(**************************************************************************)
include "ground_2/relocation/rtmap_isid.ma".
+include "ground_2/relocation/rtmap_isdiv.ma".
(* RELOCATION MAP ***********************************************************)
(* Basic properties *********************************************************)
-corec lemma sle_refl: ∀f. f ⊆ f.
-#f cases (pn_split f) * #g #H
-[ @(sle_push … H H) | @(sle_next … H H) ] -H //
+corec lemma sle_refl: ∀f1,f2. f1 ≗ f2 → f1 ⊆ f2.
+#f1 #f2 * -f1 -f2
+#f1 #f2 #g1 #g2 #H12 #H1 #H2
+[ @(sle_push … H1 H2) | @(sle_next … H1 H2) ] -H1 -H2 /2 width=1 by/
qed.
(* Basic inversion lemmas ***************************************************)
lapply (isid_inv_push … H ??) -H
/3 width=3 by isid_push/
qed-.
+
+(* Properties with isdiv ****************************************************)
+
+corec lemma sle_isdiv_dx: ∀f2. 𝛀⦃f2⦄ → ∀f1. f1 ⊆ f2.
+#f2 * -f2
+#f2 #g2 #Hf2 #H2 #f1 cases (pn_split f1) *
+/3 width=5 by sle_weak, sle_next/
+qed.
+
+(* Inversion lemmas with isdiv **********************************************)
+
+corec lemma sle_inv_isdiv_sn: ∀f1,f2. f1 ⊆ f2 → 𝛀⦃f1⦄ → 𝛀⦃f2⦄.
+#f1 #f2 * -f1 -f2
+#f1 #f2 #g1 #g2 #Hf * * #H
+[1,3: elim (isdiv_inv_push … H) // ]
+lapply (isdiv_inv_next … H ??) -H
+/3 width=3 by isdiv_next/
+qed-.