Certification Problem

Input (TPDB SRS_Relative/ICFP_2010_relative/4819)

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

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

and S is the following TRS.

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

We obtain the transformed TRS

There are 160 ruless (increase limit for explicit display).

1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

There are 960 ruless (increase limit for explicit display).

1.1.1 Rule Removal

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

There are 793 ruless (increase limit for explicit display).

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[01(x1)] = 1 · x1
[12(x1)] = 1 + 1 · x1
[23(x1)] = 1 · x1
[30(x1)] = 1 · x1
[00(x1)] = 1 · x1
[02(x1)] = 1 · x1
[22(x1)] = 1 · x1
[20(x1)] = 1 · x1
[10(x1)] = 1 · x1
[05(x1)] = 1 · x1
[55(x1)] = 1 · x1
[54(x1)] = 1 · x1
[40(x1)] = 1 · x1
[31(x1)] = 1 · x1
[41(x1)] = 1 · x1
[32(x1)] = 1 · x1
[42(x1)] = 1 · x1
[33(x1)] = 1 · x1
[43(x1)] = 1 · x1
[35(x1)] = 1 · x1
[45(x1)] = 1 · x1
[34(x1)] = 1 · x1
[44(x1)] = 1 · x1
[14(x1)] = 1 · x1
[03(x1)] = 1 · x1
[21(x1)] = 1 · x1
[04(x1)] = 1 · x1
[53(x1)] = 1 · x1
[25(x1)] = 1 + 1 · x1
[24(x1)] = 1 + 1 · x1
[50(x1)] = 1 · x1
[13(x1)] = 1 · x1
[52(x1)] = 1 + 1 · x1
[15(x1)] = 1 · x1
[11(x1)] = 1 · x1
[51(x1)] = 1 · x1
all of the following rules can be deleted.
01(12(23(30(x1)))) 00(02(22(20(01(10(05(55(54(40(x1)))))))))) (181)
01(12(23(31(x1)))) 00(02(22(20(01(10(05(55(54(41(x1)))))))))) (182)
01(12(23(32(x1)))) 00(02(22(20(01(10(05(55(54(42(x1)))))))))) (183)
01(12(23(33(x1)))) 00(02(22(20(01(10(05(55(54(43(x1)))))))))) (184)
01(12(23(35(x1)))) 00(02(22(20(01(10(05(55(54(45(x1)))))))))) (185)
01(12(23(34(x1)))) 00(02(22(20(01(10(05(55(54(44(x1)))))))))) (186)
12(23(34(40(00(x1))))) 14(44(40(03(32(20(02(21(10(00(x1)))))))))) (217)
12(23(34(40(01(x1))))) 14(44(40(03(32(20(02(21(10(01(x1)))))))))) (218)
12(23(34(40(02(x1))))) 14(44(40(03(32(20(02(21(10(02(x1)))))))))) (219)
12(23(34(40(03(x1))))) 14(44(40(03(32(20(02(21(10(03(x1)))))))))) (220)
12(23(34(40(05(x1))))) 14(44(40(03(32(20(02(21(10(05(x1)))))))))) (221)
12(23(34(40(04(x1))))) 14(44(40(03(32(20(02(21(10(04(x1)))))))))) (222)
52(25(55(55(50(00(x1)))))) 53(32(22(22(25(55(54(41(10(00(00(x1))))))))))) (541)
52(25(55(55(50(01(x1)))))) 53(32(22(22(25(55(54(41(10(00(01(x1))))))))))) (542)
52(25(55(55(50(02(x1)))))) 53(32(22(22(25(55(54(41(10(00(02(x1))))))))))) (543)
