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feat(Combinatorics/SimpleGraph): Add lemma Walk.IsCircuit.isCycle_cycleBypass #37574

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feat(Combinatorics/SimpleGraph): Add lemma Walk.IsCircuit.isCycle_cycleBypass #37574

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11 changes: 8 additions & 3 deletions Mathlib/Combinatorics/SimpleGraph/Paths.lean
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Expand Up @@ -800,14 +800,19 @@ lemma edges_cycleBypass_subset : ∀ {w : G.Walk v v}, w.cycleBypass.edges ⊆ w
gcongr
exact edges_bypass_subset _

lemma IsTrail.isCycle_cycleBypass : ∀ {w : G.Walk v v}, w ≠ .nil → w.IsTrail → w.cycleBypass.IsCycle
| .cons (v := v') hvv' w, _, hw => by
lemma IsCircuit.isCycle_cycleBypass : ∀ {w : G.Walk v v}, w.IsCircuit → w.cycleBypass.IsCycle
| .cons (v := v') hvv' w, hw => by
dsimp [cycleBypass]
refine ⟨⟨(bypass_isPath _).isTrail.cons _ fun hvv' ↦ ?_, by simp⟩, ?_⟩
· simp only [isTrail_cons] at hw
· simp only [isCircuit_def, isTrail_cons, ne_eq, reduceCtorEq, not_false_eq_true,
and_true] at hw
exact hw.2 <| edges_bypass_subset _ hvv'
· simpa using (bypass_isPath _).support_nodup

lemma IsTrail.isCycle_cycleBypass {w : G.Walk v v} (hw : w ≠ .nil) (hw' : w.IsTrail) :
w.cycleBypass.IsCycle :=
(w.isCircuit_def.mpr ⟨hw', hw⟩).isCycle_cycleBypass

end Walk

/-! ### Mapping paths -/
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