internlm/Lean-Github huggingface dataset:
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
c : α
p0 q0 : σ_1
xs : List (σ_0 ⊕ σ_1)
a2_right : Sum.inr q0 ∈ xs
x : SymbolArrow α σ_1
a3_left : x ∈ M_1.symbol_arrow_list
a3_right : x.start_state = p0 ∧ x.symbol = c ∧ List.map Sum.inr x.stop_state_list = xs
⊢ ∃ stop_state_list,
    { start_state := p0, symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
      q0 ∈ stop_state_list
Tactic:
cases a3_right
Proof state after:
case intro
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p0 q0 : σ_1
xs : List (σ_0 ⊕ σ_1)
a2_right : Sum.inr q0 ∈ xs
x : SymbolArrow α σ_1
a3_left : x ∈ M_1.symbol_arrow_list
left✝ : x.start_state = p0
right✝ : x.symbol = c ∧ List.map Sum.inr x.stop_state_list = xs
⊢ ∃ stop_state_list,
    { start_state := p0, symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
      q0 ∈ stop_state_list
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
c : α
p0 q0 : σ_1
xs : List (σ_0 ⊕ σ_1)
a2_right : Sum.inr q0 ∈ xs
x : SymbolArrow α σ_1
a3_left : x ∈ M_1.symbol_arrow_list
a3_right_left : x.start_state = p0
a3_right_right : x.symbol = c ∧ List.map Sum.inr x.stop_state_list = xs
⊢ ∃ stop_state_list,
    { start_state := p0, symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
      q0 ∈ stop_state_list
Tactic:
cases a3_right_right
Proof state after:
case intro
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p0 q0 : σ_1
xs : List (σ_0 ⊕ σ_1)
a2_right : Sum.inr q0 ∈ xs
x : SymbolArrow α σ_1
a3_left : x ∈ M_1.symbol_arrow_list
a3_right_left : x.start_state = p0
left✝ : x.symbol = c
right✝ : List.map Sum.inr x.stop_state_list = xs
⊢ ∃ stop_state_list,
    { start_state := p0, symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
      q0 ∈ stop_state_list
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
c : α
p0 q0 : σ_1
xs : List (σ_0 ⊕ σ_1)
a2_right : Sum.inr q0 ∈ xs
x : SymbolArrow α σ_1
a3_left : x ∈ M_1.symbol_arrow_list
a3_right_left : x.start_state = p0
a3_right_right_left : x.symbol = c
a3_right_right_right : List.map Sum.inr x.stop_state_list = xs
⊢ ∃ stop_state_list,
    { start_state := p0, symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
      q0 ∈ stop_state_list
Tactic:
simp only [← a3_right_right_right] at a2_right
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
p0 q0 : σ_1
xs : List (σ_0 ⊕ σ_1)
x : SymbolArrow α σ_1
a3_left : x ∈ M_1.symbol_arrow_list
a3_right_left : x.start_state = p0
a3_right_right_left : x.symbol = c
a3_right_right_right : List.map Sum.inr x.stop_state_list = xs
a2_right : Sum.inr q0 ∈ List.map Sum.inr x.stop_state_list
⊢ ∃ stop_state_list,
    { start_state := p0, symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
      q0 ∈ stop_state_list
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
c : α
p0 q0 : σ_1
xs : List (σ_0 ⊕ σ_1)
x : SymbolArrow α σ_1
a3_left : x ∈ M_1.symbol_arrow_list
a3_right_left : x.start_state = p0
a3_right_right_left : x.symbol = c
a3_right_right_right : List.map Sum.inr x.stop_state_list = xs
a2_right : Sum.inr q0 ∈ List.map Sum.inr x.stop_state_list
⊢ ∃ stop_state_list,
    { start_state := p0, symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
      q0 ∈ stop_state_list
Tactic:
clear a3_right_right_right
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
p0 q0 : σ_1
xs : List (σ_0 ⊕ σ_1)
x : SymbolArrow α σ_1
a3_left : x ∈ M_1.symbol_arrow_list
a3_right_left : x.start_state = p0
a3_right_right_left : x.symbol = c
a2_right : Sum.inr q0 ∈ List.map Sum.inr x.stop_state_list
⊢ ∃ stop_state_list,
