Certification Problem

Input (TPDB TRS_Standard/AProVE_04/Liveness_WRS)

The rewrite relation of the following TRS is considered.

active(f(x)) mark(f(f(x))) (1)
chk(no(f(x))) f(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))),x))) (2)
mat(f(x),f(y)) f(mat(x,y)) (3)
chk(no(c)) active(c) (4)
mat(f(x),c) no(c) (5)
f(active(x)) active(f(x)) (6)
f(no(x)) no(f(x)) (7)
f(mark(x)) mark(f(x)) (8)
tp(mark(x)) tp(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))),x))) (9)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by NaTT @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
chk#(no(f(x))) chk#(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))),x)) (10)
f#(no(x)) f#(x) (11)
chk#(no(f(x))) f#(f(f(f(f(f(f(f(X)))))))) (12)
chk#(no(f(x))) f#(X) (13)
chk#(no(f(x))) f#(f(f(f(f(X))))) (14)
chk#(no(f(x))) f#(f(f(X))) (15)
tp#(mark(x)) f#(f(f(f(f(X))))) (16)
tp#(mark(x)) f#(f(f(f(f(f(X)))))) (17)
chk#(no(f(x))) f#(f(X)) (18)
mat#(f(x),f(y)) mat#(x,y) (19)
f#(active(x)) active#(f(x)) (20)
mat#(f(x),f(y)) f#(mat(x,y)) (21)
tp#(mark(x)) f#(f(f(f(X)))) (22)
tp#(mark(x)) f#(f(f(f(f(f(f(f(f(X))))))))) (23)
chk#(no(f(x))) f#(f(f(f(f(f(f(X))))))) (24)
chk#(no(f(x))) f#(f(f(f(X)))) (25)
f#(mark(x)) f#(x) (26)
chk#(no(f(x))) f#(f(f(f(f(f(f(f(f(X))))))))) (27)
tp#(mark(x)) f#(X) (28)
tp#(mark(x)) f#(f(f(f(f(f(f(X))))))) (29)
chk#(no(c)) active#(c) (30)
tp#(mark(x)) f#(f(X)) (31)
tp#(mark(x)) chk#(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))),x)) (32)
active#(f(x)) f#(f(x)) (33)
tp#(mark(x)) f#(f(f(f(f(f(f(f(f(f(X)))))))))) (34)
tp#(mark(x)) f#(f(f(X))) (35)
tp#(mark(x)) f#(f(f(f(f(f(f(f(X)))))))) (36)
tp#(mark(x)) tp#(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))),x))) (37)
chk#(no(f(x))) mat#(f(f(f(f(f(f(f(f(f(f(X)))))))))),x) (38)
chk#(no(f(x))) f#(chk(mat(f(f(f(f(f(f(f(f(f(f(X)))))))))),x))) (39)
chk#(no(f(x))) f#(f(f(f(f(f(X)))))) (40)
f#(active(x)) f#(x) (41)
chk#(no(f(x))) f#(f(f(f(f(f(f(f(f(f(X)))))))))) (42)
tp#(mark(x)) mat#(f(f(f(f(f(f(f(f(f(f(X)))))))))),x) (43)

1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.