TSTP Solution File: MGT026+1 by SuperZenon---0.0.1
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- Process Solution
%------------------------------------------------------------------------------
% File : SuperZenon---0.0.1
% Problem : MGT026+1 : TPTP v8.1.0. Released v2.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_super_zenon -p0 -itptp -om -max-time %d %s
% Computer : n028.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 : Sun Jul 17 22:26:57 EDT 2022
% Result : Theorem 0.19s 0.57s
% Output : Proof 0.42s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : MGT026+1 : TPTP v8.1.0. Released v2.0.0.
% 0.07/0.13 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s
% 0.12/0.34 % Computer : n028.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 600
% 0.12/0.34 % DateTime : Thu Jun 9 12:49:17 EDT 2022
% 0.12/0.34 % CPUTime :
% 0.19/0.57 % SZS status Theorem
% 0.19/0.57 (* PROOF-FOUND *)
% 0.19/0.57 (* BEGIN-PROOF *)
% 0.19/0.57 % SZS output start Proof
% 0.19/0.57 1. (environment T_0) (-. (environment T_0)) ### Axiom
% 0.19/0.57 2. (environment T_0) (-. (environment T_0)) ### Axiom
% 0.19/0.57 3. (in_environment T_0 T_1) (-. (in_environment T_0 T_1)) ### Axiom
% 0.19/0.57 4. (-. (subpopulation (efficient_producers) T_0 T_1)) (subpopulation (efficient_producers) T_0 T_1) ### Axiom
% 0.19/0.57 5. ((subpopulation (first_movers) T_0 T_1) /\ (subpopulation (efficient_producers) T_0 T_1)) (-. (subpopulation (efficient_producers) T_0 T_1)) ### And 4
% 0.19/0.57 6. (((environment T_0) /\ (in_environment T_0 T_1)) => ((subpopulation (first_movers) T_0 T_1) /\ (subpopulation (efficient_producers) T_0 T_1))) (-. (subpopulation (efficient_producers) T_0 T_1)) (in_environment T_0 T_1) (environment T_0) ### DisjTree 2 3 5
% 0.19/0.57 7. (All T, (((environment T_0) /\ (in_environment T_0 T)) => ((subpopulation (first_movers) T_0 T) /\ (subpopulation (efficient_producers) T_0 T)))) (environment T_0) (in_environment T_0 T_1) (-. (subpopulation (efficient_producers) T_0 T_1)) ### All 6
% 0.19/0.57 8. (All E, (All T, (((environment E) /\ (in_environment E T)) => ((subpopulation (first_movers) E T) /\ (subpopulation (efficient_producers) E T))))) (-. (subpopulation (efficient_producers) T_0 T_1)) (in_environment T_0 T_1) (environment T_0) ### All 7
% 0.19/0.57 9. (environment T_0) (-. (environment T_0)) ### Axiom
% 0.19/0.57 10. (in_environment T_0 T_1) (-. (in_environment T_0 T_1)) ### Axiom
% 0.19/0.57 11. (-. (subpopulation (first_movers) T_0 T_1)) (subpopulation (first_movers) T_0 T_1) ### Axiom
% 0.19/0.57 12. ((subpopulation (first_movers) T_0 T_1) /\ (subpopulation (efficient_producers) T_0 T_1)) (-. (subpopulation (first_movers) T_0 T_1)) ### And 11
% 0.19/0.57 13. (((environment T_0) /\ (in_environment T_0 T_1)) => ((subpopulation (first_movers) T_0 T_1) /\ (subpopulation (efficient_producers) T_0 T_1))) (-. (subpopulation (first_movers) T_0 T_1)) (in_environment T_0 T_1) (environment T_0) ### DisjTree 9 10 12
% 0.19/0.57 14. (All T, (((environment T_0) /\ (in_environment T_0 T)) => ((subpopulation (first_movers) T_0 T) /\ (subpopulation (efficient_producers) T_0 T)))) (environment T_0) (in_environment T_0 T_1) (-. (subpopulation (first_movers) T_0 T_1)) ### All 13
% 0.19/0.57 15. (All E, (All T, (((environment E) /\ (in_environment E T)) => ((subpopulation (first_movers) E T) /\ (subpopulation (efficient_producers) E T))))) (-. (subpopulation (first_movers) T_0 T_1)) (in_environment T_0 T_1) (environment T_0) ### All 14
% 0.19/0.57 16. (environment T_0) (-. (environment T_0)) ### Axiom
% 0.19/0.57 17. (in_environment T_0 T_1) (-. (in_environment T_0 T_1)) ### Axiom
% 0.19/0.57 18. (greater T_1 (critical_point T_0)) (-. (greater T_1 (critical_point T_0))) ### Axiom
% 0.19/0.57 19. (greater (critical_point T_0) (appear (efficient_producers) T_0)) (-. (greater (critical_point T_0) (appear (efficient_producers) T_0))) ### Axiom
% 0.19/0.57 20. (-. (greater T_1 (appear (efficient_producers) T_0))) (greater T_1 (appear (efficient_producers) T_0)) ### Axiom
% 0.19/0.57 21. (((greater T_1 (critical_point T_0)) /\ (greater (critical_point T_0) (appear (efficient_producers) T_0))) => (greater T_1 (appear (efficient_producers) T_0))) (-. (greater T_1 (appear (efficient_producers) T_0))) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) ### DisjTree 18 19 20
% 0.19/0.57 22. (All Z, (((greater T_1 (critical_point T_0)) /\ (greater (critical_point T_0) Z)) => (greater T_1 Z))) (greater T_1 (critical_point T_0)) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (-. (greater T_1 (appear (efficient_producers) T_0))) ### All 21
% 0.19/0.57 23. (All Y, (All Z, (((greater T_1 Y) /\ (greater Y Z)) => (greater T_1 Z)))) (-. (greater T_1 (appear (efficient_producers) T_0))) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) ### All 22
% 0.19/0.57 24. (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (greater T_1 (critical_point T_0)) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (-. (greater T_1 (appear (efficient_producers) T_0))) ### All 23
% 0.19/0.57 25. (-. (greater_or_equal T_1 (appear (efficient_producers) T_0))) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) ### Extension/test/not_mp_greater_or_equal 24
% 0.19/0.57 26. (-. (greater (cardinality_at_time (efficient_producers) T_1) (zero))) (greater (cardinality_at_time (efficient_producers) T_1) (zero)) ### Axiom
% 0.19/0.57 27. (((environment T_0) /\ ((in_environment T_0 T_1) /\ (greater_or_equal T_1 (appear (efficient_producers) T_0)))) => (greater (cardinality_at_time (efficient_producers) T_1) (zero))) (-. (greater (cardinality_at_time (efficient_producers) T_1) (zero))) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (greater T_1 (critical_point T_0)) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (in_environment T_0 T_1) (environment T_0) ### DisjTree 16 17 25 26
