l3lab/ntp-mathlib huggingface dataset:
/-- The cardinals are well-ordered. We express it here by the fact that in any set of cardinals
there is an element that injects into the others.
See `Cardinal.conditionallyCompleteLinearOrderBot` for (one of) the lattice instances. -/
theorem min_injective [I : Nonempty ι] : ∃ i, Nonempty (∀ j, β i ↪ β j) :=
  let ⟨s, hs, ms⟩ :=
    show ∃ s ∈ sets β, ∀ a ∈ sets β, s ⊆ a → a = s from
      zorn_subset (sets β) fun c hc hcc =>
        ⟨⋃₀c, fun x ⟨p, hpc, hxp⟩ y ⟨q, hqc, hyq⟩ i hi =>
          (hcc.total hpc hqc).elim (fun h => hc hqc x (h hxp) y hyq i hi) fun h =>
            hc hpc x hxp y (h hyq) i hi,
          fun _ => subset_sUnion_of_mem⟩
  let ⟨i, e⟩ :=
    show ∃ i, ∀ y, ∃ x ∈ s, (x : ∀ i, β i) i = y from
      Classical.by_contradiction fun h =>
        have h : ∀ i, ∃ y, ∀ x ∈ s, (x : ∀ i, β i) i ≠ y := by
          simpa only [ne_eq, not_exists, not_forall, not_and] using h
        let ⟨f, hf⟩ := Classical.axiom_of_choice h
        have : f ∈ s :=
          have : insert f s ∈ sets β := fun x hx y hy => by
            cases' hx with hx hx <;> cases' hy with hy hy; · simp [hx, hy]
            · subst x
              exact fun i e => (hf i y hy e.symm).elim
            · subst y
              exact fun i e => (hf i x hx e).elim
            · exact hs x hx y hy
          ms _ this (subset_insert f s) ▸ mem_insert _ _
        let ⟨i⟩ := I
        hf i f this rfl
  let ⟨f, hf⟩ := Classical.axiom_of_choice e
  ⟨i,
    ⟨fun j =>
      ⟨fun a => f a j, fun a b e' => by
        

Proof state:
ι : Type u
β : ι → Type v
I : Nonempty ι
s : Set ((i : ι) → β i)
hs : s ∈ Function.Embedding.sets β
ms : ∀ a ∈ Function.Embedding.sets β, s ⊆ a → a = s
i : ι
e : ∀ (y : β i), ∃ x ∈ s, x i = y
f : β i → (i : ι) → β i
hf : ∀ (x : β i), f x ∈ s ∧ f x i = x
j : ι
a b : β i
e' : (fun a => f a j) a = (fun a => f a j) b
⊢ a = b

Next tactic:
let ⟨sa, ea⟩ := hf a
/-- The cardinals are well-ordered. We express it here by the fact that in any set of cardinals
there is an element that injects into the others.
See `Cardinal.conditionallyCompleteLinearOrderBot` for (one of) the lattice instances. -/
theorem min_injective [I : Nonempty ι] : ∃ i, Nonempty (∀ j, β i ↪ β j) :=
  let ⟨s, hs, ms⟩ :=
    show ∃ s ∈ sets β, ∀ a ∈ sets β, s ⊆ a → a = s from
      zorn_subset (sets β) fun c hc hcc =>
        ⟨⋃₀c, fun x ⟨p, hpc, hxp⟩ y ⟨q, hqc, hyq⟩ i hi =>
          (hcc.total hpc hqc).elim (fun h => hc hqc x (h hxp) y hyq i hi) fun h =>
            hc hpc x hxp y (h hyq) i hi,
          fun _ => subset_sUnion_of_mem⟩
  let ⟨i, e⟩ :=
    show ∃ i, ∀ y, ∃ x ∈ s, (x : ∀ i, β i) i = y from
      Classical.by_contradiction fun h =>
        have h : ∀ i, ∃ y, ∀ x ∈ s, (x : ∀ i, β i) i ≠ y := by
          simpa only [ne_eq, not_exists, not_forall, not_and] using h
        let ⟨f, hf⟩ := Classical.axiom_of_choice h
        have : f ∈ s :=
          have : insert f s ∈ sets β := fun x hx y hy => by
            cases' hx with hx hx <;> cases' hy with hy hy; · simp [hx, hy]
            · subst x
              exact fun i e => (hf i y hy e.symm).elim
            · subst y
              exact fun i e => (hf i x hx e).elim
            · exact hs x hx y hy
          ms _ this (subset_insert f s) ▸ mem_insert _ _
        let ⟨i⟩ := I
        hf i f this rfl
  let ⟨f, hf⟩ := Classical.axiom_of_choice e
  ⟨i,
    ⟨fun j =>
      ⟨fun a => f a j, fun a b e' => by
        let ⟨sa, ea⟩ := hf a
        

