diff --git a/template-rocq/theories/AstUtils.v b/template-rocq/theories/AstUtils.v
index 51d9608a5..91444d75a 100644
--- a/template-rocq/theories/AstUtils.v
+++ b/template-rocq/theories/AstUtils.v
@@ -713,3 +713,156 @@ Definition fold_term {Acc} (f : Acc -> term -> Acc) (acc : Acc) (t : term) : Acc
 (** Monadic variant of [fold_term]. *)
 Definition fold_termM {M} `{Monad M} {Acc} (f : Acc -> term -> M Acc) (acc : Acc) (t : term) : M Acc :=
   @fold_term_with_bindersM M _ Acc unit tt (fun _ _ => ret tt) (fun _ => f) acc t.
+
+(** * Traversal functions with a context*)
+
+
+
+  Definition fix_decls (l : mfixpoint term) :=
+    let fix aux acc ds :=
+        match ds with
+        | nil => acc
+        | d :: ds => aux (vass d.(dname) d.(dtype) :: acc) ds
+        end
+    in aux [] l.
+
+Section Lookups.
+  Context (Σ : global_env).
+
+  Definition polymorphic_constraints u :=
+    match u with
+    | Monomorphic_ctx => ConstraintSet.empty
+    | Polymorphic_ctx ctx => (AUContext.repr ctx).2.2
+    end.
+
+  Definition lookup_constant_type cst u :=
+    match lookup_env Σ cst with
+    | Some (ConstantDecl {| cst_type := ty; cst_universes := uctx |}) =>
+        Some (subst_instance u ty)
+    |  _ => None
+    end.
+
+  Definition lookup_constant_type_cstrs cst u :=
+    match lookup_env Σ cst with
+    | Some (ConstantDecl {| cst_type := ty; cst_universes := uctx |}) =>
+        let cstrs := polymorphic_constraints uctx in
+        Some (subst_instance u ty, subst_instance_cstrs u cstrs)
+      |  _ => None
+    end.
+
+  Definition lookup_ind_decl ind i :=
+    match lookup_env Σ ind with
+    | Some (InductiveDecl mdecl) =>
+      match nth_error mdecl.(ind_bodies) i with
+      | Some body => Some (mdecl, body)
+      | None => None
+      end
+    | _ => None
+    end.
+
+  Definition lookup_ind_type ind i (u : list Level.t) :=
+    match lookup_ind_decl ind i with
+    |None => None
+    |Some res =>
+       Some (subst_instance u (snd res).(ind_type))
+    end.
+
+  Definition lookup_ind_type_cstrs ind i (u : list Level.t) :=
+    match lookup_ind_decl ind i with
+    |None => None
+    |Some res =>
+    let '(mib, body) := res in
+    let uctx := mib.(ind_universes) in
+    let cstrs := polymorphic_constraints uctx in
+    Some (subst_instance u body.(ind_type), subst_instance_cstrs u cstrs)
+    end.
+
+  Definition lookup_constructor_decl ind i k :=
+    match lookup_ind_decl ind i with
+    |None => None
+    |Some res =>
+       let '(mib, body) := res in
+       match nth_error body.(ind_ctors) k with
+       | Some cdecl => Some (mib, cdecl)
+       | None => None
+       end
+    end.
+
+  Definition lookup_constructor_type ind i k u :=
+    match lookup_constructor_decl ind i k with
+    |None => None
+    |Some res =>
+    let '(mib, cdecl) := res in
+    Some (subst0 (inds ind u mib.(ind_bodies)) (subst_instance u cdecl.(cstr_type)))
+    end.
+
+  Definition lookup_constructor_type_cstrs ind i k u :=
+    match lookup_constructor_decl ind i k with
+    |None => None
+    |Some res =>
+    let '(mib, cdecl) := res in
+    let cstrs := polymorphic_constraints mib.(ind_universes) in
+    Some (subst0 (inds ind u mib.(ind_bodies)) (subst_instance u cdecl.(cstr_type)),
+        subst_instance_cstrs u cstrs)
+    end.
+End Lookups.
+
+Definition rebuild_case_predicate_ctx_with_context (Σ : global_env) ind (p : predicate term) : context :=
+  match lookup_ind_decl Σ (inductive_mind ind) (inductive_ind ind) with
+  | None => []
+  | Some (mib, oib) => case_predicate_context ind mib oib p
+  end.
+
+Definition map_context_with_context (f : context -> term -> term) (c : context) Γ : context :=
+fold_right (fun (decl : context_decl) (acc : list context_decl) => map_decl (f (Γ,,, acc)) decl :: acc) c [].
+
+
+Definition map_predicate_with_context (Σ : global_env) (f : context -> term -> term) Γ ind (p : predicate term) :=
+  let pctx := rebuild_case_predicate_ctx_with_context Σ ind p in
+  let Γparams := map_context_with_context f pctx Γ in
+  {| pparams := map (f Γ) p.(pparams);
+    puinst := p.(puinst);
+    pcontext := p.(pcontext);
+    preturn := f (Γparams  ++ Γ) (preturn p) |}.
+
+Definition rebuild_case_branch_ctx_with_context Σ ind i p br :=
+  match lookup_constructor_decl Σ (inductive_mind ind) (inductive_ind ind) i with
+  | None => []
+  | Some (mib, cdecl) => case_branch_context ind mib cdecl p br
+  end.
+
+Definition map_case_branch_with_context Σ ind i (f : context -> term -> term) Γ p br :=
+  let ctx := rebuild_case_branch_ctx_with_context Σ ind i p br in
+  map_branch (f (Γ ,,, ctx)) br.
+
+Definition map_term_with_context Σ (f : context -> term -> term) Γ (t : term) : term :=
+  match t with
+  | tRel i => t
+  | tEvar ev args => tEvar ev (List.map (f Γ) args)
+  | tLambda na T M =>
+      let T' := f Γ T in
+      tLambda na T' (f (Γ,, vass na T') M)
+  | tApp u v => tApp (f Γ u) (List.map (f Γ) v)
+  | tProd na A B =>
+      let A' := f Γ A in
+      tProd na A' (f (Γ ,, vass na A') B)
+  | tCast c kind t => tCast (f Γ c) kind (f Γ t)
+  | tLetIn na b t c =>
+      let b' := f Γ b in
+      let t' := f Γ t in
+      tLetIn na b' t' (f (Γ ,, vdef na b' t') c)
+  | tCase ci p c brs =>
+      let p' := map_predicate_with_context Σ f Γ ci.(ci_ind) p in
+      let brs' := mapi (fun i x => map_case_branch_with_context Σ ci.(ci_ind) i f Γ p' x) brs in
+      tCase ci p' (f Γ c) brs'
+  | tProj p c => tProj p (f Γ c)
+  | tFix mfix idx =>
+      let Γ' := fix_decls mfix ++ Γ in
+      let mfix' := List.map (map_def (f Γ) (f Γ')) mfix in
+      tFix mfix' idx
+  | tCoFix mfix k =>
+      let Γ' := fix_decls mfix ++ Γ in
+      let mfix' := List.map (map_def (f Γ) (f Γ')) mfix in
+      tCoFix mfix' k
+  | x => x
+  end.
diff --git a/template-rocq/theories/Checker.v b/template-rocq/theories/Checker.v
index db2cfda11..6d4caf0c8 100644
--- a/template-rocq/theories/Checker.v
+++ b/template-rocq/theories/Checker.v
@@ -106,27 +106,23 @@ Section Lookups.
     end.
 
