awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s_1 s : Seq α
H :
  match (motive := Option (Seq1 α) → Option (Seq1 α) → Prop)
    match (motive := Option (α × Seq α × Seq α) → Option (α × Seq α))
      match (motive := Option (α × Seq α) → Option (α × Seq α × Seq α)) s.destruct with
      | none => none
      | some (a, b) => some (a, nil, b) with
    | none => none
    | some (a, b) =>
      some
        (a,
          corec
            (fun (x : Seq α × Seq α) ↦
              match (motive := Option (Seq1 α) → Option (α × Seq α × Seq α)) x.1.destruct with
              | none =>
                match (motive := Option (α × Seq α) → Option (α × Seq α × Seq α)) x.2.destruct with
                | none => none
                | some (a, b) => some (a, nil, b)
              | some (a, s₁') => some (a, s₁', x.2))
            b),
    s.destruct with
  | none, none => True
  | some (a, s), some (a', s') =>
    a = a' ∧
      ∃ (s_2 : Seq α),
        s =
            corec
              (fun (x : Seq α × Seq α) ↦
                match (motive := Option (Seq1 α) → Option (α × Seq α × Seq α)) x.1.destruct with
                | none =>
                  match (motive := Option (α × Seq α) → Option (α × Seq α × Seq α)) x.2.destruct with
                  | none => none
                  | some (a, b) => some (a, nil, b)
                | some (a, s₁') => some (a, s₁', x.2))
              (nil, s_2) ∧
          s' = s_2
  | x, x_1 => False
⊢ match (motive := Option (Seq1 α) → Option (Seq1 α) → Prop)
    omap (β := Seq α × Seq α)
      (corec fun (x : Seq α × Seq α) ↦
        match (motive := Option (Seq1 α) → Option (α × Seq α × Seq α)) x.1.destruct with
        | none =>
          match (motive := Option (α × Seq α) → Option (α × Seq α × Seq α)) x.2.destruct with
          | none => none
          | some (a, b) => some (a, nil, b)
        | some (a, s₁') => some (a, s₁', x.2))
      (match (motive := Option (Seq1 α) → Option (α × Seq α × Seq α)) destruct (nil, s).1 with
      | none =>
        match (motive := Option (α × Seq α) → Option (α × Seq α × Seq α)) (nil, s).2.destruct with
        | none => none
        | some (a, b) => some (a, nil, b)
      | some (a, s₁') => some (a, s₁', (nil, s).2)),
    s.destruct with
  | none, none => True
  | some (a, s), some (a', s') =>
    a = a' ∧
      ∃ (s_2 : Seq α),
        s =
            corec
              (fun (x : Seq α × Seq α) ↦
                match (motive := Option (Seq1 α) → Option (α × Seq α × Seq α)) x.1.destruct with
                | none =>
                  match (motive := Option (α × Seq α) → Option (α × Seq α × Seq α)) x.2.destruct with
                  | none => none
                  | some (a, b) => some (a, nil, b)
                | some (a, s₁') => some (a, s₁', x.2))
              (nil, s_2) ∧
          s' = s_2
  | x, x_1 => False

Tactic:
dsimp [append]

Premises used: Stream'.Seq.append
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s : Seq α
⊢ match (motive := Option (Seq1 α) → Option (Seq1 α) → Prop)
    match (motive := Option (α × Seq α × Seq α) → Option (α × Seq α))
      match (motive := Option (α × Seq α) → Option (α × Seq α × Seq α)) destruct nil with
      | none => none
      | some (a, b) => some (a, nil, b) with
    | none => none
    | some (a, b) =>
      some
        (a,
          corec
            (fun (x : Seq α × Seq α) ↦
              match (motive := Option (Seq1 α) → Option (α × Seq α × Seq α)) x.1.destruct with
              | none =>
                match (motive := Option (α × Seq α) → Option (α × Seq α × Seq α)) x.2.destruct with
                | none => none
                | some (a, b) => some (a, nil, b)
              | some (a, s₁') => some (a, s₁', x.2))
            b),
    destruct nil with
  | none, none => True
  | some (a, s), some (a', s') =>
    a = a' ∧
      ∃ (s_1 : Seq α),
        s =
            corec
              (fun (x : Seq α × Seq α) ↦
                match (motive := Option (Seq1 α) → Option (α × Seq α × Seq α)) x.1.destruct with
                | none =>
                  match (motive := Option (α × Seq α) → Option (α × Seq α × Seq α)) x.2.destruct with
                  | none => none
                  | some (a, b) => some (a, nil, b)
                | some (a, s₁') => some (a, s₁', x.2))
              (nil, s_1) ∧
          s' = s_1
  | x, x_1 => False

