Certification Problem

Input (TPDB SRS_Relative/ICFP_2010_relative/4051)

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

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

and S is the following TRS.

2(1(1(4(2(0(x1)))))) 2(1(4(5(4(2(3(3(5(0(x1)))))))))) (51)
3(5(3(5(4(x1))))) 2(2(3(2(5(2(2(3(5(4(x1)))))))))) (52)
3(1(0(3(4(x1))))) 3(1(3(3(2(2(2(5(2(4(x1)))))))))) (53)
0(1(x1)) 2(2(2(2(2(2(3(0(4(5(x1)))))))))) (54)
5(3(1(5(5(1(5(x1))))))) 5(4(5(5(0(3(0(0(3(5(x1)))))))))) (55)
4(1(5(1(5(x1))))) 2(4(2(3(3(3(3(1(5(2(x1)))))))))) (56)
1(5(4(4(1(0(1(x1))))))) 5(4(5(0(5(3(2(5(0(1(x1)))))))))) (57)
1(1(5(x1))) 1(4(2(2(5(5(3(4(3(5(x1)))))))))) (58)
3(0(1(x1))) 4(5(0(3(0(2(0(4(3(5(x1)))))))))) (59)
5(4(0(0(4(x1))))) 5(3(4(5(4(3(2(5(5(4(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

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

We obtain the transformed TRS

There are 195 ruless (increase limit for explicit display).

1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

There are 1170 ruless (increase limit for explicit display).

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

There are 582 ruless (increase limit for explicit display).

1.1.1.1 Rule Removal

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

There are 427 ruless (increase limit for explicit display).

