Term Rewriting System R:
[x, y, z]
.(1, x) -> x
.(x, 1) -> x
.(i(x), x) -> 1
.(x, i(x)) -> 1
.(i(y), .(y, z)) -> z
.(y, .(i(y), z)) -> z
.(.(x, y), z) -> .(x, .(y, z))
i(1) -> 1
i(i(x)) -> x
i(.(x, y)) -> .(i(y), i(x))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

.'(.(x, y), z) -> .'(x, .(y, z))
.'(.(x, y), z) -> .'(y, z)
I(.(x, y)) -> .'(i(y), i(x))
I(.(x, y)) -> I(y)
I(.(x, y)) -> I(x)

Furthermore, R contains two SCCs. 


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


Dependency Pair:

.'(.(x, y), z) -> .'(y, z)


Rules:


.(1, x) -> x
.(x, 1) -> x
.(i(x), x) -> 1
.(x, i(x)) -> 1
.(i(y), .(y, z)) -> z
.(y, .(i(y), z)) -> z
.(.(x, y), z) -> .(x, .(y, z))
i(1) -> 1
i(i(x)) -> x
i(.(x, y)) -> .(i(y), i(x))


Strategy:

innermost




As we are in the innermost case, we can delete all 10 non-usable-rules.


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


Dependency Pair:

.'(.(x, y), z) -> .'(y, z)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. .'(.(x, y), z) -> .'(y, z)
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:
.(x1, x2) -> .(x1, x2)

We obtain no new DP problems. 


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


Dependency Pair:

I(.(x, y)) -> I(y)


Rules:


.(1, x) -> x
.(x, 1) -> x
.(i(x), x) -> 1
.(x, i(x)) -> 1
.(i(y), .(y, z)) -> z
.(y, .(i(y), z)) -> z
.(.(x, y), z) -> .(x, .(y, z))
i(1) -> 1
i(i(x)) -> x
i(.(x, y)) -> .(i(y), i(x))


Strategy:

innermost




As we are in the innermost case, we can delete all 10 non-usable-rules.


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


Dependency Pair:

I(.(x, y)) -> I(y)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. I(.(x, y)) -> I(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:
.(x1, x2) -> .(x1, x2)

We obtain no new DP problems.

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