Term Rewriting System R:
[N, X, Y, XS, X1, X2]
active(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(fib(X)) -> fib(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(fib1(X1, X2)) -> fib1(active(X1), X2)
active(fib1(X1, X2)) -> fib1(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fib(mark(X)) -> mark(fib(X))
fib(ok(X)) -> ok(fib(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
fib1(mark(X1), X2) -> mark(fib1(X1, X2))
fib1(X1, mark(X2)) -> mark(fib1(X1, X2))
fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fib(X)) -> fib(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
ACTIVE(fib(N)) -> SEL(N, fib1(s(0), s(0)))
ACTIVE(fib(N)) -> FIB1(s(0), s(0))
ACTIVE(fib(N)) -> S(0)
ACTIVE(fib1(X, Y)) -> CONS(X, fib1(Y, add(X, Y)))
ACTIVE(fib1(X, Y)) -> FIB1(Y, add(X, Y))
ACTIVE(fib1(X, Y)) -> ADD(X, Y)
ACTIVE(add(s(X), Y)) -> S(add(X, Y))
ACTIVE(add(s(X), Y)) -> ADD(X, Y)
ACTIVE(sel(s(N), cons(X, XS))) -> SEL(N, XS)
ACTIVE(fib(X)) -> FIB(active(X))
ACTIVE(fib(X)) -> ACTIVE(X)
ACTIVE(sel(X1, X2)) -> SEL(active(X1), X2)
ACTIVE(sel(X1, X2)) -> ACTIVE(X1)
ACTIVE(sel(X1, X2)) -> SEL(X1, active(X2))
ACTIVE(sel(X1, X2)) -> ACTIVE(X2)
ACTIVE(fib1(X1, X2)) -> FIB1(active(X1), X2)
ACTIVE(fib1(X1, X2)) -> ACTIVE(X1)
ACTIVE(fib1(X1, X2)) -> FIB1(X1, active(X2))
ACTIVE(fib1(X1, X2)) -> ACTIVE(X2)
ACTIVE(s(X)) -> S(active(X))
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(add(X1, X2)) -> ADD(active(X1), X2)
ACTIVE(add(X1, X2)) -> ACTIVE(X1)
ACTIVE(add(X1, X2)) -> ADD(X1, active(X2))
ACTIVE(add(X1, X2)) -> ACTIVE(X2)
FIB(mark(X)) -> FIB(X)
FIB(ok(X)) -> FIB(X)
SEL(mark(X1), X2) -> SEL(X1, X2)
SEL(X1, mark(X2)) -> SEL(X1, X2)
SEL(ok(X1), ok(X2)) -> SEL(X1, X2)
FIB1(mark(X1), X2) -> FIB1(X1, X2)
FIB1(X1, mark(X2)) -> FIB1(X1, X2)
FIB1(ok(X1), ok(X2)) -> FIB1(X1, X2)
S(mark(X)) -> S(X)
S(ok(X)) -> S(X)
CONS(mark(X1), X2) -> CONS(X1, X2)
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
ADD(mark(X1), X2) -> ADD(X1, X2)
ADD(X1, mark(X2)) -> ADD(X1, X2)
ADD(ok(X1), ok(X2)) -> ADD(X1, X2)
PROPER(fib(X)) -> FIB(proper(X))
PROPER(fib(X)) -> PROPER(X)
PROPER(sel(X1, X2)) -> SEL(proper(X1), proper(X2))
PROPER(sel(X1, X2)) -> PROPER(X1)
PROPER(sel(X1, X2)) -> PROPER(X2)
PROPER(fib1(X1, X2)) -> FIB1(proper(X1), proper(X2))
PROPER(fib1(X1, X2)) -> PROPER(X1)
PROPER(fib1(X1, X2)) -> PROPER(X2)
PROPER(s(X)) -> S(proper(X))
PROPER(s(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> CONS(proper(X1), proper(X2))
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(add(X1, X2)) -> ADD(proper(X1), proper(X2))
PROPER(add(X1, X2)) -> PROPER(X1)
PROPER(add(X1, X2)) -> PROPER(X2)
TOP(mark(X)) -> TOP(proper(X))
TOP(mark(X)) -> PROPER(X)
TOP(ok(X)) -> TOP(active(X))
TOP(ok(X)) -> ACTIVE(X)
Furthermore, R contains nine SCCs.
