TSTP Solution File: KLE151+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE151+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 05:36:05 EDT 2023
% Result : Theorem 14.28s 2.96s
% Output : Proof 14.70s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.13 % Problem : KLE151+1 : TPTP v8.1.2. Released v4.0.0.
% 0.08/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.35 % Computer : n002.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Tue Aug 29 11:13:05 EDT 2023
% 0.13/0.35 % CPUTime :
% 14.28/2.96 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 14.28/2.96
% 14.28/2.96 % SZS status Theorem
% 14.70/2.97
% 14.70/2.97 % SZS output start Proof
% 14.70/2.97 Take the following subset of the input axioms:
% 14.70/2.97 fof(additive_commutativity, axiom, ![A, B]: addition(A, B)=addition(B, A)).
% 14.70/2.97 fof(distributivity1, axiom, ![C, B2, A3]: multiplication(A3, addition(B2, C))=addition(multiplication(A3, B2), multiplication(A3, C))).
% 14.70/2.97 fof(goals, conjecture, ![X0, X1]: leq(multiplication(X0, strong_iteration(multiplication(X1, X0))), multiplication(strong_iteration(multiplication(X0, X1)), X0))).
% 14.70/2.97 fof(idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 14.70/2.97 fof(infty_coinduction, axiom, ![A2, B2, C2]: (leq(C2, addition(multiplication(A2, C2), B2)) => leq(C2, multiplication(strong_iteration(A2), B2)))).
% 14.70/2.97 fof(infty_unfold1, axiom, ![A3]: strong_iteration(A3)=addition(multiplication(A3, strong_iteration(A3)), one)).
% 14.70/2.97 fof(multiplicative_associativity, axiom, ![B2, A3, C2]: multiplication(A3, multiplication(B2, C2))=multiplication(multiplication(A3, B2), C2)).
% 14.70/2.97 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 14.70/2.97 fof(order, axiom, ![B2, A2_2]: (leq(A2_2, B2) <=> addition(A2_2, B2)=B2)).
% 14.70/2.97
% 14.70/2.98 Now clausify the problem and encode Horn clauses using encoding 3 of
% 14.70/2.98 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 14.70/2.98 We repeatedly replace C & s=t => u=v by the two clauses:
% 14.70/2.98 fresh(y, y, x1...xn) = u
% 14.70/2.98 C => fresh(s, t, x1...xn) = v
% 14.70/2.98 where fresh is a fresh function symbol and x1..xn are the free
% 14.70/2.98 variables of u and v.
% 14.70/2.98 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 14.70/2.98 input problem has no model of domain size 1).
% 14.70/2.99
% 14.70/2.99 The encoding turns the above axioms into the following unit equations and goals:
% 14.70/2.99
% 14.70/2.99 Axiom 1 (multiplicative_right_identity): multiplication(X, one) = X.
% 14.70/2.99 Axiom 2 (idempotence): addition(X, X) = X.
% 14.70/2.99 Axiom 3 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 14.70/2.99 Axiom 4 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 14.70/2.99 Axiom 5 (order): fresh5(X, X, Y, Z) = true.
% 14.70/2.99 Axiom 6 (infty_unfold1): strong_iteration(X) = addition(multiplication(X, strong_iteration(X)), one).
% 14.70/2.99 Axiom 7 (infty_coinduction): fresh4(X, X, Y, Z, W) = true.
% 14.70/2.99 Axiom 8 (distributivity1): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 14.70/2.99 Axiom 9 (order): fresh5(addition(X, Y), Y, X, Y) = leq(X, Y).
% 14.70/2.99 Axiom 10 (infty_coinduction): fresh4(leq(X, addition(multiplication(Y, X), Z)), true, Y, Z, X) = leq(X, multiplication(strong_iteration(Y), Z)).
% 14.70/2.99
% 14.70/2.99 Goal 1 (goals): leq(multiplication(x0, strong_iteration(multiplication(x1, x0))), multiplication(strong_iteration(multiplication(x0, x1)), x0)) = true.
