Documentation

Mathlib.Data.List.Range

Ranges of naturals as lists #

This file shows basic results about List.iota, List.range, List.range' and defines List.finRange. finRange n is the list of elements of Fin n. iota n = [n, n - 1, ..., 1] and range n = [0, ..., n - 1] are basic list constructions used for tactics. range' a b = [a, ..., a + b - 1] is there to help prove properties about them. Actual maths should use List.Ico instead.

theorem List.getElem_range'_1 {n : } {m : } (i : ) (H : i < (List.range' n m).length) :
(List.range' n m)[i] = n + i
theorem List.chain'_range_succ (r : Prop) (n : ) :
List.Chain' r (List.range n.succ) m < n, r m m.succ
theorem List.chain_range_succ (r : Prop) (n : ) (a : ) :
List.Chain r a (List.range n.succ) r a 0 m < n, r m m.succ
def List.finRange (n : ) :
List (Fin n)

All elements of Fin n, from 0 to n-1. The corresponding finset is Finset.univ.

Equations
Instances For
    @[simp]
    theorem List.mem_finRange {n : } (a : Fin n) :
    @[simp]
    theorem List.length_finRange (n : ) :
    (List.finRange n).length = n
    @[simp]
    theorem List.finRange_eq_nil {n : } :
    List.finRange n = [] n = 0
    theorem List.pairwise_lt_finRange (n : ) :
    List.Pairwise (fun (x x_1 : Fin n) => x < x_1) (List.finRange n)
    theorem List.pairwise_le_finRange (n : ) :
    List.Pairwise (fun (x x_1 : Fin n) => x x_1) (List.finRange n)
    @[simp]
    theorem List.getElem_finRange {n : } {i : } (h : i < (List.finRange n).length) :
    (List.finRange n)[i] = i,
    theorem List.get_finRange {n : } {i : } (h : i < (List.finRange n).length) :
    (List.finRange n).get i, h = i,
    @[deprecated List.get_range']
    theorem List.nthLe_range' {n : } {m : } {step : } (i : ) (H : i < (List.range' n m step).length) :
    (List.range' n m step).get i, H = n + step * i

    Alias of List.get_range'.

    @[deprecated List.getElem_range'_1]
    theorem List.nthLe_range'_1 {n : } {m : } (i : ) (H : i < (List.range' n m).length) :
    (List.range' n m)[i] = n + i

    Alias of List.getElem_range'_1.

    @[deprecated List.get_range]
    theorem List.nthLe_range {n : } (i : ) (H : i < (List.range n).length) :
    (List.range n).get i, H = i

    Alias of List.get_range.

    @[deprecated List.get_finRange]
    theorem List.nthLe_finRange {n : } {i : } (h : i < (List.finRange n).length) :
    (List.finRange n).get i, h = i,

    Alias of List.get_finRange.

    @[simp]
    theorem List.finRange_map_get {α : Type u} (l : List α) :
    List.map l.get (List.finRange l.length) = l
    @[simp]
    theorem List.indexOf_finRange {k : } (i : Fin k) :

    From l : List, construct l.ranges : List (List ℕ) such that l.ranges.map List.length = l and l.ranges.join = range l.sum

    • Example: [1,2,3].ranges = [[0],[1,2],[3,4,5]]
    Equations
    Instances For
      theorem List.ranges_disjoint (l : List ) :
      List.Pairwise List.Disjoint l.ranges

      The members of l.ranges are pairwise disjoint

      theorem List.ranges_length (l : List ) :
      List.map List.length l.ranges = l

      The lengths of the members of l.ranges are those given by l

      theorem List.ranges_join' (l : List ) :
      l.ranges.join = List.range (Nat.sum l)

      See List.ranges_join for the version about List.sum.

      theorem List.mem_mem_ranges_iff_lt_natSum (l : List ) {n : } :
      (∃ sl.ranges, n s) n < Nat.sum l

      Any entry of any member of l.ranges is strictly smaller than Nat.sum l. See List.mem_mem_ranges_iff_lt_sum for the version about List.sum.

      theorem List.ranges_nodup {l : List } {s : List } (hs : s l.ranges) :
      s.Nodup

      The members of l.ranges have no duplicate