TSTP Solution File: KLE090+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE090+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 : n027.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 05:35:51 EDT 2023
% Result : Theorem 0.18s 0.50s
% Output : Proof 0.18s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : KLE090+1 : TPTP v8.1.2. Released v4.0.0.
% 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 : n027.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
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Tue Aug 29 12:41:38 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.18/0.50 Command-line arguments: --kbo-weight0 --lhs-weight 5 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --goal-heuristic
% 0.18/0.50
% 0.18/0.50 % SZS status Theorem
% 0.18/0.50
% 0.18/0.52 % SZS output start Proof
% 0.18/0.52 Take the following subset of the input axioms:
% 0.18/0.52 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 0.18/0.52 fof(additive_commutativity, axiom, ![A2, B2]: addition(A2, B2)=addition(B2, A2)).
% 0.18/0.52 fof(additive_idempotence, axiom, ![A2]: addition(A2, A2)=A2).
% 0.18/0.52 fof(additive_identity, axiom, ![A2]: addition(A2, zero)=A2).
% 0.18/0.52 fof(domain1, axiom, ![X0]: multiplication(antidomain(X0), X0)=zero).
% 0.18/0.52 fof(domain2, axiom, ![X1, X0_2]: addition(antidomain(multiplication(X0_2, X1)), antidomain(multiplication(X0_2, antidomain(antidomain(X1)))))=antidomain(multiplication(X0_2, antidomain(antidomain(X1))))).
% 0.18/0.52 fof(domain3, axiom, ![X0_2]: addition(antidomain(antidomain(X0_2)), antidomain(X0_2))=one).
% 0.18/0.52 fof(domain4, axiom, ![X0_2]: domain(X0_2)=antidomain(antidomain(X0_2))).
% 0.18/0.52 fof(goals, conjecture, ![X0_2, X1_2]: (addition(X0_2, X1_2)=X1_2 => addition(antidomain(X1_2), antidomain(X0_2))=antidomain(X0_2))).
% 0.18/0.52 fof(left_distributivity, axiom, ![A2, B2, C2]: multiplication(addition(A2, B2), C2)=addition(multiplication(A2, C2), multiplication(B2, C2))).
% 0.18/0.52 fof(multiplicative_left_identity, axiom, ![A2]: multiplication(one, A2)=A2).
% 0.18/0.52 fof(multiplicative_right_identity, axiom, ![A2]: multiplication(A2, one)=A2).
% 0.18/0.52 fof(right_distributivity, axiom, ![A2, B2, C2]: multiplication(A2, addition(B2, C2))=addition(multiplication(A2, B2), multiplication(A2, C2))).
% 0.18/0.52
% 0.18/0.52 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.18/0.52 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.18/0.52 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.18/0.52 fresh(y, y, x1...xn) = u
% 0.18/0.52 C => fresh(s, t, x1...xn) = v
% 0.18/0.52 where fresh is a fresh function symbol and x1..xn are the free
% 0.18/0.52 variables of u and v.
% 0.18/0.52 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.18/0.52 input problem has no model of domain size 1).
% 0.18/0.52
% 0.18/0.52 The encoding turns the above axioms into the following unit equations and goals:
% 0.18/0.52
% 0.18/0.52 Axiom 1 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.18/0.52 Axiom 2 (multiplicative_left_identity): multiplication(one, X) = X.
% 0.18/0.52 Axiom 3 (additive_idempotence): addition(X, X) = X.
% 0.18/0.52 Axiom 4 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.18/0.52 Axiom 5 (additive_identity): addition(X, zero) = X.
% 0.18/0.52 Axiom 6 (goals): addition(x0, x1) = x1.
% 0.18/0.52 Axiom 7 (domain1): multiplication(antidomain(X), X) = zero.
% 0.18/0.52 Axiom 8 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 0.18/0.52 Axiom 9 (domain4): domain(X) = antidomain(antidomain(X)).
% 0.18/0.52 Axiom 10 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 0.18/0.52 Axiom 11 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 0.18/0.52 Axiom 12 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
% 0.18/0.52 Axiom 13 (domain2): addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))) = antidomain(multiplication(X, antidomain(antidomain(Y)))).
