--- /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 *)
+(* *)
+(**************************************************************************)
+
+notation "hvbox( L1 ⊢ ⬌* break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'CPConvStar $L1 $L2 }.
+
+include "basic_2/grammar/lenv_px_sn.ma".
+include "basic_2/equivalence/cpcs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON LOCAL ENVIRONMENTS *************)
+
+definition lcpcs: relation lenv ≝ lpx_sn … cpcs.
+
+interpretation "context-sensitive parallel equivalence (local environment)"
+ 'CPConvStar L1 L2 = (lcpcs L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lcpcs_inv_atom1: ∀L2. ⋆ ⊢ ⬌* L2 → L2 = ⋆.
+/2 width=2 by lpx_sn_inv_atom1/ qed-.
+
+lemma lcpcs_inv_pair1: ∀I,K1,V1,L2. K1. ⓑ{I} V1 ⊢ ⬌* L2 →
+ ∃∃K2,V2. K1 ⊢ ⬌* K2 & K1 ⊢ V1 ⬌* V2 & L2 = K2. ⓑ{I} V2.
+/2 width=1 by lpx_sn_inv_pair1/ qed-.
+
+lemma lcpcs_inv_atom2: ∀L1. L1 ⊢ ⬌* ⋆ → L1 = ⋆.
+/2 width=2 by lpx_sn_inv_atom2/ qed-.
+
+lemma lcpcs_inv_pair2: ∀I,L1,K2,V2. L1 ⊢ ⬌* K2. ⓑ{I} V2 →
+ ∃∃K1,V1. K1 ⊢ ⬌* K2 & K1 ⊢ V1 ⬌* V2 & L1 = K1. ⓑ{I} V1.
+/2 width=1 by lpx_sn_inv_pair2/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lcpcs_fwd_length: ∀L1,L2. L1 ⊢ ⬌* L2 → |L1| = |L2|.
+/2 width=2 by lpx_sn_fwd_length/ qed-.