+(**************************************************************************)
+(* ___ *)
+(* ||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 "redex_pointer.ma".
+include "multiplicity.ma".
+
+(* LABELLED SEQUENTIAL REDUCTION (ONE STEP) *********************************)
+
+(* Note: the application "(A B)" is represented by "@B.A" following:
+ F. Kamareddine and R.P. Nederpelt: "A useful λ-notation".
+ Theoretical Computer Science 155(1), Elsevier (1996), pp. 85-109.
+*)
+inductive lsred: rpointer → relation term ≝
+| lsred_beta : ∀A,D. lsred (◊) (@D.𝛌.A) ([⬐D]A)
+| lsred_abst : ∀p,A,C. lsred p A C → lsred p (𝛌.A) (𝛌.C)
+| lsred_appl_sn: ∀p,B,D,A. lsred p B D → lsred (true::p) (@B.A) (@D.A)
+| lsred_appl_dx: ∀p,B,A,C. lsred p A C → lsred (false::p) (@B.A) (@B.C)
+.
+
+interpretation "labelled sequential reduction"
+ 'LablSeqRed M p N = (lsred p M N).
+
+(* Note: we do not use → since it is reserved by CIC *)
+notation "hvbox( M break ⇀ [ term 46 p ] break term 46 N )"
+ non associative with precedence 45
+ for @{ 'LablSeqRed $M $p $N }.
+
+theorem lsred_fwd_mult: ∀p,M,N. M ⇀[p] N → #{N} < #{M} * #{M}.
+#p #M #N #H elim H -p -M -N
+[ #A #D @(le_to_lt_to_lt … (#{A}*#{D})) //
+ normalize /3 width=1 by lt_minus_to_plus_r, lt_times/ (**) (* auto: too slow without trace *)
+| //
+| #p #B #D #A #_ #IHBD
+ @(lt_to_le_to_lt … (#{B}*#{B}+#{A})) [ /2 width=1/ ] -D -p
+| #p #B #A #C #_ #IHAC
+ @(lt_to_le_to_lt … (#{B}+#{A}*#{A})) [ /2 width=1/ ] -C -p
+]
+@(transitive_le … (#{B}*#{B}+#{A}*#{A})) [ /2 width=1/ ]
+>distributive_times_plus normalize /2 width=1/
+qed-.