diff --git a/lib/inference.ml b/lib/inference.ml
index e7f78d5..416098a 100644
--- a/lib/inference.ml
+++ b/lib/inference.ml
@@ -1,19 +1,32 @@
+(** Infer the type of a given term and, if exists, returns the type of the term *)
 let rec typeof = function
-  | Term.Var _ ->
-    (* Une variable n'a pas de type *)
-    None
+  | Term.Var _ -> None
   | Term.IntConst _ -> Some Type.Int
   | Term.Binop (t1, _, t2) ->
-    (* Les 2 types de l'opération sont égaux *)
     (match typeof t1, typeof t2 with
-     | Some (_ as ty1), Some (_ as ty2) -> if ty1 = ty2 then Some ty1 else None
+     | ty1, ty2 when ty1 = ty2 -> Some Type.Int
      | _ -> None)
   | Term.Pair (t1, t2) ->
-    (* On forme le produit *)
+    (* Pair give Products *)
     (match typeof t1, typeof t2 with
      | Some ty1, Some ty2 -> Some (Type.Product (ty1, ty2))
      | _ -> None)
-  | Term.Proj (_proj, _t) -> failwith "TODO"
-  | Term.Fun (_, _) -> failwith "TODO"
-  | Term.App (_t1, _t2) -> failwith "TODO"
+  | Term.Proj (proj, t) ->
+    (* Projections returns type of product based on the projection type *)
+    (match proj, typeof t with
+     | Term.First, Some (Type.Product (ty, _)) | Term.Second, Some (Type.Product (_, ty))
+       -> Some ty
+     | _, _ -> None)
+  | Term.Fun (id, t) ->
+    (match typeof t with
+     | Some body_type -> Some (Type.Arrow (Type.Int, body_type))
+     | None -> Some (Type.Var id))
+  | Term.App (t1, t2) ->
+    (match typeof t1, typeof t2 with
+     | Some (Type.Arrow (ty_arg, ty_res)), Some ty_arg' ->
+       (* Unification for application *)
+       (match Unification.unify ty_arg ty_arg' with
+        | Some subst -> Some (TypeSubstitution.apply subst ty_res)
+        | None -> None)
+     | _ -> None)
 ;;
diff --git a/lib/typeSubstitution.ml b/lib/typeSubstitution.ml
index 9137223..efc38b4 100644
--- a/lib/typeSubstitution.ml
+++ b/lib/typeSubstitution.ml
@@ -1,4 +1,38 @@
-type t = Type.t Map.Make(Identifier).t
+(** Map of an id to a type *)
+module IdentifierMap = Map.Make (Identifier)
 
-let apply _t _tt = failwith "TODO"
-let compose _s2 _s1 = failwith "TODO"
+(** Map type *)
+type t = Type.t IdentifierMap.t
+
+(* Empty substitution *)
+let empty = IdentifierMap.empty
+
+(** Create a substitution with one element *)
+let singleton id ty = IdentifierMap.singleton id ty
+
+(** Apply substitution to a type *)
+let rec apply subst = function
+  | Type.Var id as t ->
+    (match IdentifierMap.find_opt id subst with
+     | Some ty' -> apply subst ty'
+     | None -> t)
+  | Type.Int -> Type.Int
+  | Type.Product (ty1, ty2) -> Type.Product (apply subst ty1, apply subst ty2)
+  | Type.Arrow (ty1, ty2) -> Type.Arrow (apply subst ty1, apply subst ty2)
+;;
+
+(** Compose two substitutions *)
+let compose s2 s1 =
+  IdentifierMap.merge
+    (fun _ ty1 ty2 ->
+      match ty1, ty2 with
+      (* If we have 2, we pick one of them *)
+      | Some ty1, Some _ -> Some (apply s2 ty1)
+      (* If we have 1, we pick the one we have *)
+      | Some ty1, None -> Some (apply s2 ty1)
+      | None, Some ty2 -> Some (apply s2 ty2)
+      (* If we have 0, we return nothing *)
+      | None, None -> None)
+    s1
+    s2
+;;
diff --git a/lib/typeSubstitution.mli b/lib/typeSubstitution.mli
index 92362c7..5b852aa 100644
--- a/lib/typeSubstitution.mli
+++ b/lib/typeSubstitution.mli
@@ -4,3 +4,5 @@ val apply : t -> Type.t -> Type.t
 
 (* compose s2 s1 : first s1, then s2 *)
 val compose : t -> t -> t
+val empty : t
+val singleton : Identifier.t -> Type.t -> Type.t Map.Make(Identifier).t
diff --git a/lib/unification.ml b/lib/unification.ml
index d89600c..4b70f6b 100644
--- a/lib/unification.ml
+++ b/lib/unification.ml
@@ -1 +1,16 @@
-let unify _ty1 _ty2 = failwith "TODO"
+(** Unify 2 types and, if exists, returns the substitution *)
+let rec unify ty1 ty2 =
+  match ty1, ty2 with
+  | Type.Product (p1_ty1, p1_ty2), Type.Product (p2_ty1, p2_ty2)
+  | Type.Arrow (p1_ty1, p1_ty2), Type.Arrow (p2_ty1, p2_ty2) ->
+    (match unify p1_ty1 p2_ty1 with
+     | Some s1 ->
+       (match
+          unify (TypeSubstitution.apply s1 p1_ty2) (TypeSubstitution.apply s1 p2_ty2)
+        with
+        | Some s2 -> Some (TypeSubstitution.compose s2 s1)
+        | None -> None)
+     | None -> None)
+  | ty1, ty2 when ty1 = ty2 -> Some TypeSubstitution.empty
+  | _ -> None
+;;
diff --git a/test/test_projet_pfa_23_24.ml b/test/test_projet_pfa_23_24.ml
index 591711c..c296e03 100644
--- a/test/test_projet_pfa_23_24.ml
+++ b/test/test_projet_pfa_23_24.ml
@@ -3,30 +3,38 @@ open TypeInference
 let tests_typeof =
   let x = Identifier.fresh () in
   let y = Identifier.fresh () in
-  [ (* Int Const *)
+  let z = Identifier.fresh () in
+  [ (* IntConst *)
     "0", Term.IntConst 0, Some Type.Int
-  ; (* Correct function *)
+  ; (* int -> int -> int = <fun> *)
     ( "fun x -> fun y -> x + y"
     , Term.(Fun (x, Fun (y, Binop (Var x, Plus, Var y))))
     , Some Type.(Arrow (Int, Arrow (Int, Int))) )
   ; (* Not typed variable *)
-    "x", Var "x", None
-  ; (* Operation *)
-    "1 + 2", Binop (IntConst 1, Plus, IntConst 2), Some Int
+    "x", Term.(Var "x"), None
+  ; (* Binary operation *)
+    "1 + 2", Term.(Binop (IntConst 1, Plus, IntConst 2)), Some Type.Int
   ; (* Pair *)
-    "(1, 2)", Pair (IntConst 1, IntConst 2), Some (Product (Int, Int))
+    "(1, 2)", Term.(Pair (IntConst 1, IntConst 2)), Some Type.(Product (Int, Int))
   ; (* Projection with first *)
-    "fst (1, 2)", Proj (First, Pair (IntConst 1, IntConst 2)), Some Int
+    "fst (1, 2)", Term.(Proj (First, Pair (IntConst 1, IntConst 2))), Some Type.Int
   ; (* Projection with second *)
-    "snd (1, 2)", Proj (Second, Pair (IntConst 1, IntConst 2)), Some Int
+    "snd (1, 2)", Term.(Proj (Second, Pair (IntConst 1, IntConst 2))), Some Type.Int
   ; (* Apply (int) into (fun : int -> int) *)
     ( "(fun x -> x + 1) 5"
-    , App (Fun (x, Binop (Var x, Plus, IntConst 1)), IntConst 5)
+    , Term.(App (Fun (x, Binop (Var x, Plus, IntConst 1)), IntConst 5))
     , Some Type.Int )
   ; (* Apply product (int * int) into a not compatible function (fun : int -> int) *)
     ( "(fun x -> x + 1) (1, 2)"
-    , App (Fun (x, Binop (Var x, Plus, IntConst 1)), Pair (IntConst 1, IntConst 2))
+    , Term.(App (Fun (x, Binop (Var x, Plus, IntConst 1)), Pair (IntConst 1, IntConst 2)))
     , None )
+  ; (* x -> y -> (x -> y -> z) -> z *)
+    ( "fun x y -> fun z -> z x y"
+    , Term.(Fun (x, Fun (y, Fun (z, App (Var z, App (Var x, Var y))))))
+    , Some
+        Type.(
+          Arrow (Var x, Arrow (Var y, Arrow (Arrow (Var x, Arrow (Var y, Var z)), Var z))))
+    )
   ]
 ;;