TSTP Solution File: GRP401-1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP401-1 : TPTP v8.1.2. Released v2.5.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n011.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 01:18:16 EDT 2023
% Result : Unsatisfiable 6.34s 1.20s
% Output : Proof 6.90s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.12 % Problem : GRP401-1 : TPTP v8.1.2. Released v2.5.0.
% 0.12/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.34 % Computer : n011.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Tue Aug 29 02:32:11 EDT 2023
% 0.14/0.34 % CPUTime :
% 6.34/1.20 Command-line arguments: --no-flatten-goal
% 6.34/1.20
% 6.34/1.20 % SZS status Unsatisfiable
% 6.34/1.20
% 6.90/1.25 % SZS output start Proof
% 6.90/1.25 Take the following subset of the input axioms:
% 6.90/1.25 fof(associativity_of_multiply, axiom, ![X, Y, Z]: multiply(multiply(X, Y), Z)=multiply(X, multiply(Y, Z))).
% 6.90/1.25 fof(commutator, axiom, ![A, B]: multiply(A, B)=multiply(B, multiply(A, commutator(A, B)))).
% 6.90/1.25 fof(commutator_distributes_over_product, axiom, ![C, B2, A3]: commutator(multiply(A3, B2), C)=multiply(commutator(A3, C), commutator(B2, C))).
% 6.90/1.25 fof(left_cancellation, axiom, ![B2, C2, A3]: (multiply(A3, B2)!=multiply(A3, C2) | B2=C2)).
% 6.90/1.25 fof(prove_nilpotency, negated_conjecture, multiply(commutator(a, b), c)!=multiply(c, commutator(a, b))).
% 6.90/1.25 fof(right_cancellation, axiom, ![A2, B2, C2]: (multiply(A2, B2)!=multiply(C2, B2) | A2=C2)).
% 6.90/1.25
% 6.90/1.25 Now clausify the problem and encode Horn clauses using encoding 3 of
% 6.90/1.25 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 6.90/1.25 We repeatedly replace C & s=t => u=v by the two clauses:
% 6.90/1.25 fresh(y, y, x1...xn) = u
% 6.90/1.25 C => fresh(s, t, x1...xn) = v
% 6.90/1.25 where fresh is a fresh function symbol and x1..xn are the free
% 6.90/1.25 variables of u and v.
% 6.90/1.25 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 6.90/1.25 input problem has no model of domain size 1).
% 6.90/1.25
% 6.90/1.25 The encoding turns the above axioms into the following unit equations and goals:
% 6.90/1.25
% 6.90/1.25 Axiom 1 (left_cancellation): fresh(X, X, Y, Z) = Z.
% 6.90/1.25 Axiom 2 (right_cancellation): fresh2(X, X, Y, Z) = Z.
% 6.90/1.25 Axiom 3 (associativity_of_multiply): multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)).
% 6.90/1.25 Axiom 4 (commutator): multiply(X, Y) = multiply(Y, multiply(X, commutator(X, Y))).
% 6.90/1.25 Axiom 5 (commutator_distributes_over_product): commutator(multiply(X, Y), Z) = multiply(commutator(X, Z), commutator(Y, Z)).
% 6.90/1.25 Axiom 6 (left_cancellation): fresh(multiply(X, Y), multiply(X, Z), Y, Z) = Y.
% 6.90/1.25 Axiom 7 (right_cancellation): fresh2(multiply(X, Y), multiply(Z, Y), X, Z) = X.
% 6.90/1.25
% 6.90/1.25 Lemma 8: multiply(X, commutator(multiply(Y, commutator(Y, X)), X)) = multiply(commutator(Y, X), X).
% 6.90/1.25 Proof:
% 6.90/1.25 multiply(X, commutator(multiply(Y, commutator(Y, X)), X))
% 6.90/1.25 = { by axiom 5 (commutator_distributes_over_product) }
% 6.90/1.25 multiply(X, multiply(commutator(Y, X), commutator(commutator(Y, X), X)))
% 6.90/1.25 = { by axiom 4 (commutator) R->L }
% 6.90/1.25 multiply(commutator(Y, X), X)
% 6.90/1.25
% 6.90/1.25 Lemma 9: multiply(X, commutator(X, X)) = X.
% 6.90/1.25 Proof:
% 6.90/1.25 multiply(X, commutator(X, X))
% 6.90/1.25 = { by axiom 1 (left_cancellation) R->L }
% 6.90/1.25 fresh(multiply(X, X), multiply(X, X), X, multiply(X, commutator(X, X)))
% 6.90/1.25 = { by axiom 4 (commutator) }
% 6.90/1.25 fresh(multiply(X, X), multiply(X, multiply(X, commutator(X, X))), X, multiply(X, commutator(X, X)))
% 6.90/1.25 = { by axiom 6 (left_cancellation) }
% 6.90/1.25 X
% 6.90/1.25
% 6.90/1.25 Lemma 10: multiply(commutator(X, X), X) = X.
% 6.90/1.25 Proof:
% 6.90/1.25 multiply(commutator(X, X), X)
% 6.90/1.25 = { by lemma 8 R->L }
% 6.90/1.25 multiply(X, commutator(multiply(X, commutator(X, X)), X))
% 6.90/1.25 = { by lemma 9 }
% 6.90/1.25 multiply(X, commutator(X, X))
% 6.90/1.25 = { by lemma 9 }
% 6.90/1.25 X
% 6.90/1.25
% 6.90/1.25 Lemma 11: commutator(multiply(X, Y), multiply(X, Y)) = commutator(X, X).
