Certification Problem
Input (TPDB SRS_Standard/Mixed_SRS/turing_add)
The rewrite relation of the following TRS is considered.
|
1(q0(1(x1))) |
→ |
0(1(q1(x1))) |
(1) |
|
1(q0(0(x1))) |
→ |
0(0(q1(x1))) |
(2) |
|
1(q1(1(x1))) |
→ |
1(1(q1(x1))) |
(3) |
|
1(q1(0(x1))) |
→ |
1(0(q1(x1))) |
(4) |
|
0(q1(x1)) |
→ |
q2(1(x1)) |
(5) |
|
1(q2(x1)) |
→ |
q2(1(x1)) |
(6) |
|
0(q2(x1)) |
→ |
0(q0(x1)) |
(7) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by matchbox @ termCOMP 2023)
1 Closure Under Flat Contexts
Using the flat contexts
{q1(☐), q0(☐), q2(☐), 1(☐), 0(☐)}
We obtain the transformed TRS
|
q1(1(q0(1(x1)))) |
→ |
q1(0(1(q1(x1)))) |
(8) |
|
q1(1(q0(0(x1)))) |
→ |
q1(0(0(q1(x1)))) |
(9) |
|
q1(1(q1(1(x1)))) |
→ |
q1(1(1(q1(x1)))) |
(10) |
|
q1(1(q1(0(x1)))) |
→ |
q1(1(0(q1(x1)))) |
(11) |
|
q1(0(q1(x1))) |
→ |
q1(q2(1(x1))) |
(12) |
|
q1(1(q2(x1))) |
→ |
q1(q2(1(x1))) |
(13) |
|
q1(0(q2(x1))) |
→ |
q1(0(q0(x1))) |
(14) |
|
q0(1(q0(1(x1)))) |
→ |
q0(0(1(q1(x1)))) |
(15) |
|
q0(1(q0(0(x1)))) |
→ |
q0(0(0(q1(x1)))) |
(16) |
|
q0(1(q1(1(x1)))) |
→ |
q0(1(1(q1(x1)))) |
(17) |
|
q0(1(q1(0(x1)))) |
→ |
q0(1(0(q1(x1)))) |
(18) |
|
q0(0(q1(x1))) |
→ |
q0(q2(1(x1))) |
(19) |
|
q0(1(q2(x1))) |
→ |
q0(q2(1(x1))) |
(20) |
|
q0(0(q2(x1))) |
→ |
q0(0(q0(x1))) |
(21) |
|
q2(1(q0(1(x1)))) |
→ |
q2(0(1(q1(x1)))) |
(22) |
|
q2(1(q0(0(x1)))) |
→ |
q2(0(0(q1(x1)))) |
(23) |
|
q2(1(q1(1(x1)))) |
→ |
q2(1(1(q1(x1)))) |
(24) |
|
q2(1(q1(0(x1)))) |
→ |
q2(1(0(q1(x1)))) |
(25) |
|
q2(0(q1(x1))) |
→ |
q2(q2(1(x1))) |
(26) |
|
q2(1(q2(x1))) |
→ |
q2(q2(1(x1))) |
(27) |
|
q2(0(q2(x1))) |
→ |
q2(0(q0(x1))) |
(28) |
|
1(1(q0(1(x1)))) |
→ |
1(0(1(q1(x1)))) |
(29) |
|
1(1(q0(0(x1)))) |
→ |
1(0(0(q1(x1)))) |
(30) |
|
1(1(q1(1(x1)))) |
→ |
1(1(1(q1(x1)))) |
(31) |
|
1(1(q1(0(x1)))) |
→ |
1(1(0(q1(x1)))) |
(32) |
|
1(0(q1(x1))) |
→ |
1(q2(1(x1))) |
(33) |
|
1(1(q2(x1))) |
→ |
1(q2(1(x1))) |
(34) |
|
1(0(q2(x1))) |
→ |
1(0(q0(x1))) |
(35) |
|
0(1(q0(1(x1)))) |
→ |
0(0(1(q1(x1)))) |
(36) |
|
0(1(q0(0(x1)))) |
→ |
0(0(0(q1(x1)))) |
(37) |
|
0(1(q1(1(x1)))) |
→ |
0(1(1(q1(x1)))) |
(38) |
|
0(1(q1(0(x1)))) |
→ |
0(1(0(q1(x1)))) |
(39) |
|
0(0(q1(x1))) |
→ |
0(q2(1(x1))) |
(40) |
|
0(1(q2(x1))) |
→ |
0(q2(1(x1))) |
(41) |
|
0(0(q2(x1))) |
→ |
0(0(q0(x1))) |
(42) |
1.1 Semantic Labeling
The following interpretations form a
model
of the rules.
