(* Main properties **********************************************************)
-(* Basic_1: was: ldrop_mono *)
-theorem ldrop_mono: â\88\80d,e,L,L1. â\86\93[d, e] L ≡ L1 →
- â\88\80L2. â\86\93[d, e] L ≡ L2 → L1 = L2.
+(* Basic_1: was: drop_mono *)
+theorem ldrop_mono: â\88\80d,e,L,L1. â\87©[d, e] L ≡ L1 →
+ â\88\80L2. â\87©[d, e] L ≡ L2 → L1 = L2.
#d #e #L #L1 #H elim H -d -e -L -L1
[ #d #e #L2 #H
>(ldrop_inv_atom1 … H) -L2 //
]
qed-.
-(* Basic_1: was: ldrop_conf_ge *)
-theorem ldrop_conf_ge: â\88\80d1,e1,L,L1. â\86\93[d1, e1] L ≡ L1 →
- â\88\80e2,L2. â\86\93[0, e2] L ≡ L2 → d1 + e1 ≤ e2 →
- â\86\93[0, e2 - e1] L1 ≡ L2.
+(* Basic_1: was: drop_conf_ge *)
+theorem ldrop_conf_ge: â\88\80d1,e1,L,L1. â\87©[d1, e1] L ≡ L1 →
+ â\88\80e2,L2. â\87©[0, e2] L ≡ L2 → d1 + e1 ≤ e2 →
+ â\87©[0, e2 - e1] L1 ≡ L2.
#d1 #e1 #L #L1 #H elim H -d1 -e1 -L -L1
[ #d #e #e2 #L2 #H
>(ldrop_inv_atom1 … H) -L2 //
]
qed.
-(* Basic_1: was: ldrop_conf_lt *)
-theorem ldrop_conf_lt: â\88\80d1,e1,L,L1. â\86\93[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 →
+(* Basic_1: was: drop_conf_lt *)
+theorem ldrop_conf_lt: â\88\80d1,e1,L,L1. â\87©[d1, e1] L ≡ L1 →
+ â\88\80e2,K2,I,V2. â\87©[0, e2] L â\89¡ K2. â\93\91{I} V2 →
e2 < d1 → let d ≝ d1 - e2 - 1 in
- â\88\83â\88\83K1,V1. â\86\93[0, e2] L1 â\89¡ K1. ð\9d\95\93{I} V1 &
- â\86\93[d, e1] K2 â\89¡ K1 & â\86\91[d, e1] V1 ≡ V2.
+ â\88\83â\88\83K1,V1. â\87©[0, e2] L1 â\89¡ K1. â\93\91{I} V1 &
+ â\87©[d, e1] K2 â\89¡ K1 & â\87§[d, e1] V1 ≡ V2.
#d1 #e1 #L #L1 #H elim H -d1 -e1 -L -L1
[ #d #e #e2 #K2 #I #V2 #H
lapply (ldrop_inv_atom1 … H) -H #H destruct
]
qed.
-(* Basic_1: was: ldrop_trans_le *)
-theorem ldrop_trans_le: â\88\80d1,e1,L1,L. â\86\93[d1, e1] L1 ≡ L →
- â\88\80e2,L2. â\86\93[0, e2] L ≡ L2 → e2 ≤ d1 →
- â\88\83â\88\83L0. â\86\93[0, e2] L1 â\89¡ L0 & â\86\93[d1 - e2, e1] L0 ≡ L2.
+(* Basic_1: was: drop_trans_le *)
+theorem ldrop_trans_le: â\88\80d1,e1,L1,L. â\87©[d1, e1] L1 ≡ L →
+ â\88\80e2,L2. â\87©[0, e2] L ≡ L2 → e2 ≤ d1 →
+ â\88\83â\88\83L0. â\87©[0, e2] L1 â\89¡ L0 & â\87©[d1 - e2, e1] L0 ≡ L2.
#d1 #e1 #L1 #L #H elim H -d1 -e1 -L1 -L
[ #d #e #e2 #L2 #H
>(ldrop_inv_atom1 … H) -L2 /2 width=3/
]
qed.
-(* Basic_1: was: ldrop_trans_ge *)
-theorem ldrop_trans_ge: â\88\80d1,e1,L1,L. â\86\93[d1, e1] L1 ≡ L →
- â\88\80e2,L2. â\86\93[0, e2] L â\89¡ L2 â\86\92 d1 â\89¤ e2 â\86\92 â\86\93[0, e1 + e2] L1 ≡ L2.
+(* Basic_1: was: drop_trans_ge *)
+theorem ldrop_trans_ge: â\88\80d1,e1,L1,L. â\87©[d1, e1] L1 ≡ L →
+ â\88\80e2,L2. â\87©[0, e2] L â\89¡ L2 â\86\92 d1 â\89¤ e2 â\86\92 â\87©[0, e1 + e2] L1 ≡ L2.
#d1 #e1 #L1 #L #H elim H -d1 -e1 -L1 -L
[ #d #e #e2 #L2 #H
>(ldrop_inv_atom1 … H) -H -L2 //
qed.
theorem ldrop_trans_ge_comm: ∀d1,e1,e2,L1,L2,L.
- â\86\93[d1, e1] L1 â\89¡ L â\86\92 â\86\93[0, e2] L ≡ L2 → d1 ≤ e2 →
- â\86\93[0, e2 + e1] L1 ≡ L2.
+ â\87©[d1, e1] L1 â\89¡ L â\86\92 â\87©[0, e2] L ≡ L2 → d1 ≤ e2 →
+ â\87©[0, e2 + e1] L1 ≡ L2.
#e1 #e1 #e2 >commutative_plus /2 width=5/
qed.
-(* Basic_1: was: ldrop_conf_rev *)
-axiom ldrop_div: â\88\80e1,L1,L. â\86\93[0, e1] L1 â\89¡ L â\86\92 â\88\80e2,L2. â\86\93[0, e2] L2 ≡ L →
- â\88\83â\88\83L0. â\86\93[0, e1] L0 â\89¡ L2 & â\86\93[e1, e2] L0 ≡ L1.
+(* Basic_1: was: drop_conf_rev *)
+axiom ldrop_div: â\88\80e1,L1,L. â\87©[0, e1] L1 â\89¡ L â\86\92 â\88\80e2,L2. â\87©[0, e2] L2 ≡ L →
+ â\88\83â\88\83L0. â\87©[0, e1] L0 â\89¡ L2 & â\87©[e1, e2] L0 ≡ L1.