% 6.90/1.25 Proof:
% 6.90/1.25 commutator(multiply(X, Y), multiply(X, Y))
% 6.90/1.25 = { by axiom 2 (right_cancellation) R->L }
% 6.90/1.25 fresh2(multiply(X, Y), multiply(X, Y), commutator(X, X), commutator(multiply(X, Y), multiply(X, Y)))
% 6.90/1.25 = { by lemma 10 R->L }
% 6.90/1.25 fresh2(multiply(multiply(commutator(X, X), X), Y), multiply(X, Y), commutator(X, X), commutator(multiply(X, Y), multiply(X, Y)))
% 6.90/1.25 = { by axiom 3 (associativity_of_multiply) }
% 6.90/1.25 fresh2(multiply(commutator(X, X), multiply(X, Y)), multiply(X, Y), commutator(X, X), commutator(multiply(X, Y), multiply(X, Y)))
% 6.90/1.25 = { by lemma 10 R->L }
% 6.90/1.25 fresh2(multiply(commutator(X, X), multiply(X, Y)), multiply(commutator(multiply(X, Y), multiply(X, Y)), multiply(X, Y)), commutator(X, X), commutator(multiply(X, Y), multiply(X, Y)))
% 6.90/1.25 = { by axiom 7 (right_cancellation) }
% 6.90/1.25 commutator(X, X)
% 6.90/1.25
% 6.90/1.25 Lemma 12: commutator(Y, Y) = commutator(X, X).
% 6.90/1.25 Proof:
% 6.90/1.25 commutator(Y, Y)
% 6.90/1.25 = { by lemma 11 R->L }
% 6.90/1.25 commutator(multiply(Y, X), multiply(Y, X))
% 6.90/1.25 = { by axiom 4 (commutator) }
% 6.90/1.25 commutator(multiply(Y, X), multiply(X, multiply(Y, commutator(Y, X))))
% 6.90/1.25 = { by axiom 4 (commutator) }
% 6.90/1.25 commutator(multiply(X, multiply(Y, commutator(Y, X))), multiply(X, multiply(Y, commutator(Y, X))))
% 6.90/1.25 = { by lemma 11 }
% 6.90/1.25 commutator(X, X)
% 6.90/1.25
% 6.90/1.25 Lemma 13: multiply(X, commutator(Y, Y)) = X.
% 6.90/1.25 Proof:
% 6.90/1.25 multiply(X, commutator(Y, Y))
% 6.90/1.25 = { by lemma 12 }
% 6.90/1.25 multiply(X, commutator(X, X))
% 6.90/1.25 = { by lemma 9 }
% 6.90/1.25 X
% 6.90/1.25
% 6.90/1.25 Lemma 14: multiply(commutator(X, X), Y) = Y.
% 6.90/1.25 Proof:
% 6.90/1.25 multiply(commutator(X, X), Y)
% 6.90/1.25 = { by lemma 12 }
% 6.90/1.25 multiply(commutator(Y, Y), Y)
% 6.90/1.25 = { by lemma 10 }
% 6.90/1.25 Y
% 6.90/1.25
% 6.90/1.25 Lemma 15: multiply(X, commutator(multiply(X, Y), X)) = multiply(X, commutator(Y, X)).
% 6.90/1.25 Proof:
% 6.90/1.25 multiply(X, commutator(multiply(X, Y), X))
% 6.90/1.25 = { by axiom 5 (commutator_distributes_over_product) }
% 6.90/1.25 multiply(X, multiply(commutator(X, X), commutator(Y, X)))
% 6.90/1.25 = { by axiom 3 (associativity_of_multiply) R->L }
% 6.90/1.25 multiply(multiply(X, commutator(X, X)), commutator(Y, X))
% 6.90/1.25 = { by lemma 9 }
% 6.90/1.25 multiply(X, commutator(Y, X))
% 6.90/1.25
% 6.90/1.25 Lemma 16: commutator(multiply(X, Y), Y) = commutator(X, Y).
% 6.90/1.25 Proof:
% 6.90/1.25 commutator(multiply(X, Y), Y)
% 6.90/1.25 = { by axiom 5 (commutator_distributes_over_product) }
% 6.90/1.25 multiply(commutator(X, Y), commutator(Y, Y))
% 6.90/1.25 = { by lemma 12 R->L }
% 6.90/1.25 multiply(commutator(X, Y), commutator(Z, Z))
% 6.90/1.25 = { by lemma 13 }
% 6.90/1.25 commutator(X, Y)
% 6.90/1.25
% 6.90/1.25 Lemma 17: multiply(commutator(X, Y), Y) = multiply(Y, commutator(X, Y)).
% 6.90/1.25 Proof:
% 6.90/1.25 multiply(commutator(X, Y), Y)
% 6.90/1.25 = { by lemma 8 R->L }
% 6.90/1.25 multiply(Y, commutator(multiply(X, commutator(X, Y)), Y))
% 6.90/1.25 = { by lemma 15 R->L }
% 6.90/1.25 multiply(Y, commutator(multiply(Y, multiply(X, commutator(X, Y))), Y))
% 6.90/1.25 = { by axiom 4 (commutator) R->L }
% 6.90/1.25 multiply(Y, commutator(multiply(X, Y), Y))
% 6.90/1.25 = { by lemma 16 }
% 6.90/1.25 multiply(Y, commutator(X, Y))
% 6.90/1.25
% 6.90/1.25 Lemma 18: fresh(multiply(X, multiply(Y, Z)), multiply(X, multiply(Y, W)), Z, W) = Z.
% 6.90/1.25 Proof:
% 6.90/1.25 fresh(multiply(X, multiply(Y, Z)), multiply(X, multiply(Y, W)), Z, W)
% 6.90/1.25 = { by axiom 3 (associativity_of_multiply) R->L }
% 6.90/1.25 fresh(multiply(X, multiply(Y, Z)), multiply(multiply(X, Y), W), Z, W)
% 6.90/1.25 = { by axiom 3 (associativity_of_multiply) R->L }
% 6.90/1.25 fresh(multiply(multiply(X, Y), Z), multiply(multiply(X, Y), W), Z, W)
% 6.90/1.25 = { by axiom 6 (left_cancellation) }
% 6.90/1.25 Z
% 6.90/1.25
% 6.90/1.25 Lemma 19: fresh(multiply(X, multiply(Y, Z)), multiply(Y, X), Z, commutator(Y, X)) = Z.
