)) = X.
% 8.70/1.53 Proof:
% 8.70/1.53 and(X, or(X, Y))
% 8.70/1.53 = { by lemma 9 R->L }
% 8.70/1.53 neg(f(X, or(X, Y)))
% 8.70/1.53 = { by lemma 16 R->L }
% 8.70/1.53 neg(f(X, or(X, neg(neg(Y)))))
% 8.70/1.53 = { by lemma 39 R->L }
% 8.70/1.53 neg(f(X, or(and(X, or(X, neg(neg(Y)))), X)))
% 8.70/1.53 = { by lemma 6 }
% 8.70/1.53 neg(f(X, or(and(X, or(X, or(Y, Y))), X)))
% 8.70/1.53 = { by lemma 13 }
% 8.70/1.53 neg(f(X, or(and(X, or(X, Y)), X)))
% 8.70/1.53 = { by lemma 9 }
% 8.70/1.53 and(X, or(and(X, or(X, Y)), X))
% 8.70/1.53 = { by lemma 45 }
% 8.70/1.53 and(X, X)
% 8.70/1.53 = { by lemma 25 }
% 8.70/1.53 or(X, X)
% 8.70/1.53 = { by lemma 13 }
% 8.70/1.53 X
% 8.70/1.53
% 8.70/1.53 Lemma 47: and(and(or(and(X, Y), and(Y, Z)), W), Y) = and(or(and(X, Y), and(Y, Z)), W).
% 8.70/1.53 Proof:
% 8.70/1.53 and(and(or(and(X, Y), and(Y, Z)), W), Y)
% 8.70/1.53 = { by lemma 29 R->L }
% 8.70/1.53 and(and(f(f(X, Y), f(Y, Z)), W), Y)
% 8.70/1.53 = { by lemma 8 R->L }
% 8.70/1.53 and(and(f(f(X, Y), f(Y, Z)), W), f(f(f(f(X, Y), f(Y, Z)), W), f(Y, f(f(Y, f(neg(X), X)), Z))))
% 8.70/1.53 = { by lemma 13 R->L }
% 8.70/1.53 and(and(f(f(X, Y), f(Y, Z)), W), f(or(f(f(f(X, Y), f(Y, Z)), W), f(f(f(X, Y), f(Y, Z)), W)), f(Y, f(f(Y, f(neg(X), X)), Z))))
% 8.70/1.53 = { by lemma 34 R->L }
% 8.70/1.53 and(and(f(f(X, Y), f(Y, Z)), W), f(f(and(f(f(X, Y), f(Y, Z)), W), and(f(f(X, Y), f(Y, Z)), W)), f(Y, f(f(Y, f(neg(X), X)), Z))))
% 8.70/1.53 = { by lemma 36 R->L }
% 8.70/1.53 and(and(f(f(X, Y), f(Y, Z)), W), f(or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W)), f(Y, f(f(Y, f(neg(X), X)), Z))))
% 8.70/1.53 = { by lemma 16 R->L }
% 8.70/1.53 and(neg(neg(and(f(f(X, Y), f(Y, Z)), W))), f(or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W)), f(Y, f(f(Y, f(neg(X), X)), Z))))
% 8.70/1.53 = { by lemma 37 R->L }
% 8.70/1.53 and(f(neg(and(f(f(X, Y), f(Y, Z)), W)), or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W))), f(or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W)), f(Y, f(f(Y, f(neg(X), X)), Z))))
% 8.70/1.53 = { by lemma 9 R->L }
% 8.70/1.53 neg(f(f(neg(and(f(f(X, Y), f(Y, Z)), W)), or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W))), f(or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W)), f(Y, f(f(Y, f(neg(X), X)), Z)))))
% 8.70/1.53 = { by lemma 11 R->L }
% 8.70/1.53 neg(f(f(neg(and(f(f(X, Y), f(Y, Z)), W)), or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W))), f(f(f(neg(and(f(f(X, Y), f(Y, Z)), W)), or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W))), f(or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W)), f(Y, f(f(Y, f(neg(X), X)), Z)))), or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W)))))
% 8.70/1.53 = { by lemma 9 }
% 8.70/1.53 and(f(neg(and(f(f(X, Y), f(Y, Z)), W)), or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W))), f(f(f(neg(and(f(f(X, Y), f(Y, Z)), W)), or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W))), f(or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W)), f(Y, f(f(Y, f(neg(X), X)), Z)))), or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W))))
% 8.70/1.53 = { by lemma 29 }
% 8.70/1.53 and(f(neg(and(f(f(X, Y), f(Y, Z)), W)), or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W))), f(or(and(neg(and(f(f(X, Y), f(Y, Z)), W)), or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W))), and(or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W)), f(Y, f(f(Y, f(neg(X), X)), Z)))), or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W))))
% 8.70/1.53 = { by lemma 37 }
% 8.70/1.53 and(neg(neg(and(f(f(X, Y), f(Y, Z)), W))), f(or(and(neg(and(f(f(X, Y), f(Y, Z)), W)), or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W))), and(or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W)), f(Y, f(f(Y, f(neg(X), X)), Z)))), or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W))))
% 8.70/1.53 = { by lemma 16 }
% 8.70/1.53 and(and(f(f(X, Y), f(Y, Z)), W), f(or(and(neg(and(f(f(X, Y), f(Y, Z)), W)), or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W))), and(or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W)), f(Y, f(f(Y, f(neg(X), X)), Z)))), or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W))))
% 8.70/1.53 = { by lemma 46 }
% 8.70/1.53 and(and(f(f(X, Y), f(Y, Z)), W), f(or(neg(and(f(f(X, Y), f(Y, Z)), W)), and(or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W)), f(Y, f(f(Y, f(neg(X), X)), Z)))), or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W))))
% 8.70/1.53 = { by lemma 21 }
% 8.70/1.53 and(and(f(f(X, Y), f(Y, Z)), W), f(f(and(f(f(X, Y), f(Y, Z)), W), f(or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W)), f(Y, f(f(Y, f(neg(X), X)), Z)))), or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W))))
% 8.70/1.53 = { by lemma 24 }
% 8.70/1.53 and(and(f(f(X, Y), f(Y, Z)), W), or(and(and(f(f(X, Y), f(Y, Z)), W), f(or(neg(and(f(f(X, Y), f(Y, Z)), W)), f(f(f(X, Y), f(Y, Z)), W)), f(Y, f(f(Y, f(neg(X), X)), Z)))), and(and(f(f(X, Y), f(Y, Z)), W), neg(f(f(f(X, Y), f(Y, Z)), W)))))
% 8.70/1.53 = { by lemma 36 }
% 8.70/1.53 and(and(f(f(X, Y), f(Y, Z)), W), or(and(and(f(f(X, Y), f(Y, Z)), W), f(f(and(f(f(X, Y), f(Y, Z)), W), and(f(f(X, Y), f(Y, Z)), W)), f(Y, f(f(Y, f(neg(X), X)), Z)))), and(and(f(f(X, Y), f(Y, Z)), W), neg(f(f(f(X, Y), f(Y, Z)), W)))))
% 8.70/1.53 = { by lemma 34 }
% 8.70/1.53 and(and(f(f(X, Y), f(Y, Z)), W), or(and(and(f(f(X, Y), f(Y, Z)), W), f(or(f(f(f(X, Y), f(Y, Z)), W), f(f(f(X, Y), f(Y, Z)), W)), f(Y, f(f(Y, f(neg(X), X)), Z)))), and(and(f(f(X, Y), f(Y, Z)), W), neg(f(f(f(X, Y), f(Y, Z)), W)))))
% 8.70/1.53 = { by lemma 13 }
% 8.70/1.53 and(and(f(f(X, Y), f(Y, Z)), W), or(and(and(f(f(X, Y), f(Y, Z)), W), f(f(f(f(X, Y), f(Y, Z)), W), f(Y, f(f(Y, f(neg(X), X)), Z)))), and(and(f(f(X, Y), f(Y, Z)), W), neg(f(f(f(X, Y), f(Y, Z)), W)))))
% 8.70/1.53 = { by lemma 9 }
% 8.70/1.53 and(and(f(f(X, Y), f(Y, Z)), W), or(and(and(f(f(X, Y), f(Y, Z)), W), f(f(f(f(X, Y), f(Y, Z)), W), f(Y, f(f(Y, f(neg(X), X)), Z)))), and(and(f(f(X, Y), f(Y, Z)), W), and(f(f(X, Y), f(Y, Z)), W))))
% 8.70/1.53 = { by lemma 25 }
% 8.70/1.53 and(and(f(f(X, Y), f(Y, Z)), W), or(and(and(f(f(X, Y), f(Y, Z)), W), f(f(f(f(X, Y), f(Y, Z)), W), f(Y, f(f(Y, f(neg(X), X)), Z)))), or(and(f(f(X, Y), f(Y, Z)), W), and(f(f(X, Y), f(Y, Z)), W))))
% 8.70/1.53 = { by lemma 13 }
% 8.70/1.53 and(and(f(f(X, Y), f(Y, Z)), W), or(and(and(f(f(X, Y), f(Y, Z)), W), f(f(f(f(X, Y), f(Y, Z)), W), f(Y, f(f(Y, f(neg(X), X)), Z)))), and(f(f(X, Y), f(Y, Z)), W)))
% 8.70/1.53 = { by lemma 45 }
% 8.70/1.54 and(and(f(f(X, Y), f(Y, Z)), W), and(f(f(X, Y), f(Y, Z)), W))
% 8.70/1.54 = { by lemma 25 }
% 8.70/1.54 or(and(f(f(X, Y), f(Y, Z)), W), and(f(f(X, Y), f(Y, Z)), W))
% 8.70/1.54 = { by lemma 13 }
% 8.70/1.54 and(f(f(X, Y), f(Y, Z)), W)
% 8.70/1.54 = { by lemma 29 }
% 8.70/1.54 and(or(and(X, Y), and(Y, Z)), W)
% 8.70/1.54
% 8.70/1.54 Lemma 48: and(or(and(X, Y), and(Y, Z)), Y) = or(and(X, Y), and(Y, Z)).
% 8.70/1.54 Proof:
% 8.70/1.54 and(or(and(X, Y), and(Y, Z)), Y)
% 8.70/1.54 = { by lemma 13 R->L }
% 8.70/1.54 and(or(or(and(X, Y), and(Y, Z)), or(and(X, Y), and(Y, Z))), Y)
% 8.70/1.54 = { by lemma 25 R->L }
% 8.70/1.54 and(and(or(and(X, Y), and(Y, Z)), or(and(X, Y), and(Y, Z))), Y)
% 8.70/1.54 = { by lemma 47 }
% 8.70/1.54 and(or(and(X, Y), and(Y, Z)), or(and(X, Y), and(Y, Z)))
% 8.70/1.54 = { by lemma 25 }
% 8.70/1.54 or(or(and(X, Y), and(Y, Z)), or(and(X, Y), and(Y, Z)))
% 8.70/1.54 = { by lemma 13 }
% 8.70/1.54 or(and(X, Y), and(Y, Z))
% 8.70/1.54
% 8.70/1.54 Lemma 49: and(or(and(X, Y), Y), Y) = or(and(X, Y), Y).
% 8.70/1.54 Proof:
% 8.70/1.54 and(or(and(X, Y), Y), Y)
% 8.70/1.54 = { by lemma 13 R->L }
% 8.70/1.54 and(or(and(X, Y), or(Y, Y)), Y)
% 8.70/1.54 = { by lemma 25 R->L }
% 8.70/1.54 and(or(and(X, Y), and(Y, Y)), Y)
% 8.70/1.54 = { by lemma 48 }
% 8.70/1.54 or(and(X, Y), and(Y, Y))
% 8.70/1.54 = { by lemma 25 }
% 8.70/1.54 or(and(X, Y), or(Y, Y))
% 8.70/1.54 = { by lemma 13 }
% 8.70/1.54 or(and(X, Y), Y)
% 8.70/1.54
% 8.70/1.54 Lemma 50: or(and(X, Y), Y) = Y.
% 8.70/1.54 Proof:
% 8.70/1.54 or(and(X, Y), Y)
% 8.70/1.54 = { by lemma 5 R->L }
% 8.70/1.54 f(neg(and(X, Y)), neg(Y))
% 8.70/1.54 = { by lemma 17 R->L }
% 8.70/1.54 neg(and(neg(and(X, Y)), neg(Y)))
% 8.70/1.54 = { by lemma 32 R->L }
% 8.70/1.54 neg(and(or(and(neg(and(X, Y)), neg(Y)), and(neg(Y), Z)), or(and(neg(and(X, Y)), neg(Y)), and(or(and(neg(and(X, Y)), neg(Y)), and(neg(Y), Z)), neg(or(and(neg(and(X, Y)), neg(Y)), and(neg(Y), Z)))))))
% 8.70/1.54 = { by lemma 47 R->L }
% 8.70/1.54 neg(and(and(or(and(neg(and(X, Y)), neg(Y)), and(neg(Y), Z)), or(and(neg(and(X, Y)), neg(Y)), and(or(and(neg(and(X, Y)), neg(Y)), and(neg(Y), Z)), neg(or(and(neg(and(X, Y)), neg(Y)), and(neg(Y), Z)))))), neg(Y)))
% 8.70/1.54 = { by lemma 32 }
% 8.70/1.54 neg(and(and(neg(and(X, Y)), neg(Y)), neg(Y)))
% 8.70/1.54 = { by lemma 17 }
% 8.70/1.54 f(and(neg(and(X, Y)), neg(Y)), neg(Y))
% 8.70/1.54 = { by lemma 15 }
% 8.70/1.54 f(neg(or(and(X, Y), Y)), neg(Y))
% 8.70/1.54 = { by lemma 5 }
% 8.70/1.54 or(or(and(X, Y), Y), Y)
% 8.70/1.54 = { by lemma 49 R->L }
% 8.70/1.54 or(and(or(and(X, Y), Y), Y), Y)
% 8.70/1.54 = { by lemma 40 }
% 8.70/1.54 Y
% 8.70/1.54
% 8.70/1.54 Lemma 51: and(or(X, Y), Y) = Y.
