TSTP Solution File: KLE034+2 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : KLE034+2 : 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 : n020.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 0.19s 0.54s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : KLE034+2 : TPTP v8.1.2. Released v4.0.0.
% 0.11/0.13 % 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 : n020.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 11:45:13 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.19/0.54 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.19/0.54
% 0.19/0.54 % SZS status Theorem
% 0.19/0.54
% 0.19/0.54 % SZS output start Proof
% 0.19/0.54 Take the following subset of the input axioms:
% 0.19/0.54 fof(additive_commutativity, axiom, ![A, B]: addition(A, B)=addition(B, A)).
% 0.19/0.54 fof(additive_identity, axiom, ![A3]: addition(A3, zero)=A3).
% 0.19/0.54 fof(goals, conjecture, ![X0, X1, X2, X3, X4]: ((test(X3) & (test(X2) & (test(X4) & (leq(multiplication(multiplication(X2, X0), c(X3)), zero) & leq(multiplication(multiplication(X3, X1), c(X4)), zero))))) => leq(multiplication(multiplication(multiplication(X2, X0), X1), c(X4)), zero))).
% 0.19/0.54 fof(multiplicative_associativity, axiom, ![C, A3, B2]: multiplication(A3, multiplication(B2, C))=multiplication(multiplication(A3, B2), C)).
% 0.19/0.54 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 0.19/0.54 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 0.19/0.54 fof(right_annihilation, axiom, ![A3]: multiplication(A3, zero)=zero).
% 0.19/0.54 fof(right_distributivity, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 0.19/0.54 fof(test_2, axiom, ![X0_2, X1_2]: (complement(X1_2, X0_2) <=> (multiplication(X0_2, X1_2)=zero & (multiplication(X1_2, X0_2)=zero & addition(X0_2, X1_2)=one)))).
% 0.19/0.54 fof(test_3, axiom, ![X0_2, X1_2]: (test(X0_2) => (c(X0_2)=X1_2 <=> complement(X0_2, X1_2)))).
% 0.19/0.54
% 0.19/0.54 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.54 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.54 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.54 fresh(y, y, x1...xn) = u
% 0.19/0.54 C => fresh(s, t, x1...xn) = v
% 0.19/0.54 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.54 variables of u and v.
% 0.19/0.54 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.54 input problem has no model of domain size 1).
% 0.19/0.54
% 0.19/0.54 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.54
% 0.19/0.54 Axiom 1 (goals_3): test(x3) = true.
% 0.19/0.54 Axiom 2 (right_annihilation): multiplication(X, zero) = zero.
% 0.19/0.54 Axiom 3 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.19/0.54 Axiom 4 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.19/0.54 Axiom 5 (additive_identity): addition(X, zero) = X.
% 0.19/0.54 Axiom 6 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 0.19/0.54 Axiom 7 (order): fresh15(X, X, Y, Z) = true.
% 0.19/0.55 Axiom 8 (test_2_1): fresh12(X, X, Y, Z) = one.
% 0.19/0.55 Axiom 9 (test_3): fresh9(X, X, Y, Z) = complement(Y, Z).
% 0.19/0.55 Axiom 10 (test_3): fresh8(X, X, Y, Z) = true.
% 0.19/0.55 Axiom 11 (order_1): fresh2(X, X, Y, Z) = Z.
% 0.19/0.55 Axiom 12 (test_3): fresh9(test(X), true, X, Y) = fresh8(c(X), Y, X, Y).
% 0.19/0.55 Axiom 13 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 0.19/0.55 Axiom 14 (order): fresh15(addition(X, Y), Y, X, Y) = leq(X, Y).
% 0.19/0.55 Axiom 15 (test_2_1): fresh12(complement(X, Y), true, Y, X) = addition(Y, X).
% 0.19/0.55 Axiom 16 (order_1): fresh2(leq(X, Y), true, X, Y) = addition(X, Y).
% 0.19/0.55 Axiom 17 (goals): leq(multiplication(multiplication(x2, x0), c(x3)), zero) = true.
% 0.19/0.55 Axiom 18 (goals_1): leq(multiplication(multiplication(x3, x1), c(x4)), zero) = true.
% 0.19/0.55
% 0.19/0.55 Lemma 19: addition(zero, X) = X.
% 0.19/0.55 Proof:
% 0.19/0.55 addition(zero, X)
% 0.19/0.55 = { by axiom 4 (additive_commutativity) R->L }
% 0.19/0.55 addition(X, zero)
% 0.19/0.55 = { by axiom 5 (additive_identity) }
% 0.19/0.55 X
% 0.19/0.55
% 0.19/0.55 Goal 1 (goals_5): leq(multiplication(multiplication(multiplication(x2, x0), x1), c(x4)), zero) = true.
