TSTP Solution File: GRP667+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP667+1 : TPTP v8.1.2. Released v4.0.0.
% 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 01:19:35 EDT 2023
% Result : Theorem 65.31s 8.84s
% Output : Proof 65.65s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : GRP667+1 : TPTP v8.1.2. Released v4.0.0.
% 0.07/0.14 % 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
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Mon Aug 28 21:35:09 EDT 2023
% 0.12/0.34 % CPUTime :
% 65.31/8.84 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 65.31/8.84
% 65.31/8.84 % SZS status Theorem
% 65.31/8.84
% 65.65/8.90 % SZS output start Proof
% 65.65/8.90 Take the following subset of the input axioms:
% 65.65/8.90 fof(f01, axiom, ![B, A]: mult(A, ld(A, B))=B).
% 65.65/8.90 fof(f02, axiom, ![A2, B2]: ld(A2, mult(A2, B2))=B2).
% 65.65/8.90 fof(f03, axiom, ![A2, B2]: mult(rd(A2, B2), B2)=A2).
% 65.65/8.90 fof(f04, axiom, ![A2, B2]: rd(mult(A2, B2), B2)=A2).
% 65.65/8.90 fof(f05, axiom, ![A2]: mult(A2, unit)=A2).
% 65.65/8.90 fof(f06, axiom, ![A2]: mult(unit, A2)=A2).
% 65.65/8.90 fof(f07, axiom, ![C, A2, B2]: mult(mult(A2, B2), mult(mult(C, B2), C))=mult(mult(A2, mult(mult(B2, C), B2)), C)).
% 65.65/8.90 fof(f08, axiom, ![A2, B2]: mult(mult(A2, B2), A2)=mult(A2, mult(B2, A2))).
% 65.65/8.90 fof(f09, axiom, ![A2]: mult(f(A2), f(A2))=A2).
% 65.65/8.90 fof(f10, axiom, ![X0, X1, X2]: (mult(X0, mult(X1, mult(X2, X1)))=mult(mult(mult(X0, X1), X2), X1) => mult(X1, mult(X0, mult(X1, X2)))=mult(mult(mult(X1, X0), X1), X2))).
% 65.65/8.90 fof(f12, axiom, ![X6, X7, X8]: (mult(mult(X6, X7), mult(X8, X6))=mult(X6, mult(mult(X7, X8), X6)) => mult(X6, mult(X7, mult(X6, X8)))=mult(mult(mult(X6, X7), X6), X8))).
% 65.65/8.90 fof(goals, conjecture, mult(a, mult(b, mult(a, c)))=mult(mult(mult(a, b), a), c)).
% 65.65/8.91
% 65.65/8.91 Now clausify the problem and encode Horn clauses using encoding 3 of
% 65.65/8.91 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 65.65/8.91 We repeatedly replace C & s=t => u=v by the two clauses:
% 65.65/8.91 fresh(y, y, x1...xn) = u
% 65.65/8.91 C => fresh(s, t, x1...xn) = v
% 65.65/8.91 where fresh is a fresh function symbol and x1..xn are the free
% 65.65/8.91 variables of u and v.
% 65.65/8.91 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 65.65/8.91 input problem has no model of domain size 1).
% 65.65/8.91
% 65.65/8.91 The encoding turns the above axioms into the following unit equations and goals:
% 65.65/8.91
% 65.65/8.91 Axiom 1 (f05): mult(X, unit) = X.
% 65.65/8.91 Axiom 2 (f06): mult(unit, X) = X.
% 65.65/8.91 Axiom 3 (f02): ld(X, mult(X, Y)) = Y.
% 65.65/8.91 Axiom 4 (f04): rd(mult(X, Y), Y) = X.
% 65.65/8.91 Axiom 5 (f01): mult(X, ld(X, Y)) = Y.
% 65.65/8.91 Axiom 6 (f09): mult(f(X), f(X)) = X.
% 65.65/8.91 Axiom 7 (f03): mult(rd(X, Y), Y) = X.
% 65.65/8.91 Axiom 8 (f08): mult(mult(X, Y), X) = mult(X, mult(Y, X)).
