lambda finaly moved in lib

[helm.git] / matita / matita / lib / lambda / terms / multiplicity.ma

diff --git a/matita/matita/lib/lambda/terms/multiplicity.ma b/matita/matita/lib/lambda/terms/multiplicity.ma
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+++ b/matita/matita/lib/lambda/terms/multiplicity.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 "terms/relocating_substitution.ma".
+
+(* MULTIPLICITY *************************************************************)
+
+(* Note: this gives the number of variable references in M *)
+let rec mult M on M ≝ match M with
+[ VRef i   ⇒ 1
+| Abst A   ⇒ mult A
+| Appl B A ⇒ (mult B) + (mult A)
+].
+
+interpretation "term multiplicity"
+   'Multiplicity M = (mult M).
+
+notation "hvbox( ♯{ term 46 M } )"
+   non associative with precedence 90
+   for @{ 'Multiplicity $M }.
+
+lemma mult_positive: ∀M. 0 < ♯{M}.
+#M elim M -M // /2 width=1/
+qed.
+
+lemma mult_lift: ∀h,M,d. ♯{↑[d, h] M} = ♯{M}.
+#h #M elim M -M normalize //
+qed.
+
+theorem mult_dsubst: ∀D,M,d. ♯{[d ↙ D] M} ≤ ♯{M} * ♯{D}.
+#D #M elim M -M
+[ #i #d elim (lt_or_eq_or_gt i d) #Hid
+  [ >(dsubst_vref_lt … Hid) normalize //
+  | destruct >dsubst_vref_eq normalize //
+  | >(dsubst_vref_gt … Hid) normalize //
+  ]
+| normalize //
+| normalize #B #A #IHB #IHA #d
+  >distributive_times_plus_r /2 width=1/
+]
+qed.