Certification Problem

Input (TPDB SRS_Relative/ICFP_2010_relative/3831)

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

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

and S is the following TRS.

0(3(4(4(5(x1))))) 0(2(2(2(1(2(4(2(4(5(x1)))))))))) (52)
5(5(3(5(3(4(5(x1))))))) 5(0(1(4(2(4(5(1(4(5(x1)))))))))) (53)
5(1(4(1(5(2(x1)))))) 5(2(2(4(2(2(1(0(5(2(x1)))))))))) (54)
3(0(5(4(x1)))) 5(2(2(2(5(2(2(0(5(4(x1)))))))))) (55)
3(2(3(0(3(x1))))) 3(4(3(3(4(2(1(1(3(1(x1)))))))))) (56)
3(2(5(4(4(5(0(x1))))))) 3(2(0(0(0(0(5(3(4(0(x1)))))))))) (57)
4(4(1(2(0(4(x1)))))) 5(5(4(3(4(2(3(1(3(1(x1)))))))))) (58)
1(1(3(5(4(5(0(x1))))))) 3(2(2(4(2(1(4(1(0(0(x1)))))))))) (59)
3(4(5(4(x1)))) 3(4(2(2(3(3(4(2(2(1(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
2(1(1(0(x1)))) 2(2(2(4(2(2(4(3(1(0(x1)))))))))) (61)
4(1(0(4(x1)))) 4(2(3(3(3(1(1(3(2(4(x1)))))))))) (62)
5(4(0(5(3(x1))))) 4(3(1(1(3(1(3(3(3(0(x1)))))))))) (63)
3(0(4(0(2(0(x1)))))) 1(1(3(2(2(4(3(2(2(0(x1)))))))))) (64)
2(1(1(0(4(0(x1)))))) 2(3(2(4(2(4(2(4(0(0(x1)))))))))) (65)
5(3(2(3(4(0(x1)))))) 5(4(2(2(1(2(5(0(5(5(x1)))))))))) (66)
4(0(4(4(4(0(x1)))))) 4(2(4(1(4(3(5(4(5(0(x1)))))))))) (67)
2(1(0(5(1(1(x1)))))) 2(5(4(5(0(5(2(2(2(2(x1)))))))))) (68)
3(0(2(3(2(1(x1)))))) 1(3(1(3(3(3(1(3(4(2(x1)))))))))) (69)
3(2(0(3(0(2(x1)))))) 3(2(0(5(0(0(0(1(4(2(x1)))))))))) (70)
2(3(4(4(1(2(x1)))))) 2(3(2(4(1(4(3(3(2(2(x1)))))))))) (71)
4(5(3(0(3(3(x1)))))) 4(5(1(4(2(2(2(4(3(3(x1)))))))))) (72)
1(3(4(5(3(3(x1)))))) 3(1(3(1(5(1(2(4(2(3(x1)))))))))) (73)
2(1(5(1(5(3(x1)))))) 2(2(3(4(1(1(2(2(4(3(x1)))))))))) (74)
0(1(1(3(5(3(x1)))))) 0(3(1(3(2(1(2(4(1(4(x1)))))))))) (75)
2(4(4(0(1(4(x1)))))) 2(2(4(1(3(3(2(3(2(4(x1)))))))))) (76)
3(3(0(2(3(4(x1)))))) 1(3(1(1(3(1(4(3(5(1(x1)))))))))) (77)
2(0(2(3(5(4(x1)))))) 1(3(1(5(2(1(2(1(2(4(x1)))))))))) (78)
2(3(0(2(1(1(0(x1))))))) 2(3(1(4(2(5(0(2(2(0(x1)))))))))) (79)
3(1(5(2(0(3(0(x1))))))) 3(2(3(1(1(4(5(0(5(5(x1)))))))))) (80)
4(5(0(3(0(3(0(x1))))))) 4(3(3(3(3(5(0(0(2(0(x1)))))))))) (81)
0(3(1(1(5(3(0(x1))))))) 0(1(1(4(1(5(5(0(5(5(x1)))))))))) (82)
3(5(2(1(0(4(0(x1))))))) 1(4(3(3(3(3(1(2(5(0(x1)))))))))) (83)
5(4(3(4(0(5(0(x1))))))) 5(3(2(2(4(5(0(2(5(0(x1)))))))))) (84)
