(* *)
(**************************************************************************)
+include "basic_2/relocation/lifts_lifts_bind.ma".
include "basic_2/relocation/drops.ma".
-(* GENERAL SLICING FOR LOCAL ENVIRONMENTS ***********************************)
+(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
-(* Properties on general entrywise extension of context-sensitive relations *)
+(* Properties with entrywise extension of context-sensitive relations *******)
-(* Basic_2A1: includes: lpx_sn_deliftable_dropable *)
-lemma lexs_deliftable_dropable: ∀RN,RP. d_deliftable2_sn RN → d_deliftable2_sn RP →
- dropable_sn (lexs RN RP).
-#RN #RP #HN #HP #L1 #K1 #c #f #H elim H -L1 -K1 -f
-[ #f #Hf #X #f2 #H #f1 #Hf2 >(lexs_inv_atom1 … H) -X
+(**) (* changed after commit 13218 *)
+lemma lexs_co_dropable_sn: ∀RN,RP. co_dropable_sn (lexs RN RP).
+#RN #RP #b #f #L1 #K1 #H elim H -f -L1 -K1
+[ #f #Hf #_ #f2 #X #H #f1 #Hf2 >(lexs_inv_atom1 … H) -X
/4 width=3 by lexs_atom, drops_atom, ex2_intro/
-| #I #L1 #K1 #V1 #f #_ #IH #X #f2 #H #f1 #Hf2 elim (after_inv_nxx … Hf2) -Hf2 [2,3: // ]
- #g2 #Hg2 #H2 destruct elim (lexs_inv_next1 … H) -H
- #L2 #V2 #HL12 #HV12 #H destruct elim (IH … HL12 … Hg2) -g2
- /3 width=3 by drops_drop, ex2_intro/
-| #I #L1 #K1 #V1 #W1 #f #HLK #HWV #IH #X #f2 #H #f1 #Hf2 elim (after_inv_pxx … Hf2) -Hf2 [1,3:* |*: // ]
- #g1 #g2 #Hg2 #H1 #H2 destruct
- [ elim (lexs_inv_push1 … H) | elim (lexs_inv_next1 … H) ] -H
- #L2 #V2 #HL12 #HV12 #H destruct elim (IH … HL12 … Hg2) -g2
- [ elim (HP … HV12 … HLK … HWV) | elim (HN … HV12 … HLK … HWV) ] -V1
- /3 width=5 by lexs_next, lexs_push, drops_skip, ex2_intro/
+| #f #I1 #L1 #K1 #_ #IH #Hf #f2 #X #H #f1 #Hf2
+ elim (coafter_inv_nxx … Hf2) -Hf2 [2,3: // ] #g2 #Hg2 #H2 destruct
+ elim (lexs_inv_push1 … H) -H #I2 #L2 #HL12 #HI12 #H destruct
+ elim (IH … HL12 … Hg2) -g2
+ /3 width=3 by isuni_inv_next, drops_drop, ex2_intro/
+| #f #I1 #J1 #L1 #K1 #HLK #HJI1 #IH #Hf #f2 #X #H #f1 #Hf2
+ lapply (isuni_inv_push … Hf ??) -Hf [3: |*: // ] #Hf
+ lapply (drops_fwd_isid … HLK … Hf) -HLK #H0 destruct
+ lapply (liftsb_fwd_isid … HJI1 … Hf) -HJI1 #H0 destruct
+ elim (coafter_inv_pxx … Hf2) -Hf2 [1,3:* |*: // ] #g1 #g2 #Hg2 #H1 #H2 destruct
+ [ elim (lexs_inv_push1 … H) | elim (lexs_inv_next1 … H) ] -H #I2 #L2 #HL12 #HI12 #H destruct
+ elim (IH … HL12 … Hg2) -g2 -IH /2 width=1 by isuni_isid/ #K2 #HK12 #HLK2
+ lapply (drops_fwd_isid … HLK2 … Hf) -HLK2 #H0 destruct
+ /4 width=3 by drops_refl, lexs_next, lexs_push, isid_push, ex2_intro/
]
qed-.
