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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 | import Mathlib
import OAI.Geometry.Borsuk.SphereGeometry.SphereExtension
import OAI.Geometry.Borsuk.MatrixGeometry.SymmetricCoordinates
namespace OAI
namespace BorsukNine
open scoped RealInnerProductSpace ContDiff
open scoped Manifold
open MeasureTheory Set Metric
open scoped ENNReal Pointwise Matrix
section OddSphereMap
open scoped MatrixOrder Matrix.Norms.L2Operator
variable {ν ι : Type*} [Fintype ν] [LinearOrder ν] [Nonempty ν] [Fintype ι]
local notation "E" => EuclideanSpace ℝ ν
noncomputable def sphereExtension (f : E → ι → ℝ)
(x : EuclideanSpace ℝ (UpperTriangle ν)) : EuclideanSpace ℝ ι :=
normalizedSignedExtension f (symmetricCoordinates x)
lemma Admissible.sphereExtension_norm {f : E → ι → ℝ} (hf : Admissible f)
{x : EuclideanSpace ℝ (UpperTriangle ν)} (hx : ‖x‖ = 1) : ‖sphereExtension f x‖ = 1 := by
apply hf.normalizedSignedExtension_norm (symmetricCoordinates_selfAdjoint x)
intro hh
have hn : x = 0 := symmetricCoordinates_injective (hh.trans (map_zero _).symm)
simp [hn] at hx
omit [Nonempty ν] in
lemma sphereExtension_odd (f : E → ι → ℝ) (x : EuclideanSpace ℝ (UpperTriangle ν)) :
sphereExtension f (-x) = -sphereExtension f x := by
simp only [sphereExtension, map_neg, normalizedSignedExtension_odd]
lemma Admissible.sphereExtension_continuousOn {f : E → ι → ℝ} (hf : Admissible f) :
ContinuousOn (sphereExtension f) {x : EuclideanSpace ℝ (UpperTriangle ν) | ‖x‖ = 1} := by
apply hf.normalizedSignedExtension_continuousOn.comp
(symmetricCoordinates.continuous_of_finiteDimensional.continuousOn) (fun x hx => ?_)
refine ⟨symmetricCoordinates_selfAdjoint x, ?_⟩
intro hh
have hn : x = 0 := symmetricCoordinates_injective (hh.trans (map_zero _).symm)
simp [hn] at hx
end OddSphereMap
end BorsukNine
namespace BorsukNine.SmithTheory
variable {K P Q : Type*} [Field K] [AddCommGroup P] [Module K P]
[AddCommGroup Q] [Module K Q]
lemma exists_factor_of_ker_le (N : P →ₗ[K] P) (φ : P →ₗ[K] Q)
(h : LinearMap.ker N ≤ LinearMap.ker φ) : ∃ ψ : P →ₗ[K] Q, ψ.comp N = φ := by
let ψ₀ : LinearMap.range N →ₗ[K] Q :=
(N.ker.liftQ φ h).comp N.quotKerEquivRange.symm.toLinearMap
obtain ⟨ψ, hψ⟩ := ψ₀.exists_extend
refine ⟨ψ, ?_⟩
ext x
have he := LinearMap.congr_fun hψ ⟨N x, LinearMap.mem_range_self N x⟩
simpa only [LinearMap.comp_apply, Submodule.subtype_apply, ψ₀,
LinearEquiv.coe_toLinearMap, LinearMap.quotKerEquivRange_symm_apply_image,
Submodule.mkQ_apply, Submodule.liftQ_apply] using he
lemma exists_factor_of_vanish_norm (N : P →ₗ[K] P)
(hexact : LinearMap.ker N ≤ LinearMap.range N) (φ : P →ₗ[K] Q)
(hφ : φ.comp N = 0) : ∃ ψ : P →ₗ[K] Q, ψ.comp N = φ := by
apply exists_factor_of_ker_le N φ
intro x hx
obtain ⟨y, rfl⟩ := hexact hx
exact LinearMap.congr_fun hφ y
lemma tower_augmentation_zero (d N : P →ₗ[K] P)
(hd : d.comp d = 0) (hcomm : d.comp N = N.comp d)
(hexact : LinearMap.ker N ≤ LinearMap.range N)
(n : ℕ) (φ : P →ₗ[K] Q) (hφN : φ.comp N = 0) (hφd : φ.comp d = 0)
(c : ℕ → P) (htower : ∀ i < n, d (c (i + 1)) = N (c i))
(htop : N (c n) = 0) : φ (c 0) = 0 := by
induction n generalizing φ c with
| zero =>
obtain ⟨ψ, hψ⟩ := exists_factor_of_vanish_norm N hexact φ hφN
rw [← hψ, LinearMap.comp_apply, htop, map_zero]
| succ n ih =>
obtain ⟨ψ, hψ⟩ := exists_factor_of_vanish_norm N hexact φ hφN
have hN : (ψ.comp d).comp N = 0 := by
rw [LinearMap.comp_assoc, hcomm, ← LinearMap.comp_assoc, hψ, hφd]
have hd' : (ψ.comp d).comp d = 0 := by
rw [LinearMap.comp_assoc, hd, LinearMap.comp_zero]
have hh := ih (ψ.comp d) hN hd' (fun i => c (i + 1))
(fun i hi => htower (i + 1) (Nat.add_lt_add_right hi 1)) htop
rw [← hψ, LinearMap.comp_apply, ← htower 0 (Nat.zero_lt_succ n)]
exact hh
end BorsukNine.SmithTheory
end OAI
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