(* *)
(**************************************************************************)
-include "basic_2/relocation/drops.ma".
-include "basic_2/relocation/lreq_lreq.ma".
+include "basic_2/relocation/drops_lexs.ma".
-(* GENERAL SLICING FOR LOCAL ENVIRONMENTS ***********************************)
+(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
-definition dedropable_sn: predicate (rtmap → relation lenv) ≝
- λR. ∀L1,K1,c,f. ⬇*[c, f] L1 ≡ K1 → ∀K2,f1. R f1 K1 K2 →
- ∀f2. f ⊚ f1 ≡ f2 →
- ∃∃L2. R f2 L1 L2 & ⬇*[c, f] L2 ≡ K2 & L1 ≡[f] L2.
+(* Properties with ranged equivalence for local environments ****************)
-(* Properties on equivalence ************************************************)
+lemma lreq_co_dedropable_sn: co_dedropable_sn lreq.
+@lexs_liftable_co_dedropable_sn
+/2 width=6 by cfull_lift_sn, ceq_lift_sn/ qed-.
-lemma dedropable_sn_TC: ∀R. dedropable_sn R → dedropable_sn (LTC … R).
-#R #HR #L1 #K1 #c #f #HLK1 #K2 #f1 #H elim H -K2
-[ #K2 #HK12 #f2 #Hf elim (HR … HLK1 … HK12 … Hf) -HR -K1 -f1
- /3 width=4 by inj, ex3_intro/
-| #K #K2 #_ #HK2 #IH #f2 #Hf elim (IH … Hf) -IH -K1
- #L #H1L1 #HLK #H2L1 elim (HR … HLK … HK2 … Hf) -HR -K -f1
- /3 width=6 by lreq_trans, step, ex3_intro/
-]
-qed-.
-(*
-lemma lreq_drop_trans_be: ∀L1,L2,l,k. L1 ⩬[l, k] L2 →
- ∀I,K2,W,c,i. ⬇[c, 0, i] L2 ≡ K2.ⓑ{I}W →
- l ≤ i → ∀k0. i + ⫯k0 = l + k →
- ∃∃K1. K1 ⩬[0, k0] K2 & ⬇[c, 0, i] L1 ≡ K1.ⓑ{I}W.
-#L1 #L2 #l #k #H elim H -L1 -L2 -l -k
-[ #l #k #J #K2 #W #c #i #H
- elim (drop_inv_atom1 … H) -H #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J #K2 #W #c #i #_ #_ #k0
- >yplus_succ2 #H elim (ysucc_inv_O_dx … H)
-| #I #L1 #L2 #V #k #HL12 #IHL12 #J #K2 #W #c #i #H #_ >yplus_O1 #k0 #H0
- elim (drop_inv_O1_pair1 … H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ]
- [ destruct
- /2 width=3 by drop_pair, ex2_intro/
- | lapply (ylt_inv_O1 … Hi) #H <H in H0; -H
- >yplus_succ1 #H lapply (ysucc_inv_inj … H) -H <(yplus_O1 k)
- #H0 elim (IHL12 … HLK1 … H0) -IHL12 -HLK1 -H0 //
- /3 width=3 by drop_drop_lt, ex2_intro/
- ]
-| #I1 #I2 #L1 #L2 #V1 #V2 #l #k #_ #IHL12 #J #K2 #W #c #i #HLK2 #Hli #k0
- elim (yle_inv_succ1 … Hli) -Hli
- #Hli #Hi <Hi >yplus_succ1 >yplus_succ1 #H lapply (ysucc_inv_inj … H) -H
- #H0 lapply (drop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O1/
- #HLK1 elim (IHL12 … HLK1 … H0) -IHL12 -HLK1 -H0
- /4 width=3 by ylt_O1, drop_drop_lt, ex2_intro/
-]
-qed-.
+lemma lreq_co_dropable_sn: co_dropable_sn lreq.
+@lexs_co_dropable_sn qed-.
-lemma lreq_drop_conf_be: ∀L1,L2,l,k. L1 ⩬[l, k] L2 →
- ∀I,K1,W,c,i. ⬇[c, 0, i] L1 ≡ K1.ⓑ{I}W →
- l ≤ i → ∀k0. i + ⫯k0 = l + k →
- ∃∃K2. K1 ⩬[0, k0] K2 & ⬇[c, 0, i] L2 ≡ K2.ⓑ{I}W.
