Certification Problem

Input (TPDB SRS_Relative/ICFP_2010_relative/3680)

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

0(1(1(x1))) 0(2(0(2(2(3(0(2(3(1(x1)))))))))) (1)
0(4(2(1(5(x1))))) 3(0(4(5(3(3(2(3(3(3(x1)))))))))) (2)
1(1(4(1(0(x1))))) 3(5(3(3(2(0(2(3(3(3(x1)))))))))) (3)
0(0(5(2(2(1(x1)))))) 1(3(5(3(1(2(0(2(2(3(x1)))))))))) (4)
0(4(1(4(0(0(x1)))))) 3(3(2(3(0(4(5(0(3(0(x1)))))))))) (5)
0(4(3(4(2(1(x1)))))) 0(5(3(3(3(0(0(2(0(2(x1)))))))))) (6)
1(1(2(0(5(4(x1)))))) 1(3(2(0(2(3(1(5(1(4(x1)))))))))) (7)
1(1(3(1(4(2(x1)))))) 0(2(0(2(2(2(3(0(5(2(x1)))))))))) (8)
1(3(4(0(4(1(x1)))))) 1(3(3(0(2(5(4(5(3(0(x1)))))))))) (9)
1(4(0(4(1(4(x1)))))) 5(2(5(0(5(5(4(5(0(2(x1)))))))))) (10)
1(4(1(5(4(3(x1)))))) 3(2(4(2(5(5(4(3(3(2(x1)))))))))) (11)
1(4(2(3(4(4(x1)))))) 3(0(3(3(2(5(3(2(1(2(x1)))))))))) (12)
1(4(3(1(5(1(x1)))))) 3(3(2(4(3(3(0(2(0(2(x1)))))))))) (13)
1(5(4(0(5(3(x1)))))) 3(1(3(2(0(3(3(1(3(2(x1)))))))))) (14)
1(5(5(0(1(0(x1)))))) 1(3(2(3(5(5(4(0(2(5(x1)))))))))) (15)
5(1(1(5(5(4(x1)))))) 5(1(3(3(3(0(2(0(3(2(x1)))))))))) (16)
0(0(0(0(5(5(1(x1))))))) 0(0(2(2(3(3(2(2(5(0(x1)))))))))) (17)
0(0(1(5(1(2(1(x1))))))) 1(0(2(2(0(4(5(0(2(1(x1)))))))))) (18)
0(0(3(4(0(5(4(x1))))))) 0(2(2(1(0(2(1(4(3(2(x1)))))))))) (19)
0(0(5(2(2(0(5(x1))))))) 0(2(3(3(4(2(4(0(2(1(x1)))))))))) (20)
0(1(1(1(0(0(5(x1))))))) 0(2(2(0(2(5(2(5(5(3(x1)))))))))) (21)
0(4(1(1(0(0(5(x1))))))) 1(3(2(3(4(3(0(2(5(3(x1)))))))))) (22)
0(4(3(4(0(1(0(x1))))))) 3(2(4(0(5(0(1(5(2(0(x1)))))))))) (23)
0(5(1(1(4(2(3(x1))))))) 0(2(4(2(4(4(1(5(3(2(x1)))))))))) (24)
0(5(1(4(4(0(4(x1))))))) 0(5(1(2(5(3(3(2(0(4(x1)))))))))) (25)
1(0(0(3(4(3(5(x1))))))) 3(2(4(3(3(1(3(2(1(1(x1)))))))))) (26)
1(0(1(0(0(4(2(x1))))))) 1(3(2(3(2(1(2(5(0(5(x1)))))))))) (27)
1(0(3(4(1(1(5(x1))))))) 1(0(3(5(2(4(3(1(3(2(x1)))))))))) (28)
