awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (hypothesis)
Goal:
α : Type u
R : α → α → Prop
inst : IsTrans α R
s : Seq α
motive : Seq α → Prop
base : motive s
step : ∀ (hd : α) (tl : Seq α), motive (Seq.cons hd tl) → (∀ x ∈ tl.head, R hd x) ∧ motive tl
n : ℕ
x y : α
hx : s.get? n = some x
hy : s.get? (n + 1) = some y
⊢ R x y
have hx₁ : (s.drop n).head = some x
Tactic: rw [← head_dropn] at hx
Cleared hypotheses: hx, hy
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
R : α → α → Prop
inst : IsTrans α R
s : Seq α
motive : Seq α → Prop
base : motive s
step : ∀ (hd : α) (tl : Seq α), motive (Seq.cons hd tl) → (∀ x ∈ tl.head, R hd x) ∧ motive tl
n : ℕ
x y : α
hx : (s.drop n).head = some x
hy : s.get? (n + 1) = some y
⊢ R x y
have have : ∀ (this : ∀ (n : ℕ), motive (s.drop n)), R x y
Tactic: have := all_coind_drop_motive motive base (fun hd tl ih ↦ (step hd tl ih).right)
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (hypothesis)
Goal:
α : Type u
R : α → α → Prop
inst : IsTrans α R
s : Seq α
motive : Seq α → Prop
base : motive s
step : ∀ (hd : α) (tl : Seq α), motive (Seq.cons hd tl) → (∀ x ∈ tl.head, R hd x) ∧ motive tl
⊢ Pairwise R s
have h_succ : ∀ {n : ℕ} {x y : α} (hx : s.get? n = some x) (hy : s.get? (n + 1) = some y), R x y
Tactic: have h_succ {n} {x y} (hx : s.get? n = some x) (hy : s.get? (n + 1) = some y) : R x y := by
    rw [← head_dropn] at hx
    have := all_coind_drop_motive motive base (fun hd tl ih ↦ (step hd tl ih).right)
    exact (step x (s.drop (n + 1)) (head_eq_some hx ▸ this n)).left _ (by simpa)
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
R : α → α → Prop
inst : IsTrans α R
s : Seq α
motive : Seq α → Prop
base : motive s
step : ∀ (hd : α) (tl : Seq α), motive (Seq.cons hd tl) → (∀ x ∈ tl.head, R hd x) ∧ motive tl
h_succ : ∀ {n : ℕ} {x y : α} (hx : s.get? n = some x) (hy : s.get? (n + 1) = some y), R x y
i j : ℕ
h_ij : i < j
x : α
hx : x ∈ s.get? i
y : α
hy : y ∈ s.get? j
⊢ R x y
have obtain : ∀ (k : ℕ) (h_ij : i < i + k + 1) (hy : y ∈ s.get? (i + k + 1)), R x y
Tactic: obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_lt h_ij
Cleared hypotheses: j, h_ij, hy
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
R : α → α → Prop
inst : IsTrans α R
s : Seq α
motive : Seq α → Prop
base : motive s
step : ∀ (hd : α) (tl : Seq α), motive (Seq.cons hd tl) → (∀ x ∈ tl.head, R hd x) ∧ motive tl
h_succ : ∀ {n : ℕ} {x y : α} (hx : s.get? n = some x) (hy : s.get? (n + 1) = some y), R x y
i : ℕ
x : α
hx : x ∈ s.get? i
k : ℕ
ih : ∀ (y : α) (hy : y ∈ s.get? (i + k + 1)), R x y
y : α
hy : y ∈ s.get? (i + (k + 1) + 1)
⊢ R x y
have succ : ∀ (z : α) (hz : s.get? (i + k + 1) = some z), R x y
Tactic: obtain ⟨z, hz⟩ := ge_stable (m := i + k + 1) _ (by lia) hy
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
R : α → α → Prop
inst : IsTrans α R
s : Seq α
motive : Seq α → Prop
base : motive s
step : ∀ (hd : α) (tl : Seq α), motive (Seq.cons hd tl) → (∀ x ∈ tl.head, R hd x) ∧ motive tl