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[03(x1)] = 1 + 1 · x1
[33(x1)] = 1 · x1
[31(x1)] = 1 · x1
[11(x1)] = 1 · x1
[10(x1)] = 1 + 1 · x1
[00(x1)] = 1 · x1
[34(x1)] = 1 · x1
[45(x1)] = 1 · x1
[54(x1)] = 1 · x1
[43(x1)] = 1 · x1
[35(x1)] = 1 · x1
[51(x1)] = 1 · x1
[13(x1)] = 1 · x1
[14(x1)] = 1 · x1
[15(x1)] = 1 + 1 · x1
[12(x1)] = 1 · x1
[40(x1)] = 1 · x1
[04(x1)] = 1 · x1
[41(x1)] = 1 · x1
[44(x1)] = 1 · x1
[55(x1)] = 1 · x1
[22(x1)] = 1 · x1
[20(x1)] = 1 · x1
[23(x1)] = 1 · x1
[21(x1)] = 1 · x1
[02(x1)] = 1 · x1
[25(x1)] = 1 · x1
[53(x1)] = 1 · x1
[52(x1)] = 1 · x1
[24(x1)] = 1 · x1
[42(x1)] = 1 · x1
[30(x1)] = 1 · x1
[01(x1)] = 1 · x1
[05(x1)] = 1 + 1 · x1
[50(x1)] = 1 · x1
[32(x1)] = 1 · x1
all of the following rules can be deleted.
40(04(41(11(11(10(x1)))))) 44(45(54(45(55(54(41(12(22(20(x1)))))))))) (271)
13(31(10(02(25(53(31(x1))))))) 12(25(52(22(23(35(51(12(24(41(x1)))))))))) (297)
41(15(54(40(02(25(54(x1))))))) 41(13(33(34(41(11(11(11(11(14(x1)))))))))) (358)
35(54(40(02(21(10(x1)))))) 30(02(24(43(35(51(12(25(52(25(50(x1))))))))))) (499)
55(54(40(02(21(10(x1)))))) 50(02(24(43(35(51(12(25(52(25(50(x1))))))))))) (517)
25(54(40(02(21(10(x1)))))) 20(02(24(43(35(51(12(25(52(25(50(x1))))))))))) (523)
45(54(40(02(21(10(x1)))))) 40(02(24(43(35(51(12(25(52(25(50(x1))))))))))) (529)
11(15(51(13(33(33(30(00(x1)))))))) 14(40(04(45(52(23(35(55(52(20(00(x1))))))))))) (799)
11(15(51(13(33(33(30(03(x1)))))))) 14(40(04(45(52(23(35(55(52(20(03(x1))))))))))) (800)
11(15(51(13(33(33(30(01(x1)))))))) 14(40(04(45(52(23(35(55(52(20(01(x1))))))))))) (801)
11(15(51(13(33(33(30(04(x1)))))))) 14(40(04(45(52(23(35(55(52(20(04(x1))))))))))) (802)
11(15(51(13(33(33(30(05(x1)))))))) 14(40(04(45(52(23(35(55(52(20(05(x1))))))))))) (803)
11(15(51(13(33(33(30(02(x1)))))))) 14(40(04(45(52(23(35(55(52(20(02(x1))))))))))) (804)
03(34(44(40(05(51(10(00(x1)))))))) 02(22(20(05(53(33(35(53(31(10(00(x1))))))))))) (973)
03(34(44(40(05(51(10(03(x1)))))))) 02(22(20(05(53(33(35(53(31(10(03(x1))))))))))) (974)
03(34(44(40(05(51(10(01(x1)))))))) 02(22(20(05(53(33(35(53(31(10(01(x1))))))))))) (975)
03(34(44(40(05(51(10(04(x1)))))))) 02(22(20(05(53(33(35(53(31(10(04(x1))))))))))) (976)
03(34(44(40(05(51(10(05(x1)))))))) 02(22(20(05(53(33(35(53(31(10(05(x1))))))))))) (977)
03(34(44(40(05(51(10(02(x1)))))))) 02(22(20(05(53(33(35(53(31(10(02(x1))))))))))) (978)
11(15(50(x1))) 14(42(22(25(55(53(34(43(35(50(x1)))))))))) (1201)
11(15(53(x1))) 14(42(22(25(55(53(34(43(35(53(x1)))))))))) (1202)
11(15(51(x1))) 14(42(22(25(55(53(34(43(35(51(x1)))))))))) (1203)
11(15(54(x1))) 14(42(22(25(55(53(34(43(35(54(x1)))))))))) (1204)
11(15(55(x1))) 14(42(22(25(55(53(34(43(35(55(x1)))))))))) (1205)
11(15(52(x1))) 14(42(22(25(55(53(34(43(35(52(x1)))))))))) (1206)
03(35(53(35(54(40(x1)))))) 02(22(23(32(25(52(22(23(35(54(40(x1))))))))))) (1219)
03(35(53(35(54(43(x1)))))) 02(22(23(32(25(52(22(23(35(54(43(x1))))))))))) (1220)
03(35(53(35(54(41(x1)))))) 02(22(23(32(25(52(22(23(35(54(41(x1))))))))))) (1221)
03(35(53(35(54(44(x1)))))) 02(22(23(32(25(52(22(23(35(54(44(x1))))))))))) (1222)
03(35(53(35(54(45(x1)))))) 02(22(23(32(25(52(22(23(35(54(45(x1))))))))))) (1223)
03(35(53(35(54(42(x1)))))) 02(22(23(32(25(52(22(23(35(54(42(x1))))))))))) (1224)

