url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_0
q_0 : β
xs : List (β β Ο_0 β Ο_1)
x : EpsilonArrow Ο_0
a2_left : x β M_0.epsilon_arrow_list
a2_right_left : x.start_state = p_0
a2_right_right : List.map (Sum.inr β Sum.inl) x.stop_state_list = xs
β’ Sum.inl q_0 β List.map (Sum.inr β Sum.inl) x.stop_state_list | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | cases right | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_0
q_0 : β
xs : List (β β Ο_0 β Ο_1)
right : β a β M_0.accepting_state_list, a = p_0 β§ List.map (Sum.inr β Sum.inr) M_1.starting_state_list = xs
β’ Sum.inl q_0 β xs | case intro
Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_0
q_0 : β
xs : List (β β Ο_0 β Ο_1)
wβ : Ο_0
hβ : wβ β M_0.accepting_state_list β§ wβ = p_0 β§ List.map (Sum.inr β Sum.inr) M_1.starting_state_list = xs
β’ Sum.inl q_0 β xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | case _ x a2 =>
cases a2
case _ a2_left a2_right =>
cases a2_right
case _ a2_right_left a2_right_right =>
simp only [β a2_right_right]
simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_0
q_0 : β
xs : List (β β Ο_0 β Ο_1)
x : Ο_0
a2 : x β M_0.accepting_state_list β§ x = p_0 β§ List.map (Sum.inr β Sum.inr) M_1.starting_state_list = xs
β’ Sum.inl q_0 β xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | cases a2 | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_0
q_0 : β
xs : List (β β Ο_0 β Ο_1)
x : Ο_0
a2 : x β M_0.accepting_state_list β§ x = p_0 β§ List.map (Sum.inr β Sum.inr) M_1.starting_state_list = xs
β’ Sum.inl q_0 β xs | case intro
Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_0
q_0 : β
xs : List (β β Ο_0 β Ο_1)
x : Ο_0
leftβ : x β M_0.accepting_state_list
rightβ : x = p_0 β§ List.map (Sum.inr β Sum.inr) M_1.starting_state_list = xs
β’ Sum.inl q_0 β xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | case _ a2_left a2_right =>
cases a2_right
case _ a2_right_left a2_right_right =>
simp only [β a2_right_right]
simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_0
q_0 : β
xs : List (β β Ο_0 β Ο_1)
x : Ο_0
a2_left : x β M_0.accepting_state_list
a2_right : x = p_0 β§ List.map (Sum.inr β Sum.inr) M_1.starting_state_list = xs
β’ Sum.inl q_0 β xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | cases a2_right | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_0
q_0 : β
xs : List (β β Ο_0 β Ο_1)
x : Ο_0
a2_left : x β M_0.accepting_state_list
a2_right : x = p_0 β§ List.map (Sum.inr β Sum.inr) M_1.starting_state_list = xs
β’ Sum.inl q_0 β xs | case intro
Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_0
q_0 : β
xs : List (β β Ο_0 β Ο_1)
x : Ο_0
a2_left : x β M_0.accepting_state_list
leftβ : x = p_0
rightβ : List.map (Sum.inr β Sum.inr) M_1.starting_state_list = xs
β’ Sum.inl q_0 β xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | case _ a2_right_left a2_right_right =>
simp only [β a2_right_right]
simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_0
q_0 : β
xs : List (β β Ο_0 β Ο_1)
x : Ο_0
a2_left : x β M_0.accepting_state_list
a2_right_left : x = p_0
a2_right_right : List.map (Sum.inr β Sum.inr) M_1.starting_state_list = xs
β’ Sum.inl q_0 β xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | simp only [β a2_right_right] | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_0
q_0 : β
xs : List (β β Ο_0 β Ο_1)
x : Ο_0
a2_left : x β M_0.accepting_state_list
a2_right_left : x = p_0
a2_right_right : List.map (Sum.inr β Sum.inr) M_1.starting_state_list = xs
β’ Sum.inl q_0 β xs | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_0
q_0 : β
xs : List (β β Ο_0 β Ο_1)
x : Ο_0
a2_left : x β M_0.accepting_state_list
a2_right_left : x = p_0
a2_right_right : List.map (Sum.inr β Sum.inr) M_1.starting_state_list = xs
β’ Sum.inl q_0 β List.map (Sum.inr β Sum.inr) M_1.starting_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_0
q_0 : β
xs : List (β β Ο_0 β Ο_1)
x : Ο_0
a2_left : x β M_0.accepting_state_list
a2_right_left : x = p_0
a2_right_right : List.map (Sum.inr β Sum.inr) M_1.starting_state_list = xs
β’ Sum.inl q_0 β List.map (Sum.inr β Sum.inr) M_1.starting_state_list | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | cases q_0 | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_0
q_0 : Ο_0 β Ο_1
β’ (β stop_state_list,
{ start_state := Sum.inr (Sum.inl p_0), stop_state_list := stop_state_list } β
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inl arrow.start_state,
stop_state_list := List.map Sum.inl arrow.stop_state_list })
M_0.epsilon_arrow_list) ++
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
M_1.epsilon_arrow_list) ++
List.map
(fun accepting_state =>
{ start_state := accepting_state,
stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) })
(List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) β§
Sum.inr q_0 β stop_state_list) =
match (Sum.inr (Sum.inl p_0), Sum.inr q_0) with
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_0.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_1.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' β M_0.accepting_state_list β§ q' β M_1.starting_state_list
| x => False | case inl
Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 valβ : Ο_0
β’ (β stop_state_list,
{ start_state := Sum.inr (Sum.inl p_0), stop_state_list := stop_state_list } β
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inl arrow.start_state,
stop_state_list := List.map Sum.inl arrow.stop_state_list })
M_0.epsilon_arrow_list) ++
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
M_1.epsilon_arrow_list) ++
List.map
(fun accepting_state =>
{ start_state := accepting_state,
stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) })
(List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) β§
Sum.inr (Sum.inl valβ) β stop_state_list) =
match (Sum.inr (Sum.inl p_0), Sum.inr (Sum.inl valβ)) with
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_0.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_1.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' β M_0.accepting_state_list β§ q' β M_1.starting_state_list
| x => False
case inr
Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_0
valβ : Ο_1
β’ (β stop_state_list,
{ start_state := Sum.inr (Sum.inl p_0), stop_state_list := stop_state_list } β
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inl arrow.start_state,
stop_state_list := List.map Sum.inl arrow.stop_state_list })
M_0.epsilon_arrow_list) ++
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
M_1.epsilon_arrow_list) ++
List.map
(fun accepting_state =>
{ start_state := accepting_state,
stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) })
(List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) β§
Sum.inr (Sum.inr valβ) β stop_state_list) =
match (Sum.inr (Sum.inl p_0), Sum.inr (Sum.inr valβ)) with
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_0.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_1.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' β M_0.accepting_state_list β§ q' β M_1.starting_state_list
| x => False |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | case _ q_0 =>
simp
sorry | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 q_0 : Ο_0
β’ (β stop_state_list,
{ start_state := Sum.inr (Sum.inl p_0), stop_state_list := stop_state_list } β
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inl arrow.start_state,
stop_state_list := List.map Sum.inl arrow.stop_state_list })
M_0.epsilon_arrow_list) ++
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
M_1.epsilon_arrow_list) ++
List.map
(fun accepting_state =>
{ start_state := accepting_state,
stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) })
(List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) β§
Sum.inr (Sum.inl q_0) β stop_state_list) =
match (Sum.inr (Sum.inl p_0), Sum.inr (Sum.inl q_0)) with
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_0.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_1.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' β M_0.accepting_state_list β§ q' β M_1.starting_state_list
| x => False | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | case _ q_0 =>
simp
sorry | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_0
q_0 : Ο_1
β’ (β stop_state_list,
{ start_state := Sum.inr (Sum.inl p_0), stop_state_list := stop_state_list } β
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inl arrow.start_state,
stop_state_list := List.map Sum.inl arrow.stop_state_list })
M_0.epsilon_arrow_list) ++
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
M_1.epsilon_arrow_list) ++
List.map
(fun accepting_state =>
{ start_state := accepting_state,
stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) })
(List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) β§
Sum.inr (Sum.inr q_0) β stop_state_list) =
match (Sum.inr (Sum.inl p_0), Sum.inr (Sum.inr q_0)) with
