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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 | import Mathlib
import OAI.Geometry.Borsuk.MatrixGeometry.MatrixFibre
import OAI.Geometry.Borsuk.Projectors.NegativeQuadratic
namespace OAI
open Set MeasureTheory
namespace BorsukNine
open scoped MatrixOrder RealInnerProductSpace Matrix.Norms.L2Operator
section BlockParts
variable {μ ν ι : Type*} [Fintype μ] [Fintype ν] [DecidableEq μ] [DecidableEq ν]
[Nonempty ν] [Fintype ι]
local notation "W" => EuclideanSpace ℝ μ
local notation "E" => EuclideanSpace ℝ ν
local notation "M" => Matrix ν ν ℝ
local notation "op" => Matrix.toEuclideanCLM (𝕜 := ℝ)
omit [Nonempty ν] in
lemma matrixCompress_rankOne (e : W →ₗᵢ[ℝ] E) {c : E} (he : ∀ v, ⟪c,e v⟫ = 0) :
matrixCompress e (rankOne c) = 0 := by
apply (Matrix.toEuclideanCLM (n := μ) (𝕜 := ℝ)).injective
apply ContinuousLinearMap.ext
intro v
change op (matrixCompress e (rankOne c)) v = op (0 : Matrix μ μ ℝ) v
rw [matrixCompress_action,rankOne_action,he,zero_smul,map_zero,map_zero,
zero_apply]
omit [Nonempty ν] in
lemma blockCorner_rankOne (e : W →ₗᵢ[ℝ] E) {c : E} (he : ∀ v, ⟪c,e v⟫ = 0) :
blockCorner e (rankOne c) = 0 := by
rw [blockCorner,matrixCompress_rankOne e he,map_zero]
omit [Nonempty ν] in
lemma block_decomposition {c : E} (e : W →ₗᵢ[ℝ] E) (hc : ‖c‖ = 1)
(he : ∀ v, ⟪c,e v⟫ = 0) (hes : ∀ u, ⟪c,u⟫ = 0 → ∃ v, e v = u)
{A : M} (hA : IsSelfAdjoint A) (hb : blockOff c e A = 0) :
A = blockScalar c A • rankOne c + blockCorner e A := by
have hAc : op A c = blockScalar c A • c := by
have hd := frame_decomposition c e hc he hes (op A c)
change e (blockOff c e A) = op A c - blockScalar c A • c at hd
rw [hb,map_zero] at hd
exact sub_eq_zero.mp hd.symm
have hP := (rankOne_nonneg c).isSelfAdjoint
have hB : IsSelfAdjoint (A - blockScalar c A • rankOne c) := hA.sub ((IsSelfAdjoint.all _).smul hP)
have hk : op (A - blockScalar c A • rankOne c) c = 0 := by
rw [map_sub,sub_apply,map_smul,smul_apply,
rankOne_action,real_inner_self_eq_norm_sq,hc,one_pow,one_smul,hAc,sub_self]
have hh := blockCorner_of_kernel e hes hB hk
have hC : blockCorner e (A - blockScalar c A • rankOne c) = blockCorner e A := by
simp only [blockCorner,map_sub,map_smul,matrixCompress_rankOne e he,smul_zero,sub_zero]
rw [hC] at hh
rw [hh]
abel
lemma Admissible.signedExtension_nonpos {f : E → ι → ℝ} (hf : Admissible f)
{A : M} (hA : IsSelfAdjoint A) (hs : ∀ i, signedExtension f A i ≤ 0) : A ≤ 0 := by
have hp (i : ι) : positiveExtension f A⁺ i = 0 := by
apply le_antisymm _ (hf.positiveExtension_nonneg A⁺ i)
apply le_of_not_gt
intro h
have hd := hf.positiveExtension_orthogonal (CFC.posPart_nonneg A)
(CFC.negPart_nonneg A) (CFC.posPart_mul_negPart A) i
have hn := (mul_eq_zero.mp hd).resolve_left (ne_of_gt h)
have hh := hs i
simp only [signedExtension,Pi.sub_apply,hn,sub_zero] at hh
exact (not_lt_of_ge hh) h
have hz : A⁺ = 0 := matrix_psd_trace_zero (CFC.posPart_nonneg A) (by
rw [← hf.positiveExtension_sum (CFC.posPart_nonneg A)]
simp [hp])
have he := CFC.posPart_sub_negPart A hA
rw [hz,zero_sub] at he
rw [← he]
exact neg_nonpos.mpr (CFC.negPart_nonneg A)
omit [Nonempty ν] in
lemma block_parts {c : E} (e : W →ₗᵢ[ℝ] E) (hc : ‖c‖ = 1)
(he : ∀ v, ⟪c,e v⟫ = 0) (hes : ∀ u, ⟪c,u⟫ = 0 → ∃ v, e v = u)
{A : M} (hA : IsSelfAdjoint A) (hb : blockOff c e A = 0)
(ha : 0 ≤ blockScalar c A) (hC : blockCorner e A ≤ 0) :
A⁺ = blockScalar c A • rankOne c ∧ A⁻ = -blockCorner e A := by
have hk : op (-blockCorner e A) c = 0 := by
rw [map_neg,neg_apply,blockCorner_kernel e he,neg_zero]
have hp := rankOne_split_parts (neg_nonneg.mpr hC) hk ha (show (0:ℝ) ≤ 1 by norm_num)
have heq : A = blockScalar c A • rankOne c - (1:ℝ) • (-blockCorner e A) := by
rw [one_smul,sub_neg_eq_add]
exact block_decomposition e hc he hes hA hb
rw [← heq] at hp
simpa only [one_smul] using hp
lemma Admissible.block_extension {f : E → ι → ℝ} (hf : Admissible f)
{c : E} (e : W →ₗᵢ[ℝ] E) (hc : ‖c‖ = 1)
(he : ∀ v, ⟪c,e v⟫ = 0) (hes : ∀ u, ⟪c,u⟫ = 0 → ∃ v, e v = u)
{A : M} (hA : IsSelfAdjoint A) (hb : blockOff c e A = 0)
(ha : 0 ≤ blockScalar c A) (hC : blockCorner e A ≤ 0) :
signedExtension f A = blockScalar c A • f c + signedExtension f (blockCorner e A) := by
obtain ⟨hp,hn⟩ := block_parts e hc he hes hA hb ha hC
have hCp : (blockCorner e A)⁺ = 0 := (CFC.posPart_eq_zero_iff _ (blockCorner_selfAdjoint e hA)).mpr hC
have hCn : (blockCorner e A)⁻ = -blockCorner e A := by
have h := CFC.posPart_sub_negPart (blockCorner e A) (blockCorner_selfAdjoint e hA)
rw [hCp,zero_sub] at h
exact (neg_eq_iff_eq_neg.mp h)
ext i
simp only [signedExtension,Pi.sub_apply,hp,hn,hCp,hCn,positiveExtension_smul_nonneg f (rankOne_nonneg c) ha,
hf.positiveExtension_rankOne c hc,Pi.add_apply,Pi.smul_apply,smul_eq_mul]
have hz : positiveExtension f (0:M) = 0 := by
simpa only [zero_smul] using positiveExtension_smul_nonneg f (Q := (0:M)) le_rfl (t := 0) le_rfl
rw [hz]
simp only [Pi.zero_apply,sub_eq_add_neg,zero_add]
end BlockParts
end BorsukNine
end OAI
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