03d01f4024
This PR neither adds nor removes material, but improves the organization of `Init/Data/List/*`. These files are essentially completely re-ordered, to ensure that material is developed in a consistent order between `List.Basic`, `List.Impl`, `List.BasicAux`, and `List.Lemmas`. Everything is organised in subsections, and I've added some module docs.
216 lines
6.9 KiB
Text
216 lines
6.9 KiB
Text
/-
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Copyright (c) 2016 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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Author: Leonardo de Moura, Robert Y. Lewis, Keeley Hoek, Mario Carneiro
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-/
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prelude
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import Init.Data.Nat.Bitwise.Basic
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open Nat
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namespace Fin
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instance coeToNat : CoeOut (Fin n) Nat :=
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⟨fun v => v.val⟩
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/--
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From the empty type `Fin 0`, any desired result `α` can be derived. This is simlar to `Empty.elim`.
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-/
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def elim0.{u} {α : Sort u} : Fin 0 → α
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| ⟨_, h⟩ => absurd h (not_lt_zero _)
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/--
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Returns the successor of the argument.
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The bound in the result type is increased:
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```
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(2 : Fin 3).succ = (3 : Fin 4)
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```
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This differs from addition, which wraps around:
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```
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(2 : Fin 3) + 1 = (0 : Fin 3)
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```
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-/
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def succ : Fin n → Fin n.succ
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| ⟨i, h⟩ => ⟨i+1, Nat.succ_lt_succ h⟩
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variable {n : Nat}
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/--
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Returns `a` modulo `n + 1` as a `Fin n.succ`.
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-/
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protected def ofNat {n : Nat} (a : Nat) : Fin n.succ :=
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⟨a % (n+1), Nat.mod_lt _ (Nat.zero_lt_succ _)⟩
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/--
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Returns `a` modulo `n` as a `Fin n`.
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The assumption `n > 0` ensures that `Fin n` is nonempty.
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-/
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protected def ofNat' {n : Nat} (a : Nat) (h : n > 0) : Fin n :=
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⟨a % n, Nat.mod_lt _ h⟩
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private theorem mlt {b : Nat} : {a : Nat} → a < n → b % n < n
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| 0, h => Nat.mod_lt _ h
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| _+1, h =>
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have : n > 0 := Nat.lt_trans (Nat.zero_lt_succ _) h;
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Nat.mod_lt _ this
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/-- Addition modulo `n` -/
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protected def add : Fin n → Fin n → Fin n
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| ⟨a, h⟩, ⟨b, _⟩ => ⟨(a + b) % n, mlt h⟩
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/-- Multiplication modulo `n` -/
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protected def mul : Fin n → Fin n → Fin n
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| ⟨a, h⟩, ⟨b, _⟩ => ⟨(a * b) % n, mlt h⟩
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/-- Subtraction modulo `n` -/
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protected def sub : Fin n → Fin n → Fin n
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/-
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The definition of `Fin.sub` has been updated to improve performance.
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The right-hand-side of the following `match` was originally
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```
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⟨(a + (n - b)) % n, mlt h⟩
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```
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This caused significant performance issues when testing definitional equality,
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such as `x =?= x - 1` where `x : Fin n` and `n` is a big number,
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as Lean spent a long time reducing
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```
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((n - 1) + x.val) % n
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```
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For example, this was an issue for `Fin 2^64` (i.e., `UInt64`).
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This change improves performance by leveraging the fact that `Nat.add` is defined
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using recursion on the second argument.
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See issue #4413.
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-/
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| ⟨a, h⟩, ⟨b, _⟩ => ⟨((n - b) + a) % n, mlt h⟩
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/-!
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Remark: land/lor can be defined without using (% n), but
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we are trying to minimize the number of Nat theorems
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needed to bootstrap Lean.
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-/
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protected def mod : Fin n → Fin n → Fin n
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| ⟨a, h⟩, ⟨b, _⟩ => ⟨a % b, Nat.lt_of_le_of_lt (Nat.mod_le _ _) h⟩
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protected def div : Fin n → Fin n → Fin n
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| ⟨a, h⟩, ⟨b, _⟩ => ⟨a / b, Nat.lt_of_le_of_lt (Nat.div_le_self _ _) h⟩
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def modn : Fin n → Nat → Fin n
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| ⟨a, h⟩, m => ⟨a % m, Nat.lt_of_le_of_lt (Nat.mod_le _ _) h⟩
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def land : Fin n → Fin n → Fin n
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| ⟨a, h⟩, ⟨b, _⟩ => ⟨(Nat.land a b) % n, mlt h⟩
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def lor : Fin n → Fin n → Fin n
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| ⟨a, h⟩, ⟨b, _⟩ => ⟨(Nat.lor a b) % n, mlt h⟩
