bookshelf/Common/Set/Intervals/Partition.lean

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import Mathlib.Data.Finset.Basic
import Mathlib.Data.List.Sort
import Mathlib.Data.Set.Intervals.Basic
import Common.List.Basic
/-! # Common.Set.Intervals.Partition
Additional theorems and definitions useful in the context of sets.
-/
namespace Set.Intervals
open List
/--
A `Partition` is a finite subset of `[a, b]` containing points `a` and `b`.
-/
structure Partition (α : Type _) [Preorder α] [@DecidableRel α LT.lt] where
/- The left-most endpoint of the partition. -/
a : α
/- The right-most endpoint of the partition. -/
b : α
/- The subdivision points. -/
xs : List α
/- Ensure the subdivision points are in sorted order. -/
sorted_xs : Sorted LT.lt xs
/- Ensure each subdivision point is in our defined interval. -/
within_xs : ∀ x ∈ xs, x ∈ Ioo a b
namespace Partition
/--
An object `x` is a member of a `Partition` `p` if `x` is an endpoint of `p` or a
subdivision point of `p`.
Notice that being a member of `p` is different from being a member of some
(sub)interval determined by `p`.
-/
instance [Preorder α] [@DecidableRel α LT.lt] : Membership α (Partition α) where
mem (x : α) (p : Partition α) := x = p.a x ∈ p.xs x = p.b
/--
Return the endpoints and subdivision points of a `Partition` as a sorted `List`.
-/
def toList [Preorder α] [@DecidableRel α LT.lt] (p : Partition α) : List α :=
(p.a :: p.xs) ++ [p.b]
/--
`x` is a member of `Partition` `p` **iff** `x` is a member of `p.List`.
-/
theorem mem_self_iff_mem_toList [Preorder α] [@DecidableRel α LT.lt]
(p : Partition α) : x ∈ p ↔ x ∈ p.toList := by
apply Iff.intro
· sorry
· sorry
/--
Every member of a `Partition` is greater than or equal to its left-most point.
-/
theorem left_le_mem_self [Preorder α] [@DecidableRel α LT.lt]
(p : Partition α) : ∀ x ∈ p, p.a ≤ x := by
sorry
/--
Every member of a `Partition` is less than or equal to its right-most point.
-/
theorem right_ge_mem_self [Preorder α] [@DecidableRel α LT.lt]
(p : Partition α) : ∀ x ∈ p, x ≤ p.b := by
sorry
2023-05-13 01:31:44 +00:00
/--
The closed interval determined by the endpoints of the `Partition`.
-/
abbrev toIcc [Preorder α] [@DecidableRel α LT.lt]
(p : Partition α) := Set.Icc p.a p.b
/-
Return the closed subintervals determined by the `Partition`.
-/
def closedSubintervals [Preorder α] [@DecidableRel α LT.lt]
(p : Partition α) : List (Set α) :=
p.toList.pairwise (fun x₁ x₂ => Icc x₁ x₂)
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/--
The open interval determined by the endpoints of the `Partition`.
-/
abbrev toIoo [Preorder α] [@DecidableRel α LT.lt]
(p : Partition α) := Set.Ioo p.a p.b
/-
Return the open subintervals determined by the `Partition`.
-/
def openSubintervals [Preorder α] [@DecidableRel α LT.lt]
(p : Partition α) : List (Set α) :=
p.toList.pairwise (fun x₁ x₂ => Ioo x₁ x₂)
/--
A member of an open subinterval of a `Partition` `p` is a member of the entire
open interval determined by `p`.
-/
theorem mem_open_subinterval_mem_open_interval
[Preorder α] [@DecidableRel α LT.lt] {p : Partition α}
(hI : I ∈ p.openSubintervals) (hy : y ∈ I) : y ∈ Ioo p.a p.b := by
have ⟨i, ⟨x₁, ⟨x₂, ⟨hx₁, ⟨hx₂, hI'⟩⟩⟩⟩⟩ :=
List.mem_pairwise_imp_exists_adjacent hI
have hx₁' : p.a ≤ x₁ := by
refine p.left_le_mem_self x₁ ?_
rw [p.mem_self_iff_mem_toList]
have : ↑i < p.toList.length := calc ↑i
_ < p.toList.length - 1 := i.2
_ < p.toList.length := by
unfold List.length Partition.toList
simp
exact List.mem_iff_exists_get.mpr ⟨⟨↑i, this⟩, Eq.symm hx₁⟩
have hx₂' : x₂ ≤ p.b := by
refine p.right_ge_mem_self x₂ ?_
rw [p.mem_self_iff_mem_toList]
have : ↑i + 1 < p.toList.length := add_lt_add_right i.2 1
exact List.mem_iff_exists_get.mpr ⟨⟨↑i + 1, this⟩, Eq.symm hx₂⟩
have hx_sub := Set.Ioo_subset_Ioo hx₁' hx₂'
rw [hI'] at hy
exact Set.mem_of_subset_of_mem hx_sub hy
/--
A member of an open subinterval of a `Partition` `p` is a member of the entire
closed interval determined by `p`.
-/
theorem mem_open_subinterval_mem_closed_interval
[Preorder α] [@DecidableRel α LT.lt] {p : Partition α}
(hI : I ∈ p.openSubintervals) (hy : y ∈ I) : y ∈ Icc p.a p.b := by
have := mem_open_subinterval_mem_open_interval hI hy
exact Set.mem_of_subset_of_mem Set.Ioo_subset_Icc_self this
end Partition
end Set.Intervals