Termination proof of /tmp/tmpLAMAST/poly3.itrs

Termination of the given ITRSProblem could not be shown:

↳ ITRS

  ↳ ITRStoIDPProof

ITRS problem:
The following domains are used:

z

The TRS R consists of the following rules:

eval(x, y, z) → Cond_eval1(&&(>=@z(x, 0@z), >=@z(*@z(*@z(z, z), z), y)), x, y, z)
Cond_eval1(TRUE, x, y, z) → eval(x, -@z(y, 1@z), +@z(z, y))
Cond_eval(TRUE, x, y, z) → eval(-@z(x, 1@z), -@z(y, 1@z), z)
eval(x, y, z) → Cond_eval(&&(>=@z(x, 0@z), >=@z(*@z(*@z(z, z), z), y)), x, y, z)

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval1(TRUE, x0, x1, x2)
Cond_eval(TRUE, x0, x1, x2)

Added dependency pairs

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

I DP problem:
The following domains are used:

z

The ITRS R consists of the following rules:

eval(x, y, z) → Cond_eval1(&&(>=@z(x, 0@z), >=@z(*@z(*@z(z, z), z), y)), x, y, z)
Cond_eval1(TRUE, x, y, z) → eval(x, -@z(y, 1@z), +@z(z, y))
Cond_eval(TRUE, x, y, z) → eval(-@z(x, 1@z), -@z(y, 1@z), z)
eval(x, y, z) → Cond_eval(&&(>=@z(x, 0@z), >=@z(*@z(*@z(z, z), z), y)), x, y, z)

The integer pair graph contains the following rules and edges:

(0): EVAL(x[0], y[0], z[0]) → COND_EVAL1(&&(>=@z(x[0], 0@z), >=@z(*@z(*@z(z[0], z[0]), z[0]), y[0])), x[0], y[0], z[0])
(1): COND_EVAL1(TRUE, x[1], y[1], z[1]) → EVAL(x[1], -@z(y[1], 1@z), +@z(z[1], y[1]))
(2): EVAL(x[2], y[2], z[2]) → COND_EVAL(&&(>=@z(x[2], 0@z), >=@z(*@z(*@z(z[2], z[2]), z[2]), y[2])), x[2], y[2], z[2])
(3): COND_EVAL(TRUE, x[3], y[3], z[3]) → EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])

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

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

(1) -> (2), if ((-@z(y[1], 1@z) →^* y[2])∧(+@z(z[1], y[1]) →^* z[2])∧(x[1] →^* x[2]))

(2) -> (3), if ((z[2] →^* z[3])∧(x[2] →^* x[3])∧(y[2] →^* y[3])∧(&&(>=@z(x[2], 0@z), >=@z(*@z(*@z(z[2], z[2]), z[2]), y[2])) →^* TRUE))

(3) -> (0), if ((-@z(y[3], 1@z) →^* y[0])∧(z[3] →^* z[0])∧(-@z(x[3], 1@z) →^* x[0]))

(3) -> (2), if ((-@z(y[3], 1@z) →^* y[2])∧(z[3] →^* z[2])∧(-@z(x[3], 1@z) →^* x[2]))

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval1(TRUE, x0, x1, x2)
Cond_eval(TRUE, x0, x1, x2)

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

          ↳ IDPNonInfProof

I DP problem:
The following domains are used:

z

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

(0): EVAL(x[0], y[0], z[0]) → COND_EVAL1(&&(>=@z(x[0], 0@z), >=@z(*@z(*@z(z[0], z[0]), z[0]), y[0])), x[0], y[0], z[0])
(1): COND_EVAL1(TRUE, x[1], y[1], z[1]) → EVAL(x[1], -@z(y[1], 1@z), +@z(z[1], y[1]))
(2): EVAL(x[2], y[2], z[2]) → COND_EVAL(&&(>=@z(x[2], 0@z), >=@z(*@z(*@z(z[2], z[2]), z[2]), y[2])), x[2], y[2], z[2])
(3): COND_EVAL(TRUE, x[3], y[3], z[3]) → EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])

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

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

(1) -> (2), if ((-@z(y[1], 1@z) →^* y[2])∧(+@z(z[1], y[1]) →^* z[2])∧(x[1] →^* x[2]))

(2) -> (3), if ((z[2] →^* z[3])∧(x[2] →^* x[3])∧(y[2] →^* y[3])∧(&&(>=@z(x[2], 0@z), >=@z(*@z(*@z(z[2], z[2]), z[2]), y[2])) →^* TRUE))

(3) -> (0), if ((-@z(y[3], 1@z) →^* y[0])∧(z[3] →^* z[0])∧(-@z(x[3], 1@z) →^* x[0]))

(3) -> (2), if ((-@z(y[3], 1@z) →^* y[2])∧(z[3] →^* z[2])∧(-@z(x[3], 1@z) →^* x[2]))

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval1(TRUE, x0, x1, x2)
Cond_eval(TRUE, x0, x1, x2)

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 EVAL(x, y, z) → COND_EVAL1(&&(>=@z(x, 0@z), >=@z(*@z(*@z(z, z), z), y)), x, y, z) the following chains were created:

We consider the chain EVAL(x[0], y[0], z[0]) → COND_EVAL1(&&(>=@z(x[0], 0@z), >=@z(*@z(*@z(z[0], z[0]), z[0]), y[0])), x[0], y[0], z[0]) which results in the following constraint:
(1)    (EVAL(x[0], y[0], z[0])≥NonInfC∧EVAL(x[0], y[0], z[0])≥COND_EVAL1(&&(>=@z(x[0], 0@z), >=@z(*@z(*@z(z[0], z[0]), z[0]), y[0])), x[0], y[0], z[0])∧(U^Increasing(COND_EVAL1(&&(>=@z(x[0], 0@z), >=@z(*@z(*@z(z[0], z[0]), z[0]), y[0])), x[0], y[0], z[0])), ≥))

We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2)    ((U^Increasing(COND_EVAL1(&&(>=@z(x[0], 0@z), >=@z(*@z(*@z(z[0], z[0]), z[0]), y[0])), x[0], y[0], z[0])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3)    ((U^Increasing(COND_EVAL1(&&(>=@z(x[0], 0@z), >=@z(*@z(*@z(z[0], z[0]), z[0]), y[0])), x[0], y[0], z[0])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(4)    ((U^Increasing(COND_EVAL1(&&(>=@z(x[0], 0@z), >=@z(*@z(*@z(z[0], z[0]), z[0]), y[0])), x[0], y[0], z[0])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (4) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(5)    (0 = 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL1(&&(>=@z(x[0], 0@z), >=@z(*@z(*@z(z[0], z[0]), z[0]), y[0])), x[0], y[0], z[0])), ≥)∧0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0)

For Pair COND_EVAL1(TRUE, x, y, z) → EVAL(x, -@z(y, 1@z), +@z(z, y)) the following chains were created:

We consider the chain EVAL(x[0], y[0], z[0]) → COND_EVAL1(&&(>=@z(x[0], 0@z), >=@z(*@z(*@z(z[0], z[0]), z[0]), y[0])), x[0], y[0], z[0]), COND_EVAL1(TRUE, x[1], y[1], z[1]) → EVAL(x[1], -@z(y[1], 1@z), +@z(z[1], y[1])), EVAL(x[0], y[0], z[0]) → COND_EVAL1(&&(>=@z(x[0], 0@z), >=@z(*@z(*@z(z[0], z[0]), z[0]), y[0])), x[0], y[0], z[0]) which results in the following constraint:
(6)    (+@z(z[1], y[1])=z[0]1∧x[1]=x[0]1∧y[0]=y[1]∧x[0]=x[1]∧z[0]=z[1]∧-@z(y[1], 1@z)=y[0]1∧&&(>=@z(x[0], 0@z), >=@z(*@z(*@z(z[0], z[0]), z[0]), y[0]))=TRUE ⇒ COND_EVAL1(TRUE, x[1], y[1], z[1])≥NonInfC∧COND_EVAL1(TRUE, x[1], y[1], z[1])≥EVAL(x[1], -@z(y[1], 1@z), +@z(z[1], y[1]))∧(U^Increasing(EVAL(x[1], -@z(y[1], 1@z), +@z(z[1], y[1]))), ≥))

We simplified constraint (6) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(7)    (>=@z(x[0], 0@z)=TRUE∧>=@z(*@z(*@z(z[0], z[0]), z[0]), y[0])=TRUE ⇒ COND_EVAL1(TRUE, x[0], y[0], z[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0], z[0])≥EVAL(x[0], -@z(y[0], 1@z), +@z(z[0], y[0]))∧(U^Increasing(EVAL(x[1], -@z(y[1], 1@z), +@z(z[1], y[1]))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    (x[0] ≥ 0∧z[0]³ + (-1)y[0] ≥ 0 ⇒ (U^Increasing(EVAL(x[1], -@z(y[1], 1@z), +@z(z[1], y[1]))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    (x[0] ≥ 0∧z[0]³ + (-1)y[0] ≥ 0 ⇒ (U^Increasing(EVAL(x[1], -@z(y[1], 1@z), +@z(z[1], y[1]))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    (z[0]³ + (-1)y[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[1], -@z(y[1], 1@z), +@z(z[1], y[1]))), ≥)∧0 ≥ 0)

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

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

(13)    (y[0] ≥ 0∧x[0] ≥ 0∧z[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[1], -@z(y[1], 1@z), +@z(z[1], y[1]))), ≥)∧0 ≥ 0)
We consider the chain EVAL(x[0], y[0], z[0]) → COND_EVAL1(&&(>=@z(x[0], 0@z), >=@z(*@z(*@z(z[0], z[0]), z[0]), y[0])), x[0], y[0], z[0]), COND_EVAL1(TRUE, x[1], y[1], z[1]) → EVAL(x[1], -@z(y[1], 1@z), +@z(z[1], y[1])), EVAL(x[2], y[2], z[2]) → COND_EVAL(&&(>=@z(x[2], 0@z), >=@z(*@z(*@z(z[2], z[2]), z[2]), y[2])), x[2], y[2], z[2]) which results in the following constraint:
(14)    (x[1]=x[2]∧-@z(y[1], 1@z)=y[2]∧y[0]=y[1]∧+@z(z[1], y[1])=z[2]∧x[0]=x[1]∧z[0]=z[1]∧&&(>=@z(x[0], 0@z), >=@z(*@z(*@z(z[0], z[0]), z[0]), y[0]))=TRUE ⇒ COND_EVAL1(TRUE, x[1], y[1], z[1])≥NonInfC∧COND_EVAL1(TRUE, x[1], y[1], z[1])≥EVAL(x[1], -@z(y[1], 1@z), +@z(z[1], y[1]))∧(U^Increasing(EVAL(x[1], -@z(y[1], 1@z), +@z(z[1], y[1]))), ≥))

We simplified constraint (14) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(15)    (>=@z(x[0], 0@z)=TRUE∧>=@z(*@z(*@z(z[0], z[0]), z[0]), y[0])=TRUE ⇒ COND_EVAL1(TRUE, x[0], y[0], z[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0], z[0])≥EVAL(x[0], -@z(y[0], 1@z), +@z(z[0], y[0]))∧(U^Increasing(EVAL(x[1], -@z(y[1], 1@z), +@z(z[1], y[1]))), ≥))

We simplified constraint (15) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(16)    (x[0] ≥ 0∧z[0]³ + (-1)y[0] ≥ 0 ⇒ (U^Increasing(EVAL(x[1], -@z(y[1], 1@z), +@z(z[1], y[1]))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (16) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(17)    (x[0] ≥ 0∧z[0]³ + (-1)y[0] ≥ 0 ⇒ (U^Increasing(EVAL(x[1], -@z(y[1], 1@z), +@z(z[1], y[1]))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (17) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(18)    (x[0] ≥ 0∧z[0]³ + (-1)y[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[1], -@z(y[1], 1@z), +@z(z[1], y[1]))), ≥)∧0 ≥ 0)

We simplified constraint (18) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(19)    (x[0] ≥ 0∧y[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[1], -@z(y[1], 1@z), +@z(z[1], y[1]))), ≥)∧0 ≥ 0)

We simplified constraint (19) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(20)    (x[0] ≥ 0∧y[0] ≥ 0∧z[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[1], -@z(y[1], 1@z), +@z(z[1], y[1]))), ≥)∧0 ≥ 0)

(21)    (x[0] ≥ 0∧y[0] ≥ 0∧z[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[1], -@z(y[1], 1@z), +@z(z[1], y[1]))), ≥)∧0 ≥ 0)

For Pair EVAL(x, y, z) → COND_EVAL(&&(>=@z(x, 0@z), >=@z(*@z(*@z(z, z), z), y)), x, y, z) the following chains were created:

We consider the chain EVAL(x[2], y[2], z[2]) → COND_EVAL(&&(>=@z(x[2], 0@z), >=@z(*@z(*@z(z[2], z[2]), z[2]), y[2])), x[2], y[2], z[2]) which results in the following constraint:
(22)    (EVAL(x[2], y[2], z[2])≥NonInfC∧EVAL(x[2], y[2], z[2])≥COND_EVAL(&&(>=@z(x[2], 0@z), >=@z(*@z(*@z(z[2], z[2]), z[2]), y[2])), x[2], y[2], z[2])∧(U^Increasing(COND_EVAL(&&(>=@z(x[2], 0@z), >=@z(*@z(*@z(z[2], z[2]), z[2]), y[2])), x[2], y[2], z[2])), ≥))

We simplified constraint (22) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(23)    ((U^Increasing(COND_EVAL(&&(>=@z(x[2], 0@z), >=@z(*@z(*@z(z[2], z[2]), z[2]), y[2])), x[2], y[2], z[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (23) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(24)    ((U^Increasing(COND_EVAL(&&(>=@z(x[2], 0@z), >=@z(*@z(*@z(z[2], z[2]), z[2]), y[2])), x[2], y[2], z[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (24) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(25)    (0 ≥ 0∧0 ≥ 0∧(U^Increasing(COND_EVAL(&&(>=@z(x[2], 0@z), >=@z(*@z(*@z(z[2], z[2]), z[2]), y[2])), x[2], y[2], z[2])), ≥))

We simplified constraint (25) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(26)    (0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL(&&(>=@z(x[2], 0@z), >=@z(*@z(*@z(z[2], z[2]), z[2]), y[2])), x[2], y[2], z[2])), ≥)∧0 ≥ 0∧0 = 0)

For Pair COND_EVAL(TRUE, x, y, z) → EVAL(-@z(x, 1@z), -@z(y, 1@z), z) the following chains were created:

We consider the chain EVAL(x[2], y[2], z[2]) → COND_EVAL(&&(>=@z(x[2], 0@z), >=@z(*@z(*@z(z[2], z[2]), z[2]), y[2])), x[2], y[2], z[2]), COND_EVAL(TRUE, x[3], y[3], z[3]) → EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3]), EVAL(x[0], y[0], z[0]) → COND_EVAL1(&&(>=@z(x[0], 0@z), >=@z(*@z(*@z(z[0], z[0]), z[0]), y[0])), x[0], y[0], z[0]) which results in the following constraint:
(27)    (x[2]=x[3]∧y[2]=y[3]∧&&(>=@z(x[2], 0@z), >=@z(*@z(*@z(z[2], z[2]), z[2]), y[2]))=TRUE∧-@z(y[3], 1@z)=y[0]∧z[2]=z[3]∧-@z(x[3], 1@z)=x[0]∧z[3]=z[0] ⇒ COND_EVAL(TRUE, x[3], y[3], z[3])≥NonInfC∧COND_EVAL(TRUE, x[3], y[3], z[3])≥EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])∧(U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥))

We simplified constraint (27) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(28)    (>=@z(x[2], 0@z)=TRUE∧>=@z(*@z(*@z(z[2], z[2]), z[2]), y[2])=TRUE ⇒ COND_EVAL(TRUE, x[2], y[2], z[2])≥NonInfC∧COND_EVAL(TRUE, x[2], y[2], z[2])≥EVAL(-@z(x[2], 1@z), -@z(y[2], 1@z), z[2])∧(U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥))

We simplified constraint (28) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(29)    (x[2] ≥ 0∧z[2]³ + (-1)y[2] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧-1 + (-1)Bound + x[2] ≥ 0∧0 ≥ 0)

We simplified constraint (29) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(30)    (x[2] ≥ 0∧z[2]³ + (-1)y[2] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧-1 + (-1)Bound + x[2] ≥ 0∧0 ≥ 0)

We simplified constraint (30) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(31)    (x[2] ≥ 0∧z[2]³ + (-1)y[2] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧0 ≥ 0∧-1 + (-1)Bound + x[2] ≥ 0)

We simplified constraint (31) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(32)    (x[2] ≥ 0∧y[2]³ + (3)y[2]²·z[2] + (3)y[2]·z[2]² + z[2]³ + (-1)y[2] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧0 ≥ 0∧-1 + (-1)Bound + x[2] ≥ 0)

We simplified constraint (32) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(33)    (x[2] ≥ 0∧y[2]³ + (3)y[2]²·z[2] + (3)y[2]·z[2]² + z[2]³ + (-1)y[2] ≥ 0∧z[2] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧0 ≥ 0∧-1 + (-1)Bound + x[2] ≥ 0)

(34)    (x[2] ≥ 0∧y[2]³ + (-3)y[2]²·z[2] + (3)y[2]·z[2]² + (-1)z[2]³ + (-1)y[2] ≥ 0∧z[2] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧0 ≥ 0∧-1 + (-1)Bound + x[2] ≥ 0)

We simplified constraint (33) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(35)    (x[2] ≥ 0∧y[2]³ + (3)y[2]²·z[2] + (3)y[2]·z[2]² + z[2]³ + (-1)y[2] ≥ 0∧z[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧0 ≥ 0∧-1 + (-1)Bound + x[2] ≥ 0)

(36)    (x[2] ≥ 0∧(-1)y[2]³ + (3)y[2]²·z[2] + (-3)y[2]·z[2]² + z[2]³ + y[2] ≥ 0∧z[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧0 ≥ 0∧-1 + (-1)Bound + x[2] ≥ 0)

We simplified constraint (34) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(37)    (x[2] ≥ 0∧(-1)y[2]³ + (-3)y[2]²·z[2] + (-3)y[2]·z[2]² + (-1)z[2]³ + y[2] ≥ 0∧z[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧0 ≥ 0∧-1 + (-1)Bound + x[2] ≥ 0)

(38)    (x[2] ≥ 0∧y[2]³ + (-3)y[2]²·z[2] + (3)y[2]·z[2]² + (-1)z[2]³ + (-1)y[2] ≥ 0∧z[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧0 ≥ 0∧-1 + (-1)Bound + x[2] ≥ 0)
We consider the chain EVAL(x[2], y[2], z[2]) → COND_EVAL(&&(>=@z(x[2], 0@z), >=@z(*@z(*@z(z[2], z[2]), z[2]), y[2])), x[2], y[2], z[2]), COND_EVAL(TRUE, x[3], y[3], z[3]) → EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3]), EVAL(x[2], y[2], z[2]) → COND_EVAL(&&(>=@z(x[2], 0@z), >=@z(*@z(*@z(z[2], z[2]), z[2]), y[2])), x[2], y[2], z[2]) which results in the following constraint:
(39)    (x[2]=x[3]∧y[2]=y[3]∧&&(>=@z(x[2], 0@z), >=@z(*@z(*@z(z[2], z[2]), z[2]), y[2]))=TRUE∧z[2]=z[3]∧z[3]=z[2]1∧-@z(x[3], 1@z)=x[2]1∧-@z(y[3], 1@z)=y[2]1 ⇒ COND_EVAL(TRUE, x[3], y[3], z[3])≥NonInfC∧COND_EVAL(TRUE, x[3], y[3], z[3])≥EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])∧(U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥))

We simplified constraint (39) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(40)    (>=@z(x[2], 0@z)=TRUE∧>=@z(*@z(*@z(z[2], z[2]), z[2]), y[2])=TRUE ⇒ COND_EVAL(TRUE, x[2], y[2], z[2])≥NonInfC∧COND_EVAL(TRUE, x[2], y[2], z[2])≥EVAL(-@z(x[2], 1@z), -@z(y[2], 1@z), z[2])∧(U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥))

We simplified constraint (40) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(41)    (x[2] ≥ 0∧z[2]³ + (-1)y[2] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧-1 + (-1)Bound + x[2] ≥ 0∧0 ≥ 0)

We simplified constraint (41) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(42)    (x[2] ≥ 0∧z[2]³ + (-1)y[2] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧-1 + (-1)Bound + x[2] ≥ 0∧0 ≥ 0)

We simplified constraint (42) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(43)    (x[2] ≥ 0∧z[2]³ + (-1)y[2] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧-1 + (-1)Bound + x[2] ≥ 0)

We simplified constraint (43) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(44)    (x[2] ≥ 0∧y[2]³ + (3)y[2]²·z[2] + (3)y[2]·z[2]² + z[2]³ + (-1)y[2] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧-1 + (-1)Bound + x[2] ≥ 0)

We simplified constraint (44) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(45)    (x[2] ≥ 0∧y[2]³ + (3)y[2]²·z[2] + (3)y[2]·z[2]² + z[2]³ + (-1)y[2] ≥ 0∧z[2] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧-1 + (-1)Bound + x[2] ≥ 0)

(46)    (x[2] ≥ 0∧y[2]³ + (-3)y[2]²·z[2] + (3)y[2]·z[2]² + (-1)z[2]³ + (-1)y[2] ≥ 0∧z[2] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧-1 + (-1)Bound + x[2] ≥ 0)

We simplified constraint (45) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(47)    (x[2] ≥ 0∧y[2]³ + (3)y[2]²·z[2] + (3)y[2]·z[2]² + z[2]³ + (-1)y[2] ≥ 0∧z[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧-1 + (-1)Bound + x[2] ≥ 0)

(48)    (x[2] ≥ 0∧(-1)y[2]³ + (3)y[2]²·z[2] + (-3)y[2]·z[2]² + z[2]³ + y[2] ≥ 0∧z[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧-1 + (-1)Bound + x[2] ≥ 0)

We simplified constraint (46) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(49)    (x[2] ≥ 0∧(-1)y[2]³ + (-3)y[2]²·z[2] + (-3)y[2]·z[2]² + (-1)z[2]³ + y[2] ≥ 0∧z[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧-1 + (-1)Bound + x[2] ≥ 0)

(50)    (x[2] ≥ 0∧y[2]³ + (-3)y[2]²·z[2] + (3)y[2]·z[2]² + (-1)z[2]³ + (-1)y[2] ≥ 0∧z[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧-1 + (-1)Bound + x[2] ≥ 0)