Proof state:
ι : Type u
β : ι → Type v
I : Nonempty ι
s : Set ((i : ι) → β i)
hs : s ∈ Function.Embedding.sets β
ms : ∀ a ∈ Function.Embedding.sets β, s ⊆ a → a = s
i : ι
e : ∀ (y : β i), ∃ x ∈ s, x i = y
f : β i → (i : ι) → β i
hf : ∀ (x : β i), f x ∈ s ∧ f x i = x
j : ι
a b : β i
e' : (fun a => f a j) a = (fun a => f a j) b
sa : f a ∈ s
ea : f a i = a
⊢ a = b

Next tactic:
let ⟨sb, eb⟩ := hf b
/-- The cardinals are well-ordered. We express it here by the fact that in any set of cardinals
there is an element that injects into the others.
See `Cardinal.conditionallyCompleteLinearOrderBot` for (one of) the lattice instances. -/
theorem min_injective [I : Nonempty ι] : ∃ i, Nonempty (∀ j, β i ↪ β j) :=
  let ⟨s, hs, ms⟩ :=
    show ∃ s ∈ sets β, ∀ a ∈ sets β, s ⊆ a → a = s from
      zorn_subset (sets β) fun c hc hcc =>
        ⟨⋃₀c, fun x ⟨p, hpc, hxp⟩ y ⟨q, hqc, hyq⟩ i hi =>
          (hcc.total hpc hqc).elim (fun h => hc hqc x (h hxp) y hyq i hi) fun h =>
            hc hpc x hxp y (h hyq) i hi,
          fun _ => subset_sUnion_of_mem⟩
  let ⟨i, e⟩ :=
    show ∃ i, ∀ y, ∃ x ∈ s, (x : ∀ i, β i) i = y from
      Classical.by_contradiction fun h =>
        have h : ∀ i, ∃ y, ∀ x ∈ s, (x : ∀ i, β i) i ≠ y := by
          simpa only [ne_eq, not_exists, not_forall, not_and] using h
        let ⟨f, hf⟩ := Classical.axiom_of_choice h
        have : f ∈ s :=
          have : insert f s ∈ sets β := fun x hx y hy => by
            cases' hx with hx hx <;> cases' hy with hy hy; · simp [hx, hy]
            · subst x
              exact fun i e => (hf i y hy e.symm).elim
            · subst y
              exact fun i e => (hf i x hx e).elim
            · exact hs x hx y hy
          ms _ this (subset_insert f s) ▸ mem_insert _ _
        let ⟨i⟩ := I
        hf i f this rfl
  let ⟨f, hf⟩ := Classical.axiom_of_choice e
  ⟨i,
    ⟨fun j =>
      ⟨fun a => f a j, fun a b e' => by
        let ⟨sa, ea⟩ := hf a
        let ⟨sb, eb⟩ := hf b
        

Proof state:
ι : Type u
β : ι → Type v
I : Nonempty ι
s : Set ((i : ι) → β i)
hs : s ∈ Function.Embedding.sets β
ms : ∀ a ∈ Function.Embedding.sets β, s ⊆ a → a = s
i : ι
e : ∀ (y : β i), ∃ x ∈ s, x i = y
f : β i → (i : ι) → β i
hf : ∀ (x : β i), f x ∈ s ∧ f x i = x
j : ι
a b : β i
e' : (fun a => f a j) a = (fun a => f a j) b
sa : f a ∈ s
ea : f a i = a
sb : f b ∈ s
eb : f b i = b
⊢ a = b

Next tactic:
rw [← ea, ← eb, hs _ sa _ sb _ e']
@[simp]
theorem evalFrom_append_singleton (s : σ) (x : List α) (a : α) :
    M.evalFrom s (x ++ [a]) = M.step (M.evalFrom s x) a := by
  

Proof state:
α : Type u
σ : Type v
M : DFA α σ
s : σ
x : List α
a : α
⊢ evalFrom M s (x ++ [a]) = step M (evalFrom M s x) a

