url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
| input
stringlengths 73
2.09M
|
|---|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp
|
case h
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
⊢ cs ∈ {x | ∃ r_1 ∈ if r.is_nullable then {[]} else ∅, ∃ s_1 ∈ if s.is_nullable then {[]} else ∅, r_1 ++ s_1 = x} ↔
cs ∈ if r.is_nullable ∧ s.is_nullable then {[]} else ∅
|
case h
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
⊢ (∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs) ↔
(r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
⊢ cs ∈ {x | ∃ r_1 ∈ if r.is_nullable then {[]} else ∅, ∃ s_1 ∈ if s.is_nullable then {[]} else ∅, r_1 ++ s_1 = x} ↔
cs ∈ if r.is_nullable ∧ s.is_nullable then {[]} else ∅
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
constructor
|
case h
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
⊢ (∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs) ↔
(r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
case h.mp
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
⊢ (∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs) →
(r.is_nullable ∧ s.is_nullable) ∧ cs = []
case h.mpr
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = [] →
∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
⊢ (∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs) ↔
(r.is_nullable ∧ s.is_nullable) ∧ cs = []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
intro a1
|
case h.mp
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
⊢ (∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs) →
(r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
case h.mp
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a1 : ∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
⊢ (∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs) →
(r.is_nullable ∧ s.is_nullable) ∧ cs = []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
apply Exists.elim a1
|
case h.mp
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a1 : ∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
case h.mp
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a1 : ∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs
⊢ ∀ (a : List α),
((r.is_nullable ∧ a = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ a ++ s_1 = cs) →
(r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a1 : ∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
intro xs a2
|
case h.mp
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a1 : ∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs
⊢ ∀ (a : List α),
((r.is_nullable ∧ a = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ a ++ s_1 = cs) →
(r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
case h.mp
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a1 : ∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs
xs : List α
a2 : (r.is_nullable ∧ xs = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ xs ++ s_1 = cs
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a1 : ∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs
⊢ ∀ (a : List α),
((r.is_nullable ∧ a = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ a ++ s_1 = cs) →
(r.is_nullable ∧ s.is_nullable) ∧ cs = []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
clear a1
|
case h.mp
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a1 : ∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs
xs : List α
a2 : (r.is_nullable ∧ xs = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ xs ++ s_1 = cs
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
case h.mp
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2 : (r.is_nullable ∧ xs = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ xs ++ s_1 = cs
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a1 : ∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs
xs : List α
a2 : (r.is_nullable ∧ xs = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ xs ++ s_1 = cs
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
cases a2
|
case h.mp
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2 : (r.is_nullable ∧ xs = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ xs ++ s_1 = cs
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
case h.mp.intro
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
left✝ : r.is_nullable ∧ xs = []
right✝ : ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ xs ++ s_1 = cs
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2 : (r.is_nullable ∧ xs = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ xs ++ s_1 = cs
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
case _ a2_left a2_right =>
cases a2_left
case _ a2_left_left a2_left_right =>
apply Exists.elim a2_right
intro ys a3
clear a2_right
cases a3
case _ a3_left a3_right =>
cases a3_left
case _ a3_left_left a3_left_right =>
simp only [a2_left_left]
simp only [a3_left_left]
simp only [a2_left_right] at a3_right
simp only [a3_left_right] at a3_right
simp at a3_right
tauto
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left : r.is_nullable ∧ xs = []
a2_right : ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ xs ++ s_1 = cs
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left : r.is_nullable ∧ xs = []
a2_right : ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ xs ++ s_1 = cs
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
cases a2_left
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left : r.is_nullable ∧ xs = []
a2_right : ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ xs ++ s_1 = cs
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
case intro
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_right : ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ xs ++ s_1 = cs
left✝ : r.is_nullable
right✝ : xs = []
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left : r.is_nullable ∧ xs = []
a2_right : ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ xs ++ s_1 = cs
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
case _ a2_left_left a2_left_right =>
apply Exists.elim a2_right
intro ys a3
clear a2_right
cases a3
case _ a3_left a3_right =>
cases a3_left
case _ a3_left_left a3_left_right =>
simp only [a2_left_left]
simp only [a3_left_left]
simp only [a2_left_right] at a3_right
simp only [a3_left_right] at a3_right
simp at a3_right
tauto
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_right : ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ xs ++ s_1 = cs
a2_left_left : r.is_nullable
a2_left_right : xs = []
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_right : ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ xs ++ s_1 = cs
a2_left_left : r.is_nullable
a2_left_right : xs = []
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
apply Exists.elim a2_right
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_right : ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ xs ++ s_1 = cs
a2_left_left : r.is_nullable
a2_left_right : xs = []
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_right : ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ xs ++ s_1 = cs
a2_left_left : r.is_nullable
a2_left_right : xs = []
⊢ ∀ (a : List α), (s.is_nullable ∧ a = []) ∧ xs ++ a = cs → (r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_right : ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ xs ++ s_1 = cs