-#L1 #L2 #l #k #HL12 #I #K1 #W #c #i #HLK1 #Hli #k0 #H0
-elim (lreq_drop_trans_be … (lreq_sym … HL12) … HLK1 … H0) // -L1 -Hli -H0
-/3 width=3 by lreq_sym, ex2_intro/
-qed-.
+lemma lreq_co_dropable_dx: co_dropable_dx lreq.
+@lexs_co_dropable_dx qed-.
-lemma drop_O1_ex: ∀K2,i,L1. |L1| = |K2| + i →
- ∃∃L2. L1 ⩬[0, i] L2 & ⬇[i] L2 ≡ K2.
-#K2 #i @(ynat_ind … i) -i
-[ /3 width=3 by lreq_O2, ex2_intro/
-| #i #IHi #Y >yplus_succ2 #Hi
- elim (drop_O1_lt (Ⓕ) Y 0) [2: >Hi // ]
- #I #L1 #V #H lapply (drop_inv_O2 … H) -H #H destruct
- >length_pair in Hi; #H lapply (ysucc_inv_inj … H) -H
- #HL1K2 elim (IHi L1) -IHi // -HL1K2
- /3 width=5 by lreq_pair, drop_drop, ex2_intro/
-| #L1 >yplus_Y2 #H elim (ylt_yle_false (|L1|) (∞)) //
-]
+(* Basic_2A1: includes: lreq_drop_trans_be *)
+lemma lreq_drops_trans_next: ∀f2,L1,L2. L1 ≡[f2] L2 →
+ ∀b,f,I,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I} → 𝐔⦃f⦄ →
+ ∀f1. f ~⊚ ↑f1 ≘ f2 →
+ ∃∃K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I} & K1 ≡[f1] K2.
+#f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2
+elim (lexs_drops_trans_next … HL12 … HLK2 Hf … Hf2) -f2 -L2 -Hf
+#I1 #K1 #HLK1 #HK12 #H <(ceq_ext_inv_eq … H) -I2
+/2 width=3 by ex2_intro/
qed-.
-(* Inversion lemmas on equivalence ******************************************)
+(* Basic_2A1: includes: lreq_drop_conf_be *)
+lemma lreq_drops_conf_next: ∀f2,L1,L2. L1 ≡[f2] L2 →
+ ∀b,f,I,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I} → 𝐔⦃f⦄ →
+ ∀f1. f ~⊚ ↑f1 ≘ f2 →
+ ∃∃K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I} & K1 ≡[f1] K2.
+#f2 #L1 #L2 #HL12 #b #f #I1 #K1 #HLK1 #Hf #f1 #Hf2
+elim (lreq_drops_trans_next … (lreq_sym … HL12) … HLK1 … Hf2) // -f2 -L1 -Hf
+/3 width=3 by lreq_sym, ex2_intro/
+qed-.
-lemma drop_O1_inj: ∀i,L1,L2,K. ⬇[i] L1 ≡ K → ⬇[i] L2 ≡ K → L1 ⩬[i, ∞] L2.
-#i @(ynat_ind … i) -i
-[ #L1 #L2 #K #H <(drop_inv_O2 … H) -K #H <(drop_inv_O2 … H) -L1 //
-| #i #IHi * [2: #L1 #I1 #V1 ] * [2,4: #L2 #I2 #V2 ] #K #HLK1 #HLK2 //
- lapply (drop_fwd_length … HLK1)
- <(drop_fwd_length … HLK2) [ /4 width=5 by drop_inv_drop1, lreq_succ/ ]
- #H [ elim (ysucc_inv_O_sn … H) | elim (ysucc_inv_O_dx … H) ]
-| #L1 #L2 #K #H1 lapply (drop_fwd_Y2 … H1) -H1
- #H elim (ylt_yle_false … H) //
-]
+lemma drops_lreq_trans_next: ∀f1,K1,K2. K1 ≡[f1] K2 →
+ ∀b,f,I,L1. ⬇*[b, f] L1.ⓘ{I} ≘ K1 →
+ ∀f2. f ~⊚ f1 ≘ ↑f2 →
+ ∃∃L2. ⬇*[b, f] L2.ⓘ{I} ≘ K2 & L1 ≡[f2] L2 & L1.ⓘ{I} ≡[f] L2.ⓘ{I}.
+#f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2
+elim (drops_lexs_trans_next … HK12 … HLK1 … Hf2) -f1 -K1
+/2 width=6 by cfull_lift_sn, ceq_lift_sn/
+#I2 #L2 #HLK2 #HL12 #H >(ceq_ext_inv_eq … H) -I1
+/2 width=4 by ex3_intro/
qed-.
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