feat(Combinatorics/SimpleGraph/Acyclic): characterize trees via bridges

[WangYiran01]

Add theorem isTree_iff_connected_and_forall_edge_isBridge in Mathlib/Combinatorics/SimpleGraph/Acyclic.lean.

This lemma characterizes G.IsTree as G.Connected together with the fact that every edge in G.edgeSet is a bridge.

This makes it easier to switch between tree proofs and bridge/acyclic proofs in SimpleGraph.

[github-actions]

PR summary f1a5ef6e2c

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

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Declarations diff

+ isTree_iff_connected_and_forall_edge_isBridge

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).

[SnirBroshi]

Hello 👋
Can you explain why you want to add this? Is there a reason you need to combine 2 rewrites into one theorem?

[SnirBroshi]

Suggested change

	calc
	G.IsTree ↔ G.Connected ∧ G.IsAcyclic := SimpleGraph.isTree_iff (G := G)
	_ ↔ G.Connected ∧ (∀ ⦃e : Sym2 V⦄, e ∈ G.edgeSet → G.IsBridge e) := by
	simpa using
	(and_congr_right (fun _ =>
	(SimpleGraph.isAcyclic_iff_forall_edge_isBridge (G := G))))
	rw [isTree_iff, isAcyclic_iff_forall_edge_isBridge]