le_iff_forall_lf — Mathlib

English

The simplified form of mk_le_mk expresses the same equivalence using the le_iff_forall_lf principle.

Русский

У упрощённой формы mk_le_mk выражается та же эквивалентность через принциpf le_iff_forall_lf.

LaTeX

$$$\text{mk}\ xl xr xL xR \le \text{mk}\ yl yr yL yR \iff \text{le_iff_forall_lf} (\text{mk}\ xl xr xL xR) (\text{mk}\ yl yr yL yR)$$$

Lean4

/-- Definition of `x ≤ y` on pre-games, in terms of `⧏`.

The ordering here is chosen so that `And.left` refer to moves by Left, and `And.right` refer to
moves by Right. -/
theorem le_iff_forall_lf {x y : PGame} : x ≤ y ↔ (∀ i, x.moveLeft i ⧏ y) ∧ ∀ j, x ⧏ y.moveRight j :=
  by
  unfold LE.le le
  simp only
  rw [Sym2.GameAdd.fix_eq]
  rfl

Похожие утверждения