l3lab/ntp-mathlib huggingface dataset:
theorem star_rmatch_iff (P : RegularExpression α) :
    ∀ x : List α, (star P).rmatch x ↔ ∃ S : List (List α), x
          = S.join ∧ ∀ t ∈ S, t ≠ [] ∧ P.rmatch t :=
  fun x => by
    have A : ∀ m n : ℕ, n < m + n + 1 := by
      intro m n
      convert add_lt_add_of_le_of_lt (add_le_add (zero_le m) (le_refl n)) zero_lt_one
      simp
    have IH := fun t (_h : List.length t < List.length x) => star_rmatch_iff P t
    clear star_rmatch_iff
    constructor
    · cases' x with a x
      · intro _h
        use []; dsimp; tauto
      · rw [rmatch, deriv, mul_rmatch_iff]
        rintro ⟨t, u, hs, ht, hu⟩
        have hwf : u.length < (List.cons a x).length := by
          rw [hs, List.length_cons, List.length_append]
          apply A
        rw [IH _ hwf] at hu
        rcases hu with ⟨S', hsum, helem⟩
        use (a :: t) :: S'
        constructor
        · simp [hs, hsum]
        · intro t' ht'
          cases ht'
          case head ht' =>
            simp only [ne_eq, not_false_iff, true_and, rmatch]
            exact ht
          

Proof state:
case h.right.tail
α : Type u_1
β : Type u_2
γ : Type u_3
dec : DecidableEq α
a✝¹ b : α
P : RegularExpression α
A : ∀ (m n : ℕ), n < m + n + 1
a : α
x : List α
IH :
  ∀ (t : List α),
    length t < length (a :: x) → (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ t ∈ S, t ≠ [] ∧ rmatch P t = true)
t u : List α
hs : x = t ++ u
ht : rmatch (deriv P a) t = true
hwf : length u < length (a :: x)
S' : List (List α)
hsum : u = join S'
helem : ∀ t ∈ S', t ≠ [] ∧ rmatch P t = true
t' : List α
a✝ : Mem t' S'
⊢ t' ≠ [] ∧ rmatch P t' = true

Next tactic:
case tail ht' => exact helem t' ht'
theorem star_rmatch_iff (P : RegularExpression α) :
    ∀ x : List α, (star P).rmatch x ↔ ∃ S : List (List α), x
          = S.join ∧ ∀ t ∈ S, t ≠ [] ∧ P.rmatch t :=
  fun x => by
    have A : ∀ m n : ℕ, n < m + n + 1 := by
      intro m n
      convert add_lt_add_of_le_of_lt (add_le_add (zero_le m) (le_refl n)) zero_lt_one
      simp
    have IH := fun t (_h : List.length t < List.length x) => star_rmatch_iff P t
    clear star_rmatch_iff
    constructor
    · cases' x with a x
      · intro _h
        use []; dsimp; tauto
      · rw [rmatch, deriv, mul_rmatch_iff]
        rintro ⟨t, u, hs, ht, hu⟩
        have hwf : u.length < (List.cons a x).length := by
          rw [hs, List.length_cons, List.length_append]
          apply A
        rw [IH _ hwf] at hu
        rcases hu with ⟨S', hsum, helem⟩
        use (a :: t) :: S'
        constructor
        · simp [hs, hsum]
        · intro t' ht'
          cases ht'
          case head ht' =>
            simp only [ne_eq, not_false_iff, true_and, rmatch]
            exact ht
          

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
dec : DecidableEq α
a✝ b : α
P : RegularExpression α
A : ∀ (m n : ℕ), n < m + n + 1
a : α
x : List α
IH :
  ∀ (t : List α),
    length t < length (a :: x) → (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ t ∈ S, t ≠ [] ∧ rmatch P t = true)
t u : List α
hs : x = t ++ u
ht : rmatch (deriv P a) t = true
hwf : length u < length (a :: x)
S' : List (List α)
hsum : u = join S'
helem : ∀ t ∈ S', t ≠ [] ∧ rmatch P t = true
t' : List α
ht' : Mem t' S'
⊢ t' ≠ [] ∧ rmatch P t' = true

