Partially defined possibly infinite lists

This file provides a WSeq α type representing partially defined possibly infinite lists (referred here as weak sequences).

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

Weak sequences.

While the Seq structure allows for lists which may not be finite, a weak sequence also allows the computation of each element to involve an indeterminate amount of computation, including possibly an infinite loop. This is represented as a regular Seq interspersed with none elements to indicate that computation is ongoing.

This model is appropriate for Haskell style lazy lists, and is closed under most interesting computation patterns on infinite lists, but conversely it is difficult to extract elements from it.

Equations

• Stream'.WSeq α = Stream'.Seq (Option α)

def Stream'.WSeq.ofSeq {α : Type u} : Seq α → WSeq α

Turn a sequence into a weak sequence

Equations

• Stream'.WSeq.ofSeq = (fun (x1 : α → Option α) (x2 : Stream'.Seq α) => x1 <$> x2) some

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

Turn a list into a weak sequence

Equations

• ↑l = ↑↑l

def Stream'.WSeq.ofStream {α : Type u} (l : Stream' α) : WSeq α

Turn a stream into a weak sequence

Equations

• ↑l = ↑↑l

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

Equations

• Stream'.WSeq.coeSeq = { coe := Stream'.WSeq.ofSeq }

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

Equations

• Stream'.WSeq.coeList = { coe := Stream'.WSeq.ofList }

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

Equations

• Stream'.WSeq.coeStream = { coe := Stream'.WSeq.ofStream }

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

The empty weak sequence

Equations

• Stream'.WSeq.nil = Stream'.Seq.nil

instance Stream'.WSeq.inhabited {α : Type u} : Inhabited (WSeq α)

Equations

• Stream'.WSeq.inhabited = { default := Stream'.WSeq.nil }

def Stream'.WSeq.cons {α : Type u} (a : α) : WSeq α → WSeq α

Prepend an element to a weak sequence

Equations

• Stream'.WSeq.cons a = Stream'.Seq.cons (some a)

def Stream'.WSeq.think {α : Type u} : WSeq α → WSeq α

Compute for one tick, without producing any elements

Equations

• Stream'.WSeq.think = Stream'.Seq.cons none

def Stream'.WSeq.destruct {α : Type u} : WSeq α → Computation (Option (α × WSeq α))

Destruct a weak sequence, to (eventually possibly) produce either none for nil or some (a, s) if an element is produced.

Equations

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

def Stream'.WSeq.recOn {α : Type u} {C : WSeq α → Sort v} (s : WSeq α) (h1 : C nil) (h2 : (x : α) → (s : WSeq α) → C (cons x s)) (h3 : (s : WSeq α) → C s.think) : C s

Recursion principle for weak sequences, compare with List.recOn.

Equations

• s.recOn h1 h2 h3 = Stream'.Seq.recOn s h1 fun (o : Option α) => Option.recOn (motive := fun (x : Option α) => (s : Stream'.Seq (Option α)) → C (Stream'.Seq.cons x s)) o h3 h2

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

membership for weak sequences

Equations

• s.Mem a = Stream'.Seq.Mem s (some a)

instance Stream'.WSeq.membership {α : Type u} : Membership α (WSeq α)

Equations

• Stream'.WSeq.membership = { mem := Stream'.WSeq.Mem }

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

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

Get the head of a weak sequence. This involves a possibly infinite computation.

Equations

• s.head = Computation.map (fun (x : Option (α × Stream'.WSeq α)) => Prod.fst <$> x) s.destruct

def Stream'.WSeq.flatten {α : Type u} : Computation (WSeq α) → WSeq α

Encode a computation yielding a weak sequence into additional think constructors in a weak sequence

Equations

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

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

Get the tail of a weak sequence. This doesn't need a Computation wrapper, unlike head, because flatten allows us to hide this in the construction of the weak sequence itself.

Equations

• s.tail = Stream'.WSeq.flatten ((fun (o : Option (α × Stream'.WSeq α)) => Option.recOn o Stream'.WSeq.nil Prod.snd) <$> s.destruct)

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

drop the first n elements from s.

Equations

• s.drop 0 = s
• s.drop n.succ = (s.drop n).tail

def Stream'.WSeq.get? {α : Type u} (s : WSeq α) (n : ℕ) : Computation (Option α)

Get the nth element of s.

Equations

• s.get? n = (s.drop n).head

def Stream'.WSeq.toList {α : Type u} (s : WSeq α) : Computation (List α)

Convert s to a list (if it is finite and completes in finite time).

Equations

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

def Stream'.WSeq.length {α : Type u} (s : WSeq α) : Computation ℕ

Get the length of s (if it is finite and completes in finite time).

Equations

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

class Stream'.WSeq.IsFinite {α : Type u} (s : WSeq α) : Prop

A weak sequence is finite if toList s terminates. Equivalently, it is a finite number of think and cons applied to nil.