% 6.90/1.25 Proof:
% 6.90/1.25 fresh(multiply(X, multiply(Y, Z)), multiply(Y, X), Z, commutator(Y, X))
% 6.90/1.25 = { by axiom 4 (commutator) }
% 6.90/1.25 fresh(multiply(X, multiply(Y, Z)), multiply(X, multiply(Y, commutator(Y, X))), Z, commutator(Y, X))
% 6.90/1.25 = { by lemma 18 }
% 6.90/1.25 Z
% 6.90/1.25
% 6.90/1.25 Lemma 20: fresh(multiply(X, multiply(Y, Z)), multiply(X, Y), Z, commutator(W, W)) = Z.
% 6.90/1.25 Proof:
% 6.90/1.25 fresh(multiply(X, multiply(Y, Z)), multiply(X, Y), Z, commutator(W, W))
% 6.90/1.25 = { by lemma 12 }
% 6.90/1.25 fresh(multiply(X, multiply(Y, Z)), multiply(X, Y), Z, commutator(Y, Y))
% 6.90/1.25 = { by lemma 9 R->L }
% 6.90/1.25 fresh(multiply(X, multiply(Y, Z)), multiply(X, multiply(Y, commutator(Y, Y))), Z, commutator(Y, Y))
% 6.90/1.25 = { by lemma 18 }
% 6.90/1.25 Z
% 6.90/1.25
% 6.90/1.25 Lemma 21: multiply(X, multiply(Y, multiply(commutator(Y, X), Z))) = multiply(Y, multiply(X, Z)).
% 6.90/1.25 Proof:
% 6.90/1.25 multiply(X, multiply(Y, multiply(commutator(Y, X), Z)))
% 6.90/1.25 = { by axiom 3 (associativity_of_multiply) R->L }
% 6.90/1.25 multiply(X, multiply(multiply(Y, commutator(Y, X)), Z))
% 6.90/1.25 = { by axiom 3 (associativity_of_multiply) R->L }
% 6.90/1.25 multiply(multiply(X, multiply(Y, commutator(Y, X))), Z)
% 6.90/1.25 = { by axiom 4 (commutator) R->L }
% 6.90/1.25 multiply(multiply(Y, X), Z)
% 6.90/1.25 = { by axiom 3 (associativity_of_multiply) }
% 6.90/1.25 multiply(Y, multiply(X, Z))
% 6.90/1.25
% 6.90/1.25 Lemma 22: commutator(multiply(X, Y), X) = commutator(Y, X).
% 6.90/1.25 Proof:
% 6.90/1.25 commutator(multiply(X, Y), X)
% 6.90/1.25 = { by axiom 5 (commutator_distributes_over_product) }
% 6.90/1.25 multiply(commutator(X, X), commutator(Y, X))
% 6.90/1.25 = { by lemma 12 R->L }
% 6.90/1.25 multiply(commutator(Z, Z), commutator(Y, X))
% 6.90/1.25 = { by lemma 14 }
% 6.90/1.25 commutator(Y, X)
% 6.90/1.25
% 6.90/1.25 Lemma 23: multiply(X, multiply(commutator(Y, X), commutator(X, commutator(Y, X)))) = multiply(X, commutator(Y, X)).
% 6.90/1.25 Proof:
% 6.90/1.25 multiply(X, multiply(commutator(Y, X), commutator(X, commutator(Y, X))))
% 6.90/1.25 = { by lemma 16 R->L }
% 6.90/1.25 multiply(X, multiply(commutator(multiply(Y, X), X), commutator(X, commutator(Y, X))))
% 6.90/1.25 = { by axiom 4 (commutator) }
% 6.90/1.25 multiply(X, multiply(commutator(multiply(X, multiply(Y, commutator(Y, X))), X), commutator(X, commutator(Y, X))))
% 6.90/1.25 = { by lemma 22 }
% 6.90/1.25 multiply(X, multiply(commutator(multiply(Y, commutator(Y, X)), X), commutator(X, commutator(Y, X))))
% 6.90/1.25 = { by axiom 3 (associativity_of_multiply) R->L }
% 6.90/1.25 multiply(multiply(X, commutator(multiply(Y, commutator(Y, X)), X)), commutator(X, commutator(Y, X)))
% 6.90/1.25 = { by lemma 8 }
% 6.90/1.25 multiply(multiply(commutator(Y, X), X), commutator(X, commutator(Y, X)))
% 6.90/1.25 = { by axiom 3 (associativity_of_multiply) }
% 6.90/1.25 multiply(commutator(Y, X), multiply(X, commutator(X, commutator(Y, X))))
% 6.90/1.25 = { by axiom 4 (commutator) R->L }
% 6.90/1.25 multiply(X, commutator(Y, X))
% 6.90/1.25
% 6.90/1.25 Lemma 24: commutator(X, commutator(Y, X)) = commutator(Z, Z).
% 6.90/1.25 Proof:
% 6.90/1.25 commutator(X, commutator(Y, X))
% 6.90/1.25 = { by lemma 20 R->L }
% 6.90/1.25 fresh(multiply(X, multiply(commutator(Y, X), commutator(X, commutator(Y, X)))), multiply(X, commutator(Y, X)), commutator(X, commutator(Y, X)), commutator(Z, Z))
% 6.90/1.25 = { by lemma 23 }
% 6.90/1.25 fresh(multiply(X, commutator(Y, X)), multiply(X, commutator(Y, X)), commutator(X, commutator(Y, X)), commutator(Z, Z))
% 6.90/1.25 = { by axiom 1 (left_cancellation) }
% 6.90/1.25 commutator(Z, Z)
% 6.90/1.25
% 6.90/1.25 Lemma 25: commutator(commutator(X, Y), Y) = commutator(Y, commutator(X, Y)).
