simp at a1
case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
τ : PredName → ℕ → Option (List VarName × Formula)
x : VarName
phi : Formula
phi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux τ binders phi →
      (∀ x ∉ binders, V' x = V x) →
        (Holds D
            { nonempty := ⋯, pred_const_ := I.pred_const_,
              pred_var_ := fun X ds =>
                let opt := τ X ds.length;
                if h : opt.isSome = true then
                  let val := opt.get h;
                  let zs := val.1;
                  let H := val.2;
                  if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
                else I.pred_var_ X ds }
            V E phi ↔
          Holds D I V E (replace c τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V' x = V x
d : D
v : VarName
a1 : v ∉ binders ∪ {x}
⊢ V' v = if v = x then d else V v
case h
D : Type
I : Interpretation D
V' : VarAssignment D
E : Env
c : Char
τ : PredName → ℕ → Option (List VarName × Formula)
x : VarName
phi : Formula
phi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux τ binders phi →
      (∀ x ∉ binders, V' x = V x) →
        (Holds D
            { nonempty := ⋯, pred_const_ := I.pred_const_,
              pred_var_ := fun X ds =>
                let opt := τ X ds.length;
                if h : opt.isSome = true then
                  let val := opt.get h;
                  let zs := val.1;
                  let H := val.2;
                  if ds.length = zs.length then Holds D I (Function.updateListITE V' zs ds) E H else I.pred_var_ X ds
                else I.pred_var_ X ds }
            V E phi ↔
          Holds D I V E (replace c τ phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux τ (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V' x = V x
d : D
v : VarName
a1 : v ∉ binders ∧ ¬v = x
⊢ V' v = if v = x then d else V v