TSTP Solution File: KLE145+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE145+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 : n002.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 05:36:02 EDT 2023
% Result : Theorem 13.93s 2.15s
% Output : Proof 14.47s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : KLE145+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 : n002.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 11:32:50 EDT 2023
% 0.13/0.34 % CPUTime :
% 13.93/2.15 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 13.93/2.15
% 13.93/2.15 % SZS status Theorem
% 13.93/2.15
% 13.93/2.18 % SZS output start Proof
% 13.93/2.18 Take the following subset of the input axioms:
% 13.93/2.18 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 13.93/2.18 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 13.93/2.18 fof(additive_identity, axiom, ![A3]: addition(A3, zero)=A3).
% 13.93/2.18 fof(distributivity1, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 13.93/2.18 fof(distributivity2, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 13.93/2.18 fof(goals, conjecture, ![X0]: star(strong_iteration(X0))=strong_iteration(X0)).
% 13.93/2.18 fof(idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 13.93/2.18 fof(infty_unfold1, axiom, ![A3]: strong_iteration(A3)=addition(multiplication(A3, strong_iteration(A3)), one)).
% 13.93/2.18 fof(isolation, axiom, ![A3]: strong_iteration(A3)=addition(star(A3), multiplication(strong_iteration(A3), zero))).
% 13.93/2.18 fof(left_annihilation, axiom, ![A3]: multiplication(zero, A3)=zero).
% 13.93/2.18 fof(multiplicative_associativity, axiom, ![A3, B2, C2]: multiplication(A3, multiplication(B2, C2))=multiplication(multiplication(A3, B2), C2)).
% 13.93/2.18 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 13.93/2.18 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 13.93/2.18 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 13.93/2.18 fof(star_induction1, axiom, ![B2, C2, A2_2]: (leq(addition(multiplication(A2_2, C2), B2), C2) => leq(multiplication(star(A2_2), B2), C2))).
% 13.93/2.18 fof(star_unfold1, axiom, ![A3]: addition(one, multiplication(A3, star(A3)))=star(A3)).
% 13.93/2.18 fof(star_unfold2, axiom, ![A3]: addition(one, multiplication(star(A3), A3))=star(A3)).
% 13.93/2.18
% 13.93/2.18 Now clausify the problem and encode Horn clauses using encoding 3 of
% 13.93/2.18 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 13.93/2.18 We repeatedly replace C & s=t => u=v by the two clauses:
% 13.93/2.18 fresh(y, y, x1...xn) = u
% 13.93/2.18 C => fresh(s, t, x1...xn) = v
% 13.93/2.18 where fresh is a fresh function symbol and x1..xn are the free
% 13.93/2.18 variables of u and v.
% 13.93/2.18 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 13.93/2.18 input problem has no model of domain size 1).
% 13.93/2.18
% 13.93/2.18 The encoding turns the above axioms into the following unit equations and goals:
% 13.93/2.18
% 13.93/2.18 Axiom 1 (multiplicative_right_identity): multiplication(X, one) = X.
% 13.93/2.18 Axiom 2 (multiplicative_left_identity): multiplication(one, X) = X.
% 13.93/2.18 Axiom 3 (left_annihilation): multiplication(zero, X) = zero.
% 13.93/2.18 Axiom 4 (idempotence): addition(X, X) = X.
% 13.93/2.18 Axiom 5 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 13.93/2.18 Axiom 6 (additive_identity): addition(X, zero) = X.
% 13.93/2.18 Axiom 7 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 13.93/2.18 Axiom 8 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 13.93/2.18 Axiom 9 (order_1): fresh(X, X, Y, Z) = Z.
% 13.93/2.18 Axiom 10 (order): fresh5(X, X, Y, Z) = true.
% 13.93/2.18 Axiom 11 (star_unfold1): addition(one, multiplication(X, star(X))) = star(X).
% 13.93/2.18 Axiom 12 (star_unfold2): addition(one, multiplication(star(X), X)) = star(X).
% 13.93/2.18 Axiom 13 (infty_unfold1): strong_iteration(X) = addition(multiplication(X, strong_iteration(X)), one).
% 13.93/2.18 Axiom 14 (star_induction1): fresh3(X, X, Y, Z, W) = true.
% 13.93/2.18 Axiom 15 (isolation): strong_iteration(X) = addition(star(X), multiplication(strong_iteration(X), zero)).
% 13.93/2.18 Axiom 16 (distributivity1): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 13.93/2.18 Axiom 17 (distributivity2): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 13.93/2.18 Axiom 18 (order_1): fresh(leq(X, Y), true, X, Y) = addition(X, Y).
% 13.93/2.18 Axiom 19 (order): fresh5(addition(X, Y), Y, X, Y) = leq(X, Y).
% 13.93/2.18 Axiom 20 (star_induction1): fresh3(leq(addition(multiplication(X, Y), Z), Y), true, X, Z, Y) = leq(multiplication(star(X), Z), Y).
% 13.93/2.18
% 13.93/2.18 Lemma 21: leq(X, X) = true.
% 13.93/2.18 Proof:
% 13.93/2.18 leq(X, X)
% 13.93/2.18 = { by axiom 19 (order) R->L }
% 13.93/2.18 fresh5(addition(X, X), X, X, X)
% 13.93/2.18 = { by axiom 4 (idempotence) }
% 13.93/2.18 fresh5(X, X, X, X)
% 13.93/2.18 = { by axiom 10 (order) }
% 13.93/2.18 true
% 13.93/2.18
% 13.93/2.18 Lemma 22: addition(X, addition(X, Y)) = addition(X, Y).
% 13.93/2.18 Proof:
% 13.93/2.18 addition(X, addition(X, Y))
% 13.93/2.18 = { by axiom 8 (additive_associativity) }
% 13.93/2.18 addition(addition(X, X), Y)
% 13.93/2.18 = { by axiom 4 (idempotence) }
% 13.93/2.18 addition(X, Y)
% 13.93/2.18
% 13.93/2.18 Lemma 23: addition(one, star(X)) = star(X).
% 13.93/2.18 Proof:
% 13.93/2.18 addition(one, star(X))
% 13.93/2.18 = { by axiom 11 (star_unfold1) R->L }
% 13.93/2.18 addition(one, addition(one, multiplication(X, star(X))))
% 13.93/2.18 = { by lemma 22 }
% 13.93/2.18 addition(one, multiplication(X, star(X)))
% 13.93/2.18 = { by axiom 11 (star_unfold1) }
% 13.93/2.18 star(X)
% 13.93/2.18
% 13.93/2.18 Lemma 24: addition(one, multiplication(X, strong_iteration(X))) = strong_iteration(X).
% 13.93/2.18 Proof:
% 13.93/2.18 addition(one, multiplication(X, strong_iteration(X)))
% 13.93/2.18 = { by axiom 5 (additive_commutativity) R->L }
% 13.93/2.18 addition(multiplication(X, strong_iteration(X)), one)
% 13.93/2.18 = { by axiom 13 (infty_unfold1) R->L }
% 13.93/2.18 strong_iteration(X)
% 13.93/2.18
% 13.93/2.18 Lemma 25: addition(one, strong_iteration(X)) = strong_iteration(X).
% 13.93/2.18 Proof:
% 13.93/2.18 addition(one, strong_iteration(X))
% 13.93/2.18 = { by lemma 24 R->L }
% 13.93/2.18 addition(one, addition(one, multiplication(X, strong_iteration(X))))
% 13.93/2.18 = { by lemma 22 }
% 13.93/2.18 addition(one, multiplication(X, strong_iteration(X)))
% 13.93/2.18 = { by lemma 24 }
% 13.93/2.18 strong_iteration(X)
% 13.93/2.18
% 13.93/2.18 Lemma 26: addition(X, addition(Y, X)) = addition(Y, X).
