TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60000 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (228ms).
 | – Problem 2 was processed with processor SubtermCriterion (1ms).
 | – Problem 3 was processed with processor BackwardInstantiation (3ms).
 |    | – Problem 7 was processed with processor ForwardInstantiation (5ms).
 |    |    | – Problem 9 was processed with processor Propagation (15ms).
 |    |    |    | – Problem 10 remains open; application of the following processors failed [ForwardNarrowing (1ms), BackwardInstantiation (3ms), ForwardInstantiation (3ms), Propagation (3ms), ForwardNarrowing (1ms), BackwardInstantiation (4ms), ForwardInstantiation (2ms), Propagation (2ms)].
 | – Problem 4 was processed with processor SubtermCriterion (3ms).
 | – Problem 5 was processed with processor BackwardInstantiation (4ms).
 |    | – Problem 8 was processed with processor BackwardInstantiation (5ms).
 |    |    | – Problem 11 was processed with processor Propagation (3ms).
 |    |    |    | – Problem 12 remains open; application of the following processors failed [ForwardNarrowing (0ms), BackwardInstantiation (1ms), ForwardInstantiation (2ms), Propagation (1ms)].
 | – Problem 6 was processed with processor SubtermCriterion (1ms).

The following open problems remain:



Open Dependency Pair Problem 3

Dependency Pairs

ifsum2#(false, xs, y)sum2#(cons(p(head(xs)), tail(xs)), s(y))sum2#(xs, y)ifsum#(isNil(xs), isZero(head(xs)), xs, y)
ifsum#(false, b, xs, y)ifsum2#(b, xs, y)ifsum2#(true, xs, y)sum2#(tail(xs), y)

Rewrite Rules

times(x, y)sum(generate(x, y))generate(x, y)gen(x, y, 0)
gen(x, y, z)if(ge(z, x), x, y, z)if(true, x, y, z)nil
if(false, x, y, z)cons(y, gen(x, y, s(z)))sum(xs)sum2(xs, 0)
sum2(xs, y)ifsum(isNil(xs), isZero(head(xs)), xs, y)ifsum(true, b, xs, y)y
ifsum(false, b, xs, y)ifsum2(b, xs, y)ifsum2(true, xs, y)sum2(tail(xs), y)
ifsum2(false, xs, y)sum2(cons(p(head(xs)), tail(xs)), s(y))isNil(nil)true
isNil(cons(x, xs))falsetail(nil)nil
tail(cons(x, xs))xshead(cons(x, xs))x
head(nil)errorisZero(0)true
isZero(s(0))falseisZero(s(s(x)))isZero(s(x))
p(0)s(s(0))p(s(0))0
p(s(s(x)))s(p(s(x)))ge(x, 0)true
ge(0, s(y))falsege(s(x), s(y))ge(x, y)
acad

Original Signature

Termination of terms over the following signature is verified: ifsum, d, gen, ifsum2, error, generate, c, a, sum, true, ge, tail, 0, s, times, if, sum2, p, false, head, isZero, isNil, nil, cons




Open Dependency Pair Problem 5

Dependency Pairs

if#(false, x, y, z)gen#(x, y, s(z))gen#(x, y, z)if#(ge(z, x), x, y, z)

Rewrite Rules

times(x, y)sum(generate(x, y))generate(x, y)gen(x, y, 0)
gen(x, y, z)if(ge(z, x), x, y, z)if(true, x, y, z)nil
if(false, x, y, z)cons(y, gen(x, y, s(z)))sum(xs)sum2(xs, 0)
sum2(xs, y)ifsum(isNil(xs), isZero(head(xs)), xs, y)ifsum(true, b, xs, y)y
ifsum(false, b, xs, y)ifsum2(b, xs, y)ifsum2(true, xs, y)sum2(tail(xs), y)
ifsum2(false, xs, y)sum2(cons(p(head(xs)), tail(xs)), s(y))isNil(nil)true
isNil(cons(x, xs))falsetail(nil)nil
tail(cons(x, xs))xshead(cons(x, xs))x
head(nil)errorisZero(0)true
isZero(s(0))falseisZero(s(s(x)))isZero(s(x))
p(0)s(s(0))p(s(0))0
p(s(s(x)))s(p(s(x)))ge(x, 0)true
ge(0, s(y))falsege(s(x), s(y))ge(x, y)
acad

