Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/3770)

The rewrite relation of the following TRS is considered.

1(3(3(x1))) → 3(5(3(2(5(0(2(4(5(4(x1)))))))))) (1)
0(3(3(3(x1)))) → 5(4(3(5(3(0(5(4(4(0(x1)))))))))) (2)
0(3(3(3(1(x1))))) → 5(4(4(0(3(1(0(5(1(0(x1)))))))))) (3)
1(2(3(3(3(x1))))) → 4(1(1(2(3(5(0(4(0(5(x1)))))))))) (4)
1(4(4(2(2(x1))))) → 1(1(2(0(1(1(1(0(2(2(x1)))))))))) (5)
0(3(3(1(4(3(x1)))))) → 4(4(3(0(2(3(0(3(0(0(x1)))))))))) (6)
3(3(3(3(4(0(x1)))))) → 3(0(0(2(1(0(5(3(5(4(x1)))))))))) (7)
4(0(1(3(4(0(x1)))))) → 2(2(3(0(0(0(5(0(0(0(x1)))))))))) (8)
4(1(4(4(4(1(x1)))))) → 4(1(0(3(3(5(5(5(4(1(x1)))))))))) (9)
0(1(3(5(2(2(3(x1))))))) → 0(3(0(0(5(0(0(4(4(3(x1)))))))))) (10)
0(2(3(1(3(2(5(x1))))))) → 0(4(3(1(2(3(2(3(2(0(x1)))))))))) (11)
1(1(3(3(5(3(1(x1))))))) → 3(5(0(5(3(2(5(0(0(1(x1)))))))))) (12)
3(5(2(0(1(3(3(x1))))))) → 3(4(3(2(3(2(4(4(5(5(x1)))))))))) (13)
4(1(4(2(4(0(1(x1))))))) → 5(2(2(1(0(5(5(4(5(1(x1)))))))))) (14)
4(5(1(2(4(4(4(x1))))))) → 4(1(1(4(5(3(0(1(0(4(x1)))))))))) (15)
5(1(4(5(3(3(3(x1))))))) → 5(1(4(5(3(4(4(2(3(2(x1)))))))))) (16)

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
3(3(1(x1))) → 4(5(4(2(0(5(2(3(5(3(x1)))))))))) (17)
3(3(3(0(x1)))) → 0(4(4(5(0(3(5(3(4(5(x1)))))))))) (18)
1(3(3(3(0(x1))))) → 0(1(5(0(1(3(0(4(4(5(x1)))))))))) (19)
3(3(3(2(1(x1))))) → 5(0(4(0(5(3(2(1(1(4(x1)))))))))) (20)
2(2(4(4(1(x1))))) → 2(2(0(1(1(1(0(2(1(1(x1)))))))))) (21)
3(4(1(3(3(0(x1)))))) → 0(0(3(0(3(2(0(3(4(4(x1)))))))))) (22)
0(4(3(3(3(3(x1)))))) → 4(5(3(5(0(1(2(0(0(3(x1)))))))))) (23)
0(4(3(1(0(4(x1)))))) → 0(0(0(5(0(0(0(3(2(2(x1)))))))))) (24)
1(4(4(4(1(4(x1)))))) → 1(4(5(5(5(3(3(0(1(4(x1)))))))))) (25)
3(2(2(5(3(1(0(x1))))))) → 3(4(4(0(0(5(0(0(3(0(x1)))))))))) (26)
5(2(3(1(3(2(0(x1))))))) → 0(2(3(2(3(2(1(3(4(0(x1)))))))))) (27)
1(3(5(3(3(1(1(x1))))))) → 1(0(0(5(2(3(5(0(5(3(x1)))))))))) (28)
3(3(1(0(2(5(3(x1))))))) → 5(5(4(4(2(3(2(3(4(3(x1)))))))))) (29)
1(0(4(2(4(1(4(x1))))))) → 1(5(4(5(5(0(1(2(2(5(x1)))))))))) (30)
4(4(4(2(1(5(4(x1))))))) → 4(0(1(0(3(5(4(1(1(4(x1)))))))))) (31)
3(3(3(5(4(1(5(x1))))))) → 2(3(2(4(4(3(5(4(1(5(x1)))))))))) (32)

