%------------------------------------------------------------------------------ % 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 *) %------------------------------------------------------------------------------