diff --git a/Qpf/Macro/Comp.lean b/Qpf/Macro/Comp.lean
index 8dcf5fa..35a2a4a 100644
--- a/Qpf/Macro/Comp.lean
+++ b/Qpf/Macro/Comp.lean
@@ -155,134 +155,114 @@ def DVec.toExpr {n : Nat} {αs : Q(Fin2 $n → Type u)} (xs : DVec (fun (i : Fin
 
 structure ElabQpfResult (u : Level) (arity : Nat) where
   F : Q(TypeFun.{u,u} $arity)
-  functor : Q(MvFunctor $F)
-  qpf     : Q(@MvQPF _ $F $functor)
+  functor : Q(MvFunctor $F)         := by exact q(by infer_instance)
+  qpf     : Q(@MvQPF _ $F $functor) := by exact q(by infer_instance)
 deriving Inhabited
 
+def isLiveVar (varIds : Vector FVarId n) (id : FVarId) := varIds.toList.contains id
+
 open PrettyPrinter in
 mutual
-partial def handleLiveFVar (vars : Vector Q(Type u) arity) (target : Q(Type u))  : TermElabM (ElabQpfResult u arity) := do
-  let vars' := vars.toList;
-
-  trace[QPF] f!"target {target} is a free variable"
-  let ind ← match List.indexOf' target vars' with
-  | none      => throwError "Free variable {target} is not one of the qpf arguments"
+partial def handleLiveFVar (vars : Vector FVarId arity) (target : FVarId)  : TermElabM (ElabQpfResult u arity) := do
+  trace[QPF] f!"target {Expr.fvar target} is a free variable"
+  let ind ← match List.indexOf' target vars.toList with
+  | none      => throwError "Free variable {Expr.fvar target} is not one of the qpf arguments"
   | some ind  => pure ind
 
-  let ind : Fin2 arity := cast (by simp [vars']) ind.inv
+  let ind : Fin2 arity := cast (by simp) ind.inv
   let prj := q(@Prj.{u} $arity $ind)
   trace[QPF] "represented by: {prj}"
-  pure ⟨prj, q(Prj.mvfunctor _), q(Prj.mvqpf _)⟩
 
-partial def handleConst (vars : Vector Q(Type u) arity) (target : Q(Type u))  : TermElabM (ElabQpfResult u arity) := do
+  pure { F := prj, functor := q(Prj.mvfunctor _), qpf := q(Prj.mvqpf _) }
+
+partial def handleConst (target : Q(Type u))  : TermElabM (ElabQpfResult u arity) := do
   trace[QPF] "target {target} is a constant"
   let const := q(Const.{u} $arity $target)
   trace[QPF] "represented by: {const}"
-  pure ⟨const, q(Const.MvFunctor), q(Const.mvqpf)⟩
 
-partial def handleApp (vars : Vector Q(Type u) arity) (target : Q(Type u))  : TermElabM (ElabQpfResult u arity) := do
-  let vars' := vars.toList;
+  pure { F := const, functor := q(Const.MvFunctor), qpf := q(Const.mvqpf)}
 
-  let varIds := vars'.map fun expr => expr.fvarId!
-  let isLiveVar : FVarId → Bool
-    := fun fvarId => (List.indexOf' fvarId varIds).isSome
 
-  let ⟨m, F, args⟩ ← (Comp.parseApp isLiveVar target)
+partial def handleApp (vars : Vector FVarId arity) (target : Q(Type u)) : TermElabM (ElabQpfResult u arity) := do
+  let vars' := vars.toList
+
+  let ⟨numArgs, F, args⟩ ← (Comp.parseApp (isLiveVar vars) target)
   trace[QPF] "target {target} is an application of {F} to {args.toList}"
 
