update in ground_2 and basic_2

[helm.git] / matita / matita / contribs / lambdadelta / ground_2 / relocation / rtmap_sle.ma

diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_sle.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_sle.ma
index 2fc90abd113e3dbecbac60066707a03fc983a0b0..fb59fcb786e84c360886d3642fdf2843e9ae6660 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_sle.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_sle.ma
@@ -13,6 +13,7 @@
 (**************************************************************************)
 
 include "ground_2/relocation/rtmap_isid.ma".
+include "ground_2/relocation/rtmap_isdiv.ma".
 
 (* RELOCATION MAP ***********************************************************)
 
@@ -23,15 +24,30 @@ coinductive sle: relation rtmap ≝
 .
 
 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 →
@@ -67,7 +83,7 @@ lemma sle_inv_pp: ∀g1,g2. g1 ⊆ g2 → ∀f1,f2. ↑f1 = g1 → ↑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-.
@@ -79,6 +95,13 @@ lemma sle_inv_px: ∀g1,g2. g1 ⊆ g2 → ∀f1. ↑f1 = g1 →
 /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.
@@ -100,6 +123,10 @@ 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 //
@@ -119,6 +146,12 @@ lemma sle_inv_tl_dx: ∀f1,f2. f1 ⊆ ⫱f2 → ↑f1 ⊆ f2.
 /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.
@@ -136,3 +169,21 @@ corec lemma sle_inv_isid_dx: ∀f1,f2. f1 ⊆ f2 → 𝐈⦃f2⦄ → 𝐈⦃f1
 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-.