TSTP Solution File: KLE170+1.002 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : KLE170+1.002 : 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 : n007.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 0.52s
% Output : Proof 1.01s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : KLE170+1.002 : TPTP v8.1.2. Released v5.2.0.
% 0.03/0.13 % 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 : n007.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
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Tue Aug 29 11:56:43 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.20/0.52 Command-line arguments: --flatten
% 0.20/0.52
% 0.20/0.52 % SZS status Theorem
% 0.20/0.52
% 0.20/0.53 % SZS output start Proof
% 0.20/0.53 Take the following subset of the input axioms:
% 0.20/0.53 fof(a, conjecture, leq(multiplication(a, a), star(a))).
% 0.20/0.54 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 0.20/0.54 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 0.20/0.54 fof(additive_idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 0.20/0.54 fof(left_distributivity, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 0.20/0.54 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 0.20/0.54 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 0.20/0.54 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 0.20/0.54 fof(right_distributivity, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 0.20/0.54 fof(star_unfold_left, axiom, ![A3]: leq(addition(one, multiplication(star(A3), A3)), star(A3))).
% 0.20/0.54 fof(star_unfold_right, axiom, ![A3]: leq(addition(one, multiplication(A3, star(A3))), star(A3))).
% 0.20/0.54
% 0.20/0.54 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.54 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.54 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.54 fresh(y, y, x1...xn) = u
% 0.20/0.54 C => fresh(s, t, x1...xn) = v
% 0.20/0.54 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.54 variables of u and v.
% 0.20/0.54 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.54 input problem has no model of domain size 1).
% 0.20/0.54
% 0.20/0.54 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.54
% 0.20/0.54 Axiom 1 (additive_idempotence): addition(X, X) = X.
% 0.20/0.54 Axiom 2 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.20/0.54 Axiom 3 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.20/0.54 Axiom 4 (multiplicative_left_identity): multiplication(one, X) = X.
% 0.20/0.54 Axiom 5 (order_1): fresh(X, X, Y, Z) = Z.
% 0.20/0.54 Axiom 6 (order): fresh3(X, X, Y, Z) = true.
% 0.20/0.54 Axiom 7 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 0.20/0.54 Axiom 8 (order_1): fresh(leq(X, Y), true, X, Y) = addition(X, Y).
% 0.20/0.54 Axiom 9 (order): fresh3(addition(X, Y), Y, X, Y) = leq(X, Y).
% 0.20/0.54 Axiom 10 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 0.20/0.54 Axiom 11 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 0.20/0.54 Axiom 12 (star_unfold_right): leq(addition(one, multiplication(X, star(X))), star(X)) = true.
% 0.20/0.54 Axiom 13 (star_unfold_left): leq(addition(one, multiplication(star(X), X)), star(X)) = true.
% 0.20/0.54
% 0.20/0.54 Lemma 14: multiplication(addition(one, Y), X) = addition(X, multiplication(Y, X)).
% 0.20/0.54 Proof:
% 0.20/0.54 multiplication(addition(one, Y), X)
% 0.20/0.54 = { by axiom 11 (left_distributivity) }
% 0.20/0.54 addition(multiplication(one, X), multiplication(Y, X))
% 0.20/0.54 = { by axiom 4 (multiplicative_left_identity) }
% 0.20/0.54 addition(X, multiplication(Y, X))
% 0.20/0.54
% 0.20/0.54 Lemma 15: addition(one, multiplication(addition(X, one), star(X))) = star(X).
% 0.20/0.54 Proof:
% 1.01/0.54 addition(one, multiplication(addition(X, one), star(X)))
% 1.01/0.54 = { by axiom 2 (additive_commutativity) R->L }
% 1.01/0.54 addition(one, multiplication(addition(one, X), star(X)))
% 1.01/0.54 = { by lemma 14 }
% 1.01/0.54 addition(one, addition(star(X), multiplication(X, star(X))))
% 1.01/0.54 = { by axiom 2 (additive_commutativity) R->L }
% 1.01/0.54 addition(one, addition(multiplication(X, star(X)), star(X)))
% 1.01/0.54 = { by axiom 7 (additive_associativity) }
% 1.01/0.54 addition(addition(one, multiplication(X, star(X))), star(X))
% 1.01/0.54 = { by axiom 8 (order_1) R->L }
% 1.01/0.54 fresh(leq(addition(one, multiplication(X, star(X))), star(X)), true, addition(one, multiplication(X, star(X))), star(X))
% 1.01/0.54 = { by axiom 12 (star_unfold_right) }
% 1.01/0.54 fresh(true, true, addition(one, multiplication(X, star(X))), star(X))
% 1.01/0.54 = { by axiom 5 (order_1) }
% 1.01/0.54 star(X)
% 1.01/0.54
% 1.01/0.54 Lemma 16: addition(one, star(X)) = star(X).
% 1.01/0.54 Proof:
% 1.01/0.54 addition(one, star(X))
% 1.01/0.54 = { by lemma 15 R->L }
% 1.01/0.54 addition(one, addition(one, multiplication(addition(X, one), star(X))))
% 1.01/0.54 = { by axiom 7 (additive_associativity) }
% 1.01/0.54 addition(addition(one, one), multiplication(addition(X, one), star(X)))
% 1.01/0.54 = { by axiom 1 (additive_idempotence) }
% 1.01/0.54 addition(one, multiplication(addition(X, one), star(X)))
% 1.01/0.54 = { by lemma 15 }
% 1.01/0.54 star(X)
% 1.01/0.54
% 1.01/0.54 Lemma 17: multiplication(X, addition(Y, one)) = addition(X, multiplication(X, Y)).