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_0.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_1.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' β M_0.accepting_state_list β§ q' β M_1.starting_state_list
| x => False | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 q_0 : Ο_0
β’ (β stop_state_list,
{ start_state := Sum.inr (Sum.inl p_0), stop_state_list := stop_state_list } β
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inl arrow.start_state,
stop_state_list := List.map Sum.inl arrow.stop_state_list })
M_0.epsilon_arrow_list) ++
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
M_1.epsilon_arrow_list) ++
List.map
(fun accepting_state =>
{ start_state := accepting_state,
stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) })
(List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) β§
Sum.inr (Sum.inl q_0) β stop_state_list) =
match (Sum.inr (Sum.inl p_0), Sum.inr (Sum.inl q_0)) with
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_0.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_1.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' β M_0.accepting_state_list β§ q' β M_1.starting_state_list
| x => False | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 q_0 : Ο_0
β’ (β stop_state_list,
((β a β M_0.epsilon_arrow_list,
a.start_state = p_0 β§ List.map (Sum.inr β Sum.inl) a.stop_state_list = stop_state_list) β¨
β a β M_0.accepting_state_list,
a = p_0 β§ List.map (Sum.inr β Sum.inr) M_1.starting_state_list = stop_state_list) β§
Sum.inr (Sum.inl q_0) β stop_state_list) β
β stop_state_list,
{ start_state := p_0, stop_state_list := stop_state_list } β M_0.epsilon_arrow_list β§ q_0 β stop_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | sorry | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 q_0 : Ο_0
β’ (β stop_state_list,
((β a β M_0.epsilon_arrow_list,
a.start_state = p_0 β§ List.map (Sum.inr β Sum.inl) a.stop_state_list = stop_state_list) β¨
β a β M_0.accepting_state_list,
a = p_0 β§ List.map (Sum.inr β Sum.inr) M_1.starting_state_list = stop_state_list) β§
Sum.inr (Sum.inl q_0) β stop_state_list) β
β stop_state_list,
{ start_state := p_0, stop_state_list := stop_state_list } β M_0.epsilon_arrow_list β§ q_0 β stop_state_list | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_0
q_0 : Ο_1
β’ (β stop_state_list,
{ start_state := Sum.inr (Sum.inl p_0), stop_state_list := stop_state_list } β
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inl arrow.start_state,
stop_state_list := List.map Sum.inl arrow.stop_state_list })
M_0.epsilon_arrow_list) ++
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
M_1.epsilon_arrow_list) ++
List.map
(fun accepting_state =>
{ start_state := accepting_state,
stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) })
(List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) β§
Sum.inr (Sum.inr q_0) β stop_state_list) =
match (Sum.inr (Sum.inl p_0), Sum.inr (Sum.inr q_0)) with
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_0.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_1.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' β M_0.accepting_state_list β§ q' β M_1.starting_state_list
| x => False | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_0
q_0 : Ο_1
β’ (β stop_state_list,
((β a β M_0.epsilon_arrow_list,
a.start_state = p_0 β§ List.map (Sum.inr β Sum.inl) a.stop_state_list = stop_state_list) β¨
β a β M_0.accepting_state_list,
a = p_0 β§ List.map (Sum.inr β Sum.inr) M_1.starting_state_list = stop_state_list) β§
Sum.inr (Sum.inr q_0) β stop_state_list) β
p_0 β M_0.accepting_state_list β§ q_0 β M_1.starting_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | sorry | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_0
q_0 : Ο_1
β’ (β stop_state_list,
((β a β M_0.epsilon_arrow_list,
a.start_state = p_0 β§ List.map (Sum.inr β Sum.inl) a.stop_state_list = stop_state_list) β¨
β a β M_0.accepting_state_list,
a = p_0 β§ List.map (Sum.inr β Sum.inr) M_1.starting_state_list = stop_state_list) β§
Sum.inr (Sum.inr q_0) β stop_state_list) β
p_0 β M_0.accepting_state_list β§ q_0 β M_1.starting_state_list | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | cases q | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
q : β β Ο_0 β Ο_1
p_0 : Ο_1
β’ (β stop_state_list,
{ start_state := Sum.inr (Sum.inr p_0), stop_state_list := stop_state_list } β
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inl arrow.start_state,
stop_state_list := List.map Sum.inl arrow.stop_state_list })
M_0.epsilon_arrow_list) ++
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
M_1.epsilon_arrow_list) ++
List.map
(fun accepting_state =>
{ start_state := accepting_state,
stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) })
(List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) β§
q β stop_state_list) =
match (Sum.inr (Sum.inr p_0), q) with
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_0.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_1.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' β M_0.accepting_state_list β§ q' β M_1.starting_state_list
| x => False | case inl
Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_1
valβ : β
β’ (β stop_state_list,
{ start_state := Sum.inr (Sum.inr p_0), stop_state_list := stop_state_list } β
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inl arrow.start_state,
stop_state_list := List.map Sum.inl arrow.stop_state_list })
M_0.epsilon_arrow_list) ++
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
M_1.epsilon_arrow_list) ++
List.map
(fun accepting_state =>
{ start_state := accepting_state,
stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) })
(List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) β§
Sum.inl valβ β stop_state_list) =
match (Sum.inr (Sum.inr p_0), Sum.inl valβ) with
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_0.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_1.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' β M_0.accepting_state_list β§ q' β M_1.starting_state_list
| x => False
case inr
Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_1
valβ : Ο_0 β Ο_1
β’ (β stop_state_list,
{ start_state := Sum.inr (Sum.inr p_0), stop_state_list := stop_state_list } β
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inl arrow.start_state,
stop_state_list := List.map Sum.inl arrow.stop_state_list })
M_0.epsilon_arrow_list) ++
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
M_1.epsilon_arrow_list) ++
List.map
(fun accepting_state =>
{ start_state := accepting_state,
stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) })
(List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) β§
Sum.inr valβ β stop_state_list) =
match (Sum.inr (Sum.inr p_0), Sum.inr valβ) with
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_0.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_1.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' β M_0.accepting_state_list β§ q' β M_1.starting_state_list
| x => False |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | case _ q_0 =>
simp
intro xs x a1 a2 a3
simp only [β a3]
simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_1
q_0 : β
β’ (β stop_state_list,
{ start_state := Sum.inr (Sum.inr p_0), stop_state_list := stop_state_list } β
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inl arrow.start_state,
stop_state_list := List.map Sum.inl arrow.stop_state_list })
M_0.epsilon_arrow_list) ++
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
M_1.epsilon_arrow_list) ++
List.map
(fun accepting_state =>
{ start_state := accepting_state,
stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) })
(List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) β§
Sum.inl q_0 β stop_state_list) =
match (Sum.inr (Sum.inr p_0), Sum.inl q_0) with
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_0.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_1.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' β M_0.accepting_state_list β§ q' β M_1.starting_state_list
| x => False | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | case _ q_0 =>
cases q_0
case _ q_0 =>
simp
intro xs x a1 a2 a3
simp only [β a3]
simp
case _ q_0 =>
simp
sorry | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_1
q_0 : Ο_0 β Ο_1
β’ (β stop_state_list,
{ start_state := Sum.inr (Sum.inr p_0), stop_state_list := stop_state_list } β
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inl arrow.start_state,
stop_state_list := List.map Sum.inl arrow.stop_state_list })
M_0.epsilon_arrow_list) ++
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
M_1.epsilon_arrow_list) ++
List.map
(fun accepting_state =>
{ start_state := accepting_state,
stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) })
(List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) β§
Sum.inr q_0 β stop_state_list) =