% 6.90/1.25 Proof:
% 6.90/1.25 commutator(commutator(X, Y), Y)
% 6.90/1.25 = { by axiom 1 (left_cancellation) R->L }
% 6.90/1.25 fresh(multiply(Y, commutator(X, Y)), multiply(Y, commutator(X, Y)), commutator(Y, commutator(X, Y)), commutator(commutator(X, Y), Y))
% 6.90/1.25 = { by lemma 17 R->L }
% 6.90/1.25 fresh(multiply(Y, commutator(X, Y)), multiply(commutator(X, Y), Y), commutator(Y, commutator(X, Y)), commutator(commutator(X, Y), Y))
% 6.90/1.25 = { by lemma 23 R->L }
% 6.90/1.25 fresh(multiply(Y, multiply(commutator(X, Y), commutator(Y, commutator(X, Y)))), multiply(commutator(X, Y), Y), commutator(Y, commutator(X, Y)), commutator(commutator(X, Y), Y))
% 6.90/1.25 = { by lemma 19 }
% 6.90/1.25 commutator(Y, commutator(X, Y))
% 6.90/1.25
% 6.90/1.25 Lemma 26: commutator(commutator(X, Y), X) = commutator(Z, Z).
% 6.90/1.25 Proof:
% 6.90/1.25 commutator(commutator(X, Y), X)
% 6.90/1.25 = { by lemma 20 R->L }
% 6.90/1.25 fresh(multiply(Y, multiply(X, commutator(commutator(X, Y), X))), multiply(Y, X), commutator(commutator(X, Y), X), commutator(Z, Z))
% 6.90/1.25 = { by lemma 15 R->L }
% 6.90/1.25 fresh(multiply(Y, multiply(X, commutator(multiply(X, commutator(X, Y)), X))), multiply(Y, X), commutator(commutator(X, Y), X), commutator(Z, Z))
% 6.90/1.25 = { by lemma 21 R->L }
% 6.90/1.25 fresh(multiply(X, multiply(Y, multiply(commutator(Y, X), commutator(multiply(X, commutator(X, Y)), X)))), multiply(Y, X), commutator(commutator(X, Y), X), commutator(Z, Z))
% 6.90/1.25 = { by axiom 5 (commutator_distributes_over_product) R->L }
% 6.90/1.25 fresh(multiply(X, multiply(Y, commutator(multiply(Y, multiply(X, commutator(X, Y))), X))), multiply(Y, X), commutator(commutator(X, Y), X), commutator(Z, Z))
% 6.90/1.25 = { by axiom 4 (commutator) R->L }
% 6.90/1.25 fresh(multiply(X, multiply(Y, commutator(multiply(X, Y), X))), multiply(Y, X), commutator(commutator(X, Y), X), commutator(Z, Z))
% 6.90/1.25 = { by axiom 5 (commutator_distributes_over_product) }
% 6.90/1.25 fresh(multiply(X, multiply(Y, multiply(commutator(X, X), commutator(Y, X)))), multiply(Y, X), commutator(commutator(X, Y), X), commutator(Z, Z))
% 6.90/1.25 = { by axiom 3 (associativity_of_multiply) R->L }
% 6.90/1.25 fresh(multiply(X, multiply(multiply(Y, commutator(X, X)), commutator(Y, X))), multiply(Y, X), commutator(commutator(X, Y), X), commutator(Z, Z))
% 6.90/1.25 = { by lemma 13 R->L }
% 6.90/1.25 fresh(multiply(X, multiply(multiply(Y, commutator(X, X)), multiply(commutator(Y, X), commutator(W, W)))), multiply(Y, X), commutator(commutator(X, Y), X), commutator(Z, Z))
% 6.90/1.25 = { by lemma 24 R->L }
% 6.90/1.25 fresh(multiply(X, multiply(multiply(Y, commutator(X, X)), multiply(commutator(Y, X), commutator(X, commutator(X, X))))), multiply(Y, X), commutator(commutator(X, Y), X), commutator(Z, Z))
% 6.90/1.25 = { by lemma 25 R->L }
% 6.90/1.25 fresh(multiply(X, multiply(multiply(Y, commutator(X, X)), multiply(commutator(Y, X), commutator(commutator(X, X), X)))), multiply(Y, X), commutator(commutator(X, Y), X), commutator(Z, Z))
% 6.90/1.25 = { by axiom 5 (commutator_distributes_over_product) R->L }
% 6.90/1.25 fresh(multiply(X, multiply(multiply(Y, commutator(X, X)), commutator(multiply(Y, commutator(X, X)), X))), multiply(Y, X), commutator(commutator(X, Y), X), commutator(Z, Z))
% 6.90/1.25 = { by axiom 4 (commutator) R->L }
% 6.90/1.25 fresh(multiply(multiply(Y, commutator(X, X)), X), multiply(Y, X), commutator(commutator(X, Y), X), commutator(Z, Z))
% 6.90/1.25 = { by axiom 3 (associativity_of_multiply) }
% 6.90/1.25 fresh(multiply(Y, multiply(commutator(X, X), X)), multiply(Y, X), commutator(commutator(X, Y), X), commutator(Z, Z))
% 6.90/1.25 = { by lemma 17 }
% 6.90/1.25 fresh(multiply(Y, multiply(X, commutator(X, X))), multiply(Y, X), commutator(commutator(X, Y), X), commutator(Z, Z))
% 6.90/1.25 = { by lemma 9 }
% 6.90/1.25 fresh(multiply(Y, X), multiply(Y, X), commutator(commutator(X, Y), X), commutator(Z, Z))
% 6.90/1.25 = { by axiom 1 (left_cancellation) }
% 6.90/1.26 commutator(Z, Z)
% 6.90/1.26
% 6.90/1.26 Lemma 27: commutator(X, commutator(X, Y)) = commutator(Z, Z).
