TSTP Solution File: KLE131+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE131+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 : n025.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 05:35:59 EDT 2023
% Result : Theorem 0.14s 0.43s
% Output : Proof 0.14s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.08 % Problem : KLE131+1 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.09 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.08/0.30 % Computer : n025.cluster.edu
% 0.08/0.30 % Model : x86_64 x86_64
% 0.08/0.30 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.08/0.30 % Memory : 8042.1875MB
% 0.08/0.30 % OS : Linux 3.10.0-693.el7.x86_64
% 0.08/0.30 % CPULimit : 300
% 0.08/0.30 % WCLimit : 300
% 0.08/0.30 % DateTime : Tue Aug 29 11:36:55 EDT 2023
% 0.08/0.30 % CPUTime :
% 0.14/0.43 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.14/0.43
% 0.14/0.43 % SZS status Theorem
% 0.14/0.43
% 0.14/0.44 % SZS output start Proof
% 0.14/0.44 Take the following subset of the input axioms:
% 0.14/0.44 fof(additive_commutativity, axiom, ![A, B]: addition(A, B)=addition(B, A)).
% 0.14/0.44 fof(additive_identity, axiom, ![A2]: addition(A2, zero)=A2).
% 0.14/0.44 fof(complement, axiom, ![X0]: c(X0)=antidomain(domain(X0))).
% 0.14/0.44 fof(divergence1, axiom, ![X0_2]: forward_diamond(X0_2, divergence(X0_2))=divergence(X0_2)).
% 0.14/0.44 fof(domain1, axiom, ![X0_2]: multiplication(antidomain(X0_2), X0_2)=zero).
% 0.14/0.44 fof(domain3, axiom, ![X0_2]: addition(antidomain(antidomain(X0_2)), antidomain(X0_2))=one).
% 0.14/0.44 fof(domain4, axiom, ![X0_2]: domain(X0_2)=antidomain(antidomain(X0_2))).
% 0.14/0.44 fof(domain_difference, axiom, ![X1, X0_2]: domain_difference(X0_2, X1)=multiplication(domain(X0_2), antidomain(X1))).
% 0.14/0.44 fof(forward_diamond, axiom, ![X1_2, X0_2]: forward_diamond(X0_2, X1_2)=domain(multiplication(X0_2, domain(X1_2)))).
% 0.14/0.44 fof(goals, conjecture, ![X0_2]: (divergence(X0_2)=zero <= ![X1_2]: addition(domain(X1_2), forward_diamond(star(X0_2), domain_difference(domain(X1_2), forward_diamond(X0_2, domain(X1_2)))))=forward_diamond(star(X0_2), domain_difference(domain(X1_2), forward_diamond(X0_2, domain(X1_2)))))).
% 0.14/0.44 fof(left_distributivity, axiom, ![C, A2, B2]: multiplication(addition(A2, B2), C)=addition(multiplication(A2, C), multiplication(B2, C))).
% 0.14/0.44 fof(multiplicative_left_identity, axiom, ![A2]: multiplication(one, A2)=A2).
% 0.14/0.44 fof(multiplicative_right_identity, axiom, ![A2]: multiplication(A2, one)=A2).
% 0.14/0.44 fof(right_annihilation, axiom, ![A2]: multiplication(A2, zero)=zero).
% 0.14/0.45 fof(right_distributivity, axiom, ![A2, B2, C2]: multiplication(A2, addition(B2, C2))=addition(multiplication(A2, B2), multiplication(A2, C2))).
% 0.14/0.45
% 0.14/0.45 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.14/0.45 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.14/0.45 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.14/0.45 fresh(y, y, x1...xn) = u
% 0.14/0.45 C => fresh(s, t, x1...xn) = v
% 0.14/0.45 where fresh is a fresh function symbol and x1..xn are the free
% 0.14/0.45 variables of u and v.
% 0.14/0.45 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.14/0.45 input problem has no model of domain size 1).
% 0.14/0.45
% 0.14/0.45 The encoding turns the above axioms into the following unit equations and goals:
% 0.14/0.45
% 0.14/0.45 Axiom 1 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.14/0.45 Axiom 2 (additive_identity): addition(X, zero) = X.
% 0.14/0.45 Axiom 3 (right_annihilation): multiplication(X, zero) = zero.
% 0.14/0.45 Axiom 4 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.14/0.45 Axiom 5 (multiplicative_left_identity): multiplication(one, X) = X.
% 0.14/0.45 Axiom 6 (complement): c(X) = antidomain(domain(X)).
% 0.14/0.45 Axiom 7 (domain4): domain(X) = antidomain(antidomain(X)).
