TSTP Solution File: SEU341+1 by SuperZenon---0.0.1
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- Process Solution
%------------------------------------------------------------------------------
% File : SuperZenon---0.0.1
% Problem : SEU341+1 : TPTP v8.1.0. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : run_super_zenon -p0 -itptp -om -max-time %d %s
% Computer : n006.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 : 600s
% DateTime : Tue Jul 19 14:50:23 EDT 2022
% Result : Theorem 4.00s 4.18s
% Output : Proof 4.00s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.14 % Problem : SEU341+1 : TPTP v8.1.0. Released v3.3.0.
% 0.04/0.14 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s
% 0.15/0.36 % Computer : n006.cluster.edu
% 0.15/0.36 % Model : x86_64 x86_64
% 0.15/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.36 % Memory : 8042.1875MB
% 0.15/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.36 % CPULimit : 300
% 0.15/0.36 % WCLimit : 600
% 0.15/0.36 % DateTime : Sun Jun 19 10:40:40 EDT 2022
% 0.15/0.36 % CPUTime :
% 4.00/4.18 % SZS status Theorem
% 4.00/4.18 (* PROOF-FOUND *)
% 4.00/4.18 (* BEGIN-PROOF *)
% 4.00/4.18 % SZS output start Proof
% 4.00/4.18 1. (topological_space T_0) (-. (topological_space T_0)) ### Axiom
% 4.00/4.18 2. (top_str T_0) (-. (top_str T_0)) ### Axiom
% 4.00/4.18 3. (top_str T_0) (-. (top_str T_0)) ### Axiom
% 4.00/4.18 4. (element T_1 (powerset (the_carrier T_0))) (-. (element T_1 (powerset (the_carrier T_0)))) ### Axiom
% 4.00/4.18 5. (element T_1 (powerset (the_carrier T_0))) (-. (element T_1 (powerset (the_carrier T_0)))) ### Axiom
% 4.00/4.18 6. (open_subset T_1 T_0) (-. (open_subset T_1 T_0)) ### Axiom
% 4.00/4.18 7. (-. (empty_carrier T_0)) (empty_carrier T_0) ### Axiom
% 4.00/4.18 8. (topological_space T_0) (-. (topological_space T_0)) ### Axiom
% 4.00/4.18 9. (top_str T_0) (-. (top_str T_0)) ### Axiom
% 4.00/4.18 10. (element T_2 (the_carrier T_0)) (-. (element T_2 (the_carrier T_0))) ### Axiom
% 4.00/4.18 11. (element T_1 (powerset (the_carrier T_0))) (-. (element T_1 (powerset (the_carrier T_0)))) ### Axiom
% 4.00/4.18 12. (T_2 != T_2) ### Refl(=)
% 4.00/4.18 13. ((interior T_0 T_1) = T_1) (T_1 != (interior T_0 T_1)) ### Sym(=)
% 4.00/4.18 14. (-. (in T_2 (interior T_0 T_1))) (in T_2 T_1) ((interior T_0 T_1) = T_1) ### P-NotP 12 13
% 4.00/4.18 15. (-. (point_neighbourhood T_1 T_0 T_2)) (point_neighbourhood T_1 T_0 T_2) ### Axiom
% 4.00/4.18 16. ((point_neighbourhood T_1 T_0 T_2) <=> (in T_2 (interior T_0 T_1))) (-. (point_neighbourhood T_1 T_0 T_2)) ((interior T_0 T_1) = T_1) (in T_2 T_1) ### Equiv 14 15
% 4.00/4.18 17. ((element T_1 (powerset (the_carrier T_0))) => ((point_neighbourhood T_1 T_0 T_2) <=> (in T_2 (interior T_0 T_1)))) (in T_2 T_1) ((interior T_0 T_1) = T_1) (-. (point_neighbourhood T_1 T_0 T_2)) (element T_1 (powerset (the_carrier T_0))) ### Imply 11 16
% 4.00/4.18 18. (All C, ((element C (powerset (the_carrier T_0))) => ((point_neighbourhood C T_0 T_2) <=> (in T_2 (interior T_0 C))))) (element T_1 (powerset (the_carrier T_0))) (-. (point_neighbourhood T_1 T_0 T_2)) ((interior T_0 T_1) = T_1) (in T_2 T_1) ### All 17
