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b-mehta committed Oct 16, 2023
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39 changes: 3 additions & 36 deletions src/analysis/calculus/iterated_deriv_mathlib.lean
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/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Copyright (c) 2023 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
Authors: Bhavik Mehta
-/
import analysis.calculus.iterated_deriv

/-!
# One-dimensional iterated derivatives
> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.
We define the `n`-th derivative of a function `f : 𝕜 → F` as a function
`iterated_deriv n f : 𝕜 → F`, as well as a version on domains `iterated_deriv_within n f s : 𝕜 → F`,
and prove their basic properties.
## Main definitions and results
Let `𝕜` be a nontrivially normed field, and `F` a normed vector space over `𝕜`. Let `f : 𝕜 → F`.
* `iterated_deriv n f` is the `n`-th derivative of `f`, seen as a function from `𝕜` to `F`.
It is defined as the `n`-th Fréchet derivative (which is a multilinear map) applied to the
vector `(1, ..., 1)`, to take advantage of all the existing framework, but we show that it
coincides with the naive iterative definition.
* `iterated_deriv_eq_iterate` states that the `n`-th derivative of `f` is obtained by starting
from `f` and differentiating it `n` times.
* `iterated_deriv_within n f s` is the `n`-th derivative of `f` within the domain `s`. It only
behaves well when `s` has the unique derivative property.
* `iterated_deriv_within_eq_iterate` states that the `n`-th derivative of `f` in the domain `s` is
obtained by starting from `f` and differentiating it `n` times within `s`. This only holds when
`s` has the unique derivative property.
## Implementation details
The results are deduced from the corresponding results for the more general (multilinear) iterated
Fréchet derivative. For this, we write `iterated_deriv n f` as the composition of
`iterated_fderiv 𝕜 n f` and a continuous linear equiv. As continuous linear equivs respect
differentiability and commute with differentiation, this makes it possible to prove readily that
the derivative of the `n`-th derivative is the `n+1`-th derivative in `iterated_deriv_within_succ`,
by translating the corresponding result `iterated_fderiv_within_succ_apply_left` for the
iterated Fréchet derivative.
# Stuff for analysis.calculus.iterated_deriv
-/

noncomputable theory
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9 changes: 9 additions & 0 deletions src/analysis/calculus/taylor_mathlib.lean
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/-
Copyright (c) 2023 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import analysis.calculus.taylor
import analysis.calculus.cont_diff
import data.set.intervals.unordered_interval

/-!
# Stuff for analysis.calculus.taylor
-/

open set

open_locale nat
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9 changes: 9 additions & 0 deletions src/combinatorics/simple_graph/clique_mathlib.lean
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/-
Copyright (c) 2023 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import combinatorics.simple_graph.basic_mathlib
import combinatorics.simple_graph.clique

/-!
# Stuff for combinatorics.simple_graph.clique
-/

namespace simple_graph

variables {V W : Type*} {G : simple_graph V} {H : simple_graph W} {n : ℕ}
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9 changes: 9 additions & 0 deletions src/data/fin/vec_notation_mathlib.lean
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/-
Copyright (c) 2023 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import data.fin.vec_notation

/-!
# Stuff for data.fin.vec_notation
-/

namespace function
open matrix (vec_cons)
open fin
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/-
Copyright (c) 2023 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import data.fintype.parity
import number_theory.legendre_symbol.quadratic_char.basic

/-!
# Stuff for number_theory.legendre_symbol.quadratic_char.basic
-/

open fintype (card)
open finset

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