% 0.14/0.45 Axiom 8 (domain1): multiplication(antidomain(X), X) = zero.
% 0.14/0.45 Axiom 9 (divergence1): forward_diamond(X, divergence(X)) = divergence(X).
% 0.14/0.45 Axiom 10 (domain_difference): domain_difference(X, Y) = multiplication(domain(X), antidomain(Y)).
% 0.14/0.45 Axiom 11 (forward_diamond): forward_diamond(X, Y) = domain(multiplication(X, domain(Y))).
% 0.14/0.45 Axiom 12 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
% 0.14/0.45 Axiom 13 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 0.14/0.45 Axiom 14 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 0.14/0.45 Axiom 15 (goals): addition(domain(X), forward_diamond(star(x0), domain_difference(domain(X), forward_diamond(x0, domain(X))))) = forward_diamond(star(x0), domain_difference(domain(X), forward_diamond(x0, domain(X)))).
% 0.14/0.45
% 0.14/0.45 Lemma 16: antidomain(one) = zero.
% 0.14/0.45 Proof:
% 0.14/0.45 antidomain(one)
% 0.14/0.45 = { by axiom 4 (multiplicative_right_identity) R->L }
% 0.14/0.45 multiplication(antidomain(one), one)
% 0.14/0.45 = { by axiom 8 (domain1) }
% 0.14/0.45 zero
% 0.14/0.45
% 0.14/0.45 Lemma 17: addition(domain(X), antidomain(X)) = one.
% 0.14/0.45 Proof:
% 0.14/0.45 addition(domain(X), antidomain(X))
% 0.14/0.45 = { by axiom 7 (domain4) }
% 0.14/0.45 addition(antidomain(antidomain(X)), antidomain(X))
% 0.14/0.45 = { by axiom 12 (domain3) }
% 0.14/0.45 one
% 0.14/0.45
% 0.14/0.45 Lemma 18: domain(antidomain(X)) = c(X).
% 0.14/0.45 Proof:
% 0.14/0.45 domain(antidomain(X))
% 0.14/0.45 = { by axiom 7 (domain4) }
% 0.14/0.45 antidomain(antidomain(antidomain(X)))
% 0.14/0.45 = { by axiom 7 (domain4) R->L }
% 0.14/0.45 antidomain(domain(X))
% 0.14/0.45 = { by axiom 6 (complement) R->L }
% 0.14/0.45 c(X)
% 0.14/0.45
% 0.14/0.45 Lemma 19: multiplication(antidomain(X), addition(X, Y)) = multiplication(antidomain(X), Y).
% 0.14/0.45 Proof:
% 0.14/0.45 multiplication(antidomain(X), addition(X, Y))
% 0.14/0.45 = { by axiom 1 (additive_commutativity) R->L }
% 0.14/0.45 multiplication(antidomain(X), addition(Y, X))
% 0.14/0.45 = { by axiom 13 (right_distributivity) }
% 0.14/0.45 addition(multiplication(antidomain(X), Y), multiplication(antidomain(X), X))
% 0.14/0.45 = { by axiom 8 (domain1) }
% 0.14/0.45 addition(multiplication(antidomain(X), Y), zero)
% 0.14/0.45 = { by axiom 2 (additive_identity) }
% 0.14/0.45 multiplication(antidomain(X), Y)
% 0.14/0.45
% 0.14/0.45 Lemma 20: multiplication(domain(X), X) = X.
% 0.14/0.45 Proof:
% 0.14/0.45 multiplication(domain(X), X)
% 0.14/0.45 = { by axiom 2 (additive_identity) R->L }
% 0.14/0.45 addition(multiplication(domain(X), X), zero)
% 0.14/0.45 = { by axiom 8 (domain1) R->L }
% 0.14/0.45 addition(multiplication(domain(X), X), multiplication(antidomain(X), X))
% 0.14/0.45 = { by axiom 14 (left_distributivity) R->L }
% 0.14/0.45 multiplication(addition(domain(X), antidomain(X)), X)
% 0.14/0.45 = { by lemma 17 }
% 0.14/0.45 multiplication(one, X)
% 0.14/0.45 = { by axiom 5 (multiplicative_left_identity) }
% 0.14/0.45 X
% 0.14/0.45
% 0.14/0.45 Lemma 21: domain(divergence(X)) = divergence(X).
