l3lab/ntp-mathlib huggingface dataset:
theorem dropLast_prefix : ∀ l : List α, l.dropLast <+: l
  | [] => ⟨nil, by 

Proof state:
α : Type u_1
β : Type u_2
l l₁ l₂ l₃ : List α
a b : α
m n : ℕ
⊢ dropLast [] ++ [] = []

Next tactic:
rw [dropLast, List.append_nil]
theorem tail_suffix (l : List α) : tail l <:+ l := by 

Proof state:
α : Type u_1
β : Type u_2
l✝ l₁ l₂ l₃ : List α
a b : α
m n : ℕ
l : List α
⊢ tail l <:+ l

Next tactic:
rw [← drop_one]
theorem tail_suffix (l : List α) : tail l <:+ l := by rw [← drop_one]; 

Proof state:
α : Type u_1
β : Type u_2
l✝ l₁ l₂ l₃ : List α
a b : α
m n : ℕ
l : List α
⊢ drop 1 l <:+ l

Next tactic:
apply drop_suffix
theorem prefix_iff_eq_append : l₁ <+: l₂ ↔ l₁ ++ drop (length l₁) l₂ = l₂ :=
  ⟨by 

Proof state:
α : Type u_1
β : Type u_2
l l₁ l₂ l₃ : List α
a b : α
m n : ℕ
⊢ l₁ <+: l₂ → l₁ ++ drop (length l₁) l₂ = l₂

Next tactic:
rintro ⟨r, rfl⟩
theorem prefix_iff_eq_append : l₁ <+: l₂ ↔ l₁ ++ drop (length l₁) l₂ = l₂ :=
  ⟨by rintro ⟨r, rfl⟩; 

Proof state:
case intro
α : Type u_1
β : Type u_2
l l₁ l₃ : List α
a b : α
m n : ℕ
r : List α
⊢ l₁ ++ drop (length l₁) (l₁ ++ r) = l₁ ++ r

Next tactic:
rw [drop_left]
theorem suffix_iff_eq_append : l₁ <:+ l₂ ↔ take (length l₂ - length l₁) l₂ ++ l₁ = l₂ :=
  ⟨by 

Proof state:
α : Type u_1
β : Type u_2
l l₁ l₂ l₃ : List α
a b : α
m n : ℕ
⊢ l₁ <:+ l₂ → take (length l₂ - length l₁) l₂ ++ l₁ = l₂

Next tactic:
rintro ⟨r, rfl⟩
theorem suffix_iff_eq_append : l₁ <:+ l₂ ↔ take (length l₂ - length l₁) l₂ ++ l₁ = l₂ :=
  ⟨by rintro ⟨r, rfl⟩; 

Proof state:
case intro
α : Type u_1
β : Type u_2
l l₁ l₃ : List α
a b : α
m n : ℕ
r : List α
⊢ take (length (r ++ l₁) - length l₁) (r ++ l₁) ++ l₁ = r ++ l₁

Next tactic:
simp only [length_append, add_tsub_cancel_right, take_left]
instance decidablePrefix [DecidableEq α] : ∀ l₁ l₂ : List α, Decidable (l₁ <+: l₂)
  | [], l₂ => isTrue ⟨l₂, rfl⟩
  | a :: l₁, [] => isFalse fun ⟨t, te⟩ => List.noConfusion te
  | a :: l₁, b :: l₂ =>
    if h : a = b then
      @decidable_of_decidable_of_iff _ _ (decidablePrefix l₁ l₂) (by 

Proof state:
α : Type u_1
β : Type u_2
l l₁✝ l₂✝ l₃ : List α
a✝ b✝ : α
m n : ℕ
inst✝ : DecidableEq α
a : α
l₁ : List α
b : α
l₂ : List α
h : a = b
⊢ l₁ <+: l₂ ↔ a :: l₁ <+: b :: l₂

Next tactic:
rw [← h, prefix_cons_inj]
instance decidablePrefix [DecidableEq α] : ∀ l₁ l₂ : List α, Decidable (l₁ <+: l₂)
  | [], l₂ => isTrue ⟨l₂, rfl⟩
  | a :: l₁, [] => isFalse fun ⟨t, te⟩ => List.noConfusion te
  | a :: l₁, b :: l₂ =>
    if h : a = b then
      @decidable_of_decidable_of_iff _ _ (decidablePrefix l₁ l₂) (by rw [← h, prefix_cons_inj])
    else
      isFalse fun ⟨t, te⟩ => h <| by 

