Possibly infinite lists

This file provides a Seq α type representing possibly infinite lists (referred here as sequences). It is encoded as an infinite stream of options such that if f n = none, then f m = none for all m ≥ n.

def Stream'.IsSeq {α : Type u} (s : Stream' (Option α)) : Prop

A stream s : Option α is a sequence if s.get n = none implies s.get (n + 1) = none.

def Stream'.Seq (α : Type u) : Type u

Seq α is the type of possibly infinite lists (referred here as sequences). It is encoded as an infinite stream of options such that if f n = none, then f m = none for all m ≥ n.

def Stream'.Seq1 (α : Type u_1) : Type u_1

Seq1 α is the type of nonempty sequences.

def Stream'.Seq.get? {α : Type u} : Seq α → ℕ → Option α

Get the nth element of a sequence (if it exists)

@[simp]

theorem Stream'.Seq.val_eq_get {α : Type u} (s : Seq α) (n : ℕ) : ↑s n = s.get? n

@[simp]

theorem Stream'.Seq.get?_mk {α : Type u} (f : Stream' (Option α)) (hf : f.IsSeq) : get? ⟨f, hf⟩ = f

theorem Stream'.Seq.le_stable {α : Type u} (s : Seq α) {m n : ℕ} (h : m ≤ n) : s.get? m = none → s.get? n = none

theorem Stream'.Seq.ge_stable {α : Type u} (s : Seq α) {aₙ : α} {n m : ℕ} (m_le_n : m ≤ n) (s_nth_eq_some : s.get? n = some aₙ) : ∃ (aₘ : α), s.get? m = some aₘ

If s.get? n = some aₙ for some value aₙ, then there is also some value aₘ such that s.get? = some aₘ for m ≤ n.

theorem Stream'.Seq.ext {α : Type u} {s t : Seq α} (h : ∀ (n : ℕ), s.get? n = t.get? n) : s = t

theorem Stream'.Seq.ext_iff {α : Type u} {s t : Seq α} : s = t ↔ ∀ (n : ℕ), s.get? n = t.get? n

Constructors

def Stream'.Seq.nil {α : Type u} : Seq α

The empty sequence

instance Stream'.Seq.instInhabited {α : Type u} : Inhabited (Seq α)

def Stream'.Seq.cons {α : Type u} (a : α) (s : Seq α) : Seq α

Prepend an element to a sequence

@[simp]

theorem Stream'.Seq.val_cons {α : Type u} (s : Seq α) (x : α) : ↑(cons x s) = Stream'.cons (some x) ↑s

@[simp]

theorem Stream'.Seq.get?_nil {α : Type u} (n : ℕ) : nil.get? n = none

@[simp]

theorem Stream'.Seq.get?_zero_eq_none {α : Type u} {s : Seq α} : s.get? 0 = none ↔ s = nil

@[simp]

theorem Stream'.Seq.get?_cons_zero {α : Type u} (a : α) (s : Seq α) : (cons a s).get? 0 = some a

@[simp]

theorem Stream'.Seq.get?_cons_succ {α : Type u} (a : α) (s : Seq α) (n : ℕ) : (cons a s).get? (n + 1) = s.get? n

@[simp]

theorem Stream'.Seq.cons_ne_nil {α : Type u} {x : α} {s : Seq α} : cons x s ≠ nil

@[simp]

theorem Stream'.Seq.nil_ne_cons {α : Type u} {x : α} {s : Seq α} : nil ≠ cons x s

theorem Stream'.Seq.cons_injective2 {α : Type u} : Function.Injective2 cons

theorem Stream'.Seq.cons_left_injective {α : Type u} (s : Seq α) : Function.Injective fun (x : α) => cons x s

theorem Stream'.Seq.cons_right_injective {α : Type u} (x : α) : Function.Injective (cons x)

theorem Stream'.Seq.cons_eq_cons {α : Type u} {x x' : α} {s s' : Seq α} : cons x s = cons x' s' ↔ x = x' ∧ s = s'

Destructors

def Stream'.Seq.head {α : Type u} (s : Seq α) : Option α

Get the first element of a sequence

def Stream'.Seq.tail {α : Type u} (s : Seq α) : Seq α

Get the tail of a sequence (or nil if the sequence is nil)

def Stream'.Seq.destruct {α : Type u} (s : Seq α) : Option (Seq1 α)

Destructor for a sequence, resulting in either none (for nil) or some (a, s) (for cons a s).

