TSTP Solution File: GRP769-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP769-1 : 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 : n032.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 01:20:03 EDT 2023
% Result : Unsatisfiable 99.94s 13.36s
% Output : Proof 101.76s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.09 % Problem : GRP769-1 : TPTP v8.1.2. Released v4.1.0.
% 0.08/0.10 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.10/0.29 % Computer : n032.cluster.edu
% 0.10/0.29 % Model : x86_64 x86_64
% 0.10/0.29 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.29 % Memory : 8042.1875MB
% 0.10/0.29 % OS : Linux 3.10.0-693.el7.x86_64
% 0.10/0.29 % CPULimit : 300
% 0.10/0.29 % WCLimit : 300
% 0.10/0.29 % DateTime : Mon Aug 28 19:44:16 EDT 2023
% 0.10/0.29 % CPUTime :
% 99.94/13.36 Command-line arguments: --flatten
% 99.94/13.36
% 99.94/13.36 % SZS status Unsatisfiable
% 99.94/13.36
% 100.76/13.45 % SZS output start Proof
% 100.76/13.45 Axiom 1 (sos09): i(X) = difference(X, one).
% 100.76/13.45 Axiom 2 (sos10): j(X) = quotient(one, X).
% 100.76/13.45 Axiom 3 (sos01): product(X, one) = X.
% 100.76/13.45 Axiom 4 (sos02): product(one, X) = X.
% 100.76/13.45 Axiom 5 (sos12): eta(X) = product(i(X), X).
% 100.76/13.45 Axiom 6 (sos11): product(i(X), X) = product(X, j(X)).
% 100.76/13.45 Axiom 7 (sos04): difference(X, product(X, Y)) = Y.
% 100.76/13.45 Axiom 8 (sos19): t(X, Y) = quotient(product(X, Y), X).
% 100.76/13.45 Axiom 9 (sos05): quotient(product(X, Y), Y) = X.
% 100.76/13.45 Axiom 10 (sos03): product(X, difference(X, Y)) = Y.
% 100.76/13.45 Axiom 11 (sos06): product(quotient(X, Y), Y) = X.
% 100.76/13.45 Axiom 12 (sos23): product(X, i(product(Y, X))) = i(Y).
% 100.76/13.45 Axiom 13 (sos14): product(X, product(eta(X), Y)) = product(j(j(X)), Y).
% 100.76/13.45 Axiom 14 (sos13): product(i(i(X)), Y) = product(eta(X), product(X, Y)).
% 100.76/13.45 Axiom 15 (sos24): product(j(product(X, Y)), X) = j(Y).
% 100.76/13.45 Axiom 16 (sos16): product(eta(X), product(Y, Z)) = product(product(eta(X), Y), Z).
% 100.76/13.45 Axiom 17 (sos07): difference(X, product(product(X, Y), Z)) = quotient(product(Y, product(Z, X)), X).
% 100.76/13.45 Axiom 18 (sos21): product(i(product(X, Y)), i(i(X))) = i(Y).
% 100.76/13.45 Axiom 19 (sos08): difference(product(X, Y), product(X, product(Y, Z))) = quotient(quotient(product(Z, product(X, Y)), Y), X).
% 100.76/13.45
% 100.76/13.45 Lemma 20: product(j(X), X) = one.
% 100.76/13.45 Proof:
% 100.76/13.45 product(j(X), X)
% 100.76/13.45 = { by axiom 2 (sos10) }
% 100.76/13.45 product(quotient(one, X), X)
% 100.76/13.45 = { by axiom 11 (sos06) }
% 100.76/13.45 one
% 100.76/13.45
% 100.76/13.45 Lemma 21: i(j(X)) = X.
% 100.76/13.45 Proof:
% 100.76/13.45 i(j(X))
% 100.76/13.45 = { by axiom 1 (sos09) }
% 100.76/13.45 difference(j(X), one)
% 100.76/13.45 = { by lemma 20 R->L }
% 100.76/13.45 difference(j(X), product(j(X), X))
% 100.76/13.45 = { by axiom 7 (sos04) }
% 100.76/13.45 X
% 100.76/13.45
% 100.76/13.45 Lemma 22: product(X, i(X)) = one.
% 100.76/13.45 Proof:
% 100.76/13.45 product(X, i(X))
% 100.76/13.45 = { by axiom 1 (sos09) }
% 100.76/13.45 product(X, difference(X, one))
% 100.76/13.45 = { by axiom 10 (sos03) }
% 100.76/13.45 one
% 100.76/13.45
% 100.76/13.45 Lemma 23: j(i(X)) = X.
% 100.76/13.45 Proof:
% 100.76/13.45 j(i(X))
% 100.76/13.45 = { by axiom 2 (sos10) }
% 100.76/13.45 quotient(one, i(X))
% 100.76/13.45 = { by lemma 22 R->L }
% 100.76/13.45 quotient(product(X, i(X)), i(X))
% 100.76/13.45 = { by axiom 9 (sos05) }
% 100.76/13.45 X
% 100.76/13.45
% 100.76/13.45 Lemma 24: eta(i(X)) = eta(X).
% 100.76/13.45 Proof:
% 100.76/13.45 eta(i(X))
% 100.76/13.45 = { by axiom 5 (sos12) }
% 100.76/13.45 product(i(i(X)), i(X))
% 100.76/13.45 = { by axiom 14 (sos13) }
% 100.76/13.45 product(eta(X), product(X, i(X)))
% 100.76/13.45 = { by lemma 22 }
% 100.76/13.45 product(eta(X), one)
% 100.76/13.45 = { by axiom 3 (sos01) }
% 100.76/13.45 eta(X)
% 100.76/13.45
% 100.76/13.45 Lemma 25: product(X, j(X)) = eta(X).
% 100.76/13.45 Proof:
% 100.76/13.45 product(X, j(X))
% 100.76/13.45 = { by axiom 6 (sos11) R->L }
% 100.76/13.45 product(i(X), X)
% 100.76/13.45 = { by axiom 5 (sos12) R->L }
% 100.76/13.45 eta(X)
% 100.76/13.45
% 100.76/13.45 Lemma 26: eta(j(X)) = eta(X).
