Construct a sorted list from a finset.

sort

def Finset.sort {α : Type u_1} (r : α → α → Prop) [DecidableRel r] [IsTrans α r] [IsAntisymm α r] [IsTotal α r] (s : Finset α) : List α

sort s constructs a sorted list from the unordered set s. (Uses merge sort algorithm.)

Equations

• Finset.sort r s = Multiset.sort r s.val

@[simp]

theorem Finset.sort_sorted {α : Type u_1} (r : α → α → Prop) [DecidableRel r] [IsTrans α r] [IsAntisymm α r] [IsTotal α r] (s : Finset α) : List.Sorted r (Finset.sort r s)

@[simp]

theorem Finset.sort_eq {α : Type u_1} (r : α → α → Prop) [DecidableRel r] [IsTrans α r] [IsAntisymm α r] [IsTotal α r] (s : Finset α) : ↑(Finset.sort r s) = s.val

@[simp]

theorem Finset.sort_nodup {α : Type u_1} (r : α → α → Prop) [DecidableRel r] [IsTrans α r] [IsAntisymm α r] [IsTotal α r] (s : Finset α) : List.Nodup (Finset.sort r s)

@[simp]

theorem Finset.sort_toFinset {α : Type u_1} (r : α → α → Prop) [DecidableRel r] [IsTrans α r] [IsAntisymm α r] [IsTotal α r] [DecidableEq α] (s : Finset α) : List.toFinset (Finset.sort r s) = s

@[simp]

theorem Finset.mem_sort {α : Type u_1} (r : α → α → Prop) [DecidableRel r] [IsTrans α r] [IsAntisymm α r] [IsTotal α r] {s : Finset α} {a : α} : a ∈ Finset.sort r s ↔ a ∈ s

@[simp]

theorem Finset.length_sort {α : Type u_1} (r : α → α → Prop) [DecidableRel r] [IsTrans α r] [IsAntisymm α r] [IsTotal α r] {s : Finset α} : List.length (Finset.sort r s) = s.card

@[simp]

theorem Finset.sort_empty {α : Type u_1} (r : α → α → Prop) [DecidableRel r] [IsTrans α r] [IsAntisymm α r] [IsTotal α r] : Finset.sort r ∅ = []

@[simp]

theorem Finset.sort_singleton {α : Type u_1} (r : α → α → Prop) [DecidableRel r] [IsTrans α r] [IsAntisymm α r] [IsTotal α r] (a : α) : Finset.sort r {a} = [a]

theorem Finset.sort_perm_toList {α : Type u_1} (r : α → α → Prop) [DecidableRel r] [IsTrans α r] [IsAntisymm α r] [IsTotal α r] (s : Finset α) : List.Perm (Finset.sort r s) (Finset.toList s)

theorem Finset.sort_sorted_lt {α : Type u_1} [LinearOrder α] (s : Finset α) : List.Sorted (fun (x x_1 : α) => x < x_1) (Finset.sort (fun (x x_1 : α) => x ≤ x_1) s)

theorem Finset.sort_sorted_gt {α : Type u_1} [LinearOrder α] (s : Finset α) : List.Sorted (fun (x x_1 : α) => x > x_1) (Finset.sort (fun (x x_1 : α) => x ≥ x_1) s)

theorem Finset.sorted_zero_eq_min'_aux {α : Type u_1} [LinearOrder α] (s : Finset α) (h : 0 < List.length (Finset.sort (fun (x x_1 : α) => x ≤ x_1) s)) (H : s.Nonempty) : List.nthLe (Finset.sort (fun (x x_1 : α) => x ≤ x_1) s) 0 h = Finset.min' s H

theorem Finset.sorted_zero_eq_min' {α : Type u_1} [LinearOrder α] {s : Finset α} {h : 0 < List.length (Finset.sort (fun (x x_1 : α) => x ≤ x_1) s)} : List.nthLe (Finset.sort (fun (x x_1 : α) => x ≤ x_1) s) 0 h = Finset.min' s (_ : s.Nonempty)

theorem Finset.min'_eq_sorted_zero {α : Type u_1} [LinearOrder α] {s : Finset α} {h : s.Nonempty} : Finset.min' s h = List.nthLe (Finset.sort (fun (x x_1 : α) => x ≤ x_1) s) 0 (_ : 0 < List.length (Finset.sort (fun (x x_1 : α) => x ≤ x_1) s))

