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/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 ∈ xs
this : finset_var_name_max_len xs - x.length < finset_var_name_max_len xs + 1 - x.length
⊢ (if x ∈ xs then fresh { toString := x.toString ++ c.toString } c xs else x) ∉ xs | x : VarName
c : Char
xs : Finset VarName
h : x ∈ xs
this : finset_var_name_max_len xs - x.length < finset_var_name_max_len xs + 1 - x.length
⊢ fresh { toString := x.toString ++ c.toString } c xs ∉ 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 ∈ xs
this : finset_var_name_max_len xs - x.length < finset_var_name_max_len xs + 1 - x.length
⊢ fresh { toString := x.toString ++ c.toString } c xs ∉ 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 ∉ xs
⊢ fresh x c xs ∉ xs | x : VarName
c : Char
xs : Finset VarName
h : x ∉ xs
⊢ (if h : x ∈ xs then
let_fun this := ⋯;
fresh { toString := x.toString ++ c.toString } c xs
else x) ∉
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 ∉ xs
⊢ (if h : x ∈ xs then
let_fun this := ⋯;
fresh { toString := x.toString ++ c.toString } c xs
else x) ∉
xs | x : VarName
c : Char
xs : Finset VarName
h : x ∉ xs
⊢ (if x ∈ xs then fresh { toString := x.toString ++ c.toString } c xs else x) ∉ 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 ∉ xs
⊢ (if x ∈ xs then fresh { toString := x.toString ++ c.toString } c xs else x) ∉ xs | x : VarName
c : Char
xs : Finset VarName
h : x ∉ xs
⊢ x ∉ 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 ∉ xs
⊢ x ∉ xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/NFA.lean | NFA.mem_accepts | [106, 1] | [115, 43] | rfl | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : NFA α σ
input : List α
⊢ e.accepts input ↔ ∃ s ∈ e.eval_from e.starting_state_list input, s ∈ e.accepting_state_list | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_one_no_eps_def | [111, 1] | [124, 9] | simp only [eval_one_no_eps] | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ stop_state ∈ e.eval_one_no_eps starting_state_list symbol ↔
∃ state ∈ starting_state_list, stop_state ∈ symbol_arrow_list_to_fun e.symbol_arrow_list state symbol | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ stop_state ∈
(List.map (fun state => symbol_arrow_list_to_fun e.symbol_arrow_list state symbol)
starting_state_list).join.dedup ↔
∃ state ∈ starting_state_list, stop_state ∈ symbol_arrow_list_to_fun e.symbol_arrow_list state symbol |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_one_no_eps_def | [111, 1] | [124, 9] | simp | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ stop_state ∈
(List.map (fun state => symbol_arrow_list_to_fun e.symbol_arrow_list state symbol)
starting_state_list).join.dedup ↔
∃ state ∈ starting_state_list, stop_state ∈ symbol_arrow_list_to_fun e.symbol_arrow_list state symbol | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_one_no_eps_iff | [354, 1] | [374, 53] | simp only [EpsilonNFA.eval_one_no_eps] | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ stop_state ∈ e.eval_one_no_eps starting_state_list symbol ↔
∃ state ∈ starting_state_list, e.toAbstract.symbol state symbol stop_state | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ stop_state ∈
(List.map (fun state => symbol_arrow_list_to_fun e.symbol_arrow_list state symbol)
starting_state_list).join.dedup ↔
∃ state ∈ starting_state_list, e.toAbstract.symbol state symbol stop_state |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_one_no_eps_iff | [354, 1] | [374, 53] | simp only [EpsilonNFA.toAbstract] | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ stop_state ∈
(List.map (fun state => symbol_arrow_list_to_fun e.symbol_arrow_list state symbol)
starting_state_list).join.dedup ↔
∃ state ∈ starting_state_list, e.toAbstract.symbol state symbol stop_state | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ stop_state ∈
(List.map (fun state => symbol_arrow_list_to_fun e.symbol_arrow_list state symbol)
starting_state_list).join.dedup ↔
∃ state ∈ starting_state_list,
∃ stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } ∈ e.symbol_arrow_list ∧
stop_state ∈ stop_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_one_no_eps_iff | [354, 1] | [374, 53] | simp only [symbol_arrow_list_to_fun] | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ stop_state ∈
(List.map (fun state => symbol_arrow_list_to_fun e.symbol_arrow_list state symbol)
starting_state_list).join.dedup ↔
∃ state ∈ starting_state_list,
∃ stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } ∈ e.symbol_arrow_list ∧
stop_state ∈ stop_state_list | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ stop_state ∈
(List.map
(fun state =>
(List.filterMap
(fun arrow =>
if arrow.start_state = state ∧ arrow.symbol = symbol then some arrow.stop_state_list else none)
e.symbol_arrow_list).join.dedup)
starting_state_list).join.dedup ↔
∃ state ∈ starting_state_list,
∃ stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } ∈ e.symbol_arrow_list ∧
stop_state ∈ stop_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_one_no_eps_iff | [354, 1] | [374, 53] | simp | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ stop_state ∈
(List.map
(fun state =>
(List.filterMap
(fun arrow =>
if arrow.start_state = state ∧ arrow.symbol = symbol then some arrow.stop_state_list else none)
e.symbol_arrow_list).join.dedup)
starting_state_list).join.dedup ↔
∃ state ∈ starting_state_list,
∃ stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } ∈ e.symbol_arrow_list ∧
stop_state ∈ stop_state_list | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ (∃ a ∈ starting_state_list,
∃ l,
(∃ a_1 ∈ e.symbol_arrow_list, (a_1.start_state = a ∧ a_1.symbol = symbol) ∧ a_1.stop_state_list = l) ∧
stop_state ∈ l) ↔
∃ state ∈ starting_state_list,
∃ stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } ∈ e.symbol_arrow_list ∧
stop_state ∈ stop_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_one_no_eps_iff | [354, 1] | [374, 53] | constructor | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ (∃ a ∈ starting_state_list,
∃ l,
(∃ a_1 ∈ e.symbol_arrow_list, (a_1.start_state = a ∧ a_1.symbol = symbol) ∧ a_1.stop_state_list = l) ∧
stop_state ∈ l) ↔
∃ state ∈ starting_state_list,
∃ stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } ∈ e.symbol_arrow_list ∧
stop_state ∈ stop_state_list | case mp
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ (∃ a ∈ starting_state_list,
∃ l,
(∃ a_1 ∈ e.symbol_arrow_list, (a_1.start_state = a ∧ a_1.symbol = symbol) ∧ a_1.stop_state_list = l) ∧
stop_state ∈ l) →
∃ state ∈ starting_state_list,
∃ stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } ∈ e.symbol_arrow_list ∧
stop_state ∈ stop_state_list
case mpr
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ (∃ state ∈ starting_state_list,
∃ stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } ∈ e.symbol_arrow_list ∧
stop_state ∈ stop_state_list) →
