Theorem: SetTheory.PGame.leftMoves_mul_iff
File: Mathlib/SetTheory/Game/Basic.lean

Proof steps:
state_before: x y : PGame
P : Game → Prop
⊢ (∀ (k : (x * y).LeftMoves), P ⟦(x * y).moveLeft k⟧) ↔
    (∀ (i : x.LeftMoves) (j : y.LeftMoves), P ⟦x.mulOption y i j⟧) ∧
      ∀ (i : (-x).LeftMoves) (j : (-y).LeftMoves), P ⟦(-x).mulOption (-y) i j⟧
state_after: case mk
y : PGame
P : Game → Prop
α✝ β✝ : Type u_1
a✝¹ : α✝ → PGame
a✝ : β✝ → PGame
⊢ (∀ (k : (mk α✝ β✝ a✝¹ a✝ * y).LeftMoves), P ⟦(mk α✝ β✝ a✝¹ a✝ * y).moveLeft k⟧) ↔
    (∀ (i : (mk α✝ β✝ a✝¹ a✝).LeftMoves) (j : y.LeftMoves), P ⟦(mk α✝ β✝ a✝¹ a✝).mulOption y i j⟧) ∧
      ∀ (i : (-mk α✝ β✝ a✝¹ a✝).LeftMoves) (j : (-y).LeftMoves), P ⟦(-mk α✝ β✝ a✝¹ a✝).mulOption (-y) i j⟧
tactic: cases x
state_before: case mk
y : PGame
P : Game → Prop
α✝ β✝ : Type u_1
a✝¹ : α✝ → PGame
a✝ : β✝ → PGame
⊢ (∀ (k : (mk α✝ β✝ a✝¹ a✝ * y).LeftMoves), P ⟦(mk α✝ β✝ a✝¹ a✝ * y).moveLeft k⟧) ↔
    (∀ (i : (mk α✝ β✝ a✝¹ a✝).LeftMoves) (j : y.LeftMoves), P ⟦(mk α✝ β✝ a✝¹ a✝).mulOption y i j⟧) ∧
      ∀ (i : (-mk α✝ β✝ a✝¹ a✝).LeftMoves) (j : (-y).LeftMoves), P ⟦(-mk α✝ β✝ a✝¹ a✝).mulOption (-y) i j⟧
state_after: case mk.mk
P : Game → Prop
α✝¹ β✝¹ : Type u_1
a✝³ : α✝¹ → PGame
a✝² : β✝¹ → PGame
α✝ β✝ : Type u_1
a✝¹ : α✝ → PGame
a✝ : β✝ → PGame
⊢ (∀ (k : (mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).LeftMoves), P ⟦(mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).moveLeft k⟧) ↔
    (∀ (i : (mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (mk α✝ β✝ a✝¹ a✝).LeftMoves),
        P ⟦(mk α✝¹ β✝¹ a✝³ a✝²).mulOption (mk α✝ β✝ a✝¹ a✝) i j⟧) ∧
      ∀ (i : (-mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (-mk α✝ β✝ a✝¹ a✝).LeftMoves),
        P ⟦(-mk α✝¹ β✝¹ a✝³ a✝²).mulOption (-mk α✝ β✝ a✝¹ a✝) i j⟧
tactic: cases y
state_before: case mk.mk
P : Game → Prop
α✝¹ β✝¹ : Type u_1
a✝³ : α✝¹ → PGame
a✝² : β✝¹ → PGame
α✝ β✝ : Type u_1
a✝¹ : α✝ → PGame
a✝ : β✝ → PGame
⊢ (∀ (k : (mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).LeftMoves), P ⟦(mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).moveLeft k⟧) ↔
    (∀ (i : (mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (mk α✝ β✝ a✝¹ a✝).LeftMoves),
        P ⟦(mk α✝¹ β✝¹ a✝³ a✝²).mulOption (mk α✝ β✝ a✝¹ a✝) i j⟧) ∧
      ∀ (i : (-mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (-mk α✝ β✝ a✝¹ a✝).LeftMoves),
        P ⟦(-mk α✝¹ β✝¹ a✝³ a✝²).mulOption (-mk α✝ β✝ a✝¹ a✝) i j⟧
state_after: case mk.mk.mp
P : Game → Prop
α✝¹ β✝¹ : Type u_1
a✝³ : α✝¹ → PGame
a✝² : β✝¹ → PGame
α✝ β✝ : Type u_1
a✝¹ : α✝ → PGame
a✝ : β✝ → PGame
h : ∀ (k : (mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).LeftMoves), P ⟦(mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).moveLeft k⟧
⊢ (∀ (i : (mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (mk α✝ β✝ a✝¹ a✝).LeftMoves),
      P ⟦(mk α✝¹ β✝¹ a✝³ a✝²).mulOption (mk α✝ β✝ a✝¹ a✝) i j⟧) ∧
    ∀ (i : (-mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (-mk α✝ β✝ a✝¹ a✝).LeftMoves),
      P ⟦(-mk α✝¹ β✝¹ a✝³ a✝²).mulOption (-mk α✝ β✝ a✝¹ a✝) i j⟧

