internlm/Lean-Github huggingface dataset:
Theorem: match_concat_EpsilonNFA_toAbstract
Proof state before:
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p_0 : σ_1
q_0 : σ_0 ⊕ σ_1
⊢ (∃ stop_state_list,
      { start_state := Sum.inr (Sum.inr p_0), symbol := c, stop_state_list := stop_state_list } ∈
          List.map
              (fun arrow =>
                { start_state := Sum.inr arrow.start_state, symbol := arrow.symbol,
                  stop_state_list := List.map Sum.inr arrow.stop_state_list })
              (List.map
                (fun arrow =>
                  { start_state := Sum.inl arrow.start_state, symbol := arrow.symbol,
                    stop_state_list := List.map Sum.inl arrow.stop_state_list })
                M_0.symbol_arrow_list) ++
            List.map
              (fun arrow =>
                { start_state := Sum.inr arrow.start_state, symbol := arrow.symbol,
                  stop_state_list := List.map Sum.inr arrow.stop_state_list })
              (List.map
                (fun arrow =>
                  { start_state := Sum.inr arrow.start_state, symbol := arrow.symbol,
                    stop_state_list := List.map Sum.inr arrow.stop_state_list })
                M_1.symbol_arrow_list) ∧
        Sum.inr q_0 ∈ stop_state_list) =
    match (Sum.inr (Sum.inr p_0), Sum.inr q_0) with
    | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
      ∃ stop_state_list,
        { start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_0.symbol_arrow_list ∧
          q' ∈ stop_state_list
    | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
      ∃ stop_state_list,
        { start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
          q' ∈ stop_state_list
    | x => False
Tactic:
case _ q_0 =>
  cases q_0
  case _ q_0 =>
    simp
    intro xs x a1 a2 a3 a4
    simp only [← a4]
    simp
  case _ q_0 =>
    simp
    sorry
Proof state after:
no goals
Theorem: match_concat_EpsilonNFA_toAbstract
Proof state before:
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p_0 : σ_1
q_0 : ℕ
⊢ (∃ stop_state_list,
      { start_state := Sum.inr (Sum.inr p_0), symbol := c, stop_state_list := stop_state_list } ∈
          List.map
              (fun arrow =>
                { start_state := Sum.inr arrow.start_state, symbol := arrow.symbol,
                  stop_state_list := List.map Sum.inr arrow.stop_state_list })
              (List.map
                (fun arrow =>
                  { start_state := Sum.inl arrow.start_state, symbol := arrow.symbol,
                    stop_state_list := List.map Sum.inl arrow.stop_state_list })
                M_0.symbol_arrow_list) ++
            List.map
              (fun arrow =>
                { start_state := Sum.inr arrow.start_state, symbol := arrow.symbol,
                  stop_state_list := List.map Sum.inr arrow.stop_state_list })
              (List.map
                (fun arrow =>
                  { start_state := Sum.inr arrow.start_state, symbol := arrow.symbol,
                    stop_state_list := List.map Sum.inr arrow.stop_state_list })
                M_1.symbol_arrow_list) ∧
        Sum.inl q_0 ∈ stop_state_list) =
    match (Sum.inr (Sum.inr p_0), Sum.inl q_0) with
    | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
      ∃ stop_state_list,
        { start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_0.symbol_arrow_list ∧
          q' ∈ stop_state_list
    | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
      ∃ stop_state_list,
        { start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
          q' ∈ stop_state_list
    | x => False
Tactic:
simp
Proof state after:
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p_0 : σ_1
q_0 : ℕ
⊢ ∀ (x : List (ℕ ⊕ σ_0 ⊕ σ_1)),