match (Sum.inr (Sum.inr p_0), Sum.inr q_0) with
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_0.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_1.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' β M_0.accepting_state_list β§ q' β M_1.starting_state_list
| x => False | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_1
q_0 : β
β’ (β stop_state_list,
{ start_state := Sum.inr (Sum.inr p_0), stop_state_list := stop_state_list } β
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inl arrow.start_state,
stop_state_list := List.map Sum.inl arrow.stop_state_list })
M_0.epsilon_arrow_list) ++
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
M_1.epsilon_arrow_list) ++
List.map
(fun accepting_state =>
{ start_state := accepting_state,
stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) })
(List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) β§
Sum.inl q_0 β stop_state_list) =
match (Sum.inr (Sum.inr p_0), Sum.inl q_0) with
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_0.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_1.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' β M_0.accepting_state_list β§ q' β M_1.starting_state_list
| x => False | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_1
q_0 : β
β’ β (x : List (β β Ο_0 β Ο_1)),
β x_1 β M_1.epsilon_arrow_list,
x_1.start_state = p_0 β List.map (Sum.inr β Sum.inr) x_1.stop_state_list = x β Sum.inl q_0 β x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | intro xs x a1 a2 a3 | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_1
q_0 : β
β’ β (x : List (β β Ο_0 β Ο_1)),
β x_1 β M_1.epsilon_arrow_list,
x_1.start_state = p_0 β List.map (Sum.inr β Sum.inr) x_1.stop_state_list = x β Sum.inl q_0 β x | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_1
q_0 : β
xs : List (β β Ο_0 β Ο_1)
x : EpsilonArrow Ο_1
a1 : x β M_1.epsilon_arrow_list
a2 : x.start_state = p_0
a3 : List.map (Sum.inr β Sum.inr) x.stop_state_list = xs
β’ Sum.inl q_0 β xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | simp only [β a3] | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_1
q_0 : β
xs : List (β β Ο_0 β Ο_1)
x : EpsilonArrow Ο_1
a1 : x β M_1.epsilon_arrow_list
a2 : x.start_state = p_0
a3 : List.map (Sum.inr β Sum.inr) x.stop_state_list = xs
β’ Sum.inl q_0 β xs | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_1
q_0 : β
xs : List (β β Ο_0 β Ο_1)
x : EpsilonArrow Ο_1
a1 : x β M_1.epsilon_arrow_list
a2 : x.start_state = p_0
a3 : List.map (Sum.inr β Sum.inr) x.stop_state_list = xs
β’ Sum.inl q_0 β List.map (Sum.inr β Sum.inr) x.stop_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_1
q_0 : β
xs : List (β β Ο_0 β Ο_1)
x : EpsilonArrow Ο_1
a1 : x β M_1.epsilon_arrow_list
a2 : x.start_state = p_0
a3 : List.map (Sum.inr β Sum.inr) x.stop_state_list = xs
β’ Sum.inl q_0 β List.map (Sum.inr β Sum.inr) x.stop_state_list | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | cases q_0 | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_1
q_0 : Ο_0 β Ο_1
β’ (β stop_state_list,
{ start_state := Sum.inr (Sum.inr p_0), stop_state_list := stop_state_list } β
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inl arrow.start_state,
stop_state_list := List.map Sum.inl arrow.stop_state_list })
M_0.epsilon_arrow_list) ++
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
M_1.epsilon_arrow_list) ++
List.map
(fun accepting_state =>
{ start_state := accepting_state,
stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) })
(List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) β§
Sum.inr q_0 β stop_state_list) =
match (Sum.inr (Sum.inr p_0), Sum.inr q_0) with
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_0.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_1.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' β M_0.accepting_state_list β§ q' β M_1.starting_state_list
| x => False | case inl
Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_1
valβ : Ο_0
β’ (β stop_state_list,
{ start_state := Sum.inr (Sum.inr p_0), stop_state_list := stop_state_list } β
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inl arrow.start_state,
stop_state_list := List.map Sum.inl arrow.stop_state_list })
M_0.epsilon_arrow_list) ++
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
M_1.epsilon_arrow_list) ++
List.map
(fun accepting_state =>
{ start_state := accepting_state,
stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) })
(List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) β§
Sum.inr (Sum.inl valβ) β stop_state_list) =
match (Sum.inr (Sum.inr p_0), Sum.inr (Sum.inl valβ)) with
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_0.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_1.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' β M_0.accepting_state_list β§ q' β M_1.starting_state_list
| x => False
case inr
Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 valβ : Ο_1
β’ (β stop_state_list,
{ start_state := Sum.inr (Sum.inr p_0), stop_state_list := stop_state_list } β
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inl arrow.start_state,
stop_state_list := List.map Sum.inl arrow.stop_state_list })
M_0.epsilon_arrow_list) ++
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
M_1.epsilon_arrow_list) ++
List.map
(fun accepting_state =>
{ start_state := accepting_state,
stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) })
(List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) β§
Sum.inr (Sum.inr valβ) β stop_state_list) =
match (Sum.inr (Sum.inr p_0), Sum.inr (Sum.inr valβ)) with
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_0.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_1.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' β M_0.accepting_state_list β§ q' β M_1.starting_state_list
| x => False |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | case _ q_0 =>
simp
intro xs x a1 a2 a3
simp only [β a3]
simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_1
q_0 : Ο_0
β’ (β stop_state_list,
{ start_state := Sum.inr (Sum.inr p_0), stop_state_list := stop_state_list } β
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inl arrow.start_state,
stop_state_list := List.map Sum.inl arrow.stop_state_list })
M_0.epsilon_arrow_list) ++
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
M_1.epsilon_arrow_list) ++
List.map
(fun accepting_state =>
{ start_state := accepting_state,
stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) })
(List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) β§
Sum.inr (Sum.inl q_0) β stop_state_list) =
match (Sum.inr (Sum.inr p_0), Sum.inr (Sum.inl q_0)) with
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_0.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_1.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' β M_0.accepting_state_list β§ q' β M_1.starting_state_list
| x => False | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | case _ q_0 =>
simp
sorry | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 q_0 : Ο_1
β’ (β stop_state_list,
{ start_state := Sum.inr (Sum.inr p_0), stop_state_list := stop_state_list } β
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inl arrow.start_state,
stop_state_list := List.map Sum.inl arrow.stop_state_list })
M_0.epsilon_arrow_list) ++
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
M_1.epsilon_arrow_list) ++
List.map
(fun accepting_state =>
{ start_state := accepting_state,
stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) })
(List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) β§
Sum.inr (Sum.inr q_0) β stop_state_list) =
match (Sum.inr (Sum.inr p_0), Sum.inr (Sum.inr q_0)) with
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_0.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_1.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' β M_0.accepting_state_list β§ q' β M_1.starting_state_list
| x => False | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_1
q_0 : Ο_0
β’ (β stop_state_list,
{ start_state := Sum.inr (Sum.inr p_0), stop_state_list := stop_state_list } β
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inl arrow.start_state,
stop_state_list := List.map Sum.inl arrow.stop_state_list })
M_0.epsilon_arrow_list) ++
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
M_1.epsilon_arrow_list) ++
List.map
(fun accepting_state =>
{ start_state := accepting_state,
stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) })
(List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) β§
Sum.inr (Sum.inl q_0) β stop_state_list) =
match (Sum.inr (Sum.inr p_0), Sum.inr (Sum.inl q_0)) with
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_0.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_1.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' β M_0.accepting_state_list β§ q' β M_1.starting_state_list
| x => False | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_1