% 100.76/13.45 Proof:
% 100.76/13.45 eta(j(X))
% 100.76/13.45 = { by axiom 5 (sos12) }
% 100.76/13.45 product(i(j(X)), j(X))
% 100.76/13.45 = { by lemma 21 }
% 100.76/13.45 product(X, j(X))
% 100.76/13.45 = { by lemma 25 }
% 100.76/13.45 eta(X)
% 100.76/13.45
% 100.76/13.45 Lemma 27: quotient(eta(product(X, Y)), Y) = difference(Y, eta(X)).
% 100.76/13.45 Proof:
% 100.76/13.45 quotient(eta(product(X, Y)), Y)
% 100.76/13.45 = { by axiom 5 (sos12) }
% 100.76/13.45 quotient(product(i(product(X, Y)), product(X, Y)), Y)
% 100.76/13.45 = { by axiom 17 (sos07) R->L }
% 100.76/13.45 difference(Y, product(product(Y, i(product(X, Y))), X))
% 100.76/13.45 = { by axiom 12 (sos23) }
% 100.76/13.45 difference(Y, product(i(X), X))
% 100.76/13.45 = { by axiom 5 (sos12) R->L }
% 100.76/13.45 difference(Y, eta(X))
% 100.76/13.45
% 100.76/13.45 Lemma 28: difference(X, eta(X)) = j(X).
% 100.76/13.45 Proof:
% 100.76/13.45 difference(X, eta(X))
% 100.76/13.45 = { by lemma 25 R->L }
% 100.76/13.45 difference(X, product(X, j(X)))
% 100.76/13.45 = { by axiom 7 (sos04) }
% 100.76/13.45 j(X)
% 100.76/13.45
% 100.76/13.45 Lemma 29: eta(eta(X)) = one.
% 100.76/13.45 Proof:
% 100.76/13.45 eta(eta(X))
% 100.76/13.45 = { by axiom 11 (sos06) R->L }
% 100.76/13.45 product(quotient(eta(eta(X)), X), X)
% 100.76/13.45 = { by axiom 5 (sos12) }
% 100.76/13.45 product(quotient(eta(product(i(X), X)), X), X)
% 100.76/13.45 = { by lemma 27 }
% 100.76/13.45 product(difference(X, eta(i(X))), X)
% 100.76/13.45 = { by lemma 24 }
% 100.76/13.45 product(difference(X, eta(X)), X)
% 100.76/13.45 = { by lemma 28 }
% 100.76/13.45 product(j(X), X)
% 100.76/13.45 = { by lemma 20 }
% 100.76/13.45 one
% 100.76/13.45
% 100.76/13.45 Lemma 30: product(X, i(Y)) = i(quotient(Y, X)).
% 100.76/13.45 Proof:
% 100.76/13.45 product(X, i(Y))
% 100.76/13.45 = { by axiom 11 (sos06) R->L }
% 100.76/13.45 product(X, i(product(quotient(Y, X), X)))
% 100.76/13.45 = { by axiom 12 (sos23) }
% 100.76/13.45 i(quotient(Y, X))
% 100.76/13.45
% 100.76/13.45 Lemma 31: product(j(X), Y) = j(difference(Y, X)).
% 100.76/13.45 Proof:
% 100.76/13.45 product(j(X), Y)
% 100.76/13.45 = { by axiom 10 (sos03) R->L }
% 100.76/13.45 product(j(product(Y, difference(Y, X))), Y)
% 100.76/13.45 = { by axiom 15 (sos24) }
% 100.76/13.45 j(difference(Y, X))
% 100.76/13.45
% 100.76/13.45 Lemma 32: product(eta(X), X) = i(i(X)).
% 100.76/13.45 Proof:
% 100.76/13.45 product(eta(X), X)
% 100.76/13.45 = { by axiom 3 (sos01) R->L }
% 100.76/13.45 product(eta(X), product(X, one))
% 100.76/13.45 = { by axiom 14 (sos13) R->L }
% 100.76/13.45 product(i(i(X)), one)
% 100.76/13.45 = { by axiom 3 (sos01) }
% 100.76/13.45 i(i(X))
% 100.76/13.45
% 100.76/13.45 Lemma 33: i(product(X, Y)) = difference(Y, i(X)).
% 100.76/13.45 Proof:
% 100.76/13.45 i(product(X, Y))
% 100.76/13.45 = { by axiom 7 (sos04) R->L }
% 100.76/13.45 difference(Y, product(Y, i(product(X, Y))))
% 100.76/13.45 = { by axiom 12 (sos23) }
% 100.76/13.45 difference(Y, i(X))
% 100.76/13.45
% 100.76/13.45 Lemma 34: quotient(X, difference(Y, X)) = Y.
% 100.76/13.45 Proof:
% 100.76/13.45 quotient(X, difference(Y, X))
% 100.76/13.45 = { by axiom 10 (sos03) R->L }
% 100.76/13.45 quotient(product(Y, difference(Y, X)), difference(Y, X))
% 100.76/13.45 = { by axiom 9 (sos05) }
% 100.76/13.45 Y
% 100.76/13.45
% 100.76/13.45 Lemma 35: quotient(i(X), i(Y)) = difference(X, Y).
% 100.76/13.45 Proof:
% 100.76/13.45 quotient(i(X), i(Y))
% 100.76/13.45 = { by axiom 10 (sos03) R->L }
% 100.76/13.45 quotient(i(X), i(product(X, difference(X, Y))))
% 100.76/13.45 = { by lemma 33 }
% 100.76/13.45 quotient(i(X), difference(difference(X, Y), i(X)))
% 100.76/13.45 = { by lemma 34 }
% 100.76/13.45 difference(X, Y)
% 100.76/13.45
% 100.76/13.45 Lemma 36: quotient(X, i(Y)) = difference(j(X), Y).
% 100.76/13.45 Proof:
% 100.76/13.45 quotient(X, i(Y))
% 100.76/13.45 = { by lemma 21 R->L }
% 100.76/13.45 quotient(i(j(X)), i(Y))
% 100.76/13.45 = { by lemma 35 }
% 100.76/13.45 difference(j(X), Y)
% 100.76/13.45
% 100.76/13.45 Lemma 37: quotient(i(X), Y) = difference(X, j(Y)).
