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/Fresh.lean
|
FOL.NV.fresh_not_mem
|
[69, 1]
|
[91, 59]
|
simp only [if_pos h]
|
x : VarName
c : Char
xs : Finset VarName
h : x β xs
this : finset_var_name_max_len xs - x.length < finset_var_name_max_len xs + 1 - x.length
β’ (if x β xs then fresh { toString := x.toString ++ c.toString } c xs else x) β xs
|
x : VarName
c : Char
xs : Finset VarName
h : x β xs
this : finset_var_name_max_len xs - x.length < finset_var_name_max_len xs + 1 - x.length
β’ fresh { toString := x.toString ++ c.toString } c xs β xs
|
Please generate a tactic in lean4 to solve the state.
STATE:
x : VarName
c : Char
xs : Finset VarName
h : x β xs
this : finset_var_name_max_len xs - x.length < finset_var_name_max_len xs + 1 - x.length
β’ (if x β xs then fresh { toString := x.toString ++ c.toString } c xs else x) β xs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Fresh.lean
|
FOL.NV.fresh_not_mem
|
[69, 1]
|
[91, 59]
|
apply fresh_not_mem
|
x : VarName
c : Char
xs : Finset VarName
h : x β xs
this : finset_var_name_max_len xs - x.length < finset_var_name_max_len xs + 1 - x.length
β’ fresh { toString := x.toString ++ c.toString } c xs β xs
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
x : VarName
c : Char
xs : Finset VarName
h : x β xs
this : finset_var_name_max_len xs - x.length < finset_var_name_max_len xs + 1 - x.length
β’ fresh { toString := x.toString ++ c.toString } c xs β xs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Fresh.lean
|
FOL.NV.fresh_not_mem
|
[69, 1]
|
[91, 59]
|
unfold fresh
|
x : VarName
c : Char
xs : Finset VarName
h : x β xs
β’ fresh x c xs β xs
|
x : VarName
c : Char
xs : Finset VarName
h : x β xs
β’ (if h : x β xs then
let_fun this := β―;
fresh { toString := x.toString ++ c.toString } c xs
else x) β
xs
|
Please generate a tactic in lean4 to solve the state.
STATE:
x : VarName
c : Char
xs : Finset VarName
h : x β xs
β’ fresh x c xs β xs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Fresh.lean
|
FOL.NV.fresh_not_mem
|
[69, 1]
|
[91, 59]
|
simp
|
x : VarName
c : Char
xs : Finset VarName
h : x β xs
β’ (if h : x β xs then
let_fun this := β―;
fresh { toString := x.toString ++ c.toString } c xs
else x) β
xs
|
x : VarName
c : Char
xs : Finset VarName
h : x β xs
β’ (if x β xs then fresh { toString := x.toString ++ c.toString } c xs else x) β xs
|
Please generate a tactic in lean4 to solve the state.
STATE:
x : VarName
c : Char
xs : Finset VarName
h : x β xs
β’ (if h : x β xs then
let_fun this := β―;
fresh { toString := x.toString ++ c.toString } c xs
else x) β
xs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Fresh.lean
|
FOL.NV.fresh_not_mem
|
[69, 1]
|
[91, 59]
|
simp [if_neg h]
|
x : VarName
c : Char
xs : Finset VarName
h : x β xs
β’ (if x β xs then fresh { toString := x.toString ++ c.toString } c xs else x) β xs
|
x : VarName
c : Char
xs : Finset VarName
h : x β xs
β’ x β xs
|
Please generate a tactic in lean4 to solve the state.
STATE:
x : VarName
c : Char
xs : Finset VarName
h : x β xs
β’ (if x β xs then fresh { toString := x.toString ++ c.toString } c xs else x) β xs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Fresh.lean
|
FOL.NV.fresh_not_mem
|
[69, 1]
|
[91, 59]
|
exact h
|
x : VarName
c : Char
xs : Finset VarName
h : x β xs
β’ x β xs
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
x : VarName
c : Char
xs : Finset VarName
h : x β xs
β’ x β xs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/NFA.lean
|
NFA.mem_accepts
|
[106, 1]
|
[115, 43]
|
rfl
|
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : NFA Ξ± Ο
input : List Ξ±
β’ e.accepts input β β s β e.eval_from e.starting_state_list input, s β e.accepting_state_list
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : NFA Ξ± Ο
input : List Ξ±
β’ e.accepts input β β s β e.eval_from e.starting_state_list input, s β e.accepting_state_list
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_one_no_eps_def
|
[111, 1]
|
[124, 9]
|
simp only [eval_one_no_eps]
|
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state : Ο
β’ stop_state β e.eval_one_no_eps starting_state_list symbol β
β state β starting_state_list, stop_state β symbol_arrow_list_to_fun e.symbol_arrow_list state symbol
|
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state : Ο
β’ stop_state β
(List.map (fun state => symbol_arrow_list_to_fun e.symbol_arrow_list state symbol)
starting_state_list).join.dedup β
β state β starting_state_list, stop_state β symbol_arrow_list_to_fun e.symbol_arrow_list state symbol
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state : Ο
β’ stop_state β e.eval_one_no_eps starting_state_list symbol β
β state β starting_state_list, stop_state β symbol_arrow_list_to_fun e.symbol_arrow_list state symbol
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_one_no_eps_def
|
[111, 1]
|
[124, 9]
|
simp
|
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state : Ο
β’ stop_state β
(List.map (fun state => symbol_arrow_list_to_fun e.symbol_arrow_list state symbol)
starting_state_list).join.dedup β
β state β starting_state_list, stop_state β symbol_arrow_list_to_fun e.symbol_arrow_list state symbol
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state : Ο
β’ stop_state β
(List.map (fun state => symbol_arrow_list_to_fun e.symbol_arrow_list state symbol)
starting_state_list).join.dedup β
β state β starting_state_list, stop_state β symbol_arrow_list_to_fun e.symbol_arrow_list state symbol
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_one_no_eps_iff
|
[354, 1]
|
[374, 53]
|
simp only [EpsilonNFA.eval_one_no_eps]
|
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state : Ο
β’ stop_state β e.eval_one_no_eps starting_state_list symbol β
β state β starting_state_list, e.toAbstract.symbol state symbol stop_state
|
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state : Ο
β’ stop_state β
(List.map (fun state => symbol_arrow_list_to_fun e.symbol_arrow_list state symbol)
starting_state_list).join.dedup β
β state β starting_state_list, e.toAbstract.symbol state symbol stop_state
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state : Ο
β’ stop_state β e.eval_one_no_eps starting_state_list symbol β
β state β starting_state_list, e.toAbstract.symbol state symbol stop_state
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_one_no_eps_iff
|
[354, 1]
|
[374, 53]
|
simp only [EpsilonNFA.toAbstract]
|
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state : Ο
β’ stop_state β
(List.map (fun state => symbol_arrow_list_to_fun e.symbol_arrow_list state symbol)
starting_state_list).join.dedup β
β state β starting_state_list, e.toAbstract.symbol state symbol stop_state
|
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state : Ο
β’ stop_state β
(List.map (fun state => symbol_arrow_list_to_fun e.symbol_arrow_list state symbol)
starting_state_list).join.dedup β
β state β starting_state_list,
β stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } β e.symbol_arrow_list β§
stop_state β stop_state_list
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state : Ο
β’ stop_state β
(List.map (fun state => symbol_arrow_list_to_fun e.symbol_arrow_list state symbol)
starting_state_list).join.dedup β
β state β starting_state_list, e.toAbstract.symbol state symbol stop_state
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_one_no_eps_iff
|
[354, 1]
|
[374, 53]
|
simp only [symbol_arrow_list_to_fun]
|
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state : Ο
β’ stop_state β
(List.map (fun state => symbol_arrow_list_to_fun e.symbol_arrow_list state symbol)
starting_state_list).join.dedup β
β state β starting_state_list,
β stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } β e.symbol_arrow_list β§
stop_state β stop_state_list
|
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state : Ο
β’ stop_state β
(List.map
(fun state =>
(List.filterMap
(fun arrow =>
if arrow.start_state = state β§ arrow.symbol = symbol then some arrow.stop_state_list else none)
e.symbol_arrow_list).join.dedup)
starting_state_list).join.dedup β
β state β starting_state_list,
β stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } β e.symbol_arrow_list β§
stop_state β stop_state_list
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state : Ο
β’ stop_state β
(List.map (fun state => symbol_arrow_list_to_fun e.symbol_arrow_list state symbol)
starting_state_list).join.dedup β
β state β starting_state_list,
β stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } β e.symbol_arrow_list β§
stop_state β stop_state_list
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_one_no_eps_iff
|
[354, 1]
|
[374, 53]
|
simp
|
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state : Ο
β’ stop_state β
(List.map
(fun state =>
(List.filterMap
(fun arrow =>
if arrow.start_state = state β§ arrow.symbol = symbol then some arrow.stop_state_list else none)
e.symbol_arrow_list).join.dedup)
starting_state_list).join.dedup β
β state β starting_state_list,
β stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } β e.symbol_arrow_list β§
stop_state β stop_state_list
|
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state : Ο
β’ (β a β starting_state_list,
β l,
(β a_1 β e.symbol_arrow_list, (a_1.start_state = a β§ a_1.symbol = symbol) β§ a_1.stop_state_list = l) β§
stop_state β l) β
β state β starting_state_list,
β stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } β e.symbol_arrow_list β§
stop_state β stop_state_list
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state : Ο
β’ stop_state β
(List.map
(fun state =>
(List.filterMap
(fun arrow =>
if arrow.start_state = state β§ arrow.symbol = symbol then some arrow.stop_state_list else none)
e.symbol_arrow_list).join.dedup)
starting_state_list).join.dedup β
β state β starting_state_list,
β stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } β e.symbol_arrow_list β§
stop_state β stop_state_list
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_one_no_eps_iff
|
[354, 1]
|
[374, 53]
|
constructor
|
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state : Ο
β’ (β a β starting_state_list,
β l,
(β a_1 β e.symbol_arrow_list, (a_1.start_state = a β§ a_1.symbol = symbol) β§ a_1.stop_state_list = l) β§
stop_state β l) β
β state β starting_state_list,
β stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } β e.symbol_arrow_list β§
stop_state β stop_state_list
|
case mp
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state : Ο
β’ (β a β starting_state_list,
β l,
(β a_1 β e.symbol_arrow_list, (a_1.start_state = a β§ a_1.symbol = symbol) β§ a_1.stop_state_list = l) β§
stop_state β l) β
β state β starting_state_list,
β stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } β e.symbol_arrow_list β§
stop_state β stop_state_list
case mpr
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state : Ο
β’ (β state β starting_state_list,
β stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } β e.symbol_arrow_list β§
stop_state β stop_state_list) β
β a β starting_state_list,
β l,
(β a_1 β e.symbol_arrow_list, (a_1.start_state = a β§ a_1.symbol = symbol) β§ a_1.stop_state_list = l) β§
stop_state β l
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state : Ο
β’ (β a β starting_state_list,
β l,
(β a_1 β e.symbol_arrow_list, (a_1.start_state = a β§ a_1.symbol = symbol) β§ a_1.stop_state_list = l) β§
stop_state β l) β
β state β starting_state_list,
β stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } β e.symbol_arrow_list β§
stop_state β stop_state_list
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_one_no_eps_iff
|
[354, 1]
|
[374, 53]
|
rintro β¨_, h1, _, β¨β¨β©, h2, β¨rfl, rflβ©, rflβ©, h3β©
|
case mp
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state : Ο
β’ (β a β starting_state_list,
β l,
(β a_1 β e.symbol_arrow_list, (a_1.start_state = a β§ a_1.symbol = symbol) β§ a_1.stop_state_list = l) β§
stop_state β l) β
β state β starting_state_list,
β stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } β e.symbol_arrow_list β§
stop_state β stop_state_list
|
case mp.intro.intro.intro.intro.intro.mk.intro.intro.intro
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
stop_state start_stateβ : Ο
symbolβ : Ξ±
stop_state_listβ : List Ο
h2 : { start_state := start_stateβ, symbol := symbolβ, stop_state_list := stop_state_listβ } β e.symbol_arrow_list
h1 :
{ start_state := start_stateβ, symbol := symbolβ, stop_state_list := stop_state_listβ }.start_state β
starting_state_list
h3 :
stop_state β { start_state := start_stateβ, symbol := symbolβ, stop_state_list := stop_state_listβ }.stop_state_list
β’ β state β starting_state_list,
β stop_state_list,
{ start_state := state,
symbol := { start_state := start_stateβ, symbol := symbolβ, stop_state_list := stop_state_listβ }.symbol,
stop_state_list := stop_state_list } β
e.symbol_arrow_list β§
stop_state β stop_state_list
|
Please generate a tactic in lean4 to solve the state.
STATE:
case mp
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state : Ο
β’ (β a β starting_state_list,
β l,
(β a_1 β e.symbol_arrow_list, (a_1.start_state = a β§ a_1.symbol = symbol) β§ a_1.stop_state_list = l) β§
stop_state β l) β
β state β starting_state_list,
β stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } β e.symbol_arrow_list β§
stop_state β stop_state_list
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_one_no_eps_iff
|
[354, 1]
|
[374, 53]
|
exact β¨_, h1, _, h2, h3β©
|
case mp.intro.intro.intro.intro.intro.mk.intro.intro.intro
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
stop_state start_stateβ : Ο
symbolβ : Ξ±
stop_state_listβ : List Ο
h2 : { start_state := start_stateβ, symbol := symbolβ, stop_state_list := stop_state_listβ } β e.symbol_arrow_list
h1 :
{ start_state := start_stateβ, symbol := symbolβ, stop_state_list := stop_state_listβ }.start_state β
starting_state_list
h3 :
stop_state β { start_state := start_stateβ, symbol := symbolβ, stop_state_list := stop_state_listβ }.stop_state_list
β’ β state β starting_state_list,
β stop_state_list,
{ start_state := state,
symbol := { start_state := start_stateβ, symbol := symbolβ, stop_state_list := stop_state_listβ }.symbol,
stop_state_list := stop_state_list } β
e.symbol_arrow_list β§
stop_state β stop_state_list
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro.intro.mk.intro.intro.intro
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
stop_state start_stateβ : Ο
symbolβ : Ξ±
stop_state_listβ : List Ο
h2 : { start_state := start_stateβ, symbol := symbolβ, stop_state_list := stop_state_listβ } β e.symbol_arrow_list
h1 :
{ start_state := start_stateβ, symbol := symbolβ, stop_state_list := stop_state_listβ }.start_state β
starting_state_list
h3 :
stop_state β { start_state := start_stateβ, symbol := symbolβ, stop_state_list := stop_state_listβ }.stop_state_list
β’ β state β starting_state_list,
β stop_state_list,
{ start_state := state,
symbol := { start_state := start_stateβ, symbol := symbolβ, stop_state_list := stop_state_listβ }.symbol,
stop_state_list := stop_state_list } β
e.symbol_arrow_list β§
stop_state β stop_state_list
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_one_no_eps_iff
|
[354, 1]
|
[374, 53]
|
rintro β¨_, h1, _, h2, h3β©
|
case mpr
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state : Ο
β’ (β state β starting_state_list,
β stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } β e.symbol_arrow_list β§
stop_state β stop_state_list) β
β a β starting_state_list,
β l,
(β a_1 β e.symbol_arrow_list, (a_1.start_state = a β§ a_1.symbol = symbol) β§ a_1.stop_state_list = l) β§
stop_state β l
|
case mpr.intro.intro.intro.intro
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state wβΒΉ : Ο
h1 : wβΒΉ β starting_state_list
wβ : List Ο
h2 : { start_state := wβΒΉ, symbol := symbol, stop_state_list := wβ } β e.symbol_arrow_list
h3 : stop_state β wβ
β’ β a β starting_state_list,
β l,
(β a_1 β e.symbol_arrow_list, (a_1.start_state = a β§ a_1.symbol = symbol) β§ a_1.stop_state_list = l) β§
stop_state β l
|
Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state : Ο
β’ (β state β starting_state_list,
β stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } β e.symbol_arrow_list β§
stop_state β stop_state_list) β
β a β starting_state_list,
β l,
(β a_1 β e.symbol_arrow_list, (a_1.start_state = a β§ a_1.symbol = symbol) β§ a_1.stop_state_list = l) β§
stop_state β l
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_one_no_eps_iff
|
[354, 1]
|
[374, 53]
|
exact β¨_, h1, _, β¨_, h2, β¨rfl, rflβ©, rflβ©, h3β©
|
case mpr.intro.intro.intro.intro
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state wβΒΉ : Ο
h1 : wβΒΉ β starting_state_list
wβ : List Ο
h2 : { start_state := wβΒΉ, symbol := symbol, stop_state_list := wβ } β e.symbol_arrow_list
h3 : stop_state β wβ
β’ β a β starting_state_list,
β l,
(β a_1 β e.symbol_arrow_list, (a_1.start_state = a β§ a_1.symbol = symbol) β§ a_1.stop_state_list = l) β§
stop_state β l
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
symbol : Ξ±
stop_state wβΒΉ : Ο
h1 : wβΒΉ β starting_state_list
wβ : List Ο
h2 : { start_state := wβΒΉ, symbol := symbol, stop_state_list := wβ } β e.symbol_arrow_list
h3 : stop_state β wβ
β’ β a β starting_state_list,
β l,
(β a_1 β e.symbol_arrow_list, (a_1.start_state = a β§ a_1.symbol = symbol) β§ a_1.stop_state_list = l) β§
stop_state β l
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.epsilon_closure_iff
|
[385, 1]
|
[402, 83]
|
simp only [epsilon_closure]
|
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
state : Ο
β’ state β e.epsilon_closure starting_state_list β
β start_state β starting_state_list, e.toAbstract.EpsilonClosure start_state state
|
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
state : Ο
β’ state β dft (epsilon_arrow_list_to_graph e.epsilon_arrow_list) starting_state_list β
β start_state β starting_state_list, e.toAbstract.EpsilonClosure start_state state
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
state : Ο
β’ state β e.epsilon_closure starting_state_list β
β start_state β starting_state_list, e.toAbstract.EpsilonClosure start_state state
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.epsilon_closure_iff
|
[385, 1]
|
[402, 83]
|
simp only [dft_iff]
|
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
state : Ο
β’ state β dft (epsilon_arrow_list_to_graph e.epsilon_arrow_list) starting_state_list β
β start_state β starting_state_list, e.toAbstract.EpsilonClosure start_state state
|
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
state : Ο
β’ (β s β starting_state_list,
Relation.ReflTransGen (fun a b => β l, (a, l) β epsilon_arrow_list_to_graph e.epsilon_arrow_list β§ b β l) s
state) β
β start_state β starting_state_list, e.toAbstract.EpsilonClosure start_state state
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
state : Ο
β’ state β dft (epsilon_arrow_list_to_graph e.epsilon_arrow_list) starting_state_list β
β start_state β starting_state_list, e.toAbstract.EpsilonClosure start_state state
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.epsilon_closure_iff
|
[385, 1]
|
[402, 83]
|
congr! with a b c
|
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
state : Ο
β’ (β s β starting_state_list,
Relation.ReflTransGen (fun a b => β l, (a, l) β epsilon_arrow_list_to_graph e.epsilon_arrow_list β§ b β l) s
state) β
β start_state β starting_state_list, e.toAbstract.EpsilonClosure start_state state
|
case a.h.e'_2.h.h.e'_2.h.e.h.e'_2.h.h.a
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
state a b c : Ο
β’ (β l, (b, l) β epsilon_arrow_list_to_graph e.epsilon_arrow_list β§ c β l) β e.toAbstract.epsilon b c
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
state : Ο
β’ (β s β starting_state_list,
Relation.ReflTransGen (fun a b => β l, (a, l) β epsilon_arrow_list_to_graph e.epsilon_arrow_list β§ b β l) s
state) β
β start_state β starting_state_list, e.toAbstract.EpsilonClosure start_state state
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.epsilon_closure_iff
|
[385, 1]
|
[402, 83]
|
simp [toAbstract]
|
case a.h.e'_2.h.h.e'_2.h.e.h.e'_2.h.h.a
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
state a b c : Ο
β’ (β l, (b, l) β epsilon_arrow_list_to_graph e.epsilon_arrow_list β§ c β l) β e.toAbstract.epsilon b c
|
case a.h.e'_2.h.h.e'_2.h.e.h.e'_2.h.h.a
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
state a b c : Ο
β’ (β l, (b, l) β epsilon_arrow_list_to_graph e.epsilon_arrow_list β§ c β l) β
β stop_state_list,
{ start_state := b, stop_state_list := stop_state_list } β e.epsilon_arrow_list β§ c β stop_state_list
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.h.h.e'_2.h.e.h.e'_2.h.h.a
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
state a b c : Ο
β’ (β l, (b, l) β epsilon_arrow_list_to_graph e.epsilon_arrow_list β§ c β l) β e.toAbstract.epsilon b c
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.epsilon_closure_iff
|
[385, 1]
|
[402, 83]
|
induction e.epsilon_arrow_list <;> simp [*, epsilon_arrow_list_to_graph, or_and_right, exists_or]
|
case a.h.e'_2.h.h.e'_2.h.e.h.e'_2.h.h.a
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
state a b c : Ο
β’ (β l, (b, l) β epsilon_arrow_list_to_graph e.epsilon_arrow_list β§ c β l) β
β stop_state_list,
{ start_state := b, stop_state_list := stop_state_list } β e.epsilon_arrow_list β§ c β stop_state_list
|
case a.h.e'_2.h.h.e'_2.h.e.h.e'_2.h.h.a.cons
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
state a b c : Ο
headβ : EpsilonArrow Ο
tailβ : List (EpsilonArrow Ο)
tail_ihβ :
(β l, (b, l) β epsilon_arrow_list_to_graph tailβ β§ c β l) β
β stop_state_list, { start_state := b, stop_state_list := stop_state_list } β tailβ β§ c β stop_state_list
β’ ((β x, (b = headβ.start_state β§ x = headβ.stop_state_list) β§ c β x) β¨
β stop_state_list, { start_state := b, stop_state_list := stop_state_list } β tailβ β§ c β stop_state_list) β
(β x, { start_state := b, stop_state_list := x } = headβ β§ c β x) β¨
β stop_state_list, { start_state := b, stop_state_list := stop_state_list } β tailβ β§ c β stop_state_list
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.h.h.e'_2.h.e.h.e'_2.h.h.a
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
state a b c : Ο
β’ (β l, (b, l) β epsilon_arrow_list_to_graph e.epsilon_arrow_list β§ c β l) β
β stop_state_list,
{ start_state := b, stop_state_list := stop_state_list } β e.epsilon_arrow_list β§ c β stop_state_list
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.epsilon_closure_iff
|
[385, 1]
|
[402, 83]
|
apply or_congr_left <| exists_congr fun a => and_congr_left fun _ => ?_
|
case a.h.e'_2.h.h.e'_2.h.e.h.e'_2.h.h.a.cons
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
state a b c : Ο
headβ : EpsilonArrow Ο
tailβ : List (EpsilonArrow Ο)
tail_ihβ :
(β l, (b, l) β epsilon_arrow_list_to_graph tailβ β§ c β l) β
β stop_state_list, { start_state := b, stop_state_list := stop_state_list } β tailβ β§ c β stop_state_list
β’ ((β x, (b = headβ.start_state β§ x = headβ.stop_state_list) β§ c β x) β¨
β stop_state_list, { start_state := b, stop_state_list := stop_state_list } β tailβ β§ c β stop_state_list) β
(β x, { start_state := b, stop_state_list := x } = headβ β§ c β x) β¨
β stop_state_list, { start_state := b, stop_state_list := stop_state_list } β tailβ β§ c β stop_state_list
|
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
state aβ b c : Ο
headβ : EpsilonArrow Ο
tailβ : List (EpsilonArrow Ο)
tail_ihβ :
(β l, (b, l) β epsilon_arrow_list_to_graph tailβ β§ c β l) β
β stop_state_list, { start_state := b, stop_state_list := stop_state_list } β tailβ β§ c β stop_state_list
a : List Ο
xβ : c β a
β’ b = headβ.start_state β§ a = headβ.stop_state_list β { start_state := b, stop_state_list := a } = headβ
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.h.h.e'_2.h.e.h.e'_2.h.h.a.cons
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
state a b c : Ο
headβ : EpsilonArrow Ο
tailβ : List (EpsilonArrow Ο)
tail_ihβ :
(β l, (b, l) β epsilon_arrow_list_to_graph tailβ β§ c β l) β
β stop_state_list, { start_state := b, stop_state_list := stop_state_list } β tailβ β§ c β stop_state_list
β’ ((β x, (b = headβ.start_state β§ x = headβ.stop_state_list) β§ c β x) β¨
β stop_state_list, { start_state := b, stop_state_list := stop_state_list } β tailβ β§ c β stop_state_list) β
(β x, { start_state := b, stop_state_list := x } = headβ β§ c β x) β¨
β stop_state_list, { start_state := b, stop_state_list := stop_state_list } β tailβ β§ c β stop_state_list
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.epsilon_closure_iff
|
[385, 1]
|
[402, 83]
|
constructor <;> [rintro β¨rfl, rflβ©; rintro rfl] <;> [rfl; constructor <;> rfl]
|
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
state aβ b c : Ο
headβ : EpsilonArrow Ο
tailβ : List (EpsilonArrow Ο)
tail_ihβ :
(β l, (b, l) β epsilon_arrow_list_to_graph tailβ β§ c β l) β
β stop_state_list, { start_state := b, stop_state_list := stop_state_list } β tailβ β§ c β stop_state_list
a : List Ο
xβ : c β a
β’ b = headβ.start_state β§ a = headβ.stop_state_list β { start_state := b, stop_state_list := a } = headβ
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
starting_state_list : List Ο
state aβ b c : Ο
headβ : EpsilonArrow Ο
tailβ : List (EpsilonArrow Ο)
tail_ihβ :
(β l, (b, l) β epsilon_arrow_list_to_graph tailβ β§ c β l) β
β stop_state_list, { start_state := b, stop_state_list := stop_state_list } β tailβ β§ c β stop_state_list
a : List Ο
xβ : c β a
β’ b = headβ.start_state β§ a = headβ.stop_state_list β { start_state := b, stop_state_list := a } = headβ
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
simp [epsilon_closure_iff]
|
case nil
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
β’ (β s β M.epsilon_closure S, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s []
|
case nil
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
β’ (β s, (β start_state β S, M.toAbstract.EpsilonClosure start_state s) β§ s β M.accepting_state_list) β
β s β S, M.toAbstract.eval s []
|
Please generate a tactic in lean4 to solve the state.
