Certification Problem

Input (TPDB SRS_Relative/ICFP_2010_relative/4816)

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

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

and S is the following TRS.

2(3(1(1(x1)))) 1(4(5(5(3(0(0(2(0(0(x1)))))))))) (56)
0(1(1(1(3(3(3(x1))))))) 3(5(4(5(3(2(1(5(2(3(x1)))))))))) (57)
1(1(4(5(2(1(5(x1))))))) 5(1(2(1(4(0(2(0(1(5(x1)))))))))) (58)
3(2(1(1(3(x1))))) 3(0(2(0(2(0(1(0(4(0(x1)))))))))) (59)
0(3(5(1(1(2(4(x1))))))) 0(5(4(0(0(5(2(0(0(4(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)))) 0(2(0(0(0(2(0(3(2(0(x1)))))))))) (61)
1(3(3(1(1(x1))))) 0(2(2(2(0(3(2(3(3(5(x1)))))))))) (62)
4(4(3(3(1(x1))))) 1(0(0(2(0(3(5(5(3(0(x1)))))))))) (63)
2(5(3(1(3(x1))))) 2(2(2(0(0(3(5(1(3(5(x1)))))))))) (64)
3(3(3(2(4(x1))))) 0(0(2(0(0(2(0(3(0(5(x1)))))))))) (65)
4(1(1(2(1(0(x1)))))) 4(0(2(2(0(3(2(4(0(0(x1)))))))))) (66)
2(4(3(4(4(2(x1)))))) 0(2(2(0(0(3(0(5(4(2(x1)))))))))) (67)
3(3(3(0(3(3(x1)))))) 0(2(0(3(0(4(5(4(1(5(x1)))))))))) (68)
1(4(5(2(5(3(x1)))))) 0(0(3(2(4(0(0(2(0(3(x1)))))))))) (69)
0(1(3(1(3(1(0(x1))))))) 0(1(3(4(1(0(2(0(3(2(x1)))))))))) (70)
3(5(3(3(3(1(0(x1))))))) 0(3(2(4(2(0(4(0(0(0(x1)))))))))) (71)
4(3(4(3(3(1(0(x1))))))) 4(2(0(2(4(0(2(0(3(0(x1)))))))))) (72)
3(1(5(3(3(2(0(x1))))))) 3(5(0(0(0(2(0(2(3(2(x1)))))))))) (73)
1(2(1(3(1(1(1(x1))))))) 1(5(2(2(2(0(3(5(2(1(x1)))))))))) (74)
3(3(2(1(3(1(1(x1))))))) 0(3(5(2(3(5(1(2(0(3(x1)))))))))) (75)
1(0(4(3(3(1(1(x1))))))) 2(2(3(2(1(5(0(1(2(3(x1)))))))))) (76)
3(3(4(3(4(2(1(x1))))))) 3(5(5(1(1(0(1(2(0(3(x1)))))))))) (77)
3(1(4(5(0(3(1(x1))))))) 3(5(2(2(0(0(4(2(3(1(x1)))))))))) (78)
4(2(3(0(3(3(1(x1))))))) 2(0(0(0(4(2(3(2(5(1(x1)))))))))) (79)
3(1(3(3(3(3(1(x1))))))) 3(2(2(4(1(4(2(0(3(1(x1)))))))))) (80)
4(4(3(3(3(3(1(x1))))))) 4(1(1(5(5(5(4(0(0(1(x1)))))))))) (81)
2(1(1(4(3(3(1(x1))))))) 0(2(2(0(3(4(2(5(1(3(x1)))))))))) (82)
1(1(1(5(3(3(1(x1))))))) 2(2(2(0(0(1(3(0(0(1(x1)))))))))) (83)
5(3(1(4(4(3(1(x1))))))) 5(1(2(0(2(0(3(3(3(1(x1)))))))))) (84)
4(3(2(0(5(3(1(x1))))))) 4(5(3(0(5(5(5(4(3(3(x1)))))))))) (85)
1(5(1(1(5(3(1(x1))))))) 0(0(4(0(0(4(4(0(0(1(x1)))))))))) (86)
3(4(3(1(5(3(1(x1))))))) 3(1(1(5(5(5(0(0(5(2(x1)))))))))) (87)
1(3(3(3(3(1(2(x1))))))) 1(3(0(0(4(0(2(0(3(2(x1)))))))))) (88)
0(1(1(1(1(3(2(x1))))))) 0(0(4(5(5(3(0(5(1(2(x1)))))))))) (89)
