%------------------------------------------------------------------------------ % File : Zenon---0.7.1 % Problem : SET162+3 : TPTP v8.1.0. Released v2.2.0. % Transfm : none % Format : tptp:raw % Command : run_zenon %s %d % Computer : n009.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 06:34:31 EDT 2022 % Result : Theorem 0.44s 0.62s % Output : Proof 0.44s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.11/0.12 % Problem : SET162+3 : TPTP v8.1.0. Released v2.2.0. % 0.11/0.12 % Command : run_zenon %s %d % 0.12/0.33 % Computer : n009.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 : 600 % 0.12/0.33 % DateTime : Mon Jul 11 02:43:22 EDT 2022 % 0.12/0.33 % CPUTime : % 0.44/0.62 (* PROOF-FOUND *) % 0.44/0.62 % SZS status Theorem % 0.44/0.62 (* BEGIN-PROOF *) % 0.44/0.62 % SZS output start Proof % 0.44/0.62 Theorem prove_union_empty_set : (forall B : zenon_U, ((union B (empty_set)) = B)). % 0.44/0.62 Proof. % 0.44/0.62 apply NNPP. intro zenon_G. % 0.44/0.62 apply (zenon_notallex_s (fun B : zenon_U => ((union B (empty_set)) = B)) zenon_G); [ zenon_intro zenon_H9; idtac ]. % 0.44/0.62 elim zenon_H9. zenon_intro zenon_TB_k. zenon_intro zenon_Hb. % 0.44/0.62 generalize (commutativity_of_union (empty_set)). zenon_intro zenon_Hc. % 0.44/0.62 generalize (zenon_Hc zenon_TB_k). zenon_intro zenon_Hd. % 0.44/0.62 generalize (equal_defn zenon_TB_k). zenon_intro zenon_He. % 0.44/0.62 generalize (union_defn zenon_TB_k). zenon_intro zenon_Hf. % 0.44/0.62 generalize (union_defn (empty_set)). zenon_intro zenon_H10. % 0.44/0.62 generalize (zenon_Hf (empty_set)). zenon_intro zenon_H11. % 0.44/0.62 generalize (zenon_H10 zenon_TB_k). zenon_intro zenon_H12. % 0.44/0.62 generalize (zenon_He (union zenon_TB_k (empty_set))). zenon_intro zenon_H13. % 0.44/0.62 apply (zenon_equiv_s _ _ zenon_H13); [ zenon_intro zenon_H17; zenon_intro zenon_H16 | zenon_intro zenon_H15; zenon_intro zenon_H14 ]. % 0.44/0.62 apply (zenon_notand_s _ _ zenon_H16); [ zenon_intro zenon_H19 | zenon_intro zenon_H18 ]. % 0.44/0.62 generalize (subset_defn zenon_TB_k). zenon_intro zenon_H1a. % 0.44/0.62 generalize (zenon_H1a (union zenon_TB_k (empty_set))). zenon_intro zenon_H1b. % 0.44/0.62 apply (zenon_equiv_s _ _ zenon_H1b); [ zenon_intro zenon_H19; zenon_intro zenon_H1e | zenon_intro zenon_H1d; zenon_intro zenon_H1c ]. % 0.44/0.62 apply (zenon_notallex_s (fun D : zenon_U => ((member D zenon_TB_k)->(member D (union zenon_TB_k (empty_set))))) zenon_H1e); [ zenon_intro zenon_H1f; idtac ]. % 0.44/0.62 elim zenon_H1f. zenon_intro zenon_TD_bg. zenon_intro zenon_H21. % 0.44/0.62 apply (zenon_notimply_s _ _ zenon_H21). zenon_intro zenon_H23. zenon_intro zenon_H22. % 0.44/0.62 generalize (zenon_H11 zenon_TD_bg). zenon_intro zenon_H24. % 0.44/0.62 apply (zenon_equiv_s _ _ zenon_H24); [ zenon_intro zenon_H22; zenon_intro zenon_H27 | zenon_intro zenon_H26; zenon_intro zenon_H25 ]. % 0.44/0.62 apply (zenon_notor_s _ _ zenon_H27). zenon_intro zenon_H29. zenon_intro zenon_H28. % 0.44/0.62 exact (zenon_H29 zenon_H23). % 0.44/0.62 exact (zenon_H22 zenon_H26). % 0.44/0.62 exact (zenon_H19 zenon_H1d). % 0.44/0.62 generalize (subset_defn (union zenon_TB_k (empty_set))). zenon_intro zenon_H2a. % 0.44/0.62 generalize (zenon_H2a zenon_TB_k). zenon_intro zenon_H2b. % 0.44/0.62 apply (zenon_equiv_s _ _ zenon_H2b); [ zenon_intro zenon_H18; zenon_intro zenon_H2e | zenon_intro zenon_H2d; zenon_intro zenon_H2c ]. % 0.44/0.62 apply (zenon_notallex_s (fun D : zenon_U => ((member D (union zenon_TB_k (empty_set)))->(member D zenon_TB_k))) zenon_H2e); [ zenon_intro zenon_H2f; idtac ]. % 0.44/0.62 elim zenon_H2f. zenon_intro zenon_TD_bw. zenon_intro zenon_H31. % 0.44/0.62 apply (zenon_notimply_s _ _ zenon_H31). zenon_intro zenon_H33. zenon_intro zenon_H32. % 0.44/0.62 generalize (zenon_H12 zenon_TD_bw). zenon_intro zenon_H34. % 0.44/0.62 apply (zenon_equiv_s _ _ zenon_H34); [ zenon_intro zenon_H38; zenon_intro zenon_H37 | zenon_intro zenon_H36; zenon_intro zenon_H35 ]. % 0.44/0.62 cut ((member zenon_TD_bw (union zenon_TB_k (empty_set))) = (member zenon_TD_bw (union (empty_set) zenon_TB_k))). % 0.44/0.62 intro zenon_D_pnotp. % 0.44/0.62 apply zenon_H38. % 0.44/0.62 rewrite <- zenon_D_pnotp. % 0.44/0.62 exact zenon_H33. % 0.44/0.62 cut (((union zenon_TB_k (empty_set)) = (union (empty_set) zenon_TB_k))); [idtac | apply NNPP; zenon_intro zenon_H39]. % 0.44/0.62 cut ((zenon_TD_bw = zenon_TD_bw)); [idtac | apply NNPP; zenon_intro zenon_H3a]. % 0.44/0.62 congruence. % 0.44/0.62 apply zenon_H3a. apply refl_equal. % 0.44/0.62 apply zenon_H39. apply sym_equal. exact zenon_Hd. % 0.44/0.62 apply (zenon_or_s _ _ zenon_H35); [ zenon_intro zenon_H3c | zenon_intro zenon_H3b ]. % 0.44/0.62 generalize (empty_set_defn zenon_TD_bw). zenon_intro zenon_H3d. % 0.44/0.62 exact (zenon_H3d zenon_H3c). % 0.44/0.62 exact (zenon_H32 zenon_H3b). % 0.44/0.62 exact (zenon_H18 zenon_H2d). % 0.44/0.62 apply zenon_Hb. apply sym_equal. exact zenon_H15. % 0.44/0.62 Qed. % 0.44/0.62 % SZS output end Proof % 0.44/0.62 (* END-PROOF *) % 0.44/0.62 nodes searched: 3495 % 0.44/0.62 max branch formulas: 823 % 0.44/0.62 proof nodes created: 242 % 0.44/0.62 formulas created: 10421 % 0.44/0.62 %------------------------------------------------------------------------------