Termination proof of /tmp/tmp0jUJp9/eratosthenes

Termination of the given ITRSProblem could successfully be proven:

↳ ITRS

  ↳ ITRStoIDPProof

ITRS problem:
The following domains are used:

z

The TRS R consists of the following rules:

sieve(nil) → nil
Cond_isdiv(TRUE, x, y) → isdiv(x, -@z(y, x))
Cond_isdiv1(TRUE, x) → TRUE
filter(x, cons(y, zs)) → if(isdiv(x, y), x, y, zs)
if(TRUE, x, y, zs) → filter(x, zs)
Cond_nats1(TRUE, x, y) → nil
nats(x, y) → Cond_nats1(>@z(x, y), x, y)
isdiv(x, 0@z) → Cond_isdiv1(>@z(x, 0@z), x)
nats(x, y) → Cond_nats(>=@z(y, x), x, y)
isdiv(x, y) → Cond_isdiv2(&&(>@z(x, y), >@z(y, 0@z)), x, y)
if(FALSE, x, y, zs) → cons(y, filter(x, zs))
Cond_nats(TRUE, x, y) → cons(x, nats(+@z(x, 1@z), y))
isdiv(x, y) → Cond_isdiv(&&(>=@z(y, x), >@z(x, 0@z)), x, y)
filter(x, nil) → nil
sieve(cons(x, ys)) → cons(x, sieve(filter(x, ys)))
Cond_isdiv2(TRUE, x, y) → FALSE
primes(x) → sieve(nats(2@z, x))

The set Q consists of the following terms:

sieve(nil)
Cond_isdiv(TRUE, x0, x1)
Cond_isdiv1(TRUE, x0)
filter(x0, cons(x1, x2))
if(TRUE, x0, x1, x2)
Cond_nats1(TRUE, x0, x1)
nats(x0, x1)
isdiv(x0, x1)
if(FALSE, x0, x1, x2)
Cond_nats(TRUE, x0, x1)
filter(x0, nil)
sieve(cons(x0, x1))
Cond_isdiv2(TRUE, x0, x1)
primes(x0)

Added dependency pairs

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

I DP problem:
The following domains are used:

z

The ITRS R consists of the following rules:

sieve(nil) → nil
Cond_isdiv(TRUE, x, y) → isdiv(x, -@z(y, x))
Cond_isdiv1(TRUE, x) → TRUE
filter(x, cons(y, zs)) → if(isdiv(x, y), x, y, zs)
if(TRUE, x, y, zs) → filter(x, zs)
Cond_nats1(TRUE, x, y) → nil
nats(x, y) → Cond_nats1(>@z(x, y), x, y)
isdiv(x, 0@z) → Cond_isdiv1(>@z(x, 0@z), x)
nats(x, y) → Cond_nats(>=@z(y, x), x, y)
isdiv(x, y) → Cond_isdiv2(&&(>@z(x, y), >@z(y, 0@z)), x, y)
if(FALSE, x, y, zs) → cons(y, filter(x, zs))
Cond_nats(TRUE, x, y) → cons(x, nats(+@z(x, 1@z), y))
isdiv(x, y) → Cond_isdiv(&&(>=@z(y, x), >@z(x, 0@z)), x, y)
filter(x, nil) → nil
sieve(cons(x, ys)) → cons(x, sieve(filter(x, ys)))
Cond_isdiv2(TRUE, x, y) → FALSE
primes(x) → sieve(nats(2@z, x))

The integer pair graph contains the following rules and edges:

(0): COND_NATS(TRUE, x[0], y[0]) → NATS(+@z(x[0], 1@z), y[0])
(1): NATS(x[1], y[1]) → COND_NATS(>=@z(y[1], x[1]), x[1], y[1])
(2): SIEVE(cons(x[2], ys[2])) → FILTER(x[2], ys[2])
(3): PRIMES(x[3]) → NATS(2@z, x[3])
(4): ISDIV(x[4], y[4]) → COND_ISDIV(&&(>=@z(y[4], x[4]), >@z(x[4], 0@z)), x[4], y[4])
(5): IF(FALSE, x[5], y[5], zs[5]) → FILTER(x[5], zs[5])
(6): ISDIV(x[6], 0@z) → COND_ISDIV1(>@z(x[6], 0@z), x[6])
(7): IF(TRUE, x[7], y[7], zs[7]) → FILTER(x[7], zs[7])
(8): ISDIV(x[8], y[8]) → COND_ISDIV2(&&(>@z(x[8], y[8]), >@z(y[8], 0@z)), x[8], y[8])
(9): PRIMES(x[9]) → SIEVE(nats(2@z, x[9]))
(10): FILTER(x[10], cons(y[10], zs[10])) → ISDIV(x[10], y[10])
(11): SIEVE(cons(x[11], ys[11])) → SIEVE(filter(x[11], ys[11]))
(12): FILTER(x[12], cons(y[12], zs[12])) → IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])
(13): NATS(x[13], y[13]) → COND_NATS1(>@z(x[13], y[13]), x[13], y[13])
(14): COND_ISDIV(TRUE, x[14], y[14]) → ISDIV(x[14], -@z(y[14], x[14]))

(0) -> (1), if ((y[0] →^* y[1])∧(+@z(x[0], 1@z) →^* x[1]))

(0) -> (13), if ((y[0] →^* y[13])∧(+@z(x[0], 1@z) →^* x[13]))

(1) -> (0), if ((x[1] →^* x[0])∧(y[1] →^* y[0])∧(>=@z(y[1], x[1]) →^* TRUE))

(2) -> (10), if ((ys[2] →^* cons(y[10], zs[10]))∧(x[2] →^* x[10]))

(2) -> (12), if ((ys[2] →^* cons(y[12], zs[12]))∧(x[2] →^* x[12]))

(3) -> (1), if ((x[3] →^* y[1]))

(3) -> (13), if ((x[3] →^* y[13]))

(4) -> (14), if ((x[4] →^* x[14])∧(y[4] →^* y[14])∧(&&(>=@z(y[4], x[4]), >@z(x[4], 0@z)) →^* TRUE))

(5) -> (10), if ((zs[5] →^* cons(y[10], zs[10]))∧(x[5] →^* x[10]))

(5) -> (12), if ((zs[5] →^* cons(y[12], zs[12]))∧(x[5] →^* x[12]))

(7) -> (10), if ((zs[7] →^* cons(y[10], zs[10]))∧(x[7] →^* x[10]))

(7) -> (12), if ((zs[7] →^* cons(y[12], zs[12]))∧(x[7] →^* x[12]))

(9) -> (2), if ((nats(2@z, x[9]) →^* cons(x[2], ys[2])))

(9) -> (11), if ((nats(2@z, x[9]) →^* cons(x[11], ys[11])))

(10) -> (4), if ((y[10] →^* y[4])∧(x[10] →^* x[4]))

(10) -> (6), if ((y[10] →^* 0@z)∧(x[10] →^* x[6]))

(10) -> (8), if ((y[10] →^* y[8])∧(x[10] →^* x[8]))

(11) -> (2), if ((filter(x[11], ys[11]) →^* cons(x[2], ys[2])))

(11) -> (11), if ((filter(x[11], ys[11]) →^* cons(x[11]a, ys[11]a)))

(12) -> (5), if ((zs[12] →^* zs[5])∧(x[12] →^* x[5])∧(y[12] →^* y[5])∧(isdiv(x[12], y[12]) →^* FALSE))

(12) -> (7), if ((zs[12] →^* zs[7])∧(x[12] →^* x[7])∧(y[12] →^* y[7])∧(isdiv(x[12], y[12]) →^* TRUE))

(14) -> (4), if ((-@z(y[14], x[14]) →^* y[4])∧(x[14] →^* x[4]))

(14) -> (6), if ((-@z(y[14], x[14]) →^* 0@z)∧(x[14] →^* x[6]))

(14) -> (8), if ((-@z(y[14], x[14]) →^* y[8])∧(x[14] →^* x[8]))

The set Q consists of the following terms:

sieve(nil)
Cond_isdiv(TRUE, x0, x1)
Cond_isdiv1(TRUE, x0)
filter(x0, cons(x1, x2))
if(TRUE, x0, x1, x2)
Cond_nats1(TRUE, x0, x1)
nats(x0, x1)
isdiv(x0, x1)
if(FALSE, x0, x1, x2)
Cond_nats(TRUE, x0, x1)
filter(x0, nil)
sieve(cons(x0, x1))
Cond_isdiv2(TRUE, x0, x1)
primes(x0)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDependencyGraphProof

I DP problem:
The following domains are used:

z

The ITRS R consists of the following rules:

Cond_isdiv1(TRUE, x) → TRUE
filter(x, cons(y, zs)) → if(isdiv(x, y), x, y, zs)
if(TRUE, x, y, zs) → filter(x, zs)
Cond_nats1(TRUE, x, y) → nil
nats(x, y) → Cond_nats1(>@z(x, y), x, y)
isdiv(x, 0@z) → Cond_isdiv1(>@z(x, 0@z), x)
nats(x, y) → Cond_nats(>=@z(y, x), x, y)
isdiv(x, y) → Cond_isdiv2(&&(>@z(x, y), >@z(y, 0@z)), x, y)
if(FALSE, x, y, zs) → cons(y, filter(x, zs))
Cond_nats(TRUE, x, y) → cons(x, nats(+@z(x, 1@z), y))
Cond_isdiv(TRUE, x, y) → isdiv(x, -@z(y, x))
isdiv(x, y) → Cond_isdiv(&&(>=@z(y, x), >@z(x, 0@z)), x, y)
filter(x, nil) → nil
Cond_isdiv2(TRUE, x, y) → FALSE

(0) -> (1), if ((y[0] →^* y[1])∧(+@z(x[0], 1@z) →^* x[1]))

(0) -> (13), if ((y[0] →^* y[13])∧(+@z(x[0], 1@z) →^* x[13]))

(1) -> (0), if ((x[1] →^* x[0])∧(y[1] →^* y[0])∧(>=@z(y[1], x[1]) →^* TRUE))

(2) -> (10), if ((ys[2] →^* cons(y[10], zs[10]))∧(x[2] →^* x[10]))

(2) -> (12), if ((ys[2] →^* cons(y[12], zs[12]))∧(x[2] →^* x[12]))

(3) -> (1), if ((x[3] →^* y[1]))