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

EVAL(x, y, z) → COND_EVAL1(&&(>=@z(x, 0@z), >=@z(*@z(*@z(z, z), z), y)), x, y, z)
- (0 = 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL1(&&(>=@z(x[0], 0@z), >=@z(*@z(*@z(z[0], z[0]), z[0]), y[0])), x[0], y[0], z[0])), ≥)∧0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0)
COND_EVAL1(TRUE, x, y, z) → EVAL(x, -@z(y, 1@z), +@z(z, y))
- (y[0] ≥ 0∧x[0] ≥ 0∧z[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[1], -@z(y[1], 1@z), +@z(z[1], y[1]))), ≥)∧0 ≥ 0)
- (y[0] ≥ 0∧x[0] ≥ 0∧z[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[1], -@z(y[1], 1@z), +@z(z[1], y[1]))), ≥)∧0 ≥ 0)
- (x[0] ≥ 0∧y[0] ≥ 0∧z[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[1], -@z(y[1], 1@z), +@z(z[1], y[1]))), ≥)∧0 ≥ 0)
- (x[0] ≥ 0∧y[0] ≥ 0∧z[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[1], -@z(y[1], 1@z), +@z(z[1], y[1]))), ≥)∧0 ≥ 0)
EVAL(x, y, z) → COND_EVAL(&&(>=@z(x, 0@z), >=@z(*@z(*@z(z, z), z), y)), x, y, z)
- (0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL(&&(>=@z(x[2], 0@z), >=@z(*@z(*@z(z[2], z[2]), z[2]), y[2])), x[2], y[2], z[2])), ≥)∧0 ≥ 0∧0 = 0)
COND_EVAL(TRUE, x, y, z) → EVAL(-@z(x, 1@z), -@z(y, 1@z), z)
- (x[2] ≥ 0∧y[2]³ + (3)y[2]²·z[2] + (3)y[2]·z[2]² + z[2]³ + (-1)y[2] ≥ 0∧z[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧0 ≥ 0∧-1 + (-1)Bound + x[2] ≥ 0)
- (x[2] ≥ 0∧(-1)y[2]³ + (3)y[2]²·z[2] + (-3)y[2]·z[2]² + z[2]³ + y[2] ≥ 0∧z[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧0 ≥ 0∧-1 + (-1)Bound + x[2] ≥ 0)
- (x[2] ≥ 0∧(-1)y[2]³ + (-3)y[2]²·z[2] + (-3)y[2]·z[2]² + (-1)z[2]³ + y[2] ≥ 0∧z[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧0 ≥ 0∧-1 + (-1)Bound + x[2] ≥ 0)
- (x[2] ≥ 0∧y[2]³ + (-3)y[2]²·z[2] + (3)y[2]·z[2]² + (-1)z[2]³ + (-1)y[2] ≥ 0∧z[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧0 ≥ 0∧-1 + (-1)Bound + x[2] ≥ 0)
- (x[2] ≥ 0∧y[2]³ + (3)y[2]²·z[2] + (3)y[2]·z[2]² + z[2]³ + (-1)y[2] ≥ 0∧z[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧-1 + (-1)Bound + x[2] ≥ 0)
- (x[2] ≥ 0∧(-1)y[2]³ + (3)y[2]²·z[2] + (-3)y[2]·z[2]² + z[2]³ + y[2] ≥ 0∧z[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧-1 + (-1)Bound + x[2] ≥ 0)
- (x[2] ≥ 0∧(-1)y[2]³ + (-3)y[2]²·z[2] + (-3)y[2]·z[2]² + (-1)z[2]³ + y[2] ≥ 0∧z[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧-1 + (-1)Bound + x[2] ≥ 0)
- (x[2] ≥ 0∧y[2]³ + (-3)y[2]²·z[2] + (3)y[2]·z[2]² + (-1)z[2]³ + (-1)y[2] ≥ 0∧z[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])), ≥)∧-1 + (-1)Bound + x[2] ≥ 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)) = x₁ + (-1)x₂
POL(0@z) = 0
POL(*@z(x₁, x₂)) = x₁·x₂
POL(TRUE) = 0
POL(&&(x₁, x₂)) = 0
POL(EVAL(x₁, x₂, x₃)) = -1 + x₁
POL(FALSE) = 2
POL(>=@z(x₁, x₂)) = -1
POL(COND_EVAL1(x₁, x₂, x₃, x₄)) = -1 + x₂ + (-1)x₁
POL(+@z(x₁, x₂)) = x₁ + x₂
POL(COND_EVAL(x₁, x₂, x₃, x₄)) = -1 + x₂ + (-1)x₁
POL(1@z) = 1
POL(undefined) = -1

The following pairs are in P_>:

COND_EVAL(TRUE, x[3], y[3], z[3]) → EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])

The following pairs are in P_bound:

COND_EVAL(TRUE, x[3], y[3], z[3]) → EVAL(-@z(x[3], 1@z), -@z(y[3], 1@z), z[3])

The following pairs are in P_≥:

EVAL(x[0], y[0], z[0]) → COND_EVAL1(&&(>=@z(x[0], 0@z), >=@z(*@z(*@z(z[0], z[0]), z[0]), y[0])), x[0], y[0], z[0])
COND_EVAL1(TRUE, x[1], y[1], z[1]) → EVAL(x[1], -@z(y[1], 1@z), +@z(z[1], y[1]))
EVAL(x[2], y[2], z[2]) → COND_EVAL(&&(>=@z(x[2], 0@z), >=@z(*@z(*@z(z[2], z[2]), z[2]), y[2])), x[2], y[2], z[2])

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ IDP

              ↳ IDependencyGraphProof

I DP problem:
The following domains are used:

z

(1) -> (2), if ((-@z(y[1], 1@z) →^* y[2])∧(+@z(z[1], y[1]) →^* z[2])∧(x[1] →^* x[2]))

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

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

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval1(TRUE, x0, x1, x2)
Cond_eval(TRUE, x0, x1, x2)

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ IDP

              ↳ IDependencyGraphProof

                ↳ IDP

                  ↳ IDPtoQDPProof

I DP problem:
The following domains are used:

z

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

(1): COND_EVAL1(TRUE, x[1], y[1], z[1]) → EVAL(x[1], -@z(y[1], 1@z), +@z(z[1], y[1]))
(0): EVAL(x[0], y[0], z[0]) → COND_EVAL1(&&(>=@z(x[0], 0@z), >=@z(*@z(*@z(z[0], z[0]), z[0]), y[0])), x[0], y[0], z[0])

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

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

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval1(TRUE, x0, x1, x2)
Cond_eval(TRUE, x0, x1, x2)

Represented integers and predefined function symbols by Terms

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ IDP

              ↳ IDependencyGraphProof

                ↳ IDP

                  ↳ IDPtoQDPProof

                    ↳ QDP

                      ↳ UsableRulesProof

Q DP problem:
The TRS P consists of the following rules:

COND_EVAL1(true, x[1], y[1], z[1]) → EVAL(x[1], minus_int(y[1], pos(s(0))), plus_int(z[1], y[1]))
EVAL(x[0], y[0], z[0]) → COND_EVAL1(and(greatereq_int(x[0], pos(0)), greatereq_int(mult_int(mult_int(z[0], z[0]), z[0]), y[0])), x[0], y[0], z[0])

The TRS R consists of the following rules:

minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), neg(y)) → minus_nat(y, x)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
minus_int(pos(x), neg(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
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

          ↳ IDPNonInfProof

            ↳ IDP

              ↳ IDependencyGraphProof

                ↳ IDP

                  ↳ IDPtoQDPProof

                    ↳ QDP

                      ↳ UsableRulesProof

                        ↳ QDP

                          ↳ QReductionProof

Q DP problem:
The TRS P consists of the following rules:

COND_EVAL1(true, x[1], y[1], z[1]) → EVAL(x[1], minus_int(y[1], pos(s(0))), plus_int(z[1], y[1]))
EVAL(x[0], y[0], z[0]) → COND_EVAL1(and(greatereq_int(x[0], pos(0)), greatereq_int(mult_int(mult_int(z[0], z[0]), z[0]), y[0])), x[0], y[0], z[0])

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

eval(x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)
minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

eval(x0, x1, x2)
Cond_eval1(true, x0, x1, x2)
Cond_eval(true, x0, x1, x2)

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ IDP

              ↳ IDependencyGraphProof

                ↳ IDP

                  ↳ IDPtoQDPProof

                    ↳ QDP

                      ↳ UsableRulesProof

                        ↳ QDP

                          ↳ QReductionProof

                            ↳ QDP

                              ↳ RemovalProof

                              ↳ RemovalProof

                              ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

COND_EVAL1(true, x[1], y[1], z[1]) → EVAL(x[1], minus_int(y[1], pos(s(0))), plus_int(z[1], y[1]))
EVAL(x[0], y[0], z[0]) → COND_EVAL1(and(greatereq_int(x[0], pos(0)), greatereq_int(mult_int(mult_int(z[0], z[0]), z[0]), y[0])), x[0], y[0], z[0])

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
In the following pairs the term without variables pos(s(0)) is replaced by the fresh variable x_removed.
Pair: COND_EVAL1(true, x[1], y[1], z[1]) → EVAL(x[1], minus_int(y[1], pos(s(0))), plus_int(z[1], y[1]))
Positions in right side of the pair:

[1,1]

The new variable was added to all pairs as a new argument[CONREM].

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ IDP

              ↳ IDependencyGraphProof

                ↳ IDP

                  ↳ IDPtoQDPProof

                    ↳ QDP

                      ↳ UsableRulesProof

                        ↳ QDP

                          ↳ QReductionProof

                            ↳ QDP

                              ↳ RemovalProof

                                ↳ QDP

                              ↳ RemovalProof

                              ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

COND_EVAL1(true, x[1], y[1], z[1], x_removed) → EVAL(x[1], minus_int(y[1], x_removed), plus_int(z[1], y[1]), x_removed)
EVAL(x[0], y[0], z[0], x_removed) → COND_EVAL1(and(greatereq_int(x[0], pos(0)), greatereq_int(mult_int(mult_int(z[0], z[0]), z[0]), y[0])), x[0], y[0], z[0], x_removed)

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))

[1,1]

The new variable was added to all pairs as a new argument[CONREM].

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ IDP

              ↳ IDependencyGraphProof

                ↳ IDP

                  ↳ IDPtoQDPProof

                    ↳ QDP

                      ↳ UsableRulesProof

                        ↳ QDP

                          ↳ QReductionProof

                            ↳ QDP

                              ↳ RemovalProof

                              ↳ RemovalProof

                                ↳ QDP

                              ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

COND_EVAL1(true, x[1], y[1], z[1], x_removed) → EVAL(x[1], minus_int(y[1], x_removed), plus_int(z[1], y[1]), x_removed)
EVAL(x[0], y[0], z[0], x_removed) → COND_EVAL1(and(greatereq_int(x[0], pos(0)), greatereq_int(mult_int(mult_int(z[0], z[0]), z[0]), y[0])), x[0], y[0], z[0], x_removed)

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [LPAR04] the rule EVAL(x[0], y[0], z[0]) → COND_EVAL1(and(greatereq_int(x[0], pos(0)), greatereq_int(mult_int(mult_int(z[0], z[0]), z[0]), y[0])), x[0], y[0], z[0]) at position [0] we obtained the following new rules [LPAR04]:

EVAL(y0, y1, neg(x0)) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(mult_int(pos(mult_nat(x0, x0)), neg(x0)), y1)), y0, y1, neg(x0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(mult_int(pos(mult_nat(x0, x0)), pos(x0)), y1)), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL1(and(false, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), neg(s(x0)), y1, y2)
EVAL(neg(0), y1, y2) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), neg(0), y1, y2)
EVAL(pos(x0), y1, y2) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), pos(x0), y1, y2)

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ IDP

              ↳ IDependencyGraphProof

                ↳ IDP

                  ↳ IDPtoQDPProof

                    ↳ QDP

                      ↳ UsableRulesProof

                        ↳ QDP

                          ↳ QReductionProof

                            ↳ QDP

                              ↳ RemovalProof

                              ↳ RemovalProof

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Rewriting

Q DP problem:
The TRS P consists of the following rules:

COND_EVAL1(true, x[1], y[1], z[1]) → EVAL(x[1], minus_int(y[1], pos(s(0))), plus_int(z[1], y[1]))
EVAL(y0, y1, neg(x0)) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(mult_int(pos(mult_nat(x0, x0)), neg(x0)), y1)), y0, y1, neg(x0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(mult_int(pos(mult_nat(x0, x0)), pos(x0)), y1)), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL1(and(false, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), neg(s(x0)), y1, y2)
EVAL(neg(0), y1, y2) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), neg(0), y1, y2)
EVAL(pos(x0), y1, y2) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), pos(x0), y1, y2)

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(y0, y1, neg(x0)) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(mult_int(pos(mult_nat(x0, x0)), neg(x0)), y1)), y0, y1, neg(x0)) at position [0,1,0] we obtained the following new rules [LPAR04]:

EVAL(y0, y1, neg(x0)) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, neg(x0))

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ IDP

              ↳ IDependencyGraphProof

                ↳ IDP

                  ↳ IDPtoQDPProof

                    ↳ QDP

                      ↳ UsableRulesProof

                        ↳ QDP

                          ↳ QReductionProof

                            ↳ QDP

                              ↳ RemovalProof

                              ↳ RemovalProof

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Rewriting

                                    ↳ QDP

                                      ↳ Rewriting

Q DP problem:
The TRS P consists of the following rules:

COND_EVAL1(true, x[1], y[1], z[1]) → EVAL(x[1], minus_int(y[1], pos(s(0))), plus_int(z[1], y[1]))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(mult_int(pos(mult_nat(x0, x0)), pos(x0)), y1)), y0, y1, pos(x0))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL1(and(false, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), neg(s(x0)), y1, y2)
EVAL(neg(0), y1, y2) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), neg(0), y1, y2)
EVAL(pos(x0), y1, y2) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), pos(x0), y1, y2)
EVAL(y0, y1, neg(x0)) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, neg(x0))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [LPAR04] the rule EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(mult_int(pos(mult_nat(x0, x0)), pos(x0)), y1)), y0, y1, pos(x0)) at position [0,1,0] we obtained the following new rules [LPAR04]:

EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, pos(x0))

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ IDP

              ↳ IDependencyGraphProof

                ↳ IDP

                  ↳ IDPtoQDPProof

                    ↳ QDP

                      ↳ UsableRulesProof

                        ↳ QDP

                          ↳ QReductionProof

                            ↳ QDP

                              ↳ RemovalProof

                              ↳ RemovalProof

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Rewriting

                                    ↳ QDP

                                      ↳ Rewriting

                                        ↳ QDP

                                          ↳ Instantiation

Q DP problem:
The TRS P consists of the following rules:

COND_EVAL1(true, x[1], y[1], z[1]) → EVAL(x[1], minus_int(y[1], pos(s(0))), plus_int(z[1], y[1]))
EVAL(neg(s(x0)), y1, y2) → COND_EVAL1(and(false, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), neg(s(x0)), y1, y2)
EVAL(neg(0), y1, y2) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), neg(0), y1, y2)
EVAL(pos(x0), y1, y2) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), pos(x0), y1, y2)
EVAL(y0, y1, neg(x0)) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, neg(x0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, pos(x0))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [LPAR04] the rule COND_EVAL1(true, x[1], y[1], z[1]) → EVAL(x[1], minus_int(y[1], pos(s(0))), plus_int(z[1], y[1])) we obtained the following new rules [LPAR04]:

COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), plus_int(pos(z2), z1))
COND_EVAL1(true, z0, z1, neg(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), plus_int(neg(z2), z1))
COND_EVAL1(true, neg(0), z0, z1) → EVAL(neg(0), minus_int(z0, pos(s(0))), plus_int(z1, z0))
COND_EVAL1(true, pos(z0), z1, z2) → EVAL(pos(z0), minus_int(z1, pos(s(0))), plus_int(z2, z1))
COND_EVAL1(true, neg(s(z0)), z1, z2) → EVAL(neg(s(z0)), minus_int(z1, pos(s(0))), plus_int(z2, z1))

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ IDP

              ↳ IDependencyGraphProof

                ↳ IDP

                  ↳ IDPtoQDPProof

                    ↳ QDP

                      ↳ UsableRulesProof

                        ↳ QDP

                          ↳ QReductionProof

                            ↳ QDP

                              ↳ RemovalProof

                              ↳ RemovalProof

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Rewriting

                                    ↳ QDP

                                      ↳ Rewriting

                                        ↳ QDP

                                          ↳ Instantiation

                                            ↳ QDP

                                              ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

EVAL(neg(s(x0)), y1, y2) → COND_EVAL1(and(false, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), neg(s(x0)), y1, y2)
EVAL(neg(0), y1, y2) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), neg(0), y1, y2)
EVAL(pos(x0), y1, y2) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), pos(x0), y1, y2)
EVAL(y0, y1, neg(x0)) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, neg(x0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, pos(x0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), plus_int(pos(z2), z1))
COND_EVAL1(true, z0, z1, neg(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), plus_int(neg(z2), z1))
COND_EVAL1(true, neg(0), z0, z1) → EVAL(neg(0), minus_int(z0, pos(s(0))), plus_int(z1, z0))
COND_EVAL1(true, pos(z0), z1, z2) → EVAL(pos(z0), minus_int(z1, pos(s(0))), plus_int(z2, z1))
COND_EVAL1(true, neg(s(z0)), z1, z2) → EVAL(neg(s(z0)), minus_int(z1, pos(s(0))), plus_int(z2, z1))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

EVAL(neg(s(x0)), y1, y2) → COND_EVAL1(and(false, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), neg(s(x0)), y1, y2)

The remaining pairs can at least be oriented weakly.

EVAL(neg(0), y1, y2) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), neg(0), y1, y2)
EVAL(pos(x0), y1, y2) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), pos(x0), y1, y2)
EVAL(y0, y1, neg(x0)) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, neg(x0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, pos(x0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), plus_int(pos(z2), z1))
COND_EVAL1(true, z0, z1, neg(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), plus_int(neg(z2), z1))
COND_EVAL1(true, neg(0), z0, z1) → EVAL(neg(0), minus_int(z0, pos(s(0))), plus_int(z1, z0))
COND_EVAL1(true, pos(z0), z1, z2) → EVAL(pos(z0), minus_int(z1, pos(s(0))), plus_int(z2, z1))
COND_EVAL1(true, neg(s(z0)), z1, z2) → EVAL(neg(s(z0)), minus_int(z1, pos(s(0))), plus_int(z2, z1))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(COND_EVAL1(x₁, x₂, x₃, x₄)) = x₁
POL(EVAL(x₁, x₂, x₃)) = 1
POL(and(x₁, x₂)) = x₁
POL(false) = 0
POL(greatereq_int(x₁, x₂)) = x₂
POL(minus_int(x₁, x₂)) = 0
POL(minus_nat(x₁, x₂)) = 0
POL(mult_int(x₁, x₂)) = 0
POL(mult_nat(x₁, x₂)) = 0
POL(neg(x₁)) = 0
POL(plus_int(x₁, x₂)) = 0
POL(plus_nat(x₁, x₂)) = 0
POL(pos(x₁)) = 1
POL(s(x₁)) = 0
POL(true) = 1

The following usable rules [FROCOS05] were oriented:

greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
and(true, true) → true
and(true, false) → false
and(false, true) → false
and(false, false) → false

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ IDP

              ↳ IDependencyGraphProof

                ↳ IDP

                  ↳ IDPtoQDPProof

                    ↳ QDP

                      ↳ UsableRulesProof

                        ↳ QDP

                          ↳ QReductionProof

                            ↳ QDP

                              ↳ RemovalProof

                              ↳ RemovalProof

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Rewriting

                                    ↳ QDP

                                      ↳ Rewriting

                                        ↳ QDP

                                          ↳ Instantiation

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ RemovalProof

Q DP problem:
The TRS P consists of the following rules:

EVAL(neg(0), y1, y2) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), neg(0), y1, y2)
EVAL(pos(x0), y1, y2) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), pos(x0), y1, y2)
EVAL(y0, y1, neg(x0)) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, neg(x0))
EVAL(y0, y1, pos(x0)) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, pos(x0))
COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), plus_int(pos(z2), z1))
COND_EVAL1(true, z0, z1, neg(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), plus_int(neg(z2), z1))
COND_EVAL1(true, neg(0), z0, z1) → EVAL(neg(0), minus_int(z0, pos(s(0))), plus_int(z1, z0))
COND_EVAL1(true, pos(z0), z1, z2) → EVAL(pos(z0), minus_int(z1, pos(s(0))), plus_int(z2, z1))
COND_EVAL1(true, neg(s(z0)), z1, z2) → EVAL(neg(s(z0)), minus_int(z1, pos(s(0))), plus_int(z2, z1))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
In the following pairs the term without variables pos(s(0)) is replaced by the fresh variable x_removed.
Pair: COND_EVAL1(true, z0, z1, pos(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), plus_int(pos(z2), z1))
Positions in right side of the pair:

[1,1]

Pair: COND_EVAL1(true, z0, z1, neg(z2)) → EVAL(z0, minus_int(z1, pos(s(0))), plus_int(neg(z2), z1))
Positions in right side of the pair:

[1,1]

Pair: COND_EVAL1(true, neg(0), z0, z1) → EVAL(neg(0), minus_int(z0, pos(s(0))), plus_int(z1, z0))
Positions in right side of the pair:

[1,1]

Pair: COND_EVAL1(true, pos(z0), z1, z2) → EVAL(pos(z0), minus_int(z1, pos(s(0))), plus_int(z2, z1))
Positions in right side of the pair:

[1,1]

Pair: COND_EVAL1(true, neg(s(z0)), z1, z2) → EVAL(neg(s(z0)), minus_int(z1, pos(s(0))), plus_int(z2, z1))
Positions in right side of the pair:

[1,1]

The new variable was added to all pairs as a new argument[CONREM].

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ IDP

              ↳ IDependencyGraphProof

                ↳ IDP

                  ↳ IDPtoQDPProof

                    ↳ QDP

                      ↳ UsableRulesProof

                        ↳ QDP

                          ↳ QReductionProof

                            ↳ QDP

                              ↳ RemovalProof

                              ↳ RemovalProof

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Rewriting

                                    ↳ QDP

                                      ↳ Rewriting

                                        ↳ QDP

                                          ↳ Instantiation

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ RemovalProof

                                                    ↳ QDP

                                                      ↳ NonInfProof

Q DP problem:
The TRS P consists of the following rules:

EVAL(neg(0), y1, y2, x_removed) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), neg(0), y1, y2, x_removed)
COND_EVAL1(true, z0, z1, pos(z2), x_removed) → EVAL(z0, minus_int(z1, x_removed), plus_int(pos(z2), z1), x_removed)
COND_EVAL1(true, z0, z1, neg(z2), x_removed) → EVAL(z0, minus_int(z1, x_removed), plus_int(neg(z2), z1), x_removed)
COND_EVAL1(true, neg(0), z0, z1, x_removed) → EVAL(neg(0), minus_int(z0, x_removed), plus_int(z1, z0), x_removed)
EVAL(pos(x0), y1, y2, x_removed) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), pos(x0), y1, y2, x_removed)
COND_EVAL1(true, pos(z0), z1, z2, x_removed) → EVAL(pos(z0), minus_int(z1, x_removed), plus_int(z2, z1), x_removed)
EVAL(y0, y1, neg(x0), x_removed) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, neg(x0), x_removed)
COND_EVAL1(true, neg(s(z0)), z1, z2, x_removed) → EVAL(neg(s(z0)), minus_int(z1, x_removed), plus_int(z2, z1), x_removed)
EVAL(y0, y1, pos(x0), x_removed) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, pos(x0), x_removed)

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
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 EVAL(neg(0), y1, y2, x_removed) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), neg(0), y1, y2, x_removed) the following chains were created:

We consider the chain COND_EVAL1(true, z0, z1, pos(z2), x_removed) → EVAL(z0, minus_int(z1, x_removed), plus_int(pos(z2), z1), x_removed), EVAL(neg(0), y1, y2, x_removed) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), neg(0), y1, y2, x_removed) which results in the following constraint:
(1) (EVAL(x3, minus_int(x4, x6), plus_int(pos(x5), x4), x6)=EVAL(neg(0), x7, x8, x9) ⇒ EVAL(neg(0), x7, x8, x9)≥COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(x8, x8), x8), x7)), neg(0), x7, x8, x9))

We simplified constraint (1) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
(2) (EVAL(neg(0), x7, x8, x6)≥COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(x8, x8), x8), x7)), neg(0), x7, x8, x6))
We consider the chain COND_EVAL1(true, z0, z1, neg(z2), x_removed) → EVAL(z0, minus_int(z1, x_removed), plus_int(neg(z2), z1), x_removed), EVAL(neg(0), y1, y2, x_removed) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), neg(0), y1, y2, x_removed) which results in the following constraint:
(3) (EVAL(x10, minus_int(x11, x13), plus_int(neg(x12), x11), x13)=EVAL(neg(0), x14, x15, x16) ⇒ EVAL(neg(0), x14, x15, x16)≥COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(x15, x15), x15), x14)), neg(0), x14, x15, x16))

We simplified constraint (3) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
(4) (EVAL(neg(0), x14, x15, x13)≥COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(x15, x15), x15), x14)), neg(0), x14, x15, x13))
We consider the chain COND_EVAL1(true, neg(0), z0, z1, x_removed) → EVAL(neg(0), minus_int(z0, x_removed), plus_int(z1, z0), x_removed), EVAL(neg(0), y1, y2, x_removed) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), neg(0), y1, y2, x_removed) which results in the following constraint:
(5) (EVAL(neg(0), minus_int(x17, x19), plus_int(x18, x17), x19)=EVAL(neg(0), x20, x21, x22) ⇒ EVAL(neg(0), x20, x21, x22)≥COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(x21, x21), x21), x20)), neg(0), x20, x21, x22))