% 65.65/8.91 Axiom 9 (f12): fresh(X, X, Y, Z, W) = mult(mult(mult(Y, Z), Y), W).
% 65.65/8.91 Axiom 10 (f10): fresh2(X, X, Y, Z, W) = mult(mult(mult(Z, Y), Z), W).
% 65.65/8.91 Axiom 11 (f07): mult(mult(X, Y), mult(mult(Z, Y), Z)) = mult(mult(X, mult(mult(Y, Z), Y)), Z).
% 65.65/8.91 Axiom 12 (f12): fresh(mult(mult(X, Y), mult(Z, X)), mult(X, mult(mult(Y, Z), X)), X, Y, Z) = mult(X, mult(Y, mult(X, Z))).
% 65.65/8.91 Axiom 13 (f10): fresh2(mult(X, mult(Y, mult(Z, Y))), mult(mult(mult(X, Y), Z), Y), X, Y, Z) = mult(Y, mult(X, mult(Y, Z))).
% 65.65/8.91
% 65.65/8.91 Lemma 14: ld(rd(X, Y), X) = Y.
% 65.65/8.91 Proof:
% 65.65/8.91 ld(rd(X, Y), X)
% 65.65/8.91 = { by axiom 7 (f03) R->L }
% 65.65/8.91 ld(rd(X, Y), mult(rd(X, Y), Y))
% 65.65/8.91 = { by axiom 3 (f02) }
% 65.65/8.91 Y
% 65.65/8.91
% 65.65/8.91 Lemma 15: rd(X, ld(Y, X)) = Y.
% 65.65/8.91 Proof:
% 65.65/8.91 rd(X, ld(Y, X))
% 65.65/8.91 = { by axiom 5 (f01) R->L }
% 65.65/8.91 rd(mult(Y, ld(Y, X)), ld(Y, X))
% 65.65/8.91 = { by axiom 4 (f04) }
% 65.65/8.91 Y
% 65.65/8.91
% 65.65/8.91 Lemma 16: mult(mult(X, mult(Y, mult(Z, Y))), Z) = mult(mult(X, Y), mult(Z, mult(Y, Z))).
% 65.65/8.91 Proof:
% 65.65/8.91 mult(mult(X, mult(Y, mult(Z, Y))), Z)
% 65.65/8.91 = { by axiom 8 (f08) R->L }
% 65.65/8.91 mult(mult(X, mult(mult(Y, Z), Y)), Z)
% 65.65/8.91 = { by axiom 11 (f07) R->L }
% 65.65/8.91 mult(mult(X, Y), mult(mult(Z, Y), Z))
% 65.65/8.91 = { by axiom 8 (f08) }
% 65.65/8.91 mult(mult(X, Y), mult(Z, mult(Y, Z)))
% 65.65/8.91
% 65.65/8.91 Lemma 17: rd(mult(X, mult(Y, X)), X) = mult(X, Y).
% 65.65/8.91 Proof:
% 65.65/8.91 rd(mult(X, mult(Y, X)), X)
% 65.65/8.91 = { by axiom 8 (f08) R->L }
% 65.65/8.91 rd(mult(mult(X, Y), X), X)
% 65.65/8.91 = { by axiom 4 (f04) }
% 65.65/8.91 mult(X, Y)
% 65.65/8.91
% 65.65/8.91 Lemma 18: mult(rd(unit, X), mult(X, Y)) = Y.
% 65.65/8.91 Proof:
% 65.65/8.91 mult(rd(unit, X), mult(X, Y))
% 65.65/8.91 = { by axiom 7 (f03) R->L }
% 65.65/8.91 mult(rd(unit, X), mult(X, mult(rd(Y, X), X)))
% 65.65/8.91 = { by axiom 4 (f04) R->L }
% 65.65/8.91 rd(mult(mult(rd(unit, X), mult(X, mult(rd(Y, X), X))), rd(Y, X)), rd(Y, X))
% 65.65/8.91 = { by lemma 16 }
% 65.65/8.91 rd(mult(mult(rd(unit, X), X), mult(rd(Y, X), mult(X, rd(Y, X)))), rd(Y, X))
% 65.65/8.91 = { by axiom 7 (f03) }
% 65.65/8.91 rd(mult(unit, mult(rd(Y, X), mult(X, rd(Y, X)))), rd(Y, X))
% 65.65/8.91 = { by axiom 2 (f06) }
% 65.65/8.91 rd(mult(rd(Y, X), mult(X, rd(Y, X))), rd(Y, X))
% 65.65/8.91 = { by lemma 17 }
% 65.65/8.91 mult(rd(Y, X), X)
% 65.65/8.91 = { by axiom 7 (f03) }
% 65.65/8.91 Y
% 65.65/8.91
% 65.65/8.91 Lemma 19: rd(X, mult(Y, X)) = rd(unit, Y).
