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allaun
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Defines coupling matrix A = (ε/2)·[[0,1],[1,0]], proves ‖A‖ = |ε|/2, and applies Mathlib Gronwall for the discrete-continuous coupling bound: ‖e(t)‖ ≤ ‖e(0)‖ · exp(|ε|/2 · (t-a)) Includes trajectory distance variants (interval + global).
209 lines
9.8 KiB
Text
209 lines
9.8 KiB
Text
import Mathlib.Analysis.ODE.Gronwall
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import Mathlib.Analysis.Matrix.Normed
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import Mathlib.Analysis.SpecialFunctions.ExpDeriv
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/-!
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# DiscreteContinuousBound.lean
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Exponential error bound for discrete–continuous coupling via Grönwall's inequality.
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Given the 2×2 coupling matrix A = (ε/2) · [[0,1],[1,0]],
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we prove:
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1. ‖A‖ = |ε|/2 (operator norm)
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2. ‖e(t)‖ ≤ ‖e(0)‖ · exp(|ε|/2 · t) (Grönwall bound on coupling error)
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The proof uses Mathlib's `norm_le_gronwallBound_of_norm_deriv_right_le`
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and the ∞-operator-norm on matrices (`linfty_opNorm`).
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-/
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open Matrix Set Filter Real
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open scoped Topology
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namespace Semantics.DiscreteContinuousBound
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variable {ε : ℝ}
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §1 The 2×2 Coupling Matrix
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- The off-diagonal exchange matrix M₀ = [[0,1],[1,0]]. -/
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def M₀ : Matrix (Fin 2) (Fin 2) ℝ := !![0, 1; 1, 0]
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/-- The coupling matrix A = (ε/2) · M₀ = (ε/2) · [[0,1],[1,0]]. -/
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def A (ε : ℝ) : Matrix (Fin 2) (Fin 2) ℝ := (ε / 2) • M₀
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §2 Operator Norm: ‖A‖ = |ε|/2
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- Entry (0,1) of the coupling matrix is ε/2. -/
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@[simp] lemma A_apply_zero_one : A ε 0 1 = ε / 2 := by
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simp [A, M₀, smul_apply]
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/-- Entry (1,0) of the coupling matrix is ε/2. -/
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@[simp] lemma A_apply_one_zero : A ε 1 0 = ε / 2 := by
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simp [A, M₀, smul_apply]
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/-- Entry (0,0) of the coupling matrix is 0. -/
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@[simp] lemma A_apply_zero_zero : A ε 0 0 = 0 := by
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simp [A, M₀, smul_apply]
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/-- Entry (1,1) of the coupling matrix is 0. -/
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@[simp] lemma A_apply_one_one : A ε 1 1 = 0 := by
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simp [A, M₀, smul_apply]
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/-- All entries of A satisfy ‖A i j‖ ≤ |ε/2|. -/
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lemma A_entry_bound (i j : Fin 2) : ‖A ε i j‖ ≤ |ε / 2| := by
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fin_cases i <;> fin_cases j <;> simp [abs_nonneg]
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/-- The ∞-operator norm of A equals |ε|/2.
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Strategy: antisymmetry.
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• Upper bound: every entry ≤ |ε/2|, so sup ≤ |ε/2| via `norm_le_iff`.
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• Lower bound: entry (0,1) = ε/2, so ‖A‖ ≥ |ε/2| via `norm_entry_le_entrywise_sup_norm`.
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-/
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theorem coupling_opNorm (ε : ℝ) : ‖A ε‖ = |ε| / 2 := by
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have hε2 : |ε / 2| = |ε| / 2 := by rw [abs_div, abs_of_pos (by norm_num : (0 : ℝ) < 2)]
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-- Upper bound: ‖A‖ ≤ |ε/2|
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have h_upper : ‖A ε‖ ≤ |ε / 2| := by
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rw [norm_le_iff (abs_nonneg _)]
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exact A_entry_bound
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-- Lower bound: |ε/2| ≤ ‖A‖ (witness: entry (0,1))
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have h_lower : |ε / 2| ≤ ‖A ε‖ := by
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have : ‖A ε 0 1‖ ≤ ‖A ε‖ := norm_entry_le_entrywise_sup_norm _
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simp_all
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-- Combine
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linarith [h_upper, h_lower]
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/-- Variant: ‖A‖₊ = |ε|/2 as NNReals. -/
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theorem coupling_opNNNorm (ε : ℝ) : ‖A ε‖₊ = Real.toNNReal (|ε| / 2) := by
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ext; exact coupling_opNorm ε
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §3 Lipschitz Constant of the Linear Flow x ↦ A·x
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-- ═══════════════════════════════════════════════════════════════════════════════
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/-- The linear map v ↦ A *ᵥ v is Lipschitz with constant ‖A‖₊ = |ε|/2.
