| tpssa_subst: ∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
⇩[0, i] L ≡ K. ⓓV1 → tpssa 0 (d + e - i - 1) K V1 V2 →
⇧[0, i + 1] V2 ≡ W2 → tpssa d e L (#i) W2
-| tpssa_bind : ∀L,I,V1,V2,T1,T2,d,e.
+| tpssa_bind : ∀L,a,I,V1,V2,T1,T2,d,e.
tpssa d e L V1 V2 → tpssa (d + 1) e (L. ⓑ{I} V2) T1 T2 →
- tpssa d e L (ⓑ{I} V1. T1) (ⓑ{I} V2. T2)
+ tpssa d e L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
| tpssa_flat : ∀L,I,V1,V2,T1,T2,d,e.
tpssa d e L V1 V2 → tpssa d e L T1 T2 →
tpssa d e L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
(* Basic properties *********************************************************)
-lemma tpssa_lsubs_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ▶▶* T2 →
- ∀L2. L1 [d, e] ≼ L2 → L2 ⊢ T1 [d, e] ▶▶* T2.
+lemma tpssa_lsubs_trans: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶▶* [d, e] T2 →
+ ∀L2. L2 ≼ [d, e] L1 → L2 ⊢ T1 ▶▶* [d, e] T2.
#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e
[ //
| #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
- elim (ldrop_lsubs_ldrop1_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // /3 width=6/
+ elim (ldrop_lsubs_ldrop2_abbr … HL12 … HLK1 ? ?) -HL12 -HLK1 // /3 width=6/
| /4 width=1/
| /3 width=1/
]
qed.
-lemma tpssa_refl: ∀T,L,d,e. L ⊢ T [d, e] ▶▶* T.
+lemma tpssa_refl: ∀T,L,d,e. L ⊢ T ▶▶* [d, e] T.
#T elim T -T //
#I elim I -I /2 width=1/
qed.
-lemma tpssa_tps_trans: ∀L,T1,T,d,e. L ⊢ T1 [d, e] ▶▶* T →
- ∀T2. L ⊢ T [d, e] ▶ T2 → L ⊢ T1 [d, e] ▶▶* T2.
+lemma tpssa_tps_trans: ∀L,T1,T,d,e. L ⊢ T1 ▶▶* [d, e] T →
+ ∀T2. L ⊢ T ▶ [d, e] T2 → L ⊢ T1 ▶▶* [d, e] T2.
#L #T1 #T #d #e #H elim H -L -T1 -T -d -e
[ #L #I #d #e #X #H
elim (tps_inv_atom1 … H) -H // * /2 width=6/
lapply (ldrop_fwd_ldrop2 … HLK) #H0LK
lapply (tps_weak … H 0 (d+e) ? ?) -H // #H
elim (tps_inv_lift1_be … H … H0LK … HVW2 ? ?) -H -H0LK -HVW2 // /3 width=6/
-| #L #I #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H
+| #L #a #I #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H
elim (tps_inv_bind1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
- lapply (tps_lsubs_conf … HT2 (L.ⓑ{I}V) ?) -HT2 /2 width=1/ #HT2
+ lapply (tps_lsubs_trans … HT2 (L.ⓑ{I}V) ?) -HT2 /2 width=1/ #HT2
lapply (IHV1 … HV2) -IHV1 -HV2 #HV12
lapply (IHT1 … HT2) -IHT1 -HT2 #HT12
- lapply (tpssa_lsubs_conf … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1/
+ lapply (tpssa_lsubs_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1/
| #L #I #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H
elim (tps_inv_flat1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct /3 width=1/
]
qed.
-lemma tpss_tpssa: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ▶* T2 → L ⊢ T1 [d, e] ▶▶* T2.
+lemma tpss_tpssa: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → L ⊢ T1 ▶▶* [d, e] T2.
#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 // /2 width=3/
qed.
(* Basic inversion lemmas ***************************************************)
-lemma tpssa_tpss: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ▶▶* T2 → L ⊢ T1 [d, e] ▶* T2.
+lemma tpssa_tpss: ∀L,T1,T2,d,e. L ⊢ T1 ▶▶* [d, e] T2 → L ⊢ T1 ▶* [d, e] T2.
#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e // /2 width=6/
qed-.
lemma tpss_ind_alt: ∀R:ℕ→ℕ→lenv→relation term.
(∀L,I,d,e. R d e L (⓪{I}) (⓪{I})) →
(∀L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
- ⇩[O, i] L ≡ K.ⓓV1 → K ⊢ V1 [O, d + e - i - 1] ▶* V2 →
+ ⇩[O, i] L ≡ K.ⓓV1 → K ⊢ V1 ▶* [O, d + e - i - 1] V2 →
⇧[O, i + 1] V2 ≡ W2 → R O (d+e-i-1) K V1 V2 → R d e L #i W2
) →
- (∀L,I,V1,V2,T1,T2,d,e. L ⊢ V1 [d, e] ▶* V2 →
- L.ⓑ{I}V2 ⊢ T1 [d + 1, e] ▶* T2 → R d e L V1 V2 →
- R (d+1) e (L.ⓑ{I}V2) T1 T2 → R d e L (ⓑ{I}V1.T1) (ⓑ{I}V2.T2)
+ (∀L,a,I,V1,V2,T1,T2,d,e. L ⊢ V1 ▶* [d, e] V2 →
+ L.ⓑ{I}V2 ⊢ T1 ▶* [d + 1, e] T2 → R d e L V1 V2 →
+ R (d+1) e (L.ⓑ{I}V2) T1 T2 → R d e L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
) →
- (∀L,I,V1,V2,T1,T2,d,e. L ⊢ V1 [d, e] ▶* V2 →
- L⊢ T1 [d, e] ▶* T2 → R d e L V1 V2 →
+ (∀L,I,V1,V2,T1,T2,d,e. L ⊢ V1 ▶* [d, e] V2 →
+ L ⊢ T1 ▶* [d, e] T2 → R d e L V1 V2 →
R d e L T1 T2 → R d e L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
) →
- ∀d,e,L,T1,T2. L ⊢ T1 [d, e] ▶* T2 → R d e L T1 T2.
+ ∀d,e,L,T1,T2. L ⊢ T1 ▶* [d, e] T2 → R d e L T1 T2.
#R #H1 #H2 #H3 #H4 #d #e #L #T1 #T2 #H elim (tpss_tpssa … H) -L -T1 -T2 -d -e
// /3 width=1 by tpssa_tpss/ /3 width=7 by tpssa_tpss/
qed-.