Certification Problem

Input (TPDB SRS_Relative/ICFP_2010_relative/4036)

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

0(1(0(1(x1)))) 0(1(1(1(0(0(1(2(2(2(x1)))))))))) (1)
0(3(4(4(x1)))) 0(0(0(3(1(1(4(2(2(0(x1)))))))))) (2)
1(5(1(5(4(x1))))) 1(0(0(2(2(0(5(2(2(4(x1)))))))))) (3)
3(4(5(3(4(x1))))) 3(5(0(2(1(1(1(2(1(4(x1)))))))))) (4)
4(3(4(3(4(x1))))) 2(1(1(1(4(1(2(4(2(0(x1)))))))))) (5)
5(0(1(0(1(x1))))) 5(0(2(0(1(1(4(2(1(2(x1)))))))))) (6)
1(0(1(0(1(4(x1)))))) 1(0(2(4(5(4(2(4(2(4(x1)))))))))) (7)
1(1(3(4(1(5(x1)))))) 1(1(0(0(2(2(5(2(0(0(x1)))))))))) (8)
1(3(1(1(3(3(x1)))))) 1(2(4(2(0(2(1(0(2(5(x1)))))))))) (9)
1(3(1(5(2(3(x1)))))) 1(4(3(2(1(0(0(2(4(3(x1)))))))))) (10)
1(5(0(5(5(3(x1)))))) 2(1(1(1(1(2(1(3(5(3(x1)))))))))) (11)
2(1(3(1(5(5(x1)))))) 1(1(2(2(0(5(0(0(2(2(x1)))))))))) (12)
2(2(4(3(4(5(x1)))))) 2(0(1(4(0(0(2(0(0(0(x1)))))))))) (13)
2(3(1(0(3(4(x1)))))) 0(0(1(1(5(2(4(1(1(4(x1)))))))))) (14)
2(3(3(4(1(5(x1)))))) 0(0(2(4(4(2(0(4(1(3(x1)))))))))) (15)
2(5(0(5(5(1(x1)))))) 3(0(1(4(4(0(0(0(0(1(x1)))))))))) (16)
3(0(4(3(3(4(x1)))))) 2(3(0(3(5(1(2(4(2(4(x1)))))))))) (17)
3(2(3(3(0(4(x1)))))) 5(4(2(2(0(0(4(2(4(4(x1)))))))))) (18)
3(3(5(4(3(4(x1)))))) 0(5(1(1(0(4(0(2(4(4(x1)))))))))) (19)
3(5(4(2(1(0(x1)))))) 2(5(0(0(0(0(0(4(4(0(x1)))))))))) (20)
4(1(3(4(3(1(x1)))))) 4(2(5(0(1(0(0(0(4(4(x1)))))))))) (21)
4(2(1(0(2(5(x1)))))) 4(2(5(4(2(2(2(4(2(5(x1)))))))))) (22)
4(3(3(5(1(1(x1)))))) 4(4(2(4(4(2(5(0(1(2(x1)))))))))) (23)
5(4(0(1(3(0(x1)))))) 0(2(4(2(2(1(2(0(0(0(x1)))))))))) (24)
0(4(1(5(5(3(5(x1))))))) 0(2(2(0(1(2(5(2(5(0(x1)))))))))) (25)
0(4(4(3(4(1(3(x1))))))) 0(4(2(4(1(3(2(0(2(2(x1)))))))))) (26)
1(3(3(4(5(2(5(x1))))))) 1(1(1(4(5(1(2(5(2(4(x1)))))))))) (27)
1(5(2(5(1(5(2(x1))))))) 1(2(3(0(2(3(0(1(0(2(x1)))))))))) (28)
1(5(3(2(4(5(4(x1))))))) 1(1(0(3(0(0(0(0(3(4(x1)))))))))) (29)
1(5(5(5(3(2(1(x1))))))) 1(4(1(4(2(1(3(0(1(1(x1)))))))))) (30)
2(1(2(3(1(3(3(x1))))))) 2(1(4(1(5(1(1(1(1(1(x1)))))))))) (31)
3(0(2(1(3(2(1(x1))))))) 1(2(0(0(3(3(4(2(2(1(x1)))))))))) (32)
3(0(4(4(3(4(5(x1))))))) 1(2(2(4(3(2(2(2(0(4(x1)))))))))) (33)
3(1(3(1(5(4(1(x1))))))) 0(1(2(2(5(5(5(4(2(0(x1)))))))))) (34)
3(2(5(2(1(3(4(x1))))))) 0(2(1(3(1(2(1(4(2(2(x1)))))))))) (35)
3(3(1(3(1(3(3(x1))))))) 1(2(1(3(0(5(5(1(2(1(x1)))))))))) (36)
3(3(1(5(0(3(4(x1))))))) 2(0(0(5(1(2(1(4(1(4(x1)))))))))) (37)
3(3(3(5(2(4(5(x1))))))) 3(1(0(0(1(4(2(2(0(5(x1)))))))))) (38)
3(4(3(3(4(3(5(x1))))))) 5(1(1(1(1(1(4(2(3(3(x1)))))))))) (39)
3(4(3(4(4(3(2(x1))))))) 2(5(4(5(3(4(2(4(4(0(x1)))))))))) (40)
4(1(3(4(1(0(2(x1))))))) 4(4(1(5(1(2(1(4(2(2(x1)))))))))) (41)
4(5(0(4(1(3(1(x1))))))) 1(1(1(2(0(0(4(4(5(1(x1)))))))))) (42)
4(5(0(5(3(2(1(x1))))))) 1(1(4(1(3(0(2(4(2(1(x1)))))))))) (43)
4(5(0(5(3(4(5(x1))))))) 1(1(1(0(5(4(0(2(4(5(x1)))))))))) (44)
4(5(3(1(4(4(3(x1))))))) 4(5(5(5(4(2(4(4(2(3(x1)))))))))) (45)
4(5(3(4(1(4(5(x1))))))) 4(5(4(0(2(0(1(2(1(0(x1)))))))))) (46)
4(5(4(3(4(1(0(x1))))))) 1(4(1(1(2(4(0(1(2(0(x1)))))))))) (47)
5(3(2(1(5(3(4(x1))))))) 5(1(2(1(3(3(5(1(2(4(x1)))))))))) (48)
5(3(4(3(1(3(3(x1))))))) 5(1(1(4(2(4(3(3(4(3(x1)))))))))) (49)
5(3(4(4(3(1(2(x1))))))) 5(1(0(5(0(3(5(1(1(1(x1)))))))))) (50)
5(4(3(2(3(1(3(x1))))))) 0(0(0(2(3(0(2(5(4(3(x1)))))))))) (51)
5(4(3(4(3(1(5(x1))))))) 0(2(0(0(0(0(4(1(5(0(x1)))))))))) (52)
5(5(3(3(3(5(4(x1))))))) 0(3(5(2(2(1(0(4(2(2(x1)))))))))) (53)
5(5(4(5(3(5(5(x1))))))) 5(2(4(2(2(2(4(1(5(2(x1)))))))))) (54)