% 6.90/1.26 Proof:
% 6.90/1.26 commutator(X, commutator(X, Y))
% 6.90/1.26 = { by lemma 19 R->L }
% 6.90/1.26 fresh(multiply(X, multiply(commutator(X, Y), commutator(X, commutator(X, Y)))), multiply(commutator(X, Y), X), commutator(X, commutator(X, Y)), commutator(commutator(X, Y), X))
% 6.90/1.26 = { by lemma 14 R->L }
% 6.90/1.26 fresh(multiply(X, multiply(commutator(X, Y), multiply(commutator(W, W), commutator(X, commutator(X, Y))))), multiply(commutator(X, Y), X), commutator(X, commutator(X, Y)), commutator(commutator(X, Y), X))
% 6.90/1.26 = { by lemma 26 R->L }
% 6.90/1.26 fresh(multiply(X, multiply(commutator(X, Y), multiply(commutator(commutator(X, Y), X), commutator(X, commutator(X, Y))))), multiply(commutator(X, Y), X), commutator(X, commutator(X, Y)), commutator(commutator(X, Y), X))
% 6.90/1.26 = { by lemma 21 }
% 6.90/1.26 fresh(multiply(commutator(X, Y), multiply(X, commutator(X, commutator(X, Y)))), multiply(commutator(X, Y), X), commutator(X, commutator(X, Y)), commutator(commutator(X, Y), X))
% 6.90/1.26 = { by axiom 4 (commutator) R->L }
% 6.90/1.26 fresh(multiply(X, commutator(X, Y)), multiply(commutator(X, Y), X), commutator(X, commutator(X, Y)), commutator(commutator(X, Y), X))
% 6.90/1.26 = { by axiom 4 (commutator) }
% 6.90/1.26 fresh(multiply(X, commutator(X, Y)), multiply(X, multiply(commutator(X, Y), commutator(commutator(X, Y), X))), commutator(X, commutator(X, Y)), commutator(commutator(X, Y), X))
% 6.90/1.26 = { by lemma 26 }
% 6.90/1.26 fresh(multiply(X, commutator(X, Y)), multiply(X, multiply(commutator(X, Y), commutator(V, V))), commutator(X, commutator(X, Y)), commutator(commutator(X, Y), X))
% 6.90/1.26 = { by lemma 13 }
% 6.90/1.26 fresh(multiply(X, commutator(X, Y)), multiply(X, commutator(X, Y)), commutator(X, commutator(X, Y)), commutator(commutator(X, Y), X))
% 6.90/1.26 = { by axiom 1 (left_cancellation) }
% 6.90/1.26 commutator(commutator(X, Y), X)
% 6.90/1.26 = { by lemma 26 }
% 6.90/1.26 commutator(Z, Z)
% 6.90/1.26
% 6.90/1.26 Lemma 28: multiply(commutator(X, Y), multiply(commutator(Z, Y), W)) = multiply(commutator(multiply(X, Z), Y), W).
% 6.90/1.26 Proof:
% 6.90/1.26 multiply(commutator(X, Y), multiply(commutator(Z, Y), W))
% 6.90/1.26 = { by axiom 3 (associativity_of_multiply) R->L }
% 6.90/1.26 multiply(multiply(commutator(X, Y), commutator(Z, Y)), W)
% 6.90/1.26 = { by axiom 5 (commutator_distributes_over_product) R->L }
% 6.90/1.26 multiply(commutator(multiply(X, Z), Y), W)
% 6.90/1.26
% 6.90/1.26 Lemma 29: commutator(multiply(X, Y), commutator(Z, X)) = commutator(Y, commutator(Z, X)).
% 6.90/1.26 Proof:
% 6.90/1.26 commutator(multiply(X, Y), commutator(Z, X))
% 6.90/1.26 = { by axiom 5 (commutator_distributes_over_product) }
% 6.90/1.26 multiply(commutator(X, commutator(Z, X)), commutator(Y, commutator(Z, X)))
% 6.90/1.26 = { by lemma 24 }
% 6.90/1.26 multiply(commutator(W, W), commutator(Y, commutator(Z, X)))
% 6.90/1.26 = { by lemma 14 }
% 6.90/1.26 commutator(Y, commutator(Z, X))
% 6.90/1.26
% 6.90/1.26 Lemma 30: multiply(commutator(X, Y), commutator(Y, X)) = commutator(Z, Z).
