Certification Problem

Input (TPDB SRS_Relative/ICFP_2010_relative/5130)

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

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

and S is the following TRS.

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

1.1 Closure Under Flat Contexts

Using the flat contexts

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

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

There are 808 ruless (increase limit for explicit display).

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[02(x1)] = 1 · x1
[21(x1)] = 1 + 1 · x1
[14(x1)] = 1 · x1
[45(x1)] = 1 + 1 · x1
[54(x1)] = 1 · x1
[03(x1)] = 1 · x1
[31(x1)] = 1 · x1
[40(x1)] = 1 + 1 · x1
[34(x1)] = 1 · x1
[44(x1)] = 1 · x1
[00(x1)] = 1 · x1
[04(x1)] = 1 · x1
[13(x1)] = 1 · x1
[30(x1)] = 1 · x1
[01(x1)] = 1 · x1
[10(x1)] = 1 · x1
[35(x1)] = 1 · x1
[32(x1)] = 1 · x1
[42(x1)] = 1 · x1
[41(x1)] = 1 · x1
[33(x1)] = 1 · x1
[43(x1)] = 1 · x1
[50(x1)] = 1 · x1
[24(x1)] = 1 · x1
[05(x1)] = 1 · x1
[53(x1)] = 1 + 1 · x1
[15(x1)] = 1 · x1
[22(x1)] = 1 + 1 · x1
[20(x1)] = 1 · x1
[23(x1)] = 1 + 1 · x1
[12(x1)] = 1 + 1 · x1
[51(x1)] = 1 · x1
[11(x1)] = 1 + 1 · x1
[55(x1)] = 1 · x1
[25(x1)] = 1 · x1
[52(x1)] = 1 · x1
all of the following rules can be deleted.
13(30(04(40(03(32(x1)))))) 13(30(01(10(03(35(54(44(44(42(x1)))))))))) (272)
13(30(04(40(03(31(x1)))))) 13(30(01(10(03(35(54(44(44(41(x1)))))))))) (273)
13(30(04(40(03(34(x1)))))) 13(30(01(10(03(35(54(44(44(44(x1)))))))))) (274)
13(30(04(40(03(33(x1)))))) 13(30(01(10(03(35(54(44(44(43(x1)))))))))) (276)
44(45(50(03(34(40(x1)))))) 42(24(44(41(13(35(54(43(34(40(x1)))))))))) (313)
44(45(50(03(34(42(x1)))))) 42(24(44(41(13(35(54(43(34(42(x1)))))))))) (314)
44(45(50(03(34(41(x1)))))) 42(24(44(41(13(35(54(43(34(41(x1)))))))))) (315)
44(45(50(03(34(44(x1)))))) 42(24(44(41(13(35(54(43(34(44(x1)))))))))) (316)
44(45(50(03(34(45(x1)))))) 42(24(44(41(13(35(54(43(34(45(x1)))))))))) (317)
44(45(50(03(34(43(x1)))))) 42(24(44(41(13(35(54(43(34(43(x1)))))))))) (318)
05(53(32(21(15(50(x1)))))) 03(33(34(44(42(24(44(42(22(20(x1)))))))))) (319)
05(53(32(21(15(53(x1)))))) 03(33(34(44(42(24(44(42(22(23(x1)))))))))) (324)
31(12(20(04(45(51(14(x1))))))) 30(04(44(43(33(35(51(11(10(04(x1)))))))))) (382)
53(32(22(21(14(43(30(x1))))))) 55(54(44(44(44(44(41(14(45(53(30(x1))))))))))) (649)
53(32(22(21(14(43(32(x1))))))) 55(54(44(44(44(44(41(14(45(53(32(x1))))))))))) (650)
53(32(22(21(14(43(31(x1))))))) 55(54(44(44(44(44(41(14(45(53(31(x1))))))))))) (651)
53(32(22(21(14(43(34(x1))))))) 55(54(44(44(44(44(41(14(45(53(34(x1))))))))))) (652)
53(32(22(21(14(43(35(x1))))))) 55(54(44(44(44(44(41(14(45(53(35(x1))))))))))) (653)
53(32(22(21(14(43(33(x1))))))) 55(54(44(44(44(44(41(14(45(53(33(x1))))))))))) (654)
13(35(51(14(40(03(30(00(x1)))))))) 15(50(00(03(34(44(43(33(30(00(00(x1))))))))))) (703)
13(35(51(14(40(03(30(02(x1)))))))) 15(50(00(03(34(44(43(33(30(00(02(x1))))))))))) (704)
13(35(51(14(40(03(30(01(x1)))))))) 15(50(00(03(34(44(43(33(30(00(01(x1))))))))))) (705)
13(35(51(14(40(03(30(04(x1)))))))) 15(50(00(03(34(44(43(33(30(00(04(x1))))))))))) (706)
13(35(51(14(40(03(30(05(x1)))))))) 15(50(00(03(34(44(43(33(30(00(05(x1))))))))))) (707)