and S is the following TRS.

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

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
1(0(1(0(x1)))) 2(2(2(1(0(0(1(1(1(0(x1)))))))))) (61)
4(4(3(0(x1)))) 0(2(2(4(1(1(3(0(0(0(x1)))))))))) (62)
4(5(1(5(1(x1))))) 4(2(2(5(0(2(2(0(0(1(x1)))))))))) (63)
4(3(5(4(3(x1))))) 4(1(2(1(1(1(2(0(5(3(x1)))))))))) (64)
4(3(4(3(4(x1))))) 0(2(4(2(1(4(1(1(1(2(x1)))))))))) (65)
1(0(1(0(5(x1))))) 2(1(2(4(1(1(0(2(0(5(x1)))))))))) (66)
4(1(0(1(0(1(x1)))))) 4(2(4(2(4(5(4(2(0(1(x1)))))))))) (67)
5(1(4(3(1(1(x1)))))) 0(0(2(5(2(2(0(0(1(1(x1)))))))))) (68)
3(3(1(1(3(1(x1)))))) 5(2(0(1(2(0(2(4(2(1(x1)))))))))) (69)
3(2(5(1(3(1(x1)))))) 3(4(2(0(0(1(2(3(4(1(x1)))))))))) (70)
3(5(5(0(5(1(x1)))))) 3(5(3(1(2(1(1(1(1(2(x1)))))))))) (71)
5(5(1(3(1(2(x1)))))) 2(2(0(0(5(0(2(2(1(1(x1)))))))))) (72)
5(4(3(4(2(2(x1)))))) 0(0(0(2(0(0(4(1(0(2(x1)))))))))) (73)
4(3(0(1(3(2(x1)))))) 4(1(1(4(2(5(1(1(0(0(x1)))))))))) (74)
5(1(4(3(3(2(x1)))))) 3(1(4(0(2(4(4(2(0(0(x1)))))))))) (75)
1(5(5(0(5(2(x1)))))) 1(0(0(0(0(4(4(1(0(3(x1)))))))))) (76)
4(3(3(4(0(3(x1)))))) 4(2(4(2(1(5(3(0(3(2(x1)))))))))) (77)
4(0(3(3(2(3(x1)))))) 4(4(2(4(0(0(2(2(4(5(x1)))))))))) (78)
4(3(4(5(3(3(x1)))))) 4(4(2(0(4(0(1(1(5(0(x1)))))))))) (79)
0(1(2(4(5(3(x1)))))) 0(4(4(0(0(0(0(0(5(2(x1)))))))))) (80)
1(3(4(3(1(4(x1)))))) 4(4(0(0(0(1(0(5(2(4(x1)))))))))) (81)
5(2(0(1(2(4(x1)))))) 5(2(4(2(2(2(4(5(2(4(x1)))))))))) (82)
1(1(5(3(3(4(x1)))))) 2(1(0(5(2(4(4(2(4(4(x1)))))))))) (83)
0(3(1(0(4(5(x1)))))) 0(0(0(2(1(2(2(4(2(0(x1)))))))))) (84)
5(3(5(5(1(4(0(x1))))))) 0(5(2(5(2(1(0(2(2(0(x1)))))))))) (85)
3(1(4(3(4(4(0(x1))))))) 2(2(0(2(3(1(4(2(4(0(x1)))))))))) (86)
5(2(5(4(3(3(1(x1))))))) 4(2(5(2(1(5(4(1(1(1(x1)))))))))) (87)
2(5(1(5(2(5(1(x1))))))) 2(0(1(0(3(2(0(3(2(1(x1)))))))))) (88)
4(5(4(2(3(5(1(x1))))))) 4(3(0(0(0(0(3(0(1(1(x1)))))))))) (89)
1(2(3(5(5(5(1(x1))))))) 1(1(0(3(1(2(4(1(4(1(x1)))))))))) (90)
