Term Rewriting System R:
[x, y]
g(c(x, s(y))) -> g(c(s(x), y))
f(c(s(x), y)) -> f(c(x, s(y)))
f(f(x)) -> f(d(f(x)))
f(x) -> x

Innermost Termination of R to be shown. 



   R
     ↳Removing Redundant Rules for Innermost Termination



Removing the following rules from R which left hand sides contain non normal subterms

f(f(x)) -> f(d(f(x)))


   R
     ↳RRRI
       →TRS2
         ↳Removing Redundant Rules



Removing the following rules from R which fullfill a polynomial ordering:

f(x) -> x

where the Polynomial interpretation:
  POL(c(x1, x2))=  1 + x1 + x2  
  POL(g(x1))=  1 + x1  
  POL(s(x1))=  x1  
  POL(f(x1))=  1 + x1  
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R. 


   R
     ↳RRRI
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

G(c(x, s(y))) -> G(c(s(x), y))
F(c(s(x), y)) -> F(c(x, s(y)))

Furthermore, R contains two SCCs. 


   R
     ↳RRRI
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳DPs
             ...
               →DP Problem 1
                 ↳Non-Overlappingness Check


Dependency Pair:

G(c(x, s(y))) -> G(c(s(x), y))


Rules:


g(c(x, s(y))) -> g(c(s(x), y))
f(c(s(x), y)) -> f(c(x, s(y)))





R does not overlap into P. Moreover, R is locally confluent (all critical pairs are trivially joinable).Hence we can switch to innermost.
The transformation is resulting in one subcycle: 


   R
     ↳RRRI
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳DPs
             ...
               →DP Problem 3
                 ↳Usable Rules (Innermost)


Dependency Pair:

G(c(x, s(y))) -> G(c(s(x), y))


Rules:


g(c(x, s(y))) -> g(c(s(x), y))
f(c(s(x), y)) -> f(c(x, s(y)))


Strategy:

innermost




As we are in the innermost case, we can delete all 2 non-usable-rules.


   R
     ↳RRRI
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳DPs
             ...
               →DP Problem 4
                 ↳Size-Change Principle


Dependency Pair:

G(c(x, s(y))) -> G(c(s(x), y))


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. G(c(x, s(y))) -> G(c(s(x), y))
and get the following Size-Change Graph(s):
{1} , {1}
1>1

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

DP: empty set
Oriented Rules: none

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

with Argument Filtering System:
c(x1, x2) -> c(x2)
s(x1) -> s(x1)

We obtain no new DP problems. 


   R
     ↳RRRI
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳DPs
             ...
               →DP Problem 2
                 ↳Non-Overlappingness Check


Dependency Pair:

F(c(s(x), y)) -> F(c(x, s(y)))


Rules:


g(c(x, s(y))) -> g(c(s(x), y))
f(c(s(x), y)) -> f(c(x, s(y)))





R does not overlap into P. Moreover, R is locally confluent (all critical pairs are trivially joinable).Hence we can switch to innermost.
The transformation is resulting in one subcycle: 


   R
     ↳RRRI
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳DPs
             ...
               →DP Problem 5
                 ↳Usable Rules (Innermost)


Dependency Pair:

F(c(s(x), y)) -> F(c(x, s(y)))


Rules:


g(c(x, s(y))) -> g(c(s(x), y))
f(c(s(x), y)) -> f(c(x, s(y)))


Strategy:

innermost




As we are in the innermost case, we can delete all 2 non-usable-rules.


   R
     ↳RRRI
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳DPs
             ...
               →DP Problem 6
                 ↳Size-Change Principle


Dependency Pair:

F(c(s(x), y)) -> F(c(x, s(y)))


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. F(c(s(x), y)) -> F(c(x, s(y)))
and get the following Size-Change Graph(s):
{1} , {1}
1>1

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

DP: empty set
Oriented Rules: none

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

with Argument Filtering System:
c(x1, x2) -> c(x1)
s(x1) -> s(x1)

We obtain no new DP problems.

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