Hyperbolic.lean`

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5 declarations, 0 contain sorry, 0 declared as axioms

Gromov hyperbolic pseudometric spaces

This is being upstreamed to Mathlib in leanprover-community/mathlib4#43723.

TODO: delete this file when this repository moves to a Mathlib version that contains it.

Main definitions

Metric.IsHyperbolicWith: A pseudometric space is hyperbolic with constant δ if and only if for all w x y z the four-point condition min (gromovProduct w x y) (gromovProduct w y z) - δ ≤ gromovProduct w x z holds.
Metric.IsHyperbolic: A pseudometric space is hyperbolic if it is δ-hyperbolic for some constant δ.

Main results

Metric.isHyperbolicWith_diam_univ: A bounded space is δ-hyperbolic with respect to its diameter.

Implementation notes

The constant δ in IsHyperbolicWith X δ is a real number. For a non-empty space, taking w = x = y = z in the four-point condition yields 0 ≤ δ.

Imports (2)

Imported by 2 files

def Metric.IsHyperbolicWith A pseudometric space is δ-hyperbolic if for all w x y z the four-point condition min (gromovProduct w x y) (gromovProduct w y z) - δ ≤ gromovProduct w x z holds line 56 source
Source documentation
A pseudometric space is δ-hyperbolic if for all w x y z the four-point condition min (gromovProduct w x y) (gromovProduct w y z) - δ ≤ gromovProduct w x z holds.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PseudoMetricSpace, Real
Proof uses: LE.le, HSub.hSub, Min.min, Metric.gromovProduct
4 instances, coercions and auxiliary constants left out.
```
def IsHyperbolicWith (X : Type*) [PseudoMetricSpace X] (δ : ℝ) : Prop :=
…
```
lemma Metric.isHyperbolicWith_of_forall_dist_le A pseudometric space with all distances bounded above by k is k-hyperbolic line 63 source
Source documentation
A pseudometric space with all distances bounded above by k is k-hyperbolic.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PseudoMetricSpace, Real, LE.le, Dist.dist, Metric.IsHyperbolicWith
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma isHyperbolicWith_of_forall_dist_le {k : ℝ} (hk : ∀ x y : X, dist x y ≤ k) :
    IsHyperbolicWith X k := by
…
```
Mentions: Metric.IsHyperbolicWith
theorem Metric.isHyperbolicWith_diam_univ A bounded space is δ-hyperbolic with respect to its diameter line 68 source
Source documentation
A bounded space is δ-hyperbolic with respect to its diameter.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PseudoMetricSpace, BoundedSpace, Metric.IsHyperbolicWith, Metric.diam, Set.univ
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem isHyperbolicWith_diam_univ [BoundedSpace X] :
    IsHyperbolicWith X (diam (Set.univ : Set X)) := by
…
```
Mentions: Metric.IsHyperbolicWith
class Metric.IsHyperbolic A pseudometric space is hyperbolic if it is δ-hyperbolic for some real constant δ line 75 source
Source documentation
A pseudometric space is hyperbolic if it is δ-hyperbolic for some real constant δ.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PseudoMetricSpace
```
class IsHyperbolic (X : Type*) [PseudoMetricSpace X] : Prop where
…
```
instance Metric.(instance at line 79) line 79 source
Source documentation
Every bounded pseudometric space is hyperbolic.
```
instance [BoundedSpace X] : IsHyperbolic X := ⟨_, isHyperbolicWith_diam_univ⟩
…
```
Mentions: Metric.IsHyperbolic, Metric.isHyperbolicWith_diam_univ