Certification Problem

Input (TPDB SRS_Relative/ICFP_2010_relative/4412)

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

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

and S is the following TRS.

3(0(5(4(3(1(0(x1))))))) 3(2(2(3(4(4(3(2(1(0(x1)))))))))) (52)
0(2(4(5(2(5(3(x1))))))) 3(2(3(3(5(5(1(2(3(3(x1)))))))))) (53)
0(4(0(3(1(4(0(x1))))))) 3(3(3(2(1(3(0(2(3(0(x1)))))))))) (54)
5(1(0(3(1(x1))))) 5(4(4(1(1(1(5(4(1(1(x1)))))))))) (55)
1(5(5(1(2(0(x1)))))) 1(5(5(0(2(2(3(0(2(0(x1)))))))))) (56)
0(1(0(1(2(x1))))) 0(1(1(1(1(3(2(2(3(2(x1)))))))))) (57)
4(0(2(4(2(1(3(x1))))))) 2(5(2(2(3(5(5(2(1(3(x1)))))))))) (58)
0(0(5(3(1(5(x1)))))) 2(3(2(5(5(1(1(2(3(5(x1)))))))))) (59)
2(0(1(0(5(1(5(x1))))))) 0(4(1(0(4(3(5(4(3(5(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
3(5(0(4(1(x1))))) 3(2(4(1(1(1(1(1(4(1(x1)))))))))) (61)
2(1(3(0(1(0(x1)))))) 2(4(5(3(4(5(3(4(3(3(x1)))))))))) (62)
1(1(3(2(0(1(x1)))))) 1(1(5(2(2(2(2(0(3(1(x1)))))))))) (63)
0(3(5(4(3(1(x1)))))) 0(3(1(1(4(4(2(3(3(1(x1)))))))))) (64)
2(1(3(0(1(2(x1)))))) 1(1(1(1(4(2(3(5(3(3(x1)))))))))) (65)
2(1(0(1(4(3(x1)))))) 4(2(3(2(4(5(3(4(4(3(x1)))))))))) (66)
1(5(1(0(2(5(x1)))))) 1(1(4(5(3(2(1(4(4(2(x1)))))))))) (67)
2(2(1(0(5(5(x1)))))) 3(1(4(4(2(3(3(4(4(5(x1)))))))))) (68)
0(3(4(1(0(0(0(x1))))))) 0(3(2(0(2(3(2(3(4(0(x1)))))))))) (69)
1(5(1(0(1(0(0(x1))))))) 1(1(4(4(1(0(1(5(3(2(x1)))))))))) (70)
2(5(1(2(0(1(0(x1))))))) 1(1(4(1(4(5(5(4(2(3(x1)))))))))) (71)
2(0(4(1(5(1(0(x1))))))) 2(0(0(2(0(5(3(4(0(3(x1)))))))))) (72)
1(5(2(3(5(2(0(x1))))))) 1(1(4(1(5(3(3(0(5(2(x1)))))))))) (73)
5(0(4(0(1(3(0(x1))))))) 5(5(5(4(5(4(4(1(5(3(x1)))))))))) (74)
5(1(5(1(4(3(0(x1))))))) 5(2(1(4(5(4(5(3(4(3(x1)))))))))) (75)
3(5(5(1(5(4(0(x1))))))) 3(4(5(4(5(4(5(4(3(5(x1)))))))))) (76)
4(0(4(2(0(5(0(x1))))))) 1(1(5(3(3(1(3(3(4(0(x1)))))))))) (77)
3(5(0(5(0(5(0(x1))))))) 5(5(5(3(2(1(3(4(5(3(x1)))))))))) (78)
2(4(1(5(1(5(0(x1))))))) 3(1(4(4(1(5(3(2(3(3(x1)))))))))) (79)
5(1(5(0(3(5(0(x1))))))) 5(3(3(2(1(5(2(5(3(2(x1)))))))))) (80)
2(5(4(2(4(5(0(x1))))))) 1(1(5(5(3(5(3(3(3(0(x1)))))))))) (81)
1(0(5(1(0(0(1(x1))))))) 1(0(3(0(3(2(0(0(4(3(x1)))))))))) (82)
5(2(5(0(4(1(1(x1))))))) 5(4(4(5(5(4(4(3(1(4(x1)))))))))) (83)
0(5(1(2(0(5(1(x1))))))) 0(2(0(3(2(0(3(5(2(3(x1)))))))))) (84)
3(1(0(3(1(0(2(x1))))))) 3(1(4(1(4(1(4(3(2(2(x1)))))))))) (85)
2(1(0(3(5(1(2(x1))))))) 2(5(4(5(3(5(5(5(3(2(x1)))))))))) (86)
1(3(5(0(1(2(2(x1))))))) 1(5(3(4(4(4(4(5(3(2(x1)))))))))) (87)
2(4(5(0(5(4(2(x1))))))) 1(2(3(1(1(5(5(5(3(2(x1)))))))))) (88)
5(1(5(5(0(1(3(x1))))))) 5(4(4(1(1(1(2(3(2(3(x1)))))))))) (89)
3(5(0(1(2(1(3(x1))))))) 3(5(3(2(4(3(5(3(0(3(x1)))))))))) (90)
2(5(4(0(3(1(3(x1))))))) 1(1(1(0(2(1(2(5(5(3(x1)))))))))) (91)
5(2(5(1(1(0(4(x1))))))) 5(1(1(4(1(4(2(3(0(2(x1)))))))))) (92)
2(0(3(5(2(0(4(x1))))))) 2(0(0(1(4(1(1(4(5(3(x1)))))))))) (93)
0(5(1(5(5(0(4(x1))))))) 0(3(0(3(0(2(0(0(2(4(x1)))))))))) (94)
1(0(4(5(1(3(4(x1))))))) 1(0(3(0(0(3(0(3(5(2(x1)))))))))) (95)
2(1(0(5(0(5(4(x1))))))) 1(4(0(1(4(1(2(5(4(5(x1)))))))))) (96)
1(4(3(5(1(5(4(x1))))))) 1(5(5(3(4(3(4(5(4(4(x1)))))))))) (97)
2(4(4(2(2(5(4(x1))))))) 1(1(1(1(4(1(2(4(5(4(x1)))))))))) (98)
4(0(5(5(4(0(5(x1))))))) 4(0(2(3(0(2(3(0(5(5(x1)))))))))) (99)
4(0(1(0(5(0(5(x1))))))) 0(3(0(0(0(3(2(3(1(5(x1)))))))))) (100)
5(1(3(4(5(0(5(x1))))))) 5(5(3(4(5(2(3(0(4(3(x1)))))))))) (101)
2(0(4(3(0(1(5(x1))))))) 3(2(2(0(5(3(4(0(3(0(x1)))))))))) (102)
1(0(4(1(2(1(5(x1))))))) 4(1(4(5(3(3(2(1(4(5(x1)))))))))) (103)
3(5(0(2(2(1(5(x1))))))) 3(4(5(3(2(2(4(1(1(5(x1)))))))))) (104)
4(0(2(1(5(1(5(x1))))))) 1(5(0(2(0(2(3(2(5(3(x1)))))))))) (105)
4(3(3(1(0(3(5(x1))))))) 4(3(3(1(1(4(0(0(2(0(x1)))))))))) (106)
2(5(1(3(0(3(5(x1))))))) 2(5(5(3(2(2(3(2(2(5(x1)))))))))) (107)
4(2(1(2(1(3(5(x1))))))) 4(2(3(3(2(3(2(0(3(0(x1)))))))))) (108)
2(1(3(4(2(3(5(x1))))))) 4(3(1(5(3(2(2(2(2(0(x1)))))))))) (109)
0(1(3(0(3(3(5(x1))))))) 0(4(5(3(3(1(1(4(2(3(x1)))))))))) (110)
1(3(0(3(4(3(5(x1))))))) 4(3(1(4(4(5(3(2(5(5(x1)))))))))) (111)
0(1(3(4(5(0(3(x1))))))) 0(1(2(3(4(4(3(2(2(3(x1)))))))))) (112)
3(5(2(5(4(2(0(x1))))))) 3(3(2(1(5(5(3(3(2(3(x1)))))))))) (113)
0(4(1(3(0(4(0(x1))))))) 0(3(2(0(3(1(2(3(3(3(x1)))))))))) (114)
1(3(0(1(5(x1))))) 1(1(4(5(1(1(1(4(4(5(x1)))))))))) (115)
0(2(1(5(5(1(x1)))))) 0(2(0(3(2(2(0(5(5(1(x1)))))))))) (116)
2(1(0(1(0(x1))))) 2(3(2(2(3(1(1(1(1(0(x1)))))))))) (117)
3(1(2(4(2(0(4(x1))))))) 3(1(2(5(5(3(2(2(5(2(x1)))))))))) (118)
5(1(3(5(0(0(x1)))))) 5(3(2(1(1(5(5(2(3(2(x1)))))))))) (119)
5(1(5(0(1(0(2(x1))))))) 5(3(4(5(3(4(0(1(4(0(x1)))))))))) (120)

