# mathlibdocumentation

set_theory.game.birthday

# Birthdays of games #

The birthday of a game is an ordinal that represents at which "step" the game was constructed. We define it recursively as the least ordinal larger than the birthdays of its left and right games. We prove the basic properties about these.

# Main declarations #

• pgame.birthday: The birthday of a pre-game.

# Todo #

• Define the birthdays of games and surreals.
• Characterize the birthdays of basic arithmetical operations.
noncomputable def pgame.birthday  :

The birthday of a pre-game is inductively defined as the least strict upper bound of the birthdays of its left and right games. It may be thought as the "step" in which a certain game is constructed.

Equations
theorem pgame.lt_birthday_iff {x : pgame} {o : ordinal} :
o < x.birthday (∃ (i : x.left_moves), o (x.move_left i).birthday) ∃ (i : x.right_moves), o (x.move_right i).birthday
@[simp]
theorem pgame.birthday_add_zero (x : pgame) :
@[simp]
theorem pgame.birthday_zero_add (x : pgame) :
@[simp]
theorem pgame.birthday_eq_zero (x : pgame) :
@[simp]
theorem pgame.birthday_zero  :
@[simp]
theorem pgame.birthday_one  :
@[simp]
theorem pgame.birthday_star  :
@[simp]
theorem pgame.neg_birthday (x : pgame) :
@[simp]
theorem pgame.to_pgame_birthday (o : ordinal) :
theorem pgame.le_birthday (x : pgame) :