a2_left_left : r.is_nullable
a2_left_right : xs = []
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
intro ys a3
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_right : ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ xs ++ s_1 = cs
a2_left_left : r.is_nullable
a2_left_right : xs = []
⊢ ∀ (a : List α), (s.is_nullable ∧ a = []) ∧ xs ++ a = cs → (r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_right : ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ xs ++ s_1 = cs
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3 : (s.is_nullable ∧ ys = []) ∧ xs ++ ys = cs
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_right : ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ xs ++ s_1 = cs
a2_left_left : r.is_nullable
a2_left_right : xs = []
⊢ ∀ (a : List α), (s.is_nullable ∧ a = []) ∧ xs ++ a = cs → (r.is_nullable ∧ s.is_nullable) ∧ cs = []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
clear a2_right
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_right : ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ xs ++ s_1 = cs
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3 : (s.is_nullable ∧ ys = []) ∧ xs ++ ys = cs
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3 : (s.is_nullable ∧ ys = []) ∧ xs ++ ys = cs
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_right : ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ xs ++ s_1 = cs
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3 : (s.is_nullable ∧ ys = []) ∧ xs ++ ys = cs
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
cases a3
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3 : (s.is_nullable ∧ ys = []) ∧ xs ++ ys = cs
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
case intro
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
left✝ : s.is_nullable ∧ ys = []
right✝ : xs ++ ys = cs
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3 : (s.is_nullable ∧ ys = []) ∧ xs ++ ys = cs
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
case _ a3_left a3_right =>
cases a3_left
case _ a3_left_left a3_left_right =>
simp only [a2_left_left]
simp only [a3_left_left]
simp only [a2_left_right] at a3_right
simp only [a3_left_right] at a3_right
simp at a3_right
tauto
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3_left : s.is_nullable ∧ ys = []
a3_right : xs ++ ys = cs
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3_left : s.is_nullable ∧ ys = []
a3_right : xs ++ ys = cs
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
cases a3_left
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3_left : s.is_nullable ∧ ys = []
a3_right : xs ++ ys = cs
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
case intro
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3_right : xs ++ ys = cs
left✝ : s.is_nullable
right✝ : ys = []
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3_left : s.is_nullable ∧ ys = []
a3_right : xs ++ ys = cs
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
case _ a3_left_left a3_left_right =>
simp only [a2_left_left]
simp only [a3_left_left]
simp only [a2_left_right] at a3_right
simp only [a3_left_right] at a3_right
simp at a3_right
tauto
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3_right : xs ++ ys = cs
a3_left_left : s.is_nullable
a3_left_right : ys = []
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3_right : xs ++ ys = cs
a3_left_left : s.is_nullable
a3_left_right : ys = []
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp only [a2_left_left]
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3_right : xs ++ ys = cs
a3_left_left : s.is_nullable
a3_left_right : ys = []
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3_right : xs ++ ys = cs
a3_left_left : s.is_nullable
a3_left_right : ys = []
⊢ (True ∧ s.is_nullable) ∧ cs = []
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3_right : xs ++ ys = cs
a3_left_left : s.is_nullable
a3_left_right : ys = []
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp only [a3_left_left]
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3_right : xs ++ ys = cs
a3_left_left : s.is_nullable
a3_left_right : ys = []
⊢ (True ∧ s.is_nullable) ∧ cs = []
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3_right : xs ++ ys = cs
a3_left_left : s.is_nullable
a3_left_right : ys = []
⊢ (True ∧ True) ∧ cs = []
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3_right : xs ++ ys = cs
a3_left_left : s.is_nullable
a3_left_right : ys = []
⊢ (True ∧ s.is_nullable) ∧ cs = []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp only [a2_left_right] at a3_right
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3_right : xs ++ ys = cs
a3_left_left : s.is_nullable
a3_left_right : ys = []
⊢ (True ∧ True) ∧ cs = []
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3_left_left : s.is_nullable
a3_left_right : ys = []
a3_right : [] ++ ys = cs
⊢ (True ∧ True) ∧ cs = []
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3_right : xs ++ ys = cs
a3_left_left : s.is_nullable
a3_left_right : ys = []
⊢ (True ∧ True) ∧ cs = []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp only [a3_left_right] at a3_right
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3_left_left : s.is_nullable
a3_left_right : ys = []
a3_right : [] ++ ys = cs
⊢ (True ∧ True) ∧ cs = []
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3_left_left : s.is_nullable
a3_left_right : ys = []
a3_right : [] ++ [] = cs
⊢ (True ∧ True) ∧ cs = []
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3_left_left : s.is_nullable
a3_left_right : ys = []
a3_right : [] ++ ys = cs
⊢ (True ∧ True) ∧ cs = []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp at a3_right
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3_left_left : s.is_nullable
a3_left_right : ys = []
a3_right : [] ++ [] = cs
⊢ (True ∧ True) ∧ cs = []
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3_left_left : s.is_nullable
a3_left_right : ys = []
a3_right : [] = cs
⊢ (True ∧ True) ∧ cs = []
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3_left_left : s.is_nullable
a3_left_right : ys = []
a3_right : [] ++ [] = cs
⊢ (True ∧ True) ∧ cs = []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
tauto
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3_left_left : s.is_nullable
a3_left_right : ys = []
a3_right : [] = cs
⊢ (True ∧ True) ∧ cs = []
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs xs : List α
a2_left_left : r.is_nullable
a2_left_right : xs = []
ys : List α
a3_left_left : s.is_nullable
a3_left_right : ys = []
a3_right : [] = cs
⊢ (True ∧ True) ∧ cs = []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
intro a1
|
case h.mpr
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = [] →
∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs
|
case h.mpr
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a1 : (r.is_nullable ∧ s.is_nullable) ∧ cs = []
⊢ ∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
⊢ (r.is_nullable ∧ s.is_nullable) ∧ cs = [] →
∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
cases a1
|
case h.mpr
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a1 : (r.is_nullable ∧ s.is_nullable) ∧ cs = []
