Plain-English statement
- Exact source nodes and coefficient integrals, with cardinal/constant interpolation and the sharp1/2 integrated row bound. The source logarithmic Lebesgue estimate, momentum-weight positivity and B2 error stay independent.
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Lean statement
theorem chebyshev_lobatto_coefficients {J : ℕ} (hJ : 2 ≤ J) {h : ℝ} (hh : 0 < h) :
let t := fun i : Fin J => h/2*(1-Real.cos ((i : ℝ)/(J-1 : ℝ)*Real.pi))
let ell := fun j : Fin J => Lagrange.basis Finset.univ t j
let Lambda := sSup ((fun s : ℝ => ∑ j : Fin J, |(ell j).eval s|) '' Icc 0 h)
let omega := fun i j : Fin J => ∫ s in 0..t i, (t i-s)*(ell j).eval s
let momentumWeight := fun j : Fin J => ∫ s in 0..h, (ell j).eval s
(∀ i, t i ∈ Icc 0 h) ∧ Function.Injective t ∧
t ⟨J-1,by omega⟩ = h ∧
(∀ i j, (ell i).eval (t j) = if i=j then 1 else 0) ∧
(∀ s : ℝ, ∑ j : Fin J, (ell j).eval s = 1) ∧
1 ≤ Lambda ∧
(∀ i, (∑ j : Fin J, |omega i j|) ≤ (t i)^2/2*Lambda ∧
(∑ j : Fin J, |omega i j|) ≤ h^2/2*Lambda) ∧
(∑ j : Fin J, momentumWeight j) = h := by
classical
dsimp only
let t := nodes J h
let ell := fun j : Fin J => Lagrange.basis Finset.univ t j
let F := fun s : ℝ => ∑ j : Fin J, |(ell j).eval s|
let Lambda := sSup (F '' Icc 0 h)
have ht (i : Fin J) : t i ∈ Icc 0 h := nodes_mem hh.le i
have hinj : Function.Injective t := nodes_injective hJ hh
have hcard (i j : Fin J) : (ell i).eval (t j) = if i=j then 1 else 0 := by
by_cases he : i=j
· subst j
simpa [ell] using Lagrange.eval_basis_self (s := Finset.univ) hinj.injOn
(Finset.mem_univ i)
· simpa [ell,he] using Lagrange.eval_basis_of_ne (s := Finset.univ) (v := t) he
(Finset.mem_univ j)
have hpart (s : ℝ) : ∑ j : Fin J, (ell j).eval s = 1 := by
have he : (∑ j : Fin J, ell j) = 1 :=
Lagrange.sum_basis hinj.injOn ⟨⟨0,by omega⟩,Finset.mem_univ _⟩
simpa only [Polynomial.eval_finsetSum, Polynomial.eval_one] using
congrArg (fun p : ℝ[X] => p.eval s) he
have hF : Continuous F := by
apply continuous_finsetSum
intro j _
exact (ell j).continuous.abs
have hbound : BddAbove (F '' Icc 0 h) := isCompact_Icc.bddAbove_image hF.continuousOn
have hLeb (s : ℝ) (hs : s ∈ Icc 0 h) : F s ≤ Lambda :=
le_csSup hbound (mem_image_of_mem F hs)
have hLambda : 1 ≤ Lambda := by
let i0 : Fin J := ⟨0,by omega⟩
have hval : F (t i0) = 1 := by
simp only [F,hcard]
simp_rw [apply_ite abs,abs_one,abs_zero]
simp
exact hval ▸ hLeb (t i0) (ht i0)
have hw (i : Fin J) : (∑ j : Fin J, |∫ s in 0..t i, (t i-s)*(ell j).eval s|) ≤
(t i)^2/2*Lambda := by
have hf (j : Fin J) : Continuous (fun s : ℝ => (t i-s)*(ell j).eval s) :=
(continuous_const.sub continuous_id).mul (ell j).continuous
have hfa (j : Fin J) : Continuous (fun s : ℝ => |(t i-s)*(ell j).eval s|) := (hf j).abs
calc
_ ≤ ∑ j : Fin J, ∫ s in 0..t i, |(t i-s)*(ell j).eval s| := by
apply Finset.sum_le_sum
intro j _
simpa only [Real.norm_eq_abs] using
