TSTP Solution File: SEU323+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SEU323+1 : TPTP v8.1.2. Released v3.3.0.
% 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 : n022.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 17:52:13 EDT 2023
% Result : Theorem 0.20s 0.50s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : SEU323+1 : TPTP v8.1.2. Released v3.3.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.16/0.34 % Computer : n022.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Thu Aug 24 01:15:26 EDT 2023
% 0.16/0.34 % CPUTime :
% 0.20/0.50 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.20/0.50
% 0.20/0.50 % SZS status Theorem
% 0.20/0.50
% 0.20/0.51 % SZS output start Proof
% 0.20/0.51 Take the following subset of the input axioms:
% 0.20/0.51 fof(d1_tops_1, axiom, ![A2]: (top_str(A2) => ![B]: (element(B, powerset(the_carrier(A2))) => interior(A2, B)=subset_complement(the_carrier(A2), topstr_closure(A2, subset_complement(the_carrier(A2), B)))))).
% 0.20/0.51 fof(dt_k1_tops_1, axiom, ![B2, A2_2]: ((top_str(A2_2) & element(B2, powerset(the_carrier(A2_2)))) => element(interior(A2_2, B2), powerset(the_carrier(A2_2))))).
% 0.20/0.51 fof(dt_k3_subset_1, axiom, ![B2, A2_2]: (element(B2, powerset(A2_2)) => element(subset_complement(A2_2, B2), powerset(A2_2)))).
% 0.20/0.51 fof(dt_k6_pre_topc, axiom, ![B2, A2_2]: ((top_str(A2_2) & element(B2, powerset(the_carrier(A2_2)))) => element(topstr_closure(A2_2, B2), powerset(the_carrier(A2_2))))).
% 0.20/0.51 fof(fc2_tops_1, axiom, ![B2, A2_2]: ((topological_space(A2_2) & (top_str(A2_2) & element(B2, powerset(the_carrier(A2_2))))) => closed_subset(topstr_closure(A2_2, B2), A2_2))).
% 0.20/0.51 fof(fc3_tops_1, axiom, ![B2, A2_2]: ((topological_space(A2_2) & (top_str(A2_2) & (closed_subset(B2, A2_2) & element(B2, powerset(the_carrier(A2_2)))))) => open_subset(subset_complement(the_carrier(A2_2), B2), A2_2))).
% 0.20/0.51 fof(involutiveness_k3_subset_1, axiom, ![B2, A2_2]: (element(B2, powerset(A2_2)) => subset_complement(A2_2, subset_complement(A2_2, B2))=B2)).
% 0.20/0.51 fof(t51_tops_1, conjecture, ![A]: ((topological_space(A) & top_str(A)) => ![B2]: (element(B2, powerset(the_carrier(A))) => open_subset(interior(A, B2), A)))).
% 0.20/0.51
% 0.20/0.51 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.51 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.51 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.51 fresh(y, y, x1...xn) = u
% 0.20/0.51 C => fresh(s, t, x1...xn) = v
% 0.20/0.51 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.51 variables of u and v.
% 0.20/0.51 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.51 input problem has no model of domain size 1).
% 0.20/0.51
% 0.20/0.51 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.51
% 0.20/0.51 Axiom 1 (t51_tops_1_1): top_str(a) = true.
% 0.20/0.51 Axiom 2 (t51_tops_1): topological_space(a) = true.
% 0.20/0.51 Axiom 3 (t51_tops_1_2): element(b, powerset(the_carrier(a))) = true.
% 0.20/0.51 Axiom 4 (involutiveness_k3_subset_1): fresh(X, X, Y, Z) = Z.
% 0.20/0.51 Axiom 5 (fc3_tops_1): fresh30(X, X, Y, Z) = true.
% 0.20/0.51 Axiom 6 (dt_k6_pre_topc): fresh24(X, X, Y, Z) = true.
% 0.20/0.51 Axiom 7 (fc2_tops_1): fresh22(X, X, Y, Z) = true.
% 0.20/0.51 Axiom 8 (dt_k1_tops_1): fresh16(X, X, Y, Z) = true.
% 0.20/0.51 Axiom 9 (d1_tops_1): fresh12(X, X, Y, Z) = interior(Y, Z).
