(* 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_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶▶* [d, e] T2 →
+ ∀L2. L1 ≼ [d, e] L2 → 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
]
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/
]
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 →
+ (∀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,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-.