propagating the arithmetics library, partial commit

[helm.git] / matita / matita / contribs / lambdadelta / ground / relocation / pstream_basic.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/relocation/pstream_basic.ma b/matita/matita/contribs/lambdadelta/ground/relocation/pstream_basic.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/arith/nat_lt_plus.ma".
+include "ground/relocation/rtmap_basic_at.ma".
+include "ground/relocation/pstream_after.ma".
+
+(* RELOCATION N-STREAM ******************************************************)
+
+(* Specific properties on basic relocation **********************************)
+
+lemma apply_basic_lt: ∀m,n,i. ninj i ≤ m → 𝐁❨m,n❩@❨i❩ = i.
+/3 width=1 by at_inv_total, at_basic_lt/ qed-.
+
+lemma apply_basic_ge: ∀m,n,i. m < ninj i → 𝐁❨m,n❩@❨i❩ = i+n.
+/3 width=1 by at_inv_total, at_basic_ge/ qed-.
+
+(* Specific main properties on basic relocation *****************************)
+
+theorem basic_swap: ∀d1,d2. d2 ≤ d1 →
+                    ∀h1,h2. 𝐁❨d2,h2❩∘𝐁❨d1,h1❩ ≡ 𝐁❨d1+h2,h1❩∘𝐁❨d2,h2❩.
+#d1 #d2 #Hd21 #h1 #h2
+@nstream_inv_eq
+@nstream_eq_inv_ext #i
+<compose_apply <compose_apply
+elim (nat_split_lt_ge d2 i) #Hd2
+[ elim (nat_split_lt_ge d1 i) -Hd21 #Hd1
+  [ >(apply_basic_ge … Hd1) >(apply_basic_ge … Hd2) >apply_basic_ge
+    [ >apply_basic_ge // >nrplus_inj_sn /2 width=1 by nlt_plus_bi_sn/
+    | >nrplus_inj_sn /2 width=2 by nlt_plus_dx_dx/
+    ]
+  | >(apply_basic_lt … Hd1) >(apply_basic_ge … Hd2)
+    >apply_basic_lt // >nrplus_inj_sn /2 width=1 by nle_plus_bi_dx/
+  ]
+| lapply (nle_trans … Hd2 … Hd21) -Hd21 #Hd1
+  >(apply_basic_lt … Hd1) >(apply_basic_lt … Hd2)
+  >apply_basic_lt /2 width=1 by nle_plus_dx_dx/
+]
+qed-.