drop (L1. 𝕓{I} V1) (d + 1) e (L2. 𝕓{I} V2)
.
-interpretation "dropping" 'RLift L2 d e L1 = (drop L1 d e L2).
+interpretation "dropping" 'RDrop L1 d e L2 = (drop L1 d e L2).
-(* the basic inversion lemmas ***********************************************)
+(* Basic properties *********************************************************)
-lemma drop_inv_skip2_aux: ∀d,e,L1,L2. ↑[d, e] L2 ≡ L1 → 0 < d →
+lemma drop_drop_lt: ∀L1,L2,I,V,e.
+ ↓[0, e - 1] L1 ≡ L2 → 0 < e → ↓[0, e] L1. 𝕓{I} V ≡ L2.
+#L1 #L2 #I #V #e #HL12 #He >(plus_minus_m_m e 1) /2/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma drop_inv_refl_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 → e = 0 → L1 = L2.
+#d #e #L1 #L2 #H elim H -H d e L1 L2
+[ //
+| #L1 #L2 #I #V #e #_ #_ #_ #H
+ elim (plus_S_eq_O_false … H)
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #_ #H
+ elim (plus_S_eq_O_false … H)
+]
+qed.
+
+lemma drop_inv_refl: ∀L1,L2. ↓[0, 0] L1 ≡ L2 → L1 = L2.
+/2 width=5/ qed.
+
+lemma drop_inv_O1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 →
+ ∀K,I,V. L1 = K. 𝕓{I} V →
+ (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
+ (0 < e ∧ ↓[d, e - 1] K ≡ L2).
+#d #e #L1 #L2 #H elim H -H d e L1 L2
+[ /3/
+| #L1 #L2 #I #V #e #HL12 #_ #_ #K #J #W #H destruct -L1 I V /3/
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #_ #H elim (plus_S_eq_O_false … H)
+]
+qed.
+
+lemma drop_inv_O1: ∀e,K,I,V,L2. ↓[0, e] K. 𝕓{I} V ≡ L2 →
+ (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
+ (0 < e ∧ ↓[0, e - 1] K ≡ L2).
+/2/ qed.
+
+lemma drop_inv_drop1: ∀e,K,I,V,L2.
+ ↓[0, e] K. 𝕓{I} V ≡ L2 → 0 < e → ↓[0, e - 1] K ≡ L2.
+#e #K #I #V #L2 #H #He
+elim (drop_inv_O1 … H) -H * // #H destruct -e;
+elim (lt_refl_false … He)
+qed.
+
+lemma drop_inv_skip2_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
∀I,K2,V2. L2 = K2. 𝕓{I} V2 →
- â\88\83â\88\83K1,V1. â\86\91[d - 1, e] K2 â\89¡ K1 &
+ â\88\83â\88\83K1,V1. â\86\93[d - 1, e] K1 â\89¡ K2 &
↑[d - 1, e] V2 ≡ V1 &
L1 = K1. 𝕓{I} V1.
#d #e #L1 #L2 #H elim H -H d e L1 L2
-[ #L #H elim (lt_false … H)
-| #L1 #L2 #I #V #e #_ #_ #H elim (lt_false … H)
+[ #L #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V #e #_ #_ #H elim (lt_refl_false … H)
| #L1 #X #Y #V1 #Z #d #e #HL12 #HV12 #_ #_ #I #L2 #V2 #H destruct -X Y Z;
/2 width=5/
]
qed.
-lemma drop_inv_skip2: â\88\80d,e,I,L1,K2,V2. â\86\91[d, e] K2. ð\9d\95\93{I} V2 â\89¡ L1 → 0 < d →
- â\88\83â\88\83K1,V1. â\86\91[d - 1, e] K2 â\89¡ K1 & ↑[d - 1, e] V2 ≡ V1 &
+lemma drop_inv_skip2: â\88\80d,e,I,L1,K2,V2. â\86\93[d, e] L1 â\89¡ K2. ð\9d\95\93{I} V2 → 0 < d →
+ â\88\83â\88\83K1,V1. â\86\93[d - 1, e] K1 â\89¡ K2 & ↑[d - 1, e] V2 ≡ V1 &
L1 = K1. 𝕓{I} V1.
/2/ qed.