Datasets:
internlm
/
Lean-Github

Dataset card Data Studio Files
xet
 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_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
Ξ± : Type L1 L2 L3 : Language Ξ± ⊒ {x | βˆƒ s ∈ L1, βˆƒ t ∈ {x | βˆƒ s ∈ L2, βˆƒ t ∈ L3, s ++ t = x}, s ++ t = x} = {x | βˆƒ s ∈ {x | βˆƒ s ∈ L1, βˆƒ t ∈ L2, s ++ t = x}, βˆƒ 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
Ξ± : Type L1 L2 L3 : Language Ξ± ⊒ {x | βˆƒ s ∈ L1, βˆƒ t ∈ {x | βˆƒ s ∈ L2, βˆƒ t ∈ L3, s ++ t = x}, s ++ t = x} = {x | βˆƒ s ∈ {x | βˆƒ s ∈ L1, βˆƒ t ∈ L2, s ++ t = x}, βˆƒ 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]
Ξ± : Type L1 L2 L3 : Language Ξ± ⊒ concat L1 (L2 βˆͺ L3) = concat L1 L2 βˆͺ concat L1 L3
Ξ± : Type L1 L2 L3 : Language Ξ± ⊒ {x | βˆƒ s ∈ L1, βˆƒ t ∈ L2 βˆͺ L3, s ++ t = x} = {x | βˆƒ s ∈ L1, βˆƒ t ∈ L2, s ++ t = x} βˆͺ {x | βˆƒ s ∈ L1, βˆƒ 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
Ξ± : Type L1 L2 L3 : Language Ξ± ⊒ {x | βˆƒ s ∈ L1, βˆƒ t ∈ L2 βˆͺ L3, s ++ t = x} = {x | βˆƒ s ∈ L1, βˆƒ t ∈ L2, s ++ t = x} βˆͺ {x | βˆƒ s ∈ L1, βˆƒ 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]
Ξ± : Type L : Language Ξ± n : β„• ⊒ exp L n βŠ† kleene_closure Ξ± L
Ξ± : Type L : Language Ξ± n : β„• ⊒ βˆ€ x ∈ exp L n, x ∈ kleene_closure Ξ± L
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/Text.lean
Languages.thm_4
[325, 1]
[361, 24]
intro s a1
Ξ± : Type L : Language Ξ± n : β„• ⊒ βˆ€ x ∈ exp L n, x ∈ kleene_closure Ξ± L
Ξ± : Type L : Language Ξ± n : β„• s : Str Ξ± a1 : s ∈ exp L n ⊒ s ∈ kleene_closure Ξ± L
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/Text.lean
Languages.thm_4
[325, 1]
[361, 24]
induction n generalizing s
Ξ± : Type L : Language Ξ± n : β„• s : Str Ξ± a1 : s ∈ exp L n ⊒ s ∈ kleene_closure Ξ± L
case zero Ξ± : Type L : Language Ξ± s : Str Ξ± a1 : s ∈ exp L 0 ⊒ s ∈ kleene_closure Ξ± L case succ Ξ± : Type L : Language Ξ± n✝ : β„• a✝ : βˆ€ s ∈ exp L n✝, s ∈ kleene_closure Ξ± L s : Str Ξ± a1 : s ∈ exp L (n✝ + 1) ⊒ s ∈ kleene_closure Ξ± 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
Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s : Str Ξ± a1 : s ∈ exp L (n + 1) ⊒ s ∈ kleene_closure Ξ± 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
Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s : Str Ξ± a1 : s ∈ exp L (n + 1) ⊒ s ∈ kleene_closure Ξ± L
Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s : Str Ξ± a1 : s ∈ concat (exp L n) L ⊒ s ∈ kleene_closure Ξ± L
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/Text.lean
Languages.thm_4
[325, 1]
[361, 24]
simp only [concat] at a1
Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s : Str Ξ± a1 : s ∈ concat (exp L n) L ⊒ s ∈ kleene_closure Ξ± L
Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s : Str Ξ± a1 : s ∈ {x | βˆƒ s ∈ exp L n, βˆƒ t ∈ L, s ++ t = x} ⊒ s ∈ kleene_closure Ξ± L
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/Text.lean
Languages.thm_4
[325, 1]
[361, 24]
simp at a1
Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s : Str Ξ± a1 : s ∈ {x | βˆƒ s ∈ exp L n, βˆƒ t ∈ L, s ++ t = x} ⊒ s ∈ kleene_closure Ξ± L
Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s : Str Ξ± a1 : βˆƒ s_1 ∈ exp L n, βˆƒ t ∈ L, s_1 ++ t = s ⊒ s ∈ kleene_closure Ξ± L
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/Text.lean
Languages.thm_4
[325, 1]
[361, 24]
apply Exists.elim a1
Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s : Str Ξ± a1 : βˆƒ s_1 ∈ exp L n, βˆƒ t ∈ L, s_1 ++ t = s ⊒ s ∈ kleene_closure Ξ± L
Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s : Str Ξ± a1 : βˆƒ s_1 ∈ exp L n, βˆƒ t ∈ L, s_1 ++ t = s ⊒ βˆ€ (a : Str Ξ±), (a ∈ exp L n ∧ βˆƒ t ∈ L, a ++ t = s) β†’ s ∈ kleene_closure Ξ± L
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/Text.lean
Languages.thm_4
[325, 1]
[361, 24]
intro xs a2
Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s : Str Ξ± a1 : βˆƒ s_1 ∈ exp L n, βˆƒ t ∈ L, s_1 ++ t = s ⊒ βˆ€ (a : Str Ξ±), (a ∈ exp L n ∧ βˆƒ t ∈ L, a ++ t = s) β†’ s ∈ kleene_closure Ξ± L
Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s : Str Ξ± a1 : βˆƒ s_1 ∈ exp L n, βˆƒ t ∈ L, s_1 ++ t = s xs : Str Ξ± a2 : xs ∈ exp L n ∧ βˆƒ t ∈ L, xs ++ t = s ⊒ s ∈ kleene_closure Ξ± L
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/Text.lean
Languages.thm_4
[325, 1]
[361, 24]
clear a1
Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s : Str Ξ± a1 : βˆƒ s_1 ∈ exp L n, βˆƒ t ∈ L, s_1 ++ t = s xs : Str Ξ± a2 : xs ∈ exp L n ∧ βˆƒ t ∈ L, xs ++ t = s ⊒ s ∈ kleene_closure Ξ± L
Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s xs : Str Ξ± a2 : xs ∈ exp L n ∧ βˆƒ t ∈ L, xs ++ t = s ⊒ s ∈ kleene_closure Ξ± L
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/Text.lean
Languages.thm_4
[325, 1]
[361, 24]
cases a2
Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s xs : Str Ξ± a2 : xs ∈ exp L n ∧ βˆƒ t ∈ L, xs ++ t = s ⊒ s ∈ kleene_closure Ξ± L
case intro Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s xs : Str Ξ± left✝ : xs ∈ exp L n right✝ : βˆƒ t ∈ L, xs ++ t = s ⊒ s ∈ kleene_closure Ξ± 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
Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s xs : Str Ξ± a2_left : xs ∈ exp L n a2_right : βˆƒ t ∈ L, xs ++ t = s ⊒ s ∈ kleene_closure Ξ± 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
Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s xs : Str Ξ± a2_left : xs ∈ exp L n a2_right : βˆƒ t ∈ L, xs ++ t = s ⊒ s ∈ kleene_closure Ξ± L
Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s xs : Str Ξ± a2_left : xs ∈ exp L n a2_right : βˆƒ t ∈ L, xs ++ t = s ⊒ βˆ€ (a : Str Ξ±), a ∈ L ∧ xs ++ a = s β†’ s ∈ kleene_closure Ξ± L
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/Text.lean
Languages.thm_4
[325, 1]
[361, 24]
intro ys a3
Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s xs : Str Ξ± a2_left : xs ∈ exp L n a2_right : βˆƒ t ∈ L, xs ++ t = s ⊒ βˆ€ (a : Str Ξ±), a ∈ L ∧ xs ++ a = s β†’ s ∈ kleene_closure Ξ± L
Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s xs : Str Ξ± a2_left : xs ∈ exp L n a2_right : βˆƒ t ∈ L, xs ++ t = s ys : Str Ξ± a3 : ys ∈ L ∧ xs ++ ys = s ⊒ s ∈ kleene_closure Ξ± L
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/Text.lean
Languages.thm_4
[325, 1]
[361, 24]
clear a2_right
Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s xs : Str Ξ± a2_left : xs ∈ exp L n a2_right : βˆƒ t ∈ L, xs ++ t = s ys : Str Ξ± a3 : ys ∈ L ∧ xs ++ ys = s ⊒ s ∈ kleene_closure Ξ± L
Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s xs : Str Ξ± a2_left : xs ∈ exp L n ys : Str Ξ± a3 : ys ∈ L ∧ xs ++ ys = s ⊒ s ∈ kleene_closure Ξ± L
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/Text.lean
Languages.thm_4
[325, 1]
[361, 24]
cases a3
Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s xs : Str Ξ± a2_left : xs ∈ exp L n ys : Str Ξ± a3 : ys ∈ L ∧ xs ++ ys = s ⊒ s ∈ kleene_closure Ξ± L
