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+Set Universe Polymorphism.
+Notation Type0 := Set.
+
+Definition Type1 := Eval hnf in let gt := (Set : Type@{i}) in Type@{i}.
+
+Notation compose := (fun g f x => g (f x)).
+
+Notation "g 'o' f" := (compose g f) (at level 40, left associativity) : function_scope.
+Open Scope function_scope.
+
+Definition pointwise_paths {A} {P:A->Type} (f g:forall x:A, P x)
+  := forall x:A, f x = g x.
+
+Notation "f == g" := (pointwise_paths f g) (at level 70, no associativity) : type_scope.
+
+Class Contr_internal (A : Type) := BuildContr {
+  center : A ;
+  contr : (forall y : A, center = y)
+}.
+
+Inductive trunc_index : Type :=
+| minus_two : trunc_index
+| trunc_S : trunc_index -> trunc_index.
+
+Notation "n .+1" := (trunc_S n) (at level 2, left associativity, format "n .+1") : trunc_scope.
+Local Open Scope trunc_scope.
+Notation "-2" := minus_two (at level 0) : trunc_scope.
+Notation "-1" := (-2.+1) (at level 0) : trunc_scope.
+
+Fixpoint IsTrunc_internal (n : trunc_index) (A : Type) : Type :=
+  match n with
+    | -2 => Contr_internal A
+    | n'.+1 => forall (x y : A), IsTrunc_internal n' (x = y)
+  end.
+
+Class IsTrunc (n : trunc_index) (A : Type) : Type :=
+  Trunc_is_trunc : IsTrunc_internal n A.
+
+Notation Contr := (IsTrunc -2).
+Notation IsHProp := (IsTrunc -1).
+
+Monomorphic Axiom dummy_funext_type : Type0.
+Monomorphic Class Funext := { dummy_funext_value : dummy_funext_type }.
+
+Inductive Unit : Type1 :=
+    tt : Unit.
+
+Record TruncType (n : trunc_index) := BuildTruncType {
+  trunctype_type : Type ;
+  istrunc_trunctype_type : IsTrunc n trunctype_type
+}.
+
+Arguments BuildTruncType _ _ {_}.
+
+Coercion trunctype_type : TruncType >-> Sortclass.
+
+Notation "n -Type" := (TruncType n) (at level 1) : type_scope.
+Notation hProp := (-1)-Type.
+
+Notation BuildhProp := (BuildTruncType -1).
+
+Private Inductive Trunc (n : trunc_index) (A :Type) : Type :=
+  tr : A -> Trunc n A.
+Arguments tr {n A} a.
+
+Global Instance istrunc_truncation (n : trunc_index) (A : Type@{i})
+: IsTrunc@{j} n (Trunc@{i} n A).
+Admitted.
+
+Definition Trunc_ind {n A}
+  (P : Trunc n A -> Type) {Pt : forall aa, IsTrunc n (P aa)}
+  : (forall a, P (tr a)) -> (forall aa, P aa)
+:= (fun f aa => match aa with tr a => fun _ => f a end Pt).
+Definition merely (A : Type@{i}) : hProp@{i} := BuildhProp (Trunc -1 A).
+Definition cconst_factors_contr `{Funext}  {X Y : Type} (f : X -> Y)
+           (P : Type) `{Pc : X -> Contr P}
+           (g : X -> P) (h : P -> Y) (p : h o g == f)
+: Unit.
+Proof.
+  assert (merely X -> IsHProp P) by admit.
+  refine (let g' := Trunc_ind (fun _ => P) g : merely X -> P in _);
+    [ assumption.. | ].
+  Fail pose (g' := Trunc_ind (fun _ => P) g : merely X -> P).