pkuAI4M/lean_github · Datasets at Hugging Face

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/Text.lean	Languages.thm_3_c	[255, 1]	[262, 9]	simp only [concat]	α : Type L1 L2 L3 : Language α ⊢ concat L1 (concat L2 L3) = concat (concat L1 L2) L3	α : Type L1 L2 L3 : Language α ⊢ {x \| ∃ s ∈ L1, ∃ t ∈ {x \| ∃ s ∈ L2, ∃ t ∈ L3, s ++ t = x}, s ++ t = x} = {x \| ∃ s ∈ {x \| ∃ s ∈ L1, ∃ t ∈ L2, s ++ t = x}, ∃ t ∈ L3, s ++ t = x}	Please generate a tactic in lean4 to solve the state. STATE: α : Type L1 L2 L3 : Language α ⊢ concat L1 (concat L2 L3) = concat (concat L1 L2) L3 TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_3_c	[255, 1]	[262, 9]	simp	α : Type L1 L2 L3 : Language α ⊢ {x \| ∃ s ∈ L1, ∃ t ∈ {x \| ∃ s ∈ L2, ∃ t ∈ L3, s ++ t = x}, s ++ t = x} = {x \| ∃ s ∈ {x \| ∃ s ∈ L1, ∃ t ∈ L2, s ++ t = x}, ∃ t ∈ L3, s ++ t = x}	no goals	Please generate a tactic in lean4 to solve the state. STATE: α : Type L1 L2 L3 : Language α ⊢ {x \| ∃ s ∈ L1, ∃ t ∈ {x \| ∃ s ∈ L2, ∃ t ∈ L3, s ++ t = x}, s ++ t = x} = {x \| ∃ s ∈ {x \| ∃ s ∈ L1, ∃ t ∈ L2, s ++ t = x}, ∃ t ∈ L3, s ++ t = x} TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_3_d	[265, 1]	[272, 10]	simp only [concat]	α : Type L1 L2 L3 : Language α ⊢ concat L1 (L2 ∪ L3) = concat L1 L2 ∪ concat L1 L3	α : Type L1 L2 L3 : Language α ⊢ {x \| ∃ s ∈ L1, ∃ t ∈ L2 ∪ L3, s ++ t = x} = {x \| ∃ s ∈ L1, ∃ t ∈ L2, s ++ t = x} ∪ {x \| ∃ s ∈ L1, ∃ t ∈ L3, s ++ t = x}	Please generate a tactic in lean4 to solve the state. STATE: α : Type L1 L2 L3 : Language α ⊢ concat L1 (L2 ∪ L3) = concat L1 L2 ∪ concat L1 L3 TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_3_d	[265, 1]	[272, 10]	aesop	α : Type L1 L2 L3 : Language α ⊢ {x \| ∃ s ∈ L1, ∃ t ∈ L2 ∪ L3, s ++ t = x} = {x \| ∃ s ∈ L1, ∃ t ∈ L2, s ++ t = x} ∪ {x \| ∃ s ∈ L1, ∃ t ∈ L3, s ++ t = x}	no goals	Please generate a tactic in lean4 to solve the state. STATE: α : Type L1 L2 L3 : Language α ⊢ {x \| ∃ s ∈ L1, ∃ t ∈ L2 ∪ L3, s ++ t = x} = {x \| ∃ s ∈ L1, ∃ t ∈ L2, s ++ t = x} ∪ {x \| ∃ s ∈ L1, ∃ t ∈ L3, s ++ t = x} TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_4	[325, 1]	[361, 24]	simp only [Set.subset_def]	α : Type L : Language α n : ℕ ⊢ exp L n ⊆ kleene_closure α L	α : Type L : Language α n : ℕ ⊢ ∀ x ∈ exp L n, x ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ⊢ exp L n ⊆ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_4	[325, 1]	[361, 24]	intro s a1	α : Type L : Language α n : ℕ ⊢ ∀ x ∈ exp L n, x ∈ kleene_closure α L	α : Type L : Language α n : ℕ s : Str α a1 : s ∈ exp L n ⊢ s ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ⊢ ∀ x ∈ exp L n, x ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_4	[325, 1]	[361, 24]	induction n generalizing s	α : Type L : Language α n : ℕ s : Str α a1 : s ∈ exp L n ⊢ s ∈ kleene_closure α L	case zero α : Type L : Language α s : Str α a1 : s ∈ exp L 0 ⊢ s ∈ kleene_closure α L case succ α : Type L : Language α n✝ : ℕ a✝ : ∀ s ∈ exp L n✝, s ∈ kleene_closure α L s : Str α a1 : s ∈ exp L (n✝ + 1) ⊢ s ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ s : Str α a1 : s ∈ exp L n ⊢ s ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_4	[325, 1]	[361, 24]	case zero => simp only [exp] at a1 simp at a1 simp only [a1] exact kleene_closure.eps L	α : Type L : Language α s : Str α a1 : s ∈ exp L 0 ⊢ s ∈ kleene_closure α L	no goals	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α a1 : s ∈ exp L 0 ⊢ s ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_4	[325, 1]	[361, 24]	case succ n ih => simp only [exp] at a1 simp only [concat] at a1 simp at a1 apply Exists.elim a1 intro xs a2 clear a1 cases a2 case _ a2_left a2_right => apply Exists.elim a2_right intro ys a3 clear a2_right cases a3 case _ a3_left a3_right => simp only [← a3_right] apply kleene_closure.succ L apply ih xs a2_left exact a3_left	α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : s ∈ exp L (n + 1) ⊢ s ∈ kleene_closure α L	no goals	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : s ∈ exp L (n + 1) ⊢ s ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_4	[325, 1]	[361, 24]	simp only [exp] at a1	α : Type L : Language α s : Str α a1 : s ∈ exp L 0 ⊢ s ∈ kleene_closure α L	α : Type L : Language α s : Str α a1 : s ∈ {[]} ⊢ s ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α a1 : s ∈ exp L 0 ⊢ s ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_4	[325, 1]	[361, 24]	simp at a1	α : Type L : Language α s : Str α a1 : s ∈ {[]} ⊢ s ∈ kleene_closure α L	α : Type L : Language α s : Str α a1 : s = [] ⊢ s ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α a1 : s ∈ {[]} ⊢ s ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_4	[325, 1]	[361, 24]	simp only [a1]	α : Type L : Language α s : Str α a1 : s = [] ⊢ s ∈ kleene_closure α L	α : Type L : Language α s : Str α a1 : s = [] ⊢ [] ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α a1 : s = [] ⊢ s ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_4	[325, 1]	[361, 24]	exact kleene_closure.eps L	α : Type L : Language α s : Str α a1 : s = [] ⊢ [] ∈ kleene_closure α L	no goals	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α a1 : s = [] ⊢ [] ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_4	[325, 1]	[361, 24]	simp only [exp] at a1	α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : s ∈ exp L (n + 1) ⊢ s ∈ kleene_closure α L	α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : s ∈ concat (exp L n) L ⊢ s ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : s ∈ exp L (n + 1) ⊢ s ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_4	[325, 1]	[361, 24]	simp only [concat] at a1	α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : s ∈ concat (exp L n) L ⊢ s ∈ kleene_closure α L	α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : s ∈ {x \| ∃ s ∈ exp L n, ∃ t ∈ L, s ++ t = x} ⊢ s ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : s ∈ concat (exp L n) L ⊢ s ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_4	[325, 1]	[361, 24]	simp at a1	α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : s ∈ {x \| ∃ s ∈ exp L n, ∃ t ∈ L, s ++ t = x} ⊢ s ∈ kleene_closure α L	α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : ∃ s_1 ∈ exp L n, ∃ t ∈ L, s_1 ++ t = s ⊢ s ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : s ∈ {x \| ∃ s ∈ exp L n, ∃ t ∈ L, s ++ t = x} ⊢ s ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_4	[325, 1]	[361, 24]	apply Exists.elim a1	α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : ∃ s_1 ∈ exp L n, ∃ t ∈ L, s_1 ++ t = s ⊢ s ∈ kleene_closure α L	α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : ∃ s_1 ∈ exp L n, ∃ t ∈ L, s_1 ++ t = s ⊢ ∀ (a : Str α), (a ∈ exp L n ∧ ∃ t ∈ L, a ++ t = s) → s ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : ∃ s_1 ∈ exp L n, ∃ t ∈ L, s_1 ++ t = s ⊢ s ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_4	[325, 1]	[361, 24]	intro xs a2	α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : ∃ s_1 ∈ exp L n, ∃ t ∈ L, s_1 ++ t = s ⊢ ∀ (a : Str α), (a ∈ exp L n ∧ ∃ t ∈ L, a ++ t = s) → s ∈ kleene_closure α L	α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : ∃ s_1 ∈ exp L n, ∃ t ∈ L, s_1 ++ t = s xs : Str α a2 : xs ∈ exp L n ∧ ∃ t ∈ L, xs ++ t = s ⊢ s ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : ∃ s_1 ∈ exp L n, ∃ t ∈ L, s_1 ++ t = s ⊢ ∀ (a : Str α), (a ∈ exp L n ∧ ∃ t ∈ L, a ++ t = s) → s ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_4	[325, 1]	[361, 24]	clear a1	α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : ∃ s_1 ∈ exp L n, ∃ t ∈ L, s_1 ++ t = s xs : Str α a2 : xs ∈ exp L n ∧ ∃ t ∈ L, xs ++ t = s ⊢ s ∈ kleene_closure α L	α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2 : xs ∈ exp L n ∧ ∃ t ∈ L, xs ++ t = s ⊢ s ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : ∃ s_1 ∈ exp L n, ∃ t ∈ L, s_1 ++ t = s xs : Str α a2 : xs ∈ exp L n ∧ ∃ t ∈ L, xs ++ t = s ⊢ s ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_4	[325, 1]	[361, 24]	cases a2	α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2 : xs ∈ exp L n ∧ ∃ t ∈ L, xs ++ t = s ⊢ s ∈ kleene_closure α L	case intro α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α left✝ : xs ∈ exp L n right✝ : ∃ t ∈ L, xs ++ t = s ⊢ s ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2 : xs ∈ exp L n ∧ ∃ t ∈ L, xs ++ t = s ⊢ s ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_4	[325, 1]	[361, 24]	case _ a2_left a2_right => apply Exists.elim a2_right intro ys a3 clear a2_right cases a3 case _ a3_left a3_right => simp only [← a3_right] apply kleene_closure.succ L apply ih xs a2_left exact a3_left	α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n a2_right : ∃ t ∈ L, xs ++ t = s ⊢ s ∈ kleene_closure α L	no goals	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n a2_right : ∃ t ∈ L, xs ++ t = s ⊢ s ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_4	[325, 1]	[361, 24]	apply Exists.elim a2_right	α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n a2_right : ∃ t ∈ L, xs ++ t = s ⊢ s ∈ kleene_closure α L	α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n a2_right : ∃ t ∈ L, xs ++ t = s ⊢ ∀ (a : Str α), a ∈ L ∧ xs ++ a = s → s ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n a2_right : ∃ t ∈ L, xs ++ t = s ⊢ s ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_4	[325, 1]	[361, 24]	intro ys a3	α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n a2_right : ∃ t ∈ L, xs ++ t = s ⊢ ∀ (a : Str α), a ∈ L ∧ xs ++ a = s → s ∈ kleene_closure α L	α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n a2_right : ∃ t ∈ L, xs ++ t = s ys : Str α a3 : ys ∈ L ∧ xs ++ ys = s ⊢ s ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n a2_right : ∃ t ∈ L, xs ++ t = s ⊢ ∀ (a : Str α), a ∈ L ∧ xs ++ a = s → s ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_4	[325, 1]	[361, 24]	clear a2_right	α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n a2_right : ∃ t ∈ L, xs ++ t = s ys : Str α a3 : ys ∈ L ∧ xs ++ ys = s ⊢ s ∈ kleene_closure α L	α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3 : ys ∈ L ∧ xs ++ ys = s ⊢ s ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n a2_right : ∃ t ∈ L, xs ++ t = s ys : Str α a3 : ys ∈ L ∧ xs ++ ys = s ⊢ s ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_4	[325, 1]	[361, 24]	cases a3	α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3 : ys ∈ L ∧ xs ++ ys = s ⊢ s ∈ kleene_closure α L	case intro α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α left✝ : ys ∈ L right✝ : xs ++ ys = s ⊢ s ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3 : ys ∈ L ∧ xs ++ ys = s ⊢ s ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_4	[325, 1]	[361, 24]	case _ a3_left a3_right => simp only [← a3_right] apply kleene_closure.succ L apply ih xs a2_left exact a3_left	α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ s ∈ kleene_closure α L	no goals	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ s ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_4	[325, 1]	[361, 24]	simp only [← a3_right]	α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ s ∈ kleene_closure α L	α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ xs ++ ys ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ s ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_4	[325, 1]	[361, 24]	apply kleene_closure.succ L	α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ xs ++ ys ∈ kleene_closure α L	case a α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ xs ∈ kleene_closure α L case a α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ ys ∈ L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ xs ++ ys ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_4	[325, 1]	[361, 24]	apply ih xs a2_left	case a α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ xs ∈ kleene_closure α L case a α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ ys ∈ L	case a α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ ys ∈ L	Please generate a tactic in lean4 to solve the state. STATE: case a α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ xs ∈ kleene_closure α L case a α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ ys ∈ L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_4	[325, 1]	[361, 24]	exact a3_left	case a α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ ys ∈ L	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ ys ∈ L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_left	[364, 1]	[374, 50]	simp only [Set.subset_def]	α : Type L : Language α ⊢ ⋃ n, exp L n ⊆ kleene_closure α L	α : Type L : Language α ⊢ ∀ x ∈ ⋃ n, exp L n, x ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α ⊢ ⋃ n, exp L n ⊆ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_left	[364, 1]	[374, 50]	intro s a1	α : Type L : Language α ⊢ ∀ x ∈ ⋃ n, exp L n, x ∈ kleene_closure α L	α : Type L : Language α s : Str α a1 : s ∈ ⋃ n, exp L n ⊢ s ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α ⊢ ∀ x ∈ ⋃ n, exp L n, x ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_left	[364, 1]	[374, 50]	simp at a1	α : Type L : Language α s : Str α a1 : s ∈ ⋃ n, exp L n ⊢ s ∈ kleene_closure α L	α : Type L : Language α s : Str α a1 : ∃ i, s ∈ exp L i ⊢ s ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α a1 : s ∈ ⋃ n, exp L n ⊢ s ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_left	[364, 1]	[374, 50]	apply Exists.elim a1	α : Type L : Language α s : Str α a1 : ∃ i, s ∈ exp L i ⊢ s ∈ kleene_closure α L	α : Type L : Language α s : Str α a1 : ∃ i, s ∈ exp L i ⊢ ∀ (a : ℕ), s ∈ exp L a → s ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α a1 : ∃ i, s ∈ exp L i ⊢ s ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_left	[364, 1]	[374, 50]	intro n a2	α : Type L : Language α s : Str α a1 : ∃ i, s ∈ exp L i ⊢ ∀ (a : ℕ), s ∈ exp L a → s ∈ kleene_closure α L	α : Type L : Language α s : Str α a1 : ∃ i, s ∈ exp L i n : ℕ a2 : s ∈ exp L n ⊢ s ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α a1 : ∃ i, s ∈ exp L i ⊢ ∀ (a : ℕ), s ∈ exp L a → s ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_left	[364, 1]	[374, 50]	exact Set.mem_of_subset_of_mem (thm_4 L n) a2	α : Type L : Language α s : Str α a1 : ∃ i, s ∈ exp L i n : ℕ a2 : s ∈ exp L n ⊢ s ∈ kleene_closure α L	no goals	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α a1 : ∃ i, s ∈ exp L i n : ℕ a2 : s ∈ exp L n ⊢ s ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_right	[376, 1]	[406, 19]	simp only [Set.subset_def]	α : Type L : Language α ⊢ kleene_closure α L ⊆ ⋃ n, exp L n	α : Type L : Language α ⊢ ∀ x ∈ kleene_closure α L, x ∈ ⋃ n, exp L n	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α ⊢ kleene_closure α L ⊆ ⋃ n, exp L n TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_right	[376, 1]	[406, 19]	intro s a1	α : Type L : Language α ⊢ ∀ x ∈ kleene_closure α L, x ∈ ⋃ n, exp L n	α : Type L : Language α s : Str α a1 : s ∈ kleene_closure α L ⊢ s ∈ ⋃ n, exp L n	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α ⊢ ∀ x ∈ kleene_closure α L, x ∈ ⋃ n, exp L n TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_right	[376, 1]	[406, 19]	induction a1	α : Type L : Language α s : Str α a1 : s ∈ kleene_closure α L ⊢ s ∈ ⋃ n, exp L n	case eps α : Type L : Language α s : Str α ⊢ [] ∈ ⋃ n, exp L n case succ α : Type L : Language α s s✝ t✝ : Str α a✝¹ : s✝ ∈ kleene_closure α L a✝ : t✝ ∈ L a_ih✝ : s✝ ∈ ⋃ n, exp L n ⊢ s✝ ++ t✝ ∈ ⋃ n, exp L n	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α a1 : s ∈ kleene_closure α L ⊢ s ∈ ⋃ n, exp L n TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_right	[376, 1]	[406, 19]	case eps => simp apply Exists.intro 0 simp only [exp] simp	α : Type L : Language α s : Str α ⊢ [] ∈ ⋃ n, exp L n	no goals	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α ⊢ [] ∈ ⋃ n, exp L n TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_right	[376, 1]	[406, 19]	simp	α : Type L : Language α s : Str α ⊢ [] ∈ ⋃ n, exp L n	α : Type L : Language α s : Str α ⊢ ∃ i, [] ∈ exp L i	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α ⊢ [] ∈ ⋃ n, exp L n TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_right	[376, 1]	[406, 19]	apply Exists.intro 0	α : Type L : Language α s : Str α ⊢ ∃ i, [] ∈ exp L i	α : Type L : Language α s : Str α ⊢ [] ∈ exp L 0	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α ⊢ ∃ i, [] ∈ exp L i TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_right	[376, 1]	[406, 19]	simp only [exp]	α : Type L : Language α s : Str α ⊢ [] ∈ exp L 0	α : Type L : Language α s : Str α ⊢ [] ∈ {[]}	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α ⊢ [] ∈ exp L 0 TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_right	[376, 1]	[406, 19]	simp	α : Type L : Language α s : Str α ⊢ [] ∈ {[]}	no goals	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α ⊢ [] ∈ {[]} TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_right	[376, 1]	[406, 19]	simp at ih_3	α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L ih_3 : xs ∈ ⋃ n, exp L n ⊢ xs ++ ys ∈ ⋃ n, exp L n	α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L ih_3 : ∃ i, xs ∈ exp L i ⊢ xs ++ ys ∈ ⋃ n, exp L n	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L ih_3 : xs ∈ ⋃ n, exp L n ⊢ xs ++ ys ∈ ⋃ n, exp L n TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_right	[376, 1]	[406, 19]	apply Exists.elim ih_3	α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L ih_3 : ∃ i, xs ∈ exp L i ⊢ xs ++ ys ∈ ⋃ n, exp L n	α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L ih_3 : ∃ i, xs ∈ exp L i ⊢ ∀ (a : ℕ), xs ∈ exp L a → xs ++ ys ∈ ⋃ n, exp L n	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L ih_3 : ∃ i, xs ∈ exp L i ⊢ xs ++ ys ∈ ⋃ n, exp L n TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_right	[376, 1]	[406, 19]	intro n a2	α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L ih_3 : ∃ i, xs ∈ exp L i ⊢ ∀ (a : ℕ), xs ∈ exp L a → xs ++ ys ∈ ⋃ n, exp L n	α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L ih_3 : ∃ i, xs ∈ exp L i n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ ⋃ n, exp L n	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L ih_3 : ∃ i, xs ∈ exp L i ⊢ ∀ (a : ℕ), xs ∈ exp L a → xs ++ ys ∈ ⋃ n, exp L n TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_right	[376, 1]	[406, 19]	clear ih_3	α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L ih_3 : ∃ i, xs ∈ exp L i n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ ⋃ n, exp L n	α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ ⋃ n, exp L n	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L ih_3 : ∃ i, xs ∈ exp L i n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ ⋃ n, exp L n TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_right	[376, 1]	[406, 19]	simp	α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ ⋃ n, exp L n	α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ∃ i, xs ++ ys ∈ exp L i	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ ⋃ n, exp L n TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_right	[376, 1]	[406, 19]	apply Exists.intro (n + 1)	α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ∃ i, xs ++ ys ∈ exp L i	α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ exp L (n + 1)	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ∃ i, xs ++ ys ∈ exp L i TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_right	[376, 1]	[406, 19]	simp only [exp]	α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ exp L (n + 1)	α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ concat (exp L n) L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ exp L (n + 1) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_right	[376, 1]	[406, 19]	simp only [concat]	α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ concat (exp L n) L	α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ {x \| ∃ s ∈ exp L n, ∃ t ∈ L, s ++ t = x}	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ concat (exp L n) L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_right	[376, 1]	[406, 19]	simp	α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ {x \| ∃ s ∈ exp L n, ∃ t ∈ L, s ++ t = x}	α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ∃ s ∈ exp L n, ∃ t ∈ L, s ++ t = xs ++ ys	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ {x \| ∃ s ∈ exp L n, ∃ t ∈ L, s ++ t = x} TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_right	[376, 1]	[406, 19]	apply Exists.intro xs	α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ∃ s ∈ exp L n, ∃ t ∈ L, s ++ t = xs ++ ys	α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ∈ exp L n ∧ ∃ t ∈ L, xs ++ t = xs ++ ys	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ∃ s ∈ exp L n, ∃ t ∈ L, s ++ t = xs ++ ys TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_right	[376, 1]	[406, 19]	constructor	α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ∈ exp L n ∧ ∃ t ∈ L, xs ++ t = xs ++ ys	case left α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ∈ exp L n case right α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ∃ t ∈ L, xs ++ t = xs ++ ys	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ∈ exp L n ∧ ∃ t ∈ L, xs ++ t = xs ++ ys TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_right	[376, 1]	[406, 19]	exact a2	case left α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ∈ exp L n	no goals	Please generate a tactic in lean4 to solve the state. STATE: case left α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ∈ exp L n TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_right	[376, 1]	[406, 19]	apply Exists.intro ys	case right α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ∃ t ∈ L, xs ++ t = xs ++ ys	case right α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ys ∈ L ∧ xs ++ ys = xs ++ ys	Please generate a tactic in lean4 to solve the state. STATE: case right α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ∃ t ∈ L, xs ++ t = xs ++ ys TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_right	[376, 1]	[406, 19]	simp	case right α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ys ∈ L ∧ xs ++ ys = xs ++ ys	case right α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ys ∈ L	Please generate a tactic in lean4 to solve the state. STATE: case right α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ys ∈ L ∧ xs ++ ys = xs ++ ys TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5_right	[376, 1]	[406, 19]	exact ih_2	case right α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ys ∈ L	no goals	Please generate a tactic in lean4 to solve the state. STATE: case right α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ys ∈ L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5	[409, 1]	[416, 25]	apply Set.eq_of_subset_of_subset	α : Type L : Language α ⊢ kleene_closure α L = ⋃ n, exp L n	case a α : Type L : Language α ⊢ kleene_closure α L ⊆ ⋃ n, exp L n case a α : Type L : Language α ⊢ ⋃ n, exp L n ⊆ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α ⊢ kleene_closure α L = ⋃ n, exp L n TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5	[409, 1]	[416, 25]	exact thm_5_right L	case a α : Type L : Language α ⊢ kleene_closure α L ⊆ ⋃ n, exp L n	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a α : Type L : Language α ⊢ kleene_closure α L ⊆ ⋃ n, exp L n TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_5	[409, 1]	[416, 25]	exact thm_5_left L	case a α : Type L : Language α ⊢ ⋃ n, exp L n ⊆ kleene_closure α L	no goals	Please generate a tactic in lean4 to solve the state. STATE: case a α : Type L : Language α ⊢ ⋃ n, exp L n ⊆ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.concat_exp_comm	[419, 1]	[433, 26]	induction n	α : Type L : Language α n : ℕ ⊢ concat (exp L n) L = concat L (exp L n)	case zero α : Type L : Language α ⊢ concat (exp L 0) L = concat L (exp L 0) case succ α : Type L : Language α n✝ : ℕ a✝ : concat (exp L n✝) L = concat L (exp L n✝) ⊢ concat (exp L (n✝ + 1)) L = concat L (exp L (n✝ + 1))	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ⊢ concat (exp L n) L = concat L (exp L n) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.concat_exp_comm	[419, 1]	[433, 26]	case zero => simp only [exp] simp only [concat] simp	α : Type L : Language α ⊢ concat (exp L 0) L = concat L (exp L 0)	no goals	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α ⊢ concat (exp L 0) L = concat L (exp L 0) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.concat_exp_comm	[419, 1]	[433, 26]	case succ k ih => simp only [exp] conv => left; simp only [ih] simp only [thm_3_c]	α : Type L : Language α k : ℕ ih : concat (exp L k) L = concat L (exp L k) ⊢ concat (exp L (k + 1)) L = concat L (exp L (k + 1))	no goals	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α k : ℕ ih : concat (exp L k) L = concat L (exp L k) ⊢ concat (exp L (k + 1)) L = concat L (exp L (k + 1)) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.concat_exp_comm	[419, 1]	[433, 26]	simp only [exp]	α : Type L : Language α ⊢ concat (exp L 0) L = concat L (exp L 0)	α : Type L : Language α ⊢ concat {[]} L = concat L {[]}	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α ⊢ concat (exp L 0) L = concat L (exp L 0) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.concat_exp_comm	[419, 1]	[433, 26]	simp only [concat]	α : Type L : Language α ⊢ concat {[]} L = concat L {[]}	α : Type L : Language α ⊢ {x \| ∃ s ∈ {[]}, ∃ t ∈ L, s ++ t = x} = {x \| ∃ s ∈ L, ∃ t ∈ {[]}, s ++ t = x}	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α ⊢ concat {[]} L = concat L {[]} TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.concat_exp_comm	[419, 1]	[433, 26]	simp	α : Type L : Language α ⊢ {x \| ∃ s ∈ {[]}, ∃ t ∈ L, s ++ t = x} = {x \| ∃ s ∈ L, ∃ t ∈ {[]}, s ++ t = x}	no goals	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α ⊢ {x \| ∃ s ∈ {[]}, ∃ t ∈ L, s ++ t = x} = {x \| ∃ s ∈ L, ∃ t ∈ {[]}, s ++ t = x} TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.concat_exp_comm	[419, 1]	[433, 26]	simp only [exp]	α : Type L : Language α k : ℕ ih : concat (exp L k) L = concat L (exp L k) ⊢ concat (exp L (k + 1)) L = concat L (exp L (k + 1))	α : Type L : Language α k : ℕ ih : concat (exp L k) L = concat L (exp L k) ⊢ concat (concat (exp L k) L) L = concat L (concat (exp L k) L)	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α k : ℕ ih : concat (exp L k) L = concat L (exp L k) ⊢ concat (exp L (k + 1)) L = concat L (exp L (k + 1)) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.concat_exp_comm	[419, 1]	[433, 26]	conv => left; simp only [ih]	α : Type L : Language α k : ℕ ih : concat (exp L k) L = concat L (exp L k) ⊢ concat (concat (exp L k) L) L = concat L (concat (exp L k) L)	α : Type L : Language α k : ℕ ih : concat (exp L k) L = concat L (exp L k) ⊢ concat (concat L (exp L k)) L = concat L (concat (exp L k) L)	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α k : ℕ ih : concat (exp L k) L = concat L (exp L k) ⊢ concat (concat (exp L k) L) L = concat L (concat (exp L k) L) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.concat_exp_comm	[419, 1]	[433, 26]	simp only [thm_3_c]	α : Type L : Language α k : ℕ ih : concat (exp L k) L = concat L (exp L k) ⊢ concat (concat L (exp L k)) L = concat L (concat (exp L k) L)	no goals	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α k : ℕ ih : concat (exp L k) L = concat L (exp L k) ⊢ concat (concat L (exp L k)) L = concat L (concat (exp L k) L) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_6	[436, 1]	[459, 14]	apply Set.eq_of_subset_of_subset	α : Type L : Language α ⊢ kleene_closure α L = {[]} ∪ concat L (kleene_closure α L)	case a α : Type L : Language α ⊢ kleene_closure α L ⊆ {[]} ∪ concat L (kleene_closure α L) case a α : Type L : Language α ⊢ {[]} ∪ concat L (kleene_closure α L) ⊆ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α ⊢ kleene_closure α L = {[]} ∪ concat L (kleene_closure α L) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_6	[436, 1]	[459, 14]	simp only [Set.subset_def]	case a α : Type L : Language α ⊢ kleene_closure α L ⊆ {[]} ∪ concat L (kleene_closure α L)	case a α : Type L : Language α ⊢ ∀ x ∈ kleene_closure α L, x ∈ {[]} ∪ concat L (kleene_closure α L)	Please generate a tactic in lean4 to solve the state. STATE: case a α : Type L : Language α ⊢ kleene_closure α L ⊆ {[]} ∪ concat L (kleene_closure α L) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_6	[436, 1]	[459, 14]	intro s a1	case a α : Type L : Language α ⊢ ∀ x ∈ kleene_closure α L, x ∈ {[]} ∪ concat L (kleene_closure α L)	case a α : Type L : Language α s : Str α a1 : s ∈ kleene_closure α L ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L)	Please generate a tactic in lean4 to solve the state. STATE: case a α : Type L : Language α ⊢ ∀ x ∈ kleene_closure α L, x ∈ {[]} ∪ concat L (kleene_closure α L) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_6	[436, 1]	[459, 14]	simp only [thm_5 L] at a1	case a α : Type L : Language α s : Str α a1 : s ∈ kleene_closure α L ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L)	case a α : Type L : Language α s : Str α a1 : s ∈ ⋃ n, exp L n ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L)	Please generate a tactic in lean4 to solve the state. STATE: case a α : Type