Tangent cone

In this file, we define two predicates UniqueDiffWithinAt 𝕜 s x and UniqueDiffOn 𝕜 s ensuring that, if a function has two derivatives, then they have to coincide. As a direct definition of this fact (quantifying on all target types and all functions) would depend on universes, we use a more intrinsic definition: if all the possible tangent directions to the set s at the point x span a dense subset of the whole subset, it is easy to check that the derivative has to be unique.

Therefore, we introduce the set of all tangent directions, named tangentConeAt, and express UniqueDiffWithinAt and UniqueDiffOn in terms of it. One should however think of this definition as an implementation detail: the only reason to introduce the predicates UniqueDiffWithinAt and UniqueDiffOn is to ensure the uniqueness of the derivative. This is why their names reflect their uses, and not how they are defined.

Implementation details

Note that this file is imported by Mathlib/Analysis/Calculus/FDeriv/Basic.lean. Hence, derivatives are not defined yet. The property of uniqueness of the derivative is therefore proved in Mathlib/Analysis/Calculus/FDeriv/Basic.lean, but based on the properties of the tangent cone we prove here.

def tangentConeAt (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] (s : Set E) (x : E) : Set E

The set of all tangent directions to the set s at the point x.

Equations

• One or more equations did not get rendered due to their size.

structure UniqueDiffWithinAt (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] (s : Set E) (x : E) : Prop

A property ensuring that the tangent cone to s at x spans a dense subset of the whole space. The main role of this property is to ensure that the differential within s at x is unique, hence this name. The uniqueness it asserts is proved in UniqueDiffWithinAt.eq in Mathlib/Analysis/Calculus/FDeriv/Basic.lean. To avoid pathologies in dimension 0, we also require that x belongs to the closure of s (which is automatic when E is not 0-dimensional).

• dense_tangentConeAt : Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x))
• mem_closure : x ∈ closure s

theorem uniqueDiffWithinAt_iff (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] (s : Set E) (x : E) : UniqueDiffWithinAt 𝕜 s x ↔ Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x)) ∧ x ∈ closure s

@[deprecated UniqueDiffWithinAt.dense_tangentConeAt (since := "2025-04-27")]

theorem UniqueDiffWithinAt.dense_tangentCone {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] {s : Set E} {x : E} (self : UniqueDiffWithinAt 𝕜 s x) : Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x))

Alias of UniqueDiffWithinAt.dense_tangentConeAt.

def UniqueDiffOn (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] (s : Set E) : Prop

A property ensuring that the tangent cone to s at any of its points spans a dense subset of the whole space. The main role of this property is to ensure that the differential along s is unique, hence this name. The uniqueness it asserts is proved in UniqueDiffOn.eq in Mathlib/Analysis/Calculus/FDeriv/Basic.lean.

Equations

• UniqueDiffOn 𝕜 s = ∀ x ∈ s, UniqueDiffWithinAt 𝕜 s x