52(25(55(55(50(03(x1)))))) 53(32(22(22(25(55(54(41(10(00(03(x1))))))))))) (544)
52(25(55(55(50(05(x1)))))) 53(32(22(22(25(55(54(41(10(00(05(x1))))))))))) (545)
52(25(55(55(50(04(x1)))))) 53(32(22(22(25(55(54(41(10(00(04(x1))))))))))) (546)
02(25(55(53(32(25(55(x1))))))) 00(00(00(03(30(02(21(14(42(23(35(x1))))))))))) (629)
22(25(55(53(32(25(55(x1))))))) 20(00(00(03(30(02(21(14(42(23(35(x1))))))))))) (641)
32(25(55(53(32(25(55(x1))))))) 30(00(00(03(30(02(21(14(42(23(35(x1))))))))))) (647)
52(25(55(53(32(25(55(x1))))))) 50(00(00(03(30(02(21(14(42(23(35(x1))))))))))) (653)
42(25(55(53(32(25(55(x1))))))) 40(00(00(03(30(02(21(14(42(23(35(x1))))))))))) (659)
15(52(25(55(52(21(10(x1))))))) 11(15(54(45(53(32(23(32(20(01(10(x1))))))))))) (775)
15(52(25(55(52(21(11(x1))))))) 11(15(54(45(53(32(23(32(20(01(11(x1))))))))))) (776)
15(52(25(55(52(21(12(x1))))))) 11(15(54(45(53(32(23(32(20(01(12(x1))))))))))) (777)
15(52(25(55(52(21(13(x1))))))) 11(15(54(45(53(32(23(32(20(01(13(x1))))))))))) (778)
15(52(25(55(52(21(15(x1))))))) 11(15(54(45(53(32(23(32(20(01(15(x1))))))))))) (779)
15(52(25(55(52(21(14(x1))))))) 11(15(54(45(53(32(23(32(20(01(14(x1))))))))))) (780)
55(52(25(55(52(21(10(x1))))))) 51(15(54(45(53(32(23(32(20(01(10(x1))))))))))) (793)
55(52(25(55(52(21(11(x1))))))) 51(15(54(45(53(32(23(32(20(01(11(x1))))))))))) (794)
55(52(25(55(52(21(12(x1))))))) 51(15(54(45(53(32(23(32(20(01(12(x1))))))))))) (795)
55(52(25(55(52(21(13(x1))))))) 51(15(54(45(53(32(23(32(20(01(13(x1))))))))))) (796)
55(52(25(55(52(21(15(x1))))))) 51(15(54(45(53(32(23(32(20(01(15(x1))))))))))) (797)
55(52(25(55(52(21(14(x1))))))) 51(15(54(45(53(32(23(32(20(01(14(x1))))))))))) (798)
12(23(33(33(30(00(x1)))))) 15(53(31(13(35(51(15(54(42(20(x1)))))))))) (1099)
12(23(33(33(30(01(x1)))))) 15(53(31(13(35(51(15(54(42(21(x1)))))))))) (1100)
12(23(33(33(30(02(x1)))))) 15(53(31(13(35(51(15(54(42(22(x1)))))))))) (1101)
12(23(33(33(30(03(x1)))))) 15(53(31(13(35(51(15(54(42(23(x1)))))))))) (1102)
22(24(45(50(x1)))) 21(10(02(23(35(51(15(55(55(50(x1)))))))))) (1111)
22(24(45(51(x1)))) 21(10(02(23(35(51(15(55(55(51(x1)))))))))) (1112)
22(24(45(52(x1)))) 21(10(02(23(35(51(15(55(55(52(x1)))))))))) (1113)
22(24(45(53(x1)))) 21(10(02(23(35(51(15(55(55(53(x1)))))))))) (1114)
22(24(45(55(x1)))) 21(10(02(23(35(51(15(55(55(55(x1)))))))))) (1115)
22(24(45(54(x1)))) 21(10(02(23(35(51(15(55(55(54(x1)))))))))) (1116)
42(25(x1)) 43(31(15(54(42(21(11(15(54(45(x1)))))))))) (1127)
42(24(x1)) 43(31(15(54(42(21(11(15(54(44(x1)))))))))) (1128)

