--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground/relocation/pr_tl_eq.ma".
+include "ground/relocation/pr_id.ma".
+
+(* IDENTITY ELEMENT FOR PARTIAL RELOCATION MAPS *****************************)
+
+(* Constructions with pr_eq *************************************************)
+
+corec lemma pr_id_eq (f): ⫯f ≡ f → 𝐢 ≡ f.
+cases pr_id_unfold #Hf
+cases (pr_eq_inv_push_sn … Hf) [|*: // ] #_ #H
+cases H in Hf; -H #Hf
+@pr_eq_push [3:|*: // ]
+/3 width=5 by pr_eq_inv_push_bi/
+qed.
+
+(* Inversions with pr_eq ****************************************************)
+
+(* Note: this has the same proof of the previous *)
+corec lemma pr_id_inv_eq (f): 𝐢 ≡ f → ⫯f ≡ f.
+cases pr_id_unfold #Hf
+cases (pr_eq_inv_push_sn … Hf) [|*: // ] #_ #H
+cases H in Hf; -H #Hf
+@pr_eq_push [3:|*: // ]
+/3 width=5 by pr_eq_inv_push_bi/
+qed-.