R
↳DPs
→DP Problem 1
↳Size-Change Principle
→DP Problem 2
↳SCP
→DP Problem 3
↳SCP
→DP Problem 4
↳SCP
→DP Problem 5
↳SCP
→DP Problem 6
↳SCP
→DP Problem 7
↳SCP
→DP Problem 8
↳SCP
→DP Problem 9
↳Nar
Dependency Pairs:
SEL(ok(X1), ok(X2)) -> SEL(X1, X2)
SEL(X1, mark(X2)) -> SEL(X1, X2)
SEL(mark(X1), X2) -> SEL(X1, X2)
Rules:
active(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(fib(X)) -> fib(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(fib1(X1, X2)) -> fib1(active(X1), X2)
active(fib1(X1, X2)) -> fib1(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fib(mark(X)) -> mark(fib(X))
fib(ok(X)) -> ok(fib(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
fib1(mark(X1), X2) -> mark(fib1(X1, X2))
fib1(X1, mark(X2)) -> mark(fib1(X1, X2))
fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fib(X)) -> fib(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
We number the DPs as follows:
- SEL(ok(X1), ok(X2)) -> SEL(X1, X2)
- SEL(X1, mark(X2)) -> SEL(X1, X2)
- SEL(mark(X1), X2) -> SEL(X1, X2)
and get the following Size-Change Graph(s): {3, 2, 1} | , | {3, 2, 1} |
---|
1 | > | 1 |
2 | > | 2 |
|
{3, 2, 1} | , | {3, 2, 1} |
---|
1 | = | 1 |
2 | > | 2 |
|
{3, 2, 1} | , | {3, 2, 1} |
---|
1 | > | 1 |
2 | = | 2 |
|
which lead(s) to this/these maximal multigraph(s): {3, 2, 1} | , | {3, 2, 1} |
---|
1 | > | 1 |
2 | = | 2 |
|
{3, 2, 1} | , | {3, 2, 1} |
---|
1 | = | 1 |
2 | > | 2 |
|
{3, 2, 1} | , | {3, 2, 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:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)
We obtain no new DP problems.
R
↳DPs
→DP Problem 1
↳SCP
→DP Problem 2
↳Size-Change Principle
→DP Problem 3
↳SCP
→DP Problem 4
↳SCP
→DP Problem 5
↳SCP
→DP Problem 6
↳SCP
→DP Problem 7
↳SCP
→DP Problem 8
↳SCP
→DP Problem 9
↳Nar
Dependency Pairs:
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
CONS(mark(X1), X2) -> CONS(X1, X2)
Rules:
active(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(fib(X)) -> fib(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(fib1(X1, X2)) -> fib1(active(X1), X2)
active(fib1(X1, X2)) -> fib1(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fib(mark(X)) -> mark(fib(X))
fib(ok(X)) -> ok(fib(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
fib1(mark(X1), X2) -> mark(fib1(X1, X2))
fib1(X1, mark(X2)) -> mark(fib1(X1, X2))
fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fib(X)) -> fib(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
We number the DPs as follows:
- CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
- CONS(mark(X1), X2) -> CONS(X1, X2)
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:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)
We obtain no new DP problems.
R
↳DPs
→DP Problem 1
↳SCP
→DP Problem 2
↳SCP
→DP Problem 3
↳Size-Change Principle
→DP Problem 4
↳SCP
→DP Problem 5
↳SCP
→DP Problem 6
↳SCP
→DP Problem 7
↳SCP
→DP Problem 8
↳SCP
→DP Problem 9
↳Nar
Dependency Pairs:
FIB1(ok(X1), ok(X2)) -> FIB1(X1, X2)
FIB1(X1, mark(X2)) -> FIB1(X1, X2)
FIB1(mark(X1), X2) -> FIB1(X1, X2)
Rules:
active(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(fib(X)) -> fib(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(fib1(X1, X2)) -> fib1(active(X1), X2)
active(fib1(X1, X2)) -> fib1(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fib(mark(X)) -> mark(fib(X))
fib(ok(X)) -> ok(fib(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
fib1(mark(X1), X2) -> mark(fib1(X1, X2))
fib1(X1, mark(X2)) -> mark(fib1(X1, X2))
fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fib(X)) -> fib(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
We number the DPs as follows:
- FIB1(ok(X1), ok(X2)) -> FIB1(X1, X2)
- FIB1(X1, mark(X2)) -> FIB1(X1, X2)
- FIB1(mark(X1), X2) -> FIB1(X1, X2)
and get the following Size-Change Graph(s): {3, 2, 1} | , | {3, 2, 1} |
---|
1 | > | 1 |
2 | > | 2 |
|
{3, 2, 1} | , | {3, 2, 1} |
---|
1 | = | 1 |
2 | > | 2 |
|
{3, 2, 1} | , | {3, 2, 1} |
---|
1 | > | 1 |
2 | = | 2 |
|
which lead(s) to this/these maximal multigraph(s): {3, 2, 1} | , | {3, 2, 1} |
---|
1 | > | 1 |
2 | = | 2 |
|
{3, 2, 1} | , | {3, 2, 1} |
---|
1 | = | 1 |
2 | > | 2 |
|
{3, 2, 1} | , | {3, 2, 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:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)
We obtain no new DP problems.