% 0.19/0.57 28. (All T, (((environment T_0) /\ ((in_environment T_0 T) /\ (greater_or_equal T (appear (efficient_producers) T_0)))) => (greater (cardinality_at_time (efficient_producers) T) (zero)))) (environment T_0) (in_environment T_0 T_1) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (-. (greater (cardinality_at_time (efficient_producers) T_1) (zero))) ### All 27
% 0.19/0.57 29. (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (-. (greater (cardinality_at_time (efficient_producers) T_1) (zero))) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (greater T_1 (critical_point T_0)) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (in_environment T_0 T_1) (environment T_0) ### All 28
% 0.19/0.57 30. (environment T_0) (-. (environment T_0)) ### Axiom
% 0.19/0.57 31. (in_environment T_0 T_1) (-. (in_environment T_0 T_1)) ### Axiom
% 0.19/0.57 32. (environment T_0) (-. (environment T_0)) ### Axiom
% 0.19/0.57 33. (environment T_0) (-. (environment T_0)) ### Axiom
% 0.19/0.57 34. (in_environment T_0 T_1) (-. (in_environment T_0 T_1)) ### Axiom
% 0.19/0.57 35. (greater (cardinality_at_time (first_movers) T_1) (zero)) (-. (greater (cardinality_at_time (first_movers) T_1) (zero))) ### Axiom
% 0.19/0.57 36. (-. (subpopulations (first_movers) (efficient_producers) T_0 T_1)) (subpopulations (first_movers) (efficient_producers) T_0 T_1) ### Axiom
% 0.19/0.57 37. (((environment T_0) /\ ((in_environment T_0 T_1) /\ ((greater (cardinality_at_time (first_movers) T_1) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T_1) (zero))))) => (subpopulations (first_movers) (efficient_producers) T_0 T_1)) (-. (subpopulations (first_movers) (efficient_producers) T_0 T_1)) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (greater (cardinality_at_time (first_movers) T_1) (zero)) (in_environment T_0 T_1) (environment T_0) ### DisjTree 33 34 35 29 36
% 0.19/0.57 38. (All T, (((environment T_0) /\ ((in_environment T_0 T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) T_0 T))) (environment T_0) (in_environment T_0 T_1) (greater (cardinality_at_time (first_movers) T_1) (zero)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (greater T_1 (critical_point T_0)) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (-. (subpopulations (first_movers) (efficient_producers) T_0 T_1)) ### All 37
% 0.19/0.57 39. (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (-. (subpopulations (first_movers) (efficient_producers) T_0 T_1)) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (greater (cardinality_at_time (first_movers) T_1) (zero)) (in_environment T_0 T_1) (environment T_0) ### All 38
% 0.19/0.57 40. (environment T_0) (-. (environment T_0)) ### Axiom
% 0.19/0.57 41. ((critical_point T_0) != (critical_point T_0)) ### Refl(=)
% 0.19/0.57 42. (greater T_1 (critical_point T_0)) (-. (greater T_1 (critical_point T_0))) ### Axiom
% 0.19/0.57 43. (-. (greater (growth_rate (efficient_producers) T_1) (growth_rate (first_movers) T_1))) (greater (growth_rate (efficient_producers) T_1) (growth_rate (first_movers) T_1)) ### Axiom
% 0.19/0.57 44. (((subpopulations (first_movers) (efficient_producers) T_0 T_1) /\ (greater T_1 (critical_point T_0))) => (greater (growth_rate (efficient_producers) T_1) (growth_rate (first_movers) T_1))) (-. (greater (growth_rate (efficient_producers) T_1) (growth_rate (first_movers) T_1))) (environment T_0) (in_environment T_0 T_1) (greater (cardinality_at_time (first_movers) T_1) (zero)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (greater T_1 (critical_point T_0)) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) ### DisjTree 39 42 43
% 0.19/0.57 45. (All T, (((subpopulations (first_movers) (efficient_producers) T_0 T) /\ (greater T (critical_point T_0))) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (greater (cardinality_at_time (first_movers) T_1) (zero)) (in_environment T_0 T_1) (environment T_0) (-. (greater (growth_rate (efficient_producers) T_1) (growth_rate (first_movers) T_1))) ### All 44
% 0.19/0.57 46. ((-. (greater (growth_rate (efficient_producers) (critical_point T_0)) (growth_rate (first_movers) (critical_point T_0)))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) T_0 T) /\ (greater T (critical_point T_0))) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T))))) (-. (greater (growth_rate (efficient_producers) T_1) (growth_rate (first_movers) T_1))) (environment T_0) (in_environment T_0 T_1) (greater (cardinality_at_time (first_movers) T_1) (zero)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (greater T_1 (critical_point T_0)) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) ### And 45
% 0.19/0.57 47. (((environment T_0) /\ ((critical_point T_0) = (critical_point T_0))) => ((-. (greater (growth_rate (efficient_producers) (critical_point T_0)) (growth_rate (first_movers) (critical_point T_0)))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) T_0 T) /\ (greater T (critical_point T_0))) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (greater (cardinality_at_time (first_movers) T_1) (zero)) (in_environment T_0 T_1) (-. (greater (growth_rate (efficient_producers) T_1) (growth_rate (first_movers) T_1))) (environment T_0) ### DisjTree 40 41 46
% 0.19/0.57 48. (All Tc, (((environment T_0) /\ (Tc = (critical_point T_0))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) T_0 T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T))))))) (environment T_0) (-. (greater (growth_rate (efficient_producers) T_1) (growth_rate (first_movers) T_1))) (in_environment T_0 T_1) (greater (cardinality_at_time (first_movers) T_1) (zero)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (greater T_1 (critical_point T_0)) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) ### All 47
% 0.19/0.57 49. (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (greater (cardinality_at_time (first_movers) T_1) (zero)) (in_environment T_0 T_1) (-. (greater (growth_rate (efficient_producers) T_1) (growth_rate (first_movers) T_1))) (environment T_0) ### All 48
% 0.19/0.57 50. (-. (selection_favors (efficient_producers) (first_movers) T_1)) (selection_favors (efficient_producers) (first_movers) T_1) ### Axiom