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This follows from the ∞-operator-norm submultiplicativity:
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‖A *ᵥ v‖ ≤ ‖A‖ · ‖v‖
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Combined with `ContinuousLinearMap.lipschitzWith_opNorm` or a direct proof.
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-/
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theorem coupling_lipschitzWith (ε : ℝ) :
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LipschitzWith (Real.toNNReal (|ε| / 2)) (fun v : Fin 2 → ℝ => A ε *ᵥ v) := by
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-- Step 1: LipschitzWith ‖mulVecLin (A ε)‖₊
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have h_lip : LipschitzWith ‖mulVecLin (A ε)‖₊ (fun v => A ε *ᵥ v) :=
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(mulVecLin (A ε)).lipschitz
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-- Step 2: ‖mulVecLin (A ε)‖₊ = ‖A ε‖₊ (linfty_opNNNorm_eq_opNNNorm)
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have h_norm : ‖mulVecLin (A ε)‖₊ = ‖A ε‖₊ := by
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rw [← linfty_opNNNorm_eq_opNNNorm]
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-- Step 3: ‖A ε‖₊ = Real.toNNReal (|ε|/2)
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rw [coupling_opNNNorm] at h_norm
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rwa [h_norm] at h_lip
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-- ═══════════════════════════════════════════════════════════════════════════════
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-- §4 Grönwall Exponential Error Bound
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-- ═══════════════════════════════════════════════════════════════════════════════
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section GronwallBound
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variable {a b : ℝ}
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/-- **Discrete–Continuous Coupling Bound (Grönwall form).**
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If `e : ℝ → Fin 2 → ℝ` satisfies the linear ODE e' = A · e on `[a, b]`,
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and the initial error is bounded by δ, then:
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‖e(t)‖ ≤ δ · exp(|ε|/2 · (t - a)) for all t ∈ [a, b].
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This is a direct application of `norm_le_gronwallBound_of_norm_deriv_right_le`
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with K = |ε|/2 and ε_g = 0 (exact ODE, no perturbation).
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-/
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theorem norm_le_gronwallBound_of_coupling
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{e e' : ℝ → Fin 2 → ℝ} {δ : ℝ} (hδ : 0 ≤ δ)
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(he_cont : ContinuousOn e (Icc a b))
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(he_deriv : ∀ t ∈ Ico a b, HasDerivWithinAt e (e' t) (Ici t) t)
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(he_ode : ∀ t ∈ Ico a b, e' t = A ε *ᵥ e t)
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(he_init : ‖e a‖ ≤ δ) :
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∀ t ∈ Icc a b, ‖e t‖ ≤ gronwallBound δ (|ε| / 2) 0 (t - a) := by
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apply norm_le_gronwallBound_of_norm_deriv_right_le he_cont he_deriv he_init
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intro t ht
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-- Goal: ‖e' t‖ ≤ (|ε|/2) · ‖e t‖ + 0
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rw [he_ode t ht, add_zero]
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-- Goal: ‖A ε *ᵥ e t‖ ≤ (|ε|/2) · ‖e t‖
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have := linfty_opNorm_mulVec (A ε) (e t)
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rw [coupling_opNorm] at this
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exact this
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/-- **Clean exponential form** (using `gronwallBound_ε0`).
