-- Mathlib.Data.WSeq.Relation.BisimO (L62-L64)
def BisimO (R : WSeq α → WSeq α → Prop) : Option (α × WSeq α) → Option (α × WSeq α) → Prop :=
LiftRelO (· = ·) R

-- Mathlib.Data.WSeq.Relation.Equiv (L159-L161)
def Equiv : WSeq α → WSeq α → Prop :=
LiftRel (· = ·)

-- Mathlib.Data.WSeq.Relation.Equiv (L165-L167)
theorem Equiv.refl : ∀ s : WSeq α, s ~ʷ s :=
LiftRel.refl (· = ·) Eq.refl

-- Mathlib.Data.WSeq.Relation.Equiv (L169-L171)
theorem Equiv.symm : ∀ {s t : WSeq α}, s ~ʷ t → t ~ʷ s :=
@(LiftRel.symm (· = ·) (@Eq.symm _))

-- Mathlib.Data.WSeq.Relation.Equiv (L176-L178)
theorem Equiv.equivalence : Equivalence (@Equiv α) :=
⟨@Equiv.refl _, @Equiv.symm _, @Equiv.trans _⟩

-- Mathlib.Data.WSeq.Relation.liftRel_nil (L212-L214)
theorem liftRel_nil (R : α → β → Prop) : LiftRel R nil nil := by
simp [liftRel_destruct_iff]

-- Mathlib.Data.WSeq.Relation.think_equiv (L231-L232)
theorem think_equiv (s : WSeq α) : think s ~ʷ s := by unfold Equiv; simpa using Equiv.refl _

-- Mathlib.Data.WSeq.Relation.think_congr (L233-L235)
theorem think_congr {s t : WSeq α} (h : s ~ʷ t) : think s ~ʷ think t := by
unfold Equiv; simpa using h

-- Mathlib.Data.WSeq.Basic.cons_append (L310-L312)
theorem cons_append (a : α) (s t) : append (cons a s) t = cons a (append s t) :=
Seq.cons_append _ _ _

-- Mathlib.Data.WSeq.Basic.ofList (L51-L54)
def ofList (l : List α) : WSeq α :=
ofSeq l

/-- Turn a stream into a weak sequence -/

-- Mathlib.Data.WSeq.Basic.monad (L177-L183)
instance monad : Monad WSeq where
  map := @map
pure := @ret
  bind := @bind

open Computation

-- Mathlib.Data.WSeq.Basic.head_cons (L212-L213)
theorem head_cons (a : α) (s) : head (cons a s) = Computation.pure (some a) := by simp [head]

-- Mathlib.Data.WSeq.Basic.append_nil (L318-L320)
theorem append_nil (s : WSeq α) : append s nil = s :=
Seq.append_nil _

-- Mathlib.Data.WSeq.Basic.dropn_nil (L261-L262)
theorem dropn_nil (n) : drop (nil : WSeq α) n = nil := by induction n <;> simp [*, drop]

-- Mathlib.Data.WSeq.Basic.mem_cons_of_mem (L450-L452)
theorem mem_cons_of_mem {s : WSeq α} (b) {a} (h : a ∈ s) : a ∈ cons b s :=
(mem_cons_iff _ _).2 (Or.inr h)

-- Mathlib.Data.WSeq.Basic.head_nil (L209-L210)
theorem head_nil : head (nil : WSeq α) = Computation.pure none := by simp [head]

-- Mathlib.Data.WSeq.Basic.mem_cons_iff (L447-L449)
theorem mem_cons_iff (s : WSeq α) (b) {a} : a ∈ cons b s ↔ a = b ∨ a ∈ s :=
eq_or_mem_iff_mem <| by simp

-- Mathlib.Data.WSeq.Basic.coeList (L62-L64)
instance coeList : Coe (List α) (WSeq α) :=
⟨ofList⟩

-- Mathlib.Data.WSeq.Basic.tail_nil (L252-L253)
theorem tail_nil : tail (nil : WSeq α) = nil := by simp [tail]

-- Mathlib.Data.WSeq.Basic.inhabited (L72-L75)
instance inhabited : Inhabited (WSeq α) :=
⟨nil⟩

/-- Prepend an element to a weak sequence -/

-- Mathlib.Data.WSeq.Basic.join_nil (L290-L292)
theorem join_nil : join nil = (nil : WSeq α) :=
Seq.join_nil

-- Mathlib.Data.WSeq.Basic.seq_destruct_think (L205-L207)
theorem seq_destruct_think (s : WSeq α) : Seq.destruct (think s) = some (none, s) :=
Seq.destruct_cons _ _

-- Mathlib.Data.WSeq.Basic.coeStream (L65-L68)
instance coeStream : Coe (Stream' α) (WSeq α) :=
⟨ofStream⟩

/-- The empty weak sequence -/

-- Mathlib.Data.WSeq.Basic.nil (L69-L71)
def nil : WSeq α :=
Seq.nil

-- Mathlib.Data.WSeq.Basic.cons (L76-L79)
def cons (a : α) : WSeq α → WSeq α :=
Seq.cons (some a)

/-- Compute for one tick, without producing any elements -/

-- Mathlib.Data.WSeq.Basic.ofList_nil (L537-L539)
theorem ofList_nil : ofList [] = (nil : WSeq α) :=
rfl

-- Mathlib.Data.WSeq.Basic.seq_destruct_nil (L197-L199)
theorem seq_destruct_nil : Seq.destruct (nil : WSeq α) = none :=
rfl

-- Mathlib.Data.WSeq.Basic.destruct_nil (L185-L187)
theorem destruct_nil : destruct (nil : WSeq α) = Computation.pure none :=
Computation.destruct_eq_pure rfl

-- Mathlib.Data.WSeq.Basic.coeSeq (L59-L61)
instance coeSeq : Coe (Seq α) (WSeq α) :=
⟨ofSeq⟩

-- Mathlib.Data.WSeq.Basic.head_think (L215-L216)
theorem head_think (s : WSeq α) : head (think s) = (head s).think := by simp [head]

-- Mathlib.Data.WSeq.Basic.ret (L166-L169)
def ret (a : α) : WSeq α :=
ofList [a]

/-- Monadic bind operator for weak sequences -/

-- Mathlib.Data.WSeq.Basic.think_append (L314-L316)
theorem think_append (s t : WSeq α) : append (think s) t = think (append s t) :=
Seq.cons_append _ _ _

-- Mathlib.Data.WSeq.Basic.seq_destruct_cons (L201-L203)
theorem seq_destruct_cons (a : α) (s) : Seq.destruct (cons a s) = some (some a, s) :=
Seq.destruct_cons _ _

-- Mathlib.Data.WSeq.Basic.of_mem_append (L522-L524)
theorem of_mem_append {s₁ s₂ : WSeq α} {a : α} : a ∈ append s₁ s₂ → a ∈ s₁ ∨ a ∈ s₂ :=
Seq.of_mem_append

-- Mathlib.Data.WSeq.Basic.ofSeq (L46-L49)
def ofSeq : Seq α → WSeq α :=
(· <$> ·) some

/-- Turn a list into a weak sequence -/

-- Mathlib.Data.WSeq.Basic.membership (L102-L104)
instance membership : Membership α (WSeq α) :=
⟨WSeq.Mem⟩

-- Mathlib.Data.WSeq.Basic.dropn_think (L271-L273)
theorem dropn_think (s : WSeq α) (n) : drop (think s) n = (drop s n).think := by
induction n <;> simp [*, drop]

-- Mathlib.Data.WSeq.Basic.mem_append_left (L525-L527)
theorem mem_append_left {s₁ s₂ : WSeq α} {a : α} : a ∈ s₁ → a ∈ append s₁ s₂ :=
Seq.mem_append_left

-- Mathlib.Data.WSeq.Basic.nil_append (L306-L308)
theorem nil_append (s : WSeq α) : append nil s = s :=
Seq.nil_append _

-- Mathlib.Data.WSeq.Basic.tail_cons (L255-L256)
theorem tail_cons (a : α) (s) : tail (cons a s) = s := by simp [tail]

-- Mathlib.Data.WSeq.Basic.tail_think (L258-L259)
theorem tail_think (s : WSeq α) : tail (think s) = (tail s).think := by simp [tail]

-- Mathlib.Data.WSeq.Basic.ofStream (L56-L58)
def ofStream (l : Stream' α) : WSeq α :=
ofSeq l

theorem LiftRel.refl (R : α → α → Prop) (H : Reflexive R) : Reflexive (LiftRel R)

[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.WSeq.Relation).

[User]
Prove the following theorem. The retrieved context below may help.

```lean
theorem LiftRel.refl (R : α → α → Prop) (H : Reflexive R) : Reflexive (LiftRel R)
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Data.WSeq.Relation.LiftRelO (L37-L41)
def LiftRelO (R : α → β → Prop) (C : WSeq α → WSeq β → Prop) :
    Option (α × WSeq α) → Option (β × WSeq β) → Prop
  | none, none => True
  | some (a, s), some (b, t) => R a b ∧ C s t
  | _, _ => False
```

-- snippet 2
```lean
-- Mathlib.Data.WSeq.Relation.LiftRelO (L44-L50)
theorem LiftRelO.imp {R S : α → β → Prop} {C D : WSeq α → WSeq β → Prop} (H1 : ∀ a b, R a b → S a b)
    (H2 : ∀ s t, C s t → D s t) : ∀ {o p}, LiftRelO R C o p → LiftRelO S D o p
  | none, none, _ => trivial
  | some (_, _), some (_, _), h => And.imp (H1 _ _) (H2 _ _) h
  | none, some _, h => False.elim h
  | some (_, _), none, h => False.elim h
```

-- snippet 3
```lean
-- Mathlib.Data.WSeq.Relation.LiftRelO (L51-L54)
theorem LiftRelO.imp_right (R : α → β → Prop) {C D : WSeq α → WSeq β → Prop}
    (H : ∀ s t, C s t → D s t) {o p} : LiftRelO R C o p → LiftRelO R D o p :=
LiftRelO.imp (fun _ _ => id) H
```

-- snippet 4
```lean
-- Mathlib.Data.WSeq.Relation.LiftRelO (L55-L60)
theorem LiftRelO.swap (R : α → β → Prop) (C) :
    swap (LiftRelO R C) = LiftRelO (swap R) (swap C) := by
funext x y
  rcases x with ⟨⟩ | ⟨hx, jx⟩ <;> rcases y with ⟨⟩ | ⟨hy, jy⟩ <;> rfl

/-- Definition of bisimilarity for weak sequences -/
```

-- snippet 5
```lean
-- Mathlib.Data.WSeq.Relation.BisimO (L62-L64)
def BisimO (R : WSeq α → WSeq α → Prop) : Option (α × WSeq α) → Option (α × WSeq α) → Prop :=
LiftRelO (· = ·) R
```

-- snippet 6
```lean
-- Mathlib.Data.WSeq.Relation.BisimO (L65-L71)
theorem BisimO.imp {R S : WSeq α → WSeq α → Prop} (H : ∀ s t, R s t → S s t) {o p} :
    BisimO R o p → BisimO S o p :=
LiftRelO.imp_right _ H

/-- 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.) -/
```

-- snippet 7
```lean
-- Mathlib.Data.WSeq.Relation.liftRel_destruct (L76-L82)
theorem liftRel_destruct {R : α → β → Prop} {s : WSeq α} {t : WSeq β} :
    LiftRel R s t → Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)
  | ⟨R, h1, h2⟩ => by
    refine Computation.LiftRel.imp ?_ _ _ (h2 h1)
    apply LiftRelO.imp_right
    exact fun s' t' h' => ⟨R, h', @h2⟩
```

-- snippet 8
```lean
-- Mathlib.Data.WSeq.Relation.liftRel_destruct_iff (L83-L97)
theorem liftRel_destruct_iff {R : α → β → Prop} {s : WSeq α} {t : WSeq β} :
    LiftRel R s t ↔ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t) :=
⟨liftRel_destruct, fun h =>
    ⟨fun s t =>
      LiftRel R s t ∨ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t),
      Or.inr h, fun {s t} h => by
      have h : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t) := by
        obtain h | h := h
        · exact liftRel_destruct h
        · assumption
      apply Computation.LiftRel.imp _ _ _ h
      apply LiftRelO.imp_right
      intro s t
      apply Or.inl⟩⟩
```

-- snippet 9
```lean
-- Mathlib.Data.WSeq.Relation.Equiv (L159-L161)
def Equiv : WSeq α → WSeq α → Prop :=
LiftRel (· = ·)
```

-- snippet 10
```lean
-- Mathlib.Data.WSeq.Relation.Equiv (L165-L167)
theorem Equiv.refl : ∀ s : WSeq α, s ~ʷ s :=
LiftRel.refl (· = ·) Eq.refl
```

-- snippet 11
```lean
-- Mathlib.Data.WSeq.Relation.Equiv (L169-L171)
theorem Equiv.symm : ∀ {s t : WSeq α}, s ~ʷ t → t ~ʷ s :=
@(LiftRel.symm (· = ·) (@Eq.symm _))
```

-- snippet 12
```lean
-- Mathlib.Data.WSeq.Relation.Equiv (L173-L175)
theorem Equiv.trans : ∀ {s t u : WSeq α}, s ~ʷ t → t ~ʷ u → s ~ʷ u :=
LiftRel.trans (· = ·) inferInstance |>.trans _ _ _
```

-- snippet 13
```lean
-- Mathlib.Data.WSeq.Relation.Equiv (L176-L178)
theorem Equiv.equivalence : Equivalence (@Equiv α) :=
⟨@Equiv.refl _, @Equiv.symm _, @Equiv.trans _⟩
```

-- snippet 14
```lean
-- Mathlib.Data.WSeq.Relation.destruct_congr (L179-L182)
theorem destruct_congr {s t : WSeq α} :
    s ~ʷ t → Computation.LiftRel (BisimO (· ~ʷ ·)) (destruct s) (destruct t) :=
liftRel_destruct
```

-- snippet 15
```lean
-- Mathlib.Data.WSeq.Relation.destruct_congr_iff (L183-L188)
theorem destruct_congr_iff {s t : WSeq α} :
    s ~ʷ t ↔ Computation.LiftRel (BisimO (· ~ʷ ·)) (destruct s) (destruct t) :=
liftRel_destruct_iff

open Computation
```

-- snippet 16
```lean
-- Mathlib.Data.WSeq.Relation.liftRel_dropn_destruct (L189-L200)
theorem liftRel_dropn_destruct {R : α → β → Prop} {s t} (H : LiftRel R s t) :
    ∀ n, Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct (drop s n)) (destruct (drop t n))
  | 0 => liftRel_destruct H
  | n + 1 => by
    simp only [drop, destruct_tail]
    apply liftRel_bind
    · apply liftRel_dropn_destruct H n
    exact fun {a b} o =>
      match a, b, o with
      | none, none, _ => by simp
      | some (a, s), some (b, t), ⟨_, h2⟩ => by simpa [tail.aux] using liftRel_destruct h2
```