theorem Finset.sorted_last_eq_max'_aux {α : Type u_1} [LinearOrder α] (s : Finset α) (h : List.length (Finset.sort (fun (x x_1 : α) => x ≤ x_1) s) - 1 < List.length (Finset.sort (fun (x x_1 : α) => x ≤ x_1) s)) (H : s.Nonempty) : List.nthLe (Finset.sort (fun (x x_1 : α) => x ≤ x_1) s) (List.length (Finset.sort (fun (x x_1 : α) => x ≤ x_1) s) - 1) h = Finset.max' s H

theorem Finset.sorted_last_eq_max' {α : Type u_1} [LinearOrder α] {s : Finset α} {h : List.length (Finset.sort (fun (x x_1 : α) => x ≤ x_1) s) - 1 < List.length (Finset.sort (fun (x x_1 : α) => x ≤ x_1) s)} : List.nthLe (Finset.sort (fun (x x_1 : α) => x ≤ x_1) s) (List.length (Finset.sort (fun (x x_1 : α) => x ≤ x_1) s) - 1) h = Finset.max' s (_ : s.Nonempty)

theorem Finset.max'_eq_sorted_last {α : Type u_1} [LinearOrder α] {s : Finset α} {h : s.Nonempty} : Finset.max' s h = List.nthLe (Finset.sort (fun (x x_1 : α) => x ≤ x_1) s) (List.length (Finset.sort (fun (x x_1 : α) => x ≤ x_1) s) - 1) (_ : List.length (Finset.sort (fun (x x_1 : α) => x ≤ x_1) s) - 1 < List.length (Finset.sort (fun (x x_1 : α) => x ≤ x_1) s))

def Finset.orderIsoOfFin {α : Type u_1} [LinearOrder α] (s : Finset α) {k : ℕ} (h : s.card = k) : Fin k ≃o { x : α // x ∈ s }

Given a finset s of cardinality k in a linear order α, the map orderIsoOfFin s h is the increasing bijection between Fin k and s as an OrderIso. Here, h is a proof that the cardinality of s is k. We use this instead of an iso Fin s.card ≃o s to avoid casting issues in further uses of this function.

Equations

• One or more equations did not get rendered due to their size.

def Finset.orderEmbOfFin {α : Type u_1} [LinearOrder α] (s : Finset α) {k : ℕ} (h : s.card = k) : Fin k ↪o α

Given a finset s of cardinality k in a linear order α, the map orderEmbOfFin s h is the increasing bijection between Fin k and s as an order embedding into α. Here, h is a proof that the cardinality of s is k. We use this instead of an embedding Fin s.card ↪o α to avoid casting issues in further uses of this function.

Equations

• Finset.orderEmbOfFin s h = RelEmbedding.trans (OrderIso.toOrderEmbedding (Finset.orderIsoOfFin s h)) (OrderEmbedding.subtype fun (x : α) => x ∈ s)

@[simp]

theorem Finset.coe_orderIsoOfFin_apply {α : Type u_1} [LinearOrder α] (s : Finset α) {k : ℕ} (h : s.card = k) (i : Fin k) : ↑((Finset.orderIsoOfFin s h) i) = (Finset.orderEmbOfFin s h) i

theorem Finset.orderIsoOfFin_symm_apply {α : Type u_1} [LinearOrder α] (s : Finset α) {k : ℕ} (h : s.card = k) (x : { x : α // x ∈ s }) : ↑((OrderIso.symm (Finset.orderIsoOfFin s h)) x) = List.indexOf (↑x) (Finset.sort (fun (x x_1 : α) => x ≤ x_1) s)

theorem Finset.orderEmbOfFin_apply {α : Type u_1} [LinearOrder α] (s : Finset α) {k : ℕ} (h : s.card = k) (i : Fin k) : (Finset.orderEmbOfFin s h) i = List.nthLe (Finset.sort (fun (x x_1 : α) => x ≤ x_1) s) ↑i (_ : ↑i < List.length (Finset.sort (fun (x x_1 : α) => x ≤ x_1) s))