h_succ : ∀ {n : ℕ} {x y : α} (hx : s.get? n = some x) (hy : s.get? (n + 1) = some y), R x y
i : ℕ
x : α
hx : x ∈ s.get? i
y : α
k : ℕ
hy : y ∈ s.get? (i + k + 1)
⊢ R x y
have zero : ∀ (y : α) (hy : y ∈ s.get? (i + 0 + 1)), R x y
Tactic: induction k generalizing y with
  | zero => exact h_succ hx hy
  | succ k ih =>
    obtain ⟨z, hz⟩ := ge_stable (m := i + k + 1) _ (by lia) hy
    exact _root_.trans (ih z hz) <| h_succ hz hy
Cleared hypotheses: y, k, hy
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
R : α → α → Prop
inst : IsTrans α R
s : Seq α
motive : Seq α → Prop
base : motive s
step : ∀ (hd : α) (tl : Seq α), motive (Seq.cons hd tl) → (∀ x ∈ tl.head, R hd x) ∧ motive tl
h_succ : ∀ {n : ℕ} {x y : α} (hx : s.get? n = some x) (hy : s.get? (n + 1) = some y), R x y
i : ℕ
x : α
hx : x ∈ s.get? i
y : α
k : ℕ
hy : y ∈ s.get? (i + k + 1)
zero : ∀ (y : α) (hy : y ∈ s.get? (i + 0 + 1)), R x y
⊢ R x y
have succ : ∀ (k : ℕ) (ih : ∀ (y : α) (hy : y ∈ s.get? (i + k + 1)), R x y) (y : α) (hy : y ∈ s.get? (i + (k + 1) + 1)), R x y
Tactic: induction k generalizing y with
  | zero => exact h_succ hx hy
  | succ k ih =>
    obtain ⟨z, hz⟩ := ge_stable (m := i + k + 1) _ (by lia) hy
    exact _root_.trans (ih z hz) <| h_succ hz hy
Cleared hypotheses: y, k, hy
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
R : α → α → Prop
s : Seq α
h : Pairwise R s
⊢ Pairwise R s.tail
have nil : ∀ (h : Pairwise R nil), Pairwise R (tail nil)
Tactic: cases s
Cleared hypotheses: s, h
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
R : α → α → Prop
s : Seq α
h : Pairwise R s
nil : ∀ (h : Pairwise R Seq.nil), Pairwise R (tail Seq.nil)
⊢ Pairwise R s.tail
have cons : ∀ (x : α) (s : Seq α) (h : Pairwise R (cons x s)), Pairwise R (cons x s).tail
Tactic: cases s
Cleared hypotheses: s, h
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
R : α → α → Prop
s : Seq α
h : Pairwise R s
n : ℕ
⊢ Pairwise R (s.drop n)
have zero : Pairwise R (s.drop 0)
Tactic: induction n with
  | zero => simpa
  | succ m ih => simp [drop, Pairwise_tail ih]
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
R : α → α → Prop
s : Seq α
h : Pairwise R s
n : ℕ
zero : Pairwise R (s.drop 0)
⊢ Pairwise R (s.drop n)
have succ : ∀ (m : ℕ) (ih : Pairwise R (s.drop m)), Pairwise R (s.drop (m + 1))
Tactic: induction n with
  | zero => simpa
  | succ m ih => simp [drop, Pairwise_tail ih]
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
β : Type v
a : Seq α
b : Seq β
motive : Seq α → Seq β → Prop
base : motive a b
step :
  ∀ (a : Seq α) (b : Seq β),
    motive a b →
      ∀ (b_hd : β) (b_tl : Seq β),
        b = cons b_hd b_tl → ∃ (a_hd : α) (a_tl : Seq α), a = cons a_hd a_tl ∧ motive a_tl b_tl
m : ℕ
ih : ∀ (hb : b.drop m ≠ nil), motive (a.drop m) (b.drop m)
hb : b.drop (m + 1) ≠ nil
⊢ motive (a.drop (m + 1)) (b.drop (m + 1))
have succ : motive (a.drop m).tail (b.drop m).tail
Tactic: simp only [drop] at hb ⊢