-- snippet 17
```lean
-- Mathlib.Data.WSeq.Relation.exists_of_liftRel_left (L201-L207)
theorem exists_of_liftRel_left {R : α → β → Prop} {s t} (H : LiftRel R s t) {a} (h : a ∈ s) :
    ∃ b, b ∈ t ∧ R a b := by
let ⟨n, h⟩ := exists_get?_of_mem h
  let ⟨some (_, s'), sd, rfl⟩ := Computation.exists_of_mem_map h
  let ⟨some (b, t'), td, ⟨ab, _⟩⟩ := (liftRel_dropn_destruct H n).left sd
  exact ⟨b, get?_mem (Computation.mem_map (Prod.fst.{v, v} <$> ·) td), ab⟩
```

-- snippet 18
```lean
-- Mathlib.Data.WSeq.Relation.exists_of_liftRel_right (L208-L210)
theorem exists_of_liftRel_right {R : α → β → Prop} {s t} (H : LiftRel R s t) {b} (h : b ∈ t) :
    ∃ a, a ∈ s ∧ R a b := by rw [← LiftRel.swap] at H; exact exists_of_liftRel_left H h
```

-- snippet 19
```lean
-- Mathlib.Data.WSeq.Relation.liftRel_nil (L212-L214)
theorem liftRel_nil (R : α → β → Prop) : LiftRel R nil nil := by
simp [liftRel_destruct_iff]
```

-- snippet 20
```lean
-- Mathlib.Data.WSeq.Relation.liftRel_cons (L216-L219)
theorem liftRel_cons (R : α → β → Prop) (a b s t) :
    LiftRel R (cons a s) (cons b t) ↔ R a b ∧ LiftRel R s t := by
simp [liftRel_destruct_iff]
```

-- snippet 21
```lean
-- Mathlib.Data.WSeq.Relation.liftRel_think_left (L221-L223)
theorem liftRel_think_left (R : α → β → Prop) (s t) : LiftRel R (think s) t ↔ LiftRel R s t := by
rw [liftRel_destruct_iff, liftRel_destruct_iff]; simp
```

-- snippet 22
```lean
-- Mathlib.Data.WSeq.Relation.liftRel_think_right (L225-L227)
theorem liftRel_think_right (R : α → β → Prop) (s t) : LiftRel R s (think t) ↔ LiftRel R s t := by
rw [liftRel_destruct_iff, liftRel_destruct_iff]; simp
```

-- snippet 23
```lean
-- Mathlib.Data.WSeq.Relation.cons_congr (L228-L230)
theorem cons_congr {s t : WSeq α} (a : α) (h : s ~ʷ t) : cons a s ~ʷ cons a t := by
unfold Equiv; simpa using h
```

-- snippet 24
```lean
-- Mathlib.Data.WSeq.Relation.think_equiv (L231-L232)
theorem think_equiv (s : WSeq α) : think s ~ʷ s := by unfold Equiv; simpa using Equiv.refl _
```

-- snippet 25
```lean
-- Mathlib.Data.WSeq.Relation.think_congr (L233-L235)
theorem think_congr {s t : WSeq α} (h : s ~ʷ t) : think s ~ʷ think t := by
unfold Equiv; simpa using h
```

-- snippet 26
```lean
-- Mathlib.Data.WSeq.Basic.head_terminates_iff (L248-L250)
theorem head_terminates_iff (s : WSeq α) : Terminates (head s) ↔ Terminates (destruct s) :=
terminates_map_iff _ (destruct s)
```

-- snippet 27
```lean
-- Mathlib.Data.WSeq.Basic.flatten_pure (L218-L228)
theorem flatten_pure (s : WSeq α) : flatten (Computation.pure s) = s := by
refine Seq.eq_of_bisim (fun s1 s2 => flatten (Computation.pure s2) = s1) ?_ rfl
  intro s' s h
  rw [← h]
  simp only [Seq.BisimO, flatten, Seq.omap, pure_def, Seq.corec_eq, destruct_pure]
  cases Seq.destruct s with
  | none => simp
  | some val =>
    obtain ⟨o, s'⟩ := val
    simp
```

-- snippet 28
```lean
-- Mathlib.Data.WSeq.Basic.tail (L124-L127)
def tail (s : WSeq α) : WSeq α :=
flatten <| (fun o => Option.recOn o nil Prod.snd) <$> destruct s

/-- drop the first `n` elements from `s`. -/
```

-- snippet 29
```lean
-- Mathlib.Data.WSeq.Basic.eq_or_mem_iff_mem (L425-L445)
theorem eq_or_mem_iff_mem {s : WSeq α} {a a' s'} :
    some (a', s') ∈ destruct s → (a ∈ s ↔ a = a' ∨ a ∈ s') := by
generalize e : destruct s = c; intro h
  revert s
  apply Computation.memRecOn h <;> [skip; intro c IH] <;> intro s <;>
    induction s using WSeq.recOn <;>
    intro m <;>
    have := congr_arg Computation.destruct m <;>
    simp at this
  · obtain ⟨i1, i2⟩ := this
    rw [i1, i2]
    dsimp only [cons, Membership.mem, WSeq.Mem, Seq.Mem, Seq.cons]
    have h_a_eq_a' : a = a' ↔ some (some a) = some (some a') := by simp
    rw [h_a_eq_a']
    refine ⟨Stream'.eq_or_mem_of_mem_cons, fun o => ?_⟩
    · rcases o with e | m
      · rw [e]
        apply Stream'.mem_cons
      · exact Stream'.mem_cons_of_mem _ m
  · simp [IH this]
```

-- snippet 30
```lean
-- Mathlib.Data.WSeq.Basic.mem_think (L415-L424)
theorem mem_think (s : WSeq α) (a) : a ∈ think s ↔ a ∈ s := by
obtain ⟨f, al⟩ := s
  change (some (some a) ∈ some none::f) ↔ some (some a) ∈ f
  constructor <;> intro h
  · apply (Stream'.eq_or_mem_of_mem_cons h).resolve_left
    intro
    injections
  · apply Stream'.mem_cons_of_mem _ h

set_option linter.flexible false in -- TODO: fix non-terminal simp
```

-- snippet 31
```lean
-- Mathlib.Data.WSeq.Basic.cons_append (L310-L312)
theorem cons_append (a : α) (s t) : append (cons a s) t = cons a (append s t) :=
Seq.cons_append _ _ _
```

-- snippet 32
```lean
-- Mathlib.Data.WSeq.Basic.toList'_cons (L554-L566)
theorem toList'_cons (l : List α) (s : WSeq α) (a : α) :
    Computation.corec (fun ⟨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 ⟨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 :=
destruct_eq_think <| by simp [cons]
```

-- snippet 33
```lean
-- Mathlib.Data.WSeq.Basic.WSeq (L37-L44)
def WSeq (α) :=
Seq (Option α)

namespace WSeq

variable {α : Type u} {β : Type v} {γ : Type w}

/-- Turn a sequence into a weak sequence -/
```

-- snippet 34
```lean
-- Mathlib.Data.WSeq.Basic.ofList (L51-L54)
def ofList (l : List α) : WSeq α :=
ofSeq l

/-- Turn a stream into a weak sequence -/
```

-- snippet 35
```lean
-- Mathlib.Data.WSeq.Basic.head (L110-L114)
def head (s : WSeq α) : Computation (Option α) :=
Computation.map (Prod.fst <$> ·) (destruct s)

/-- Encode a computation yielding a weak sequence into additional
  `think` constructors in a weak sequence -/
```

-- snippet 36
```lean
-- Mathlib.Data.WSeq.Basic.head_terminates_of_head_tail_terminates (L353-L360)
theorem head_terminates_of_head_tail_terminates (s : WSeq α) [T : Terminates (head (tail s))] :
    Terminates (head s) :=
(head_terminates_iff _).2 <| by
    rcases (head_terminates_iff _).1 T with ⟨⟨a, h⟩⟩
    simp? [tail] at h says simp only [tail, destruct_flatten, bind_map_left] at h
    rcases exists_of_mem_bind h with ⟨s', h1, _⟩
    exact terminates_of_mem h1
```

-- snippet 37
```lean
-- Mathlib.Data.WSeq.Basic.destruct_terminates_of_get (L397-L400)
theorem destruct_terminates_of_get?_terminates {s : WSeq α} {n} (T : Terminates (get? s n)) :
    Terminates (destruct s) :=
(head_terminates_iff _).1 <| head_terminates_of_get?_terminates T
```

-- snippet 38
```lean
-- Mathlib.Data.WSeq.Basic.monad (L177-L183)
instance monad : Monad WSeq where
  map := @map
pure := @ret
  bind := @bind

open Computation
```

-- snippet 39
```lean
-- Mathlib.Data.WSeq.Basic.exists_get (L487-L503)
theorem exists_get?_of_mem {s : WSeq α} {a} (h : a ∈ s) : ∃ n, some a ∈ get? s n := by
apply mem_rec_on h
  · intro a' s' h
    rcases h with h | h
    · exists 0
      simp only [get?, drop, head_cons]
      rw [h]
      apply ret_mem
    · obtain ⟨n, h⟩ := h
      exists n + 1
      simpa [get?]
  · intro s' h
    obtain ⟨n, h⟩ := h
    exists n
    simp only [get?, dropn_think, head_think]
    apply think_mem h
```

-- snippet 40
```lean
-- Mathlib.Data.WSeq.Basic.join_cons (L300-L304)
theorem join_cons (s : WSeq α) (S) : join (cons s S) = think (append s (join S)) := by
simp only [join, think]
  dsimp only [(· <$> ·)]
  simp [cons, append]
```

-- snippet 41
```lean
-- Mathlib.Data.WSeq.Basic.dropn_tail (L278-L282)
theorem dropn_tail (s : WSeq α) (n) : drop (tail s) n = drop s (n + 1) := by
rw [Nat.add_comm]
  symm
  apply dropn_add
```

-- snippet 42
```lean
-- Mathlib.Data.WSeq.Basic.toList'_think (L568-L580)
theorem toList'_think (l : List α) (s : WSeq α) :
    Computation.corec (fun ⟨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, think s) =
      (Computation.corec (fun ⟨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 :=
destruct_eq_think <| by simp [think]
```

-- snippet 43
```lean
-- Mathlib.Data.WSeq.Basic.head_cons (L212-L213)
theorem head_cons (a : α) (s) : head (cons a s) = Computation.pure (some a) := by simp [head]
```

-- snippet 44
```lean
-- Mathlib.Data.WSeq.Basic.append_nil (L318-L320)
theorem append_nil (s : WSeq α) : append s nil = s :=
Seq.append_nil _
```

-- snippet 45
```lean
-- Mathlib.Data.WSeq.Basic.mem_of_mem_tail (L457-L468)
theorem mem_of_mem_tail {s : WSeq α} {a} : a ∈ tail s → a ∈ s := by
intro h; have := h; obtain ⟨n, e⟩ := h; revert s; simp only [Stream'.get]
  induction n <;> intro s <;> induction s using WSeq.recOn <;>
    simp <;> intro m e <;> injections
  · exact Or.inr m
  · exact Or.inr m
  case succ.think n IH s =>
    apply IH m
    rw [e]
    cases tail s
    rfl
```

-- snippet 46
```lean
-- Mathlib.Data.WSeq.Basic.head_some_of_get (L379-L386)
theorem head_some_of_get?_some {s : WSeq α} {a n} (h : some a ∈ get? s n) :
    ∃ a', some a' ∈ head s := by
induction n generalizing a with
  | zero => exact ⟨_, h⟩
  | succ n IH =>
      let ⟨a', h'⟩ := head_some_of_head_tail_some h
      exact IH h'
```

-- snippet 47
```lean
-- Mathlib.Data.WSeq.Basic.dropn_nil (L261-L262)
theorem dropn_nil (n) : drop (nil : WSeq α) n = nil := by induction n <;> simp [*, drop]
```

-- snippet 48
```lean
-- Mathlib.Data.WSeq.Basic.mem_cons_of_mem (L450-L452)
theorem mem_cons_of_mem {s : WSeq α} (b) {a} (h : a ∈ s) : a ∈ cons b s :=
(mem_cons_iff _ _).2 (Or.inr h)
```

-- snippet 49
```lean
-- Mathlib.Data.WSeq.Basic.head_nil (L209-L210)
theorem head_nil : head (nil : WSeq α) = Computation.pure none := by simp [head]
```

-- snippet 50
```lean
-- Mathlib.Data.WSeq.Basic.mem_cons_iff (L447-L449)
theorem mem_cons_iff (s : WSeq α) (b) {a} : a ∈ cons b s ↔ a = b ∨ a ∈ s :=
eq_or_mem_iff_mem <| by simp
```

-- snippet 51
```lean
-- Mathlib.Data.WSeq.Basic.coeList (L62-L64)
instance coeList : Coe (List α) (WSeq α) :=
⟨ofList⟩
```

-- snippet 52
```lean
-- Mathlib.Data.WSeq.Basic.tail_nil (L252-L253)
theorem tail_nil : tail (nil : WSeq α) = nil := by simp [tail]
```

-- snippet 53
```lean
-- Mathlib.Data.WSeq.Basic.join_think (L294-L298)
theorem join_think (S : WSeq (WSeq α)) : join (think S) = think (join S) := by
simp only [join, think]
  dsimp only [(· <$> ·)]
  simp [Seq1.ret]
```

-- snippet 54
```lean
-- Mathlib.Data.WSeq.Basic.map (L162-L165)
def map (f : α → β) : WSeq α → WSeq β :=
Seq.map (Option.map f)

/-- The monadic `return a` is a singleton list containing `a`. -/
```

-- snippet 55
```lean
-- Mathlib.Data.WSeq.Basic.head_terminates_of_mem (L518-L521)
theorem head_terminates_of_mem {s : WSeq α} {a} (h : a ∈ s) : Terminates (head s) :=
let ⟨_, h⟩ := exists_get?_of_mem h
  head_terminates_of_get?_terminates ⟨⟨_, h⟩⟩
```

-- snippet 56
```lean
-- Mathlib.Data.WSeq.Basic.head_terminates_of_get (L393-L396)
theorem head_terminates_of_get?_terminates {s : WSeq α} {n} :
    Terminates (get? s n) → Terminates (head s) :=
get?_terminates_le (Nat.zero_le n)
```

-- snippet 57
```lean
-- Mathlib.Data.WSeq.Basic.append_assoc (L322-L325)
theorem append_assoc (s t u : WSeq α) : append (append s t) u = append s (append t u) :=
Seq.append_assoc _ _ _

/-- auxiliary definition of tail over weak sequences -/
```

-- snippet 58
```lean
-- Mathlib.Data.WSeq.Basic.inhabited (L72-L75)
instance inhabited : Inhabited (WSeq α) :=
⟨nil⟩

/-- Prepend an element to a weak sequence -/
```

