YES Termination proof of rt2-1.trs

Termination of the given RelTRS could be proven:

0 RelTRS
↳1 RelTRStoRelADPProof (⇔, 0 ms)
↳2 RelADPP
↳3 RelADPReductionPairProof (⇔, 0 ms)
↳4 RelADPP
↳5 DAbsisEmptyProof (⇔, 1 ms)
↳6 YES

(0) Obligation:

Relative term rewrite system:
The relative TRS consists of the following R rules:

T(I(x), y) → T(x, y)

The relative TRS consists of the following S rules:

T(x, y) → T(x, I(y))

(1) RelTRStoRelADPProof (EQUIVALENT transformation)

We upgrade the RelTRS problem to an equivalent Relative ADP Problem [IJCAR24].

(2) Obligation:

Relative ADP Problem with
absolute ADPs:

T(I(x), y) → T1(x, y)

and relative ADPs:

T(x, y) → T1(x, I(y))

(3) RelADPReductionPairProof (EQUIVALENT transformation)

We use the reduction pair processor [IJCAR24].
The following rules can be oriented strictly (l^# > ann(r))
and therefore we can remove all of its annotations in the right-hand side:
Absolute ADPs:


T(I(x), y) → T1(x, y)


The remaining rules can at least be oriented weakly:

Ordered with Polynomial interpretation [POLO]:

POL(I(x1)) = 3 + 2·x1   
POL(T(x1, x2)) = x1   
POL(T1(x1, x2)) = 2·x1   

(4) Obligation:

Relative ADP Problem with
No absolute ADPs, and relative ADPs:

T(I(x), y) → T(x, y)
T(x, y) → T1(x, I(y))

(5) DAbsisEmptyProof (EQUIVALENT transformation)

The RDT Problem has an empty P_abs. Hence, no infinite chain exists.

(6) YES