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 (sex_inv_push1 … H) | elim (sex_inv_next1 … H) ] -H #I2 #L2 #HL12 #HI12 #H destruct
+ [ elim (sex_inv_push1 … H) | elim (sex_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, sex_next, sex_push, isid_push, ex2_intro/
qed-.
lemma sex_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 →
- â\88\80b,f. â¬\87*[b, f] L1 â\89\98 K1 â\86\92 â¬\87*[b, f] L2 ≘ K2 →
- f ~⊚ f1 ≘ f2 → L1 ⪤[RN, RP, f2] L2.
+ ∀f2,L1,L2. L1 ⪤[cfull,RP,f2] L2 → ∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 →
+ â\88\80b,f. â\87©*[b,f] L1 â\89\98 K1 â\86\92 â\87©*[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
]
qed-.
-fact sex_dropable_dx_aux: â\88\80RN,RP,b,f,L2,K2. â¬\87*[b, f] L2 â\89\98 K2 â\86\92 ð\9d\90\94â¦\83fâ¦\84 →
- ∀f2,L1. L1 ⪤[RN, RP, f2] L2 → ∀f1. f ~⊚ f1 ≘ f2 →
- â\88\83â\88\83K1. â¬\87*[b, f] L1 â\89\98 K1 & K1 ⪤[RN, RP, f1] K2.
+fact sex_dropable_dx_aux: â\88\80RN,RP,b,f,L2,K2. â\87©*[b,f] L2 â\89\98 K2 â\86\92 ð\9d\90\94â\9dªfâ\9d« →
+ ∀f2,L1. L1 ⪤[RN,RP,f2] L2 → ∀f1. f ~⊚ f1 ≘ f2 →
+ â\88\83â\88\83K1. â\87©*[b,f] L1 â\89\98 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 (sex_inv_atom2 … H) -H
#H destruct /4 width=3 by sex_atom, drops_atom, ex2_intro/
/2 width=5 by sex_dropable_dx_aux/ qed-.
lemma sex_drops_conf_next: ∀RN,RP.
- ∀f2,L1,L2. L1 ⪤[RN, RP, f2] L2 →
- â\88\80b,f,I1,K1. â¬\87*[b, f] L1 â\89\98 K1.â\93\98{I1} â\86\92 ð\9d\90\94â¦\83fâ¦\84 →
+ ∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
+ â\88\80b,f,I1,K1. â\87©*[b,f] L1 â\89\98 K1.â\93\98[I1] â\86\92 ð\9d\90\94â\9dªfâ\9d« →
∀f1. f ~⊚ ↑f1 ≘ f2 →
- â\88\83â\88\83I2,K2. â¬\87*[b, f] L2 â\89\98 K2.â\93\98{I2} & K1 ⪤[RN, RP, f1] K2 & RN K1 I1 I2.
+ â\88\83â\88\83I2,K2. â\87©*[b,f] L2 â\89\98 K2.â\93\98[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 (sex_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf
#X #HX #HLK2 elim (sex_inv_next1 … HX) -HX
qed-.
lemma sex_drops_conf_push: ∀RN,RP.
- ∀f2,L1,L2. L1 ⪤[RN, RP, f2] L2 →
- â\88\80b,f,I1,K1. â¬\87*[b, f] L1 â\89\98 K1.â\93\98{I1} â\86\92 ð\9d\90\94â¦\83fâ¦\84 →
+ ∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
+ â\88\80b,f,I1,K1. â\87©*[b,f] L1 â\89\98 K1.â\93\98[I1] â\86\92 ð\9d\90\94â\9dªfâ\9d« →
∀f1. f ~⊚ ⫯f1 ≘ f2 →
- â\88\83â\88\83I2,K2. â¬\87*[b, f] L2 â\89\98 K2.â\93\98{I2} & K1 ⪤[RN, RP, f1] K2 & RP K1 I1 I2.
+ â\88\83â\88\83I2,K2. â\87©*[b,f] L2 â\89\98 K2.â\93\98[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 (sex_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf
#X #HX #HLK2 elim (sex_inv_push1 … HX) -HX
#I2 #K2 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/
qed-.
-lemma sex_drops_trans_next: ∀RN,RP,f2,L1,L2. L1 ⪤[RN, RP, f2] L2 →
- â\88\80b,f,I2,K2. â¬\87*[b, f] L2 â\89\98 K2.â\93\98{I2} â\86\92 ð\9d\90\94â¦\83fâ¦\84 →
+lemma sex_drops_trans_next: ∀RN,RP,f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
+ â\88\80b,f,I2,K2. â\87©*[b,f] L2 â\89\98 K2.â\93\98[I2] â\86\92 ð\9d\90\94â\9dªfâ\9d« →
∀f1. f ~⊚ ↑f1 ≘ f2 →
- â\88\83â\88\83I1,K1. â¬\87*[b, f] L1 â\89\98 K1.â\93\98{I1} & K1 ⪤[RN, RP, f1] K2 & RN K1 I1 I2.
+ â\88\83â\88\83I1,K1. â\87©*[b,f] L1 â\89\98 K1.â\93\98[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 (sex_co_dropable_dx … HL12 … HLK2 … Hf … Hf2) -L2 -f2 -Hf
#X #HLK1 #HX elim (sex_inv_next2 … HX) -HX
#I1 #K1 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/
qed-.
-lemma sex_drops_trans_push: ∀RN,RP,f2,L1,L2. L1 ⪤[RN, RP, f2] L2 →
- â\88\80b,f,I2,K2. â¬\87*[b, f] L2 â\89\98 K2.â\93\98{I2} â\86\92 ð\9d\90\94â¦\83fâ¦\84 →
+lemma sex_drops_trans_push: ∀RN,RP,f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
+ â\88\80b,f,I2,K2. â\87©*[b,f] L2 â\89\98 K2.â\93\98[I2] â\86\92 ð\9d\90\94â\9dªfâ\9d« →
∀f1. f ~⊚ ⫯f1 ≘ f2 →
- â\88\83â\88\83I1,K1. â¬\87*[b, f] L1 â\89\98 K1.â\93\98{I1} & K1 ⪤[RN, RP, f1] K2 & RP K1 I1 I2.
+ â\88\83â\88\83I1,K1. â\87©*[b,f] L1 â\89\98 K1.â\93\98[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 (sex_co_dropable_dx … HL12 … HLK2 … Hf … Hf2) -L2 -f2 -Hf
#X #HLK1 #HX elim (sex_inv_push2 … HX) -HX
lemma drops_sex_trans_next: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
- ∀f1,K1,K2. K1 ⪤[RN, RP, f1] K2 →
- â\88\80b,f,I1,L1. â¬\87*[b, f] L1.â\93\98{I1} ≘ K1 →
+ ∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 →
+ â\88\80b,f,I1,L1. â\87©*[b,f] L1.â\93\98[I1] ≘ K1 →
∀f2. f ~⊚ f1 ≘ ↑f2 →
- â\88\83â\88\83I2,L2. â¬\87*[b, f] L2.â\93\98{I2} â\89\98 K2 & L1 ⪤[RN, RP, f2] L2 & RN L1 I1 I2 & L1.â\93\98{I1} â\89¡[f] L2.â\93\98{I2}.
+ â\88\83â\88\83I2,L2. â\87©*[b,f] L2.â\93\98[I2] â\89\98 K2 & L1 ⪤[RN,RP,f2] L2 & RN L1 I1 I2 & L1.â\93\98[I1] â\89¡[f] L2.â\93\98[I2].
#RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2
elim (sex_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
#X #HX #HLK2 #H1L12 elim (sex_inv_next1 … HX) -HX
lemma drops_sex_trans_push: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
- ∀f1,K1,K2. K1 ⪤[RN, RP, f1] K2 →
- â\88\80b,f,I1,L1. â¬\87*[b, f] L1.â\93\98{I1} ≘ K1 →
+ ∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 →
+ â\88\80b,f,I1,L1. â\87©*[b,f] L1.â\93\98[I1] ≘ K1 →
∀f2. f ~⊚ f1 ≘ ⫯f2 →
- â\88\83â\88\83I2,L2. â¬\87*[b, f] L2.â\93\98{I2} â\89\98 K2 & L1 ⪤[RN, RP, f2] L2 & RP L1 I1 I2 & L1.â\93\98{I1} â\89¡[f] L2.â\93\98{I2}.
+ â\88\83â\88\83I2,L2. â\87©*[b,f] L2.â\93\98[I2] â\89\98 K2 & L1 ⪤[RN,RP,f2] L2 & RP L1 I1 I2 & L1.â\93\98[I1] â\89¡[f] L2.â\93\98[I2].
#RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2
elim (sex_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
#X #HX #HLK2 #H1L12 elim (sex_inv_push1 … HX) -HX
#I2 #L2 #H2L12 #HI12 #H destruct /2 width=6 by ex4_2_intro/
qed-.
-lemma drops_atom2_sex_conf: â\88\80RN,RP,b,f1,L1. â¬\87*[b, f1] L1 â\89\98 â\8b\86 â\86\92 ð\9d\90\94â¦\83f1â¦\84 →
- ∀f,L2. L1 ⪤[RN, RP, f] L2 →
- â\88\80f2. f1 ~â\8a\9a f2 â\89\98f â\86\92 â¬\87*[b, f1] L2 ≘ ⋆.
+lemma drops_atom2_sex_conf: â\88\80RN,RP,b,f1,L1. â\87©*[b,f1] L1 â\89\98 â\8b\86 â\86\92 ð\9d\90\94â\9dªf1â\9d« →
+ ∀f,L2. L1 ⪤[RN,RP,f] L2 →
+ â\88\80f2. f1 ~â\8a\9a f2 â\89\98f â\86\92 â\87©*[b,f1] L2 ≘ ⋆.
#RN #RP #b #f1 #L1 #H1 #Hf1 #f #L2 #H2 #f2 #H3
elim (sex_co_dropable_sn … H1 … H2 … H3) // -H1 -H2 -H3 -Hf1
#L #H #HL2 lapply (sex_inv_atom1 … H) -H //