Datasets:
internlm
/
Lean-Github

Dataset card Data Studio Files Community
1
url
stringclasses
147 values
commit
stringclasses
147 values
file_path
stringlengths
7
101
full_name
stringlengths
1
94
start
stringlengths
6
10
end
stringlengths
6
11
tactic
stringlengths
1
11.2k
state_before
stringlengths
3
2.09M
state_after
stringlengths
6
2.09M
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/DFT.lean
dft_iff
[884, 1]
[900, 50]
induction h2 with | refl => exact reachable_from_list.base _ h1 | tail _ h ih => have ⟨_, h3, h4⟩ := h exact reachable_from_list.step _ _ h3 h4 ih
case mpr α : Type inst✝ : DecidableEq α g : Graph α S : List α s' s : α h1 : s ∈ S h2 : Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s s' ⊢ s' ∈ reachable_from_list g S
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/DFT.lean
dft_iff
[884, 1]
[900, 50]
exact reachable_from_list.base _ h1
case mpr.refl α : Type inst✝ : DecidableEq α g : Graph α S : List α s' s : α h1 : s ∈ S ⊢ s ∈ reachable_from_list g S
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/DFT.lean
dft_iff
[884, 1]
[900, 50]
have ⟨_, h3, h4⟩ := h
case mpr.tail α : Type inst✝ : DecidableEq α g : Graph α S : List α s' s : α h1 : s ∈ S b✝ c✝ : α a✝ : Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s b✝ h : ∃ l, (b✝, l) ∈ g ∧ c✝ ∈ l ih : b✝ ∈ reachable_from_list g S ⊢ c✝ ∈ reachable_from_list g S
case mpr.tail α : Type inst✝ : DecidableEq α g : Graph α S : List α s' s : α h1 : s ∈ S b✝ c✝ : α a✝ : Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s b✝ h : ∃ l, (b✝, l) ∈ g ∧ c✝ ∈ l ih : b✝ ∈ reachable_from_list g S w✝ : List α h3 : (b✝, w✝) ∈ g h4 : c✝ ∈ w✝ ⊢ c✝ ∈ reachable_from_list g S
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/Parsing/DFT.lean
dft_iff
[884, 1]
[900, 50]
exact reachable_from_list.step _ _ h3 h4 ih
case mpr.tail α : Type inst✝ : DecidableEq α g : Graph α S : List α s' s : α h1 : s ∈ S b✝ c✝ : α a✝ : Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ g ∧ b ∈ l) s b✝ h : ∃ l, (b✝, l) ∈ g ∧ c✝ ∈ l ih : b✝ ∈ reachable_from_list g S w✝ : List α h3 : (b✝, w✝) ∈ g h4 : c✝ ∈ w✝ ⊢ c✝ ∈ reachable_from_list g S
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
induction F
v : VarName F : Formula ⊢ occursIn v F ↔ v ∈ F.varSet
case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName ⊢ occursIn v (pred_const_ a✝¹ a✝) ↔ v ∈ (pred_const_ a✝¹ a✝).varSet case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName ⊢ occursIn v (pred_var_ a✝¹ a✝) ↔ v ∈ (pred_var_ a✝¹ a✝).varSet case eq_ v a✝¹ a✝ : VarName ⊢ occursIn v (eq_ a✝¹ a✝) ↔ v ∈ (eq_ a✝¹ a✝).varSet case true_ v : VarName ⊢ occursIn v true_ ↔ v ∈ true_.varSet case false_ v : VarName ⊢ occursIn v false_ ↔ v ∈ false_.varSet case not_ v : VarName a✝ : Formula a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v a✝.not_ ↔ v ∈ a✝.not_.varSet case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (a✝¹.imp_ a✝) ↔ v ∈ (a✝¹.imp_ a✝).varSet case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (a✝¹.and_ a✝) ↔ v ∈ (a✝¹.and_ a✝).varSet case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (a✝¹.or_ a✝) ↔ v ∈ (a✝¹.or_ a✝).varSet case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (a✝¹.iff_ a✝) ↔ v ∈ (a✝¹.iff_ a✝).varSet case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (forall_ a✝¹ a✝) ↔ v ∈ (forall_ a✝¹ a✝).varSet case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (exists_ a✝¹ a✝) ↔ v ∈ (exists_ a✝¹ a✝).varSet case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ occursIn v (def_ a✝¹ a✝) ↔ v ∈ (def_ a✝¹ a✝).varSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
all_goals simp only [occursIn] simp only [Formula.varSet]
case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName ⊢ occursIn v (pred_const_ a✝¹ a✝) ↔ v ∈ (pred_const_ a✝¹ a✝).varSet case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName ⊢ occursIn v (pred_var_ a✝¹ a✝) ↔ v ∈ (pred_var_ a✝¹ a✝).varSet case eq_ v a✝¹ a✝ : VarName ⊢ occursIn v (eq_ a✝¹ a✝) ↔ v ∈ (eq_ a✝¹ a✝).varSet case true_ v : VarName ⊢ occursIn v true_ ↔ v ∈ true_.varSet case false_ v : VarName ⊢ occursIn v false_ ↔ v ∈ false_.varSet case not_ v : VarName a✝ : Formula a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v a✝.not_ ↔ v ∈ a✝.not_.varSet case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (a✝¹.imp_ a✝) ↔ v ∈ (a✝¹.imp_ a✝).varSet case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (a✝¹.and_ a✝) ↔ v ∈ (a✝¹.and_ a✝).varSet case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (a✝¹.or_ a✝) ↔ v ∈ (a✝¹.or_ a✝).varSet case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (a✝¹.iff_ a✝) ↔ v ∈ (a✝¹.iff_ a✝).varSet case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (forall_ a✝¹ a✝) ↔ v ∈ (forall_ a✝¹ a✝).varSet case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v (exists_ a✝¹ a✝) ↔ v ∈ (exists_ a✝¹ a✝).varSet case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ occursIn v (def_ a✝¹ a✝) ↔ v ∈ (def_ a✝¹ a✝).varSet
case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName ⊢ v ∈ a✝ ↔ v ∈ a✝.toFinset case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName ⊢ v ∈ a✝ ↔ v ∈ a✝.toFinset case eq_ v a✝¹ a✝ : VarName ⊢ v = a✝¹ ∨ v = a✝ ↔ v ∈ {a✝¹, a✝} case true_ v : VarName ⊢ False ↔ v ∈ ∅ case false_ v : VarName ⊢ False ↔ v ∈ ∅ case not_ v : VarName a✝ : Formula a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v a✝ ↔ v ∈ a✝.varSet case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v a✝¹ ∨ occursIn v a✝ ↔ v ∈ a✝¹.varSet ∪ a✝.varSet case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v a✝¹ ∨ occursIn v a✝ ↔ v ∈ a✝¹.varSet ∪ a✝.varSet case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v a✝¹ ∨ occursIn v a✝ ↔ v ∈ a✝¹.varSet ∪ a✝.varSet case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ↔ v ∈ a✝¹.varSet a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ occursIn v a✝¹ ∨ occursIn v a✝ ↔ v ∈ a✝¹.varSet ∪ a✝.varSet case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ v = a✝¹ ∨ occursIn v a✝ ↔ v ∈ a✝.varSet ∪ {a✝¹} case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : occursIn v a✝ ↔ v ∈ a✝.varSet ⊢ v = a✝¹ ∨ occursIn v a✝ ↔ v ∈ a✝.varSet ∪ {a✝¹} case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ v ∈ a✝ ↔ v ∈ a✝.toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
