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 : T... | 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... |
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... |
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... |
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... | 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... | 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 ... |
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... | 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 ... | 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 ... |
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... | 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 ... |
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... | 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 ... |
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... | 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 ... |
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... | 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... | 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 ... |
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... | 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 t... | 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 ... |
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 t... | 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 t... | 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) ↔
... |
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 t... | 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 t... | 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) ↔
... |
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 t... | 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 t... | 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) ↔
... |
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 t... | 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 t... | 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) ↔
... |
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 t... | 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 t... | 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) ↔
... |
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 t... | 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 t... | 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) ↔
... |
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 t... | 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 t... | 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) ↔
... |
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 t... | 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 ... | 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) ↔
... |
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... | 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 ... |
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... | 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 ... |
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 t... | 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 t... | 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) ↔
... |
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 t... | 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 = ... | 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) ↔
... |
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... | 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 ... |
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... | 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 ... |
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 = ... | 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 = ... | 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) ↔... |
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 = ... | 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 = ... | 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) ↔... |
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 = ... | 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) ↔... |
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 = ... | 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 = ... | 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) ↔... |
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 = ... | 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) ↔... |
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... | 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... | 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 ... |
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... | 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 ... |
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... | 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... | 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 ... |
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... | 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 ... |
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... | 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 ... |
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... | 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 +... |
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.... | 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 ... | 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.epsilo... |
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 RegEx... |
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 ... | 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.lan... | 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.epsilo... |
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.lan... | 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.un... |
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.language... |
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... |
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_nulla... |
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... |
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 {[]} ... | 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_nul... |
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 {[]} ... | 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 {[]} e... |
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.i... |
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 ... | 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.l... | 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.epsilo... |
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.l... | 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.co... |
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.langua... |
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 =... |
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 {[]} e... | 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 ++... |
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 {[]} e... | 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 {... | 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 ∅, ... |
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 {... | 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_nu... | 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 ∅, ... |
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