%------------------------------------------------------------------------------ % File : Zenon---0.7.1 % Problem : MGT036+2 : TPTP v9.3.0. Released v2.0.0. % Transfm : none % Format : tptp:raw % Command : run_zenon %s %d % Computer : n015.cluster.edu % Model : x86_64 x86_64 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz % Memory : 8046.5625MB % OS : Linux 6.8.0-71-generic % CPULimit : 300s % WCLimit : 300s % DateTime : Tue Sep 8 06:24:21 AM UTC 2026 % Result : Theorem 0.63s 0.91s % Output : Proof 0.63s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.00/0.02 % Problem : MGT036+2 : TPTP v9.3.0. Released v2.0.0. % 0.00/0.03 % Command : run_zenon %s %d % 0.09/0.35 % Computer : n015.cluster.edu % 0.09/0.35 % Model : x86_64 x86_64 % 0.09/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.09/0.35 % Memory : 8046.5625MB % 0.09/0.35 % OS : Linux 6.8.0-71-generic % 0.09/0.35 % CPULimit : 300 % 0.09/0.35 % WCLimit : 300 % 0.09/0.35 % DateTime : Mon Sep 7 12:43:45 UTC 2026 % 0.09/0.35 % CPUTime : % 0.63/0.91 (* PROOF-FOUND *) % 0.63/0.91 % SZS status Theorem % 0.63/0.91 (* BEGIN-PROOF *) % 0.63/0.91 % SZS output start Proof % 0.63/0.91 Theorem prove_t5 : (forall E : zenon_U, (forall T : zenon_U, (((environment E)/\(subpopulations (first_movers) (efficient_producers) E T))->(~(outcompetes (first_movers) (efficient_producers) T))))). % 0.63/0.91 Proof. % 0.63/0.91 assert (zenon_L1_ : forall (zenon_TT_n : zenon_U) (zenon_TE_o : zenon_U), (forall T : zenon_U, (((environment zenon_TE_o)/\(subpopulations (first_movers) (efficient_producers) zenon_TE_o T))->(subpopulations (efficient_producers) (first_movers) zenon_TE_o T))) -> (environment zenon_TE_o) -> (subpopulations (first_movers) (efficient_producers) zenon_TE_o zenon_TT_n) -> (~(subpopulations (efficient_producers) (first_movers) zenon_TE_o zenon_TT_n)) -> False). % 0.63/0.91 do 2 intro. intros zenon_H9 zenon_Ha zenon_Hb zenon_Hc. % 0.63/0.91 generalize (zenon_H9 zenon_TT_n). zenon_intro zenon_Hf. % 0.63/0.91 apply (zenon_imply_s _ _ zenon_Hf); [ zenon_intro zenon_H11 | zenon_intro zenon_H10 ]. % 0.63/0.91 apply (zenon_notand_s _ _ zenon_H11); [ zenon_intro zenon_H13 | zenon_intro zenon_H12 ]. % 0.63/0.91 exact (zenon_H13 zenon_Ha). % 0.63/0.91 exact (zenon_H12 zenon_Hb). % 0.63/0.91 exact (zenon_Hc zenon_H10). % 0.63/0.91 (* end of lemma zenon_L1_ *) % 0.63/0.91 assert (zenon_L2_ : forall (zenon_TT_n : zenon_U) (zenon_TE_o : zenon_U), (~(subpopulations (efficient_producers) (first_movers) zenon_TE_o zenon_TT_n)) -> (subpopulations (first_movers) (efficient_producers) zenon_TE_o zenon_TT_n) -> (environment zenon_TE_o) -> False). % 0.63/0.91 do 2 intro. intros zenon_Hc zenon_Hb zenon_Ha. % 0.63/0.91 generalize (mp_symmetry_of_subpopulations zenon_TE_o). zenon_intro zenon_H14. % 0.63/0.91 generalize (zenon_H14 (first_movers)). zenon_intro zenon_H15. % 0.63/0.91 generalize (zenon_H15 (efficient_producers)). zenon_intro zenon_H9. % 0.63/0.91 apply (zenon_L1_ zenon_TT_n zenon_TE_o); trivial. % 0.63/0.91 (* end of lemma zenon_L2_ *) % 0.63/0.91 assert (zenon_L3_ : forall (zenon_TT_n : zenon_U), (((greater_or_equal (growth_rate (first_movers) zenon_TT_n) (zero))/\(greater (zero) (growth_rate (efficient_producers) zenon_TT_n)))<->(outcompetes (first_movers) (efficient_producers) zenon_TT_n)) -> (~(greater_or_equal (growth_rate (first_movers) zenon_TT_n) (zero))) -> (outcompetes (first_movers) (efficient_producers) zenon_TT_n) -> False). % 0.63/0.91 do 1 intro. intros zenon_H16 zenon_H17 zenon_H18. % 0.63/0.91 apply (zenon_equiv_s _ _ zenon_H16); [ zenon_intro zenon_H1b; zenon_intro zenon_H1a | zenon_intro zenon_H19; zenon_intro zenon_H18 ]. % 0.63/0.91 exact (zenon_H1a zenon_H18). % 0.63/0.91 apply (zenon_and_s _ _ zenon_H19). zenon_intro zenon_H1d. zenon_intro zenon_H1c. % 0.63/0.91 exact (zenon_H17 zenon_H1d). % 0.63/0.91 (* end of lemma zenon_L3_ *) % 0.63/0.91 assert (zenon_L4_ : forall (zenon_TE_o : zenon_U) (zenon_TT_n : zenon_U), (~(((environment zenon_E)/\(subpopulations (first_movers) zenon_E zenon_E zenon_TT_n))->(greater_or_equal (growth_rate (first_movers) zenon_TT_n) (zero)))) -> (outcompetes (first_movers) (efficient_producers) zenon_TT_n) -> (subpopulations (first_movers) (efficient_producers) zenon_TE_o zenon_TT_n) -> (environment zenon_TE_o) -> False). % 0.63/0.91 do 2 intro. intros zenon_H1e zenon_H18 zenon_Hb zenon_Ha. % 0.63/0.91 apply (zenon_notimply_s _ _ zenon_H1e). zenon_intro zenon_H1f. zenon_intro zenon_H17. % 0.63/0.91 generalize (d2 zenon_TE_o). zenon_intro zenon_H20. % 0.63/0.91 generalize (zenon_H20 (efficient_producers)). zenon_intro zenon_H21. % 0.63/0.91 generalize (zenon_H21 (first_movers)). zenon_intro zenon_H22. % 0.63/0.91 generalize (zenon_H22 zenon_TT_n). zenon_intro zenon_H23. % 0.63/0.91 apply (zenon_imply_s _ _ zenon_H23); [ zenon_intro zenon_H24 | zenon_intro zenon_H16 ]. % 0.63/0.91 apply (zenon_notand_s _ _ zenon_H24); [ zenon_intro zenon_H13 | zenon_intro zenon_Hc ]. % 0.63/0.91 exact (zenon_H13 zenon_Ha). % 0.63/0.91 apply (zenon_L2_ zenon_TT_n zenon_TE_o); trivial. % 0.63/0.91 apply (zenon_L3_ zenon_TT_n); trivial. % 0.63/0.91 (* end of lemma zenon_L4_ *) % 0.63/0.91 assert (zenon_L5_ : forall (zenon_TT_n : zenon_U) (zenon_TE_o : zenon_U), (forall T : zenon_U, ((((environment zenon_E)/\(subpopulations (first_movers) zenon_E zenon_E T))->(greater_or_equal (growth_rate (first_movers) T) (zero)))<->(~(greater (zero) (growth_rate (first_movers) T))))) -> (environment zenon_TE_o) -> (subpopulations (first_movers) (efficient_producers) zenon_TE_o zenon_TT_n) -> (outcompetes (first_movers) (efficient_producers) zenon_TT_n) -> (forall S1 : zenon_U, (forall S2 : zenon_U, (forall T : zenon_U, (((environment zenon_TE_o)/\((in_environment zenon_TE_o T)/\((~(greater (zero) (growth_rate S1 T)))/\(greater (resilience S2) (resilience S1)))))->(~(greater (zero) (growth_rate S2 T))))))) -> (in_environment zenon_TE_o zenon_TT_n) -> (greater (zero) (growth_rate (efficient_producers) zenon_TT_n)) -> False). % 0.63/0.91 do 2 intro. intros zenon_H25 zenon_Ha zenon_Hb zenon_H18 zenon_H26 zenon_H27 zenon_H1c. % 0.63/0.91 generalize (zenon_H25 zenon_TT_n). zenon_intro zenon_H28. % 0.63/0.91 apply (zenon_equiv_s _ _ zenon_H28); [ zenon_intro