L : Language α s : Str α a1 : s ∈ kleene_closure α L ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_6	[436, 1]	[459, 14]	simp at a1	case a α : Type L : Language α s : Str α a1 : s ∈ ⋃ n, exp L n ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L)	case a α : Type L : Language α s : Str α a1 : ∃ i, s ∈ exp L i ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L)	Please generate a tactic in lean4 to solve the state. STATE: case a α : Type L : Language α s : Str α a1 : s ∈ ⋃ n, exp L n ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_6	[436, 1]	[459, 14]	cases a1	case a α : Type L : Language α s : Str α a1 : ∃ i, s ∈ exp L i ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L)	case a.intro α : Type L : Language α s : Str α w✝ : ℕ h✝ : s ∈ exp L w✝ ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L)	Please generate a tactic in lean4 to solve the state. STATE: case a α : Type L : Language α s : Str α a1 : ∃ i, s ∈ exp L i ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_6	[436, 1]	[459, 14]	case _ i a2 => simp sorry	α : Type L : Language α s : Str α i : ℕ a2 : s ∈ exp L i ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L)	no goals	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α i : ℕ a2 : s ∈ exp L i ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_6	[436, 1]	[459, 14]	simp	α : Type L : Language α s : Str α i : ℕ a2 : s ∈ exp L i ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L)	α : Type L : Language α s : Str α i : ℕ a2 : s ∈ exp L i ⊢ s = [] ∨ s ∈ concat L (kleene_closure α L)	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α i : ℕ a2 : s ∈ exp L i ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_6	[436, 1]	[459, 14]	sorry	α : Type L : Language α s : Str α i : ℕ a2 : s ∈ exp L i ⊢ s = [] ∨ s ∈ concat L (kleene_closure α L)	no goals	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α i : ℕ a2 : s ∈ exp L i ⊢ s = [] ∨ s ∈ concat L (kleene_closure α L) TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_6	[436, 1]	[459, 14]	simp only [Set.subset_def]	case a α : Type L : Language α ⊢ {[]} ∪ concat L (kleene_closure α L) ⊆ kleene_closure α L	case a α : Type L : Language α ⊢ ∀ x ∈ {[]} ∪ concat L (kleene_closure α L), x ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: case a α : Type L : Language α ⊢ {[]} ∪ concat L (kleene_closure α L) ⊆ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_6	[436, 1]	[459, 14]	intro x a1	case a α : Type L : Language α ⊢ ∀ x ∈ {[]} ∪ concat L (kleene_closure α L), x ∈ kleene_closure α L	case a α : Type L : Language α x : Str α a1 : x ∈ {[]} ∪ concat L (kleene_closure α L) ⊢ x ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: case a α : Type L : Language α ⊢ ∀ x ∈ {[]} ∪ concat L (kleene_closure α L), x ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_6	[436, 1]	[459, 14]	simp at a1	case a α : Type L : Language α x : Str α a1 : x ∈ {[]} ∪ concat L (kleene_closure α L) ⊢ x ∈ kleene_closure α L	case a α : Type L : Language α x : Str α a1 : x = [] ∨ x ∈ concat L (kleene_closure α L) ⊢ x ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: case a α : Type L : Language α x : Str α a1 : x ∈ {[]} ∪ concat L (kleene_closure α L) ⊢ x ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_6	[436, 1]	[459, 14]	cases a1	case a α : Type L : Language α x : Str α a1 : x = [] ∨ x ∈ concat L (kleene_closure α L) ⊢ x ∈ kleene_closure α L	case a.inl α : Type L : Language α x : Str α h✝ : x = [] ⊢ x ∈ kleene_closure α L case a.inr α : Type L : Language α x : Str α h✝ : x ∈ concat L (kleene_closure α L) ⊢ x ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: case a α : Type L : Language α x : Str α a1 : x = [] ∨ x ∈ concat L (kleene_closure α L) ⊢ x ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_6	[436, 1]	[459, 14]	case _ a1_left => simp only [a1_left] exact kleene_closure.eps L	α : Type L : Language α x : Str α a1_left : x = [] ⊢ x ∈ kleene_closure α L	no goals	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α x : Str α a1_left : x = [] ⊢ x ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_6	[436, 1]	[459, 14]	case _ a1_right => simp only [thm_5 L] at a1_right sorry	α : Type L : Language α x : Str α a1_right : x ∈ concat L (kleene_closure α L) ⊢ x ∈ kleene_closure α L	no goals	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α x : Str α a1_right : x ∈ concat L (kleene_closure α L) ⊢ x ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_6	[436, 1]	[459, 14]	simp only [a1_left]	α : Type L : Language α x : Str α a1_left : x = [] ⊢ x ∈ kleene_closure α L	α : Type L : Language α x : Str α a1_left : x = [] ⊢ [] ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α x : Str α a1_left : x = [] ⊢ x ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_6	[436, 1]	[459, 14]	exact kleene_closure.eps L	α : Type L : Language α x : Str α a1_left : x = [] ⊢ [] ∈ kleene_closure α L	no goals	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α x : Str α a1_left : x = [] ⊢ [] ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_6	[436, 1]	[459, 14]	simp only [thm_5 L] at a1_right	α : Type L : Language α x : Str α a1_right : x ∈ concat L (kleene_closure α L) ⊢ x ∈ kleene_closure α L	α : Type L : Language α x : Str α a1_right : x ∈ concat L (⋃ n, exp L n) ⊢ x ∈ kleene_closure α L	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α x : Str α a1_right : x ∈ concat L (kleene_closure α L) ⊢ x ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/Text.lean	Languages.thm_6	[436, 1]	[459, 14]	sorry	α : Type L : Language α x : Str α a1_right : x ∈ concat L (⋃ n, exp L n) ⊢ x ∈ kleene_closure α L	no goals	Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α x : Str α a1_right : x ∈ concat L (⋃ n, exp L n) ⊢ x ∈ kleene_closure α L TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.occursIn_iff_mem_varSet	[186, 1]	[208, 17]	induction F	v : Var F : Formula ⊢ occursIn v F ↔ v ∈ F.varSet	case pred_ v : Var a✝¹ : String a✝ : List Var ⊢ occursIn v (pred_ a✝¹ a✝) ↔ v ∈ (pred_ a✝¹ a✝).varSet case not_ v : Var a✝ : Formula a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v a✝.not_ ↔ v ∈ a✝.not_.varSet case imp_ v : Var a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (a✝¹.imp_ a✝) ↔ v ∈ (a✝¹.imp_ a✝).varSet case forall_ v : Var a✝¹ : String a✝ : Formula a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (forall_ a✝¹ a✝) ↔ v ∈ (forall_ a✝¹ a✝).varSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var F : Formula ⊢ occursIn v F ↔ v ∈ F.varSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.occursIn_iff_mem_varSet	[186, 1]	[208, 17]	case pred_ X vs => simp only [occursIn] simp only [varSet] simp	v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ↔ v ∈ (pred_ X vs).varSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ↔ v ∈ (pred_ X vs).varSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.occursIn_iff_mem_varSet	[186, 1]	[208, 17]	case not_ phi phi_ih => simp only [occursIn] simp only [varSet] exact phi_ih	v : Var phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi.not_ ↔ v ∈ phi.not_.varSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: v : Var phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi.not_ ↔ v ∈ phi.not_.varSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.occursIn_iff_mem_varSet	[186, 1]	[208, 17]	case imp_ phi psi phi_ih psi_ih => simp only [occursIn] simp only [varSet] simp congr!	v : Var phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v (phi.imp_ psi) ↔ v ∈ (phi.imp_ psi).varSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v (phi.imp_ psi) ↔ v ∈ (phi.imp_ psi).varSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.occursIn_iff_mem_varSet	[186, 1]	[208, 17]	case forall_ _ phi phi_ih => simp only [occursIn] simp only [varSet] exact phi_ih	v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v (forall_ a✝ phi) ↔ v ∈ (forall_ a✝ phi).varSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v (forall_ a✝ phi) ↔ v ∈ (forall_ a✝ phi).varSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.occursIn_iff_mem_varSet	[186, 1]	[208, 17]	simp only [occursIn]	v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ↔ v ∈ (pred_ X vs).varSet	v : Var X : String vs : List Var ⊢ v ∈ vs ↔ v ∈ (pred_ X vs).varSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ↔ v ∈ (pred_ X vs).varSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.occursIn_iff_mem_varSet	[186, 1]	[208, 17]	simp only [varSet]	v : Var X : String vs : List Var ⊢ v ∈ vs ↔ v ∈ (pred_ X vs).varSet	v : Var X : String vs : List Var ⊢ v ∈ vs ↔ v ∈ vs.toFinset	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var ⊢ v ∈ vs ↔ v ∈ (pred_ X vs).varSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.occursIn_iff_mem_varSet	[186, 1]	[208, 17]	simp	v : Var X : String vs : List Var ⊢ v ∈ vs ↔ v ∈ vs.toFinset	no goals	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var ⊢ v ∈ vs ↔ v ∈ vs.toFinset TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.occursIn_iff_mem_varSet	[186, 1]	[208, 17]	simp only [occursIn]	v : Var phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi.not_ ↔ v ∈ phi.not_.varSet	v : Var phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi ↔ v ∈ phi.not_.varSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi.not_ ↔ v ∈ phi.not_.varSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.occursIn_iff_mem_varSet	[186, 1]	[208, 17]	simp only [varSet]	v : Var phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi ↔ v ∈ phi.not_.varSet	v : Var phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi ↔ v ∈ phi.varSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi ↔ v ∈ phi.not_.varSet TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_3_c

