mathlib docs: data.seq.seq

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def stream.is_seq {α : Type u} :

stream (option α) → Prop

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

Equations

s.is_seq = ∀ {n : ℕ}, s n = option.none → s (n + 1) = option.none

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def seq :

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.

Equations

seq α = {f // f.is_seq}

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def seq1 :

Type u_1 → Type u_1

seq1 α is the type of nonempty sequences.

Equations

seq1 α = (α × seq α)

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def seq.nil {α : Type u} :

seq α

The empty sequence

Equations

seq.nil = ⟨stream.const option.none, _⟩

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@[instance]

def seq.inhabited {α : Type u} :

inhabited (seq α)

Equations

seq.inhabited = {default := seq.nil α}

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def seq.cons {α : Type u} :

α → seq α → seq α

Prepend an element to a sequence

Equations

seq.cons a ⟨f, al⟩ = ⟨option.some a :: f, _⟩

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def seq.nth {α : Type u} :

seq α → ℕ → option α

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

Equations

seq.nth = subtype.val

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def seq.terminated_at {α : Type u} :

seq α → ℕ → Prop

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

Equations

s.terminated_at n = (s.nth n = option.none)

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@[instance]

def seq.terminated_at_decidable {α : Type u} (s : seq α) (n : ℕ) :

decidable (s.terminated_at n)

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

Equations

s.terminated_at_decidable n = decidable_of_iff' ↥((s.nth n).is_none) _

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def seq.terminates {α : Type u} :

seq α → Prop

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

Equations

s.terminates = ∃ (n : ℕ), s.terminated_at n

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@[simp]

def seq.omap {α : Type u} {β : Type v} {γ : Type w} :

(β → γ) → option (α × β) → option (α × γ)

Functorial action of the functor option (α × _)

Equations

seq.omap f (option.some (a, b)) = option.some (a, f b)
seq.omap f option.none = option.none

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def seq.head {α : Type u} :

seq α → option α

Get the first element of a sequence

Equations

s.head = s.nth 0

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def seq.tail {α : Type u} :

seq α → seq α

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

Equations

seq.tail ⟨f, al⟩ = ⟨f.tail, _⟩

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def seq.mem {α : Type u} :

α → seq α → Prop

Equations

seq.mem a s = (option.some a ∈ s.val)

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@[instance]

def seq.has_mem {α : Type u} :

has_mem α (seq α)

Equations

seq.has_mem = {mem := seq.mem α}

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theorem seq.le_stable {α : Type u} (s : seq α) {m n : ℕ} :

m ≤ n → s.val m = option.none → s.val n = option.none

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theorem seq.terminated_stable {α : Type u} (s : seq α) {m n : ℕ} :

m ≤ n → s.terminated_at m → s.terminated_at n

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

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theorem seq.ge_stable {α : Type u} (s : seq α) {aₙ : α} {n m : ℕ} :

m ≤ n → s.nth n = option.some aₙ → (∃ (aₘ : α), s.nth m = option.some aₘ)

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

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theorem seq.not_mem_nil {α : Type u} (a : α) :

a ∉ seq.nil

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theorem seq.mem_cons {α : Type u} (a : α) (s : seq α) :

a ∈ seq.cons a s

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theorem seq.mem_cons_of_mem {α : Type u} (y : α) {a : α} {s : seq α} :

a ∈ s → a ∈ seq.cons y s

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theorem seq.eq_or_mem_of_mem_cons {α : Type u} {a b : α} {s : seq α} :

a ∈ seq.cons b s → a = b ∨ a ∈ s

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@[simp]

theorem seq.mem_cons_iff {α : Type u} {a b : α} {s : seq α} :

a ∈ seq.cons b s ↔ a = b ∨ a ∈ s

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def seq.destruct {α : Type u} :

seq α → option (seq1 α)

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

Equations

s.destruct = (λ (a' : α), (a', s.tail)) <$> s.nth 0

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theorem seq.destruct_eq_nil {α : Type u} {s : seq α} :

s.destruct = option.none → s = seq.nil

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theorem seq.destruct_eq_cons {α : Type u} {s : seq α} {a : α} {s' : seq α} :

s.destruct = option.some (a, s') → s = seq.cons a s'

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@[simp]

theorem seq.destruct_nil {α : Type u} :

seq.nil.destruct = option.none

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@[simp]

theorem seq.destruct_cons {α : Type u} (a : α) (s : seq α) :

(seq.cons a s).destruct = option.some (a, s)