As carrier we take the set
{0,...,4}.
Symbols are labeled by the interpretation of their arguments using the interpretations
(modulo 5):
| [q1(x1)] |
= |
5x1 + 0 |
| [q0(x1)] |
= |
5x1 + 1 |
| [q2(x1)] |
= |
5x1 + 2 |
| [1(x1)] |
= |
5x1 + 3 |
| [0(x1)] |
= |
5x1 + 4 |
We obtain the labeled TRS
There are 175 ruless (increase limit for explicit display).
1.1.1 Rule Removal
Using the
matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [q10(x1)] |
= |
x1 +
|
| [q11(x1)] |
= |
x1 +
|
| [q12(x1)] |
= |
x1 +
|
| [q13(x1)] |
= |
x1 +
|
| [q14(x1)] |
= |
x1 +
|
| [q00(x1)] |
= |
x1 +
|
| [q01(x1)] |
= |
x1 +
|
| [q02(x1)] |
= |
x1 +
|
| [q03(x1)] |
= |
x1 +
|
| [q04(x1)] |
= |
x1 +
|
| [q20(x1)] |
= |
x1 +
|
| [q21(x1)] |
= |
x1 +
|
| [q22(x1)] |
= |
x1 +
|
| [q23(x1)] |
= |
x1 +
|
| [q24(x1)] |
= |
x1 +
|
| [10(x1)] |
= |
x1 +
|
| [11(x1)] |
= |
x1 +
|
| [12(x1)] |
= |
x1 +
|
| [13(x1)] |
= |
x1 +
|
| [14(x1)] |
= |
x1 +
|
| [00(x1)] |
= |
x1 +
|
| [01(x1)] |
= |
x1 +
|
| [02(x1)] |
= |
x1 +
|
| [03(x1)] |
= |
x1 +
|
| [04(x1)] |
= |
x1 +
|
all of the following rules can be deleted.
There are 123 ruless (increase limit for explicit display).
1.1.1.1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
|
13(q03(11(13(x1)))) |
→ |
q13(10(03(14(x1)))) |
(218) |
|
14(q03(11(13(x1)))) |
→ |
q14(10(03(14(x1)))) |
(219) |
|
13(q03(11(q03(x1)))) |
→ |
q13(10(03(q04(x1)))) |
(220) |
|
14(q03(11(q03(x1)))) |
→ |
q14(10(03(q04(x1)))) |
(221) |
|
13(q03(11(03(x1)))) |
→ |
q13(10(03(04(x1)))) |
(222) |
|
14(q03(11(03(x1)))) |
→ |
q14(10(03(04(x1)))) |
(223) |
|
13(q03(11(q13(x1)))) |
→ |
q13(10(03(q14(x1)))) |
(224) |
|
14(q03(11(q13(x1)))) |
→ |
q14(10(03(q14(x1)))) |
(225) |
|
13(q03(11(q23(x1)))) |
→ |
q13(10(03(q24(x1)))) |
(226) |
|
14(q03(11(q23(x1)))) |
→ |
q14(10(03(q24(x1)))) |
(227) |
|
01(q04(11(13(x1)))) |
→ |
q11(00(04(14(x1)))) |
(228) |
|
01(q04(11(q03(x1)))) |
→ |
q11(00(04(q04(x1)))) |
(229) |
|
01(q04(11(03(x1)))) |
→ |
q11(00(04(04(x1)))) |
(230) |
|
01(q04(11(q13(x1)))) |
→ |
q11(00(04(q14(x1)))) |
(231) |
|
