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

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 = b ∧ x = t ++ join U
⊢ ∃ t u, x = t ++ u ∧ rmatch (deriv P a) t = true ∧ rmatch (star P) u = true

Next tactic:
refine' ⟨t, U.join, hsum.2, _, _⟩
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
          simp only [List.join, List.cons_append, List.cons_eq_cons] at hsum
          refine' ⟨t, U.join, hsum.2, _, _⟩
          · 

Proof state:
case mpr.intro.intro.cons.cons.cons.refine'_1
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 = b ∧ x = t ++ join U
⊢ rmatch (deriv P a) t = true

Next tactic:
specialize helem (b :: t) (by simp)
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
          simp only [List.join, List.cons_append, List.cons_eq_cons] at hsum
          refine' ⟨t, U.join, hsum.2, _, _⟩
          · specialize helem (b :: t) (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)
U : List (List α)
b : α
t : List α
helem : ∀ t_1 ∈ (b :: t) :: U, t_1 ≠ [] ∧ rmatch P t_1 = true
hsum : a = b ∧ x = t ++ join U
⊢ b :: t ∈ (b :: t) :: U

Next tactic:
simp
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
          simp only [List.join, List.cons_append, List.cons_eq_cons] at hsum
          refine' ⟨t, U.join, hsum.2, _, _⟩
          · specialize helem (b :: t) (by simp)
            

Proof state:
case mpr.intro.intro.cons.cons.cons.refine'_1
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
hsum : a = b ∧ x = t ++ join U
helem : b :: t ≠ [] ∧ rmatch P (b :: t) = true
⊢ rmatch (deriv P a) t = true

Next tactic:
rw [rmatch] 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
          simp only [List.join, List.cons_append, List.cons_eq_cons] at hsum
          refine' ⟨t, U.join, hsum.2, _, _⟩
          · specialize helem (b :: t) (by simp)
            rw [rmatch] at helem
            

Proof state:
case mpr.intro.intro.cons.cons.cons.refine'_1
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
hsum : a = b ∧ x = t ++ join U
helem : b :: t ≠ [] ∧ rmatch (deriv P b) t = true
⊢ rmatch (deriv P a) t = true

Next tactic:
convert helem.2
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
          simp only [List.join, List.cons_append, List.cons_eq_cons] at hsum
          refine' ⟨t, U.join, hsum.2, _, _⟩
          · specialize helem (b :: t) (by simp)
            rw [rmatch] at helem
            convert helem.2
            

Proof state:
case h.e'_2.h.e'_3.h.e'_4
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
hsum : a = b ∧ x = t ++ join U
helem : b :: t ≠ [] ∧ rmatch (deriv P b) t = true
⊢ a = b

Next tactic:
exact hsum.1
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
          simp only [List.join, List.cons_append, List.cons_eq_cons] at hsum
          refine' ⟨t, U.join, hsum.2, _, _⟩
          · specialize helem (b :: t) (by simp)
            rw [rmatch] at helem
            convert helem.2
            exact hsum.1
          · 

Proof state:
case mpr.intro.intro.cons.cons.cons.refine'_2
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 = b ∧ x = t ++ join U
⊢ rmatch (star P) (join U) = true

Next tactic:
have hwf : U.join.length < (List.cons a x).length := by
              rw [hsum.1, hsum.2]
              simp only [List.length_append, List.length_join, List.length]
              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
        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
          simp only [List.join, List.cons_append, List.cons_eq_cons] at hsum
          refine' ⟨t, U.join, hsum.2, _, _⟩
          · specialize helem (b :: t) (by simp)
            rw [rmatch] at helem
            convert helem.2
            exact hsum.1
          · have hwf : U.join.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)
U : List (List α)
b : α
t : List α
helem : ∀ t_1 ∈ (b :: t) :: U, t_1 ≠ [] ∧ rmatch P t_1 = true
hsum : a = b ∧ x = t ++ join U
⊢ length (join U) < length (a :: x)

Next tactic:
rw [hsum.1, hsum.2]
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
          simp only [List.join, List.cons_append, List.cons_eq_cons] at hsum
          refine' ⟨t, U.join, hsum.2, _, _⟩
          · specialize helem (b :: t) (by simp)
            rw [rmatch] at helem
            convert helem.2
            exact hsum.1
          · have hwf : U.join.length < (List.cons a x).length := by
              rw [hsum.1, hsum.2]
              

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)
U : List (List α)
b : α
t : List α
helem : ∀ t_1 ∈ (b :: t) :: U, t_1 ≠ [] ∧ rmatch P t_1 = true
hsum : a = b ∧ x = t ++ join U
⊢ length (join U) < length (b :: (t ++ join U))

Next tactic:
simp only [List.length_append, List.length_join, List.length]
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
          simp only [List.join, List.cons_append, List.cons_eq_cons] at hsum
          refine' ⟨t, U.join, hsum.2, _, _⟩
          · specialize helem (b :: t) (by simp)
            rw [rmatch] at helem
            convert helem.2
            exact hsum.1
          · have hwf : U.join.length < (List.cons a x).length := by
              rw [hsum.1, hsum.2]
              simp only [List.length_append, List.length_join, List.length]
              

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)
U : List (List α)
b : α
t : List α
helem : ∀ t_1 ∈ (b :: t) :: U, t_1 ≠ [] ∧ rmatch P t_1 = true
hsum : a = b ∧ x = t ++ join U
⊢ sum (map length U) < length t + sum (map length U) + 1

Next tactic:
apply A