Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/85834)

The rewrite relation of the following TRS is considered.

0(1(2(x1))) → 1(3(2(x1))) (1)
1(1(4(2(3(x1))))) → 1(5(4(4(x1)))) (2)
2(2(1(3(5(x1))))) → 1(2(0(5(x1)))) (3)
3(0(0(2(1(x1))))) → 0(5(4(0(x1)))) (4)
4(3(2(5(0(x1))))) → 3(4(1(5(5(x1))))) (5)
0(0(0(3(2(5(x1)))))) → 2(0(5(5(4(5(x1)))))) (6)
2(2(1(1(3(2(x1)))))) → 2(1(4(0(0(x1))))) (7)
3(3(1(1(2(2(x1)))))) → 3(3(0(0(2(x1))))) (8)
0(2(5(1(0(4(2(2(x1)))))))) → 0(0(3(5(1(5(4(x1))))))) (9)
4(0(5(3(5(1(3(5(x1)))))))) → 4(0(0(1(3(0(1(5(x1)))))))) (10)
4(4(4(3(5(1(4(0(x1)))))))) → 3(0(2(2(2(2(3(2(x1)))))))) (11)
0(0(5(3(2(2(5(0(3(x1))))))))) → 2(5(5(4(2(2(5(0(3(x1))))))))) (12)
3(2(2(0(0(0(0(3(1(0(5(x1))))))))))) → 0(4(3(0(4(4(2(4(1(4(0(5(x1)))))))))))) (13)
4(3(3(3(5(0(0(3(2(4(4(1(2(x1))))))))))))) → 4(4(5(2(2(0(5(0(1(4(3(0(x1)))))))))))) (14)
4(4(3(3(1(2(2(5(3(5(3(2(3(x1))))))))))))) → 4(1(4(0(0(2(5(4(4(2(0(3(x1)))))))))))) (15)
0(0(5(0(1(4(4(3(5(2(0(0(3(3(x1)))))))))))))) → 2(1(2(1(2(0(4(0(2(2(4(3(3(5(4(x1))))))))))))))) (16)
3(0(1(5(5(1(0(4(0(0(2(1(0(3(x1)))))))))))))) → 3(4(0(1(2(5(2(2(0(3(0(4(5(1(x1)))))))))))))) (17)
5(5(0(5(4(4(4(3(4(0(5(4(3(3(x1)))))))))))))) → 2(4(5(0(2(0(3(0(2(5(3(1(3(3(x1)))))))))))))) (18)
1(0(4(3(2(1(1(1(2(4(4(5(5(0(1(x1))))))))))))))) → 1(0(2(4(5(5(0(1(1(4(4(5(0(1(x1)))))))))))))) (19)
3(4(1(1(4(4(0(4(4(2(4(1(0(0(5(3(2(x1))))))))))))))))) → 3(5(5(5(0(1(3(2(4(2(0(3(5(3(0(x1))))))))))))))) (20)
0(2(1(3(5(3(4(1(1(4(4(0(4(3(4(1(0(2(x1)))))))))))))))))) → 1(5(2(2(0(3(2(3(4(2(0(1(1(1(3(2(1(x1))))))))))))))))) (21)
3(3(0(0(1(2(3(5(3(0(5(2(0(0(2(4(4(1(x1)))))))))))))))))) → 2(2(4(1(4(4(2(5(2(2(5(1(4(2(5(2(0(4(1(x1))))))))))))))))))) (22)
3(3(0(5(2(3(1(3(0(0(3(1(5(2(2(1(2(2(x1)))))))))))))))))) → 1(0(3(0(4(2(4(3(2(0(4(2(1(5(5(2(2(x1))))))))))))))))) (23)
4(5(4(0(1(1(5(5(4(5(3(2(1(3(2(4(4(2(x1)))))))))))))))))) → 3(4(5(5(3(0(4(3(3(3(0(5(3(2(2(5(0(x1))))))))))))))))) (24)
0(0(4(1(2(3(3(5(5(2(0(3(1(2(2(4(0(1(5(x1))))))))))))))))))) → 2(2(0(1(0(1(5(0(1(0(0(5(0(1(1(4(5(x1))))))))))))))))) (25)
0(5(0(3(2(3(2(3(0(1(5(5(5(3(4(0(0(2(2(x1))))))))))))))))))) → 2(2(5(4(0(0(1(5(5(3(2(2(5(0(0(5(4(0(x1)))))))))))))))))) (26)
4(0(3(4(1(3(2(0(0(0(2(1(0(1(1(3(1(5(1(x1))))))))))))))))))) → 0(3(5(3(4(5(1(0(0(3(1(0(2(4(1(3(3(0(x1)))))))))))))))))) (27)
4(4(5(4(5(4(3(1(2(2(0(2(5(4(2(0(4(1(2(x1))))))))))))))))))) → 0(3(5(5(2(0(3(4(2(4(5(0(2(1(3(2(2(1(x1)))))))))))))))))) (28)
3(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1)))))))))))))))))))) → 3(4(5(0(5(3(1(3(5(5(5(4(0(4(1(2(5(5(0(x1))))))))))))))))))) (29)
