⬆*[1] V2 ≡ W2 → cpg Rt h c G (L.ⓓV1) (#0) W2
| cpg_ell : ∀c,G,L,V1,V2,W2. cpg Rt h c G L V1 V2 →
⬆*[1] V2 ≡ W2 → cpg Rt h (c+𝟘𝟙) G (L.ⓛV1) (#0) W2
-| cpg_lref : ∀c,I,G,L,V,T,U,i. cpg Rt h c G L (#i) T →
- â¬\86*[1] T â\89¡ U â\86\92 cpg Rt h c G (L.â\93\91{I}V) (#⫯i) U
+| cpg_lref : ∀c,I,G,L,T,U,i. cpg Rt h c G L (#i) T →
+ â¬\86*[1] T â\89¡ U â\86\92 cpg Rt h c G (L.â\93\98{I}) (#⫯i) U
| cpg_bind : ∀cV,cT,p,I,G,L,V1,V2,T1,T2.
cpg Rt h cV G L V1 V2 → cpg Rt h cT G (L.ⓑ{I}V1) T1 T2 →
cpg Rt h ((↓cV)∨cT) G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
L = K.ⓓV1 & J = LRef 0 & c = cV
| ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≡ T2 &
L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙
- | ∃∃I,K,V,T,i. ⦃G, K⦄ ⊢ #i ⬈[Rt, c, h] T & ⬆*[1] T ≡ T2 &
- L = K.ⓑ{I}V & J = LRef (⫯i).
+ | ∃∃I,K,T,i. ⦃G, K⦄ ⊢ #i ⬈[Rt, c, h] T & ⬆*[1] T ≡ T2 &
+ L = K.ⓘ{I} & J = LRef (⫯i).
#Rt #c #h #G #L #T1 #T2 * -c -G -L -T1 -T2
[ #I #G #L #J #H destruct /3 width=1 by or5_intro0, conj/
| #G #L #s #J #H destruct /3 width=3 by or5_intro1, ex3_intro/
| #c #G #L #V1 #V2 #W2 #HV12 #VW2 #J #H destruct /3 width=8 by or5_intro2, ex5_4_intro/
| #c #G #L #V1 #V2 #W2 #HV12 #VW2 #J #H destruct /3 width=8 by or5_intro3, ex5_4_intro/
-| #c #I #G #L #V #T #U #i #HT #HTU #J #H destruct /3 width=9 by or5_intro4, ex4_5_intro/
+| #c #I #G #L #T #U #i #HT #HTU #J #H destruct /3 width=8 by or5_intro4, ex4_4_intro/
| #cV #cT #p #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
| #cV #cT #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
| #cU #cT #G #L #U1 #U2 #T1 #T2 #_ #_ #_ #J #H destruct
L = K.ⓓV1 & J = LRef 0 & c = cV
| ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≡ T2 &
L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙
- | ∃∃I,K,V,T,i. ⦃G, K⦄ ⊢ #i ⬈[Rt, c, h] T & ⬆*[1] T ≡ T2 &
- L = K.ⓑ{I}V & J = LRef (⫯i).
+ | ∃∃I,K,T,i. ⦃G, K⦄ ⊢ #i ⬈[Rt, c, h] T & ⬆*[1] T ≡ T2 &
+ L = K.ⓘ{I} & J = LRef (⫯i).
/2 width=3 by cpg_inv_atom1_aux/ qed-.
lemma cpg_inv_sort1: ∀Rt,c,h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ⬈[Rt, c, h] T2 →
elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/
[ #s0 #H destruct /3 width=1 by or_intror, conj/
|2,3: #cV #K #V1 #V2 #_ #_ #_ #H destruct
-| #I #K #V1 #V2 #i #_ #_ #_ #H destruct
+| #I #K #T #i #_ #_ #_ #H destruct
]
qed-.
elim (cpg_inv_atom1 … H) -H * /3 width=1 by or3_intro0, conj/
[ #s #H destruct
|2,3: #cV #K #V1 #V2 #HV12 #HVT2 #H1 #_ #H2 destruct /3 width=8 by or3_intro1, or3_intro2, ex4_4_intro/
-| #I #K #V1 #V2 #i #_ #_ #_ #H destruct
+| #I #K #T #i #_ #_ #_ #H destruct
]
qed-.
lemma cpg_inv_lref1: ∀Rt,c,h,G,L,T2,i. ⦃G, L⦄ ⊢ #⫯i ⬈[Rt, c, h] T2 →
(T2 = #(⫯i) ∧ c = 𝟘𝟘) ∨
- ∃∃I,K,V,T. ⦃G, K⦄ ⊢ #i ⬈[Rt, c, h] T & ⬆*[1] T ≡ T2 & L = K.ⓑ{I}V.
