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.

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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.

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Instances For
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    @[simp]
    theorem SetTheory.PGame.powHalf_zero :
    SetTheory.PGame.powHalf 0 = 1
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    theorem SetTheory.PGame.powHalf_leftMoves (n : ℕ) :
    SetTheory.PGame.LeftMoves (SetTheory.PGame.powHalf n) = PUnit.{u_1 + 1}
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    theorem SetTheory.PGame.powHalf_zero_rightMoves :
    SetTheory.PGame.RightMoves (SetTheory.PGame.powHalf 0) = PEmpty.{u_1 + 1}
    source
    theorem SetTheory.PGame.powHalf_succ_rightMoves (n : ℕ) :
    SetTheory.PGame.RightMoves (SetTheory.PGame.powHalf (n + 1)) = PUnit.{u_1 + 1}
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    @[simp]
    theorem SetTheory.PGame.powHalf_moveLeft (n : ℕ) (i : SetTheory.PGame.LeftMoves (SetTheory.PGame.powHalf n)) :
    SetTheory.PGame.moveLeft (SetTheory.PGame.powHalf n) i = 0
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    @[simp]
    theorem SetTheory.PGame.powHalf_succ_moveRight (n : ℕ) (i : SetTheory.PGame.RightMoves (SetTheory.PGame.powHalf (n + 1))) :
    SetTheory.PGame.moveRight (SetTheory.PGame.powHalf (n + 1)) i = SetTheory.PGame.powHalf n
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    instance SetTheory.PGame.uniquePowHalfLeftMoves (n : ℕ) :
    Unique (SetTheory.PGame.LeftMoves (SetTheory.PGame.powHalf n))
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    instance SetTheory.PGame.isEmpty_powHalf_zero_rightMoves :
    IsEmpty (SetTheory.PGame.RightMoves (SetTheory.PGame.powHalf 0))
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    instance SetTheory.PGame.uniquePowHalfSuccRightMoves (n : ℕ) :
    Unique (SetTheory.PGame.RightMoves (SetTheory.PGame.powHalf (n + 1)))
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    @[simp]
    theorem SetTheory.PGame.birthday_half :
    SetTheory.PGame.birthday (SetTheory.PGame.powHalf 1) = 2
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    theorem SetTheory.PGame.numeric_powHalf (n : ℕ) :
    SetTheory.PGame.Numeric (SetTheory.PGame.powHalf n)

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

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    theorem SetTheory.PGame.powHalf_succ_lt_powHalf (n : ℕ) :
    SetTheory.PGame.powHalf (n + 1) < SetTheory.PGame.powHalf n
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    theorem SetTheory.PGame.powHalf_succ_le_powHalf (n : ℕ) :
    SetTheory.PGame.powHalf (n + 1) ≤ SetTheory.PGame.powHalf n
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    theorem SetTheory.PGame.powHalf_le_one (n : ℕ) :
    SetTheory.PGame.powHalf n ≤ 1
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    theorem SetTheory.PGame.powHalf_succ_lt_one (n : ℕ) :
    SetTheory.PGame.powHalf (n + 1) < 1
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    theorem SetTheory.PGame.powHalf_pos (n : ℕ) :
    0 < SetTheory.PGame.powHalf n
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    theorem SetTheory.PGame.zero_le_powHalf (n : ℕ) :
    0 ≤ SetTheory.PGame.powHalf n
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    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
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    theorem SetTheory.PGame.half_add_half_equiv_one :
    SetTheory.PGame.powHalf 1 + SetTheory.PGame.powHalf 1 ≈ 1
    source
    def Surreal.powHalf (n : ℕ) :
    Surreal

    Powers of the surreal number half.

    Instances For
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      @[simp]
      theorem Surreal.powHalf_zero :
      Surreal.powHalf 0 = 1
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      @[simp]
      theorem Surreal.double_powHalf_succ_eq_powHalf (n : ℕ) :
      2 • Surreal.powHalf (Nat.succ n) = Surreal.powHalf n
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      @[simp]
      theorem Surreal.nsmul_pow_two_powHalf (n : ℕ) :
      2 ^ n • Surreal.powHalf n = 1
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      @[simp]
      theorem Surreal.nsmul_pow_two_powHalf' (n : ℕ) (k : ℕ) :
      2 ^ n • Surreal.powHalf (n + k) = Surreal.powHalf k
      source
      theorem Surreal.zsmul_pow_two_powHalf (m : ℤ) (n : ℕ) (k : ℕ) :
      (m * 2 ^ n) • Surreal.powHalf (n + k) = m • Surreal.powHalf k
      source
      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
      source
      def Surreal.dyadicMap :
      Localization.Away 2 →+ Surreal

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

      Instances For
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        @[simp]
        theorem Surreal.dyadicMap_apply (m : ℤ) (p : { x // x ∈ Submonoid.powers 2 }) :
        ↑Surreal.dyadicMap (IsLocalization.mk' (Localization (Submonoid.powers 2)) m p) = m • Surreal.powHalf (Submonoid.log p)
        source
        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
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        @[simp]
        theorem Surreal.dyadicMap_apply_pow' (m : ℤ) (n : ℕ) :
        m • Surreal.powHalf (Submonoid.log (Submonoid.pow 2 n)) = m • Surreal.powHalf n
        source
        def Surreal.dyadic :
        Set Surreal

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

        Instances For