TSTP Solution File: GRP777+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP777+1 : TPTP v8.1.2. Released v4.1.0.
% 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 : n022.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 01:20:04 EDT 2023
% Result : Theorem 2.34s 0.70s
% Output : Proof 3.29s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : GRP777+1 : TPTP v8.1.2. Released v4.1.0.
% 0.12/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.33 % Computer : n022.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Mon Aug 28 20:23:54 EDT 2023
% 0.12/0.34 % CPUTime :
% 2.34/0.70 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 2.34/0.70
% 2.34/0.70 % SZS status Theorem
% 2.34/0.70
% 3.03/0.76 % SZS output start Proof
% 3.03/0.76 Axiom 1 (sos06): product(X, X) = X.
% 3.03/0.76 Axiom 2 (sos02): product(X, difference(X, Y)) = Y.
% 3.03/0.76 Axiom 3 (sos04): product(quotient(X, Y), Y) = X.
% 3.03/0.76 Axiom 4 (sos01): difference(X, product(X, Y)) = Y.
% 3.03/0.76 Axiom 5 (sos03): quotient(product(X, Y), Y) = X.
% 3.03/0.76 Axiom 6 (sos08): bigC(X, Y, Z) = product(product(X, Y), product(Z, X)).
% 3.03/0.76 Axiom 7 (sos05): product(product(X, Y), product(Z, W)) = product(product(X, Z), product(Y, W)).
% 3.03/0.76 Axiom 8 (sos09): product(product(a, c), product(c, b)) = product(a, b).
% 3.03/0.76 Axiom 9 (sos07): product(product(product(X, Y), Y), product(Y, product(Y, X))) = Y.
% 3.03/0.76
% 3.03/0.76 Lemma 10: bigC(X, Z, Y) = bigC(X, Y, Z).
% 3.03/0.76 Proof:
% 3.03/0.76 bigC(X, Z, Y)
% 3.03/0.76 = { by axiom 6 (sos08) }
% 3.03/0.76 product(product(X, Z), product(Y, X))
% 3.03/0.76 = { by axiom 7 (sos05) R->L }
% 3.03/0.76 product(product(X, Y), product(Z, X))
% 3.03/0.76 = { by axiom 6 (sos08) R->L }
% 3.03/0.76 bigC(X, Y, Z)
% 3.03/0.76
% 3.03/0.76 Lemma 11: product(X, product(Y, X)) = bigC(X, X, Y).
% 3.03/0.76 Proof:
% 3.03/0.76 product(X, product(Y, X))
% 3.03/0.76 = { by axiom 1 (sos06) R->L }
% 3.03/0.76 product(product(X, X), product(Y, X))
% 3.03/0.76 = { by axiom 6 (sos08) R->L }
% 3.03/0.76 bigC(X, X, Y)
% 3.03/0.76
% 3.03/0.76 Lemma 12: product(product(X, Y), X) = bigC(X, X, Y).
% 3.03/0.76 Proof:
% 3.03/0.76 product(product(X, Y), X)
% 3.03/0.76 = { by axiom 1 (sos06) R->L }
% 3.03/0.76 product(product(X, Y), product(X, X))
% 3.03/0.76 = { by axiom 6 (sos08) R->L }
% 3.03/0.76 bigC(X, Y, X)
% 3.03/0.76 = { by lemma 10 R->L }
% 3.03/0.76 bigC(X, X, Y)
% 3.03/0.76
% 3.03/0.76 Lemma 13: bigC(X, Y, difference(X, Z)) = product(Z, product(Y, X)).
% 3.03/0.76 Proof:
% 3.03/0.76 bigC(X, Y, difference(X, Z))
% 3.03/0.76 = { by lemma 10 }
% 3.03/0.76 bigC(X, difference(X, Z), Y)
% 3.03/0.76 = { by axiom 6 (sos08) }
% 3.03/0.76 product(product(X, difference(X, Z)), product(Y, X))
% 3.03/0.76 = { by axiom 2 (sos02) }
% 3.03/0.76 product(Z, product(Y, X))
% 3.03/0.76
% 3.03/0.76 Lemma 14: product(Z, product(difference(Y, X), Y)) = product(X, product(difference(Y, Z), Y)).
% 3.03/0.76 Proof:
% 3.03/0.76 product(Z, product(difference(Y, X), Y))
% 3.03/0.76 = { by lemma 13 R->L }
% 3.03/0.76 bigC(Y, difference(Y, X), difference(Y, Z))
% 3.03/0.76 = { by lemma 10 R->L }
% 3.03/0.76 bigC(Y, difference(Y, Z), difference(Y, X))
% 3.03/0.76 = { by lemma 13 }
% 3.03/0.76 product(X, product(difference(Y, Z), Y))
% 3.03/0.76
% 3.03/0.76 Lemma 15: difference(X, product(Y, X)) = product(difference(X, Y), X).
% 3.03/0.76 Proof:
% 3.03/0.76 difference(X, product(Y, X))
% 3.03/0.76 = { by axiom 1 (sos06) R->L }
% 3.03/0.76 difference(X, product(Y, product(X, X)))
% 3.03/0.76 = { by lemma 13 R->L }
% 3.03/0.76 difference(X, bigC(X, X, difference(X, Y)))
% 3.03/0.76 = { by lemma 11 R->L }
% 3.03/0.76 difference(X, product(X, product(difference(X, Y), X)))
% 3.03/0.76 = { by axiom 4 (sos01) }
% 3.03/0.76 product(difference(X, Y), X)
% 3.03/0.76
% 3.03/0.76 Lemma 16: product(product(difference(X, Y), X), X) = quotient(X, product(X, Y)).
