TSTP Solution File: KLE048+1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE048+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 : n029.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 05:35:40 EDT 2023
% Result : Theorem 2.26s 0.69s
% Output : Proof 2.91s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.12 % Problem : KLE048+1 : TPTP v8.1.2. Released v4.0.0.
% 0.10/0.13 % 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 : n029.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 12:07:11 EDT 2023
% 0.13/0.34 % CPUTime :
% 2.26/0.69 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 2.26/0.69
% 2.26/0.69 % SZS status Theorem
% 2.26/0.69
% 2.26/0.71 % SZS output start Proof
% 2.26/0.71 Take the following subset of the input axioms:
% 2.26/0.71 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 2.26/0.71 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 2.26/0.71 fof(additive_idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 2.26/0.71 fof(goals, conjecture, ![X0]: (test(X0) => star(X0)=one)).
% 2.26/0.71 fof(left_distributivity, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 2.26/0.71 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 2.26/0.71 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 2.26/0.71 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 2.26/0.71 fof(right_distributivity, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 2.26/0.71 fof(star_induction_left, axiom, ![B2, C2, A2_2]: (leq(addition(multiplication(A2_2, B2), C2), B2) => leq(multiplication(star(A2_2), C2), B2))).
% 2.26/0.71 fof(star_induction_right, axiom, ![B2, C2, A2_2]: (leq(addition(multiplication(A2_2, B2), C2), A2_2) => leq(multiplication(C2, star(B2)), A2_2))).
% 2.26/0.71 fof(star_unfold_right, axiom, ![A3]: leq(addition(one, multiplication(A3, star(A3))), star(A3))).
% 2.26/0.71 fof(test_1, axiom, ![X0_2]: (test(X0_2) <=> ?[X1]: complement(X1, X0_2))).
% 2.26/0.71 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)))).
% 2.26/0.71
% 2.26/0.71 Now clausify the problem and encode Horn clauses using encoding 3 of
% 2.26/0.71 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 2.26/0.71 We repeatedly replace C & s=t => u=v by the two clauses:
% 2.26/0.71 fresh(y, y, x1...xn) = u
% 2.26/0.71 C => fresh(s, t, x1...xn) = v
% 2.26/0.71 where fresh is a fresh function symbol and x1..xn are the free
% 2.26/0.71 variables of u and v.
% 2.26/0.71 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 2.26/0.71 input problem has no model of domain size 1).
% 2.26/0.71
% 2.26/0.71 The encoding turns the above axioms into the following unit equations and goals:
% 2.26/0.71
% 2.26/0.71 Axiom 1 (goals): test(x0) = true.
% 2.26/0.71 Axiom 2 (multiplicative_right_identity): multiplication(X, one) = X.
% 2.26/0.71 Axiom 3 (multiplicative_left_identity): multiplication(one, X) = X.
% 2.26/0.71 Axiom 4 (additive_idempotence): addition(X, X) = X.
% 2.26/0.71 Axiom 5 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 2.26/0.71 Axiom 6 (test_1): fresh11(X, X, Y) = true.
% 2.26/0.71 Axiom 7 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 2.26/0.71 Axiom 8 (order): fresh13(X, X, Y, Z) = true.
% 2.26/0.71 Axiom 9 (test_1): fresh11(test(X), true, X) = complement(x1(X), X).
% 2.26/0.71 Axiom 10 (test_2_1): fresh8(X, X, Y, Z) = one.
% 2.26/0.71 Axiom 11 (order_1): fresh2(X, X, Y, Z) = Z.
% 2.26/0.71 Axiom 12 (star_induction_left): fresh14(X, X, Y, Z, W) = true.
% 2.26/0.71 Axiom 13 (star_induction_right): fresh12(X, X, Y, Z, W) = true.
% 2.26/0.71 Axiom 14 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 2.26/0.71 Axiom 15 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 2.26/0.71 Axiom 16 (order): fresh13(addition(X, Y), Y, X, Y) = leq(X, Y).
% 2.26/0.71 Axiom 17 (test_2_1): fresh8(complement(X, Y), true, Y, X) = addition(Y, X).
% 2.26/0.71 Axiom 18 (order_1): fresh2(leq(X, Y), true, X, Y) = addition(X, Y).
% 2.26/0.71 Axiom 19 (star_unfold_right): leq(addition(one, multiplication(X, star(X))), star(X)) = true.
% 2.26/0.71 Axiom 20 (star_induction_left): fresh14(leq(addition(multiplication(X, Y), Z), Y), true, X, Y, Z) = leq(multiplication(star(X), Z), Y).
% 2.26/0.71 Axiom 21 (star_induction_right): fresh12(leq(addition(multiplication(X, Y), Z), X), true, X, Y, Z) = leq(multiplication(Z, star(Y)), X).
