# Basic definitions about impartial (pre-)games #

We will define an impartial game, one in which left and right can make exactly the same moves. Our definition differs slightly by saying that the game is always equivalent to its negative, no matter what moves are played. This allows for games such as poker-nim to be classified as impartial.

@[irreducible]

The definition for an impartial game, defined using Conway induction.

Equations
• x.ImpartialAux = (x -x (∀ (i : x.LeftMoves), (x.moveLeft i).ImpartialAux) ∀ (j : x.RightMoves), (x.moveRight j).ImpartialAux)
Instances For
theorem SetTheory.PGame.impartialAux_def {G : SetTheory.PGame} :
G.ImpartialAux G -G (∀ (i : G.LeftMoves), (G.moveLeft i).ImpartialAux) ∀ (j : G.RightMoves), (G.moveRight j).ImpartialAux

A typeclass on impartial games.

• out : G.ImpartialAux
Instances
theorem SetTheory.PGame.Impartial.out {G : SetTheory.PGame} [self : G.Impartial] :
G.ImpartialAux
theorem SetTheory.PGame.impartial_iff_aux {G : SetTheory.PGame} :
G.Impartial G.ImpartialAux
theorem SetTheory.PGame.impartial_def {G : SetTheory.PGame} :
G.Impartial G -G (∀ (i : G.LeftMoves), (G.moveLeft i).Impartial) ∀ (j : G.RightMoves), (G.moveRight j).Impartial
@[simp]
theorem SetTheory.PGame.Impartial.mk'_neg_equiv_self (G : SetTheory.PGame) [G.Impartial] :
-G = G
instance SetTheory.PGame.Impartial.moveLeft_impartial {G : SetTheory.PGame} [h : G.Impartial] (i : G.LeftMoves) :
(G.moveLeft i).Impartial
Equations
• =
instance SetTheory.PGame.Impartial.moveRight_impartial {G : SetTheory.PGame} [h : G.Impartial] (j : G.RightMoves) :
(G.moveRight j).Impartial
Equations
• =
@[irreducible]
theorem SetTheory.PGame.Impartial.impartial_congr {G : SetTheory.PGame} {H : SetTheory.PGame} :
G.Relabelling H∀ [inst : G.Impartial], H.Impartial
@[irreducible]
instance SetTheory.PGame.Impartial.impartial_add (G : SetTheory.PGame) (H : SetTheory.PGame) [G.Impartial] [H.Impartial] :
(G + H).Impartial
Equations
• =
@[irreducible]
instance SetTheory.PGame.Impartial.impartial_neg (G : SetTheory.PGame) [G.Impartial] :
(-G).Impartial
Equations
• =
theorem SetTheory.PGame.Impartial.equiv_or_fuzzy_zero (G : SetTheory.PGame) [G.Impartial] :
G 0 G.Fuzzy 0

In an impartial game, either the first player always wins, or the second player always wins.

@[simp]
theorem SetTheory.PGame.Impartial.not_equiv_zero_iff (G : SetTheory.PGame) [G.Impartial] :
¬G 0 G.Fuzzy 0
@[simp]
theorem SetTheory.PGame.Impartial.not_fuzzy_zero_iff (G : SetTheory.PGame) [G.Impartial] :
¬G.Fuzzy 0 G 0
@[simp]
theorem SetTheory.PGame.Impartial.mk'_add_self (G : SetTheory.PGame) [G.Impartial] :
G + G = 0

This lemma doesn't require H to be impartial.

This lemma doesn't require H to be impartial.

theorem SetTheory.PGame.Impartial.fuzzy_zero_iff_lf (G : SetTheory.PGame) [G.Impartial] :
G.Fuzzy 0 G.LF 0
theorem SetTheory.PGame.Impartial.forall_leftMoves_fuzzy_iff_equiv_zero (G : SetTheory.PGame) [G.Impartial] :
(∀ (i : G.LeftMoves), (G.moveLeft i).Fuzzy 0) G 0
theorem SetTheory.PGame.Impartial.forall_rightMoves_fuzzy_iff_equiv_zero (G : SetTheory.PGame) [G.Impartial] :
(∀ (j : G.RightMoves), (G.moveRight j).Fuzzy 0) G 0
theorem SetTheory.PGame.Impartial.exists_left_move_equiv_iff_fuzzy_zero (G : SetTheory.PGame) [G.Impartial] :
(∃ (i : G.LeftMoves), G.moveLeft i 0) G.Fuzzy 0
theorem SetTheory.PGame.Impartial.exists_right_move_equiv_iff_fuzzy_zero (G : SetTheory.PGame) [G.Impartial] :
(∃ (j : G.RightMoves), G.moveRight j 0) G.Fuzzy 0