Derive a more constructive `StepFunction` definition.
parent
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@ -1,4 +1,5 @@
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import Common.Combinator
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import Common.Finset
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import Common.List
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import Common.Real
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import Common.Set
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@ -0,0 +1,17 @@
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import Mathlib.Data.Finset.Basic
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/-! # Common.Finset
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Additional theorems and definitions useful in the context of `Finset`s.
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-/
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namespace Finset
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/--
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An alternative `Finset.range` function that returns `Fin` indices instead of `ℕ`
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indices.
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-/
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def finRange (n : ℕ) : Finset (Fin n) :=
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⟨sorry, sorry⟩
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end Finset
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@ -1,5 +1,5 @@
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import Common.Geometry.Basic
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import Common.Geometry.Rectangle.Skew
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import Common.Geometry.Rectangle.Orthogonal
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import Common.Geometry.StepFunction
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/-! # Common.Geometry.Area
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@ -46,6 +46,13 @@ The bottom-right corner of the `Orthogonal` rectangle.
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-/
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def br (r : Orthogonal) : Point := ⟨r.bl.x + r.width, r.bl.y⟩
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/--
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The `Set` of `Point`s enclosed in the region determined by the edges of the
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`Orthogonal` rectangle. Edges of the rectangle are included in the result set.
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-/
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def toSet (r : Orthogonal) : Set Point :=
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{ p | r.bl.x ≤ p.x ∧ p.x ≤ r.br.x ∧ r.bl.y ≤ p.y ∧ p.y ≤ r.tl.y }
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/--
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An `Orthogonal` rectangle's top side is equal in length to its bottom side.
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-/
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@ -1,4 +1,7 @@
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import Common.Finset
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import Common.Geometry.Point
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import Common.Geometry.Rectangle.Orthogonal
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import Common.List.Basic
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import Common.Set.Partition
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/-! # Common.Geometry.StepFunction
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@ -14,20 +17,24 @@ open Set Partition
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A function `f`, whose domain is a closed interval `[a, b]`, is a `StepFunction`
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if there exists a `Partition` `P = {x₀, x₁, …, xₙ}` of `[a, b]` such that `f` is
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constant on each open subinterval of `P`.
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Instead of maintaining a function from `[a, b]` to `ℝ`, we instead maintain a
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function that maps each `Partition` index to some constant value.
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-/
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structure StepFunction where
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p : Partition ℝ
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toFun : ∀ x ∈ p.toIcc, ℝ
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const_open_subintervals :
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∀ (hI : I ∈ p.openSubintervals), ∃ c, ∀ (hy : y ∈ I),
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toFun y (mem_open_subinterval_mem_closed_interval hI hy) = c
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toFun : Fin p.ivls.length → ℝ
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namespace StepFunction
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/--
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The ordinate set of the function.
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The ordinate set of the `StepFunction`.
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-/
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def toSet (s : StepFunction) : Set Point := sorry
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def toSet (sf : StepFunction) : Set Point :=
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⋃ i ∈ Finset.finRange sf.p.ivls.length,
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let I := sf.p.ivls[i]
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Rectangle.Orthogonal.toSet
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⟨{ x := I.left, y := 0 }, { x := I.right, y := sf.toFun i }⟩
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end StepFunction
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@ -1 +1,2 @@
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import Common.List.Basic
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import Common.List.NonEmpty
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@ -0,0 +1,81 @@
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import Mathlib.Algebra.Group.Defs
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import Mathlib.Data.Nat.Basic
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import Mathlib.Tactic.NormNum
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/-! # Common.List.NonEmpty
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A `List` with at least one member.
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-/
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namespace List
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structure NonEmpty (α : Type _) : Type _ where
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head : α
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tail : List α
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instance : Coe (NonEmpty α) (List α) where
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coe (xs : NonEmpty α) := xs.head :: xs.tail
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instance : CoeDep (List α) (cons x xs : List α) (NonEmpty α) where
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coe := { head := x, tail := xs }
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namespace NonEmpty
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/--
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The length of a `List.NonEmpty`.
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-/
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def length (xs : NonEmpty α) : Nat := 1 + xs.tail.length
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/--
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The length of a `List.NonEmpty` is always one plus the length of its tail.
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-/
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theorem length_self_eq_one_add_length_tail (xs : NonEmpty α)
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: xs.length = 1 + xs.tail.length := rfl
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/--
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A proof that an index is within the bounds of the `List.NonEmpty`.
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-/
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abbrev inBounds (xs : NonEmpty α) (i : Nat) : Prop :=
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i < xs.length
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/--
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Retrieves the member of the `List.NonEmpty` at the specified index.
