(* 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.
-#d #e #L #L1 #H elim H -H d e L L1
+(* 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) -H L2 //
+ >(ldrop_inv_atom1 … H) -L2 //
| #K #I #V #L2 #HL12
- <(ldrop_inv_refl … HL12) -HL12 L2 //
+ <(ldrop_inv_refl … HL12) -L2 //
| #L #K #I #V #e #_ #IHLK #L2 #H
- lapply (ldrop_inv_ldrop1 … H ?) -H /2/
+ lapply (ldrop_inv_ldrop1 … H ?) -H // /2 width=1/
| #L #K1 #I #T #V1 #d #e #_ #HVT1 #IHLK1 #X #H
- elim (ldrop_inv_skip1 … H ?) -H // <minus_plus_m_m #K2 #V2 #HLK2 #HVT2 #H destruct -X
- >(lift_inj … HVT1 … HVT2) -HVT1 HVT2
- >(IHLK1 … HLK2) -IHLK1 HLK2 //
+ elim (ldrop_inv_skip1 … H ?) -H // <minus_plus_m_m #K2 #V2 #HLK2 #HVT2 #H destruct
+ >(lift_inj … HVT1 … HVT2) -HVT1 -HVT2
+ >(IHLK1 … HLK2) -IHLK1 -HLK2 //
]
-qed.
+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.
-#d1 #e1 #L #L1 #H elim H -H d1 e1 L L1
+(* 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) -H L2 //
+ >(ldrop_inv_atom1 … H) -L2 //
| //
| #L #K #I #V #e #_ #IHLK #e2 #L2 #H #He2
- lapply (ldrop_inv_ldrop1 … H ?) -H /2/ #HL2
- <minus_plus_comm /3/
+ lapply (ldrop_inv_ldrop1 … H ?) -H /2 width=2/ #HL2
+ <minus_plus >minus_minus_comm /3 width=1/
| #L #K #I #V1 #V2 #d #e #_ #_ #IHLK #e2 #L2 #H #Hdee2
lapply (transitive_le 1 … Hdee2) // #He2
lapply (ldrop_inv_ldrop1 … H ?) -H // -He2 #HL2
lapply (transitive_le (1 + e) … Hdee2) // #Hee2
- @ldrop_ldrop_lt >minus_minus_comm /3/ (**) (* explicit constructor *)
+ @ldrop_ldrop_lt >minus_minus_comm /3 width=1/ (**) (* explicit constructor *)
]
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.
-#d1 #e1 #L #L1 #H elim H -H d1 e1 L L1
+ 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
| #L #I #V #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 (ldrop_inv_O1 … H) -H *
- [ -IHL12 He2d #H1 #H2 destruct -e2 K2 J V /2 width=5/
+ [ -IHL12 -He2d #H1 #H2 destruct /2 width=5/
| -HL12 -HV12 #He #HLK
- elim (IHL12 … HLK ?) -IHL12 HLK [ <minus_minus /3 width=5/ | /2/ ] (**) (* a bit slow *)
+ elim (IHL12 … HLK ?) -IHL12 -HLK [ <minus_minus /3 width=5/ | /2 width=1/ ] (**) (* a bit slow *)
]
]
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.
-#d1 #e1 #L1 #L #H elim H -H d1 e1 L1 L
+(* 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) -H L2 /2/
+ >(ldrop_inv_atom1 … H) -L2 /2 width=3/
| #K #I #V #e2 #L2 #HL2 #H
- lapply (le_O_to_eq_O … H) -H #H destruct -e2 /2/
+ lapply (le_n_O_to_eq … H) -H #H destruct /2 width=3/
| #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 (ldrop_inv_refl … H) -H #H destruct -L1 /3 width=5/
+ lapply (le_n_O_to_eq … H) -H #H destruct
+ elim (IHL12 … HL2 ?) -IHL12 -HL2 // #L0 #H #HL0
+ lapply (ldrop_inv_refl … H) -H #H destruct /3 width=5/
| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #IHL12 #e2 #L #H #He2d
elim (ldrop_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/ ]
+ [ -He2d -IHL12 #H1 #H2 destruct /3 width=5/
+ | -HL12 -HV12 #He2 #HL2
+ elim (IHL12 … HL2 ?) -L2 [ >minus_le_minus_minus_comm // /3 width=3/ | /2 width=1/ ]
]
]
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.
-#d1 #e1 #L1 #L #H elim H -H d1 e1 L1 L
+(* 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 //
+ >(ldrop_inv_atom1 … H) -H -L2 //
| //
-| /3/
+| /3 width=1/
| #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 (ldrop_inv_ldrop1 … H ?) -H // #HL2
- @ldrop_ldrop_lt // >le_plus_minus // @IHL12 /2/ (**) (* explicit constructor *)
+ @ldrop_ldrop_lt // >le_plus_minus // @IHL12 /2 width=1/ (**) (* explicit constructor *)
]
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.