1.1 Closure Under Flat Contexts

Using the flat contexts

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

We obtain the transformed TRS

There are 150 ruless (increase limit for explicit display).

1.1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

There are 900 ruless (increase limit for explicit display).

1.1.1.1 Rule Removal

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

There are 630 ruless (increase limit for explicit display).

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[11(x1)] = 1 · x1
[13(x1)] = 1 · x1
[32(x1)] = 1 · x1
[20(x1)] = 1 · x1
[01(x1)] = 1 · x1
[15(x1)] = 1 · x1
[52(x1)] = 1 · x1
[22(x1)] = 1 · x1
[03(x1)] = 1 · x1
[31(x1)] = 1 · x1
[10(x1)] = 1 · x1
[14(x1)] = 1 · x1
[12(x1)] = 1 · x1
[35(x1)] = 1 + 1 · x1
[54(x1)] = 1 · x1
[43(x1)] = 1 · x1
[44(x1)] = 1 · x1
[42(x1)] = 1 · x1
[23(x1)] = 1 · x1
[33(x1)] = 1 · x1
[34(x1)] = 1 · x1
[41(x1)] = 1 · x1
[00(x1)] = 1 · x1
[02(x1)] = 1 · x1
[40(x1)] = 1 · x1
[05(x1)] = 1 · x1
[04(x1)] = 1 · x1
[53(x1)] = 1 · x1
[30(x1)] = 1 · x1
[25(x1)] = 1 · x1
[24(x1)] = 1 · x1
[21(x1)] = 1 · x1
[55(x1)] = 1 · x1
[51(x1)] = 1 · x1
[45(x1)] = 1 · x1
[50(x1)] = 1 · x1
all of the following rules can be deleted.
03(35(54(43(31(13(x1)))))) 03(31(11(14(44(42(23(33(31(13(x1)))))))))) (247)
03(35(54(43(31(15(x1)))))) 03(31(11(14(44(42(23(33(31(15(x1)))))))))) (248)
03(35(54(43(31(10(x1)))))) 03(31(11(14(44(42(23(33(31(10(x1)))))))))) (249)
03(35(54(43(31(14(x1)))))) 03(31(11(14(44(42(23(33(31(14(x1)))))))))) (250)
03(35(54(43(31(11(x1)))))) 03(31(11(14(44(42(23(33(31(11(x1)))))))))) (251)
03(35(54(43(31(12(x1)))))) 03(31(11(14(44(42(23(33(31(12(x1)))))))))) (252)
15(52(23(35(52(20(03(x1))))))) 11(14(41(15(53(33(30(05(52(23(x1)))))))))) (277)
15(52(23(35(52(20(05(x1))))))) 11(14(41(15(53(33(30(05(52(25(x1)))))))))) (278)
15(52(23(35(52(20(00(x1))))))) 11(14(41(15(53(33(30(05(52(20(x1)))))))))) (279)
15(52(23(35(52(20(04(x1))))))) 11(14(41(15(53(33(30(05(52(24(x1)))))))))) (280)
15(52(23(35(52(20(01(x1))))))) 11(14(41(15(53(33(30(05(52(21(x1)))))))))) (281)
15(52(23(35(52(20(02(x1))))))) 11(14(41(15(53(33(30(05(52(22(x1)))))))))) (282)
20(03(35(52(20(04(44(x1))))))) 20(00(01(14(41(11(14(45(53(34(x1)))))))))) (364)
20(03(35(52(20(04(41(x1))))))) 20(00(01(14(41(11(14(45(53(31(x1)))))))))) (365)
14(43(35(51(15(54(43(x1))))))) 15(55(53(34(43(34(45(54(44(43(x1)))))))))) (379)
14(43(35(51(15(54(45(x1))))))) 15(55(53(34(43(34(45(54(44(45(x1)))))))))) (380)
14(43(35(51(15(54(40(x1))))))) 15(55(53(34(43(34(45(54(44(40(x1)))))))))) (381)
14(43(35(51(15(54(44(x1))))))) 15(55(53(34(43(34(45(54(44(44(x1)))))))))) (382)
14(43(35(51(15(54(41(x1))))))) 15(55(53(34(43(34(45(54(44(41(x1)))))))))) (383)
14(43(35(51(15(54(42(x1))))))) 15(55(53(34(43(34(45(54(44(42(x1)))))))))) (384)
43(33(31(10(03(35(53(x1))))))) 43(33(31(11(14(40(00(02(20(03(x1)))))))))) (403)
43(33(31(10(03(35(55(x1))))))) 43(33(31(11(14(40(00(02(20(05(x1)))))))))) (404)
43(33(31(10(03(35(54(x1))))))) 43(33(31(11(14(40(00(02(20(04(x1)))))))))) (406)
43(33(31(10(03(35(51(x1))))))) 43(33(31(11(14(40(00(02(20(01(x1)))))))))) (407)
43(33(31(10(03(35(52(x1))))))) 43(33(31(11(14(40(00(02(20(02(x1)))))))))) (408)
01(13(30(03(33(35(53(x1))))))) 04(45(53(33(31(11(14(42(23(33(x1)))))))))) (421)
01(13(30(03(33(35(54(x1))))))) 04(45(53(33(31(11(14(42(23(34(x1)))))))))) (424)
01(13(30(03(33(35(51(x1))))))) 04(45(53(33(31(11(14(42(23(31(x1)))))))))) (425)
01(13(30(03(33(35(52(x1))))))) 04(45(53(33(31(11(14(42(23(32(x1)))))))))) (426)
32(21(13(34(42(23(35(53(x1)))))))) 34(43(31(15(53(32(22(22(22(20(03(x1))))))))))) (1003)
32(21(13(34(42(23(35(55(x1)))))))) 34(43(31(15(53(32(22(22(22(20(05(x1))))))))))) (1004)
32(21(13(34(42(23(35(54(x1)))))))) 34(43(31(15(53(32(22(22(22(20(04(x1))))))))))) (1006)
32(21(13(34(42(23(35(51(x1)))))))) 34(43(31(15(53(32(22(22(22(20(01(x1))))))))))) (1007)
32(21(13(34(42(23(35(52(x1)))))))) 34(43(31(15(53(32(22(22(22(20(02(x1))))))))))) (1008)
52(21(13(34(42(23(35(53(x1)))))))) 54(43(31(15(53(32(22(22(22(20(03(x1))))))))))) (1009)
52(21(13(34(42(23(35(55(x1)))))))) 54(43(31(15(53(32(22(22(22(20(05(x1))))))))))) (1010)
52(21(13(34(42(23(35(54(x1)))))))) 54(43(31(15(53(32(22(22(22(20(04(x1))))))))))) (1012)
52(21(13(34(42(23(35(51(x1)))))))) 54(43(31(15(53(32(22(22(22(20(01(x1))))))))))) (1013)
52(21(13(34(42(23(35(52(x1)))))))) 54(43(31(15(53(32(22(22(22(20(02(x1))))))))))) (1014)
02(21(13(34(42(23(35(53(x1)))))))) 04(43(31(15(53(32(22(22(22(20(03(x1))))))))))) (1015)
02(21(13(34(42(23(35(55(x1)))))))) 04(43(31(15(53(32(22(22(22(20(05(x1))))))))))) (1016)
02(21(13(34(42(23(35(54(x1)))))))) 04(43(31(15(53(32(22(22(22(20(04(x1))))))))))) (1018)
02(21(13(34(42(23(35(51(x1)))))))) 04(43(31(15(53(32(22(22(22(20(01(x1))))))))))) (1019)
02(21(13(34(42(23(35(52(x1)))))))) 04(43(31(15(53(32(22(22(22(20(02(x1))))))))))) (1020)
22(21(13(34(42(23(35(53(x1)))))))) 24(43(31(15(53(32(22(22(22(20(03(x1))))))))))) (1033)
22(21(13(34(42(23(35(55(x1)))))))) 24(43(31(15(53(32(22(22(22(20(05(x1))))))))))) (1034)
22(21(13(34(42(23(35(54(x1)))))))) 24(43(31(15(53(32(22(22(22(20(04(x1))))))))))) (1036)
22(21(13(34(42(23(35(51(x1)))))))) 24(43(31(15(53(32(22(22(22(20(01(x1))))))))))) (1037)
22(21(13(34(42(23(35(52(x1)))))))) 24(43(31(15(53(32(22(22(22(20(02(x1))))))))))) (1038)
35(52(25(54(42(20(03(x1))))))) 33(32(21(15(55(53(33(32(23(33(x1)))))))))) (1081)
35(52(25(54(42(20(00(x1))))))) 33(32(21(15(55(53(33(32(23(30(x1)))))))))) (1083)
35(52(25(54(42(20(04(x1))))))) 33(32(21(15(55(53(33(32(23(34(x1)))))))))) (1084)
35(52(25(54(42(20(01(x1))))))) 33(32(21(15(55(53(33(32(23(31(x1)))))))))) (1085)
35(52(25(54(42(20(02(x1))))))) 33(32(21(15(55(53(33(32(23(32(x1)))))))))) (1086)

