- paths and left residuals: third case of the equivalence proved!

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

diff --git a/matita/matita/contribs/lambda/paths/standard_precedence.ma b/matita/matita/contribs/lambda/paths/standard_precedence.ma
index e8f41eecf8c4f57ba7c3abf1c2b31457d82c3f07..a4ff9e9fbdb7f3e297a551e6af456f5c1985c30f 100644 (file)
--- a/matita/matita/contribs/lambda/paths/standard_precedence.ma
+++ b/matita/matita/contribs/lambda/paths/standard_precedence.ma
@@ -19,10 +19,10 @@ include "paths/path.ma".
 (* 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_nil : ∀o,q.   sprec (◊) (o::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_comp: ∀o,p,q. sprec p q → sprec (o::p) (o::q)
 | sprec_skip:         sprec (dx::◊) ◊
 .
 
@@ -32,18 +32,56 @@ interpretation "standard 'precedes' on paths"
 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
+[ #o #q #p0 #H destruct
 | #p #q #p0 #H destruct
 | #p #q #p0 #H destruct
-| #c #p #q #Hpq #p0 #H destruct /2 width=3/
+| #o #p #q #Hpq #p0 #H destruct /2 width=3/
 | #p0 #H destruct
 ]
 qed-.
 
+lemma sprec_inv_dx: ∀p,q. p ≺ q → ∀q0. dx::q0 = q →
+                    ◊ = p ∨ ∃∃p0. p0 ≺ q0 & dx::p0 = p.
+#p #q * -p -q
+[ #o #q #q0 #H destruct /2 width=1/
+| #p #q #q0 #H destruct
+| #p #q #q0 #H destruct
+| #o #p #q #Hpq #q0 #H destruct /3 width=3/
+| #q0 #H destruct
+]
+qed-.
+
+lemma sprec_inv_rc: ∀p,q. p ≺ q → ∀p0. rc::p0 = p →
+                    (∃∃q0. p0 ≺ q0 & rc::q0 = q) ∨
+                    ∃q0. sn::q0 = q.
+#p #q * -p -q
+[ #o #q #p0 #H destruct
+| #p #q #p0 #H destruct
+| #p #q #p0 #H destruct /3 width=2/
+| #o #p #q #Hpq #p0 #H destruct /3 width=3/
+| #p0 #H destruct
+]
+qed-.
+
+lemma sprec_inv_comp: ∀p1,p2. p1 ≺ p2 →
+                      ∀o,q1,q2. o::q1 = p1 → o::q2 = p2 → q1 ≺ q2.
+#p1 #p2 * -p1 -p2
+[ #o #q #o0 #q1 #q2 #H destruct
+| #p #q #o0 #q1 #q2 #H1 #H2 destruct
+| #p #q #o0 #q1 #q2 #H1 #H2 destruct
+| #o #p #q #Hpq #o0 #q1 #q2 #H1 #H2 destruct //
+| #o0 #q1 #q2 #_ #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/
+| #o #p #q #_ #IHpq * #H destruct /3 width=1/
 ]
 qed-.
+
+lemma sprec_fwd_in_inner: ∀p,q. p ≺ q → in_inner p → in_inner q.
+/3 width=3 by sprec_fwd_in_whd/
+qed-.