%------------------------------------------------------------------------------ % File : Zenon---0.7.1 % Problem : MGT026+1 : TPTP v9.3.0. Released v2.0.0. % Transfm : none % Format : tptp:raw % Command : run_zenon %s %d % Computer : n003.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 06:24:19 AM UTC 2026 % Result : Theorem 0.21s 0.47s % Output : Proof 0.21s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.00/0.02 % Problem : MGT026+1 : TPTP v9.3.0. Released v2.0.0. % 0.00/0.03 % Command : run_zenon %s %d % 0.06/0.34 % Computer : n003.cluster.edu % 0.06/0.34 % Model : x86_64 x86_64 % 0.06/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.06/0.34 % Memory : 8046.5625MB % 0.06/0.34 % OS : Linux 6.8.0-71-generic % 0.06/0.34 % CPULimit : 300 % 0.06/0.34 % WCLimit : 300 % 0.06/0.34 % DateTime : Mon Sep 7 12:12:53 UTC 2026 % 0.06/0.35 % CPUTime : % 0.21/0.47 (* PROOF-FOUND *) % 0.21/0.47 % SZS status Theorem % 0.21/0.47 (* BEGIN-PROOF *) % 0.21/0.47 % SZS output start Proof % 0.21/0.47 Theorem prove_l8 : (forall E : zenon_U, (forall T : zenon_U, (((environment E)/\((in_environment E T)/\(greater T (critical_point E))))->(selection_favors (efficient_producers) (first_movers) T)))). % 0.21/0.47 Proof. % 0.21/0.47 assert (zenon_L1_ : forall (zenon_TE_q : zenon_U) (zenon_TT_r : zenon_U), (forall x : zenon_U, (forall y : zenon_U, (forall z : zenon_U, ((greater x y)->((greater y z)->(greater x z)))))) -> (~(greater (cardinality_at_time (efficient_producers) zenon_TT_r) (zero))) -> (greater zenon_TT_r (critical_point zenon_TE_q)) -> (in_environment zenon_TE_q zenon_TT_r) -> (environment zenon_TE_q) -> False). % 0.21/0.47 do 2 intro. intros zenon_Hb zenon_Hc zenon_Hd zenon_He zenon_Hf. % 0.21/0.47 generalize (t6 zenon_TE_q). zenon_intro zenon_H12. % 0.21/0.47 generalize (zenon_H12 zenon_TT_r). zenon_intro zenon_H13. % 0.21/0.47 apply (zenon_imply_s _ _ zenon_H13); [ zenon_intro zenon_H15 | zenon_intro zenon_H14 ]. % 0.21/0.47 apply (zenon_notand_s _ _ zenon_H15); [ zenon_intro zenon_H17 | zenon_intro zenon_H16 ]. % 0.21/0.47 exact (zenon_H17 zenon_Hf). % 0.21/0.47 apply (zenon_notand_s _ _ zenon_H16); [ zenon_intro zenon_H19 | zenon_intro zenon_H18 ]. % 0.21/0.47 exact (zenon_H19 zenon_He). % 0.21/0.47 generalize (mp_greater_or_equal zenon_TT_r). zenon_intro zenon_H1a. % 0.21/0.47 generalize (zenon_H1a (appear (efficient_producers) zenon_TE_q)). zenon_intro zenon_H1b. % 0.21/0.47 apply (zenon_equiv_s _ _ zenon_H1b); [ zenon_intro zenon_H18; zenon_intro zenon_H1e | zenon_intro zenon_H1d; zenon_intro zenon_H1c ]. % 0.21/0.47 apply (zenon_notor_s _ _ zenon_H1e). zenon_intro zenon_H20. zenon_intro zenon_H1f. % 0.21/0.47 elim (classic ((critical_point zenon_TE_q) = (appear (efficient_producers) zenon_TE_q))); [ zenon_intro zenon_H21 | zenon_intro zenon_H22 ]. % 0.21/0.47 cut ((greater zenon_TT_r (critical_point zenon_TE_q)) = (greater zenon_TT_r (appear (efficient_producers) zenon_TE_q))). % 0.21/0.47 intro zenon_D_pnotp. % 0.21/0.47 apply zenon_H20. % 