1(1(3(3(1(3(2(x1))))))) 1(5(5(5(0(2(2(2(2(2(x1)))))))))) (90)
3(3(1(5(3(3(2(x1))))))) 3(2(3(4(3(5(5(4(4(2(x1)))))))))) (91)
5(2(4(3(3(4(2(x1))))))) 5(0(1(5(2(0(4(2(4(0(x1)))))))))) (92)
1(5(3(0(2(5(2(x1))))))) 2(0(0(2(0(4(0(2(5(2(x1)))))))))) (93)
4(1(1(1(3(1(3(x1))))))) 4(1(0(4(0(0(3(5(3(5(x1)))))))))) (94)
0(4(4(1(3(1(3(x1))))))) 2(4(0(0(2(0(4(1(2(3(x1)))))))))) (95)
4(2(4(4(5(2(3(x1))))))) 4(0(0(2(2(2(0(1(3(5(x1)))))))))) (96)
2(5(4(3(3(3(3(x1))))))) 1(2(0(0(4(2(1(1(5(5(x1)))))))))) (97)
1(1(5(1(5(3(3(x1))))))) 1(3(4(5(5(4(2(0(3(5(x1)))))))))) (98)
4(0(5(2(5(3(3(x1))))))) 2(0(3(5(5(1(0(1(5(5(x1)))))))))) (99)
1(1(0(3(5(3(3(x1))))))) 1(5(4(5(3(5(5(4(1(5(x1)))))))))) (100)
1(1(3(1(1(4(3(x1))))))) 1(5(5(5(3(5(4(2(2(1(x1)))))))))) (101)
1(1(4(0(2(4(3(x1))))))) 1(5(5(2(0(2(0(3(3(5(x1)))))))))) (102)
3(2(0(5(2(4(3(x1))))))) 3(5(5(3(5(0(0(1(5(5(x1)))))))))) (103)
1(5(4(1(1(1(4(x1))))))) 2(0(4(2(0(3(4(0(3(5(x1)))))))))) (104)
4(3(3(3(1(2(4(x1))))))) 5(4(1(5(0(0(0(2(4(5(x1)))))))))) (105)
1(4(1(5(3(3(4(x1))))))) 4(0(2(3(5(0(0(5(3(5(x1)))))))))) (106)
3(3(3(4(0(4(4(x1))))))) 3(1(2(3(2(2(0(3(5(4(x1)))))))))) (107)
1(4(3(3(3(4(4(x1))))))) 2(0(2(0(1(2(0(1(0(4(x1)))))))))) (108)
3(4(2(5(1(5(4(x1))))))) 3(0(0(0(0(2(0(3(3(4(x1)))))))))) (109)
5(2(3(0(2(5(4(x1))))))) 5(0(0(0(4(5(4(3(0(5(x1)))))))))) (110)
3(1(4(5(2(5(4(x1))))))) 4(2(0(3(4(1(2(3(0(5(x1)))))))))) (111)
4(5(2(3(1(0(5(x1))))))) 1(5(2(0(0(0(0(3(2(4(x1)))))))))) (112)
0(5(1(1(1(1(5(x1))))))) 0(5(0(0(2(4(5(5(4(3(x1)))))))))) (113)
1(1(1(2(1(1(5(x1))))))) 1(2(0(2(1(5(0(0(5(0(x1)))))))))) (114)
2(2(5(3(1(2(5(x1))))))) 0(0(0(0(0(3(2(1(2(4(x1)))))))))) (115)
1(1(3(2(x1)))) 0(0(2(0(0(3(5(5(4(1(x1)))))))))) (116)
3(3(3(1(1(1(0(x1))))))) 3(2(5(1(2(3(5(4(5(3(x1)))))))))) (117)
5(1(2(5(4(1(1(x1))))))) 5(1(0(2(0(4(1(2(1(5(x1)))))))))) (118)
3(1(1(2(3(x1))))) 0(4(0(1(0(2(0(2(0(3(x1)))))))))) (119)
4(2(1(1(5(3(0(x1))))))) 4(0(0(2(5(0(0(4(5(0(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 195 ruless (increase limit for explicit display).

1.1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

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

1.1.1.1 Rule Removal

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

There are 1084 ruless (increase limit for explicit display).

