--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/lpss_cpss.ma".
+
+(* SN PARALLEL SUBSTITUTION ON LOCAL ENVIRONMENTS ***************************)
+
+(* Main properties on context-sensitive parallel substitution for terms *****)
+
+fact cpss_conf_lpss_atom_atom:
+ ∀I,L1,L2. ∃∃T. L1 ⊢ ⓪{I} ▶* T & L2 ⊢ ⓪{I} ▶* T.
+/2 width=3/ qed-.
+
+fact cpss_conf_lpss_atom_delta:
+ ∀L0,i. (
+ ∀L,T. ⦃L0, #i⦄ ⊃+ ⦃L, T⦄ →
+ ∀T1. L ⊢ T ▶* T1 → ∀T2. L ⊢ T ▶* T2 →
+ ∀L1. L ⊢ ▶* L1 → ∀L2. L ⊢ ▶* L2 →
+ ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ▶* T0
+ ) →
+ ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
+ ∀V2. K0 ⊢ V0 ▶* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
+ ∀L1. L0 ⊢ ▶* L1 → ∀L2. L0 ⊢ ▶* L2 →
+ ∃∃T. L1 ⊢ #i ▶* T & L2 ⊢ T2 ▶* T.
+#L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
+elim (lpss_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
+elim (lpss_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct
+elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
+elim (lpss_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
+lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
+lapply (fsupp_lref … HLK0) -HLK0 #HLK0
+elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
+elim (lift_total V 0 (i+1)) #T #HVT
+lapply (cpss_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 /3 width=6/
+qed-.
+
+fact cpss_conf_lpss_delta_delta:
+ ∀L0,i. (
+ ∀L,T. ⦃L0, #i⦄ ⊃+ ⦃L, T⦄ →
+ ∀T1. L ⊢ T ▶* T1 → ∀T2. L ⊢ T ▶* T2 →
+ ∀L1. L ⊢ ▶* L1 → ∀L2. L ⊢ ▶* L2 →
+ ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ▶* T0
+ ) →
+ ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
+ ∀V1. K0 ⊢ V0 ▶* V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
+ ∀KX,VX. ⇩[O, i] L0 ≡ KX.ⓓVX →
+ ∀V2. KX ⊢ VX ▶* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
+ ∀L1. L0 ⊢ ▶* L1 → ∀L2. L0 ⊢ ▶* L2 →
+ ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ T2 ▶* T.
+#L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
+#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
+lapply (ldrop_mono … H … HLK0) -H #H destruct
+elim (lpss_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
+elim (lpss_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
+lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1
+elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
+elim (lpss_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
+lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
+lapply (fsupp_lref … HLK0) -HLK0 #HLK0
+elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
+elim (lift_total V 0 (i+1)) #T #HVT
+lapply (cpss_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1
+lapply (cpss_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/
+qed-.
+
+fact cpss_conf_lpss_bind_bind:
+ ∀a,I,L0,V0,T0. (
+ ∀L,T. ⦃L0,ⓑ{a,I}V0.T0⦄ ⊃+ ⦃L,T⦄ →
+ ∀T1. L ⊢ T ▶* T1 → ∀T2. L ⊢ T ▶* T2 →
+ ∀L1. L ⊢ ▶* L1 → ∀L2. L ⊢ ▶* L2 →
+ ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ▶* T0
+ ) →
+ ∀V1. L0 ⊢ V0 ▶* V1 → ∀T1. L0.ⓑ{I}V0 ⊢ T0 ▶* T1 →
+ ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0.ⓑ{I}V0 ⊢ T0 ▶* T2 →
+ ∀L1. L0 ⊢ ▶* L1 → ∀L2. L0 ⊢ ▶* L2 →
+ ∃∃T. L1 ⊢ ⓑ{a,I}V1.T1 ▶* T & L2 ⊢ ⓑ{a,I}V2.T2 ▶* T.
+#a #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) //
+elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH // /2 width=1/ /3 width=5/
+qed-.