zenon_H1e; zenon_intro zenon_H2b | zenon_intro zenon_H2a; zenon_intro zenon_H29 ]. % 0.63/0.91 apply (zenon_L4_ zenon_TE_o zenon_TT_n); trivial. % 0.63/0.91 generalize (zenon_H26 (first_movers)). zenon_intro zenon_H2c. % 0.63/0.91 generalize (zenon_H2c (efficient_producers)). zenon_intro zenon_H2d. % 0.63/0.91 generalize (zenon_H2d zenon_TT_n). zenon_intro zenon_H2e. % 0.63/0.91 apply (zenon_imply_s _ _ zenon_H2e); [ zenon_intro zenon_H30 | zenon_intro zenon_H2f ]. % 0.63/0.91 apply (zenon_notand_s _ _ zenon_H30); [ zenon_intro zenon_H13 | zenon_intro zenon_H31 ]. % 0.63/0.91 exact (zenon_H13 zenon_Ha). % 0.63/0.91 apply (zenon_notand_s _ _ zenon_H31); [ zenon_intro zenon_H33 | zenon_intro zenon_H32 ]. % 0.63/0.91 exact (zenon_H33 zenon_H27). % 0.63/0.91 apply (zenon_notand_s _ _ zenon_H32); [ zenon_intro zenon_H2b | zenon_intro zenon_H34 ]. % 0.63/0.91 exact (zenon_H2b zenon_H29). % 0.63/0.91 exact (zenon_H34 a2). % 0.63/0.91 exact (zenon_H2f zenon_H1c). % 0.63/0.91 (* end of lemma zenon_L5_ *) % 0.63/0.91 assert (zenon_L6_ : forall (zenon_TT_n : zenon_U), (((greater_or_equal (growth_rate (first_movers) zenon_TT_n) (zero))/\(greater (zero) (growth_rate (efficient_producers) zenon_TT_n)))<->(outcompetes (first_movers) (efficient_producers) zenon_TT_n)) -> (~(greater (zero) (growth_rate (efficient_producers) zenon_TT_n))) -> (outcompetes (first_movers) (efficient_producers) zenon_TT_n) -> False). % 0.63/0.91 do 1 intro. intros zenon_H16 zenon_H2f zenon_H18. % 0.63/0.91 apply (zenon_equiv_s _ _ zenon_H16); [ zenon_intro zenon_H1b; zenon_intro zenon_H1a | zenon_intro zenon_H19; zenon_intro zenon_H18 ]. % 0.63/0.91 exact (zenon_H1a zenon_H18). % 0.63/0.91 apply (zenon_and_s _ _ zenon_H19). zenon_intro zenon_H1d. zenon_intro zenon_H1c. % 0.63/0.91 exact (zenon_H2f zenon_H1c). % 0.63/0.91 (* end of lemma zenon_L6_ *) % 0.63/0.91 assert (zenon_L7_ : forall (zenon_TT_n : zenon_U) (zenon_TE_o : zenon_U), (forall S1 : zenon_U, (forall S2 : zenon_U, (forall T : zenon_U, (((environment zenon_TE_o)/\(subpopulations S1 S2 zenon_TE_o T))->(((greater_or_equal (growth_rate S2 T) (zero))/\(greater (zero) (growth_rate S1 T)))<->(outcompetes S2 S1 T)))))) -> (environment zenon_TE_o) -> (subpopulations (first_movers) (efficient_producers) zenon_TE_o zenon_TT_n) -> (~(greater (zero) (growth_rate (efficient_producers) zenon_TT_n))) -> (outcompetes (first_movers) (efficient_producers) zenon_TT_n) -> False). % 0.63/0.91 do 2 intro. intros zenon_H20 zenon_Ha zenon_Hb zenon_H2f zenon_H18. % 0.63/0.91 generalize (zenon_H20 (efficient_producers)). zenon_intro zenon_H21. % 0.63/0.91 generalize (zenon_H21 (first_movers)). zenon_intro zenon_H22. % 0.63/0.91 generalize (zenon_H22 zenon_TT_n). zenon_intro zenon_H23. % 0.63/0.91 apply (zenon_imply_s _ _ zenon_H23); [ zenon_intro zenon_H24 | zenon_intro zenon_H16 ]. % 0.63/0.91 apply (zenon_notand_s _ _ zenon_H24); [ zenon_intro zenon_H13 | zenon_intro zenon_Hc ]. % 0.63/0.91 exact (zenon_H13 zenon_Ha). % 0.63/0.91 apply (zenon_L2_ zenon_TT_n zenon_TE_o); trivial. % 0.63/0.91 apply (zenon_L6_ zenon_TT_n); trivial. % 0.63/0.91 (* end of