Next tactic:
case tail ht' => exact helem t' ht'
theorem star_rmatch_iff (P : RegularExpression α) :
    ∀ x : List α, (star P).rmatch x ↔ ∃ S : List (List α), x
          = S.join ∧ ∀ t ∈ S, t ≠ [] ∧ P.rmatch t :=
  fun x => by
    have A : ∀ m n : ℕ, n < m + n + 1 := by
      intro m n
      convert add_lt_add_of_le_of_lt (add_le_add (zero_le m) (le_refl n)) zero_lt_one
      simp
    have IH := fun t (_h : List.length t < List.length x) => star_rmatch_iff P t
    clear star_rmatch_iff
    constructor
    · cases' x with a x
      · intro _h
        use []; dsimp; tauto
      · rw [rmatch, deriv, mul_rmatch_iff]
        rintro ⟨t, u, hs, ht, hu⟩
        have hwf : u.length < (List.cons a x).length := by
          rw [hs, List.length_cons, List.length_append]
          apply A
        rw [IH _ hwf] at hu
        rcases hu with ⟨S', hsum, helem⟩
        use (a :: t) :: S'
        constructor
        · simp [hs, hsum]
        · intro t' ht'
          cases ht'
          case head ht' =>
            simp only [ne_eq, not_false_iff, true_and, rmatch]
            exact ht
          case tail ht' => 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
dec : DecidableEq α
a✝ b : α
P : RegularExpression α
A : ∀ (m n : ℕ), n < m + n + 1
a : α
x : List α
IH :
  ∀ (t : List α),
    length t < length (a :: x) → (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ t ∈ S, t ≠ [] ∧ rmatch P t = true)
t u : List α
hs : x = t ++ u
ht : rmatch (deriv P a) t = true
hwf : length u < length (a :: x)
S' : List (List α)
hsum : u = join S'
helem : ∀ t ∈ S', t ≠ [] ∧ rmatch P t = true
t' : List α
ht' : Mem t' S'
⊢ t' ≠ [] ∧ rmatch P t' = true

Next tactic:
exact helem t' ht'
theorem star_rmatch_iff (P : RegularExpression α) :
    ∀ x : List α, (star P).rmatch x ↔ ∃ S : List (List α), x
          = S.join ∧ ∀ t ∈ S, t ≠ [] ∧ P.rmatch t :=
  fun x => by
    have A : ∀ m n : ℕ, n < m + n + 1 := by
      intro m n
      convert add_lt_add_of_le_of_lt (add_le_add (zero_le m) (le_refl n)) zero_lt_one
      simp
    have IH := fun t (_h : List.length t < List.length x) => star_rmatch_iff P t
    clear star_rmatch_iff
    constructor
    · cases' x with a x
      · intro _h
        use []; dsimp; tauto
      · rw [rmatch, deriv, mul_rmatch_iff]
        rintro ⟨t, u, hs, ht, hu⟩
        have hwf : u.length < (List.cons a x).length := by
          rw [hs, List.length_cons, List.length_append]
          apply A
        rw [IH _ hwf] at hu
        rcases hu with ⟨S', hsum, helem⟩
        use (a :: t) :: S'
        constructor
        · simp [hs, hsum]
        · intro t' ht'
          cases ht'
          case head ht' =>
            simp only [ne_eq, not_false_iff, true_and, rmatch]
            exact ht
          case tail ht' => exact helem t' ht'
    · 

Proof state:
case mpr
α : Type u_1
β : Type u_2
γ : Type u_3
dec : DecidableEq α
a b : α
P : RegularExpression α
x : List α
A : ∀ (m n : ℕ), n < m + n + 1
IH :
  ∀ (t : List α),
    length t < length x → (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ t ∈ S, t ≠ [] ∧ rmatch P t = true)
⊢ (∃ S, x = join S ∧ ∀ t ∈ S, t ≠ [] ∧ rmatch P t = true) → rmatch (star P) x = true