% 6.90/1.26 Proof:
% 6.90/1.26 multiply(commutator(X, Y), commutator(Y, X))
% 6.90/1.26 = { by lemma 20 R->L }
% 6.90/1.26 fresh(multiply(X, multiply(Y, multiply(commutator(X, Y), commutator(Y, X)))), multiply(X, Y), multiply(commutator(X, Y), commutator(Y, X)), commutator(Z, Z))
% 6.90/1.26 = { by axiom 4 (commutator) }
% 6.90/1.26 fresh(multiply(X, multiply(Y, multiply(commutator(Y, X), multiply(commutator(X, Y), commutator(commutator(X, Y), commutator(Y, X)))))), multiply(X, Y), multiply(commutator(X, Y), commutator(Y, X)), commutator(Z, Z))
% 6.90/1.26 = { by lemma 21 }
% 6.90/1.26 fresh(multiply(Y, multiply(X, multiply(commutator(X, Y), commutator(commutator(X, Y), commutator(Y, X))))), multiply(X, Y), multiply(commutator(X, Y), commutator(Y, X)), commutator(Z, Z))
% 6.90/1.26 = { by lemma 21 }
% 6.90/1.26 fresh(multiply(X, multiply(Y, commutator(commutator(X, Y), commutator(Y, X)))), multiply(X, Y), multiply(commutator(X, Y), commutator(Y, X)), commutator(Z, Z))
% 6.90/1.26 = { by lemma 14 R->L }
% 6.90/1.26 fresh(multiply(X, multiply(Y, multiply(commutator(W, W), commutator(commutator(X, Y), commutator(Y, X))))), multiply(X, Y), multiply(commutator(X, Y), commutator(Y, X)), commutator(Z, Z))
% 6.90/1.26 = { by lemma 27 R->L }
% 6.90/1.26 fresh(multiply(X, multiply(Y, multiply(commutator(Y, commutator(Y, X)), commutator(commutator(X, Y), commutator(Y, X))))), multiply(X, Y), multiply(commutator(X, Y), commutator(Y, X)), commutator(Z, Z))
% 6.90/1.26 = { by lemma 29 R->L }
% 6.90/1.26 fresh(multiply(X, multiply(Y, multiply(commutator(Y, commutator(Y, X)), commutator(multiply(X, commutator(X, Y)), commutator(Y, X))))), multiply(X, Y), multiply(commutator(X, Y), commutator(Y, X)), commutator(Z, Z))
% 6.90/1.26 = { by axiom 5 (commutator_distributes_over_product) R->L }
% 6.90/1.26 fresh(multiply(X, multiply(Y, commutator(multiply(Y, multiply(X, commutator(X, Y))), commutator(Y, X)))), multiply(X, Y), multiply(commutator(X, Y), commutator(Y, X)), commutator(Z, Z))
% 6.90/1.26 = { by axiom 4 (commutator) R->L }
% 6.90/1.26 fresh(multiply(X, multiply(Y, commutator(multiply(X, Y), commutator(Y, X)))), multiply(X, Y), multiply(commutator(X, Y), commutator(Y, X)), commutator(Z, Z))
% 6.90/1.26 = { by lemma 29 }
% 6.90/1.26 fresh(multiply(X, multiply(Y, commutator(Y, commutator(Y, X)))), multiply(X, Y), multiply(commutator(X, Y), commutator(Y, X)), commutator(Z, Z))
% 6.90/1.26 = { by lemma 27 }
% 6.90/1.26 fresh(multiply(X, multiply(Y, commutator(V, V))), multiply(X, Y), multiply(commutator(X, Y), commutator(Y, X)), commutator(Z, Z))
% 6.90/1.26 = { by lemma 13 }
% 6.90/1.26 fresh(multiply(X, Y), multiply(X, Y), multiply(commutator(X, Y), commutator(Y, X)), commutator(Z, Z))
% 6.90/1.26 = { by axiom 1 (left_cancellation) }
% 6.90/1.26 commutator(Z, Z)
% 6.90/1.26
% 6.90/1.26 Lemma 31: multiply(commutator(multiply(X, Y), Z), commutator(Z, Y)) = commutator(X, Z).
% 6.90/1.26 Proof:
% 6.90/1.26 multiply(commutator(multiply(X, Y), Z), commutator(Z, Y))
% 6.90/1.26 = { by lemma 28 R->L }
% 6.90/1.26 multiply(commutator(X, Z), multiply(commutator(Y, Z), commutator(Z, Y)))
% 6.90/1.26 = { by lemma 30 }
% 6.90/1.26 multiply(commutator(X, Z), commutator(W, W))
% 6.90/1.26 = { by lemma 13 }
% 6.90/1.26 commutator(X, Z)
% 6.90/1.26
% 6.90/1.26 Lemma 32: commutator(commutator(X, Y), Z) = commutator(Z, commutator(Y, X)).
% 6.90/1.26 Proof:
% 6.90/1.26 commutator(commutator(X, Y), Z)
% 6.90/1.26 = { by lemma 31 R->L }
% 6.90/1.26 multiply(commutator(multiply(commutator(X, Y), commutator(Y, X)), Z), commutator(Z, commutator(Y, X)))
% 6.90/1.26 = { by lemma 30 }
% 6.90/1.26 multiply(commutator(commutator(Z, Z), Z), commutator(Z, commutator(Y, X)))
% 6.90/1.26 = { by lemma 26 }
% 6.90/1.26 multiply(commutator(W, W), commutator(Z, commutator(Y, X)))
% 6.90/1.26 = { by lemma 14 }
% 6.90/1.26 commutator(Z, commutator(Y, X))
% 6.90/1.26
% 6.90/1.26 Lemma 33: commutator(X, commutator(multiply(X, Y), Z)) = commutator(Y, commutator(Z, X)).