   Definition lookup_constant_type cst u :=
-    match lookup_env Σ cst with
-    | Some (ConstantDecl {| cst_type := ty; cst_universes := uctx |}) =>
-      ret (subst_instance u ty)
+    match lookup_constant_type Σ cst u with
+    | Some res =>
+        ret res
     |  _ => raise (UndeclaredConstant cst)
     end.
 
   Definition lookup_constant_type_cstrs cst u :=
-    match lookup_env Σ cst with
-    | Some (ConstantDecl {| cst_type := ty; cst_universes := uctx |}) =>
-      let cstrs := polymorphic_constraints uctx in
-      ret (subst_instance u ty, subst_instance_cstrs u cstrs)
-      |  _ => raise (UndeclaredConstant cst)
+    match lookup_constant_type_cstrs Σ cst u with
+    | Some res =>
+        ret res
+    |  _ => raise (UndeclaredConstant cst)
     end.
 
   Definition lookup_ind_decl ind i :=
-    match lookup_env Σ ind with
-    | Some (InductiveDecl mdecl) =>
-      match nth_error mdecl.(ind_bodies) i with
-      | Some body => ret (mdecl, body)
-      | None => raise (UndeclaredInductive (mkInd ind i))
-      end
+    match lookup_ind_decl Σ ind i with
+    | Some res =>
+        ret res
     | _ => raise (UndeclaredInductive (mkInd ind i))
     end.
 
