TSTP Solution File: KLE035+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE035+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 : n017.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 05:35:36 EDT 2023
% Result : Theorem 0.17s 0.41s
% 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
% 0.05/0.11 % Problem : KLE035+1 : TPTP v8.1.2. Released v4.0.0.
% 0.05/0.12 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.10/0.32 % Computer : n017.cluster.edu
% 0.10/0.32 % Model : x86_64 x86_64
% 0.10/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.32 % Memory : 8042.1875MB
% 0.10/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.10/0.32 % CPULimit : 300
% 0.10/0.32 % WCLimit : 300
% 0.10/0.32 % DateTime : Tue Aug 29 11:22:13 EDT 2023
% 0.10/0.32 % CPUTime :
% 0.17/0.41 Command-line arguments: --kbo-weight0 --lhs-weight 5 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --goal-heuristic
% 0.17/0.41
% 0.17/0.41 % SZS status Theorem
% 0.17/0.41
% 0.17/0.42 % SZS output start Proof
% 0.17/0.42 Take the following subset of the input axioms:
% 0.17/0.42 fof(additive_commutativity, axiom, ![A, B]: addition(A, B)=addition(B, A)).
% 0.17/0.42 fof(additive_identity, axiom, ![A3]: addition(A3, zero)=A3).
% 0.17/0.42 fof(goals, conjecture, ![X0, X1, X2, X3]: ((test(X3) & (test(X2) & (leq(multiplication(multiplication(X2, X0), c(X3)), zero) & leq(multiplication(multiplication(X2, X1), c(X3)), zero)))) => leq(multiplication(multiplication(X2, addition(X0, X1)), c(X3)), zero))).
% 0.17/0.42 fof(left_distributivity, axiom, ![C, A3, B2]: multiplication(addition(A3, B2), C)=addition(multiplication(A3, C), multiplication(B2, C))).
% 0.17/0.42 fof(multiplicative_associativity, axiom, ![A3, B2, C2]: multiplication(A3, multiplication(B2, C2))=multiplication(multiplication(A3, B2), C2)).
% 0.17/0.42 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 0.17/0.42 fof(right_distributivity, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 0.17/0.42
% 0.17/0.42 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.17/0.42 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.17/0.42 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.17/0.42 fresh(y, y, x1...xn) = u
% 0.17/0.42 C => fresh(s, t, x1...xn) = v
% 0.17/0.42 where fresh is a fresh function symbol and x1..xn are the free
% 0.17/0.42 variables of u and v.
% 0.17/0.42 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.17/0.42 input problem has no model of domain size 1).
% 0.17/0.42
% 0.17/0.42 The encoding turns the above axioms into the following unit equations and goals:
% 0.17/0.42
% 0.17/0.42 Axiom 1 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.17/0.42 Axiom 2 (additive_identity): addition(X, zero) = X.
% 0.17/0.42 Axiom 3 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 0.17/0.42 Axiom 4 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 0.17/0.42 Axiom 5 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 0.17/0.42 Axiom 6 (order): fresh11(X, X, Y, Z) = true.
% 0.17/0.42 Axiom 7 (order_1): fresh2(X, X, Y, Z) = Z.
% 0.17/0.42 Axiom 8 (goals): leq(multiplication(multiplication(x2, x1), c(x3)), zero) = true.
% 0.17/0.42 Axiom 9 (goals_1): leq(multiplication(multiplication(x2, x0), c(x3)), zero) = true.
% 0.17/0.42 Axiom 10 (order): fresh11(addition(X, Y), Y, X, Y) = leq(X, Y).
% 0.17/0.42 Axiom 11 (order_1): fresh2(leq(X, Y), true, X, Y) = addition(X, Y).
% 0.17/0.42
% 0.17/0.42 Lemma 12: addition(zero, X) = X.
% 0.17/0.42 Proof:
% 0.17/0.42 addition(zero, X)
% 0.17/0.42 = { by axiom 1 (additive_commutativity) R->L }
% 0.17/0.42 addition(X, zero)
% 0.17/0.42 = { by axiom 2 (additive_identity) }
% 0.17/0.42 X
% 0.17/0.42
% 0.17/0.42 Goal 1 (goals_4): leq(multiplication(multiplication(x2, addition(x0, x1)), c(x3)), zero) = true.