% 0.19/0.57 51. (((environment T_0) /\ ((subpopulations (first_movers) (efficient_producers) T_0 T_1) /\ (greater (growth_rate (efficient_producers) T_1) (growth_rate (first_movers) T_1)))) => (selection_favors (efficient_producers) (first_movers) T_1)) (-. (selection_favors (efficient_producers) (first_movers) T_1)) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (in_environment T_0 T_1) (greater (cardinality_at_time (first_movers) T_1) (zero)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (greater T_1 (critical_point T_0)) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (environment T_0) ### DisjTree 32 39 49 50
% 0.19/0.57 52. (All T, (((environment T_0) /\ ((subpopulations (first_movers) (efficient_producers) T_0 T) /\ (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))) => (selection_favors (efficient_producers) (first_movers) T))) (environment T_0) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (greater (cardinality_at_time (first_movers) T_1) (zero)) (in_environment T_0 T_1) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (-. (selection_favors (efficient_producers) (first_movers) T_1)) ### All 51
% 0.19/0.57 53. (All S2, (All T, (((environment T_0) /\ ((subpopulations (first_movers) S2 T_0 T) /\ (greater (growth_rate S2 T) (growth_rate (first_movers) T)))) => (selection_favors S2 (first_movers) T)))) (-. (selection_favors (efficient_producers) (first_movers) T_1)) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (in_environment T_0 T_1) (greater (cardinality_at_time (first_movers) T_1) (zero)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (greater T_1 (critical_point T_0)) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (environment T_0) ### All 52
% 0.19/0.57 54. (All S1, (All S2, (All T, (((environment T_0) /\ ((subpopulations S1 S2 T_0 T) /\ (greater (growth_rate S2 T) (growth_rate S1 T)))) => (selection_favors S2 S1 T))))) (environment T_0) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (greater (cardinality_at_time (first_movers) T_1) (zero)) (in_environment T_0 T_1) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (-. (selection_favors (efficient_producers) (first_movers) T_1)) ### All 53
% 0.19/0.57 55. ((cardinality_at_time (first_movers) T_1) != (zero)) ((cardinality_at_time (first_movers) T_1) = (zero)) ### Axiom
% 0.19/0.57 56. (greater_or_equal (cardinality_at_time (first_movers) T_1) (zero)) ((cardinality_at_time (first_movers) T_1) != (zero)) (-. (selection_favors (efficient_producers) (first_movers) T_1)) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (in_environment T_0 T_1) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (greater T_1 (critical_point T_0)) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (environment T_0) (All S1, (All S2, (All T, (((environment T_0) /\ ((subpopulations S1 S2 T_0 T) /\ (greater (growth_rate S2 T) (growth_rate S1 T)))) => (selection_favors S2 S1 T))))) ### Extension/test/mp_greater_or_equal 54 55
% 0.19/0.57 57. (((environment T_0) /\ (in_environment T_0 T_1)) => (greater_or_equal (cardinality_at_time (first_movers) T_1) (zero))) (All S1, (All S2, (All T, (((environment T_0) /\ ((subpopulations S1 S2 T_0 T) /\ (greater (growth_rate S2 T) (growth_rate S1 T)))) => (selection_favors S2 S1 T))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (-. (selection_favors (efficient_producers) (first_movers) T_1)) ((cardinality_at_time (first_movers) T_1) != (zero)) (in_environment T_0 T_1) (environment T_0) ### DisjTree 30 31 56
% 0.19/0.58 58. (All T, (((environment T_0) /\ (in_environment T_0 T)) => (greater_or_equal (cardinality_at_time (first_movers) T) (zero)))) (environment T_0) (in_environment T_0 T_1) ((cardinality_at_time (first_movers) T_1) != (zero)) (-. (selection_favors (efficient_producers) (first_movers) T_1)) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (greater T_1 (critical_point T_0)) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (All S1, (All S2, (All T, (((environment T_0) /\ ((subpopulations S1 S2 T_0 T) /\ (greater (growth_rate S2 T) (growth_rate S1 T)))) => (selection_favors S2 S1 T))))) ### All 57
% 0.19/0.58 59. (All E, (All T, (((environment E) /\ (in_environment E T)) => (greater_or_equal (cardinality_at_time (first_movers) T) (zero))))) (All S1, (All S2, (All T, (((environment T_0) /\ ((subpopulations S1 S2 T_0 T) /\ (greater (growth_rate S2 T) (growth_rate S1 T)))) => (selection_favors S2 S1 T))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (-. (selection_favors (efficient_producers) (first_movers) T_1)) ((cardinality_at_time (first_movers) T_1) != (zero)) (in_environment T_0 T_1) (environment T_0) ### All 58
% 0.19/0.58 60. (-. (selection_favors (efficient_producers) (first_movers) T_1)) (selection_favors (efficient_producers) (first_movers) T_1) ### Axiom
% 0.19/0.58 61. (((environment T_0) /\ ((subpopulation (efficient_producers) T_0 T_1) /\ ((subpopulation (first_movers) T_0 T_1) /\ ((greater (cardinality_at_time (efficient_producers) T_1) (zero)) /\ ((cardinality_at_time (first_movers) T_1) = (zero)))))) => (selection_favors (efficient_producers) (first_movers) T_1)) (-. (selection_favors (efficient_producers) (first_movers) T_1)) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (All S1, (All S2, (All T, (((environment T_0) /\ ((subpopulations S1 S2 T_0 T) /\ (greater (growth_rate S2 T) (growth_rate S1 T)))) => (selection_favors S2 S1 T))))) (All E, (All T, (((environment E) /\ (in_environment E T)) => (greater_or_equal (cardinality_at_time (first_movers) T) (zero))))) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (in_environment T_0 T_1) (All E, (All T, (((environment E) /\ (in_environment E T)) => ((subpopulation (first_movers) E T) /\ (subpopulation (efficient_producers) E T))))) (environment T_0) ### DisjTree 1 8 15 29 59 60