% 0.14/0.45 Proof:
% 0.14/0.45 domain(divergence(X))
% 0.14/0.45 = { by axiom 7 (domain4) }
% 0.14/0.45 antidomain(antidomain(divergence(X)))
% 0.14/0.45 = { by axiom 4 (multiplicative_right_identity) R->L }
% 0.14/0.45 multiplication(antidomain(antidomain(divergence(X))), one)
% 0.14/0.45 = { by lemma 17 R->L }
% 0.14/0.45 multiplication(antidomain(antidomain(divergence(X))), addition(domain(antidomain(multiplication(X, domain(divergence(X))))), antidomain(antidomain(multiplication(X, domain(divergence(X)))))))
% 0.14/0.45 = { by axiom 7 (domain4) R->L }
% 0.14/0.45 multiplication(antidomain(antidomain(divergence(X))), addition(domain(antidomain(multiplication(X, domain(divergence(X))))), domain(multiplication(X, domain(divergence(X))))))
% 0.14/0.45 = { by lemma 18 }
% 0.14/0.45 multiplication(antidomain(antidomain(divergence(X))), addition(c(multiplication(X, domain(divergence(X)))), domain(multiplication(X, domain(divergence(X))))))
% 0.14/0.45 = { by axiom 1 (additive_commutativity) }
% 0.14/0.45 multiplication(antidomain(antidomain(divergence(X))), addition(domain(multiplication(X, domain(divergence(X)))), c(multiplication(X, domain(divergence(X))))))
% 0.14/0.45 = { by axiom 6 (complement) }
% 0.14/0.45 multiplication(antidomain(antidomain(divergence(X))), addition(domain(multiplication(X, domain(divergence(X)))), antidomain(domain(multiplication(X, domain(divergence(X)))))))
% 0.14/0.45 = { by axiom 11 (forward_diamond) R->L }
% 0.14/0.45 multiplication(antidomain(antidomain(divergence(X))), addition(domain(multiplication(X, domain(divergence(X)))), antidomain(forward_diamond(X, divergence(X)))))
% 0.14/0.45 = { by axiom 11 (forward_diamond) R->L }
% 0.14/0.45 multiplication(antidomain(antidomain(divergence(X))), addition(forward_diamond(X, divergence(X)), antidomain(forward_diamond(X, divergence(X)))))
% 0.14/0.45 = { by axiom 9 (divergence1) }
% 0.14/0.45 multiplication(antidomain(antidomain(divergence(X))), addition(divergence(X), antidomain(forward_diamond(X, divergence(X)))))
% 0.14/0.45 = { by axiom 9 (divergence1) }
% 0.14/0.45 multiplication(antidomain(antidomain(divergence(X))), addition(divergence(X), antidomain(divergence(X))))
% 0.14/0.45 = { by axiom 1 (additive_commutativity) R->L }
% 0.14/0.45 multiplication(antidomain(antidomain(divergence(X))), addition(antidomain(divergence(X)), divergence(X)))
% 0.14/0.45 = { by lemma 19 }
% 0.14/0.45 multiplication(antidomain(antidomain(divergence(X))), divergence(X))
% 0.14/0.45 = { by axiom 7 (domain4) R->L }
% 0.14/0.45 multiplication(domain(divergence(X)), divergence(X))
% 0.14/0.45 = { by lemma 20 }
% 0.14/0.45 divergence(X)
% 0.14/0.45
% 0.14/0.45 Lemma 22: forward_diamond(X, zero) = zero.
% 0.14/0.45 Proof:
% 0.14/0.45 forward_diamond(X, zero)
% 0.14/0.45 = { by lemma 16 R->L }
% 0.14/0.46 forward_diamond(X, antidomain(one))
% 0.14/0.46 = { by axiom 11 (forward_diamond) }
% 0.14/0.46 domain(multiplication(X, domain(antidomain(one))))
% 0.14/0.46 = { by lemma 18 }
% 0.14/0.46 domain(multiplication(X, c(one)))
% 0.14/0.46 = { by axiom 6 (complement) }
% 0.14/0.46 domain(multiplication(X, antidomain(domain(one))))
% 0.14/0.46 = { by axiom 4 (multiplicative_right_identity) R->L }
% 0.14/0.46 domain(multiplication(X, multiplication(antidomain(domain(one)), one)))
% 0.14/0.46 = { by lemma 17 R->L }
% 0.14/0.46 domain(multiplication(X, multiplication(antidomain(domain(one)), addition(domain(one), antidomain(one)))))
% 0.14/0.46 = { by lemma 19 }
% 0.14/0.46 domain(multiplication(X, multiplication(antidomain(domain(one)), antidomain(one))))
% 0.14/0.46 = { by axiom 6 (complement) R->L }
% 0.14/0.46 domain(multiplication(X, multiplication(c(one), antidomain(one))))
% 0.14/0.46 = { by lemma 18 R->L }
% 0.14/0.46 domain(multiplication(X, multiplication(domain(antidomain(one)), antidomain(one))))
% 0.14/0.46 = { by lemma 20 }
% 0.14/0.46 domain(multiplication(X, antidomain(one)))
% 0.14/0.46 = { by lemma 16 }
% 0.14/0.46 domain(multiplication(X, zero))
% 0.14/0.46 = { by axiom 3 (right_annihilation) }
% 0.14/0.46 domain(zero)
% 0.14/0.46 = { by lemma 16 R->L }
% 0.14/0.46 domain(antidomain(one))
% 0.14/0.46 = { by lemma 18 }
% 0.14/0.46 c(one)
% 0.14/0.46 = { by axiom 6 (complement) }
% 0.14/0.46 antidomain(domain(one))
% 0.14/0.46 = { by axiom 2 (additive_identity) R->L }
% 0.14/0.46 antidomain(addition(domain(one), zero))
% 0.14/0.46 = { by lemma 16 R->L }
% 0.14/0.46 antidomain(addition(domain(one), antidomain(one)))
% 0.14/0.46 = { by lemma 17 }
% 0.14/0.46 antidomain(one)
% 0.14/0.46 = { by lemma 16 }
% 0.14/0.46 zero
% 0.14/0.46
% 0.14/0.46 Lemma 23: domain_difference(X, X) = zero.