2(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1))))))))))))))))))))) → 5(4(5(4(1(3(4(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x1))))))))))))))))))))) (30)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Rule Removal

Using the linear polynomial interpretation over the naturals
[0(x1)] = 1 · x1 + 17
[1(x1)] = 1 · x1 + 13
[2(x1)] = 1 · x1 + 13
[3(x1)] = 1 · x1 + 13
[4(x1)] = 1 · x1 + 12
[5(x1)] = 1 · x1 + 17
all of the following rules can be deleted.
0(1(2(x1))) → 1(3(2(x1))) (1)
1(1(4(2(3(x1))))) → 1(5(4(4(x1)))) (2)
2(2(1(3(5(x1))))) → 1(2(0(5(x1)))) (3)
3(0(0(2(1(x1))))) → 0(5(4(0(x1)))) (4)
0(0(0(3(2(5(x1)))))) → 2(0(5(5(4(5(x1)))))) (6)
2(2(1(1(3(2(x1)))))) → 2(1(4(0(0(x1))))) (7)
3(3(1(1(2(2(x1)))))) → 3(3(0(0(2(x1))))) (8)
0(2(5(1(0(4(2(2(x1)))))))) → 0(0(3(5(1(5(4(x1))))))) (9)
0(0(5(3(2(2(5(0(3(x1))))))))) → 2(5(5(4(2(2(5(0(3(x1))))))))) (12)
4(3(3(3(5(0(0(3(2(4(4(1(2(x1))))))))))))) → 4(4(5(2(2(0(5(0(1(4(3(0(x1)))))))))))) (14)
4(4(3(3(1(2(2(5(3(5(3(2(3(x1))))))))))))) → 4(1(4(0(0(2(5(4(4(2(0(3(x1)))))))))))) (15)
0(0(5(0(1(4(4(3(5(2(0(0(3(3(x1)))))))))))))) → 2(1(2(1(2(0(4(0(2(2(4(3(3(5(4(x1))))))))))))))) (16)
3(0(1(5(5(1(0(4(0(0(2(1(0(3(x1)))))))))))))) → 3(4(0(1(2(5(2(2(0(3(0(4(5(1(x1)))))))))))))) (17)
1(0(4(3(2(1(1(1(2(4(4(5(5(0(1(x1))))))))))))))) → 1(0(2(4(5(5(0(1(1(4(4(5(0(1(x1)))))))))))))) (19)
3(4(1(1(4(4(0(4(4(2(4(1(0(0(5(3(2(x1))))))))))))))))) → 3(5(5(5(0(1(3(2(4(2(0(3(5(3(0(x1))))))))))))))) (20)
0(2(1(3(5(3(4(1(1(4(4(0(4(3(4(1(0(2(x1)))))))))))))))))) → 1(5(2(2(0(3(2(3(4(2(0(1(1(1(3(2(1(x1))))))))))))))))) (21)
3(3(0(0(1(2(3(5(3(0(5(2(0(0(2(4(4(1(x1)))))))))))))))))) → 2(2(4(1(4(4(2(5(2(2(5(1(4(2(5(2(0(4(1(x1))))))))))))))))))) (22)
3(3(0(5(2(3(1(3(0(0(3(1(5(2(2(1(2(2(x1)))))))))))))))))) → 1(0(3(0(4(2(4(3(2(0(4(2(1(5(5(2(2(x1))))))))))))))))) (23)
4(5(4(0(1(1(5(5(4(5(3(2(1(3(2(4(4(2(x1)))))))))))))))))) → 3(4(5(5(3(0(4(3(3(3(0(5(3(2(2(5(0(x1))))))))))))))))) (24)
0(0(4(1(2(3(3(5(5(2(0(3(1(2(2(4(0(1(5(x1))))))))))))))))))) → 2(2(0(1(0(1(5(0(1(0(0(5(0(1(1(4(5(x1))))))))))))))))) (25)
0(5(0(3(2(3(2(3(0(1(5(5(5(3(4(0(0(2(2(x1))))))))))))))))))) → 2(2(5(4(0(0(1(5(5(3(2(2(5(0(0(5(4(0(x1)))))))))))))))))) (26)
4(0(3(4(1(3(2(0(0(0(2(1(0(1(1(3(1(5(1(x1))))))))))))))))))) → 0(3(5(3(4(5(1(0(0(3(1(0(2(4(1(3(3(0(x1)))))))))))))))))) (27)
4(4(5(4(5(4(3(1(2(2(0(2(5(4(2(0(4(1(2(x1))))))))))))))))))) → 0(3(5(5(2(0(3(4(2(4(5(0(2(1(3(2(2(1(x1)))))))))))))))))) (28)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
4#(3(2(5(0(x1))))) → 3#(4(1(5(5(x1))))) (31)
4#(3(2(5(0(x1))))) → 4#(1(5(5(x1)))) (32)
4#(3(2(5(0(x1))))) → 5#(5(x1)) (33)
4#(3(2(5(0(x1))))) → 5#(x1) (34)
4#(0(5(3(5(1(3(5(x1)))))))) → 4#(0(0(1(3(0(1(5(x1)))))))) (35)
4#(0(5(3(5(1(3(5(x1)))))))) → 3#(0(1(5(x1)))) (36)
4#(4(4(3(5(1(4(0(x1)))))))) → 3#(0(2(2(2(2(3(2(x1)))))))) (37)
4#(4(4(3(5(1(4(0(x1)))))))) → 2#(2(2(2(3(2(x1)))))) (38)
4#(4(4(3(5(1(4(0(x1)))))))) → 2#(2(2(3(2(x1))))) (39)
4#(4(4(3(5(1(4(0(x1)))))))) → 2#(2(3(2(x1)))) (40)
4#(4(4(3(5(1(4(0(x1)))))))) → 2#(3(2(x1))) (41)
4#(4(4(3(5(1(4(0(x1)))))))) → 3#(2(x1)) (42)
4#(4(4(3(5(1(4(0(x1)))))))) → 2#(x1) (43)
3#(2(2(0(0(0(0(3(1(0(5(x1))))))))))) → 4#(3(0(4(4(2(4(1(4(0(5(x1))))))))))) (44)
3#(2(2(0(0(0(0(3(1(0(5(x1))))))))))) → 3#(0(4(4(2(4(1(4(0(5(x1)))))))))) (45)
3#(2(2(0(0(0(0(3(1(0(5(x1))))))))))) → 4#(4(2(4(1(4(0(5(x1)))))))) (46)
3#(2(2(0(0(0(0(3(1(0(5(x1))))))))))) → 4#(2(4(1(4(0(5(x1))))))) (47)
3#(2(2(0(0(0(0(3(1(0(5(x1))))))))))) → 2#(4(1(4(0(5(x1)))))) (48)
3#(2(2(0(0(0(0(3(1(0(5(x1))))))))))) → 4#(1(4(0(5(x1))))) (49)
3#(2(2(0(0(0(0(3(1(0(5(x1))))))))))) → 4#(0(5(x1))) (50)
5#(5(0(5(4(4(4(3(4(0(5(4(3(3(x1)))))))))))))) → 2#(4(5(0(2(0(3(0(2(5(3(1(3(3(x1)))))))))))))) (51)
5#(5(0(5(4(4(4(3(4(0(5(4(3(3(x1)))))))))))))) → 4#(5(0(2(0(3(0(2(5(3(1(3(3(x1))))))))))))) (52)
5#(5(0(5(4(4(4(3(4(0(5(4(3(3(x1)))))))))))))) → 5#(0(2(0(3(0(2(5(3(1(3(3(x1)))))))))))) (53)
5#(5(0(5(4(4(4(3(4(0(5(4(3(3(x1)))))))))))))) → 2#(0(3(0(2(5(3(1(3(3(x1)))))))))) (54)
5#(5(0(5(4(4(4(3(4(0(5(4(3(3(x1)))))))))))))) → 3#(0(2(5(3(1(3(3(x1)))))))) (55)
5#(5(0(5(4(4(4(3(4(0(5(4(3(3(x1)))))))))))))) → 2#(5(3(1(3(3(x1)))))) (56)
5#(5(0(5(4(4(4(3(4(0(5(4(3(3(x1)))))))))))))) → 5#(3(1(3(3(x1))))) (57)
5#(5(0(5(4(4(4(3(4(0(5(4(3(3(x1)))))))))))))) → 3#(1(3(3(x1)))) (58)
3#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1)))))))))))))))))))) → 3#(4(5(0(5(3(1(3(5(5(5(4(0(4(1(2(5(5(0(x1))))))))))))))))))) (59)
3#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1)))))))))))))))))))) → 4#(5(0(5(3(1(3(5(5(5(4(0(4(1(2(5(5(0(x1)))))))))))))))))) (60)
3#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1)))))))))))))))))))) → 5#(0(5(3(1(3(5(5(5(4(0(4(1(2(5(5(0(x1))))))))))))))))) (61)
3#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1)))))))))))))))))))) → 5#(3(1(3(5(5(5(4(0(4(1(2(5(5(0(x1))))))))))))))) (62)
3#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1)))))))))))))))))))) → 3#(1(3(5(5(5(4(0(4(1(2(5(5(0(x1)))))))))))))) (63)
3#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1)))))))))))))))))))) → 3#(5(5(5(4(0(4(1(2(5(5(0(x1)))))))))))) (64)
3#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1)))))))))))))))))))) → 5#(5(5(4(0(4(1(2(5(5(0(x1))))))))))) (65)
