include "basic_2/grammar/cl_weight.ma".
include "basic_2/substitution/lift.ma".
-include "basic_2/substitution/lsubs.ma".
+include "basic_2/substitution/lsubr.ma".
(* LOCAL ENVIRONMENT SLICING ************************************************)
/2 width=5/ qed-.
fact ldrop_inv_O1_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → d = 0 →
- ∀K,I,V. L1 = K. ⓑ{I} V →
+ ∀K,I,V. L1 = K. ⓑ{I} V →
(e = 0 ∧ L2 = K. ⓑ{I} V) ∨
(0 < e ∧ ⇩[d, e - 1] K ≡ L2).
#d #e #L1 #L2 * -d -e -L1 -L2
fact ldrop_inv_skip1_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → 0 < d →
∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
∃∃K2,V2. ⇩[d - 1, e] K1 ≡ K2 &
- ⇧[d - 1, e] V2 ≡ V1 &
+ ⇧[d - 1, e] V2 ≡ V1 &
L2 = K2. ⓑ{I} V2.
#d #e #L1 #L2 * -d -e -L1 -L2
[ #d #e #_ #I #K #V #H destruct
(* Basic_1: was: drop_gen_skip_l *)
lemma ldrop_inv_skip1: ∀d,e,I,K1,V1,L2. ⇩[d, e] K1. ⓑ{I} V1 ≡ L2 → 0 < d →
∃∃K2,V2. ⇩[d - 1, e] K1 ≡ K2 &
- ⇧[d - 1, e] V2 ≡ V1 &
+ ⇧[d - 1, e] V2 ≡ V1 &
L2 = K2. ⓑ{I} V2.
/2 width=3/ qed-.
+lemma ldrop_inv_O1_pair2: ∀I,K,V,e,L1. ⇩[0, e] L1 ≡ K. ⓑ{I} V →
+ (e = 0 ∧ L1 = K. ⓑ{I} V) ∨
+ ∃∃I1,K1,V1. ⇩[0, e - 1] K1 ≡ K. ⓑ{I} V & L1 = K1.ⓑ{I1}V1 & 0 < e.
+#I #K #V #e *
+[ #H lapply (ldrop_inv_atom1 … H) -H #H destruct
+| #L1 #I1 #V1 #H
+ elim (ldrop_inv_O1 … H) -H *
+ [ #H1 #H2 destruct /3 width=1/
+ | /3 width=5/
+ ]
+]
+qed-.
+
fact ldrop_inv_skip2_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → 0 < d →
∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
∃∃K1,V1. ⇩[d - 1, e] K1 ≡ K2 &
- ⇧[d - 1, e] V2 ≡ V1 &
+ ⇧[d - 1, e] V2 ≡ V1 &
L1 = K1. ⓑ{I} V1.
#d #e #L1 #L2 * -d -e -L1 -L2
[ #d #e #_ #I #K #V #H destruct
lemma ldrop_O1_le: ∀i,L. i ≤ |L| → ∃K. ⇩[0, i] L ≡ K.
#i @(nat_ind_plus … i) -i /2 width=2/
#i #IHi *
-[ #H lapply (le_n_O_to_eq … H) -H >commutative_plus normalize #H destruct
+[ #H lapply (le_n_O_to_eq … H) -H >commutative_plus normalize #H destruct
| #L #I #V normalize #H
elim (IHi L ?) -IHi /2 width=1/ -H /3 width=2/
]
]
qed.
-lemma ldrop_lsubs_ldrop2_abbr: ∀L1,L2,d,e. L1 ≼ [d, e] L2 →
+lemma ldrop_lsubr_ldrop2_abbr: ∀L1,L2,d,e. L1 ⊑ [d, e] L2 →
∀K2,V,i. ⇩[0, i] L2 ≡ K2. ⓓV →
d ≤ i → i < d + e →
- â\88\83â\88\83K1. K1 â\89¼ [0, d + e - i - 1] K2 &
+ â\88\83â\88\83K1. K1 â\8a\91 [0, d + e - i - 1] K2 &
⇩[0, i] L1 ≡ K1. ⓓV.
#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
[ #d #e #K1 #V #i #H
]
qed-.
-lemma ldrop_fwd_lw: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → #{L2} ≤ #{L1}.
+lemma ldrop_fwd_length: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → |L2| ≤ |L1|.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e // normalize /2 width=1/
+qed-.
+
+lemma ldrop_fwd_lw: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ♯{L2} ≤ ♯{L1}.
#L1 #L2 #d #e #H elim H -L1 -L2 -d -e // normalize
[ /2 width=3/
| #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12
- >(tw_lift … HV21) -HV21 /2 width=1/
+ >(lift_fwd_tw … HV21) -HV21 /2 width=1/
]
-qed-.
+qed-.
lemma ldrop_pair2_fwd_fw: ∀I,L,K,V,d,e. ⇩[d, e] L ≡ K. ⓑ{I} V →
- ∀T. #{K, V} < #{L, T}.
+ ∀T. ♯{K, V} < ♯{L, T}.
#I #L #K #V #d #e #H #T
lapply (ldrop_fwd_lw … H) -H #H
@(le_to_lt_to_lt … H) -H /3 width=1/
qed-.
(* Basic_1: removed theorems 50:
- drop_ctail drop_skip_flat
+ drop_ctail drop_skip_flat
cimp_flat_sx cimp_flat_dx cimp_bind cimp_getl_conf
drop_clear drop_clear_O drop_clear_S
clear_gen_sort clear_gen_bind clear_gen_flat clear_gen_flat_r