% 0.18/0.52
% 0.18/0.52 Lemma 14: antidomain(one) = zero.
% 0.18/0.52 Proof:
% 0.18/0.52 antidomain(one)
% 0.18/0.52 = { by axiom 1 (multiplicative_right_identity) R->L }
% 0.18/0.52 multiplication(antidomain(one), one)
% 0.18/0.52 = { by axiom 7 (domain1) }
% 0.18/0.52 zero
% 0.18/0.52
% 0.18/0.52 Lemma 15: addition(zero, X) = X.
% 0.18/0.52 Proof:
% 0.18/0.52 addition(zero, X)
% 0.18/0.52 = { by axiom 4 (additive_commutativity) R->L }
% 0.18/0.52 addition(X, zero)
% 0.18/0.52 = { by axiom 5 (additive_identity) }
% 0.18/0.52 X
% 0.18/0.52
% 0.18/0.52 Lemma 16: addition(antidomain(X), domain(X)) = one.
% 0.18/0.52 Proof:
% 0.18/0.52 addition(antidomain(X), domain(X))
% 0.18/0.52 = { by axiom 4 (additive_commutativity) R->L }
% 0.18/0.52 addition(domain(X), antidomain(X))
% 0.18/0.52 = { by axiom 9 (domain4) }
% 0.18/0.52 addition(antidomain(antidomain(X)), antidomain(X))
% 0.18/0.52 = { by axiom 12 (domain3) }
% 0.18/0.52 one
% 0.18/0.52
% 0.18/0.52 Lemma 17: addition(antidomain(X), antidomain(antidomain(X))) = one.
% 0.18/0.52 Proof:
% 0.18/0.52 addition(antidomain(X), antidomain(antidomain(X)))
% 0.18/0.52 = { by axiom 9 (domain4) R->L }
% 0.18/0.52 addition(antidomain(X), domain(X))
% 0.18/0.52 = { by lemma 16 }
% 0.18/0.52 one
% 0.18/0.52
% 0.18/0.52 Lemma 18: addition(X, addition(X, Y)) = addition(X, Y).
% 0.18/0.52 Proof:
% 0.18/0.52 addition(X, addition(X, Y))
% 0.18/0.52 = { by axiom 8 (additive_associativity) }
% 0.18/0.52 addition(addition(X, X), Y)
% 0.18/0.52 = { by axiom 3 (additive_idempotence) }
% 0.18/0.52 addition(X, Y)
% 0.18/0.52
% 0.18/0.52 Lemma 19: addition(one, antidomain(X)) = one.
% 0.18/0.52 Proof:
% 0.18/0.52 addition(one, antidomain(X))
% 0.18/0.52 = { by axiom 4 (additive_commutativity) R->L }
% 0.18/0.52 addition(antidomain(X), one)
% 0.18/0.52 = { by lemma 17 R->L }
% 0.18/0.52 addition(antidomain(X), addition(antidomain(X), antidomain(antidomain(X))))
% 0.18/0.52 = { by lemma 18 }
% 0.18/0.52 addition(antidomain(X), antidomain(antidomain(X)))
% 0.18/0.52 = { by lemma 17 }
% 0.18/0.52 one
% 0.18/0.52
% 0.18/0.52 Lemma 20: multiplication(addition(X, antidomain(Y)), Y) = multiplication(X, Y).
% 0.18/0.52 Proof:
% 0.18/0.52 multiplication(addition(X, antidomain(Y)), Y)
% 0.18/0.52 = { by axiom 11 (left_distributivity) }
% 0.18/0.52 addition(multiplication(X, Y), multiplication(antidomain(Y), Y))
% 0.18/0.52 = { by axiom 7 (domain1) }
% 0.18/0.52 addition(multiplication(X, Y), zero)
% 0.18/0.52 = { by axiom 5 (additive_identity) }
% 0.18/0.52 multiplication(X, Y)
% 0.18/0.52
% 0.18/0.52 Goal 1 (goals_1): addition(antidomain(x1), antidomain(x0)) = antidomain(x0).