• out : s.toList.Terminates

instance Stream'.WSeq.toList_terminates {α : Type u} (s : WSeq α) [h : s.IsFinite] : s.toList.Terminates

def Stream'.WSeq.get {α : Type u} (s : WSeq α) [s.IsFinite] : List α

Get the list corresponding to a finite weak sequence.

Equations

• s.get = s.toList.get

class Stream'.WSeq.Productive {α : Type u} (s : WSeq α) : Prop

A weak sequence is productive if it never stalls forever - there are always a finite number of thinks between cons constructors. The sequence itself is allowed to be infinite though.

• get?_terminates (n : ℕ) : (s.get? n).Terminates

theorem Stream'.WSeq.productive_iff {α : Type u} (s : WSeq α) : s.Productive ↔ ∀ (n : ℕ), (s.get? n).Terminates

instance Stream'.WSeq.get?_terminates {α : Type u} (s : WSeq α) [h : s.Productive] (n : ℕ) : (s.get? n).Terminates

instance Stream'.WSeq.head_terminates {α : Type u} (s : WSeq α) [s.Productive] : s.head.Terminates

def Stream'.WSeq.updateNth {α : Type u} (s : WSeq α) (n : ℕ) (a : α) : WSeq α

Replace the nth element of s with a.

Equations

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

def Stream'.WSeq.removeNth {α : Type u} (s : WSeq α) (n : ℕ) : WSeq α

Remove the nth element of s.

Equations

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

def Stream'.WSeq.filterMap {α : Type u} {β : Type v} (f : α → Option β) : WSeq α → WSeq β

Map the elements of s over f, removing any values that yield none.

Equations

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

def Stream'.WSeq.filter {α : Type u} (p : α → Prop) [DecidablePred p] : WSeq α → WSeq α

Select the elements of s that satisfy p.

Equations

• Stream'.WSeq.filter p = Stream'.WSeq.filterMap fun (a : α) => if p a then some a else none

def Stream'.WSeq.find {α : Type u} (p : α → Prop) [DecidablePred p] (s : WSeq α) : Computation (Option α)

Get the first element of s satisfying p.

Equations

• Stream'.WSeq.find p s = (Stream'.WSeq.filter p s).head

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

Zip a function over two weak sequences

Equations

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

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

Zip two weak sequences into a single sequence of pairs

Equations

• Stream'.WSeq.zip = Stream'.WSeq.zipWith Prod.mk

def Stream'.WSeq.findIndexes {α : Type u} (p : α → Prop) [DecidablePred p] (s : WSeq α) : WSeq ℕ

Get the list of indexes of elements of s satisfying p

Equations

• Stream'.WSeq.findIndexes p s = Stream'.WSeq.filterMap (fun (x : α × ℕ) => match x with | (a, n) => if p a then some n else none) (s.zip ↑Stream'.nats)

def Stream'.WSeq.findIndex {α : Type u} (p : α → Prop) [DecidablePred p] (s : WSeq α) : Computation ℕ

Get the index of the first element of s satisfying p

Equations

• Stream'.WSeq.findIndex p s = (fun (o : Option ℕ) => o.getD 0) <$> (Stream'.WSeq.findIndexes p s).head

def Stream'.WSeq.indexOf {α : Type u} [DecidableEq α] (a : α) : WSeq α → Computation ℕ

Get the index of the first occurrence of a in s

Equations

• Stream'.WSeq.indexOf a = Stream'.WSeq.findIndex (Eq a)

def Stream'.WSeq.indexesOf {α : Type u} [DecidableEq α] (a : α) : WSeq α → WSeq ℕ

Get the indexes of occurrences of a in s

Equations

• Stream'.WSeq.indexesOf a = Stream'.WSeq.findIndexes (Eq a)

def Stream'.WSeq.union {α : Type u} (s1 s2 : WSeq α) : WSeq α

union s1 s2 is a weak sequence which interleaves s1 and s2 in some order (nondeterministically).

Equations

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

def Stream'.WSeq.isEmpty {α : Type u} (s : WSeq α) : Computation Bool

Returns true if s is nil and false if s has an element

Equations

• s.isEmpty = Computation.map Option.isNone s.head

def Stream'.WSeq.compute {α : Type u} (s : WSeq α) : WSeq α

Calculate one step of computation

Equations

• s.compute = match Stream'.Seq.destruct s with | some (none, s') => s' | x => s

def Stream'.WSeq.take {α : Type u} (s : WSeq α) (n : ℕ) : WSeq α

Get the first n elements of a weak sequence

Equations

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

def Stream'.WSeq.splitAt {α : Type u} (s : WSeq α) (n : ℕ) : Computation (List α × WSeq α)

Split the sequence at position n into a finite initial segment and the weak sequence tail

Equations

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

def Stream'.WSeq.any {α : Type u} (s : WSeq α) (p : α → Bool) : Computation Bool

Returns true if any element of s satisfies p

Equations

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

def Stream'.WSeq.all {α : Type u} (s : WSeq α) (p : α → Bool) : Computation Bool

Returns true if every element of s satisfies p

Equations

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

def Stream'.WSeq.scanl {α : Type u} {β : Type v} (f : α → β → α) (a : α) (s : WSeq β) : WSeq α

Apply a function to the elements of the sequence to produce a sequence of partial results. (There is no scanr because this would require working from the end of the sequence, which may not exist.)