% 3.03/0.76 Proof:
% 3.03/0.76 product(product(difference(X, Y), X), X)
% 3.03/0.76 = { by axiom 5 (sos03) R->L }
% 3.03/0.76 quotient(product(product(product(difference(X, Y), X), X), product(X, product(X, difference(X, Y)))), product(X, product(X, difference(X, Y))))
% 3.03/0.76 = { by axiom 9 (sos07) }
% 3.03/0.76 quotient(X, product(X, product(X, difference(X, Y))))
% 3.03/0.76 = { by axiom 2 (sos02) }
% 3.03/0.76 quotient(X, product(X, Y))
% 3.03/0.76
% 3.03/0.76 Lemma 17: quotient(X, product(X, Y)) = product(X, quotient(X, Y)).
% 3.03/0.76 Proof:
% 3.03/0.76 quotient(X, product(X, Y))
% 3.03/0.76 = { by lemma 16 R->L }
% 3.03/0.76 product(product(difference(X, Y), X), X)
% 3.03/0.76 = { by axiom 1 (sos06) R->L }
% 3.03/0.76 product(product(difference(X, Y), product(X, X)), X)
% 3.03/0.76 = { by lemma 13 R->L }
% 3.03/0.76 product(bigC(X, X, difference(X, difference(X, Y))), X)
% 3.03/0.76 = { by lemma 11 R->L }
% 3.03/0.76 product(product(X, product(difference(X, difference(X, Y)), X)), X)
% 3.03/0.76 = { by lemma 12 }
% 3.03/0.76 bigC(X, X, product(difference(X, difference(X, Y)), X))
% 3.03/0.76 = { by lemma 11 R->L }
% 3.03/0.76 product(X, product(product(difference(X, difference(X, Y)), X), X))
% 3.03/0.76 = { by lemma 16 }
% 3.03/0.76 product(X, quotient(X, product(X, difference(X, Y))))
% 3.03/0.76 = { by axiom 2 (sos02) }
% 3.03/0.76 product(X, quotient(X, Y))
% 3.03/0.76
% 3.03/0.76 Lemma 18: product(difference(X, Y), X) = product(Y, quotient(X, Y)).
% 3.03/0.76 Proof:
% 3.03/0.76 product(difference(X, Y), X)
% 3.03/0.76 = { by axiom 1 (sos06) R->L }
% 3.03/0.76 product(product(difference(X, Y), X), product(difference(X, Y), X))
% 3.03/0.76 = { by lemma 14 R->L }
% 3.03/0.76 product(Y, product(difference(X, product(difference(X, Y), X)), X))
% 3.03/0.76 = { by lemma 15 R->L }
% 3.03/0.76 product(Y, difference(X, product(product(difference(X, Y), X), X)))
% 3.03/0.76 = { by lemma 16 }
% 3.03/0.76 product(Y, difference(X, quotient(X, product(X, Y))))
% 3.03/0.76 = { by lemma 17 }
% 3.03/0.76 product(Y, difference(X, product(X, quotient(X, Y))))
% 3.03/0.76 = { by axiom 4 (sos01) }
% 3.03/0.76 product(Y, quotient(X, Y))
% 3.03/0.76
% 3.03/0.76 Lemma 19: product(product(X, Y), product(Z, Y)) = product(product(X, Z), Y).
% 3.03/0.76 Proof:
% 3.03/0.76 product(product(X, Y), product(Z, Y))
% 3.03/0.76 = { by axiom 7 (sos05) R->L }
% 3.03/0.76 product(product(X, Z), product(Y, Y))
% 3.03/0.76 = { by axiom 1 (sos06) }
% 3.03/0.76 product(product(X, Z), Y)
% 3.03/0.76
% 3.03/0.76 Lemma 20: quotient(product(X, product(Y, Z)), Z) = product(quotient(X, Z), Y).
% 3.03/0.76 Proof:
% 3.03/0.76 quotient(product(X, product(Y, Z)), Z)
% 3.03/0.76 = { by axiom 3 (sos04) R->L }
% 3.03/0.76 quotient(product(product(quotient(X, Z), Z), product(Y, Z)), Z)
% 3.03/0.76 = { by lemma 19 }
% 3.03/0.76 quotient(product(product(quotient(X, Z), Y), Z), Z)
% 3.03/0.76 = { by axiom 5 (sos03) }
% 3.03/0.76 product(quotient(X, Z), Y)
% 3.03/0.76
% 3.03/0.76 Lemma 21: product(product(X, Y), product(X, Z)) = product(X, product(Y, Z)).
% 3.03/0.76 Proof:
% 3.03/0.76 product(product(X, Y), product(X, Z))
% 3.03/0.76 = { by axiom 7 (sos05) R->L }
% 3.03/0.76 product(product(X, X), product(Y, Z))
% 3.03/0.76 = { by axiom 1 (sos06) }
% 3.03/0.76 product(X, product(Y, Z))
% 3.03/0.76
% 3.03/0.76 Lemma 22: product(X, product(Y, difference(X, Z))) = product(product(X, Y), Z).