01(q04(11(q23(x1)))) |
→ |
q11(00(04(q24(x1)))) |
(232) |
|
13(q13(10(13(x1)))) |
→ |
q13(10(13(13(x1)))) |
(233) |
|
14(q13(10(13(x1)))) |
→ |
q14(10(13(13(x1)))) |
(234) |
|
13(q13(10(q03(x1)))) |
→ |
q13(10(13(q03(x1)))) |
(235) |
|
14(q13(10(q03(x1)))) |
→ |
q14(10(13(q03(x1)))) |
(236) |
|
13(q13(10(03(x1)))) |
→ |
q13(10(13(03(x1)))) |
(237) |
|
14(q13(10(03(x1)))) |
→ |
q14(10(13(03(x1)))) |
(238) |
|
13(q13(10(q13(x1)))) |
→ |
q13(10(13(q13(x1)))) |
(239) |
|
14(q13(10(q13(x1)))) |
→ |
q14(10(13(q13(x1)))) |
(240) |
|
13(q13(10(q23(x1)))) |
→ |
q13(10(13(q23(x1)))) |
(241) |
|
14(q13(10(q23(x1)))) |
→ |
q14(10(13(q23(x1)))) |
(242) |
|
01(q14(10(13(x1)))) |
→ |
q11(00(14(13(x1)))) |
(243) |
|
01(q14(10(q03(x1)))) |
→ |
q11(00(14(q03(x1)))) |
(244) |
|
01(q14(10(03(x1)))) |
→ |
q11(00(14(03(x1)))) |
(245) |
|
01(q14(10(q13(x1)))) |
→ |
q11(00(14(q13(x1)))) |
(246) |
|
01(q14(10(q23(x1)))) |
→ |
q11(00(14(q23(x1)))) |
(247) |
|
q11(00(14(x1))) |
→ |
11(q23(12(x1))) |
(248) |
|
q11(00(04(x1))) |
→ |
11(q23(02(x1))) |
(249) |
|
q11(00(q24(x1))) |
→ |
11(q23(q22(x1))) |
(250) |
|
q23(12(13(x1))) |
→ |
13(q23(12(x1))) |
(251) |
|
q24(12(13(x1))) |
→ |
14(q23(12(x1))) |
(252) |
|
q22(12(13(x1))) |
→ |
12(q23(12(x1))) |
(253) |
|
q23(12(03(x1))) |
→ |
13(q23(02(x1))) |
(254) |
|
q24(12(03(x1))) |
→ |
14(q23(02(x1))) |
(255) |
|
q22(12(03(x1))) |
→ |
12(q23(02(x1))) |
(256) |
|
q23(12(q23(x1))) |
→ |
13(q23(q22(x1))) |
(257) |
|
q24(12(q23(x1))) |
→ |
14(q23(q22(x1))) |
(258) |
|
q22(12(q23(x1))) |
→ |
12(q23(q22(x1))) |
(259) |
|
q23(02(14(x1))) |
→ |
q03(01(14(x1))) |
(260) |
|
q24(02(14(x1))) |
→ |
q04(01(14(x1))) |
(261) |
|
q23(02(q04(x1))) |
→ |
q03(01(q04(x1))) |
(262) |
|
q24(02(q04(x1))) |
→ |
q04(01(q04(x1))) |
(263) |
|
q23(02(04(x1))) |
→ |
q03(01(04(x1))) |
(264) |
|
q24(02(04(x1))) |
→ |
q04(01(04(x1))) |
(265) |
|
q23(02(q14(x1))) |
→ |
q03(01(q14(x1))) |
(266) |
|
q24(02(q14(x1))) |
→ |
q04(01(q14(x1))) |
(267) |
|
q23(02(q24(x1))) |
→ |
q03(01(q24(x1))) |
(268) |
|
q24(02(q24(x1))) |
→ |
q04(01(q24(x1))) |
(269) |