   /-
-    Optimization: check if the application is of the form `F α β γ .. = F α β γ ..`.
-    In such cases, we can directly return `F`, rather than generate a composition of projections.
+    Optimization: check if the application is of the form `F α β γ .. = G α β γ ..`.
+    In such cases, we can directly return `G`, rather than generate a composition of projections.
   -/
+
+  -- O(n²), equivalent to a zipping and would be O(n) and much more readable
   let is_trivial :=
     args.length == arity
     && args.toList.enum.all fun ⟨i, arg⟩ =>
-        arg.isFVar && isLiveVar arg.fvarId! && vars'.indexOf arg == i
+        arg.isFVar && isLiveVar vars arg.fvarId! && vars'.indexOf arg.fvarId! == i
+  -- Equivalent (?), O(n), and more readable
+  -- TODO: is this really equivalent
+  /- let is_trivial := (args.toList.mapM Expr.fvarId?).any (· == vars') -/
+
   if is_trivial then
     trace[QPF] "The application is trivial"
-    let mvFunctor ← synthInstanceQ q(MvFunctor $F)
-    let mvQPF ← synthInstanceQ q(MvQPF $F)
-    pure ⟨F, mvFunctor, mvQPF⟩
-  else
-    let G ← args.toList.mapM fun a =>
-      elabQpf vars a none false
-
-    /-
-      HACK: We know that `m`, which was defines as `args.length`, is equal to `G.length`.
-      It's a bit difficult to prove this. Thus, we simply assert it
-    -/
-    if hm : m ≠ G.length then
-      throwError "This shouldn't happen" -- TODO: come up with a better error message
-    else
-      have hm : m = G.length := by simpa using hm
-
-
-
-      let Ffunctor ← synthInstanceQ q(MvFunctor $F)
-      let Fqpf ← synthInstanceQ q(@MvQPF _ $F $Ffunctor)
-
-      let G : Vec _ m := fun i => G.get (hm ▸ i.inv)
-      let GExpr : Q(Vec (TypeFun.{u,u} $arity) $m) :=
-        Vec.toExpr (fun i => (G i).F)
-      let GFunctor : Q(∀ i, MvFunctor.{u,u} ($GExpr i)) :=
-        let αs := q(fun i => MvFunctor.{u,u} ($GExpr i))
-        @DVec.toExpr _ _ αs (fun i => (G i).functor)
-      let GQpf : Q(∀ i, @MvQPF.{u,u} _ _ ($GFunctor i)) :=
-        let αs := q(fun i => @MvQPF.{u,u} _ _ ($GFunctor i))
-        @DVec.toExpr _ _ αs (fun i => (G i).qpf)
-
-      let comp := q(@Comp $m _ $F $GExpr)
-      trace[QPF] "G := {GExpr}"
-      trace[QPF] "comp := {comp}"
-
-      let functor := q(Comp.instMvFunctorComp)
-      let qpf := q(Comp.instMvQPFCompInstMvFunctorCompFin2
-        (fF := $Ffunctor) (q := $Fqpf) (fG := _) (q' := $GQpf)
-      )
-      pure ⟨comp, functor, qpf⟩
+    let functor ← synthInstanceQ q(MvFunctor $F)
+    let qpf ← synthInstanceQ q(MvQPF $F)
 
+    return { F, functor, qpf }
+  else
+    let G ← args.mmap (elabQpf vars · none false)
+
+    let Ffunctor ← synthInstanceQ q(MvFunctor $F)
+    let Fqpf ← synthInstanceQ q(@MvQPF _ $F $Ffunctor)
+
+    let G : Vec _ numArgs := fun i => G.get i.inv
+    let GExpr : Q(Vec (TypeFun.{u,u} $arity) $numArgs) :=
+      Vec.toExpr (fun i => (G i).F)
+    let GFunctor : Q(∀ i, MvFunctor.{u,u} ($GExpr i)) :=
+      let αs := q(fun i => MvFunctor.{u,u} ($GExpr i))
+      @DVec.toExpr _ _ αs (fun i => (G i).functor)
+    let GQpf : Q(∀ i, @MvQPF.{u,u} _ _ ($GFunctor i)) :=
+      let αs := q(fun i => @MvQPF.{u,u} _ _ ($GFunctor i))
+      @DVec.toExpr _ _ αs (fun i => (G i).qpf)
+
+    let comp := q(@Comp $numArgs _ $F $GExpr)
+    trace[QPF] "G := {GExpr}"
+    trace[QPF] "comp := {comp}"
+
+    let functor := q(Comp.instMvFunctorComp)
+    let qpf := q(Comp.instMvQPFCompInstMvFunctorCompFin2
+      (fF := $Ffunctor) (q := $Fqpf) (fG := _) (q' := $GQpf)
+    )
+    return { F := comp, functor, qpf }
 
