Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/142142)

The rewrite relation of the following TRS is considered.

0(0(0(1(1(0(0(0(0(1(2(0(1(x1))))))))))))) 1(1(1(1(0(0(0(1(0(0(1(1(0(1(1(0(0(x1))))))))))))))))) (1)
0(0(0(1(1(2(2(1(1(0(0(1(1(x1))))))))))))) 0(1(1(0(2(0(0(0(1(0(1(1(0(0(0(1(0(x1))))))))))))))))) (2)
0(1(0(0(2(1(0(0(1(0(0(1(0(x1))))))))))))) 0(1(1(0(0(0(1(1(0(1(0(0(1(1(1(0(0(x1))))))))))))))))) (3)
0(1(0(1(1(0(0(0(0(2(0(1(2(x1))))))))))))) 1(1(1(0(0(0(1(0(1(1(0(1(1(1(0(0(2(x1))))))))))))))))) (4)
0(1(0(1(1(0(2(1(0(1(0(0(0(x1))))))))))))) 0(1(1(0(1(1(0(0(1(0(0(1(0(1(1(1(0(x1))))))))))))))))) (5)
0(1(1(0(0(1(0(2(0(0(0(2(1(x1))))))))))))) 0(0(1(1(0(1(0(1(1(0(0(1(1(2(1(1(1(x1))))))))))))))))) (6)
0(1(2(1(0(0(0(0(0(2(1(1(1(x1))))))))))))) 0(1(0(1(1(2(0(1(0(0(0(1(1(0(0(0(1(x1))))))))))))))))) (7)
0(2(0(0(2(1(0(0(0(0(0(0(1(x1))))))))))))) 1(0(0(1(0(1(1(1(0(0(0(1(0(0(0(0(1(x1))))))))))))))))) (8)
0(2(1(0(1(2(1(2(2(0(1(1(1(x1))))))))))))) 0(2(1(1(0(0(2(1(1(0(0(1(2(1(1(1(1(x1))))))))))))))))) (9)
0(2(1(0(2(1(0(1(2(0(0(0(1(x1))))))))))))) 0(0(2(1(1(1(1(0(0(1(0(2(0(0(0(1(0(x1))))))))))))))))) (10)
0(2(1(1(0(0(0(0(2(1(0(1(1(x1))))))))))))) 0(0(1(0(1(0(1(0(2(1(1(1(0(0(0(0(0(x1))))))))))))))))) (11)
0(2(2(0(0(1(1(0(1(0(1(0(1(x1))))))))))))) 0(0(2(1(0(0(0(1(1(0(1(1(0(1(1(0(0(x1))))))))))))))))) (12)
1(0(0(0(0(1(1(1(0(0(2(1(1(x1))))))))))))) 1(1(1(1(1(1(1(1(1(1(0(1(1(0(1(0(0(x1))))))))))))))))) (13)
1(0(0(0(1(1(1(2(0(0(1(0(2(x1))))))))))))) 0(1(1(0(1(0(0(1(0(0(0(0(0(2(0(1(2(x1))))))))))))))))) (14)
1(0(0(0(1(2(0(1(0(0(0(1(0(x1))))))))))))) 1(1(1(0(0(0(1(1(0(1(1(0(1(1(1(0(1(x1))))))))))))))))) (15)
1(0(0(0(2(1(0(0(2(0(1(0(1(x1))))))))))))) 2(1(0(0(1(1(0(1(0(0(0(0(0(1(0(0(0(x1))))))))))))))))) (16)
1(0(0(1(0(1(2(1(0(2(1(1(0(x1))))))))))))) 0(1(1(1(0(0(0(0(0(2(1(1(1(1(1(1(0(x1))))))))))))))))) (17)
1(0(0(1(0(2(1(1(0(1(1(0(1(x1))))))))))))) 1(1(1(1(0(0(1(0(1(0(1(1(1(0(0(1(1(x1))))))))))))))))) (18)
1(0(0(1(2(0(0(0(2(2(1(1(2(x1))))))))))))) 1(0(0(0(1(0(2(2(0(1(1(1(0(1(2(1(2(x1))))))))))))))))) (19)
1(0(0(1(2(1(1(1(0(2(0(2(1(x1))))))))))))) 1(0(0(1(0(0(1(2(0(1(0(1(2(2(1(0(0(x1))))))))))))))))) (20)
1(0(0(2(0(1(0(1(0(2(1(0(2(x1))))))))))))) 0(1(0(0(0(2(0(1(1(0(1(0(1(1(0(0(2(x1))))))))))))))))) (21)
1(0(1(0(1(2(1(1(2(1(0(0(1(x1))))))))))))) 1(1(0(0(1(0(0(2(2(0(0(0(0(1(1(0(0(x1))))))))))))))))) (22)
