X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frelocation%2Fdrops_lexs.ma;h=11b08f3e665021fc6f4729c03e764a319f1f17ed;hb=5c186c72f508da0849058afeecc6877cd9ed6303;hp=f639424f4c3a68ebf01ae8d6c1892ac0cbd5c42e;hpb=629687db8a55432e95c82f0c79e3f51c023e65a6;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_lexs.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_lexs.ma index f639424f4..11b08f3e6 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_lexs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/drops_lexs.ma @@ -12,140 +12,176 @@ (* *) (**************************************************************************) +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-.