Next tactic:
rintro ⟨S, hsum, helem⟩
theorem star_rmatch_iff (P : RegularExpression α) :
    ∀ x : List α, (star P).rmatch x ↔ ∃ S : List (List α), x
          = S.join ∧ ∀ t ∈ S, t ≠ [] ∧ P.rmatch t :=
  fun x => by
    have A : ∀ m n : ℕ, n < m + n + 1 := by
      intro m n
      convert add_lt_add_of_le_of_lt (add_le_add (zero_le m) (le_refl n)) zero_lt_one
      simp
    have IH := fun t (_h : List.length t < List.length x) => star_rmatch_iff P t
    clear star_rmatch_iff
    constructor
    · cases' x with a x
      · intro _h
        use []; dsimp; tauto
      · rw [rmatch, deriv, mul_rmatch_iff]
        rintro ⟨t, u, hs, ht, hu⟩
        have hwf : u.length < (List.cons a x).length := by
          rw [hs, List.length_cons, List.length_append]
          apply A
        rw [IH _ hwf] at hu
        rcases hu with ⟨S', hsum, helem⟩
        use (a :: t) :: S'
        constructor
        · simp [hs, hsum]
        · intro t' ht'
          cases ht'
          case head ht' =>
            simp only [ne_eq, not_false_iff, true_and, rmatch]
            exact ht
          case tail ht' => exact helem t' ht'
    · rintro ⟨S, hsum, helem⟩
      

Proof state:
case mpr.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
dec : DecidableEq α
a b : α
P : RegularExpression α
x : List α
A : ∀ (m n : ℕ), n < m + n + 1
IH :
  ∀ (t : List α),
    length t < length x → (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ t ∈ S, t ≠ [] ∧ rmatch P t = true)
S : List (List α)
hsum : x = join S
helem : ∀ t ∈ S, t ≠ [] ∧ rmatch P t = true
⊢ rmatch (star P) x = true

Next tactic:
cases' x with a x
theorem star_rmatch_iff (P : RegularExpression α) :
    ∀ x : List α, (star P).rmatch x ↔ ∃ S : List (List α), x
          = S.join ∧ ∀ t ∈ S, t ≠ [] ∧ P.rmatch t :=
  fun x => by
    have A : ∀ m n : ℕ, n < m + n + 1 := by
      intro m n
      convert add_lt_add_of_le_of_lt (add_le_add (zero_le m) (le_refl n)) zero_lt_one
      simp
    have IH := fun t (_h : List.length t < List.length x) => star_rmatch_iff P t
    clear star_rmatch_iff
    constructor
    · cases' x with a x
      · intro _h
        use []; dsimp; tauto
      · rw [rmatch, deriv, mul_rmatch_iff]
        rintro ⟨t, u, hs, ht, hu⟩
        have hwf : u.length < (List.cons a x).length := by
          rw [hs, List.length_cons, List.length_append]
          apply A
        rw [IH _ hwf] at hu
        rcases hu with ⟨S', hsum, helem⟩
        use (a :: t) :: S'
        constructor
        · simp [hs, hsum]
        · intro t' ht'
          cases ht'
          case head ht' =>
            simp only [ne_eq, not_false_iff, true_and, rmatch]
            exact ht
          case tail ht' => exact helem t' ht'
    · rintro ⟨S, hsum, helem⟩
      cases' x with a x
      · 

Proof state:
case mpr.intro.intro.nil
α : Type u_1
β : Type u_2
γ : Type u_3
dec : DecidableEq α
a b : α
P : RegularExpression α
A : ∀ (m n : ℕ), n < m + n + 1
S : List (List α)
helem : ∀ t ∈ S, t ≠ [] ∧ rmatch P t = true
IH :
  ∀ (t : List α),
    length t < length [] → (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ t ∈ S, t ≠ [] ∧ rmatch P t = true)
hsum : [] = join S
⊢ rmatch (star P) [] = true

Next tactic:
rfl
theorem star_rmatch_iff (P : RegularExpression α) :
    ∀ x : List α, (star P).rmatch x ↔ ∃ S : List (List α), x
          = S.join ∧ ∀ t ∈ S, t ≠ [] ∧ P.rmatch t :=
  fun x => by
    have A : ∀ m n : ℕ, n < m + n + 1 := by
      intro m n
      convert add_lt_add_of_le_of_lt (add_le_add (zero_le m) (le_refl n)) zero_lt_one
      simp
    have IH := fun t (_h : List.length t < List.length x) => star_rmatch_iff P t
    clear star_rmatch_iff
    constructor
    · cases' x with a x
      · intro _h
        use []; dsimp; tauto
      · rw [rmatch, deriv, mul_rmatch_iff]
        rintro ⟨t, u, hs, ht, hu⟩
        have hwf : u.length < (List.cons a x).length := by
          rw [hs, List.length_cons, List.length_append]
          apply A
        rw [IH _ hwf] at hu
        rcases hu with ⟨S', hsum, helem⟩
        use (a :: t) :: S'
        constructor
        · simp [hs, hsum]
        · intro t' ht'
          cases ht'
          case head ht' =>
            simp only [ne_eq, not_false_iff, true_and, rmatch]
            exact ht
          case tail ht' => exact helem t' ht'
    · rintro ⟨S, hsum, helem⟩
      cases' x with a x
      · rfl
      · 

