url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
| input
stringlengths 73
2.09M
|
|---|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
induction E
|
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (def_ h1_X h1_xs) ↔
Holds D I V E (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
|
case nil
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) [] (def_ h1_X h1_xs) ↔
Holds D I V [] (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
case cons
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
Holds D I (Function.updateITE V h1_v (V h1_t)) tail✝ (def_ h1_X h1_xs) ↔
Holds D I V tail✝ (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) (head✝ :: tail✝) (def_ h1_X h1_xs) ↔
Holds D I V (head✝ :: tail✝) (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (def_ h1_X h1_xs) ↔
Holds D I V E (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
case nil =>
simp only [Holds]
|
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) [] (def_ h1_X h1_xs) ↔
Holds D I V [] (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) [] (def_ h1_X h1_xs) ↔
Holds D I V [] (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
simp only [Holds]
|
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) [] (def_ h1_X h1_xs) ↔
Holds D I V [] (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) [] (def_ h1_X h1_xs) ↔
Holds D I V [] (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
simp only [Holds]
|
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) (hd :: tl) (def_ h1_X h1_xs) ↔
Holds D I V (hd :: tl) (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
|
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
⊢ (if h1_X = hd.name ∧ h1_xs.length = hd.args.length then
Holds D I
(Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs))
tl hd.q
else Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs)) ↔
if h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length then
Holds D I (Function.updateListITE V hd.args (List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)))
tl hd.q
else Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) (hd :: tl) (def_ h1_X h1_xs) ↔
Holds D I V (hd :: tl) (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
split_ifs
|
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
⊢ (if h1_X = hd.name ∧ h1_xs.length = hd.args.length then
Holds D I
(Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs))
tl hd.q
else Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs)) ↔
if h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length then
Holds D I (Function.updateListITE V hd.args (List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)))
tl hd.q
else Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
|
case pos
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
h✝¹ : h1_X = hd.name ∧ h1_xs.length = hd.args.length
h✝ : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
⊢ Holds D I
(Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs))
tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)))
tl hd.q
case neg
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
h✝¹ : h1_X = hd.name ∧ h1_xs.length = hd.args.length
h✝ : ¬(h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length)
⊢ Holds D I
(Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs))
tl hd.q ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
case pos
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
h✝¹ : ¬(h1_X = hd.name ∧ h1_xs.length = hd.args.length)
h✝ : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I (Function.updateListITE V hd.args (List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)))
tl hd.q
case neg
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
h✝¹ : ¬(h1_X = hd.name ∧ h1_xs.length = hd.args.length)
h✝ : ¬(h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length)
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
⊢ (if h1_X = hd.name ∧ h1_xs.length = hd.args.length then
Holds D I
(Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs))
tl hd.q
else Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs)) ↔
if h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length then
Holds D I (Function.updateListITE V hd.args (List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)))
tl hd.q
else Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
case _ c1 c2 =>
simp only [List.length_map] at c2
contradiction
|
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : ¬(h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length)
⊢ Holds D I
(Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs))
tl hd.q ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : ¬(h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length)
⊢ Holds D I
(Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs))
tl hd.q ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
case _ c1 c2 =>
simp at c2
contradiction
|
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : ¬(h1_X = hd.name ∧ h1_xs.length = hd.args.length)
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I (Function.updateListITE V hd.args (List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)))
tl hd.q
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : ¬(h1_X = hd.name ∧ h1_xs.length = hd.args.length)
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I (Function.updateListITE V hd.args (List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)))
tl hd.q
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
case _ c1 c2 =>
exact ih
|
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : ¬(h1_X = hd.name ∧ h1_xs.length = hd.args.length)
c2 : ¬(h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length)
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : ¬(h1_X = hd.name ∧ h1_xs.length = hd.args.length)
c2 : ¬(h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length)
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
simp
|
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
⊢ Holds D I
(Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs))
tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)))
tl hd.q
|
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
⊢ Holds D I
(Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs))
tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs)) tl hd.q
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
⊢ Holds D I
(Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs))
tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)))
tl hd.q
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
apply Holds_coincide_Var
|
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
⊢ Holds D I
(Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs))
tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs)) tl hd.q
|
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
⊢ ∀ (v : VarName),
isFreeIn v hd.q →
Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
⊢ Holds D I
(Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs))
tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs)) tl hd.q
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
intro v' a1
|
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
⊢ ∀ (v : VarName),
isFreeIn v hd.q →
Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v
|
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v'
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
⊢ ∀ (v : VarName),
isFreeIn v hd.q →
Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
have s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun (x : VarName) => if h1_v = x then h1_t else x) h1_xs
|
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v'
|
case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v'
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
simp only [List.map_eq_map_iff]
|
case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v'
|
case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ ∀ x ∈ h1_xs, Function.updateITE V h1_v (V h1_t) x = (V ∘ fun x => if h1_v = x then h1_t else x) x
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v'
|
Please generate a tactic in lean4 to solve the state.
STATE:
case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
intro x _
|
case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ ∀ x ∈ h1_xs, Function.updateITE V h1_v (V h1_t) x = (V ∘ fun x => if h1_v = x then h1_t else x) x
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v'
|
case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
⊢ Function.updateITE V h1_v (V h1_t) x = (V ∘ fun x => if h1_v = x then h1_t else x) x
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v'
|
Please generate a tactic in lean4 to solve the state.
STATE:
case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ ∀ x ∈ h1_xs, Function.updateITE V h1_v (V h1_t) x = (V ∘ fun x => if h1_v = x then h1_t else x) x
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
simp only [Function.updateITE]
|
case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
⊢ Function.updateITE V h1_v (V h1_t) x = (V ∘ fun x => if h1_v = x then h1_t else x) x
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v'
|
case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
⊢ (if x = h1_v then V h1_t else V x) = (V ∘ fun x => if h1_v = x then h1_t else x) x
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v'
|
Please generate a tactic in lean4 to solve the state.
