induction F
F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ fastReplaceFree t v (fastReplaceFree v t F) = F
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : ¬occursIn t (pred_const_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (pred_const_ a✝¹ a✝)) = pred_const_ a✝¹ a✝

case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
h1 : ¬occursIn t (pred_var_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (pred_var_ a✝¹ a✝)) = pred_var_ a✝¹ a✝

case eq_
v t a✝¹ a✝ : VarName
h1 : ¬occursIn t (eq_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (eq_ a✝¹ a✝)) = eq_ a✝¹ a✝

case true_
v t : VarName
h1 : ¬occursIn t true_
⊢ fastReplaceFree t v (fastReplaceFree v t true_) = true_

case false_
v t : VarName
h1 : ¬occursIn t false_
⊢ fastReplaceFree t v (fastReplaceFree v t false_) = false_

case not_
v t : VarName
a✝ : Formula
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t a✝.not_
⊢ fastReplaceFree t v (fastReplaceFree v t a✝.not_) = a✝.not_

case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (a✝¹.imp_ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (a✝¹.imp_ a✝)) = a✝¹.imp_ a✝

case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (a✝¹.and_ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (a✝¹.and_ a✝)) = a✝¹.and_ a✝

case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (a✝¹.or_ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (a✝¹.or_ a✝)) = a✝¹.or_ a✝

case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ¬occursIn t a✝¹ → fastReplaceFree t v (fastReplaceFree v t a✝¹) = a✝¹
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (a✝¹.iff_ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (a✝¹.iff_ a✝)) = a✝¹.iff_ a✝

case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (forall_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (forall_ a✝¹ a✝)) = forall_ a✝¹ a✝

case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ¬occursIn t a✝ → fastReplaceFree t v (fastReplaceFree v t a✝) = a✝
h1 : ¬occursIn t (exists_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (exists_ a✝¹ a✝)) = exists_ a✝¹ a✝

case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
h1 : ¬occursIn t (def_ a✝¹ a✝)
⊢ fastReplaceFree t v (fastReplaceFree v t (def_ a✝¹ a✝)) = def_ a✝¹ a✝