Term Rewriting System R:
[x, y]
sqr(0) -> 0
sqr(s(x)) -> +(sqr(x), s(double(x)))
sqr(s(x)) -> s(+(sqr(x), double(x)))
double(0) -> 0
double(s(x)) -> s(s(double(x)))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
Innermost Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
SQR(s(x)) -> +'(sqr(x), s(double(x)))
SQR(s(x)) -> SQR(x)
SQR(s(x)) -> DOUBLE(x)
SQR(s(x)) -> +'(sqr(x), double(x))
DOUBLE(s(x)) -> DOUBLE(x)
+'(x, s(y)) -> +'(x, y)
Furthermore, R contains three SCCs.
R
↳DPs
→DP Problem 1
↳Usable Rules (Innermost)
→DP Problem 2
↳UsableRules
→DP Problem 3
↳UsableRules
Dependency Pair:
+'(x, s(y)) -> +'(x, y)
Rules:
sqr(0) -> 0
sqr(s(x)) -> +(sqr(x), s(double(x)))
sqr(s(x)) -> s(+(sqr(x), double(x)))
double(0) -> 0
double(s(x)) -> s(s(double(x)))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
Strategy:
innermost
As we are in the innermost case, we can delete all 7 non-usable-rules.
R
↳DPs
→DP Problem 1
↳UsableRules
→DP Problem 4
↳Size-Change Principle
→DP Problem 2
↳UsableRules
→DP Problem 3
↳UsableRules
Dependency Pair:
+'(x, s(y)) -> +'(x, y)
Rule:
none
Strategy:
innermost
We number the DPs as follows:
- +'(x, s(y)) -> +'(x, y)
and get the following Size-Change Graph(s):
which lead(s) to this/these maximal multigraph(s):
D_{P}: empty set
Oriented Rules: none
We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial
with Argument Filtering System:
s(x_{1}) -> s(x_{1})
We obtain no new DP problems.
R
↳DPs
→DP Problem 1
↳UsableRules
→DP Problem 2
↳Usable Rules (Innermost)
→DP Problem 3
↳UsableRules
Dependency Pair:
DOUBLE(s(x)) -> DOUBLE(x)
Rules:
sqr(0) -> 0
sqr(s(x)) -> +(sqr(x), s(double(x)))
sqr(s(x)) -> s(+(sqr(x), double(x)))
double(0) -> 0
double(s(x)) -> s(s(double(x)))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
Strategy:
innermost
As we are in the innermost case, we can delete all 7 non-usable-rules.
R
↳DPs
→DP Problem 1
↳UsableRules
→DP Problem 2
↳UsableRules
→DP Problem 5
↳Size-Change Principle
→DP Problem 3
↳UsableRules
Dependency Pair:
DOUBLE(s(x)) -> DOUBLE(x)
Rule:
none
Strategy:
innermost
We number the DPs as follows:
- DOUBLE(s(x)) -> DOUBLE(x)
and get the following Size-Change Graph(s):
which lead(s) to this/these maximal multigraph(s):
D_{P}: empty set
Oriented Rules: none
We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial
with Argument Filtering System:
s(x_{1}) -> s(x_{1})
We obtain no new DP problems.
R
↳DPs
→DP Problem 1
↳UsableRules
→DP Problem 2
↳UsableRules
→DP Problem 3
↳Usable Rules (Innermost)
Dependency Pair:
SQR(s(x)) -> SQR(x)
Rules:
sqr(0) -> 0
sqr(s(x)) -> +(sqr(x), s(double(x)))
sqr(s(x)) -> s(+(sqr(x), double(x)))
double(0) -> 0
double(s(x)) -> s(s(double(x)))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
Strategy:
innermost
As we are in the innermost case, we can delete all 7 non-usable-rules.
R
↳DPs
→DP Problem 1
↳UsableRules
→DP Problem 2
↳UsableRules
→DP Problem 3
↳UsableRules
→DP Problem 6
↳Size-Change Principle
Dependency Pair:
SQR(s(x)) -> SQR(x)
Rule:
none
Strategy:
innermost
We number the DPs as follows:
- SQR(s(x)) -> SQR(x)
and get the following Size-Change Graph(s):
which lead(s) to this/these maximal multigraph(s):
D_{P}: empty set
Oriented Rules: none
We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial
with Argument Filtering System:
s(x_{1}) -> s(x_{1})
We obtain no new DP problems.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes