TSTP Solution File: KLE062+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE062+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 : 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 05:35:44 EDT 2023
% Result : Theorem 2.28s 0.68s
% Output : Proof 2.28s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : KLE062+1 : TPTP v8.1.2. Released v4.0.0.
% 0.07/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.34 % Computer : n007.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Tue Aug 29 11:51:58 EDT 2023
% 0.14/0.35 % CPUTime :
% 2.28/0.68 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 2.28/0.68
% 2.28/0.68 % SZS status Theorem
% 2.28/0.68
% 2.28/0.70 % SZS output start Proof
% 2.28/0.70 Take the following subset of the input axioms:
% 2.28/0.70 fof(additive_commutativity, axiom, ![A, B]: addition(A, B)=addition(B, A)).
% 2.28/0.70 fof(domain1, axiom, ![X0]: addition(X0, multiplication(domain(X0), X0))=multiplication(domain(X0), X0)).
% 2.28/0.70 fof(domain2, axiom, ![X1, X0_2]: domain(multiplication(X0_2, X1))=domain(multiplication(X0_2, domain(X1)))).
% 2.28/0.70 fof(domain3, axiom, ![X0_2]: addition(domain(X0_2), one)=one).
% 2.28/0.70 fof(domain5, axiom, ![X0_2, X1_2]: domain(addition(X0_2, X1_2))=addition(domain(X0_2), domain(X1_2))).
% 2.28/0.70 fof(goals, conjecture, ![X0_2, X1_2]: multiplication(domain(X0_2), domain(X1_2))=multiplication(domain(X1_2), domain(X0_2))).
% 2.28/0.70 fof(left_distributivity, axiom, ![C, A2, B2]: multiplication(addition(A2, B2), C)=addition(multiplication(A2, C), multiplication(B2, C))).
% 2.28/0.70 fof(multiplicative_associativity, axiom, ![A2, B2, C2]: multiplication(A2, multiplication(B2, C2))=multiplication(multiplication(A2, B2), C2)).
% 2.28/0.70 fof(multiplicative_left_identity, axiom, ![A2]: multiplication(one, A2)=A2).
% 2.28/0.70 fof(multiplicative_right_identity, axiom, ![A2]: multiplication(A2, one)=A2).
% 2.28/0.70 fof(right_distributivity, axiom, ![A2, B2, C2]: multiplication(A2, addition(B2, C2))=addition(multiplication(A2, B2), multiplication(A2, C2))).
% 2.28/0.70
% 2.28/0.70 Now clausify the problem and encode Horn clauses using encoding 3 of
% 2.28/0.70 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 2.28/0.70 We repeatedly replace C & s=t => u=v by the two clauses:
% 2.28/0.70 fresh(y, y, x1...xn) = u
% 2.28/0.70 C => fresh(s, t, x1...xn) = v
% 2.28/0.70 where fresh is a fresh function symbol and x1..xn are the free
% 2.28/0.70 variables of u and v.
% 2.28/0.70 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 2.28/0.70 input problem has no model of domain size 1).
% 2.28/0.70
% 2.28/0.70 The encoding turns the above axioms into the following unit equations and goals:
% 2.28/0.70
% 2.28/0.70 Axiom 1 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 2.28/0.70 Axiom 2 (multiplicative_right_identity): multiplication(X, one) = X.
% 2.28/0.70 Axiom 3 (multiplicative_left_identity): multiplication(one, X) = X.
% 2.28/0.70 Axiom 4 (domain3): addition(domain(X), one) = one.
% 2.28/0.70 Axiom 5 (domain2): domain(multiplication(X, Y)) = domain(multiplication(X, domain(Y))).
% 2.28/0.70 Axiom 6 (domain5): domain(addition(X, Y)) = addition(domain(X), domain(Y)).
% 2.28/0.70 Axiom 7 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 2.28/0.70 Axiom 8 (domain1): addition(X, multiplication(domain(X), X)) = multiplication(domain(X), X).
% 2.28/0.70 Axiom 9 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 2.28/0.70 Axiom 10 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 2.28/0.70
% 2.28/0.70 Lemma 11: domain(domain(X)) = domain(X).
% 2.28/0.70 Proof:
% 2.28/0.70 domain(domain(X))
% 2.28/0.70 = { by axiom 3 (multiplicative_left_identity) R->L }
% 2.28/0.70 domain(multiplication(one, domain(X)))
% 2.28/0.70 = { by axiom 5 (domain2) R->L }
% 2.28/0.70 domain(multiplication(one, X))
% 2.28/0.70 = { by axiom 3 (multiplicative_left_identity) }
% 2.28/0.70 domain(X)
% 2.28/0.70
% 2.28/0.70 Lemma 12: addition(one, domain(X)) = one.