4(5(2(3(0(2(1(x1))))))) 4(2(1(2(0(0(5(5(1(1(x1)))))))))) (85)
5(1(1(0(0(3(1(x1))))))) 5(5(2(5(5(2(2(4(1(1(x1)))))))))) (86)
2(0(4(0(4(3(1(x1))))))) 2(2(2(5(2(4(1(4(4(3(x1)))))))))) (87)
2(3(2(5(1(5(1(x1))))))) 2(1(1(1(3(3(2(0(2(5(x1)))))))))) (88)
0(5(0(3(2(5(1(x1))))))) 0(3(1(2(4(2(4(2(5(1(x1)))))))))) (89)
5(3(5(2(3(5(1(x1))))))) 5(0(0(2(2(5(3(4(4(2(x1)))))))))) (90)
2(3(5(4(4(5(1(x1))))))) 2(5(0(2(2(2(5(0(2(1(x1)))))))))) (91)
4(4(5(0(3(0(2(x1))))))) 1(3(3(0(1(1(2(2(5(2(x1)))))))))) (92)
4(4(4(4(3(0(2(x1))))))) 4(3(4(3(1(3(2(2(2(5(x1)))))))))) (93)
4(3(0(2(0(3(2(x1))))))) 1(2(4(1(0(0(0(5(3(2(x1)))))))))) (94)
5(0(0(1(5(3(2(x1))))))) 2(2(5(0(5(0(0(0(2(1(x1)))))))))) (95)
4(5(0(3(2(0(3(x1))))))) 4(2(2(5(0(0(0(3(2(2(x1)))))))))) (96)
2(1(0(2(3(0(3(x1))))))) 2(1(2(4(3(4(2(5(3(1(x1)))))))))) (97)
2(0(4(4(4(0(3(x1))))))) 2(2(2(2(1(1(3(3(1(3(x1)))))))))) (98)
0(3(3(1(0(2(3(x1))))))) 0(2(1(2(4(2(0(5(2(2(x1)))))))))) (99)
0(4(4(0(3(4(3(x1))))))) 0(4(3(0(2(1(2(2(4(3(x1)))))))))) (100)
5(3(4(4(4(4(3(x1))))))) 5(1(1(2(4(1(5(4(3(3(x1)))))))))) (101)
4(2(3(0(0(5(3(x1))))))) 4(3(5(5(5(0(0(0(1(2(x1)))))))))) (102)
4(3(3(0(4(5(3(x1))))))) 1(2(4(2(2(3(3(1(3(1(x1)))))))))) (103)
0(3(1(1(0(2(4(x1))))))) 0(2(4(1(3(2(2(5(2(4(x1)))))))))) (104)
0(4(3(5(4(4(4(x1))))))) 0(0(4(2(4(2(4(2(5(1(x1)))))))))) (105)
4(2(3(5(5(4(4(x1))))))) 1(2(1(2(4(5(4(1(2(5(x1)))))))))) (106)
0(3(0(3(0(5(4(x1))))))) 1(5(2(0(0(0(1(4(1(4(x1)))))))))) (107)
0(0(3(4(0(5(4(x1))))))) 0(1(0(3(3(1(1(2(2(4(x1)))))))))) (108)
3(5(3(1(1(3(5(x1))))))) 1(4(2(4(2(4(5(5(0(0(x1)))))))))) (109)
3(1(0(2(2(3(5(x1))))))) 1(3(3(3(5(5(5(5(0(0(x1)))))))))) (110)
0(2(4(0(1(4(5(x1))))))) 1(4(2(4(1(0(0(0(1(5(x1)))))))))) (111)
5(4(4(3(0(x1))))) 5(4(2(4(2(1(2(2(2(0(x1)))))))))) (112)
5(4(3(5(3(5(5(x1))))))) 5(4(1(5(4(2(4(1(0(5(x1)))))))))) (113)
2(5(1(4(1(5(x1)))))) 2(5(0(1(2(2(4(2(2(5(x1)))))))))) (114)
4(5(0(3(x1)))) 4(5(0(2(2(5(2(2(2(5(x1)))))))))) (115)
3(0(3(2(3(x1))))) 1(3(1(1(2(4(3(3(4(3(x1)))))))))) (116)
0(5(4(4(5(2(3(x1))))))) 0(4(3(5(0(0(0(0(2(3(x1)))))))))) (117)
4(0(2(1(4(4(x1)))))) 1(3(1(3(2(4(3(4(5(5(x1)))))))))) (118)
0(5(4(5(3(1(1(x1))))))) 0(0(1(4(1(2(4(2(2(3(x1)))))))))) (119)
4(5(4(3(x1)))) 1(2(2(4(3(3(2(2(4(3(x1)))))))))) (120)