STATE:
case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
⊢ Function.updateITE V h1_v (V h1_t) x = (V ∘ fun x => if h1_v = x then h1_t else x) x
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
simp only [eq_comm]
|
case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
⊢ (if x = h1_v then V h1_t else V x) = (V ∘ fun x => if h1_v = x then h1_t else x) x
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v'
|
case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
⊢ (if h1_v = x then V h1_t else V x) = (V ∘ fun x => if h1_v = x then h1_t else x) x
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v'
|
Please generate a tactic in lean4 to solve the state.
STATE:
case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
⊢ (if x = h1_v then V h1_t else V x) = (V ∘ fun x => if h1_v = x then h1_t else x) x
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
simp
|
case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
⊢ (if h1_v = x then V h1_t else V x) = (V ∘ fun x => if h1_v = x then h1_t else x) x
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v'
|
case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
⊢ (if h1_v = x then V h1_t else V x) = V (if h1_v = x then h1_t else x)
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v'
|
Please generate a tactic in lean4 to solve the state.
STATE:
case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
⊢ (if h1_v = x then V h1_t else V x) = (V ∘ fun x => if h1_v = x then h1_t else x) x
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
split_ifs
|
case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
⊢ (if h1_v = x then V h1_t else V x) = V (if h1_v = x then h1_t else x)
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v'
|
case pos
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
h✝ : h1_v = x
⊢ V h1_t = V h1_t
case neg
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
h✝ : ¬h1_v = x
⊢ V x = V x
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v'
|
Please generate a tactic in lean4 to solve the state.
STATE:
case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
⊢ (if h1_v = x then V h1_t else V x) = V (if h1_v = x then h1_t else x)
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
case _ c3 =>
simp only [if_pos c3]
|
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
c3 : h1_v = x
⊢ V h1_t = V h1_t
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
c3 : h1_v = x
⊢ V h1_t = V h1_t
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
case _ c3 =>
simp only [if_neg c3]
|
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
c3 : ¬h1_v = x
⊢ V x = V x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
c3 : ¬h1_v = x
⊢ V x = V x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
simp only [s1]
|
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v'
|
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v'
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
apply Function.updateListITE_mem_eq_len
|
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v'
|
case h1.h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ v' ∈ hd.args
case h1.h2
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ hd.args.length = (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs).length
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
simp only [if_pos c3]
|
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
c3 : h1_v = x
⊢ V h1_t = V h1_t
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
c3 : h1_v = x
⊢ V h1_t = V h1_t
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
simp only [if_neg c3]
|
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
c3 : ¬h1_v = x
⊢ V x = V x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
c3 : ¬h1_v = x
⊢ V x = V x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
simp only [isFreeIn_iff_mem_freeVarSet] at a1
|
case h1.h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ v' ∈ hd.args
|
case h1.h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
a1 : v' ∈ hd.q.freeVarSet
⊢ v' ∈ hd.args
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ v' ∈ hd.args
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
simp only [← List.mem_toFinset]
|
case h1.h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
a1 : v' ∈ hd.q.freeVarSet
⊢ v' ∈ hd.args
|
case h1.h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
a1 : v' ∈ hd.q.freeVarSet
⊢ v' ∈ hd.args.toFinset
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
a1 : v' ∈ hd.q.freeVarSet
⊢ v' ∈ hd.args
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
apply Finset.mem_of_subset hd.h1 a1
|
case h1.h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
a1 : v' ∈ hd.q.freeVarSet
⊢ v' ∈ hd.args.toFinset
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
a1 : v' ∈ hd.q.freeVarSet
⊢ v' ∈ hd.args.toFinset
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
simp
|
case h1.h2
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ hd.args.length = (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs).length
|
case h1.h2
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ hd.args.length = h1_xs.length
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h2
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ hd.args.length = (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs).length
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
tauto
|
case h1.h2
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ hd.args.length = h1_xs.length
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h2
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ hd.args.length = h1_xs.length
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
simp only [List.length_map] at c2
|
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : ¬(h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length)
⊢ Holds D I
(Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs))
tl hd.q ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
|