13(35(51(14(40(03(30(03(x1)))))))) 15(50(00(03(34(44(43(33(30(00(03(x1))))))))))) (708)
03(35(51(14(40(03(30(00(x1)))))))) 05(50(00(03(34(44(43(33(30(00(00(x1))))))))))) (709)
03(35(51(14(40(03(30(02(x1)))))))) 05(50(00(03(34(44(43(33(30(00(02(x1))))))))))) (710)
03(35(51(14(40(03(30(01(x1)))))))) 05(50(00(03(34(44(43(33(30(00(01(x1))))))))))) (711)
03(35(51(14(40(03(30(04(x1)))))))) 05(50(00(03(34(44(43(33(30(00(04(x1))))))))))) (712)
03(35(51(14(40(03(30(05(x1)))))))) 05(50(00(03(34(44(43(33(30(00(05(x1))))))))))) (713)
03(35(51(14(40(03(30(03(x1)))))))) 05(50(00(03(34(44(43(33(30(00(03(x1))))))))))) (714)
53(35(51(14(40(03(30(00(x1)))))))) 55(50(00(03(34(44(43(33(30(00(00(x1))))))))))) (721)
53(35(51(14(40(03(30(02(x1)))))))) 55(50(00(03(34(44(43(33(30(00(02(x1))))))))))) (722)
53(35(51(14(40(03(30(01(x1)))))))) 55(50(00(03(34(44(43(33(30(00(01(x1))))))))))) (723)
53(35(51(14(40(03(30(04(x1)))))))) 55(50(00(03(34(44(43(33(30(00(04(x1))))))))))) (724)
53(35(51(14(40(03(30(05(x1)))))))) 55(50(00(03(34(44(43(33(30(00(05(x1))))))))))) (725)
53(35(51(14(40(03(30(03(x1)))))))) 55(50(00(03(34(44(43(33(30(00(03(x1))))))))))) (726)
22(22(21(14(42(21(14(40(x1)))))))) 25(50(04(44(41(11(13(35(54(43(30(x1))))))))))) (877)
22(22(21(14(42(21(14(42(x1)))))))) 25(50(04(44(41(11(13(35(54(43(32(x1))))))))))) (878)
22(22(21(14(42(21(14(41(x1)))))))) 25(50(04(44(41(11(13(35(54(43(31(x1))))))))))) (879)
22(22(21(14(42(21(14(44(x1)))))))) 25(50(04(44(41(11(13(35(54(43(34(x1))))))))))) (880)
22(22(21(14(42(21(14(45(x1)))))))) 25(50(04(44(41(11(13(35(54(43(35(x1))))))))))) (881)
22(22(21(14(42(21(14(43(x1)))))))) 25(50(04(44(41(11(13(35(54(43(33(x1))))))))))) (882)
12(22(21(14(42(21(14(40(x1)))))))) 15(50(04(44(41(11(13(35(54(43(30(x1))))))))))) (883)
12(22(21(14(42(21(14(42(x1)))))))) 15(50(04(44(41(11(13(35(54(43(32(x1))))))))))) (884)
12(22(21(14(42(21(14(41(x1)))))))) 15(50(04(44(41(11(13(35(54(43(31(x1))))))))))) (885)
12(22(21(14(42(21(14(44(x1)))))))) 15(50(04(44(41(11(13(35(54(43(34(x1))))))))))) (886)
12(22(21(14(42(21(14(45(x1)))))))) 15(50(04(44(41(11(13(35(54(43(35(x1))))))))))) (887)
12(22(21(14(42(21(14(43(x1)))))))) 15(50(04(44(41(11(13(35(54(43(33(x1))))))))))) (888)
02(22(21(14(42(21(14(40(x1)))))))) 05(50(04(44(41(11(13(35(54(43(30(x1))))))))))) (889)
02(22(21(14(42(21(14(42(x1)))))))) 05(50(04(44(41(11(13(35(54(43(32(x1))))))))))) (890)
02(22(21(14(42(21(14(41(x1)))))))) 05(50(04(44(41(11(13(35(54(43(31(x1))))))))))) (891)
02(22(21(14(42(21(14(44(x1)))))))) 05(50(04(44(41(11(13(35(54(43(34(x1))))))))))) (892)
02(22(21(14(42(21(14(45(x1)))))))) 05(50(04(44(41(11(13(35(54(43(35(x1))))))))))) (893)
02(22(21(14(42(21(14(43(x1)))))))) 05(50(04(44(41(11(13(35(54(43(33(x1))))))))))) (894)
32(22(21(14(42(21(14(40(x1)))))))) 35(50(04(44(41(11(13(35(54(43(30(x1))))))))))) (895)
32(22(21(14(42(21(14(42(x1)))))))) 35(50(04(44(41(11(13(35(54(43(32(x1))))))))))) (896)
32(22(21(14(42(21(14(41(x1)))))))) 35(50(04(44(41(11(13(35(54(43(31(x1))))))))))) (897)
32(22(21(14(42(21(14(44(x1)))))))) 35(50(04(44(41(11(13(35(54(43(34(x1))))))))))) (898)
32(22(21(14(42(21(14(45(x1)))))))) 35(50(04(44(41(11(13(35(54(43(35(x1))))))))))) (899)