We simplified constraint (5) using rules (I), (II), (III), (IV) which results in the following new constraint:
(6) (EVAL(neg(0), x20, x21, x19)≥COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(x21, x21), x21), x20)), neg(0), x20, x21, x19))

For Pair COND_EVAL1(true, z0, z1, pos(z2), x_removed) → EVAL(z0, minus_int(z1, x_removed), plus_int(pos(z2), z1), x_removed) the following chains were created:

We consider the chain EVAL(neg(0), y1, y2, x_removed) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), neg(0), y1, y2, x_removed), COND_EVAL1(true, z0, z1, pos(z2), x_removed) → EVAL(z0, minus_int(z1, x_removed), plus_int(pos(z2), z1), x_removed) which results in the following constraint:
(7)    (COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(x44, x44), x44), x43)), neg(0), x43, x44, x45)=COND_EVAL1(true, x46, x47, pos(x48), x49) ⇒ COND_EVAL1(true, x46, x47, pos(x48), x49)≥EVAL(x46, minus_int(x47, x49), plus_int(pos(x48), x47), x49))

We simplified constraint (7) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
(8)    (true=x422∧greatereq_int(x424, x43)=x423∧and(x422, x423)=true ⇒ COND_EVAL1(true, neg(0), x43, pos(x48), x45)≥EVAL(neg(0), minus_int(x43, x45), plus_int(pos(x48), x43), x45))

We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on and(x422, x423)=true which results in the following new constraint:
(9)    (true=true∧greatereq_int(x424, x43)=true ⇒ COND_EVAL1(true, neg(0), x43, pos(x48), x45)≥EVAL(neg(0), minus_int(x43, x45), plus_int(pos(x48), x43), x45))

We simplified constraint (9) using rules (I), (II) which results in the following new constraint:
(10)    (greatereq_int(x424, x43)=true ⇒ COND_EVAL1(true, neg(0), x43, pos(x48), x45)≥EVAL(neg(0), minus_int(x43, x45), plus_int(pos(x48), x43), x45))

We simplified constraint (10) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x424, x43)=true which results in the following new constraints:
(11)    (true=true ⇒ COND_EVAL1(true, neg(0), pos(0), pos(x48), x45)≥EVAL(neg(0), minus_int(pos(0), x45), plus_int(pos(x48), pos(0)), x45))

(12)    (true=true ⇒ COND_EVAL1(true, neg(0), pos(0), pos(x48), x45)≥EVAL(neg(0), minus_int(pos(0), x45), plus_int(pos(x48), pos(0)), x45))

(13)    (true=true ⇒ COND_EVAL1(true, neg(0), neg(x431), pos(x48), x45)≥EVAL(neg(0), minus_int(neg(x431), x45), plus_int(pos(x48), neg(x431)), x45))

(14)    (true=true ⇒ COND_EVAL1(true, neg(0), neg(x432), pos(x48), x45)≥EVAL(neg(0), minus_int(neg(x432), x45), plus_int(pos(x48), neg(x432)), x45))

(15)    (greatereq_int(pos(x439), pos(x438))=true∧(∀x440,x441:greatereq_int(pos(x439), pos(x438))=true ⇒ COND_EVAL1(true, neg(0), pos(x438), pos(x440), x441)≥EVAL(neg(0), minus_int(pos(x438), x441), plus_int(pos(x440), pos(x438)), x441)) ⇒ COND_EVAL1(true, neg(0), pos(s(x438)), pos(x48), x45)≥EVAL(neg(0), minus_int(pos(s(x438)), x45), plus_int(pos(x48), pos(s(x438))), x45))

(16)    (greatereq_int(neg(x443), neg(x442))=true∧(∀x444,x445:greatereq_int(neg(x443), neg(x442))=true ⇒ COND_EVAL1(true, neg(0), neg(x442), pos(x444), x445)≥EVAL(neg(0), minus_int(neg(x442), x445), plus_int(pos(x444), neg(x442)), x445)) ⇒ COND_EVAL1(true, neg(0), neg(s(x442)), pos(x48), x45)≥EVAL(neg(0), minus_int(neg(s(x442)), x45), plus_int(pos(x48), neg(s(x442))), x45))

We simplified constraint (11) using rules (I), (II) which results in the following new constraint:
(17)    (COND_EVAL1(true, neg(0), pos(0), pos(x48), x45)≥EVAL(neg(0), minus_int(pos(0), x45), plus_int(pos(x48), pos(0)), x45))

We simplified constraint (12) using rules (I), (II) which results in the following new constraint:
(18)    (COND_EVAL1(true, neg(0), pos(0), pos(x48), x45)≥EVAL(neg(0), minus_int(pos(0), x45), plus_int(pos(x48), pos(0)), x45))

We simplified constraint (13) using rules (I), (II) which results in the following new constraint:
(19)    (COND_EVAL1(true, neg(0), neg(x431), pos(x48), x45)≥EVAL(neg(0), minus_int(neg(x431), x45), plus_int(pos(x48), neg(x431)), x45))

We simplified constraint (14) using rules (I), (II) which results in the following new constraint:
(20)    (COND_EVAL1(true, neg(0), neg(x432), pos(x48), x45)≥EVAL(neg(0), minus_int(neg(x432), x45), plus_int(pos(x48), neg(x432)), x45))

We simplified constraint (15) using rule (VI) where we applied the induction hypothesis (∀x440,x441:greatereq_int(pos(x439), pos(x438))=true ⇒ COND_EVAL1(true, neg(0), pos(x438), pos(x440), x441)≥EVAL(neg(0), minus_int(pos(x438), x441), plus_int(pos(x440), pos(x438)), x441)) with σ = [x440 / x48, x441 / x45] which results in the following new constraint:
(21)    (COND_EVAL1(true, neg(0), pos(x438), pos(x48), x45)≥EVAL(neg(0), minus_int(pos(x438), x45), plus_int(pos(x48), pos(x438)), x45) ⇒ COND_EVAL1(true, neg(0), pos(s(x438)), pos(x48), x45)≥EVAL(neg(0), minus_int(pos(s(x438)), x45), plus_int(pos(x48), pos(s(x438))), x45))

We simplified constraint (16) using rule (VI) where we applied the induction hypothesis (∀x444,x445:greatereq_int(neg(x443), neg(x442))=true ⇒ COND_EVAL1(true, neg(0), neg(x442), pos(x444), x445)≥EVAL(neg(0), minus_int(neg(x442), x445), plus_int(pos(x444), neg(x442)), x445)) with σ = [x444 / x48, x445 / x45] which results in the following new constraint:
(22)    (COND_EVAL1(true, neg(0), neg(x442), pos(x48), x45)≥EVAL(neg(0), minus_int(neg(x442), x45), plus_int(pos(x48), neg(x442)), x45) ⇒ COND_EVAL1(true, neg(0), neg(s(x442)), pos(x48), x45)≥EVAL(neg(0), minus_int(neg(s(x442)), x45), plus_int(pos(x48), neg(s(x442))), x45))
We consider the chain EVAL(pos(x0), y1, y2, x_removed) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), pos(x0), y1, y2, x_removed), COND_EVAL1(true, z0, z1, pos(z2), x_removed) → EVAL(z0, minus_int(z1, x_removed), plus_int(pos(z2), z1), x_removed) which results in the following constraint:
(23)    (COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(x63, x63), x63), x62)), pos(x61), x62, x63, x64)=COND_EVAL1(true, x65, x66, pos(x67), x68) ⇒ COND_EVAL1(true, x65, x66, pos(x67), x68)≥EVAL(x65, minus_int(x66, x68), plus_int(pos(x67), x66), x68))

We simplified constraint (23) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
(24)    (true=x446∧greatereq_int(x448, x62)=x447∧and(x446, x447)=true ⇒ COND_EVAL1(true, pos(x61), x62, pos(x67), x64)≥EVAL(pos(x61), minus_int(x62, x64), plus_int(pos(x67), x62), x64))

We simplified constraint (24) using rule (V) (with possible (I) afterwards) using induction on and(x446, x447)=true which results in the following new constraint:
(25)    (true=true∧greatereq_int(x448, x62)=true ⇒ COND_EVAL1(true, pos(x61), x62, pos(x67), x64)≥EVAL(pos(x61), minus_int(x62, x64), plus_int(pos(x67), x62), x64))

We simplified constraint (25) using rules (I), (II) which results in the following new constraint:
(26)    (greatereq_int(x448, x62)=true ⇒ COND_EVAL1(true, pos(x61), x62, pos(x67), x64)≥EVAL(pos(x61), minus_int(x62, x64), plus_int(pos(x67), x62), x64))

We simplified constraint (26) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x448, x62)=true which results in the following new constraints:
(27)    (true=true ⇒ COND_EVAL1(true, pos(x61), pos(0), pos(x67), x64)≥EVAL(pos(x61), minus_int(pos(0), x64), plus_int(pos(x67), pos(0)), x64))

(28)    (true=true ⇒ COND_EVAL1(true, pos(x61), pos(0), pos(x67), x64)≥EVAL(pos(x61), minus_int(pos(0), x64), plus_int(pos(x67), pos(0)), x64))

(29)    (true=true ⇒ COND_EVAL1(true, pos(x61), neg(x455), pos(x67), x64)≥EVAL(pos(x61), minus_int(neg(x455), x64), plus_int(pos(x67), neg(x455)), x64))

(30)    (true=true ⇒ COND_EVAL1(true, pos(x61), neg(x456), pos(x67), x64)≥EVAL(pos(x61), minus_int(neg(x456), x64), plus_int(pos(x67), neg(x456)), x64))

(31)    (greatereq_int(pos(x463), pos(x462))=true∧(∀x464,x465,x466:greatereq_int(pos(x463), pos(x462))=true ⇒ COND_EVAL1(true, pos(x464), pos(x462), pos(x465), x466)≥EVAL(pos(x464), minus_int(pos(x462), x466), plus_int(pos(x465), pos(x462)), x466)) ⇒ COND_EVAL1(true, pos(x61), pos(s(x462)), pos(x67), x64)≥EVAL(pos(x61), minus_int(pos(s(x462)), x64), plus_int(pos(x67), pos(s(x462))), x64))

(32)    (greatereq_int(neg(x468), neg(x467))=true∧(∀x469,x470,x471:greatereq_int(neg(x468), neg(x467))=true ⇒ COND_EVAL1(true, pos(x469), neg(x467), pos(x470), x471)≥EVAL(pos(x469), minus_int(neg(x467), x471), plus_int(pos(x470), neg(x467)), x471)) ⇒ COND_EVAL1(true, pos(x61), neg(s(x467)), pos(x67), x64)≥EVAL(pos(x61), minus_int(neg(s(x467)), x64), plus_int(pos(x67), neg(s(x467))), x64))

We simplified constraint (27) using rules (I), (II) which results in the following new constraint:
(33)    (COND_EVAL1(true, pos(x61), pos(0), pos(x67), x64)≥EVAL(pos(x61), minus_int(pos(0), x64), plus_int(pos(x67), pos(0)), x64))

We simplified constraint (28) using rules (I), (II) which results in the following new constraint:
(34)    (COND_EVAL1(true, pos(x61), pos(0), pos(x67), x64)≥EVAL(pos(x61), minus_int(pos(0), x64), plus_int(pos(x67), pos(0)), x64))

We simplified constraint (29) using rules (I), (II) which results in the following new constraint:
(35)    (COND_EVAL1(true, pos(x61), neg(x455), pos(x67), x64)≥EVAL(pos(x61), minus_int(neg(x455), x64), plus_int(pos(x67), neg(x455)), x64))

We simplified constraint (30) using rules (I), (II) which results in the following new constraint:
(36)    (COND_EVAL1(true, pos(x61), neg(x456), pos(x67), x64)≥EVAL(pos(x61), minus_int(neg(x456), x64), plus_int(pos(x67), neg(x456)), x64))

We simplified constraint (31) using rule (VI) where we applied the induction hypothesis (∀x464,x465,x466:greatereq_int(pos(x463), pos(x462))=true ⇒ COND_EVAL1(true, pos(x464), pos(x462), pos(x465), x466)≥EVAL(pos(x464), minus_int(pos(x462), x466), plus_int(pos(x465), pos(x462)), x466)) with σ = [x465 / x67, x464 / x61, x466 / x64] which results in the following new constraint:
(37)    (COND_EVAL1(true, pos(x61), pos(x462), pos(x67), x64)≥EVAL(pos(x61), minus_int(pos(x462), x64), plus_int(pos(x67), pos(x462)), x64) ⇒ COND_EVAL1(true, pos(x61), pos(s(x462)), pos(x67), x64)≥EVAL(pos(x61), minus_int(pos(s(x462)), x64), plus_int(pos(x67), pos(s(x462))), x64))

We simplified constraint (32) using rule (VI) where we applied the induction hypothesis (∀x469,x470,x471:greatereq_int(neg(x468), neg(x467))=true ⇒ COND_EVAL1(true, pos(x469), neg(x467), pos(x470), x471)≥EVAL(pos(x469), minus_int(neg(x467), x471), plus_int(pos(x470), neg(x467)), x471)) with σ = [x469 / x61, x471 / x64, x470 / x67] which results in the following new constraint:
(38)    (COND_EVAL1(true, pos(x61), neg(x467), pos(x67), x64)≥EVAL(pos(x61), minus_int(neg(x467), x64), plus_int(pos(x67), neg(x467)), x64) ⇒ COND_EVAL1(true, pos(x61), neg(s(x467)), pos(x67), x64)≥EVAL(pos(x61), minus_int(neg(s(x467)), x64), plus_int(pos(x67), neg(s(x467))), x64))
We consider the chain EVAL(y0, y1, pos(x0), x_removed) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, pos(x0), x_removed), COND_EVAL1(true, z0, z1, pos(z2), x_removed) → EVAL(z0, minus_int(z1, x_removed), plus_int(pos(z2), z1), x_removed) which results in the following constraint:
(39)    (COND_EVAL1(and(greatereq_int(x81, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x83, x83), x83)), x82)), x81, x82, pos(x83), x84)=COND_EVAL1(true, x85, x86, pos(x87), x88) ⇒ COND_EVAL1(true, x85, x86, pos(x87), x88)≥EVAL(x85, minus_int(x86, x88), plus_int(pos(x87), x86), x88))

We simplified constraint (39) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
(40)    (pos(0)=x474∧greatereq_int(x81, x474)=x472∧greatereq_int(x475, x82)=x473∧and(x472, x473)=true ⇒ COND_EVAL1(true, x81, x82, pos(x83), x84)≥EVAL(x81, minus_int(x82, x84), plus_int(pos(x83), x82), x84))

We simplified constraint (40) using rule (V) (with possible (I) afterwards) using induction on and(x472, x473)=true which results in the following new constraint:
(41)    (true=true∧pos(0)=x474∧greatereq_int(x81, x474)=true∧greatereq_int(x475, x82)=true ⇒ COND_EVAL1(true, x81, x82, pos(x83), x84)≥EVAL(x81, minus_int(x82, x84), plus_int(pos(x83), x82), x84))

We simplified constraint (41) using rules (I), (II) which results in the following new constraint:
(42)    (pos(0)=x474∧greatereq_int(x81, x474)=true∧greatereq_int(x475, x82)=true ⇒ COND_EVAL1(true, x81, x82, pos(x83), x84)≥EVAL(x81, minus_int(x82, x84), plus_int(pos(x83), x82), x84))

We simplified constraint (42) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x81, x474)=true which results in the following new constraints:
(43)    (true=true∧pos(0)=pos(0)∧greatereq_int(x475, x82)=true ⇒ COND_EVAL1(true, pos(x476), x82, pos(x83), x84)≥EVAL(pos(x476), minus_int(x82, x84), plus_int(pos(x83), x82), x84))

(44)    (true=true∧pos(0)=pos(0)∧greatereq_int(x475, x82)=true ⇒ COND_EVAL1(true, neg(0), x82, pos(x83), x84)≥EVAL(neg(0), minus_int(x82, x84), plus_int(pos(x83), x82), x84))

(45)    (true=true∧pos(0)=neg(x478)∧greatereq_int(x475, x82)=true ⇒ COND_EVAL1(true, neg(0), x82, pos(x83), x84)≥EVAL(neg(0), minus_int(x82, x84), plus_int(pos(x83), x82), x84))

(46)    (true=true∧pos(0)=neg(x479)∧greatereq_int(x475, x82)=true ⇒ COND_EVAL1(true, pos(x480), x82, pos(x83), x84)≥EVAL(pos(x480), minus_int(x82, x84), plus_int(pos(x83), x82), x84))

(47)    (greatereq_int(pos(x486), pos(x485))=true∧pos(0)=pos(s(x485))∧greatereq_int(x475, x82)=true∧(∀x487,x488,x489,x490:greatereq_int(pos(x486), pos(x485))=true∧pos(0)=pos(x485)∧greatereq_int(x487, x488)=true ⇒ COND_EVAL1(true, pos(x486), x488, pos(x489), x490)≥EVAL(pos(x486), minus_int(x488, x490), plus_int(pos(x489), x488), x490)) ⇒ COND_EVAL1(true, pos(s(x486)), x82, pos(x83), x84)≥EVAL(pos(s(x486)), minus_int(x82, x84), plus_int(pos(x83), x82), x84))

(48)    (greatereq_int(neg(x492), neg(x491))=true∧pos(0)=neg(s(x491))∧greatereq_int(x475, x82)=true∧(∀x493,x494,x495,x496:greatereq_int(neg(x492), neg(x491))=true∧pos(0)=neg(x491)∧greatereq_int(x493, x494)=true ⇒ COND_EVAL1(true, neg(x492), x494, pos(x495), x496)≥EVAL(neg(x492), minus_int(x494, x496), plus_int(pos(x495), x494), x496)) ⇒ COND_EVAL1(true, neg(s(x492)), x82, pos(x83), x84)≥EVAL(neg(s(x492)), minus_int(x82, x84), plus_int(pos(x83), x82), x84))

We simplified constraint (43) using rules (I), (II) which results in the following new constraint:
(49)    (greatereq_int(x475, x82)=true ⇒ COND_EVAL1(true, pos(x476), x82, pos(x83), x84)≥EVAL(pos(x476), minus_int(x82, x84), plus_int(pos(x83), x82), x84))

We simplified constraint (44) using rules (I), (II) which results in the following new constraint:
(50)    (greatereq_int(x475, x82)=true ⇒ COND_EVAL1(true, neg(0), x82, pos(x83), x84)≥EVAL(neg(0), minus_int(x82, x84), plus_int(pos(x83), x82), x84))

We solved constraint (45) using rules (I), (II).We solved constraint (46) using rules (I), (II).We solved constraint (47) using rules (I), (II).We solved constraint (48) using rules (I), (II).We simplified constraint (49) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x475, x82)=true which results in the following new constraints:
(51)    (true=true ⇒ COND_EVAL1(true, pos(x476), pos(0), pos(x83), x84)≥EVAL(pos(x476), minus_int(pos(0), x84), plus_int(pos(x83), pos(0)), x84))

(52)    (true=true ⇒ COND_EVAL1(true, pos(x476), pos(0), pos(x83), x84)≥EVAL(pos(x476), minus_int(pos(0), x84), plus_int(pos(x83), pos(0)), x84))

(53)    (true=true ⇒ COND_EVAL1(true, pos(x476), neg(x499), pos(x83), x84)≥EVAL(pos(x476), minus_int(neg(x499), x84), plus_int(pos(x83), neg(x499)), x84))

(54)    (true=true ⇒ COND_EVAL1(true, pos(x476), neg(x500), pos(x83), x84)≥EVAL(pos(x476), minus_int(neg(x500), x84), plus_int(pos(x83), neg(x500)), x84))

(55)    (greatereq_int(pos(x507), pos(x506))=true∧(∀x508,x509,x510:greatereq_int(pos(x507), pos(x506))=true ⇒ COND_EVAL1(true, pos(x508), pos(x506), pos(x509), x510)≥EVAL(pos(x508), minus_int(pos(x506), x510), plus_int(pos(x509), pos(x506)), x510)) ⇒ COND_EVAL1(true, pos(x476), pos(s(x506)), pos(x83), x84)≥EVAL(pos(x476), minus_int(pos(s(x506)), x84), plus_int(pos(x83), pos(s(x506))), x84))

(56)    (greatereq_int(neg(x512), neg(x511))=true∧(∀x513,x514,x515:greatereq_int(neg(x512), neg(x511))=true ⇒ COND_EVAL1(true, pos(x513), neg(x511), pos(x514), x515)≥EVAL(pos(x513), minus_int(neg(x511), x515), plus_int(pos(x514), neg(x511)), x515)) ⇒ COND_EVAL1(true, pos(x476), neg(s(x511)), pos(x83), x84)≥EVAL(pos(x476), minus_int(neg(s(x511)), x84), plus_int(pos(x83), neg(s(x511))), x84))

We simplified constraint (51) using rules (I), (II) which results in the following new constraint:
(57)    (COND_EVAL1(true, pos(x476), pos(0), pos(x83), x84)≥EVAL(pos(x476), minus_int(pos(0), x84), plus_int(pos(x83), pos(0)), x84))

We simplified constraint (52) using rules (I), (II) which results in the following new constraint:
(58)    (COND_EVAL1(true, pos(x476), pos(0), pos(x83), x84)≥EVAL(pos(x476), minus_int(pos(0), x84), plus_int(pos(x83), pos(0)), x84))

We simplified constraint (53) using rules (I), (II) which results in the following new constraint:
(59)    (COND_EVAL1(true, pos(x476), neg(x499), pos(x83), x84)≥EVAL(pos(x476), minus_int(neg(x499), x84), plus_int(pos(x83), neg(x499)), x84))

We simplified constraint (54) using rules (I), (II) which results in the following new constraint:
(60)    (COND_EVAL1(true, pos(x476), neg(x500), pos(x83), x84)≥EVAL(pos(x476), minus_int(neg(x500), x84), plus_int(pos(x83), neg(x500)), x84))

We simplified constraint (55) using rule (VI) where we applied the induction hypothesis (∀x508,x509,x510:greatereq_int(pos(x507), pos(x506))=true ⇒ COND_EVAL1(true, pos(x508), pos(x506), pos(x509), x510)≥EVAL(pos(x508), minus_int(pos(x506), x510), plus_int(pos(x509), pos(x506)), x510)) with σ = [x508 / x476, x509 / x83, x510 / x84] which results in the following new constraint:
(61)    (COND_EVAL1(true, pos(x476), pos(x506), pos(x83), x84)≥EVAL(pos(x476), minus_int(pos(x506), x84), plus_int(pos(x83), pos(x506)), x84) ⇒ COND_EVAL1(true, pos(x476), pos(s(x506)), pos(x83), x84)≥EVAL(pos(x476), minus_int(pos(s(x506)), x84), plus_int(pos(x83), pos(s(x506))), x84))

We simplified constraint (56) using rule (VI) where we applied the induction hypothesis (∀x513,x514,x515:greatereq_int(neg(x512), neg(x511))=true ⇒ COND_EVAL1(true, pos(x513), neg(x511), pos(x514), x515)≥EVAL(pos(x513), minus_int(neg(x511), x515), plus_int(pos(x514), neg(x511)), x515)) with σ = [x513 / x476, x515 / x84, x514 / x83] which results in the following new constraint:
(62)    (COND_EVAL1(true, pos(x476), neg(x511), pos(x83), x84)≥EVAL(pos(x476), minus_int(neg(x511), x84), plus_int(pos(x83), neg(x511)), x84) ⇒ COND_EVAL1(true, pos(x476), neg(s(x511)), pos(x83), x84)≥EVAL(pos(x476), minus_int(neg(s(x511)), x84), plus_int(pos(x83), neg(s(x511))), x84))

We simplified constraint (50) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x475, x82)=true which results in the following new constraints:
(63)    (true=true ⇒ COND_EVAL1(true, neg(0), pos(0), pos(x83), x84)≥EVAL(neg(0), minus_int(pos(0), x84), plus_int(pos(x83), pos(0)), x84))

(64)    (true=true ⇒ COND_EVAL1(true, neg(0), pos(0), pos(x83), x84)≥EVAL(neg(0), minus_int(pos(0), x84), plus_int(pos(x83), pos(0)), x84))

