| import Mathlib | |
| namespace OAI | |
| /-! A derivative comparison obtained from pointwise increments. -/ | |
| noncomputable section | |
| open Filter | |
| open scoped Topology | |
| namespace InvariantIsing | |
| theorem norm_derivative_le_of_increment_bound | |
| {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| [NormedAddCommGroup F] [NormedSpace ℝ F] | |
| {f : ℝ → E} {g : ℝ → F} {f' : E} {g' : F} {x L : ℝ} | |
| (hf : HasDerivAt f f' x) (hg : HasDerivAt g g' x) | |
| (h : ∀ t, ‖f t-f x‖ ≤ L*‖g t-g x‖) : ‖f'‖ ≤ L*‖g'‖ := by | |
| apply le_of_tendsto_of_tendsto hf.tendsto_slope.norm (hg.tendsto_slope.norm.const_mul L) | |
| apply Eventually.of_forall | |
| intro t | |
| simp only [slope_def_module,norm_smul] | |
| calc | |
| ‖(t-x)⁻¹‖*‖f t-f x‖ ≤ ‖(t-x)⁻¹‖*(L*‖g t-g x‖) := | |
| mul_le_mul_of_nonneg_left (h t) (norm_nonneg _) | |
| _ = L*(‖(t-x)⁻¹‖*‖g t-g x‖) := by ring | |
| end InvariantIsing | |
| end | |
| end OAI | |