Sorting tuples by their values

Given an n-tuple f : Fin n → α where α is ordered, we may want to turn it into a sorted n-tuple. This file provides an API for doing so, with the sorted n-tuple given by f ∘ Tuple.sort f.

Main declarations

• Tuple.sort: given f : Fin n → α, produces a permutation on Fin n
• Tuple.monotone_sort: f ∘ Tuple.sort f is Monotone

def Tuple.graph {n : ℕ} {α : Type u_1} [LinearOrder α] (f : Fin n → α) : Finset (Lex (α × Fin n))

graph f produces the finset of pairs (f i, i) equipped with the lexicographic order.

Equations

• Tuple.graph f = Finset.image (fun (i : Fin n) => (f i, i)) Finset.univ

def Tuple.graph.proj {n : ℕ} {α : Type u_1} [LinearOrder α] {f : Fin n → α} : { x : Lex (α × Fin n) // x ∈ graph f } → α

Given p : α ×ₗ (Fin n) := (f i, i) with p ∈ graph f, graph.proj p is defined to be f i.

Equations

• Tuple.graph.proj p = (↑p).1

@[simp]

theorem Tuple.graph.card {n : ℕ} {α : Type u_1} [LinearOrder α] (f : Fin n → α) : (graph f).card = n

def Tuple.graphEquiv₁ {n : ℕ} {α : Type u_1} [LinearOrder α] (f : Fin n → α) : Fin n ≃ { x : Lex (α × Fin n) // x ∈ graph f }

graphEquiv₁ f is the natural equivalence between Fin n and graph f, mapping i to (f i, i).

Equations

• Tuple.graphEquiv₁ f = { toFun := fun (i : Fin n) => ⟨(f i, i), ⋯⟩, invFun := fun (p : { x : Lex (α × Fin n) // x ∈ Tuple.graph f }) => (↑p).2, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem Tuple.proj_equiv₁' {n : ℕ} {α : Type u_1} [LinearOrder α] (f : Fin n → α) : graph.proj ∘ ⇑(graphEquiv₁ f) = f

def Tuple.graphEquiv₂ {n : ℕ} {α : Type u_1} [LinearOrder α] (f : Fin n → α) : Fin n ≃o { x : Lex (α × Fin n) // x ∈ graph f }

graphEquiv₂ f is an equivalence between Fin n and graph f that respects the order.

Equations

• Tuple.graphEquiv₂ f = (Tuple.graph f).orderIsoOfFin ⋯

def Tuple.sort {n : ℕ} {α : Type u_1} [LinearOrder α] (f : Fin n → α) : Equiv.Perm (Fin n)

sort f is the permutation that orders Fin n according to the order of the outputs of f.

Equations

• Tuple.sort f = (Tuple.graphEquiv₂ f).trans (Tuple.graphEquiv₁ f).symm

theorem Tuple.graphEquiv₂_apply {n : ℕ} {α : Type u_1} [LinearOrder α] (f : Fin n → α) (i : Fin n) : (graphEquiv₂ f) i = (graphEquiv₁ f) ((sort f) i)

theorem Tuple.self_comp_sort {n : ℕ} {α : Type u_1} [LinearOrder α] (f : Fin n → α) : f ∘ ⇑(sort f) = graph.proj ∘ ⇑(graphEquiv₂ f)

theorem Tuple.monotone_proj {n : ℕ} {α : Type u_1} [LinearOrder α] (f : Fin n → α) : Monotone graph.proj

theorem Tuple.monotone_sort {n : ℕ} {α : Type u_1} [LinearOrder α] (f : Fin n → α) : Monotone (f ∘ ⇑(sort f))

theorem Tuple.lt_card_le_iff_apply_le_of_monotone {α : Type u_1} [Preorder α] [DecidableLE α] {m : ℕ} (f : Fin m → α) (a : α) (h_sorted : Monotone f) (j : Fin m) : ↑j < Fintype.card { i : Fin m // f i ≤ a } ↔ f j ≤ a

If f₀ ≤ f₁ ≤ f₂ ≤ ⋯ is a sorted m-tuple of elements of α, then for any j : Fin m and a : α we have j < #{i | fᵢ ≤ a} iff fⱼ ≤ a.

