bookshelf/Bookshelf/Real/Set/Partition.lean

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import Bookshelf.List.Basic
import Bookshelf.Real.Set.Interval
import Mathlib.Data.List.Sort
/-! # Bookshelf.Real.Set.Partition
A description of a partition as defined in the context of stepwise functions.
Refer to [^1] for more information.
[^1]: Apostol, Tom M. Calculus, Vol. 1: One-Variable Calculus, with an
Introduction to Linear Algebra. 2nd ed. Vol. 1. 2 vols. Wiley, 1991.
-/
namespace Real
open List
/--
A `Partition` is some finite subset of `[a, b]` containing points `a` and `b`.
It is assumed that the points of the `Partition` are distinct and sorted. The
use of a `List` ensures finite-ness.
-/
structure Partition where
xs : List
sorted : Sorted LT.lt xs
has_min_length : xs.length ≥ 2
/--
The length of any list associated with a `Partition` is `> 0`.
-/
private lemma length_gt_zero (p : Partition) : p.xs.length > 0 :=
calc p.xs.length
_ ≥ 2 := p.has_min_length
_ > 0 := by simp
/--
The length of any list associated with a `Partition` is `≠ 0`.
-/
instance (p : Partition) : NeZero (length p.xs) where
out := LT.lt.ne' (length_gt_zero p)
namespace Partition
/--
The left-most subdivision point of the `Partition`.
-/
def left (p : Partition) : :=
p.xs.head (neq_nil_iff_length_gt_zero.mpr (length_gt_zero p))
/--
The right-most subdivision point of the `Partition`.
-/
def right (p : Partition) : :=
p.xs.getLast (neq_nil_iff_length_gt_zero.mpr (length_gt_zero p))
/--
Define `∈` syntax for a `Partition`. We say a real is a member of a partition
provided it lies somewhere in closed interval `[a, b]`.
-/
instance : Membership Partition where
mem (x : ) (p : Partition) := p.left ≤ x ∧ x ≤ p.right
/--
Every subdivision point is `≥` the left-most point of the partition.
-/
theorem subdivision_point_geq_left {p : Partition} (h : x ∈ p.xs)
: p.left ≤ x := by
unfold left
rw [head_eq_get_zero (exists_mem_iff_neq_nil.mp ⟨x, h⟩)]
have ⟨i, hi⟩ := mem_iff_exists_get.mp h
conv => rhs; rw [← hi]
by_cases hz : i = (0 : Fin (length p.xs))
· rw [hz]
simp
· refine le_of_lt (Sorted.rel_get_of_lt p.sorted ?_)
rwa [← ne_eq, ← Fin.pos_iff_ne_zero i] at hz
/--
Every subdivision point is `≤` the right-most point of the partition.
-/
theorem subdivision_point_leq_right {p : Partition} (h : x ∈ p.xs)
: x ≤ p.right := by
unfold right
have hx := exists_mem_iff_neq_nil.mp ⟨x, h⟩
rw [getLast_eq_get_length_sub_one hx]
have ⟨i, hi⟩ := mem_iff_exists_get.mp h
conv => lhs; rw [← hi]
have ⟨_, ⟨_, hs⟩⟩ := self_neq_nil_imp_exists_mem.mp hx
by_cases hz : i = ⟨p.xs.length - 1, by rw [hs]; simp⟩
· rw [hz]
· refine le_of_lt (Sorted.rel_get_of_lt p.sorted ?_)
rw [← ne_eq, Fin.ne_iff_vne] at hz
rw [Fin.lt_iff_val_lt_val]
exact lt_of_le_of_ne (le_tsub_of_add_le_right i.2) hz
/--
Every subdivision point of a `Partition` is itself a member of the `Partition`.
-/
theorem subdivision_point_mem_partition {p : Partition} (h : x ∈ p.xs)
: x ∈ p := ⟨subdivision_point_geq_left h, subdivision_point_leq_right h⟩
end Partition
end Real