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Database GRAPH THEORY Walks, paths and cycles Closed walks as words Closed walks of a fixed length as words erclwwlkn
Description: .~ is an equivalence relation over the set of closed walks (defined
as words) with a fixed length. (Contributed by Alexander van der
Vekens , 10-Apr-2018) (Revised by AV , 30-Apr-2021)
Ref
Expression
Hypotheses
erclwwlkn.w ⊢ 𝑊 = ( 𝑁 ClWWalksN 𝐺 )
erclwwlkn.r ⊢ ∼ = { ⟨ 𝑡 , 𝑢 ⟩ ∣ ( 𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑡 = ( 𝑢 cyclShift 𝑛 ) ) }
Assertion
erclwwlkn ⊢ ∼ Er 𝑊
Proof
Step
Hyp
Ref
Expression
1
erclwwlkn.w ⊢ 𝑊 = ( 𝑁 ClWWalksN 𝐺 )
2
erclwwlkn.r ⊢ ∼ = { ⟨ 𝑡 , 𝑢 ⟩ ∣ ( 𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃ 𝑛 ∈ ( 0 ... 𝑁 ) 𝑡 = ( 𝑢 cyclShift 𝑛 ) ) }
3
1 2
erclwwlknrel ⊢ Rel ∼
4
1 2
erclwwlknsym ⊢ ( 𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥 )
5
1 2
erclwwlkntr ⊢ ( ( 𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧 ) → 𝑥 ∼ 𝑧 )
6
1 2
erclwwlknref ⊢ ( 𝑥 ∈ 𝑊 ↔ 𝑥 ∼ 𝑥 )
7
3 4 5 6
iseri ⊢ ∼ Er 𝑊