% 13.93/2.18 Proof:
% 13.93/2.18 addition(X, addition(Y, X))
% 13.93/2.18 = { by lemma 22 R->L }
% 13.93/2.18 addition(X, addition(Y, addition(Y, X)))
% 13.93/2.18 = { by axiom 5 (additive_commutativity) R->L }
% 13.93/2.18 addition(X, addition(Y, addition(X, Y)))
% 13.93/2.18 = { by axiom 8 (additive_associativity) }
% 13.93/2.18 addition(addition(X, Y), addition(X, Y))
% 13.93/2.18 = { by axiom 4 (idempotence) }
% 13.93/2.18 addition(X, Y)
% 13.93/2.18 = { by axiom 5 (additive_commutativity) }
% 13.93/2.18 addition(Y, X)
% 13.93/2.18
% 13.93/2.18 Lemma 27: addition(X, multiplication(X, Y)) = multiplication(X, addition(Y, one)).
% 13.93/2.18 Proof:
% 13.93/2.18 addition(X, multiplication(X, Y))
% 13.93/2.18 = { by axiom 1 (multiplicative_right_identity) R->L }
% 13.93/2.18 addition(multiplication(X, one), multiplication(X, Y))
% 13.93/2.18 = { by axiom 16 (distributivity1) R->L }
% 13.93/2.18 multiplication(X, addition(one, Y))
% 13.93/2.18 = { by axiom 5 (additive_commutativity) }
% 13.93/2.18 multiplication(X, addition(Y, one))
% 13.93/2.18
% 13.93/2.18 Lemma 28: multiplication(star(strong_iteration(X)), strong_iteration(X)) = star(strong_iteration(X)).
% 13.93/2.18 Proof:
% 13.93/2.18 multiplication(star(strong_iteration(X)), strong_iteration(X))
% 13.93/2.18 = { by lemma 25 R->L }
% 13.93/2.18 multiplication(star(strong_iteration(X)), addition(one, strong_iteration(X)))
% 13.93/2.18 = { by axiom 5 (additive_commutativity) R->L }
% 13.93/2.18 multiplication(star(strong_iteration(X)), addition(strong_iteration(X), one))
% 13.93/2.18 = { by lemma 27 R->L }
% 13.93/2.18 addition(star(strong_iteration(X)), multiplication(star(strong_iteration(X)), strong_iteration(X)))
% 13.93/2.18 = { by axiom 5 (additive_commutativity) R->L }
% 13.93/2.18 addition(multiplication(star(strong_iteration(X)), strong_iteration(X)), star(strong_iteration(X)))
% 13.93/2.18 = { by axiom 12 (star_unfold2) R->L }
% 13.93/2.18 addition(multiplication(star(strong_iteration(X)), strong_iteration(X)), addition(one, multiplication(star(strong_iteration(X)), strong_iteration(X))))
% 13.93/2.18 = { by lemma 26 }
% 13.93/2.18 addition(one, multiplication(star(strong_iteration(X)), strong_iteration(X)))
% 13.93/2.18 = { by axiom 12 (star_unfold2) }
% 13.93/2.18 star(strong_iteration(X))
% 13.93/2.18
% 13.93/2.18 Goal 1 (goals): star(strong_iteration(x0)) = strong_iteration(x0).
