url
stringclasses
147 values
commit
stringclasses
147 values
file_path
stringlengths
7
101
full_name
stringlengths
1
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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/LN/Paper.lean
LN.free_var_list_to_string_list
[186, 1]
[210, 24]
case nil => apply Exists.intro [] simp
h1 : ∀ v ∈ [], Var.lc_at 0 v ⊢ ∃ xs, [] = List.map free_ xs
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.free_var_list_to_string_list
[186, 1]
[210, 24]
case cons hd tl ih => simp at h1 cases h1 case intro h1_left h1_right => specialize ih h1_right apply Exists.elim ih intro xs a1 cases hd case free_ x => apply Exists.intro (x :: xs) simp exact a1 case bound_ i => simp only [Var.lc_at] at h1_left simp at h1_left
hd : Var tl : List Var ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs h1 : ∀ v ∈ hd :: tl, Var.lc_at 0 v ⊢ ∃ xs, hd :: tl = List.map free_ xs
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.free_var_list_to_string_list
[186, 1]
[210, 24]
apply Exists.intro []
h1 : ∀ v ∈ [], Var.lc_at 0 v ⊢ ∃ xs, [] = List.map free_ xs
h1 : ∀ v ∈ [], Var.lc_at 0 v ⊢ [] = List.map free_ []
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.free_var_list_to_string_list
[186, 1]
[210, 24]
simp
h1 : ∀ v ∈ [], Var.lc_at 0 v ⊢ [] = List.map free_ []
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.free_var_list_to_string_list
[186, 1]
[210, 24]
simp at h1
hd : Var tl : List Var ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs h1 : ∀ v ∈ hd :: tl, Var.lc_at 0 v ⊢ ∃ xs, hd :: tl = List.map free_ xs
hd : Var tl : List Var ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs h1 : Var.lc_at 0 hd ∧ ∀ a ∈ tl, Var.lc_at 0 a ⊢ ∃ xs, hd :: tl = List.map free_ xs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.free_var_list_to_string_list
[186, 1]
[210, 24]
cases h1
hd : Var tl : List Var ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs h1 : Var.lc_at 0 hd ∧ ∀ a ∈ tl, Var.lc_at 0 a ⊢ ∃ xs, hd :: tl = List.map free_ xs
case intro hd : Var tl : List Var ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs left✝ : Var.lc_at 0 hd right✝ : ∀ a ∈ tl, Var.lc_at 0 a ⊢ ∃ xs, hd :: tl = List.map free_ xs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.free_var_list_to_string_list
[186, 1]
[210, 24]
case intro h1_left h1_right => specialize ih h1_right apply Exists.elim ih intro xs a1 cases hd case free_ x => apply Exists.intro (x :: xs) simp exact a1 case bound_ i => simp only [Var.lc_at] at h1_left simp at h1_left
hd : Var tl : List Var ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs h1_left : Var.lc_at 0 hd h1_right : ∀ a ∈ tl, Var.lc_at 0 a ⊢ ∃ xs, hd :: tl = List.map free_ xs
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.free_var_list_to_string_list
[186, 1]
[210, 24]
specialize ih h1_right
hd : Var tl : List Var ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs h1_left : Var.lc_at 0 hd h1_right : ∀ a ∈ tl, Var.lc_at 0 a ⊢ ∃ xs, hd :: tl = List.map free_ xs
hd : Var tl : List Var h1_left : Var.lc_at 0 hd h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs ⊢ ∃ xs, hd :: tl = List.map free_ xs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.free_var_list_to_string_list
[186, 1]
[210, 24]
apply Exists.elim ih
hd : Var tl : List Var h1_left : Var.lc_at 0 hd h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs ⊢ ∃ xs, hd :: tl = List.map free_ xs
hd : Var tl : List Var h1_left : Var.lc_at 0 hd h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs ⊢ ∀ (a : List String), tl = List.map free_ a → ∃ xs, hd :: tl = List.map free_ xs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.free_var_list_to_string_list
[186, 1]
[210, 24]
intro xs a1
hd : Var tl : List Var h1_left : Var.lc_at 0 hd h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs ⊢ ∀ (a : List String), tl = List.map free_ a → ∃ xs, hd :: tl = List.map free_ xs
hd : Var tl : List Var h1_left : Var.lc_at 0 hd h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs ⊢ ∃ xs, hd :: tl = List.map free_ xs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.free_var_list_to_string_list
[186, 1]
[210, 24]
cases hd
hd : Var tl : List Var h1_left : Var.lc_at 0 hd h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs ⊢ ∃ xs, hd :: tl = List.map free_ xs
case free_ tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs a✝ : String h1_left : Var.lc_at 0 (free_ a✝) ⊢ ∃ xs, free_ a✝ :: tl = List.map free_ xs case bound_ tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs a✝ : ℕ h1_left : Var.lc_at 0 (bound_ a✝) ⊢ ∃ xs, bound_ a✝ :: tl = List.map free_ xs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.free_var_list_to_string_list
[186, 1]
[210, 24]
case free_ x => apply Exists.intro (x :: xs) simp exact a1
tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs x : String h1_left : Var.lc_at 0 (free_ x) ⊢ ∃ xs, free_ x :: tl = List.map free_ xs
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.free_var_list_to_string_list
[186, 1]
[210, 24]
case bound_ i => simp only [Var.lc_at] at h1_left simp at h1_left
tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs i : ℕ h1_left : Var.lc_at 0 (bound_ i) ⊢ ∃ xs, bound_ i :: tl = List.map free_ xs
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.free_var_list_to_string_list
[186, 1]
[210, 24]
apply Exists.intro (x :: xs)
tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs x : String h1_left : Var.lc_at 0 (free_ x) ⊢ ∃ xs, free_ x :: tl = List.map free_ xs
tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs x : String h1_left : Var.lc_at 0 (free_ x) ⊢ free_ x :: tl = List.map free_ (x :: xs)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.free_var_list_to_string_list
[186, 1]
[210, 24]
simp
tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs x : String h1_left : Var.lc_at 0 (free_ x) ⊢ free_ x :: tl = List.map free_ (x :: xs)
tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs x : String h1_left : Var.lc_at 0 (free_ x) ⊢ tl = List.map free_ xs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.free_var_list_to_string_list
[186, 1]
[210, 24]
exact a1
tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs x : String h1_left : Var.lc_at 0 (free_ x) ⊢ tl = List.map free_ xs
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.free_var_list_to_string_list
[186, 1]
[210, 24]
simp only [Var.lc_at] at h1_left
tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs i : ℕ h1_left : Var.lc_at 0 (bound_ i) ⊢ ∃ xs, bound_ i :: tl = List.map free_ xs
tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs i : ℕ h1_left : i < 0 ⊢ ∃ xs, bound_ i :: tl = List.map free_ xs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.free_var_list_to_string_list
[186, 1]
[210, 24]
simp at h1_left
tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs i : ℕ h1_left : i < 0 ⊢ ∃ xs, bound_ i :: tl = List.map free_ xs
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenFreeVarSet
[216, 1]
[238, 11]
cases v
j : ℕ z : String v : Var ⊢ (Var.open j (free_ z) v).freeVarSet ⊆ v.freeVarSet ∪ {free_ z}
case free_ j : ℕ z a✝ : String ⊢ (Var.open j (free_ z) (free_ a✝)).freeVarSet ⊆ (free_ a✝).freeVarSet ∪ {free_ z} case bound_ j : ℕ z : String a✝ : ℕ ⊢ (Var.open j (free_ z) (bound_ a✝)).freeVarSet ⊆ (bound_ a✝).freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenFreeVarSet
[216, 1]
[238, 11]
case free_ x => simp only [Var.open] simp only [Var.freeVarSet] simp
j : ℕ z x : String ⊢ (Var.open j (free_ z) (free_ x)).freeVarSet ⊆ (free_ x).freeVarSet ∪ {free_ z}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenFreeVarSet
[216, 1]
[238, 11]
case bound_ i => simp only [Var.open] split_ifs case _ c1 => simp only [Var.freeVarSet] simp case _ c1 c2 => simp only [Var.freeVarSet] simp case _ c1 c2 => simp only [Var.freeVarSet] simp
j : ℕ z : String i : ℕ ⊢ (Var.open j (free_ z) (bound_ i)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenFreeVarSet
[216, 1]
[238, 11]
simp only [Var.open]
j : ℕ z x : String ⊢ (Var.open j (free_ z) (free_ x)).freeVarSet ⊆ (free_ x).freeVarSet ∪ {free_ z}
j : ℕ z x : String ⊢ (free_ x).freeVarSet ⊆ (free_ x).freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenFreeVarSet
[216, 1]
[238, 11]
simp only [Var.freeVarSet]
j : ℕ z x : String ⊢ (free_ x).freeVarSet ⊆ (free_ x).freeVarSet ∪ {free_ z}