% 2.26/0.71
% 2.26/0.71 Lemma 22: addition(X, multiplication(Y, X)) = multiplication(addition(Y, one), X).
% 2.26/0.71 Proof:
% 2.26/0.71 addition(X, multiplication(Y, X))
% 2.26/0.71 = { by axiom 3 (multiplicative_left_identity) R->L }
% 2.26/0.71 addition(multiplication(one, X), multiplication(Y, X))
% 2.26/0.71 = { by axiom 15 (left_distributivity) R->L }
% 2.26/0.71 multiplication(addition(one, Y), X)
% 2.26/0.71 = { by axiom 5 (additive_commutativity) }
% 2.26/0.71 multiplication(addition(Y, one), X)
% 2.26/0.71
% 2.26/0.71 Lemma 23: addition(one, multiplication(addition(X, one), star(X))) = star(X).
% 2.26/0.71 Proof:
% 2.26/0.71 addition(one, multiplication(addition(X, one), star(X)))
% 2.26/0.71 = { by lemma 22 R->L }
% 2.26/0.71 addition(one, addition(star(X), multiplication(X, star(X))))
% 2.26/0.71 = { by axiom 5 (additive_commutativity) R->L }
% 2.26/0.71 addition(one, addition(multiplication(X, star(X)), star(X)))
% 2.26/0.71 = { by axiom 7 (additive_associativity) }
% 2.26/0.71 addition(addition(one, multiplication(X, star(X))), star(X))
% 2.26/0.71 = { by axiom 18 (order_1) R->L }
% 2.26/0.71 fresh2(leq(addition(one, multiplication(X, star(X))), star(X)), true, addition(one, multiplication(X, star(X))), star(X))
% 2.26/0.71 = { by axiom 19 (star_unfold_right) }
% 2.26/0.71 fresh2(true, true, addition(one, multiplication(X, star(X))), star(X))
% 2.26/0.71 = { by axiom 11 (order_1) }
% 2.26/0.71 star(X)
% 2.26/0.71
% 2.26/0.71 Lemma 24: addition(X, addition(X, Y)) = addition(X, Y).
% 2.26/0.71 Proof:
% 2.26/0.71 addition(X, addition(X, Y))
% 2.26/0.71 = { by axiom 7 (additive_associativity) }
% 2.26/0.71 addition(addition(X, X), Y)
% 2.26/0.71 = { by axiom 4 (additive_idempotence) }
% 2.26/0.71 addition(X, Y)
% 2.26/0.71
% 2.26/0.71 Lemma 25: addition(one, star(X)) = star(X).
% 2.26/0.71 Proof:
% 2.26/0.71 addition(one, star(X))
% 2.26/0.71 = { by lemma 23 R->L }
% 2.26/0.71 addition(one, addition(one, multiplication(addition(X, one), star(X))))
% 2.91/0.71 = { by lemma 24 }
% 2.91/0.71 addition(one, multiplication(addition(X, one), star(X)))
% 2.91/0.71 = { by lemma 23 }
% 2.91/0.71 star(X)
% 2.91/0.71
% 2.91/0.71 Lemma 26: addition(x0, x1(x0)) = one.
% 2.91/0.71 Proof:
% 2.91/0.71 addition(x0, x1(x0))
% 2.91/0.71 = { by axiom 17 (test_2_1) R->L }
% 2.91/0.71 fresh8(complement(x1(x0), x0), true, x0, x1(x0))
% 2.91/0.71 = { by axiom 9 (test_1) R->L }
% 2.91/0.71 fresh8(fresh11(test(x0), true, x0), true, x0, x1(x0))
% 2.91/0.71 = { by axiom 1 (goals) }
% 2.91/0.71 fresh8(fresh11(true, true, x0), true, x0, x1(x0))
% 2.91/0.71 = { by axiom 6 (test_1) }
% 2.91/0.71 fresh8(true, true, x0, x1(x0))
% 2.91/0.71 = { by axiom 10 (test_2_1) }
% 2.91/0.71 one
% 2.91/0.71
% 2.91/0.71 Lemma 27: star(star(x0)) = one.