-- snippet 59
```lean
-- Mathlib.Data.WSeq.Basic.join_nil (L290-L292)
theorem join_nil : join nil = (nil : WSeq α) :=
Seq.join_nil
```

-- snippet 60
```lean
-- Mathlib.Data.WSeq.Basic.bind (L170-L176)
def bind (s : WSeq α) (f : α → WSeq β) : WSeq β :=
join (map f s)

/-- Unfortunately, `WSeq` is not a lawful monad, because it does not satisfy the monad laws exactly,
only up to sequence equivalence. Furthermore, even quotienting by the equivalence is not sufficient,
because the join operation involves lists of quotient elements, with a lifted equivalence relation,
and pure quotients cannot handle this type of construction. -/
```

-- snippet 61
```lean
-- Mathlib.Data.WSeq.Basic.exists_of_mem_map (L528-L535)
theorem exists_of_mem_map {f} {b : β} : ∀ {s : WSeq α}, b ∈ map f s → ∃ a, a ∈ s ∧ f a = b
  | ⟨g, al⟩, h => by
    let ⟨o, om, oe⟩ := Seq.exists_of_mem_map h
rcases o with - | a
    · injection oe
    injection oe with h'
    exact ⟨a, om, h'⟩
```

-- snippet 62
```lean
-- Mathlib.Data.WSeq.Basic.seq_destruct_think (L205-L207)
theorem seq_destruct_think (s : WSeq α) : Seq.destruct (think s) = some (none, s) :=
Seq.destruct_cons _ _
```

-- snippet 63
```lean
-- Mathlib.Data.WSeq.Basic.coeStream (L65-L68)
instance coeStream : Coe (Stream' α) (WSeq α) :=
⟨ofStream⟩

/-- The empty weak sequence -/
```

-- snippet 64
```lean
-- Mathlib.Data.WSeq.Basic.nil (L69-L71)
def nil : WSeq α :=
Seq.nil
```

-- snippet 65
```lean
-- Mathlib.Data.WSeq.Basic.head_some_of_head_tail_some (L371-L378)
theorem head_some_of_head_tail_some {s : WSeq α} {a} (h : some a ∈ head (tail s)) :
    ∃ a', some a' ∈ head s := by
unfold head at h
  rcases Computation.exists_of_mem_map h with ⟨o, md, e⟩; clear h
  rcases o with - | o <;> [injection e; injection e with h']; clear h'
  obtain ⟨a, am⟩ := destruct_some_of_destruct_tail_some md
  exact ⟨_, Computation.mem_map (@Prod.fst α (WSeq α) <$> ·) am⟩
```

-- snippet 66
```lean
-- Mathlib.Data.WSeq.Basic.cons (L76-L79)
def cons (a : α) : WSeq α → WSeq α :=
Seq.cons (some a)

/-- Compute for one tick, without producing any elements -/
```

-- snippet 67
```lean
-- Mathlib.Data.WSeq.Basic.mem_cons (L453-L456)
theorem mem_cons (s : WSeq α) (a) : a ∈ cons a s :=
(mem_cons_iff _ _).2 (Or.inl rfl)

set_option linter.flexible false in -- TODO: fix non-terminal simp
```

-- snippet 68
```lean
-- Mathlib.Data.WSeq.Basic.destruct_cons (L189-L191)
theorem destruct_cons (a : α) (s) : destruct (cons a s) = Computation.pure (some (a, s)) :=
Computation.destruct_eq_pure <| by simp [destruct, cons, Computation.rmap]
```

-- snippet 69
```lean
-- Mathlib.Data.WSeq.Basic.get (L133-L136)
def get? (s : WSeq α) (n : ℕ) : Computation (Option α) :=
head (drop s n)

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

-- snippet 70
```lean
-- Mathlib.Data.WSeq.Basic.notMem_nil (L105-L109)
theorem notMem_nil (a : α) : a ∉ @nil α :=
Seq.notMem_nil (some a)

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

-- snippet 71
```lean
-- Mathlib.Data.WSeq.Basic.ofList_nil (L537-L539)
theorem ofList_nil : ofList [] = (nil : WSeq α) :=
rfl
```

-- snippet 72
```lean
-- Mathlib.Data.WSeq.Basic.dropn_cons (L264-L269)
theorem dropn_cons (a : α) (s) (n) : drop (cons a s) (n + 1) = drop s n := by
induction n with
  | zero => simp [drop]
  | succ n n_ih =>
    simp [drop, ← n_ih]
```

-- snippet 73
```lean
-- Mathlib.Data.WSeq.Basic.mem_rec_on (L401-L413)
theorem mem_rec_on {C : WSeq α → Prop} {a s} (M : a ∈ s) (h1 : ∀ b s', a = b ∨ C s' → C (cons b s'))
    (h2 : ∀ s, C s → C (think s)) : C s := by
apply Seq.mem_rec_on M
  intro o s' h; rcases o with - | b
  · apply h2
    cases h
    · contradiction
    · assumption
  · apply h1
    apply Or.imp_left _ h
    intro h
    injection h
```

-- snippet 74
```lean
-- Mathlib.Data.WSeq.Basic.join (L153-L161)
def join (S : WSeq (WSeq α)) : WSeq α :=
Seq.join
    ((fun o : Option (WSeq α) =>
        match o with
        | none => Seq1.ret none
        | some s => (none, s)) <$>
      S)

/-- Map a function over a weak sequence -/
```

-- snippet 75
```lean
-- Mathlib.Data.WSeq.Basic.destruct_think (L193-L195)
theorem destruct_think (s : WSeq α) : destruct (think s) = (destruct s).think :=
Computation.destruct_eq_think <| by simp [destruct, think, Computation.rmap]
```

-- snippet 76
```lean
-- Mathlib.Data.WSeq.Basic.think (L80-L84)
def think : WSeq α → WSeq α :=
Seq.cons none

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

-- snippet 77
```lean
-- Mathlib.Data.WSeq.Basic.ofList_cons (L541-L543)
theorem ofList_cons (a : α) (l) : ofList (a::l) = cons a (ofList l) :=
show Seq.map some (Seq.ofList (a::l)) = Seq.cons (some a) (Seq.map some (Seq.ofList l)) by simp
```

-- snippet 78
```lean
-- Mathlib.Data.WSeq.Basic.flatten (L115-L123)
def flatten : Computation (WSeq α) → WSeq α :=
Seq.corec fun c =>
    match Computation.destruct c with
    | Sum.inl s => Seq.omap (return ·) (Seq.destruct s)
    | Sum.inr c' => some (none, c')

/-- 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. -/
```

-- snippet 79
```lean
-- Mathlib.Data.WSeq.Basic.seq_destruct_nil (L197-L199)
theorem seq_destruct_nil : Seq.destruct (nil : WSeq α) = none :=
rfl
```

-- snippet 80
```lean
-- Mathlib.Data.WSeq.Basic.destruct_nil (L185-L187)
theorem destruct_nil : destruct (nil : WSeq α) = Computation.pure none :=
Computation.destruct_eq_pure rfl
```

-- snippet 81
```lean
-- Mathlib.Data.WSeq.Basic.recOn (L94-L101)
def recOn {motive : WSeq α → Sort v} (s : WSeq α) (nil : motive nil)
    (cons : ∀ x s, motive (cons x s)) (think : ∀ s, motive (think s)) : motive s :=
Seq.recOn s nil fun o => Option.recOn o think cons

/-- membership for weak sequences -/
protected def Mem (s : WSeq α) (a : α) :=
  Seq.Mem s (some a)
```

-- snippet 82
```lean
-- Mathlib.Data.WSeq.Basic.coeSeq (L59-L61)
instance coeSeq : Coe (Seq α) (WSeq α) :=
⟨ofSeq⟩
```

-- snippet 83
```lean
-- Mathlib.Data.WSeq.Basic.head_think (L215-L216)
theorem head_think (s : WSeq α) : head (think s) = (head s).think := by simp [head]
```

-- snippet 84
```lean
-- Mathlib.Data.WSeq.Basic.ret (L166-L169)
def ret (a : α) : WSeq α :=
ofList [a]

/-- Monadic bind operator for weak sequences -/
```

-- snippet 85
```lean
-- Mathlib.Data.WSeq.Basic.destruct_some_of_destruct_tail_some (L361-L370)
theorem destruct_some_of_destruct_tail_some {s : WSeq α} {a} (h : some a ∈ destruct (tail s)) :
    ∃ a', some a' ∈ destruct s := by
unfold tail Functor.map at h; simp only [destruct_flatten] at h
  rcases exists_of_mem_bind h with ⟨t, tm, td⟩; clear h
  rcases Computation.exists_of_mem_map tm with ⟨t', ht', ht2⟩; clear tm
  rcases t' with - | t' <;> rw [← ht2] at td <;> simp only [destruct_nil] at td
  · have := mem_unique td (ret_mem _)
    contradiction
  · exact ⟨_, ht'⟩
```

-- snippet 86
```lean
-- Mathlib.Data.WSeq.Basic.think_append (L314-L316)
theorem think_append (s t : WSeq α) : append (think s) t = think (append s t) :=
Seq.cons_append _ _ _
```

-- snippet 87
```lean
-- Mathlib.Data.WSeq.Basic.seq_destruct_cons (L201-L203)
theorem seq_destruct_cons (a : α) (s) : Seq.destruct (cons a s) = some (some a, s) :=
Seq.destruct_cons _ _
```

-- snippet 88
```lean
-- Mathlib.Data.WSeq.Basic.destruct (L85-L92)
def destruct : WSeq α → Computation (Option (α × WSeq α)) :=
Computation.corec fun s =>
    match Seq.destruct s with
    | none => Sum.inl none
    | some (none, s') => Sum.inr s'
    | some (some a, s') => Sum.inl (some (a, s'))

/-- Recursion principle for weak sequences, compare with `List.recOn`. -/
```

-- snippet 89
```lean
-- Mathlib.Data.WSeq.Basic.of_mem_append (L522-L524)
theorem of_mem_append {s₁ s₂ : WSeq α} {a : α} : a ∈ append s₁ s₂ → a ∈ s₁ ∨ a ∈ s₂ :=
Seq.of_mem_append
```

-- snippet 90
```lean
-- Mathlib.Data.WSeq.Basic.flatten_think (L230-L232)
theorem flatten_think (c : Computation (WSeq α)) : flatten c.think = think (flatten c) :=
Seq.destruct_eq_cons <| by simp [flatten]
```

-- snippet 91
```lean
-- Mathlib.Data.WSeq.Basic.toList'_map (L581-L607)
theorem toList'_map (l : List α) (s : WSeq α) :
    Computation.corec (fun ⟨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) = (l.reverse ++ ·) <$> toList s := by
refine
    Computation.eq_of_bisim
      (fun c1 c2 =>
        ∃ (l' : List α) (s : WSeq α),
          c1 = Computation.corec (fun ⟨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' ++ l, s) ∧
            c2 = Computation.map (l.reverse ++ ·) (Computation.corec (fun ⟨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)))
      ?_ ⟨[], s, rfl, rfl⟩
  intro s1 s2 h; rcases h with ⟨l', s, h⟩; rw [h.left, h.right]
  induction s using WSeq.recOn
  case nil => simp
  case cons a s => refine ⟨a :: l', s, ?_, ?_⟩ <;> simp
  case think s => refine ⟨l', s, ?_, ?_⟩ <;> simp
```

-- snippet 92
```lean
-- Mathlib.Data.WSeq.Basic.destruct_flatten (L234-L247)
theorem destruct_flatten (c : Computation (WSeq α)) : destruct (flatten c) = c >>= destruct := by
refine
    Computation.eq_of_bisim
      (fun c1 c2 => c1 = c2 ∨ ∃ c, c1 = destruct (flatten c) ∧ c2 = Computation.bind c destruct) ?_
      (Or.inr ⟨c, rfl, rfl⟩)
  intro c1 c2 h
  exact
    match c1, c2, h with
    | c, _, Or.inl rfl => by cases c.destruct <;> simp
    | _, _, Or.inr ⟨c, rfl, rfl⟩ => by
      induction c using Computation.recOn with
      | pure a => simp; cases (destruct a).destruct <;> simp
      | think c' => simpa using Or.inr ⟨c', rfl, rfl⟩
```

-- snippet 93
```lean
-- Mathlib.Data.WSeq.Basic.ofSeq (L46-L49)
def ofSeq : Seq α → WSeq α :=
(· <$> ·) some

/-- Turn a list into a weak sequence -/
```

-- snippet 94
```lean
-- Mathlib.Data.WSeq.Basic.membership (L102-L104)
instance membership : Membership α (WSeq α) :=
⟨WSeq.Mem⟩
```

-- snippet 95
```lean
-- Mathlib.Data.WSeq.Basic.dropn_think (L271-L273)
theorem dropn_think (s : WSeq α) (n) : drop (think s) n = (drop s n).think := by
induction n <;> simp [*, drop]
```

-- snippet 96
```lean
-- Mathlib.Data.WSeq.Basic.toList (L137-L147)
def toList (s : WSeq α) : Computation (List α) :=
@Computation.corec (List α) (List α × WSeq α)
    (fun ⟨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'))
    ([], s)

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

-- snippet 97
```lean
-- Mathlib.Data.WSeq.Basic.destruct_tail (L331-L335)
theorem destruct_tail (s : WSeq α) : destruct (tail s) = destruct s >>= tail.aux := by
simp only [tail, destruct_flatten]; rw [← bind_pure_comp, LawfulMonad.bind_assoc]
  apply congr_arg; ext1 (_ | ⟨a, s⟩) <;> apply (@pure_bind Computation _ _ _ _ _ _).trans _ <;> simp

/-- auxiliary definition of drop over weak sequences -/
```

-- snippet 98
```lean
-- Mathlib.Data.WSeq.Basic.toList'_nil (L545-L552)
theorem toList'_nil (l : List α) :
    Computation.corec (fun ⟨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 :=
destruct_eq_pure rfl
```

-- snippet 99
```lean
-- Mathlib.Data.WSeq.Basic.mem_append_left (L525-L527)
theorem mem_append_left {s₁ s₂ : WSeq α} {a : α} : a ∈ s₁ → a ∈ append s₁ s₂ :=
Seq.mem_append_left
```

-- snippet 100
```lean
-- Mathlib.Data.WSeq.Basic.nil_append (L306-L308)
theorem nil_append (s : WSeq α) : append nil s = s :=
Seq.nil_append _
```

-- snippet 101
```lean
-- Mathlib.Data.WSeq.Basic.tail_cons (L255-L256)
theorem tail_cons (a : α) (s) : tail (cons a s) = s := by simp [tail]
```

-- snippet 102
```lean
-- Mathlib.Data.WSeq.Basic.exists_dropn_of_mem (L504-L517)
theorem exists_dropn_of_mem {s : WSeq α} {a} (h : a ∈ s) :
    ∃ n s', some (a, s') ∈ destruct (drop s n) :=
let ⟨n, h⟩ := exists_get?_of_mem h
  ⟨n, by
    rcases (head_terminates_iff _).1 ⟨⟨_, h⟩⟩ with ⟨⟨o, om⟩⟩
    have := Computation.mem_unique (Computation.mem_map _ om) h
    rcases o with - | o
    · injection this
    injection this with i
    obtain ⟨a', s'⟩ := o
    dsimp at i
    rw [i] at om
    exact ⟨_, om⟩⟩
```

-- snippet 103
```lean
-- Mathlib.Data.WSeq.Basic.tail_think (L258-L259)
theorem tail_think (s : WSeq α) : tail (think s) = (tail s).think := by simp [tail]
```

-- snippet 104
```lean
-- Mathlib.Data.WSeq.Basic.ofStream (L56-L58)
def ofStream (l : Stream' α) : WSeq α :=
ofSeq l
```

-- snippet 105
```lean
-- Mathlib.Data.WSeq.Basic.append (L148-L152)
def append : WSeq α → WSeq α → WSeq α :=
Seq.append

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

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Data.WSeq.Relation).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Prove the following theorem. The retrieved context below may help.