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theorem seq.head_eq_destruct {α : Type u} (s : seq α) :

s.head = prod.fst <$> s.destruct

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@[simp]

theorem seq.head_nil {α : Type u} :

seq.nil.head = option.none

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@[simp]

theorem seq.head_cons {α : Type u} (a : α) (s : seq α) :

(seq.cons a s).head = option.some a

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@[simp]

theorem seq.tail_nil {α : Type u} :

seq.nil.tail = seq.nil

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@[simp]

theorem seq.tail_cons {α : Type u} (a : α) (s : seq α) :

(seq.cons a s).tail = s

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def seq.cases_on {α : Type u} {C : seq α → Sort v} (s : seq α) :

C seq.nil → (Π (x : α) (s : seq α), C (seq.cons x s)) → C s

Equations

s.cases_on h1 h2 = (λ (_x : option (seq1 α)) (H : s.destruct = _x), option.rec (λ (H : s.destruct = option.none), _.mpr h1) (λ (v : seq1 α) (H : s.destruct = option.some v), prod.cases_on v (λ (a : α) (s' : seq α) (H : s.destruct = option.some (a, s')), _.mpr (h2 a s')) H) _x H) s.destruct _

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theorem seq.mem_rec_on {α : Type u} {C : seq α → Prop} {a : α} {s : seq α} :

a ∈ s → (∀ (b : α) (s' : seq α), a = b ∨ C s' → C (seq.cons b s')) → C s

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def seq.corec.F {α : Type u} {β : Type v} :

(β → option (α × β)) → option β → option α × option β

Equations

seq.corec.F f (option.some b) = seq.corec.F._match_1 (f b)
seq.corec.F f option.none = (option.none α, option.none β)
seq.corec.F._match_1 (option.some (a, b')) = (option.some a, option.some b')
seq.corec.F._match_1 option.none = (option.none α, option.none β)

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def seq.corec {α : Type u} {β : Type v} :

(β → option (α × β)) → β → seq α

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

Equations

seq.corec f b = ⟨stream.corec' (seq.corec.F f) (option.some b), _⟩

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@[simp]

theorem seq.corec_eq {α : Type u} {β : Type v} (f : β → option (α × β)) (b : β) :

(seq.corec f b).destruct = seq.omap (seq.corec f) (f b)

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def seq.of_list {α : Type u} :

list α → seq α

Embed a list as a sequence

Equations

seq.of_list l = ⟨l.nth, _⟩

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@[instance]

def seq.coe_list {α : Type u} :

has_coe (list α) (seq α)

Equations

seq.coe_list = {coe := seq.of_list α}

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@[simp]

def seq.bisim_o {α : Type u} :

(seq α → seq α → Prop) → option (seq1 α) → option (seq1 α) → Prop

Equations

seq.bisim_o R (option.some (a, s)) (option.some (a', s')) = (a = a' ∧ R s s')
seq.bisim_o R (option.some (fst, snd)) option.none = false
seq.bisim_o R option.none (option.some val) = false
seq.bisim_o R option.none option.none = true

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def seq.is_bisimulation {α : Type u} :

(seq α → seq α → Prop) → Prop

Equations

seq.is_bisimulation R = ∀ ⦃s₁ s₂ : seq α⦄, R s₁ s₂ → seq.bisim_o R s₁.destruct s₂.destruct

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theorem seq.eq_of_bisim {α : Type u} (R : seq α → seq α → Prop) (bisim : seq.is_bisimulation R) {s₁ s₂ : seq α} :

R s₁ s₂ → s₁ = s₂

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theorem 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

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theorem seq.coinduction2 {α : Type u} {β : Type v} (s : seq α) (f g : seq α → seq β) :

(∀ (s : seq α), seq.bisim_o (λ (s1 s2 : seq β), ∃ (s : seq α), s1 = f s ∧ s2 = g s) (f s).destruct (g s).destruct) → f s = g s

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def seq.of_stream {α : Type u} :

stream α → seq α

Embed an infinite stream as a sequence

Equations

seq.of_stream s = ⟨stream.map option.some s, _⟩

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@[instance]

def seq.coe_stream {α : Type u} :

has_coe (stream α) (seq α)

Equations

seq.coe_stream = {coe := seq.of_stream α}

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def seq.of_lazy_list {α : Type u} :

lazy_list α → seq α

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

Equations

seq.of_lazy_list = seq.corec (λ (l : lazy_list α), seq.of_lazy_list._match_1 l)
seq.of_lazy_list._match_1 (lazy_list.cons a l') = option.some (a, l' ())
seq.of_lazy_list._match_1 lazy_list.nil = option.none

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@[instance]

def seq.coe_lazy_list {α : Type u} :

has_coe (lazy_list α) (seq α)

Equations

seq.coe_lazy_list = {coe := seq.of_lazy_list α}

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meta def seq.to_lazy_list {α : Type u} :

seq α → lazy_list α

Translate a sequence into a lazy_list. Since lazy_list and list are isomorphic as non-meta types, this function is necessarily meta.