% 6.90/1.26 Proof:
% 6.90/1.26 commutator(X, commutator(multiply(X, Y), Z))
% 6.90/1.26 = { by lemma 13 R->L }
% 6.90/1.26 commutator(X, multiply(commutator(multiply(X, Y), Z), commutator(W, W)))
% 6.90/1.26 = { by lemma 28 R->L }
% 6.90/1.26 commutator(X, multiply(commutator(X, Z), multiply(commutator(Y, Z), commutator(W, W))))
% 6.90/1.26 = { by lemma 30 R->L }
% 6.90/1.26 commutator(X, multiply(commutator(X, Z), multiply(commutator(Y, Z), multiply(commutator(commutator(X, Z), multiply(commutator(Y, Z), X)), commutator(multiply(commutator(Y, Z), X), commutator(X, Z))))))
% 6.90/1.26 = { by lemma 14 R->L }
% 6.90/1.26 commutator(X, multiply(commutator(X, Z), multiply(commutator(Y, Z), multiply(commutator(multiply(commutator(V, V), commutator(X, Z)), multiply(commutator(Y, Z), X)), commutator(multiply(commutator(Y, Z), X), commutator(X, Z))))))
% 6.90/1.26 = { by lemma 24 R->L }
% 6.90/1.26 commutator(X, multiply(commutator(X, Z), multiply(commutator(Y, Z), multiply(commutator(multiply(commutator(Z, commutator(Y, Z)), commutator(X, Z)), multiply(commutator(Y, Z), X)), commutator(multiply(commutator(Y, Z), X), commutator(X, Z))))))
% 6.90/1.26 = { by lemma 25 R->L }
% 6.90/1.26 commutator(X, multiply(commutator(X, Z), multiply(commutator(Y, Z), multiply(commutator(multiply(commutator(commutator(Y, Z), Z), commutator(X, Z)), multiply(commutator(Y, Z), X)), commutator(multiply(commutator(Y, Z), X), commutator(X, Z))))))
% 6.90/1.26 = { by axiom 5 (commutator_distributes_over_product) R->L }
% 6.90/1.26 commutator(X, multiply(commutator(X, Z), multiply(commutator(Y, Z), multiply(commutator(commutator(multiply(commutator(Y, Z), X), Z), multiply(commutator(Y, Z), X)), commutator(multiply(commutator(Y, Z), X), commutator(X, Z))))))
% 6.90/1.26 = { by lemma 26 }
% 6.90/1.26 commutator(X, multiply(commutator(X, Z), multiply(commutator(Y, Z), multiply(commutator(U, U), commutator(multiply(commutator(Y, Z), X), commutator(X, Z))))))
% 6.90/1.26 = { by lemma 14 }
% 6.90/1.26 commutator(X, multiply(commutator(X, Z), multiply(commutator(Y, Z), commutator(multiply(commutator(Y, Z), X), commutator(X, Z)))))
% 6.90/1.26 = { by axiom 5 (commutator_distributes_over_product) }
% 6.90/1.26 commutator(X, multiply(commutator(X, Z), multiply(commutator(Y, Z), multiply(commutator(commutator(Y, Z), commutator(X, Z)), commutator(X, commutator(X, Z))))))
% 6.90/1.26 = { by lemma 27 }
% 6.90/1.26 commutator(X, multiply(commutator(X, Z), multiply(commutator(Y, Z), multiply(commutator(commutator(Y, Z), commutator(X, Z)), commutator(T, T)))))
% 6.90/1.26 = { by lemma 13 }
% 6.90/1.26 commutator(X, multiply(commutator(X, Z), multiply(commutator(Y, Z), commutator(commutator(Y, Z), commutator(X, Z)))))
% 6.90/1.26 = { by axiom 4 (commutator) R->L }
% 6.90/1.26 commutator(X, multiply(commutator(Y, Z), commutator(X, Z)))
% 6.90/1.26 = { by axiom 5 (commutator_distributes_over_product) R->L }
% 6.90/1.26 commutator(X, commutator(multiply(Y, X), Z))
% 6.90/1.26 = { by lemma 32 R->L }
% 6.90/1.26 commutator(commutator(Z, multiply(Y, X)), X)
% 6.90/1.26 = { by lemma 14 R->L }
% 6.90/1.26 multiply(commutator(S, S), commutator(commutator(Z, multiply(Y, X)), X))
% 6.90/1.26 = { by lemma 24 R->L }
% 6.90/1.26 multiply(commutator(multiply(Y, X), commutator(Z, multiply(Y, X))), commutator(commutator(Z, multiply(Y, X)), X))
% 6.90/1.26 = { by lemma 31 }
% 6.90/1.26 commutator(Y, commutator(Z, multiply(Y, X)))
% 6.90/1.26 = { by lemma 32 R->L }
% 6.90/1.26 commutator(commutator(multiply(Y, X), Z), Y)
% 6.90/1.26 = { by axiom 5 (commutator_distributes_over_product) }
% 6.90/1.26 commutator(multiply(commutator(Y, Z), commutator(X, Z)), Y)
% 6.90/1.26 = { by axiom 5 (commutator_distributes_over_product) }
% 6.90/1.26 multiply(commutator(commutator(Y, Z), Y), commutator(commutator(X, Z), Y))
% 6.90/1.26 = { by lemma 26 }
% 6.90/1.26 multiply(commutator(X2, X2), commutator(commutator(X, Z), Y))
% 6.90/1.26 = { by lemma 14 }
% 6.90/1.26 commutator(commutator(X, Z), Y)
% 6.90/1.26 = { by lemma 32 }
% 6.90/1.26 commutator(Y, commutator(Z, X))
% 6.90/1.26
% 6.90/1.26 Lemma 34: commutator(X, commutator(Y, Z)) = commutator(Z, commutator(X, Y)).
% 6.90/1.26 Proof:
% 6.90/1.26 commutator(X, commutator(Y, Z))
% 6.90/1.26 = { by lemma 33 R->L }
% 6.90/1.26 commutator(Z, commutator(multiply(Z, X), Y))
% 6.90/1.26 = { by lemma 22 R->L }
% 6.90/1.26 commutator(Z, commutator(multiply(Y, multiply(Z, X)), Y))
% 6.90/1.26 = { by axiom 3 (associativity_of_multiply) R->L }
% 6.90/1.26 commutator(Z, commutator(multiply(multiply(Y, Z), X), Y))
% 6.90/1.26 = { by lemma 33 R->L }
% 6.90/1.26 commutator(Y, commutator(multiply(Y, Z), multiply(multiply(Y, Z), X)))
% 6.90/1.26 = { by lemma 31 R->L }
% 6.90/1.26 commutator(Y, multiply(commutator(multiply(multiply(Y, Z), X), multiply(multiply(Y, Z), X)), commutator(multiply(multiply(Y, Z), X), X)))
% 6.90/1.26 = { by lemma 14 }
% 6.90/1.26 commutator(Y, commutator(multiply(multiply(Y, Z), X), X))
% 6.90/1.26 = { by lemma 16 }
% 6.90/1.26 commutator(Y, commutator(multiply(Y, Z), X))
% 6.90/1.26 = { by lemma 33 }
% 6.90/1.26 commutator(Z, commutator(X, Y))
% 6.90/1.26
% 6.90/1.26 Lemma 35: multiply(commutator(X, Y), multiply(commutator(Y, X), Z)) = Z.
