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110f4f3 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 | import OAI.Analysis.BackwardIntertwiners.WeightSeparation
namespace OAI
namespace BackwardIntertwiners
open Filter
open scoped Topology
noncomputable section
abbrev Fiber := lp (fun _ : ℤ => ℂ) 2
def coordinateTail (h : ℤ) : Submodule ℂ Fiber where
carrier := {v | ∀ j : ℤ, j < h → v j = 0}
zero_mem' := by simp
add_mem' := by intro v z hv hz j hj; simp [hv j hj, hz j hj]
smul_mem' := by intro a v hv j hj; simp [hv j hj]
lemma fiber_exists_coordinate {v : Fiber} (hv : v ≠ 0) : ∃ j : ℤ, v j ≠ 0 := by
by_contra! h
apply hv
ext j
exact h j
lemma weighted_shift_pow {w : ℤ → ℝ} (hw : ∀ j, 0 < w j)
{T : Fiber →L[ℂ] Fiber}
(hT : ∀ v j, T v j = (w j / w (j-1) : ℂ) * v (j-1))
(n : ℕ) (v : Fiber) (j : ℤ) :
(T^n) v j = (w j / w (j-(n:ℤ)) : ℂ) * v (j-(n:ℤ)) := by
have hwC (k : ℤ) : (w k:ℂ) ≠ 0 := by exact_mod_cast (ne_of_gt (hw k))
induction n generalizing j with
| zero => simp [hwC]
| succ n ih =>
rw [pow_succ']
change T ((T^n) v) j = _
rw [hT, ih]
have hi : j-1-(n:ℤ) = j-((n+1:ℕ):ℤ) := by omega
rw [hi]
field_simp [hwC]
lemma invariant_pow {T : Fiber →L[ℂ] Fiber} {L : Submodule ℂ Fiber}
(hi : ∀ v ∈ L, T v ∈ L) {v : Fiber} (hv : v ∈ L) (n : ℕ) : (T^n) v ∈ L := by
induction n with
| zero => simpa using hv
| succ n ih => simpa only [pow_succ', mul_apply_eq_comp] using hi _ ih
lemma SupportSeparation.strict_order {w : ℤ → ℝ} (hsep : SupportSeparation w)
{b c : ℤ → ℂ} (hbn : ∃ j, b j ≠ 0) (hcn : ∃ j, c j ≠ 0)
{B C : ℝ} (hB : 0 < B) (hC : 0 < C)
(hb : ∀ j, ‖b j‖ ≤ B*w j) (hc : ∀ j, ‖c j‖*w j ≤ C)
(hz : ∀ r : ℕ, (∑' j : ℤ, b j * c (j-r)) = 0)
{i j : ℤ} (hbi : b i ≠ 0) (hcj : c j ≠ 0) : i < j := by
obtain ⟨t, h, hbt, hch, hbmax, hcmin, hth⟩ :=
hsep.of_bounded hbn hcn hB hC hb hc (fun r _ => hz r)
have hit : i ≤ t := by by_contra! hh; exact hbi (hbmax i hh)
have hhj : h ≤ j := by by_contra! hh; exact hcj (hcmin j hh)
have hstrict : t < h := by
by_contra! hht
have he : t = h := le_antisymm hth hht
have hsingle : (∑' k : ℤ, b k*c k) = b t*c h := by
rw [tsum_eq_single t, he]
intro k hk
by_cases hkt : t < k
· simp [hbmax k hkt]
· rw [hcmin k (by omega), mul_zero]
have hzero := hz 0
simp only [Nat.cast_zero, sub_zero] at hzero
rw [hsingle] at hzero
exact (mul_ne_zero hbt hch) hzero
omega
lemma fiber_orthogonal_order {w : ℤ → ℝ} (hw : QuadraticWeights w)
{T : Fiber →L[ℂ] Fiber}
(hT : ∀ v j, T v j = (w j / w (j-1) : ℂ) * v (j-1))
{L : Submodule ℂ Fiber} (hinvariant : ∀ v ∈ L, T v ∈ L)
{v z : Fiber} (hv : v ∈ L) (hz : z ∈ L.orthogonal)
{i j : ℤ} (hzi : z i ≠ 0) (hvj : v j ≠ 0) : i < j := by
let b := fun k : ℤ => star (z k) * (w k:ℂ)
let c := fun k : ℤ => v k / (w k:ℂ)
have hwC (k : ℤ) : (w k:ℂ) ≠ 0 := by exact_mod_cast (ne_of_gt (hw.positive k))
have hbi : b i ≠ 0 := mul_ne_zero (by simpa using hzi) (hwC i)