% 14.70/2.99 Proof:
% 14.70/2.99 leq(multiplication(x0, strong_iteration(multiplication(x1, x0))), multiplication(strong_iteration(multiplication(x0, x1)), x0))
% 14.70/2.99 = { by axiom 10 (infty_coinduction) R->L }
% 14.70/2.99 fresh4(leq(multiplication(x0, strong_iteration(multiplication(x1, x0))), addition(multiplication(multiplication(x0, x1), multiplication(x0, strong_iteration(multiplication(x1, x0)))), x0)), true, multiplication(x0, x1), x0, multiplication(x0, strong_iteration(multiplication(x1, x0))))
% 14.70/2.99 = { by axiom 3 (additive_commutativity) }
% 14.70/2.99 fresh4(leq(multiplication(x0, strong_iteration(multiplication(x1, x0))), addition(x0, multiplication(multiplication(x0, x1), multiplication(x0, strong_iteration(multiplication(x1, x0)))))), true, multiplication(x0, x1), x0, multiplication(x0, strong_iteration(multiplication(x1, x0))))
% 14.70/2.99 = { by axiom 4 (multiplicative_associativity) R->L }
% 14.70/2.99 fresh4(leq(multiplication(x0, strong_iteration(multiplication(x1, x0))), addition(x0, multiplication(x0, multiplication(x1, multiplication(x0, strong_iteration(multiplication(x1, x0))))))), true, multiplication(x0, x1), x0, multiplication(x0, strong_iteration(multiplication(x1, x0))))
% 14.70/2.99 = { by axiom 1 (multiplicative_right_identity) R->L }
% 14.70/2.99 fresh4(leq(multiplication(x0, strong_iteration(multiplication(x1, x0))), addition(multiplication(x0, one), multiplication(x0, multiplication(x1, multiplication(x0, strong_iteration(multiplication(x1, x0))))))), true, multiplication(x0, x1), x0, multiplication(x0, strong_iteration(multiplication(x1, x0))))
% 14.70/2.99 = { by axiom 8 (distributivity1) R->L }
% 14.70/2.99 fresh4(leq(multiplication(x0, strong_iteration(multiplication(x1, x0))), multiplication(x0, addition(one, multiplication(x1, multiplication(x0, strong_iteration(multiplication(x1, x0))))))), true, multiplication(x0, x1), x0, multiplication(x0, strong_iteration(multiplication(x1, x0))))
% 14.70/2.99 = { by axiom 4 (multiplicative_associativity) }
% 14.70/2.99 fresh4(leq(multiplication(x0, strong_iteration(multiplication(x1, x0))), multiplication(x0, addition(one, multiplication(multiplication(x1, x0), strong_iteration(multiplication(x1, x0)))))), true, multiplication(x0, x1), x0, multiplication(x0, strong_iteration(multiplication(x1, x0))))
% 14.70/2.99 = { by axiom 3 (additive_commutativity) R->L }
% 14.70/2.99 fresh4(leq(multiplication(x0, strong_iteration(multiplication(x1, x0))), multiplication(x0, addition(multiplication(multiplication(x1, x0), strong_iteration(multiplication(x1, x0))), one))), true, multiplication(x0, x1), x0, multiplication(x0, strong_iteration(multiplication(x1, x0))))
% 14.70/2.99 = { by axiom 6 (infty_unfold1) R->L }
% 14.70/2.99 fresh4(leq(multiplication(x0, strong_iteration(multiplication(x1, x0))), multiplication(x0, strong_iteration(multiplication(x1, x0)))), true, multiplication(x0, x1), x0, multiplication(x0, strong_iteration(multiplication(x1, x0))))
% 14.70/2.99 = { by axiom 9 (order) R->L }
% 14.70/2.99 fresh4(fresh5(addition(multiplication(x0, strong_iteration(multiplication(x1, x0))), multiplication(x0, strong_iteration(multiplication(x1, x0)))), multiplication(x0, strong_iteration(multiplication(x1, x0))), multiplication(x0, strong_iteration(multiplication(x1, x0))), multiplication(x0, strong_iteration(multiplication(x1, x0)))), true, multiplication(x0, x1), x0, multiplication(x0, strong_iteration(multiplication(x1, x0))))
% 14.70/2.99 = { by axiom 2 (idempotence) }
% 14.70/2.99 fresh4(fresh5(multiplication(x0, strong_iteration(multiplication(x1, x0))), multiplication(x0, strong_iteration(multiplication(x1, x0))), multiplication(x0, strong_iteration(multiplication(x1, x0))), multiplication(x0, strong_iteration(multiplication(x1, x0)))), true, multiplication(x0, x1), x0, multiplication(x0, strong_iteration(multiplication(x1, x0))))
% 14.70/2.99 = { by axiom 5 (order) }
% 14.70/2.99 fresh4(true, true, multiplication(x0, x1), x0, multiplication(x0, strong_iteration(multiplication(x1, x0))))
% 14.70/2.99 = { by axiom 7 (infty_coinduction) }
% 14.70/2.99 true
% 14.70/2.99 % SZS output end Proof
% 14.70/2.99
% 14.70/2.99 RESULT: Theorem (the conjecture is true).
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