extend_agrees — Mathlib

English

The extension of a compatible family to the generated sieve is compatible.

Русский

Расширение совместимой семьи к/generated sieve совместимо.

LaTeX

$$$x.Compatible \\Rightarrow x.sieveExtend.Compatible$$$

Lean4

/-- The extension of a family agrees with the original family. -/
theorem extend_agrees {x : FamilyOfElements P R} (t : x.Compatible) {f : Y ⟶ X} (hf : R f) :
    x.sieveExtend f (le_generate R Y hf) = x f hf :=
  by
  have h := (le_generate R Y hf).choose_spec
  unfold FamilyOfElements.sieveExtend
  rw [t h.choose (𝟙 _) _ hf _]
  · simp
  · rw [id_comp]
    exact h.choose_spec.choose_spec.2

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