mathlib documentation

set_theory.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 pgame.pow_half  :
ℕ → pgame

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

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@[simp]
theorem pgame.pow_half_left_moves {n : ℕ} :
(pgame.pow_half n).left_moves = punit
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@[simp]
theorem pgame.pow_half_right_moves {n : ℕ} :
(pgame.pow_half (n + 1)).right_moves = punit
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@[simp]
theorem pgame.pow_half_move_left {n : ℕ} {i : (pgame.pow_half n).left_moves} :
(pgame.pow_half n).move_left i = 0
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@[simp]
theorem pgame.pow_half_move_right {n : ℕ} {i : (pgame.pow_half (n + 1)).right_moves} :
(pgame.pow_half (n + 1)).move_right i = pgame.pow_half n
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theorem pgame.pow_half_move_left' (n : ℕ) :
(pgame.pow_half n).move_left (⇑(equiv.cast _) punit.star) = 0
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theorem pgame.pow_half_move_right' (n : ℕ) :
(pgame.pow_half (n + 1)).move_right (⇑(equiv.cast _) punit.star) = pgame.pow_half n
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theorem pgame.numeric_pow_half {n : ℕ} :
(pgame.pow_half n).numeric

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

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theorem pgame.pow_half_succ_lt_pow_half {n : ℕ} :
pgame.pow_half (n + 1) < pgame.pow_half n
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theorem pgame.pow_half_succ_le_pow_half {n : ℕ} :
pgame.pow_half (n + 1) ≤ pgame.pow_half n
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theorem pgame.zero_lt_pow_half {n : ℕ} :
0 < pgame.pow_half n
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theorem pgame.zero_le_pow_half {n : ℕ} :
0 ≤ pgame.pow_half n
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theorem pgame.add_pow_half_succ_self_eq_pow_half {n : ℕ} :
(pgame.pow_half (n + 1) + pgame.pow_half (n + 1)).equiv (pgame.pow_half n)
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def surreal.half  :
surreal

The surreal number half.

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def surreal.pow_half (n : ℕ) :
surreal

Powers of the surreal number half.

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@[simp]
theorem surreal.pow_half_zero  :
surreal.pow_half 0 = 1
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@[simp]
theorem surreal.pow_half_one  :
surreal.pow_half 1 = surreal.half
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@[simp]
theorem surreal.add_half_self_eq_one  :
surreal.half + surreal.half = 1
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theorem surreal.double_pow_half_succ_eq_pow_half (n : ℕ) :
2 • surreal.pow_half n.succ = surreal.pow_half n
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theorem surreal.nsmul_pow_two_pow_half (n : ℕ) :
2 ^ n • surreal.pow_half n = 1
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theorem surreal.nsmul_pow_two_pow_half' (n k : ℕ) :
2 ^ n • surreal.pow_half (n + k) = surreal.pow_half k
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theorem surreal.zsmul_pow_two_pow_half (m : ℤ) (n k : ℕ) :
(m * 2 ^ n) • surreal.pow_half (n + k) = m • surreal.pow_half k
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theorem surreal.dyadic_aux {m₁ m₂ : ℤ} {y₁ y₂ : ℕ} (h₂ : m₁ * 2 ^ y₁ = m₂ * 2 ^ y₂) :
m₁ • surreal.pow_half y₂ = m₂ • surreal.pow_half y₁
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def surreal.dyadic_map  :
localization.away 2 →+ surreal

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

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@[simp]
theorem surreal.dyadic_map_apply (m : ℤ) (p : ↥(submonoid.powers 2)) :
⇑surreal.dyadic_map (is_localization.mk' (localization (submonoid.powers 2)) m p) = m • surreal.pow_half (submonoid.log p)
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@[simp]
theorem surreal.dyadic_map_apply_pow (m : ℤ) (n : ℕ) :
⇑surreal.dyadic_map (is_localization.mk' (localization (submonoid.powers 2)) m (submonoid.pow 2 n)) = m • surreal.pow_half n
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def surreal.dyadic  :
set surreal

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

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