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
β : Type v
a : Seq α
b : Seq β
motive : Seq α → Seq β → Prop
base : motive a b
step :
  ∀ (a : Seq α) (b : Seq β),
    motive a b →
      ∀ (b_hd : β) (b_tl : Seq β),
        b = cons b_hd b_tl → ∃ (a_hd : α) (a_tl : Seq α), a = cons a_hd a_tl ∧ motive a_tl b_tl
m : ℕ
ih : ∀ (hb : b.drop m ≠ nil), motive (a.drop m) (b.drop m)
hb : (b.drop m).tail ≠ nil
⊢ motive (a.drop m).tail (b.drop m).tail
have succ : ∀ (tb : Seq β) (ih : ∀ (hb : tb ≠ nil), motive (a.drop m) tb) (hb : tb.tail ≠ nil), motive (a.drop m).tail tb.tail
Tactic: generalize b.drop m = tb at *
Cleared hypotheses: ih, hb
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (hypothesis)
Goal:
α : Type u
β : Type v
a : Seq α
b : Seq β
motive : Seq α → Seq β → Prop
base : motive a b
step :
  ∀ (a : Seq α) (b : Seq β),
    motive a b →
      ∀ (b_hd : β) (b_tl : Seq β),
        b = cons b_hd b_tl → ∃ (a_hd : α) (a_tl : Seq α), a = cons a_hd a_tl ∧ motive a_tl b_tl
m : ℕ
tb_hd : β
tb_tl : Seq β
ih : ∀ (hb : cons tb_hd tb_tl ≠ nil), motive (a.drop m) (cons tb_hd tb_tl)
hb : (cons tb_hd tb_tl).tail ≠ nil
⊢ motive (a.drop m).tail (cons tb_hd tb_tl).tail
have ih₁ : motive (a.drop m) (cons tb_hd tb_tl)
Tactic: simp only [ne_eq, cons_ne_nil, not_false_eq_true, forall_const] at ih
Cleared hypotheses: ih
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
β : Type v
a : Seq α
b : Seq β
motive : Seq α → Seq β → Prop
base : motive a b
step :
  ∀ (a : Seq α) (b : Seq β),
    motive a b →
      ∀ (b_hd : β) (b_tl : Seq β),
        b = cons b_hd b_tl → ∃ (a_hd : α) (a_tl : Seq α), a = cons a_hd a_tl ∧ motive a_tl b_tl
m : ℕ
tb_hd : β
tb_tl : Seq β
hb : (cons tb_hd tb_tl).tail ≠ nil
ih : motive (a.drop m) (cons tb_hd tb_tl)
⊢ motive (a.drop m).tail (cons tb_hd tb_tl).tail
have succ : ∀ (a_hd : α) (a_tl : Seq α) (ha : a.drop m = cons a_hd a_tl) (h_tail : motive a_tl tb_tl),
  motive (a.drop m).tail (cons tb_hd tb_tl).tail
Tactic: obtain ⟨a_hd, a_tl, ha, h_tail⟩ := step (a.drop m) (.cons tb_hd tb_tl) ih _ _ rfl
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
β : Type v
a : Seq α
b : Seq β
motive : Seq α → Seq β → Prop
base : motive a b
step :
  ∀ (a : Seq α) (b : Seq β),
    motive a b →
      ∀ (b_hd : β) (b_tl : Seq β),
        b = cons b_hd b_tl → ∃ (a_hd : α) (a_tl : Seq α), a = cons a_hd a_tl ∧ motive a_tl b_tl
m : ℕ
tb : Seq β
ih : ∀ (hb : tb ≠ nil), motive (a.drop m) tb
hb : tb.tail ≠ nil
⊢ motive (a.drop m).tail tb.tail
have succ : ∀ (ih : ∀ (hb : Ne.{v + 1} (α := Seq β) nil nil), motive (a.drop m) nil) (hb : Ne.{v + 1} (α := Seq β) (tail nil) nil),
  motive (a.drop m).tail (tail nil)
Tactic: cases tb with
      | nil => simp at hb
      | cons tb_hd tb_tl =>
        simp only [ne_eq, cons_ne_nil, not_false_eq_true, forall_const] at ih
        obtain ⟨a_hd, a_tl, ha, h_tail⟩ := step (a.drop m) (.cons tb_hd tb_tl) ih _ _ rfl
        simpa [ha]
Cleared hypotheses: tb, ih, hb
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
