Surreal numbers

The basic theory of surreal numbers, built on top of the theory of combinatorial (pre-)games.

A pregame is Numeric if all the Left options are strictly smaller than all the Right options, and all those options are themselves numeric. In terms of combinatorial games, the numeric games have "frozen"; you can only make your position worse by playing, and Left is some definite "number" of moves ahead (or behind) Right.

A surreal number is an equivalence class of numeric pregames.

In fact, the surreals form a complete ordered field, containing a copy of the reals (and much else besides!) but we do not yet have a complete development.

Order properties

Surreal numbers inherit the relations ≤ and < from games (Surreal.instLE and Surreal.instLT), and these relations satisfy the axioms of a partial order.

Algebraic operations

We show that the surreals form a linear ordered commutative group.

One can also map all the ordinals into the surreals!

Multiplication of surreal numbers

The proof that multiplication lifts to surreal numbers is surprisingly difficult and is currently missing in the library. A sample proof can be found in Theorem 3.8 in the second reference below. The difficulty lies in the length of the proof and the number of theorems that need to proven simultaneously. This will make for a fun and challenging project.

The branch surreal_mul contains some progress on this proof.

Todo

• Define the field structure on the surreals.

References

• [Conway, On numbers and games][conway2001]
• [Schleicher, Stoll, An introduction to Conway's games and numbers][schleicher_stoll]

def SetTheory.PGame.Numeric : SetTheory.PGame → Prop

A pre-game is numeric if everything in the L set is less than everything in the R set, and all the elements of L and R are also numeric.

Equations

• SetTheory.PGame.Numeric (SetTheory.PGame.mk α β L R) = ((∀ (i : α) (j : β), L i < R j) ∧ (∀ (i : α), SetTheory.PGame.Numeric (L i)) ∧ ∀ (j : β), SetTheory.PGame.Numeric (R j))

theorem SetTheory.PGame.numeric_def {x : SetTheory.PGame} : SetTheory.PGame.Numeric x ↔ (∀ (i : SetTheory.PGame.LeftMoves x) (j : SetTheory.PGame.RightMoves x), SetTheory.PGame.moveLeft x i < SetTheory.PGame.moveRight x j) ∧ (∀ (i : SetTheory.PGame.LeftMoves x), SetTheory.PGame.Numeric (SetTheory.PGame.moveLeft x i)) ∧ ∀ (j : SetTheory.PGame.RightMoves x), SetTheory.PGame.Numeric (SetTheory.PGame.moveRight x j)

theorem SetTheory.PGame.Numeric.mk {x : SetTheory.PGame} (h₁ : ∀ (i : SetTheory.PGame.LeftMoves x) (j : SetTheory.PGame.RightMoves x), SetTheory.PGame.moveLeft x i < SetTheory.PGame.moveRight x j) (h₂ : ∀ (i : SetTheory.PGame.LeftMoves x), SetTheory.PGame.Numeric (SetTheory.PGame.moveLeft x i)) (h₃ : ∀ (j : SetTheory.PGame.RightMoves x), SetTheory.PGame.Numeric (SetTheory.PGame.moveRight x j)) : SetTheory.PGame.Numeric x

theorem SetTheory.PGame.Numeric.left_lt_right {x : SetTheory.PGame} (o : SetTheory.PGame.Numeric x) (i : SetTheory.PGame.LeftMoves x) (j : SetTheory.PGame.RightMoves x) : SetTheory.PGame.moveLeft x i < SetTheory.PGame.moveRight x j

theorem SetTheory.PGame.Numeric.moveLeft {x : SetTheory.PGame} (o : SetTheory.PGame.Numeric x) (i : SetTheory.PGame.LeftMoves x) : SetTheory.PGame.Numeric (SetTheory.PGame.moveLeft x i)

theorem SetTheory.PGame.Numeric.moveRight {x : SetTheory.PGame} (o : SetTheory.PGame.Numeric x) (j : SetTheory.PGame.RightMoves x) : SetTheory.PGame.Numeric (SetTheory.PGame.moveRight x j)