intervalIntegral.norm_integral_le_integral_norm (f := fun s => (t i-s)*(ell j).eval s)
(ht i).1
_ = ∫ s in 0..t i, ∑ j : Fin J, |(t i-s)*(ell j).eval s| :=
(intervalIntegral.integral_finsetSum (fun j _ => (hfa j).intervalIntegrable _ _)).symm
_ ≤ ∫ s in 0..t i, (t i-s)*Lambda := by
apply intervalIntegral.integral_mono_on (ht i).1
((continuous_finsetSum Finset.univ (fun j _ => hfa j)).intervalIntegrable _ _)
(((continuous_const.sub continuous_id).mul continuous_const).intervalIntegrable _ _)
intro s hs
have hnon : 0 ≤ t i-s := sub_nonneg.mpr hs.2
have hs' : s ∈ Icc 0 h := ⟨hs.1,hs.2.trans (ht i).2⟩
calc
_ = (t i-s)*F s := by
simp only [abs_mul,abs_of_nonneg hnon,F,Finset.mul_sum]
_ ≤ _ := mul_le_mul_of_nonneg_left (hLeb s hs') hnon
_ = (t i)^2/2*Lambda := by
rw [intervalIntegral.integral_mul_const,
intervalIntegral.integral_sub (f := fun _ : ℝ => t i) (g := fun s : ℝ => s)
(continuous_const.intervalIntegrable _ _)
(continuous_id.intervalIntegrable _ _),intervalIntegral.integral_const,
integral_id]
simp only [sub_zero]
ring
refine ⟨ht,hinj,nodes_last hJ h,hcard,hpart,hLambda,?_,?_⟩
· intro i
refine ⟨hw i,(hw i).trans ?_⟩
have hsq : (t i)^2 ≤ h^2 := sq_le_sq₀ (ht i).1 hh.le |>.mpr (ht i).2
exact mul_le_mul_of_nonneg_right (by linarith : (t i)^2/2 ≤ h^2/2)
(by linarith : 0 ≤ Lambda)
· change (∑ j : Fin J, ∫ s in 0..h, (ell j).eval s) = h
rw [← intervalIntegral.integral_finsetSum (s := Finset.univ)
(f := fun j s => (ell j).eval s)
(fun j _ => (ell j).continuous.intervalIntegrable 0 h)]
simp only [hpart,intervalIntegral.integral_const]
simp
end AutoSamplingTheory.TechnicalLemmas.Analysis.ChebyshevLobattoQuadratureOpen AutoSamplingTheory/TechnicalLemmas/Analysis/ChebyshevLobattoQuadrature.lean:64published source at ed709a7f320d
Proof architecture
actual SPHMC counted phase source coefficients; depth-K exact Picard integrator (planned consumer)
Lean proof walkthrough
- Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
- `intro` introduces quantified hypotheses into the local proof context.
- `have` creates a named intermediate mathematical fact.
- `calc` records an equality or inequality chain matching a paper calculation.
- `rw` rewrites by an established identity.
- `simp` normalizes through registered definitional and theorem rewrites.
- `apply` reduces the goal to the hypotheses of a reusable theorem.
- `refine` instantiates a reusable theorem while leaving explicit subgoals.
Why the statement has this shape
The declaration is kept at the reusable level recorded by its Registry tags and direct consumers. Explicit measures, spaces, wrappers, and regularity hypotheses expose interfaces that paper notation often infers. A theorem card explains those interfaces but never widens the compiled statement.
Hidden assumptions and non-claims
- Integrability is an input or proved output; a displayed integral alone does not supply it.