% 0.19/0.58 62. (All T, (((environment T_0) /\ ((subpopulation (efficient_producers) T_0 T) /\ ((subpopulation (first_movers) T_0 T) /\ ((greater (cardinality_at_time (efficient_producers) T) (zero)) /\ ((cardinality_at_time (first_movers) T) = (zero)))))) => (selection_favors (efficient_producers) (first_movers) T))) (environment T_0) (All E, (All T, (((environment E) /\ (in_environment E T)) => ((subpopulation (first_movers) E T) /\ (subpopulation (efficient_producers) E T))))) (in_environment T_0 T_1) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (greater T_1 (critical_point T_0)) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (All E, (All T, (((environment E) /\ (in_environment E T)) => (greater_or_equal (cardinality_at_time (first_movers) T) (zero))))) (All S1, (All S2, (All T, (((environment T_0) /\ ((subpopulations S1 S2 T_0 T) /\ (greater (growth_rate S2 T) (growth_rate S1 T)))) => (selection_favors S2 S1 T))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (-. (selection_favors (efficient_producers) (first_movers) T_1)) ### All 61
% 0.19/0.58 63. (All S2, (All T, (((environment T_0) /\ ((subpopulation (efficient_producers) T_0 T) /\ ((subpopulation S2 T_0 T) /\ ((greater (cardinality_at_time (efficient_producers) T) (zero)) /\ ((cardinality_at_time S2 T) = (zero)))))) => (selection_favors (efficient_producers) S2 T)))) (-. (selection_favors (efficient_producers) (first_movers) T_1)) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (All S1, (All S2, (All T, (((environment T_0) /\ ((subpopulations S1 S2 T_0 T) /\ (greater (growth_rate S2 T) (growth_rate S1 T)))) => (selection_favors S2 S1 T))))) (All E, (All T, (((environment E) /\ (in_environment E T)) => (greater_or_equal (cardinality_at_time (first_movers) T) (zero))))) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (in_environment T_0 T_1) (All E, (All T, (((environment E) /\ (in_environment E T)) => ((subpopulation (first_movers) E T) /\ (subpopulation (efficient_producers) E T))))) (environment T_0) ### All 62
% 0.19/0.58 64. (All E, (All S1, (All S2, (All T, (((environment E) /\ ((subpopulations S1 S2 E T) /\ (greater (growth_rate S2 T) (growth_rate S1 T)))) => (selection_favors S2 S1 T)))))) (environment T_0) (All E, (All T, (((environment E) /\ (in_environment E T)) => ((subpopulation (first_movers) E T) /\ (subpopulation (efficient_producers) E T))))) (in_environment T_0 T_1) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (greater T_1 (critical_point T_0)) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (All E, (All T, (((environment E) /\ (in_environment E T)) => (greater_or_equal (cardinality_at_time (first_movers) T) (zero))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (-. (selection_favors (efficient_producers) (first_movers) T_1)) (All S2, (All T, (((environment T_0) /\ ((subpopulation (efficient_producers) T_0 T) /\ ((subpopulation S2 T_0 T) /\ ((greater (cardinality_at_time (efficient_producers) T) (zero)) /\ ((cardinality_at_time S2 T) = (zero)))))) => (selection_favors (efficient_producers) S2 T)))) ### All 63
% 0.19/0.58 65. (All S1, (All S2, (All T, (((environment T_0) /\ ((subpopulation S1 T_0 T) /\ ((subpopulation S2 T_0 T) /\ ((greater (cardinality_at_time S1 T) (zero)) /\ ((cardinality_at_time S2 T) = (zero)))))) => (selection_favors S1 S2 T))))) (-. (selection_favors (efficient_producers) (first_movers) T_1)) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (All E, (All T, (((environment E) /\ (in_environment E T)) => (greater_or_equal (cardinality_at_time (first_movers) T) (zero))))) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (in_environment T_0 T_1) (All E, (All T, (((environment E) /\ (in_environment E T)) => ((subpopulation (first_movers) E T) /\ (subpopulation (efficient_producers) E T))))) (environment T_0) (All E, (All S1, (All S2, (All T, (((environment E) /\ ((subpopulations S1 S2 E T) /\ (greater (growth_rate S2 T) (growth_rate S1 T)))) => (selection_favors S2 S1 T)))))) ### All 64
% 0.19/0.58 66. (All E, (All S1, (All S2, (All T, (((environment E) /\ ((subpopulation S1 E T) /\ ((subpopulation S2 E T) /\ ((greater (cardinality_at_time S1 T) (zero)) /\ ((cardinality_at_time S2 T) = (zero)))))) => (selection_favors S1 S2 T)))))) (All E, (All S1, (All S2, (All T, (((environment E) /\ ((subpopulations S1 S2 E T) /\ (greater (growth_rate S2 T) (growth_rate S1 T)))) => (selection_favors S2 S1 T)))))) (environment T_0) (All E, (All T, (((environment E) /\ (in_environment E T)) => ((subpopulation (first_movers) E T) /\ (subpopulation (efficient_producers) E T))))) (in_environment T_0 T_1) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (greater T_1 (critical_point T_0)) (greater (critical_point T_0) (appear (efficient_producers) T_0)) (All E, (All T, (((environment E) /\ (in_environment E T)) => (greater_or_equal (cardinality_at_time (first_movers) T) (zero))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (-. (selection_favors (efficient_producers) (first_movers) T_1)) ### All 65
% 0.19/0.58 67. (environment T_0) (-. (environment T_0)) ### Axiom
% 0.19/0.58 68. (environment T_0) (-. (environment T_0)) ### Axiom
% 0.19/0.58 69. (in_environment T_0 T_1) (-. (in_environment T_0 T_1)) ### Axiom
% 0.19/0.58 70. (T_1 != T_1) ### Refl(=)
% 0.19/0.58 71. ((critical_point T_0) = (appear (efficient_producers) T_0)) ((critical_point T_0) != (appear (efficient_producers) T_0)) ### Axiom
% 0.19/0.58 72. (-. (greater T_1 (appear (efficient_producers) T_0))) (greater T_1 (critical_point T_0)) ((critical_point T_0) = (appear (efficient_producers) T_0)) ### P-NotP 70 71
% 0.19/0.58 73. (-. (greater_or_equal T_1 (appear (efficient_producers) T_0))) ((critical_point T_0) = (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) ### Extension/test/not_mp_greater_or_equal 72
% 0.19/0.58 74. (-. (greater (cardinality_at_time (efficient_producers) T_1) (zero))) (greater (cardinality_at_time (efficient_producers) T_1) (zero)) ### Axiom
% 0.19/0.58 75. (((environment T_0) /\ ((in_environment T_0 T_1) /\ (greater_or_equal T_1 (appear (efficient_producers) T_0)))) => (greater (cardinality_at_time (efficient_producers) T_1) (zero))) (-. (greater (cardinality_at_time (efficient_producers) T_1) (zero))) (greater T_1 (critical_point T_0)) ((critical_point T_0) = (appear (efficient_producers) T_0)) (in_environment T_0 T_1) (environment T_0) ### DisjTree 68 69 73 74