1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[11(x1)] = 1 · x1
[13(x1)] = 1 + 1 · x1
[32(x1)] = 1 · x1
[20(x1)] = 1 · x1
[01(x1)] = 1 · x1
[15(x1)] = 1 · x1
[52(x1)] = 1 · x1
[22(x1)] = 1 · x1
[03(x1)] = 1 · x1
[31(x1)] = 1 · x1
[10(x1)] = 1 + 1 · x1
[14(x1)] = 1 · x1
[12(x1)] = 1 · x1
[34(x1)] = 1 · x1
[41(x1)] = 1 · x1
[00(x1)] = 1 + 1 · x1
[02(x1)] = 1 · x1
[23(x1)] = 1 · x1
[40(x1)] = 1 · x1
[05(x1)] = 1 · x1
[04(x1)] = 1 + 1 · x1
[35(x1)] = 1 · x1
[55(x1)] = 1 · x1
[51(x1)] = 1 · x1
[54(x1)] = 1 · x1
[45(x1)] = 1 · x1
[43(x1)] = 1 · x1
[50(x1)] = 1 · x1
[30(x1)] = 1 · x1
[25(x1)] = 1 · x1
[44(x1)] = 1 · x1
[42(x1)] = 1 · x1
[24(x1)] = 1 · x1
[21(x1)] = 1 · x1
[53(x1)] = 1 · x1
[33(x1)] = 1 · x1
all of the following rules can be deleted.
11(13(32(20(01(13(x1)))))) 11(15(52(22(22(22(20(03(31(13(x1)))))))))) (241)
11(13(32(20(01(15(x1)))))) 11(15(52(22(22(22(20(03(31(15(x1)))))))))) (242)
11(13(32(20(01(10(x1)))))) 11(15(52(22(22(22(20(03(31(10(x1)))))))))) (243)
11(13(32(20(01(14(x1)))))) 11(15(52(22(22(22(20(03(31(14(x1)))))))))) (244)
11(13(32(20(01(11(x1)))))) 11(15(52(22(22(22(20(03(31(11(x1)))))))))) (245)
11(13(32(20(01(12(x1)))))) 11(15(52(22(22(22(20(03(31(12(x1)))))))))) (246)
03(34(41(10(00(00(03(x1))))))) 03(32(20(02(23(32(23(34(40(03(x1)))))))))) (259)
03(34(41(10(00(00(05(x1))))))) 03(32(20(02(23(32(23(34(40(05(x1)))))))))) (260)
03(34(41(10(00(00(00(x1))))))) 03(32(20(02(23(32(23(34(40(00(x1)))))))))) (261)
03(34(41(10(00(00(04(x1))))))) 03(32(20(02(23(32(23(34(40(04(x1)))))))))) (262)
03(34(41(10(00(00(01(x1))))))) 03(32(20(02(23(32(23(34(40(01(x1)))))))))) (263)
03(34(41(10(00(00(02(x1))))))) 03(32(20(02(23(32(23(34(40(02(x1)))))))))) (264)
35(55(51(15(54(40(00(x1))))))) 34(45(54(45(54(45(54(43(35(50(x1)))))))))) (297)
52(25(50(04(41(11(13(x1))))))) 54(44(45(55(54(44(43(31(14(43(x1)))))))))) (313)
52(25(50(04(41(11(10(x1))))))) 54(44(45(55(54(44(43(31(14(40(x1)))))))))) (315)
52(25(50(04(41(11(14(x1))))))) 54(44(45(55(54(44(43(31(14(44(x1)))))))))) (316)
52(25(50(04(41(11(11(x1))))))) 54(44(45(55(54(44(43(31(14(41(x1)))))))))) (317)
52(25(50(04(41(11(12(x1))))))) 54(44(45(55(54(44(43(31(14(42(x1)))))))))) (318)
52(25(51(11(10(04(45(x1))))))) 51(11(14(41(14(42(23(30(02(25(x1)))))))))) (356)
52(25(51(11(10(04(44(x1))))))) 51(11(14(41(14(42(23(30(02(24(x1)))))))))) (358)
52(25(51(11(10(04(41(x1))))))) 51(11(14(41(14(42(23(30(02(21(x1)))))))))) (359)
32(21(10(01(14(43(33(x1))))))) 34(42(23(32(24(45(53(34(44(43(33(x1))))))))))) (463)
32(21(10(01(14(43(35(x1))))))) 34(42(23(32(24(45(53(34(44(43(35(x1))))))))))) (464)
32(21(10(01(14(43(30(x1))))))) 34(42(23(32(24(45(53(34(44(43(30(x1))))))))))) (465)
32(21(10(01(14(43(34(x1))))))) 34(42(23(32(24(45(53(34(44(43(34(x1))))))))))) (466)
32(21(10(01(14(43(31(x1))))))) 34(42(23(32(24(45(53(34(44(43(31(x1))))))))))) (467)
32(21(10(01(14(43(32(x1))))))) 34(42(23(32(24(45(53(34(44(43(32(x1))))))))))) (468)
52(21(10(01(14(43(33(x1))))))) 54(42(23(32(24(45(53(34(44(43(33(x1))))))))))) (469)
52(21(10(01(14(43(35(x1))))))) 54(42(23(32(24(45(53(34(44(43(35(x1))))))))))) (470)
52(21(10(01(14(43(30(x1))))))) 54(42(23(32(24(45(53(34(44(43(30(x1))))))))))) (471)
52(21(10(01(14(43(34(x1))))))) 54(42(23(32(24(45(53(34(44(43(34(x1))))))))))) (472)
52(21(10(01(14(43(31(x1))))))) 54(42(23(32(24(45(53(34(44(43(31(x1))))))))))) (473)
52(21(10(01(14(43(32(x1))))))) 54(42(23(32(24(45(53(34(44(43(32(x1))))))))))) (474)
22(21(10(01(14(43(33(x1))))))) 24(42(23(32(24(45(53(34(44(43(33(x1))))))))))) (493)
22(21(10(01(14(43(35(x1))))))) 24(42(23(32(24(45(53(34(44(43(35(x1))))))))))) (494)
22(21(10(01(14(43(30(x1))))))) 24(42(23(32(24(45(53(34(44(43(30(x1))))))))))) (495)
22(21(10(01(14(43(34(x1))))))) 24(42(23(32(24(45(53(34(44(43(34(x1))))))))))) (496)
22(21(10(01(14(43(31(x1))))))) 24(42(23(32(24(45(53(34(44(43(31(x1))))))))))) (497)
22(21(10(01(14(43(32(x1))))))) 24(42(23(32(24(45(53(34(44(43(32(x1))))))))))) (498)
32(22(21(10(05(55(53(x1))))))) 33(31(14(44(42(23(33(34(44(45(53(x1))))))))))) (499)
32(22(21(10(05(55(55(x1))))))) 33(31(14(44(42(23(33(34(44(45(55(x1))))))))))) (500)
32(22(21(10(05(55(50(x1))))))) 33(31(14(44(42(23(33(34(44(45(50(x1))))))))))) (501)
32(22(21(10(05(55(54(x1))))))) 33(31(14(44(42(23(33(34(44(45(54(x1))))))))))) (502)
32(22(21(10(05(55(51(x1))))))) 33(31(14(44(42(23(33(34(44(45(51(x1))))))))))) (503)
32(22(21(10(05(55(52(x1))))))) 33(31(14(44(42(23(33(34(44(45(52(x1))))))))))) (504)
52(22(21(10(05(55(53(x1))))))) 53(31(14(44(42(23(33(34(44(45(53(x1))))))))))) (505)
52(22(21(10(05(55(55(x1))))))) 53(31(14(44(42(23(33(34(44(45(55(x1))))))))))) (506)
52(22(21(10(05(55(50(x1))))))) 53(31(14(44(42(23(33(34(44(45(50(x1))))))))))) (507)
52(22(21(10(05(55(54(x1))))))) 53(31(14(44(42(23(33(34(44(45(54(x1))))))))))) (508)
52(22(21(10(05(55(51(x1))))))) 53(31(14(44(42(23(33(34(44(45(51(x1))))))))))) (509)
52(22(21(10(05(55(52(x1))))))) 53(31(14(44(42(23(33(34(44(45(52(x1))))))))))) (510)
02(22(21(10(05(55(53(x1))))))) 03(31(14(44(42(23(33(34(44(45(53(x1))))))))))) (511)
02(22(21(10(05(55(55(x1))))))) 03(31(14(44(42(23(33(34(44(45(55(x1))))))))))) (512)
02(22(21(10(05(55(50(x1))))))) 03(31(14(44(42(23(33(34(44(45(50(x1))))))))))) (513)
02(22(21(10(05(55(54(x1))))))) 03(31(14(44(42(23(33(34(44(45(54(x1))))))))))) (514)
02(22(21(10(05(55(51(x1))))))) 03(31(14(44(42(23(33(34(44(45(51(x1))))))))))) (515)
02(22(21(10(05(55(52(x1))))))) 03(31(14(44(42(23(33(34(44(45(52(x1))))))))))) (516)
42(22(21(10(05(55(53(x1))))))) 43(31(14(44(42(23(33(34(44(45(53(x1))))))))))) (517)
42(22(21(10(05(55(55(x1))))))) 43(31(14(44(42(23(33(34(44(45(55(x1))))))))))) (518)
42(22(21(10(05(55(50(x1))))))) 43(31(14(44(42(23(33(34(44(45(50(x1))))))))))) (519)
42(22(21(10(05(55(54(x1))))))) 43(31(14(44(42(23(33(34(44(45(54(x1))))))))))) (520)
42(22(21(10(05(55(51(x1))))))) 43(31(14(44(42(23(33(34(44(45(51(x1))))))))))) (521)
42(22(21(10(05(55(52(x1))))))) 43(31(14(44(42(23(33(34(44(45(52(x1))))))))))) (522)
22(22(21(10(05(55(53(x1))))))) 23(31(14(44(42(23(33(34(44(45(53(x1))))))))))) (529)
22(22(21(10(05(55(55(x1))))))) 23(31(14(44(42(23(33(34(44(45(55(x1))))))))))) (530)
22(22(21(10(05(55(50(x1))))))) 23(31(14(44(42(23(33(34(44(45(50(x1))))))))))) (531)
22(22(21(10(05(55(54(x1))))))) 23(31(14(44(42(23(33(34(44(45(54(x1))))))))))) (532)
22(22(21(10(05(55(51(x1))))))) 23(31(14(44(42(23(33(34(44(45(51(x1))))))))))) (533)
22(22(21(10(05(55(52(x1))))))) 23(31(14(44(42(23(33(34(44(45(52(x1))))))))))) (534)
04(40(04(42(20(05(50(03(x1)))))))) 01(11(15(53(33(31(13(33(34(40(03(x1))))))))))) (583)
04(40(04(42(20(05(50(05(x1)))))))) 01(11(15(53(33(31(13(33(34(40(05(x1))))))))))) (584)
04(40(04(42(20(05(50(00(x1)))))))) 01(11(15(53(33(31(13(33(34(40(00(x1))))))))))) (585)
04(40(04(42(20(05(50(04(x1)))))))) 01(11(15(53(33(31(13(33(34(40(04(x1))))))))))) (586)
04(40(04(42(20(05(50(01(x1)))))))) 01(11(15(53(33(31(13(33(34(40(01(x1))))))))))) (587)
04(40(04(42(20(05(50(02(x1)))))))) 01(11(15(53(33(31(13(33(34(40(02(x1))))))))))) (588)
32(21(10(05(50(05(54(45(x1)))))))) 31(14(40(01(14(41(12(25(54(45(55(x1))))))))))) (788)
32(21(10(05(50(05(54(40(x1)))))))) 31(14(40(01(14(41(12(25(54(45(50(x1))))))))))) (789)
32(21(10(05(50(05(54(44(x1)))))))) 31(14(40(01(14(41(12(25(54(45(54(x1))))))))))) (790)
32(21(10(05(50(05(54(41(x1)))))))) 31(14(40(01(14(41(12(25(54(45(51(x1))))))))))) (791)
52(21(10(05(50(05(54(45(x1)))))))) 51(14(40(01(14(41(12(25(54(45(55(x1))))))))))) (794)
52(21(10(05(50(05(54(40(x1)))))))) 51(14(40(01(14(41(12(25(54(45(50(x1))))))))))) (795)
52(21(10(05(50(05(54(44(x1)))))))) 51(14(40(01(14(41(12(25(54(45(54(x1))))))))))) (796)
52(21(10(05(50(05(54(41(x1)))))))) 51(14(40(01(14(41(12(25(54(45(51(x1))))))))))) (797)
02(21(10(05(50(05(54(45(x1)))))))) 01(14(40(01(14(41(12(25(54(45(55(x1))))))))))) (800)
02(21(10(05(50(05(54(40(x1)))))))) 01(14(40(01(14(41(12(25(54(45(50(x1))))))))))) (801)
02(21(10(05(50(05(54(44(x1)))))))) 01(14(40(01(14(41(12(25(54(45(54(x1))))))))))) (802)
02(21(10(05(50(05(54(41(x1)))))))) 01(14(40(01(14(41(12(25(54(45(51(x1))))))))))) (803)
22(21(10(05(50(05(54(45(x1)))))))) 21(14(40(01(14(41(12(25(54(45(55(x1))))))))))) (818)
22(21(10(05(50(05(54(40(x1)))))))) 21(14(40(01(14(41(12(25(54(45(50(x1))))))))))) (819)
22(21(10(05(50(05(54(44(x1)))))))) 21(14(40(01(14(41(12(25(54(45(54(x1))))))))))) (820)
22(21(10(05(50(05(54(41(x1)))))))) 21(14(40(01(14(41(12(25(54(45(51(x1))))))))))) (821)
01(13(34(45(50(03(33(x1))))))) 01(12(23(34(44(43(32(22(23(33(x1)))))))))) (1075)
01(13(34(45(50(03(35(x1))))))) 01(12(23(34(44(43(32(22(23(35(x1)))))))))) (1076)
01(13(34(45(50(03(30(x1))))))) 01(12(23(34(44(43(32(22(23(30(x1)))))))))) (1077)
01(13(34(45(50(03(34(x1))))))) 01(12(23(34(44(43(32(22(23(34(x1)))))))))) (1078)
01(13(34(45(50(03(31(x1))))))) 01(12(23(34(44(43(32(22(23(31(x1)))))))))) (1079)
01(13(34(45(50(03(32(x1))))))) 01(12(23(34(44(43(32(22(23(32(x1)))))))))) (1080)

