+lemma sle_inv_px: ∀g1,g2. g1 ⊆ g2 → ∀f1. ⫯f1 = g1 →
+ (∃∃f2. f1 ⊆ f2 & ⫯f2 = g2) ∨ ∃∃f2. f1 ⊆ f2 & ↑f2 = g2.
+#g1 #g2 elim (pn_split g2) * #f2 #H2 #H #f1 #H1
+[ lapply (sle_inv_pp … H … H1 H2) | lapply (sle_inv_pn … H … H1 H2) ] -H -H1
+/3 width=3 by ex2_intro, or_introl, or_intror/
+qed-.
+
+lemma sle_inv_xn: ∀g1,g2. g1 ⊆ g2 → ∀f2. ↑f2 = g2 →
+ (∃∃f1. f1 ⊆ f2 & ⫯f1 = g1) ∨ ∃∃f1. f1 ⊆ f2 & ↑f1 = g1.
+#g1 #g2 elim (pn_split g1) * #f1 #H1 #H #f2 #H2
+[ lapply (sle_inv_pn … H … H1 H2) | lapply (sle_inv_nn … H … H1 H2) ] -H -H2
+/3 width=3 by ex2_intro, or_introl, or_intror/
+qed-.
+
+(* Main properties **********************************************************)
+
+corec theorem sle_trans: Transitive … sle.
+#f1 #f * -f1 -f
+#f1 #f #g1 #g #Hf #H1 #H #g2 #H0
+[ cases (sle_inv_px … H0 … H) * |*: cases (sle_inv_nx … H0 … H) ] -g
+/3 width=5 by sle_push, sle_next, sle_weak/
+qed-.
+
+(* Properties with iteraded push ********************************************)
+
+lemma sle_pushs: ∀f1,f2. f1 ⊆ f2 → ∀i. ⫯*[i] f1 ⊆ ⫯*[i] f2.
+#f1 #f2 #Hf12 #i elim i -i /2 width=5 by sle_push/
+qed.
+
+(* Properties with tail *****************************************************)
+
+lemma sle_px_tl: ∀g1,g2. g1 ⊆ g2 → ∀f1. ⫯f1 = g1 → f1 ⊆ ⫱g2.
+#g1 #g2 #H #f1 #H1 elim (sle_inv_px … H … H1) -H -H1 * //
+qed.
+
+lemma sle_xn_tl: ∀g1,g2. g1 ⊆ g2 → ∀f2. ↑f2 = g2 → ⫱g1 ⊆ f2.
+#g1 #g2 #H #f2 #H2 elim (sle_inv_xn … H … H2) -H -H2 * //
+qed.
+
+lemma sle_tl: ∀f1,f2. f1 ⊆ f2 → ⫱f1 ⊆ ⫱f2.
+#f1 elim (pn_split f1) * #g1 #H1 #f2 #H
+[ lapply (sle_px_tl … H … H1) -H //
+| elim (sle_inv_nx … H … H1) -H //
+]
+qed.
+
+(* Inversion lemmas with tail ***********************************************)
+
+lemma sle_inv_tl_sn: ∀f1,f2. ⫱f1 ⊆ f2 → f1 ⊆ ↑f2.
+#f1 elim (pn_split f1) * #g1 #H destruct
+/2 width=5 by sle_next, sle_weak/
+qed-.
+
+lemma sle_inv_tl_dx: ∀f1,f2. f1 ⊆ ⫱f2 → ⫯f1 ⊆ f2.
+#f1 #f2 elim (pn_split f2) * #g2 #H destruct
+/2 width=5 by sle_push, sle_weak/
+qed-.
+
+(* Properties with iteraded tail ********************************************)