R
↳DPs
→DP Problem 1
↳SCP
→DP Problem 2
↳SCP
→DP Problem 3
↳SCP
→DP Problem 4
↳Size-Change Principle
→DP Problem 5
↳SCP
→DP Problem 6
↳SCP
→DP Problem 7
↳SCP
→DP Problem 8
↳SCP
→DP Problem 9
↳Nar
Dependency Pairs:
ADD(ok(X1), ok(X2)) -> ADD(X1, X2)
ADD(X1, mark(X2)) -> ADD(X1, X2)
ADD(mark(X1), X2) -> ADD(X1, X2)
Rules:
active(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(fib(X)) -> fib(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(fib1(X1, X2)) -> fib1(active(X1), X2)
active(fib1(X1, X2)) -> fib1(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fib(mark(X)) -> mark(fib(X))
fib(ok(X)) -> ok(fib(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
fib1(mark(X1), X2) -> mark(fib1(X1, X2))
fib1(X1, mark(X2)) -> mark(fib1(X1, X2))
fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fib(X)) -> fib(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
We number the DPs as follows:
- ADD(ok(X1), ok(X2)) -> ADD(X1, X2)
- ADD(X1, mark(X2)) -> ADD(X1, X2)
- ADD(mark(X1), X2) -> ADD(X1, X2)
and get the following Size-Change Graph(s): {3, 2, 1} | , | {3, 2, 1} |
---|
1 | > | 1 |
2 | > | 2 |
|
{3, 2, 1} | , | {3, 2, 1} |
---|
1 | = | 1 |
2 | > | 2 |
|
{3, 2, 1} | , | {3, 2, 1} |
---|
1 | > | 1 |
2 | = | 2 |
|
which lead(s) to this/these maximal multigraph(s): {3, 2, 1} | , | {3, 2, 1} |
---|
1 | > | 1 |
2 | > | 2 |
|
{3, 2, 1} | , | {3, 2, 1} |
---|
1 | > | 1 |
2 | = | 2 |
|
{3, 2, 1} | , | {3, 2, 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:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)
We obtain no new DP problems.
R
↳DPs
→DP Problem 1
↳SCP
→DP Problem 2
↳SCP
→DP Problem 3
↳SCP
→DP Problem 4
↳SCP
→DP Problem 5
↳Size-Change Principle
→DP Problem 6
↳SCP
→DP Problem 7
↳SCP
→DP Problem 8
↳SCP
→DP Problem 9
↳Nar
Dependency Pairs:
S(ok(X)) -> S(X)
S(mark(X)) -> S(X)
Rules:
active(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(fib(X)) -> fib(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(fib1(X1, X2)) -> fib1(active(X1), X2)
active(fib1(X1, X2)) -> fib1(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fib(mark(X)) -> mark(fib(X))
fib(ok(X)) -> ok(fib(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
fib1(mark(X1), X2) -> mark(fib1(X1, X2))
fib1(X1, mark(X2)) -> mark(fib1(X1, X2))
fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fib(X)) -> fib(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
We number the DPs as follows:
- S(ok(X)) -> S(X)
- S(mark(X)) -> S(X)
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:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)
We obtain no new DP problems.
R
↳DPs
→DP Problem 1
↳SCP
→DP Problem 2
↳SCP
→DP Problem 3
↳SCP
→DP Problem 4
↳SCP
→DP Problem 5
↳SCP
→DP Problem 6
↳Size-Change Principle
→DP Problem 7
↳SCP
→DP Problem 8
↳SCP
→DP Problem 9
↳Nar
Dependency Pairs:
FIB(ok(X)) -> FIB(X)
FIB(mark(X)) -> FIB(X)
Rules:
active(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(fib(X)) -> fib(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(fib1(X1, X2)) -> fib1(active(X1), X2)
active(fib1(X1, X2)) -> fib1(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fib(mark(X)) -> mark(fib(X))
fib(ok(X)) -> ok(fib(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
fib1(mark(X1), X2) -> mark(fib1(X1, X2))
fib1(X1, mark(X2)) -> mark(fib1(X1, X2))
fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fib(X)) -> fib(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
We number the DPs as follows:
- FIB(ok(X)) -> FIB(X)
- FIB(mark(X)) -> FIB(X)
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:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)
We obtain no new DP problems.
R
↳DPs
→DP Problem 1
↳SCP
→DP Problem 2
↳SCP
→DP Problem 3
↳SCP
→DP Problem 4
↳SCP
→DP Problem 5
↳SCP
→DP Problem 6
↳SCP
→DP Problem 7
↳Size-Change Principle
→DP Problem 8
↳SCP
→DP Problem 9
↳Nar
Dependency Pairs:
ACTIVE(add(X1, X2)) -> ACTIVE(X2)
ACTIVE(add(X1, X2)) -> ACTIVE(X1)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(fib1(X1, X2)) -> ACTIVE(X2)
ACTIVE(fib1(X1, X2)) -> ACTIVE(X1)
ACTIVE(sel(X1, X2)) -> ACTIVE(X2)
ACTIVE(sel(X1, X2)) -> ACTIVE(X1)
ACTIVE(fib(X)) -> ACTIVE(X)
Rules:
active(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(fib(X)) -> fib(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(fib1(X1, X2)) -> fib1(active(X1), X2)
active(fib1(X1, X2)) -> fib1(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fib(mark(X)) -> mark(fib(X))
fib(ok(X)) -> ok(fib(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
fib1(mark(X1), X2) -> mark(fib1(X1, X2))
fib1(X1, mark(X2)) -> mark(fib1(X1, X2))
fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fib(X)) -> fib(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
We number the DPs as follows:
- ACTIVE(add(X1, X2)) -> ACTIVE(X2)
- ACTIVE(add(X1, X2)) -> ACTIVE(X1)
- ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
- ACTIVE(s(X)) -> ACTIVE(X)
- ACTIVE(fib1(X1, X2)) -> ACTIVE(X2)
- ACTIVE(fib1(X1, X2)) -> ACTIVE(X1)
- ACTIVE(sel(X1, X2)) -> ACTIVE(X2)
- ACTIVE(sel(X1, X2)) -> ACTIVE(X1)
- ACTIVE(fib(X)) -> ACTIVE(X)
and get the following Size-Change Graph(s): {9, 8, 7, 6, 5, 4, 3, 2, 1} | , | {9, 8, 7, 6, 5, 4, 3, 2, 1} |
---|
1 | > | 1 |
|
which lead(s) to this/these maximal multigraph(s): {9, 8, 7, 6, 5, 4, 3, 2, 1} | , | {9, 8, 7, 6, 5, 4, 3, 2, 1} |
---|
1 | > | 1 |
|
DP: empty set
Oriented Rules: none
We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial
with Argument Filtering System:
cons(x1, x2) -> cons(x1, x2)
fib(x1) -> fib(x1)
s(x1) -> s(x1)
sel(x1, x2) -> sel(x1, x2)
fib1(x1, x2) -> fib1(x1, x2)
add(x1, x2) -> add(x1, x2)
We obtain no new DP problems.
