simp only [Interpretation.usingPred]
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
    admitsAux P zs H binders F →
      (∀ x ∉ binders, V x = V' x) →
        let E_ref := tail✝;
        Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 :
  if X = P ∧ xs.length = zs.length then
    Var.All.Rec.admits (Function.updateListITE id zs xs) H ∧ ∀ x ∈ binders, ¬(isFreeIn x H ∧ x ∉ zs)
  else True
h2 : ∀ x ∉ binders, V x = V' x
⊢ (Interpretation.usingPred D I fun Q ds =>
          if Q = P ∧ ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E_ref H
          else I.pred_var_ Q ds).pred_var_
      X (List.map V xs) ↔
    Holds D I V E_ref
      (if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
      else pred_var_ X xs)
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
    admitsAux P zs H binders F →
      (∀ x ∉ binders, V x = V' x) →
        let E_ref := tail✝;
        Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 :
  if X = P ∧ xs.length = zs.length then
    Var.All.Rec.admits (Function.updateListITE id zs xs) H ∧ ∀ x ∈ binders, ¬(isFreeIn x H ∧ x ∉ zs)
  else True
h2 : ∀ x ∉ binders, V x = V' x
⊢ (if X = P ∧ (List.map V xs).length = zs.length then Holds D I (Function.updateListITE V' zs (List.map V xs)) E_ref H
    else I.pred_var_ X (List.map V xs)) ↔
    Holds D I V E_ref
      (if X = P ∧ xs.length = zs.length then Var.All.Rec.fastReplaceFree (Function.updateListITE id zs xs) H
      else pred_var_ X xs)