(* *)
(**************************************************************************)
-include "basic_2/grammar/lenv_weight.ma".
-include "basic_2/grammar/lsubs.ma".
+include "basic_2/grammar/cl_weight.ma".
include "basic_2/substitution/lift.ma".
+include "basic_2/substitution/lsubs.ma".
(* LOCAL ENVIRONMENT SLICING ************************************************)
interpretation "local slicing" 'RDrop d e L1 L2 = (ldrop d e L1 L2).
+definition l_liftable: (lenv → relation term) → Prop ≝
+ λR. ∀K,T1,T2. R K T1 T2 → ∀L,d,e. ⇩[d, e] L ≡ K →
+ ∀U1. ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 → R L U1 U2.
+
+definition l_deliftable_sn: (lenv → relation term) → Prop ≝
+ λR. ∀L,U1,U2. R L U1 U2 → ∀K,d,e. ⇩[d, e] L ≡ K →
+ ∀T1. ⇧[d, e] T1 ≡ U1 →
+ ∃∃T2. ⇧[d, e] T2 ≡ U2 & R K T1 T2.
+
+definition dropable_sn: relation lenv → Prop ≝
+ λR. ∀L1,K1,d,e. ⇩[d, e] L1 ≡ K1 → ∀L2. R L1 L2 →
+ ∃∃K2. R K1 K2 & ⇩[d, e] L2 ≡ K2.
+
+definition dedropable_sn: relation lenv → Prop ≝
+ λR. ∀L1,K1,d,e. ⇩[d, e] L1 ≡ K1 → ∀K2. R K1 K2 →
+ ∃∃L2. R L1 L2 & ⇩[d, e] L2 ≡ K2.
+
+definition dropable_dx: relation lenv → Prop ≝
+ λR. ∀L1,L2. R L1 L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 →
+ ∃∃K1. ⇩[0, e] L1 ≡ K1 & R K1 K2.
+
(* Basic inversion lemmas ***************************************************)
fact ldrop_inv_refl_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → d = 0 → e = 0 → L1 = L2.
(0 < e ∧ ⇩[0, e - 1] K ≡ L2).
/2 width=3/ qed-.
+lemma ldrop_inv_pair1: ∀K,I,V,L2. ⇩[0, 0] K. ⓑ{I} V ≡ L2 → L2 = K. ⓑ{I} V.
+#K #I #V #L2 #H
+elim (ldrop_inv_O1 … H) -H * // #H destruct
+elim (lt_refl_false … H)
+qed-.
+
(* Basic_1: was: drop_gen_drop *)
lemma ldrop_inv_ldrop1: ∀e,K,I,V,L2.
⇩[0, e] K. ⓑ{I} V ≡ L2 → 0 < e → ⇩[0, e - 1] K ≡ L2.
#L1 #L2 #I #V #e #HL12 #He >(plus_minus_m_m e 1) // /2 width=1/
qed.
-lemma ldrop_lsubs_ldrop1_abbr: ∀L1,L2,d,e. L1 [d, e] ≼ L2 →
- ∀K1,V,i. ⇩[0, i] L1 ≡ K1. ⓓV →
+lemma ldrop_skip_lt: ∀L1,L2,I,V1,V2,d,e.
+ ⇩[d - 1, e] L1 ≡ L2 → ⇧[d - 1, e] V2 ≡ V1 → 0 < d →
+ ⇩[d, e] L1. ⓑ{I} V1 ≡ L2. ⓑ{I} V2.
+#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV21 #Hd >(plus_minus_m_m d 1) // /2 width=1/
+qed.
+
+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
+| #L #I #V normalize #H
+ elim (IHi L ?) -IHi /2 width=1/ -H /3 width=2/
+]
+qed.
+
+lemma ldrop_O1_lt: ∀L,i. i < |L| → ∃∃I,K,V. ⇩[0, i] L ≡ K.ⓑ{I}V.
+#L elim L -L
+[ #i #H elim (lt_zero_false … H)
+| #L #I #V #IHL #i @(nat_ind_plus … i) -i /2 width=4/
+ #i #_ normalize #H
+ elim (IHL i ? ) -IHL /2 width=1/ -H /3 width=4/
+]
+qed.
+
+lemma ldrop_lsubs_ldrop2_abbr: ∀L1,L2,d,e. L1 ≼ [d, e] L2 →
+ ∀K2,V,i. ⇩[0, i] L2 ≡ K2. ⓓV →
d ≤ i → i < d + e →
- ∃∃K2. K1 [0, d + e - i - 1] ≼ K2 &
- ⇩[0, i] L2 ≡ K2. ⓓV.
+ ∃∃K1. K1 ≼ [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
lapply (ldrop_inv_atom1 … H) -H #H destruct
]
qed.
+lemma dropable_sn_TC: ∀R. dropable_sn R → dropable_sn (TC … R).
+#R #HR #L1 #K1 #d #e #HLK1 #L2 #H elim H -L2
+[ #L2 #HL12
+ elim (HR … HLK1 … HL12) -HR -L1 /3 width=3/
+| #L #L2 #_ #HL2 * #K #HK1 #HLK
+ elim (HR … HLK … HL2) -HR -L /3 width=3/
+]
+qed.
+
+lemma dedropable_sn_TC: ∀R. dedropable_sn R → dedropable_sn (TC … R).
+#R #HR #L1 #K1 #d #e #HLK1 #K2 #H elim H -K2
+[ #K2 #HK12
+ elim (HR … HLK1 … HK12) -HR -K1 /3 width=3/
+| #K #K2 #_ #HK2 * #L #HL1 #HLK
+ elim (HR … HLK … HK2) -HR -K /3 width=3/
+]
+qed.
+
+lemma dropable_dx_TC: ∀R. dropable_dx R → dropable_dx (TC … R).
+#R #HR #L1 #L2 #H elim H -L2
+[ #L2 #HL12 #K2 #e #HLK2
+ elim (HR … HL12 … HLK2) -HR -L2 /3 width=3/
+| #L #L2 #_ #HL2 #IHL1 #K2 #e #HLK2
+ elim (HR … HL2 … HLK2) -HR -L2 #K #HLK #HK2
+ elim (IHL1 … HLK) -L /3 width=5/
+]
+qed.
+
(* Basic forvard lemmas *****************************************************)
(* Basic_1: was: drop_S *)
]
qed-.
-lemma ldrop_fwd_lw: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → #[L2] ≤ #[L1].
+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
]
qed-.
+lemma ldrop_pair2_fwd_cw: ∀I,L,K,V,d,e. ⇩[d, e] L ≡ K. ⓑ{I} V →
+ ∀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-.
+
lemma ldrop_fwd_ldrop2_length: ∀L1,I2,K2,V2,e.
⇩[0, e] L1 ≡ K2. ⓑ{I2} V2 → e < |L1|.
#L1 elim L1 -L1
]
qed-.
-(* Basic_1: removed theorems 49:
- drop_skip_flat
+(* Basic_1: removed theorems 50:
+ 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