(3) -> (13), if ((x[3] →^* y[13]))

(4) -> (14), if ((x[4] →^* x[14])∧(y[4] →^* y[14])∧(&&(>=@z(y[4], x[4]), >@z(x[4], 0@z)) →^* TRUE))

(5) -> (10), if ((zs[5] →^* cons(y[10], zs[10]))∧(x[5] →^* x[10]))

(5) -> (12), if ((zs[5] →^* cons(y[12], zs[12]))∧(x[5] →^* x[12]))

(7) -> (10), if ((zs[7] →^* cons(y[10], zs[10]))∧(x[7] →^* x[10]))

(7) -> (12), if ((zs[7] →^* cons(y[12], zs[12]))∧(x[7] →^* x[12]))

(9) -> (2), if ((nats(2@z, x[9]) →^* cons(x[2], ys[2])))

(9) -> (11), if ((nats(2@z, x[9]) →^* cons(x[11], ys[11])))

(10) -> (4), if ((y[10] →^* y[4])∧(x[10] →^* x[4]))

(10) -> (6), if ((y[10] →^* 0@z)∧(x[10] →^* x[6]))

(10) -> (8), if ((y[10] →^* y[8])∧(x[10] →^* x[8]))

(11) -> (2), if ((filter(x[11], ys[11]) →^* cons(x[2], ys[2])))

(11) -> (11), if ((filter(x[11], ys[11]) →^* cons(x[11]a, ys[11]a)))

(12) -> (5), if ((zs[12] →^* zs[5])∧(x[12] →^* x[5])∧(y[12] →^* y[5])∧(isdiv(x[12], y[12]) →^* FALSE))

(12) -> (7), if ((zs[12] →^* zs[7])∧(x[12] →^* x[7])∧(y[12] →^* y[7])∧(isdiv(x[12], y[12]) →^* TRUE))

(14) -> (4), if ((-@z(y[14], x[14]) →^* y[4])∧(x[14] →^* x[4]))

(14) -> (6), if ((-@z(y[14], x[14]) →^* 0@z)∧(x[14] →^* x[6]))

(14) -> (8), if ((-@z(y[14], x[14]) →^* y[8])∧(x[14] →^* x[8]))

The set Q consists of the following terms:

sieve(nil)
Cond_isdiv(TRUE, x0, x1)
Cond_isdiv1(TRUE, x0)
filter(x0, cons(x1, x2))
if(TRUE, x0, x1, x2)
Cond_nats1(TRUE, x0, x1)
nats(x0, x1)
isdiv(x0, x1)
if(FALSE, x0, x1, x2)
Cond_nats(TRUE, x0, x1)
filter(x0, nil)
sieve(cons(x0, x1))
Cond_isdiv2(TRUE, x0, x1)
primes(x0)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 7 less nodes.

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDependencyGraphProof

            ↳ AND

              ↳ IDP

                ↳ UsableRulesProof

              ↳ IDP

              ↳ IDP

              ↳ IDP

I DP problem:
The following domains are used:

z

The ITRS R consists of the following rules:

Cond_isdiv1(TRUE, x) → TRUE
filter(x, cons(y, zs)) → if(isdiv(x, y), x, y, zs)
if(TRUE, x, y, zs) → filter(x, zs)
Cond_nats1(TRUE, x, y) → nil
nats(x, y) → Cond_nats1(>@z(x, y), x, y)
isdiv(x, 0@z) → Cond_isdiv1(>@z(x, 0@z), x)
nats(x, y) → Cond_nats(>=@z(y, x), x, y)
isdiv(x, y) → Cond_isdiv2(&&(>@z(x, y), >@z(y, 0@z)), x, y)
if(FALSE, x, y, zs) → cons(y, filter(x, zs))
Cond_nats(TRUE, x, y) → cons(x, nats(+@z(x, 1@z), y))
Cond_isdiv(TRUE, x, y) → isdiv(x, -@z(y, x))
isdiv(x, y) → Cond_isdiv(&&(>=@z(y, x), >@z(x, 0@z)), x, y)
filter(x, nil) → nil
Cond_isdiv2(TRUE, x, y) → FALSE

The integer pair graph contains the following rules and edges:

(1): NATS(x[1], y[1]) → COND_NATS(>=@z(y[1], x[1]), x[1], y[1])
(0): COND_NATS(TRUE, x[0], y[0]) → NATS(+@z(x[0], 1@z), y[0])

(0) -> (1), if ((y[0] →^* y[1])∧(+@z(x[0], 1@z) →^* x[1]))

(1) -> (0), if ((x[1] →^* x[0])∧(y[1] →^* y[0])∧(>=@z(y[1], x[1]) →^* TRUE))

The set Q consists of the following terms:

sieve(nil)
Cond_isdiv(TRUE, x0, x1)
Cond_isdiv1(TRUE, x0)
filter(x0, cons(x1, x2))
if(TRUE, x0, x1, x2)
Cond_nats1(TRUE, x0, x1)
nats(x0, x1)
isdiv(x0, x1)
if(FALSE, x0, x1, x2)
Cond_nats(TRUE, x0, x1)
filter(x0, nil)
sieve(cons(x0, x1))
Cond_isdiv2(TRUE, x0, x1)
primes(x0)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDependencyGraphProof

            ↳ AND

              ↳ IDP

                ↳ UsableRulesProof

                  ↳ IDP

                    ↳ IDPNonInfProof

              ↳ IDP

              ↳ IDP

              ↳ IDP

I DP problem:
The following domains are used:

z

R is empty.
The integer pair graph contains the following rules and edges:

(1): NATS(x[1], y[1]) → COND_NATS(>=@z(y[1], x[1]), x[1], y[1])
(0): COND_NATS(TRUE, x[0], y[0]) → NATS(+@z(x[0], 1@z), y[0])

(0) -> (1), if ((y[0] →^* y[1])∧(+@z(x[0], 1@z) →^* x[1]))

(1) -> (0), if ((x[1] →^* x[0])∧(y[1] →^* y[0])∧(>=@z(y[1], x[1]) →^* TRUE))

The set Q consists of the following terms:

sieve(nil)
Cond_isdiv(TRUE, x0, x1)
Cond_isdiv1(TRUE, x0)
filter(x0, cons(x1, x2))
if(TRUE, x0, x1, x2)
Cond_nats1(TRUE, x0, x1)
nats(x0, x1)
isdiv(x0, x1)
if(FALSE, x0, x1, x2)
Cond_nats(TRUE, x0, x1)
filter(x0, nil)
sieve(cons(x0, x1))
Cond_isdiv2(TRUE, x0, x1)
primes(x0)

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.

For Pair NATS(x[1], y[1]) → COND_NATS(>=@z(y[1], x[1]), x[1], y[1]) the following chains were created:

We consider the chain NATS(x[1], y[1]) → COND_NATS(>=@z(y[1], x[1]), x[1], y[1]) which results in the following constraint:
(1)    (NATS(x[1], y[1])≥NonInfC∧NATS(x[1], y[1])≥COND_NATS(>=@z(y[1], x[1]), x[1], y[1])∧(U^Increasing(COND_NATS(>=@z(y[1], x[1]), x[1], y[1])), ≥))

We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2)    ((U^Increasing(COND_NATS(>=@z(y[1], x[1]), x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3)    ((U^Increasing(COND_NATS(>=@z(y[1], x[1]), x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(4)    (0 ≥ 0∧0 ≥ 0∧(U^Increasing(COND_NATS(>=@z(y[1], x[1]), x[1], y[1])), ≥))

We simplified constraint (4) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(5)    (0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 ≥ 0∧(U^Increasing(COND_NATS(>=@z(y[1], x[1]), x[1], y[1])), ≥)∧0 = 0)

For Pair COND_NATS(TRUE, x[0], y[0]) → NATS(+@z(x[0], 1@z), y[0]) the following chains were created:

We consider the chain NATS(x[1], y[1]) → COND_NATS(>=@z(y[1], x[1]), x[1], y[1]), COND_NATS(TRUE, x[0], y[0]) → NATS(+@z(x[0], 1@z), y[0]), NATS(x[1], y[1]) → COND_NATS(>=@z(y[1], x[1]), x[1], y[1]) which results in the following constraint:
(6)    (y[0]=y[1]1∧+@z(x[0], 1@z)=x[1]1∧y[1]=y[0]∧>=@z(y[1], x[1])=TRUE∧x[1]=x[0] ⇒ COND_NATS(TRUE, x[0], y[0])≥NonInfC∧COND_NATS(TRUE, x[0], y[0])≥NATS(+@z(x[0], 1@z), y[0])∧(U^Increasing(NATS(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (6) using rules (III), (IV) which results in the following new constraint:
(7)    (>=@z(y[1], x[1])=TRUE ⇒ COND_NATS(TRUE, x[1], y[1])≥NonInfC∧COND_NATS(TRUE, x[1], y[1])≥NATS(+@z(x[1], 1@z), y[1])∧(U^Increasing(NATS(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    (y[1] + (-1)x[1] ≥ 0 ⇒ (U^Increasing(NATS(+@z(x[0], 1@z), y[0])), ≥)∧-1 + (-1)Bound + y[1] + (-1)x[1] ≥ 0∧0 ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    (y[1] + (-1)x[1] ≥ 0 ⇒ (U^Increasing(NATS(+@z(x[0], 1@z), y[0])), ≥)∧-1 + (-1)Bound + y[1] + (-1)x[1] ≥ 0∧0 ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    (y[1] + (-1)x[1] ≥ 0 ⇒ -1 + (-1)Bound + y[1] + (-1)x[1] ≥ 0∧0 ≥ 0∧(U^Increasing(NATS(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(11)    (x[1] ≥ 0 ⇒ -1 + (-1)Bound + x[1] ≥ 0∧0 ≥ 0∧(U^Increasing(NATS(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(12)    (x[1] ≥ 0∧y[1] ≥ 0 ⇒ -1 + (-1)Bound + x[1] ≥ 0∧0 ≥ 0∧(U^Increasing(NATS(+@z(x[0], 1@z), y[0])), ≥))

(13)    (x[1] ≥ 0∧y[1] ≥ 0 ⇒ -1 + (-1)Bound + x[1] ≥ 0∧0 ≥ 0∧(U^Increasing(NATS(+@z(x[0], 1@z), y[0])), ≥))

To summarize, we get the following constraints P_≥ for the following pairs.