@[simp]

theorem Finset.orderEmbOfFin_mem {α : Type u_1} [LinearOrder α] (s : Finset α) {k : ℕ} (h : s.card = k) (i : Fin k) : (Finset.orderEmbOfFin s h) i ∈ s

@[simp]

theorem Finset.range_orderEmbOfFin {α : Type u_1} [LinearOrder α] (s : Finset α) {k : ℕ} (h : s.card = k) : Set.range ⇑(Finset.orderEmbOfFin s h) = ↑s

theorem Finset.orderEmbOfFin_zero {α : Type u_1} [LinearOrder α] {s : Finset α} {k : ℕ} (h : s.card = k) (hz : 0 < k) : (Finset.orderEmbOfFin s h) { val := 0, isLt := hz } = Finset.min' s (_ : s.Nonempty)

The bijection orderEmbOfFin s h sends 0 to the minimum of s.

theorem Finset.orderEmbOfFin_last {α : Type u_1} [LinearOrder α] {s : Finset α} {k : ℕ} (h : s.card = k) (hz : 0 < k) : (Finset.orderEmbOfFin s h) { val := k - 1, isLt := (_ : k - 1 < k) } = Finset.max' s (_ : s.Nonempty)

The bijection orderEmbOfFin s h sends k-1 to the maximum of s.

@[simp]

theorem Finset.orderEmbOfFin_singleton {α : Type u_1} [LinearOrder α] (a : α) (i : Fin 1) : (Finset.orderEmbOfFin {a} (_ : {a}.card = 1)) i = a

orderEmbOfFin {a} h sends any argument to a.

theorem Finset.orderEmbOfFin_unique {α : Type u_1} [LinearOrder α] {s : Finset α} {k : ℕ} (h : s.card = k) {f : Fin k → α} (hfs : ∀ (x : Fin k), f x ∈ s) (hmono : StrictMono f) : f = ⇑(Finset.orderEmbOfFin s h)

Any increasing map f from Fin k to a finset of cardinality k has to coincide with the increasing bijection orderEmbOfFin s h.

theorem Finset.orderEmbOfFin_unique' {α : Type u_1} [LinearOrder α] {s : Finset α} {k : ℕ} (h : s.card = k) {f : Fin k ↪o α} (hfs : ∀ (x : Fin k), f x ∈ s) : f = Finset.orderEmbOfFin s h

An order embedding f from Fin k to a finset of cardinality k has to coincide with the increasing bijection orderEmbOfFin s h.

@[simp]

theorem Finset.orderEmbOfFin_eq_orderEmbOfFin_iff {α : Type u_1} [LinearOrder α] {k : ℕ} {l : ℕ} {s : Finset α} {i : Fin k} {j : Fin l} {h : s.card = k} {h' : s.card = l} : (Finset.orderEmbOfFin s h) i = (Finset.orderEmbOfFin s h') j ↔ ↑i = ↑j

Two parametrizations orderEmbOfFin of the same set take the same value on i and j if and only if i = j. Since they can be defined on a priori not defeq types Fin k and Fin l (although necessarily k = l), the conclusion is rather written (i : ℕ) = (j : ℕ).

def Finset.orderEmbOfCardLe {α : Type u_1} [LinearOrder α] (s : Finset α) {k : ℕ} (h : k ≤ s.card) : Fin k ↪o α

Given a finset s of size at least k in a linear order α, the map orderEmbOfCardLe is an order embedding from Fin k to α whose image is contained in s. Specifically, it maps Fin k to an initial segment of s.

Equations

• Finset.orderEmbOfCardLe s h = RelEmbedding.trans (Fin.castLEEmb h) (Finset.orderEmbOfFin s (_ : s.card = s.card))

theorem Finset.orderEmbOfCardLe_mem {α : Type u_1} [LinearOrder α] (s : Finset α) {k : ℕ} (h : k ≤ s.card) (a : Fin k) : (Finset.orderEmbOfCardLe s h) a ∈ s

unsafe instance Finset.instReprFinset {α : Type u_1} [Repr α] : Repr (Finset α)

Equations

• Finset.instReprFinset = { reprPrec := fun (s : Finset α) (x : ℕ) => if s.card = 0 then Std.Format.text "∅" else repr s.val }

theorem Fin.sort_univ (n : ℕ) : Finset.sort (fun (x y : Fin n) => x ≤ y) Finset.univ = List.finRange n