∃ a ∈ starting_state_list,
∃ l,
(∃ a_1 ∈ e.symbol_arrow_list, (a_1.start_state = a ∧ a_1.symbol = symbol) ∧ a_1.stop_state_list = l) ∧
stop_state ∈ l |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_one_no_eps_iff | [354, 1] | [374, 53] | rintro ⟨_, h1, _, ⟨⟨⟩, h2, ⟨rfl, rfl⟩, rfl⟩, h3⟩ | case mp
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ (∃ a ∈ starting_state_list,
∃ l,
(∃ a_1 ∈ e.symbol_arrow_list, (a_1.start_state = a ∧ a_1.symbol = symbol) ∧ a_1.stop_state_list = l) ∧
stop_state ∈ l) →
∃ state ∈ starting_state_list,
∃ stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } ∈ e.symbol_arrow_list ∧
stop_state ∈ stop_state_list | case mp.intro.intro.intro.intro.intro.mk.intro.intro.intro
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
stop_state start_state✝ : σ
symbol✝ : α
stop_state_list✝ : List σ
h2 : { start_state := start_state✝, symbol := symbol✝, stop_state_list := stop_state_list✝ } ∈ e.symbol_arrow_list
h1 :
{ start_state := start_state✝, symbol := symbol✝, stop_state_list := stop_state_list✝ }.start_state ∈
starting_state_list
h3 :
stop_state ∈ { start_state := start_state✝, symbol := symbol✝, stop_state_list := stop_state_list✝ }.stop_state_list
⊢ ∃ state ∈ starting_state_list,
∃ stop_state_list,
{ start_state := state,
symbol := { start_state := start_state✝, symbol := symbol✝, stop_state_list := stop_state_list✝ }.symbol,
stop_state_list := stop_state_list } ∈
e.symbol_arrow_list ∧
stop_state ∈ stop_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_one_no_eps_iff | [354, 1] | [374, 53] | exact ⟨_, h1, _, h2, h3⟩ | case mp.intro.intro.intro.intro.intro.mk.intro.intro.intro
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
stop_state start_state✝ : σ
symbol✝ : α
stop_state_list✝ : List σ
h2 : { start_state := start_state✝, symbol := symbol✝, stop_state_list := stop_state_list✝ } ∈ e.symbol_arrow_list
h1 :
{ start_state := start_state✝, symbol := symbol✝, stop_state_list := stop_state_list✝ }.start_state ∈
starting_state_list
h3 :
stop_state ∈ { start_state := start_state✝, symbol := symbol✝, stop_state_list := stop_state_list✝ }.stop_state_list
⊢ ∃ state ∈ starting_state_list,
∃ stop_state_list,
{ start_state := state,
symbol := { start_state := start_state✝, symbol := symbol✝, stop_state_list := stop_state_list✝ }.symbol,
stop_state_list := stop_state_list } ∈
e.symbol_arrow_list ∧
stop_state ∈ stop_state_list | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_one_no_eps_iff | [354, 1] | [374, 53] | rintro ⟨_, h1, _, h2, h3⟩ | case mpr
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ (∃ state ∈ starting_state_list,
∃ stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } ∈ e.symbol_arrow_list ∧
stop_state ∈ stop_state_list) →
∃ a ∈ starting_state_list,
∃ l,
(∃ a_1 ∈ e.symbol_arrow_list, (a_1.start_state = a ∧ a_1.symbol = symbol) ∧ a_1.stop_state_list = l) ∧
stop_state ∈ l | case mpr.intro.intro.intro.intro
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state w✝¹ : σ
h1 : w✝¹ ∈ starting_state_list
w✝ : List σ
h2 : { start_state := w✝¹, symbol := symbol, stop_state_list := w✝ } ∈ e.symbol_arrow_list
h3 : stop_state ∈ w✝
⊢ ∃ a ∈ starting_state_list,
∃ l,
(∃ a_1 ∈ e.symbol_arrow_list, (a_1.start_state = a ∧ a_1.symbol = symbol) ∧ a_1.stop_state_list = l) ∧
stop_state ∈ l |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_one_no_eps_iff | [354, 1] | [374, 53] | exact ⟨_, h1, _, ⟨_, h2, ⟨rfl, rfl⟩, rfl⟩, h3⟩ | case mpr.intro.intro.intro.intro
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state w✝¹ : σ
h1 : w✝¹ ∈ starting_state_list
w✝ : List σ
h2 : { start_state := w✝¹, symbol := symbol, stop_state_list := w✝ } ∈ e.symbol_arrow_list
h3 : stop_state ∈ w✝
⊢ ∃ a ∈ starting_state_list,
∃ l,
(∃ a_1 ∈ e.symbol_arrow_list, (a_1.start_state = a ∧ a_1.symbol = symbol) ∧ a_1.stop_state_list = l) ∧
stop_state ∈ l | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.epsilon_closure_iff | [385, 1] | [402, 83] | simp only [epsilon_closure] | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
state : σ
⊢ state ∈ e.epsilon_closure starting_state_list ↔
∃ start_state ∈ starting_state_list, e.toAbstract.EpsilonClosure start_state state | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
state : σ
⊢ state ∈ dft (epsilon_arrow_list_to_graph e.epsilon_arrow_list) starting_state_list ↔
∃ start_state ∈ starting_state_list, e.toAbstract.EpsilonClosure start_state state |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.epsilon_closure_iff | [385, 1] | [402, 83] | simp only [dft_iff] | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
state : σ
⊢ state ∈ dft (epsilon_arrow_list_to_graph e.epsilon_arrow_list) starting_state_list ↔
∃ start_state ∈ starting_state_list, e.toAbstract.EpsilonClosure start_state state | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
state : σ
⊢ (∃ s ∈ starting_state_list,
Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ epsilon_arrow_list_to_graph e.epsilon_arrow_list ∧ b ∈ l) s
state) ↔
∃ start_state ∈ starting_state_list, e.toAbstract.EpsilonClosure start_state state |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.epsilon_closure_iff | [385, 1] | [402, 83] | congr! with a b c | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
state : σ
⊢ (∃ s ∈ starting_state_list,
Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ epsilon_arrow_list_to_graph e.epsilon_arrow_list ∧ b ∈ l) s
state) ↔
∃ start_state ∈ starting_state_list, e.toAbstract.EpsilonClosure start_state state | case a.h.e'_2.h.h.e'_2.h.e.h.e'_2.h.h.a
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
state a b c : σ
⊢ (∃ l, (b, l) ∈ epsilon_arrow_list_to_graph e.epsilon_arrow_list ∧ c ∈ l) ↔ e.toAbstract.epsilon b c |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.epsilon_closure_iff | [385, 1] | [402, 83] | simp [toAbstract] | case a.h.e'_2.h.h.e'_2.h.e.h.e'_2.h.h.a
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
state a b c : σ
⊢ (∃ l, (b, l) ∈ epsilon_arrow_list_to_graph e.epsilon_arrow_list ∧ c ∈ l) ↔ e.toAbstract.epsilon b c | case a.h.e'_2.h.h.e'_2.h.e.h.e'_2.h.h.a
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
state a b c : σ
⊢ (∃ l, (b, l) ∈ epsilon_arrow_list_to_graph e.epsilon_arrow_list ∧ c ∈ l) ↔
∃ stop_state_list,
{ start_state := b, stop_state_list := stop_state_list } ∈ e.epsilon_arrow_list ∧ c ∈ stop_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.epsilon_closure_iff | [385, 1] | [402, 83] | induction e.epsilon_arrow_list <;> simp [*, epsilon_arrow_list_to_graph, or_and_right, exists_or] | case a.h.e'_2.h.h.e'_2.h.e.h.e'_2.h.h.a
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
state a b c : σ
⊢ (∃ l, (b, l) ∈ epsilon_arrow_list_to_graph e.epsilon_arrow_list ∧ c ∈ l) ↔
∃ stop_state_list,
{ start_state := b, stop_state_list := stop_state_list } ∈ e.epsilon_arrow_list ∧ c ∈ stop_state_list | case a.h.e'_2.h.h.e'_2.h.e.h.e'_2.h.h.a.cons
