Term Rewriting System R:
[x, y, z]
+(x, 0) -> x
+(x, i(x)) -> 0
+(+(x, y), z) -> +(x, +(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
*(+(x, y), z) -> +(*(x, z), *(y, z))

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

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:

+(x, 0) -> x
+(x, i(x)) -> 0
+(+(x, y), z) -> +(x, +(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
*(+(x, y), z) -> +(*(x, z), *(y, z))

Strategy:

innermost

As we are in the innermost case, we can delete all 5 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 Pairs:

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

Rules:

+(x, 0) -> x
+(x, i(x)) -> 0
+(+(x, y), z) -> +(x, +(y, z))
*(x, +(y, z)) -> +(*(x, y), *(x, z))
*(+(x, y), z) -> +(*(x, z), *(y, z))

Strategy:

innermost

As we are in the innermost case, we can delete all 5 non-usable-rules.

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

Dependency Pairs:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:
1. *'(+(x, y), z) -> *'(y, z)
2. *'(x, +(y, z)) -> *'(x, z)
3. *'(+(x, y), z) -> *'(x, z)
4. *'(x, +(y, z)) -> *'(x, y)
and get the following Size-Change Graph(s):
{1} , {1}
1>1
2=2
{2} , {2}
1=1
2>2
{3} , {3}
1>1
2=2
{4} , {4}
1=1
2>2

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

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