theorem Stream'.Seq.head_eq_destruct {α : Type u} (s : Seq α) : s.head = Prod.fst <$> s.destruct

@[simp]

theorem Stream'.Seq.get?_tail {α : Type u} (s : Seq α) (n : ℕ) : s.tail.get? n = s.get? (n + 1)

@[simp]

theorem Stream'.Seq.destruct_nil {α : Type u} : nil.destruct = none

@[simp]

theorem Stream'.Seq.destruct_cons {α : Type u} (a : α) (s : Seq α) : (cons a s).destruct = some (a, s)

theorem Stream'.Seq.destruct_eq_none {α : Type u} {s : Seq α} : s.destruct = none → s = nil

theorem Stream'.Seq.destruct_eq_cons {α : Type u} {s : Seq α} {a : α} {s' : Seq α} : s.destruct = some (a, s') → s = cons a s'

@[simp]

theorem Stream'.Seq.head_nil {α : Type u} : nil.head = none

@[simp]

theorem Stream'.Seq.head_cons {α : Type u} (a : α) (s : Seq α) : (cons a s).head = some a

@[simp]

theorem Stream'.Seq.tail_nil {α : Type u} : nil.tail = nil

@[simp]

theorem Stream'.Seq.tail_cons {α : Type u} (a : α) (s : Seq α) : (cons a s).tail = s

theorem Stream'.Seq.head_eq_some {α : Type u} {s : Seq α} {x : α} (h : s.head = some x) : s = cons x s.tail

theorem Stream'.Seq.head_eq_none {α : Type u} {s : Seq α} (h : s.head = none) : s = nil

@[simp]

theorem Stream'.Seq.head_eq_none_iff {α : Type u} {s : Seq α} : s.head = none ↔ s = nil

Recursion and corecursion principles

def Stream'.Seq.recOn {α : Type u} {motive : Seq α → Sort v} (s : Seq α) (nil : motive nil) (cons : (x : α) → (s : Seq α) → motive (cons x s)) : motive s

Recursion principle for sequences, compare with List.recOn.

def Stream'.Seq.omap {α : Type u} {β : Type v} {γ : Type w} (f : β → γ) : Option (α × β) → Option (α × γ)

Functorial action of the functor Option (α × _)

def Stream'.Seq.Corec.f {α : Type u} {β : Type v} (f : β → Option (α × β)) : Option β → Option α × Option β

Corecursor over pairs of Option values

def Stream'.Seq.corec {α : Type u} {β : Type v} (f : β → Option (α × β)) (b : β) : Seq α

Corecursor for Seq α as a coinductive type. Iterates f to produce new elements of the sequence until none is obtained.

@[simp]

theorem Stream'.Seq.corec_eq {α : Type u} {β : Type v} (f : β → Option (α × β)) (b : β) : (corec f b).destruct = omap (corec f) (f b)

theorem Stream'.Seq.corec_nil {α : Type u} {β : Type v} (f : β → Option (α × β)) (b : β) (h : f b = none) : corec f b = nil

theorem Stream'.Seq.corec_cons {α : Type u} {β : Type v} {f : β → Option (α × β)} {b : β} {x : α} {s : β} (h : f b = some (x, s)) : corec f b = cons x (corec f s)

Bisimulation

def Stream'.Seq.BisimO {α : Type u} (R : Seq α → Seq α → Prop) : Option (Seq1 α) → Option (Seq1 α) → Prop

Bisimilarity relation over Option of Seq1 α

def Stream'.Seq.IsBisimulation {α : Type u} (R : Seq α → Seq α → Prop) : Prop

a relation is bisimilar if it meets the BisimO test

theorem Stream'.Seq.eq_of_bisim {α : Type u} (R : Seq α → Seq α → Prop) (bisim : IsBisimulation R) {s₁ s₂ : Seq α} (r : R s₁ s₂) : s₁ = s₂

theorem Stream'.Seq.coinduction {α : Type u} {s₁ s₂ : Seq α} : s₁.head = s₂.head → (∀ (β : Type u) (fr : Seq α → β), fr s₁ = fr s₂ → fr s₁.tail = fr s₂.tail) → s₁ = s₂

theorem Stream'.Seq.coinduction2 {α : Type u} {β : Type v} (s : Seq α) (f g : Seq α → Seq β) (H : ∀ (s : Seq α), BisimO (fun (s1 s2 : Seq β) => ∃ (s : Seq α), s1 = f s ∧ s2 = g s) (f s).destruct (g s).destruct) : f s = g s