32(22(21(14(42(21(14(43(x1)))))))) 35(50(04(44(41(11(13(35(54(43(33(x1))))))))))) (900)
52(22(21(14(42(21(14(40(x1)))))))) 55(50(04(44(41(11(13(35(54(43(30(x1))))))))))) (901)
52(22(21(14(42(21(14(42(x1)))))))) 55(50(04(44(41(11(13(35(54(43(32(x1))))))))))) (902)
52(22(21(14(42(21(14(41(x1)))))))) 55(50(04(44(41(11(13(35(54(43(31(x1))))))))))) (903)
52(22(21(14(42(21(14(44(x1)))))))) 55(50(04(44(41(11(13(35(54(43(34(x1))))))))))) (904)
52(22(21(14(42(21(14(45(x1)))))))) 55(50(04(44(41(11(13(35(54(43(35(x1))))))))))) (905)
52(22(21(14(42(21(14(43(x1)))))))) 55(50(04(44(41(11(13(35(54(43(33(x1))))))))))) (906)
42(22(21(14(42(21(14(40(x1)))))))) 45(50(04(44(41(11(13(35(54(43(30(x1))))))))))) (907)
42(22(21(14(42(21(14(42(x1)))))))) 45(50(04(44(41(11(13(35(54(43(32(x1))))))))))) (908)
42(22(21(14(42(21(14(41(x1)))))))) 45(50(04(44(41(11(13(35(54(43(31(x1))))))))))) (909)
42(22(21(14(42(21(14(44(x1)))))))) 45(50(04(44(41(11(13(35(54(43(34(x1))))))))))) (910)
42(22(21(14(42(21(14(45(x1)))))))) 45(50(04(44(41(11(13(35(54(43(35(x1))))))))))) (911)
42(22(21(14(42(21(14(43(x1)))))))) 45(50(04(44(41(11(13(35(54(43(33(x1))))))))))) (912)
43(33(31(12(21(10(x1)))))) 42(23(30(03(31(14(42(24(41(10(x1)))))))))) (1057)
43(33(31(12(21(12(x1)))))) 42(23(30(03(31(14(42(24(41(12(x1)))))))))) (1058)
43(33(31(12(21(11(x1)))))) 42(23(30(03(31(14(42(24(41(11(x1)))))))))) (1059)
43(33(31(12(21(14(x1)))))) 42(23(30(03(31(14(42(24(41(14(x1)))))))))) (1060)
43(33(31(12(21(15(x1)))))) 42(23(30(03(31(14(42(24(41(15(x1)))))))))) (1061)
43(33(31(12(21(13(x1)))))) 42(23(30(03(31(14(42(24(41(13(x1)))))))))) (1062)
50(03(34(40(x1)))) 50(03(33(31(14(43(34(43(33(30(x1)))))))))) (1075)
50(03(34(45(x1)))) 50(03(33(31(14(43(34(43(33(35(x1)))))))))) (1079)
00(04(45(52(21(10(x1)))))) 04(43(30(03(35(54(44(45(51(10(x1)))))))))) (1081)
00(04(45(52(21(12(x1)))))) 04(43(30(03(35(54(44(45(51(12(x1)))))))))) (1082)
00(04(45(52(21(11(x1)))))) 04(43(30(03(35(54(44(45(51(11(x1)))))))))) (1083)
00(04(45(52(21(14(x1)))))) 04(43(30(03(35(54(44(45(51(14(x1)))))))))) (1084)
00(04(45(52(21(15(x1)))))) 04(43(30(03(35(54(44(45(51(15(x1)))))))))) (1085)
00(04(45(52(21(13(x1)))))) 04(43(30(03(35(54(44(45(51(13(x1)))))))))) (1086)
22(22(21(12(23(31(11(x1))))))) 25(52(20(03(33(34(45(54(44(45(51(x1))))))))))) (1095)
12(22(21(12(23(31(11(x1))))))) 15(52(20(03(33(34(45(54(44(45(51(x1))))))))))) (1101)
02(22(21(12(23(31(11(x1))))))) 05(52(20(03(33(34(45(54(44(45(51(x1))))))))))) (1107)
32(22(21(12(23(31(11(x1))))))) 35(52(20(03(33(34(45(54(44(45(51(x1))))))))))) (1113)
52(22(21(12(23(31(11(x1))))))) 55(52(20(03(33(34(45(54(44(45(51(x1))))))))))) (1119)
42(22(21(12(23(31(11(x1))))))) 45(52(20(03(33(34(45(54(44(45(51(x1))))))))))) (1125)
22(23(35(50(03(33(32(21(x1)))))))) 25(50(00(00(01(14(40(01(13(35(51(x1))))))))))) (1131)
12(23(35(50(03(33(32(21(x1)))))))) 15(50(00(00(01(14(40(01(13(35(51(x1))))))))))) (1137)
02(23(35(50(03(33(32(21(x1)))))))) 05(50(00(00(01(14(40(01(13(35(51(x1))))))))))) (1143)
32(23(35(50(03(33(32(21(x1)))))))) 35(50(00(00(01(14(40(01(13(35(51(x1))))))))))) (1149)
52(23(35(50(03(33(32(21(x1)))))))) 55(50(00(00(01(14(40(01(13(35(51(x1))))))))))) (1155)

