cons_injective2 — Mathlib

English

The binary function cons is injective in both arguments: if cons x0 x = cons y0 y, then x0 = y0 and x = y.

Русский

Двойная инъективность конc: если конc x0 x = конc y0 y, то x0 = y0 и x = y.

LaTeX

$$$\\forall x_0\\, x\\, y_0\\, y,\\ (\\\\mathrm{cons}(x_0,x) = \\\\mathrm{cons}(y_0,y) \\\\Rightarrow x_0 = y_0 \\land x = y)$$$

Lean4

/-- As a binary function, `Fin.cons` is injective. -/
theorem cons_injective2 : Function.Injective2 (@cons n α) := fun x₀ y₀ x y h ↦
  ⟨congr_fun h 0, funext fun i ↦ by simpa using congr_fun h (Fin.succ i)⟩

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