% 4.00/4.18 19. ((element T_2 (the_carrier T_0)) => (All C, ((element C (powerset (the_carrier T_0))) => ((point_neighbourhood C T_0 T_2) <=> (in T_2 (interior T_0 C)))))) (in T_2 T_1) ((interior T_0 T_1) = T_1) (-. (point_neighbourhood T_1 T_0 T_2)) (element T_1 (powerset (the_carrier T_0))) (element T_2 (the_carrier T_0)) ### Imply 10 18
% 4.00/4.18 20. (All B, ((element B (the_carrier T_0)) => (All C, ((element C (powerset (the_carrier T_0))) => ((point_neighbourhood C T_0 B) <=> (in B (interior T_0 C))))))) (element T_2 (the_carrier T_0)) (element T_1 (powerset (the_carrier T_0))) (-. (point_neighbourhood T_1 T_0 T_2)) ((interior T_0 T_1) = T_1) (in T_2 T_1) ### All 19
% 4.00/4.18 21. (((-. (empty_carrier T_0)) /\ ((topological_space T_0) /\ (top_str T_0))) => (All B, ((element B (the_carrier T_0)) => (All C, ((element C (powerset (the_carrier T_0))) => ((point_neighbourhood C T_0 B) <=> (in B (interior T_0 C)))))))) (in T_2 T_1) ((interior T_0 T_1) = T_1) (-. (point_neighbourhood T_1 T_0 T_2)) (element T_1 (powerset (the_carrier T_0))) (element T_2 (the_carrier T_0)) (top_str T_0) (topological_space T_0) (-. (empty_carrier T_0)) ### DisjTree 7 8 9 20
% 4.00/4.18 22. (All A, (((-. (empty_carrier A)) /\ ((topological_space A) /\ (top_str A))) => (All B, ((element B (the_carrier A)) => (All C, ((element C (powerset (the_carrier A))) => ((point_neighbourhood C A B) <=> (in B (interior A C))))))))) (-. (empty_carrier T_0)) (topological_space T_0) (top_str T_0) (element T_2 (the_carrier T_0)) (element T_1 (powerset (the_carrier T_0))) (-. (point_neighbourhood T_1 T_0 T_2)) ((interior T_0 T_1) = T_1) (in T_2 T_1) ### All 21
% 4.00/4.18 23. ((open_subset T_1 T_0) => ((interior T_0 T_1) = T_1)) (in T_2 T_1) (-. (point_neighbourhood T_1 T_0 T_2)) (element T_1 (powerset (the_carrier T_0))) (element T_2 (the_carrier T_0)) (top_str T_0) (topological_space T_0) (-. (empty_carrier T_0)) (All A, (((-. (empty_carrier A)) /\ ((topological_space A) /\ (top_str A))) => (All B, ((element B (the_carrier A)) => (All C, ((element C (powerset (the_carrier A))) => ((point_neighbourhood C A B) <=> (in B (interior A C))))))))) (open_subset T_1 T_0) ### Imply 6 22
% 4.00/4.18 24. (((open_subset T_1 T_0) => ((interior T_0 T_1) = T_1)) /\ (((interior T_0 T_1) = T_1) => (open_subset T_1 T_0))) (open_subset T_1 T_0) (All A, (((-. (empty_carrier A)) /\ ((topological_space A) /\ (top_str A))) => (All B, ((element B (the_carrier A)) => (All C, ((element C (powerset (the_carrier A))) => ((point_neighbourhood C A B) <=> (in B (interior A C))))))))) (-. (empty_carrier T_0)) (topological_space T_0) (top_str T_0) (element T_2 (the_carrier T_0)) (element T_1 (powerset (the_carrier T_0))) (-. (point_neighbourhood T_1 T_0 T_2)) (in T_2 T_1) ### And 23
% 4.00/4.18 25. ((element T_1 (powerset (the_carrier T_0))) => (((open_subset T_1 T_0) => ((interior T_0 T_1) = T_1)) /\ (((interior T_0 T_1) = T_1) => (open_subset T_1 T_0)))) (in T_2 T_1) (-. (point_neighbourhood T_1 T_0 T_2)) (element T_2 (the_carrier T_0)) (top_str T_0) (topological_space T_0) (-. (empty_carrier T_0)) (All A, (((-. (empty_carrier A)) /\ ((topological_space A) /\ (top_str A))) => (All B, ((element B (the_carrier A)) => (All C, ((element C (powerset (the_carrier A))) => ((point_neighbourhood C A B) <=> (in B (interior A C))))))))) (open_subset T_1 T_0) (element T_1 (powerset (the_carrier T_0))) ### Imply 5 24