(65)    (true=true ⇒ COND_EVAL1(true, neg(0), neg(x518), pos(x83), x84)≥EVAL(neg(0), minus_int(neg(x518), x84), plus_int(pos(x83), neg(x518)), x84))

(66)    (true=true ⇒ COND_EVAL1(true, neg(0), neg(x519), pos(x83), x84)≥EVAL(neg(0), minus_int(neg(x519), x84), plus_int(pos(x83), neg(x519)), x84))

(67)    (greatereq_int(pos(x526), pos(x525))=true∧(∀x527,x528:greatereq_int(pos(x526), pos(x525))=true ⇒ COND_EVAL1(true, neg(0), pos(x525), pos(x527), x528)≥EVAL(neg(0), minus_int(pos(x525), x528), plus_int(pos(x527), pos(x525)), x528)) ⇒ COND_EVAL1(true, neg(0), pos(s(x525)), pos(x83), x84)≥EVAL(neg(0), minus_int(pos(s(x525)), x84), plus_int(pos(x83), pos(s(x525))), x84))

(68)    (greatereq_int(neg(x530), neg(x529))=true∧(∀x531,x532:greatereq_int(neg(x530), neg(x529))=true ⇒ COND_EVAL1(true, neg(0), neg(x529), pos(x531), x532)≥EVAL(neg(0), minus_int(neg(x529), x532), plus_int(pos(x531), neg(x529)), x532)) ⇒ COND_EVAL1(true, neg(0), neg(s(x529)), pos(x83), x84)≥EVAL(neg(0), minus_int(neg(s(x529)), x84), plus_int(pos(x83), neg(s(x529))), x84))

We simplified constraint (63) using rules (I), (II) which results in the following new constraint:
(69)    (COND_EVAL1(true, neg(0), pos(0), pos(x83), x84)≥EVAL(neg(0), minus_int(pos(0), x84), plus_int(pos(x83), pos(0)), x84))

We simplified constraint (64) using rules (I), (II) which results in the following new constraint:
(70)    (COND_EVAL1(true, neg(0), pos(0), pos(x83), x84)≥EVAL(neg(0), minus_int(pos(0), x84), plus_int(pos(x83), pos(0)), x84))

We simplified constraint (65) using rules (I), (II) which results in the following new constraint:
(71)    (COND_EVAL1(true, neg(0), neg(x518), pos(x83), x84)≥EVAL(neg(0), minus_int(neg(x518), x84), plus_int(pos(x83), neg(x518)), x84))

We simplified constraint (66) using rules (I), (II) which results in the following new constraint:
(72)    (COND_EVAL1(true, neg(0), neg(x519), pos(x83), x84)≥EVAL(neg(0), minus_int(neg(x519), x84), plus_int(pos(x83), neg(x519)), x84))

We simplified constraint (67) using rule (VI) where we applied the induction hypothesis (∀x527,x528:greatereq_int(pos(x526), pos(x525))=true ⇒ COND_EVAL1(true, neg(0), pos(x525), pos(x527), x528)≥EVAL(neg(0), minus_int(pos(x525), x528), plus_int(pos(x527), pos(x525)), x528)) with σ = [x527 / x83, x528 / x84] which results in the following new constraint:
(73)    (COND_EVAL1(true, neg(0), pos(x525), pos(x83), x84)≥EVAL(neg(0), minus_int(pos(x525), x84), plus_int(pos(x83), pos(x525)), x84) ⇒ COND_EVAL1(true, neg(0), pos(s(x525)), pos(x83), x84)≥EVAL(neg(0), minus_int(pos(s(x525)), x84), plus_int(pos(x83), pos(s(x525))), x84))

We simplified constraint (68) using rule (VI) where we applied the induction hypothesis (∀x531,x532:greatereq_int(neg(x530), neg(x529))=true ⇒ COND_EVAL1(true, neg(0), neg(x529), pos(x531), x532)≥EVAL(neg(0), minus_int(neg(x529), x532), plus_int(pos(x531), neg(x529)), x532)) with σ = [x532 / x84, x531 / x83] which results in the following new constraint:
(74)    (COND_EVAL1(true, neg(0), neg(x529), pos(x83), x84)≥EVAL(neg(0), minus_int(neg(x529), x84), plus_int(pos(x83), neg(x529)), x84) ⇒ COND_EVAL1(true, neg(0), neg(s(x529)), pos(x83), x84)≥EVAL(neg(0), minus_int(neg(s(x529)), x84), plus_int(pos(x83), neg(s(x529))), x84))

For Pair COND_EVAL1(true, z0, z1, neg(z2), x_removed) → EVAL(z0, minus_int(z1, x_removed), plus_int(neg(z2), z1), x_removed) the following chains were created:

We consider the chain EVAL(neg(0), y1, y2, x_removed) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), neg(0), y1, y2, x_removed), COND_EVAL1(true, z0, z1, neg(z2), x_removed) → EVAL(z0, minus_int(z1, x_removed), plus_int(neg(z2), z1), x_removed) which results in the following constraint:
(75)    (COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(x90, x90), x90), x89)), neg(0), x89, x90, x91)=COND_EVAL1(true, x92, x93, neg(x94), x95) ⇒ COND_EVAL1(true, x92, x93, neg(x94), x95)≥EVAL(x92, minus_int(x93, x95), plus_int(neg(x94), x93), x95))

We simplified constraint (75) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
(76)    (true=x533∧greatereq_int(x535, x89)=x534∧and(x533, x534)=true ⇒ COND_EVAL1(true, neg(0), x89, neg(x94), x91)≥EVAL(neg(0), minus_int(x89, x91), plus_int(neg(x94), x89), x91))

We simplified constraint (76) using rule (V) (with possible (I) afterwards) using induction on and(x533, x534)=true which results in the following new constraint:
(77)    (true=true∧greatereq_int(x535, x89)=true ⇒ COND_EVAL1(true, neg(0), x89, neg(x94), x91)≥EVAL(neg(0), minus_int(x89, x91), plus_int(neg(x94), x89), x91))

We simplified constraint (77) using rules (I), (II) which results in the following new constraint:
(78)    (greatereq_int(x535, x89)=true ⇒ COND_EVAL1(true, neg(0), x89, neg(x94), x91)≥EVAL(neg(0), minus_int(x89, x91), plus_int(neg(x94), x89), x91))

We simplified constraint (78) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x535, x89)=true which results in the following new constraints:
(79)    (true=true ⇒ COND_EVAL1(true, neg(0), pos(0), neg(x94), x91)≥EVAL(neg(0), minus_int(pos(0), x91), plus_int(neg(x94), pos(0)), x91))

(80)    (true=true ⇒ COND_EVAL1(true, neg(0), pos(0), neg(x94), x91)≥EVAL(neg(0), minus_int(pos(0), x91), plus_int(neg(x94), pos(0)), x91))

(81)    (true=true ⇒ COND_EVAL1(true, neg(0), neg(x542), neg(x94), x91)≥EVAL(neg(0), minus_int(neg(x542), x91), plus_int(neg(x94), neg(x542)), x91))

(82)    (true=true ⇒ COND_EVAL1(true, neg(0), neg(x543), neg(x94), x91)≥EVAL(neg(0), minus_int(neg(x543), x91), plus_int(neg(x94), neg(x543)), x91))

(83)    (greatereq_int(pos(x550), pos(x549))=true∧(∀x551,x552:greatereq_int(pos(x550), pos(x549))=true ⇒ COND_EVAL1(true, neg(0), pos(x549), neg(x551), x552)≥EVAL(neg(0), minus_int(pos(x549), x552), plus_int(neg(x551), pos(x549)), x552)) ⇒ COND_EVAL1(true, neg(0), pos(s(x549)), neg(x94), x91)≥EVAL(neg(0), minus_int(pos(s(x549)), x91), plus_int(neg(x94), pos(s(x549))), x91))

(84)    (greatereq_int(neg(x554), neg(x553))=true∧(∀x555,x556:greatereq_int(neg(x554), neg(x553))=true ⇒ COND_EVAL1(true, neg(0), neg(x553), neg(x555), x556)≥EVAL(neg(0), minus_int(neg(x553), x556), plus_int(neg(x555), neg(x553)), x556)) ⇒ COND_EVAL1(true, neg(0), neg(s(x553)), neg(x94), x91)≥EVAL(neg(0), minus_int(neg(s(x553)), x91), plus_int(neg(x94), neg(s(x553))), x91))

We simplified constraint (79) using rules (I), (II) which results in the following new constraint:
(85)    (COND_EVAL1(true, neg(0), pos(0), neg(x94), x91)≥EVAL(neg(0), minus_int(pos(0), x91), plus_int(neg(x94), pos(0)), x91))

We simplified constraint (80) using rules (I), (II) which results in the following new constraint:
(86)    (COND_EVAL1(true, neg(0), pos(0), neg(x94), x91)≥EVAL(neg(0), minus_int(pos(0), x91), plus_int(neg(x94), pos(0)), x91))

We simplified constraint (81) using rules (I), (II) which results in the following new constraint:
(87)    (COND_EVAL1(true, neg(0), neg(x542), neg(x94), x91)≥EVAL(neg(0), minus_int(neg(x542), x91), plus_int(neg(x94), neg(x542)), x91))

We simplified constraint (82) using rules (I), (II) which results in the following new constraint:
(88)    (COND_EVAL1(true, neg(0), neg(x543), neg(x94), x91)≥EVAL(neg(0), minus_int(neg(x543), x91), plus_int(neg(x94), neg(x543)), x91))

We simplified constraint (83) using rule (VI) where we applied the induction hypothesis (∀x551,x552:greatereq_int(pos(x550), pos(x549))=true ⇒ COND_EVAL1(true, neg(0), pos(x549), neg(x551), x552)≥EVAL(neg(0), minus_int(pos(x549), x552), plus_int(neg(x551), pos(x549)), x552)) with σ = [x552 / x91, x551 / x94] which results in the following new constraint:
(89)    (COND_EVAL1(true, neg(0), pos(x549), neg(x94), x91)≥EVAL(neg(0), minus_int(pos(x549), x91), plus_int(neg(x94), pos(x549)), x91) ⇒ COND_EVAL1(true, neg(0), pos(s(x549)), neg(x94), x91)≥EVAL(neg(0), minus_int(pos(s(x549)), x91), plus_int(neg(x94), pos(s(x549))), x91))

We simplified constraint (84) using rule (VI) where we applied the induction hypothesis (∀x555,x556:greatereq_int(neg(x554), neg(x553))=true ⇒ COND_EVAL1(true, neg(0), neg(x553), neg(x555), x556)≥EVAL(neg(0), minus_int(neg(x553), x556), plus_int(neg(x555), neg(x553)), x556)) with σ = [x556 / x91, x555 / x94] which results in the following new constraint:
(90)    (COND_EVAL1(true, neg(0), neg(x553), neg(x94), x91)≥EVAL(neg(0), minus_int(neg(x553), x91), plus_int(neg(x94), neg(x553)), x91) ⇒ COND_EVAL1(true, neg(0), neg(s(x553)), neg(x94), x91)≥EVAL(neg(0), minus_int(neg(s(x553)), x91), plus_int(neg(x94), neg(s(x553))), x91))
We consider the chain EVAL(pos(x0), y1, y2, x_removed) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), pos(x0), y1, y2, x_removed), COND_EVAL1(true, z0, z1, neg(z2), x_removed) → EVAL(z0, minus_int(z1, x_removed), plus_int(neg(z2), z1), x_removed) which results in the following constraint:
(91)    (COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(x109, x109), x109), x108)), pos(x107), x108, x109, x110)=COND_EVAL1(true, x111, x112, neg(x113), x114) ⇒ COND_EVAL1(true, x111, x112, neg(x113), x114)≥EVAL(x111, minus_int(x112, x114), plus_int(neg(x113), x112), x114))

We simplified constraint (91) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
(92)    (true=x557∧greatereq_int(x559, x108)=x558∧and(x557, x558)=true ⇒ COND_EVAL1(true, pos(x107), x108, neg(x113), x110)≥EVAL(pos(x107), minus_int(x108, x110), plus_int(neg(x113), x108), x110))

We simplified constraint (92) using rule (V) (with possible (I) afterwards) using induction on and(x557, x558)=true which results in the following new constraint:
(93)    (true=true∧greatereq_int(x559, x108)=true ⇒ COND_EVAL1(true, pos(x107), x108, neg(x113), x110)≥EVAL(pos(x107), minus_int(x108, x110), plus_int(neg(x113), x108), x110))

We simplified constraint (93) using rules (I), (II) which results in the following new constraint:
(94)    (greatereq_int(x559, x108)=true ⇒ COND_EVAL1(true, pos(x107), x108, neg(x113), x110)≥EVAL(pos(x107), minus_int(x108, x110), plus_int(neg(x113), x108), x110))

We simplified constraint (94) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x559, x108)=true which results in the following new constraints:
(95)    (true=true ⇒ COND_EVAL1(true, pos(x107), pos(0), neg(x113), x110)≥EVAL(pos(x107), minus_int(pos(0), x110), plus_int(neg(x113), pos(0)), x110))

(96)    (true=true ⇒ COND_EVAL1(true, pos(x107), pos(0), neg(x113), x110)≥EVAL(pos(x107), minus_int(pos(0), x110), plus_int(neg(x113), pos(0)), x110))

(97)    (true=true ⇒ COND_EVAL1(true, pos(x107), neg(x566), neg(x113), x110)≥EVAL(pos(x107), minus_int(neg(x566), x110), plus_int(neg(x113), neg(x566)), x110))

(98)    (true=true ⇒ COND_EVAL1(true, pos(x107), neg(x567), neg(x113), x110)≥EVAL(pos(x107), minus_int(neg(x567), x110), plus_int(neg(x113), neg(x567)), x110))

(99)    (greatereq_int(pos(x574), pos(x573))=true∧(∀x575,x576,x577:greatereq_int(pos(x574), pos(x573))=true ⇒ COND_EVAL1(true, pos(x575), pos(x573), neg(x576), x577)≥EVAL(pos(x575), minus_int(pos(x573), x577), plus_int(neg(x576), pos(x573)), x577)) ⇒ COND_EVAL1(true, pos(x107), pos(s(x573)), neg(x113), x110)≥EVAL(pos(x107), minus_int(pos(s(x573)), x110), plus_int(neg(x113), pos(s(x573))), x110))

(100)    (greatereq_int(neg(x579), neg(x578))=true∧(∀x580,x581,x582:greatereq_int(neg(x579), neg(x578))=true ⇒ COND_EVAL1(true, pos(x580), neg(x578), neg(x581), x582)≥EVAL(pos(x580), minus_int(neg(x578), x582), plus_int(neg(x581), neg(x578)), x582)) ⇒ COND_EVAL1(true, pos(x107), neg(s(x578)), neg(x113), x110)≥EVAL(pos(x107), minus_int(neg(s(x578)), x110), plus_int(neg(x113), neg(s(x578))), x110))

We simplified constraint (95) using rules (I), (II) which results in the following new constraint:
(101)    (COND_EVAL1(true, pos(x107), pos(0), neg(x113), x110)≥EVAL(pos(x107), minus_int(pos(0), x110), plus_int(neg(x113), pos(0)), x110))

We simplified constraint (96) using rules (I), (II) which results in the following new constraint:
(102)    (COND_EVAL1(true, pos(x107), pos(0), neg(x113), x110)≥EVAL(pos(x107), minus_int(pos(0), x110), plus_int(neg(x113), pos(0)), x110))

We simplified constraint (97) using rules (I), (II) which results in the following new constraint:
(103)    (COND_EVAL1(true, pos(x107), neg(x566), neg(x113), x110)≥EVAL(pos(x107), minus_int(neg(x566), x110), plus_int(neg(x113), neg(x566)), x110))

We simplified constraint (98) using rules (I), (II) which results in the following new constraint:
(104)    (COND_EVAL1(true, pos(x107), neg(x567), neg(x113), x110)≥EVAL(pos(x107), minus_int(neg(x567), x110), plus_int(neg(x113), neg(x567)), x110))

We simplified constraint (99) using rule (VI) where we applied the induction hypothesis (∀x575,x576,x577:greatereq_int(pos(x574), pos(x573))=true ⇒ COND_EVAL1(true, pos(x575), pos(x573), neg(x576), x577)≥EVAL(pos(x575), minus_int(pos(x573), x577), plus_int(neg(x576), pos(x573)), x577)) with σ = [x577 / x110, x576 / x113, x575 / x107] which results in the following new constraint:
(105)    (COND_EVAL1(true, pos(x107), pos(x573), neg(x113), x110)≥EVAL(pos(x107), minus_int(pos(x573), x110), plus_int(neg(x113), pos(x573)), x110) ⇒ COND_EVAL1(true, pos(x107), pos(s(x573)), neg(x113), x110)≥EVAL(pos(x107), minus_int(pos(s(x573)), x110), plus_int(neg(x113), pos(s(x573))), x110))

We simplified constraint (100) using rule (VI) where we applied the induction hypothesis (∀x580,x581,x582:greatereq_int(neg(x579), neg(x578))=true ⇒ COND_EVAL1(true, pos(x580), neg(x578), neg(x581), x582)≥EVAL(pos(x580), minus_int(neg(x578), x582), plus_int(neg(x581), neg(x578)), x582)) with σ = [x580 / x107, x582 / x110, x581 / x113] which results in the following new constraint:
(106)    (COND_EVAL1(true, pos(x107), neg(x578), neg(x113), x110)≥EVAL(pos(x107), minus_int(neg(x578), x110), plus_int(neg(x113), neg(x578)), x110) ⇒ COND_EVAL1(true, pos(x107), neg(s(x578)), neg(x113), x110)≥EVAL(pos(x107), minus_int(neg(s(x578)), x110), plus_int(neg(x113), neg(s(x578))), x110))
We consider the chain EVAL(y0, y1, neg(x0), x_removed) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, neg(x0), x_removed), COND_EVAL1(true, z0, z1, neg(z2), x_removed) → EVAL(z0, minus_int(z1, x_removed), plus_int(neg(z2), z1), x_removed) which results in the following constraint:
(107)    (COND_EVAL1(and(greatereq_int(x119, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x121, x121), x121)), x120)), x119, x120, neg(x121), x122)=COND_EVAL1(true, x123, x124, neg(x125), x126) ⇒ COND_EVAL1(true, x123, x124, neg(x125), x126)≥EVAL(x123, minus_int(x124, x126), plus_int(neg(x125), x124), x126))

We simplified constraint (107) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
(108)    (pos(0)=x585∧greatereq_int(x119, x585)=x583∧greatereq_int(x586, x120)=x584∧and(x583, x584)=true ⇒ COND_EVAL1(true, x119, x120, neg(x121), x122)≥EVAL(x119, minus_int(x120, x122), plus_int(neg(x121), x120), x122))

We simplified constraint (108) using rule (V) (with possible (I) afterwards) using induction on and(x583, x584)=true which results in the following new constraint:
(109)    (true=true∧pos(0)=x585∧greatereq_int(x119, x585)=true∧greatereq_int(x586, x120)=true ⇒ COND_EVAL1(true, x119, x120, neg(x121), x122)≥EVAL(x119, minus_int(x120, x122), plus_int(neg(x121), x120), x122))

We simplified constraint (109) using rules (I), (II) which results in the following new constraint:
(110)    (pos(0)=x585∧greatereq_int(x119, x585)=true∧greatereq_int(x586, x120)=true ⇒ COND_EVAL1(true, x119, x120, neg(x121), x122)≥EVAL(x119, minus_int(x120, x122), plus_int(neg(x121), x120), x122))

We simplified constraint (110) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x119, x585)=true which results in the following new constraints:
(111)    (true=true∧pos(0)=pos(0)∧greatereq_int(x586, x120)=true ⇒ COND_EVAL1(true, pos(x587), x120, neg(x121), x122)≥EVAL(pos(x587), minus_int(x120, x122), plus_int(neg(x121), x120), x122))

(112)    (true=true∧pos(0)=pos(0)∧greatereq_int(x586, x120)=true ⇒ COND_EVAL1(true, neg(0), x120, neg(x121), x122)≥EVAL(neg(0), minus_int(x120, x122), plus_int(neg(x121), x120), x122))

(113)    (true=true∧pos(0)=neg(x589)∧greatereq_int(x586, x120)=true ⇒ COND_EVAL1(true, neg(0), x120, neg(x121), x122)≥EVAL(neg(0), minus_int(x120, x122), plus_int(neg(x121), x120), x122))

(114)    (true=true∧pos(0)=neg(x590)∧greatereq_int(x586, x120)=true ⇒ COND_EVAL1(true, pos(x591), x120, neg(x121), x122)≥EVAL(pos(x591), minus_int(x120, x122), plus_int(neg(x121), x120), x122))

(115)    (greatereq_int(pos(x597), pos(x596))=true∧pos(0)=pos(s(x596))∧greatereq_int(x586, x120)=true∧(∀x598,x599,x600,x601:greatereq_int(pos(x597), pos(x596))=true∧pos(0)=pos(x596)∧greatereq_int(x598, x599)=true ⇒ COND_EVAL1(true, pos(x597), x599, neg(x600), x601)≥EVAL(pos(x597), minus_int(x599, x601), plus_int(neg(x600), x599), x601)) ⇒ COND_EVAL1(true, pos(s(x597)), x120, neg(x121), x122)≥EVAL(pos(s(x597)), minus_int(x120, x122), plus_int(neg(x121), x120), x122))

(116)    (greatereq_int(neg(x603), neg(x602))=true∧pos(0)=neg(s(x602))∧greatereq_int(x586, x120)=true∧(∀x604,x605,x606,x607:greatereq_int(neg(x603), neg(x602))=true∧pos(0)=neg(x602)∧greatereq_int(x604, x605)=true ⇒ COND_EVAL1(true, neg(x603), x605, neg(x606), x607)≥EVAL(neg(x603), minus_int(x605, x607), plus_int(neg(x606), x605), x607)) ⇒ COND_EVAL1(true, neg(s(x603)), x120, neg(x121), x122)≥EVAL(neg(s(x603)), minus_int(x120, x122), plus_int(neg(x121), x120), x122))

We simplified constraint (111) using rules (I), (II) which results in the following new constraint:
(117)    (greatereq_int(x586, x120)=true ⇒ COND_EVAL1(true, pos(x587), x120, neg(x121), x122)≥EVAL(pos(x587), minus_int(x120, x122), plus_int(neg(x121), x120), x122))

We simplified constraint (112) using rules (I), (II) which results in the following new constraint:
(118)    (greatereq_int(x586, x120)=true ⇒ COND_EVAL1(true, neg(0), x120, neg(x121), x122)≥EVAL(neg(0), minus_int(x120, x122), plus_int(neg(x121), x120), x122))

We solved constraint (113) using rules (I), (II).We solved constraint (114) using rules (I), (II).We solved constraint (115) using rules (I), (II).We solved constraint (116) using rules (I), (II).We simplified constraint (117) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x586, x120)=true which results in the following new constraints:
(119)    (true=true ⇒ COND_EVAL1(true, pos(x587), pos(0), neg(x121), x122)≥EVAL(pos(x587), minus_int(pos(0), x122), plus_int(neg(x121), pos(0)), x122))

(120)    (true=true ⇒ COND_EVAL1(true, pos(x587), pos(0), neg(x121), x122)≥EVAL(pos(x587), minus_int(pos(0), x122), plus_int(neg(x121), pos(0)), x122))

(121)    (true=true ⇒ COND_EVAL1(true, pos(x587), neg(x610), neg(x121), x122)≥EVAL(pos(x587), minus_int(neg(x610), x122), plus_int(neg(x121), neg(x610)), x122))

(122)    (true=true ⇒ COND_EVAL1(true, pos(x587), neg(x611), neg(x121), x122)≥EVAL(pos(x587), minus_int(neg(x611), x122), plus_int(neg(x121), neg(x611)), x122))

(123)    (greatereq_int(pos(x618), pos(x617))=true∧(∀x619,x620,x621:greatereq_int(pos(x618), pos(x617))=true ⇒ COND_EVAL1(true, pos(x619), pos(x617), neg(x620), x621)≥EVAL(pos(x619), minus_int(pos(x617), x621), plus_int(neg(x620), pos(x617)), x621)) ⇒ COND_EVAL1(true, pos(x587), pos(s(x617)), neg(x121), x122)≥EVAL(pos(x587), minus_int(pos(s(x617)), x122), plus_int(neg(x121), pos(s(x617))), x122))