1(0(1(2(0(1(2(1(0(0(1(0(0(x1))))))))))))) 0(0(0(0(0(1(1(0(1(1(1(1(0(2(1(0(1(x1))))))))))))))))) (23)
1(0(2(2(1(0(1(0(1(0(1(0(2(x1))))))))))))) 1(1(1(0(0(0(0(0(0(1(1(0(0(0(2(1(2(x1))))))))))))))))) (24)
1(1(0(1(0(2(2(2(1(0(0(0(0(x1))))))))))))) 1(1(0(1(1(0(1(0(1(0(1(0(0(2(0(0(1(x1))))))))))))))))) (25)
1(1(0(2(2(0(0(1(2(1(2(1(2(x1))))))))))))) 1(1(2(0(0(0(1(1(0(2(2(1(1(0(1(0(2(x1))))))))))))))))) (26)
1(1(1(1(0(1(0(2(1(1(0(0(2(x1))))))))))))) 1(1(1(0(0(1(0(1(1(0(1(0(0(2(1(0(2(x1))))))))))))))))) (27)
1(1(1(2(1(1(0(2(0(2(0(1(0(x1))))))))))))) 1(1(1(0(0(1(0(2(2(0(0(1(2(1(0(1(0(x1))))))))))))))))) (28)
1(1(2(1(1(0(1(1(0(1(1(0(2(x1))))))))))))) 1(1(2(0(1(0(1(0(0(0(0(0(1(0(0(1(2(x1))))))))))))))))) (29)
1(1(2(2(1(1(1(1(0(1(1(0(0(x1))))))))))))) 1(0(1(1(1(2(1(1(0(1(0(1(1(1(0(0(0(x1))))))))))))))))) (30)
1(1(2(2(2(1(0(0(1(1(0(0(0(x1))))))))))))) 0(1(0(1(1(1(1(1(0(2(1(0(1(1(0(0(0(x1))))))))))))))))) (31)
1(2(0(1(0(1(1(1(0(1(1(1(1(x1))))))))))))) 1(0(0(1(1(1(0(1(0(0(1(1(0(1(1(0(1(x1))))))))))))))))) (32)
1(2(1(0(0(0(0(1(0(1(1(2(1(x1))))))))))))) 0(1(0(0(1(0(0(0(1(1(0(0(2(1(0(0(0(x1))))))))))))))))) (33)
1(2(1(0(1(0(0(1(0(2(0(1(1(x1))))))))))))) 1(0(0(1(0(0(0(2(2(0(0(1(0(0(1(1(0(x1))))))))))))))))) (34)
1(2(2(0(1(1(1(1(1(1(0(1(0(x1))))))))))))) 1(0(0(2(2(0(0(1(0(0(1(0(0(0(1(1(0(x1))))))))))))))))) (35)
2(1(0(0(1(1(0(1(2(1(0(1(1(x1))))))))))))) 1(0(0(0(1(0(0(0(0(2(1(0(0(1(1(0(1(x1))))))))))))))))) (36)
2(1(0(1(2(0(0(2(0(1(0(1(2(x1))))))))))))) 2(1(0(0(1(0(0(0(0(1(0(2(2(2(0(1(2(x1))))))))))))))))) (37)
2(1(0(2(2(2(2(1(0(0(0(0(1(x1))))))))))))) 1(1(1(0(0(0(1(1(2(1(0(0(1(2(0(1(0(x1))))))))))))))))) (38)
2(1(1(0(1(0(1(1(1(2(1(1(0(x1))))))))))))) 1(1(2(1(1(0(0(1(1(0(0(1(1(0(1(0(0(x1))))))))))))))))) (39)
2(1(1(0(2(1(0(2(1(0(0(0(1(x1))))))))))))) 0(1(0(1(1(1(1(2(0(1(0(0(1(0(0(1(0(x1))))))))))))))))) (40)
2(1(1(1(1(1(1(1(0(0(2(0(1(x1))))))))))))) 1(1(1(0(1(1(1(0(1(0(0(1(0(2(0(1(0(x1))))))))))))))))) (41)
2(2(1(0(0(2(1(1(1(1(1(0(1(x1))))))))))))) 2(2(2(0(0(0(1(0(1(0(1(1(0(1(1(1(0(x1))))))))))))))))) (42)
2(2(1(0(0(2(1(2(0(0(1(0(2(x1))))))))))))) 2(1(0(0(0(2(1(0(0(1(2(0(0(1(1(1(2(x1))))))))))))))))) (43)
2(2(2(0(0(0(1(0(0(0(0(1(0(x1))))))))))))) 0(2(0(0(1(1(1(0(1(1(0(0(0(1(0(1(0(x1))))))))))))))))) (44)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by matchbox @ termCOMP 2023)