NATS(x[1], y[1]) → COND_NATS(>=@z(y[1], x[1]), x[1], y[1])
- (0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 ≥ 0∧(U^Increasing(COND_NATS(>=@z(y[1], x[1]), x[1], y[1])), ≥)∧0 = 0)
COND_NATS(TRUE, x[0], y[0]) → NATS(+@z(x[0], 1@z), y[0])
- (x[1] ≥ 0∧y[1] ≥ 0 ⇒ -1 + (-1)Bound + x[1] ≥ 0∧0 ≥ 0∧(U^Increasing(NATS(+@z(x[0], 1@z), y[0])), ≥))
- (x[1] ≥ 0∧y[1] ≥ 0 ⇒ -1 + (-1)Bound + x[1] ≥ 0∧0 ≥ 0∧(U^Increasing(NATS(+@z(x[0], 1@z), y[0])), ≥))

The constraints for P_> respective P_bound are constructed from P_≥ where we just replace every occurence of "t ≥ s" in P_≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for P_bound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(>=@z(x₁, x₂)) = -1
POL(TRUE) = -1
POL(COND_NATS(x₁, x₂, x₃)) = -1 + x₃ + (-1)x₂
POL(+@z(x₁, x₂)) = x₁ + x₂
POL(NATS(x₁, x₂)) = -1 + x₂ + (-1)x₁
POL(FALSE) = -1
POL(1@z) = 1
POL(undefined) = -1

The following pairs are in P_>:

COND_NATS(TRUE, x[0], y[0]) → NATS(+@z(x[0], 1@z), y[0])

The following pairs are in P_bound:

COND_NATS(TRUE, x[0], y[0]) → NATS(+@z(x[0], 1@z), y[0])

The following pairs are in P_≥:

NATS(x[1], y[1]) → COND_NATS(>=@z(y[1], x[1]), x[1], y[1])

At least the following rules have been oriented under context sensitive arithmetic replacement:

+@z¹ ↔

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDependencyGraphProof

            ↳ AND

              ↳ IDP

                ↳ UsableRulesProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ IDP

                        ↳ IDependencyGraphProof

              ↳ IDP

              ↳ IDP

              ↳ IDP

I DP problem:
The following domains are used:

z

R is empty.
The integer pair graph contains the following rules and edges:

(1): NATS(x[1], y[1]) → COND_NATS(>=@z(y[1], x[1]), x[1], y[1])

The set Q consists of the following terms:

sieve(nil)
Cond_isdiv(TRUE, x0, x1)
Cond_isdiv1(TRUE, x0)
filter(x0, cons(x1, x2))
if(TRUE, x0, x1, x2)
Cond_nats1(TRUE, x0, x1)
nats(x0, x1)
isdiv(x0, x1)
if(FALSE, x0, x1, x2)
Cond_nats(TRUE, x0, x1)
filter(x0, nil)
sieve(cons(x0, x1))
Cond_isdiv2(TRUE, x0, x1)
primes(x0)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDependencyGraphProof

            ↳ AND

              ↳ IDP

              ↳ IDP

                ↳ UsableRulesProof

              ↳ IDP

              ↳ IDP

I DP problem:
The following domains are used:

z

The ITRS R consists of the following rules:

Cond_isdiv1(TRUE, x) → TRUE
filter(x, cons(y, zs)) → if(isdiv(x, y), x, y, zs)
if(TRUE, x, y, zs) → filter(x, zs)
Cond_nats1(TRUE, x, y) → nil
nats(x, y) → Cond_nats1(>@z(x, y), x, y)
isdiv(x, 0@z) → Cond_isdiv1(>@z(x, 0@z), x)
nats(x, y) → Cond_nats(>=@z(y, x), x, y)
isdiv(x, y) → Cond_isdiv2(&&(>@z(x, y), >@z(y, 0@z)), x, y)
if(FALSE, x, y, zs) → cons(y, filter(x, zs))
Cond_nats(TRUE, x, y) → cons(x, nats(+@z(x, 1@z), y))
Cond_isdiv(TRUE, x, y) → isdiv(x, -@z(y, x))
isdiv(x, y) → Cond_isdiv(&&(>=@z(y, x), >@z(x, 0@z)), x, y)
filter(x, nil) → nil
Cond_isdiv2(TRUE, x, y) → FALSE

The integer pair graph contains the following rules and edges:

(14): COND_ISDIV(TRUE, x[14], y[14]) → ISDIV(x[14], -@z(y[14], x[14]))
(4): ISDIV(x[4], y[4]) → COND_ISDIV(&&(>=@z(y[4], x[4]), >@z(x[4], 0@z)), x[4], y[4])

(4) -> (14), if ((x[4] →^* x[14])∧(y[4] →^* y[14])∧(&&(>=@z(y[4], x[4]), >@z(x[4], 0@z)) →^* TRUE))

(14) -> (4), if ((-@z(y[14], x[14]) →^* y[4])∧(x[14] →^* x[4]))

The set Q consists of the following terms:

sieve(nil)
Cond_isdiv(TRUE, x0, x1)
Cond_isdiv1(TRUE, x0)
filter(x0, cons(x1, x2))
if(TRUE, x0, x1, x2)
Cond_nats1(TRUE, x0, x1)
nats(x0, x1)
isdiv(x0, x1)
if(FALSE, x0, x1, x2)
Cond_nats(TRUE, x0, x1)
filter(x0, nil)
sieve(cons(x0, x1))
Cond_isdiv2(TRUE, x0, x1)
primes(x0)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDependencyGraphProof

            ↳ AND

              ↳ IDP

              ↳ IDP

                ↳ UsableRulesProof

                  ↳ IDP

                    ↳ IDPNonInfProof

              ↳ IDP

              ↳ IDP

I DP problem:
The following domains are used:

z

R is empty.
The integer pair graph contains the following rules and edges:

(14): COND_ISDIV(TRUE, x[14], y[14]) → ISDIV(x[14], -@z(y[14], x[14]))
(4): ISDIV(x[4], y[4]) → COND_ISDIV(&&(>=@z(y[4], x[4]), >@z(x[4], 0@z)), x[4], y[4])

(4) -> (14), if ((x[4] →^* x[14])∧(y[4] →^* y[14])∧(&&(>=@z(y[4], x[4]), >@z(x[4], 0@z)) →^* TRUE))

(14) -> (4), if ((-@z(y[14], x[14]) →^* y[4])∧(x[14] →^* x[4]))

The set Q consists of the following terms:

sieve(nil)
Cond_isdiv(TRUE, x0, x1)
Cond_isdiv1(TRUE, x0)
filter(x0, cons(x1, x2))
if(TRUE, x0, x1, x2)
Cond_nats1(TRUE, x0, x1)
nats(x0, x1)
isdiv(x0, x1)
if(FALSE, x0, x1, x2)
Cond_nats(TRUE, x0, x1)
filter(x0, nil)
sieve(cons(x0, x1))
Cond_isdiv2(TRUE, x0, x1)
primes(x0)

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.

For Pair COND_ISDIV(TRUE, x[14], y[14]) → ISDIV(x[14], -@z(y[14], x[14])) the following chains were created:

We consider the chain ISDIV(x[4], y[4]) → COND_ISDIV(&&(>=@z(y[4], x[4]), >@z(x[4], 0@z)), x[4], y[4]), COND_ISDIV(TRUE, x[14], y[14]) → ISDIV(x[14], -@z(y[14], x[14])), ISDIV(x[4], y[4]) → COND_ISDIV(&&(>=@z(y[4], x[4]), >@z(x[4], 0@z)), x[4], y[4]) which results in the following constraint:
(1)    (x[4]=x[14]∧-@z(y[14], x[14])=y[4]1∧x[14]=x[4]1∧&&(>=@z(y[4], x[4]), >@z(x[4], 0@z))=TRUE∧y[4]=y[14] ⇒ COND_ISDIV(TRUE, x[14], y[14])≥NonInfC∧COND_ISDIV(TRUE, x[14], y[14])≥ISDIV(x[14], -@z(y[14], x[14]))∧(U^Increasing(ISDIV(x[14], -@z(y[14], x[14]))), ≥))

We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>=@z(y[4], x[4])=TRUE∧>@z(x[4], 0@z)=TRUE ⇒ COND_ISDIV(TRUE, x[4], y[4])≥NonInfC∧COND_ISDIV(TRUE, x[4], y[4])≥ISDIV(x[4], -@z(y[4], x[4]))∧(U^Increasing(ISDIV(x[14], -@z(y[14], x[14]))), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (y[4] + (-1)x[4] ≥ 0∧-1 + x[4] ≥ 0 ⇒ (U^Increasing(ISDIV(x[14], -@z(y[14], x[14]))), ≥)∧-1 + (-1)Bound + y[4] + (-1)x[4] ≥ 0∧-1 + x[4] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (y[4] + (-1)x[4] ≥ 0∧-1 + x[4] ≥ 0 ⇒ (U^Increasing(ISDIV(x[14], -@z(y[14], x[14]))), ≥)∧-1 + (-1)Bound + y[4] + (-1)x[4] ≥ 0∧-1 + x[4] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (-1 + x[4] ≥ 0∧y[4] + (-1)x[4] ≥ 0 ⇒ -1 + x[4] ≥ 0∧-1 + (-1)Bound + y[4] + (-1)x[4] ≥ 0∧(U^Increasing(ISDIV(x[14], -@z(y[14], x[14]))), ≥))

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x[4] ≥ 0∧-1 + y[4] + (-1)x[4] ≥ 0 ⇒ x[4] ≥ 0∧-2 + (-1)Bound + y[4] + (-1)x[4] ≥ 0∧(U^Increasing(ISDIV(x[14], -@z(y[14], x[14]))), ≥))

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x[4] ≥ 0∧y[4] ≥ 0 ⇒ x[4] ≥ 0∧-1 + (-1)Bound + y[4] ≥ 0∧(U^Increasing(ISDIV(x[14], -@z(y[14], x[14]))), ≥))

For Pair ISDIV(x[4], y[4]) → COND_ISDIV(&&(>=@z(y[4], x[4]), >@z(x[4], 0@z)), x[4], y[4]) the following chains were created:

We consider the chain ISDIV(x[4], y[4]) → COND_ISDIV(&&(>=@z(y[4], x[4]), >@z(x[4], 0@z)), x[4], y[4]) which results in the following constraint:
(8)    (ISDIV(x[4], y[4])≥NonInfC∧ISDIV(x[4], y[4])≥COND_ISDIV(&&(>=@z(y[4], x[4]), >@z(x[4], 0@z)), x[4], y[4])∧(U^Increasing(COND_ISDIV(&&(>=@z(y[4], x[4]), >@z(x[4], 0@z)), x[4], y[4])), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    ((U^Increasing(COND_ISDIV(&&(>=@z(y[4], x[4]), >@z(x[4], 0@z)), x[4], y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10)    ((U^Increasing(COND_ISDIV(&&(>=@z(y[4], x[4]), >@z(x[4], 0@z)), x[4], y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11)    (0 ≥ 0∧0 ≥ 0∧(U^Increasing(COND_ISDIV(&&(>=@z(y[4], x[4]), >@z(x[4], 0@z)), x[4], y[4])), ≥))

We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(12)    (0 = 0∧0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧(U^Increasing(COND_ISDIV(&&(>=@z(y[4], x[4]), >@z(x[4], 0@z)), x[4], y[4])), ≥))

To summarize, we get the following constraints P_≥ for the following pairs.