lemma zenon_L7_ *) % 0.63/0.91 assert (zenon_L8_ : forall (zenon_TE_o : zenon_U) (zenon_TT_n : zenon_U), (outcompetes (first_movers) (efficient_producers) zenon_TT_n) -> (~(greater (zero) (growth_rate (efficient_producers) zenon_TT_n))) -> (subpopulations (first_movers) (efficient_producers) zenon_TE_o zenon_TT_n) -> (environment zenon_TE_o) -> False). % 0.63/0.91 do 2 intro. intros zenon_H18 zenon_H2f zenon_Hb zenon_Ha. % 0.63/0.91 generalize (d2 zenon_TE_o). zenon_intro zenon_H20. % 0.63/0.91 apply (zenon_L7_ zenon_TT_n zenon_TE_o); trivial. % 0.63/0.91 (* end of lemma zenon_L8_ *) % 0.63/0.91 apply NNPP. intro zenon_G. % 0.63/0.91 apply (zenon_notallex_s (fun E : zenon_U => (forall T : zenon_U, (((environment E)/\(subpopulations (first_movers) (efficient_producers) E T))->(~(outcompetes (first_movers) (efficient_producers) T))))) zenon_G); [ zenon_intro zenon_H35; idtac ]. % 0.63/0.91 elim zenon_H35. zenon_intro zenon_TE_o. zenon_intro zenon_H36. % 0.63/0.92 apply (zenon_notallex_s (fun T : zenon_U => (((environment zenon_TE_o)/\(subpopulations (first_movers) (efficient_producers) zenon_TE_o T))->(~(outcompetes (first_movers) (efficient_producers) T)))) zenon_H36); [ zenon_intro zenon_H37; idtac ]. % 0.63/0.92 elim zenon_H37. zenon_intro zenon_TT_n. zenon_intro zenon_H38. % 0.63/0.92 apply (zenon_notimply_s _ _ zenon_H38). zenon_intro zenon_H3a. zenon_intro zenon_H39. % 0.63/0.92 apply zenon_H39. zenon_intro zenon_H18. % 0.63/0.92 apply (zenon_and_s _ _ zenon_H3a). zenon_intro zenon_Ha. zenon_intro zenon_Hb. % 0.63/0.92 generalize (mp_growth_rate_relationships zenon_E). zenon_intro zenon_H3b. % 0.63/0.92 generalize (mp_time_point_occur zenon_TE_o). zenon_intro zenon_H3c. % 0.63/0.92 generalize (zenon_H3c zenon_TT_n). zenon_intro zenon_H3d. % 0.63/0.92 apply (zenon_imply_s _ _ zenon_H3d); [ zenon_intro zenon_H11 | zenon_intro zenon_H27 ]. % 0.63/0.92 apply (zenon_notand_s _ _ zenon_H11); [ zenon_intro zenon_H13 | zenon_intro zenon_H12 ]. % 0.63/0.92 exact (zenon_H13 zenon_Ha). % 0.63/0.92 exact (zenon_H12 zenon_Hb). % 0.63/0.92 generalize (a13 zenon_TE_o). zenon_intro zenon_H26. % 0.63/0.92 generalize (zenon_H3b (efficient_producers)). zenon_intro zenon_H3e. % 0.63/0.92 generalize (zenon_H3e zenon_E). zenon_intro zenon_H3f. % 0.63/0.92 generalize (zenon_H3b (first_movers)). zenon_intro zenon_H40. % 0.63/0.92 generalize (zenon_H40 zenon_E). zenon_intro zenon_H25. % 0.63/0.92 generalize (zenon_H3f zenon_TT_n). zenon_intro zenon_H41. % 0.63/0.92 apply (zenon_equiv_s _ _ zenon_H41); [ zenon_intro zenon_H44; zenon_intro zenon_H43 | zenon_intro zenon_H42; zenon_intro zenon_H2f ]. % 0.63/0.92 apply zenon_H43. zenon_intro zenon_H1c. % 0.63/0.92 apply (zenon_L5_ zenon_TT_n zenon_TE_o); trivial. % 0.63/0.92 apply (zenon_L8_ zenon_TE_o zenon_TT_n); trivial. % 0.63/0.92 Qed. % 0.63/0.92 % SZS output end Proof % 0.63/0.92 (* END-PROOF *) % 0.63/0.92 nodes searched: 20795 % 0.63/0.92 max branch formulas: 2797 % 0.63/0.92 proof nodes created: 672 % 0.63/0.92 formulas created: 66704 % 0.63/0.92 %------------------------------------------------------------------------------