⊢ ∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs
|
case h.mpr.intro
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
left✝ : r.is_nullable ∧ s.is_nullable
right✝ : cs = []
⊢ ∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a1 : (r.is_nullable ∧ s.is_nullable) ∧ cs = []
⊢ ∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
case _ a1_left a2_right =>
cases a1_left
case _ a1_left_left a1_left_right =>
simp only [a1_left_left]
simp only [a1_left_right]
simp
simp only [a2_right]
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a1_left : r.is_nullable ∧ s.is_nullable
a2_right : cs = []
⊢ ∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a1_left : r.is_nullable ∧ s.is_nullable
a2_right : cs = []
⊢ ∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
cases a1_left
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a1_left : r.is_nullable ∧ s.is_nullable
a2_right : cs = []
⊢ ∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs
|
case intro
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a2_right : cs = []
left✝ : r.is_nullable
right✝ : s.is_nullable
⊢ ∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a1_left : r.is_nullable ∧ s.is_nullable
a2_right : cs = []
⊢ ∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
case _ a1_left_left a1_left_right =>
simp only [a1_left_left]
simp only [a1_left_right]
simp
simp only [a2_right]
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a2_right : cs = []
a1_left_left : r.is_nullable
a1_left_right : s.is_nullable
⊢ ∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a2_right : cs = []
a1_left_left : r.is_nullable
a1_left_right : s.is_nullable
⊢ ∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp only [a1_left_left]
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a2_right : cs = []
a1_left_left : r.is_nullable
a1_left_right : s.is_nullable
⊢ ∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a2_right : cs = []
a1_left_left : r.is_nullable
a1_left_right : s.is_nullable
⊢ ∃ r, (True ∧ r = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r ++ s_1 = cs
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a2_right : cs = []
a1_left_left : r.is_nullable
a1_left_right : s.is_nullable
⊢ ∃ r_1, (r.is_nullable ∧ r_1 = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r_1 ++ s_1 = cs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp only [a1_left_right]
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a2_right : cs = []
a1_left_left : r.is_nullable
a1_left_right : s.is_nullable
⊢ ∃ r, (True ∧ r = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r ++ s_1 = cs
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a2_right : cs = []
a1_left_left : r.is_nullable
a1_left_right : s.is_nullable
⊢ ∃ r, (True ∧ r = []) ∧ ∃ s, (True ∧ s = []) ∧ r ++ s = cs
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a2_right : cs = []
a1_left_left : r.is_nullable
a1_left_right : s.is_nullable
⊢ ∃ r, (True ∧ r = []) ∧ ∃ s_1, (s.is_nullable ∧ s_1 = []) ∧ r ++ s_1 = cs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a2_right : cs = []
a1_left_left : r.is_nullable
a1_left_right : s.is_nullable
⊢ ∃ r, (True ∧ r = []) ∧ ∃ s, (True ∧ s = []) ∧ r ++ s = cs
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a2_right : cs = []
a1_left_left : r.is_nullable
a1_left_right : s.is_nullable
⊢ [] = cs
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a2_right : cs = []
a1_left_left : r.is_nullable
a1_left_right : s.is_nullable
⊢ ∃ r, (True ∧ r = []) ∧ ∃ s, (True ∧ s = []) ∧ r ++ s = cs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp only [a2_right]
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a2_right : cs = []
a1_left_left : r.is_nullable
a1_left_right : s.is_nullable
⊢ [] = cs
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
a2_right : cs = []
a1_left_left : r.is_nullable
a1_left_right : s.is_nullable
⊢ [] = cs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp only [RegExp.languageOf]
|
α : Type
e : RegExp α
a_ih✝ :
RegExp.languageOf α e.delta =
if e.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
⊢ RegExp.languageOf α e.closure.delta =
if e.closure.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
|
α : Type
e : RegExp α
a_ih✝ :
RegExp.languageOf α e.delta =
if e.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
⊢ {[]} = if e.closure.is_nullable then {[]} else ∅
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
e : RegExp α
a_ih✝ :
RegExp.languageOf α e.delta =
if e.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
⊢ RegExp.languageOf α e.closure.delta =
if e.closure.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp only [RegExp.is_nullable]
|
α : Type
e : RegExp α
a_ih✝ :
RegExp.languageOf α e.delta =
if e.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
⊢ {[]} = if e.closure.is_nullable then {[]} else ∅
|
α : Type
e : RegExp α
a_ih✝ :
RegExp.languageOf α e.delta =
if e.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
⊢ {[]} = if True then {[]} else ∅
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
e : RegExp α
a_ih✝ :
RegExp.languageOf α e.delta =
if e.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
⊢ {[]} = if e.closure.is_nullable then {[]} else ∅
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp
|
α : Type
e : RegExp α
a_ih✝ :
RegExp.languageOf α e.delta =
if e.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
⊢ {[]} = if True then {[]} else ∅
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
e : RegExp α
a_ih✝ :
RegExp.languageOf α e.delta =
if e.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
⊢ {[]} = if True then {[]} else ∅
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
induction e generalizing w
|
α : Type
inst✝ : DecidableEq α
a : α
w : List α
e : RegExp α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
|
case char
α : Type
inst✝ : DecidableEq α
a a✝ : α
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a (RegExp.char a✝)) ↔ a :: w ∈ RegExp.languageOf α (RegExp.char a✝)
case epsilon
α : Type
inst✝ : DecidableEq α
a : α
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a RegExp.epsilon) ↔ a :: w ∈ RegExp.languageOf α RegExp.epsilon
case zero
α : Type
inst✝ : DecidableEq α
a : α
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a RegExp.zero) ↔ a :: w ∈ RegExp.languageOf α RegExp.zero
case union
α : Type
inst✝ : DecidableEq α
a : α
a✝¹ a✝ : RegExp α
a_ih✝¹ : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a a✝¹) ↔ a :: w ∈ RegExp.languageOf α a✝¹
a_ih✝ : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a a✝) ↔ a :: w ∈ RegExp.languageOf α a✝
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a (a✝¹.union a✝)) ↔ a :: w ∈ RegExp.languageOf α (a✝¹.union a✝)
case concat
α : Type
inst✝ : DecidableEq α
a : α
a✝¹ a✝ : RegExp α
a_ih✝¹ : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a a✝¹) ↔ a :: w ∈ RegExp.languageOf α a✝¹
a_ih✝ : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a a✝) ↔ a :: w ∈ RegExp.languageOf α a✝
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a (a✝¹.concat a✝)) ↔ a :: w ∈ RegExp.languageOf α (a✝¹.concat a✝)
case closure
α : Type
inst✝ : DecidableEq α
a : α
a✝ : RegExp α
a_ih✝ : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a a✝) ↔ a :: w ∈ RegExp.languageOf α a✝
