Dyadic numbers

Dyadic numbers are obtained by localizing ℤ away from 2. They are the initial object in the category of rings with no 2-torsion.

Dyadic surreal numbers

We construct dyadic surreal numbers using the canonical map from ℤ[2 ^ {-1}] to surreals. As we currently do not have a ring structure on Surreal we construct this map explicitly. Once we have the ring structure, this map can be constructed directly by sending 2 ^ {-1} to half.

Embeddings

The above construction gives us an abelian group embedding of ℤ into Surreal. The goal is to extend this to an embedding of dyadic rationals into Surreal and use Cauchy sequences of dyadic rational numbers to construct an ordered field embedding of ℝ into Surreal.

def SetTheory.PGame.powHalf : ℕ → PGame

For a natural number n, the pre-game powHalf (n + 1) is recursively defined as {0 | powHalf n}. These are the explicit expressions of powers of 1 / 2. By definition, we have powHalf 0 = 1 and powHalf 1 ≈ 1 / 2 and we prove later on that powHalf (n + 1) + powHalf (n + 1) ≈ powHalf n.

Equations

• SetTheory.PGame.powHalf 0 = 1
• SetTheory.PGame.powHalf n.succ = SetTheory.PGame.mk PUnit.{?u.89 + 1} PUnit.{?u.89 + 1} 0 fun (x : PUnit.{?u.89 + 1}) => SetTheory.PGame.powHalf n

@[simp]

theorem SetTheory.PGame.powHalf_zero : powHalf 0 = 1

theorem SetTheory.PGame.powHalf_leftMoves (n : ℕ) : (powHalf n).LeftMoves = PUnit.{u_1 + 1}

theorem SetTheory.PGame.powHalf_zero_rightMoves : (powHalf 0).RightMoves = PEmpty.{u_1 + 1}

theorem SetTheory.PGame.powHalf_succ_rightMoves (n : ℕ) : (powHalf (n + 1)).RightMoves = PUnit.{u_1 + 1}

@[simp]

theorem SetTheory.PGame.powHalf_moveLeft (n : ℕ) (i : (powHalf n).LeftMoves) : (powHalf n).moveLeft i = 0

@[simp]

theorem SetTheory.PGame.powHalf_succ_moveRight (n : ℕ) (i : (powHalf (n + 1)).RightMoves) : (powHalf (n + 1)).moveRight i = powHalf n

instance SetTheory.PGame.uniquePowHalfLeftMoves (n : ℕ) : Unique (powHalf n).LeftMoves

Equations

• One or more equations did not get rendered due to their size.

instance SetTheory.PGame.isEmpty_powHalf_zero_rightMoves : IsEmpty (powHalf 0).RightMoves

instance SetTheory.PGame.uniquePowHalfSuccRightMoves (n : ℕ) : Unique (powHalf (n + 1)).RightMoves

Equations

• SetTheory.PGame.uniquePowHalfSuccRightMoves n = PUnit.instUnique

@[simp]

theorem SetTheory.PGame.birthday_half : (powHalf 1).birthday = 2

theorem SetTheory.PGame.numeric_powHalf (n : ℕ) : (powHalf n).Numeric

For all natural numbers n, the pre-games powHalf n are numeric.

theorem SetTheory.PGame.powHalf_succ_lt_powHalf (n : ℕ) : powHalf (n + 1) < powHalf n

theorem SetTheory.PGame.powHalf_succ_le_powHalf (n : ℕ) : powHalf (n + 1) ≤ powHalf n

theorem SetTheory.PGame.powHalf_le_one (n : ℕ) : powHalf n ≤ 1

theorem SetTheory.PGame.powHalf_succ_lt_one (n : ℕ) : powHalf (n + 1) < 1

theorem SetTheory.PGame.powHalf_pos (n : ℕ) : 0 < powHalf n

theorem SetTheory.PGame.zero_le_powHalf (n : ℕ) : 0 ≤ powHalf n

theorem SetTheory.PGame.add_powHalf_succ_self_eq_powHalf (n : ℕ) : powHalf (n + 1) + powHalf (n + 1) ≈ powHalf n

theorem SetTheory.PGame.half_add_half_equiv_one : powHalf 1 + powHalf 1 ≈ 1

def Surreal.powHalf (n : ℕ) : Surreal

Powers of the surreal number half.

Equations

• Surreal.powHalf n = ⟦⟨SetTheory.PGame.powHalf n, ⋯⟩⟧

@[simp]

theorem Surreal.powHalf_zero : powHalf 0 = 1

@[simp]

theorem Surreal.double_powHalf_succ_eq_powHalf (n : ℕ) : 2 * powHalf (n + 1) = powHalf n

@[simp]

theorem Surreal.nsmul_pow_two_powHalf (n : ℕ) : 2 ^ n * powHalf n = 1

@[simp]

theorem Surreal.nsmul_pow_two_powHalf' (n k : ℕ) : 2 ^ n * powHalf (n + k) = powHalf k

theorem Surreal.zsmul_pow_two_powHalf (m : ℤ) (n k : ℕ) : ↑m * 2 ^ n * powHalf (n + k) = ↑m * powHalf k

theorem Surreal.dyadic_aux {m₁ m₂ : ℤ} {y₁ y₂ : ℕ} (h₂ : m₁ * 2 ^ y₁ = m₂ * 2 ^ y₂) : ↑m₁ * powHalf y₂ = ↑m₂ * powHalf y₁

noncomputable def Surreal.dyadicMap : Localization.Away 2 →+ Surreal

The additive monoid morphism dyadicMap sends ⟦⟨m, 2^n⟩⟧ to m • half ^ n.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Surreal.dyadicMap_apply (m : ℤ) (p : ↥(Submonoid.powers 2)) : dyadicMap (IsLocalization.mk' (Localization (Submonoid.powers 2)) m p) = ↑m * powHalf (Submonoid.log p)

theorem Surreal.dyadicMap_apply_pow (m : ℤ) (n : ℕ) : dyadicMap (IsLocalization.mk' (Localization (Submonoid.powers 2)) m (Submonoid.pow 2 n)) = m • powHalf n

@[simp]

theorem Surreal.dyadicMap_apply_pow' (m : ℤ) (n : ℕ) : ↑m * powHalf (Submonoid.log (Submonoid.pow 2 n)) = ↑m * powHalf n

def Surreal.dyadic : Set Surreal

We define dyadic surreals as the range of the map dyadicMap.

Equations

• Surreal.dyadic = Set.range ⇑Surreal.dyadicMap