1.1 Closure Under Flat Contexts

Using the flat contexts

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

We obtain the transformed TRS

There are 155 ruless (increase limit for explicit display).

1.1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

There are 930 ruless (increase limit for explicit display).

1.1.1.1 Rule Removal

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

There are 799 ruless (increase limit for explicit display).

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[21(x1)] = 1 + 1 · x1
[11(x1)] = 1 + 1 · x1
[10(x1)] = 1 + 1 · x1
[02(x1)] = 1 · x1
[22(x1)] = 1 · x1
[24(x1)] = 1 · x1
[42(x1)] = 1 · x1
[43(x1)] = 1 · x1
[31(x1)] = 1 · x1
[01(x1)] = 1 · x1
[00(x1)] = 1 · x1
[04(x1)] = 1 · x1
[03(x1)] = 1 · x1
[05(x1)] = 1 + 1 · x1
[51(x1)] = 1 · x1
[12(x1)] = 1 + 1 · x1
[25(x1)] = 1 · x1
[54(x1)] = 1 · x1
[45(x1)] = 1 + 1 · x1
[50(x1)] = 1 · x1
[52(x1)] = 1 · x1
[20(x1)] = 1 · x1
[14(x1)] = 1 · x1
[13(x1)] = 1 · x1
[23(x1)] = 1 + 1 · x1
[15(x1)] = 1 · x1
[30(x1)] = 1 + 1 · x1
[55(x1)] = 1 · x1
[32(x1)] = 1 · x1
[34(x1)] = 1 + 1 · x1
[53(x1)] = 1 · x1
[33(x1)] = 1 · x1
[40(x1)] = 1 + 1 · x1
[44(x1)] = 1 + 1 · x1
[41(x1)] = 1 · x1
[35(x1)] = 1 · x1
all of the following rules can be deleted.
21(11(10(02(x1)))) 22(22(24(42(22(24(43(31(10(02(x1)))))))))) (235)
21(11(10(01(x1)))) 22(22(24(42(22(24(43(31(10(01(x1)))))))))) (236)
21(11(10(00(x1)))) 22(22(24(42(22(24(43(31(10(00(x1)))))))))) (237)
21(11(10(04(x1)))) 22(22(24(42(22(24(43(31(10(04(x1)))))))))) (238)
21(11(10(03(x1)))) 22(22(24(42(22(24(43(31(10(03(x1)))))))))) (239)
21(11(10(05(x1)))) 22(22(24(42(22(24(43(31(10(05(x1)))))))))) (240)
21(10(05(51(11(12(x1)))))) 25(54(45(50(05(52(22(22(22(22(x1)))))))))) (265)
21(10(05(51(11(11(x1)))))) 25(54(45(50(05(52(22(22(22(21(x1)))))))))) (266)
21(10(05(51(11(10(x1)))))) 25(54(45(50(05(52(22(22(22(20(x1)))))))))) (267)
21(10(05(51(11(14(x1)))))) 25(54(45(50(05(52(22(22(22(24(x1)))))))))) (268)
21(10(05(51(11(13(x1)))))) 25(54(45(50(05(52(22(22(22(23(x1)))))))))) (269)
21(10(05(51(11(15(x1)))))) 25(54(45(50(05(52(22(22(22(25(x1)))))))))) (270)
21(10(02(23(30(03(33(x1))))))) 21(12(24(43(34(42(25(53(31(13(x1)))))))))) (395)
05(54(40(05(53(33(x1)))))) 04(43(31(11(13(31(13(33(33(30(03(x1))))))))))) (461)
23(30(02(23(32(21(12(x1))))))) 21(13(31(13(33(33(31(13(34(42(22(x1))))))))))) (517)
23(30(02(23(32(21(11(x1))))))) 21(13(31(13(33(33(31(13(34(42(21(x1))))))))))) (518)
23(30(02(23(32(21(10(x1))))))) 21(13(31(13(33(33(31(13(34(42(20(x1))))))))))) (519)
23(30(02(23(32(21(14(x1))))))) 21(13(31(13(33(33(31(13(34(42(24(x1))))))))))) (520)
23(30(02(23(32(21(13(x1))))))) 21(13(31(13(33(33(31(13(34(42(23(x1))))))))))) (521)
23(30(02(23(32(21(15(x1))))))) 21(13(31(13(33(33(31(13(34(42(25(x1))))))))))) (522)
13(30(02(23(32(21(12(x1))))))) 11(13(31(13(33(33(31(13(34(42(22(x1))))))))))) (523)
13(30(02(23(32(21(11(x1))))))) 11(13(31(13(33(33(31(13(34(42(21(x1))))))))))) (524)
13(30(02(23(32(21(10(x1))))))) 11(13(31(13(33(33(31(13(34(42(20(x1))))))))))) (525)
13(30(02(23(32(21(14(x1))))))) 11(13(31(13(33(33(31(13(34(42(24(x1))))))))))) (526)
13(30(02(23(32(21(15(x1))))))) 11(13(31(13(33(33(31(13(34(42(25(x1))))))))))) (528)
