Term Rewriting System R:
[x, u, v, z, y]
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

Termination of R to be shown. 



   R
     ↳Overlay and local confluence Check



The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost. 


   R
     ↳OC
       →TRS2
         ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

ADMIT(x, .(u, .(v, .(w, z)))) -> COND(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
ADMIT(x, .(u, .(v, .(w, z)))) -> ADMIT(carry(x, u, v), z)

Furthermore, R contains one SCC. 


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Usable Rules (Innermost)


Dependency Pair:

ADMIT(x, .(u, .(v, .(w, z)))) -> ADMIT(carry(x, u, v), z)


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


Strategy:

innermost




As we are in the innermost case, we can delete all 3 non-usable-rules.


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
             ...
               →DP Problem 2
                 ↳Size-Change Principle


Dependency Pair:

ADMIT(x, .(u, .(v, .(w, z)))) -> ADMIT(carry(x, u, v), z)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. ADMIT(x, .(u, .(v, .(w, z)))) -> ADMIT(carry(x, u, v), z)
and get the following Size-Change Graph(s):
{1} , {1}
2>2

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
carry(x1, x2, x3) -> carry(x1, x2, x3)
.(x1, x2) -> .(x1, x2)

We obtain no new DP problems.

Termination of R successfully shown.
Duration:
0:00 minutes