1.1 Closure Under Flat Contexts

Using the flat contexts

{3(☐), 1(☐), 4(☐), 5(☐), 2(☐), 0(☐)}

We obtain the transformed TRS
2(2(4(4(1(x1))))) → 2(2(0(1(1(1(0(2(1(1(x1)))))))))) (21)
0(4(3(1(0(4(x1)))))) → 0(0(0(5(0(0(0(3(2(2(x1)))))))))) (24)
1(4(4(4(1(4(x1)))))) → 1(4(5(5(5(3(3(0(1(4(x1)))))))))) (25)
3(2(2(5(3(1(0(x1))))))) → 3(4(4(0(0(5(0(0(3(0(x1)))))))))) (26)
1(3(5(3(3(1(1(x1))))))) → 1(0(0(5(2(3(5(0(5(3(x1)))))))))) (28)
1(0(4(2(4(1(4(x1))))))) → 1(5(4(5(5(0(1(2(2(5(x1)))))))))) (30)
4(4(4(2(1(5(4(x1))))))) → 4(0(1(0(3(5(4(1(1(4(x1)))))))))) (31)
3(3(3(1(x1)))) → 3(4(5(4(2(0(5(2(3(5(3(x1))))))))))) (33)
1(3(3(1(x1)))) → 1(4(5(4(2(0(5(2(3(5(3(x1))))))))))) (34)
4(3(3(1(x1)))) → 4(4(5(4(2(0(5(2(3(5(3(x1))))))))))) (35)
5(3(3(1(x1)))) → 5(4(5(4(2(0(5(2(3(5(3(x1))))))))))) (36)
2(3(3(1(x1)))) → 2(4(5(4(2(0(5(2(3(5(3(x1))))))))))) (37)
0(3(3(1(x1)))) → 0(4(5(4(2(0(5(2(3(5(3(x1))))))))))) (38)
3(3(3(3(0(x1))))) → 3(0(4(4(5(0(3(5(3(4(5(x1))))))))))) (39)
1(3(3(3(0(x1))))) → 1(0(4(4(5(0(3(5(3(4(5(x1))))))))))) (40)
4(3(3(3(0(x1))))) → 4(0(4(4(5(0(3(5(3(4(5(x1))))))))))) (41)
5(3(3(3(0(x1))))) → 5(0(4(4(5(0(3(5(3(4(5(x1))))))))))) (42)
2(3(3(3(0(x1))))) → 2(0(4(4(5(0(3(5(3(4(5(x1))))))))))) (43)
0(3(3(3(0(x1))))) → 0(0(4(4(5(0(3(5(3(4(5(x1))))))))))) (44)
3(1(3(3(3(0(x1)))))) → 3(0(1(5(0(1(3(0(4(4(5(x1))))))))))) (45)
1(1(3(3(3(0(x1)))))) → 1(0(1(5(0(1(3(0(4(4(5(x1))))))))))) (46)
4(1(3(3(3(0(x1)))))) → 4(0(1(5(0(1(3(0(4(4(5(x1))))))))))) (47)
5(1(3(3(3(0(x1)))))) → 5(0(1(5(0(1(3(0(4(4(5(x1))))))))))) (48)
2(1(3(3(3(0(x1)))))) → 2(0(1(5(0(1(3(0(4(4(5(x1))))))))))) (49)
0(1(3(3(3(0(x1)))))) → 0(0(1(5(0(1(3(0(4(4(5(x1))))))))))) (50)
3(3(3(3(2(1(x1)))))) → 3(5(0(4(0(5(3(2(1(1(4(x1))))))))))) (51)
1(3(3(3(2(1(x1)))))) → 1(5(0(4(0(5(3(2(1(1(4(x1))))))))))) (52)
4(3(3(3(2(1(x1)))))) → 4(5(0(4(0(5(3(2(1(1(4(x1))))))))))) (53)
5(3(3(3(2(1(x1)))))) → 5(5(0(4(0(5(3(2(1(1(4(x1))))))))))) (54)
2(3(3(3(2(1(x1)))))) → 2(5(0(4(0(5(3(2(1(1(4(x1))))))))))) (55)
0(3(3(3(2(1(x1)))))) → 0(5(0(4(0(5(3(2(1(1(4(x1))))))))))) (56)
3(3(4(1(3(3(0(x1))))))) → 3(0(0(3(0(3(2(0(3(4(4(x1))))))))))) (57)
1(3(4(1(3(3(0(x1))))))) → 1(0(0(3(0(3(2(0(3(4(4(x1))))))))))) (58)
4(3(4(1(3(3(0(x1))))))) → 4(0(0(3(0(3(2(0(3(4(4(x1))))))))))) (59)
5(3(4(1(3(3(0(x1))))))) → 5(0(0(3(0(3(2(0(3(4(4(x1))))))))))) (60)
2(3(4(1(3(3(0(x1))))))) → 2(0(0(3(0(3(2(0(3(4(4(x1))))))))))) (61)
0(3(4(1(3(3(0(x1))))))) → 0(0(0(3(0(3(2(0(3(4(4(x1))))))))))) (62)
3(0(4(3(3(3(3(x1))))))) → 3(4(5(3(5(0(1(2(0(0(3(x1))))))))))) (63)
1(0(4(3(3(3(3(x1))))))) → 1(4(5(3(5(0(1(2(0(0(3(x1))))))))))) (64)
4(0(4(3(3(3(3(x1))))))) → 4(4(5(3(5(0(1(2(0(0(3(x1))))))))))) (65)
5(0(4(3(3(3(3(x1))))))) → 5(4(5(3(5(0(1(2(0(0(3(x1))))))))))) (66)
2(0(4(3(3(3(3(x1))))))) → 2(4(5(3(5(0(1(2(0(0(3(x1))))))))))) (67)
0(0(4(3(3(3(3(x1))))))) → 0(4(5(3(5(0(1(2(0(0(3(x1))))))))))) (68)
3(5(2(3(1(3(2(0(x1)))))))) → 3(0(2(3(2(3(2(1(3(4(0(x1))))))))))) (69)
1(5(2(3(1(3(2(0(x1)))))))) → 1(0(2(3(2(3(2(1(3(4(0(x1))))))))))) (70)
4(5(2(3(1(3(2(0(x1)))))))) → 4(0(2(3(2(3(2(1(3(4(0(x1))))))))))) (71)
5(5(2(3(1(3(2(0(x1)))))))) → 5(0(2(3(2(3(2(1(3(4(0(x1))))))))))) (72)
2(5(2(3(1(3(2(0(x1)))))))) → 2(0(2(3(2(3(2(1(3(4(0(x1))))))))))) (73)
0(5(2(3(1(3(2(0(x1)))))))) → 0(0(2(3(2(3(2(1(3(4(0(x1))))))))))) (74)
3(3(3(1(0(2(5(3(x1)))))))) → 3(5(5(4(4(2(3(2(3(4(3(x1))))))))))) (75)
1(3(3(1(0(2(5(3(x1)))))))) → 1(5(5(4(4(2(3(2(3(4(3(x1))))))))))) (76)
4(3(3(1(0(2(5(3(x1)))))))) → 4(5(5(4(4(2(3(2(3(4(3(x1))))))))))) (77)
5(3(3(1(0(2(5(3(x1)))))))) → 5(5(5(4(4(2(3(2(3(4(3(x1))))))))))) (78)
2(3(3(1(0(2(5(3(x1)))))))) → 2(5(5(4(4(2(3(2(3(4(3(x1))))))))))) (79)
0(3(3(1(0(2(5(3(x1)))))))) → 0(5(5(4(4(2(3(2(3(4(3(x1))))))))))) (80)
3(3(3(3(5(4(1(5(x1)))))))) → 3(2(3(2(4(4(3(5(4(1(5(x1))))))))))) (81)
1(3(3(3(5(4(1(5(x1)))))))) → 1(2(3(2(4(4(3(5(4(1(5(x1))))))))))) (82)
4(3(3(3(5(4(1(5(x1)))))))) → 4(2(3(2(4(4(3(5(4(1(5(x1))))))))))) (83)
5(3(3(3(5(4(1(5(x1)))))))) → 5(2(3(2(4(4(3(5(4(1(5(x1))))))))))) (84)
2(3(3(3(5(4(1(5(x1)))))))) → 2(2(3(2(4(4(3(5(4(1(5(x1))))))))))) (85)
0(3(3(3(5(4(1(5(x1)))))))) → 0(2(3(2(4(4(3(5(4(1(5(x1))))))))))) (86)

