Datasets:
internlm
/
Lean-Github

Dataset card Data Studio Files Community
1
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