STATE:
case nil
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
β’ (β s β M.epsilon_closure S, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
constructor
|
case nil
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
β’ (β s, (β start_state β S, M.toAbstract.EpsilonClosure start_state s) β§ s β M.accepting_state_list) β
β s β S, M.toAbstract.eval s []
|
case nil.mp
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
β’ (β s, (β start_state β S, M.toAbstract.EpsilonClosure start_state s) β§ s β M.accepting_state_list) β
β s β S, M.toAbstract.eval s []
case nil.mpr
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
β’ (β s β S, M.toAbstract.eval s []) β
β s, (β start_state β S, M.toAbstract.EpsilonClosure start_state s) β§ s β M.accepting_state_list
|
Please generate a tactic in lean4 to solve the state.
STATE:
case nil
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
β’ (β s, (β start_state β S, M.toAbstract.EpsilonClosure start_state s) β§ s β M.accepting_state_list) β
β s β S, M.toAbstract.eval s []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
intro β¨s, β¨s', h1, h2β©, h3β©
|
case nil.mp
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
β’ (β s, (β start_state β S, M.toAbstract.EpsilonClosure start_state s) β§ s β M.accepting_state_list) β
β s β S, M.toAbstract.eval s []
|
case nil.mp
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s s' : Ο
h1 : s' β S
h2 : M.toAbstract.EpsilonClosure s' s
h3 : s β M.accepting_state_list
β’ β s β S, M.toAbstract.eval s []
|
Please generate a tactic in lean4 to solve the state.
STATE:
case nil.mp
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
β’ (β s, (β start_state β S, M.toAbstract.EpsilonClosure start_state s) β§ s β M.accepting_state_list) β
β s β S, M.toAbstract.eval s []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
refine β¨_, h1, ?_β©
|
case nil.mp
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s s' : Ο
h1 : s' β S
h2 : M.toAbstract.EpsilonClosure s' s
h3 : s β M.accepting_state_list
β’ β s β S, M.toAbstract.eval s []
|
case nil.mp
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s s' : Ο
h1 : s' β S
h2 : M.toAbstract.EpsilonClosure s' s
h3 : s β M.accepting_state_list
β’ M.toAbstract.eval s' []
|
Please generate a tactic in lean4 to solve the state.
STATE:
case nil.mp
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s s' : Ο
h1 : s' β S
h2 : M.toAbstract.EpsilonClosure s' s
h3 : s β M.accepting_state_list
β’ β s β S, M.toAbstract.eval s []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
clear h1
|
case nil.mp
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s s' : Ο
h1 : s' β S
h2 : M.toAbstract.EpsilonClosure s' s
h3 : s β M.accepting_state_list
β’ M.toAbstract.eval s' []
|
case nil.mp
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s s' : Ο
h2 : M.toAbstract.EpsilonClosure s' s
h3 : s β M.accepting_state_list
β’ M.toAbstract.eval s' []
|
Please generate a tactic in lean4 to solve the state.
STATE:
case nil.mp
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s s' : Ο
h1 : s' β S
h2 : M.toAbstract.EpsilonClosure s' s
h3 : s β M.accepting_state_list
β’ M.toAbstract.eval s' []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
induction h2 using Relation.ReflTransGen.head_induction_on with
| refl => exact .accept _ h3
| head h _ ih => exact .eps _ _ _ ih h
|
case nil.mp
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s s' : Ο
h2 : M.toAbstract.EpsilonClosure s' s
h3 : s β M.accepting_state_list
β’ M.toAbstract.eval s' []
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case nil.mp
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s s' : Ο
h2 : M.toAbstract.EpsilonClosure s' s
h3 : s β M.accepting_state_list
β’ M.toAbstract.eval s' []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
exact .accept _ h3
|
case nil.mp.refl
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s s' : Ο
h3 : s β M.accepting_state_list
β’ M.toAbstract.eval s []
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case nil.mp.refl
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s s' : Ο
h3 : s β M.accepting_state_list
β’ M.toAbstract.eval s []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
exact .eps _ _ _ ih h
|
case nil.mp.head
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s s' : Ο
h3 : s β M.accepting_state_list
aβ cβ : Ο
h : M.toAbstract.epsilon aβ cβ
hβ : Relation.ReflTransGen M.toAbstract.epsilon cβ s
ih : M.toAbstract.eval cβ []
β’ M.toAbstract.eval aβ []
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case nil.mp.head
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s s' : Ο
h3 : s β M.accepting_state_list
aβ cβ : Ο
h : M.toAbstract.epsilon aβ cβ
hβ : Relation.ReflTransGen M.toAbstract.epsilon cβ s
ih : M.toAbstract.eval cβ []
β’ M.toAbstract.eval aβ []
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
intro β¨s, h1, h2β©
|
case nil.mpr
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
β’ (β s β S, M.toAbstract.eval s []) β
β s, (β start_state β S, M.toAbstract.EpsilonClosure start_state s) β§ s β M.accepting_state_list
|
case nil.mpr
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s : Ο
h1 : s β S
h2 : M.toAbstract.eval s []
β’ β s, (β start_state β S, M.toAbstract.EpsilonClosure start_state s) β§ s β M.accepting_state_list
|
Please generate a tactic in lean4 to solve the state.
STATE:
case nil.mpr
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
β’ (β s β S, M.toAbstract.eval s []) β
β s, (β start_state β S, M.toAbstract.EpsilonClosure start_state s) β§ s β M.accepting_state_list
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
obtain β¨s', h3, h4β© : β s', M.toAbstract.EpsilonClosure s s' β§ s' β M.accepting_state_list := by
clear h1; generalize e : [] = l at h2
induction h2 with cases e
| accept _ h' => exact β¨_, .refl, h'β©
| eps _ _ _ _ h2 ih =>
have β¨s', h3, h4β© := ih rfl
exact β¨_, .head h2 h3, h4β©
|
case nil.mpr
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s : Ο
h1 : s β S
h2 : M.toAbstract.eval s []
β’ β s, (β start_state β S, M.toAbstract.EpsilonClosure start_state s) β§ s β M.accepting_state_list
|
case nil.mpr.intro.intro
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s : Ο
h1 : s β S
h2 : M.toAbstract.eval s []
s' : Ο
h3 : M.toAbstract.EpsilonClosure s s'
h4 : s' β M.accepting_state_list
β’ β s, (β start_state β S, M.toAbstract.EpsilonClosure start_state s) β§ s β M.accepting_state_list
|
Please generate a tactic in lean4 to solve the state.
STATE:
case nil.mpr
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s : Ο
h1 : s β S
h2 : M.toAbstract.eval s []
β’ β s, (β start_state β S, M.toAbstract.EpsilonClosure start_state s) β§ s β M.accepting_state_list
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
exact β¨_, β¨_, h1, h3β©, h4β©
|
case nil.mpr.intro.intro
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s : Ο
h1 : s β S
h2 : M.toAbstract.eval s []
s' : Ο
h3 : M.toAbstract.EpsilonClosure s s'
h4 : s' β M.accepting_state_list
β’ β s, (β start_state β S, M.toAbstract.EpsilonClosure start_state s) β§ s β M.accepting_state_list
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case nil.mpr.intro.intro
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s : Ο
h1 : s β S
h2 : M.toAbstract.eval s []
s' : Ο
h3 : M.toAbstract.EpsilonClosure s s'
h4 : s' β M.accepting_state_list
β’ β s, (β start_state β S, M.toAbstract.EpsilonClosure start_state s) β§ s β M.accepting_state_list
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
clear h1
|
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s : Ο
h1 : s β S
h2 : M.toAbstract.eval s []
β’ β s', M.toAbstract.EpsilonClosure s s' β§ s' β M.accepting_state_list
|
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s : Ο
h2 : M.toAbstract.eval s []
β’ β s', M.toAbstract.EpsilonClosure s s' β§ s' β M.accepting_state_list
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s : Ο
h1 : s β S
h2 : M.toAbstract.eval s []
β’ β s', M.toAbstract.EpsilonClosure s s' β§ s' β M.accepting_state_list
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
generalize e : [] = l at h2
|
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s : Ο
h2 : M.toAbstract.eval s []
β’ β s', M.toAbstract.EpsilonClosure s s' β§ s' β M.accepting_state_list
|
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s : Ο
l : List Ξ±
e : [] = l
h2 : M.toAbstract.eval s l
β’ β s', M.toAbstract.EpsilonClosure s s' β§ s' β M.accepting_state_list
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s : Ο
h2 : M.toAbstract.eval s []
β’ β s', M.toAbstract.EpsilonClosure s s' β§ s' β M.accepting_state_list
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
induction h2 with cases e
| accept _ h' => exact β¨_, .refl, h'β©
| eps _ _ _ _ h2 ih =>
have β¨s', h3, h4β© := ih rfl
exact β¨_, .head h2 h3, h4β©
|
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s : Ο
l : List Ξ±
e : [] = l
h2 : M.toAbstract.eval s l
β’ β s', M.toAbstract.EpsilonClosure s s' β§ s' β M.accepting_state_list
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s : Ο
l : List Ξ±
e : [] = l
h2 : M.toAbstract.eval s l
β’ β s', M.toAbstract.EpsilonClosure s s' β§ s' β M.accepting_state_list
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
exact β¨_, .refl, h'β©
|
case accept.refl
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s : Ο
l : List Ξ±
start_stateβ : Ο
h' : M.toAbstract.accepting start_stateβ
β’ β s', M.toAbstract.EpsilonClosure start_stateβ s' β§ s' β M.accepting_state_list
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case accept.refl
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s : Ο
l : List Ξ±
start_stateβ : Ο
h' : M.toAbstract.accepting start_stateβ
β’ β s', M.toAbstract.EpsilonClosure start_stateβ s' β§ s' β M.accepting_state_list
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
have β¨s', h3, h4β© := ih rfl
|
case eps.refl
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s : Ο
l : List Ξ±
start_stateβ stop_stateβ : Ο
h2 : M.toAbstract.epsilon start_stateβ stop_stateβ
aβ : M.toAbstract.eval stop_stateβ []
ih : [] = [] β β s', M.toAbstract.EpsilonClosure stop_stateβ s' β§ s' β M.accepting_state_list
β’ β s', M.toAbstract.EpsilonClosure start_stateβ s' β§ s' β M.accepting_state_list
|
case eps.refl
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s : Ο
l : List Ξ±
start_stateβ stop_stateβ : Ο
h2 : M.toAbstract.epsilon start_stateβ stop_stateβ
aβ : M.toAbstract.eval stop_stateβ []
ih : [] = [] β β s', M.toAbstract.EpsilonClosure stop_stateβ s' β§ s' β M.accepting_state_list
s' : Ο
h3 : M.toAbstract.EpsilonClosure stop_stateβ s'
h4 : s' β M.accepting_state_list
β’ β s', M.toAbstract.EpsilonClosure start_stateβ s' β§ s' β M.accepting_state_list
|
Please generate a tactic in lean4 to solve the state.
STATE:
case eps.refl
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s : Ο
l : List Ξ±
start_stateβ stop_stateβ : Ο
h2 : M.toAbstract.epsilon start_stateβ stop_stateβ
aβ : M.toAbstract.eval stop_stateβ []
ih : [] = [] β β s', M.toAbstract.EpsilonClosure stop_stateβ s' β§ s' β M.accepting_state_list
β’ β s', M.toAbstract.EpsilonClosure start_stateβ s' β§ s' β M.accepting_state_list
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
exact β¨_, .head h2 h3, h4β©
|
case eps.refl
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s : Ο
l : List Ξ±
start_stateβ stop_stateβ : Ο
h2 : M.toAbstract.epsilon start_stateβ stop_stateβ
aβ : M.toAbstract.eval stop_stateβ []
ih : [] = [] β β s', M.toAbstract.EpsilonClosure stop_stateβ s' β§ s' β M.accepting_state_list
s' : Ο
h3 : M.toAbstract.EpsilonClosure stop_stateβ s'
h4 : s' β M.accepting_state_list
β’ β s', M.toAbstract.EpsilonClosure start_stateβ s' β§ s' β M.accepting_state_list
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case eps.refl
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
S : List Ο
s : Ο
l : List Ξ±
start_stateβ stop_stateβ : Ο
h2 : M.toAbstract.epsilon start_stateβ stop_stateβ
aβ : M.toAbstract.eval stop_stateβ []
ih : [] = [] β β s', M.toAbstract.EpsilonClosure stop_stateβ s' β§ s' β M.accepting_state_list
s' : Ο
h3 : M.toAbstract.EpsilonClosure stop_stateβ s'
h4 : s' β M.accepting_state_list
β’ β s', M.toAbstract.EpsilonClosure start_stateβ s' β§ s' β M.accepting_state_list
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
simp [IH, epsilon_closure_iff, eval_one_no_eps_iff]
|
case cons
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
β’ (β s β M.eval_from (M.eval_one_no_eps (M.epsilon_closure S) a) as, s β M.accepting_state_list) β
β s β S, M.toAbstract.eval s (a :: as)
|
case cons
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
β’ (β s,
(β state, (β start_state β S, M.toAbstract.EpsilonClosure start_state state) β§ M.toAbstract.symbol state a s) β§
M.toAbstract.eval s as) β
β s β S, M.toAbstract.eval s (a :: as)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case cons
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
β’ (β s β M.eval_from (M.eval_one_no_eps (M.epsilon_closure S) a) as, s β M.accepting_state_list) β
β s β S, M.toAbstract.eval s (a :: as)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
constructor
|
case cons
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
β’ (β s,
(β state, (β start_state β S, M.toAbstract.EpsilonClosure start_state state) β§ M.toAbstract.symbol state a s) β§
M.toAbstract.eval s as) β
β s β S, M.toAbstract.eval s (a :: as)
|
case cons.mp
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
β’ (β s,
(β state, (β start_state β S, M.toAbstract.EpsilonClosure start_state state) β§ M.toAbstract.symbol state a s) β§
M.toAbstract.eval s as) β
β s β S, M.toAbstract.eval s (a :: as)
case cons.mpr
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
β’ (β s β S, M.toAbstract.eval s (a :: as)) β
β s,
(β state, (β start_state β S, M.toAbstract.EpsilonClosure start_state state) β§ M.toAbstract.symbol state a s) β§
M.toAbstract.eval s as
|
Please generate a tactic in lean4 to solve the state.
STATE:
case cons
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
β’ (β s,
(β state, (β start_state β S, M.toAbstract.EpsilonClosure start_state state) β§ M.toAbstract.symbol state a s) β§
M.toAbstract.eval s as) β
β s β S, M.toAbstract.eval s (a :: as)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
intro β¨sβ, β¨sβ, β¨sβ, h1, h2β©, h3β©, h4β©
|
case cons.mp
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
β’ (β s,
(β state, (β start_state β S, M.toAbstract.EpsilonClosure start_state state) β§ M.toAbstract.symbol state a s) β§
M.toAbstract.eval s as) β
β s β S, M.toAbstract.eval s (a :: as)
|
case cons.mp
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
sβ sβ sβ : Ο
h1 : sβ β S
h2 : M.toAbstract.EpsilonClosure sβ sβ
h3 : M.toAbstract.symbol sβ a sβ
h4 : M.toAbstract.eval sβ as
β’ β s β S, M.toAbstract.eval s (a :: as)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case cons.mp
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
β’ (β s,
(β state, (β start_state β S, M.toAbstract.EpsilonClosure start_state state) β§ M.toAbstract.symbol state a s) β§
M.toAbstract.eval s as) β
β s β S, M.toAbstract.eval s (a :: as)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
refine β¨_, h1, ?_β©
|
case cons.mp
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
sβ sβ sβ : Ο
h1 : sβ β S
h2 : M.toAbstract.EpsilonClosure sβ sβ
h3 : M.toAbstract.symbol sβ a sβ
h4 : M.toAbstract.eval sβ as
β’ β s β S, M.toAbstract.eval s (a :: as)
|
case cons.mp
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
sβ sβ sβ : Ο
h1 : sβ β S
h2 : M.toAbstract.EpsilonClosure sβ sβ
h3 : M.toAbstract.symbol sβ a sβ
h4 : M.toAbstract.eval sβ as
β’ M.toAbstract.eval sβ (a :: as)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case cons.mp
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
sβ sβ sβ : Ο
h1 : sβ β S
h2 : M.toAbstract.EpsilonClosure sβ sβ
h3 : M.toAbstract.symbol sβ a sβ
h4 : M.toAbstract.eval sβ as
β’ β s β S, M.toAbstract.eval s (a :: as)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
clear h1
|
case cons.mp
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
sβ sβ sβ : Ο
h1 : sβ β S
h2 : M.toAbstract.EpsilonClosure sβ sβ
h3 : M.toAbstract.symbol sβ a sβ
h4 : M.toAbstract.eval sβ as
β’ M.toAbstract.eval sβ (a :: as)
|
case cons.mp
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
sβ sβ sβ : Ο
h2 : M.toAbstract.EpsilonClosure sβ sβ
h3 : M.toAbstract.symbol sβ a sβ
h4 : M.toAbstract.eval sβ as
β’ M.toAbstract.eval sβ (a :: as)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case cons.mp
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
sβ sβ sβ : Ο
h1 : sβ β S
h2 : M.toAbstract.EpsilonClosure sβ sβ
h3 : M.toAbstract.symbol sβ a sβ
h4 : M.toAbstract.eval sβ as
β’ M.toAbstract.eval sβ (a :: as)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
induction h2 using Relation.ReflTransGen.head_induction_on with
| refl => {apply AbstractEpsilonNFA.eval.sym _ _ _ _ h4 h3; }
| head h _ ih => exact .eps _ _ _ ih h
|
case cons.mp
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
sβ sβ sβ : Ο
h2 : M.toAbstract.EpsilonClosure sβ sβ
h3 : M.toAbstract.symbol sβ a sβ
h4 : M.toAbstract.eval sβ as
β’ M.toAbstract.eval sβ (a :: as)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case cons.mp
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
sβ sβ sβ : Ο
h2 : M.toAbstract.EpsilonClosure sβ sβ
h3 : M.toAbstract.symbol sβ a sβ
h4 : M.toAbstract.eval sβ as
β’ M.toAbstract.eval sβ (a :: as)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
apply AbstractEpsilonNFA.eval.sym _ _ _ _ h4 h3
|
case cons.mp.refl
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
sβ sβ sβ : Ο
h3 : M.toAbstract.symbol sβ a sβ
h4 : M.toAbstract.eval sβ as
β’ M.toAbstract.eval sβ (a :: as)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case cons.mp.refl
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
sβ sβ sβ : Ο
h3 : M.toAbstract.symbol sβ a sβ
h4 : M.toAbstract.eval sβ as
β’ M.toAbstract.eval sβ (a :: as)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
exact .eps _ _ _ ih h
|
case cons.mp.head
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
sβ sβ sβ : Ο
h3 : M.toAbstract.symbol sβ a sβ
h4 : M.toAbstract.eval sβ as
aβ cβ : Ο
h : M.toAbstract.epsilon aβ cβ
hβ : Relation.ReflTransGen M.toAbstract.epsilon cβ sβ
ih : M.toAbstract.eval cβ (a :: as)
β’ M.toAbstract.eval aβ (a :: as)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case cons.mp.head
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
sβ sβ sβ : Ο
h3 : M.toAbstract.symbol sβ a sβ
h4 : M.toAbstract.eval sβ as
aβ cβ : Ο
h : M.toAbstract.epsilon aβ cβ
hβ : Relation.ReflTransGen M.toAbstract.epsilon cβ sβ
ih : M.toAbstract.eval cβ (a :: as)
β’ M.toAbstract.eval aβ (a :: as)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
intro β¨s, h1, h2β©
|
case cons.mpr
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
β’ (β s β S, M.toAbstract.eval s (a :: as)) β
β s,
(β state, (β start_state β S, M.toAbstract.EpsilonClosure start_state state) β§ M.toAbstract.symbol state a s) β§
M.toAbstract.eval s as
|
case cons.mpr
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
s : Ο
h1 : s β S
h2 : M.toAbstract.eval s (a :: as)
β’ β s,
(β state, (β start_state β S, M.toAbstract.EpsilonClosure start_state state) β§ M.toAbstract.symbol state a s) β§
M.toAbstract.eval s as
|
Please generate a tactic in lean4 to solve the state.
STATE:
case cons.mpr
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
β’ (β s β S, M.toAbstract.eval s (a :: as)) β
β s,
(β state, (β start_state β S, M.toAbstract.EpsilonClosure start_state state) β§ M.toAbstract.symbol state a s) β§
M.toAbstract.eval s as
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
obtain β¨sβ, sβ, h3, h4, h5β© :
β sβ sβ, M.toAbstract.EpsilonClosure s sβ β§
M.toAbstract.symbol sβ a sβ β§ M.toAbstract.eval sβ as := by
clear h1; generalize e : a::as = l at h2
induction h2 with cases e
| sym _ _ _ _ h1 h2 => exact β¨_, _, .refl, h2, h1β©
| eps _ _ _ h1 h2 ih =>
have β¨sβ, sβ, h3, h4, h5β© := ih rfl
exact β¨_, _, .head h2 h3, h4, h5β©
|
case cons.mpr
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
s : Ο
h1 : s β S
h2 : M.toAbstract.eval s (a :: as)
β’ β s,
(β state, (β start_state β S, M.toAbstract.EpsilonClosure start_state state) β§ M.toAbstract.symbol state a s) β§
M.toAbstract.eval s as
|
case cons.mpr.intro.intro.intro.intro
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
s : Ο
h1 : s β S
h2 : M.toAbstract.eval s (a :: as)
sβ sβ : Ο
h3 : M.toAbstract.EpsilonClosure s sβ
h4 : M.toAbstract.symbol sβ a sβ
h5 : M.toAbstract.eval sβ as
β’ β s,
(β state, (β start_state β S, M.toAbstract.EpsilonClosure start_state state) β§ M.toAbstract.symbol state a s) β§
M.toAbstract.eval s as
|
Please generate a tactic in lean4 to solve the state.
