Documentation

Mathlib.SetTheory.Surreal.Dyadic

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 → SetTheory.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.147 + 1} PUnit.{?u.147 + 1} 0 fun (x : PUnit.{?u.147 + 1} ) => SetTheory.PGame.powHalf n

Instances For

@[simp]

theorem SetTheory.PGame.powHalf_zero :

SetTheory.PGame.powHalf 0 = 1

theorem SetTheory.PGame.powHalf_leftMoves (n : ℕ) :

(SetTheory.PGame.powHalf n).LeftMoves = PUnit.{u_1 + 1}

theorem SetTheory.PGame.powHalf_zero_rightMoves :

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

theorem SetTheory.PGame.powHalf_succ_rightMoves (n : ℕ) :

(SetTheory.PGame.powHalf (n + 1)).RightMoves = PUnit.{u_1 + 1}

@[simp]

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

(SetTheory.PGame.powHalf n).moveLeft i = 0

@[simp]

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

(SetTheory.PGame.powHalf (n + 1)).moveRight i = SetTheory.PGame.powHalf n

instance SetTheory.PGame.uniquePowHalfLeftMoves (n : ℕ) :

Unique (SetTheory.PGame.powHalf n).LeftMoves

Equations

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

instance SetTheory.PGame.isEmpty_powHalf_zero_rightMoves :

IsEmpty (SetTheory.PGame.powHalf 0).RightMoves

Equations

SetTheory.PGame.isEmpty_powHalf_zero_rightMoves = ⋯

instance SetTheory.PGame.uniquePowHalfSuccRightMoves (n : ℕ) :

Unique (SetTheory.PGame.powHalf (n + 1)).RightMoves

Equations

SetTheory.PGame.uniquePowHalfSuccRightMoves n = PUnit.unique

@[simp]

theorem SetTheory.PGame.birthday_half :

(SetTheory.PGame.powHalf 1).birthday = 2

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

(SetTheory.PGame.powHalf n).Numeric

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

theorem SetTheory.PGame.powHalf_succ_lt_powHalf (n : ℕ) :

SetTheory.PGame.powHalf (n + 1) < SetTheory.PGame.powHalf n

theorem SetTheory.PGame.powHalf_succ_le_powHalf (n : ℕ) :

SetTheory.PGame.powHalf (n + 1) ≤ SetTheory.PGame.powHalf n

theorem SetTheory.PGame.powHalf_le_one (n : ℕ) :

SetTheory.PGame.powHalf n ≤ 1

theorem SetTheory.PGame.powHalf_succ_lt_one (n : ℕ) :

SetTheory.PGame.powHalf (n + 1) < 1

theorem SetTheory.PGame.powHalf_pos (n : ℕ) :

0 < SetTheory.PGame.powHalf n

theorem SetTheory.PGame.zero_le_powHalf (n : ℕ) :

0 ≤ SetTheory.PGame.powHalf n

theorem SetTheory.PGame.add_powHalf_succ_self_eq_powHalf (n : ℕ) :

SetTheory.PGame.powHalf (n + 1) + SetTheory.PGame.powHalf (n + 1) ≈ SetTheory.PGame.powHalf n

theorem SetTheory.PGame.half_add_half_equiv_one :

SetTheory.PGame.powHalf 1 + SetTheory.PGame.powHalf 1 ≈ 1

def Surreal.powHalf (n : ℕ) :

Powers of the surreal number half.

Equations

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

Instances For

@[simp]

theorem Surreal.powHalf_zero :

Surreal.powHalf 0 = 1

@[simp]

theorem Surreal.double_powHalf_succ_eq_powHalf (n : ℕ) :

2 * Surreal.powHalf (n + 1) = Surreal.powHalf n

@[simp]

theorem Surreal.nsmul_pow_two_powHalf (n : ℕ) :

2 ^ n * Surreal.powHalf n = 1

@[simp]

theorem Surreal.nsmul_pow_two_powHalf' (n : ℕ) (k : ℕ) :

2 ^ n * Surreal.powHalf (n + k) = Surreal.powHalf k

theorem Surreal.zsmul_pow_two_powHalf (m : ℤ) (n : ℕ) (k : ℕ) :

↑m * 2 ^ n * Surreal.powHalf (n + k) = ↑m * Surreal.powHalf k

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

↑m₁ * Surreal.powHalf y₂ = ↑m₂ * Surreal.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.

Instances For

@[simp]

theorem Surreal.dyadicMap_apply (m : ℤ) (p : ↥(Submonoid.powers 2)) :

Surreal.dyadicMap (IsLocalization.mk' (Localization (Submonoid.powers 2)) m p) = ↑m * Surreal.powHalf (Submonoid.log p)

theorem Surreal.dyadicMap_apply_pow (m : ℤ) (n : ℕ) :

Surreal.dyadicMap (IsLocalization.mk' (Localization (Submonoid.powers 2)) m (Submonoid.pow 2 n)) = m • Surreal.powHalf n

@[simp]

theorem Surreal.dyadicMap_apply_pow' (m : ℤ) (n : ℕ) :

↑m * Surreal.powHalf (Submonoid.log (Submonoid.pow 2 n)) = ↑m * Surreal.powHalf n

def Surreal.dyadic :

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

Equations

Surreal.dyadic = Set.range ⇑Surreal.dyadicMap

Instances For