Original Signature

Termination of terms over the following signature is verified: ifsum, d, gen, ifsum2, error, generate, c, a, sum, true, ge, tail, 0, s, times, if, sum2, p, false, head, isZero, isNil, nil, cons


Problem 1: DependencyGraph



Dependency Pair Problem

Dependency Pairs

if#(false, x, y, z)gen#(x, y, s(z))ifsum2#(false, xs, y)p#(head(xs))
ifsum2#(false, xs, y)sum2#(cons(p(head(xs)), tail(xs)), s(y))sum2#(xs, y)isNil#(xs)
sum#(xs)sum2#(xs, 0)times#(x, y)sum#(generate(x, y))
sum2#(xs, y)isZero#(head(xs))ifsum#(false, b, xs, y)ifsum2#(b, xs, y)
ifsum2#(true, xs, y)tail#(xs)ifsum2#(true, xs, y)sum2#(tail(xs), y)
ifsum2#(false, xs, y)head#(xs)gen#(x, y, z)if#(ge(z, x), x, y, z)
gen#(x, y, z)ge#(z, x)generate#(x, y)gen#(x, y, 0)
ifsum2#(false, xs, y)tail#(xs)isZero#(s(s(x)))isZero#(s(x))
sum2#(xs, y)head#(xs)ge#(s(x), s(y))ge#(x, y)
times#(x, y)generate#(x, y)sum2#(xs, y)ifsum#(isNil(xs), isZero(head(xs)), xs, y)
p#(s(s(x)))p#(s(x))

Rewrite Rules

times(x, y)sum(generate(x, y))generate(x, y)gen(x, y, 0)
gen(x, y, z)if(ge(z, x), x, y, z)if(true, x, y, z)nil
if(false, x, y, z)cons(y, gen(x, y, s(z)))sum(xs)sum2(xs, 0)
sum2(xs, y)ifsum(isNil(xs), isZero(head(xs)), xs, y)ifsum(true, b, xs, y)y
ifsum(false, b, xs, y)ifsum2(b, xs, y)ifsum2(true, xs, y)sum2(tail(xs), y)
ifsum2(false, xs, y)sum2(cons(p(head(xs)), tail(xs)), s(y))isNil(nil)true
isNil(cons(x, xs))falsetail(nil)nil
tail(cons(x, xs))xshead(cons(x, xs))x
head(nil)errorisZero(0)true
isZero(s(0))falseisZero(s(s(x)))isZero(s(x))
p(0)s(s(0))p(s(0))0
p(s(s(x)))s(p(s(x)))ge(x, 0)true
ge(0, s(y))falsege(s(x), s(y))ge(x, y)
acad

Original Signature

Termination of terms over the following signature is verified: ifsum, d, gen, ifsum2, error, generate, c, a, sum, true, ge, tail, 0, s, times, sum2, if, p, false, head, cons, nil, isNil, isZero

Strategy


The following SCCs where found

ifsum2#(false, xs, y) → sum2#(cons(p(head(xs)), tail(xs)), s(y))sum2#(xs, y) → ifsum#(isNil(xs), isZero(head(xs)), xs, y)
ifsum#(false, b, xs, y) → ifsum2#(b, xs, y)ifsum2#(true, xs, y) → sum2#(tail(xs), y)

isZero#(s(s(x))) → isZero#(s(x))

if#(false, x, y, z) → gen#(x, y, s(z))gen#(x, y, z) → if#(ge(z, x), x, y, z)

ge#(s(x), s(y)) → ge#(x, y)

p#(s(s(x))) → p#(s(x))

Problem 2: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

ge#(s(x), s(y))ge#(x, y)

Rewrite Rules

times(x, y)sum(generate(x, y))generate(x, y)gen(x, y, 0)
gen(x, y, z)if(ge(z, x), x, y, z)if(true, x, y, z)nil
if(false, x, y, z)cons(y, gen(x, y, s(z)))sum(xs)sum2(xs, 0)
sum2(xs, y)ifsum(isNil(xs), isZero(head(xs)), xs, y)ifsum(true, b, xs, y)y
ifsum(false, b, xs, y)ifsum2(b, xs, y)ifsum2(true, xs, y)sum2(tail(xs), y)
ifsum2(false, xs, y)sum2(cons(p(head(xs)), tail(xs)), s(y))isNil(nil)true
isNil(cons(x, xs))falsetail(nil)nil
tail(cons(x, xs))xshead(cons(x, xs))x
head(nil)errorisZero(0)true
isZero(s(0))falseisZero(s(s(x)))isZero(s(x))
p(0)s(s(0))p(s(0))0
p(s(s(x)))s(p(s(x)))ge(x, 0)true
ge(0, s(y))falsege(s(x), s(y))ge(x, y)
acad