(124)    (greatereq_int(neg(x623), neg(x622))=true∧(∀x624,x625,x626:greatereq_int(neg(x623), neg(x622))=true ⇒ COND_EVAL1(true, pos(x624), neg(x622), neg(x625), x626)≥EVAL(pos(x624), minus_int(neg(x622), x626), plus_int(neg(x625), neg(x622)), x626)) ⇒ COND_EVAL1(true, pos(x587), neg(s(x622)), neg(x121), x122)≥EVAL(pos(x587), minus_int(neg(s(x622)), x122), plus_int(neg(x121), neg(s(x622))), x122))

We simplified constraint (119) using rules (I), (II) which results in the following new constraint:
(125)    (COND_EVAL1(true, pos(x587), pos(0), neg(x121), x122)≥EVAL(pos(x587), minus_int(pos(0), x122), plus_int(neg(x121), pos(0)), x122))

We simplified constraint (120) using rules (I), (II) which results in the following new constraint:
(126)    (COND_EVAL1(true, pos(x587), pos(0), neg(x121), x122)≥EVAL(pos(x587), minus_int(pos(0), x122), plus_int(neg(x121), pos(0)), x122))

We simplified constraint (121) using rules (I), (II) which results in the following new constraint:
(127)    (COND_EVAL1(true, pos(x587), neg(x610), neg(x121), x122)≥EVAL(pos(x587), minus_int(neg(x610), x122), plus_int(neg(x121), neg(x610)), x122))

We simplified constraint (122) using rules (I), (II) which results in the following new constraint:
(128)    (COND_EVAL1(true, pos(x587), neg(x611), neg(x121), x122)≥EVAL(pos(x587), minus_int(neg(x611), x122), plus_int(neg(x121), neg(x611)), x122))

We simplified constraint (123) using rule (VI) where we applied the induction hypothesis (∀x619,x620,x621:greatereq_int(pos(x618), pos(x617))=true ⇒ COND_EVAL1(true, pos(x619), pos(x617), neg(x620), x621)≥EVAL(pos(x619), minus_int(pos(x617), x621), plus_int(neg(x620), pos(x617)), x621)) with σ = [x621 / x122, x620 / x121, x619 / x587] which results in the following new constraint:
(129)    (COND_EVAL1(true, pos(x587), pos(x617), neg(x121), x122)≥EVAL(pos(x587), minus_int(pos(x617), x122), plus_int(neg(x121), pos(x617)), x122) ⇒ COND_EVAL1(true, pos(x587), pos(s(x617)), neg(x121), x122)≥EVAL(pos(x587), minus_int(pos(s(x617)), x122), plus_int(neg(x121), pos(s(x617))), x122))

We simplified constraint (124) using rule (VI) where we applied the induction hypothesis (∀x624,x625,x626:greatereq_int(neg(x623), neg(x622))=true ⇒ COND_EVAL1(true, pos(x624), neg(x622), neg(x625), x626)≥EVAL(pos(x624), minus_int(neg(x622), x626), plus_int(neg(x625), neg(x622)), x626)) with σ = [x625 / x121, x624 / x587, x626 / x122] which results in the following new constraint:
(130)    (COND_EVAL1(true, pos(x587), neg(x622), neg(x121), x122)≥EVAL(pos(x587), minus_int(neg(x622), x122), plus_int(neg(x121), neg(x622)), x122) ⇒ COND_EVAL1(true, pos(x587), neg(s(x622)), neg(x121), x122)≥EVAL(pos(x587), minus_int(neg(s(x622)), x122), plus_int(neg(x121), neg(s(x622))), x122))

We simplified constraint (118) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x586, x120)=true which results in the following new constraints:
(131)    (true=true ⇒ COND_EVAL1(true, neg(0), pos(0), neg(x121), x122)≥EVAL(neg(0), minus_int(pos(0), x122), plus_int(neg(x121), pos(0)), x122))

(132)    (true=true ⇒ COND_EVAL1(true, neg(0), pos(0), neg(x121), x122)≥EVAL(neg(0), minus_int(pos(0), x122), plus_int(neg(x121), pos(0)), x122))

(133)    (true=true ⇒ COND_EVAL1(true, neg(0), neg(x629), neg(x121), x122)≥EVAL(neg(0), minus_int(neg(x629), x122), plus_int(neg(x121), neg(x629)), x122))

(134)    (true=true ⇒ COND_EVAL1(true, neg(0), neg(x630), neg(x121), x122)≥EVAL(neg(0), minus_int(neg(x630), x122), plus_int(neg(x121), neg(x630)), x122))

(135)    (greatereq_int(pos(x637), pos(x636))=true∧(∀x638,x639:greatereq_int(pos(x637), pos(x636))=true ⇒ COND_EVAL1(true, neg(0), pos(x636), neg(x638), x639)≥EVAL(neg(0), minus_int(pos(x636), x639), plus_int(neg(x638), pos(x636)), x639)) ⇒ COND_EVAL1(true, neg(0), pos(s(x636)), neg(x121), x122)≥EVAL(neg(0), minus_int(pos(s(x636)), x122), plus_int(neg(x121), pos(s(x636))), x122))

(136)    (greatereq_int(neg(x641), neg(x640))=true∧(∀x642,x643:greatereq_int(neg(x641), neg(x640))=true ⇒ COND_EVAL1(true, neg(0), neg(x640), neg(x642), x643)≥EVAL(neg(0), minus_int(neg(x640), x643), plus_int(neg(x642), neg(x640)), x643)) ⇒ COND_EVAL1(true, neg(0), neg(s(x640)), neg(x121), x122)≥EVAL(neg(0), minus_int(neg(s(x640)), x122), plus_int(neg(x121), neg(s(x640))), x122))

We simplified constraint (131) using rules (I), (II) which results in the following new constraint:
(137)    (COND_EVAL1(true, neg(0), pos(0), neg(x121), x122)≥EVAL(neg(0), minus_int(pos(0), x122), plus_int(neg(x121), pos(0)), x122))

We simplified constraint (132) using rules (I), (II) which results in the following new constraint:
(138)    (COND_EVAL1(true, neg(0), pos(0), neg(x121), x122)≥EVAL(neg(0), minus_int(pos(0), x122), plus_int(neg(x121), pos(0)), x122))

We simplified constraint (133) using rules (I), (II) which results in the following new constraint:
(139)    (COND_EVAL1(true, neg(0), neg(x629), neg(x121), x122)≥EVAL(neg(0), minus_int(neg(x629), x122), plus_int(neg(x121), neg(x629)), x122))

We simplified constraint (134) using rules (I), (II) which results in the following new constraint:
(140)    (COND_EVAL1(true, neg(0), neg(x630), neg(x121), x122)≥EVAL(neg(0), minus_int(neg(x630), x122), plus_int(neg(x121), neg(x630)), x122))

We simplified constraint (135) using rule (VI) where we applied the induction hypothesis (∀x638,x639:greatereq_int(pos(x637), pos(x636))=true ⇒ COND_EVAL1(true, neg(0), pos(x636), neg(x638), x639)≥EVAL(neg(0), minus_int(pos(x636), x639), plus_int(neg(x638), pos(x636)), x639)) with σ = [x638 / x121, x639 / x122] which results in the following new constraint:
(141)    (COND_EVAL1(true, neg(0), pos(x636), neg(x121), x122)≥EVAL(neg(0), minus_int(pos(x636), x122), plus_int(neg(x121), pos(x636)), x122) ⇒ COND_EVAL1(true, neg(0), pos(s(x636)), neg(x121), x122)≥EVAL(neg(0), minus_int(pos(s(x636)), x122), plus_int(neg(x121), pos(s(x636))), x122))

We simplified constraint (136) using rule (VI) where we applied the induction hypothesis (∀x642,x643:greatereq_int(neg(x641), neg(x640))=true ⇒ COND_EVAL1(true, neg(0), neg(x640), neg(x642), x643)≥EVAL(neg(0), minus_int(neg(x640), x643), plus_int(neg(x642), neg(x640)), x643)) with σ = [x643 / x122, x642 / x121] which results in the following new constraint:
(142)    (COND_EVAL1(true, neg(0), neg(x640), neg(x121), x122)≥EVAL(neg(0), minus_int(neg(x640), x122), plus_int(neg(x121), neg(x640)), x122) ⇒ COND_EVAL1(true, neg(0), neg(s(x640)), neg(x121), x122)≥EVAL(neg(0), minus_int(neg(s(x640)), x122), plus_int(neg(x121), neg(s(x640))), x122))

For Pair COND_EVAL1(true, neg(0), z0, z1, x_removed) → EVAL(neg(0), minus_int(z0, x_removed), plus_int(z1, z0), x_removed) the following chains were created:

We consider the chain EVAL(neg(0), y1, y2, x_removed) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), neg(0), y1, y2, x_removed), COND_EVAL1(true, neg(0), z0, z1, x_removed) → EVAL(neg(0), minus_int(z0, x_removed), plus_int(z1, z0), x_removed) which results in the following constraint:
(143)    (COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(x136, x136), x136), x135)), neg(0), x135, x136, x137)=COND_EVAL1(true, neg(0), x138, x139, x140) ⇒ COND_EVAL1(true, neg(0), x138, x139, x140)≥EVAL(neg(0), minus_int(x138, x140), plus_int(x139, x138), x140))

We simplified constraint (143) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
(144)    (true=x644∧greatereq_int(x646, x135)=x645∧and(x644, x645)=true ⇒ COND_EVAL1(true, neg(0), x135, x136, x137)≥EVAL(neg(0), minus_int(x135, x137), plus_int(x136, x135), x137))

We simplified constraint (144) using rule (V) (with possible (I) afterwards) using induction on and(x644, x645)=true which results in the following new constraint:
(145)    (true=true∧greatereq_int(x646, x135)=true ⇒ COND_EVAL1(true, neg(0), x135, x136, x137)≥EVAL(neg(0), minus_int(x135, x137), plus_int(x136, x135), x137))

We simplified constraint (145) using rules (I), (II) which results in the following new constraint:
(146)    (greatereq_int(x646, x135)=true ⇒ COND_EVAL1(true, neg(0), x135, x136, x137)≥EVAL(neg(0), minus_int(x135, x137), plus_int(x136, x135), x137))

We simplified constraint (146) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x646, x135)=true which results in the following new constraints:
(147)    (true=true ⇒ COND_EVAL1(true, neg(0), pos(0), x136, x137)≥EVAL(neg(0), minus_int(pos(0), x137), plus_int(x136, pos(0)), x137))

(148)    (true=true ⇒ COND_EVAL1(true, neg(0), pos(0), x136, x137)≥EVAL(neg(0), minus_int(pos(0), x137), plus_int(x136, pos(0)), x137))

(149)    (true=true ⇒ COND_EVAL1(true, neg(0), neg(x651), x136, x137)≥EVAL(neg(0), minus_int(neg(x651), x137), plus_int(x136, neg(x651)), x137))

(150)    (true=true ⇒ COND_EVAL1(true, neg(0), neg(x652), x136, x137)≥EVAL(neg(0), minus_int(neg(x652), x137), plus_int(x136, neg(x652)), x137))

(151)    (greatereq_int(pos(x659), pos(x658))=true∧(∀x660,x661:greatereq_int(pos(x659), pos(x658))=true ⇒ COND_EVAL1(true, neg(0), pos(x658), x660, x661)≥EVAL(neg(0), minus_int(pos(x658), x661), plus_int(x660, pos(x658)), x661)) ⇒ COND_EVAL1(true, neg(0), pos(s(x658)), x136, x137)≥EVAL(neg(0), minus_int(pos(s(x658)), x137), plus_int(x136, pos(s(x658))), x137))

(152)    (greatereq_int(neg(x663), neg(x662))=true∧(∀x664,x665:greatereq_int(neg(x663), neg(x662))=true ⇒ COND_EVAL1(true, neg(0), neg(x662), x664, x665)≥EVAL(neg(0), minus_int(neg(x662), x665), plus_int(x664, neg(x662)), x665)) ⇒ COND_EVAL1(true, neg(0), neg(s(x662)), x136, x137)≥EVAL(neg(0), minus_int(neg(s(x662)), x137), plus_int(x136, neg(s(x662))), x137))

We simplified constraint (147) using rules (I), (II) which results in the following new constraint:
(153)    (COND_EVAL1(true, neg(0), pos(0), x136, x137)≥EVAL(neg(0), minus_int(pos(0), x137), plus_int(x136, pos(0)), x137))

We simplified constraint (148) using rules (I), (II) which results in the following new constraint:
(154)    (COND_EVAL1(true, neg(0), pos(0), x136, x137)≥EVAL(neg(0), minus_int(pos(0), x137), plus_int(x136, pos(0)), x137))

We simplified constraint (149) using rules (I), (II) which results in the following new constraint:
(155)    (COND_EVAL1(true, neg(0), neg(x651), x136, x137)≥EVAL(neg(0), minus_int(neg(x651), x137), plus_int(x136, neg(x651)), x137))

We simplified constraint (150) using rules (I), (II) which results in the following new constraint:
(156)    (COND_EVAL1(true, neg(0), neg(x652), x136, x137)≥EVAL(neg(0), minus_int(neg(x652), x137), plus_int(x136, neg(x652)), x137))

We simplified constraint (151) using rule (VI) where we applied the induction hypothesis (∀x660,x661:greatereq_int(pos(x659), pos(x658))=true ⇒ COND_EVAL1(true, neg(0), pos(x658), x660, x661)≥EVAL(neg(0), minus_int(pos(x658), x661), plus_int(x660, pos(x658)), x661)) with σ = [x660 / x136, x661 / x137] which results in the following new constraint:
(157)    (COND_EVAL1(true, neg(0), pos(x658), x136, x137)≥EVAL(neg(0), minus_int(pos(x658), x137), plus_int(x136, pos(x658)), x137) ⇒ COND_EVAL1(true, neg(0), pos(s(x658)), x136, x137)≥EVAL(neg(0), minus_int(pos(s(x658)), x137), plus_int(x136, pos(s(x658))), x137))

We simplified constraint (152) using rule (VI) where we applied the induction hypothesis (∀x664,x665:greatereq_int(neg(x663), neg(x662))=true ⇒ COND_EVAL1(true, neg(0), neg(x662), x664, x665)≥EVAL(neg(0), minus_int(neg(x662), x665), plus_int(x664, neg(x662)), x665)) with σ = [x664 / x136, x665 / x137] which results in the following new constraint:
(158)    (COND_EVAL1(true, neg(0), neg(x662), x136, x137)≥EVAL(neg(0), minus_int(neg(x662), x137), plus_int(x136, neg(x662)), x137) ⇒ COND_EVAL1(true, neg(0), neg(s(x662)), x136, x137)≥EVAL(neg(0), minus_int(neg(s(x662)), x137), plus_int(x136, neg(s(x662))), x137))
We consider the chain EVAL(y0, y1, neg(x0), x_removed) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, neg(x0), x_removed), COND_EVAL1(true, neg(0), z0, z1, x_removed) → EVAL(neg(0), minus_int(z0, x_removed), plus_int(z1, z0), x_removed) which results in the following constraint:
(159)    (COND_EVAL1(and(greatereq_int(x160, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x162, x162), x162)), x161)), x160, x161, neg(x162), x163)=COND_EVAL1(true, neg(0), x164, x165, x166) ⇒ COND_EVAL1(true, neg(0), x164, x165, x166)≥EVAL(neg(0), minus_int(x164, x166), plus_int(x165, x164), x166))

We simplified constraint (159) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
(160)    (neg(0)=x668∧pos(0)=x669∧greatereq_int(x668, x669)=x666∧greatereq_int(x670, x161)=x667∧and(x666, x667)=true ⇒ COND_EVAL1(true, neg(0), x161, neg(x162), x163)≥EVAL(neg(0), minus_int(x161, x163), plus_int(neg(x162), x161), x163))

We simplified constraint (160) using rule (V) (with possible (I) afterwards) using induction on and(x666, x667)=true which results in the following new constraint:
(161)    (true=true∧neg(0)=x668∧pos(0)=x669∧greatereq_int(x668, x669)=true∧greatereq_int(x670, x161)=true ⇒ COND_EVAL1(true, neg(0), x161, neg(x162), x163)≥EVAL(neg(0), minus_int(x161, x163), plus_int(neg(x162), x161), x163))

We simplified constraint (161) using rules (I), (II) which results in the following new constraint:
(162)    (neg(0)=x668∧pos(0)=x669∧greatereq_int(x668, x669)=true∧greatereq_int(x670, x161)=true ⇒ COND_EVAL1(true, neg(0), x161, neg(x162), x163)≥EVAL(neg(0), minus_int(x161, x163), plus_int(neg(x162), x161), x163))

We simplified constraint (162) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x668, x669)=true which results in the following new constraints:
(163)    (true=true∧neg(0)=pos(x671)∧pos(0)=pos(0)∧greatereq_int(x670, x161)=true ⇒ COND_EVAL1(true, neg(0), x161, neg(x162), x163)≥EVAL(neg(0), minus_int(x161, x163), plus_int(neg(x162), x161), x163))

(164)    (true=true∧neg(0)=neg(0)∧pos(0)=pos(0)∧greatereq_int(x670, x161)=true ⇒ COND_EVAL1(true, neg(0), x161, neg(x162), x163)≥EVAL(neg(0), minus_int(x161, x163), plus_int(neg(x162), x161), x163))

(165)    (true=true∧neg(0)=neg(0)∧pos(0)=neg(x673)∧greatereq_int(x670, x161)=true ⇒ COND_EVAL1(true, neg(0), x161, neg(x162), x163)≥EVAL(neg(0), minus_int(x161, x163), plus_int(neg(x162), x161), x163))

(166)    (true=true∧neg(0)=pos(x675)∧pos(0)=neg(x674)∧greatereq_int(x670, x161)=true ⇒ COND_EVAL1(true, neg(0), x161, neg(x162), x163)≥EVAL(neg(0), minus_int(x161, x163), plus_int(neg(x162), x161), x163))

(167)    (greatereq_int(pos(x681), pos(x680))=true∧neg(0)=pos(s(x681))∧pos(0)=pos(s(x680))∧greatereq_int(x670, x161)=true∧(∀x682,x683,x684,x685:greatereq_int(pos(x681), pos(x680))=true∧neg(0)=pos(x681)∧pos(0)=pos(x680)∧greatereq_int(x682, x683)=true ⇒ COND_EVAL1(true, neg(0), x683, neg(x684), x685)≥EVAL(neg(0), minus_int(x683, x685), plus_int(neg(x684), x683), x685)) ⇒ COND_EVAL1(true, neg(0), x161, neg(x162), x163)≥EVAL(neg(0), minus_int(x161, x163), plus_int(neg(x162), x161), x163))

(168)    (greatereq_int(neg(x687), neg(x686))=true∧neg(0)=neg(s(x687))∧pos(0)=neg(s(x686))∧greatereq_int(x670, x161)=true∧(∀x688,x689,x690,x691:greatereq_int(neg(x687), neg(x686))=true∧neg(0)=neg(x687)∧pos(0)=neg(x686)∧greatereq_int(x688, x689)=true ⇒ COND_EVAL1(true, neg(0), x689, neg(x690), x691)≥EVAL(neg(0), minus_int(x689, x691), plus_int(neg(x690), x689), x691)) ⇒ COND_EVAL1(true, neg(0), x161, neg(x162), x163)≥EVAL(neg(0), minus_int(x161, x163), plus_int(neg(x162), x161), x163))

We solved constraint (163) using rules (I), (II).We simplified constraint (164) using rules (I), (II) which results in the following new constraint:
(169)    (greatereq_int(x670, x161)=true ⇒ COND_EVAL1(true, neg(0), x161, neg(x162), x163)≥EVAL(neg(0), minus_int(x161, x163), plus_int(neg(x162), x161), x163))

We solved constraint (165) using rules (I), (II).We solved constraint (166) using rules (I), (II).We solved constraint (167) using rules (I), (II).We solved constraint (168) using rules (I), (II).We simplified constraint (169) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x670, x161)=true which results in the following new constraints:
(170)    (true=true ⇒ COND_EVAL1(true, neg(0), pos(0), neg(x162), x163)≥EVAL(neg(0), minus_int(pos(0), x163), plus_int(neg(x162), pos(0)), x163))

(171)    (true=true ⇒ COND_EVAL1(true, neg(0), pos(0), neg(x162), x163)≥EVAL(neg(0), minus_int(pos(0), x163), plus_int(neg(x162), pos(0)), x163))

(172)    (true=true ⇒ COND_EVAL1(true, neg(0), neg(x694), neg(x162), x163)≥EVAL(neg(0), minus_int(neg(x694), x163), plus_int(neg(x162), neg(x694)), x163))

(173)    (true=true ⇒ COND_EVAL1(true, neg(0), neg(x695), neg(x162), x163)≥EVAL(neg(0), minus_int(neg(x695), x163), plus_int(neg(x162), neg(x695)), x163))

(174)    (greatereq_int(pos(x702), pos(x701))=true∧(∀x703,x704:greatereq_int(pos(x702), pos(x701))=true ⇒ COND_EVAL1(true, neg(0), pos(x701), neg(x703), x704)≥EVAL(neg(0), minus_int(pos(x701), x704), plus_int(neg(x703), pos(x701)), x704)) ⇒ COND_EVAL1(true, neg(0), pos(s(x701)), neg(x162), x163)≥EVAL(neg(0), minus_int(pos(s(x701)), x163), plus_int(neg(x162), pos(s(x701))), x163))

(175)    (greatereq_int(neg(x706), neg(x705))=true∧(∀x707,x708:greatereq_int(neg(x706), neg(x705))=true ⇒ COND_EVAL1(true, neg(0), neg(x705), neg(x707), x708)≥EVAL(neg(0), minus_int(neg(x705), x708), plus_int(neg(x707), neg(x705)), x708)) ⇒ COND_EVAL1(true, neg(0), neg(s(x705)), neg(x162), x163)≥EVAL(neg(0), minus_int(neg(s(x705)), x163), plus_int(neg(x162), neg(s(x705))), x163))

We simplified constraint (170) using rules (I), (II) which results in the following new constraint:
(176)    (COND_EVAL1(true, neg(0), pos(0), neg(x162), x163)≥EVAL(neg(0), minus_int(pos(0), x163), plus_int(neg(x162), pos(0)), x163))

We simplified constraint (171) using rules (I), (II) which results in the following new constraint:
(177)    (COND_EVAL1(true, neg(0), pos(0), neg(x162), x163)≥EVAL(neg(0), minus_int(pos(0), x163), plus_int(neg(x162), pos(0)), x163))

We simplified constraint (172) using rules (I), (II) which results in the following new constraint:
(178)    (COND_EVAL1(true, neg(0), neg(x694), neg(x162), x163)≥EVAL(neg(0), minus_int(neg(x694), x163), plus_int(neg(x162), neg(x694)), x163))

We simplified constraint (173) using rules (I), (II) which results in the following new constraint:
(179)    (COND_EVAL1(true, neg(0), neg(x695), neg(x162), x163)≥EVAL(neg(0), minus_int(neg(x695), x163), plus_int(neg(x162), neg(x695)), x163))

We simplified constraint (174) using rule (VI) where we applied the induction hypothesis (∀x703,x704:greatereq_int(pos(x702), pos(x701))=true ⇒ COND_EVAL1(true, neg(0), pos(x701), neg(x703), x704)≥EVAL(neg(0), minus_int(pos(x701), x704), plus_int(neg(x703), pos(x701)), x704)) with σ = [x704 / x163, x703 / x162] which results in the following new constraint:
(180)    (COND_EVAL1(true, neg(0), pos(x701), neg(x162), x163)≥EVAL(neg(0), minus_int(pos(x701), x163), plus_int(neg(x162), pos(x701)), x163) ⇒ COND_EVAL1(true, neg(0), pos(s(x701)), neg(x162), x163)≥EVAL(neg(0), minus_int(pos(s(x701)), x163), plus_int(neg(x162), pos(s(x701))), x163))