1 Closure Under Flat Contexts

Using the flat contexts

{2(), 1(), 0()}

We obtain the transformed TRS

There are 132 ruless (increase limit for explicit display).

1.1 Semantic Labeling

The following interpretations form a model of the rules.

As carrier we take the set {0,1,2}. Symbols are labeled by the interpretation of their arguments using the interpretations (modulo 3):

[2(x1)] = 3x1 + 0
[1(x1)] = 3x1 + 1
[0(x1)] = 3x1 + 2

We obtain the labeled TRS

There are 396 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
[20(x1)] = x1 +
4
[21(x1)] = x1 +
7
[22(x1)] = x1 +
5
[10(x1)] = x1 +
5
[11(x1)] = x1 +
0
[12(x1)] = x1 +
0
[00(x1)] = x1 +
4
[01(x1)] = x1 +
0
[02(x1)] = x1 +
0
all of the following rules can be deleted.

There are 372 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
22(00(12(01(02(22(10(11(11(01(02(02(12(01(x1)))))))))))))) 22(10(01(22(00(02(02(02(02(12(01(02(12(01(12(11(01(02(x1)))))))))))))))))) (573)
21(00(12(01(02(22(10(11(11(01(02(02(12(01(x1)))))))))))))) 21(10(01(22(00(02(02(02(02(12(01(02(12(01(12(11(01(02(x1)))))))))))))))))) (574)
20(00(12(01(02(22(10(11(11(01(02(02(12(01(x1)))))))))))))) 20(10(01(22(00(02(02(02(02(12(01(02(12(01(12(11(01(02(x1)))))))))))))))))) (575)
22(00(12(01(02(22(10(11(11(01(02(02(12(11(x1)))))))))))))) 22(10(01(22(00(02(02(02(02(12(01(02(12(01(12(11(01(12(x1)))))))))))))))))) (576)
21(00(12(01(02(22(10(11(11(01(02(02(12(11(x1)))))))))))))) 21(10(01(22(00(02(02(02(02(12(01(02(12(01(12(11(01(12(x1)))))))))))))))))) (577)
20(00(12(01(02(22(10(11(11(01(02(02(12(11(x1)))))))))))))) 20(10(01(22(00(02(02(02(02(12(01(02(12(01(12(11(01(12(x1)))))))))))))))))) (578)
22(10(11(21(20(00(02(02(22(10(01(02(12(01(x1)))))))))))))) 22(10(21(10(01(12(11(11(01(22(20(00(12(01(02(02(12(01(x1)))))))))))))))))) (579)
21(10(11(21(20(00(02(02(22(10(01(02(12(01(x1)))))))))))))) 21(10(21(10(01(12(11(11(01(22(20(00(12(01(02(02(12(01(x1)))))))))))))))))) (580)
20(10(11(21(20(00(02(02(22(10(01(02(12(01(x1)))))))))))))) 20(10(21(10(01(12(11(11(01(22(20(00(12(01(02(02(12(01(x1)))))))))))))))))) (581)
22(10(11(21(20(00(02(02(22(10(01(02(12(11(x1)))))))))))))) 22(10(21(10(01(12(11(11(01(22(20(00(12(01(02(02(12(11(x1)))))))))))))))))) (582)
21(10(11(21(20(00(02(02(22(10(01(02(12(11(x1)))))))))))))) 21(10(21(10(01(12(11(11(01(22(20(00(12(01(02(02(12(11(x1)))))))))))))))))) (583)
20(10(11(21(20(00(02(02(22(10(01(02(12(11(x1)))))))))))))) 20(10(21(10(01(12(11(11(01(22(20(00(12(01(02(02(12(11(x1)))))))))))))))))) (584)
22(10(11(21(20(00(02(02(22(10(01(02(12(21(x1)))))))))))))) 22(10(21(10(01(12(11(11(01(22(20(00(12(01(02(02(12(21(x1)))))))))))))))))) (585)
21(10(11(21(20(00(02(02(22(10(01(02(12(21(x1)))))))))))))) 21(10(21(10(01(12(11(11(01(22(20(00(12(01(02(02(12(21(x1)))))))))))))))))) (586)
20(10(11(21(20(00(02(02(22(10(01(02(12(21(x1)))))))))))))) 20(10(21(10(01(12(11(11(01(22(20(00(12(01(02(02(12(21(x1)))))))))))))))))) (587)
22(00(02(12(11(21(00(12(01(12(11(11(11(01(x1)))))))))))))) 22(00(12(21(00(02(12(01(12(11(01(12(01(02(12(11(11(01(x1)))))))))))))))))) (588)
21(00(02(12(11(21(00(12(01(12(11(11(11(01(x1)))))))))))))) 21(00(12(21(00(02(12(01(12(11(01(12(01(02(12(11(11(01(x1)))))))))))))))))) (589)
20(00(02(12(11(21(00(12(01(12(11(11(11(01(x1)))))))))))))) 20(00(12(21(00(02(12(01(12(11(01(12(01(02(12(11(11(01(x1)))))))))))))))))) (590)
22(00(02(12(11(21(00(12(01(12(11(11(11(11(x1)))))))))))))) 22(00(12(21(00(02(12(01(12(11(01(12(01(02(12(11(11(11(x1)))))))))))))))))) (591)
21(00(02(12(11(21(00(12(01(12(11(11(11(11(x1)))))))))))))) 21(00(12(21(00(02(12(01(12(11(01(12(01(02(12(11(11(11(x1)))))))))))))))))) (592)
20(00(02(12(11(21(00(12(01(12(11(11(11(11(x1)))))))))))))) 20(00(12(21(00(02(12(01(12(11(01(12(01(02(12(11(11(11(x1)))))))))))))))))) (593)
22(00(02(12(11(21(00(12(01(12(11(11(11(21(x1)))))))))))))) 22(00(12(21(00(02(12(01(12(11(01(12(01(02(12(11(11(21(x1)))))))))))))))))) (594)
21(00(02(12(11(21(00(12(01(12(11(11(11(21(x1)))))))))))))) 21(00(12(21(00(02(12(01(12(11(01(12(01(02(12(11(11(21(x1)))))))))))))))))) (595)
20(00(02(12(11(21(00(12(01(12(11(11(11(21(x1)))))))))))))) 20(00(12(21(00(02(12(01(12(11(01(12(01(02(12(11(11(21(x1)))))))))))))))))) (596)