case pred_const_ X xs | pred_var_ X xs | def_ X xs => simp
v : VarName X : DefName xs : List VarName ⊢ v ∈ xs ↔ v ∈ xs.toFinset
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
case eq_ x y => simp
v x y : VarName ⊢ v = x ∨ v = y ↔ v ∈ {x, y}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
case true_ | false_ => tauto
v : VarName ⊢ False ↔ v ∈ ∅
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
case not_ phi phi_ih => tauto
v : VarName phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi ↔ v ∈ phi.varSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
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 => simp tauto
v : VarName phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ phi.varSet ∪ psi.varSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
case forall_ x phi phi_ih | exists_ x phi phi_ih => simp tauto
v x : VarName phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ v = x ∨ occursIn v phi ↔ v ∈ phi.varSet ∪ {x}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
simp only [occursIn]
case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ occursIn v (def_ a✝¹ a✝) ↔ v ∈ (def_ a✝¹ a✝).varSet
case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ v ∈ a✝ ↔ v ∈ (def_ a✝¹ a✝).varSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
simp only [Formula.varSet]
case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ v ∈ a✝ ↔ v ∈ (def_ a✝¹ a✝).varSet
case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ v ∈ a✝ ↔ v ∈ a✝.toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
simp
v : VarName X : DefName xs : List VarName ⊢ v ∈ xs ↔ v ∈ xs.toFinset
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
simp
v x y : VarName ⊢ v = x ∨ v = y ↔ v ∈ {x, y}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
tauto
v : VarName ⊢ False ↔ v ∈ ∅
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
tauto
v : VarName phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi ↔ v ∈ phi.varSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
simp
v : VarName phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ phi.varSet ∪ psi.varSet
v : VarName phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ phi.varSet ∨ v ∈ psi.varSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
tauto
v : VarName phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ phi.varSet ∨ v ∈ psi.varSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
simp
v x : VarName phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ v = x ∨ occursIn v phi ↔ v ∈ phi.varSet ∪ {x}
v x : VarName phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ v = x ∨ occursIn v phi ↔ v ∈ phi.varSet ∨ v = x
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.occursIn_iff_mem_varSet
[295, 1]
[321, 10]
tauto
v x : VarName phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ v = x ∨ occursIn v phi ↔ v ∈ phi.varSet ∨ v = x
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
induction F
v : VarName F : Formula ⊢ isBoundIn v F ↔ v ∈ F.boundVarSet
case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName ⊢ isBoundIn v (pred_const_ a✝¹ a✝) ↔ v ∈ (pred_const_ a✝¹ a✝).boundVarSet case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName ⊢ isBoundIn v (pred_var_ a✝¹ a✝) ↔ v ∈ (pred_var_ a✝¹ a✝).boundVarSet case eq_ v a✝¹ a✝ : VarName ⊢ isBoundIn v (eq_ a✝¹ a✝) ↔ v ∈ (eq_ a✝¹ a✝).boundVarSet case true_ v : VarName ⊢ isBoundIn v true_ ↔ v ∈ true_.boundVarSet case false_ v : VarName ⊢ isBoundIn v false_ ↔ v ∈ false_.boundVarSet case not_ v : VarName a✝ : Formula a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v a✝.not_ ↔ v ∈ a✝.not_.boundVarSet case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v (a✝¹.imp_ a✝) ↔ v ∈ (a✝¹.imp_ a✝).boundVarSet case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v (a✝¹.and_ a✝) ↔ v ∈ (a✝¹.and_ a✝).boundVarSet case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v (a✝¹.or_ a✝) ↔ v ∈ (a✝¹.or_ a✝).boundVarSet case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v (a✝¹.iff_ a✝) ↔ v ∈ (a✝¹.iff_ a✝).boundVarSet case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v (forall_ a✝¹ a✝) ↔ v ∈ (forall_ a✝¹ a✝).boundVarSet case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v (exists_ a✝¹ a✝) ↔ v ∈ (exists_ a✝¹ a✝).boundVarSet case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ isBoundIn v (def_ a✝¹ a✝) ↔ v ∈ (def_ a✝¹ a✝).boundVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
all_goals simp only [isBoundIn] simp only [Formula.boundVarSet]
case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName ⊢ isBoundIn v (pred_const_ a✝¹ a✝) ↔ v ∈ (pred_const_ a✝¹ a✝).boundVarSet case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName ⊢ isBoundIn v (pred_var_ a✝¹ a✝) ↔ v ∈ (pred_var_ a✝¹ a✝).boundVarSet case eq_ v a✝¹ a✝ : VarName ⊢ isBoundIn v (eq_ a✝¹ a✝) ↔ v ∈ (eq_ a✝¹ a✝).boundVarSet case true_ v : VarName ⊢ isBoundIn v true_ ↔ v ∈ true_.boundVarSet case false_ v : VarName ⊢ isBoundIn v false_ ↔ v ∈ false_.boundVarSet case not_ v : VarName a✝ : Formula a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v a✝.not_ ↔ v ∈ a✝.not_.boundVarSet case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v (a✝¹.imp_ a✝) ↔ v ∈ (a✝¹.imp_ a✝).boundVarSet case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v (a✝¹.and_ a✝) ↔ v ∈ (a✝¹.and_ a✝).boundVarSet case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v (a✝¹.or_ a✝) ↔ v ∈ (a✝¹.or_ a✝).boundVarSet case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v (a✝¹.iff_ a✝) ↔ v ∈ (a✝¹.iff_ a✝).boundVarSet case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v (forall_ a✝¹ a✝) ↔ v ∈ (forall_ a✝¹ a✝).boundVarSet case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v (exists_ a✝¹ a✝) ↔ v ∈ (exists_ a✝¹ a✝).boundVarSet case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ isBoundIn v (def_ a✝¹ a✝) ↔ v ∈ (def_ a✝¹ a✝).boundVarSet