Next tactic:
simp only [evalFrom, List.foldl_append, List.foldl_cons, List.foldl_nil]
theorem mem_accepts (x : List α) : x ∈ M.accepts ↔ M.evalFrom M.start x ∈ M.accept := by 

Proof state:
α : Type u
σ : Type v
M : DFA α σ
x : List α
⊢ x ∈ accepts M ↔ evalFrom M M.start x ∈ M.accept

Next tactic:
rfl
theorem evalFrom_split [Fintype σ] {x : List α} {s t : σ} (hlen : Fintype.card σ ≤ x.length)
    (hx : M.evalFrom s x = t) :
    ∃ q a b c,
      x = a ++ b ++ c ∧
        a.length + b.length ≤ Fintype.card σ ∧
          b ≠ [] ∧ M.evalFrom s a = q ∧ M.evalFrom q b = q ∧ M.evalFrom q c = t := by
  

Proof state:
α : Type u
σ : Type v
M : DFA α σ
inst✝ : Fintype σ
x : List α
s t : σ
hlen : Fintype.card σ ≤ List.length x
hx : evalFrom M s x = t
⊢ ∃ q a b c,
    x = a ++ b ++ c ∧
      List.length a + List.length b ≤ Fintype.card σ ∧
        b ≠ [] ∧ evalFrom M s a = q ∧ evalFrom M q b = q ∧ evalFrom M q c = t

Next tactic:
obtain ⟨n, m, hneq, heq⟩ :=
    Fintype.exists_ne_map_eq_of_card_lt
      (fun n : Fin (Fintype.card σ + 1) => M.evalFrom s (x.take n)) (by norm_num)
theorem evalFrom_split [Fintype σ] {x : List α} {s t : σ} (hlen : Fintype.card σ ≤ x.length)
    (hx : M.evalFrom s x = t) :
    ∃ q a b c,
      x = a ++ b ++ c ∧
        a.length + b.length ≤ Fintype.card σ ∧
          b ≠ [] ∧ M.evalFrom s a = q ∧ M.evalFrom q b = q ∧ M.evalFrom q c = t := by
  obtain ⟨n, m, hneq, heq⟩ :=
    Fintype.exists_ne_map_eq_of_card_lt
      (fun n : Fin (Fintype.card σ + 1) => M.evalFrom s (x.take n)) (by 

Proof state:
α : Type u
σ : Type v
M : DFA α σ
inst✝ : Fintype σ
x : List α
s t : σ
hlen : Fintype.card σ ≤ List.length x
hx : evalFrom M s x = t
⊢ Fintype.card σ < Fintype.card (Fin (Fintype.card σ + 1))

Next tactic:
norm_num
theorem evalFrom_split [Fintype σ] {x : List α} {s t : σ} (hlen : Fintype.card σ ≤ x.length)
    (hx : M.evalFrom s x = t) :
    ∃ q a b c,
      x = a ++ b ++ c ∧
        a.length + b.length ≤ Fintype.card σ ∧
          b ≠ [] ∧ M.evalFrom s a = q ∧ M.evalFrom q b = q ∧ M.evalFrom q c = t := by
  obtain ⟨n, m, hneq, heq⟩ :=
    Fintype.exists_ne_map_eq_of_card_lt
      (fun n : Fin (Fintype.card σ + 1) => M.evalFrom s (x.take n)) (by norm_num)
  

Proof state:
case intro.intro.intro
α : Type u
σ : Type v
M : DFA α σ
inst✝ : Fintype σ
x : List α
s t : σ
hlen : Fintype.card σ ≤ List.length x
hx : evalFrom M s x = t
n m : Fin (Fintype.card σ + 1)
hneq : n ≠ m
heq : evalFrom M s (List.take (↑n) x) = evalFrom M s (List.take (↑m) x)
⊢ ∃ q a b c,
    x = a ++ b ++ c ∧
      List.length a + List.length b ≤ Fintype.card σ ∧
        b ≠ [] ∧ evalFrom M s a = q ∧ evalFrom M q b = q ∧ evalFrom M q c = t