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When the ODE is exact (ε_g = 0), the Grönwall bound simplifies to:
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‖e(t)‖ ≤ ‖e(0)‖ · exp(|ε|/2 · (t - a))
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-/
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theorem discrete_continuous_bound
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{e e' : ℝ → Fin 2 → ℝ} {δ : ℝ} (hδ : 0 ≤ δ)
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(he_cont : ContinuousOn e (Icc a b))
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(he_deriv : ∀ t ∈ Ico a b, HasDerivWithinAt e (e' t) (Ici t) t)
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(he_ode : ∀ t ∈ Ico a b, e' t = A ε *ᵥ e t)
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(he_init : ‖e a‖ ≤ δ) :
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∀ t ∈ Icc a b, ‖e t‖ ≤ δ * exp (|ε| / 2 * (t - a)) := by
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intro t ht
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have h := norm_le_gronwallBound_of_coupling hδ he_cont he_deriv he_ode he_init t ht
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rwa [gronwallBound_ε0] at h
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/-- **ODE-trajectory version** via Mathlib's `dist_le_of_trajectories_ODE`.
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If `f` and `g` are two trajectories of the linear ODE ẋ = A · x on `[a, b]`,
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then their distance grows at most exponentially:
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dist(f(t), g(t)) ≤ dist(f(a), g(a)) · exp(|ε|/2 · (t - a))
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-/
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theorem trajectory_dist_bound
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{f g : ℝ → Fin 2 → ℝ}
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(hf_cont : ContinuousOn f (Icc a b))
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(hf_deriv : ∀ t ∈ Ico a b, HasDerivWithinAt f (A ε *ᵥ f t) (Ici t) t)
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(hg_cont : ContinuousOn g (Icc a b))
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(hg_deriv : ∀ t ∈ Ico a b, HasDerivWithinAt g (A ε *ᵥ g t) (Ici t) t)
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{δ : ℝ} (hδ : 0 ≤ δ) (h_init : dist (f a) (g a) ≤ δ) :
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∀ t ∈ Icc a b, dist (f t) (g t) ≤ δ * exp (|ε| / 2 * (t - a)) := by
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intro t ht
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-- Apply `dist_le_of_trajectories_ODE` with K = |ε|/2
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have hv : ∀ t, LipschitzWith (Real.toNNReal (|ε| / 2)) (fun x => A ε *ᵥ x) :=
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fun _ => coupling_lipschitzWith ε
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have h := dist_le_of_trajectories_ODE hv hf_cont hf_deriv hg_cont hg_deriv h_init t ht
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-- Rewrite: the K in `dist_le_of_trajectories_ODE` is a `ℝ≥0`, and
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-- it appears as (K : ℝ) * (t - a) in the exponent.
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-- After the LipschitzWith substitution we need: ↑(Real.toNNReal (|ε|/2)) = |ε|/2
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have hnn : |ε| / 2 ≥ 0 := div_nonneg (abs_nonneg _) (by norm_num)
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rw [Real.coe_toNNReal _ hnn] at h
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exact h
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/-- **Global version**: on all of ℝ, two trajectories of ẋ = A · x satisfy
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dist(f(t), g(t)) ≤ dist(f(a), g(a)) · exp(|ε|/2 · |t - a|)
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-/
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theorem trajectory_dist_bound_univ
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{f g : ℝ → Fin 2 → ℝ}
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(hf : ∀ t, HasDerivAt f (A ε *ᵥ f t) t)
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(hg : ∀ t, HasDerivAt g (A ε *ᵥ g t) t)
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(a t : ℝ) :
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dist (f t) (g t) ≤ dist (f a) (g a) * exp (|ε| / 2 * |t - a|) := by
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wlog hle : a ≤ t with h
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-- t < a: swap and use symmetry
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have := h hg hf t a (le_of_not_le hle)
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rw [dist_comm (f t) (g t), dist_comm (f a) (g a), abs_sub_comm] at this
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exact this
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-- a ≤ t: use the interval version on [a, t]
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have hnn : |ε| / 2 ≥ 0 := div_nonneg (abs_nonneg _) (by norm_num)
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have hlip : ∀ s, LipschitzWith (Real.toNNReal (|ε| / 2)) (fun x => A ε *ᵥ x) :=
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fun _ => coupling_lipschitzWith ε
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have h :=
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dist_le_of_trajectories_ODE hlip
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(hf.continuousOn) (fun s _ => (hf s).hasDerivWithinAt)
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(hg.continuousOn) (fun s _ => (hg s).hasDerivWithinAt)
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le_rfl t ⟨hle, le_rfl⟩
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rw [Real.coe_toNNReal _ hnn, sub_nonneg.mpr hle, abs_of_nonneg (sub_nonneg.mpr hle)] at h
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exact h
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end GronwallBound
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end Semantics.DiscreteContinuousBound
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