Termination

def Stream'.Seq.TerminatedAt {α : Type u} (s : Seq α) (n : ℕ) : Prop

A sequence has terminated at position n if the value at position n equals none.

instance Stream'.Seq.terminatedAtDecidable {α : Type u} (s : Seq α) (n : ℕ) : Decidable (s.TerminatedAt n)

It is decidable whether a sequence terminates at a given position.

def Stream'.Seq.Terminates {α : Type u} (s : Seq α) : Prop

A sequence terminates if there is some position n at which it has terminated.

def Stream'.Seq.length {α : Type u} (s : Seq α) (h : s.Terminates) : ℕ

The length of a terminating sequence.

theorem Stream'.Seq.terminated_stable {α : Type u} (s : Seq α) {m n : ℕ} : m ≤ n → s.TerminatedAt m → s.TerminatedAt n

If a sequence terminated at position n, it also terminated at m ≥ n.

theorem Stream'.Seq.not_terminates_iff {α : Type u} {s : Seq α} : ¬s.Terminates ↔ ∀ (n : ℕ), (s.get? n).isSome = true

theorem Stream'.Seq.terminatedAt_nil {α : Type u} {n : ℕ} : nil.TerminatedAt n

@[simp]

theorem Stream'.Seq.cons_not_terminatedAt_zero {α : Type u} {x : α} {s : Seq α} : ¬(cons x s).TerminatedAt 0

@[simp]

theorem Stream'.Seq.cons_terminatedAt_succ_iff {α : Type u} {x : α} {s : Seq α} {n : ℕ} : (cons x s).TerminatedAt (n + 1) ↔ s.TerminatedAt n

@[simp]

theorem Stream'.Seq.terminates_nil {α : Type u} : nil.Terminates

@[simp]

theorem Stream'.Seq.terminates_cons_iff {α : Type u} {x : α} {s : Seq α} : (cons x s).Terminates ↔ s.Terminates

@[simp]

theorem Stream'.Seq.length_nil {α : Type u} : nil.length ⋯ = 0

@[simp]

theorem Stream'.Seq.length_eq_zero {α : Type u} {s : Seq α} {h : s.Terminates} : s.length h = 0 ↔ s = nil

theorem Stream'.Seq.terminatedAt_zero_iff {α : Type u} {s : Seq α} : s.TerminatedAt 0 ↔ s = nil

theorem Stream'.Seq.length_le_iff' {α : Type u} {s : Seq α} {n : ℕ} : (∃ (h : s.Terminates), s.length h ≤ n) ↔ s.TerminatedAt n

The statement of length_le_iff' does not assume that the sequence terminates. For a simpler statement of the theorem where the sequence is known to terminate see length_le_iff

theorem Stream'.Seq.length_le_iff {α : Type u} {s : Seq α} {n : ℕ} {h : s.Terminates} : s.length h ≤ n ↔ s.TerminatedAt n

The statement of length_le_iff assumes that the sequence terminates. For a statement of the where the sequence is not known to terminate see length_le_iff'

theorem Stream'.Seq.lt_length_iff' {α : Type u} {s : Seq α} {n : ℕ} : (∀ (h : s.Terminates), n < s.length h) ↔ ∃ (a : α), a ∈ s.get? n

The statement of lt_length_iff' does not assume that the sequence terminates. For a simpler statement of the theorem where the sequence is known to terminate see lt_length_iff

theorem Stream'.Seq.lt_length_iff {α : Type u} {s : Seq α} {n : ℕ} {h : s.Terminates} : n < s.length h ↔ ∃ (a : α), a ∈ s.get? n

The statement of length_le_iff assumes that the sequence terminates. For a statement of the where the sequence is not known to terminate see length_le_iff'

Membership

def Stream'.Seq.Mem {α : Type u} (s : Seq α) (a : α) : Prop

member definition for Seq

instance Stream'.Seq.instMembership {α : Type u} : Membership α (Seq α)

theorem Stream'.Seq.get?_mem {α : Type u} {s : Seq α} {n : ℕ} {x : α} (h : s.get? n = some x) : x ∈ s

theorem Stream'.Seq.notMem_nil {α : Type u} (a : α) : ¬a ∈ nil

@[deprecated Stream'.Seq.notMem_nil (since := "2025-05-23")]

theorem Stream'.Seq.not_mem_nil {α : Type u} (a : α) : ¬a ∈ nil

Alias of Stream'.Seq.notMem_nil.