% 100.76/13.45 Proof:
% 100.76/13.45 quotient(i(X), Y)
% 100.76/13.45 = { by lemma 21 R->L }
% 100.76/13.45 quotient(i(X), i(j(Y)))
% 100.76/13.45 = { by lemma 35 }
% 100.76/13.45 difference(X, j(Y))
% 100.76/13.45
% 100.76/13.45 Lemma 38: j(product(X, Y)) = quotient(j(Y), X).
% 100.76/13.45 Proof:
% 100.76/13.45 j(product(X, Y))
% 100.76/13.45 = { by axiom 9 (sos05) R->L }
% 100.76/13.45 quotient(product(j(product(X, Y)), X), X)
% 100.76/13.45 = { by axiom 15 (sos24) }
% 100.76/13.45 quotient(j(Y), X)
% 100.76/13.45
% 100.76/13.45 Lemma 39: difference(quotient(X, Y), X) = Y.
% 100.76/13.45 Proof:
% 100.76/13.45 difference(quotient(X, Y), X)
% 100.76/13.45 = { by axiom 11 (sos06) R->L }
% 100.76/13.45 difference(quotient(X, Y), product(quotient(X, Y), Y))
% 100.76/13.45 = { by axiom 7 (sos04) }
% 100.76/13.45 Y
% 100.76/13.45
% 100.76/13.45 Lemma 40: product(eta(X), product(i(eta(X)), Y)) = Y.
% 100.76/13.45 Proof:
% 100.76/13.45 product(eta(X), product(i(eta(X)), Y))
% 100.76/13.45 = { by axiom 16 (sos16) }
% 100.76/13.45 product(product(eta(X), i(eta(X))), Y)
% 100.76/13.45 = { by lemma 22 }
% 100.76/13.45 product(one, Y)
% 100.76/13.45 = { by axiom 4 (sos02) }
% 100.76/13.45 Y
% 100.76/13.45
% 100.76/13.45 Lemma 41: i(difference(eta(X), Y)) = difference(Y, eta(X)).
% 100.76/13.45 Proof:
% 100.76/13.45 i(difference(eta(X), Y))
% 100.76/13.45 = { by lemma 35 R->L }
% 100.76/13.45 i(quotient(i(eta(X)), i(Y)))
% 100.76/13.45 = { by lemma 30 R->L }
% 100.76/13.45 product(i(Y), i(i(eta(X))))
% 100.76/13.45 = { by lemma 40 R->L }
% 100.76/13.45 product(i(product(eta(X), product(i(eta(X)), Y))), i(i(eta(X))))
% 100.76/13.45 = { by axiom 18 (sos21) }
% 100.76/13.45 i(product(i(eta(X)), Y))
% 100.76/13.45 = { by lemma 33 }
% 100.76/13.45 difference(Y, i(i(eta(X))))
% 100.76/13.45 = { by lemma 32 R->L }
% 100.76/13.45 difference(Y, product(eta(eta(X)), eta(X)))
% 100.76/13.45 = { by lemma 29 }
% 100.76/13.45 difference(Y, product(one, eta(X)))
% 100.76/13.45 = { by axiom 4 (sos02) }
% 100.76/13.45 difference(Y, eta(X))
% 100.76/13.45
% 100.76/13.45 Lemma 42: i(quotient(j(X), Y)) = product(Y, X).
% 100.76/13.45 Proof:
% 100.76/13.45 i(quotient(j(X), Y))
% 100.76/13.45 = { by lemma 30 R->L }
% 100.76/13.45 product(Y, i(j(X)))
% 100.76/13.45 = { by axiom 15 (sos24) R->L }
% 100.76/13.45 product(Y, i(product(j(product(Y, X)), Y)))
% 100.76/13.46 = { by axiom 12 (sos23) }
% 100.76/13.46 i(j(product(Y, X)))
% 100.76/13.46 = { by lemma 21 }
% 100.76/13.46 product(Y, X)
% 100.76/13.46
% 100.76/13.46 Lemma 43: j(difference(X, i(Y))) = product(Y, X).
% 100.76/13.46 Proof:
% 100.76/13.46 j(difference(X, i(Y)))
% 100.76/13.46 = { by lemma 31 R->L }
% 100.76/13.46 product(j(i(Y)), X)
% 100.76/13.46 = { by axiom 12 (sos23) R->L }
% 100.76/13.46 product(j(product(X, i(product(Y, X)))), X)
% 100.76/13.46 = { by axiom 15 (sos24) }
% 100.76/13.46 j(i(product(Y, X)))
% 100.76/13.46 = { by lemma 23 }
% 100.76/13.46 product(Y, X)
% 100.76/13.46
% 100.76/13.46 Lemma 44: product(eta(X), product(difference(eta(X), Y), Z)) = product(Y, Z).
% 100.76/13.46 Proof:
% 100.76/13.46 product(eta(X), product(difference(eta(X), Y), Z))
% 100.76/13.46 = { by axiom 16 (sos16) }
% 100.76/13.46 product(product(eta(X), difference(eta(X), Y)), Z)
% 100.76/13.46 = { by axiom 10 (sos03) }
% 100.76/13.46 product(Y, Z)
% 100.76/13.46
% 100.76/13.46 Lemma 45: difference(product(X, Y), eta(Z)) = difference(Y, difference(X, eta(Z))).
% 100.76/13.46 Proof:
% 100.76/13.46 difference(product(X, Y), eta(Z))
% 100.76/13.46 = { by lemma 41 R->L }
% 100.76/13.46 i(difference(eta(Z), product(X, Y)))
% 100.76/13.46 = { by lemma 35 R->L }
% 100.76/13.46 i(quotient(i(eta(Z)), i(product(X, Y))))
% 100.76/13.46 = { by lemma 30 R->L }
% 100.76/13.46 product(i(product(X, Y)), i(i(eta(Z))))
% 100.76/13.46 = { by lemma 44 R->L }
% 100.76/13.46 product(i(product(eta(Z), product(difference(eta(Z), X), Y))), i(i(eta(Z))))
% 100.76/13.46 = { by axiom 18 (sos21) }
% 100.76/13.46 i(product(difference(eta(Z), X), Y))
% 100.76/13.46 = { by lemma 33 }
% 100.76/13.46 difference(Y, i(difference(eta(Z), X)))
% 100.76/13.46 = { by lemma 41 }
% 100.76/13.46 difference(Y, difference(X, eta(Z)))
% 100.76/13.46
% 100.76/13.46 Lemma 46: quotient(j(X), quotient(Y, X)) = j(Y).
