--- /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_sle.ma".
+(* * this should go first *)
+include "ground/relocation/pr_tl_eq.ma".
+
+(* INCLUSION FOR PARTIAL RELOCATION MAPS ************************************)
+
+(* Constructions with pr_eq *************************************************)
+
+(*** sle_eq_repl_back1 *)
+corec lemma pr_sle_eq_repl_back_sn:
+ ∀f2:pr_map. pr_eq_repl_back … (λf1:pr_map. f1 ⊆ f2).
+#f2 #f1 * -f2 -f1
+#f1 #f2 #g1 #g2 #Hf #H1 #H2 #g #H0
+[ cases (pr_eq_inv_push_sn … H0 … H1) -g1 /3 width=5 by pr_sle_push/
+| cases (pr_eq_inv_next_sn … H0 … H1) -g1 /3 width=5 by pr_sle_next/
+| cases (pr_eq_inv_push_sn … H0 … H1) -g1 /3 width=5 by pr_sle_weak/
+]
+qed-.
+
+(*** sle_eq_repl_fwd1 *)
+lemma pr_sle_eq_repl_fwd_sn:
+ ∀f2. pr_eq_repl_fwd … (λf1. f1 ⊆ f2).
+#f2 @pr_eq_repl_sym /2 width=3 by pr_sle_eq_repl_back_sn/
+qed-.
+
+(*** sle_eq_repl_back2 *)
+corec lemma pr_sle_eq_repl_back_dx:
+ ∀f1. pr_eq_repl_back … (λf2. f1 ⊆ f2).
+#f1 #f2 * -f1 -f2
+#f1 #f2 #g1 #g2 #Hf #H1 #H2 #g #H0
+[ cases (pr_eq_inv_push_sn … H0 … H2) -g2 /3 width=5 by pr_sle_push/
+| cases (pr_eq_inv_next_sn … H0 … H2) -g2 /3 width=5 by pr_sle_next/
+| cases (pr_eq_inv_next_sn … H0 … H2) -g2 /3 width=5 by pr_sle_weak/
+]
+qed-.
+
+(*** sle_eq_repl_fwd2 *)
+lemma pr_sle_eq_repl_fwd_dx:
+ ∀f1. pr_eq_repl_fwd … (λf2. f1 ⊆ f2).
+#f1 @pr_eq_repl_sym /2 width=3 by pr_sle_eq_repl_back_dx/
+qed-.
+
+(*** sle_refl_eq *)
+lemma pr_sle_refl_eq:
+ ∀f1,f2. f1 ≡ f2 → f1 ⊆ f2.
+/2 width=3 by pr_sle_eq_repl_back_dx/ qed.