1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[10(x1)] = 1 · x1
[05(x1)] = 1 + 1 · x1
[51(x1)] = 1 · x1
[00(x1)] = 1 · x1
[01(x1)] = 1 · x1
[14(x1)] = 1 · x1
[03(x1)] = 1 · x1
[30(x1)] = 1 + 1 · x1
[32(x1)] = 1 · x1
[20(x1)] = 1 · x1
[04(x1)] = 1 · x1
[43(x1)] = 1 · x1
[34(x1)] = 1 · x1
[11(x1)] = 1 · x1
[31(x1)] = 1 · x1
[35(x1)] = 1 · x1
[52(x1)] = 1 · x1
[45(x1)] = 1 · x1
[41(x1)] = 1 · x1
[53(x1)] = 1 · x1
[13(x1)] = 1 · x1
[33(x1)] = 1 · x1
[55(x1)] = 1 · x1
[42(x1)] = 1 · x1
[23(x1)] = 1 · x1
[02(x1)] = 1 + 1 · x1
[21(x1)] = 1 · x1
[24(x1)] = 1 · x1
[44(x1)] = 1 · x1
[12(x1)] = 1 · x1
[22(x1)] = 1 · x1
[50(x1)] = 1 · x1
[54(x1)] = 1 · x1
[40(x1)] = 1 + 1 · x1
[15(x1)] = 1 · x1
[25(x1)] = 1 · x1
all of the following rules can be deleted.
01(13(30(03(33(35(55(x1))))))) 04(45(53(33(31(11(14(42(23(35(x1)))))))))) (422)
02(21(10(01(14(43(33(x1))))))) 04(42(23(32(24(45(53(34(44(43(33(x1))))))))))) (475)
02(21(10(01(14(43(35(x1))))))) 04(42(23(32(24(45(53(34(44(43(35(x1))))))))))) (476)
02(21(10(01(14(43(30(x1))))))) 04(42(23(32(24(45(53(34(44(43(30(x1))))))))))) (477)
02(21(10(01(14(43(34(x1))))))) 04(42(23(32(24(45(53(34(44(43(34(x1))))))))))) (478)
02(21(10(01(14(43(31(x1))))))) 04(42(23(32(24(45(53(34(44(43(31(x1))))))))))) (479)
02(21(10(01(14(43(32(x1))))))) 04(42(23(32(24(45(53(34(44(43(32(x1))))))))))) (480)
12(22(21(10(05(55(53(x1))))))) 13(31(14(44(42(23(33(34(44(45(53(x1))))))))))) (523)
12(22(21(10(05(55(55(x1))))))) 13(31(14(44(42(23(33(34(44(45(55(x1))))))))))) (524)
12(22(21(10(05(55(50(x1))))))) 13(31(14(44(42(23(33(34(44(45(50(x1))))))))))) (525)
12(22(21(10(05(55(54(x1))))))) 13(31(14(44(42(23(33(34(44(45(54(x1))))))))))) (526)
12(22(21(10(05(55(51(x1))))))) 13(31(14(44(42(23(33(34(44(45(51(x1))))))))))) (527)
12(22(21(10(05(55(52(x1))))))) 13(31(14(44(42(23(33(34(44(45(52(x1))))))))))) (528)
34(40(04(42(20(05(50(03(x1)))))))) 31(11(15(53(33(31(13(33(34(40(03(x1))))))))))) (571)
34(40(04(42(20(05(50(05(x1)))))))) 31(11(15(53(33(31(13(33(34(40(05(x1))))))))))) (572)
34(40(04(42(20(05(50(00(x1)))))))) 31(11(15(53(33(31(13(33(34(40(00(x1))))))))))) (573)
34(40(04(42(20(05(50(04(x1)))))))) 31(11(15(53(33(31(13(33(34(40(04(x1))))))))))) (574)
34(40(04(42(20(05(50(01(x1)))))))) 31(11(15(53(33(31(13(33(34(40(01(x1))))))))))) (575)
34(40(04(42(20(05(50(02(x1)))))))) 31(11(15(53(33(31(13(33(34(40(02(x1))))))))))) (576)
54(40(04(42(20(05(50(03(x1)))))))) 51(11(15(53(33(31(13(33(34(40(03(x1))))))))))) (577)
54(40(04(42(20(05(50(05(x1)))))))) 51(11(15(53(33(31(13(33(34(40(05(x1))))))))))) (578)
54(40(04(42(20(05(50(00(x1)))))))) 51(11(15(53(33(31(13(33(34(40(00(x1))))))))))) (579)
54(40(04(42(20(05(50(04(x1)))))))) 51(11(15(53(33(31(13(33(34(40(04(x1))))))))))) (580)
54(40(04(42(20(05(50(01(x1)))))))) 51(11(15(53(33(31(13(33(34(40(01(x1))))))))))) (581)
54(40(04(42(20(05(50(02(x1)))))))) 51(11(15(53(33(31(13(33(34(40(02(x1))))))))))) (582)
44(40(04(42(20(05(50(03(x1)))))))) 41(11(15(53(33(31(13(33(34(40(03(x1))))))))))) (589)
44(40(04(42(20(05(50(05(x1)))))))) 41(11(15(53(33(31(13(33(34(40(05(x1))))))))))) (590)
44(40(04(42(20(05(50(00(x1)))))))) 41(11(15(53(33(31(13(33(34(40(00(x1))))))))))) (591)
44(40(04(42(20(05(50(04(x1)))))))) 41(11(15(53(33(31(13(33(34(40(04(x1))))))))))) (592)
44(40(04(42(20(05(50(01(x1)))))))) 41(11(15(53(33(31(13(33(34(40(01(x1))))))))))) (593)
44(40(04(42(20(05(50(02(x1)))))))) 41(11(15(53(33(31(13(33(34(40(02(x1))))))))))) (594)
14(40(04(42(20(05(50(03(x1)))))))) 11(11(15(53(33(31(13(33(34(40(03(x1))))))))))) (595)
14(40(04(42(20(05(50(05(x1)))))))) 11(11(15(53(33(31(13(33(34(40(05(x1))))))))))) (596)
14(40(04(42(20(05(50(00(x1)))))))) 11(11(15(53(33(31(13(33(34(40(00(x1))))))))))) (597)
14(40(04(42(20(05(50(04(x1)))))))) 11(11(15(53(33(31(13(33(34(40(04(x1))))))))))) (598)
14(40(04(42(20(05(50(01(x1)))))))) 11(11(15(53(33(31(13(33(34(40(01(x1))))))))))) (599)
14(40(04(42(20(05(50(02(x1)))))))) 11(11(15(53(33(31(13(33(34(40(02(x1))))))))))) (600)
24(40(04(42(20(05(50(03(x1)))))))) 21(11(15(53(33(31(13(33(34(40(03(x1))))))))))) (601)
24(40(04(42(20(05(50(05(x1)))))))) 21(11(15(53(33(31(13(33(34(40(05(x1))))))))))) (602)
24(40(04(42(20(05(50(00(x1)))))))) 21(11(15(53(33(31(13(33(34(40(00(x1))))))))))) (603)
24(40(04(42(20(05(50(04(x1)))))))) 21(11(15(53(33(31(13(33(34(40(04(x1))))))))))) (604)
24(40(04(42(20(05(50(01(x1)))))))) 21(11(15(53(33(31(13(33(34(40(01(x1))))))))))) (605)
24(40(04(42(20(05(50(02(x1)))))))) 21(11(15(53(33(31(13(33(34(40(02(x1))))))))))) (606)
32(24(45(50(05(54(42(23(x1)))))))) 31(12(23(31(11(15(55(55(53(32(23(x1))))))))))) (715)
32(24(45(50(05(54(42(25(x1)))))))) 31(12(23(31(11(15(55(55(53(32(25(x1))))))))))) (716)
32(24(45(50(05(54(42(20(x1)))))))) 31(12(23(31(11(15(55(55(53(32(20(x1))))))))))) (717)
32(24(45(50(05(54(42(24(x1)))))))) 31(12(23(31(11(15(55(55(53(32(24(x1))))))))))) (718)
32(24(45(50(05(54(42(21(x1)))))))) 31(12(23(31(11(15(55(55(53(32(21(x1))))))))))) (719)
32(24(45(50(05(54(42(22(x1)))))))) 31(12(23(31(11(15(55(55(53(32(22(x1))))))))))) (720)
52(24(45(50(05(54(42(23(x1)))))))) 51(12(23(31(11(15(55(55(53(32(23(x1))))))))))) (721)
52(24(45(50(05(54(42(25(x1)))))))) 51(12(23(31(11(15(55(55(53(32(25(x1))))))))))) (722)
52(24(45(50(05(54(42(20(x1)))))))) 51(12(23(31(11(15(55(55(53(32(20(x1))))))))))) (723)
52(24(45(50(05(54(42(24(x1)))))))) 51(12(23(31(11(15(55(55(53(32(24(x1))))))))))) (724)
52(24(45(50(05(54(42(21(x1)))))))) 51(12(23(31(11(15(55(55(53(32(21(x1))))))))))) (725)
52(24(45(50(05(54(42(22(x1)))))))) 51(12(23(31(11(15(55(55(53(32(22(x1))))))))))) (726)
02(24(45(50(05(54(42(23(x1)))))))) 01(12(23(31(11(15(55(55(53(32(23(x1))))))))))) (727)
02(24(45(50(05(54(42(25(x1)))))))) 01(12(23(31(11(15(55(55(53(32(25(x1))))))))))) (728)
02(24(45(50(05(54(42(20(x1)))))))) 01(12(23(31(11(15(55(55(53(32(20(x1))))))))))) (729)
02(24(45(50(05(54(42(24(x1)))))))) 01(12(23(31(11(15(55(55(53(32(24(x1))))))))))) (730)
02(24(45(50(05(54(42(21(x1)))))))) 01(12(23(31(11(15(55(55(53(32(21(x1))))))))))) (731)
02(24(45(50(05(54(42(22(x1)))))))) 01(12(23(31(11(15(55(55(53(32(22(x1))))))))))) (732)
22(24(45(50(05(54(42(23(x1)))))))) 21(12(23(31(11(15(55(55(53(32(23(x1))))))))))) (745)
22(24(45(50(05(54(42(25(x1)))))))) 21(12(23(31(11(15(55(55(53(32(25(x1))))))))))) (746)
22(24(45(50(05(54(42(20(x1)))))))) 21(12(23(31(11(15(55(55(53(32(20(x1))))))))))) (747)
22(24(45(50(05(54(42(24(x1)))))))) 21(12(23(31(11(15(55(55(53(32(24(x1))))))))))) (748)
22(24(45(50(05(54(42(21(x1)))))))) 21(12(23(31(11(15(55(55(53(32(21(x1))))))))))) (749)
22(24(45(50(05(54(42(22(x1)))))))) 21(12(23(31(11(15(55(55(53(32(22(x1))))))))))) (750)
02(25(54(40(03(31(13(33(x1)))))))) 01(11(11(10(02(21(12(25(55(53(33(x1))))))))))) (763)
02(25(54(40(03(31(13(35(x1)))))))) 01(11(11(10(02(21(12(25(55(53(35(x1))))))))))) (764)
02(25(54(40(03(31(13(30(x1)))))))) 01(11(11(10(02(21(12(25(55(53(30(x1))))))))))) (765)
02(25(54(40(03(31(13(34(x1)))))))) 01(11(11(10(02(21(12(25(55(53(34(x1))))))))))) (766)
02(25(54(40(03(31(13(31(x1)))))))) 01(11(11(10(02(21(12(25(55(53(31(x1))))))))))) (767)
02(25(54(40(03(31(13(32(x1)))))))) 01(11(11(10(02(21(12(25(55(53(32(x1))))))))))) (768)
02(24(44(42(22(25(54(43(x1)))))))) 01(11(11(11(14(41(12(24(45(54(43(x1))))))))))) (835)
02(24(44(42(22(25(54(45(x1)))))))) 01(11(11(11(14(41(12(24(45(54(45(x1))))))))))) (836)
02(24(44(42(22(25(54(40(x1)))))))) 01(11(11(11(14(41(12(24(45(54(40(x1))))))))))) (837)
02(24(44(42(22(25(54(44(x1)))))))) 01(11(11(11(14(41(12(24(45(54(44(x1))))))))))) (838)
02(24(44(42(22(25(54(41(x1)))))))) 01(11(11(11(14(41(12(24(45(54(41(x1))))))))))) (839)
02(24(44(42(22(25(54(42(x1)))))))) 01(11(11(11(14(41(12(24(45(54(42(x1))))))))))) (840)
35(52(25(54(42(20(05(x1))))))) 33(32(21(15(55(53(33(32(23(35(x1)))))))))) (1082)