Next tactic:
wlog hle : (n : ℕ) ≤ m
theorem evalFrom_split [Fintype σ] {x : List α} {s t : σ} (hlen : Fintype.card σ ≤ x.length)
    (hx : M.evalFrom s x = t) :
    ∃ q a b c,
      x = a ++ b ++ c ∧
        a.length + b.length ≤ Fintype.card σ ∧
          b ≠ [] ∧ M.evalFrom s a = q ∧ M.evalFrom q b = q ∧ M.evalFrom q c = t := by
  obtain ⟨n, m, hneq, heq⟩ :=
    Fintype.exists_ne_map_eq_of_card_lt
      (fun n : Fin (Fintype.card σ + 1) => M.evalFrom s (x.take n)) (by norm_num)
  wlog hle : (n : ℕ) ≤ m
  · 

Proof state:
case intro.intro.intro.inr
α : Type u
σ : Type v
M : DFA α σ
inst✝ : Fintype σ
x : List α
s t : σ
hlen : Fintype.card σ ≤ List.length x
hx : evalFrom M s x = t
n m : Fin (Fintype.card σ + 1)
hneq : n ≠ m
heq : evalFrom M s (List.take (↑n) x) = evalFrom M s (List.take (↑m) x)
this :
  ∀ {α : Type u} {σ : Type v} (M : DFA α σ) [inst : Fintype σ] {x : List α} {s t : σ},
    Fintype.card σ ≤ List.length x →
      evalFrom M s x = t →
        ∀ (n m : Fin (Fintype.card σ + 1)),
          n ≠ m →
            evalFrom M s (List.take (↑n) x) = evalFrom M s (List.take (↑m) x) →
              ↑n ≤ ↑m →
                ∃ q a b c,
                  x = a ++ b ++ c ∧
                    List.length a + List.length b ≤ Fintype.card σ ∧
                      b ≠ [] ∧ evalFrom M s a = q ∧ evalFrom M q b = q ∧ evalFrom M q c = t
hle : ¬↑n ≤ ↑m
⊢ ∃ q a b c,
    x = a ++ b ++ c ∧
      List.length a + List.length b ≤ Fintype.card σ ∧
        b ≠ [] ∧ evalFrom M s a = q ∧ evalFrom M q b = q ∧ evalFrom M q c = t

Next tactic:
exact this _ hlen hx _ _ hneq.symm heq.symm (le_of_not_le hle)
theorem evalFrom_split [Fintype σ] {x : List α} {s t : σ} (hlen : Fintype.card σ ≤ x.length)
    (hx : M.evalFrom s x = t) :
    ∃ q a b c,
      x = a ++ b ++ c ∧
        a.length + b.length ≤ Fintype.card σ ∧
          b ≠ [] ∧ M.evalFrom s a = q ∧ M.evalFrom q b = q ∧ M.evalFrom q c = t := by
  obtain ⟨n, m, hneq, heq⟩ :=
    Fintype.exists_ne_map_eq_of_card_lt
      (fun n : Fin (Fintype.card σ + 1) => M.evalFrom s (x.take n)) (by norm_num)
  wlog hle : (n : ℕ) ≤ m
  · exact this _ hlen hx _ _ hneq.symm heq.symm (le_of_not_le hle)
  

Proof state:
α✝ : Type u
σ✝ : Type v
M✝ : DFA α✝ σ✝
α : Type u
σ : Type v
M : DFA α σ
inst✝ : Fintype σ
x : List α
s t : σ
hlen : Fintype.card σ ≤ List.length x
hx : evalFrom M s x = t
n m : Fin (Fintype.card σ + 1)
hneq : n ≠ m
heq : evalFrom M s (List.take (↑n) x) = evalFrom M s (List.take (↑m) x)
hle : ↑n ≤ ↑m
⊢ ∃ q a b c,
    x = a ++ b ++ c ∧
      List.length a + List.length b ≤ Fintype.card σ ∧
        b ≠ [] ∧ evalFrom M s a = q ∧ evalFrom M q b = q ∧ evalFrom M q c = t

Next tactic:
have hm : (m : ℕ) ≤ Fintype.card σ := Fin.is_le m
theorem evalFrom_split [Fintype σ] {x : List α} {s t : σ} (hlen : Fintype.card σ ≤ x.length)
    (hx : M.evalFrom s x = t) :
    ∃ q a b c,
      x = a ++ b ++ c ∧
        a.length + b.length ≤ Fintype.card σ ∧
          b ≠ [] ∧ M.evalFrom s a = q ∧ M.evalFrom q b = q ∧ M.evalFrom q c = t := by
  obtain ⟨n, m, hneq, heq⟩ :=
    Fintype.exists_ne_map_eq_of_card_lt
      (fun n : Fin (Fintype.card σ + 1) => M.evalFrom s (x.take n)) (by norm_num)
  wlog hle : (n : ℕ) ≤ m
  · exact this _ hlen hx _ _ hneq.symm heq.symm (le_of_not_le hle)
  have hm : (m : ℕ) ≤ Fintype.card σ := Fin.is_le m
  

