TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60003 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (223ms).
 | – Problem 2 was processed with processor SubtermCriterion (1ms).
 | – Problem 3 was processed with processor PolynomialLinearRange4iUR (1509ms).
 |    | – Problem 9 was processed with processor PolynomialLinearRange4iUR (759ms).
 |    |    | – Problem 10 was processed with processor DependencyGraph (1ms).
 | – Problem 4 was processed with processor SubtermCriterion (0ms).
 | – Problem 5 was processed with processor SubtermCriterion (2ms).
 |    | – Problem 8 was processed with processor DependencyGraph (1ms).
 | – Problem 6 was processed with processor ForwardInstantiation (8ms).
 |    | – Problem 11 remains open; application of the following processors failed [Propagation (10ms), ForwardNarrowing (1ms), BackwardInstantiation (7ms), ForwardInstantiation (6ms), Propagation (10ms)].
 | – Problem 7 was processed with processor SubtermCriterion (1ms).

The following open problems remain:



Open Dependency Pair Problem 6

Dependency Pairs

minIter#(nil, add(n, y), m)minIter#(add(n, y), add(n, y), s(m))if_min#(false, x, y, m)minIter#(x, y, m)
minIter#(add(n, x), y, m)if_min#(le(n, m), x, y, m)

Rewrite Rules

eq(0, 0)trueeq(0, s(x))false
eq(s(x), 0)falseeq(s(x), s(y))eq(x, y)
le(0, y)truele(s(x), 0)false
le(s(x), s(y))le(x, y)app(nil, y)y
app(add(n, x), y)add(n, app(x, y))min(nil)0
min(add(n, x))minIter(add(n, x), add(n, x), 0)minIter(nil, add(n, y), m)minIter(add(n, y), add(n, y), s(m))
minIter(add(n, x), y, m)if_min(le(n, m), x, y, m)if_min(true, x, y, m)m
if_min(false, x, y, m)minIter(x, y, m)head(add(n, x))n
tail(add(n, x))xtail(nil)nil
null(nil)truenull(add(n, x))false
rm(n, nil)nilrm(n, add(m, x))if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x))rm(n, x)if_rm(false, n, add(m, x))add(m, rm(n, x))
minsort(nil, nil)nilminsort(add(n, x), y)if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y)add(n, minsort(app(rm(n, x), y), nil))if_minsort(false, add(n, x), y)minsort(x, add(n, y))

Original Signature

Termination of terms over the following signature is verified: minsort, min, app, rm, true, add, tail, minIter, if_min, 0, if_minsort, s, le, if_rm, false, head, null, eq, nil


Problem 1: DependencyGraph



Dependency Pair Problem

Dependency Pairs

app#(add(n, x), y)app#(x, y)rm#(n, add(m, x))eq#(n, m)
if_rm#(false, n, add(m, x))rm#(n, x)minsort#(add(n, x), y)eq#(n, min(add(n, x)))
if_minsort#(true, add(n, x), y)rm#(n, x)minsort#(add(n, x), y)min#(add(n, x))
if_min#(false, x, y, m)minIter#(x, y, m)if_rm#(true, n, add(m, x))rm#(n, x)
min#(add(n, x))minIter#(add(n, x), add(n, x), 0)if_minsort#(true, add(n, x), y)app#(rm(n, x), y)
minIter#(add(n, x), y, m)if_min#(le(n, m), x, y, m)le#(s(x), s(y))le#(x, y)
minIter#(add(n, x), y, m)le#(n, m)minIter#(nil, add(n, y), m)minIter#(add(n, y), add(n, y), s(m))
if_minsort#(true, add(n, x), y)minsort#(app(rm(n, x), y), nil)minsort#(add(n, x), y)if_minsort#(eq(n, min(add(n, x))), add(n, x), y)
rm#(n, add(m, x))if_rm#(eq(n, m), n, add(m, x))eq#(s(x), s(y))eq#(x, y)
if_minsort#(false, add(n, x), y)minsort#(x, add(n, y))