q_0 : Ο_0
β’ β (x : List (β β Ο_0 β Ο_1)),
β x_1 β M_1.epsilon_arrow_list,
x_1.start_state = p_0 β List.map (Sum.inr β Sum.inr) x_1.stop_state_list = x β Sum.inr (Sum.inl q_0) β x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | intro xs x a1 a2 a3 | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_1
q_0 : Ο_0
β’ β (x : List (β β Ο_0 β Ο_1)),
β x_1 β M_1.epsilon_arrow_list,
x_1.start_state = p_0 β List.map (Sum.inr β Sum.inr) x_1.stop_state_list = x β Sum.inr (Sum.inl q_0) β x | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_1
q_0 : Ο_0
xs : List (β β Ο_0 β Ο_1)
x : EpsilonArrow Ο_1
a1 : x β M_1.epsilon_arrow_list
a2 : x.start_state = p_0
a3 : List.map (Sum.inr β Sum.inr) x.stop_state_list = xs
β’ Sum.inr (Sum.inl q_0) β xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | simp only [β a3] | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_1
q_0 : Ο_0
xs : List (β β Ο_0 β Ο_1)
x : EpsilonArrow Ο_1
a1 : x β M_1.epsilon_arrow_list
a2 : x.start_state = p_0
a3 : List.map (Sum.inr β Sum.inr) x.stop_state_list = xs
β’ Sum.inr (Sum.inl q_0) β xs | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_1
q_0 : Ο_0
xs : List (β β Ο_0 β Ο_1)
x : EpsilonArrow Ο_1
a1 : x β M_1.epsilon_arrow_list
a2 : x.start_state = p_0
a3 : List.map (Sum.inr β Sum.inr) x.stop_state_list = xs
β’ Sum.inr (Sum.inl q_0) β List.map (Sum.inr β Sum.inr) x.stop_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_1
q_0 : Ο_0
xs : List (β β Ο_0 β Ο_1)
x : EpsilonArrow Ο_1
a1 : x β M_1.epsilon_arrow_list
a2 : x.start_state = p_0
a3 : List.map (Sum.inr β Sum.inr) x.stop_state_list = xs
β’ Sum.inr (Sum.inl q_0) β List.map (Sum.inr β Sum.inr) x.stop_state_list | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 q_0 : Ο_1
β’ (β stop_state_list,
{ start_state := Sum.inr (Sum.inr p_0), stop_state_list := stop_state_list } β
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inl arrow.start_state,
stop_state_list := List.map Sum.inl arrow.stop_state_list })
M_0.epsilon_arrow_list) ++
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
(List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
M_1.epsilon_arrow_list) ++
List.map
(fun accepting_state =>
{ start_state := accepting_state,
stop_state_list := List.map Sum.inr (List.map Sum.inr M_1.starting_state_list) })
(List.map Sum.inr (List.map Sum.inl M_0.accepting_state_list)) β§
Sum.inr (Sum.inr q_0) β stop_state_list) =
match (Sum.inr (Sum.inr p_0), Sum.inr (Sum.inr q_0)) with
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inl q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_0.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inr p'), Sum.inr (Sum.inr q')) =>
β stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } β M_1.epsilon_arrow_list β§ q' β stop_state_list
| (Sum.inr (Sum.inl p'), Sum.inr (Sum.inr q')) => p' β M_0.accepting_state_list β§ q' β M_1.starting_state_list
| x => False | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 q_0 : Ο_1
β’ (β stop_state_list,
(β a β M_1.epsilon_arrow_list,
a.start_state = p_0 β§ List.map (Sum.inr β Sum.inr) a.stop_state_list = stop_state_list) β§
Sum.inr (Sum.inr q_0) β stop_state_list) β
β stop_state_list,
{ start_state := p_0, stop_state_list := stop_state_list } β M_1.epsilon_arrow_list β§ q_0 β stop_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | sorry | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 q_0 : Ο_1
β’ (β stop_state_list,
(β a β M_1.epsilon_arrow_list,
a.start_state = p_0 β§ List.map (Sum.inr β Sum.inr) a.stop_state_list = stop_state_list) β§
Sum.inr (Sum.inr q_0) β stop_state_list) β
β stop_state_list,
{ start_state := p_0, stop_state_list := stop_state_list } β M_1.epsilon_arrow_list β§ q_0 β stop_state_list | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | constructor | case right.right
Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
β’ ((fun state => state β List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) = fun p =>
match p with
| Sum.inr (Sum.inl p') => p' β M_0.starting_state_list
| x => False) β§
(fun state => state β List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) = fun p =>
match p with
| Sum.inr (Sum.inr p') => p' β M_1.accepting_state_list
| x => False | case right.right.left
Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
β’ (fun state => state β List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) = fun p =>
match p with
| Sum.inr (Sum.inl p') => p' β M_0.starting_state_list
| x => False
case right.right.right
Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
β’ (fun state => state β List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) = fun p =>
match p with
| Sum.inr (Sum.inr p') => p' β M_1.accepting_state_list
| x => False |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | funext p | case right.right.left
Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
β’ (fun state => state β List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) = fun p =>
match p with
| Sum.inr (Sum.inl p') => p' β M_0.starting_state_list
| x => False | case right.right.left.h
Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p : β β Ο_0 β Ο_1
β’ (p β List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) =
match p with
| Sum.inr (Sum.inl p') => p' β M_0.starting_state_list
| x => False |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | cases p | case right.right.left.h
Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p : β β Ο_0 β Ο_1
β’ (p β List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) =
match p with
| Sum.inr (Sum.inl p') => p' β M_0.starting_state_list
| x => False | case right.right.left.h.inl
Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
valβ : β
β’ (Sum.inl valβ β List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) =
match Sum.inl valβ with
| Sum.inr (Sum.inl p') => p' β M_0.starting_state_list
| x => False
case right.right.left.h.inr
Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
valβ : Ο_0 β Ο_1
β’ (Sum.inr valβ β List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) =
match Sum.inr valβ with
| Sum.inr (Sum.inl p') => p' β M_0.starting_state_list
| x => False |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | case _ p_0 =>
simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : β
β’ (Sum.inl p_0 β List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) =
match Sum.inl p_0 with
| Sum.inr (Sum.inl p') => p' β M_0.starting_state_list
| x => False | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | case _ p_0 =>
cases p_0
case _ p_0 =>
simp
case _ p_0 =>
simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_0 β Ο_1
β’ (Sum.inr p_0 β List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) =
match Sum.inr p_0 with
| Sum.inr (Sum.inl p') => p' β M_0.starting_state_list
| x => False | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : β
β’ (Sum.inl p_0 β List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) =
match Sum.inl p_0 with
| Sum.inr (Sum.inl p') => p' β M_0.starting_state_list
| x => False | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | cases p_0 | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_0 β Ο_1
β’ (Sum.inr p_0 β List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) =
match Sum.inr p_0 with
| Sum.inr (Sum.inl p') => p' β M_0.starting_state_list
| x => False | case inl
Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
valβ : Ο_0
β’ (Sum.inr (Sum.inl valβ) β List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) =
match Sum.inr (Sum.inl valβ) with
| Sum.inr (Sum.inl p') => p' β M_0.starting_state_list
| x => False
case inr
Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
valβ : Ο_1
β’ (Sum.inr (Sum.inr valβ) β List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) =
match Sum.inr (Sum.inr valβ) with
| Sum.inr (Sum.inl p') => p' β M_0.starting_state_list
| x => False |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | case _ p_0 =>
simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_0
β’ (Sum.inr (Sum.inl p_0) β List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) =
match Sum.inr (Sum.inl p_0) with
| Sum.inr (Sum.inl p') => p' β M_0.starting_state_list
| x => False | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | case _ p_0 =>
simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_1
β’ (Sum.inr (Sum.inr p_0) β List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) =
match Sum.inr (Sum.inr p_0) with
| Sum.inr (Sum.inl p') => p' β M_0.starting_state_list
| x => False | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_0
β’ (Sum.inr (Sum.inl p_0) β List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) =
match Sum.inr (Sum.inl p_0) with
| Sum.inr (Sum.inl p') => p' β M_0.starting_state_list
| x => False | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_1