0.21/0.47 rewrite <- zenon_D_pnotp. % 0.21/0.47 exact zenon_Hd. % 0.21/0.47 cut (((critical_point zenon_TE_q) = (appear (efficient_producers) zenon_TE_q))); [idtac | apply NNPP; zenon_intro zenon_H22]. % 0.21/0.47 cut ((zenon_TT_r = zenon_TT_r)); [idtac | apply NNPP; zenon_intro zenon_H23]. % 0.21/0.47 congruence. % 0.21/0.47 apply zenon_H23. apply refl_equal. % 0.21/0.47 exact (zenon_H22 zenon_H21). % 0.21/0.47 elim (classic (greater (critical_point zenon_TE_q) (appear (efficient_producers) zenon_TE_q))); [ zenon_intro zenon_H24 | zenon_intro zenon_H25 ]. % 0.21/0.47 generalize (zenon_Hb zenon_TT_r). zenon_intro zenon_H26. % 0.21/0.47 generalize (zenon_H26 (critical_point zenon_TE_q)). zenon_intro zenon_H27. % 0.21/0.47 generalize (zenon_H27 (appear (efficient_producers) zenon_TE_q)). zenon_intro zenon_H28. % 0.21/0.47 apply (zenon_imply_s _ _ zenon_H28); [ zenon_intro zenon_H2a | zenon_intro zenon_H29 ]. % 0.21/0.47 exact (zenon_H2a zenon_Hd). % 0.21/0.47 apply (zenon_imply_s _ _ zenon_H29); [ zenon_intro zenon_H25 | zenon_intro zenon_H2b ]. % 0.21/0.47 exact (zenon_H25 zenon_H24). % 0.21/0.47 exact (zenon_H20 zenon_H2b). % 0.21/0.47 generalize (mp_critical_point_after_EP zenon_TE_q). zenon_intro zenon_H2c. % 0.21/0.47 apply (zenon_imply_s _ _ zenon_H2c); [ zenon_intro zenon_H17 | zenon_intro zenon_H2d ]. % 0.21/0.47 exact (zenon_H17 zenon_Hf). % 0.21/0.47 generalize (mp_greater_or_equal (critical_point zenon_TE_q)). zenon_intro zenon_H2e. % 0.21/0.47 generalize (zenon_H2e (appear (efficient_producers) zenon_TE_q)). zenon_intro zenon_H2f. % 0.21/0.47 apply (zenon_equiv_s _ _ zenon_H2f); [ zenon_intro zenon_H32; zenon_intro zenon_H31 | zenon_intro zenon_H2d; zenon_intro zenon_H30 ]. % 0.21/0.47 exact (zenon_H32 zenon_H2d). % 0.21/0.47 apply (zenon_or_s _ _ zenon_H30); [ zenon_intro zenon_H24 | zenon_intro zenon_H21 ]. % 0.21/0.47 exact (zenon_H25 zenon_H24). % 0.21/0.47 exact (zenon_H22 zenon_H21). % 0.21/0.47 exact (zenon_H18 zenon_H1d). % 0.21/0.47 exact (zenon_Hc zenon_H14). % 0.21/0.47 (* end of lemma zenon_L1_ *) % 0.21/0.47 assert (zenon_L2_ : forall (zenon_TT_r : zenon_U) (zenon_TE_q : zenon_U), (forall x : zenon_U, (forall y : zenon_U, (forall z : zenon_U, ((greater x y)->((greater y z)->(greater x z)))))) -> (forall T : zenon_U, (((environment zenon_TE_q)/\((in_environment zenon_TE_q T)/\((greater (cardinality_at_time (first_movers) T) (zero))/\(greater (cardinality_at_time (efficient_producers) T) (zero)))))->(subpopulations (first_movers) (efficient_producers) zenon_TE_q T))) -> (environment zenon_TE_q) -> (in_environment zenon_TE_q zenon_TT_r) -> (greater (cardinality_at_time (first_movers) zenon_TT_r) (zero)) -> (greater zenon_TT_r (critical_point zenon_TE_q)) -> (~(subpopulations (first_movers) (efficient_producers) zenon_TE_q zenon_TT_r)) -> False). % 0.21/0.47 do 2 intro. intros zenon_Hb zenon_H33 zenon_Hf zenon_He zenon_H34 zenon_Hd zenon_H35. % 0.21/0.47 generalize (zenon_H33 zenon_TT_r). zenon_intro zenon_H36. % 0.21/0.47 apply (zenon_imply_s _ _ zenon_H36); [ zenon_intro zenon_H38 | zenon_intro zenon_H37 ]. % 0.21/0.47 apply (zenon_notand_s _ _ zenon_H38); [ zenon_intro zenon_H17 | zenon_intro zenon_H39 ]. % 0.21/0.47 exact (zenon_H17 zenon_Hf). % 0.21/0.47 apply (zenon_notand_s _ _ zenon_H39); [ zenon_intro zenon_H19 | zenon_intro zenon_H3a ]. % 0.21/0.47 exact (zenon_H19 zenon_He). % 0.21/0.47 apply (zenon_notand_s _ _ zenon_H3a); [ zenon_intro zenon_H3b | zenon_intro zenon_Hc ]. % 0.21/0.47 exact (zenon_H3b zenon_H34). % 0.21/0.47 apply (zenon_L1_ zenon_TE_q zenon_TT_r); trivial. % 0.21/0.47 exact (zenon_H35 zenon_H37). % 0.21/0.47 (* end of lemma zenon_L2_ *) % 0.21/0.47 assert (zenon_L3_ : forall (zenon_TE_q : zenon_U) (zenon_TT_r : zenon_U), (forall x : zenon_U, (forall y : zenon_U, (forall z : zenon_U, ((greater x y)->((greater y z)->(greater x z)))))) -> (greater_or_equal (cardinality_at_time (first_movers) zenon_TT_r) (zero)) -> (~((cardinality_at_time (first_movers) zenon_TT_r) = (zero))) -> (~(selection_favors (efficient_producers) (first_movers) zenon_TT_r)) -> (in_environment zenon_TE_q zenon_TT_r) -> (greater zenon_TT_r (critical_point zenon_TE_q)) -> (environment zenon_TE_q) -> (forall S1 : zenon_U, (forall S2 : zenon_U, (forall T : zenon_U, (((environment zenon_TE_q)/\((subpopulations S1 S2 zenon_TE_q T)/\(greater (growth_rate S2 T) (growth_rate S1 T))))->(selection_favors S2 S1 T))))) -> False). % 0.21/0.47 do 2 intro. intros zenon_Hb zenon_H3c zenon_H3d zenon_H3e zenon_He zenon_Hd zenon_Hf zenon_H3f. % 0.21/0.47 generalize (mp_greater_or_equal (cardinality_at_time (first_movers) zenon_TT_r)). zenon_intro zenon_H40. % 0.21/0.47 generalize (zenon_H40 (zero)). zenon_intro zenon_H41. % 0.21/0.47 apply (zenon_equiv_s _ _ zenon_H41); [ zenon_intro zenon_H44; zenon_intro zenon_H43 | zenon_intro zenon_H3c; zenon_intro zenon_H42 ]. % 0.21/0.47 exact (zenon_H44 zenon_H3c). % 0.21/0.47 apply (zenon_or_s _ _ zenon_H42); [ zenon_intro zenon_H34 | zenon_intro zenon_H45 ]. % 0.21/0.47 generalize (zenon_H3f (first_movers)). zenon_intro zenon_H46. % 0.21/0.47 generalize (zenon_H46 (efficient_producers)). zenon_intro zenon_H47. % 0.21/0.47 generalize (zenon_H47 zenon_TT_r). zenon_intro zenon_H48. % 0.21/0.47 apply (zenon_imply_s _ _ zenon_H48); [ zenon_intro zenon_H4a | zenon_intro zenon_H49 ]. % 0.21/0.47 apply (zenon_notand_s _ _ zenon_H4a); [ zenon_intro zenon_H17 | zenon_intro zenon_H4b ]. % 0.21/0.47 exact (zenon_H17 zenon_Hf). % 0.21/0.47 apply (zenon_notand_s _ _ zenon_H4b); [ zenon_intro zenon_H35 | zenon_intro zenon_H4c ]. % 0.21/0.47 generalize (mp_non_empty_fm_and_ep zenon_TE_q). zenon_intro zenon_H33. % 0.21/0.47 apply (zenon_L2_ zenon_TT_r zenon_TE_q); trivial. % 0.21/0.47 generalize (mp_non_empty_fm_and_ep zenon_TE_q). zenon_intro zenon_H33. % 0.21/0.47 generalize (d1 zenon_TE_q). zenon_intro zenon_H4d. % 0.21/0.47 generalize (zenon_H4d (critical_point zenon_TE_q)). zenon_intro zenon_H4e. % 0.21/0.47 apply (zenon_imply_s _ _ zenon_H4e); [ zenon_intro zenon_H50 | zenon_intro zenon_H4f ]. % 0.21/0.47 apply (zenon_notand_s _ _ zenon_H50); [ zenon_intro zenon_H17 | zenon_intro zenon_H51 ]. % 0.21/0.47 exact (zenon_H17 zenon_Hf). % 0.21/0.47 apply zenon_H51. apply refl_equal. % 0.21/0.47 apply (zenon_and_s _ _ zenon_H4f). zenon_intro zenon_H53. zenon_intro zenon_H52. % 0.21/0.47 generalize (zenon_H52 zenon_TT_r). zenon_intro zenon_H54. % 0.21/0.47 apply (zenon_imply_s _ _ zenon_H54); [ zenon_intro zenon_H56 | zenon_intro zenon_H55 ]. % 0.21/0.47 apply (zenon_notand_s _ _ zenon_H56); [ zenon_intro zenon_H35 | zenon_intro zenon_H2a ]. % 0.21/0.47 apply (zenon_L2_ zenon_TT_r zenon_TE_q); trivial. % 0.21/0.47 exact (zenon_H2a zenon_Hd). % 0.21/0.47 exact (zenon_H4c zenon_H55). % 0.21/0.47 exact (zenon_H3e zenon_H49). % 0.21/0.47 exact (zenon_H3d zenon_H45). % 0.21/0.47 (* end of lemma zenon_L3_ *) % 0.21/0.47 apply NNPP. intro zenon_G. % 0.21/0.47 elim (classic (forall x : zenon_U, (forall y : zenon_U, (forall z : zenon_U, ((greater x y)->((greater y z)->(greater x z))))))); [ zenon_intro zenon_Hb | zenon_intro zenon_H57 ]. % 0.21/0.47 apply (zenon_notallex_s (fun E : zenon_U => (forall T : zenon_U, (((environment E)/\((in_environment E T)/\(greater T (critical_point E))))->(selection_favors (efficient_producers) (first_movers) T)))) zenon_G); [ zenon_intro zenon_H58; idtac ]. % 0.21/0.47 elim zenon_H58. zenon_intro zenon_TE_q. zenon_intro zenon_H59. % 0.21/0.47 apply (zenon_notallex_s (fun T : zenon_U => (((environment zenon_TE_q)/\((in_environment zenon_TE_q T)/\(greater T (critical_point zenon_TE_q))))->(selection_favors (efficient_producers) (first_movers) T))) zenon_H59); [ zenon_intro zenon_H5a; idtac ]. % 0.21/0.47 elim zenon_H5a. zenon_intro zenon_TT_r. zenon_intro zenon_H5b. % 0.21/0.47 apply (zenon_notimply_s _ _ zenon_H5b). zenon_intro zenon_H5c. zenon_intro zenon_H3e. % 0.21/0.47 apply (zenon_and_s _ _ zenon_H5c). zenon_intro zenon_Hf. zenon_intro zenon_H5d. % 0.21/0.47 apply (zenon_and_s _ _ zenon_H5d). zenon_intro zenon_He. zenon_intro zenon_Hd. % 0.21/0.47 generalize (mp2_favour_members zenon_TE_q). zenon_intro zenon_H5e. % 0.21/0.47 generalize (zenon_H5e (efficient_producers)). zenon_intro zenon_H5f. % 0.21/0.47 generalize (mp1_high_growth_rates zenon_TE_q). zenon_intro zenon_H3f. % 0.21/0.47 generalize (zenon_H5f (first_movers)). zenon_intro zenon_H60. % 0.21/0.47 generalize (zenon_H60 zenon_TT_r). zenon_intro zenon_H61. % 0.21/0.47 apply (zenon_imply_s _ _ zenon_H61); [ zenon_intro zenon_H62 | zenon_intro zenon_H49 ]. % 0.21/0.47 apply (zenon_notand_s _ _ zenon_H62); [ zenon_intro zenon_H17 | zenon_intro zenon_H63 ]. % 0.21/0.47 exact (zenon_H17 zenon_Hf). % 0.21/0.47 apply (zenon_notand_s _ _ zenon_H63); [ zenon_intro zenon_H65 | zenon_intro zenon_H64 ]. % 0.21/0.47 generalize (mp_subpopulations zenon_TE_q). zenon_intro zenon_H66. % 0.21/0.47 generalize (zenon_H66 zenon_TT_r). zenon_intro zenon_H67. % 0.21/0.47 apply (zenon_imply_s _ _ zenon_H67); [ zenon_intro zenon_H69 | zenon_intro zenon_H68 ]. % 