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Database GRAPH THEORY Walks, paths and cycles Closed walks as words Closed walks as words erclwwlk
Description: .~ is an equivalence relation over the set of closed walks (defined
as words). (Contributed by Alexander van der Vekens , 10-Apr-2018)
(Revised by AV , 30-Apr-2021)
Ref
Expression
Hypothesis
erclwwlk.r ⊢ ∼ = { ⟨ 𝑢 , 𝑤 ⟩ ∣ ( 𝑢 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑤 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) 𝑢 = ( 𝑤 cyclShift 𝑛 ) ) }
Assertion
erclwwlk ⊢ ∼ Er ( ClWWalks ‘ 𝐺 )
Proof
Step
Hyp
Ref
Expression
1
erclwwlk.r ⊢ ∼ = { ⟨ 𝑢 , 𝑤 ⟩ ∣ ( 𝑢 ∈ ( ClWWalks ‘ 𝐺 ) ∧ 𝑤 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ∃ 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) 𝑢 = ( 𝑤 cyclShift 𝑛 ) ) }
2
1
erclwwlkrel ⊢ Rel ∼
3
1
erclwwlksym ⊢ ( 𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥 )
4
1
erclwwlktr ⊢ ( ( 𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧 ) → 𝑥 ∼ 𝑧 )
5
1
erclwwlkref ⊢ ( 𝑥 ∈ ( ClWWalks ‘ 𝐺 ) ↔ 𝑥 ∼ 𝑥 )
6
2 3 4 5
iseri ⊢ ∼ Er ( ClWWalks ‘ 𝐺 )