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[33(x1)] = 1 · x1
[34(x1)] = 1 · x1
[43(x1)] = 1 · x1 + 1
[42(x1)] = 1 · x1 + 7
[21(x1)] = 1 · x1 + 25
[15(x1)] = 1 · x1 + 17
[35(x1)] = 1 · x1
[55(x1)] = 1 · x1
[51(x1)] = 1 · x1 + 10
[11(x1)] = 1 · x1 + 10
[10(x1)] = 1 · x1
[01(x1)] = 1 · x1
[12(x1)] = 1 · x1 + 6
[20(x1)] = 1 · x1
[03(x1)] = 1 · x1 + 1
[13(x1)] = 1 · x1
[31(x1)] = 1 · x1 + 15
[14(x1)] = 1 · x1
[45(x1)] = 1 · x1 + 9
[50(x1)] = 1 · x1
[52(x1)] = 1 · x1 + 18
[22(x1)] = 1 · x1
[00(x1)] = 1 · x1
[04(x1)] = 1 · x1
[23(x1)] = 1 · x1
[53(x1)] = 1 · x1
[44(x1)] = 1 · x1 + 1
[02(x1)] = 1 · x1
[32(x1)] = 1 · x1
[05(x1)] = 1 · x1 + 8
[30(x1)] = 1 · x1
[54(x1)] = 1 · x1
[24(x1)] = 1 · x1 + 9
[40(x1)] = 1 · x1 + 21
[41(x1)] = 1 · x1 + 11
[25(x1)] = 1 · x1
all of the following rules can be deleted.
33(34(43(34(42(21(15(x1))))))) 35(55(51(11(10(01(12(20(03(35(x1)))))))))) (320)
33(34(43(34(42(21(13(x1))))))) 35(55(51(11(10(01(12(20(03(33(x1)))))))))) (321)
33(34(43(34(42(21(11(x1))))))) 35(55(51(11(10(01(12(20(03(31(x1)))))))))) (322)
33(34(43(34(42(21(14(x1))))))) 35(55(51(11(10(01(12(20(03(34(x1)))))))))) (324)
43(32(20(05(53(31(15(x1))))))) 45(53(30(05(55(55(54(43(33(35(x1)))))))))) (350)
43(32(20(05(53(31(13(x1))))))) 45(53(30(05(55(55(54(43(33(33(x1)))))))))) (351)
43(32(20(05(53(31(11(x1))))))) 45(53(30(05(55(55(54(43(33(31(x1)))))))))) (352)
43(32(20(05(53(31(14(x1))))))) 45(53(30(05(55(55(54(43(33(34(x1)))))))))) (354)
42(24(44(45(52(23(32(x1))))))) 40(00(02(22(22(20(01(13(35(52(x1)))))))))) (397)
42(24(44(45(52(23(35(x1))))))) 40(00(02(22(22(20(01(13(35(55(x1)))))))))) (398)
42(24(44(45(52(23(33(x1))))))) 40(00(02(22(22(20(01(13(35(53(x1)))))))))) (399)
42(24(44(45(52(23(30(x1))))))) 40(00(02(22(22(20(01(13(35(50(x1)))))))))) (401)
42(24(44(45(52(23(34(x1))))))) 40(00(02(22(22(20(01(13(35(54(x1)))))))))) (402)
32(20(05(52(24(43(32(x1))))))) 35(55(53(35(50(00(01(15(55(52(x1)))))))))) (427)
32(20(05(52(24(43(35(x1))))))) 35(55(53(35(50(00(01(15(55(55(x1)))))))))) (428)
32(20(05(52(24(43(33(x1))))))) 35(55(53(35(50(00(01(15(55(53(x1)))))))))) (429)
32(20(05(52(24(43(30(x1))))))) 35(55(53(35(50(00(01(15(55(50(x1)))))))))) (431)
32(20(05(52(24(43(34(x1))))))) 35(55(53(35(50(00(01(15(55(54(x1)))))))))) (432)
52(23(30(02(25(54(45(x1))))))) 50(00(00(04(45(54(43(30(05(55(x1)))))))))) (446)
52(23(30(02(25(54(43(x1))))))) 50(00(00(04(45(54(43(30(05(53(x1)))))))))) (447)
52(23(30(02(25(54(41(x1))))))) 50(00(00(04(45(54(43(30(05(51(x1)))))))))) (448)
52(23(30(02(25(54(40(x1))))))) 50(00(00(04(45(54(43(30(05(50(x1)))))))))) (449)
52(23(30(02(25(54(44(x1))))))) 50(00(00(04(45(54(43(30(05(54(x1)))))))))) (450)
14(44(43(33(31(15(x1)))))) 11(10(00(02(20(03(35(55(53(30(05(x1))))))))))) (542)
14(44(43(33(31(14(x1)))))) 11(10(00(02(20(03(35(55(53(30(04(x1))))))))))) (546)
34(44(43(33(31(15(x1)))))) 31(10(00(02(20(03(35(55(53(30(05(x1))))))))))) (554)
34(44(43(33(31(14(x1)))))) 31(10(00(02(20(03(35(55(53(30(04(x1))))))))))) (558)
41(15(54(41(11(11(14(45(x1)))))))) 42(20(04(42(20(03(34(40(03(35(55(x1))))))))))) (1142)
41(15(54(41(11(11(14(43(x1)))))))) 42(20(04(42(20(03(34(40(03(35(53(x1))))))))))) (1143)
41(15(54(41(11(11(14(41(x1)))))))) 42(20(04(42(20(03(34(40(03(35(51(x1))))))))))) (1144)
41(15(54(41(11(11(14(40(x1)))))))) 42(20(04(42(20(03(34(40(03(35(50(x1))))))))))) (1145)
41(15(54(41(11(11(14(44(x1)))))))) 42(20(04(42(20(03(34(40(03(35(54(x1))))))))))) (1146)
23(31(14(45(52(25(54(45(x1)))))))) 24(42(20(03(34(41(12(23(30(05(55(x1))))))))))) (1256)
23(31(14(45(52(25(54(43(x1)))))))) 24(42(20(03(34(41(12(23(30(05(53(x1))))))))))) (1257)
23(31(14(45(52(25(54(41(x1)))))))) 24(42(20(03(34(41(12(23(30(05(51(x1))))))))))) (1258)
23(31(14(45(52(25(54(40(x1)))))))) 24(42(20(03(34(41(12(23(30(05(50(x1))))))))))) (1259)
23(31(14(45(52(25(54(44(x1)))))))) 24(42(20(03(34(41(12(23(30(05(54(x1))))))))))) (1260)
03(31(14(45(52(25(54(45(x1)))))))) 04(42(20(03(34(41(12(23(30(05(55(x1))))))))))) (1268)
03(31(14(45(52(25(54(43(x1)))))))) 04(42(20(03(34(41(12(23(30(05(53(x1))))))))))) (1269)
03(31(14(45(52(25(54(41(x1)))))))) 04(42(20(03(34(41(12(23(30(05(51(x1))))))))))) (1270)
03(31(14(45(52(25(54(40(x1)))))))) 04(42(20(03(34(41(12(23(30(05(50(x1))))))))))) (1271)
03(31(14(45(52(25(54(44(x1)))))))) 04(42(20(03(34(41(12(23(30(05(54(x1))))))))))) (1272)
14(45(52(23(31(10(05(52(x1)))))))) 11(15(52(20(00(00(00(03(32(24(42(x1))))))))))) (1297)
34(45(52(23(31(10(05(52(x1)))))))) 31(15(52(20(00(00(00(03(32(24(42(x1))))))))))) (1309)

