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f1f7953 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 | import OAI.Analysis.IrrationalRotation.Rotation.CircleParameterBlocks
import OAI.Analysis.IrrationalRotation.Rotation.CircleParameterConsequences
import Mathlib.Topology.Algebra.InfiniteSum.Real
namespace OAI
namespace Invariant233
open scoped BigOperators ENNReal
namespace CircleParameterLaws
variable {θ : ℝ} {p : CircleParameterData} (hp : CircleParameterLaws θ p)
include hp
theorem block_phase_sum_le (j : ℕ) (hj : 1 ≤ j) :
(∑ i ∈ p.block j, p.alpha θ i) ≤ 2 / (p.K j : ℝ) := by
have hm : (p.m j : ℝ) ≠ 0 := by exact_mod_cast (hp.m_pos j hj).ne'
have hK : (p.K j : ℝ) ≠ 0 := by exact_mod_cast (hp.K_pos j hj).ne'
calc
(∑ i ∈ p.block j, p.alpha θ i) ≤ ∑ i ∈ p.block j, 2 / p.N j := by
apply Finset.sum_le_sum
intro i hi
exact (hp.phase_bounds j hj i hi).2.le
_ = ((p.block j).card : ℝ) * (2 / p.N j) := by simp
_ = 2 / (p.K j : ℝ) := by
rw [p.card_block j hj]
simp only [CircleParameterData.N]
field_simp
theorem block_phase_sum_lower (j : ℕ) (hj : 1 ≤ j) :
1 / (p.K j : ℝ) ≤ ∑ i ∈ p.block j, p.alpha θ i := by
have hm : (p.m j : ℝ) ≠ 0 := by exact_mod_cast (hp.m_pos j hj).ne'
have hK : (p.K j : ℝ) ≠ 0 := by exact_mod_cast (hp.K_pos j hj).ne'
calc
1 / (p.K j : ℝ) = ((p.block j).card : ℝ) * (1 / p.N j) := by
rw [p.card_block j hj]
simp only [CircleParameterData.N]
field_simp
_ = ∑ i ∈ p.block j, 1 / p.N j := by simp
_ ≤ ∑ i ∈ p.block j, p.alpha θ i := by
apply Finset.sum_le_sum
intro i hi
exact (hp.phase_bounds j hj i hi).1.le
theorem one_le_block_log_cost (j : ℕ) (hj : 1 ≤ j) :
1 ≤ p.C j * ∑ i ∈ p.block j, p.alpha θ i := by
have hK : 0 < (p.K j : ℝ) := by exact_mod_cast hp.K_pos j hj
have hK1 : 1 ≤ (p.K j : ℝ) := by exact_mod_cast hp.K_pos j hj
have hC : (p.K j : ℝ) ^ 2 ≤ p.C j := by
rw [hp.C_eq j hj]
exact le_max_right _ _
calc
1 ≤ (p.K j : ℝ) := hK1
_ ≤ p.C j / (p.K j : ℝ) := by
apply (le_div_iff₀ hK).mpr
nlinarith
_ = p.C j * (1 / (p.K j : ℝ)) := by ring
_ ≤ p.C j * ∑ i ∈ p.block j, p.alpha θ i :=
mul_le_mul_of_nonneg_left (hp.block_phase_sum_lower j hj) (hp.C_pos j hj).le
theorem tsum_block_log_cost_eq_top :
(∑' n : ℕ, ENNReal.ofReal
(p.C (n + 1) * ∑ i ∈ p.block (n + 1), p.alpha θ i)) = ∞ := by
apply top_unique
calc
(∞ : ENNReal) = ∑' _ : ℕ, (1 : ENNReal) :=
(ENNReal.tsum_const_eq_top_of_ne_zero one_ne_zero).symm
_ ≤ ∑' n : ℕ, ENNReal.ofReal
(p.C (n + 1) * ∑ i ∈ p.block (n + 1), p.alpha θ i) := by
apply ENNReal.tsum_le_tsum
intro n
simpa only [ENNReal.ofReal_one] using
ENNReal.ofReal_le_ofReal (hp.one_le_block_log_cost (n + 1) (by omega))
theorem summable_block_phases :
Summable (fun n : ℕ => ∑ i ∈ p.block (n + 1), p.alpha θ i) := by
apply Summable.of_nonneg_of_le
(fun n => Finset.sum_nonneg (fun i _ => Int.fract_nonneg _))
(fun n => hp.block_phase_sum_le (n + 1) (by omega))
simpa only [mul_one_div] using hp.summable_reciprocal.mul_left 2
private def blocksWithZero (p : CircleParameterData) (j : ℕ) : Finset ℕ :=
if j = 0 then {0} else p.block j
private theorem blocksWithZero_partition :
∀ i : ℕ, ∃! j : ℕ, i ∈ blocksWithZero p j := by
intro i
by_cases hi : i = 0
· subst i
refine ⟨0, by simp [blocksWithZero], ?_⟩
intro j hj
by_contra h
have hb : 0 ∈ p.block j := by simpa [blocksWithZero, h] using hj
have hb' := (p.mem_block_iff j 0).mp hb
omega
· obtain ⟨j, hj, hij⟩ := p.exists_mem_block hp.m_pos i (by omega)
have hj0 : j ≠ 0 := by omega
refine ⟨j, by simpa [blocksWithZero, hj0] using hij, ?_⟩
intro k hik
have hk0 : k ≠ 0 := by
intro hk
subst k
simp [blocksWithZero, hi] at hik
have hk : i ∈ p.block k := by simpa [blocksWithZero, hk0] using hik
exact p.block_index_unique hp.m_pos hk hij
theorem summable_phases_all : Summable (p.alpha θ) := by
classical
apply (summable_partition (fun i => Int.fract_nonneg _) hp.blocksWithZero_partition).mpr
constructor
· intro j
exact (hasSum_fintype _).summable
· apply (summable_nat_add_iff 1).mp
apply hp.summable_block_phases.congr
intro n
simp only [blocksWithZero,
tsum_fintype, CircleParameterData.alpha]
exact (Finset.sum_attach (p.block (n + 1)) (fun i => Int.fract ((p.q i : ℝ) * θ))).symm
theorem summable_phases : Summable (fun i : ℕ => p.alpha θ (i + 1)) :=
(summable_nat_add_iff 1).mpr hp.summable_phases_all
end CircleParameterLaws
end Invariant233
end OAI
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