% 0.19/0.58 76. (All T, (((environment T_0) /\ ((in_environment T_0 T) /\ (greater_or_equal T (appear (efficient_producers) T_0)))) => (greater (cardinality_at_time (efficient_producers) T) (zero)))) (environment T_0) (in_environment T_0 T_1) ((critical_point T_0) = (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) (-. (greater (cardinality_at_time (efficient_producers) T_1) (zero))) ### All 75
% 0.19/0.58 77. (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (-. (greater (cardinality_at_time (efficient_producers) T_1) (zero))) (greater T_1 (critical_point T_0)) ((critical_point T_0) = (appear (efficient_producers) T_0)) (in_environment T_0 T_1) (environment T_0) ### All 76
% 0.19/0.58 78. (environment T_0) (-. (environment T_0)) ### Axiom
% 0.19/0.58 79. (in_environment T_0 T_1) (-. (in_environment T_0 T_1)) ### Axiom
% 0.19/0.58 80. (environment T_0) (-. (environment T_0)) ### Axiom
% 0.19/0.58 81. (environment T_0) (-. (environment T_0)) ### Axiom
% 0.19/0.58 82. (in_environment T_0 T_1) (-. (in_environment T_0 T_1)) ### Axiom
% 0.19/0.58 83. (greater (cardinality_at_time (first_movers) T_1) (zero)) (-. (greater (cardinality_at_time (first_movers) T_1) (zero))) ### Axiom
% 0.19/0.58 84. (-. (subpopulations (first_movers) (efficient_producers) T_0 T_1)) (subpopulations (first_movers) (efficient_producers) T_0 T_1) ### Axiom
% 0.42/0.59 85. (((environment T_0) /\ ((in_environment T_0 T_1) /\ ((greater (cardinality_at_time (first_movers) T_1) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T_1) (zero))))) => (subpopulations (first_movers) (efficient_producers) T_0 T_1)) (-. (subpopulations (first_movers) (efficient_producers) T_0 T_1)) ((critical_point T_0) = (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (greater (cardinality_at_time (first_movers) T_1) (zero)) (in_environment T_0 T_1) (environment T_0) ### DisjTree 81 82 83 77 84
% 0.42/0.59 86. (All T, (((environment T_0) /\ ((in_environment T_0 T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) T_0 T))) (environment T_0) (in_environment T_0 T_1) (greater (cardinality_at_time (first_movers) T_1) (zero)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (greater T_1 (critical_point T_0)) ((critical_point T_0) = (appear (efficient_producers) T_0)) (-. (subpopulations (first_movers) (efficient_producers) T_0 T_1)) ### All 85
% 0.42/0.59 87. (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (-. (subpopulations (first_movers) (efficient_producers) T_0 T_1)) ((critical_point T_0) = (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (greater (cardinality_at_time (first_movers) T_1) (zero)) (in_environment T_0 T_1) (environment T_0) ### All 86
% 0.42/0.59 88. (environment T_0) (-. (environment T_0)) ### Axiom
% 0.42/0.59 89. ((critical_point T_0) != (critical_point T_0)) ### Refl(=)
% 0.42/0.59 90. (greater T_1 (critical_point T_0)) (-. (greater T_1 (critical_point T_0))) ### Axiom
% 0.42/0.59 91. (-. (greater (growth_rate (efficient_producers) T_1) (growth_rate (first_movers) T_1))) (greater (growth_rate (efficient_producers) T_1) (growth_rate (first_movers) T_1)) ### Axiom
% 0.42/0.59 92. (((subpopulations (first_movers) (efficient_producers) T_0 T_1) /\ (greater T_1 (critical_point T_0))) => (greater (growth_rate (efficient_producers) T_1) (growth_rate (first_movers) T_1))) (-. (greater (growth_rate (efficient_producers) T_1) (growth_rate (first_movers) T_1))) (environment T_0) (in_environment T_0 T_1) (greater (cardinality_at_time (first_movers) T_1) (zero)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (greater T_1 (critical_point T_0)) ((critical_point T_0) = (appear (efficient_producers) T_0)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) ### DisjTree 87 90 91
% 0.42/0.59 93. (All T, (((subpopulations (first_movers) (efficient_producers) T_0 T) /\ (greater T (critical_point T_0))) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) ((critical_point T_0) = (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (greater (cardinality_at_time (first_movers) T_1) (zero)) (in_environment T_0 T_1) (environment T_0) (-. (greater (growth_rate (efficient_producers) T_1) (growth_rate (first_movers) T_1))) ### All 92
% 0.42/0.59 94. ((-. (greater (growth_rate (efficient_producers) (critical_point T_0)) (growth_rate (first_movers) (critical_point T_0)))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) T_0 T) /\ (greater T (critical_point T_0))) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T))))) (-. (greater (growth_rate (efficient_producers) T_1) (growth_rate (first_movers) T_1))) (environment T_0) (in_environment T_0 T_1) (greater (cardinality_at_time (first_movers) T_1) (zero)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (greater T_1 (critical_point T_0)) ((critical_point T_0) = (appear (efficient_producers) T_0)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) ### And 93
% 0.42/0.59 95. (((environment T_0) /\ ((critical_point T_0) = (critical_point T_0))) => ((-. (greater (growth_rate (efficient_producers) (critical_point T_0)) (growth_rate (first_movers) (critical_point T_0)))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) T_0 T) /\ (greater T (critical_point T_0))) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) ((critical_point T_0) = (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (greater (cardinality_at_time (first_movers) T_1) (zero)) (in_environment T_0 T_1) (-. (greater (growth_rate (efficient_producers) T_1) (growth_rate (first_movers) T_1))) (environment T_0) ### DisjTree 88 89 94
% 0.42/0.59 96. (All Tc, (((environment T_0) /\ (Tc = (critical_point T_0))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) T_0 T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T))))))) (environment T_0) (-. (greater (growth_rate (efficient_producers) T_1) (growth_rate (first_movers) T_1))) (in_environment T_0 T_1) (greater (cardinality_at_time (first_movers) T_1) (zero)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (greater T_1 (critical_point T_0)) ((critical_point T_0) = (appear (efficient_producers) T_0)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) ### All 95
