TSTP Solution File: KLE063+1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : KLE063+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 : n023.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 05:35:44 EDT 2023
% Result : Theorem 0.16s 0.38s
% Output : Proof 0.16s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.09/0.10 % Problem : KLE063+1 : TPTP v8.1.2. Released v4.0.0.
% 0.09/0.11 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.10/0.31 % Computer : n023.cluster.edu
% 0.10/0.31 % Model : x86_64 x86_64
% 0.10/0.31 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.31 % Memory : 8042.1875MB
% 0.10/0.31 % OS : Linux 3.10.0-693.el7.x86_64
% 0.10/0.31 % CPULimit : 300
% 0.10/0.31 % WCLimit : 300
% 0.10/0.31 % DateTime : Tue Aug 29 11:59:10 EDT 2023
% 0.10/0.31 % CPUTime :
% 0.16/0.38 Command-line arguments: --no-flatten-goal
% 0.16/0.38
% 0.16/0.38 % SZS status Theorem
% 0.16/0.38
% 0.16/0.39 % SZS output start Proof
% 0.16/0.39 Take the following subset of the input axioms:
% 0.16/0.39 fof(additive_commutativity, axiom, ![A, B]: addition(A, B)=addition(B, A)).
% 0.16/0.39 fof(domain1, axiom, ![X0]: addition(X0, multiplication(domain(X0), X0))=multiplication(domain(X0), X0)).
% 0.16/0.39 fof(domain2, axiom, ![X1, X0_2]: domain(multiplication(X0_2, X1))=domain(multiplication(X0_2, domain(X1)))).
% 0.16/0.39 fof(domain3, axiom, ![X0_2]: addition(domain(X0_2), one)=one).
% 0.16/0.39 fof(domain5, axiom, ![X0_2, X1_2]: domain(addition(X0_2, X1_2))=addition(domain(X0_2), domain(X1_2))).
% 0.16/0.39 fof(goals, conjecture, ![X0_2, X1_2]: (addition(X0_2, multiplication(domain(X1_2), X0_2))=multiplication(domain(X1_2), X0_2) => addition(domain(X0_2), domain(X1_2))=domain(X1_2))).
% 0.16/0.39 fof(left_distributivity, axiom, ![C, A2, B2]: multiplication(addition(A2, B2), C)=addition(multiplication(A2, C), multiplication(B2, C))).
% 0.16/0.39 fof(multiplicative_left_identity, axiom, ![A2]: multiplication(one, A2)=A2).
% 0.16/0.39 fof(multiplicative_right_identity, axiom, ![A2]: multiplication(A2, one)=A2).
% 0.16/0.39 fof(right_distributivity, axiom, ![A2, B2, C2]: multiplication(A2, addition(B2, C2))=addition(multiplication(A2, B2), multiplication(A2, C2))).
% 0.16/0.39
% 0.16/0.39 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.16/0.39 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.16/0.39 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.16/0.39 fresh(y, y, x1...xn) = u
% 0.16/0.39 C => fresh(s, t, x1...xn) = v
% 0.16/0.39 where fresh is a fresh function symbol and x1..xn are the free
% 0.16/0.39 variables of u and v.
% 0.16/0.39 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.16/0.39 input problem has no model of domain size 1).
% 0.16/0.39
% 0.16/0.39 The encoding turns the above axioms into the following unit equations and goals:
% 0.16/0.39
% 0.16/0.39 Axiom 1 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.16/0.39 Axiom 2 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.16/0.39 Axiom 3 (multiplicative_left_identity): multiplication(one, X) = X.
% 0.16/0.39 Axiom 4 (domain3): addition(domain(X), one) = one.
% 0.16/0.39 Axiom 5 (domain2): domain(multiplication(X, Y)) = domain(multiplication(X, domain(Y))).
% 0.16/0.39 Axiom 6 (domain5): domain(addition(X, Y)) = addition(domain(X), domain(Y)).
% 0.16/0.39 Axiom 7 (domain1): addition(X, multiplication(domain(X), X)) = multiplication(domain(X), X).
% 0.16/0.39 Axiom 8 (goals): addition(x0, multiplication(domain(x1), x0)) = multiplication(domain(x1), x0).
% 0.16/0.39 Axiom 9 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 0.16/0.39 Axiom 10 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 0.16/0.39
% 0.16/0.39 Lemma 11: domain(domain(X)) = domain(X).
% 0.16/0.39 Proof:
% 0.16/0.39 domain(domain(X))
% 0.16/0.39 = { by axiom 3 (multiplicative_left_identity) R->L }
% 0.16/0.39 domain(multiplication(one, domain(X)))
% 0.16/0.39 = { by axiom 5 (domain2) R->L }
% 0.16/0.39 domain(multiplication(one, X))
% 0.16/0.39 = { by axiom 3 (multiplicative_left_identity) }
% 0.16/0.39 domain(X)
% 0.16/0.39
% 0.16/0.39 Lemma 12: addition(one, domain(X)) = one.
