(* Advanced properties ******************************************************)
lemma cpg_delta_drops (Rs) (Rk) (c) (G) (K):
- â\88\80V,V2,i,L,T2. â\87©[i] L â\89\98 K.â\93\93V â\86\92 â\9dªG,Kâ\9d« ⊢ V ⬈[Rs,Rk,c] V2 →
- â\87§[â\86\91i] V2 â\89\98 T2 â\86\92 â\9dªG,Lâ\9d« ⊢ #i ⬈[Rs,Rk,c] T2.
+ â\88\80V,V2,i,L,T2. â\87©[i] L â\89\98 K.â\93\93V â\86\92 â\9d¨G,Kâ\9d© ⊢ V ⬈[Rs,Rk,c] V2 →
+ â\87§[â\86\91i] V2 â\89\98 T2 â\86\92 â\9d¨G,Lâ\9d© ⊢ #i ⬈[Rs,Rk,c] T2.
#Rs #Rk #c #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 (Rs) (Rk) (c) (G) (K):
- â\88\80V,V2,i,L,T2. â\87©[i] L â\89\98 K.â\93\9bV â\86\92 â\9dªG,Kâ\9d« ⊢ V ⬈[Rs,Rk,c] V2 →
- â\87§[â\86\91i] V2 â\89\98 T2 â\86\92 â\9dªG,Lâ\9d« ⊢ #i ⬈[Rs,Rk,c+𝟘𝟙] T2.
+ â\88\80V,V2,i,L,T2. â\87©[i] L â\89\98 K.â\93\9bV â\86\92 â\9d¨G,Kâ\9d© ⊢ V ⬈[Rs,Rk,c] V2 →
+ â\87§[â\86\91i] V2 â\89\98 T2 â\86\92 â\9d¨G,Lâ\9d© ⊢ #i ⬈[Rs,Rk,c+𝟘𝟙] T2.
#Rs #Rk #c #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 (Rs) (Rk) (c) (G):
- â\88\80i,L,T2. â\9dªG,Lâ\9d« ⊢ #i ⬈[Rs,Rk,c] T2 →
+ â\88\80i,L,T2. â\9d¨G,Lâ\9d© ⊢ #i ⬈[Rs,Rk,c] 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\9dªG,Kâ\9d« ⊢ V ⬈[Rs,Rk,cV] 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\9dªG,Kâ\9d« ⊢ V ⬈[Rs,Rk,cV] V2 & ⇧[↑i] V2 ≘ T2 & c = cV + 𝟘𝟙.
+ | â\88\83â\88\83cV,K,V,V2. â\87©[i] L â\89\98 K.â\93\93V & â\9d¨G,Kâ\9d© ⊢ V ⬈[Rs,Rk,cV] 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\9d¨G,Kâ\9d© ⊢ V ⬈[Rs,Rk,cV] V2 & ⇧[↑i] V2 ≘ T2 & c = cV + 𝟘𝟙.
#Rs #Rk #c #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 (Rs) (Rk) (c) (G) (L):
- â\88\80I,T2. â\9dªG,Lâ\9d« ⊢ ⓪[I] ⬈[Rs,Rk,c] T2 →
+ â\88\80I,T2. â\9d¨G,Lâ\9d© ⊢ ⓪[I] ⬈[Rs,Rk,c] T2 →
∨∨ ∧∧ T2 = ⓪[I] & c = 𝟘𝟘
| ∃∃s1,s2. Rs s1 s2 & T2 = ⋆s2 & I = Sort s1 & c = 𝟘𝟙
- | 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 ⬈[Rs,Rk,cV] 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\9dªG,Kâ\9d« ⊢ V ⬈[Rs,Rk,cV] 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\98 K.â\93\93V & â\9d¨G,Kâ\9d© ⊢ V ⬈[Rs,Rk,cV] 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\9d¨G,Kâ\9d© ⊢ V ⬈[Rs,Rk,cV] V2 & ⇧[↑i] V2 ≘ T2 & I = LRef i & c = cV + 𝟘𝟙.
#Rs #Rk #c #G #L * #x #T2 #H
[ elim (cpg_inv_sort1 … H) -H *
/3 width=5 by or4_intro0, or4_intro1, ex4_2_intro, conj/