url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
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| start
stringlengths 6
10
| end
stringlengths 6
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stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
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stringlengths 6
2.09M
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|---|---|---|---|---|---|---|---|---|
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 |