Original Signature

Termination of terms over the following signature is verified: ifsum, d, gen, ifsum2, error, generate, c, a, sum, true, ge, tail, 0, s, times, sum2, if, p, false, head, cons, nil, isNil, isZero

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

ge#(s(x), s(y))ge#(x, y)

Problem 3: BackwardInstantiation



Dependency Pair Problem

Dependency Pairs

ifsum2#(false, xs, y)sum2#(cons(p(head(xs)), tail(xs)), s(y))sum2#(xs, y)ifsum#(isNil(xs), isZero(head(xs)), xs, y)
ifsum#(false, b, xs, y)ifsum2#(b, xs, y)ifsum2#(true, xs, y)sum2#(tail(xs), y)

Rewrite Rules

times(x, y)sum(generate(x, y))generate(x, y)gen(x, y, 0)
gen(x, y, z)if(ge(z, x), x, y, z)if(true, x, y, z)nil
if(false, x, y, z)cons(y, gen(x, y, s(z)))sum(xs)sum2(xs, 0)
sum2(xs, y)ifsum(isNil(xs), isZero(head(xs)), xs, y)ifsum(true, b, xs, y)y
ifsum(false, b, xs, y)ifsum2(b, xs, y)ifsum2(true, xs, y)sum2(tail(xs), y)
ifsum2(false, xs, y)sum2(cons(p(head(xs)), tail(xs)), s(y))isNil(nil)true
isNil(cons(x, xs))falsetail(nil)nil
tail(cons(x, xs))xshead(cons(x, xs))x
head(nil)errorisZero(0)true
isZero(s(0))falseisZero(s(s(x)))isZero(s(x))
p(0)s(s(0))p(s(0))0
p(s(s(x)))s(p(s(x)))ge(x, 0)true
ge(0, s(y))falsege(s(x), s(y))ge(x, y)
acad

Original Signature

Termination of terms over the following signature is verified: ifsum, d, gen, ifsum2, error, generate, c, a, sum, true, ge, tail, 0, s, times, sum2, if, p, false, head, cons, nil, isNil, isZero

Strategy


Instantiation

For all potential predecessors l → r of the rule sum2#(xs, y) → ifsum#(isNil(xs), isZero(head(xs)), xs, y) on dependency pair chains it holds that: Thus, sum2#(xs, y) → ifsum#(isNil(xs), isZero(head(xs)), xs, y) is replaced by instances determined through the above matching. These instances are:
sum2#(tail(_xs), _y) → ifsum#(isNil(tail(_xs)), isZero(head(tail(_xs))), tail(_xs), _y)sum2#(cons(p(head(_xs)), tail(_xs)), s(_y)) → ifsum#(isNil(cons(p(head(_xs)), tail(_xs))), isZero(head(cons(p(head(_xs)), tail(_xs)))), cons(p(head(_xs)), tail(_xs)), s(_y))

Problem 7: ForwardInstantiation



Dependency Pair Problem

Dependency Pairs

sum2#(tail(_xs), _y)ifsum#(isNil(tail(_xs)), isZero(head(tail(_xs))), tail(_xs), _y)ifsum2#(false, xs, y)sum2#(cons(p(head(xs)), tail(xs)), s(y))
sum2#(cons(p(head(_xs)), tail(_xs)), s(_y))ifsum#(isNil(cons(p(head(_xs)), tail(_xs))), isZero(head(cons(p(head(_xs)), tail(_xs)))), cons(p(head(_xs)), tail(_xs)), s(_y))ifsum#(false, b, xs, y)ifsum2#(b, xs, y)
ifsum2#(true, xs, y)sum2#(tail(xs), y)