% 0.42/0.59 97. (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) ((critical_point T_0) = (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (greater (cardinality_at_time (first_movers) T_1) (zero)) (in_environment T_0 T_1) (-. (greater (growth_rate (efficient_producers) T_1) (growth_rate (first_movers) T_1))) (environment T_0) ### All 96
% 0.42/0.59 98. (-. (selection_favors (efficient_producers) (first_movers) T_1)) (selection_favors (efficient_producers) (first_movers) T_1) ### Axiom
% 0.42/0.59 99. (((environment T_0) /\ ((subpopulations (first_movers) (efficient_producers) T_0 T_1) /\ (greater (growth_rate (efficient_producers) T_1) (growth_rate (first_movers) T_1)))) => (selection_favors (efficient_producers) (first_movers) T_1)) (-. (selection_favors (efficient_producers) (first_movers) T_1)) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (in_environment T_0 T_1) (greater (cardinality_at_time (first_movers) T_1) (zero)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (greater T_1 (critical_point T_0)) ((critical_point T_0) = (appear (efficient_producers) T_0)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (environment T_0) ### DisjTree 80 87 97 98
% 0.42/0.59 100. (All T, (((environment T_0) /\ ((subpopulations (first_movers) (efficient_producers) T_0 T) /\ (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))) => (selection_favors (efficient_producers) (first_movers) T))) (environment T_0) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) ((critical_point T_0) = (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (greater (cardinality_at_time (first_movers) T_1) (zero)) (in_environment T_0 T_1) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (-. (selection_favors (efficient_producers) (first_movers) T_1)) ### All 99
% 0.42/0.59 101. (All S2, (All T, (((environment T_0) /\ ((subpopulations (first_movers) S2 T_0 T) /\ (greater (growth_rate S2 T) (growth_rate (first_movers) T)))) => (selection_favors S2 (first_movers) T)))) (-. (selection_favors (efficient_producers) (first_movers) T_1)) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (in_environment T_0 T_1) (greater (cardinality_at_time (first_movers) T_1) (zero)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (greater T_1 (critical_point T_0)) ((critical_point T_0) = (appear (efficient_producers) T_0)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (environment T_0) ### All 100
% 0.42/0.59 102. (All S1, (All S2, (All T, (((environment T_0) /\ ((subpopulations S1 S2 T_0 T) /\ (greater (growth_rate S2 T) (growth_rate S1 T)))) => (selection_favors S2 S1 T))))) (environment T_0) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) ((critical_point T_0) = (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (greater (cardinality_at_time (first_movers) T_1) (zero)) (in_environment T_0 T_1) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (-. (selection_favors (efficient_producers) (first_movers) T_1)) ### All 101
% 0.42/0.59 103. ((cardinality_at_time (first_movers) T_1) != (zero)) ((cardinality_at_time (first_movers) T_1) = (zero)) ### Axiom
% 0.42/0.59 104. (greater_or_equal (cardinality_at_time (first_movers) T_1) (zero)) ((cardinality_at_time (first_movers) T_1) != (zero)) (-. (selection_favors (efficient_producers) (first_movers) T_1)) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (in_environment T_0 T_1) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (greater T_1 (critical_point T_0)) ((critical_point T_0) = (appear (efficient_producers) T_0)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (environment T_0) (All S1, (All S2, (All T, (((environment T_0) /\ ((subpopulations S1 S2 T_0 T) /\ (greater (growth_rate S2 T) (growth_rate S1 T)))) => (selection_favors S2 S1 T))))) ### Extension/test/mp_greater_or_equal 102 103
% 0.42/0.59 105. (((environment T_0) /\ (in_environment T_0 T_1)) => (greater_or_equal (cardinality_at_time (first_movers) T_1) (zero))) (All S1, (All S2, (All T, (((environment T_0) /\ ((subpopulations S1 S2 T_0 T) /\ (greater (growth_rate S2 T) (growth_rate S1 T)))) => (selection_favors S2 S1 T))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) ((critical_point T_0) = (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (-. (selection_favors (efficient_producers) (first_movers) T_1)) ((cardinality_at_time (first_movers) T_1) != (zero)) (in_environment T_0 T_1) (environment T_0) ### DisjTree 78 79 104
% 0.42/0.59 106. (All T, (((environment T_0) /\ (in_environment T_0 T)) => (greater_or_equal (cardinality_at_time (first_movers) T) (zero)))) (environment T_0) (in_environment T_0 T_1) ((cardinality_at_time (first_movers) T_1) != (zero)) (-. (selection_favors (efficient_producers) (first_movers) T_1)) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (greater T_1 (critical_point T_0)) ((critical_point T_0) = (appear (efficient_producers) T_0)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (All S1, (All S2, (All T, (((environment T_0) /\ ((subpopulations S1 S2 T_0 T) /\ (greater (growth_rate S2 T) (growth_rate S1 T)))) => (selection_favors S2 S1 T))))) ### All 105
% 0.42/0.59 107. (All E, (All T, (((environment E) /\ (in_environment E T)) => (greater_or_equal (cardinality_at_time (first_movers) T) (zero))))) (All S1, (All S2, (All T, (((environment T_0) /\ ((subpopulations S1 S2 T_0 T) /\ (greater (growth_rate S2 T) (growth_rate S1 T)))) => (selection_favors S2 S1 T))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) ((critical_point T_0) = (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (-. (selection_favors (efficient_producers) (first_movers) T_1)) ((cardinality_at_time (first_movers) T_1) != (zero)) (in_environment T_0 T_1) (environment T_0) ### All 106
% 0.42/0.59 108. (-. (selection_favors (efficient_producers) (first_movers) T_1)) (selection_favors (efficient_producers) (first_movers) T_1) ### Axiom