Proof state:
α : Type u_1
β : Type u_2
l l₁✝ l₂✝ l₃ : List α
a✝ b✝ : α
m n : ℕ
inst✝ : DecidableEq α
a : α
l₁ : List α
b : α
l₂ : List α
h : ¬a = b
x✝ : a :: l₁ <+: b :: l₂
t : List α
te : a :: l₁ ++ t = b :: l₂
⊢ a = b

Next tactic:
injection te
instance decidableSuffix [DecidableEq α] : ∀ l₁ l₂ : List α, Decidable (l₁ <:+ l₂)
  | [], l₂ => isTrue ⟨l₂, append_nil _⟩
  | a :: l₁, [] => isFalse <| mt (Sublist.length_le ∘ IsSuffix.sublist) (by 

Proof state:
α : Type u_1
β : Type u_2
l l₁✝ l₂ l₃ : List α
a✝ b : α
m n : ℕ
inst✝ : DecidableEq α
a : α
l₁ : List α
⊢ ¬length (a :: l₁) ≤ length []

Next tactic:
simp
instance decidableInfix [DecidableEq α] : ∀ l₁ l₂ : List α, Decidable (l₁ <:+: l₂)
  | [], l₂ => isTrue ⟨[], l₂, rfl⟩
  | a :: l₁, [] => isFalse fun ⟨s, t, te⟩ => by 

Proof state:
α : Type u_1
β : Type u_2
l l₁✝ l₂ l₃ : List α
a✝ b : α
m n : ℕ
inst✝ : DecidableEq α
a : α
l₁ : List α
x✝ : a :: l₁ <:+: []
s t : List α
te : s ++ a :: l₁ ++ t = []
⊢ False

Next tactic:
simp at te
theorem prefix_take_le_iff {L : List (List (Option α))} (hm : m < L.length) :
    L.take m <+: L.take n ↔ m ≤ n := by
  

Proof state:
α : Type u_1
β : Type u_2
l l₁ l₂ l₃ : List α
a b : α
m n : ℕ
L : List (List (Option α))
hm : m < length L
⊢ take m L <+: take n L ↔ m ≤ n

Next tactic:
simp only [prefix_iff_eq_take, length_take]
theorem prefix_take_le_iff {L : List (List (Option α))} (hm : m < L.length) :
    L.take m <+: L.take n ↔ m ≤ n := by
  simp only [prefix_iff_eq_take, length_take]
  

Proof state:
α : Type u_1
β : Type u_2
l l₁ l₂ l₃ : List α
a b : α
m n : ℕ
L : List (List (Option α))
hm : m < length L
⊢ take m L = take (min m (length L)) (take n L) ↔ m ≤ n

Next tactic:
induction m generalizing L n with
  | zero => simp [min_eq_left, eq_self_iff_true, Nat.zero_le, take]
  | succ m IH =>
    cases L with
    | nil => exact (not_lt_bot hm).elim
    | cons l ls =>
      cases n with
      | zero =>
        refine' iff_of_false _ (zero_lt_succ _).not_le
        rw [take_zero, take_nil]
        simp only [take, not_false_eq_true]
      | succ n =>
        simp only [length] at hm
        have specializedIH := @IH n ls (Nat.lt_of_succ_lt_succ hm)
        simp only [le_of_lt (Nat.lt_of_succ_lt_succ hm), min_eq_left] at specializedIH
        simp only [take, length, ge_iff_le, le_of_lt hm, min_eq_left, take_cons_succ, cons.injEq,
          specializedIH, true_and]
        exact ⟨Nat.succ_le_succ, Nat.le_of_succ_le_succ⟩
theorem prefix_take_le_iff {L : List (List (Option α))} (hm : m < L.length) :
    L.take m <+: L.take n ↔ m ≤ n := by
  simp only [prefix_iff_eq_take, length_take]
  induction m generalizing L n with
  

Proof state:
case zero
α : Type u_1
β : Type u_2
l l₁ l₂ l₃ : List α
a b : α
n : ℕ
L : List (List (Option α))
hm : zero < length L
⊢ take zero L = take (min zero (length L)) (take n L) ↔ zero ≤ n