Equations

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

def Stream'.WSeq.inits {α : Type u} (s : WSeq α) : WSeq (List α)

Get the weak sequence of initial segments of the input sequence

Equations

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

def Stream'.WSeq.collect {α : Type u} (s : WSeq α) (n : ℕ) : List α

Like take, but does not wait for a result. Calculates n steps of computation and returns the sequence computed so far

Equations

• s.collect n = List.filterMap id (Stream'.Seq.take n s)

def Stream'.WSeq.append {α : Type u} : WSeq α → WSeq α → WSeq α

Append two weak sequences. As with Seq.append, this may not use the second sequence if the first one takes forever to compute

Equations

• Stream'.WSeq.append = Stream'.Seq.append

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

Map a function over a weak sequence

Equations

• Stream'.WSeq.map f = Stream'.Seq.map (Option.map f)

def Stream'.WSeq.join {α : Type u} (S : WSeq (WSeq α)) : WSeq α

Flatten a sequence of weak sequences. (Note that this allows empty sequences, unlike Seq.join.)

Equations

• S.join = ((fun (o : Option (Stream'.WSeq α)) => match o with | none => Stream'.Seq1.ret none | some s => (none, s)) <$> S).join

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

Monadic bind operator for weak sequences

Equations

• s.bind f = (Stream'.WSeq.map f s).join

def Stream'.WSeq.LiftRelO {α : Type u} {β : Type v} (R : α → β → Prop) (C : WSeq α → WSeq β → Prop) : Option (α × WSeq α) → Option (β × WSeq β) → Prop

lift a relation to a relation over weak sequences

Equations

• Stream'.WSeq.LiftRelO R C none none = True
• Stream'.WSeq.LiftRelO R C (some (a, s)) (some (b, t)) = (R a b ∧ C s t)
• Stream'.WSeq.LiftRelO R C x✝¹ x✝ = False

theorem Stream'.WSeq.LiftRelO.imp {α : Type u} {β : Type v} {R S : α → β → Prop} {C D : WSeq α → WSeq β → Prop} (H1 : ∀ (a : α) (b : β), R a b → S a b) (H2 : ∀ (s : WSeq α) (t : WSeq β), C s t → D s t) {o : Option (α × WSeq α)} {p : Option (β × WSeq β)} : LiftRelO R C o p → LiftRelO S D o p

theorem Stream'.WSeq.LiftRelO.imp_right {α : Type u} {β : Type v} (R : α → β → Prop) {C D : WSeq α → WSeq β → Prop} (H : ∀ (s : WSeq α) (t : WSeq β), C s t → D s t) {o : Option (α × WSeq α)} {p : Option (β × WSeq β)} : LiftRelO R C o p → LiftRelO R D o p

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

Definition of bisimilarity for weak sequences

Equations

• Stream'.WSeq.BisimO R = Stream'.WSeq.LiftRelO (fun (x1 x2 : α) => x1 = x2) R

theorem Stream'.WSeq.BisimO.imp {α : Type u} {R S : WSeq α → WSeq α → Prop} (H : ∀ (s t : WSeq α), R s t → S s t) {o p : Option (α × WSeq α)} : BisimO R o p → BisimO S o p

def Stream'.WSeq.LiftRel {α : Type u} {β : Type v} (R : α → β → Prop) (s : WSeq α) (t : WSeq β) : Prop

Two weak sequences are LiftRel R related if they are either both empty, or they are both nonempty and the heads are R related and the tails are LiftRel R related. (This is a coinductive definition.)

Equations

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

def Stream'.WSeq.Equiv {α : Type u} : WSeq α → WSeq α → Prop

If two sequences are equivalent, then they have the same values and the same computational behavior (i.e. if one loops forever then so does the other), although they may differ in the number of thinks needed to arrive at the answer.

Equations

• Stream'.WSeq.Equiv = Stream'.WSeq.LiftRel fun (x1 x2 : α) => x1 = x2

theorem Stream'.WSeq.liftRel_destruct {α : Type u} {β : Type v} {R : α → β → Prop} {s : WSeq α} {t : WSeq β} : LiftRel R s t → Computation.LiftRel (LiftRelO R (LiftRel R)) s.destruct t.destruct

theorem Stream'.WSeq.liftRel_destruct_iff {α : Type u} {β : Type v} {R : α → β → Prop} {s : WSeq α} {t : WSeq β} : LiftRel R s t ↔ Computation.LiftRel (LiftRelO R (LiftRel R)) s.destruct t.destruct

def Stream'.WSeq.«term_~ʷ_» : Lean.TrailingParserDescr

If two sequences are equivalent, then they have the same values and the same computational behavior (i.e. if one loops forever then so does the other), although they may differ in the number of thinks needed to arrive at the answer.

