Certification Problem

Input (TPDB SRS_Relative/ICFP_2010_relative/4893)

The relative rewrite relation R/S is considered where R is the following TRS

3(5(3(x1))) 1(0(3(4(4(3(1(4(5(4(x1)))))))))) (1)
5(0(0(x1))) 5(0(3(2(1(2(3(4(0(0(x1)))))))))) (2)
0(1(0(4(x1)))) 4(2(1(2(3(4(4(3(0(4(x1)))))))))) (3)
4(5(0(4(x1)))) 4(0(4(0(4(0(3(4(4(4(x1)))))))))) (4)
5(5(5(5(x1)))) 5(2(0(0(3(2(1(4(4(3(x1)))))))))) (5)
5(5(5(5(x1)))) 5(4(0(0(1(1(1(4(3(5(x1)))))))))) (6)
1(5(3(1(3(x1))))) 1(3(4(4(2(0(2(3(1(2(x1)))))))))) (7)
2(3(5(5(0(x1))))) 4(0(1(4(2(1(2(0(1(0(x1)))))))))) (8)
2(5(5(5(5(x1))))) 2(0(5(1(2(0(3(2(1(5(x1)))))))))) (9)
4(1(0(0(0(x1))))) 4(1(4(1(0(5(1(2(4(2(x1)))))))))) (10)
4(1(5(5(3(x1))))) 2(1(2(3(0(3(2(5(1(2(x1)))))))))) (11)
4(3(1(5(5(x1))))) 4(0(3(1(2(3(2(1(1(5(x1)))))))))) (12)
5(0(2(4(2(x1))))) 5(2(1(2(3(0(0(3(4(2(x1)))))))))) (13)
5(3(1(5(1(x1))))) 3(2(1(1(0(3(0(1(5(1(x1)))))))))) (14)
5(5(3(0(2(x1))))) 3(2(3(4(5(2(1(2(0(0(x1)))))))))) (15)
5(5(5(5(4(x1))))) 3(4(5(4(0(0(4(0(3(4(x1)))))))))) (16)
0(1(5(1(0(4(x1)))))) 3(2(0(4(0(1(4(3(0(4(x1)))))))))) (17)
0(1(5(1(5(0(x1)))))) 4(4(4(2(5(5(0(1(4(0(x1)))))))))) (18)
0(1(5(3(0(1(x1)))))) 4(4(5(1(1(1(4(0(3(1(x1)))))))))) (19)
0(3(1(3(0(5(x1)))))) 4(4(3(3(4(4(0(5(2(3(x1)))))))))) (20)
1(5(5(5(5(3(x1)))))) 1(2(5(1(4(0(0(2(5(4(x1)))))))))) (21)
2(0(5(3(4(1(x1)))))) 2(3(4(0(3(3(0(1(2(1(x1)))))))))) (22)
2(4(0(1(3(5(x1)))))) 2(3(2(0(2(0(3(1(4(3(x1)))))))))) (23)
2(5(4(5(5(5(x1)))))) 2(4(2(1(2(4(4(4(0(5(x1)))))))))) (24)
3(5(3(1(5(5(x1)))))) 3(2(4(4(1(1(4(0(2(4(x1)))))))))) (25)
4(1(3(1(5(0(x1)))))) 0(2(2(2(1(1(2(3(1(4(x1)))))))))) (26)
4(3(5(0(4(2(x1)))))) 4(3(4(3(4(1(3(1(4(2(x1)))))))))) (27)
5(0(5(0(4(2(x1)))))) 5(4(4(4(4(0(4(4(3(0(x1)))))))))) (28)
5(5(1(5(3(0(x1)))))) 2(2(0(1(4(3(0(3(3(4(x1)))))))))) (29)
5(5(3(2(3(5(x1)))))) 1(4(2(4(1(2(2(4(1(5(x1)))))))))) (30)
5(5(5(1(3(1(x1)))))) 5(1(2(4(4(4(1(3(3(4(x1)))))))))) (31)
1(0(4(5(3(5(0(x1))))))) 1(0(2(1(2(5(2(0(5(2(x1)))))))))) (32)
1(5(1(3(5(5(5(x1))))))) 5(1(2(3(0(2(2(0(5(5(x1)))))))))) (33)
3(0(1(5(1(3(0(x1))))))) 3(0(0(3(1(2(1(5(4(2(x1)))))))))) (34)
3(2(4(1(5(5(1(x1))))))) 0(5(0(3(4(1(0(3(5(1(x1)))))))))) (35)
3(5(5(5(1(4(1(x1))))))) 1(4(4(0(0(1(0(1(3(1(x1)))))))))) (36)
4(0(5(5(5(5(4(x1))))))) 2(1(4(3(5(4(0(1(5(4(x1)))))))))) (37)
4(1(5(0(0(1(3(x1))))))) 0(2(1(1(1(1(4(5(1(3(x1)))))))))) (38)
4(1(5(0(0(1(5(x1))))))) 4(3(0(5(3(5(2(1(4(5(x1)))))))))) (39)
4(2(3(5(0(5(0(x1))))))) 4(4(4(0(0(1(5(2(3(4(x1)))))))))) (40)
4(2(5(5(0(2(2(x1))))))) 4(2(1(0(0(1(4(5(1(2(x1)))))))))) (41)
4(5(5(3(5(5(3(x1))))))) 0(1(4(0(1(3(2(4(1(4(x1)))))))))) (42)
4(5(5(5(0(4(3(x1))))))) 0(3(3(3(2(0(2(3(2(3(x1)))))))))) (43)
5(0(5(5(3(5(4(x1))))))) 5(3(5(5(0(1(1(2(3(4(x1)))))))))) (44)
5(2(5(5(0(0(3(x1))))))) 1(4(4(3(1(0(1(3(0(3(x1)))))))))) (45)
5(3(5(5(5(3(0(x1))))))) 5(4(3(4(5(2(5(5(5(0(x1)))))))))) (46)
5(4(5(3(2(5(3(x1))))))) 5(4(3(1(1(4(2(0(3(5(x1)))))))))) (47)
5(5(0(5(5(0(1(x1))))))) 2(3(4(5(4(2(1(2(5(1(x1)))))))))) (48)
5(5(0(5(5(5(3(x1))))))) 5(0(4(4(0(1(2(3(3(3(x1)))))))))) (49)
5(5(2(5(3(5(0(x1))))))) 5(2(0(4(0(3(2(3(3(0(x1)))))))))) (50)
5(5(5(3(0(4(2(x1))))))) 5(5(2(1(5(4(3(4(4(2(x1)))))))))) (51)

and S is the following TRS.

