(* Main properties **********************************************************)
(* Basic_1: was: ldrop_mono *)
-theorem ldrop_mono: â\88\80d,e,L,L1. â\86\93[d, e] L â\89¡ L1 â\86\92
- â\88\80L2. â\86\93[d, e] L â\89¡ L2 â\86\92 L1 = L2.
+theorem ldrop_mono: â\88\80d,e,L,L1. â\87\93[d, e] L â\89¡ L1 â\86\92
+ â\88\80L2. â\87\93[d, e] L â\89¡ L2 â\86\92 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 â\89¡ L1 â\86\92
- â\88\80e2,L2. â\86\93[0, e2] L â\89¡ L2 â\86\92 d1 + e1 â\89¤ e2 â\86\92
- â\86\93[0, e2 - e1] L1 â\89¡ L2.
+theorem ldrop_conf_ge: â\88\80d1,e1,L,L1. â\87\93[d1, e1] L â\89¡ L1 â\86\92
+ â\88\80e2,L2. â\87\93[0, e2] L â\89¡ L2 â\86\92 d1 + e1 â\89¤ e2 â\86\92
+ â\87\93[0, e2 - e1] L1 â\89¡ 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 â\89¡ L1 â\86\92
- â\88\80e2,K2,I,V2. â\86\93[0, e2] L â\89¡ K2. ð\9d\95\93{I} V2 â\86\92
+theorem ldrop_conf_lt: â\88\80d1,e1,L,L1. â\87\93[d1, e1] L â\89¡ L1 â\86\92
+ â\88\80e2,K2,I,V2. â\87\93[0, e2] L â\89¡ K2. ð\9d\95\93{I} V2 â\86\92
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 â\89¡ V2.
+ â\88\83â\88\83K1,V1. â\87\93[0, e2] L1 â\89¡ K1. ð\9d\95\93{I} V1 &
+ â\87\93[d, e1] K2 â\89¡ K1 & â\87\91[d, e1] V1 â\89¡ 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 â\89¡ L â\86\92
- â\88\80e2,L2. â\86\93[0, e2] L â\89¡ L2 â\86\92 e2 â\89¤ d1 â\86\92
- â\88\83â\88\83L0. â\86\93[0, e2] L1 â\89¡ L0 & â\86\93[d1 - e2, e1] L0 â\89¡ L2.
+theorem ldrop_trans_le: â\88\80d1,e1,L1,L. â\87\93[d1, e1] L1 â\89¡ L â\86\92
+ â\88\80e2,L2. â\87\93[0, e2] L â\89¡ L2 â\86\92 e2 â\89¤ d1 â\86\92
+ â\88\83â\88\83L0. â\87\93[0, e2] L1 â\89¡ L0 & â\87\93[d1 - e2, e1] L0 â\89¡ 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 â\89¡ L â\86\92
- 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 â\89¡ L2.
+theorem ldrop_trans_ge: â\88\80d1,e1,L1,L. â\87\93[d1, e1] L1 â\89¡ L â\86\92
+ â\88\80e2,L2. â\87\93[0, e2] L â\89¡ L2 â\86\92 d1 â\89¤ e2 â\86\92 â\87\93[0, e1 + e2] L1 â\89¡ 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 â\89¡ L2 â\86\92 d1 â\89¤ e2 â\86\92
- â\86\93[0, e2 + e1] L1 â\89¡ L2.
+ â\87\93[d1, e1] L1 â\89¡ L â\86\92 â\87\93[0, e2] L â\89¡ L2 â\86\92 d1 â\89¤ e2 â\86\92
+ â\87\93[0, e2 + e1] L1 â\89¡ 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 â\89¡ L â\86\92
- â\88\83â\88\83L0. â\86\93[0, e1] L0 â\89¡ L2 & â\86\93[e1, e2] L0 â\89¡ L1.
+axiom ldrop_div: â\88\80e1,L1,L. â\87\93[0, e1] L1 â\89¡ L â\86\92 â\88\80e2,L2. â\87\93[0, e2] L2 â\89¡ L â\86\92
+ â\88\83â\88\83L0. â\87\93[0, e1] L0 â\89¡ L2 & â\87\93[e1, e2] L0 â\89¡ L1.