% 100.76/13.46 Proof:
% 100.76/13.46 quotient(j(X), quotient(Y, X))
% 100.76/13.46 = { by lemma 38 R->L }
% 100.76/13.46 j(product(quotient(Y, X), X))
% 100.76/13.46 = { by axiom 11 (sos06) }
% 100.76/13.46 j(Y)
% 100.76/13.46
% 100.76/13.46 Lemma 47: quotient(difference(X, product(Y, Z)), Z) = difference(product(Z, X), product(Z, Y)).
% 100.76/13.46 Proof:
% 100.76/13.46 quotient(difference(X, product(Y, Z)), Z)
% 100.76/13.46 = { by axiom 10 (sos03) R->L }
% 100.76/13.46 quotient(difference(X, product(product(X, difference(X, Y)), Z)), Z)
% 100.76/13.46 = { by axiom 17 (sos07) }
% 100.76/13.46 quotient(quotient(product(difference(X, Y), product(Z, X)), X), Z)
% 100.76/13.46 = { by axiom 19 (sos08) R->L }
% 100.76/13.46 difference(product(Z, X), product(Z, product(X, difference(X, Y))))
% 100.76/13.46 = { by axiom 10 (sos03) }
% 100.76/13.46 difference(product(Z, X), product(Z, Y))
% 100.76/13.46
% 100.76/13.46 Lemma 48: j(difference(X, product(Y, X))) = t(j(X), j(Y)).
% 100.76/13.46 Proof:
% 100.76/13.46 j(difference(X, product(Y, X)))
% 100.76/13.46 = { by lemma 34 R->L }
% 100.76/13.46 j(difference(X, product(quotient(X, difference(Y, X)), X)))
% 100.76/13.46 = { by lemma 43 R->L }
% 100.76/13.46 j(difference(X, j(difference(X, i(quotient(X, difference(Y, X)))))))
% 100.76/13.46 = { by lemma 37 R->L }
% 100.76/13.46 j(quotient(i(X), difference(X, i(quotient(X, difference(Y, X))))))
% 100.76/13.46 = { by axiom 10 (sos03) R->L }
% 100.76/13.46 j(quotient(i(X), difference(X, i(quotient(X, product(X, difference(X, difference(Y, X))))))))
% 100.76/13.46 = { by lemma 30 R->L }
% 100.76/13.46 j(quotient(i(X), difference(X, product(product(X, difference(X, difference(Y, X))), i(X)))))
% 100.76/13.46 = { by axiom 17 (sos07) }
% 100.76/13.46 j(quotient(i(X), quotient(product(difference(X, difference(Y, X)), product(i(X), X)), X)))
% 100.76/13.46 = { by axiom 5 (sos12) R->L }
% 100.76/13.46 j(quotient(i(X), quotient(product(difference(X, difference(Y, X)), eta(X)), X)))
% 100.76/13.46 = { by lemma 39 R->L }
% 100.76/13.46 j(quotient(i(X), quotient(difference(quotient(eta(X), product(difference(X, difference(Y, X)), eta(X))), eta(X)), X)))
% 100.76/13.46 = { by axiom 5 (sos12) }
% 100.76/13.46 j(quotient(i(X), quotient(difference(quotient(eta(X), product(difference(X, difference(Y, X)), eta(X))), product(i(X), X)), X)))
% 100.76/13.46 = { by lemma 47 }
% 100.76/13.46 j(quotient(i(X), difference(product(X, quotient(eta(X), product(difference(X, difference(Y, X)), eta(X)))), product(X, i(X)))))
% 100.76/13.46 = { by lemma 22 }
% 100.76/13.46 j(quotient(i(X), difference(product(X, quotient(eta(X), product(difference(X, difference(Y, X)), eta(X)))), one)))
% 100.76/13.46 = { by axiom 1 (sos09) R->L }
% 100.76/13.46 j(quotient(i(X), i(product(X, quotient(eta(X), product(difference(X, difference(Y, X)), eta(X)))))))
% 100.76/13.46 = { by lemma 33 }
% 100.76/13.46 j(quotient(i(X), difference(quotient(eta(X), product(difference(X, difference(Y, X)), eta(X))), i(X))))
% 100.76/13.46 = { by lemma 43 R->L }
% 100.76/13.46 j(quotient(i(X), difference(quotient(eta(X), j(difference(eta(X), i(difference(X, difference(Y, X)))))), i(X))))
% 100.76/13.46 = { by axiom 10 (sos03) R->L }
% 100.76/13.46 j(quotient(i(X), difference(product(eta(X), difference(eta(X), quotient(eta(X), j(difference(eta(X), i(difference(X, difference(Y, X)))))))), i(X))))
% 100.76/13.46 = { by lemma 40 R->L }
% 100.76/13.46 j(quotient(i(X), difference(product(eta(X), difference(eta(X), product(eta(X), product(i(eta(X)), quotient(eta(X), j(difference(eta(X), i(difference(X, difference(Y, X)))))))))), i(X))))
% 100.76/13.46 = { by axiom 7 (sos04) }
% 100.76/13.46 j(quotient(i(X), difference(product(eta(X), product(i(eta(X)), quotient(eta(X), j(difference(eta(X), i(difference(X, difference(Y, X)))))))), i(X))))
% 100.76/13.46 = { by lemma 21 R->L }
% 100.76/13.46 j(quotient(i(X), difference(product(eta(X), product(i(i(j(eta(X)))), quotient(eta(X), j(difference(eta(X), i(difference(X, difference(Y, X)))))))), i(X))))
% 100.76/13.46 = { by axiom 14 (sos13) }
% 100.76/13.46 j(quotient(i(X), difference(product(eta(X), product(eta(j(eta(X))), product(j(eta(X)), quotient(eta(X), j(difference(eta(X), i(difference(X, difference(Y, X))))))))), i(X))))