theorem Stream'.Seq.mem_cons {α : Type u} (a : α) (s : Seq α) : a ∈ cons a s

theorem Stream'.Seq.mem_cons_of_mem {α : Type u} (y : α) {a : α} {s : Seq α} : a ∈ s → a ∈ cons y s

theorem Stream'.Seq.eq_or_mem_of_mem_cons {α : Type u} {a b : α} {s : Seq α} : a ∈ cons b s → a = b ∨ a ∈ s

@[simp]

theorem Stream'.Seq.mem_cons_iff {α : Type u} {a b : α} {s : Seq α} : a ∈ cons b s ↔ a = b ∨ a ∈ s

theorem Stream'.Seq.mem_rec_on {α : Type u} {C : Seq α → Prop} {a : α} {s : Seq α} (M : a ∈ s) (h1 : ∀ (b : α) (s' : Seq α), a = b ∨ C s' → C (cons b s')) : C s

Converting from/to other types

def Stream'.Seq.ofList {α : Type u} (l : List α) : Seq α

Embed a list as a sequence

instance Stream'.Seq.coeList {α : Type u} : Coe (List α) (Seq α)

@[simp]

theorem Stream'.Seq.ofList_nil {α : Type u} : ↑[] = nil

@[simp]

theorem Stream'.Seq.ofList_get? {α : Type u} (l : List α) (n : ℕ) : (↑l).get? n = l[n]?

@[deprecated Stream'.Seq.ofList_get? (since := "2025-02-21")]

theorem Stream'.Seq.ofList_get {α : Type u} (l : List α) (n : ℕ) : (↑l).get? n = l[n]?

Alias of Stream'.Seq.ofList_get?.

@[simp]

theorem Stream'.Seq.ofList_cons {α : Type u} (a : α) (l : List α) : ↑(a :: l) = cons a ↑l

theorem Stream'.Seq.ofList_injective {α : Type u} : Function.Injective ofList

def Stream'.Seq.ofStream {α : Type u} (s : Stream' α) : Seq α

Embed an infinite stream as a sequence

instance Stream'.Seq.coeStream {α : Type u} : Coe (Stream' α) (Seq α)

def Stream'.Seq.ofMLList {α : Type u} : MLList Id α → Seq α

Embed a MLList α as a sequence. Note that even though this is non-meta, it will produce infinite sequences if used with cyclic MLLists created by meta constructions.

instance Stream'.Seq.coeMLList {α : Type u} : Coe (MLList Id α) (Seq α)

unsafe def Stream'.Seq.toMLList {α : Type u} : Seq α → MLList Id α

Translate a sequence into a MLList.

unsafe def Stream'.Seq.forceToList {α : Type u} (s : Seq α) : List α

Translate a sequence to a list. This function will run forever if run on an infinite sequence.

def Stream'.Seq.take {α : Type u} : ℕ → Seq α → List α

Take the first n elements of the sequence (producing a list)

def Stream'.Seq.toList {α : Type u} (s : Seq α) (h : s.Terminates) : List α

Convert a sequence which is known to terminate into a list

def Stream'.Seq.toStream {α : Type u} (s : Seq α) (h : ¬s.Terminates) : Stream' α

Convert a sequence which is known not to terminate into a stream

def Stream'.Seq.toListOrStream {α : Type u} (s : Seq α) [Decidable s.Terminates] : List α ⊕ Stream' α

Convert a sequence into either a list or a stream depending on whether it is finite or infinite. (Without decidability of the infiniteness predicate, this is not constructively possible.)

def Stream'.Seq.toList' {α : Type u_1} (s : Seq α) : Computation (List α)

Convert a sequence into a list, embedded in a computation to allow for the possibility of infinite sequences (in which case the computation never returns anything).