STATE:
case cons.mpr
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
s : Ο
h1 : s β S
h2 : M.toAbstract.eval s (a :: as)
β’ β s,
(β state, (β start_state β S, M.toAbstract.EpsilonClosure start_state state) β§ M.toAbstract.symbol state a s) β§
M.toAbstract.eval s as
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
exact β¨_, β¨_, β¨_, h1, h3β©, h4β©, h5β©
|
case cons.mpr.intro.intro.intro.intro
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
s : Ο
h1 : s β S
h2 : M.toAbstract.eval s (a :: as)
sβ sβ : Ο
h3 : M.toAbstract.EpsilonClosure s sβ
h4 : M.toAbstract.symbol sβ a sβ
h5 : M.toAbstract.eval sβ as
β’ β s,
(β state, (β start_state β S, M.toAbstract.EpsilonClosure start_state state) β§ M.toAbstract.symbol state a s) β§
M.toAbstract.eval s as
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case cons.mpr.intro.intro.intro.intro
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
s : Ο
h1 : s β S
h2 : M.toAbstract.eval s (a :: as)
sβ sβ : Ο
h3 : M.toAbstract.EpsilonClosure s sβ
h4 : M.toAbstract.symbol sβ a sβ
h5 : M.toAbstract.eval sβ as
β’ β s,
(β state, (β start_state β S, M.toAbstract.EpsilonClosure start_state state) β§ M.toAbstract.symbol state a s) β§
M.toAbstract.eval s as
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
clear h1
|
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
s : Ο
h1 : s β S
h2 : M.toAbstract.eval s (a :: as)
β’ β sβ sβ, M.toAbstract.EpsilonClosure s sβ β§ M.toAbstract.symbol sβ a sβ β§ M.toAbstract.eval sβ as
|
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
s : Ο
h2 : M.toAbstract.eval s (a :: as)
β’ β sβ sβ, M.toAbstract.EpsilonClosure s sβ β§ M.toAbstract.symbol sβ a sβ β§ M.toAbstract.eval sβ as
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
s : Ο
h1 : s β S
h2 : M.toAbstract.eval s (a :: as)
β’ β sβ sβ, M.toAbstract.EpsilonClosure s sβ β§ M.toAbstract.symbol sβ a sβ β§ M.toAbstract.eval sβ as
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
generalize e : a::as = l at h2
|
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
s : Ο
h2 : M.toAbstract.eval s (a :: as)
β’ β sβ sβ, M.toAbstract.EpsilonClosure s sβ β§ M.toAbstract.symbol sβ a sβ β§ M.toAbstract.eval sβ as
|
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
s : Ο
l : List Ξ±
e : a :: as = l
h2 : M.toAbstract.eval s l
β’ β sβ sβ, M.toAbstract.EpsilonClosure s sβ β§ M.toAbstract.symbol sβ a sβ β§ M.toAbstract.eval sβ as
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
s : Ο
h2 : M.toAbstract.eval s (a :: as)
β’ β sβ sβ, M.toAbstract.EpsilonClosure s sβ β§ M.toAbstract.symbol sβ a sβ β§ M.toAbstract.eval sβ as
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
induction h2 with cases e
| sym _ _ _ _ h1 h2 => exact β¨_, _, .refl, h2, h1β©
| eps _ _ _ h1 h2 ih =>
have β¨sβ, sβ, h3, h4, h5β© := ih rfl
exact β¨_, _, .head h2 h3, h4, h5β©
|
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
s : Ο
l : List Ξ±
e : a :: as = l
h2 : M.toAbstract.eval s l
β’ β sβ sβ, M.toAbstract.EpsilonClosure s sβ β§ M.toAbstract.symbol sβ a sβ β§ M.toAbstract.eval sβ as
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
s : Ο
l : List Ξ±
e : a :: as = l
h2 : M.toAbstract.eval s l
β’ β sβ sβ, M.toAbstract.EpsilonClosure s sβ β§ M.toAbstract.symbol sβ a sβ β§ M.toAbstract.eval sβ as
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
exact β¨_, _, .refl, h2, h1β©
|
case sym.refl
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
s : Ο
l : List Ξ±
start_stateβ stop_stateβ : Ο
h2 : M.toAbstract.symbol start_stateβ a stop_stateβ
h1 : M.toAbstract.eval stop_stateβ as
a_ihβ :
a :: as = as β
β sβ sβ, M.toAbstract.EpsilonClosure stop_stateβ sβ β§ M.toAbstract.symbol sβ a sβ β§ M.toAbstract.eval sβ as
β’ β sβ sβ, M.toAbstract.EpsilonClosure start_stateβ sβ β§ M.toAbstract.symbol sβ a sβ β§ M.toAbstract.eval sβ as
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case sym.refl
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
s : Ο
l : List Ξ±
start_stateβ stop_stateβ : Ο
h2 : M.toAbstract.symbol start_stateβ a stop_stateβ
h1 : M.toAbstract.eval stop_stateβ as
a_ihβ :
a :: as = as β
β sβ sβ, M.toAbstract.EpsilonClosure stop_stateβ sβ β§ M.toAbstract.symbol sβ a sβ β§ M.toAbstract.eval sβ as
β’ β sβ sβ, M.toAbstract.EpsilonClosure start_stateβ sβ β§ M.toAbstract.symbol sβ a sβ β§ M.toAbstract.eval sβ as
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
have β¨sβ, sβ, h3, h4, h5β© := ih rfl
|
case eps.refl
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
s : Ο
l : List Ξ±
start_stateβ stop_stateβ : Ο
h2 : M.toAbstract.epsilon start_stateβ stop_stateβ
h1 : M.toAbstract.eval stop_stateβ (a :: as)
ih :
a :: as = a :: as β
β sβ sβ, M.toAbstract.EpsilonClosure stop_stateβ sβ β§ M.toAbstract.symbol sβ a sβ β§ M.toAbstract.eval sβ as
β’ β sβ sβ, M.toAbstract.EpsilonClosure start_stateβ sβ β§ M.toAbstract.symbol sβ a sβ β§ M.toAbstract.eval sβ as
|
case eps.refl
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
s : Ο
l : List Ξ±
start_stateβ stop_stateβ : Ο
h2 : M.toAbstract.epsilon start_stateβ stop_stateβ
h1 : M.toAbstract.eval stop_stateβ (a :: as)
ih :
a :: as = a :: as β
β sβ sβ, M.toAbstract.EpsilonClosure stop_stateβ sβ β§ M.toAbstract.symbol sβ a sβ β§ M.toAbstract.eval sβ as
sβ sβ : Ο
h3 : M.toAbstract.EpsilonClosure stop_stateβ sβ
h4 : M.toAbstract.symbol sβ a sβ
h5 : M.toAbstract.eval sβ as
β’ β sβ sβ, M.toAbstract.EpsilonClosure start_stateβ sβ β§ M.toAbstract.symbol sβ a sβ β§ M.toAbstract.eval sβ as
|
Please generate a tactic in lean4 to solve the state.
STATE:
case eps.refl
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
s : Ο
l : List Ξ±
start_stateβ stop_stateβ : Ο
h2 : M.toAbstract.epsilon start_stateβ stop_stateβ
h1 : M.toAbstract.eval stop_stateβ (a :: as)
ih :
a :: as = a :: as β
β sβ sβ, M.toAbstract.EpsilonClosure stop_stateβ sβ β§ M.toAbstract.symbol sβ a sβ β§ M.toAbstract.eval sβ as
β’ β sβ sβ, M.toAbstract.EpsilonClosure start_stateβ sβ β§ M.toAbstract.symbol sβ a sβ β§ M.toAbstract.eval sβ as
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_from_iff
|
[405, 1]
|
[454, 44]
|
exact β¨_, _, .head h2 h3, h4, h5β©
|
case eps.refl
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
s : Ο
l : List Ξ±
start_stateβ stop_stateβ : Ο
h2 : M.toAbstract.epsilon start_stateβ stop_stateβ
h1 : M.toAbstract.eval stop_stateβ (a :: as)
ih :
a :: as = a :: as β
β sβ sβ, M.toAbstract.EpsilonClosure stop_stateβ sβ β§ M.toAbstract.symbol sβ a sβ β§ M.toAbstract.eval sβ as
sβ sβ : Ο
h3 : M.toAbstract.EpsilonClosure stop_stateβ sβ
h4 : M.toAbstract.symbol sβ a sβ
h5 : M.toAbstract.eval sβ as
β’ β sβ sβ, M.toAbstract.EpsilonClosure start_stateβ sβ β§ M.toAbstract.symbol sβ a sβ β§ M.toAbstract.eval sβ as
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case eps.refl
Ξ± Ο : Type
instβΒΉ : DecidableEq Ξ±
instβ : DecidableEq Ο
M : EpsilonNFA Ξ± Ο
a : Ξ±
as : List Ξ±
IH : β (S : List Ο), (β s β M.eval_from S as, s β M.accepting_state_list) β β s β S, M.toAbstract.eval s as
S : List Ο
s : Ο
l : List Ξ±
start_stateβ stop_stateβ : Ο
h2 : M.toAbstract.epsilon start_stateβ stop_stateβ
h1 : M.toAbstract.eval stop_stateβ (a :: as)
ih :
a :: as = a :: as β
β sβ sβ, M.toAbstract.EpsilonClosure stop_stateβ sβ β§ M.toAbstract.symbol sβ a sβ β§ M.toAbstract.eval sβ as
sβ sβ : Ο
h3 : M.toAbstract.EpsilonClosure stop_stateβ sβ
h4 : M.toAbstract.symbol sβ a sβ
h5 : M.toAbstract.eval sβ as
β’ β sβ sβ, M.toAbstract.EpsilonClosure start_stateβ sβ β§ M.toAbstract.symbol sβ a sβ β§ M.toAbstract.eval sβ as
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.accepts_iff
|
[457, 1]
|
[468, 8]
|
simp [accepts, eval]
|
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
input : List Ξ±
β’ e.accepts input β e.toAbstract.accepts input
|
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
input : List Ξ±
β’ (β s β e.eval_from e.starting_state_list input, s β e.accepting_state_list) β e.toAbstract.accepts input
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
input : List Ξ±
β’ e.accepts input β e.toAbstract.accepts input
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.accepts_iff
|
[457, 1]
|
[468, 8]
|
rw [EpsilonNFA.eval_from_iff]
|
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
input : List Ξ±
β’ (β s β e.eval_from e.starting_state_list input, s β e.accepting_state_list) β e.toAbstract.accepts input
|
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
input : List Ξ±
β’ (β s β e.starting_state_list, e.toAbstract.eval s input) β e.toAbstract.accepts input
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
input : List Ξ±
β’ (β s β e.eval_from e.starting_state_list input, s β e.accepting_state_list) β e.toAbstract.accepts input
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.accepts_iff
|
[457, 1]
|
[468, 8]
|
rfl
|
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
input : List Ξ±
β’ (β s β e.starting_state_list, e.toAbstract.eval s input) β e.toAbstract.accepts input
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
instβΒΉ : DecidableEq Ξ±
Ο : Type
instβ : DecidableEq Ο
e : EpsilonNFA Ξ± Ο
input : List Ξ±
β’ (β s β e.starting_state_list, e.toAbstract.eval s input) β e.toAbstract.accepts input
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
induction F
|
F : Formula
β’ fastReplaceFree id F = F
|
case pred_const_
aβΒΉ : PredName
aβ : List VarName
β’ fastReplaceFree id (pred_const_ aβΒΉ aβ) = pred_const_ aβΒΉ aβ
case pred_var_
aβΒΉ : PredName
aβ : List VarName
β’ fastReplaceFree id (pred_var_ aβΒΉ aβ) = pred_var_ aβΒΉ aβ
case eq_
aβΒΉ aβ : VarName
β’ fastReplaceFree id (eq_ aβΒΉ aβ) = eq_ aβΒΉ aβ
case true_
β’ fastReplaceFree id true_ = true_
case false_
β’ fastReplaceFree id false_ = false_
case not_