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : ¬(h1_X = hd.name ∧ h1_xs.length = hd.args.length)
⊢ Holds D I
(Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs))
tl hd.q ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : ¬(h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length)
⊢ Holds D I
(Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs))
tl hd.q ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
contradiction
|
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : ¬(h1_X = hd.name ∧ h1_xs.length = hd.args.length)
⊢ Holds D I
(Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs))
tl hd.q ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : ¬(h1_X = hd.name ∧ h1_xs.length = hd.args.length)
⊢ Holds D I
(Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs))
tl hd.q ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
simp at c2
|
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : ¬(h1_X = hd.name ∧ h1_xs.length = hd.args.length)
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I (Function.updateListITE V hd.args (List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)))
tl hd.q
|
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : ¬(h1_X = hd.name ∧ h1_xs.length = hd.args.length)
c2 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I (Function.updateListITE V hd.args (List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)))
tl hd.q
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : ¬(h1_X = hd.name ∧ h1_xs.length = hd.args.length)
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I (Function.updateListITE V hd.args (List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)))
tl hd.q
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
contradiction
|
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : ¬(h1_X = hd.name ∧ h1_xs.length = hd.args.length)
c2 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I (Function.updateListITE V hd.args (List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)))
tl hd.q
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : ¬(h1_X = hd.name ∧ h1_xs.length = hd.args.length)
c2 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I (Function.updateListITE V hd.args (List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)))
tl hd.q
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_theorem
|
[298, 1]
|
[424, 17]
|
exact ih
|
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : ¬(h1_X = hd.name ∧ h1_xs.length = hd.args.length)
c2 : ¬(h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length)
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : ¬(h1_X = hd.name ∧ h1_xs.length = hd.args.length)
c2 : ¬(h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length)
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_is_valid
|
[427, 1]
|
[439, 11]
|
simp only [IsValid] at h2
|
v t : VarName
F F' : Formula
h1 : IsSub F v t F'
h2 : F.IsValid
⊢ F'.IsValid
|
v t : VarName
F F' : Formula
h1 : IsSub F v t F'
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ F'.IsValid
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
F F' : Formula
h1 : IsSub F v t F'
h2 : F.IsValid
⊢ F'.IsValid
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_is_valid
|
[427, 1]
|
[439, 11]
|
simp only [IsValid]
|
v t : VarName
F F' : Formula
h1 : IsSub F v t F'
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ F'.IsValid
|
v t : VarName
F F' : Formula
h1 : IsSub F v t F'
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F'
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
F F' : Formula
h1 : IsSub F v t F'
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ F'.IsValid
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_is_valid
|
[427, 1]
|
[439, 11]
|
intro D I V E
|
v t : VarName
F F' : Formula
h1 : IsSub F v t F'
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F'
|
v t : VarName
F F' : Formula
h1 : IsSub F v t F'
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D I V E F'
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
F F' : Formula
h1 : IsSub F v t F'
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_is_valid
|
[427, 1]
|
[439, 11]
|
simp only [← substitution_theorem D I V E v t F F' h1]
|
v t : VarName
F F' : Formula
h1 : IsSub F v t F'
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D I V E F'
|
v t : VarName
F F' : Formula
h1 : IsSub F v t F'
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D I (Function.updateITE V v (V t)) E F
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
F F' : Formula
h1 : IsSub F v t F'
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D I V E F'
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/One/Ind/Sub.lean
|
FOL.NV.Sub.Var.One.Ind.substitution_is_valid
|
[427, 1]
|
[439, 11]
|
apply h2
|
v t : VarName
F F' : Formula
h1 : IsSub F v t F'
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D I (Function.updateITE V v (V t)) E F
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
v t : VarName
F F' : Formula
h1 : IsSub F v t F'
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D I (Function.updateITE V v (V t)) E F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
induction e
|
α : Type
e : RegExp α
⊢ e.is_nullable ↔ [] ∈ RegExp.languageOf α e
|
case char
α : Type
a✝ : α
⊢ (RegExp.char a✝).is_nullable ↔ [] ∈ RegExp.languageOf α (RegExp.char a✝)
case epsilon
α : Type
⊢ RegExp.epsilon.is_nullable ↔ [] ∈ RegExp.languageOf α RegExp.epsilon
case zero
α : Type
⊢ RegExp.zero.is_nullable ↔ [] ∈ RegExp.languageOf α RegExp.zero
case union
α : Type
a✝¹ a✝ : RegExp α
a_ih✝¹ : a✝¹.is_nullable ↔ [] ∈ RegExp.languageOf α a✝¹
a_ih✝ : a✝.is_nullable ↔ [] ∈ RegExp.languageOf α a✝
⊢ (a✝¹.union a✝).is_nullable ↔ [] ∈ RegExp.languageOf α (a✝¹.union a✝)
case concat
α : Type
a✝¹ a✝ : RegExp α
a_ih✝¹ : a✝¹.is_nullable ↔ [] ∈ RegExp.languageOf α a✝¹
a_ih✝ : a✝.is_nullable ↔ [] ∈ RegExp.languageOf α a✝
⊢ (a✝¹.concat a✝).is_nullable ↔ [] ∈ RegExp.languageOf α (a✝¹.concat a✝)