% 0.17/0.42 Proof:
% 0.17/0.42 leq(multiplication(multiplication(x2, addition(x0, x1)), c(x3)), zero)
% 0.17/0.42 = { by axiom 3 (multiplicative_associativity) R->L }
% 0.17/0.42 leq(multiplication(x2, multiplication(addition(x0, x1), c(x3))), zero)
% 0.17/0.42 = { by axiom 1 (additive_commutativity) }
% 0.17/0.42 leq(multiplication(x2, multiplication(addition(x1, x0), c(x3))), zero)
% 0.17/0.42 = { by axiom 5 (left_distributivity) }
% 0.17/0.42 leq(multiplication(x2, addition(multiplication(x1, c(x3)), multiplication(x0, c(x3)))), zero)
% 0.17/0.42 = { by axiom 1 (additive_commutativity) R->L }
% 0.17/0.42 leq(multiplication(x2, addition(multiplication(x0, c(x3)), multiplication(x1, c(x3)))), zero)
% 0.17/0.42 = { by axiom 4 (right_distributivity) }
% 0.17/0.42 leq(addition(multiplication(x2, multiplication(x0, c(x3))), multiplication(x2, multiplication(x1, c(x3)))), zero)
% 0.17/0.42 = { by axiom 2 (additive_identity) R->L }
% 0.17/0.42 leq(addition(addition(multiplication(x2, multiplication(x0, c(x3))), zero), multiplication(x2, multiplication(x1, c(x3)))), zero)
% 0.17/0.42 = { by axiom 11 (order_1) R->L }
% 0.17/0.42 leq(addition(fresh2(leq(multiplication(x2, multiplication(x0, c(x3))), zero), true, multiplication(x2, multiplication(x0, c(x3))), zero), multiplication(x2, multiplication(x1, c(x3)))), zero)
% 0.17/0.42 = { by axiom 3 (multiplicative_associativity) }
% 0.17/0.42 leq(addition(fresh2(leq(multiplication(multiplication(x2, x0), c(x3)), zero), true, multiplication(x2, multiplication(x0, c(x3))), zero), multiplication(x2, multiplication(x1, c(x3)))), zero)
% 0.17/0.42 = { by axiom 9 (goals_1) }
% 0.17/0.42 leq(addition(fresh2(true, true, multiplication(x2, multiplication(x0, c(x3))), zero), multiplication(x2, multiplication(x1, c(x3)))), zero)
% 0.17/0.42 = { by axiom 7 (order_1) }
% 0.17/0.42 leq(addition(zero, multiplication(x2, multiplication(x1, c(x3)))), zero)
% 0.17/0.42 = { by lemma 12 }
% 0.17/0.42 leq(multiplication(x2, multiplication(x1, c(x3))), zero)
% 0.17/0.42 = { by axiom 2 (additive_identity) R->L }
% 0.17/0.42 leq(addition(multiplication(x2, multiplication(x1, c(x3))), zero), zero)
% 0.17/0.42 = { by axiom 11 (order_1) R->L }
% 0.17/0.42 leq(fresh2(leq(multiplication(x2, multiplication(x1, c(x3))), zero), true, multiplication(x2, multiplication(x1, c(x3))), zero), zero)
% 0.17/0.42 = { by axiom 3 (multiplicative_associativity) }
% 0.17/0.42 leq(fresh2(leq(multiplication(multiplication(x2, x1), c(x3)), zero), true, multiplication(x2, multiplication(x1, c(x3))), zero), zero)
% 0.17/0.42 = { by axiom 8 (goals) }
% 0.17/0.42 leq(fresh2(true, true, multiplication(x2, multiplication(x1, c(x3))), zero), zero)
% 0.17/0.42 = { by axiom 7 (order_1) }
% 0.17/0.42 leq(zero, zero)
% 0.17/0.42 = { by axiom 10 (order) R->L }
% 0.17/0.42 fresh11(addition(zero, zero), zero, zero, zero)
% 0.17/0.42 = { by lemma 12 }
% 0.17/0.42 fresh11(zero, zero, zero, zero)
% 0.17/0.42 = { by axiom 6 (order) }
% 0.17/0.42 true
% 0.17/0.42 % SZS output end Proof
% 0.17/0.42
% 0.17/0.42 RESULT: Theorem (the conjecture is true).
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