YES

The TRS could be proven terminating. The proof took 29341 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (921ms).
 | – Problem 2 was processed with processor SubtermCriterion (2ms).
 |    | – Problem 10 was processed with processor SubtermCriterion (1ms).
 | – Problem 3 was processed with processor SubtermCriterion (3ms).
 |    | – Problem 11 was processed with processor SubtermCriterion (0ms).
 | – Problem 4 was processed with processor PolynomialLinearRange4 (264ms).
 |    | – Problem 16 was processed with processor PolynomialLinearRange4 (417ms).
 |    |    | – Problem 18 was processed with processor PolynomialLinearRange4 (201ms).
 |    |    |    | – Problem 19 was processed with processor PolynomialLinearRange4 (186ms).
 |    |    |    |    | – Problem 20 was processed with processor PolynomialLinearRange4 (194ms).
 |    |    |    |    |    | – Problem 21 was processed with processor PolynomialLinearRange4 (187ms).
 |    |    |    |    |    |    | – Problem 22 was processed with processor PolynomialLinearRange4 (197ms).
 |    |    |    |    |    |    |    | – Problem 23 was processed with processor PolynomialLinearRange4 (160ms).
 |    |    |    |    |    |    |    |    | – Problem 24 was processed with processor PolynomialLinearRange4 (170ms).
 |    |    |    |    |    |    |    |    |    | – Problem 25 was processed with processor PolynomialLinearRange4 (134ms).
 |    |    |    |    |    |    |    |    |    |    | – Problem 26 was processed with processor ReductionPairSAT (3064ms).
 |    |    |    |    |    |    |    |    |    |    |    | – Problem 27 was processed with processor ReductionPairSAT (2459ms).
 |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 28 was processed with processor DependencyGraph (0ms).
 | – Problem 5 was processed with processor SubtermCriterion (2ms).
 |    | – Problem 12 was processed with processor SubtermCriterion (1ms).
 | – Problem 6 was processed with processor SubtermCriterion (3ms).
 | – Problem 7 was processed with processor SubtermCriterion (2ms).
 |    | – Problem 13 was processed with processor SubtermCriterion (0ms).
 | – Problem 8 was processed with processor SubtermCriterion (2ms).
 |    | – Problem 14 was processed with processor SubtermCriterion (0ms).
 |    |    | – Problem 15 was processed with processor PolynomialLinearRange4 (21ms).
 |    |    |    | – Problem 17 was processed with processor PolynomialLinearRange4 (127ms).
 | – Problem 9 was processed with processor SubtermCriterion (2ms).

Problem 1: DependencyGraph



Dependency Pair Problem

Dependency Pairs

active#(first(0, X))mark#(nil)active#(and(false, Y))mark#(false)
add#(X1, mark(X2))add#(X1, X2)active#(add(s(X), Y))mark#(s(add(X, Y)))
if#(X1, X2, active(X3))if#(X1, X2, X3)active#(first(s(X), cons(Y, Z)))cons#(Y, first(X, Z))
first#(X1, active(X2))first#(X1, X2)active#(if(true, X, Y))mark#(X)
active#(first(s(X), cons(Y, Z)))mark#(cons(Y, first(X, Z)))first#(mark(X1), X2)first#(X1, X2)
mark#(add(X1, X2))mark#(X1)mark#(true)active#(true)
mark#(and(X1, X2))and#(mark(X1), X2)active#(add(0, X))mark#(X)
add#(mark(X1), X2)add#(X1, X2)and#(mark(X1), X2)and#(X1, X2)
active#(from(X))mark#(cons(X, from(s(X))))mark#(s(X))active#(s(X))
active#(from(X))s#(X)mark#(and(X1, X2))mark#(X1)
cons#(X1, active(X2))cons#(X1, X2)active#(add(s(X), Y))add#(X, Y)
if#(X1, mark(X2), X3)if#(X1, X2, X3)first#(active(X1), X2)first#(X1, X2)
from#(active(X))from#(X)if#(X1, X2, mark(X3))if#(X1, X2, X3)
add#(active(X1), X2)add#(X1, X2)mark#(first(X1, X2))mark#(X1)
if#(mark(X1), X2, X3)if#(X1, X2, X3)mark#(if(X1, X2, X3))mark#(X1)
and#(active(X1), X2)and#(X1, X2)mark#(false)active#(false)
mark#(add(X1, X2))active#(add(mark(X1), X2))mark#(if(X1, X2, X3))active#(if(mark(X1), X2, X3))
active#(if(false, X, Y))mark#(Y)and#(X1, active(X2))and#(X1, X2)
cons#(mark(X1), X2)cons#(X1, X2)mark#(add(X1, X2))add#(mark(X1), X2)
mark#(cons(X1, X2))cons#(X1, X2)if#(X1, active(X2), X3)if#(X1, X2, X3)
from#(mark(X))from#(X)active#(and(true, X))mark#(X)
mark#(from(X))from#(X)active#(first(s(X), cons(Y, Z)))first#(X, Z)
mark#(nil)active#(nil)active#(add(s(X), Y))s#(add(X, Y))
active#(from(X))cons#(X, from(s(X)))mark#(and(X1, X2))active#(and(mark(X1), X2))
add#(X1, active(X2))add#(X1, X2)cons#(X1, mark(X2))cons#(X1, X2)
mark#(first(X1, X2))first#(mark(X1), mark(X2))and#(X1, mark(X2))and#(X1, X2)
first#(X1, mark(X2))first#(X1, X2)mark#(if(X1, X2, X3))if#(mark(X1), X2, X3)
mark#(first(X1, X2))mark#(X2)mark#(0)active#(0)
mark#(first(X1, X2))active#(first(mark(X1), mark(X2)))mark#(from(X))active#(from(X))
cons#(active(X1), X2)cons#(X1, X2)mark#(cons(X1, X2))active#(cons(X1, X2))
if#(active(X1), X2, X3)if#(X1, X2, X3)mark#(s(X))s#(X)
s#(mark(X))s#(X)s#(active(X))s#(X)
active#(from(X))from#(s(X))