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
[11(x1)] = 1 · x1
[10(x1)] = 1 + 1 · x1
[00(x1)] = 1 · x1
[34(x1)] = 1 · x1
[45(x1)] = 1 · x1
[54(x1)] = 1 · x1
[43(x1)] = 1 · x1
[35(x1)] = 1 · x1
[51(x1)] = 1 · x1
[13(x1)] = 1 + 1 · x1
[14(x1)] = 1 + 1 · x1
[15(x1)] = 1 · x1
[12(x1)] = 1 · x1
[40(x1)] = 1 + 1 · x1
[04(x1)] = 1 + 1 · x1
[41(x1)] = 1 · x1
[44(x1)] = 1 + 1 · x1
[55(x1)] = 1 · x1
[22(x1)] = 1 · x1
[23(x1)] = 1 · x1
[21(x1)] = 1 · x1
[02(x1)] = 1 · x1
[25(x1)] = 1 · x1
[24(x1)] = 1 · x1
[52(x1)] = 1 · x1
[42(x1)] = 1 · x1
[53(x1)] = 1 + 1 · x1
[30(x1)] = 1 · x1
[01(x1)] = 1 · x1
[05(x1)] = 1 + 1 · x1
[50(x1)] = 1 · x1
[20(x1)] = 1 · x1
[32(x1)] = 1 · x1
all of the following rules can be deleted.
40(04(41(11(11(13(x1)))))) 44(45(54(45(55(54(41(12(22(23(x1)))))))))) (272)
40(04(41(11(11(11(x1)))))) 44(45(54(45(55(54(41(12(22(21(x1)))))))))) (273)
40(04(41(11(11(12(x1)))))) 44(45(54(45(55(54(41(12(22(22(x1)))))))))) (276)
35(54(40(02(21(13(x1)))))) 30(02(24(43(35(51(12(25(52(25(53(x1))))))))))) (500)
35(54(40(02(21(11(x1)))))) 30(02(24(43(35(51(12(25(52(25(51(x1))))))))))) (501)
35(54(40(02(21(12(x1)))))) 30(02(24(43(35(51(12(25(52(25(52(x1))))))))))) (504)
55(54(40(02(21(13(x1)))))) 50(02(24(43(35(51(12(25(52(25(53(x1))))))))))) (518)
55(54(40(02(21(11(x1)))))) 50(02(24(43(35(51(12(25(52(25(51(x1))))))))))) (519)
55(54(40(02(21(12(x1)))))) 50(02(24(43(35(51(12(25(52(25(52(x1))))))))))) (522)
25(54(40(02(21(13(x1)))))) 20(02(24(43(35(51(12(25(52(25(53(x1))))))))))) (524)
25(54(40(02(21(11(x1)))))) 20(02(24(43(35(51(12(25(52(25(51(x1))))))))))) (525)
25(54(40(02(21(12(x1)))))) 20(02(24(43(35(51(12(25(52(25(52(x1))))))))))) (528)
13(34(44(40(05(51(10(00(x1)))))))) 12(22(20(05(53(33(35(53(31(10(00(x1))))))))))) (979)
13(34(44(40(05(51(10(03(x1)))))))) 12(22(20(05(53(33(35(53(31(10(03(x1))))))))))) (980)
13(34(44(40(05(51(10(01(x1)))))))) 12(22(20(05(53(33(35(53(31(10(01(x1))))))))))) (981)
13(34(44(40(05(51(10(04(x1)))))))) 12(22(20(05(53(33(35(53(31(10(04(x1))))))))))) (982)
13(34(44(40(05(51(10(05(x1)))))))) 12(22(20(05(53(33(35(53(31(10(05(x1))))))))))) (983)
13(34(44(40(05(51(10(02(x1)))))))) 12(22(20(05(53(33(35(53(31(10(02(x1))))))))))) (984)