% 0.20/0.51 Axiom 10 (dt_k3_subset_1): fresh10(X, X, Y, Z) = true.
% 0.20/0.51 Axiom 11 (fc2_tops_1): fresh8(X, X, Y, Z) = closed_subset(topstr_closure(Y, Z), Y).
% 0.20/0.51 Axiom 12 (fc3_tops_1): fresh28(X, X, Y, Z) = open_subset(subset_complement(the_carrier(Y), Z), Y).
% 0.20/0.51 Axiom 13 (fc3_tops_1): fresh29(X, X, Y, Z) = fresh30(topological_space(Y), true, Y, Z).
% 0.20/0.51 Axiom 14 (fc3_tops_1): fresh27(X, X, Y, Z) = fresh28(top_str(Y), true, Y, Z).
% 0.20/0.51 Axiom 15 (dt_k6_pre_topc): fresh23(X, X, Y, Z) = fresh24(top_str(Y), true, Y, Z).
% 0.20/0.51 Axiom 16 (fc2_tops_1): fresh21(X, X, Y, Z) = fresh22(topological_space(Y), true, Y, Z).
% 0.20/0.51 Axiom 17 (dt_k1_tops_1): fresh15(X, X, Y, Z) = fresh16(top_str(Y), true, Y, Z).
% 0.20/0.51 Axiom 18 (d1_tops_1): fresh11(X, X, Y, Z) = fresh12(top_str(Y), true, Y, Z).
% 0.20/0.51 Axiom 19 (involutiveness_k3_subset_1): fresh(element(X, powerset(Y)), true, Y, X) = subset_complement(Y, subset_complement(Y, X)).
% 0.20/0.51 Axiom 20 (dt_k3_subset_1): fresh10(element(X, powerset(Y)), true, Y, X) = element(subset_complement(Y, X), powerset(Y)).
% 0.20/0.51 Axiom 21 (fc3_tops_1): fresh27(element(X, powerset(the_carrier(Y))), true, Y, X) = fresh29(closed_subset(X, Y), true, Y, X).
% 0.20/0.51 Axiom 22 (dt_k6_pre_topc): fresh23(element(X, powerset(the_carrier(Y))), true, Y, X) = element(topstr_closure(Y, X), powerset(the_carrier(Y))).
% 0.20/0.51 Axiom 23 (fc2_tops_1): fresh21(element(X, powerset(the_carrier(Y))), true, Y, X) = fresh8(top_str(Y), true, Y, X).
% 0.20/0.51 Axiom 24 (dt_k1_tops_1): fresh15(element(X, powerset(the_carrier(Y))), true, Y, X) = element(interior(Y, X), powerset(the_carrier(Y))).
% 0.20/0.51 Axiom 25 (d1_tops_1): fresh11(element(X, powerset(the_carrier(Y))), true, Y, X) = subset_complement(the_carrier(Y), topstr_closure(Y, subset_complement(the_carrier(Y), X))).
% 0.20/0.51
% 0.20/0.51 Lemma 26: element(subset_complement(the_carrier(a), b), powerset(the_carrier(a))) = true.
% 0.20/0.51 Proof:
% 0.20/0.51 element(subset_complement(the_carrier(a), b), powerset(the_carrier(a)))
% 0.20/0.51 = { by axiom 20 (dt_k3_subset_1) R->L }
% 0.20/0.51 fresh10(element(b, powerset(the_carrier(a))), true, the_carrier(a), b)
% 0.20/0.51 = { by axiom 3 (t51_tops_1_2) }
% 0.20/0.51 fresh10(true, true, the_carrier(a), b)
% 0.20/0.51 = { by axiom 10 (dt_k3_subset_1) }
% 0.20/0.51 true
% 0.20/0.51
% 0.20/0.51 Lemma 27: element(topstr_closure(a, subset_complement(the_carrier(a), b)), powerset(the_carrier(a))) = true.
