(* *)
(**************************************************************************)
-include "Basic_2/unfold/ltpss_ltpss.ma".
-include "Basic_2/reducibility/ltpr_ldrop.ma".
+include "basic_2/unfold/ltpss_ltpss.ma".
+include "basic_2/reducibility/ltpr_ldrop.ma".
(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
(* Basic_1: was: pr0_subst1 *)
lemma tpr_tps_ltpr: ∀T1,T2. T1 ➡ T2 →
- ∀L1,d,e,U1. L1 ⊢ T1 [d, e] ▶ U1 →
+ ∀L1,d,e,U1. L1 ⊢ T1 ▶ [d, e] U1 →
∀L2. L1 ➡ L2 →
- ∃∃U2. U1 ➡ U2 & L2 ⊢ T2 [d, e] ▶* U2.
+ ∃∃U2. U1 ➡ U2 & L2 ⊢ T2 ▶* [d, e] U2.
#T1 #T2 #H elim H -T1 -T2
[ #I #L1 #d #e #X #H
elim (tps_inv_atom1 … H) -H
elim (tpss_strip_neq … HTT2 … HTU2 ?) -T2 /2 width=1/ #T2 #HTT2 #HUT2
lapply (tps_lsubs_conf … HTT2 (L2. ⓑ{I} V2) ?) -HTT2 /2 width=1/ #HTT2
elim (ltpss_tps_conf … HTT2 (L2. ⓑ{I} VV2) (d + 1) e ?) -HTT2 /2 width=1/ #W2 #HTTW2 #HTW2
- lapply (tpss_lsubs_conf … HTTW2 (⋆. ⓑ{I} VV2) ?) -HTTW2 /2 width=1/ #HTTW2
- lapply (tpss_tps … HTTW2) -HTTW2 #HTTW2
+ lapply (tps_lsubs_conf … HTTW2 (⋆. ⓑ{I} VV2) ?) -HTTW2 /2 width=1/ #HTTW2
lapply (tpss_lsubs_conf … HTW2 (L2. ⓑ{I} VV2) ?) -HTW2 /2 width=1/ #HTW2
lapply (tpss_trans_eq … HUT2 … HTW2) -T2 /3 width=5/
| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #L1 #d #e #X #H #L2 #HL12
qed.
lemma tpr_tps_bind: ∀I,V1,V2,T1,T2,U1. V1 ➡ V2 → T1 ➡ T2 →
- ⋆. ⓑ{I} V1 ⊢ T1 [0, 1] ▶ U1 →
- ∃∃U2. U1 ➡ U2 & ⋆. ⓑ{I} V2 ⊢ T2 [0, 1] ▶ U2.
+ ⋆. ⓑ{I} V1 ⊢ T1 ▶ [0, 1] U1 →
+ ∃∃U2. U1 ➡ U2 & ⋆. ⓑ{I} V2 ⊢ T2 ▶ [0, 1] U2.
#I #V1 #V2 #T1 #T2 #U1 #HV12 #HT12 #HTU1
elim (tpr_tps_ltpr … HT12 … HTU1 (⋆. ⓑ{I} V2) ?) -T1 /2 width=1/ /3 width=3/
qed.
lemma tpr_tpss_ltpr: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. T1 ➡ T2 →
- ∀d,e,U1. L1 ⊢ T1 [d, e] ▶* U1 →
- ∃∃U2. U1 ➡ U2 & L2 ⊢ T2 [d, e] ▶* U2.
+ ∀d,e,U1. L1 ⊢ T1 ▶* [d, e] U1 →
+ ∃∃U2. U1 ➡ U2 & L2 ⊢ T2 ▶* [d, e] U2.
#L1 #L2 #HL12 #T1 #T2 #HT12 #d #e #U1 #HTU1 @(tpss_ind … HTU1) -U1
[ /2 width=3/
| -HT12 #U #U1 #_ #HU1 * #T #HUT #HT2
lapply (tpss_trans_eq … HT2 … HTU2) -T /2 width=3/
]
qed.
+
+lemma tpr_tpss_conf: ∀T1,T2. T1 ➡ T2 →
+ ∀L,U1,d,e. L ⊢ T1 ▶* [d, e] U1 →
+ ∃∃U2. U1 ➡ U2 & L ⊢ T2 ▶* [d, e] U2.
+/2 width=5/ qed.