```lean
theorem LiftRel.refl (R : α → α → Prop) (H : Reflexive R) : Reflexive (LiftRel R)
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Data.WSeq.Relation.LiftRelO (L37-L41)
def LiftRelO (R : α → β → Prop) (C : WSeq α → WSeq β → Prop) :
    Option (α × WSeq α) → Option (β × WSeq β) → Prop
  | none, none => True
  | some (a, s), some (b, t) => R a b ∧ C s t
  | _, _ => False
```

-- snippet 2
```lean
-- Mathlib.Data.WSeq.Relation.LiftRelO (L44-L50)
theorem LiftRelO.imp {R S : α → β → Prop} {C D : WSeq α → WSeq β → Prop} (H1 : ∀ a b, R a b → S a b)
    (H2 : ∀ s t, C s t → D s t) : ∀ {o p}, LiftRelO R C o p → LiftRelO S D o p
  | none, none, _ => trivial
  | some (_, _), some (_, _), h => And.imp (H1 _ _) (H2 _ _) h
  | none, some _, h => False.elim h
  | some (_, _), none, h => False.elim h
```

-- snippet 3
```lean
-- Mathlib.Data.WSeq.Relation.LiftRelO (L51-L54)
theorem LiftRelO.imp_right (R : α → β → Prop) {C D : WSeq α → WSeq β → Prop}
    (H : ∀ s t, C s t → D s t) {o p} : LiftRelO R C o p → LiftRelO R D o p :=
LiftRelO.imp (fun _ _ => id) H
```

-- snippet 4
```lean
-- Mathlib.Data.WSeq.Relation.LiftRelO (L55-L60)
theorem LiftRelO.swap (R : α → β → Prop) (C) :
    swap (LiftRelO R C) = LiftRelO (swap R) (swap C) := by
funext x y
  rcases x with ⟨⟩ | ⟨hx, jx⟩ <;> rcases y with ⟨⟩ | ⟨hy, jy⟩ <;> rfl

/-- Definition of bisimilarity for weak sequences -/
```

-- snippet 5
```lean
-- Mathlib.Data.WSeq.Relation.BisimO (L62-L64)
def BisimO (R : WSeq α → WSeq α → Prop) : Option (α × WSeq α) → Option (α × WSeq α) → Prop :=
LiftRelO (· = ·) R
```

-- snippet 6
```lean
-- Mathlib.Data.WSeq.Relation.BisimO (L65-L71)
theorem BisimO.imp {R S : WSeq α → WSeq α → Prop} (H : ∀ s t, R s t → S s t) {o p} :
    BisimO R o p → BisimO S o p :=
LiftRelO.imp_right _ H

/-- 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.) -/
```

-- snippet 7
```lean
-- Mathlib.Data.WSeq.Relation.liftRel_destruct (L76-L82)
theorem liftRel_destruct {R : α → β → Prop} {s : WSeq α} {t : WSeq β} :
    LiftRel R s t → Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)
  | ⟨R, h1, h2⟩ => by
    refine Computation.LiftRel.imp ?_ _ _ (h2 h1)
    apply LiftRelO.imp_right
    exact fun s' t' h' => ⟨R, h', @h2⟩
```

-- snippet 8
```lean
-- Mathlib.Data.WSeq.Relation.liftRel_destruct_iff (L83-L97)
theorem liftRel_destruct_iff {R : α → β → Prop} {s : WSeq α} {t : WSeq β} :
    LiftRel R s t ↔ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t) :=
⟨liftRel_destruct, fun h =>
    ⟨fun s t =>
      LiftRel R s t ∨ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t),
      Or.inr h, fun {s t} h => by
      have h : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t) := by
        obtain h | h := h
        · exact liftRel_destruct h
        · assumption
      apply Computation.LiftRel.imp _ _ _ h
      apply LiftRelO.imp_right
      intro s t
      apply Or.inl⟩⟩
```

-- snippet 9
```lean
-- Mathlib.Data.WSeq.Relation.Equiv (L159-L161)
def Equiv : WSeq α → WSeq α → Prop :=
LiftRel (· = ·)
```

-- snippet 10
```lean
-- Mathlib.Data.WSeq.Relation.Equiv (L165-L167)
theorem Equiv.refl : ∀ s : WSeq α, s ~ʷ s :=
LiftRel.refl (· = ·) Eq.refl
```

-- snippet 11
```lean
-- Mathlib.Data.WSeq.Relation.Equiv (L169-L171)
theorem Equiv.symm : ∀ {s t : WSeq α}, s ~ʷ t → t ~ʷ s :=
@(LiftRel.symm (· = ·) (@Eq.symm _))
```

-- snippet 12
```lean
-- Mathlib.Data.WSeq.Relation.Equiv (L173-L175)
theorem Equiv.trans : ∀ {s t u : WSeq α}, s ~ʷ t → t ~ʷ u → s ~ʷ u :=
LiftRel.trans (· = ·) inferInstance |>.trans _ _ _
```

-- snippet 13
```lean
-- Mathlib.Data.WSeq.Relation.Equiv (L176-L178)
theorem Equiv.equivalence : Equivalence (@Equiv α) :=
⟨@Equiv.refl _, @Equiv.symm _, @Equiv.trans _⟩
```

-- snippet 14
```lean
-- Mathlib.Data.WSeq.Relation.destruct_congr (L179-L182)
theorem destruct_congr {s t : WSeq α} :
    s ~ʷ t → Computation.LiftRel (BisimO (· ~ʷ ·)) (destruct s) (destruct t) :=
liftRel_destruct
```

-- snippet 15
```lean
-- Mathlib.Data.WSeq.Relation.destruct_congr_iff (L183-L188)
theorem destruct_congr_iff {s t : WSeq α} :
    s ~ʷ t ↔ Computation.LiftRel (BisimO (· ~ʷ ·)) (destruct s) (destruct t) :=
liftRel_destruct_iff

open Computation
```

-- snippet 16
```lean
-- Mathlib.Data.WSeq.Relation.liftRel_dropn_destruct (L189-L200)
theorem liftRel_dropn_destruct {R : α → β → Prop} {s t} (H : LiftRel R s t) :
    ∀ n, Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct (drop s n)) (destruct (drop t n))
  | 0 => liftRel_destruct H
  | n + 1 => by
    simp only [drop, destruct_tail]
    apply liftRel_bind
    · apply liftRel_dropn_destruct H n
    exact fun {a b} o =>
      match a, b, o with
      | none, none, _ => by simp
      | some (a, s), some (b, t), ⟨_, h2⟩ => by simpa [tail.aux] using liftRel_destruct h2
```

-- snippet 17
```lean
-- Mathlib.Data.WSeq.Relation.exists_of_liftRel_left (L201-L207)
theorem exists_of_liftRel_left {R : α → β → Prop} {s t} (H : LiftRel R s t) {a} (h : a ∈ s) :
    ∃ b, b ∈ t ∧ R a b := by
let ⟨n, h⟩ := exists_get?_of_mem h
  let ⟨some (_, s'), sd, rfl⟩ := Computation.exists_of_mem_map h
  let ⟨some (b, t'), td, ⟨ab, _⟩⟩ := (liftRel_dropn_destruct H n).left sd
  exact ⟨b, get?_mem (Computation.mem_map (Prod.fst.{v, v} <$> ·) td), ab⟩
```

-- snippet 18
```lean
-- Mathlib.Data.WSeq.Relation.exists_of_liftRel_right (L208-L210)
theorem exists_of_liftRel_right {R : α → β → Prop} {s t} (H : LiftRel R s t) {b} (h : b ∈ t) :
    ∃ a, a ∈ s ∧ R a b := by rw [← LiftRel.swap] at H; exact exists_of_liftRel_left H h
```

-- snippet 19
```lean
-- Mathlib.Data.WSeq.Relation.liftRel_nil (L212-L214)
theorem liftRel_nil (R : α → β → Prop) : LiftRel R nil nil := by
simp [liftRel_destruct_iff]
```

-- snippet 20
```lean
-- Mathlib.Data.WSeq.Relation.liftRel_cons (L216-L219)
theorem liftRel_cons (R : α → β → Prop) (a b s t) :
    LiftRel R (cons a s) (cons b t) ↔ R a b ∧ LiftRel R s t := by
simp [liftRel_destruct_iff]
```

-- snippet 21
```lean
-- Mathlib.Data.WSeq.Relation.liftRel_think_left (L221-L223)
theorem liftRel_think_left (R : α → β → Prop) (s t) : LiftRel R (think s) t ↔ LiftRel R s t := by
rw [liftRel_destruct_iff, liftRel_destruct_iff]; simp
```

-- snippet 22
```lean
-- Mathlib.Data.WSeq.Relation.liftRel_think_right (L225-L227)
theorem liftRel_think_right (R : α → β → Prop) (s t) : LiftRel R s (think t) ↔ LiftRel R s t := by
rw [liftRel_destruct_iff, liftRel_destruct_iff]; simp
```

-- snippet 23
```lean
-- Mathlib.Data.WSeq.Relation.cons_congr (L228-L230)
theorem cons_congr {s t : WSeq α} (a : α) (h : s ~ʷ t) : cons a s ~ʷ cons a t := by
unfold Equiv; simpa using h
```

-- snippet 24
```lean
-- Mathlib.Data.WSeq.Relation.think_equiv (L231-L232)
theorem think_equiv (s : WSeq α) : think s ~ʷ s := by unfold Equiv; simpa using Equiv.refl _
```

-- snippet 25
```lean
-- Mathlib.Data.WSeq.Relation.think_congr (L233-L235)
theorem think_congr {s t : WSeq α} (h : s ~ʷ t) : think s ~ʷ think t := by
unfold Equiv; simpa using h
```

-- snippet 26
```lean
-- Mathlib.Data.WSeq.Basic.head_terminates_iff (L248-L250)
theorem head_terminates_iff (s : WSeq α) : Terminates (head s) ↔ Terminates (destruct s) :=
terminates_map_iff _ (destruct s)
```

-- snippet 27
```lean
-- Mathlib.Data.WSeq.Basic.flatten_pure (L218-L228)
theorem flatten_pure (s : WSeq α) : flatten (Computation.pure s) = s := by
refine Seq.eq_of_bisim (fun s1 s2 => flatten (Computation.pure s2) = s1) ?_ rfl
  intro s' s h
  rw [← h]
  simp only [Seq.BisimO, flatten, Seq.omap, pure_def, Seq.corec_eq, destruct_pure]
  cases Seq.destruct s with
  | none => simp
  | some val =>
    obtain ⟨o, s'⟩ := val
    simp
```

-- snippet 28
```lean
-- Mathlib.Data.WSeq.Basic.tail (L124-L127)
def tail (s : WSeq α) : WSeq α :=
flatten <| (fun o => Option.recOn o nil Prod.snd) <$> destruct s

/-- drop the first `n` elements from `s`. -/
```

-- snippet 29
```lean
-- Mathlib.Data.WSeq.Basic.eq_or_mem_iff_mem (L425-L445)
theorem eq_or_mem_iff_mem {s : WSeq α} {a a' s'} :
    some (a', s') ∈ destruct s → (a ∈ s ↔ a = a' ∨ a ∈ s') := by
generalize e : destruct s = c; intro h
  revert s
  apply Computation.memRecOn h <;> [skip; intro c IH] <;> intro s <;>
    induction s using WSeq.recOn <;>
    intro m <;>
    have := congr_arg Computation.destruct m <;>
    simp at this
  · obtain ⟨i1, i2⟩ := this
    rw [i1, i2]
    dsimp only [cons, Membership.mem, WSeq.Mem, Seq.Mem, Seq.cons]
    have h_a_eq_a' : a = a' ↔ some (some a) = some (some a') := by simp
    rw [h_a_eq_a']
    refine ⟨Stream'.eq_or_mem_of_mem_cons, fun o => ?_⟩
    · rcases o with e | m
      · rw [e]
        apply Stream'.mem_cons
      · exact Stream'.mem_cons_of_mem _ m
  · simp [IH this]
```

-- snippet 30
```lean
-- Mathlib.Data.WSeq.Basic.mem_think (L415-L424)
theorem mem_think (s : WSeq α) (a) : a ∈ think s ↔ a ∈ s := by
obtain ⟨f, al⟩ := s
  change (some (some a) ∈ some none::f) ↔ some (some a) ∈ f
  constructor <;> intro h
  · apply (Stream'.eq_or_mem_of_mem_cons h).resolve_left
    intro
    injections
  · apply Stream'.mem_cons_of_mem _ h

set_option linter.flexible false in -- TODO: fix non-terminal simp
```

-- snippet 31
```lean
-- Mathlib.Data.WSeq.Basic.cons_append (L310-L312)
theorem cons_append (a : α) (s t) : append (cons a s) t = cons a (append s t) :=
Seq.cons_append _ _ _
```

-- snippet 32
```lean
-- Mathlib.Data.WSeq.Basic.toList'_cons (L554-L566)
theorem toList'_cons (l : List α) (s : WSeq α) (a : α) :
    Computation.corec (fun ⟨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 ⟨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 :=
destruct_eq_think <| by simp [cons]
```

-- snippet 33
```lean
-- Mathlib.Data.WSeq.Basic.WSeq (L37-L44)
def WSeq (α) :=
Seq (Option α)

namespace WSeq

variable {α : Type u} {β : Type v} {γ : Type w}

/-- Turn a sequence into a weak sequence -/
```

-- snippet 34
```lean
-- Mathlib.Data.WSeq.Basic.ofList (L51-L54)
def ofList (l : List α) : WSeq α :=
ofSeq l

/-- Turn a stream into a weak sequence -/
```

-- snippet 35
```lean
-- Mathlib.Data.WSeq.Basic.head (L110-L114)
def head (s : WSeq α) : Computation (Option α) :=
Computation.map (Prod.fst <$> ·) (destruct s)