    ∀ x_1 ∈ M_1.symbol_arrow_list,
      x_1.start_state = p_0 → x_1.symbol = c → List.map (Sum.inr ∘ Sum.inr) x_1.stop_state_list = x → Sum.inl q_0 ∉ x
Theorem: match_concat_EpsilonNFA_toAbstract
Proof state before:
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p_0 : σ_1
q_0 : ℕ
⊢ ∀ (x : List (ℕ ⊕ σ_0 ⊕ σ_1)),
    ∀ x_1 ∈ M_1.symbol_arrow_list,
      x_1.start_state = p_0 → x_1.symbol = c → List.map (Sum.inr ∘ Sum.inr) x_1.stop_state_list = x → Sum.inl q_0 ∉ x
Tactic:
intro xs x a1 a2 a3 a4
Proof state after:
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p_0 : σ_1
q_0 : ℕ
xs : List (ℕ ⊕ σ_0 ⊕ σ_1)
x : SymbolArrow α σ_1
a1 : x ∈ M_1.symbol_arrow_list
a2 : x.start_state = p_0
a3 : x.symbol = c
a4 : List.map (Sum.inr ∘ Sum.inr) x.stop_state_list = xs
⊢ Sum.inl q_0 ∉ xs
Theorem: match_concat_EpsilonNFA_toAbstract
Proof state before:
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p_0 : σ_1
q_0 : ℕ
xs : List (ℕ ⊕ σ_0 ⊕ σ_1)
x : SymbolArrow α σ_1
a1 : x ∈ M_1.symbol_arrow_list
a2 : x.start_state = p_0
a3 : x.symbol = c
a4 : List.map (Sum.inr ∘ Sum.inr) x.stop_state_list = xs
⊢ Sum.inl q_0 ∉ xs
Tactic:
simp only [← a4]
Proof state after:
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p_0 : σ_1
q_0 : ℕ
xs : List (ℕ ⊕ σ_0 ⊕ σ_1)
x : SymbolArrow α σ_1
a1 : x ∈ M_1.symbol_arrow_list
a2 : x.start_state = p_0
a3 : x.symbol = c
a4 : List.map (Sum.inr ∘ Sum.inr) x.stop_state_list = xs
⊢ Sum.inl q_0 ∉ List.map (Sum.inr ∘ Sum.inr) x.stop_state_list
Theorem: match_concat_EpsilonNFA_toAbstract
Proof state before:
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p_0 : σ_1
q_0 : ℕ
xs : List (ℕ ⊕ σ_0 ⊕ σ_1)
x : SymbolArrow α σ_1
a1 : x ∈ M_1.symbol_arrow_list
a2 : x.start_state = p_0
a3 : x.symbol = c
a4 : List.map (Sum.inr ∘ Sum.inr) x.stop_state_list = xs
⊢ Sum.inl q_0 ∉ List.map (Sum.inr ∘ Sum.inr) x.stop_state_list
Tactic:
simp
Proof state after:
no goals
Theorem: match_concat_EpsilonNFA_toAbstract
Proof state before:
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p_0 : σ_1
q_0 : σ_0
⊢ (∃ stop_state_list,
      { start_state := Sum.inr (Sum.inr p_0), symbol := c, stop_state_list := stop_state_list } ∈
          List.map
              (fun arrow =>
                { start_state := Sum.inr arrow.start_state, symbol := arrow.symbol,
                  stop_state_list := List.map Sum.inr arrow.stop_state_list })
              (List.map
                (fun arrow =>
                  { start_state := Sum.inl arrow.start_state, symbol := arrow.symbol,
                    stop_state_list := List.map Sum.inl arrow.stop_state_list })
                M_0.symbol_arrow_list) ++
            List.map
              (fun arrow =>
                { start_state := Sum.inr arrow.start_state, symbol := arrow.symbol,
                  stop_state_list := List.map Sum.inr arrow.stop_state_list })
              (List.map
                (fun arrow =>
                  { start_state := Sum.inr arrow.start_state, symbol := arrow.symbol,
                    stop_state_list := List.map Sum.inr arrow.stop_state_list })
                M_1.symbol_arrow_list) ∧
        Sum.inr (Sum.inl q_0) ∈ stop_state_list) =
    match (Sum.inr (Sum.inr p_0), Sum.inr (Sum.inl q_0)) with
    | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
      ∃ stop_state_list,
        { start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_0.symbol_arrow_list ∧
          q' ∈ stop_state_list
    | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
      ∃ stop_state_list,
        { start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
          q' ∈ stop_state_list
    | x => False
Tactic:
case _ q_0 =>
  simp
  intro xs x a1 a2 a3 a4
  simp only [← a4]
  simp
Proof state after:
no goals
Theorem: match_concat_EpsilonNFA_toAbstract
Proof state before:
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p_0 q_0 : σ_1
⊢ (∃ stop_state_list,
      { start_state := Sum.inr (Sum.inr p_0), symbol := c, stop_state_list := stop_state_list } ∈
          List.map
              (fun arrow =>
                { start_state := Sum.inr arrow.start_state, symbol := arrow.symbol,
                  stop_state_list := List.map Sum.inr arrow.stop_state_list })
              (List.map
                (fun arrow =>
                  { start_state := Sum.inl arrow.start_state, symbol := arrow.symbol,
                    stop_state_list := List.map Sum.inl arrow.stop_state_list })
                M_0.symbol_arrow_list) ++
            List.map
              (fun arrow =>
                { start_state := Sum.inr arrow.start_state, symbol := arrow.symbol,
                  stop_state_list := List.map Sum.inr arrow.stop_state_list })
              (List.map
                (fun arrow =>
                  { start_state := Sum.inr arrow.start_state, symbol := arrow.symbol,
                    stop_state_list := List.map Sum.inr arrow.stop_state_list })
                M_1.symbol_arrow_list) ∧
        Sum.inr (Sum.inr q_0) ∈ stop_state_list) =
    match (Sum.inr (Sum.inr p_0), Sum.inr (Sum.inr q_0)) with
    | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
      ∃ stop_state_list,
        { start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_0.symbol_arrow_list ∧
          q' ∈ stop_state_list
    | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
      ∃ stop_state_list,
        { start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
          q' ∈ stop_state_list
    | x => False
Tactic:
case _ q_0 =>
  simp
  sorry
Proof state after:
no goals
Theorem: match_concat_EpsilonNFA_toAbstract
Proof state before:
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p_0 : σ_1
q_0 : σ_0
⊢ (∃ stop_state_list,
      { start_state := Sum.inr (Sum.inr p_0), symbol := c, stop_state_list := stop_state_list } ∈
          List.map
              (fun arrow =>
                { start_state := Sum.inr arrow.start_state, symbol := arrow.symbol,
                  stop_state_list := List.map Sum.inr arrow.stop_state_list })
              (List.map
                (fun arrow =>
                  { start_state := Sum.inl arrow.start_state, symbol := arrow.symbol,
                    stop_state_list := List.map Sum.inl arrow.stop_state_list })
                M_0.symbol_arrow_list) ++
            List.map
              (fun arrow =>
                { start_state := Sum.inr arrow.start_state, symbol := arrow.symbol,
                  stop_state_list := List.map Sum.inr arrow.stop_state_list })
              (List.map
                (fun arrow =>
                  { start_state := Sum.inr arrow.start_state, symbol := arrow.symbol,
                    stop_state_list := List.map Sum.inr arrow.stop_state_list })
                M_1.symbol_arrow_list) ∧
        Sum.inr (Sum.inl q_0) ∈ stop_state_list) =
    match (Sum.inr (Sum.inr p_0), Sum.inr (Sum.inl q_0)) with
    | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
      ∃ stop_state_list,
        { start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_0.symbol_arrow_list ∧
          q' ∈ stop_state_list
    | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
      ∃ stop_state_list,
        { start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
          q' ∈ stop_state_list
    | x => False
Tactic:
simp
Proof state after:
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p_0 : σ_1
q_0 : σ_0
⊢ ∀ (x : List (ℕ ⊕ σ_0 ⊕ σ_1)),
    ∀ x_1 ∈ M_1.symbol_arrow_list,
      x_1.start_state = p_0 →
        x_1.symbol = c → List.map (Sum.inr ∘ Sum.inr) x_1.stop_state_list = x → Sum.inr (Sum.inl q_0) ∉ x
Theorem: match_concat_EpsilonNFA_toAbstract
Proof state before:
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p_0 : σ_1
q_0 : σ_0
⊢ ∀ (x : List (ℕ ⊕ σ_0 ⊕ σ_1)),
    ∀ x_1 ∈ M_1.symbol_arrow_list,
      x_1.start_state = p_0 →
        x_1.symbol = c → List.map (Sum.inr ∘ Sum.inr) x_1.stop_state_list = x → Sum.inr (Sum.inl q_0) ∉ x
Tactic:
intro xs x a1 a2 a3 a4
Proof state after:
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p_0 : σ_1
q_0 : σ_0
xs : List (ℕ ⊕ σ_0 ⊕ σ_1)
x : SymbolArrow α σ_1
a1 : x ∈ M_1.symbol_arrow_list
a2 : x.start_state = p_0
a3 : x.symbol = c
a4 : List.map (Sum.inr ∘ Sum.inr) x.stop_state_list = xs
⊢ Sum.inr (Sum.inl q_0) ∉ xs
Theorem: match_concat_EpsilonNFA_toAbstract
Proof state before:
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p_0 : σ_1
q_0 : σ_0
xs : List (ℕ ⊕ σ_0 ⊕ σ_1)
x : SymbolArrow α σ_1
a1 : x ∈ M_1.symbol_arrow_list
a2 : x.start_state = p_0
a3 : x.symbol = c
a4 : List.map (Sum.inr ∘ Sum.inr) x.stop_state_list = xs
⊢ Sum.inr (Sum.inl q_0) ∉ xs
Tactic:
simp only [← a4]
Proof state after:
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p_0 : σ_1
q_0 : σ_0
xs : List (ℕ ⊕ σ_0 ⊕ σ_1)
x : SymbolArrow α σ_1
a1 : x ∈ M_1.symbol_arrow_list
a2 : x.start_state = p_0
a3 : x.symbol = c
a4 : List.map (Sum.inr ∘ Sum.inr) x.stop_state_list = xs
⊢ Sum.inr (Sum.inl q_0) ∉ List.map (Sum.inr ∘ Sum.inr) x.stop_state_list
Theorem: match_concat_EpsilonNFA_toAbstract
Proof state before:
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p_0 : σ_1
q_0 : σ_0
xs : List (ℕ ⊕ σ_0 ⊕ σ_1)
x : SymbolArrow α σ_1
a1 : x ∈ M_1.symbol_arrow_list
a2 : x.start_state = p_0
a3 : x.symbol = c
a4 : List.map (Sum.inr ∘ Sum.inr) x.stop_state_list = xs
⊢ Sum.inr (Sum.inl q_0) ∉ List.map (Sum.inr ∘ Sum.inr) x.stop_state_list
Tactic:
simp
Proof state after:
no goals
Theorem: match_concat_EpsilonNFA_toAbstract
Proof state before:
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p_0 q_0 : σ_1
⊢ (∃ stop_state_list,
      { start_state := Sum.inr (Sum.inr p_0), symbol := c, stop_state_list := stop_state_list } ∈
          List.map
              (fun arrow =>
                { start_state := Sum.inr arrow.start_state, symbol := arrow.symbol,
                  stop_state_list := List.map Sum.inr arrow.stop_state_list })
              (List.map
                (fun arrow =>
                  { start_state := Sum.inl arrow.start_state, symbol := arrow.symbol,
                    stop_state_list := List.map Sum.inl arrow.stop_state_list })
                M_0.symbol_arrow_list) ++
            List.map
              (fun arrow =>
                { start_state := Sum.inr arrow.start_state, symbol := arrow.symbol,
                  stop_state_list := List.map Sum.inr arrow.stop_state_list })
              (List.map
                (fun arrow =>
                  { start_state := Sum.inr arrow.start_state, symbol := arrow.symbol,
                    stop_state_list := List.map Sum.inr arrow.stop_state_list })
                M_1.symbol_arrow_list) ∧
        Sum.inr (Sum.inr q_0) ∈ stop_state_list) =
    match (Sum.inr (Sum.inr p_0), Sum.inr (Sum.inr q_0)) with
    | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
      ∃ stop_state_list,
        { start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_0.symbol_arrow_list ∧
          q' ∈ stop_state_list
    | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
      ∃ stop_state_list,
        { start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
          q' ∈ stop_state_list
    | x => False
Tactic:
simp
Proof state after:
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p_0 q_0 : σ_1
⊢ (∃ stop_state_list,
      (∃ a ∈ M_1.symbol_arrow_list,
          a.start_state = p_0 ∧ a.symbol = c ∧ List.map (Sum.inr ∘ Sum.inr) a.stop_state_list = stop_state_list) ∧
        Sum.inr (Sum.inr q_0) ∈ stop_state_list) ↔
    ∃ stop_state_list,
      { start_state := p_0, symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
        q_0 ∈ stop_state_list
Theorem: match_concat_EpsilonNFA_toAbstract
Proof state before:
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p_0 q_0 : σ_1
⊢ (∃ stop_state_list,
      (∃ a ∈ M_1.symbol_arrow_list,
          a.start_state = p_0 ∧ a.symbol = c ∧ List.map (Sum.inr ∘ Sum.inr) a.stop_state_list = stop_state_list) ∧
        Sum.inr (Sum.inr q_0) ∈ stop_state_list) ↔
    ∃ stop_state_list,
      { start_state := p_0, symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
        q_0 ∈ stop_state_list
Tactic:
sorry
Proof state after:
no goals
Theorem: match_concat_EpsilonNFA_toAbstract
Proof state before:
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
q : ℕ ⊕ σ_0 ⊕ σ_1
p_0 : ℕ
⊢ (∃ stop_state_list,
      { start_state := Sum.inl p_0, stop_state_list := stop_state_list } ∈
          List.map
                (fun arrow =>
                  { start_state := Sum.inr arrow.start_state,
                    stop_state_list := List.map Sum.inr arrow.stop_state_list })
                (List.map
                  (fun arrow =>
                    { start_state := Sum.inl arrow.start_state,
                      stop_state_list := List.map Sum.inl arrow.stop_state_list })
                  M_0.epsilon_arrow_list) ++
              List.map
                (fun arrow =>
                  { start_state := Sum.inr arrow.start_state,
                    stop_state_list := List.map Sum.inr arrow.stop_state_list })
                (List.map
                  (fun arrow =>
                    { start_state := Sum.inr arrow.start_state,
                      stop_state_list := List.map Sum.inr arrow.stop_state_list })
                  M_1.epsilon_arrow_list) ++
            List.map
              (fun accepting_state =>
                { start_state := accepting_state,
                  stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) })
              (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧
        q ∈ stop_state_list) =
    match (Sum.inl p_0, q) with
    | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
      ∃ stop_state_list,
        { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list
    | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
      ∃ stop_state_list,
        { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list
    | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list
    | x => False
Tactic:
case _ p_0 =>
  simp
Proof state after:
no goals
Theorem: match_concat_EpsilonNFA_toAbstract
Proof state before:
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
q : ℕ ⊕ σ_0 ⊕ σ_1
p_0 : σ_0 ⊕ σ_1
⊢ (∃ stop_state_list,
      { start_state := Sum.inr p_0, stop_state_list := stop_state_list } ∈
          List.map
                (fun arrow =>
                  { start_state := Sum.inr arrow.start_state,
                    stop_state_list := List.map Sum.inr arrow.stop_state_list })
                (List.map
                  (fun arrow =>
                    { start_state := Sum.inl arrow.start_state,
                      stop_state_list := List.map Sum.inl arrow.stop_state_list })
                  M_0.epsilon_arrow_list) ++
              List.map
                (fun arrow =>
                  { start_state := Sum.inr arrow.start_state,
                    stop_state_list := List.map Sum.inr arrow.stop_state_list })
                (List.map
                  (fun arrow =>
                    { start_state := Sum.inr arrow.start_state,
                      stop_state_list := List.map Sum.inr arrow.stop_state_list })
                  M_1.epsilon_arrow_list) ++
            List.map
              (fun accepting_state =>
                { start_state := accepting_state,
                  stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) })
              (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧
        q ∈ stop_state_list) =
    match (Sum.inr p_0, q) with
    | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
      ∃ stop_state_list,
        { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list
    | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
      ∃ stop_state_list,
        { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list
    | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list
    | x => False
Tactic:
case _ p_0 =>
  cases p_0
  case _ p_0 =>
    cases q
    case _ q_0 =>
      simp
      intro xs a1
      cases a1
      case _ left =>
        cases left
        case _ x a2 =>
          cases a2
          case _ a2_left a2_right =>
            cases a2_right
            case _ a2_right_left a2_right_right =>
              simp only [← a2_right_right]
              simp
      case _ right =>
        cases right
        case _ x a2 =>
          cases a2
          case _ a2_left a2_right =>
            cases a2_right
            case _ a2_right_left a2_right_right =>
              simp only [← a2_right_right]
              simp
    case _ q_0 =>
      cases q_0
      case _ q_0 =>
        simp
        sorry
      case _ q_0 =>
        simp
        sorry
  case _ p_0 =>
    cases q
    case _ q_0 =>
      simp
      intro xs x a1 a2 a3
      simp only [← a3]
      simp
    case _ q_0 =>
      cases q_0
      case _ q_0 =>
        simp
        intro xs x a1 a2 a3
        simp only [← a3]
        simp
      case _ q_0 =>
        simp
        sorry
Proof state after:
no goals
Theorem: match_concat_EpsilonNFA_toAbstract
Proof state before:
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
q : ℕ ⊕ σ_0 ⊕ σ_1
p_0 : ℕ
⊢ (∃ stop_state_list,
      { start_state := Sum.inl p_0, stop_state_list := stop_state_list } ∈
          List.map
                (fun arrow =>
                  { start_state := Sum.inr arrow.start_state,
                    stop_state_list := List.map Sum.inr arrow.stop_state_list })
                (List.map
                  (fun arrow =>
                    { start_state := Sum.inl arrow.start_state,
                      stop_state_list := List.map Sum.inl arrow.stop_state_list })
                  M_0.epsilon_arrow_list) ++
              List.map
                (fun arrow =>
                  { start_state := Sum.inr arrow.start_state,
                    stop_state_list := List.map Sum.inr arrow.stop_state_list })
                (List.map
                  (fun arrow =>
                    { start_state := Sum.inr arrow.start_state,
                      stop_state_list := List.map Sum.inr arrow.stop_state_list })
                  M_1.epsilon_arrow_list) ++
            List.map
              (fun accepting_state =>
                { start_state := accepting_state,
                  stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) })
              (List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) ∧
        q ∈ stop_state_list) =
    match (Sum.inl p_0, q) with
    | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
      ∃ stop_state_list,
        { start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list
    | (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
      ∃ stop_state_list,
        { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list
    | (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' ∈ M_0.accepting_state_list ∧ q' ∈ M_1.starting_state_list
    | x => False
Tactic:
simp
Proof state after:
no goals