j : ℕ z x : String ⊢ {free_ x} ⊆ {free_ x} ∪ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenFreeVarSet
[216, 1]
[238, 11]
simp
j : ℕ z x : String ⊢ {free_ x} ⊆ {free_ x} ∪ {free_ z}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenFreeVarSet
[216, 1]
[238, 11]
simp only [Var.open]
j : ℕ z : String i : ℕ ⊢ (Var.open j (free_ z) (bound_ i)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}
j : ℕ z : String i : ℕ ⊢ (if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenFreeVarSet
[216, 1]
[238, 11]
split_ifs
j : ℕ z : String i : ℕ ⊢ (if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}
case pos j : ℕ z : String i : ℕ h✝ : i < j ⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z} case pos j : ℕ z : String i : ℕ h✝¹ : ¬i < j h✝ : i = j ⊢ (free_ z).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z} case neg j : ℕ z : String i : ℕ h✝¹ : ¬i < j h✝ : ¬i = j ⊢ (bound_ (i - 1)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenFreeVarSet
[216, 1]
[238, 11]
case _ c1 => simp only [Var.freeVarSet] simp
j : ℕ z : String i : ℕ c1 : i < j ⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenFreeVarSet
[216, 1]
[238, 11]
case _ c1 c2 => simp only [Var.freeVarSet] simp
j : ℕ z : String i : ℕ c1 : ¬i < j c2 : i = j ⊢ (free_ z).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenFreeVarSet
[216, 1]
[238, 11]
case _ c1 c2 => simp only [Var.freeVarSet] simp
j : ℕ z : String i : ℕ c1 : ¬i < j c2 : ¬i = j ⊢ (bound_ (i - 1)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenFreeVarSet
[216, 1]
[238, 11]
simp only [Var.freeVarSet]
j : ℕ z : String i : ℕ c1 : i < j ⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}
j : ℕ z : String i : ℕ c1 : i < j ⊢ ∅ ⊆ ∅ ∪ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenFreeVarSet
[216, 1]
[238, 11]
simp
j : ℕ z : String i : ℕ c1 : i < j ⊢ ∅ ⊆ ∅ ∪ {free_ z}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenFreeVarSet
[216, 1]
[238, 11]
simp only [Var.freeVarSet]
j : ℕ z : String i : ℕ c1 : ¬i < j c2 : i = j ⊢ (free_ z).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}
j : ℕ z : String i : ℕ c1 : ¬i < j c2 : i = j ⊢ {free_ z} ⊆ ∅ ∪ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenFreeVarSet
[216, 1]
[238, 11]
simp
j : ℕ z : String i : ℕ c1 : ¬i < j c2 : i = j ⊢ {free_ z} ⊆ ∅ ∪ {free_ z}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenFreeVarSet
[216, 1]
[238, 11]
simp only [Var.freeVarSet]
j : ℕ z : String i : ℕ c1 : ¬i < j c2 : ¬i = j ⊢ (bound_ (i - 1)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}
j : ℕ z : String i : ℕ c1 : ¬i < j c2 : ¬i = j ⊢ ∅ ⊆ ∅ ∪ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenFreeVarSet
[216, 1]
[238, 11]
simp
j : ℕ z : String i : ℕ c1 : ¬i < j c2 : ¬i = j ⊢ ∅ ⊆ ∅ ∪ {free_ z}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenFreeVarSet
[241, 1]
[273, 17]
induction F generalizing j
j : ℕ z : String F : Formula ⊢ (Formula.open j (free_ z) F).freeVarSet ⊆ F.freeVarSet ∪ {free_ z}
case pred_ z a✝¹ : String a✝ : List Var j : ℕ ⊢ (Formula.open j (free_ z) (pred_ a✝¹ a✝)).freeVarSet ⊆ (pred_ a✝¹ a✝).freeVarSet ∪ {free_ z} case not_ z : String a✝ : Formula a_ih✝ : ∀ (j : ℕ), (Formula.open j (free_ z) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) a✝.not_).freeVarSet ⊆ a✝.not_.freeVarSet ∪ {free_ z} case imp_ z : String a✝¹ a✝ : Formula a_ih✝¹ : ∀ (j : ℕ), (Formula.open j (free_ z) a✝¹).freeVarSet ⊆ a✝¹.freeVarSet ∪ {free_ z} a_ih✝ : ∀ (j : ℕ), (Formula.open j (free_ z) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) (a✝¹.imp_ a✝)).freeVarSet ⊆ (a✝¹.imp_ a✝).freeVarSet ∪ {free_ z} case forall_ z a✝¹ : String a✝ : Formula a_ih✝ : ∀ (j : ℕ), (Formula.open j (free_ z) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) (forall_ a✝¹ a✝)).freeVarSet ⊆ (forall_ a✝¹ a✝).freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenFreeVarSet
[241, 1]
[273, 17]
case not_ phi phi_ih => simp only [Formula.open] simp only [Formula.freeVarSet] apply phi_ih
z : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi.not_).freeVarSet ⊆ phi.not_.freeVarSet ∪ {free_ z}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenFreeVarSet
[241, 1]
[273, 17]
