.
interpretation "inclusion (rtmap)"
- 'subseteq t1 t2 = (sle t1 t2).
+ 'subseteq f1 f2 = (sle f1 f2).
(* Basic properties *********************************************************)
#x1 #H #Hx1 destruct lapply (injective_push … Hx1) -Hx1 //
qed-.
-lemma sle_inv_nn: ∀g1,g2. g1 ⊆ g2 → ∀f1,f2. ⫯f1 = g1 → ⫯f2 = g2 → f1 ⊆ f2.
+lemma sle_inv_nn: ∀g1,g2. g1 ⊆ g2 → ∀f1,f2. ⫯f1 = g1 → ⫯f2 = g2 → f1 ⊆ f2.
#g1 #g2 #H #f1 #f2 #H1 #H2 elim (sle_inv_nx … H … H1) -g1
#x2 #H #Hx2 destruct lapply (injective_next … Hx2) -Hx2 //
qed-.
/2 width=5 by sle_push, sle_weak/
qed-.
+(* Properties with iteraded tail ********************************************)
+
+lemma sle_tls: ∀f1,f2. f1 ⊆ f2 → ∀i. ⫱*[i] f1 ⊆ ⫱*[i] f2.
+#f1 #f2 #Hf12 #i elim i -i /2 width=5 by sle_tl/
+qed.
+
(* Properties with isid *****************************************************)
corec lemma sle_isid_sn: ∀f1. 𝐈⦃f1⦄ → ∀f2. f1 ⊆ f2.