Proof state:
case mpr.intro.intro.cons
α : Type u_1
β : Type u_2
γ : Type u_3
dec : DecidableEq α
a✝ b : α
P : RegularExpression α
A : ∀ (m n : ℕ), n < m + n + 1
S : List (List α)
helem : ∀ t ∈ S, t ≠ [] ∧ rmatch P t = true
a : α
x : List α
IH :
  ∀ (t : List α),
    length t < length (a :: x) → (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ t ∈ S, t ≠ [] ∧ rmatch P t = true)
hsum : a :: x = join S
⊢ rmatch (star P) (a :: x) = true

Next tactic:
rw [rmatch, deriv, mul_rmatch_iff]
theorem star_rmatch_iff (P : RegularExpression α) :
    ∀ x : List α, (star P).rmatch x ↔ ∃ S : List (List α), x
          = S.join ∧ ∀ t ∈ S, t ≠ [] ∧ P.rmatch t :=
  fun x => by
    have A : ∀ m n : ℕ, n < m + n + 1 := by
      intro m n
      convert add_lt_add_of_le_of_lt (add_le_add (zero_le m) (le_refl n)) zero_lt_one
      simp
    have IH := fun t (_h : List.length t < List.length x) => star_rmatch_iff P t
    clear star_rmatch_iff
    constructor
    · cases' x with a x
      · intro _h
        use []; dsimp; tauto
      · rw [rmatch, deriv, mul_rmatch_iff]
        rintro ⟨t, u, hs, ht, hu⟩
        have hwf : u.length < (List.cons a x).length := by
          rw [hs, List.length_cons, List.length_append]
          apply A
        rw [IH _ hwf] at hu
        rcases hu with ⟨S', hsum, helem⟩
        use (a :: t) :: S'
        constructor
        · simp [hs, hsum]
        · intro t' ht'
          cases ht'
          case head ht' =>
            simp only [ne_eq, not_false_iff, true_and, rmatch]
            exact ht
          case tail ht' => exact helem t' ht'
    · rintro ⟨S, hsum, helem⟩
      cases' x with a x
      · rfl
      · rw [rmatch, deriv, mul_rmatch_iff]
        

Proof state:
case mpr.intro.intro.cons
α : Type u_1
β : Type u_2
γ : Type u_3
dec : DecidableEq α
a✝ b : α
P : RegularExpression α
A : ∀ (m n : ℕ), n < m + n + 1
S : List (List α)
helem : ∀ t ∈ S, t ≠ [] ∧ rmatch P t = true
a : α
x : List α
IH :
  ∀ (t : List α),
    length t < length (a :: x) → (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ t ∈ S, t ≠ [] ∧ rmatch P t = true)
hsum : a :: x = join S
⊢ ∃ t u, x = t ++ u ∧ rmatch (deriv P a) t = true ∧ rmatch (star P) u = true

Next tactic:
cases' S with t' U
theorem star_rmatch_iff (P : RegularExpression α) :
    ∀ x : List α, (star P).rmatch x ↔ ∃ S : List (List α), x
          = S.join ∧ ∀ t ∈ S, t ≠ [] ∧ P.rmatch t :=
  fun x => by
    have A : ∀ m n : ℕ, n < m + n + 1 := by
      intro m n
      convert add_lt_add_of_le_of_lt (add_le_add (zero_le m) (le_refl n)) zero_lt_one
      simp
    have IH := fun t (_h : List.length t < List.length x) => star_rmatch_iff P t
    clear star_rmatch_iff
    constructor
    · cases' x with a x
      · intro _h
        use []; dsimp; tauto
      · rw [rmatch, deriv, mul_rmatch_iff]
        rintro ⟨t, u, hs, ht, hu⟩
        have hwf : u.length < (List.cons a x).length := by
          rw [hs, List.length_cons, List.length_append]
          apply A
        rw [IH _ hwf] at hu
        rcases hu with ⟨S', hsum, helem⟩
        use (a :: t) :: S'
        constructor
        · simp [hs, hsum]
        · intro t' ht'
          cases ht'
          case head ht' =>
            simp only [ne_eq, not_false_iff, true_and, rmatch]
            exact ht
          case tail ht' => exact helem t' ht'
    · rintro ⟨S, hsum, helem⟩
      cases' x with a x
      · rfl
      · rw [rmatch, deriv, mul_rmatch_iff]
        cases' S with t' U
        · 

