(* *)
(**************************************************************************)
-include "ground_2/xoa/ex_3_2.ma".
include "ground_2/xoa/ex_6_8.ma".
-include "ground_2/steps/rtc_shift.ma".
+include "ground_2/steps/rtc.ma".
(* RT-TRANSITION COUNTER ****************************************************)
* #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: โn1,n2,c1,c2. ๐๐โชn1,c1โซ โ ๐๐โชn2,c2โซ โ ๐๐โชn1โจn2,c1โจc2โซ.
-#n1 #n2 #c1 #c2 * #ri1 #rs1 #H1 * #ri2 #rs2 #H2 destruct
-/2 width=3 by ex1_2_intro/
-qed.
-
-lemma isrt_max_O1: โn,c1,c2. ๐๐โช0,c1โซ โ ๐๐โชn,c2โซ โ ๐๐โชn,c1โจc2โซ.
-/2 width=1 by isrt_max/ qed.
-
-lemma isrt_max_O2: โn,c1,c2. ๐๐โชn,c1โซ โ ๐๐โช0,c2โซ โ ๐๐โชn,c1โจc2โซ.
-#n #c1 #c2 #H1 #H2 >(max_O2 n) /2 width=1 by isrt_max/
-qed.
-
-lemma isrt_max_idem1: โn,c1,c2. ๐๐โชn,c1โซ โ ๐๐โชn,c2โซ โ ๐๐โชn,c1โจc2โซ.
-#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: โn,c1,c2. ๐๐โชn,c1 โจ c2โซ โ
- โโn1,n2. ๐๐โชn1,c1โซ & ๐๐โชn2,c2โซ & (n1 โจ n2) = n.
-#n #c1 #c2 * #ri #rs #H
-elim (max_inv_dx โฆ H) -H #ri1 #rs1 #ti1 #ts1 #ri2 #rs2 #ti2 #ts2 #_ #_ #H1 #H2 #H3 #H4
-elim (max_inv_O3 โฆ H1) -H1 /3 width=5 by ex3_2_intro, ex1_2_intro/
-qed-.
-
-lemma isrt_O_inv_max: โc1,c2. ๐๐โช0,c1 โจ c2โซ โ โงโง ๐๐โช0,c1โซ & ๐๐โช0,c2โซ.
-#c1 #c2 #H
-elim (isrt_inv_max โฆ H) -H #n1 #n2 #Hn1 #Hn2 #H
-elim (max_inv_O3 โฆ H) -H #H1 #H2 destruct
-/2 width=1 by conj/
-qed-.
-
-lemma isrt_inv_max_O_dx: โn,c1,c2. ๐๐โชn,c1 โจ c2โซ โ ๐๐โช0,c2โซ โ ๐๐โชn,c1โซ.
-#n #c1 #c2 #H #H2
-elim (isrt_inv_max โฆ H) -H #n1 #n2 #Hn1 #Hn2 #H destruct
-lapply (isrt_inj โฆ Hn2 H2) -c2 #H destruct //
-qed-.
-
-lemma isrt_inv_max_eq_t: โn,c1,c2. ๐๐โชn,c1 โจ c2โซ โ eq_t c1 c2 โ
- โงโง ๐๐โชn,c1โซ & ๐๐โชn,c2โซ.
-#n #c1 #c2 #H #Hc12
-elim (isrt_inv_max โฆ H) -H #n1 #n2 #Hc1 #Hc2 #H destruct
-lapply (isrt_eq_t_trans โฆ Hc1 โฆ Hc12) -Hc12 #H
-<(isrt_inj โฆ H โฆ Hc2) -Hc2
-<idempotent_max /2 width=1 by conj/
-qed-.
-
-(* Properties with shift ****************************************************)
-
-lemma max_shift: โc1,c2. ((โ*c1) โจ (โ*c2)) = โ*(c1โจc2).
-* #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: โn,c1,c2. ๐๐โชn,โ*c1 โจ c2โซ โ
- โงโง ๐๐โช0,c1โซ & ๐๐โชn,c2โซ.
-#n #c1 #c2 #H
-elim (isrt_inv_max โฆ H) -H #n1 #n2 #Hc1 #Hc2 #H destruct
-elim (isrt_inv_shift โฆ Hc1) -Hc1 #Hc1 * -n1
-/2 width=1 by conj/
-qed-.