Next tactic:
| zero => simp [min_eq_left, eq_self_iff_true, Nat.zero_le, take]
theorem prefix_take_le_iff {L : List (List (Option α))} (hm : m < L.length) :
    L.take m <+: L.take n ↔ m ≤ n := by
  simp only [prefix_iff_eq_take, length_take]
  induction m generalizing L n with
  | zero => 

Proof state:
case zero
α : Type u_1
β : Type u_2
l l₁ l₂ l₃ : List α
a b : α
n : ℕ
L : List (List (Option α))
hm : zero < length L
⊢ take zero L = take (min zero (length L)) (take n L) ↔ zero ≤ n

Next tactic:
simp [min_eq_left, eq_self_iff_true, Nat.zero_le, take]
theorem prefix_take_le_iff {L : List (List (Option α))} (hm : m < L.length) :
    L.take m <+: L.take n ↔ m ≤ n := by
  simp only [prefix_iff_eq_take, length_take]
  induction m generalizing L n with
  | zero => simp [min_eq_left, eq_self_iff_true, Nat.zero_le, take]
  

Proof state:
case succ
α : Type u_1
β : Type u_2
l l₁ l₂ l₃ : List α
a b : α
m : ℕ
IH : ∀ {n : ℕ} {L : List (List (Option α))}, m < length L → (take m L = take (min m (length L)) (take n L) ↔ m ≤ n)
n : ℕ
L : List (List (Option α))
hm : succ m < length L
⊢ take (succ m) L = take (min (succ m) (length L)) (take n L) ↔ succ m ≤ n

Next tactic:
| succ m IH =>
    cases L with
    | nil => exact (not_lt_bot hm).elim
    | cons l ls =>
      cases n with
      | zero =>
        refine' iff_of_false _ (zero_lt_succ _).not_le
        rw [take_zero, take_nil]
        simp only [take, not_false_eq_true]
      | succ n =>
        simp only [length] at hm
        have specializedIH := @IH n ls (Nat.lt_of_succ_lt_succ hm)
        simp only [le_of_lt (Nat.lt_of_succ_lt_succ hm), min_eq_left] at specializedIH
        simp only [take, length, ge_iff_le, le_of_lt hm, min_eq_left, take_cons_succ, cons.injEq,
          specializedIH, true_and]
        exact ⟨Nat.succ_le_succ, Nat.le_of_succ_le_succ⟩
theorem prefix_take_le_iff {L : List (List (Option α))} (hm : m < L.length) :
    L.take m <+: L.take n ↔ m ≤ n := by
  simp only [prefix_iff_eq_take, length_take]
  induction m generalizing L n with
  | zero => simp [min_eq_left, eq_self_iff_true, Nat.zero_le, take]
  | succ m IH =>
    

Proof state:
case succ
α : Type u_1
β : Type u_2
l l₁ l₂ l₃ : List α
a b : α
m : ℕ
IH : ∀ {n : ℕ} {L : List (List (Option α))}, m < length L → (take m L = take (min m (length L)) (take n L) ↔ m ≤ n)
n : ℕ
L : List (List (Option α))
hm : succ m < length L
⊢ take (succ m) L = take (min (succ m) (length L)) (take n L) ↔ succ m ≤ n

Next tactic:
cases L with
    | nil => exact (not_lt_bot hm).elim
    | cons l ls =>
      cases n with
      | zero =>
        refine' iff_of_false _ (zero_lt_succ _).not_le
        rw [take_zero, take_nil]
        simp only [take, not_false_eq_true]
      | succ n =>
        simp only [length] at hm
        have specializedIH := @IH n ls (Nat.lt_of_succ_lt_succ hm)
        simp only [le_of_lt (Nat.lt_of_succ_lt_succ hm), min_eq_left] at specializedIH
        simp only [take, length, ge_iff_le, le_of_lt hm, min_eq_left, take_cons_succ, cons.injEq,
          specializedIH, true_and]
        exact ⟨Nat.succ_le_succ, Nat.le_of_succ_le_succ⟩
theorem prefix_take_le_iff {L : List (List (Option α))} (hm : m < L.length) :
    L.take m <+: L.take n ↔ m ≤ n := by
  simp only [prefix_iff_eq_take, length_take]
  induction m generalizing L n with
  | zero => simp [min_eq_left, eq_self_iff_true, Nat.zero_le, take]
  | succ m IH =>
    cases L with
    