Operations on sequences

def Stream'.Seq.append {α : Type u} (s₁ s₂ : Seq α) : Seq α

Append two sequences. If s₁ is infinite, then s₁ ++ s₂ = s₁, otherwise it puts s₂ at the location of the nil in s₁.

def Stream'.Seq.map {α : Type u} {β : Type v} (f : α → β) : Seq α → Seq β

Map a function over a sequence.

def Stream'.Seq.join {α : Type u} : Seq (Seq1 α) → Seq α

Flatten a sequence of sequences. (It is required that the sequences be nonempty to ensure productivity; in the case of an infinite sequence of nil, the first element is never generated.)

def Stream'.Seq.drop {α : Type u} (s : Seq α) : ℕ → Seq α

Remove the first n elements from the sequence.

def Stream'.Seq.splitAt {α : Type u} : ℕ → Seq α → List α × Seq α

Split a sequence at n, producing a finite initial segment and an infinite tail.

def Stream'.Seq.zipWith {α : Type u} {β : Type v} {γ : Type w} (f : α → β → γ) (s₁ : Seq α) (s₂ : Seq β) : Seq γ

Combine two sequences with a function

def Stream'.Seq.zip {α : Type u} {β : Type v} : Seq α → Seq β → Seq (α × β)

Pair two sequences into a sequence of pairs

def Stream'.Seq.unzip {α : Type u} {β : Type v} (s : Seq (α × β)) : Seq α × Seq β

Separate a sequence of pairs into two sequences

def Stream'.Seq.nats : Seq ℕ

The sequence of natural numbers some 0, some 1, ...

def Stream'.Seq.enum {α : Type u} (s : Seq α) : Seq (ℕ × α)

Enumerate a sequence by tagging each element with its index.

def Stream'.Seq.fold {α : Type u} {β : Type v} (s : Seq α) (init : β) (f : β → α → β) : Seq β

Folds a sequence using f, producing a sequence of intermediate values, i.e. [init, f init s.head, f (f init s.head) s.tail.head, ...].

@[simp]

theorem Stream'.Seq.ofStream_cons {α : Type u} (a : α) (s : Stream' α) : ↑(Stream'.cons a s) = cons a ↑s

theorem Stream'.Seq.terminatedAt_ofList {α : Type u} (l : List α) : (↑l).TerminatedAt l.length

theorem Stream'.Seq.terminates_ofList {α : Type u} (l : List α) : (↑l).Terminates

@[simp]

theorem Stream'.Seq.take_nil {α : Type u} {n : ℕ} : take n nil = []

@[simp]

theorem Stream'.Seq.take_zero {α : Type u} {s : Seq α} : take 0 s = []

@[simp]

theorem Stream'.Seq.take_succ_cons {α : Type u} {n : ℕ} {x : α} {s : Seq α} : take (n + 1) (cons x s) = x :: take n s

@[simp]

theorem Stream'.Seq.getElem?_take {α : Type u} (n k : ℕ) (s : Seq α) : (take k s)[n]? = if n < k then s.get? n else none

theorem Stream'.Seq.get?_mem_take {α : Type u} {s : Seq α} {m n : ℕ} (h_mn : m < n) {x : α} (h_get : s.get? m = some x) : x ∈ take n s

theorem Stream'.Seq.length_take_le {α : Type u} {s : Seq α} {n : ℕ} : (take n s).length ≤ n

theorem Stream'.Seq.length_take_of_le_length {α : Type u} {s : Seq α} {n : ℕ} (hle : ∀ (h : s.Terminates), n ≤ s.length h) : (take n s).length = n

@[simp]

theorem Stream'.Seq.length_toList {α : Type u} (s : Seq α) (h : s.Terminates) : (s.toList h).length = s.length h

@[simp]

theorem Stream'.Seq.getElem?_toList {α : Type u} (s : Seq α) (h : s.Terminates) (n : ℕ) : (s.toList h)[n]? = s.get? n

@[simp]

theorem Stream'.Seq.ofList_toList {α : Type u} (s : Seq α) (h : s.Terminates) : ↑(s.toList h) = s

@[simp]

theorem Stream'.Seq.toList_ofList {α : Type u} (l : List α) : (↑l).toList ⋯ = l

@[simp]

theorem Stream'.Seq.toList_nil {α : Type u} : nil.toList ⋯ = []

theorem Stream'.Seq.getLast?_toList {α : Type u} (s : Seq α) (h : s.Terminates) : (s.toList h).getLast? = s.get? (s.length h - 1)

@[simp]

theorem Stream'.Seq.cons_append {α : Type u} (a : α) (s t : Seq α) : (cons a s).append t = cons a (s.append t)

@[simp]

theorem Stream'.Seq.nil_append {α : Type u} (s : Seq α) : nil.append s = s

@[simp]

theorem Stream'.Seq.append_nil {α : Type u} (s : Seq α) : s.append nil = s

@[simp]

theorem Stream'.Seq.append_assoc {α : Type u} (s t u : Seq α) : (s.append t).append u = s.append (t.append u)

theorem Stream'.Seq.of_mem_append {α : Type u} {s₁ s₂ : Seq α} {a : α} (h : a ∈ s₁.append s₂) : a ∈ s₁ ∨ a ∈ s₂

theorem Stream'.Seq.mem_append_left {α : Type u} {s₁ s₂ : Seq α} {a : α} (h : a ∈ s₁) : a ∈ s₁.append s₂