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


The following SCCs where found

from#(active(X)) → from#(X)from#(mark(X)) → from#(X)

mark#(first(X1, X2)) → mark#(X2)mark#(if(X1, X2, X3)) → active#(if(mark(X1), X2, X3))
mark#(add(X1, X2)) → active#(add(mark(X1), X2))active#(if(false, X, Y)) → mark#(Y)
mark#(first(X1, X2)) → active#(first(mark(X1), mark(X2)))active#(add(s(X), Y)) → mark#(s(add(X, Y)))
active#(add(0, X)) → mark#(X)mark#(from(X)) → active#(from(X))
active#(from(X)) → mark#(cons(X, from(s(X))))active#(and(true, X)) → mark#(X)
mark#(s(X)) → active#(s(X))mark#(cons(X1, X2)) → active#(cons(X1, X2))
active#(if(true, X, Y)) → mark#(X)mark#(and(X1, X2)) → mark#(X1)
active#(first(s(X), cons(Y, Z))) → mark#(cons(Y, first(X, Z)))mark#(and(X1, X2)) → active#(and(mark(X1), X2))
mark#(first(X1, X2)) → mark#(X1)mark#(if(X1, X2, X3)) → mark#(X1)
mark#(add(X1, X2)) → mark#(X1)

s#(mark(X)) → s#(X)s#(active(X)) → s#(X)

cons#(X1, active(X2)) → cons#(X1, X2)cons#(mark(X1), X2) → cons#(X1, X2)
cons#(X1, mark(X2)) → cons#(X1, X2)cons#(active(X1), X2) → cons#(X1, X2)

if#(X1, mark(X2), X3) → if#(X1, X2, X3)if#(X1, X2, mark(X3)) → if#(X1, X2, X3)
if#(mark(X1), X2, X3) → if#(X1, X2, X3)if#(X1, active(X2), X3) → if#(X1, X2, X3)
if#(X1, X2, active(X3)) → if#(X1, X2, X3)if#(active(X1), X2, X3) → if#(X1, X2, X3)

and#(active(X1), X2) → and#(X1, X2)and#(X1, active(X2)) → and#(X1, X2)
and#(mark(X1), X2) → and#(X1, X2)and#(X1, mark(X2)) → and#(X1, X2)

first#(active(X1), X2) → first#(X1, X2)first#(mark(X1), X2) → first#(X1, X2)
first#(X1, mark(X2)) → first#(X1, X2)first#(X1, active(X2)) → first#(X1, X2)

add#(active(X1), X2) → add#(X1, X2)add#(X1, mark(X2)) → add#(X1, X2)
add#(X1, active(X2)) → add#(X1, X2)add#(mark(X1), X2) → add#(X1, X2)

Problem 2: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

cons#(X1, active(X2))cons#(X1, X2)cons#(mark(X1), X2)cons#(X1, X2)
cons#(X1, mark(X2))cons#(X1, X2)cons#(active(X1), X2)cons#(X1, X2)

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

cons#(mark(X1), X2)cons#(X1, X2)cons#(active(X1), X2)cons#(X1, X2)

Problem 10: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

cons#(X1, active(X2))cons#(X1, X2)cons#(X1, mark(X2))cons#(X1, X2)

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

cons#(X1, active(X2))cons#(X1, X2)cons#(X1, mark(X2))cons#(X1, X2)

Problem 3: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

and#(active(X1), X2)and#(X1, X2)and#(X1, active(X2))and#(X1, X2)
and#(mark(X1), X2)and#(X1, X2)and#(X1, mark(X2))and#(X1, X2)

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

and#(active(X1), X2)and#(X1, X2)and#(mark(X1), X2)and#(X1, X2)

Problem 11: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

and#(X1, active(X2))and#(X1, X2)and#(X1, mark(X2))and#(X1, X2)

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

and#(X1, active(X2))and#(X1, X2)and#(X1, mark(X2))and#(X1, X2)

Problem 4: PolynomialLinearRange4



Dependency Pair Problem

Dependency Pairs

mark#(first(X1, X2))mark#(X2)mark#(add(X1, X2))active#(add(mark(X1), X2))
mark#(if(X1, X2, X3))active#(if(mark(X1), X2, X3))active#(if(false, X, Y))mark#(Y)
mark#(first(X1, X2))active#(first(mark(X1), mark(X2)))active#(add(s(X), Y))mark#(s(add(X, Y)))
active#(add(0, X))mark#(X)mark#(from(X))active#(from(X))
active#(from(X))mark#(cons(X, from(s(X))))active#(and(true, X))mark#(X)
mark#(s(X))active#(s(X))mark#(cons(X1, X2))active#(cons(X1, X2))
active#(if(true, X, Y))mark#(X)mark#(and(X1, X2))mark#(X1)
active#(first(s(X), cons(Y, Z)))mark#(cons(Y, first(X, Z)))mark#(and(X1, X2))active#(and(mark(X1), X2))
mark#(first(X1, X2))mark#(X1)mark#(if(X1, X2, X3))mark#(X1)
mark#(add(X1, X2))mark#(X1)

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


Polynomial Interpretation

Standard Usable rules

cons(active(X1), X2)cons(X1, X2)from(mark(X))from(X)
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(add(X1, X2))active(add(mark(X1), X2))
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))active(and(false, Y))mark(false)
active(first(0, X))mark(nil)mark(cons(X1, X2))active(cons(X1, X2))
mark(s(X))active(s(X))if(active(X1), X2, X3)if(X1, X2, X3)
and(active(X1), X2)and(X1, X2)add(X1, mark(X2))add(X1, X2)
active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))cons(X1, mark(X2))cons(X1, X2)
first(X1, mark(X2))first(X1, X2)and(X1, mark(X2))and(X1, X2)
mark(true)active(true)first(active(X1), X2)first(X1, X2)
mark(and(X1, X2))active(and(mark(X1), X2))first(X1, active(X2))first(X1, X2)
if(X1, X2, active(X3))if(X1, X2, X3)add(X1, active(X2))add(X1, X2)
active(and(true, X))mark(X)if(X1, X2, mark(X3))if(X1, X2, X3)
active(from(X))mark(cons(X, from(s(X))))add(mark(X1), X2)add(X1, X2)
mark(nil)active(nil)s(active(X))s(X)
mark(0)active(0)from(active(X))from(X)
cons(X1, active(X2))cons(X1, X2)if(X1, active(X2), X3)if(X1, X2, X3)
if(mark(X1), X2, X3)if(X1, X2, X3)first(mark(X1), X2)first(X1, X2)
add(active(X1), X2)add(X1, X2)active(add(s(X), Y))mark(s(add(X, Y)))
active(if(false, X, Y))mark(Y)mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)if(X1, mark(X2), X3)if(X1, X2, X3)
cons(mark(X1), X2)cons(X1, X2)mark(false)active(false)
active(add(0, X))mark(X)s(mark(X))s(X)
and(X1, active(X2))and(X1, X2)active(if(true, X, Y))mark(X)