case intro Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s xs : Str Ξ± a2_left : xs ∈ exp L n ys : Str Ξ± left✝ : ys ∈ L right✝ : xs ++ ys = s ⊒ s ∈ kleene_closure Ξ± 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
Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s xs : Str Ξ± a2_left : xs ∈ exp L n ys : Str Ξ± a3_left : ys ∈ L a3_right : xs ++ ys = s ⊒ s ∈ kleene_closure Ξ± 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]
Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s xs : Str Ξ± a2_left : xs ∈ exp L n ys : Str Ξ± a3_left : ys ∈ L a3_right : xs ++ ys = s ⊒ s ∈ kleene_closure Ξ± L
Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s xs : Str Ξ± a2_left : xs ∈ exp L n ys : Str Ξ± a3_left : ys ∈ L a3_right : xs ++ ys = s ⊒ xs ++ ys ∈ kleene_closure Ξ± L
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/Text.lean
Languages.thm_4
[325, 1]
[361, 24]
apply kleene_closure.succ L
Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s xs : Str Ξ± a2_left : xs ∈ exp L n ys : Str Ξ± a3_left : ys ∈ L a3_right : xs ++ ys = s ⊒ xs ++ ys ∈ kleene_closure Ξ± L
case a Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s xs : Str Ξ± a2_left : xs ∈ exp L n ys : Str Ξ± a3_left : ys ∈ L a3_right : xs ++ ys = s ⊒ xs ∈ kleene_closure Ξ± L case a Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s xs : Str Ξ± a2_left : xs ∈ exp L n ys : Str Ξ± 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 Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s xs : Str Ξ± a2_left : xs ∈ exp L n ys : Str Ξ± a3_left : ys ∈ L a3_right : xs ++ ys = s ⊒ xs ∈ kleene_closure Ξ± L case a Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s xs : Str Ξ± a2_left : xs ∈ exp L n ys : Str Ξ± a3_left : ys ∈ L a3_right : xs ++ ys = s ⊒ ys ∈ L
case a Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s xs : Str Ξ± a2_left : xs ∈ exp L n ys : Str Ξ± 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 Ξ± : Type L : Language Ξ± n : β„• ih : βˆ€ s ∈ exp L n, s ∈ kleene_closure Ξ± L s xs : Str Ξ± a2_left : xs ∈ exp L n ys : Str Ξ± 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]
Ξ± : Type L : Language Ξ± ⊒ ⋃ n, exp L n βŠ† kleene_closure Ξ± L
Ξ± : Type L : Language Ξ± ⊒ βˆ€ x ∈ ⋃ n, exp L n, x ∈ kleene_closure Ξ± L
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/Text.lean
Languages.thm_5_left
[364, 1]
[374, 50]
intro s a1
Ξ± : Type L : Language Ξ± ⊒ βˆ€ x ∈ ⋃ n, exp L n, x ∈ kleene_closure Ξ± L
Ξ± : Type L : Language Ξ± s : Str Ξ± a1 : s ∈ ⋃ n, exp L n ⊒ s ∈ kleene_closure Ξ± L
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/Text.lean
Languages.thm_5_left
[364, 1]
[374, 50]
simp at a1
Ξ± : Type L : Language Ξ± s : Str Ξ± a1 : s ∈ ⋃ n, exp L n ⊒ s ∈ kleene_closure Ξ± L
Ξ± : Type L : Language Ξ± s : Str Ξ± a1 : βˆƒ i, s ∈ exp L i ⊒ s ∈ kleene_closure Ξ± L
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/Text.lean
Languages.thm_5_left
[364, 1]
[374, 50]
apply Exists.elim a1
Ξ± : Type L : Language Ξ± s : Str Ξ± a1 : βˆƒ i, s ∈ exp L i ⊒ s ∈ kleene_closure Ξ± L
Ξ± : Type L : Language Ξ± s : Str Ξ± a1 : βˆƒ i, s ∈ exp L i ⊒ βˆ€ (a : β„•), s ∈ exp L a β†’ s ∈ kleene_closure Ξ± L
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/Text.lean
Languages.thm_5_left
[364, 1]
[374, 50]
intro n a2
Ξ± : Type L : Language Ξ± s : Str Ξ± a1 : βˆƒ i, s ∈ exp L i ⊒ βˆ€ (a : β„•), s ∈ exp L a β†’ s ∈ kleene_closure Ξ± L
Ξ± : Type L : Language Ξ± s : Str Ξ± a1 : βˆƒ i, s ∈ exp L i n : β„• a2 : s ∈ exp L n ⊒ s ∈ kleene_closure Ξ± 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