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
state a b c : σ
head✝ : EpsilonArrow σ
tail✝ : List (EpsilonArrow σ)
tail_ih✝ :
(∃ l, (b, l) ∈ epsilon_arrow_list_to_graph tail✝ ∧ c ∈ l) ↔
∃ stop_state_list, { start_state := b, stop_state_list := stop_state_list } ∈ tail✝ ∧ c ∈ stop_state_list
⊢ ((∃ x, (b = head✝.start_state ∧ x = head✝.stop_state_list) ∧ c ∈ x) ∨
∃ stop_state_list, { start_state := b, stop_state_list := stop_state_list } ∈ tail✝ ∧ c ∈ stop_state_list) ↔
(∃ x, { start_state := b, stop_state_list := x } = head✝ ∧ c ∈ x) ∨
∃ stop_state_list, { start_state := b, stop_state_list := stop_state_list } ∈ tail✝ ∧ c ∈ stop_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.epsilon_closure_iff | [385, 1] | [402, 83] | apply or_congr_left <| exists_congr fun a => and_congr_left fun _ => ?_ | case a.h.e'_2.h.h.e'_2.h.e.h.e'_2.h.h.a.cons
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
state a b c : σ
head✝ : EpsilonArrow σ
tail✝ : List (EpsilonArrow σ)
tail_ih✝ :
(∃ l, (b, l) ∈ epsilon_arrow_list_to_graph tail✝ ∧ c ∈ l) ↔
∃ stop_state_list, { start_state := b, stop_state_list := stop_state_list } ∈ tail✝ ∧ c ∈ stop_state_list
⊢ ((∃ x, (b = head✝.start_state ∧ x = head✝.stop_state_list) ∧ c ∈ x) ∨
∃ stop_state_list, { start_state := b, stop_state_list := stop_state_list } ∈ tail✝ ∧ c ∈ stop_state_list) ↔
(∃ x, { start_state := b, stop_state_list := x } = head✝ ∧ c ∈ x) ∨
∃ stop_state_list, { start_state := b, stop_state_list := stop_state_list } ∈ tail✝ ∧ c ∈ stop_state_list | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
state a✝ b c : σ
head✝ : EpsilonArrow σ
tail✝ : List (EpsilonArrow σ)
tail_ih✝ :
(∃ l, (b, l) ∈ epsilon_arrow_list_to_graph tail✝ ∧ c ∈ l) ↔
∃ stop_state_list, { start_state := b, stop_state_list := stop_state_list } ∈ tail✝ ∧ c ∈ stop_state_list
a : List σ
x✝ : c ∈ a
⊢ b = head✝.start_state ∧ a = head✝.stop_state_list ↔ { start_state := b, stop_state_list := a } = head✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.epsilon_closure_iff | [385, 1] | [402, 83] | constructor <;> [rintro ⟨rfl, rfl⟩; rintro rfl] <;> [rfl; constructor <;> rfl] | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
state a✝ b c : σ
head✝ : EpsilonArrow σ
tail✝ : List (EpsilonArrow σ)
tail_ih✝ :
(∃ l, (b, l) ∈ epsilon_arrow_list_to_graph tail✝ ∧ c ∈ l) ↔
∃ stop_state_list, { start_state := b, stop_state_list := stop_state_list } ∈ tail✝ ∧ c ∈ stop_state_list
a : List σ
x✝ : c ∈ a
⊢ b = head✝.start_state ∧ a = head✝.stop_state_list ↔ { start_state := b, stop_state_list := a } = head✝ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | simp [epsilon_closure_iff] | case nil
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
⊢ (∃ s ∈ M.epsilon_closure S, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s [] | case nil
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
⊢ (∃ s, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state s) ∧ s ∈ M.accepting_state_list) ↔
∃ s ∈ S, M.toAbstract.eval s [] |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | constructor | case nil
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
⊢ (∃ s, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state s) ∧ s ∈ M.accepting_state_list) ↔
∃ s ∈ S, M.toAbstract.eval s [] | case nil.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
⊢ (∃ s, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state s) ∧ s ∈ M.accepting_state_list) →
∃ s ∈ S, M.toAbstract.eval s []
case nil.mpr
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
⊢ (∃ s ∈ S, M.toAbstract.eval s []) →
∃ s, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state s) ∧ s ∈ M.accepting_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | intro ⟨s, ⟨s', h1, h2⟩, h3⟩ | case nil.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
⊢ (∃ s, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state s) ∧ s ∈ M.accepting_state_list) →
∃ s ∈ S, M.toAbstract.eval s [] | case nil.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s s' : σ
h1 : s' ∈ S
h2 : M.toAbstract.EpsilonClosure s' s
h3 : s ∈ M.accepting_state_list
⊢ ∃ s ∈ S, M.toAbstract.eval s [] |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | refine ⟨_, h1, ?_⟩ | case nil.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s s' : σ
h1 : s' ∈ S
h2 : M.toAbstract.EpsilonClosure s' s
h3 : s ∈ M.accepting_state_list
⊢ ∃ s ∈ S, M.toAbstract.eval s [] | case nil.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s s' : σ
h1 : s' ∈ S
h2 : M.toAbstract.EpsilonClosure s' s
h3 : s ∈ M.accepting_state_list
⊢ M.toAbstract.eval s' [] |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | clear h1 | case nil.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s s' : σ
h1 : s' ∈ S
h2 : M.toAbstract.EpsilonClosure s' s
h3 : s ∈ M.accepting_state_list
⊢ M.toAbstract.eval s' [] | case nil.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s s' : σ
h2 : M.toAbstract.EpsilonClosure s' s
h3 : s ∈ M.accepting_state_list
⊢ M.toAbstract.eval s' [] |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | induction h2 using Relation.ReflTransGen.head_induction_on with
| refl => exact .accept _ h3
| head h _ ih => exact .eps _ _ _ ih h | case nil.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s s' : σ
h2 : M.toAbstract.EpsilonClosure s' s
h3 : s ∈ M.accepting_state_list
⊢ M.toAbstract.eval s' [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | exact .accept _ h3 | case nil.mp.refl
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s s' : σ
h3 : s ∈ M.accepting_state_list
⊢ M.toAbstract.eval s [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | exact .eps _ _ _ ih h | case nil.mp.head
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s s' : σ
h3 : s ∈ M.accepting_state_list
a✝ c✝ : σ
h : M.toAbstract.epsilon a✝ c✝
h✝ : Relation.ReflTransGen M.toAbstract.epsilon c✝ s
ih : M.toAbstract.eval c✝ []
⊢ M.toAbstract.eval a✝ [] | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | intro ⟨s, h1, h2⟩ | case nil.mpr
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
⊢ (∃ s ∈ S, M.toAbstract.eval s []) →
∃ s, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state s) ∧ s ∈ M.accepting_state_list | case nil.mpr
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s : σ
h1 : s ∈ S
h2 : M.toAbstract.eval s []
⊢ ∃ s, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state s) ∧ s ∈ M.accepting_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | obtain ⟨s', h3, h4⟩ : ∃ s', M.toAbstract.EpsilonClosure s s' ∧ s' ∈ M.accepting_state_list := by
clear h1; generalize e : [] = l at h2
induction h2 with cases e
| accept _ h' => exact ⟨_, .refl, h'⟩
| eps _ _ _ _ h2 ih =>
have ⟨s', h3, h4⟩ := ih rfl
exact ⟨_, .head h2 h3, h4⟩ | case nil.mpr
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s : σ
h1 : s ∈ S
h2 : M.toAbstract.eval s []