% 6.90/1.26 Proof:
% 6.90/1.26 multiply(commutator(X, Y), multiply(commutator(Y, X), Z))
% 6.90/1.26 = { by axiom 3 (associativity_of_multiply) R->L }
% 6.90/1.26 multiply(multiply(commutator(X, Y), commutator(Y, X)), Z)
% 6.90/1.26 = { by lemma 30 }
% 6.90/1.26 multiply(commutator(W, W), Z)
% 6.90/1.26 = { by lemma 14 }
% 6.90/1.26 Z
% 6.90/1.26
% 6.90/1.26 Goal 1 (prove_nilpotency): multiply(commutator(a, b), c) = multiply(c, commutator(a, b)).
% 6.90/1.26 Proof:
% 6.90/1.26 multiply(commutator(a, b), c)
% 6.90/1.26 = { by lemma 13 R->L }
% 6.90/1.26 multiply(commutator(a, b), multiply(c, commutator(X, X)))
% 6.90/1.26 = { by lemma 30 R->L }
% 6.90/1.26 multiply(commutator(a, b), multiply(c, multiply(commutator(c, multiply(a, commutator(a, b))), commutator(multiply(a, commutator(a, b)), c))))
% 6.90/1.26 = { by lemma 13 R->L }
% 6.90/1.26 multiply(commutator(a, b), multiply(c, multiply(commutator(c, multiply(a, commutator(a, b))), multiply(commutator(multiply(a, commutator(a, b)), c), commutator(Y, Y)))))
% 6.90/1.26 = { by lemma 27 R->L }
% 6.90/1.26 multiply(commutator(a, b), multiply(c, multiply(commutator(c, multiply(a, commutator(a, b))), multiply(commutator(multiply(a, commutator(a, b)), c), commutator(b, commutator(b, b))))))
% 6.90/1.26 = { by lemma 24 R->L }
% 6.90/1.26 multiply(commutator(a, b), multiply(c, multiply(commutator(c, multiply(a, commutator(a, b))), multiply(commutator(multiply(a, commutator(a, b)), c), commutator(b, commutator(c, commutator(multiply(a, commutator(a, b)), c)))))))
% 6.90/1.26 = { by lemma 25 R->L }
% 6.90/1.26 multiply(commutator(a, b), multiply(c, multiply(commutator(c, multiply(a, commutator(a, b))), multiply(commutator(multiply(a, commutator(a, b)), c), commutator(b, commutator(commutator(multiply(a, commutator(a, b)), c), c))))))
% 6.90/1.26 = { by lemma 34 }
% 6.90/1.26 multiply(commutator(a, b), multiply(c, multiply(commutator(c, multiply(a, commutator(a, b))), multiply(commutator(multiply(a, commutator(a, b)), c), commutator(c, commutator(b, commutator(multiply(a, commutator(a, b)), c)))))))
% 6.90/1.26 = { by lemma 34 }
% 6.90/1.26 multiply(commutator(a, b), multiply(c, multiply(commutator(c, multiply(a, commutator(a, b))), multiply(commutator(multiply(a, commutator(a, b)), c), commutator(commutator(multiply(a, commutator(a, b)), c), commutator(c, b))))))
% 6.90/1.26 = { by lemma 35 R->L }
% 6.90/1.26 multiply(commutator(a, b), multiply(c, multiply(commutator(c, multiply(a, commutator(a, b))), multiply(commutator(b, c), multiply(commutator(c, b), multiply(commutator(multiply(a, commutator(a, b)), c), commutator(commutator(multiply(a, commutator(a, b)), c), commutator(c, b))))))))
% 6.90/1.26 = { by axiom 4 (commutator) R->L }
% 6.90/1.26 multiply(commutator(a, b), multiply(c, multiply(commutator(c, multiply(a, commutator(a, b))), multiply(commutator(b, c), multiply(commutator(multiply(a, commutator(a, b)), c), commutator(c, b))))))
% 6.90/1.26 = { by lemma 28 }
% 6.90/1.26 multiply(commutator(a, b), multiply(c, multiply(commutator(c, multiply(a, commutator(a, b))), multiply(commutator(multiply(b, multiply(a, commutator(a, b))), c), commutator(c, b)))))
% 6.90/1.26 = { by axiom 4 (commutator) R->L }
% 6.90/1.26 multiply(commutator(a, b), multiply(c, multiply(commutator(c, multiply(a, commutator(a, b))), multiply(commutator(multiply(a, b), c), commutator(c, b)))))
% 6.90/1.27 = { by lemma 31 }
% 6.90/1.27 multiply(commutator(a, b), multiply(c, multiply(commutator(c, multiply(a, commutator(a, b))), commutator(a, c))))
% 6.90/1.27 = { by lemma 31 R->L }
% 6.90/1.27 multiply(commutator(a, b), multiply(c, multiply(commutator(c, multiply(a, commutator(a, b))), multiply(commutator(multiply(a, commutator(a, b)), c), commutator(c, commutator(a, b))))))
% 6.90/1.27 = { by lemma 35 }
% 6.90/1.27 multiply(commutator(a, b), multiply(c, commutator(c, commutator(a, b))))
% 6.90/1.27 = { by axiom 4 (commutator) R->L }
% 6.90/1.27 multiply(c, commutator(a, b))
% 6.90/1.27 % SZS output end Proof
% 6.90/1.27
% 6.90/1.27 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------