Proof state:
case succ.nil
α : Type u_1
β : Type u_2
l l₁ l₂ l₃ : List α
a b : α
m : ℕ
IH : ∀ {n : ℕ} {L : List (List (Option α))}, m < length L → (take m L = take (min m (length L)) (take n L) ↔ m ≤ n)
n : ℕ
hm : succ m < length []
⊢ take (succ m) [] = take (min (succ m) (length [])) (take n []) ↔ succ m ≤ n

Next tactic:
| nil => exact (not_lt_bot hm).elim
theorem prefix_take_le_iff {L : List (List (Option α))} (hm : m < L.length) :
    L.take m <+: L.take n ↔ m ≤ n := by
  simp only [prefix_iff_eq_take, length_take]
  induction m generalizing L n with
  | zero => simp [min_eq_left, eq_self_iff_true, Nat.zero_le, take]
  | succ m IH =>
    cases L with
    | nil => 

Proof state:
case succ.nil
α : Type u_1
β : Type u_2
l l₁ l₂ l₃ : List α
a b : α
m : ℕ
IH : ∀ {n : ℕ} {L : List (List (Option α))}, m < length L → (take m L = take (min m (length L)) (take n L) ↔ m ≤ n)
n : ℕ
hm : succ m < length []
⊢ take (succ m) [] = take (min (succ m) (length [])) (take n []) ↔ succ m ≤ n

Next tactic:
exact (not_lt_bot hm).elim
theorem prefix_take_le_iff {L : List (List (Option α))} (hm : m < L.length) :
    L.take m <+: L.take n ↔ m ≤ n := by
  simp only [prefix_iff_eq_take, length_take]
  induction m generalizing L n with
  | zero => simp [min_eq_left, eq_self_iff_true, Nat.zero_le, take]
  | succ m IH =>
    cases L with
    | nil => exact (not_lt_bot hm).elim
    

Proof state:
case succ.cons
α : Type u_1
β : Type u_2
l✝ l₁ l₂ l₃ : List α
a b : α
m : ℕ
IH : ∀ {n : ℕ} {L : List (List (Option α))}, m < length L → (take m L = take (min m (length L)) (take n L) ↔ m ≤ n)
n : ℕ
l : List (Option α)
ls : List (List (Option α))
hm : succ m < length (l :: ls)
⊢ take (succ m) (l :: ls) = take (min (succ m) (length (l :: ls))) (take n (l :: ls)) ↔ succ m ≤ n

Next tactic:
| cons l ls =>
      cases n with
      | zero =>
        refine' iff_of_false _ (zero_lt_succ _).not_le
        rw [take_zero, take_nil]
        simp only [take, not_false_eq_true]
      | succ n =>
        simp only [length] at hm
        have specializedIH := @IH n ls (Nat.lt_of_succ_lt_succ hm)
        simp only [le_of_lt (Nat.lt_of_succ_lt_succ hm), min_eq_left] at specializedIH
        simp only [take, length, ge_iff_le, le_of_lt hm, min_eq_left, take_cons_succ, cons.injEq,
          specializedIH, true_and]
        exact ⟨Nat.succ_le_succ, Nat.le_of_succ_le_succ⟩
theorem prefix_take_le_iff {L : List (List (Option α))} (hm : m < L.length) :
    L.take m <+: L.take n ↔ m ≤ n := by
  simp only [prefix_iff_eq_take, length_take]
  induction m generalizing L n with
  | zero => simp [min_eq_left, eq_self_iff_true, Nat.zero_le, take]
  | succ m IH =>
    cases L with
    | nil => exact (not_lt_bot hm).elim
    | cons l ls =>
      

Proof state:
case succ.cons
α : Type u_1
β : Type u_2
l✝ l₁ l₂ l₃ : List α
a b : α
m : ℕ
IH : ∀ {n : ℕ} {L : List (List (Option α))}, m < length L → (take m L = take (min m (length L)) (take n L) ↔ m ≤ n)
n : ℕ
l : List (Option α)
ls : List (List (Option α))
hm : succ m < length (l :: ls)
⊢ take (succ m) (l :: ls) = take (min (succ m) (length (l :: ls))) (take n (l :: ls)) ↔ succ m ≤ n