case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName ⊢ False ↔ v ∈ ∅ case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName ⊢ False ↔ v ∈ ∅ case eq_ v a✝¹ a✝ : VarName ⊢ False ↔ v ∈ ∅ case true_ v : VarName ⊢ False ↔ v ∈ ∅ case false_ v : VarName ⊢ False ↔ v ∈ ∅ case not_ v : VarName a✝ : Formula a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v a✝¹ ∨ isBoundIn v a✝ ↔ v ∈ a✝¹.boundVarSet ∪ a✝.boundVarSet case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v a✝¹ ∨ isBoundIn v a✝ ↔ v ∈ a✝¹.boundVarSet ∪ a✝.boundVarSet case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v a✝¹ ∨ isBoundIn v a✝ ↔ v ∈ a✝¹.boundVarSet ∪ a✝.boundVarSet case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isBoundIn v a✝¹ ↔ v ∈ a✝¹.boundVarSet a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ isBoundIn v a✝¹ ∨ isBoundIn v a✝ ↔ v ∈ a✝¹.boundVarSet ∪ a✝.boundVarSet case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ v = a✝¹ ∨ isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ∪ {a✝¹} case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ⊢ v = a✝¹ ∨ isBoundIn v a✝ ↔ v ∈ a✝.boundVarSet ∪ {a✝¹} case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ False ↔ v ∈ ∅
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
case pred_const_ X xs | pred_var_ X xs | def_ X xs => simp
v : VarName X : DefName xs : List VarName ⊢ False ↔ v ∈ ∅
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
case eq_ x y => simp
v x y : VarName ⊢ False ↔ v ∈ ∅
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
case true_ | false_ => tauto
v : VarName ⊢ False ↔ v ∈ ∅
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
case not_ phi phi_ih => tauto
v : VarName phi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet ⊢ isBoundIn v phi ↔ v ∈ phi.boundVarSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
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 => simp tauto
v : VarName phi psi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet psi_ih : isBoundIn v psi ↔ v ∈ psi.boundVarSet ⊢ isBoundIn v phi ∨ isBoundIn v psi ↔ v ∈ phi.boundVarSet ∪ psi.boundVarSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
case forall_ x phi phi_ih | exists_ x phi phi_ih => simp tauto
v x : VarName phi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet ⊢ v = x ∨ isBoundIn v phi ↔ v ∈ phi.boundVarSet ∪ {x}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
simp only [isBoundIn]
case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ isBoundIn v (def_ a✝¹ a✝) ↔ v ∈ (def_ a✝¹ a✝).boundVarSet
case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ False ↔ v ∈ (def_ a✝¹ a✝).boundVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
simp only [Formula.boundVarSet]
case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ False ↔ v ∈ (def_ a✝¹ a✝).boundVarSet
case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ False ↔ v ∈ ∅
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
simp
v : VarName X : DefName xs : List VarName ⊢ False ↔ v ∈ ∅
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
simp
v x y : VarName ⊢ False ↔ v ∈ ∅
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
tauto
v : VarName ⊢ False ↔ v ∈ ∅
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
tauto
v : VarName phi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet ⊢ isBoundIn v phi ↔ v ∈ phi.boundVarSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
simp
v : VarName phi psi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet psi_ih : isBoundIn v psi ↔ v ∈ psi.boundVarSet ⊢ isBoundIn v phi ∨ isBoundIn v psi ↔ v ∈ phi.boundVarSet ∪ psi.boundVarSet
v : VarName phi psi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet psi_ih : isBoundIn v psi ↔ v ∈ psi.boundVarSet ⊢ isBoundIn v phi ∨ isBoundIn v psi ↔ v ∈ phi.boundVarSet ∨ v ∈ psi.boundVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
tauto
v : VarName phi psi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet psi_ih : isBoundIn v psi ↔ v ∈ psi.boundVarSet ⊢ isBoundIn v phi ∨ isBoundIn v psi ↔ v ∈ phi.boundVarSet ∨ v ∈ psi.boundVarSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
simp
v x : VarName phi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet ⊢ v = x ∨ isBoundIn v phi ↔ v ∈ phi.boundVarSet ∪ {x}
v x : VarName phi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet ⊢ v = x ∨ isBoundIn v phi ↔ v ∈ phi.boundVarSet ∨ v = x
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isBoundIn_iff_mem_boundVarSet
[324, 1]
[350, 10]
tauto
v x : VarName phi : Formula phi_ih : isBoundIn v phi ↔ v ∈ phi.boundVarSet ⊢ v = x ∨ isBoundIn v phi ↔ v ∈ phi.boundVarSet ∨ v = x
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
induction F
v : VarName F : Formula ⊢ isFreeIn v F ↔ v ∈ F.freeVarSet