β : Type v
a : Seq α
b : Seq β
motive : Seq α → Seq β → Prop
base : motive a b
step :
  ∀ (a : Seq α) (b : Seq β),
    motive a b →
      ∀ (b_hd : β) (b_tl : Seq β),
        b = cons b_hd b_tl → ∃ (a_hd : α) (a_tl : Seq α), a = cons a_hd a_tl ∧ motive a_tl b_tl
m : ℕ
tb : Seq β
ih : ∀ (hb : tb ≠ nil), motive (a.drop m) tb
hb : tb.tail ≠ nil
succ :
  ∀ (ih : ∀ (hb : Ne.{v + 1} (α := Seq β) nil nil), motive (a.drop m) nil)
    (hb : Ne.{v + 1} (α := Seq β) (tail nil) nil), motive (a.drop m).tail (tail nil)
⊢ motive (a.drop m).tail tb.tail
have succ₁ : ∀ (tb_hd : β) (tb_tl : Seq β) (ih : ∀ (hb : cons tb_hd tb_tl ≠ nil), motive (a.drop m) (cons tb_hd tb_tl))
  (hb : (cons tb_hd tb_tl).tail ≠ nil), motive (a.drop m).tail (cons tb_hd tb_tl).tail
Tactic: cases tb with
      | nil => simp at hb
      | cons tb_hd tb_tl =>
        simp only [ne_eq, cons_ne_nil, not_false_eq_true, forall_const] at ih
        obtain ⟨a_hd, a_tl, ha, h_tail⟩ := step (a.drop m) (.cons tb_hd tb_tl) ih _ _ rfl
        simpa [ha]
Cleared hypotheses: tb, ih, hb
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
β : Type v
a : Seq α
b : Seq β
motive : Seq α → Seq β → Prop
base : motive a b
step :
  ∀ (a : Seq α) (b : Seq β),
    motive a b →
      ∀ (b_hd : β) (b_tl : Seq β),
        b = cons b_hd b_tl → ∃ (a_hd : α) (a_tl : Seq α), a = cons a_hd a_tl ∧ motive a_tl b_tl
n : ℕ
hb : b.drop n ≠ nil
⊢ motive (a.drop n) (b.drop n)
have zero : ∀ (hb : b.drop 0 ≠ nil), motive (a.drop 0) (b.drop 0)
Tactic: induction n with
    | zero => simpa
    | succ m ih =>
      simp only [drop] at hb ⊢
      generalize b.drop m = tb at *
      cases tb with
      | nil => simp at hb
      | cons tb_hd tb_tl =>
        simp only [ne_eq, cons_ne_nil, not_false_eq_true, forall_const] at ih
        obtain ⟨a_hd, a_tl, ha, h_tail⟩ := step (a.drop m) (.cons tb_hd tb_tl) ih _ _ rfl
        simpa [ha]
Cleared hypotheses: n, hb
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
β : Type v
a : Seq α
b : Seq β
motive : Seq α → Seq β → Prop
base : motive a b
step :
  ∀ (a : Seq α) (b : Seq β),
    motive a b →
      ∀ (b_hd : β) (b_tl : Seq β),
        b = cons b_hd b_tl → ∃ (a_hd : α) (a_tl : Seq α), a = cons a_hd a_tl ∧ motive a_tl b_tl
n : ℕ
hb : b.drop n ≠ nil
zero : ∀ (hb : b.drop 0 ≠ nil), motive (a.drop 0) (b.drop 0)
⊢ motive (a.drop n) (b.drop n)
have succ : ∀ (m : ℕ) (ih : ∀ (hb : b.drop m ≠ nil), motive (a.drop m) (b.drop m)) (hb : b.drop (m + 1) ≠ nil),
  motive (a.drop (m + 1)) (b.drop (m + 1))
Tactic: induction n with
    | zero => simpa
    | succ m ih =>
      simp only [drop] at hb ⊢
      generalize b.drop m = tb at *
      cases tb with
      | nil => simp at hb
      | cons tb_hd tb_tl =>
        simp only [ne_eq, cons_ne_nil, not_false_eq_true, forall_const] at ih
        obtain ⟨a_hd, a_tl, ha, h_tail⟩ := step (a.drop m) (.cons tb_hd tb_tl) ih _ _ rfl
        simpa [ha]
Cleared hypotheses: n, hb
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (hypothesis)