1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[10(x1)] = 1 · x1
[05(x1)] = 1 · x1 + 2
[51(x1)] = 1 · x1
[00(x1)] = 1 · x1
[01(x1)] = 1 · x1
[14(x1)] = 1 · x1
[03(x1)] = 1 · x1
[30(x1)] = 1 · x1
[32(x1)] = 1 · x1
[20(x1)] = 1 · x1
[04(x1)] = 1 · x1 + 1
[43(x1)] = 1 · x1
[34(x1)] = 1 · x1
[11(x1)] = 1 · x1
[31(x1)] = 1 · x1
[35(x1)] = 1 · x1
[52(x1)] = 1 · x1
[45(x1)] = 1 · x1
[41(x1)] = 1 · x1
[53(x1)] = 1 · x1
[25(x1)] = 1 · x1
[54(x1)] = 1 · x1
[40(x1)] = 1 · x1
[13(x1)] = 1 · x1
[33(x1)] = 1 · x1
[02(x1)] = 1 · x1
[21(x1)] = 1 · x1
[12(x1)] = 1 · x1
[55(x1)] = 1 · x1
[22(x1)] = 1 · x1
[24(x1)] = 1 · x1
[44(x1)] = 1 · x1
[42(x1)] = 1 · x1
all of the following rules can be deleted.
10(05(51(10(00(01(14(x1))))))) 10(03(30(03(32(20(00(04(43(34(x1)))))))))) (310)
10(05(51(10(00(01(11(x1))))))) 10(03(30(03(32(20(00(04(43(31(x1)))))))))) (311)
20(03(35(52(20(04(45(x1))))))) 20(00(01(14(41(11(14(45(53(35(x1)))))))))) (362)