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

mark#(first(X1, X2))mark#(X2)active#(if(false, X, Y))mark#(Y)
active#(and(true, X))mark#(X)mark#(and(X1, X2))mark#(X1)
mark#(first(X1, X2))mark#(X1)

Problem 16: PolynomialLinearRange4



Dependency Pair Problem

Dependency Pairs

mark#(add(X1, X2))active#(add(mark(X1), X2))mark#(if(X1, X2, X3))active#(if(mark(X1), X2, X3))
mark#(first(X1, X2))active#(first(mark(X1), mark(X2)))active#(add(s(X), Y))mark#(s(add(X, Y)))
active#(add(0, X))mark#(X)mark#(from(X))active#(from(X))
active#(from(X))mark#(cons(X, from(s(X))))mark#(s(X))active#(s(X))
mark#(cons(X1, X2))active#(cons(X1, X2))active#(if(true, X, Y))mark#(X)
active#(first(s(X), cons(Y, Z)))mark#(cons(Y, first(X, Z)))mark#(and(X1, X2))active#(and(mark(X1), X2))
mark#(if(X1, X2, X3))mark#(X1)mark#(add(X1, X2))mark#(X1)

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


Polynomial Interpretation

Standard Usable rules

cons(active(X1), X2)cons(X1, X2)from(mark(X))from(X)
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(add(X1, X2))active(add(mark(X1), X2))
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))active(and(false, Y))mark(false)
active(first(0, X))mark(nil)mark(cons(X1, X2))active(cons(X1, X2))
mark(s(X))active(s(X))if(active(X1), X2, X3)if(X1, X2, X3)
and(active(X1), X2)and(X1, X2)add(X1, mark(X2))add(X1, X2)
active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))cons(X1, mark(X2))cons(X1, X2)
first(X1, mark(X2))first(X1, X2)and(X1, mark(X2))and(X1, X2)
mark(true)active(true)first(active(X1), X2)first(X1, X2)
mark(and(X1, X2))active(and(mark(X1), X2))first(X1, active(X2))first(X1, X2)
if(X1, X2, active(X3))if(X1, X2, X3)add(X1, active(X2))add(X1, X2)
active(and(true, X))mark(X)if(X1, X2, mark(X3))if(X1, X2, X3)
active(from(X))mark(cons(X, from(s(X))))add(mark(X1), X2)add(X1, X2)
mark(nil)active(nil)s(active(X))s(X)
mark(0)active(0)from(active(X))from(X)
cons(X1, active(X2))cons(X1, X2)if(X1, active(X2), X3)if(X1, X2, X3)
if(mark(X1), X2, X3)if(X1, X2, X3)first(mark(X1), X2)first(X1, X2)
add(active(X1), X2)add(X1, X2)active(add(s(X), Y))mark(s(add(X, Y)))
active(if(false, X, Y))mark(Y)mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)if(X1, mark(X2), X3)if(X1, X2, X3)
cons(mark(X1), X2)cons(X1, X2)mark(false)active(false)
active(add(0, X))mark(X)s(mark(X))s(X)
and(X1, active(X2))and(X1, X2)active(if(true, X, Y))mark(X)

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

active#(add(0, X))mark#(X)

Problem 18: PolynomialLinearRange4



Dependency Pair Problem

Dependency Pairs

mark#(add(X1, X2))active#(add(mark(X1), X2))mark#(if(X1, X2, X3))active#(if(mark(X1), X2, X3))
mark#(first(X1, X2))active#(first(mark(X1), mark(X2)))active#(add(s(X), Y))mark#(s(add(X, Y)))
mark#(from(X))active#(from(X))active#(from(X))mark#(cons(X, from(s(X))))
mark#(s(X))active#(s(X))mark#(cons(X1, X2))active#(cons(X1, X2))
active#(if(true, X, Y))mark#(X)active#(first(s(X), cons(Y, Z)))mark#(cons(Y, first(X, Z)))
mark#(and(X1, X2))active#(and(mark(X1), X2))mark#(if(X1, X2, X3))mark#(X1)
mark#(add(X1, X2))mark#(X1)

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


Polynomial Interpretation

Standard Usable rules

cons(active(X1), X2)cons(X1, X2)from(mark(X))from(X)
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(add(X1, X2))active(add(mark(X1), X2))
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))active(and(false, Y))mark(false)
active(first(0, X))mark(nil)mark(cons(X1, X2))active(cons(X1, X2))
mark(s(X))active(s(X))if(active(X1), X2, X3)if(X1, X2, X3)
and(active(X1), X2)and(X1, X2)add(X1, mark(X2))add(X1, X2)
active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))cons(X1, mark(X2))cons(X1, X2)
first(X1, mark(X2))first(X1, X2)and(X1, mark(X2))and(X1, X2)
mark(true)active(true)first(active(X1), X2)first(X1, X2)
mark(and(X1, X2))active(and(mark(X1), X2))first(X1, active(X2))first(X1, X2)
if(X1, X2, active(X3))if(X1, X2, X3)add(X1, active(X2))add(X1, X2)
active(and(true, X))mark(X)if(X1, X2, mark(X3))if(X1, X2, X3)
active(from(X))mark(cons(X, from(s(X))))add(mark(X1), X2)add(X1, X2)
mark(nil)active(nil)s(active(X))s(X)
mark(0)active(0)from(active(X))from(X)
cons(X1, active(X2))cons(X1, X2)if(X1, active(X2), X3)if(X1, X2, X3)
if(mark(X1), X2, X3)if(X1, X2, X3)first(mark(X1), X2)first(X1, X2)
add(active(X1), X2)add(X1, X2)active(add(s(X), Y))mark(s(add(X, Y)))
active(if(false, X, Y))mark(Y)mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)if(X1, mark(X2), X3)if(X1, X2, X3)
cons(mark(X1), X2)cons(X1, X2)mark(false)active(false)
active(add(0, X))mark(X)s(mark(X))s(X)
and(X1, active(X2))and(X1, X2)active(if(true, X, Y))mark(X)