2(5(3(0(5(0(1(x1))))))) 4(0(4(1(4(0(5(4(2(1(x1)))))))))) (52)
2(0(0(2(4(x1))))) 4(4(0(4(3(1(2(4(0(4(x1)))))))))) (53)
5(3(0(1(x1)))) 3(4(4(1(4(4(3(2(0(3(x1)))))))))) (54)
2(5(5(5(0(x1))))) 2(5(0(2(3(0(3(0(5(0(x1)))))))))) (55)
0(0(2(x1))) 1(0(3(4(3(1(2(0(0(2(x1)))))))))) (56)
4(0(1(x1))) 4(4(1(1(4(0(3(4(0(1(x1)))))))))) (57)
5(5(5(1(0(x1))))) 3(3(5(2(4(4(1(2(1(2(x1)))))))))) (58)
5(5(0(x1))) 0(2(0(3(1(1(1(2(2(2(x1)))))))))) (59)
0(0(0(x1))) 1(2(3(4(4(2(3(3(2(2(x1)))))))))) (60)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Closure Under Flat Contexts

Using the flat contexts

{3(), 5(), 1(), 0(), 4(), 2()}

We obtain the transformed TRS

There are 210 ruless (increase limit for explicit display).

1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

There are 1260 ruless (increase limit for explicit display).

1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[50(x1)] = 1 + 1 · x1
[00(x1)] = 1 · x1
[05(x1)] = 1 · x1
[03(x1)] = 1 · x1
[32(x1)] = 1 · x1
[21(x1)] = 1 · x1
[12(x1)] = 1 · x1
[23(x1)] = 1 · x1
[34(x1)] = 1 · x1
[40(x1)] = 1 · x1
[02(x1)] = 1 · x1
[01(x1)] = 1 · x1
[04(x1)] = 1 · x1
[45(x1)] = 1 · x1
[44(x1)] = 1 · x1
[43(x1)] = 1 · x1
[42(x1)] = 1 · x1
[41(x1)] = 1 · x1
[55(x1)] = 1 + 1 · x1
[52(x1)] = 1 + 1 · x1
[20(x1)] = 1 · x1
[14(x1)] = 1 · x1
[35(x1)] = 1 · x1
[30(x1)] = 1 · x1
[53(x1)] = 1 + 1 · x1
[33(x1)] = 1 · x1
[51(x1)] = 1 · x1
[31(x1)] = 1 · x1
[54(x1)] = 1 · x1
[11(x1)] = 1 · x1
[15(x1)] = 1 + 1 · x1
[13(x1)] = 1 + 1 · x1
[25(x1)] = 1 · x1
[22(x1)] = 1 · x1
[24(x1)] = 1 · x1
[10(x1)] = 1 · x1
all of the following rules can be deleted.

There are 1064 ruless (increase limit for explicit display).

