Term Rewriting System R:
[x, y, u, v]
s(a) -> a
s(s(x)) -> x
s(f(x, y)) -> f(s(y), s(x))
s(g(x, y)) -> g(s(x), s(y))
f(x, a) -> x
f(a, y) -> y
f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v))
g(a, a) -> a

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

S(f(x, y)) -> F(s(y), s(x))
S(f(x, y)) -> S(y)
S(f(x, y)) -> S(x)
S(g(x, y)) -> G(s(x), s(y))
S(g(x, y)) -> S(x)
S(g(x, y)) -> S(y)
F(g(x, y), g(u, v)) -> G(f(x, u), f(y, v))
F(g(x, y), g(u, v)) -> F(x, u)
F(g(x, y), g(u, v)) -> F(y, v)

Furthermore, R contains two SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Usable Rules (Innermost)
       →DP Problem 2
         ↳UsableRules


Dependency Pairs:

F(g(x, y), g(u, v)) -> F(y, v)
F(g(x, y), g(u, v)) -> F(x, u)


Rules:


s(a) -> a
s(s(x)) -> x
s(f(x, y)) -> f(s(y), s(x))
s(g(x, y)) -> g(s(x), s(y))
f(x, a) -> x
f(a, y) -> y
f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v))
g(a, a) -> a


Strategy:

innermost




As we are in the innermost case, we can delete all 8 non-usable-rules.


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


Dependency Pairs:

F(g(x, y), g(u, v)) -> F(y, v)
F(g(x, y), g(u, v)) -> F(x, u)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. F(g(x, y), g(u, v)) -> F(y, v)
  2. F(g(x, y), g(u, v)) -> F(x, u)
and get the following Size-Change Graph(s):
{1, 2} , {1, 2}
1>1
2>2

which lead(s) to this/these maximal multigraph(s):
{1, 2} , {1, 2}
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:
g(x1, x2) -> g(x1, x2)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳Usable Rules (Innermost)


Dependency Pairs:

S(g(x, y)) -> S(y)
S(g(x, y)) -> S(x)
S(f(x, y)) -> S(x)
S(f(x, y)) -> S(y)


Rules:


s(a) -> a
s(s(x)) -> x
s(f(x, y)) -> f(s(y), s(x))
s(g(x, y)) -> g(s(x), s(y))
f(x, a) -> x
f(a, y) -> y
f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v))
g(a, a) -> a


Strategy:

innermost




As we are in the innermost case, we can delete all 8 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
           →DP Problem 4
             ↳Size-Change Principle


Dependency Pairs:

S(g(x, y)) -> S(y)
S(g(x, y)) -> S(x)
S(f(x, y)) -> S(x)
S(f(x, y)) -> S(y)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. S(g(x, y)) -> S(y)
  2. S(g(x, y)) -> S(x)
  3. S(f(x, y)) -> S(x)
  4. S(f(x, y)) -> S(y)
and get the following Size-Change Graph(s):
{1, 2, 3, 4} , {1, 2, 3, 4}
1>1

which lead(s) to this/these maximal multigraph(s):
{1, 2, 3, 4} , {1, 2, 3, 4}
1>1

DP: empty set
Oriented Rules: none

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

with Argument Filtering System:
g(x1, x2) -> g(x1, x2)
f(x1, x2) -> f(x1, x2)

We obtain no new DP problems.

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