-(* Basic_2A1: includes: lpx_sn_liftable_dedropable *)
-lemma lexs_liftable_dedropable: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
- d_liftable2 RN → d_liftable2 RP → dedropable_sn (lexs RN RP).
-#RN #RP #H1RN #H1RP #H2RN #H2RP #L1 #K1 #c #f #H elim H -L1 -K1 -f
+lemma lexs_liftable_co_dedropable_bi: ∀RN,RP. d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
+ ∀f2,L1,L2. L1 ⪤*[cfull, RP, f2] L2 → ∀f1,K1,K2. K1 ⪤*[RN, RP, f1] K2 →
+ ∀b,f. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 →
+ f ~⊚ f1 ≘ f2 → L1 ⪤*[RN, RP, f2] L2.
+#RN #RP #HRN #HRP #f2 #L1 #L2 #H elim H -f2 -L1 -L2 //
+#g2 #I1 #I2 #L1 #L2 #HL12 #HI12 #IH #f1 #Y1 #Y2 #HK12 #b #f #HY1 #HY2 #H
+[ elim (coafter_inv_xxn … H) [ |*: // ] -H #g #g1 #Hg2 #H1 #H2 destruct
+ elim (drops_inv_skip1 … HY1) -HY1 #J1 #K1 #HLK1 #HJI1 #H destruct
+ elim (drops_inv_skip1 … HY2) -HY2 #J2 #K2 #HLK2 #HJI2 #H destruct
+ elim (lexs_inv_next … HK12) -HK12 #HK12 #HJ12
+ elim (HRN … HJ12 … HLK1 … HJI1) -HJ12 -HJI1 #Z #Hz
+ >(liftsb_mono … Hz … HJI2) -Z /3 width=9 by lexs_next/
+| elim (coafter_inv_xxp … H) [1,2: |*: // ] -H *
+ [ #g #g1 #Hg2 #H1 #H2 destruct
+ elim (drops_inv_skip1 … HY1) -HY1 #J1 #K1 #HLK1 #HJI1 #H destruct
+ elim (drops_inv_skip1 … HY2) -HY2 #J2 #K2 #HLK2 #HJI2 #H destruct
+ elim (lexs_inv_push … HK12) -HK12 #HK12 #HJ12
+ elim (HRP … HJ12 … HLK1 … HJI1) -HJ12 -HJI1 #Z #Hz
+ >(liftsb_mono … Hz … HJI2) -Z /3 width=9 by lexs_push/
+ | #g #Hg2 #H destruct
+ lapply (drops_inv_drop1 … HY1) -HY1 #HLK1
+ lapply (drops_inv_drop1 … HY2) -HY2 #HLK2
+ /3 width=9 by lexs_push/
+ ]
+]
+qed-.
+
+lemma lexs_liftable_co_dedropable_sn: ∀RN,RP. (∀L. reflexive … (RN L)) → (∀L. reflexive … (RP L)) →
+ d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
+ co_dedropable_sn (lexs RN RP).
