Term Rewriting System R:
[x, y]
sqr(0) -> 0
sqr(s(x)) -> +(sqr(x), s(double(x)))
sqr(s(x)) -> s(+(sqr(x), double(x)))
double(0) -> 0
double(s(x)) -> s(s(double(x)))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

SQR(s(x)) -> +'(sqr(x), s(double(x)))
SQR(s(x)) -> SQR(x)
SQR(s(x)) -> DOUBLE(x)
SQR(s(x)) -> +'(sqr(x), double(x))
DOUBLE(s(x)) -> DOUBLE(x)
+'(x, s(y)) -> +'(x, y)

Furthermore, R contains three SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Size-Change Principle
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP


Dependency Pair:

+'(x, s(y)) -> +'(x, y)


Rules:


sqr(0) -> 0
sqr(s(x)) -> +(sqr(x), s(double(x)))
sqr(s(x)) -> s(+(sqr(x), double(x)))
double(0) -> 0
double(s(x)) -> s(s(double(x)))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))





We number the DPs as follows:
  1. +'(x, s(y)) -> +'(x, y)
and get the following Size-Change Graph(s):
{1} , {1}
1=1
2>2

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
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:
s(x1) -> s(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳Size-Change Principle
       →DP Problem 3
         ↳SCP


Dependency Pair:

DOUBLE(s(x)) -> DOUBLE(x)


Rules:


sqr(0) -> 0
sqr(s(x)) -> +(sqr(x), s(double(x)))
sqr(s(x)) -> s(+(sqr(x), double(x)))
double(0) -> 0
double(s(x)) -> s(s(double(x)))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))





We number the DPs as follows:
  1. DOUBLE(s(x)) -> DOUBLE(x)
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:
s(x1) -> s(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳Size-Change Principle


Dependency Pair:

SQR(s(x)) -> SQR(x)


Rules:


sqr(0) -> 0
sqr(s(x)) -> +(sqr(x), s(double(x)))
sqr(s(x)) -> s(+(sqr(x), double(x)))
double(0) -> 0
double(s(x)) -> s(s(double(x)))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))





We number the DPs as follows:
  1. SQR(s(x)) -> SQR(x)
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:
s(x1) -> s(x1)

We obtain no new DP problems.

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