Equations

• Stream'.WSeq.«term_~ʷ_» = Lean.ParserDescr.trailingNode `Stream'.WSeq.«term_~ʷ_» 50 50 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " ~ʷ ") (Lean.ParserDescr.cat `term 51))

theorem Stream'.WSeq.destruct_congr {α : Type u} {s t : WSeq α} : s ~ʷ t → Computation.LiftRel (BisimO fun (x1 x2 : WSeq α) => x1 ~ʷ x2) s.destruct t.destruct

theorem Stream'.WSeq.destruct_congr_iff {α : Type u} {s t : WSeq α} : s ~ʷ t ↔ Computation.LiftRel (BisimO fun (x1 x2 : WSeq α) => x1 ~ʷ x2) s.destruct t.destruct

theorem Stream'.WSeq.LiftRel.refl {α : Type u} (R : α → α → Prop) (H : Reflexive R) : Reflexive (LiftRel R)

theorem Stream'.WSeq.LiftRelO.swap {α : Type u} {β : Type v} (R : α → β → Prop) (C : WSeq α → WSeq β → Prop) : Function.swap (LiftRelO R C) = LiftRelO (Function.swap R) (Function.swap C)

theorem Stream'.WSeq.LiftRel.swap_lem {α : Type u} {β : Type v} {R : α → β → Prop} {s1 : WSeq α} {s2 : WSeq β} (h : LiftRel R s1 s2) : LiftRel (Function.swap R) s2 s1

theorem Stream'.WSeq.LiftRel.swap {α : Type u} {β : Type v} (R : α → β → Prop) : Function.swap (LiftRel R) = LiftRel (Function.swap R)

theorem Stream'.WSeq.LiftRel.symm {α : Type u} (R : α → α → Prop) (H : Symmetric R) : Symmetric (LiftRel R)

theorem Stream'.WSeq.LiftRel.trans {α : Type u} (R : α → α → Prop) (H : Transitive R) : Transitive (LiftRel R)

theorem Stream'.WSeq.LiftRel.equiv {α : Type u} (R : α → α → Prop) : Equivalence R → Equivalence (LiftRel R)

theorem Stream'.WSeq.Equiv.refl {α : Type u} (s : WSeq α) : s ~ʷ s

theorem Stream'.WSeq.Equiv.symm {α : Type u} {s t : WSeq α} : s ~ʷ t → t ~ʷ s

theorem Stream'.WSeq.Equiv.trans {α : Type u} {s t u : WSeq α} : s ~ʷ t → t ~ʷ u → s ~ʷ u

theorem Stream'.WSeq.Equiv.equivalence {α : Type u} : Equivalence Equiv

@[simp]

theorem Stream'.WSeq.destruct_nil {α : Type u} : nil.destruct = Computation.pure none

@[simp]

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

@[simp]

theorem Stream'.WSeq.destruct_think {α : Type u} (s : WSeq α) : s.think.destruct = s.destruct.think

@[simp]

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

@[simp]

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

@[simp]

theorem Stream'.WSeq.seq_destruct_think {α : Type u} (s : WSeq α) : Seq.destruct s.think = some (none, s)

@[simp]

theorem Stream'.WSeq.head_nil {α : Type u} : nil.head = Computation.pure none

@[simp]

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

@[simp]

theorem Stream'.WSeq.head_think {α : Type u} (s : WSeq α) : s.think.head = s.head.think

@[simp]

theorem Stream'.WSeq.flatten_pure {α : Type u} (s : WSeq α) : flatten (Computation.pure s) = s

@[simp]

theorem Stream'.WSeq.flatten_think {α : Type u} (c : Computation (WSeq α)) : flatten c.think = (flatten c).think

@[simp]

theorem Stream'.WSeq.destruct_flatten {α : Type u} (c : Computation (WSeq α)) : (flatten c).destruct = c >>= destruct

theorem Stream'.WSeq.head_terminates_iff {α : Type u} (s : WSeq α) : s.head.Terminates ↔ s.destruct.Terminates

@[simp]

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

@[simp]

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

@[simp]

theorem Stream'.WSeq.tail_think {α : Type u} (s : WSeq α) : s.think.tail = s.tail.think

@[simp]

theorem Stream'.WSeq.dropn_nil {α : Type u} (n : ℕ) : nil.drop n = nil

@[simp]

theorem Stream'.WSeq.dropn_cons {α : Type u} (a : α) (s : WSeq α) (n : ℕ) : (cons a s).drop (n + 1) = s.drop n

@[simp]

theorem Stream'.WSeq.dropn_think {α : Type u} (s : WSeq α) (n : ℕ) : s.think.drop n = (s.drop n).think

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

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

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

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

@[simp]

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

@[simp]

theorem Stream'.WSeq.join_think {α : Type u} (S : WSeq (WSeq α)) : S.think.join = S.join.think

@[simp]

theorem Stream'.WSeq.join_cons {α : Type u} (s : WSeq α) (S : WSeq (WSeq α)) : (cons s S).join = (s.append S.join).think

@[simp]

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

@[simp]

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

@[simp]

theorem Stream'.WSeq.think_append {α : Type u} (s t : WSeq α) : s.think.append t = (s.append t).think