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meta def seq.force_to_list {α : Type u} :

seq α → list α

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

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def seq.nats :

seq ℕ

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

Equations

seq.nats = ↑stream.nats

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@[simp]

theorem seq.nats_nth (n : ℕ) :

seq.nats.nth n = option.some n

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def seq.append {α : Type u} :

seq α → 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.

Equations

s₁.append s₂ = seq.corec (λ (_x : seq α × seq α), seq.append._match_2 _x) (s₁, s₂)
seq.append._match_2 (s₁, s₂) = seq.append._match_1 s₂ s₁.destruct
seq.append._match_1 s₂ (option.some (a, s₁')) = option.some (a, s₁', s₂)
seq.append._match_1 s₂ option.none = seq.omap (λ (s₂ : seq α), (seq.nil α, s₂)) s₂.destruct

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def seq.map {α : Type u} {β : Type v} :

(α → β) → seq α → seq β

Map a function over a sequence.

Equations

seq.map f ⟨s, al⟩ = ⟨stream.map (option.map f) s, _⟩

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def 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.)

Equations

seq.join = seq.corec (λ (S : seq (seq1 α)), seq.join._match_1 S.destruct)
seq.join._match_1 (option.some ((a, s), S')) = option.some (a, seq.join._match_2 S' s.destruct)
seq.join._match_1 option.none = option.none
seq.join._match_2 S' (option.some s') = seq.cons s' S'
seq.join._match_2 S' option.none = S'

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@[simp]

def seq.drop {α : Type u} :

seq α → ℕ → seq α

Remove the first n elements from the sequence.

Equations

s.drop (n + 1) = (s.drop n).tail
s.drop 0 = s

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def seq.take {α : Type u} :

ℕ → seq α → list α

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

Equations

seq.take (n + 1) s = seq.take._match_1 (λ (r : seq α), seq.take n r) s.destruct
seq.take 0 s = list.nil
seq.take._match_1 _f_1 (option.some (x, r)) = x :: _f_1 r
seq.take._match_1 _f_1 option.none = list.nil

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def seq.split_at {α : Type u} :

ℕ → seq α → list α × seq α

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

Equations

seq.split_at (n + 1) s = seq.split_at._match_1 (λ (s' : seq α), seq.split_at n s') s.destruct
seq.split_at 0 s = (list.nil α, s)
seq.split_at._match_1 _f_1 (option.some (x, s')) = seq.split_at._match_2 x (_f_1 s')
seq.split_at._match_1 _f_1 option.none = (list.nil α, seq.nil α)
seq.split_at._match_2 x (l, r) = (x :: l, r)

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def seq.zip_with {α : Type u} {β : Type v} {γ : Type w} :

(α → β → γ) → seq α → seq β → seq γ

Combine two sequences with a function

Equations

seq.zip_with f ⟨f₁, a₁⟩ ⟨f₂, a₂⟩ = ⟨λ (n : ℕ), seq.zip_with._match_1 f (f₁ n) (f₂ n), _⟩
seq.zip_with._match_1 f (option.some a) (option.some b) = option.some (f a b)
seq.zip_with._match_1 f (option.some val) option.none = option.none
seq.zip_with._match_1 f option.none _x = option.none

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theorem seq.zip_with_nth_some {α : Type u} {β : Type v} {γ : Type w} {s : seq α} {s' : seq β} {n : ℕ} {a : α} {b : β} (s_nth_eq_some : s.nth n = option.some a) (s_nth_eq_some' : s'.nth n = option.some b) (f : α → β → γ) :

(seq.zip_with f s s').nth n = option.some (f a b)

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theorem seq.zip_with_nth_none {α : Type u} {β : Type v} {γ : Type w} {s : seq α} {s' : seq β} {n : ℕ} (s_nth_eq_none : s.nth n = option.none) (f : α → β → γ) :

(seq.zip_with f s s').nth n = option.none

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theorem seq.zip_with_nth_none' {α : Type u} {β : Type v} {γ : Type w} {s : seq α} {s' : seq β} {n : ℕ} (s'_nth_eq_none : s'.nth n = option.none) (f : α → β → γ) :