⊢ ∃ s, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state s) ∧ s ∈ M.accepting_state_list | case nil.mpr.intro.intro
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s : σ
h1 : s ∈ S
h2 : M.toAbstract.eval s []
s' : σ
h3 : M.toAbstract.EpsilonClosure s s'
h4 : s' ∈ M.accepting_state_list
⊢ ∃ s, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state s) ∧ s ∈ M.accepting_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | exact ⟨_, ⟨_, h1, h3⟩, h4⟩ | case nil.mpr.intro.intro
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s : σ
h1 : s ∈ S
h2 : M.toAbstract.eval s []
s' : σ
h3 : M.toAbstract.EpsilonClosure s s'
h4 : s' ∈ M.accepting_state_list
⊢ ∃ s, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state s) ∧ s ∈ M.accepting_state_list | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | clear h1 | α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s : σ
h1 : s ∈ S
h2 : M.toAbstract.eval s []
⊢ ∃ s', M.toAbstract.EpsilonClosure s s' ∧ s' ∈ M.accepting_state_list | α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s : σ
h2 : M.toAbstract.eval s []
⊢ ∃ s', M.toAbstract.EpsilonClosure s s' ∧ s' ∈ M.accepting_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | generalize e : [] = l at h2 | α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s : σ
h2 : M.toAbstract.eval s []
⊢ ∃ s', M.toAbstract.EpsilonClosure s s' ∧ s' ∈ M.accepting_state_list | α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s : σ
l : List α
e : [] = l
h2 : M.toAbstract.eval s l
⊢ ∃ s', M.toAbstract.EpsilonClosure s s' ∧ s' ∈ M.accepting_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | induction h2 with cases e
| accept _ h' => exact ⟨_, .refl, h'⟩
| eps _ _ _ _ h2 ih =>
have ⟨s', h3, h4⟩ := ih rfl
exact ⟨_, .head h2 h3, h4⟩ | α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s : σ
l : List α
e : [] = l
h2 : M.toAbstract.eval s l
⊢ ∃ s', M.toAbstract.EpsilonClosure s s' ∧ s' ∈ M.accepting_state_list | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | exact ⟨_, .refl, h'⟩ | case accept.refl
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s : σ
l : List α
start_state✝ : σ
h' : M.toAbstract.accepting start_state✝
⊢ ∃ s', M.toAbstract.EpsilonClosure start_state✝ s' ∧ s' ∈ M.accepting_state_list | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | have ⟨s', h3, h4⟩ := ih rfl | case eps.refl
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s : σ
l : List α
start_state✝ stop_state✝ : σ
h2 : M.toAbstract.epsilon start_state✝ stop_state✝
a✝ : M.toAbstract.eval stop_state✝ []
ih : [] = [] → ∃ s', M.toAbstract.EpsilonClosure stop_state✝ s' ∧ s' ∈ M.accepting_state_list
⊢ ∃ s', M.toAbstract.EpsilonClosure start_state✝ s' ∧ s' ∈ M.accepting_state_list | case eps.refl
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s : σ
l : List α
start_state✝ stop_state✝ : σ
h2 : M.toAbstract.epsilon start_state✝ stop_state✝
a✝ : M.toAbstract.eval stop_state✝ []
ih : [] = [] → ∃ s', M.toAbstract.EpsilonClosure stop_state✝ s' ∧ s' ∈ M.accepting_state_list
s' : σ
h3 : M.toAbstract.EpsilonClosure stop_state✝ s'
h4 : s' ∈ M.accepting_state_list
⊢ ∃ s', M.toAbstract.EpsilonClosure start_state✝ s' ∧ s' ∈ M.accepting_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | exact ⟨_, .head h2 h3, h4⟩ | case eps.refl
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
S : List σ
s : σ
l : List α
start_state✝ stop_state✝ : σ
h2 : M.toAbstract.epsilon start_state✝ stop_state✝
a✝ : M.toAbstract.eval stop_state✝ []
ih : [] = [] → ∃ s', M.toAbstract.EpsilonClosure stop_state✝ s' ∧ s' ∈ M.accepting_state_list
s' : σ
h3 : M.toAbstract.EpsilonClosure stop_state✝ s'
h4 : s' ∈ M.accepting_state_list
⊢ ∃ s', M.toAbstract.EpsilonClosure start_state✝ s' ∧ s' ∈ M.accepting_state_list | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | simp [IH, epsilon_closure_iff, eval_one_no_eps_iff] | case cons
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
⊢ (∃ s ∈ M.eval_from (M.eval_one_no_eps (M.epsilon_closure S) a) as, s ∈ M.accepting_state_list) ↔
∃ s ∈ S, M.toAbstract.eval s (a :: as) | case cons
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
⊢ (∃ s,
(∃ state, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state state) ∧ M.toAbstract.symbol state a s) ∧
M.toAbstract.eval s as) ↔
∃ s ∈ S, M.toAbstract.eval s (a :: as) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | constructor | case cons
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
⊢ (∃ s,
(∃ state, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state state) ∧ M.toAbstract.symbol state a s) ∧
M.toAbstract.eval s as) ↔
∃ s ∈ S, M.toAbstract.eval s (a :: as) | case cons.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
⊢ (∃ s,
(∃ state, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state state) ∧ M.toAbstract.symbol state a s) ∧
M.toAbstract.eval s as) →
∃ s ∈ S, M.toAbstract.eval s (a :: as)
case cons.mpr
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
⊢ (∃ s ∈ S, M.toAbstract.eval s (a :: as)) →
∃ s,
(∃ state, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state state) ∧ M.toAbstract.symbol state a s) ∧
M.toAbstract.eval s as |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | intro ⟨s₁, ⟨s₂, ⟨s₃, h1, h2⟩, h3⟩, h4⟩ | case cons.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
⊢ (∃ s,
(∃ state, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state state) ∧ M.toAbstract.symbol state a s) ∧
M.toAbstract.eval s as) →
∃ s ∈ S, M.toAbstract.eval s (a :: as) | case cons.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s₁ s₂ s₃ : σ
h1 : s₃ ∈ S
h2 : M.toAbstract.EpsilonClosure s₃ s₂
h3 : M.toAbstract.symbol s₂ a s₁
h4 : M.toAbstract.eval s₁ as
⊢ ∃ s ∈ S, M.toAbstract.eval s (a :: as) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | refine ⟨_, h1, ?_⟩ | case cons.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s₁ s₂ s₃ : σ
h1 : s₃ ∈ S
h2 : M.toAbstract.EpsilonClosure s₃ s₂
h3 : M.toAbstract.symbol s₂ a s₁
h4 : M.toAbstract.eval s₁ as
⊢ ∃ s ∈ S, M.toAbstract.eval s (a :: as) | case cons.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s₁ s₂ s₃ : σ
h1 : s₃ ∈ S
h2 : M.toAbstract.EpsilonClosure s₃ s₂
h3 : M.toAbstract.symbol s₂ a s₁
h4 : M.toAbstract.eval s₁ as
⊢ M.toAbstract.eval s₃ (a :: as) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | clear h1 | case cons.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s₁ s₂ s₃ : σ
h1 : s₃ ∈ S
h2 : M.toAbstract.EpsilonClosure s₃ s₂
h3 : M.toAbstract.symbol s₂ a s₁
h4 : M.toAbstract.eval s₁ as
⊢ M.toAbstract.eval s₃ (a :: as) | case cons.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s₁ s₂ s₃ : σ
h2 : M.toAbstract.EpsilonClosure s₃ s₂
h3 : M.toAbstract.symbol s₂ a s₁
h4 : M.toAbstract.eval s₁ as
⊢ M.toAbstract.eval s₃ (a :: as) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | induction h2 using Relation.ReflTransGen.head_induction_on with
| refl => {apply AbstractEpsilonNFA.eval.sym _ _ _ _ h4 h3; }
