mathlib documentation

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 classifed as impartial.

theorem pgame.impartial_def {G : pgame} :
G.impartial G.equiv (-G) (∀ (i : G.left_moves), (G.move_left i).impartial) ∀ (j : G.right_moves), (G.move_right j).impartial


theorem pgame.impartial.lt_zero_iff {G : pgame} [G.impartial] :
G < 0 0 < G