YES
Termination Proof using AProVETerm Rewriting System R:
[X, N, M]
and(tt, X) -> X
plus(N, 0) -> N
plus(N, s(M)) -> s(plus(N, M))
x(N, 0) -> 0
x(N, s(M)) -> plus(x(N, M), N)
Termination of R to be shown.
R
↳Orthogonal Check
This CSR is orthogonal, so it is sufficient to show innermost termination.
R
↳OrthoCSR
→TRS1
↳CSR to TRS Transformation
Zantema-Transformation successful.
Replacement map:
and(1, 2)
plus(1, 2)
x(1, 2)
s(1)
Old CSR:
and(tt, X) -> X
plus(N, 0) -> N
plus(N, s(M)) -> s(plus(N, M))
x(N, 0) -> 0
x(N, s(M)) -> plus(x(N, M), N)
new TRS:
and(tt, X) -> a(X)
plus(N, 0) -> N
plus(N, s(M)) -> s(plus(N, M))
x(N, 0) -> 0
x(N, s(M)) -> plus(x(N, M), N)
R
↳OrthoCSR
→TRS1
↳CSRtoTRS
→TRS2
↳Overlay and local confluence Check
The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost.
R
↳OrthoCSR
→TRS1
↳CSRtoTRS
→TRS2
↳OC
...
→TRS3
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
PLUS(N, s(M)) -> PLUS(N, M)
X(N, s(M)) -> PLUS(x(N, M), N)
X(N, s(M)) -> X(N, M)
Furthermore, R contains two SCCs.
R
↳OrthoCSR
→TRS1
↳CSRtoTRS
→TRS2
↳OC
...
→DP Problem 1
↳Usable Rules (Innermost)
Dependency Pair:
PLUS(N, s(M)) -> PLUS(N, M)
Rules:
and(tt, X) -> a(X)
plus(N, 0) -> N
plus(N, s(M)) -> s(plus(N, M))
x(N, 0) -> 0
x(N, s(M)) -> plus(x(N, M), N)
Strategy:
innermost
As we are in the innermost case, we can delete all 5 non-usable-rules.
R
↳OrthoCSR
→TRS1
↳CSRtoTRS
→TRS2
↳OC
...
→DP Problem 3
↳Size-Change Principle
Dependency Pair:
PLUS(N, s(M)) -> PLUS(N, M)
Rule:
none
Strategy:
innermost
We number the DPs as follows:
- PLUS(N, s(M)) -> PLUS(N, M)
and get the following Size-Change Graph(s):
which lead(s) to this/these maximal multigraph(s):
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
↳OrthoCSR
→TRS1
↳CSRtoTRS
→TRS2
↳OC
...
→DP Problem 2
↳Usable Rules (Innermost)
Dependency Pair:
X(N, s(M)) -> X(N, M)
Rules:
and(tt, X) -> a(X)
plus(N, 0) -> N
plus(N, s(M)) -> s(plus(N, M))
x(N, 0) -> 0
x(N, s(M)) -> plus(x(N, M), N)
Strategy:
innermost
As we are in the innermost case, we can delete all 5 non-usable-rules.
R
↳OrthoCSR
→TRS1
↳CSRtoTRS
→TRS2
↳OC
...
→DP Problem 4
↳Size-Change Principle
Dependency Pair:
X(N, s(M)) -> X(N, M)
Rule:
none
Strategy:
innermost
We number the DPs as follows:
- X(N, s(M)) -> X(N, M)
and get the following Size-Change Graph(s):
which lead(s) to this/these maximal multigraph(s):
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.
Termination of R successfully shown.
Duration:
0:01 minutes