case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName ⊢ isFreeIn v (pred_const_ a✝¹ a✝) ↔ v ∈ (pred_const_ a✝¹ a✝).freeVarSet case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName ⊢ isFreeIn v (pred_var_ a✝¹ a✝) ↔ v ∈ (pred_var_ a✝¹ a✝).freeVarSet case eq_ v a✝¹ a✝ : VarName ⊢ isFreeIn v (eq_ a✝¹ a✝) ↔ v ∈ (eq_ a✝¹ a✝).freeVarSet case true_ v : VarName ⊢ isFreeIn v true_ ↔ v ∈ true_.freeVarSet case false_ v : VarName ⊢ isFreeIn v false_ ↔ v ∈ false_.freeVarSet case not_ v : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v a✝.not_ ↔ v ∈ a✝.not_.freeVarSet case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ ↔ v ∈ a✝¹.freeVarSet a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v (a✝¹.imp_ a✝) ↔ v ∈ (a✝¹.imp_ a✝).freeVarSet case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ ↔ v ∈ a✝¹.freeVarSet a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v (a✝¹.and_ a✝) ↔ v ∈ (a✝¹.and_ a✝).freeVarSet case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ ↔ v ∈ a✝¹.freeVarSet a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v (a✝¹.or_ a✝) ↔ v ∈ (a✝¹.or_ a✝).freeVarSet case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ ↔ v ∈ a✝¹.freeVarSet a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v (a✝¹.iff_ a✝) ↔ v ∈ (a✝¹.iff_ a✝).freeVarSet case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v (forall_ a✝¹ a✝) ↔ v ∈ (forall_ a✝¹ a✝).freeVarSet case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v (exists_ a✝¹ a✝) ↔ v ∈ (exists_ a✝¹ a✝).freeVarSet case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ isFreeIn v (def_ a✝¹ a✝) ↔ v ∈ (def_ a✝¹ a✝).freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
all_goals simp only [isFreeIn] simp only [Formula.freeVarSet]
case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName ⊢ isFreeIn v (pred_const_ a✝¹ a✝) ↔ v ∈ (pred_const_ a✝¹ a✝).freeVarSet case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName ⊢ isFreeIn v (pred_var_ a✝¹ a✝) ↔ v ∈ (pred_var_ a✝¹ a✝).freeVarSet case eq_ v a✝¹ a✝ : VarName ⊢ isFreeIn v (eq_ a✝¹ a✝) ↔ v ∈ (eq_ a✝¹ a✝).freeVarSet case true_ v : VarName ⊢ isFreeIn v true_ ↔ v ∈ true_.freeVarSet case false_ v : VarName ⊢ isFreeIn v false_ ↔ v ∈ false_.freeVarSet case not_ v : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v a✝.not_ ↔ v ∈ a✝.not_.freeVarSet case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ ↔ v ∈ a✝¹.freeVarSet a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v (a✝¹.imp_ a✝) ↔ v ∈ (a✝¹.imp_ a✝).freeVarSet case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ ↔ v ∈ a✝¹.freeVarSet a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v (a✝¹.and_ a✝) ↔ v ∈ (a✝¹.and_ a✝).freeVarSet case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ ↔ v ∈ a✝¹.freeVarSet a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v (a✝¹.or_ a✝) ↔ v ∈ (a✝¹.or_ a✝).freeVarSet case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ ↔ v ∈ a✝¹.freeVarSet a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v (a✝¹.iff_ a✝) ↔ v ∈ (a✝¹.iff_ a✝).freeVarSet case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v (forall_ a✝¹ a✝) ↔ v ∈ (forall_ a✝¹ a✝).freeVarSet case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v (exists_ a✝¹ a✝) ↔ v ∈ (exists_ a✝¹ a✝).freeVarSet case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ isFreeIn v (def_ a✝¹ a✝) ↔ v ∈ (def_ a✝¹ a✝).freeVarSet
case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName ⊢ v ∈ a✝ ↔ v ∈ a✝.toFinset case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName ⊢ v ∈ a✝ ↔ v ∈ a✝.toFinset case eq_ v a✝¹ a✝ : VarName ⊢ v = a✝¹ ∨ v = a✝ ↔ v ∈ {a✝¹, a✝} case true_ v : VarName ⊢ False ↔ v ∈ ∅ case false_ v : VarName ⊢ False ↔ v ∈ ∅ case not_ v : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ ↔ v ∈ a✝¹.freeVarSet a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v a✝¹ ∨ isFreeIn v a✝ ↔ v ∈ a✝¹.freeVarSet ∪ a✝.freeVarSet case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ ↔ v ∈ a✝¹.freeVarSet a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v a✝¹ ∨ isFreeIn v a✝ ↔ v ∈ a✝¹.freeVarSet ∪ a✝.freeVarSet case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ ↔ v ∈ a✝¹.freeVarSet a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v a✝¹ ∨ isFreeIn v a✝ ↔ v ∈ a✝¹.freeVarSet ∪ a✝.freeVarSet case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ ↔ v ∈ a✝¹.freeVarSet a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ isFreeIn v a✝¹ ∨ isFreeIn v a✝ ↔ v ∈ a✝¹.freeVarSet ∪ a✝.freeVarSet case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ ¬v = a✝¹ ∧ isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet \ {a✝¹} case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet ⊢ ¬v = a✝¹ ∧ isFreeIn v a✝ ↔ v ∈ a✝.freeVarSet \ {a✝¹} case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ v ∈ a✝ ↔ v ∈ a✝.toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
case pred_const_ X xs | pred_var_ X xs | def_ X xs => simp
v : VarName X : DefName xs : List VarName ⊢ v ∈ xs ↔ v ∈ xs.toFinset
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
case eq_ x y => simp
v x y : VarName ⊢ v = x ∨ v = y ↔ v ∈ {x, y}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
case true_ | false_ => tauto
v : VarName ⊢ False ↔ v ∈ ∅
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
case not_ phi phi_ih => tauto
v : VarName phi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet ⊢ isFreeIn v phi ↔ v ∈ phi.freeVarSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
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 => simp tauto
v : VarName phi psi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet psi_ih : isFreeIn v psi ↔ v ∈ psi.freeVarSet ⊢ isFreeIn v phi ∨ isFreeIn v psi ↔ v ∈ phi.freeVarSet ∪ psi.freeVarSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
case forall_ x phi phi_ih | exists_ x phi phi_ih => simp tauto
v x : VarName phi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet ⊢ ¬v = x ∧ isFreeIn v phi ↔ v ∈ phi.freeVarSet \ {x}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
simp only [isFreeIn]
case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ isFreeIn v (def_ a✝¹ a✝) ↔ v ∈ (def_ a✝¹ a✝).freeVarSet
case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ v ∈ a✝ ↔ v ∈ (def_ a✝¹ a✝).freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
simp only [Formula.freeVarSet]
case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ v ∈ a✝ ↔ v ∈ (def_ a✝¹ a✝).freeVarSet
case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ v ∈ a✝ ↔ v ∈ a✝.toFinset
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
simp
v : VarName X : DefName xs : List VarName ⊢ v ∈ xs ↔ v ∈ xs.toFinset
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
simp