We simplified constraint (175) using rule (VI) where we applied the induction hypothesis (∀x707,x708:greatereq_int(neg(x706), neg(x705))=true ⇒ COND_EVAL1(true, neg(0), neg(x705), neg(x707), x708)≥EVAL(neg(0), minus_int(neg(x705), x708), plus_int(neg(x707), neg(x705)), x708)) with σ = [x708 / x163, x707 / x162] which results in the following new constraint:
(181)    (COND_EVAL1(true, neg(0), neg(x705), neg(x162), x163)≥EVAL(neg(0), minus_int(neg(x705), x163), plus_int(neg(x162), neg(x705)), x163) ⇒ COND_EVAL1(true, neg(0), neg(s(x705)), neg(x162), x163)≥EVAL(neg(0), minus_int(neg(s(x705)), x163), plus_int(neg(x162), neg(s(x705))), x163))
We consider the chain EVAL(y0, y1, pos(x0), x_removed) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, pos(x0), x_removed), COND_EVAL1(true, neg(0), z0, z1, x_removed) → EVAL(neg(0), minus_int(z0, x_removed), plus_int(z1, z0), x_removed) which results in the following constraint:
(182)    (COND_EVAL1(and(greatereq_int(x171, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x173, x173), x173)), x172)), x171, x172, pos(x173), x174)=COND_EVAL1(true, neg(0), x175, x176, x177) ⇒ COND_EVAL1(true, neg(0), x175, x176, x177)≥EVAL(neg(0), minus_int(x175, x177), plus_int(x176, x175), x177))

We simplified constraint (182) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
(183)    (neg(0)=x711∧pos(0)=x712∧greatereq_int(x711, x712)=x709∧greatereq_int(x713, x172)=x710∧and(x709, x710)=true ⇒ COND_EVAL1(true, neg(0), x172, pos(x173), x174)≥EVAL(neg(0), minus_int(x172, x174), plus_int(pos(x173), x172), x174))

We simplified constraint (183) using rule (V) (with possible (I) afterwards) using induction on and(x709, x710)=true which results in the following new constraint:
(184)    (true=true∧neg(0)=x711∧pos(0)=x712∧greatereq_int(x711, x712)=true∧greatereq_int(x713, x172)=true ⇒ COND_EVAL1(true, neg(0), x172, pos(x173), x174)≥EVAL(neg(0), minus_int(x172, x174), plus_int(pos(x173), x172), x174))

We simplified constraint (184) using rules (I), (II) which results in the following new constraint:
(185)    (neg(0)=x711∧pos(0)=x712∧greatereq_int(x711, x712)=true∧greatereq_int(x713, x172)=true ⇒ COND_EVAL1(true, neg(0), x172, pos(x173), x174)≥EVAL(neg(0), minus_int(x172, x174), plus_int(pos(x173), x172), x174))

We simplified constraint (185) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x711, x712)=true which results in the following new constraints:
(186)    (true=true∧neg(0)=pos(x714)∧pos(0)=pos(0)∧greatereq_int(x713, x172)=true ⇒ COND_EVAL1(true, neg(0), x172, pos(x173), x174)≥EVAL(neg(0), minus_int(x172, x174), plus_int(pos(x173), x172), x174))

(187)    (true=true∧neg(0)=neg(0)∧pos(0)=pos(0)∧greatereq_int(x713, x172)=true ⇒ COND_EVAL1(true, neg(0), x172, pos(x173), x174)≥EVAL(neg(0), minus_int(x172, x174), plus_int(pos(x173), x172), x174))

(188)    (true=true∧neg(0)=neg(0)∧pos(0)=neg(x716)∧greatereq_int(x713, x172)=true ⇒ COND_EVAL1(true, neg(0), x172, pos(x173), x174)≥EVAL(neg(0), minus_int(x172, x174), plus_int(pos(x173), x172), x174))

(189)    (true=true∧neg(0)=pos(x718)∧pos(0)=neg(x717)∧greatereq_int(x713, x172)=true ⇒ COND_EVAL1(true, neg(0), x172, pos(x173), x174)≥EVAL(neg(0), minus_int(x172, x174), plus_int(pos(x173), x172), x174))

(190)    (greatereq_int(pos(x724), pos(x723))=true∧neg(0)=pos(s(x724))∧pos(0)=pos(s(x723))∧greatereq_int(x713, x172)=true∧(∀x725,x726,x727,x728:greatereq_int(pos(x724), pos(x723))=true∧neg(0)=pos(x724)∧pos(0)=pos(x723)∧greatereq_int(x725, x726)=true ⇒ COND_EVAL1(true, neg(0), x726, pos(x727), x728)≥EVAL(neg(0), minus_int(x726, x728), plus_int(pos(x727), x726), x728)) ⇒ COND_EVAL1(true, neg(0), x172, pos(x173), x174)≥EVAL(neg(0), minus_int(x172, x174), plus_int(pos(x173), x172), x174))

(191)    (greatereq_int(neg(x730), neg(x729))=true∧neg(0)=neg(s(x730))∧pos(0)=neg(s(x729))∧greatereq_int(x713, x172)=true∧(∀x731,x732,x733,x734:greatereq_int(neg(x730), neg(x729))=true∧neg(0)=neg(x730)∧pos(0)=neg(x729)∧greatereq_int(x731, x732)=true ⇒ COND_EVAL1(true, neg(0), x732, pos(x733), x734)≥EVAL(neg(0), minus_int(x732, x734), plus_int(pos(x733), x732), x734)) ⇒ COND_EVAL1(true, neg(0), x172, pos(x173), x174)≥EVAL(neg(0), minus_int(x172, x174), plus_int(pos(x173), x172), x174))

We solved constraint (186) using rules (I), (II).We simplified constraint (187) using rules (I), (II) which results in the following new constraint:
(192)    (greatereq_int(x713, x172)=true ⇒ COND_EVAL1(true, neg(0), x172, pos(x173), x174)≥EVAL(neg(0), minus_int(x172, x174), plus_int(pos(x173), x172), x174))

We solved constraint (188) using rules (I), (II).We solved constraint (189) using rules (I), (II).We solved constraint (190) using rules (I), (II).We solved constraint (191) using rules (I), (II).We simplified constraint (192) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x713, x172)=true which results in the following new constraints:
(193)    (true=true ⇒ COND_EVAL1(true, neg(0), pos(0), pos(x173), x174)≥EVAL(neg(0), minus_int(pos(0), x174), plus_int(pos(x173), pos(0)), x174))

(194)    (true=true ⇒ COND_EVAL1(true, neg(0), pos(0), pos(x173), x174)≥EVAL(neg(0), minus_int(pos(0), x174), plus_int(pos(x173), pos(0)), x174))

(195)    (true=true ⇒ COND_EVAL1(true, neg(0), neg(x737), pos(x173), x174)≥EVAL(neg(0), minus_int(neg(x737), x174), plus_int(pos(x173), neg(x737)), x174))

(196)    (true=true ⇒ COND_EVAL1(true, neg(0), neg(x738), pos(x173), x174)≥EVAL(neg(0), minus_int(neg(x738), x174), plus_int(pos(x173), neg(x738)), x174))

(197)    (greatereq_int(pos(x745), pos(x744))=true∧(∀x746,x747:greatereq_int(pos(x745), pos(x744))=true ⇒ COND_EVAL1(true, neg(0), pos(x744), pos(x746), x747)≥EVAL(neg(0), minus_int(pos(x744), x747), plus_int(pos(x746), pos(x744)), x747)) ⇒ COND_EVAL1(true, neg(0), pos(s(x744)), pos(x173), x174)≥EVAL(neg(0), minus_int(pos(s(x744)), x174), plus_int(pos(x173), pos(s(x744))), x174))

(198)    (greatereq_int(neg(x749), neg(x748))=true∧(∀x750,x751:greatereq_int(neg(x749), neg(x748))=true ⇒ COND_EVAL1(true, neg(0), neg(x748), pos(x750), x751)≥EVAL(neg(0), minus_int(neg(x748), x751), plus_int(pos(x750), neg(x748)), x751)) ⇒ COND_EVAL1(true, neg(0), neg(s(x748)), pos(x173), x174)≥EVAL(neg(0), minus_int(neg(s(x748)), x174), plus_int(pos(x173), neg(s(x748))), x174))

We simplified constraint (193) using rules (I), (II) which results in the following new constraint:
(199)    (COND_EVAL1(true, neg(0), pos(0), pos(x173), x174)≥EVAL(neg(0), minus_int(pos(0), x174), plus_int(pos(x173), pos(0)), x174))

We simplified constraint (194) using rules (I), (II) which results in the following new constraint:
(200)    (COND_EVAL1(true, neg(0), pos(0), pos(x173), x174)≥EVAL(neg(0), minus_int(pos(0), x174), plus_int(pos(x173), pos(0)), x174))

We simplified constraint (195) using rules (I), (II) which results in the following new constraint:
(201)    (COND_EVAL1(true, neg(0), neg(x737), pos(x173), x174)≥EVAL(neg(0), minus_int(neg(x737), x174), plus_int(pos(x173), neg(x737)), x174))

We simplified constraint (196) using rules (I), (II) which results in the following new constraint:
(202)    (COND_EVAL1(true, neg(0), neg(x738), pos(x173), x174)≥EVAL(neg(0), minus_int(neg(x738), x174), plus_int(pos(x173), neg(x738)), x174))

We simplified constraint (197) using rule (VI) where we applied the induction hypothesis (∀x746,x747:greatereq_int(pos(x745), pos(x744))=true ⇒ COND_EVAL1(true, neg(0), pos(x744), pos(x746), x747)≥EVAL(neg(0), minus_int(pos(x744), x747), plus_int(pos(x746), pos(x744)), x747)) with σ = [x747 / x174, x746 / x173] which results in the following new constraint:
(203)    (COND_EVAL1(true, neg(0), pos(x744), pos(x173), x174)≥EVAL(neg(0), minus_int(pos(x744), x174), plus_int(pos(x173), pos(x744)), x174) ⇒ COND_EVAL1(true, neg(0), pos(s(x744)), pos(x173), x174)≥EVAL(neg(0), minus_int(pos(s(x744)), x174), plus_int(pos(x173), pos(s(x744))), x174))

We simplified constraint (198) using rule (VI) where we applied the induction hypothesis (∀x750,x751:greatereq_int(neg(x749), neg(x748))=true ⇒ COND_EVAL1(true, neg(0), neg(x748), pos(x750), x751)≥EVAL(neg(0), minus_int(neg(x748), x751), plus_int(pos(x750), neg(x748)), x751)) with σ = [x751 / x174, x750 / x173] which results in the following new constraint:
(204)    (COND_EVAL1(true, neg(0), neg(x748), pos(x173), x174)≥EVAL(neg(0), minus_int(neg(x748), x174), plus_int(pos(x173), neg(x748)), x174) ⇒ COND_EVAL1(true, neg(0), neg(s(x748)), pos(x173), x174)≥EVAL(neg(0), minus_int(neg(s(x748)), x174), plus_int(pos(x173), neg(s(x748))), x174))

For Pair EVAL(pos(x0), y1, y2, x_removed) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), pos(x0), y1, y2, x_removed) the following chains were created:

We consider the chain COND_EVAL1(true, z0, z1, pos(z2), x_removed) → EVAL(z0, minus_int(z1, x_removed), plus_int(pos(z2), z1), x_removed), EVAL(pos(x0), y1, y2, x_removed) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), pos(x0), y1, y2, x_removed) which results in the following constraint:
(205) (EVAL(x181, minus_int(x182, x184), plus_int(pos(x183), x182), x184)=EVAL(pos(x185), x186, x187, x188) ⇒ EVAL(pos(x185), x186, x187, x188)≥COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(x187, x187), x187), x186)), pos(x185), x186, x187, x188))

We simplified constraint (205) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
(206) (EVAL(pos(x185), x186, x187, x184)≥COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(x187, x187), x187), x186)), pos(x185), x186, x187, x184))
We consider the chain COND_EVAL1(true, z0, z1, neg(z2), x_removed) → EVAL(z0, minus_int(z1, x_removed), plus_int(neg(z2), z1), x_removed), EVAL(pos(x0), y1, y2, x_removed) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), pos(x0), y1, y2, x_removed) which results in the following constraint:
(207) (EVAL(x189, minus_int(x190, x192), plus_int(neg(x191), x190), x192)=EVAL(pos(x193), x194, x195, x196) ⇒ EVAL(pos(x193), x194, x195, x196)≥COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(x195, x195), x195), x194)), pos(x193), x194, x195, x196))

We simplified constraint (207) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
(208) (EVAL(pos(x193), x194, x195, x192)≥COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(x195, x195), x195), x194)), pos(x193), x194, x195, x192))
We consider the chain COND_EVAL1(true, pos(z0), z1, z2, x_removed) → EVAL(pos(z0), minus_int(z1, x_removed), plus_int(z2, z1), x_removed), EVAL(pos(x0), y1, y2, x_removed) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), pos(x0), y1, y2, x_removed) which results in the following constraint:
(209) (EVAL(pos(x204), minus_int(x205, x207), plus_int(x206, x205), x207)=EVAL(pos(x208), x209, x210, x211) ⇒ EVAL(pos(x208), x209, x210, x211)≥COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(x210, x210), x210), x209)), pos(x208), x209, x210, x211))

We simplified constraint (209) using rules (I), (II), (III), (IV) which results in the following new constraint:
(210) (EVAL(pos(x204), x209, x210, x207)≥COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(x210, x210), x210), x209)), pos(x204), x209, x210, x207))

For Pair COND_EVAL1(true, pos(z0), z1, z2, x_removed) → EVAL(pos(z0), minus_int(z1, x_removed), plus_int(z2, z1), x_removed) the following chains were created:

We consider the chain EVAL(pos(x0), y1, y2, x_removed) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), pos(x0), y1, y2, x_removed), COND_EVAL1(true, pos(z0), z1, z2, x_removed) → EVAL(pos(z0), minus_int(z1, x_removed), plus_int(z2, z1), x_removed) which results in the following constraint:
(211)    (COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(x240, x240), x240), x239)), pos(x238), x239, x240, x241)=COND_EVAL1(true, pos(x242), x243, x244, x245) ⇒ COND_EVAL1(true, pos(x242), x243, x244, x245)≥EVAL(pos(x242), minus_int(x243, x245), plus_int(x244, x243), x245))

We simplified constraint (211) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
(212)    (true=x754∧greatereq_int(x756, x239)=x755∧and(x754, x755)=true ⇒ COND_EVAL1(true, pos(x238), x239, x240, x241)≥EVAL(pos(x238), minus_int(x239, x241), plus_int(x240, x239), x241))

We simplified constraint (212) using rule (V) (with possible (I) afterwards) using induction on and(x754, x755)=true which results in the following new constraint:
(213)    (true=true∧greatereq_int(x756, x239)=true ⇒ COND_EVAL1(true, pos(x238), x239, x240, x241)≥EVAL(pos(x238), minus_int(x239, x241), plus_int(x240, x239), x241))

We simplified constraint (213) using rules (I), (II) which results in the following new constraint:
(214)    (greatereq_int(x756, x239)=true ⇒ COND_EVAL1(true, pos(x238), x239, x240, x241)≥EVAL(pos(x238), minus_int(x239, x241), plus_int(x240, x239), x241))

We simplified constraint (214) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x756, x239)=true which results in the following new constraints:
(215)    (true=true ⇒ COND_EVAL1(true, pos(x238), pos(0), x240, x241)≥EVAL(pos(x238), minus_int(pos(0), x241), plus_int(x240, pos(0)), x241))

(216)    (true=true ⇒ COND_EVAL1(true, pos(x238), pos(0), x240, x241)≥EVAL(pos(x238), minus_int(pos(0), x241), plus_int(x240, pos(0)), x241))

(217)    (true=true ⇒ COND_EVAL1(true, pos(x238), neg(x761), x240, x241)≥EVAL(pos(x238), minus_int(neg(x761), x241), plus_int(x240, neg(x761)), x241))

(218)    (true=true ⇒ COND_EVAL1(true, pos(x238), neg(x762), x240, x241)≥EVAL(pos(x238), minus_int(neg(x762), x241), plus_int(x240, neg(x762)), x241))

(219)    (greatereq_int(pos(x769), pos(x768))=true∧(∀x770,x771,x772:greatereq_int(pos(x769), pos(x768))=true ⇒ COND_EVAL1(true, pos(x770), pos(x768), x771, x772)≥EVAL(pos(x770), minus_int(pos(x768), x772), plus_int(x771, pos(x768)), x772)) ⇒ COND_EVAL1(true, pos(x238), pos(s(x768)), x240, x241)≥EVAL(pos(x238), minus_int(pos(s(x768)), x241), plus_int(x240, pos(s(x768))), x241))

(220)    (greatereq_int(neg(x774), neg(x773))=true∧(∀x775,x776,x777:greatereq_int(neg(x774), neg(x773))=true ⇒ COND_EVAL1(true, pos(x775), neg(x773), x776, x777)≥EVAL(pos(x775), minus_int(neg(x773), x777), plus_int(x776, neg(x773)), x777)) ⇒ COND_EVAL1(true, pos(x238), neg(s(x773)), x240, x241)≥EVAL(pos(x238), minus_int(neg(s(x773)), x241), plus_int(x240, neg(s(x773))), x241))

We simplified constraint (215) using rules (I), (II) which results in the following new constraint:
(221)    (COND_EVAL1(true, pos(x238), pos(0), x240, x241)≥EVAL(pos(x238), minus_int(pos(0), x241), plus_int(x240, pos(0)), x241))

We simplified constraint (216) using rules (I), (II) which results in the following new constraint:
(222)    (COND_EVAL1(true, pos(x238), pos(0), x240, x241)≥EVAL(pos(x238), minus_int(pos(0), x241), plus_int(x240, pos(0)), x241))

We simplified constraint (217) using rules (I), (II) which results in the following new constraint:
(223)    (COND_EVAL1(true, pos(x238), neg(x761), x240, x241)≥EVAL(pos(x238), minus_int(neg(x761), x241), plus_int(x240, neg(x761)), x241))

We simplified constraint (218) using rules (I), (II) which results in the following new constraint:
(224)    (COND_EVAL1(true, pos(x238), neg(x762), x240, x241)≥EVAL(pos(x238), minus_int(neg(x762), x241), plus_int(x240, neg(x762)), x241))

We simplified constraint (219) using rule (VI) where we applied the induction hypothesis (∀x770,x771,x772:greatereq_int(pos(x769), pos(x768))=true ⇒ COND_EVAL1(true, pos(x770), pos(x768), x771, x772)≥EVAL(pos(x770), minus_int(pos(x768), x772), plus_int(x771, pos(x768)), x772)) with σ = [x772 / x241, x770 / x238, x771 / x240] which results in the following new constraint:
(225)    (COND_EVAL1(true, pos(x238), pos(x768), x240, x241)≥EVAL(pos(x238), minus_int(pos(x768), x241), plus_int(x240, pos(x768)), x241) ⇒ COND_EVAL1(true, pos(x238), pos(s(x768)), x240, x241)≥EVAL(pos(x238), minus_int(pos(s(x768)), x241), plus_int(x240, pos(s(x768))), x241))

We simplified constraint (220) using rule (VI) where we applied the induction hypothesis (∀x775,x776,x777:greatereq_int(neg(x774), neg(x773))=true ⇒ COND_EVAL1(true, pos(x775), neg(x773), x776, x777)≥EVAL(pos(x775), minus_int(neg(x773), x777), plus_int(x776, neg(x773)), x777)) with σ = [x775 / x238, x776 / x240, x777 / x241] which results in the following new constraint:
(226)    (COND_EVAL1(true, pos(x238), neg(x773), x240, x241)≥EVAL(pos(x238), minus_int(neg(x773), x241), plus_int(x240, neg(x773)), x241) ⇒ COND_EVAL1(true, pos(x238), neg(s(x773)), x240, x241)≥EVAL(pos(x238), minus_int(neg(s(x773)), x241), plus_int(x240, neg(s(x773))), x241))
We consider the chain EVAL(y0, y1, neg(x0), x_removed) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, neg(x0), x_removed), COND_EVAL1(true, pos(z0), z1, z2, x_removed) → EVAL(pos(z0), minus_int(z1, x_removed), plus_int(z2, z1), x_removed) which results in the following constraint:
(227)    (COND_EVAL1(and(greatereq_int(x250, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x252, x252), x252)), x251)), x250, x251, neg(x252), x253)=COND_EVAL1(true, pos(x254), x255, x256, x257) ⇒ COND_EVAL1(true, pos(x254), x255, x256, x257)≥EVAL(pos(x254), minus_int(x255, x257), plus_int(x256, x255), x257))

We simplified constraint (227) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
(228)    (pos(x254)=x780∧pos(0)=x781∧greatereq_int(x780, x781)=x778∧greatereq_int(x782, x251)=x779∧and(x778, x779)=true ⇒ COND_EVAL1(true, pos(x254), x251, neg(x252), x253)≥EVAL(pos(x254), minus_int(x251, x253), plus_int(neg(x252), x251), x253))

We simplified constraint (228) using rule (V) (with possible (I) afterwards) using induction on and(x778, x779)=true which results in the following new constraint:
(229)    (true=true∧pos(x254)=x780∧pos(0)=x781∧greatereq_int(x780, x781)=true∧greatereq_int(x782, x251)=true ⇒ COND_EVAL1(true, pos(x254), x251, neg(x252), x253)≥EVAL(pos(x254), minus_int(x251, x253), plus_int(neg(x252), x251), x253))

We simplified constraint (229) using rules (I), (II) which results in the following new constraint:
(230)    (pos(x254)=x780∧pos(0)=x781∧greatereq_int(x780, x781)=true∧greatereq_int(x782, x251)=true ⇒ COND_EVAL1(true, pos(x254), x251, neg(x252), x253)≥EVAL(pos(x254), minus_int(x251, x253), plus_int(neg(x252), x251), x253))

We simplified constraint (230) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x780, x781)=true which results in the following new constraints:
(231)    (true=true∧pos(x254)=pos(x783)∧pos(0)=pos(0)∧greatereq_int(x782, x251)=true ⇒ COND_EVAL1(true, pos(x254), x251, neg(x252), x253)≥EVAL(pos(x254), minus_int(x251, x253), plus_int(neg(x252), x251), x253))

(232)    (true=true∧pos(x254)=neg(0)∧pos(0)=pos(0)∧greatereq_int(x782, x251)=true ⇒ COND_EVAL1(true, pos(x254), x251, neg(x252), x253)≥EVAL(pos(x254), minus_int(x251, x253), plus_int(neg(x252), x251), x253))

(233)    (true=true∧pos(x254)=neg(0)∧pos(0)=neg(x785)∧greatereq_int(x782, x251)=true ⇒ COND_EVAL1(true, pos(x254), x251, neg(x252), x253)≥EVAL(pos(x254), minus_int(x251, x253), plus_int(neg(x252), x251), x253))

(234)    (true=true∧pos(x254)=pos(x787)∧pos(0)=neg(x786)∧greatereq_int(x782, x251)=true ⇒ COND_EVAL1(true, pos(x254), x251, neg(x252), x253)≥EVAL(pos(x254), minus_int(x251, x253), plus_int(neg(x252), x251), x253))

(235)    (greatereq_int(pos(x793), pos(x792))=true∧pos(x254)=pos(s(x793))∧pos(0)=pos(s(x792))∧greatereq_int(x782, x251)=true∧(∀x794,x795,x796,x797,x798:greatereq_int(pos(x793), pos(x792))=true∧pos(x794)=pos(x793)∧pos(0)=pos(x792)∧greatereq_int(x795, x796)=true ⇒ COND_EVAL1(true, pos(x794), x796, neg(x797), x798)≥EVAL(pos(x794), minus_int(x796, x798), plus_int(neg(x797), x796), x798)) ⇒ COND_EVAL1(true, pos(x254), x251, neg(x252), x253)≥EVAL(pos(x254), minus_int(x251, x253), plus_int(neg(x252), x251), x253))

(236)    (greatereq_int(neg(x800), neg(x799))=true∧pos(x254)=neg(s(x800))∧pos(0)=neg(s(x799))∧greatereq_int(x782, x251)=true∧(∀x801,x802,x803,x804,x805:greatereq_int(neg(x800), neg(x799))=true∧pos(x801)=neg(x800)∧pos(0)=neg(x799)∧greatereq_int(x802, x803)=true ⇒ COND_EVAL1(true, pos(x801), x803, neg(x804), x805)≥EVAL(pos(x801), minus_int(x803, x805), plus_int(neg(x804), x803), x805)) ⇒ COND_EVAL1(true, pos(x254), x251, neg(x252), x253)≥EVAL(pos(x254), minus_int(x251, x253), plus_int(neg(x252), x251), x253))

