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