v x y : VarName ⊢ v = x ∨ v = y ↔ v ∈ {x, y}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
tauto
v : VarName ⊢ False ↔ v ∈ ∅
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
tauto
v : VarName phi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet ⊢ isFreeIn v phi ↔ v ∈ phi.freeVarSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
simp
v : VarName phi psi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet psi_ih : isFreeIn v psi ↔ v ∈ psi.freeVarSet ⊢ isFreeIn v phi ∨ isFreeIn v psi ↔ v ∈ phi.freeVarSet ∪ psi.freeVarSet
v : VarName phi psi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet psi_ih : isFreeIn v psi ↔ v ∈ psi.freeVarSet ⊢ isFreeIn v phi ∨ isFreeIn v psi ↔ v ∈ phi.freeVarSet ∨ v ∈ psi.freeVarSet
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
tauto
v : VarName phi psi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet psi_ih : isFreeIn v psi ↔ v ∈ psi.freeVarSet ⊢ isFreeIn v phi ∨ isFreeIn v psi ↔ v ∈ phi.freeVarSet ∨ v ∈ psi.freeVarSet
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
simp
v x : VarName phi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet ⊢ ¬v = x ∧ isFreeIn v phi ↔ v ∈ phi.freeVarSet \ {x}
v x : VarName phi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet ⊢ ¬v = x ∧ isFreeIn v phi ↔ v ∈ phi.freeVarSet ∧ ¬v = x
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_mem_freeVarSet
[353, 1]
[379, 10]
tauto
v x : VarName phi : Formula phi_ih : isFreeIn v phi ↔ v ∈ phi.freeVarSet ⊢ ¬v = x ∧ isFreeIn v phi ↔ v ∈ phi.freeVarSet ∧ ¬v = x
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
induction F
v : VarName F : Formula h1 : isFreeIn v F ⊢ isFreeInInd v F
case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName h1 : isFreeIn v (pred_const_ a✝¹ a✝) ⊢ isFreeInInd v (pred_const_ a✝¹ a✝) case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName h1 : isFreeIn v (pred_var_ a✝¹ a✝) ⊢ isFreeInInd v (pred_var_ a✝¹ a✝) case eq_ v a✝¹ a✝ : VarName h1 : isFreeIn v (eq_ a✝¹ a✝) ⊢ isFreeInInd v (eq_ a✝¹ a✝) case true_ v : VarName h1 : isFreeIn v true_ ⊢ isFreeInInd v true_ case false_ v : VarName h1 : isFreeIn v false_ ⊢ isFreeInInd v false_ case not_ v : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v a✝.not_ ⊢ isFreeInInd v a✝.not_ case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ → isFreeInInd v a✝¹ a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v (a✝¹.imp_ a✝) ⊢ isFreeInInd v (a✝¹.imp_ a✝) case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ → isFreeInInd v a✝¹ a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v (a✝¹.and_ a✝) ⊢ isFreeInInd v (a✝¹.and_ a✝) case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ → isFreeInInd v a✝¹ a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v (a✝¹.or_ a✝) ⊢ isFreeInInd v (a✝¹.or_ a✝) case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ → isFreeInInd v a✝¹ a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v (a✝¹.iff_ a✝) ⊢ isFreeInInd v (a✝¹.iff_ a✝) case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v (forall_ a✝¹ a✝) ⊢ isFreeInInd v (forall_ a✝¹ a✝) case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v (exists_ a✝¹ a✝) ⊢ isFreeInInd v (exists_ a✝¹ a✝) case def_ v : VarName a✝¹ : DefName a✝ : List VarName h1 : isFreeIn v (def_ a✝¹ a✝) ⊢ isFreeInInd v (def_ a✝¹ a✝)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
any_goals simp only [isFreeIn] at h1
case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName h1 : isFreeIn v (pred_const_ a✝¹ a✝) ⊢ isFreeInInd v (pred_const_ a✝¹ a✝) case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName h1 : isFreeIn v (pred_var_ a✝¹ a✝) ⊢ isFreeInInd v (pred_var_ a✝¹ a✝) case eq_ v a✝¹ a✝ : VarName h1 : isFreeIn v (eq_ a✝¹ a✝) ⊢ isFreeInInd v (eq_ a✝¹ a✝) case true_ v : VarName h1 : isFreeIn v true_ ⊢ isFreeInInd v true_ case false_ v : VarName h1 : isFreeIn v false_ ⊢ isFreeInInd v false_ case not_ v : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v a✝.not_ ⊢ isFreeInInd v a✝.not_ case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ → isFreeInInd v a✝¹ a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v (a✝¹.imp_ a✝) ⊢ isFreeInInd v (a✝¹.imp_ a✝) case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ → isFreeInInd v a✝¹ a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v (a✝¹.and_ a✝) ⊢ isFreeInInd v (a✝¹.and_ a✝) case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ → isFreeInInd v a✝¹ a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v (a✝¹.or_ a✝) ⊢ isFreeInInd v (a✝¹.or_ a✝) case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ → isFreeInInd v a✝¹ a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v (a✝¹.iff_ a✝) ⊢ isFreeInInd v (a✝¹.iff_ a✝) case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v (forall_ a✝¹ a✝) ⊢ isFreeInInd v (forall_ a✝¹ a✝) case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v (exists_ a✝¹ a✝) ⊢ isFreeInInd v (exists_ a✝¹ a✝) case def_ v : VarName a✝¹ : DefName a✝ : List VarName h1 : isFreeIn v (def_ a✝¹ a✝) ⊢ isFreeInInd v (def_ a✝¹ a✝)
case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName h1 : v ∈ a✝ ⊢ isFreeInInd v (pred_const_ a✝¹ a✝) case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName h1 : v ∈ a✝ ⊢ isFreeInInd v (pred_var_ a✝¹ a✝) case eq_ v a✝¹ a✝ : VarName h1 : v = a✝¹ ∨ v = a✝ ⊢ isFreeInInd v (eq_ a✝¹ a✝) case not_ v : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v a✝ ⊢ isFreeInInd v a✝.not_ case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ → isFreeInInd v a✝¹ a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v a✝¹ ∨ isFreeIn v a✝ ⊢ isFreeInInd v (a✝¹.imp_ a✝) case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ → isFreeInInd v a✝¹ a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v a✝¹ ∨ isFreeIn v a✝ ⊢ isFreeInInd v (a✝¹.and_ a✝) case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ → isFreeInInd v a✝¹ a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v a✝¹ ∨ isFreeIn v a✝ ⊢ isFreeInInd v (a✝¹.or_ a✝) case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : isFreeIn v a✝¹ → isFreeInInd v a✝¹ a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : isFreeIn v a✝¹ ∨ isFreeIn v a✝ ⊢ isFreeInInd v (a✝¹.iff_ a✝) case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : ¬v = a✝¹ ∧ isFreeIn v a✝ ⊢ isFreeInInd v (forall_ a✝¹ a✝) case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : isFreeIn v a✝ → isFreeInInd v a✝ h1 : ¬v = a✝¹ ∧ isFreeIn v a✝ ⊢ isFreeInInd v (exists_ a✝¹ a✝) case def_ v : VarName a✝¹ : DefName a✝ : List VarName h1 : v ∈ a✝ ⊢ isFreeInInd v (def_ a✝¹ a✝)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
case pred_const_ X xs | pred_var_ X xs | eq_ x y | def_ X xs => first | exact isFreeInInd.pred_const_ X xs h1 | exact isFreeInInd.pred_var_ X xs h1 | exact isFreeInInd.eq_ x y h1 | exact isFreeInInd.def_ X xs h1
v : VarName X : DefName xs : List VarName h1 : v ∈ xs ⊢ isFreeInInd v (def_ X xs)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
case not_ phi phi_ih => apply isFreeInInd.not_ exact phi_ih h1
v : VarName phi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi h1 : isFreeIn v phi ⊢ isFreeInInd v phi.not_
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
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 => cases h1 case inl c1 => first | apply isFreeInInd.imp_left_ | apply isFreeInInd.and_left_ | apply isFreeInInd.or_left_ | apply isFreeInInd.iff_left_ exact phi_ih c1 case inr c1 => first | apply isFreeInInd.imp_right_ | apply isFreeInInd.and_right_ | apply isFreeInInd.or_right_ | apply isFreeInInd.iff_right_ exact psi_ih c1
v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi h1 : isFreeIn v phi ∨ isFreeIn v psi ⊢ isFreeInInd v (phi.iff_ psi)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
simp only [isFreeIn] at h1
case def_ v : VarName a✝¹ : DefName a✝ : List VarName h1 : isFreeIn v (def_ a✝¹ a✝) ⊢ isFreeInInd v (def_ a✝¹ a✝)
case def_ v : VarName a✝¹ : DefName a✝ : List VarName h1 : v ∈ a✝ ⊢ isFreeInInd v (def_ a✝¹ a✝)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
first | exact isFreeInInd.pred_const_ X xs h1 | exact isFreeInInd.pred_var_ X xs h1 | exact isFreeInInd.eq_ x y h1 | exact isFreeInInd.def_ X xs h1
v : VarName X : DefName xs : List VarName h1 : v ∈ xs ⊢ isFreeInInd v (def_ X xs)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
exact isFreeInInd.pred_const_ X xs h1
v : VarName X : PredName xs : List VarName h1 : v ∈ xs ⊢ isFreeInInd v (pred_const_ X xs)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
exact isFreeInInd.pred_var_ X xs h1
v : VarName X : PredName xs : List VarName h1 : v ∈ xs ⊢ isFreeInInd v (pred_var_ X xs)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
exact isFreeInInd.eq_ x y h1
v x y : VarName h1 : v = x ∨ v = y ⊢ isFreeInInd v (eq_ x y)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
exact isFreeInInd.def_ X xs h1
v : VarName X : DefName xs : List VarName h1 : v ∈ xs ⊢ isFreeInInd v (def_ X xs)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
apply isFreeInInd.not_
v : VarName phi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi h1 : isFreeIn v phi ⊢ isFreeInInd v phi.not_
case a v : VarName phi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi h1 : isFreeIn v phi ⊢ isFreeInInd v phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
exact phi_ih h1
case a v : VarName phi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi h1 : isFreeIn v phi ⊢ isFreeInInd v phi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
cases h1
v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi h1 : isFreeIn v phi ∨ isFreeIn v psi ⊢ isFreeInInd v (phi.iff_ psi)
case inl v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi h✝ : isFreeIn v phi ⊢ isFreeInInd v (phi.iff_ psi) case inr v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi h✝ : isFreeIn v psi ⊢ isFreeInInd v (phi.iff_ psi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
case inl c1 => first | apply isFreeInInd.imp_left_ | apply isFreeInInd.and_left_ | apply isFreeInInd.or_left_ | apply isFreeInInd.iff_left_ exact phi_ih c1
v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v phi ⊢ isFreeInInd v (phi.iff_ psi)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
case inr c1 => first | apply isFreeInInd.imp_right_ | apply isFreeInInd.and_right_ | apply isFreeInInd.or_right_ | apply isFreeInInd.iff_right_ exact psi_ih c1
v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v psi ⊢ isFreeInInd v (phi.iff_ psi)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
first | apply isFreeInInd.imp_left_ | apply isFreeInInd.and_left_ | apply isFreeInInd.or_left_ | apply isFreeInInd.iff_left_
v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v phi ⊢ isFreeInInd v (phi.iff_ psi)
case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v phi ⊢ isFreeInInd v phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
exact phi_ih c1
case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v phi ⊢ isFreeInInd v phi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
apply isFreeInInd.imp_left_
v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v phi ⊢ isFreeInInd v (phi.imp_ psi)
case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v phi ⊢ isFreeInInd v phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
apply isFreeInInd.and_left_
v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v phi ⊢ isFreeInInd v (phi.and_ psi)
case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v phi ⊢ isFreeInInd v phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
apply isFreeInInd.or_left_
v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v phi ⊢ isFreeInInd v (phi.or_ psi)
case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v phi ⊢ isFreeInInd v phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
apply isFreeInInd.iff_left_