% 0.20/0.51 Proof:
% 0.20/0.51 element(topstr_closure(a, subset_complement(the_carrier(a), b)), powerset(the_carrier(a)))
% 0.20/0.51 = { by axiom 22 (dt_k6_pre_topc) R->L }
% 0.20/0.51 fresh23(element(subset_complement(the_carrier(a), b), powerset(the_carrier(a))), true, a, subset_complement(the_carrier(a), b))
% 0.20/0.51 = { by lemma 26 }
% 0.20/0.51 fresh23(true, true, a, subset_complement(the_carrier(a), b))
% 0.20/0.51 = { by axiom 15 (dt_k6_pre_topc) }
% 0.20/0.51 fresh24(top_str(a), true, a, subset_complement(the_carrier(a), b))
% 0.20/0.51 = { by axiom 1 (t51_tops_1_1) }
% 0.20/0.51 fresh24(true, true, a, subset_complement(the_carrier(a), b))
% 0.20/0.52 = { by axiom 6 (dt_k6_pre_topc) }
% 0.20/0.52 true
% 0.20/0.52
% 0.20/0.52 Goal 1 (t51_tops_1_3): open_subset(interior(a, b), a) = true.
% 0.20/0.52 Proof:
% 0.20/0.52 open_subset(interior(a, b), a)
% 0.20/0.52 = { by axiom 4 (involutiveness_k3_subset_1) R->L }
% 0.20/0.52 open_subset(fresh(true, true, the_carrier(a), interior(a, b)), a)
% 0.20/0.52 = { by axiom 8 (dt_k1_tops_1) R->L }
% 0.20/0.52 open_subset(fresh(fresh16(true, true, a, b), true, the_carrier(a), interior(a, b)), a)
% 0.20/0.52 = { by axiom 1 (t51_tops_1_1) R->L }
% 0.20/0.52 open_subset(fresh(fresh16(top_str(a), true, a, b), true, the_carrier(a), interior(a, b)), a)
% 0.20/0.52 = { by axiom 17 (dt_k1_tops_1) R->L }
% 0.20/0.52 open_subset(fresh(fresh15(true, true, a, b), true, the_carrier(a), interior(a, b)), a)
% 0.20/0.52 = { by axiom 3 (t51_tops_1_2) R->L }
% 0.20/0.52 open_subset(fresh(fresh15(element(b, powerset(the_carrier(a))), true, a, b), true, the_carrier(a), interior(a, b)), a)
% 0.20/0.52 = { by axiom 24 (dt_k1_tops_1) }
% 0.20/0.52 open_subset(fresh(element(interior(a, b), powerset(the_carrier(a))), true, the_carrier(a), interior(a, b)), a)
% 0.20/0.52 = { by axiom 19 (involutiveness_k3_subset_1) }
% 0.20/0.52 open_subset(subset_complement(the_carrier(a), subset_complement(the_carrier(a), interior(a, b))), a)
% 0.20/0.52 = { by axiom 12 (fc3_tops_1) R->L }
% 0.20/0.52 fresh28(true, true, a, subset_complement(the_carrier(a), interior(a, b)))
% 0.20/0.52 = { by axiom 9 (d1_tops_1) R->L }
% 0.20/0.52 fresh28(true, true, a, subset_complement(the_carrier(a), fresh12(true, true, a, b)))
% 0.20/0.52 = { by axiom 1 (t51_tops_1_1) R->L }
% 0.20/0.52 fresh28(true, true, a, subset_complement(the_carrier(a), fresh12(top_str(a), true, a, b)))
% 0.20/0.52 = { by axiom 18 (d1_tops_1) R->L }
% 0.20/0.52 fresh28(true, true, a, subset_complement(the_carrier(a), fresh11(true, true, a, b)))
% 0.20/0.52 = { by axiom 3 (t51_tops_1_2) R->L }
% 0.20/0.52 fresh28(true, true, a, subset_complement(the_carrier(a), fresh11(element(b, powerset(the_carrier(a))), true, a, b)))
% 0.20/0.52 = { by axiom 25 (d1_tops_1) }
% 0.20/0.52 fresh28(true, true, a, subset_complement(the_carrier(a), subset_complement(the_carrier(a), topstr_closure(a, subset_complement(the_carrier(a), b)))))
% 0.20/0.52 = { by axiom 19 (involutiveness_k3_subset_1) R->L }