% 0.42/0.59 109. (((environment T_0) /\ ((subpopulation (efficient_producers) T_0 T_1) /\ ((subpopulation (first_movers) T_0 T_1) /\ ((greater (cardinality_at_time (efficient_producers) T_1) (zero)) /\ ((cardinality_at_time (first_movers) T_1) = (zero)))))) => (selection_favors (efficient_producers) (first_movers) T_1)) (-. (selection_favors (efficient_producers) (first_movers) T_1)) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (All S1, (All S2, (All T, (((environment T_0) /\ ((subpopulations S1 S2 T_0 T) /\ (greater (growth_rate S2 T) (growth_rate S1 T)))) => (selection_favors S2 S1 T))))) (All E, (All T, (((environment E) /\ (in_environment E T)) => (greater_or_equal (cardinality_at_time (first_movers) T) (zero))))) ((critical_point T_0) = (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (in_environment T_0 T_1) (All E, (All T, (((environment E) /\ (in_environment E T)) => ((subpopulation (first_movers) E T) /\ (subpopulation (efficient_producers) E T))))) (environment T_0) ### DisjTree 67 8 15 77 107 108
% 0.42/0.59 110. (All T, (((environment T_0) /\ ((subpopulation (efficient_producers) T_0 T) /\ ((subpopulation (first_movers) T_0 T) /\ ((greater (cardinality_at_time (efficient_producers) T) (zero)) /\ ((cardinality_at_time (first_movers) T) = (zero)))))) => (selection_favors (efficient_producers) (first_movers) T))) (environment T_0) (All E, (All T, (((environment E) /\ (in_environment E T)) => ((subpopulation (first_movers) E T) /\ (subpopulation (efficient_producers) E T))))) (in_environment T_0 T_1) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (greater T_1 (critical_point T_0)) ((critical_point T_0) = (appear (efficient_producers) T_0)) (All E, (All T, (((environment E) /\ (in_environment E T)) => (greater_or_equal (cardinality_at_time (first_movers) T) (zero))))) (All S1, (All S2, (All T, (((environment T_0) /\ ((subpopulations S1 S2 T_0 T) /\ (greater (growth_rate S2 T) (growth_rate S1 T)))) => (selection_favors S2 S1 T))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (-. (selection_favors (efficient_producers) (first_movers) T_1)) ### All 109
% 0.42/0.59 111. (All S2, (All T, (((environment T_0) /\ ((subpopulation (efficient_producers) T_0 T) /\ ((subpopulation S2 T_0 T) /\ ((greater (cardinality_at_time (efficient_producers) T) (zero)) /\ ((cardinality_at_time S2 T) = (zero)))))) => (selection_favors (efficient_producers) S2 T)))) (-. (selection_favors (efficient_producers) (first_movers) T_1)) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (All S1, (All S2, (All T, (((environment T_0) /\ ((subpopulations S1 S2 T_0 T) /\ (greater (growth_rate S2 T) (growth_rate S1 T)))) => (selection_favors S2 S1 T))))) (All E, (All T, (((environment E) /\ (in_environment E T)) => (greater_or_equal (cardinality_at_time (first_movers) T) (zero))))) ((critical_point T_0) = (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (in_environment T_0 T_1) (All E, (All T, (((environment E) /\ (in_environment E T)) => ((subpopulation (first_movers) E T) /\ (subpopulation (efficient_producers) E T))))) (environment T_0) ### All 110
% 0.42/0.59 112. (All E, (All S1, (All S2, (All T, (((environment E) /\ ((subpopulations S1 S2 E T) /\ (greater (growth_rate S2 T) (growth_rate S1 T)))) => (selection_favors S2 S1 T)))))) (environment T_0) (All E, (All T, (((environment E) /\ (in_environment E T)) => ((subpopulation (first_movers) E T) /\ (subpopulation (efficient_producers) E T))))) (in_environment T_0 T_1) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (greater T_1 (critical_point T_0)) ((critical_point T_0) = (appear (efficient_producers) T_0)) (All E, (All T, (((environment E) /\ (in_environment E T)) => (greater_or_equal (cardinality_at_time (first_movers) T) (zero))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (-. (selection_favors (efficient_producers) (first_movers) T_1)) (All S2, (All T, (((environment T_0) /\ ((subpopulation (efficient_producers) T_0 T) /\ ((subpopulation S2 T_0 T) /\ ((greater (cardinality_at_time (efficient_producers) T) (zero)) /\ ((cardinality_at_time S2 T) = (zero)))))) => (selection_favors (efficient_producers) S2 T)))) ### All 111
% 0.42/0.59 113. (All S1, (All S2, (All T, (((environment T_0) /\ ((subpopulation S1 T_0 T) /\ ((subpopulation S2 T_0 T) /\ ((greater (cardinality_at_time S1 T) (zero)) /\ ((cardinality_at_time S2 T) = (zero)))))) => (selection_favors S1 S2 T))))) (-. (selection_favors (efficient_producers) (first_movers) T_1)) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (All E, (All T, (((environment E) /\ (in_environment E T)) => (greater_or_equal (cardinality_at_time (first_movers) T) (zero))))) ((critical_point T_0) = (appear (efficient_producers) T_0)) (greater T_1 (critical_point T_0)) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (in_environment T_0 T_1) (All E, (All T, (((environment E) /\ (in_environment E T)) => ((subpopulation (first_movers) E T) /\ (subpopulation (efficient_producers) E T))))) (environment T_0) (All E, (All S1, (All S2, (All T, (((environment E) /\ ((subpopulations S1 S2 E T) /\ (greater (growth_rate S2 T) (growth_rate S1 T)))) => (selection_favors S2 S1 T)))))) ### All 112
% 0.42/0.59 114. (All E, (All S1, (All S2, (All T, (((environment E) /\ ((subpopulation S1 E T) /\ ((subpopulation S2 E T) /\ ((greater (cardinality_at_time S1 T) (zero)) /\ ((cardinality_at_time S2 T) = (zero)))))) => (selection_favors S1 S2 T)))))) (All E, (All S1, (All S2, (All T, (((environment E) /\ ((subpopulations S1 S2 E T) /\ (greater (growth_rate S2 T) (growth_rate S1 T)))) => (selection_favors S2 S1 T)))))) (environment T_0) (All E, (All T, (((environment E) /\ (in_environment E T)) => ((subpopulation (first_movers) E T) /\ (subpopulation (efficient_producers) E T))))) (in_environment T_0 T_1) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (greater T_1 (critical_point T_0)) ((critical_point T_0) = (appear (efficient_producers) T_0)) (All E, (All T, (((environment E) /\ (in_environment E T)) => (greater_or_equal (cardinality_at_time (first_movers) T) (zero))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (-. (selection_favors (efficient_producers) (first_movers) T_1)) ### All 113
