GromovProduct.lean`

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11 declarations, 0 contain sorry, 0 declared as axioms

The Gromov product

The Gromov product of y and z with respect to x in a pseudometric space is (y, z)_x = (dist x y + dist x z - dist y z) / 2.

This file is copied from Mathlib.Topology.MetricSpace.GromovProduct (leanprover-community/mathlib4#43641). It is in Mathlib from v4.34.0.

TODO: delete this file when this repository moves to Mathlib v4.34.0.

Main definitions

Metric.gromovProduct x y z: the Gromov product of y and z with respect to x.

Imports (1)

Mathlib.Topology.MetricSpace.Pseudo.Defs

Imported by 2 files

def Metric.gromovProduct The Gromov product of y and z with respect to x line 44 source
Source documentation
The Gromov product of y and z with respect to x.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PseudoMetricSpace, Real
Proof uses: HDiv.hDiv, HSub.hSub, HAdd.hAdd, Dist.dist, OfNat.ofNat
11 instances, coercions and auxiliary constants left out.
```
noncomputable def gromovProduct : ℝ := (dist x y + dist x z - dist y z) / 2
```
lemma Metric.gromovProduct_eq line 46 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PseudoMetricSpace, Eq, Real, Metric.gromovProduct, HDiv.hDiv, HSub.hSub, HAdd.hAdd, Dist.dist, OfNat.ofNat, Nat
Proof uses: Eq.refl
13 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Metric.gromovProduct_comm line 48 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PseudoMetricSpace, Eq, Real, Metric.gromovProduct
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Metric.gromovProduct_nonneg line 54 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PseudoMetricSpace, LE.le, Real, OfNat.ofNat, Metric.gromovProduct
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Metric.gromovProduct_le_dist_left line 57 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PseudoMetricSpace, LE.le, Real, Metric.gromovProduct, Dist.dist
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Metric.gromovProduct_le_dist_right line 60 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PseudoMetricSpace, LE.le, Real, Metric.gromovProduct, Dist.dist
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Metric.gromovProduct_self_left line 64 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PseudoMetricSpace, Eq, Real, Metric.gromovProduct, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, HDiv.hDiv, HSub.hSub, HAdd.hAdd, Dist.dist, Nat, Metric.gromovProduct_eq, dist_self, zero_add, sub_self, zero_div, eq_self
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Metric.gromovProduct_self_right line 68 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PseudoMetricSpace, Eq, Real, Metric.gromovProduct, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, HDiv.hDiv, HSub.hSub, HAdd.hAdd, Dist.dist, Nat, Metric.gromovProduct_eq, congr, dist_self, add_zero, dist_comm, sub_self, zero_div, eq_self
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Metric.gromovProduct_self line 72 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PseudoMetricSpace, Eq, Real, Metric.gromovProduct, Dist.dist
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, HDiv.hDiv, HSub.hSub, HAdd.hAdd, OfNat.ofNat, Nat, Metric.gromovProduct_eq, dist_self, sub_zero, add_self_div_two, eq_self
39 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Metric.gromovProduct_add_gromovProduct₁₃ line 75 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PseudoMetricSpace, Eq, Real, HAdd.hAdd, Metric.gromovProduct, Dist.dist
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Metric.gromovProduct_add_gromovProduct₁₂ line 78 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PseudoMetricSpace, Eq, Real, HAdd.hAdd, Metric.gromovProduct, Dist.dist
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.