| 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 | |