TSTP Solution File: KLE147+1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE147+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 : n028.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 05:36:03 EDT 2023
% Result : Theorem 29.77s 4.41s
% Output : Proof 29.77s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : KLE147+1 : TPTP v8.1.2. Released v4.0.0.
% 0.07/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n028.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Tue Aug 29 11:40:11 EDT 2023
% 0.13/0.34 % CPUTime :
% 29.77/4.41 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 29.77/4.41
% 29.77/4.41 % SZS status Theorem
% 29.77/4.41
% 29.77/4.43 % SZS output start Proof
% 29.77/4.43 Take the following subset of the input axioms:
% 29.77/4.43 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 29.77/4.43 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 29.77/4.43 fof(distributivity1, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 29.77/4.43 fof(distributivity2, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 29.77/4.43 fof(goals, conjecture, ![X0, X1]: strong_iteration(multiplication(star(X0), X1))=multiplication(star(X1), strong_iteration(multiplication(star(X0), X1)))).
% 29.77/4.43 fof(idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 29.77/4.43 fof(infty_coinduction, axiom, ![A2, B2, C2]: (leq(C2, addition(multiplication(A2, C2), B2)) => leq(C2, multiplication(strong_iteration(A2), B2)))).
% 29.77/4.43 fof(infty_unfold1, axiom, ![A3]: strong_iteration(A3)=addition(multiplication(A3, strong_iteration(A3)), one)).
% 29.77/4.43 fof(multiplicative_associativity, axiom, ![A3, B2, C2]: multiplication(A3, multiplication(B2, C2))=multiplication(multiplication(A3, B2), C2)).
% 29.77/4.43 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 29.77/4.43 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 29.77/4.43 fof(order, axiom, ![B2, A2_2]: (leq(A2_2, B2) <=> addition(A2_2, B2)=B2)).
% 29.77/4.43 fof(star_unfold1, axiom, ![A3]: addition(one, multiplication(A3, star(A3)))=star(A3)).
% 29.77/4.43
% 29.77/4.43 Now clausify the problem and encode Horn clauses using encoding 3 of
% 29.77/4.43 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 29.77/4.43 We repeatedly replace C & s=t => u=v by the two clauses:
% 29.77/4.43 fresh(y, y, x1...xn) = u
% 29.77/4.43 C => fresh(s, t, x1...xn) = v
% 29.77/4.43 where fresh is a fresh function symbol and x1..xn are the free
% 29.77/4.43 variables of u and v.
% 29.77/4.43 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 29.77/4.43 input problem has no model of domain size 1).
% 29.77/4.43
% 29.77/4.43 The encoding turns the above axioms into the following unit equations and goals:
% 29.77/4.43
% 29.77/4.43 Axiom 1 (multiplicative_right_identity): multiplication(X, one) = X.
% 29.77/4.43 Axiom 2 (multiplicative_left_identity): multiplication(one, X) = X.
% 29.77/4.43 Axiom 3 (idempotence): addition(X, X) = X.
% 29.77/4.43 Axiom 4 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 29.77/4.43 Axiom 5 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 29.77/4.43 Axiom 6 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 29.77/4.43 Axiom 7 (order_1): fresh(X, X, Y, Z) = Z.
% 29.77/4.43 Axiom 8 (order): fresh5(X, X, Y, Z) = true.
% 29.77/4.43 Axiom 9 (star_unfold1): addition(one, multiplication(X, star(X))) = star(X).
% 29.77/4.43 Axiom 10 (infty_unfold1): strong_iteration(X) = addition(multiplication(X, strong_iteration(X)), one).
% 29.77/4.43 Axiom 11 (infty_coinduction): fresh4(X, X, Y, Z, W) = true.
% 29.77/4.43 Axiom 12 (distributivity1): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 29.77/4.43 Axiom 13 (distributivity2): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 29.77/4.43 Axiom 14 (order_1): fresh(leq(X, Y), true, X, Y) = addition(X, Y).
% 29.77/4.43 Axiom 15 (order): fresh5(addition(X, Y), Y, X, Y) = leq(X, Y).
% 29.77/4.43 Axiom 16 (infty_coinduction): fresh4(leq(X, addition(multiplication(Y, X), Z)), true, Y, Z, X) = leq(X, multiplication(strong_iteration(Y), Z)).
% 29.77/4.43
% 29.77/4.43 Lemma 17: addition(X, addition(X, Y)) = addition(X, Y).
% 29.77/4.43 Proof:
% 29.77/4.43 addition(X, addition(X, Y))
% 29.77/4.43 = { by axiom 6 (additive_associativity) }
% 29.77/4.43 addition(addition(X, X), Y)
% 29.77/4.43 = { by axiom 3 (idempotence) }
% 29.77/4.43 addition(X, Y)
% 29.77/4.43
% 29.77/4.43 Lemma 18: addition(one, star(X)) = star(X).