53(34(44(40(05(51(10(00(x1)))))))) 52(22(20(05(53(33(35(53(31(10(00(x1))))))))))) (985)
53(34(44(40(05(51(10(03(x1)))))))) 52(22(20(05(53(33(35(53(31(10(03(x1))))))))))) (986)
53(34(44(40(05(51(10(01(x1)))))))) 52(22(20(05(53(33(35(53(31(10(01(x1))))))))))) (987)
53(34(44(40(05(51(10(04(x1)))))))) 52(22(20(05(53(33(35(53(31(10(04(x1))))))))))) (988)
53(34(44(40(05(51(10(05(x1)))))))) 52(22(20(05(53(33(35(53(31(10(05(x1))))))))))) (989)
53(34(44(40(05(51(10(02(x1)))))))) 52(22(20(05(53(33(35(53(31(10(02(x1))))))))))) (990)
04(40(03(31(14(40(03(30(x1)))))))) 05(50(02(20(00(04(43(31(10(02(20(x1))))))))))) (1045)
04(40(03(31(14(40(03(33(x1)))))))) 05(50(02(20(00(04(43(31(10(02(23(x1))))))))))) (1046)
04(40(03(31(14(40(03(31(x1)))))))) 05(50(02(20(00(04(43(31(10(02(21(x1))))))))))) (1047)
04(40(03(31(14(40(03(34(x1)))))))) 05(50(02(20(00(04(43(31(10(02(24(x1))))))))))) (1048)
04(40(03(31(14(40(03(35(x1)))))))) 05(50(02(20(00(04(43(31(10(02(25(x1))))))))))) (1049)
04(40(03(31(14(40(03(32(x1)))))))) 05(50(02(20(00(04(43(31(10(02(22(x1))))))))))) (1050)
53(31(15(55(51(15(50(x1))))))) 54(45(55(50(03(30(00(03(35(50(x1)))))))))) (1195)
53(31(15(55(51(15(53(x1))))))) 54(45(55(50(03(30(00(03(35(53(x1)))))))))) (1196)
53(31(15(55(51(15(51(x1))))))) 54(45(55(50(03(30(00(03(35(51(x1)))))))))) (1197)
53(31(15(55(51(15(54(x1))))))) 54(45(55(50(03(30(00(03(35(54(x1)))))))))) (1198)
53(31(15(55(51(15(55(x1))))))) 54(45(55(50(03(30(00(03(35(55(x1)))))))))) (1199)
53(31(15(55(51(15(52(x1))))))) 54(45(55(50(03(30(00(03(35(52(x1)))))))))) (1200)
54(40(00(04(40(x1))))) 53(34(45(54(43(32(25(55(54(40(x1)))))))))) (1207)
54(40(00(04(43(x1))))) 53(34(45(54(43(32(25(55(54(43(x1)))))))))) (1208)
54(40(00(04(41(x1))))) 53(34(45(54(43(32(25(55(54(41(x1)))))))))) (1209)
54(40(00(04(44(x1))))) 53(34(45(54(43(32(25(55(54(44(x1)))))))))) (1210)
54(40(00(04(45(x1))))) 53(34(45(54(43(32(25(55(54(45(x1)))))))))) (1211)
54(40(00(04(42(x1))))) 53(34(45(54(43(32(25(55(54(42(x1)))))))))) (1212)
33(35(53(35(54(40(x1)))))) 32(22(23(32(25(52(22(23(35(54(40(x1))))))))))) (1213)
33(35(53(35(54(43(x1)))))) 32(22(23(32(25(52(22(23(35(54(43(x1))))))))))) (1214)
33(35(53(35(54(41(x1)))))) 32(22(23(32(25(52(22(23(35(54(41(x1))))))))))) (1215)