1.1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

There are 366 ruless (increase limit for explicit display).

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[22(x1)] = 1 · x1
[24(x1)] = 1 · x1 + 1
[44(x1)] = 1 · x1 + 56
[41(x1)] = 1 · x1 + 34
[12(x1)] = 1 · x1 + 82
[20(x1)] = 1 · x1
[01(x1)] = 1 · x1 + 6
[11(x1)] = 1 · x1 + 5
[10(x1)] = 1 · x1 + 6
[02(x1)] = 1 · x1
[21(x1)] = 1 · x1
[14(x1)] = 1 · x1 + 99
[13(x1)] = 1 · x1 + 82
[15(x1)] = 1 · x1 + 82
[04(x1)] = 1 · x1
[43(x1)] = 1 · x1 + 39
[31(x1)] = 1 · x1 + 147
[42(x1)] = 1 · x1 + 39
[00(x1)] = 1 · x1
[05(x1)] = 1 · x1
[50(x1)] = 1 · x1
[03(x1)] = 1 · x1
[32(x1)] = 1 · x1 + 96
[40(x1)] = 1 · x1 + 35
[23(x1)] = 1 · x1
[45(x1)] = 1 · x1 + 39
[25(x1)] = 1 · x1
[55(x1)] = 1 · x1 + 1
[53(x1)] = 1 · x1 + 1
[33(x1)] = 1 · x1 + 96
[30(x1)] = 1 · x1
[34(x1)] = 1 · x1 + 17
[35(x1)] = 1 · x1 + 1
[52(x1)] = 1 · x1 + 1
[54(x1)] = 1 · x1 + 2
[51(x1)] = 1 · x1
all of the following rules can be deleted.

There are 286 ruless (increase limit for explicit display).