% 1.01/0.54 Proof:
% 1.01/0.54 multiplication(X, addition(Y, one))
% 1.01/0.54 = { by axiom 2 (additive_commutativity) R->L }
% 1.01/0.54 multiplication(X, addition(one, Y))
% 1.01/0.54 = { by axiom 10 (right_distributivity) }
% 1.01/0.54 addition(multiplication(X, one), multiplication(X, Y))
% 1.01/0.54 = { by axiom 3 (multiplicative_right_identity) }
% 1.01/0.54 addition(X, multiplication(X, Y))
% 1.01/0.54
% 1.01/0.54 Lemma 18: multiplication(star(X), addition(X, one)) = star(X).
% 1.01/0.54 Proof:
% 1.01/0.54 multiplication(star(X), addition(X, one))
% 1.01/0.54 = { by lemma 17 }
% 1.01/0.54 addition(star(X), multiplication(star(X), X))
% 1.01/0.54 = { by lemma 16 R->L }
% 1.01/0.54 addition(addition(one, star(X)), multiplication(star(X), X))
% 1.01/0.54 = { by axiom 7 (additive_associativity) R->L }
% 1.01/0.54 addition(one, addition(star(X), multiplication(star(X), X)))
% 1.01/0.54 = { by axiom 2 (additive_commutativity) }
% 1.01/0.54 addition(one, addition(multiplication(star(X), X), star(X)))
% 1.01/0.54 = { by axiom 7 (additive_associativity) }
% 1.01/0.54 addition(addition(one, multiplication(star(X), X)), star(X))
% 1.01/0.54 = { by axiom 8 (order_1) R->L }
% 1.01/0.54 fresh(leq(addition(one, multiplication(star(X), X)), star(X)), true, addition(one, multiplication(star(X), X)), star(X))
% 1.01/0.54 = { by axiom 13 (star_unfold_left) }
% 1.01/0.54 fresh(true, true, addition(one, multiplication(star(X), X)), star(X))
% 1.01/0.54 = { by axiom 5 (order_1) }
% 1.01/0.54 star(X)
% 1.01/0.54
% 1.01/0.54 Lemma 19: addition(star(X), multiplication(star(X), X)) = star(X).
% 1.01/0.54 Proof:
% 1.01/0.54 addition(star(X), multiplication(star(X), X))
% 1.01/0.54 = { by lemma 17 R->L }
% 1.01/0.54 multiplication(star(X), addition(X, one))
% 1.01/0.54 = { by lemma 18 }
% 1.01/0.54 star(X)
% 1.01/0.54
% 1.01/0.54 Goal 1 (a): leq(multiplication(a, a), star(a)) = true.
% 1.01/0.54 Proof:
% 1.01/0.54 leq(multiplication(a, a), star(a))
% 1.01/0.54 = { by axiom 9 (order) R->L }
% 1.01/0.54 fresh3(addition(multiplication(a, a), star(a)), star(a), multiplication(a, a), star(a))
% 1.01/0.54 = { by axiom 2 (additive_commutativity) R->L }
% 1.01/0.54 fresh3(addition(star(a), multiplication(a, a)), star(a), multiplication(a, a), star(a))
% 1.01/0.54 = { by lemma 19 R->L }
% 1.01/0.54 fresh3(addition(addition(star(a), multiplication(star(a), a)), multiplication(a, a)), star(a), multiplication(a, a), star(a))
% 1.01/0.54 = { by axiom 7 (additive_associativity) R->L }
% 1.01/0.54 fresh3(addition(star(a), addition(multiplication(star(a), a), multiplication(a, a))), star(a), multiplication(a, a), star(a))
% 1.01/0.54 = { by axiom 2 (additive_commutativity) }
% 1.01/0.54 fresh3(addition(star(a), addition(multiplication(a, a), multiplication(star(a), a))), star(a), multiplication(a, a), star(a))
% 1.01/0.54 = { by axiom 11 (left_distributivity) R->L }
% 1.01/0.54 fresh3(addition(star(a), multiplication(addition(a, star(a)), a)), star(a), multiplication(a, a), star(a))
% 1.01/0.54 = { by lemma 16 R->L }
% 1.01/0.54 fresh3(addition(star(a), multiplication(addition(a, addition(one, star(a))), a)), star(a), multiplication(a, a), star(a))
% 1.01/0.54 = { by axiom 7 (additive_associativity) }
% 1.01/0.54 fresh3(addition(star(a), multiplication(addition(addition(a, one), star(a)), a)), star(a), multiplication(a, a), star(a))
% 1.01/0.54 = { by lemma 18 R->L }
% 1.01/0.54 fresh3(addition(star(a), multiplication(addition(addition(a, one), multiplication(star(a), addition(a, one))), a)), star(a), multiplication(a, a), star(a))
% 1.01/0.54 = { by lemma 14 R->L }
% 1.01/0.54 fresh3(addition(star(a), multiplication(multiplication(addition(one, star(a)), addition(a, one)), a)), star(a), multiplication(a, a), star(a))
% 1.01/0.54 = { by lemma 16 }
% 1.01/0.54 fresh3(addition(star(a), multiplication(multiplication(star(a), addition(a, one)), a)), star(a), multiplication(a, a), star(a))
% 1.01/0.54 = { by lemma 18 }
% 1.01/0.54 fresh3(addition(star(a), multiplication(star(a), a)), star(a), multiplication(a, a), star(a))
% 1.01/0.54 = { by lemma 19 }
% 1.01/0.54 fresh3(star(a), star(a), multiplication(a, a), star(a))
% 1.01/0.54 = { by axiom 6 (order) }
% 1.01/0.54 true
% 1.01/0.54 % SZS output end Proof
% 1.01/0.54
% 1.01/0.54 RESULT: Theorem (the conjecture is true).
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