Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/28838)

The rewrite relation of the following TRS is considered.

0(1(2(1(x1)))) → 1(2(1(1(0(1(2(0(1(2(x1)))))))))) (1)
0(1(2(1(x1)))) → 1(2(1(1(0(1(2(0(1(2(0(1(2(x1))))))))))))) (2)
0(1(2(1(x1)))) → 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))) (3)
0(1(2(1(x1)))) → 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))) (4)
0(1(2(1(x1)))) → 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))) (5)
0(1(2(1(x1)))) → 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))) (6)
0(1(2(1(x1)))) → 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))))))))) (7)
0(1(2(1(x1)))) → 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))))))))) (8)
0(1(2(1(x1)))) → 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))))))))))))))) (9)
0(1(2(1(x1)))) → 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))))))))))))))) (10)
0(1(2(1(x1)))) → 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))))))))))))))))))))) (11)
0(1(2(1(x1)))) → 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))))))))))))))))))))) (12)
0(1(2(1(x1)))) → 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))))))))))))))))))))))))))))) (13)
0(1(2(1(x1)))) → 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))))))))))))))))))))))))))))))))))) (14)
0(1(2(1(x1)))) → 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (15)
0(1(2(1(x1)))) → 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (16)
0(1(2(1(x1)))) → 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (17)
0(1(2(1(x1)))) → 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (18)
0(1(2(1(x1)))) → 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (19)
0(1(2(1(x1)))) → 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (20)
0(1(2(1(x1)))) → 1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (21)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by matchbox @ termCOMP 2023)

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
1(2(1(0(x1)))) → 2(1(0(2(1(0(1(1(2(1(x1)))))))))) (22)
1(2(1(0(x1)))) → 2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))) (23)
1(2(1(0(x1)))) → 2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))) (24)
1(2(1(0(x1)))) → 2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))) (25)
1(2(1(0(x1)))) → 2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))) (26)
1(2(1(0(x1)))) → 2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))) (27)
1(2(1(0(x1)))) → 2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))))) (28)
1(2(1(0(x1)))) → 2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))))))) (29)
1(2(1(0(x1)))) → 2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))))))))))) (30)
1(2(1(0(x1)))) → 2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))))))))))))) (31)
1(2(1(0(x1)))) → 2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))))))))))))))))) (32)
1(2(1(0(x1)))) → 2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))))))))))))))))))) (33)
1(2(1(0(x1)))) → 2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))))))))))))))))))))))) (34)
1(2(1(0(x1)))) → 2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))))))))))))))))))))))))) (35)
1(2(1(0(x1)))) → 2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (36)
1(2(1(0(x1)))) → 2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (37)
1(2(1(0(x1)))) → 2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (38)
1(2(1(0(x1)))) → 2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (39)
1(2(1(0(x1)))) → 2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (40)
1(2(1(0(x1)))) → 2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (41)
1(2(1(0(x1)))) → 2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (42)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
1#(2(1(0(x1)))) → 1#(x1) (43)
1#(2(1(0(x1)))) → 1#(2(1(x1))) (44)
1#(2(1(0(x1)))) → 1#(1(2(1(x1)))) (45)
1#(2(1(0(x1)))) → 1#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (46)
1#(2(1(0(x1)))) → 1#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (47)
1#(2(1(0(x1)))) → 1#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (48)
1#(2(1(0(x1)))) → 1#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (49)
1#(2(1(0(x1)))) → 1#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (50)
1#(2(1(0(x1)))) → 1#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (51)
1#(2(1(0(x1)))) → 1#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(...display limit reached...))))))))))))))))))))))))))))))))))))))))))))))))))) (52)
1#(2(1(0(x1)))) → 1#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))))))))))))))))))))))))) (53)
1#(2(1(0(x1)))) → 1#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))))))))))))))))))))) (54)
1#(2(1(0(x1)))) → 1#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))))))))))))))))))) (55)
1#(2(1(0(x1)))) → 1#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))))))))))))))) (56)
1#(2(1(0(x1)))) → 1#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))))))))))))) (57)
1#(2(1(0(x1)))) → 1#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))))))))) (58)
1#(2(1(0(x1)))) → 1#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))))))))) (59)
1#(2(1(0(x1)))) → 1#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))))))))) (60)
1#(2(1(0(x1)))) → 1#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))))))))) (61)
1#(2(1(0(x1)))) → 1#(0(2(1(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))))))))) (62)
1#(2(1(0(x1)))) → 1#(0(2(1(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))))))))) (63)
1#(2(1(0(x1)))) → 1#(0(2(1(0(2(1(0(2(1(0(1(1(2(1(x1))))))))))))))) (64)
1#(2(1(0(x1)))) → 1#(0(2(1(0(2(1(0(1(1(2(1(x1)))))))))))) (65)
1#(2(1(0(x1)))) → 1#(0(2(1(0(1(1(2(1(x1))))))))) (66)
1#(2(1(0(x1)))) → 1#(0(1(1(2(1(x1)))))) (67)

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.