v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v phi ⊢ isFreeInInd v (phi.iff_ psi)
case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v phi ⊢ isFreeInInd v phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
first | apply isFreeInInd.imp_right_ | apply isFreeInInd.and_right_ | apply isFreeInInd.or_right_ | apply isFreeInInd.iff_right_
v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v psi ⊢ isFreeInInd v (phi.iff_ psi)
case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v psi ⊢ isFreeInInd v psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
exact psi_ih c1
case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v psi ⊢ isFreeInInd v psi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
apply isFreeInInd.imp_right_
v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v psi ⊢ isFreeInInd v (phi.imp_ psi)
case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v psi ⊢ isFreeInInd v psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
apply isFreeInInd.and_right_
v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v psi ⊢ isFreeInInd v (phi.and_ psi)
case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v psi ⊢ isFreeInInd v psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
apply isFreeInInd.or_right_
v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v psi ⊢ isFreeInInd v (phi.or_ psi)
case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v psi ⊢ isFreeInInd v psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
apply isFreeInInd.iff_right_
v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v psi ⊢ isFreeInInd v (phi.iff_ psi)
case a v : VarName phi psi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi psi_ih : isFreeIn v psi → isFreeInInd v psi c1 : isFreeIn v psi ⊢ isFreeInInd v psi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
cases h1
v x : VarName phi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi h1 : ¬v = x ∧ isFreeIn v phi ⊢ isFreeInInd v (exists_ x phi)
case intro v x : VarName phi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi left✝ : ¬v = x right✝ : isFreeIn v phi ⊢ isFreeInInd v (exists_ x phi)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
first | apply isFreeInInd.forall_ | apply isFreeInInd.exists_
v x : VarName phi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi h1_left : ¬v = x h1_right : isFreeIn v phi ⊢ isFreeInInd v (exists_ x phi)
case a v x : VarName phi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi h1_left : ¬v = x h1_right : isFreeIn v phi ⊢ ¬v = x case a v x : VarName phi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi h1_left : ¬v = x h1_right : isFreeIn v phi ⊢ isFreeInInd v phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
apply isFreeInInd.forall_
v x : VarName phi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi h1_left : ¬v = x h1_right : isFreeIn v phi ⊢ isFreeInInd v (forall_ x phi)
case a v x : VarName phi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi h1_left : ¬v = x h1_right : isFreeIn v phi ⊢ ¬v = x case a v x : VarName phi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi h1_left : ¬v = x h1_right : isFreeIn v phi ⊢ isFreeInInd v phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
apply isFreeInInd.exists_
v x : VarName phi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi h1_left : ¬v = x h1_right : isFreeIn v phi ⊢ isFreeInInd v (exists_ x phi)
case a v x : VarName phi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi h1_left : ¬v = x h1_right : isFreeIn v phi ⊢ ¬v = x case a v x : VarName phi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi h1_left : ¬v = x h1_right : isFreeIn v phi ⊢ isFreeInInd v phi
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
exact h1_left
case a v x : VarName phi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi h1_left : ¬v = x h1_right : isFreeIn v phi ⊢ ¬v = x
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_imp_isFreeInInd
[382, 1]
[413, 30]
exact phi_ih h1_right
case a v x : VarName phi : Formula phi_ih : isFreeIn v phi → isFreeInInd v phi h1_left : ¬v = x h1_right : isFreeIn v phi ⊢ isFreeInInd v phi
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeInInd_imp_isFreeIn
[416, 1]
[425, 10]
induction h1
v : VarName F : Formula h1 : isFreeInInd v F ⊢ isFreeIn v F
case pred_const_ v : VarName F : Formula X✝ : PredName xs✝ : List VarName a✝ : v ∈ xs✝ ⊢ isFreeIn v (pred_const_ X✝ xs✝) case pred_var_ v : VarName F : Formula X✝ : PredName xs✝ : List VarName a✝ : v ∈ xs✝ ⊢ isFreeIn v (pred_var_ X✝ xs✝) case eq_ v : VarName F : Formula x✝ y✝ : VarName a✝ : v = x✝ ∨ v = y✝ ⊢ isFreeIn v (eq_ x✝ y✝) case not_ v : VarName F phi✝ : Formula a✝ : isFreeInInd v phi✝ a_ih✝ : isFreeIn v phi✝ ⊢ isFreeIn v phi✝.not_ case imp_left_ v : VarName F phi✝ psi✝ : Formula a✝ : isFreeInInd v phi✝ a_ih✝ : isFreeIn v phi✝ ⊢ isFreeIn v (phi✝.imp_ psi✝) case imp_right_ v : VarName F phi✝ psi✝ : Formula a✝ : isFreeInInd v psi✝ a_ih✝ : isFreeIn v psi✝ ⊢ isFreeIn v (phi✝.imp_ psi✝) case and_left_ v : VarName F phi✝ psi✝ : Formula a✝ : isFreeInInd v phi✝ a_ih✝ : isFreeIn v phi✝ ⊢ isFreeIn v (phi✝.and_ psi✝) case and_right_ v : VarName F phi✝ psi✝ : Formula a✝ : isFreeInInd v psi✝ a_ih✝ : isFreeIn v psi✝ ⊢ isFreeIn v (phi✝.and_ psi✝) case or_left_ v : VarName F phi✝ psi✝ : Formula a✝ : isFreeInInd v phi✝ a_ih✝ : isFreeIn v phi✝ ⊢ isFreeIn v (phi✝.or_ psi✝) case or_right_ v : VarName F phi✝ psi✝ : Formula a✝ : isFreeInInd v psi✝ a_ih✝ : isFreeIn v psi✝ ⊢ isFreeIn v (phi✝.or_ psi✝) case iff_left_ v : VarName F phi✝ psi✝ : Formula a✝ : isFreeInInd v phi✝ a_ih✝ : isFreeIn v phi✝ ⊢ isFreeIn v (phi✝.iff_ psi✝) case iff_right_ v : VarName F phi✝ psi✝ : Formula a✝ : isFreeInInd v psi✝ a_ih✝ : isFreeIn v psi✝ ⊢ isFreeIn v (phi✝.iff_ psi✝) case forall_ v : VarName F : Formula x✝ : VarName phi✝ : Formula a✝¹ : ¬v = x✝ a✝ : isFreeInInd v phi✝ a_ih✝ : isFreeIn v phi✝ ⊢ isFreeIn v (forall_ x✝ phi✝) case exists_ v : VarName F : Formula x✝ : VarName phi✝ : Formula a✝¹ : ¬v = x✝ a✝ : isFreeInInd v phi✝ a_ih✝ : isFreeIn v phi✝ ⊢ isFreeIn v (exists_ x✝ phi✝) case def_ v : VarName F : Formula X✝ : DefName xs✝ : List VarName a✝ : v ∈ xs✝ ⊢ isFreeIn v (def_ X✝ xs✝)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeInInd_imp_isFreeIn
