60 lines
2.1 KiB
Text
60 lines
2.1 KiB
Text
/-
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Copyright (c) 2016 Robert Y. Lewis. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Robert Y. Lewis
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Derivatives on ℝ
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-/
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import .bounded_linear_operator
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open real nat classical topology analysis set
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noncomputable theory
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namespace real
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-- move this to group
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theorem add_sub_self (a b : ℝ) : a + b - a = b :=
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by rewrite [add_sub_assoc, add.comm, sub_add_cancel]
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-- make instance of const mul bdd lin op?
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definition derivative_at (f : ℝ → ℝ) (d x : ℝ) := is_frechet_deriv_at f (λ t, d • t) x
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theorem derivative_at_intro (f : ℝ → ℝ) (d x : ℝ) (H : (λ h, (f (x + h) - f x) / h) ⟶ d [at 0]) :
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derivative_at f d x :=
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begin
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apply is_frechet_deriv_at_intro,
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intros ε Hε,
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cases approaches_at_dest H Hε with δ Hδ,
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existsi δ,
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split,
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exact and.left Hδ,
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intro y Hy,
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rewrite [-sub_zero y at Hy{2}],
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note Hδ' := and.right Hδ y (and.right Hy) (and.left Hy),
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have Hδ'' : abs ((f (x + y) - f x - d * y) / y) < ε,
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by rewrite [-div_sub_div_same, mul_div_cancel _ (and.left Hy)]; apply Hδ',
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show abs (f (x + y) - f x - d * y) / abs y < ε, by rewrite -abs_div; apply Hδ''
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end
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theorem derivative_at_of_frechet_derivative_at {f g : ℝ → ℝ} [is_bdd_linear_map g] {d x : ℝ}
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(H : is_frechet_deriv_at f g x) (Hg : g = λ x, d * x) :
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derivative_at f d x :=
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by apply is_frechet_deriv_at_of_eq H Hg
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theorem deriv_at_const (c x : ℝ) : derivative_at (λ t, c) 0 x :=
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derivative_at_of_frechet_derivative_at
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(@frechet_deriv_at_const ℝ ℝ _ _ _ c)
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(funext (λ v, by rewrite zero_mul))
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theorem deriv_at_id (x : ℝ) : derivative_at (λ t, t) 1 x :=
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derivative_at_of_frechet_derivative_at
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(@frechet_deriv_at_id ℝ ℝ _ _ _)
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(funext (λ v, by rewrite one_mul))
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theorem deriv_at_mul {f : ℝ → ℝ} {d x : ℝ} (H : derivative_at f d x) (c : ℝ) :
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derivative_at (λ t, c * f t) (c * d) x :=
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derivative_at_of_frechet_derivative_at
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(frechet_deriv_at_smul _ _ c H)
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(funext (λ v, by rewrite mul.assoc))
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end real
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