% 3.03/0.76 Proof:
% 3.03/0.76 product(X, product(Y, difference(X, Z)))
% 3.03/0.76 = { by lemma 21 R->L }
% 3.03/0.76 product(product(X, Y), product(X, difference(X, Z)))
% 3.03/0.76 = { by axiom 2 (sos02) }
% 3.03/0.76 product(product(X, Y), Z)
% 3.03/0.76
% 3.03/0.76 Lemma 23: difference(X, product(product(X, Y), Z)) = product(Y, difference(X, Z)).
% 3.03/0.76 Proof:
% 3.03/0.76 difference(X, product(product(X, Y), Z))
% 3.03/0.76 = { by lemma 22 R->L }
% 3.03/0.76 difference(X, product(X, product(Y, difference(X, Z))))
% 3.03/0.76 = { by axiom 4 (sos01) }
% 3.03/0.76 product(Y, difference(X, Z))
% 3.03/0.76
% 3.03/0.76 Lemma 24: difference(a, product(c, b)) = difference(c, b).
% 3.03/0.76 Proof:
% 3.03/0.76 difference(a, product(c, b))
% 3.03/0.76 = { by axiom 4 (sos01) R->L }
% 3.03/0.76 difference(c, product(c, difference(a, product(c, b))))
% 3.03/0.76 = { by lemma 23 R->L }
% 3.03/0.76 difference(c, difference(a, product(product(a, c), product(c, b))))
% 3.03/0.76 = { by axiom 8 (sos09) }
% 3.03/0.76 difference(c, difference(a, product(a, b)))
% 3.03/0.76 = { by axiom 4 (sos01) }
% 3.03/0.76 difference(c, b)
% 3.03/0.76
% 3.03/0.76 Lemma 25: product(quotient(c, b), c) = product(c, a).
% 3.03/0.76 Proof:
% 3.03/0.76 product(quotient(c, b), c)
% 3.03/0.76 = { by lemma 20 R->L }
% 3.03/0.76 quotient(product(c, product(c, b)), b)
% 3.03/0.76 = { by axiom 2 (sos02) R->L }
% 3.03/0.76 quotient(product(c, product(a, difference(a, product(c, b)))), b)
% 3.03/0.76 = { by lemma 24 }
% 3.03/0.76 quotient(product(c, product(a, difference(c, b))), b)
% 3.03/0.76 = { by lemma 22 }
% 3.03/0.76 quotient(product(product(c, a), b), b)
% 3.03/0.76 = { by axiom 5 (sos03) }
% 3.03/0.76 product(c, a)
% 3.03/0.76
% 3.03/0.76 Lemma 26: product(X, product(X, quotient(Y, X))) = difference(product(Y, X), X).
% 3.03/0.76 Proof:
% 3.03/0.76 product(X, product(X, quotient(Y, X)))
% 3.03/0.76 = { by axiom 4 (sos01) R->L }
% 3.03/0.76 difference(product(product(quotient(Y, X), X), X), product(product(product(quotient(Y, X), X), X), product(X, product(X, quotient(Y, X)))))
% 3.03/0.76 = { by axiom 9 (sos07) }
% 3.03/0.76 difference(product(product(quotient(Y, X), X), X), X)
% 3.03/0.76 = { by axiom 3 (sos04) }
% 3.03/0.76 difference(product(Y, X), X)
% 3.03/0.76
% 3.03/0.76 Lemma 27: bigC(X, Y, quotient(Z, X)) = product(product(X, Y), Z).
% 3.03/0.76 Proof:
% 3.03/0.76 bigC(X, Y, quotient(Z, X))
% 3.03/0.76 = { by axiom 6 (sos08) }
% 3.03/0.76 product(product(X, Y), product(quotient(Z, X), X))
% 3.03/0.76 = { by axiom 3 (sos04) }
% 3.03/0.76 product(product(X, Y), Z)
% 3.03/0.76
% 3.03/0.76 Lemma 28: product(difference(product(X, Y), Y), Y) = product(Y, product(Y, X)).
% 3.03/0.76 Proof:
% 3.03/0.76 product(difference(product(X, Y), Y), Y)
% 3.03/0.76 = { by lemma 26 R->L }
% 3.03/0.76 product(product(Y, product(Y, quotient(X, Y))), Y)
% 3.03/0.76 = { by lemma 12 }
% 3.03/0.76 bigC(Y, Y, product(Y, quotient(X, Y)))
% 3.03/0.76 = { by lemma 11 R->L }
% 3.03/0.76 product(Y, product(product(Y, quotient(X, Y)), Y))
% 3.03/0.76 = { by lemma 12 }
% 3.03/0.76 product(Y, bigC(Y, Y, quotient(X, Y)))
% 3.03/0.76 = { by lemma 27 }
% 3.03/0.76 product(Y, product(product(Y, Y), X))
% 3.03/0.76 = { by axiom 1 (sos06) }
% 3.03/0.76 product(Y, product(Y, X))
% 3.03/0.76
% 3.03/0.76 Lemma 29: difference(product(X, Y), Y) = product(difference(X, Y), Y).