@[simp]

theorem Stream'.Seq.ofList_append {α : Type u} (l l' : List α) : ↑(l ++ l') = (↑l).append ↑l'

@[simp]

theorem Stream'.Seq.ofStream_append {α : Type u} (l : List α) (s : Stream' α) : ↑(l ++ₛ s) = (↑l).append ↑s

@[simp]

theorem Stream'.Seq.map_get? {α : Type u} {β : Type v} (f : α → β) (s : Seq α) (n : ℕ) : (map f s).get? n = Option.map f (s.get? n)

@[simp]

theorem Stream'.Seq.map_nil {α : Type u} {β : Type v} (f : α → β) : map f nil = nil

@[simp]

theorem Stream'.Seq.map_cons {α : Type u} {β : Type v} (f : α → β) (a : α) (s : Seq α) : map f (cons a s) = cons (f a) (map f s)

@[simp]

theorem Stream'.Seq.map_id {α : Type u} (s : Seq α) : map id s = s

@[simp]

theorem Stream'.Seq.map_tail {α : Type u} {β : Type v} (f : α → β) (s : Seq α) : map f s.tail = (map f s).tail

theorem Stream'.Seq.map_comp {α : Type u} {β : Type v} {γ : Type w} (f : α → β) (g : β → γ) (s : Seq α) : map (g ∘ f) s = map g (map f s)

@[simp]

theorem Stream'.Seq.terminatedAt_map_iff {α : Type u} {β : Type v} {f : α → β} {s : Seq α} {n : ℕ} : (map f s).TerminatedAt n ↔ s.TerminatedAt n

@[simp]

theorem Stream'.Seq.terminates_map_iff {α : Type u} {β : Type v} {f : α → β} {s : Seq α} : (map f s).Terminates ↔ s.Terminates

@[simp]

theorem Stream'.Seq.length_map {α : Type u} {β : Type v} {s : Seq α} {f : α → β} (h : (map f s).Terminates) : (map f s).length h = s.length ⋯

theorem Stream'.Seq.mem_map {α : Type u} {β : Type v} (f : α → β) {a : α} {s : Seq α} : a ∈ s → f a ∈ map f s

theorem Stream'.Seq.exists_of_mem_map {α : Type u} {β : Type v} {f : α → β} {b : β} {s : Seq α} : b ∈ map f s → ∃ (a : α), a ∈ s ∧ f a = b

@[simp]

theorem Stream'.Seq.map_append {α : Type u} {β : Type v} (f : α → β) (s t : Seq α) : map f (s.append t) = (map f s).append (map f t)

@[simp]

theorem Stream'.Seq.join_nil {α : Type u} : nil.join = nil

theorem Stream'.Seq.join_cons_nil {α : Type u} (a : α) (S : Seq (Seq1 α)) : (cons (a, nil) S).join = cons a S.join

theorem Stream'.Seq.join_cons_cons {α : Type u} (a b : α) (s : Seq α) (S : Seq (Seq1 α)) : (cons (a, cons b s) S).join = cons a (cons (b, s) S).join

@[simp]

theorem Stream'.Seq.join_cons {α : Type u} (a : α) (s : Seq α) (S : Seq (Seq1 α)) : (cons (a, s) S).join = cons a (s.append S.join)

@[simp]

theorem Stream'.Seq.join_append {α : Type u} (S T : Seq (Seq1 α)) : (S.append T).join = S.join.append T.join

@[simp]

theorem Stream'.Seq.drop_get? {α : Type u} {n m : ℕ} {s : Seq α} : (s.drop n).get? m = s.get? (n + m)

theorem Stream'.Seq.dropn_add {α : Type u} (s : Seq α) (m n : ℕ) : s.drop (m + n) = (s.drop m).drop n

theorem Stream'.Seq.dropn_tail {α : Type u} (s : Seq α) (n : ℕ) : s.tail.drop n = s.drop (n + 1)

@[simp]

theorem Stream'.Seq.head_dropn {α : Type u} (s : Seq α) (n : ℕ) : (s.drop n).head = s.get? n