% 13.93/2.18 Proof:
% 13.93/2.18 star(strong_iteration(x0))
% 13.93/2.19 = { by axiom 11 (star_unfold1) R->L }
% 13.93/2.19 addition(one, multiplication(strong_iteration(x0), star(strong_iteration(x0))))
% 13.93/2.19 = { by lemma 23 R->L }
% 13.93/2.19 addition(one, multiplication(strong_iteration(x0), addition(one, star(strong_iteration(x0)))))
% 13.93/2.19 = { by axiom 5 (additive_commutativity) R->L }
% 13.93/2.19 addition(one, multiplication(strong_iteration(x0), addition(star(strong_iteration(x0)), one)))
% 13.93/2.19 = { by lemma 27 R->L }
% 13.93/2.19 addition(one, addition(strong_iteration(x0), multiplication(strong_iteration(x0), star(strong_iteration(x0)))))
% 13.93/2.19 = { by axiom 5 (additive_commutativity) R->L }
% 13.93/2.19 addition(one, addition(multiplication(strong_iteration(x0), star(strong_iteration(x0))), strong_iteration(x0)))
% 13.93/2.19 = { by axiom 8 (additive_associativity) }
% 13.93/2.19 addition(addition(one, multiplication(strong_iteration(x0), star(strong_iteration(x0)))), strong_iteration(x0))
% 13.93/2.19 = { by axiom 11 (star_unfold1) }
% 13.93/2.19 addition(star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by axiom 18 (order_1) R->L }
% 13.93/2.19 fresh(leq(star(strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by lemma 28 R->L }
% 13.93/2.19 fresh(leq(multiplication(star(strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by lemma 28 R->L }
% 13.93/2.19 fresh(leq(multiplication(multiplication(star(strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by axiom 7 (multiplicative_associativity) R->L }
% 13.93/2.19 fresh(leq(multiplication(star(strong_iteration(x0)), multiplication(strong_iteration(x0), strong_iteration(x0))), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by axiom 20 (star_induction1) R->L }
% 13.93/2.19 fresh(fresh3(leq(addition(multiplication(strong_iteration(x0), strong_iteration(x0)), multiplication(strong_iteration(x0), strong_iteration(x0))), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by axiom 4 (idempotence) }
% 13.93/2.19 fresh(fresh3(leq(multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by lemma 25 R->L }
% 13.93/2.19 fresh(fresh3(leq(multiplication(strong_iteration(x0), addition(one, strong_iteration(x0))), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by axiom 5 (additive_commutativity) R->L }
% 13.93/2.19 fresh(fresh3(leq(multiplication(strong_iteration(x0), addition(strong_iteration(x0), one)), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by axiom 2 (multiplicative_left_identity) R->L }
% 13.93/2.19 fresh(fresh3(leq(multiplication(strong_iteration(x0), multiplication(one, addition(strong_iteration(x0), one))), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by axiom 6 (additive_identity) R->L }
% 13.93/2.19 fresh(fresh3(leq(multiplication(strong_iteration(x0), multiplication(addition(one, zero), addition(strong_iteration(x0), one))), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by axiom 5 (additive_commutativity) }
% 13.93/2.19 fresh(fresh3(leq(multiplication(strong_iteration(x0), multiplication(addition(zero, one), addition(strong_iteration(x0), one))), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by axiom 7 (multiplicative_associativity) }
% 13.93/2.19 fresh(fresh3(leq(multiplication(multiplication(strong_iteration(x0), addition(zero, one)), addition(strong_iteration(x0), one)), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by lemma 27 R->L }
% 13.93/2.19 fresh(fresh3(leq(multiplication(addition(strong_iteration(x0), multiplication(strong_iteration(x0), zero)), addition(strong_iteration(x0), one)), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by axiom 5 (additive_commutativity) R->L }
% 13.93/2.19 fresh(fresh3(leq(multiplication(addition(multiplication(strong_iteration(x0), zero), strong_iteration(x0)), addition(strong_iteration(x0), one)), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by axiom 15 (isolation) }
