Term Rewriting System R:
[x, y]
times(x, 0) -> 0
times(x, s(y)) -> plus(times(x, y), x)
plus(x, 0) -> x
plus(0, x) -> x
plus(x, s(y)) -> s(plus(x, y))
plus(s(x), y) -> s(plus(x, y))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

TIMES(x, s(y)) -> PLUS(times(x, y), x)
TIMES(x, s(y)) -> TIMES(x, y)
PLUS(x, s(y)) -> PLUS(x, y)
PLUS(s(x), y) -> PLUS(x, y)

Furthermore, R contains two SCCs. 


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


Dependency Pairs:

PLUS(s(x), y) -> PLUS(x, y)
PLUS(x, s(y)) -> PLUS(x, y)


Rules:


times(x, 0) -> 0
times(x, s(y)) -> plus(times(x, y), x)
plus(x, 0) -> x
plus(0, x) -> x
plus(x, s(y)) -> s(plus(x, y))
plus(s(x), y) -> s(plus(x, y))


Strategy:

innermost




As we are in the innermost case, we can delete all 6 non-usable-rules.


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


Dependency Pairs:

PLUS(s(x), y) -> PLUS(x, y)
PLUS(x, s(y)) -> PLUS(x, y)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. PLUS(s(x), y) -> PLUS(x, y)
  2. PLUS(x, s(y)) -> PLUS(x, y)
and get the following Size-Change Graph(s):
{1, 2} , {1, 2}
1>1
2=2
{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
{1, 2} , {1, 2}
1>1
2=2
{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:
s(x1) -> s(x1)

We obtain no new DP problems. 


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


Dependency Pair:

TIMES(x, s(y)) -> TIMES(x, y)


Rules:


times(x, 0) -> 0
times(x, s(y)) -> plus(times(x, y), x)
plus(x, 0) -> x
plus(0, x) -> x
plus(x, s(y)) -> s(plus(x, y))
plus(s(x), y) -> s(plus(x, y))


Strategy:

innermost




As we are in the innermost case, we can delete all 6 non-usable-rules.


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


Dependency Pair:

TIMES(x, s(y)) -> TIMES(x, y)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. TIMES(x, s(y)) -> TIMES(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.

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