theorem drop_mono: ∀d,e,L,L1. ↓[d, e] L ≡ L1 →
∀L2. ↓[d, e] L ≡ L2 → L1 = L2.
#d #e #L #L1 #H elim H -H d e L L1
[ #d #e #L2 #H
theorem drop_mono: ∀d,e,L,L1. ↓[d, e] L ≡ L1 →
∀L2. ↓[d, e] L ≡ L2 → L1 = L2.
#d #e #L #L1 #H elim H -H d e L L1
[ #d #e #L2 #H
<(drop_inv_refl … HL12) -HL12 L2 //
| #L #K #I #V #e #_ #IHLK #L2 #H
lapply (drop_inv_drop1 … H ?) -H /2/
<(drop_inv_refl … HL12) -HL12 L2 //
| #L #K #I #V #e #_ #IHLK #L2 #H
lapply (drop_inv_drop1 … H ?) -H /2/
theorem drop_conf_ge: ∀d1,e1,L,L1. ↓[d1, e1] L ≡ L1 →
∀e2,L2. ↓[0, e2] L ≡ L2 → d1 + e1 ≤ e2 →
↓[0, e2 - e1] L1 ≡ L2.
#d1 #e1 #L #L1 #H elim H -H d1 e1 L L1
[ #d #e #e2 #L2 #H
theorem drop_conf_ge: ∀d1,e1,L,L1. ↓[d1, e1] L ≡ L1 →
∀e2,L2. ↓[0, e2] L ≡ L2 → d1 + e1 ≤ e2 →
↓[0, e2 - e1] L1 ≡ L2.
#d1 #e1 #L #L1 #H elim H -H d1 e1 L L1
[ #d #e #e2 #L2 #H
theorem drop_conf_lt: ∀d1,e1,L,L1. ↓[d1, e1] L ≡ L1 →
∀e2,K2,I,V2. ↓[0, e2] L ≡ K2. 𝕓{I} V2 →
e2 < d1 → let d ≝ d1 - e2 - 1 in
theorem drop_conf_lt: ∀d1,e1,L,L1. ↓[d1, e1] L ≡ L1 →
∀e2,K2,I,V2. ↓[0, e2] L ≡ K2. 𝕓{I} V2 →
e2 < d1 → let d ≝ d1 - e2 - 1 in
↓[d, e1] K2 ≡ K1 & ↑[d, e1] V1 ≡ V2.
#d1 #e1 #L #L1 #H elim H -H d1 e1 L L1
[ #d #e #e2 #K2 #I #V2 #H
↓[d, e1] K2 ≡ K1 & ↑[d, e1] V1 ≡ V2.
#d1 #e1 #L #L1 #H elim H -H d1 e1 L L1
[ #d #e #e2 #K2 #I #V2 #H
theorem drop_trans_le: ∀d1,e1,L1,L. ↓[d1, e1] L1 ≡ L →
∀e2,L2. ↓[0, e2] L ≡ L2 → e2 ≤ d1 →
∃∃L0. ↓[0, e2] L1 ≡ L0 & ↓[d1 - e2, e1] L0 ≡ L2.
#d1 #e1 #L1 #L #H elim H -H d1 e1 L1 L
[ #d #e #e2 #L2 #H
theorem drop_trans_le: ∀d1,e1,L1,L. ↓[d1, e1] L1 ≡ L →
∀e2,L2. ↓[0, e2] L ≡ L2 → e2 ≤ d1 →
∃∃L0. ↓[0, e2] L1 ≡ L0 & ↓[d1 - e2, e1] L0 ≡ L2.
#d1 #e1 #L1 #L #H elim H -H d1 e1 L1 L
[ #d #e #e2 #L2 #H
- >(drop_inv_sort1 … H) -H L2 /2/
-| #K1 #K2 #I #V #HK12 #_ #e2 #L2 #HL2 #H
- >(drop_inv_refl … HK12) -HK12 K1;
+ >(drop_inv_atom1 … H) -H L2 /2/
+| #K #I #V #e2 #L2 #HL2 #H
lapply (le_O_to_eq_O … H) -H #H destruct -e2 /2/
| #L1 #L2 #I #V #e #_ #IHL12 #e2 #L #HL2 #H
lapply (le_O_to_eq_O … H) -H #H destruct -e2;
lapply (le_O_to_eq_O … H) -H #H destruct -e2 /2/
| #L1 #L2 #I #V #e #_ #IHL12 #e2 #L #HL2 #H
lapply (le_O_to_eq_O … H) -H #H destruct -e2;
theorem drop_trans_ge: ∀d1,e1,L1,L. ↓[d1, e1] L1 ≡ L →
∀e2,L2. ↓[0, e2] L ≡ L2 → d1 ≤ e2 → ↓[0, e1 + e2] L1 ≡ L2.
#d1 #e1 #L1 #L #H elim H -H d1 e1 L1 L
[ #d #e #e2 #L2 #H
theorem drop_trans_ge: ∀d1,e1,L1,L. ↓[d1, e1] L1 ≡ L →
∀e2,L2. ↓[0, e2] L ≡ L2 → d1 ≤ e2 → ↓[0, e1 + e2] L1 ≡ L2.
#d1 #e1 #L1 #L #H elim H -H d1 e1 L1 L
[ #d #e #e2 #L2 #H
axiom drop_div: ∀e1,L1,L. ↓[0, e1] L1 ≡ L → ∀e2,L2. ↓[0, e2] L2 ≡ L →
∃∃L0. ↓[0, e1] L0 ≡ L2 & ↓[e1, e2] L0 ≡ L1.
axiom drop_div: ∀e1,L1,L. ↓[0, e1] L1 ≡ L → ∀e2,L2. ↓[0, e2] L2 ≡ L →
∃∃L0. ↓[0, e1] L0 ≡ L2 & ↓[e1, e2] L0 ≡ L1.