@[simp]

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

@[simp]

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

def Stream'.WSeq.tail.aux {α : Type u} : Option (α × WSeq α) → Computation (Option (α × WSeq α))

auxiliary definition of tail over weak sequences

Equations

• Stream'.WSeq.tail.aux none = Computation.pure none
• Stream'.WSeq.tail.aux (some (fst, s)) = s.destruct

theorem Stream'.WSeq.destruct_tail {α : Type u} (s : WSeq α) : s.tail.destruct = s.destruct >>= tail.aux

def Stream'.WSeq.drop.aux {α : Type u} : ℕ → Option (α × WSeq α) → Computation (Option (α × WSeq α))

auxiliary definition of drop over weak sequences

Equations

• Stream'.WSeq.drop.aux 0 = Computation.pure
• Stream'.WSeq.drop.aux n.succ = fun (a : Option (α × Stream'.WSeq α)) => Stream'.WSeq.tail.aux a >>= Stream'.WSeq.drop.aux n

theorem Stream'.WSeq.drop.aux_none {α : Type u} (n : ℕ) : aux n none = Computation.pure none

theorem Stream'.WSeq.destruct_dropn {α : Type u} (s : WSeq α) (n : ℕ) : (s.drop n).destruct = s.destruct >>= drop.aux n

theorem Stream'.WSeq.head_terminates_of_head_tail_terminates {α : Type u} (s : WSeq α) [T : s.tail.head.Terminates] : s.head.Terminates

theorem Stream'.WSeq.destruct_some_of_destruct_tail_some {α : Type u} {s : WSeq α} {a : α × WSeq α} (h : some a ∈ s.tail.destruct) : ∃ (a' : α × WSeq α), some a' ∈ s.destruct

theorem Stream'.WSeq.head_some_of_head_tail_some {α : Type u} {s : WSeq α} {a : α} (h : some a ∈ s.tail.head) : ∃ (a' : α), some a' ∈ s.head

theorem Stream'.WSeq.head_some_of_get?_some {α : Type u} {s : WSeq α} {a : α} {n : ℕ} (h : some a ∈ s.get? n) : ∃ (a' : α), some a' ∈ s.head

instance Stream'.WSeq.productive_tail {α : Type u} (s : WSeq α) [s.Productive] : s.tail.Productive

instance Stream'.WSeq.productive_dropn {α : Type u} (s : WSeq α) [s.Productive] (n : ℕ) : (s.drop n).Productive

def Stream'.WSeq.toSeq {α : Type u} (s : WSeq α) [s.Productive] : Seq α

Given a productive weak sequence, we can collapse all the thinks to produce a sequence.

Equations

• s.toSeq = ⟨fun (n : ℕ) => (s.get? n).get, ⋯⟩

theorem Stream'.WSeq.get?_terminates_le {α : Type u} {s : WSeq α} {m n : ℕ} (h : m ≤ n) : (s.get? n).Terminates → (s.get? m).Terminates

theorem Stream'.WSeq.head_terminates_of_get?_terminates {α : Type u} {s : WSeq α} {n : ℕ} : (s.get? n).Terminates → s.head.Terminates

theorem Stream'.WSeq.destruct_terminates_of_get?_terminates {α : Type u} {s : WSeq α} {n : ℕ} (T : (s.get? n).Terminates) : s.destruct.Terminates

theorem Stream'.WSeq.mem_rec_on {α : Type u} {C : WSeq α → Prop} {a : α} {s : WSeq α} (M : a ∈ s) (h1 : ∀ (b : α) (s' : WSeq α), a = b ∨ C s' → C (cons b s')) (h2 : ∀ (s : WSeq α), C s → C s.think) : C s

@[simp]

theorem Stream'.WSeq.mem_think {α : Type u} (s : WSeq α) (a : α) : a ∈ s.think ↔ a ∈ s

theorem Stream'.WSeq.eq_or_mem_iff_mem {α : Type u} {s : WSeq α} {a a' : α} {s' : WSeq α} : some (a', s') ∈ s.destruct → (a ∈ s ↔ a = a' ∨ a ∈ s')

@[simp]

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

theorem Stream'.WSeq.mem_cons_of_mem {α : Type u} {s : WSeq α} (b : α) {a : α} (h : a ∈ s) : a ∈ cons b s