+#RN #RP #H1RN #H1RP #H2RN #H2RP #b #f #L1 #K1 #H elim H -f -L1 -K1
[ #f #Hf #X #f1 #H #f2 #Hf2 >(lexs_inv_atom1 … H) -X
/4 width=4 by drops_atom, lexs_atom, ex3_intro/
-| #I #L1 #K1 #V1 #f #_ #IHLK1 #K2 #f1 #HK12 #f2 #Hf2
- elim (after_inv_nxx … Hf2) -Hf2 [2,3: // ] #g2 #Hg2 #H destruct
+| #f #I1 #L1 #K1 #_ #IHLK1 #K2 #f1 #HK12 #f2 #Hf2
+ elim (coafter_inv_nxx … Hf2) -Hf2 [2,3: // ] #g2 #Hg2 #H destruct
elim (IHLK1 … HK12 … Hg2) -K1
- /3 width=6 by drops_drop, lexs_next, ex3_intro/
-| #I #L1 #K1 #V1 #W1 #f #HLK1 #HWV1 #IHLK1 #X #f1 #H #f2 #Hf2
- elim (after_inv_pxx … Hf2) -Hf2 [1,3: * |*: // ] #g1 #g2 #Hg2 #H1 #H2 destruct
- [ elim (lexs_inv_push1 … H) | elim (lexs_inv_next1 … H) ] -H #K2 #W2 #HK12 #HW12 #H destruct
- [ elim (H2RP … HW12 … HLK1 … HWV1) | elim (H2RN … HW12 … HLK1 … HWV1) ] -W1
+ /3 width=6 by drops_drop, lexs_next, lexs_push, ex3_intro/
+| #f #I1 #J1 #L1 #K1 #HLK1 #HJI1 #IHLK1 #X #f1 #H #f2 #Hf2
+ elim (coafter_inv_pxx … Hf2) -Hf2 [1,3: * |*: // ] #g1 #g2 #Hg2 #H1 #H2 destruct
+ [ elim (lexs_inv_push1 … H) | elim (lexs_inv_next1 … H) ] -H #J2 #K2 #HK12 #HJ12 #H destruct
+ [ elim (H2RP … HJ12 … HLK1 … HJI1) | elim (H2RN … HJ12 … HLK1 … HJI1) ] -J1
elim (IHLK1 … HK12 … Hg2) -K1
/3 width=6 by drops_skip, lexs_next, lexs_push, ex3_intro/
]
qed-.
-fact lexs_dropable_aux: ∀RN,RP,L2,K2,c,f. ⬇*[c, f] L2 ≡ K2 → 𝐔⦃f⦄ →
- ∀L1,f2. L1 ⦻*[RN, RP, f2] L2 → ∀f1. f ⊚ f1 ≡ f2 →
- ∃∃K1. ⬇*[c, f] L1 ≡ K1 & K1 ⦻*[RN, RP, f1] K2.
-#RN #RP #L2 #K2 #c #f #H elim H -L2 -K2 -f
-[ #f #Hf #_ #X #f2 #H #f1 #Hf2 lapply (lexs_inv_atom2 … H) -H
+fact lexs_dropable_dx_aux: ∀RN,RP,b,f,L2,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ →
+ ∀f2,L1. L1 ⪤*[RN, RP, f2] L2 → ∀f1. f ~⊚ f1 ≘ f2 →
+ ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤*[RN, RP, f1] K2.
+#RN #RP #b #f #L2 #K2 #H elim H -f -L2 -K2
+[ #f #Hf #_ #f2 #X #H #f1 #Hf2 lapply (lexs_inv_atom2 … H) -H
#H destruct /4 width=3 by lexs_atom, drops_atom, ex2_intro/
-| #I #L2 #K2 #V2 #f #_ #IH #Hf #X #f2 #HX #f1 #Hf2
- elim (after_inv_nxx … Hf2) -Hf2 [2,3: // ] #g2 #Hg2 #H destruct
- elim (lexs_inv_next2 … HX) -HX #L1 #V1 #HL12 #HV12 #H destruct
- elim (IH … HL12 … Hg2) -L2 -V2 -g2
+| #f #I2 #L2 #K2 #_ #IH #Hf #f2 #X #HX #f1 #Hf2
+ elim (coafter_inv_nxx … Hf2) -Hf2 [2,3: // ] #g2 #Hg2 #H destruct
+ elim (lexs_inv_push2 … HX) -HX #I1 #L1 #HL12 #HI12 #H destruct
+ elim (IH … HL12 … Hg2) -L2 -I2 -g2
/3 width=3 by drops_drop, isuni_inv_next, ex2_intro/
-| #I #L2 #K2 #V2 #W2 #f #_ #HWV2 #IH #Hf #X #f2 #HX #f1 #Hf2
- elim (after_inv_pxx … Hf2) -Hf2 [1,3: * |*: // ] #g1 #g2 #Hg2 #H1 #H2 destruct
- [ elim (lexs_inv_push2 … HX) | elim (lexs_inv_next2 … HX) ] -HX #L1 #V1 #HL12 #HV12 #H destruct
+| #f #I2 #J2 #L2 #K2 #_ #HJI2 #IH #Hf #f2 #X #HX #f1 #Hf2
+ elim (coafter_inv_pxx … Hf2) -Hf2 [1,3: * |*: // ] #g1 #g2 #Hg2 #H1 #H2 destruct
+ [ elim (lexs_inv_push2 … HX) | elim (lexs_inv_next2 … HX) ] -HX #I1 #L1 #HL12 #HI12 #H destruct
elim (IH … HL12 … Hg2) -L2 -g2 /2 width=3 by isuni_fwd_push/ #K1 #HLK1 #HK12
lapply (isuni_inv_push … Hf ??) -Hf [3,6: |*: // ] #Hf
- lapply (lifts_fwd_isid … HWV2 … Hf) #H destruct -HWV2
+ lapply (liftsb_fwd_isid … HJI2 … Hf) #H destruct -HJI2
lapply (drops_fwd_isid … HLK1 … Hf) #H destruct -HLK1
/4 width=5 by lexs_next, lexs_push, drops_refl, isid_push, ex2_intro/
]
qed-.
