TSTP Solution File: KLE169+1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE169+1 : TPTP v8.1.2. Released v5.2.0.
% 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 : n010.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 05:36:09 EDT 2023
% Result : Theorem 0.20s 0.47s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : KLE169+1 : TPTP v8.1.2. Released v5.2.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.33 % Computer : n010.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Tue Aug 29 11:29:18 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.20/0.47 Command-line arguments: --ground-connectedness --complete-subsets
% 0.20/0.47
% 0.20/0.47 % SZS status Theorem
% 0.20/0.47
% 0.20/0.48 % SZS output start Proof
% 0.20/0.48 Take the following subset of the input axioms:
% 0.20/0.48 fof(a, conjecture, leq(multiplication(a, multiplication(b, a)), multiplication(star(sigma), multiplication(a, multiplication(sigma, a))))).
% 0.20/0.48 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 0.20/0.48 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 0.20/0.48 fof(additive_idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 0.20/0.48 fof(an, axiom, sigma=addition(a, b)).
% 0.20/0.48 fof(left_distributivity, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 0.20/0.48 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 0.20/0.48 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 0.20/0.48 fof(right_distributivity, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 0.20/0.48 fof(star_unfold_right, axiom, ![A3]: leq(addition(one, multiplication(A3, star(A3))), star(A3))).
% 0.20/0.48
% 0.20/0.48 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.48 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.48 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.48 fresh(y, y, x1...xn) = u
% 0.20/0.48 C => fresh(s, t, x1...xn) = v
% 0.20/0.48 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.48 variables of u and v.
% 0.20/0.48 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.48 input problem has no model of domain size 1).
% 0.20/0.48
% 0.20/0.48 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.48
% 0.20/0.48 Axiom 1 (additive_idempotence): addition(X, X) = X.
% 0.20/0.48 Axiom 2 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.20/0.48 Axiom 3 (an): sigma = addition(a, b).
% 0.20/0.48 Axiom 4 (multiplicative_left_identity): multiplication(one, X) = X.
% 0.20/0.48 Axiom 5 (order_1): fresh(X, X, Y, Z) = Z.
% 0.20/0.48 Axiom 6 (order): fresh3(X, X, Y, Z) = true.
% 0.20/0.48 Axiom 7 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 0.20/0.48 Axiom 8 (order_1): fresh(leq(X, Y), true, X, Y) = addition(X, Y).
% 0.20/0.48 Axiom 9 (order): fresh3(addition(X, Y), Y, X, Y) = leq(X, Y).
% 0.20/0.48 Axiom 10 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 0.20/0.48 Axiom 11 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 0.20/0.48 Axiom 12 (star_unfold_right): leq(addition(one, multiplication(X, star(X))), star(X)) = true.
% 0.20/0.48
% 0.20/0.48 Lemma 13: addition(b, a) = sigma.
% 0.20/0.48 Proof:
% 0.20/0.48 addition(b, a)
% 0.20/0.48 = { by axiom 2 (additive_commutativity) R->L }
% 0.20/0.48 addition(a, b)
% 0.20/0.48 = { by axiom 3 (an) R->L }
% 0.20/0.48 sigma
% 0.20/0.48
% 0.20/0.48 Lemma 14: addition(X, addition(X, Y)) = addition(X, Y).
% 0.20/0.48 Proof:
% 0.20/0.48 addition(X, addition(X, Y))
% 0.20/0.48 = { by axiom 7 (additive_associativity) }
% 0.20/0.48 addition(addition(X, X), Y)
% 0.20/0.48 = { by axiom 1 (additive_idempotence) }
% 0.20/0.48 addition(X, Y)
% 0.20/0.48
% 0.20/0.48 Lemma 15: addition(Y, addition(X, Z)) = addition(X, addition(Y, Z)).
