finished semantics for termination of match machine

[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 8ce5b879053ed9dc9db83d97aa7aa74b1bf812eb..5ff11ea34c74ba5dc8d72deb867c7a19450a0345 100644 (file)
--- a/matita/matita/contribs/lambda/paths/standard_order.ma
+++ b/matita/matita/contribs/lambda/paths/standard_order.ma
@@ -46,12 +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.
 
@@ -67,12 +67,37 @@ 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_dx: ∀p,q. p ≤ q → ∀q0. dx::q0 = q →
+                  in_whd p ∨ ∃∃p0. p0 ≤ q0 & dx::p0 = p.
+#p #q #H @(star_ind_l … p H) -p [ /3 width=3/ ]
+#p0 #p #Hp0 #_ #IHpq #q1 #H destruct
+elim (IHpq ??) -IHpq [4: // |3: skip ] (**) (* simplify line *)
+[ lapply (sprec_fwd_in_whd … Hp0) -Hp0 /3 width=1/
+| * #p1 #Hpq1 #H elim (sprec_inv_dx … Hp0 … H) -p
+  [ #H destruct /2 width=1/
+  | * /4 width=3/
+  ]
+]
+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/
@@ -91,9 +116,17 @@ 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-.
+
+lemma sle_fwd_in_inner: ∀p,q. p ≤ q → in_inner p → in_inner q.
+/3 width=3 by sle_fwd_in_whd/
+qed-.