qed.
lemma cpg_ell_drops: ∀c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≡ K.ⓛV → ⦃G, K⦄ ⊢ V ⬈[c, h] V2 →
- ⬆*[⫯i] V2 ≡ T2 → ⦃G, L⦄ ⊢ #i ⬈[(↓c)+𝟘𝟙, h] T2.
+ ⬆*[⫯i] V2 ≡ T2 → ⦃G, L⦄ ⊢ #i ⬈[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
| ∃∃cV,K,V,V2. ⬇*[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ⬈[cV, h] V2 &
⬆*[⫯i] V2 ≡ T2 & c = cV
| ∃∃cV,K,V,V2. ⬇*[i] L ≡ K.ⓛV & ⦃G, K⦄ ⊢ V ⬈[cV, h] V2 &
- ⬆*[⫯i] V2 ≡ T2 & c = (↓cV) + 𝟘𝟙.
+ ⬆*[⫯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/
| ∃∃cV,i,K,V,V2. ⬇*[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ⬈[cV, h] V2 &
⬆*[⫯i] V2 ≡ T2 & I = LRef i & c = cV
| ∃∃cV,i,K,V,V2. ⬇*[i] L ≡ K.ⓛV & ⦃G, K⦄ ⊢ V ⬈[cV, h] V2 &
- ⬆*[⫯i] V2 ≡ T2 & I = LRef i & c = (↓cV) + 𝟘𝟙.
+ ⬆*[⫯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/