(seq.zip_with f s s').nth n = option.none

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def seq.zip {α : Type u} {β : Type v} :

seq α → seq β → seq (α × β)

Pair two sequences into a sequence of pairs

Equations

seq.zip = seq.zip_with prod.mk

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def seq.unzip {α : Type u} {β : Type v} :

seq (α × β) → seq α × seq β

Separate a sequence of pairs into two sequences

Equations

s.unzip = (seq.map prod.fst s, seq.map prod.snd s)

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def seq.to_list {α : Type u} (s : seq α) :

(∃ (n : ℕ), ¬↥((s.nth n).is_some)) → list α

Convert a sequence which is known to terminate into a list

Equations

s.to_list h = seq.take (nat.find h) s

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def seq.to_stream {α : Type u} (s : seq α) :

(∀ (n : ℕ), ↥((s.nth n).is_some)) → stream α

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

Equations

s.to_stream h = λ (n : ℕ), option.get _

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def seq.to_list_or_stream {α : Type u} (s : seq α) [decidable (∃ (n : ℕ), ¬↥((s.nth n).is_some))] :

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.)

Equations

s.to_list_or_stream = dite (∃ (n : ℕ), ¬↥((s.nth n).is_some)) (λ (h : ∃ (n : ℕ), ¬↥((s.nth n).is_some)), sum.inl (s.to_list h)) (λ (h : ¬∃ (n : ℕ), ¬↥((s.nth n).is_some)), sum.inr (s.to_stream _))

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@[simp]

theorem seq.nil_append {α : Type u} (s : seq α) :

seq.nil.append s = s

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@[simp]

theorem seq.cons_append {α : Type u} (a : α) (s t : seq α) :

(seq.cons a s).append t = seq.cons a (s.append t)

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@[simp]

theorem seq.append_nil {α : Type u} (s : seq α) :

s.append seq.nil = s

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@[simp]

theorem seq.append_assoc {α : Type u} (s t u : seq α) :

(s.append t).append u = s.append (t.append u)

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@[simp]

theorem seq.map_nil {α : Type u} {β : Type v} (f : α → β) :

seq.map f seq.nil = seq.nil

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@[simp]

theorem seq.map_cons {α : Type u} {β : Type v} (f : α → β) (a : α) (s : seq α) :

seq.map f (seq.cons a s) = seq.cons (f a) (seq.map f s)

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@[simp]

theorem seq.map_id {α : Type u} (s : seq α) :

seq.map id s = s

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@[simp]

theorem seq.map_tail {α : Type u} {β : Type v} (f : α → β) (s : seq α) :

seq.map f s.tail = (seq.map f s).tail

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theorem seq.map_comp {α : Type u} {β : Type v} {γ : Type w} (f : α → β) (g : β → γ) (s : seq α) :

seq.map (g ∘ f) s = seq.map g (seq.map f s)

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@[simp]

theorem seq.map_append {α : Type u} {β : Type v} (f : α → β) (s t : seq α) :

seq.map f (s.append t) = (seq.map f s).append (seq.map f t)

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@[simp]

theorem seq.map_nth {α : Type u} {β : Type v} (f : α → β) (s : seq α) (n : ℕ) :

(seq.map f s).nth n = option.map f (s.nth n)

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@[instance]

def seq.functor :

functor seq

Equations

seq.functor = {map := seq.map, map_const := λ (α β : Type u_1), seq.map ∘ function.const β}

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@[instance]

def seq.is_lawful_functor :

is_lawful_functor seq

Equations

seq.is_lawful_functor = _

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@[simp]

theorem seq.join_nil {α : Type u} :

seq.nil.join = seq.nil

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@[simp]

theorem seq.join_cons_nil {α : Type u} (a : α) (S : seq (seq1 α)) :

(seq.cons (a, seq.nil α) S).join = seq.cons a S.join

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@[simp]

theorem seq.join_cons_cons {α : Type u} (a b : α) (s : seq α) (S : seq (seq1 α)) :

(seq.cons (a, seq.cons b s) S).join = seq.cons a (seq.cons (b, s) S).join

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@[simp]

theorem seq.join_cons {α : Type u} (a : α) (s : seq α) (S : seq (seq1 α)) :

(seq.cons (a, s) S).join = seq.cons a (s.append S.join)

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@[simp]

theorem seq.join_append {α : Type u} (S T : seq (seq1 α)) :

(S.append T).join = S.join.append T.join

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@[simp]

theorem seq.of_list_nil {α : Type u} :

seq.of_list list.nil = seq.nil

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@[simp]

theorem seq.of_list_cons {α : Type u} (a : α) (l : list α) :

seq.of_list (a :: l) = seq.cons a (seq.of_list l)