Proof state:
α✝ : Type u
σ✝ : Type v
M✝ : DFA α✝ σ✝
α : Type u
σ : Type v
M : DFA α σ
inst✝ : Fintype σ
x : List α
s t : σ
hlen : Fintype.card σ ≤ List.length x
hx : evalFrom M s x = t
n m : Fin (Fintype.card σ + 1)
hneq : n ≠ m
heq : evalFrom M s (List.take (↑n) x) = evalFrom M s (List.take (↑m) x)
hle : ↑n ≤ ↑m
hm : ↑m ≤ Fintype.card σ
⊢ ∃ q a b c,
    x = a ++ b ++ c ∧
      List.length a + List.length b ≤ Fintype.card σ ∧
        b ≠ [] ∧ evalFrom M s a = q ∧ evalFrom M q b = q ∧ evalFrom M q c = t

Next tactic:
refine'
    ⟨M.evalFrom s ((x.take m).take n), (x.take m).take n, (x.take m).drop n, x.drop m, _, _, _, by
      rfl, _⟩
theorem evalFrom_split [Fintype σ] {x : List α} {s t : σ} (hlen : Fintype.card σ ≤ x.length)
    (hx : M.evalFrom s x = t) :
    ∃ q a b c,
      x = a ++ b ++ c ∧
        a.length + b.length ≤ Fintype.card σ ∧
          b ≠ [] ∧ M.evalFrom s a = q ∧ M.evalFrom q b = q ∧ M.evalFrom q c = t := by
  obtain ⟨n, m, hneq, heq⟩ :=
    Fintype.exists_ne_map_eq_of_card_lt
      (fun n : Fin (Fintype.card σ + 1) => M.evalFrom s (x.take n)) (by norm_num)
  wlog hle : (n : ℕ) ≤ m
  · exact this _ hlen hx _ _ hneq.symm heq.symm (le_of_not_le hle)
  have hm : (m : ℕ) ≤ Fintype.card σ := Fin.is_le m
  refine'
    ⟨M.evalFrom s ((x.take m).take n), (x.take m).take n, (x.take m).drop n, x.drop m, _, _, _, by
      

Proof state:
α✝ : Type u
σ✝ : Type v
M✝ : DFA α✝ σ✝
α : Type u
σ : Type v
M : DFA α σ
inst✝ : Fintype σ
x : List α
s t : σ
hlen : Fintype.card σ ≤ List.length x
hx : evalFrom M s x = t
n m : Fin (Fintype.card σ + 1)
hneq : n ≠ m
heq : evalFrom M s (List.take (↑n) x) = evalFrom M s (List.take (↑m) x)
hle : ↑n ≤ ↑m
hm : ↑m ≤ Fintype.card σ
⊢ evalFrom M s (List.take (↑n) (List.take (↑m) x)) = evalFrom M s (List.take (↑n) (List.take (↑m) x))

Next tactic:
rfl
theorem evalFrom_split [Fintype σ] {x : List α} {s t : σ} (hlen : Fintype.card σ ≤ x.length)
    (hx : M.evalFrom s x = t) :
    ∃ q a b c,
      x = a ++ b ++ c ∧
        a.length + b.length ≤ Fintype.card σ ∧
          b ≠ [] ∧ M.evalFrom s a = q ∧ M.evalFrom q b = q ∧ M.evalFrom q c = t := by
  obtain ⟨n, m, hneq, heq⟩ :=
    Fintype.exists_ne_map_eq_of_card_lt
      (fun n : Fin (Fintype.card σ + 1) => M.evalFrom s (x.take n)) (by norm_num)
  wlog hle : (n : ℕ) ≤ m
  · exact this _ hlen hx _ _ hneq.symm heq.symm (le_of_not_le hle)
  have hm : (m : ℕ) ≤ Fintype.card σ := Fin.is_le m
  refine'
    ⟨M.evalFrom s ((x.take m).take n), (x.take m).take n, (x.take m).drop n, x.drop m, _, _, _, by
      rfl, _⟩
  · 