case closure
α : Type
a✝ : RegExp α
a_ih✝ : a✝.is_nullable ↔ [] ∈ RegExp.languageOf α a✝
⊢ a✝.closure.is_nullable ↔ [] ∈ RegExp.languageOf α a✝.closure
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
e : RegExp α
⊢ e.is_nullable ↔ [] ∈ RegExp.languageOf α e
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
case char a =>
simp only [RegExp.languageOf]
simp only [RegExp.is_nullable]
simp
|
α : Type
a : α
⊢ (RegExp.char a).is_nullable ↔ [] ∈ RegExp.languageOf α (RegExp.char a)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
a : α
⊢ (RegExp.char a).is_nullable ↔ [] ∈ RegExp.languageOf α (RegExp.char a)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
case epsilon =>
simp only [RegExp.languageOf]
simp only [RegExp.is_nullable]
simp
|
α : Type
⊢ RegExp.epsilon.is_nullable ↔ [] ∈ RegExp.languageOf α RegExp.epsilon
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
⊢ RegExp.epsilon.is_nullable ↔ [] ∈ RegExp.languageOf α RegExp.epsilon
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
case zero =>
simp only [RegExp.languageOf]
simp only [RegExp.is_nullable]
simp
|
α : Type
⊢ RegExp.zero.is_nullable ↔ [] ∈ RegExp.languageOf α RegExp.zero
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
⊢ RegExp.zero.is_nullable ↔ [] ∈ RegExp.languageOf α RegExp.zero
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
case union r s r_ih s_ih =>
simp only [RegExp.languageOf]
simp only [RegExp.is_nullable]
simp
simp only [r_ih]
simp only [s_ih]
|
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ (r.union s).is_nullable ↔ [] ∈ RegExp.languageOf α (r.union s)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ (r.union s).is_nullable ↔ [] ∈ RegExp.languageOf α (r.union s)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
case concat r s r_ih s_ih =>
simp only [RegExp.languageOf]
simp only [RegExp.is_nullable]
simp
simp only [r_ih]
simp only [s_ih]
|
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ (r.concat s).is_nullable ↔ [] ∈ RegExp.languageOf α (r.concat s)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ (r.concat s).is_nullable ↔ [] ∈ RegExp.languageOf α (r.concat s)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
case closure e _ =>
simp only [equiv_language_of_closure]
simp only [RegExp.languageOf]
simp only [RegExp.is_nullable]
simp
|
α : Type
e : RegExp α
a_ih✝ : e.is_nullable ↔ [] ∈ RegExp.languageOf α e
⊢ e.closure.is_nullable ↔ [] ∈ RegExp.languageOf α e.closure
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
e : RegExp α
a_ih✝ : e.is_nullable ↔ [] ∈ RegExp.languageOf α e
⊢ e.closure.is_nullable ↔ [] ∈ RegExp.languageOf α e.closure
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
simp only [RegExp.languageOf]
|
α : Type
a : α
⊢ (RegExp.char a).is_nullable ↔ [] ∈ RegExp.languageOf α (RegExp.char a)
|
α : Type
a : α
⊢ (RegExp.char a).is_nullable ↔ [] ∈ {[a]}
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
a : α
⊢ (RegExp.char a).is_nullable ↔ [] ∈ RegExp.languageOf α (RegExp.char a)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
simp only [RegExp.is_nullable]
|
α : Type
a : α
⊢ (RegExp.char a).is_nullable ↔ [] ∈ {[a]}
|
α : Type
a : α
⊢ False ↔ [] ∈ {[a]}
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
a : α
⊢ (RegExp.char a).is_nullable ↔ [] ∈ {[a]}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
simp
|
α : Type
a : α
⊢ False ↔ [] ∈ {[a]}
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
a : α
⊢ False ↔ [] ∈ {[a]}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
simp only [RegExp.languageOf]
|
α : Type
⊢ RegExp.epsilon.is_nullable ↔ [] ∈ RegExp.languageOf α RegExp.epsilon
|
α : Type
⊢ RegExp.epsilon.is_nullable ↔ [] ∈ {[]}
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
⊢ RegExp.epsilon.is_nullable ↔ [] ∈ RegExp.languageOf α RegExp.epsilon
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
simp only [RegExp.is_nullable]
|
α : Type
⊢ RegExp.epsilon.is_nullable ↔ [] ∈ {[]}
|
α : Type
⊢ True ↔ [] ∈ {[]}
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
⊢ RegExp.epsilon.is_nullable ↔ [] ∈ {[]}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
simp
|
α : Type
⊢ True ↔ [] ∈ {[]}
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
⊢ True ↔ [] ∈ {[]}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
simp only [RegExp.languageOf]
|
α : Type
⊢ RegExp.zero.is_nullable ↔ [] ∈ RegExp.languageOf α RegExp.zero
|
α : Type
⊢ RegExp.zero.is_nullable ↔ [] ∈ ∅
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
⊢ RegExp.zero.is_nullable ↔ [] ∈ RegExp.languageOf α RegExp.zero
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
simp only [RegExp.is_nullable]
|
α : Type
⊢ RegExp.zero.is_nullable ↔ [] ∈ ∅
|
α : Type
⊢ False ↔ [] ∈ ∅
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
⊢ RegExp.zero.is_nullable ↔ [] ∈ ∅
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
simp
|
α : Type
⊢ False ↔ [] ∈ ∅
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
⊢ False ↔ [] ∈ ∅
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
simp only [RegExp.languageOf]
|
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ (r.union s).is_nullable ↔ [] ∈ RegExp.languageOf α (r.union s)
|
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ (r.union s).is_nullable ↔ [] ∈ RegExp.languageOf α r ∪ RegExp.languageOf α s
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ (r.union s).is_nullable ↔ [] ∈ RegExp.languageOf α (r.union s)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
simp only [RegExp.is_nullable]
|
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ (r.union s).is_nullable ↔ [] ∈ RegExp.languageOf α r ∪ RegExp.languageOf α s
|
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ r.is_nullable ∨ s.is_nullable ↔ [] ∈ RegExp.languageOf α r ∪ RegExp.languageOf α s
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ (r.union s).is_nullable ↔ [] ∈ RegExp.languageOf α r ∪ RegExp.languageOf α s
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
simp
|
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ r.is_nullable ∨ s.is_nullable ↔ [] ∈ RegExp.languageOf α r ∪ RegExp.languageOf α s
|
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ r.is_nullable ∨ s.is_nullable ↔ [] ∈ RegExp.languageOf α r ∨ [] ∈ RegExp.languageOf α s
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ r.is_nullable ∨ s.is_nullable ↔ [] ∈ RegExp.languageOf α r ∪ RegExp.languageOf α s
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
simp only [r_ih]
|
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ r.is_nullable ∨ s.is_nullable ↔ [] ∈ RegExp.languageOf α r ∨ [] ∈ RegExp.languageOf α s
|
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ [] ∈ RegExp.languageOf α r ∨ s.is_nullable ↔ [] ∈ RegExp.languageOf α r ∨ [] ∈ RegExp.languageOf α s
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ r.is_nullable ∨ s.is_nullable ↔ [] ∈ RegExp.languageOf α r ∨ [] ∈ RegExp.languageOf α s
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
simp only [s_ih]
|
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ [] ∈ RegExp.languageOf α r ∨ s.is_nullable ↔ [] ∈ RegExp.languageOf α r ∨ [] ∈ RegExp.languageOf α s
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ [] ∈ RegExp.languageOf α r ∨ s.is_nullable ↔ [] ∈ RegExp.languageOf α r ∨ [] ∈ RegExp.languageOf α s
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
simp only [RegExp.languageOf]
|
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ (r.concat s).is_nullable ↔ [] ∈ RegExp.languageOf α (r.concat s)
|
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ (r.concat s).is_nullable ↔ [] ∈ {x | ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = x}
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ (r.concat s).is_nullable ↔ [] ∈ RegExp.languageOf α (r.concat s)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
simp only [RegExp.is_nullable]
|
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ (r.concat s).is_nullable ↔ [] ∈ {x | ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = x}
|
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ r.is_nullable ∧ s.is_nullable ↔
[] ∈ {x | ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = x}
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ (r.concat s).is_nullable ↔ [] ∈ {x | ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = x}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
simp
|
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ r.is_nullable ∧ s.is_nullable ↔
[] ∈ {x | ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = x}
|
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ r.is_nullable ∧ s.is_nullable ↔ [] ∈ RegExp.languageOf α r ∧ [] ∈ RegExp.languageOf α s
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ r.is_nullable ∧ s.is_nullable ↔
[] ∈ {x | ∃ r_1 ∈ RegExp.languageOf α r, ∃ s_1 ∈ RegExp.languageOf α s, r_1 ++ s_1 = x}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
simp only [r_ih]
|
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ r.is_nullable ∧ s.is_nullable ↔ [] ∈ RegExp.languageOf α r ∧ [] ∈ RegExp.languageOf α s
|
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ [] ∈ RegExp.languageOf α r ∧ s.is_nullable ↔ [] ∈ RegExp.languageOf α r ∧ [] ∈ RegExp.languageOf α s
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ r.is_nullable ∧ s.is_nullable ↔ [] ∈ RegExp.languageOf α r ∧ [] ∈ RegExp.languageOf α s
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
simp only [s_ih]
|
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ [] ∈ RegExp.languageOf α r ∧ s.is_nullable ↔ [] ∈ RegExp.languageOf α r ∧ [] ∈ RegExp.languageOf α s
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : r.is_nullable ↔ [] ∈ RegExp.languageOf α r
s_ih : s.is_nullable ↔ [] ∈ RegExp.languageOf α s
⊢ [] ∈ RegExp.languageOf α r ∧ s.is_nullable ↔ [] ∈ RegExp.languageOf α r ∧ [] ∈ RegExp.languageOf α s
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
simp only [equiv_language_of_closure]
|
α : Type
e : RegExp α
a_ih✝ : e.is_nullable ↔ [] ∈ RegExp.languageOf α e
⊢ e.closure.is_nullable ↔ [] ∈ RegExp.languageOf α e.closure
|
α : Type
e : RegExp α
a_ih✝ : e.is_nullable ↔ [] ∈ RegExp.languageOf α e
⊢ e.closure.is_nullable ↔ [] ∈ RegExp.languageOf α (RegExp.epsilon.union (e.concat e.closure))
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
e : RegExp α
a_ih✝ : e.is_nullable ↔ [] ∈ RegExp.languageOf α e
⊢ e.closure.is_nullable ↔ [] ∈ RegExp.languageOf α e.closure
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
simp only [RegExp.languageOf]
|
α : Type
e : RegExp α
a_ih✝ : e.is_nullable ↔ [] ∈ RegExp.languageOf α e
⊢ e.closure.is_nullable ↔ [] ∈ RegExp.languageOf α (RegExp.epsilon.union (e.concat e.closure))
|
α : Type
e : RegExp α
a_ih✝ : e.is_nullable ↔ [] ∈ RegExp.languageOf α e
⊢ e.closure.is_nullable ↔
[] ∈
{[]} ∪
{x |
∃ r ∈ RegExp.languageOf α e,
∃ s ∈ {l | ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = l}, r ++ s = x}
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
e : RegExp α
a_ih✝ : e.is_nullable ↔ [] ∈ RegExp.languageOf α e
⊢ e.closure.is_nullable ↔ [] ∈ RegExp.languageOf α (RegExp.epsilon.union (e.concat e.closure))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
simp only [RegExp.is_nullable]
|
α : Type
e : RegExp α
a_ih✝ : e.is_nullable ↔ [] ∈ RegExp.languageOf α e
⊢ e.closure.is_nullable ↔
[] ∈
{[]} ∪
{x |
∃ r ∈ RegExp.languageOf α e,
∃ s ∈ {l | ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = l}, r ++ s = x}
|
α : Type
e : RegExp α
a_ih✝ : e.is_nullable ↔ [] ∈ RegExp.languageOf α e
⊢ True ↔
[] ∈
{[]} ∪
{x |
∃ r ∈ RegExp.languageOf α e,
∃ s ∈ {l | ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = l}, r ++ s = x}
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
e : RegExp α
a_ih✝ : e.is_nullable ↔ [] ∈ RegExp.languageOf α e
⊢ e.closure.is_nullable ↔
[] ∈
{[]} ∪
{x |
∃ r ∈ RegExp.languageOf α e,
∃ s ∈ {l | ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = l}, r ++ s = x}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