theorem Tuple.lt_card_ge_iff_apply_ge_of_antitone {α : Type u_1} [Preorder α] [DecidableLE α] {m : ℕ} (f : Fin m → α) (a : α) (h_sorted : Antitone f) (j : Fin m) : ↑j < Fintype.card { i : Fin m // a ≤ f i } ↔ a ≤ f j

theorem Tuple.unique_monotone {n : ℕ} {α : Type u_1} [PartialOrder α] {f : Fin n → α} {σ τ : Equiv.Perm (Fin n)} (hfσ : Monotone (f ∘ ⇑σ)) (hfτ : Monotone (f ∘ ⇑τ)) : f ∘ ⇑σ = f ∘ ⇑τ

If two permutations of a tuple f are both monotone, then they are equal.

theorem Tuple.unique_antitone {n : ℕ} {α : Type u_1} [PartialOrder α] {f : Fin n → α} {σ τ : Equiv.Perm (Fin n)} (hfσ : Antitone (f ∘ ⇑σ)) (hfτ : Antitone (f ∘ ⇑τ)) : f ∘ ⇑σ = f ∘ ⇑τ

If two permutations of a tuple f are both antitone, then they are equal.

theorem Tuple.eq_sort_iff' {n : ℕ} {α : Type u_1} [LinearOrder α] {f : Fin n → α} {σ : Equiv.Perm (Fin n)} : σ = sort f ↔ StrictMono ⇑(Equiv.trans σ (graphEquiv₁ f))

A permutation σ equals sort f if and only if the map i ↦ (f (σ i), σ i) is strictly monotone (w.r.t. the lexicographic ordering on the target).

theorem Tuple.eq_sort_iff {n : ℕ} {α : Type u_1} [LinearOrder α] {f : Fin n → α} {σ : Equiv.Perm (Fin n)} : σ = sort f ↔ Monotone (f ∘ ⇑σ) ∧ ∀ (i j : Fin n), i < j → f (σ i) = f (σ j) → σ i < σ j

A permutation σ equals sort f if and only if f ∘ σ is monotone and whenever i < j and f (σ i) = f (σ j), then σ i < σ j. This means that sort f is the lexicographically smallest permutation σ such that f ∘ σ is monotone.

theorem Tuple.sort_eq_refl_iff_monotone {n : ℕ} {α : Type u_1} [LinearOrder α] {f : Fin n → α} : sort f = Equiv.refl (Fin n) ↔ Monotone f

The permutation that sorts f is the identity if and only if f is monotone.

theorem Tuple.comp_sort_eq_comp_iff_monotone {n : ℕ} {α : Type u_1} [LinearOrder α] {f : Fin n → α} {σ : Equiv.Perm (Fin n)} : f ∘ ⇑σ = f ∘ ⇑(sort f) ↔ Monotone (f ∘ ⇑σ)

A permutation of a tuple f is f sorted if and only if it is monotone.

theorem Tuple.comp_perm_comp_sort_eq_comp_sort {n : ℕ} {α : Type u_1} [LinearOrder α] {f : Fin n → α} {σ : Equiv.Perm (Fin n)} : (f ∘ ⇑σ) ∘ ⇑(sort (f ∘ ⇑σ)) = f ∘ ⇑(sort f)

The sorted versions of a tuple f and of any permutation of f agree.

theorem Tuple.antitone_pair_of_not_sorted' {n : ℕ} {α : Type u_1} [LinearOrder α] {f : Fin n → α} {σ : Equiv.Perm (Fin n)} (h : f ∘ ⇑σ ≠ f ∘ ⇑(sort f)) : ∃ (i : Fin n) (j : Fin n), i < j ∧ (f ∘ ⇑σ) j < (f ∘ ⇑σ) i

If a permutation f ∘ σ of the tuple f is not the same as f ∘ sort f, then f ∘ σ has a pair of strictly decreasing entries.

theorem Tuple.antitone_pair_of_not_sorted {n : ℕ} {α : Type u_1} [LinearOrder α] {f : Fin n → α} (h : f ≠ f ∘ ⇑(sort f)) : ∃ (i : Fin n) (j : Fin n), i < j ∧ f j < f i

If the tuple f is not the same as f ∘ sort f, then f has a pair of strictly decreasing entries.