smul_floorDiv — Mathlib

English

For ha>0 and any b, a ⋅ b ⌊/⌋ a = b holds under positivity and monotonicity assumptions.

Русский

При ha>0 и подходящих условиях монотонности выполняется a ⋅ b ⌊/⌋ a = b.

LaTeX

$$$ ha > 0 \\\\Rightarrow (a \\cdot b \\\\ll floor / a \\\\rr) = b.$$$

Lean4

@[simp]
theorem smul_floorDiv [PosSMulMono α β] [PosSMulReflectLE α β] (ha : 0 < a) (b : β) : a • b ⌊/⌋ a = b :=
  eq_of_forall_le_iff <| by simp [smul_le_smul_iff_of_pos_left, ha]

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