β’ (Sum.inr (Sum.inr p_0) β List.map Sum.inr (List.map Sum.inl M_0.starting_state_list)) =
match Sum.inr (Sum.inr p_0) with
| Sum.inr (Sum.inl p') => p' β M_0.starting_state_list
| x => False | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | funext p | case right.right.right
Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
β’ (fun state => state β List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) = fun p =>
match p with
| Sum.inr (Sum.inr p') => p' β M_1.accepting_state_list
| x => False | case right.right.right.h
Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p : β β Ο_0 β Ο_1
β’ (p β List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) =
match p with
| Sum.inr (Sum.inr p') => p' β M_1.accepting_state_list
| x => False |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | cases p | case right.right.right.h
Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p : β β Ο_0 β Ο_1
β’ (p β List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) =
match p with
| Sum.inr (Sum.inr p') => p' β M_1.accepting_state_list
| x => False | case right.right.right.h.inl
Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
valβ : β
β’ (Sum.inl valβ β List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) =
match Sum.inl valβ with
| Sum.inr (Sum.inr p') => p' β M_1.accepting_state_list
| x => False
case right.right.right.h.inr
Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
valβ : Ο_0 β Ο_1
β’ (Sum.inr valβ β List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) =
match Sum.inr valβ with
| Sum.inr (Sum.inr p') => p' β M_1.accepting_state_list
| x => False |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | case _ p_0 =>
simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : β
β’ (Sum.inl p_0 β List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) =
match Sum.inl p_0 with
| Sum.inr (Sum.inr p') => p' β M_1.accepting_state_list
| x => False | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | case _ p_0 =>
cases p_0
case _ p_0 =>
simp
case _ p_0 =>
simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_0 β Ο_1
β’ (Sum.inr p_0 β List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) =
match Sum.inr p_0 with
| Sum.inr (Sum.inr p') => p' β M_1.accepting_state_list
| x => False | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : β
β’ (Sum.inl p_0 β List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) =
match Sum.inl p_0 with
| Sum.inr (Sum.inr p') => p' β M_1.accepting_state_list
| x => False | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | cases p_0 | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_0 β Ο_1
β’ (Sum.inr p_0 β List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) =
match Sum.inr p_0 with
| Sum.inr (Sum.inr p') => p' β M_1.accepting_state_list
| x => False | case inl
Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
valβ : Ο_0
β’ (Sum.inr (Sum.inl valβ) β List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) =
match Sum.inr (Sum.inl valβ) with
| Sum.inr (Sum.inr p') => p' β M_1.accepting_state_list
| x => False
case inr
Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
valβ : Ο_1
β’ (Sum.inr (Sum.inr valβ) β List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) =
match Sum.inr (Sum.inr valβ) with
| Sum.inr (Sum.inr p') => p' β M_1.accepting_state_list
| x => False |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | case _ p_0 =>
simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_0
β’ (Sum.inr (Sum.inl p_0) β List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) =
match Sum.inr (Sum.inl p_0) with
| Sum.inr (Sum.inr p') => p' β M_1.accepting_state_list
| x => False | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | case _ p_0 =>
simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_1
β’ (Sum.inr (Sum.inr p_0) β List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) =
match Sum.inr (Sum.inr p_0) with
| Sum.inr (Sum.inr p') => p' β M_1.accepting_state_list
| x => False | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_0
β’ (Sum.inr (Sum.inl p_0) β List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) =
match Sum.inr (Sum.inl p_0) with
| Sum.inr (Sum.inr p') => p' β M_1.accepting_state_list
| x => False | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_concat_EpsilonNFA_toAbstract | [557, 1] | [690, 19] | simp | Ξ± : Type
instβ : DecidableEq Ξ±
Ο_0 Ο_1 : Type
M_0 : EpsilonNFA Ξ± Ο_0
M_1 : EpsilonNFA Ξ± Ο_1
p_0 : Ο_1
β’ (Sum.inr (Sum.inr p_0) β List.map Sum.inr (List.map Sum.inr M_1.accepting_state_list)) =
match Sum.inr (Sum.inr p_0) with
| Sum.inr (Sum.inr p') => p' β M_1.accepting_state_list
| x => False | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Fresh.lean | FOL.NV.finset_var_name_max_len_mem | [23, 1] | [46, 18] | induction xs using Finset.induction_on | x : VarName
xs : Finset VarName
h1 : x β xs
β’ x.length β€ finset_var_name_max_len xs | case empty
x : VarName
h1 : x β β
β’ x.length β€ finset_var_name_max_len β
case insert
x aβΒ² : VarName
sβ : Finset VarName
aβΒΉ : aβΒ² β sβ
aβ : x β sβ β x.length β€ finset_var_name_max_len sβ
h1 : x β insert aβΒ² sβ
β’ x.length β€ finset_var_name_max_len (insert aβΒ² sβ) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Fresh.lean | FOL.NV.finset_var_name_max_len_mem | [23, 1] | [46, 18] | case empty =>
simp at h1 | x : VarName
h1 : x β β
β’ x.length β€ finset_var_name_max_len β
| no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Fresh.lean | FOL.NV.finset_var_name_max_len_mem | [23, 1] | [46, 18] | case insert hd tl a1 ih =>
simp at h1
cases h1
case inl c1 =>
subst c1
simp only [finset_var_name_max_len]
simp
case inr c1 =>
simp only [finset_var_name_max_len] at ih
simp only [finset_var_name_max_len]
simp
right
exact ih c1 | x hd : VarName
tl : Finset VarName
a1 : hd β tl
ih : x β tl β x.length β€ finset_var_name_max_len tl
h1 : x β insert hd tl
β’ x.length β€ finset_var_name_max_len (insert hd tl) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Fresh.lean | FOL.NV.finset_var_name_max_len_mem | [23, 1] | [46, 18] | simp at h1 | x : VarName
h1 : x β β
β’ x.length β€ finset_var_name_max_len β
| no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Fresh.lean | FOL.NV.finset_var_name_max_len_mem | [23, 1] | [46, 18] | simp at h1 | x hd : VarName
tl : Finset VarName
a1 : hd β tl
ih : x β tl β x.length β€ finset_var_name_max_len tl
h1 : x β insert hd tl
β’ x.length β€ finset_var_name_max_len (insert hd tl) | x hd : VarName
tl : Finset VarName
a1 : hd β tl
ih : x β tl β x.length β€ finset_var_name_max_len tl
h1 : x = hd β¨ x β tl
β’ x.length β€ finset_var_name_max_len (insert hd tl) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Fresh.lean | FOL.NV.finset_var_name_max_len_mem | [23, 1] | [46, 18] | cases h1 | x hd : VarName
tl : Finset VarName
a1 : hd β tl
ih : x β tl β x.length β€ finset_var_name_max_len tl
h1 : x = hd β¨ x β tl
β’ x.length β€ finset_var_name_max_len (insert hd tl) | case inl
x hd : VarName
tl : Finset VarName
a1 : hd β tl
ih : x β tl β x.length β€ finset_var_name_max_len tl
hβ : x = hd
β’ x.length β€ finset_var_name_max_len (insert hd tl)
case inr
x hd : VarName
tl : Finset VarName
a1 : hd β tl
ih : x β tl β x.length β€ finset_var_name_max_len tl
hβ : x β tl
β’ x.length β€ finset_var_name_max_len (insert hd tl) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Fresh.lean | FOL.NV.finset_var_name_max_len_mem | [23, 1] | [46, 18] | case inl c1 =>
subst c1
simp only [finset_var_name_max_len]
simp | x hd : VarName
tl : Finset VarName
a1 : hd β tl
ih : x β tl β x.length β€ finset_var_name_max_len tl
c1 : x = hd
β’ x.length β€ finset_var_name_max_len (insert hd tl) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Fresh.lean | FOL.NV.finset_var_name_max_len_mem | [23, 1] | [46, 18] | case inr c1 =>
simp only [finset_var_name_max_len] at ih
simp only [finset_var_name_max_len]
simp
right
exact ih c1 | x hd : VarName
tl : Finset VarName
a1 : hd β tl
ih : x β tl β x.length β€ finset_var_name_max_len tl
c1 : x β tl
β’ x.length β€ finset_var_name_max_len (insert hd tl) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Fresh.lean | FOL.NV.finset_var_name_max_len_mem | [23, 1] | [46, 18] | subst c1 | x hd : VarName
tl : Finset VarName
a1 : hd β tl
ih : x β tl β x.length β€ finset_var_name_max_len tl
c1 : x = hd
β’ x.length β€ finset_var_name_max_len (insert hd tl) | x : VarName
tl : Finset VarName
ih : x β tl β x.length β€ finset_var_name_max_len tl
a1 : x β tl
β’ x.length β€ finset_var_name_max_len (insert x tl) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Fresh.lean | FOL.NV.finset_var_name_max_len_mem | [23, 1] | [46, 18] | simp only [finset_var_name_max_len] | x : VarName
tl : Finset VarName
ih : x β tl β x.length β€ finset_var_name_max_len tl
a1 : x β tl
β’ x.length β€ finset_var_name_max_len (insert x tl) | x : VarName
tl : Finset VarName
