(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

admit(x, nil) → nil
admit(x, .(u, .(v, .(w, z)))) → cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
cond(true, y) → y

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic path order with status [LPO].
Quasi-Precedence:
[admit2, w, =2] > [.2, cond2, carry3] > sum3
nil > sum3
true > sum3

Status:
carry3: [1,3,2]
w: []
=2: [2,1]
sum3: [2,1,3]
true: []
cond2: [2,1]
admit2: [2,1]
.2: [1,2]
nil: []

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

admit(x, nil) → nil
admit(x, .(u, .(v, .(w, z)))) → cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
cond(true, y) → y


(2) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(3) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(4) TRUE