1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[02(x1)] = 1 + 1 · x1
[21(x1)] = 1 · x1
[14(x1)] = 1 · x1
[45(x1)] = 1 · x1
[54(x1)] = 1 · x1
[03(x1)] = 1 · x1
[31(x1)] = 1 · x1
[40(x1)] = 1 · x1
[34(x1)] = 1 · x1
[44(x1)] = 1 · x1
[00(x1)] = 1 · x1
[04(x1)] = 1 + 1 · x1
[13(x1)] = 1 · x1
[30(x1)] = 1 + 1 · x1
[01(x1)] = 1 · x1
[10(x1)] = 1 + 1 · x1
[35(x1)] = 1 + 1 · x1
[33(x1)] = 1 · x1
[50(x1)] = 1 · x1
[41(x1)] = 1 + 1 · x1
[42(x1)] = 1 · x1
[20(x1)] = 1 · x1
[53(x1)] = 1 · x1
[23(x1)] = 1 + 1 · x1
[15(x1)] = 1 · x1
[11(x1)] = 1 + 1 · x1
[51(x1)] = 1 · x1
[32(x1)] = 1 · x1
[22(x1)] = 1 + 1 · x1
[43(x1)] = 1 · x1
[05(x1)] = 1 · x1
all of the following rules can be deleted.
33(31(10(03(35(50(x1)))))) 33(31(13(34(44(41(14(44(42(20(x1)))))))))) (325)
21(15(50(04(41(11(x1)))))) 20(03(33(33(34(45(54(44(44(45(51(x1))))))))))) (555)
11(15(50(04(41(11(x1)))))) 10(03(33(33(34(45(54(44(44(45(51(x1))))))))))) (561)
50(03(34(41(x1)))) 50(03(33(31(14(43(34(43(33(31(x1)))))))))) (1077)
42(23(35(50(03(33(32(21(x1)))))))) 45(50(00(00(01(14(40(01(13(35(51(x1))))))))))) (1161)