| head h _ ih => exact .eps _ _ _ ih h | case cons.mp
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s₁ s₂ s₃ : σ
h2 : M.toAbstract.EpsilonClosure s₃ s₂
h3 : M.toAbstract.symbol s₂ a s₁
h4 : M.toAbstract.eval s₁ as
⊢ M.toAbstract.eval s₃ (a :: as) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | apply AbstractEpsilonNFA.eval.sym _ _ _ _ h4 h3 | case cons.mp.refl
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s₁ s₂ s₃ : σ
h3 : M.toAbstract.symbol s₂ a s₁
h4 : M.toAbstract.eval s₁ as
⊢ M.toAbstract.eval s₂ (a :: as) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | exact .eps _ _ _ ih h | case cons.mp.head
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s₁ s₂ s₃ : σ
h3 : M.toAbstract.symbol s₂ a s₁
h4 : M.toAbstract.eval s₁ as
a✝ c✝ : σ
h : M.toAbstract.epsilon a✝ c✝
h✝ : Relation.ReflTransGen M.toAbstract.epsilon c✝ s₂
ih : M.toAbstract.eval c✝ (a :: as)
⊢ M.toAbstract.eval a✝ (a :: as) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | intro ⟨s, h1, h2⟩ | case cons.mpr
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
⊢ (∃ s ∈ S, M.toAbstract.eval s (a :: as)) →
∃ s,
(∃ state, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state state) ∧ M.toAbstract.symbol state a s) ∧
M.toAbstract.eval s as | case cons.mpr
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s : σ
h1 : s ∈ S
h2 : M.toAbstract.eval s (a :: as)
⊢ ∃ s,
(∃ state, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state state) ∧ M.toAbstract.symbol state a s) ∧
M.toAbstract.eval s as |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | obtain ⟨s₁, s₂, h3, h4, h5⟩ :
∃ s₁ s₂, M.toAbstract.EpsilonClosure s s₂ ∧
M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as := by
clear h1; generalize e : a::as = l at h2
induction h2 with cases e
| sym _ _ _ _ h1 h2 => exact ⟨_, _, .refl, h2, h1⟩
| eps _ _ _ h1 h2 ih =>
have ⟨s₁, s₂, h3, h4, h5⟩ := ih rfl
exact ⟨_, _, .head h2 h3, h4, h5⟩ | case cons.mpr
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s : σ
h1 : s ∈ S
h2 : M.toAbstract.eval s (a :: as)
⊢ ∃ s,
(∃ state, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state state) ∧ M.toAbstract.symbol state a s) ∧
M.toAbstract.eval s as | case cons.mpr.intro.intro.intro.intro
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s : σ
h1 : s ∈ S
h2 : M.toAbstract.eval s (a :: as)
s₁ s₂ : σ
h3 : M.toAbstract.EpsilonClosure s s₂
h4 : M.toAbstract.symbol s₂ a s₁
h5 : M.toAbstract.eval s₁ as
⊢ ∃ s,
(∃ state, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state state) ∧ M.toAbstract.symbol state a s) ∧
M.toAbstract.eval s as |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | exact ⟨_, ⟨_, ⟨_, h1, h3⟩, h4⟩, h5⟩ | case cons.mpr.intro.intro.intro.intro
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s : σ
h1 : s ∈ S
h2 : M.toAbstract.eval s (a :: as)
s₁ s₂ : σ
h3 : M.toAbstract.EpsilonClosure s s₂
h4 : M.toAbstract.symbol s₂ a s₁
h5 : M.toAbstract.eval s₁ as
⊢ ∃ s,
(∃ state, (∃ start_state ∈ S, M.toAbstract.EpsilonClosure start_state state) ∧ M.toAbstract.symbol state a s) ∧
M.toAbstract.eval s as | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | clear h1 | α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s : σ
h1 : s ∈ S
h2 : M.toAbstract.eval s (a :: as)
⊢ ∃ s₁ s₂, M.toAbstract.EpsilonClosure s s₂ ∧ M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as | α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s : σ
h2 : M.toAbstract.eval s (a :: as)
⊢ ∃ s₁ s₂, M.toAbstract.EpsilonClosure s s₂ ∧ M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | generalize e : a::as = l at h2 | α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s : σ
h2 : M.toAbstract.eval s (a :: as)
⊢ ∃ s₁ s₂, M.toAbstract.EpsilonClosure s s₂ ∧ M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as | α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s : σ
l : List α
e : a :: as = l
h2 : M.toAbstract.eval s l
⊢ ∃ s₁ s₂, M.toAbstract.EpsilonClosure s s₂ ∧ M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | induction h2 with cases e
| sym _ _ _ _ h1 h2 => exact ⟨_, _, .refl, h2, h1⟩
| eps _ _ _ h1 h2 ih =>
have ⟨s₁, s₂, h3, h4, h5⟩ := ih rfl
exact ⟨_, _, .head h2 h3, h4, h5⟩ | α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s : σ
l : List α
e : a :: as = l
h2 : M.toAbstract.eval s l
⊢ ∃ s₁ s₂, M.toAbstract.EpsilonClosure s s₂ ∧ M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | exact ⟨_, _, .refl, h2, h1⟩ | case sym.refl
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s : σ
l : List α
start_state✝ stop_state✝ : σ
h2 : M.toAbstract.symbol start_state✝ a stop_state✝
h1 : M.toAbstract.eval stop_state✝ as
a_ih✝ :
a :: as = as →
∃ s₁ s₂, M.toAbstract.EpsilonClosure stop_state✝ s₂ ∧ M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as
⊢ ∃ s₁ s₂, M.toAbstract.EpsilonClosure start_state✝ s₂ ∧ M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | have ⟨s₁, s₂, h3, h4, h5⟩ := ih rfl | case eps.refl
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s : σ
l : List α
start_state✝ stop_state✝ : σ
h2 : M.toAbstract.epsilon start_state✝ stop_state✝
h1 : M.toAbstract.eval stop_state✝ (a :: as)
ih :
a :: as = a :: as →
∃ s₁ s₂, M.toAbstract.EpsilonClosure stop_state✝ s₂ ∧ M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as
⊢ ∃ s₁ s₂, M.toAbstract.EpsilonClosure start_state✝ s₂ ∧ M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as | case eps.refl
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s : σ
l : List α
start_state✝ stop_state✝ : σ
h2 : M.toAbstract.epsilon start_state✝ stop_state✝
h1 : M.toAbstract.eval stop_state✝ (a :: as)
ih :
a :: as = a :: as →
∃ s₁ s₂, M.toAbstract.EpsilonClosure stop_state✝ s₂ ∧ M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as
s₁ s₂ : σ
h3 : M.toAbstract.EpsilonClosure stop_state✝ s₂
h4 : M.toAbstract.symbol s₂ a s₁
h5 : M.toAbstract.eval s₁ as
⊢ ∃ s₁ s₂, M.toAbstract.EpsilonClosure start_state✝ s₂ ∧ M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.eval_from_iff | [405, 1] | [454, 44] | exact ⟨_, _, .head h2 h3, h4, h5⟩ | case eps.refl
α σ : Type
inst✝¹ : DecidableEq α
inst✝ : DecidableEq σ
M : EpsilonNFA α σ
a : α
as : List α
IH : ∀ (S : List σ), (∃ s ∈ M.eval_from S as, s ∈ M.accepting_state_list) ↔ ∃ s ∈ S, M.toAbstract.eval s as
S : List σ
s : σ
l : List α
start_state✝ stop_state✝ : σ
h2 : M.toAbstract.epsilon start_state✝ stop_state✝
h1 : M.toAbstract.eval stop_state✝ (a :: as)
ih :
a :: as = a :: as →
∃ s₁ s₂, M.toAbstract.EpsilonClosure stop_state✝ s₂ ∧ M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as
s₁ s₂ : σ
h3 : M.toAbstract.EpsilonClosure stop_state✝ s₂
h4 : M.toAbstract.symbol s₂ a s₁
h5 : M.toAbstract.eval s₁ as
⊢ ∃ s₁ s₂, M.toAbstract.EpsilonClosure start_state✝ s₂ ∧ M.toAbstract.symbol s₂ a s₁ ∧ M.toAbstract.eval s₁ as | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.accepts_iff | [457, 1] | [468, 8] | simp [accepts, eval] | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
