File size: 933 Bytes
a89d282 | 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 | 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
|