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

theorem Stream'.WSeq.mem_of_mem_tail {α : Type u} {s : WSeq α} {a : α} : a ∈ s.tail → a ∈ s

theorem Stream'.WSeq.mem_of_mem_dropn {α : Type u} {s : WSeq α} {a : α} {n : ℕ} : a ∈ s.drop n → a ∈ s

theorem Stream'.WSeq.get?_mem {α : Type u} {s : WSeq α} {a : α} {n : ℕ} : some a ∈ s.get? n → a ∈ s

theorem Stream'.WSeq.exists_get?_of_mem {α : Type u} {s : WSeq α} {a : α} (h : a ∈ s) : ∃ (n : ℕ), some a ∈ s.get? n

theorem Stream'.WSeq.exists_dropn_of_mem {α : Type u} {s : WSeq α} {a : α} (h : a ∈ s) : ∃ (n : ℕ), ∃ (s' : WSeq α), some (a, s') ∈ (s.drop n).destruct

theorem Stream'.WSeq.liftRel_dropn_destruct {α : Type u} {β : Type v} {R : α → β → Prop} {s : WSeq α} {t : WSeq β} (H : LiftRel R s t) (n : ℕ) : Computation.LiftRel (LiftRelO R (LiftRel R)) (s.drop n).destruct (t.drop n).destruct

theorem Stream'.WSeq.exists_of_liftRel_left {α : Type u} {β : Type v} {R : α → β → Prop} {s : WSeq α} {t : WSeq β} (H : LiftRel R s t) {a : α} (h : a ∈ s) : ∃ (b : β), b ∈ t ∧ R a b

theorem Stream'.WSeq.exists_of_liftRel_right {α : Type u} {β : Type v} {R : α → β → Prop} {s : WSeq α} {t : WSeq β} (H : LiftRel R s t) {b : β} (h : b ∈ t) : ∃ (a : α), a ∈ s ∧ R a b

theorem Stream'.WSeq.head_terminates_of_mem {α : Type u} {s : WSeq α} {a : α} (h : a ∈ s) : s.head.Terminates

theorem Stream'.WSeq.of_mem_append {α : Type u} {s₁ s₂ : WSeq α} {a : α} : a ∈ s₁.append s₂ → a ∈ s₁ ∨ a ∈ s₂

theorem Stream'.WSeq.mem_append_left {α : Type u} {s₁ s₂ : WSeq α} {a : α} : a ∈ s₁ → a ∈ s₁.append s₂

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

@[simp]

theorem Stream'.WSeq.liftRel_nil {α : Type u} {β : Type v} (R : α → β → Prop) : LiftRel R nil nil

@[simp]

theorem Stream'.WSeq.liftRel_cons {α : Type u} {β : Type v} (R : α → β → Prop) (a : α) (b : β) (s : WSeq α) (t : WSeq β) : LiftRel R (cons a s) (cons b t) ↔ R a b ∧ LiftRel R s t

@[simp]

theorem Stream'.WSeq.liftRel_think_left {α : Type u} {β : Type v} (R : α → β → Prop) (s : WSeq α) (t : WSeq β) : LiftRel R s.think t ↔ LiftRel R s t

@[simp]

theorem Stream'.WSeq.liftRel_think_right {α : Type u} {β : Type v} (R : α → β → Prop) (s : WSeq α) (t : WSeq β) : LiftRel R s t.think ↔ LiftRel R s t

theorem Stream'.WSeq.cons_congr {α : Type u} {s t : WSeq α} (a : α) (h : s ~ʷ t) : cons a s ~ʷ cons a t

theorem Stream'.WSeq.think_equiv {α : Type u} (s : WSeq α) : s.think ~ʷ s

theorem Stream'.WSeq.think_congr {α : Type u} {s t : WSeq α} (h : s ~ʷ t) : s.think ~ʷ t.think

theorem Stream'.WSeq.head_congr {α : Type u} {s t : WSeq α} : s ~ʷ t → s.head.Equiv t.head

theorem Stream'.WSeq.flatten_equiv {α : Type u} {c : Computation (WSeq α)} {s : WSeq α} (h : s ∈ c) : flatten c ~ʷ s

theorem Stream'.WSeq.liftRel_flatten {α : Type u} {β : Type v} {R : α → β → Prop} {c1 : Computation (WSeq α)} {c2 : Computation (WSeq β)} (h : Computation.LiftRel (LiftRel R) c1 c2) : LiftRel R (flatten c1) (flatten c2)

theorem Stream'.WSeq.flatten_congr {α : Type u} {c1 c2 : Computation (WSeq α)} : Computation.LiftRel Equiv c1 c2 → flatten c1 ~ʷ flatten c2

theorem Stream'.WSeq.tail_congr {α : Type u} {s t : WSeq α} (h : s ~ʷ t) : s.tail ~ʷ t.tail

theorem Stream'.WSeq.dropn_congr {α : Type u} {s t : WSeq α} (h : s ~ʷ t) (n : ℕ) : s.drop n ~ʷ t.drop n

theorem Stream'.WSeq.get?_congr {α : Type u} {s t : WSeq α} (h : s ~ʷ t) (n : ℕ) : (s.get? n).Equiv (t.get? n)

theorem Stream'.WSeq.mem_congr {α : Type u} {s t : WSeq α} (h : s ~ʷ t) (a : α) : a ∈ s ↔ a ∈ t

theorem Stream'.WSeq.productive_congr {α : Type u} {s t : WSeq α} (h : s ~ʷ t) : s.Productive ↔ t.Productive

theorem Stream'.WSeq.Equiv.ext {α : Type u} {s t : WSeq α} (h : ∀ (n : ℕ), (s.get? n).Equiv (t.get? n)) : s ~ʷ t

theorem Stream'.WSeq.length_eq_map {α : Type u} (s : WSeq α) : s.length = Computation.map List.length s.toList