% 0.20/0.48 Proof:
% 0.20/0.48 addition(Y, addition(X, Z))
% 0.20/0.48 = { by axiom 2 (additive_commutativity) R->L }
% 0.20/0.48 addition(addition(X, Z), Y)
% 0.20/0.48 = { by axiom 7 (additive_associativity) R->L }
% 0.20/0.48 addition(X, addition(Z, Y))
% 0.20/0.48 = { by axiom 2 (additive_commutativity) }
% 0.20/0.48 addition(X, addition(Y, Z))
% 0.20/0.48
% 0.20/0.48 Lemma 16: leq(X, addition(Y, addition(X, Z))) = true.
% 0.20/0.48 Proof:
% 0.20/0.48 leq(X, addition(Y, addition(X, Z)))
% 0.20/0.48 = { by lemma 15 }
% 0.20/0.48 leq(X, addition(X, addition(Y, Z)))
% 0.20/0.48 = { by axiom 9 (order) R->L }
% 0.20/0.48 fresh3(addition(X, addition(X, addition(Y, Z))), addition(X, addition(Y, Z)), X, addition(X, addition(Y, Z)))
% 0.20/0.48 = { by lemma 14 }
% 0.20/0.48 fresh3(addition(X, addition(Y, Z)), addition(X, addition(Y, Z)), X, addition(X, addition(Y, Z)))
% 0.20/0.48 = { by axiom 6 (order) }
% 0.20/0.48 true
% 0.20/0.48
% 0.20/0.48 Goal 1 (a): leq(multiplication(a, multiplication(b, a)), multiplication(star(sigma), multiplication(a, multiplication(sigma, a)))) = true.
% 0.20/0.48 Proof:
% 0.20/0.48 leq(multiplication(a, multiplication(b, a)), multiplication(star(sigma), multiplication(a, multiplication(sigma, a))))
% 0.20/0.48 = { by axiom 5 (order_1) R->L }
% 0.20/0.48 leq(multiplication(a, multiplication(b, a)), multiplication(fresh(true, true, one, star(sigma)), multiplication(a, multiplication(sigma, a))))
% 0.20/0.48 = { by lemma 16 R->L }
% 0.20/0.48 leq(multiplication(a, multiplication(b, a)), multiplication(fresh(leq(one, addition(star(sigma), addition(one, multiplication(sigma, star(sigma))))), true, one, star(sigma)), multiplication(a, multiplication(sigma, a))))
% 0.20/0.48 = { by lemma 15 }
% 0.20/0.48 leq(multiplication(a, multiplication(b, a)), multiplication(fresh(leq(one, addition(one, addition(star(sigma), multiplication(sigma, star(sigma))))), true, one, star(sigma)), multiplication(a, multiplication(sigma, a))))
% 0.20/0.48 = { by axiom 2 (additive_commutativity) R->L }
% 0.20/0.48 leq(multiplication(a, multiplication(b, a)), multiplication(fresh(leq(one, addition(one, addition(multiplication(sigma, star(sigma)), star(sigma)))), true, one, star(sigma)), multiplication(a, multiplication(sigma, a))))
% 0.20/0.48 = { by axiom 7 (additive_associativity) }
% 0.20/0.48 leq(multiplication(a, multiplication(b, a)), multiplication(fresh(leq(one, addition(addition(one, multiplication(sigma, star(sigma))), star(sigma))), true, one, star(sigma)), multiplication(a, multiplication(sigma, a))))
% 0.20/0.48 = { by axiom 8 (order_1) R->L }
% 0.20/0.48 leq(multiplication(a, multiplication(b, a)), multiplication(fresh(leq(one, fresh(leq(addition(one, multiplication(sigma, star(sigma))), star(sigma)), true, addition(one, multiplication(sigma, star(sigma))), star(sigma))), true, one, star(sigma)), multiplication(a, multiplication(sigma, a))))
% 0.20/0.48 = { by axiom 12 (star_unfold_right) }
% 0.20/0.48 leq(multiplication(a, multiplication(b, a)), multiplication(fresh(leq(one, fresh(true, true, addition(one, multiplication(sigma, star(sigma))), star(sigma))), true, one, star(sigma)), multiplication(a, multiplication(sigma, a))))