Proof state:
case refine'_1
α✝ : Type u
σ✝ : Type v
M✝ : DFA α✝ σ✝
α : Type u
σ : Type v
M : DFA α σ
inst✝ : Fintype σ
x : List α
s t : σ
hlen : Fintype.card σ ≤ List.length x
hx : evalFrom M s x = t
n m : Fin (Fintype.card σ + 1)
hneq : n ≠ m
heq : evalFrom M s (List.take (↑n) x) = evalFrom M s (List.take (↑m) x)
hle : ↑n ≤ ↑m
hm : ↑m ≤ Fintype.card σ
⊢ x = List.take (↑n) (List.take (↑m) x) ++ List.drop (↑n) (List.take (↑m) x) ++ List.drop (↑m) x

Next tactic:
rw [List.take_append_drop, List.take_append_drop]
theorem evalFrom_split [Fintype σ] {x : List α} {s t : σ} (hlen : Fintype.card σ ≤ x.length)
    (hx : M.evalFrom s x = t) :
    ∃ q a b c,
      x = a ++ b ++ c ∧
        a.length + b.length ≤ Fintype.card σ ∧
          b ≠ [] ∧ M.evalFrom s a = q ∧ M.evalFrom q b = q ∧ M.evalFrom q c = t := by
  obtain ⟨n, m, hneq, heq⟩ :=
    Fintype.exists_ne_map_eq_of_card_lt
      (fun n : Fin (Fintype.card σ + 1) => M.evalFrom s (x.take n)) (by norm_num)
  wlog hle : (n : ℕ) ≤ m
  · exact this _ hlen hx _ _ hneq.symm heq.symm (le_of_not_le hle)
  have hm : (m : ℕ) ≤ Fintype.card σ := Fin.is_le m
  refine'
    ⟨M.evalFrom s ((x.take m).take n), (x.take m).take n, (x.take m).drop n, x.drop m, _, _, _, by
      rfl, _⟩
  · rw [List.take_append_drop, List.take_append_drop]
  · 

Proof state:
case refine'_2
α✝ : Type u
σ✝ : Type v
M✝ : DFA α✝ σ✝
α : Type u
σ : Type v
M : DFA α σ
inst✝ : Fintype σ
x : List α
s t : σ
hlen : Fintype.card σ ≤ List.length x
hx : evalFrom M s x = t
n m : Fin (Fintype.card σ + 1)
hneq : n ≠ m
heq : evalFrom M s (List.take (↑n) x) = evalFrom M s (List.take (↑m) x)
hle : ↑n ≤ ↑m
hm : ↑m ≤ Fintype.card σ
⊢ List.length (List.take (↑n) (List.take (↑m) x)) + List.length (List.drop (↑n) (List.take (↑m) x)) ≤ Fintype.card σ

Next tactic:
simp only [List.length_drop, List.length_take]
theorem evalFrom_split [Fintype σ] {x : List α} {s t : σ} (hlen : Fintype.card σ ≤ x.length)
    (hx : M.evalFrom s x = t) :
    ∃ q a b c,
      x = a ++ b ++ c ∧
        a.length + b.length ≤ Fintype.card σ ∧
          b ≠ [] ∧ M.evalFrom s a = q ∧ M.evalFrom q b = q ∧ M.evalFrom q c = t := by
  obtain ⟨n, m, hneq, heq⟩ :=
    Fintype.exists_ne_map_eq_of_card_lt
      (fun n : Fin (Fintype.card σ + 1) => M.evalFrom s (x.take n)) (by norm_num)
  wlog hle : (n : ℕ) ≤ m
  · exact this _ hlen hx _ _ hneq.symm heq.symm (le_of_not_le hle)
  have hm : (m : ℕ) ≤ Fintype.card σ := Fin.is_le m
  refine'
    ⟨M.evalFrom s ((x.take m).take n), (x.take m).take n, (x.take m).drop n, x.drop m, _, _, _, by
      rfl, _⟩
  · rw [List.take_append_drop, List.take_append_drop]
  · simp only [List.length_drop, List.length_take]
    

Proof state:
case refine'_2
α✝ : Type u
σ✝ : Type v
M✝ : DFA α✝ σ✝
α : Type u
σ : Type v
M : DFA α σ
inst✝ : Fintype σ
x : List α
s t : σ
hlen : Fintype.card σ ≤ List.length x
hx : evalFrom M s x = t
n m : Fin (Fintype.card σ + 1)
hneq : n ≠ m
heq : evalFrom M s (List.take (↑n) x) = evalFrom M s (List.take (↑m) x)
hle : ↑n ≤ ↑m
hm : ↑m ≤ Fintype.card σ
⊢ min (↑n) (min (↑m) (List.length x)) + (min (↑m) (List.length x) - ↑n) ≤ Fintype.card σ