1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[32(x1)] = 1 · x1 + 1
[25(x1)] = 1 · x1
[54(x1)] = 1 · x1
[40(x1)] = 1 · x1
[03(x1)] = 1 · x1
[31(x1)] = 1 · x1
[13(x1)] = 1 · x1
[33(x1)] = 1 · x1
[11(x1)] = 1 · x1
[10(x1)] = 1 · x1
[02(x1)] = 1 · x1
[21(x1)] = 1 · x1
[12(x1)] = 1 · x1
[55(x1)] = 1 · x1
[53(x1)] = 1 · x1
[35(x1)] = 1 · x1
[30(x1)] = 1 · x1
[34(x1)] = 1 · x1
[52(x1)] = 1 · x1 + 1
[51(x1)] = 1 · x1
[22(x1)] = 1 · x1 + 1
[24(x1)] = 1 · x1
[44(x1)] = 1 · x1
[42(x1)] = 1 · x1
[43(x1)] = 1 · x1
[14(x1)] = 1 · x1 + 2
[41(x1)] = 1 · x1
[45(x1)] = 1 · x1
all of the following rules can be deleted.
32(25(54(40(03(31(13(33(x1)))))))) 31(11(11(10(02(21(12(25(55(53(33(x1))))))))))) (751)
32(25(54(40(03(31(13(35(x1)))))))) 31(11(11(10(02(21(12(25(55(53(35(x1))))))))))) (752)
32(25(54(40(03(31(13(30(x1)))))))) 31(11(11(10(02(21(12(25(55(53(30(x1))))))))))) (753)
32(25(54(40(03(31(13(34(x1)))))))) 31(11(11(10(02(21(12(25(55(53(34(x1))))))))))) (754)
32(25(54(40(03(31(13(31(x1)))))))) 31(11(11(10(02(21(12(25(55(53(31(x1))))))))))) (755)
32(25(54(40(03(31(13(32(x1)))))))) 31(11(11(10(02(21(12(25(55(53(32(x1))))))))))) (756)
52(25(54(40(03(31(13(33(x1)))))))) 51(11(11(10(02(21(12(25(55(53(33(x1))))))))))) (757)
52(25(54(40(03(31(13(35(x1)))))))) 51(11(11(10(02(21(12(25(55(53(35(x1))))))))))) (758)
52(25(54(40(03(31(13(30(x1)))))))) 51(11(11(10(02(21(12(25(55(53(30(x1))))))))))) (759)
52(25(54(40(03(31(13(34(x1)))))))) 51(11(11(10(02(21(12(25(55(53(34(x1))))))))))) (760)
52(25(54(40(03(31(13(31(x1)))))))) 51(11(11(10(02(21(12(25(55(53(31(x1))))))))))) (761)
52(25(54(40(03(31(13(32(x1)))))))) 51(11(11(10(02(21(12(25(55(53(32(x1))))))))))) (762)
22(25(54(40(03(31(13(33(x1)))))))) 21(11(11(10(02(21(12(25(55(53(33(x1))))))))))) (781)
22(25(54(40(03(31(13(35(x1)))))))) 21(11(11(10(02(21(12(25(55(53(35(x1))))))))))) (782)
22(25(54(40(03(31(13(30(x1)))))))) 21(11(11(10(02(21(12(25(55(53(30(x1))))))))))) (783)
22(25(54(40(03(31(13(34(x1)))))))) 21(11(11(10(02(21(12(25(55(53(34(x1))))))))))) (784)
22(25(54(40(03(31(13(31(x1)))))))) 21(11(11(10(02(21(12(25(55(53(31(x1))))))))))) (785)
22(25(54(40(03(31(13(32(x1)))))))) 21(11(11(10(02(21(12(25(55(53(32(x1))))))))))) (786)

