feat(Combinatorics/SimpleGraph/Paths): existence of longest paths/trails/cycles/circuits

[SnirBroshi]

We already have the existence of a longest path/trail. This adds:

• the existence of a longest cycle/circuit in non-acyclic graphs
• the existence of a longest path/trail from a specific vertex
• the existence of a longest path/trail between specific endpoints
• the existence of a longest cycle/circuit from a specific vertex

---

exists_isTrail_forall_length_le_of_pred is the common argument that works for any predicate on trails, which is then used to prove the 10 versions mentioned above.
I know this looks a bit ugly, but taking a longest path/trail/cycle/circuit is a useful proof technique.

[github-actions]

PR summary 7d3065d46a

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

+ Reachable.exists_isPath_forall_isPath_length_le_length
+ Reachable.exists_isTrail_forall_isTrail_length_le_length
+ _root_.SimpleGraph.exists_isPath_forall_isPath_length_le_length
+ _root_.SimpleGraph.exists_isPath_forall_isPath_length_le_length'
+ _root_.SimpleGraph.exists_isTrail_forall_isTrail_length_le_length
+ _root_.SimpleGraph.exists_isTrail_forall_isTrail_length_le_length'
+ _root_.SimpleGraph.exists_isTrail_forall_length_le_of_pred
+ exists_isCircuit_forall_isCircuit_length_le_length
+ exists_isCircuit_forall_isCircuit_length_le_length'
+ exists_isCycle_forall_isCycle_length_le_length
+ exists_isCycle_forall_isCycle_length_le_length'
+ isAcyclic_iff_forall_not_isCircuit
+ isAcyclic_iff_forall_not_isCycle
+ not_isAcyclic_iff_exists_isCircuit
+ not_isAcyclic_iff_exists_isCycle

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

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

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit 7d3065d).

• +19 new declarations
• −0 removed declarations

+SimpleGraph.Reachable.exists_isPath_forall_isPath_length_le_length
+SimpleGraph.Reachable.exists_isTrail_forall_isTrail_length_le_length
+SimpleGraph.Walk.IsCircuit.hcongr_4
+SimpleGraph.Walk.IsCycle.hcongr_4
+SimpleGraph.Walk.IsPath.hcongr_5
+SimpleGraph.Walk.IsTrail.hcongr_5
+SimpleGraph.exists_isCircuit_forall_isCircuit_length_le_length
+SimpleGraph.exists_isCircuit_forall_isCircuit_length_le_length'
+SimpleGraph.exists_isCycle_forall_isCycle_length_le_length
+SimpleGraph.exists_isCycle_forall_isCycle_length_le_length'
+SimpleGraph.exists_isPath_forall_isPath_length_le_length
+SimpleGraph.exists_isPath_forall_isPath_length_le_length'
+SimpleGraph.exists_isTrail_forall_isTrail_length_le_length
+SimpleGraph.exists_isTrail_forall_isTrail_length_le_length'
+SimpleGraph.exists_isTrail_forall_length_le_of_pred
+SimpleGraph.isAcyclic_iff_forall_not_isCircuit
+SimpleGraph.isAcyclic_iff_forall_not_isCycle
+SimpleGraph.not_isAcyclic_iff_exists_isCircuit
+SimpleGraph.not_isAcyclic_iff_exists_isCycle

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 7d3065d46a
Reference commit 80ffd59621

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/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).