(* *)
(**************************************************************************)
+include "basic_2/notation/relations/rdropstar_3.ma".
include "basic_2/relocation/ldrop.ma".
include "basic_2/substitution/gr2_minus.ma".
include "basic_2/substitution/lifts.ma".
(* GENERIC LOCAL ENVIRONMENT SLICING ****************************************)
inductive ldrops: list2 nat nat → relation lenv ≝
-| ldrops_nil : ∀L. ldrops ⟠ L L
+| ldrops_nil : ∀L. ldrops (⟠) L L
| ldrops_cons: ∀L1,L,L2,des,d,e.
ldrops des L1 L → ⇩[d,e] L ≡ L2 → ldrops ({d, e} @ des) L1 L2
.
fact ldrops_inv_nil_aux: ∀L1,L2,des. ⇩*[des] L1 ≡ L2 → des = ⟠ → L1 = L2.
#L1 #L2 #des * -L1 -L2 -des //
#L1 #L #L2 #d #e #des #_ #_ #H destruct
-qed.
+qed-.
(* Basic_1: was: drop1_gen_pnil *)
lemma ldrops_inv_nil: ∀L1,L2. ⇩*[⟠] L1 ≡ L2 → L1 = L2.
-/2 width=3/ qed-.
+/2 width=3 by ldrops_inv_nil_aux/ qed-.
fact ldrops_inv_cons_aux: ∀L1,L2,des. ⇩*[des] L1 ≡ L2 →
∀d,e,tl. des = {d, e} @ tl →
#L1 #L2 #des * -L1 -L2 -des
[ #L #d #e #tl #H destruct
| #L1 #L #L2 #des #d #e #HT1 #HT2 #hd #he #tl #H destruct
- /2 width=3/
-qed.
+ /2 width=3 by ex2_intro/
+qed-.
(* Basic_1: was: drop1_gen_pcons *)
lemma ldrops_inv_cons: ∀L1,L2,d,e,des. ⇩*[{d, e} @ des] L1 ≡ L2 →
∃∃L. ⇩*[des] L1 ≡ L & ⇩[d, e] L ≡ L2.
-/2 width=3/ qed-.
+/2 width=3 by ldrops_inv_cons_aux/ qed-.
lemma ldrops_inv_skip2: ∀I,des,i,des2. des ▭ i ≡ des2 →
∀L1,K2,V2. ⇩*[des2] L1 ≡ K2. ⓑ{I} V2 →
L1 = K1. ⓑ{I} V1.
#I #des #i #des2 #H elim H -des -i -des2
[ #i #L1 #K2 #V2 #H
- >(ldrops_inv_nil … H) -L1 /2 width=7/
+ >(ldrops_inv_nil … H) -L1 /2 width=7 by lifts_nil, minuss_nil, ex4_3_intro, ldrops_nil/
| #des #des2 #d #e #i #Hid #_ #IHdes2 #L1 #K2 #V2 #H
elim (ldrops_inv_cons … H) -H #L #HL1 #H
- elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ #K #V >minus_plus #HK2 #HV2 #H destruct
+ elim (ldrop_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
- @(ex4_3_intro … K1 V1 … ) // [3,4: /2 width=7/ | skip ]
- normalize >plus_minus // @minuss_lt // /2 width=1/ (**) (* explicit constructors, /3 width=1/ is a bit slow *)
+ @(ex4_3_intro … K1 V1 … ) // [3,4: /2 width=7 by lifts_cons, ldrops_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/
+ /4 width=7 by minuss_ge, ex4_3_intro, le_S_S/
]
qed-.
[ #L #V1 #V2 #HV12 #I
>(lifts_inv_nil … HV12) -HV12 //
| #L1 #L #L2 #des #d #e #_ #HL2 #IHL #V1 #V2 #H #I
- elim (lifts_inv_cons … H) -H /3 width=5/
+ elim (lifts_inv_cons … H) -H /3 width=5 by ldrop_skip, ldrops_cons/
].
qed.