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[03(x1)] = 1 · x1
[32(x1)] = 1 · x1
[21(x1)] = 1 · x1
[14(x1)] = 1 + 1 · x1
[42(x1)] = 1 + 1 · x1
[20(x1)] = 1 · x1
[01(x1)] = 1 · x1
[10(x1)] = 1 · x1
[00(x1)] = 1 · x1
[02(x1)] = 1 · x1
[35(x1)] = 1 + 1 · x1
[53(x1)] = 1 · x1
[22(x1)] = 1 + 1 · x1
[23(x1)] = 1 · x1
[25(x1)] = 1 · x1
[24(x1)] = 1 · x1
[33(x1)] = 1 · x1
[50(x1)] = 1 + 1 · x1
[04(x1)] = 1 · x1
[30(x1)] = 1 + 1 · x1
[41(x1)] = 1 · x1
[44(x1)] = 1 · x1
[31(x1)] = 1 · x1
[13(x1)] = 1 · x1
[52(x1)] = 1 · x1
[05(x1)] = 1 · x1
[55(x1)] = 1 + 1 · x1
[54(x1)] = 1 · x1
[43(x1)] = 1 · x1
[12(x1)] = 1 + 1 · x1
[15(x1)] = 1 · x1
[45(x1)] = 1 · x1
[11(x1)] = 1 · x1
[51(x1)] = 1 · x1
[34(x1)] = 1 · x1
[40(x1)] = 1 · x1
all of the following rules can be deleted.
03(32(21(14(42(20(x1)))))) 01(10(00(02(20(03(35(53(32(20(x1)))))))))) (265)
03(32(21(14(42(21(x1)))))) 01(10(00(02(20(03(35(53(32(21(x1)))))))))) (266)
03(32(21(14(42(22(x1)))))) 01(10(00(02(20(03(35(53(32(22(x1)))))))))) (267)
03(32(21(14(42(23(x1)))))) 01(10(00(02(20(03(35(53(32(23(x1)))))))))) (268)
03(32(21(14(42(25(x1)))))) 01(10(00(02(20(03(35(53(32(25(x1)))))))))) (269)
03(32(21(14(42(24(x1)))))) 01(10(00(02(20(03(35(53(32(24(x1)))))))))) (270)
22(25(55(55(50(00(x1)))))) 23(32(22(22(25(55(54(41(10(00(00(x1))))))))))) (529)
22(25(55(55(50(01(x1)))))) 23(32(22(22(25(55(54(41(10(00(01(x1))))))))))) (530)
22(25(55(55(50(02(x1)))))) 23(32(22(22(25(55(54(41(10(00(02(x1))))))))))) (531)
22(25(55(55(50(03(x1)))))) 23(32(22(22(25(55(54(41(10(00(03(x1))))))))))) (532)
22(25(55(55(50(05(x1)))))) 23(32(22(22(25(55(54(41(10(00(05(x1))))))))))) (533)
22(25(55(55(50(04(x1)))))) 23(32(22(22(25(55(54(41(10(00(04(x1))))))))))) (534)
42(25(55(55(50(00(x1)))))) 43(32(22(22(25(55(54(41(10(00(00(x1))))))))))) (547)
42(25(55(55(50(01(x1)))))) 43(32(22(22(25(55(54(41(10(00(01(x1))))))))))) (548)
42(25(55(55(50(02(x1)))))) 43(32(22(22(25(55(54(41(10(00(02(x1))))))))))) (549)
42(25(55(55(50(03(x1)))))) 43(32(22(22(25(55(54(41(10(00(03(x1))))))))))) (550)
42(25(55(55(50(05(x1)))))) 43(32(22(22(25(55(54(41(10(00(05(x1))))))))))) (551)
