(* Advanced properties ******************************************************)
-lemma cpg_delta_drops: â\88\80c,h,G,K,V,V2,i,L,T2. â¬\87*[i] L â\89¡ K.â\93\93V â\86\92 â¦\83G, Kâ¦\84 â\8a¢ V â\9e¡[c, h] V2 →
- â¬\86*[⫯i] V2 â\89¡ T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ #i â\9e¡[c, h] T2.
+lemma cpg_delta_drops: â\88\80c,h,G,K,V,V2,i,L,T2. â¬\87*[i] L â\89¡ K.â\93\93V â\86\92 â¦\83G, Kâ¦\84 â\8a¢ V â¬\88[c, h] V2 →
+ â¬\86*[⫯i] V2 â\89¡ T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ #i â¬\88[c, h] T2.
#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
]
qed.
-lemma cpg_ell_drops: â\88\80c,h,G,K,V,V2,i,L,T2. â¬\87*[i] L â\89¡ K.â\93\9bV â\86\92 â¦\83G, Kâ¦\84 â\8a¢ V â\9e¡[c, h] V2 →
- â¬\86*[⫯i] V2 â\89¡ T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ #i â\9e¡[(â\86\93c)+𝟘𝟙, h] T2.
+lemma cpg_ell_drops: â\88\80c,h,G,K,V,V2,i,L,T2. â¬\87*[i] L â\89¡ K.â\93\9bV â\86\92 â¦\83G, Kâ¦\84 â\8a¢ V â¬\88[c, h] V2 →
+ â¬\86*[⫯i] V2 â\89¡ T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ #i â¬\88[c+𝟘𝟙, h] T2.
#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
(* Advanced inversion lemmas ************************************************)
-lemma cpg_inv_lref1_drops: â\88\80c,h,G,i,L,T2. â¦\83G, Lâ¦\84 â\8a¢ #i â\9e¡[c, h] T2 →
+lemma cpg_inv_lref1_drops: â\88\80c,h,G,i,L,T2. â¦\83G, Lâ¦\84 â\8a¢ #i â¬\88[c, h] T2 →
∨∨ T2 = #i ∧ c = 𝟘𝟘
- | â\88\83â\88\83cV,K,V,V2. â¬\87*[i] L â\89¡ K.â\93\93V & â¦\83G, Kâ¦\84 â\8a¢ V â\9e¡[cV, h] V2 &
+ | â\88\83â\88\83cV,K,V,V2. â¬\87*[i] L â\89¡ K.â\93\93V & â¦\83G, Kâ¦\84 â\8a¢ V â¬\88[cV, h] V2 &
⬆*[⫯i] V2 ≡ T2 & c = cV
- | â\88\83â\88\83cV,K,V,V2. â¬\87*[i] L â\89¡ K.â\93\9bV & â¦\83G, Kâ¦\84 â\8a¢ V â\9e¡[cV, h] V2 &
- ⬆*[⫯i] V2 ≡ T2 & c = (↓cV) + 𝟘𝟙.
+ | â\88\83â\88\83cV,K,V,V2. â¬\87*[i] L â\89¡ K.â\93\9bV & â¦\83G, Kâ¦\84 â\8a¢ V â¬\88[cV, h] V2 &
+ ⬆*[⫯i] V2 ≡ T2 & c = cV + 𝟘𝟙.
#c #h #G #i elim i -i
[ #L #T2 #H elim (cpg_inv_zero1 … H) -H * /3 width=1 by or3_intro0, conj/
/4 width=8 by drops_refl, ex4_4_intro, or3_intro2, or3_intro1/
]
qed-.
-lemma cpg_inv_atom1_drops: â\88\80c,h,I,G,L,T2. â¦\83G, Lâ¦\84 â\8a¢ â\93ª{I} â\9e¡[c, h] T2 →
+lemma cpg_inv_atom1_drops: â\88\80c,h,I,G,L,T2. â¦\83G, Lâ¦\84 â\8a¢ â\93ª{I} â¬\88[c, h] T2 →
∨∨ T2 = ⓪{I} ∧ c = 𝟘𝟘
| ∃∃s. T2 = ⋆(next h s) & I = Sort s & c = 𝟘𝟙
- | â\88\83â\88\83cV,i,K,V,V2. â¬\87*[i] L â\89¡ K.â\93\93V & â¦\83G, Kâ¦\84 â\8a¢ V â\9e¡[cV, h] V2 &
+ | â\88\83â\88\83cV,i,K,V,V2. â¬\87*[i] L â\89¡ K.â\93\93V & â¦\83G, Kâ¦\84 â\8a¢ V â¬\88[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¡ K.â\93\9bV & â¦\83G, Kâ¦\84 â\8a¢ V â\9e¡[cV, h] V2 &
- ⬆*[⫯i] V2 ≡ T2 & I = LRef i & c = (↓cV) + 𝟘𝟙.
+ | â\88\83â\88\83cV,i,K,V,V2. â¬\87*[i] L â\89¡ K.â\93\9bV & â¦\83G, Kâ¦\84 â\8a¢ V â¬\88[cV, h] V2 &
+ ⬆*[⫯i] V2 ≡ T2 & I = LRef i & c = cV + 𝟘𝟙.
#c #h * #n #G #L #T2 #H
[ elim (cpg_inv_sort1 … H) -H *
/3 width=3 by or4_intro0, or4_intro1, ex3_intro, conj/
(* Properties with generic slicing for local environments *******************)
+(* Note: it should use drops_split_trans_pair2 *)
lemma cpg_lifts: ∀c,h,G. d_liftable2 (cpg h c G).
#c #h #G #K #T generalize in match c; -c
@(fqup_wf_ind_eq … G K T) -G -K -T #G0 #K0 #T0 #IH #G #K * *
elim (lifts_inv_lref1 … H1) -H1 #i2 #Hf #H destruct
lapply (drops_trans … HLK … HK0 ??) -HLK [3,6: |*: // ] #H
elim (drops_split_trans … H) -H [1,6: |*: /2 width=6 by after_uni_dx/ ] #Y #HL0 #HY
- lapply (drops_inv_tls_at … Hf … HY) -HY #HY
+ lapply (drops_tls_at … Hf … HY) -HY #HY
elim (drops_inv_skip2 … HY) -HY #L0 #W #HLK0 #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 (𝐔❴⫯i2❵)) #U2 #HWU2
elim (lifts_inv_lref2 … H1) -H1 #i1 #Hf #H destruct
lapply (drops_split_div … HLK (𝐔❴i1❵) ???) -HLK [4,8: * |*: // ] #Y0 #HK0 #HLY0
lapply (drops_conf … HL0 … HLY0 ??) -HLY0 [3,6: |*: /2 width=6 by after_uni_dx/ ] #HLY0
- lapply (drops_inv_tls_at … Hf … HLY0) -HLY0 #HLY0
+ lapply (drops_tls_at … Hf … HLY0) -HLY0 #HLY0
elim (drops_inv_skip1 … HLY0) -HLY0 #K0 #V #HLK0 #HVW #H destruct
elim (IH … HW2 … HLK0 … HVW) -IH /2 width=2 by fqup_lref/ -L -L0 -W #V2 #HVW2 #HV2
lapply (lifts_trans … HVW2 … HWU2 ??) -W2 [3,6: |*: // ] #HVU2