% 0.16/0.39 Proof:
% 0.16/0.39 addition(one, domain(X))
% 0.16/0.39 = { by axiom 1 (additive_commutativity) R->L }
% 0.16/0.39 addition(domain(X), one)
% 0.16/0.39 = { by axiom 4 (domain3) }
% 0.16/0.39 one
% 0.16/0.39
% 0.16/0.39 Goal 1 (goals_1): addition(domain(x0), domain(x1)) = domain(x1).
% 0.16/0.39 Proof:
% 0.16/0.39 addition(domain(x0), domain(x1))
% 0.16/0.39 = { by axiom 6 (domain5) R->L }
% 0.16/0.39 domain(addition(x0, x1))
% 0.16/0.39 = { by axiom 1 (additive_commutativity) R->L }
% 0.16/0.39 domain(addition(x1, x0))
% 0.16/0.39 = { by axiom 6 (domain5) }
% 0.16/0.39 addition(domain(x1), domain(x0))
% 0.16/0.39 = { by lemma 11 R->L }
% 0.16/0.39 addition(domain(domain(x1)), domain(x0))
% 0.16/0.39 = { by axiom 6 (domain5) R->L }
% 0.16/0.39 domain(addition(domain(x1), x0))
% 0.16/0.39 = { by axiom 3 (multiplicative_left_identity) R->L }
% 0.16/0.39 domain(addition(domain(x1), multiplication(one, x0)))
% 0.16/0.39 = { by lemma 12 R->L }
% 0.16/0.39 domain(addition(domain(x1), multiplication(addition(one, domain(x1)), x0)))
% 0.16/0.39 = { by axiom 10 (left_distributivity) }
% 0.16/0.39 domain(addition(domain(x1), addition(multiplication(one, x0), multiplication(domain(x1), x0))))
% 0.16/0.39 = { by axiom 3 (multiplicative_left_identity) }
% 0.16/0.39 domain(addition(domain(x1), addition(x0, multiplication(domain(x1), x0))))
% 0.16/0.39 = { by axiom 8 (goals) }
% 0.16/0.39 domain(addition(domain(x1), multiplication(domain(x1), x0)))
% 0.16/0.39 = { by axiom 2 (multiplicative_right_identity) R->L }
% 0.16/0.39 domain(addition(multiplication(domain(x1), one), multiplication(domain(x1), x0)))
% 0.16/0.39 = { by axiom 9 (right_distributivity) R->L }
% 0.16/0.39 domain(multiplication(domain(x1), addition(one, x0)))
% 0.16/0.39 = { by axiom 1 (additive_commutativity) }
% 0.16/0.39 domain(multiplication(domain(x1), addition(x0, one)))
% 0.16/0.39 = { by axiom 5 (domain2) }
% 0.16/0.39 domain(multiplication(domain(x1), domain(addition(x0, one))))
% 0.16/0.39 = { by axiom 1 (additive_commutativity) R->L }
% 0.16/0.39 domain(multiplication(domain(x1), domain(addition(one, x0))))
% 0.16/0.39 = { by axiom 6 (domain5) }
% 0.16/0.39 domain(multiplication(domain(x1), addition(domain(one), domain(x0))))
% 0.16/0.39 = { by axiom 2 (multiplicative_right_identity) R->L }
% 0.16/0.39 domain(multiplication(domain(x1), addition(multiplication(domain(one), one), domain(x0))))
% 0.16/0.39 = { by axiom 7 (domain1) R->L }
% 0.16/0.39 domain(multiplication(domain(x1), addition(addition(one, multiplication(domain(one), one)), domain(x0))))
% 0.16/0.39 = { by axiom 2 (multiplicative_right_identity) }
% 0.16/0.39 domain(multiplication(domain(x1), addition(addition(one, domain(one)), domain(x0))))
% 0.16/0.39 = { by lemma 12 }
% 0.16/0.39 domain(multiplication(domain(x1), addition(one, domain(x0))))
% 0.16/0.39 = { by lemma 12 }
% 0.16/0.39 domain(multiplication(domain(x1), one))
% 0.16/0.39 = { by axiom 2 (multiplicative_right_identity) }
% 0.16/0.39 domain(domain(x1))
% 0.16/0.39 = { by lemma 11 }
% 0.16/0.39 domain(x1)
% 0.16/0.39 % SZS output end Proof
% 0.16/0.39
% 0.16/0.39 RESULT: Theorem (the conjecture is true).
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