/-- Encode a computation yielding a weak sequence into additional
  `think` constructors in a weak sequence -/
```

-- snippet 36
```lean
-- Mathlib.Data.WSeq.Basic.head_terminates_of_head_tail_terminates (L353-L360)
theorem head_terminates_of_head_tail_terminates (s : WSeq α) [T : Terminates (head (tail s))] :
    Terminates (head s) :=
(head_terminates_iff _).2 <| by
    rcases (head_terminates_iff _).1 T with ⟨⟨a, h⟩⟩
    simp? [tail] at h says simp only [tail, destruct_flatten, bind_map_left] at h
    rcases exists_of_mem_bind h with ⟨s', h1, _⟩
    exact terminates_of_mem h1
```

-- snippet 37
```lean
-- Mathlib.Data.WSeq.Basic.destruct_terminates_of_get (L397-L400)
theorem destruct_terminates_of_get?_terminates {s : WSeq α} {n} (T : Terminates (get? s n)) :
    Terminates (destruct s) :=
(head_terminates_iff _).1 <| head_terminates_of_get?_terminates T
```

-- snippet 38
```lean
-- Mathlib.Data.WSeq.Basic.monad (L177-L183)
instance monad : Monad WSeq where
  map := @map
pure := @ret
  bind := @bind

open Computation
```

-- snippet 39
```lean
-- Mathlib.Data.WSeq.Basic.exists_get (L487-L503)
theorem exists_get?_of_mem {s : WSeq α} {a} (h : a ∈ s) : ∃ n, some a ∈ get? s n := by
apply mem_rec_on h
  · intro a' s' h
    rcases h with h | h
    · exists 0
      simp only [get?, drop, head_cons]
      rw [h]
      apply ret_mem
    · obtain ⟨n, h⟩ := h
      exists n + 1
      simpa [get?]
  · intro s' h
    obtain ⟨n, h⟩ := h
    exists n
    simp only [get?, dropn_think, head_think]
    apply think_mem h
```

-- snippet 40
```lean
-- Mathlib.Data.WSeq.Basic.join_cons (L300-L304)
theorem join_cons (s : WSeq α) (S) : join (cons s S) = think (append s (join S)) := by
simp only [join, think]
  dsimp only [(· <$> ·)]
  simp [cons, append]
```

-- snippet 41
```lean
-- Mathlib.Data.WSeq.Basic.dropn_tail (L278-L282)
theorem dropn_tail (s : WSeq α) (n) : drop (tail s) n = drop s (n + 1) := by
rw [Nat.add_comm]
  symm
  apply dropn_add
```

-- snippet 42
```lean
-- Mathlib.Data.WSeq.Basic.toList'_think (L568-L580)
theorem toList'_think (l : List α) (s : WSeq α) :
    Computation.corec (fun ⟨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, think s) =
      (Computation.corec (fun ⟨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 :=
destruct_eq_think <| by simp [think]
```

-- snippet 43
```lean
-- Mathlib.Data.WSeq.Basic.head_cons (L212-L213)
theorem head_cons (a : α) (s) : head (cons a s) = Computation.pure (some a) := by simp [head]
```

-- snippet 44
```lean
-- Mathlib.Data.WSeq.Basic.append_nil (L318-L320)
theorem append_nil (s : WSeq α) : append s nil = s :=
Seq.append_nil _
```

-- snippet 45
```lean
-- Mathlib.Data.WSeq.Basic.mem_of_mem_tail (L457-L468)
theorem mem_of_mem_tail {s : WSeq α} {a} : a ∈ tail s → a ∈ s := by
intro h; have := h; obtain ⟨n, e⟩ := h; revert s; simp only [Stream'.get]
  induction n <;> intro s <;> induction s using WSeq.recOn <;>
    simp <;> intro m e <;> injections
  · exact Or.inr m
  · exact Or.inr m
  case succ.think n IH s =>
    apply IH m
    rw [e]
    cases tail s
    rfl
```

-- snippet 46
```lean
-- Mathlib.Data.WSeq.Basic.head_some_of_get (L379-L386)
theorem head_some_of_get?_some {s : WSeq α} {a n} (h : some a ∈ get? s n) :
    ∃ a', some a' ∈ head s := by
induction n generalizing a with
  | zero => exact ⟨_, h⟩
  | succ n IH =>
      let ⟨a', h'⟩ := head_some_of_head_tail_some h
      exact IH h'
```

-- snippet 47
```lean
-- Mathlib.Data.WSeq.Basic.dropn_nil (L261-L262)
theorem dropn_nil (n) : drop (nil : WSeq α) n = nil := by induction n <;> simp [*, drop]
```

-- snippet 48
```lean
-- Mathlib.Data.WSeq.Basic.mem_cons_of_mem (L450-L452)
theorem mem_cons_of_mem {s : WSeq α} (b) {a} (h : a ∈ s) : a ∈ cons b s :=
(mem_cons_iff _ _).2 (Or.inr h)
```

-- snippet 49
```lean
-- Mathlib.Data.WSeq.Basic.head_nil (L209-L210)
theorem head_nil : head (nil : WSeq α) = Computation.pure none := by simp [head]
```

-- snippet 50
```lean
-- Mathlib.Data.WSeq.Basic.mem_cons_iff (L447-L449)
theorem mem_cons_iff (s : WSeq α) (b) {a} : a ∈ cons b s ↔ a = b ∨ a ∈ s :=
eq_or_mem_iff_mem <| by simp
```

-- snippet 51
```lean
-- Mathlib.Data.WSeq.Basic.coeList (L62-L64)
instance coeList : Coe (List α) (WSeq α) :=
⟨ofList⟩
```

-- snippet 52
```lean
-- Mathlib.Data.WSeq.Basic.tail_nil (L252-L253)
theorem tail_nil : tail (nil : WSeq α) = nil := by simp [tail]
```

-- snippet 53
```lean
-- Mathlib.Data.WSeq.Basic.join_think (L294-L298)
theorem join_think (S : WSeq (WSeq α)) : join (think S) = think (join S) := by
simp only [join, think]
  dsimp only [(· <$> ·)]
  simp [Seq1.ret]
```

-- snippet 54
```lean
-- Mathlib.Data.WSeq.Basic.map (L162-L165)
def map (f : α → β) : WSeq α → WSeq β :=
Seq.map (Option.map f)

/-- The monadic `return a` is a singleton list containing `a`. -/
```

-- snippet 55
```lean
-- Mathlib.Data.WSeq.Basic.head_terminates_of_mem (L518-L521)
theorem head_terminates_of_mem {s : WSeq α} {a} (h : a ∈ s) : Terminates (head s) :=
let ⟨_, h⟩ := exists_get?_of_mem h
  head_terminates_of_get?_terminates ⟨⟨_, h⟩⟩
```

-- snippet 56
```lean
-- Mathlib.Data.WSeq.Basic.head_terminates_of_get (L393-L396)
theorem head_terminates_of_get?_terminates {s : WSeq α} {n} :
    Terminates (get? s n) → Terminates (head s) :=
get?_terminates_le (Nat.zero_le n)
```

-- snippet 57
```lean
-- Mathlib.Data.WSeq.Basic.append_assoc (L322-L325)
theorem append_assoc (s t u : WSeq α) : append (append s t) u = append s (append t u) :=
Seq.append_assoc _ _ _

/-- auxiliary definition of tail over weak sequences -/
```

-- snippet 58
```lean
-- Mathlib.Data.WSeq.Basic.inhabited (L72-L75)
instance inhabited : Inhabited (WSeq α) :=
⟨nil⟩

/-- Prepend an element to a weak sequence -/
```

-- snippet 59
```lean
-- Mathlib.Data.WSeq.Basic.join_nil (L290-L292)
theorem join_nil : join nil = (nil : WSeq α) :=
Seq.join_nil
```

-- snippet 60
```lean
-- Mathlib.Data.WSeq.Basic.bind (L170-L176)
def bind (s : WSeq α) (f : α → WSeq β) : WSeq β :=
join (map f s)

/-- Unfortunately, `WSeq` is not a lawful monad, because it does not satisfy the monad laws exactly,
only up to sequence equivalence. Furthermore, even quotienting by the equivalence is not sufficient,
because the join operation involves lists of quotient elements, with a lifted equivalence relation,
and pure quotients cannot handle this type of construction. -/
```

-- snippet 61
```lean
-- Mathlib.Data.WSeq.Basic.exists_of_mem_map (L528-L535)
theorem exists_of_mem_map {f} {b : β} : ∀ {s : WSeq α}, b ∈ map f s → ∃ a, a ∈ s ∧ f a = b
  | ⟨g, al⟩, h => by
    let ⟨o, om, oe⟩ := Seq.exists_of_mem_map h
rcases o with - | a
    · injection oe
    injection oe with h'
    exact ⟨a, om, h'⟩
```

-- snippet 62
```lean
-- Mathlib.Data.WSeq.Basic.seq_destruct_think (L205-L207)
theorem seq_destruct_think (s : WSeq α) : Seq.destruct (think s) = some (none, s) :=
Seq.destruct_cons _ _
```

-- snippet 63
```lean
-- Mathlib.Data.WSeq.Basic.coeStream (L65-L68)
instance coeStream : Coe (Stream' α) (WSeq α) :=
⟨ofStream⟩

/-- The empty weak sequence -/
```

-- snippet 64
```lean
-- Mathlib.Data.WSeq.Basic.nil (L69-L71)
def nil : WSeq α :=
Seq.nil
```

-- snippet 65
```lean
-- Mathlib.Data.WSeq.Basic.head_some_of_head_tail_some (L371-L378)
theorem head_some_of_head_tail_some {s : WSeq α} {a} (h : some a ∈ head (tail s)) :
    ∃ a', some a' ∈ head s := by
unfold head at h
  rcases Computation.exists_of_mem_map h with ⟨o, md, e⟩; clear h
  rcases o with - | o <;> [injection e; injection e with h']; clear h'
  obtain ⟨a, am⟩ := destruct_some_of_destruct_tail_some md
  exact ⟨_, Computation.mem_map (@Prod.fst α (WSeq α) <$> ·) am⟩
```

-- snippet 66
```lean
-- Mathlib.Data.WSeq.Basic.cons (L76-L79)
def cons (a : α) : WSeq α → WSeq α :=
Seq.cons (some a)

/-- Compute for one tick, without producing any elements -/
```

-- snippet 67
```lean
-- Mathlib.Data.WSeq.Basic.mem_cons (L453-L456)
theorem mem_cons (s : WSeq α) (a) : a ∈ cons a s :=
(mem_cons_iff _ _).2 (Or.inl rfl)

set_option linter.flexible false in -- TODO: fix non-terminal simp
```

-- snippet 68
```lean
-- Mathlib.Data.WSeq.Basic.destruct_cons (L189-L191)
theorem destruct_cons (a : α) (s) : destruct (cons a s) = Computation.pure (some (a, s)) :=
Computation.destruct_eq_pure <| by simp [destruct, cons, Computation.rmap]
```

-- snippet 69
```lean
-- Mathlib.Data.WSeq.Basic.get (L133-L136)
def get? (s : WSeq α) (n : ℕ) : Computation (Option α) :=
head (drop s n)

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

-- snippet 70
```lean
-- Mathlib.Data.WSeq.Basic.notMem_nil (L105-L109)
theorem notMem_nil (a : α) : a ∉ @nil α :=
Seq.notMem_nil (some a)

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

-- snippet 71
```lean
-- Mathlib.Data.WSeq.Basic.ofList_nil (L537-L539)
theorem ofList_nil : ofList [] = (nil : WSeq α) :=
rfl
```

-- snippet 72
```lean
-- Mathlib.Data.WSeq.Basic.dropn_cons (L264-L269)
theorem dropn_cons (a : α) (s) (n) : drop (cons a s) (n + 1) = drop s n := by
induction n with
  | zero => simp [drop]
  | succ n n_ih =>
    simp [drop, ← n_ih]
```

-- snippet 73
```lean
-- Mathlib.Data.WSeq.Basic.mem_rec_on (L401-L413)
theorem mem_rec_on {C : WSeq α → Prop} {a s} (M : a ∈ s) (h1 : ∀ b s', a = b ∨ C s' → C (cons b s'))
    (h2 : ∀ s, C s → C (think s)) : C s := by
apply Seq.mem_rec_on M
  intro o s' h; rcases o with - | b
  · apply h2
    cases h
    · contradiction
    · assumption
  · apply h1
    apply Or.imp_left _ h
    intro h
    injection h
```

-- snippet 74
```lean
-- Mathlib.Data.WSeq.Basic.join (L153-L161)
def join (S : WSeq (WSeq α)) : WSeq α :=
Seq.join
    ((fun o : Option (WSeq α) =>
        match o with
        | none => Seq1.ret none
        | some s => (none, s)) <$>
      S)

/-- Map a function over a weak sequence -/
```

-- snippet 75
```lean
-- Mathlib.Data.WSeq.Basic.destruct_think (L193-L195)
theorem destruct_think (s : WSeq α) : destruct (think s) = (destruct s).think :=
Computation.destruct_eq_think <| by simp [destruct, think, Computation.rmap]
```

-- snippet 76
```lean
-- Mathlib.Data.WSeq.Basic.think (L80-L84)
def think : WSeq α → WSeq α :=
Seq.cons none

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

-- snippet 77
```lean
-- Mathlib.Data.WSeq.Basic.ofList_cons (L541-L543)
theorem ofList_cons (a : α) (l) : ofList (a::l) = cons a (ofList l) :=
show Seq.map some (Seq.ofList (a::l)) = Seq.cons (some a) (Seq.map some (Seq.ofList l)) by simp
```

-- snippet 78
```lean
-- Mathlib.Data.WSeq.Basic.flatten (L115-L123)
def flatten : Computation (WSeq α) → WSeq α :=
Seq.corec fun c =>
    match Computation.destruct c with
    | Sum.inl s => Seq.omap (return ·) (Seq.destruct s)
    | Sum.inr c' => some (none, c')

/-- 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. -/
```

-- snippet 79
```lean
-- Mathlib.Data.WSeq.Basic.seq_destruct_nil (L197-L199)
theorem seq_destruct_nil : Seq.destruct (nil : WSeq α) = none :=
rfl
```

-- snippet 80
```lean
-- Mathlib.Data.WSeq.Basic.destruct_nil (L185-L187)
theorem destruct_nil : destruct (nil : WSeq α) = Computation.pure none :=
Computation.destruct_eq_pure rfl
```

-- snippet 81
```lean
-- Mathlib.Data.WSeq.Basic.recOn (L94-L101)
def recOn {motive : WSeq α → Sort v} (s : WSeq α) (nil : motive nil)
    (cons : ∀ x s, motive (cons x s)) (think : ∀ s, motive (think s)) : motive s :=
Seq.recOn s nil fun o => Option.recOn o think cons

/-- membership for weak sequences -/
protected def Mem (s : WSeq α) (a : α) :=
  Seq.Mem s (some a)
```