Ξ± : Type L : Language Ξ± s : Str Ξ± a1 : βˆƒ i, s ∈ exp L i n : β„• a2 : s ∈ exp L n ⊒ s ∈ kleene_closure Ξ± 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]
Ξ± : Type L : Language Ξ± ⊒ kleene_closure Ξ± L βŠ† ⋃ n, exp L n
Ξ± : Type L : Language Ξ± ⊒ βˆ€ x ∈ kleene_closure Ξ± 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
Ξ± : Type L : Language Ξ± ⊒ βˆ€ x ∈ kleene_closure Ξ± L, x ∈ ⋃ n, exp L n
Ξ± : Type L : Language Ξ± s : Str Ξ± a1 : s ∈ kleene_closure Ξ± 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
Ξ± : Type L : Language Ξ± s : Str Ξ± a1 : s ∈ kleene_closure Ξ± L ⊒ s ∈ ⋃ n, exp L n
case eps Ξ± : Type L : Language Ξ± s : Str Ξ± ⊒ [] ∈ ⋃ n, exp L n case succ Ξ± : Type L : Language Ξ± s s✝ t✝ : Str Ξ± a✝¹ : s✝ ∈ kleene_closure Ξ± 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
Ξ± : Type L : Language Ξ± s : Str Ξ± ⊒ [] ∈ ⋃ 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
Ξ± : Type L : Language Ξ± s : Str Ξ± ⊒ [] ∈ ⋃ n, exp L n
Ξ± : Type L : Language Ξ± s : Str Ξ± ⊒ βˆƒ 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
Ξ± : Type L : Language Ξ± s : Str Ξ± ⊒ βˆƒ 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
Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L ih_3 : xs ∈ ⋃ n, exp L n ⊒ xs ++ ys ∈ ⋃ n, exp L n
Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L ih_3 : βˆƒ 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
Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L ih_3 : βˆƒ i, xs ∈ exp L i ⊒ xs ++ ys ∈ ⋃ n, exp L n
Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L ih_3 : βˆƒ i, xs ∈ exp L i ⊒ βˆ€ (a : β„•), xs ∈ exp L a β†’ 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
Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L ih_3 : βˆƒ i, xs ∈ exp L i ⊒ βˆ€ (a : β„•), xs ∈ exp L a β†’ xs ++ ys ∈ ⋃ n, exp L n
Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L ih_3 : βˆƒ i, xs ∈ exp L i n : β„• 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
Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L ih_3 : βˆƒ i, xs ∈ exp L i n : β„• a2 : xs ∈ exp L n ⊒ xs ++ ys ∈ ⋃ n, exp L n
Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L n : β„• 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
Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L n : β„• a2 : xs ∈ exp L n ⊒ xs ++ ys ∈ ⋃ n, exp L n
Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L n : β„• a2 : xs ∈ exp L n ⊒ βˆƒ 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)
Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L n : β„• a2 : xs ∈ exp L n ⊒ βˆƒ i, xs ++ ys ∈ exp L i
Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L n : β„• 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]
Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L n : β„• a2 : xs ∈ exp L n ⊒ xs ++ ys ∈ exp L (n + 1)
Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L n : β„• 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]
Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L n : β„• a2 : xs ∈ exp L n ⊒ xs ++ ys ∈ concat (exp L n) L
Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L n : β„• a2 : xs ∈ exp L n ⊒ xs ++ ys ∈ {x | βˆƒ s ∈ exp L n, βˆƒ 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
Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L n : β„• a2 : xs ∈ exp L n ⊒ xs ++ ys ∈ {x | βˆƒ s ∈ exp L n, βˆƒ t ∈ L, s ++ t = x}
Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L n : β„• a2 : xs ∈ exp L n ⊒ βˆƒ s ∈ exp L n, βˆƒ 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
Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L n : β„• a2 : xs ∈ exp L n ⊒ βˆƒ s ∈ exp L n, βˆƒ t ∈ L, s ++ t = xs ++ ys
Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L n : β„• a2 : xs ∈ exp L n ⊒ xs ∈ exp L n ∧ βˆƒ 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
Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L n : β„• a2 : xs ∈ exp L n ⊒ xs ∈ exp L n ∧ βˆƒ t ∈ L, xs ++ t = xs ++ ys
case left Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L n : β„• a2 : xs ∈ exp L n ⊒ xs ∈ exp L n case right Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L n : β„• a2 : xs ∈ exp L n ⊒ βˆƒ 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 Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L n : β„• 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 Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L n : β„• a2 : xs ∈ exp L n ⊒ βˆƒ t ∈ L, xs ++ t = xs ++ ys
case right Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L n : β„• 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 Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L n : β„• a2 : xs ∈ exp L n ⊒ ys ∈ L ∧ xs ++ ys = xs ++ ys
case right Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L n : β„• 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 Ξ± : Type L : Language Ξ± s xs ys : Str Ξ± a✝ : xs ∈ kleene_closure Ξ± L ih_2 : ys ∈ L n : β„• 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
Ξ± : Type L : Language Ξ± ⊒ kleene_closure Ξ± L = ⋃ n, exp L n
case a Ξ± : Type L : Language Ξ± ⊒ kleene_closure Ξ± L βŠ† ⋃ n, exp L n case a Ξ± : Type L : Language Ξ± ⊒ ⋃ n, exp L n βŠ† kleene_closure Ξ± 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 Ξ± : Type L : Language Ξ± ⊒ kleene_closure Ξ± L βŠ† ⋃ 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 Ξ± : Type L : Language Ξ± ⊒ ⋃ n, exp L n βŠ† kleene_closure Ξ± L
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/Text.lean
Languages.concat_exp_comm
[419, 1]
[433, 26]
induction n
Ξ± : Type L : Language Ξ± n : β„• ⊒ concat (exp L n) L = concat L (exp L n)
case zero Ξ± : Type L : Language Ξ± ⊒ concat (exp L 0) L = concat L (exp L 0) case succ Ξ± : Type L : Language Ξ± n✝ : β„• 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]
Ξ± : Type L : Language Ξ± k : β„• 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 {[]}
Ξ± : Type L : Language Ξ± ⊒ {x | βˆƒ s ∈ {[]}, βˆƒ t ∈ L, s ++ t = x} = {x | βˆƒ s ∈ L, βˆƒ t ∈ {[]}, s ++ t = x}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/Text.lean
Languages.concat_exp_comm
[419, 1]
[433, 26]
simp
Ξ± : Type L : Language Ξ± ⊒ {x | βˆƒ s ∈ {[]}, βˆƒ t ∈ L, s ++ t = x} = {x | βˆƒ s ∈ L, βˆƒ 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]
Ξ± : Type L : Language Ξ± k : β„• ih : concat (exp L k) L = concat L (exp L k) ⊒ concat (exp L (k + 1)) L = concat L (exp L (k + 1))
Ξ± : Type L : Language Ξ± k : β„• 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]
Ξ± : Type L : Language Ξ± k : β„• 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)
Ξ± : Type L : Language Ξ± k : β„• 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]
Ξ± : Type L : Language Ξ± k : β„• 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
Ξ± : Type L : Language Ξ± ⊒ kleene_closure Ξ± L = {[]} βˆͺ concat L (kleene_closure Ξ± L)