Proof state:
case mpr.intro.intro.cons.nil
α : Type u_1
β : Type u_2
γ : Type u_3
dec : DecidableEq α
a✝ b : α
P : RegularExpression α
A : ∀ (m n : ℕ), n < m + n + 1
a : α
x : List α
IH :
  ∀ (t : List α),
    length t < length (a :: x) → (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ t ∈ S, t ≠ [] ∧ rmatch P t = true)
helem : ∀ t ∈ [], t ≠ [] ∧ rmatch P t = true
hsum : a :: x = join []
⊢ ∃ t u, x = t ++ u ∧ rmatch (deriv P a) t = true ∧ rmatch (star P) u = true

Next tactic:
exact ⟨[], [], by tauto⟩
theorem star_rmatch_iff (P : RegularExpression α) :
    ∀ x : List α, (star P).rmatch x ↔ ∃ S : List (List α), x
          = S.join ∧ ∀ t ∈ S, t ≠ [] ∧ P.rmatch t :=
  fun x => by
    have A : ∀ m n : ℕ, n < m + n + 1 := by
      intro m n
      convert add_lt_add_of_le_of_lt (add_le_add (zero_le m) (le_refl n)) zero_lt_one
      simp
    have IH := fun t (_h : List.length t < List.length x) => star_rmatch_iff P t
    clear star_rmatch_iff
    constructor
    · cases' x with a x
      · intro _h
        use []; dsimp; tauto
      · rw [rmatch, deriv, mul_rmatch_iff]
        rintro ⟨t, u, hs, ht, hu⟩
        have hwf : u.length < (List.cons a x).length := by
          rw [hs, List.length_cons, List.length_append]
          apply A
        rw [IH _ hwf] at hu
        rcases hu with ⟨S', hsum, helem⟩
        use (a :: t) :: S'
        constructor
        · simp [hs, hsum]
        · intro t' ht'
          cases ht'
          case head ht' =>
            simp only [ne_eq, not_false_iff, true_and, rmatch]
            exact ht
          case tail ht' => exact helem t' ht'
    · rintro ⟨S, hsum, helem⟩
      cases' x with a x
      · rfl
      · rw [rmatch, deriv, mul_rmatch_iff]
        cases' S with t' U
        · exact ⟨[], [], by 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
dec : DecidableEq α
a✝ b : α
P : RegularExpression α
A : ∀ (m n : ℕ), n < m + n + 1
a : α
x : List α
IH :
  ∀ (t : List α),
    length t < length (a :: x) → (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ t ∈ S, t ≠ [] ∧ rmatch P t = true)
helem : ∀ t ∈ [], t ≠ [] ∧ rmatch P t = true
hsum : a :: x = join []
⊢ x = [] ++ [] ∧ rmatch (deriv P a) [] = true ∧ rmatch (star P) [] = true

Next tactic:
tauto
theorem star_rmatch_iff (P : RegularExpression α) :
    ∀ x : List α, (star P).rmatch x ↔ ∃ S : List (List α), x
          = S.join ∧ ∀ t ∈ S, t ≠ [] ∧ P.rmatch t :=
  fun x => by
    have A : ∀ m n : ℕ, n < m + n + 1 := by
      intro m n
      convert add_lt_add_of_le_of_lt (add_le_add (zero_le m) (le_refl n)) zero_lt_one
      simp
    have IH := fun t (_h : List.length t < List.length x) => star_rmatch_iff P t
    clear star_rmatch_iff
    constructor
    · cases' x with a x
      · intro _h
        use []; dsimp; tauto
      · rw [rmatch, deriv, mul_rmatch_iff]
        rintro ⟨t, u, hs, ht, hu⟩
        have hwf : u.length < (List.cons a x).length := by
          rw [hs, List.length_cons, List.length_append]
          apply A
        rw [IH _ hwf] at hu
        rcases hu with ⟨S', hsum, helem⟩
        use (a :: t) :: S'
        constructor
        · simp [hs, hsum]
        · intro t' ht'
          cases ht'
          case head ht' =>
            simp only [ne_eq, not_false_iff, true_and, rmatch]
            exact ht
          case tail ht' => exact helem t' ht'
    · rintro ⟨S, hsum, helem⟩
      cases' x with a x
      · rfl
      · rw [rmatch, deriv, mul_rmatch_iff]
        cases' S with t' U
        · exact ⟨[], [], by tauto⟩
        · 

