awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
s_1 s : Seq α
⊢ match (motive := Option (Seq1 α) → Option (Seq1 α) → Prop)
    destruct
      (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.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
have H : 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: rw [corec_eq]
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
s_1 s : Seq α
⊢ 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
have 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
Tactic: dsimp [append]
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
s_1 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)) 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
have 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)) 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_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: cases s
Cleared hypotheses: s
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
s : Seq α
x : α
s_1 : 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)) (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
have 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
Tactic: rw [destruct_cons]
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
s : Seq α
⊢ s.append nil = s
have 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
Tactic: apply coinduction2 s
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
s_1 s : Seq α
⊢ BisimO (fun (s1 s2 : Seq α) ↦ ∃ (s : Seq α), s1 = s.append nil ∧ s2 = s) (s.append nil).destruct s.destruct
have 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)
Tactic: cases s
Cleared hypotheses: s
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
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)
⊢ BisimO (fun (s1 s2 : Seq α) ↦ ∃ (s : Seq α), s1 = s.append nil ∧ s2 = s) (s.append nil).destruct s.destruct
have 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
Tactic: cases s
Cleared hypotheses: s
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
s : Seq α
x : α
s_1 : Seq α
⊢ 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
have 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))
Tactic: rw [cons_append, destruct_cons, destruct_cons]
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
s t u : Seq α
⊢ (s.append t).append u = s.append (t.append u)
have 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)
Tactic: apply eq_of_bisim fun s1 s2 => ∃ s t u, s1 = append (append s t) u ∧ s2 = append s (append t u)
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
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)
⊢ (s.append t).append u = s.append (t.append u)
have 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)
Tactic: apply eq_of_bisim fun s1 s2 => ∃ s t u, s1 = append (append s t) u ∧ s2 = append s (append t u)
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
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
⊢ 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
have 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 nil t).append u).destruct (append nil (t.append u)).destruct
Tactic: cases s
Cleared hypotheses: s
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
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
⊢ 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
have 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))
    (((cons x s).append t).append u).destruct ((cons x s).append (t.append u)).destruct
Tactic: cases s
Cleared hypotheses: s
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
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
⊢ 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
have 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
Tactic: simp
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
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
⊢ 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))
    (((cons x s_1).append t).append u).destruct ((cons x s_1).append (t.append u)).destruct
have 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)
Tactic: simp
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
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
have nil : 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: cases t
Cleared hypotheses: t
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
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
⊢ 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
have cons : ∀ (x : α) (s : Seq α),
  match (motive := Option (Seq1 α) → Option (Seq1 α) → Prop) ((cons x s).append u).destruct,
    ((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
Tactic: cases t
Cleared hypotheses: t
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
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) (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
have 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
Tactic: simp
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
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
⊢ match (motive := Option (Seq1 α) → Option (Seq1 α) → Prop) ((cons x s_1).append u).destruct,
    ((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
have 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)
Tactic: simp
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
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
have nil : 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: cases u
Cleared hypotheses: u
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
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
⊢ 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
have cons : ∀ (x : α) (s : Seq α),
  match (motive := Option (Seq1 α) → Option (Seq1 α) → Prop) (cons x s).destruct, (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
Tactic: cases u
Cleared hypotheses: u
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
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
⊢ match (motive := Option (Seq1 α) → Option (Seq1 α) → Prop) (cons x s_1).destruct, (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
have cons : ∃ (s : Seq α) (t : Seq α) (u : Seq α), s_1 = (s.append t).append u ∧ s_1 = s.append (t.append u)
Tactic: simp
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
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)
have refine_1 : u = append (append nil nil) u
Tactic: refine ⟨nil, nil, u, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
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
⊢ ∃ (s : Seq α) (t : Seq α) (u_2 : Seq α), u = (s.append t).append u_2 ∧ u = s.append (t.append u_2)
have refine_2 : u = append nil (append nil u)
Tactic: refine ⟨nil, nil, u, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
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)
have refine_1 : t.append u = (append nil t).append u
Tactic: refine ⟨nil, t, u, ?_, ?_⟩