Plain-language statement
The integral appearing in corrugations is periodic.
Exact Lean statement
theorem per_corrugation (γ : Loop F) (hγ : ∀ s t, IntervalIntegrable γ volume s t) :
OnePeriodic fun s ↦ ∫ t in (0)..s, γ t - γ.averageFormal artifact
Lean source
theorem per_corrugation (γ : Loop F) (hγ : ∀ s t, IntervalIntegrable γ volume s t) : OnePeriodic fun s ↦ ∫ t in (0)..s, γ t - γ.average := by have int_avg : ∀ s t, IntervalIntegrable (fun _ : ℝ ↦ γ.average) volume s t := fun s t ↦ intervalIntegrable_const intro s have int₁ : IntervalIntegrable (fun t ↦ γ t - γ.average) volume 0 s := (hγ _ _).sub (int_avg _ _) have int₂ : IntervalIntegrable (fun t ↦ γ t - γ.average) volume s (s + 1) := (hγ _ _).sub (int_avg _ _) have int₃ : IntervalIntegrable γ volume s (s + 1) := hγ _ _ have int₄ : IntervalIntegrable (fun _ ↦ γ.average) volume s (s + 1) := int_avg _ _ dsimp only /- Rmk: Lean doesn't want to rewrite using `intervalIntegral.integral_sub` without being given the integrability assumptions :-( -/ rw [← intervalIntegral.integral_add_adjacent_intervals int₁ int₂, intervalIntegral.integral_sub int₃ int₄, γ.periodic.intervalIntegral_add_eq s 0, zero_add, Loop.average] simp only [add_zero, add_tsub_cancel_left, intervalIntegral.integral_const, one_smul, sub_self]- Project
- Sphere eversion
- License
- Apache-2.0
- Commit
- ded8fda5e76b
- Source
- SphereEversion/Local/Corrugation.lean:53-69
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Related declarations
Continuous of Path
Continuous.ofPath
Plain-language statement
Loop.ofPath is continuous, general version.
Source project: Sphere eversion
Person-level attribution pending.
Injective update iff
DualPair.injective_update_iff
Project documentation
Map a dual pair under a linear equivalence. -/ @[simps] def map (p : DualPair E) (L : E ≃L[ℝ] E') : DualPair E' := ⟨p.π ∘L ↑L.symm, L p.v, (congr_arg p.π <| L.symm_apply_apply p.v).trans p.pairing⟩ theorem update_comp_left (p : DualPair E) (ψ' : F →L[ℝ] G) (φ : E →L[ℝ] F) (w : F) : p.update (ψ' ∘L φ) (ψ' w) = ψ' ∘L p.update φ w := by ext1 u simp only [upd...
Source project: Sphere eversion
Person-level attribution pending.
Extend loops
extend_loops
Project documentation
A more precise version of sfHomotopy_in. -/ theorem sfHomotopy_in' {ι} (h₀ : SurroundingFamily g b γ₀ U) (h₁ : SurroundingFamily g b γ₁ U) (τ : ι → ℝ) (x : ι → E) (i : ι) {V : Set E} (hx : x i ∈ V) {t : ℝ} (ht : t ∈ I) {s : ℝ} (h_in₀ : ∀ i, x i ∈ V → ∀ t ∈ I, ∀ (s : ℝ), τ i ≠ 1 → (x i, γ₀ (x i) t s) ∈ Ω) (h_in₁ : ∀ i, x i ∈ V → ∀ t ∈ I, ∀ (s : ℝ), τ i ≠...
Source project: Sphere eversion
Person-level attribution pending.