Tactic:
trivial
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s : Seq α
x : α
s_1 : Seq α
H :
  match (motive := Option (Seq1 α) → Option (Seq1 α) → Prop)
    match (motive := Option (α × Seq α × Seq α) → Option (α × Seq α))
      match (motive := Option (α × Seq α) → Option (α × Seq α × Seq α)) some (x, s_1) with
      | none => none
      | some (a, b) => some (a, nil, b) with
    | none => none
    | some (a, b) =>
      some
        (a,
          corec
            (fun (x : Seq α × Seq α) ↦
              match (motive := Option (Seq1 α) → Option (α × Seq α × Seq α)) x.1.destruct with
              | none =>
                match (motive := Option (α × Seq α) → Option (α × Seq α × Seq α)) x.2.destruct with
                | none => none
                | some (a, b) => some (a, nil, b)
              | some (a, s₁') => some (a, s₁', x.2))
            b),
    some (x, s_1) with
  | none, none => True
  | some (a, s), some (a', s') =>
    a = a' ∧
      ∃ (s_2 : Seq α),
        s =
            corec
              (fun (x : Seq α × Seq α) ↦
                match (motive := Option (Seq1 α) → Option (α × Seq α × Seq α)) x.1.destruct with
                | none =>
                  match (motive := Option (α × Seq α) → Option (α × Seq α × Seq α)) x.2.destruct with
                  | none => none
                  | some (a, b) => some (a, nil, b)
                | some (a, s₁') => some (a, s₁', x.2))
              (nil, s_2) ∧
          s' = s_2
  | x, x_1 => False
⊢ match (motive := Option (Seq1 α) → Option (Seq1 α) → Prop)
    match (motive := Option (α × Seq α × Seq α) → Option (α × Seq α))
      match (motive := Option (α × Seq α) → Option (α × Seq α × Seq α)) (cons x s_1).destruct with
      | none => none
      | some (a, b) => some (a, nil, b) with
    | none => none
    | some (a, b) =>
      some
        (a,
          corec
            (fun (x : Seq α × Seq α) ↦
              match (motive := Option (Seq1 α) → Option (α × Seq α × Seq α)) x.1.destruct with
              | none =>
                match (motive := Option (α × Seq α) → Option (α × Seq α × Seq α)) x.2.destruct with
                | none => none
                | some (a, b) => some (a, nil, b)
              | some (a, s₁') => some (a, s₁', x.2))
            b),
    (cons x s_1).destruct with
  | none, none => True
  | some (a, s), some (a', s') =>
    a = a' ∧
      ∃ (s_2 : Seq α),
        s =
            corec
              (fun (x : Seq α × Seq α) ↦
                match (motive := Option (Seq1 α) → Option (α × Seq α × Seq α)) x.1.destruct with
                | none =>
                  match (motive := Option (α × Seq α) → Option (α × Seq α × Seq α)) x.2.destruct with
                  | none => none
                  | some (a, b) => some (a, nil, b)
                | some (a, s₁') => some (a, s₁', x.2))
              (nil, s_2) ∧
          s' = s_2
  | x, x_1 => False

Tactic:
rw [destruct_cons]

Premises used: Stream'.Seq.destruct_cons
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s : Seq α
x : α
s_1 : Seq α
⊢ x = x ∧
    ∃ (s : Seq α),
      Eq (α := Seq α)
          (corec
            (fun (x : Seq α × Seq α) ↦
              match (motive := Option (Seq1 α) → Option (α × Seq α × Seq α)) x.1.destruct with
              | none =>
                match (motive := Option (α × Seq α) → Option (α × Seq α × Seq α)) x.2.destruct with
                | none => none
                | some (a, b) => some (a, nil, b)
              | some (a, s₁') => some (a, s₁', x.2))
            (nil, s_1))
          (corec
            (fun (x : Seq α × Seq α) ↦
              match (motive := Option (Seq1 α) → Option (α × Seq α × Seq α)) x.1.destruct with
              | none =>
                match (motive := Option (α × Seq α) → Option (α × Seq α × Seq α)) x.2.destruct with
                | none => none
                | some (a, b) => some (a, nil, b)
              | some (a, s₁') => some (a, s₁', x.2))
            (nil, s)) ∧
        s_1 = s