1.1.1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
q11#(00(q24(x1))) |
→ |
q22#(x1) |
(270) |
|
q11#(00(q24(x1))) |
→ |
q23#(q22(x1)) |
(271) |
|
q11#(00(14(x1))) |
→ |
q23#(12(x1)) |
(272) |
|
q11#(00(04(x1))) |
→ |
q23#(02(x1)) |
(273) |
|
q22#(12(q23(x1))) |
→ |
q22#(x1) |
(274) |
|
q22#(12(q23(x1))) |
→ |
q23#(q22(x1)) |
(275) |
|
q22#(12(13(x1))) |
→ |
q23#(12(x1)) |
(276) |
|
q22#(12(03(x1))) |
→ |
q23#(02(x1)) |
(277) |
|
q23#(12(q23(x1))) |
→ |
q22#(x1) |
(278) |
|
q23#(12(q23(x1))) |
→ |
q23#(q22(x1)) |
(279) |
|
q23#(12(q23(x1))) |
→ |
13#(q23(q22(x1))) |
(280) |
|
q23#(12(13(x1))) |
→ |
q23#(12(x1)) |
(281) |
|
q23#(12(13(x1))) |
→ |
13#(q23(12(x1))) |
(282) |
|
q23#(12(03(x1))) |
→ |
q23#(02(x1)) |
(283) |
|
q23#(12(03(x1))) |
→ |
13#(q23(02(x1))) |
(284) |
|
q23#(02(q14(x1))) |
→ |
01#(q14(x1)) |
(285) |
|
q23#(02(q04(x1))) |
→ |
01#(q04(x1)) |
(286) |
|
q23#(02(q24(x1))) |
→ |
01#(q24(x1)) |
(287) |
|
q23#(02(14(x1))) |
→ |
01#(14(x1)) |
(288) |
|
q23#(02(04(x1))) |
→ |
01#(04(x1)) |
(289) |
|
q24#(12(q23(x1))) |
→ |
q22#(x1) |
(290) |
|
q24#(12(q23(x1))) |
→ |
q23#(q22(x1)) |
(291) |
|
q24#(12(q23(x1))) |
→ |
14#(q23(q22(x1))) |
(292) |
|
q24#(12(13(x1))) |
→ |
q23#(12(x1)) |
(293) |
|
q24#(12(13(x1))) |
→ |
14#(q23(12(x1))) |
(294) |
|
q24#(12(03(x1))) |
→ |
q23#(02(x1)) |
(295) |
|
q24#(12(03(x1))) |
→ |
14#(q23(02(x1))) |
(296) |
|
q24#(02(q14(x1))) |
→ |
01#(q14(x1)) |
(297) |
|
q24#(02(q04(x1))) |
→ |
01#(q04(x1)) |
(298) |
|
q24#(02(q24(x1))) |
→ |
01#(q24(x1)) |
(299) |
|
q24#(02(14(x1))) |
→ |
01#(14(x1)) |
(300) |
|
q24#(02(04(x1))) |
→ |
01#(04(x1)) |
(301) |
|
13#(q13(10(q13(x1)))) |
→ |
13#(q13(x1)) |
(302) |
|
13#(q13(10(q03(x1)))) |
→ |
13#(q03(x1)) |
(303) |
|
13#(q13(10(q23(x1)))) |
→ |
13#(q23(x1)) |
(304) |
|
13#(q13(10(13(x1)))) |
→ |
13#(13(x1)) |
(305) |
|
13#(q13(10(03(x1)))) |
→ |
13#(03(x1)) |
(306) |
|
13#(q03(11(q23(x1)))) |
→ |
q24#(x1) |
(307) |
|
13#(q03(11(13(x1)))) |
→ |
14#(x1) |
(308) |
|
14#(q13(10(q13(x1)))) |
→ |
13#(q13(x1)) |
(309) |
|
14#(q13(10(q03(x1)))) |