% 13.93/2.19 fresh(fresh3(leq(multiplication(addition(multiplication(strong_iteration(x0), zero), addition(star(x0), multiplication(strong_iteration(x0), zero))), addition(strong_iteration(x0), one)), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by axiom 5 (additive_commutativity) R->L }
% 13.93/2.19 fresh(fresh3(leq(multiplication(addition(multiplication(strong_iteration(x0), zero), addition(multiplication(strong_iteration(x0), zero), star(x0))), addition(strong_iteration(x0), one)), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by axiom 8 (additive_associativity) }
% 13.93/2.19 fresh(fresh3(leq(multiplication(addition(addition(multiplication(strong_iteration(x0), zero), multiplication(strong_iteration(x0), zero)), star(x0)), addition(strong_iteration(x0), one)), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by axiom 16 (distributivity1) R->L }
% 13.93/2.19 fresh(fresh3(leq(multiplication(addition(multiplication(strong_iteration(x0), addition(zero, zero)), star(x0)), addition(strong_iteration(x0), one)), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by axiom 5 (additive_commutativity) }
% 13.93/2.19 fresh(fresh3(leq(multiplication(addition(star(x0), multiplication(strong_iteration(x0), addition(zero, zero))), addition(strong_iteration(x0), one)), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by axiom 6 (additive_identity) }
% 13.93/2.19 fresh(fresh3(leq(multiplication(addition(star(x0), multiplication(strong_iteration(x0), zero)), addition(strong_iteration(x0), one)), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by axiom 5 (additive_commutativity) R->L }
% 13.93/2.19 fresh(fresh3(leq(multiplication(addition(multiplication(strong_iteration(x0), zero), star(x0)), addition(strong_iteration(x0), one)), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by axiom 17 (distributivity2) }
% 13.93/2.19 fresh(fresh3(leq(addition(multiplication(multiplication(strong_iteration(x0), zero), addition(strong_iteration(x0), one)), multiplication(star(x0), addition(strong_iteration(x0), one))), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by axiom 7 (multiplicative_associativity) R->L }
% 13.93/2.19 fresh(fresh3(leq(addition(multiplication(strong_iteration(x0), multiplication(zero, addition(strong_iteration(x0), one))), multiplication(star(x0), addition(strong_iteration(x0), one))), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by axiom 3 (left_annihilation) }
% 13.93/2.19 fresh(fresh3(leq(addition(multiplication(strong_iteration(x0), zero), multiplication(star(x0), addition(strong_iteration(x0), one))), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by axiom 5 (additive_commutativity) }
% 13.93/2.19 fresh(fresh3(leq(addition(multiplication(star(x0), addition(strong_iteration(x0), one)), multiplication(strong_iteration(x0), zero)), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by lemma 27 R->L }
% 13.93/2.19 fresh(fresh3(leq(addition(addition(star(x0), multiplication(star(x0), strong_iteration(x0))), multiplication(strong_iteration(x0), zero)), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by axiom 8 (additive_associativity) R->L }
% 13.93/2.19 fresh(fresh3(leq(addition(star(x0), addition(multiplication(star(x0), strong_iteration(x0)), multiplication(strong_iteration(x0), zero))), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by axiom 5 (additive_commutativity) R->L }
% 13.93/2.19 fresh(fresh3(leq(addition(star(x0), addition(multiplication(strong_iteration(x0), zero), multiplication(star(x0), strong_iteration(x0)))), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by axiom 8 (additive_associativity) }
% 13.93/2.19 fresh(fresh3(leq(addition(addition(star(x0), multiplication(strong_iteration(x0), zero)), multiplication(star(x0), strong_iteration(x0))), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 13.93/2.19 = { by axiom 15 (isolation) R->L }
% 14.47/2.19 fresh(fresh3(leq(addition(strong_iteration(x0), multiplication(star(x0), strong_iteration(x0))), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 14.47/2.19 = { by lemma 23 R->L }
% 14.47/2.19 fresh(fresh3(leq(addition(strong_iteration(x0), multiplication(addition(one, star(x0)), strong_iteration(x0))), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 14.47/2.19 = { by axiom 17 (distributivity2) }
