4575799f8e
This PR cleans up the style of the library in anticipation of a future PR that requires strict indentation for tactic sequences.
49 lines
1.2 KiB
Text
49 lines
1.2 KiB
Text
/-
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Copyright (c) 2021 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Gabriel Ebner
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-/
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module
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prelude
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public import Init.Data.Nat.Linear
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public section
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namespace Nat
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theorem log2_terminates : ∀ n, n ≥ 2 → n / 2 < n
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| 2, _ => by decide
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| 3, _ => by decide
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| n+4, _ => by
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rw [div_eq, if_pos]
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refine succ_lt_succ (Nat.lt_trans ?_ (lt_succ_self _))
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exact log2_terminates (n+2) (by simp)
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simp
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/--
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Base-two logarithm of natural numbers. Returns `⌊max 0 (log₂ n)⌋`.
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This function is overridden at runtime with an efficient implementation. This definition is
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the logical model.
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Examples:
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* `Nat.log2 0 = 0`
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* `Nat.log2 1 = 0`
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* `Nat.log2 2 = 1`
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* `Nat.log2 4 = 2`
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* `Nat.log2 7 = 2`
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* `Nat.log2 8 = 3`
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-/
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@[extern "lean_nat_log2"]
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def log2 (n : @& Nat) : Nat :=
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if n ≥ 2 then log2 (n / 2) + 1 else 0
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decreasing_by exact log2_terminates _ ‹_›
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theorem log2_le_self (n : Nat) : Nat.log2 n ≤ n := by
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unfold Nat.log2; split
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next h =>
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have := log2_le_self (n / 2)
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exact Nat.lt_of_le_of_lt this (Nat.div_lt_self (Nat.le_of_lt h) (by decide))
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· apply Nat.zero_le
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decreasing_by exact Nat.log2_terminates _ ‹_›
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