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[11(x1)] = 1 · x1 + 1
[13(x1)] = 1 · x1
[33(x1)] = 1 · x1
[30(x1)] = 1 · x1
[04(x1)] = 1 · x1
[10(x1)] = 1 · x1
[01(x1)] = 1 · x1
[15(x1)] = 1 · x1
[50(x1)] = 1 · x1
[44(x1)] = 1 · x1
[45(x1)] = 1 · x1
[54(x1)] = 1 · x1
[41(x1)] = 1 · x1 + 1
[40(x1)] = 1 · x1
[25(x1)] = 1 · x1
[52(x1)] = 1 · x1
[23(x1)] = 1 · x1
[31(x1)] = 1 · x1
[32(x1)] = 1 · x1
[20(x1)] = 1 · x1
[02(x1)] = 1 · x1
[21(x1)] = 1 · x1
[34(x1)] = 1 · x1
[00(x1)] = 1 · x1
[03(x1)] = 1 · x1
[05(x1)] = 1 · x1
[53(x1)] = 1 · x1
[55(x1)] = 1 · x1
[42(x1)] = 1 · x1
[43(x1)] = 1 · x1
[35(x1)] = 1 · x1
[24(x1)] = 1 · x1
[51(x1)] = 1 · x1
[12(x1)] = 1 · x1
[22(x1)] = 1 · x1
all of the following rules can be deleted.
11(13(33(33(30(04(x1)))))) → 10(01(15(50(01(13(30(04(44(45(54(x1))))))))))) (208)
41(13(33(33(30(04(x1)))))) → 40(01(15(50(01(13(30(04(44(45(54(x1))))))))))) (214)

1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[25(x1)] = 1 · x1
[52(x1)] = 1 · x1
[23(x1)] = 1 · x1
[31(x1)] = 1 · x1 + 1
[13(x1)] = 1 · x1
[32(x1)] = 1 · x1
[20(x1)] = 1 · x1
[02(x1)] = 1 · x1
[21(x1)] = 1 · x1
[34(x1)] = 1 · x1
[40(x1)] = 1 · x1
[04(x1)] = 1 · x1
[01(x1)] = 1 · x1
[00(x1)] = 1 · x1 + 1
[03(x1)] = 1 · x1
[05(x1)] = 1 · x1
[33(x1)] = 1 · x1
[10(x1)] = 1 · x1
[53(x1)] = 1 · x1
[15(x1)] = 1 · x1
[55(x1)] = 1 · x1
[54(x1)] = 1 · x1 + 1
[44(x1)] = 1 · x1
[42(x1)] = 1 · x1
[43(x1)] = 1 · x1
[30(x1)] = 1 · x1
[35(x1)] = 1 · x1
[45(x1)] = 1 · x1
[41(x1)] = 1 · x1
[24(x1)] = 1 · x1
[51(x1)] = 1 · x1
[50(x1)] = 1 · x1
[12(x1)] = 1 · x1
[22(x1)] = 1 · x1
all of the following rules can be deleted.
25(52(23(31(13(32(20(02(x1)))))))) → 20(02(23(32(23(32(21(13(34(40(02(x1))))))))))) (369)
25(52(23(31(13(32(20(04(x1)))))))) → 20(02(23(32(23(32(21(13(34(40(04(x1))))))))))) (370)
25(52(23(31(13(32(20(01(x1)))))))) → 20(02(23(32(23(32(21(13(34(40(01(x1))))))))))) (371)
25(52(23(31(13(32(20(00(x1)))))))) → 20(02(23(32(23(32(21(13(34(40(00(x1))))))))))) (372)
25(52(23(31(13(32(20(03(x1)))))))) → 20(02(23(32(23(32(21(13(34(40(03(x1))))))))))) (373)
25(52(23(31(13(32(20(05(x1)))))))) → 20(02(23(32(23(32(21(13(34(40(05(x1))))))))))) (374)

1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[05(x1)] = 1 · x1
[52(x1)] = 1 · x1
[23(x1)] = 1 · x1
[31(x1)] = 1 · x1
[13(x1)] = 1 · x1
[32(x1)] = 1 · x1
[20(x1)] = 1 · x1 + 1
[02(x1)] = 1 · x1
[00(x1)] = 1 · x1
[21(x1)] = 1 · x1
[34(x1)] = 1 · x1
[40(x1)] = 1 · x1
[04(x1)] = 1 · x1
[01(x1)] = 1 · x1
[03(x1)] = 1 · x1
[33(x1)] = 1 · x1
[10(x1)] = 1 · x1
[25(x1)] = 1 · x1
[53(x1)] = 1 · x1
[15(x1)] = 1 · x1
[55(x1)] = 1 · x1
[54(x1)] = 1 · x1
[44(x1)] = 1 · x1
[42(x1)] = 1 · x1
[43(x1)] = 1 · x1
[30(x1)] = 1 · x1
[35(x1)] = 1 · x1
[45(x1)] = 1 · x1
[41(x1)] = 1 · x1
[24(x1)] = 1 · x1
[51(x1)] = 1 · x1
[50(x1)] = 1 · x1
[12(x1)] = 1 · x1
[22(x1)] = 1 · x1
all of the following rules can be deleted.
05(52(23(31(13(32(20(02(x1)))))))) → 00(02(23(32(23(32(21(13(34(40(02(x1))))))))))) (375)
05(52(23(31(13(32(20(04(x1)))))))) → 00(02(23(32(23(32(21(13(34(40(04(x1))))))))))) (376)
05(52(23(31(13(32(20(01(x1)))))))) → 00(02(23(32(23(32(21(13(34(40(01(x1))))))))))) (377)
05(52(23(31(13(32(20(00(x1)))))))) → 00(02(23(32(23(32(21(13(34(40(00(x1))))))))))) (378)
05(52(23(31(13(32(20(03(x1)))))))) → 00(02(23(32(23(32(21(13(34(40(03(x1))))))))))) (379)
05(52(23(31(13(32(20(05(x1)))))))) → 00(02(23(32(23(32(21(13(34(40(05(x1))))))))))) (380)

