TSTP Solution File: KLE002+1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : KLE002+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 : n001.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 05:35:23 EDT 2023
% Result : Theorem 0.17s 0.37s
% Output : Proof 0.17s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : KLE002+1 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.11 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.11/0.32 % Computer : n001.cluster.edu
% 0.11/0.32 % Model : x86_64 x86_64
% 0.11/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.32 % Memory : 8042.1875MB
% 0.11/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.32 % CPULimit : 300
% 0.11/0.32 % WCLimit : 300
% 0.11/0.32 % DateTime : Tue Aug 29 12:33:37 EDT 2023
% 0.11/0.32 % CPUTime :
% 0.17/0.37 Command-line arguments: --kbo-weight0 --lhs-weight 5 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --goal-heuristic
% 0.17/0.37
% 0.17/0.37 % SZS status Theorem
% 0.17/0.37
% 0.17/0.38 % SZS output start Proof
% 0.17/0.38 Take the following subset of the input axioms:
% 0.17/0.38 fof(additive_commutativity, axiom, ![A, B]: addition(A, B)=addition(B, A)).
% 0.17/0.38 fof(goals, conjecture, ![X0, X1, X2]: (leq(X0, X1) => leq(multiplication(X0, X2), multiplication(X1, X2)))).
% 0.17/0.38 fof(left_distributivity, axiom, ![C, B2, A3]: multiplication(addition(A3, B2), C)=addition(multiplication(A3, C), multiplication(B2, C))).
% 0.17/0.38 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 0.17/0.38
% 0.17/0.38 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.17/0.38 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.17/0.38 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.17/0.38 fresh(y, y, x1...xn) = u
% 0.17/0.38 C => fresh(s, t, x1...xn) = v
% 0.17/0.38 where fresh is a fresh function symbol and x1..xn are the free
% 0.17/0.38 variables of u and v.
% 0.17/0.38 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.17/0.38 input problem has no model of domain size 1).
% 0.17/0.38
% 0.17/0.38 The encoding turns the above axioms into the following unit equations and goals:
% 0.17/0.38
% 0.17/0.38 Axiom 1 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.17/0.38 Axiom 2 (goals): leq(x0, x1) = true.
% 0.17/0.38 Axiom 3 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 0.17/0.38 Axiom 4 (order): fresh(X, X, Y, Z) = true.
% 0.17/0.38 Axiom 5 (order_1): fresh2(X, X, Y, Z) = Z.
% 0.17/0.38 Axiom 6 (order): fresh(addition(X, Y), Y, X, Y) = leq(X, Y).
% 0.17/0.38 Axiom 7 (order_1): fresh2(leq(X, Y), true, X, Y) = addition(X, Y).
% 0.17/0.38
% 0.17/0.38 Goal 1 (goals_1): leq(multiplication(x0, x2), multiplication(x1, x2)) = true.
% 0.17/0.38 Proof:
% 0.17/0.38 leq(multiplication(x0, x2), multiplication(x1, x2))
% 0.17/0.38 = { by axiom 6 (order) R->L }
% 0.17/0.38 fresh(addition(multiplication(x0, x2), multiplication(x1, x2)), multiplication(x1, x2), multiplication(x0, x2), multiplication(x1, x2))
% 0.17/0.38 = { by axiom 1 (additive_commutativity) R->L }
% 0.17/0.38 fresh(addition(multiplication(x1, x2), multiplication(x0, x2)), multiplication(x1, x2), multiplication(x0, x2), multiplication(x1, x2))
% 0.17/0.38 = { by axiom 3 (left_distributivity) R->L }
% 0.17/0.38 fresh(multiplication(addition(x1, x0), x2), multiplication(x1, x2), multiplication(x0, x2), multiplication(x1, x2))
% 0.17/0.38 = { by axiom 1 (additive_commutativity) }
% 0.17/0.38 fresh(multiplication(addition(x0, x1), x2), multiplication(x1, x2), multiplication(x0, x2), multiplication(x1, x2))
% 0.17/0.38 = { by axiom 7 (order_1) R->L }
% 0.17/0.38 fresh(multiplication(fresh2(leq(x0, x1), true, x0, x1), x2), multiplication(x1, x2), multiplication(x0, x2), multiplication(x1, x2))
% 0.17/0.38 = { by axiom 2 (goals) }
% 0.17/0.38 fresh(multiplication(fresh2(true, true, x0, x1), x2), multiplication(x1, x2), multiplication(x0, x2), multiplication(x1, x2))
% 0.17/0.38 = { by axiom 5 (order_1) }
% 0.17/0.38 fresh(multiplication(x1, x2), multiplication(x1, x2), multiplication(x0, x2), multiplication(x1, x2))
% 0.17/0.38 = { by axiom 4 (order) }
% 0.17/0.38 true
% 0.17/0.38 % SZS output end Proof
% 0.17/0.38
% 0.17/0.38 RESULT: Theorem (the conjecture is true).
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