% 0.20/0.52 fresh28(true, true, a, fresh(element(topstr_closure(a, subset_complement(the_carrier(a), b)), powerset(the_carrier(a))), true, the_carrier(a), topstr_closure(a, subset_complement(the_carrier(a), b))))
% 0.20/0.52 = { by lemma 27 }
% 0.20/0.52 fresh28(true, true, a, fresh(true, true, the_carrier(a), topstr_closure(a, subset_complement(the_carrier(a), b))))
% 0.20/0.52 = { by axiom 4 (involutiveness_k3_subset_1) }
% 0.20/0.52 fresh28(true, true, a, topstr_closure(a, subset_complement(the_carrier(a), b)))
% 0.20/0.52 = { by axiom 1 (t51_tops_1_1) R->L }
% 0.20/0.52 fresh28(top_str(a), true, a, topstr_closure(a, subset_complement(the_carrier(a), b)))
% 0.20/0.52 = { by axiom 14 (fc3_tops_1) R->L }
% 0.20/0.52 fresh27(true, true, a, topstr_closure(a, subset_complement(the_carrier(a), b)))
% 0.20/0.52 = { by lemma 27 R->L }
% 0.20/0.52 fresh27(element(topstr_closure(a, subset_complement(the_carrier(a), b)), powerset(the_carrier(a))), true, a, topstr_closure(a, subset_complement(the_carrier(a), b)))
% 0.20/0.52 = { by axiom 21 (fc3_tops_1) }
% 0.20/0.52 fresh29(closed_subset(topstr_closure(a, subset_complement(the_carrier(a), b)), a), true, a, topstr_closure(a, subset_complement(the_carrier(a), b)))
% 0.20/0.52 = { by axiom 11 (fc2_tops_1) R->L }
% 0.20/0.52 fresh29(fresh8(true, true, a, subset_complement(the_carrier(a), b)), true, a, topstr_closure(a, subset_complement(the_carrier(a), b)))
% 0.20/0.52 = { by axiom 1 (t51_tops_1_1) R->L }
% 0.20/0.52 fresh29(fresh8(top_str(a), true, a, subset_complement(the_carrier(a), b)), true, a, topstr_closure(a, subset_complement(the_carrier(a), b)))
% 0.20/0.52 = { by axiom 23 (fc2_tops_1) R->L }
% 0.20/0.52 fresh29(fresh21(element(subset_complement(the_carrier(a), b), powerset(the_carrier(a))), true, a, subset_complement(the_carrier(a), b)), true, a, topstr_closure(a, subset_complement(the_carrier(a), b)))
% 0.20/0.52 = { by lemma 26 }
% 0.20/0.52 fresh29(fresh21(true, true, a, subset_complement(the_carrier(a), b)), true, a, topstr_closure(a, subset_complement(the_carrier(a), b)))
% 0.20/0.52 = { by axiom 16 (fc2_tops_1) }
% 0.20/0.52 fresh29(fresh22(topological_space(a), true, a, subset_complement(the_carrier(a), b)), true, a, topstr_closure(a, subset_complement(the_carrier(a), b)))
% 0.20/0.52 = { by axiom 2 (t51_tops_1) }
% 0.20/0.52 fresh29(fresh22(true, true, a, subset_complement(the_carrier(a), b)), true, a, topstr_closure(a, subset_complement(the_carrier(a), b)))
% 0.20/0.52 = { by axiom 7 (fc2_tops_1) }
% 0.20/0.52 fresh29(true, true, a, topstr_closure(a, subset_complement(the_carrier(a), b)))
% 0.20/0.52 = { by axiom 13 (fc3_tops_1) }
% 0.20/0.52 fresh30(topological_space(a), true, a, topstr_closure(a, subset_complement(the_carrier(a), b)))
% 0.20/0.52 = { by axiom 2 (t51_tops_1) }
% 0.20/0.52 fresh30(true, true, a, topstr_closure(a, subset_complement(the_carrier(a), b)))
% 0.20/0.52 = { by axiom 5 (fc3_tops_1) }
% 0.20/0.52 true
% 0.20/0.52 % SZS output end Proof
% 0.20/0.52
% 0.20/0.52 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------