induction F
F : Formula
v : VarName
⊢ fastReplaceFree v v F = F
case pred_const_
v : VarName
a✝¹ : PredName
a✝ : List VarName
⊢ fastReplaceFree v v (pred_const_ a✝¹ a✝) = pred_const_ a✝¹ a✝

case pred_var_
v : VarName
a✝¹ : PredName
a✝ : List VarName
⊢ fastReplaceFree v v (pred_var_ a✝¹ a✝) = pred_var_ a✝¹ a✝

case eq_
v a✝¹ a✝ : VarName
⊢ fastReplaceFree v v (eq_ a✝¹ a✝) = eq_ a✝¹ a✝

case true_
v : VarName
⊢ fastReplaceFree v v true_ = true_

case false_
v : VarName
⊢ fastReplaceFree v v false_ = false_

case not_
v : VarName
a✝ : Formula
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v a✝.not_ = a✝.not_

case imp_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v (a✝¹.imp_ a✝) = a✝¹.imp_ a✝

case and_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v (a✝¹.and_ a✝) = a✝¹.and_ a✝

case or_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v (a✝¹.or_ a✝) = a✝¹.or_ a✝

case iff_
v : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v (a✝¹.iff_ a✝) = a✝¹.iff_ a✝

case forall_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v (forall_ a✝¹ a✝) = forall_ a✝¹ a✝

case exists_
v a✝¹ : VarName
a✝ : Formula
a_ih✝ : fastReplaceFree v v a✝ = a✝
⊢ fastReplaceFree v v (exists_ a✝¹ a✝) = exists_ a✝¹ a✝

case def_
v : VarName
a✝¹ : DefName
a✝ : List VarName
⊢ fastReplaceFree v v (def_ a✝¹ a✝) = def_ a✝¹ a✝