COND_ISDIV(TRUE, x[14], y[14]) → ISDIV(x[14], -@z(y[14], x[14]))
- (x[4] ≥ 0∧y[4] ≥ 0 ⇒ x[4] ≥ 0∧-1 + (-1)Bound + y[4] ≥ 0∧(U^Increasing(ISDIV(x[14], -@z(y[14], x[14]))), ≥))
ISDIV(x[4], y[4]) → COND_ISDIV(&&(>=@z(y[4], x[4]), >@z(x[4], 0@z)), x[4], y[4])
- (0 = 0∧0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧(U^Increasing(COND_ISDIV(&&(>=@z(y[4], x[4]), >@z(x[4], 0@z)), x[4], y[4])), ≥))

POL(-@z(x₁, x₂)) = x₁ + (-1)x₂
POL(>=@z(x₁, x₂)) = -1
POL(0@z) = 0
POL(ISDIV(x₁, x₂)) = -1 + x₂ + (-1)x₁
POL(TRUE) = 0
POL(&&(x₁, x₂)) = 0
POL(FALSE) = 0
POL(COND_ISDIV(x₁, x₂, x₃)) = -1 + x₃ + (-1)x₂ + (-1)x₁
POL(undefined) = -1
POL(>@z(x₁, x₂)) = -1

The following pairs are in P_>:

COND_ISDIV(TRUE, x[14], y[14]) → ISDIV(x[14], -@z(y[14], x[14]))

The following pairs are in P_bound:

COND_ISDIV(TRUE, x[14], y[14]) → ISDIV(x[14], -@z(y[14], x[14]))

The following pairs are in P_≥:

ISDIV(x[4], y[4]) → COND_ISDIV(&&(>=@z(y[4], x[4]), >@z(x[4], 0@z)), x[4], y[4])

At least the following rules have been oriented under context sensitive arithmetic replacement:

&&(FALSE, FALSE)¹ ↔ FALSE¹
-@z¹ ↔
&&(TRUE, TRUE)¹ ↔ TRUE¹
&&(FALSE, TRUE)¹ ↔ FALSE¹
&&(TRUE, FALSE)¹ ↔ FALSE¹

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDependencyGraphProof

            ↳ AND

              ↳ IDP

              ↳ IDP

                ↳ UsableRulesProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ IDP

                        ↳ IDependencyGraphProof

              ↳ IDP

              ↳ IDP

I DP problem:
The following domains are used:

z

R is empty.
The integer pair graph contains the following rules and edges:

(4): ISDIV(x[4], y[4]) → COND_ISDIV(&&(>=@z(y[4], x[4]), >@z(x[4], 0@z)), x[4], y[4])

The set Q consists of the following terms:

sieve(nil)
Cond_isdiv(TRUE, x0, x1)
Cond_isdiv1(TRUE, x0)
filter(x0, cons(x1, x2))
if(TRUE, x0, x1, x2)
Cond_nats1(TRUE, x0, x1)
nats(x0, x1)
isdiv(x0, x1)
if(FALSE, x0, x1, x2)
Cond_nats(TRUE, x0, x1)
filter(x0, nil)
sieve(cons(x0, x1))
Cond_isdiv2(TRUE, x0, x1)
primes(x0)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDependencyGraphProof

            ↳ AND

              ↳ IDP

              ↳ IDP

              ↳ IDP

                ↳ UsableRulesProof

              ↳ IDP

I DP problem:
The following domains are used:

z

The ITRS R consists of the following rules:

Cond_isdiv1(TRUE, x) → TRUE
filter(x, cons(y, zs)) → if(isdiv(x, y), x, y, zs)
if(TRUE, x, y, zs) → filter(x, zs)
Cond_nats1(TRUE, x, y) → nil
nats(x, y) → Cond_nats1(>@z(x, y), x, y)
isdiv(x, 0@z) → Cond_isdiv1(>@z(x, 0@z), x)
nats(x, y) → Cond_nats(>=@z(y, x), x, y)
isdiv(x, y) → Cond_isdiv2(&&(>@z(x, y), >@z(y, 0@z)), x, y)
if(FALSE, x, y, zs) → cons(y, filter(x, zs))
Cond_nats(TRUE, x, y) → cons(x, nats(+@z(x, 1@z), y))
Cond_isdiv(TRUE, x, y) → isdiv(x, -@z(y, x))
isdiv(x, y) → Cond_isdiv(&&(>=@z(y, x), >@z(x, 0@z)), x, y)
filter(x, nil) → nil
Cond_isdiv2(TRUE, x, y) → FALSE

The integer pair graph contains the following rules and edges:

(7): IF(TRUE, x[7], y[7], zs[7]) → FILTER(x[7], zs[7])
(5): IF(FALSE, x[5], y[5], zs[5]) → FILTER(x[5], zs[5])
(12): FILTER(x[12], cons(y[12], zs[12])) → IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])

(7) -> (12), if ((zs[7] →^* cons(y[12], zs[12]))∧(x[7] →^* x[12]))

(12) -> (5), if ((zs[12] →^* zs[5])∧(x[12] →^* x[5])∧(y[12] →^* y[5])∧(isdiv(x[12], y[12]) →^* FALSE))

(5) -> (12), if ((zs[5] →^* cons(y[12], zs[12]))∧(x[5] →^* x[12]))

(12) -> (7), if ((zs[12] →^* zs[7])∧(x[12] →^* x[7])∧(y[12] →^* y[7])∧(isdiv(x[12], y[12]) →^* TRUE))

The set Q consists of the following terms:

sieve(nil)
Cond_isdiv(TRUE, x0, x1)
Cond_isdiv1(TRUE, x0)
filter(x0, cons(x1, x2))
if(TRUE, x0, x1, x2)
Cond_nats1(TRUE, x0, x1)
nats(x0, x1)
isdiv(x0, x1)
if(FALSE, x0, x1, x2)
Cond_nats(TRUE, x0, x1)
filter(x0, nil)
sieve(cons(x0, x1))
Cond_isdiv2(TRUE, x0, x1)
primes(x0)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDependencyGraphProof

            ↳ AND

              ↳ IDP

              ↳ IDP

              ↳ IDP

                ↳ UsableRulesProof

                  ↳ IDP

                    ↳ IDPNonInfProof

              ↳ IDP

I DP problem:
The following domains are used:

z

The ITRS R consists of the following rules:

Cond_isdiv2(TRUE, x, y) → FALSE
Cond_isdiv(TRUE, x, y) → isdiv(x, -@z(y, x))
isdiv(x, y) → Cond_isdiv(&&(>=@z(y, x), >@z(x, 0@z)), x, y)
isdiv(x, y) → Cond_isdiv2(&&(>@z(x, y), >@z(y, 0@z)), x, y)
isdiv(x, 0@z) → Cond_isdiv1(>@z(x, 0@z), x)
Cond_isdiv1(TRUE, x) → TRUE

(7) -> (12), if ((zs[7] →^* cons(y[12], zs[12]))∧(x[7] →^* x[12]))

(12) -> (5), if ((zs[12] →^* zs[5])∧(x[12] →^* x[5])∧(y[12] →^* y[5])∧(isdiv(x[12], y[12]) →^* FALSE))

(5) -> (12), if ((zs[5] →^* cons(y[12], zs[12]))∧(x[5] →^* x[12]))

(12) -> (7), if ((zs[12] →^* zs[7])∧(x[12] →^* x[7])∧(y[12] →^* y[7])∧(isdiv(x[12], y[12]) →^* TRUE))

The set Q consists of the following terms:

sieve(nil)
Cond_isdiv(TRUE, x0, x1)
Cond_isdiv1(TRUE, x0)
filter(x0, cons(x1, x2))
if(TRUE, x0, x1, x2)
Cond_nats1(TRUE, x0, x1)
nats(x0, x1)
isdiv(x0, x1)
if(FALSE, x0, x1, x2)
Cond_nats(TRUE, x0, x1)
filter(x0, nil)
sieve(cons(x0, x1))
Cond_isdiv2(TRUE, x0, x1)
primes(x0)

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.