% 29.77/4.43 Proof:
% 29.77/4.43 addition(one, star(X))
% 29.77/4.43 = { by axiom 9 (star_unfold1) R->L }
% 29.77/4.43 addition(one, addition(one, multiplication(X, star(X))))
% 29.77/4.43 = { by lemma 17 }
% 29.77/4.43 addition(one, multiplication(X, star(X)))
% 29.77/4.43 = { by axiom 9 (star_unfold1) }
% 29.77/4.43 star(X)
% 29.77/4.43
% 29.77/4.43 Lemma 19: addition(X, multiplication(Y, X)) = multiplication(addition(Y, one), X).
% 29.77/4.43 Proof:
% 29.77/4.43 addition(X, multiplication(Y, X))
% 29.77/4.43 = { by axiom 2 (multiplicative_left_identity) R->L }
% 29.77/4.43 addition(multiplication(one, X), multiplication(Y, X))
% 29.77/4.43 = { by axiom 13 (distributivity2) R->L }
% 29.77/4.43 multiplication(addition(one, Y), X)
% 29.77/4.43 = { by axiom 4 (additive_commutativity) }
% 29.77/4.43 multiplication(addition(Y, one), X)
% 29.77/4.43
% 29.77/4.43 Lemma 20: addition(multiplication(X, Y), multiplication(Z, Y)) = multiplication(addition(Z, X), Y).
% 29.77/4.43 Proof:
% 29.77/4.43 addition(multiplication(X, Y), multiplication(Z, Y))
% 29.77/4.43 = { by axiom 13 (distributivity2) R->L }
% 29.77/4.43 multiplication(addition(X, Z), Y)
% 29.77/4.43 = { by axiom 4 (additive_commutativity) }
% 29.77/4.43 multiplication(addition(Z, X), Y)
% 29.77/4.43
% 29.77/4.43 Lemma 21: addition(multiplication(star(X), Y), Y) = multiplication(star(X), Y).
% 29.77/4.43 Proof:
% 29.77/4.43 addition(multiplication(star(X), Y), Y)
% 29.77/4.43 = { by axiom 2 (multiplicative_left_identity) R->L }
% 29.77/4.43 addition(multiplication(star(X), Y), multiplication(one, Y))
% 29.77/4.43 = { by lemma 20 }
% 29.77/4.43 multiplication(addition(one, star(X)), Y)
% 29.77/4.43 = { by lemma 18 }
% 29.77/4.44 multiplication(star(X), Y)
% 29.77/4.44
% 29.77/4.44 Goal 1 (goals): strong_iteration(multiplication(star(x0), x1)) = multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))).
% 29.77/4.44 Proof:
% 29.77/4.44 strong_iteration(multiplication(star(x0), x1))
% 29.77/4.44 = { by axiom 7 (order_1) R->L }
% 29.77/4.44 fresh(true, true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.44 = { by axiom 11 (infty_coinduction) R->L }
% 29.77/4.44 fresh(fresh4(true, true, multiplication(star(x0), x1), one, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.44 = { by axiom 8 (order) R->L }
% 29.77/4.44 fresh(fresh4(fresh5(addition(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), multiplication(multiplication(star(x0), x1), multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))))), addition(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), multiplication(multiplication(star(x0), x1), multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))))), multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), addition(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), multiplication(multiplication(star(x0), x1), multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))))), true, multiplication(star(x0), x1), one, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.44 = { by lemma 17 R->L }
% 29.77/4.44 fresh(fresh4(fresh5(addition(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), addition(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), multiplication(multiplication(star(x0), x1), multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))))), addition(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), multiplication(multiplication(star(x0), x1), multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))))), multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), addition(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), multiplication(multiplication(star(x0), x1), multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))))), true, multiplication(star(x0), x1), one, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.44 = { by axiom 15 (order) }
% 29.77/4.44 fresh(fresh4(leq(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), addition(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), multiplication(multiplication(star(x0), x1), multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))))), true, multiplication(star(x0), x1), one, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.44 = { by lemma 19 }
% 29.77/4.44 fresh(fresh4(leq(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), multiplication(addition(multiplication(star(x0), x1), one), multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))))), true, multiplication(star(x0), x1), one, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.44 = { by axiom 4 (additive_commutativity) }
% 29.77/4.44 fresh(fresh4(leq(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), multiplication(addition(one, multiplication(star(x0), x1)), multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))))), true, multiplication(star(x0), x1), one, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.44 = { by axiom 5 (multiplicative_associativity) }
% 29.77/4.44 fresh(fresh4(leq(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), multiplication(multiplication(addition(one, multiplication(star(x0), x1)), star(x1)), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x0), x1), one, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.44 = { by axiom 4 (additive_commutativity) R->L }
% 29.77/4.44 fresh(fresh4(leq(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), multiplication(multiplication(addition(multiplication(star(x0), x1), one), star(x1)), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x0), x1), one, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.44 = { by lemma 19 R->L }
% 29.77/4.44 fresh(fresh4(leq(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), multiplication(addition(star(x1), multiplication(multiplication(star(x0), x1), star(x1))), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x0), x1), one, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.44 = { by axiom 9 (star_unfold1) R->L }
% 29.77/4.44 fresh(fresh4(leq(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), multiplication(addition(addition(one, multiplication(x1, star(x1))), multiplication(multiplication(star(x0), x1), star(x1))), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x0), x1), one, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.44 = { by axiom 6 (additive_associativity) R->L }
% 29.77/4.44 fresh(fresh4(leq(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), multiplication(addition(one, addition(multiplication(x1, star(x1)), multiplication(multiplication(star(x0), x1), star(x1)))), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x0), x1), one, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.44 = { by axiom 4 (additive_commutativity) }
% 29.77/4.44 fresh(fresh4(leq(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), multiplication(addition(one, addition(multiplication(multiplication(star(x0), x1), star(x1)), multiplication(x1, star(x1)))), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x0), x1), one, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.44 = { by axiom 13 (distributivity2) R->L }
% 29.77/4.44 fresh(fresh4(leq(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), multiplication(addition(one, multiplication(addition(multiplication(star(x0), x1), x1), star(x1))), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x0), x1), one, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.44 = { by axiom 2 (multiplicative_left_identity) R->L }
% 29.77/4.44 fresh(fresh4(leq(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), multiplication(addition(one, multiplication(addition(multiplication(star(x0), x1), multiplication(one, x1)), star(x1))), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x0), x1), one, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.44 = { by lemma 20 }
% 29.77/4.44 fresh(fresh4(leq(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), multiplication(addition(one, multiplication(multiplication(addition(one, star(x0)), x1), star(x1))), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x0), x1), one, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.44 = { by lemma 18 }
% 29.77/4.44 fresh(fresh4(leq(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), multiplication(addition(one, multiplication(multiplication(star(x0), x1), star(x1))), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x0), x1), one, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.44 = { by axiom 4 (additive_commutativity) R->L }
% 29.77/4.44 fresh(fresh4(leq(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), multiplication(addition(multiplication(multiplication(star(x0), x1), star(x1)), one), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x0), x1), one, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.44 = { by lemma 19 R->L }
% 29.77/4.44 fresh(fresh4(leq(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), addition(strong_iteration(multiplication(star(x0), x1)), multiplication(multiplication(multiplication(star(x0), x1), star(x1)), strong_iteration(multiplication(star(x0), x1))))), true, multiplication(star(x0), x1), one, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.44 = { by axiom 5 (multiplicative_associativity) R->L }
% 29.77/4.44 fresh(fresh4(leq(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), addition(strong_iteration(multiplication(star(x0), x1)), multiplication(multiplication(star(x0), x1), multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))))), true, multiplication(star(x0), x1), one, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.44 = { by axiom 4 (additive_commutativity) R->L }
% 29.77/4.44 fresh(fresh4(leq(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), addition(multiplication(multiplication(star(x0), x1), multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x0), x1), one, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.44 = { by axiom 10 (infty_unfold1) }
% 29.77/4.44 fresh(fresh4(leq(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), addition(multiplication(multiplication(star(x0), x1), multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), addition(multiplication(multiplication(star(x0), x1), strong_iteration(multiplication(star(x0), x1))), one))), true, multiplication(star(x0), x1), one, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.44 = { by axiom 6 (additive_associativity) }
% 29.77/4.45 fresh(fresh4(leq(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), addition(addition(multiplication(multiplication(star(x0), x1), multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), multiplication(multiplication(star(x0), x1), strong_iteration(multiplication(star(x0), x1)))), one)), true, multiplication(star(x0), x1), one, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.45 = { by axiom 12 (distributivity1) R->L }
% 29.77/4.45 fresh(fresh4(leq(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), addition(multiplication(multiplication(star(x0), x1), addition(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))), one)), true, multiplication(star(x0), x1), one, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.45 = { by axiom 4 (additive_commutativity) }
% 29.77/4.45 fresh(fresh4(leq(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), addition(one, multiplication(multiplication(star(x0), x1), addition(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))))), true, multiplication(star(x0), x1), one, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.45 = { by lemma 21 }
% 29.77/4.45 fresh(fresh4(leq(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), addition(one, multiplication(multiplication(star(x0), x1), multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))))), true, multiplication(star(x0), x1), one, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.45 = { by axiom 4 (additive_commutativity) R->L }
% 29.77/4.45 fresh(fresh4(leq(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), addition(multiplication(multiplication(star(x0), x1), multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), one)), true, multiplication(star(x0), x1), one, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.45 = { by axiom 16 (infty_coinduction) }
% 29.77/4.45 fresh(leq(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), multiplication(strong_iteration(multiplication(star(x0), x1)), one)), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.45 = { by axiom 1 (multiplicative_right_identity) }
% 29.77/4.45 fresh(leq(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1))), true, multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.45 = { by axiom 14 (order_1) }
% 29.77/4.45 addition(multiplication(star(x1), strong_iteration(multiplication(star(x0), x1))), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.45 = { by lemma 21 }
% 29.77/4.45 multiplication(star(x1), strong_iteration(multiplication(star(x0), x1)))
% 29.77/4.45 % SZS output end Proof
% 29.77/4.45
% 29.77/4.45 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------