% 0.42/0.59 115. (greater_or_equal (critical_point T_0) (appear (efficient_producers) T_0)) (-. (selection_favors (efficient_producers) (first_movers) T_1)) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (All E, (All T, (((environment E) /\ (in_environment E T)) => (greater_or_equal (cardinality_at_time (first_movers) T) (zero))))) (greater T_1 (critical_point T_0)) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (in_environment T_0 T_1) (All E, (All T, (((environment E) /\ (in_environment E T)) => ((subpopulation (first_movers) E T) /\ (subpopulation (efficient_producers) E T))))) (environment T_0) (All E, (All S1, (All S2, (All T, (((environment E) /\ ((subpopulations S1 S2 E T) /\ (greater (growth_rate S2 T) (growth_rate S1 T)))) => (selection_favors S2 S1 T)))))) (All E, (All S1, (All S2, (All T, (((environment E) /\ ((subpopulation S1 E T) /\ ((subpopulation S2 E T) /\ ((greater (cardinality_at_time S1 T) (zero)) /\ ((cardinality_at_time S2 T) = (zero)))))) => (selection_favors S1 S2 T)))))) ### Extension/test/mp_greater_or_equal 66 114
% 0.42/0.59 116. (All E, (All S1, (All S2, (All T, (((environment E) /\ ((subpopulation S1 E T) /\ ((subpopulation S2 E T) /\ ((greater (cardinality_at_time S1 T) (zero)) /\ ((cardinality_at_time S2 T) = (zero)))))) => (selection_favors S1 S2 T)))))) (All E, (All S1, (All S2, (All T, (((environment E) /\ ((subpopulations S1 S2 E T) /\ (greater (growth_rate S2 T) (growth_rate S1 T)))) => (selection_favors S2 S1 T)))))) (environment T_0) (All E, (All T, (((environment E) /\ (in_environment E T)) => ((subpopulation (first_movers) E T) /\ (subpopulation (efficient_producers) E T))))) (in_environment T_0 T_1) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (greater T_1 (critical_point T_0)) (All E, (All T, (((environment E) /\ (in_environment E T)) => (greater_or_equal (cardinality_at_time (first_movers) T) (zero))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (-. (selection_favors (efficient_producers) (first_movers) T_1)) ### Extension/test/mp_critical_point_after_EP 115
% 0.42/0.60 117. (-. (((environment T_0) /\ ((in_environment T_0 T_1) /\ (greater T_1 (critical_point T_0)))) => (selection_favors (efficient_producers) (first_movers) T_1))) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (All E, (All T, (((environment E) /\ (in_environment E T)) => (greater_or_equal (cardinality_at_time (first_movers) T) (zero))))) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (All E, (All T, (((environment E) /\ (in_environment E T)) => ((subpopulation (first_movers) E T) /\ (subpopulation (efficient_producers) E T))))) (All E, (All S1, (All S2, (All T, (((environment E) /\ ((subpopulations S1 S2 E T) /\ (greater (growth_rate S2 T) (growth_rate S1 T)))) => (selection_favors S2 S1 T)))))) (All E, (All S1, (All S2, (All T, (((environment E) /\ ((subpopulation S1 E T) /\ ((subpopulation S2 E T) /\ ((greater (cardinality_at_time S1 T) (zero)) /\ ((cardinality_at_time S2 T) = (zero)))))) => (selection_favors S1 S2 T)))))) ### ConjTree 116
% 0.42/0.60 118. (-. (All T, (((environment T_0) /\ ((in_environment T_0 T) /\ (greater T (critical_point T_0)))) => (selection_favors (efficient_producers) (first_movers) T)))) (All E, (All S1, (All S2, (All T, (((environment E) /\ ((subpopulation S1 E T) /\ ((subpopulation S2 E T) /\ ((greater (cardinality_at_time S1 T) (zero)) /\ ((cardinality_at_time S2 T) = (zero)))))) => (selection_favors S1 S2 T)))))) (All E, (All S1, (All S2, (All T, (((environment E) /\ ((subpopulations S1 S2 E T) /\ (greater (growth_rate S2 T) (growth_rate S1 T)))) => (selection_favors S2 S1 T)))))) (All E, (All T, (((environment E) /\ (in_environment E T)) => ((subpopulation (first_movers) E T) /\ (subpopulation (efficient_producers) E T))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (All E, (All T, (((environment E) /\ (in_environment E T)) => (greater_or_equal (cardinality_at_time (first_movers) T) (zero))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) ### NotAllEx 117
% 0.42/0.60 119. (-. (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater T (critical_point E)))) => (selection_favors (efficient_producers) (first_movers) T))))) (All E, (All Tc, (((environment E) /\ (Tc = (critical_point E))) => ((-. (greater (growth_rate (efficient_producers) Tc) (growth_rate (first_movers) Tc))) /\ (All T, (((subpopulations (first_movers) (efficient_producers) E T) /\ (greater T Tc)) => (greater (growth_rate (efficient_producers) T) (growth_rate (first_movers) T)))))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ ((greater (cardinality_at_time (first_movers) T) (zero)) /\ (greater (cardinality_at_time (efficient_producers) T) (zero))))) => (subpopulations (first_movers) (efficient_producers) E T)))) (All E, (All T, (((environment E) /\ (in_environment E T)) => (greater_or_equal (cardinality_at_time (first_movers) T) (zero))))) (All X, (All Y, (All Z, (((greater X Y) /\ (greater Y Z)) => (greater X Z))))) (All E, (All T, (((environment E) /\ ((in_environment E T) /\ (greater_or_equal T (appear (efficient_producers) E)))) => (greater (cardinality_at_time (efficient_producers) T) (zero))))) (All E, (All T, (((environment E) /\ (in_environment E T)) => ((subpopulation (first_movers) E T) /\ (subpopulation (efficient_producers) E T))))) (All E, (All S1, (All S2, (All T, (((environment E) /\ ((subpopulations S1 S2 E T) /\ (greater (growth_rate S2 T) (growth_rate S1 T)))) => (selection_favors S2 S1 T)))))) (All E, (All S1, (All S2, (All T, (((environment E) /\ ((subpopulation S1 E T) /\ ((subpopulation S2 E T) /\ ((greater (cardinality_at_time S1 T) (zero)) /\ ((cardinality_at_time S2 T) = (zero)))))) => (selection_favors S1 S2 T)))))) ### NotAllEx 118
% 0.42/0.60 % SZS output end Proof
% 0.42/0.60 (* END-PROOF *)
%------------------------------------------------------------------------------