Proof of normalization of permutations

Proposition (Normalization of permutations). For any permutation p : X₁,X,X₂ Y we can find Y₁,X,Y₂ = Y and q : X₁,X₂ Y₁,Y₂ such that:

=

Proof. Induction on p.

• If p = id then let q = id, and:
  =
• If p = p1 ; p2 then by induction we can find appropriate q1 ; q2 such that: = =
• If p = p1 p2 then either:
  • we can induct on p1, and: = = = =
  • or we can induct on p2, and: =