Term Rewriting System R:
[x, y, z]
rev(nil) -> nil
rev(rev(x)) -> x
rev(++(x, y)) -> ++(rev(y), rev(x))
++(nil, y) -> y
++(x, nil) -> x
++(.(x, y), z) -> .(x, ++(y, z))
++(x, ++(y, z)) -> ++(++(x, y), z)
make(x) -> .(x, nil)

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

REV(++(x, y)) -> ++'(rev(y), rev(x))
REV(++(x, y)) -> REV(y)
REV(++(x, y)) -> REV(x)
++'(.(x, y), z) -> ++'(y, z)
++'(x, ++(y, z)) -> ++'(++(x, y), z)
++'(x, ++(y, z)) -> ++'(x, y)

Furthermore, R contains two SCCs. 


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


Dependency Pairs:

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


Rules:


rev(nil) -> nil
rev(rev(x)) -> x
rev(++(x, y)) -> ++(rev(y), rev(x))
++(nil, y) -> y
++(x, nil) -> x
++(.(x, y), z) -> .(x, ++(y, z))
++(x, ++(y, z)) -> ++(++(x, y), z)
make(x) -> .(x, nil)


Strategy:

innermost




As we are in the innermost case, we can delete all 4 non-usable-rules.


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


Dependency Pairs:

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


Rules:


++(nil, y) -> y
++(x, nil) -> x
++(.(x, y), z) -> .(x, ++(y, z))
++(x, ++(y, z)) -> ++(++(x, y), z)


Strategy:

innermost




We number the DPs as follows:
  1. ++'(x, ++(y, z)) -> ++'(x, y)
  2. ++'(x, ++(y, z)) -> ++'(++(x, y), z)
  3. ++'(.(x, y), z) -> ++'(y, z)
and get the following Size-Change Graph(s):
{1} , {1}
1=1
2>2
{2} , {2}
2>2
{3} , {3}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1=1
2>2
{3} , {3}
1>1
2=2
{2} , {3}
2>2
{3} , {1}
1>1
2>2
{1} , {3}
1>1
2>2
{3} , {2}
2>2
{3} , {3}
1>1
2>2
{3} , {3}
2>2
{1} , {3}
2>2
{1} , {1}
1>1
2>2
{2} , {1}
2>2
{3} , {1}
2>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:
.(x1, x2) -> .(x1, x2)

We obtain no new DP problems. 


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


Dependency Pair:

REV(++(x, y)) -> REV(x)


Rules:


rev(nil) -> nil
rev(rev(x)) -> x
rev(++(x, y)) -> ++(rev(y), rev(x))
++(nil, y) -> y
++(x, nil) -> x
++(.(x, y), z) -> .(x, ++(y, z))
++(x, ++(y, z)) -> ++(++(x, y), z)
make(x) -> .(x, nil)


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 Pair:

REV(++(x, y)) -> REV(x)


Rule:

none


Strategy:

innermost




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

We obtain no new DP problems.

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