Rewrite Rules

times(x, y)sum(generate(x, y))generate(x, y)gen(x, y, 0)
gen(x, y, z)if(ge(z, x), x, y, z)if(true, x, y, z)nil
if(false, x, y, z)cons(y, gen(x, y, s(z)))sum(xs)sum2(xs, 0)
sum2(xs, y)ifsum(isNil(xs), isZero(head(xs)), xs, y)ifsum(true, b, xs, y)y
ifsum(false, b, xs, y)ifsum2(b, xs, y)ifsum2(true, xs, y)sum2(tail(xs), y)
ifsum2(false, xs, y)sum2(cons(p(head(xs)), tail(xs)), s(y))isNil(nil)true
isNil(cons(x, xs))falsetail(nil)nil
tail(cons(x, xs))xshead(cons(x, xs))x
head(nil)errorisZero(0)true
isZero(s(0))falseisZero(s(s(x)))isZero(s(x))
p(0)s(s(0))p(s(0))0
p(s(s(x)))s(p(s(x)))ge(x, 0)true
ge(0, s(y))falsege(s(x), s(y))ge(x, y)
acad

Original Signature

Termination of terms over the following signature is verified: ifsum, d, gen, ifsum2, error, generate, c, a, sum, true, ge, tail, 0, s, times, if, sum2, p, false, head, isZero, isNil, nil, cons

Strategy


Instantiation

For all potential successors l → r of the rule ifsum#(false, b, xs, y) → ifsum2#(b, xs, y) on dependency pair chains it holds that: Thus, ifsum#(false, b, xs, y) → ifsum2#(b, xs, y) is replaced by instances determined through the above matching. These instances are:
ifsum#(false, true, xs, y) → ifsum2#(true, xs, y)ifsum#(false, false, xs, y) → ifsum2#(false, xs, y)

Problem 9: Propagation



Dependency Pair Problem

Dependency Pairs

sum2#(tail(_xs), _y)ifsum#(isNil(tail(_xs)), isZero(head(tail(_xs))), tail(_xs), _y)ifsum#(false, true, xs, y)ifsum2#(true, xs, y)
ifsum2#(false, xs, y)sum2#(cons(p(head(xs)), tail(xs)), s(y))ifsum#(false, false, xs, y)ifsum2#(false, xs, y)
sum2#(cons(p(head(_xs)), tail(_xs)), s(_y))ifsum#(isNil(cons(p(head(_xs)), tail(_xs))), isZero(head(cons(p(head(_xs)), tail(_xs)))), cons(p(head(_xs)), tail(_xs)), s(_y))ifsum2#(true, xs, y)sum2#(tail(xs), y)

Rewrite Rules

times(x, y)sum(generate(x, y))generate(x, y)gen(x, y, 0)
gen(x, y, z)if(ge(z, x), x, y, z)if(true, x, y, z)nil
if(false, x, y, z)cons(y, gen(x, y, s(z)))sum(xs)sum2(xs, 0)
sum2(xs, y)ifsum(isNil(xs), isZero(head(xs)), xs, y)ifsum(true, b, xs, y)y
ifsum(false, b, xs, y)ifsum2(b, xs, y)ifsum2(true, xs, y)sum2(tail(xs), y)
ifsum2(false, xs, y)sum2(cons(p(head(xs)), tail(xs)), s(y))isNil(nil)true
isNil(cons(x, xs))falsetail(nil)nil
tail(cons(x, xs))xshead(cons(x, xs))x
head(nil)errorisZero(0)true
isZero(s(0))falseisZero(s(s(x)))isZero(s(x))
p(0)s(s(0))p(s(0))0
p(s(s(x)))s(p(s(x)))ge(x, 0)true
ge(0, s(y))falsege(s(x), s(y))ge(x, y)
acad

Original Signature

Termination of terms over the following signature is verified: ifsum, d, gen, ifsum2, error, generate, c, a, sum, true, ge, tail, 0, s, times, sum2, if, p, false, head, cons, nil, isNil, isZero

Strategy


The dependency pairs ifsum#(false, true, xs, y) → ifsum2#(true, xs, y) and ifsum2#(true, xs, y) → sum2#(tail(xs), y) are consolidated into the rule ifsum#(false, true, xs, y) → sum2#(tail(xs), y) .

This is possible as

The dependency pairs ifsum#(false, true, xs, y) → ifsum2#(true, xs, y) and ifsum2#(true, xs, y) → sum2#(tail(xs), y) are consolidated into the rule ifsum#(false, true, xs, y) → sum2#(tail(xs), y) .

This is possible as

The dependency pairs ifsum#(false, true, xs, y) → ifsum2#(true, xs, y) and ifsum2#(true, xs, y) → sum2#(tail(xs), y) are consolidated into the rule ifsum#(false, true, xs, y) → sum2#(tail(xs), y) .