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

active#(if(true, X, Y))mark#(X)

Problem 19: PolynomialLinearRange4



Dependency Pair Problem

Dependency Pairs

mark#(if(X1, X2, X3))active#(if(mark(X1), X2, X3))mark#(add(X1, X2))active#(add(mark(X1), X2))
active#(first(s(X), cons(Y, Z)))mark#(cons(Y, first(X, Z)))mark#(first(X1, X2))active#(first(mark(X1), mark(X2)))
mark#(and(X1, X2))active#(and(mark(X1), X2))active#(add(s(X), Y))mark#(s(add(X, Y)))
mark#(if(X1, X2, X3))mark#(X1)mark#(from(X))active#(from(X))
active#(from(X))mark#(cons(X, from(s(X))))mark#(add(X1, X2))mark#(X1)
mark#(s(X))active#(s(X))mark#(cons(X1, X2))active#(cons(X1, X2))

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


Polynomial Interpretation

Standard Usable rules

cons(active(X1), X2)cons(X1, X2)from(mark(X))from(X)
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(add(X1, X2))active(add(mark(X1), X2))
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))active(and(false, Y))mark(false)
active(first(0, X))mark(nil)mark(cons(X1, X2))active(cons(X1, X2))
mark(s(X))active(s(X))if(active(X1), X2, X3)if(X1, X2, X3)
and(active(X1), X2)and(X1, X2)add(X1, mark(X2))add(X1, X2)
active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))cons(X1, mark(X2))cons(X1, X2)
first(X1, mark(X2))first(X1, X2)and(X1, mark(X2))and(X1, X2)
mark(true)active(true)first(active(X1), X2)first(X1, X2)
mark(and(X1, X2))active(and(mark(X1), X2))first(X1, active(X2))first(X1, X2)
if(X1, X2, active(X3))if(X1, X2, X3)add(X1, active(X2))add(X1, X2)
active(and(true, X))mark(X)if(X1, X2, mark(X3))if(X1, X2, X3)
active(from(X))mark(cons(X, from(s(X))))add(mark(X1), X2)add(X1, X2)
mark(nil)active(nil)s(active(X))s(X)
mark(0)active(0)from(active(X))from(X)
cons(X1, active(X2))cons(X1, X2)if(X1, active(X2), X3)if(X1, X2, X3)
if(mark(X1), X2, X3)if(X1, X2, X3)first(mark(X1), X2)first(X1, X2)
add(active(X1), X2)add(X1, X2)active(add(s(X), Y))mark(s(add(X, Y)))
active(if(false, X, Y))mark(Y)mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)if(X1, mark(X2), X3)if(X1, X2, X3)
cons(mark(X1), X2)cons(X1, X2)mark(false)active(false)
active(add(0, X))mark(X)s(mark(X))s(X)
and(X1, active(X2))and(X1, X2)active(if(true, X, Y))mark(X)

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

mark#(and(X1, X2))active#(and(mark(X1), X2))

Problem 20: PolynomialLinearRange4



Dependency Pair Problem

Dependency Pairs

mark#(add(X1, X2))active#(add(mark(X1), X2))mark#(if(X1, X2, X3))active#(if(mark(X1), X2, X3))
active#(first(s(X), cons(Y, Z)))mark#(cons(Y, first(X, Z)))mark#(first(X1, X2))active#(first(mark(X1), mark(X2)))
active#(add(s(X), Y))mark#(s(add(X, Y)))mark#(if(X1, X2, X3))mark#(X1)
mark#(from(X))active#(from(X))active#(from(X))mark#(cons(X, from(s(X))))
mark#(add(X1, X2))mark#(X1)mark#(s(X))active#(s(X))
mark#(cons(X1, X2))active#(cons(X1, X2))

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


Polynomial Interpretation

Standard Usable rules

cons(active(X1), X2)cons(X1, X2)from(mark(X))from(X)
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(add(X1, X2))active(add(mark(X1), X2))
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))active(and(false, Y))mark(false)
active(first(0, X))mark(nil)mark(cons(X1, X2))active(cons(X1, X2))
mark(s(X))active(s(X))if(active(X1), X2, X3)if(X1, X2, X3)
and(active(X1), X2)and(X1, X2)add(X1, mark(X2))add(X1, X2)
and(X1, mark(X2))and(X1, X2)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
cons(X1, mark(X2))cons(X1, X2)first(X1, mark(X2))first(X1, X2)
mark(true)active(true)first(active(X1), X2)first(X1, X2)
mark(and(X1, X2))active(and(mark(X1), X2))first(X1, active(X2))first(X1, X2)
if(X1, X2, active(X3))if(X1, X2, X3)add(X1, active(X2))add(X1, X2)
active(and(true, X))mark(X)if(X1, X2, mark(X3))if(X1, X2, X3)
active(from(X))mark(cons(X, from(s(X))))add(mark(X1), X2)add(X1, X2)
mark(nil)active(nil)mark(0)active(0)
s(active(X))s(X)from(active(X))from(X)
cons(X1, active(X2))cons(X1, X2)if(X1, active(X2), X3)if(X1, X2, X3)
if(mark(X1), X2, X3)if(X1, X2, X3)add(active(X1), X2)add(X1, X2)
first(mark(X1), X2)first(X1, X2)active(add(s(X), Y))mark(s(add(X, Y)))
active(if(false, X, Y))mark(Y)mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)if(X1, mark(X2), X3)if(X1, X2, X3)
cons(mark(X1), X2)cons(X1, X2)mark(false)active(false)
active(add(0, X))mark(X)s(mark(X))s(X)
and(X1, active(X2))and(X1, X2)active(if(true, X, Y))mark(X)

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

mark#(cons(X1, X2))active#(cons(X1, X2))

