feat(Topology,Analysis): discrete topology metric space and normed groups

[pechersky]

Explicit construction of the discrete topology metric space and normed groups where dist x y = 1 for all x != y
Provide PseudoMetricSpace, MetricSpace, Seminormed(Add)Group, and Normed(Add)Group constructions

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[github-actions]

PR summary bd4da70be5

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff

+ NormedGroup.ofDiscreteTopology
+ NormedGroup.ofDiscreteTopology_norm_def
+ SeminormedGroup.ofDiscreteTopology
+ SeminormedGroup.ofDiscreteTopology_norm_def
++ ofDiscreteTopology
++ ofDiscreteTopology_discreteUniformity
++ ofDiscreteTopology_dist_def
++ ofDiscreteTopology_uniformSpace_eq_bot

You can run this locally as follows

## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for script/declarations_diff.sh contains some details about this script.

---

No changes to technical debt.

You can run this locally as

./scripts/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[kckennylau]

Why do we need this?

[pechersky]

Because I don't want people installing this discrete metric space unless they really mean to.

[plp127]

Why do you replace the topology but not the uniformity?

[kckennylau]

Does this one imply SeminormedGroup.ofDiscreteTopology?

[kckennylau]

I guess I don't understand why we need to do everything for MetricSpace etc. and then do it again for PseudoMetricSpace. Does MetricSpace not imply PseudoMetricSpace?

[mathlib4-merge-conflict-bot]

This pull request has conflicts, please merge master and resolve them.

[pechersky]

Why do it for PseudoMetricSpace when we did it for MetricSpace? Because I want the construction to be available earlier in import tree without having to bring in metric spaces.

One could also ask, why have it for MetricSpace if we have it for PseudoMetricSpace? And that's so that there are shortcut constructions that give as rich of an object as possible.

[kckennylau]

I see, I missed the fact that MS depends on PMS because github orders files alphabetically

[kckennylau]

Suggested change

	dist_self := by simp
	dist_comm := by intros; split_ifs <;> simp_all
	dist_triangle := by intros; split_ifs <;> simp_all
	edist_dist := by intros; split_ifs <;> simp_all
	dist_self := by grind
	dist_comm := by grind
	dist_triangle := by grind
	edist_dist := by aesop

if you want

[kckennylau]

Suggested change

	eq_of_dist_eq_zero := by simp [PseudoMetricSpace.ofDiscreteTopology_dist_def]
	eq_of_dist_eq_zero := by grind

[kckennylau]

Suggested change

	variable [TopologicalSpace X] [DiscreteTopology X] [DecidableEq X]
	lemma ofDiscreteTopology_dist_def (x y : X) :
	letI := ofDiscreteTopology (X := X)
	dist x y = if x = y then 0 else 1 :=
	rfl
	lemma ofDiscreteTopology_uniformSpace_eq_bot :
	(MetricSpace.ofDiscreteTopology (X := X)).toUniformSpace = ⊥ :=
	PseudoMetricSpace.ofDiscreteTopology_uniformSpace_eq_bot
	lemma ofDiscreteTopology_discreteUniformity :
	@DiscreteUniformity X (ofDiscreteTopology (X := X)).toUniformSpace :=
	PseudoMetricSpace.ofDiscreteTopology_discreteUniformity
	section
	variable [TopologicalSpace X] [DiscreteTopology X] [DecidableEq X]
	attribute [local instance] ofDiscreteTopology
	@[grind =] lemma ofDiscreteTopology_dist_def (x y : X) : dist x y = if x = y then 0 else 1 :=
	rfl
	lemma ofDiscreteTopology_uniformSpace_eq_bot : (ofDiscreteTopology (X := X)).toUniformSpace = ⊥ :=
	PseudoMetricSpace.ofDiscreteTopology_uniformSpace_eq_bot
	lemma ofDiscreteTopology_discreteUniformity : DiscreteUniformity X :=
	PseudoMetricSpace.ofDiscreteTopology_discreteUniformity
	end

[mathlib-merge-conflicts]

This pull request has conflicts, please merge master and resolve them.