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a a✝.closure) ↔ a :: w ∈ RegExp.languageOf α a✝.closure
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
w : List α
e : RegExp α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a e) ↔ a :: w ∈ RegExp.languageOf α e
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
case char b =>
simp only [RegExp.derivative]
split_ifs
case pos c1 =>
simp only [c1]
simp only [RegExp.languageOf]
simp
case neg c1 =>
simp only [RegExp.languageOf]
simp
intro a1
contradiction
|
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a (RegExp.char b)) ↔ a :: w ∈ RegExp.languageOf α (RegExp.char b)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a (RegExp.char b)) ↔ a :: w ∈ RegExp.languageOf α (RegExp.char b)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
case epsilon =>
simp only [RegExp.derivative]
simp only [RegExp.languageOf]
simp
|
α : Type
inst✝ : DecidableEq α
a : α
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a RegExp.epsilon) ↔ a :: w ∈ RegExp.languageOf α RegExp.epsilon
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a RegExp.epsilon) ↔ a :: w ∈ RegExp.languageOf α RegExp.epsilon
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
case zero =>
simp only [RegExp.derivative]
simp only [RegExp.languageOf]
simp
|
α : Type
inst✝ : DecidableEq α
a : α
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a RegExp.zero) ↔ a :: w ∈ RegExp.languageOf α RegExp.zero
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a RegExp.zero) ↔ a :: w ∈ RegExp.languageOf α RegExp.zero
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
case union r s r_ih s_ih =>
simp only [RegExp.derivative]
simp only [RegExp.languageOf]
specialize r_ih w
specialize s_ih w
simp
tauto
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a (r.union s)) ↔ a :: w ∈ RegExp.languageOf α (r.union s)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a (r.union s)) ↔ a :: w ∈ RegExp.languageOf α (r.union s)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
simp only [RegExp.derivative]
|
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a (RegExp.char b)) ↔ a :: w ∈ RegExp.languageOf α (RegExp.char b)
|
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
⊢ w ∈ RegExp.languageOf α (if a = b then RegExp.epsilon else RegExp.zero) ↔ a :: w ∈ RegExp.languageOf α (RegExp.char b)
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a (RegExp.char b)) ↔ a :: w ∈ RegExp.languageOf α (RegExp.char b)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
split_ifs
|
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
⊢ w ∈ RegExp.languageOf α (if a = b then RegExp.epsilon else RegExp.zero) ↔ a :: w ∈ RegExp.languageOf α (RegExp.char b)
|
case pos
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
h✝ : a = b
⊢ w ∈ RegExp.languageOf α RegExp.epsilon ↔ a :: w ∈ RegExp.languageOf α (RegExp.char b)
case neg
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
h✝ : ¬a = b
⊢ w ∈ RegExp.languageOf α RegExp.zero ↔ a :: w ∈ RegExp.languageOf α (RegExp.char b)
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
⊢ w ∈ RegExp.languageOf α (if a = b then RegExp.epsilon else RegExp.zero) ↔ a :: w ∈ RegExp.languageOf α (RegExp.char b)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
case pos c1 =>
simp only [c1]
simp only [RegExp.languageOf]
simp
|
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
c1 : a = b
⊢ w ∈ RegExp.languageOf α RegExp.epsilon ↔ a :: w ∈ RegExp.languageOf α (RegExp.char b)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
c1 : a = b
⊢ w ∈ RegExp.languageOf α RegExp.epsilon ↔ a :: w ∈ RegExp.languageOf α (RegExp.char b)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
case neg c1 =>
simp only [RegExp.languageOf]
simp
intro a1
contradiction
|
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
c1 : ¬a = b
⊢ w ∈ RegExp.languageOf α RegExp.zero ↔ a :: w ∈ RegExp.languageOf α (RegExp.char b)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
c1 : ¬a = b
⊢ w ∈ RegExp.languageOf α RegExp.zero ↔ a :: w ∈ RegExp.languageOf α (RegExp.char b)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
simp only [c1]
|
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
c1 : a = b
⊢ w ∈ RegExp.languageOf α RegExp.epsilon ↔ a :: w ∈ RegExp.languageOf α (RegExp.char b)
|
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
c1 : a = b
⊢ w ∈ RegExp.languageOf α RegExp.epsilon ↔ b :: w ∈ RegExp.languageOf α (RegExp.char b)
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
c1 : a = b
⊢ w ∈ RegExp.languageOf α RegExp.epsilon ↔ a :: w ∈ RegExp.languageOf α (RegExp.char b)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
simp only [RegExp.languageOf]
|
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
c1 : a = b
⊢ w ∈ RegExp.languageOf α RegExp.epsilon ↔ b :: w ∈ RegExp.languageOf α (RegExp.char b)
|
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
c1 : a = b
⊢ w ∈ {[]} ↔ b :: w ∈ {[b]}
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
c1 : a = b
⊢ w ∈ RegExp.languageOf α RegExp.epsilon ↔ b :: w ∈ RegExp.languageOf α (RegExp.char b)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
simp
|
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
c1 : a = b
⊢ w ∈ {[]} ↔ b :: w ∈ {[b]}
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
c1 : a = b
⊢ w ∈ {[]} ↔ b :: w ∈ {[b]}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
simp only [RegExp.languageOf]
|
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
c1 : ¬a = b
⊢ w ∈ RegExp.languageOf α RegExp.zero ↔ a :: w ∈ RegExp.languageOf α (RegExp.char b)
|
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
c1 : ¬a = b
⊢ w ∈ ∅ ↔ a :: w ∈ {[b]}
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
c1 : ¬a = b
⊢ w ∈ RegExp.languageOf α RegExp.zero ↔ a :: w ∈ RegExp.languageOf α (RegExp.char b)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
simp
|
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
c1 : ¬a = b
⊢ w ∈ ∅ ↔ a :: w ∈ {[b]}
|
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
c1 : ¬a = b
⊢ a = b → ¬w = []
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
c1 : ¬a = b
⊢ w ∈ ∅ ↔ a :: w ∈ {[b]}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
intro a1
|
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
c1 : ¬a = b
⊢ a = b → ¬w = []
|
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
c1 : ¬a = b
a1 : a = b
⊢ ¬w = []
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
c1 : ¬a = b
⊢ a = b → ¬w = []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
contradiction
|
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
c1 : ¬a = b
a1 : a = b
⊢ ¬w = []
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a b : α
w : List α
c1 : ¬a = b
a1 : a = b
⊢ ¬w = []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
simp only [RegExp.derivative]
|
α : Type
inst✝ : DecidableEq α
a : α
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a RegExp.epsilon) ↔ a :: w ∈ RegExp.languageOf α RegExp.epsilon
|
α : Type
inst✝ : DecidableEq α
a : α
w : List α
⊢ w ∈ RegExp.languageOf α RegExp.zero ↔ a :: w ∈ RegExp.languageOf α RegExp.epsilon
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a RegExp.epsilon) ↔ a :: w ∈ RegExp.languageOf α RegExp.epsilon
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