% 65.65/8.91 Proof:
% 65.65/8.91 rd(X, mult(Y, X))
% 65.65/8.91 = { by lemma 18 R->L }
% 65.65/8.91 rd(mult(rd(unit, Y), mult(Y, X)), mult(Y, X))
% 65.65/8.91 = { by axiom 4 (f04) }
% 65.65/8.91 rd(unit, Y)
% 65.65/8.91
% 65.65/8.91 Lemma 20: mult(mult(X, mult(Y, Z)), ld(Z, Y)) = mult(mult(X, Z), mult(ld(Z, Y), Y)).
% 65.65/8.91 Proof:
% 65.65/8.91 mult(mult(X, mult(Y, Z)), ld(Z, Y))
% 65.65/8.91 = { by axiom 5 (f01) R->L }
% 65.65/8.91 mult(mult(X, mult(mult(Z, ld(Z, Y)), Z)), ld(Z, Y))
% 65.65/8.91 = { by axiom 8 (f08) }
% 65.65/8.91 mult(mult(X, mult(Z, mult(ld(Z, Y), Z))), ld(Z, Y))
% 65.65/8.91 = { by lemma 16 }
% 65.65/8.91 mult(mult(X, Z), mult(ld(Z, Y), mult(Z, ld(Z, Y))))
% 65.65/8.91 = { by axiom 5 (f01) }
% 65.65/8.91 mult(mult(X, Z), mult(ld(Z, Y), Y))
% 65.65/8.91
% 65.65/8.91 Lemma 21: mult(mult(X, Y), Y) = mult(X, mult(Y, Y)).
% 65.65/8.91 Proof:
% 65.65/8.91 mult(mult(X, Y), Y)
% 65.65/8.91 = { by axiom 1 (f05) R->L }
% 65.65/8.91 mult(mult(X, Y), mult(Y, unit))
% 65.65/8.91 = { by axiom 2 (f06) R->L }
% 65.65/8.91 mult(mult(X, Y), mult(unit, mult(Y, unit)))
% 65.65/8.91 = { by lemma 16 R->L }
% 65.65/8.91 mult(mult(X, mult(Y, mult(unit, Y))), unit)
% 65.65/8.91 = { by axiom 1 (f05) }
% 65.65/8.91 mult(X, mult(Y, mult(unit, Y)))
% 65.65/8.91 = { by axiom 2 (f06) }
% 65.65/8.91 mult(X, mult(Y, Y))
% 65.65/8.91
% 65.65/8.91 Lemma 22: mult(X, rd(X, Y)) = rd(mult(X, X), Y).
% 65.65/8.91 Proof:
% 65.65/8.91 mult(X, rd(X, Y))
% 65.65/8.91 = { by axiom 4 (f04) R->L }
% 65.65/8.91 rd(mult(mult(X, rd(X, Y)), Y), Y)
% 65.65/8.91 = { by lemma 14 R->L }
% 65.65/8.91 rd(mult(mult(X, rd(X, Y)), ld(rd(X, Y), X)), Y)
% 65.65/8.91 = { by lemma 15 R->L }
% 65.65/8.91 rd(mult(mult(X, rd(unit, ld(rd(X, Y), unit))), ld(rd(X, Y), X)), Y)
% 65.65/8.91 = { by lemma 18 R->L }
% 65.65/8.91 rd(mult(mult(X, rd(unit, ld(rd(X, Y), unit))), ld(rd(X, Y), mult(rd(unit, ld(rd(X, Y), unit)), mult(ld(rd(X, Y), unit), X)))), Y)
% 65.65/8.91 = { by lemma 15 }
% 65.65/8.91 rd(mult(mult(X, rd(unit, ld(rd(X, Y), unit))), ld(rd(X, Y), mult(rd(X, Y), mult(ld(rd(X, Y), unit), X)))), Y)
% 65.65/8.91 = { by axiom 3 (f02) }
% 65.65/8.91 rd(mult(mult(X, rd(unit, ld(rd(X, Y), unit))), mult(ld(rd(X, Y), unit), X)), Y)
% 65.65/8.91 = { by lemma 19 R->L }
% 65.65/8.91 rd(mult(mult(X, rd(X, mult(ld(rd(X, Y), unit), X))), mult(ld(rd(X, Y), unit), X)), Y)
% 65.65/8.91 = { by lemma 14 R->L }
% 65.65/8.91 rd(mult(mult(X, rd(X, mult(ld(rd(X, Y), unit), X))), ld(rd(X, mult(ld(rd(X, Y), unit), X)), X)), Y)
% 65.65/8.91 = { by axiom 2 (f06) R->L }
% 65.65/8.91 rd(mult(mult(unit, mult(X, rd(X, mult(ld(rd(X, Y), unit), X)))), ld(rd(X, mult(ld(rd(X, Y), unit), X)), X)), Y)
% 65.65/8.91 = { by lemma 20 }
% 65.65/8.91 rd(mult(mult(unit, rd(X, mult(ld(rd(X, Y), unit), X))), mult(ld(rd(X, mult(ld(rd(X, Y), unit), X)), X), X)), Y)
% 65.65/8.91 = { by axiom 2 (f06) }
% 65.65/8.91 rd(mult(rd(X, mult(ld(rd(X, Y), unit), X)), mult(ld(rd(X, mult(ld(rd(X, Y), unit), X)), X), X)), Y)
% 65.65/8.91 = { by lemma 14 }
% 65.65/8.91 rd(mult(rd(X, mult(ld(rd(X, Y), unit), X)), mult(mult(ld(rd(X, Y), unit), X), X)), Y)
% 65.65/8.91 = { by lemma 19 }
% 65.65/8.91 rd(mult(rd(unit, ld(rd(X, Y), unit)), mult(mult(ld(rd(X, Y), unit), X), X)), Y)
% 65.65/8.91 = { by lemma 21 }
% 65.65/8.91 rd(mult(rd(unit, ld(rd(X, Y), unit)), mult(ld(rd(X, Y), unit), mult(X, X))), Y)
% 65.65/8.91 = { by lemma 18 }
% 65.65/8.91 rd(mult(X, X), Y)
% 65.65/8.91
% 65.65/8.91 Lemma 23: mult(mult(X, rd(unit, Y)), mult(Y, mult(Y, Y))) = mult(X, mult(Y, Y)).
% 65.65/8.91 Proof:
% 65.65/8.91 mult(mult(X, rd(unit, Y)), mult(Y, mult(Y, Y)))
% 65.65/8.91 = { by axiom 2 (f06) R->L }
% 65.65/8.91 mult(mult(X, rd(unit, Y)), mult(Y, mult(unit, mult(Y, Y))))
% 65.65/8.91 = { by axiom 12 (f12) R->L }
% 65.65/8.91 mult(mult(X, rd(unit, Y)), fresh(mult(mult(Y, unit), mult(Y, Y)), mult(Y, mult(mult(unit, Y), Y)), Y, unit, Y))