→ |
13#(q03(x1)) |
(310) |
|
14#(q13(10(q23(x1)))) |
→ |
13#(q23(x1)) |
(311) |
|
14#(q13(10(13(x1)))) |
→ |
13#(13(x1)) |
(312) |
|
14#(q13(10(03(x1)))) |
→ |
13#(03(x1)) |
(313) |
|
14#(q03(11(q23(x1)))) |
→ |
q24#(x1) |
(314) |
|
14#(q03(11(13(x1)))) |
→ |
14#(x1) |
(315) |
|
01#(q14(10(q13(x1)))) |
→ |
q11#(00(14(q13(x1)))) |
(316) |
|
01#(q14(10(q13(x1)))) |
→ |
14#(q13(x1)) |
(317) |
|
01#(q14(10(q03(x1)))) |
→ |
q11#(00(14(q03(x1)))) |
(318) |
|
01#(q14(10(q03(x1)))) |
→ |
14#(q03(x1)) |
(319) |
|
01#(q14(10(q23(x1)))) |
→ |
q11#(00(14(q23(x1)))) |
(320) |
|
01#(q14(10(q23(x1)))) |
→ |
14#(q23(x1)) |
(321) |
|
01#(q14(10(13(x1)))) |
→ |
q11#(00(14(13(x1)))) |
(322) |
|
01#(q14(10(13(x1)))) |
→ |
14#(13(x1)) |
(323) |
|
01#(q14(10(03(x1)))) |
→ |
q11#(00(14(03(x1)))) |
(324) |
|
01#(q14(10(03(x1)))) |
→ |
14#(03(x1)) |
(325) |
|
01#(q04(11(q13(x1)))) |
→ |
q11#(00(04(q14(x1)))) |
(326) |
|
01#(q04(11(q03(x1)))) |
→ |
q11#(00(04(q04(x1)))) |
(327) |
|
01#(q04(11(q23(x1)))) |
→ |
q11#(00(04(q24(x1)))) |
(328) |
|
01#(q04(11(q23(x1)))) |
→ |
q24#(x1) |
(329) |
|
01#(q04(11(13(x1)))) |
→ |
q11#(00(04(14(x1)))) |
(330) |
|
01#(q04(11(13(x1)))) |
→ |
14#(x1) |
(331) |
|
01#(q04(11(03(x1)))) |
→ |
q11#(00(04(04(x1)))) |
(332) |
1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [q11(x1)] |
= |
x1 +
|
| [q13(x1)] |
= |
x1 +
|
| [q14(x1)] |
= |
x1 +
|
| [q03(x1)] |
= |
x1 +
|
| [q04(x1)] |
= |
x1 +
|
| [q22(x1)] |
= |
x1 +
|
| [q23(x1)] |
= |
x1 +
|
| [q24(x1)] |
= |
x1 +
|
| [10(x1)] |
= |
x1 +
|
| [11(x1)] |
= |
x1 +
|
| [12(x1)] |
= |
x1 +
|
| [13(x1)] |
= |
x1 +
|
| [14(x1)] |
= |
x1 +
|
| [00(x1)] |
= |
x1 +
|
| [01(x1)] |
= |
x1 +
|
| [02(x1)] |
= |
x1 +
|
| [03(x1)] |
= |
x1 +
|
| [04(x1)] |
= |
x1 +
|
| [q11#(x1)] |
= |
x1 +
|
| [q22#(x1)] |
= |
x1 +
|
| [q23#(x1)] |
= |
x1 +
|
| [q24#(x1)] |
= |
x1 +
|
| [13#(x1)] |
= |
x1 +
|
| [14#(x1)] |
= |
x1 +
|
| [01#(x1)] |
= |
x1 +
|
together with the usable
rules
|
13(q03(11(13(x1)))) |
→ |
q13(10(03(14(x1)))) |
(218) |
|
14(q03(11(13(x1)))) |
→ |
q14(10(03(14(x1)))) |
(219) |
|
13(q03(11(q03(x1)))) |
→ |
q13(10(03(q04(x1)))) |
(220) |
|
14(q03(11(q03(x1)))) |
→ |
q14(10(03(q04(x1)))) |
(221) |
|
13(q03(11(03(x1)))) |
→ |
q13(10(03(04(x1)))) |
(222) |
|
14(q03(11(03(x1)))) |
→ |
q14(10(03(04(x1)))) |
(223) |
|
13(q03(11(q13(x1)))) |
→ |
q13(10(03(q14(x1)))) |
(224) |
|
14(q03(11(q13(x1)))) |
→ |
q14(10(03(q14(x1)))) |
(225) |
|
13(q03(11(q23(x1)))) |
→ |
q13(10(03(q24(x1)))) |
(226) |
|
14(q03(11(q23(x1)))) |
→ |
q14(10(03(q24(x1)))) |
(227) |
|
01(q04(11(13(x1)))) |
→ |
q11(00(04(14(x1)))) |
(228) |
|
01(q04(11(q03(x1)))) |
→ |
q11(00(04(q04(x1)))) |
(229) |
|
01(q04(11(03(x1)))) |
→ |
q11(00(04(04(x1)))) |
(230) |
|
01(q04(11(q13(x1)))) |
→ |
q11(00(04(q14(x1)))) |
(231) |
|
01(q04(11(q23(x1)))) |
→ |
q11(00(04(q24(x1)))) |
(232) |
|
13(q13(10(13(x1)))) |
→ |
q13(10(13(13(x1)))) |
(233) |
|
14(q13(10(13(x1)))) |
→ |
q14(10(13(13(x1)))) |
(234) |
|
13(q13(10(q03(x1)))) |
→ |
q13(10(13(q03(x1)))) |
(235) |
|
14(q13(10(q03(x1)))) |
→ |
q14(10(13(q03(x1)))) |
(236) |
|
13(q13(10(03(x1)))) |
→ |
q13(10(13(03(x1)))) |
(237) |
|
14(q13(10(03(x1)))) |
→ |
q14(10(13(03(x1)))) |
(238) |
|
13(q13(10(q13(x1)))) |
→ |
q13(10(13(q13(x1)))) |
(239) |
|
14(q13(10(q13(x1)))) |
→ |
q14(10(13(q13(x1)))) |
(240) |
|
13(q13(10(q23(x1)))) |
→ |
q13(10(13(q23(x1)))) |
(241) |
|
14(q13(10(q23(x1)))) |
→ |
q14(10(13(q23(x1)))) |
(242) |
|
01(q14(10(13(x1)))) |
→ |
q11(00(14(13(x1)))) |
(243) |
|
01(q14(10(q03(x1)))) |
→ |
q11(00(14(q03(x1)))) |
(244) |
|
01(q14(10(03(x1)))) |
→ |
q11(00(14(03(x1)))) |
(245) |
|
01(q14(10(q13(x1)))) |
→ |
q11(00(14(q13(x1)))) |
(246) |
|
01(q14(10(q23(x1)))) |
→ |
q11(00(14(q23(x1)))) |
(247) |
|
q11(00(14(x1))) |
→ |
11(q23(12(x1))) |
(248) |
|
q11(00(04(x1))) |
→ |
11(q23(02(x1))) |
(249) |
|
q11(00(q24(x1))) |
→ |
11(q23(q22(x1))) |
(250) |
|
q23(12(13(x1))) |
→ |
13(q23(12(x1))) |
(251) |
|
q24(12(13(x1))) |
→ |
14(q23(12(x1))) |
(252) |
|
q22(12(13(x1))) |
→ |
12(q23(12(x1))) |
(253) |
|
q23(12(03(x1))) |
→ |
13(q23(02(x1))) |
(254) |
|
q24(12(03(x1))) |
→ |
14(q23(02(x1))) |
(255) |
|
q22(12(03(x1))) |
→ |
12(q23(02(x1))) |
(256) |
|
q23(12(q23(x1))) |
→ |
13(q23(q22(x1))) |
(257) |
|
q24(12(q23(x1))) |
→ |
14(q23(q22(x1))) |
(258) |
|
q22(12(q23(x1))) |
→ |
12(q23(q22(x1))) |
(259) |
|
q23(02(14(x1))) |
→ |
q03(01(14(x1))) |
(260) |
|
q24(02(14(x1))) |
→ |
q04(01(14(x1))) |
(261) |
|
q23(02(q04(x1))) |
→ |
q03(01(q04(x1))) |
(262) |
|
q24(02(q04(x1))) |
→ |
q04(01(q04(x1))) |
(263) |
|
q23(02(04(x1))) |
→ |
q03(01(04(x1))) |
(264) |
|
q24(02(04(x1))) |
→ |
q04(01(04(x1))) |
(265) |
|
q23(02(q14(x1))) |
→ |
q03(01(q14(x1))) |
(266) |
|
q24(02(q14(x1))) |
→ |
q04(01(q14(x1))) |
(267) |
|
q23(02(q24(x1))) |
→ |
q03(01(q24(x1))) |
(268) |
|
q24(02(q24(x1))) |
→ |
q04(01(q24(x1))) |
(269) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
q11#(00(q24(x1))) |
→ |
q22#(x1) |
(270) |
|
q11#(00(q24(x1))) |
→ |
q23#(q22(x1)) |
(271) |
|
q11#(00(14(x1))) |
→ |
q23#(12(x1)) |
(272) |
|
q11#(00(04(x1))) |
→ |
q23#(02(x1)) |
(273) |
|
q22#(12(q23(x1))) |
→ |
q22#(x1) |
(274) |
|
q22#(12(q23(x1))) |
→ |
q23#(q22(x1)) |
(275) |
|
q22#(12(13(x1))) |
→ |
q23#(12(x1)) |
(276) |
|
q22#(12(03(x1))) |
→ |
q23#(02(x1)) |
(277) |
|
q23#(12(q23(x1))) |
→ |
q22#(x1) |
(278) |
|
q23#(12(q23(x1))) |
→ |
q23#(q22(x1)) |
(279) |
|
q23#(12(q23(x1))) |
→ |
13#(q23(q22(x1))) |
(280) |
|
q23#(12(13(x1))) |
→ |
q23#(12(x1)) |
(281) |
|
q23#(12(13(x1))) |
→ |
13#(q23(12(x1))) |
(282) |
|
q23#(12(03(x1))) |
→ |
q23#(02(x1)) |
(283) |
|
q23#(12(03(x1))) |
→ |
13#(q23(02(x1))) |
(284) |
|
q23#(02(q14(x1))) |
→ |
01#(q14(x1)) |
(285) |
|
q23#(02(q04(x1))) |
→ |
01#(q04(x1)) |
(286) |
|
q23#(02(q24(x1))) |
→ |
01#(q24(x1)) |
(287) |
|
q23#(02(14(x1))) |
→ |
01#(14(x1)) |
(288) |
|
q23#(02(04(x1))) |
→ |
01#(04(x1)) |
(289) |
|
q24#(12(q23(x1))) |
→ |
q22#(x1) |
(290) |
|
q24#(12(q23(x1))) |
→ |
q23#(q22(x1)) |
(291) |
|
q24#(12(q23(x1))) |
→ |
14#(q23(q22(x1))) |
(292) |
|
q24#(12(13(x1))) |
→ |
q23#(12(x1)) |
(293) |
|
q24#(12(13(x1))) |
→ |
14#(q23(12(x1))) |
(294) |
|
q24#(12(03(x1))) |
→ |
q23#(02(x1)) |
(295) |
|
