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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 | import Mathlib
namespace OAI
section
section
open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal
open Filter Set
open scoped Topology NNReal
open Set Filter MeasureTheory TopologicalSpace
open scoped Topology ENNReal
open MeasureTheory Filter Set Metric
open scoped Topology Pointwise NNReal
open Set MeasureTheory
open scoped RealInnerProductSpace
open Matrix
open scoped RealInnerProductSpace MatrixOrder
namespace CAT0Fillings
open Matrix
open scoped BigOperators
variable {ι : Type*} [Fintype ι] [DecidableEq ι]
lemma det_updateRow_identity (j : ι) (v : ι → ℝ) :
(Matrix.updateRow (1 : Matrix ι ι ℝ) j v).det = v j := by
have he : (∑ i : ι, v i • (1 : Matrix ι ι ℝ) i) = v := by
ext l
simp [Matrix.one_apply]
rw [←he,Matrix.det_updateRow_sum]
simp [Matrix.one_apply]
theorem hasDerivAt_det_at_identity {A : ℝ → Matrix ι ι ℝ}
{B : Matrix ι ι ℝ} {t : ℝ} (hA : A t = 1) (hd : HasDerivAt A B t) :
HasDerivAt (fun s => (A s).det) B.trace t := by
let D : ContinuousMultilinearMap ℝ (fun _ : ι => ι → ℝ) ℝ :=
⟨Matrix.detRowAlternating.toMultilinearMap,continuous_id.matrix_det⟩
have hD := (D.hasFDerivAt (A t)).comp_hasDerivAt t hd
change HasDerivAt (fun s => (A s).det) (D.linearDeriv (A t) B) t at hD
have hv : D.linearDeriv (fun i j => A t i j) (fun i j => B i j) = B.trace := by
rw [ContinuousMultilinearMap.linearDeriv_apply]
change (∑ i, (Matrix.updateRow (A t) i (B i)).det) = B.trace
rw [hA]
simp only [det_updateRow_identity,Matrix.trace,Matrix.diag]
exact hD.congr_deriv hv
def spatialRadialForm (g r t : ℝ) (α γ : ι → ℝ) : Matrix ι ι ℝ :=
Matrix.of fun i j => (1-t*g)^2 * (if i=j then 1 else 0) -
(1-t*g)*t*r*(α i*γ j+γ i*α j) + t^2*r^2*γ i*γ j
omit [Fintype ι] in
lemma spatialRadialForm_zero (g r : ℝ) (α γ : ι → ℝ) :
spatialRadialForm g r 0 α γ = 1 := by
ext i j
simp [spatialRadialForm,Matrix.one_apply]
lemma spatialRadialForm_hasDerivAt_zero (g r : ℝ) (α γ : ι → ℝ) :
HasDerivAt (fun t => spatialRadialForm g r t α γ)
(Matrix.of fun i j => -2*g*(if i=j then 1 else 0)-r*(α i*γ j+γ i*α j)) 0 := by
apply hasDerivAt_pi.2
intro i
apply hasDerivAt_pi.2
intro j
have hl := (hasDerivAt_const (0:ℝ) 1).sub ((hasDerivAt_id (0:ℝ)).mul_const g)
have he := (((hl.fun_pow 2).mul_const (if i=j then 1 else 0)).sub
(((hl.mul (hasDerivAt_id (0:ℝ))).mul_const r).mul_const (α i*γ j+γ i*α j))).add
((((hasDerivAt_id (0:ℝ)).fun_pow 2).mul_const (r^2)).mul_const (γ i*γ j))
convert he using 1
· ext t
simp only [spatialRadialForm,Matrix.of_apply,Pi.sub_apply,Pi.add_apply,Pi.mul_apply,id_eq]
ring
· simp
theorem spatialRadialForm_sqrt_det_hasDerivAt_zero (g r : ℝ) (α γ : ι → ℝ) :
HasDerivAt (fun t => Real.sqrt (spatialRadialForm g r t α γ).det)
(-((Fintype.card ι : ℝ)*g+r*∑ i, α i*γ i)) 0 := by
have hd := hasDerivAt_det_at_identity (spatialRadialForm_zero g r α γ)
(spatialRadialForm_hasDerivAt_zero g r α γ)
have hs := hd.sqrt (by simp [spatialRadialForm_zero])
convert hs using 1
simp only [spatialRadialForm_zero,Matrix.det_one,Real.sqrt_one,Matrix.trace,
Matrix.diag,Matrix.of_apply,ite_true, mul_one, Finset.sum_sub_distrib,
Finset.sum_const, nsmul_eq_mul, Finset.card_univ]
rw [←Finset.mul_sum]
have hh : (∑ i, (α i*γ i+γ i*α i)) = 2*(∑ i, α i*γ i) := by
rw [Finset.sum_add_distrib]
simp only [mul_comm (γ _) (α _)]
ring
rw [hh]
ring
end CAT0Fillings
open Matrix
open scoped RealInnerProductSpace MatrixOrder
end
end
end OAI
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