1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[13(x1)] = 1 · x1 + 1
[33(x1)] = 1 · x1
[31(x1)] = 1 · x1
[10(x1)] = 1 · x1
[02(x1)] = 1 · x1
[25(x1)] = 1 · x1
[53(x1)] = 1 · x1
[32(x1)] = 1 · x1
[15(x1)] = 1 · x1
[55(x1)] = 1 · x1
[54(x1)] = 1 · x1
[44(x1)] = 1 · x1
[42(x1)] = 1 · x1
[23(x1)] = 1 · x1
[34(x1)] = 1 · x1
[43(x1)] = 1 · x1
[30(x1)] = 1 · x1
[35(x1)] = 1 · x1
[45(x1)] = 1 · x1
[03(x1)] = 1 · x1
[05(x1)] = 1 · x1
[41(x1)] = 1 · x1
[52(x1)] = 1 · x1
[24(x1)] = 1 · x1
[51(x1)] = 1 · x1
[50(x1)] = 1 · x1
[12(x1)] = 1 · x1 + 1
[22(x1)] = 1 · x1
all of the following rules can be deleted.
13(33(31(10(02(25(53(32(x1)))))))) → 15(55(54(44(42(23(32(23(34(43(32(x1))))))))))) (387)
13(33(31(10(02(25(53(34(x1)))))))) → 15(55(54(44(42(23(32(23(34(43(34(x1))))))))))) (388)
13(33(31(10(02(25(53(31(x1)))))))) → 15(55(54(44(42(23(32(23(34(43(31(x1))))))))))) (389)
13(33(31(10(02(25(53(30(x1)))))))) → 15(55(54(44(42(23(32(23(34(43(30(x1))))))))))) (390)
13(33(31(10(02(25(53(33(x1)))))))) → 15(55(54(44(42(23(32(23(34(43(33(x1))))))))))) (391)
13(33(31(10(02(25(53(35(x1)))))))) → 15(55(54(44(42(23(32(23(34(43(35(x1))))))))))) (392)

1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[43(x1)] = 1 · x1
[33(x1)] = 1 · x1
[31(x1)] = 1 · x1 + 1
[10(x1)] = 1 · x1
[02(x1)] = 1 · x1
[25(x1)] = 1 · x1 + 1
[53(x1)] = 1 · x1
[32(x1)] = 1 · x1
[45(x1)] = 1 · x1
[55(x1)] = 1 · x1 + 1
[54(x1)] = 1 · x1
[44(x1)] = 1 · x1
[42(x1)] = 1 · x1
[23(x1)] = 1 · x1
[34(x1)] = 1 · x1
[30(x1)] = 1 · x1
[35(x1)] = 1 · x1
[03(x1)] = 1 · x1
[05(x1)] = 1 · x1 + 1
[41(x1)] = 1 · x1
[15(x1)] = 1 · x1
[52(x1)] = 1 · x1
[24(x1)] = 1 · x1
[51(x1)] = 1 · x1
[50(x1)] = 1 · x1
[13(x1)] = 1 · x1
[12(x1)] = 1 · x1
[22(x1)] = 1 · x1
all of the following rules can be deleted.
43(33(31(10(02(25(53(32(x1)))))))) → 45(55(54(44(42(23(32(23(34(43(32(x1))))))))))) (393)
43(33(31(10(02(25(53(34(x1)))))))) → 45(55(54(44(42(23(32(23(34(43(34(x1))))))))))) (394)
43(33(31(10(02(25(53(31(x1)))))))) → 45(55(54(44(42(23(32(23(34(43(31(x1))))))))))) (395)
43(33(31(10(02(25(53(30(x1)))))))) → 45(55(54(44(42(23(32(23(34(43(30(x1))))))))))) (396)
43(33(31(10(02(25(53(33(x1)))))))) → 45(55(54(44(42(23(32(23(34(43(33(x1))))))))))) (397)
43(33(31(10(02(25(53(35(x1)))))))) → 45(55(54(44(42(23(32(23(34(43(35(x1))))))))))) (398)

