(* *)
(**************************************************************************)
-include "basic_2/grammar/cl_weight.ma".
+include "basic_2/grammar/lenv_length.ma".
+include "basic_2/grammar/lenv_weight.ma".
include "basic_2/relocation/lift.ma".
-include "basic_2/relocation/lsubr.ma".
(* LOCAL ENVIRONMENT SLICING ************************************************)
interpretation "local slicing" 'RDrop d e L1 L2 = (ldrop d e L1 L2).
-definition l_liftable: (lenv → relation term) → Prop ≝
+definition l_liftable: predicate (lenv → relation term) ≝
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 ≝
+definition l_deliftable_sn: predicate (lenv → relation term) ≝
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 ≝
+definition dropable_sn: predicate (relation lenv) ≝
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 ≝
+definition dedropable_sn: predicate (relation lenv) ≝
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 ≝
+definition dropable_dx: predicate (relation lenv) ≝
λR. ∀L1,L2. R L1 L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 →
∃∃K1. ⇩[0, e] L1 ≡ K1 & R K1 K2.
]
qed.
-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 →
- ∃∃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
-| #L1 #L2 #K1 #V #i #_ #_ #H
- elim (lt_zero_false … H)
-| #L1 #L2 #V #e #HL12 #IHL12 #K1 #W #i #H #_ #Hie
- elim (ldrop_inv_O1 … H) -H * #Hi #HLK1
- [ -IHL12 -Hie destruct
- <minus_n_O <minus_plus_m_m // /2 width=3/
- | -HL12
- elim (IHL12 … HLK1 ? ?) -IHL12 -HLK1 // /2 width=1/ -Hie >minus_minus_comm >arith_b1 // /4 width=3/
- ]
-| #L1 #L2 #I #V1 #V2 #e #_ #IHL12 #K1 #W #i #H #_ #Hie
- elim (ldrop_inv_O1 … H) -H * #Hi #HLK1
- [ -IHL12 -Hie -Hi destruct
- | elim (IHL12 … HLK1 ? ?) -IHL12 -HLK1 // /2 width=1/ -Hie >minus_minus_comm >arith_b1 // /3 width=3/
- ]
-| #L1 #L2 #I1 #I2 #V1 #V2 #d #e #_ #IHL12 #K1 #V #i #H #Hdi >plus_plus_comm_23 #Hide
- elim (le_inv_plus_l … Hdi) #Hdim #Hi
- lapply (ldrop_inv_ldrop1 … H ?) -H // #HLK1
- elim (IHL12 … HLK1 ? ?) -IHL12 -HLK1 // /2 width=1/ -Hdi -Hide >minus_minus_comm >arith_b1 // /3 width=3/
+lemma l_liftable_LTC: ∀R. l_liftable R → l_liftable (LTC … R).
+#R #HR #K #T1 #T2 #H elim H -T2
+[ /3 width=9/
+| #T #T2 #_ #HT2 #IHT1 #L #d #e #HLK #U1 #HTU1 #U2 #HTU2
+ elim (lift_total T d e) /4 width=11 by step/ (**) (* auto too slow without trace *)
+]
+qed.
+
+lemma l_deliftable_sn_LTC: ∀R. l_deliftable_sn R → l_deliftable_sn (LTC … R).
+#R #HR #L #U1 #U2 #H elim H -U2
+[ #U2 #HU12 #K #d #e #HLK #T1 #HTU1
+ elim (HR … HU12 … HLK … HTU1) -HR -L -U1 /3 width=3/
+| #U #U2 #_ #HU2 #IHU1 #K #d #e #HLK #T1 #HTU1
+ elim (IHU1 … HLK … HTU1) -IHU1 -U1 #T #HTU #HT1
+ elim (HR … HU2 … HLK … HTU) -HR -L -U /3 width=5/
]
qed.
(* Basic forvard lemmas *****************************************************)
+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
+ >(lift_fwd_tw … HV21) -HV21 /2 width=1/
+]
+qed-.
+
(* Basic_1: was: drop_S *)
lemma ldrop_fwd_ldrop2: ∀L1,I2,K2,V2,e. ⇩[O, e] L1 ≡ K2. ⓑ{I2} V2 →
⇩[O, e + 1] L1 ≡ K2.
#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
- >(lift_fwd_tw … HV21) -HV21 /2 width=1/
-]
-qed-.
-
-lemma ldrop_pair2_fwd_fw: ∀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-.
+lemma ldrop_fwd_lw_eq: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 →
+ |L1| = |L2| → ♯{L2} = ♯{L1}.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e //
+[ #L1 #L2 #I #V #e #HL12 #_
+ lapply (ldrop_fwd_O1_length … HL12) -HL12 #HL21 >HL21 -HL21 normalize #H -I
+ lapply (discr_plus_xy_minus_xz … H) -e #H destruct
+| #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #HL12 normalize in ⊢ (??%%→??%%); #H -I
+ >(lift_fwd_tw … HV21) -V2 /3 width=1 by eq_f2/ (**) (* auto is a bit slow without trace *)
+]
+qed-.
+
+lemma ldrop_fwd_lw_lt: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 →
+ |L2| < |L1| → ♯{L2} < ♯{L1}.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e //
+[ #L #I #V #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V #e #HL12 #_ #_
+ lapply (ldrop_fwd_lw … HL12) -HL12 #HL12
+ @(le_to_lt_to_lt … HL12) -HL12 //
+| #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12 normalize in ⊢ (?%%→?%%); #H -I
+ >(lift_fwd_tw … HV21) -V2 /4 width=2 by lt_minus_to_plus, lt_plus_to_lt_l/ (**) (* auto too slow without trace *)
+]
+qed-.
+
(* Basic_1: removed theorems 50:
drop_ctail drop_skip_flat
cimp_flat_sx cimp_flat_dx cimp_bind cimp_getl_conf
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