3(3(1(3(2(1(2(x1))))))) 1(1(1(1(1(5(1(4(1(2(x1)))))))))) (91)
1(2(3(1(2(0(3(x1))))))) 1(2(2(4(3(3(0(0(2(1(x1)))))))))) (92)
5(4(3(4(4(0(3(x1))))))) 4(0(2(2(2(3(4(2(2(1(x1)))))))))) (93)
1(4(5(1(3(1(3(x1))))))) 0(2(4(5(5(5(2(2(1(0(x1)))))))))) (94)
4(3(1(2(5(2(3(x1))))))) 2(2(4(1(2(1(3(1(2(0(x1)))))))))) (95)
3(3(1(3(1(3(3(x1))))))) 1(2(1(5(5(0(3(1(2(1(x1)))))))))) (96)
4(3(0(5(1(3(3(x1))))))) 4(1(4(1(2(1(5(0(0(2(x1)))))))))) (97)
5(4(2(5(3(3(3(x1))))))) 5(0(2(2(4(1(0(0(1(3(x1)))))))))) (98)
5(3(4(3(3(4(3(x1))))))) 3(3(2(4(1(1(1(1(1(5(x1)))))))))) (99)
2(3(4(4(3(4(3(x1))))))) 0(4(4(2(4(3(5(4(5(2(x1)))))))))) (100)
2(0(1(4(3(1(4(x1))))))) 2(2(4(1(2(1(5(1(4(4(x1)))))))))) (101)
1(3(1(4(0(5(4(x1))))))) 1(5(4(4(0(0(2(1(1(1(x1)))))))))) (102)
1(2(3(5(0(5(4(x1))))))) 1(2(4(2(0(3(1(4(1(1(x1)))))))))) (103)
5(4(3(5(0(5(4(x1))))))) 5(4(2(0(4(5(0(1(1(1(x1)))))))))) (104)
3(4(4(1(3(5(4(x1))))))) 3(2(4(4(2(4(5(5(5(4(x1)))))))))) (105)
5(4(1(4(3(5(4(x1))))))) 0(1(2(1(0(2(0(4(5(4(x1)))))))))) (106)
0(1(4(3(4(5(4(x1))))))) 0(2(1(0(4(2(1(1(4(1(x1)))))))))) (107)
4(3(5(1(2(3(5(x1))))))) 4(2(1(5(3(3(1(2(1(5(x1)))))))))) (108)
3(3(1(3(4(3(5(x1))))))) 3(4(3(3(4(2(4(1(1(5(x1)))))))))) (109)
2(1(3(4(4(3(5(x1))))))) 1(1(1(5(3(0(5(0(1(5(x1)))))))))) (110)
3(1(3(2(3(4(5(x1))))))) 3(4(5(2(0(3(2(0(0(0(x1)))))))))) (111)
5(1(3(4(3(4(5(x1))))))) 0(5(1(4(0(0(0(0(2(0(x1)))))))))) (112)
4(5(3(3(3(5(5(x1))))))) 2(2(4(0(1(2(2(5(3(0(x1)))))))))) (113)
5(5(3(5(4(5(5(x1))))))) 2(5(1(4(2(2(2(4(2(5(x1)))))))))) (114)
1(3(4(3(4(0(x1)))))) 0(0(4(2(2(0(3(0(5(2(x1)))))))))) (115)
3(5(5(2(3(2(x1)))))) 4(0(1(2(2(1(3(2(0(0(x1)))))))))) (116)
2(1(5(4(2(3(3(x1))))))) 2(2(2(1(4(1(5(1(1(5(x1)))))))))) (117)
1(4(5(3(1(3(1(x1))))))) 4(5(0(1(5(0(0(4(0(1(x1)))))))))) (118)
5(1(5(1(0(5(x1)))))) 2(0(0(1(1(0(4(1(0(5(x1)))))))))) (119)
0(1(1(3(5(4(4(x1))))))) 2(0(2(2(2(1(0(5(0(4(x1)))))))))) (120)