Tactic:
⟨rfl, _, rfl, rfl⟩

Premises used: And.intro, Exists.intro, rfl
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s : Seq α
x : α
s_1 : Seq α
⊢ x = x ∧
    ∃ (s : Seq α),
      Eq (α := Seq α)
          (corec
            (fun (x : Seq α × Seq α) ↦
              match (motive := Option (Seq1 α) → Option (α × Seq α × Seq α)) x.1.destruct with
              | none =>
                match (motive := Option (α × Seq α) → Option (α × Seq α × Seq α)) x.2.destruct with
                | none => none
                | some (a, b) => some (a, nil, b)
              | some (a, s₁') => some (a, s₁', x.2))
            (nil, s_1))
          (corec
            (fun (x : Seq α × Seq α) ↦
              match (motive := Option (Seq1 α) → Option (α × Seq α × Seq α)) x.1.destruct with
              | none =>
                match (motive := Option (α × Seq α) → Option (α × Seq α × Seq α)) x.2.destruct with
                | none => none
                | some (a, b) => some (a, nil, b)
              | some (a, s₁') => some (a, s₁', x.2))
            (nil, s)) ∧
        s_1 = s

Tactic:
exact ⟨rfl, _, rfl, rfl⟩
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s : Seq α
H :
  ∀ (s : Seq α),
    BisimO (fun (s1 s2 : Seq α) ↦ ∃ (s : Seq α), s1 = s.append nil ∧ s2 = s) (s.append nil).destruct s.destruct
⊢ s.append nil = s

Tactic:
apply coinduction2 s

Premises used: Stream'.Seq.coinduction2
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s_1 s : Seq α
H :
  BisimO.{u} (α := α) (fun (s1 s2 : Seq α) ↦ ∃ (s : Seq α), s1 = s.append nil ∧ s2 = s) (destruct (append nil nil))
    (destruct nil)
H₁ :
  ∀ (x : α) (s : Seq α),
    BisimO (fun (s1 s2 : Seq α) ↦ ∃ (s : Seq α), s1 = s.append nil ∧ s2 = s) ((cons x s).append nil).destruct
      (cons x s).destruct
⊢ BisimO (fun (s1 s2 : Seq α) ↦ ∃ (s : Seq α), s1 = s.append nil ∧ s2 = s) (s.append nil).destruct s.destruct

Tactic:
cases s
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s : Seq α
⊢ BisimO.{u} (α := α) (fun (s1 s2 : Seq α) ↦ ∃ (s : Seq α), s1 = s.append nil ∧ s2 = s) (destruct (append nil nil))
    (destruct nil)

Tactic:
trivial
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s : Seq α
x : α
s_1 : Seq α
H : BisimO (fun (s1 s2 : Seq α) ↦ ∃ (s : Seq α), s1 = s.append nil ∧ s2 = s) (some (x, s_1.append nil)) (some (x, s_1))
⊢ BisimO (fun (s1 s2 : Seq α) ↦ ∃ (s : Seq α), s1 = s.append nil ∧ s2 = s) ((cons x s_1).append nil).destruct
    (cons x s_1).destruct

Tactic:
rw [cons_append, destruct_cons, destruct_cons]

Premises used: Stream'.Seq.destruct_cons, Stream'.Seq.cons_append
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s : Seq α
x : α
s_1 : Seq α
⊢ x = x ∧ ∃ (s : Seq α), s_1.append nil = s.append nil ∧ s_1 = s

Tactic:
⟨rfl, _, rfl, rfl⟩

Premises used: And.intro, Exists.intro, rfl
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s : Seq α
x : α
s_1 : Seq α
⊢ x = x ∧ ∃ (s : Seq α), s_1.append nil = s.append nil ∧ s_1 = s