is_nullable_def
|
[73, 1]
|
[107, 11]
|
simp
|
α : Type
e : RegExp α
a_ih✝ : e.is_nullable ↔ [] ∈ RegExp.languageOf α e
⊢ True ↔
[] ∈
{[]} ∪
{x |
∃ r ∈ RegExp.languageOf α e,
∃ s ∈ {l | ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = l}, r ++ s = x}
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
e : RegExp α
a_ih✝ : e.is_nullable ↔ [] ∈ RegExp.languageOf α e
⊢ True ↔
[] ∈
{[]} ∪
{x |
∃ r ∈ RegExp.languageOf α e,
∃ s ∈ {l | ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α e) ∧ rs.join = l}, r ++ s = x}
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
induction e
|
α : Type
e : RegExp α
⊢ RegExp.languageOf α e.delta =
if e.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
|
case char
α : Type
a✝ : α
⊢ RegExp.languageOf α (RegExp.char a✝).delta =
if (RegExp.char a✝).is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
case epsilon
α : Type
⊢ RegExp.languageOf α RegExp.epsilon.delta =
if RegExp.epsilon.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
case zero
α : Type
⊢ RegExp.languageOf α RegExp.zero.delta =
if RegExp.zero.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
case union
α : Type
a✝¹ a✝ : RegExp α
a_ih✝¹ :
RegExp.languageOf α a✝¹.delta =
if a✝¹.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
a_ih✝ :
RegExp.languageOf α a✝.delta =
if a✝.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
⊢ RegExp.languageOf α (a✝¹.union a✝).delta =
if (a✝¹.union a✝).is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
case concat
α : Type
a✝¹ a✝ : RegExp α
a_ih✝¹ :
RegExp.languageOf α a✝¹.delta =
if a✝¹.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
a_ih✝ :
RegExp.languageOf α a✝.delta =
if a✝.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
⊢ RegExp.languageOf α (a✝¹.concat a✝).delta =
if (a✝¹.concat a✝).is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
case closure
α : Type
a✝ : RegExp α
a_ih✝ :
RegExp.languageOf α a✝.delta =
if a✝.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
⊢ RegExp.languageOf α a✝.closure.delta =
if a✝.closure.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
e : RegExp α
⊢ RegExp.languageOf α e.delta =
if e.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
case char a =>
simp only [RegExp.is_nullable]
simp only [RegExp.languageOf]
simp
|
α : Type
a : α
⊢ RegExp.languageOf α (RegExp.char a).delta =
if (RegExp.char a).is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
a : α
⊢ RegExp.languageOf α (RegExp.char a).delta =
if (RegExp.char a).is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
case epsilon =>
simp only [RegExp.is_nullable]
simp only [RegExp.languageOf]
simp
|
α : Type
⊢ RegExp.languageOf α RegExp.epsilon.delta =
if RegExp.epsilon.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
⊢ RegExp.languageOf α RegExp.epsilon.delta =
if RegExp.epsilon.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
case zero =>
simp only [RegExp.is_nullable]
simp only [RegExp.languageOf]
simp
|
α : Type
⊢ RegExp.languageOf α RegExp.zero.delta =
if RegExp.zero.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
⊢ RegExp.languageOf α RegExp.zero.delta =
if RegExp.zero.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
case union r s r_ih s_ih =>
simp only [RegExp.languageOf] at r_ih
simp only [RegExp.languageOf] at s_ih
simp only [RegExp.languageOf]
simp only [r_ih]
simp only [s_ih]
simp only [RegExp.is_nullable]
ext cs
simp
tauto
|
α : Type
r s : RegExp α
r_ih :
RegExp.languageOf α r.delta =
if r.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
s_ih :
RegExp.languageOf α s.delta =
if s.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
⊢ RegExp.languageOf α (r.union s).delta =
if (r.union s).is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih :
RegExp.languageOf α r.delta =
if r.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
s_ih :
RegExp.languageOf α s.delta =
if s.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
⊢ RegExp.languageOf α (r.union s).delta =
if (r.union s).is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
case closure e _ =>
simp only [RegExp.languageOf]
simp only [RegExp.is_nullable]
simp
|
α : Type
e : RegExp α
a_ih✝ :
RegExp.languageOf α e.delta =
if e.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
⊢ RegExp.languageOf α e.closure.delta =
if e.closure.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
e : RegExp α
a_ih✝ :
RegExp.languageOf α e.delta =
if e.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
⊢ RegExp.languageOf α e.closure.delta =
if e.closure.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp only [RegExp.is_nullable]
|
α : Type
a : α
⊢ RegExp.languageOf α (RegExp.char a).delta =
if (RegExp.char a).is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
|
α : Type
a : α
⊢ RegExp.languageOf α (RegExp.char a).delta =
if False then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
a : α
⊢ RegExp.languageOf α (RegExp.char a).delta =
if (RegExp.char a).is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp only [RegExp.languageOf]
|
α : Type
a : α
⊢ RegExp.languageOf α (RegExp.char a).delta =
if False then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
|
α : Type
a : α
⊢ ∅ = if False then {[]} else ∅
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
a : α
⊢ RegExp.languageOf α (RegExp.char a).delta =
if False then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp
|
α : Type
a : α
⊢ ∅ = if False then {[]} else ∅
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
a : α
⊢ ∅ = if False then {[]} else ∅
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp only [RegExp.is_nullable]
|
α : Type
⊢ RegExp.languageOf α RegExp.epsilon.delta =
if RegExp.epsilon.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
|
α : Type
⊢ RegExp.languageOf α RegExp.epsilon.delta =
if True then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
⊢ RegExp.languageOf α RegExp.epsilon.delta =