+ ∃∃I,K,T. ⦃G, K⦄ ⊢ #i ⬈[Rt, c, h] T & ⬆*[1] T ≡ T2 & L = K.ⓘ{I}.
#Rt #c #h #G #L #T2 #i #H
elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/
[ #s #H destruct
|2,3: #cV #K #V1 #V2 #_ #_ #_ #H destruct
-| #I #K #V1 #V2 #j #HV2 #HVT2 #H1 #H2 destruct /3 width=7 by ex3_4_intro, or_intror/
+| #I #K #T #j #HT #HT2 #H1 #H2 destruct /3 width=6 by ex3_3_intro, or_intror/
]
qed-.
elim (cpg_inv_atom1 … H) -H * /2 width=1 by conj/
[ #s #H destruct
|2,3: #cV #K #V1 #V2 #_ #_ #_ #H destruct
-| #I #K #V1 #V2 #i #_ #_ #_ #H destruct
+| #I #K #T #i #_ #_ #_ #H destruct
]
qed-.
| #G #L #s #q #J #W #U1 #H destruct
| #c #G #L #V1 #V2 #W2 #_ #_ #q #J #W #U1 #H destruct
| #c #G #L #V1 #V2 #W2 #_ #_ #q #J #W #U1 #H destruct
-| #c #I #G #L #V #T #U #i #_ #_ #q #J #W #U1 #H destruct
+| #c #I #G #L #T #U #i #_ #_ #q #J #W #U1 #H destruct
| #cV #cT #p #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #q #J #W #U1 #H destruct /3 width=8 by ex4_4_intro, or_introl/
| #cV #cT #G #L #V1 #V2 #T1 #T2 #_ #_ #q #J #W #U1 #H destruct
| #cU #cT #G #L #U1 #U2 #T1 #T2 #_ #_ #_ #q #J #W #U1 #H destruct
| #G #L #s #W #U1 #H destruct
| #c #G #L #V1 #V2 #W2 #_ #_ #W #U1 #H destruct
| #c #G #L #V1 #V2 #W2 #_ #_ #W #U1 #H destruct
-| #c #I #G #L #V #T #U #i #_ #_ #W #U1 #H destruct
+| #c #I #G #L #T #U #i #_ #_ #W #U1 #H destruct
| #cV #cT #p #I #G #L #V1 #V2 #T1 #T2 #_ #_ #W #U1 #H destruct
| #cV #cT #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #W #U1 #H destruct /3 width=8 by or3_intro0, ex4_4_intro/
| #cV #cT #G #L #V1 #V2 #T1 #T2 #_ #_ #_ #W #U1 #H destruct
| #G #L #s #W #U1 #H destruct
| #c #G #L #V1 #V2 #W2 #_ #_ #W #U1 #H destruct
| #c #G #L #V1 #V2 #W2 #_ #_ #W #U1 #H destruct
-| #c #I #G #L #V #T #U #i #_ #_ #W #U1 #H destruct
+| #c #I #G #L #T #U #i #_ #_ #W #U1 #H destruct
| #cV #cT #p #I #G #L #V1 #V2 #T1 #T2 #_ #_ #W #U1 #H destruct
| #cV #cT #G #L #V1 #V2 #T1 #T2 #_ #_ #W #U1 #H destruct
| #cV #cT #G #L #V1 #V2 #T1 #T2 #HRt #HV12 #HT12 #W #U1 #H destruct /3 width=9 by or3_intro0, ex5_4_intro/
* #z #Y #X1 #X2 #HX12 #HXT2 #H1 #H2 destruct /3 width=5 by or3_intro1, or3_intro2, ex4_2_intro/
qed-.
-lemma cpg_inv_lref1_pair: ∀Rt,c,h,I,G,K,V,T2,i. ⦃G, K.ⓑ{I}V⦄ ⊢ #⫯i ⬈[Rt, c, h] T2 →
+lemma cpg_inv_lref1_bind: ∀Rt,c,h,I,G,K,T2,i. ⦃G, K.ⓘ{I}⦄ ⊢ #⫯i ⬈[Rt, c, h] T2 →
(T2 = #(⫯i) ∧ c = 𝟘𝟘) ∨
∃∃T. ⦃G, K⦄ ⊢ #i ⬈[Rt, c, h] T & ⬆*[1] T ≡ T2.
-#Rt #c #h #I #G #L #V #T2 #i #H elim (cpg_inv_lref1 … H) -H /2 width=1 by or_introl/
-* #Z #Y #X #T #HT #HT2 #H destruct /3 width=3 by ex2_intro, or_intror/
+#Rt #c #h #I #G #L #T2 #i #H elim (cpg_inv_lref1 … H) -H /2 width=1 by or_introl/
+* #Z #Y #T #HT #HT2 #H destruct /3 width=3 by ex2_intro, or_intror/
qed-.
(* Basic forward lemmas *****************************************************)