Problem 21: PolynomialLinearRange4



Dependency Pair Problem

Dependency Pairs

mark#(if(X1, X2, X3))active#(if(mark(X1), X2, X3))mark#(add(X1, X2))active#(add(mark(X1), X2))
active#(first(s(X), cons(Y, Z)))mark#(cons(Y, first(X, Z)))mark#(first(X1, X2))active#(first(mark(X1), mark(X2)))
active#(add(s(X), Y))mark#(s(add(X, Y)))mark#(if(X1, X2, X3))mark#(X1)
mark#(from(X))active#(from(X))active#(from(X))mark#(cons(X, from(s(X))))
mark#(add(X1, X2))mark#(X1)mark#(s(X))active#(s(X))

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


Polynomial Interpretation

Standard Usable rules

cons(active(X1), X2)cons(X1, X2)from(mark(X))from(X)
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(add(X1, X2))active(add(mark(X1), X2))
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))active(and(false, Y))mark(false)
active(first(0, X))mark(nil)mark(cons(X1, X2))active(cons(X1, X2))
mark(s(X))active(s(X))if(active(X1), X2, X3)if(X1, X2, X3)
and(active(X1), X2)and(X1, X2)add(X1, mark(X2))add(X1, X2)
and(X1, mark(X2))and(X1, X2)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
cons(X1, mark(X2))cons(X1, X2)first(X1, mark(X2))first(X1, X2)
mark(true)active(true)first(active(X1), X2)first(X1, X2)
mark(and(X1, X2))active(and(mark(X1), X2))first(X1, active(X2))first(X1, X2)
if(X1, X2, active(X3))if(X1, X2, X3)add(X1, active(X2))add(X1, X2)
active(and(true, X))mark(X)if(X1, X2, mark(X3))if(X1, X2, X3)
active(from(X))mark(cons(X, from(s(X))))add(mark(X1), X2)add(X1, X2)
mark(nil)active(nil)mark(0)active(0)
s(active(X))s(X)from(active(X))from(X)
cons(X1, active(X2))cons(X1, X2)if(X1, active(X2), X3)if(X1, X2, X3)
if(mark(X1), X2, X3)if(X1, X2, X3)add(active(X1), X2)add(X1, X2)
first(mark(X1), X2)first(X1, X2)active(add(s(X), Y))mark(s(add(X, Y)))
active(if(false, X, Y))mark(Y)mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)if(X1, mark(X2), X3)if(X1, X2, X3)
cons(mark(X1), X2)cons(X1, X2)mark(false)active(false)
active(add(0, X))mark(X)s(mark(X))s(X)
and(X1, active(X2))and(X1, X2)active(if(true, X, Y))mark(X)

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

mark#(if(X1, X2, X3))active#(if(mark(X1), X2, X3))mark#(s(X))active#(s(X))

Problem 22: PolynomialLinearRange4



Dependency Pair Problem

Dependency Pairs

mark#(add(X1, X2))active#(add(mark(X1), X2))active#(first(s(X), cons(Y, Z)))mark#(cons(Y, first(X, Z)))
mark#(first(X1, X2))active#(first(mark(X1), mark(X2)))active#(add(s(X), Y))mark#(s(add(X, Y)))
mark#(if(X1, X2, X3))mark#(X1)mark#(from(X))active#(from(X))
active#(from(X))mark#(cons(X, from(s(X))))mark#(add(X1, X2))mark#(X1)

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


Polynomial Interpretation

Standard Usable rules

cons(active(X1), X2)cons(X1, X2)from(mark(X))from(X)
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(add(X1, X2))active(add(mark(X1), X2))
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))active(and(false, Y))mark(false)
active(first(0, X))mark(nil)mark(cons(X1, X2))active(cons(X1, X2))
mark(s(X))active(s(X))if(active(X1), X2, X3)if(X1, X2, X3)
and(active(X1), X2)and(X1, X2)add(X1, mark(X2))add(X1, X2)
and(X1, mark(X2))and(X1, X2)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
first(X1, mark(X2))first(X1, X2)cons(X1, mark(X2))cons(X1, X2)
mark(true)active(true)mark(and(X1, X2))active(and(mark(X1), X2))
first(active(X1), X2)first(X1, X2)if(X1, X2, active(X3))if(X1, X2, X3)
first(X1, active(X2))first(X1, X2)add(X1, active(X2))add(X1, X2)
active(and(true, X))mark(X)if(X1, X2, mark(X3))if(X1, X2, X3)
active(from(X))mark(cons(X, from(s(X))))add(mark(X1), X2)add(X1, X2)
mark(nil)active(nil)mark(0)active(0)
s(active(X))s(X)from(active(X))from(X)
cons(X1, active(X2))cons(X1, X2)if(X1, active(X2), X3)if(X1, X2, X3)
if(mark(X1), X2, X3)if(X1, X2, X3)add(active(X1), X2)add(X1, X2)
first(mark(X1), X2)first(X1, X2)active(add(s(X), Y))mark(s(add(X, Y)))
active(if(false, X, Y))mark(Y)mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)if(X1, mark(X2), X3)if(X1, X2, X3)
mark(false)active(false)cons(mark(X1), X2)cons(X1, X2)
active(add(0, X))mark(X)s(mark(X))s(X)
and(X1, active(X2))and(X1, X2)active(if(true, X, Y))mark(X)

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

mark#(first(X1, X2))active#(first(mark(X1), mark(X2)))mark#(from(X))active#(from(X))

Problem 23: PolynomialLinearRange4



Dependency Pair Problem

Dependency Pairs

mark#(add(X1, X2))active#(add(mark(X1), X2))active#(first(s(X), cons(Y, Z)))mark#(cons(Y, first(X, Z)))
active#(add(s(X), Y))mark#(s(add(X, Y)))mark#(if(X1, X2, X3))mark#(X1)
active#(from(X))mark#(cons(X, from(s(X))))mark#(add(X1, X2))mark#(X1)

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


Polynomial Interpretation

Standard Usable rules

cons(active(X1), X2)cons(X1, X2)from(mark(X))from(X)
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(add(X1, X2))active(add(mark(X1), X2))
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))active(and(false, Y))mark(false)
active(first(0, X))mark(nil)mark(cons(X1, X2))active(cons(X1, X2))
mark(s(X))active(s(X))if(active(X1), X2, X3)if(X1, X2, X3)
and(active(X1), X2)and(X1, X2)add(X1, mark(X2))add(X1, X2)
and(X1, mark(X2))and(X1, X2)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
first(X1, mark(X2))first(X1, X2)cons(X1, mark(X2))cons(X1, X2)
mark(true)active(true)mark(and(X1, X2))active(and(mark(X1), X2))
first(active(X1), X2)first(X1, X2)if(X1, X2, active(X3))if(X1, X2, X3)
first(X1, active(X2))first(X1, X2)add(X1, active(X2))add(X1, X2)
active(and(true, X))mark(X)if(X1, X2, mark(X3))if(X1, X2, X3)
active(from(X))mark(cons(X, from(s(X))))add(mark(X1), X2)add(X1, X2)
mark(nil)active(nil)mark(0)active(0)
s(active(X))s(X)from(active(X))from(X)
cons(X1, active(X2))cons(X1, X2)if(X1, active(X2), X3)if(X1, X2, X3)
if(mark(X1), X2, X3)if(X1, X2, X3)add(active(X1), X2)add(X1, X2)
first(mark(X1), X2)first(X1, X2)active(add(s(X), Y))mark(s(add(X, Y)))
active(if(false, X, Y))mark(Y)mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)if(X1, mark(X2), X3)if(X1, X2, X3)
mark(false)active(false)cons(mark(X1), X2)cons(X1, X2)
active(add(0, X))mark(X)s(mark(X))s(X)
and(X1, active(X2))and(X1, X2)active(if(true, X, Y))mark(X)

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

active#(from(X))mark#(cons(X, from(s(X))))

Problem 24: PolynomialLinearRange4



Dependency Pair Problem

Dependency Pairs

mark#(add(X1, X2))active#(add(mark(X1), X2))active#(first(s(X), cons(Y, Z)))mark#(cons(Y, first(X, Z)))
active#(add(s(X), Y))mark#(s(add(X, Y)))mark#(if(X1, X2, X3))mark#(X1)
mark#(add(X1, X2))mark#(X1)

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


Polynomial Interpretation

Standard Usable rules

cons(active(X1), X2)cons(X1, X2)from(mark(X))from(X)
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(add(X1, X2))active(add(mark(X1), X2))
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))active(and(false, Y))mark(false)
active(first(0, X))mark(nil)mark(cons(X1, X2))active(cons(X1, X2))
mark(s(X))active(s(X))if(active(X1), X2, X3)if(X1, X2, X3)
and(active(X1), X2)and(X1, X2)add(X1, mark(X2))add(X1, X2)
and(X1, mark(X2))and(X1, X2)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
first(X1, mark(X2))first(X1, X2)cons(X1, mark(X2))cons(X1, X2)
mark(true)active(true)mark(and(X1, X2))active(and(mark(X1), X2))
first(active(X1), X2)first(X1, X2)if(X1, X2, active(X3))if(X1, X2, X3)
first(X1, active(X2))first(X1, X2)add(X1, active(X2))add(X1, X2)
active(and(true, X))mark(X)if(X1, X2, mark(X3))if(X1, X2, X3)
active(from(X))mark(cons(X, from(s(X))))add(mark(X1), X2)add(X1, X2)
mark(nil)active(nil)mark(0)active(0)
s(active(X))s(X)from(active(X))from(X)
cons(X1, active(X2))cons(X1, X2)if(X1, active(X2), X3)if(X1, X2, X3)
if(mark(X1), X2, X3)if(X1, X2, X3)add(active(X1), X2)add(X1, X2)
first(mark(X1), X2)first(X1, X2)active(add(s(X), Y))mark(s(add(X, Y)))
active(if(false, X, Y))mark(Y)mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)if(X1, mark(X2), X3)if(X1, X2, X3)
mark(false)active(false)cons(mark(X1), X2)cons(X1, X2)
active(add(0, X))mark(X)s(mark(X))s(X)
and(X1, active(X2))and(X1, X2)active(if(true, X, Y))mark(X)

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

active#(first(s(X), cons(Y, Z)))mark#(cons(Y, first(X, Z)))

Problem 25: PolynomialLinearRange4



Dependency Pair Problem

Dependency Pairs

mark#(add(X1, X2))active#(add(mark(X1), X2))active#(add(s(X), Y))mark#(s(add(X, Y)))
mark#(if(X1, X2, X3))mark#(X1)mark#(add(X1, X2))mark#(X1)

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


Polynomial Interpretation

Standard Usable rules

cons(active(X1), X2)cons(X1, X2)from(mark(X))from(X)
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(add(X1, X2))active(add(mark(X1), X2))
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))active(and(false, Y))mark(false)
mark(cons(X1, X2))active(cons(X1, X2))active(first(0, X))mark(nil)
mark(s(X))active(s(X))if(active(X1), X2, X3)if(X1, X2, X3)
add(X1, mark(X2))add(X1, X2)and(active(X1), X2)and(X1, X2)
cons(X1, mark(X2))cons(X1, X2)first(X1, mark(X2))first(X1, X2)
and(X1, mark(X2))and(X1, X2)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
mark(true)active(true)first(active(X1), X2)first(X1, X2)
mark(and(X1, X2))active(and(mark(X1), X2))first(X1, active(X2))first(X1, X2)
if(X1, X2, active(X3))if(X1, X2, X3)add(X1, active(X2))add(X1, X2)
if(X1, X2, mark(X3))if(X1, X2, X3)active(and(true, X))mark(X)
active(from(X))mark(cons(X, from(s(X))))add(mark(X1), X2)add(X1, X2)
mark(nil)active(nil)mark(0)active(0)
s(active(X))s(X)cons(X1, active(X2))cons(X1, X2)
from(active(X))from(X)if(X1, active(X2), X3)if(X1, X2, X3)
if(mark(X1), X2, X3)if(X1, X2, X3)first(mark(X1), X2)first(X1, X2)
add(active(X1), X2)add(X1, X2)active(add(s(X), Y))mark(s(add(X, Y)))
active(if(false, X, Y))mark(Y)mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)if(X1, mark(X2), X3)if(X1, X2, X3)
cons(mark(X1), X2)cons(X1, X2)mark(false)active(false)
active(add(0, X))mark(X)s(mark(X))s(X)
and(X1, active(X2))and(X1, X2)active(if(true, X, Y))mark(X)

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

mark#(if(X1, X2, X3))mark#(X1)

Problem 26: ReductionPairSAT



Dependency Pair Problem

Dependency Pairs

mark#(add(X1, X2))active#(add(mark(X1), X2))active#(add(s(X), Y))mark#(s(add(X, Y)))
mark#(add(X1, X2))mark#(X1)

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


Function Precedence

true < mark = false < and < first < nil < from = add = mark# = 0 = s = if = active = active# = cons

Argument Filtering

true: all arguments are removed from true
mark: collapses to 1
from: all arguments are removed from from
add: 1 2
mark#: collapses to 1
and: 1 2
0: all arguments are removed from 0
s: collapses to 1
if: 1 2 3
active: collapses to 1
false: all arguments are removed from false
active#: collapses to 1
first: all arguments are removed from first
cons: all arguments are removed from cons
nil: all arguments are removed from nil

Status

true: multiset
from: multiset
add: lexicographic with permutation 1 → 1 2 → 2
and: lexicographic with permutation 1 → 2 2 → 1
0: multiset
if: lexicographic with permutation 1 → 3 2 → 2 3 → 1
false: multiset
first: multiset
cons: multiset
nil: multiset

Usable Rules

cons(active(X1), X2) → cons(X1, X2)from(mark(X)) → from(X)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))mark(add(X1, X2)) → active(add(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))active(and(false, Y)) → mark(false)
mark(cons(X1, X2)) → active(cons(X1, X2))active(first(0, X)) → mark(nil)
mark(s(X)) → active(s(X))if(active(X1), X2, X3) → if(X1, X2, X3)
add(X1, mark(X2)) → add(X1, X2)and(active(X1), X2) → and(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)first(X1, mark(X2)) → first(X1, X2)
and(X1, mark(X2)) → and(X1, X2)active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
mark(true) → active(true)first(active(X1), X2) → first(X1, X2)
mark(and(X1, X2)) → active(and(mark(X1), X2))first(X1, active(X2)) → first(X1, X2)
if(X1, X2, active(X3)) → if(X1, X2, X3)add(X1, active(X2)) → add(X1, X2)
if(X1, X2, mark(X3)) → if(X1, X2, X3)active(and(true, X)) → mark(X)
active(from(X)) → mark(cons(X, from(s(X))))add(mark(X1), X2) → add(X1, X2)
mark(nil) → active(nil)mark(0) → active(0)
s(active(X)) → s(X)cons(X1, active(X2)) → cons(X1, X2)
from(active(X)) → from(X)if(X1, active(X2), X3) → if(X1, X2, X3)
if(mark(X1), X2, X3) → if(X1, X2, X3)first(mark(X1), X2) → first(X1, X2)
add(active(X1), X2) → add(X1, X2)active(add(s(X), Y)) → mark(s(add(X, Y)))
active(if(false, X, Y)) → mark(Y)mark(from(X)) → active(from(X))
and(mark(X1), X2) → and(X1, X2)if(X1, mark(X2), X3) → if(X1, X2, X3)
cons(mark(X1), X2) → cons(X1, X2)mark(false) → active(false)
active(add(0, X)) → mark(X)s(mark(X)) → s(X)
and(X1, active(X2)) → and(X1, X2)active(if(true, X, Y)) → mark(X)

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

The following dependency pairs (at least) can be eliminated according to the given precedence.

mark#(add(X1, X2)) → mark#(X1)

Problem 27: ReductionPairSAT



Dependency Pair Problem

Dependency Pairs

mark#(add(X1, X2))active#(add(mark(X1), X2))active#(add(s(X), Y))mark#(s(add(X, Y)))

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


Function Precedence

active < mark = from = add = mark# = and = 0 = if = false = active# = first < true = s = cons = nil

Argument Filtering

true: all arguments are removed from true
mark: all arguments are removed from mark
from: all arguments are removed from from
add: all arguments are removed from add
mark#: collapses to 1
and: all arguments are removed from and
0: all arguments are removed from 0
s: all arguments are removed from s
if: all arguments are removed from if
active: collapses to 1
false: all arguments are removed from false
active#: all arguments are removed from active#
first: all arguments are removed from first
cons: all arguments are removed from cons
nil: all arguments are removed from nil

Status

true: multiset
mark: multiset
from: multiset
add: multiset
and: multiset
0: multiset
s: multiset
if: multiset
false: multiset
active#: multiset
first: multiset
cons: multiset
nil: multiset

Usable Rules

cons(active(X1), X2) → cons(X1, X2)from(mark(X)) → from(X)
mark(first(X1, X2)) → active(first(mark(X1), mark(X2)))mark(add(X1, X2)) → active(add(mark(X1), X2))
mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3))active(and(false, Y)) → mark(false)
mark(cons(X1, X2)) → active(cons(X1, X2))active(first(0, X)) → mark(nil)
mark(s(X)) → active(s(X))if(active(X1), X2, X3) → if(X1, X2, X3)
add(X1, mark(X2)) → add(X1, X2)and(active(X1), X2) → and(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)first(X1, mark(X2)) → first(X1, X2)
and(X1, mark(X2)) → and(X1, X2)active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
mark(true) → active(true)first(active(X1), X2) → first(X1, X2)
mark(and(X1, X2)) → active(and(mark(X1), X2))first(X1, active(X2)) → first(X1, X2)
if(X1, X2, active(X3)) → if(X1, X2, X3)add(X1, active(X2)) → add(X1, X2)
if(X1, X2, mark(X3)) → if(X1, X2, X3)active(and(true, X)) → mark(X)
active(from(X)) → mark(cons(X, from(s(X))))add(mark(X1), X2) → add(X1, X2)
mark(nil) → active(nil)mark(0) → active(0)
s(active(X)) → s(X)cons(X1, active(X2)) → cons(X1, X2)
from(active(X)) → from(X)if(X1, active(X2), X3) → if(X1, X2, X3)
if(mark(X1), X2, X3) → if(X1, X2, X3)first(mark(X1), X2) → first(X1, X2)
add(active(X1), X2) → add(X1, X2)active(add(s(X), Y)) → mark(s(add(X, Y)))
active(if(false, X, Y)) → mark(Y)mark(from(X)) → active(from(X))
and(mark(X1), X2) → and(X1, X2)if(X1, mark(X2), X3) → if(X1, X2, X3)
cons(mark(X1), X2) → cons(X1, X2)mark(false) → active(false)
active(add(0, X)) → mark(X)s(mark(X)) → s(X)
and(X1, active(X2)) → and(X1, X2)active(if(true, X, Y)) → mark(X)

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

The following dependency pairs (at least) can be eliminated according to the given precedence.

active#(add(s(X), Y)) → mark#(s(add(X, Y)))

Problem 28: DependencyGraph



Dependency Pair Problem

Dependency Pairs

mark#(add(X1, X2))active#(add(mark(X1), X2))

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


There are no SCCs!

Problem 5: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

first#(active(X1), X2)first#(X1, X2)first#(mark(X1), X2)first#(X1, X2)
first#(X1, mark(X2))first#(X1, X2)first#(X1, active(X2))first#(X1, X2)

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

first#(active(X1), X2)first#(X1, X2)first#(mark(X1), X2)first#(X1, X2)

Problem 12: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

first#(X1, active(X2))first#(X1, X2)first#(X1, mark(X2))first#(X1, X2)

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

first#(X1, mark(X2))first#(X1, X2)first#(X1, active(X2))first#(X1, X2)

Problem 6: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

from#(active(X))from#(X)from#(mark(X))from#(X)

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

from#(active(X))from#(X)from#(mark(X))from#(X)

Problem 7: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

add#(active(X1), X2)add#(X1, X2)add#(X1, mark(X2))add#(X1, X2)
add#(X1, active(X2))add#(X1, X2)add#(mark(X1), X2)add#(X1, X2)

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

add#(active(X1), X2)add#(X1, X2)add#(mark(X1), X2)add#(X1, X2)

Problem 13: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

add#(X1, mark(X2))add#(X1, X2)add#(X1, active(X2))add#(X1, X2)

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

add#(X1, mark(X2))add#(X1, X2)add#(X1, active(X2))add#(X1, X2)

Problem 8: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

if#(X1, mark(X2), X3)if#(X1, X2, X3)if#(X1, X2, mark(X3))if#(X1, X2, X3)
if#(mark(X1), X2, X3)if#(X1, X2, X3)if#(X1, active(X2), X3)if#(X1, X2, X3)
if#(X1, X2, active(X3))if#(X1, X2, X3)if#(active(X1), X2, X3)if#(X1, X2, X3)

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

if#(mark(X1), X2, X3)if#(X1, X2, X3)if#(active(X1), X2, X3)if#(X1, X2, X3)

Problem 14: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

if#(X1, mark(X2), X3)if#(X1, X2, X3)if#(X1, X2, mark(X3))if#(X1, X2, X3)
if#(X1, active(X2), X3)if#(X1, X2, X3)if#(X1, X2, active(X3))if#(X1, X2, X3)

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

if#(X1, mark(X2), X3)if#(X1, X2, X3)if#(X1, active(X2), X3)if#(X1, X2, X3)

Problem 15: PolynomialLinearRange4



Dependency Pair Problem

Dependency Pairs

if#(X1, X2, mark(X3))if#(X1, X2, X3)if#(X1, X2, active(X3))if#(X1, X2, X3)

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


Polynomial Interpretation

There are no usable rules

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

if#(X1, X2, mark(X3))if#(X1, X2, X3)

Problem 17: PolynomialLinearRange4



Dependency Pair Problem

Dependency Pairs

if#(X1, X2, active(X3))if#(X1, X2, X3)

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


Polynomial Interpretation

There are no usable rules

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

if#(X1, X2, active(X3))if#(X1, X2, X3)

Problem 9: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

s#(mark(X))s#(X)s#(active(X))s#(X)

Rewrite Rules

active(and(true, X))mark(X)active(and(false, Y))mark(false)
active(if(true, X, Y))mark(X)active(if(false, X, Y))mark(Y)
active(add(0, X))mark(X)active(add(s(X), Y))mark(s(add(X, Y)))
active(first(0, X))mark(nil)active(first(s(X), cons(Y, Z)))mark(cons(Y, first(X, Z)))
active(from(X))mark(cons(X, from(s(X))))mark(and(X1, X2))active(and(mark(X1), X2))
mark(true)active(true)mark(false)active(false)
mark(if(X1, X2, X3))active(if(mark(X1), X2, X3))mark(add(X1, X2))active(add(mark(X1), X2))
mark(0)active(0)mark(s(X))active(s(X))
mark(first(X1, X2))active(first(mark(X1), mark(X2)))mark(nil)active(nil)
mark(cons(X1, X2))active(cons(X1, X2))mark(from(X))active(from(X))
and(mark(X1), X2)and(X1, X2)and(X1, mark(X2))and(X1, X2)
and(active(X1), X2)and(X1, X2)and(X1, active(X2))and(X1, X2)
if(mark(X1), X2, X3)if(X1, X2, X3)if(X1, mark(X2), X3)if(X1, X2, X3)
if(X1, X2, mark(X3))if(X1, X2, X3)if(active(X1), X2, X3)if(X1, X2, X3)
if(X1, active(X2), X3)if(X1, X2, X3)if(X1, X2, active(X3))if(X1, X2, X3)
add(mark(X1), X2)add(X1, X2)add(X1, mark(X2))add(X1, X2)
add(active(X1), X2)add(X1, X2)add(X1, active(X2))add(X1, X2)
s(mark(X))s(X)s(active(X))s(X)
first(mark(X1), X2)first(X1, X2)first(X1, mark(X2))first(X1, X2)
first(active(X1), X2)first(X1, X2)first(X1, active(X2))first(X1, X2)
cons(mark(X1), X2)cons(X1, X2)cons(X1, mark(X2))cons(X1, X2)
cons(active(X1), X2)cons(X1, X2)cons(X1, active(X2))cons(X1, X2)
from(mark(X))from(X)from(active(X))from(X)

Original Signature

Termination of terms over the following signature is verified: true, mark, from, add, and, 0, s, if, active, false, first, nil, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

s#(mark(X))s#(X)s#(active(X))s#(X)