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8e4d533 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 | import OAI.Geometry.BundleTests.Multiplicities
import OAI.Geometry.ComplexBundle.ProjectionBundle
import Mathlib.Topology.VectorBundle.Constructions
namespace OAI
/-! The compact bundle-test base, prefix dimensions, and retained multiplicities. -/
noncomputable section
open scoped BigOperators ComplexOrder Matrix.Norms.L2Operator
open Matrix Bundle Topology
namespace Kaplansky
def testProjectiveDimension (m R : ℕ) (i : Fin R) : ℕ :=
320 * m^2 * 4^(2*i.val) * (4^(R-i.val)-1)
theorem testProjectiveDimension_pos {m R : ℕ} (hm : 0 < m) (i : Fin R) :
0 < testProjectiveDimension m R i := by
have hn : R-i.val ≠ 0 := by omega
obtain ⟨k,hk⟩ := Nat.exists_eq_succ_of_ne_zero hn
have hp : 0 < 4^k := pow_pos (by omega) _
have hh : 0 < 4^(R-i.val)-1 := by rw [hk, pow_succ]; omega
unfold testProjectiveDimension
positivity
/-- A concrete basepoint in every projective factor. -/
def coordinateLineProjection (k : ℕ) : projectionSpace (k+1) 1 := by
let f : Fin (k+1) → ℂ := fun j => if j = 0 then 1 else 0
refine ⟨Matrix.diagonal f, ?_, ?_, ?_⟩
· apply Matrix.isHermitian_diagonal_of_self_adjoint
funext j
simp [f]
· rw [Matrix.diagonal_mul_diagonal]
congr 1
funext j
simp [f]
· simp [Matrix.trace_diagonal, f]
abbrev BundleTestBase (m R : ℕ) := ∀ i : Fin R,
projectionSpace (testProjectiveDimension m R i + 1) 1
instance (m R : ℕ) : CompactSpace (BundleTestBase m R) := inferInstance
def bundleTestBasePoint (m R : ℕ) : BundleTestBase m R :=
fun i => coordinateLineProjection (testProjectiveDimension m R i)
instance (m R : ℕ) : Nonempty (BundleTestBase m R) := ⟨bundleTestBasePoint m R⟩
def bundleTestProjection (m R : ℕ) (i : Fin R) :
C(BundleTestBase m R, projectionSpace (testProjectiveDimension m R i + 1) 1) :=
⟨fun y => y i, continuous_apply i⟩
abbrev TestTautologicalLine (m R : ℕ) (i : Fin R) :=
(bundleTestProjection m R i) *ᵖ (projectionLineCore (testProjectiveDimension m R i + 1)).Fiber
instance (m R : ℕ) (i : Fin R) : FiberBundle ℂ (TestTautologicalLine m R i) := inferInstance
instance (m R : ℕ) (i : Fin R) : VectorBundle ℂ ℂ (TestTautologicalLine m R i) := inferInstance
theorem testTautologicalLine_finrank (m R : ℕ) (i : Fin R) (y : BundleTestBase m R) :
Module.finrank ℂ (TestTautologicalLine m R i y) = 1 := by
change Module.finrank ℂ ℂ = 1
exact Module.finrank_self ℂ
/-- The actual dual hyperplane-line candidate on the projection product. -/
abbrev TestHyperplaneLine (m R : ℕ) (i : Fin R) :=
(bundleTestProjection m R i) *ᵖ (projectionDualLineCore (testProjectiveDimension m R i + 1)).Fiber
instance (m R : ℕ) (i : Fin R) : FiberBundle ℂ (TestHyperplaneLine m R i) := inferInstance
instance (m R : ℕ) (i : Fin R) : VectorBundle ℂ ℂ (TestHyperplaneLine m R i) := inferInstance
theorem testHyperplaneLine_finrank (m R : ℕ) (i : Fin R) (y : BundleTestBase m R) :
Module.finrank ℂ (TestHyperplaneLine m R i y) = 1 := by
change Module.finrank ℂ ℂ = 1
exact Module.finrank_self ℂ
end Kaplansky
end
namespace Kaplansky
open scoped BigOperators
theorem three_sum_pow_four (n : ℕ) : 3*(∑ i : Fin n,4^i.val)+1=4^n := by
induction n with
| zero => simp
| succ n ih =>
rw [Fin.sum_univ_castSucc,pow_succ]
simp only [Fin.val_castSucc,Fin.val_last]
omega
