- pointer structure simplified

[helm.git] / matita / matita / contribs / lambda / pointer_order.ma

diff --git a/matita/matita/contribs/lambda/pointer_order.ma b/matita/matita/contribs/lambda/pointer_order.ma
index ffd1e725d409351ec5908f4ee4e2280069a68515..24dcad4fa953a14824c8c4f610287967fa850e69 100644 (file)
--- a/matita/matita/contribs/lambda/pointer_order.ma
+++ b/matita/matita/contribs/lambda/pointer_order.ma
@@ -20,8 +20,7 @@ include "pointer.ma".
          to serve as base for the order relations: p < q and p ≤ q *)
 inductive pprec: relation ptr ≝
 | pprec_nil : ∀c,q.   pprec (◊) (c::q)
-| ppprc_rc  : ∀p,q.   pprec (dx::p) (rc::q)
-| ppprc_sn  : ∀p,q.   pprec (dx::p) (sn::q)
+| ppprc_cons: ∀p,q.   pprec (dx::p) (sn::q)
 | pprec_comp: ∀c,p,q. pprec p q → pprec (c::p) (c::q)
 | pprec_skip:         pprec (dx::◊) ◊
 .
@@ -37,7 +36,6 @@ notation "hvbox(a break ≺ b)"
 lemma pprec_fwd_in_head: ∀p,q. p ≺ q → in_head q → in_head p.
 #p #q #H elim H -p -q // /2 width=1/
 [ #p #q * #H destruct
-| #p #q * #H destruct
 | #c #p #q #_ #IHpq * #H destruct /3 width=1/
 ]
 qed-.
@@ -48,6 +46,31 @@ definition plt: relation ptr ≝ TC … pprec.
 interpretation "'less than' on redex pointers"
    'lt p q = (plt p q).
 
+lemma plt_step_rc: ∀p,q. p ≺ q → p < q.
+/2 width=1/
+qed.
+
+lemma plt_nil: ∀c,p. ◊ < c::p.
+/2 width=1/
+qed.
+
+lemma plt_skip: dx::◊ < ◊.
+/2 width=1/
+qed.
+
+lemma plt_comp: ∀c,p,q. p < q → c::p < c::q.
+#c #p #q #H elim H -q /3 width=1/ /3 width=3/
+qed.
+
+theorem plt_trans: transitive … plt.
+/2 width=3/
+qed-.
+
+lemma plt_refl: ∀p. p < p.
+#p elim p -p /2 width=1/
+@(plt_trans … (dx::◊)) //
+qed.
+
 (* Note: this is p ≤ q *)
 definition ple: relation ptr ≝ star … pprec.
 
@@ -83,6 +106,15 @@ lemma ple_skip_ple: ∀p. p ≤ ◊ → dx::p ≤ ◊.
 #p #q #Hpq #_ #H @(ple_step_sn … H) -H /2 width=1/
 qed.
 
+lemma ple_dichotomy: ∀p1,p2:ptr. p1 ≤ p2 ∨ p2 ≤ p1.
+#p1 elim p1 -p1
+[ * /2 width=1/
+| #c1 #p1 #IHp1 * /2 width=1/
+  * #p2 cases c1 -c1 /2 width=1/
+  elim (IHp1 p2) -IHp1 /3 width=1/
+]
+qed-.
+
 lemma in_head_ple_nil: ∀p. in_head p → p ≤ ◊.
 #p #H @(in_head_ind … H) -p // /2 width=1/
 qed.