1(0(4(1(1(4(1(x1))))))) 0(0(5(0(2(4(2(0(2(3(x1)))))))))) (29)
1(1(0(3(0(1(5(x1))))))) 0(2(0(2(0(2(0(4(5(1(x1)))))))))) (30)
1(1(1(3(1(1(4(x1))))))) 3(1(2(3(3(0(2(0(5(2(x1)))))))))) (31)
1(1(1(4(0(5(0(x1))))))) 3(5(5(2(2(4(0(2(0(0(x1)))))))))) (32)
1(1(1(5(4(0(5(x1))))))) 3(4(3(5(3(3(2(5(3(3(x1)))))))))) (33)
1(1(4(0(5(1(4(x1))))))) 4(2(2(3(0(3(2(5(0(2(x1)))))))))) (34)
1(1(4(2(0(4(3(x1))))))) 3(1(3(4(4(3(0(2(3(3(x1)))))))))) (35)
1(3(4(0(5(1(5(x1))))))) 1(3(3(5(0(2(0(3(3(1(x1)))))))))) (36)
1(3(5(0(0(0(0(x1))))))) 3(3(3(3(5(5(3(2(0(1(x1)))))))))) (37)
1(4(0(0(0(1(5(x1))))))) 3(3(0(4(4(0(3(1(1(3(x1)))))))))) (38)
1(4(0(0(5(4(4(x1))))))) 2(2(4(0(4(2(5(3(3(2(x1)))))))))) (39)
1(4(1(1(1(1(1(x1))))))) 4(1(0(2(2(1(2(5(1(3(x1)))))))))) (40)
1(4(2(1(1(1(1(x1))))))) 1(5(2(4(0(2(4(5(0(1(x1)))))))))) (41)
1(4(2(1(3(4(3(x1))))))) 3(1(3(2(3(3(5(2(5(1(x1)))))))))) (42)
1(4(3(0(0(4(1(x1))))))) 5(5(2(4(2(5(2(2(4(3(x1)))))))))) (43)
2(0(0(1(1(1(1(x1))))))) 2(1(5(4(5(5(0(2(2(1(x1)))))))))) (44)
2(0(4(0(0(0(0(x1))))))) 2(5(2(2(2(5(4(2(0(0(x1)))))))))) (45)
2(1(1(3(5(1(4(x1))))))) 2(1(1(3(2(2(3(5(0(2(x1)))))))))) (46)
2(1(4(0(1(4(5(x1))))))) 2(4(5(3(3(2(3(3(3(5(x1)))))))))) (47)
3(0(0(1(1(4(3(x1))))))) 0(5(3(1(3(2(0(2(4(3(x1)))))))))) (48)
3(0(0(5(4(4(4(x1))))))) 3(3(1(3(2(3(0(3(3(1(x1)))))))))) (49)
3(0(4(1(4(0(0(x1))))))) 0(4(0(2(2(2(0(5(5(0(x1)))))))))) (50)
3(4(3(4(3(4(0(x1))))))) 3(5(5(1(0(2(4(3(2(0(x1)))))))))) (51)
3(5(1(4(0(1(4(x1))))))) 0(3(2(5(0(2(2(2(0(2(x1)))))))))) (52)
4(0(1(1(0(5(1(x1))))))) 4(3(2(5(2(1(1(3(3(2(x1)))))))))) (53)
4(1(1(2(0(4(1(x1))))))) 4(0(2(4(0(0(2(0(2(3(x1)))))))))) (54)
4(1(5(0(1(0(1(x1))))))) 4(3(0(3(5(5(2(4(0(2(x1)))))))))) (55)
5(1(2(3(4(4(5(x1))))))) 5(4(0(2(0(2(1(2(4(5(x1)))))))))) (56)
5(3(1(4(4(1(1(x1))))))) 5(0(2(5(2(5(0(2(4(1(x1)))))))))) (57)

