(* Properties with unbound parallel rt-transition on all entries ************)
-lemma csx_lpx_conf (h) (G):
- ∀L1,T. ⦃G,L1⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ →
- â\88\80L2. â¦\83G,L1â¦\84 â\8a¢ â¬\88[h] L2 â\86\92 â¦\83G,L2â¦\84 â\8a¢ â¬\88*[h] ð\9d\90\92â¦\83Tâ¦\84.
+lemma csx_lpx_conf (h) (G) (L1):
+ ∀T. ❪G,L1❫ ⊢ ⬈*𝐒[h] T →
+ â\88\80L2. â\9dªG,L1â\9d« â\8a¢ â¬\88[h] L2 â\86\92 â\9dªG,L2â\9d« â\8a¢ â¬\88*ð\9d\90\92[h] T.
#h #G #L1 #T #H @(csx_ind_cpxs … H) -T
/4 width=3 by csx_intro, lpx_cpx_trans/
qed-.
(* Advanced properties ******************************************************)
-lemma csx_abst (h) (G):
- ∀p,L,W. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃W⦄ →
- â\88\80T. â¦\83G,L.â\93\9bWâ¦\84 â\8a¢ â¬\88*[h] ð\9d\90\92â¦\83Tâ¦\84 â\86\92 â¦\83G,Lâ¦\84 â\8a¢ â¬\88*[h] ð\9d\90\92â¦\83â\93\9b{p}W.Tâ¦\84.
-#h #G #p #L #W #HW
+lemma csx_abst (h) (G) (L):
+ ∀p,W. ❪G,L❫ ⊢ ⬈*𝐒[h] W →
+ â\88\80T. â\9dªG,L.â\93\9bWâ\9d« â\8a¢ â¬\88*ð\9d\90\92[h] T â\86\92 â\9dªG,Lâ\9d« â\8a¢ â¬\88*ð\9d\90\92[h] â\93\9b[p]W.T.
+#h #G #L #p #W #HW
@(csx_ind … HW) -W #W #_ #IHW #T #HT
@(csx_ind … HT) -T #T #HT #IHT
@csx_intro #X #H1 #H2
]
qed.
-lemma csx_abbr (h) (G):
- ∀p,L,V. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ →
- â\88\80T. â¦\83G,L.â\93\93Vâ¦\84 â\8a¢ â¬\88*[h] ð\9d\90\92â¦\83Tâ¦\84 â\86\92 â¦\83G,Lâ¦\84 â\8a¢ â¬\88*[h] ð\9d\90\92â¦\83â\93\93{p}V.Tâ¦\84.
-#h #G #p #L #V #HV
+lemma csx_abbr (h) (G) (L):
+ ∀p,V. ❪G,L❫ ⊢ ⬈*𝐒[h] V →
+ â\88\80T. â\9dªG,L.â\93\93Vâ\9d« â\8a¢ â¬\88*ð\9d\90\92[h] T â\86\92 â\9dªG,Lâ\9d« â\8a¢ â¬\88*ð\9d\90\92[h] â\93\93[p]V.T.
+#h #G #L #p #V #HV
@(csx_ind … HV) -V #V #_ #IHV #T #HT
@(csx_ind_cpxs … HT) -T #T #HT #IHT
@csx_intro #X #H1 #H2
]
qed.
-lemma csx_bind (h) (G):
- ∀p,I,L,V. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ →
- â\88\80T. â¦\83G,L.â\93\91{I}Vâ¦\84 â\8a¢ â¬\88*[h] ð\9d\90\92â¦\83Tâ¦\84 â\86\92 â¦\83G,Lâ¦\84 â\8a¢ â¬\88*[h] ð\9d\90\92â¦\83â\93\91{p,I}V.Tâ¦\84.
-#h #G #p * #L #V #HV #T #HT
+lemma csx_bind (h) (G) (L):
+ ∀p,I,V. ❪G,L❫ ⊢ ⬈*𝐒[h] V →
+ â\88\80T. â\9dªG,L.â\93\91[I]Vâ\9d« â\8a¢ â¬\88*ð\9d\90\92[h] T â\86\92 â\9dªG,Lâ\9d« â\8a¢ â¬\88*ð\9d\90\92[h] â\93\91[p,I]V.T.
+#h #G #L #p * #V #HV #T #HT
/2 width=1 by csx_abbr, csx_abst/
qed.
-fact csx_appl_theta_aux (h) (G):
- ∀p,L,U. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ → ∀V1,V2. ⇧*[1] V1 ≘ V2 →
- ∀V,T. U = ⓓ{p}V.ⓐV2.T → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV1.ⓓ{p}V.T⦄.
-#h #G #p #L #X #H
+fact csx_appl_theta_aux (h) (G) (L):
+ ∀p,U. ❪G,L❫ ⊢ ⬈*𝐒[h] U → ∀V1,V2. ⇧[1] V1 ≘ V2 →
+ ∀V,T. U = ⓓ[p]V.ⓐV2.T → ❪G,L❫ ⊢ ⬈*𝐒[h] ⓐV1.ⓓ[p]V.T.
+#h #G #L #p #X #H
@(csx_ind_cpxs … H) -X #X #HVT #IHVT #V1 #V2 #HV12 #V #T #H destruct
lapply (csx_fwd_pair_sn … HVT) #HV
lapply (csx_fwd_bind_dx … HVT) -HVT #HVT
elim (cpx_inv_abbr1 … HL) -HL *
[ #V3 #T3 #HV3 #HLT3 #H0 destruct
elim (cpx_lifts_sn … HLV10 (Ⓣ) … (L.ⓓV) … HV12) -HLV10 /3 width=1 by drops_refl, drops_drop/ #V4 #HV04 #HV24
- elim (teqx_dec (ⓓ{p}V.ⓐV2.T) (ⓓ{p}V3.ⓐV4.T3)) #H0
+ elim (teqx_dec (ⓓ[p]V.ⓐV2.T) (ⓓ[p]V3.ⓐV4.T3)) #H0
[ -IHVT -HV3 -HV24 -HLT3
elim (teqx_inv_pair … H0) -H0 #_ #HV3 #H0
elim (teqx_inv_pair … H0) -H0 #_ #HV24 #HT3
]
qed-.
-lemma csx_appl_theta (h) (G):
- ∀p,L,V,V2,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓓ{p}V.ⓐV2.T⦄ →
- ∀V1. ⇧*[1] V1 ≘ V2 → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV1.ⓓ{p}V.T⦄.
+lemma csx_appl_theta (h) (G) (L):
+ ∀p,V,V2,T. ❪G,L❫ ⊢ ⬈*𝐒[h] ⓓ[p]V.ⓐV2.T →
+ ∀V1. ⇧[1] V1 ≘ V2 → ❪G,L❫ ⊢ ⬈*𝐒[h] ⓐV1.ⓓ[p]V.T.
/2 width=5 by csx_appl_theta_aux/ qed.