% 8.70/1.54 Proof:
% 8.70/1.54 and(or(X, Y), Y)
% 8.70/1.54 = { by lemma 9 R->L }
% 8.70/1.54 neg(f(or(X, Y), Y))
% 8.70/1.54 = { by lemma 14 R->L }
% 8.70/1.54 neg(or(neg(or(X, Y)), neg(Y)))
% 8.70/1.54 = { by lemma 15 R->L }
% 8.70/1.54 neg(or(and(neg(X), neg(Y)), neg(Y)))
% 8.70/1.54 = { by lemma 50 }
% 8.70/1.54 neg(neg(Y))
% 8.70/1.54 = { by lemma 16 }
% 8.70/1.54 Y
% 8.70/1.54
% 8.70/1.54 Lemma 52: and(neg(X), f(or(X, and(X, Y)), X)) = and(neg(X), f(X, Y)).
% 8.70/1.54 Proof:
% 8.70/1.54 and(neg(X), f(or(X, and(X, Y)), X))
% 8.70/1.54 = { by lemma 9 R->L }
% 8.70/1.54 and(neg(X), f(or(X, neg(f(X, Y))), X))
% 8.70/1.54 = { by lemma 20 R->L }
% 8.70/1.54 and(neg(X), f(f(neg(X), f(X, Y)), X))
% 8.70/1.54 = { by lemma 9 R->L }
% 8.70/1.54 neg(f(neg(X), f(f(neg(X), f(X, Y)), X)))
% 8.70/1.54 = { by axiom 1 (sos_003) }
% 8.70/1.54 neg(f(neg(X), f(f(f(X, X), f(X, Y)), X)))
% 8.70/1.54 = { by axiom 1 (sos_003) }
% 8.70/1.54 neg(f(f(X, X), f(f(f(X, X), f(X, Y)), X)))
% 8.70/1.54 = { by lemma 11 }
% 8.70/1.54 neg(f(f(X, X), f(X, Y)))
% 8.70/1.54 = { by axiom 1 (sos_003) R->L }
% 8.70/1.54 neg(f(neg(X), f(X, Y)))
% 8.70/1.54 = { by lemma 9 }
% 8.70/1.54 and(neg(X), f(X, Y))
% 8.70/1.54
% 8.70/1.54 Lemma 53: and(and(X, Y), or(and(and(X, Y), X), and(X, Y))) = and(and(X, Y), or(and(X, Y), X)).
% 8.70/1.54 Proof:
% 8.70/1.54 and(and(X, Y), or(and(and(X, Y), X), and(X, Y)))
% 8.70/1.54 = { by lemma 29 R->L }
% 8.70/1.54 and(and(X, Y), f(f(and(X, Y), X), f(X, Y)))
% 8.70/1.54 = { by lemma 13 R->L }
% 8.70/1.54 and(and(X, Y), f(f(and(X, Y), or(X, X)), f(X, Y)))
% 8.70/1.54 = { by lemma 6 R->L }
% 8.70/1.54 and(and(X, Y), f(f(and(X, Y), neg(neg(X))), f(X, Y)))
% 8.70/1.54 = { by lemma 33 }
% 8.70/1.54 and(and(X, Y), f(or(f(X, Y), neg(X)), f(X, Y)))
% 8.70/1.54 = { by lemma 16 R->L }
% 8.70/1.54 and(and(neg(neg(X)), Y), f(or(f(X, Y), neg(X)), f(X, Y)))
% 8.70/1.54 = { by lemma 16 R->L }
% 8.70/1.54 and(and(neg(neg(X)), Y), f(or(f(neg(neg(X)), Y), neg(X)), f(X, Y)))
% 8.70/1.54 = { by lemma 16 R->L }
% 8.70/1.54 and(and(neg(neg(X)), Y), f(or(f(neg(neg(X)), Y), neg(X)), f(neg(neg(X)), Y)))
% 8.70/1.54 = { by lemma 9 R->L }
% 8.70/1.54 and(neg(f(neg(neg(X)), Y)), f(or(f(neg(neg(X)), Y), neg(X)), f(neg(neg(X)), Y)))
% 8.70/1.54 = { by lemma 28 R->L }
% 8.70/1.54 and(neg(f(neg(neg(X)), Y)), f(or(f(neg(neg(X)), Y), and(f(neg(neg(X)), Y), f(X, f(f(X, f(neg(X), X)), X)))), f(neg(neg(X)), Y)))
% 8.70/1.54 = { by lemma 52 }
% 8.70/1.54 and(neg(f(neg(neg(X)), Y)), f(f(neg(neg(X)), Y), f(X, f(f(X, f(neg(X), X)), X))))
% 8.70/1.54 = { by lemma 9 }
% 8.70/1.54 and(and(neg(neg(X)), Y), f(f(neg(neg(X)), Y), f(X, f(f(X, f(neg(X), X)), X))))
% 8.70/1.54 = { by lemma 16 }
% 8.70/1.54 and(and(X, Y), f(f(neg(neg(X)), Y), f(X, f(f(X, f(neg(X), X)), X))))
% 8.70/1.54 = { by lemma 16 }
% 8.70/1.54 and(and(X, Y), f(f(X, Y), f(X, f(f(X, f(neg(X), X)), X))))
% 8.70/1.54 = { by lemma 29 }
% 8.70/1.54 and(and(X, Y), or(and(X, Y), and(X, f(f(X, f(neg(X), X)), X))))
% 8.70/1.54 = { by lemma 20 }
% 8.70/1.54 and(and(X, Y), or(and(X, Y), and(X, f(f(X, or(X, neg(X))), X))))
% 8.70/1.54 = { by lemma 43 }
% 8.70/1.54 and(and(X, Y), or(and(X, Y), X))
% 8.70/1.54
% 8.70/1.54 Lemma 54: or(and(X, Y), and(or(and(X, Y), Y), Y)) = or(and(X, Y), Y).
% 8.70/1.54 Proof:
% 8.70/1.54 or(and(X, Y), and(or(and(X, Y), Y), Y))
% 8.70/1.54 = { by lemma 29 R->L }
% 8.70/1.54 f(f(X, Y), f(or(and(X, Y), Y), Y))
% 8.70/1.54 = { by lemma 17 R->L }
% 8.70/1.54 neg(and(f(X, Y), f(or(and(X, Y), Y), Y)))
% 8.70/1.54 = { by lemma 9 R->L }
% 8.70/1.54 neg(and(f(X, Y), f(or(neg(f(X, Y)), Y), Y)))
% 8.70/1.54 = { by lemma 22 R->L }
% 8.70/1.54 neg(and(f(X, Y), f(f(f(X, Y), neg(Y)), Y)))
% 8.70/1.54 = { by lemma 9 R->L }
% 8.70/1.54 neg(neg(f(f(X, Y), f(f(f(X, Y), neg(Y)), Y))))
% 8.70/1.54 = { by axiom 1 (sos_003) }
% 8.70/1.54 neg(neg(f(f(X, Y), f(f(f(X, Y), f(Y, Y)), Y))))
% 8.70/1.54 = { by lemma 11 }
% 8.70/1.54 neg(neg(f(f(X, Y), f(Y, Y))))
% 8.70/1.54 = { by axiom 1 (sos_003) R->L }
% 8.70/1.54 neg(neg(f(f(X, Y), neg(Y))))
% 8.70/1.54 = { by lemma 9 }
% 8.70/1.54 neg(and(f(X, Y), neg(Y)))
% 8.70/1.54 = { by lemma 17 }
% 8.70/1.54 f(f(X, Y), neg(Y))
% 8.70/1.54 = { by lemma 22 }
% 8.70/1.54 or(neg(f(X, Y)), Y)
% 8.70/1.54 = { by lemma 9 }
% 8.70/1.54 or(and(X, Y), Y)
% 8.70/1.54
% 8.70/1.54 Lemma 55: or(or(X, Y), or(Y, X)) = or(Y, X).
