(* Properties concerning partial unfold on terms ****************************)
-lemma ltpss_tpss_conf_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 [d2, e2] ▶* U2 →
- ∀L1,d1,e1. L0 [d1, e1] ▶* L1 → d1 + e1 ≤ d2 →
- L1 ⊢ T2 [d2, e2] ▶* U2.
+lemma ltpss_tpss_conf_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
+ ∀L1,d1,e1. L0 ▶* [d1, e1] L1 → d1 + e1 ≤ d2 →
+ L1 ⊢ T2 ▶* [d2, e2] U2.
#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind … H) -U2 //
#U #U2 #_ #HU2 #IHU
lapply (ltpss_tps_conf_ge … HU2 … HL01 ?) -L0 // -Hde1d2 /2 width=3/
qed.
(* Basic_1: was: subst1_subst1_back *)
-lemma ltpss_tps_conf: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 [d2, e2] ▶ U2 →
- ∀L1,d1,e1. L0 [d1, e1] ▶* L1 →
- ∃∃T. L1 ⊢ T2 [d2, e2] ▶ T &
- L1 ⊢ U2 [d1, e1] ▶* T.
+lemma ltpss_tps_conf: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
+ ∀L1,d1,e1. L0 ▶* [d1, e1] L1 →
+ ∃∃T. L1 ⊢ T2 ▶ [d2, e2] T &
+ L1 ⊢ U2 ▶* [d1, e1] T.
#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
[ /2 width=3/
| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01
| lapply (ltpss_ldrop_conf_ge … HL01 … HLK0 ?) -L0 // /3 width=4/
]
]
-| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
+| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
elim (IHVW2 … HL01) -IHVW2 #V #HV2 #HVW2
elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL01 /3 width=5/
| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
]
qed.
-lemma ltpss_tpss_trans_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 [d2, e2] ▶* U2 →
- ∀L1,d1,e1. L1 [d1, e1] ▶* L0 → d1 + e1 ≤ d2 →
- L1 ⊢ T2 [d2, e2] ▶* U2.
+lemma ltpss_tpss_trans_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶* [d2, e2] U2 →
+ ∀L1,d1,e1. L1 ▶* [d1, e1] L0 → d1 + e1 ≤ d2 →
+ L1 ⊢ T2 ▶* [d2, e2] U2.
#L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 #Hde1d2 @(tpss_ind … H) -U2 //
#U #U2 #_ #HU2 #IHU
lapply (ltpss_tps_trans_ge … HU2 … HL01 ?) -L0 // -Hde1d2 /2 width=3/
qed.
(* Basic_1: was: subst1_subst1 *)
-lemma ltpss_tps_trans: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 [d2, e2] ▶ U2 →
- ∀L1,d1,e1. L1 [d1, e1] ▶* L0 →
- ∃∃T. L1 ⊢ T2 [d2, e2] ▶ T &
- L0 ⊢ T [d1, e1] ▶* U2.
+lemma ltpss_tps_trans: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 ▶ [d2, e2] U2 →
+ ∀L1,d1,e1. L1 ▶* [d1, e1] L0 →
+ ∃∃T. L1 ⊢ T2 ▶ [d2, e2] T &
+ L0 ⊢ T ▶* [d1, e1] U2.
#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
[ /2 width=3/
| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10
| lapply (ltpss_ldrop_trans_ge … HL10 … HLK0 ?) -HL10 -HLK0 // /3 width=4/
]
]
-| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10
+| #L0 #a #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10
elim (IHVW2 … HL10) -IHVW2 #V #HV2 #HVW2
elim (IHTU2 (L1. ⓑ{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL10 /3 width=5/
| #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10