1.1 Closure Under Flat Contexts

Using the flat contexts

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

We obtain the transformed TRS

There are 215 ruless (increase limit for explicit display).

1.1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

There are 1290 ruless (increase limit for explicit display).

1.1.1.1 Rule Removal

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

There are 1136 ruless (increase limit for explicit display).

1.1.1.1.1 Rule Removal

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

There are 103 ruless (increase limit for explicit display).

1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[45(x1)] = 1 + 1 · x1
[54(x1)] = 1 + 1 · x1
[42(x1)] = 1 · x1
[23(x1)] = 1 + 1 · x1
[35(x1)] = 1 + 1 · x1
[51(x1)] = 1 + 1 · x1
[14(x1)] = 1 + 1 · x1
[43(x1)] = 1 · x1
[30(x1)] = 1 · x1
[00(x1)] = 1 · x1
[03(x1)] = 1 · x1
[01(x1)] = 1 · x1
[11(x1)] = 1 · x1
[15(x1)] = 1 · x1
[12(x1)] = 1 · x1
[10(x1)] = 1 + 1 · x1
[13(x1)] = 1 · x1
[31(x1)] = 1 + 1 · x1
[20(x1)] = 1 · x1
[32(x1)] = 1 · x1
[22(x1)] = 1 · x1
[24(x1)] = 1 · x1
[33(x1)] = 1 · x1
[02(x1)] = 1 · x1
[21(x1)] = 1 · x1
[34(x1)] = 1 + 1 · x1
[44(x1)] = 1 · x1
[41(x1)] = 1 + 1 · x1
[55(x1)] = 1 · x1
[40(x1)] = 1 · x1
[04(x1)] = 1 · x1
[05(x1)] = 1 · x1
[52(x1)] = 1 + 1 · x1
[53(x1)] = 1 · x1
[50(x1)] = 1 · x1
[25(x1)] = 1 · x1
all of the following rules can be deleted.
45(54(42(23(35(51(14(x1))))))) 43(30(00(00(00(03(30(01(11(14(x1)))))))))) (391)
45(54(42(23(35(51(15(x1))))))) 43(30(00(00(00(03(30(01(11(15(x1)))))))))) (392)
45(54(42(23(35(51(11(x1))))))) 43(30(00(00(00(03(30(01(11(11(x1)))))))))) (393)
45(54(42(23(35(51(12(x1))))))) 43(30(00(00(00(03(30(01(11(12(x1)))))))))) (394)
45(54(42(23(35(51(10(x1))))))) 43(30(00(00(00(03(30(01(11(10(x1)))))))))) (395)
45(54(42(23(35(51(13(x1))))))) 43(30(00(00(00(03(30(01(11(13(x1)))))))))) (396)
12(23(31(12(20(03(32(x1))))))) 12(22(24(43(33(30(00(02(21(12(x1)))))))))) (406)
34(44(41(13(35(54(44(x1))))))) 32(24(44(42(24(45(55(55(54(44(x1)))))))))) (445)
34(44(41(13(35(54(45(x1))))))) 32(24(44(42(24(45(55(55(54(45(x1)))))))))) (446)
34(44(41(13(35(54(41(x1))))))) 32(24(44(42(24(45(55(55(54(41(x1)))))))))) (447)
34(44(41(13(35(54(42(x1))))))) 32(24(44(42(24(45(55(55(54(42(x1)))))))))) (448)