Proof state:
case mpr.intro.intro.cons.cons
α : Type u_1
β : Type u_2
γ : Type u_3
dec : DecidableEq α
a✝ b : α
P : RegularExpression α
A : ∀ (m n : ℕ), n < m + n + 1
a : α
x : List α
IH :
  ∀ (t : List α),
    length t < length (a :: x) → (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ t ∈ S, t ≠ [] ∧ rmatch P t = true)
t' : List α
U : List (List α)
helem : ∀ t ∈ t' :: U, t ≠ [] ∧ rmatch P t = true
hsum : a :: x = join (t' :: U)
⊢ ∃ t u, x = t ++ u ∧ rmatch (deriv P a) t = true ∧ rmatch (star P) u = true

Next tactic:
cases' t' with b t
theorem star_rmatch_iff (P : RegularExpression α) :
    ∀ x : List α, (star P).rmatch x ↔ ∃ S : List (List α), x
          = S.join ∧ ∀ t ∈ S, t ≠ [] ∧ P.rmatch t :=
  fun x => by
    have A : ∀ m n : ℕ, n < m + n + 1 := by
      intro m n
      convert add_lt_add_of_le_of_lt (add_le_add (zero_le m) (le_refl n)) zero_lt_one
      simp
    have IH := fun t (_h : List.length t < List.length x) => star_rmatch_iff P t
    clear star_rmatch_iff
    constructor
    · cases' x with a x
      · intro _h
        use []; dsimp; tauto
      · rw [rmatch, deriv, mul_rmatch_iff]
        rintro ⟨t, u, hs, ht, hu⟩
        have hwf : u.length < (List.cons a x).length := by
          rw [hs, List.length_cons, List.length_append]
          apply A
        rw [IH _ hwf] at hu
        rcases hu with ⟨S', hsum, helem⟩
        use (a :: t) :: S'
        constructor
        · simp [hs, hsum]
        · intro t' ht'
          cases ht'
          case head ht' =>
            simp only [ne_eq, not_false_iff, true_and, rmatch]
            exact ht
          case tail ht' => exact helem t' ht'
    · rintro ⟨S, hsum, helem⟩
      cases' x with a x
      · rfl
      · rw [rmatch, deriv, mul_rmatch_iff]
        cases' S with t' U
        · exact ⟨[], [], by tauto⟩
        · cases' t' with b t
          · 

Proof state:
case mpr.intro.intro.cons.cons.nil
α : Type u_1
β : Type u_2
γ : Type u_3
dec : DecidableEq α
a✝ b : α
P : RegularExpression α
A : ∀ (m n : ℕ), n < m + n + 1
a : α
x : List α
IH :
  ∀ (t : List α),
    length t < length (a :: x) → (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ t ∈ S, t ≠ [] ∧ rmatch P t = true)
U : List (List α)
helem : ∀ t ∈ [] :: U, t ≠ [] ∧ rmatch P t = true
hsum : a :: x = join ([] :: U)
⊢ ∃ t u, x = t ++ u ∧ rmatch (deriv P a) t = true ∧ rmatch (star P) u = true

Next tactic:
simp only [forall_eq_or_imp, List.mem_cons] at helem
theorem star_rmatch_iff (P : RegularExpression α) :
    ∀ x : List α, (star P).rmatch x ↔ ∃ S : List (List α), x
          = S.join ∧ ∀ t ∈ S, t ≠ [] ∧ P.rmatch t :=
  fun x => by
    have A : ∀ m n : ℕ, n < m + n + 1 := by
      intro m n
      convert add_lt_add_of_le_of_lt (add_le_add (zero_le m) (le_refl n)) zero_lt_one
      simp
    have IH := fun t (_h : List.length t < List.length x) => star_rmatch_iff P t
    clear star_rmatch_iff
    constructor
    · cases' x with a x
      · intro _h
        use []; dsimp; tauto
      · rw [rmatch, deriv, mul_rmatch_iff]
        rintro ⟨t, u, hs, ht, hu⟩
        have hwf : u.length < (List.cons a x).length := by
          rw [hs, List.length_cons, List.length_append]
          apply A
        rw [IH _ hwf] at hu
        rcases hu with ⟨S', hsum, helem⟩
        use (a :: t) :: S'
        constructor
        · simp [hs, hsum]
        · intro t' ht'
          cases ht'
          case head ht' =>
            simp only [ne_eq, not_false_iff, true_and, rmatch]
            exact ht
          case tail ht' => exact helem t' ht'
    · rintro ⟨S, hsum, helem⟩
      cases' x with a x
      · rfl
      · rw [rmatch, deriv, mul_rmatch_iff]
        cases' S with t' U
        · exact ⟨[], [], by tauto⟩
        · cases' t' with b t
          · simp only [forall_eq_or_imp, List.mem_cons] at helem
            