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-/
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def get (xs : NonEmpty α) : (i : Nat) → (h : xs.inBounds i) → α
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| 0, _ => xs.head
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| n + 1, h =>
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have : n < xs.tail.length := by
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unfold inBounds at h
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rw [length_self_eq_one_add_length_tail, add_comm] at h
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norm_num at h
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exact h
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xs.tail[n]
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/--
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Variant of `get` that returns an `Option α` in the case of an invalid index.
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-/
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def get? : NonEmpty α → Nat → Option α
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| xs, 0 => some xs.head
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| xs, n + 1 => xs.tail.get? n
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/--
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Type class instance for allowing direct indexing notation.
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-/
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instance : GetElem (NonEmpty α) Nat α inBounds where
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getElem := get
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/--
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Convert a `List.NonEmpty` into a plain `List`.
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-/
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def toList (xs : NonEmpty α) : List α := xs
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/--
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Retrieve the last member of the `List.NonEmpty`.
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-/
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def last : NonEmpty α → α
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| ⟨head, []⟩ => head
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| ⟨_, cons x xs⟩ => last (cons x xs)
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end NonEmpty
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end List
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@ -2,7 +2,7 @@ import Mathlib.Data.Set.Basic
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/-! # Common.Set.Basic
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Additional theorems and definitions useful in the context of sets.
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Additional theorems and definitions useful in the context of `Set`s.
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-/
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namespace Set
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@ -0,0 +1,58 @@
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import Mathlib.Data.Real.Basic
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import Mathlib.Data.Set.Intervals.Basic
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import Common.List.Basic
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/-! # Common.Set.Interval
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A representation of a range of values.
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-/
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namespace Set
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/--
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An interval defines a range of values, characterized by a "left" value and a
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"right" value. We require these values to be distinct; we do not support the
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notion of an empty interval.
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-/
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structure Interval (α : Type _) [LT α] where
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left : α
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right : α
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h : left < right
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namespace Interval
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/--
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Computes the size of the interval.
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-/
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def size [LT α] [Sub α] (i : Interval α) : α := i.right - i.left
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/--
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Computes the midpoint of the interval.
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-/
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def midpoint [LT α] [Add α] [HDiv α ℝ α] (i : Interval α) : α :=
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(i.left + i.right) / (2 : ℝ)
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/--
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Convert an `Interval` into a `Set.Ico`.
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-/
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def toIco [Preorder α] (i : Interval α) : Set α := Set.Ico i.left i.right
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/--
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Convert an `Interval` into a `Set.Ioc`.
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-/
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def toIoc [Preorder α] (i : Interval α) : Set α := Set.Ioc i.left i.right
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/--
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Convert an `Interval` into a `Set.Icc`.
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-/
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def toIcc [Preorder α] (i : Interval α) : Set α := Set.Icc i.left i.right
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/--
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Convert an `Interval` into a `Set.Ioo`.
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-/
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def toIoo [Preorder α] (i : Interval α) : Set α := Set.Ioo i.left i.right
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end Interval
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end Set
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@ -3,6 +3,8 @@ import Mathlib.Data.List.Sort
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import Mathlib.Data.Set.Intervals.Basic
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import Common.List.Basic
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import Common.List.NonEmpty
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import Common.Set.Interval
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/-! # Common.Set.Partition
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@ -11,126 +13,28 @@ Additional theorems and definitions useful in the context of sets.
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namespace Set
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open List
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/--
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A `Partition` is a finite subset of `[a, b]` containing points `a` and `b`.
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We use a `List.NonEmpty` internally to ensure the existence of at least one
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`Interval`, which cannot be empty. Thus our `Partition` can never be empty.
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The intervals are sorted via the `connected` proposition.
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-/
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structure Partition (α : Type _) [Preorder α] [@DecidableRel α LT.lt] where
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/- The left-most endpoint of the partition. -/
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a : α
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/- The right-most endpoint of the partition. -/
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b : α
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/- The subdivision points. -/
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xs : List α
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/- Ensure the subdivision points are in sorted order. -/
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sorted_xs : Sorted LT.lt xs
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/- Ensure each subdivision point is in our defined interval. -/
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within_xs : ∀ x ∈ xs, x ∈ Ioo a b
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structure Partition (α : Type _) [LT α] where
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ivls : List.NonEmpty (Interval α)
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connected : ∀ I ∈ ivls.toList.pairwise (·.right = ·.left), I
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namespace Partition
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/--
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An object `x` is a member of a `Partition` `p` if `x` is an endpoint of `p` or a
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subdivision point of `p`.
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Notice that being a member of `p` is different from being a member of some
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(sub)interval determined by `p`.
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Return the left-most endpoint of the `Partition`.
