url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Text.lean
|
Languages.thm_3_b
|
[246, 1]
|
[252, 9]
|
simp
|
α : Type
L : Language α
⊢ {x | ∃ s ∈ ∅, ∃ t ∈ L, s ++ t = x} = ∅
|
no goals
|
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}
|
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
|
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}
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
⊢ [] ∈ {[]}
|
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
|
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
|
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
|
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
|
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
|
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
|
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)
|
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
|
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}
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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))
|
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
|
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
|
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 {[]}
|
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}
|
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
|
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)
|
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)
|
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
|
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
|
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)
|
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)
|
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)
|
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)
|
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)
|
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
|
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)
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.lc_at_zero_iff_is_free
|
[171, 1]
|
[182, 9]
|
cases v
|
v : Var
⊢ Var.lc_at 0 v ↔ v.isFree
|
case free_
a✝ : String
⊢ Var.lc_at 0 (free_ a✝) ↔ (free_ a✝).isFree
case bound_
a✝ : ℕ
⊢ Var.lc_at 0 (bound_ a✝) ↔ (bound_ a✝).isFree
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.lc_at_zero_iff_is_free
|
[171, 1]
|
[182, 9]
|
case free_ x =>
simp only [Var.lc_at]
simp only [isFree]
|
x : String
⊢ Var.lc_at 0 (free_ x) ↔ (free_ x).isFree
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.lc_at_zero_iff_is_free
|
[171, 1]
|
[182, 9]
|
case bound_ i =>
simp only [Var.lc_at]
simp only [isFree]
simp
|
i : ℕ
⊢ Var.lc_at 0 (bound_ i) ↔ (bound_ i).isFree
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.lc_at_zero_iff_is_free
|
[171, 1]
|
[182, 9]
|
simp only [Var.lc_at]
|
x : String
⊢ Var.lc_at 0 (free_ x) ↔ (free_ x).isFree
|
x : String
⊢ True ↔ (free_ x).isFree
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.lc_at_zero_iff_is_free
|
[171, 1]
|
[182, 9]
|
simp only [isFree]
|
x : String
⊢ True ↔ (free_ x).isFree
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.lc_at_zero_iff_is_free
|
[171, 1]
|
[182, 9]
|
simp only [Var.lc_at]
|
i : ℕ
⊢ Var.lc_at 0 (bound_ i) ↔ (bound_ i).isFree
|
i : ℕ
⊢ i < 0 ↔ (bound_ i).isFree
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.lc_at_zero_iff_is_free
|
[171, 1]
|
[182, 9]
|
simp only [isFree]
|
i : ℕ
⊢ i < 0 ↔ (bound_ i).isFree
|
i : ℕ
⊢ i < 0 ↔ False
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.lc_at_zero_iff_is_free
|
[171, 1]
|
[182, 9]
|
simp
|
i : ℕ
⊢ i < 0 ↔ False
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/LN/Paper.lean
|
LN.free_var_list_to_string_list
|
[186, 1]
|
[210, 24]
|
induction vs
|
vs : List Var
h1 : ∀ v ∈ vs, Var.lc_at 0 v
⊢ ∃ xs, vs = List.map free_ xs
|
case nil
h1 : ∀ v ∈ [], Var.lc_at 0 v
⊢ ∃ xs, [] = List.map free_ xs
case cons
head✝ : Var
tail✝ : List Var
tail_ih✝ : (∀ v ∈ tail✝, Var.lc_at 0 v) → ∃ xs, tail✝ = List.map free_ xs
h1 : ∀ v ∈ head✝ :: tail✝, Var.lc_at 0 v
⊢ ∃ xs, head✝ :: tail✝ = List.map free_ xs
|
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