1.1.1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
20#(10(11(21(20(00(02(02(22(10(01(02(12(21(x1)))))))))))))) 20#(10(21(10(01(12(11(11(01(22(20(00(12(01(02(02(12(21(x1)))))))))))))))))) (597)
20#(10(11(21(20(00(02(02(22(10(01(02(12(21(x1)))))))))))))) 20#(00(12(01(02(02(12(21(x1)))))))) (598)
20#(10(11(21(20(00(02(02(22(10(01(02(12(21(x1)))))))))))))) 21#(10(01(12(11(11(01(22(20(00(12(01(02(02(12(21(x1)))))))))))))))) (599)
20#(10(11(21(20(00(02(02(22(10(01(02(12(21(x1)))))))))))))) 22#(20(00(12(01(02(02(12(21(x1))))))))) (600)
20#(10(11(21(20(00(02(02(22(10(01(02(12(11(x1)))))))))))))) 20#(10(21(10(01(12(11(11(01(22(20(00(12(01(02(02(12(11(x1)))))))))))))))))) (601)
20#(10(11(21(20(00(02(02(22(10(01(02(12(11(x1)))))))))))))) 20#(00(12(01(02(02(12(11(x1)))))))) (602)
20#(10(11(21(20(00(02(02(22(10(01(02(12(11(x1)))))))))))))) 21#(10(01(12(11(11(01(22(20(00(12(01(02(02(12(11(x1)))))))))))))))) (603)
20#(10(11(21(20(00(02(02(22(10(01(02(12(11(x1)))))))))))))) 22#(20(00(12(01(02(02(12(11(x1))))))))) (604)
20#(10(11(21(20(00(02(02(22(10(01(02(12(01(x1)))))))))))))) 20#(10(21(10(01(12(11(11(01(22(20(00(12(01(02(02(12(01(x1)))))))))))))))))) (605)
20#(10(11(21(20(00(02(02(22(10(01(02(12(01(x1)))))))))))))) 20#(00(12(01(02(02(12(01(x1)))))))) (606)
20#(10(11(21(20(00(02(02(22(10(01(02(12(01(x1)))))))))))))) 21#(10(01(12(11(11(01(22(20(00(12(01(02(02(12(01(x1)))))))))))))))) (607)
20#(10(11(21(20(00(02(02(22(10(01(02(12(01(x1)))))))))))))) 22#(20(00(12(01(02(02(12(01(x1))))))))) (608)
20#(00(12(01(02(22(10(11(11(01(02(02(12(11(x1)))))))))))))) 20#(10(01(22(00(02(02(02(02(12(01(02(12(01(12(11(01(12(x1)))))))))))))))))) (609)
20#(00(12(01(02(22(10(11(11(01(02(02(12(11(x1)))))))))))))) 22#(00(02(02(02(02(12(01(02(12(01(12(11(01(12(x1))))))))))))))) (610)
20#(00(12(01(02(22(10(11(11(01(02(02(12(01(x1)))))))))))))) 20#(10(01(22(00(02(02(02(02(12(01(02(12(01(12(11(01(02(x1)))))))))))))))))) (611)
20#(00(12(01(02(22(10(11(11(01(02(02(12(01(x1)))))))))))))) 22#(00(02(02(02(02(12(01(02(12(01(12(11(01(02(x1))))))))))))))) (612)
20#(00(02(12(11(21(00(12(01(12(11(11(11(21(x1)))))))))))))) 20#(00(12(21(00(02(12(01(12(11(01(12(01(02(12(11(11(21(x1)))))))))))))))))) (613)
20#(00(02(12(11(21(00(12(01(12(11(11(11(21(x1)))))))))))))) 21#(00(02(12(01(12(11(01(12(01(02(12(11(11(21(x1))))))))))))))) (614)
20#(00(02(12(11(21(00(12(01(12(11(11(11(11(x1)))))))))))))) 20#(00(12(21(00(02(12(01(12(11(01(12(01(02(12(11(11(11(x1)))))))))))))))))) (615)
20#(00(02(12(11(21(00(12(01(12(11(11(11(11(x1)))))))))))))) 21#(00(02(12(01(12(11(01(12(01(02(12(11(11(11(x1))))))))))))))) (616)
20#(00(02(12(11(21(00(12(01(12(11(11(11(01(x1)))))))))))))) 20#(00(12(21(00(02(12(01(12(11(01(12(01(02(12(11(11(01(x1)))))))))))))))))) (617)
20#(00(02(12(11(21(00(12(01(12(11(11(11(01(x1)))))))))))))) 21#(00(02(12(01(12(11(01(12(01(02(12(11(11(01(x1))))))))))))))) (618)
21#(10(11(21(20(00(02(02(22(10(01(02(12(21(x1)))))))))))))) 20#(00(12(01(02(02(12(21(x1)))))))) (619)
21#(10(11(21(20(00(02(02(22(10(01(02(12(21(x1)))))))))))))) 21#(10(21(10(01(12(11(11(01(22(20(00(12(01(02(02(12(21(x1)))))))))))))))))) (620)
21#(10(11(21(20(00(02(02(22(10(01(02(12(21(x1)))))))))))))) 21#(10(01(12(11(11(01(22(20(00(12(01(02(02(12(21(x1)))))))))))))))) (621)
21#(10(11(21(20(00(02(02(22(10(01(02(12(21(x1)))))))))))))) 22#(20(00(12(01(02(02(12(21(x1))))))))) (622)
21#(10(11(21(20(00(02(02(22(10(01(02(12(11(x1)))))))))))))) 20#(00(12(01(02(02(12(11(x1)))))))) (623)
21#(10(11(21(20(00(02(02(22(10(01(02(12(11(x1)))))))))))))) 21#(10(21(10(01(12(11(11(01(22(20(00(12(01(02(02(12(11(x1)))))))))))))))))) (624)
21#(10(11(21(20(00(02(02(22(10(01(02(12(11(x1)))))))))))))) 21#(10(01(12(11(11(01(22(20(00(12(01(02(02(12(11(x1)))))))))))))))) (625)
21#(10(11(21(20(00(02(02(22(10(01(02(12(11(x1)))))))))))))) 22#(20(00(12(01(02(02(12(11(x1))))))))) (626)
21#(10(11(21(20(00(02(02(22(10(01(02(12(01(x1)))))))))))))) 20#(00(12(01(02(02(12(01(x1)))))))) (627)
21#(10(11(21(20(00(02(02(22(10(01(02(12(01(x1)))))))))))))) 21#(10(21(10(01(12(11(11(01(22(20(00(12(01(02(02(12(01(x1)))))))))))))))))) (628)
21#(10(11(21(20(00(02(02(22(10(01(02(12(01(x1)))))))))))))) 21#(10(01(12(11(11(01(22(20(00(12(01(02(02(12(01(x1)))))))))))))))) (629)
21#(10(11(21(20(00(02(02(22(10(01(02(12(01(x1)))))))))))))) 22#(20(00(12(01(02(02(12(01(x1))))))))) (630)
21#(00(12(01(02(22(10(11(11(01(02(02(12(11(x1)))))))))))))) 21#(10(01(22(00(02(02(02(02(12(01(02(12(01(12(11(01(12(x1)))))))))))))))))) (631)
21#(00(12(01(02(22(10(11(11(01(02(02(12(11(x1)))))))))))))) 22#(00(02(02(02(02(12(01(02(12(01(12(11(01(12(x1))))))))))))))) (632)
21#(00(12(01(02(22(10(11(11(01(02(02(12(01(x1)))))))))))))) 21#(10(01(22(00(02(02(02(02(12(01(02(12(01(12(11(01(02(x1)))))))))))))))))) (633)
21#(00(12(01(02(22(10(11(11(01(02(02(12(01(x1)))))))))))))) 22#(00(02(02(02(02(12(01(02(12(01(12(11(01(02(x1))))))))))))))) (634)
21#(00(02(12(11(21(00(12(01(12(11(11(11(21(x1)))))))))))))) 21#(00(12(21(00(02(12(01(12(11(01(12(01(02(12(11(11(21(x1)))))))))))))))))) (635)
21#(00(02(12(11(21(00(12(01(12(11(11(11(21(x1)))))))))))))) 21#(00(02(12(01(12(11(01(12(01(02(12(11(11(21(x1))))))))))))))) (636)
21#(00(02(12(11(21(00(12(01(12(11(11(11(11(x1)))))))))))))) 21#(00(12(21(00(02(12(01(12(11(01(12(01(02(12(11(11(11(x1)))))))))))))))))) (637)
21#(00(02(12(11(21(00(12(01(12(11(11(11(11(x1)))))))))))))) 21#(00(02(12(01(12(11(01(12(01(02(12(11(11(11(x1))))))))))))))) (638)
21#(00(02(12(11(21(00(12(01(12(11(11(11(01(x1)))))))))))))) 21#(00(12(21(00(02(12(01(12(11(01(12(01(02(12(11(11(01(x1)))))))))))))))))) (639)
21#(00(02(12(11(21(00(12(01(12(11(11(11(01(x1)))))))))))))) 21#(00(02(12(01(12(11(01(12(01(02(12(11(11(01(x1))))))))))))))) (640)
22#(10(11(21(20(00(02(02(22(10(01(02(12(21(x1)))))))))))))) 20#(00(12(01(02(02(12(21(x1)))))))) (641)
22#(10(11(21(20(00(02(02(22(10(01(02(12(21(x1)))))))))))))) 21#(10(01(12(11(11(01(22(20(00(12(01(02(02(12(21(x1)))))))))))))))) (642)
22#(10(11(21(20(00(02(02(22(10(01(02(12(21(x1)))))))))))))) 22#(20(00(12(01(02(02(12(21(x1))))))))) (643)
22#(10(11(21(20(00(02(02(22(10(01(02(12(21(x1)))))))))))))) 22#(10(21(10(01(12(11(11(01(22(20(00(12(01(02(02(12(21(x1)))))))))))))))))) (644)
22#(10(11(21(20(00(02(02(22(10(01(02(12(11(x1)))))))))))))) 20#(00(12(01(02(02(12(11(x1)))))))) (645)
22#(10(11(21(20(00(02(02(22(10(01(02(12(11(x1)))))))))))))) 21#(10(01(12(11(11(01(22(20(00(12(01(02(02(12(11(x1)))))))))))))))) (646)
22#(10(11(21(20(00(02(02(22(10(01(02(12(11(x1)))))))))))))) 22#(20(00(12(01(02(02(12(11(x1))))))))) (647)
22#(10(11(21(20(00(02(02(22(10(01(02(12(11(x1)))))))))))))) 22#(10(21(10(01(12(11(11(01(22(20(00(12(01(02(02(12(11(x1)))))))))))))))))) (648)
22#(10(11(21(20(00(02(02(22(10(01(02(12(01(x1)))))))))))))) 20#(00(12(01(02(02(12(01(x1)))))))) (649)
22#(10(11(21(20(00(02(02(22(10(01(02(12(01(x1)))))))))))))) 21#(10(01(12(11(11(01(22(20(00(12(01(02(02(12(01(x1)))))))))))))))) (650)
22#(10(11(21(20(00(02(02(22(10(01(02(12(01(x1)))))))))))))) 22#(20(00(12(01(02(02(12(01(x1))))))))) (651)
22#(10(11(21(20(00(02(02(22(10(01(02(12(01(x1)))))))))))))) 22#(10(21(10(01(12(11(11(01(22(20(00(12(01(02(02(12(01(x1)))))))))))))))))) (652)
22#(00(12(01(02(22(10(11(11(01(02(02(12(11(x1)))))))))))))) 22#(10(01(22(00(02(02(02(02(12(01(02(12(01(12(11(01(12(x1)))))))))))))))))) (653)
22#(00(12(01(02(22(10(11(11(01(02(02(12(11(x1)))))))))))))) 22#(00(02(02(02(02(12(01(02(12(01(12(11(01(12(x1))))))))))))))) (654)
22#(00(12(01(02(22(10(11(11(01(02(02(12(01(x1)))))))))))))) 22#(10(01(22(00(02(02(02(02(12(01(02(12(01(12(11(01(02(x1)))))))))))))))))) (655)
22#(00(12(01(02(22(10(11(11(01(02(02(12(01(x1)))))))))))))) 22#(00(02(02(02(02(12(01(02(12(01(12(11(01(02(x1))))))))))))))) (656)
22#(00(02(12(11(21(00(12(01(12(11(11(11(21(x1)))))))))))))) 21#(00(02(12(01(12(11(01(12(01(02(12(11(11(21(x1))))))))))))))) (657)
22#(00(02(12(11(21(00(12(01(12(11(11(11(21(x1)))))))))))))) 22#(00(12(21(00(02(12(01(12(11(01(12(01(02(12(11(11(21(x1)))))))))))))))))) (658)
22#(00(02(12(11(21(00(12(01(12(11(11(11(11(x1)))))))))))))) 21#(00(02(12(01(12(11(01(12(01(02(12(11(11(11(x1))))))))))))))) (659)
22#(00(02(12(11(21(00(12(01(12(11(11(11(11(x1)))))))))))))) 22#(00(12(21(00(02(12(01(12(11(01(12(01(02(12(11(11(11(x1)))))))))))))))))) (660)
22#(00(02(12(11(21(00(12(01(12(11(11(11(01(x1)))))))))))))) 21#(00(02(12(01(12(11(01(12(01(02(12(11(11(01(x1))))))))))))))) (661)
22#(00(02(12(11(21(00(12(01(12(11(11(11(01(x1)))))))))))))) 22#(00(12(21(00(02(12(01(12(11(01(12(01(02(12(11(11(01(x1)))))))))))))))))) (662)

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
[20(x1)] = x1 +
1
[21(x1)] = x1 +
1
[22(x1)] = x1 +
1
[10(x1)] = x1 +
1
[11(x1)] = x1 +
0
[12(x1)] = x1 +
0
[00(x1)] = x1 +
1
[01(x1)] = x1 +
0
[02(x1)] = x1 +
0
[20#(x1)] = x1 +
0
[21#(x1)] = x1 +
0
[22#(x1)] = x1 +
0
together with the usable rules
22(00(12(01(02(22(10(11(11(01(02(02(12(01(x1)))))))))))))) 22(10(01(22(00(02(02(02(02(12(01(02(12(01(12(11(01(02(x1)))))))))))))))))) (573)
21(00(12(01(02(22(10(11(11(01(02(02(12(01(x1)))))))))))))) 21(10(01(22(00(02(02(02(02(12(01(02(12(01(12(11(01(02(x1)))))))))))))))))) (574)
20(00(12(01(02(22(10(11(11(01(02(02(12(01(x1)))))))))))))) 20(10(01(22(00(02(02(02(02(12(01(02(12(01(12(11(01(02(x1)))))))))))))))))) (575)
22(00(12(01(02(22(10(11(11(01(02(02(12(11(x1)))))))))))))) 22(10(01(22(00(02(02(02(02(12(01(02(12(01(12(11(01(12(x1)))))))))))))))))) (576)
21(00(12(01(02(22(10(11(11(01(02(02(12(11(x1)))))))))))))) 21(10(01(22(00(02(02(02(02(12(01(02(12(01(12(11(01(12(x1)))))))))))))))))) (577)
20(00(12(01(02(22(10(11(11(01(02(02(12(11(x1)))))))))))))) 20(10(01(22(00(02(02(02(02(12(01(02(12(01(12(11(01(12(x1)))))))))))))))))) (578)
22(10(11(21(20(00(02(02(22(10(01(02(12(01(x1)))))))))))))) 22(10(21(10(01(12(11(11(01(22(20(00(12(01(02(02(12(01(x1)))))))))))))))))) (579)
21(10(11(21(20(00(02(02(22(10(01(02(12(01(x1)))))))))))))) 21(10(21(10(01(12(11(11(01(22(20(00(12(01(02(02(12(01(x1)))))))))))))))))) (580)
20(10(11(21(20(00(02(02(22(10(01(02(12(01(x1)))))))))))))) 20(10(21(10(01(12(11(11(01(22(20(00(12(01(02(02(12(01(x1)))))))))))))))))) (581)
22(10(11(21(20(00(02(02(22(10(01(02(12(11(x1)))))))))))))) 22(10(21(10(01(12(11(11(01(22(20(00(12(01(02(02(12(11(x1)))))))))))))))))) (582)
21(10(11(21(20(00(02(02(22(10(01(02(12(11(x1)))))))))))))) 21(10(21(10(01(12(11(11(01(22(20(00(12(01(02(02(12(11(x1)))))))))))))))))) (583)
20(10(11(21(20(00(02(02(22(10(01(02(12(11(x1)))))))))))))) 20(10(21(10(01(12(11(11(01(22(20(00(12(01(02(02(12(11(x1)))))))))))))))))) (584)
22(10(11(21(20(00(02(02(22(10(01(02(12(21(x1)))))))))))))) 22(10(21(10(01(12(11(11(01(22(20(00(12(01(02(02(12(21(x1)))))))))))))))))) (585)
21(10(11(21(20(00(02(02(22(10(01(02(12(21(x1)))))))))))))) 21(10(21(10(01(12(11(11(01(22(20(00(12(01(02(02(12(21(x1)))))))))))))))))) (586)
20(10(11(21(20(00(02(02(22(10(01(02(12(21(x1)))))))))))))) 20(10(21(10(01(12(11(11(01(22(20(00(12(01(02(02(12(21(x1)))))))))))))))))) (587)
22(00(02(12(11(21(00(12(01(12(11(11(11(01(x1)))))))))))))) 22(00(12(21(00(02(12(01(12(11(01(12(01(02(12(11(11(01(x1)))))))))))))))))) (588)
21(00(02(12(11(21(00(12(01(12(11(11(11(01(x1)))))))))))))) 21(00(12(21(00(02(12(01(12(11(01(12(01(02(12(11(11(01(x1)))))))))))))))))) (589)
20(00(02(12(11(21(00(12(01(12(11(11(11(01(x1)))))))))))))) 20(00(12(21(00(02(12(01(12(11(01(12(01(02(12(11(11(01(x1)))))))))))))))))) (590)
22(00(02(12(11(21(00(12(01(12(11(11(11(11(x1)))))))))))))) 22(00(12(21(00(02(12(01(12(11(01(12(01(02(12(11(11(11(x1)))))))))))))))))) (591)
21(00(02(12(11(21(00(12(01(12(11(11(11(11(x1)))))))))))))) 21(00(12(21(00(02(12(01(12(11(01(12(01(02(12(11(11(11(x1)))))))))))))))))) (592)
20(00(02(12(11(21(00(12(01(12(11(11(11(11(x1)))))))))))))) 20(00(12(21(00(02(12(01(12(11(01(12(01(02(12(11(11(11(x1)))))))))))))))))) (593)
22(00(02(12(11(21(00(12(01(12(11(11(11(21(x1)))))))))))))) 22(00(12(21(00(02(12(01(12(11(01(12(01(02(12(11(11(21(x1)))))))))))))))))) (594)
21(00(02(12(11(21(00(12(01(12(11(11(11(21(x1)))))))))))))) 21(00(12(21(00(02(12(01(12(11(01(12(01(02(12(11(11(21(x1)))))))))))))))))) (595)
20(00(02(12(11(21(00(12(01(12(11(11(11(21(x1)))))))))))))) 20(00(12(21(00(02(12(01(12(11(01(12(01(02(12(11(11(21(x1)))))))))))))))))) (596)
(w.r.t. the implicit argument filter of the reduction pair), the pairs
20#(10(11(21(20(00(02(02(22(10(01(02(12(21(x1)))))))))))))) 20#(00(12(01(02(02(12(21(x1)))))))) (598)
20#(10(11(21(20(00(02(02(22(10(01(02(12(21(x1)))))))))))))) 21#(10(01(12(11(11(01(22(20(00(12(01(02(02(12(21(x1)))))))))))))))) (599)
20#(10(11(21(20(00(02(02(22(10(01(02(12(21(x1)))))))))))))) 22#(20(00(12(01(02(02(12(21(x1))))))))) (600)
20#(10(11(21(20(00(02(02(22(10(01(02(12(11(x1)))))))))))))) 20#(00(12(01(02(02(12(11(x1)))))))) (602)
20#(10(11(21(20(00(02(02(22(10(01(02(12(11(x1)))))))))))))) 21#(10(01(12(11(11(01(22(20(00(12(01(02(02(12(11(x1)))))))))))))))) (603)
20#(10(11(21(20(00(02(02(22(10(01(02(12(11(x1)))))))))))))) 22#(20(00(12(01(02(02(12(11(x1))))))))) (604)
20#(10(11(21(20(00(02(02(22(10(01(02(12(01(x1)))))))))))))) 20#(00(12(01(02(02(12(01(x1)))))))) (606)
20#(10(11(21(20(00(02(02(22(10(01(02(12(01(x1)))))))))))))) 21#(10(01(12(11(11(01(22(20(00(12(01(02(02(12(01(x1)))))))))))))))) (607)
20#(10(11(21(20(00(02(02(22(10(01(02(12(01(x1)))))))))))))) 22#(20(00(12(01(02(02(12(01(x1))))))))) (608)
20#(00(12(01(02(22(10(11(11(01(02(02(12(11(x1)))))))))))))) 22#(00(02(02(02(02(12(01(02(12(01(12(11(01(12(x1))))))))))))))) (610)
20#(00(12(01(02(22(10(11(11(01(02(02(12(01(x1)))))))))))))) 22#(00(02(02(02(02(12(01(02(12(01(12(11(01(02(x1))))))))))))))) (612)
20#(00(02(12(11(21(00(12(01(12(11(11(11(21(x1)))))))))))))) 21#(00(02(12(01(12(11(01(12(01(02(12(11(11(21(x1))))))))))))))) (614)
20#(00(02(12(11(21(00(12(01(12(11(11(11(11(x1)))))))))))))) 21#(00(02(12(01(12(11(01(12(01(02(12(11(11(11(x1))))))))))))))) (616)
20#(00(02(12(11(21(00(12(01(12(11(11(11(01(x1)))))))))))))) 21#(00(02(12(01(12(11(01(12(01(02(12(11(11(01(x1))))))))))))))) (618)
21#(10(11(21(20(00(02(02(22(10(01(02(12(21(x1)))))))))))))) 20#(00(12(01(02(02(12(21(x1)))))))) (619)
21#(10(11(21(20(00(02(02(22(10(01(02(12(21(x1)))))))))))))) 21#(10(01(12(11(11(01(22(20(00(12(01(02(02(12(21(x1)))))))))))))))) (621)
21#(10(11(21(20(00(02(02(22(10(01(02(12(21(x1)))))))))))))) 22#(20(00(12(01(02(02(12(21(x1))))))))) (622)
21#(10(11(21(20(00(02(02(22(10(01(02(12(11(x1)))))))))))))) 20#(00(12(01(02(02(12(11(x1)))))))) (623)
21#(10(11(21(20(00(02(02(22(10(01(02(12(11(x1)))))))))))))) 21#(10(01(12(11(11(01(22(20(00(12(01(02(02(12(11(x1)))))))))))))))) (625)
21#(10(11(21(20(00(02(02(22(10(01(02(12(11(x1)))))))))))))) 22#(20(00(12(01(02(02(12(11(x1))))))))) (626)
21#(10(11(21(20(00(02(02(22(10(01(02(12(01(x1)))))))))))))) 20#(00(12(01(02(02(12(01(x1)))))))) (627)
21#(10(11(21(20(00(02(02(22(10(01(02(12(01(x1)))))))))))))) 21#(10(01(12(11(11(01(22(20(00(12(01(02(02(12(01(x1)))))))))))))))) (629)
21#(10(11(21(20(00(02(02(22(10(01(02(12(01(x1)))))))))))))) 22#(20(00(12(01(02(02(12(01(x1))))))))) (630)
21#(00(12(01(02(22(10(11(11(01(02(02(12(11(x1)))))))))))))) 22#(00(02(02(02(02(12(01(02(12(01(12(11(01(12(x1))))))))))))))) (632)
21#(00(12(01(02(22(10(11(11(01(02(02(12(01(x1)))))))))))))) 22#(00(02(02(02(02(12(01(02(12(01(12(11(01(02(x1))))))))))))))) (634)
21#(00(02(12(11(21(00(12(01(12(11(11(11(21(x1)))))))))))))) 21#(00(02(12(01(12(11(01(12(01(02(12(11(11(21(x1))))))))))))))) (636)
21#(00(02(12(11(21(00(12(01(12(11(11(11(11(x1)))))))))))))) 21#(00(02(12(01(12(11(01(12(01(02(12(11(11(11(x1))))))))))))))) (638)
21#(00(02(12(11(21(00(12(01(12(11(11(11(01(x1)))))))))))))) 21#(00(02(12(01(12(11(01(12(01(02(12(11(11(01(x1))))))))))))))) (640)
22#(10(11(21(20(00(02(02(22(10(01(02(12(21(x1)))))))))))))) 20#(00(12(01(02(02(12(21(x1)))))))) (641)
22#(10(11(21(20(00(02(02(22(10(01(02(12(21(x1)))))))))))))) 21#(10(01(12(11(11(01(22(20(00(12(01(02(02(12(21(x1)))))))))))))))) (642)
22#(10(11(21(20(00(02(02(22(10(01(02(12(21(x1)))))))))))))) 22#(20(00(12(01(02(02(12(21(x1))))))))) (643)
22#(10(11(21(20(00(02(02(22(10(01(02(12(11(x1)))))))))))))) 20#(00(12(01(02(02(12(11(x1)))))))) (645)
22#(10(11(21(20(00(02(02(22(10(01(02(12(11(x1)))))))))))))) 21#(10(01(12(11(11(01(22(20(00(12(01(02(02(12(11(x1)))))))))))))))) (646)
22#(10(11(21(20(00(02(02(22(10(01(02(12(11(x1)))))))))))))) 22#(20(00(12(01(02(02(12(11(x1))))))))) (647)
22#(10(11(21(20(00(02(02(22(10(01(02(12(01(x1)))))))))))))) 20#(00(12(01(02(02(12(01(x1)))))))) (649)
22#(10(11(21(20(00(02(02(22(10(01(02(12(01(x1)))))))))))))) 21#(10(01(12(11(11(01(22(20(00(12(01(02(02(12(01(x1)))))))))))))))) (650)
22#(10(11(21(20(00(02(02(22(10(01(02(12(01(x1)))))))))))))) 22#(20(00(12(01(02(02(12(01(x1))))))))) (651)
22#(00(12(01(02(22(10(11(11(01(02(02(12(11(x1)))))))))))))) 22#(00(02(02(02(02(12(01(02(12(01(12(11(01(12(x1))))))))))))))) (654)
22#(00(12(01(02(22(10(11(11(01(02(02(12(01(x1)))))))))))))) 22#(00(02(02(02(02(12(01(02(12(01(12(11(01(02(x1))))))))))))))) (656)
22#(00(02(12(11(21(00(12(01(12(11(11(11(21(x1)))))))))))))) 21#(00(02(12(01(12(11(01(12(01(02(12(11(11(21(x1))))))))))))))) (657)
22#(00(02(12(11(21(00(12(01(12(11(11(11(11(x1)))))))))))))) 21#(00(02(12(01(12(11(01(12(01(02(12(11(11(11(x1))))))))))))))) (659)
22#(00(02(12(11(21(00(12(01(12(11(11(11(01(x1)))))))))))))) 21#(00(02(12(01(12(11(01(12(01(02(12(11(11(01(x1))))))))))))))) (661)
and no rules could be deleted.

1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.