1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[53(x1)] = 1 · x1
[33(x1)] = 1 · x1
[31(x1)] = 1 · x1
[10(x1)] = 1 · x1
[02(x1)] = 1 · x1
[25(x1)] = 1 · x1 + 1
[32(x1)] = 1 · x1
[55(x1)] = 1 · x1
[54(x1)] = 1 · x1
[44(x1)] = 1 · x1
[42(x1)] = 1 · x1
[23(x1)] = 1 · x1
[34(x1)] = 1 · x1
[43(x1)] = 1 · x1
[30(x1)] = 1 · x1
[35(x1)] = 1 · x1
[03(x1)] = 1 · x1
[05(x1)] = 1 · x1 + 1
[41(x1)] = 1 · x1
[15(x1)] = 1 · x1
[52(x1)] = 1 · x1
[24(x1)] = 1 · x1
[51(x1)] = 1 · x1
[50(x1)] = 1 · x1
[13(x1)] = 1 · x1
[12(x1)] = 1 · x1
[22(x1)] = 1 · x1
all of the following rules can be deleted.
53(33(31(10(02(25(53(32(x1)))))))) → 55(55(54(44(42(23(32(23(34(43(32(x1))))))))))) (399)
53(33(31(10(02(25(53(34(x1)))))))) → 55(55(54(44(42(23(32(23(34(43(34(x1))))))))))) (400)
53(33(31(10(02(25(53(31(x1)))))))) → 55(55(54(44(42(23(32(23(34(43(31(x1))))))))))) (401)
53(33(31(10(02(25(53(30(x1)))))))) → 55(55(54(44(42(23(32(23(34(43(30(x1))))))))))) (402)
53(33(31(10(02(25(53(33(x1)))))))) → 55(55(54(44(42(23(32(23(34(43(33(x1))))))))))) (403)
53(33(31(10(02(25(53(35(x1)))))))) → 55(55(54(44(42(23(32(23(34(43(35(x1))))))))))) (404)

1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[23(x1)] = 1 · x1
[33(x1)] = 1 · x1
[31(x1)] = 1 · x1
[10(x1)] = 1 · x1 + 1
[02(x1)] = 1 · x1
[25(x1)] = 1 · x1
[53(x1)] = 1 · x1
[32(x1)] = 1 · x1
[55(x1)] = 1 · x1
[54(x1)] = 1 · x1
[44(x1)] = 1 · x1
[42(x1)] = 1 · x1
[34(x1)] = 1 · x1
[43(x1)] = 1 · x1
[30(x1)] = 1 · x1
[35(x1)] = 1 · x1
[03(x1)] = 1 · x1
[05(x1)] = 1 · x1 + 1
[41(x1)] = 1 · x1
[15(x1)] = 1 · x1
[52(x1)] = 1 · x1
[24(x1)] = 1 · x1
[51(x1)] = 1 · x1
[50(x1)] = 1 · x1
[13(x1)] = 1 · x1
[12(x1)] = 1 · x1
[22(x1)] = 1 · x1
all of the following rules can be deleted.
23(33(31(10(02(25(53(32(x1)))))))) → 25(55(54(44(42(23(32(23(34(43(32(x1))))))))))) (405)
23(33(31(10(02(25(53(34(x1)))))))) → 25(55(54(44(42(23(32(23(34(43(34(x1))))))))))) (406)
23(33(31(10(02(25(53(31(x1)))))))) → 25(55(54(44(42(23(32(23(34(43(31(x1))))))))))) (407)
23(33(31(10(02(25(53(30(x1)))))))) → 25(55(54(44(42(23(32(23(34(43(30(x1))))))))))) (408)
23(33(31(10(02(25(53(33(x1)))))))) → 25(55(54(44(42(23(32(23(34(43(33(x1))))))))))) (409)
23(33(31(10(02(25(53(35(x1)))))))) → 25(55(54(44(42(23(32(23(34(43(35(x1))))))))))) (410)

1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[03(x1)] = 1 · x1
[33(x1)] = 1 · x1
[31(x1)] = 1 · x1
[10(x1)] = 1 · x1 + 1
[02(x1)] = 1 · x1
[25(x1)] = 1 · x1
[53(x1)] = 1 · x1
[32(x1)] = 1 · x1
[05(x1)] = 1 · x1
[55(x1)] = 1 · x1
[54(x1)] = 1 · x1
[44(x1)] = 1 · x1
[42(x1)] = 1 · x1
[23(x1)] = 1 · x1
[34(x1)] = 1 · x1
[43(x1)] = 1 · x1
[30(x1)] = 1 · x1
[35(x1)] = 1 · x1
[41(x1)] = 1 · x1
[15(x1)] = 1 · x1
[52(x1)] = 1 · x1
[24(x1)] = 1 · x1
[51(x1)] = 1 · x1
[50(x1)] = 1 · x1
[13(x1)] = 1 · x1
[12(x1)] = 1 · x1
[22(x1)] = 1 · x1
all of the following rules can be deleted.
03(33(31(10(02(25(53(32(x1)))))))) → 05(55(54(44(42(23(32(23(34(43(32(x1))))))))))) (411)
03(33(31(10(02(25(53(34(x1)))))))) → 05(55(54(44(42(23(32(23(34(43(34(x1))))))))))) (412)
03(33(31(10(02(25(53(31(x1)))))))) → 05(55(54(44(42(23(32(23(34(43(31(x1))))))))))) (413)
03(33(31(10(02(25(53(30(x1)))))))) → 05(55(54(44(42(23(32(23(34(43(30(x1))))))))))) (414)
03(33(31(10(02(25(53(33(x1)))))))) → 05(55(54(44(42(23(32(23(34(43(33(x1))))))))))) (415)
03(33(31(10(02(25(53(35(x1)))))))) → 05(55(54(44(42(23(32(23(34(43(35(x1))))))))))) (416)