simp only [RegExp.languageOf]
|
α : Type
inst✝ : DecidableEq α
a : α
w : List α
⊢ w ∈ RegExp.languageOf α RegExp.zero ↔ a :: w ∈ RegExp.languageOf α RegExp.epsilon
|
α : Type
inst✝ : DecidableEq α
a : α
w : List α
⊢ w ∈ ∅ ↔ a :: w ∈ {[]}
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
w : List α
⊢ w ∈ RegExp.languageOf α RegExp.zero ↔ a :: w ∈ RegExp.languageOf α RegExp.epsilon
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
simp
|
α : Type
inst✝ : DecidableEq α
a : α
w : List α
⊢ w ∈ ∅ ↔ a :: w ∈ {[]}
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
w : List α
⊢ w ∈ ∅ ↔ a :: w ∈ {[]}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
simp only [RegExp.derivative]
|
α : Type
inst✝ : DecidableEq α
a : α
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a RegExp.zero) ↔ a :: w ∈ RegExp.languageOf α RegExp.zero
|
α : Type
inst✝ : DecidableEq α
a : α
w : List α
⊢ w ∈ RegExp.languageOf α RegExp.zero ↔ a :: w ∈ RegExp.languageOf α RegExp.zero
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a RegExp.zero) ↔ a :: w ∈ RegExp.languageOf α RegExp.zero
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
simp only [RegExp.languageOf]
|
α : Type
inst✝ : DecidableEq α
a : α
w : List α
⊢ w ∈ RegExp.languageOf α RegExp.zero ↔ a :: w ∈ RegExp.languageOf α RegExp.zero
|
α : Type
inst✝ : DecidableEq α
a : α
w : List α
⊢ w ∈ ∅ ↔ a :: w ∈ ∅
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
w : List α
⊢ w ∈ RegExp.languageOf α RegExp.zero ↔ a :: w ∈ RegExp.languageOf α RegExp.zero
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
simp
|
α : Type
inst✝ : DecidableEq α
a : α
w : List α
⊢ w ∈ ∅ ↔ a :: w ∈ ∅
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
w : List α
⊢ w ∈ ∅ ↔ a :: w ∈ ∅
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
simp only [RegExp.derivative]
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a (r.union s)) ↔ a :: w ∈ RegExp.languageOf α (r.union s)
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
⊢ w ∈ RegExp.languageOf α ((RegExp.derivative a r).union (RegExp.derivative a s)) ↔
a :: w ∈ RegExp.languageOf α (r.union s)
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a (r.union s)) ↔ a :: w ∈ RegExp.languageOf α (r.union s)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
simp only [RegExp.languageOf]
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
⊢ w ∈ RegExp.languageOf α ((RegExp.derivative a r).union (RegExp.derivative a s)) ↔
a :: w ∈ RegExp.languageOf α (r.union s)
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a r) ∪ RegExp.languageOf α (RegExp.derivative a s) ↔
a :: w ∈ RegExp.languageOf α r ∪ RegExp.languageOf α s
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
⊢ w ∈ RegExp.languageOf α ((RegExp.derivative a r).union (RegExp.derivative a s)) ↔
a :: w ∈ RegExp.languageOf α (r.union s)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
specialize r_ih w
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a r) ∪ RegExp.languageOf α (RegExp.derivative a s) ↔
a :: w ∈ RegExp.languageOf α r ∪ RegExp.languageOf α s
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
r_ih : w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a r) ∪ RegExp.languageOf α (RegExp.derivative a s) ↔
a :: w ∈ RegExp.languageOf α r ∪ RegExp.languageOf α s
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a r) ∪ RegExp.languageOf α (RegExp.derivative a s) ↔
a :: w ∈ RegExp.languageOf α r ∪ RegExp.languageOf α s
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
specialize s_ih w
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
r_ih : w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a r) ∪ RegExp.languageOf α (RegExp.derivative a s) ↔
a :: w ∈ RegExp.languageOf α r ∪ RegExp.languageOf α s
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
w : List α
r_ih : w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a r) ∪ RegExp.languageOf α (RegExp.derivative a s) ↔
a :: w ∈ RegExp.languageOf α r ∪ RegExp.languageOf α s
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
r_ih : w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a r) ∪ RegExp.languageOf α (RegExp.derivative a s) ↔
a :: w ∈ RegExp.languageOf α r ∪ RegExp.languageOf α s
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
simp
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
w : List α
r_ih : w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a r) ∪ RegExp.languageOf α (RegExp.derivative a s) ↔
a :: w ∈ RegExp.languageOf α r ∪ RegExp.languageOf α s
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
w : List α
r_ih : w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a r) ∨ w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔
a :: w ∈ RegExp.languageOf α r ∨ a :: w ∈ RegExp.languageOf α s
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
w : List α
r_ih : w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a r) ∪ RegExp.languageOf α (RegExp.derivative a s) ↔
a :: w ∈ RegExp.languageOf α r ∪ RegExp.languageOf α s
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
tauto
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
w : List α
r_ih : w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a r) ∨ w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔
a :: w ∈ RegExp.languageOf α r ∨ a :: w ∈ RegExp.languageOf α s
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
w : List α
r_ih : w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a r) ∨ w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔
a :: w ∈ RegExp.languageOf α r ∨ a :: w ∈ RegExp.languageOf α s
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
simp only [RegExp.derivative]
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a (r.concat s)) ↔ a :: w ∈ RegExp.languageOf α (r.concat s)
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
⊢ w ∈
RegExp.languageOf α
(if r.is_nullable then (RegExp.derivative a r).concat s else r.delta.concat (RegExp.derivative a s)) ↔
a :: w ∈ RegExp.languageOf α (r.concat s)
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
⊢ w ∈ RegExp.languageOf α (RegExp.derivative a (r.concat s)) ↔ a :: w ∈ RegExp.languageOf α (r.concat s)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
split_ifs
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
⊢ w ∈
RegExp.languageOf α
(if r.is_nullable then (RegExp.derivative a r).concat s else r.delta.concat (RegExp.derivative a s)) ↔
a :: w ∈ RegExp.languageOf α (r.concat s)
|
case pos
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
h✝ : r.is_nullable
⊢ w ∈ RegExp.languageOf α ((RegExp.derivative a r).concat s) ↔ a :: w ∈ RegExp.languageOf α (r.concat s)
case neg
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
h✝ : ¬r.is_nullable
⊢ w ∈ RegExp.languageOf α (r.delta.concat (RegExp.derivative a s)) ↔ a :: w ∈ RegExp.languageOf α (r.concat s)
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
⊢ w ∈
RegExp.languageOf α
(if r.is_nullable then (RegExp.derivative a r).concat s else r.delta.concat (RegExp.derivative a s)) ↔
a :: w ∈ RegExp.languageOf α (r.concat s)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
simp only [RegExp.languageOf]
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
⊢ w ∈ RegExp.languageOf α ((RegExp.derivative a r).concat s) ↔ a :: w ∈ RegExp.languageOf α (r.concat s)
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
⊢ w ∈ {x | ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = x} ↔
a :: w ∈ {x | ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = x}
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
⊢ w ∈ RegExp.languageOf α ((RegExp.derivative a r).concat s) ↔ a :: w ∈ RegExp.languageOf α (r.concat s)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
simp
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
⊢ w ∈ {x | ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = x} ↔
a :: w ∈ {x | ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = x}
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
⊢ (∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w) ↔
∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
⊢ w ∈ {x | ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = x} ↔
a :: w ∈ {x | ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = x}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
constructor
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
⊢ (∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w) ↔
∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
|
case mp
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
⊢ (∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w) →
∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
case mpr
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
⊢ (∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w) →
∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
⊢ (∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w) ↔
∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
intro a1
|
case mp
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
⊢ (∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w) →
∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
|
case mp
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
a1 : ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
⊢ ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
|
Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
⊢ (∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w) →
∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
apply Exists.elim a1
|
case mp
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
a1 : ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
⊢ ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
|
case mp
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
a1 : ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
⊢ ∀ (a_1 : List α),
(a_1 ∈ RegExp.languageOf α (RegExp.derivative a r) ∧ ∃ s_1 ∈ RegExp.languageOf α s, a_1 ++ s_1 = w) →
∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
|
Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
a1 : ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
⊢ ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
intro xs a2
|
case mp
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
a1 : ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
⊢ ∀ (a_1 : List α),
(a_1 ∈ RegExp.languageOf α (RegExp.derivative a r) ∧ ∃ s_1 ∈ RegExp.languageOf α s, a_1 ++ s_1 = w) →
∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
|
case mp
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
a1 : ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
xs : List α
a2 : xs ∈ RegExp.languageOf α (RegExp.derivative a r) ∧ ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = w
⊢ ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
|
Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
a1 : ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
⊢ ∀ (a_1 : List α),
(a_1 ∈ RegExp.languageOf α (RegExp.derivative a r) ∧ ∃ s_1 ∈ RegExp.languageOf α s, a_1 ++ s_1 = w) →
∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
clear a1
|
case mp
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
a1 : ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
xs : List α
a2 : xs ∈ RegExp.languageOf α (RegExp.derivative a r) ∧ ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = w
⊢ ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
|
case mp
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2 : xs ∈ RegExp.languageOf α (RegExp.derivative a r) ∧ ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = w
⊢ ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
|
Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
a1 : ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
xs : List α
a2 : xs ∈ RegExp.languageOf α (RegExp.derivative a r) ∧ ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = w
⊢ ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
cases a2
|
case mp
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2 : xs ∈ RegExp.languageOf α (RegExp.derivative a r) ∧ ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = w
⊢ ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
|
case mp.intro
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
left✝ : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
right✝ : ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = w
⊢ ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
|
Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2 : xs ∈ RegExp.languageOf α (RegExp.derivative a r) ∧ ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = w
⊢ ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
apply Exists.elim a2_right
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
a2_right : ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = w
⊢ ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
a2_right : ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = w
⊢ ∀ (a_1 : List α),
a_1 ∈ RegExp.languageOf α s ∧ xs ++ a_1 = w →
∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
a2_right : ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = w
⊢ ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
intro ys a3
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
a2_right : ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = w
⊢ ∀ (a_1 : List α),
a_1 ∈ RegExp.languageOf α s ∧ xs ++ a_1 = w →
∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
a2_right : ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = w
ys : List α
a3 : ys ∈ RegExp.languageOf α s ∧ xs ++ ys = w
⊢ ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
a2_right : ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = w
⊢ ∀ (a_1 : List α),
a_1 ∈ RegExp.languageOf α s ∧ xs ++ a_1 = w →
∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
clear a2_right
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
a2_right : ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = w
ys : List α
a3 : ys ∈ RegExp.languageOf α s ∧ xs ++ ys = w
⊢ ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3 : ys ∈ RegExp.languageOf α s ∧ xs ++ ys = w
⊢ ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
a2_right : ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = w
ys : List α
a3 : ys ∈ RegExp.languageOf α s ∧ xs ++ ys = w
⊢ ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
cases a3
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3 : ys ∈ RegExp.languageOf α s ∧ xs ++ ys = w
⊢ ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
|
case intro
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
left✝ : ys ∈ RegExp.languageOf α s
right✝ : xs ++ ys = w
⊢ ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3 : ys ∈ RegExp.languageOf α s ∧ xs ++ ys = w
⊢ ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
apply Exists.intro (a :: xs)
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = w
⊢ ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = w
⊢ a :: xs ∈ RegExp.languageOf α r ∧ ∃ s_1 ∈ RegExp.languageOf α s, a :: xs ++ s_1 = a :: w
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = w
⊢ ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
constructor
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = w
⊢ a :: xs ∈ RegExp.languageOf α r ∧ ∃ s_1 ∈ RegExp.languageOf α s, a :: xs ++ s_1 = a :: w
|
case left
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = w
⊢ a :: xs ∈ RegExp.languageOf α r
case right
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = w
⊢ ∃ s_1 ∈ RegExp.languageOf α s, a :: xs ++ s_1 = a :: w
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = w
⊢ a :: xs ∈ RegExp.languageOf α r ∧ ∃ s_1 ∈ RegExp.languageOf α s, a :: xs ++ s_1 = a :: w
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
simp only [← r_ih]
|
case left
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = w
⊢ a :: xs ∈ RegExp.languageOf α r
|
case left
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = w
⊢ xs ∈ RegExp.languageOf α (RegExp.derivative a r)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case left
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = w
⊢ a :: xs ∈ RegExp.languageOf α r
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
exact a2_left
|
case left
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = w
⊢ xs ∈ RegExp.languageOf α (RegExp.derivative a r)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case left
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = w
⊢ xs ∈ RegExp.languageOf α (RegExp.derivative a r)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
apply Exists.intro ys
|
case right
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = w
⊢ ∃ s_1 ∈ RegExp.languageOf α s, a :: xs ++ s_1 = a :: w
|
case right
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = w
⊢ ys ∈ RegExp.languageOf α s ∧ a :: xs ++ ys = a :: w
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = w
⊢ ∃ s_1 ∈ RegExp.languageOf α s, a :: xs ++ s_1 = a :: w
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
constructor
|
case right
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = w
⊢ ys ∈ RegExp.languageOf α s ∧ a :: xs ++ ys = a :: w
|
case right.left
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = w
⊢ ys ∈ RegExp.languageOf α s
case right.right
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = w
⊢ a :: xs ++ ys = a :: w
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = w
⊢ ys ∈ RegExp.languageOf α s ∧ a :: xs ++ ys = a :: w
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
exact a3_left
|
case right.left
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = w
⊢ ys ∈ RegExp.languageOf α s
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right.left
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = w
⊢ ys ∈ RegExp.languageOf α s
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
simp only [← a3_right]
|
case right.right
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = w
⊢ a :: xs ++ ys = a :: w
|
case right.right
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = w
⊢ a :: xs ++ ys = a :: (xs ++ ys)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right.right
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = w
⊢ a :: xs ++ ys = a :: w
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
simp
|
case right.right
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = w
⊢ a :: xs ++ ys = a :: (xs ++ ys)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right.right
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α (RegExp.derivative a r)
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = w
⊢ a :: xs ++ ys = a :: (xs ++ ys)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
intro a1
|
case mpr
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
⊢ (∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w) →
∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
|
case mpr
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
a1 : ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
⊢ ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
|
Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
⊢ (∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w) →
∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
apply Exists.elim a1
|
case mpr
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
a1 : ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
⊢ ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
|
case mpr
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
a1 : ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
⊢ ∀ (a_1 : List α),
(a_1 ∈ RegExp.languageOf α r ∧ ∃ s_1 ∈ RegExp.languageOf α s, a_1 ++ s_1 = a :: w) →
∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
|
Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
a1 : ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
⊢ ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
intro xs a2
|
case mpr
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
a1 : ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
⊢ ∀ (a_1 : List α),
(a_1 ∈ RegExp.languageOf α r ∧ ∃ s_1 ∈ RegExp.languageOf α s, a_1 ++ s_1 = a :: w) →
∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
|
case mpr
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
a1 : ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
xs : List α
a2 : xs ∈ RegExp.languageOf α r ∧ ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = a :: w
⊢ ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
|
Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
a1 : ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
⊢ ∀ (a_1 : List α),
(a_1 ∈ RegExp.languageOf α r ∧ ∃ s_1 ∈ RegExp.languageOf α s, a_1 ++ s_1 = a :: w) →
∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
clear a1
|
case mpr
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
a1 : ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
xs : List α
a2 : xs ∈ RegExp.languageOf α r ∧ ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = a :: w
⊢ ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
|
case mpr
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2 : xs ∈ RegExp.languageOf α r ∧ ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = a :: w
⊢ ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
|
Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
a1 : ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = a :: w
xs : List α
a2 : xs ∈ RegExp.languageOf α r ∧ ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = a :: w
⊢ ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
cases a2
|
case mpr
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2 : xs ∈ RegExp.languageOf α r ∧ ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = a :: w
⊢ ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
|
case mpr.intro
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
left✝ : xs ∈ RegExp.languageOf α r
right✝ : ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = a :: w
⊢ ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
|
Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2 : xs ∈ RegExp.languageOf α r ∧ ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = a :: w
⊢ ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
apply Exists.elim a2_right
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α r
a2_right : ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = a :: w
⊢ ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α r
a2_right : ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = a :: w
⊢ ∀ (a_1 : List α),
a_1 ∈ RegExp.languageOf α s ∧ xs ++ a_1 = a :: w →
∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α r
a2_right : ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = a :: w
⊢ ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
intro ys a3
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α r
a2_right : ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = a :: w
⊢ ∀ (a_1 : List α),
a_1 ∈ RegExp.languageOf α s ∧ xs ++ a_1 = a :: w →
∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α r
a2_right : ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = a :: w
ys : List α
a3 : ys ∈ RegExp.languageOf α s ∧ xs ++ ys = a :: w
⊢ ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α r
a2_right : ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = a :: w
⊢ ∀ (a_1 : List α),
a_1 ∈ RegExp.languageOf α s ∧ xs ++ a_1 = a :: w →
∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
clear a2_right
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α r
a2_right : ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = a :: w
ys : List α
a3 : ys ∈ RegExp.languageOf α s ∧ xs ++ ys = a :: w
⊢ ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α r
ys : List α
a3 : ys ∈ RegExp.languageOf α s ∧ xs ++ ys = a :: w
⊢ ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α r
a2_right : ∃ s_1 ∈ RegExp.languageOf α s, xs ++ s_1 = a :: w
ys : List α
a3 : ys ∈ RegExp.languageOf α s ∧ xs ++ ys = a :: w
⊢ ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
cases a3
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α r
ys : List α
a3 : ys ∈ RegExp.languageOf α s ∧ xs ++ ys = a :: w
⊢ ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
|
case intro
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α r
ys : List α
left✝ : ys ∈ RegExp.languageOf α s
right✝ : xs ++ ys = a :: w
⊢ ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α r
ys : List α
a3 : ys ∈ RegExp.languageOf α s ∧ xs ++ ys = a :: w
⊢ ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
simp only [is_nullable_def] at c1
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α r
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = a :: w
⊢ ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w xs : List α
a2_left : xs ∈ RegExp.languageOf α r
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = a :: w
c1 : [] ∈ RegExp.languageOf α r
⊢ ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w : List α
c1 : r.is_nullable
xs : List α
a2_left : xs ∈ RegExp.languageOf α r
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = a :: w
⊢ ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
apply Exists.intro (a :: w)
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w xs : List α
a2_left : xs ∈ RegExp.languageOf α r
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = a :: w
c1 : [] ∈ RegExp.languageOf α r
⊢ ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w xs : List α
a2_left : xs ∈ RegExp.languageOf α r
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = a :: w
c1 : [] ∈ RegExp.languageOf α r
⊢ a :: w ∈ RegExp.languageOf α (RegExp.derivative a r) ∧ ∃ s_1 ∈ RegExp.languageOf α s, a :: w ++ s_1 = w
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w xs : List α
a2_left : xs ∈ RegExp.languageOf α r
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = a :: w
c1 : [] ∈ RegExp.languageOf α r
⊢ ∃ r_1 ∈ RegExp.languageOf α (RegExp.derivative a r), ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = w
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
constructor
|
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w xs : List α
a2_left : xs ∈ RegExp.languageOf α r
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = a :: w
c1 : [] ∈ RegExp.languageOf α r
⊢ a :: w ∈ RegExp.languageOf α (RegExp.derivative a r) ∧ ∃ s_1 ∈ RegExp.languageOf α s, a :: w ++ s_1 = w
|
case left
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w xs : List α
a2_left : xs ∈ RegExp.languageOf α r
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = a :: w
c1 : [] ∈ RegExp.languageOf α r
⊢ a :: w ∈ RegExp.languageOf α (RegExp.derivative a r)
case right
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w xs : List α
a2_left : xs ∈ RegExp.languageOf α r
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = a :: w
c1 : [] ∈ RegExp.languageOf α r
⊢ ∃ s_1 ∈ RegExp.languageOf α s, a :: w ++ s_1 = w
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w xs : List α
a2_left : xs ∈ RegExp.languageOf α r
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = a :: w
c1 : [] ∈ RegExp.languageOf α r
⊢ a :: w ∈ RegExp.languageOf α (RegExp.derivative a r) ∧ ∃ s_1 ∈ RegExp.languageOf α s, a :: w ++ s_1 = w
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
simp only [← a3_right]
|
case left
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w xs : List α
a2_left : xs ∈ RegExp.languageOf α r
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = a :: w
c1 : [] ∈ RegExp.languageOf α r
⊢ a :: w ∈ RegExp.languageOf α (RegExp.derivative a r)
|
case left
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w xs : List α
a2_left : xs ∈ RegExp.languageOf α r
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = a :: w
c1 : [] ∈ RegExp.languageOf α r
⊢ xs ++ ys ∈ RegExp.languageOf α (RegExp.derivative a r)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case left
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w xs : List α
a2_left : xs ∈ RegExp.languageOf α r
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = a :: w
c1 : [] ∈ RegExp.languageOf α r
⊢ a :: w ∈ RegExp.languageOf α (RegExp.derivative a r)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
derivative_def
|
[295, 1]
|
[407, 12]
|
sorry
|
case left
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w xs : List α
a2_left : xs ∈ RegExp.languageOf α r
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = a :: w
c1 : [] ∈ RegExp.languageOf α r
⊢ xs ++ ys ∈ RegExp.languageOf α (RegExp.derivative a r)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case left
α : Type
inst✝ : DecidableEq α
a : α
r s : RegExp α
r_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a r) ↔ a :: w ∈ RegExp.languageOf α r
s_ih : ∀ (w : List α), w ∈ RegExp.languageOf α (RegExp.derivative a s) ↔ a :: w ∈ RegExp.languageOf α s
w xs : List α
a2_left : xs ∈ RegExp.languageOf α r
ys : List α
a3_left : ys ∈ RegExp.languageOf α s
a3_right : xs ++ ys = a :: w
c1 : [] ∈ RegExp.languageOf α r
⊢ xs ++ ys ∈ RegExp.languageOf α (RegExp.derivative a r)
TACTIC:
|
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