1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[31(x1)] = 1 · x1
[14(x1)] = 1 · x1
[45(x1)] = 1 · x1
[50(x1)] = 1 · x1 + 1
[03(x1)] = 1 · x1
[12(x1)] = 1 · x1
[35(x1)] = 1 · x1
[52(x1)] = 1 · x1
[22(x1)] = 1 · x1
[20(x1)] = 1 · x1
[00(x1)] = 1 · x1
[04(x1)] = 1 · x1
[42(x1)] = 1 · x1
[23(x1)] = 1 · x1
[15(x1)] = 1 · x1
[13(x1)] = 1 · x1
[11(x1)] = 1 · x1
[10(x1)] = 1 · x1
[53(x1)] = 1 · x1
[44(x1)] = 1 · x1
[43(x1)] = 1 · x1
[51(x1)] = 1 · x1
[02(x1)] = 1 · x1
[33(x1)] = 1 · x1
[34(x1)] = 1 · x1
[40(x1)] = 1 · x1
[32(x1)] = 1 · x1
[54(x1)] = 1 · x1
[41(x1)] = 1 · x1
[30(x1)] = 1 · x1
[25(x1)] = 1 · x1
[21(x1)] = 1 · x1
[24(x1)] = 1 · x1
[01(x1)] = 1 · x1
all of the following rules can be deleted.
31(14(45(50(03(31(12(x1))))))) 35(52(22(20(00(04(42(23(31(12(x1)))))))))) (325)
31(14(45(50(03(31(15(x1))))))) 35(52(22(20(00(04(42(23(31(15(x1)))))))))) (326)
31(14(45(50(03(31(13(x1))))))) 35(52(22(20(00(04(42(23(31(13(x1)))))))))) (327)
31(14(45(50(03(31(11(x1))))))) 35(52(22(20(00(04(42(23(31(11(x1)))))))))) (328)
31(14(45(50(03(31(10(x1))))))) 35(52(22(20(00(04(42(23(31(10(x1)))))))))) (329)
31(14(45(50(03(31(14(x1))))))) 35(52(22(20(00(04(42(23(31(14(x1)))))))))) (330)