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[50(x1)] = 1 + 1 · x1
[00(x1)] = 1 + 1 · x1
[05(x1)] = 1 · x1
[03(x1)] = 1 · x1
[32(x1)] = 1 · x1
[21(x1)] = 1 · x1
[12(x1)] = 1 · x1
[23(x1)] = 1 · x1
[34(x1)] = 1 · x1
[40(x1)] = 1 · x1
[02(x1)] = 1 + 1 · x1
[01(x1)] = 1 + 1 · x1
[04(x1)] = 1 + 1 · x1
[41(x1)] = 1 · x1
[10(x1)] = 1 · x1
[14(x1)] = 1 · x1
[51(x1)] = 1 + 1 · x1
[24(x1)] = 1 · x1
[42(x1)] = 1 + 1 · x1
[25(x1)] = 1 · x1
[20(x1)] = 1 · x1
[22(x1)] = 1 · x1
[52(x1)] = 1 · x1
[30(x1)] = 1 · x1
[43(x1)] = 1 · x1
[35(x1)] = 1 · x1
[13(x1)] = 1 + 1 · x1
[31(x1)] = 1 · x1
[45(x1)] = 1 · x1
[53(x1)] = 1 + 1 · x1
[44(x1)] = 1 · x1
[15(x1)] = 1 + 1 · x1
[55(x1)] = 1 + 1 · x1
[54(x1)] = 1 + 1 · x1
[11(x1)] = 1 · x1
[33(x1)] = 1 + 1 · x1
all of the following rules can be deleted.
41(10(00(00(00(x1))))) 41(14(41(10(05(51(12(24(42(20(x1)))))))))) (278)
41(10(00(00(02(x1))))) 41(14(41(10(05(51(12(24(42(22(x1)))))))))) (280)
41(10(00(00(01(x1))))) 41(14(41(10(05(51(12(24(42(21(x1)))))))))) (281)
41(10(00(00(04(x1))))) 41(14(41(10(05(51(12(24(42(24(x1)))))))))) (282)
50(02(24(42(25(x1))))) 52(21(12(23(30(00(03(34(42(25(x1)))))))))) (289)
50(02(24(42(20(x1))))) 52(21(12(23(30(00(03(34(42(20(x1)))))))))) (290)
50(02(24(42(23(x1))))) 52(21(12(23(30(00(03(34(42(23(x1)))))))))) (291)
50(02(24(42(22(x1))))) 52(21(12(23(30(00(03(34(42(22(x1)))))))))) (292)
50(02(24(42(21(x1))))) 52(21(12(23(30(00(03(34(42(21(x1)))))))))) (293)
50(02(24(42(24(x1))))) 52(21(12(23(30(00(03(34(42(24(x1)))))))))) (294)
43(35(50(04(42(25(x1)))))) 43(34(43(34(41(13(31(14(42(25(x1)))))))))) (325)
43(35(50(04(42(20(x1)))))) 43(34(43(34(41(13(31(14(42(20(x1)))))))))) (326)
43(35(50(04(42(23(x1)))))) 43(34(43(34(41(13(31(14(42(23(x1)))))))))) (327)
43(35(50(04(42(22(x1)))))) 43(34(43(34(41(13(31(14(42(22(x1)))))))))) (328)
43(35(50(04(42(21(x1)))))) 43(34(43(34(41(13(31(14(42(21(x1)))))))))) (329)
43(35(50(04(42(24(x1)))))) 43(34(43(34(41(13(31(14(42(24(x1)))))))))) (330)
10(04(45(53(35(50(05(x1))))))) 10(02(21(12(25(52(20(05(52(25(x1)))))))))) (343)
10(04(45(53(35(50(00(x1))))))) 10(02(21(12(25(52(20(05(52(20(x1)))))))))) (344)
10(04(45(53(35(50(03(x1))))))) 10(02(21(12(25(52(20(05(52(23(x1)))))))))) (345)
10(04(45(53(35(50(02(x1))))))) 10(02(21(12(25(52(20(05(52(22(x1)))))))))) (346)
10(04(45(53(35(50(01(x1))))))) 10(02(21(12(25(52(20(05(52(21(x1)))))))))) (347)
10(04(45(53(35(50(04(x1))))))) 10(02(21(12(25(52(20(05(52(24(x1)))))))))) (348)
42(23(35(50(05(50(00(x1))))))) 44(44(40(00(01(15(52(23(34(40(x1)))))))))) (362)
42(23(35(50(05(50(01(x1))))))) 44(44(40(00(01(15(52(23(34(41(x1)))))))))) (365)
42(23(35(50(05(50(04(x1))))))) 44(44(40(00(01(15(52(23(34(44(x1)))))))))) (366)
55(55(53(30(04(42(25(x1))))))) 55(52(21(15(54(43(34(44(42(25(x1)))))))))) (403)
55(55(53(30(04(42(20(x1))))))) 55(52(21(15(54(43(34(44(42(20(x1)))))))))) (404)
55(55(53(30(04(42(23(x1))))))) 55(52(21(15(54(43(34(44(42(23(x1)))))))))) (405)
55(55(53(30(04(42(22(x1))))))) 55(52(21(15(54(43(34(44(42(22(x1)))))))))) (406)
55(55(53(30(04(42(21(x1))))))) 55(52(21(15(54(43(34(44(42(21(x1)))))))))) (407)
55(55(53(30(04(42(24(x1))))))) 55(52(21(15(54(43(34(44(42(24(x1)))))))))) (408)
30(03(31(13(30(05(51(x1))))))) 34(44(43(33(34(44(40(05(52(23(31(x1))))))))))) (773)
30(03(31(13(30(05(54(x1))))))) 34(44(43(33(34(44(40(05(52(23(34(x1))))))))))) (774)
10(03(31(13(30(05(51(x1))))))) 14(44(43(33(34(44(40(05(52(23(31(x1))))))))))) (785)
10(03(31(13(30(05(54(x1))))))) 14(44(43(33(34(44(40(05(52(23(34(x1))))))))))) (786)
00(03(31(13(30(05(51(x1))))))) 04(44(43(33(34(44(40(05(52(23(31(x1))))))))))) (791)
00(03(31(13(30(05(54(x1))))))) 04(44(43(33(34(44(40(05(52(23(34(x1))))))))))) (792)
40(03(31(13(30(05(51(x1))))))) 44(44(43(33(34(44(40(05(52(23(31(x1))))))))))) (797)
40(03(31(13(30(05(54(x1))))))) 44(44(43(33(34(44(40(05(52(23(34(x1))))))))))) (798)
20(03(31(13(30(05(51(x1))))))) 24(44(43(33(34(44(40(05(52(23(31(x1))))))))))) (803)
20(03(31(13(30(05(54(x1))))))) 24(44(43(33(34(44(40(05(52(23(34(x1))))))))))) (804)
42(20(00(02(24(45(x1)))))) 44(44(40(04(43(31(12(24(40(04(45(x1))))))))))) (1309)
42(20(00(02(24(40(x1)))))) 44(44(40(04(43(31(12(24(40(04(40(x1))))))))))) (1310)
42(20(00(02(24(43(x1)))))) 44(44(40(04(43(31(12(24(40(04(43(x1))))))))))) (1311)
42(20(00(02(24(42(x1)))))) 44(44(40(04(43(31(12(24(40(04(42(x1))))))))))) (1312)
42(20(00(02(24(41(x1)))))) 44(44(40(04(43(31(12(24(40(04(41(x1))))))))))) (1313)
42(20(00(02(24(44(x1)))))) 44(44(40(04(43(31(12(24(40(04(44(x1))))))))))) (1314)
30(00(00(00(x1)))) 31(12(23(34(44(42(23(33(32(22(20(x1))))))))))) (1466)
30(00(00(02(x1)))) 31(12(23(34(44(42(23(33(32(22(22(x1))))))))))) (1468)
30(00(00(01(x1)))) 31(12(23(34(44(42(23(33(32(22(21(x1))))))))))) (1469)
30(00(00(04(x1)))) 31(12(23(34(44(42(23(33(32(22(24(x1))))))))))) (1470)
10(00(00(00(x1)))) 11(12(23(34(44(42(23(33(32(22(20(x1))))))))))) (1478)
10(00(00(02(x1)))) 11(12(23(34(44(42(23(33(32(22(22(x1))))))))))) (1480)
10(00(00(01(x1)))) 11(12(23(34(44(42(23(33(32(22(21(x1))))))))))) (1481)
10(00(00(04(x1)))) 11(12(23(34(44(42(23(33(32(22(24(x1))))))))))) (1482)
00(00(00(00(x1)))) 01(12(23(34(44(42(23(33(32(22(20(x1))))))))))) (1484)
00(00(00(02(x1)))) 01(12(23(34(44(42(23(33(32(22(22(x1))))))))))) (1486)
00(00(00(01(x1)))) 01(12(23(34(44(42(23(33(32(22(21(x1))))))))))) (1487)
00(00(00(04(x1)))) 01(12(23(34(44(42(23(33(32(22(24(x1))))))))))) (1488)
40(00(00(00(x1)))) 41(12(23(34(44(42(23(33(32(22(20(x1))))))))))) (1490)
40(00(00(02(x1)))) 41(12(23(34(44(42(23(33(32(22(22(x1))))))))))) (1492)
40(00(00(01(x1)))) 41(12(23(34(44(42(23(33(32(22(21(x1))))))))))) (1493)
40(00(00(04(x1)))) 41(12(23(34(44(42(23(33(32(22(24(x1))))))))))) (1494)
20(00(00(00(x1)))) 21(12(23(34(44(42(23(33(32(22(20(x1))))))))))) (1496)
20(00(00(02(x1)))) 21(12(23(34(44(42(23(33(32(22(22(x1))))))))))) (1498)
20(00(00(01(x1)))) 21(12(23(34(44(42(23(33(32(22(21(x1))))))))))) (1499)
20(00(00(04(x1)))) 21(12(23(34(44(42(23(33(32(22(24(x1))))))))))) (1500)