input : List α
⊢ e.accepts input ↔ e.toAbstract.accepts input | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
input : List α
⊢ (∃ s ∈ e.eval_from e.starting_state_list input, s ∈ e.accepting_state_list) ↔ e.toAbstract.accepts input |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.accepts_iff | [457, 1] | [468, 8] | rw [EpsilonNFA.eval_from_iff] | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
input : List α
⊢ (∃ s ∈ e.eval_from e.starting_state_list input, s ∈ e.accepting_state_list) ↔ e.toAbstract.accepts input | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
input : List α
⊢ (∃ s ∈ e.starting_state_list, e.toAbstract.eval s input) ↔ e.toAbstract.accepts input |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/EpsilonNFA.lean | EpsilonNFA.accepts_iff | [457, 1] | [468, 8] | rfl | α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
input : List α
⊢ (∃ s ∈ e.starting_state_list, e.toAbstract.eval s input) ↔ e.toAbstract.accepts input | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id | [98, 1] | [121, 17] | induction F | F : Formula
⊢ fastReplaceFree id F = F | case pred_const_
a✝¹ : PredName
a✝ : List VarName
⊢ fastReplaceFree id (pred_const_ a✝¹ a✝) = pred_const_ a✝¹ a✝
case pred_var_
a✝¹ : PredName
a✝ : List VarName
⊢ fastReplaceFree id (pred_var_ a✝¹ a✝) = pred_var_ a✝¹ a✝
case eq_
a✝¹ a✝ : VarName
⊢ fastReplaceFree id (eq_ a✝¹ a✝) = eq_ a✝¹ a✝
case true_
⊢ fastReplaceFree id true_ = true_
case false_
⊢ fastReplaceFree id false_ = false_
case not_
a✝ : Formula
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id a✝.not_ = a✝.not_
case imp_
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree id a✝¹ = a✝¹
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id (a✝¹.imp_ a✝) = a✝¹.imp_ a✝
case and_
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree id a✝¹ = a✝¹
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id (a✝¹.and_ a✝) = a✝¹.and_ a✝
case or_
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree id a✝¹ = a✝¹
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id (a✝¹.or_ a✝) = a✝¹.or_ a✝
case iff_
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree id a✝¹ = a✝¹
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id (a✝¹.iff_ a✝) = a✝¹.iff_ a✝
case forall_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id (forall_ a✝¹ a✝) = forall_ a✝¹ a✝
case exists_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id (exists_ a✝¹ a✝) = exists_ a✝¹ a✝
case def_
a✝¹ : DefName
a✝ : List VarName
⊢ fastReplaceFree id (def_ a✝¹ a✝) = def_ a✝¹ a✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id | [98, 1] | [121, 17] | all_goals
simp only [fastReplaceFree] | case pred_const_
a✝¹ : PredName
a✝ : List VarName
⊢ fastReplaceFree id (pred_const_ a✝¹ a✝) = pred_const_ a✝¹ a✝
case pred_var_
a✝¹ : PredName
a✝ : List VarName
⊢ fastReplaceFree id (pred_var_ a✝¹ a✝) = pred_var_ a✝¹ a✝
case eq_
a✝¹ a✝ : VarName
⊢ fastReplaceFree id (eq_ a✝¹ a✝) = eq_ a✝¹ a✝
case true_
⊢ fastReplaceFree id true_ = true_
case false_
⊢ fastReplaceFree id false_ = false_
case not_
a✝ : Formula
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id a✝.not_ = a✝.not_
case imp_
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree id a✝¹ = a✝¹
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id (a✝¹.imp_ a✝) = a✝¹.imp_ a✝
case and_
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree id a✝¹ = a✝¹
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id (a✝¹.and_ a✝) = a✝¹.and_ a✝
case or_
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree id a✝¹ = a✝¹
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id (a✝¹.or_ a✝) = a✝¹.or_ a✝
case iff_
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree id a✝¹ = a✝¹
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id (a✝¹.iff_ a✝) = a✝¹.iff_ a✝
case forall_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id (forall_ a✝¹ a✝) = forall_ a✝¹ a✝
case exists_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ fastReplaceFree id (exists_ a✝¹ a✝) = exists_ a✝¹ a✝
case def_
a✝¹ : DefName
a✝ : List VarName
⊢ fastReplaceFree id (def_ a✝¹ a✝) = def_ a✝¹ a✝ | case pred_const_
a✝¹ : PredName
a✝ : List VarName
⊢ pred_const_ a✝¹ (List.map id a✝) = pred_const_ a✝¹ a✝
case pred_var_
a✝¹ : PredName
a✝ : List VarName
⊢ pred_var_ a✝¹ (List.map id a✝) = pred_var_ a✝¹ a✝
case eq_
a✝¹ a✝ : VarName
⊢ eq_ (id a✝¹) (id a✝) = eq_ a✝¹ a✝
case not_
a✝ : Formula
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ (fastReplaceFree id a✝).not_ = a✝.not_
case imp_
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree id a✝¹ = a✝¹
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ (fastReplaceFree id a✝¹).imp_ (fastReplaceFree id a✝) = a✝¹.imp_ a✝
case and_
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree id a✝¹ = a✝¹
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ (fastReplaceFree id a✝¹).and_ (fastReplaceFree id a✝) = a✝¹.and_ a✝
case or_
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree id a✝¹ = a✝¹
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ (fastReplaceFree id a✝¹).or_ (fastReplaceFree id a✝) = a✝¹.or_ a✝
case iff_
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree id a✝¹ = a✝¹
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ (fastReplaceFree id a✝¹).iff_ (fastReplaceFree id a✝) = a✝¹.iff_ a✝
case forall_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ forall_ a✝¹ (fastReplaceFree (Function.updateITE id a✝¹ a✝¹) a✝) = forall_ a✝¹ a✝
case exists_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : fastReplaceFree id a✝ = a✝
⊢ exists_ a✝¹ (fastReplaceFree (Function.updateITE id a✝¹ a✝¹) a✝) = exists_ a✝¹ a✝
case def_
a✝¹ : DefName
a✝ : List VarName
⊢ def_ a✝¹ (List.map id a✝) = def_ a✝¹ a✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id | [98, 1] | [121, 17] | case pred_const_ X xs | pred_var_ X xs | def_ X xs =>
congr!
simp | X : DefName
xs : List VarName
⊢ def_ X (List.map id xs) = def_ X xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id | [98, 1] | [121, 17] | case eq_ x y =>
congr! | x y : VarName
⊢ eq_ (id x) (id y) = eq_ x y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id | [98, 1] | [121, 17] | case not_ phi phi_ih =>
congr! | phi : Formula
phi_ih : fastReplaceFree id phi = phi
⊢ (fastReplaceFree id phi).not_ = phi.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id | [98, 1] | [121, 17] | case
imp_ phi psi phi_ih psi_ih
| and_ phi psi phi_ih psi_ih
| or_ phi psi phi_ih psi_ih
| iff_ phi psi phi_ih psi_ih =>
congr! | phi psi : Formula
phi_ih : fastReplaceFree id phi = phi
psi_ih : fastReplaceFree id psi = psi
⊢ (fastReplaceFree id phi).iff_ (fastReplaceFree id psi) = phi.iff_ psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id | [98, 1] | [121, 17] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
congr!
