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