Next tactic:
cases n with
      | zero =>
        refine' iff_of_false _ (zero_lt_succ _).not_le
        rw [take_zero, take_nil]
        simp only [take, not_false_eq_true]
      | succ n =>
        simp only [length] at hm
        have specializedIH := @IH n ls (Nat.lt_of_succ_lt_succ hm)
        simp only [le_of_lt (Nat.lt_of_succ_lt_succ hm), min_eq_left] at specializedIH
        simp only [take, length, ge_iff_le, le_of_lt hm, min_eq_left, take_cons_succ, cons.injEq,
          specializedIH, true_and]
        exact ⟨Nat.succ_le_succ, Nat.le_of_succ_le_succ⟩
theorem prefix_take_le_iff {L : List (List (Option α))} (hm : m < L.length) :
    L.take m <+: L.take n ↔ m ≤ n := by
  simp only [prefix_iff_eq_take, length_take]
  induction m generalizing L n with
  | zero => simp [min_eq_left, eq_self_iff_true, Nat.zero_le, take]
  | succ m IH =>
    cases L with
    | nil => exact (not_lt_bot hm).elim
    | cons l ls =>
      cases n with
      

Proof state:
case succ.cons.zero
α : Type u_1
β : Type u_2
l✝ l₁ l₂ l₃ : List α
a b : α
m : ℕ
IH : ∀ {n : ℕ} {L : List (List (Option α))}, m < length L → (take m L = take (min m (length L)) (take n L) ↔ m ≤ n)
l : List (Option α)
ls : List (List (Option α))
hm : succ m < length (l :: ls)
⊢ take (succ m) (l :: ls) = take (min (succ m) (length (l :: ls))) (take zero (l :: ls)) ↔ succ m ≤ zero

Next tactic:
| zero =>
        refine' iff_of_false _ (zero_lt_succ _).not_le
        rw [take_zero, take_nil]
        simp only [take, not_false_eq_true]
theorem prefix_take_le_iff {L : List (List (Option α))} (hm : m < L.length) :
    L.take m <+: L.take n ↔ m ≤ n := by
  simp only [prefix_iff_eq_take, length_take]
  induction m generalizing L n with
  | zero => simp [min_eq_left, eq_self_iff_true, Nat.zero_le, take]
  | succ m IH =>
    cases L with
    | nil => exact (not_lt_bot hm).elim
    | cons l ls =>
      cases n with
      | zero =>
        

Proof state:
case succ.cons.zero
α : Type u_1
β : Type u_2
l✝ l₁ l₂ l₃ : List α
a b : α
m : ℕ
IH : ∀ {n : ℕ} {L : List (List (Option α))}, m < length L → (take m L = take (min m (length L)) (take n L) ↔ m ≤ n)
l : List (Option α)
ls : List (List (Option α))
hm : succ m < length (l :: ls)
⊢ take (succ m) (l :: ls) = take (min (succ m) (length (l :: ls))) (take zero (l :: ls)) ↔ succ m ≤ zero

Next tactic:
refine' iff_of_false _ (zero_lt_succ _).not_le
theorem prefix_take_le_iff {L : List (List (Option α))} (hm : m < L.length) :
    L.take m <+: L.take n ↔ m ≤ n := by
  simp only [prefix_iff_eq_take, length_take]
  induction m generalizing L n with
  | zero => simp [min_eq_left, eq_self_iff_true, Nat.zero_le, take]
  | succ m IH =>
    cases L with
    | nil => exact (not_lt_bot hm).elim
    | cons l ls =>
      cases n with
      | zero =>
        refine' iff_of_false _ (zero_lt_succ _).not_le
        

Proof state:
case succ.cons.zero
α : Type u_1
β : Type u_2
l✝ l₁ l₂ l₃ : List α
a b : α
m : ℕ
IH : ∀ {n : ℕ} {L : List (List (Option α))}, m < length L → (take m L = take (min m (length L)) (take n L) ↔ m ≤ n)
l : List (Option α)
ls : List (List (Option α))
hm : succ m < length (l :: ls)
⊢ ¬take (succ m) (l :: ls) = take (min (succ m) (length (l :: ls))) (take zero (l :: ls))

