Term Rewriting System R:
[X, Y, Z]
plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))
times(X, s(Y)) -> plus(X, times(Y, X))
Innermost Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
PLUS(plus(X, Y), Z) -> PLUS(X, plus(Y, Z))
PLUS(plus(X, Y), Z) -> PLUS(Y, Z)
TIMES(X, s(Y)) -> PLUS(X, times(Y, X))
TIMES(X, s(Y)) -> TIMES(Y, X)
Furthermore, R contains two SCCs.
R
↳DPs
→DP Problem 1
↳Usable Rules (Innermost)
→DP Problem 2
↳UsableRules
Dependency Pair:
PLUS(plus(X, Y), Z) -> PLUS(Y, Z)
Rules:
plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))
times(X, s(Y)) -> plus(X, times(Y, X))
Strategy:
innermost
As we are in the innermost case, we can delete all 2 non-usable-rules.
R
↳DPs
→DP Problem 1
↳UsableRules
→DP Problem 3
↳Size-Change Principle
→DP Problem 2
↳UsableRules
Dependency Pair:
PLUS(plus(X, Y), Z) -> PLUS(Y, Z)
Rule:
none
Strategy:
innermost
We number the DPs as follows:
- PLUS(plus(X, Y), Z) -> PLUS(Y, Z)
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:
plus(x_{1}, x_{2}) -> plus(x_{1}, x_{2})
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(Y, X)
Rules:
plus(plus(X, Y), Z) -> plus(X, plus(Y, Z))
times(X, s(Y)) -> plus(X, times(Y, X))
Strategy:
innermost
As we are in the innermost case, we can delete all 2 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(Y, X)
Rule:
none
Strategy:
innermost
We number the DPs as follows:
- TIMES(X, s(Y)) -> TIMES(Y, 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