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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 | import Mathlib
import OAI.Geometry.Borsuk.Support.PurityChart
namespace OAI
namespace BorsukNine
open scoped RealInnerProductSpace MatrixOrder Matrix.Norms.L2Operator
section BalancedChart
variable {ν : Type*} [Fintype ν] [DecidableEq ν] [Nonempty ν]
local notation "E" => EuclideanSpace ℝ ν
local notation "M" => Matrix ν ν ℝ
local notation "op" => Matrix.toEuclideanCLM (n := ν) (𝕜 := ℝ)
lemma split_unit_of_fibre_unit {u : E} (hu : ‖u‖ = 1) {Q : M} (hQ : 0 ≤ Q)
(hQu : op Q u = 0) {lam mu : ℝ} (hl : 0 < lam) (hm : 0 < mu)
(hunit : IsUnit (rankOne u + Q)) : IsUnit (lam • rankOne u - mu • Q) := by
apply (isUnit_map_iff (op) _).mp
apply ContinuousLinearMap.isUnit_iff_bijective.mpr
have hi : Function.Injective (op (lam • rankOne u - mu • Q)) := by
apply (LinearMap.ker_eq_bot).mp
rw [Submodule.eq_bot_iff]
intro v hv
have hv0 : op (lam • rankOne u - mu • Q) v = 0 := hv
have hi := congrArg (fun b : E => ⟪u,b⟫) hv0
have hQv : ⟪u,op Q v⟫ = 0 := matrix_image_orthogonal_kernel hQ.isSelfAdjoint hQu v
simp only [map_sub,map_smul,sub_apply,smul_apply,
rankOne_action,inner_sub_right,inner_smul_right,real_inner_self_eq_norm_sq,hu,one_pow,
mul_one,hQv,mul_zero,sub_zero,inner_zero_right] at hi
have huv : ⟪u,v⟫ = 0 := (mul_eq_zero.mp hi).resolve_left (ne_of_gt hl)
have hzero : op Q v = 0 := by
simp only [map_sub,map_smul,sub_apply,smul_apply,
rankOne_action,huv,zero_smul,smul_zero,zero_sub,neg_eq_zero] at hv0
exact (smul_eq_zero.mp hv0).resolve_left (ne_of_gt hm)
have huop := ContinuousLinearMap.isUnit_iff_bijective.mp (hunit.map (op).toMonoidHom)
apply huop.1
change op (rankOne u + Q) v = op (rankOne u + Q) 0
simp only [map_add,add_apply,rankOne_action,huv,zero_smul,hzero,add_zero,map_zero]
exact ⟨hi,(LinearMap.injective_iff_surjective).mp hi⟩
omit [Nonempty ν] in
lemma lineRepresentative_split {u w : E} (hu : ‖u‖ = 1) {A : M} {lam : ℝ}
(hl : 0 < lam) (he : A⁺ = lam • rankOne u) (hwu : 0 < ⟪w,u⟫) :
lineRepresentative w A = u := by
have hp : 0 < lam * ⟪u,w⟫ := mul_pos hl (by simpa only [real_inner_comm] using hwu)
have hop : op A⁺ w = (lam * ⟪u,w⟫) • u := by
rw [he,map_smul,smul_apply,rankOne_action,smul_smul]
rw [lineRepresentative,hop,norm_smul,Real.norm_eq_abs,abs_of_pos hp,hu,mul_one,smul_smul,inv_mul_cancel₀ (ne_of_gt hp),one_smul]
lemma balancedFibre_chart {ι : Type*} [Fintype ι] {f : E → ι → ℝ} {I : Finset ι}
{w c u : E} (hu : ‖u‖ = 1) (hwu : 0 < ⟪w,u⟫)
(huc : u ∈ commonPositiveCone f I)
(hch : u ∈ interior (signChamber (activeVectors f I) c)) {Q : M}
(hQ : Q ∈ matrixFibre u) (hunit : IsUnit (rankOne u + Q)) :
((1/2:ℝ) • rankOne u - (1/2:ℝ) • Q) ∈ purityChart f I w c := by
have hp : (0:ℝ) < 1/2 := by norm_num
obtain ⟨he,hn⟩ := rankOne_split_parts hQ.1 hQ.2.2 hp.le hp.le
have hline : HasPositiveLine ((1/2:ℝ) • rankOne u - (1/2:ℝ) • Q) := ⟨u,hu,1/2,hp,he⟩
have hrep := lineRepresentative_split hu hp he hwu
refine ⟨split_unit_of_fibre_unit hu hQ.1 hQ.2.2 hp hp hunit,hline,
unit_rankOne_inner_nonzero hline hu hp he (ne_of_gt hwu),?_,?_,?_⟩
· rw [hn,Matrix.trace_smul,hQ.2.1,smul_eq_mul,mul_one]; exact hp
· rwa [hrep]
· rwa [hrep]
end BalancedChart
end BorsukNine
end OAI
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