Next tactic:
rw [take_zero, take_nil]
theorem prefix_take_le_iff {L : List (List (Option α))} (hm : m < L.length) :
    L.take m <+: L.take n ↔ m ≤ n := by
  simp only [prefix_iff_eq_take, length_take]
  induction m generalizing L n with
  | zero => simp [min_eq_left, eq_self_iff_true, Nat.zero_le, take]
  | succ m IH =>
    cases L with
    | nil => exact (not_lt_bot hm).elim
    | cons l ls =>
      cases n with
      | zero =>
        refine' iff_of_false _ (zero_lt_succ _).not_le
        rw [take_zero, take_nil]
        

Proof state:
case succ.cons.zero
α : Type u_1
β : Type u_2
l✝ l₁ l₂ l₃ : List α
a b : α
m : ℕ
IH : ∀ {n : ℕ} {L : List (List (Option α))}, m < length L → (take m L = take (min m (length L)) (take n L) ↔ m ≤ n)
l : List (Option α)
ls : List (List (Option α))
hm : succ m < length (l :: ls)
⊢ ¬take (succ m) (l :: ls) = []

Next tactic:
simp only [take, not_false_eq_true]
theorem prefix_take_le_iff {L : List (List (Option α))} (hm : m < L.length) :
    L.take m <+: L.take n ↔ m ≤ n := by
  simp only [prefix_iff_eq_take, length_take]
  induction m generalizing L n with
  | zero => simp [min_eq_left, eq_self_iff_true, Nat.zero_le, take]
  | succ m IH =>
    cases L with
    | nil => exact (not_lt_bot hm).elim
    | cons l ls =>
      cases n with
      | zero =>
        refine' iff_of_false _ (zero_lt_succ _).not_le
        rw [take_zero, take_nil]
        simp only [take, not_false_eq_true]
      

Proof state:
case succ.cons.succ
α : Type u_1
β : Type u_2
l✝ l₁ l₂ l₃ : List α
a b : α
m : ℕ
IH : ∀ {n : ℕ} {L : List (List (Option α))}, m < length L → (take m L = take (min m (length L)) (take n L) ↔ m ≤ n)
l : List (Option α)
ls : List (List (Option α))
hm : succ m < length (l :: ls)
n : ℕ
⊢ take (succ m) (l :: ls) = take (min (succ m) (length (l :: ls))) (take (succ n) (l :: ls)) ↔ succ m ≤ succ n

Next tactic:
| succ n =>
        simp only [length] at hm
        have specializedIH := @IH n ls (Nat.lt_of_succ_lt_succ hm)
        simp only [le_of_lt (Nat.lt_of_succ_lt_succ hm), min_eq_left] at specializedIH
        simp only [take, length, ge_iff_le, le_of_lt hm, min_eq_left, take_cons_succ, cons.injEq,
          specializedIH, true_and]
        exact ⟨Nat.succ_le_succ, Nat.le_of_succ_le_succ⟩
theorem prefix_take_le_iff {L : List (List (Option α))} (hm : m < L.length) :
    L.take m <+: L.take n ↔ m ≤ n := by
  simp only [prefix_iff_eq_take, length_take]
  induction m generalizing L n with
  | zero => simp [min_eq_left, eq_self_iff_true, Nat.zero_le, take]
  | succ m IH =>
    cases L with
    | nil => exact (not_lt_bot hm).elim
    | cons l ls =>
      cases n with
      | zero =>
        refine' iff_of_false _ (zero_lt_succ _).not_le
        rw [take_zero, take_nil]
        simp only [take, not_false_eq_true]
      | succ n =>
        

Proof state:
case succ.cons.succ
α : Type u_1
β : Type u_2
l✝ l₁ l₂ l₃ : List α
a b : α
m : ℕ
IH : ∀ {n : ℕ} {L : List (List (Option α))}, m < length L → (take m L = take (min m (length L)) (take n L) ↔ m ≤ n)
l : List (Option α)
ls : List (List (Option α))
hm : succ m < length (l :: ls)
n : ℕ
⊢ take (succ m) (l :: ls) = take (min (succ m) (length (l :: ls))) (take (succ n) (l :: ls)) ↔ succ m ≤ succ n

