(* 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 *********************************************************)
[ @(sle_push … H H) | @(sle_next … H H) ] -H //
qed.
-lemma sle_refl_eq: â\88\80f1,f2. f1 â\89\97 f2 → f1 ⊆ f2.
+lemma sle_refl_eq: â\88\80f1,f2. f1 â\89¡ 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)
(* 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-.
-lemma sle_inv_px: 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) â\88¨ â\88\83â\88\83f2. f1 â\8a\86 f2 & ⫯f2 = g2.
+lemma sle_inv_px: 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) â\88¨ â\88\83â\88\83f2. f1 â\8a\86 f2 & â\86\91f2 = 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: 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 & â\86\91f1 = g1) â\88¨ â\88\83â\88\83f1. f1 â\8a\86 f2 & ⫯f1 = g1.
+lemma sle_inv_xn: 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 & ⫯f1 = g1) â\88¨ â\88\83â\88\83f1. f1 â\8a\86 f2 & â\86\91f1 = 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/
(* Properties with iteraded push ********************************************)
-lemma sle_pushs: â\88\80f1,f2. f1 â\8a\86 f2 â\86\92 â\88\80i. â\86\91*[i] f1 â\8a\86 â\86\91*[i] f2.
+lemma sle_pushs: â\88\80f1,f2. f1 â\8a\86 f2 â\86\92 â\88\80i. ⫯*[i] f1 â\8a\86 ⫯*[i] f2.
#f1 #f2 #Hf12 #i elim i -i /2 width=5 by sle_push/
qed.
(* Properties with tail *****************************************************)
-lemma sle_px_tl: â\88\80g1,g2. g1 â\8a\86 g2 â\86\92 â\88\80f1. â\86\91f1 = g1 → f1 ⊆ ⫱g2.
+lemma sle_px_tl: â\88\80g1,g2. g1 â\8a\86 g2 â\86\92 â\88\80f1. ⫯f1 = g1 → f1 ⊆ ⫱g2.
#g1 #g2 #H #f1 #H1 elim (sle_inv_px … H … H1) -H -H1 * //
qed.
-lemma sle_xn_tl: â\88\80g1,g2. g1 â\8a\86 g2 â\86\92 â\88\80f2. ⫯f2 = g2 → ⫱g1 ⊆ f2.
+lemma sle_xn_tl: â\88\80g1,g2. g1 â\8a\86 g2 â\86\92 â\88\80f2. â\86\91f2 = g2 → ⫱g1 ⊆ f2.
#g1 #g2 #H #f2 #H2 elim (sle_inv_xn … H … H2) -H -H2 * //
qed.
(* Inversion lemmas with tail ***********************************************)
-lemma sle_inv_tl_sn: â\88\80f1,f2. ⫱f1 â\8a\86 f2 â\86\92 f1 â\8a\86 ⫯f2.
+lemma sle_inv_tl_sn: â\88\80f1,f2. ⫱f1 â\8a\86 f2 â\86\92 f1 â\8a\86 â\86\91f2.
#f1 elim (pn_split f1) * #g1 #H destruct
/2 width=5 by sle_next, sle_weak/
qed-.
-lemma sle_inv_tl_dx: â\88\80f1,f2. f1 â\8a\86 ⫱f2 â\86\92 â\86\91f1 ⊆ f2.
+lemma sle_inv_tl_dx: â\88\80f1,f2. f1 â\8a\86 ⫱f2 â\86\92 ⫯f1 ⊆ f2.
#f1 #f2 elim (pn_split f2) * #g2 #H destruct
/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: â\88\80f1. ð\9d\90\88â¦\83f1â¦\84 → ∀f2. f1 ⊆ f2.
+corec lemma sle_isid_sn: â\88\80f1. ð\9d\90\88â\9dªf1â\9d« → ∀f2. f1 ⊆ f2.
#f1 * -f1
#f1 #g1 #Hf1 #H1 #f2 cases (pn_split f2) *
/3 width=5 by sle_weak, sle_push/
(* Inversion lemmas with isid ***********************************************)
-corec lemma sle_inv_isid_dx: â\88\80f1,f2. f1 â\8a\86 f2 â\86\92 ð\9d\90\88â¦\83f2â¦\84 â\86\92 ð\9d\90\88â¦\83f1â¦\84.
+corec lemma sle_inv_isid_dx: â\88\80f1,f2. f1 â\8a\86 f2 â\86\92 ð\9d\90\88â\9dªf2â\9d« â\86\92 ð\9d\90\88â\9dªf1â\9d«.
#f1 #f2 * -f1 -f2
#f1 #f2 #g1 #g2 #Hf * * #H
[2,3: elim (isid_inv_next … H) // ]
(* Properties with isdiv ****************************************************)
-corec lemma sle_isdiv_dx: â\88\80f2. ð\9d\9b\80â¦\83f2â¦\84 → ∀f1. f1 ⊆ f2.
+corec lemma sle_isdiv_dx: â\88\80f2. ð\9d\9b\80â\9dªf2â\9d« → ∀f1. f1 ⊆ f2.
#f2 * -f2
#f2 #g2 #Hf2 #H2 #f1 cases (pn_split f1) *
/3 width=5 by sle_weak, sle_next/
(* Inversion lemmas with isdiv **********************************************)
-corec lemma sle_inv_isdiv_sn: â\88\80f1,f2. f1 â\8a\86 f2 â\86\92 ð\9d\9b\80â¦\83f1â¦\84 â\86\92 ð\9d\9b\80â¦\83f2â¦\84.
+corec lemma sle_inv_isdiv_sn: â\88\80f1,f2. f1 â\8a\86 f2 â\86\92 ð\9d\9b\80â\9dªf1â\9d« â\86\92 ð\9d\9b\80â\9dªf2â\9d«.
#f1 #f2 * -f1 -f2
#f1 #f2 #g1 #g2 #Hf * * #H
[1,3: elim (isdiv_inv_push … H) // ]