1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[53(x1)] = 1 · x1
[31(x1)] = 1 · x1
[14(x1)] = 1 · x1
[44(x1)] = 1 · x1
[43(x1)] = 1 · x1 + 1
[12(x1)] = 1 · x1
[51(x1)] = 1 · x1
[20(x1)] = 1 · x1
[02(x1)] = 1 · x1
[03(x1)] = 1 · x1
[33(x1)] = 1 · x1
[15(x1)] = 1 · x1
[13(x1)] = 1 · x1
[11(x1)] = 1 · x1
[10(x1)] = 1 · x1 + 1
[34(x1)] = 1 · x1
[40(x1)] = 1 · x1
[04(x1)] = 1 · x1
[42(x1)] = 1 · x1
[23(x1)] = 1 · x1
[32(x1)] = 1 · x1
[22(x1)] = 1 · x1
[35(x1)] = 1 · x1
[54(x1)] = 1 · x1
[45(x1)] = 1 · x1
[41(x1)] = 1 · x1
[30(x1)] = 1 · x1
[00(x1)] = 1 · x1
[25(x1)] = 1 · x1
[52(x1)] = 1 · x1
[21(x1)] = 1 · x1
[24(x1)] = 1 · x1
[01(x1)] = 1 · x1
all of the following rules can be deleted.
53(31(14(44(43(31(12(x1))))))) 51(12(20(02(20(03(33(33(31(12(x1)))))))))) (343)
53(31(14(44(43(31(15(x1))))))) 51(12(20(02(20(03(33(33(31(15(x1)))))))))) (344)
53(31(14(44(43(31(13(x1))))))) 51(12(20(02(20(03(33(33(31(13(x1)))))))))) (345)
53(31(14(44(43(31(11(x1))))))) 51(12(20(02(20(03(33(33(31(11(x1)))))))))) (346)
53(31(14(44(43(31(10(x1))))))) 51(12(20(02(20(03(33(33(31(10(x1)))))))))) (347)
53(31(14(44(43(31(14(x1))))))) 51(12(20(02(20(03(33(33(31(14(x1)))))))))) (348)

1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[33(x1)] = 1 · x1
[34(x1)] = 1 · x1
[40(x1)] = 1 · x1 + 1
[04(x1)] = 1 · x1
[44(x1)] = 1 · x1
[42(x1)] = 1 · x1
[31(x1)] = 1 · x1
[12(x1)] = 1 · x1
[23(x1)] = 1 · x1
[32(x1)] = 1 · x1
[22(x1)] = 1 · x1
[20(x1)] = 1 · x1
[03(x1)] = 1 · x1
[35(x1)] = 1 · x1
[54(x1)] = 1 · x1
[45(x1)] = 1 · x1
[43(x1)] = 1 · x1
[41(x1)] = 1 · x1
[14(x1)] = 1 · x1
[30(x1)] = 1 · x1
[00(x1)] = 1 · x1
[25(x1)] = 1 · x1
[51(x1)] = 1 · x1
[15(x1)] = 1 · x1 + 1
[13(x1)] = 1 · x1
[11(x1)] = 1 · x1
[10(x1)] = 1 · x1
[53(x1)] = 1 · x1
[02(x1)] = 1 · x1
[52(x1)] = 1 · x1
[21(x1)] = 1 · x1
[24(x1)] = 1 · x1
[01(x1)] = 1 · x1
all of the following rules can be deleted.
33(33(34(40(04(44(42(x1))))))) 31(12(23(32(22(20(03(35(54(42(x1)))))))))) (433)
33(33(34(40(04(44(45(x1))))))) 31(12(23(32(22(20(03(35(54(45(x1)))))))))) (434)
33(33(34(40(04(44(43(x1))))))) 31(12(23(32(22(20(03(35(54(43(x1)))))))))) (435)
33(33(34(40(04(44(41(x1))))))) 31(12(23(32(22(20(03(35(54(41(x1)))))))))) (436)
33(33(34(40(04(44(40(x1))))))) 31(12(23(32(22(20(03(35(54(40(x1)))))))))) (437)
33(33(34(40(04(44(44(x1))))))) 31(12(23(32(22(20(03(35(54(44(x1)))))))))) (438)