% 2.91/0.71 Proof:
% 2.91/0.71 star(star(x0))
% 2.91/0.71 = { by lemma 25 R->L }
% 2.91/0.71 addition(one, star(star(x0)))
% 2.91/0.71 = { by axiom 5 (additive_commutativity) R->L }
% 2.91/0.71 addition(star(star(x0)), one)
% 2.91/0.71 = { by axiom 18 (order_1) R->L }
% 2.91/0.71 fresh2(leq(star(star(x0)), one), true, star(star(x0)), one)
% 2.91/0.71 = { by axiom 2 (multiplicative_right_identity) R->L }
% 2.91/0.71 fresh2(leq(multiplication(star(star(x0)), one), one), true, star(star(x0)), one)
% 2.91/0.71 = { by axiom 20 (star_induction_left) R->L }
% 2.91/0.71 fresh2(fresh14(leq(addition(multiplication(star(x0), one), one), one), true, star(x0), one, one), true, star(star(x0)), one)
% 2.91/0.71 = { by axiom 2 (multiplicative_right_identity) }
% 2.91/0.72 fresh2(fresh14(leq(addition(star(x0), one), one), true, star(x0), one, one), true, star(star(x0)), one)
% 2.91/0.72 = { by axiom 5 (additive_commutativity) }
% 2.91/0.72 fresh2(fresh14(leq(addition(one, star(x0)), one), true, star(x0), one, one), true, star(star(x0)), one)
% 2.91/0.72 = { by lemma 25 }
% 2.91/0.72 fresh2(fresh14(leq(star(x0), one), true, star(x0), one, one), true, star(star(x0)), one)
% 2.91/0.72 = { by axiom 4 (additive_idempotence) R->L }
% 2.91/0.72 fresh2(fresh14(leq(star(x0), addition(one, one)), true, star(x0), one, one), true, star(star(x0)), one)
% 2.91/0.72 = { by axiom 3 (multiplicative_left_identity) R->L }
% 2.91/0.72 fresh2(fresh14(leq(multiplication(one, star(x0)), addition(one, one)), true, star(x0), one, one), true, star(star(x0)), one)
% 2.91/0.72 = { by axiom 21 (star_induction_right) R->L }
% 2.91/0.72 fresh2(fresh14(fresh12(leq(addition(multiplication(addition(one, one), x0), one), addition(one, one)), true, addition(one, one), x0, one), true, star(x0), one, one), true, star(star(x0)), one)
% 2.91/0.72 = { by axiom 5 (additive_commutativity) }
% 2.91/0.72 fresh2(fresh14(fresh12(leq(addition(one, multiplication(addition(one, one), x0)), addition(one, one)), true, addition(one, one), x0, one), true, star(x0), one, one), true, star(star(x0)), one)
% 2.91/0.72 = { by axiom 2 (multiplicative_right_identity) R->L }
% 2.91/0.72 fresh2(fresh14(fresh12(leq(addition(multiplication(one, one), multiplication(addition(one, one), x0)), addition(one, one)), true, addition(one, one), x0, one), true, star(x0), one, one), true, star(star(x0)), one)
% 2.91/0.72 = { by lemma 22 R->L }
% 2.91/0.72 fresh2(fresh14(fresh12(leq(addition(multiplication(one, one), addition(x0, multiplication(one, x0))), addition(one, one)), true, addition(one, one), x0, one), true, star(x0), one, one), true, star(star(x0)), one)
% 2.91/0.72 = { by axiom 5 (additive_commutativity) R->L }
% 2.91/0.72 fresh2(fresh14(fresh12(leq(addition(multiplication(one, one), addition(multiplication(one, x0), x0)), addition(one, one)), true, addition(one, one), x0, one), true, star(x0), one, one), true, star(star(x0)), one)
% 2.91/0.72 = { by axiom 7 (additive_associativity) }
% 2.91/0.72 fresh2(fresh14(fresh12(leq(addition(addition(multiplication(one, one), multiplication(one, x0)), x0), addition(one, one)), true, addition(one, one), x0, one), true, star(x0), one, one), true, star(star(x0)), one)
% 2.91/0.72 = { by axiom 14 (right_distributivity) R->L }
% 2.91/0.72 fresh2(fresh14(fresh12(leq(addition(multiplication(one, addition(one, x0)), x0), addition(one, one)), true, addition(one, one), x0, one), true, star(x0), one, one), true, star(star(x0)), one)
% 2.91/0.72 = { by axiom 5 (additive_commutativity) }
% 2.91/0.72 fresh2(fresh14(fresh12(leq(addition(x0, multiplication(one, addition(one, x0))), addition(one, one)), true, addition(one, one), x0, one), true, star(x0), one, one), true, star(star(x0)), one)
% 2.91/0.72 = { by axiom 5 (additive_commutativity) R->L }
% 2.91/0.72 fresh2(fresh14(fresh12(leq(addition(x0, multiplication(one, addition(x0, one))), addition(one, one)), true, addition(one, one), x0, one), true, star(x0), one, one), true, star(star(x0)), one)