[255, 1]

[262, 9]

simp only [concat]

α : Type L1 L2 L3 : Language α ⊢ concat L1 (concat L2 L3) = concat (concat L1 L2) L3

α : Type L1 L2 L3 : Language α ⊢ {x | ∃ s ∈ L1, ∃ t ∈ {x | ∃ s ∈ L2, ∃ t ∈ L3, s ++ t = x}, s ++ t = x} = {x | ∃ s ∈ {x | ∃ s ∈ L1, ∃ t ∈ L2, s ++ t = x}, ∃ t ∈ L3, s ++ t = x}

Please generate a tactic in lean4 to solve the state. STATE: α : Type L1 L2 L3 : Language α ⊢ concat L1 (concat L2 L3) = concat (concat L1 L2) L3 TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_3_c

[255, 1]

[262, 9]

simp

α : Type L1 L2 L3 : Language α ⊢ {x | ∃ s ∈ L1, ∃ t ∈ {x | ∃ s ∈ L2, ∃ t ∈ L3, s ++ t = x}, s ++ t = x} = {x | ∃ s ∈ {x | ∃ s ∈ L1, ∃ t ∈ L2, s ++ t = x}, ∃ t ∈ L3, s ++ t = x}

no goals

Please generate a tactic in lean4 to solve the state. STATE: α : Type L1 L2 L3 : Language α ⊢ {x | ∃ s ∈ L1, ∃ t ∈ {x | ∃ s ∈ L2, ∃ t ∈ L3, s ++ t = x}, s ++ t = x} = {x | ∃ s ∈ {x | ∃ s ∈ L1, ∃ t ∈ L2, s ++ t = x}, ∃ t ∈ L3, s ++ t = x} TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_3_d

[265, 1]

[272, 10]

simp only [concat]

α : Type L1 L2 L3 : Language α ⊢ concat L1 (L2 ∪ L3) = concat L1 L2 ∪ concat L1 L3

α : Type L1 L2 L3 : Language α ⊢ {x | ∃ s ∈ L1, ∃ t ∈ L2 ∪ L3, s ++ t = x} = {x | ∃ s ∈ L1, ∃ t ∈ L2, s ++ t = x} ∪ {x | ∃ s ∈ L1, ∃ t ∈ L3, s ++ t = x}

Please generate a tactic in lean4 to solve the state. STATE: α : Type L1 L2 L3 : Language α ⊢ concat L1 (L2 ∪ L3) = concat L1 L2 ∪ concat L1 L3 TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_3_d

[265, 1]

[272, 10]

aesop

α : Type L1 L2 L3 : Language α ⊢ {x | ∃ s ∈ L1, ∃ t ∈ L2 ∪ L3, s ++ t = x} = {x | ∃ s ∈ L1, ∃ t ∈ L2, s ++ t = x} ∪ {x | ∃ s ∈ L1, ∃ t ∈ L3, s ++ t = x}

no goals

Please generate a tactic in lean4 to solve the state. STATE: α : Type L1 L2 L3 : Language α ⊢ {x | ∃ s ∈ L1, ∃ t ∈ L2 ∪ L3, s ++ t = x} = {x | ∃ s ∈ L1, ∃ t ∈ L2, s ++ t = x} ∪ {x | ∃ s ∈ L1, ∃ t ∈ L3, s ++ t = x} TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_4

[325, 1]

[361, 24]

simp only [Set.subset_def]

α : Type L : Language α n : ℕ ⊢ exp L n ⊆ kleene_closure α L

α : Type L : Language α n : ℕ ⊢ ∀ x ∈ exp L n, x ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ⊢ exp L n ⊆ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_4

[325, 1]

[361, 24]

intro s a1

α : Type L : Language α n : ℕ ⊢ ∀ x ∈ exp L n, x ∈ kleene_closure α L

α : Type L : Language α n : ℕ s : Str α a1 : s ∈ exp L n ⊢ s ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ⊢ ∀ x ∈ exp L n, x ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_4

[325, 1]

[361, 24]

induction n generalizing s

α : Type L : Language α n : ℕ s : Str α a1 : s ∈ exp L n ⊢ s ∈ kleene_closure α L

case zero α : Type L : Language α s : Str α a1 : s ∈ exp L 0 ⊢ s ∈ kleene_closure α L case succ α : Type L : Language α n✝ : ℕ a✝ : ∀ s ∈ exp L n✝, s ∈ kleene_closure α L s : Str α a1 : s ∈ exp L (n✝ + 1) ⊢ s ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ s : Str α a1 : s ∈ exp L n ⊢ s ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_4

[325, 1]

[361, 24]

case zero => simp only [exp] at a1 simp at a1 simp only [a1] exact kleene_closure.eps L

α : Type L : Language α s : Str α a1 : s ∈ exp L 0 ⊢ s ∈ kleene_closure α L

no goals

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α a1 : s ∈ exp L 0 ⊢ s ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_4

[325, 1]

[361, 24]

case succ n ih => simp only [exp] at a1 simp only [concat] at a1 simp at a1 apply Exists.elim a1 intro xs a2 clear a1 cases a2 case _ a2_left a2_right => apply Exists.elim a2_right intro ys a3 clear a2_right cases a3 case _ a3_left a3_right => simp only [← a3_right] apply kleene_closure.succ L apply ih xs a2_left exact a3_left

α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : s ∈ exp L (n + 1) ⊢ s ∈ kleene_closure α L

no goals

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : s ∈ exp L (n + 1) ⊢ s ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_4

[325, 1]

[361, 24]

simp only [exp] at a1

α : Type L : Language α s : Str α a1 : s ∈ exp L 0 ⊢ s ∈ kleene_closure α L

α : Type L : Language α s : Str α a1 : s ∈ {[]} ⊢ s ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α a1 : s ∈ exp L 0 ⊢ s ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_4

[325, 1]

[361, 24]

simp at a1

α : Type L : Language α s : Str α a1 : s ∈ {[]} ⊢ s ∈ kleene_closure α L

α : Type L : Language α s : Str α a1 : s = [] ⊢ s ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α a1 : s ∈ {[]} ⊢ s ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_4

[325, 1]

[361, 24]

simp only [a1]

α : Type L : Language α s : Str α a1 : s = [] ⊢ s ∈ kleene_closure α L

α : Type L : Language α s : Str α a1 : s = [] ⊢ [] ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α a1 : s = [] ⊢ s ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_4

[325, 1]

[361, 24]

exact kleene_closure.eps L

α : Type L : Language α s : Str α a1 : s = [] ⊢ [] ∈ kleene_closure α L

no goals

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α a1 : s = [] ⊢ [] ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_4

[325, 1]

[361, 24]

simp only [exp] at a1

α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : s ∈ exp L (n + 1) ⊢ s ∈ kleene_closure α L

α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : s ∈ concat (exp L n) L ⊢ s ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : s ∈ exp L (n + 1) ⊢ s ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_4

[325, 1]

[361, 24]

simp only [concat] at a1

α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : s ∈ concat (exp L n) L ⊢ s ∈ kleene_closure α L

α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : s ∈ {x | ∃ s ∈ exp L n, ∃ t ∈ L, s ++ t = x} ⊢ s ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : s ∈ concat (exp L n) L ⊢ s ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_4

[325, 1]

[361, 24]

simp at a1

α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : s ∈ {x | ∃ s ∈ exp L n, ∃ t ∈ L, s ++ t = x} ⊢ s ∈ kleene_closure α L

α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : ∃ s_1 ∈ exp L n, ∃ t ∈ L, s_1 ++ t = s ⊢ s ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : s ∈ {x | ∃ s ∈ exp L n, ∃ t ∈ L, s ++ t = x} ⊢ s ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_4

[325, 1]

[361, 24]

apply Exists.elim a1

α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : ∃ s_1 ∈ exp L n, ∃ t ∈ L, s_1 ++ t = s ⊢ s ∈ kleene_closure α L

α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : ∃ s_1 ∈ exp L n, ∃ t ∈ L, s_1 ++ t = s ⊢ ∀ (a : Str α), (a ∈ exp L n ∧ ∃ t ∈ L, a ++ t = s) → s ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : ∃ s_1 ∈ exp L n, ∃ t ∈ L, s_1 ++ t = s ⊢ s ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_4

[325, 1]

[361, 24]

intro xs a2

α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : ∃ s_1 ∈ exp L n, ∃ t ∈ L, s_1 ++ t = s ⊢ ∀ (a : Str α), (a ∈ exp L n ∧ ∃ t ∈ L, a ++ t = s) → s ∈ kleene_closure α L

α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : ∃ s_1 ∈ exp L n, ∃ t ∈ L, s_1 ++ t = s xs : Str α a2 : xs ∈ exp L n ∧ ∃ t ∈ L, xs ++ t = s ⊢ s ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : ∃ s_1 ∈ exp L n, ∃ t ∈ L, s_1 ++ t = s ⊢ ∀ (a : Str α), (a ∈ exp L n ∧ ∃ t ∈ L, a ++ t = s) → s ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_4

[325, 1]

[361, 24]

clear a1

α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : ∃ s_1 ∈ exp L n, ∃ t ∈ L, s_1 ++ t = s xs : Str α a2 : xs ∈ exp L n ∧ ∃ t ∈ L, xs ++ t = s ⊢ s ∈ kleene_closure α L

α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2 : xs ∈ exp L n ∧ ∃ t ∈ L, xs ++ t = s ⊢ s ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s : Str α a1 : ∃ s_1 ∈ exp L n, ∃ t ∈ L, s_1 ++ t = s xs : Str α a2 : xs ∈ exp L n ∧ ∃ t ∈ L, xs ++ t = s ⊢ s ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_4

[325, 1]

[361, 24]

cases a2

α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2 : xs ∈ exp L n ∧ ∃ t ∈ L, xs ++ t = s ⊢ s ∈ kleene_closure α L

case intro α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α left✝ : xs ∈ exp L n right✝ : ∃ t ∈ L, xs ++ t = s ⊢ s ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2 : xs ∈ exp L n ∧ ∃ t ∈ L, xs ++ t = s ⊢ s ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_4

[325, 1]

[361, 24]

case _ a2_left a2_right => apply Exists.elim a2_right intro ys a3 clear a2_right cases a3 case _ a3_left a3_right => simp only [← a3_right] apply kleene_closure.succ L apply ih xs a2_left exact a3_left

