Documentation

Mathlib.Data.Nat.Size

Lemmas about size.

shiftLeft and shiftRight #

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theorem Nat.shiftLeft_eq_mul_pow (m : ℕ) (n : ℕ) :
m <<< n = m * 2 ^ n
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theorem Nat.shiftLeft'_tt_eq_mul_pow (m : ℕ) (n : ℕ) :
Nat.shiftLeft' true m n + 1 = (m + 1) * 2 ^ n
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theorem Nat.shiftLeft'_ne_zero_left (b : Bool) {m : ℕ} (h : m ≠ 0) (n : ℕ) :
Nat.shiftLeft' b m n ≠ 0
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theorem Nat.shiftLeft'_tt_ne_zero (m : ℕ) {n : ℕ} :
n ≠ 0 → Nat.shiftLeft' true m n ≠ 0

size #

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@[simp]
theorem Nat.size_zero :
Nat.size 0 = 0
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@[simp]
theorem Nat.size_bit {b : Bool} {n : ℕ} (h : Nat.bit b n ≠ 0) :
Nat.size (Nat.bit b n) = Nat.succ (Nat.size n)
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@[simp]
theorem Nat.size_bit0 {n : ℕ} (h : n ≠ 0) :
Nat.size (bit0 n) = Nat.succ (Nat.size n)
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@[simp]
theorem Nat.size_bit1 (n : ℕ) :
Nat.size (bit1 n) = Nat.succ (Nat.size n)
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@[simp]
theorem Nat.size_one :
Nat.size 1 = 1
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@[simp]
theorem Nat.size_shiftLeft' {b : Bool} {m : ℕ} {n : ℕ} (h : Nat.shiftLeft' b m n ≠ 0) :
Nat.size (Nat.shiftLeft' b m n) = Nat.size m + n
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@[simp]
theorem Nat.size_shiftLeft {m : ℕ} (h : m ≠ 0) (n : ℕ) :
Nat.size (m <<< n) = Nat.size m + n
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theorem Nat.lt_size_self (n : ℕ) :
n < 2 ^ Nat.size n
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theorem Nat.size_le {m : ℕ} {n : ℕ} :
Nat.size m ≤ n ↔ m < 2 ^ n
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theorem Nat.lt_size {m : ℕ} {n : ℕ} :
m < Nat.size n ↔ 2 ^ m ≤ n
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theorem Nat.size_pos {n : ℕ} :
0 < Nat.size n ↔ 0 < n
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theorem Nat.size_eq_zero {n : ℕ} :
Nat.size n = 0 ↔ n = 0
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theorem Nat.size_pow {n : ℕ} :
Nat.size (2 ^ n) = n + 1
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theorem Nat.size_le_size {m : ℕ} {n : ℕ} (h : m ≤ n) :
Nat.size m ≤ Nat.size n
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theorem Nat.size_eq_bits_len (n : ℕ) :
List.length (Nat.bits n) = Nat.size n