Goal:
α : Type u
β : Type v
a : Seq α
b : Seq β
motive : Seq α → Seq β → Prop
base : motive a b
step :
  ∀ (a : Seq α) (b : Seq β),
    motive a b →
      ∀ (b_hd : β) (b_tl : Seq β),
        b = cons b_hd b_tl → ∃ (a_hd : α) (a_tl : Seq α), a = cons a_hd a_tl ∧ motive a_tl b_tl
⊢ b.length' ≤ a.length'
have this : ∀ (n : ℕ) (hb : b.drop n ≠ nil), motive (a.drop n) (b.drop n)
Tactic: have (n) (hb : b.drop n ≠ .nil) : motive (a.drop n) (b.drop n) := by
    induction n with
    | zero => simpa
    | succ m ih =>
      simp only [drop] at hb ⊢
      generalize b.drop m = tb at *
      cases tb with
      | nil => simp at hb
      | cons tb_hd tb_tl =>
        simp only [ne_eq, cons_ne_nil, not_false_eq_true, forall_const] at ih
        obtain ⟨a_hd, a_tl, ha, h_tail⟩ := step (a.drop m) (.cons tb_hd tb_tl) ih _ _ rfl
        simpa [ha]
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
β : Type v
a : Seq α
b : Seq β
motive : Seq α → Seq β → Prop
base : motive a b
step :
  ∀ (a : Seq α) (b : Seq β),
    motive a b →
      ∀ (b_hd : β) (b_tl : Seq β),
        b = cons b_hd b_tl → ∃ (a_hd : α) (a_tl : Seq α), a = cons a_hd a_tl ∧ motive a_tl b_tl
this : ∀ (n : ℕ) (hb : b.drop n ≠ nil), motive (a.drop n) (b.drop n)
⊢ b.length' ≤ a.length'
have pos : ∀ (ha : a.Terminates), b.length' ≤ a.length'
Tactic: by_cases ha : a.Terminates
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
β : Type v
a : Seq α
b : Seq β
motive : Seq α → Seq β → Prop
base : motive a b
step :
  ∀ (a : Seq α) (b : Seq β),
    motive a b →
      ∀ (b_hd : β) (b_tl : Seq β),
        b = cons b_hd b_tl → ∃ (a_hd : α) (a_tl : Seq α), a = cons a_hd a_tl ∧ motive a_tl b_tl
this : ∀ (n : ℕ) (hb : b.drop n ≠ nil), motive (a.drop n) (b.drop n)
pos : ∀ (ha : a.Terminates), b.length' ≤ a.length'
⊢ b.length' ≤ a.length'
have neg : ∀ (ha : ¬a.Terminates), b.length' ≤ a.length'
Tactic: by_cases ha : a.Terminates
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
β : Type v
a : Seq α
b : Seq β
motive : Seq α → Seq β → Prop
base : motive a b
step :
  ∀ (a : Seq α) (b : Seq β),
    motive a b →
      ∀ (b_hd : β) (b_tl : Seq β),
        b = cons b_hd b_tl → ∃ (a_hd : α) (a_tl : Seq α), a = cons a_hd a_tl ∧ motive a_tl b_tl
this : ∀ (n : ℕ) (hb : b.drop n ≠ nil), motive (a.drop n) (b.drop n)
ha : a.Terminates
⊢ b.length' ≤ a.length'
have pos : b.TerminatedAt (a.length ha)
Tactic: simp only [length'_of_terminates ha, length'_le_iff]
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
β : Type v
a : Seq α
b : Seq β
motive : Seq α → Seq β → Prop
base : motive a b
step :
  ∀ (a : Seq α) (b : Seq β),
    motive a b →
      ∀ (b_hd : β) (b_tl : Seq β),
        b = cons b_hd b_tl → ∃ (a_hd : α) (a_tl : Seq α), a = cons a_hd a_tl ∧ motive a_tl b_tl
this : ∀ (n : ℕ) (hb : b.drop n ≠ nil), motive (a.drop n) (b.drop n)
ha : a.Terminates
⊢ b.TerminatedAt (a.length ha)
have pos : ∀ (hb : ¬b.TerminatedAt (a.length ha)), False