% 4.00/4.18 26. (All D, ((element D (powerset (the_carrier T_0))) => (((open_subset D T_0) => ((interior T_0 D) = D)) /\ (((interior T_0 T_1) = T_1) => (open_subset T_1 T_0))))) (element T_1 (powerset (the_carrier T_0))) (open_subset T_1 T_0) (All A, (((-. (empty_carrier A)) /\ ((topological_space A) /\ (top_str A))) => (All B, ((element B (the_carrier A)) => (All C, ((element C (powerset (the_carrier A))) => ((point_neighbourhood C A B) <=> (in B (interior A C))))))))) (-. (empty_carrier T_0)) (topological_space T_0) (top_str T_0) (element T_2 (the_carrier T_0)) (-. (point_neighbourhood T_1 T_0 T_2)) (in T_2 T_1) ### All 25
% 4.00/4.18 27. ((element T_1 (powerset (the_carrier T_0))) => (All D, ((element D (powerset (the_carrier T_0))) => (((open_subset D T_0) => ((interior T_0 D) = D)) /\ (((interior T_0 T_1) = T_1) => (open_subset T_1 T_0)))))) (in T_2 T_1) (-. (point_neighbourhood T_1 T_0 T_2)) (element T_2 (the_carrier T_0)) (top_str T_0) (topological_space T_0) (-. (empty_carrier T_0)) (All A, (((-. (empty_carrier A)) /\ ((topological_space A) /\ (top_str A))) => (All B, ((element B (the_carrier A)) => (All C, ((element C (powerset (the_carrier A))) => ((point_neighbourhood C A B) <=> (in B (interior A C))))))))) (open_subset T_1 T_0) (element T_1 (powerset (the_carrier T_0))) ### Imply 4 26
% 4.00/4.18 28. (All C, ((element C (powerset (the_carrier T_0))) => (All D, ((element D (powerset (the_carrier T_0))) => (((open_subset D T_0) => ((interior T_0 D) = D)) /\ (((interior T_0 C) = C) => (open_subset C T_0))))))) (element T_1 (powerset (the_carrier T_0))) (open_subset T_1 T_0) (All A, (((-. (empty_carrier A)) /\ ((topological_space A) /\ (top_str A))) => (All B, ((element B (the_carrier A)) => (All C, ((element C (powerset (the_carrier A))) => ((point_neighbourhood C A B) <=> (in B (interior A C))))))))) (-. (empty_carrier T_0)) (topological_space T_0) (top_str T_0) (element T_2 (the_carrier T_0)) (-. (point_neighbourhood T_1 T_0 T_2)) (in T_2 T_1) ### All 27
% 4.00/4.18 29. ((top_str T_0) => (All C, ((element C (powerset (the_carrier T_0))) => (All D, ((element D (powerset (the_carrier T_0))) => (((open_subset D T_0) => ((interior T_0 D) = D)) /\ (((interior T_0 C) = C) => (open_subset C T_0)))))))) (in T_2 T_1) (-. (point_neighbourhood T_1 T_0 T_2)) (element T_2 (the_carrier T_0)) (topological_space T_0) (-. (empty_carrier T_0)) (All A, (((-. (empty_carrier A)) /\ ((topological_space A) /\ (top_str A))) => (All B, ((element B (the_carrier A)) => (All C, ((element C (powerset (the_carrier A))) => ((point_neighbourhood C A B) <=> (in B (interior A C))))))))) (open_subset T_1 T_0) (element T_1 (powerset (the_carrier T_0))) (top_str T_0) ### Imply 3 28
% 4.00/4.18 30. (All B, ((top_str B) => (All C, ((element C (powerset (the_carrier T_0))) => (All D, ((element D (powerset (the_carrier B))) => (((open_subset D B) => ((interior B D) = D)) /\ (((interior T_0 C) = C) => (open_subset C T_0))))))))) (top_str T_0) (element T_1 (powerset (the_carrier T_0))) (open_subset T_1 T_0) (All A, (((-. (empty_carrier A)) /\ ((topological_space A) /\ (top_str A))) => (All B, ((element B (the_carrier A)) => (All C, ((element C (powerset (the_carrier A))) => ((point_neighbourhood C A B) <=> (in B (interior A C))))))))) (-. (empty_carrier T_0)) (topological_space T_0) (element T_2 (the_carrier T_0)) (-. (point_neighbourhood T_1 T_0 T_2)) (in T_2 T_1) ### All 29