1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[02(x1)] = 1 + 1 · x1
[21(x1)] = 1 · x1
[14(x1)] = 1 · x1
[45(x1)] = 1 · x1
[54(x1)] = 1 · x1
[03(x1)] = 1 · x1
[31(x1)] = 1 · x1
[40(x1)] = 1 · x1
[34(x1)] = 1 · x1
[44(x1)] = 1 · x1
[00(x1)] = 1 · x1
[04(x1)] = 1 + 1 · x1
[13(x1)] = 1 · x1
[30(x1)] = 1 + 1 · x1
[01(x1)] = 1 · x1
[10(x1)] = 1 + 1 · x1
[35(x1)] = 1 · x1
[33(x1)] = 1 · x1
[53(x1)] = 1 · x1
[41(x1)] = 1 · x1
[42(x1)] = 1 · x1
[23(x1)] = 1 · x1
[32(x1)] = 1 · x1
[22(x1)] = 1 + 1 · x1
[43(x1)] = 1 · x1
[15(x1)] = 1 · x1
[05(x1)] = 1 · x1
[50(x1)] = 1 · x1
all of the following rules can be deleted.
13(30(04(40(03(30(x1)))))) 13(30(01(10(03(35(54(44(44(40(x1)))))))))) (271)
33(31(10(03(35(53(x1)))))) 33(31(13(34(44(41(14(44(42(23(x1)))))))))) (330)
13(32(22(21(14(43(30(x1))))))) 15(54(44(44(44(44(41(14(45(53(30(x1))))))))))) (631)
13(32(22(21(14(43(32(x1))))))) 15(54(44(44(44(44(41(14(45(53(32(x1))))))))))) (632)
13(32(22(21(14(43(31(x1))))))) 15(54(44(44(44(44(41(14(45(53(31(x1))))))))))) (633)
13(32(22(21(14(43(34(x1))))))) 15(54(44(44(44(44(41(14(45(53(34(x1))))))))))) (634)
13(32(22(21(14(43(35(x1))))))) 15(54(44(44(44(44(41(14(45(53(35(x1))))))))))) (635)
13(32(22(21(14(43(33(x1))))))) 15(54(44(44(44(44(41(14(45(53(33(x1))))))))))) (636)
03(32(22(21(14(43(30(x1))))))) 05(54(44(44(44(44(41(14(45(53(30(x1))))))))))) (637)
03(32(22(21(14(43(32(x1))))))) 05(54(44(44(44(44(41(14(45(53(32(x1))))))))))) (638)
03(32(22(21(14(43(31(x1))))))) 05(54(44(44(44(44(41(14(45(53(31(x1))))))))))) (639)
03(32(22(21(14(43(34(x1))))))) 05(54(44(44(44(44(41(14(45(53(34(x1))))))))))) (640)
03(32(22(21(14(43(35(x1))))))) 05(54(44(44(44(44(41(14(45(53(35(x1))))))))))) (641)
03(32(22(21(14(43(33(x1))))))) 05(54(44(44(44(44(41(14(45(53(33(x1))))))))))) (642)