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def xor : Fin n → Fin n → Fin n
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| ⟨a, h⟩, ⟨b, _⟩ => ⟨(Nat.xor a b) % n, mlt h⟩
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def shiftLeft : Fin n → Fin n → Fin n
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| ⟨a, h⟩, ⟨b, _⟩ => ⟨(a <<< b) % n, mlt h⟩
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def shiftRight : Fin n → Fin n → Fin n
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| ⟨a, h⟩, ⟨b, _⟩ => ⟨(a >>> b) % n, mlt h⟩
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instance : Add (Fin n) where
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add := Fin.add
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instance : Sub (Fin n) where
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sub := Fin.sub
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instance : Mul (Fin n) where
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mul := Fin.mul
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instance : Mod (Fin n) where
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mod := Fin.mod
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instance : Div (Fin n) where
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div := Fin.div
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instance : AndOp (Fin n) where
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and := Fin.land
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instance : OrOp (Fin n) where
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or := Fin.lor
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instance : Xor (Fin n) where
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xor := Fin.xor
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instance : ShiftLeft (Fin n) where
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shiftLeft := Fin.shiftLeft
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instance : ShiftRight (Fin n) where
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shiftRight := Fin.shiftRight
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instance instOfNat : OfNat (Fin (no_index (n+1))) i where
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ofNat := Fin.ofNat i
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instance : Inhabited (Fin (no_index (n+1))) where
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default := 0
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@[simp] theorem zero_eta : (⟨0, Nat.zero_lt_succ _⟩ : Fin (n + 1)) = 0 := rfl
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theorem val_ne_of_ne {i j : Fin n} (h : i ≠ j) : val i ≠ val j :=
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fun h' => absurd (eq_of_val_eq h') h
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theorem modn_lt : ∀ {m : Nat} (i : Fin n), m > 0 → (modn i m).val < m
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| _, ⟨_, _⟩, hp => by simp [modn]; apply Nat.mod_lt; assumption
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theorem val_lt_of_le (i : Fin b) (h : b ≤ n) : i.val < n :=
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Nat.lt_of_lt_of_le i.isLt h
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protected theorem pos (i : Fin n) : 0 < n :=
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Nat.lt_of_le_of_lt (Nat.zero_le _) i.2
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/-- The greatest value of `Fin (n+1)`. -/
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@[inline] def last (n : Nat) : Fin (n + 1) := ⟨n, n.lt_succ_self⟩
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/-- `castLT i h` embeds `i` into a `Fin` where `h` proves it belongs into. -/
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@[inline] def castLT (i : Fin m) (h : i.1 < n) : Fin n := ⟨i.1, h⟩
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/-- `castLE h i` embeds `i` into a larger `Fin` type. -/
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@[inline] def castLE (h : n ≤ m) (i : Fin n) : Fin m := ⟨i, Nat.lt_of_lt_of_le i.2 h⟩
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/-- `cast eq i` embeds `i` into an equal `Fin` type. -/
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@[inline] def cast (eq : n = m) (i : Fin n) : Fin m := ⟨i, eq ▸ i.2⟩
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/-- `castAdd m i` embeds `i : Fin n` in `Fin (n+m)`. See also `Fin.natAdd` and `Fin.addNat`. -/
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@[inline] def castAdd (m) : Fin n → Fin (n + m) :=
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castLE <| Nat.le_add_right n m
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/-- `castSucc i` embeds `i : Fin n` in `Fin (n+1)`. -/
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@[inline] def castSucc : Fin n → Fin (n + 1) := castAdd 1
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/-- `addNat m i` adds `m` to `i`, generalizes `Fin.succ`. -/
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def addNat (i : Fin n) (m) : Fin (n + m) := ⟨i + m, Nat.add_lt_add_right i.2 _⟩
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/-- `natAdd n i` adds `n` to `i` "on the left". -/
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def natAdd (n) (i : Fin m) : Fin (n + m) := ⟨n + i, Nat.add_lt_add_left i.2 _⟩
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/-- Maps `0` to `n-1`, `1` to `n-2`, ..., `n-1` to `0`. -/
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@[inline] def rev (i : Fin n) : Fin n := ⟨n - (i + 1), Nat.sub_lt i.pos (Nat.succ_pos _)⟩
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/-- `subNat i h` subtracts `m` from `i`, generalizes `Fin.pred`. -/
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@[inline] def subNat (m) (i : Fin (n + m)) (h : m ≤ i) : Fin n :=
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⟨i - m, Nat.sub_lt_right_of_lt_add h i.2⟩
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/-- Predecessor of a nonzero element of `Fin (n+1)`. -/
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@[inline] def pred {n : Nat} (i : Fin (n + 1)) (h : i ≠ 0) : Fin n :=
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subNat 1 i <| Nat.pos_of_ne_zero <| mt (Fin.eq_of_val_eq (j := 0)) h
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theorem val_inj {a b : Fin n} : a.1 = b.1 ↔ a = b := ⟨Fin.eq_of_val_eq, Fin.val_eq_of_eq⟩
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theorem val_congr {n : Nat} {a b : Fin n} (h : a = b) : (a : Nat) = (b : Nat) :=
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Fin.val_inj.mpr h
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theorem val_le_of_le {n : Nat} {a b : Fin n} (h : a ≤ b) : (a : Nat) ≤ (b : Nat) := h
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theorem val_le_of_ge {n : Nat} {a b : Fin n} (h : a ≥ b) : (b : Nat) ≤ (a : Nat) := h
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theorem val_add_one_le_of_lt {n : Nat} {a b : Fin n} (h : a < b) : (a : Nat) + 1 ≤ (b : Nat) := h
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theorem val_add_one_le_of_gt {n : Nat} {a b : Fin n} (h : a > b) : (b : Nat) + 1 ≤ (a : Nat) := h
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theorem exists_iff {p : Fin n → Prop} : (Exists fun i => p i) ↔ Exists fun i => Exists fun h => p ⟨i, h⟩ :=
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⟨fun ⟨⟨i, hi⟩, hpi⟩ => ⟨i, hi, hpi⟩, fun ⟨i, hi, hpi⟩ => ⟨⟨i, hi⟩, hpi⟩⟩
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end Fin
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