simp only [Function.updateITE_id]
exact phi_ih | x : VarName
phi : Formula
phi_ih : fastReplaceFree id phi = phi
⊢ exists_ x (fastReplaceFree (Function.updateITE id x x) phi) = exists_ x phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id | [98, 1] | [121, 17] | simp only [fastReplaceFree] | case def_
a✝¹ : DefName
a✝ : List VarName
⊢ fastReplaceFree id (def_ a✝¹ a✝) = def_ a✝¹ a✝ | case def_
a✝¹ : DefName
a✝ : List VarName
⊢ def_ a✝¹ (List.map id a✝) = def_ a✝¹ a✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id | [98, 1] | [121, 17] | congr! | X : DefName
xs : List VarName
⊢ def_ X (List.map id xs) = def_ X xs | case h.e'_2
X : DefName
xs : List VarName
⊢ List.map id xs = xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id | [98, 1] | [121, 17] | simp | case h.e'_2
X : DefName
xs : List VarName
⊢ List.map id xs = xs | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id | [98, 1] | [121, 17] | congr! | x y : VarName
⊢ eq_ (id x) (id y) = eq_ x y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id | [98, 1] | [121, 17] | congr! | phi : Formula
phi_ih : fastReplaceFree id phi = phi
⊢ (fastReplaceFree id phi).not_ = phi.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id | [98, 1] | [121, 17] | congr! | phi psi : Formula
phi_ih : fastReplaceFree id phi = phi
psi_ih : fastReplaceFree id psi = psi
⊢ (fastReplaceFree id phi).iff_ (fastReplaceFree id psi) = phi.iff_ psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id | [98, 1] | [121, 17] | congr! | x : VarName
phi : Formula
phi_ih : fastReplaceFree id phi = phi
⊢ exists_ x (fastReplaceFree (Function.updateITE id x x) phi) = exists_ x phi | case h.e'_2
x : VarName
phi : Formula
phi_ih : fastReplaceFree id phi = phi
⊢ fastReplaceFree (Function.updateITE id x x) phi = phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id | [98, 1] | [121, 17] | simp only [Function.updateITE_id] | case h.e'_2
x : VarName
phi : Formula
phi_ih : fastReplaceFree id phi = phi
⊢ fastReplaceFree (Function.updateITE id x x) phi = phi | case h.e'_2
x : VarName
phi : Formula
phi_ih : fastReplaceFree id phi = phi
⊢ fastReplaceFree id phi = phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_id | [98, 1] | [121, 17] | exact phi_ih | case h.e'_2
x : VarName
phi : Formula
phi_ih : fastReplaceFree id phi = phi
⊢ fastReplaceFree id phi = phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | induction F generalizing σ σ' | F : Formula
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v F → σ v = σ' v
⊢ fastReplaceFree σ F = fastReplaceFree σ' F | case pred_const_
a✝¹ : PredName
a✝ : List VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (pred_const_ a✝¹ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (pred_const_ a✝¹ a✝) = fastReplaceFree σ' (pred_const_ a✝¹ a✝)
case pred_var_
a✝¹ : PredName
a✝ : List VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (pred_var_ a✝¹ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (pred_var_ a✝¹ a✝) = fastReplaceFree σ' (pred_var_ a✝¹ a✝)
case eq_
a✝¹ a✝ : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (eq_ a✝¹ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (eq_ a✝¹ a✝) = fastReplaceFree σ' (eq_ a✝¹ a✝)
case true_
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v true_ → σ v = σ' v
⊢ fastReplaceFree σ true_ = fastReplaceFree σ' true_
case false_
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v false_ → σ v = σ' v
⊢ fastReplaceFree σ false_ = fastReplaceFree σ' false_
case not_
a✝ : Formula
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v a✝.not_ → σ v = σ' v
⊢ fastReplaceFree σ a✝.not_ = fastReplaceFree σ' a✝.not_
case imp_
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝¹ → σ v = σ' v) → fastReplaceFree σ a✝¹ = fastReplaceFree σ' a✝¹
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.imp_ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (a✝¹.imp_ a✝) = fastReplaceFree σ' (a✝¹.imp_ a✝)
case and_
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝¹ → σ v = σ' v) → fastReplaceFree σ a✝¹ = fastReplaceFree σ' a✝¹
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.and_ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (a✝¹.and_ a✝) = fastReplaceFree σ' (a✝¹.and_ a✝)
case or_
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝¹ → σ v = σ' v) → fastReplaceFree σ a✝¹ = fastReplaceFree σ' a✝¹
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.or_ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (a✝¹.or_ a✝) = fastReplaceFree σ' (a✝¹.or_ a✝)
case iff_
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝¹ → σ v = σ' v) → fastReplaceFree σ a✝¹ = fastReplaceFree σ' a✝¹
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.iff_ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (a✝¹.iff_ a✝) = fastReplaceFree σ' (a✝¹.iff_ a✝)
case forall_
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (forall_ a✝¹ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (forall_ a✝¹ a✝) = fastReplaceFree σ' (forall_ a✝¹ a✝)
case exists_
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (exists_ a✝¹ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (exists_ a✝¹ a✝) = fastReplaceFree σ' (exists_ a✝¹ a✝)
case def_
a✝¹ : DefName
a✝ : List VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (def_ a✝¹ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (def_ a✝¹ a✝) = fastReplaceFree σ' (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | all_goals
simp only [isFreeIn] at h1
simp only [fastReplaceFree] | case pred_const_
a✝¹ : PredName
a✝ : List VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (pred_const_ a✝¹ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (pred_const_ a✝¹ a✝) = fastReplaceFree σ' (pred_const_ a✝¹ a✝)
case pred_var_
a✝¹ : PredName
a✝ : List VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (pred_var_ a✝¹ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (pred_var_ a✝¹ a✝) = fastReplaceFree σ' (pred_var_ a✝¹ a✝)
case eq_
a✝¹ a✝ : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (eq_ a✝¹ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (eq_ a✝¹ a✝) = fastReplaceFree σ' (eq_ a✝¹ a✝)
case true_
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v true_ → σ v = σ' v
⊢ fastReplaceFree σ true_ = fastReplaceFree σ' true_
case false_
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v false_ → σ v = σ' v
⊢ fastReplaceFree σ false_ = fastReplaceFree σ' false_
case not_
a✝ : Formula
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v a✝.not_ → σ v = σ' v
⊢ fastReplaceFree σ a✝.not_ = fastReplaceFree σ' a✝.not_
case imp_
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝¹ → σ v = σ' v) → fastReplaceFree σ a✝¹ = fastReplaceFree σ' a✝¹
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.imp_ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (a✝¹.imp_ a✝) = fastReplaceFree σ' (a✝¹.imp_ a✝)
case and_
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝¹ → σ v = σ' v) → fastReplaceFree σ a✝¹ = fastReplaceFree σ' a✝¹
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.and_ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (a✝¹.and_ a✝) = fastReplaceFree σ' (a✝¹.and_ a✝)
case or_
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝¹ → σ v = σ' v) → fastReplaceFree σ a✝¹ = fastReplaceFree σ' a✝¹
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.or_ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (a✝¹.or_ a✝) = fastReplaceFree σ' (a✝¹.or_ a✝)
case iff_
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝¹ → σ v = σ' v) → fastReplaceFree σ a✝¹ = fastReplaceFree σ' a✝¹
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.iff_ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (a✝¹.iff_ a✝) = fastReplaceFree σ' (a✝¹.iff_ a✝)
case forall_
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (forall_ a✝¹ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (forall_ a✝¹ a✝) = fastReplaceFree σ' (forall_ a✝¹ a✝)
case exists_
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (exists_ a✝¹ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (exists_ a✝¹ a✝) = fastReplaceFree σ' (exists_ a✝¹ a✝)
case def_
a✝¹ : DefName
a✝ : List VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (def_ a✝¹ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (def_ a✝¹ a✝) = fastReplaceFree σ' (def_ a✝¹ a✝) | case pred_const_
a✝¹ : PredName
a✝ : List VarName
σ σ' : VarName → VarName
h1 : ∀ v ∈ a✝, σ v = σ' v
⊢ pred_const_ a✝¹ (List.map σ a✝) = pred_const_ a✝¹ (List.map σ' a✝)
case pred_var_
a✝¹ : PredName
a✝ : List VarName
σ σ' : VarName → VarName
h1 : ∀ v ∈ a✝, σ v = σ' v
⊢ pred_var_ a✝¹ (List.map σ a✝) = pred_var_ a✝¹ (List.map σ' a✝)
case eq_
a✝¹ a✝ : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = a✝¹ ∨ v = a✝ → σ v = σ' v
