Term Rewriting System R:
[y, x, z]
le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
eq(0, 0) -> true
eq(0, s(y)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
minsort(nil) -> nil
minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) -> x
min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
del(x, nil) -> nil
del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z)))

Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

LE(s(x), s(y)) -> LE(x, y)
EQ(s(x), s(y)) -> EQ(x, y)
MINSORT(cons(x, y)) -> MIN(x, y)
MINSORT(cons(x, y)) -> MINSORT(del(min(x, y), cons(x, y)))
MINSORT(cons(x, y)) -> DEL(min(x, y), cons(x, y))
MIN(x, cons(y, z)) -> IF(le(x, y), min(x, z), min(y, z))
MIN(x, cons(y, z)) -> LE(x, y)
MIN(x, cons(y, z)) -> MIN(x, z)
MIN(x, cons(y, z)) -> MIN(y, z)
DEL(x, cons(y, z)) -> IF(eq(x, y), z, cons(y, del(x, z)))
DEL(x, cons(y, z)) -> EQ(x, y)
DEL(x, cons(y, z)) -> DEL(x, z)

Furthermore, R contains five SCCs.


   R
DPs
       →DP Problem 1
Argument Filtering and Ordering
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
Remaining


Dependency Pair:

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


Rules:


le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
eq(0, 0) -> true
eq(0, s(y)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
minsort(nil) -> nil
minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) -> x
min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
del(x, nil) -> nil
del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z)))





The following dependency pair can be strictly oriented:

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


There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
LE(x1, x2) -> LE(x1, x2)
s(x1) -> s(x1)


   R
DPs
       →DP Problem 1
AFS
           →DP Problem 6
Dependency Graph
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
Remaining


Dependency Pair:


Rules:


le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
eq(0, 0) -> true
eq(0, s(y)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
minsort(nil) -> nil
minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) -> x
min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
del(x, nil) -> nil
del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z)))





Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
Argument Filtering and Ordering
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
Remaining


Dependency Pair:

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


Rules:


le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
eq(0, 0) -> true
eq(0, s(y)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
minsort(nil) -> nil
minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) -> x
min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
del(x, nil) -> nil
del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z)))





The following dependency pair can be strictly oriented:

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


There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
EQ(x1, x2) -> EQ(x1, x2)
s(x1) -> s(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
           →DP Problem 7
Dependency Graph
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
Remaining


Dependency Pair:


Rules:


le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
eq(0, 0) -> true
eq(0, s(y)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
minsort(nil) -> nil
minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) -> x
min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
del(x, nil) -> nil
del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z)))





Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
Argument Filtering and Ordering
       →DP Problem 4
AFS
       →DP Problem 5
Remaining


Dependency Pairs:

MIN(x, cons(y, z)) -> MIN(y, z)
MIN(x, cons(y, z)) -> MIN(x, z)


Rules:


le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
eq(0, 0) -> true
eq(0, s(y)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
minsort(nil) -> nil
minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) -> x
min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
del(x, nil) -> nil
del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z)))





The following dependency pairs can be strictly oriented:

MIN(x, cons(y, z)) -> MIN(y, z)
MIN(x, cons(y, z)) -> MIN(x, z)


There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
MIN(x1, x2) -> x2
cons(x1, x2) -> cons(x1, x2)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
           →DP Problem 8
Dependency Graph
       →DP Problem 4
AFS
       →DP Problem 5
Remaining


Dependency Pair:


Rules:


le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
eq(0, 0) -> true
eq(0, s(y)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
minsort(nil) -> nil
minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) -> x
min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
del(x, nil) -> nil
del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z)))





Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
Argument Filtering and Ordering
       →DP Problem 5
Remaining


Dependency Pair:

DEL(x, cons(y, z)) -> DEL(x, z)


Rules:


le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
eq(0, 0) -> true
eq(0, s(y)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
minsort(nil) -> nil
minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) -> x
min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
del(x, nil) -> nil
del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z)))





The following dependency pair can be strictly oriented:

DEL(x, cons(y, z)) -> DEL(x, z)


There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
DEL(x1, x2) -> DEL(x1, x2)
cons(x1, x2) -> cons(x1, x2)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
           →DP Problem 9
Dependency Graph
       →DP Problem 5
Remaining


Dependency Pair:


Rules:


le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
eq(0, 0) -> true
eq(0, s(y)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
minsort(nil) -> nil
minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) -> x
min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
del(x, nil) -> nil
del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z)))





Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

MINSORT(cons(x, y)) -> MINSORT(del(min(x, y), cons(x, y)))


Rules:


le(0, y) -> true
le(s(x), 0) -> false
le(s(x), s(y)) -> le(x, y)
eq(0, 0) -> true
eq(0, s(y)) -> false
eq(s(x), 0) -> false
eq(s(x), s(y)) -> eq(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
minsort(nil) -> nil
minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) -> x
min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
del(x, nil) -> nil
del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, z)))




Termination of R could not be shown.
Duration:
0:00 minutes