TSTP Solution File: KLE055+1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : KLE055+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 05:35:42 EDT 2023
% Result : Theorem 0.18s 0.41s
% Output : Proof 0.18s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : KLE055+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
% 0.12/0.33 % Computer : n007.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 11:32:58 EDT 2023
% 0.12/0.33 % CPUTime :
% 0.18/0.41 Command-line arguments: --no-flatten-goal
% 0.18/0.41
% 0.18/0.41 % SZS status Theorem
% 0.18/0.41
% 0.18/0.41 % SZS output start Proof
% 0.18/0.41 Take the following subset of the input axioms:
% 0.18/0.41 fof(additive_commutativity, axiom, ![A, B]: addition(A, B)=addition(B, A)).
% 0.18/0.41 fof(domain1, axiom, ![X0]: addition(X0, multiplication(domain(X0), X0))=multiplication(domain(X0), X0)).
% 0.18/0.41 fof(domain3, axiom, ![X0_2]: addition(domain(X0_2), one)=one).
% 0.18/0.41 fof(goals, conjecture, ![X0_2]: (addition(X0_2, one)=one => addition(X0_2, domain(X0_2))=domain(X0_2))).
% 0.18/0.41 fof(left_distributivity, axiom, ![C, A2, B2]: multiplication(addition(A2, B2), C)=addition(multiplication(A2, C), multiplication(B2, C))).
% 0.18/0.41 fof(multiplicative_left_identity, axiom, ![A2]: multiplication(one, A2)=A2).
% 0.18/0.41 fof(multiplicative_right_identity, axiom, ![A2]: multiplication(A2, one)=A2).
% 0.18/0.41 fof(right_distributivity, axiom, ![A2, B2, C2]: multiplication(A2, addition(B2, C2))=addition(multiplication(A2, B2), multiplication(A2, C2))).
% 0.18/0.41
% 0.18/0.41 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.18/0.41 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.18/0.41 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.18/0.41 fresh(y, y, x1...xn) = u
% 0.18/0.41 C => fresh(s, t, x1...xn) = v
% 0.18/0.41 where fresh is a fresh function symbol and x1..xn are the free
% 0.18/0.41 variables of u and v.
% 0.18/0.41 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.18/0.41 input problem has no model of domain size 1).
% 0.18/0.41
% 0.18/0.41 The encoding turns the above axioms into the following unit equations and goals:
% 0.18/0.41
% 0.18/0.41 Axiom 1 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.18/0.41 Axiom 2 (multiplicative_left_identity): multiplication(one, X) = X.
% 0.18/0.41 Axiom 3 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.18/0.41 Axiom 4 (goals): addition(x0, one) = one.
% 0.18/0.41 Axiom 5 (domain3): addition(domain(X), one) = one.
% 0.18/0.41 Axiom 6 (domain1): addition(X, multiplication(domain(X), X)) = multiplication(domain(X), X).
% 0.18/0.41 Axiom 7 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 0.18/0.41 Axiom 8 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 0.18/0.41
% 0.18/0.42 Goal 1 (goals_1): addition(x0, domain(x0)) = domain(x0).
% 0.18/0.42 Proof:
% 0.18/0.42 addition(x0, domain(x0))
% 0.18/0.42 = { by axiom 3 (additive_commutativity) R->L }
% 0.18/0.42 addition(domain(x0), x0)
% 0.18/0.42 = { by axiom 2 (multiplicative_left_identity) R->L }
% 0.18/0.42 addition(domain(x0), multiplication(one, x0))
% 0.18/0.42 = { by axiom 5 (domain3) R->L }
% 0.18/0.42 addition(domain(x0), multiplication(addition(domain(x0), one), x0))
% 0.18/0.42 = { by axiom 3 (additive_commutativity) R->L }
% 0.18/0.42 addition(domain(x0), multiplication(addition(one, domain(x0)), x0))
% 0.18/0.42 = { by axiom 8 (left_distributivity) }
% 0.18/0.42 addition(domain(x0), addition(multiplication(one, x0), multiplication(domain(x0), x0)))
% 0.18/0.42 = { by axiom 2 (multiplicative_left_identity) }
% 0.18/0.42 addition(domain(x0), addition(x0, multiplication(domain(x0), x0)))
% 0.18/0.42 = { by axiom 6 (domain1) }
% 0.18/0.42 addition(domain(x0), multiplication(domain(x0), x0))
% 0.18/0.42 = { by axiom 1 (multiplicative_right_identity) R->L }
% 0.18/0.42 addition(multiplication(domain(x0), one), multiplication(domain(x0), x0))
% 0.18/0.42 = { by axiom 7 (right_distributivity) R->L }
% 0.18/0.42 multiplication(domain(x0), addition(one, x0))
% 0.18/0.42 = { by axiom 3 (additive_commutativity) }
% 0.18/0.42 multiplication(domain(x0), addition(x0, one))
% 0.18/0.42 = { by axiom 4 (goals) }
% 0.18/0.42 multiplication(domain(x0), one)
% 0.18/0.42 = { by axiom 1 (multiplicative_right_identity) }
% 0.18/0.42 domain(x0)
% 0.18/0.42 % SZS output end Proof
% 0.18/0.42
% 0.18/0.42 RESULT: Theorem (the conjecture is true).
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