+
+fact cpss_conf_lpss_flat_flat:
+ ∀I,L0,V0,T0. (
+ ∀L,T. ⦃L0,ⓕ{I}V0.T0⦄ ⊃+ ⦃L,T⦄ →
+ ∀T1. L ⊢ T ▶* T1 → ∀T2. L ⊢ T ▶* T2 →
+ ∀L1. L ⊢ ▶* L1 → ∀L2. L ⊢ ▶* L2 →
+ ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ▶* T0
+ ) →
+ ∀V1. L0 ⊢ V0 ▶* V1 → ∀T1. L0 ⊢ T0 ▶* T1 →
+ ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ T0 ▶* T2 →
+ ∀L1. L0 ⊢ ▶* L1 → ∀L2. L0 ⊢ ▶* L2 →
+ ∃∃T. L1 ⊢ ⓕ{I}V1.T1 ▶* T & L2 ⊢ ⓕ{I}V2.T2 ▶* T.
+#I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+elim (IH … HV01 … HV02 … HL01 … HL02) //
+elim (IH … HT01 … HT02 … HL01 … HL02) // /3 width=5/
+qed-.
+
+theorem cpss_conf_lpss: lpx_sn_confluent cpss cpss.
+#L0 #T0 @(fsupp_wf_ind … L0 T0) -L0 -T0 #L #T #IH #L0 * [|*]
+[ #I0 #HL #HT #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpss_inv_atom1 … H1) -H1
+ elim (cpss_inv_atom1 … H2) -H2
+ [ #H2 #H1 destruct
+ /2 width=1 by cpss_conf_lpss_atom_atom/
+ | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
+ /3 width=10 by cpss_conf_lpss_atom_delta/
+ | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
+ /4 width=10 by ex2_commute, cpss_conf_lpss_atom_delta/
+ | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
+ * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
+ /3 width=17 by cpss_conf_lpss_delta_delta/
+ ]
+| #a #I #V0 #T0 #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpss_inv_bind1 … H1) -H1 #V1 #T1 #HV01 #HT01 #H destruct
+ elim (cpss_inv_bind1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H destruct
+ /3 width=10 by cpss_conf_lpss_bind_bind/
+| #I #V0 #T0 #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
+ elim (cpss_inv_flat1 … H1) -H1 #V1 #T1 #HV01 #HT01 #H destruct
+ elim (cpss_inv_flat1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H destruct
+ /3 width=10 by cpss_conf_lpss_flat_flat/
+]
+qed-.
+
+(* Basic_1: was only: subst1_confluence_eq *)
+theorem cpss_conf: ∀L. confluent … (cpss L).
+/2 width=6 by cpss_conf_lpss/ qed-.
+
+(* Properties on context-sensitive parallel substitution for terms **********)
+
+(* Basic_1: was only: subst1_subst1_back *)
+lemma lpss_cpss_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ▶* T1 → ∀L1. L0 ⊢ ▶* L1 →
+ ∃∃T. L1 ⊢ T0 ▶* T & L1 ⊢ T1 ▶* T.
+#L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpss_conf_lpss … HT01 T0 … HL01 … HL01) // -L0 /2 width=3/
+qed-.
+
+lemma lpss_cpss_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ▶* T1 → ∀L1. L0 ⊢ ▶* L1 →
+ ∃∃T. L1 ⊢ T0 ▶* T & L0 ⊢ T1 ▶* T.
+#L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpss_conf_lpss … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem lpss_conf: confluent … lpss.
+/3 width=6 by lpx_sn_conf, cpss_conf_lpss/
+qed-.
+
+theorem lpss_trans: Transitive … lpss.
+/3 width=5 by lpx_sn_trans, cpss_trans_lpss/
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma cpss_fwd_shift1: ∀L1,L,T1,T. L ⊢ L1 @@ T1 ▶* T →
+ ∃∃L2,T2. L @@ L1 ⊢ ▶* L @@ L2 & L @@ L1 ⊢ T1 ▶* T2 &
+ T = L2 @@ T2.
+#L1 @(lenv_ind_dx … L1) -L1
+[ #L #T1 #T #HT1
+ @ex3_2_intro [3: // |4,5: // |1,2: skip ] (**) (* /2 width=4/ does not work *)
+| #I #L1 #V1 #IH #L #T1 #T >shift_append_assoc #H <append_assoc
+ elim (cpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ elim (IH … HT12) -IH -HT12 #L2 #T #HL12 #HT1 #H destruct
+ lapply (lpss_trans … HL12 (L.ⓑ{I}V2@@L2) ?) -HL12 /3 width=1/ #HL12
+ @(ex3_2_intro … (⋆.ⓑ{I}V2@@L2)) [4: /2 width=3/ | skip ] <append_assoc // (**) (* explicit constructor *)
+]
+qed-.
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