have hcj : c j ≠ 0 := div_ne_zero hvj (hwC j)
have hb (k : ℤ) : ‖b k‖ ≤ (‖z‖+1)*w k := by
dsimp [b]
rw [norm_mul, RCLike.norm_conj, Complex.norm_real, Real.norm_of_nonneg (hw.positive k).le]
apply mul_le_mul_of_nonneg_right _ (hw.positive k).le
exact (lp.norm_apply_le_norm (by norm_num : (2 : ENNReal) ≠ 0) z k).trans (by linarith)
have hc (k : ℤ) : ‖c k‖*w k ≤ ‖v‖+1 := by
dsimp [c]
rw [norm_div, Complex.norm_real, Real.norm_of_nonneg (hw.positive k).le,
div_mul_cancel₀ _ (ne_of_gt (hw.positive k))]
exact (lp.norm_apply_le_norm (by norm_num : (2 : ENNReal) ≠ 0) v k).trans (by linarith)
have hcor (r : ℕ) : (∑' k : ℤ, b k*c (k-r)) = 0 := by
calc (∑' k : ℤ, b k*c (k-r)) = inner ℂ z ((T^r) v) := by
rw [lp.inner_eq_tsum]
apply tsum_congr
intro k
rw [RCLike.inner_apply, weighted_shift_pow hw.positive hT]
dsimp [b, c]
simp only [← Complex.star_def]
ring
_ = 0 := Submodule.inner_left_of_mem_orthogonal (invariant_pow hinvariant hv r) hz
exact (quadratic_support_separation w hw).strict_order ⟨i,hbi⟩ ⟨j,hcj⟩
(by positivity : 0 < ‖z‖+1) (by positivity : 0 < ‖v‖+1) hb hc hcor hbi hcj
theorem quadratic_fiber_tails (w : ℤ → ℝ) (hw : QuadraticWeights w)
(T : Fiber →L[ℂ] Fiber)
(hT : ∀ v j, T v j = (w j / w (j-1) : ℂ) * v (j-1))
(L : Submodule ℂ Fiber) (hclosed : IsClosed (L : Set Fiber))
(hinvariant : ∀ v ∈ L, T v ∈ L) :
L = ⊥ ∨ L = ⊤ ∨ ∃ h : ℤ, L = coordinateTail h := by
by_cases hbot : L = ⊥
· exact Or.inl hbot
by_cases htop : L = ⊤
· exact Or.inr (Or.inl htop)
have hOO : L.orthogonal.orthogonal = L := by
rw [Submodule.orthogonal_orthogonal_eq_closure, hclosed.submodule_topologicalClosure_eq]
have hnotOrth : L.orthogonal ≠ ⊥ := by
intro hh
rw [hh, Submodule.bot_orthogonal_eq_top] at hOO
exact htop hOO.symm
obtain ⟨z, hz, hzne⟩ := (Submodule.ne_bot_iff L.orthogonal).mp hnotOrth
obtain ⟨i, hzi⟩ := fiber_exists_coordinate hzne
obtain ⟨v, hv, hvne⟩ := (Submodule.ne_bot_iff L).mp hbot
obtain ⟨j, hvj⟩ := fiber_exists_coordinate hvne
let P := fun k : ℤ => ∃ c : Fiber, c ∈ L ∧ c k ≠ 0
have hPnonempty : ∃ k, P k := ⟨j, v, hv, hvj⟩
have hPbound : ∃ a : ℤ, ∀ k : ℤ, P k → a ≤ k := by
refine ⟨i+1, ?_⟩
rintro k ⟨c, hc, hck⟩
have hh := fiber_orthogonal_order hw hT hinvariant hc hz hzi hck
omega
obtain ⟨h, hhP, hhmin⟩ := Int.exists_least_of_bdd hPbound hPnonempty
obtain ⟨v₀, hv₀, hv₀h⟩ := hhP
refine Or.inr (Or.inr ⟨h, le_antisymm ?_ ?_⟩)
· intro c hc k hk
by_contra hck
have hh := hhmin k ⟨c, hc, hck⟩
omega
· intro c hc
rw [← hOO]
apply (Submodule.mem_orthogonal _ c).mpr
intro u hu
rw [lp.inner_eq_tsum]
apply Eq.trans (tsum_congr (g := fun _ : ℤ => (0:ℂ)) ?_) tsum_zero
intro k
rw [RCLike.inner_apply]
by_cases hkh : k < h
· rw [hc k hkh, zero_mul]
· have huk : u k = 0 := by
by_contra hne
have hh := fiber_orthogonal_order hw hT hinvariant hv₀ hu hne hv₀h
exact hkh hh
simp [huk]
end
end BackwardIntertwiners
end OAI
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