1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[32(x1)] = 1 · x1 + 1
[24(x1)] = 1 · x1
[44(x1)] = 1 · x1
[42(x1)] = 1 · x1
[22(x1)] = 1 · x1
[25(x1)] = 1 · x1
[54(x1)] = 1 · x1
[43(x1)] = 1 · x1
[31(x1)] = 1 · x1
[11(x1)] = 1 · x1
[14(x1)] = 1 · x1
[41(x1)] = 1 · x1
[12(x1)] = 1 · x1
[45(x1)] = 1 · x1
[40(x1)] = 1 · x1
[52(x1)] = 1 · x1
[51(x1)] = 1 · x1
[21(x1)] = 1 · x1
all of the following rules can be deleted.
32(24(44(42(22(25(54(43(x1)))))))) 31(11(11(11(14(41(12(24(45(54(43(x1))))))))))) (823)
32(24(44(42(22(25(54(45(x1)))))))) 31(11(11(11(14(41(12(24(45(54(45(x1))))))))))) (824)
32(24(44(42(22(25(54(40(x1)))))))) 31(11(11(11(14(41(12(24(45(54(40(x1))))))))))) (825)
32(24(44(42(22(25(54(44(x1)))))))) 31(11(11(11(14(41(12(24(45(54(44(x1))))))))))) (826)
32(24(44(42(22(25(54(41(x1)))))))) 31(11(11(11(14(41(12(24(45(54(41(x1))))))))))) (827)
32(24(44(42(22(25(54(42(x1)))))))) 31(11(11(11(14(41(12(24(45(54(42(x1))))))))))) (828)