% 3.03/0.76 Proof:
% 3.03/0.76 difference(product(X, Y), Y)
% 3.03/0.76 = { by lemma 26 R->L }
% 3.03/0.76 product(Y, product(Y, quotient(X, Y)))
% 3.03/0.76 = { by lemma 28 R->L }
% 3.03/0.76 product(difference(product(quotient(X, Y), Y), Y), Y)
% 3.03/0.77 = { by axiom 3 (sos04) }
% 3.03/0.77 product(difference(X, Y), Y)
% 3.03/0.77
% 3.03/0.77 Lemma 30: difference(X, product(Y, product(X, Z))) = product(difference(X, Y), Z).
% 3.03/0.77 Proof:
% 3.03/0.77 difference(X, product(Y, product(X, Z)))
% 3.03/0.77 = { by axiom 2 (sos02) R->L }
% 3.03/0.77 difference(X, product(product(X, difference(X, Y)), product(X, Z)))
% 3.03/0.77 = { by lemma 21 }
% 3.03/0.77 difference(X, product(X, product(difference(X, Y), Z)))
% 3.03/0.77 = { by axiom 4 (sos01) }
% 3.03/0.77 product(difference(X, Y), Z)
% 3.03/0.77
% 3.03/0.77 Lemma 31: product(difference(X, Y), quotient(Y, X)) = Y.
% 3.03/0.77 Proof:
% 3.03/0.77 product(difference(X, Y), quotient(Y, X))
% 3.03/0.77 = { by lemma 30 R->L }
% 3.03/0.77 difference(X, product(Y, product(X, quotient(Y, X))))
% 3.03/0.77 = { by lemma 21 R->L }
% 3.03/0.77 difference(X, product(product(Y, X), product(Y, quotient(Y, X))))
% 3.03/0.77 = { by lemma 17 R->L }
% 3.03/0.77 difference(X, product(product(Y, X), quotient(Y, product(Y, X))))
% 3.03/0.77 = { by lemma 18 R->L }
% 3.03/0.77 difference(X, product(difference(Y, product(Y, X)), Y))
% 3.03/0.77 = { by axiom 4 (sos01) }
% 3.03/0.77 difference(X, product(X, Y))
% 3.03/0.77 = { by axiom 4 (sos01) }
% 3.03/0.77 Y
% 3.03/0.77
% 3.03/0.77 Lemma 32: difference(difference(X, Y), Y) = quotient(Y, X).
% 3.03/0.77 Proof:
% 3.03/0.77 difference(difference(X, Y), Y)
% 3.03/0.77 = { by lemma 31 R->L }
% 3.03/0.77 difference(difference(X, Y), product(difference(X, Y), quotient(Y, X)))
% 3.03/0.77 = { by axiom 4 (sos01) }
% 3.03/0.77 quotient(Y, X)
% 3.03/0.77
% 3.03/0.77 Lemma 33: difference(quotient(X, Y), X) = Y.
% 3.03/0.77 Proof:
% 3.03/0.77 difference(quotient(X, Y), X)
% 3.03/0.77 = { by axiom 3 (sos04) R->L }
% 3.03/0.77 difference(quotient(X, Y), product(quotient(X, Y), Y))
% 3.03/0.77 = { by axiom 4 (sos01) }
% 3.03/0.77 Y
% 3.03/0.77
% 3.03/0.77 Lemma 34: quotient(X, difference(Y, X)) = Y.
% 3.03/0.77 Proof:
% 3.03/0.77 quotient(X, difference(Y, X))
% 3.03/0.77 = { by axiom 2 (sos02) R->L }
% 3.03/0.77 quotient(product(Y, difference(Y, X)), difference(Y, X))
% 3.03/0.77 = { by axiom 5 (sos03) }
% 3.03/0.77 Y
% 3.03/0.77
% 3.03/0.77 Lemma 35: quotient(X, product(Y, X)) = product(quotient(X, Y), X).
% 3.03/0.77 Proof:
% 3.03/0.77 quotient(X, product(Y, X))
% 3.03/0.77 = { by lemma 33 R->L }
% 3.03/0.77 quotient(X, product(difference(quotient(X, Y), X), X))
% 3.03/0.77 = { by lemma 29 R->L }
% 3.03/0.77 quotient(X, difference(product(quotient(X, Y), X), X))
% 3.03/0.77 = { by lemma 34 }
% 3.03/0.77 product(quotient(X, Y), X)
% 3.03/0.77
% 3.03/0.77 Lemma 36: quotient(X, quotient(X, Y)) = difference(Y, X).
% 3.03/0.77 Proof:
% 3.03/0.77 quotient(X, quotient(X, Y))
% 3.03/0.77 = { by lemma 31 R->L }
% 3.03/0.77 quotient(product(difference(Y, X), quotient(X, Y)), quotient(X, Y))
% 3.03/0.77 = { by axiom 5 (sos03) }
% 3.03/0.77 difference(Y, X)
% 3.03/0.77
% 3.03/0.77 Lemma 37: quotient(product(X, Y), X) = product(X, quotient(Y, X)).