Tactic:
exact ⟨rfl, _, rfl, rfl⟩
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s t u : Seq α
bisim :
  IsBisimulation.{u} (α := α) fun (s1 s2 : Seq α) ↦
    ∃ (s : Seq α) (t : Seq α) (u : Seq α), s1 = (s.append t).append u ∧ s2 = s.append (t.append u)
r :
  ∃ (s_1 : Seq α) (t_1 : Seq α) (u_1 : Seq α),
    (s.append t).append u = (s_1.append t_1).append u_1 ∧ s.append (t.append u) = s_1.append (t_1.append u_1)
⊢ (s.append t).append u = s.append (t.append u)

Tactic:
apply eq_of_bisim fun s1 s2 => ∃ s t u, s1 = append (append s t) u ∧ s2 = append s (append t u)

Premises used: Stream'.Seq.eq_of_bisim, Exists, And, Stream'.Seq.append, Eq
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s_1 t_1 u_1 s1 s2 : Seq α
h : ∃ (s : Seq α) (t : Seq α) (u : Seq α), s1 = (s.append t).append u ∧ s2 = s.append (t.append u)
s t u : Seq α
nil :
  BisimO
    (fun (s1 s2 : Seq α) ↦
      ∃ (s : Seq α) (t : Seq α) (u : Seq α), s1 = (s.append t).append u ∧ s2 = s.append (t.append u))
    ((append Seq.nil t).append u).destruct (append Seq.nil (t.append u)).destruct
cons :
  ∀ (x : α) (s : Seq α),
    BisimO
      (fun (s1 s2 : Seq α) ↦
        ∃ (s : Seq α) (t : Seq α) (u : Seq α), s1 = (s.append t).append u ∧ s2 = s.append (t.append u))
      (((Seq.cons x s).append t).append u).destruct ((Seq.cons x s).append (t.append u)).destruct
⊢ BisimO
    (fun (s1 s2 : Seq α) ↦
      ∃ (s : Seq α) (t : Seq α) (u : Seq α), s1 = (s.append t).append u ∧ s2 = s.append (t.append u))
    ((s.append t).append u).destruct (s.append (t.append u)).destruct

Tactic:
cases s
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s t_1 u_1 s1 s2 : Seq α
h : ∃ (s : Seq α) (t : Seq α) (u : Seq α), s1 = (s.append t).append u ∧ s2 = s.append (t.append u)
t u : Seq α
simp :
  match (motive := Option (Seq1 α) → Option (Seq1 α) → Prop) (t.append u).destruct, (t.append u).destruct with
  | none, none => True
  | some (a, s), some (a', s') =>
    a = a' ∧ ∃ (s_1 : Seq α) (t : Seq α) (u : Seq α), s = (s_1.append t).append u ∧ s' = s_1.append (t.append u)
  | x, x_1 => False
⊢ BisimO
    (fun (s1 s2 : Seq α) ↦
      ∃ (s : Seq α) (t : Seq α) (u : Seq α), s1 = (s.append t).append u ∧ s2 = s.append (t.append u))
    ((append nil t).append u).destruct (append nil (t.append u)).destruct

Tactic:
simp
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s t_1 u_1 s1 s2 : Seq α
h : ∃ (s : Seq α) (t : Seq α) (u : Seq α), s1 = (s.append t).append u ∧ s2 = s.append (t.append u)
t u : Seq α
x : α
s_1 : Seq α
cons :
  ∃ (s : Seq α) (t_2 : Seq α) (u_2 : Seq α),
    (s_1.append t).append u = (s.append t_2).append u_2 ∧ s_1.append (t.append u) = s.append (t_2.append u_2)
⊢ BisimO
    (fun (s1 s2 : Seq α) ↦
      ∃ (s : Seq α) (t : Seq α) (u : Seq α), s1 = (s.append t).append u ∧ s2 = s.append (t.append u))
    (((Seq.cons x s_1).append t).append u).destruct ((Seq.cons x s_1).append (t.append u)).destruct

Tactic:
simp
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s t_1 u_1 s1 s2 : Seq α
h : ∃ (s : Seq α) (t : Seq α) (u : Seq α), s1 = (s.append t).append u ∧ s2 = s.append (t.append u)
t u : Seq α
nil :
  match (motive := Option (Seq1 α) → Option (Seq1 α) → Prop) (append Seq.nil u).destruct,
    (append Seq.nil u).destruct with
  | none, none => True
  | some (a, s), some (a', s') =>
    a = a' ∧ ∃ (s_1 : Seq α) (t : Seq α) (u : Seq α), s = (s_1.append t).append u ∧ s' = s_1.append (t.append u)
  | x, x_1 => False
cons :
  ∀ (x : α) (s : Seq α),
    match (motive := Option (Seq1 α) → Option (Seq1 α) → Prop) ((Seq.cons x s).append u).destruct,
      ((Seq.cons x s).append u).destruct with
    | none, none => True
    | some (a, s), some (a', s') =>
      a = a' ∧ ∃ (s_1 : Seq α) (t : Seq α) (u : Seq α), s = (s_1.append t).append u ∧ s' = s_1.append (t.append u)
    | x, x_1 => False
⊢ match (motive := Option (Seq1 α) → Option (Seq1 α) → Prop) (t.append u).destruct, (t.append u).destruct with
  | none, none => True
  | some (a, s), some (a', s') =>
    a = a' ∧ ∃ (s_1 : Seq α) (t : Seq α) (u : Seq α), s = (s_1.append t).append u ∧ s' = s_1.append (t.append u)
  | x, x_1 => False

Tactic:
cases t
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s t u_1 s1 s2 : Seq α
h : ∃ (s : Seq α) (t : Seq α) (u : Seq α), s1 = (s.append t).append u ∧ s2 = s.append (t.append u)
u : Seq α
simp :
  match (motive := Option (Seq1 α) → Option (Seq1 α) → Prop) u.destruct, u.destruct with
  | none, none => True
  | some (a, s), some (a', s') =>
    a = a' ∧ ∃ (s_1 : Seq α) (t : Seq α) (u : Seq α), s = (s_1.append t).append u ∧ s' = s_1.append (t.append u)
  | x, x_1 => False
⊢ match (motive := Option (Seq1 α) → Option (Seq1 α) → Prop) (append nil u).destruct, (append nil u).destruct with
  | none, none => True
  | some (a, s), some (a', s') =>
    a = a' ∧ ∃ (s_1 : Seq α) (t : Seq α) (u : Seq α), s = (s_1.append t).append u ∧ s' = s_1.append (t.append u)
  | x, x_1 => False

Tactic:
simp
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s t u_1 s1 s2 : Seq α
h : ∃ (s : Seq α) (t : Seq α) (u : Seq α), s1 = (s.append t).append u ∧ s2 = s.append (t.append u)
u : Seq α
x : α
s_1 : Seq α
cons :
  ∃ (s : Seq α) (t : Seq α) (u_2 : Seq α),
    s_1.append u = (s.append t).append u_2 ∧ s_1.append u = s.append (t.append u_2)
⊢ match (motive := Option (Seq1 α) → Option (Seq1 α) → Prop) ((Seq.cons x s_1).append u).destruct,
    ((Seq.cons x s_1).append u).destruct with
  | none, none => True
  | some (a, s), some (a', s') =>
    a = a' ∧ ∃ (s_2 : Seq α) (t : Seq α) (u : Seq α), s = (s_2.append t).append u ∧ s' = s_2.append (t.append u)
  | x, x_1 => False

Tactic:
simp
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s t u_1 s1 s2 : Seq α
h : ∃ (s : Seq α) (t : Seq α) (u : Seq α), s1 = (s.append t).append u ∧ s2 = s.append (t.append u)
u : Seq α
nil :
  match (motive := Option (Seq1 α) → Option (Seq1 α) → Prop) destruct Seq.nil, destruct Seq.nil with
  | none, none => True
  | some (a, s), some (a', s') =>
    a = a' ∧ ∃ (s_1 : Seq α) (t : Seq α) (u : Seq α), s = (s_1.append t).append u ∧ s' = s_1.append (t.append u)
  | x, x_1 => False
cons :
  ∀ (x : α) (s : Seq α),
    match (motive := Option (Seq1 α) → Option (Seq1 α) → Prop) (Seq.cons x s).destruct, (Seq.cons x s).destruct with
    | none, none => True
    | some (a, s), some (a', s') =>
      a = a' ∧ ∃ (s_1 : Seq α) (t : Seq α) (u : Seq α), s = (s_1.append t).append u ∧ s' = s_1.append (t.append u)
    | x, x_1 => False
⊢ match (motive := Option (Seq1 α) → Option (Seq1 α) → Prop) u.destruct, u.destruct with
  | none, none => True
  | some (a, s), some (a', s') =>
    a = a' ∧ ∃ (s_1 : Seq α) (t : Seq α) (u : Seq α), s = (s_1.append t).append u ∧ s' = s_1.append (t.append u)
  | x, x_1 => False

Tactic:
cases u
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s t u s1 s2 : Seq α
h : ∃ (s : Seq α) (t : Seq α) (u : Seq α), s1 = (s.append t).append u ∧ s2 = s.append (t.append u)
⊢ match (motive := Option (Seq1 α) → Option (Seq1 α) → Prop) destruct nil, destruct nil with
  | none, none => True
  | some (a, s), some (a', s') =>
    a = a' ∧ ∃ (s_1 : Seq α) (t : Seq α) (u : Seq α), s = (s_1.append t).append u ∧ s' = s_1.append (t.append u)
  | x, x_1 => False

Tactic:
simp
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s t u s1 s2 : Seq α
h : ∃ (s : Seq α) (t : Seq α) (u : Seq α), s1 = (s.append t).append u ∧ s2 = s.append (t.append u)
x : α
s_1 : Seq α
cons : ∃ (s : Seq α) (t : Seq α) (u : Seq α), s_1 = (s.append t).append u ∧ s_1 = s.append (t.append u)
⊢ match (motive := Option (Seq1 α) → Option (Seq1 α) → Prop) (Seq.cons x s_1).destruct, (Seq.cons x s_1).destruct with
  | none, none => True
  | some (a, s), some (a', s') =>
    a = a' ∧ ∃ (s_2 : Seq α) (t : Seq α) (u : Seq α), s = (s_2.append t).append u ∧ s' = s_2.append (t.append u)
  | x, x_1 => False

Tactic:
simp
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s t u_1 s1 s2 : Seq α
h : ∃ (s : Seq α) (t : Seq α) (u : Seq α), s1 = (s.append t).append u ∧ s2 = s.append (t.append u)
x : α
u : Seq α
⊢ ∃ (s : Seq α) (t : Seq α) (u_2 : Seq α), u = (s.append t).append u_2 ∧ u = s.append (t.append u_2)

Tactic:
⟨nil, nil, u, ?_, ?_⟩

Premises used: Stream'.Seq.nil, And.intro, Exists.intro
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s t u_1 s1 s2 : Seq α
h : ∃ (s : Seq α) (t : Seq α) (u : Seq α), s1 = (s.append t).append u ∧ s2 = s.append (t.append u)
x : α
u : Seq α
refine_1 : u = append (append nil nil) u
refine_2 : u = append nil (append nil u)
⊢ ∃ (s : Seq α) (t : Seq α) (u_2 : Seq α), u = (s.append t).append u_2 ∧ u = s.append (t.append u_2)

Tactic:
refine ⟨nil, nil, u, ?_, ?_⟩
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s t u_1 s1 s2 : Seq α
h : ∃ (s : Seq α) (t : Seq α) (u : Seq α), s1 = (s.append t).append u ∧ s2 = s.append (t.append u)
x : α
u : Seq α
⊢ u = append (append nil nil) u

Tactic:
simp
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s t u_1 s1 s2 : Seq α
h : ∃ (s : Seq α) (t : Seq α) (u : Seq α), s1 = (s.append t).append u ∧ s2 = s.append (t.append u)
x : α
u : Seq α
⊢ u = append nil (append nil u)

Tactic:
simp
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s t u_1 s1 s2 : Seq α
h : ∃ (s : Seq α) (t : Seq α) (u : Seq α), s1 = (s.append t).append u ∧ s2 = s.append (t.append u)
u : Seq α
⊢ match (motive := Option (Seq1 α) → Option (Seq1 α) → Prop) u.destruct, u.destruct with
  | none, none => True
  | some (a, s), some (a', s') =>
    a = a' ∧ ∃ (s_1 : Seq α) (t : Seq α) (u : Seq α), s = (s_1.append t).append u ∧ s' = s_1.append (t.append u)
  | x, x_1 => False

Tactic:
case nil =>
            cases u <;> simp
            case cons _ u => refine ⟨nil, nil, u, ?_, ?_⟩ <;> simp

Premises used: Stream'.Seq.nil, Stream'.Seq.cons
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s t_1 u_1 s1 s2 : Seq α
h : ∃ (s : Seq α) (t : Seq α) (u : Seq α), s1 = (s.append t).append u ∧ s2 = s.append (t.append u)
u : Seq α
x : α
t : Seq α
⊢ ∃ (s : Seq α) (t_2 : Seq α) (u_2 : Seq α),
    t.append u = (s.append t_2).append u_2 ∧ t.append u = s.append (t_2.append u_2)

Tactic:
⟨nil, t, u, ?_, ?_⟩

Premises used: Stream'.Seq.nil, And.intro, Exists.intro
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s t_1 u_1 s1 s2 : Seq α
h : ∃ (s : Seq α) (t : Seq α) (u : Seq α), s1 = (s.append t).append u ∧ s2 = s.append (t.append u)
u : Seq α
x : α
t : Seq α
refine_1 : t.append u = (append nil t).append u
refine_2 : t.append u = append nil (t.append u)
⊢ ∃ (s : Seq α) (t_2 : Seq α) (u_2 : Seq α),
    t.append u = (s.append t_2).append u_2 ∧ t.append u = s.append (t_2.append u_2)

Tactic:
refine ⟨nil, t, u, ?_, ?_⟩
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s t_1 u_1 s1 s2 : Seq α
h : ∃ (s : Seq α) (t : Seq α) (u : Seq α), s1 = (s.append t).append u ∧ s2 = s.append (t.append u)
u : Seq α
x : α
t : Seq α
⊢ t.append u = (append nil t).append u

Tactic:
simp
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s t_1 u_1 s1 s2 : Seq α
h : ∃ (s : Seq α) (t : Seq α) (u : Seq α), s1 = (s.append t).append u ∧ s2 = s.append (t.append u)
u : Seq α
x : α
t : Seq α
⊢ t.append u = append nil (t.append u)

Tactic:
simp
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s t_1 u_1 s1 s2 : Seq α
h : ∃ (s : Seq α) (t : Seq α) (u : Seq α), s1 = (s.append t).append u ∧ s2 = s.append (t.append u)
t u : Seq α
⊢ match (motive := Option (Seq1 α) → Option (Seq1 α) → Prop) (t.append u).destruct, (t.append u).destruct with
  | none, none => True
  | some (a, s), some (a', s') =>
    a = a' ∧ ∃ (s_1 : Seq α) (t : Seq α) (u : Seq α), s = (s_1.append t).append u ∧ s' = s_1.append (t.append u)
  | x, x_1 => False

Tactic:
case nil =>
          cases t <;> simp
          case nil =>
            cases u <;> simp
            case cons _ u => refine ⟨nil, nil, u, ?_, ?_⟩ <;> simp
          case cons _ t => refine ⟨nil, t, u, ?_, ?_⟩ <;> simp

Premises used: Stream'.Seq.nil, Stream'.Seq.cons
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s_1 t_1 u_1 s1 s2 : Seq α
h : ∃ (s : Seq α) (t : Seq α) (u : Seq α), s1 = (s.append t).append u ∧ s2 = s.append (t.append u)
t u : Seq α
x : α
s : Seq α
⊢ ∃ (s_2 : Seq α) (t_2 : Seq α) (u_2 : Seq α),
    (s.append t).append u = (s_2.append t_2).append u_2 ∧ s.append (t.append u) = s_2.append (t_2.append u_2)

Tactic:
⟨s, t, u, rfl, rfl⟩

Premises used: And.intro, Exists.intro, rfl
-- Module: Mathlib.Data.Seq.Basic
Goal:
α : Type u
s_1 t_1 u_1 s1 s2 : Seq α
h : ∃ (s : Seq α) (t : Seq α) (u : Seq α), s1 = (s.append t).append u ∧ s2 = s.append (t.append u)
t u : Seq α
x : α
s : Seq α
⊢ ∃ (s_2 : Seq α) (t_2 : Seq α) (u_2 : Seq α),
    (s.append t).append u = (s_2.append t_2).append u_2 ∧ s.append (t.append u) = s_2.append (t_2.append u_2)

Tactic:
exact ⟨s, t, u, rfl, rfl⟩