-- snippet 82
```lean
-- Mathlib.Data.WSeq.Basic.coeSeq (L59-L61)
instance coeSeq : Coe (Seq α) (WSeq α) :=
⟨ofSeq⟩
```

-- snippet 83
```lean
-- Mathlib.Data.WSeq.Basic.head_think (L215-L216)
theorem head_think (s : WSeq α) : head (think s) = (head s).think := by simp [head]
```

-- snippet 84
```lean
-- Mathlib.Data.WSeq.Basic.ret (L166-L169)
def ret (a : α) : WSeq α :=
ofList [a]

/-- Monadic bind operator for weak sequences -/
```

-- snippet 85
```lean
-- Mathlib.Data.WSeq.Basic.destruct_some_of_destruct_tail_some (L361-L370)
theorem destruct_some_of_destruct_tail_some {s : WSeq α} {a} (h : some a ∈ destruct (tail s)) :
    ∃ a', some a' ∈ destruct s := by
unfold tail Functor.map at h; simp only [destruct_flatten] at h
  rcases exists_of_mem_bind h with ⟨t, tm, td⟩; clear h
  rcases Computation.exists_of_mem_map tm with ⟨t', ht', ht2⟩; clear tm
  rcases t' with - | t' <;> rw [← ht2] at td <;> simp only [destruct_nil] at td
  · have := mem_unique td (ret_mem _)
    contradiction
  · exact ⟨_, ht'⟩
```

-- snippet 86
```lean
-- Mathlib.Data.WSeq.Basic.think_append (L314-L316)
theorem think_append (s t : WSeq α) : append (think s) t = think (append s t) :=
Seq.cons_append _ _ _
```

-- snippet 87
```lean
-- Mathlib.Data.WSeq.Basic.seq_destruct_cons (L201-L203)
theorem seq_destruct_cons (a : α) (s) : Seq.destruct (cons a s) = some (some a, s) :=
Seq.destruct_cons _ _
```

-- snippet 88
```lean
-- Mathlib.Data.WSeq.Basic.destruct (L85-L92)
def destruct : WSeq α → Computation (Option (α × WSeq α)) :=
Computation.corec fun s =>
    match Seq.destruct s with
    | none => Sum.inl none
    | some (none, s') => Sum.inr s'
    | some (some a, s') => Sum.inl (some (a, s'))

/-- Recursion principle for weak sequences, compare with `List.recOn`. -/
```

-- snippet 89
```lean
-- Mathlib.Data.WSeq.Basic.of_mem_append (L522-L524)
theorem of_mem_append {s₁ s₂ : WSeq α} {a : α} : a ∈ append s₁ s₂ → a ∈ s₁ ∨ a ∈ s₂ :=
Seq.of_mem_append
```

-- snippet 90
```lean
-- Mathlib.Data.WSeq.Basic.flatten_think (L230-L232)
theorem flatten_think (c : Computation (WSeq α)) : flatten c.think = think (flatten c) :=
Seq.destruct_eq_cons <| by simp [flatten]
```

-- snippet 91
```lean
-- Mathlib.Data.WSeq.Basic.toList'_map (L581-L607)
theorem toList'_map (l : List α) (s : WSeq α) :
    Computation.corec (fun ⟨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) = (l.reverse ++ ·) <$> toList s := by
refine
    Computation.eq_of_bisim
      (fun c1 c2 =>
        ∃ (l' : List α) (s : WSeq α),
          c1 = Computation.corec (fun ⟨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' ++ l, s) ∧
            c2 = Computation.map (l.reverse ++ ·) (Computation.corec (fun ⟨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)))
      ?_ ⟨[], s, rfl, rfl⟩
  intro s1 s2 h; rcases h with ⟨l', s, h⟩; rw [h.left, h.right]
  induction s using WSeq.recOn
  case nil => simp
  case cons a s => refine ⟨a :: l', s, ?_, ?_⟩ <;> simp
  case think s => refine ⟨l', s, ?_, ?_⟩ <;> simp
```

-- snippet 92
```lean
-- Mathlib.Data.WSeq.Basic.destruct_flatten (L234-L247)
theorem destruct_flatten (c : Computation (WSeq α)) : destruct (flatten c) = c >>= destruct := by
refine
    Computation.eq_of_bisim
      (fun c1 c2 => c1 = c2 ∨ ∃ c, c1 = destruct (flatten c) ∧ c2 = Computation.bind c destruct) ?_
      (Or.inr ⟨c, rfl, rfl⟩)
  intro c1 c2 h
  exact
    match c1, c2, h with
    | c, _, Or.inl rfl => by cases c.destruct <;> simp
    | _, _, Or.inr ⟨c, rfl, rfl⟩ => by
      induction c using Computation.recOn with
      | pure a => simp; cases (destruct a).destruct <;> simp
      | think c' => simpa using Or.inr ⟨c', rfl, rfl⟩
```

-- snippet 93
```lean
-- Mathlib.Data.WSeq.Basic.ofSeq (L46-L49)
def ofSeq : Seq α → WSeq α :=
(· <$> ·) some

/-- Turn a list into a weak sequence -/
```

-- snippet 94
```lean
-- Mathlib.Data.WSeq.Basic.membership (L102-L104)
instance membership : Membership α (WSeq α) :=
⟨WSeq.Mem⟩
```

-- snippet 95
```lean
-- Mathlib.Data.WSeq.Basic.dropn_think (L271-L273)
theorem dropn_think (s : WSeq α) (n) : drop (think s) n = (drop s n).think := by
induction n <;> simp [*, drop]
```

-- snippet 96
```lean
-- Mathlib.Data.WSeq.Basic.toList (L137-L147)
def toList (s : WSeq α) : Computation (List α) :=
@Computation.corec (List α) (List α × WSeq α)
    (fun ⟨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'))
    ([], s)

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

-- snippet 97
```lean
-- Mathlib.Data.WSeq.Basic.destruct_tail (L331-L335)
theorem destruct_tail (s : WSeq α) : destruct (tail s) = destruct s >>= tail.aux := by
simp only [tail, destruct_flatten]; rw [← bind_pure_comp, LawfulMonad.bind_assoc]
  apply congr_arg; ext1 (_ | ⟨a, s⟩) <;> apply (@pure_bind Computation _ _ _ _ _ _).trans _ <;> simp

/-- auxiliary definition of drop over weak sequences -/
```

-- snippet 98
```lean
-- Mathlib.Data.WSeq.Basic.toList'_nil (L545-L552)
theorem toList'_nil (l : List α) :
    Computation.corec (fun ⟨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 :=
destruct_eq_pure rfl
```

-- snippet 99
```lean
-- Mathlib.Data.WSeq.Basic.mem_append_left (L525-L527)
theorem mem_append_left {s₁ s₂ : WSeq α} {a : α} : a ∈ s₁ → a ∈ append s₁ s₂ :=
Seq.mem_append_left
```

-- snippet 100
```lean
-- Mathlib.Data.WSeq.Basic.nil_append (L306-L308)
theorem nil_append (s : WSeq α) : append nil s = s :=
Seq.nil_append _
```

-- snippet 101
```lean
-- Mathlib.Data.WSeq.Basic.tail_cons (L255-L256)
theorem tail_cons (a : α) (s) : tail (cons a s) = s := by simp [tail]
```

-- snippet 102
```lean
-- Mathlib.Data.WSeq.Basic.exists_dropn_of_mem (L504-L517)
theorem exists_dropn_of_mem {s : WSeq α} {a} (h : a ∈ s) :
    ∃ n s', some (a, s') ∈ destruct (drop s n) :=
let ⟨n, h⟩ := exists_get?_of_mem h
  ⟨n, by
    rcases (head_terminates_iff _).1 ⟨⟨_, h⟩⟩ with ⟨⟨o, om⟩⟩
    have := Computation.mem_unique (Computation.mem_map _ om) h
    rcases o with - | o
    · injection this
    injection this with i
    obtain ⟨a', s'⟩ := o
    dsimp at i
    rw [i] at om
    exact ⟨_, om⟩⟩
```

-- snippet 103
```lean
-- Mathlib.Data.WSeq.Basic.tail_think (L258-L259)
theorem tail_think (s : WSeq α) : tail (think s) = (tail s).think := by simp [tail]
```

-- snippet 104
```lean
-- Mathlib.Data.WSeq.Basic.ofStream (L56-L58)
def ofStream (l : Stream' α) : WSeq α :=
ofSeq l
```

-- snippet 105
```lean
-- Mathlib.Data.WSeq.Basic.append (L148-L152)
def append : WSeq α → WSeq α → WSeq α :=
Seq.append

/-- Flatten a sequence of weak sequences. (Note that this allows
  empty sequences, unlike `Seq.join`.) -/
```
[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Algebra.Support).

[User]
Prove the following theorem. The retrieved context below may help.

```lean
theorem isClosed_mulTSupport (f : X → α) : IsClosed (mulTSupport f)
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Topology.Algebra.Support.mulTSupport (L49-L50)
def mulTSupport (f : X → α) : Set X := closure (mulSupport f)
```

-- snippet 2
```lean
-- Mathlib.Topology.Algebra.Support.subset_mulTSupport (L52-L54)
theorem subset_mulTSupport (f : X → α) : mulSupport f ⊆ mulTSupport f :=
subset_closure
```

-- snippet 3
```lean
-- Mathlib.Topology.Algebra.Support.mulTSupport_eq_empty_iff (L60-L62)
theorem mulTSupport_eq_empty_iff {f : X → α} : mulTSupport f = ∅ ↔ f = 1 := by
rw [mulTSupport, closure_empty_iff, mulSupport_eq_empty_iff]
```

-- snippet 4
```lean
-- Mathlib.Topology.Algebra.Support.mulTSupport_fun_one (L64-L66)
theorem mulTSupport_fun_one : mulTSupport (fun _ ↦ 1 : X → α) = ∅ := by
rw [mulTSupport, mulSupport_fun_one, closure_empty]
```

-- snippet 5
```lean
-- Mathlib.Topology.Algebra.Support.mulTSupport_one (L68-L70)
theorem mulTSupport_one : mulTSupport (1 : X → α) = ∅ := by
rw [mulTSupport, mulSupport_one, closure_empty]
```

-- snippet 6
```lean
-- Mathlib.Topology.Algebra.Support.mulTSupport_binop_subset (L72-L76)
theorem mulTSupport_binop_subset [One β] [One γ] (op : α → β → γ)
    (op1 : op 1 1 = 1) (f : X → α) (g : X → β) :
    mulTSupport (fun x ↦ op (f x) (g x)) ⊆ mulTSupport f ∪ mulTSupport g :=
closure_mono (mulSupport_binop_subset op op1 f g) |>.trans closure_union.subset
```

-- snippet 7
```lean
-- Mathlib.Topology.Algebra.Support.mulTSupport_comp_subset (L78-L81)
lemma mulTSupport_comp_subset [One β] {g : α → β} (hg : g 1 = 1) (f : X → α) :
    mulTSupport (g ∘ f) ⊆ mulTSupport f :=
closure_mono (mulSupport_comp_subset hg f)
```

-- snippet 8
```lean
-- Mathlib.Topology.Algebra.Support.mulTSupport_subset_comp (L83-L86)
lemma mulTSupport_subset_comp [One β] {g : α → β} (hg : ∀ {x}, g x = 1 → x = 1) (f : X → α) :
    mulTSupport f ⊆ mulTSupport (g ∘ f) :=
closure_mono (mulSupport_subset_comp hg f)
```

-- snippet 9
```lean
-- Mathlib.Topology.Algebra.Support.mulTSupport_comp_eq (L88-L91)
lemma mulTSupport_comp_eq [One β] {g : α → β} (hg : ∀ {x}, g x = 1 ↔ x = 1) (f : X → α) :
    mulTSupport (g ∘ f) = mulTSupport f := by
rw [mulTSupport, mulTSupport, mulSupport_comp_eq g hg]
```

-- snippet 10
```lean
-- Mathlib.Topology.Algebra.Support.mulTSupport_comp_eq_of_range_subset (L93-L97)
lemma mulTSupport_comp_eq_of_range_subset [One β] {g : α → β} {f : X → α}
    (hg : ∀ {x}, x ∈ range f → (g x = 1 ↔ x = 1)) :
    mulTSupport (g ∘ f) = mulTSupport f := by
rw [mulTSupport, mulTSupport, mulSupport_comp_eq_of_range_subset hg]
```

-- snippet 11
```lean
-- Mathlib.Topology.Algebra.Support.mulTSupport_comp_subset_preimage (L99-L104)
lemma mulTSupport_comp_subset_preimage {Y : Type*} [TopologicalSpace Y] (g : Y → α) {f : X → Y}
    (hf : Continuous f) :
    mulTSupport (g ∘ f) ⊆ f ⁻¹' mulTSupport g := by
rw [mulTSupport, mulTSupport, mulSupport_comp_eq_preimage]
  exact hf.closure_preimage_subset _
```

-- snippet 12
```lean
-- Mathlib.Topology.Algebra.Support.mulTSupport_comp_eq_preimage (L106-L109)
lemma mulTSupport_comp_eq_preimage {Y : Type*} [TopologicalSpace Y] (g : Y → α) (f : X ≃ₜ Y) :
    mulTSupport (g ∘ f) = f ⁻¹' mulTSupport g := by
rw [mulTSupport, mulTSupport, mulSupport_comp_eq_preimage, Homeomorph.preimage_closure]
```

-- snippet 13
```lean
-- Mathlib.Topology.Algebra.Support.image_eq_one_of_notMem_mulTSupport (L111-L113)
theorem image_eq_one_of_notMem_mulTSupport {f : X → α} {x : X} (hx : x ∉ mulTSupport f) : f x = 1 :=
mulSupport_subset_iff'.mp (subset_mulTSupport f) x hx
```

-- snippet 14
```lean
-- Mathlib.Topology.Algebra.Support.range_subset_insert_image_mulTSupport (L115-L118)
theorem range_subset_insert_image_mulTSupport (f : X → α) :
    range f ⊆ insert 1 (f '' mulTSupport f) := by
grw [← subset_mulTSupport f]; exact range_subset_insert_image_mulSupport f
```

-- snippet 15
```lean
-- Mathlib.Topology.Algebra.Support.range_eq_image_mulTSupport_or (L120-L123)
theorem range_eq_image_mulTSupport_or (f : X → α) :
    range f = f '' mulTSupport f ∨ range f = insert 1 (f '' mulTSupport f) :=
(wcovBy_insert _ _).eq_or_eq (image_subset_range _ _) (range_subset_insert_image_mulTSupport f)
```

-- snippet 16
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.toAlgEquiv_injective (L106-L109)
theorem toAlgEquiv_injective : Function.Injective (toAlgEquiv : (A ≃A[R] B) → A ≃ₐ[R] B) := by
rintro ⟨e, _, _⟩ ⟨e', _, _⟩ rfl
  rfl
```