% 100.76/13.46 = { by lemma 31 }
% 100.76/13.46 j(quotient(i(X), difference(product(eta(X), product(eta(j(eta(X))), j(difference(quotient(eta(X), j(difference(eta(X), i(difference(X, difference(Y, X)))))), eta(X))))), i(X))))
% 100.76/13.46 = { by lemma 26 }
% 100.76/13.46 j(quotient(i(X), difference(product(eta(X), product(eta(eta(X)), j(difference(quotient(eta(X), j(difference(eta(X), i(difference(X, difference(Y, X)))))), eta(X))))), i(X))))
% 101.76/13.46 = { by lemma 39 }
% 101.76/13.46 j(quotient(i(X), difference(product(eta(X), product(eta(eta(X)), j(j(difference(eta(X), i(difference(X, difference(Y, X)))))))), i(X))))
% 101.76/13.46 = { by lemma 29 }
% 101.76/13.46 j(quotient(i(X), difference(product(eta(X), product(one, j(j(difference(eta(X), i(difference(X, difference(Y, X)))))))), i(X))))
% 101.76/13.46 = { by axiom 4 (sos02) }
% 101.76/13.46 j(quotient(i(X), difference(product(eta(X), j(j(difference(eta(X), i(difference(X, difference(Y, X))))))), i(X))))
% 101.76/13.46 = { by axiom 3 (sos01) R->L }
% 101.76/13.46 j(quotient(i(X), difference(product(eta(X), product(j(j(difference(eta(X), i(difference(X, difference(Y, X)))))), one)), i(X))))
% 101.76/13.46 = { by axiom 13 (sos14) R->L }
% 101.76/13.46 j(quotient(i(X), difference(product(eta(X), product(difference(eta(X), i(difference(X, difference(Y, X)))), product(eta(difference(eta(X), i(difference(X, difference(Y, X))))), one))), i(X))))
% 101.76/13.46 = { by axiom 3 (sos01) }
% 101.76/13.46 j(quotient(i(X), difference(product(eta(X), product(difference(eta(X), i(difference(X, difference(Y, X)))), eta(difference(eta(X), i(difference(X, difference(Y, X))))))), i(X))))
% 101.76/13.46 = { by lemma 44 }
% 101.76/13.46 j(quotient(i(X), difference(product(i(difference(X, difference(Y, X))), eta(difference(eta(X), i(difference(X, difference(Y, X)))))), i(X))))
% 101.76/13.46 = { by lemma 24 R->L }
% 101.76/13.46 j(quotient(i(X), difference(product(i(difference(X, difference(Y, X))), eta(i(difference(eta(X), i(difference(X, difference(Y, X))))))), i(X))))
% 101.76/13.46 = { by lemma 39 R->L }
% 101.76/13.46 j(quotient(i(X), difference(product(i(difference(X, difference(Y, X))), difference(quotient(i(difference(eta(X), i(difference(X, difference(Y, X))))), eta(i(difference(eta(X), i(difference(X, difference(Y, X))))))), i(difference(eta(X), i(difference(X, difference(Y, X))))))), i(X))))
% 101.76/13.46 = { by lemma 24 R->L }
% 101.76/13.46 j(quotient(i(X), difference(product(i(difference(X, difference(Y, X))), difference(quotient(i(difference(eta(X), i(difference(X, difference(Y, X))))), eta(i(i(difference(eta(X), i(difference(X, difference(Y, X)))))))), i(difference(eta(X), i(difference(X, difference(Y, X))))))), i(X))))
% 101.76/13.46 = { by lemma 23 R->L }
% 101.76/13.47 j(quotient(i(X), difference(product(i(difference(X, difference(Y, X))), difference(quotient(j(i(i(difference(eta(X), i(difference(X, difference(Y, X))))))), eta(i(i(difference(eta(X), i(difference(X, difference(Y, X)))))))), i(difference(eta(X), i(difference(X, difference(Y, X))))))), i(X))))
% 101.76/13.47 = { by lemma 38 R->L }
% 101.76/13.47 j(quotient(i(X), difference(product(i(difference(X, difference(Y, X))), difference(j(product(eta(i(i(difference(eta(X), i(difference(X, difference(Y, X))))))), i(i(difference(eta(X), i(difference(X, difference(Y, X)))))))), i(difference(eta(X), i(difference(X, difference(Y, X))))))), i(X))))
% 101.76/13.47 = { by lemma 32 }
% 101.76/13.47 j(quotient(i(X), difference(product(i(difference(X, difference(Y, X))), difference(j(i(i(i(i(difference(eta(X), i(difference(X, difference(Y, X))))))))), i(difference(eta(X), i(difference(X, difference(Y, X))))))), i(X))))
% 101.76/13.47 = { by lemma 23 }
% 101.76/13.47 j(quotient(i(X), difference(product(i(difference(X, difference(Y, X))), difference(i(i(i(difference(eta(X), i(difference(X, difference(Y, X))))))), i(difference(eta(X), i(difference(X, difference(Y, X))))))), i(X))))
% 101.76/13.47 = { by lemma 33 R->L }
% 101.76/13.47 j(quotient(i(X), difference(product(i(difference(X, difference(Y, X))), i(product(difference(eta(X), i(difference(X, difference(Y, X)))), i(i(i(difference(eta(X), i(difference(X, difference(Y, X)))))))))), i(X))))
