update in ground_2, static_2, basic_2, apps_2, alpha_1

[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 b27488644c533fecc5ccd7340ff0410629332a04..c08d574cd83fe438d6a2b90e110803304a905502 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_sle.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_sle.ma
@@ -13,37 +13,60 @@
 (**************************************************************************)
 
 include "ground_2/relocation/rtmap_isid.ma".
+include "ground_2/relocation/rtmap_isdiv.ma".
 
 (* RELOCATION MAP ***********************************************************)
 
 coinductive sle: relation rtmap ≝
-| sle_push: â\88\80f1,f2,g1,g2. sle f1 f2 â\86\92 â\86\91f1 = g1 â\86\92 â\86\91f2 = g2 → sle g1 g2
-| sle_next: â\88\80f1,f2,g1,g2. sle f1 f2 â\86\92 ⫯f1 = g1 â\86\92 ⫯f2 = g2 → sle g1 g2
-| sle_weak: â\88\80f1,f2,g1,g2. sle f1 f2 â\86\92 â\86\91f1 = g1 â\86\92 ⫯f2 = g2 → sle g1 g2
+| sle_push: â\88\80f1,f2,g1,g2. sle f1 f2 â\86\92 ⫯f1 = g1 â\86\92 ⫯f2 = g2 → sle g1 g2
+| sle_next: â\88\80f1,f2,g1,g2. sle f1 f2 â\86\92 â\86\91f1 = g1 â\86\92 â\86\91f2 = g2 → sle g1 g2
+| sle_weak: â\88\80f1,f2,g1,g2. sle f1 f2 â\86\92 ⫯f1 = g1 â\86\92 â\86\91f2 = g2 → sle g1 g2
 .
 
 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: â\88\80g1,g2. g1 â\8a\86 g2 â\86\92 â\88\80f2. â\86\91f2 = g2 →
-                  â\88\83â\88\83f1. f1 â\8a\86 f2 & â\86\91f1 = g1.
+lemma sle_inv_xp: â\88\80g1,g2. g1 â\8a\86 g2 â\86\92 â\88\80f2. ⫯f2 = g2 →
+                  â\88\83â\88\83f1. f1 â\8a\86 f2 & ⫯f1 = g1.
 #g1 #g2 * -g1 -g2
 #f1 #f2 #g1 #g2 #H #H1 #H2 #x2 #Hx2 destruct
 [ lapply (injective_push … Hx2) -Hx2 /2 width=3 by ex2_intro/ ]
 elim (discr_push_next … Hx2)
 qed-.
 
-lemma sle_inv_nx: â\88\80g1,g2. g1 â\8a\86 g2 â\86\92 â\88\80f1. ⫯f1 = g1 →
-                  â\88\83â\88\83f2. f1 â\8a\86 f2 & ⫯f2 = g2.
+lemma sle_inv_nx: â\88\80g1,g2. g1 â\8a\86 g2 â\86\92 â\88\80f1. â\86\91f1 = g1 →
+                  â\88\83â\88\83f2. f1 â\8a\86 f2 & â\86\91f2 = g2.
 #g1 #g2 * -g1 -g2
 #f1 #f2 #g1 #g2 #H #H1 #H2 #x1 #Hx1 destruct
 [2: lapply (injective_next … Hx1) -Hx1 /2 width=3 by ex2_intro/ ]
 elim (discr_next_push … Hx1)
 qed-.
 
-lemma sle_inv_pn: â\88\80g1,g2. g1 â\8a\86 g2 â\86\92 â\88\80f1,f2. â\86\91f1 = g1 â\86\92 ⫯f2 = g2 → f1 ⊆ f2.
+lemma sle_inv_pn: â\88\80g1,g2. g1 â\8a\86 g2 â\86\92 â\88\80f1,f2. ⫯f1 = g1 â\86\92 â\86\91f2 = g2 → f1 ⊆ f2.
 #g1 #g2 * -g1 -g2
 #f1 #f2 #g1 #g2 #H #H1 #H2 #x1 #x2 #Hx1 #Hx2 destruct
 [ elim (discr_next_push … Hx2)
@@ -55,30 +78,112 @@ qed-.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma sle_inv_pp: â\88\80g1,g2. g1 â\8a\86 g2 â\86\92 â\88\80f1,f2. â\86\91f1 = g1 â\86\92 â\86\91f2 = g2 → f1 ⊆ f2.
+lemma sle_inv_pp: â\88\80g1,g2. g1 â\8a\86 g2 â\86\92 â\88\80f1,f2. ⫯f1 = g1 â\86\92 ⫯f2 = g2 → f1 ⊆ f2.
 #g1 #g2 #H #f1 #f2 #H1 #H2 elim (sle_inv_xp … H … H2) -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-.
 
-(* properties on isid *******************************************************)
+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 ********************************************)
 
-let corec sle_isid_sn: ∀f1. 𝐈⦃f1⦄ → ∀f2. f1 ⊆ f2 ≝ ?.
+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.
 #f1 * -f1
 #f1 #g1 #Hf1 #H1 #f2 cases (pn_split f2) *
 /3 width=5 by sle_weak, sle_push/
 qed.
 
-(* inversion lemmas on isid *************************************************)
+(* Inversion lemmas with isid ***********************************************)
 
-let corec sle_inv_isid_dx: ∀f1,f2. f1 ⊆ f2 → 𝐈⦃f2⦄ → 𝐈⦃f1⦄ ≝ ?.
+corec lemma sle_inv_isid_dx: ∀f1,f2. f1 ⊆ f2 → 𝐈❪f2❫ → 𝐈❪f1❫.
 #f1 #f2 * -f1 -f2
 #f1 #f2 #g1 #g2 #Hf * * #H
 [2,3: elim (isid_inv_next … H) // ]
 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-.