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

Innermost Termination of R to be shown. 



   R
     ↳Removing Redundant Rules for Innermost Termination



Removing the following rules from R which left hand sides contain non normal subterms

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


   R
     ↳RRRI
       →TRS2
         ↳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)

Furthermore, R contains two SCCs. 


   R
     ↳RRRI
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Non-Overlappingness Check
           →DP Problem 2
             ↳NOC


Dependency Pair:

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





R does not overlap into P. Moreover, R is locally confluent (all critical pairs are trivially joinable).Hence we can switch to innermost.
The transformation is resulting in one subcycle: 


   R
     ↳RRRI
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳NOC
             ...
               →DP Problem 3
                 ↳Usable Rules (Innermost)
           →DP Problem 2
             ↳NOC


Dependency Pair:

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


Strategy:

innermost




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


   R
     ↳RRRI
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳NOC
             ...
               →DP Problem 4
                 ↳Size-Change Principle
           →DP Problem 2
             ↳NOC


Dependency Pair:

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


Rule:

none


Strategy:

innermost




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


   R
     ↳RRRI
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳NOC
           →DP Problem 2
             ↳Non-Overlappingness Check


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





R does not overlap into P. Moreover, R is locally confluent (all critical pairs are trivially joinable).Hence we can switch to innermost.
The transformation is resulting in one subcycle: 


   R
     ↳RRRI
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳NOC
           →DP Problem 2
             ↳NOC
             ...
               →DP Problem 5
                 ↳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(x, s(y)) -> s(plus(x, y))


Strategy:

innermost




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


   R
     ↳RRRI
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳NOC
           →DP Problem 2
             ↳NOC
             ...
               →DP Problem 6
                 ↳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