ih : x β tl β x.length β€ finset_var_name_max_len tl
a1 : x β tl
β’ x.length β€ Finset.fold (fun m n => max m n) 0 (String.length β VarName.toString) (insert x tl) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Fresh.lean | FOL.NV.finset_var_name_max_len_mem | [23, 1] | [46, 18] | simp | x : VarName
tl : Finset VarName
ih : x β tl β x.length β€ finset_var_name_max_len tl
a1 : x β tl
β’ x.length β€ Finset.fold (fun m n => max m n) 0 (String.length β VarName.toString) (insert x tl) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Fresh.lean | FOL.NV.finset_var_name_max_len_mem | [23, 1] | [46, 18] | simp only [finset_var_name_max_len] at ih | x hd : VarName
tl : Finset VarName
a1 : hd β tl
ih : x β tl β x.length β€ finset_var_name_max_len tl
c1 : x β tl
β’ x.length β€ finset_var_name_max_len (insert hd tl) | x hd : VarName
tl : Finset VarName
a1 : hd β tl
ih : x β tl β x.length β€ Finset.fold (fun m n => max m n) 0 (String.length β VarName.toString) tl
c1 : x β tl
β’ x.length β€ finset_var_name_max_len (insert hd tl) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Fresh.lean | FOL.NV.finset_var_name_max_len_mem | [23, 1] | [46, 18] | simp only [finset_var_name_max_len] | x hd : VarName
tl : Finset VarName
a1 : hd β tl
ih : x β tl β x.length β€ Finset.fold (fun m n => max m n) 0 (String.length β VarName.toString) tl
c1 : x β tl
β’ x.length β€ finset_var_name_max_len (insert hd tl) | x hd : VarName
tl : Finset VarName
a1 : hd β tl
ih : x β tl β x.length β€ Finset.fold (fun m n => max m n) 0 (String.length β VarName.toString) tl
c1 : x β tl
β’ x.length β€ Finset.fold (fun m n => max m n) 0 (String.length β VarName.toString) (insert hd tl) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Fresh.lean | FOL.NV.finset_var_name_max_len_mem | [23, 1] | [46, 18] | simp | x hd : VarName
tl : Finset VarName
a1 : hd β tl
ih : x β tl β x.length β€ Finset.fold (fun m n => max m n) 0 (String.length β VarName.toString) tl
c1 : x β tl
β’ x.length β€ Finset.fold (fun m n => max m n) 0 (String.length β VarName.toString) (insert hd tl) | x hd : VarName
tl : Finset VarName
a1 : hd β tl
ih : x β tl β x.length β€ Finset.fold (fun m n => max m n) 0 (String.length β VarName.toString) tl
c1 : x β tl
β’ x.length β€ hd.length β¨ x.length β€ Finset.fold (fun m n => max m n) 0 (fun x => x.length) tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Fresh.lean | FOL.NV.finset_var_name_max_len_mem | [23, 1] | [46, 18] | right | x hd : VarName
tl : Finset VarName
a1 : hd β tl
ih : x β tl β x.length β€ Finset.fold (fun m n => max m n) 0 (String.length β VarName.toString) tl
c1 : x β tl
β’ x.length β€ hd.length β¨ x.length β€ Finset.fold (fun m n => max m n) 0 (fun x => x.length) tl | case h
x hd : VarName
tl : Finset VarName
a1 : hd β tl
ih : x β tl β x.length β€ Finset.fold (fun m n => max m n) 0 (String.length β VarName.toString) tl
c1 : x β tl
β’ x.length β€ Finset.fold (fun m n => max m n) 0 (fun x => x.length) tl |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Fresh.lean | FOL.NV.finset_var_name_max_len_mem | [23, 1] | [46, 18] | exact ih c1 | case h
x hd : VarName
tl : Finset VarName
a1 : hd β tl
ih : x β tl β x.length β€ Finset.fold (fun m n => max m n) 0 (String.length β VarName.toString) tl
c1 : x β tl
β’ x.length β€ Finset.fold (fun m n => max m n) 0 (fun x => x.length) tl | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Fresh.lean | FOL.NV.fresh_not_mem | [69, 1] | [91, 59] | obtain s1 := finset_var_name_max_len_mem x xs h | x : VarName
c : Char
xs : Finset VarName
h : x β xs
β’ finset_var_name_max_len xs - x.length < finset_var_name_max_len xs + 1 - x.length | x : VarName
c : Char
xs : Finset VarName
h : x β xs
s1 : x.length β€ finset_var_name_max_len xs
β’ finset_var_name_max_len xs - x.length < finset_var_name_max_len xs + 1 - x.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Fresh.lean | FOL.NV.fresh_not_mem | [69, 1] | [91, 59] | simp only [tsub_lt_tsub_iff_right s1] | x : VarName
c : Char
xs : Finset VarName
h : x β xs
s1 : x.length β€ finset_var_name_max_len xs
β’ finset_var_name_max_len xs - x.length < finset_var_name_max_len xs + 1 - x.length | x : VarName
c : Char
xs : Finset VarName
h : x β xs
s1 : x.length β€ finset_var_name_max_len xs
β’ finset_var_name_max_len xs < finset_var_name_max_len xs + 1 |
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
s1 : x.length β€ finset_var_name_max_len xs
β’ finset_var_name_max_len xs < finset_var_name_max_len xs + 1 | no goals |
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
this : finset_var_name_max_len xs - x.length < finset_var_name_max_len xs + 1 - x.length
β’ fresh x c xs 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’ (if h : x β xs then
let_fun this := β―;
fresh { toString := x.toString ++ c.toString } c xs
else x) β
xs |
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
this : finset_var_name_max_len xs - x.length < finset_var_name_max_len xs + 1 - x.length
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
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 |
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 |
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 |
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 |
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 |
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 |
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 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | induction F generalizing binders V | D : Type
I : Interpretation D
V V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
binders : Finset VarName
F : Formula
h1 : admitsAux Ο binders F
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E F β
Holds D I V E (replace c Ο F) | case pred_const_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
aβΒΉ : PredName
aβ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (pred_const_ aβΒΉ aβ)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E (pred_const_ aβΒΉ aβ) β
Holds D I V E (replace c Ο (pred_const_ aβΒΉ aβ))
case pred_var_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
aβΒΉ : PredName
aβ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (pred_var_ aβΒΉ aβ)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E (pred_var_ aβΒΉ aβ) β
Holds D I V E (replace c Ο (pred_var_ aβΒΉ aβ))
case eq_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
aβΒΉ aβ : VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (eq_ aβΒΉ aβ)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E (eq_ aβΒΉ aβ) β
Holds D I V E (replace c Ο (eq_ aβΒΉ aβ))
case true_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders true_
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E true_ β
Holds D I V E (replace c Ο true_)
case false_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders false_
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E false_ β
Holds D I V E (replace c Ο false_)
case not_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
aβ : Formula
a_ihβ :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders aβ β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E aβ β
Holds D I V E (replace c Ο aβ))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders aβ.not_
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E aβ.not_ β
Holds D I V E (replace c Ο aβ.not_)
case imp_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
aβΒΉ aβ : Formula
a_ihβΒΉ :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders aβΒΉ β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E aβΒΉ β
Holds D I V E (replace c Ο aβΒΉ))
a_ihβ :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders aβ β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E aβ β
Holds D I V E (replace c Ο aβ))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (aβΒΉ.imp_ aβ)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E (aβΒΉ.imp_ aβ) β
Holds D I V E (replace c Ο (aβΒΉ.imp_ aβ))
case and_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
aβΒΉ aβ : Formula
a_ihβΒΉ :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders aβΒΉ β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E aβΒΉ β
Holds D I V E (replace c Ο aβΒΉ))
a_ihβ :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders aβ β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E aβ β
Holds D I V E (replace c Ο aβ))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (aβΒΉ.and_ aβ)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E (aβΒΉ.and_ aβ) β
Holds D I V E (replace c Ο (aβΒΉ.and_ aβ))
case or_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
aβΒΉ aβ : Formula
a_ihβΒΉ :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders aβΒΉ β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E aβΒΉ β
Holds D I V E (replace c Ο aβΒΉ))
a_ihβ :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders aβ β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E aβ β
