(* Properties on partial unfold on terms ************************************)
-lemma delift_tpss_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 [d, e] ▶* U2 →
- ∀T1,dd,ee. L ⊢ U1 [dd, ee] ≡ T1 →
+lemma delift_tpss_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
∀K. ⇩[dd, ee] L ≡ K → d + e ≤ dd →
- ∃∃T2. K ⊢ T1 [d, e] ▶* T2 & L ⊢ U2 [dd, ee] ≡ T2.
+ ∃∃T2. K ⊢ T1 ▶* [d, e] T2 & L ⊢ ▼*[dd, ee] U2 ≡ T2.
#L #U1 #U2 #d #e #HU12 #T1 #dd #ee * #X1 #HUX1 #HTX1 #K #HLK #H1
elim (tpss_conf_eq … HU12 … HUX1) -U1 #U1 #HU21 #HXU1
elim (tpss_inv_lift1_le … HXU1 … HLK … HTX1 ?) -X1 -HLK // -H1 /3 width=5/
qed.
-lemma delift_tps_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 [d, e] ▶ U2 →
- ∀T1,dd,ee. L ⊢ U1 [dd, ee] ≡ T1 →
+lemma delift_tps_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
∀K. ⇩[dd, ee] L ≡ K → d + e ≤ dd →
- ∃∃T2. K ⊢ T1 [d, e] ▶* T2 & L ⊢ U2 [dd, ee] ≡ T2.
+ ∃∃T2. K ⊢ T1 ▶* [d, e] T2 & L ⊢ ▼*[dd, ee] U2 ≡ T2.
/3 width=3/ qed.
-lemma delift_tpss_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 [d, e] ▶* U2 →
- ∀T1,dd,ee. L ⊢ U1 [dd, ee] ≡ T1 →
+lemma delift_tpss_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
∀K. ⇩[dd, ee] L ≡ K →
d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
- ∃∃T2. K ⊢ T1 [d, dd - d] ▶* T2 &
- L ⊢ U2 [dd, ee] ≡ T2.
+ ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
#L #U1 #U2 #d #e #HU12 #T1 #dd #ee * #X1 #HUX1 #HTX1 #K #HLK #H1 #H2 #H3
elim (tpss_conf_eq … HU12 … HUX1) -U1 #U1 #HU21 #HXU1
elim (tpss_inv_lift1_le_up … HXU1 … HLK … HTX1 ? ? ?) -X1 -HLK // -H1 -H2 -H3 /3 width=5/
qed.
-lemma delift_tps_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 [d, e] ▶ U2 →
- ∀T1,dd,ee. L ⊢ U1 [dd, ee] ≡ T1 →
+lemma delift_tps_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
∀K. ⇩[dd, ee] L ≡ K →
d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
- ∃∃T2. K ⊢ T1 [d, dd - d] ▶* T2 &
- L ⊢ U2 [dd, ee] ≡ T2.
+ ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
/3 width=6/ qed.
-lemma delift_tpss_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 [d, e] ▶* U2 →
- ∀T1,dd,ee. L ⊢ U1 [dd, ee] ≡ T1 →
+lemma delift_tpss_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
∀K. ⇩[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
- ∃∃T2. K ⊢ T1 [d, e - ee] ▶* T2 &
- L ⊢ U2 [dd, ee] ≡ T2.
+ ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
#L #U1 #U2 #d #e #HU12 #T1 #dd #ee * #X1 #HUX1 #HTX1 #K #HLK #H1 #H2
elim (tpss_conf_eq … HU12 … HUX1) -U1 #U1 #HU21 #HXU1
elim (tpss_inv_lift1_be … HXU1 … HLK … HTX1 ? ?) -X1 -HLK // -H1 -H2 /3 width=5/
qed.
-lemma delift_tps_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 [d, e] ▶ U2 →
- ∀T1,dd,ee. L ⊢ U1 [dd, ee] ≡ T1 →
+lemma delift_tps_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
+ ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
∀K. ⇩[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
- ∃∃T2. K ⊢ T1 [d, e - ee] ▶* T2 &
- L ⊢ U2 [dd, ee] ≡ T2.
+ ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
+ L ⊢ ▼*[dd, ee] U2 ≡ T2.
/3 width=3/ qed.
-lemma delift_tpss_conf_eq: ∀L,U1,U2,d,e. L ⊢ U1 [d, e] ▶* U2 →
- ∀T. L ⊢ U1 [d, e] ≡ T → L ⊢ U2 [d, e] ≡ T.
+lemma delift_tpss_conf_eq: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
+ ∀T. L ⊢ ▼*[d, e] U1 ≡ T → L ⊢ ▼*[d, e] U2 ≡ T.
#L #U1 #U2 #d #e #HU12 #T * #X1 #HUX1 #HTX1
elim (tpss_conf_eq … HU12 … HUX1) -U1 #U1 #HU21 #HXU1
lapply (tpss_inv_lift1_eq … HXU1 … HTX1) -HXU1 #H destruct /2 width=3/
qed.
-lemma delift_tps_conf_eq: ∀L,U1,U2,d,e. L ⊢ U1 [d, e] ▶ U2 →
- ∀T. L ⊢ U1 [d, e] ≡ T → L ⊢ U2 [d, e] ≡ T.
+lemma delift_tps_conf_eq: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
+ ∀T. L ⊢ ▼*[d, e] U1 ≡ T → L ⊢ ▼*[d, e] U2 ≡ T.
/3 width=3/ qed.
-lemma tpss_delift_trans_eq: ∀L,U1,U2,d,e. L ⊢ U1 [d, e] ▶* U2 →
- ∀T. L ⊢ U2 [d, e] ≡ T → L ⊢ U1 [d, e] ≡ T.
+lemma tpss_delift_trans_eq: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
+ ∀T. L ⊢ ▼*[d, e] U2 ≡ T → L ⊢ ▼*[d, e] U1 ≡ T.
#L #U1 #U2 #d #e #HU12 #T * #X1 #HUX1 #HTX1
lapply (tpss_trans_eq … HU12 … HUX1) -U2 /2 width=3/
qed.
-lemma tps_delift_trans_eq: ∀L,U1,U2,d,e. L ⊢ U1 [d, e] ▶ U2 →
- ∀T. L ⊢ U2 [d, e] ≡ T → L ⊢ U1 [d, e] ≡ T.
+lemma tps_delift_trans_eq: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
+ ∀T. L ⊢ ▼*[d, e] U2 ≡ T → L ⊢ ▼*[d, e] U1 ≡ T.
/3 width=3/ qed.