theorem SetTheory.PGame.numeric_rec {C : SetTheory.PGame → Prop} (H : (l r : Type u_1) → (L : l → SetTheory.PGame) → (R : r → SetTheory.PGame) → (∀ (i : l) (j : r), L i < R j) → (∀ (i : l), SetTheory.PGame.Numeric (L i)) → (∀ (i : r), SetTheory.PGame.Numeric (R i)) → ((i : l) → C (L i)) → ((i : r) → C (R i)) → C (SetTheory.PGame.mk l r L R)) (x : SetTheory.PGame) : SetTheory.PGame.Numeric x → C x

theorem SetTheory.PGame.Relabelling.numeric_imp {x : SetTheory.PGame} {y : SetTheory.PGame} (r : SetTheory.PGame.Relabelling x y) (ox : SetTheory.PGame.Numeric x) : SetTheory.PGame.Numeric y

theorem SetTheory.PGame.Relabelling.numeric_congr {x : SetTheory.PGame} {y : SetTheory.PGame} (r : SetTheory.PGame.Relabelling x y) : SetTheory.PGame.Numeric x ↔ SetTheory.PGame.Numeric y

Relabellings preserve being numeric.

theorem SetTheory.PGame.lf_asymm {x : SetTheory.PGame} {y : SetTheory.PGame} (ox : SetTheory.PGame.Numeric x) (oy : SetTheory.PGame.Numeric y) : SetTheory.PGame.LF x y → ¬SetTheory.PGame.LF y x

theorem SetTheory.PGame.le_of_lf {x : SetTheory.PGame} {y : SetTheory.PGame} (h : SetTheory.PGame.LF x y) (ox : SetTheory.PGame.Numeric x) (oy : SetTheory.PGame.Numeric y) : x ≤ y

theorem SetTheory.PGame.LF.le {x : SetTheory.PGame} {y : SetTheory.PGame} (h : SetTheory.PGame.LF x y) (ox : SetTheory.PGame.Numeric x) (oy : SetTheory.PGame.Numeric y) : x ≤ y

Alias of SetTheory.PGame.le_of_lf.

theorem SetTheory.PGame.lt_of_lf {x : SetTheory.PGame} {y : SetTheory.PGame} (h : SetTheory.PGame.LF x y) (ox : SetTheory.PGame.Numeric x) (oy : SetTheory.PGame.Numeric y) : x < y

theorem SetTheory.PGame.LF.lt {x : SetTheory.PGame} {y : SetTheory.PGame} (h : SetTheory.PGame.LF x y) (ox : SetTheory.PGame.Numeric x) (oy : SetTheory.PGame.Numeric y) : x < y

Alias of SetTheory.PGame.lt_of_lf.

theorem SetTheory.PGame.lf_iff_lt {x : SetTheory.PGame} {y : SetTheory.PGame} (ox : SetTheory.PGame.Numeric x) (oy : SetTheory.PGame.Numeric y) : SetTheory.PGame.LF x y ↔ x < y

theorem SetTheory.PGame.le_iff_forall_lt {x : SetTheory.PGame} {y : SetTheory.PGame} (ox : SetTheory.PGame.Numeric x) (oy : SetTheory.PGame.Numeric y) : x ≤ y ↔ (∀ (i : SetTheory.PGame.LeftMoves x), SetTheory.PGame.moveLeft x i < y) ∧ ∀ (j : SetTheory.PGame.RightMoves y), x < SetTheory.PGame.moveRight y j

Definition of x ≤ y on numeric pre-games, in terms of <

theorem SetTheory.PGame.lt_iff_exists_le {x : SetTheory.PGame} {y : SetTheory.PGame} (ox : SetTheory.PGame.Numeric x) (oy : SetTheory.PGame.Numeric y) : x < y ↔ (∃ i, x ≤ SetTheory.PGame.moveLeft y i) ∨ ∃ j, SetTheory.PGame.moveRight x j ≤ y