Rewrite Rules

eq(0, 0)trueeq(0, s(x))false
eq(s(x), 0)falseeq(s(x), s(y))eq(x, y)
le(0, y)truele(s(x), 0)false
le(s(x), s(y))le(x, y)app(nil, y)y
app(add(n, x), y)add(n, app(x, y))min(nil)0
min(add(n, x))minIter(add(n, x), add(n, x), 0)minIter(nil, add(n, y), m)minIter(add(n, y), add(n, y), s(m))
minIter(add(n, x), y, m)if_min(le(n, m), x, y, m)if_min(true, x, y, m)m
if_min(false, x, y, m)minIter(x, y, m)head(add(n, x))n
tail(add(n, x))xtail(nil)nil
null(nil)truenull(add(n, x))false
rm(n, nil)nilrm(n, add(m, x))if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x))rm(n, x)if_rm(false, n, add(m, x))add(m, rm(n, x))
minsort(nil, nil)nilminsort(add(n, x), y)if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y)add(n, minsort(app(rm(n, x), y), nil))if_minsort(false, add(n, x), y)minsort(x, add(n, y))

Original Signature

Termination of terms over the following signature is verified: minsort, min, app, rm, true, add, tail, minIter, if_min, 0, if_minsort, s, le, false, if_rm, head, null, nil, eq

Strategy


The following SCCs where found

if_minsort#(true, add(n, x), y) → minsort#(app(rm(n, x), y), nil)minsort#(add(n, x), y) → if_minsort#(eq(n, min(add(n, x))), add(n, x), y)
if_minsort#(false, add(n, x), y) → minsort#(x, add(n, y))

le#(s(x), s(y)) → le#(x, y)

app#(add(n, x), y) → app#(x, y)

if_rm#(false, n, add(m, x)) → rm#(n, x)rm#(n, add(m, x)) → if_rm#(eq(n, m), n, add(m, x))
if_rm#(true, n, add(m, x)) → rm#(n, x)

minIter#(nil, add(n, y), m) → minIter#(add(n, y), add(n, y), s(m))if_min#(false, x, y, m) → minIter#(x, y, m)
minIter#(add(n, x), y, m) → if_min#(le(n, m), x, y, m)

eq#(s(x), s(y)) → eq#(x, y)

Problem 2: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

app#(add(n, x), y)app#(x, y)

Rewrite Rules

eq(0, 0)trueeq(0, s(x))false
eq(s(x), 0)falseeq(s(x), s(y))eq(x, y)
le(0, y)truele(s(x), 0)false
le(s(x), s(y))le(x, y)app(nil, y)y
app(add(n, x), y)add(n, app(x, y))min(nil)0
min(add(n, x))minIter(add(n, x), add(n, x), 0)minIter(nil, add(n, y), m)minIter(add(n, y), add(n, y), s(m))
minIter(add(n, x), y, m)if_min(le(n, m), x, y, m)if_min(true, x, y, m)m
if_min(false, x, y, m)minIter(x, y, m)head(add(n, x))n
tail(add(n, x))xtail(nil)nil
null(nil)truenull(add(n, x))false
rm(n, nil)nilrm(n, add(m, x))if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x))rm(n, x)if_rm(false, n, add(m, x))add(m, rm(n, x))
minsort(nil, nil)nilminsort(add(n, x), y)if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y)add(n, minsort(app(rm(n, x), y), nil))if_minsort(false, add(n, x), y)minsort(x, add(n, y))

Original Signature

Termination of terms over the following signature is verified: minsort, min, app, rm, true, add, tail, minIter, if_min, 0, if_minsort, s, le, false, if_rm, head, null, nil, eq

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

app#(add(n, x), y)app#(x, y)

Problem 3: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

if_minsort#(true, add(n, x), y)minsort#(app(rm(n, x), y), nil)minsort#(add(n, x), y)if_minsort#(eq(n, min(add(n, x))), add(n, x), y)
if_minsort#(false, add(n, x), y)minsort#(x, add(n, y))

Rewrite Rules

eq(0, 0)trueeq(0, s(x))false
eq(s(x), 0)falseeq(s(x), s(y))eq(x, y)
le(0, y)truele(s(x), 0)false
le(s(x), s(y))le(x, y)app(nil, y)y
app(add(n, x), y)add(n, app(x, y))min(nil)0
min(add(n, x))minIter(add(n, x), add(n, x), 0)minIter(nil, add(n, y), m)minIter(add(n, y), add(n, y), s(m))
minIter(add(n, x), y, m)if_min(le(n, m), x, y, m)if_min(true, x, y, m)m
if_min(false, x, y, m)minIter(x, y, m)head(add(n, x))n
tail(add(n, x))xtail(nil)nil
null(nil)truenull(add(n, x))false
rm(n, nil)nilrm(n, add(m, x))if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x))rm(n, x)if_rm(false, n, add(m, x))add(m, rm(n, x))
minsort(nil, nil)nilminsort(add(n, x), y)if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y)add(n, minsort(app(rm(n, x), y), nil))if_minsort(false, add(n, x), y)minsort(x, add(n, y))

Original Signature

Termination of terms over the following signature is verified: minsort, min, app, rm, true, add, tail, minIter, if_min, 0, if_minsort, s, le, false, if_rm, head, null, nil, eq

Strategy


Polynomial Interpretation

Improved Usable rules

app(nil, y)yif_rm(false, n, add(m, x))add(m, rm(n, x))
if_rm(true, n, add(m, x))rm(n, x)rm(n, add(m, x))if_rm(eq(n, m), n, add(m, x))
rm(n, nil)nilapp(add(n, x), y)add(n, app(x, y))

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

if_minsort#(true, add(n, x), y)minsort#(app(rm(n, x), y), nil)

Problem 9: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

minsort#(add(n, x), y)if_minsort#(eq(n, min(add(n, x))), add(n, x), y)if_minsort#(false, add(n, x), y)minsort#(x, add(n, y))

Rewrite Rules

eq(0, 0)trueeq(0, s(x))false
eq(s(x), 0)falseeq(s(x), s(y))eq(x, y)
le(0, y)truele(s(x), 0)false
le(s(x), s(y))le(x, y)app(nil, y)y
app(add(n, x), y)add(n, app(x, y))min(nil)0
min(add(n, x))minIter(add(n, x), add(n, x), 0)minIter(nil, add(n, y), m)minIter(add(n, y), add(n, y), s(m))
minIter(add(n, x), y, m)if_min(le(n, m), x, y, m)if_min(true, x, y, m)m
if_min(false, x, y, m)minIter(x, y, m)head(add(n, x))n
tail(add(n, x))xtail(nil)nil
null(nil)truenull(add(n, x))false
rm(n, nil)nilrm(n, add(m, x))if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x))rm(n, x)if_rm(false, n, add(m, x))add(m, rm(n, x))
minsort(nil, nil)nilminsort(add(n, x), y)if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y)add(n, minsort(app(rm(n, x), y), nil))if_minsort(false, add(n, x), y)minsort(x, add(n, y))

Original Signature

Termination of terms over the following signature is verified: minsort, min, app, rm, true, add, tail, minIter, if_min, 0, if_minsort, s, le, if_rm, false, head, null, eq, nil

Strategy


Polynomial Interpretation

Improved Usable rules

minIter(add(n, x), y, m)if_min(le(n, m), x, y, m)min(nil)0
eq(0, 0)trueeq(0, s(x))false
eq(s(x), s(y))eq(x, y)minIter(nil, add(n, y), m)minIter(add(n, y), add(n, y), s(m))
if_min(true, x, y, m)meq(s(x), 0)false
min(add(n, x))minIter(add(n, x), add(n, x), 0)if_min(false, x, y, m)minIter(x, y, m)

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

if_minsort#(false, add(n, x), y)minsort#(x, add(n, y))

Problem 10: DependencyGraph



Dependency Pair Problem

Dependency Pairs

minsort#(add(n, x), y)if_minsort#(eq(n, min(add(n, x))), add(n, x), y)

Rewrite Rules

eq(0, 0)trueeq(0, s(x))false
eq(s(x), 0)falseeq(s(x), s(y))eq(x, y)
le(0, y)truele(s(x), 0)false
le(s(x), s(y))le(x, y)app(nil, y)y
app(add(n, x), y)add(n, app(x, y))min(nil)0
min(add(n, x))minIter(add(n, x), add(n, x), 0)minIter(nil, add(n, y), m)minIter(add(n, y), add(n, y), s(m))
minIter(add(n, x), y, m)if_min(le(n, m), x, y, m)if_min(true, x, y, m)m
if_min(false, x, y, m)minIter(x, y, m)head(add(n, x))n
tail(add(n, x))xtail(nil)nil
null(nil)truenull(add(n, x))false
rm(n, nil)nilrm(n, add(m, x))if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x))rm(n, x)if_rm(false, n, add(m, x))add(m, rm(n, x))
minsort(nil, nil)nilminsort(add(n, x), y)if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y)add(n, minsort(app(rm(n, x), y), nil))if_minsort(false, add(n, x), y)minsort(x, add(n, y))

Original Signature

Termination of terms over the following signature is verified: minsort, min, app, rm, true, add, tail, minIter, if_min, 0, if_minsort, s, le, false, if_rm, head, null, nil, eq

Strategy


There are no SCCs!

Problem 4: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

eq#(s(x), s(y))eq#(x, y)

Rewrite Rules

eq(0, 0)trueeq(0, s(x))false
eq(s(x), 0)falseeq(s(x), s(y))eq(x, y)
le(0, y)truele(s(x), 0)false
le(s(x), s(y))le(x, y)app(nil, y)y
app(add(n, x), y)add(n, app(x, y))min(nil)0
min(add(n, x))minIter(add(n, x), add(n, x), 0)minIter(nil, add(n, y), m)minIter(add(n, y), add(n, y), s(m))
minIter(add(n, x), y, m)if_min(le(n, m), x, y, m)if_min(true, x, y, m)m
if_min(false, x, y, m)minIter(x, y, m)head(add(n, x))n
tail(add(n, x))xtail(nil)nil
null(nil)truenull(add(n, x))false
rm(n, nil)nilrm(n, add(m, x))if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x))rm(n, x)if_rm(false, n, add(m, x))add(m, rm(n, x))
minsort(nil, nil)nilminsort(add(n, x), y)if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y)add(n, minsort(app(rm(n, x), y), nil))if_minsort(false, add(n, x), y)minsort(x, add(n, y))

Original Signature

Termination of terms over the following signature is verified: minsort, min, app, rm, true, add, tail, minIter, if_min, 0, if_minsort, s, le, false, if_rm, head, null, nil, eq

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

eq#(s(x), s(y))eq#(x, y)

Problem 5: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

if_rm#(false, n, add(m, x))rm#(n, x)rm#(n, add(m, x))if_rm#(eq(n, m), n, add(m, x))
if_rm#(true, n, add(m, x))rm#(n, x)

Rewrite Rules

eq(0, 0)trueeq(0, s(x))false
eq(s(x), 0)falseeq(s(x), s(y))eq(x, y)
le(0, y)truele(s(x), 0)false
le(s(x), s(y))le(x, y)app(nil, y)y
app(add(n, x), y)add(n, app(x, y))min(nil)0
min(add(n, x))minIter(add(n, x), add(n, x), 0)minIter(nil, add(n, y), m)minIter(add(n, y), add(n, y), s(m))
minIter(add(n, x), y, m)if_min(le(n, m), x, y, m)if_min(true, x, y, m)m
if_min(false, x, y, m)minIter(x, y, m)head(add(n, x))n
tail(add(n, x))xtail(nil)nil
null(nil)truenull(add(n, x))false
rm(n, nil)nilrm(n, add(m, x))if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x))rm(n, x)if_rm(false, n, add(m, x))add(m, rm(n, x))
minsort(nil, nil)nilminsort(add(n, x), y)if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y)add(n, minsort(app(rm(n, x), y), nil))if_minsort(false, add(n, x), y)minsort(x, add(n, y))

Original Signature

Termination of terms over the following signature is verified: minsort, min, app, rm, true, add, tail, minIter, if_min, 0, if_minsort, s, le, false, if_rm, head, null, nil, eq

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

if_rm#(false, n, add(m, x))rm#(n, x)if_rm#(true, n, add(m, x))rm#(n, x)

Problem 8: DependencyGraph



Dependency Pair Problem

Dependency Pairs

rm#(n, add(m, x))if_rm#(eq(n, m), n, add(m, x))

Rewrite Rules

eq(0, 0)trueeq(0, s(x))false
eq(s(x), 0)falseeq(s(x), s(y))eq(x, y)
le(0, y)truele(s(x), 0)false
le(s(x), s(y))le(x, y)app(nil, y)y
app(add(n, x), y)add(n, app(x, y))min(nil)0
min(add(n, x))minIter(add(n, x), add(n, x), 0)minIter(nil, add(n, y), m)minIter(add(n, y), add(n, y), s(m))
minIter(add(n, x), y, m)if_min(le(n, m), x, y, m)if_min(true, x, y, m)m
if_min(false, x, y, m)minIter(x, y, m)head(add(n, x))n
tail(add(n, x))xtail(nil)nil
null(nil)truenull(add(n, x))false
rm(n, nil)nilrm(n, add(m, x))if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x))rm(n, x)if_rm(false, n, add(m, x))add(m, rm(n, x))
minsort(nil, nil)nilminsort(add(n, x), y)if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y)add(n, minsort(app(rm(n, x), y), nil))if_minsort(false, add(n, x), y)minsort(x, add(n, y))

Original Signature

Termination of terms over the following signature is verified: minsort, min, app, rm, true, add, tail, minIter, if_min, 0, if_minsort, s, le, if_rm, false, head, null, eq, nil

Strategy


There are no SCCs!

Problem 6: ForwardInstantiation



Dependency Pair Problem

Dependency Pairs

minIter#(nil, add(n, y), m)minIter#(add(n, y), add(n, y), s(m))if_min#(false, x, y, m)minIter#(x, y, m)
minIter#(add(n, x), y, m)if_min#(le(n, m), x, y, m)

Rewrite Rules

eq(0, 0)trueeq(0, s(x))false
eq(s(x), 0)falseeq(s(x), s(y))eq(x, y)
le(0, y)truele(s(x), 0)false
le(s(x), s(y))le(x, y)app(nil, y)y
app(add(n, x), y)add(n, app(x, y))min(nil)0
min(add(n, x))minIter(add(n, x), add(n, x), 0)minIter(nil, add(n, y), m)minIter(add(n, y), add(n, y), s(m))
minIter(add(n, x), y, m)if_min(le(n, m), x, y, m)if_min(true, x, y, m)m
if_min(false, x, y, m)minIter(x, y, m)head(add(n, x))n
tail(add(n, x))xtail(nil)nil
null(nil)truenull(add(n, x))false
rm(n, nil)nilrm(n, add(m, x))if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x))rm(n, x)if_rm(false, n, add(m, x))add(m, rm(n, x))
minsort(nil, nil)nilminsort(add(n, x), y)if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y)add(n, minsort(app(rm(n, x), y), nil))if_minsort(false, add(n, x), y)minsort(x, add(n, y))

Original Signature

Termination of terms over the following signature is verified: minsort, min, app, rm, true, add, tail, minIter, if_min, 0, if_minsort, s, le, false, if_rm, head, null, nil, eq

Strategy


Instantiation

For all potential successors l → r of the rule if_min#(false, x, y, m) → minIter#(x, y, m) on dependency pair chains it holds that: Thus, if_min#(false, x, y, m) → minIter#(x, y, m) is replaced by instances determined through the above matching. These instances are:
if_min#(false, add(_n, _x), y, m) → minIter#(add(_n, _x), y, m)if_min#(false, nil, add(_n, _y), m) → minIter#(nil, add(_n, _y), m)

Problem 7: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

le#(s(x), s(y))le#(x, y)

Rewrite Rules

eq(0, 0)trueeq(0, s(x))false
eq(s(x), 0)falseeq(s(x), s(y))eq(x, y)
le(0, y)truele(s(x), 0)false
le(s(x), s(y))le(x, y)app(nil, y)y
app(add(n, x), y)add(n, app(x, y))min(nil)0
min(add(n, x))minIter(add(n, x), add(n, x), 0)minIter(nil, add(n, y), m)minIter(add(n, y), add(n, y), s(m))
minIter(add(n, x), y, m)if_min(le(n, m), x, y, m)if_min(true, x, y, m)m
if_min(false, x, y, m)minIter(x, y, m)head(add(n, x))n
tail(add(n, x))xtail(nil)nil
null(nil)truenull(add(n, x))false
rm(n, nil)nilrm(n, add(m, x))if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x))rm(n, x)if_rm(false, n, add(m, x))add(m, rm(n, x))
minsort(nil, nil)nilminsort(add(n, x), y)if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y)add(n, minsort(app(rm(n, x), y), nil))if_minsort(false, add(n, x), y)minsort(x, add(n, y))

Original Signature

Termination of terms over the following signature is verified: minsort, min, app, rm, true, add, tail, minIter, if_min, 0, if_minsort, s, le, false, if_rm, head, null, nil, eq

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

le#(s(x), s(y))le#(x, y)