aβ : Formula
a_ihβ : fastReplaceFree id aβ = aβ
β’ fastReplaceFree id aβ.not_ = aβ.not_
case imp_
aβΒΉ aβ : Formula
a_ihβΒΉ : fastReplaceFree id aβΒΉ = aβΒΉ
a_ihβ : fastReplaceFree id aβ = aβ
β’ fastReplaceFree id (aβΒΉ.imp_ aβ) = aβΒΉ.imp_ aβ
case and_
aβΒΉ aβ : Formula
a_ihβΒΉ : fastReplaceFree id aβΒΉ = aβΒΉ
a_ihβ : fastReplaceFree id aβ = aβ
β’ fastReplaceFree id (aβΒΉ.and_ aβ) = aβΒΉ.and_ aβ
case or_
aβΒΉ aβ : Formula
a_ihβΒΉ : fastReplaceFree id aβΒΉ = aβΒΉ
a_ihβ : fastReplaceFree id aβ = aβ
β’ fastReplaceFree id (aβΒΉ.or_ aβ) = aβΒΉ.or_ aβ
case iff_
aβΒΉ aβ : Formula
a_ihβΒΉ : fastReplaceFree id aβΒΉ = aβΒΉ
a_ihβ : fastReplaceFree id aβ = aβ
β’ fastReplaceFree id (aβΒΉ.iff_ aβ) = aβΒΉ.iff_ aβ
case forall_
aβΒΉ : VarName
aβ : Formula
a_ihβ : fastReplaceFree id aβ = aβ
β’ fastReplaceFree id (forall_ aβΒΉ aβ) = forall_ aβΒΉ aβ
case exists_
aβΒΉ : VarName
aβ : Formula
a_ihβ : fastReplaceFree id aβ = aβ
β’ fastReplaceFree id (exists_ aβΒΉ aβ) = exists_ aβΒΉ aβ
case def_
aβΒΉ : DefName
aβ : List VarName
β’ fastReplaceFree id (def_ aβΒΉ aβ) = def_ aβΒΉ aβ
|
Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
β’ fastReplaceFree id F = F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
all_goals
simp only [fastReplaceFree]
|
case pred_const_
aβΒΉ : PredName
aβ : List VarName
β’ fastReplaceFree id (pred_const_ aβΒΉ aβ) = pred_const_ aβΒΉ aβ
case pred_var_
aβΒΉ : PredName
aβ : List VarName
β’ fastReplaceFree id (pred_var_ aβΒΉ aβ) = pred_var_ aβΒΉ aβ
case eq_
aβΒΉ aβ : VarName
β’ fastReplaceFree id (eq_ aβΒΉ aβ) = eq_ aβΒΉ aβ
case true_
β’ fastReplaceFree id true_ = true_
case false_
β’ fastReplaceFree id false_ = false_
case not_
aβ : Formula
a_ihβ : fastReplaceFree id aβ = aβ
β’ fastReplaceFree id aβ.not_ = aβ.not_
case imp_
aβΒΉ aβ : Formula
a_ihβΒΉ : fastReplaceFree id aβΒΉ = aβΒΉ
a_ihβ : fastReplaceFree id aβ = aβ
β’ fastReplaceFree id (aβΒΉ.imp_ aβ) = aβΒΉ.imp_ aβ
case and_
aβΒΉ aβ : Formula
a_ihβΒΉ : fastReplaceFree id aβΒΉ = aβΒΉ
a_ihβ : fastReplaceFree id aβ = aβ
β’ fastReplaceFree id (aβΒΉ.and_ aβ) = aβΒΉ.and_ aβ
case or_
aβΒΉ aβ : Formula
a_ihβΒΉ : fastReplaceFree id aβΒΉ = aβΒΉ
a_ihβ : fastReplaceFree id aβ = aβ
β’ fastReplaceFree id (aβΒΉ.or_ aβ) = aβΒΉ.or_ aβ
case iff_
aβΒΉ aβ : Formula
a_ihβΒΉ : fastReplaceFree id aβΒΉ = aβΒΉ
a_ihβ : fastReplaceFree id aβ = aβ
β’ fastReplaceFree id (aβΒΉ.iff_ aβ) = aβΒΉ.iff_ aβ
case forall_
aβΒΉ : VarName
aβ : Formula
a_ihβ : fastReplaceFree id aβ = aβ
β’ fastReplaceFree id (forall_ aβΒΉ aβ) = forall_ aβΒΉ aβ
case exists_
aβΒΉ : VarName
aβ : Formula
a_ihβ : fastReplaceFree id aβ = aβ
β’ fastReplaceFree id (exists_ aβΒΉ aβ) = exists_ aβΒΉ aβ
case def_
aβΒΉ : DefName
aβ : List VarName
β’ fastReplaceFree id (def_ aβΒΉ aβ) = def_ aβΒΉ aβ
|
case pred_const_
aβΒΉ : PredName
aβ : List VarName
β’ pred_const_ aβΒΉ (List.map id aβ) = pred_const_ aβΒΉ aβ
case pred_var_
aβΒΉ : PredName
aβ : List VarName
β’ pred_var_ aβΒΉ (List.map id aβ) = pred_var_ aβΒΉ aβ
case eq_
aβΒΉ aβ : VarName
β’ eq_ (id aβΒΉ) (id aβ) = eq_ aβΒΉ aβ
case not_
aβ : Formula
a_ihβ : fastReplaceFree id aβ = aβ
β’ (fastReplaceFree id aβ).not_ = aβ.not_
case imp_
aβΒΉ aβ : Formula
a_ihβΒΉ : fastReplaceFree id aβΒΉ = aβΒΉ
a_ihβ : fastReplaceFree id aβ = aβ
β’ (fastReplaceFree id aβΒΉ).imp_ (fastReplaceFree id aβ) = aβΒΉ.imp_ aβ
case and_
aβΒΉ aβ : Formula
a_ihβΒΉ : fastReplaceFree id aβΒΉ = aβΒΉ
a_ihβ : fastReplaceFree id aβ = aβ
β’ (fastReplaceFree id aβΒΉ).and_ (fastReplaceFree id aβ) = aβΒΉ.and_ aβ
case or_
aβΒΉ aβ : Formula
a_ihβΒΉ : fastReplaceFree id aβΒΉ = aβΒΉ
a_ihβ : fastReplaceFree id aβ = aβ
β’ (fastReplaceFree id aβΒΉ).or_ (fastReplaceFree id aβ) = aβΒΉ.or_ aβ
case iff_
aβΒΉ aβ : Formula
a_ihβΒΉ : fastReplaceFree id aβΒΉ = aβΒΉ
a_ihβ : fastReplaceFree id aβ = aβ
β’ (fastReplaceFree id aβΒΉ).iff_ (fastReplaceFree id aβ) = aβΒΉ.iff_ aβ
case forall_
aβΒΉ : VarName
aβ : Formula
a_ihβ : fastReplaceFree id aβ = aβ
β’ forall_ aβΒΉ (fastReplaceFree (Function.updateITE id aβΒΉ aβΒΉ) aβ) = forall_ aβΒΉ aβ
case exists_
aβΒΉ : VarName
aβ : Formula
a_ihβ : fastReplaceFree id aβ = aβ
β’ exists_ aβΒΉ (fastReplaceFree (Function.updateITE id aβΒΉ aβΒΉ) aβ) = exists_ aβΒΉ aβ
case def_
aβΒΉ : DefName
aβ : List VarName
β’ def_ aβΒΉ (List.map id aβ) = def_ aβΒΉ aβ
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
aβΒΉ : PredName
aβ : List VarName
β’ fastReplaceFree id (pred_const_ aβΒΉ aβ) = pred_const_ aβΒΉ aβ
case pred_var_
aβΒΉ : PredName
aβ : List VarName
β’ fastReplaceFree id (pred_var_ aβΒΉ aβ) = pred_var_ aβΒΉ aβ
case eq_
aβΒΉ aβ : VarName
β’ fastReplaceFree id (eq_ aβΒΉ aβ) = eq_ aβΒΉ aβ
case true_
β’ fastReplaceFree id true_ = true_
case false_
β’ fastReplaceFree id false_ = false_
case not_
aβ : Formula
a_ihβ : fastReplaceFree id aβ = aβ
β’ fastReplaceFree id aβ.not_ = aβ.not_
case imp_
aβΒΉ aβ : Formula
a_ihβΒΉ : fastReplaceFree id aβΒΉ = aβΒΉ
a_ihβ : fastReplaceFree id aβ = aβ
β’ fastReplaceFree id (aβΒΉ.imp_ aβ) = aβΒΉ.imp_ aβ
case and_
aβΒΉ aβ : Formula
a_ihβΒΉ : fastReplaceFree id aβΒΉ = aβΒΉ
a_ihβ : fastReplaceFree id aβ = aβ
β’ fastReplaceFree id (aβΒΉ.and_ aβ) = aβΒΉ.and_ aβ
case or_
aβΒΉ aβ : Formula
a_ihβΒΉ : fastReplaceFree id aβΒΉ = aβΒΉ
a_ihβ : fastReplaceFree id aβ = aβ
β’ fastReplaceFree id (aβΒΉ.or_ aβ) = aβΒΉ.or_ aβ
case iff_
aβΒΉ aβ : Formula
a_ihβΒΉ : fastReplaceFree id aβΒΉ = aβΒΉ
a_ihβ : fastReplaceFree id aβ = aβ
β’ fastReplaceFree id (aβΒΉ.iff_ aβ) = aβΒΉ.iff_ aβ
case forall_
aβΒΉ : VarName
aβ : Formula
a_ihβ : fastReplaceFree id aβ = aβ
β’ fastReplaceFree id (forall_ aβΒΉ aβ) = forall_ aβΒΉ aβ
case exists_
aβΒΉ : VarName
aβ : Formula
a_ihβ : fastReplaceFree id aβ = aβ
β’ fastReplaceFree id (exists_ aβΒΉ aβ) = exists_ aβΒΉ aβ
case def_
aβΒΉ : DefName
aβ : List VarName
β’ fastReplaceFree id (def_ aβΒΉ aβ) = def_ aβΒΉ aβ
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
case pred_const_ X xs | pred_var_ X xs | def_ X xs =>
congr!
simp
|
X : DefName
xs : List VarName
β’ def_ X (List.map id xs) = def_ X xs
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : DefName
xs : List VarName
β’ def_ X (List.map id xs) = def_ X xs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
case eq_ x y =>
congr!
|
x y : VarName
β’ eq_ (id x) (id y) = eq_ x y
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
x y : VarName
β’ eq_ (id x) (id y) = eq_ x y
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
case not_ phi phi_ih =>
congr!
|
phi : Formula
phi_ih : fastReplaceFree id phi = phi
β’ (fastReplaceFree id phi).not_ = phi.not_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
phi : Formula
phi_ih : fastReplaceFree id phi = phi
β’ (fastReplaceFree id phi).not_ = phi.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
case
imp_ phi psi phi_ih psi_ih
| and_ phi psi phi_ih psi_ih
| or_ phi psi phi_ih psi_ih
| iff_ phi psi phi_ih psi_ih =>
congr!
|
phi psi : Formula
phi_ih : fastReplaceFree id phi = phi
psi_ih : fastReplaceFree id psi = psi
β’ (fastReplaceFree id phi).iff_ (fastReplaceFree id psi) = phi.iff_ psi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
phi psi : Formula
phi_ih : fastReplaceFree id phi = phi
psi_ih : fastReplaceFree id psi = psi
β’ (fastReplaceFree id phi).iff_ (fastReplaceFree id psi) = phi.iff_ psi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
case forall_ x phi phi_ih | exists_ x phi phi_ih =>
congr!
simp only [Function.updateITE_id]
exact phi_ih
|
x : VarName
phi : Formula
phi_ih : fastReplaceFree id phi = phi
β’ exists_ x (fastReplaceFree (Function.updateITE id x x) phi) = exists_ x phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
x : VarName
phi : Formula
phi_ih : fastReplaceFree id phi = phi
β’ exists_ x (fastReplaceFree (Function.updateITE id x x) phi) = exists_ x phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
simp only [fastReplaceFree]
|
case def_
aβΒΉ : DefName
aβ : List VarName
β’ fastReplaceFree id (def_ aβΒΉ aβ) = def_ aβΒΉ aβ
|
case def_
aβΒΉ : DefName
aβ : List VarName
β’ def_ aβΒΉ (List.map id aβ) = def_ aβΒΉ aβ
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
aβΒΉ : DefName
aβ : List VarName
β’ fastReplaceFree id (def_ aβΒΉ aβ) = def_ aβΒΉ aβ
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
congr!
|
X : DefName
xs : List VarName
β’ def_ X (List.map id xs) = def_ X xs
|
case h.e'_2
X : DefName
xs : List VarName
β’ List.map id xs = xs
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : DefName
xs : List VarName
β’ def_ X (List.map id xs) = def_ X xs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
simp
|
case h.e'_2
X : DefName
xs : List VarName
β’ List.map id xs = xs
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
X : DefName
xs : List VarName
β’ List.map id xs = xs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
congr!
|
x y : VarName
β’ eq_ (id x) (id y) = eq_ x y
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
x y : VarName
β’ eq_ (id x) (id y) = eq_ x y
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
congr!
|
phi : Formula
phi_ih : fastReplaceFree id phi = phi
β’ (fastReplaceFree id phi).not_ = phi.not_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
phi : Formula
phi_ih : fastReplaceFree id phi = phi
β’ (fastReplaceFree id phi).not_ = phi.not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
congr!
|
phi psi : Formula
phi_ih : fastReplaceFree id phi = phi
psi_ih : fastReplaceFree id psi = psi
β’ (fastReplaceFree id phi).iff_ (fastReplaceFree id psi) = phi.iff_ psi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
phi psi : Formula
phi_ih : fastReplaceFree id phi = phi
psi_ih : fastReplaceFree id psi = psi
β’ (fastReplaceFree id phi).iff_ (fastReplaceFree id psi) = phi.iff_ psi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
congr!
