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 ≝
+inductive drops (s:bool): list2 ynat nat → relation lenv ≝
| drops_nil : ∀L. drops s (◊) L L
-| drops_cons: ∀L1,L,L2,des,d,e.
- drops s des L1 L → ⇩[s, d, e] L ≡ L2 → drops s ({d, e} @ des) L1 L2
+| drops_cons: ∀L1,L,L2,cs,l,m.
+ drops s cs L1 L → ⬇[s, l, m] L ≡ L2 → drops s ({l, m} @ cs) L1 L2
.
interpretation "iterated slicing (local environment) abstract"
- 'RDropStar s des T1 T2 = (drops s des T1 T2).
+ 'RDropStar s cs T1 T2 = (drops s cs T1 T2).
(*
interpretation "iterated slicing (local environment) general"
'RDropStar des T1 T2 = (drops true des T1 T2).
*)
+definition d_liftable1: relation2 lenv term → predicate bool ≝
+ λR,s. ∀K,T. R K T → ∀L,l,m. ⬇[s, l, m] L ≡ K →
+ ∀U. ⬆[l, m] T ≡ U → R L U.
+
+definition d_liftables1: relation2 lenv term → predicate bool ≝
+ λR,s. ∀L,K,cs. ⬇*[s, cs] L ≡ K →
+ ∀T,U. ⬆*[cs] T ≡ U → R K T → R L U.
+
+definition d_liftables1_all: relation2 lenv term → predicate bool ≝
+ λR,s. ∀L,K,cs. ⬇*[s, cs] L ≡ K →
+ ∀Ts,Us. ⬆*[cs] Ts ≡ Us →
+ all … (R K) Ts → all … (R L) Us.
+
(* Basic inversion lemmas ***************************************************)
-fact drops_inv_nil_aux: ∀L1,L2,s,des. ⇩*[s, des] L1 ≡ L2 → des = ◊ → L1 = L2.
-#L1 #L2 #s #des * -L1 -L2 -des //
-#L1 #L #L2 #d #e #des #_ #_ #H destruct
+fact drops_inv_nil_aux: ∀L1,L2,s,cs. ⬇*[s, cs] L1 ≡ L2 → cs = ◊ → L1 = L2.
+#L1 #L2 #s #cs * -L1 -L2 -cs //
+#L1 #L #L2 #l #m #cs #_ #_ #H destruct
qed-.
(* Basic_1: was: drop1_gen_pnil *)
-lemma drops_inv_nil: â\88\80L1,L2,s. â\87©*[s, ◊] L1 ≡ L2 → L1 = L2.
+lemma drops_inv_nil: â\88\80L1,L2,s. â¬\87*[s, ◊] L1 ≡ L2 → L1 = L2.
/2 width=4 by drops_inv_nil_aux/ qed-.
-fact drops_inv_cons_aux: ∀L1,L2,s,des. ⇩*[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.
-#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
+fact drops_inv_cons_aux: ∀L1,L2,s,cs. ⬇*[s, cs] L1 ≡ L2 →
+ ∀l,m,tl. cs = {l, m} @ tl →
+ â\88\83â\88\83L. â¬\87*[s, tl] L1 â\89¡ L & â¬\87[s, l, m] L ≡ L2.
+#L1 #L2 #s #cs * -L1 -L2 -cs
+[ #L #l #m #tl #H destruct
+| #L1 #L #L2 #cs #l #m #HT1 #HT2 #l0 #m0 #tl #H destruct
/2 width=3 by ex2_intro/
]
qed-.
(* Basic_1: was: drop1_gen_pcons *)
-lemma drops_inv_cons: ∀L1,L2,s,d,e,des. ⇩*[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: ∀L1,L2,s,l,m,cs. ⬇*[s, {l, m} @ cs] L1 ≡ L2 →
+ â\88\83â\88\83L. â¬\87*[s, cs] L1 â\89¡ L & â¬\87[s, l, m] 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 →
- ∃∃K1,V1,des1. des + 1 ▭ i + 1 ≡ des1 + 1 &
- â\87©*[s, des1] K1 ≡ K2 &
- â\87§*[des1] V2 ≡ V1 &
+lemma drops_inv_skip2: ∀I,s,cs,cs2,i. cs ▭ i ≡ cs2 →
+ â\88\80L1,K2,V2. â¬\87*[s, cs2] L1 ≡ K2. ⓑ{I} V2 →
+ ∃∃K1,V1,cs1. cs + 1 ▭ i + 1 ≡ cs1 + 1 &
+ â¬\87*[s, cs1] K1 ≡ K2 &
+ â¬\86*[cs1] V2 ≡ V1 &
L1 = K1. ⓑ{I} V1.