Proof state:
case mpr.intro.intro.cons.cons.nil
α : Type u_1
β : Type u_2
γ : Type u_3
dec : DecidableEq α
a✝ b : α
P : RegularExpression α
A : ∀ (m n : ℕ), n < m + n + 1
a : α
x : List α
IH :
  ∀ (t : List α),
    length t < length (a :: x) → (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ t ∈ S, t ≠ [] ∧ rmatch P t = true)
U : List (List α)
hsum : a :: x = join ([] :: U)
helem : ([] ≠ [] ∧ rmatch P [] = true) ∧ ∀ a ∈ U, a ≠ [] ∧ rmatch P a = true
⊢ ∃ t u, x = t ++ u ∧ rmatch (deriv P a) t = true ∧ rmatch (star P) u = true

Next tactic:
simp only [eq_self_iff_true, not_true, Ne.def, false_and_iff] at helem
theorem star_rmatch_iff (P : RegularExpression α) :
    ∀ x : List α, (star P).rmatch x ↔ ∃ S : List (List α), x
          = S.join ∧ ∀ t ∈ S, t ≠ [] ∧ P.rmatch t :=
  fun x => by
    have A : ∀ m n : ℕ, n < m + n + 1 := by
      intro m n
      convert add_lt_add_of_le_of_lt (add_le_add (zero_le m) (le_refl n)) zero_lt_one
      simp
    have IH := fun t (_h : List.length t < List.length x) => star_rmatch_iff P t
    clear star_rmatch_iff
    constructor
    · cases' x with a x
      · intro _h
        use []; dsimp; tauto
      · rw [rmatch, deriv, mul_rmatch_iff]
        rintro ⟨t, u, hs, ht, hu⟩
        have hwf : u.length < (List.cons a x).length := by
          rw [hs, List.length_cons, List.length_append]
          apply A
        rw [IH _ hwf] at hu
        rcases hu with ⟨S', hsum, helem⟩
        use (a :: t) :: S'
        constructor
        · simp [hs, hsum]
        · intro t' ht'
          cases ht'
          case head ht' =>
            simp only [ne_eq, not_false_iff, true_and, rmatch]
            exact ht
          case tail ht' => exact helem t' ht'
    · rintro ⟨S, hsum, helem⟩
      cases' x with a x
      · rfl
      · rw [rmatch, deriv, mul_rmatch_iff]
        cases' S with t' U
        · exact ⟨[], [], by tauto⟩
        · cases' t' with b t
          · simp only [forall_eq_or_imp, List.mem_cons] at helem
            simp only [eq_self_iff_true, not_true, Ne.def, false_and_iff] at helem
          

Proof state:
case mpr.intro.intro.cons.cons.cons
α : Type u_1
β : Type u_2
γ : Type u_3
dec : DecidableEq α
a✝ b✝ : α
P : RegularExpression α
A : ∀ (m n : ℕ), n < m + n + 1
a : α
x : List α
IH :
  ∀ (t : List α),
    length t < length (a :: x) → (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ t ∈ S, t ≠ [] ∧ rmatch P t = true)
U : List (List α)
b : α
t : List α
helem : ∀ t_1 ∈ (b :: t) :: U, t_1 ≠ [] ∧ rmatch P t_1 = true
hsum : a :: x = join ((b :: t) :: U)
⊢ ∃ t u, x = t ++ u ∧ rmatch (deriv P a) t = true ∧ rmatch (star P) u = true

Next tactic:
simp only [List.join, List.cons_append, List.cons_eq_cons] at hsum