% 2.28/0.70 Proof:
% 2.28/0.70 addition(one, domain(X))
% 2.28/0.70 = { by axiom 1 (additive_commutativity) R->L }
% 2.28/0.70 addition(domain(X), one)
% 2.28/0.70 = { by axiom 4 (domain3) }
% 2.28/0.70 one
% 2.28/0.70
% 2.28/0.70 Lemma 13: multiplication(addition(X, one), Y) = addition(Y, multiplication(X, Y)).
% 2.28/0.70 Proof:
% 2.28/0.70 multiplication(addition(X, one), Y)
% 2.28/0.70 = { by axiom 1 (additive_commutativity) R->L }
% 2.28/0.70 multiplication(addition(one, X), Y)
% 2.28/0.70 = { by axiom 10 (left_distributivity) }
% 2.28/0.70 addition(multiplication(one, Y), multiplication(X, Y))
% 2.28/0.70 = { by axiom 3 (multiplicative_left_identity) }
% 2.28/0.70 addition(Y, multiplication(X, Y))
% 2.28/0.70
% 2.28/0.70 Lemma 14: multiplication(domain(X), X) = X.
% 2.28/0.70 Proof:
% 2.28/0.70 multiplication(domain(X), X)
% 2.28/0.70 = { by axiom 8 (domain1) R->L }
% 2.28/0.70 addition(X, multiplication(domain(X), X))
% 2.28/0.70 = { by lemma 13 R->L }
% 2.28/0.70 multiplication(addition(domain(X), one), X)
% 2.28/0.70 = { by axiom 1 (additive_commutativity) }
% 2.28/0.70 multiplication(addition(one, domain(X)), X)
% 2.28/0.70 = { by lemma 12 }
% 2.28/0.70 multiplication(one, X)
% 2.28/0.70 = { by axiom 3 (multiplicative_left_identity) }
% 2.28/0.70 X
% 2.28/0.70
% 2.28/0.70 Lemma 15: multiplication(X, addition(Y, one)) = addition(X, multiplication(X, Y)).
% 2.28/0.71 Proof:
% 2.28/0.71 multiplication(X, addition(Y, one))
% 2.28/0.71 = { by axiom 1 (additive_commutativity) R->L }
% 2.28/0.71 multiplication(X, addition(one, Y))
% 2.28/0.71 = { by axiom 9 (right_distributivity) }
% 2.28/0.71 addition(multiplication(X, one), multiplication(X, Y))
% 2.28/0.71 = { by axiom 2 (multiplicative_right_identity) }
% 2.28/0.71 addition(X, multiplication(X, Y))
% 2.28/0.71
% 2.28/0.71 Lemma 16: domain(multiplication(X, addition(Y, one))) = domain(X).
% 2.28/0.71 Proof:
% 2.28/0.71 domain(multiplication(X, addition(Y, one)))
% 2.28/0.71 = { by axiom 5 (domain2) }
% 2.28/0.71 domain(multiplication(X, domain(addition(Y, one))))
% 2.28/0.71 = { by axiom 1 (additive_commutativity) R->L }
% 2.28/0.71 domain(multiplication(X, domain(addition(one, Y))))
% 2.28/0.71 = { by axiom 6 (domain5) }
% 2.28/0.71 domain(multiplication(X, addition(domain(one), domain(Y))))
% 2.28/0.71 = { by axiom 2 (multiplicative_right_identity) R->L }
% 2.28/0.71 domain(multiplication(X, addition(multiplication(domain(one), one), domain(Y))))
% 2.28/0.71 = { by axiom 8 (domain1) R->L }
% 2.28/0.71 domain(multiplication(X, addition(addition(one, multiplication(domain(one), one)), domain(Y))))
% 2.28/0.71 = { by axiom 2 (multiplicative_right_identity) }
% 2.28/0.71 domain(multiplication(X, addition(addition(one, domain(one)), domain(Y))))
% 2.28/0.71 = { by lemma 12 }
% 2.28/0.71 domain(multiplication(X, addition(one, domain(Y))))
% 2.28/0.71 = { by lemma 12 }
% 2.28/0.71 domain(multiplication(X, one))
% 2.28/0.71 = { by axiom 2 (multiplicative_right_identity) }
% 2.28/0.71 domain(X)
% 2.28/0.71
% 2.28/0.71 Lemma 17: multiplication(domain(X), addition(X, Y)) = addition(X, multiplication(domain(X), Y)).
