Theorem: match_union_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
⊢ ((fun start_state symbol stop_state =>
        ∃ stop_state_list,
          { start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈
              (EpsilonNFA.map Sum.inl M_0).symbol_arrow_list ++ (EpsilonNFA.map Sum.inr M_1).symbol_arrow_list ∧
            stop_state ∈ stop_state_list) =
      fun p c q =>
      match (p, q) with
      | (Sum.inl p', 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 p', 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) ∧
    ((fun start_state stop_state =>
          ∃ stop_state_list,
            { start_state := start_state, stop_state_list := stop_state_list } ∈
                (EpsilonNFA.map Sum.inl M_0).epsilon_arrow_list ++ (EpsilonNFA.map Sum.inr M_1).epsilon_arrow_list ∧
              stop_state ∈ stop_state_list) =
        fun p q =>
        match (p, q) with
        | (Sum.inl p', 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 p', Sum.inr q') =>
          ∃ stop_state_list,
            { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list
        | x => False) ∧
      ((fun state =>
            state ∈
              (EpsilonNFA.map Sum.inl M_0).starting_state_list ++ (EpsilonNFA.map Sum.inr M_1).starting_state_list) =
          fun p =>
          match p with
          | Sum.inl p' => p' ∈ M_0.starting_state_list
          | Sum.inr p' => p' ∈ M_1.starting_state_list) ∧
        (fun state =>
            state ∈
              (EpsilonNFA.map Sum.inl M_0).accepting_state_list ++ (EpsilonNFA.map Sum.inr M_1).accepting_state_list) =
          fun p =>
          match p with
          | Sum.inl p' => p' ∈ M_0.accepting_state_list
          | Sum.inr p' => p' ∈ M_1.accepting_state_list
Tactic:
simp only [EpsilonNFA.map]
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
⊢ ((fun start_state symbol stop_state =>
        ∃ stop_state_list,
          { start_state := start_state, symbol := symbol, stop_state_list := 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 })
                  M_1.symbol_arrow_list ∧
            stop_state ∈ stop_state_list) =
      fun p c q =>
      match (p, q) with
      | (Sum.inl p', 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 p', 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) ∧
    ((fun start_state stop_state =>
          ∃ stop_state_list,
            { start_state := start_state, stop_state_list := 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 })
                    M_1.epsilon_arrow_list ∧
              stop_state ∈ stop_state_list) =
        fun p q =>
        match (p, q) with
        | (Sum.inl p', 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 p', Sum.inr q') =>
          ∃ stop_state_list,
            { start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list
        | x => False) ∧
      ((fun state => state ∈ List.map Sum.inl M_0.starting_state_list ++ List.map Sum.inr M_1.starting_state_list) =
          fun p =>
          match p with
          | Sum.inl p' => p' ∈ M_0.starting_state_list
          | Sum.inr p' => p' ∈ M_1.starting_state_list) ∧
        (fun state => state ∈ List.map Sum.inl M_0.accepting_state_list ++ List.map Sum.inr M_1.accepting_state_list) =
          fun p =>
          match p with
          | Sum.inl p' => p' ∈ M_0.accepting_state_list
          | Sum.inr p' => p' ∈ M_1.accepting_state_list