% 0.20/0.48 = { by axiom 5 (order_1) }
% 0.20/0.48 leq(multiplication(a, multiplication(b, a)), multiplication(fresh(leq(one, star(sigma)), true, one, star(sigma)), multiplication(a, multiplication(sigma, a))))
% 0.20/0.48 = { by axiom 8 (order_1) }
% 0.20/0.48 leq(multiplication(a, multiplication(b, a)), multiplication(addition(one, star(sigma)), multiplication(a, multiplication(sigma, a))))
% 0.20/0.49 = { by axiom 2 (additive_commutativity) }
% 0.20/0.49 leq(multiplication(a, multiplication(b, a)), multiplication(addition(star(sigma), one), multiplication(a, multiplication(sigma, a))))
% 0.20/0.49 = { by axiom 11 (left_distributivity) }
% 0.20/0.49 leq(multiplication(a, multiplication(b, a)), addition(multiplication(star(sigma), multiplication(a, multiplication(sigma, a))), multiplication(one, multiplication(a, multiplication(sigma, a)))))
% 0.20/0.49 = { by axiom 4 (multiplicative_left_identity) }
% 0.20/0.49 leq(multiplication(a, multiplication(b, a)), addition(multiplication(star(sigma), multiplication(a, multiplication(sigma, a))), multiplication(a, multiplication(sigma, a))))
% 0.20/0.49 = { by lemma 13 R->L }
% 0.20/0.49 leq(multiplication(a, multiplication(b, a)), addition(multiplication(star(sigma), multiplication(a, multiplication(sigma, a))), multiplication(a, multiplication(addition(b, a), a))))
% 0.20/0.49 = { by lemma 14 R->L }
% 0.20/0.49 leq(multiplication(a, multiplication(b, a)), addition(multiplication(star(sigma), multiplication(a, multiplication(sigma, a))), multiplication(a, multiplication(addition(b, addition(b, a)), a))))
% 0.20/0.49 = { by lemma 13 }
% 0.20/0.49 leq(multiplication(a, multiplication(b, a)), addition(multiplication(star(sigma), multiplication(a, multiplication(sigma, a))), multiplication(a, multiplication(addition(b, sigma), a))))
% 0.20/0.49 = { by axiom 2 (additive_commutativity) R->L }
% 0.20/0.49 leq(multiplication(a, multiplication(b, a)), addition(multiplication(star(sigma), multiplication(a, multiplication(sigma, a))), multiplication(a, multiplication(addition(sigma, b), a))))
% 0.20/0.49 = { by axiom 11 (left_distributivity) }
% 0.20/0.49 leq(multiplication(a, multiplication(b, a)), addition(multiplication(star(sigma), multiplication(a, multiplication(sigma, a))), multiplication(a, addition(multiplication(sigma, a), multiplication(b, a)))))
% 0.20/0.49 = { by axiom 2 (additive_commutativity) R->L }
% 0.20/0.49 leq(multiplication(a, multiplication(b, a)), addition(multiplication(star(sigma), multiplication(a, multiplication(sigma, a))), multiplication(a, addition(multiplication(b, a), multiplication(sigma, a)))))
% 0.20/0.49 = { by axiom 10 (right_distributivity) }
% 0.20/0.49 leq(multiplication(a, multiplication(b, a)), addition(multiplication(star(sigma), multiplication(a, multiplication(sigma, a))), addition(multiplication(a, multiplication(b, a)), multiplication(a, multiplication(sigma, a)))))
% 0.20/0.49 = { by lemma 16 }
% 0.20/0.49 true
% 0.20/0.49 % SZS output end Proof
% 0.20/0.49
% 0.20/0.49 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------