q24#(12(03(x1))) |
→ |
14#(q23(02(x1))) |
(296) |
|
q24#(02(q14(x1))) |
→ |
01#(q14(x1)) |
(297) |
|
q24#(02(q04(x1))) |
→ |
01#(q04(x1)) |
(298) |
|
q24#(02(q24(x1))) |
→ |
01#(q24(x1)) |
(299) |
|
q24#(02(14(x1))) |
→ |
01#(14(x1)) |
(300) |
|
q24#(02(04(x1))) |
→ |
01#(04(x1)) |
(301) |
|
13#(q13(10(q13(x1)))) |
→ |
13#(q13(x1)) |
(302) |
|
13#(q13(10(q03(x1)))) |
→ |
13#(q03(x1)) |
(303) |
|
13#(q13(10(q23(x1)))) |
→ |
13#(q23(x1)) |
(304) |
|
13#(q13(10(13(x1)))) |
→ |
13#(13(x1)) |
(305) |
|
13#(q13(10(03(x1)))) |
→ |
13#(03(x1)) |
(306) |
|
13#(q03(11(q23(x1)))) |
→ |
q24#(x1) |
(307) |
|
13#(q03(11(13(x1)))) |
→ |
14#(x1) |
(308) |
|
14#(q13(10(q13(x1)))) |
→ |
13#(q13(x1)) |
(309) |
|
14#(q13(10(q03(x1)))) |
→ |
13#(q03(x1)) |
(310) |
|
14#(q13(10(q23(x1)))) |
→ |
13#(q23(x1)) |
(311) |
|
14#(q13(10(13(x1)))) |
→ |
13#(13(x1)) |
(312) |
|
14#(q13(10(03(x1)))) |
→ |
13#(03(x1)) |
(313) |
|
14#(q03(11(q23(x1)))) |
→ |
q24#(x1) |
(314) |
|
14#(q03(11(13(x1)))) |
→ |
14#(x1) |
(315) |
|
01#(q14(10(q13(x1)))) |
→ |
q11#(00(14(q13(x1)))) |
(316) |
|
01#(q14(10(q13(x1)))) |
→ |
14#(q13(x1)) |
(317) |
|
01#(q14(10(q03(x1)))) |
→ |
q11#(00(14(q03(x1)))) |
(318) |
|
01#(q14(10(q03(x1)))) |
→ |
14#(q03(x1)) |
(319) |
|
01#(q14(10(q23(x1)))) |
→ |
q11#(00(14(q23(x1)))) |
(320) |
|
01#(q14(10(q23(x1)))) |
→ |
14#(q23(x1)) |
(321) |
|
01#(q14(10(13(x1)))) |
→ |
q11#(00(14(13(x1)))) |
(322) |
|
01#(q14(10(13(x1)))) |
→ |
14#(13(x1)) |
(323) |
|
01#(q14(10(03(x1)))) |
→ |
q11#(00(14(03(x1)))) |
(324) |
|
01#(q14(10(03(x1)))) |
→ |
14#(03(x1)) |
(325) |
|
01#(q04(11(q13(x1)))) |
→ |
q11#(00(04(q14(x1)))) |
(326) |
|
01#(q04(11(q03(x1)))) |
→ |
q11#(00(04(q04(x1)))) |
(327) |
|
01#(q04(11(q23(x1)))) |
→ |
q11#(00(04(q24(x1)))) |
(328) |
|
01#(q04(11(q23(x1)))) |
→ |
q24#(x1) |
(329) |
|
01#(q04(11(13(x1)))) |
→ |
q11#(00(04(14(x1)))) |
(330) |
|
01#(q04(11(13(x1)))) |
→ |
14#(x1) |
(331) |
|
01#(q04(11(03(x1)))) |
→ |
q11#(00(04(04(x1)))) |
(332) |
and
no rules
could be deleted.
1.1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.