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@[simp]

theorem seq.of_stream_cons {α : Type u} (a : α) (s : stream α) :

seq.of_stream (a :: s) = seq.cons a (seq.of_stream s)

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@[simp]

theorem seq.of_list_append {α : Type u} (l l' : list α) :

seq.of_list (l ++ l') = (seq.of_list l).append (seq.of_list l')

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@[simp]

theorem seq.of_stream_append {α : Type u} (l : list α) (s : stream α) :

seq.of_stream (l++ₛs) = (seq.of_list l).append (seq.of_stream s)

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def seq.to_list' {α : Type u_1} :

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).

Equations

s.to_list' = computation.corec (λ (_x : list α × seq α), seq.to_list'._match_2 _x) (list.nil α, s)
seq.to_list'._match_2 (l, s) = seq.to_list'._match_1 l s.destruct
seq.to_list'._match_1 l (option.some (a, s')) = sum.inr (a :: l, s')
seq.to_list'._match_1 l option.none = sum.inl l.reverse

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theorem seq.dropn_add {α : Type u} (s : seq α) (m n : ℕ) :

s.drop (m + n) = (s.drop m).drop n

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theorem seq.dropn_tail {α : Type u} (s : seq α) (n : ℕ) :

s.tail.drop n = s.drop (n + 1)

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theorem seq.nth_tail {α : Type u} (s : seq α) (n : ℕ) :

s.tail.nth n = s.nth (n + 1)

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@[ext]

theorem seq.ext {α : Type u} (s s' : seq α) :

(∀ (n : ℕ), s.nth n = s'.nth n) → s = s'

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@[simp]

theorem seq.head_dropn {α : Type u} (s : seq α) (n : ℕ) :

(s.drop n).head = s.nth n

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theorem seq.mem_map {α : Type u} {β : Type v} (f : α → β) {a : α} {s : seq α} :

a ∈ s → f a ∈ seq.map f s

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theorem seq.exists_of_mem_map {α : Type u} {β : Type v} {f : α → β} {b : β} {s : seq α} :

b ∈ seq.map f s → (∃ (a : α), a ∈ s ∧ f a = b)

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theorem seq.of_mem_append {α : Type u} {s₁ s₂ : seq α} {a : α} :

a ∈ s₁.append s₂ → a ∈ s₁ ∨ a ∈ s₂

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theorem seq.mem_append_left {α : Type u} {s₁ s₂ : seq α} {a : α} :

a ∈ s₁ → a ∈ s₁.append s₂

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def seq1.to_seq {α : Type u} :

seq1 α → seq α

Convert a seq1 to a sequence.

Equations

seq1.to_seq (a, s) = seq.cons a s

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@[instance]

def seq1.coe_seq {α : Type u} :

has_coe (seq1 α) (seq α)

Equations

seq1.coe_seq = {coe := seq1.to_seq α}

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def seq1.map {α : Type u} {β : Type v} :

(α → β) → seq1 α → seq1 β

Map a function on a seq1

Equations

seq1.map f (a, s) = (f a, seq.map f s)

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theorem seq1.map_id {α : Type u} (s : seq1 α) :

seq1.map id s = s

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def seq1.join {α : Type u} :

seq1 (seq1 α) → seq1 α

Flatten a nonempty sequence of nonempty sequences

Equations

seq1.join ((a, s), S) = seq1.join._match_3 a S s.destruct
seq1.join._match_3 a S (option.some s') = (a, (seq.cons s' S).join)
seq1.join._match_3 a S option.none = (a, S.join)

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@[simp]

theorem seq1.join_nil {α : Type u} (a : α) (S : seq (seq1 α)) :

seq1.join ((a, seq.nil α), S) = (a, S.join)

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@[simp]

theorem seq1.join_cons {α : Type u} (a b : α) (s : seq α) (S : seq (seq1 α)) :

seq1.join ((a, seq.cons b s), S) = (a, (seq.cons (b, s) S).join)

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def seq1.ret {α : Type u} :

α → seq1 α

The return operator for the seq1 monad, which produces a singleton sequence.

Equations

seq1.ret a = (a, seq.nil α)

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@[instance]

def seq1.inhabited {α : Type u} [inhabited α] :

inhabited (seq1 α)

Equations

seq1.inhabited = {default := seq1.ret (inhabited.default α)}

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def seq1.bind {α : Type u} {β : Type v} :

seq1 α → (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.)

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