|
x : VarName
phi : Formula
phi_ih : fastReplaceFree id phi = phi
β’ exists_ x (fastReplaceFree (Function.updateITE id x x) phi) = exists_ x phi
|
case h.e'_2
x : VarName
phi : Formula
phi_ih : fastReplaceFree id phi = phi
β’ fastReplaceFree (Function.updateITE id x x) phi = phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
x : VarName
phi : Formula
phi_ih : fastReplaceFree id phi = phi
β’ exists_ x (fastReplaceFree (Function.updateITE id x x) phi) = exists_ x phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
simp only [Function.updateITE_id]
|
case h.e'_2
x : VarName
phi : Formula
phi_ih : fastReplaceFree id phi = phi
β’ fastReplaceFree (Function.updateITE id x x) phi = phi
|
case h.e'_2
x : VarName
phi : Formula
phi_ih : fastReplaceFree id phi = phi
β’ fastReplaceFree id phi = phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
x : VarName
phi : Formula
phi_ih : fastReplaceFree id phi = phi
β’ fastReplaceFree (Function.updateITE id x x) phi = phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id
|
[98, 1]
|
[121, 17]
|
exact phi_ih
|
case h.e'_2
x : VarName
phi : Formula
phi_ih : fastReplaceFree id phi = phi
β’ fastReplaceFree id phi = phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
x : VarName
phi : Formula
phi_ih : fastReplaceFree id phi = phi
β’ fastReplaceFree id phi = phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
induction F generalizing Ο Ο'
|
F : Formula
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v F β Ο v = Ο' v
β’ fastReplaceFree Ο F = fastReplaceFree Ο' F
|
case pred_const_
aβΒΉ : PredName
aβ : List VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (pred_const_ aβΒΉ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (pred_const_ aβΒΉ aβ) = fastReplaceFree Ο' (pred_const_ aβΒΉ aβ)
case pred_var_
aβΒΉ : PredName
aβ : List VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (pred_var_ aβΒΉ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (pred_var_ aβΒΉ aβ) = fastReplaceFree Ο' (pred_var_ aβΒΉ aβ)
case eq_
aβΒΉ aβ : VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (eq_ aβΒΉ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (eq_ aβΒΉ aβ) = fastReplaceFree Ο' (eq_ aβΒΉ aβ)
case true_
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v true_ β Ο v = Ο' v
β’ fastReplaceFree Ο true_ = fastReplaceFree Ο' true_
case false_
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v false_ β Ο v = Ο' v
β’ fastReplaceFree Ο false_ = fastReplaceFree Ο' false_
case not_
aβ : Formula
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v aβ.not_ β Ο v = Ο' v
β’ fastReplaceFree Ο aβ.not_ = fastReplaceFree Ο' aβ.not_
case imp_
aβΒΉ aβ : Formula
a_ihβΒΉ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβΒΉ β Ο v = Ο' v) β fastReplaceFree Ο aβΒΉ = fastReplaceFree Ο' aβΒΉ
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (aβΒΉ.imp_ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (aβΒΉ.imp_ aβ) = fastReplaceFree Ο' (aβΒΉ.imp_ aβ)
case and_
aβΒΉ aβ : Formula
a_ihβΒΉ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβΒΉ β Ο v = Ο' v) β fastReplaceFree Ο aβΒΉ = fastReplaceFree Ο' aβΒΉ
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (aβΒΉ.and_ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (aβΒΉ.and_ aβ) = fastReplaceFree Ο' (aβΒΉ.and_ aβ)
case or_
aβΒΉ aβ : Formula
a_ihβΒΉ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβΒΉ β Ο v = Ο' v) β fastReplaceFree Ο aβΒΉ = fastReplaceFree Ο' aβΒΉ
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (aβΒΉ.or_ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (aβΒΉ.or_ aβ) = fastReplaceFree Ο' (aβΒΉ.or_ aβ)
case iff_
aβΒΉ aβ : Formula
a_ihβΒΉ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβΒΉ β Ο v = Ο' v) β fastReplaceFree Ο aβΒΉ = fastReplaceFree Ο' aβΒΉ
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (aβΒΉ.iff_ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (aβΒΉ.iff_ aβ) = fastReplaceFree Ο' (aβΒΉ.iff_ aβ)
case forall_
aβΒΉ : VarName
aβ : Formula
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (forall_ aβΒΉ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (forall_ aβΒΉ aβ) = fastReplaceFree Ο' (forall_ aβΒΉ aβ)
case exists_
aβΒΉ : VarName
aβ : Formula
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (exists_ aβΒΉ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (exists_ aβΒΉ aβ) = fastReplaceFree Ο' (exists_ aβΒΉ aβ)
case def_
aβΒΉ : DefName
aβ : List VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (def_ aβΒΉ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (def_ aβΒΉ aβ) = fastReplaceFree Ο' (def_ aβΒΉ aβ)
|
Please generate a tactic in lean4 to solve the state.
STATE:
F : Formula
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v F β Ο v = Ο' v
β’ fastReplaceFree Ο F = fastReplaceFree Ο' F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
all_goals
simp only [isFreeIn] at h1
simp only [fastReplaceFree]
|
case pred_const_
aβΒΉ : PredName
aβ : List VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (pred_const_ aβΒΉ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (pred_const_ aβΒΉ aβ) = fastReplaceFree Ο' (pred_const_ aβΒΉ aβ)
case pred_var_
aβΒΉ : PredName
aβ : List VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (pred_var_ aβΒΉ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (pred_var_ aβΒΉ aβ) = fastReplaceFree Ο' (pred_var_ aβΒΉ aβ)
case eq_
aβΒΉ aβ : VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (eq_ aβΒΉ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (eq_ aβΒΉ aβ) = fastReplaceFree Ο' (eq_ aβΒΉ aβ)
case true_
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v true_ β Ο v = Ο' v
β’ fastReplaceFree Ο true_ = fastReplaceFree Ο' true_
case false_
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v false_ β Ο v = Ο' v
β’ fastReplaceFree Ο false_ = fastReplaceFree Ο' false_
case not_
aβ : Formula
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v aβ.not_ β Ο v = Ο' v
β’ fastReplaceFree Ο aβ.not_ = fastReplaceFree Ο' aβ.not_
case imp_
aβΒΉ aβ : Formula
a_ihβΒΉ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβΒΉ β Ο v = Ο' v) β fastReplaceFree Ο aβΒΉ = fastReplaceFree Ο' aβΒΉ
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (aβΒΉ.imp_ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (aβΒΉ.imp_ aβ) = fastReplaceFree Ο' (aβΒΉ.imp_ aβ)
case and_
aβΒΉ aβ : Formula
a_ihβΒΉ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβΒΉ β Ο v = Ο' v) β fastReplaceFree Ο aβΒΉ = fastReplaceFree Ο' aβΒΉ
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (aβΒΉ.and_ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (aβΒΉ.and_ aβ) = fastReplaceFree Ο' (aβΒΉ.and_ aβ)
case or_
aβΒΉ aβ : Formula
a_ihβΒΉ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβΒΉ β Ο v = Ο' v) β fastReplaceFree Ο aβΒΉ = fastReplaceFree Ο' aβΒΉ
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (aβΒΉ.or_ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (aβΒΉ.or_ aβ) = fastReplaceFree Ο' (aβΒΉ.or_ aβ)
case iff_
aβΒΉ aβ : Formula
a_ihβΒΉ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβΒΉ β Ο v = Ο' v) β fastReplaceFree Ο aβΒΉ = fastReplaceFree Ο' aβΒΉ
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (aβΒΉ.iff_ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (aβΒΉ.iff_ aβ) = fastReplaceFree Ο' (aβΒΉ.iff_ aβ)
case forall_
aβΒΉ : VarName
aβ : Formula
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (forall_ aβΒΉ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (forall_ aβΒΉ aβ) = fastReplaceFree Ο' (forall_ aβΒΉ aβ)
case exists_
aβΒΉ : VarName
aβ : Formula
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (exists_ aβΒΉ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (exists_ aβΒΉ aβ) = fastReplaceFree Ο' (exists_ aβΒΉ aβ)
case def_
aβΒΉ : DefName
aβ : List VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (def_ aβΒΉ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (def_ aβΒΉ aβ) = fastReplaceFree Ο' (def_ aβΒΉ aβ)
|
case pred_const_
aβΒΉ : PredName
aβ : List VarName
Ο Ο' : VarName β VarName
h1 : β v β aβ, Ο v = Ο' v
β’ pred_const_ aβΒΉ (List.map Ο aβ) = pred_const_ aβΒΉ (List.map Ο' aβ)
case pred_var_
aβΒΉ : PredName
aβ : List VarName
Ο Ο' : VarName β VarName
h1 : β v β aβ, Ο v = Ο' v
β’ pred_var_ aβΒΉ (List.map Ο aβ) = pred_var_ aβΒΉ (List.map Ο' aβ)
case eq_
aβΒΉ aβ : VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), v = aβΒΉ β¨ v = aβ β Ο v = Ο' v
β’ eq_ (Ο aβΒΉ) (Ο aβ) = eq_ (Ο' aβΒΉ) (Ο' aβ)
case not_
aβ : Formula
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v aβ β Ο v = Ο' v
β’ (fastReplaceFree Ο aβ).not_ = (fastReplaceFree Ο' aβ).not_
case imp_
aβΒΉ aβ : Formula
a_ihβΒΉ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβΒΉ β Ο v = Ο' v) β fastReplaceFree Ο aβΒΉ = fastReplaceFree Ο' aβΒΉ
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v aβΒΉ β¨ isFreeIn v aβ β Ο v = Ο' v
β’ (fastReplaceFree Ο aβΒΉ).imp_ (fastReplaceFree Ο aβ) = (fastReplaceFree Ο' aβΒΉ).imp_ (fastReplaceFree Ο' aβ)
case and_
aβΒΉ aβ : Formula
a_ihβΒΉ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβΒΉ β Ο v = Ο' v) β fastReplaceFree Ο aβΒΉ = fastReplaceFree Ο' aβΒΉ
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v aβΒΉ β¨ isFreeIn v aβ β Ο v = Ο' v
β’ (fastReplaceFree Ο aβΒΉ).and_ (fastReplaceFree Ο aβ) = (fastReplaceFree Ο' aβΒΉ).and_ (fastReplaceFree Ο' aβ)
case or_
aβΒΉ aβ : Formula
a_ihβΒΉ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβΒΉ β Ο v = Ο' v) β fastReplaceFree Ο aβΒΉ = fastReplaceFree Ο' aβΒΉ
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v aβΒΉ β¨ isFreeIn v aβ β Ο v = Ο' v
β’ (fastReplaceFree Ο aβΒΉ).or_ (fastReplaceFree Ο aβ) = (fastReplaceFree Ο' aβΒΉ).or_ (fastReplaceFree Ο' aβ)
case iff_
aβΒΉ aβ : Formula
a_ihβΒΉ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβΒΉ β Ο v = Ο' v) β fastReplaceFree Ο aβΒΉ = fastReplaceFree Ο' aβΒΉ
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v aβΒΉ β¨ isFreeIn v aβ β Ο v = Ο' v
β’ (fastReplaceFree Ο aβΒΉ).iff_ (fastReplaceFree Ο aβ) = (fastReplaceFree Ο' aβΒΉ).iff_ (fastReplaceFree Ο' aβ)
case forall_
aβΒΉ : VarName
aβ : Formula
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = aβΒΉ β§ isFreeIn v aβ β Ο v = Ο' v
β’ forall_ aβΒΉ (fastReplaceFree (Function.updateITE Ο aβΒΉ aβΒΉ) aβ) =
forall_ aβΒΉ (fastReplaceFree (Function.updateITE Ο' aβΒΉ aβΒΉ) aβ)
case exists_
aβΒΉ : VarName
aβ : Formula
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = aβΒΉ β§ isFreeIn v aβ β Ο v = Ο' v
β’ exists_ aβΒΉ (fastReplaceFree (Function.updateITE Ο aβΒΉ aβΒΉ) aβ) =
exists_ aβΒΉ (fastReplaceFree (Function.updateITE Ο' aβΒΉ aβΒΉ) aβ)
case def_
aβΒΉ : DefName
aβ : List VarName
Ο Ο' : VarName β VarName
h1 : β v β aβ, Ο v = Ο' v
β’ def_ aβΒΉ (List.map Ο aβ) = def_ aβΒΉ (List.map Ο' aβ)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
aβΒΉ : PredName
aβ : List VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (pred_const_ aβΒΉ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (pred_const_ aβΒΉ aβ) = fastReplaceFree Ο' (pred_const_ aβΒΉ aβ)
case pred_var_
aβΒΉ : PredName
aβ : List VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (pred_var_ aβΒΉ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (pred_var_ aβΒΉ aβ) = fastReplaceFree Ο' (pred_var_ aβΒΉ aβ)
case eq_
aβΒΉ aβ : VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (eq_ aβΒΉ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (eq_ aβΒΉ aβ) = fastReplaceFree Ο' (eq_ aβΒΉ aβ)
case true_
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v true_ β Ο v = Ο' v
β’ fastReplaceFree Ο true_ = fastReplaceFree Ο' true_
case false_
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v false_ β Ο v = Ο' v
β’ fastReplaceFree Ο false_ = fastReplaceFree Ο' false_
case not_
aβ : Formula
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v aβ.not_ β Ο v = Ο' v
β’ fastReplaceFree Ο aβ.not_ = fastReplaceFree Ο' aβ.not_
case imp_
aβΒΉ aβ : Formula
a_ihβΒΉ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβΒΉ β Ο v = Ο' v) β fastReplaceFree Ο aβΒΉ = fastReplaceFree Ο' aβΒΉ
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (aβΒΉ.imp_ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (aβΒΉ.imp_ aβ) = fastReplaceFree Ο' (aβΒΉ.imp_ aβ)
case and_
aβΒΉ aβ : Formula
a_ihβΒΉ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβΒΉ β Ο v = Ο' v) β fastReplaceFree Ο aβΒΉ = fastReplaceFree Ο' aβΒΉ
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (aβΒΉ.and_ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (aβΒΉ.and_ aβ) = fastReplaceFree Ο' (aβΒΉ.and_ aβ)
case or_
aβΒΉ aβ : Formula
a_ihβΒΉ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβΒΉ β Ο v = Ο' v) β fastReplaceFree Ο aβΒΉ = fastReplaceFree Ο' aβΒΉ
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (aβΒΉ.or_ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (aβΒΉ.or_ aβ) = fastReplaceFree Ο' (aβΒΉ.or_ aβ)