This is possible as

The dependency pairs ifsum#(false, true, xs, y) → ifsum2#(true, xs, y) and ifsum2#(true, xs, y) → sum2#(tail(xs), y) are consolidated into the rule ifsum#(false, true, xs, y) → sum2#(tail(xs), y) .

This is possible as

The dependency pairs ifsum#(false, true, xs, y) → ifsum2#(true, xs, y) and ifsum2#(true, xs, y) → sum2#(tail(xs), y) are consolidated into the rule ifsum#(false, true, xs, y) → sum2#(tail(xs), y) .

This is possible as

The dependency pairs ifsum#(false, true, xs, y) → ifsum2#(true, xs, y) and ifsum2#(true, xs, y) → sum2#(tail(xs), y) are consolidated into the rule ifsum#(false, true, xs, y) → sum2#(tail(xs), y) .

This is possible as

The dependency pairs ifsum#(false, false, xs, y) → ifsum2#(false, xs, y) and ifsum2#(false, xs, y) → sum2#(cons(p(head(xs)), tail(xs)), s(y)) are consolidated into the rule ifsum#(false, false, xs, y) → sum2#(cons(p(head(xs)), tail(xs)), s(y)) .

This is possible as

The dependency pairs ifsum#(false, false, xs, y) → ifsum2#(false, xs, y) and ifsum2#(false, xs, y) → sum2#(cons(p(head(xs)), tail(xs)), s(y)) are consolidated into the rule ifsum#(false, false, xs, y) → sum2#(cons(p(head(xs)), tail(xs)), s(y)) .

This is possible as

The dependency pairs ifsum#(false, false, xs, y) → ifsum2#(false, xs, y) and ifsum2#(false, xs, y) → sum2#(cons(p(head(xs)), tail(xs)), s(y)) are consolidated into the rule ifsum#(false, false, xs, y) → sum2#(cons(p(head(xs)), tail(xs)), s(y)) .

This is possible as

The dependency pairs ifsum#(false, false, xs, y) → ifsum2#(false, xs, y) and ifsum2#(false, xs, y) → sum2#(cons(p(head(xs)), tail(xs)), s(y)) are consolidated into the rule ifsum#(false, false, xs, y) → sum2#(cons(p(head(xs)), tail(xs)), s(y)) .

This is possible as

The dependency pairs ifsum#(false, false, xs, y) → ifsum2#(false, xs, y) and ifsum2#(false, xs, y) → sum2#(cons(p(head(xs)), tail(xs)), s(y)) are consolidated into the rule ifsum#(false, false, xs, y) → sum2#(cons(p(head(xs)), tail(xs)), s(y)) .

This is possible as

The dependency pairs ifsum#(false, false, xs, y) → ifsum2#(false, xs, y) and ifsum2#(false, xs, y) → sum2#(cons(p(head(xs)), tail(xs)), s(y)) are consolidated into the rule ifsum#(false, false, xs, y) → sum2#(cons(p(head(xs)), tail(xs)), s(y)) .

This is possible as


Summary

Removed Dependency PairsAdded Dependency Pairs
ifsum#(false, true, xs, y) → ifsum2#(true, xs, y)ifsum#(false, false, xs, y) → sum2#(cons(p(head(xs)), tail(xs)), s(y))
ifsum2#(false, xs, y) → sum2#(cons(p(head(xs)), tail(xs)), s(y))ifsum#(false, true, xs, y) → sum2#(tail(xs), y)
ifsum#(false, false, xs, y) → ifsum2#(false, xs, y) 
ifsum2#(true, xs, y) → sum2#(tail(xs), y) 

Problem 4: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

p#(s(s(x)))p#(s(x))

Rewrite Rules

times(x, y)sum(generate(x, y))generate(x, y)gen(x, y, 0)
gen(x, y, z)if(ge(z, x), x, y, z)if(true, x, y, z)nil
if(false, x, y, z)cons(y, gen(x, y, s(z)))sum(xs)sum2(xs, 0)
sum2(xs, y)ifsum(isNil(xs), isZero(head(xs)), xs, y)ifsum(true, b, xs, y)y
ifsum(false, b, xs, y)ifsum2(b, xs, y)ifsum2(true, xs, y)sum2(tail(xs), y)
ifsum2(false, xs, y)sum2(cons(p(head(xs)), tail(xs)), s(y))isNil(nil)true
isNil(cons(x, xs))falsetail(nil)nil
tail(cons(x, xs))xshead(cons(x, xs))x
head(nil)errorisZero(0)true
isZero(s(0))falseisZero(s(s(x)))isZero(s(x))
p(0)s(s(0))p(s(0))0
p(s(s(x)))s(p(s(x)))ge(x, 0)true
ge(0, s(y))falsege(s(x), s(y))ge(x, y)
acad

Original Signature

Termination of terms over the following signature is verified: ifsum, d, gen, ifsum2, error, generate, c, a, sum, true, ge, tail, 0, s, times, sum2, if, p, false, head, cons, nil, isNil, isZero

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

p#(s(s(x)))p#(s(x))

Problem 5: BackwardInstantiation



Dependency Pair Problem

Dependency Pairs

if#(false, x, y, z)gen#(x, y, s(z))gen#(x, y, z)if#(ge(z, x), x, y, z)

Rewrite Rules

times(x, y)sum(generate(x, y))generate(x, y)gen(x, y, 0)
gen(x, y, z)if(ge(z, x), x, y, z)if(true, x, y, z)nil
if(false, x, y, z)cons(y, gen(x, y, s(z)))sum(xs)sum2(xs, 0)
sum2(xs, y)ifsum(isNil(xs), isZero(head(xs)), xs, y)ifsum(true, b, xs, y)y
ifsum(false, b, xs, y)ifsum2(b, xs, y)ifsum2(true, xs, y)sum2(tail(xs), y)
ifsum2(false, xs, y)sum2(cons(p(head(xs)), tail(xs)), s(y))isNil(nil)true
isNil(cons(x, xs))falsetail(nil)nil
tail(cons(x, xs))xshead(cons(x, xs))x
head(nil)errorisZero(0)true
isZero(s(0))falseisZero(s(s(x)))isZero(s(x))
p(0)s(s(0))p(s(0))0
p(s(s(x)))s(p(s(x)))ge(x, 0)true
ge(0, s(y))falsege(s(x), s(y))ge(x, y)
acad

Original Signature

Termination of terms over the following signature is verified: ifsum, d, gen, ifsum2, error, generate, c, a, sum, true, ge, tail, 0, s, times, sum2, if, p, false, head, cons, nil, isNil, isZero

Strategy


Instantiation

For all potential predecessors l → r of the rule gen#(x, y, z) → if#(ge(z, x), x, y, z) on dependency pair chains it holds that: Thus, gen#(x, y, z) → if#(ge(z, x), x, y, z) is replaced by instances determined through the above matching. These instances are:
gen#(_x, _y, s(_z)) → if#(ge(s(_z), _x), _x, _y, s(_z))

Problem 8: BackwardInstantiation



Dependency Pair Problem

Dependency Pairs

if#(false, x, y, z)gen#(x, y, s(z))gen#(_x, _y, s(_z))if#(ge(s(_z), _x), _x, _y, s(_z))

Rewrite Rules

times(x, y)sum(generate(x, y))generate(x, y)gen(x, y, 0)
gen(x, y, z)if(ge(z, x), x, y, z)if(true, x, y, z)nil
if(false, x, y, z)cons(y, gen(x, y, s(z)))sum(xs)sum2(xs, 0)
sum2(xs, y)ifsum(isNil(xs), isZero(head(xs)), xs, y)ifsum(true, b, xs, y)y
ifsum(false, b, xs, y)ifsum2(b, xs, y)ifsum2(true, xs, y)sum2(tail(xs), y)
ifsum2(false, xs, y)sum2(cons(p(head(xs)), tail(xs)), s(y))isNil(nil)true
isNil(cons(x, xs))falsetail(nil)nil
tail(cons(x, xs))xshead(cons(x, xs))x
head(nil)errorisZero(0)true
isZero(s(0))falseisZero(s(s(x)))isZero(s(x))
p(0)s(s(0))p(s(0))0
p(s(s(x)))s(p(s(x)))ge(x, 0)true
ge(0, s(y))falsege(s(x), s(y))ge(x, y)
acad

Original Signature

Termination of terms over the following signature is verified: ifsum, d, gen, ifsum2, error, generate, c, a, sum, true, ge, tail, 0, s, times, if, sum2, p, false, head, isZero, isNil, nil, cons