% 2.28/0.71 Proof:
% 2.28/0.71 multiplication(domain(X), addition(X, Y))
% 2.28/0.71 = { by axiom 9 (right_distributivity) }
% 2.28/0.71 addition(multiplication(domain(X), X), multiplication(domain(X), Y))
% 2.28/0.71 = { by lemma 14 }
% 2.28/0.71 addition(X, multiplication(domain(X), Y))
% 2.28/0.71
% 2.28/0.71 Lemma 18: addition(X, multiplication(domain(Y), X)) = X.
% 2.28/0.71 Proof:
% 2.28/0.71 addition(X, multiplication(domain(Y), X))
% 2.28/0.71 = { by lemma 13 R->L }
% 2.28/0.71 multiplication(addition(domain(Y), one), X)
% 2.28/0.71 = { by axiom 1 (additive_commutativity) }
% 2.28/0.71 multiplication(addition(one, domain(Y)), X)
% 2.28/0.71 = { by lemma 12 }
% 2.28/0.71 multiplication(one, X)
% 2.28/0.71 = { by axiom 3 (multiplicative_left_identity) }
% 2.28/0.71 X
% 2.28/0.71
% 2.28/0.71 Lemma 19: multiplication(domain(X), domain(addition(Y, X))) = domain(X).
% 2.28/0.71 Proof:
% 2.28/0.71 multiplication(domain(X), domain(addition(Y, X)))
% 2.28/0.71 = { by axiom 1 (additive_commutativity) R->L }
% 2.28/0.71 multiplication(domain(X), domain(addition(X, Y)))
% 2.28/0.71 = { by lemma 11 R->L }
% 2.28/0.71 multiplication(domain(domain(X)), domain(addition(X, Y)))
% 2.28/0.71 = { by axiom 6 (domain5) }
% 2.28/0.71 multiplication(domain(domain(X)), addition(domain(X), domain(Y)))
% 2.28/0.71 = { by lemma 17 }
% 2.28/0.71 addition(domain(X), multiplication(domain(domain(X)), domain(Y)))
% 2.28/0.71 = { by lemma 11 }
% 2.28/0.71 addition(domain(X), multiplication(domain(X), domain(Y)))
% 2.28/0.71 = { by lemma 15 R->L }
% 2.28/0.71 multiplication(domain(X), addition(domain(Y), one))
% 2.28/0.71 = { by axiom 1 (additive_commutativity) }
% 2.28/0.71 multiplication(domain(X), addition(one, domain(Y)))
% 2.28/0.71 = { by lemma 12 }
% 2.28/0.71 multiplication(domain(X), one)
% 2.28/0.71 = { by axiom 2 (multiplicative_right_identity) }
% 2.28/0.71 domain(X)
% 2.28/0.71
% 2.28/0.71 Lemma 20: multiplication(domain(X), domain(Y)) = domain(multiplication(domain(X), Y)).
% 2.28/0.71 Proof:
% 2.28/0.71 multiplication(domain(X), domain(Y))
% 2.28/0.71 = { by lemma 11 R->L }
% 2.28/0.71 multiplication(domain(domain(X)), domain(Y))
% 2.28/0.71 = { by lemma 16 R->L }
% 2.28/0.71 multiplication(domain(multiplication(domain(X), addition(domain(Y), one))), domain(Y))
% 2.28/0.71 = { by lemma 15 }
% 2.28/0.71 multiplication(domain(addition(domain(X), multiplication(domain(X), domain(Y)))), domain(Y))
% 2.28/0.71 = { by axiom 6 (domain5) }
% 2.28/0.71 multiplication(addition(domain(domain(X)), domain(multiplication(domain(X), domain(Y)))), domain(Y))
% 2.28/0.71 = { by lemma 11 }
% 2.28/0.71 multiplication(addition(domain(X), domain(multiplication(domain(X), domain(Y)))), domain(Y))
% 2.28/0.71 = { by axiom 10 (left_distributivity) }
% 2.28/0.71 addition(multiplication(domain(X), domain(Y)), multiplication(domain(multiplication(domain(X), domain(Y))), domain(Y)))
% 2.28/0.71 = { by lemma 17 R->L }
% 2.28/0.71 multiplication(domain(multiplication(domain(X), domain(Y))), addition(multiplication(domain(X), domain(Y)), domain(Y)))
% 2.28/0.71 = { by axiom 1 (additive_commutativity) }
% 2.28/0.71 multiplication(domain(multiplication(domain(X), domain(Y))), addition(domain(Y), multiplication(domain(X), domain(Y))))
% 2.28/0.71 = { by lemma 18 }
% 2.28/0.71 multiplication(domain(multiplication(domain(X), domain(Y))), domain(Y))
% 2.28/0.71 = { by axiom 5 (domain2) R->L }
% 2.28/0.71 multiplication(domain(multiplication(domain(X), Y)), domain(Y))
% 2.28/0.71 = { by lemma 18 R->L }
% 2.28/0.71 multiplication(domain(multiplication(domain(X), Y)), domain(addition(Y, multiplication(domain(X), Y))))
% 2.28/0.71 = { by lemma 19 }
% 2.28/0.71 domain(multiplication(domain(X), Y))
% 2.28/0.71
% 2.28/0.71 Goal 1 (goals): multiplication(domain(x0), domain(x1)) = multiplication(domain(x1), domain(x0)).