% 101.76/13.47 = { by lemma 30 }
% 101.76/13.47 j(quotient(i(X), difference(i(quotient(product(difference(eta(X), i(difference(X, difference(Y, X)))), i(i(i(difference(eta(X), i(difference(X, difference(Y, X)))))))), i(difference(X, difference(Y, X))))), i(X))))
% 101.76/13.47 = { by lemma 30 }
% 101.76/13.47 j(quotient(i(X), difference(i(quotient(i(quotient(i(i(difference(eta(X), i(difference(X, difference(Y, X)))))), difference(eta(X), i(difference(X, difference(Y, X)))))), i(difference(X, difference(Y, X))))), i(X))))
% 101.76/13.47 = { by lemma 37 }
% 101.76/13.47 j(quotient(i(X), difference(i(difference(quotient(i(i(difference(eta(X), i(difference(X, difference(Y, X)))))), difference(eta(X), i(difference(X, difference(Y, X))))), j(i(difference(X, difference(Y, X)))))), i(X))))
% 101.76/13.47 = { by lemma 32 R->L }
% 101.76/13.47 j(quotient(i(X), difference(i(difference(quotient(product(eta(difference(eta(X), i(difference(X, difference(Y, X))))), difference(eta(X), i(difference(X, difference(Y, X))))), difference(eta(X), i(difference(X, difference(Y, X))))), j(i(difference(X, difference(Y, X)))))), i(X))))
% 101.76/13.47 = { by axiom 9 (sos05) }
% 101.76/13.47 j(quotient(i(X), difference(i(difference(eta(difference(eta(X), i(difference(X, difference(Y, X))))), j(i(difference(X, difference(Y, X)))))), i(X))))
% 101.76/13.47 = { by lemma 41 }
% 101.76/13.47 j(quotient(i(X), difference(difference(j(i(difference(X, difference(Y, X)))), eta(difference(eta(X), i(difference(X, difference(Y, X)))))), i(X))))
% 101.76/13.47 = { by lemma 21 R->L }
% 101.76/13.47 j(quotient(i(X), difference(difference(j(i(difference(X, difference(Y, X)))), eta(i(j(difference(eta(X), i(difference(X, difference(Y, X)))))))), i(X))))
% 101.76/13.47 = { by axiom 1 (sos09) }
% 101.76/13.47 j(quotient(i(X), difference(difference(j(i(difference(X, difference(Y, X)))), eta(difference(j(difference(eta(X), i(difference(X, difference(Y, X))))), one))), i(X))))
% 101.76/13.47 = { by lemma 20 R->L }
% 101.76/13.47 j(quotient(i(X), difference(difference(j(i(difference(X, difference(Y, X)))), eta(difference(j(difference(eta(X), i(difference(X, difference(Y, X))))), product(j(i(difference(X, difference(Y, X)))), i(difference(X, difference(Y, X))))))), i(X))))
% 101.76/13.47 = { by lemma 31 R->L }
% 101.76/13.47 j(quotient(i(X), difference(difference(j(i(difference(X, difference(Y, X)))), eta(difference(product(j(i(difference(X, difference(Y, X)))), eta(X)), product(j(i(difference(X, difference(Y, X)))), i(difference(X, difference(Y, X))))))), i(X))))
% 101.76/13.47 = { by lemma 47 R->L }
% 101.76/13.47 j(quotient(i(X), difference(difference(j(i(difference(X, difference(Y, X)))), eta(quotient(difference(eta(X), product(i(difference(X, difference(Y, X))), j(i(difference(X, difference(Y, X)))))), j(i(difference(X, difference(Y, X))))))), i(X))))
% 101.76/13.47 = { by lemma 25 }
% 101.76/13.47 j(quotient(i(X), difference(difference(j(i(difference(X, difference(Y, X)))), eta(quotient(difference(eta(X), eta(i(difference(X, difference(Y, X))))), j(i(difference(X, difference(Y, X))))))), i(X))))
% 101.76/13.47 = { by lemma 27 R->L }
% 101.76/13.47 j(quotient(i(X), difference(quotient(eta(product(quotient(difference(eta(X), eta(i(difference(X, difference(Y, X))))), j(i(difference(X, difference(Y, X))))), j(i(difference(X, difference(Y, X)))))), j(i(difference(X, difference(Y, X))))), i(X))))
% 101.76/13.47 = { by axiom 11 (sos06) }
% 101.76/13.47 j(quotient(i(X), difference(quotient(eta(difference(eta(X), eta(i(difference(X, difference(Y, X)))))), j(i(difference(X, difference(Y, X))))), i(X))))
% 101.76/13.47 = { by lemma 26 R->L }
% 101.76/13.47 j(quotient(i(X), difference(quotient(eta(j(difference(eta(X), eta(i(difference(X, difference(Y, X))))))), j(i(difference(X, difference(Y, X))))), i(X))))
% 101.76/13.47 = { by lemma 31 R->L }
% 101.76/13.47 j(quotient(i(X), difference(quotient(eta(product(j(eta(i(difference(X, difference(Y, X))))), eta(X))), j(i(difference(X, difference(Y, X))))), i(X))))
% 101.76/13.47 = { by axiom 11 (sos06) R->L }
