Certification Problem

Input (TPDB SRS_Standard/Secret_05_SRS/torpa4)

The rewrite relation of the following TRS is considered.

a(b(c(a(x1)))) → b(a(c(b(a(b(x1)))))) (1)
a(d(x1)) → c(x1) (2)
a(f(f(x1))) → g(x1) (3)
b(g(x1)) → g(b(x1)) (4)
c(x1) → f(f(x1)) (5)
c(a(c(x1))) → b(c(a(b(c(x1))))) (6)
c(d(x1)) → a(a(x1)) (7)
g(x1) → c(a(x1)) (8)
g(x1) → d(d(d(d(x1)))) (9)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by ttt2 @ termCOMP 2023)

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
a(c(b(a(x1)))) → b(a(b(c(a(b(x1)))))) (10)
d(a(x1)) → c(x1) (11)
f(f(a(x1))) → g(x1) (12)
g(b(x1)) → b(g(x1)) (13)
c(x1) → f(f(x1)) (5)
c(a(c(x1))) → c(b(a(c(b(x1))))) (14)
d(c(x1)) → a(a(x1)) (15)
g(x1) → a(c(x1)) (16)
g(x1) → d(d(d(d(x1)))) (9)

1.1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers
[f(x1)] = 3 · x1 + -∞
[c(x1)] = 6 · x1 + -∞
[d(x1)] = 2 · x1 + -∞
[g(x1)] = 10 · x1 + -∞
[a(x1)] = 4 · x1 + -∞
[b(x1)] = 0 · x1 + -∞
all of the following rules can be deleted.
g(x1) → d(d(d(d(x1)))) (9)

1.1.1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers
[f(x1)] = 0 · x1 + -∞
[c(x1)] = 0 · x1 + -∞
[d(x1)] = 8 · x1 + -∞
[g(x1)] = 0 · x1 + -∞
[a(x1)] = 0 · x1 + -∞
[b(x1)] = 0 · x1 + -∞
all of the following rules can be deleted.
d(a(x1)) → c(x1) (11)
d(c(x1)) → a(a(x1)) (15)

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[f(x1)] =
1 1 0
0 0 1
0 1 0
· x1 +
0 0 0
0 0 0
0 0 0
[c(x1)] =
1 1 1
0 1 0
0 0 1
· x1 +
0 0 0
0 0 0
0 0 0
[g(x1)] =
1 1 1
0 1 1
0 0 0
· x1 +
0 0 0
0 0 0
1 0 0
[a(x1)] =
1 0 0
0 1 1
0 0 0
· x1 +
0 0 0
0 0 0
1 0 0
[b(x1)] =
1 0 0
0 1 0
0 0 1
· x1 +
0 0 0
0 0 0
0 0 0
all of the following rules can be deleted.
f(f(a(x1))) → g(x1) (12)

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers
[f(x1)] = 0 · x1 + -∞
[c(x1)] = 0 · x1 + -∞
[g(x1)] = 15 · x1 + -∞
[a(x1)] = 8 · x1 + -∞
[b(x1)] = 0 · x1 + -∞
all of the following rules can be deleted.
g(x1) → a(c(x1)) (16)

1.1.1.1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
a#(c(b(a(x1)))) → a#(b(x1)) (17)
a#(c(b(a(x1)))) → c#(a(b(x1))) (18)
a#(c(b(a(x1)))) → a#(b(c(a(b(x1))))) (19)
g#(b(x1)) → g#(x1) (20)
c#(a(c(x1))) → c#(b(x1)) (21)
c#(a(c(x1))) → a#(c(b(x1))) (22)
c#(a(c(x1))) → c#(b(a(c(b(x1))))) (23)

1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.