-(* Basic_2A1: includes: lpx_sn_dropable *)
-lemma lexs_dropable: ∀RN,RP. dropable_dx (lexs RN RP).
-/2 width=5 by lexs_dropable_aux/ qed-.
+lemma lexs_co_dropable_dx: ∀RN,RP. co_dropable_dx (lexs RN RP).
+/2 width=5 by lexs_dropable_dx_aux/ qed-.
-(* Basic_2A1: includes: lpx_sn_drop_conf *)
-lemma lexs_drops_conf_next: ∀RN,RP. d_deliftable2_sn RN → d_deliftable2_sn RP →
- ∀L1,L2,f2. L1 ⦻*[RN, RP, f2] L2 →
- ∀I,K1,V1,c,f. ⬇*[c,f] L1 ≡ K1.ⓑ{I}V1 →
- ∀f1. f ⊚ ⫯f1 ≡ f2 →
- ∃∃K2,V2. ⬇*[c,f] L2 ≡ K2.ⓑ{I}V2 & K1 ⦻*[RN, RP, f1] K2 & RN K1 V1 V2.
-#RN #RP #HRN #HRP #L1 #L2 #f2 #HL12 #I #K1 #V1 #c #f #HLK1 #f1 #Hf2
-elim (lexs_deliftable_dropable … HRN HRP … HLK1 … HL12 … Hf2) -L1 -f2 -HRN -HRP
+lemma lexs_drops_conf_next: ∀RN,RP.
+ ∀f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
+ ∀b,f,I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} → 𝐔⦃f⦄ →
+ ∀f1. f ~⊚ ↑f1 ≘ f2 →
+ ∃∃I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} & K1 ⪤*[RN, RP, f1] K2 & RN K1 I1 I2.
+#RN #RP #f2 #L1 #L2 #HL12 #b #f #I1 #K1 #HLK1 #Hf #f1 #Hf2
+elim (lexs_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf
#X #HX #HLK2 elim (lexs_inv_next1 … HX) -HX
-#K2 #V2 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
+#I2 #K2 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/
qed-.
-lemma lexs_drops_conf_push: ∀RN,RP. d_deliftable2_sn RN → d_deliftable2_sn RP →
- ∀L1,L2,f2. L1 ⦻*[RN, RP, f2] L2 →
- ∀I,K1,V1,c,f. ⬇*[c,f] L1 ≡ K1.ⓑ{I}V1 →
- ∀f1. f ⊚ ↑f1 ≡ f2 →
- ∃∃K2,V2. ⬇*[c,f] L2 ≡ K2.ⓑ{I}V2 & K1 ⦻*[RN, RP, f1] K2 & RP K1 V1 V2.
-#RN #RP #HRN #HRP #L1 #L2 #f2 #HL12 #I #K1 #V1 #c #f #HLK1 #f1 #Hf2
-elim (lexs_deliftable_dropable … HRN HRP … HLK1 … HL12 … Hf2) -L1 -f2 -HRN -HRP
+lemma lexs_drops_conf_push: ∀RN,RP.