1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[02(x1)] = 1 + 1 · x1
[21(x1)] = 1 + 1 · x1
[14(x1)] = 1 · x1
[45(x1)] = 1 · x1
[54(x1)] = 1 · x1
[03(x1)] = 1 · x1
[31(x1)] = 1 · x1
[40(x1)] = 1 · x1
[34(x1)] = 1 · x1
[44(x1)] = 1 · x1
[00(x1)] = 1 · x1
[04(x1)] = 1 · x1
[13(x1)] = 1 · x1
[30(x1)] = 1 · x1
[35(x1)] = 1 · x1
[01(x1)] = 1 · x1
[10(x1)] = 1 · x1
[50(x1)] = 1 · x1
[42(x1)] = 1 + 1 · x1
[33(x1)] = 1 · x1
[43(x1)] = 1 · x1
[32(x1)] = 1 + 1 · x1
all of the following rules can be deleted.
02(21(14(45(54(x1))))) 03(31(14(40(03(34(44(40(00(04(x1)))))))))) (238)

1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[13(x1)] = 1 · x1
[30(x1)] = 1 · x1
[04(x1)] = 1 + 1 · x1
[40(x1)] = 1 + 1 · x1
[03(x1)] = 1 · x1
[35(x1)] = 1 · x1
[01(x1)] = 1 + 1 · x1
[10(x1)] = 1 + 1 · x1
[54(x1)] = 1 · x1
[44(x1)] = 1 · x1
[45(x1)] = 1 · x1
[50(x1)] = 1 · x1
[34(x1)] = 1 · x1
[42(x1)] = 1 + 1 · x1
[33(x1)] = 1 · x1
[31(x1)] = 1 · x1
[14(x1)] = 1 · x1
[43(x1)] = 1 · x1
[32(x1)] = 1 · x1
all of the following rules can be deleted.
50(03(34(42(x1)))) 50(03(33(31(14(43(34(43(33(32(x1)))))))))) (1076)

1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[13(x1)] = 1 · x1
[30(x1)] = 1 · x1
[04(x1)] = 1 + 1 · x1
[40(x1)] = 1 + 1 · x1
[03(x1)] = 1 · x1
[35(x1)] = 1 · x1
[01(x1)] = 1 · x1
[10(x1)] = 1 · x1
[54(x1)] = 1 · x1
[44(x1)] = 1 · x1
[45(x1)] = 1 · x1
[50(x1)] = 1 · x1
[34(x1)] = 1 · x1
[33(x1)] = 1 · x1
[31(x1)] = 1 · x1
[14(x1)] = 1 · x1
[43(x1)] = 1 · x1
all of the following rules can be deleted.
13(30(04(40(03(35(x1)))))) 13(30(01(10(03(35(54(44(44(45(x1)))))))))) (275)

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.