(**************************************************************************)
include "basic_2/relocation/lifts_lifts.ma".
-include "basic_2/relocation/drops.ma".
+include "basic_2/relocation/drops_weight.ma".
(* GENERAL SLICING FOR LOCAL ENVIRONMENTS ***********************************)
[ #f1 #_ #L2 #c2 #f #HL2 #f2 #Hf12 elim (drops_inv_atom1 … HL2) -c1 -HL2
#H #Hf destruct @drops_atom
#H elim (after_inv_isid3 … Hf12) -Hf12 /2 width=1 by/
-| #I #K1 #K #V1 #f1 #_ #IH #L2 #c2 #f #HL2 #f2 #Hf elim (after_inv_Sxx … Hf) -Hf
+| #I #K1 #K #V1 #f1 #_ #IH #L2 #c2 #f #HL2 #f2 #Hf elim (after_inv_nxx … Hf) -Hf [2,3: // ]
#g #Hg #H destruct /3 width=3 by drops_inv_drop1/
-| #I #K1 #K #V1 #V #f1 #_ #HV1 #IH #L2 #c2 #f #HL2 #f2 #Hf elim (after_inv_Oxx … Hf) -Hf *
+| #I #K1 #K #V1 #V #f1 #_ #HV1 #IH #L2 #c2 #f #HL2 #f2 #Hf elim (after_inv_pxx … Hf) -Hf [1,3: * |*:// ]
#g2 #g #Hf #H1 #H2 destruct
[ elim (drops_inv_skip1 … HL2) -HL2 /3 width=6 by drops_skip, lifts_div/
| /4 width=3 by drops_inv_drop1, drops_drop/
#H #Hf2 destruct @drops_atom #H elim (andb_inv_true_dx … H) -H
#H1 #H2 lapply (after_isid_inv_sn … Hf ?) -Hf
/3 width=3 by isid_eq_repl_back/
-| #I #K1 #K #V1 #f1 #_ #IH #L2 #c2 #f2 #HL2 #f #Hf elim (after_inv_Sxx … Hf) -Hf
+| #I #K1 #K #V1 #f1 #_ #IH #L2 #c2 #f2 #HL2 #f #Hf elim (after_inv_nxx … Hf) -Hf
/3 width=3 by drops_drop/
-| #I #K1 #K #V1 #V #f1 #_ #HV1 #IH #L2 #c2 #f2 #HL2 #f #Hf elim (after_inv_Oxx … Hf) -Hf *
+| #I #K1 #K #V1 #V #f1 #_ #HV1 #IH #L2 #c2 #f2 #HL2 #f #Hf elim (after_inv_pxx … Hf) -Hf [1,3: * |*: // ]
#g2 #g #Hg #H1 #H2 destruct
[ elim (drops_inv_skip1 … HL2) -HL2 /3 width=6 by drops_skip, lifts_trans/
| /4 width=3 by drops_inv_drop1, drops_drop/
]
qed-.
+theorem drops_conf_div: ∀L,K,f1. ⬇*[Ⓣ,f1] L ≡ K → ∀f2. ⬇*[Ⓣ,f2] L ≡ K →
+ 𝐔⦃f1⦄ → 𝐔⦃f2⦄ → f1 ≗ f2.
+#L #K #f1 #H elim H -L -K -f1
+[ #f1 #Hf1 #f2 #Hf2 elim (drops_inv_atom1 … Hf2) -Hf2
+ /3 width=1 by isid_inv_eq_repl/
+| #I #L #K #V #f1 #Hf1 #IH #f2 elim (pn_split f2) *
+ #g2 #H2 #Hf2 #HU1 #HU2 destruct
+ [ elim (drops_inv_skip1 … Hf2) -IH -HU1 -Hf2 #Y2 #X2 #HY2 #_ #H destruct
+ lapply (drops_fwd_isid … HY2 ?) -HY2 /2 width=3 by isuni_inv_push/ -HU2
+ #H destruct elim (drops_inv_x_pair_xy … Hf1)
+ | /4 width=5 by drops_inv_drop1, isuni_inv_next, eq_next/
+ ]
+| #I #L #K #V #W #f1 #Hf1 #_ #IH #f2 elim (pn_split f2) *
+ #g2 #H2 #Hf2 #HU1 #HU2 destruct
+ [ elim (drops_inv_skip1 … Hf2) -Hf2 #Y2 #X2 #HY2 #_ #H destruct -Hf1
+ /4 width=5 by isuni_fwd_push, eq_push/
+ | lapply (drops_inv_drop1 … Hf2) -Hf2 -IH -HU2 #Hg2
+ lapply (drops_fwd_isid … Hf1 ?) -Hf1 /2 width=3 by isuni_inv_push/ -HU1
+ #H destruct elim (drops_inv_x_pair_xy … Hg2)
+ ]
+]
+qed-.
+
(* Advanced properties ******************************************************)
(* Basic_2A1: includes: drop_mono *)
lemma drops_mono: ∀L,L1,c1,f. ⬇*[c1, f] L ≡ L1 →
∀L2,c2. ⬇*[c2, f] L ≡ L2 → L1 = L2.
-#L #L1 #c1 #f lapply (isid_after_dx 𝐈𝐝 f ?)
+#L #L1 #c1 #f lapply (isid_after_dx 𝐈𝐝 … f)
/3 width=8 by drops_conf, drops_fwd_isid/
qed-.
lapply (drops_trans … H1 … H2 … Hf) -L -Hf
#H elim (drops_inv_skip2 … H) -H /2 width=5 by ex3_2_intro/
qed-.
+
+(* Basic_2A1: includes: drops_conf_div *)
+lemma drops_conf_div_pair: ∀I1,I2,L,K,V1,V2,f1,f2.
+ ⬇*[Ⓣ,f1] L ≡ K.ⓑ{I1}V1 → ⬇*[Ⓣ,f2] L ≡ K.ⓑ{I2}V2 →
+ 𝐔⦃f1⦄ → 𝐔⦃f2⦄ → ∧∧ f1 ≗ f2 & I1 = I2 & V1 = V2.
+#I1 #I2 #L #K #V1 #V2 #f1 #f2 #Hf1 #Hf2 #HU1 #HU2
+lapply (drops_isuni_fwd_drop2 … Hf1) // #H1
+lapply (drops_isuni_fwd_drop2 … Hf2) // #H2
+lapply (drops_conf_div … H1 … H2 ??) /2 width=3 by isuni_next/ -H1 -H2 -HU1 -HU2 #H
+lapply (eq_inv_nn … H ????) -H [5: |*: // ] #H12
+lapply (drops_eq_repl_back … Hf1 … H12) -Hf1 #H0
+lapply (drops_mono … H0 … Hf2) -L #H
+destruct /2 width=1 by and3_intro/
+qed-.