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[50(x1)] = 1 + 1 · x1
[00(x1)] = 1 + 1 · x1
[05(x1)] = 1 · x1
[03(x1)] = 1 · x1
[32(x1)] = 1 · x1
[21(x1)] = 1 · x1
[12(x1)] = 1 · x1
[23(x1)] = 1 · x1
[34(x1)] = 1 · x1
[40(x1)] = 1 · x1
[02(x1)] = 1 + 1 · x1
[01(x1)] = 1 + 1 · x1
[04(x1)] = 1 · x1
[41(x1)] = 1 · x1
[10(x1)] = 1 · x1
[14(x1)] = 1 · x1
[51(x1)] = 1 + 1 · x1
[24(x1)] = 1 · x1
[42(x1)] = 1 · x1
[25(x1)] = 1 · x1
[35(x1)] = 1 + 1 · x1
[44(x1)] = 1 · x1
[15(x1)] = 1 + 1 · x1
[52(x1)] = 1 · x1
[45(x1)] = 1 · x1
[43(x1)] = 1 · x1
[55(x1)] = 1 · x1
[53(x1)] = 1 · x1
[54(x1)] = 1 + 1 · x1
[11(x1)] = 1 · x1
[30(x1)] = 1 + 1 · x1
[20(x1)] = 1 · x1
[13(x1)] = 1 · x1
[31(x1)] = 1 · x1
[22(x1)] = 1 · x1
[33(x1)] = 1 + 1 · x1
all of the following rules can be deleted.
41(10(00(00(05(x1))))) 41(14(41(10(05(51(12(24(42(25(x1)))))))))) (277)
41(10(00(00(03(x1))))) 41(14(41(10(05(51(12(24(42(23(x1)))))))))) (279)
42(23(35(50(05(50(02(x1))))))) 44(44(40(00(01(15(52(23(34(42(x1)))))))))) (364)
30(01(10(04(45(x1))))) 34(42(21(12(23(34(44(43(30(04(45(x1))))))))))) (445)
30(01(10(04(40(x1))))) 34(42(21(12(23(34(44(43(30(04(40(x1))))))))))) (446)
30(01(10(04(43(x1))))) 34(42(21(12(23(34(44(43(30(04(43(x1))))))))))) (447)
30(01(10(04(42(x1))))) 34(42(21(12(23(34(44(43(30(04(42(x1))))))))))) (448)
30(01(10(04(41(x1))))) 34(42(21(12(23(34(44(43(30(04(41(x1))))))))))) (449)
30(01(10(04(44(x1))))) 34(42(21(12(23(34(44(43(30(04(44(x1))))))))))) (450)
00(01(10(04(45(x1))))) 04(42(21(12(23(34(44(43(30(04(45(x1))))))))))) (463)
00(01(10(04(40(x1))))) 04(42(21(12(23(34(44(43(30(04(40(x1))))))))))) (464)
00(01(10(04(43(x1))))) 04(42(21(12(23(34(44(43(30(04(43(x1))))))))))) (465)
00(01(10(04(42(x1))))) 04(42(21(12(23(34(44(43(30(04(42(x1))))))))))) (466)
00(01(10(04(41(x1))))) 04(42(21(12(23(34(44(43(30(04(41(x1))))))))))) (467)
00(01(10(04(44(x1))))) 04(42(21(12(23(34(44(43(30(04(44(x1))))))))))) (468)
10(01(15(51(10(04(45(x1))))))) 13(32(20(04(40(01(14(43(30(04(45(x1))))))))))) (673)
10(01(15(51(10(04(40(x1))))))) 13(32(20(04(40(01(14(43(30(04(40(x1))))))))))) (674)
10(01(15(51(10(04(43(x1))))))) 13(32(20(04(40(01(14(43(30(04(43(x1))))))))))) (675)
10(01(15(51(10(04(42(x1))))))) 13(32(20(04(40(01(14(43(30(04(42(x1))))))))))) (676)
10(01(15(51(10(04(41(x1))))))) 13(32(20(04(40(01(14(43(30(04(41(x1))))))))))) (677)
10(01(15(51(10(04(44(x1))))))) 13(32(20(04(40(01(14(43(30(04(44(x1))))))))))) (678)
32(20(00(02(24(45(x1)))))) 34(44(40(04(43(31(12(24(40(04(45(x1))))))))))) (1285)
32(20(00(02(24(40(x1)))))) 34(44(40(04(43(31(12(24(40(04(40(x1))))))))))) (1286)
32(20(00(02(24(43(x1)))))) 34(44(40(04(43(31(12(24(40(04(43(x1))))))))))) (1287)
32(20(00(02(24(42(x1)))))) 34(44(40(04(43(31(12(24(40(04(42(x1))))))))))) (1288)
32(20(00(02(24(41(x1)))))) 34(44(40(04(43(31(12(24(40(04(41(x1))))))))))) (1289)
32(20(00(02(24(44(x1)))))) 34(44(40(04(43(31(12(24(40(04(44(x1))))))))))) (1290)
12(20(00(02(24(45(x1)))))) 14(44(40(04(43(31(12(24(40(04(45(x1))))))))))) (1297)
12(20(00(02(24(40(x1)))))) 14(44(40(04(43(31(12(24(40(04(40(x1))))))))))) (1298)
12(20(00(02(24(43(x1)))))) 14(44(40(04(43(31(12(24(40(04(43(x1))))))))))) (1299)
12(20(00(02(24(42(x1)))))) 14(44(40(04(43(31(12(24(40(04(42(x1))))))))))) (1300)
12(20(00(02(24(41(x1)))))) 14(44(40(04(43(31(12(24(40(04(41(x1))))))))))) (1301)
12(20(00(02(24(44(x1)))))) 14(44(40(04(43(31(12(24(40(04(44(x1))))))))))) (1302)
02(20(00(02(24(45(x1)))))) 04(44(40(04(43(31(12(24(40(04(45(x1))))))))))) (1303)
02(20(00(02(24(40(x1)))))) 04(44(40(04(43(31(12(24(40(04(40(x1))))))))))) (1304)
02(20(00(02(24(43(x1)))))) 04(44(40(04(43(31(12(24(40(04(43(x1))))))))))) (1305)
02(20(00(02(24(42(x1)))))) 04(44(40(04(43(31(12(24(40(04(42(x1))))))))))) (1306)
02(20(00(02(24(41(x1)))))) 04(44(40(04(43(31(12(24(40(04(41(x1))))))))))) (1307)
02(20(00(02(24(44(x1)))))) 04(44(40(04(43(31(12(24(40(04(44(x1))))))))))) (1308)
22(20(00(02(24(45(x1)))))) 24(44(40(04(43(31(12(24(40(04(45(x1))))))))))) (1315)
22(20(00(02(24(40(x1)))))) 24(44(40(04(43(31(12(24(40(04(40(x1))))))))))) (1316)
22(20(00(02(24(43(x1)))))) 24(44(40(04(43(31(12(24(40(04(43(x1))))))))))) (1317)
22(20(00(02(24(42(x1)))))) 24(44(40(04(43(31(12(24(40(04(42(x1))))))))))) (1318)
22(20(00(02(24(41(x1)))))) 24(44(40(04(43(31(12(24(40(04(41(x1))))))))))) (1319)
22(20(00(02(24(44(x1)))))) 24(44(40(04(43(31(12(24(40(04(44(x1))))))))))) (1320)
30(00(02(25(x1)))) 31(10(03(34(43(31(12(20(00(02(25(x1))))))))))) (1357)
30(00(02(20(x1)))) 31(10(03(34(43(31(12(20(00(02(20(x1))))))))))) (1358)
30(00(02(23(x1)))) 31(10(03(34(43(31(12(20(00(02(23(x1))))))))))) (1359)
30(00(02(22(x1)))) 31(10(03(34(43(31(12(20(00(02(22(x1))))))))))) (1360)
30(00(02(21(x1)))) 31(10(03(34(43(31(12(20(00(02(21(x1))))))))))) (1361)
30(00(02(24(x1)))) 31(10(03(34(43(31(12(20(00(02(24(x1))))))))))) (1362)
30(00(00(05(x1)))) 31(12(23(34(44(42(23(33(32(22(25(x1))))))))))) (1465)
30(00(00(03(x1)))) 31(12(23(34(44(42(23(33(32(22(23(x1))))))))))) (1467)
10(00(00(05(x1)))) 11(12(23(34(44(42(23(33(32(22(25(x1))))))))))) (1477)
10(00(00(03(x1)))) 11(12(23(34(44(42(23(33(32(22(23(x1))))))))))) (1479)
00(00(00(05(x1)))) 01(12(23(34(44(42(23(33(32(22(25(x1))))))))))) (1483)
00(00(00(03(x1)))) 01(12(23(34(44(42(23(33(32(22(23(x1))))))))))) (1485)
40(00(00(05(x1)))) 41(12(23(34(44(42(23(33(32(22(25(x1))))))))))) (1489)
40(00(00(03(x1)))) 41(12(23(34(44(42(23(33(32(22(23(x1))))))))))) (1491)
20(00(00(05(x1)))) 21(12(23(34(44(42(23(33(32(22(25(x1))))))))))) (1495)
20(00(00(03(x1)))) 21(12(23(34(44(42(23(33(32(22(23(x1))))))))))) (1497)

1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[50(x1)] = 1 + 1 · x1
[00(x1)] = 1 + 1 · x1
[05(x1)] = 1 + 1 · x1
[03(x1)] = 1 · x1
[32(x1)] = 1 · x1
[21(x1)] = 1 · x1
[12(x1)] = 1 · x1
[23(x1)] = 1 · x1
[34(x1)] = 1 · x1
[40(x1)] = 1 · x1
[02(x1)] = 1 · x1
[01(x1)] = 1 + 1 · x1
[04(x1)] = 1 · x1
[42(x1)] = 1 + 1 · x1
[35(x1)] = 1 · x1
[44(x1)] = 1 · x1
[15(x1)] = 1 · x1
[52(x1)] = 1 · x1
[45(x1)] = 1 · x1
[43(x1)] = 1 · x1
[55(x1)] = 1 · x1
[53(x1)] = 1 + 1 · x1
[54(x1)] = 1 · x1
[11(x1)] = 1 · x1
[41(x1)] = 1 · x1
[30(x1)] = 1 · x1
[25(x1)] = 1 · x1
[10(x1)] = 1 · x1
[14(x1)] = 1 · x1
[20(x1)] = 1 · x1
[24(x1)] = 1 · x1
[13(x1)] = 1 · x1
[31(x1)] = 1 · x1
[22(x1)] = 1 · x1
all of the following rules can be deleted.
42(23(35(50(05(50(05(x1))))))) 44(44(40(00(01(15(52(23(34(45(x1)))))))))) (361)
42(23(35(50(05(50(03(x1))))))) 44(44(40(00(01(15(52(23(34(43(x1)))))))))) (363)
53(35(55(55(53(30(05(x1))))))) 54(43(34(45(52(25(55(55(50(05(x1)))))))))) (379)
53(35(55(55(53(30(00(x1))))))) 54(43(34(45(52(25(55(55(50(00(x1)))))))))) (380)
53(35(55(55(53(30(03(x1))))))) 54(43(34(45(52(25(55(55(50(03(x1)))))))))) (381)
53(35(55(55(53(30(02(x1))))))) 54(43(34(45(52(25(55(55(50(02(x1)))))))))) (382)
53(35(55(55(53(30(01(x1))))))) 54(43(34(45(52(25(55(55(50(01(x1)))))))))) (383)
53(35(55(55(53(30(04(x1))))))) 54(43(34(45(52(25(55(55(50(04(x1)))))))))) (384)

1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[50(x1)] = 1 · x1
[00(x1)] = 1 · x1
[05(x1)] = 1 + 1 · x1
[03(x1)] = 1 · x1
[32(x1)] = 1 · x1
[21(x1)] = 1 · x1
[12(x1)] = 1 · x1
[23(x1)] = 1 · x1
[34(x1)] = 1 · x1
[40(x1)] = 1 · x1
[02(x1)] = 1 · x1
[01(x1)] = 1 · x1
[04(x1)] = 1 · x1
[55(x1)] = 1 · x1
[53(x1)] = 1 · x1
[35(x1)] = 1 · x1
[54(x1)] = 1 + 1 · x1
[45(x1)] = 1 · x1
[11(x1)] = 1 · x1
[43(x1)] = 1 · x1
[42(x1)] = 1 · x1
[41(x1)] = 1 · x1
[44(x1)] = 1 · x1
[10(x1)] = 1 · x1
[14(x1)] = 1 · x1
[30(x1)] = 1 · x1
[20(x1)] = 1 · x1
[24(x1)] = 1 · x1
[15(x1)] = 1 · x1
[13(x1)] = 1 · x1
[25(x1)] = 1 · x1
[31(x1)] = 1 · x1
[22(x1)] = 1 · x1
all of the following rules can be deleted.
50(05(55(53(35(54(45(x1))))))) 53(35(55(50(01(11(12(23(34(45(x1)))))))))) (373)
50(05(55(53(35(54(40(x1))))))) 53(35(55(50(01(11(12(23(34(40(x1)))))))))) (374)
50(05(55(53(35(54(43(x1))))))) 53(35(55(50(01(11(12(23(34(43(x1)))))))))) (375)
50(05(55(53(35(54(42(x1))))))) 53(35(55(50(01(11(12(23(34(42(x1)))))))))) (376)
50(05(55(53(35(54(41(x1))))))) 53(35(55(50(01(11(12(23(34(41(x1)))))))))) (377)
50(05(55(53(35(54(44(x1))))))) 53(35(55(50(01(11(12(23(34(44(x1)))))))))) (378)

1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[50(x1)] = 1 · x1
[00(x1)] = 1 + 1 · x1
[05(x1)] = 1 · x1
[03(x1)] = 1 · x1
[32(x1)] = 1 · x1
[21(x1)] = 1 · x1
[12(x1)] = 1 · x1
[23(x1)] = 1 · x1
[34(x1)] = 1 · x1
[40(x1)] = 1 · x1
[02(x1)] = 1 · x1
[01(x1)] = 1 + 1 · x1
[04(x1)] = 1 · x1
[10(x1)] = 1 · x1
[45(x1)] = 1 · x1
[14(x1)] = 1 · x1
[42(x1)] = 1 · x1
[44(x1)] = 1 · x1
[43(x1)] = 1 · x1
[30(x1)] = 1 · x1
[41(x1)] = 1 · x1
[20(x1)] = 1 · x1
[24(x1)] = 1 + 1 · x1
[15(x1)] = 1 · x1
[11(x1)] = 1 · x1
[13(x1)] = 1 · x1
[25(x1)] = 1 · x1
[31(x1)] = 1 · x1
[22(x1)] = 1 · x1
all of the following rules can be deleted.
10(01(10(04(45(x1))))) 14(42(21(12(23(34(44(43(30(04(45(x1))))))))))) (457)
10(01(10(04(40(x1))))) 14(42(21(12(23(34(44(43(30(04(40(x1))))))))))) (458)
10(01(10(04(43(x1))))) 14(42(21(12(23(34(44(43(30(04(43(x1))))))))))) (459)
10(01(10(04(42(x1))))) 14(42(21(12(23(34(44(43(30(04(42(x1))))))))))) (460)
10(01(10(04(41(x1))))) 14(42(21(12(23(34(44(43(30(04(41(x1))))))))))) (461)
10(01(10(04(44(x1))))) 14(42(21(12(23(34(44(43(30(04(44(x1))))))))))) (462)
40(01(10(04(45(x1))))) 44(42(21(12(23(34(44(43(30(04(45(x1))))))))))) (469)
40(01(10(04(40(x1))))) 44(42(21(12(23(34(44(43(30(04(40(x1))))))))))) (470)
40(01(10(04(43(x1))))) 44(42(21(12(23(34(44(43(30(04(43(x1))))))))))) (471)
40(01(10(04(42(x1))))) 44(42(21(12(23(34(44(43(30(04(42(x1))))))))))) (472)
40(01(10(04(41(x1))))) 44(42(21(12(23(34(44(43(30(04(41(x1))))))))))) (473)
40(01(10(04(44(x1))))) 44(42(21(12(23(34(44(43(30(04(44(x1))))))))))) (474)

1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[50(x1)] = 1 · x1
[00(x1)] = 1 + 1 · x1
[05(x1)] = 1 · x1
[03(x1)] = 1 · x1
[32(x1)] = 1 · x1
[21(x1)] = 1 · x1
[12(x1)] = 1 · x1
[23(x1)] = 1 · x1
[34(x1)] = 1 · x1
[40(x1)] = 1 · x1
[02(x1)] = 1 · x1
[01(x1)] = 1 + 1 · x1
[04(x1)] = 1 · x1
[20(x1)] = 1 · x1
[10(x1)] = 1 · x1
[45(x1)] = 1 · x1
[24(x1)] = 1 · x1
[42(x1)] = 1 · x1
[44(x1)] = 1 · x1
[43(x1)] = 1 · x1
[30(x1)] = 1 · x1
[41(x1)] = 1 · x1
[15(x1)] = 1 · x1
[11(x1)] = 1 · x1
[14(x1)] = 1 · x1
[13(x1)] = 1 · x1
[25(x1)] = 1 · x1
[31(x1)] = 1 · x1
[22(x1)] = 1 · x1
all of the following rules can be deleted.
20(01(10(04(45(x1))))) 24(42(21(12(23(34(44(43(30(04(45(x1))))))))))) (475)
20(01(10(04(40(x1))))) 24(42(21(12(23(34(44(43(30(04(40(x1))))))))))) (476)
20(01(10(04(43(x1))))) 24(42(21(12(23(34(44(43(30(04(43(x1))))))))))) (477)
20(01(10(04(42(x1))))) 24(42(21(12(23(34(44(43(30(04(42(x1))))))))))) (478)
20(01(10(04(41(x1))))) 24(42(21(12(23(34(44(43(30(04(41(x1))))))))))) (479)
20(01(10(04(44(x1))))) 24(42(21(12(23(34(44(43(30(04(44(x1))))))))))) (480)

1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[50(x1)] = 1 · x1
[00(x1)] = 1 + 1 · x1
[05(x1)] = 1 · x1
[03(x1)] = 1 · x1
[32(x1)] = 1 · x1
[21(x1)] = 1 · x1
[12(x1)] = 1 · x1
[23(x1)] = 1 · x1
[34(x1)] = 1 · x1
[40(x1)] = 1 · x1
[02(x1)] = 1 · x1
[01(x1)] = 1 · x1
[04(x1)] = 1 · x1
[15(x1)] = 1 · x1
[44(x1)] = 1 · x1
[41(x1)] = 1 · x1
[11(x1)] = 1 · x1
[14(x1)] = 1 · x1
[10(x1)] = 1 · x1
[13(x1)] = 1 · x1
[25(x1)] = 1 · x1
[43(x1)] = 1 · x1
[31(x1)] = 1 · x1
[20(x1)] = 1 · x1
[22(x1)] = 1 · x1
[24(x1)] = 1 · x1
all of the following rules can be deleted.
00(00(02(25(x1)))) 01(10(03(34(43(31(12(20(00(02(25(x1))))))))))) (1375)
00(00(02(20(x1)))) 01(10(03(34(43(31(12(20(00(02(20(x1))))))))))) (1376)
00(00(02(23(x1)))) 01(10(03(34(43(31(12(20(00(02(23(x1))))))))))) (1377)
00(00(02(22(x1)))) 01(10(03(34(43(31(12(20(00(02(22(x1))))))))))) (1378)
00(00(02(21(x1)))) 01(10(03(34(43(31(12(20(00(02(21(x1))))))))))) (1379)
00(00(02(24(x1)))) 01(10(03(34(43(31(12(20(00(02(24(x1))))))))))) (1380)

1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the matrix interpretations of dimension 2 with strict dimension 1 over the integers
[50(x1)] =
0
0
+
2 2
0 0
· x1
[00(x1)] =
0
0
+
1 2
2 2
· x1
[05(x1)] =
2
0
+
2 0
1 0
· x1
[03(x1)] =
0
0
+
1 0
0 0
· x1
[32(x1)] =
0
0
+
2 0
0 0
· x1
[21(x1)] =
0
0
+
1 0
0 0
· x1
[12(x1)] =
0
0
+
1 0
0 0
· x1
[23(x1)] =
0
0
+
1 0
0 0
· x1
[34(x1)] =
0
0
+
1 0
0 0
· x1
[40(x1)] =
0
0
+
1 0
0 0
· x1
[02(x1)] =
0
0
+
1 0
0 0
· x1
[01(x1)] =
0
0
+
1 0
0 0
· x1
[04(x1)] =
0
0
+
2 0
0 0
· x1
[15(x1)] =
0
0
+
2 0
0 0
· x1
[44(x1)] =
0
0
+
1 0
0 0
· x1
[41(x1)] =
0
0
+
1 0
0 0
· x1
[11(x1)] =
0
0
+
1 0
0 0
· x1
[14(x1)] =
0
0
+
1 0
0 0
· x1
[10(x1)] =
0
0
+
1 0
0 0
· x1
[13(x1)] =
0
0
+
2 0
0 0
· x1
[25(x1)] =
0
0
+
1 0
0 0
· x1
[43(x1)] =
0
0
+
1 0
0 0
· x1
[31(x1)] =
0
0
+
1 0
0 0
· x1
[20(x1)] =
0
0
+
1 0
0 0
· x1
[22(x1)] =
0
0
+
2 0
0 0
· x1
[24(x1)] =
0
0
+
1 0
0 0
· x1
all of the following rules can be deleted.
50(00(05(x1))) 50(03(32(21(12(23(34(40(00(05(x1)))))))))) (241)

1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the matrix interpretations of dimension 2 with strict dimension 1 over the integers
[50(x1)] =
0
0
+
1 2
0 0
· x1
[00(x1)] =
0
0
+
1 0
1 0
· x1
[03(x1)] =
0
0
+
1 0
0 0
· x1
[32(x1)] =
0
0
+
1 0
0 0
· x1
[21(x1)] =
0
0
+
1 0
0 0
· x1
[12(x1)] =
0
0
+
1 0
0 0
· x1
[23(x1)] =
0
0
+
1 0
0 0
· x1
[34(x1)] =
0
0
+
1 0
0 0
· x1
[40(x1)] =
0
0
+
1 1
0 0
· x1
[02(x1)] =
0
0
+
2 0
0 0
· x1
[01(x1)] =
0
0
+
1 0
0 0
· x1
[04(x1)] =
2
0
+
1 0
0 0
· x1
[15(x1)] =
0
0
+
2 0
0 0
· x1
[44(x1)] =
0
0
+
1 0
0 0
· x1
[41(x1)] =
0
0
+
1 0
0 0
· x1
[11(x1)] =
0
0
+
1 0
0 0
· x1
[14(x1)] =
0
0
+
1 0
0 0
· x1
[10(x1)] =
0
0
+
1 0
0 0
· x1
[13(x1)] =
0
0
+
1 0
0 0
· x1
[25(x1)] =
0
0
+
2 0
0 0
· x1
[43(x1)] =
0
0
+
1 0
0 0
· x1
[31(x1)] =
0
0
+
1 0
0 0
· x1
[20(x1)] =
0
0
+
1 0
0 0
· x1
[22(x1)] =
0
0
+
1 0
0 0
· x1
[24(x1)] =
0
0
+
2 0
0 0
· x1
all of the following rules can be deleted.
50(00(04(x1))) 50(03(32(21(12(23(34(40(00(04(x1)))))))))) (246)

1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the matrix interpretations of dimension 2 with strict dimension 1 over the integers
[50(x1)] =
0
0
+
1 2
0 0
· x1
[00(x1)] =
0
0
+
1 0
2 0
· x1
[03(x1)] =
0
0
+
1 0
0 0
· x1
[32(x1)] =
0
0
+
1 0
0 0
· x1
[21(x1)] =
0
2
+
1 0
0 0
· x1
[12(x1)] =
0
0
+
1 0
0 0
· x1
[23(x1)] =
0
0
+
2 0
0 0
· x1
[34(x1)] =
0
0
+
1 0
0 0
· x1
[40(x1)] =
0
0
+
1 0
0 0
· x1
[02(x1)] =
2
0
+
1 0
0 0
· x1
[01(x1)] =
0
0
+
1 1
0 0
· x1
[15(x1)] =
0
0
+
1 0
0 0
· x1
[44(x1)] =
0
0
+
1 0
0 0
· x1
[41(x1)] =
0
0
+
1 0
0 0
· x1
[11(x1)] =
0
0
+
1 0
1 0
· x1
[14(x1)] =
0
0
+
1 0
0 0
· x1
[10(x1)] =
0
0
+
1 0
0 1
· x1
[13(x1)] =
0
0
+
2 0
2 0
· x1
[25(x1)] =
0
0
+
2 0
0 0
· x1
[43(x1)] =
0
0
+
1 0
0 0
· x1
[31(x1)] =
0
0
+
1 0
0 0
· x1
[20(x1)] =
0
0
+
1 0
0 1
· x1
[22(x1)] =
2
0
+
2 0
0 0
· x1
[24(x1)] =
0
0
+
1 0
0 0
· x1
all of the following rules can be deleted.
50(00(02(x1))) 50(03(32(21(12(23(34(40(00(02(x1)))))))))) (244)

1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the matrix interpretations of dimension 2 with strict dimension 1 over the integers
[50(x1)] =
0
0
+
2 2
0 2
· x1
[00(x1)] =
0
0
+
1 0
1 2
· x1
[03(x1)] =
0
0
+
1 0
0 0
· x1
[32(x1)] =
0
0
+
1 0
0 0
· x1
[21(x1)] =
0
0
+
1 0
0 0
· x1
[12(x1)] =
0
0
+
1 0
0 0
· x1
[23(x1)] =
0
0
+
2 0
0 0
· x1
[34(x1)] =
0
0
+
1 0
0 0
· x1
[40(x1)] =
0
0
+
1 0
0 0
· x1
[01(x1)] =
0
2
+
2 0
0 0
· x1
[15(x1)] =
0
0
+
1 0
0 0
· x1
[44(x1)] =
0
0
+
1 0
0 0
· x1
[41(x1)] =
0
0
+
1 0
0 0
· x1
[11(x1)] =
0
0
+
1 0
0 0
· x1
[14(x1)] =
0
0
+
1 0
0 0
· x1
[10(x1)] =
0
0
+
1 0
0 0
· x1
[13(x1)] =
0
0
+
2 0
0 0
· x1
[02(x1)] =
0
0
+
1 0
0 0
· x1
[25(x1)] =
0
0
+
1 0
0 0
· x1
[43(x1)] =
0
0
+
1 0
0 0
· x1
[31(x1)] =
0
0
+
1 0
0 0
· x1
[20(x1)] =
0
0
+
1 0
0 2
· x1
[22(x1)] =
0
0
+
1 0
0 0
· x1
[24(x1)] =
0
0
+
1 0
0 0
· x1
all of the following rules can be deleted.
50(00(01(x1))) 50(03(32(21(12(23(34(40(00(01(x1)))))))))) (245)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the matrix interpretations of dimension 2 with strict dimension 1 over the integers
[50(x1)] =
0
0
+
2 0
0 0
· x1
[00(x1)] =
0
0
+
1 2
0 1
· x1
[03(x1)] =
0
0
+
1 0
0 0
· x1
[32(x1)] =
0
0
+
1 0
0 0
· x1
[21(x1)] =
0
0
+
1 0
0 0
· x1
[12(x1)] =
0
0
+
1 0
0 0
· x1
[23(x1)] =
0
0
+
1 0
0 0
· x1
[34(x1)] =
0
0
+
1 0
0 0
· x1
[40(x1)] =
0
0
+
1 0
0 0
· x1
[01(x1)] =
0
0
+
2 0
0 0
· x1
[15(x1)] =
0
0
+
2 0
0 0
· x1
[44(x1)] =
0
0
+
1 0
0 0
· x1
[41(x1)] =
0
0
+
1 0
0 0
· x1
[11(x1)] =
0
0
+
1 0
0 0
· x1
[14(x1)] =
0
0
+
1 0
0 0
· x1
[10(x1)] =
0
0
+
1 1
0 0
· x1
[13(x1)] =
0
0
+
2 0
0 0
· x1
[02(x1)] =
0
0
+
2 2
2 2
· x1
[25(x1)] =
0
0
+
2 0
2 0
· x1
[43(x1)] =
0
0
+
1 0
0 0
· x1
[31(x1)] =
0
0
+
1 0
0 0
· x1
[20(x1)] =
0
0
+
1 0
1 0
· x1
[22(x1)] =
2
2
+
2 0
0 0
· x1
[24(x1)] =
0
0
+
1 0
0 0
· x1
all of the following rules can be deleted.
10(00(02(22(x1)))) 11(10(03(34(43(31(12(20(00(02(22(x1))))))))))) (1372)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the matrix interpretations of dimension 2 with strict dimension 1 over the integers
[50(x1)] =
0
0
+
2 2
0 0
· x1
[00(x1)] =
0
0
+
1 0
1 1
· x1
[03(x1)] =
0
0
+
1 0
0 0
· x1
[32(x1)] =
0
0
+
2 0
0 0
· x1
[21(x1)] =
0
0
+
1 0
0 0
· x1
[12(x1)] =
0
0
+
1 0
0 0
· x1
[23(x1)] =
0
1
+
1 0
0 0
· x1
[34(x1)] =
0
0
+
1 0
0 0
· x1
[40(x1)] =
0
0
+
1 0
0 0
· x1
[01(x1)] =
0
0
+
1 0
0 0
· x1
[15(x1)] =
0
0
+
2 0
0 0
· x1
[44(x1)] =
0
0
+
1 0
0 0
· x1
[41(x1)] =
0
0
+
1 0
0 0
· x1
[11(x1)] =
0
0
+
1 0
0 0
· x1
[14(x1)] =
0
0
+
1 0
0 0
· x1
[10(x1)] =
0
0
+
1 1
0 0
· x1
[13(x1)] =
0
0
+
1 0
0 0
· x1
[02(x1)] =
0
0
+
1 0
0 1
· x1
[25(x1)] =
0
0
+
1 0
1 0
· x1
[43(x1)] =
0
0
+
1 0
0 0
· x1
[31(x1)] =
0
0
+
1 0
0 0
· x1
[20(x1)] =
0
0
+
1 0
0 0
· x1
[24(x1)] =
0
0
+
1 0
0 0
· x1
[22(x1)] =
0
0
+
2 0
0 0
· x1
all of the following rules can be deleted.
10(00(02(23(x1)))) 11(10(03(34(43(31(12(20(00(02(23(x1))))))))))) (1371)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the matrix interpretations of dimension 2 with strict dimension 1 over the integers
[50(x1)] =
0
2
+
2 2
0 2
· x1
[00(x1)] =
0
1
+
2 2
2 2
· x1
[03(x1)] =
0
0
+
1 0
0 0
· x1
[32(x1)] =
0
0
+
1 0
0 0
· x1
[21(x1)] =
0
0
+
1 0
0 0
· x1
[12(x1)] =
0
0
+
1 0
0 0
· x1
[23(x1)] =
0
0
+
2 0
0 0
· x1
[34(x1)] =
0
0
+
1 0
0 0
· x1
[40(x1)] =
0
0
+
1 0
0 0
· x1
[01(x1)] =
0
0
+
2 0
0 0
· x1
[15(x1)] =
0
0
+
1 0
0 0
· x1
[44(x1)] =
0
0
+
1 0
0 0
· x1
[41(x1)] =
0
0
+
1 0
0 0
· x1
[11(x1)] =
0
0
+
1 0
0 0
· x1
[14(x1)] =
0
0
+
1 0
0 0
· x1
[10(x1)] =
0
0
+
1 0
0 0
· x1
[13(x1)] =
0
0
+
1 0
0 0
· x1
[02(x1)] =
0
0
+
2 0
0 0
· x1
[25(x1)] =
0
0
+
2 0
0 0
· x1
[43(x1)] =
0
0
+
1 0
0 0
· x1
[31(x1)] =
0
0
+
1 0
0 0
· x1
[20(x1)] =
0
0
+
1 0
0 0
· x1
[24(x1)] =
0
0
+
1 0
0 0
· x1
[22(x1)] =
0
0
+
2 0
0 0
· x1
all of the following rules can be deleted.
50(00(00(x1))) 50(03(32(21(12(23(34(40(00(00(x1)))))))))) (242)
50(00(03(x1))) 50(03(32(21(12(23(34(40(00(03(x1)))))))))) (243)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 R is empty

There are no rules in the TRS. Hence, it is terminating.