| import Mathlib |
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| namespace OAI |
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| section |
| section |
| section |
| noncomputable section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| section |
| open Finset Filter Topology |
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| namespace CannonFlow |
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| theorem positive_capacity_transitions {ι : Type*} (S : Finset ι) (t : ι → ℝ) |
| {L Q : ℝ} (hL : 0 < L) (hQ : 0 < Q) (ht : ∀ i ∈ S, 0 ≤ t i) |
| (hc : 1 ≤ ∑ i ∈ S, min (L*t i) (Q*(t i)^2)) : |
| ∃ T : Finset ι, T ⊆ S ∧ (∀ i ∈ T, 0 < t i) ∧ ∃ p : T → ℝ, |
| (∀ i, 0 < p i) ∧ (∑ i, p i = 1) ∧ |
| (∀ i, p i ≤ L*t i) ∧ ∀ i, p i ≤ Q*(t i)^2 := by |
| classical |
| let T := S.filter (fun i => 0 < t i) |
| let cap : ι → ℝ := fun i => min (L*t i) (Q*(t i)^2) |
| have hsum : (∑ i : T, cap i) = ∑ i ∈ S, cap i := by |
| rw [Finset.sum_coe_sort] |
| change (∑ i ∈ S.filter (fun i => 0 < t i), cap i) = ∑ i ∈ S, cap i |
| apply sum_subset (filter_subset _ _) (fun i hi hit => ?_) |
| have hn : ¬ 0 < t i := fun hh => hit (mem_filter.mpr ⟨hi,hh⟩) |
| have hz : t i=0 := le_antisymm (le_of_not_gt hn) (ht i hi) |
| simp [cap,hz] |
| let Z := ∑ i : T, cap i |
| have hZ : 1 ≤ Z := by rw [show Z = _ from hsum]; exact hc |
| have hZpos : 0 < Z := lt_of_lt_of_le zero_lt_one hZ |
| refine ⟨T,filter_subset _ _,fun i hi => (mem_filter.mp hi).2, |
| fun i => cap i/Z,?_,?_,?_,?_⟩ |
| · intro i |
| have hti := (mem_filter.mp i.property).2 |
| exact div_pos (lt_min (mul_pos hL hti) (mul_pos hQ (sq_pos_of_pos hti))) hZpos |
| · rw [← Finset.sum_div] |
| exact div_self hZpos.ne' |
| · intro i |
| have hti := (mem_filter.mp i.property).2 |
| have hcap : 0 ≤ cap i := le_min (mul_nonneg hL.le hti.le) (mul_nonneg hQ.le (sq_nonneg _)) |
| have hh : cap i/Z ≤ cap i := (div_le_iff₀ hZpos).mpr (by nlinarith only [hZ,hcap]) |
| exact hh.trans (min_le_left _ _) |
| · intro i |
| have hti := (mem_filter.mp i.property).2 |
| have hcap : 0 ≤ cap i := le_min (mul_nonneg hL.le hti.le) (mul_nonneg hQ.le (sq_nonneg _)) |
| have hh : cap i/Z ≤ cap i := (div_le_iff₀ hZpos).mpr (by nlinarith only [hZ,hcap]) |
| exact hh.trans (min_le_right _ _) |
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| theorem uniform_bad_transitions {ι : Type*} (S : Finset ι) (t : ι → ℝ) |
| {L Q : ℝ} (hL : 0 < L) (hQ : 0 < Q) |
| (ht : ∀ i ∈ S, (1:ℝ)/8 ≤ t i) |
| (hcardL : 8 ≤ L*(S.card:ℝ)) (hcardQ : 64 ≤ Q*(S.card:ℝ)) : |
| ∃ p : S → ℝ, (∀ i, 0 < p i) ∧ (∑ i, p i = 1) ∧ |
| (∀ i, p i ≤ L*t i) ∧ ∀ i, p i ≤ Q*(t i)^2 := by |
| classical |
| have hcard : 0 < (S.card:ℝ) := by |
| have hc : (0:ℝ) ≤ S.card := Nat.cast_nonneg _ |
| by_contra hn |
| have hz : (S.card:ℝ)=0 := le_antisymm (le_of_not_gt hn) hc |
| rw [hz,mul_zero] at hcardL |
| norm_num at hcardL |
| refine ⟨fun _ => 1/(S.card:ℝ),fun _ => div_pos zero_lt_one hcard,?_,?_,?_⟩ |
| · simp only [sum_const,nsmul_eq_mul,Finset.card_univ,Fintype.card_coe] |
| exact mul_div_cancel₀ 1 hcard.ne' |
| · intro i |
| apply (div_le_iff₀ hcard).mpr |
| have hh := mul_le_mul_of_nonneg_right (ht i i.property) |
| (mul_nonneg hL.le hcard.le) |
| nlinarith only [hh,hcardL] |
| · intro i |
| apply (div_le_iff₀ hcard).mpr |
| have hh := pow_le_pow_left₀ (by norm_num : (0:ℝ) ≤ 1/8) (ht i i.property) 2 |
| have hh' := mul_le_mul_of_nonneg_right hh (mul_nonneg hQ.le hcard.le) |
| nlinarith only [hh',hcardQ] |
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| end CannonFlow |
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| end |
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| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
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| end OAI |
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