% 3.03/0.77 Proof:
% 3.03/0.77 quotient(product(X, Y), X)
% 3.03/0.77 = { by axiom 1 (sos06) R->L }
% 3.03/0.77 quotient(product(product(X, X), Y), X)
% 3.03/0.77 = { by lemma 27 R->L }
% 3.03/0.77 quotient(bigC(X, X, quotient(Y, X)), X)
% 3.03/0.77 = { by lemma 12 R->L }
% 3.03/0.77 quotient(product(product(X, quotient(Y, X)), X), X)
% 3.03/0.77 = { by axiom 5 (sos03) }
% 3.03/0.77 product(X, quotient(Y, X))
% 3.03/0.77
% 3.03/0.77 Lemma 38: quotient(difference(X, Y), Y) = difference(quotient(X, Y), Y).
% 3.03/0.77 Proof:
% 3.03/0.77 quotient(difference(X, Y), Y)
% 3.03/0.77 = { by axiom 5 (sos03) R->L }
% 3.03/0.77 quotient(quotient(product(difference(X, Y), Y), Y), Y)
% 3.03/0.77 = { by lemma 29 R->L }
% 3.03/0.77 quotient(quotient(difference(product(X, Y), Y), Y), Y)
% 3.03/0.77 = { by lemma 26 R->L }
% 3.03/0.77 quotient(quotient(product(Y, product(Y, quotient(X, Y))), Y), Y)
% 3.03/0.77 = { by lemma 37 }
% 3.03/0.77 quotient(product(Y, quotient(product(Y, quotient(X, Y)), Y)), Y)
% 3.03/0.77 = { by lemma 37 }
% 3.03/0.77 quotient(product(Y, product(Y, quotient(quotient(X, Y), Y))), Y)
% 3.03/0.77 = { by lemma 26 }
% 3.03/0.77 quotient(difference(product(quotient(X, Y), Y), Y), Y)
% 3.03/0.77 = { by lemma 29 }
% 3.03/0.77 quotient(product(difference(quotient(X, Y), Y), Y), Y)
% 3.03/0.77 = { by axiom 5 (sos03) }
% 3.03/0.77 difference(quotient(X, Y), Y)
% 3.03/0.77
% 3.03/0.77 Lemma 39: quotient(quotient(X, Y), X) = quotient(X, quotient(Y, X)).
% 3.03/0.77 Proof:
% 3.03/0.77 quotient(quotient(X, Y), X)
% 3.03/0.77 = { by lemma 32 R->L }
% 3.03/0.77 quotient(difference(difference(Y, X), X), X)
% 3.03/0.77 = { by lemma 38 }
% 3.03/0.77 difference(quotient(difference(Y, X), X), X)
% 3.03/0.77 = { by lemma 38 }
% 3.03/0.77 difference(difference(quotient(Y, X), X), X)
% 3.03/0.77 = { by lemma 32 }
% 3.03/0.77 quotient(X, quotient(Y, X))
% 3.03/0.77
% 3.03/0.77 Lemma 40: bigC(X, quotient(Y, X), Z) = product(product(X, Z), Y).
% 3.03/0.77 Proof:
% 3.03/0.77 bigC(X, quotient(Y, X), Z)
% 3.03/0.77 = { by lemma 10 }
% 3.03/0.77 bigC(X, Z, quotient(Y, X))
% 3.03/0.77 = { by lemma 27 }
% 3.03/0.77 product(product(X, Z), Y)
% 3.03/0.77
% 3.03/0.77 Lemma 41: product(product(X, difference(Y, Z)), Y) = product(Z, product(Y, quotient(Y, X))).
% 3.03/0.77 Proof:
% 3.03/0.77 product(product(X, difference(Y, Z)), Y)
% 3.03/0.77 = { by lemma 19 R->L }
% 3.03/0.77 product(product(X, Y), product(difference(Y, Z), Y))
% 3.03/0.77 = { by lemma 14 }
% 3.03/0.77 product(Z, product(difference(Y, product(X, Y)), Y))
% 3.03/0.77 = { by lemma 15 }
% 3.03/0.77 product(Z, product(product(difference(Y, X), Y), Y))
% 3.03/0.77 = { by lemma 16 }
% 3.03/0.77 product(Z, quotient(Y, product(Y, X)))
% 3.03/0.77 = { by lemma 17 }
% 3.03/0.77 product(Z, product(Y, quotient(Y, X)))
% 3.03/0.77
% 3.03/0.77 Lemma 42: product(product(X, quotient(Y, Z)), Z) = product(product(X, Z), Y).
% 3.03/0.77 Proof:
% 3.03/0.77 product(product(X, quotient(Y, Z)), Z)
% 3.03/0.77 = { by lemma 19 R->L }
% 3.03/0.77 product(product(X, Z), product(quotient(Y, Z), Z))
% 3.03/0.77 = { by axiom 3 (sos04) }
% 3.03/0.77 product(product(X, Z), Y)
% 3.03/0.77
% 3.03/0.77 Lemma 43: product(quotient(product(a, c), b), c) = a.
% 3.03/0.77 Proof:
% 3.03/0.77 product(quotient(product(a, c), b), c)
% 3.03/0.77 = { by lemma 20 R->L }
% 3.03/0.77 quotient(product(product(a, c), product(c, b)), b)
% 3.03/0.77 = { by axiom 8 (sos09) }
% 3.03/0.77 quotient(product(a, b), b)
% 3.03/0.77 = { by axiom 5 (sos03) }
% 3.03/0.77 a
% 3.03/0.77
% 3.03/0.77 Lemma 44: difference(X, difference(product(Y, X), X)) = product(X, quotient(Y, X)).