% 2.91/0.72 = { by lemma 26 R->L }
% 2.91/0.72 fresh2(fresh14(fresh12(leq(addition(x0, multiplication(one, addition(x0, addition(x0, x1(x0))))), addition(one, one)), true, addition(one, one), x0, one), true, star(x0), one, one), true, star(star(x0)), one)
% 2.91/0.72 = { by lemma 24 }
% 2.91/0.72 fresh2(fresh14(fresh12(leq(addition(x0, multiplication(one, addition(x0, x1(x0)))), addition(one, one)), true, addition(one, one), x0, one), true, star(x0), one, one), true, star(star(x0)), one)
% 2.91/0.72 = { by lemma 26 }
% 2.91/0.72 fresh2(fresh14(fresh12(leq(addition(x0, multiplication(one, one)), addition(one, one)), true, addition(one, one), x0, one), true, star(x0), one, one), true, star(star(x0)), one)
% 2.91/0.72 = { by axiom 2 (multiplicative_right_identity) }
% 2.91/0.72 fresh2(fresh14(fresh12(leq(addition(x0, one), addition(one, one)), true, addition(one, one), x0, one), true, star(x0), one, one), true, star(star(x0)), one)
% 2.91/0.72 = { by axiom 5 (additive_commutativity) }
% 2.91/0.72 fresh2(fresh14(fresh12(leq(addition(one, x0), addition(one, one)), true, addition(one, one), x0, one), true, star(x0), one, one), true, star(star(x0)), one)
% 2.91/0.72 = { by lemma 26 R->L }
% 2.91/0.72 fresh2(fresh14(fresh12(leq(addition(one, x0), addition(one, addition(x0, x1(x0)))), true, addition(one, one), x0, one), true, star(x0), one, one), true, star(star(x0)), one)
% 2.91/0.72 = { by axiom 7 (additive_associativity) }
% 2.91/0.72 fresh2(fresh14(fresh12(leq(addition(one, x0), addition(addition(one, x0), x1(x0))), true, addition(one, one), x0, one), true, star(x0), one, one), true, star(star(x0)), one)
% 2.91/0.72 = { by axiom 16 (order) R->L }
% 2.91/0.72 fresh2(fresh14(fresh12(fresh13(addition(addition(one, x0), addition(addition(one, x0), x1(x0))), addition(addition(one, x0), x1(x0)), addition(one, x0), addition(addition(one, x0), x1(x0))), true, addition(one, one), x0, one), true, star(x0), one, one), true, star(star(x0)), one)
% 2.91/0.72 = { by lemma 24 }
% 2.91/0.72 fresh2(fresh14(fresh12(fresh13(addition(addition(one, x0), x1(x0)), addition(addition(one, x0), x1(x0)), addition(one, x0), addition(addition(one, x0), x1(x0))), true, addition(one, one), x0, one), true, star(x0), one, one), true, star(star(x0)), one)
% 2.91/0.72 = { by axiom 8 (order) }
% 2.91/0.72 fresh2(fresh14(fresh12(true, true, addition(one, one), x0, one), true, star(x0), one, one), true, star(star(x0)), one)
% 2.91/0.72 = { by axiom 13 (star_induction_right) }
% 2.91/0.72 fresh2(fresh14(true, true, star(x0), one, one), true, star(star(x0)), one)
% 2.91/0.72 = { by axiom 12 (star_induction_left) }
% 2.91/0.72 fresh2(true, true, star(star(x0)), one)
% 2.91/0.72 = { by axiom 11 (order_1) }
% 2.91/0.72 one
% 2.91/0.72
% 2.91/0.72 Goal 1 (goals_1): star(x0) = one.
% 2.91/0.72 Proof:
% 2.91/0.72 star(x0)
% 2.91/0.72 = { by lemma 25 R->L }
% 2.91/0.72 addition(one, star(x0))
% 2.91/0.72 = { by axiom 5 (additive_commutativity) R->L }
% 2.91/0.72 addition(star(x0), one)
% 2.91/0.72 = { by lemma 25 R->L }
% 2.91/0.72 addition(addition(one, star(x0)), one)
% 2.91/0.72 = { by axiom 2 (multiplicative_right_identity) R->L }
% 2.91/0.72 multiplication(addition(addition(one, star(x0)), one), one)
% 2.91/0.72 = { by lemma 22 R->L }
% 2.91/0.72 addition(one, multiplication(addition(one, star(x0)), one))
% 2.91/0.72 = { by lemma 27 R->L }
% 2.91/0.72 addition(one, multiplication(addition(one, star(x0)), star(star(x0))))
% 2.91/0.72 = { by axiom 5 (additive_commutativity) R->L }
% 2.91/0.72 addition(one, multiplication(addition(star(x0), one), star(star(x0))))
% 2.91/0.72 = { by lemma 23 }
% 2.91/0.72 star(star(x0))
% 2.91/0.72 = { by lemma 27 }
% 2.91/0.72 one
% 2.91/0.72 % SZS output end Proof
% 2.91/0.72
% 2.91/0.72 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------