Definition of x < y on numeric pre-games, in terms of ≤

theorem SetTheory.PGame.lt_of_exists_le {x : SetTheory.PGame} {y : SetTheory.PGame} (ox : SetTheory.PGame.Numeric x) (oy : SetTheory.PGame.Numeric y) : ((∃ i, x ≤ SetTheory.PGame.moveLeft y i) ∨ ∃ j, SetTheory.PGame.moveRight x j ≤ y) → x < y

theorem SetTheory.PGame.lt_def {x : SetTheory.PGame} {y : SetTheory.PGame} (ox : SetTheory.PGame.Numeric x) (oy : SetTheory.PGame.Numeric y) : x < y ↔ (∃ i, (∀ (i' : SetTheory.PGame.LeftMoves x), SetTheory.PGame.moveLeft x i' < SetTheory.PGame.moveLeft y i) ∧ ∀ (j : SetTheory.PGame.RightMoves (SetTheory.PGame.moveLeft y i)), x < SetTheory.PGame.moveRight (SetTheory.PGame.moveLeft y i) j) ∨ ∃ j, (∀ (i : SetTheory.PGame.LeftMoves (SetTheory.PGame.moveRight x j)), SetTheory.PGame.moveLeft (SetTheory.PGame.moveRight x j) i < y) ∧ ∀ (j' : SetTheory.PGame.RightMoves y), SetTheory.PGame.moveRight x j < SetTheory.PGame.moveRight y j'

The definition of x < y on numeric pre-games, in terms of < two moves later.

theorem SetTheory.PGame.not_fuzzy {x : SetTheory.PGame} {y : SetTheory.PGame} (ox : SetTheory.PGame.Numeric x) (oy : SetTheory.PGame.Numeric y) : ¬SetTheory.PGame.Fuzzy x y

theorem SetTheory.PGame.lt_or_equiv_or_gt {x : SetTheory.PGame} {y : SetTheory.PGame} (ox : SetTheory.PGame.Numeric x) (oy : SetTheory.PGame.Numeric y) : x < y ∨ x ≈ y ∨ y < x

theorem SetTheory.PGame.numeric_of_isEmpty (x : SetTheory.PGame) [IsEmpty (SetTheory.PGame.LeftMoves x)] [IsEmpty (SetTheory.PGame.RightMoves x)] : SetTheory.PGame.Numeric x

theorem SetTheory.PGame.numeric_of_isEmpty_leftMoves (x : SetTheory.PGame) [IsEmpty (SetTheory.PGame.LeftMoves x)] : (∀ (j : SetTheory.PGame.RightMoves x), SetTheory.PGame.Numeric (SetTheory.PGame.moveRight x j)) → SetTheory.PGame.Numeric x

theorem SetTheory.PGame.numeric_of_isEmpty_rightMoves (x : SetTheory.PGame) [IsEmpty (SetTheory.PGame.RightMoves x)] (H : ∀ (i : SetTheory.PGame.LeftMoves x), SetTheory.PGame.Numeric (SetTheory.PGame.moveLeft x i)) : SetTheory.PGame.Numeric x

theorem SetTheory.PGame.numeric_zero : SetTheory.PGame.Numeric 0

theorem SetTheory.PGame.numeric_one : SetTheory.PGame.Numeric 1

theorem SetTheory.PGame.Numeric.neg {x : SetTheory.PGame} : SetTheory.PGame.Numeric x → SetTheory.PGame.Numeric (-x)

theorem SetTheory.PGame.Numeric.moveLeft_lt {x : SetTheory.PGame} (o : SetTheory.PGame.Numeric x) (i : SetTheory.PGame.LeftMoves x) : SetTheory.PGame.moveLeft x i < x

theorem SetTheory.PGame.Numeric.moveLeft_le {x : SetTheory.PGame} (o : SetTheory.PGame.Numeric x) (i : SetTheory.PGame.LeftMoves x) : SetTheory.PGame.moveLeft x i ≤ x

theorem SetTheory.PGame.Numeric.lt_moveRight {x : SetTheory.PGame} (o : SetTheory.PGame.Numeric x) (j : SetTheory.PGame.RightMoves x) : x < SetTheory.PGame.moveRight x j

theorem SetTheory.PGame.Numeric.le_moveRight {x : SetTheory.PGame} (o : SetTheory.PGame.Numeric x) (j : SetTheory.PGame.RightMoves x) : x ≤ SetTheory.PGame.moveRight x j

