x : PGame
⊢ ((∀ (i : LeftMoves x), (∃ i', moveLeft x i ≤ moveLeft 0 i') ∨ ∃ j, moveRight (moveLeft x i) j ≤ 0) ∧
      ∀ (j : RightMoves 0), (∃ i, x ≤ moveLeft (moveRight 0 j) i) ∨ ∃ j', moveRight x j' ≤ moveRight 0 j) ↔
    ∀ (i : LeftMoves x), ∃ j, moveRight (moveLeft x i) j ≤ 0
x : PGame
⊢ x ≤ 0 ↔ ∀ (i : LeftMoves x), ∃ j, moveRight (moveLeft x i) j ≤ 0
rw [le_def]
no goals
x : PGame
⊢ ((∀ (i : LeftMoves x), (∃ i', moveLeft x i ≤ moveLeft 0 i') ∨ ∃ j, moveRight (moveLeft x i) j ≤ 0) ∧
      ∀ (j : RightMoves 0), (∃ i, x ≤ moveLeft (moveRight 0 j) i) ∨ ∃ j', moveRight x j' ≤ moveRight 0 j) ↔
    ∀ (i : LeftMoves x), ∃ j, moveRight (moveLeft x i) j ≤ 0
simp