(**************************************************************************)
include "ground_2/relocation/rtmap_isid.ma".
+include "ground_2/relocation/rtmap_isdiv.ma".
(* RELOCATION MAP ***********************************************************)
.
interpretation "inclusion (rtmap)"
- 'subseteq t1 t2 = (sle t1 t2).
+ 'subseteq f1 f2 = (sle f1 f2).
(* Basic properties *********************************************************)
+axiom sle_eq_repl_back1: ∀f2. eq_repl_back … (λf1. f1 ⊆ f2).
+
+lemma sle_eq_repl_fwd1: ∀f2. eq_repl_fwd … (λf1. f1 ⊆ f2).
+#f2 @eq_repl_sym /2 width=3 by sle_eq_repl_back1/
+qed-.
+
+axiom sle_eq_repl_back2: ∀f1. eq_repl_back … (λf2. f1 ⊆ f2).
+
+lemma sle_eq_repl_fwd2: ∀f1. eq_repl_fwd … (λf2. f1 ⊆ f2).
+#f1 @eq_repl_sym /2 width=3 by sle_eq_repl_back2/
+qed-.
+
corec lemma sle_refl: ∀f. f ⊆ f.
#f cases (pn_split f) * #g #H
[ @(sle_push … H H) | @(sle_next … H H) ] -H //
qed.
+lemma sle_refl_eq: ∀f1,f2. f1 ≗ f2 → f1 ⊆ f2.
+/2 width=3 by sle_eq_repl_back2/ qed.
+
(* Basic inversion lemmas ***************************************************)
lemma sle_inv_xp: ∀g1,g2. g1 ⊆ g2 → ∀f2. ↑f2 = g2 →
#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.
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-.