For Pair IF(TRUE, x[7], y[7], zs[7]) → FILTER(x[7], zs[7]) the following chains were created:

We consider the chain FILTER(x[12], cons(y[12], zs[12])) → IF(isdiv(x[12], y[12]), x[12], y[12], zs[12]), IF(TRUE, x[7], y[7], zs[7]) → FILTER(x[7], zs[7]), FILTER(x[12], cons(y[12], zs[12])) → IF(isdiv(x[12], y[12]), x[12], y[12], zs[12]) which results in the following constraint:
(1)    (isdiv(x[12], y[12])=TRUE∧zs[7]=cons(y[12]1, zs[12]1)∧zs[12]=zs[7]∧x[7]=x[12]1∧x[12]=x[7]∧y[12]=y[7] ⇒ IF(TRUE, x[7], y[7], zs[7])≥FILTER(x[7], zs[7])∧(U^Increasing(FILTER(x[7], zs[7])), ≥))

We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
(2)    (isdiv(x[12], y[12])=TRUE ⇒ IF(TRUE, x[12], y[12], cons(y[12]1, zs[12]1))≥FILTER(x[12], cons(y[12]1, zs[12]1))∧(U^Increasing(FILTER(x[7], zs[7])), ≥))

We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on isdiv(x[12], y[12])=TRUE which results in the following new constraints:
(3)    (Cond_isdiv(&&(>=@z(x0, x1), >@z(x1, 0@z)), x1, x0)=TRUE ⇒ IF(TRUE, x1, x0, cons(y[12]1, zs[12]1))≥FILTER(x1, cons(y[12]1, zs[12]1))∧(U^Increasing(FILTER(x[7], zs[7])), ≥))

(4)    (Cond_isdiv2(&&(>@z(x3, x2), >@z(x2, 0@z)), x3, x2)=TRUE ⇒ IF(TRUE, x3, x2, cons(y[12]1, zs[12]1))≥FILTER(x3, cons(y[12]1, zs[12]1))∧(U^Increasing(FILTER(x[7], zs[7])), ≥))

(5)    (Cond_isdiv1(>@z(x4, 0@z), x4)=TRUE ⇒ IF(TRUE, x4, 0@z, cons(y[12]1, zs[12]1))≥FILTER(x4, cons(y[12]1, zs[12]1))∧(U^Increasing(FILTER(x[7], zs[7])), ≥))

We simplified constraint (3) using rules (III), (VII) which results in the following new constraint:
(6)    (Cond_isdiv(&&(>=@z(x0, x1), >@z(x1, 0@z)), x1, x0)=TRUE ⇒ IF(TRUE, x1, x0, cons(y[12]1, zs[12]1))≥FILTER(x1, cons(y[12]1, zs[12]1))∧(U^Increasing(FILTER(x[7], zs[7])), ≥))

We simplified constraint (4) using rules (III), (VII) which results in the following new constraint:
(7)    (Cond_isdiv2(&&(>@z(x3, x2), >@z(x2, 0@z)), x3, x2)=TRUE ⇒ IF(TRUE, x3, x2, cons(y[12]1, zs[12]1))≥FILTER(x3, cons(y[12]1, zs[12]1))∧(U^Increasing(FILTER(x[7], zs[7])), ≥))

We simplified constraint (5) using rules (III), (VII) which results in the following new constraint:
(8)    (Cond_isdiv1(>@z(x4, 0@z), x4)=TRUE ⇒ IF(TRUE, x4, 0@z, cons(y[12]1, zs[12]1))≥FILTER(x4, cons(y[12]1, zs[12]1))∧(U^Increasing(FILTER(x[7], zs[7])), ≥))

We simplified constraint (6) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    ((U^Increasing(FILTER(x[7], zs[7])), ≥)∧0 ≥ 0)

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    ((U^Increasing(FILTER(x[7], zs[7])), ≥)∧0 ≥ 0)

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(11)    ((U^Increasing(FILTER(x[7], zs[7])), ≥)∧0 ≥ 0)

We simplified constraint (11) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(12)    ((U^Increasing(FILTER(x[7], zs[7])), ≥)∧0 ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(13)    ((U^Increasing(FILTER(x[7], zs[7])), ≥)∧0 ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(14)    ((U^Increasing(FILTER(x[7], zs[7])), ≥)∧0 ≥ 0)

We simplified constraint (12) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(15)    (0 ≥ 0∧(U^Increasing(FILTER(x[7], zs[7])), ≥))

We simplified constraint (13) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(16)    (0 ≥ 0∧(U^Increasing(FILTER(x[7], zs[7])), ≥))

We simplified constraint (14) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(17)    (0 ≥ 0∧(U^Increasing(FILTER(x[7], zs[7])), ≥))

We simplified constraint (15) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(18)    (0 = 0∧0 ≥ 0∧0 = 0∧(U^Increasing(FILTER(x[7], zs[7])), ≥)∧0 = 0)

We simplified constraint (16) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(19)    (0 ≥ 0∧0 = 0∧(U^Increasing(FILTER(x[7], zs[7])), ≥)∧0 = 0∧0 = 0∧0 = 0)

We simplified constraint (17) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(20)    (0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧(U^Increasing(FILTER(x[7], zs[7])), ≥)∧0 = 0)

For Pair IF(FALSE, x[5], y[5], zs[5]) → FILTER(x[5], zs[5]) the following chains were created:

We consider the chain FILTER(x[12], cons(y[12], zs[12])) → IF(isdiv(x[12], y[12]), x[12], y[12], zs[12]), IF(FALSE, x[5], y[5], zs[5]) → FILTER(x[5], zs[5]), FILTER(x[12], cons(y[12], zs[12])) → IF(isdiv(x[12], y[12]), x[12], y[12], zs[12]) which results in the following constraint:
(21)    (x[12]=x[5]∧zs[12]=zs[5]∧zs[5]=cons(y[12]1, zs[12]1)∧y[12]=y[5]∧x[5]=x[12]1∧isdiv(x[12], y[12])=FALSE ⇒ IF(FALSE, x[5], y[5], zs[5])≥FILTER(x[5], zs[5])∧(U^Increasing(FILTER(x[5], zs[5])), ≥))

We simplified constraint (21) using rules (III), (IV) which results in the following new constraint:
(22)    (isdiv(x[12], y[12])=FALSE ⇒ IF(FALSE, x[12], y[12], cons(y[12]1, zs[12]1))≥FILTER(x[12], cons(y[12]1, zs[12]1))∧(U^Increasing(FILTER(x[5], zs[5])), ≥))

We simplified constraint (22) using rule (V) (with possible (I) afterwards) using induction on isdiv(x[12], y[12])=FALSE which results in the following new constraints:
(23)    (Cond_isdiv(&&(>=@z(x8, x9), >@z(x9, 0@z)), x9, x8)=FALSE ⇒ IF(FALSE, x9, x8, cons(y[12]1, zs[12]1))≥FILTER(x9, cons(y[12]1, zs[12]1))∧(U^Increasing(FILTER(x[5], zs[5])), ≥))

(24)    (Cond_isdiv2(&&(>@z(x11, x10), >@z(x10, 0@z)), x11, x10)=FALSE ⇒ IF(FALSE, x11, x10, cons(y[12]1, zs[12]1))≥FILTER(x11, cons(y[12]1, zs[12]1))∧(U^Increasing(FILTER(x[5], zs[5])), ≥))

(25)    (Cond_isdiv1(>@z(x12, 0@z), x12)=FALSE ⇒ IF(FALSE, x12, 0@z, cons(y[12]1, zs[12]1))≥FILTER(x12, cons(y[12]1, zs[12]1))∧(U^Increasing(FILTER(x[5], zs[5])), ≥))

We simplified constraint (23) using rules (III), (VII) which results in the following new constraint:
(26)    (Cond_isdiv(&&(>=@z(x8, x9), >@z(x9, 0@z)), x9, x8)=FALSE ⇒ IF(FALSE, x9, x8, cons(y[12]1, zs[12]1))≥FILTER(x9, cons(y[12]1, zs[12]1))∧(U^Increasing(FILTER(x[5], zs[5])), ≥))

We simplified constraint (24) using rules (III), (VII) which results in the following new constraint:
(27)    (Cond_isdiv2(&&(>@z(x11, x10), >@z(x10, 0@z)), x11, x10)=FALSE ⇒ IF(FALSE, x11, x10, cons(y[12]1, zs[12]1))≥FILTER(x11, cons(y[12]1, zs[12]1))∧(U^Increasing(FILTER(x[5], zs[5])), ≥))

We simplified constraint (25) using rules (III), (VII) which results in the following new constraint:
(28)    (Cond_isdiv1(>@z(x12, 0@z), x12)=FALSE ⇒ IF(FALSE, x12, 0@z, cons(y[12]1, zs[12]1))≥FILTER(x12, cons(y[12]1, zs[12]1))∧(U^Increasing(FILTER(x[5], zs[5])), ≥))

We simplified constraint (26) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(29)    ((U^Increasing(FILTER(x[5], zs[5])), ≥)∧0 ≥ 0)

We simplified constraint (27) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(30)    ((U^Increasing(FILTER(x[5], zs[5])), ≥)∧0 ≥ 0)

We simplified constraint (28) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(31)    ((U^Increasing(FILTER(x[5], zs[5])), ≥)∧0 ≥ 0)

We simplified constraint (31) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(32)    ((U^Increasing(FILTER(x[5], zs[5])), ≥)∧0 ≥ 0)

We simplified constraint (30) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(33)    ((U^Increasing(FILTER(x[5], zs[5])), ≥)∧0 ≥ 0)

We simplified constraint (29) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(34)    ((U^Increasing(FILTER(x[5], zs[5])), ≥)∧0 ≥ 0)

We simplified constraint (32) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(35)    (0 ≥ 0∧(U^Increasing(FILTER(x[5], zs[5])), ≥))

We simplified constraint (33) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(36)    (0 ≥ 0∧(U^Increasing(FILTER(x[5], zs[5])), ≥))

We simplified constraint (34) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(37)    (0 ≥ 0∧(U^Increasing(FILTER(x[5], zs[5])), ≥))

We simplified constraint (35) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(38)    (0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧(U^Increasing(FILTER(x[5], zs[5])), ≥))

We simplified constraint (36) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(39)    (0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(FILTER(x[5], zs[5])), ≥))

We simplified constraint (37) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(40)    (0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧(U^Increasing(FILTER(x[5], zs[5])), ≥))

For Pair FILTER(x[12], cons(y[12], zs[12])) → IF(isdiv(x[12], y[12]), x[12], y[12], zs[12]) the following chains were created:

We consider the chain IF(FALSE, x[5], y[5], zs[5]) → FILTER(x[5], zs[5]), FILTER(x[12], cons(y[12], zs[12])) → IF(isdiv(x[12], y[12]), x[12], y[12], zs[12]), IF(FALSE, x[5], y[5], zs[5]) → FILTER(x[5], zs[5]) which results in the following constraint:
(41)    (x[5]=x[12]∧zs[5]=cons(y[12], zs[12])∧zs[12]=zs[5]1∧y[12]=y[5]1∧x[12]=x[5]1∧isdiv(x[12], y[12])=FALSE ⇒ FILTER(x[12], cons(y[12], zs[12]))≥IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (41) using rules (III), (IV) which results in the following new constraint:
(42)    (isdiv(x[12], y[12])=FALSE ⇒ FILTER(x[12], cons(y[12], zs[12]))≥IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (42) using rule (V) (with possible (I) afterwards) using induction on isdiv(x[12], y[12])=FALSE which results in the following new constraints:
(43)    (Cond_isdiv(&&(>=@z(x16, x17), >@z(x17, 0@z)), x17, x16)=FALSE ⇒ FILTER(x17, cons(x16, zs[12]))≥IF(isdiv(x17, x16), x17, x16, zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