R
↳DPs
→DP Problem 1
↳SCP
→DP Problem 2
↳SCP
→DP Problem 3
↳SCP
→DP Problem 4
↳SCP
→DP Problem 5
↳SCP
→DP Problem 6
↳SCP
→DP Problem 7
↳SCP
→DP Problem 8
↳Size-Change Principle
→DP Problem 9
↳Nar
Dependency Pairs:
PROPER(add(X1, X2)) -> PROPER(X2)
PROPER(add(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(s(X)) -> PROPER(X)
PROPER(fib1(X1, X2)) -> PROPER(X2)
PROPER(fib1(X1, X2)) -> PROPER(X1)
PROPER(sel(X1, X2)) -> PROPER(X2)
PROPER(sel(X1, X2)) -> PROPER(X1)
PROPER(fib(X)) -> PROPER(X)
Rules:
active(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(fib(X)) -> fib(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(fib1(X1, X2)) -> fib1(active(X1), X2)
active(fib1(X1, X2)) -> fib1(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fib(mark(X)) -> mark(fib(X))
fib(ok(X)) -> ok(fib(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
fib1(mark(X1), X2) -> mark(fib1(X1, X2))
fib1(X1, mark(X2)) -> mark(fib1(X1, X2))
fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fib(X)) -> fib(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
We number the DPs as follows:
- PROPER(add(X1, X2)) -> PROPER(X2)
- PROPER(add(X1, X2)) -> PROPER(X1)
- PROPER(cons(X1, X2)) -> PROPER(X2)
- PROPER(cons(X1, X2)) -> PROPER(X1)
- PROPER(s(X)) -> PROPER(X)
- PROPER(fib1(X1, X2)) -> PROPER(X2)
- PROPER(fib1(X1, X2)) -> PROPER(X1)
- PROPER(sel(X1, X2)) -> PROPER(X2)
- PROPER(sel(X1, X2)) -> PROPER(X1)
- PROPER(fib(X)) -> PROPER(X)
and get the following Size-Change Graph(s): {10, 9, 8, 7, 6, 5, 4, 3, 2, 1} | , | {10, 9, 8, 7, 6, 5, 4, 3, 2, 1} |
---|
1 | > | 1 |
|
which lead(s) to this/these maximal multigraph(s): {10, 9, 8, 7, 6, 5, 4, 3, 2, 1} | , | {10, 9, 8, 7, 6, 5, 4, 3, 2, 1} |
---|
1 | > | 1 |
|
DP: empty set
Oriented Rules: none
We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial
with Argument Filtering System:
cons(x1, x2) -> cons(x1, x2)
fib(x1) -> fib(x1)
sel(x1, x2) -> sel(x1, x2)
s(x1) -> s(x1)
fib1(x1, x2) -> fib1(x1, x2)
add(x1, x2) -> add(x1, x2)
We obtain no new DP problems.
R
↳DPs
→DP Problem 1
↳SCP
→DP Problem 2
↳SCP
→DP Problem 3
↳SCP
→DP Problem 4
↳SCP
→DP Problem 5
↳SCP
→DP Problem 6
↳SCP
→DP Problem 7
↳SCP
→DP Problem 8
↳SCP
→DP Problem 9
↳Narrowing Transformation
Dependency Pairs:
TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))
Rules:
active(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(fib(X)) -> fib(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(fib1(X1, X2)) -> fib1(active(X1), X2)
active(fib1(X1, X2)) -> fib1(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fib(mark(X)) -> mark(fib(X))
fib(ok(X)) -> ok(fib(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
fib1(mark(X1), X2) -> mark(fib1(X1, X2))
fib1(X1, mark(X2)) -> mark(fib1(X1, X2))
fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fib(X)) -> fib(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule
TOP(mark(X)) -> TOP(proper(X))
seven new Dependency Pairs
are created:
TOP(mark(fib(X''))) -> TOP(fib(proper(X'')))
TOP(mark(sel(X1', X2'))) -> TOP(sel(proper(X1'), proper(X2')))
TOP(mark(fib1(X1', X2'))) -> TOP(fib1(proper(X1'), proper(X2')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(0)) -> TOP(ok(0))
TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(mark(add(X1', X2'))) -> TOP(add(proper(X1'), proper(X2')))
The transformation is resulting in one new DP problem:
R
↳DPs
→DP Problem 1
↳SCP
→DP Problem 2
↳SCP
→DP Problem 3
↳SCP
→DP Problem 4
↳SCP
→DP Problem 5
↳SCP
→DP Problem 6
↳SCP
→DP Problem 7
↳SCP
→DP Problem 8
↳SCP
→DP Problem 9
↳Nar
→DP Problem 10
↳Narrowing Transformation
Dependency Pairs:
TOP(mark(add(X1', X2'))) -> TOP(add(proper(X1'), proper(X2')))
TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(mark(0)) -> TOP(ok(0))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(fib1(X1', X2'))) -> TOP(fib1(proper(X1'), proper(X2')))