Holds D I V E (replace c Ο aβ))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (aβΒΉ.or_ aβ)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E (aβΒΉ.or_ aβ) β
Holds D I V E (replace c Ο (aβΒΉ.or_ aβ))
case iff_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
aβΒΉ aβ : Formula
a_ihβΒΉ :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders aβΒΉ β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E aβΒΉ β
Holds D I V E (replace c Ο aβΒΉ))
a_ihβ :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders aβ β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E aβ β
Holds D I V E (replace c Ο aβ))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (aβΒΉ.iff_ aβ)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E (aβΒΉ.iff_ aβ) β
Holds D I V E (replace c Ο (aβΒΉ.iff_ aβ))
case forall_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
aβΒΉ : VarName
aβ : Formula
a_ihβ :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders aβ β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E aβ β
Holds D I V E (replace c Ο aβ))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (forall_ aβΒΉ aβ)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E (forall_ aβΒΉ aβ) β
Holds D I V E (replace c Ο (forall_ aβΒΉ aβ))
case exists_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
aβΒΉ : VarName
aβ : Formula
a_ihβ :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders aβ β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E aβ β
Holds D I V E (replace c Ο aβ))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (exists_ aβΒΉ aβ)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E (exists_ aβΒΉ aβ) β
Holds D I V E (replace c Ο (exists_ aβΒΉ aβ))
case def_
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
aβΒΉ : DefName
aβ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (def_ aβΒΉ aβ)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E (def_ aβΒΉ aβ) β
Holds D I V E (replace c Ο (def_ aβΒΉ aβ)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | case pred_const_ X xs =>
simp only [replace]
simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (pred_const_ X xs)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E (pred_const_ X xs) β
Holds D I V E (replace c Ο (pred_const_ X xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | case eq_ x y =>
simp only [replace]
simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (eq_ x y)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E (eq_ x y) β
Holds D I V E (replace c Ο (eq_ x y)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | case true_ | false_ =>
simp only [replace]
simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders false_
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E false_ β
Holds D I V E (replace c Ο false_) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | case not_ phi phi_ih =>
simp only [admitsAux] at h1
simp only [replace]
simp only [Holds]
congr! 1
exact phi_ih V binders h1 h2 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E phi β
Holds D I V E (replace c Ο phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders phi.not_
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E phi.not_ β
Holds D I V E (replace c Ο phi.not_) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
simp only [admitsAux] at h1
simp only [replace]
simp only [Holds]
first | apply forall_congr' | apply exists_congr
intro d
apply phi_ih (Function.updateITE V x d) (binders βͺ {x}) h1
intro v a1
simp only [Function.updateITE]
simp at a1
push_neg at a1
cases a1
case h.intro a1_left a1_right =>
simp only [if_neg a1_right]
exact h2 v a1_left | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
x : VarName
phi : Formula
phi_ih :
β (V : VarAssignment D) (binders : Finset VarName),
admitsAux Ο binders phi β
(β x β binders, V' x = V x) β
(Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E phi β
Holds D I V E (replace c Ο phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (exists_ x phi)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E (exists_ x phi) β
Holds D I V E (replace c Ο (exists_ x phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (pred_const_ X xs)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E (pred_const_ X xs) β
Holds D I V E (replace c Ο (pred_const_ X xs)) | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (pred_const_ X xs)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V' ((Ο X ds.length).get β―).1 ds) E ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds }
V E (pred_const_ X xs) β
Holds D I V E (pred_const_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (pred_const_ X xs)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V' ((Ο X ds.length).get β―).1 ds) E ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds }
V E (pred_const_ X xs) β
Holds D I V E (pred_const_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [admitsAux] at h1 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux Ο binders (pred_var_ X xs)
h2 : β x β binders, V' x = V x
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E (pred_var_ X xs) β
Holds D I V E (replace c Ο (pred_var_ X xs)) | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
if xs.length = ((Ο X xs.length).get β―).1.length then
binders β© (((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset) = β
else True
else True
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E (pred_var_ X xs) β
Holds D I V E (replace c Ο (pred_var_ X xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp at h1 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
if xs.length = ((Ο X xs.length).get β―).1.length then
binders β© (((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset) = β
else True
else True
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E (pred_var_ X xs) β
Holds D I V E (replace c Ο (pred_var_ X xs)) | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.length β
binders β© (((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset) = β
else True
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E (pred_var_ X xs) β
Holds D I V E (replace c Ο (pred_var_ X xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [replace] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.length β
binders β© (((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset) = β
else True
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
let opt := Ο X ds.length;
if h : opt.isSome = true then
let val := opt.get h;
let zs := val.1;
let H := val.2;
if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
else I.pred_var_ X ds }
V E (pred_var_ X xs) β
Holds D I V E (replace c Ο (pred_var_ X xs)) | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.length β
binders β© (((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset) = β
else True
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V' ((Ο X ds.length).get β―).1 ds) E ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds }
V E (pred_var_ X xs) β
Holds D I V E
(if h : (Ο X xs.length).isSome = true then
if xs.length = ((Ο X xs.length).get β―).1.length then
Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 xs) c ((Ο X xs.length).get β―).2
else pred_var_ X xs
else pred_var_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.length β
binders β© (((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset) = β
else True
β’ Holds D
{ nonempty := β―, pred_const_ := I.pred_const_,
pred_var_ := fun X ds =>
if h : (Ο X ds.length).isSome = true then
if ds.length = ((Ο X ds.length).get β―).1.length then
Holds D I (Function.updateListITE V' ((Ο X ds.length).get β―).1 ds) E ((Ο X ds.length).get β―).2
else I.pred_var_ X ds
else I.pred_var_ X ds }
V E (pred_var_ X xs) β
Holds D I V E
(if h : (Ο X xs.length).isSome = true then
if xs.length = ((Ο X xs.length).get β―).1.length then
Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 xs) c ((Ο X xs.length).get β―).2
else pred_var_ X xs
else pred_var_ X xs) | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.length β
binders β© (((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset) = β
else True
β’ (if h : (Ο X (List.map V xs).length).isSome = true then
if (List.map V xs).length = ((Ο X (List.map V xs).length).get β―).1.length then
Holds D I (Function.updateListITE V' ((Ο X (List.map V xs).length).get β―).1 (List.map V xs)) E
((Ο X (List.map V xs).length).get β―).2
else I.pred_var_ X (List.map V xs)
else I.pred_var_ X (List.map V xs)) β
Holds D I V E
(if h : (Ο X xs.length).isSome = true then
if xs.length = ((Ο X xs.length).get β―).1.length then
Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 xs) c ((Ο X xs.length).get β―).2
else pred_var_ X xs
else pred_var_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | simp | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.length β