if RegExp.epsilon.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp only [RegExp.languageOf]
|
α : Type
⊢ RegExp.languageOf α RegExp.epsilon.delta =
if True then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
|
α : Type
⊢ {[]} = if True then {[]} else ∅
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
⊢ RegExp.languageOf α RegExp.epsilon.delta =
if True then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp
|
α : Type
⊢ {[]} = if True then {[]} else ∅
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
⊢ {[]} = if True then {[]} else ∅
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp only [RegExp.is_nullable]
|
α : Type
⊢ RegExp.languageOf α RegExp.zero.delta =
if RegExp.zero.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
|
α : Type
⊢ RegExp.languageOf α RegExp.zero.delta =
if False then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
⊢ RegExp.languageOf α RegExp.zero.delta =
if RegExp.zero.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp only [RegExp.languageOf]
|
α : Type
⊢ RegExp.languageOf α RegExp.zero.delta =
if False then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
|
α : Type
⊢ ∅ = if False then {[]} else ∅
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
⊢ RegExp.languageOf α RegExp.zero.delta =
if False then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp
|
α : Type
⊢ ∅ = if False then {[]} else ∅
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
⊢ ∅ = if False then {[]} else ∅
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp only [RegExp.languageOf] at r_ih
|
α : Type
r s : RegExp α
r_ih :
RegExp.languageOf α r.delta =
if r.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
s_ih :
RegExp.languageOf α s.delta =
if s.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
⊢ RegExp.languageOf α (r.union s).delta =
if (r.union s).is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
|
α : Type
r s : RegExp α
s_ih :
RegExp.languageOf α s.delta =
if s.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
⊢ RegExp.languageOf α (r.union s).delta =
if (r.union s).is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih :
RegExp.languageOf α r.delta =
if r.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
s_ih :
RegExp.languageOf α s.delta =
if s.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
⊢ RegExp.languageOf α (r.union s).delta =
if (r.union s).is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp only [RegExp.languageOf] at s_ih
|
α : Type
r s : RegExp α
s_ih :
RegExp.languageOf α s.delta =
if s.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
⊢ RegExp.languageOf α (r.union s).delta =
if (r.union s).is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ RegExp.languageOf α (r.union s).delta =
if (r.union s).is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
s_ih :
RegExp.languageOf α s.delta =
if s.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
⊢ RegExp.languageOf α (r.union s).delta =
if (r.union s).is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp only [RegExp.languageOf]
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ RegExp.languageOf α (r.union s).delta =
if (r.union s).is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ RegExp.languageOf α r.delta ∪ RegExp.languageOf α s.delta = if (r.union s).is_nullable then {[]} else ∅
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ RegExp.languageOf α (r.union s).delta =
if (r.union s).is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp only [r_ih]
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ RegExp.languageOf α r.delta ∪ RegExp.languageOf α s.delta = if (r.union s).is_nullable then {[]} else ∅
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ (if r.is_nullable then {[]} else ∅) ∪ RegExp.languageOf α s.delta = if (r.union s).is_nullable then {[]} else ∅
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ RegExp.languageOf α r.delta ∪ RegExp.languageOf α s.delta = if (r.union s).is_nullable then {[]} else ∅
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp only [s_ih]
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ (if r.is_nullable then {[]} else ∅) ∪ RegExp.languageOf α s.delta = if (r.union s).is_nullable then {[]} else ∅
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ ((if r.is_nullable then {[]} else ∅) ∪ if s.is_nullable then {[]} else ∅) =
if (r.union s).is_nullable then {[]} else ∅
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ (if r.is_nullable then {[]} else ∅) ∪ RegExp.languageOf α s.delta = if (r.union s).is_nullable then {[]} else ∅
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp only [RegExp.is_nullable]
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ ((if r.is_nullable then {[]} else ∅) ∪ if s.is_nullable then {[]} else ∅) =
if (r.union s).is_nullable then {[]} else ∅
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ ((if r.is_nullable then {[]} else ∅) ∪ if s.is_nullable then {[]} else ∅) =
if r.is_nullable ∨ s.is_nullable then {[]} else ∅
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ ((if r.is_nullable then {[]} else ∅) ∪ if s.is_nullable then {[]} else ∅) =
if (r.union s).is_nullable then {[]} else ∅
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
ext cs
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ ((if r.is_nullable then {[]} else ∅) ∪ if s.is_nullable then {[]} else ∅) =
if r.is_nullable ∨ s.is_nullable then {[]} else ∅
|
case h
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
⊢ (cs ∈ (if r.is_nullable then {[]} else ∅) ∪ if s.is_nullable then {[]} else ∅) ↔
cs ∈ if r.is_nullable ∨ s.is_nullable then {[]} else ∅
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ ((if r.is_nullable then {[]} else ∅) ∪ if s.is_nullable then {[]} else ∅) =
if r.is_nullable ∨ s.is_nullable then {[]} else ∅
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp
|
case h
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
⊢ (cs ∈ (if r.is_nullable then {[]} else ∅) ∪ if s.is_nullable then {[]} else ∅) ↔