-partial def handleArrow (vars : Vector Q(Type u) arity) (target : Q(Type u)) (targetStx : Option Term := none) (normalized := false) : TermElabM (ElabQpfResult u arity) := do
-  match target with
-  | Expr.forallE _ e₁ e₂ .. =>
-    let newTarget ← mkAppM ``MvQPF.Arrow.Arrow #[e₁, e₂]
-    elabQpf vars newTarget targetStx normalized
-  | _ => unreachable!
 
+partial def handleArrow (binderType body : Expr) (vars : Vector FVarId arity) (targetStx : Option Term := none) (normalized := false) : TermElabM (ElabQpfResult u arity) := do
+  let newTarget ← mkAppM ``MvQPF.Arrow.Arrow #[binderType, body]
+  elabQpf vars newTarget targetStx normalized
 /--
   Elaborate the body of a qpf
 -/
-partial def elabQpf {arity : Nat} (vars : Vector Q(Type u) arity) (target : Q(Type u)) (targetStx : Option Term := none) (normalized := false) :
+partial def elabQpf {arity : Nat} (vars : Vector FVarId arity) (target : Q(Type u)) (targetStx : Option Term := none) (normalized := false) :
     TermElabM (ElabQpfResult u arity) := do
-  trace[QPF] "elabQPF :: {vars.toList} -> {target}"
-  let vars' := vars.toList;
-
-  let varIds := vars'.map fun expr => expr.fvarId!
-  let isLiveVar : FVarId → Bool
-    := fun fvarId => (List.indexOf' fvarId varIds).isSome
+  trace[QPF] "elabQPF :: {(vars.map Expr.fvar).toList} -> {target}"
+  let isLiveVar := isLiveVar vars
 
-  if target.isFVar && isLiveVar target.fvarId! then
+  if let some target := target.fvarId?.filter isLiveVar then
     handleLiveFVar vars target
   else if !target.hasAnyFVar isLiveVar then
-    handleConst vars target
-  else if target.isApp then
+    handleConst target
+  else if target.isApp then -- Could be pattern-matched here as well
     handleApp vars target
-  else if target.isArrow then
-    handleArrow vars target (targetStx := targetStx) (normalized := normalized)
+  else if let .forallE _ binderType body .. := target then
+    handleArrow binderType body vars (normalized := normalized) (targetStx := targetStx)
+  else if !normalized then
+    let target ← whnfR target
+    elabQpf vars target targetStx true
   else
-    if !normalized then
-      let target ← whnfR target
-      elabQpf vars target targetStx true
-    else
-      let extra :=
-        if target.isForall then
-          "Dependent arrows / forall are not supported"
-        else
-          ""
-      throwError f!"Unexpected target expression :\n {target}\n{extra}\nNote that the expression contains live variables, hence, must be functorial"
+    let extra := if target.isForall then "Dependent arrows / forall are not supported" else ""
+    throwError f!"Unexpected target expression :\n {target}\n{extra}\nNote that the expression contains live variables, hence, must be functorial"
 end
 
-
 structure QpfCompositionBodyView where
   (type?  : Option Syntax := none)
   (target : Term)
@@ -323,6 +303,10 @@ def elabQpfCompositionBody (view: QpfCompositionBodyView) :
             let target_expr ← elabTermEnsuringTypeQ (u:=u.succ.succ) view.target q(Type u)
             let arity := vars.toList.length
             let vars : Vector _ arity := ⟨vars.toList, rfl⟩
+
+            let some vars := vars.mmap Expr.fvarId? |
+              throwError "Expected all args to be fvars" -- TODO: throwErrorAt
+
             let res ← elabQpf vars target_expr view.target
 
             res.F.check