Next tactic:
rw [min_eq_left (hm.trans hlen), min_eq_left hle, add_tsub_cancel_of_le hle]
theorem evalFrom_split [Fintype σ] {x : List α} {s t : σ} (hlen : Fintype.card σ ≤ x.length)
    (hx : M.evalFrom s x = t) :
    ∃ q a b c,
      x = a ++ b ++ c ∧
        a.length + b.length ≤ Fintype.card σ ∧
          b ≠ [] ∧ M.evalFrom s a = q ∧ M.evalFrom q b = q ∧ M.evalFrom q c = t := by
  obtain ⟨n, m, hneq, heq⟩ :=
    Fintype.exists_ne_map_eq_of_card_lt
      (fun n : Fin (Fintype.card σ + 1) => M.evalFrom s (x.take n)) (by norm_num)
  wlog hle : (n : ℕ) ≤ m
  · exact this _ hlen hx _ _ hneq.symm heq.symm (le_of_not_le hle)
  have hm : (m : ℕ) ≤ Fintype.card σ := Fin.is_le m
  refine'
    ⟨M.evalFrom s ((x.take m).take n), (x.take m).take n, (x.take m).drop n, x.drop m, _, _, _, by
      rfl, _⟩
  · rw [List.take_append_drop, List.take_append_drop]
  · simp only [List.length_drop, List.length_take]
    rw [min_eq_left (hm.trans hlen), min_eq_left hle, add_tsub_cancel_of_le hle]
    

Proof state:
case refine'_2
α✝ : Type u
σ✝ : Type v
M✝ : DFA α✝ σ✝
α : Type u
σ : Type v
M : DFA α σ
inst✝ : Fintype σ
x : List α
s t : σ
hlen : Fintype.card σ ≤ List.length x
hx : evalFrom M s x = t
n m : Fin (Fintype.card σ + 1)
hneq : n ≠ m
heq : evalFrom M s (List.take (↑n) x) = evalFrom M s (List.take (↑m) x)
hle : ↑n ≤ ↑m
hm : ↑m ≤ Fintype.card σ
⊢ ↑m ≤ Fintype.card σ

Next tactic:
exact hm
theorem evalFrom_split [Fintype σ] {x : List α} {s t : σ} (hlen : Fintype.card σ ≤ x.length)
    (hx : M.evalFrom s x = t) :
    ∃ q a b c,
      x = a ++ b ++ c ∧
        a.length + b.length ≤ Fintype.card σ ∧
          b ≠ [] ∧ M.evalFrom s a = q ∧ M.evalFrom q b = q ∧ M.evalFrom q c = t := by
  obtain ⟨n, m, hneq, heq⟩ :=
    Fintype.exists_ne_map_eq_of_card_lt
      (fun n : Fin (Fintype.card σ + 1) => M.evalFrom s (x.take n)) (by norm_num)
  wlog hle : (n : ℕ) ≤ m
  · exact this _ hlen hx _ _ hneq.symm heq.symm (le_of_not_le hle)
  have hm : (m : ℕ) ≤ Fintype.card σ := Fin.is_le m
  refine'
    ⟨M.evalFrom s ((x.take m).take n), (x.take m).take n, (x.take m).drop n, x.drop m, _, _, _, by
      rfl, _⟩
  · rw [List.take_append_drop, List.take_append_drop]
  · simp only [List.length_drop, List.length_take]
    rw [min_eq_left (hm.trans hlen), min_eq_left hle, add_tsub_cancel_of_le hle]
    exact hm
  · 

Proof state:
case refine'_3
α✝ : Type u
σ✝ : Type v
M✝ : DFA α✝ σ✝
α : Type u
σ : Type v
M : DFA α σ
inst✝ : Fintype σ
x : List α
s t : σ
hlen : Fintype.card σ ≤ List.length x
hx : evalFrom M s x = t
n m : Fin (Fintype.card σ + 1)
hneq : n ≠ m
heq : evalFrom M s (List.take (↑n) x) = evalFrom M s (List.take (↑m) x)
hle : ↑n ≤ ↑m
hm : ↑m ≤ Fintype.card σ
⊢ List.drop (↑n) (List.take (↑m) x) ≠ []