case iff_
aβΒΉ aβ : Formula
a_ihβΒΉ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβΒΉ β Ο v = Ο' v) β fastReplaceFree Ο aβΒΉ = fastReplaceFree Ο' aβΒΉ
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (aβΒΉ.iff_ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (aβΒΉ.iff_ aβ) = fastReplaceFree Ο' (aβΒΉ.iff_ aβ)
case forall_
aβΒΉ : VarName
aβ : Formula
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (forall_ aβΒΉ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (forall_ aβΒΉ aβ) = fastReplaceFree Ο' (forall_ aβΒΉ aβ)
case exists_
aβΒΉ : VarName
aβ : Formula
a_ihβ :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v aβ β Ο v = Ο' v) β fastReplaceFree Ο aβ = fastReplaceFree Ο' aβ
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (exists_ aβΒΉ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (exists_ aβΒΉ aβ) = fastReplaceFree Ο' (exists_ aβΒΉ aβ)
case def_
aβΒΉ : DefName
aβ : List VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (def_ aβΒΉ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (def_ aβΒΉ aβ) = fastReplaceFree Ο' (def_ aβΒΉ aβ)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
case pred_const_ X xs | pred_var_ X xs | def_ X xs =>
congr! 1
simp only [List.map_eq_map_iff]
exact h1
|
X : DefName
xs : List VarName
Ο Ο' : VarName β VarName
h1 : β v β xs, Ο v = Ο' v
β’ def_ X (List.map Ο xs) = def_ X (List.map Ο' xs)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : DefName
xs : List VarName
Ο Ο' : VarName β VarName
h1 : β v β xs, Ο v = Ο' v
β’ def_ X (List.map Ο xs) = def_ X (List.map Ο' xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
case not_ phi phi_ih =>
congr! 1
exact phi_ih Ο Ο' h1
|
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v phi β Ο v = Ο' v
β’ (fastReplaceFree Ο phi).not_ = (fastReplaceFree Ο' phi).not_
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v phi β Ο v = Ο' v
β’ (fastReplaceFree Ο phi).not_ = (fastReplaceFree Ο' phi).not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
case forall_ x phi phi_ih | exists_ x phi phi_ih =>
congr! 1
apply phi_ih
intro v a1
simp only [Function.updateITE]
split_ifs
case _ c1 =>
rfl
case _ c1 =>
apply h1
tauto
|
x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v = Ο' v
β’ exists_ x (fastReplaceFree (Function.updateITE Ο x x) phi) =
exists_ x (fastReplaceFree (Function.updateITE Ο' x x) phi)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
x : VarName
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), Β¬v = x β§ isFreeIn v phi β Ο v = Ο' v
β’ exists_ x (fastReplaceFree (Function.updateITE Ο x x) phi) =
exists_ x (fastReplaceFree (Function.updateITE Ο' x x) phi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
simp only [isFreeIn] at h1
|
case def_
aβΒΉ : DefName
aβ : List VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (def_ aβΒΉ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (def_ aβΒΉ aβ) = fastReplaceFree Ο' (def_ aβΒΉ aβ)
|
case def_
aβΒΉ : DefName
aβ : List VarName
Ο Ο' : VarName β VarName
h1 : β v β aβ, Ο v = Ο' v
β’ fastReplaceFree Ο (def_ aβΒΉ aβ) = fastReplaceFree Ο' (def_ aβΒΉ aβ)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
aβΒΉ : DefName
aβ : List VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v (def_ aβΒΉ aβ) β Ο v = Ο' v
β’ fastReplaceFree Ο (def_ aβΒΉ aβ) = fastReplaceFree Ο' (def_ aβΒΉ aβ)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
simp only [fastReplaceFree]
|
case def_
aβΒΉ : DefName
aβ : List VarName
Ο Ο' : VarName β VarName
h1 : β v β aβ, Ο v = Ο' v
β’ fastReplaceFree Ο (def_ aβΒΉ aβ) = fastReplaceFree Ο' (def_ aβΒΉ aβ)
|
case def_
aβΒΉ : DefName
aβ : List VarName
Ο Ο' : VarName β VarName
h1 : β v β aβ, Ο v = Ο' v
β’ def_ aβΒΉ (List.map Ο aβ) = def_ aβΒΉ (List.map Ο' aβ)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
aβΒΉ : DefName
aβ : List VarName
Ο Ο' : VarName β VarName
h1 : β v β aβ, Ο v = Ο' v
β’ fastReplaceFree Ο (def_ aβΒΉ aβ) = fastReplaceFree Ο' (def_ aβΒΉ aβ)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
congr! 1
|
X : DefName
xs : List VarName
Ο Ο' : VarName β VarName
h1 : β v β xs, Ο v = Ο' v
β’ def_ X (List.map Ο xs) = def_ X (List.map Ο' xs)
|
case h.e'_2
X : DefName
xs : List VarName
Ο Ο' : VarName β VarName
h1 : β v β xs, Ο v = Ο' v
β’ List.map Ο xs = List.map Ο' xs
|
Please generate a tactic in lean4 to solve the state.
STATE:
X : DefName
xs : List VarName
Ο Ο' : VarName β VarName
h1 : β v β xs, Ο v = Ο' v
β’ def_ X (List.map Ο xs) = def_ X (List.map Ο' xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
simp only [List.map_eq_map_iff]
|
case h.e'_2
X : DefName
xs : List VarName
Ο Ο' : VarName β VarName
h1 : β v β xs, Ο v = Ο' v
β’ List.map Ο xs = List.map Ο' xs
|
case h.e'_2
X : DefName
xs : List VarName
Ο Ο' : VarName β VarName
h1 : β v β xs, Ο v = Ο' v
β’ β x β xs, Ο x = Ο' x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
X : DefName
xs : List VarName
Ο Ο' : VarName β VarName
h1 : β v β xs, Ο v = Ο' v
β’ List.map Ο xs = List.map Ο' xs
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
exact h1
|
case h.e'_2
X : DefName
xs : List VarName
Ο Ο' : VarName β VarName
h1 : β v β xs, Ο v = Ο' v
β’ β x β xs, Ο x = Ο' x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
X : DefName
xs : List VarName
Ο Ο' : VarName β VarName
h1 : β v β xs, Ο v = Ο' v
β’ β x β xs, Ο x = Ο' x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
congr! 1
|
x y : VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), v = x β¨ v = y β Ο v = Ο' v
β’ eq_ (Ο x) (Ο y) = eq_ (Ο' x) (Ο' y)
|
case h.e'_1
x y : VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), v = x β¨ v = y β Ο v = Ο' v
β’ Ο x = Ο' x
case h.e'_2
x y : VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), v = x β¨ v = y β Ο v = Ο' v
β’ Ο y = Ο' y
|
Please generate a tactic in lean4 to solve the state.
STATE:
x y : VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), v = x β¨ v = y β Ο v = Ο' v
β’ eq_ (Ο x) (Ο y) = eq_ (Ο' x) (Ο' y)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
apply h1
|
case h.e'_1
x y : VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), v = x β¨ v = y β Ο v = Ο' v
β’ Ο x = Ο' x
|
case h.e'_1.a
x y : VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), v = x β¨ v = y β Ο v = Ο' v
β’ x = x β¨ x = y
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1
x y : VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), v = x β¨ v = y β Ο v = Ο' v
β’ Ο x = Ο' x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
left
|
case h.e'_1.a
x y : VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), v = x β¨ v = y β Ο v = Ο' v
β’ x = x β¨ x = y
|
case h.e'_1.a.h
x y : VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), v = x β¨ v = y β Ο v = Ο' v
β’ x = x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1.a
x y : VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), v = x β¨ v = y β Ο v = Ο' v
β’ x = x β¨ x = y
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
rfl
|
case h.e'_1.a.h
x y : VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), v = x β¨ v = y β Ο v = Ο' v
β’ x = x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1.a.h
x y : VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), v = x β¨ v = y β Ο v = Ο' v
β’ x = x
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
apply h1
|
case h.e'_2
x y : VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), v = x β¨ v = y β Ο v = Ο' v
β’ Ο y = Ο' y
|
case h.e'_2.a
x y : VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), v = x β¨ v = y β Ο v = Ο' v
β’ y = x β¨ y = y
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
x y : VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), v = x β¨ v = y β Ο v = Ο' v
β’ Ο y = Ο' y
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
right
|
case h.e'_2.a
x y : VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), v = x β¨ v = y β Ο v = Ο' v
β’ y = x β¨ y = y
|
case h.e'_2.a.h
x y : VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), v = x β¨ v = y β Ο v = Ο' v
β’ y = y
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.a
x y : VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), v = x β¨ v = y β Ο v = Ο' v
β’ y = x β¨ y = y
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
rfl
|
case h.e'_2.a.h
x y : VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), v = x β¨ v = y β Ο v = Ο' v
β’ y = y
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.a.h
x y : VarName
Ο Ο' : VarName β VarName
h1 : β (v : VarName), v = x β¨ v = y β Ο v = Ο' v
β’ y = y
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
congr! 1
|
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v phi β Ο v = Ο' v
β’ (fastReplaceFree Ο phi).not_ = (fastReplaceFree Ο' phi).not_
|
case h.e'_1
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v phi β Ο v = Ο' v
β’ fastReplaceFree Ο phi = fastReplaceFree Ο' phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v phi β Ο v = Ο' v
β’ (fastReplaceFree Ο phi).not_ = (fastReplaceFree Ο' phi).not_
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
exact phi_ih Ο Ο' h1
|
case h.e'_1
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v phi β Ο v = Ο' v
β’ fastReplaceFree Ο phi = fastReplaceFree Ο' phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1
phi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v phi β Ο v = Ο' v
β’ fastReplaceFree Ο phi = fastReplaceFree Ο' phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
congr! 1
|
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v = Ο' v) β fastReplaceFree Ο psi = fastReplaceFree Ο' psi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v phi β¨ isFreeIn v psi β Ο v = Ο' v
β’ (fastReplaceFree Ο phi).iff_ (fastReplaceFree Ο psi) = (fastReplaceFree Ο' phi).iff_ (fastReplaceFree Ο' psi)
|
case h.e'_1
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v = Ο' v) β fastReplaceFree Ο psi = fastReplaceFree Ο' psi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v phi β¨ isFreeIn v psi β Ο v = Ο' v
β’ fastReplaceFree Ο phi = fastReplaceFree Ο' phi
case h.e'_2
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v = Ο' v) β fastReplaceFree Ο psi = fastReplaceFree Ο' psi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v phi β¨ isFreeIn v psi β Ο v = Ο' v
β’ fastReplaceFree Ο psi = fastReplaceFree Ο' psi
|
Please generate a tactic in lean4 to solve the state.
STATE:
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v = Ο' v) β fastReplaceFree Ο psi = fastReplaceFree Ο' psi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v phi β¨ isFreeIn v psi β Ο v = Ο' v
β’ (fastReplaceFree Ο phi).iff_ (fastReplaceFree Ο psi) = (fastReplaceFree Ο' phi).iff_ (fastReplaceFree Ο' psi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
apply phi_ih
|
case h.e'_1
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v = Ο' v) β fastReplaceFree Ο psi = fastReplaceFree Ο' psi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v phi β¨ isFreeIn v psi β Ο v = Ο' v
β’ fastReplaceFree Ο phi = fastReplaceFree Ο' phi
|
case h.e'_1.h1
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v = Ο' v) β fastReplaceFree Ο psi = fastReplaceFree Ο' psi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v phi β¨ isFreeIn v psi β Ο v = Ο' v
β’ β (v : VarName), isFreeIn v phi β Ο v = Ο' v
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v = Ο' v) β fastReplaceFree Ο psi = fastReplaceFree Ο' psi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v phi β¨ isFreeIn v psi β Ο v = Ο' v
β’ fastReplaceFree Ο phi = fastReplaceFree Ο' phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean
|
FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free
|
[171, 1]
|
[223, 12]
|
intro v a1
|
case h.e'_1.h1
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v = Ο' v) β fastReplaceFree Ο psi = fastReplaceFree Ο' psi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v phi β¨ isFreeIn v psi β Ο v = Ο' v
β’ β (v : VarName), isFreeIn v phi β Ο v = Ο' v
|
case h.e'_1.h1
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v = Ο' v) β fastReplaceFree Ο psi = fastReplaceFree Ο' psi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v phi β¨ isFreeIn v psi β Ο v = Ο' v
v : VarName
a1 : isFreeIn v phi
β’ Ο v = Ο' v
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1.h1
phi psi : Formula
phi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v phi β Ο v = Ο' v) β fastReplaceFree Ο phi = fastReplaceFree Ο' phi
psi_ih :
β (Ο Ο' : VarName β VarName),
(β (v : VarName), isFreeIn v psi β Ο v = Ο' v) β fastReplaceFree Ο psi = fastReplaceFree Ο' psi
Ο Ο' : VarName β VarName
h1 : β (v : VarName), isFreeIn v phi β¨ isFreeIn v psi β Ο v = Ο' v
β’ β (v : VarName), isFreeIn v phi β Ο v = Ο' v
TACTIC:
|
Subsets and Splits
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