(* the main properties ******************************************************)
-lemma drop_conf_ge: â\88\80d1,e1,L,L1. â\86\91[d1, e1] L1 â\89¡ L →
- â\88\80e2,L2. â\86\91[0, e2] L2 â\89¡ L → d1 + e1 ≤ e2 →
- â\86\91[0, e2 - e1] L2 â\89¡ L1.
+lemma drop_conf_ge: â\88\80d1,e1,L,L1. â\86\93[d1, e1] L â\89¡ L1 →
+ â\88\80e2,L2. â\86\93[0, e2] L â\89¡ L2 → d1 + e1 ≤ e2 →
+ â\86\93[0, e2 - e1] L1 â\89¡ L2.
#d1 #e1 #L #L1 #H elim H -H d1 e1 L L1
[ //
| #L #K #I #V #e #_ #IHLK #e2 #L2 #H #He2
lapply (transitive_le 1 … Hdee2) // #He2
lapply (drop_inv_drop1 … H ?) -H // -He2 #HL2
lapply (transitive_le (1+e) … Hdee2) // #Hee2
- >(plus_minus_m_m (e2-e) 1 ?) [ @drop_drop >minus_minus_comm /3/ | /2/ ]
+ @drop_drop_lt >minus_minus_comm /3/ (**) (* explicit constructor *)
]
qed.
-axiom drop_conf_lt: ∀d1,e1,L,L1. ↑[d1, e1] L1 ≡ L →
- â\88\80e2,K2,I,V2. â\86\91[0, e2] K2. ð\9d\95\93{I} V2 â\89¡ L →
+lemma drop_conf_lt: ∀d1,e1,L,L1. ↓[d1, e1] L ≡ L1 →
+ â\88\80e2,K2,I,V2. â\86\93[0, e2] L â\89¡ K2. ð\9d\95\93{I} V2 →
e2 < d1 → let d ≝ d1 - e2 - 1 in
- ∃∃K1,V1. ↑[0, e2] K1. 𝕓{I} V1 ≡ L1 &
- ↑[d, e1] K1 ≡ K2 & ↑[d,e1] V1 ≡ V2.
+ ∃∃K1,V1. ↓[0, e2] L1 ≡ K1. 𝕓{I} V1 &
+ ↓[d, e1] K2 ≡ K1 & ↑[d, e1] V1 ≡ V2.
+#d1 #e1 #L #L1 #H elim H -H d1 e1 L L1
+[ #L0 #e2 #K2 #I #V2 #_ #H
+ elim (lt_zero_false … H)
+| #L1 #L2 #I #V #e #_ #_ #e2 #K2 #J #V2 #_ #H
+ elim (lt_zero_false … H)
+| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #IHL12 #e2 #K2 #J #V #H #He2d
+ elim (drop_inv_O1 … H) -H *
+ [ -IHL12 He2d #H1 #H2 destruct -e2 K2 J V /2 width=5/
+ | -HL12 -HV12 #He #HLK
+ elim (IHL12 … HLK ?) -IHL12 HLK [ <minus_minus /3 width=5/ | /2/ ] (**) (* a bit slow *)
+ ]
+]
+qed.
-axiom drop_trans_le: ∀d1,e1,L1. ∀L:lenv. ↑[d1, e1] L ≡ L1 →
- ∀e2,L2. ↑[0, e2] L2 ≡ L → e2 ≤ d1 →
- ∃∃L0. ↑[0, e2] L0 ≡ L1 & ↑[d1 - e2, e1] L2 ≡ L0.
+lemma 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
+[ #L #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;
+ elim (IHL12 … HL2 ?) -IHL12 HL2 // #L0 #H #HL0
+ lapply (drop_inv_refl … H) -H #H destruct -L1 /3 width=5/
+| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #IHL12 #e2 #L #H #He2d
+ elim (drop_inv_O1 … H) -H *
+ [ -He2d IHL12 #H1 #H2 destruct -e2 L /3 width=5/
+ | -HL12 HV12 #He2 #HL2
+ elim (IHL12 … HL2 ?) -IHL12 HL2 L2
+ [ <minus_le_minus_minus_comm /3/ | /2/ ]
+ ]
+]
+qed.
+
+lemma 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
+[ //
+| /3/
+| #L1 #L2 #I #V1 #V2 #d #e #H_ #_ #IHL12 #e2 #L #H #Hde2
+ lapply (lt_to_le_to_lt 0 … Hde2) // #He2
+ lapply (lt_to_le_to_lt … (e + e2) He2 ?) // #Hee2
+ lapply (drop_inv_drop1 … H ?) -H // #HL2
+ @drop_drop_lt // >le_plus_minus // @IHL12 /2/ (**) (* explicit constructor *)
+]
+qed.
-axiom drop_trans_ge: ∀d1,e1,L1,L. ↑[d1, e1] L ≡ L1 →
- ∀e2,L2. ↑[0, e2] L2 ≡ L → d1 ≤ e2 → ↑[0, e1 + e2] L2 ≡ 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.