@[simp]

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

@[simp]

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

@[simp]

theorem Stream'.WSeq.toList'_nil {α : Type u} (l : List α) : Computation.corec (fun (x : List α × WSeq α) => match x with | (l, s) => match Seq.destruct s with | none => Sum.inl l.reverse | some (none, s') => Sum.inr (l, s') | some (some a, s') => Sum.inr (a :: l, s')) (l, nil) = Computation.pure l.reverse

@[simp]

theorem Stream'.WSeq.toList'_cons {α : Type u} (l : List α) (s : WSeq α) (a : α) : Computation.corec (fun (x : List α × WSeq α) => match x with | (l, s) => match Seq.destruct s with | none => Sum.inl l.reverse | some (none, s') => Sum.inr (l, s') | some (some a, s') => Sum.inr (a :: l, s')) (l, cons a s) = (Computation.corec (fun (x : List α × WSeq α) => match x with | (l, s) => match Seq.destruct s with | none => Sum.inl l.reverse | some (none, s') => Sum.inr (l, s') | some (some a, s') => Sum.inr (a :: l, s')) (a :: l, s)).think

@[simp]

theorem Stream'.WSeq.toList'_think {α : Type u} (l : List α) (s : WSeq α) : Computation.corec (fun (x : List α × WSeq α) => match x with | (l, s) => match Seq.destruct s with | none => Sum.inl l.reverse | some (none, s') => Sum.inr (l, s') | some (some a, s') => Sum.inr (a :: l, s')) (l, s.think) = (Computation.corec (fun (x : List α × WSeq α) => match x with | (l, s) => match Seq.destruct s with | none => Sum.inl l.reverse | some (none, s') => Sum.inr (l, s') | some (some a, s') => Sum.inr (a :: l, s')) (l, s)).think

theorem Stream'.WSeq.toList'_map {α : Type u} (l : List α) (s : WSeq α) : Computation.corec (fun (x : List α × WSeq α) => match x with | (l, s) => match Seq.destruct s with | none => Sum.inl l.reverse | some (none, s') => Sum.inr (l, s') | some (some a, s') => Sum.inr (a :: l, s')) (l, s) = (fun (x : List α) => l.reverse ++ x) <$> s.toList

@[simp]

theorem Stream'.WSeq.toList_cons {α : Type u} (a : α) (s : WSeq α) : (cons a s).toList = (List.cons a <$> s.toList).think

@[simp]

theorem Stream'.WSeq.toList_nil {α : Type u} : nil.toList = Computation.pure []

theorem Stream'.WSeq.toList_ofList {α : Type u} (l : List α) : l ∈ (↑l).toList

@[simp]

theorem Stream'.WSeq.destruct_ofSeq {α : Type u} (s : Seq α) : (↑s).destruct = Computation.pure (Option.map (fun (a : α) => (a, ↑s.tail)) s.head)

@[simp]

theorem Stream'.WSeq.head_ofSeq {α : Type u} (s : Seq α) : (↑s).head = Computation.pure s.head

@[simp]

theorem Stream'.WSeq.tail_ofSeq {α : Type u} (s : Seq α) : (↑s).tail = ↑s.tail

@[simp]

theorem Stream'.WSeq.dropn_ofSeq {α : Type u} (s : Seq α) (n : ℕ) : (↑s).drop n = ↑(s.drop n)

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

instance Stream'.WSeq.productive_ofSeq {α : Type u} (s : Seq α) : (↑s).Productive

theorem Stream'.WSeq.toSeq_ofSeq {α : Type u} (s : Seq α) : (↑s).toSeq = s

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

The monadic return a is a singleton list containing a.

Equations

• Stream'.WSeq.ret a = ↑[a]

@[simp]

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

@[simp]

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

@[simp]

theorem Stream'.WSeq.map_think {α : Type u} {β : Type v} (f : α → β) (s : WSeq α) : map f s.think = (map f s).think

@[simp]

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

@[simp]

theorem Stream'.WSeq.map_ret {α : Type u} {β : Type v} (f : α → β) (a : α) : map f (ret a) = ret (f a)

@[simp]

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

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

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

theorem Stream'.WSeq.exists_of_mem_join {α : Type u} {a : α} {S : WSeq (WSeq α)} : a ∈ S.join → ∃ (s : WSeq α), s ∈ S ∧ a ∈ s

theorem Stream'.WSeq.exists_of_mem_bind {α : Type u} {β : Type v} {s : WSeq α} {f : α → WSeq β} {b : β} (h : b ∈ s.bind f) : ∃ (a : α), a ∈ s ∧ b ∈ f a

theorem Stream'.WSeq.destruct_map {α : Type u} {β : Type v} (f : α → β) (s : WSeq α) : (map f s).destruct = Computation.map (Option.map (Prod.map f (map f))) s.destruct