% 101.76/13.47 j(quotient(i(X), difference(quotient(product(quotient(eta(product(j(eta(i(difference(X, difference(Y, X))))), eta(X))), eta(X)), eta(X)), j(i(difference(X, difference(Y, X))))), i(X))))
% 101.76/13.47 = { by lemma 27 }
% 101.76/13.47 j(quotient(i(X), difference(quotient(product(difference(eta(X), eta(j(eta(i(difference(X, difference(Y, X))))))), eta(X)), j(i(difference(X, difference(Y, X))))), i(X))))
% 101.76/13.47 = { by lemma 26 }
% 101.76/13.47 j(quotient(i(X), difference(quotient(product(difference(eta(X), eta(eta(i(difference(X, difference(Y, X)))))), eta(X)), j(i(difference(X, difference(Y, X))))), i(X))))
% 101.76/13.47 = { by lemma 29 }
% 101.76/13.47 j(quotient(i(X), difference(quotient(product(difference(eta(X), one), eta(X)), j(i(difference(X, difference(Y, X))))), i(X))))
% 101.76/13.47 = { by axiom 1 (sos09) R->L }
% 101.76/13.47 j(quotient(i(X), difference(quotient(product(i(eta(X)), eta(X)), j(i(difference(X, difference(Y, X))))), i(X))))
% 101.76/13.47 = { by axiom 5 (sos12) R->L }
% 101.76/13.47 j(quotient(i(X), difference(quotient(eta(eta(X)), j(i(difference(X, difference(Y, X))))), i(X))))
% 101.76/13.47 = { by lemma 29 }
% 101.76/13.47 j(quotient(i(X), difference(quotient(one, j(i(difference(X, difference(Y, X))))), i(X))))
% 101.76/13.47 = { by axiom 2 (sos10) R->L }
% 101.76/13.47 j(quotient(i(X), difference(j(j(i(difference(X, difference(Y, X))))), i(X))))
% 101.76/13.47 = { by lemma 23 }
% 101.76/13.47 j(quotient(i(X), difference(j(difference(X, difference(Y, X))), i(X))))
% 101.76/13.47 = { by lemma 34 }
% 101.76/13.47 j(j(difference(X, difference(Y, X))))
% 101.76/13.47 = { by axiom 7 (sos04) R->L }
% 101.76/13.47 j(j(difference(X, difference(Y, difference(i(X), product(i(X), X))))))
% 101.76/13.47 = { by axiom 5 (sos12) R->L }
% 101.76/13.47 j(j(difference(X, difference(Y, difference(i(X), eta(X))))))
% 101.76/13.47 = { by lemma 45 R->L }
% 101.76/13.47 j(j(difference(X, difference(product(i(X), Y), eta(X)))))
% 101.76/13.47 = { by lemma 23 R->L }
% 101.76/13.47 j(j(difference(j(i(X)), difference(product(i(X), Y), eta(X)))))
% 101.76/13.47 = { by lemma 45 R->L }
% 101.76/13.47 j(j(difference(product(product(i(X), Y), j(i(X))), eta(X))))
% 101.76/13.47 = { by axiom 10 (sos03) R->L }
% 101.76/13.47 j(j(difference(product(i(X), difference(i(X), product(product(i(X), Y), j(i(X))))), eta(X))))
% 101.76/13.47 = { by axiom 5 (sos12) }
% 101.76/13.47 j(j(difference(product(i(X), difference(i(X), product(product(i(X), Y), j(i(X))))), product(i(X), X))))
% 101.76/13.47 = { by lemma 47 R->L }
% 101.76/13.47 j(j(quotient(difference(difference(i(X), product(product(i(X), Y), j(i(X)))), product(X, i(X))), i(X))))
% 101.76/13.47 = { by lemma 22 }
% 101.76/13.47 j(j(quotient(difference(difference(i(X), product(product(i(X), Y), j(i(X)))), one), i(X))))
% 101.76/13.47 = { by lemma 36 }
% 101.76/13.47 j(j(difference(j(difference(difference(i(X), product(product(i(X), Y), j(i(X)))), one)), X)))
% 101.76/13.47 = { by axiom 1 (sos09) R->L }
% 101.76/13.47 j(j(difference(j(i(difference(i(X), product(product(i(X), Y), j(i(X)))))), X)))
% 101.76/13.47 = { by lemma 23 }
% 101.76/13.47 j(j(difference(difference(i(X), product(product(i(X), Y), j(i(X)))), X)))
% 101.76/13.47 = { by axiom 17 (sos07) }
% 101.76/13.47 j(j(difference(quotient(product(Y, product(j(i(X)), i(X))), i(X)), X)))
% 101.76/13.47 = { by lemma 20 }
% 101.76/13.47 j(j(difference(quotient(product(Y, one), i(X)), X)))
% 101.76/13.47 = { by axiom 3 (sos01) }
% 101.76/13.47 j(j(difference(quotient(Y, i(X)), X)))
% 101.76/13.47 = { by lemma 36 }
% 101.76/13.47 j(j(difference(difference(j(Y), X), X)))
% 101.76/13.47 = { by lemma 31 R->L }
% 101.76/13.47 j(product(j(X), difference(j(Y), X)))
% 101.76/13.47 = { by lemma 42 R->L }
% 101.76/13.47 j(i(quotient(j(difference(j(Y), X)), j(X))))
% 101.76/13.47 = { by lemma 31 R->L }
% 101.76/13.47 j(i(quotient(product(j(X), j(Y)), j(X))))
% 101.76/13.48 = { by axiom 8 (sos19) R->L }
% 101.76/13.48 j(i(t(j(X), j(Y))))
% 101.76/13.48 = { by lemma 23 }
% 101.76/13.48 t(j(X), j(Y))
% 101.76/13.48
% 101.76/13.48 Goal 1 (goals): product(product(x0, x1), x2) = product(product(x0, x2), difference(x2, product(x1, x2))).