binders β© (((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset) = β
else True
β’ (if h : (Ο X (List.map V xs).length).isSome = true then
if (List.map V xs).length = ((Ο X (List.map V xs).length).get β―).1.length then
Holds D I (Function.updateListITE V' ((Ο X (List.map V xs).length).get β―).1 (List.map V xs)) E
((Ο X (List.map V xs).length).get β―).2
else I.pred_var_ X (List.map V xs)
else I.pred_var_ X (List.map V xs)) β
Holds D I V E
(if h : (Ο X xs.length).isSome = true then
if xs.length = ((Ο X xs.length).get β―).1.length then
Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 xs) c ((Ο X xs.length).get β―).2
else pred_var_ X xs
else pred_var_ X xs) | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.length β
binders β© (((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset) = β
else True
β’ (if h : (Ο X xs.length).isSome = true then
if xs.length = ((Ο X xs.length).get β―).1.length then
Holds D I (Function.updateListITE V' ((Ο X xs.length).get β―).1 (List.map V xs)) E ((Ο X xs.length).get β―).2
else I.pred_var_ X (List.map V xs)
else I.pred_var_ X (List.map V xs)) β
Holds D I V E
(if h : (Ο X xs.length).isSome = true then
if xs.length = ((Ο X xs.length).get β―).1.length then
Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 xs) c ((Ο X xs.length).get β―).2
else pred_var_ X xs
else pred_var_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | split_ifs | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.length β
binders β© (((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset) = β
else True
β’ (if h : (Ο X xs.length).isSome = true then
if xs.length = ((Ο X xs.length).get β―).1.length then
Holds D I (Function.updateListITE V' ((Ο X xs.length).get β―).1 (List.map V xs)) E ((Ο X xs.length).get β―).2
else I.pred_var_ X (List.map V xs)
else I.pred_var_ X (List.map V xs)) β
Holds D I V E
(if h : (Ο X xs.length).isSome = true then
if xs.length = ((Ο X xs.length).get β―).1.length then
Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 xs) c ((Ο X xs.length).get β―).2
else pred_var_ X xs
else pred_var_ X xs) | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.length β
binders β© (((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset) = β
else True
hβΒΉ : (Ο X xs.length).isSome = true
hβ : xs.length = ((Ο X xs.length).get β―).1.length
β’ Holds D I (Function.updateListITE V' ((Ο X xs.length).get β―).1 (List.map V xs)) E ((Ο X xs.length).get β―).2 β
Holds D I V E
(Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 xs) c ((Ο X xs.length).get β―).2)
case neg
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.length β
binders β© (((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset) = β
else True
hβΒΉ : (Ο X xs.length).isSome = true
hβ : Β¬xs.length = ((Ο X xs.length).get β―).1.length
β’ I.pred_var_ X (List.map V xs) β Holds D I V E (pred_var_ X xs)
case neg
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.length β
binders β© (((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset) = β
else True
hβ : Β¬(Ο X xs.length).isSome = true
β’ I.pred_var_ X (List.map V xs) β Holds D I V E (pred_var_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | case _ c1 c2 =>
simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.length β
binders β© (((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset) = β
else True
c1 : (Ο X xs.length).isSome = true
c2 : Β¬xs.length = ((Ο X xs.length).get β―).1.length
β’ I.pred_var_ X (List.map V xs) β Holds D I V E (pred_var_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | case _ c1 =>
simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.length β
binders β© (((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset) = β
else True
c1 : Β¬(Ο X xs.length).isSome = true
β’ I.pred_var_ X (List.map V xs) β Holds D I V E (pred_var_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | let opt := Ο X xs.length | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.length β
binders β© (((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset) = β
else True
c1 : (Ο X xs.length).isSome = true
c2 : xs.length = ((Ο X xs.length).get β―).1.length
β’ Holds D I (Function.updateListITE V' ((Ο X xs.length).get β―).1 (List.map V xs)) E ((Ο X xs.length).get β―).2 β
Holds D I V E
(Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 xs) c ((Ο X xs.length).get β―).2) | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.length β
binders β© (((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset) = β
else True
c1 : (Ο X xs.length).isSome = true
c2 : xs.length = ((Ο X xs.length).get β―).1.length
opt : Option (List VarName Γ Formula) := Ο X xs.length
β’ Holds D I (Function.updateListITE V' ((Ο X xs.length).get β―).1 (List.map V xs)) E ((Ο X xs.length).get β―).2 β
Holds D I V E
(Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 xs) c ((Ο X xs.length).get β―).2) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | let val := Option.get opt c1 | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.length β
binders β© (((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset) = β
else True
c1 : (Ο X xs.length).isSome = true
c2 : xs.length = ((Ο X xs.length).get β―).1.length
opt : Option (List VarName Γ Formula) := Ο X xs.length
β’ Holds D I (Function.updateListITE V' ((Ο X xs.length).get β―).1 (List.map V xs)) E ((Ο X xs.length).get β―).2 β
Holds D I V E
(Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 xs) c ((Ο X xs.length).get β―).2) | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.length β
binders β© (((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset) = β
else True
c1 : (Ο X xs.length).isSome = true
c2 : xs.length = ((Ο X xs.length).get β―).1.length
opt : Option (List VarName Γ Formula) := Ο X xs.length
val : List VarName Γ Formula := opt.get c1
β’ Holds D I (Function.updateListITE V' ((Ο X xs.length).get β―).1 (List.map V xs)) E ((Ο X xs.length).get β―).2 β
Holds D I V E
(Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 xs) c ((Ο X xs.length).get β―).2) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | let zs := val.fst | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.length β
binders β© (((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset) = β
else True
c1 : (Ο X xs.length).isSome = true
c2 : xs.length = ((Ο X xs.length).get β―).1.length
opt : Option (List VarName Γ Formula) := Ο X xs.length
val : List VarName Γ Formula := opt.get c1
β’ Holds D I (Function.updateListITE V' ((Ο X xs.length).get β―).1 (List.map V xs)) E ((Ο X xs.length).get β―).2 β
Holds D I V E
(Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 xs) c ((Ο X xs.length).get β―).2) | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.length β
binders β© (((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset) = β
else True
c1 : (Ο X xs.length).isSome = true
c2 : xs.length = ((Ο X xs.length).get β―).1.length
opt : Option (List VarName Γ Formula) := Ο X xs.length
val : List VarName Γ Formula := opt.get c1
zs : List VarName := val.1
β’ Holds D I (Function.updateListITE V' ((Ο X xs.length).get β―).1 (List.map V xs)) E ((Ο X xs.length).get β―).2 β
Holds D I V E
(Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 xs) c ((Ο X xs.length).get β―).2) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Pred/All/Rec/Option/Sub.lean | FOL.NV.Sub.Pred.All.Rec.Option.substitution_theorem_aux | [90, 1] | [226, 15] | let H := val.snd | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.length β
binders β© (((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset) = β
else True
c1 : (Ο X xs.length).isSome = true
c2 : xs.length = ((Ο X xs.length).get β―).1.length
opt : Option (List VarName Γ Formula) := Ο X xs.length
val : List VarName Γ Formula := opt.get c1
zs : List VarName := val.1
β’ Holds D I (Function.updateListITE V' ((Ο X xs.length).get β―).1 (List.map V xs)) E ((Ο X xs.length).get β―).2 β
Holds D I V E
(Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 xs) c ((Ο X xs.length).get β―).2) | D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
Ο : PredName β β β Option (List VarName Γ Formula)
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : β x β binders, V' x = V x
h1 :
if h : (Ο X xs.length).isSome = true then
xs.length = ((Ο X xs.length).get β―).1.length β
binders β© (((Ο X xs.length).get β―).2.freeVarSet \ ((Ο X xs.length).get β―).1.toFinset) = β
else True
c1 : (Ο X xs.length).isSome = true
c2 : xs.length = ((Ο X xs.length).get β―).1.length
opt : Option (List VarName Γ Formula) := Ο X xs.length
val : List VarName Γ Formula := opt.get c1
zs : List VarName := val.1
H : Formula := val.2
β’ Holds D I (Function.updateListITE V' ((Ο X xs.length).get β―).1 (List.map V xs)) E ((Ο X xs.length).get β―).2 β
Holds D I V E
(Var.All.Rec.Fresh.sub (Function.updateListITE id ((Ο X xs.length).get β―).1 xs) c ((Ο X xs.length).get β―).2) |
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