case forall_ x phi phi_ih => simp only [Formula.open] simp only [Formula.freeVarSet] apply phi_ih
z x : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) (forall_ x phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ {free_ z}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenFreeVarSet
[241, 1]
[273, 17]
simp only [Formula.open]
z X : String vs : List Var j : ℕ ⊢ (Formula.open j (free_ z) (pred_ X vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet ∪ {free_ z}
z X : String vs : List Var j : ℕ ⊢ (pred_ X (List.map (Var.open j (free_ z)) vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenFreeVarSet
[241, 1]
[273, 17]
simp only [Formula.freeVarSet]
z X : String vs : List Var j : ℕ ⊢ (pred_ X (List.map (Var.open j (free_ z)) vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet ∪ {free_ z}
z X : String vs : List Var j : ℕ ⊢ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenFreeVarSet
[241, 1]
[273, 17]
simp
z X : String vs : List Var j : ℕ ⊢ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}
z X : String vs : List Var j : ℕ ⊢ ∀ a ∈ vs, (Var.open j (free_ z) a).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenFreeVarSet
[241, 1]
[273, 17]
intro v a1
z X : String vs : List Var j : ℕ ⊢ ∀ a ∈ vs, (Var.open j (free_ z) a).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}
z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ (Var.open j (free_ z) v).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenFreeVarSet
[241, 1]
[273, 17]
trans Var.freeVarSet v ∪ {free_ z}
z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ (Var.open j (free_ z) v).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}
z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ (Var.open j (free_ z) v).freeVarSet ⊆ v.freeVarSet ∪ {free_ z} z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v.freeVarSet ∪ {free_ z} ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenFreeVarSet
[241, 1]
[273, 17]
exact VarOpenFreeVarSet j z v
z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ (Var.open j (free_ z) v).freeVarSet ⊆ v.freeVarSet ∪ {free_ z}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenFreeVarSet
[241, 1]
[273, 17]
apply Finset.union_subset_union_left
z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v.freeVarSet ∪ {free_ z} ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}
case h z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenFreeVarSet
[241, 1]
[273, 17]
apply Finset.subset_biUnion_of_mem
case h z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet
case h.xs z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v ∈ vs.toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenFreeVarSet
[241, 1]
[273, 17]
simp
case h.xs z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v ∈ vs.toFinset
case h.xs z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v ∈ vs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenFreeVarSet
[241, 1]
[273, 17]
exact a1
case h.xs z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v ∈ vs
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenFreeVarSet
[241, 1]
[273, 17]
simp only [Formula.open]
z : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi.not_).freeVarSet ⊆ phi.not_.freeVarSet ∪ {free_ z}
z : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi).not_.freeVarSet ⊆ phi.not_.freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenFreeVarSet
[241, 1]
[273, 17]
simp only [Formula.freeVarSet]
z : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi).not_.freeVarSet ⊆ phi.not_.freeVarSet ∪ {free_ z}
z : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenFreeVarSet
[241, 1]
[273, 17]
apply phi_ih
z : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenFreeVarSet
[241, 1]
[273, 17]
simp only [Formula.open]
z : String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) (phi.imp_ psi)).freeVarSet ⊆ (phi.imp_ psi).freeVarSet ∪ {free_ z}
z : String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z} j : ℕ ⊢ ((Formula.open j (free_ z) phi).imp_ (Formula.open j (free_ z) psi)).freeVarSet ⊆ (phi.imp_ psi).freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenFreeVarSet
[241, 1]
[273, 17]