We simplified constraint (231) using rules (I), (II), (IV) which results in the following new constraint:
(237)    (greatereq_int(x782, x251)=true ⇒ COND_EVAL1(true, pos(x254), x251, neg(x252), x253)≥EVAL(pos(x254), minus_int(x251, x253), plus_int(neg(x252), x251), x253))

We solved constraint (232) using rules (I), (II).We solved constraint (233) using rules (I), (II).We solved constraint (234) using rules (I), (II).We solved constraint (235) using rules (I), (II).We solved constraint (236) using rules (I), (II).We simplified constraint (237) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x782, x251)=true which results in the following new constraints:
(238)    (true=true ⇒ COND_EVAL1(true, pos(x254), pos(0), neg(x252), x253)≥EVAL(pos(x254), minus_int(pos(0), x253), plus_int(neg(x252), pos(0)), x253))

(239)    (true=true ⇒ COND_EVAL1(true, pos(x254), pos(0), neg(x252), x253)≥EVAL(pos(x254), minus_int(pos(0), x253), plus_int(neg(x252), pos(0)), x253))

(240)    (true=true ⇒ COND_EVAL1(true, pos(x254), neg(x808), neg(x252), x253)≥EVAL(pos(x254), minus_int(neg(x808), x253), plus_int(neg(x252), neg(x808)), x253))

(241)    (true=true ⇒ COND_EVAL1(true, pos(x254), neg(x809), neg(x252), x253)≥EVAL(pos(x254), minus_int(neg(x809), x253), plus_int(neg(x252), neg(x809)), x253))

(242)    (greatereq_int(pos(x816), pos(x815))=true∧(∀x817,x818,x819:greatereq_int(pos(x816), pos(x815))=true ⇒ COND_EVAL1(true, pos(x817), pos(x815), neg(x818), x819)≥EVAL(pos(x817), minus_int(pos(x815), x819), plus_int(neg(x818), pos(x815)), x819)) ⇒ COND_EVAL1(true, pos(x254), pos(s(x815)), neg(x252), x253)≥EVAL(pos(x254), minus_int(pos(s(x815)), x253), plus_int(neg(x252), pos(s(x815))), x253))

(243)    (greatereq_int(neg(x821), neg(x820))=true∧(∀x822,x823,x824:greatereq_int(neg(x821), neg(x820))=true ⇒ COND_EVAL1(true, pos(x822), neg(x820), neg(x823), x824)≥EVAL(pos(x822), minus_int(neg(x820), x824), plus_int(neg(x823), neg(x820)), x824)) ⇒ COND_EVAL1(true, pos(x254), neg(s(x820)), neg(x252), x253)≥EVAL(pos(x254), minus_int(neg(s(x820)), x253), plus_int(neg(x252), neg(s(x820))), x253))

We simplified constraint (238) using rules (I), (II) which results in the following new constraint:
(244)    (COND_EVAL1(true, pos(x254), pos(0), neg(x252), x253)≥EVAL(pos(x254), minus_int(pos(0), x253), plus_int(neg(x252), pos(0)), x253))

We simplified constraint (239) using rules (I), (II) which results in the following new constraint:
(245)    (COND_EVAL1(true, pos(x254), pos(0), neg(x252), x253)≥EVAL(pos(x254), minus_int(pos(0), x253), plus_int(neg(x252), pos(0)), x253))

We simplified constraint (240) using rules (I), (II) which results in the following new constraint:
(246)    (COND_EVAL1(true, pos(x254), neg(x808), neg(x252), x253)≥EVAL(pos(x254), minus_int(neg(x808), x253), plus_int(neg(x252), neg(x808)), x253))

We simplified constraint (241) using rules (I), (II) which results in the following new constraint:
(247)    (COND_EVAL1(true, pos(x254), neg(x809), neg(x252), x253)≥EVAL(pos(x254), minus_int(neg(x809), x253), plus_int(neg(x252), neg(x809)), x253))

We simplified constraint (242) using rule (VI) where we applied the induction hypothesis (∀x817,x818,x819:greatereq_int(pos(x816), pos(x815))=true ⇒ COND_EVAL1(true, pos(x817), pos(x815), neg(x818), x819)≥EVAL(pos(x817), minus_int(pos(x815), x819), plus_int(neg(x818), pos(x815)), x819)) with σ = [x817 / x254, x818 / x252, x819 / x253] which results in the following new constraint:
(248)    (COND_EVAL1(true, pos(x254), pos(x815), neg(x252), x253)≥EVAL(pos(x254), minus_int(pos(x815), x253), plus_int(neg(x252), pos(x815)), x253) ⇒ COND_EVAL1(true, pos(x254), pos(s(x815)), neg(x252), x253)≥EVAL(pos(x254), minus_int(pos(s(x815)), x253), plus_int(neg(x252), pos(s(x815))), x253))

We simplified constraint (243) using rule (VI) where we applied the induction hypothesis (∀x822,x823,x824:greatereq_int(neg(x821), neg(x820))=true ⇒ COND_EVAL1(true, pos(x822), neg(x820), neg(x823), x824)≥EVAL(pos(x822), minus_int(neg(x820), x824), plus_int(neg(x823), neg(x820)), x824)) with σ = [x824 / x253, x823 / x252, x822 / x254] which results in the following new constraint:
(249)    (COND_EVAL1(true, pos(x254), neg(x820), neg(x252), x253)≥EVAL(pos(x254), minus_int(neg(x820), x253), plus_int(neg(x252), neg(x820)), x253) ⇒ COND_EVAL1(true, pos(x254), neg(s(x820)), neg(x252), x253)≥EVAL(pos(x254), minus_int(neg(s(x820)), x253), plus_int(neg(x252), neg(s(x820))), x253))
We consider the chain EVAL(y0, y1, pos(x0), x_removed) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, pos(x0), x_removed), COND_EVAL1(true, pos(z0), z1, z2, x_removed) → EVAL(pos(z0), minus_int(z1, x_removed), plus_int(z2, z1), x_removed) which results in the following constraint:
(250)    (COND_EVAL1(and(greatereq_int(x262, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x264, x264), x264)), x263)), x262, x263, pos(x264), x265)=COND_EVAL1(true, pos(x266), x267, x268, x269) ⇒ COND_EVAL1(true, pos(x266), x267, x268, x269)≥EVAL(pos(x266), minus_int(x267, x269), plus_int(x268, x267), x269))

We simplified constraint (250) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
(251)    (pos(x266)=x827∧pos(0)=x828∧greatereq_int(x827, x828)=x825∧greatereq_int(x829, x263)=x826∧and(x825, x826)=true ⇒ COND_EVAL1(true, pos(x266), x263, pos(x264), x265)≥EVAL(pos(x266), minus_int(x263, x265), plus_int(pos(x264), x263), x265))

We simplified constraint (251) using rule (V) (with possible (I) afterwards) using induction on and(x825, x826)=true which results in the following new constraint:
(252)    (true=true∧pos(x266)=x827∧pos(0)=x828∧greatereq_int(x827, x828)=true∧greatereq_int(x829, x263)=true ⇒ COND_EVAL1(true, pos(x266), x263, pos(x264), x265)≥EVAL(pos(x266), minus_int(x263, x265), plus_int(pos(x264), x263), x265))

We simplified constraint (252) using rules (I), (II) which results in the following new constraint:
(253)    (pos(x266)=x827∧pos(0)=x828∧greatereq_int(x827, x828)=true∧greatereq_int(x829, x263)=true ⇒ COND_EVAL1(true, pos(x266), x263, pos(x264), x265)≥EVAL(pos(x266), minus_int(x263, x265), plus_int(pos(x264), x263), x265))

We simplified constraint (253) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x827, x828)=true which results in the following new constraints:
(254)    (true=true∧pos(x266)=pos(x830)∧pos(0)=pos(0)∧greatereq_int(x829, x263)=true ⇒ COND_EVAL1(true, pos(x266), x263, pos(x264), x265)≥EVAL(pos(x266), minus_int(x263, x265), plus_int(pos(x264), x263), x265))

(255)    (true=true∧pos(x266)=neg(0)∧pos(0)=pos(0)∧greatereq_int(x829, x263)=true ⇒ COND_EVAL1(true, pos(x266), x263, pos(x264), x265)≥EVAL(pos(x266), minus_int(x263, x265), plus_int(pos(x264), x263), x265))

(256)    (true=true∧pos(x266)=neg(0)∧pos(0)=neg(x832)∧greatereq_int(x829, x263)=true ⇒ COND_EVAL1(true, pos(x266), x263, pos(x264), x265)≥EVAL(pos(x266), minus_int(x263, x265), plus_int(pos(x264), x263), x265))

(257)    (true=true∧pos(x266)=pos(x834)∧pos(0)=neg(x833)∧greatereq_int(x829, x263)=true ⇒ COND_EVAL1(true, pos(x266), x263, pos(x264), x265)≥EVAL(pos(x266), minus_int(x263, x265), plus_int(pos(x264), x263), x265))

(258)    (greatereq_int(pos(x840), pos(x839))=true∧pos(x266)=pos(s(x840))∧pos(0)=pos(s(x839))∧greatereq_int(x829, x263)=true∧(∀x841,x842,x843,x844,x845:greatereq_int(pos(x840), pos(x839))=true∧pos(x841)=pos(x840)∧pos(0)=pos(x839)∧greatereq_int(x842, x843)=true ⇒ COND_EVAL1(true, pos(x841), x843, pos(x844), x845)≥EVAL(pos(x841), minus_int(x843, x845), plus_int(pos(x844), x843), x845)) ⇒ COND_EVAL1(true, pos(x266), x263, pos(x264), x265)≥EVAL(pos(x266), minus_int(x263, x265), plus_int(pos(x264), x263), x265))

(259)    (greatereq_int(neg(x847), neg(x846))=true∧pos(x266)=neg(s(x847))∧pos(0)=neg(s(x846))∧greatereq_int(x829, x263)=true∧(∀x848,x849,x850,x851,x852:greatereq_int(neg(x847), neg(x846))=true∧pos(x848)=neg(x847)∧pos(0)=neg(x846)∧greatereq_int(x849, x850)=true ⇒ COND_EVAL1(true, pos(x848), x850, pos(x851), x852)≥EVAL(pos(x848), minus_int(x850, x852), plus_int(pos(x851), x850), x852)) ⇒ COND_EVAL1(true, pos(x266), x263, pos(x264), x265)≥EVAL(pos(x266), minus_int(x263, x265), plus_int(pos(x264), x263), x265))

We simplified constraint (254) using rules (I), (II), (IV) which results in the following new constraint:
(260)    (greatereq_int(x829, x263)=true ⇒ COND_EVAL1(true, pos(x266), x263, pos(x264), x265)≥EVAL(pos(x266), minus_int(x263, x265), plus_int(pos(x264), x263), x265))

We solved constraint (255) using rules (I), (II).We solved constraint (256) using rules (I), (II).We solved constraint (257) using rules (I), (II).We solved constraint (258) using rules (I), (II).We solved constraint (259) using rules (I), (II).We simplified constraint (260) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x829, x263)=true which results in the following new constraints:
(261)    (true=true ⇒ COND_EVAL1(true, pos(x266), pos(0), pos(x264), x265)≥EVAL(pos(x266), minus_int(pos(0), x265), plus_int(pos(x264), pos(0)), x265))

(262)    (true=true ⇒ COND_EVAL1(true, pos(x266), pos(0), pos(x264), x265)≥EVAL(pos(x266), minus_int(pos(0), x265), plus_int(pos(x264), pos(0)), x265))

(263)    (true=true ⇒ COND_EVAL1(true, pos(x266), neg(x855), pos(x264), x265)≥EVAL(pos(x266), minus_int(neg(x855), x265), plus_int(pos(x264), neg(x855)), x265))

(264)    (true=true ⇒ COND_EVAL1(true, pos(x266), neg(x856), pos(x264), x265)≥EVAL(pos(x266), minus_int(neg(x856), x265), plus_int(pos(x264), neg(x856)), x265))

(265)    (greatereq_int(pos(x863), pos(x862))=true∧(∀x864,x865,x866:greatereq_int(pos(x863), pos(x862))=true ⇒ COND_EVAL1(true, pos(x864), pos(x862), pos(x865), x866)≥EVAL(pos(x864), minus_int(pos(x862), x866), plus_int(pos(x865), pos(x862)), x866)) ⇒ COND_EVAL1(true, pos(x266), pos(s(x862)), pos(x264), x265)≥EVAL(pos(x266), minus_int(pos(s(x862)), x265), plus_int(pos(x264), pos(s(x862))), x265))

(266)    (greatereq_int(neg(x868), neg(x867))=true∧(∀x869,x870,x871:greatereq_int(neg(x868), neg(x867))=true ⇒ COND_EVAL1(true, pos(x869), neg(x867), pos(x870), x871)≥EVAL(pos(x869), minus_int(neg(x867), x871), plus_int(pos(x870), neg(x867)), x871)) ⇒ COND_EVAL1(true, pos(x266), neg(s(x867)), pos(x264), x265)≥EVAL(pos(x266), minus_int(neg(s(x867)), x265), plus_int(pos(x264), neg(s(x867))), x265))

We simplified constraint (261) using rules (I), (II) which results in the following new constraint:
(267)    (COND_EVAL1(true, pos(x266), pos(0), pos(x264), x265)≥EVAL(pos(x266), minus_int(pos(0), x265), plus_int(pos(x264), pos(0)), x265))

We simplified constraint (262) using rules (I), (II) which results in the following new constraint:
(268)    (COND_EVAL1(true, pos(x266), pos(0), pos(x264), x265)≥EVAL(pos(x266), minus_int(pos(0), x265), plus_int(pos(x264), pos(0)), x265))

We simplified constraint (263) using rules (I), (II) which results in the following new constraint:
(269)    (COND_EVAL1(true, pos(x266), neg(x855), pos(x264), x265)≥EVAL(pos(x266), minus_int(neg(x855), x265), plus_int(pos(x264), neg(x855)), x265))

We simplified constraint (264) using rules (I), (II) which results in the following new constraint:
(270)    (COND_EVAL1(true, pos(x266), neg(x856), pos(x264), x265)≥EVAL(pos(x266), minus_int(neg(x856), x265), plus_int(pos(x264), neg(x856)), x265))

We simplified constraint (265) using rule (VI) where we applied the induction hypothesis (∀x864,x865,x866:greatereq_int(pos(x863), pos(x862))=true ⇒ COND_EVAL1(true, pos(x864), pos(x862), pos(x865), x866)≥EVAL(pos(x864), minus_int(pos(x862), x866), plus_int(pos(x865), pos(x862)), x866)) with σ = [x866 / x265, x864 / x266, x865 / x264] which results in the following new constraint:
(271)    (COND_EVAL1(true, pos(x266), pos(x862), pos(x264), x265)≥EVAL(pos(x266), minus_int(pos(x862), x265), plus_int(pos(x264), pos(x862)), x265) ⇒ COND_EVAL1(true, pos(x266), pos(s(x862)), pos(x264), x265)≥EVAL(pos(x266), minus_int(pos(s(x862)), x265), plus_int(pos(x264), pos(s(x862))), x265))

We simplified constraint (266) using rule (VI) where we applied the induction hypothesis (∀x869,x870,x871:greatereq_int(neg(x868), neg(x867))=true ⇒ COND_EVAL1(true, pos(x869), neg(x867), pos(x870), x871)≥EVAL(pos(x869), minus_int(neg(x867), x871), plus_int(pos(x870), neg(x867)), x871)) with σ = [x871 / x265, x869 / x266, x870 / x264] which results in the following new constraint:
(272)    (COND_EVAL1(true, pos(x266), neg(x867), pos(x264), x265)≥EVAL(pos(x266), minus_int(neg(x867), x265), plus_int(pos(x264), neg(x867)), x265) ⇒ COND_EVAL1(true, pos(x266), neg(s(x867)), pos(x264), x265)≥EVAL(pos(x266), minus_int(neg(s(x867)), x265), plus_int(pos(x264), neg(s(x867))), x265))

For Pair EVAL(y0, y1, neg(x0), x_removed) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, neg(x0), x_removed) the following chains were created:

We consider the chain COND_EVAL1(true, z0, z1, pos(z2), x_removed) → EVAL(z0, minus_int(z1, x_removed), plus_int(pos(z2), z1), x_removed), EVAL(y0, y1, neg(x0), x_removed) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, neg(x0), x_removed) which results in the following constraint:
(273)    (EVAL(x273, minus_int(x274, x276), plus_int(pos(x275), x274), x276)=EVAL(x277, x278, neg(x279), x280) ⇒ EVAL(x277, x278, neg(x279), x280)≥COND_EVAL1(and(greatereq_int(x277, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x279, x279), x279)), x278)), x277, x278, neg(x279), x280))

We simplified constraint (273) using rules (I), (II), (III), (VII) which results in the following new constraint:
(274)    (minus_int(x274, x276)=x278∧pos(x275)=x872∧plus_int(x872, x274)=neg(x279) ⇒ EVAL(x273, x278, neg(x279), x276)≥COND_EVAL1(and(greatereq_int(x273, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x279, x279), x279)), x278)), x273, x278, neg(x279), x276))

We simplified constraint (274) using rule (V) (with possible (I) afterwards) using induction on plus_int(x872, x274)=neg(x279) which results in the following new constraints:
(275)    (minus_nat(x874, x873)=neg(x279)∧minus_int(neg(x873), x276)=x278∧pos(x275)=pos(x874) ⇒ EVAL(x273, x278, neg(x279), x276)≥COND_EVAL1(and(greatereq_int(x273, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x279, x279), x279)), x278)), x273, x278, neg(x279), x276))

(276)    (minus_nat(x875, x876)=neg(x279)∧minus_int(pos(x875), x276)=x278∧pos(x275)=neg(x876) ⇒ EVAL(x273, x278, neg(x279), x276)≥COND_EVAL1(and(greatereq_int(x273, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x279, x279), x279)), x278)), x273, x278, neg(x279), x276))

(277)    (neg(plus_nat(x878, x877))=neg(x279)∧minus_int(neg(x877), x276)=x278∧pos(x275)=neg(x878) ⇒ EVAL(x273, x278, neg(x279), x276)≥COND_EVAL1(and(greatereq_int(x273, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x279, x279), x279)), x278)), x273, x278, neg(x279), x276))

We simplified constraint (275) using rules (I), (II), (III), (VII) which results in the following new constraint:
(278)    (minus_nat(x874, x873)=neg(x279)∧neg(x873)=x881∧minus_int(x881, x276)=x278 ⇒ EVAL(x273, x278, neg(x279), x276)≥COND_EVAL1(and(greatereq_int(x273, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x279, x279), x279)), x278)), x273, x278, neg(x279), x276))

We solved constraint (276) using rules (I), (II).We solved constraint (277) using rules (I), (II).We simplified constraint (278) using rule (V) (with possible (I) afterwards) using induction on minus_nat(x874, x873)=neg(x279) which results in the following new constraints:
(279)    (neg(s(x882))=neg(x279)∧neg(s(x882))=x881∧minus_int(x881, x276)=x278 ⇒ EVAL(x273, x278, neg(x279), x276)≥COND_EVAL1(and(greatereq_int(x273, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x279, x279), x279)), x278)), x273, x278, neg(x279), x276))

(280)    (minus_nat(x885, x884)=neg(x279)∧neg(s(x884))=x881∧minus_int(x881, x276)=x278∧(∀x886,x887,x888,x889,x890:minus_nat(x885, x884)=neg(x886)∧neg(x884)=x887∧minus_int(x887, x888)=x889 ⇒ EVAL(x890, x889, neg(x886), x888)≥COND_EVAL1(and(greatereq_int(x890, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x886, x886), x886)), x889)), x890, x889, neg(x886), x888)) ⇒ EVAL(x273, x278, neg(x279), x276)≥COND_EVAL1(and(greatereq_int(x273, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x279, x279), x279)), x278)), x273, x278, neg(x279), x276))

We simplified constraint (279) using rules (I), (II), (III), (IV) which results in the following new constraint:
(281)    (EVAL(x273, x278, neg(s(x882)), x276)≥COND_EVAL1(and(greatereq_int(x273, pos(0)), greatereq_int(neg(mult_nat(mult_nat(s(x882), s(x882)), s(x882))), x278)), x273, x278, neg(s(x882)), x276))

We simplified constraint (280) using rules (III), (IV) which results in the following new constraint:
(282)    (EVAL(x273, x278, neg(x279), x276)≥COND_EVAL1(and(greatereq_int(x273, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x279, x279), x279)), x278)), x273, x278, neg(x279), x276))
We consider the chain COND_EVAL1(true, z0, z1, neg(z2), x_removed) → EVAL(z0, minus_int(z1, x_removed), plus_int(neg(z2), z1), x_removed), EVAL(y0, y1, neg(x0), x_removed) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, neg(x0), x_removed) which results in the following constraint:
(283)    (EVAL(x281, minus_int(x282, x284), plus_int(neg(x283), x282), x284)=EVAL(x285, x286, neg(x287), x288) ⇒ EVAL(x285, x286, neg(x287), x288)≥COND_EVAL1(and(greatereq_int(x285, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x287, x287), x287)), x286)), x285, x286, neg(x287), x288))

We simplified constraint (283) using rules (I), (II), (III), (VII) which results in the following new constraint:
(284)    (minus_int(x282, x284)=x286∧neg(x283)=x891∧plus_int(x891, x282)=neg(x287) ⇒ EVAL(x281, x286, neg(x287), x284)≥COND_EVAL1(and(greatereq_int(x281, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x287, x287), x287)), x286)), x281, x286, neg(x287), x284))

We simplified constraint (284) using rule (V) (with possible (I) afterwards) using induction on plus_int(x891, x282)=neg(x287) which results in the following new constraints:
(285)    (minus_nat(x893, x892)=neg(x287)∧minus_int(neg(x892), x284)=x286∧neg(x283)=pos(x893) ⇒ EVAL(x281, x286, neg(x287), x284)≥COND_EVAL1(and(greatereq_int(x281, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x287, x287), x287)), x286)), x281, x286, neg(x287), x284))

(286)    (minus_nat(x894, x895)=neg(x287)∧minus_int(pos(x894), x284)=x286∧neg(x283)=neg(x895) ⇒ EVAL(x281, x286, neg(x287), x284)≥COND_EVAL1(and(greatereq_int(x281, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x287, x287), x287)), x286)), x281, x286, neg(x287), x284))

(287)    (neg(plus_nat(x897, x896))=neg(x287)∧minus_int(neg(x896), x284)=x286∧neg(x283)=neg(x897) ⇒ EVAL(x281, x286, neg(x287), x284)≥COND_EVAL1(and(greatereq_int(x281, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x287, x287), x287)), x286)), x281, x286, neg(x287), x284))

We solved constraint (285) using rules (I), (II).We simplified constraint (286) using rules (I), (II), (III), (VII) which results in the following new constraint:
(288)    (minus_nat(x894, x895)=neg(x287)∧pos(x894)=x900∧minus_int(x900, x284)=x286 ⇒ EVAL(x281, x286, neg(x287), x284)≥COND_EVAL1(and(greatereq_int(x281, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x287, x287), x287)), x286)), x281, x286, neg(x287), x284))

We solved constraint (287) using rules (I), (II), (III), (IV), (VII).We simplified constraint (288) using rule (V) (with possible (I) afterwards) using induction on minus_nat(x894, x895)=neg(x287) which results in the following new constraints:
(289)    (neg(s(x901))=neg(x287)∧pos(0)=x900∧minus_int(x900, x284)=x286 ⇒ EVAL(x281, x286, neg(x287), x284)≥COND_EVAL1(and(greatereq_int(x281, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x287, x287), x287)), x286)), x281, x286, neg(x287), x284))