% 4.00/4.20 31. (((topological_space T_0) /\ (top_str T_0)) => (All B, ((top_str B) => (All C, ((element C (powerset (the_carrier T_0))) => (All D, ((element D (powerset (the_carrier B))) => (((open_subset D B) => ((interior B D) = D)) /\ (((interior T_0 C) = C) => (open_subset C T_0)))))))))) (in T_2 T_1) (-. (point_neighbourhood T_1 T_0 T_2)) (element T_2 (the_carrier T_0)) (-. (empty_carrier T_0)) (All A, (((-. (empty_carrier A)) /\ ((topological_space A) /\ (top_str A))) => (All B, ((element B (the_carrier A)) => (All C, ((element C (powerset (the_carrier A))) => ((point_neighbourhood C A B) <=> (in B (interior A C))))))))) (open_subset T_1 T_0) (element T_1 (powerset (the_carrier T_0))) (top_str T_0) (topological_space T_0) ### DisjTree 1 2 30
% 4.00/4.20 32. (All A, (((topological_space A) /\ (top_str A)) => (All B, ((top_str B) => (All C, ((element C (powerset (the_carrier A))) => (All D, ((element D (powerset (the_carrier B))) => (((open_subset D B) => ((interior B D) = D)) /\ (((interior A C) = C) => (open_subset C A))))))))))) (topological_space T_0) (top_str T_0) (element T_1 (powerset (the_carrier T_0))) (open_subset T_1 T_0) (All A, (((-. (empty_carrier A)) /\ ((topological_space A) /\ (top_str A))) => (All B, ((element B (the_carrier A)) => (All C, ((element C (powerset (the_carrier A))) => ((point_neighbourhood C A B) <=> (in B (interior A C))))))))) (-. (empty_carrier T_0)) (element T_2 (the_carrier T_0)) (-. (point_neighbourhood T_1 T_0 T_2)) (in T_2 T_1) ### All 31
% 4.00/4.20 33. (-. ((element T_2 (the_carrier T_0)) => (((open_subset T_1 T_0) /\ (in T_2 T_1)) => (point_neighbourhood T_1 T_0 T_2)))) (-. (empty_carrier T_0)) (All A, (((-. (empty_carrier A)) /\ ((topological_space A) /\ (top_str A))) => (All B, ((element B (the_carrier A)) => (All C, ((element C (powerset (the_carrier A))) => ((point_neighbourhood C A B) <=> (in B (interior A C))))))))) (element T_1 (powerset (the_carrier T_0))) (top_str T_0) (topological_space T_0) (All A, (((topological_space A) /\ (top_str A)) => (All B, ((top_str B) => (All C, ((element C (powerset (the_carrier A))) => (All D, ((element D (powerset (the_carrier B))) => (((open_subset D B) => ((interior B D) = D)) /\ (((interior A C) = C) => (open_subset C A))))))))))) ### ConjTree 32
% 4.00/4.20 34. (-. (All C, ((element C (the_carrier T_0)) => (((open_subset T_1 T_0) /\ (in C T_1)) => (point_neighbourhood T_1 T_0 C))))) (All A, (((topological_space A) /\ (top_str A)) => (All B, ((top_str B) => (All C, ((element C (powerset (the_carrier A))) => (All D, ((element D (powerset (the_carrier B))) => (((open_subset D B) => ((interior B D) = D)) /\ (((interior A C) = C) => (open_subset C A))))))))))) (topological_space T_0) (top_str T_0) (element T_1 (powerset (the_carrier T_0))) (All A, (((-. (empty_carrier A)) /\ ((topological_space A) /\ (top_str A))) => (All B, ((element B (the_carrier A)) => (All C, ((element C (powerset (the_carrier A))) => ((point_neighbourhood C A B) <=> (in B (interior A C))))))))) (-. (empty_carrier T_0)) ### NotAllEx 33