theorem testProjectiveDimension_bound (m R : ℕ) (i : Fin R) :
testProjectiveDimension m R i ≤ 5*m*testRank m R 0*4^i.val := by
have h := Nat.mul_le_mul_left (320*m^2*4^(2*i.val)) (Nat.sub_le (4^(R-i.val)) 1)
change testProjectiveDimension m R i ≤ _ at h
have he : 2*i.val+(R-i.val)=R+i.val := by omega
calc
_ ≤ 320*m^2*4^(2*i.val)*4^(R-i.val) := h
_ = 320*m^2*4^(R+i.val) := by rw [mul_assoc,← pow_add,he]
_ = _ := by rw [testRank_eq]; simp only [Nat.add_zero,pow_add]; ring
/-- Exact prefix dimensions, retaining every later occurrence through R. -/
def testPrefixDimension (m R j : ℕ) (hj : j<R) : ℕ :=
∑ i : Fin (j+1), testProjectiveDimension m R (i.castLE hj)
theorem testPrefixDimension_margin {m R j : ℕ} (hm : 0<m) (hj : j<R) :
3*testPrefixDimension m R j hj < 20*m*testRank m R j := by
have h : testPrefixDimension m R j hj ≤
5*m*testRank m R 0*(∑ i : Fin (j+1),4^i.val) := by
unfold testPrefixDimension
rw [Finset.mul_sum]
exact Finset.sum_le_sum (fun i _ => testProjectiveDimension_bound m R (i.castLE hj))
have hg := three_sum_pow_four (j+1)
have hd : 0<testRank m R 0 := by unfold testRank; positivity
have he : testRank m R j = testRank m R 0*4^j := by
simp only [testRank,pow_zero,one_mul]
ring
rw [pow_succ] at hg
rw [he]
nlinarith [Nat.mul_le_mul_left 3 h,
congrArg (fun n : ℕ => 5*m*testRank m R 0*n) hg]
/-- Count all copies of L_(i+1) in m copies of E_0⊕...⊕E_R.
The stages are enumerated in descending order. -/
def retainedHyperplaneMultiplicity (m R : ℕ) (i : Fin R) : ℕ :=
m * ∑ t : Fin (R-i.val), testMultiplicity m R (R-t.val) ⟨i.val+1,by omega⟩
theorem retainedHyperplaneMultiplicity_eq (m R : ℕ) (i : Fin R) :
retainedHyperplaneMultiplicity m R i = testProjectiveDimension m R i := by
have ht : ∀ t : Fin (R-i.val),
testMultiplicity m R (R-t.val) ⟨i.val+1,by omega⟩ =
960*m*4^(2*i.val)*4^t.val := by
intro t
unfold testMultiplicity
rw [ite_eq_right (by simp)]
have he : R+2*(i.val+1)-(R-t.val)-2=2*i.val+t.val := by omega
dsimp only
rw [he,pow_add]
ring
unfold retainedHyperplaneMultiplicity
simp_rw [ht]
rw [← Finset.mul_sum]
have hg : 3*(∑ t : Fin (R-i.val),4^t.val)=4^(R-i.val)-1 := by
have h := three_sum_pow_four (R-i.val)
omega
unfold testProjectiveDimension
rw [← hg]
ring
/-- The exact rank of the retained trivial block, in descending stage order. -/
def retainedTrivialMultiplicity (m R : ℕ) : ℕ :=
m * ∑ t : Fin (R+1), testMultiplicity m R (R-t.val) 0
theorem retainedTrivialMultiplicity_eq (m R : ℕ) :
retainedTrivialMultiplicity m R = 64*m^2*(∑ t : Fin (R+1),4^t.val) := by
have ht : ∀ t : Fin (R+1), testMultiplicity m R (R-t.val) 0=64*m*4^t.val := by
intro t
change 64*m*4^(R-(R-t.val))=64*m*4^t.val
have he : R-(R-t.val)=t.val := by omega
rw [he]
unfold retainedTrivialMultiplicity
simp_rw [ht]
rw [← Finset.mul_sum]
ring
theorem retainedTrivialMultiplicity_bound {m R : ℕ} (hm : 0<m) :
retainedTrivialMultiplicity m R < 2*m*testRank m R 0 := by
rw [retainedTrivialMultiplicity_eq,testRank_eq]
have h := three_sum_pow_four (R+1)
rw [pow_succ] at h
have hp : 0<64*m^2 := by positivity
have hsum : (∑ t : Fin (R+1),4^t.val)<2*4^R := by omega
have hh := Nat.mul_lt_mul_of_pos_left hsum hp
simpa only [Nat.add_zero] using (show 64*m^2*(∑ t : Fin (R+1),4^t.val)<2*m*(64*m*4^R) from by nlinarith [hh])
end Kaplansky
end OAI
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