+(**************************************************************************)
+(* ___ *)
+(* ||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 "ground_2/notation/relations/isid_1.ma".
+include "ground_2/relocation/trace_after.ma".
+
+(* RELOCATION TRACE *********************************************************)
+
+definition isid: predicate trace ≝ λcs. ∥cs∥ = |cs|.
+
+interpretation "test for identity (trace)"
+ 'IsId cs = (isid cs).
+
+(* Basic properties *********************************************************)
+
+lemma isid_empty: 𝐈⦃◊⦄.
+// qed.
+
+lemma isid_true: ∀cs. 𝐈⦃cs⦄ → 𝐈⦃Ⓣ @ cs⦄.
+// qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma isid_inv_true: ∀cs. 𝐈⦃Ⓣ @ cs⦄ → 𝐈⦃cs⦄.
+/2 width=1 by injective_S/ qed-.
+
+lemma isid_inv_false: ∀cs. 𝐈⦃Ⓕ @ cs⦄ → ⊥.
+/3 width=4 by colength_le, lt_le_false/ qed-.
+
+lemma isid_inv_cons: ∀cs,b. 𝐈⦃b @ cs⦄ → 𝐈⦃cs⦄ ∧ b = Ⓣ.
+#cs * #H /3 width=1 by isid_inv_true, conj/
+elim (isid_inv_false … H)
+qed-.
+
+(* Properties on application ************************************************)
+
+lemma isid_at: ∀cs. (∀i1,i2. @⦃i1, cs⦄ ≡ i2 → i1 = i2) → 𝐈⦃cs⦄.
+#cs elim cs -cs // * /2 width=1 by/
+qed.
+
+(* Inversion lemmas on application ******************************************)
+
+lemma isid_inv_at: ∀cs,i1,i2. @⦃i1, cs⦄ ≡ i2 → 𝐈⦃cs⦄ → i1 = i2.
+#cs #i1 #i2 #H elim H -cs -i1 -i2 /4 width=1 by isid_inv_true, eq_f/
+#cs #i1 #i2 #_ #IH #H elim (isid_inv_false … H)
+qed-.
+
+(* Properties on composition ************************************************)
+
+lemma isid_after_sn: ∀cs1,cs2. cs1 ⊚ cs2 ≡ cs2 → 𝐈⦃cs1⦄ .
+#cs1 #cs2 #H elim (after_inv_length … H) -H //
+qed.
+
+lemma isid_after_dx: ∀cs1,cs2. cs1 ⊚ cs2 ≡ cs1 → 𝐈⦃cs2⦄ .
+#cs1 #cs2 #H elim (after_inv_length … H) -H //
+qed.
+
+(* Inversion lemmas on composition ******************************************)
+
+lemma isid_inv_after_sn: ∀cs1,cs2,cs. cs1 ⊚ cs2 ≡ cs → 𝐈⦃cs1⦄ → cs = cs2.
+#cs1 #cs2 #cs #H elim H -cs1 -cs2 -cs //
+#cs1 #cs2 #cs #_ [ #b ] #IH #H
+[ >IH -IH // | elim (isid_inv_false … H) ]
+qed-.
+
+lemma isid_inv_after_dx: ∀cs1,cs2,cs. cs1 ⊚ cs2 ≡ cs → 𝐈⦃cs2⦄ → cs = cs1.
+#cs1 #cs2 #cs #H elim H -cs1 -cs2 -cs //
+#cs1 #cs2 #cs #_ [ #b ] #IH #H
+[ elim (isid_inv_cons … H) -H #H >IH -IH // | >IH -IH // ]
+qed-.