33(35(53(35(54(44(x1)))))) 32(22(23(32(25(52(22(23(35(54(44(x1))))))))))) (1216)
33(35(53(35(54(45(x1)))))) 32(22(23(32(25(52(22(23(35(54(45(x1))))))))))) (1217)
33(35(53(35(54(42(x1)))))) 32(22(23(32(25(52(22(23(35(54(42(x1))))))))))) (1218)
13(35(53(35(54(40(x1)))))) 12(22(23(32(25(52(22(23(35(54(40(x1))))))))))) (1225)
13(35(53(35(54(43(x1)))))) 12(22(23(32(25(52(22(23(35(54(43(x1))))))))))) (1226)
13(35(53(35(54(41(x1)))))) 12(22(23(32(25(52(22(23(35(54(41(x1))))))))))) (1227)
13(35(53(35(54(44(x1)))))) 12(22(23(32(25(52(22(23(35(54(44(x1))))))))))) (1228)
13(35(53(35(54(45(x1)))))) 12(22(23(32(25(52(22(23(35(54(45(x1))))))))))) (1229)
13(35(53(35(54(42(x1)))))) 12(22(23(32(25(52(22(23(35(54(42(x1))))))))))) (1230)
53(35(53(35(54(40(x1)))))) 52(22(23(32(25(52(22(23(35(54(40(x1))))))))))) (1231)
53(35(53(35(54(43(x1)))))) 52(22(23(32(25(52(22(23(35(54(43(x1))))))))))) (1232)
53(35(53(35(54(41(x1)))))) 52(22(23(32(25(52(22(23(35(54(41(x1))))))))))) (1233)
53(35(53(35(54(44(x1)))))) 52(22(23(32(25(52(22(23(35(54(44(x1))))))))))) (1234)
53(35(53(35(54(45(x1)))))) 52(22(23(32(25(52(22(23(35(54(45(x1))))))))))) (1235)
53(35(53(35(54(42(x1)))))) 52(22(23(32(25(52(22(23(35(54(42(x1))))))))))) (1236)
23(35(53(35(54(40(x1)))))) 22(22(23(32(25(52(22(23(35(54(40(x1))))))))))) (1237)
23(35(53(35(54(43(x1)))))) 22(22(23(32(25(52(22(23(35(54(43(x1))))))))))) (1238)
23(35(53(35(54(41(x1)))))) 22(22(23(32(25(52(22(23(35(54(41(x1))))))))))) (1239)
23(35(53(35(54(44(x1)))))) 22(22(23(32(25(52(22(23(35(54(44(x1))))))))))) (1240)
23(35(53(35(54(45(x1)))))) 22(22(23(32(25(52(22(23(35(54(45(x1))))))))))) (1241)
23(35(53(35(54(42(x1)))))) 22(22(23(32(25(52(22(23(35(54(42(x1))))))))))) (1242)
43(35(53(35(54(40(x1)))))) 42(22(23(32(25(52(22(23(35(54(40(x1))))))))))) (1243)
43(35(53(35(54(43(x1)))))) 42(22(23(32(25(52(22(23(35(54(43(x1))))))))))) (1244)
43(35(53(35(54(41(x1)))))) 42(22(23(32(25(52(22(23(35(54(41(x1))))))))))) (1245)
43(35(53(35(54(44(x1)))))) 42(22(23(32(25(52(22(23(35(54(44(x1))))))))))) (1246)
43(35(53(35(54(45(x1)))))) 42(22(23(32(25(52(22(23(35(54(45(x1))))))))))) (1247)
43(35(53(35(54(42(x1)))))) 42(22(23(32(25(52(22(23(35(54(42(x1))))))))))) (1248)