Next tactic:
simp only [length] at hm
theorem prefix_take_le_iff {L : List (List (Option α))} (hm : m < L.length) :
    L.take m <+: L.take n ↔ m ≤ n := by
  simp only [prefix_iff_eq_take, length_take]
  induction m generalizing L n with
  | zero => simp [min_eq_left, eq_self_iff_true, Nat.zero_le, take]
  | succ m IH =>
    cases L with
    | nil => exact (not_lt_bot hm).elim
    | cons l ls =>
      cases n with
      | zero =>
        refine' iff_of_false _ (zero_lt_succ _).not_le
        rw [take_zero, take_nil]
        simp only [take, not_false_eq_true]
      | succ n =>
        simp only [length] at hm
        

Proof state:
case succ.cons.succ
α : Type u_1
β : Type u_2
l✝ l₁ l₂ l₃ : List α
a b : α
m : ℕ
IH : ∀ {n : ℕ} {L : List (List (Option α))}, m < length L → (take m L = take (min m (length L)) (take n L) ↔ m ≤ n)
l : List (Option α)
ls : List (List (Option α))
hm : succ m < length ls + 1
n : ℕ
⊢ take (succ m) (l :: ls) = take (min (succ m) (length (l :: ls))) (take (succ n) (l :: ls)) ↔ succ m ≤ succ n

Next tactic:
have specializedIH := @IH n ls (Nat.lt_of_succ_lt_succ hm)
theorem prefix_take_le_iff {L : List (List (Option α))} (hm : m < L.length) :
    L.take m <+: L.take n ↔ m ≤ n := by
  simp only [prefix_iff_eq_take, length_take]
  induction m generalizing L n with
  | zero => simp [min_eq_left, eq_self_iff_true, Nat.zero_le, take]
  | succ m IH =>
    cases L with
    | nil => exact (not_lt_bot hm).elim
    | cons l ls =>
      cases n with
      | zero =>
        refine' iff_of_false _ (zero_lt_succ _).not_le
        rw [take_zero, take_nil]
        simp only [take, not_false_eq_true]
      | succ n =>
        simp only [length] at hm
        have specializedIH := @IH n ls (Nat.lt_of_succ_lt_succ hm)
        

Proof state:
case succ.cons.succ
α : Type u_1
β : Type u_2
l✝ l₁ l₂ l₃ : List α
a b : α
m : ℕ
IH : ∀ {n : ℕ} {L : List (List (Option α))}, m < length L → (take m L = take (min m (length L)) (take n L) ↔ m ≤ n)
l : List (Option α)
ls : List (List (Option α))
hm : succ m < length ls + 1
n : ℕ
specializedIH : take m ls = take (min m (length ls)) (take n ls) ↔ m ≤ n
⊢ take (succ m) (l :: ls) = take (min (succ m) (length (l :: ls))) (take (succ n) (l :: ls)) ↔ succ m ≤ succ n

Next tactic:
simp only [le_of_lt (Nat.lt_of_succ_lt_succ hm), min_eq_left] at specializedIH
theorem prefix_take_le_iff {L : List (List (Option α))} (hm : m < L.length) :
    L.take m <+: L.take n ↔ m ≤ n := by
  simp only [prefix_iff_eq_take, length_take]
  induction m generalizing L n with
  | zero => simp [min_eq_left, eq_self_iff_true, Nat.zero_le, take]
  | succ m IH =>
    cases L with
    | nil => exact (not_lt_bot hm).elim
    | cons l ls =>
      cases n with
      | zero =>
        refine' iff_of_false _ (zero_lt_succ _).not_le
        rw [take_zero, take_nil]
        simp only [take, not_false_eq_true]
      | succ n =>
        simp only [length] at hm
        have specializedIH := @IH n ls (Nat.lt_of_succ_lt_succ hm)
        simp only [le_of_lt (Nat.lt_of_succ_lt_succ hm), min_eq_left] at specializedIH
        

