cfg_to_obj completed (modulo daemons)

[helm.git] / matita / matita / contribs / lambda / paths / standard_order.ma

diff --git a/matita/matita/contribs/lambda/paths/standard_order.ma b/matita/matita/contribs/lambda/paths/standard_order.ma
index 563cdce3488a49074b89742c81291f796655f8ee..672ba56f1f8df144ac82393e69365bc7cdc3fd0a 100644 (file)
--- a/matita/matita/contribs/lambda/paths/standard_order.ma
+++ b/matita/matita/contribs/lambda/paths/standard_order.ma
@@ -12,73 +12,10 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "paths/path.ma".
+include "paths/standard_precedence.ma".
 
 (* STANDARD ORDER ************************************************************)
 
-(* Note: standard precedence relation on paths: p ≺ q
-         to serve as base for the order relations: p < q and p ≤ q *)
-inductive sprec: relation path ≝
-| sprec_nil : ∀c,q.   sprec (◊) (c::q)
-| sprec_rc  : ∀p,q.   sprec (dx::p) (rc::q)
-| sprec_sn  : ∀p,q.   sprec (rc::p) (sn::q)
-| sprec_comp: ∀c,p,q. sprec p q → sprec (c::p) (c::q)
-| sprec_skip:         sprec (dx::◊) ◊
-.
-
-interpretation "standard 'precedes' on paths"
-   'prec p q = (sprec p q).
-
-lemma sprec_inv_sn: ∀p,q. p ≺ q → ∀p0. sn::p0 = p →
-                    ∃∃q0. p0 ≺ q0 & sn::q0 = q.
-#p #q * -p -q
-[ #c #q #p0 #H destruct
-| #p #q #p0 #H destruct
-| #p #q #p0 #H destruct
-| #c #p #q #Hpq #p0 #H destruct /2 width=3/
-| #p0 #H destruct
-]
-qed-.
-
-lemma sprec_fwd_in_whd: ∀p,q. p ≺ q → in_whd q → in_whd 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-.
-
-(* Note: this is p < q *)
-definition slt: relation path ≝ TC … sprec.
-
-interpretation "standard 'less than' on paths"
-   'lt p q = (slt p q).
-
-lemma slt_step_rc: ∀p,q. p ≺ q → p < q.
-/2 width=1/
-qed.
-
-lemma slt_nil: ∀c,p. ◊ < c::p.
-/2 width=1/
-qed.
-
-lemma slt_skip: dx::◊ < ◊.
-/2 width=1/
-qed.
-
-lemma slt_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 slt_trans: transitive … slt.
-/2 width=3/
-qed-.
-
-lemma slt_refl: ∀p. p < p.
-#p elim p -p /2 width=1/
-@(slt_trans … (dx::◊)) //
-qed.
-
 (* Note: this is p ≤ q *)
 definition sle: relation path ≝ star … sprec.
 
@@ -109,12 +46,12 @@ lemma sle_nil: ∀p. ◊ ≤ p.
 * // /2 width=1/
 qed.
 
-lemma sle_comp: ∀p1,p2. p1 ≤ p2 → ∀c. (c::p1) ≤ (c::p2).
+lemma sle_comp: ∀p1,p2. p1 ≤ p2 → ∀o. (o::p1) ≤ (o::p2).
 #p1 #p2 #H elim H -p2 // /3 width=3/
 qed.
 
 lemma sle_skip_sle: ∀p. p ≤ ◊ → dx::p ≤ ◊.
-#p #H @(star_ind_l ??????? H) -p //
+#p #H @(star_ind_l … p H) -p //
 #p #q #Hpq #_ #H @(sle_step_sn … H) -H /2 width=1/
 qed.
 
@@ -130,12 +67,24 @@ qed.
 lemma sle_dichotomy: ∀p1,p2:path. p1 ≤ p2 ∨ p2 ≤ p1.
 #p1 elim p1 -p1
 [ * /2 width=1/
-| #c1 #p1 #IHp1 * /2 width=1/
-  * #p2 cases c1 -c1 /2 width=1/
+| #o1 #p1 #IHp1 * /2 width=1/
+  * #p2 cases o1 -o1 /2 width=1/
   elim (IHp1 p2) -IHp1 /3 width=1/
 ]
 qed-.
 
+lemma sle_inv_rc: ∀p,q. p ≤ q → ∀p0. rc::p0 = p →
+                  (∃∃q0. p0 ≤ q0 & rc::q0 = q) ∨
+                  ∃q0. sn::q0 = q.
+#p #q #H elim H -q /3 width=3/
+#q #q0 #_ #Hq0 #IHpq #p0 #H destruct
+elim (IHpq p0 ?) -IHpq // *
+[ #p1 #Hp01 #H
+  elim (sprec_inv_rc … Hq0 … H) -q * /3 width=3/ /4 width=3/
+| #p1 #H elim (sprec_inv_sn … Hq0 … H) -q /3 width=2/
+]
+qed-.
+
 lemma sle_inv_sn: ∀p,q. p ≤ q → ∀p0. sn::p0 = p →
                   ∃∃q0. p0 ≤ q0 & sn::q0 = q.
 #p #q #H elim H -q /2 width=3/
@@ -154,9 +103,13 @@ theorem in_whd_sle: ∀p. in_whd p → ∀q. p ≤ q.
 qed.
 
 lemma sle_nil_inv_in_whd: ∀p. p ≤ ◊ → in_whd p.
-#p #H @(star_ind_l ??????? H) -p // /2 width=3 by sprec_fwd_in_whd/
+#p #H @(star_ind_l … p H) -p // /2 width=3 by sprec_fwd_in_whd/
 qed-.
 
 lemma sle_inv_in_whd: ∀p. (∀q. p ≤ q) → in_whd p.
 /2 width=1 by sle_nil_inv_in_whd/
 qed-.
+
+lemma sle_fwd_in_whd: ∀p,q. p ≤ q → in_whd q → in_whd p.
+#p #q #H @(star_ind_l … p H) -p // /3 width=3 by sprec_fwd_in_whd/
+qed-.