% 3.03/0.77 Proof:
% 3.03/0.77 difference(X, difference(product(Y, X), X))
% 3.03/0.77 = { by lemma 26 R->L }
% 3.03/0.77 difference(X, product(X, product(X, quotient(Y, X))))
% 3.03/0.77 = { by axiom 4 (sos01) }
% 3.03/0.77 product(X, quotient(Y, X))
% 3.03/0.77
% 3.03/0.77 Goal 1 (goals): bigC(a, b, x0) = bigC(c, c, x0).
% 3.03/0.77 Proof:
% 3.03/0.77 bigC(a, b, x0)
% 3.03/0.77 = { by axiom 6 (sos08) }
% 3.03/0.77 product(product(a, b), product(x0, a))
% 3.03/0.77 = { by axiom 5 (sos03) R->L }
% 3.03/0.77 product(product(a, b), product(x0, quotient(product(a, b), b)))
% 3.03/0.77 = { by lemma 21 R->L }
% 3.03/0.77 product(product(product(a, b), x0), product(product(a, b), quotient(product(a, b), b)))
% 3.03/0.77 = { by lemma 41 R->L }
% 3.03/0.77 product(product(b, difference(product(a, b), product(product(a, b), x0))), product(a, b))
% 3.03/0.77 = { by axiom 4 (sos01) }
% 3.03/0.77 product(product(b, x0), product(a, b))
% 3.03/0.77 = { by axiom 2 (sos02) R->L }
% 3.03/0.77 product(product(b, x0), product(product(c, difference(c, a)), b))
% 3.03/0.77 = { by lemma 40 R->L }
% 3.03/0.77 product(product(b, x0), bigC(c, quotient(b, c), difference(c, a)))
% 3.03/0.77 = { by axiom 3 (sos04) R->L }
% 3.03/0.77 product(product(b, x0), bigC(c, quotient(b, c), difference(c, product(quotient(a, c), c))))
% 3.03/0.77 = { by lemma 15 }
% 3.03/0.77 product(product(b, x0), bigC(c, quotient(b, c), product(difference(c, quotient(a, c)), c)))
% 3.03/0.77 = { by axiom 5 (sos03) R->L }
% 3.03/0.77 product(product(b, x0), bigC(c, quotient(b, c), quotient(product(product(difference(c, quotient(a, c)), c), c), c)))
% 3.03/0.77 = { by lemma 16 }
% 3.03/0.77 product(product(b, x0), bigC(c, quotient(b, c), quotient(quotient(c, product(c, quotient(a, c))), c)))
% 3.03/0.77 = { by lemma 37 R->L }
% 3.03/0.77 product(product(b, x0), bigC(c, quotient(b, c), quotient(quotient(c, quotient(product(c, a), c)), c)))
% 3.03/0.77 = { by lemma 25 R->L }
% 3.03/0.77 product(product(b, x0), bigC(c, quotient(b, c), quotient(quotient(c, quotient(product(quotient(c, b), c), c)), c)))
% 3.03/0.77 = { by axiom 5 (sos03) }
% 3.03/0.77 product(product(b, x0), bigC(c, quotient(b, c), quotient(quotient(c, quotient(c, b)), c)))
% 3.03/0.77 = { by lemma 39 }
% 3.03/0.77 product(product(b, x0), bigC(c, quotient(b, c), quotient(c, quotient(quotient(c, b), c))))
% 3.03/0.77 = { by lemma 39 }
% 3.03/0.77 product(product(b, x0), bigC(c, quotient(b, c), quotient(c, quotient(c, quotient(b, c)))))
% 3.03/0.77 = { by lemma 36 }
% 3.03/0.77 product(product(b, x0), bigC(c, quotient(b, c), difference(quotient(b, c), c)))
% 3.03/0.77 = { by axiom 6 (sos08) }
% 3.03/0.77 product(product(b, x0), product(product(c, quotient(b, c)), product(difference(quotient(b, c), c), c)))
% 3.03/0.77 = { by lemma 29 R->L }
% 3.03/0.77 product(product(b, x0), product(product(c, quotient(b, c)), difference(product(quotient(b, c), c), c)))
% 3.03/0.77 = { by axiom 1 (sos06) R->L }
% 3.03/0.77 product(product(b, x0), product(product(product(c, c), quotient(b, c)), difference(product(quotient(b, c), c), c)))
% 3.03/0.77 = { by lemma 42 R->L }
% 3.03/0.77 product(product(b, x0), product(product(product(c, quotient(quotient(b, c), c)), c), difference(product(quotient(b, c), c), c)))
% 3.03/0.77 = { by lemma 44 R->L }
% 3.03/0.77 product(product(b, x0), product(product(difference(c, difference(product(quotient(b, c), c), c)), c), difference(product(quotient(b, c), c), c)))
% 3.03/0.77 = { by lemma 19 R->L }
% 3.03/0.77 product(product(b, x0), product(product(difference(c, difference(product(quotient(b, c), c), c)), difference(product(quotient(b, c), c), c)), product(c, difference(product(quotient(b, c), c), c))))
% 3.03/0.77 = { by lemma 29 R->L }
% 3.03/0.77 product(product(b, x0), product(difference(product(c, difference(product(quotient(b, c), c), c)), difference(product(quotient(b, c), c), c)), product(c, difference(product(quotient(b, c), c), c))))
% 3.03/0.77 = { by lemma 18 }
% 3.03/0.77 product(product(b, x0), product(difference(product(quotient(b, c), c), c), quotient(product(c, difference(product(quotient(b, c), c), c)), difference(product(quotient(b, c), c), c))))
% 3.03/0.78 = { by axiom 5 (sos03) }
% 3.03/0.78 product(product(b, x0), product(difference(product(quotient(b, c), c), c), c))
% 3.03/0.78 = { by lemma 28 }
% 3.03/0.78 product(product(b, x0), product(c, product(c, quotient(b, c))))
% 3.03/0.78 = { by lemma 26 }
% 3.03/0.78 product(product(b, x0), difference(product(b, c), c))
% 3.03/0.78 = { by lemma 29 }
% 3.03/0.78 product(product(b, x0), product(difference(b, c), c))
% 3.03/0.78 = { by lemma 36 R->L }
% 3.03/0.78 product(product(b, x0), product(quotient(c, quotient(c, b)), c))
% 3.03/0.78 = { by lemma 35 R->L }
% 3.03/0.78 product(product(b, x0), quotient(c, product(quotient(c, b), c)))
% 3.03/0.78 = { by lemma 25 }
% 3.03/0.78 product(product(b, x0), quotient(c, product(c, a)))
% 3.03/0.78 = { by lemma 17 }
% 3.03/0.78 product(product(b, x0), product(c, quotient(c, a)))
% 3.03/0.78 = { by lemma 41 R->L }
% 3.03/0.78 product(product(a, difference(c, product(b, x0))), c)
% 3.03/0.78 = { by lemma 23 R->L }
% 3.03/0.78 product(difference(c, product(product(c, a), product(b, x0))), c)
% 3.03/0.78 = { by lemma 40 R->L }
% 3.03/0.78 product(difference(c, bigC(c, quotient(product(b, x0), c), a)), c)
% 3.03/0.78 = { by lemma 43 R->L }
% 3.03/0.78 product(difference(c, bigC(c, quotient(product(b, x0), c), product(quotient(product(a, c), b), c))), c)
% 3.03/0.78 = { by lemma 10 }
% 3.03/0.78 product(difference(c, bigC(c, product(quotient(product(a, c), b), c), quotient(product(b, x0), c))), c)
% 3.03/0.78 = { by axiom 6 (sos08) }
% 3.03/0.78 product(difference(c, product(product(c, product(quotient(product(a, c), b), c)), product(quotient(product(b, x0), c), c))), c)
% 3.03/0.78 = { by lemma 23 }
% 3.03/0.78 product(product(product(quotient(product(a, c), b), c), difference(c, product(quotient(product(b, x0), c), c))), c)
% 3.03/0.78 = { by lemma 15 }
% 3.03/0.78 product(product(product(quotient(product(a, c), b), c), product(difference(c, quotient(product(b, x0), c)), c)), c)
% 3.03/0.78 = { by lemma 19 }
% 3.03/0.78 product(product(product(quotient(product(a, c), b), difference(c, quotient(product(b, x0), c))), c), c)
% 3.03/0.78 = { by lemma 41 }
% 3.29/0.78 product(product(quotient(product(b, x0), c), product(c, quotient(c, quotient(product(a, c), b)))), c)
% 3.29/0.78 = { by axiom 5 (sos03) R->L }
% 3.29/0.78 product(product(quotient(product(b, x0), c), product(c, quotient(c, quotient(product(quotient(product(a, c), b), c), c)))), c)
% 3.29/0.78 = { by lemma 43 }
% 3.29/0.78 product(product(quotient(product(b, x0), c), product(c, quotient(c, quotient(a, c)))), c)
% 3.29/0.78 = { by lemma 39 R->L }
% 3.29/0.78 product(product(quotient(product(b, x0), c), product(c, quotient(quotient(c, a), c))), c)
% 3.29/0.78 = { by axiom 5 (sos03) R->L }
% 3.29/0.78 product(product(quotient(product(b, x0), c), product(c, quotient(quotient(product(quotient(c, a), c), c), c))), c)
% 3.29/0.78 = { by lemma 35 R->L }
% 3.29/0.78 product(product(quotient(product(b, x0), c), product(c, quotient(quotient(quotient(c, product(a, c)), c), c))), c)
% 3.29/0.78 = { by axiom 2 (sos02) R->L }
% 3.29/0.78 product(product(quotient(product(b, x0), c), product(c, quotient(quotient(quotient(product(a, difference(a, c)), product(a, c)), c), c))), c)
% 3.29/0.78 = { by axiom 4 (sos01) R->L }
% 3.29/0.78 product(product(quotient(product(b, x0), c), product(c, quotient(quotient(quotient(product(a, difference(c, product(c, difference(a, c)))), product(a, c)), c), c))), c)
% 3.29/0.78 = { by lemma 33 R->L }
% 3.29/0.78 product(product(quotient(product(b, x0), c), product(c, quotient(quotient(quotient(product(a, difference(c, difference(quotient(c, product(c, difference(a, c))), c))), product(a, c)), c), c))), c)
% 3.29/0.78 = { by lemma 17 }
% 3.29/0.78 product(product(quotient(product(b, x0), c), product(c, quotient(quotient(quotient(product(a, difference(c, difference(product(c, quotient(c, difference(a, c))), c))), product(a, c)), c), c))), c)
% 3.29/0.78 = { by lemma 34 }
% 3.29/0.78 product(product(quotient(product(b, x0), c), product(c, quotient(quotient(quotient(product(a, difference(c, difference(product(c, a), c))), product(a, c)), c), c))), c)
% 3.29/0.78 = { by lemma 25 R->L }
% 3.29/0.78 product(product(quotient(product(b, x0), c), product(c, quotient(quotient(quotient(product(a, difference(c, difference(product(quotient(c, b), c), c))), product(a, c)), c), c))), c)
% 3.29/0.78 = { by lemma 29 }
% 3.29/0.78 product(product(quotient(product(b, x0), c), product(c, quotient(quotient(quotient(product(a, difference(c, product(difference(quotient(c, b), c), c))), product(a, c)), c), c))), c)
% 3.29/0.78 = { by lemma 33 }
% 3.29/0.78 product(product(quotient(product(b, x0), c), product(c, quotient(quotient(quotient(product(a, difference(c, product(b, c))), product(a, c)), c), c))), c)
% 3.29/0.78 = { by lemma 15 }
% 3.29/0.78 product(product(quotient(product(b, x0), c), product(c, quotient(quotient(quotient(product(a, product(difference(c, b), c)), product(a, c)), c), c))), c)
% 3.29/0.78 = { by lemma 24 R->L }
% 3.29/0.78 product(product(quotient(product(b, x0), c), product(c, quotient(quotient(quotient(product(a, product(difference(a, product(c, b)), c)), product(a, c)), c), c))), c)
% 3.29/0.78 = { by lemma 30 R->L }
% 3.29/0.78 product(product(quotient(product(b, x0), c), product(c, quotient(quotient(quotient(product(a, difference(a, product(product(c, b), product(a, c)))), product(a, c)), c), c))), c)
% 3.29/0.78 = { by axiom 6 (sos08) R->L }
% 3.29/0.78 product(product(quotient(product(b, x0), c), product(c, quotient(quotient(quotient(product(a, difference(a, bigC(c, b, a))), product(a, c)), c), c))), c)
% 3.29/0.78 = { by lemma 10 R->L }
% 3.29/0.78 product(product(quotient(product(b, x0), c), product(c, quotient(quotient(quotient(product(a, difference(a, bigC(c, a, b))), product(a, c)), c), c))), c)
% 3.29/0.78 = { by axiom 2 (sos02) }
% 3.29/0.78 product(product(quotient(product(b, x0), c), product(c, quotient(quotient(quotient(bigC(c, a, b), product(a, c)), c), c))), c)
% 3.29/0.78 = { by lemma 10 }
% 3.29/0.78 product(product(quotient(product(b, x0), c), product(c, quotient(quotient(quotient(bigC(c, b, a), product(a, c)), c), c))), c)
% 3.29/0.78 = { by axiom 6 (sos08) }
% 3.29/0.78 product(product(quotient(product(b, x0), c), product(c, quotient(quotient(quotient(product(product(c, b), product(a, c)), product(a, c)), c), c))), c)
% 3.29/0.78 = { by axiom 5 (sos03) }
% 3.29/0.78 product(product(quotient(product(b, x0), c), product(c, quotient(quotient(product(c, b), c), c))), c)
% 3.29/0.78 = { by lemma 37 }
% 3.29/0.78 product(product(quotient(product(b, x0), c), product(c, quotient(product(c, quotient(b, c)), c))), c)
% 3.29/0.78 = { by lemma 37 }
% 3.29/0.78 product(product(quotient(product(b, x0), c), product(c, product(c, quotient(quotient(b, c), c)))), c)
% 3.29/0.78 = { by lemma 44 R->L }
% 3.29/0.78 product(product(quotient(product(b, x0), c), product(c, difference(c, difference(product(quotient(b, c), c), c)))), c)
% 3.29/0.78 = { by axiom 3 (sos04) }
% 3.29/0.78 product(product(quotient(product(b, x0), c), product(c, difference(c, difference(b, c)))), c)
% 3.29/0.78 = { by axiom 2 (sos02) }
% 3.29/0.78 product(product(quotient(product(b, x0), c), difference(b, c)), c)
% 3.29/0.78 = { by lemma 23 R->L }
% 3.29/0.78 product(difference(b, product(product(b, quotient(product(b, x0), c)), c)), c)
% 3.29/0.78 = { by lemma 42 }
% 3.29/0.78 product(difference(b, product(product(b, c), product(b, x0))), c)
% 3.29/0.78 = { by lemma 23 }
% 3.29/0.78 product(product(c, difference(b, product(b, x0))), c)
% 3.29/0.78 = { by lemma 12 }
% 3.29/0.78 bigC(c, c, difference(b, product(b, x0)))
% 3.29/0.78 = { by axiom 4 (sos01) }
% 3.29/0.78 bigC(c, c, x0)
% 3.29/0.78 % SZS output end Proof
% 3.29/0.78
% 3.29/0.78 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------