% 65.65/8.91 = { by axiom 1 (f05) }
% 65.65/8.91 mult(mult(X, rd(unit, Y)), fresh(mult(Y, mult(Y, Y)), mult(Y, mult(mult(unit, Y), Y)), Y, unit, Y))
% 65.65/8.91 = { by axiom 2 (f06) }
% 65.65/8.91 mult(mult(X, rd(unit, Y)), fresh(mult(Y, mult(Y, Y)), mult(Y, mult(Y, Y)), Y, unit, Y))
% 65.65/8.91 = { by axiom 9 (f12) }
% 65.65/8.91 mult(mult(X, rd(unit, Y)), mult(mult(mult(Y, unit), Y), Y))
% 65.65/8.91 = { by axiom 8 (f08) }
% 65.65/8.91 mult(mult(X, rd(unit, Y)), mult(mult(Y, mult(unit, Y)), Y))
% 65.65/8.91 = { by axiom 2 (f06) }
% 65.65/8.91 mult(mult(X, rd(unit, Y)), mult(mult(Y, Y), Y))
% 65.65/8.91 = { by lemma 19 R->L }
% 65.65/8.91 mult(mult(X, rd(Y, mult(Y, Y))), mult(mult(Y, Y), Y))
% 65.65/8.91 = { by lemma 14 R->L }
% 65.65/8.91 mult(mult(X, rd(Y, mult(Y, Y))), mult(ld(rd(Y, mult(Y, Y)), Y), Y))
% 65.65/8.91 = { by lemma 20 R->L }
% 65.65/8.91 mult(mult(X, mult(Y, rd(Y, mult(Y, Y)))), ld(rd(Y, mult(Y, Y)), Y))
% 65.65/8.91 = { by lemma 14 }
% 65.65/8.92 mult(mult(X, mult(Y, rd(Y, mult(Y, Y)))), mult(Y, Y))
% 65.65/8.92 = { by lemma 19 }
% 65.65/8.92 mult(mult(X, mult(Y, rd(unit, Y))), mult(Y, Y))
% 65.65/8.92 = { by lemma 17 R->L }
% 65.65/8.92 mult(mult(X, rd(mult(Y, mult(rd(unit, Y), Y)), Y)), mult(Y, Y))
% 65.65/8.92 = { by axiom 7 (f03) }
% 65.65/8.92 mult(mult(X, rd(mult(Y, unit), Y)), mult(Y, Y))
% 65.65/8.92 = { by axiom 1 (f05) }
% 65.65/8.92 mult(mult(X, rd(Y, Y)), mult(Y, Y))
% 65.65/8.92 = { by axiom 2 (f06) R->L }
% 65.65/8.92 mult(mult(X, rd(mult(unit, Y), Y)), mult(Y, Y))
% 65.65/8.92 = { by axiom 4 (f04) }
% 65.65/8.92 mult(mult(X, unit), mult(Y, Y))
% 65.65/8.92 = { by axiom 1 (f05) }
% 65.65/8.92 mult(X, mult(Y, Y))
% 65.65/8.92
% 65.65/8.92 Lemma 24: mult(mult(rd(X, mult(Y, mult(Z, Y))), Y), mult(Z, mult(Y, Z))) = mult(X, Z).
% 65.65/8.92 Proof:
% 65.65/8.92 mult(mult(rd(X, mult(Y, mult(Z, Y))), Y), mult(Z, mult(Y, Z)))
% 65.65/8.92 = { by lemma 16 R->L }
% 65.65/8.92 mult(mult(rd(X, mult(Y, mult(Z, Y))), mult(Y, mult(Z, Y))), Z)
% 65.65/8.92 = { by axiom 7 (f03) }
% 65.65/8.92 mult(X, Z)
% 65.65/8.92
% 65.65/8.92 Lemma 25: mult(rd(unit, mult(X, Y)), mult(X, mult(Y, X))) = X.
% 65.65/8.92 Proof:
% 65.65/8.92 mult(rd(unit, mult(X, Y)), mult(X, mult(Y, X)))
% 65.65/8.92 = { by axiom 8 (f08) R->L }
% 65.65/8.92 mult(rd(unit, mult(X, Y)), mult(mult(X, Y), X))
% 65.65/8.92 = { by lemma 18 }
% 65.65/8.92 X
% 65.65/8.92
% 65.65/8.92 Lemma 26: mult(rd(unit, mult(X, Y)), X) = rd(unit, Y).
