(* Properties with entrywise extension of context-sensitive relations *******)
-(* Basic_2A1: includes: lpx_sn_deliftable_dropable *) (**) (* changed after commit 13218 *)
+(**) (* 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
]
qed-.
-(* Basic_2A1: includes: lpx_sn_liftable_dedropable *)
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).
]
qed-.
-(* Basic_2A1: includes: lpx_sn_dropable *)
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.
∀f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
∀b,f,I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} → 𝐔⦃f⦄ →
- â\88\80f1. f ~â\8a\9a ⫯f1 ≘ f2 →
+ â\88\80f1. f ~â\8a\9a â\86\91f1 ≘ 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
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⦄ →
- â\88\80f1. f ~â\8a\9a â\86\91f1 ≘ f2 →
+ â\88\80f1. f ~â\8a\9a ⫯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
#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,f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
∀b,f,I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} → 𝐔⦃f⦄ →
- â\88\80f1. f ~â\8a\9a ⫯f1 ≘ f2 →
+ â\88\80f1. f ~â\8a\9a â\86\91f1 ≘ 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
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⦄ →
- â\88\80f1. f ~â\8a\9a â\86\91f1 ≘ f2 →
+ â\88\80f1. f ~â\8a\9a ⫯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
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 →
- â\88\80f2. f ~â\8a\9a f1 â\89\98 ⫯f2 →
+ â\88\80f2. f ~â\8a\9a f1 â\89\98 â\86\91f2 →
∃∃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
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 →
- â\88\80f2. f ~â\8a\9a f1 â\89\98 â\86\91f2 →
+ â\88\80f2. f ~â\8a\9a f1 â\89\98 ⫯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