case a Ξ± : Type L : Language Ξ± ⊒ kleene_closure Ξ± L βŠ† {[]} βˆͺ concat L (kleene_closure Ξ± L) case a Ξ± : Type L : Language Ξ± ⊒ {[]} βˆͺ concat L (kleene_closure Ξ± L) βŠ† kleene_closure Ξ± 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 Ξ± : Type L : Language Ξ± ⊒ kleene_closure Ξ± L βŠ† {[]} βˆͺ concat L (kleene_closure Ξ± L)
case a Ξ± : Type L : Language Ξ± ⊒ βˆ€ x ∈ kleene_closure Ξ± L, x ∈ {[]} βˆͺ concat L (kleene_closure Ξ± L)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/Text.lean
Languages.thm_6
[436, 1]
[459, 14]
intro s a1
case a Ξ± : Type L : Language Ξ± ⊒ βˆ€ x ∈ kleene_closure Ξ± L, x ∈ {[]} βˆͺ concat L (kleene_closure Ξ± L)
case a Ξ± : Type L : Language Ξ± s : Str Ξ± a1 : s ∈ kleene_closure Ξ± L ⊒ s ∈ {[]} βˆͺ concat L (kleene_closure Ξ± 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 Ξ± : Type L : Language Ξ± s : Str Ξ± a1 : s ∈ kleene_closure Ξ± L ⊒ s ∈ {[]} βˆͺ concat L (kleene_closure Ξ± L)
case a Ξ± : Type L : Language Ξ± s : Str Ξ± a1 : s ∈ ⋃ n, exp L n ⊒ s ∈ {[]} βˆͺ concat L (kleene_closure Ξ± L)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/Text.lean
Languages.thm_6
[436, 1]
[459, 14]
simp at a1
case a Ξ± : Type L : Language Ξ± s : Str Ξ± a1 : s ∈ ⋃ n, exp L n ⊒ s ∈ {[]} βˆͺ concat L (kleene_closure Ξ± L)
case a Ξ± : Type L : Language Ξ± s : Str Ξ± a1 : βˆƒ i, s ∈ exp L i ⊒ s ∈ {[]} βˆͺ concat L (kleene_closure Ξ± L)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/Text.lean
Languages.thm_6
[436, 1]
[459, 14]
cases a1
case a Ξ± : Type L : Language Ξ± s : Str Ξ± a1 : βˆƒ i, s ∈ exp L i ⊒ s ∈ {[]} βˆͺ concat L (kleene_closure Ξ± L)
case a.intro Ξ± : Type L : Language Ξ± s : Str Ξ± w✝ : β„• h✝ : s ∈ exp L w✝ ⊒ s ∈ {[]} βˆͺ concat L (kleene_closure Ξ± L)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/Text.lean
Languages.thm_6
[436, 1]
[459, 14]
case _ i a2 => simp sorry
Ξ± : Type L : Language Ξ± s : Str Ξ± i : β„• a2 : s ∈ exp L i ⊒ s ∈ {[]} βˆͺ concat L (kleene_closure Ξ± L)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/Text.lean
Languages.thm_6
[436, 1]
[459, 14]
simp
Ξ± : Type L : Language Ξ± s : Str Ξ± i : β„• a2 : s ∈ exp L i ⊒ s ∈ {[]} βˆͺ concat L (kleene_closure Ξ± L)
Ξ± : Type L : Language Ξ± s : Str Ξ± i : β„• a2 : s ∈ exp L i ⊒ s = [] ∨ s ∈ concat L (kleene_closure Ξ± L)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/Text.lean
Languages.thm_6
[436, 1]
[459, 14]
sorry
Ξ± : Type L : Language Ξ± s : Str Ξ± i : β„• a2 : s ∈ exp L i ⊒ s = [] ∨ s ∈ concat L (kleene_closure Ξ± 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 Ξ± : Type L : Language Ξ± ⊒ {[]} βˆͺ concat L (kleene_closure Ξ± L) βŠ† kleene_closure Ξ± L
case a Ξ± : Type L : Language Ξ± ⊒ βˆ€ x ∈ {[]} βˆͺ concat L (kleene_closure Ξ± L), x ∈ kleene_closure Ξ± L
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/Text.lean
Languages.thm_6
[436, 1]
[459, 14]
intro x a1
case a Ξ± : Type L : Language Ξ± ⊒ βˆ€ x ∈ {[]} βˆͺ concat L (kleene_closure Ξ± L), x ∈ kleene_closure Ξ± L
case a Ξ± : Type L : Language Ξ± x : Str Ξ± a1 : x ∈ {[]} βˆͺ concat L (kleene_closure Ξ± L) ⊒ x ∈ kleene_closure Ξ± L
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/Text.lean
Languages.thm_6
[436, 1]
[459, 14]
simp at a1
case a Ξ± : Type L : Language Ξ± x : Str Ξ± a1 : x ∈ {[]} βˆͺ concat L (kleene_closure Ξ± L) ⊒ x ∈ kleene_closure Ξ± 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
Ξ± : Type L : Language Ξ± x : Str Ξ± a1_right : x ∈ concat L (⋃ n, exp L n) ⊒ x ∈ kleene_closure Ξ± L