34(44(41(13(35(54(40(x1))))))) 32(24(44(42(24(45(55(55(54(40(x1)))))))))) (449)
34(44(41(13(35(54(43(x1))))))) 32(24(44(42(24(45(55(55(54(43(x1)))))))))) (450)
41(10(01(10(04(x1))))) 42(22(22(21(10(00(01(11(11(10(04(x1))))))))))) (493)
41(10(01(10(05(x1))))) 42(22(22(21(10(00(01(11(11(10(05(x1))))))))))) (494)
41(10(01(10(01(x1))))) 42(22(22(21(10(00(01(11(11(10(01(x1))))))))))) (495)
41(10(01(10(02(x1))))) 42(22(22(21(10(00(01(11(11(10(02(x1))))))))))) (496)
41(10(01(10(00(x1))))) 42(22(22(21(10(00(01(11(11(10(00(x1))))))))))) (497)
41(10(01(10(03(x1))))) 42(22(22(21(10(00(01(11(11(10(03(x1))))))))))) (498)
11(14(45(51(13(31(13(31(x1)))))))) 10(02(24(45(55(55(52(22(21(10(01(x1))))))))))) (1053)
21(14(45(51(13(31(13(31(x1)))))))) 20(02(24(45(55(55(52(22(21(10(01(x1))))))))))) (1065)
41(14(45(51(13(31(13(31(x1)))))))) 40(02(24(45(55(55(52(22(21(10(01(x1))))))))))) (1071)
31(14(45(51(13(31(13(31(x1)))))))) 30(02(24(45(55(55(52(22(21(10(01(x1))))))))))) (1077)
02(21(13(34(44(43(35(54(x1)))))))) 01(11(11(15(53(30(05(50(01(15(54(x1))))))))))) (1273)
02(21(13(34(44(43(35(55(x1)))))))) 01(11(11(15(53(30(05(50(01(15(55(x1))))))))))) (1274)
02(21(13(34(44(43(35(51(x1)))))))) 01(11(11(15(53(30(05(50(01(15(51(x1))))))))))) (1275)
02(21(13(34(44(43(35(52(x1)))))))) 01(11(11(15(53(30(05(50(01(15(52(x1))))))))))) (1276)
02(21(13(34(44(43(35(50(x1)))))))) 01(11(11(15(53(30(05(50(01(15(50(x1))))))))))) (1277)
02(21(13(34(44(43(35(53(x1)))))))) 01(11(11(15(53(30(05(50(01(15(53(x1))))))))))) (1278)
52(21(13(34(44(43(35(54(x1)))))))) 51(11(11(15(53(30(05(50(01(15(54(x1))))))))))) (1297)
52(21(13(34(44(43(35(55(x1)))))))) 51(11(11(15(53(30(05(50(01(15(55(x1))))))))))) (1298)
52(21(13(34(44(43(35(51(x1)))))))) 51(11(11(15(53(30(05(50(01(15(51(x1))))))))))) (1299)
52(21(13(34(44(43(35(52(x1)))))))) 51(11(11(15(53(30(05(50(01(15(52(x1))))))))))) (1300)
52(21(13(34(44(43(35(50(x1)))))))) 51(11(11(15(53(30(05(50(01(15(50(x1))))))))))) (1301)
52(21(13(34(44(43(35(53(x1)))))))) 51(11(11(15(53(30(05(50(01(15(53(x1))))))))))) (1302)
21(13(34(43(34(40(05(x1))))))) 20(00(04(42(22(20(03(30(05(52(25(x1))))))))))) (1430)
41(13(34(43(34(40(05(x1))))))) 40(00(04(42(22(20(03(30(05(52(25(x1))))))))))) (1436)
31(13(34(43(34(40(05(x1))))))) 30(00(04(42(22(20(03(30(05(52(25(x1))))))))))) (1442)