1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[52(x1)] = 1 · x1 + 1
[24(x1)] = 1 · x1
[44(x1)] = 1 · x1
[42(x1)] = 1 · x1
[22(x1)] = 1 · x1
[25(x1)] = 1 · x1
[54(x1)] = 1 · x1
[43(x1)] = 1 · x1
[51(x1)] = 1 · x1
[11(x1)] = 1 · x1
[14(x1)] = 1 · x1
[41(x1)] = 1 · x1
[12(x1)] = 1 · x1
[45(x1)] = 1 · x1
[40(x1)] = 1 · x1
[21(x1)] = 1 · x1
all of the following rules can be deleted.
52(24(44(42(22(25(54(43(x1)))))))) 51(11(11(11(14(41(12(24(45(54(43(x1))))))))))) (829)
52(24(44(42(22(25(54(45(x1)))))))) 51(11(11(11(14(41(12(24(45(54(45(x1))))))))))) (830)
52(24(44(42(22(25(54(40(x1)))))))) 51(11(11(11(14(41(12(24(45(54(40(x1))))))))))) (831)
52(24(44(42(22(25(54(44(x1)))))))) 51(11(11(11(14(41(12(24(45(54(44(x1))))))))))) (832)
52(24(44(42(22(25(54(41(x1)))))))) 51(11(11(11(14(41(12(24(45(54(41(x1))))))))))) (833)
52(24(44(42(22(25(54(42(x1)))))))) 51(11(11(11(14(41(12(24(45(54(42(x1))))))))))) (834)

1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[22(x1)] = 1 · x1
[24(x1)] = 1 · x1
[44(x1)] = 1 · x1 + 1
[42(x1)] = 1 · x1
[25(x1)] = 1 · x1
[54(x1)] = 1 · x1
[43(x1)] = 1 · x1
[21(x1)] = 1 · x1
[11(x1)] = 1 · x1
[14(x1)] = 1 · x1
[41(x1)] = 1 · x1
[12(x1)] = 1 · x1
[45(x1)] = 1 · x1
[40(x1)] = 1 · x1
all of the following rules can be deleted.
22(24(44(42(22(25(54(43(x1)))))))) 21(11(11(11(14(41(12(24(45(54(43(x1))))))))))) (853)
22(24(44(42(22(25(54(45(x1)))))))) 21(11(11(11(14(41(12(24(45(54(45(x1))))))))))) (854)
22(24(44(42(22(25(54(40(x1)))))))) 21(11(11(11(14(41(12(24(45(54(40(x1))))))))))) (855)
22(24(44(42(22(25(54(44(x1)))))))) 21(11(11(11(14(41(12(24(45(54(44(x1))))))))))) (856)
22(24(44(42(22(25(54(41(x1)))))))) 21(11(11(11(14(41(12(24(45(54(41(x1))))))))))) (857)
22(24(44(42(22(25(54(42(x1)))))))) 21(11(11(11(14(41(12(24(45(54(42(x1))))))))))) (858)

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.