% 0.19/0.55 Proof:
% 0.19/0.55 leq(multiplication(multiplication(multiplication(x2, x0), x1), c(x4)), zero)
% 0.19/0.55 = { by axiom 6 (multiplicative_associativity) R->L }
% 0.19/0.55 leq(multiplication(multiplication(x2, x0), multiplication(x1, c(x4))), zero)
% 0.19/0.55 = { by axiom 3 (multiplicative_right_identity) R->L }
% 0.19/0.55 leq(multiplication(multiplication(multiplication(x2, x0), one), multiplication(x1, c(x4))), zero)
% 0.19/0.55 = { by axiom 8 (test_2_1) R->L }
% 0.19/0.55 leq(multiplication(multiplication(multiplication(x2, x0), fresh12(true, true, c(x3), x3)), multiplication(x1, c(x4))), zero)
% 0.19/0.55 = { by axiom 10 (test_3) R->L }
% 0.19/0.55 leq(multiplication(multiplication(multiplication(x2, x0), fresh12(fresh8(c(x3), c(x3), x3, c(x3)), true, c(x3), x3)), multiplication(x1, c(x4))), zero)
% 0.19/0.55 = { by axiom 12 (test_3) R->L }
% 0.19/0.55 leq(multiplication(multiplication(multiplication(x2, x0), fresh12(fresh9(test(x3), true, x3, c(x3)), true, c(x3), x3)), multiplication(x1, c(x4))), zero)
% 0.19/0.55 = { by axiom 1 (goals_3) }
% 0.19/0.55 leq(multiplication(multiplication(multiplication(x2, x0), fresh12(fresh9(true, true, x3, c(x3)), true, c(x3), x3)), multiplication(x1, c(x4))), zero)
% 0.19/0.55 = { by axiom 9 (test_3) }
% 0.19/0.55 leq(multiplication(multiplication(multiplication(x2, x0), fresh12(complement(x3, c(x3)), true, c(x3), x3)), multiplication(x1, c(x4))), zero)
% 0.19/0.55 = { by axiom 15 (test_2_1) }
% 0.19/0.55 leq(multiplication(multiplication(multiplication(x2, x0), addition(c(x3), x3)), multiplication(x1, c(x4))), zero)
% 0.19/0.55 = { by axiom 13 (right_distributivity) }
% 0.19/0.55 leq(multiplication(addition(multiplication(multiplication(x2, x0), c(x3)), multiplication(multiplication(x2, x0), x3)), multiplication(x1, c(x4))), zero)
% 0.19/0.55 = { by axiom 6 (multiplicative_associativity) R->L }
% 0.19/0.55 leq(multiplication(addition(multiplication(x2, multiplication(x0, c(x3))), multiplication(multiplication(x2, x0), x3)), multiplication(x1, c(x4))), zero)
% 0.19/0.55 = { by axiom 5 (additive_identity) R->L }
% 0.19/0.55 leq(multiplication(addition(addition(multiplication(x2, multiplication(x0, c(x3))), zero), multiplication(multiplication(x2, x0), x3)), multiplication(x1, c(x4))), zero)
% 0.19/0.55 = { by axiom 16 (order_1) R->L }
% 0.19/0.55 leq(multiplication(addition(fresh2(leq(multiplication(x2, multiplication(x0, c(x3))), zero), true, multiplication(x2, multiplication(x0, c(x3))), zero), multiplication(multiplication(x2, x0), x3)), multiplication(x1, c(x4))), zero)
% 0.19/0.55 = { by axiom 6 (multiplicative_associativity) }
% 0.19/0.55 leq(multiplication(addition(fresh2(leq(multiplication(multiplication(x2, x0), c(x3)), zero), true, multiplication(x2, multiplication(x0, c(x3))), zero), multiplication(multiplication(x2, x0), x3)), multiplication(x1, c(x4))), zero)
% 0.19/0.55 = { by axiom 17 (goals) }
% 0.19/0.55 leq(multiplication(addition(fresh2(true, true, multiplication(x2, multiplication(x0, c(x3))), zero), multiplication(multiplication(x2, x0), x3)), multiplication(x1, c(x4))), zero)
% 0.19/0.55 = { by axiom 11 (order_1) }
% 0.19/0.55 leq(multiplication(addition(zero, multiplication(multiplication(x2, x0), x3)), multiplication(x1, c(x4))), zero)
% 0.19/0.55 = { by lemma 19 }
% 0.19/0.55 leq(multiplication(multiplication(multiplication(x2, x0), x3), multiplication(x1, c(x4))), zero)
% 0.19/0.55 = { by axiom 6 (multiplicative_associativity) R->L }
% 0.19/0.55 leq(multiplication(multiplication(x2, x0), multiplication(x3, multiplication(x1, c(x4)))), zero)
% 0.19/0.55 = { by axiom 5 (additive_identity) R->L }
% 0.19/0.55 leq(multiplication(multiplication(x2, x0), addition(multiplication(x3, multiplication(x1, c(x4))), zero)), zero)
% 0.19/0.55 = { by axiom 16 (order_1) R->L }
% 0.19/0.55 leq(multiplication(multiplication(x2, x0), fresh2(leq(multiplication(x3, multiplication(x1, c(x4))), zero), true, multiplication(x3, multiplication(x1, c(x4))), zero)), zero)
% 0.19/0.55 = { by axiom 6 (multiplicative_associativity) }
% 0.19/0.55 leq(multiplication(multiplication(x2, x0), fresh2(leq(multiplication(multiplication(x3, x1), c(x4)), zero), true, multiplication(x3, multiplication(x1, c(x4))), zero)), zero)
% 0.19/0.55 = { by axiom 18 (goals_1) }
% 0.19/0.55 leq(multiplication(multiplication(x2, x0), fresh2(true, true, multiplication(x3, multiplication(x1, c(x4))), zero)), zero)
% 0.19/0.55 = { by axiom 11 (order_1) }
% 0.19/0.55 leq(multiplication(multiplication(x2, x0), zero), zero)
% 0.19/0.55 = { by axiom 2 (right_annihilation) }
% 0.19/0.55 leq(zero, zero)
% 0.19/0.55 = { by axiom 14 (order) R->L }
% 0.19/0.55 fresh15(addition(zero, zero), zero, zero, zero)
% 0.19/0.55 = { by lemma 19 }
% 0.19/0.55 fresh15(zero, zero, zero, zero)
% 0.19/0.55 = { by axiom 7 (order) }
% 0.19/0.55 true
% 0.19/0.55 % SZS output end Proof
% 0.19/0.55
% 0.19/0.55 RESULT: Theorem (the conjecture is true).
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