% 65.65/8.92 Proof:
% 65.65/8.92 mult(rd(unit, mult(X, Y)), X)
% 65.65/8.92 = { by axiom 7 (f03) R->L }
% 65.65/8.92 mult(rd(unit, mult(X, mult(rd(Y, X), X))), X)
% 65.65/8.92 = { by axiom 4 (f04) R->L }
% 65.65/8.92 rd(mult(mult(rd(unit, mult(X, mult(rd(Y, X), X))), X), mult(rd(Y, X), mult(X, rd(Y, X)))), mult(rd(Y, X), mult(X, rd(Y, X))))
% 65.65/8.92 = { by lemma 24 }
% 65.65/8.92 rd(mult(unit, rd(Y, X)), mult(rd(Y, X), mult(X, rd(Y, X))))
% 65.65/8.92 = { by axiom 2 (f06) }
% 65.65/8.92 rd(rd(Y, X), mult(rd(Y, X), mult(X, rd(Y, X))))
% 65.65/8.92 = { by lemma 25 R->L }
% 65.65/8.92 rd(mult(rd(unit, mult(rd(Y, X), X)), mult(rd(Y, X), mult(X, rd(Y, X)))), mult(rd(Y, X), mult(X, rd(Y, X))))
% 65.65/8.92 = { by axiom 4 (f04) }
% 65.65/8.92 rd(unit, mult(rd(Y, X), X))
% 65.65/8.92 = { by axiom 7 (f03) }
% 65.65/8.92 rd(unit, Y)
% 65.65/8.92
% 65.65/8.92 Lemma 27: mult(X, rd(unit, Y)) = rd(X, Y).
% 65.65/8.92 Proof:
% 65.65/8.92 mult(X, rd(unit, Y))
% 65.65/8.92 = { by axiom 4 (f04) R->L }
% 65.65/8.92 rd(mult(mult(X, rd(unit, Y)), mult(Y, Y)), mult(Y, Y))
% 65.65/8.92 = { by axiom 4 (f04) R->L }
% 65.65/8.92 rd(rd(mult(mult(mult(X, rd(unit, Y)), mult(Y, Y)), Y), Y), mult(Y, Y))
% 65.65/8.92 = { by lemma 21 R->L }
% 65.65/8.92 rd(rd(mult(mult(mult(mult(X, rd(unit, Y)), Y), Y), Y), Y), mult(Y, Y))
% 65.65/8.92 = { by lemma 21 }
% 65.65/8.92 rd(rd(mult(mult(mult(X, rd(unit, Y)), Y), mult(Y, Y)), Y), mult(Y, Y))
% 65.65/8.92 = { by lemma 23 R->L }
% 65.65/8.92 rd(rd(mult(mult(mult(mult(X, rd(unit, Y)), Y), rd(unit, Y)), mult(Y, mult(Y, Y))), Y), mult(Y, Y))
% 65.65/8.92 = { by lemma 26 R->L }
% 65.65/8.92 rd(rd(mult(mult(mult(mult(X, rd(unit, Y)), Y), mult(rd(unit, mult(mult(X, rd(unit, Y)), Y)), mult(X, rd(unit, Y)))), mult(Y, mult(Y, Y))), Y), mult(Y, Y))
% 65.65/8.92 = { by axiom 4 (f04) R->L }
% 65.65/8.92 rd(rd(mult(mult(rd(mult(mult(mult(X, rd(unit, Y)), Y), mult(Z, mult(mult(X, rd(unit, Y)), Y))), mult(Z, mult(mult(X, rd(unit, Y)), Y))), mult(rd(unit, mult(mult(X, rd(unit, Y)), Y)), mult(X, rd(unit, Y)))), mult(Y, mult(Y, Y))), Y), mult(Y, Y))
% 65.65/8.92 = { by lemma 25 R->L }
% 65.65/8.92 rd(rd(mult(mult(rd(mult(mult(mult(X, rd(unit, Y)), Y), mult(Z, mult(mult(X, rd(unit, Y)), Y))), mult(mult(rd(unit, mult(Z, mult(mult(X, rd(unit, Y)), Y))), mult(Z, mult(mult(mult(X, rd(unit, Y)), Y), Z))), mult(mult(X, rd(unit, Y)), Y))), mult(rd(unit, mult(mult(X, rd(unit, Y)), Y)), mult(X, rd(unit, Y)))), mult(Y, mult(Y, Y))), Y), mult(Y, Y))