+ ∀f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
+ ∀b,f,I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} → 𝐔⦃f⦄ →
+ ∀f1. f ~⊚ ⫯f1 ≘ f2 →
+ ∃∃I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} & K1 ⪤*[RN, RP, f1] K2 & RP K1 I1 I2.
+#RN #RP #f2 #L1 #L2 #HL12 #b #f #I1 #K1 #HLK1 #Hf #f1 #Hf2
+elim (lexs_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf
#X #HX #HLK2 elim (lexs_inv_push1 … HX) -HX
-#K2 #V2 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
+#I2 #K2 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/
qed-.
-(* Basic_2A1: includes: lpx_sn_drop_trans *)
-lemma lexs_drops_trans_next: ∀RN,RP,L1,L2,f2. L1 ⦻*[RN, RP, f2] L2 →
- ∀I,K2,V2,c,f. ⬇*[c,f] L2 ≡ K2.ⓑ{I}V2 → 𝐔⦃f⦄ →
- ∀f1. f ⊚ ⫯f1 ≡ f2 →
- ∃∃K1,V1. ⬇*[c,f] L1 ≡ K1.ⓑ{I}V1 & K1 ⦻*[RN, RP, f1] K2 & RN K1 V1 V2.
-#RN #RP #L1 #L2 #f2 #HL12 #I #K1 #V1 #c #f #HLK1 #Hf #f1 #Hf2
-elim (lexs_dropable … HL12 … HLK1 … Hf … Hf2) -L2 -f2 -Hf
+lemma lexs_drops_trans_next: ∀RN,RP,f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
+ ∀b,f,I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} → 𝐔⦃f⦄ →
+ ∀f1. f ~⊚ ↑f1 ≘ f2 →
+ ∃∃I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} & K1 ⪤*[RN, RP, f1] K2 & RN K1 I1 I2.
+#RN #RP #f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2
+elim (lexs_co_dropable_dx … HL12 … HLK2 … Hf … Hf2) -L2 -f2 -Hf
#X #HLK1 #HX elim (lexs_inv_next2 … HX) -HX
-#K1 #V1 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
+#I1 #K1 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/
qed-.
-lemma lexs_drops_trans_push: ∀RN,RP,L1,L2,f2. L1 ⦻*[RN, RP, f2] L2 →
- ∀I,K2,V2,c,f. ⬇*[c,f] L2 ≡ K2.ⓑ{I}V2 → 𝐔⦃f⦄ →
- ∀f1. f ⊚ ↑f1 ≡ f2 →
- ∃∃K1,V1. ⬇*[c,f] L1 ≡ K1.ⓑ{I}V1 & K1 ⦻*[RN, RP, f1] K2 & RP K1 V1 V2.
-#RN #RP #L1 #L2 #f2 #HL12 #I #K1 #V1 #c #f #HLK1 #Hf #f1 #Hf2
-elim (lexs_dropable … HL12 … HLK1 … Hf … Hf2) -L2 -f2 -Hf
+lemma lexs_drops_trans_push: ∀RN,RP,f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
+ ∀b,f,I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} → 𝐔⦃f⦄ →
+ ∀f1. f ~⊚ ⫯f1 ≘ f2 →
+ ∃∃I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} & K1 ⪤*[RN, RP, f1] K2 & RP K1 I1 I2.
+#RN #RP #f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2
+elim (lexs_co_dropable_dx … HL12 … HLK2 … Hf … Hf2) -L2 -f2 -Hf
#X #HLK1 #HX elim (lexs_inv_push2 … HX) -HX
-#K1 #V1 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
+#I1 #K1 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/
qed-.
lemma drops_lexs_trans_next: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
- d_liftable2 RN → d_liftable2 RP →
- ∀K1,K2,f1. K1 ⦻*[RN, RP, f1] K2 →
- ∀I,L1,V1,c,f. ⬇*[c,f] L1.ⓑ{I}V1 ≡ K1 →
- ∀f2. f ⊚ f1 ≡ ⫯f2 →
- ∃∃L2,V2. ⬇*[c,f] L2.ⓑ{I}V2 ≡ K2 & L1 ⦻*[RN, RP, f2] L2 & RN L1 V1 V2 & L1.ⓑ{I}V1≡[f]L2.ⓑ{I}V2.
