Datasets:

internlm
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Split (1)

train · 219k rows

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
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 = []
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
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 = []
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 = []
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 = []
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 = []
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 = []
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
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 = []
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
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 = []
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 = []
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 = []
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 = []
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
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 = []
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
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 = []
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 = []
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 = []
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 = []
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 = []
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
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
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
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
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
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
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
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
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
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
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 ∅
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 ∅
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
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
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
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
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
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
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)
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)
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
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
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)
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]}
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
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]}
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 = []
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 = []
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
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
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 ∈ {[]}
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
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
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 ∈ ∅
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
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)
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
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
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
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
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
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)
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)
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}
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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

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