α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n a2_right : ∃ t ∈ L, xs ++ t = s ⊢ s ∈ kleene_closure α L

no goals

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n a2_right : ∃ t ∈ L, xs ++ t = s ⊢ s ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_4

[325, 1]

[361, 24]

apply Exists.elim a2_right

α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n a2_right : ∃ t ∈ L, xs ++ t = s ⊢ s ∈ kleene_closure α L

α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n a2_right : ∃ t ∈ L, xs ++ t = s ⊢ ∀ (a : Str α), a ∈ L ∧ xs ++ a = s → s ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n a2_right : ∃ t ∈ L, xs ++ t = s ⊢ s ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_4

[325, 1]

[361, 24]

intro ys a3

α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n a2_right : ∃ t ∈ L, xs ++ t = s ⊢ ∀ (a : Str α), a ∈ L ∧ xs ++ a = s → s ∈ kleene_closure α L

α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n a2_right : ∃ t ∈ L, xs ++ t = s ys : Str α a3 : ys ∈ L ∧ xs ++ ys = s ⊢ s ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n a2_right : ∃ t ∈ L, xs ++ t = s ⊢ ∀ (a : Str α), a ∈ L ∧ xs ++ a = s → s ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_4

[325, 1]

[361, 24]

clear a2_right

α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n a2_right : ∃ t ∈ L, xs ++ t = s ys : Str α a3 : ys ∈ L ∧ xs ++ ys = s ⊢ s ∈ kleene_closure α L

α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3 : ys ∈ L ∧ xs ++ ys = s ⊢ s ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n a2_right : ∃ t ∈ L, xs ++ t = s ys : Str α a3 : ys ∈ L ∧ xs ++ ys = s ⊢ s ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_4

[325, 1]

[361, 24]

cases a3

α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3 : ys ∈ L ∧ xs ++ ys = s ⊢ s ∈ kleene_closure α L

case intro α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α left✝ : ys ∈ L right✝ : xs ++ ys = s ⊢ s ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3 : ys ∈ L ∧ xs ++ ys = s ⊢ s ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_4

[325, 1]

[361, 24]

case _ a3_left a3_right => simp only [← a3_right] apply kleene_closure.succ L apply ih xs a2_left exact a3_left

α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ s ∈ kleene_closure α L

no goals

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ s ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_4

[325, 1]

[361, 24]

simp only [← a3_right]

α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ s ∈ kleene_closure α L

α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ xs ++ ys ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ s ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_4

[325, 1]

[361, 24]

apply kleene_closure.succ L

α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ xs ++ ys ∈ kleene_closure α L

case a α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ xs ∈ kleene_closure α L case a α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ ys ∈ L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ xs ++ ys ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_4

[325, 1]

[361, 24]

apply ih xs a2_left

case a α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ xs ∈ kleene_closure α L case a α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ ys ∈ L

case a α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ ys ∈ L

Please generate a tactic in lean4 to solve the state. STATE: case a α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ xs ∈ kleene_closure α L case a α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ ys ∈ L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_4

[325, 1]

[361, 24]

exact a3_left

case a α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ ys ∈ L

no goals

Please generate a tactic in lean4 to solve the state. STATE: case a α : Type L : Language α n : ℕ ih : ∀ s ∈ exp L n, s ∈ kleene_closure α L s xs : Str α a2_left : xs ∈ exp L n ys : Str α a3_left : ys ∈ L a3_right : xs ++ ys = s ⊢ ys ∈ L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_left

[364, 1]

[374, 50]

simp only [Set.subset_def]

α : Type L : Language α ⊢ ⋃ n, exp L n ⊆ kleene_closure α L

α : Type L : Language α ⊢ ∀ x ∈ ⋃ n, exp L n, x ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α ⊢ ⋃ n, exp L n ⊆ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_left

[364, 1]

[374, 50]

intro s a1

α : Type L : Language α ⊢ ∀ x ∈ ⋃ n, exp L n, x ∈ kleene_closure α L

α : Type L : Language α s : Str α a1 : s ∈ ⋃ n, exp L n ⊢ s ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α ⊢ ∀ x ∈ ⋃ n, exp L n, x ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_left

[364, 1]

[374, 50]

simp at a1

α : Type L : Language α s : Str α a1 : s ∈ ⋃ n, exp L n ⊢ s ∈ kleene_closure α L

α : Type L : Language α s : Str α a1 : ∃ i, s ∈ exp L i ⊢ s ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α a1 : s ∈ ⋃ n, exp L n ⊢ s ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_left

[364, 1]

[374, 50]

apply Exists.elim a1

α : Type L : Language α s : Str α a1 : ∃ i, s ∈ exp L i ⊢ s ∈ kleene_closure α L

α : Type L : Language α s : Str α a1 : ∃ i, s ∈ exp L i ⊢ ∀ (a : ℕ), s ∈ exp L a → s ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α a1 : ∃ i, s ∈ exp L i ⊢ s ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_left

[364, 1]

[374, 50]

intro n a2

α : Type L : Language α s : Str α a1 : ∃ i, s ∈ exp L i ⊢ ∀ (a : ℕ), s ∈ exp L a → s ∈ kleene_closure α L

α : Type L : Language α s : Str α a1 : ∃ i, s ∈ exp L i n : ℕ a2 : s ∈ exp L n ⊢ s ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α a1 : ∃ i, s ∈ exp L i ⊢ ∀ (a : ℕ), s ∈ exp L a → s ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_left

[364, 1]

[374, 50]

exact Set.mem_of_subset_of_mem (thm_4 L n) a2

α : Type L : Language α s : Str α a1 : ∃ i, s ∈ exp L i n : ℕ a2 : s ∈ exp L n ⊢ s ∈ kleene_closure α L

no goals

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α a1 : ∃ i, s ∈ exp L i n : ℕ a2 : s ∈ exp L n ⊢ s ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_right

[376, 1]

[406, 19]

simp only [Set.subset_def]

α : Type L : Language α ⊢ kleene_closure α L ⊆ ⋃ n, exp L n

α : Type L : Language α ⊢ ∀ x ∈ kleene_closure α L, x ∈ ⋃ n, exp L n

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α ⊢ kleene_closure α L ⊆ ⋃ n, exp L n TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_right

[376, 1]

[406, 19]

intro s a1

α : Type L : Language α ⊢ ∀ x ∈ kleene_closure α L, x ∈ ⋃ n, exp L n

α : Type L : Language α s : Str α a1 : s ∈ kleene_closure α L ⊢ s ∈ ⋃ n, exp L n

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α ⊢ ∀ x ∈ kleene_closure α L, x ∈ ⋃ n, exp L n TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_right

[376, 1]

[406, 19]

induction a1

α : Type L : Language α s : Str α a1 : s ∈ kleene_closure α L ⊢ s ∈ ⋃ n, exp L n

case eps α : Type L : Language α s : Str α ⊢ [] ∈ ⋃ n, exp L n case succ α : Type L : Language α s s✝ t✝ : Str α a✝¹ : s✝ ∈ kleene_closure α L a✝ : t✝ ∈ L a_ih✝ : s✝ ∈ ⋃ n, exp L n ⊢ s✝ ++ t✝ ∈ ⋃ n, exp L n

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α a1 : s ∈ kleene_closure α L ⊢ s ∈ ⋃ n, exp L n TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_right

[376, 1]

[406, 19]

case eps => simp apply Exists.intro 0 simp only [exp] simp

α : Type L : Language α s : Str α ⊢ [] ∈ ⋃ n, exp L n

no goals

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α ⊢ [] ∈ ⋃ n, exp L n TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_right

[376, 1]

[406, 19]

simp

α : Type L : Language α s : Str α ⊢ [] ∈ ⋃ n, exp L n

α : Type L : Language α s : Str α ⊢ ∃ i, [] ∈ exp L i

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α ⊢ [] ∈ ⋃ n, exp L n TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_right

[376, 1]

[406, 19]

apply Exists.intro 0

α : Type L : Language α s : Str α ⊢ ∃ i, [] ∈ exp L i

α : Type L : Language α s : Str α ⊢ [] ∈ exp L 0

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α ⊢ ∃ i, [] ∈ exp L i TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_right

[376, 1]

[406, 19]

simp only [exp]

α : Type L : Language α s : Str α ⊢ [] ∈ exp L 0

α : Type L : Language α s : Str α ⊢ [] ∈ {[]}

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α ⊢ [] ∈ exp L 0 TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_right

[376, 1]

[406, 19]

simp

α : Type L : Language α s : Str α ⊢ [] ∈ {[]}

no goals

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α ⊢ [] ∈ {[]} TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_right

[376, 1]

[406, 19]

simp at ih_3

α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L ih_3 : xs ∈ ⋃ n, exp L n ⊢ xs ++ ys ∈ ⋃ n, exp L n

α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L ih_3 : ∃ i, xs ∈ exp L i ⊢ xs ++ ys ∈ ⋃ n, exp L n

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L ih_3 : xs ∈ ⋃ n, exp L n ⊢ xs ++ ys ∈ ⋃ n, exp L n TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_right

[376, 1]

[406, 19]

apply Exists.elim ih_3

α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L ih_3 : ∃ i, xs ∈ exp L i ⊢ xs ++ ys ∈ ⋃ n, exp L n

α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L ih_3 : ∃ i, xs ∈ exp L i ⊢ ∀ (a : ℕ), xs ∈ exp L a → xs ++ ys ∈ ⋃ n, exp L n

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L ih_3 : ∃ i, xs ∈ exp L i ⊢ xs ++ ys ∈ ⋃ n, exp L n TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_right

[376, 1]

[406, 19]

intro n a2

α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L ih_3 : ∃ i, xs ∈ exp L i ⊢ ∀ (a : ℕ), xs ∈ exp L a → xs ++ ys ∈ ⋃ n, exp L n

α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L ih_3 : ∃ i, xs ∈ exp L i n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ ⋃ n, exp L n

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L ih_3 : ∃ i, xs ∈ exp L i ⊢ ∀ (a : ℕ), xs ∈ exp L a → xs ++ ys ∈ ⋃ n, exp L n TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_right

[376, 1]

[406, 19]

clear ih_3

α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L ih_3 : ∃ i, xs ∈ exp L i n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ ⋃ n, exp L n