-#RN #RP #H1RN #H1RP #H2RN #H2RP #K1 #K2 #f1 #HK12 #I #L1 #V1 #c #f #HLK1 #f2 #Hf2
-elim (lexs_liftable_dedropable … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
+ d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
+ ∀f1,K1,K2. K1 ⪤*[RN, RP, f1] K2 →
+ ∀b,f,I1,L1. ⬇*[b, f] L1.ⓘ{I1} ≘ K1 →
+ ∀f2. f ~⊚ f1 ≘ ↑f2 →
+ ∃∃I2,L2. ⬇*[b, f] L2.ⓘ{I2} ≘ K2 & L1 ⪤*[RN, RP, f2] L2 & RN L1 I1 I2 & L1.ⓘ{I1} ≡[f] L2.ⓘ{I2}.
+#RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2
+elim (lexs_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
#X #HX #HLK2 #H1L12 elim (lexs_inv_next1 … HX) -HX
-#L2 #V2 #H2L12 #HV12 #H destruct /2 width=6 by ex4_2_intro/
+#I2 #L2 #H2L12 #HI12 #H destruct /2 width=6 by ex4_2_intro/
qed-.
lemma drops_lexs_trans_push: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
- d_liftable2 RN → d_liftable2 RP →
- ∀K1,K2,f1. K1 ⦻*[RN, RP, f1] K2 →
- ∀I,L1,V1,c,f. ⬇*[c,f] L1.ⓑ{I}V1 ≡ K1 →
- ∀f2. f ⊚ f1 ≡ ↑f2 →
- ∃∃L2,V2. ⬇*[c,f] L2.ⓑ{I}V2 ≡ K2 & L1 ⦻*[RN, RP, f2] L2 & RP L1 V1 V2 & L1.ⓑ{I}V1≡[f]L2.ⓑ{I}V2.
-#RN #RP #H1RN #H1RP #H2RN #H2RP #K1 #K2 #f1 #HK12 #I #L1 #V1 #c #f #HLK1 #f2 #Hf2
-elim (lexs_liftable_dedropable … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
+ d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
+ ∀f1,K1,K2. K1 ⪤*[RN, RP, f1] K2 →
+ ∀b,f,I1,L1. ⬇*[b, f] L1.ⓘ{I1} ≘ K1 →
+ ∀f2. f ~⊚ f1 ≘ ⫯f2 →
+ ∃∃I2,L2. ⬇*[b, f] L2.ⓘ{I2} ≘ K2 & L1 ⪤*[RN, RP, f2] L2 & RP L1 I1 I2 & L1.ⓘ{I1} ≡[f] L2.ⓘ{I2}.
+#RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2
+elim (lexs_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
#X #HX #HLK2 #H1L12 elim (lexs_inv_push1 … HX) -HX
-#L2 #V2 #H2L12 #HV12 #H destruct /2 width=6 by ex4_2_intro/
+#I2 #L2 #H2L12 #HI12 #H destruct /2 width=6 by ex4_2_intro/
+qed-.
+
+lemma drops_atom2_lexs_conf: ∀RN,RP,b,f1,L1. ⬇*[b, f1] L1 ≘ ⋆ → 𝐔⦃f1⦄ →
+ ∀f,L2. L1 ⪤*[RN, RP, f] L2 →
+ ∀f2. f1 ~⊚ f2 ≘f → ⬇*[b, f1] L2 ≘ ⋆.
+#RN #RP #b #f1 #L1 #H1 #Hf1 #f #L2 #H2 #f2 #H3
+elim (lexs_co_dropable_sn … H1 … H2 … H3) // -H1 -H2 -H3 -Hf1
+#L #H #HL2 lapply (lexs_inv_atom1 … H) -H //
qed-.
projects / helm.git / blobdiff