realize_iAlls — Mathlib

English

For [Finite β], and {φ : L.Formula (Sum α β)} {v : α → M}, the equality between realizations holds with v' dummy of Fin 0.

Русский

Для [ конечного β ], и {φ : L.Formula (Sum α β)} {v : α → M}, равенство реализаций выполняется при фиксации p-аргумента Fin 0.

LaTeX

$$$\text{BoundedFormula.Realize} (\varphi.iAlls β) v v' \iff \forall i : β → M, \varphi.Realize (fun a => Sum.elim v i a)$$$

Lean4

@[simp]
theorem realize_iAlls [Finite β] {φ : L.Formula (α ⊕ β)} {v : α → M} {v' : Fin 0 → M} :
    BoundedFormula.Realize (φ.iAlls β) v v' ↔ ∀ (i : β → M), φ.Realize (fun a => Sum.elim v i a) := by
  rw [← Formula.realize_iAlls, iff_iff_eq]; congr; simp [eq_iff_true_of_subsingleton]

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