TOP(mark(sel(X1', X2'))) -> TOP(sel(proper(X1'), proper(X2')))
TOP(mark(fib(X''))) -> TOP(fib(proper(X'')))
TOP(ok(X)) -> TOP(active(X))
Rules:
active(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(fib(X)) -> fib(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(fib1(X1, X2)) -> fib1(active(X1), X2)
active(fib1(X1, X2)) -> fib1(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fib(mark(X)) -> mark(fib(X))
fib(ok(X)) -> ok(fib(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
fib1(mark(X1), X2) -> mark(fib1(X1, X2))
fib1(X1, mark(X2)) -> mark(fib1(X1, X2))
fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fib(X)) -> fib(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule
TOP(ok(X)) -> TOP(active(X))
15 new Dependency Pairs
are created:
TOP(ok(fib(N'))) -> TOP(mark(sel(N', fib1(s(0), s(0)))))
TOP(ok(fib1(X'', Y'))) -> TOP(mark(cons(X'', fib1(Y', add(X'', Y')))))
TOP(ok(add(0, X''))) -> TOP(mark(X''))
TOP(ok(add(s(X''), Y'))) -> TOP(mark(s(add(X'', Y'))))
TOP(ok(sel(0, cons(X'', XS')))) -> TOP(mark(X''))
TOP(ok(sel(s(N'), cons(X'', XS')))) -> TOP(mark(sel(N', XS')))
TOP(ok(fib(X''))) -> TOP(fib(active(X'')))
TOP(ok(sel(X1', X2'))) -> TOP(sel(active(X1'), X2'))
TOP(ok(sel(X1', X2'))) -> TOP(sel(X1', active(X2')))
TOP(ok(fib1(X1', X2'))) -> TOP(fib1(active(X1'), X2'))
TOP(ok(fib1(X1', X2'))) -> TOP(fib1(X1', active(X2')))
TOP(ok(s(X''))) -> TOP(s(active(X'')))
TOP(ok(cons(X1', X2'))) -> TOP(cons(active(X1'), X2'))
TOP(ok(add(X1', X2'))) -> TOP(add(active(X1'), X2'))
TOP(ok(add(X1', X2'))) -> TOP(add(X1', active(X2')))
The transformation is resulting in one new DP problem:
R
↳DPs
→DP Problem 1
↳SCP
→DP Problem 2
↳SCP
→DP Problem 3
↳SCP
→DP Problem 4
↳SCP
→DP Problem 5
↳SCP
→DP Problem 6
↳SCP
→DP Problem 7
↳SCP
→DP Problem 8
↳SCP
→DP Problem 9
↳Nar
→DP Problem 10
↳Nar
...
→DP Problem 11
↳Remaining Obligation(s)
The following remains to be proven:
Dependency Pairs:
TOP(ok(add(X1', X2'))) -> TOP(add(X1', active(X2')))
TOP(ok(add(X1', X2'))) -> TOP(add(active(X1'), X2'))
TOP(ok(cons(X1', X2'))) -> TOP(cons(active(X1'), X2'))
TOP(ok(s(X''))) -> TOP(s(active(X'')))
TOP(ok(fib1(X1', X2'))) -> TOP(fib1(X1', active(X2')))
TOP(ok(fib1(X1', X2'))) -> TOP(fib1(active(X1'), X2'))
TOP(ok(sel(X1', X2'))) -> TOP(sel(X1', active(X2')))
TOP(ok(sel(X1', X2'))) -> TOP(sel(active(X1'), X2'))
TOP(ok(fib(X''))) -> TOP(fib(active(X'')))
TOP(ok(sel(s(N'), cons(X'', XS')))) -> TOP(mark(sel(N', XS')))
TOP(ok(sel(0, cons(X'', XS')))) -> TOP(mark(X''))
TOP(ok(add(s(X''), Y'))) -> TOP(mark(s(add(X'', Y'))))
TOP(ok(add(0, X''))) -> TOP(mark(X''))
TOP(ok(fib1(X'', Y'))) -> TOP(mark(cons(X'', fib1(Y', add(X'', Y')))))
TOP(ok(fib(N'))) -> TOP(mark(sel(N', fib1(s(0), s(0)))))
TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(fib1(X1', X2'))) -> TOP(fib1(proper(X1'), proper(X2')))
TOP(mark(sel(X1', X2'))) -> TOP(sel(proper(X1'), proper(X2')))
TOP(mark(fib(X''))) -> TOP(fib(proper(X'')))
TOP(mark(add(X1', X2'))) -> TOP(add(proper(X1'), proper(X2')))
Rules:
active(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(fib(X)) -> fib(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(fib1(X1, X2)) -> fib1(active(X1), X2)
active(fib1(X1, X2)) -> fib1(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
fib(mark(X)) -> mark(fib(X))
fib(ok(X)) -> ok(fib(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
fib1(mark(X1), X2) -> mark(fib1(X1, X2))
fib1(X1, mark(X2)) -> mark(fib1(X1, X2))
fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
proper(fib(X)) -> fib(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
The Proof could not be continued due to a Timeout.
Termination of R could not be shown.
Duration:
1:00 minutes