and S is the following TRS.

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

1.1 Closure Under Flat Contexts

Using the flat contexts

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

We obtain the transformed TRS

There are 270 ruless (increase limit for explicit display).

1.1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

There are 1620 ruless (increase limit for explicit display).

1.1.1.1 Rule Removal

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

There are 1511 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
[50(x1)] = 1 + 1 · x1
[02(x1)] = 1 · x1
[21(x1)] = 1 · x1
[11(x1)] = 1 · x1
[41(x1)] = 1 · x1
[15(x1)] = 1 · x1
[51(x1)] = 1 · x1
[13(x1)] = 1 · x1
[32(x1)] = 1 + 1 · x1
[20(x1)] = 1 · x1
[23(x1)] = 1 · x1
[31(x1)] = 1 · x1
[10(x1)] = 1 · x1
[12(x1)] = 1 · x1
[14(x1)] = 1 · x1
[00(x1)] = 1 · x1
[01(x1)] = 1 · x1
[05(x1)] = 1 · x1
[54(x1)] = 1 · x1
[40(x1)] = 1 · x1
[22(x1)] = 1 · x1
[03(x1)] = 1 · x1
[04(x1)] = 1 · x1
[44(x1)] = 1 · x1
[43(x1)] = 1 · x1
[42(x1)] = 1 · x1
[53(x1)] = 1 + 1 · x1
[52(x1)] = 1 · x1
[55(x1)] = 1 + 1 · x1
[30(x1)] = 1 · x1
[34(x1)] = 1 · x1
[25(x1)] = 1 · x1
[24(x1)] = 1 + 1 · x1
[33(x1)] = 1 · x1
[35(x1)] = 1 + 1 · x1
all of the following rules can be deleted.
54(44(43(32(21(15(51(x1))))))) 54(42(21(12(20(02(20(04(45(51(x1)))))))))) (469)
54(44(43(32(21(15(50(x1))))))) 54(42(21(12(20(02(20(04(45(50(x1)))))))))) (470)
54(44(43(32(21(15(53(x1))))))) 54(42(21(12(20(02(20(04(45(53(x1)))))))))) (471)
54(44(43(32(21(15(52(x1))))))) 54(42(21(12(20(02(20(04(45(52(x1)))))))))) (472)
54(44(43(32(21(15(54(x1))))))) 54(42(21(12(20(02(20(04(45(54(x1)))))))))) (473)
54(44(43(32(21(15(55(x1))))))) 54(42(21(12(20(02(20(04(45(55(x1)))))))))) (474)
54(45(50(04(43(30(00(01(x1)))))))) 52(23(34(41(12(20(01(12(22(20(01(x1))))))))))) (973)
54(45(50(04(43(30(00(00(x1)))))))) 52(23(34(41(12(20(01(12(22(20(00(x1))))))))))) (974)
54(45(50(04(43(30(00(03(x1)))))))) 52(23(34(41(12(20(01(12(22(20(03(x1))))))))))) (975)
54(45(50(04(43(30(00(02(x1)))))))) 52(23(34(41(12(20(01(12(22(20(02(x1))))))))))) (976)
54(45(50(04(43(30(00(04(x1)))))))) 52(23(34(41(12(20(01(12(22(20(04(x1))))))))))) (977)
54(45(50(04(43(30(00(05(x1)))))))) 52(23(34(41(12(20(01(12(22(20(05(x1))))))))))) (978)
