include "basic_2/notation/relations/rdropstar_4.ma".
include "basic_2/substitution/drop.ma".
include "basic_2/multiple/mr2_minus.ma".
-include "basic_2/multiple/lifts.ma".
+include "basic_2/multiple/lifts_vector.ma".
(* ITERATED LOCAL ENVIRONMENT SLICING ***************************************)
inductive drops (s:bool): list2 nat nat → relation lenv ≝
-| drops_nil : â\88\80L. drops s (â\9f ) L L
+| drops_nil : â\88\80L. drops s (â\97\8a) L L
| drops_cons: ∀L1,L,L2,des,d,e.
- drops s des L1 L â\86\92 â\87©[s, d, e] L ≡ L2 → drops s ({d, e} @ des) L1 L2
+ drops s des L1 L â\86\92 â¬\87[s, d, e] L ≡ L2 → drops s ({d, e} @ des) L1 L2
.
interpretation "iterated slicing (local environment) abstract"
'RDropStar des T1 T2 = (drops true des T1 T2).
*)
+definition l_liftable1: relation2 lenv term → predicate bool ≝
+ λR,s. ∀K,T. R K T → ∀L,d,e. ⬇[s, d, e] L ≡ K →
+ ∀U. ⬆[d, e] T ≡ U → R L U.
+
+definition l_liftables1: relation2 lenv term → predicate bool ≝
+ λR,s. ∀L,K,des. ⬇*[s, des] L ≡ K →
+ ∀T,U. ⬆*[des] T ≡ U → R K T → R L U.
+
+definition l_liftables1_all: relation2 lenv term → predicate bool ≝
+ λR,s. ∀L,K,des. ⬇*[s, des] L ≡ K →
+ ∀Ts,Us. ⬆*[des] Ts ≡ Us →
+ all … (R K) Ts → all … (R L) Us.
+
(* Basic inversion lemmas ***************************************************)
-fact drops_inv_nil_aux: â\88\80L1,L2,s,des. â\87©*[s, des] L1 â\89¡ L2 â\86\92 des = â\9f → L1 = L2.
+fact drops_inv_nil_aux: â\88\80L1,L2,s,des. â¬\87*[s, des] L1 â\89¡ L2 â\86\92 des = â\97\8a → L1 = L2.
#L1 #L2 #s #des * -L1 -L2 -des //
#L1 #L #L2 #d #e #des #_ #_ #H destruct
qed-.
(* Basic_1: was: drop1_gen_pnil *)
-lemma drops_inv_nil: â\88\80L1,L2,s. â\87©*[s, â\9f ] L1 ≡ L2 → L1 = L2.
+lemma drops_inv_nil: â\88\80L1,L2,s. â¬\87*[s, â\97\8a] L1 ≡ L2 → L1 = L2.
/2 width=4 by drops_inv_nil_aux/ qed-.
-fact drops_inv_cons_aux: â\88\80L1,L2,s,des. â\87©*[s, des] L1 ≡ L2 →
+fact drops_inv_cons_aux: â\88\80L1,L2,s,des. â¬\87*[s, des] L1 ≡ L2 →
∀d,e,tl. des = {d, e} @ tl →
- â\88\83â\88\83L. â\87©*[s, tl] L1 â\89¡ L & â\87©[s, d, e] L ≡ L2.
+ â\88\83â\88\83L. â¬\87*[s, tl] L1 â\89¡ L & â¬\87[s, d, e] L ≡ L2.
#L1 #L2 #s #des * -L1 -L2 -des
[ #L #d #e #tl #H destruct
| #L1 #L #L2 #des #d #e #HT1 #HT2 #hd #he #tl #H destruct
qed-.
(* Basic_1: was: drop1_gen_pcons *)
-lemma drops_inv_cons: â\88\80L1,L2,s,d,e,des. â\87©*[s, {d, e} @ des] L1 ≡ L2 →
- â\88\83â\88\83L. â\87©*[s, des] L1 â\89¡ L & â\87©[s, d, e] L ≡ L2.
+lemma drops_inv_cons: â\88\80L1,L2,s,d,e,des. â¬\87*[s, {d, e} @ des] L1 ≡ L2 →
+ â\88\83â\88\83L. â¬\87*[s, des] L1 â\89¡ L & â¬\87[s, d, e] L ≡ L2.
/2 width=3 by drops_inv_cons_aux/ qed-.
lemma drops_inv_skip2: ∀I,s,des,des2,i. des ▭ i ≡ des2 →
- â\88\80L1,K2,V2. â\87©*[s, des2] L1 ≡ K2. ⓑ{I} V2 →
+ â\88\80L1,K2,V2. â¬\87*[s, des2] L1 ≡ K2. ⓑ{I} V2 →
∃∃K1,V1,des1. des + 1 ▭ i + 1 ≡ des1 + 1 &
- â\87©*[s, des1] K1 ≡ K2 &
- â\87§*[des1] V2 ≡ V1 &
+ â¬\87*[s, des1] K1 ≡ K2 &
+ â¬\86*[des1] V2 ≡ V1 &
L1 = K1. ⓑ{I} V1.
#I #s #des #des2 #i #H elim H -des -des2 -i
[ #i #L1 #K2 #V2 #H
(* Basic properties *********************************************************)
(* Basic_1: was: drop1_skip_bind *)
-lemma drops_skip: â\88\80L1,L2,s,des. â\87©*[s, des] L1 â\89¡ L2 â\86\92 â\88\80V1,V2. â\87§*[des] V2 ≡ V1 →
- â\88\80I. â\87©*[s, des + 1] L1.ⓑ{I}V1 ≡ L2.ⓑ{I}V2.
+lemma drops_skip: â\88\80L1,L2,s,des. â¬\87*[s, des] L1 â\89¡ L2 â\86\92 â\88\80V1,V2. â¬\86*[des] V2 ≡ V1 →
+ â\88\80I. â¬\87*[s, des + 1] L1.ⓑ{I}V1 ≡ L2.ⓑ{I}V2.
#L1 #L2 #s #des #H elim H -L1 -L2 -des
[ #L #V1 #V2 #HV12 #I
>(lifts_inv_nil … HV12) -HV12 //
].
qed.
+lemma l1_liftable_liftables: ∀R,s. l_liftable1 R s → l_liftables1 R s.
+#R #s #HR #L #K #des #H elim H -L -K -des
+[ #L #T #U #H #HT <(lifts_inv_nil … H) -H //
+| #L1 #L #L2 #des #d #e #_ #HL2 #IHL #T2 #T1 #H #HLT2
+ elim (lifts_inv_cons … H) -H /3 width=10 by/
+]
+qed.
+
+lemma l1_liftables_liftables_all: ∀R,s. l_liftables1 R s → l_liftables1_all R s.
+#R #s #HR #L #K #des #HLK #Ts #Us #H elim H -Ts -Us normalize //
+#Ts #Us #T #U #HTU #_ #IHTUs * /3 width=7 by conj/
+qed.
+
(* Basic_1: removed theorems 1: drop1_getl_trans *)