(* *)
(**************************************************************************)
-include "basic_2/multiple/drops_drop.ma".
+include "basic_2/relocation/lifts_lifts.ma".
+include "basic_2/relocation/drops_weight.ma".
-(* ITERATED LOCAL ENVIRONMENT SLICING ***************************************)
+(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
(* Main properties **********************************************************)
+(* Basic_2A1: includes: drop_conf_ge drop_conf_be drop_conf_le *)
+theorem drops_conf: ∀L1,L,c1,f1. ⬇*[c1, f1] L1 ≡ L →
+ ∀L2,c2,f. ⬇*[c2, f] L1 ≡ L2 →
+ ∀f2. f1 ⊚ f2 ≡ f → ⬇*[c2, f2] L ≡ L2.
+#L1 #L #c1 #f1 #H elim H -L1 -L -f1
+[ #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_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_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/
+ ]
+]
+qed-.
+
(* Basic_1: was: drop1_trans *)
-theorem drops_trans: ∀L,L2,s,cs2. ⬇*[s, cs2] L ≡ L2 → ∀L1,cs1. ⬇*[s, cs1] L1 ≡ L →
- ⬇*[s, cs2 @@ cs1] L1 ≡ L2.
-#L #L2 #s #cs2 #H elim H -L -L2 -cs2 /3 width=3 by drops_cons/
+(* Basic_2A1: includes: drop_trans_ge drop_trans_le drop_trans_ge_comm
+ drops_drop_trans
+*)
+theorem drops_trans: ∀L1,L,c1,f1. ⬇*[c1, f1] L1 ≡ L →
+ ∀L2,c2,f2. ⬇*[c2, f2] L ≡ L2 →
+ ∀f. f1 ⊚ f2 ≡ f → ⬇*[c1∧c2, f] L1 ≡ L2.
+#L1 #L #c1 #f1 #H elim H -L1 -L -f1
+[ #f1 #Hf1 #L2 #c2 #f2 #HL2 #f #Hf elim (drops_inv_atom1 … HL2) -HL2
+ #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_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_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)
+/3 width=8 by drops_conf, drops_fwd_isid/
+qed-.
+
+(* Basic_2A1: includes: drop_conf_lt *)
+lemma drops_conf_skip1: ∀L,L2,c2,f. ⬇*[c2, f] L ≡ L2 →
+ ∀I,K1,V1,c1,f1. ⬇*[c1, f1] L ≡ K1.ⓑ{I}V1 →
+ ∀f2. f1 ⊚ ↑f2 ≡ f →
+ ∃∃K2,V2. L2 = K2.ⓑ{I}V2 &
+ ⬇*[c2, f2] K1 ≡ K2 & ⬆*[f2] V2 ≡ V1.
+#L #L2 #c2 #f #H2 #I #K1 #V1 #c1 #f1 #H1 #f2 #Hf lapply (drops_conf … H1 … H2 … Hf) -L -Hf
+#H elim (drops_inv_skip1 … H) -H /2 width=5 by ex3_2_intro/
+qed-.
+
+(* Basic_2A1: includes: drop_trans_lt *)
+lemma drops_trans_skip2: ∀L1,L,c1,f1. ⬇*[c1, f1] L1 ≡ L →
+ ∀I,K2,V2,c2,f2. ⬇*[c2, f2] L ≡ K2.ⓑ{I}V2 →
+ ∀f. f1 ⊚ f2 ≡ ↑f →
+ ∃∃K1,V1. L1 = K1.ⓑ{I}V1 &
+ ⬇*[c1∧c2, f] K1 ≡ K2 & ⬆*[f] V2 ≡ V1.
+#L1 #L #c1 #f1 #H1 #I #K2 #V2 #c2 #f2 #H2 #f #Hf
+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-.
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