* #ri #rs #ti #ts <max_rew //
qed.
+lemma max_idem: βc. c = (c β¨ c).
+* #ri #rs #ti #ts <max_rew //
+qed.
+
(* Basic inversion properties ***********************************************)
lemma max_inv_dx: βri,rs,ti,ts,c1,c2. β©ri,rs,ti,tsβͺ = (c1 β¨ c2) β
<max_rew #H destruct /2 width=14 by ex6_8_intro/
qed-.
+(* Main Properties **********************************************************)
+
+theorem max_assoc: associative β¦ max.
+* #ri1 #rs1 #ti1 #ts1 * #ri2 #rs2 #ti2 #ts2 * #ri3 #rs3 #ti3 #ts3
+<max_rew <max_rew //
+qed.
+
(* Properties with test for constrained rt-transition counter ***************)
lemma isrt_max: βn1,n2,c1,c2. ππβ¦n1, c1β¦ β ππβ¦n2, c2β¦ β ππβ¦n1β¨n2, c1β¨c2β¦.
#n #c1 #c2 #H1 #H2 >(max_O2 n) /2 width=1 by isrt_max/
qed.
+lemma isrt_max_idem1: βn,c1,c2. ππβ¦n, c1β¦ β ππβ¦n, c2β¦ β ππβ¦n, c1β¨c2β¦.
+#n #c1 #c2 #H1 #H2 >(idempotent_max n) /2 width=1 by isrt_max/
+qed.
+
(* Inversion properties with test for constrained rt-transition counter *****)
lemma isrt_inv_max: βn,c1,c2. ππβ¦n, c1 β¨ c2β¦ β
elim (max_inv_O3 β¦ H1) -H1 /3 width=5 by ex3_2_intro, ex1_2_intro/
qed-.
+lemma isrt_O_inv_max: βc1,c2. ππβ¦0, c1 β¨ c2β¦ β β§β§ ππβ¦0, c1β¦ & ππβ¦0, c2β¦.
+#c1 #c2 #H
+elim (isrt_inv_max β¦ H) -H #n1 #n2 #Hn1 #Hn2 #H
+elim (max_inv_O3 β¦ H) -H #H1 #H2 destruct
+/2 width=1 by conj/
+qed-.
+
lemma isrt_inv_max_O_dx: βn,c1,c2. ππβ¦n, c1 β¨ c2β¦ β ππβ¦0, c2β¦ β ππβ¦n, c1β¦.
#n #c1 #c2 #H #H2
elim (isrt_inv_max β¦ H) -H #n1 #n2 #Hn1 #Hn2 #H destruct
-lapply (isrt_mono β¦ Hn2 H2) -c2 #H destruct //
+lapply (isrt_inj β¦ Hn2 H2) -c2 #H destruct //
+qed-.
+
+lemma isrt_inv_max_eq_t: βn,c1,c2. ππβ¦n, c1 β¨ c2β¦ β eq_t c1 c2 β
+ β§β§ ππβ¦n, c1β¦ & ππβ¦n, c2β¦.
+#n #c1 #c2 #H #Hc12
+elim (isrt_inv_max β¦ H) -H #n1 #n2 #Hc1 #Hc2 #H destruct
+lapply (isrt_eq_t_trans β¦ Hc1 β¦ Hc12) -Hc12 #H
+<(isrt_inj β¦ H β¦ Hc2) -Hc2
+<idempotent_max /2 width=1 by conj/
qed-.
(* Properties with shift ****************************************************)
-(*
-lemma max_shift: βc1,c2. (βc1) β¨ (βc2) = β(c1β¨c2).
+
+lemma max_shift: βc1,c2. ((β*c1) β¨ (β*c2)) = β*(c1β¨c2).
* #ri1 #rs1 #ti1 #ts1 * #ri2 #rs2 #ti2 #ts2
<shift_rew <shift_rew <shift_rew <max_rew //
qed.
-*)
+
+(* Inversion lemmaswith shift ***********************************************)
+
+lemma isrt_inv_max_shift_sn: βn,c1,c2. ππβ¦n, β*c1 β¨ c2β¦ β
+ β§β§ ππβ¦0, c1β¦ & ππβ¦n, c2β¦.
+#n #c1 #c2 #H
+elim (isrt_inv_max β¦ H) -H #n1 #n2 #Hc1 #Hc2 #H destruct
+elim (isrt_inv_shift β¦ Hc1) -Hc1 #Hc1 * -n1
+/2 width=1 by conj/
+qed-.