1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[14(x1)] = 1 · x1
[42(x1)] = 1 · x1 + 1
[23(x1)] = 1 · x1
[30(x1)] = 1 · x1
[03(x1)] = 1 · x1 + 1
[33(x1)] = 1 · x1
[31(x1)] = 1 · x1
[12(x1)] = 1 · x1
[20(x1)] = 1 · x1
[00(x1)] = 1 · x1
[04(x1)] = 1 · x1
[32(x1)] = 1 · x1
[25(x1)] = 1 · x1
[51(x1)] = 1 · x1
[15(x1)] = 1 · x1 + 1
[13(x1)] = 1 · x1
[11(x1)] = 1 · x1
[10(x1)] = 1 · x1
[44(x1)] = 1 · x1
[41(x1)] = 1 · x1
[53(x1)] = 1 · x1
[02(x1)] = 1 · x1
[52(x1)] = 1 · x1
[22(x1)] = 1 · x1
[40(x1)] = 1 · x1
[21(x1)] = 1 · x1
[24(x1)] = 1 · x1
[43(x1)] = 1 · x1
[34(x1)] = 1 · x1 + 1
[01(x1)] = 1 · x1
[45(x1)] = 1 · x1
all of the following rules can be deleted.
14(42(23(30(03(33(31(12(x1)))))))) 12(20(00(00(04(42(23(32(25(51(12(x1))))))))))) (829)
14(42(23(30(03(33(31(15(x1)))))))) 12(20(00(00(04(42(23(32(25(51(15(x1))))))))))) (830)
14(42(23(30(03(33(31(13(x1)))))))) 12(20(00(00(04(42(23(32(25(51(13(x1))))))))))) (831)
14(42(23(30(03(33(31(11(x1)))))))) 12(20(00(00(04(42(23(32(25(51(11(x1))))))))))) (832)
14(42(23(30(03(33(31(10(x1)))))))) 12(20(00(00(04(42(23(32(25(51(10(x1))))))))))) (833)
14(42(23(30(03(33(31(14(x1)))))))) 12(20(00(00(04(42(23(32(25(51(14(x1))))))))))) (834)

1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[44(x1)] = 1 · x1
[42(x1)] = 1 · x1
[23(x1)] = 1 · x1
[30(x1)] = 1 · x1
[03(x1)] = 1 · x1 + 1
[33(x1)] = 1 · x1
[31(x1)] = 1 · x1
[12(x1)] = 1 · x1
[20(x1)] = 1 · x1
[00(x1)] = 1 · x1
[04(x1)] = 1 · x1
[32(x1)] = 1 · x1
[25(x1)] = 1 · x1
[51(x1)] = 1 · x1
[15(x1)] = 1 · x1
[13(x1)] = 1 · x1
[11(x1)] = 1 · x1
[10(x1)] = 1 · x1
[14(x1)] = 1 · x1
[41(x1)] = 1 · x1
[53(x1)] = 1 · x1
[02(x1)] = 1 · x1
[52(x1)] = 1 · x1
[22(x1)] = 1 · x1
[40(x1)] = 1 · x1
[21(x1)] = 1 · x1
[24(x1)] = 1 · x1
[43(x1)] = 1 · x1
[34(x1)] = 1 · x1
[01(x1)] = 1 · x1
[45(x1)] = 1 · x1
all of the following rules can be deleted.
44(42(23(30(03(33(31(12(x1)))))))) 42(20(00(00(04(42(23(32(25(51(12(x1))))))))))) (853)
44(42(23(30(03(33(31(15(x1)))))))) 42(20(00(00(04(42(23(32(25(51(15(x1))))))))))) (854)
44(42(23(30(03(33(31(13(x1)))))))) 42(20(00(00(04(42(23(32(25(51(13(x1))))))))))) (855)
44(42(23(30(03(33(31(11(x1)))))))) 42(20(00(00(04(42(23(32(25(51(11(x1))))))))))) (856)
44(42(23(30(03(33(31(10(x1)))))))) 42(20(00(00(04(42(23(32(25(51(10(x1))))))))))) (857)
44(42(23(30(03(33(31(14(x1)))))))) 42(20(00(00(04(42(23(32(25(51(14(x1))))))))))) (858)

