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 []; 

Proof state:
case h
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
IH :
  ∀ (t : List α),
    length t < length [] → (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ t ∈ S, t ≠ [] ∧ rmatch P t = true)
_h : rmatch (star P) [] = true
⊢ [] = join [] ∧ ∀ t ∈ [], t ≠ [] ∧ rmatch P t = true

Next tactic:
dsimp
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; 

Proof state:
case h
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
IH :
  ∀ (t : List α),
    length t < length [] → (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ t ∈ S, t ≠ [] ∧ rmatch P t = true)
_h : rmatch (star P) [] = true
⊢ [] = [] ∧ ∀ t ∈ [], ¬t = [] ∧ rmatch P t = 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
      · 

Proof state:
case mp.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)
⊢ rmatch (star P) (a :: x) = true → ∃ S, a :: x = join S ∧ ∀ t ∈ S, t ≠ [] ∧ rmatch P t = 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]
        

Proof state:
case mp.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 u, x = t ++ u ∧ rmatch (deriv P a) t = true ∧ rmatch (star P) u = true) →
    ∃ S, a :: x = join S ∧ ∀ t ∈ S, t ≠ [] ∧ rmatch P t = true

Next tactic:
rintro ⟨t, u, hs, ht, hu⟩
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⟩
        

Proof state:
case mp.cons.intro.intro.intro.intro
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
hu : rmatch (star P) u = true
⊢ ∃ S, a :: x = join S ∧ ∀ t ∈ S, t ≠ [] ∧ rmatch P t = true

Next tactic:
have hwf : u.length < (List.cons a x).length := by
          rw [hs, List.length_cons, List.length_append]
          apply A
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
          

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
hu : rmatch (star P) u = true
⊢ length u < length (a :: x)

Next tactic:
rw [hs, List.length_cons, List.length_append]
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]
          

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
hu : rmatch (star P) u = true
⊢ length u < Nat.succ (length t + length u)

Next tactic:
apply A
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
        

Proof state:
case mp.cons.intro.intro.intro.intro
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
hu : rmatch (star P) u = true
hwf : length u < length (a :: x)
⊢ ∃ S, a :: x = join S ∧ ∀ t ∈ S, t ≠ [] ∧ rmatch P t = true

Next tactic:
rw [IH _ hwf] at hu
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
        

Proof state:
case mp.cons.intro.intro.intro.intro
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
hu : ∃ S, u = join S ∧ ∀ t ∈ S, t ≠ [] ∧ rmatch P t = true
hwf : length u < length (a :: x)
⊢ ∃ S, a :: x = join S ∧ ∀ t ∈ S, t ≠ [] ∧ rmatch P t = true

Next tactic:
rcases hu with ⟨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⟩
        

Proof state:
case mp.cons.intro.intro.intro.intro.intro.intro
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
⊢ ∃ S, a :: x = join S ∧ ∀ t ∈ S, t ≠ [] ∧ rmatch P t = true

Next tactic:
use (a :: t) :: S'
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'
        

Proof state:
case h
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
⊢ a :: x = join ((a :: t) :: S') ∧ ∀ t_1 ∈ (a :: t) :: S', t_1 ≠ [] ∧ rmatch P t_1 = true

Next tactic:
constructor
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
        · 

Proof state:
case h.left
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
⊢ a :: x = join ((a :: t) :: S')

Next tactic:
simp [hs, hsum]
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]
        · 

Proof state:
case h.right
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_1 ∈ (a :: t) :: S', t_1 ≠ [] ∧ rmatch P t_1 = true

Next tactic:
intro 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'
          

Proof state:
case h.right
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' : t' ∈ (a :: t) :: S'
⊢ t' ≠ [] ∧ rmatch P t' = true

Next tactic:
cases 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'
          

Proof state:
case h.right.head
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
⊢ a :: t ≠ [] ∧ rmatch P (a :: t) = true
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 head ht' =>
            simp only [ne_eq, not_false_iff, true_and, rmatch]
            exact 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'
          

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
dec : DecidableEq α
ht' 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
⊢ a :: t ≠ [] ∧ rmatch P (a :: t) = true

Next tactic:
case head ht' =>
            simp only [ne_eq, not_false_iff, true_and, rmatch]
            exact 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' =>
            

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
dec : DecidableEq α
ht' 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
⊢ a :: t ≠ [] ∧ rmatch P (a :: t) = true

Next tactic:
simp only [ne_eq, not_false_iff, true_and, rmatch]
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]
            

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
dec : DecidableEq α
ht' 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
⊢ rmatch (deriv P a) t = true

Next tactic:
exact ht