simp only [Formula.freeVarSet]
z : String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z} j : ℕ ⊢ ((Formula.open j (free_ z) phi).imp_ (Formula.open j (free_ z) psi)).freeVarSet ⊆ (phi.imp_ psi).freeVarSet ∪ {free_ z}
z : String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi).freeVarSet ∪ (Formula.open j (free_ z) psi).freeVarSet ⊆ phi.freeVarSet ∪ psi.freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenFreeVarSet
[241, 1]
[273, 17]
apply Finset.union_subset_union_left_right
z : String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi).freeVarSet ∪ (Formula.open j (free_ z) psi).freeVarSet ⊆ phi.freeVarSet ∪ psi.freeVarSet ∪ {free_ z}
case h1 z : String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} case h2 z : String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenFreeVarSet
[241, 1]
[273, 17]
exact phi_ih j
case h1 z : String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenFreeVarSet
[241, 1]
[273, 17]
exact psi_ih j
case h2 z : String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenFreeVarSet
[241, 1]
[273, 17]
simp only [Formula.open]
z x : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) (forall_ x phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ {free_ z}
z x : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (forall_ x (Formula.open (j + 1) (free_ z) phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenFreeVarSet
[241, 1]
[273, 17]
simp only [Formula.freeVarSet]
z x : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (forall_ x (Formula.open (j + 1) (free_ z) phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ {free_ z}
z x : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open (j + 1) (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenFreeVarSet
[241, 1]
[273, 17]
apply phi_ih
z x : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open (j + 1) (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenListFreeVarSet
[279, 1]
[303, 11]
cases v
j : ℕ zs : List String v : Var ⊢ (Var.openList j (List.map free_ zs) v).freeVarSet ⊆ v.freeVarSet ∪ (List.map free_ zs).toFinset
case free_ j : ℕ zs : List String a✝ : String ⊢ (Var.openList j (List.map free_ zs) (free_ a✝)).freeVarSet ⊆ (free_ a✝).freeVarSet ∪ (List.map free_ zs).toFinset case bound_ j : ℕ zs : List String a✝ : ℕ ⊢ (Var.openList j (List.map free_ zs) (bound_ a✝)).freeVarSet ⊆ (bound_ a✝).freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenListFreeVarSet
[279, 1]
[303, 11]
case free_ x => simp only [Var.openList] simp only [Var.freeVarSet] simp
j : ℕ zs : List String x : String ⊢ (Var.openList j (List.map free_ zs) (free_ x)).freeVarSet ⊆ (free_ x).freeVarSet ∪ (List.map free_ zs).toFinset
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenListFreeVarSet
[279, 1]
[303, 11]
case bound_ i => simp only [Var.openList] split_ifs case pos c1 => simp only [Var.freeVarSet] simp case pos c1 c2 => simp simp only [Var.freeVarSet] simp apply List.get_mem case neg c1 c2 => simp only [Var.freeVarSet] simp
j : ℕ zs : List String i : ℕ ⊢ (Var.openList j (List.map free_ zs) (bound_ i)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenListFreeVarSet
[279, 1]
[303, 11]
simp only [Var.openList]
j : ℕ zs : List String x : String ⊢ (Var.openList j (List.map free_ zs) (free_ x)).freeVarSet ⊆ (free_ x).freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ zs : List String x : String ⊢ (free_ x).freeVarSet ⊆ (free_ x).freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenListFreeVarSet
[279, 1]
[303, 11]
simp only [Var.freeVarSet]
j : ℕ zs : List String x : String ⊢ (free_ x).freeVarSet ⊆ (free_ x).freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ zs : List String x : String ⊢ {free_ x} ⊆ {free_ x} ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenListFreeVarSet
[279, 1]
[303, 11]
simp
j : ℕ zs : List String x : String ⊢ {free_ x} ⊆ {free_ x} ∪ (List.map free_ zs).toFinset
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenListFreeVarSet
[279, 1]
[303, 11]
simp only [Var.openList]