% 65.65/8.92 = { by lemma 16 }
% 65.65/8.92 rd(rd(mult(mult(rd(mult(mult(mult(X, rd(unit, Y)), Y), mult(Z, mult(mult(X, rd(unit, Y)), Y))), mult(mult(rd(unit, mult(Z, mult(mult(X, rd(unit, Y)), Y))), Z), mult(mult(mult(X, rd(unit, Y)), Y), mult(Z, mult(mult(X, rd(unit, Y)), Y))))), mult(rd(unit, mult(mult(X, rd(unit, Y)), Y)), mult(X, rd(unit, Y)))), mult(Y, mult(Y, Y))), Y), mult(Y, Y))
% 65.65/8.92 = { by lemma 19 }
% 65.65/8.92 rd(rd(mult(mult(rd(unit, mult(rd(unit, mult(Z, mult(mult(X, rd(unit, Y)), Y))), Z)), mult(rd(unit, mult(mult(X, rd(unit, Y)), Y)), mult(X, rd(unit, Y)))), mult(Y, mult(Y, Y))), Y), mult(Y, Y))
% 65.65/8.92 = { by lemma 26 }
% 65.65/8.92 rd(rd(mult(mult(rd(unit, rd(unit, mult(mult(X, rd(unit, Y)), Y))), mult(rd(unit, mult(mult(X, rd(unit, Y)), Y)), mult(X, rd(unit, Y)))), mult(Y, mult(Y, Y))), Y), mult(Y, Y))
% 65.65/8.92 = { by lemma 18 }
% 65.65/8.92 rd(rd(mult(mult(X, rd(unit, Y)), mult(Y, mult(Y, Y))), Y), mult(Y, Y))
% 65.65/8.92 = { by lemma 23 }
% 65.65/8.92 rd(rd(mult(X, mult(Y, Y)), Y), mult(Y, Y))
% 65.65/8.92 = { by lemma 21 R->L }
% 65.65/8.92 rd(rd(mult(mult(X, Y), Y), Y), mult(Y, Y))
% 65.65/8.92 = { by axiom 4 (f04) }
% 65.65/8.92 rd(mult(X, Y), mult(Y, Y))
% 65.65/8.92 = { by axiom 7 (f03) R->L }
% 65.65/8.92 rd(mult(mult(rd(X, Y), Y), Y), mult(Y, Y))
% 65.65/8.92 = { by lemma 21 }
% 65.65/8.92 rd(mult(rd(X, Y), mult(Y, Y)), mult(Y, Y))
% 65.65/8.92 = { by axiom 4 (f04) }
% 65.65/8.92 rd(X, Y)
% 65.65/8.92
% 65.65/8.92 Lemma 28: mult(mult(X, mult(Y, Z)), rd(Z, Y)) = mult(mult(X, Y), rd(mult(Z, Z), Y)).
% 65.65/8.92 Proof:
% 65.65/8.92 mult(mult(X, mult(Y, Z)), rd(Z, Y))
% 65.65/8.92 = { by lemma 24 R->L }
% 65.65/8.92 mult(mult(rd(mult(X, mult(Y, Z)), mult(Y, mult(rd(Z, Y), Y))), Y), mult(rd(Z, Y), mult(Y, rd(Z, Y))))
% 65.65/8.92 = { by axiom 7 (f03) }
% 65.65/8.92 mult(mult(rd(mult(X, mult(Y, Z)), mult(Y, Z)), Y), mult(rd(Z, Y), mult(Y, rd(Z, Y))))
% 65.65/8.92 = { by axiom 8 (f08) R->L }
% 65.65/8.92 mult(mult(rd(mult(X, mult(Y, Z)), mult(Y, Z)), Y), mult(mult(rd(Z, Y), Y), rd(Z, Y)))
% 65.65/8.92 = { by axiom 7 (f03) }
% 65.65/8.92 mult(mult(rd(mult(X, mult(Y, Z)), mult(Y, Z)), Y), mult(Z, rd(Z, Y)))
% 65.65/8.92 = { by lemma 22 }
% 65.65/8.92 mult(mult(rd(mult(X, mult(Y, Z)), mult(Y, Z)), Y), rd(mult(Z, Z), Y))
% 65.65/8.92 = { by axiom 4 (f04) }
% 65.65/8.92 mult(mult(X, Y), rd(mult(Z, Z), Y))
% 65.65/8.92
% 65.65/8.92 Goal 1 (goals): mult(a, mult(b, mult(a, c))) = mult(mult(mult(a, b), a), c).