theorem Stream'.WSeq.liftRel_map {α : Type u} {β : Type v} {γ : Type w} {δ : Type u_1} (R : α → β → Prop) (S : γ → δ → Prop) {s1 : WSeq α} {s2 : WSeq β} {f1 : α → γ} {f2 : β → δ} (h1 : LiftRel R s1 s2) (h2 : ∀ {a : α} {b : β}, R a b → S (f1 a) (f2 b)) : LiftRel S (map f1 s1) (map f2 s2)

theorem Stream'.WSeq.map_congr {α : Type u} {β : Type v} (f : α → β) {s t : WSeq α} (h : s ~ʷ t) : map f s ~ʷ map f t

def Stream'.WSeq.destruct_append.aux {α : Type u} (t : WSeq α) : Option (α × WSeq α) → Computation (Option (α × WSeq α))

auxiliary definition of destruct_append over weak sequences

Equations

• Stream'.WSeq.destruct_append.aux t none = t.destruct
• Stream'.WSeq.destruct_append.aux t (some (fst, s)) = Computation.pure (some (fst, s.append t))

theorem Stream'.WSeq.destruct_append {α : Type u} (s t : WSeq α) : (s.append t).destruct = s.destruct.bind (destruct_append.aux t)

def Stream'.WSeq.destruct_join.aux {α : Type u} : Option (WSeq α × WSeq (WSeq α)) → Computation (Option (α × WSeq α))

auxiliary definition of destruct_join over weak sequences

Equations

• Stream'.WSeq.destruct_join.aux none = Computation.pure none
• Stream'.WSeq.destruct_join.aux (some (s, S)) = (s.append S.join).destruct.think

theorem Stream'.WSeq.destruct_join {α : Type u} (S : WSeq (WSeq α)) : S.join.destruct = S.destruct.bind destruct_join.aux

theorem Stream'.WSeq.liftRel_append {α : Type u} {β : Type v} (R : α → β → Prop) {s1 s2 : WSeq α} {t1 t2 : WSeq β} (h1 : LiftRel R s1 t1) (h2 : LiftRel R s2 t2) : LiftRel R (s1.append s2) (t1.append t2)

theorem Stream'.WSeq.liftRel_join.lem {α : Type u} {β : Type v} (R : α → β → Prop) {S : WSeq (WSeq α)} {T : WSeq (WSeq β)} {U : WSeq α → WSeq β → Prop} (ST : LiftRel (LiftRel R) S T) (HU : ∀ (s1 : WSeq α) (s2 : WSeq β), (∃ (s : WSeq α), ∃ (t : WSeq β), ∃ (S : WSeq (WSeq α)), ∃ (T : WSeq (WSeq β)), s1 = s.append S.join ∧ s2 = t.append T.join ∧ LiftRel R s t ∧ LiftRel (LiftRel R) S T) → U s1 s2) {a : Option (α × WSeq α)} (ma : a ∈ S.join.destruct) : ∃ (b : Option (β × WSeq β)), b ∈ T.join.destruct ∧ LiftRelO R U a b

theorem Stream'.WSeq.liftRel_join {α : Type u} {β : Type v} (R : α → β → Prop) {S : WSeq (WSeq α)} {T : WSeq (WSeq β)} (h : LiftRel (LiftRel R) S T) : LiftRel R S.join T.join

theorem Stream'.WSeq.join_congr {α : Type u} {S T : WSeq (WSeq α)} (h : LiftRel Equiv S T) : S.join ~ʷ T.join

theorem Stream'.WSeq.liftRel_bind {α : Type u} {β : Type v} {γ : Type w} {δ : Type u_1} (R : α → β → Prop) (S : γ → δ → Prop) {s1 : WSeq α} {s2 : WSeq β} {f1 : α → WSeq γ} {f2 : β → WSeq δ} (h1 : LiftRel R s1 s2) (h2 : ∀ {a : α} {b : β}, R a b → LiftRel S (f1 a) (f2 b)) : LiftRel S (s1.bind f1) (s2.bind f2)

theorem Stream'.WSeq.bind_congr {α : Type u} {β : Type v} {s1 s2 : WSeq α} {f1 f2 : α → WSeq β} (h1 : s1 ~ʷ s2) (h2 : ∀ (a : α), f1 a ~ʷ f2 a) : s1.bind f1 ~ʷ s2.bind f2

@[simp]

theorem Stream'.WSeq.join_ret {α : Type u} (s : WSeq α) : (ret s).join ~ʷ s

@[simp]

theorem Stream'.WSeq.join_map_ret {α : Type u} (s : WSeq α) : (map ret s).join ~ʷ s

@[simp]

theorem Stream'.WSeq.join_append {α : Type u} (S T : WSeq (WSeq α)) : (S.append T).join ~ʷ S.join.append T.join

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

theorem Stream'.WSeq.join_join {α : Type u} (SS : WSeq (WSeq (WSeq α))) : SS.join.join ~ʷ (map join SS).join

@[simp]

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

instance Stream'.WSeq.monad : Monad WSeq

Equations

• Stream'.WSeq.monad = Monad.mk