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
[11(x1)] = 1 + 1 · x1
[10(x1)] = 1 + 1 · x1
[00(x1)] = 1 · x1
[34(x1)] = 1 · x1
[45(x1)] = 1 + 1 · x1
[54(x1)] = 1 · x1
[43(x1)] = 1 · x1
[35(x1)] = 1 · x1
[51(x1)] = 1 · x1
[13(x1)] = 1 · x1
[14(x1)] = 1 · x1
[15(x1)] = 1 + 1 · x1
[12(x1)] = 1 · x1
[41(x1)] = 1 · x1
[40(x1)] = 1 + 1 · x1
[02(x1)] = 1 + 1 · x1
[25(x1)] = 1 + 1 · x1
[55(x1)] = 1 + 1 · x1
[24(x1)] = 1 · x1
[52(x1)] = 1 · x1
[23(x1)] = 1 · x1
[42(x1)] = 1 · x1
[53(x1)] = 1 · x1
[30(x1)] = 1 · x1
[01(x1)] = 1 · x1
[04(x1)] = 1 + 1 · x1
[05(x1)] = 1 · x1
[21(x1)] = 1 + 1 · x1
[50(x1)] = 1 · x1
[22(x1)] = 1 · x1
[44(x1)] = 1 · x1
[32(x1)] = 1 · x1
[20(x1)] = 1 + 1 · x1
all of the following rules can be deleted.
55(55(51(12(24(40(00(x1))))))) 52(23(34(42(24(45(53(33(30(00(x1)))))))))) (385)
55(55(51(12(24(40(03(x1))))))) 52(23(34(42(24(45(53(33(30(03(x1)))))))))) (386)
55(55(51(12(24(40(01(x1))))))) 52(23(34(42(24(45(53(33(30(01(x1)))))))))) (387)
55(55(51(12(24(40(04(x1))))))) 52(23(34(42(24(45(53(33(30(04(x1)))))))))) (388)
55(55(51(12(24(40(05(x1))))))) 52(23(34(42(24(45(53(33(30(05(x1)))))))))) (389)
55(55(51(12(24(40(02(x1))))))) 52(23(34(42(24(45(53(33(30(02(x1)))))))))) (390)
45(54(40(02(21(11(x1)))))) 40(02(24(43(35(51(12(25(52(25(51(x1))))))))))) (531)
04(43(30(02(21(15(50(x1))))))) 05(54(41(12(21(11(12(23(35(55(50(x1))))))))))) (685)
04(43(30(02(21(15(53(x1))))))) 05(54(41(12(21(11(12(23(35(55(53(x1))))))))))) (686)
04(43(30(02(21(15(51(x1))))))) 05(54(41(12(21(11(12(23(35(55(51(x1))))))))))) (687)
04(43(30(02(21(15(54(x1))))))) 05(54(41(12(21(11(12(23(35(55(54(x1))))))))))) (688)
04(43(30(02(21(15(55(x1))))))) 05(54(41(12(21(11(12(23(35(55(55(x1))))))))))) (689)
04(43(30(02(21(15(52(x1))))))) 05(54(41(12(21(11(12(23(35(55(52(x1))))))))))) (690)
21(11(14(42(20(00(x1)))))) 21(14(45(54(42(23(33(35(50(00(x1)))))))))) (1183)
21(11(14(42(20(03(x1)))))) 21(14(45(54(42(23(33(35(50(03(x1)))))))))) (1184)
21(11(14(42(20(01(x1)))))) 21(14(45(54(42(23(33(35(50(01(x1)))))))))) (1185)
21(11(14(42(20(04(x1)))))) 21(14(45(54(42(23(33(35(50(04(x1)))))))))) (1186)
21(11(14(42(20(05(x1)))))) 21(14(45(54(42(23(33(35(50(05(x1)))))))))) (1187)
21(11(14(42(20(02(x1)))))) 21(14(45(54(42(23(33(35(50(02(x1)))))))))) (1188)