theorem SetTheory.PGame.Numeric.add {x : SetTheory.PGame} {y : SetTheory.PGame} : SetTheory.PGame.Numeric x → SetTheory.PGame.Numeric y → SetTheory.PGame.Numeric (x + y)

theorem SetTheory.PGame.Numeric.sub {x : SetTheory.PGame} {y : SetTheory.PGame} (ox : SetTheory.PGame.Numeric x) (oy : SetTheory.PGame.Numeric y) : SetTheory.PGame.Numeric (x - y)

theorem SetTheory.PGame.numeric_nat (n : ℕ) : SetTheory.PGame.Numeric ↑n

Pre-games defined by natural numbers are numeric.

theorem SetTheory.PGame.numeric_toPGame (o : Ordinal.{u_1}) : SetTheory.PGame.Numeric (Ordinal.toPGame o)

Ordinal games are numeric.

def Surreal : Type (u_1 + 1)

The type of surreal numbers. These are the numeric pre-games quotiented by the equivalence relation x ≈ y ↔ x ≤ y ∧ y ≤ x. In the quotient, the order becomes a total order.

def Surreal.mk (x : SetTheory.PGame) (h : SetTheory.PGame.Numeric x) : Surreal

Construct a surreal number from a numeric pre-game.

instance Surreal.instZeroSurreal : Zero Surreal

instance Surreal.instOneSurreal : One Surreal

instance Surreal.instInhabitedSurreal : Inhabited Surreal

def Surreal.lift {α : Sort u_1} (f : (x : SetTheory.PGame) → SetTheory.PGame.Numeric x → α) (H : ∀ {x y : SetTheory.PGame} (hx : SetTheory.PGame.Numeric x) (hy : SetTheory.PGame.Numeric y), SetTheory.PGame.Equiv x y → f x hx = f y hy) : Surreal → α

Lift an equivalence-respecting function on pre-games to surreals.

def Surreal.lift₂ {α : Sort u_1} (f : (x : SetTheory.PGame) → (y : SetTheory.PGame) → SetTheory.PGame.Numeric x → SetTheory.PGame.Numeric y → α) (H : ∀ {x₁ : SetTheory.PGame} {y₁ : SetTheory.PGame} {x₂ : SetTheory.PGame} {y₂ : SetTheory.PGame} (ox₁ : SetTheory.PGame.Numeric x₁) (oy₁ : SetTheory.PGame.Numeric y₁) (ox₂ : SetTheory.PGame.Numeric x₂) (oy₂ : SetTheory.PGame.Numeric y₂), SetTheory.PGame.Equiv x₁ x₂ → SetTheory.PGame.Equiv y₁ y₂ → f x₁ y₁ ox₁ oy₁ = f x₂ y₂ ox₂ oy₂) : Surreal → Surreal → α

Lift a binary equivalence-respecting function on pre-games to surreals.

instance Surreal.instLE : LE Surreal

instance Surreal.instLT : LT Surreal

instance Surreal.instAddSurreal : Add Surreal

Addition on surreals is inherited from pre-game addition: the sum of x = {xL | xR} and y = {yL | yR} is {xL + y, x + yL | xR + y, x + yR}.

instance Surreal.instNegSurreal : Neg Surreal

Negation for surreal numbers is inherited from pre-game negation: the negation of {L | R} is {-R | -L}.

instance Surreal.orderedAddCommGroup : OrderedAddCommGroup Surreal

noncomputable instance Surreal.instLinearOrderedAddCommGroupSurreal : LinearOrderedAddCommGroup Surreal

instance Surreal.instAddMonoidWithOneSurreal : AddMonoidWithOne Surreal

def Surreal.toGame : Surreal →+o SetTheory.Game

Casts a Surreal number into a Game.

theorem Surreal.zero_toGame : ↑Surreal.toGame 0 = 0

@[simp]

theorem Surreal.one_toGame : ↑Surreal.toGame 1 = 1

@[simp]

theorem Surreal.nat_toGame (n : ℕ) : ↑Surreal.toGame ↑n = ↑n

noncomputable def Ordinal.toSurreal : Ordinal.{u_1} ↪o Surreal

Converts an ordinal into the corresponding surreal.