1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[41(x1)] = 1 · x1
[15(x1)] = 1 · x1
[53(x1)] = 1 · x1 + 1
[30(x1)] = 1 · x1
[02(x1)] = 1 · x1
[25(x1)] = 1 · x1
[52(x1)] = 1 · x1
[22(x1)] = 1 · x1
[42(x1)] = 1 · x1
[20(x1)] = 1 · x1
[00(x1)] = 1 · x1
[04(x1)] = 1 · x1
[40(x1)] = 1 · x1
[23(x1)] = 1 · x1
[21(x1)] = 1 · x1
[24(x1)] = 1 · x1
[14(x1)] = 1 · x1
[43(x1)] = 1 · x1
[33(x1)] = 1 · x1
[34(x1)] = 1 · x1
[44(x1)] = 1 · x1
[01(x1)] = 1 · x1
[12(x1)] = 1 · x1
[10(x1)] = 1 · x1
[45(x1)] = 1 · x1
all of the following rules can be deleted.
41(15(53(30(02(25(52(22(x1)))))))) 42(20(00(02(20(04(40(02(25(52(22(x1))))))))))) (997)
41(15(53(30(02(25(52(25(x1)))))))) 42(20(00(02(20(04(40(02(25(52(25(x1))))))))))) (998)
41(15(53(30(02(25(52(23(x1)))))))) 42(20(00(02(20(04(40(02(25(52(23(x1))))))))))) (999)
41(15(53(30(02(25(52(21(x1)))))))) 42(20(00(02(20(04(40(02(25(52(21(x1))))))))))) (1000)
41(15(53(30(02(25(52(20(x1)))))))) 42(20(00(02(20(04(40(02(25(52(20(x1))))))))))) (1001)
41(15(53(30(02(25(52(24(x1)))))))) 42(20(00(02(20(04(40(02(25(52(24(x1))))))))))) (1002)

1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[41(x1)] = 1 · x1
[14(x1)] = 1 · x1
[43(x1)] = 1 · x1
[33(x1)] = 1 · x1
[34(x1)] = 1 · x1
[44(x1)] = 1 · x1 + 1
[42(x1)] = 1 · x1
[20(x1)] = 1 · x1
[02(x1)] = 1 · x1
[01(x1)] = 1 · x1
[12(x1)] = 1 · x1
[10(x1)] = 1 · x1
[04(x1)] = 1 · x1
[45(x1)] = 1 · x1
[40(x1)] = 1 · x1
all of the following rules can be deleted.
41(14(43(33(33(34(44(42(x1)))))))) 42(20(02(20(01(12(20(01(10(04(42(x1))))))))))) (1249)
41(14(43(33(33(34(44(45(x1)))))))) 42(20(02(20(01(12(20(01(10(04(45(x1))))))))))) (1250)
41(14(43(33(33(34(44(43(x1)))))))) 42(20(02(20(01(12(20(01(10(04(43(x1))))))))))) (1251)
41(14(43(33(33(34(44(41(x1)))))))) 42(20(02(20(01(12(20(01(10(04(41(x1))))))))))) (1252)
41(14(43(33(33(34(44(40(x1)))))))) 42(20(02(20(01(12(20(01(10(04(40(x1))))))))))) (1253)
41(14(43(33(33(34(44(44(x1)))))))) 42(20(02(20(01(12(20(01(10(04(44(x1))))))))))) (1254)

1.1.1.1.1.1.1.1.1.1.1.1.1 R is empty

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