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/Text.lean
Languages.thm_6
[436, 1]
[459, 14]
sorry
Ξ± : Type L : Language Ξ± x : Str Ξ± a1_right : x ∈ concat L (⋃ n, exp L n) ⊒ x ∈ kleene_closure Ξ± L
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Binders.lean
LN.occursIn_iff_mem_varSet
[186, 1]
[208, 17]
induction F
v : Var F : Formula ⊒ occursIn v F ↔ v ∈ F.varSet
case pred_ v : Var a✝¹ : String a✝ : List Var ⊒ occursIn v (pred_ a✝¹ a✝) ↔ v ∈ (pred_ a✝¹ a✝).varSet case not_ v : Var a✝ : Formula a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊒ occursIn v a✝.not_ ↔ v ∈ a✝.not_.varSet case imp_ v : Var a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊒ occursIn v (a✝¹.imp_ a✝) ↔ v ∈ (a✝¹.imp_ a✝).varSet case forall_ v : Var a✝¹ : String a✝ : Formula a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊒ occursIn v (forall_ a✝¹ a✝) ↔ v ∈ (forall_ a✝¹ a✝).varSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Binders.lean
LN.occursIn_iff_mem_varSet
[186, 1]
[208, 17]
case pred_ X vs => simp only [occursIn] simp only [varSet] simp
v : Var X : String vs : List Var ⊒ occursIn v (pred_ X vs) ↔ v ∈ (pred_ X vs).varSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Binders.lean
LN.occursIn_iff_mem_varSet
[186, 1]
[208, 17]
case not_ phi phi_ih => simp only [occursIn] simp only [varSet] exact phi_ih
v : Var phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊒ occursIn v phi.not_ ↔ v ∈ phi.not_.varSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Binders.lean
LN.occursIn_iff_mem_varSet
[186, 1]
[208, 17]
case imp_ phi psi phi_ih psi_ih => simp only [occursIn] simp only [varSet] simp congr!
v : Var phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊒ occursIn v (phi.imp_ psi) ↔ v ∈ (phi.imp_ psi).varSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Binders.lean
LN.occursIn_iff_mem_varSet
[186, 1]
[208, 17]
case forall_ _ phi phi_ih => simp only [occursIn] simp only [varSet] exact phi_ih
v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊒ occursIn v (forall_ a✝ phi) ↔ v ∈ (forall_ a✝ phi).varSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Binders.lean
LN.occursIn_iff_mem_varSet
[186, 1]
[208, 17]
simp only [occursIn]
v : Var X : String vs : List Var ⊒ occursIn v (pred_ X vs) ↔ v ∈ (pred_ X vs).varSet
v : Var X : String vs : List Var ⊒ v ∈ vs ↔ v ∈ (pred_ X vs).varSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Binders.lean
LN.occursIn_iff_mem_varSet
[186, 1]
[208, 17]
simp only [varSet]
v : Var X : String vs : List Var ⊒ v ∈ vs ↔ v ∈ (pred_ X vs).varSet
v : Var X : String vs : List Var ⊒ v ∈ vs ↔ v ∈ vs.toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Binders.lean
LN.occursIn_iff_mem_varSet
[186, 1]
[208, 17]
simp
v : Var X : String vs : List Var ⊒ v ∈ vs ↔ v ∈ vs.toFinset
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Binders.lean
LN.occursIn_iff_mem_varSet
[186, 1]
[208, 17]
simp only [occursIn]
v : Var phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊒ occursIn v phi.not_ ↔ v ∈ phi.not_.varSet
v : Var phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊒ occursIn v phi ↔ v ∈ phi.not_.varSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Binders.lean
LN.occursIn_iff_mem_varSet
[186, 1]
[208, 17]
simp only [varSet]
v : Var phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊒ occursIn v phi ↔ v ∈ phi.not_.varSet
v : Var phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊒ occursIn v phi ↔ v ∈ phi.varSet