Tactic: by_contra hb
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (hypothesis)
Goal:
α : Type u
β : Type v
a : Seq α
b : Seq β
motive : Seq α → Seq β → Prop
base : motive a b
step :
  ∀ (a : Seq α) (b : Seq β),
    motive a b →
      ∀ (b_hd : β) (b_tl : Seq β),
        b = cons b_hd b_tl → ∃ (a_hd : α) (a_tl : Seq α), a = cons a_hd a_tl ∧ motive a_tl b_tl
this : ∀ (n : ℕ) (hb : b.drop n ≠ nil), motive (a.drop n) (b.drop n)
ha : a.Terminates
hb : ¬b.TerminatedAt (a.length ha)
hb' : b.drop (a.length ha) = nil
⊢ False
have hb'₁ : b.TerminatedAt (a.length ha)
Tactic: simp only [← length'_eq_zero_iff_nil, drop_length', tsub_eq_zero_iff_le, length'_le_iff] at hb'
Cleared hypotheses: hb'
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (hypothesis)
Goal:
α : Type u
β : Type v
a : Seq α
b : Seq β
motive : Seq α → Seq β → Prop
base : motive a b
step :
  ∀ (a : Seq α) (b : Seq β),
    motive a b →
      ∀ (b_hd : β) (b_tl : Seq β),
        b = cons b_hd b_tl → ∃ (a_hd : α) (a_tl : Seq α), a = cons a_hd a_tl ∧ motive a_tl b_tl
this : ∀ (n : ℕ) (hb : b.drop n ≠ nil), motive (a.drop n) (b.drop n)
ha : a.Terminates
hb : ¬b.TerminatedAt (a.length ha)
⊢ False
have hb_cons : b.drop (a.length ha) ≠ nil
Tactic: have hb_cons : b.drop (a.length ha) ≠ .nil := by
    intro hb'
    simp only [← length'_eq_zero_iff_nil, drop_length', tsub_eq_zero_iff_le, length'_le_iff] at hb'
    contradiction
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (hypothesis)
Goal:
α : Type u
β : Type v
a : Seq α
b : Seq β
motive : Seq α → Seq β → Prop
base : motive a b
step :
  ∀ (a : Seq α) (b : Seq β),
    motive a b →
      ∀ (b_hd : β) (b_tl : Seq β),
        b = cons b_hd b_tl → ∃ (a_hd : α) (a_tl : Seq α), a = cons a_hd a_tl ∧ motive a_tl b_tl
this : ∀ (n : ℕ) (hb : b.drop n ≠ nil), motive (a.drop n) (b.drop n)
ha : a.Terminates
hb : ¬b.TerminatedAt (a.length ha)
hb_cons : b.drop (a.length ha) ≠ nil
⊢ False
have this₁ : motive (a.drop (a.length ha)) (b.drop (a.length ha))
Tactic: specialize this (a.length ha) hb_cons
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
β : Type v
a : Seq α
b : Seq β
motive : Seq α → Seq β → Prop
base : motive a b
step :
  ∀ (a : Seq α) (b : Seq β),
    motive a b →
      ∀ (b_hd : β) (b_tl : Seq β),
        b = cons b_hd b_tl → ∃ (a_hd : α) (a_tl : Seq α), a = cons a_hd a_tl ∧ motive a_tl b_tl
ha : a.Terminates
hb : ¬b.TerminatedAt (a.length ha)
hb_cons : b.drop (a.length ha) ≠ nil
this : motive (a.drop (a.length ha)) (b.drop (a.length ha))
⊢ False
have pos : ∀ (b' : Seq β) (hb_cons : b' ≠ nil) (this : motive (a.drop (a.length ha)) b'), False
Tactic: generalize b.drop (a.length ha) = b' at *
Cleared hypotheses: hb_cons, this
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
β : Type v
a : Seq α
b : Seq β
motive : Seq α → Seq β → Prop
base : motive a b
step :
  ∀ (a : Seq α) (b : Seq β),
    motive a b →
      ∀ (b_hd : β) (b_tl : Seq β),
        b = cons b_hd b_tl → ∃ (a_hd : α) (a_tl : Seq α), a = cons a_hd a_tl ∧ motive a_tl b_tl
