Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex25_Luc06_GM)

The rewrite relation of the following TRS is considered.

a__f(f(X)) → a__c(f(g(f(X)))) (1)
a__c(X) → d(X) (2)
a__h(X) → a__c(d(X)) (3)
mark(f(X)) → a__f(mark(X)) (4)
mark(c(X)) → a__c(X) (5)
mark(h(X)) → a__h(mark(X)) (6)
mark(g(X)) → g(X) (7)
mark(d(X)) → d(X) (8)
a__f(X) → f(X) (9)
a__c(X) → c(X) (10)
a__h(X) → h(X) (11)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by ttt2 @ termCOMP 2023)

1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[d(x1)] =
1 0 0
0 0 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
[h(x1)] =
1 0 0
0 0 0
0 0 1
· x1 +
1 0 0
0 0 0
0 0 0
[a__f(x1)] =
1 0 0
0 0 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
[a__c(x1)] =
1 0 0
0 0 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
[a__h(x1)] =
1 1 0
0 0 1
0 1 1
· x1 +
1 0 0
0 0 0
0 0 0
[f(x1)] =
1 0 0
0 0 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
[c(x1)] =
1 0 0
0 0 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
[mark(x1)] =
1 0 0
0 0 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
[g(x1)] =
1 0 0
0 0 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
all of the following rules can be deleted.
a__h(X) → a__c(d(X)) (3)

1.1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers
[d(x1)] = 0 · x1 + -∞
[h(x1)] = 0 · x1 + -∞
[a__f(x1)] = 0 · x1 + -∞
[a__c(x1)] = 0 · x1 + -∞
[a__h(x1)] = 0 · x1 + -∞
[f(x1)] = 0 · x1 + -∞
[c(x1)] = 0 · x1 + -∞
[mark(x1)] = 1 · x1 + -∞
[g(x1)] = 0 · x1 + -∞
all of the following rules can be deleted.
mark(c(X)) → a__c(X) (5)
mark(g(X)) → g(X) (7)
mark(d(X)) → d(X) (8)

1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[d(x1)] =
1 0 0
0 0 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
[h(x1)] =
1 0 0
0 0 0
0 0 1
· x1 +
0 0 0
0 0 0
0 0 0
[a__f(x1)] =
1 1 1
0 1 0
0 0 0
· x1 +
0 0 0
0 0 0
1 0 0
[a__c(x1)] =
1 0 0
0 0 0
0 0 1
· x1 +
1 0 0
0 0 0
0 0 0
[a__h(x1)] =
1 1 0
0 0 0
0 0 1
· x1 +
0 0 0
0 0 0
0 0 0
[f(x1)] =
1 1 1
0 0 0
0 0 0
· x1 +
0 0 0
0 0 0
1 0 0
[c(x1)] =
1 0 0
0 0 0
0 0 0
· x1 +
1 0 0
0 0 0
0 0 0
[mark(x1)] =
1 0 1
0 0 0
0 0 0
· x1 +
0 0 0
0 0 0
1 0 0
[g(x1)] =
1 0 0
0 0 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
all of the following rules can be deleted.
a__c(X) → d(X) (2)

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[h(x1)] =
1 1 1
1 1 1
0 0 0
· x1 +
0 0 0
1 0 0
1 0 0
[a__f(x1)] =
1 1 1
1 1 1
0 0 0
· x1 +
1 0 0
1 0 0
1 0 0
[a__c(x1)] =
1 0 0
1 0 1
0 0 1
· x1 +
1 0 0
0 0 0
0 0 0
[a__h(x1)] =
1 1 1
1 1 1
0 0 0
· x1 +
1 0 0
1 0 0
1 0 0
[f(x1)] =
1 1 1
1 1 1
0 0 0
· x1 +
0 0 0
1 0 0
1 0 0
[c(x1)] =
1 0 0
0 0 0
0 0 1
· x1 +
0 0 0
0 0 0
0 0 0
[mark(x1)] =
1 1 1
1 1 1
0 0 0
· x1 +
0 0 0
0 0 0
1 0 0
[g(x1)] =
1 1 0
0 0 0
0 0 0
· x1 +
1 0 0
0 0 0
0 0 0
all of the following rules can be deleted.
a__f(X) → f(X) (9)
a__c(X) → c(X) (10)
a__h(X) → h(X) (11)

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[h(x1)] =
1 1 0
0 0 0
0 0 0
· x1 +
1 0 0
1 0 0
0 0 0
[a__f(x1)] =
1 1 1
0 0 0
0 0 0
· x1 +
1 0 0
0 0 0
0 0 0
[a__c(x1)] =
1 1 0
0 0 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
[a__h(x1)] =
1 0 0
0 0 0
0 0 0
· x1 +
1 0 0
0 0 0
0 0 0
[f(x1)] =
1 0 0
0 1 0
0 1 0
· x1 +
0 0 0
1 0 0
1 0 0
[mark(x1)] =
1 1 0
0 0 0
0 0 0
· x1 +
1 0 0
0 0 0
0 0 0
[g(x1)] =
1 0 0
0 1 1
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
all of the following rules can be deleted.
mark(h(X)) → a__h(mark(X)) (6)

1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[a__f(x1)] =
1 1 0
1 1 0
0 0 0
· x1 +
0 0 0
1 0 0
0 0 0
[a__c(x1)] =
1 1 0
1 1 0
0 0 0
· x1 +
0 0 0
1 0 0
0 0 0
[f(x1)] =
1 0 1
1 0 1
0 1 1
· x1 +
0 0 0
0 0 0
1 0 0
[mark(x1)] =
1 0 1
0 1 1
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
[g(x1)] =
1 0 0
0 0 1
0 0 0
· x1 +
0 0 0
1 0 0
0 0 0
all of the following rules can be deleted.
mark(f(X)) → a__f(mark(X)) (4)

1.1.1.1.1.1.1 Rule Removal

Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(a__c) = 1 weight(a__c) = 1
prec(g) = 3 weight(g) = 0
prec(a__f) = 2 weight(a__f) = 2
prec(f) = 0 weight(f) = 1
all of the following rules can be deleted.
a__f(f(X)) → a__c(f(g(f(X)))) (1)

1.1.1.1.1.1.1.1 R is empty

There are no rules in the TRS. Hence, it is terminating.