% 0.14/0.46 Proof:
% 0.14/0.46 domain_difference(X, X)
% 0.14/0.46 = { by axiom 10 (domain_difference) }
% 0.14/0.46 multiplication(domain(X), antidomain(X))
% 0.14/0.46 = { by axiom 7 (domain4) }
% 0.14/0.46 multiplication(antidomain(antidomain(X)), antidomain(X))
% 0.14/0.46 = { by axiom 8 (domain1) }
% 0.14/0.46 zero
% 0.14/0.46
% 0.14/0.46 Goal 1 (goals_1): divergence(x0) = zero.
% 0.14/0.46 Proof:
% 0.14/0.46 divergence(x0)
% 0.14/0.46 = { by axiom 2 (additive_identity) R->L }
% 0.14/0.46 addition(divergence(x0), zero)
% 0.14/0.46 = { by lemma 22 R->L }
% 0.14/0.46 addition(divergence(x0), forward_diamond(star(x0), zero))
% 0.14/0.46 = { by lemma 23 R->L }
% 0.14/0.46 addition(divergence(x0), forward_diamond(star(x0), domain_difference(divergence(x0), divergence(x0))))
% 0.14/0.46 = { by axiom 9 (divergence1) R->L }
% 0.14/0.46 addition(divergence(x0), forward_diamond(star(x0), domain_difference(divergence(x0), forward_diamond(x0, divergence(x0)))))
% 0.14/0.46 = { by lemma 21 R->L }
% 0.14/0.46 addition(divergence(x0), forward_diamond(star(x0), domain_difference(domain(divergence(x0)), forward_diamond(x0, divergence(x0)))))
% 0.14/0.46 = { by lemma 21 R->L }
% 0.14/0.46 addition(divergence(x0), forward_diamond(star(x0), domain_difference(domain(divergence(x0)), forward_diamond(x0, domain(divergence(x0))))))
% 0.14/0.46 = { by lemma 21 R->L }
% 0.14/0.46 addition(domain(divergence(x0)), forward_diamond(star(x0), domain_difference(domain(divergence(x0)), forward_diamond(x0, domain(divergence(x0))))))
% 0.14/0.46 = { by axiom 15 (goals) }
% 0.14/0.46 forward_diamond(star(x0), domain_difference(domain(divergence(x0)), forward_diamond(x0, domain(divergence(x0)))))
% 0.14/0.46 = { by lemma 21 }
% 0.14/0.46 forward_diamond(star(x0), domain_difference(divergence(x0), forward_diamond(x0, domain(divergence(x0)))))
% 0.14/0.46 = { by lemma 21 }
% 0.14/0.46 forward_diamond(star(x0), domain_difference(divergence(x0), forward_diamond(x0, divergence(x0))))
% 0.14/0.46 = { by axiom 9 (divergence1) }
% 0.14/0.46 forward_diamond(star(x0), domain_difference(divergence(x0), divergence(x0)))
% 0.14/0.46 = { by lemma 23 }
% 0.14/0.46 forward_diamond(star(x0), zero)
% 0.14/0.46 = { by lemma 22 }
% 0.14/0.46 zero
% 0.14/0.46 % SZS output end Proof
% 0.14/0.46
% 0.14/0.46 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------