1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[33(x1)] = 1 · x1 + 1
[35(x1)] = 1 · x1
[54(x1)] = 1 · x1
[41(x1)] = 1 · x1
[15(x1)] = 1 · x1
[52(x1)] = 1 · x1
[32(x1)] = 1 · x1
[23(x1)] = 1 · x1
[24(x1)] = 1 · x1 + 2
[44(x1)] = 1 · x1
[43(x1)] = 1 · x1
[51(x1)] = 1 · x1
[50(x1)] = 1 · x1
[53(x1)] = 1 · x1
[55(x1)] = 1 · x1
[13(x1)] = 1 · x1
[12(x1)] = 1 · x1
[42(x1)] = 1 · x1
[22(x1)] = 1 · x1
[03(x1)] = 1 · x1
[02(x1)] = 1 · x1
all of the following rules can be deleted.
33(33(33(35(54(41(15(52(x1)))))))) → 32(23(32(24(44(43(35(54(41(15(52(x1))))))))))) (417)
33(33(33(35(54(41(15(54(x1)))))))) → 32(23(32(24(44(43(35(54(41(15(54(x1))))))))))) (418)
33(33(33(35(54(41(15(51(x1)))))))) → 32(23(32(24(44(43(35(54(41(15(51(x1))))))))))) (419)
33(33(33(35(54(41(15(50(x1)))))))) → 32(23(32(24(44(43(35(54(41(15(50(x1))))))))))) (420)
33(33(33(35(54(41(15(53(x1)))))))) → 32(23(32(24(44(43(35(54(41(15(53(x1))))))))))) (421)
33(33(33(35(54(41(15(55(x1)))))))) → 32(23(32(24(44(43(35(54(41(15(55(x1))))))))))) (422)

1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[13(x1)] = 1 · x1 + 1
[33(x1)] = 1 · x1
[35(x1)] = 1 · x1
[54(x1)] = 1 · x1
[41(x1)] = 1 · x1
[15(x1)] = 1 · x1
[52(x1)] = 1 · x1
[12(x1)] = 1 · x1
[23(x1)] = 1 · x1
[32(x1)] = 1 · x1
[24(x1)] = 1 · x1
[44(x1)] = 1 · x1
[43(x1)] = 1 · x1
[51(x1)] = 1 · x1
[50(x1)] = 1 · x1
[53(x1)] = 1 · x1
[55(x1)] = 1 · x1
[42(x1)] = 1 · x1
[22(x1)] = 1 · x1
[03(x1)] = 1 · x1
[02(x1)] = 1 · x1
all of the following rules can be deleted.
13(33(33(35(54(41(15(52(x1)))))))) → 12(23(32(24(44(43(35(54(41(15(52(x1))))))))))) (423)
13(33(33(35(54(41(15(54(x1)))))))) → 12(23(32(24(44(43(35(54(41(15(54(x1))))))))))) (424)
13(33(33(35(54(41(15(51(x1)))))))) → 12(23(32(24(44(43(35(54(41(15(51(x1))))))))))) (425)
13(33(33(35(54(41(15(50(x1)))))))) → 12(23(32(24(44(43(35(54(41(15(50(x1))))))))))) (426)
13(33(33(35(54(41(15(53(x1)))))))) → 12(23(32(24(44(43(35(54(41(15(53(x1))))))))))) (427)
13(33(33(35(54(41(15(55(x1)))))))) → 12(23(32(24(44(43(35(54(41(15(55(x1))))))))))) (428)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[43(x1)] = 1 · x1
[33(x1)] = 1 · x1 + 1
[35(x1)] = 1 · x1
[54(x1)] = 1 · x1
[41(x1)] = 1 · x1
[15(x1)] = 1 · x1
[52(x1)] = 1 · x1 + 1
[42(x1)] = 1 · x1
[23(x1)] = 1 · x1
[32(x1)] = 1 · x1 + 1
[24(x1)] = 1 · x1
[44(x1)] = 1 · x1
[51(x1)] = 1 · x1
[50(x1)] = 1 · x1
[53(x1)] = 1 · x1
[55(x1)] = 1 · x1
[22(x1)] = 1 · x1 + 1
[03(x1)] = 1 · x1
[02(x1)] = 1 · x1 + 1
all of the following rules can be deleted.
43(33(33(35(54(41(15(52(x1)))))))) → 42(23(32(24(44(43(35(54(41(15(52(x1))))))))))) (429)
43(33(33(35(54(41(15(54(x1)))))))) → 42(23(32(24(44(43(35(54(41(15(54(x1))))))))))) (430)
43(33(33(35(54(41(15(51(x1)))))))) → 42(23(32(24(44(43(35(54(41(15(51(x1))))))))))) (431)
43(33(33(35(54(41(15(50(x1)))))))) → 42(23(32(24(44(43(35(54(41(15(50(x1))))))))))) (432)
43(33(33(35(54(41(15(53(x1)))))))) → 42(23(32(24(44(43(35(54(41(15(53(x1))))))))))) (433)
43(33(33(35(54(41(15(55(x1)))))))) → 42(23(32(24(44(43(35(54(41(15(55(x1))))))))))) (434)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[53(x1)] = 1 · x1 + 1
[33(x1)] = 1 · x1
[35(x1)] = 1 · x1
[54(x1)] = 1 · x1
[41(x1)] = 1 · x1
[15(x1)] = 1 · x1
[52(x1)] = 1 · x1
[23(x1)] = 1 · x1
[32(x1)] = 1 · x1
[24(x1)] = 1 · x1
[44(x1)] = 1 · x1
[43(x1)] = 1 · x1
[51(x1)] = 1 · x1
[50(x1)] = 1 · x1
[55(x1)] = 1 · x1
[22(x1)] = 1 · x1
[03(x1)] = 1 · x1
[02(x1)] = 1 · x1
all of the following rules can be deleted.
53(33(33(35(54(41(15(52(x1)))))))) → 52(23(32(24(44(43(35(54(41(15(52(x1))))))))))) (435)
53(33(33(35(54(41(15(54(x1)))))))) → 52(23(32(24(44(43(35(54(41(15(54(x1))))))))))) (436)
53(33(33(35(54(41(15(51(x1)))))))) → 52(23(32(24(44(43(35(54(41(15(51(x1))))))))))) (437)
53(33(33(35(54(41(15(50(x1)))))))) → 52(23(32(24(44(43(35(54(41(15(50(x1))))))))))) (438)
53(33(33(35(54(41(15(53(x1)))))))) → 52(23(32(24(44(43(35(54(41(15(53(x1))))))))))) (439)
53(33(33(35(54(41(15(55(x1)))))))) → 52(23(32(24(44(43(35(54(41(15(55(x1))))))))))) (440)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[23(x1)] = 1 · x1
[33(x1)] = 1 · x1 + 1
[35(x1)] = 1 · x1
[54(x1)] = 1 · x1
[41(x1)] = 1 · x1
[15(x1)] = 1 · x1
[52(x1)] = 1 · x1
[22(x1)] = 1 · x1
[32(x1)] = 1 · x1
[24(x1)] = 1 · x1
[44(x1)] = 1 · x1
[43(x1)] = 1 · x1 + 1
[51(x1)] = 1 · x1
[50(x1)] = 1 · x1
[53(x1)] = 1 · x1
[55(x1)] = 1 · x1
[03(x1)] = 1 · x1
[02(x1)] = 1 · x1 + 1
all of the following rules can be deleted.
23(33(33(35(54(41(15(52(x1)))))))) → 22(23(32(24(44(43(35(54(41(15(52(x1))))))))))) (441)
23(33(33(35(54(41(15(54(x1)))))))) → 22(23(32(24(44(43(35(54(41(15(54(x1))))))))))) (442)
23(33(33(35(54(41(15(51(x1)))))))) → 22(23(32(24(44(43(35(54(41(15(51(x1))))))))))) (443)
23(33(33(35(54(41(15(50(x1)))))))) → 22(23(32(24(44(43(35(54(41(15(50(x1))))))))))) (444)
23(33(33(35(54(41(15(53(x1)))))))) → 22(23(32(24(44(43(35(54(41(15(53(x1))))))))))) (445)
23(33(33(35(54(41(15(55(x1)))))))) → 22(23(32(24(44(43(35(54(41(15(55(x1))))))))))) (446)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[03(x1)] = 1 · x1 + 1
[33(x1)] = 1 · x1
[35(x1)] = 1 · x1
[54(x1)] = 1 · x1
[41(x1)] = 1 · x1
[15(x1)] = 1 · x1
[52(x1)] = 1 · x1
[02(x1)] = 1 · x1
[23(x1)] = 1 · x1
[32(x1)] = 1 · x1
[24(x1)] = 1 · x1
[44(x1)] = 1 · x1
[43(x1)] = 1 · x1
[51(x1)] = 1 · x1
[50(x1)] = 1 · x1
[53(x1)] = 1 · x1
[55(x1)] = 1 · x1
all of the following rules can be deleted.
03(33(33(35(54(41(15(52(x1)))))))) → 02(23(32(24(44(43(35(54(41(15(52(x1))))))))))) (447)
03(33(33(35(54(41(15(54(x1)))))))) → 02(23(32(24(44(43(35(54(41(15(54(x1))))))))))) (448)
03(33(33(35(54(41(15(51(x1)))))))) → 02(23(32(24(44(43(35(54(41(15(51(x1))))))))))) (449)
03(33(33(35(54(41(15(50(x1)))))))) → 02(23(32(24(44(43(35(54(41(15(50(x1))))))))))) (450)
03(33(33(35(54(41(15(53(x1)))))))) → 02(23(32(24(44(43(35(54(41(15(53(x1))))))))))) (451)
03(33(33(35(54(41(15(55(x1)))))))) → 02(23(32(24(44(43(35(54(41(15(55(x1))))))))))) (452)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 R is empty

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