09a5b34931
This PR adjusts the experimental module system to make `private` the default visibility modifier in `module`s, introducing `public` as a new modifier instead. `public section` can be used to revert the default for an entire section, though this is more intended to ease gradual adoption of the new semantics such as in `Init` (and soon `Std`) where they should be replaced by a future decl-by-decl re-review of visibilities.
72 lines
2.1 KiB
Text
72 lines
2.1 KiB
Text
/-
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Copyright (c) 2024 Lean FRO, LLC. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joachim Breitner
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-/
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module
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prelude
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public import Init.PropLemmas
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public section
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@[expose] section
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namespace Lean
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/--
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A `RArray` can model `Fin n → α` or `Array α`, but is optimized for a fast kernel-reducible `get`
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operation.
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The primary intended use case is the “denote” function of a typical proof by reflection proof, where
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only the `get` operation is necessary. It is not suitable as a general-purpose data structure.
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There is no well-formedness invariant attached to this data structure, to keep it concise; it's
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semantics is given through `RArray.get`. In that way one can also view an `RArray` as a decision
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tree implementing `Nat → α`.
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See `RArray.ofFn` and `RArray.ofArray` in module `Lean.Data.RArray` for functions that construct an
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`RArray`.
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-/
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inductive RArray (α : Type u) : Type u where
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| leaf : α → RArray α
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| branch : Nat → RArray α → RArray α → RArray α
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variable {α : Type u}
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/-- The crucial operation, written with very little abstractional overhead -/
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noncomputable def RArray.get (a : RArray α) (n : Nat) : α :=
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RArray.rec (fun x => x) (fun p _ _ l r => (Nat.ble p n).rec l r) a
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private theorem RArray.get_eq_def (a : RArray α) (n : Nat) :
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a.get n = match a with
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| .leaf x => x
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| .branch p l r => (Nat.ble p n).rec (l.get n) (r.get n) := by
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conv => lhs; unfold RArray.get
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split <;> rfl
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/-- `RArray.get`, implemented conventionally -/
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def RArray.getImpl (a : RArray α) (n : Nat) : α :=
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match a with
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| .leaf x => x
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| .branch p l r => if n < p then l.getImpl n else r.getImpl n
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@[csimp]
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theorem RArray.get_eq_getImpl : @RArray.get = @RArray.getImpl := by
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funext α a n
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induction a with
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| leaf _ => rfl
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| branch p l r ihl ihr =>
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rw [RArray.getImpl, RArray.get_eq_def]
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simp only [ihl, ihr, ← Nat.not_le, ← Nat.ble_eq, ite_not]
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cases hnp : Nat.ble p n <;> rfl
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instance : GetElem (RArray α) Nat α (fun _ _ => True) where
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getElem a n _ := a.get n
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def RArray.size : RArray α → Nat
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| leaf _ => 1
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| branch _ l r => l.size + r.size
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end Lean
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