42(25(55(55(50(04(x1)))))) 43(32(22(22(25(55(54(41(10(00(04(x1))))))))))) (552)
04(43(34(42(25(55(51(10(x1)))))))) 05(52(20(04(41(13(30(01(11(11(10(x1))))))))))) (1057)
04(43(34(42(25(55(51(11(x1)))))))) 05(52(20(04(41(13(30(01(11(11(11(x1))))))))))) (1058)
04(43(34(42(25(55(51(12(x1)))))))) 05(52(20(04(41(13(30(01(11(11(12(x1))))))))))) (1059)
04(43(34(42(25(55(51(13(x1)))))))) 05(52(20(04(41(13(30(01(11(11(13(x1))))))))))) (1060)
04(43(34(42(25(55(51(15(x1)))))))) 05(52(20(04(41(13(30(01(11(11(15(x1))))))))))) (1061)
04(43(34(42(25(55(51(14(x1)))))))) 05(52(20(04(41(13(30(01(11(11(14(x1))))))))))) (1062)
24(43(34(42(25(55(51(10(x1)))))))) 25(52(20(04(41(13(30(01(11(11(10(x1))))))))))) (1069)
24(43(34(42(25(55(51(11(x1)))))))) 25(52(20(04(41(13(30(01(11(11(11(x1))))))))))) (1070)
24(43(34(42(25(55(51(12(x1)))))))) 25(52(20(04(41(13(30(01(11(11(12(x1))))))))))) (1071)
24(43(34(42(25(55(51(13(x1)))))))) 25(52(20(04(41(13(30(01(11(11(13(x1))))))))))) (1072)
24(43(34(42(25(55(51(15(x1)))))))) 25(52(20(04(41(13(30(01(11(11(15(x1))))))))))) (1073)
24(43(34(42(25(55(51(14(x1)))))))) 25(52(20(04(41(13(30(01(11(11(14(x1))))))))))) (1074)
44(43(34(42(25(55(51(10(x1)))))))) 45(52(20(04(41(13(30(01(11(11(10(x1))))))))))) (1087)
44(43(34(42(25(55(51(11(x1)))))))) 45(52(20(04(41(13(30(01(11(11(11(x1))))))))))) (1088)
44(43(34(42(25(55(51(12(x1)))))))) 45(52(20(04(41(13(30(01(11(11(12(x1))))))))))) (1089)
44(43(34(42(25(55(51(13(x1)))))))) 45(52(20(04(41(13(30(01(11(11(13(x1))))))))))) (1090)
44(43(34(42(25(55(51(15(x1)))))))) 45(52(20(04(41(13(30(01(11(11(15(x1))))))))))) (1091)
44(43(34(42(25(55(51(14(x1)))))))) 45(52(20(04(41(13(30(01(11(11(14(x1))))))))))) (1092)
41(14(43(32(23(30(x1)))))) 45(54(41(11(10(03(32(20(03(30(x1)))))))))) (1117)
41(14(43(32(23(31(x1)))))) 45(54(41(11(10(03(32(20(03(31(x1)))))))))) (1118)
41(14(43(32(23(32(x1)))))) 45(54(41(11(10(03(32(20(03(32(x1)))))))))) (1119)
41(14(43(32(23(33(x1)))))) 45(54(41(11(10(03(32(20(03(33(x1)))))))))) (1120)
41(14(43(32(23(35(x1)))))) 45(54(41(11(10(03(32(20(03(35(x1)))))))))) (1121)
41(14(43(32(23(34(x1)))))) 45(54(41(11(10(03(32(20(03(34(x1)))))))))) (1122)

1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[33(x1)] = 1 + 1 · x1
[35(x1)] = 1 + 1 · x1
[50(x1)] = 1 + 1 · x1
[04(x1)] = 1 + 1 · x1
[42(x1)] = 1 · x1
[20(x1)] = 1 · x1
[30(x1)] = 1 · x1
[00(x1)] = 1 · x1
[02(x1)] = 1 · x1
[41(x1)] = 1 · x1
[14(x1)] = 1 · x1
[44(x1)] = 1 · x1
[21(x1)] = 1 · x1
[22(x1)] = 1 · x1
[23(x1)] = 1 · x1
[25(x1)] = 1 · x1
[24(x1)] = 1 · x1
[53(x1)] = 1 + 1 · x1
[31(x1)] = 1 · x1
[13(x1)] = 1 + 1 · x1
[52(x1)] = 1 · x1
[01(x1)] = 1 · x1
[10(x1)] = 1 · x1
[05(x1)] = 1 + 1 · x1
[55(x1)] = 1 + 1 · x1
[03(x1)] = 1 + 1 · x1
[32(x1)] = 1 · x1
[54(x1)] = 1 · x1
[12(x1)] = 1 + 1 · x1
[15(x1)] = 1 · x1
[43(x1)] = 1 · x1
[45(x1)] = 1 + 1 · x1
[11(x1)] = 1 · x1
[51(x1)] = 1 + 1 · x1
[34(x1)] = 1 + 1 · x1
[40(x1)] = 1 · x1
all of the following rules can be deleted.
33(35(50(04(42(20(x1)))))) 30(00(02(20(04(41(14(44(42(20(x1)))))))))) (277)
33(35(50(04(42(21(x1)))))) 30(00(02(20(04(41(14(44(42(21(x1)))))))))) (278)
33(35(50(04(42(22(x1)))))) 30(00(02(20(04(41(14(44(42(22(x1)))))))))) (279)
33(35(50(04(42(23(x1)))))) 30(00(02(20(04(41(14(44(42(23(x1)))))))))) (280)
33(35(50(04(42(25(x1)))))) 30(00(02(20(04(41(14(44(42(25(x1)))))))))) (281)
33(35(50(04(42(24(x1)))))) 30(00(02(20(04(41(14(44(42(24(x1)))))))))) (282)
53(31(13(35(52(20(x1)))))) 50(01(10(02(20(00(00(05(52(20(x1)))))))))) (301)
53(31(13(35(52(21(x1)))))) 50(01(10(02(20(00(00(05(52(21(x1)))))))))) (302)
53(31(13(35(52(22(x1)))))) 50(01(10(02(20(00(00(05(52(22(x1)))))))))) (303)
53(31(13(35(52(23(x1)))))) 50(01(10(02(20(00(00(05(52(23(x1)))))))))) (304)
53(31(13(35(52(25(x1)))))) 50(01(10(02(20(00(00(05(52(25(x1)))))))))) (305)
53(31(13(35(52(24(x1)))))) 50(01(10(02(20(00(00(05(52(24(x1)))))))))) (306)
02(25(55(55(50(00(x1)))))) 03(32(22(22(25(55(54(41(10(00(00(x1))))))))))) (517)
02(25(55(55(50(01(x1)))))) 03(32(22(22(25(55(54(41(10(00(01(x1))))))))))) (518)
02(25(55(55(50(02(x1)))))) 03(32(22(22(25(55(54(41(10(00(02(x1))))))))))) (519)
02(25(55(55(50(03(x1)))))) 03(32(22(22(25(55(54(41(10(00(03(x1))))))))))) (520)
02(25(55(55(50(05(x1)))))) 03(32(22(22(25(55(54(41(10(00(05(x1))))))))))) (521)
02(25(55(55(50(04(x1)))))) 03(32(22(22(25(55(54(41(10(00(04(x1))))))))))) (522)
32(25(55(55(50(00(x1)))))) 33(32(22(22(25(55(54(41(10(00(00(x1))))))))))) (535)
32(25(55(55(50(01(x1)))))) 33(32(22(22(25(55(54(41(10(00(01(x1))))))))))) (536)
32(25(55(55(50(02(x1)))))) 33(32(22(22(25(55(54(41(10(00(02(x1))))))))))) (537)
32(25(55(55(50(03(x1)))))) 33(32(22(22(25(55(54(41(10(00(03(x1))))))))))) (538)
32(25(55(55(50(05(x1)))))) 33(32(22(22(25(55(54(41(10(00(05(x1))))))))))) (539)
32(25(55(55(50(04(x1)))))) 33(32(22(22(25(55(54(41(10(00(04(x1))))))))))) (540)
03(35(53(33(31(12(20(x1))))))) 01(15(54(43(33(35(54(45(52(22(20(x1))))))))))) (697)
03(35(53(33(31(12(21(x1))))))) 01(15(54(43(33(35(54(45(52(22(21(x1))))))))))) (698)
03(35(53(33(31(12(22(x1))))))) 01(15(54(43(33(35(54(45(52(22(22(x1))))))))))) (699)
03(35(53(33(31(12(23(x1))))))) 01(15(54(43(33(35(54(45(52(22(23(x1))))))))))) (700)
03(35(53(33(31(12(25(x1))))))) 01(15(54(43(33(35(54(45(52(22(25(x1))))))))))) (701)
03(35(53(33(31(12(24(x1))))))) 01(15(54(43(33(35(54(45(52(22(24(x1))))))))))) (702)
13(35(53(33(31(12(20(x1))))))) 11(15(54(43(33(35(54(45(52(22(20(x1))))))))))) (703)
13(35(53(33(31(12(21(x1))))))) 11(15(54(43(33(35(54(45(52(22(21(x1))))))))))) (704)
13(35(53(33(31(12(22(x1))))))) 11(15(54(43(33(35(54(45(52(22(22(x1))))))))))) (705)
13(35(53(33(31(12(23(x1))))))) 11(15(54(43(33(35(54(45(52(22(23(x1))))))))))) (706)
13(35(53(33(31(12(25(x1))))))) 11(15(54(43(33(35(54(45(52(22(25(x1))))))))))) (707)
13(35(53(33(31(12(24(x1))))))) 11(15(54(43(33(35(54(45(52(22(24(x1))))))))))) (708)
23(35(53(33(31(12(20(x1))))))) 21(15(54(43(33(35(54(45(52(22(20(x1))))))))))) (709)
23(35(53(33(31(12(21(x1))))))) 21(15(54(43(33(35(54(45(52(22(21(x1))))))))))) (710)
23(35(53(33(31(12(22(x1))))))) 21(15(54(43(33(35(54(45(52(22(22(x1))))))))))) (711)
23(35(53(33(31(12(23(x1))))))) 21(15(54(43(33(35(54(45(52(22(23(x1))))))))))) (712)
23(35(53(33(31(12(25(x1))))))) 21(15(54(43(33(35(54(45(52(22(25(x1))))))))))) (713)
23(35(53(33(31(12(24(x1))))))) 21(15(54(43(33(35(54(45(52(22(24(x1))))))))))) (714)
33(35(53(33(31(12(20(x1))))))) 31(15(54(43(33(35(54(45(52(22(20(x1))))))))))) (715)
33(35(53(33(31(12(21(x1))))))) 31(15(54(43(33(35(54(45(52(22(21(x1))))))))))) (716)
33(35(53(33(31(12(22(x1))))))) 31(15(54(43(33(35(54(45(52(22(22(x1))))))))))) (717)
33(35(53(33(31(12(23(x1))))))) 31(15(54(43(33(35(54(45(52(22(23(x1))))))))))) (718)
33(35(53(33(31(12(25(x1))))))) 31(15(54(43(33(35(54(45(52(22(25(x1))))))))))) (719)
33(35(53(33(31(12(24(x1))))))) 31(15(54(43(33(35(54(45(52(22(24(x1))))))))))) (720)
53(35(53(33(31(12(20(x1))))))) 51(15(54(43(33(35(54(45(52(22(20(x1))))))))))) (721)
53(35(53(33(31(12(21(x1))))))) 51(15(54(43(33(35(54(45(52(22(21(x1))))))))))) (722)
53(35(53(33(31(12(22(x1))))))) 51(15(54(43(33(35(54(45(52(22(22(x1))))))))))) (723)
53(35(53(33(31(12(23(x1))))))) 51(15(54(43(33(35(54(45(52(22(23(x1))))))))))) (724)
53(35(53(33(31(12(25(x1))))))) 51(15(54(43(33(35(54(45(52(22(25(x1))))))))))) (725)
53(35(53(33(31(12(24(x1))))))) 51(15(54(43(33(35(54(45(52(22(24(x1))))))))))) (726)
43(35(53(33(31(12(20(x1))))))) 41(15(54(43(33(35(54(45(52(22(20(x1))))))))))) (727)
43(35(53(33(31(12(21(x1))))))) 41(15(54(43(33(35(54(45(52(22(21(x1))))))))))) (728)
43(35(53(33(31(12(22(x1))))))) 41(15(54(43(33(35(54(45(52(22(22(x1))))))))))) (729)
43(35(53(33(31(12(23(x1))))))) 41(15(54(43(33(35(54(45(52(22(23(x1))))))))))) (730)
43(35(53(33(31(12(25(x1))))))) 41(15(54(43(33(35(54(45(52(22(25(x1))))))))))) (731)
43(35(53(33(31(12(24(x1))))))) 41(15(54(43(33(35(54(45(52(22(24(x1))))))))))) (732)
34(43(34(42(25(55(51(10(x1)))))))) 35(52(20(04(41(13(30(01(11(11(10(x1))))))))))) (1075)
34(43(34(42(25(55(51(11(x1)))))))) 35(52(20(04(41(13(30(01(11(11(11(x1))))))))))) (1076)
34(43(34(42(25(55(51(12(x1)))))))) 35(52(20(04(41(13(30(01(11(11(12(x1))))))))))) (1077)
34(43(34(42(25(55(51(13(x1)))))))) 35(52(20(04(41(13(30(01(11(11(13(x1))))))))))) (1078)
34(43(34(42(25(55(51(15(x1)))))))) 35(52(20(04(41(13(30(01(11(11(15(x1))))))))))) (1079)
34(43(34(42(25(55(51(14(x1)))))))) 35(52(20(04(41(13(30(01(11(11(14(x1))))))))))) (1080)

1.1.1.1.1.1.1 R is empty

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