Documentation

Mathlib.SetTheory.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 #

Todo #

source
noncomputable def SetTheory.PGame.birthday :
SetTheory.PGame → Ordinal.{u}

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
Instances For
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    theorem SetTheory.PGame.birthday_def (x : SetTheory.PGame) :
    x.birthday = max (Ordinal.lsub fun (i : x.LeftMoves) => (x.moveLeft i).birthday) (Ordinal.lsub fun (i : x.RightMoves) => (x.moveRight i).birthday)
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    theorem SetTheory.PGame.birthday_moveLeft_lt {x : SetTheory.PGame} (i : x.LeftMoves) :
    (x.moveLeft i).birthday < x.birthday
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    theorem SetTheory.PGame.birthday_moveRight_lt {x : SetTheory.PGame} (i : x.RightMoves) :
    (x.moveRight i).birthday < x.birthday
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    theorem SetTheory.PGame.lt_birthday_iff {x : SetTheory.PGame} {o : Ordinal.{u_1}} :
    o < x.birthday ↔ (∃ (i : x.LeftMoves), o ≤ (x.moveLeft i).birthday) ∨ ∃ (i : x.RightMoves), o ≤ (x.moveRight i).birthday
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    theorem SetTheory.PGame.Relabelling.birthday_congr {x : SetTheory.PGame} {y : SetTheory.PGame} :
    x.Relabelling y → x.birthday = y.birthday
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    @[simp]
    theorem SetTheory.PGame.birthday_eq_zero {x : SetTheory.PGame} :
    x.birthday = 0 ↔ IsEmpty x.LeftMoves ∧ IsEmpty x.RightMoves
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    @[simp]
    theorem SetTheory.PGame.birthday_zero :
    SetTheory.PGame.birthday 0 = 0
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    @[simp]
    theorem SetTheory.PGame.birthday_one :
    SetTheory.PGame.birthday 1 = 1
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    @[simp]
    theorem SetTheory.PGame.birthday_star :
    SetTheory.PGame.star.birthday = 1
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    @[simp]
    theorem SetTheory.PGame.neg_birthday (x : SetTheory.PGame) :
    (-x).birthday = x.birthday
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    @[simp]
    theorem SetTheory.PGame.toPGame_birthday (o : Ordinal.{u_1}) :
    o.toPGame.birthday = o
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    theorem SetTheory.PGame.le_birthday (x : SetTheory.PGame) :
    x ≤ x.birthday.toPGame
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    theorem SetTheory.PGame.neg_birthday_le (x : SetTheory.PGame) :
    -x.birthday.toPGame ≤ x
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    @[simp]
    theorem SetTheory.PGame.birthday_add (x : SetTheory.PGame) (y : SetTheory.PGame) :
    (x + y).birthday = x.birthday.nadd y.birthday
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    theorem SetTheory.PGame.birthday_add_zero (a : SetTheory.PGame) :
    (a + 0).birthday = a.birthday
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    theorem SetTheory.PGame.birthday_zero_add (a : SetTheory.PGame) :
    (0 + a).birthday = a.birthday
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    theorem SetTheory.PGame.birthday_add_one (a : SetTheory.PGame) :
    (a + 1).birthday = Order.succ a.birthday
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    theorem SetTheory.PGame.birthday_one_add (a : SetTheory.PGame) :
    (1 + a).birthday = Order.succ a.birthday
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    @[simp]
    theorem SetTheory.PGame.birthday_natCast (n : ℕ) :
    (↑n).birthday = ↑n
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    theorem SetTheory.PGame.birthday_add_nat (a : SetTheory.PGame) (n : ℕ) :
    (a + ↑n).birthday = a.birthday + ↑n
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    theorem SetTheory.PGame.birthday_nat_add (a : SetTheory.PGame) (n : ℕ) :
    (↑n + a).birthday = a.birthday + ↑n