Strategy


Instantiation

For all potential predecessors l → r of the rule gen#(_x, _y, s(_z)) → if#(ge(s(_z), _x), _x, _y, s(_z)) on dependency pair chains it holds that: Thus, gen#(_x, _y, s(_z)) → if#(ge(s(_z), _x), _x, _y, s(_z)) is replaced by instances determined through the above matching. These instances are:
gen#(x, y, s(z)) → if#(ge(s(z), x), x, y, s(z))

Problem 11: Propagation



Dependency Pair Problem

Dependency Pairs

if#(false, x, y, z)gen#(x, y, s(z))gen#(x, y, s(z))if#(ge(s(z), x), x, y, s(z))

Rewrite Rules

times(x, y)sum(generate(x, y))generate(x, y)gen(x, y, 0)
gen(x, y, z)if(ge(z, x), x, y, z)if(true, x, y, z)nil
if(false, x, y, z)cons(y, gen(x, y, s(z)))sum(xs)sum2(xs, 0)
sum2(xs, y)ifsum(isNil(xs), isZero(head(xs)), xs, y)ifsum(true, b, xs, y)y
ifsum(false, b, xs, y)ifsum2(b, xs, y)ifsum2(true, xs, y)sum2(tail(xs), y)
ifsum2(false, xs, y)sum2(cons(p(head(xs)), tail(xs)), s(y))isNil(nil)true
isNil(cons(x, xs))falsetail(nil)nil
tail(cons(x, xs))xshead(cons(x, xs))x
head(nil)errorisZero(0)true
isZero(s(0))falseisZero(s(s(x)))isZero(s(x))
p(0)s(s(0))p(s(0))0
p(s(s(x)))s(p(s(x)))ge(x, 0)true
ge(0, s(y))falsege(s(x), s(y))ge(x, y)
acad

Original Signature

Termination of terms over the following signature is verified: ifsum, d, gen, ifsum2, error, generate, c, a, sum, true, ge, tail, 0, s, times, sum2, if, p, false, head, cons, nil, isNil, isZero

Strategy


The dependency pairs if#(false, x, y, z) → gen#(x, y, s(z)) and gen#(x, y, s(z)) → if#(ge(s(z), x), x, y, s(z)) are consolidated into the rule if#(false, x, y, z) → if#(ge(s(z), x), x, y, s(z)) .

This is possible as

The dependency pairs if#(false, x, y, z) → gen#(x, y, s(z)) and gen#(x, y, s(z)) → if#(ge(s(z), x), x, y, s(z)) are consolidated into the rule if#(false, x, y, z) → if#(ge(s(z), x), x, y, s(z)) .

This is possible as


Summary

Removed Dependency PairsAdded Dependency Pairs
if#(false, x, y, z) → gen#(x, y, s(z))if#(false, x, y, z) → if#(ge(s(z), x), x, y, s(z))
gen#(x, y, s(z)) → if#(ge(s(z), x), x, y, s(z)) 

Problem 6: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

isZero#(s(s(x)))isZero#(s(x))

Rewrite Rules

times(x, y)sum(generate(x, y))generate(x, y)gen(x, y, 0)
gen(x, y, z)if(ge(z, x), x, y, z)if(true, x, y, z)nil
if(false, x, y, z)cons(y, gen(x, y, s(z)))sum(xs)sum2(xs, 0)
sum2(xs, y)ifsum(isNil(xs), isZero(head(xs)), xs, y)ifsum(true, b, xs, y)y
ifsum(false, b, xs, y)ifsum2(b, xs, y)ifsum2(true, xs, y)sum2(tail(xs), y)
ifsum2(false, xs, y)sum2(cons(p(head(xs)), tail(xs)), s(y))isNil(nil)true
isNil(cons(x, xs))falsetail(nil)nil
tail(cons(x, xs))xshead(cons(x, xs))x
head(nil)errorisZero(0)true
isZero(s(0))falseisZero(s(s(x)))isZero(s(x))
p(0)s(s(0))p(s(0))0
p(s(s(x)))s(p(s(x)))ge(x, 0)true
ge(0, s(y))falsege(s(x), s(y))ge(x, y)
acad

Original Signature

Termination of terms over the following signature is verified: ifsum, d, gen, ifsum2, error, generate, c, a, sum, true, ge, tail, 0, s, times, sum2, if, p, false, head, cons, nil, isNil, isZero

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

isZero#(s(s(x)))isZero#(s(x))