-- snippet 17
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.coe (L78-L79)
instance coe : Coe (A ≃A[R] B) (A →A[R] B) := ⟨toContinuousAlgHom⟩
```

-- snippet 18
```lean
-- Mathlib.Topology.Algebra.Affine.lineMap_continuous (L66-L75)
theorem lineMap_continuous {p q : P} :
    Continuous (lineMap p q : R →ᵃ[R] P) := by
fun_prop

open Topology Filter

section Tendsto

variable {α : Type*} {l : Filter α}
```

-- snippet 19
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.coe_inj (L118-L120)
theorem coe_inj {f g : A ≃A[R] B} : (f : A →A[R] B) = g ↔ f = g :=
coe_injective.eq_iff
```

-- snippet 20
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.equivLike (L80-L91)
instance equivLike : EquivLike (A ≃A[R] B) A B where
  coe f := f.toFun
inv f := f.invFun
  coe_injective' f g h₁ h₂ := by
    obtain ⟨f', _⟩ := f
    obtain ⟨g', _⟩ := g
    rcases f' with ⟨⟨_, _⟩, _⟩
    rcases g' with ⟨⟨_, _⟩, _⟩
    congr
  left_inv f := f.left_inv
  right_inv f := f.right_inv
```

-- snippet 21
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.coe_injective (L114-L116)
theorem coe_injective : Function.Injective ((↑) : (A ≃A[R] B) → A →A[R] B) :=
fun _ _ h => ext <| funext <| ContinuousAlgHom.ext_iff.1 h
```

-- snippet 22
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.toContinuousLinearEquiv_apply (L137-L139)
theorem toContinuousLinearEquiv_apply (e : A ≃A[R] B) (a : A) :
    e.toContinuousLinearEquiv a = e a := rfl
```

-- snippet 23
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.isOpenMap (L148-L150)
theorem isOpenMap (e : A ≃A[R] B) : IsOpenMap e :=
e.toHomeomorph.isOpenMap
```

-- snippet 24
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.toContinuousAlgHom (L74-L77)
def toContinuousAlgHom (e : A ≃A[R] B) : A →A[R] B where
  __ := e.toAlgHom
cont := e.continuous_toFun
```

-- snippet 25
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.image_closure (L151-L153)
theorem image_closure (e : A ≃A[R] B) (S : Set A) : e '' closure S = closure (e '' S) :=
e.toHomeomorph.image_closure S
```

-- snippet 26
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.coe_coe (L104-L105)
theorem coe_coe (e : A ≃A[R] B) : ⇑(e : A →A[R] B) = e := rfl
```

-- snippet 27
```lean
-- Mathlib.Topology.Algebra.AffineSubspace.isClosed_direction_iff (L54-L61)
theorem isClosed_direction_iff [T1Space V] (s : AffineSubspace R P) :
    IsClosed (s.direction : Set V) ↔ IsClosed (s : Set P) := by
rcases s.eq_bot_or_nonempty with (rfl | ⟨x, hx⟩); · simp
  rw [← (Homeomorph.vaddConst x).symm.isClosed_image,
    AffineSubspace.coe_direction_eq_vsub_set_right hx]
  simp only [Homeomorph.vaddConst_symm_apply]

end AffineSubspace
```

-- snippet 28
```lean
-- Mathlib.Topology.Algebra.Affine.continuous_linear_iff (L35-L45)
theorem continuous_linear_iff {f : P →ᵃ[R] Q} : Continuous f.linear ↔ Continuous f := by
inhabit P
  have :
    (f.linear : V → W) =
      (Homeomorph.vaddConst <| f default).symm ∘ f ∘ (Homeomorph.vaddConst default) := by
    ext v
    simp
  rw [this]
  simp only [Homeomorph.comp_continuous_iff, Homeomorph.comp_continuous_iff']

/-- If `f` is an affine map, then its linear part is an open map iff `f` is an open map. -/
```

-- snippet 29
```lean
-- Mathlib.Topology.Algebra.AffineSubspace.subtypeA (L33-L36)
def subtypeA (s : AffineSubspace R P) [Nonempty s] : s →ᴬ[R] P where
  toAffineMap := s.subtype
cont := continuous_subtype_val
```

-- snippet 30
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.continuousAlgEquivClass (L92-L98)
instance continuousAlgEquivClass : ContinuousAlgEquivClass (A ≃A[R] B) R A B where
  map_add f := f.map_add'
map_mul f := f.map_mul'
  commutes f := f.commutes'
  map_continuous := continuous_toFun
  inv_continuous := continuous_invFun
```

-- snippet 31
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.coe_apply (L99-L100)
theorem coe_apply (e : A ≃A[R] B) (a : A) : (e : A →A[R] B) a = e a := rfl
```

-- snippet 32
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.toContinuousLinearEquiv_toLinearEquiv_eq (L144-L147)
theorem toContinuousLinearEquiv_toLinearEquiv_eq (e : A ≃A[R] B) :
    e.toContinuousLinearEquiv.toLinearEquiv
    = e.toAlgEquiv.toLinearEquiv := rfl
```

-- snippet 33
```lean
-- Mathlib.Topology.Algebra.Affine.homothety_isOpenMap (L146-L152)
theorem homothety_isOpenMap (x : P) (t : R) (ht : t ≠ 0) : IsOpenMap <| homothety x t := by
apply IsOpenMap.of_inverse (homothety_continuous x t⁻¹) <;> intro e <;>
    simp [← AffineMap.comp_apply, ← homothety_mul, ht]

end Field

end AffineMap
```

-- snippet 34
```lean
-- Mathlib.Topology.Algebra.Affine.homothety_continuous (L118-L125)
theorem homothety_continuous (x : P) (t : R) : Continuous <| homothety x t := by
rw [coe_homothety]
  fun_prop

variable (R) [TopologicalSpace R] [Module R W] [ContinuousSMul R W] (x : Q) {s : Set Q}

open Topology
```

-- snippet 35
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.coe_toAlgEquiv (L122-L125)
theorem coe_toAlgEquiv (e : A ≃A[R] B) : ⇑e.toAlgEquiv = e := rfl
/-- The natural coercion from a continuous algebra isomorphism
to a continuous linear isomorphism. -/
```

-- snippet 36
```lean
-- Mathlib.Topology.Algebra.Affine.lineMap_continuous_uncurry (L60-L65)
theorem lineMap_continuous_uncurry :
    Continuous (fun pqt : P × P × R ↦ lineMap pqt.1 pqt.2.1 pqt.2.2) := by
simp only [coe_lineMap]
  fun_prop

/-- The line map is continuous. -/
```

-- snippet 37
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.ext (L111-L113)
theorem ext {f g : A ≃A[R] B} (h : ⇑f = ⇑g) : f = g :=
toAlgEquiv_injective <| AlgEquiv.ext <| congr_fun h
```

-- snippet 38
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.toContinuousLinearMap_toContinuousLinearEquiv_eq (L140-L143)
theorem toContinuousLinearMap_toContinuousLinearEquiv_eq (e : A ≃A[R] B) :
    e.toContinuousLinearEquiv.toContinuousLinearMap
    = e.toContinuousAlgHom.toContinuousLinearMap := rfl
```

-- snippet 39
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.toContinuousLinearEquiv (L127-L129)
def toContinuousLinearEquiv (e : A ≃A[R] B) : A ≃L[R] B :=
{ e with __ := e.toLinearEquiv }
```

-- snippet 40
```lean
-- Mathlib.Topology.Algebra.Affine.isOpenMap_linear_iff (L46-L58)
theorem isOpenMap_linear_iff {f : P →ᵃ[R] Q} : IsOpenMap f.linear ↔ IsOpenMap f := by
inhabit P
  have :
    (f.linear : V → W) =
      (Homeomorph.vaddConst <| f default).symm ∘ f ∘ (Homeomorph.vaddConst default) := by
    ext v
    simp
  rw [this]
  simp only [Homeomorph.comp_isOpenMap_iff, Homeomorph.comp_isOpenMap_iff']

variable [TopologicalSpace R] [ContinuousSMul R V]

/-- The line map is continuous in all arguments. -/
```

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Topology.Algebra.Support).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Prove the following theorem. The retrieved context below may help.

```lean
theorem isClosed_mulTSupport (f : X → α) : IsClosed (mulTSupport f)
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Topology.Algebra.Support.mulTSupport (L49-L50)
def mulTSupport (f : X → α) : Set X := closure (mulSupport f)
```

-- snippet 2
```lean
-- Mathlib.Topology.Algebra.Support.subset_mulTSupport (L52-L54)
theorem subset_mulTSupport (f : X → α) : mulSupport f ⊆ mulTSupport f :=
subset_closure
```

-- snippet 3
```lean
-- Mathlib.Topology.Algebra.Support.mulTSupport_eq_empty_iff (L60-L62)
theorem mulTSupport_eq_empty_iff {f : X → α} : mulTSupport f = ∅ ↔ f = 1 := by
rw [mulTSupport, closure_empty_iff, mulSupport_eq_empty_iff]
```

-- snippet 4
```lean
-- Mathlib.Topology.Algebra.Support.mulTSupport_fun_one (L64-L66)
theorem mulTSupport_fun_one : mulTSupport (fun _ ↦ 1 : X → α) = ∅ := by
rw [mulTSupport, mulSupport_fun_one, closure_empty]
```

-- snippet 5
```lean
-- Mathlib.Topology.Algebra.Support.mulTSupport_one (L68-L70)
theorem mulTSupport_one : mulTSupport (1 : X → α) = ∅ := by
rw [mulTSupport, mulSupport_one, closure_empty]
```

-- snippet 6
```lean
-- Mathlib.Topology.Algebra.Support.mulTSupport_binop_subset (L72-L76)
theorem mulTSupport_binop_subset [One β] [One γ] (op : α → β → γ)
    (op1 : op 1 1 = 1) (f : X → α) (g : X → β) :
    mulTSupport (fun x ↦ op (f x) (g x)) ⊆ mulTSupport f ∪ mulTSupport g :=
closure_mono (mulSupport_binop_subset op op1 f g) |>.trans closure_union.subset
```

-- snippet 7
```lean
-- Mathlib.Topology.Algebra.Support.mulTSupport_comp_subset (L78-L81)
lemma mulTSupport_comp_subset [One β] {g : α → β} (hg : g 1 = 1) (f : X → α) :
    mulTSupport (g ∘ f) ⊆ mulTSupport f :=
closure_mono (mulSupport_comp_subset hg f)
```

-- snippet 8
```lean
-- Mathlib.Topology.Algebra.Support.mulTSupport_subset_comp (L83-L86)
lemma mulTSupport_subset_comp [One β] {g : α → β} (hg : ∀ {x}, g x = 1 → x = 1) (f : X → α) :
    mulTSupport f ⊆ mulTSupport (g ∘ f) :=
closure_mono (mulSupport_subset_comp hg f)
```

-- snippet 9
```lean
-- Mathlib.Topology.Algebra.Support.mulTSupport_comp_eq (L88-L91)
lemma mulTSupport_comp_eq [One β] {g : α → β} (hg : ∀ {x}, g x = 1 ↔ x = 1) (f : X → α) :
    mulTSupport (g ∘ f) = mulTSupport f := by
rw [mulTSupport, mulTSupport, mulSupport_comp_eq g hg]
```

-- snippet 10
```lean
-- Mathlib.Topology.Algebra.Support.mulTSupport_comp_eq_of_range_subset (L93-L97)
lemma mulTSupport_comp_eq_of_range_subset [One β] {g : α → β} {f : X → α}
    (hg : ∀ {x}, x ∈ range f → (g x = 1 ↔ x = 1)) :
    mulTSupport (g ∘ f) = mulTSupport f := by
rw [mulTSupport, mulTSupport, mulSupport_comp_eq_of_range_subset hg]
```

-- snippet 11
```lean
-- Mathlib.Topology.Algebra.Support.mulTSupport_comp_subset_preimage (L99-L104)
lemma mulTSupport_comp_subset_preimage {Y : Type*} [TopologicalSpace Y] (g : Y → α) {f : X → Y}
    (hf : Continuous f) :
    mulTSupport (g ∘ f) ⊆ f ⁻¹' mulTSupport g := by
rw [mulTSupport, mulTSupport, mulSupport_comp_eq_preimage]
  exact hf.closure_preimage_subset _
```

-- snippet 12
```lean
-- Mathlib.Topology.Algebra.Support.mulTSupport_comp_eq_preimage (L106-L109)
lemma mulTSupport_comp_eq_preimage {Y : Type*} [TopologicalSpace Y] (g : Y → α) (f : X ≃ₜ Y) :
    mulTSupport (g ∘ f) = f ⁻¹' mulTSupport g := by
rw [mulTSupport, mulTSupport, mulSupport_comp_eq_preimage, Homeomorph.preimage_closure]
```

-- snippet 13
```lean
-- Mathlib.Topology.Algebra.Support.image_eq_one_of_notMem_mulTSupport (L111-L113)
theorem image_eq_one_of_notMem_mulTSupport {f : X → α} {x : X} (hx : x ∉ mulTSupport f) : f x = 1 :=
mulSupport_subset_iff'.mp (subset_mulTSupport f) x hx
```

-- snippet 14
```lean
-- Mathlib.Topology.Algebra.Support.range_subset_insert_image_mulTSupport (L115-L118)
theorem range_subset_insert_image_mulTSupport (f : X → α) :
    range f ⊆ insert 1 (f '' mulTSupport f) := by
grw [← subset_mulTSupport f]; exact range_subset_insert_image_mulSupport f
```

-- snippet 15
```lean
-- Mathlib.Topology.Algebra.Support.range_eq_image_mulTSupport_or (L120-L123)
theorem range_eq_image_mulTSupport_or (f : X → α) :
    range f = f '' mulTSupport f ∨ range f = insert 1 (f '' mulTSupport f) :=
(wcovBy_insert _ _).eq_or_eq (image_subset_range _ _) (range_subset_insert_image_mulTSupport f)
```

-- snippet 16
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.toAlgEquiv_injective (L106-L109)
theorem toAlgEquiv_injective : Function.Injective (toAlgEquiv : (A ≃A[R] B) → A ≃ₐ[R] B) := by
rintro ⟨e, _, _⟩ ⟨e', _, _⟩ rfl
  rfl
```

-- snippet 17
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.coe (L78-L79)
instance coe : Coe (A ≃A[R] B) (A →A[R] B) := ⟨toContinuousAlgHom⟩
```

-- snippet 18
```lean
-- Mathlib.Topology.Algebra.Affine.lineMap_continuous (L66-L75)
theorem lineMap_continuous {p q : P} :
    Continuous (lineMap p q : R →ᵃ[R] P) := by
fun_prop

open Topology Filter

section Tendsto

variable {α : Type*} {l : Filter α}
```

-- snippet 19
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.coe_inj (L118-L120)
theorem coe_inj {f g : A ≃A[R] B} : (f : A →A[R] B) = g ↔ f = g :=
coe_injective.eq_iff
```

-- snippet 20
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.equivLike (L80-L91)
instance equivLike : EquivLike (A ≃A[R] B) A B where
  coe f := f.toFun
inv f := f.invFun
  coe_injective' f g h₁ h₂ := by
    obtain ⟨f', _⟩ := f
    obtain ⟨g', _⟩ := g
    rcases f' with ⟨⟨_, _⟩, _⟩
    rcases g' with ⟨⟨_, _⟩, _⟩
    congr
  left_inv f := f.left_inv
  right_inv f := f.right_inv
```

-- snippet 21
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.coe_injective (L114-L116)
theorem coe_injective : Function.Injective ((↑) : (A ≃A[R] B) → A →A[R] B) :=
fun _ _ h => ext <| funext <| ContinuousAlgHom.ext_iff.1 h
```

-- snippet 22
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.toContinuousLinearEquiv_apply (L137-L139)
theorem toContinuousLinearEquiv_apply (e : A ≃A[R] B) (a : A) :
    e.toContinuousLinearEquiv a = e a := rfl
```

-- snippet 23
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.isOpenMap (L148-L150)
theorem isOpenMap (e : A ≃A[R] B) : IsOpenMap e :=
e.toHomeomorph.isOpenMap
```

-- snippet 24
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.toContinuousAlgHom (L74-L77)
def toContinuousAlgHom (e : A ≃A[R] B) : A →A[R] B where
  __ := e.toAlgHom
cont := e.continuous_toFun
```

-- snippet 25
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.image_closure (L151-L153)
theorem image_closure (e : A ≃A[R] B) (S : Set A) : e '' closure S = closure (e '' S) :=
e.toHomeomorph.image_closure S
```

-- snippet 26
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.coe_coe (L104-L105)
theorem coe_coe (e : A ≃A[R] B) : ⇑(e : A →A[R] B) = e := rfl
```

-- snippet 27
```lean
-- Mathlib.Topology.Algebra.AffineSubspace.isClosed_direction_iff (L54-L61)
theorem isClosed_direction_iff [T1Space V] (s : AffineSubspace R P) :
    IsClosed (s.direction : Set V) ↔ IsClosed (s : Set P) := by
rcases s.eq_bot_or_nonempty with (rfl | ⟨x, hx⟩); · simp
  rw [← (Homeomorph.vaddConst x).symm.isClosed_image,
    AffineSubspace.coe_direction_eq_vsub_set_right hx]
  simp only [Homeomorph.vaddConst_symm_apply]

end AffineSubspace
```

-- snippet 28
```lean
-- Mathlib.Topology.Algebra.Affine.continuous_linear_iff (L35-L45)
theorem continuous_linear_iff {f : P →ᵃ[R] Q} : Continuous f.linear ↔ Continuous f := by
inhabit P
  have :
    (f.linear : V → W) =
      (Homeomorph.vaddConst <| f default).symm ∘ f ∘ (Homeomorph.vaddConst default) := by
    ext v
    simp
  rw [this]
  simp only [Homeomorph.comp_continuous_iff, Homeomorph.comp_continuous_iff']

/-- If `f` is an affine map, then its linear part is an open map iff `f` is an open map. -/
```

-- snippet 29
```lean
-- Mathlib.Topology.Algebra.AffineSubspace.subtypeA (L33-L36)
def subtypeA (s : AffineSubspace R P) [Nonempty s] : s →ᴬ[R] P where
  toAffineMap := s.subtype
cont := continuous_subtype_val
```

-- snippet 30
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.continuousAlgEquivClass (L92-L98)
instance continuousAlgEquivClass : ContinuousAlgEquivClass (A ≃A[R] B) R A B where
  map_add f := f.map_add'
map_mul f := f.map_mul'
  commutes f := f.commutes'
  map_continuous := continuous_toFun
  inv_continuous := continuous_invFun
```

-- snippet 31
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.coe_apply (L99-L100)
theorem coe_apply (e : A ≃A[R] B) (a : A) : (e : A →A[R] B) a = e a := rfl
```

-- snippet 32
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.toContinuousLinearEquiv_toLinearEquiv_eq (L144-L147)
theorem toContinuousLinearEquiv_toLinearEquiv_eq (e : A ≃A[R] B) :
    e.toContinuousLinearEquiv.toLinearEquiv
    = e.toAlgEquiv.toLinearEquiv := rfl
```

-- snippet 33
```lean
-- Mathlib.Topology.Algebra.Affine.homothety_isOpenMap (L146-L152)
theorem homothety_isOpenMap (x : P) (t : R) (ht : t ≠ 0) : IsOpenMap <| homothety x t := by
apply IsOpenMap.of_inverse (homothety_continuous x t⁻¹) <;> intro e <;>
    simp [← AffineMap.comp_apply, ← homothety_mul, ht]

end Field

end AffineMap
```

-- snippet 34
```lean
-- Mathlib.Topology.Algebra.Affine.homothety_continuous (L118-L125)
theorem homothety_continuous (x : P) (t : R) : Continuous <| homothety x t := by
rw [coe_homothety]
  fun_prop

variable (R) [TopologicalSpace R] [Module R W] [ContinuousSMul R W] (x : Q) {s : Set Q}

open Topology
```

-- snippet 35
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.coe_toAlgEquiv (L122-L125)
theorem coe_toAlgEquiv (e : A ≃A[R] B) : ⇑e.toAlgEquiv = e := rfl
/-- The natural coercion from a continuous algebra isomorphism
to a continuous linear isomorphism. -/
```

-- snippet 36
```lean
-- Mathlib.Topology.Algebra.Affine.lineMap_continuous_uncurry (L60-L65)
theorem lineMap_continuous_uncurry :
    Continuous (fun pqt : P × P × R ↦ lineMap pqt.1 pqt.2.1 pqt.2.2) := by
simp only [coe_lineMap]
  fun_prop

/-- The line map is continuous. -/
```

-- snippet 37
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.ext (L111-L113)
theorem ext {f g : A ≃A[R] B} (h : ⇑f = ⇑g) : f = g :=
toAlgEquiv_injective <| AlgEquiv.ext <| congr_fun h
```

-- snippet 38
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.toContinuousLinearMap_toContinuousLinearEquiv_eq (L140-L143)
theorem toContinuousLinearMap_toContinuousLinearEquiv_eq (e : A ≃A[R] B) :
    e.toContinuousLinearEquiv.toContinuousLinearMap
    = e.toContinuousAlgHom.toContinuousLinearMap := rfl
```

-- snippet 39
```lean
-- Mathlib.Topology.Algebra.Algebra.Equiv.toContinuousLinearEquiv (L127-L129)
def toContinuousLinearEquiv (e : A ≃A[R] B) : A ≃L[R] B :=
{ e with __ := e.toLinearEquiv }
```

-- snippet 40
```lean
-- Mathlib.Topology.Algebra.Affine.isOpenMap_linear_iff (L46-L58)
theorem isOpenMap_linear_iff {f : P →ᵃ[R] Q} : IsOpenMap f.linear ↔ IsOpenMap f := by
inhabit P
  have :
    (f.linear : V → W) =
      (Homeomorph.vaddConst <| f default).symm ∘ f ∘ (Homeomorph.vaddConst default) := by
    ext v
    simp
  rw [this]
  simp only [Homeomorph.comp_isOpenMap_iff, Homeomorph.comp_isOpenMap_iff']

variable [TopologicalSpace R] [ContinuousSMul R V]

/-- The line map is continuous in all arguments. -/
```
[/INST]


Instruction: Edit `ofLp_bijective` so it still compiles and the proof is simplified where possible. Return the full updated `ofLp_bijective` block.

[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Normed.Lp.WithLp).

[User]
You are editing a file in `Mathlib/Analysis/Normed/Lp/WithLp.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Analysis.Normed.Lp.WithLp.ofLp_bijective`, L116-L118)
```lean
lemma ofLp_bijective : Function.Bijective (@ofLp p V) := ⟨ofLp_injective p, ofLp_surjective p⟩
```

## Edit instruction
Edit `ofLp_bijective` so it still compiles and the proof is simplified where possible. Return the full updated `ofLp_bijective` block.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Normed.Lp.WithLp).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/Analysis/Normed/Lp/WithLp.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.Analysis.Normed.Lp.WithLp.ofLp_bijective`, L116-L118)
```lean
lemma ofLp_bijective : Function.Bijective (@ofLp p V) := ⟨ofLp_injective p, ofLp_surjective p⟩
```

## Edit instruction
Edit `ofLp_bijective` so it still compiles and the proof is simplified where possible. Return the full updated `ofLp_bijective` block.
[/INST]


Instruction: Explain the Lean statement below in plain mathematical language, then formalize a short related example in Lean 4.

[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Algebra.NonUnitalSubalgebra).

[User]
Explain the Lean statement below in plain mathematical language, then formalize a short related example in Lean 4.

```lean
def adjoin (s : Set A) : NonUnitalSubalgebra R A
```

Declaration: `Mathlib.Algebra.Algebra.NonUnitalSubalgebra.adjoin` (module Mathlib.Algebra.Algebra.NonUnitalSubalgebra).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.Algebra.NonUnitalSubalgebra).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Explain the Lean statement below in plain mathematical language, then formalize a short related example in Lean 4.

```lean
def adjoin (s : Set A) : NonUnitalSubalgebra R A
```

Declaration: `Mathlib.Algebra.Algebra.NonUnitalSubalgebra.adjoin` (module Mathlib.Algebra.Algebra.NonUnitalSubalgebra).[/INST]


Instruction: Edit `innerRegularWRT_isCompact_isClosed_measure_ne_top_of_group` so it still compiles and the proof is simplified where possible. Return the full updated `innerRegularWRT_isCompact_isClosed_measure_ne_top_of_group` block.

[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.MeasureTheory.Group.Measure).

[User]
You are editing a file in `Mathlib/MeasureTheory/Group/Measure.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.MeasureTheory.Group.Measure.innerRegularWRT_isCompact_isClosed_measure_ne_top_of_group`, L718-L734)
```lean
lemma innerRegularWRT_isCompact_isClosed_measure_ne_top_of_group [h : InnerRegularCompactLTTop μ] :
    InnerRegularWRT μ (fun s ↦ IsCompact s ∧ IsClosed s) (fun s ↦ MeasurableSet s ∧ μ s ≠ ∞) := by
intro s ⟨s_meas, μs⟩ r hr
  rcases h.innerRegular ⟨s_meas, μs⟩ r hr with ⟨K, Ks, K_comp, hK⟩
  refine ⟨closure K, ?_, ⟨K_comp.closure, isClosed_closure⟩, ?_⟩
  · exact IsCompact.closure_subset_measurableSet K_comp s_meas Ks
  · rwa [K_comp.measure_closure]

end IsTopologicalGroup

section CommSemigroup

variable [CommSemigroup G]

/-- In an abelian group every left invariant measure is also right-invariant.
  We don't declare the converse as an instance, since that would loop type-class inference, and
  we use `IsMulLeftInvariant` as the default hypothesis in abelian groups. -/
```

## Edit instruction
Edit `innerRegularWRT_isCompact_isClosed_measure_ne_top_of_group` so it still compiles and the proof is simplified where possible. Return the full updated `innerRegularWRT_isCompact_isClosed_measure_ne_top_of_group` block.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.MeasureTheory.Group.Measure).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]You are editing a file in `Mathlib/MeasureTheory/Group/Measure.lean` (mathlib commit 5e932f97dd25535344f80f9dd8da3aab83df0fe6).

## Current excerpt (around `Mathlib.MeasureTheory.Group.Measure.innerRegularWRT_isCompact_isClosed_measure_ne_top_of_group`, L718-L734)
```lean
lemma innerRegularWRT_isCompact_isClosed_measure_ne_top_of_group [h : InnerRegularCompactLTTop μ] :
    InnerRegularWRT μ (fun s ↦ IsCompact s ∧ IsClosed s) (fun s ↦ MeasurableSet s ∧ μ s ≠ ∞) := by
intro s ⟨s_meas, μs⟩ r hr
  rcases h.innerRegular ⟨s_meas, μs⟩ r hr with ⟨K, Ks, K_comp, hK⟩
  refine ⟨closure K, ?_, ⟨K_comp.closure, isClosed_closure⟩, ?_⟩
  · exact IsCompact.closure_subset_measurableSet K_comp s_meas Ks
  · rwa [K_comp.measure_closure]

end IsTopologicalGroup

section CommSemigroup

variable [CommSemigroup G]

/-- In an abelian group every left invariant measure is also right-invariant.
  We don't declare the converse as an instance, since that would loop type-class inference, and
  we use `IsMulLeftInvariant` as the default hypothesis in abelian groups. -/
```

## Edit instruction
Edit `innerRegularWRT_isCompact_isClosed_measure_ne_top_of_group` so it still compiles and the proof is simplified where possible. Return the full updated `innerRegularWRT_isCompact_isClosed_measure_ne_top_of_group` block.
[/INST]


Instruction: Explain the Lean statement below in plain mathematical language, then formalize a short related example in Lean 4.

[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Combinatorics.SimpleGraph.ConcreteColorings).

[User]
Explain the Lean statement below in plain mathematical language, then formalize a short related example in Lean 4.

```lean
def Coloring.completeEquipartiteGraph :
  (completeEquipartiteGraph r t).Coloring (Fin r)
```

Declaration: `Mathlib.Combinatorics.SimpleGraph.ConcreteColorings.Coloring` (module Mathlib.Combinatorics.SimpleGraph.ConcreteColorings).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Combinatorics.SimpleGraph.ConcreteColorings).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Explain the Lean statement below in plain mathematical language, then formalize a short related example in Lean 4.

```lean
def Coloring.completeEquipartiteGraph :
  (completeEquipartiteGraph r t).Coloring (Fin r)
```

Declaration: `Mathlib.Combinatorics.SimpleGraph.ConcreteColorings.Coloring` (module Mathlib.Combinatorics.SimpleGraph.ConcreteColorings).[/INST]


Instruction: Explain the Lean statement below in plain mathematical language, then formalize a short related example in Lean 4.

[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.LinearAlgebra.Finsupp.Supported).

[User]
Explain the Lean statement below in plain mathematical language, then formalize a short related example in Lean 4.

```lean
theorem supported_comap_lmapDomain (f : α → α') (s : Set α') :
    supported M R (f ⁻¹' s) ≤ (supported M R s).comap (lmapDomain M R f)
```

Declaration: `Mathlib.LinearAlgebra.Finsupp.Supported.supported_comap_lmapDomain` (module Mathlib.LinearAlgebra.Finsupp.Supported).

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.LinearAlgebra.Finsupp.Supported).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Explain the Lean statement below in plain mathematical language, then formalize a short related example in Lean 4.

```lean
theorem supported_comap_lmapDomain (f : α → α') (s : Set α') :
    supported M R (f ⁻¹' s) ≤ (supported M R s).comap (lmapDomain M R f)
```

Declaration: `Mathlib.LinearAlgebra.Finsupp.Supported.supported_comap_lmapDomain` (module Mathlib.LinearAlgebra.Finsupp.Supported).[/INST]