equiv — Mathlib

English

Another form of the compatibility of prod with smul for the inductive definition.

Русский

Еще одна форма совместимости prod со смул для индуктивного определения.

LaTeX

$$$$ prod\\_smul\\_alt(m,w) $$$$

Lean4

/-- Each element of the free product corresponds to a unique reduced word. -/
def equiv : CoprodI M ≃ Word M where
  toFun m := m • empty
  invFun w := prod w
  left_inv m := by dsimp only; rw [prod_smul, prod_empty, mul_one]
  right_inv := by
    apply smul_induction
    · dsimp only
      rw [prod_empty, one_smul]
    · dsimp only
      intro i m w ih
      rw [prod_smul, mul_smul, ih]

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