(* Advanced properties ******************************************************)
-lemma cpg_delta_drops: â\88\80Rt,c,h,G,K,V,V2,i,L,T2. â\87©*[i] L â\89\98 K.â\93\93V â\86\92 â¦\83G,Kâ¦\84 ⊢ V ⬈[Rt,c,h] V2 →
- â\87§*[â\86\91i] V2 â\89\98 T2 â\86\92 â¦\83G,Lâ¦\84 ⊢ #i ⬈[Rt,c,h] T2.
+lemma cpg_delta_drops: â\88\80Rt,c,h,G,K,V,V2,i,L,T2. â\87©*[i] L â\89\98 K.â\93\93V â\86\92 â\9dªG,Kâ\9d« ⊢ V ⬈[Rt,c,h] V2 →
+ â\87§*[â\86\91i] V2 â\89\98 T2 â\86\92 â\9dªG,Lâ\9d« ⊢ #i ⬈[Rt,c,h] T2.
#Rt #c #h #G #K #V #V2 #i elim i -i
[ #L #T2 #HLK lapply (drops_fwd_isid … HLK ?) // #H destruct /3 width=3 by cpg_delta/
| #i #IH #L0 #T0 #H0 #HV2 #HVT2
elim (drops_inv_succ … H0) -H0 #I #L #HLK #H destruct
- elim (lifts_split_trans â\80¦ HVT2 (ð\9d\90\94â\9d´â\86\91iâ\9dµ) (ð\9d\90\94â\9d´1â\9dµ) ?) -HVT2 /3 width=3 by cpg_lref/
+ elim (lifts_split_trans â\80¦ HVT2 (ð\9d\90\94â\9d¨â\86\91iâ\9d©) (ð\9d\90\94â\9d¨1â\9d©) ?) -HVT2 /3 width=3 by cpg_lref/
]
qed.
-lemma cpg_ell_drops: â\88\80Rt,c,h,G,K,V,V2,i,L,T2. â\87©*[i] L â\89\98 K.â\93\9bV â\86\92 â¦\83G,Kâ¦\84 ⊢ V ⬈[Rt,c,h] V2 →
- â\87§*[â\86\91i] V2 â\89\98 T2 â\86\92 â¦\83G,Lâ¦\84 ⊢ #i ⬈[Rt,c+𝟘𝟙,h] T2.
+lemma cpg_ell_drops: â\88\80Rt,c,h,G,K,V,V2,i,L,T2. â\87©*[i] L â\89\98 K.â\93\9bV â\86\92 â\9dªG,Kâ\9d« ⊢ V ⬈[Rt,c,h] V2 →
+ â\87§*[â\86\91i] V2 â\89\98 T2 â\86\92 â\9dªG,Lâ\9d« ⊢ #i ⬈[Rt,c+𝟘𝟙,h] T2.
#Rt #c #h #G #K #V #V2 #i elim i -i
[ #L #T2 #HLK lapply (drops_fwd_isid … HLK ?) // #H destruct /3 width=3 by cpg_ell/
| #i #IH #L0 #T0 #H0 #HV2 #HVT2
elim (drops_inv_succ … H0) -H0 #I #L #HLK #H destruct
- elim (lifts_split_trans â\80¦ HVT2 (ð\9d\90\94â\9d´â\86\91iâ\9dµ) (ð\9d\90\94â\9d´1â\9dµ) ?) -HVT2 /3 width=3 by cpg_lref/
+ elim (lifts_split_trans â\80¦ HVT2 (ð\9d\90\94â\9d¨â\86\91iâ\9d©) (ð\9d\90\94â\9d¨1â\9d©) ?) -HVT2 /3 width=3 by cpg_lref/
]
qed.
(* Advanced inversion lemmas ************************************************)
-lemma cpg_inv_lref1_drops: â\88\80Rt,c,h,G,i,L,T2. â¦\83G,Lâ¦\84 ⊢ #i ⬈[Rt,c,h] T2 →
+lemma cpg_inv_lref1_drops: â\88\80Rt,c,h,G,i,L,T2. â\9dªG,Lâ\9d« ⊢ #i ⬈[Rt,c,h] T2 →
∨∨ T2 = #i ∧ c = 𝟘𝟘
- | â\88\83â\88\83cV,K,V,V2. â\87©*[i] L â\89\98 K.â\93\93V & â¦\83G,Kâ¦\84 ⊢ V ⬈[Rt,cV,h] V2 &
+ | â\88\83â\88\83cV,K,V,V2. â\87©*[i] L â\89\98 K.â\93\93V & â\9dªG,Kâ\9d« ⊢ V ⬈[Rt,cV,h] V2 &
⇧*[↑i] V2 ≘ T2 & c = cV
- | â\88\83â\88\83cV,K,V,V2. â\87©*[i] L â\89\98 K.â\93\9bV & â¦\83G,Kâ¦\84 ⊢ V ⬈[Rt,cV,h] V2 &
+ | â\88\83â\88\83cV,K,V,V2. â\87©*[i] L â\89\98 K.â\93\9bV & â\9dªG,Kâ\9d« ⊢ V ⬈[Rt,cV,h] V2 &
⇧*[↑i] V2 ≘ T2 & c = cV + 𝟘𝟙.