Proof state:
case succ.cons.succ
α : Type u_1
β : Type u_2
l✝ l₁ l₂ l₃ : List α
a b : α
m : ℕ
IH : ∀ {n : ℕ} {L : List (List (Option α))}, m < length L → (take m L = take (min m (length L)) (take n L) ↔ m ≤ n)
l : List (Option α)
ls : List (List (Option α))
hm : succ m < length ls + 1
n : ℕ
specializedIH : take m ls = take m (take n ls) ↔ m ≤ n
⊢ take (succ m) (l :: ls) = take (min (succ m) (length (l :: ls))) (take (succ n) (l :: ls)) ↔ succ m ≤ succ n

Next tactic:
simp only [take, length, ge_iff_le, le_of_lt hm, min_eq_left, take_cons_succ, cons.injEq,
          specializedIH, true_and]
theorem prefix_take_le_iff {L : List (List (Option α))} (hm : m < L.length) :
    L.take m <+: L.take n ↔ m ≤ n := by
  simp only [prefix_iff_eq_take, length_take]
  induction m generalizing L n with
  | zero => simp [min_eq_left, eq_self_iff_true, Nat.zero_le, take]
  | succ m IH =>
    cases L with
    | nil => exact (not_lt_bot hm).elim
    | cons l ls =>
      cases n with
      | zero =>
        refine' iff_of_false _ (zero_lt_succ _).not_le
        rw [take_zero, take_nil]
        simp only [take, not_false_eq_true]
      | succ n =>
        simp only [length] at hm
        have specializedIH := @IH n ls (Nat.lt_of_succ_lt_succ hm)
        simp only [le_of_lt (Nat.lt_of_succ_lt_succ hm), min_eq_left] at specializedIH
        simp only [take, length, ge_iff_le, le_of_lt hm, min_eq_left, take_cons_succ, cons.injEq,
          specializedIH, true_and]
        

Proof state:
case succ.cons.succ
α : Type u_1
β : Type u_2
l✝ l₁ l₂ l₃ : List α
a b : α
m : ℕ
IH : ∀ {n : ℕ} {L : List (List (Option α))}, m < length L → (take m L = take (min m (length L)) (take n L) ↔ m ≤ n)
l : List (Option α)
ls : List (List (Option α))
hm : succ m < length ls + 1
n : ℕ
specializedIH : take m ls = take m (take n ls) ↔ m ≤ n
⊢ m ≤ n ↔ succ m ≤ succ n

Next tactic:
exact ⟨Nat.succ_le_succ, Nat.le_of_succ_le_succ⟩
theorem cons_prefix_iff : a :: l₁ <+: b :: l₂ ↔ a = b ∧ l₁ <+: l₂ := by
  

Proof state:
α : Type u_1
β : Type u_2
l l₁ l₂ l₃ : List α
a b : α
m n : ℕ
⊢ a :: l₁ <+: b :: l₂ ↔ a = b ∧ l₁ <+: l₂

Next tactic:
constructor
theorem cons_prefix_iff : a :: l₁ <+: b :: l₂ ↔ a = b ∧ l₁ <+: l₂ := by
  constructor
  · 

Proof state:
case mp
α : Type u_1
β : Type u_2
l l₁ l₂ l₃ : List α
a b : α
m n : ℕ
⊢ a :: l₁ <+: b :: l₂ → a = b ∧ l₁ <+: l₂

Next tactic:
rintro ⟨L, hL⟩
theorem cons_prefix_iff : a :: l₁ <+: b :: l₂ ↔ a = b ∧ l₁ <+: l₂ := by
  constructor
  · rintro ⟨L, hL⟩
    

Proof state:
case mp.intro
α : Type u_1
β : Type u_2
l l₁ l₂ l₃ : List α
a b : α
m n : ℕ
L : List α
hL : a :: l₁ ++ L = b :: l₂
⊢ a = b ∧ l₁ <+: l₂

Next tactic:
simp only [cons_append] at hL
theorem cons_prefix_iff : a :: l₁ <+: b :: l₂ ↔ a = b ∧ l₁ <+: l₂ := by
  constructor
  · rintro ⟨L, hL⟩
    simp only [cons_append] at hL
    

Proof state:
case mp.intro
α : Type u_1
β : Type u_2
l l₁ l₂ l₃ : List α
a b : α
m n : ℕ
L : List α
hL : a :: (l₁ ++ L) = b :: l₂
⊢ a = b ∧ l₁ <+: l₂

Next tactic:
injection hL with hLLeft hLRight