% 65.65/8.92 Proof:
% 65.65/8.92 mult(a, mult(b, mult(a, c)))
% 65.65/8.92 = { by axiom 13 (f10) R->L }
% 65.65/8.92 fresh2(mult(b, mult(a, mult(c, a))), mult(mult(mult(b, a), c), a), b, a, c)
% 65.65/8.92 = { by axiom 4 (f04) R->L }
% 65.65/8.92 fresh2(mult(b, mult(a, mult(c, a))), mult(mult(mult(b, a), rd(mult(c, a), a)), a), b, a, c)
% 65.65/8.92 = { by axiom 6 (f09) R->L }
% 65.65/8.92 fresh2(mult(b, mult(a, mult(c, a))), mult(mult(mult(b, a), rd(mult(f(mult(c, a)), f(mult(c, a))), a)), a), b, a, c)
% 65.65/8.92 = { by lemma 28 R->L }
% 65.65/8.92 fresh2(mult(b, mult(a, mult(c, a))), mult(mult(mult(b, mult(a, f(mult(c, a)))), rd(f(mult(c, a)), a)), a), b, a, c)
% 65.65/8.92 = { by lemma 27 R->L }
% 65.65/8.92 fresh2(mult(b, mult(a, mult(c, a))), mult(mult(mult(b, mult(a, f(mult(c, a)))), mult(f(mult(c, a)), rd(unit, a))), a), b, a, c)
% 65.65/8.92 = { by lemma 19 R->L }
% 65.65/8.92 fresh2(mult(b, mult(a, mult(c, a))), mult(mult(mult(b, mult(a, f(mult(c, a)))), mult(f(mult(c, a)), rd(f(mult(c, a)), mult(a, f(mult(c, a)))))), a), b, a, c)
% 65.65/8.93 = { by lemma 22 }
% 65.65/8.93 fresh2(mult(b, mult(a, mult(c, a))), mult(mult(mult(b, mult(a, f(mult(c, a)))), rd(mult(f(mult(c, a)), f(mult(c, a))), mult(a, f(mult(c, a))))), a), b, a, c)
% 65.65/8.93 = { by lemma 28 R->L }
% 65.65/8.93 fresh2(mult(b, mult(a, mult(c, a))), mult(mult(mult(b, mult(mult(a, f(mult(c, a))), f(mult(c, a)))), rd(f(mult(c, a)), mult(a, f(mult(c, a))))), a), b, a, c)
% 65.65/8.93 = { by lemma 21 }
% 65.65/8.93 fresh2(mult(b, mult(a, mult(c, a))), mult(mult(mult(b, mult(a, mult(f(mult(c, a)), f(mult(c, a))))), rd(f(mult(c, a)), mult(a, f(mult(c, a))))), a), b, a, c)
% 65.65/8.93 = { by lemma 19 }
% 65.65/8.93 fresh2(mult(b, mult(a, mult(c, a))), mult(mult(mult(b, mult(a, mult(f(mult(c, a)), f(mult(c, a))))), rd(unit, a)), a), b, a, c)
% 65.65/8.93 = { by lemma 27 }
% 65.65/8.93 fresh2(mult(b, mult(a, mult(c, a))), mult(rd(mult(b, mult(a, mult(f(mult(c, a)), f(mult(c, a))))), a), a), b, a, c)
% 65.65/8.93 = { by axiom 6 (f09) }
% 65.65/8.93 fresh2(mult(b, mult(a, mult(c, a))), mult(rd(mult(b, mult(a, mult(c, a))), a), a), b, a, c)
% 65.65/8.93 = { by axiom 7 (f03) }
% 65.65/8.93 fresh2(mult(b, mult(a, mult(c, a))), mult(b, mult(a, mult(c, a))), b, a, c)
% 65.65/8.93 = { by axiom 10 (f10) }
% 65.65/8.93 mult(mult(mult(a, b), a), c)
% 65.65/8.93 % SZS output end Proof
% 65.65/8.93
% 65.65/8.93 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------