#Rt #c #h #G #i elim i -i
[ #L #T2 #H elim (cpg_inv_zero1 … H) -H * /3 width=1 by or3_intro0, conj/
]
qed-.
-lemma cpg_inv_atom1_drops: â\88\80Rt,c,h,I,G,L,T2. â¦\83G,Lâ¦\84 â\8a¢ â\93ª{I} ⬈[Rt,c,h] T2 →
- ∨∨ T2 = ⓪{I} ∧ c = 𝟘𝟘
+lemma cpg_inv_atom1_drops: â\88\80Rt,c,h,I,G,L,T2. â\9dªG,Lâ\9d« â\8a¢ â\93ª[I] ⬈[Rt,c,h] T2 →
+ ∨∨ T2 = ⓪[I] ∧ c = 𝟘𝟘
| ∃∃s. T2 = ⋆(⫯[h]s) & I = Sort s & c = 𝟘𝟙
- | â\88\83â\88\83cV,i,K,V,V2. â\87©*[i] L â\89\98 K.â\93\93V & â¦\83G,Kâ¦\84 ⊢ V ⬈[Rt,cV,h] V2 &
+ | â\88\83â\88\83cV,i,K,V,V2. â\87©*[i] L â\89\98 K.â\93\93V & â\9dªG,Kâ\9d« ⊢ V ⬈[Rt,cV,h] V2 &
⇧*[↑i] V2 ≘ T2 & I = LRef i & c = cV
- | â\88\83â\88\83cV,i,K,V,V2. â\87©*[i] L â\89\98 K.â\93\9bV & â¦\83G,Kâ¦\84 ⊢ V ⬈[Rt,cV,h] V2 &
+ | â\88\83â\88\83cV,i,K,V,V2. â\87©*[i] L â\89\98 K.â\93\9bV & â\9dªG,Kâ\9d« ⊢ V ⬈[Rt,cV,h] V2 &
⇧*[↑i] V2 ≘ T2 & I = LRef i & c = cV + 𝟘𝟙.
#Rt #c #h * #n #G #L #T2 #H
[ elim (cpg_inv_sort1 … H) -H *
elim (drops_inv_skip2 … HY) -HY #Z #L0 #HLK0 #HZ #H destruct
elim (liftsb_inv_pair_sn … HZ) -HZ #W #HVW #H destruct
elim (IH … HV2 … HLK0 … HVW) -IH /2 width=2 by fqup_lref/ -K -K0 -V #W2 #HVW2 #HW2
- elim (lifts_total W2 (ð\9d\90\94â\9d´â\86\91i2â\9dµ)) #U2 #HWU2
+ elim (lifts_total W2 (ð\9d\90\94â\9d¨â\86\91i2â\9d©)) #U2 #HWU2
lapply (lifts_trans … HVW2 … HWU2 ??) -HVW2 [3,6: |*: // ] #HVU2
lapply (lifts_conf … HVT2 … HVU2 f ?) -V2 [1,3: /2 width=3 by after_uni_succ_sn/ ]
/4 width=8 by cpg_ell_drops, cpg_delta_drops, drops_inv_gen, ex2_intro/
elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 // #W2 #HVW2 #HW12
elim (IH … HY12 … HLK … HYZ1) -HY12 //
elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
- elim (lifts_total W2 (ð\9d\90\94â\9d´1â\9dµ)) #W20 #HW20
+ elim (lifts_total W2 (ð\9d\90\94â\9d¨1â\9d©)) #W20 #HW20
lapply (lifts_trans … HVW2 … HW20 ??) -HVW2 [3: |*: // ] #H
lapply (lifts_conf … HV20 … H (⫯f) ?) -V2 /2 width=3 by after_uni_one_sn/
/4 width=9 by cpg_theta, lifts_bind, lifts_flat, ex2_intro/
[ #H1 #H2 destruct /3 width=3 by cpg_refl, ex2_intro/ ]
#cW #L0 #W #W2 #HL0 #HW2 #HWU2 #H destruct
elim (lifts_inv_lref2 … H1) -H1 #i1 #Hf #H destruct
- lapply (drops_split_div â\80¦ HLK (ð\9d\90\94â\9d´i1â\9dµ) ???) -HLK [4,8: * |*: // ] #Y0 #HK0 #HLY0
+ lapply (drops_split_div â\80¦ HLK (ð\9d\90\94â\9d¨i1â\9d©) ???) -HLK [4,8: * |*: // ] #Y0 #HK0 #HLY0
lapply (drops_conf … HL0 … HLY0 ??) -HLY0 [3,6: |*: /2 width=6 by after_uni_dx/ ] #HLY0
lapply (drops_tls_at … Hf … HLY0) -HLY0 #HLY0
elim (drops_inv_skip1 … HLY0) -HLY0 #Z #K0 #HLK0 #HZ #H destruct