⊢ eq_ (σ a✝¹) (σ a✝) = eq_ (σ' a✝¹) (σ' a✝)
case not_
a✝ : Formula
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v
⊢ (fastReplaceFree σ a✝).not_ = (fastReplaceFree σ' a✝).not_
case imp_
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝¹ → σ v = σ' v) → fastReplaceFree σ a✝¹ = fastReplaceFree σ' a✝¹
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v a✝¹ ∨ isFreeIn v a✝ → σ v = σ' v
⊢ (fastReplaceFree σ a✝¹).imp_ (fastReplaceFree σ a✝) = (fastReplaceFree σ' a✝¹).imp_ (fastReplaceFree σ' a✝)
case and_
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝¹ → σ v = σ' v) → fastReplaceFree σ a✝¹ = fastReplaceFree σ' a✝¹
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v a✝¹ ∨ isFreeIn v a✝ → σ v = σ' v
⊢ (fastReplaceFree σ a✝¹).and_ (fastReplaceFree σ a✝) = (fastReplaceFree σ' a✝¹).and_ (fastReplaceFree σ' a✝)
case or_
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝¹ → σ v = σ' v) → fastReplaceFree σ a✝¹ = fastReplaceFree σ' a✝¹
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v a✝¹ ∨ isFreeIn v a✝ → σ v = σ' v
⊢ (fastReplaceFree σ a✝¹).or_ (fastReplaceFree σ a✝) = (fastReplaceFree σ' a✝¹).or_ (fastReplaceFree σ' a✝)
case iff_
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝¹ → σ v = σ' v) → fastReplaceFree σ a✝¹ = fastReplaceFree σ' a✝¹
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v a✝¹ ∨ isFreeIn v a✝ → σ v = σ' v
⊢ (fastReplaceFree σ a✝¹).iff_ (fastReplaceFree σ a✝) = (fastReplaceFree σ' a✝¹).iff_ (fastReplaceFree σ' a✝)
case forall_
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = a✝¹ ∧ isFreeIn v a✝ → σ v = σ' v
⊢ forall_ a✝¹ (fastReplaceFree (Function.updateITE σ a✝¹ a✝¹) a✝) =
forall_ a✝¹ (fastReplaceFree (Function.updateITE σ' a✝¹ a✝¹) a✝)
case exists_
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v a✝ → σ v = σ' v) → fastReplaceFree σ a✝ = fastReplaceFree σ' a✝
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = a✝¹ ∧ isFreeIn v a✝ → σ v = σ' v
⊢ exists_ a✝¹ (fastReplaceFree (Function.updateITE σ a✝¹ a✝¹) a✝) =
exists_ a✝¹ (fastReplaceFree (Function.updateITE σ' a✝¹ a✝¹) a✝)
case def_
a✝¹ : DefName
a✝ : List VarName
σ σ' : VarName → VarName
h1 : ∀ v ∈ a✝, σ v = σ' v
⊢ def_ a✝¹ (List.map σ a✝) = def_ a✝¹ (List.map σ' a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | case pred_const_ X xs | pred_var_ X xs | def_ X xs =>
congr! 1
simp only [List.map_eq_map_iff]
exact h1 | X : DefName
xs : List VarName
σ σ' : VarName → VarName
h1 : ∀ v ∈ xs, σ v = σ' v
⊢ def_ X (List.map σ xs) = def_ X (List.map σ' xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | case not_ phi phi_ih =>
congr! 1
exact phi_ih σ σ' h1 | phi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi → σ v = σ' v
⊢ (fastReplaceFree σ phi).not_ = (fastReplaceFree σ' phi).not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
congr! 1
apply phi_ih
intro v a1
simp only [Function.updateITE]
split_ifs
case _ c1 =>
rfl
case _ c1 =>
apply h1
tauto | x : VarName
phi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v
⊢ exists_ x (fastReplaceFree (Function.updateITE σ x x) phi) =
exists_ x (fastReplaceFree (Function.updateITE σ' x x) phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | simp only [isFreeIn] at h1 | case def_
a✝¹ : DefName
a✝ : List VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v (def_ a✝¹ a✝) → σ v = σ' v
⊢ fastReplaceFree σ (def_ a✝¹ a✝) = fastReplaceFree σ' (def_ a✝¹ a✝) | case def_
a✝¹ : DefName
a✝ : List VarName
σ σ' : VarName → VarName
h1 : ∀ v ∈ a✝, σ v = σ' v
⊢ fastReplaceFree σ (def_ a✝¹ a✝) = fastReplaceFree σ' (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | simp only [fastReplaceFree] | case def_
a✝¹ : DefName
a✝ : List VarName
σ σ' : VarName → VarName
h1 : ∀ v ∈ a✝, σ v = σ' v
⊢ fastReplaceFree σ (def_ a✝¹ a✝) = fastReplaceFree σ' (def_ a✝¹ a✝) | case def_
a✝¹ : DefName
a✝ : List VarName
σ σ' : VarName → VarName
h1 : ∀ v ∈ a✝, σ v = σ' v
⊢ def_ a✝¹ (List.map σ a✝) = def_ a✝¹ (List.map σ' a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | congr! 1 | X : DefName
xs : List VarName
σ σ' : VarName → VarName
h1 : ∀ v ∈ xs, σ v = σ' v
⊢ def_ X (List.map σ xs) = def_ X (List.map σ' xs) | case h.e'_2
X : DefName
xs : List VarName
σ σ' : VarName → VarName
h1 : ∀ v ∈ xs, σ v = σ' v
⊢ List.map σ xs = List.map σ' xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | simp only [List.map_eq_map_iff] | case h.e'_2
X : DefName
xs : List VarName
σ σ' : VarName → VarName
h1 : ∀ v ∈ xs, σ v = σ' v
⊢ List.map σ xs = List.map σ' xs | case h.e'_2
X : DefName
xs : List VarName
σ σ' : VarName → VarName
h1 : ∀ v ∈ xs, σ v = σ' v
⊢ ∀ x ∈ xs, σ x = σ' x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | exact h1 | case h.e'_2
X : DefName
xs : List VarName
σ σ' : VarName → VarName
h1 : ∀ v ∈ xs, σ v = σ' v
⊢ ∀ x ∈ xs, σ x = σ' x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | congr! 1 | x y : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = x ∨ v = y → σ v = σ' v
⊢ eq_ (σ x) (σ y) = eq_ (σ' x) (σ' y) | case h.e'_1
x y : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = x ∨ v = y → σ v = σ' v
⊢ σ x = σ' x
case h.e'_2
x y : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = x ∨ v = y → σ v = σ' v
⊢ σ y = σ' y |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | apply h1 | case h.e'_1
x y : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = x ∨ v = y → σ v = σ' v
⊢ σ x = σ' x | case h.e'_1.a
x y : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = x ∨ v = y → σ v = σ' v
⊢ x = x ∨ x = y |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | left | case h.e'_1.a
x y : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = x ∨ v = y → σ v = σ' v
⊢ x = x ∨ x = y | case h.e'_1.a.h
x y : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = x ∨ v = y → σ v = σ' v
⊢ x = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | rfl | case h.e'_1.a.h
x y : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = x ∨ v = y → σ v = σ' v
⊢ x = x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | apply h1 | case h.e'_2
x y : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = x ∨ v = y → σ v = σ' v
⊢ σ y = σ' y | case h.e'_2.a
x y : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = x ∨ v = y → σ v = σ' v
⊢ y = x ∨ y = y |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | right | case h.e'_2.a
x y : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = x ∨ v = y → σ v = σ' v
⊢ y = x ∨ y = y | case h.e'_2.a.h
x y : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = x ∨ v = y → σ v = σ' v
⊢ y = y |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | rfl | case h.e'_2.a.h
x y : VarName
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), v = x ∨ v = y → σ v = σ' v
⊢ y = y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | congr! 1 | phi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi → σ v = σ' v
⊢ (fastReplaceFree σ phi).not_ = (fastReplaceFree σ' phi).not_ | case h.e'_1
phi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi → σ v = σ' v
⊢ fastReplaceFree σ phi = fastReplaceFree σ' phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | exact phi_ih σ σ' h1 | case h.e'_1
phi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi → σ v = σ' v
⊢ fastReplaceFree σ phi = fastReplaceFree σ' phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | congr! 1 | phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
⊢ (fastReplaceFree σ phi).iff_ (fastReplaceFree σ psi) = (fastReplaceFree σ' phi).iff_ (fastReplaceFree σ' psi) | case h.e'_1
phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
⊢ fastReplaceFree σ phi = fastReplaceFree σ' phi
case h.e'_2
phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
⊢ fastReplaceFree σ psi = fastReplaceFree σ' psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | apply phi_ih | case h.e'_1
phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
⊢ fastReplaceFree σ phi = fastReplaceFree σ' phi | case h.e'_1.h1
phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
⊢ ∀ (v : VarName), isFreeIn v phi → σ v = σ' v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean | FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free | [171, 1] | [223, 12] | intro v a1 | case h.e'_1.h1
phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
⊢ ∀ (v : VarName), isFreeIn v phi → σ v = σ' v | case h.e'_1.h1
phi psi : Formula
phi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi
psi_ih :
∀ (σ σ' : VarName → VarName),
(∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi
σ σ' : VarName → VarName
h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v
v : VarName
a1 : isFreeIn v phi
⊢ σ v = σ' v |
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