case mk.mk.mpr
P : Game → Prop
α✝¹ β✝¹ : Type u_1
a✝³ : α✝¹ → PGame
a✝² : β✝¹ → PGame
α✝ β✝ : Type u_1
a✝¹ : α✝ → PGame
a✝ : β✝ → PGame
h :
  (∀ (i : (mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (mk α✝ β✝ a✝¹ a✝).LeftMoves),
      P ⟦(mk α✝¹ β✝¹ a✝³ a✝²).mulOption (mk α✝ β✝ a✝¹ a✝) i j⟧) ∧
    ∀ (i : (-mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (-mk α✝ β✝ a✝¹ a✝).LeftMoves),
      P ⟦(-mk α✝¹ β✝¹ a✝³ a✝²).mulOption (-mk α✝ β✝ a✝¹ a✝) i j⟧
⊢ ∀ (k : (mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).LeftMoves), P ⟦(mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).moveLeft k⟧
tactic: constructor <;> intro h
state_before: case h.e'_1
P : Game → Prop
α✝¹ β✝¹ : Type u_1
a✝³ : α✝¹ → PGame
a✝² : β✝¹ → PGame
α✝ β✝ : Type u_1
a✝¹ : α✝ → PGame
a✝ : β✝ → PGame
h : ∀ (k : (mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).LeftMoves), P ⟦(mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).moveLeft k⟧
i : (-mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves
j : (-mk α✝ β✝ a✝¹ a✝).LeftMoves
⊢ ⟦(-mk α✝¹ β✝¹ a✝³ a✝²).mulOption (-mk α✝ β✝ a✝¹ a✝) i j⟧ =
    ⟦(mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).moveLeft (Sum.inr (i, j))⟧

case h.e'_1
P : Game → Prop
α✝¹ β✝¹ : Type u_1
a✝³ : α✝¹ → PGame
a✝² : β✝¹ → PGame
α✝ β✝ : Type u_1
a✝¹ : α✝ → PGame
a✝ : β✝ → PGame
h :
  (∀ (i : (mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (mk α✝ β✝ a✝¹ a✝).LeftMoves),
      P ⟦(mk α✝¹ β✝¹ a✝³ a✝²).mulOption (mk α✝ β✝ a✝¹ a✝) i j⟧) ∧
    ∀ (i : (-mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (-mk α✝ β✝ a✝¹ a✝).LeftMoves),
      P ⟦(-mk α✝¹ β✝¹ a✝³ a✝²).mulOption (-mk α✝ β✝ a✝¹ a✝) i j⟧
i : β✝¹
j : β✝
⊢ ⟦(mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).moveLeft (Sum.inr (i, j))⟧ =
    ⟦(-mk α✝¹ β✝¹ a✝³ a✝²).mulOption (-mk α✝ β✝ a✝¹ a✝) i j⟧
state_after: no goals
tactic: all_goals
    dsimp only [mk_mul_moveLeft_inr, quot_sub, quot_add, neg_def, mulOption, moveLeft_mk]
    rw [← neg_def, ← neg_def]
    congr 1
    on_goal 1 => congr 1
    all_goals rw [quot_neg_mul_neg]