cs ∈ if r.is_nullable ∨ s.is_nullable then {[]} else ∅
|
case h
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
⊢ r.is_nullable ∧ cs = [] ∨ s.is_nullable ∧ cs = [] ↔ (r.is_nullable ∨ s.is_nullable) ∧ cs = []
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
⊢ (cs ∈ (if r.is_nullable then {[]} else ∅) ∪ if s.is_nullable then {[]} else ∅) ↔
cs ∈ if r.is_nullable ∨ s.is_nullable then {[]} else ∅
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
tauto
|
case h
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
⊢ r.is_nullable ∧ cs = [] ∨ s.is_nullable ∧ cs = [] ↔ (r.is_nullable ∨ s.is_nullable) ∧ cs = []
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
⊢ r.is_nullable ∧ cs = [] ∨ s.is_nullable ∧ cs = [] ↔ (r.is_nullable ∨ s.is_nullable) ∧ cs = []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp only [RegExp.languageOf] at r_ih
|
α : Type
r s : RegExp α
r_ih :
RegExp.languageOf α r.delta =
if r.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
s_ih :
RegExp.languageOf α s.delta =
if s.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
⊢ RegExp.languageOf α (r.concat s).delta =
if (r.concat s).is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
|
α : Type
r s : RegExp α
s_ih :
RegExp.languageOf α s.delta =
if s.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
⊢ RegExp.languageOf α (r.concat s).delta =
if (r.concat s).is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih :
RegExp.languageOf α r.delta =
if r.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
s_ih :
RegExp.languageOf α s.delta =
if s.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
⊢ RegExp.languageOf α (r.concat s).delta =
if (r.concat s).is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp only [RegExp.languageOf] at s_ih
|
α : Type
r s : RegExp α
s_ih :
RegExp.languageOf α s.delta =
if s.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
⊢ RegExp.languageOf α (r.concat s).delta =
if (r.concat s).is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ RegExp.languageOf α (r.concat s).delta =
if (r.concat s).is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
s_ih :
RegExp.languageOf α s.delta =
if s.is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
⊢ RegExp.languageOf α (r.concat s).delta =
if (r.concat s).is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp only [RegExp.languageOf]
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ RegExp.languageOf α (r.concat s).delta =
if (r.concat s).is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ {x | ∃ r_1 ∈ RegExp.languageOf α r.delta, ∃ s_1 ∈ RegExp.languageOf α s.delta, r_1 ++ s_1 = x} =
if (r.concat s).is_nullable then {[]} else ∅
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ RegExp.languageOf α (r.concat s).delta =
if (r.concat s).is_nullable then RegExp.languageOf α RegExp.epsilon else RegExp.languageOf α RegExp.zero
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp only [r_ih]
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ {x | ∃ r_1 ∈ RegExp.languageOf α r.delta, ∃ s_1 ∈ RegExp.languageOf α s.delta, r_1 ++ s_1 = x} =
if (r.concat s).is_nullable then {[]} else ∅
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ {x | ∃ r_1 ∈ if r.is_nullable then {[]} else ∅, ∃ s_1 ∈ RegExp.languageOf α s.delta, r_1 ++ s_1 = x} =
if (r.concat s).is_nullable then {[]} else ∅
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ {x | ∃ r_1 ∈ RegExp.languageOf α r.delta, ∃ s_1 ∈ RegExp.languageOf α s.delta, r_1 ++ s_1 = x} =
if (r.concat s).is_nullable then {[]} else ∅
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp only [s_ih]
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ {x | ∃ r_1 ∈ if r.is_nullable then {[]} else ∅, ∃ s_1 ∈ RegExp.languageOf α s.delta, r_1 ++ s_1 = x} =
if (r.concat s).is_nullable then {[]} else ∅
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ {x | ∃ r_1 ∈ if r.is_nullable then {[]} else ∅, ∃ s_1 ∈ if s.is_nullable then {[]} else ∅, r_1 ++ s_1 = x} =
if (r.concat s).is_nullable then {[]} else ∅
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ {x | ∃ r_1 ∈ if r.is_nullable then {[]} else ∅, ∃ s_1 ∈ RegExp.languageOf α s.delta, r_1 ++ s_1 = x} =
if (r.concat s).is_nullable then {[]} else ∅
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
simp only [RegExp.is_nullable]
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ {x | ∃ r_1 ∈ if r.is_nullable then {[]} else ∅, ∃ s_1 ∈ if s.is_nullable then {[]} else ∅, r_1 ++ s_1 = x} =
if (r.concat s).is_nullable then {[]} else ∅
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ {x | ∃ r_1 ∈ if r.is_nullable then {[]} else ∅, ∃ s_1 ∈ if s.is_nullable then {[]} else ∅, r_1 ++ s_1 = x} =
if r.is_nullable ∧ s.is_nullable then {[]} else ∅
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ {x | ∃ r_1 ∈ if r.is_nullable then {[]} else ∅, ∃ s_1 ∈ if s.is_nullable then {[]} else ∅, r_1 ++ s_1 = x} =
if (r.concat s).is_nullable then {[]} else ∅
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Brzozowski.lean
|
language_of_delta
|
[121, 1]
|
[197, 11]
|
ext cs
|
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ {x | ∃ r_1 ∈ if r.is_nullable then {[]} else ∅, ∃ s_1 ∈ if s.is_nullable then {[]} else ∅, r_1 ++ s_1 = x} =
if r.is_nullable ∧ s.is_nullable then {[]} else ∅
|
case h
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
cs : List α
⊢ cs ∈ {x | ∃ r_1 ∈ if r.is_nullable then {[]} else ∅, ∃ s_1 ∈ if s.is_nullable then {[]} else ∅, r_1 ++ s_1 = x} ↔
cs ∈ if r.is_nullable ∧ s.is_nullable then {[]} else ∅
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
r s : RegExp α
r_ih : RegExp.languageOf α r.delta = if r.is_nullable then {[]} else ∅
s_ih : RegExp.languageOf α s.delta = if s.is_nullable then {[]} else ∅
⊢ {x | ∃ r_1 ∈ if r.is_nullable then {[]} else ∅, ∃ s_1 ∈ if s.is_nullable then {[]} else ∅, r_1 ++ s_1 = x} =
if r.is_nullable ∧ s.is_nullable then {[]} else ∅
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.