gameAdd_iff — Mathlib

English

For a GaloisCoinsertion, l is strictly monotone when composed with dualization.

Русский

Для коинсерции Галуа отображение l явно строго монотонно после дуализации.

LaTeX

$$$\\text{StrictMono } l$$$

Lean4

theorem gameAdd_iff {rα rβ} {x y : α × β} : GameAdd rα rβ x y ↔ rα x.1 y.1 ∧ x.2 = y.2 ∨ rβ x.2 y.2 ∧ x.1 = y.1 :=
  by
  constructor
  · rintro (@⟨a₁, a₂, b, h⟩ | @⟨a, b₁, b₂, h⟩)
    exacts [Or.inl ⟨h, rfl⟩, Or.inr ⟨h, rfl⟩]
  · revert x y
    rintro ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ (⟨h, rfl : b₁ = b₂⟩ | ⟨h, rfl : a₁ = a₂⟩)
    exacts [GameAdd.fst h, GameAdd.snd h]

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