% 8.70/1.54 Proof:
% 8.70/1.54 or(or(X, Y), or(Y, X))
% 8.70/1.54 = { by lemma 51 R->L }
% 8.70/1.54 or(or(and(or(Y, X), X), Y), or(Y, X))
% 8.70/1.54 = { by lemma 13 R->L }
% 8.70/1.54 or(or(or(and(or(Y, X), X), and(or(Y, X), X)), Y), or(Y, X))
% 8.70/1.54 = { by lemma 25 R->L }
% 8.70/1.54 or(or(and(and(or(Y, X), X), and(or(Y, X), X)), Y), or(Y, X))
% 8.70/1.54 = { by lemma 45 R->L }
% 8.70/1.54 or(or(and(and(or(Y, X), X), or(and(and(or(Y, X), X), or(Y, X)), and(or(Y, X), X))), Y), or(Y, X))
% 8.70/1.54 = { by lemma 53 }
% 8.70/1.54 or(or(and(and(or(Y, X), X), or(and(or(Y, X), X), or(Y, X))), Y), or(Y, X))
% 8.70/1.54 = { by lemma 51 }
% 8.70/1.54 or(or(and(X, or(and(or(Y, X), X), or(Y, X))), Y), or(Y, X))
% 8.70/1.54 = { by lemma 45 }
% 8.70/1.54 or(or(and(X, or(Y, X)), Y), or(Y, X))
% 8.70/1.54 = { by lemma 32 R->L }
% 8.70/1.54 or(or(and(X, or(Y, X)), and(or(Y, X), or(Y, and(or(Y, X), neg(or(Y, X)))))), or(Y, X))
% 8.70/1.54 = { by lemma 50 R->L }
% 8.70/1.54 or(or(and(X, or(Y, X)), and(or(Y, X), or(Y, and(or(Y, X), neg(or(Y, X)))))), or(and(or(and(X, or(Y, X)), and(or(Y, X), or(Y, and(or(Y, X), neg(or(Y, X)))))), or(Y, X)), or(Y, X)))
% 8.70/1.54 = { by lemma 49 R->L }
% 8.70/1.54 or(or(and(X, or(Y, X)), and(or(Y, X), or(Y, and(or(Y, X), neg(or(Y, X)))))), and(or(and(or(and(X, or(Y, X)), and(or(Y, X), or(Y, and(or(Y, X), neg(or(Y, X)))))), or(Y, X)), or(Y, X)), or(Y, X)))
% 8.70/1.54 = { by lemma 48 R->L }
% 8.70/1.54 or(and(or(and(X, or(Y, X)), and(or(Y, X), or(Y, and(or(Y, X), neg(or(Y, X)))))), or(Y, X)), and(or(and(or(and(X, or(Y, X)), and(or(Y, X), or(Y, and(or(Y, X), neg(or(Y, X)))))), or(Y, X)), or(Y, X)), or(Y, X)))
% 8.70/1.54 = { by lemma 54 }
% 8.70/1.54 or(and(or(and(X, or(Y, X)), and(or(Y, X), or(Y, and(or(Y, X), neg(or(Y, X)))))), or(Y, X)), or(Y, X))
% 8.70/1.54 = { by lemma 50 }
% 8.70/1.54 or(Y, X)
% 8.70/1.54
% 8.70/1.54 Lemma 56: and(neg(X), f(X, and(X, Y))) = and(neg(X), f(X, Y)).
% 8.70/1.54 Proof:
% 8.70/1.54 and(neg(X), f(X, and(X, Y)))
% 8.70/1.54 = { by lemma 16 R->L }
% 8.70/1.54 and(neg(X), f(X, and(neg(neg(X)), Y)))
% 8.70/1.54 = { by lemma 36 R->L }
% 8.70/1.54 and(neg(X), or(neg(X), f(neg(neg(X)), Y)))
% 8.70/1.54 = { by lemma 28 R->L }
% 8.70/1.54 and(and(f(neg(neg(X)), Y), f(X, f(f(X, f(neg(X), X)), X))), or(neg(X), f(neg(neg(X)), Y)))
% 8.70/1.54 = { by lemma 28 R->L }
% 8.70/1.54 and(and(f(neg(neg(X)), Y), f(X, f(f(X, f(neg(X), X)), X))), or(and(f(neg(neg(X)), Y), f(X, f(f(X, f(neg(X), X)), X))), f(neg(neg(X)), Y)))
% 8.70/1.54 = { by lemma 53 R->L }
% 8.70/1.54 and(and(f(neg(neg(X)), Y), f(X, f(f(X, f(neg(X), X)), X))), or(and(and(f(neg(neg(X)), Y), f(X, f(f(X, f(neg(X), X)), X))), f(neg(neg(X)), Y)), and(f(neg(neg(X)), Y), f(X, f(f(X, f(neg(X), X)), X)))))
% 8.70/1.54 = { by lemma 28 }
% 8.70/1.54 and(neg(X), or(and(and(f(neg(neg(X)), Y), f(X, f(f(X, f(neg(X), X)), X))), f(neg(neg(X)), Y)), and(f(neg(neg(X)), Y), f(X, f(f(X, f(neg(X), X)), X)))))
% 8.70/1.54 = { by lemma 28 }
% 8.70/1.54 and(neg(X), or(and(neg(X), f(neg(neg(X)), Y)), and(f(neg(neg(X)), Y), f(X, f(f(X, f(neg(X), X)), X)))))
% 8.70/1.54 = { by lemma 28 }
% 8.70/1.54 and(neg(X), or(and(neg(X), f(neg(neg(X)), Y)), neg(X)))
% 8.70/1.54 = { by lemma 23 }
% 8.70/1.54 and(neg(X), f(f(neg(X), f(neg(neg(X)), Y)), X))
% 8.70/1.54 = { by lemma 20 }
% 8.70/1.54 and(neg(X), f(or(X, neg(f(neg(neg(X)), Y))), X))
% 8.70/1.54 = { by lemma 9 }
% 8.70/1.54 and(neg(X), f(or(X, and(neg(neg(X)), Y)), X))
% 8.70/1.54 = { by lemma 16 }
% 8.70/1.54 and(neg(X), f(or(X, and(X, Y)), X))
% 8.70/1.54 = { by lemma 52 }
% 8.70/1.54 and(neg(X), f(X, Y))
% 8.70/1.54
% 8.70/1.54 Lemma 57: neg(or(and(X, Y), Z)) = and(f(X, Y), neg(Z)).
% 8.70/1.54 Proof:
% 8.70/1.54 neg(or(and(X, Y), Z))
% 8.70/1.54 = { by lemma 9 R->L }
% 8.70/1.54 neg(or(neg(f(X, Y)), Z))
% 8.70/1.54 = { by lemma 19 }
% 8.70/1.54 and(f(X, Y), neg(Z))
% 8.70/1.54
% 8.70/1.54 Lemma 58: neg(or(X, neg(Y))) = and(neg(X), Y).
% 8.70/1.54 Proof:
% 8.70/1.54 neg(or(X, neg(Y)))
% 8.70/1.54 = { by lemma 15 R->L }
% 8.70/1.54 and(neg(X), neg(neg(Y)))
% 8.70/1.54 = { by lemma 6 }
% 8.70/1.54 and(neg(X), or(Y, Y))
% 8.70/1.54 = { by lemma 13 }
% 8.70/1.54 and(neg(X), Y)
% 8.70/1.54
% 8.70/1.54 Lemma 59: and(or(X, Y), X) = X.
% 8.70/1.54 Proof:
% 8.70/1.54 and(or(X, Y), X)
% 8.70/1.54 = { by lemma 5 R->L }
% 8.70/1.54 and(f(neg(X), neg(Y)), X)
% 8.70/1.54 = { by lemma 17 R->L }
% 8.70/1.54 and(neg(and(neg(X), neg(Y))), X)
% 8.70/1.54 = { by lemma 58 R->L }
% 8.70/1.54 neg(or(and(neg(X), neg(Y)), neg(X)))
% 8.70/1.54 = { by lemma 44 }
% 8.70/1.54 neg(neg(X))
% 8.70/1.54 = { by lemma 16 }
% 8.70/1.55 X
% 8.70/1.55
% 8.70/1.55 Lemma 60: or(Y, X) = or(X, Y).
% 8.70/1.55 Proof:
% 8.70/1.55 or(Y, X)
% 8.70/1.55 = { by lemma 55 R->L }
% 8.70/1.55 or(or(X, Y), or(Y, X))
% 8.70/1.55 = { by lemma 55 R->L }
% 8.70/1.55 or(or(or(Y, X), or(X, Y)), or(Y, X))
% 8.70/1.55 = { by lemma 5 R->L }
% 8.70/1.55 or(f(neg(or(Y, X)), neg(or(X, Y))), or(Y, X))
% 8.70/1.55 = { by lemma 33 R->L }
% 8.70/1.55 f(and(neg(or(Y, X)), neg(or(X, Y))), neg(or(Y, X)))
% 8.70/1.55 = { by lemma 17 R->L }
% 8.70/1.55 neg(and(and(neg(or(Y, X)), neg(or(X, Y))), neg(or(Y, X))))
% 8.70/1.55 = { by lemma 16 R->L }
% 8.70/1.55 neg(and(and(neg(or(Y, X)), neg(neg(neg(or(X, Y))))), neg(or(Y, X))))
% 8.70/1.55 = { by lemma 41 R->L }
% 8.70/1.55 neg(and(and(neg(or(Y, X)), neg(neg(neg(or(X, Y))))), f(or(neg(neg(or(Y, X))), neg(neg(or(X, Y)))), f(neg(or(Y, X)), f(f(neg(or(Y, X)), f(neg(neg(or(Y, X))), neg(or(Y, X)))), neg(or(Y, X)))))))
% 8.70/1.55 = { by lemma 19 R->L }
% 8.70/1.55 neg(and(neg(or(neg(neg(or(Y, X))), neg(neg(or(X, Y))))), f(or(neg(neg(or(Y, X))), neg(neg(or(X, Y)))), f(neg(or(Y, X)), f(f(neg(or(Y, X)), f(neg(neg(or(Y, X))), neg(or(Y, X)))), neg(or(Y, X)))))))
% 8.70/1.55 = { by lemma 56 R->L }
% 8.70/1.55 neg(and(neg(or(neg(neg(or(Y, X))), neg(neg(or(X, Y))))), f(or(neg(neg(or(Y, X))), neg(neg(or(X, Y)))), and(or(neg(neg(or(Y, X))), neg(neg(or(X, Y)))), f(neg(or(Y, X)), f(f(neg(or(Y, X)), f(neg(neg(or(Y, X))), neg(or(Y, X)))), neg(or(Y, X))))))))
% 8.70/1.55 = { by lemma 42 }
% 8.70/1.55 neg(and(neg(or(neg(neg(or(Y, X))), neg(neg(or(X, Y))))), f(or(neg(neg(or(Y, X))), neg(neg(or(X, Y)))), neg(neg(or(Y, X))))))
% 8.70/1.55 = { by lemma 19 }
% 8.70/1.55 neg(and(and(neg(or(Y, X)), neg(neg(neg(or(X, Y))))), f(or(neg(neg(or(Y, X))), neg(neg(or(X, Y)))), neg(neg(or(Y, X))))))
% 8.70/1.55 = { by lemma 22 }
% 8.70/1.55 neg(and(and(neg(or(Y, X)), neg(neg(neg(or(X, Y))))), or(neg(or(neg(neg(or(Y, X))), neg(neg(or(X, Y))))), neg(or(Y, X)))))
% 8.70/1.55 = { by lemma 19 }
% 8.70/1.55 neg(and(and(neg(or(Y, X)), neg(neg(neg(or(X, Y))))), or(and(neg(or(Y, X)), neg(neg(neg(or(X, Y))))), neg(or(Y, X)))))
% 8.70/1.55 = { by lemma 6 }
% 8.70/1.55 neg(and(and(neg(or(Y, X)), or(neg(or(X, Y)), neg(or(X, Y)))), or(and(neg(or(Y, X)), neg(neg(neg(or(X, Y))))), neg(or(Y, X)))))
% 8.70/1.55 = { by lemma 13 }
% 8.70/1.55 neg(and(and(neg(or(Y, X)), neg(or(X, Y))), or(and(neg(or(Y, X)), neg(neg(neg(or(X, Y))))), neg(or(Y, X)))))
% 8.70/1.55 = { by lemma 16 }
% 8.70/1.55 neg(and(and(neg(or(Y, X)), neg(or(X, Y))), or(and(neg(or(Y, X)), neg(or(X, Y))), neg(or(Y, X)))))
% 8.70/1.55 = { by lemma 17 }
% 8.70/1.55 f(and(neg(or(Y, X)), neg(or(X, Y))), or(and(neg(or(Y, X)), neg(or(X, Y))), neg(or(Y, X))))
% 8.70/1.55 = { by lemma 39 R->L }
% 8.70/1.55 f(and(neg(or(Y, X)), neg(or(X, Y))), or(and(and(neg(or(Y, X)), neg(or(X, Y))), or(and(neg(or(Y, X)), neg(or(X, Y))), neg(or(Y, X)))), and(neg(or(Y, X)), neg(or(X, Y)))))
% 8.70/1.55 = { by lemma 44 }
% 8.70/1.55 f(and(neg(or(Y, X)), neg(or(X, Y))), or(and(and(neg(or(Y, X)), neg(or(X, Y))), neg(or(Y, X))), and(neg(or(Y, X)), neg(or(X, Y)))))
% 8.70/1.55 = { by lemma 15 }
% 8.70/1.55 f(neg(or(or(Y, X), or(X, Y))), or(and(and(neg(or(Y, X)), neg(or(X, Y))), neg(or(Y, X))), and(neg(or(Y, X)), neg(or(X, Y)))))
% 8.70/1.55 = { by lemma 15 }
% 8.70/1.55 f(neg(or(or(Y, X), or(X, Y))), or(and(neg(or(or(Y, X), or(X, Y))), neg(or(Y, X))), and(neg(or(Y, X)), neg(or(X, Y)))))
% 8.70/1.55 = { by lemma 15 }
% 8.70/1.55 f(neg(or(or(Y, X), or(X, Y))), or(and(neg(or(or(Y, X), or(X, Y))), neg(or(Y, X))), neg(or(or(Y, X), or(X, Y)))))
% 8.70/1.55 = { by lemma 20 }
% 8.70/1.55 or(or(or(Y, X), or(X, Y)), neg(or(and(neg(or(or(Y, X), or(X, Y))), neg(or(Y, X))), neg(or(or(Y, X), or(X, Y))))))
% 8.70/1.55 = { by lemma 57 }
% 8.70/1.55 or(or(or(Y, X), or(X, Y)), and(f(neg(or(or(Y, X), or(X, Y))), neg(or(Y, X))), neg(neg(or(or(Y, X), or(X, Y))))))
% 8.70/1.55 = { by lemma 5 }
% 8.70/1.55 or(or(or(Y, X), or(X, Y)), and(or(or(or(Y, X), or(X, Y)), or(Y, X)), neg(neg(or(or(Y, X), or(X, Y))))))
% 8.70/1.55 = { by lemma 16 }
% 8.70/1.55 or(or(or(Y, X), or(X, Y)), and(or(or(or(Y, X), or(X, Y)), or(Y, X)), or(or(Y, X), or(X, Y))))
% 8.70/1.55 = { by lemma 59 }
% 8.70/1.55 or(or(or(Y, X), or(X, Y)), or(or(Y, X), or(X, Y)))
% 8.70/1.55 = { by lemma 13 }
% 8.70/1.55 or(or(Y, X), or(X, Y))
% 8.70/1.55 = { by lemma 55 }
% 8.70/1.55 or(X, Y)
% 8.70/1.55
% 8.70/1.55 Lemma 61: f(Y, X) = f(X, Y).
% 8.70/1.55 Proof:
% 8.70/1.55 f(Y, X)
% 8.70/1.55 = { by lemma 14 R->L }
% 8.70/1.55 or(neg(Y), neg(X))
% 8.70/1.55 = { by lemma 60 }
% 8.70/1.55 or(neg(X), neg(Y))
% 8.70/1.55 = { by lemma 14 }
% 8.70/1.55 f(X, Y)
% 8.70/1.55
% 8.70/1.55 Lemma 62: and(Y, X) = and(X, Y).
% 8.70/1.55 Proof:
% 8.70/1.55 and(Y, X)
% 8.70/1.55 = { by lemma 9 R->L }
% 8.70/1.55 neg(f(Y, X))
% 8.70/1.55 = { by lemma 61 }
% 8.70/1.55 neg(f(X, Y))
% 8.70/1.55 = { by lemma 9 }
% 8.70/1.55 and(X, Y)
% 8.70/1.55
% 8.70/1.55 Lemma 63: or(X, or(X, Y)) = or(X, Y).
% 8.70/1.55 Proof:
% 8.70/1.55 or(X, or(X, Y))
% 8.70/1.55 = { by lemma 59 R->L }
% 8.70/1.55 or(and(or(X, Y), X), or(X, Y))
% 8.70/1.55 = { by lemma 45 }
% 8.70/1.55 or(X, Y)
% 8.70/1.55
% 8.70/1.55 Lemma 64: or(or(X, neg(Y)), neg(Z)) = f(and(neg(X), Y), Z).
% 8.70/1.55 Proof:
% 8.70/1.55 or(or(X, neg(Y)), neg(Z))
% 8.70/1.55 = { by lemma 20 R->L }
% 8.70/1.55 f(neg(or(X, neg(Y))), Z)
% 8.70/1.55 = { by lemma 58 }
% 8.70/1.55 f(and(neg(X), Y), Z)
% 8.70/1.55
% 8.70/1.55 Lemma 65: or(X, and(or(X, Y), or(X, neg(X)))) = or(X, Y).
% 8.70/1.55 Proof:
% 8.70/1.55 or(X, and(or(X, Y), or(X, neg(X))))
% 8.70/1.55 = { by lemma 5 R->L }
% 8.70/1.55 or(X, and(or(X, Y), f(neg(X), neg(neg(X)))))
% 8.70/1.55 = { by lemma 17 R->L }
% 8.70/1.55 or(X, and(or(X, Y), neg(and(neg(X), neg(neg(X))))))
% 8.70/1.55 = { by lemma 19 R->L }
% 8.70/1.55 or(X, neg(or(neg(or(X, Y)), and(neg(X), neg(neg(X))))))
% 8.70/1.55 = { by lemma 20 R->L }
% 8.70/1.55 f(neg(X), or(neg(or(X, Y)), and(neg(X), neg(neg(X)))))
% 8.70/1.55 = { by lemma 15 R->L }
% 8.70/1.55 f(neg(X), or(and(neg(X), neg(Y)), and(neg(X), neg(neg(X)))))
% 8.70/1.55 = { by lemma 31 }
% 8.70/1.55 f(neg(X), neg(Y))
% 8.70/1.55 = { by lemma 5 }
% 8.70/1.55 or(X, Y)
% 8.70/1.55
% 8.70/1.55 Lemma 66: and(X, or(X, or(X, Y))) = and(X, or(X, Y)).
% 8.70/1.55 Proof:
% 8.70/1.55 and(X, or(X, or(X, Y)))
% 8.70/1.55 = { by lemma 5 R->L }
% 8.70/1.55 and(X, f(neg(X), neg(or(X, Y))))
% 8.70/1.55 = { by lemma 16 R->L }
% 8.70/1.55 and(neg(neg(X)), f(neg(X), neg(or(X, Y))))
% 8.70/1.55 = { by lemma 15 R->L }
% 8.70/1.55 and(neg(neg(X)), f(neg(X), and(neg(X), neg(Y))))
% 8.70/1.55 = { by lemma 56 }
% 8.70/1.55 and(neg(neg(X)), f(neg(X), neg(Y)))
% 8.70/1.55 = { by lemma 16 }
% 8.70/1.55 and(X, f(neg(X), neg(Y)))
% 8.70/1.55 = { by lemma 5 }
% 8.70/1.55 and(X, or(X, Y))
% 8.70/1.55
% 8.70/1.55 Lemma 67: or(or(X, Y), or(X, neg(X))) = or(X, neg(X)).
% 8.70/1.55 Proof:
% 8.70/1.55 or(or(X, Y), or(X, neg(X)))
% 8.70/1.55 = { by lemma 50 R->L }
% 8.70/1.55 or(or(X, Y), or(and(or(X, Y), or(X, neg(X))), or(X, neg(X))))
% 8.70/1.55 = { by lemma 49 R->L }
% 8.70/1.55 or(or(X, Y), and(or(and(or(X, Y), or(X, neg(X))), or(X, neg(X))), or(X, neg(X))))
% 8.70/1.55 = { by lemma 65 R->L }
% 8.70/1.55 or(or(X, and(or(X, Y), or(X, neg(X)))), and(or(and(or(X, Y), or(X, neg(X))), or(X, neg(X))), or(X, neg(X))))
% 8.70/1.55 = { by lemma 62 }
% 8.70/1.55 or(or(X, and(or(X, neg(X)), or(X, Y))), and(or(and(or(X, Y), or(X, neg(X))), or(X, neg(X))), or(X, neg(X))))
% 8.70/1.55 = { by lemma 63 R->L }
% 8.70/1.55 or(or(X, and(or(X, or(X, neg(X))), or(X, Y))), and(or(and(or(X, Y), or(X, neg(X))), or(X, neg(X))), or(X, neg(X))))
% 8.70/1.55 = { by lemma 46 R->L }
% 8.70/1.55 or(or(and(X, or(X, neg(X))), and(or(X, or(X, neg(X))), or(X, Y))), and(or(and(or(X, Y), or(X, neg(X))), or(X, neg(X))), or(X, neg(X))))
% 8.70/1.55 = { by lemma 66 R->L }
% 8.70/1.55 or(or(and(X, or(X, or(X, neg(X)))), and(or(X, or(X, neg(X))), or(X, Y))), and(or(and(or(X, Y), or(X, neg(X))), or(X, neg(X))), or(X, neg(X))))
% 8.70/1.55 = { by lemma 48 R->L }
% 8.70/1.55 or(and(or(and(X, or(X, or(X, neg(X)))), and(or(X, or(X, neg(X))), or(X, Y))), or(X, or(X, neg(X)))), and(or(and(or(X, Y), or(X, neg(X))), or(X, neg(X))), or(X, neg(X))))
% 8.70/1.55 = { by lemma 66 }
% 8.70/1.55 or(and(or(and(X, or(X, neg(X))), and(or(X, or(X, neg(X))), or(X, Y))), or(X, or(X, neg(X)))), and(or(and(or(X, Y), or(X, neg(X))), or(X, neg(X))), or(X, neg(X))))
% 8.70/1.55 = { by lemma 46 }
% 8.70/1.55 or(and(or(X, and(or(X, or(X, neg(X))), or(X, Y))), or(X, or(X, neg(X)))), and(or(and(or(X, Y), or(X, neg(X))), or(X, neg(X))), or(X, neg(X))))
% 8.70/1.55 = { by lemma 63 }
% 8.70/1.55 or(and(or(X, and(or(X, or(X, neg(X))), or(X, Y))), or(X, neg(X))), and(or(and(or(X, Y), or(X, neg(X))), or(X, neg(X))), or(X, neg(X))))
% 8.70/1.55 = { by lemma 63 }
% 8.70/1.55 or(and(or(X, and(or(X, neg(X)), or(X, Y))), or(X, neg(X))), and(or(and(or(X, Y), or(X, neg(X))), or(X, neg(X))), or(X, neg(X))))
% 8.70/1.55 = { by lemma 62 R->L }
% 8.70/1.55 or(and(or(X, neg(X)), or(X, and(or(X, neg(X)), or(X, Y)))), and(or(and(or(X, Y), or(X, neg(X))), or(X, neg(X))), or(X, neg(X))))
% 8.70/1.55 = { by lemma 62 R->L }
% 8.70/1.55 or(and(or(X, neg(X)), or(X, and(or(X, Y), or(X, neg(X))))), and(or(and(or(X, Y), or(X, neg(X))), or(X, neg(X))), or(X, neg(X))))
% 8.70/1.55 = { by lemma 65 }
% 8.70/1.55 or(and(or(X, neg(X)), or(X, Y)), and(or(and(or(X, Y), or(X, neg(X))), or(X, neg(X))), or(X, neg(X))))
% 8.70/1.55 = { by lemma 62 R->L }
% 8.70/1.55 or(and(or(X, Y), or(X, neg(X))), and(or(and(or(X, Y), or(X, neg(X))), or(X, neg(X))), or(X, neg(X))))
% 8.70/1.55 = { by lemma 54 }
% 8.70/1.55 or(and(or(X, Y), or(X, neg(X))), or(X, neg(X)))
% 8.70/1.55 = { by lemma 50 }
% 8.70/1.55 or(X, neg(X))
% 8.70/1.55
% 8.70/1.55 Lemma 68: f(X, and(Y, neg(Y))) = or(Y, neg(Y)).
% 8.70/1.55 Proof:
% 8.70/1.55 f(X, and(Y, neg(Y)))
% 8.70/1.55 = { by lemma 61 }
% 8.70/1.55 f(and(Y, neg(Y)), X)
% 8.70/1.55 = { by lemma 62 }
% 8.70/1.55 f(and(neg(Y), Y), X)
% 8.70/1.55 = { by lemma 64 R->L }
% 8.70/1.55 or(or(Y, neg(Y)), neg(X))
% 8.70/1.55 = { by lemma 20 R->L }
% 8.70/1.55 f(neg(or(Y, neg(Y))), X)
% 8.70/1.55 = { by lemma 30 R->L }
% 8.70/1.55 f(neg(or(Y, neg(Y))), or(and(neg(or(Y, neg(Y))), X), and(and(neg(or(Y, Z)), neg(or(Y, neg(Y)))), f(neg(or(Y, Z)), neg(or(Y, neg(Y)))))))
% 8.70/1.55 = { by lemma 5 }
% 8.70/1.55 f(neg(or(Y, neg(Y))), or(and(neg(or(Y, neg(Y))), X), and(and(neg(or(Y, Z)), neg(or(Y, neg(Y)))), or(or(Y, Z), or(Y, neg(Y))))))
% 8.70/1.56 = { by lemma 20 }
% 8.70/1.56 or(or(Y, neg(Y)), neg(or(and(neg(or(Y, neg(Y))), X), and(and(neg(or(Y, Z)), neg(or(Y, neg(Y)))), or(or(Y, Z), or(Y, neg(Y)))))))
% 8.70/1.56 = { by lemma 57 }
% 8.70/1.56 or(or(Y, neg(Y)), and(f(neg(or(Y, neg(Y))), X), neg(and(and(neg(or(Y, Z)), neg(or(Y, neg(Y)))), or(or(Y, Z), or(Y, neg(Y)))))))
% 8.70/1.56 = { by lemma 20 }
% 8.70/1.56 or(or(Y, neg(Y)), and(or(or(Y, neg(Y)), neg(X)), neg(and(and(neg(or(Y, Z)), neg(or(Y, neg(Y)))), or(or(Y, Z), or(Y, neg(Y)))))))
% 8.70/1.56 = { by lemma 17 }
% 8.70/1.56 or(or(Y, neg(Y)), and(or(or(Y, neg(Y)), neg(X)), f(and(neg(or(Y, Z)), neg(or(Y, neg(Y)))), or(or(Y, Z), or(Y, neg(Y))))))
% 8.70/1.56 = { by lemma 15 }
% 8.70/1.56 or(or(Y, neg(Y)), and(or(or(Y, neg(Y)), neg(X)), f(neg(or(or(Y, Z), or(Y, neg(Y)))), or(or(Y, Z), or(Y, neg(Y))))))
% 8.70/1.56 = { by lemma 20 }
% 8.70/1.56 or(or(Y, neg(Y)), and(or(or(Y, neg(Y)), neg(X)), or(or(or(Y, Z), or(Y, neg(Y))), neg(or(or(Y, Z), or(Y, neg(Y)))))))
% 8.70/1.56 = { by lemma 67 }
% 8.70/1.56 or(or(Y, neg(Y)), and(or(or(Y, neg(Y)), neg(X)), or(or(Y, neg(Y)), neg(or(or(Y, Z), or(Y, neg(Y)))))))
% 8.70/1.56 = { by lemma 64 }
% 8.70/1.56 or(or(Y, neg(Y)), and(f(and(neg(Y), Y), X), or(or(Y, neg(Y)), neg(or(or(Y, Z), or(Y, neg(Y)))))))
% 8.70/1.56 = { by lemma 64 }
% 8.70/1.56 or(or(Y, neg(Y)), and(f(and(neg(Y), Y), X), f(and(neg(Y), Y), or(or(Y, Z), or(Y, neg(Y))))))
% 8.70/1.56 = { by lemma 67 }
% 8.70/1.56 or(or(Y, neg(Y)), and(f(and(neg(Y), Y), X), f(and(neg(Y), Y), or(Y, neg(Y)))))
% 8.70/1.56 = { by lemma 59 R->L }
% 8.70/1.56 or(or(Y, neg(Y)), and(f(and(neg(Y), Y), X), f(and(or(and(neg(Y), Y), Y), and(neg(Y), Y)), or(Y, neg(Y)))))
% 8.70/1.56 = { by lemma 50 }
% 8.70/1.56 or(or(Y, neg(Y)), and(f(and(neg(Y), Y), X), f(and(Y, and(neg(Y), Y)), or(Y, neg(Y)))))
% 8.70/1.56 = { by lemma 17 R->L }
% 8.70/1.56 or(or(Y, neg(Y)), and(f(and(neg(Y), Y), X), neg(and(and(Y, and(neg(Y), Y)), or(Y, neg(Y))))))
% 8.70/1.56 = { by lemma 59 R->L }
% 8.70/1.56 or(or(Y, neg(Y)), and(f(and(neg(Y), Y), X), neg(and(and(and(or(Y, neg(Y)), Y), and(neg(Y), Y)), or(Y, neg(Y))))))
% 8.70/1.56 = { by lemma 45 R->L }
% 8.70/1.56 or(or(Y, neg(Y)), and(f(and(neg(Y), Y), X), neg(and(and(or(and(and(or(Y, neg(Y)), Y), or(Y, neg(Y))), and(or(Y, neg(Y)), Y)), and(neg(Y), Y)), or(Y, neg(Y))))))
% 8.70/1.56 = { by lemma 47 }
% 8.70/1.56 or(or(Y, neg(Y)), and(f(and(neg(Y), Y), X), neg(and(or(and(and(or(Y, neg(Y)), Y), or(Y, neg(Y))), and(or(Y, neg(Y)), Y)), and(neg(Y), Y)))))
% 8.70/1.56 = { by lemma 45 }
% 8.70/1.56 or(or(Y, neg(Y)), and(f(and(neg(Y), Y), X), neg(and(and(or(Y, neg(Y)), Y), and(neg(Y), Y)))))
% 8.70/1.56 = { by lemma 59 }
% 8.70/1.56 or(or(Y, neg(Y)), and(f(and(neg(Y), Y), X), neg(and(Y, and(neg(Y), Y)))))
% 8.70/1.56 = { by lemma 17 }
% 8.70/1.56 or(or(Y, neg(Y)), and(f(and(neg(Y), Y), X), f(Y, and(neg(Y), Y))))
% 8.70/1.56 = { by lemma 50 R->L }
% 8.70/1.56 or(or(Y, neg(Y)), and(f(and(neg(Y), Y), X), f(or(and(neg(Y), Y), Y), and(neg(Y), Y))))
% 8.70/1.56 = { by lemma 14 R->L }
% 8.70/1.56 or(or(Y, neg(Y)), and(f(and(neg(Y), Y), X), or(neg(or(and(neg(Y), Y), Y)), neg(and(neg(Y), Y)))))
% 8.70/1.56 = { by lemma 15 R->L }
% 8.70/1.56 or(or(Y, neg(Y)), and(f(and(neg(Y), Y), X), or(and(neg(and(neg(Y), Y)), neg(Y)), neg(and(neg(Y), Y)))))
% 8.70/1.56 = { by lemma 44 }
% 8.70/1.56 or(or(Y, neg(Y)), and(f(and(neg(Y), Y), X), neg(and(neg(Y), Y))))
% 8.70/1.56 = { by lemma 17 }
% 8.70/1.56 or(or(Y, neg(Y)), and(f(and(neg(Y), Y), X), f(neg(Y), Y)))
% 8.70/1.56 = { by lemma 17 R->L }
% 8.70/1.56 or(or(Y, neg(Y)), and(neg(and(and(neg(Y), Y), X)), f(neg(Y), Y)))
% 8.70/1.56 = { by lemma 58 R->L }
% 8.70/1.56 or(or(Y, neg(Y)), neg(or(and(and(neg(Y), Y), X), neg(f(neg(Y), Y)))))
% 8.70/1.56 = { by lemma 9 }
% 8.70/1.56 or(or(Y, neg(Y)), neg(or(and(and(neg(Y), Y), X), and(neg(Y), Y))))
% 8.70/1.56 = { by lemma 45 }
% 8.70/1.56 or(or(Y, neg(Y)), neg(and(neg(Y), Y)))
% 8.70/1.56 = { by lemma 17 }
% 8.70/1.56 or(or(Y, neg(Y)), f(neg(Y), Y))
% 8.70/1.56 = { by lemma 20 }
% 8.70/1.56 or(or(Y, neg(Y)), or(Y, neg(Y)))
% 8.70/1.56 = { by lemma 67 }
% 8.70/1.56 or(Y, neg(Y))
% 8.70/1.56
% 8.70/1.56 Goal 1 (goals): f(x0, f(x0, x0)) = f(x1, f(x1, x1)).
% 8.70/1.56 Proof:
% 8.70/1.56 f(x0, f(x0, x0))
% 8.70/1.56 = { by axiom 1 (sos_003) R->L }
% 8.70/1.56 f(x0, neg(x0))
% 8.70/1.56 = { by lemma 22 }
% 8.70/1.56 or(neg(x0), x0)
% 8.70/1.56 = { by lemma 60 R->L }
% 8.70/1.56 or(x0, neg(x0))
% 8.70/1.56 = { by lemma 68 R->L }
% 8.70/1.56 f(and(x1, neg(x1)), and(x0, neg(x0)))
% 8.70/1.56 = { by lemma 62 }
% 8.70/1.56 f(and(x1, neg(x1)), and(neg(x0), x0))
% 8.70/1.56 = { by lemma 61 }
% 8.70/1.56 f(and(neg(x0), x0), and(x1, neg(x1)))
% 8.70/1.56 = { by lemma 68 }
% 8.70/1.56 or(x1, neg(x1))
% 8.70/1.56 = { by lemma 60 }
% 8.70/1.56 or(neg(x1), x1)
% 8.70/1.56 = { by lemma 22 R->L }
% 8.70/1.56 f(x1, neg(x1))
% 8.70/1.56 = { by axiom 1 (sos_003) }
% 8.70/1.56 f(x1, f(x1, x1))
% 8.70/1.56 % SZS output end Proof
% 8.70/1.56
% 8.70/1.56 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------