(44)    (Cond_isdiv2(&&(>@z(x19, x18), >@z(x18, 0@z)), x19, x18)=FALSE ⇒ FILTER(x19, cons(x18, zs[12]))≥IF(isdiv(x19, x18), x19, x18, zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

(45)    (Cond_isdiv1(>@z(x20, 0@z), x20)=FALSE ⇒ FILTER(x20, cons(0@z, zs[12]))≥IF(isdiv(x20, 0@z), x20, 0@z, zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (43) using rules (III), (VII) which results in the following new constraint:
(46)    (Cond_isdiv(&&(>=@z(x16, x17), >@z(x17, 0@z)), x17, x16)=FALSE ⇒ FILTER(x17, cons(x16, zs[12]))≥IF(isdiv(x17, x16), x17, x16, zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (44) using rules (III), (VII) which results in the following new constraint:
(47)    (Cond_isdiv2(&&(>@z(x19, x18), >@z(x18, 0@z)), x19, x18)=FALSE ⇒ FILTER(x19, cons(x18, zs[12]))≥IF(isdiv(x19, x18), x19, x18, zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (45) using rules (III), (VII) which results in the following new constraint:
(48)    (Cond_isdiv1(>@z(x20, 0@z), x20)=FALSE ⇒ FILTER(x20, cons(0@z, zs[12]))≥IF(isdiv(x20, 0@z), x20, 0@z, zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (46) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(49)    ((U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (47) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(50)    ((U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (48) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(51)    ((U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (51) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(52)    ((U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (50) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(53)    ((U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (49) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(54)    ((U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (52) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(55)    (0 ≥ 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (53) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(56)    (0 ≥ 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (54) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(57)    (0 ≥ 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (55) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(58)    (0 = 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0∧0 = 0)

We simplified constraint (56) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(59)    (0 = 0∧0 ≥ 0∧0 = 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 = 0)

We simplified constraint (57) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(60)    (0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))
We consider the chain IF(FALSE, x[5], y[5], zs[5]) → FILTER(x[5], zs[5]), FILTER(x[12], cons(y[12], zs[12])) → IF(isdiv(x[12], y[12]), x[12], y[12], zs[12]), IF(TRUE, x[7], y[7], zs[7]) → FILTER(x[7], zs[7]) which results in the following constraint:
(61)    (isdiv(x[12], y[12])=TRUE∧x[5]=x[12]∧zs[12]=zs[7]∧zs[5]=cons(y[12], zs[12])∧x[12]=x[7]∧y[12]=y[7] ⇒ FILTER(x[12], cons(y[12], zs[12]))≥IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (61) using rules (III), (IV) which results in the following new constraint:
(62)    (isdiv(x[12], y[12])=TRUE ⇒ FILTER(x[12], cons(y[12], zs[12]))≥IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (62) using rule (V) (with possible (I) afterwards) using induction on isdiv(x[12], y[12])=TRUE which results in the following new constraints:
(63)    (Cond_isdiv(&&(>=@z(x24, x25), >@z(x25, 0@z)), x25, x24)=TRUE ⇒ FILTER(x25, cons(x24, zs[12]))≥IF(isdiv(x25, x24), x25, x24, zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

(64)    (Cond_isdiv2(&&(>@z(x27, x26), >@z(x26, 0@z)), x27, x26)=TRUE ⇒ FILTER(x27, cons(x26, zs[12]))≥IF(isdiv(x27, x26), x27, x26, zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

(65)    (Cond_isdiv1(>@z(x28, 0@z), x28)=TRUE ⇒ FILTER(x28, cons(0@z, zs[12]))≥IF(isdiv(x28, 0@z), x28, 0@z, zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (63) using rules (III), (VII) which results in the following new constraint:
(66)    (Cond_isdiv(&&(>=@z(x24, x25), >@z(x25, 0@z)), x25, x24)=TRUE ⇒ FILTER(x25, cons(x24, zs[12]))≥IF(isdiv(x25, x24), x25, x24, zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (64) using rules (III), (VII) which results in the following new constraint:
(67)    (Cond_isdiv2(&&(>@z(x27, x26), >@z(x26, 0@z)), x27, x26)=TRUE ⇒ FILTER(x27, cons(x26, zs[12]))≥IF(isdiv(x27, x26), x27, x26, zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (65) using rules (III), (VII) which results in the following new constraint:
(68)    (Cond_isdiv1(>@z(x28, 0@z), x28)=TRUE ⇒ FILTER(x28, cons(0@z, zs[12]))≥IF(isdiv(x28, 0@z), x28, 0@z, zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (66) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(69)    ((U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (67) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(70)    ((U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (68) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(71)    ((U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (71) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(72)    ((U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (70) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(73)    ((U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (69) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(74)    ((U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (72) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(75)    ((U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (73) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(76)    (0 ≥ 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (74) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(77)    (0 ≥ 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (75) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(78)    (0 = 0∧0 ≥ 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 = 0)

We simplified constraint (76) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(79)    (0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (77) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(80)    (0 = 0∧0 = 0∧0 ≥ 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 = 0)
We consider the chain IF(TRUE, x[7], y[7], zs[7]) → FILTER(x[7], zs[7]), FILTER(x[12], cons(y[12], zs[12])) → IF(isdiv(x[12], y[12]), x[12], y[12], zs[12]), IF(TRUE, x[7], y[7], zs[7]) → FILTER(x[7], zs[7]) which results in the following constraint:
(81)    (isdiv(x[12], y[12])=TRUE∧zs[12]=zs[7]1∧zs[7]=cons(y[12], zs[12])∧x[7]=x[12]∧x[12]=x[7]1∧y[12]=y[7]1 ⇒ FILTER(x[12], cons(y[12], zs[12]))≥IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (81) using rules (III), (IV) which results in the following new constraint:
(82)    (isdiv(x[12], y[12])=TRUE ⇒ FILTER(x[12], cons(y[12], zs[12]))≥IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (82) using rule (V) (with possible (I) afterwards) using induction on isdiv(x[12], y[12])=TRUE which results in the following new constraints:
(83)    (Cond_isdiv(&&(>=@z(x32, x33), >@z(x33, 0@z)), x33, x32)=TRUE ⇒ FILTER(x33, cons(x32, zs[12]))≥IF(isdiv(x33, x32), x33, x32, zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

(84)    (Cond_isdiv2(&&(>@z(x35, x34), >@z(x34, 0@z)), x35, x34)=TRUE ⇒ FILTER(x35, cons(x34, zs[12]))≥IF(isdiv(x35, x34), x35, x34, zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

(85)    (Cond_isdiv1(>@z(x36, 0@z), x36)=TRUE ⇒ FILTER(x36, cons(0@z, zs[12]))≥IF(isdiv(x36, 0@z), x36, 0@z, zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (83) using rules (III), (VII) which results in the following new constraint:
(86)    (Cond_isdiv(&&(>=@z(x32, x33), >@z(x33, 0@z)), x33, x32)=TRUE ⇒ FILTER(x33, cons(x32, zs[12]))≥IF(isdiv(x33, x32), x33, x32, zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (84) using rules (III), (VII) which results in the following new constraint:
(87)    (Cond_isdiv2(&&(>@z(x35, x34), >@z(x34, 0@z)), x35, x34)=TRUE ⇒ FILTER(x35, cons(x34, zs[12]))≥IF(isdiv(x35, x34), x35, x34, zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (85) using rules (III), (VII) which results in the following new constraint:
(88)    (Cond_isdiv1(>@z(x36, 0@z), x36)=TRUE ⇒ FILTER(x36, cons(0@z, zs[12]))≥IF(isdiv(x36, 0@z), x36, 0@z, zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (86) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(89)    ((U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (87) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(90)    ((U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (88) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(91)    ((U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (91) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(92)    ((U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (90) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(93)    ((U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (89) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(94)    ((U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (92) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(95)    ((U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (93) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(96)    (0 ≥ 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (94) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(97)    (0 ≥ 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (95) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(98)    (0 = 0∧0 ≥ 0∧0 = 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (96) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(99)    (0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (97) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(100)    (0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))
We consider the chain IF(TRUE, x[7], y[7], zs[7]) → FILTER(x[7], zs[7]), FILTER(x[12], cons(y[12], zs[12])) → IF(isdiv(x[12], y[12]), x[12], y[12], zs[12]), IF(FALSE, x[5], y[5], zs[5]) → FILTER(x[5], zs[5]) which results in the following constraint:
(101)    (x[12]=x[5]∧zs[12]=zs[5]∧zs[7]=cons(y[12], zs[12])∧x[7]=x[12]∧y[12]=y[5]∧isdiv(x[12], y[12])=FALSE ⇒ FILTER(x[12], cons(y[12], zs[12]))≥IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (101) using rules (III), (IV) which results in the following new constraint:
(102)    (isdiv(x[12], y[12])=FALSE ⇒ FILTER(x[12], cons(y[12], zs[12]))≥IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (102) using rule (V) (with possible (I) afterwards) using induction on isdiv(x[12], y[12])=FALSE which results in the following new constraints:
(103)    (Cond_isdiv(&&(>=@z(x40, x41), >@z(x41, 0@z)), x41, x40)=FALSE ⇒ FILTER(x41, cons(x40, zs[12]))≥IF(isdiv(x41, x40), x41, x40, zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

(104)    (Cond_isdiv2(&&(>@z(x43, x42), >@z(x42, 0@z)), x43, x42)=FALSE ⇒ FILTER(x43, cons(x42, zs[12]))≥IF(isdiv(x43, x42), x43, x42, zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

(105)    (Cond_isdiv1(>@z(x44, 0@z), x44)=FALSE ⇒ FILTER(x44, cons(0@z, zs[12]))≥IF(isdiv(x44, 0@z), x44, 0@z, zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (103) using rules (III), (VII) which results in the following new constraint:
(106)    (Cond_isdiv(&&(>=@z(x40, x41), >@z(x41, 0@z)), x41, x40)=FALSE ⇒ FILTER(x41, cons(x40, zs[12]))≥IF(isdiv(x41, x40), x41, x40, zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (104) using rules (III), (VII) which results in the following new constraint:
(107)    (Cond_isdiv2(&&(>@z(x43, x42), >@z(x42, 0@z)), x43, x42)=FALSE ⇒ FILTER(x43, cons(x42, zs[12]))≥IF(isdiv(x43, x42), x43, x42, zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (105) using rules (III), (VII) which results in the following new constraint:
(108)    (Cond_isdiv1(>@z(x44, 0@z), x44)=FALSE ⇒ FILTER(x44, cons(0@z, zs[12]))≥IF(isdiv(x44, 0@z), x44, 0@z, zs[12])∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (106) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(109)    ((U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (107) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(110)    ((U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (108) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(111)    ((U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (111) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(112)    ((U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (110) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(113)    ((U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (109) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(114)    ((U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (112) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(115)    ((U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (113) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(116)    (0 ≥ 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (114) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(117)    (0 ≥ 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (115) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(118)    (0 ≥ 0∧0 = 0∧0 = 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

We simplified constraint (116) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(119)    (0 = 0∧0 = 0∧0 = 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (117) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(120)    (0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

To summarize, we get the following constraints P_≥ for the following pairs.

IF(TRUE, x[7], y[7], zs[7]) → FILTER(x[7], zs[7])
- (0 = 0∧0 ≥ 0∧0 = 0∧(U^Increasing(FILTER(x[7], zs[7])), ≥)∧0 = 0)
- (0 ≥ 0∧0 = 0∧(U^Increasing(FILTER(x[7], zs[7])), ≥)∧0 = 0∧0 = 0∧0 = 0)
- (0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧(U^Increasing(FILTER(x[7], zs[7])), ≥)∧0 = 0)
IF(FALSE, x[5], y[5], zs[5]) → FILTER(x[5], zs[5])
- (0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧(U^Increasing(FILTER(x[5], zs[5])), ≥))
- (0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(FILTER(x[5], zs[5])), ≥))
- (0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧(U^Increasing(FILTER(x[5], zs[5])), ≥))
FILTER(x[12], cons(y[12], zs[12])) → IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])
- (0 = 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0∧0 = 0)
- (0 = 0∧0 ≥ 0∧0 = 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 = 0)
- (0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))
- (0 = 0∧0 ≥ 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 = 0)
- (0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))
- (0 = 0∧0 = 0∧0 ≥ 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 = 0)
- (0 = 0∧0 ≥ 0∧0 = 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))
- (0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))
- (0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))
- (0 ≥ 0∧0 = 0∧0 = 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))
- (0 = 0∧0 = 0∧0 = 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥)∧0 ≥ 0)
- (0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧(U^Increasing(IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])), ≥))

POL(-@z(x₁, x₂)) = 0
POL(Cond_isdiv2(x₁, x₂, x₃)) = (2)x₁
POL(0@z) = 0
POL(TRUE) = 0
POL(&&(x₁, x₂)) = 0
POL(FALSE) = 0
POL(>@z(x₁, x₂)) = 0
POL(isdiv(x₁, x₂)) = 0
POL(cons(x₁, x₂)) = 1 + x₂
POL(>=@z(x₁, x₂)) = 0
POL(Cond_isdiv(x₁, x₂, x₃)) = (3)x₁
POL(Cond_isdiv1(x₁, x₂)) = 0
POL(undefined) = 0
POL(IF(x₁, x₂, x₃, x₄)) = -1 + x₄ + (2)x₂ + (-1)x₁
POL(FILTER(x₁, x₂)) = -1 + x₂ + (2)x₁

The following pairs are in P_>:

FILTER(x[12], cons(y[12], zs[12])) → IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])

The following pairs are in P_bound:

IF(TRUE, x[7], y[7], zs[7]) → FILTER(x[7], zs[7])
IF(FALSE, x[5], y[5], zs[5]) → FILTER(x[5], zs[5])
FILTER(x[12], cons(y[12], zs[12])) → IF(isdiv(x[12], y[12]), x[12], y[12], zs[12])

The following pairs are in P_≥:

IF(TRUE, x[7], y[7], zs[7]) → FILTER(x[7], zs[7])
IF(FALSE, x[5], y[5], zs[5]) → FILTER(x[5], zs[5])

At least the following rules have been oriented under context sensitive arithmetic replacement:

&&(FALSE, FALSE)¹ ↔ FALSE¹
Cond_isdiv(TRUE, x, y)¹ ↔ isdiv(x, -@z(y, x))¹
Cond_isdiv1(TRUE, x)¹ ↔ TRUE¹
-@z¹ ↔
isdiv(x, y)¹ ↔ Cond_isdiv(&&(>=@z(y, x), >@z(x, 0@z)), x, y)¹
isdiv(x, 0@z)¹ ↔ Cond_isdiv1(>@z(x, 0@z), x)¹
&&(TRUE, TRUE)¹ ↔ TRUE¹
&&(TRUE, FALSE)¹ ↔ FALSE¹
&&(FALSE, TRUE)¹ ↔ FALSE¹
Cond_isdiv2(TRUE, x, y)¹ ↔ FALSE¹
isdiv(x, y)¹ ↔ Cond_isdiv2(&&(>@z(x, y), >@z(y, 0@z)), x, y)¹

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDependencyGraphProof

            ↳ AND

              ↳ IDP

              ↳ IDP

              ↳ IDP

                ↳ UsableRulesProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ AND

                        ↳ IDP

                          ↳ IDependencyGraphProof

                        ↳ IDP

              ↳ IDP

I DP problem:
The following domains are used:

z

The ITRS R consists of the following rules:

Cond_isdiv2(TRUE, x, y) → FALSE
Cond_isdiv(TRUE, x, y) → isdiv(x, -@z(y, x))
isdiv(x, y) → Cond_isdiv(&&(>=@z(y, x), >@z(x, 0@z)), x, y)
isdiv(x, y) → Cond_isdiv2(&&(>@z(x, y), >@z(y, 0@z)), x, y)
isdiv(x, 0@z) → Cond_isdiv1(>@z(x, 0@z), x)
Cond_isdiv1(TRUE, x) → TRUE

The integer pair graph is empty.
The set Q consists of the following terms:

sieve(nil)
Cond_isdiv(TRUE, x0, x1)
Cond_isdiv1(TRUE, x0)
filter(x0, cons(x1, x2))
if(TRUE, x0, x1, x2)
Cond_nats1(TRUE, x0, x1)
nats(x0, x1)
isdiv(x0, x1)
if(FALSE, x0, x1, x2)
Cond_nats(TRUE, x0, x1)
filter(x0, nil)
sieve(cons(x0, x1))
Cond_isdiv2(TRUE, x0, x1)
primes(x0)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs.

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDependencyGraphProof

            ↳ AND

              ↳ IDP

              ↳ IDP

              ↳ IDP

                ↳ UsableRulesProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ AND

                        ↳ IDP

                        ↳ IDP

                          ↳ IDependencyGraphProof

              ↳ IDP

I DP problem:
The following domains are used:

z

The ITRS R consists of the following rules:

Cond_isdiv2(TRUE, x, y) → FALSE
Cond_isdiv(TRUE, x, y) → isdiv(x, -@z(y, x))
isdiv(x, y) → Cond_isdiv(&&(>=@z(y, x), >@z(x, 0@z)), x, y)
isdiv(x, y) → Cond_isdiv2(&&(>@z(x, y), >@z(y, 0@z)), x, y)
isdiv(x, 0@z) → Cond_isdiv1(>@z(x, 0@z), x)
Cond_isdiv1(TRUE, x) → TRUE

The integer pair graph contains the following rules and edges:

(7): IF(TRUE, x[7], y[7], zs[7]) → FILTER(x[7], zs[7])
(5): IF(FALSE, x[5], y[5], zs[5]) → FILTER(x[5], zs[5])

The set Q consists of the following terms:

sieve(nil)
Cond_isdiv(TRUE, x0, x1)
Cond_isdiv1(TRUE, x0)
filter(x0, cons(x1, x2))
if(TRUE, x0, x1, x2)
Cond_nats1(TRUE, x0, x1)
nats(x0, x1)
isdiv(x0, x1)
if(FALSE, x0, x1, x2)
Cond_nats(TRUE, x0, x1)
filter(x0, nil)
sieve(cons(x0, x1))
Cond_isdiv2(TRUE, x0, x1)
primes(x0)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDependencyGraphProof

            ↳ AND

              ↳ IDP

              ↳ IDP

              ↳ IDP

              ↳ IDP

                ↳ UsableRulesProof

I DP problem:
The following domains are used:

z

The ITRS R consists of the following rules:

Cond_isdiv1(TRUE, x) → TRUE
filter(x, cons(y, zs)) → if(isdiv(x, y), x, y, zs)
if(TRUE, x, y, zs) → filter(x, zs)
Cond_nats1(TRUE, x, y) → nil
nats(x, y) → Cond_nats1(>@z(x, y), x, y)
isdiv(x, 0@z) → Cond_isdiv1(>@z(x, 0@z), x)
nats(x, y) → Cond_nats(>=@z(y, x), x, y)
isdiv(x, y) → Cond_isdiv2(&&(>@z(x, y), >@z(y, 0@z)), x, y)
if(FALSE, x, y, zs) → cons(y, filter(x, zs))
Cond_nats(TRUE, x, y) → cons(x, nats(+@z(x, 1@z), y))
Cond_isdiv(TRUE, x, y) → isdiv(x, -@z(y, x))
isdiv(x, y) → Cond_isdiv(&&(>=@z(y, x), >@z(x, 0@z)), x, y)
filter(x, nil) → nil
Cond_isdiv2(TRUE, x, y) → FALSE

The integer pair graph contains the following rules and edges:

(11): SIEVE(cons(x[11], ys[11])) → SIEVE(filter(x[11], ys[11]))

(11) -> (11), if ((filter(x[11], ys[11]) →^* cons(x[11]a, ys[11]a)))

The set Q consists of the following terms:

sieve(nil)
Cond_isdiv(TRUE, x0, x1)
Cond_isdiv1(TRUE, x0)
filter(x0, cons(x1, x2))
if(TRUE, x0, x1, x2)
Cond_nats1(TRUE, x0, x1)
nats(x0, x1)
isdiv(x0, x1)
if(FALSE, x0, x1, x2)
Cond_nats(TRUE, x0, x1)
filter(x0, nil)
sieve(cons(x0, x1))
Cond_isdiv2(TRUE, x0, x1)
primes(x0)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDependencyGraphProof

            ↳ AND

              ↳ IDP

              ↳ IDP

              ↳ IDP

              ↳ IDP

                ↳ UsableRulesProof

                  ↳ IDP

                    ↳ IDPNonInfProof

I DP problem:
The following domains are used:

z

The ITRS R consists of the following rules:

isdiv(x, 0@z) → Cond_isdiv1(>@z(x, 0@z), x)
filter(x, cons(y, zs)) → if(isdiv(x, y), x, y, zs)
if(TRUE, x, y, zs) → filter(x, zs)
Cond_isdiv1(TRUE, x) → TRUE
Cond_isdiv2(TRUE, x, y) → FALSE
filter(x, nil) → nil
Cond_isdiv(TRUE, x, y) → isdiv(x, -@z(y, x))
isdiv(x, y) → Cond_isdiv(&&(>=@z(y, x), >@z(x, 0@z)), x, y)
if(FALSE, x, y, zs) → cons(y, filter(x, zs))
isdiv(x, y) → Cond_isdiv2(&&(>@z(x, y), >@z(y, 0@z)), x, y)

The integer pair graph contains the following rules and edges:

(11): SIEVE(cons(x[11], ys[11])) → SIEVE(filter(x[11], ys[11]))

(11) -> (11), if ((filter(x[11], ys[11]) →^* cons(x[11]a, ys[11]a)))

The set Q consists of the following terms:

sieve(nil)
Cond_isdiv(TRUE, x0, x1)
Cond_isdiv1(TRUE, x0)
filter(x0, cons(x1, x2))
if(TRUE, x0, x1, x2)
Cond_nats1(TRUE, x0, x1)
nats(x0, x1)
isdiv(x0, x1)
if(FALSE, x0, x1, x2)
Cond_nats(TRUE, x0, x1)
filter(x0, nil)
sieve(cons(x0, x1))
Cond_isdiv2(TRUE, x0, x1)
primes(x0)

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.

For Pair SIEVE(cons(x[11], ys[11])) → SIEVE(filter(x[11], ys[11])) the following chains were created:

We consider the chain SIEVE(cons(x[11], ys[11])) → SIEVE(filter(x[11], ys[11])), SIEVE(cons(x[11], ys[11])) → SIEVE(filter(x[11], ys[11])), SIEVE(cons(x[11], ys[11])) → SIEVE(filter(x[11], ys[11])) which results in the following constraint:
(1)    (filter(x[11], ys[11])=cons(x[11]1, ys[11]1)∧filter(x[11]1, ys[11]1)=cons(x[11]2, ys[11]2) ⇒ SIEVE(cons(x[11]1, ys[11]1))≥SIEVE(filter(x[11]1, ys[11]1))∧(U^Increasing(SIEVE(filter(x[11]1, ys[11]1))), ≥))

We simplified constraint (1) using rule (V) (with possible (I) afterwards) using induction on filter(x[11], ys[11])=cons(x[11]1, ys[11]1) which results in the following new constraint:
(2)    (if(isdiv(x2, x1), x2, x1, x0)=cons(x[11]1, ys[11]1)∧filter(x[11]1, ys[11]1)=cons(x[11]2, ys[11]2) ⇒ SIEVE(cons(x[11]1, ys[11]1))≥SIEVE(filter(x[11]1, ys[11]1))∧(U^Increasing(SIEVE(filter(x[11]1, ys[11]1))), ≥))

We simplified constraint (2) using rule (VII) which results in the following new constraint:
(3)    (isdiv(x2, x1)=x4∧if(x4, x2, x1, x0)=cons(x[11]1, ys[11]1)∧filter(x[11]1, ys[11]1)=cons(x[11]2, ys[11]2) ⇒ SIEVE(cons(x[11]1, ys[11]1))≥SIEVE(filter(x[11]1, ys[11]1))∧(U^Increasing(SIEVE(filter(x[11]1, ys[11]1))), ≥))

We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on filter(x[11]1, ys[11]1)=cons(x[11]2, ys[11]2) which results in the following new constraint:
(4)    (if(isdiv(x7, x6), x7, x6, x5)=cons(x[11]2, ys[11]2)∧isdiv(x2, x1)=x4∧if(x4, x2, x1, x0)=cons(x7, cons(x6, x5)) ⇒ SIEVE(cons(x7, cons(x6, x5)))≥SIEVE(filter(x7, cons(x6, x5)))∧(U^Increasing(SIEVE(filter(x[11]1, ys[11]1))), ≥))

We simplified constraint (4) using rule (VII) which results in the following new constraint:
(5)    (isdiv(x7, x6)=x9∧if(x9, x7, x6, x5)=cons(x[11]2, ys[11]2)∧isdiv(x2, x1)=x4∧if(x4, x2, x1, x0)=cons(x7, cons(x6, x5)) ⇒ SIEVE(cons(x7, cons(x6, x5)))≥SIEVE(filter(x7, cons(x6, x5)))∧(U^Increasing(SIEVE(filter(x[11]1, ys[11]1))), ≥))

We simplified constraint (5) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(6)    ((U^Increasing(SIEVE(filter(x[11]1, ys[11]1))), ≥)∧0 ≥ 0)

We simplified constraint (6) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(7)    ((U^Increasing(SIEVE(filter(x[11]1, ys[11]1))), ≥)∧0 ≥ 0)

We simplified constraint (7) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(8)    ((U^Increasing(SIEVE(filter(x[11]1, ys[11]1))), ≥)∧0 ≥ 0)

We simplified constraint (8) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(9)    ((U^Increasing(SIEVE(filter(x[11]1, ys[11]1))), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0)

To summarize, we get the following constraints P_≥ for the following pairs.

SIEVE(cons(x[11], ys[11])) → SIEVE(filter(x[11], ys[11]))
- ((U^Increasing(SIEVE(filter(x[11]1, ys[11]1))), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0)

POL(-@z(x₁, x₂)) = 0
POL(SIEVE(x₁)) = -1 + x₁
POL(if(x₁, x₂, x₃, x₄)) = 1 + x₄ + (2)x₁
POL(Cond_isdiv2(x₁, x₂, x₃)) = 0
POL(0@z) = 0
POL(TRUE) = 0
POL(&&(x₁, x₂)) = 0
POL(FALSE) = 0
POL(isdiv(x₁, x₂)) = 0
POL(>@z(x₁, x₂)) = 0
POL(cons(x₁, x₂)) = 1 + x₂
POL(>=@z(x₁, x₂)) = 0
POL(Cond_isdiv(x₁, x₂, x₃)) = (3)x₁
POL(filter(x₁, x₂)) = x₂
POL(Cond_isdiv1(x₁, x₂)) = 0
POL(nil) = 0
POL(undefined) = 0

The following pairs are in P_>:

SIEVE(cons(x[11], ys[11])) → SIEVE(filter(x[11], ys[11]))

The following pairs are in P_bound:

SIEVE(cons(x[11], ys[11])) → SIEVE(filter(x[11], ys[11]))

The following pairs are in P_≥:
none

At least the following rules have been oriented under context sensitive arithmetic replacement:

Cond_isdiv(TRUE, x, y)¹ ↔ isdiv(x, -@z(y, x))¹
Cond_isdiv1(TRUE, x)¹ ↔ TRUE¹
filter(x, cons(y, zs))¹ ↔ if(isdiv(x, y), x, y, zs)¹
if(TRUE, x, y, zs)¹ → filter(x, zs)¹
isdiv(x, 0@z)¹ ↔ Cond_isdiv1(>@z(x, 0@z), x)¹
&&(TRUE, FALSE)¹ ↔ FALSE¹
&&(FALSE, TRUE)¹ ↔ FALSE¹
isdiv(x, y)¹ ↔ Cond_isdiv2(&&(>@z(x, y), >@z(y, 0@z)), x, y)¹
&&(FALSE, FALSE)¹ ↔ FALSE¹
if(FALSE, x, y, zs)¹ ↔ cons(y, filter(x, zs))¹
-@z¹ ↔
isdiv(x, y)¹ ↔ Cond_isdiv(&&(>=@z(y, x), >@z(x, 0@z)), x, y)¹
filter(x, nil)¹ ↔ nil¹
&&(TRUE, TRUE)¹ ↔ TRUE¹
Cond_isdiv2(TRUE, x, y)¹ ↔ FALSE¹

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDependencyGraphProof

            ↳ AND

              ↳ IDP

              ↳ IDP

              ↳ IDP

              ↳ IDP

                ↳ UsableRulesProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ IDP

                        ↳ IDependencyGraphProof

I DP problem:
The following domains are used:

z

The ITRS R consists of the following rules:

isdiv(x, 0@z) → Cond_isdiv1(>@z(x, 0@z), x)
filter(x, cons(y, zs)) → if(isdiv(x, y), x, y, zs)
if(TRUE, x, y, zs) → filter(x, zs)
Cond_isdiv1(TRUE, x) → TRUE
Cond_isdiv2(TRUE, x, y) → FALSE
filter(x, nil) → nil
Cond_isdiv(TRUE, x, y) → isdiv(x, -@z(y, x))
isdiv(x, y) → Cond_isdiv(&&(>=@z(y, x), >@z(x, 0@z)), x, y)
if(FALSE, x, y, zs) → cons(y, filter(x, zs))
isdiv(x, y) → Cond_isdiv2(&&(>@z(x, y), >@z(y, 0@z)), x, y)

The integer pair graph is empty.
The set Q consists of the following terms:

sieve(nil)
Cond_isdiv(TRUE, x0, x1)
Cond_isdiv1(TRUE, x0)
filter(x0, cons(x1, x2))
if(TRUE, x0, x1, x2)
Cond_nats1(TRUE, x0, x1)
nats(x0, x1)
isdiv(x0, x1)
if(FALSE, x0, x1, x2)
Cond_nats(TRUE, x0, x1)
filter(x0, nil)
sieve(cons(x0, x1))
Cond_isdiv2(TRUE, x0, x1)
primes(x0)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs.