Certification Problem

Input (TPDB SRS_Relative/ICFP_2010_relative/5109)

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

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

and S is the following TRS.

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

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

We obtain the transformed TRS

There are 220 ruless (increase limit for explicit display).

1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

There are 1320 ruless (increase limit for explicit display).

1.1.1 Rule Removal

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

There are 1250 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
31(33(13(41(54(35(13(x1))))))) 31(33(53(15(11(01(20(42(44(14(x1)))))))))) (1573)
30(33(13(41(54(35(13(x1))))))) 30(33(53(15(11(01(20(42(44(14(x1)))))))))) (1574)
35(33(13(41(54(35(13(x1))))))) 35(33(53(15(11(01(20(42(44(14(x1)))))))))) (1575)
32(33(13(41(54(35(13(x1))))))) 32(33(53(15(11(01(20(42(44(14(x1)))))))))) (1576)
34(33(13(41(54(35(13(x1))))))) 34(33(53(15(11(01(20(42(44(14(x1)))))))))) (1577)
33(33(13(41(54(35(13(x1))))))) 33(33(53(15(11(01(20(42(44(14(x1)))))))))) (1578)
03(10(01(40(04(00(x1)))))) 53(35(33(13(11(11(11(51(05(20(02(x1))))))))))) (1579)
03(10(01(40(04(10(x1)))))) 53(35(33(13(11(11(11(51(05(20(12(x1))))))))))) (1580)
03(10(01(40(04(20(x1)))))) 53(35(33(13(11(11(11(51(05(20(22(x1))))))))))) (1581)
03(10(01(40(04(30(x1)))))) 53(35(33(13(11(11(11(51(05(20(32(x1))))))))))) (1582)
03(10(01(40(04(40(x1)))))) 53(35(33(13(11(11(11(51(05(20(42(x1))))))))))) (1583)
03(10(01(40(04(50(x1)))))) 53(35(33(13(11(11(11(51(05(20(52(x1))))))))))) (1584)
41(14(51(45(04(00(10(x1))))))) 41(54(05(20(42(14(41(24(42(34(13(x1))))))))))) (1585)
40(14(51(45(04(00(10(x1))))))) 40(54(05(20(42(14(41(24(42(34(13(x1))))))))))) (1586)
45(14(51(45(04(00(10(x1))))))) 45(54(05(20(42(14(41(24(42(34(13(x1))))))))))) (1587)
42(14(51(45(04(00(10(x1))))))) 42(54(05(20(42(14(41(24(42(34(13(x1))))))))))) (1588)
44(14(51(45(04(00(10(x1))))))) 44(54(05(20(42(14(41(24(42(34(13(x1))))))))))) (1589)
43(14(51(45(04(00(10(x1))))))) 43(54(05(20(42(14(41(24(42(34(13(x1))))))))))) (1590)
41(14(51(45(04(00(50(x1))))))) 41(54(05(20(42(14(41(24(42(34(53(x1))))))))))) (1591)
40(14(51(45(04(00(50(x1))))))) 40(54(05(20(42(14(41(24(42(34(53(x1))))))))))) (1592)
45(14(51(45(04(00(50(x1))))))) 45(54(05(20(42(14(41(24(42(34(53(x1))))))))))) (1593)
42(14(51(45(04(00(50(x1))))))) 42(54(05(20(42(14(41(24(42(34(53(x1))))))))))) (1594)
44(14(51(45(04(00(50(x1))))))) 44(54(05(20(42(14(41(24(42(34(53(x1))))))))))) (1595)
43(14(51(45(04(00(50(x1))))))) 43(54(05(20(42(14(41(24(42(34(53(x1))))))))))) (1596)
44(04(50(45(34(03(00(10(x1)))))))) 34(53(35(53(45(54(15(11(21(32(13(x1))))))))))) (1597)
44(04(50(45(34(03(00(50(x1)))))))) 34(53(35(53(45(54(15(11(21(32(53(x1))))))))))) (1598)
11(01(40(04(20(32(03(10(x1)))))))) 11(31(13(41(24(42(04(00(20(32(13(x1))))))))))) (1599)
10(01(40(04(20(32(03(10(x1)))))))) 10(31(13(41(24(42(04(00(20(32(13(x1))))))))))) (1600)
15(01(40(04(20(32(03(10(x1)))))))) 15(31(13(41(24(42(04(00(20(32(13(x1))))))))))) (1601)
12(01(40(04(20(32(03(10(x1)))))))) 12(31(13(41(24(42(04(00(20(32(13(x1))))))))))) (1602)
14(01(40(04(20(32(03(10(x1)))))))) 14(31(13(41(24(42(04(00(20(32(13(x1))))))))))) (1603)
13(01(40(04(20(32(03(10(x1)))))))) 13(31(13(41(24(42(04(00(20(32(13(x1))))))))))) (1604)
11(01(40(04(20(32(03(50(x1)))))))) 11(31(13(41(24(42(04(00(20(32(53(x1))))))))))) (1605)
10(01(40(04(20(32(03(50(x1)))))))) 10(31(13(41(24(42(04(00(20(32(53(x1))))))))))) (1606)
15(01(40(04(20(32(03(50(x1)))))))) 15(31(13(41(24(42(04(00(20(32(53(x1))))))))))) (1607)
12(01(40(04(20(32(03(50(x1)))))))) 12(31(13(41(24(42(04(00(20(32(53(x1))))))))))) (1608)
14(01(40(04(20(32(03(50(x1)))))))) 14(31(13(41(24(42(04(00(20(32(53(x1))))))))))) (1609)
13(01(40(04(20(32(03(50(x1)))))))) 13(31(13(41(24(42(04(00(20(32(53(x1))))))))))) (1610)
21(12(11(41(04(00(10(11(x1)))))))) 21(12(41(54(05(20(02(40(24(22(12(x1))))))))))) (1611)
20(12(11(41(04(00(10(11(x1)))))))) 20(12(41(54(05(20(02(40(24(22(12(x1))))))))))) (1612)
25(12(11(41(04(00(10(11(x1)))))))) 25(12(41(54(05(20(02(40(24(22(12(x1))))))))))) (1613)
22(12(11(41(04(00(10(11(x1)))))))) 22(12(41(54(05(20(02(40(24(22(12(x1))))))))))) (1614)
24(12(11(41(04(00(10(11(x1)))))))) 24(12(41(54(05(20(02(40(24(22(12(x1))))))))))) (1615)
23(12(11(41(04(00(10(11(x1)))))))) 23(12(41(54(05(20(02(40(24(22(12(x1))))))))))) (1616)
21(12(11(41(04(00(10(21(x1)))))))) 21(12(41(54(05(20(02(40(24(22(22(x1))))))))))) (1617)
20(12(11(41(04(00(10(21(x1)))))))) 20(12(41(54(05(20(02(40(24(22(22(x1))))))))))) (1618)
25(12(11(41(04(00(10(21(x1)))))))) 25(12(41(54(05(20(02(40(24(22(22(x1))))))))))) (1619)
22(12(11(41(04(00(10(21(x1)))))))) 22(12(41(54(05(20(02(40(24(22(22(x1))))))))))) (1620)
24(12(11(41(04(00(10(21(x1)))))))) 24(12(41(54(05(20(02(40(24(22(22(x1))))))))))) (1621)
23(12(11(41(04(00(10(21(x1)))))))) 23(12(41(54(05(20(02(40(24(22(22(x1))))))))))) (1622)
21(12(11(41(04(00(10(31(x1)))))))) 21(12(41(54(05(20(02(40(24(22(32(x1))))))))))) (1623)
20(12(11(41(04(00(10(31(x1)))))))) 20(12(41(54(05(20(02(40(24(22(32(x1))))))))))) (1624)
25(12(11(41(04(00(10(31(x1)))))))) 25(12(41(54(05(20(02(40(24(22(32(x1))))))))))) (1625)
22(12(11(41(04(00(10(31(x1)))))))) 22(12(41(54(05(20(02(40(24(22(32(x1))))))))))) (1626)
24(12(11(41(04(00(10(31(x1)))))))) 24(12(41(54(05(20(02(40(24(22(32(x1))))))))))) (1627)
23(12(11(41(04(00(10(31(x1)))))))) 23(12(41(54(05(20(02(40(24(22(32(x1))))))))))) (1628)
21(12(11(41(04(00(10(41(x1)))))))) 21(12(41(54(05(20(02(40(24(22(42(x1))))))))))) (1629)
20(12(11(41(04(00(10(41(x1)))))))) 20(12(41(54(05(20(02(40(24(22(42(x1))))))))))) (1630)
25(12(11(41(04(00(10(41(x1)))))))) 25(12(41(54(05(20(02(40(24(22(42(x1))))))))))) (1631)
22(12(11(41(04(00(10(41(x1)))))))) 22(12(41(54(05(20(02(40(24(22(42(x1))))))))))) (1632)
24(12(11(41(04(00(10(41(x1)))))))) 24(12(41(54(05(20(02(40(24(22(42(x1))))))))))) (1633)
23(12(11(41(04(00(10(41(x1)))))))) 23(12(41(54(05(20(02(40(24(22(42(x1))))))))))) (1634)
21(12(11(41(04(00(10(51(x1)))))))) 21(12(41(54(05(20(02(40(24(22(52(x1))))))))))) (1635)
20(12(11(41(04(00(10(51(x1)))))))) 20(12(41(54(05(20(02(40(24(22(52(x1))))))))))) (1636)
25(12(11(41(04(00(10(51(x1)))))))) 25(12(41(54(05(20(02(40(24(22(52(x1))))))))))) (1637)
22(12(11(41(04(00(10(51(x1)))))))) 22(12(41(54(05(20(02(40(24(22(52(x1))))))))))) (1638)
24(12(11(41(04(00(10(51(x1)))))))) 24(12(41(54(05(20(02(40(24(22(52(x1))))))))))) (1639)
23(12(11(41(04(00(10(51(x1)))))))) 23(12(41(54(05(20(02(40(24(22(52(x1))))))))))) (1640)
43(04(10(31(03(00(50(15(x1)))))))) 13(21(42(34(43(34(43(24(12(31(13(x1))))))))))) (1641)
43(04(10(31(03(00(50(55(x1)))))))) 13(21(42(34(43(34(43(24(12(31(53(x1))))))))))) (1642)

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[31(x1)] = 1 · x1
[33(x1)] = 1 · x1
[13(x1)] = 1 · x1
[41(x1)] = 1 · x1
[54(x1)] = 1 · x1
[35(x1)] = 1 · x1 + 1
[53(x1)] = 1 · x1
[15(x1)] = 1 · x1
[11(x1)] = 1 · x1
[01(x1)] = 1 · x1 + 1
[20(x1)] = 1 · x1
[42(x1)] = 1 · x1
[44(x1)] = 1 · x1
[14(x1)] = 1 · x1
[30(x1)] = 1 · x1
[32(x1)] = 1 · x1
[34(x1)] = 1 · x1
[03(x1)] = 1 · x1
[10(x1)] = 1 · x1
[40(x1)] = 1 · x1
[04(x1)] = 1 · x1 + 2
[00(x1)] = 1 · x1
[51(x1)] = 1 · x1
[05(x1)] = 1 · x1 + 1
[02(x1)] = 1 · x1
[12(x1)] = 1 · x1
[22(x1)] = 1 · x1
[50(x1)] = 1 · x1
[52(x1)] = 1 · x1
[45(x1)] = 1 · x1
[24(x1)] = 1 · x1 + 1
[43(x1)] = 1 · x1
[21(x1)] = 1 · x1
[25(x1)] = 1 · x1
[23(x1)] = 1 · x1
[55(x1)] = 1 · x1
all of the following rules can be deleted.
03(10(01(40(04(00(x1)))))) 53(35(33(13(11(11(11(51(05(20(02(x1))))))))))) (1579)
03(10(01(40(04(10(x1)))))) 53(35(33(13(11(11(11(51(05(20(12(x1))))))))))) (1580)
03(10(01(40(04(20(x1)))))) 53(35(33(13(11(11(11(51(05(20(22(x1))))))))))) (1581)
03(10(01(40(04(30(x1)))))) 53(35(33(13(11(11(11(51(05(20(32(x1))))))))))) (1582)
03(10(01(40(04(40(x1)))))) 53(35(33(13(11(11(11(51(05(20(42(x1))))))))))) (1583)
03(10(01(40(04(50(x1)))))) 53(35(33(13(11(11(11(51(05(20(52(x1))))))))))) (1584)
44(04(50(45(34(03(00(10(x1)))))))) 34(53(35(53(45(54(15(11(21(32(13(x1))))))))))) (1597)
44(04(50(45(34(03(00(50(x1)))))))) 34(53(35(53(45(54(15(11(21(32(53(x1))))))))))) (1598)
43(04(10(31(03(00(50(15(x1)))))))) 13(21(42(34(43(34(43(24(12(31(13(x1))))))))))) (1641)
43(04(10(31(03(00(50(55(x1)))))))) 13(21(42(34(43(34(43(24(12(31(53(x1))))))))))) (1642)

1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[31(x1)] = 1 · x1
[33(x1)] = 1 · x1
[13(x1)] = 1 · x1
[41(x1)] = 1 · x1
[54(x1)] = 1 · x1
[35(x1)] = 1 · x1 + 1
[53(x1)] = 1 · x1
[15(x1)] = 1 · x1
[11(x1)] = 1 · x1
[01(x1)] = 1 · x1
[20(x1)] = 1 · x1
[42(x1)] = 1 · x1
[44(x1)] = 1 · x1
[14(x1)] = 1 · x1
[30(x1)] = 1 · x1
[32(x1)] = 1 · x1
[34(x1)] = 1 · x1
[51(x1)] = 1 · x1
[45(x1)] = 1 · x1
[04(x1)] = 1 · x1
[00(x1)] = 1 · x1
[10(x1)] = 1 · x1
[05(x1)] = 1 · x1
[24(x1)] = 1 · x1
[40(x1)] = 1 · x1
[43(x1)] = 1 · x1
[50(x1)] = 1 · x1
[03(x1)] = 1 · x1
[12(x1)] = 1 · x1
[21(x1)] = 1 · x1
[02(x1)] = 1 · x1
[22(x1)] = 1 · x1
[25(x1)] = 1 · x1
[23(x1)] = 1 · x1
[52(x1)] = 1 · x1
all of the following rules can be deleted.
31(33(13(41(54(35(13(x1))))))) 31(33(53(15(11(01(20(42(44(14(x1)))))))))) (1573)
30(33(13(41(54(35(13(x1))))))) 30(33(53(15(11(01(20(42(44(14(x1)))))))))) (1574)
35(33(13(41(54(35(13(x1))))))) 35(33(53(15(11(01(20(42(44(14(x1)))))))))) (1575)
32(33(13(41(54(35(13(x1))))))) 32(33(53(15(11(01(20(42(44(14(x1)))))))))) (1576)
34(33(13(41(54(35(13(x1))))))) 34(33(53(15(11(01(20(42(44(14(x1)))))))))) (1577)
33(33(13(41(54(35(13(x1))))))) 33(33(53(15(11(01(20(42(44(14(x1)))))))))) (1578)

1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[41(x1)] = 1 · x1
[14(x1)] = 1 · x1
[51(x1)] = 1 · x1
[45(x1)] = 1 · x1 + 1
[04(x1)] = 1 · x1
[00(x1)] = 1 · x1
[10(x1)] = 1 · x1
[54(x1)] = 1 · x1
[05(x1)] = 1 · x1
[20(x1)] = 1 · x1
[42(x1)] = 1 · x1
[24(x1)] = 1 · x1
[34(x1)] = 1 · x1
[13(x1)] = 1 · x1
[40(x1)] = 1 · x1
[44(x1)] = 1 · x1
[43(x1)] = 1 · x1
[50(x1)] = 1 · x1
[53(x1)] = 1 · x1
[11(x1)] = 1 · x1
[01(x1)] = 1 · x1
[32(x1)] = 1 · x1
[03(x1)] = 1 · x1
[31(x1)] = 1 · x1
[15(x1)] = 1 · x1
[12(x1)] = 1 · x1
[21(x1)] = 1 · x1
[02(x1)] = 1 · x1
[22(x1)] = 1 · x1
[25(x1)] = 1 · x1
[23(x1)] = 1 · x1
[52(x1)] = 1 · x1
all of the following rules can be deleted.
41(14(51(45(04(00(10(x1))))))) 41(54(05(20(42(14(41(24(42(34(13(x1))))))))))) (1585)
40(14(51(45(04(00(10(x1))))))) 40(54(05(20(42(14(41(24(42(34(13(x1))))))))))) (1586)
45(14(51(45(04(00(10(x1))))))) 45(54(05(20(42(14(41(24(42(34(13(x1))))))))))) (1587)
42(14(51(45(04(00(10(x1))))))) 42(54(05(20(42(14(41(24(42(34(13(x1))))))))))) (1588)
44(14(51(45(04(00(10(x1))))))) 44(54(05(20(42(14(41(24(42(34(13(x1))))))))))) (1589)
43(14(51(45(04(00(10(x1))))))) 43(54(05(20(42(14(41(24(42(34(13(x1))))))))))) (1590)
41(14(51(45(04(00(50(x1))))))) 41(54(05(20(42(14(41(24(42(34(53(x1))))))))))) (1591)
40(14(51(45(04(00(50(x1))))))) 40(54(05(20(42(14(41(24(42(34(53(x1))))))))))) (1592)
45(14(51(45(04(00(50(x1))))))) 45(54(05(20(42(14(41(24(42(34(53(x1))))))))))) (1593)
42(14(51(45(04(00(50(x1))))))) 42(54(05(20(42(14(41(24(42(34(53(x1))))))))))) (1594)
44(14(51(45(04(00(50(x1))))))) 44(54(05(20(42(14(41(24(42(34(53(x1))))))))))) (1595)
43(14(51(45(04(00(50(x1))))))) 43(54(05(20(42(14(41(24(42(34(53(x1))))))))))) (1596)

1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[11(x1)] = 1 · x1
[01(x1)] = 1 · x1
[40(x1)] = 1 · x1
[04(x1)] = 1 · x1
[20(x1)] = 1 · x1
[32(x1)] = 1 · x1
[03(x1)] = 1 · x1 + 1
[10(x1)] = 1 · x1
[31(x1)] = 1 · x1
[13(x1)] = 1 · x1
[41(x1)] = 1 · x1
[24(x1)] = 1 · x1
[42(x1)] = 1 · x1
[00(x1)] = 1 · x1
[15(x1)] = 1 · x1
[12(x1)] = 1 · x1
[14(x1)] = 1 · x1
[50(x1)] = 1 · x1
[53(x1)] = 1 · x1 + 1
[21(x1)] = 1 · x1
[54(x1)] = 1 · x1
[05(x1)] = 1 · x1
[02(x1)] = 1 · x1
[22(x1)] = 1 · x1
[25(x1)] = 1 · x1
[23(x1)] = 1 · x1
[51(x1)] = 1 · x1
[52(x1)] = 1 · x1
all of the following rules can be deleted.
11(01(40(04(20(32(03(10(x1)))))))) 11(31(13(41(24(42(04(00(20(32(13(x1))))))))))) (1599)
10(01(40(04(20(32(03(10(x1)))))))) 10(31(13(41(24(42(04(00(20(32(13(x1))))))))))) (1600)
15(01(40(04(20(32(03(10(x1)))))))) 15(31(13(41(24(42(04(00(20(32(13(x1))))))))))) (1601)
12(01(40(04(20(32(03(10(x1)))))))) 12(31(13(41(24(42(04(00(20(32(13(x1))))))))))) (1602)
14(01(40(04(20(32(03(10(x1)))))))) 14(31(13(41(24(42(04(00(20(32(13(x1))))))))))) (1603)
13(01(40(04(20(32(03(10(x1)))))))) 13(31(13(41(24(42(04(00(20(32(13(x1))))))))))) (1604)

1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[11(x1)] = 1 · x1
[01(x1)] = 1 · x1
[40(x1)] = 1 · x1
[04(x1)] = 1 · x1
[20(x1)] = 1 · x1
[32(x1)] = 1 · x1
[03(x1)] = 1 · x1 + 1
[50(x1)] = 1 · x1
[31(x1)] = 1 · x1
[13(x1)] = 1 · x1
[41(x1)] = 1 · x1
[24(x1)] = 1 · x1
[42(x1)] = 1 · x1
[00(x1)] = 1 · x1
[53(x1)] = 1 · x1
[10(x1)] = 1 · x1
[15(x1)] = 1 · x1
[12(x1)] = 1 · x1
[14(x1)] = 1 · x1
[21(x1)] = 1 · x1
[54(x1)] = 1 · x1
[05(x1)] = 1 · x1
[02(x1)] = 1 · x1
[22(x1)] = 1 · x1
[25(x1)] = 1 · x1
[23(x1)] = 1 · x1
[51(x1)] = 1 · x1
[52(x1)] = 1 · x1
all of the following rules can be deleted.
11(01(40(04(20(32(03(50(x1)))))))) 11(31(13(41(24(42(04(00(20(32(53(x1))))))))))) (1605)
10(01(40(04(20(32(03(50(x1)))))))) 10(31(13(41(24(42(04(00(20(32(53(x1))))))))))) (1606)
15(01(40(04(20(32(03(50(x1)))))))) 15(31(13(41(24(42(04(00(20(32(53(x1))))))))))) (1607)
12(01(40(04(20(32(03(50(x1)))))))) 12(31(13(41(24(42(04(00(20(32(53(x1))))))))))) (1608)
14(01(40(04(20(32(03(50(x1)))))))) 14(31(13(41(24(42(04(00(20(32(53(x1))))))))))) (1609)
13(01(40(04(20(32(03(50(x1)))))))) 13(31(13(41(24(42(04(00(20(32(53(x1))))))))))) (1610)

1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[21(x1)] = 1 · x1
[12(x1)] = 1 · x1
[11(x1)] = 1 · x1
[41(x1)] = 1 · x1
[04(x1)] = 1 · x1 + 2
[00(x1)] = 1 · x1
[10(x1)] = 1 · x1
[54(x1)] = 1 · x1
[05(x1)] = 1 · x1
[20(x1)] = 1 · x1
[02(x1)] = 1 · x1
[40(x1)] = 1 · x1
[24(x1)] = 1 · x1
[22(x1)] = 1 · x1 + 1
[25(x1)] = 1 · x1
[23(x1)] = 1 · x1
[31(x1)] = 1 · x1
[32(x1)] = 1 · x1 + 1
[42(x1)] = 1 · x1 + 1
[51(x1)] = 1 · x1
[52(x1)] = 1 · x1 + 1
all of the following rules can be deleted.
21(12(11(41(04(00(10(11(x1)))))))) 21(12(41(54(05(20(02(40(24(22(12(x1))))))))))) (1611)
20(12(11(41(04(00(10(11(x1)))))))) 20(12(41(54(05(20(02(40(24(22(12(x1))))))))))) (1612)
25(12(11(41(04(00(10(11(x1)))))))) 25(12(41(54(05(20(02(40(24(22(12(x1))))))))))) (1613)
22(12(11(41(04(00(10(11(x1)))))))) 22(12(41(54(05(20(02(40(24(22(12(x1))))))))))) (1614)
24(12(11(41(04(00(10(11(x1)))))))) 24(12(41(54(05(20(02(40(24(22(12(x1))))))))))) (1615)
23(12(11(41(04(00(10(11(x1)))))))) 23(12(41(54(05(20(02(40(24(22(12(x1))))))))))) (1616)

1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[21(x1)] = 1 · x1 + 1
[12(x1)] = 1 · x1
[11(x1)] = 1 · x1
[41(x1)] = 1 · x1
[04(x1)] = 1 · x1
[00(x1)] = 1 · x1
[10(x1)] = 1 · x1
[54(x1)] = 1 · x1
[05(x1)] = 1 · x1
[20(x1)] = 1 · x1
[02(x1)] = 1 · x1
[40(x1)] = 1 · x1
[24(x1)] = 1 · x1
[22(x1)] = 1 · x1
[25(x1)] = 1 · x1
[23(x1)] = 1 · x1
[31(x1)] = 1 · x1
[32(x1)] = 1 · x1
[42(x1)] = 1 · x1
[51(x1)] = 1 · x1
[52(x1)] = 1 · x1
all of the following rules can be deleted.
21(12(11(41(04(00(10(21(x1)))))))) 21(12(41(54(05(20(02(40(24(22(22(x1))))))))))) (1617)
20(12(11(41(04(00(10(21(x1)))))))) 20(12(41(54(05(20(02(40(24(22(22(x1))))))))))) (1618)
25(12(11(41(04(00(10(21(x1)))))))) 25(12(41(54(05(20(02(40(24(22(22(x1))))))))))) (1619)
22(12(11(41(04(00(10(21(x1)))))))) 22(12(41(54(05(20(02(40(24(22(22(x1))))))))))) (1620)
24(12(11(41(04(00(10(21(x1)))))))) 24(12(41(54(05(20(02(40(24(22(22(x1))))))))))) (1621)
23(12(11(41(04(00(10(21(x1)))))))) 23(12(41(54(05(20(02(40(24(22(22(x1))))))))))) (1622)

1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[21(x1)] = 1 · x1
[12(x1)] = 1 · x1
[11(x1)] = 1 · x1
[41(x1)] = 1 · x1
[04(x1)] = 1 · x1
[00(x1)] = 1 · x1
[10(x1)] = 1 · x1
[31(x1)] = 1 · x1 + 1
[54(x1)] = 1 · x1
[05(x1)] = 1 · x1
[20(x1)] = 1 · x1
[02(x1)] = 1 · x1
[40(x1)] = 1 · x1
[24(x1)] = 1 · x1
[22(x1)] = 1 · x1
[32(x1)] = 1 · x1
[25(x1)] = 1 · x1
[23(x1)] = 1 · x1
[42(x1)] = 1 · x1
[51(x1)] = 1 · x1
[52(x1)] = 1 · x1
all of the following rules can be deleted.
21(12(11(41(04(00(10(31(x1)))))))) 21(12(41(54(05(20(02(40(24(22(32(x1))))))))))) (1623)
20(12(11(41(04(00(10(31(x1)))))))) 20(12(41(54(05(20(02(40(24(22(32(x1))))))))))) (1624)
25(12(11(41(04(00(10(31(x1)))))))) 25(12(41(54(05(20(02(40(24(22(32(x1))))))))))) (1625)
22(12(11(41(04(00(10(31(x1)))))))) 22(12(41(54(05(20(02(40(24(22(32(x1))))))))))) (1626)
24(12(11(41(04(00(10(31(x1)))))))) 24(12(41(54(05(20(02(40(24(22(32(x1))))))))))) (1627)
23(12(11(41(04(00(10(31(x1)))))))) 23(12(41(54(05(20(02(40(24(22(32(x1))))))))))) (1628)

1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[21(x1)] = 1 · x1
[12(x1)] = 1 · x1
[11(x1)] = 1 · x1
[41(x1)] = 1 · x1 + 1
[04(x1)] = 1 · x1
[00(x1)] = 1 · x1
[10(x1)] = 1 · x1
[54(x1)] = 1 · x1
[05(x1)] = 1 · x1
[20(x1)] = 1 · x1
[02(x1)] = 1 · x1
[40(x1)] = 1 · x1
[24(x1)] = 1 · x1
[22(x1)] = 1 · x1
[42(x1)] = 1 · x1
[25(x1)] = 1 · x1
[23(x1)] = 1 · x1
[51(x1)] = 1 · x1
[52(x1)] = 1 · x1
all of the following rules can be deleted.
21(12(11(41(04(00(10(41(x1)))))))) 21(12(41(54(05(20(02(40(24(22(42(x1))))))))))) (1629)
20(12(11(41(04(00(10(41(x1)))))))) 20(12(41(54(05(20(02(40(24(22(42(x1))))))))))) (1630)
25(12(11(41(04(00(10(41(x1)))))))) 25(12(41(54(05(20(02(40(24(22(42(x1))))))))))) (1631)
22(12(11(41(04(00(10(41(x1)))))))) 22(12(41(54(05(20(02(40(24(22(42(x1))))))))))) (1632)
24(12(11(41(04(00(10(41(x1)))))))) 24(12(41(54(05(20(02(40(24(22(42(x1))))))))))) (1633)
23(12(11(41(04(00(10(41(x1)))))))) 23(12(41(54(05(20(02(40(24(22(42(x1))))))))))) (1634)

1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[21(x1)] = 1 · x1
[12(x1)] = 1 · x1
[11(x1)] = 1 · x1
[41(x1)] = 1 · x1
[04(x1)] = 1 · x1 + 1
[00(x1)] = 1 · x1
[10(x1)] = 1 · x1
[51(x1)] = 1 · x1
[54(x1)] = 1 · x1
[05(x1)] = 1 · x1
[20(x1)] = 1 · x1
[02(x1)] = 1 · x1
[40(x1)] = 1 · x1
[24(x1)] = 1 · x1
[22(x1)] = 1 · x1
[52(x1)] = 1 · x1
[25(x1)] = 1 · x1
[23(x1)] = 1 · x1
all of the following rules can be deleted.
21(12(11(41(04(00(10(51(x1)))))))) 21(12(41(54(05(20(02(40(24(22(52(x1))))))))))) (1635)
20(12(11(41(04(00(10(51(x1)))))))) 20(12(41(54(05(20(02(40(24(22(52(x1))))))))))) (1636)
25(12(11(41(04(00(10(51(x1)))))))) 25(12(41(54(05(20(02(40(24(22(52(x1))))))))))) (1637)
22(12(11(41(04(00(10(51(x1)))))))) 22(12(41(54(05(20(02(40(24(22(52(x1))))))))))) (1638)
24(12(11(41(04(00(10(51(x1)))))))) 24(12(41(54(05(20(02(40(24(22(52(x1))))))))))) (1639)
23(12(11(41(04(00(10(51(x1)))))))) 23(12(41(54(05(20(02(40(24(22(52(x1))))))))))) (1640)

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.