% 0.18/0.52 Proof:
% 0.18/0.52 addition(antidomain(x1), antidomain(x0))
% 0.18/0.52 = { by axiom 1 (multiplicative_right_identity) R->L }
% 0.18/0.52 multiplication(addition(antidomain(x1), antidomain(x0)), one)
% 0.18/0.52 = { by lemma 17 R->L }
% 0.18/0.52 multiplication(addition(antidomain(x1), antidomain(x0)), addition(antidomain(x0), antidomain(antidomain(x0))))
% 0.18/0.53 = { by axiom 4 (additive_commutativity) R->L }
% 0.18/0.53 multiplication(addition(antidomain(x1), antidomain(x0)), addition(antidomain(antidomain(x0)), antidomain(x0)))
% 0.18/0.53 = { by axiom 10 (right_distributivity) }
% 0.18/0.53 addition(multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(antidomain(x0))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(x0)))
% 0.18/0.53 = { by axiom 2 (multiplicative_left_identity) R->L }
% 0.18/0.53 addition(multiplication(one, multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(antidomain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(x0)))
% 0.18/0.53 = { by lemma 19 R->L }
% 0.18/0.53 addition(multiplication(addition(one, antidomain(multiplication(addition(antidomain(x1), antidomain(x0)), domain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(antidomain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(x0)))
% 0.18/0.53 = { by lemma 16 R->L }
% 0.18/0.53 addition(multiplication(addition(addition(antidomain(one), domain(one)), antidomain(multiplication(addition(antidomain(x1), antidomain(x0)), domain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(antidomain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(x0)))
% 0.18/0.53 = { by axiom 9 (domain4) }
% 0.18/0.53 addition(multiplication(addition(addition(antidomain(one), antidomain(antidomain(one))), antidomain(multiplication(addition(antidomain(x1), antidomain(x0)), domain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(antidomain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(x0)))
% 0.18/0.53 = { by lemma 14 }
% 0.18/0.53 addition(multiplication(addition(addition(antidomain(one), antidomain(zero)), antidomain(multiplication(addition(antidomain(x1), antidomain(x0)), domain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(antidomain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(x0)))
% 0.18/0.53 = { by lemma 14 }
% 0.18/0.53 addition(multiplication(addition(addition(zero, antidomain(zero)), antidomain(multiplication(addition(antidomain(x1), antidomain(x0)), domain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(antidomain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(x0)))
% 0.18/0.53 = { by lemma 15 }
% 0.18/0.53 addition(multiplication(addition(antidomain(zero), antidomain(multiplication(addition(antidomain(x1), antidomain(x0)), domain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(antidomain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(x0)))
% 0.18/0.53 = { by axiom 7 (domain1) R->L }
% 0.18/0.53 addition(multiplication(addition(antidomain(multiplication(antidomain(x1), x1)), antidomain(multiplication(addition(antidomain(x1), antidomain(x0)), domain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(antidomain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(x0)))
% 0.18/0.53 = { by axiom 6 (goals) R->L }
% 0.18/0.53 addition(multiplication(addition(antidomain(multiplication(antidomain(x1), addition(x0, x1))), antidomain(multiplication(addition(antidomain(x1), antidomain(x0)), domain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(antidomain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(x0)))
% 0.18/0.53 = { by axiom 4 (additive_commutativity) R->L }
% 0.18/0.53 addition(multiplication(addition(antidomain(multiplication(antidomain(x1), addition(x1, x0))), antidomain(multiplication(addition(antidomain(x1), antidomain(x0)), domain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(antidomain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(x0)))
% 0.18/0.53 = { by axiom 10 (right_distributivity) }
% 0.18/0.53 addition(multiplication(addition(antidomain(addition(multiplication(antidomain(x1), x1), multiplication(antidomain(x1), x0))), antidomain(multiplication(addition(antidomain(x1), antidomain(x0)), domain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(antidomain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(x0)))