j : ℕ zs : List String i : ℕ ⊢ (Var.openList j (List.map free_ zs) (bound_ i)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ zs : List String i : ℕ ⊢ (if i < j then bound_ i else if x : i - j < (List.map free_ zs).length then (List.map free_ zs)[i - j] else bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenListFreeVarSet
[279, 1]
[303, 11]
split_ifs
j : ℕ zs : List String i : ℕ ⊢ (if i < j then bound_ i else if x : i - j < (List.map free_ zs).length then (List.map free_ zs)[i - j] else bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
case pos j : ℕ zs : List String i : ℕ h✝ : i < j ⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset case pos j : ℕ zs : List String i : ℕ h✝¹ : ¬i < j h✝ : i - j < (List.map free_ zs).length ⊢ (List.map free_ zs)[i - j].freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset case neg j : ℕ zs : List String i : ℕ h✝¹ : ¬i < j h✝ : ¬i - j < (List.map free_ zs).length ⊢ (bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenListFreeVarSet
[279, 1]
[303, 11]
case pos c1 => simp only [Var.freeVarSet] simp
j : ℕ zs : List String i : ℕ c1 : i < j ⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenListFreeVarSet
[279, 1]
[303, 11]
case pos c1 c2 => simp simp only [Var.freeVarSet] simp apply List.get_mem
j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : i - j < (List.map free_ zs).length ⊢ (List.map free_ zs)[i - j].freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenListFreeVarSet
[279, 1]
[303, 11]
case neg c1 c2 => simp only [Var.freeVarSet] simp
j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : ¬i - j < (List.map free_ zs).length ⊢ (bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenListFreeVarSet
[279, 1]
[303, 11]
simp only [Var.freeVarSet]
j : ℕ zs : List String i : ℕ c1 : i < j ⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ zs : List String i : ℕ c1 : i < j ⊢ ∅ ⊆ ∅ ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenListFreeVarSet
[279, 1]
[303, 11]
simp
j : ℕ zs : List String i : ℕ c1 : i < j ⊢ ∅ ⊆ ∅ ∪ (List.map free_ zs).toFinset
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenListFreeVarSet
[279, 1]
[303, 11]
simp
j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : i - j < (List.map free_ zs).length ⊢ (List.map free_ zs)[i - j].freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : i - j < (List.map free_ zs).length ⊢ (free_ (zs.get ⟨i - j, ⋯⟩)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenListFreeVarSet
[279, 1]
[303, 11]
simp only [Var.freeVarSet]
j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : i - j < (List.map free_ zs).length ⊢ (free_ (zs.get ⟨i - j, ⋯⟩)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : i - j < (List.map free_ zs).length ⊢ {free_ (zs.get ⟨i - j, ⋯⟩)} ⊆ ∅ ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenListFreeVarSet
[279, 1]
[303, 11]
simp
j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : i - j < (List.map free_ zs).length ⊢ {free_ (zs.get ⟨i - j, ⋯⟩)} ⊆ ∅ ∪ (List.map free_ zs).toFinset
j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : i - j < (List.map free_ zs).length ⊢ zs.get ⟨i - j, ⋯⟩ ∈ zs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenListFreeVarSet
[279, 1]
[303, 11]
apply List.get_mem
j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : i - j < (List.map free_ zs).length ⊢ zs.get ⟨i - j, ⋯⟩ ∈ zs
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenListFreeVarSet
[279, 1]
[303, 11]
simp only [Var.freeVarSet]
j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : ¬i - j < (List.map free_ zs).length ⊢ (bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : ¬i - j < (List.map free_ zs).length ⊢ ∅ ⊆ ∅ ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.VarOpenListFreeVarSet
[279, 1]
[303, 11]
simp
j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : ¬i - j < (List.map free_ zs).length ⊢ ∅ ⊆ ∅ ∪ (List.map free_ zs).toFinset
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenListFreeVarSet
[306, 1]
[338, 17]
induction F generalizing j
j : ℕ zs : List String F : Formula ⊢ (Formula.openList j (List.map free_ zs) F).freeVarSet ⊆ F.freeVarSet ∪ (List.map free_ zs).toFinset
case pred_ zs : List String a✝¹ : String a✝ : List Var j : ℕ ⊢ (Formula.openList j (List.map free_ zs) (pred_ a✝¹ a✝)).freeVarSet ⊆ (pred_ a✝¹ a✝).freeVarSet ∪ (List.map free_ zs).toFinset case not_ zs : List String a✝ : Formula a_ih✝ : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) a✝.not_).freeVarSet ⊆ a✝.not_.freeVarSet ∪ (List.map free_ zs).toFinset case imp_ zs : List String a✝¹ a✝ : Formula a_ih✝¹ : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) a✝¹).freeVarSet ⊆ a✝¹.freeVarSet ∪ (List.map free_ zs).toFinset a_ih✝ : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) (a✝¹.imp_ a✝)).freeVarSet ⊆ (a✝¹.imp_ a✝).freeVarSet ∪ (List.map free_ zs).toFinset case forall_ zs : List String a✝¹ : String a✝ : Formula a_ih✝ : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) (forall_ a✝¹ a✝)).freeVarSet ⊆ (forall_ a✝¹ a✝).freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenListFreeVarSet
[306, 1]
[338, 17]
case not_ phi phi_ih => simp only [Formula.openList] simp only [Formula.freeVarSet] apply phi_ih
zs : List String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi.not_).freeVarSet ⊆ phi.not_.freeVarSet ∪ (List.map free_ zs).toFinset
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenListFreeVarSet
[306, 1]
[338, 17]
case forall_ x phi phi_ih => simp only [Formula.openList] simp only [Formula.freeVarSet] apply phi_ih
zs : List String x : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) (forall_ x phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ (List.map free_ zs).toFinset
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenListFreeVarSet
[306, 1]
[338, 17]
simp only [Formula.openList]
zs : List String X : String vs : List Var j : ℕ ⊢ (Formula.openList j (List.map free_ zs) (pred_ X vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet ∪ (List.map free_ zs).toFinset
zs : List String X : String vs : List Var j : ℕ ⊢ (pred_ X (List.map (Var.openList j (List.map free_ zs)) vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenListFreeVarSet
[306, 1]
[338, 17]
simp only [Formula.freeVarSet]
zs : List String X : String vs : List Var j : ℕ ⊢ (pred_ X (List.map (Var.openList j (List.map free_ zs)) vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet ∪ (List.map free_ zs).toFinset
zs : List String X : String vs : List Var j : ℕ ⊢ (List.map (Var.openList j (List.map free_ zs)) vs).toFinset.biUnion Var.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenListFreeVarSet
[306, 1]
[338, 17]
simp
zs : List String X : String vs : List Var j : ℕ ⊢ (List.map (Var.openList j (List.map free_ zs)) vs).toFinset.biUnion Var.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset
zs : List String X : String vs : List Var j : ℕ ⊢ ∀ a ∈ vs, (Var.openList j (List.map free_ zs) a).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenListFreeVarSet
[306, 1]
[338, 17]
intro v a1
zs : List String X : String vs : List Var j : ℕ ⊢ ∀ a ∈ vs, (Var.openList j (List.map free_ zs) a).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset
zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ (Var.openList j (List.map free_ zs) v).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenListFreeVarSet
[306, 1]
[338, 17]
trans v.freeVarSet ∪ (zs.map free_).toFinset
zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ (Var.openList j (List.map free_ zs) v).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset
zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ (Var.openList j (List.map free_ zs) v).freeVarSet ⊆ v.freeVarSet ∪ (List.map free_ zs).toFinset zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v.freeVarSet ∪ (List.map free_ zs).toFinset ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenListFreeVarSet
[306, 1]
[338, 17]
exact VarOpenListFreeVarSet j zs v
zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ (Var.openList j (List.map free_ zs) v).freeVarSet ⊆ v.freeVarSet ∪ (List.map free_ zs).toFinset
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenListFreeVarSet
[306, 1]
[338, 17]
apply Finset.union_subset_union_left
zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v.freeVarSet ∪ (List.map free_ zs).toFinset ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset
case h zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenListFreeVarSet
[306, 1]
[338, 17]
apply Finset.subset_biUnion_of_mem
case h zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet
case h.xs zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v ∈ vs.toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenListFreeVarSet
[306, 1]
[338, 17]
simp
case h.xs zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v ∈ vs.toFinset
case h.xs zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v ∈ vs
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenListFreeVarSet
[306, 1]
[338, 17]
exact a1
case h.xs zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v ∈ vs
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenListFreeVarSet
[306, 1]
[338, 17]
simp only [Formula.openList]
zs : List String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi.not_).freeVarSet ⊆ phi.not_.freeVarSet ∪ (List.map free_ zs).toFinset
zs : List String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi).not_.freeVarSet ⊆ phi.not_.freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenListFreeVarSet
[306, 1]
[338, 17]
simp only [Formula.freeVarSet]
zs : List String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi).not_.freeVarSet ⊆ phi.not_.freeVarSet ∪ (List.map free_ zs).toFinset
zs : List String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenListFreeVarSet
[306, 1]
[338, 17]
apply phi_ih
zs : List String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenListFreeVarSet
[306, 1]
[338, 17]
simp only [Formula.openList]
zs : List String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset psi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) (phi.imp_ psi)).freeVarSet ⊆ (phi.imp_ psi).freeVarSet ∪ (List.map free_ zs).toFinset
zs : List String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset psi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ ((Formula.openList j (List.map free_ zs) phi).imp_ (Formula.openList j (List.map free_ zs) psi)).freeVarSet ⊆ (phi.imp_ psi).freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenListFreeVarSet
[306, 1]
[338, 17]
simp only [Formula.freeVarSet]
zs : List String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset psi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ ((Formula.openList j (List.map free_ zs) phi).imp_ (Formula.openList j (List.map free_ zs) psi)).freeVarSet ⊆ (phi.imp_ psi).freeVarSet ∪ (List.map free_ zs).toFinset
zs : List String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset psi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi).freeVarSet ∪ (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ phi.freeVarSet ∪ psi.freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenListFreeVarSet
[306, 1]
[338, 17]
apply Finset.union_subset_union_left_right
zs : List String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset psi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi).freeVarSet ∪ (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ phi.freeVarSet ∪ psi.freeVarSet ∪ (List.map free_ zs).toFinset
case h1 zs : List String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset psi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset case h2 zs : List String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset psi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenListFreeVarSet
[306, 1]
[338, 17]
exact phi_ih j
case h1 zs : List String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset psi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenListFreeVarSet
[306, 1]
[338, 17]
exact psi_ih j
case h2 zs : List String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset psi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/LN/Paper.lean
LN.FormulaOpenListFreeVarSet
[306, 1]
[338, 17]
simp only [Formula.openList]
zs : List String x : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) (forall_ x phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ (List.map free_ zs).toFinset
zs : List String x : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (forall_ x (Formula.openList (j + 1) (List.map free_ zs) phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ (List.map free_ zs).toFinset