% 14.47/2.19 fresh(fresh3(leq(addition(strong_iteration(x0), addition(multiplication(one, strong_iteration(x0)), multiplication(star(x0), strong_iteration(x0)))), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 14.47/2.19 = { by axiom 2 (multiplicative_left_identity) }
% 14.47/2.19 fresh(fresh3(leq(addition(strong_iteration(x0), addition(strong_iteration(x0), multiplication(star(x0), strong_iteration(x0)))), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 14.47/2.19 = { by axiom 5 (additive_commutativity) R->L }
% 14.47/2.19 fresh(fresh3(leq(addition(strong_iteration(x0), addition(multiplication(star(x0), strong_iteration(x0)), strong_iteration(x0))), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 14.47/2.19 = { by axiom 18 (order_1) R->L }
% 14.47/2.19 fresh(fresh3(leq(addition(strong_iteration(x0), fresh(leq(multiplication(star(x0), strong_iteration(x0)), strong_iteration(x0)), true, multiplication(star(x0), strong_iteration(x0)), strong_iteration(x0))), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 14.47/2.19 = { by axiom 20 (star_induction1) R->L }
% 14.47/2.19 fresh(fresh3(leq(addition(strong_iteration(x0), fresh(fresh3(leq(addition(multiplication(x0, strong_iteration(x0)), strong_iteration(x0)), strong_iteration(x0)), true, x0, strong_iteration(x0), strong_iteration(x0)), true, multiplication(star(x0), strong_iteration(x0)), strong_iteration(x0))), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 14.47/2.19 = { by lemma 24 R->L }
% 14.47/2.19 fresh(fresh3(leq(addition(strong_iteration(x0), fresh(fresh3(leq(addition(multiplication(x0, strong_iteration(x0)), addition(one, multiplication(x0, strong_iteration(x0)))), strong_iteration(x0)), true, x0, strong_iteration(x0), strong_iteration(x0)), true, multiplication(star(x0), strong_iteration(x0)), strong_iteration(x0))), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 14.47/2.19 = { by lemma 26 }
% 14.47/2.19 fresh(fresh3(leq(addition(strong_iteration(x0), fresh(fresh3(leq(addition(one, multiplication(x0, strong_iteration(x0))), strong_iteration(x0)), true, x0, strong_iteration(x0), strong_iteration(x0)), true, multiplication(star(x0), strong_iteration(x0)), strong_iteration(x0))), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 14.47/2.19 = { by lemma 24 }
% 14.47/2.19 fresh(fresh3(leq(addition(strong_iteration(x0), fresh(fresh3(leq(strong_iteration(x0), strong_iteration(x0)), true, x0, strong_iteration(x0), strong_iteration(x0)), true, multiplication(star(x0), strong_iteration(x0)), strong_iteration(x0))), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 14.47/2.19 = { by lemma 21 }
% 14.47/2.19 fresh(fresh3(leq(addition(strong_iteration(x0), fresh(fresh3(true, true, x0, strong_iteration(x0), strong_iteration(x0)), true, multiplication(star(x0), strong_iteration(x0)), strong_iteration(x0))), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 14.47/2.19 = { by axiom 14 (star_induction1) }
% 14.47/2.19 fresh(fresh3(leq(addition(strong_iteration(x0), fresh(true, true, multiplication(star(x0), strong_iteration(x0)), strong_iteration(x0))), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 14.47/2.19 = { by axiom 9 (order_1) }
% 14.47/2.19 fresh(fresh3(leq(addition(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 14.47/2.19 = { by axiom 4 (idempotence) }
% 14.47/2.19 fresh(fresh3(leq(strong_iteration(x0), strong_iteration(x0)), true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 14.47/2.19 = { by lemma 21 }
% 14.47/2.19 fresh(fresh3(true, true, strong_iteration(x0), multiplication(strong_iteration(x0), strong_iteration(x0)), strong_iteration(x0)), true, star(strong_iteration(x0)), strong_iteration(x0))
% 14.47/2.19 = { by axiom 14 (star_induction1) }
% 14.47/2.19 fresh(true, true, star(strong_iteration(x0)), strong_iteration(x0))
% 14.47/2.19 = { by axiom 9 (order_1) }
% 14.47/2.19 strong_iteration(x0)
% 14.47/2.19 % SZS output end Proof
% 14.47/2.19
% 14.47/2.19 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------