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-/
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instance [Preorder α] [@DecidableRel α LT.lt] : Membership α (Partition α) where
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mem (x : α) (p : Partition α) := x = p.a ∨ x ∈ p.xs ∨ x = p.b
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def left [LT α] (p : Partition α) := p.ivls.head.left
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/--
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Return the endpoints and subdivision points of a `Partition` as a sorted `List`.
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Return the right-most endpoint of the `Partition`.
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-/
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def toList [Preorder α] [@DecidableRel α LT.lt] (p : Partition α) : List α :=
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(p.a :: p.xs) ++ [p.b]
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/--
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`x` is a member of `Partition` `p` **iff** `x` is a member of `p.List`.
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-/
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theorem mem_self_iff_mem_toList [Preorder α] [@DecidableRel α LT.lt]
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(p : Partition α) : x ∈ p ↔ x ∈ p.toList := by
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apply Iff.intro
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· sorry
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· sorry
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/--
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Every member of a `Partition` is greater than or equal to its left-most point.
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-/
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theorem left_le_mem_self [Preorder α] [@DecidableRel α LT.lt]
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(p : Partition α) : ∀ x ∈ p, p.a ≤ x := by
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sorry
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/--
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Every member of a `Partition` is less than or equal to its right-most point.
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-/
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theorem right_ge_mem_self [Preorder α] [@DecidableRel α LT.lt]
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(p : Partition α) : ∀ x ∈ p, x ≤ p.b := by
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sorry
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/--
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The closed interval determined by the endpoints of the `Partition`.
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-/
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abbrev toIcc [Preorder α] [@DecidableRel α LT.lt]
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(p : Partition α) := Set.Icc p.a p.b
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/-
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Return the closed subintervals determined by the `Partition`.
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-/
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def closedSubintervals [Preorder α] [@DecidableRel α LT.lt]
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(p : Partition α) : List (Set α) :=
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p.toList.pairwise (fun x₁ x₂ => Icc x₁ x₂)
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/--
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The open interval determined by the endpoints of the `Partition`.
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-/
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abbrev toIoo [Preorder α] [@DecidableRel α LT.lt]
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(p : Partition α) := Set.Ioo p.a p.b
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/-
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Return the open subintervals determined by the `Partition`.
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-/
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def openSubintervals [Preorder α] [@DecidableRel α LT.lt]
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(p : Partition α) : List (Set α) :=
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p.toList.pairwise (fun x₁ x₂ => Ioo x₁ x₂)
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/--
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A member of an open subinterval of a `Partition` `p` is a member of the entire
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open interval determined by `p`.
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-/
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theorem mem_open_subinterval_mem_open_interval
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[Preorder α] [@DecidableRel α LT.lt] {p : Partition α}
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(hI : I ∈ p.openSubintervals) (hy : y ∈ I) : y ∈ Ioo p.a p.b := by
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have ⟨i, ⟨x₁, ⟨x₂, ⟨hx₁, ⟨hx₂, hI'⟩⟩⟩⟩⟩ :=
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List.mem_pairwise_imp_exists_adjacent hI
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have hx₁' : p.a ≤ x₁ := by
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refine p.left_le_mem_self x₁ ?_
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rw [p.mem_self_iff_mem_toList]
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have : ↑i < p.toList.length := calc ↑i
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_ < p.toList.length - 1 := i.2
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_ < p.toList.length := by
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unfold List.length Partition.toList
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simp
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exact List.mem_iff_exists_get.mpr ⟨⟨↑i, this⟩, Eq.symm hx₁⟩
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have hx₂' : x₂ ≤ p.b := by
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refine p.right_ge_mem_self x₂ ?_
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rw [p.mem_self_iff_mem_toList]
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have : ↑i + 1 < p.toList.length := add_lt_add_right i.2 1
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exact List.mem_iff_exists_get.mpr ⟨⟨↑i + 1, this⟩, Eq.symm hx₂⟩
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have hx_sub := Set.Ioo_subset_Ioo hx₁' hx₂'
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rw [hI'] at hy
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exact Set.mem_of_subset_of_mem hx_sub hy
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/--
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A member of an open subinterval of a `Partition` `p` is a member of the entire
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closed interval determined by `p`.
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-/
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theorem mem_open_subinterval_mem_closed_interval
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[Preorder α] [@DecidableRel α LT.lt] {p : Partition α}
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(hI : I ∈ p.openSubintervals) (hy : y ∈ I) : y ∈ Icc p.a p.b := by
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have := mem_open_subinterval_mem_open_interval hI hy
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exact Set.mem_of_subset_of_mem Set.Ioo_subset_Icc_self this
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def right [LT α] (p : Partition α) := p.ivls.last.right
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end Partition
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