1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[11(x1)] = 1 · x1
[10(x1)] = 1 + 1 · x1
[01(x1)] = 1 · x1
[04(x1)] = 1 + 1 · x1
[12(x1)] = 1 · x1
[22(x1)] = 1 · x1
[21(x1)] = 1 · x1
[00(x1)] = 1 · x1
[05(x1)] = 1 · x1
[02(x1)] = 1 · x1
[03(x1)] = 1 + 1 · x1
[13(x1)] = 1 + 1 · x1
[34(x1)] = 1 + 1 · x1
[43(x1)] = 1 + 1 · x1
[40(x1)] = 1 + 1 · x1
[42(x1)] = 1 · x1
[20(x1)] = 1 · x1
[30(x1)] = 1 · x1
[52(x1)] = 1 + 1 · x1
[25(x1)] = 1 · x1
all of the following rules can be deleted.
11(13(34(43(34(40(05(x1))))))) 10(00(04(42(22(20(03(30(05(52(25(x1))))))))))) (1418)

1.1.1.1.1.1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
04(10(01(10(11(x1))))) 04(10(11(11(01(00(10(21(22(22(12(x1))))))))))) (1597)
05(10(01(10(11(x1))))) 05(10(11(11(01(00(10(21(22(22(12(x1))))))))))) (1598)
01(10(01(10(11(x1))))) 01(10(11(11(01(00(10(21(22(22(12(x1))))))))))) (1599)
02(10(01(10(11(x1))))) 02(10(11(11(01(00(10(21(22(22(12(x1))))))))))) (1600)
00(10(01(10(11(x1))))) 00(10(11(11(01(00(10(21(22(22(12(x1))))))))))) (1601)
03(10(01(10(11(x1))))) 03(10(11(11(01(00(10(21(22(22(12(x1))))))))))) (1602)
04(10(01(10(21(x1))))) 04(10(11(11(01(00(10(21(22(22(22(x1))))))))))) (1603)
05(10(01(10(21(x1))))) 05(10(11(11(01(00(10(21(22(22(22(x1))))))))))) (1604)
01(10(01(10(21(x1))))) 01(10(11(11(01(00(10(21(22(22(22(x1))))))))))) (1605)
02(10(01(10(21(x1))))) 02(10(11(11(01(00(10(21(22(22(22(x1))))))))))) (1606)
00(10(01(10(21(x1))))) 00(10(11(11(01(00(10(21(22(22(22(x1))))))))))) (1607)
03(10(01(10(21(x1))))) 03(10(11(11(01(00(10(21(22(22(22(x1))))))))))) (1608)

1.1.1.1.1.1.1.1.1 Switch to Innermost Termination

The TRS is overlay and locally confluent:

10

Hence, it suffices to show innermost termination in the following.