43(30(02(23(32(21(12(x1))))))) 41(13(31(13(33(33(31(13(34(42(22(x1))))))))))) (535)
43(30(02(23(32(21(11(x1))))))) 41(13(31(13(33(33(31(13(34(42(21(x1))))))))))) (536)
43(30(02(23(32(21(10(x1))))))) 41(13(31(13(33(33(31(13(34(42(20(x1))))))))))) (537)
43(30(02(23(32(21(14(x1))))))) 41(13(31(13(33(33(31(13(34(42(24(x1))))))))))) (538)
43(30(02(23(32(21(13(x1))))))) 41(13(31(13(33(33(31(13(34(42(23(x1))))))))))) (539)
43(30(02(23(32(21(15(x1))))))) 41(13(31(13(33(33(31(13(34(42(25(x1))))))))))) (540)
33(30(02(23(32(21(12(x1))))))) 31(13(31(13(33(33(31(13(34(42(22(x1))))))))))) (541)
33(30(02(23(32(21(11(x1))))))) 31(13(31(13(33(33(31(13(34(42(21(x1))))))))))) (542)
33(30(02(23(32(21(10(x1))))))) 31(13(31(13(33(33(31(13(34(42(20(x1))))))))))) (543)
33(30(02(23(32(21(14(x1))))))) 31(13(31(13(33(33(31(13(34(42(24(x1))))))))))) (544)
33(30(02(23(32(21(13(x1))))))) 31(13(31(13(33(33(31(13(34(42(23(x1))))))))))) (545)
33(30(02(23(32(21(15(x1))))))) 31(13(31(13(33(33(31(13(34(42(25(x1))))))))))) (546)
53(30(02(23(32(21(12(x1))))))) 51(13(31(13(33(33(31(13(34(42(22(x1))))))))))) (547)
53(30(02(23(32(21(11(x1))))))) 51(13(31(13(33(33(31(13(34(42(21(x1))))))))))) (548)
53(30(02(23(32(21(10(x1))))))) 51(13(31(13(33(33(31(13(34(42(20(x1))))))))))) (549)
53(30(02(23(32(21(14(x1))))))) 51(13(31(13(33(33(31(13(34(42(24(x1))))))))))) (550)
53(30(02(23(32(21(13(x1))))))) 51(13(31(13(33(33(31(13(34(42(23(x1))))))))))) (551)
53(30(02(23(32(21(15(x1))))))) 51(13(31(13(33(33(31(13(34(42(25(x1))))))))))) (552)
23(33(30(02(23(34(42(x1))))))) 21(13(31(11(13(31(14(43(35(51(12(x1))))))))))) (589)
23(33(30(02(23(34(41(x1))))))) 21(13(31(11(13(31(14(43(35(51(11(x1))))))))))) (590)
23(33(30(02(23(34(43(x1))))))) 21(13(31(11(13(31(14(43(35(51(13(x1))))))))))) (593)
23(33(30(02(23(34(45(x1))))))) 21(13(31(11(13(31(14(43(35(51(15(x1))))))))))) (594)
13(33(30(02(23(34(43(x1))))))) 11(13(31(11(13(31(14(43(35(51(13(x1))))))))))) (599)
13(33(30(02(23(34(45(x1))))))) 11(13(31(11(13(31(14(43(35(51(15(x1))))))))))) (600)
43(33(30(02(23(34(42(x1))))))) 41(13(31(11(13(31(14(43(35(51(12(x1))))))))))) (607)
43(33(30(02(23(34(41(x1))))))) 41(13(31(11(13(31(14(43(35(51(11(x1))))))))))) (608)
43(33(30(02(23(34(43(x1))))))) 41(13(31(11(13(31(14(43(35(51(13(x1))))))))))) (611)
43(33(30(02(23(34(45(x1))))))) 41(13(31(11(13(31(14(43(35(51(15(x1))))))))))) (612)
33(33(30(02(23(34(42(x1))))))) 31(13(31(11(13(31(14(43(35(51(12(x1))))))))))) (613)
33(33(30(02(23(34(41(x1))))))) 31(13(31(11(13(31(14(43(35(51(11(x1))))))))))) (614)
33(33(30(02(23(34(43(x1))))))) 31(13(31(11(13(31(14(43(35(51(13(x1))))))))))) (617)
33(33(30(02(23(34(45(x1))))))) 31(13(31(11(13(31(14(43(35(51(15(x1))))))))))) (618)
53(33(30(02(23(34(42(x1))))))) 51(13(31(11(13(31(14(43(35(51(12(x1))))))))))) (619)
53(33(30(02(23(34(41(x1))))))) 51(13(31(11(13(31(14(43(35(51(11(x1))))))))))) (620)
53(33(30(02(23(34(43(x1))))))) 51(13(31(11(13(31(14(43(35(51(13(x1))))))))))) (623)
53(33(30(02(23(34(45(x1))))))) 51(13(31(11(13(31(14(43(35(51(15(x1))))))))))) (624)
23(31(10(02(22(23(35(52(x1)))))))) 21(13(33(33(35(55(55(55(50(00(02(x1))))))))))) (949)
23(31(10(02(22(23(35(50(x1)))))))) 21(13(33(33(35(55(55(55(50(00(00(x1))))))))))) (951)
23(31(10(02(22(23(35(54(x1)))))))) 21(13(33(33(35(55(55(55(50(00(04(x1))))))))))) (952)
23(31(10(02(22(23(35(53(x1)))))))) 21(13(33(33(35(55(55(55(50(00(03(x1))))))))))) (953)
23(31(10(02(22(23(35(55(x1)))))))) 21(13(33(33(35(55(55(55(50(00(05(x1))))))))))) (954)
13(31(10(02(22(23(35(52(x1)))))))) 11(13(33(33(35(55(55(55(50(00(02(x1))))))))))) (955)
13(31(10(02(22(23(35(50(x1)))))))) 11(13(33(33(35(55(55(55(50(00(00(x1))))))))))) (957)
13(31(10(02(22(23(35(54(x1)))))))) 11(13(33(33(35(55(55(55(50(00(04(x1))))))))))) (958)
13(31(10(02(22(23(35(53(x1)))))))) 11(13(33(33(35(55(55(55(50(00(03(x1))))))))))) (959)
43(31(10(02(22(23(35(52(x1)))))))) 41(13(33(33(35(55(55(55(50(00(02(x1))))))))))) (967)
43(31(10(02(22(23(35(50(x1)))))))) 41(13(33(33(35(55(55(55(50(00(00(x1))))))))))) (969)
43(31(10(02(22(23(35(54(x1)))))))) 41(13(33(33(35(55(55(55(50(00(04(x1))))))))))) (970)
43(31(10(02(22(23(35(53(x1)))))))) 41(13(33(33(35(55(55(55(50(00(03(x1))))))))))) (971)
43(31(10(02(22(23(35(55(x1)))))))) 41(13(33(33(35(55(55(55(50(00(05(x1))))))))))) (972)
33(31(10(02(22(23(35(52(x1)))))))) 31(13(33(33(35(55(55(55(50(00(02(x1))))))))))) (973)
33(31(10(02(22(23(35(50(x1)))))))) 31(13(33(33(35(55(55(55(50(00(00(x1))))))))))) (975)
33(31(10(02(22(23(35(54(x1)))))))) 31(13(33(33(35(55(55(55(50(00(04(x1))))))))))) (976)
33(31(10(02(22(23(35(53(x1)))))))) 31(13(33(33(35(55(55(55(50(00(03(x1))))))))))) (977)
33(31(10(02(22(23(35(55(x1)))))))) 31(13(33(33(35(55(55(55(50(00(05(x1))))))))))) (978)
53(31(10(02(22(23(35(52(x1)))))))) 51(13(33(33(35(55(55(55(50(00(02(x1))))))))))) (979)
53(31(10(02(22(23(35(50(x1)))))))) 51(13(33(33(35(55(55(55(50(00(00(x1))))))))))) (981)
53(31(10(02(22(23(35(54(x1)))))))) 51(13(33(33(35(55(55(55(50(00(04(x1))))))))))) (982)
53(31(10(02(22(23(35(53(x1)))))))) 51(13(33(33(35(55(55(55(50(00(03(x1))))))))))) (983)
53(31(10(02(22(23(35(55(x1)))))))) 51(13(33(33(35(55(55(55(50(00(05(x1))))))))))) (984)
50(02(24(40(01(14(45(52(x1)))))))) 51(14(42(24(41(10(00(00(01(15(52(x1))))))))))) (1015)
50(02(24(40(01(14(45(51(x1)))))))) 51(14(42(24(41(10(00(00(01(15(51(x1))))))))))) (1016)
50(02(24(40(01(14(45(50(x1)))))))) 51(14(42(24(41(10(00(00(01(15(50(x1))))))))))) (1017)
50(02(24(40(01(14(45(54(x1)))))))) 51(14(42(24(41(10(00(00(01(15(54(x1))))))))))) (1018)
50(02(24(40(01(14(45(53(x1)))))))) 51(14(42(24(41(10(00(00(01(15(53(x1))))))))))) (1019)
50(02(24(40(01(14(45(55(x1)))))))) 51(14(42(24(41(10(00(00(01(15(55(x1))))))))))) (1020)
05(54(44(45(52(23(32(x1))))))) 04(43(35(50(00(00(00(02(23(32(x1)))))))))) (1045)
05(54(44(45(52(23(31(x1))))))) 04(43(35(50(00(00(00(02(23(31(x1)))))))))) (1046)
05(54(44(45(52(23(30(x1))))))) 04(43(35(50(00(00(00(02(23(30(x1)))))))))) (1047)
05(54(44(45(52(23(34(x1))))))) 04(43(35(50(00(00(00(02(23(34(x1)))))))))) (1048)
05(54(44(45(52(23(33(x1))))))) 04(43(35(50(00(00(00(02(23(33(x1)))))))))) (1049)
05(54(44(45(52(23(35(x1))))))) 04(43(35(50(00(00(00(02(23(35(x1)))))))))) (1050)
54(40(02(21(14(44(41(x1))))))) 51(13(31(13(32(24(43(34(45(55(51(x1))))))))))) (1124)
54(40(02(21(14(44(43(x1))))))) 51(13(31(13(32(24(43(34(45(55(53(x1))))))))))) (1127)

1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[45(x1)] = 1 · x1
[52(x1)] = 1 · x1
[23(x1)] = 1 · x1
[30(x1)] = 1 · x1
[02(x1)] = 1 · x1
[21(x1)] = 1 · x1
[12(x1)] = 1 · x1
[42(x1)] = 1 · x1
[20(x1)] = 1 · x1
[00(x1)] = 1 · x1
[05(x1)] = 1 · x1
[55(x1)] = 1 · x1
[51(x1)] = 1 · x1
[11(x1)] = 1 · x1
[10(x1)] = 1 · x1
[14(x1)] = 1 · x1
[13(x1)] = 1 · x1
[15(x1)] = 1 · x1
[50(x1)] = 1 · x1
[03(x1)] = 1 · x1
[32(x1)] = 1 · x1
[25(x1)] = 1 · x1
[31(x1)] = 1 · x1
[24(x1)] = 1 · x1
[43(x1)] = 1 · x1
[34(x1)] = 1 · x1
[53(x1)] = 1 · x1
[54(x1)] = 1 · x1
[40(x1)] = 1 + 1 · x1
[33(x1)] = 1 · x1
[44(x1)] = 1 · x1
[01(x1)] = 1 + 1 · x1
[35(x1)] = 1 · x1
[41(x1)] = 1 · x1
[22(x1)] = 1 · x1
all of the following rules can be deleted.
45(54(40(05(53(33(x1)))))) 44(43(31(11(13(31(13(33(33(30(03(x1))))))))))) (467)
55(54(40(05(53(33(x1)))))) 54(43(31(11(13(31(13(33(33(30(03(x1))))))))))) (479)
01(13(34(45(53(33(32(x1))))))) 03(31(13(31(15(51(12(24(42(23(32(x1))))))))))) (565)
01(13(34(45(53(33(31(x1))))))) 03(31(13(31(15(51(12(24(42(23(31(x1))))))))))) (566)
01(13(34(45(53(33(30(x1))))))) 03(31(13(31(15(51(12(24(42(23(30(x1))))))))))) (567)
01(13(34(45(53(33(34(x1))))))) 03(31(13(31(15(51(12(24(42(23(34(x1))))))))))) (568)
01(13(34(45(53(33(33(x1))))))) 03(31(13(31(15(51(12(24(42(23(33(x1))))))))))) (569)
01(13(34(45(53(33(35(x1))))))) 03(31(13(31(15(51(12(24(42(23(35(x1))))))))))) (570)
24(40(02(21(14(44(41(x1))))))) 21(13(31(13(32(24(43(34(45(55(51(x1))))))))))) (1094)
24(40(02(21(14(44(43(x1))))))) 21(13(31(13(32(24(43(34(45(55(53(x1))))))))))) (1097)
14(40(02(21(14(44(41(x1))))))) 11(13(31(13(32(24(43(34(45(55(51(x1))))))))))) (1100)
14(40(02(21(14(44(43(x1))))))) 11(13(31(13(32(24(43(34(45(55(53(x1))))))))))) (1103)

1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[45(x1)] = 1 + 1 · x1
[52(x1)] = 1 · x1
[23(x1)] = 1 + 1 · x1
[30(x1)] = 1 + 1 · x1
[02(x1)] = 1 + 1 · x1
[21(x1)] = 1 + 1 · x1
[12(x1)] = 1 · x1
[42(x1)] = 1 · x1
[20(x1)] = 1 + 1 · x1
[00(x1)] = 1 · x1
[05(x1)] = 1 · x1
[55(x1)] = 1 · x1
[51(x1)] = 1 · x1
[11(x1)] = 1 · x1
[10(x1)] = 1 · x1
[14(x1)] = 1 · x1
[13(x1)] = 1 · x1
[15(x1)] = 1 · x1
[50(x1)] = 1 · x1
[03(x1)] = 1 + 1 · x1
[32(x1)] = 1 + 1 · x1
[25(x1)] = 1 · x1
[31(x1)] = 1 + 1 · x1
[24(x1)] = 1 · x1
[43(x1)] = 1 · x1
[34(x1)] = 1 · x1
[53(x1)] = 1 · x1
[33(x1)] = 1 · x1
[35(x1)] = 1 · x1
[41(x1)] = 1 · x1
[22(x1)] = 1 · x1
all of the following rules can be deleted.
45(52(23(30(02(21(12(x1))))))) 42(21(12(20(00(05(55(51(11(12(x1)))))))))) (337)
45(52(23(30(02(21(11(x1))))))) 42(21(12(20(00(05(55(51(11(11(x1)))))))))) (338)
45(52(23(30(02(21(10(x1))))))) 42(21(12(20(00(05(55(51(11(10(x1)))))))))) (339)
45(52(23(30(02(21(14(x1))))))) 42(21(12(20(00(05(55(51(11(14(x1)))))))))) (340)
45(52(23(30(02(21(13(x1))))))) 42(21(12(20(00(05(55(51(11(13(x1)))))))))) (341)
45(52(23(30(02(21(15(x1))))))) 42(21(12(20(00(05(55(51(11(15(x1)))))))))) (342)
21(10(02(23(30(03(32(x1))))))) 21(12(24(43(34(42(25(53(31(12(x1)))))))))) (391)
21(10(02(23(30(03(31(x1))))))) 21(12(24(43(34(42(25(53(31(11(x1)))))))))) (392)
13(30(02(23(32(21(13(x1))))))) 11(13(31(13(33(33(31(13(34(42(23(x1))))))))))) (527)
13(33(30(02(23(34(42(x1))))))) 11(13(31(11(13(31(14(43(35(51(12(x1))))))))))) (595)
13(33(30(02(23(34(41(x1))))))) 11(13(31(11(13(31(14(43(35(51(11(x1))))))))))) (596)
13(31(10(02(22(23(35(55(x1)))))))) 11(13(33(33(35(55(55(55(50(00(05(x1))))))))))) (960)
45(50(03(31(x1)))) 45(50(02(22(25(52(22(22(25(51(x1)))))))))) (1040)

1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[05(x1)] = 1 + 1 · x1
[50(x1)] = 1 + 1 · x1
[03(x1)] = 1 + 1 · x1
[32(x1)] = 1 + 1 · x1
[25(x1)] = 1 · x1
[51(x1)] = 1 · x1
[12(x1)] = 1 + 1 · x1
[31(x1)] = 1 + 1 · x1
[24(x1)] = 1 · x1
[42(x1)] = 1 · x1
[11(x1)] = 1 · x1
[10(x1)] = 1 · x1
[14(x1)] = 1 · x1
[13(x1)] = 1 · x1
[15(x1)] = 1 · x1
[45(x1)] = 1 · x1
[33(x1)] = 1 + 1 · x1
[02(x1)] = 1 + 1 · x1
[22(x1)] = 1 · x1
[52(x1)] = 1 + 1 · x1
[53(x1)] = 1 · x1
all of the following rules can be deleted.
05(50(03(32(25(51(12(x1))))))) 03(31(12(24(42(24(42(25(51(12(x1)))))))))) (361)
05(50(03(32(25(51(11(x1))))))) 03(31(12(24(42(24(42(25(51(11(x1)))))))))) (362)
05(50(03(32(25(51(10(x1))))))) 03(31(12(24(42(24(42(25(51(10(x1)))))))))) (363)
05(50(03(32(25(51(14(x1))))))) 03(31(12(24(42(24(42(25(51(14(x1)))))))))) (364)
05(50(03(32(25(51(13(x1))))))) 03(31(12(24(42(24(42(25(51(13(x1)))))))))) (365)
05(50(03(32(25(51(15(x1))))))) 03(31(12(24(42(24(42(25(51(15(x1)))))))))) (366)

1.1.1.1.1.1.1.1.1 R is empty

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