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 + 1 · x1
[31(x1)] = 1 + 1 · x1
[11(x1)] = 1 · x1
[10(x1)] = 1 + 1 · x1
[00(x1)] = 1 · x1
[34(x1)] = 1 · x1
[45(x1)] = 1 + 1 · x1
[54(x1)] = 1 + 1 · x1
[43(x1)] = 1 · x1
[35(x1)] = 1 · x1
[51(x1)] = 1 · x1
[13(x1)] = 1 · x1
[14(x1)] = 1 · x1
[15(x1)] = 1 · x1
[12(x1)] = 1 · x1
[41(x1)] = 1 · x1
[40(x1)] = 1 + 1 · x1
[02(x1)] = 1 · x1
[25(x1)] = 1 · x1
[55(x1)] = 1 · x1
[21(x1)] = 1 + 1 · x1
[24(x1)] = 1 · x1
[52(x1)] = 1 · x1
[53(x1)] = 1 · x1
[50(x1)] = 1 · x1
[05(x1)] = 1 + 1 · x1
[42(x1)] = 1 · x1
[22(x1)] = 1 · x1
[23(x1)] = 1 · x1
[44(x1)] = 1 + 1 · x1
[32(x1)] = 1 · x1
[20(x1)] = 1 · x1
[01(x1)] = 1 · x1
[04(x1)] = 1 · x1
all of the following rules can be deleted.
41(15(54(40(02(25(55(x1))))))) 41(13(33(34(41(11(11(11(11(15(x1)))))))))) (359)
45(54(40(02(21(13(x1)))))) 40(02(24(43(35(51(12(25(52(25(53(x1))))))))))) (530)
45(54(40(02(21(12(x1)))))) 40(02(24(43(35(51(12(25(52(25(52(x1))))))))))) (534)
12(25(55(54(45(50(05(50(x1)))))))) 15(55(53(34(42(25(52(22(23(35(50(x1))))))))))) (907)
12(25(55(54(45(50(05(53(x1)))))))) 15(55(53(34(42(25(52(22(23(35(53(x1))))))))))) (908)
12(25(55(54(45(50(05(51(x1)))))))) 15(55(53(34(42(25(52(22(23(35(51(x1))))))))))) (909)
12(25(55(54(45(50(05(54(x1)))))))) 15(55(53(34(42(25(52(22(23(35(54(x1))))))))))) (910)
12(25(55(54(45(50(05(55(x1)))))))) 15(55(53(34(42(25(52(22(23(35(55(x1))))))))))) (911)
12(25(55(54(45(50(05(52(x1)))))))) 15(55(53(34(42(25(52(22(23(35(52(x1))))))))))) (912)
33(34(44(40(05(51(10(00(x1)))))))) 32(22(20(05(53(33(35(53(31(10(00(x1))))))))))) (967)
33(34(44(40(05(51(10(03(x1)))))))) 32(22(20(05(53(33(35(53(31(10(03(x1))))))))))) (968)
33(34(44(40(05(51(10(01(x1)))))))) 32(22(20(05(53(33(35(53(31(10(01(x1))))))))))) (969)
33(34(44(40(05(51(10(04(x1)))))))) 32(22(20(05(53(33(35(53(31(10(04(x1))))))))))) (970)
33(34(44(40(05(51(10(05(x1)))))))) 32(22(20(05(53(33(35(53(31(10(05(x1))))))))))) (971)
33(34(44(40(05(51(10(02(x1)))))))) 32(22(20(05(53(33(35(53(31(10(02(x1))))))))))) (972)