1.1.1.1.1.1.1.1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
04#(10(01(10(11(x1))))) 04#(10(11(11(01(00(10(21(22(22(12(x1))))))))))) (1609)
04#(10(01(10(11(x1))))) 01#(00(10(21(22(22(12(x1))))))) (1610)
04#(10(01(10(11(x1))))) 00#(10(21(22(22(12(x1)))))) (1611)
05#(10(01(10(11(x1))))) 05#(10(11(11(01(00(10(21(22(22(12(x1))))))))))) (1612)
05#(10(01(10(11(x1))))) 01#(00(10(21(22(22(12(x1))))))) (1613)
05#(10(01(10(11(x1))))) 00#(10(21(22(22(12(x1)))))) (1614)
01#(10(01(10(11(x1))))) 01#(10(11(11(01(00(10(21(22(22(12(x1))))))))))) (1615)
01#(10(01(10(11(x1))))) 01#(00(10(21(22(22(12(x1))))))) (1616)
01#(10(01(10(11(x1))))) 00#(10(21(22(22(12(x1)))))) (1617)
02#(10(01(10(11(x1))))) 02#(10(11(11(01(00(10(21(22(22(12(x1))))))))))) (1618)
02#(10(01(10(11(x1))))) 01#(00(10(21(22(22(12(x1))))))) (1619)
02#(10(01(10(11(x1))))) 00#(10(21(22(22(12(x1)))))) (1620)
00#(10(01(10(11(x1))))) 00#(10(11(11(01(00(10(21(22(22(12(x1))))))))))) (1621)
00#(10(01(10(11(x1))))) 01#(00(10(21(22(22(12(x1))))))) (1622)
00#(10(01(10(11(x1))))) 00#(10(21(22(22(12(x1)))))) (1623)
03#(10(01(10(11(x1))))) 03#(10(11(11(01(00(10(21(22(22(12(x1))))))))))) (1624)
03#(10(01(10(11(x1))))) 01#(00(10(21(22(22(12(x1))))))) (1625)
03#(10(01(10(11(x1))))) 00#(10(21(22(22(12(x1)))))) (1626)
04#(10(01(10(21(x1))))) 04#(10(11(11(01(00(10(21(22(22(22(x1))))))))))) (1627)
04#(10(01(10(21(x1))))) 01#(00(10(21(22(22(22(x1))))))) (1628)
04#(10(01(10(21(x1))))) 00#(10(21(22(22(22(x1)))))) (1629)
05#(10(01(10(21(x1))))) 05#(10(11(11(01(00(10(21(22(22(22(x1))))))))))) (1630)
05#(10(01(10(21(x1))))) 01#(00(10(21(22(22(22(x1))))))) (1631)
05#(10(01(10(21(x1))))) 00#(10(21(22(22(22(x1)))))) (1632)
01#(10(01(10(21(x1))))) 01#(10(11(11(01(00(10(21(22(22(22(x1))))))))))) (1633)
01#(10(01(10(21(x1))))) 01#(00(10(21(22(22(22(x1))))))) (1634)
01#(10(01(10(21(x1))))) 00#(10(21(22(22(22(x1)))))) (1635)
02#(10(01(10(21(x1))))) 02#(10(11(11(01(00(10(21(22(22(22(x1))))))))))) (1636)
02#(10(01(10(21(x1))))) 01#(00(10(21(22(22(22(x1))))))) (1637)
02#(10(01(10(21(x1))))) 00#(10(21(22(22(22(x1)))))) (1638)
00#(10(01(10(21(x1))))) 00#(10(11(11(01(00(10(21(22(22(22(x1))))))))))) (1639)
00#(10(01(10(21(x1))))) 01#(00(10(21(22(22(22(x1))))))) (1640)
00#(10(01(10(21(x1))))) 00#(10(21(22(22(22(x1)))))) (1641)
03#(10(01(10(21(x1))))) 03#(10(11(11(01(00(10(21(22(22(22(x1))))))))))) (1642)
03#(10(01(10(21(x1))))) 01#(00(10(21(22(22(22(x1))))))) (1643)
03#(10(01(10(21(x1))))) 00#(10(21(22(22(22(x1)))))) (1644)

1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.