% 4.00/4.20 35. (-. ((element T_1 (powerset (the_carrier T_0))) => (All C, ((element C (the_carrier T_0)) => (((open_subset T_1 T_0) /\ (in C T_1)) => (point_neighbourhood T_1 T_0 C)))))) (-. (empty_carrier T_0)) (All A, (((-. (empty_carrier A)) /\ ((topological_space A) /\ (top_str A))) => (All B, ((element B (the_carrier A)) => (All C, ((element C (powerset (the_carrier A))) => ((point_neighbourhood C A B) <=> (in B (interior A C))))))))) (top_str T_0) (topological_space T_0) (All A, (((topological_space A) /\ (top_str A)) => (All B, ((top_str B) => (All C, ((element C (powerset (the_carrier A))) => (All D, ((element D (powerset (the_carrier B))) => (((open_subset D B) => ((interior B D) = D)) /\ (((interior A C) = C) => (open_subset C A))))))))))) ### NotImply 34
% 4.00/4.20 36. (-. (All B, ((element B (powerset (the_carrier T_0))) => (All C, ((element C (the_carrier T_0)) => (((open_subset B T_0) /\ (in C B)) => (point_neighbourhood B T_0 C))))))) (All A, (((topological_space A) /\ (top_str A)) => (All B, ((top_str B) => (All C, ((element C (powerset (the_carrier A))) => (All D, ((element D (powerset (the_carrier B))) => (((open_subset D B) => ((interior B D) = D)) /\ (((interior A C) = C) => (open_subset C A))))))))))) (topological_space T_0) (top_str T_0) (All A, (((-. (empty_carrier A)) /\ ((topological_space A) /\ (top_str A))) => (All B, ((element B (the_carrier A)) => (All C, ((element C (powerset (the_carrier A))) => ((point_neighbourhood C A B) <=> (in B (interior A C))))))))) (-. (empty_carrier T_0)) ### NotAllEx 35
% 4.00/4.20 37. (-. (((-. (empty_carrier T_0)) /\ ((topological_space T_0) /\ (top_str T_0))) => (All B, ((element B (powerset (the_carrier T_0))) => (All C, ((element C (the_carrier T_0)) => (((open_subset B T_0) /\ (in C B)) => (point_neighbourhood B T_0 C)))))))) (All A, (((-. (empty_carrier A)) /\ ((topological_space A) /\ (top_str A))) => (All B, ((element B (the_carrier A)) => (All C, ((element C (powerset (the_carrier A))) => ((point_neighbourhood C A B) <=> (in B (interior A C))))))))) (All A, (((topological_space A) /\ (top_str A)) => (All B, ((top_str B) => (All C, ((element C (powerset (the_carrier A))) => (All D, ((element D (powerset (the_carrier B))) => (((open_subset D B) => ((interior B D) = D)) /\ (((interior A C) = C) => (open_subset C A))))))))))) ### ConjTree 36
% 4.00/4.20 38. (-. (All A, (((-. (empty_carrier A)) /\ ((topological_space A) /\ (top_str A))) => (All B, ((element B (powerset (the_carrier A))) => (All C, ((element C (the_carrier A)) => (((open_subset B A) /\ (in C B)) => (point_neighbourhood B A C))))))))) (All A, (((topological_space A) /\ (top_str A)) => (All B, ((top_str B) => (All C, ((element C (powerset (the_carrier A))) => (All D, ((element D (powerset (the_carrier B))) => (((open_subset D B) => ((interior B D) = D)) /\ (((interior A C) = C) => (open_subset C A))))))))))) (All A, (((-. (empty_carrier A)) /\ ((topological_space A) /\ (top_str A))) => (All B, ((element B (the_carrier A)) => (All C, ((element C (powerset (the_carrier A))) => ((point_neighbourhood C A B) <=> (in B (interior A C))))))))) ### NotAllEx 37
% 4.00/4.20 % SZS output end Proof
% 4.00/4.20 (* END-PROOF *)
%------------------------------------------------------------------------------