(* *)
(**************************************************************************)
-include "ground_2/notation/functions/drop_1.ma".
+include "ground_2/notation/functions/updownarrowstar_1.ma".
include "ground_2/steps/rtc_isrt.ma".
(* RT-TRANSITION COUNTER ****************************************************)
definition shift (c:rtc): rtc ≝ match c with
-[ mk_rtc ri rs ti ts ⇒ 〈ri+rs, 0, ti+ts, 0〉 ].
+[ mk_rtc ri rs ti ts ⇒ 〈ri∨rs, 0, ti∨ts, 0〉 ].
interpretation "shift (rtc)"
- 'Drop c = (shift c).
+ 'UpDownArrowStar c = (shift c).
(* Basic properties *********************************************************)
-lemma shift_rew: ∀ri,rs,ti,ts. 〈ri+rs, 0, ti+ts, 0〉 = ↓〈ri, rs, ti, ts〉.
+lemma shift_rew: ∀ri,rs,ti,ts. 〈ri∨rs, 0, ti∨ts, 0〉 = ↕*〈ri, rs, ti, ts〉.
normalize //
qed.
-lemma shift_O: ð\9d\9f\98ð\9d\9f\98 = â\86\93𝟘𝟘.
+lemma shift_O: ð\9d\9f\98ð\9d\9f\98 = â\86\95*𝟘𝟘.
// qed.
(* Basic inversion properties ***********************************************)
-lemma shift_inv_dx: â\88\80ri,rs,ti,ts,c. â\8c©ri, rs, ti, tsâ\8cª = â\86\93c →
- ∃∃ri0,rs0,ti0,ts0. ri0+rs0 = ri & 0 = rs & ti0+ts0 = ti & 0 = ts &
+lemma shift_inv_dx: â\88\80ri,rs,ti,ts,c. â\8c©ri, rs, ti, tsâ\8cª = â\86\95*c →
+ ∃∃ri0,rs0,ti0,ts0. (ri0∨rs0) = ri & 0 = rs & (ti0∨ts0) = ti & 0 = ts &
〈ri0, rs0, ti0, ts0〉 = c.
#ri #rs #ti #ts * #ri0 #rs0 #ti0 #ts0 <shift_rew #H destruct
/2 width=7 by ex5_4_intro/
(* Properties with test for costrained rt-transition counter ****************)
-lemma isr_shift: â\88\80c. ð\9d\90\91ð\9d\90\93â¦\830, câ¦\84 â\86\92 ð\9d\90\91ð\9d\90\93â¦\830, â\86\93c⦄.
+lemma isr_shift: â\88\80c. ð\9d\90\91ð\9d\90\93â¦\830, câ¦\84 â\86\92 ð\9d\90\91ð\9d\90\93â¦\830, â\86\95*c⦄.
#c * #ri #rs #H destruct /2 width=3 by ex1_2_intro/
qed.
(* Inversion properties with test for costrained rt-counter *****************)
-lemma isrt_inv_shift: â\88\80n,c. ð\9d\90\91ð\9d\90\93â¦\83n, â\86\93c⦄ → 𝐑𝐓⦃0, c⦄ ∧ 0 = n.
+lemma isrt_inv_shift: â\88\80n,c. ð\9d\90\91ð\9d\90\93â¦\83n, â\86\95*c⦄ → 𝐑𝐓⦃0, c⦄ ∧ 0 = n.
#n #c * #ri #rs #H
elim (shift_inv_dx … H) -H #rt0 #rs0 #ti0 #ts0 #_ #_ #H1 #H2 #H3
-elim (plus_inv_O3 … H1) -H1 /3 width=3 by ex1_2_intro, conj/
+elim (max_inv_O3 … H1) -H1 /3 width=3 by ex1_2_intro, conj/
qed-.
-lemma isr_inv_shift: â\88\80c. ð\9d\90\91ð\9d\90\93â¦\830, â\86\93c⦄ → 𝐑𝐓⦃0, c⦄.
+lemma isr_inv_shift: â\88\80c. ð\9d\90\91ð\9d\90\93â¦\830, â\86\95*c⦄ → 𝐑𝐓⦃0, c⦄.
#c #H elim (isrt_inv_shift … H) -H //
qed-.