@[simp]

theorem Stream'.Seq.drop_succ_cons {α : Type u} {x : α} {s : Seq α} {n : ℕ} : (cons x s).drop (n + 1) = s.drop n

@[simp]

theorem Stream'.Seq.drop_nil {α : Type u} {n : ℕ} : nil.drop n = nil

theorem Stream'.Seq.take_drop {α : Type u} {s : Seq α} {n m : ℕ} : List.drop m (take n s) = take (n - m) (s.drop m)

@[simp]

theorem Stream'.Seq.get?_zipWith {α : Type u} {β : Type v} {γ : Type w} (f : α → β → γ) (s : Seq α) (s' : Seq β) (n : ℕ) : (zipWith f s s').get? n = Option.map₂ f (s.get? n) (s'.get? n)

@[simp]

theorem Stream'.Seq.get?_zip {α : Type u} {β : Type v} (s : Seq α) (t : Seq β) (n : ℕ) : (s.zip t).get? n = Option.map₂ Prod.mk (s.get? n) (t.get? n)

@[simp]

theorem Stream'.Seq.nats_get? (n : ℕ) : nats.get? n = some n

@[simp]

theorem Stream'.Seq.get?_enum {α : Type u} (s : Seq α) (n : ℕ) : s.enum.get? n = Option.map (Prod.mk n) (s.get? n)

@[simp]

theorem Stream'.Seq.zipWith_nil_left {α : Type u} {β : Type v} {γ : Type w} {f : α → β → γ} {s : Seq β} : zipWith f nil s = nil

@[simp]

theorem Stream'.Seq.zipWith_nil_right {α : Type u} {β : Type v} {γ : Type w} {f : α → β → γ} {s : Seq α} : zipWith f s nil = nil

@[simp]

theorem Stream'.Seq.zipWith_cons_cons {α : Type u} {β : Type v} {γ : Type w} {f : α → β → γ} {x : α} {s : Seq α} {x' : β} {s' : Seq β} : zipWith f (cons x s) (cons x' s') = cons (f x x') (zipWith f s s')

@[simp]

theorem Stream'.Seq.zip_nil_left {α : Type u} {s : Seq α} : nil.zip s = nil

@[simp]

theorem Stream'.Seq.zip_nil_right {α : Type u} {s : Seq α} : s.zip nil = nil

@[simp]

theorem Stream'.Seq.zip_cons_cons {α : Type u} {s s' : Seq α} {x x' : α} : (cons x s).zip (cons x' s') = cons (x, x') (s.zip s')

@[simp]

theorem Stream'.Seq.enum_nil {α : Type u} : nil.enum = nil

@[simp]

theorem Stream'.Seq.enum_cons {α : Type u} (s : Seq α) (x : α) : (cons x s).enum = cons (0, x) (map (Prod.map Nat.succ id) s.enum)

theorem Stream'.Seq.zipWith_map {α : Type u} {β : Type v} {γ : Type w} {α' : Type u'} {β' : Type v'} (s₁ : Seq α) (s₂ : Seq β) (f₁ : α → α') (f₂ : β → β') (g : α' → β' → γ) : zipWith g (map f₁ s₁) (map f₂ s₂) = zipWith (fun (a : α) (b : β) => g (f₁ a) (f₂ b)) s₁ s₂

theorem Stream'.Seq.zipWith_map_left {α : Type u} {β : Type v} {γ : Type w} {α' : Type u'} (s₁ : Seq α) (s₂ : Seq β) (f : α → α') (g : α' → β → γ) : zipWith g (map f s₁) s₂ = zipWith (fun (a : α) (b : β) => g (f a) b) s₁ s₂

theorem Stream'.Seq.zipWith_map_right {α : Type u} {β : Type v} {γ : Type w} {β' : Type v'} (s₁ : Seq α) (s₂ : Seq β) (f : β → β') (g : α → β' → γ) : zipWith g s₁ (map f s₂) = zipWith (fun (a : α) (b : β) => g a (f b)) s₁ s₂

theorem Stream'.Seq.zip_map {α : Type u} {β : Type v} {α' : Type u'} {β' : Type v'} (s₁ : Seq α) (s₂ : Seq β) (f₁ : α → α') (f₂ : β → β') : (map f₁ s₁).zip (map f₂ s₂) = map (Prod.map f₁ f₂) (s₁.zip s₂)

theorem Stream'.Seq.zip_map_left {α : Type u} {β : Type v} {α' : Type u'} (s₁ : Seq α) (s₂ : Seq β) (f : α → α') : (map f s₁).zip s₂ = map (Prod.map f id) (s₁.zip s₂)

theorem Stream'.Seq.zip_map_right {α : Type u} {β : Type v} {β' : Type v'} (s₁ : Seq α) (s₂ : Seq β) (f : β → β') : s₁.zip (map f s₂) = map (Prod.map id f) (s₁.zip s₂)