% 0.18/0.53 = { by axiom 7 (domain1) }
% 0.18/0.53 addition(multiplication(addition(antidomain(addition(zero, multiplication(antidomain(x1), x0))), antidomain(multiplication(addition(antidomain(x1), antidomain(x0)), domain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(antidomain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(x0)))
% 0.18/0.53 = { by lemma 15 }
% 0.18/0.53 addition(multiplication(addition(antidomain(multiplication(antidomain(x1), x0)), antidomain(multiplication(addition(antidomain(x1), antidomain(x0)), domain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(antidomain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(x0)))
% 0.18/0.53 = { by lemma 20 R->L }
% 0.18/0.53 addition(multiplication(addition(antidomain(multiplication(addition(antidomain(x1), antidomain(x0)), x0)), antidomain(multiplication(addition(antidomain(x1), antidomain(x0)), domain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(antidomain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(x0)))
% 0.18/0.53 = { by axiom 9 (domain4) }
% 0.18/0.54 addition(multiplication(addition(antidomain(multiplication(addition(antidomain(x1), antidomain(x0)), x0)), antidomain(multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(antidomain(x0))))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(antidomain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(x0)))
% 0.18/0.54 = { by axiom 13 (domain2) }
% 0.18/0.54 addition(multiplication(antidomain(multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(antidomain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(antidomain(x0)))), multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(x0)))
% 0.18/0.54 = { by axiom 7 (domain1) }
% 0.18/0.54 addition(zero, multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(x0)))
% 0.18/0.54 = { by lemma 15 }
% 0.18/0.54 multiplication(addition(antidomain(x1), antidomain(x0)), antidomain(x0))
% 0.18/0.54 = { by lemma 20 R->L }
% 0.18/0.54 multiplication(addition(addition(antidomain(x1), antidomain(x0)), antidomain(antidomain(x0))), antidomain(x0))
% 0.18/0.54 = { by axiom 4 (additive_commutativity) }
% 0.18/0.54 multiplication(addition(addition(antidomain(x0), antidomain(x1)), antidomain(antidomain(x0))), antidomain(x0))
% 0.18/0.54 = { by lemma 18 R->L }
% 0.18/0.54 multiplication(addition(addition(antidomain(x0), addition(antidomain(x0), antidomain(x1))), antidomain(antidomain(x0))), antidomain(x0))
% 0.18/0.54 = { by axiom 4 (additive_commutativity) R->L }
% 0.18/0.54 multiplication(addition(addition(antidomain(x0), addition(antidomain(x1), antidomain(x0))), antidomain(antidomain(x0))), antidomain(x0))
% 0.18/0.54 = { by axiom 4 (additive_commutativity) }
% 0.18/0.54 multiplication(addition(addition(addition(antidomain(x1), antidomain(x0)), antidomain(x0)), antidomain(antidomain(x0))), antidomain(x0))
% 0.18/0.54 = { by axiom 8 (additive_associativity) R->L }
% 0.18/0.54 multiplication(addition(addition(antidomain(x1), antidomain(x0)), addition(antidomain(x0), antidomain(antidomain(x0)))), antidomain(x0))
% 0.18/0.54 = { by lemma 17 }
% 0.18/0.54 multiplication(addition(addition(antidomain(x1), antidomain(x0)), one), antidomain(x0))
% 0.18/0.54 = { by axiom 4 (additive_commutativity) }
% 0.18/0.54 multiplication(addition(addition(antidomain(x0), antidomain(x1)), one), antidomain(x0))
% 0.18/0.54 = { by axiom 8 (additive_associativity) R->L }
% 0.18/0.54 multiplication(addition(antidomain(x0), addition(antidomain(x1), one)), antidomain(x0))
% 0.18/0.54 = { by axiom 4 (additive_commutativity) }
% 0.18/0.54 multiplication(addition(antidomain(x0), addition(one, antidomain(x1))), antidomain(x0))
% 0.18/0.54 = { by lemma 19 }
% 0.18/0.54 multiplication(addition(antidomain(x0), one), antidomain(x0))
% 0.18/0.54 = { by axiom 4 (additive_commutativity) }
% 0.18/0.54 multiplication(addition(one, antidomain(x0)), antidomain(x0))
% 0.18/0.54 = { by lemma 19 }
% 0.18/0.54 multiplication(one, antidomain(x0))
% 0.18/0.54 = { by axiom 2 (multiplicative_left_identity) }
% 0.18/0.54 antidomain(x0)
% 0.18/0.54 % SZS output end Proof
% 0.18/0.54
% 0.18/0.54 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------