--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A|| This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ \ /
+ V_______________________________________________________________ *)
+
+include "lambda-delta/substitution/drop_defs.ma".
+
+(* PARALLEL SUBSTITUTION ****************************************************)
+
+inductive ps: lenv → term → nat → nat → term → Prop ≝
+| ps_sort : ∀L,k,d,e. ps L (⋆k) d e (⋆k)
+| ps_lref : ∀L,i,d,e. ps L (#i) d e (#i)
+| ps_subst: ∀L,K,V,U1,U2,i,d,e.
+ d ≤ i → i < d + e →
+ ↓[0, i] L ≡ K. 𝕓{Abbr} V → ps K V 0 (d + e - i - 1) U1 →
+ ↑[0, i + 1] U1 ≡ U2 → ps L (#i) d e U2
+| ps_bind : ∀L,I,V1,V2,T1,T2,d,e.
+ ps L V1 d e V2 → ps (L. 𝕓{I} V1) T1 (d + 1) e T2 →
+ ps L (𝕓{I} V1. T1) d e (𝕓{I} V2. T2)
+| ps_flat : ∀L,I,V1,V2,T1,T2,d,e.
+ ps L V1 d e V2 → ps L T1 d e T2 →
+ ps L (𝕗{I} V1. T1) d e (𝕗{I} V2. T2)
+.
+
+interpretation "parallel substritution" 'PSubst L T1 d e T2 = (ps L T1 d e T2).
+
+(* Basic properties *********************************************************)
+
+lemma subst_refl: ∀T,L,d,e. L ⊢ T [d, e] ≫ T.
+#T elim T -T //
+#I elim I -I /2/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma ps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
+ ∀I,V1,T1. U1 = 𝕓{I} V1. T1 →
+ ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 &
+ L. 𝕓{I} V1 ⊢ T1 [d + 1, e] ≫ T2 &
+ U2 = 𝕓{I} V2. T2.
+#d #e #L #U1 #U2 #H elim H -H d e L U1 U2
+[ #L #k #d #e #I #V1 #T1 #H destruct
+| #L #i #d #e #I #V1 #T1 #H destruct
+| #L #K #V #U1 #U2 #i #d #e #_ #_ #_ #_ #_ #_ #I #V1 #T1 #H destruct
+| #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #_ #_ #I #V #T #H destruct /2 width=5/
+| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #I #V #T #H destruct
+]
+qed.
+
+lemma subst_inv_bind1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕓{I} V1. T1 [d, e] ≫ U2 →
+ ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 &
+ L. 𝕓{I} V1 ⊢ T1 [d + 1, e] ≫ T2 &
+ U2 = 𝕓{I} V2. T2.
+/2/ qed.
+
+lemma subst_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
+ ∀I,V1,T1. U1 = 𝕗{I} V1. T1 →
+ ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 &
+ U2 = 𝕗{I} V2. T2.
+#d #e #L #U1 #U2 #H elim H -H d e L U1 U2
+[ #L #k #d #e #I #V1 #T1 #H destruct
+| #L #i #d #e #I #V1 #T1 #H destruct
+| #L #K #V #U1 #U2 #i #d #e #_ #_ #_ #_ #_ #_ #I #V1 #T1 #H destruct
+| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #I #V #T #H destruct
+| #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #_ #_ #I #V #T #H destruct /2 width=5/
+]
+qed.
+
+lemma subst_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕗{I} V1. T1 [d, e] ≫ U2 →
+ ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 &
+ U2 = 𝕗{I} V2. T2.
+/2/ qed.