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 /-- 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) /-- 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