@[simp]

theorem Stream'.Seq.fold_nil {α : Type u} {β : Type v} (init : β) (f : β → α → β) : nil.fold init f = cons init nil

@[simp]

theorem Stream'.Seq.fold_cons {α : Type u} {β : Type v} (init : β) (f : β → α → β) (x : α) (s : Seq α) : (cons x s).fold init f = cons init (s.fold (f init x) f)

@[simp]

theorem Stream'.Seq.fold_head {α : Type u} {β : Type v} (init : β) (f : β → α → β) (s : Seq α) : (s.fold init f).head = some init

instance Stream'.Seq.instFunctor : Functor Seq

instance Stream'.Seq.instLawfulFunctor : LawfulFunctor Seq

def Stream'.Seq1.toSeq {α : Type u} : Seq1 α → Seq α

Convert a Seq1 to a sequence.

instance Stream'.Seq1.coeSeq {α : Type u} : Coe (Seq1 α) (Seq α)

def Stream'.Seq1.map {α : Type u} {β : Type v} (f : α → β) : Seq1 α → Seq1 β

Map a function on a Seq1

theorem Stream'.Seq1.map_pair {α : Type u} {β : Type v} {f : α → β} {a : α} {s : Seq α} : map f (a, s) = (f a, Seq.map f s)

theorem Stream'.Seq1.map_id {α : Type u} (s : Seq1 α) : map id s = s

def Stream'.Seq1.join {α : Type u} : Seq1 (Seq1 α) → Seq1 α

Flatten a nonempty sequence of nonempty sequences

@[simp]

theorem Stream'.Seq1.join_nil {α : Type u} (a : α) (S : Seq (Seq1 α)) : join ((a, Seq.nil), S) = (a, S.join)

@[simp]

theorem Stream'.Seq1.join_cons {α : Type u} (a b : α) (s : Seq α) (S : Seq (Seq1 α)) : join ((a, Seq.cons b s), S) = (a, (Seq.cons (b, s) S).join)

def Stream'.Seq1.ret {α : Type u} (a : α) : Seq1 α

The return operator for the Seq1 monad, which produces a singleton sequence.

instance Stream'.Seq1.instInhabited {α : Type u} [Inhabited α] : Inhabited (Seq1 α)

def Stream'.Seq1.bind {α : Type u} {β : Type v} (s : Seq1 α) (f : α → Seq1 β) : Seq1 β

The bind operator for the Seq1 monad, which maps f on each element of s and appends the results together. (Not all of s may be evaluated, because the first few elements of s may already produce an infinite result.)

@[simp]

theorem Stream'.Seq1.join_map_ret {α : Type u} (s : Seq α) : (Seq.map ret s).join = s

@[simp]

theorem Stream'.Seq1.bind_ret {α : Type u} {β : Type v} (f : α → β) (s : Seq1 α) : s.bind (ret ∘ f) = map f s

@[simp]

theorem Stream'.Seq1.ret_bind {α : Type u} {β : Type v} (a : α) (f : α → Seq1 β) : (ret a).bind f = f a

@[simp]

theorem Stream'.Seq1.map_join' {α : Type u} {β : Type v} (f : α → β) (S : Seq (Seq1 α)) : Seq.map f S.join = (Seq.map (map f) S).join

@[simp]

theorem Stream'.Seq1.map_join {α : Type u} {β : Type v} (f : α → β) (S : Seq1 (Seq1 α)) : map f S.join = (map (map f) S).join

@[simp]

theorem Stream'.Seq1.join_join {α : Type u} (SS : Seq (Seq1 (Seq1 α))) : SS.join.join = (Seq.map join SS).join

@[simp]

theorem Stream'.Seq1.bind_assoc {α : Type u} {β : Type v} {γ : Type w} (s : Seq1 α) (f : α → Seq1 β) (g : β → Seq1 γ) : (s.bind f).bind g = s.bind fun (x : α) => (f x).bind g

instance Stream'.Seq1.monad : Monad Seq1

instance Stream'.Seq1.lawfulMonad : LawfulMonad Seq1