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
[11(x1)] = 1 + 1 · x1
[10(x1)] = 1 · x1
[00(x1)] = 1 · x1
[34(x1)] = 1 · x1
[45(x1)] = 1 · x1
[54(x1)] = 1 · x1
[43(x1)] = 1 + 1 · x1
[35(x1)] = 1 · x1
[51(x1)] = 1 · x1
[13(x1)] = 1 · x1
[14(x1)] = 1 · x1
[15(x1)] = 1 · x1
[12(x1)] = 1 · x1
[23(x1)] = 1 + 1 · x1
[44(x1)] = 1 + 1 · x1
[40(x1)] = 1 + 1 · x1
[05(x1)] = 1 · x1
[22(x1)] = 1 · x1
[20(x1)] = 1 · x1
[53(x1)] = 1 · x1
[01(x1)] = 1 · x1
[04(x1)] = 1 · x1
[02(x1)] = 1 · x1
[42(x1)] = 1 · x1
[41(x1)] = 1 + 1 · x1
[50(x1)] = 1 · x1
[32(x1)] = 1 · x1
[25(x1)] = 1 · x1
all of the following rules can be deleted.
23(34(44(40(05(51(10(00(x1)))))))) 22(22(20(05(53(33(35(53(31(10(00(x1))))))))))) (991)
23(34(44(40(05(51(10(03(x1)))))))) 22(22(20(05(53(33(35(53(31(10(03(x1))))))))))) (992)
23(34(44(40(05(51(10(01(x1)))))))) 22(22(20(05(53(33(35(53(31(10(01(x1))))))))))) (993)
23(34(44(40(05(51(10(04(x1)))))))) 22(22(20(05(53(33(35(53(31(10(04(x1))))))))))) (994)
23(34(44(40(05(51(10(05(x1)))))))) 22(22(20(05(53(33(35(53(31(10(05(x1))))))))))) (995)
23(34(44(40(05(51(10(02(x1)))))))) 22(22(20(05(53(33(35(53(31(10(02(x1))))))))))) (996)
43(34(44(40(05(51(10(00(x1)))))))) 42(22(20(05(53(33(35(53(31(10(00(x1))))))))))) (997)
43(34(44(40(05(51(10(03(x1)))))))) 42(22(20(05(53(33(35(53(31(10(03(x1))))))))))) (998)
43(34(44(40(05(51(10(01(x1)))))))) 42(22(20(05(53(33(35(53(31(10(01(x1))))))))))) (999)
43(34(44(40(05(51(10(04(x1)))))))) 42(22(20(05(53(33(35(53(31(10(04(x1))))))))))) (1000)
43(34(44(40(05(51(10(05(x1)))))))) 42(22(20(05(53(33(35(53(31(10(05(x1))))))))))) (1001)
43(34(44(40(05(51(10(02(x1)))))))) 42(22(20(05(53(33(35(53(31(10(02(x1))))))))))) (1002)
11(15(54(44(41(10(01(10(x1)))))))) 15(54(45(50(05(53(32(25(50(01(10(x1))))))))))) (1333)
11(15(54(44(41(10(01(13(x1)))))))) 15(54(45(50(05(53(32(25(50(01(13(x1))))))))))) (1334)
11(15(54(44(41(10(01(11(x1)))))))) 15(54(45(50(05(53(32(25(50(01(11(x1))))))))))) (1335)
11(15(54(44(41(10(01(14(x1)))))))) 15(54(45(50(05(53(32(25(50(01(14(x1))))))))))) (1336)
11(15(54(44(41(10(01(15(x1)))))))) 15(54(45(50(05(53(32(25(50(01(15(x1))))))))))) (1337)
11(15(54(44(41(10(01(12(x1)))))))) 15(54(45(50(05(53(32(25(50(01(12(x1))))))))))) (1338)

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 + 1
[11(x1)] = 1 · x1
[10(x1)] = 1 · x1
[00(x1)] = 1 · x1
[34(x1)] = 1 · x1
[45(x1)] = 1 · x1
[54(x1)] = 1 · x1
[43(x1)] = 1 · x1
[35(x1)] = 1 · x1
[51(x1)] = 1 · x1
[13(x1)] = 1 · x1
[14(x1)] = 1 · x1
[15(x1)] = 1 · x1
[12(x1)] = 1 · x1
all of the following rules can be deleted.
03(33(31(11(10(x1))))) 00(00(03(34(45(54(43(35(51(10(x1)))))))))) (223)
03(33(31(11(13(x1))))) 00(00(03(34(45(54(43(35(51(13(x1)))))))))) (224)
03(33(31(11(11(x1))))) 00(00(03(34(45(54(43(35(51(11(x1)))))))))) (225)
03(33(31(11(14(x1))))) 00(00(03(34(45(54(43(35(51(14(x1)))))))))) (226)
03(33(31(11(15(x1))))) 00(00(03(34(45(54(43(35(51(15(x1)))))))))) (227)
03(33(31(11(12(x1))))) 00(00(03(34(45(54(43(35(51(12(x1)))))))))) (228)

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.