Theoretical Computer Science 155(1), Elsevier (1996), pp. 85-109.
*)
inductive lsred: ptr → relation term ≝
-| lsred_beta : â\88\80B,A. lsred (â\97\8a) (@B.ð\9d\9b\8c.A) ([â¬\90B]A)
+| lsred_beta : â\88\80B,A. lsred (â\97\8a) (@B.ð\9d\9b\8c.A) ([â\86\99B]A)
| lsred_abst : ∀p,A1,A2. lsred p A1 A2 → lsred (sn::p) (𝛌.A1) (𝛌.A2)
| lsred_appl_sn: ∀p,B1,B2,A. lsred p B1 B2 → lsred (sn::p) (@B1.A) (@B2.A)
| lsred_appl_dx: ∀p,B,A1,A2. lsred p A1 A2 → lsred (dx::p) (@B.A1) (@B.A2)
'SeqRed M p N = (lsred p M N).
(* Note: we do not use → since it is reserved by CIC *)
-notation "hvbox( M break â\87\80 [ term 46 p ] break term 46 N )"
+notation "hvbox( M break â\86¦ [ term 46 p ] break term 46 N )"
non associative with precedence 45
for @{ 'SeqRed $M $p $N }.
-lemma lsred_inv_vref: â\88\80p,M,N. M â\87\80[p] N → ∀i. #i = M → ⊥.
+lemma lsred_inv_vref: â\88\80p,M,N. M â\86¦[p] N → ∀i. #i = M → ⊥.
#p #M #N * -p -M -N
[ #B #A #i #H destruct
| #p #A1 #A2 #_ #i #H destruct
]
qed-.
-lemma lsred_inv_nil: â\88\80p,M,N. M â\87\80[p] N → ◊ = p →
- â\88\83â\88\83B,A. @B. ð\9d\9b\8c.A = M & [â¬\90B] A = N.
+lemma lsred_inv_nil: â\88\80p,M,N. M â\86¦[p] N → ◊ = p →
+ â\88\83â\88\83B,A. @B. ð\9d\9b\8c.A = M & [â\86\99B] A = N.
#p #M #N * -p -M -N
[ #B #A #_ destruct /2 width=4/
| #p #A1 #A2 #_ #H destruct
]
qed-.
-lemma lsred_inv_sn: â\88\80p,M,N. M â\87\80[p] N → ∀q. sn::q = p →
- (â\88\83â\88\83A1,A2. A1 â\87\80[q] A2 & 𝛌.A1 = M & 𝛌.A2 = N) ∨
- â\88\83â\88\83B1,B2,A. B1 â\87\80[q] B2 & @B1.A = M & @B2.A = N.
+lemma lsred_inv_sn: â\88\80p,M,N. M â\86¦[p] N → ∀q. sn::q = p →
+ (â\88\83â\88\83A1,A2. A1 â\86¦[q] A2 & 𝛌.A1 = M & 𝛌.A2 = N) ∨
+ â\88\83â\88\83B1,B2,A. B1 â\86¦[q] B2 & @B1.A = M & @B2.A = N.
#p #M #N * -p -M -N
[ #B #A #q #H destruct
| #p #A1 #A2 #HA12 #q #H destruct /3 width=5/
]
qed-.
-lemma lsred_inv_dx: â\88\80p,M,N. M â\87\80[p] N → ∀q. dx::q = p →
- â\88\83â\88\83B,A1,A2. A1 â\87\80[q] A2 & @B.A1 = M & @B.A2 = N.
+lemma lsred_inv_dx: â\88\80p,M,N. M â\86¦[p] N → ∀q. dx::q = p →
+ â\88\83â\88\83B,A1,A2. A1 â\86¦[q] A2 & @B.A1 = M & @B.A2 = N.
#p #M #N * -p -M -N
[ #B #A #q #H destruct
| #p #A1 #A2 #_ #q #H destruct
]
qed-.
-lemma lsred_fwd_mult: â\88\80p,M,N. M â\87\80[p] N → #{N} < #{M} * #{M}.
+lemma lsred_fwd_mult: â\88\80p,M,N. M â\86¦[p] N → #{N} < #{M} * #{M}.
#p #M #N #H elim H -p -M -N
[ #B #A @(le_to_lt_to_lt … (#{A}*#{B})) //
normalize /3 width=1 by lt_minus_to_plus_r, lt_times/ (**) (* auto: too slow without trace *)