3#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1)))))))))))))))))))) → 5#(5(4(0(4(1(2(5(5(0(x1)))))))))) (66)
3#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1)))))))))))))))))))) → 5#(4(0(4(1(2(5(5(0(x1))))))))) (67)
3#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1)))))))))))))))))))) → 4#(0(4(1(2(5(5(0(x1)))))))) (68)
3#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1)))))))))))))))))))) → 4#(1(2(5(5(0(x1)))))) (69)
3#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1)))))))))))))))))))) → 2#(5(5(0(x1)))) (70)
3#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1)))))))))))))))))))) → 5#(5(0(x1))) (71)
3#(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x1)))))))))))))))))))) → 5#(0(x1)) (72)
2#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1))))))))))))))))))))) → 5#(4(5(4(1(3(4(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x1))))))))))))))))))))) (73)
2#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1))))))))))))))))))))) → 4#(5(4(1(3(4(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x1)))))))))))))))))))) (74)
2#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1))))))))))))))))))))) → 5#(4(1(3(4(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x1))))))))))))))))))) (75)
2#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1))))))))))))))))))))) → 4#(1(3(4(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x1)))))))))))))))))) (76)
2#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1))))))))))))))))))))) → 3#(4(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x1)))))))))))))))) (77)
2#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1))))))))))))))))))))) → 4#(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x1))))))))))))))) (78)
2#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1))))))))))))))))))))) → 3#(0(3(3(1(1(4(4(2(1(2(0(1(0(x1)))))))))))))) (79)
2#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1))))))))))))))))))))) → 3#(3(1(1(4(4(2(1(2(0(1(0(x1)))))))))))) (80)
2#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1))))))))))))))))))))) → 3#(1(1(4(4(2(1(2(0(1(0(x1))))))))))) (81)
2#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1))))))))))))))))))))) → 4#(4(2(1(2(0(1(0(x1)))))))) (82)
2#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1))))))))))))))))))))) → 4#(2(1(2(0(1(0(x1))))))) (83)
2#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1))))))))))))))))))))) → 2#(1(2(0(1(0(x1)))))) (84)
2#(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x1))))))))))))))))))))) → 2#(0(1(0(x1)))) (85)

1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.