% 101.76/13.48 Proof:
% 101.76/13.48 product(product(x0, x1), x2)
% 101.76/13.48 = { by lemma 21 R->L }
% 101.76/13.48 i(j(product(product(x0, x1), x2)))
% 101.76/13.48 = { by axiom 10 (sos03) R->L }
% 101.76/13.48 i(j(product(product(x0, x2), difference(product(x0, x2), product(product(x0, x1), x2)))))
% 101.76/13.48 = { by lemma 38 }
% 101.76/13.48 i(quotient(j(difference(product(x0, x2), product(product(x0, x1), x2))), product(x0, x2)))
% 101.76/13.48 = { by axiom 7 (sos04) R->L }
% 101.76/13.48 i(quotient(j(difference(x2, product(x2, difference(product(x0, x2), product(product(x0, x1), x2))))), product(x0, x2)))
% 101.76/13.48 = { by lemma 31 R->L }
% 101.76/13.48 i(quotient(product(j(product(x2, difference(product(x0, x2), product(product(x0, x1), x2)))), x2), product(x0, x2)))
% 101.76/13.48 = { by lemma 42 R->L }
% 101.76/13.48 i(quotient(i(quotient(j(x2), j(product(x2, difference(product(x0, x2), product(product(x0, x1), x2)))))), product(x0, x2)))
% 101.76/13.48 = { by lemma 46 R->L }
% 101.76/13.48 i(quotient(i(quotient(j(x2), quotient(j(x2), quotient(product(x2, difference(product(x0, x2), product(product(x0, x1), x2))), x2)))), product(x0, x2)))
% 101.76/13.48 = { by lemma 42 }
% 101.76/13.48 i(quotient(product(quotient(j(x2), quotient(product(x2, difference(product(x0, x2), product(product(x0, x1), x2))), x2)), x2), product(x0, x2)))
% 101.76/13.48 = { by lemma 38 R->L }
% 101.76/13.48 i(quotient(product(j(product(quotient(product(x2, difference(product(x0, x2), product(product(x0, x1), x2))), x2), x2)), x2), product(x0, x2)))
% 101.76/13.49 = { by lemma 31 }
% 101.76/13.49 i(quotient(j(difference(x2, product(quotient(product(x2, difference(product(x0, x2), product(product(x0, x1), x2))), x2), x2))), product(x0, x2)))
% 101.76/13.49 = { by lemma 48 }
% 101.76/13.49 i(quotient(t(j(x2), j(quotient(product(x2, difference(product(x0, x2), product(product(x0, x1), x2))), x2))), product(x0, x2)))
% 101.76/13.49 = { by axiom 8 (sos19) R->L }
% 101.76/13.49 i(quotient(t(j(x2), j(t(x2, difference(product(x0, x2), product(product(x0, x1), x2))))), product(x0, x2)))
% 101.76/13.49 = { by lemma 21 R->L }
% 101.76/13.49 i(quotient(t(j(x2), j(t(x2, difference(product(x0, x2), i(j(product(product(x0, x1), x2))))))), product(x0, x2)))
% 101.76/13.49 = { by lemma 33 R->L }
% 101.76/13.49 i(quotient(t(j(x2), j(t(x2, i(product(j(product(product(x0, x1), x2)), product(x0, x2)))))), product(x0, x2)))
% 101.76/13.49 = { by axiom 8 (sos19) }
% 101.76/13.49 i(quotient(t(j(x2), j(quotient(product(x2, i(product(j(product(product(x0, x1), x2)), product(x0, x2)))), x2))), product(x0, x2)))
% 101.76/13.49 = { by lemma 30 }
% 101.76/13.49 i(quotient(t(j(x2), j(quotient(i(quotient(product(j(product(product(x0, x1), x2)), product(x0, x2)), x2)), x2))), product(x0, x2)))
% 101.76/13.49 = { by lemma 37 }
% 101.76/13.49 i(quotient(t(j(x2), j(difference(quotient(product(j(product(product(x0, x1), x2)), product(x0, x2)), x2), j(x2)))), product(x0, x2)))
% 101.76/13.49 = { by axiom 17 (sos07) R->L }
% 101.76/13.49 i(quotient(t(j(x2), j(difference(difference(x2, product(product(x2, j(product(product(x0, x1), x2))), x0)), j(x2)))), product(x0, x2)))
% 101.76/13.49 = { by lemma 37 R->L }
% 101.76/13.49 i(quotient(t(j(x2), j(quotient(i(difference(x2, product(product(x2, j(product(product(x0, x1), x2))), x0))), x2))), product(x0, x2)))
% 101.76/13.49 = { by axiom 1 (sos09) }
% 101.76/13.49 i(quotient(t(j(x2), j(quotient(difference(difference(x2, product(product(x2, j(product(product(x0, x1), x2))), x0)), one), x2))), product(x0, x2)))
% 101.76/13.49 = { by lemma 20 R->L }
% 101.76/13.49 i(quotient(t(j(x2), j(quotient(difference(difference(x2, product(product(x2, j(product(product(x0, x1), x2))), x0)), product(j(x2), x2)), x2))), product(x0, x2)))
% 101.76/13.49 = { by lemma 47 }
% 101.76/13.49 i(quotient(t(j(x2), j(difference(product(x2, difference(x2, product(product(x2, j(product(product(x0, x1), x2))), x0))), product(x2, j(x2))))), product(x0, x2)))
% 101.76/13.49 = { by lemma 25 }
% 101.76/13.49 i(quotient(t(j(x2), j(difference(product(x2, difference(x2, product(product(x2, j(product(product(x0, x1), x2))), x0))), eta(x2)))), product(x0, x2)))
% 101.76/13.49 = { by axiom 10 (sos03) }
% 101.76/13.49 i(quotient(t(j(x2), j(difference(product(product(x2, j(product(product(x0, x1), x2))), x0), eta(x2)))), product(x0, x2)))
% 101.76/13.49 = { by lemma 45 }
% 101.76/13.49 i(quotient(t(j(x2), j(difference(x0, difference(product(x2, j(product(product(x0, x1), x2))), eta(x2))))), product(x0, x2)))
% 101.76/13.49 = { by lemma 45 }
% 101.76/13.49 i(quotient(t(j(x2), j(difference(x0, difference(j(product(product(x0, x1), x2)), difference(x2, eta(x2)))))), product(x0, x2)))
% 101.76/13.49 = { by lemma 28 }
% 101.76/13.49 i(quotient(t(j(x2), j(difference(x0, difference(j(product(product(x0, x1), x2)), j(x2))))), product(x0, x2)))
% 101.76/13.49 = { by lemma 46 R->L }
% 101.76/13.49 i(quotient(t(j(x2), j(difference(x0, difference(quotient(j(x2), quotient(product(product(x0, x1), x2), x2)), j(x2))))), product(x0, x2)))
% 101.76/13.49 = { by lemma 39 }
% 101.76/13.49 i(quotient(t(j(x2), j(difference(x0, quotient(product(product(x0, x1), x2), x2)))), product(x0, x2)))
% 101.76/13.49 = { by axiom 9 (sos05) }
% 101.76/13.49 i(quotient(t(j(x2), j(difference(x0, product(x0, x1)))), product(x0, x2)))
% 101.76/13.49 = { by axiom 7 (sos04) }
% 101.76/13.49 i(quotient(t(j(x2), j(x1)), product(x0, x2)))
% 101.76/13.49 = { by lemma 48 R->L }
% 101.76/13.49 i(quotient(j(difference(x2, product(x1, x2))), product(x0, x2)))
% 101.76/13.49 = { by lemma 42 }
% 101.76/13.49 product(product(x0, x2), difference(x2, product(x1, x2)))
% 101.76/13.49 % SZS output end Proof
% 101.76/13.49
% 101.76/13.49 RESULT: Unsatisfiable (the axioms are contradictory).
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