α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ ⋃ n, exp L n

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L ih_3 : ∃ i, xs ∈ exp L i n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ ⋃ n, exp L n TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_right

[376, 1]

[406, 19]

simp

α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ ⋃ n, exp L n

α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ∃ i, xs ++ ys ∈ exp L i

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ ⋃ n, exp L n TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_right

[376, 1]

[406, 19]

apply Exists.intro (n + 1)

α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ∃ i, xs ++ ys ∈ exp L i

α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ exp L (n + 1)

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ∃ i, xs ++ ys ∈ exp L i TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_right

[376, 1]

[406, 19]

simp only [exp]

α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ exp L (n + 1)

α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ concat (exp L n) L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ exp L (n + 1) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_right

[376, 1]

[406, 19]

simp only [concat]

α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ concat (exp L n) L

α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ {x | ∃ s ∈ exp L n, ∃ t ∈ L, s ++ t = x}

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ concat (exp L n) L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_right

[376, 1]

[406, 19]

simp

α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ {x | ∃ s ∈ exp L n, ∃ t ∈ L, s ++ t = x}

α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ∃ s ∈ exp L n, ∃ t ∈ L, s ++ t = xs ++ ys

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ++ ys ∈ {x | ∃ s ∈ exp L n, ∃ t ∈ L, s ++ t = x} TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_right

[376, 1]

[406, 19]

apply Exists.intro xs

α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ∃ s ∈ exp L n, ∃ t ∈ L, s ++ t = xs ++ ys

α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ∈ exp L n ∧ ∃ t ∈ L, xs ++ t = xs ++ ys

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ∃ s ∈ exp L n, ∃ t ∈ L, s ++ t = xs ++ ys TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_right

[376, 1]

[406, 19]

constructor

α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ∈ exp L n ∧ ∃ t ∈ L, xs ++ t = xs ++ ys

case left α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ∈ exp L n case right α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ∃ t ∈ L, xs ++ t = xs ++ ys

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ∈ exp L n ∧ ∃ t ∈ L, xs ++ t = xs ++ ys TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_right

[376, 1]

[406, 19]

exact a2

case left α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ∈ exp L n

no goals

Please generate a tactic in lean4 to solve the state. STATE: case left α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ xs ∈ exp L n TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_right

[376, 1]

[406, 19]

apply Exists.intro ys

case right α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ∃ t ∈ L, xs ++ t = xs ++ ys

case right α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ys ∈ L ∧ xs ++ ys = xs ++ ys

Please generate a tactic in lean4 to solve the state. STATE: case right α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ∃ t ∈ L, xs ++ t = xs ++ ys TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_right

[376, 1]

[406, 19]

simp

case right α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ys ∈ L ∧ xs ++ ys = xs ++ ys

case right α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ys ∈ L

Please generate a tactic in lean4 to solve the state. STATE: case right α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ys ∈ L ∧ xs ++ ys = xs ++ ys TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5_right

[376, 1]

[406, 19]

exact ih_2

case right α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ys ∈ L

no goals

Please generate a tactic in lean4 to solve the state. STATE: case right α : Type L : Language α s xs ys : Str α a✝ : xs ∈ kleene_closure α L ih_2 : ys ∈ L n : ℕ a2 : xs ∈ exp L n ⊢ ys ∈ L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5

[409, 1]

[416, 25]

apply Set.eq_of_subset_of_subset

α : Type L : Language α ⊢ kleene_closure α L = ⋃ n, exp L n

case a α : Type L : Language α ⊢ kleene_closure α L ⊆ ⋃ n, exp L n case a α : Type L : Language α ⊢ ⋃ n, exp L n ⊆ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α ⊢ kleene_closure α L = ⋃ n, exp L n TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5

[409, 1]

[416, 25]

exact thm_5_right L

case a α : Type L : Language α ⊢ kleene_closure α L ⊆ ⋃ n, exp L n

no goals

Please generate a tactic in lean4 to solve the state. STATE: case a α : Type L : Language α ⊢ kleene_closure α L ⊆ ⋃ n, exp L n TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_5

[409, 1]

[416, 25]

exact thm_5_left L

case a α : Type L : Language α ⊢ ⋃ n, exp L n ⊆ kleene_closure α L

no goals

Please generate a tactic in lean4 to solve the state. STATE: case a α : Type L : Language α ⊢ ⋃ n, exp L n ⊆ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.concat_exp_comm

[419, 1]

[433, 26]

induction n

α : Type L : Language α n : ℕ ⊢ concat (exp L n) L = concat L (exp L n)

case zero α : Type L : Language α ⊢ concat (exp L 0) L = concat L (exp L 0) case succ α : Type L : Language α n✝ : ℕ a✝ : concat (exp L n✝) L = concat L (exp L n✝) ⊢ concat (exp L (n✝ + 1)) L = concat L (exp L (n✝ + 1))

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α n : ℕ ⊢ concat (exp L n) L = concat L (exp L n) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.concat_exp_comm

[419, 1]

[433, 26]

case zero => simp only [exp] simp only [concat] simp

α : Type L : Language α ⊢ concat (exp L 0) L = concat L (exp L 0)

no goals

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α ⊢ concat (exp L 0) L = concat L (exp L 0) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.concat_exp_comm

[419, 1]

[433, 26]

case succ k ih => simp only [exp] conv => left; simp only [ih] simp only [thm_3_c]

α : Type L : Language α k : ℕ ih : concat (exp L k) L = concat L (exp L k) ⊢ concat (exp L (k + 1)) L = concat L (exp L (k + 1))

no goals

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α k : ℕ ih : concat (exp L k) L = concat L (exp L k) ⊢ concat (exp L (k + 1)) L = concat L (exp L (k + 1)) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.concat_exp_comm

[419, 1]

[433, 26]

simp only [exp]

α : Type L : Language α ⊢ concat (exp L 0) L = concat L (exp L 0)

α : Type L : Language α ⊢ concat {[]} L = concat L {[]}

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α ⊢ concat (exp L 0) L = concat L (exp L 0) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.concat_exp_comm

[419, 1]

[433, 26]

simp only [concat]

α : Type L : Language α ⊢ concat {[]} L = concat L {[]}

α : Type L : Language α ⊢ {x | ∃ s ∈ {[]}, ∃ t ∈ L, s ++ t = x} = {x | ∃ s ∈ L, ∃ t ∈ {[]}, s ++ t = x}

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α ⊢ concat {[]} L = concat L {[]} TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.concat_exp_comm

[419, 1]

[433, 26]

simp

α : Type L : Language α ⊢ {x | ∃ s ∈ {[]}, ∃ t ∈ L, s ++ t = x} = {x | ∃ s ∈ L, ∃ t ∈ {[]}, s ++ t = x}

no goals

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α ⊢ {x | ∃ s ∈ {[]}, ∃ t ∈ L, s ++ t = x} = {x | ∃ s ∈ L, ∃ t ∈ {[]}, s ++ t = x} TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.concat_exp_comm

[419, 1]

[433, 26]

simp only [exp]

α : Type L : Language α k : ℕ ih : concat (exp L k) L = concat L (exp L k) ⊢ concat (exp L (k + 1)) L = concat L (exp L (k + 1))

α : Type L : Language α k : ℕ ih : concat (exp L k) L = concat L (exp L k) ⊢ concat (concat (exp L k) L) L = concat L (concat (exp L k) L)

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α k : ℕ ih : concat (exp L k) L = concat L (exp L k) ⊢ concat (exp L (k + 1)) L = concat L (exp L (k + 1)) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.concat_exp_comm

[419, 1]

[433, 26]

conv => left; simp only [ih]

α : Type L : Language α k : ℕ ih : concat (exp L k) L = concat L (exp L k) ⊢ concat (concat (exp L k) L) L = concat L (concat (exp L k) L)

α : Type L : Language α k : ℕ ih : concat (exp L k) L = concat L (exp L k) ⊢ concat (concat L (exp L k)) L = concat L (concat (exp L k) L)

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α k : ℕ ih : concat (exp L k) L = concat L (exp L k) ⊢ concat (concat (exp L k) L) L = concat L (concat (exp L k) L) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.concat_exp_comm

[419, 1]

[433, 26]

simp only [thm_3_c]

α : Type L : Language α k : ℕ ih : concat (exp L k) L = concat L (exp L k) ⊢ concat (concat L (exp L k)) L = concat L (concat (exp L k) L)

no goals

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α k : ℕ ih : concat (exp L k) L = concat L (exp L k) ⊢ concat (concat L (exp L k)) L = concat L (concat (exp L k) L) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_6

[436, 1]

[459, 14]

apply Set.eq_of_subset_of_subset

α : Type L : Language α ⊢ kleene_closure α L = {[]} ∪ concat L (kleene_closure α L)

case a α : Type L : Language α ⊢ kleene_closure α L ⊆ {[]} ∪ concat L (kleene_closure α L) case a α : Type L : Language α ⊢ {[]} ∪ concat L (kleene_closure α L) ⊆ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α ⊢ kleene_closure α L = {[]} ∪ concat L (kleene_closure α L) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_6

[436, 1]

[459, 14]

simp only [Set.subset_def]

case a α : Type L : Language α ⊢ kleene_closure α L ⊆ {[]} ∪ concat L (kleene_closure α L)

case a α : Type L : Language α ⊢ ∀ x ∈ kleene_closure α L, x ∈ {[]} ∪ concat L (kleene_closure α L)

Please generate a tactic in lean4 to solve the state. STATE: case a α : Type L : Language α ⊢ kleene_closure α L ⊆ {[]} ∪ concat L (kleene_closure α L) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_6

[436, 1]

[459, 14]

intro s a1

case a α : Type L : Language α ⊢ ∀ x ∈ kleene_closure α L, x ∈ {[]} ∪ concat L (kleene_closure α L)

case a α : Type L : Language α s : Str α a1 : s ∈ kleene_closure α L ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L)

Please generate a tactic in lean4 to solve the state. STATE: case a α : Type L : Language α ⊢ ∀ x ∈ kleene_closure α L, x ∈ {[]} ∪ concat L (kleene_closure α L) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_6

[436, 1]