Next tactic:
intro h
theorem evalFrom_split [Fintype σ] {x : List α} {s t : σ} (hlen : Fintype.card σ ≤ x.length)
    (hx : M.evalFrom s x = t) :
    ∃ q a b c,
      x = a ++ b ++ c ∧
        a.length + b.length ≤ Fintype.card σ ∧
          b ≠ [] ∧ M.evalFrom s a = q ∧ M.evalFrom q b = q ∧ M.evalFrom q c = t := by
  obtain ⟨n, m, hneq, heq⟩ :=
    Fintype.exists_ne_map_eq_of_card_lt
      (fun n : Fin (Fintype.card σ + 1) => M.evalFrom s (x.take n)) (by norm_num)
  wlog hle : (n : ℕ) ≤ m
  · exact this _ hlen hx _ _ hneq.symm heq.symm (le_of_not_le hle)
  have hm : (m : ℕ) ≤ Fintype.card σ := Fin.is_le m
  refine'
    ⟨M.evalFrom s ((x.take m).take n), (x.take m).take n, (x.take m).drop n, x.drop m, _, _, _, by
      rfl, _⟩
  · rw [List.take_append_drop, List.take_append_drop]
  · simp only [List.length_drop, List.length_take]
    rw [min_eq_left (hm.trans hlen), min_eq_left hle, add_tsub_cancel_of_le hle]
    exact hm
  · intro h
    

Proof state:
case refine'_3
α✝ : Type u
σ✝ : Type v
M✝ : DFA α✝ σ✝
α : Type u
σ : Type v
M : DFA α σ
inst✝ : Fintype σ
x : List α
s t : σ
hlen : Fintype.card σ ≤ List.length x
hx : evalFrom M s x = t
n m : Fin (Fintype.card σ + 1)
hneq : n ≠ m
heq : evalFrom M s (List.take (↑n) x) = evalFrom M s (List.take (↑m) x)
hle : ↑n ≤ ↑m
hm : ↑m ≤ Fintype.card σ
h : List.drop (↑n) (List.take (↑m) x) = []
⊢ False

Next tactic:
have hlen' := congr_arg List.length h
theorem evalFrom_split [Fintype σ] {x : List α} {s t : σ} (hlen : Fintype.card σ ≤ x.length)
    (hx : M.evalFrom s x = t) :
    ∃ q a b c,
      x = a ++ b ++ c ∧
        a.length + b.length ≤ Fintype.card σ ∧
          b ≠ [] ∧ M.evalFrom s a = q ∧ M.evalFrom q b = q ∧ M.evalFrom q c = t := by
  obtain ⟨n, m, hneq, heq⟩ :=
    Fintype.exists_ne_map_eq_of_card_lt
      (fun n : Fin (Fintype.card σ + 1) => M.evalFrom s (x.take n)) (by norm_num)
  wlog hle : (n : ℕ) ≤ m
  · exact this _ hlen hx _ _ hneq.symm heq.symm (le_of_not_le hle)
  have hm : (m : ℕ) ≤ Fintype.card σ := Fin.is_le m
  refine'
    ⟨M.evalFrom s ((x.take m).take n), (x.take m).take n, (x.take m).drop n, x.drop m, _, _, _, by
      rfl, _⟩
  · rw [List.take_append_drop, List.take_append_drop]
  · simp only [List.length_drop, List.length_take]
    rw [min_eq_left (hm.trans hlen), min_eq_left hle, add_tsub_cancel_of_le hle]
    exact hm
  · intro h
    have hlen' := congr_arg List.length h
    

Proof state:
case refine'_3
α✝ : Type u
σ✝ : Type v
M✝ : DFA α✝ σ✝
α : Type u
σ : Type v
M : DFA α σ
inst✝ : Fintype σ
x : List α
s t : σ
hlen : Fintype.card σ ≤ List.length x
hx : evalFrom M s x = t
n m : Fin (Fintype.card σ + 1)
hneq : n ≠ m
heq : evalFrom M s (List.take (↑n) x) = evalFrom M s (List.take (↑m) x)
hle : ↑n ≤ ↑m
hm : ↑m ≤ Fintype.card σ
h : List.drop (↑n) (List.take (↑m) x) = []
hlen' : List.length (List.drop (↑n) (List.take (↑m) x)) = List.length []
⊢ False

Next tactic:
simp only [List.length_drop, List.length, List.length_take] at hlen'