[416, 1]
[425, 10]
all_goals simp only [isFreeIn] tauto
case pred_const_ v : VarName F : Formula X✝ : PredName xs✝ : List VarName a✝ : v ∈ xs✝ ⊢ isFreeIn v (pred_const_ X✝ xs✝) case pred_var_ v : VarName F : Formula X✝ : PredName xs✝ : List VarName a✝ : v ∈ xs✝ ⊢ isFreeIn v (pred_var_ X✝ xs✝) case eq_ v : VarName F : Formula x✝ y✝ : VarName a✝ : v = x✝ ∨ v = y✝ ⊢ isFreeIn v (eq_ x✝ y✝) case not_ v : VarName F phi✝ : Formula a✝ : isFreeInInd v phi✝ a_ih✝ : isFreeIn v phi✝ ⊢ isFreeIn v phi✝.not_ case imp_left_ v : VarName F phi✝ psi✝ : Formula a✝ : isFreeInInd v phi✝ a_ih✝ : isFreeIn v phi✝ ⊢ isFreeIn v (phi✝.imp_ psi✝) case imp_right_ v : VarName F phi✝ psi✝ : Formula a✝ : isFreeInInd v psi✝ a_ih✝ : isFreeIn v psi✝ ⊢ isFreeIn v (phi✝.imp_ psi✝) case and_left_ v : VarName F phi✝ psi✝ : Formula a✝ : isFreeInInd v phi✝ a_ih✝ : isFreeIn v phi✝ ⊢ isFreeIn v (phi✝.and_ psi✝) case and_right_ v : VarName F phi✝ psi✝ : Formula a✝ : isFreeInInd v psi✝ a_ih✝ : isFreeIn v psi✝ ⊢ isFreeIn v (phi✝.and_ psi✝) case or_left_ v : VarName F phi✝ psi✝ : Formula a✝ : isFreeInInd v phi✝ a_ih✝ : isFreeIn v phi✝ ⊢ isFreeIn v (phi✝.or_ psi✝) case or_right_ v : VarName F phi✝ psi✝ : Formula a✝ : isFreeInInd v psi✝ a_ih✝ : isFreeIn v psi✝ ⊢ isFreeIn v (phi✝.or_ psi✝) case iff_left_ v : VarName F phi✝ psi✝ : Formula a✝ : isFreeInInd v phi✝ a_ih✝ : isFreeIn v phi✝ ⊢ isFreeIn v (phi✝.iff_ psi✝) case iff_right_ v : VarName F phi✝ psi✝ : Formula a✝ : isFreeInInd v psi✝ a_ih✝ : isFreeIn v psi✝ ⊢ isFreeIn v (phi✝.iff_ psi✝) case forall_ v : VarName F : Formula x✝ : VarName phi✝ : Formula a✝¹ : ¬v = x✝ a✝ : isFreeInInd v phi✝ a_ih✝ : isFreeIn v phi✝ ⊢ isFreeIn v (forall_ x✝ phi✝) case exists_ v : VarName F : Formula x✝ : VarName phi✝ : Formula a✝¹ : ¬v = x✝ a✝ : isFreeInInd v phi✝ a_ih✝ : isFreeIn v phi✝ ⊢ isFreeIn v (exists_ x✝ phi✝) case def_ v : VarName F : Formula X✝ : DefName xs✝ : List VarName a✝ : v ∈ xs✝ ⊢ isFreeIn v (def_ X✝ xs✝)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeInInd_imp_isFreeIn
[416, 1]
[425, 10]
simp only [isFreeIn]
case def_ v : VarName F : Formula X✝ : DefName xs✝ : List VarName a✝ : v ∈ xs✝ ⊢ isFreeIn v (def_ X✝ xs✝)
case def_ v : VarName F : Formula X✝ : DefName xs✝ : List VarName a✝ : v ∈ xs✝ ⊢ v ∈ xs✝
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeInInd_imp_isFreeIn
[416, 1]
[425, 10]
tauto
case def_ v : VarName F : Formula X✝ : DefName xs✝ : List VarName a✝ : v ∈ xs✝ ⊢ v ∈ xs✝
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_isFreeInInd
[428, 1]
[435, 39]
constructor
v : VarName F : Formula ⊢ isFreeIn v F ↔ isFreeInInd v F
case mp v : VarName F : Formula ⊢ isFreeIn v F → isFreeInInd v F case mpr v : VarName F : Formula ⊢ isFreeInInd v F → isFreeIn v F
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_isFreeInInd
[428, 1]
[435, 39]
exact isFreeIn_imp_isFreeInInd v F
case mp v : VarName F : Formula ⊢ isFreeIn v F → isFreeInInd v F
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.isFreeIn_iff_isFreeInInd
[428, 1]
[435, 39]
exact isFreeInInd_imp_isFreeIn v F
case mpr v : VarName F : Formula ⊢ isFreeInInd v F → isFreeIn v F
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Binders.lean
FOL.NV.predVarOccursIn_iff_mem_predVarSet
[438, 1]
[464, 10]
induction F
P : PredName n : ℕ F : Formula ⊢ predVarOccursIn P n F ↔ (P, n) ∈ F.predVarSet
case pred_const_ P : PredName n : ℕ a✝¹ : PredName a✝ : List VarName ⊢ predVarOccursIn P n (pred_const_ a✝¹ a✝) ↔ (P, n) ∈ (pred_const_ a✝¹ a✝).predVarSet case pred_var_ P : PredName n : ℕ a✝¹ : PredName a✝ : List VarName ⊢ predVarOccursIn P n (pred_var_ a✝¹ a✝) ↔ (P, n) ∈ (pred_var_ a✝¹ a✝).predVarSet case eq_ P : PredName n : ℕ a✝¹ a✝ : VarName ⊢ predVarOccursIn P n (eq_ a✝¹ a✝) ↔ (P, n) ∈ (eq_ a✝¹ a✝).predVarSet case true_ P : PredName n : ℕ ⊢ predVarOccursIn P n true_ ↔ (P, n) ∈ true_.predVarSet case false_ P : PredName n : ℕ ⊢ predVarOccursIn P n false_ ↔ (P, n) ∈ false_.predVarSet case not_ P : PredName n : ℕ a✝ : Formula a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet ⊢ predVarOccursIn P n a✝.not_ ↔ (P, n) ∈ a✝.not_.predVarSet case imp_ P : PredName n : ℕ a✝¹ a✝ : Formula a_ih✝¹ : predVarOccursIn P n a✝¹ ↔ (P, n) ∈ a✝¹.predVarSet a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet ⊢ predVarOccursIn P n (a✝¹.imp_ a✝) ↔ (P, n) ∈ (a✝¹.imp_ a✝).predVarSet case and_ P : PredName n : ℕ a✝¹ a✝ : Formula a_ih✝¹ : predVarOccursIn P n a✝¹ ↔ (P, n) ∈ a✝¹.predVarSet a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet ⊢ predVarOccursIn P n (a✝¹.and_ a✝) ↔ (P, n) ∈ (a✝¹.and_ a✝).predVarSet case or_ P : PredName n : ℕ a✝¹ a✝ : Formula a_ih✝¹ : predVarOccursIn P n a✝¹ ↔ (P, n) ∈ a✝¹.predVarSet a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet ⊢ predVarOccursIn P n (a✝¹.or_ a✝) ↔ (P, n) ∈ (a✝¹.or_ a✝).predVarSet case iff_ P : PredName n : ℕ a✝¹ a✝ : Formula a_ih✝¹ : predVarOccursIn P n a✝¹ ↔ (P, n) ∈ a✝¹.predVarSet a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet ⊢ predVarOccursIn P n (a✝¹.iff_ a✝) ↔ (P, n) ∈ (a✝¹.iff_ a✝).predVarSet case forall_ P : PredName n : ℕ a✝¹ : VarName a✝ : Formula a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet ⊢ predVarOccursIn P n (forall_ a✝¹ a✝) ↔ (P, n) ∈ (forall_ a✝¹ a✝).predVarSet case exists_ P : PredName n : ℕ a✝¹ : VarName a✝ : Formula a_ih✝ : predVarOccursIn P n a✝ ↔ (P, n) ∈ a✝.predVarSet ⊢ predVarOccursIn P n (exists_ a✝¹ a✝) ↔ (P, n) ∈ (exists_ a✝¹ a✝).predVarSet case def_ P : PredName n : ℕ a✝¹ : DefName a✝ : List VarName ⊢ predVarOccursIn P n (def_ a✝¹ a✝) ↔ (P, n) ∈ (def_ a✝¹ a✝).predVarSet