[459, 14]

simp only [thm_5 L] at a1

case a α : Type L : Language α s : Str α a1 : s ∈ kleene_closure α L ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L)

case a α : Type L : Language α s : Str α a1 : s ∈ ⋃ n, exp L n ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L)

Please generate a tactic in lean4 to solve the state. STATE: case a α : Type L : Language α s : Str α a1 : s ∈ kleene_closure α L ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_6

[436, 1]

[459, 14]

simp at a1

case a α : Type L : Language α s : Str α a1 : s ∈ ⋃ n, exp L n ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L)

case a α : Type L : Language α s : Str α a1 : ∃ i, s ∈ exp L i ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L)

Please generate a tactic in lean4 to solve the state. STATE: case a α : Type L : Language α s : Str α a1 : s ∈ ⋃ n, exp L n ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_6

[436, 1]

[459, 14]

cases a1

case a α : Type L : Language α s : Str α a1 : ∃ i, s ∈ exp L i ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L)

case a.intro α : Type L : Language α s : Str α w✝ : ℕ h✝ : s ∈ exp L w✝ ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L)

Please generate a tactic in lean4 to solve the state. STATE: case a α : Type L : Language α s : Str α a1 : ∃ i, s ∈ exp L i ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_6

[436, 1]

[459, 14]

case _ i a2 => simp sorry

α : Type L : Language α s : Str α i : ℕ a2 : s ∈ exp L i ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L)

no goals

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α i : ℕ a2 : s ∈ exp L i ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_6

[436, 1]

[459, 14]

simp

α : Type L : Language α s : Str α i : ℕ a2 : s ∈ exp L i ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L)

α : Type L : Language α s : Str α i : ℕ a2 : s ∈ exp L i ⊢ s = [] ∨ s ∈ concat L (kleene_closure α L)

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α i : ℕ a2 : s ∈ exp L i ⊢ s ∈ {[]} ∪ concat L (kleene_closure α L) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_6

[436, 1]

[459, 14]

sorry

α : Type L : Language α s : Str α i : ℕ a2 : s ∈ exp L i ⊢ s = [] ∨ s ∈ concat L (kleene_closure α L)

no goals

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α s : Str α i : ℕ a2 : s ∈ exp L i ⊢ s = [] ∨ s ∈ concat L (kleene_closure α L) TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_6

[436, 1]

[459, 14]

simp only [Set.subset_def]

case a α : Type L : Language α ⊢ {[]} ∪ concat L (kleene_closure α L) ⊆ kleene_closure α L

case a α : Type L : Language α ⊢ ∀ x ∈ {[]} ∪ concat L (kleene_closure α L), x ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: case a α : Type L : Language α ⊢ {[]} ∪ concat L (kleene_closure α L) ⊆ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_6

[436, 1]

[459, 14]

intro x a1

case a α : Type L : Language α ⊢ ∀ x ∈ {[]} ∪ concat L (kleene_closure α L), x ∈ kleene_closure α L

case a α : Type L : Language α x : Str α a1 : x ∈ {[]} ∪ concat L (kleene_closure α L) ⊢ x ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: case a α : Type L : Language α ⊢ ∀ x ∈ {[]} ∪ concat L (kleene_closure α L), x ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_6

[436, 1]

[459, 14]

simp at a1

case a α : Type L : Language α x : Str α a1 : x ∈ {[]} ∪ concat L (kleene_closure α L) ⊢ x ∈ kleene_closure α L

case a α : Type L : Language α x : Str α a1 : x = [] ∨ x ∈ concat L (kleene_closure α L) ⊢ x ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: case a α : Type L : Language α x : Str α a1 : x ∈ {[]} ∪ concat L (kleene_closure α L) ⊢ x ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_6

[436, 1]

[459, 14]

cases a1

case a α : Type L : Language α x : Str α a1 : x = [] ∨ x ∈ concat L (kleene_closure α L) ⊢ x ∈ kleene_closure α L

case a.inl α : Type L : Language α x : Str α h✝ : x = [] ⊢ x ∈ kleene_closure α L case a.inr α : Type L : Language α x : Str α h✝ : x ∈ concat L (kleene_closure α L) ⊢ x ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: case a α : Type L : Language α x : Str α a1 : x = [] ∨ x ∈ concat L (kleene_closure α L) ⊢ x ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_6

[436, 1]

[459, 14]

case _ a1_left => simp only [a1_left] exact kleene_closure.eps L

α : Type L : Language α x : Str α a1_left : x = [] ⊢ x ∈ kleene_closure α L

no goals

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α x : Str α a1_left : x = [] ⊢ x ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_6

[436, 1]

[459, 14]

case _ a1_right => simp only [thm_5 L] at a1_right sorry

α : Type L : Language α x : Str α a1_right : x ∈ concat L (kleene_closure α L) ⊢ x ∈ kleene_closure α L

no goals

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α x : Str α a1_right : x ∈ concat L (kleene_closure α L) ⊢ x ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_6

[436, 1]

[459, 14]

simp only [a1_left]

α : Type L : Language α x : Str α a1_left : x = [] ⊢ x ∈ kleene_closure α L

α : Type L : Language α x : Str α a1_left : x = [] ⊢ [] ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α x : Str α a1_left : x = [] ⊢ x ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_6

[436, 1]

[459, 14]

exact kleene_closure.eps L

α : Type L : Language α x : Str α a1_left : x = [] ⊢ [] ∈ kleene_closure α L

no goals

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α x : Str α a1_left : x = [] ⊢ [] ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_6

[436, 1]

[459, 14]

simp only [thm_5 L] at a1_right

α : Type L : Language α x : Str α a1_right : x ∈ concat L (kleene_closure α L) ⊢ x ∈ kleene_closure α L

α : Type L : Language α x : Str α a1_right : x ∈ concat L (⋃ n, exp L n) ⊢ x ∈ kleene_closure α L

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α x : Str α a1_right : x ∈ concat L (kleene_closure α L) ⊢ x ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/Parsing/Text.lean

Languages.thm_6

[436, 1]

[459, 14]

sorry

α : Type L : Language α x : Str α a1_right : x ∈ concat L (⋃ n, exp L n) ⊢ x ∈ kleene_closure α L

no goals

Please generate a tactic in lean4 to solve the state. STATE: α : Type L : Language α x : Str α a1_right : x ∈ concat L (⋃ n, exp L n) ⊢ x ∈ kleene_closure α L TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.occursIn_iff_mem_varSet

[186, 1]

[208, 17]

induction F

v : Var F : Formula ⊢ occursIn v F ↔ v ∈ F.varSet

case pred_ v : Var a✝¹ : String a✝ : List Var ⊢ occursIn v (pred_ a✝¹ a✝) ↔ v ∈ (pred_ a✝¹ a✝).varSet case not_ v : Var a✝ : Formula a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v a✝.not_ ↔ v ∈ a✝.not_.varSet case imp_ v : Var a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (a✝¹.imp_ a✝) ↔ v ∈ (a✝¹.imp_ a✝).varSet case forall_ v : Var a✝¹ : String a✝ : Formula a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (forall_ a✝¹ a✝) ↔ v ∈ (forall_ a✝¹ a✝).varSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var F : Formula ⊢ occursIn v F ↔ v ∈ F.varSet TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.occursIn_iff_mem_varSet

[186, 1]

[208, 17]

case pred_ X vs => simp only [occursIn] simp only [varSet] simp

v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ↔ v ∈ (pred_ X vs).varSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ↔ v ∈ (pred_ X vs).varSet TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.occursIn_iff_mem_varSet

[186, 1]

[208, 17]

case not_ phi phi_ih => simp only [occursIn] simp only [varSet] exact phi_ih

v : Var phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi.not_ ↔ v ∈ phi.not_.varSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: v : Var phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi.not_ ↔ v ∈ phi.not_.varSet TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.occursIn_iff_mem_varSet

[186, 1]

[208, 17]

case imp_ phi psi phi_ih psi_ih => simp only [occursIn] simp only [varSet] simp congr!

v : Var phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v (phi.imp_ psi) ↔ v ∈ (phi.imp_ psi).varSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v (phi.imp_ psi) ↔ v ∈ (phi.imp_ psi).varSet TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.occursIn_iff_mem_varSet

[186, 1]

[208, 17]

case forall_ _ phi phi_ih => simp only [occursIn] simp only [varSet] exact phi_ih

v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v (forall_ a✝ phi) ↔ v ∈ (forall_ a✝ phi).varSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v (forall_ a✝ phi) ↔ v ∈ (forall_ a✝ phi).varSet TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.occursIn_iff_mem_varSet

[186, 1]

[208, 17]

simp only [occursIn]

v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ↔ v ∈ (pred_ X vs).varSet

v : Var X : String vs : List Var ⊢ v ∈ vs ↔ v ∈ (pred_ X vs).varSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ↔ v ∈ (pred_ X vs).varSet TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.occursIn_iff_mem_varSet

[186, 1]

[208, 17]

simp only [varSet]

v : Var X : String vs : List Var ⊢ v ∈ vs ↔ v ∈ (pred_ X vs).varSet

v : Var X : String vs : List Var ⊢ v ∈ vs ↔ v ∈ vs.toFinset

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var ⊢ v ∈ vs ↔ v ∈ (pred_ X vs).varSet TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.occursIn_iff_mem_varSet

[186, 1]

[208, 17]

simp

v : Var X : String vs : List Var ⊢ v ∈ vs ↔ v ∈ vs.toFinset

no goals

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var ⊢ v ∈ vs ↔ v ∈ vs.toFinset TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.occursIn_iff_mem_varSet

[186, 1]

[208, 17]

simp only [occursIn]

v : Var phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi.not_ ↔ v ∈ phi.not_.varSet

v : Var phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi ↔ v ∈ phi.not_.varSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi.not_ ↔ v ∈ phi.not_.varSet TACTIC:

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.occursIn_iff_mem_varSet

[186, 1]

[208, 17]

simp only [varSet]

v : Var phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi ↔ v ∈ phi.not_.varSet

v : Var phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi ↔ v ∈ phi.varSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi ↔ v ∈ phi.not_.varSet TACTIC: