diff --git a/theories/numbers.v b/theories/numbers.v
index 029154f2f7ab4c48e41e47e5201831eaa0d3920f..80975a32a23d419adb4864276f2cbd16dab7ec70 100644
--- a/theories/numbers.v
+++ b/theories/numbers.v
@@ -347,6 +347,14 @@ Proof.
 Qed.
 Close Scope Z_scope.
 
+(** * Injectivity of casts *)
+Instance N_of_nat_inj: Inj (=) (=) N.of_nat := Nat2N.inj.
+Instance nat_of_N_inj: Inj (=) (=) N.to_nat := N2Nat.inj.
+Instance nat_of_pos_inj: Inj (=) (=) Pos.to_nat := Pos2Nat.inj.
+Instance pos_of_Snat_inj: Inj (=) (=) Pos.of_succ_nat := SuccNat2Pos.inj.
+Instance Z_of_N_inj: Inj (=) (=) Z.of_N := N2Z.inj.
+(* Add others here. *)
+
 (** * Notations and properties of [Qc] *)
 Open Scope Qc_scope.
 Delimit Scope Qc_scope with Qc.
diff --git a/theories/pretty.v b/theories/pretty.v
index 541e72a23363bf660fcb8496874fb7414e8ab614..a1e85a4642318e669e66fbdb09eff3ae80e7723c 100644
--- a/theories/pretty.v
+++ b/theories/pretty.v
@@ -1,7 +1,7 @@
 (* Copyright (c) 2012-2017, Coq-std++ developers. *)
 (* This file is distributed under the terms of the BSD license. *)
 From stdpp Require Export strings.
-From stdpp Require Import relations.
+From stdpp Require Import relations numbers.
 From Coq Require Import Ascii.
 Set Default Proof Using "Type".
 
@@ -70,3 +70,7 @@ Instance pretty_Z : Pretty Z := λ x,
   match x with
   | Z0 => "" | Zpos x => pretty (Npos x) | Zneg x => "-" +:+ pretty (Npos x)
   end%string.
+Instance pretty_nat : Pretty nat := λ x, pretty (N.of_nat x).
+Instance pretty_nat_inj : Inj (@eq nat) (=) pretty.
+Proof. apply _. Qed.
+