    { start_state := p0, symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
      q0 ∈ stop_state_list
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
c : α
p0 q0 : σ_1
xs : List (σ_0 ⊕ σ_1)
x : SymbolArrow α σ_1
a3_left : x ∈ M_1.symbol_arrow_list
a3_right_left : x.start_state = p0
a3_right_right_left : x.symbol = c
a2_right : Sum.inr q0 ∈ List.map Sum.inr x.stop_state_list
⊢ ∃ stop_state_list,
    { start_state := p0, symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
      q0 ∈ stop_state_list
Tactic:
simp only [← a3_right_left]
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
p0 q0 : σ_1
xs : List (σ_0 ⊕ σ_1)
x : SymbolArrow α σ_1
a3_left : x ∈ M_1.symbol_arrow_list
a3_right_left : x.start_state = p0
a3_right_right_left : x.symbol = c
a2_right : Sum.inr q0 ∈ List.map Sum.inr x.stop_state_list
⊢ ∃ stop_state_list,
    { start_state := x.start_state, symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
      q0 ∈ stop_state_list
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
c : α
p0 q0 : σ_1
xs : List (σ_0 ⊕ σ_1)
x : SymbolArrow α σ_1
a3_left : x ∈ M_1.symbol_arrow_list
a3_right_left : x.start_state = p0
a3_right_right_left : x.symbol = c
a2_right : Sum.inr q0 ∈ List.map Sum.inr x.stop_state_list
⊢ ∃ stop_state_list,
    { start_state := x.start_state, symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
      q0 ∈ stop_state_list
Tactic:
simp only [← a3_right_right_left]
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
p0 q0 : σ_1
xs : List (σ_0 ⊕ σ_1)
x : SymbolArrow α σ_1
a3_left : x ∈ M_1.symbol_arrow_list
a3_right_left : x.start_state = p0
a3_right_right_left : x.symbol = c
a2_right : Sum.inr q0 ∈ List.map Sum.inr x.stop_state_list
⊢ ∃ stop_state_list,
    { start_state := x.start_state, symbol := x.symbol, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
      q0 ∈ stop_state_list
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
c : α
p0 q0 : σ_1
xs : List (σ_0 ⊕ σ_1)
x : SymbolArrow α σ_1
a3_left : x ∈ M_1.symbol_arrow_list
a3_right_left : x.start_state = p0
a3_right_right_left : x.symbol = c
a2_right : Sum.inr q0 ∈ List.map Sum.inr x.stop_state_list
⊢ ∃ stop_state_list,
    { start_state := x.start_state, symbol := x.symbol, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
      q0 ∈ stop_state_list
Tactic:
simp at a2_right
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
p0 q0 : σ_1
xs : List (σ_0 ⊕ σ_1)
x : SymbolArrow α σ_1
a3_left : x ∈ M_1.symbol_arrow_list
a3_right_left : x.start_state = p0
a3_right_right_left : x.symbol = c
a2_right : q0 ∈ x.stop_state_list
⊢ ∃ stop_state_list,
    { start_state := x.start_state, symbol := x.symbol, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
      q0 ∈ stop_state_list
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
c : α
p0 q0 : σ_1
xs : List (σ_0 ⊕ σ_1)
x : SymbolArrow α σ_1
a3_left : x ∈ M_1.symbol_arrow_list
a3_right_left : x.start_state = p0
a3_right_right_left : x.symbol = c
a2_right : q0 ∈ x.stop_state_list
⊢ ∃ stop_state_list,
    { start_state := x.start_state, symbol := x.symbol, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
      q0 ∈ stop_state_list
Tactic:
apply Exists.intro x.stop_state_list
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
p0 q0 : σ_1
xs : List (σ_0 ⊕ σ_1)
x : SymbolArrow α σ_1
a3_left : x ∈ M_1.symbol_arrow_list
a3_right_left : x.start_state = p0
a3_right_right_left : x.symbol = c
a2_right : q0 ∈ x.stop_state_list
⊢ { start_state := x.start_state, symbol := x.symbol, stop_state_list := x.stop_state_list } ∈ M_1.symbol_arrow_list ∧
    q0 ∈ x.stop_state_list
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
c : α
p0 q0 : σ_1
xs : List (σ_0 ⊕ σ_1)
x : SymbolArrow α σ_1
a3_left : x ∈ M_1.symbol_arrow_list
a3_right_left : x.start_state = p0
a3_right_right_left : x.symbol = c
a2_right : q0 ∈ x.stop_state_list
⊢ { start_state := x.start_state, symbol := x.symbol, stop_state_list := x.stop_state_list } ∈ M_1.symbol_arrow_list ∧
    q0 ∈ x.stop_state_list
Tactic:
constructor
Proof state after:
case left
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p0 q0 : σ_1
xs : List (σ_0 ⊕ σ_1)
x : SymbolArrow α σ_1
a3_left : x ∈ M_1.symbol_arrow_list
a3_right_left : x.start_state = p0
a3_right_right_left : x.symbol = c
a2_right : q0 ∈ x.stop_state_list
⊢ { start_state := x.start_state, symbol := x.symbol, stop_state_list := x.stop_state_list } ∈ M_1.symbol_arrow_list

case right
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p0 q0 : σ_1
xs : List (σ_0 ⊕ σ_1)
x : SymbolArrow α σ_1
a3_left : x ∈ M_1.symbol_arrow_list
a3_right_left : x.start_state = p0
a3_right_right_left : x.symbol = c
a2_right : q0 ∈ x.stop_state_list
⊢ q0 ∈ x.stop_state_list
Theorem: match_union_EpsilonNFA_toAbstract
Proof state before:
case left
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p0 q0 : σ_1
xs : List (σ_0 ⊕ σ_1)
x : SymbolArrow α σ_1
a3_left : x ∈ M_1.symbol_arrow_list
a3_right_left : x.start_state = p0
a3_right_right_left : x.symbol = c
a2_right : q0 ∈ x.stop_state_list
⊢ { start_state := x.start_state, symbol := x.symbol, stop_state_list := x.stop_state_list } ∈ M_1.symbol_arrow_list
Tactic:
exact a3_left
Proof state after:
no goals
Theorem: match_union_EpsilonNFA_toAbstract
Proof state before:
case right
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p0 q0 : σ_1
xs : List (σ_0 ⊕ σ_1)
x : SymbolArrow α σ_1
a3_left : x ∈ M_1.symbol_arrow_list
a3_right_left : x.start_state = p0
a3_right_right_left : x.symbol = c
a2_right : q0 ∈ x.stop_state_list
⊢ q0 ∈ x.stop_state_list
Tactic:
exact a2_right
Proof state after:
no goals
Theorem: match_union_EpsilonNFA_toAbstract
Proof state before:
case mpr
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p0 q0 : σ_1
⊢ (∃ stop_state_list,
      { start_state := p0, symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
        q0 ∈ stop_state_list) →
    ∃ stop_state_list,
      (∃ a ∈ M_1.symbol_arrow_list,
          a.start_state = p0 ∧ a.symbol = c ∧ List.map Sum.inr a.stop_state_list = stop_state_list) ∧
        Sum.inr q0 ∈ stop_state_list
Tactic:
intro a1
Proof state after:
case mpr
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p0 q0 : σ_1
a1 :
  ∃ stop_state_list,
    { start_state := p0, symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
      q0 ∈ stop_state_list
⊢ ∃ stop_state_list,
    (∃ a ∈ M_1.symbol_arrow_list,
        a.start_state = p0 ∧ a.symbol = c ∧ List.map Sum.inr a.stop_state_list = stop_state_list) ∧
      Sum.inr q0 ∈ stop_state_list
Theorem: match_union_EpsilonNFA_toAbstract
Proof state before:
case mpr
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p0 q0 : σ_1
a1 :
  ∃ stop_state_list,
    { start_state := p0, symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
      q0 ∈ stop_state_list
⊢ ∃ stop_state_list,
    (∃ a ∈ M_1.symbol_arrow_list,
        a.start_state = p0 ∧ a.symbol = c ∧ List.map Sum.inr a.stop_state_list = stop_state_list) ∧
      Sum.inr q0 ∈ stop_state_list
Tactic:
cases a1
Proof state after:
case mpr.intro
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p0 q0 : σ_1
w✝ : List σ_1
h✝ : { start_state := p0, symbol := c, stop_state_list := w✝ } ∈ M_1.symbol_arrow_list ∧ q0 ∈ w✝
⊢ ∃ stop_state_list,
    (∃ a ∈ M_1.symbol_arrow_list,
        a.start_state = p0 ∧ a.symbol = c ∧ List.map Sum.inr a.stop_state_list = stop_state_list) ∧
      Sum.inr q0 ∈ stop_state_list
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
c : α
p0 q0 : σ_1
xs : List σ_1
a2 : { start_state := p0, symbol := c, stop_state_list := xs } ∈ M_1.symbol_arrow_list ∧ q0 ∈ xs
⊢ ∃ stop_state_list,
    (∃ a ∈ M_1.symbol_arrow_list,
        a.start_state = p0 ∧ a.symbol = c ∧ List.map Sum.inr a.stop_state_list = stop_state_list) ∧
      Sum.inr q0 ∈ stop_state_list
Tactic:
cases a2
Proof state after:
case intro
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p0 q0 : σ_1
xs : List σ_1
left✝ : { start_state := p0, symbol := c, stop_state_list := xs } ∈ M_1.symbol_arrow_list
right✝ : q0 ∈ xs
⊢ ∃ stop_state_list,
    (∃ a ∈ M_1.symbol_arrow_list,
        a.start_state = p0 ∧ a.symbol = c ∧ List.map Sum.inr a.stop_state_list = stop_state_list) ∧
      Sum.inr q0 ∈ stop_state_list
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
c : α
p0 q0 : σ_1
xs : List σ_1
a2_left : { start_state := p0, symbol := c, stop_state_list := xs } ∈ M_1.symbol_arrow_list
a2_right : q0 ∈ xs
⊢ ∃ stop_state_list,
    (∃ a ∈ M_1.symbol_arrow_list,
        a.start_state = p0 ∧ a.symbol = c ∧ List.map Sum.inr a.stop_state_list = stop_state_list) ∧
      Sum.inr q0 ∈ stop_state_list
Tactic:
apply Exists.intro (xs.map Sum.inr)
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
p0 q0 : σ_1
xs : List σ_1
a2_left : { start_state := p0, symbol := c, stop_state_list := xs } ∈ M_1.symbol_arrow_list
a2_right : q0 ∈ xs
⊢ (∃ a ∈ M_1.symbol_arrow_list,
      a.start_state = p0 ∧ a.symbol = c ∧ List.map Sum.inr a.stop_state_list = List.map Sum.inr xs) ∧
    Sum.inr q0 ∈ List.map Sum.inr xs
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
c : α
p0 q0 : σ_1
xs : List σ_1
a2_left : { start_state := p0, symbol := c, stop_state_list := xs } ∈ M_1.symbol_arrow_list
a2_right : q0 ∈ xs
⊢ (∃ a ∈ M_1.symbol_arrow_list,
      a.start_state = p0 ∧ a.symbol = c ∧ List.map Sum.inr a.stop_state_list = List.map Sum.inr xs) ∧
    Sum.inr q0 ∈ List.map Sum.inr xs
Tactic:
constructor
Proof state after:
case left
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p0 q0 : σ_1
xs : List σ_1
a2_left : { start_state := p0, symbol := c, stop_state_list := xs } ∈ M_1.symbol_arrow_list
a2_right : q0 ∈ xs
⊢ ∃ a ∈ M_1.symbol_arrow_list,
    a.start_state = p0 ∧ a.symbol = c ∧ List.map Sum.inr a.stop_state_list = List.map Sum.inr xs

case right
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p0 q0 : σ_1
xs : List σ_1
a2_left : { start_state := p0, symbol := c, stop_state_list := xs } ∈ M_1.symbol_arrow_list
a2_right : q0 ∈ xs
⊢ Sum.inr q0 ∈ List.map Sum.inr xs
Theorem: match_union_EpsilonNFA_toAbstract
Proof state before:
case left
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p0 q0 : σ_1
xs : List σ_1
a2_left : { start_state := p0, symbol := c, stop_state_list := xs } ∈ M_1.symbol_arrow_list
a2_right : q0 ∈ xs
⊢ ∃ a ∈ M_1.symbol_arrow_list,
    a.start_state = p0 ∧ a.symbol = c ∧ List.map Sum.inr a.stop_state_list = List.map Sum.inr xs
Tactic:
apply Exists.intro { start_state := p0, symbol := c, stop_state_list := xs }
Proof state after:
case left
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p0 q0 : σ_1
xs : List σ_1
a2_left : { start_state := p0, symbol := c, stop_state_list := xs } ∈ M_1.symbol_arrow_list
a2_right : q0 ∈ xs
⊢ { start_state := p0, symbol := c, stop_state_list := xs } ∈ M_1.symbol_arrow_list ∧
    { start_state := p0, symbol := c, stop_state_list := xs }.start_state = p0 ∧
      { start_state := p0, symbol := c, stop_state_list := xs }.symbol = c ∧
        List.map Sum.inr { start_state := p0, symbol := c, stop_state_list := xs }.stop_state_list = List.map Sum.inr xs
Theorem: match_union_EpsilonNFA_toAbstract
Proof state before:
case left
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p0 q0 : σ_1
xs : List σ_1
a2_left : { start_state := p0, symbol := c, stop_state_list := xs } ∈ M_1.symbol_arrow_list
a2_right : q0 ∈ xs
⊢ { start_state := p0, symbol := c, stop_state_list := xs } ∈ M_1.symbol_arrow_list ∧
    { start_state := p0, symbol := c, stop_state_list := xs }.start_state = p0 ∧
      { start_state := p0, symbol := c, stop_state_list := xs }.symbol = c ∧
        List.map Sum.inr { start_state := p0, symbol := c, stop_state_list := xs }.stop_state_list = List.map Sum.inr xs
Tactic:
simp
Proof state after:
case left
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p0 q0 : σ_1
xs : List σ_1
a2_left : { start_state := p0, symbol := c, stop_state_list := xs } ∈ M_1.symbol_arrow_list
a2_right : q0 ∈ xs
⊢ { start_state := p0, symbol := c, stop_state_list := xs } ∈ M_1.symbol_arrow_list
Theorem: match_union_EpsilonNFA_toAbstract
Proof state before:
case left
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p0 q0 : σ_1
xs : List σ_1
a2_left : { start_state := p0, symbol := c, stop_state_list := xs } ∈ M_1.symbol_arrow_list
a2_right : q0 ∈ xs
⊢ { start_state := p0, symbol := c, stop_state_list := xs } ∈ M_1.symbol_arrow_list
Tactic:
exact a2_left
Proof state after:
no goals
Theorem: match_union_EpsilonNFA_toAbstract
Proof state before:
case right
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p0 q0 : σ_1
xs : List σ_1
a2_left : { start_state := p0, symbol := c, stop_state_list := xs } ∈ M_1.symbol_arrow_list
a2_right : q0 ∈ xs
⊢ Sum.inr q0 ∈ List.map Sum.inr xs
Tactic:
simp
Proof state after:
case right
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p0 q0 : σ_1
xs : List σ_1
a2_left : { start_state := p0, symbol := c, stop_state_list := xs } ∈ M_1.symbol_arrow_list
a2_right : q0 ∈ xs
⊢ q0 ∈ xs
Theorem: match_union_EpsilonNFA_toAbstract
Proof state before:
case right
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
c : α
p0 q0 : σ_1
xs : List σ_1
a2_left : { start_state := p0, symbol := c, stop_state_list := xs } ∈ M_1.symbol_arrow_list
a2_right : q0 ∈ xs
⊢ q0 ∈ xs
Tactic:
exact a2_right
Proof state after:
no goals
Theorem: match_union_EpsilonNFA_toAbstract
Proof state before:
case right.left
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 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
Tactic:
funext p q
Proof state after:
case right.left.h.h
α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
p q : σ_0 ⊕ σ_1
⊢ (∃ stop_state_list,
      { start_state := p, 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 ∧
        q ∈ stop_state_list) =
    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
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
p_0 : σ_0
q_0 : σ_1
⊢ (∃ stop_state_list,
      { start_state := Sum.inl p_0, 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 ∧
        Sum.inr q_0 ∈ stop_state_list) =
    match (Sum.inl p_0, Sum.inr q_0) 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
Tactic:
case _ q_0 =>
  simp
  intro xs x _ _ a3
  simp only [← a3]
  simp
Proof state after:
no goals