0.21/0.47 apply (zenon_notand_s _ _ zenon_H69); [ zenon_intro zenon_H17 | zenon_intro zenon_H19 ]. % 0.21/0.47 exact (zenon_H17 zenon_Hf). % 0.21/0.47 exact (zenon_H19 zenon_He). % 0.21/0.47 apply (zenon_and_s _ _ zenon_H68). zenon_intro zenon_H6b. zenon_intro zenon_H6a. % 0.21/0.47 exact (zenon_H65 zenon_H6a). % 0.21/0.47 apply (zenon_notand_s _ _ zenon_H64); [ zenon_intro zenon_H6d | zenon_intro zenon_H6c ]. % 0.21/0.47 generalize (mp_subpopulations zenon_TE_q). zenon_intro zenon_H66. % 0.21/0.47 generalize (zenon_H66 zenon_TT_r). zenon_intro zenon_H67. % 0.21/0.47 apply (zenon_imply_s _ _ zenon_H67); [ zenon_intro zenon_H69 | zenon_intro zenon_H68 ]. % 0.21/0.47 apply (zenon_notand_s _ _ zenon_H69); [ zenon_intro zenon_H17 | zenon_intro zenon_H19 ]. % 0.21/0.47 exact (zenon_H17 zenon_Hf). % 0.21/0.47 exact (zenon_H19 zenon_He). % 0.21/0.47 apply (zenon_and_s _ _ zenon_H68). zenon_intro zenon_H6b. zenon_intro zenon_H6a. % 0.21/0.47 exact (zenon_H6d zenon_H6b). % 0.21/0.47 apply (zenon_notand_s _ _ zenon_H6c); [ zenon_intro zenon_Hc | zenon_intro zenon_H3d ]. % 0.21/0.47 apply (zenon_L1_ zenon_TE_q zenon_TT_r); trivial. % 0.21/0.47 generalize (mp_first_movers_exist zenon_TE_q). zenon_intro zenon_H6e. % 0.21/0.47 generalize (zenon_H6e zenon_TT_r). zenon_intro zenon_H6f. % 0.21/0.47 apply (zenon_imply_s _ _ zenon_H6f); [ zenon_intro zenon_H69 | zenon_intro zenon_H3c ]. % 0.21/0.47 apply (zenon_notand_s _ _ zenon_H69); [ zenon_intro zenon_H17 | zenon_intro zenon_H19 ]. % 0.21/0.47 exact (zenon_H17 zenon_Hf). % 0.21/0.47 exact (zenon_H19 zenon_He). % 0.21/0.47 apply (zenon_L3_ zenon_TE_q zenon_TT_r); trivial. % 0.21/0.47 exact (zenon_H3e zenon_H49). % 0.21/0.47 apply zenon_H57. zenon_intro zenon_Tx_ei. apply NNPP. zenon_intro zenon_H71. % 0.21/0.47 apply zenon_H71. zenon_intro zenon_Ty_ek. apply NNPP. zenon_intro zenon_H73. % 0.21/0.47 apply zenon_H73. zenon_intro zenon_Tz_em. apply NNPP. zenon_intro zenon_H75. % 0.21/0.47 apply (zenon_notimply_s _ _ zenon_H75). zenon_intro zenon_H77. zenon_intro zenon_H76. % 0.21/0.47 apply (zenon_notimply_s _ _ zenon_H76). zenon_intro zenon_H79. zenon_intro zenon_H78. % 0.21/0.47 generalize (mp_greater_transitivity zenon_Tx_ei). zenon_intro zenon_H7a. % 0.21/0.47 generalize (zenon_H7a zenon_Ty_ek). zenon_intro zenon_H7b. % 0.21/0.47 generalize (zenon_H7b zenon_Tz_em). zenon_intro zenon_H7c. % 0.21/0.47 apply (zenon_imply_s _ _ zenon_H7c); [ zenon_intro zenon_H7e | zenon_intro zenon_H7d ]. % 0.21/0.47 apply (zenon_notand_s _ _ zenon_H7e); [ zenon_intro zenon_H80 | zenon_intro zenon_H7f ]. % 0.21/0.47 exact (zenon_H80 zenon_H77). % 0.21/0.47 exact (zenon_H7f zenon_H79). % 0.21/0.47 exact (zenon_H78 zenon_H7d). % 0.21/0.47 Qed. % 0.21/0.47 % SZS output end Proof % 0.21/0.47 (* END-PROOF *) % 0.21/0.47 nodes searched: 1001 % 0.21/0.47 max branch formulas: 286 % 0.21/0.47 proof nodes created: 127 % 0.21/0.47 formulas created: 3671 % 0.21/0.47 %------------------------------------------------------------------------------