@@ -262,70 +258,6 @@ Section Reduce.
     res <- reduce_stack Γ n c [] ;;
     ret (zip res).
 
-  Definition fix_decls (l : mfixpoint term) :=
-    let fix aux acc ds :=
-        match ds with
-        | nil => acc
-        | d :: ds => aux (vass d.(dname) d.(dtype) :: acc) ds
-        end
-    in aux [] l.
-
-  Definition rebuild_case_predicate_ctx ind (p : predicate term) : context :=
-    match lookup_ind_decl Σ (inductive_mind ind) (inductive_ind ind) with
-    | TypeError _ => []
-    | Checked (mib, oib) => case_predicate_context ind mib oib p
-    end.
-
-  Definition map_context_with_binders (f : context -> term -> term) (c : context) Γ : context :=
-    fold_left (fun acc decl => map_decl (f (Γ ,,, acc)) decl :: acc) (List.rev c) [].
-
-  Definition map_predicate_with_binders (f : context -> term -> term) Γ ind (p : predicate term) :=
-    let pctx := rebuild_case_predicate_ctx ind p in
-    let Γparams := map_context_with_binders f pctx Γ in
-    {| pparams := map (f Γ) p.(pparams);
-       puinst := p.(puinst);
-       pcontext := p.(pcontext);
-       preturn := f Γparams (preturn p) |}.
-
-  Definition rebuild_case_branch_ctx ind i p br :=
-    match lookup_constructor_decl Σ (inductive_mind ind) (inductive_ind ind) i with
-    | TypeError _ => []
-    | Checked (mib, cdecl) => case_branch_context ind mib cdecl p br
-    end.
-
-  Definition map_case_branch_with_binders ind i (f : context -> term -> term) Γ p br :=
-    let ctx := rebuild_case_branch_ctx ind i p br in
-    map_branch (f (Γ ,,, ctx)) br.
-
-  Definition map_constr_with_binders (f : context -> term -> term) Γ (t : term) : term :=
-    match t with
-    | tRel i => t
-    | tEvar ev args => tEvar ev (List.map (f Γ) args)
-    | tLambda na T M => tLambda na (f Γ T) (f Γ M)
-    | tApp u v => tApp (f Γ u) (List.map (f Γ) v)
-    | tProd na A B =>
-      let A' := f Γ A in
-      tProd na A' (f (Γ ,, vass na A') B)
-    | tCast c kind t => tCast (f Γ c) kind (f Γ t)
-    | tLetIn na b t c =>
-      let b' := f Γ b in
-      let t' := f Γ t in
-      tLetIn na b' t' (f (Γ ,, vdef na b' t') c)
-    | tCase ci p c brs =>
-      let p' := map_predicate_with_binders f Γ ci.(ci_ind) p in
-      let brs' := mapi (fun i x => map_case_branch_with_binders ci.(ci_ind) i f Γ p' x) brs in
-      tCase ci p' (f Γ c) brs'
-    | tProj p c => tProj p (f Γ c)
-    | tFix mfix idx =>
-      let Γ' := fix_decls mfix ++ Γ in
-      let mfix' := List.map (map_def (f Γ) (f Γ')) mfix in
-      tFix mfix' idx
-    | tCoFix mfix k =>
-      let Γ' := fix_decls mfix ++ Γ in
-      let mfix' := List.map (map_def (f Γ) (f Γ')) mfix in
-      tCoFix mfix' k
-    | x => x
-    end.
 
   Fixpoint reduce_opt Γ n c :=
     match n with
@@ -333,7 +265,7 @@ Section Reduce.
     | S n =>
       match reduce_stack_term Γ n c with
       | Some c' =>
-        Some (map_constr_with_binders
+        Some (map_term_with_context Σ
                 (fun Γ t => match reduce_opt Γ n t with
                             | Some t => t
                             | None => t end) Γ c')
@@ -774,7 +706,7 @@ Section Typecheck.
       (** TODO check branches *)
       let '(ind, u, args) := indargs in
       if eq_inductive ind ci.(ci_ind) then
-        let pctx := rebuild_case_predicate_ctx Σ ind p in
+        let pctx := rebuild_case_predicate_ctx_with_context Σ ind p in
         let ptm := it_mkLambda_or_LetIn pctx p.(preturn) in
         ret (tApp ptm (List.skipn ci.(ci_npar) args ++ [c]))
       else