-#I #s #des #des2 #i #H elim H -des -des2 -i
+#I #s #cs #cs2 #i #H elim H -cs -cs2 -i
[ #i #L1 #K2 #V2 #H
>(drops_inv_nil … H) -L1 /2 width=7 by lifts_nil, minuss_nil, ex4_3_intro, drops_nil/
-| #des #des2 #d #e #i #Hid #_ #IHdes2 #L1 #K2 #V2 #H
+| #cs #cs2 #l #m #i #Hil #_ #IHcs2 #L1 #K2 #V2 #H
elim (drops_inv_cons … H) -H #L #HL1 #H
- elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ #K #V >minus_plus #HK2 #HV2 #H destruct
- elim (IHdes2 … HL1) -IHdes2 -HL1 #K1 #V1 #des1 #Hdes1 #HK1 #HV1 #X destruct
+ elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ #K #V <yminus_succ2 #HK2 #HV2 #H destruct
+ elim (IHcs2 … HL1) -IHcs2 -HL1 #K1 #V1 #cs1 #Hcs1 #HK1 #HV1 #X destruct
@(ex4_3_intro … K1 V1 … ) // [3,4: /2 width=7 by lifts_cons, drops_cons/ | skip ]
- normalize >plus_minus /3 width=1 by minuss_lt, lt_minus_to_plus/ (**) (* explicit constructors *)
-| #des #des2 #d #e #i #Hid #_ #IHdes2 #L1 #K2 #V2 #H
- elim (IHdes2 … H) -IHdes2 -H #K1 #V1 #des1 #Hdes1 #HK1 #HV1 #X destruct
- /4 width=7 by minuss_ge, ex4_3_intro, le_S_S/
+ >pluss_SO2 >pluss_SO2
+ >yminus_succ2 >ylt_inv_O1 /2 width=1 by ylt_to_minus/ <yminus_succ >commutative_plus (**) (* <yminus_succ1_inj does not work *)
+ /3 width=1 by minuss_lt, ylt_succ/
+| #cs #cs2 #l #m #i #Hil #_ #IHcs2 #L1 #K2 #V2 #H
+ elim (IHcs2 … H) -IHcs2 -H #K1 #V1 #cs1 #Hcs1 #HK1 #HV1 #X destruct
+ /4 width=7 by minuss_ge, yle_succ, ex4_3_intro/
]
qed-.
(* Basic properties *********************************************************)
(* Basic_1: was: drop1_skip_bind *)
-lemma drops_skip: ∀L1,L2,s,des. ⇩*[s, des] L1 ≡ L2 → ∀V1,V2. ⇧*[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
+lemma drops_skip: ∀L1,L2,s,cs. ⬇*[s, cs] L1 ≡ L2 → ∀V1,V2. ⬆*[cs] V2 ≡ V1 →
+ â\88\80I. â¬\87*[s, cs + 1] L1.ⓑ{I}V1 ≡ L2.ⓑ{I}V2.
+#L1 #L2 #s #cs #H elim H -L1 -L2 -cs
[ #L #V1 #V2 #HV12 #I
>(lifts_inv_nil … HV12) -HV12 //
-| #L1 #L #L2 #des #d #e #_ #HL2 #IHL #V1 #V2 #H #I
+| #L1 #L #L2 #cs #l #m #_ #HL2 #IHL #V1 #V2 #H #I
elim (lifts_inv_cons … H) -H /3 width=5 by drop_skip, drops_cons/
].
qed.
+lemma d1_liftable_liftables: ∀R,s. d_liftable1 R s → d_liftables1 R s.
+#R #s #HR #L #K #cs #H elim H -L -K -cs
+[ #L #T #U #H #HT <(lifts_inv_nil … H) -H //
+| #L1 #L #L2 #cs #l #m #_ #HL2 #IHL #T2 #T1 #H #HLT2
+ elim (lifts_inv_cons … H) -H /3 width=10 by/
+]
+qed.
+
+lemma d1_liftables_liftables_all: ∀R,s. d_liftables1 R s → d_liftables1_all R s.
+#R #s #HR #L #K #cs #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 *)
projects / helm.git / blobdiff