55(50(02(22(25(50(00(01(x1)))))))) 51(12(20(04(42(24(43(33(32(20(01(x1))))))))))) (1009)
55(50(02(22(25(50(00(00(x1)))))))) 51(12(20(04(42(24(43(33(32(20(00(x1))))))))))) (1010)
55(50(02(22(25(50(00(03(x1)))))))) 51(12(20(04(42(24(43(33(32(20(03(x1))))))))))) (1011)
55(50(02(22(25(50(00(02(x1)))))))) 51(12(20(04(42(24(43(33(32(20(02(x1))))))))))) (1012)
55(50(02(22(25(50(00(04(x1)))))))) 51(12(20(04(42(24(43(33(32(20(04(x1))))))))))) (1013)
55(50(02(22(25(50(00(05(x1)))))))) 51(12(20(04(42(24(43(33(32(20(05(x1))))))))))) (1014)
13(32(24(41(11(15(50(01(x1)))))))) 12(23(35(51(14(44(42(24(42(20(01(x1))))))))))) (1093)
13(32(24(41(11(15(50(00(x1)))))))) 12(23(35(51(14(44(42(24(42(20(00(x1))))))))))) (1094)
13(32(24(41(11(15(50(03(x1)))))))) 12(23(35(51(14(44(42(24(42(20(03(x1))))))))))) (1095)
13(32(24(41(11(15(50(02(x1)))))))) 12(23(35(51(14(44(42(24(42(20(02(x1))))))))))) (1096)
13(32(24(41(11(15(50(04(x1)))))))) 12(23(35(51(14(44(42(24(42(20(04(x1))))))))))) (1097)
13(32(24(41(11(15(50(05(x1)))))))) 12(23(35(51(14(44(42(24(42(20(05(x1))))))))))) (1098)
03(32(24(41(11(15(50(01(x1)))))))) 02(23(35(51(14(44(42(24(42(20(01(x1))))))))))) (1099)
03(32(24(41(11(15(50(00(x1)))))))) 02(23(35(51(14(44(42(24(42(20(00(x1))))))))))) (1100)
03(32(24(41(11(15(50(03(x1)))))))) 02(23(35(51(14(44(42(24(42(20(03(x1))))))))))) (1101)
03(32(24(41(11(15(50(02(x1)))))))) 02(23(35(51(14(44(42(24(42(20(02(x1))))))))))) (1102)
03(32(24(41(11(15(50(04(x1)))))))) 02(23(35(51(14(44(42(24(42(20(04(x1))))))))))) (1103)
03(32(24(41(11(15(50(05(x1)))))))) 02(23(35(51(14(44(42(24(42(20(05(x1))))))))))) (1104)
23(32(24(41(11(15(50(01(x1)))))))) 22(23(35(51(14(44(42(24(42(20(01(x1))))))))))) (1111)
23(32(24(41(11(15(50(00(x1)))))))) 22(23(35(51(14(44(42(24(42(20(00(x1))))))))))) (1112)
23(32(24(41(11(15(50(03(x1)))))))) 22(23(35(51(14(44(42(24(42(20(03(x1))))))))))) (1113)
23(32(24(41(11(15(50(02(x1)))))))) 22(23(35(51(14(44(42(24(42(20(02(x1))))))))))) (1114)
23(32(24(41(11(15(50(04(x1)))))))) 22(23(35(51(14(44(42(24(42(20(04(x1))))))))))) (1115)
23(32(24(41(11(15(50(05(x1)))))))) 22(23(35(51(14(44(42(24(42(20(05(x1))))))))))) (1116)
53(32(24(41(11(15(50(01(x1)))))))) 52(23(35(51(14(44(42(24(42(20(01(x1))))))))))) (1117)
53(32(24(41(11(15(50(00(x1)))))))) 52(23(35(51(14(44(42(24(42(20(00(x1))))))))))) (1118)
53(32(24(41(11(15(50(03(x1)))))))) 52(23(35(51(14(44(42(24(42(20(03(x1))))))))))) (1119)
53(32(24(41(11(15(50(02(x1)))))))) 52(23(35(51(14(44(42(24(42(20(02(x1))))))))))) (1120)
53(32(24(41(11(15(50(04(x1)))))))) 52(23(35(51(14(44(42(24(42(20(04(x1))))))))))) (1121)
53(32(24(41(11(15(50(05(x1)))))))) 52(23(35(51(14(44(42(24(42(20(05(x1))))))))))) (1122)
43(32(24(41(11(15(50(01(x1)))))))) 42(23(35(51(14(44(42(24(42(20(01(x1))))))))))) (1123)
43(32(24(41(11(15(50(00(x1)))))))) 42(23(35(51(14(44(42(24(42(20(00(x1))))))))))) (1124)
43(32(24(41(11(15(50(03(x1)))))))) 42(23(35(51(14(44(42(24(42(20(03(x1))))))))))) (1125)
43(32(24(41(11(15(50(02(x1)))))))) 42(23(35(51(14(44(42(24(42(20(02(x1))))))))))) (1126)
43(32(24(41(11(15(50(04(x1)))))))) 42(23(35(51(14(44(42(24(42(20(04(x1))))))))))) (1127)
43(32(24(41(11(15(50(05(x1)))))))) 42(23(35(51(14(44(42(24(42(20(05(x1))))))))))) (1128)
15(53(34(43(30(00(01(10(x1)))))))) 11(11(12(23(31(13(33(34(42(23(30(x1))))))))))) (1130)
15(53(34(43(30(00(01(13(x1)))))))) 11(11(12(23(31(13(33(34(42(23(33(x1))))))))))) (1131)
15(53(34(43(30(00(01(14(x1)))))))) 11(11(12(23(31(13(33(34(42(23(34(x1))))))))))) (1133)
05(53(34(43(30(00(01(10(x1)))))))) 01(11(12(23(31(13(33(34(42(23(30(x1))))))))))) (1136)
05(53(34(43(30(00(01(13(x1)))))))) 01(11(12(23(31(13(33(34(42(23(33(x1))))))))))) (1137)
05(53(34(43(30(00(01(14(x1)))))))) 01(11(12(23(31(13(33(34(42(23(34(x1))))))))))) (1139)
55(53(34(43(30(00(01(10(x1)))))))) 51(11(12(23(31(13(33(34(42(23(30(x1))))))))))) (1154)
55(53(34(43(30(00(01(13(x1)))))))) 51(11(12(23(31(13(33(34(42(23(33(x1))))))))))) (1155)
55(53(34(43(30(00(01(12(x1)))))))) 51(11(12(23(31(13(33(34(42(23(32(x1))))))))))) (1156)
55(53(34(43(30(00(01(14(x1)))))))) 51(11(12(23(31(13(33(34(42(23(34(x1))))))))))) (1157)
55(53(34(43(30(00(01(15(x1)))))))) 51(11(12(23(31(13(33(34(42(23(35(x1))))))))))) (1158)
45(53(34(43(30(00(01(10(x1)))))))) 41(11(12(23(31(13(33(34(42(23(30(x1))))))))))) (1160)
45(53(34(43(30(00(01(13(x1)))))))) 41(11(12(23(31(13(33(34(42(23(33(x1))))))))))) (1161)
45(53(34(43(30(00(01(14(x1)))))))) 41(11(12(23(31(13(33(34(42(23(34(x1))))))))))) (1163)
55(51(10(03(30(01(11(10(x1)))))))) 51(15(54(40(02(20(02(20(02(20(00(x1))))))))))) (1298)

1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[45(x1)] = 1 + 1 · x1
[50(x1)] = 1 + 1 · x1
[02(x1)] = 1 · x1
[21(x1)] = 1 + 1 · x1
[11(x1)] = 1 · x1
[41(x1)] = 1 · x1
[15(x1)] = 1 + 1 · x1
[51(x1)] = 1 + 1 · x1
[13(x1)] = 1 + 1 · x1
[32(x1)] = 1 · x1
[20(x1)] = 1 · x1
[23(x1)] = 1 · x1
[31(x1)] = 1 · x1
[10(x1)] = 1 + 1 · x1
[12(x1)] = 1 · x1
[14(x1)] = 1 · x1
[00(x1)] = 1 + 1 · x1
[01(x1)] = 1 · x1
[05(x1)] = 1 + 1 · x1
[54(x1)] = 1 + 1 · x1
[40(x1)] = 1 · x1
[22(x1)] = 1 · x1
[03(x1)] = 1 + 1 · x1
[04(x1)] = 1 · x1
[25(x1)] = 1 + 1 · x1
[42(x1)] = 1 · x1
[24(x1)] = 1 + 1 · x1
[43(x1)] = 1 + 1 · x1
[33(x1)] = 1 · x1
[35(x1)] = 1 · x1
[44(x1)] = 1 · x1
[53(x1)] = 1 · x1
[34(x1)] = 1 · x1
[30(x1)] = 1 + 1 · x1
all of the following rules can be deleted.
12(21(15(51(10(00(01(x1))))))) 12(20(05(54(40(02(22(20(01(11(x1)))))))))) (397)
12(21(15(51(10(00(03(x1))))))) 12(20(05(54(40(02(22(20(01(13(x1)))))))))) (399)
12(21(15(51(10(00(02(x1))))))) 12(20(05(54(40(02(22(20(01(12(x1)))))))))) (400)
12(21(15(51(10(00(04(x1))))))) 12(20(05(54(40(02(22(20(01(14(x1)))))))))) (401)
12(21(15(51(10(00(05(x1))))))) 12(20(05(54(40(02(22(20(01(15(x1)))))))))) (402)
15(50(02(22(25(50(00(01(x1)))))))) 11(12(20(04(42(24(43(33(32(20(01(x1))))))))))) (985)
15(50(02(22(25(50(00(00(x1)))))))) 11(12(20(04(42(24(43(33(32(20(00(x1))))))))))) (986)
15(50(02(22(25(50(00(03(x1)))))))) 11(12(20(04(42(24(43(33(32(20(03(x1))))))))))) (987)
15(50(02(22(25(50(00(02(x1)))))))) 11(12(20(04(42(24(43(33(32(20(02(x1))))))))))) (988)
15(50(02(22(25(50(00(04(x1)))))))) 11(12(20(04(42(24(43(33(32(20(04(x1))))))))))) (989)
15(50(02(22(25(50(00(05(x1)))))))) 11(12(20(04(42(24(43(33(32(20(05(x1))))))))))) (990)
05(50(02(22(25(50(00(01(x1)))))))) 01(12(20(04(42(24(43(33(32(20(01(x1))))))))))) (991)
05(50(02(22(25(50(00(00(x1)))))))) 01(12(20(04(42(24(43(33(32(20(00(x1))))))))))) (992)
05(50(02(22(25(50(00(03(x1)))))))) 01(12(20(04(42(24(43(33(32(20(03(x1))))))))))) (993)
05(50(02(22(25(50(00(02(x1)))))))) 01(12(20(04(42(24(43(33(32(20(02(x1))))))))))) (994)
05(50(02(22(25(50(00(04(x1)))))))) 01(12(20(04(42(24(43(33(32(20(04(x1))))))))))) (995)
05(50(02(22(25(50(00(05(x1)))))))) 01(12(20(04(42(24(43(33(32(20(05(x1))))))))))) (996)
45(50(02(22(25(50(00(01(x1)))))))) 41(12(20(04(42(24(43(33(32(20(01(x1))))))))))) (1015)
45(50(02(22(25(50(00(00(x1)))))))) 41(12(20(04(42(24(43(33(32(20(00(x1))))))))))) (1016)
45(50(02(22(25(50(00(03(x1)))))))) 41(12(20(04(42(24(43(33(32(20(03(x1))))))))))) (1017)
45(50(02(22(25(50(00(02(x1)))))))) 41(12(20(04(42(24(43(33(32(20(02(x1))))))))))) (1018)
45(50(02(22(25(50(00(04(x1)))))))) 41(12(20(04(42(24(43(33(32(20(04(x1))))))))))) (1019)
45(50(02(22(25(50(00(05(x1)))))))) 41(12(20(04(42(24(43(33(32(20(05(x1))))))))))) (1020)
33(32(24(41(11(15(50(01(x1)))))))) 32(23(35(51(14(44(42(24(42(20(01(x1))))))))))) (1105)
33(32(24(41(11(15(50(00(x1)))))))) 32(23(35(51(14(44(42(24(42(20(00(x1))))))))))) (1106)
33(32(24(41(11(15(50(03(x1)))))))) 32(23(35(51(14(44(42(24(42(20(03(x1))))))))))) (1107)
33(32(24(41(11(15(50(02(x1)))))))) 32(23(35(51(14(44(42(24(42(20(02(x1))))))))))) (1108)
33(32(24(41(11(15(50(04(x1)))))))) 32(23(35(51(14(44(42(24(42(20(04(x1))))))))))) (1109)
33(32(24(41(11(15(50(05(x1)))))))) 32(23(35(51(14(44(42(24(42(20(05(x1))))))))))) (1110)
15(53(34(43(30(00(01(12(x1)))))))) 11(11(12(23(31(13(33(34(42(23(32(x1))))))))))) (1132)
15(53(34(43(30(00(01(15(x1)))))))) 11(11(12(23(31(13(33(34(42(23(35(x1))))))))))) (1134)
05(53(34(43(30(00(01(12(x1)))))))) 01(11(12(23(31(13(33(34(42(23(32(x1))))))))))) (1138)
05(53(34(43(30(00(01(15(x1)))))))) 01(11(12(23(31(13(33(34(42(23(35(x1))))))))))) (1140)
45(53(34(43(30(00(01(12(x1)))))))) 41(11(12(23(31(13(33(34(42(23(32(x1))))))))))) (1162)
45(53(34(43(30(00(01(15(x1)))))))) 41(11(12(23(31(13(33(34(42(23(35(x1))))))))))) (1164)
15(51(10(03(30(01(11(10(x1)))))))) 11(15(54(40(02(20(02(20(02(20(00(x1))))))))))) (1274)
05(51(10(03(30(01(11(10(x1)))))))) 01(15(54(40(02(20(02(20(02(20(00(x1))))))))))) (1280)
45(51(10(03(30(01(11(10(x1)))))))) 41(15(54(40(02(20(02(20(02(20(00(x1))))))))))) (1304)

1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[45(x1)] = 1 + 1 · x1
[50(x1)] = 1 · x1
[02(x1)] = 1 + 1 · x1
[21(x1)] = 1 + 1 · x1
[11(x1)] = 1 · x1
[41(x1)] = 1 · x1
[15(x1)] = 1 · x1
[51(x1)] = 1 · x1
[13(x1)] = 1 · x1
[32(x1)] = 1 · x1
[20(x1)] = 1 · x1
[23(x1)] = 1 · x1
[31(x1)] = 1 + 1 · x1
[10(x1)] = 1 · x1
[12(x1)] = 1 · x1
[14(x1)] = 1 · x1
[54(x1)] = 1 + 1 · x1
[40(x1)] = 1 + 1 · x1
[04(x1)] = 1 · x1
[43(x1)] = 1 + 1 · x1
[33(x1)] = 1 + 1 · x1
[22(x1)] = 1 · x1
[01(x1)] = 1 · x1
[30(x1)] = 1 + 1 · x1
[00(x1)] = 1 · x1
all of the following rules can be deleted.
45(50(02(21(11(11(x1)))))) 41(15(51(13(32(20(02(23(31(11(x1)))))))))) (385)
45(50(02(21(11(10(x1)))))) 41(15(51(13(32(20(02(23(31(10(x1)))))))))) (386)
45(50(02(21(11(13(x1)))))) 41(15(51(13(32(20(02(23(31(13(x1)))))))))) (387)
45(50(02(21(11(12(x1)))))) 41(15(51(13(32(20(02(23(31(12(x1)))))))))) (388)
45(50(02(21(11(14(x1)))))) 41(15(51(13(32(20(02(23(31(14(x1)))))))))) (389)
45(50(02(21(11(15(x1)))))) 41(15(51(13(32(20(02(23(31(15(x1)))))))))) (390)
54(40(04(43(33(31(x1)))))) 51(12(22(22(22(20(02(21(12(20(01(x1))))))))))) (1909)
54(40(04(43(33(30(x1)))))) 51(12(22(22(22(20(02(21(12(20(00(x1))))))))))) (1910)

1.1.1.1.1.1.1.1 R is empty

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