% 2.28/0.71 Proof:
% 2.28/0.71 multiplication(domain(x0), domain(x1))
% 2.28/0.71 = { by lemma 20 }
% 2.28/0.71 domain(multiplication(domain(x0), x1))
% 2.28/0.71 = { by lemma 18 R->L }
% 2.28/0.71 domain(addition(multiplication(domain(x0), x1), multiplication(domain(x1), multiplication(domain(x0), x1))))
% 2.28/0.71 = { by lemma 14 R->L }
% 2.28/0.71 domain(addition(multiplication(domain(multiplication(domain(x0), x1)), multiplication(domain(x0), x1)), multiplication(domain(x1), multiplication(domain(x0), x1))))
% 2.28/0.71 = { by axiom 10 (left_distributivity) R->L }
% 2.28/0.71 domain(multiplication(addition(domain(multiplication(domain(x0), x1)), domain(x1)), multiplication(domain(x0), x1)))
% 2.28/0.71 = { by axiom 1 (additive_commutativity) }
% 2.28/0.71 domain(multiplication(addition(domain(x1), domain(multiplication(domain(x0), x1))), multiplication(domain(x0), x1)))
% 2.28/0.71 = { by axiom 6 (domain5) R->L }
% 2.28/0.71 domain(multiplication(domain(addition(x1, multiplication(domain(x0), x1))), multiplication(domain(x0), x1)))
% 2.28/0.71 = { by lemma 18 }
% 2.28/0.71 domain(multiplication(domain(x1), multiplication(domain(x0), x1)))
% 2.28/0.71 = { by axiom 5 (domain2) }
% 2.28/0.71 domain(multiplication(domain(x1), domain(multiplication(domain(x0), x1))))
% 2.28/0.71 = { by axiom 5 (domain2) }
% 2.28/0.71 domain(multiplication(domain(x1), domain(multiplication(domain(x0), domain(x1)))))
% 2.28/0.71 = { by axiom 5 (domain2) R->L }
% 2.28/0.71 domain(multiplication(domain(x1), multiplication(domain(x0), domain(x1))))
% 2.28/0.71 = { by lemma 20 R->L }
% 2.28/0.71 multiplication(domain(x1), domain(multiplication(domain(x0), domain(x1))))
% 2.28/0.71 = { by lemma 20 R->L }
% 2.28/0.71 multiplication(domain(x1), multiplication(domain(x0), domain(domain(x1))))
% 2.28/0.71 = { by axiom 7 (multiplicative_associativity) }
% 2.28/0.71 multiplication(multiplication(domain(x1), domain(x0)), domain(domain(x1)))
% 2.28/0.71 = { by lemma 20 }
% 2.28/0.71 multiplication(domain(multiplication(domain(x1), x0)), domain(domain(x1)))
% 2.28/0.71 = { by lemma 16 R->L }
% 2.28/0.71 multiplication(domain(multiplication(domain(x1), x0)), domain(multiplication(domain(x1), addition(x0, one))))
% 2.28/0.71 = { by lemma 15 }
% 2.28/0.71 multiplication(domain(multiplication(domain(x1), x0)), domain(addition(domain(x1), multiplication(domain(x1), x0))))
% 2.28/0.71 = { by lemma 19 }
% 2.28/0.71 domain(multiplication(domain(x1), x0))
% 2.28/0.71 = { by lemma 20 R->L }
% 2.28/0.71 multiplication(domain(x1), domain(x0))
% 2.28/0.71 % SZS output end Proof
% 2.28/0.71
% 2.28/0.71 RESULT: Theorem (the conjecture is true).
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