ha : a.Terminates
hb : ¬b.TerminatedAt (a.length ha)
b_hd : β
b_tl : Seq β
hb_cons : cons b_hd b_tl ≠ nil
this : motive (a.drop (a.length ha)) (cons b_hd b_tl)
⊢ False
have pos : ∀ (a_hd : α) (a_tl : Seq α) (ha' : a.drop (a.length ha) = cons a_hd a_tl), motive a_tl b_tl → False
Tactic: obtain ⟨a_hd, a_tl, ha', _⟩ := step _ _ this _ _ rfl
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (hypothesis)
Goal:
α : Type u
β : Type v
a : Seq α
b : Seq β
motive : Seq α → Seq β → Prop
base : motive a b
step :
  ∀ (a : Seq α) (b : Seq β),
    motive a b →
      ∀ (b_hd : β) (b_tl : Seq β),
        b = cons b_hd b_tl → ∃ (a_hd : α) (a_tl : Seq α), a = cons a_hd a_tl ∧ motive a_tl b_tl
ha : a.Terminates
hb : ¬b.TerminatedAt (a.length ha)
b_hd : β
b_tl : Seq β
hb_cons : cons b_hd b_tl ≠ nil
this : motive (a.drop (a.length ha)) (cons b_hd b_tl)
a_hd : α
a_tl : Seq α
ha' : a.drop (a.length ha) = cons a_hd a_tl
right : motive a_tl b_tl
⊢ False
have ha'₁ : (a.drop (a.length ha)).length' = (cons a_hd a_tl).length'
Tactic: apply_fun length' at ha'
Cleared hypotheses: ha'
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (hypothesis)
Goal:
α : Type u
β : Type v
a : Seq α
b : Seq β
motive : Seq α → Seq β → Prop
base : motive a b
step :
  ∀ (a : Seq α) (b : Seq β),
    motive a b →
      ∀ (b_hd : β) (b_tl : Seq β),
        b = cons b_hd b_tl → ∃ (a_hd : α) (a_tl : Seq α), a = cons a_hd a_tl ∧ motive a_tl b_tl
ha : a.Terminates
hb : ¬b.TerminatedAt (a.length ha)
b_hd : β
b_tl : Seq β
hb_cons : cons b_hd b_tl ≠ nil
this : motive (a.drop (a.length ha)) (cons b_hd b_tl)
a_hd : α
a_tl : Seq α
right : motive a_tl b_tl
ha' : (a.drop (a.length ha)).length' = (cons a_hd a_tl).length'
⊢ False
have ha'₁ : 0 = a_tl.length' + 1
Tactic: simp only [drop_length', length'_of_terminates ha, tsub_self, length'_cons] at ha'
Cleared hypotheses: ha'
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
β : Type v
a : Seq α
b : Seq β
motive : Seq α → Seq β → Prop
base : motive a b
step :
  ∀ (a : Seq α) (b : Seq β),
    motive a b →
      ∀ (b_hd : β) (b_tl : Seq β),
        b = cons b_hd b_tl → ∃ (a_hd : α) (a_tl : Seq α), a = cons a_hd a_tl ∧ motive a_tl b_tl
ha : a.Terminates
hb : ¬b.TerminatedAt (a.length ha)
b_hd : β
b_tl : Seq β
hb_cons : cons b_hd b_tl ≠ nil
this : motive (a.drop (a.length ha)) (cons b_hd b_tl)
a_hd : α
a_tl : Seq α
right : motive a_tl b_tl
ha' : 0 = a_tl.length' + 1
⊢ False
have pos : ∀ (u : ℕ∞) (ha' : 0 = u + 1), False
Tactic: generalize a_tl.length' = u at ha'
Cleared hypotheses: ha'
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
β : Type v
a : Seq α
b : Seq β
motive : Seq α → Seq β → Prop
base : motive a b
step :
  ∀ (a : Seq α) (b : Seq β),
    motive a b →
      ∀ (b_hd : β) (b_tl : Seq β),
        b = cons b_hd b_tl → ∃ (a_hd : α) (a_tl : Seq α), a = cons a_hd a_tl ∧ motive a_tl b_tl
ha : a.Terminates
hb : ¬b.TerminatedAt (a.length ha)
b_hd : β
b_tl : Seq β
hb_cons : cons b_hd b_tl ≠ nil
this : motive (a.drop (a.length ha)) (cons b_hd b_tl)
a_hd : α
a_tl : Seq α
right : motive a_tl b_tl
u : ℕ∞
ha' : 0 = u + 1
⊢ False
have pos : ∀ (u : ℕ) (ha' : 0 = u + 1), False
Tactic: enat_to_nat
Cleared hypotheses: u, ha'
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
β : Type v
a : Seq α
b : Seq β
motive : Seq α → Seq β → Prop
base : motive a b
step :
  ∀ (a : Seq α) (b : Seq β),
    motive a b →
      ∀ (b_hd : β) (b_tl : Seq β),
        b = cons b_hd b_tl → ∃ (a_hd : α) (a_tl : Seq α), a = cons a_hd a_tl ∧ motive a_tl b_tl
ha : a.Terminates
hb : ¬b.TerminatedAt (a.length ha)
b' : Seq β
hb_cons : b' ≠ nil
this : motive (a.drop (a.length ha)) b'
⊢ False
have pos : ∀ (hb_cons : Ne.{v + 1} (α := Seq β) nil nil) (this : motive (a.drop (a.length ha)) nil), False
Tactic: cases b' with
  | nil =>
    contradiction
  | cons b_hd b_tl =>
    obtain ⟨a_hd, a_tl, ha', _⟩ := step _ _ this _ _ rfl
    apply_fun length' at ha'
    simp only [drop_length', length'_of_terminates ha, tsub_self, length'_cons] at ha'
    generalize a_tl.length' = u at ha'
    enat_to_nat
    lia
Cleared hypotheses: b', hb_cons, this
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
β : Type v
a : Seq α
b : Seq β
motive : Seq α → Seq β → Prop
base : motive a b
step :
  ∀ (a : Seq α) (b : Seq β),
    motive a b →
      ∀ (b_hd : β) (b_tl : Seq β),
        b = cons b_hd b_tl → ∃ (a_hd : α) (a_tl : Seq α), a = cons a_hd a_tl ∧ motive a_tl b_tl
ha : a.Terminates
hb : ¬b.TerminatedAt (a.length ha)
b' : Seq β
hb_cons : b' ≠ nil
this : motive (a.drop (a.length ha)) b'
pos : ∀ (hb_cons : Ne.{v + 1} (α := Seq β) nil nil) (this : motive (a.drop (a.length ha)) nil), False
⊢ False
have pos₁ : ∀ (b_hd : β) (b_tl : Seq β) (hb_cons : cons b_hd b_tl ≠ nil) (this : motive (a.drop (a.length ha)) (cons b_hd b_tl)),
  False
Tactic: cases b' with
  | nil =>
    contradiction
  | cons b_hd b_tl =>
    obtain ⟨a_hd, a_tl, ha', _⟩ := step _ _ this _ _ rfl
    apply_fun length' at ha'
    simp only [drop_length', length'_of_terminates ha, tsub_self, length'_cons] at ha'
    generalize a_tl.length' = u at ha'
    enat_to_nat
    lia
Cleared hypotheses: b', hb_cons, this
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
a b : α
s : Seq α
S : Seq (Seq1 α)
⊢ (match (motive := Option (Seq1 α) → Seq1 α) (Seq.cons b s).destruct with
    | none => (a, S.join)
    | some s' => (a, (Seq.cons s' S).join)) =
    (a, (Seq.cons (b, s) S).join)
have rw : (match (motive := Option (Seq1 α) → Seq1 α) some (b, s) with
  | none => (a, S.join)
  | some s' => (a, (Seq.cons s' S).join)) =
  (a, (Seq.cons (b, s) S).join)
Tactic: rw [destruct_cons]
-- Lean 4 proof step from Mathlib.Data.Seq.Basic (goal)
Goal:
α : Type u
s : Seq α
⊢ (Seq.map ret s).join = s
have H : ∀ (s : Seq α),
  BisimO (fun (s1 s2 : Seq α) ↦ ∃ (s : Seq α), s1 = (Seq.map ret s).join ∧ s2 = s) (Seq.map ret s).join.destruct
    s.destruct
Tactic: apply coinduction2 s