(290)    (minus_nat(x904, x903)=neg(x287)∧pos(s(x904))=x900∧minus_int(x900, x284)=x286∧(∀x905,x906,x907,x908,x909:minus_nat(x904, x903)=neg(x905)∧pos(x904)=x906∧minus_int(x906, x907)=x908 ⇒ EVAL(x909, x908, neg(x905), x907)≥COND_EVAL1(and(greatereq_int(x909, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x905, x905), x905)), x908)), x909, x908, neg(x905), x907)) ⇒ EVAL(x281, x286, neg(x287), x284)≥COND_EVAL1(and(greatereq_int(x281, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x287, x287), x287)), x286)), x281, x286, neg(x287), x284))

We simplified constraint (289) using rules (I), (II), (III), (IV) which results in the following new constraint:
(291)    (EVAL(x281, x286, neg(s(x901)), x284)≥COND_EVAL1(and(greatereq_int(x281, pos(0)), greatereq_int(neg(mult_nat(mult_nat(s(x901), s(x901)), s(x901))), x286)), x281, x286, neg(s(x901)), x284))

We simplified constraint (290) using rules (III), (IV) which results in the following new constraint:
(292)    (EVAL(x281, x286, neg(x287), x284)≥COND_EVAL1(and(greatereq_int(x281, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x287, x287), x287)), x286)), x281, x286, neg(x287), x284))
We consider the chain COND_EVAL1(true, neg(0), z0, z1, x_removed) → EVAL(neg(0), minus_int(z0, x_removed), plus_int(z1, z0), x_removed), EVAL(y0, y1, neg(x0), x_removed) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, neg(x0), x_removed) which results in the following constraint:
(293)    (EVAL(neg(0), minus_int(x289, x291), plus_int(x290, x289), x291)=EVAL(x292, x293, neg(x294), x295) ⇒ EVAL(x292, x293, neg(x294), x295)≥COND_EVAL1(and(greatereq_int(x292, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x294, x294), x294)), x293)), x292, x293, neg(x294), x295))

We simplified constraint (293) using rules (I), (II), (III) which results in the following new constraint:
(294)    (minus_int(x289, x291)=x293∧plus_int(x290, x289)=neg(x294) ⇒ EVAL(neg(0), x293, neg(x294), x291)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x294, x294), x294)), x293)), neg(0), x293, neg(x294), x291))

We simplified constraint (294) using rule (V) (with possible (I) afterwards) using induction on plus_int(x290, x289)=neg(x294) which results in the following new constraints:
(295)    (minus_nat(x912, x911)=neg(x294)∧minus_int(neg(x911), x291)=x293 ⇒ EVAL(neg(0), x293, neg(x294), x291)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x294, x294), x294)), x293)), neg(0), x293, neg(x294), x291))

(296)    (minus_nat(x913, x914)=neg(x294)∧minus_int(pos(x913), x291)=x293 ⇒ EVAL(neg(0), x293, neg(x294), x291)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x294, x294), x294)), x293)), neg(0), x293, neg(x294), x291))

(297)    (neg(plus_nat(x916, x915))=neg(x294)∧minus_int(neg(x915), x291)=x293 ⇒ EVAL(neg(0), x293, neg(x294), x291)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x294, x294), x294)), x293)), neg(0), x293, neg(x294), x291))

We simplified constraint (295) using rule (VII) which results in the following new constraint:
(298)    (minus_nat(x912, x911)=neg(x294)∧neg(x911)=x919∧minus_int(x919, x291)=x293 ⇒ EVAL(neg(0), x293, neg(x294), x291)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x294, x294), x294)), x293)), neg(0), x293, neg(x294), x291))

We simplified constraint (296) using rule (VII) which results in the following new constraint:
(299)    (minus_nat(x913, x914)=neg(x294)∧pos(x913)=x928∧minus_int(x928, x291)=x293 ⇒ EVAL(neg(0), x293, neg(x294), x291)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x294, x294), x294)), x293)), neg(0), x293, neg(x294), x291))

We solved constraint (297) using rules (I), (II), (III), (IV), (VII).We simplified constraint (298) using rule (V) (with possible (I) afterwards) using induction on minus_nat(x912, x911)=neg(x294) which results in the following new constraints:
(300)    (neg(s(x920))=neg(x294)∧neg(s(x920))=x919∧minus_int(x919, x291)=x293 ⇒ EVAL(neg(0), x293, neg(x294), x291)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x294, x294), x294)), x293)), neg(0), x293, neg(x294), x291))

(301)    (minus_nat(x923, x922)=neg(x294)∧neg(s(x922))=x919∧minus_int(x919, x291)=x293∧(∀x924,x925,x926,x927:minus_nat(x923, x922)=neg(x924)∧neg(x922)=x925∧minus_int(x925, x926)=x927 ⇒ EVAL(neg(0), x927, neg(x924), x926)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x924, x924), x924)), x927)), neg(0), x927, neg(x924), x926)) ⇒ EVAL(neg(0), x293, neg(x294), x291)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x294, x294), x294)), x293)), neg(0), x293, neg(x294), x291))

We simplified constraint (300) using rules (I), (II), (III), (IV) which results in the following new constraint:
(302)    (EVAL(neg(0), x293, neg(s(x920)), x291)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(neg(mult_nat(mult_nat(s(x920), s(x920)), s(x920))), x293)), neg(0), x293, neg(s(x920)), x291))

We simplified constraint (301) using rules (III), (IV) which results in the following new constraint:
(303)    (EVAL(neg(0), x293, neg(x294), x291)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x294, x294), x294)), x293)), neg(0), x293, neg(x294), x291))

We simplified constraint (299) using rule (V) (with possible (I) afterwards) using induction on minus_nat(x913, x914)=neg(x294) which results in the following new constraints:
(304)    (neg(s(x929))=neg(x294)∧pos(0)=x928∧minus_int(x928, x291)=x293 ⇒ EVAL(neg(0), x293, neg(x294), x291)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x294, x294), x294)), x293)), neg(0), x293, neg(x294), x291))

(305)    (minus_nat(x932, x931)=neg(x294)∧pos(s(x932))=x928∧minus_int(x928, x291)=x293∧(∀x933,x934,x935,x936:minus_nat(x932, x931)=neg(x933)∧pos(x932)=x934∧minus_int(x934, x935)=x936 ⇒ EVAL(neg(0), x936, neg(x933), x935)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x933, x933), x933)), x936)), neg(0), x936, neg(x933), x935)) ⇒ EVAL(neg(0), x293, neg(x294), x291)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x294, x294), x294)), x293)), neg(0), x293, neg(x294), x291))

We simplified constraint (304) using rules (I), (II), (III), (IV) which results in the following new constraint:
(306)    (EVAL(neg(0), x293, neg(s(x929)), x291)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(neg(mult_nat(mult_nat(s(x929), s(x929)), s(x929))), x293)), neg(0), x293, neg(s(x929)), x291))

We simplified constraint (305) using rules (III), (IV) which results in the following new constraint:
(307)    (EVAL(neg(0), x293, neg(x294), x291)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x294, x294), x294)), x293)), neg(0), x293, neg(x294), x291))
We consider the chain COND_EVAL1(true, pos(z0), z1, z2, x_removed) → EVAL(pos(z0), minus_int(z1, x_removed), plus_int(z2, z1), x_removed), EVAL(y0, y1, neg(x0), x_removed) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, neg(x0), x_removed) which results in the following constraint:
(308)    (EVAL(pos(x300), minus_int(x301, x303), plus_int(x302, x301), x303)=EVAL(x304, x305, neg(x306), x307) ⇒ EVAL(x304, x305, neg(x306), x307)≥COND_EVAL1(and(greatereq_int(x304, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x306, x306), x306)), x305)), x304, x305, neg(x306), x307))

We simplified constraint (308) using rules (I), (II), (III) which results in the following new constraint:
(309)    (minus_int(x301, x303)=x305∧plus_int(x302, x301)=neg(x306) ⇒ EVAL(pos(x300), x305, neg(x306), x303)≥COND_EVAL1(and(greatereq_int(pos(x300), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x306, x306), x306)), x305)), pos(x300), x305, neg(x306), x303))

We simplified constraint (309) using rule (V) (with possible (I) afterwards) using induction on plus_int(x302, x301)=neg(x306) which results in the following new constraints:
(310)    (minus_nat(x939, x938)=neg(x306)∧minus_int(neg(x938), x303)=x305 ⇒ EVAL(pos(x300), x305, neg(x306), x303)≥COND_EVAL1(and(greatereq_int(pos(x300), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x306, x306), x306)), x305)), pos(x300), x305, neg(x306), x303))

(311)    (minus_nat(x940, x941)=neg(x306)∧minus_int(pos(x940), x303)=x305 ⇒ EVAL(pos(x300), x305, neg(x306), x303)≥COND_EVAL1(and(greatereq_int(pos(x300), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x306, x306), x306)), x305)), pos(x300), x305, neg(x306), x303))

(312)    (neg(plus_nat(x943, x942))=neg(x306)∧minus_int(neg(x942), x303)=x305 ⇒ EVAL(pos(x300), x305, neg(x306), x303)≥COND_EVAL1(and(greatereq_int(pos(x300), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x306, x306), x306)), x305)), pos(x300), x305, neg(x306), x303))

We simplified constraint (310) using rule (VII) which results in the following new constraint:
(313)    (minus_nat(x939, x938)=neg(x306)∧neg(x938)=x946∧minus_int(x946, x303)=x305 ⇒ EVAL(pos(x300), x305, neg(x306), x303)≥COND_EVAL1(and(greatereq_int(pos(x300), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x306, x306), x306)), x305)), pos(x300), x305, neg(x306), x303))

We simplified constraint (311) using rule (VII) which results in the following new constraint:
(314)    (minus_nat(x940, x941)=neg(x306)∧pos(x940)=x956∧minus_int(x956, x303)=x305 ⇒ EVAL(pos(x300), x305, neg(x306), x303)≥COND_EVAL1(and(greatereq_int(pos(x300), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x306, x306), x306)), x305)), pos(x300), x305, neg(x306), x303))

We solved constraint (312) using rules (I), (II), (III), (IV), (VII).We simplified constraint (313) using rule (V) (with possible (I) afterwards) using induction on minus_nat(x939, x938)=neg(x306) which results in the following new constraints:
(315)    (neg(s(x947))=neg(x306)∧neg(s(x947))=x946∧minus_int(x946, x303)=x305 ⇒ EVAL(pos(x300), x305, neg(x306), x303)≥COND_EVAL1(and(greatereq_int(pos(x300), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x306, x306), x306)), x305)), pos(x300), x305, neg(x306), x303))

(316)    (minus_nat(x950, x949)=neg(x306)∧neg(s(x949))=x946∧minus_int(x946, x303)=x305∧(∀x951,x952,x953,x954,x955:minus_nat(x950, x949)=neg(x951)∧neg(x949)=x952∧minus_int(x952, x953)=x954 ⇒ EVAL(pos(x955), x954, neg(x951), x953)≥COND_EVAL1(and(greatereq_int(pos(x955), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x951, x951), x951)), x954)), pos(x955), x954, neg(x951), x953)) ⇒ EVAL(pos(x300), x305, neg(x306), x303)≥COND_EVAL1(and(greatereq_int(pos(x300), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x306, x306), x306)), x305)), pos(x300), x305, neg(x306), x303))

We simplified constraint (315) using rules (I), (II), (III), (IV) which results in the following new constraint:
(317)    (EVAL(pos(x300), x305, neg(s(x947)), x303)≥COND_EVAL1(and(greatereq_int(pos(x300), pos(0)), greatereq_int(neg(mult_nat(mult_nat(s(x947), s(x947)), s(x947))), x305)), pos(x300), x305, neg(s(x947)), x303))

We simplified constraint (316) using rules (III), (IV) which results in the following new constraint:
(318)    (EVAL(pos(x300), x305, neg(x306), x303)≥COND_EVAL1(and(greatereq_int(pos(x300), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x306, x306), x306)), x305)), pos(x300), x305, neg(x306), x303))

We simplified constraint (314) using rule (V) (with possible (I) afterwards) using induction on minus_nat(x940, x941)=neg(x306) which results in the following new constraints:
(319)    (neg(s(x957))=neg(x306)∧pos(0)=x956∧minus_int(x956, x303)=x305 ⇒ EVAL(pos(x300), x305, neg(x306), x303)≥COND_EVAL1(and(greatereq_int(pos(x300), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x306, x306), x306)), x305)), pos(x300), x305, neg(x306), x303))

(320)    (minus_nat(x960, x959)=neg(x306)∧pos(s(x960))=x956∧minus_int(x956, x303)=x305∧(∀x961,x962,x963,x964,x965:minus_nat(x960, x959)=neg(x961)∧pos(x960)=x962∧minus_int(x962, x963)=x964 ⇒ EVAL(pos(x965), x964, neg(x961), x963)≥COND_EVAL1(and(greatereq_int(pos(x965), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x961, x961), x961)), x964)), pos(x965), x964, neg(x961), x963)) ⇒ EVAL(pos(x300), x305, neg(x306), x303)≥COND_EVAL1(and(greatereq_int(pos(x300), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x306, x306), x306)), x305)), pos(x300), x305, neg(x306), x303))

We simplified constraint (319) using rules (I), (II), (III), (IV) which results in the following new constraint:
(321)    (EVAL(pos(x300), x305, neg(s(x957)), x303)≥COND_EVAL1(and(greatereq_int(pos(x300), pos(0)), greatereq_int(neg(mult_nat(mult_nat(s(x957), s(x957)), s(x957))), x305)), pos(x300), x305, neg(s(x957)), x303))

We simplified constraint (320) using rules (III), (IV) which results in the following new constraint:
(322)    (EVAL(pos(x300), x305, neg(x306), x303)≥COND_EVAL1(and(greatereq_int(pos(x300), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x306, x306), x306)), x305)), pos(x300), x305, neg(x306), x303))
We consider the chain COND_EVAL1(true, neg(s(z0)), z1, z2, x_removed) → EVAL(neg(s(z0)), minus_int(z1, x_removed), plus_int(z2, z1), x_removed), EVAL(y0, y1, neg(x0), x_removed) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, neg(x0), x_removed) which results in the following constraint:
(323)    (EVAL(neg(s(x312)), minus_int(x313, x315), plus_int(x314, x313), x315)=EVAL(x316, x317, neg(x318), x319) ⇒ EVAL(x316, x317, neg(x318), x319)≥COND_EVAL1(and(greatereq_int(x316, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x318, x318), x318)), x317)), x316, x317, neg(x318), x319))

We simplified constraint (323) using rules (I), (II), (III) which results in the following new constraint:
(324)    (minus_int(x313, x315)=x317∧plus_int(x314, x313)=neg(x318) ⇒ EVAL(neg(s(x312)), x317, neg(x318), x315)≥COND_EVAL1(and(greatereq_int(neg(s(x312)), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x318, x318), x318)), x317)), neg(s(x312)), x317, neg(x318), x315))

We simplified constraint (324) using rule (V) (with possible (I) afterwards) using induction on plus_int(x314, x313)=neg(x318) which results in the following new constraints:
(325)    (minus_nat(x968, x967)=neg(x318)∧minus_int(neg(x967), x315)=x317 ⇒ EVAL(neg(s(x312)), x317, neg(x318), x315)≥COND_EVAL1(and(greatereq_int(neg(s(x312)), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x318, x318), x318)), x317)), neg(s(x312)), x317, neg(x318), x315))

(326)    (minus_nat(x969, x970)=neg(x318)∧minus_int(pos(x969), x315)=x317 ⇒ EVAL(neg(s(x312)), x317, neg(x318), x315)≥COND_EVAL1(and(greatereq_int(neg(s(x312)), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x318, x318), x318)), x317)), neg(s(x312)), x317, neg(x318), x315))

(327)    (neg(plus_nat(x972, x971))=neg(x318)∧minus_int(neg(x971), x315)=x317 ⇒ EVAL(neg(s(x312)), x317, neg(x318), x315)≥COND_EVAL1(and(greatereq_int(neg(s(x312)), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x318, x318), x318)), x317)), neg(s(x312)), x317, neg(x318), x315))

We simplified constraint (325) using rule (VII) which results in the following new constraint:
(328)    (minus_nat(x968, x967)=neg(x318)∧neg(x967)=x975∧minus_int(x975, x315)=x317 ⇒ EVAL(neg(s(x312)), x317, neg(x318), x315)≥COND_EVAL1(and(greatereq_int(neg(s(x312)), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x318, x318), x318)), x317)), neg(s(x312)), x317, neg(x318), x315))

We simplified constraint (326) using rule (VII) which results in the following new constraint:
(329)    (minus_nat(x969, x970)=neg(x318)∧pos(x969)=x985∧minus_int(x985, x315)=x317 ⇒ EVAL(neg(s(x312)), x317, neg(x318), x315)≥COND_EVAL1(and(greatereq_int(neg(s(x312)), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x318, x318), x318)), x317)), neg(s(x312)), x317, neg(x318), x315))

We solved constraint (327) using rules (I), (II), (III), (IV), (VII).We simplified constraint (328) using rule (V) (with possible (I) afterwards) using induction on minus_nat(x968, x967)=neg(x318) which results in the following new constraints:
(330)    (neg(s(x976))=neg(x318)∧neg(s(x976))=x975∧minus_int(x975, x315)=x317 ⇒ EVAL(neg(s(x312)), x317, neg(x318), x315)≥COND_EVAL1(and(greatereq_int(neg(s(x312)), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x318, x318), x318)), x317)), neg(s(x312)), x317, neg(x318), x315))

(331)    (minus_nat(x979, x978)=neg(x318)∧neg(s(x978))=x975∧minus_int(x975, x315)=x317∧(∀x980,x981,x982,x983,x984:minus_nat(x979, x978)=neg(x980)∧neg(x978)=x981∧minus_int(x981, x982)=x983 ⇒ EVAL(neg(s(x984)), x983, neg(x980), x982)≥COND_EVAL1(and(greatereq_int(neg(s(x984)), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x980, x980), x980)), x983)), neg(s(x984)), x983, neg(x980), x982)) ⇒ EVAL(neg(s(x312)), x317, neg(x318), x315)≥COND_EVAL1(and(greatereq_int(neg(s(x312)), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x318, x318), x318)), x317)), neg(s(x312)), x317, neg(x318), x315))

We simplified constraint (330) using rules (I), (II), (III), (IV) which results in the following new constraint:
(332)    (EVAL(neg(s(x312)), x317, neg(s(x976)), x315)≥COND_EVAL1(and(greatereq_int(neg(s(x312)), pos(0)), greatereq_int(neg(mult_nat(mult_nat(s(x976), s(x976)), s(x976))), x317)), neg(s(x312)), x317, neg(s(x976)), x315))

We simplified constraint (331) using rules (III), (IV) which results in the following new constraint:
(333)    (EVAL(neg(s(x312)), x317, neg(x318), x315)≥COND_EVAL1(and(greatereq_int(neg(s(x312)), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x318, x318), x318)), x317)), neg(s(x312)), x317, neg(x318), x315))

We simplified constraint (329) using rule (V) (with possible (I) afterwards) using induction on minus_nat(x969, x970)=neg(x318) which results in the following new constraints:
(334)    (neg(s(x986))=neg(x318)∧pos(0)=x985∧minus_int(x985, x315)=x317 ⇒ EVAL(neg(s(x312)), x317, neg(x318), x315)≥COND_EVAL1(and(greatereq_int(neg(s(x312)), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x318, x318), x318)), x317)), neg(s(x312)), x317, neg(x318), x315))

(335)    (minus_nat(x989, x988)=neg(x318)∧pos(s(x989))=x985∧minus_int(x985, x315)=x317∧(∀x990,x991,x992,x993,x994:minus_nat(x989, x988)=neg(x990)∧pos(x989)=x991∧minus_int(x991, x992)=x993 ⇒ EVAL(neg(s(x994)), x993, neg(x990), x992)≥COND_EVAL1(and(greatereq_int(neg(s(x994)), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x990, x990), x990)), x993)), neg(s(x994)), x993, neg(x990), x992)) ⇒ EVAL(neg(s(x312)), x317, neg(x318), x315)≥COND_EVAL1(and(greatereq_int(neg(s(x312)), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x318, x318), x318)), x317)), neg(s(x312)), x317, neg(x318), x315))

We simplified constraint (334) using rules (I), (II), (III), (IV) which results in the following new constraint:
(336)    (EVAL(neg(s(x312)), x317, neg(s(x986)), x315)≥COND_EVAL1(and(greatereq_int(neg(s(x312)), pos(0)), greatereq_int(neg(mult_nat(mult_nat(s(x986), s(x986)), s(x986))), x317)), neg(s(x312)), x317, neg(s(x986)), x315))

We simplified constraint (335) using rules (III), (IV) which results in the following new constraint:
(337)    (EVAL(neg(s(x312)), x317, neg(x318), x315)≥COND_EVAL1(and(greatereq_int(neg(s(x312)), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x318, x318), x318)), x317)), neg(s(x312)), x317, neg(x318), x315))

For Pair COND_EVAL1(true, neg(s(z0)), z1, z2, x_removed) → EVAL(neg(s(z0)), minus_int(z1, x_removed), plus_int(z2, z1), x_removed) the following chains were created:

We consider the chain EVAL(y0, y1, neg(x0), x_removed) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, neg(x0), x_removed), COND_EVAL1(true, neg(s(z0)), z1, z2, x_removed) → EVAL(neg(s(z0)), minus_int(z1, x_removed), plus_int(z2, z1), x_removed) which results in the following constraint:
(338)    (COND_EVAL1(and(greatereq_int(x346, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x348, x348), x348)), x347)), x346, x347, neg(x348), x349)=COND_EVAL1(true, neg(s(x350)), x351, x352, x353) ⇒ COND_EVAL1(true, neg(s(x350)), x351, x352, x353)≥EVAL(neg(s(x350)), minus_int(x351, x353), plus_int(x352, x351), x353))

We simplified constraint (338) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
(339)    (neg(s(x350))=x998∧pos(0)=x999∧greatereq_int(x998, x999)=x996∧greatereq_int(x1000, x347)=x997∧and(x996, x997)=true ⇒ COND_EVAL1(true, neg(s(x350)), x347, neg(x348), x349)≥EVAL(neg(s(x350)), minus_int(x347, x349), plus_int(neg(x348), x347), x349))

We simplified constraint (339) using rule (V) (with possible (I) afterwards) using induction on and(x996, x997)=true which results in the following new constraint:
(340)    (true=true∧neg(s(x350))=x998∧pos(0)=x999∧greatereq_int(x998, x999)=true∧greatereq_int(x1000, x347)=true ⇒ COND_EVAL1(true, neg(s(x350)), x347, neg(x348), x349)≥EVAL(neg(s(x350)), minus_int(x347, x349), plus_int(neg(x348), x347), x349))

We simplified constraint (340) using rules (I), (II) which results in the following new constraint:
(341)    (neg(s(x350))=x998∧pos(0)=x999∧greatereq_int(x998, x999)=true∧greatereq_int(x1000, x347)=true ⇒ COND_EVAL1(true, neg(s(x350)), x347, neg(x348), x349)≥EVAL(neg(s(x350)), minus_int(x347, x349), plus_int(neg(x348), x347), x349))

We simplified constraint (341) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x998, x999)=true which results in the following new constraints:
(342)    (true=true∧neg(s(x350))=pos(x1001)∧pos(0)=pos(0)∧greatereq_int(x1000, x347)=true ⇒ COND_EVAL1(true, neg(s(x350)), x347, neg(x348), x349)≥EVAL(neg(s(x350)), minus_int(x347, x349), plus_int(neg(x348), x347), x349))

(343)    (true=true∧neg(s(x350))=neg(0)∧pos(0)=pos(0)∧greatereq_int(x1000, x347)=true ⇒ COND_EVAL1(true, neg(s(x350)), x347, neg(x348), x349)≥EVAL(neg(s(x350)), minus_int(x347, x349), plus_int(neg(x348), x347), x349))

(344)    (true=true∧neg(s(x350))=neg(0)∧pos(0)=neg(x1003)∧greatereq_int(x1000, x347)=true ⇒ COND_EVAL1(true, neg(s(x350)), x347, neg(x348), x349)≥EVAL(neg(s(x350)), minus_int(x347, x349), plus_int(neg(x348), x347), x349))

(345)    (true=true∧neg(s(x350))=pos(x1005)∧pos(0)=neg(x1004)∧greatereq_int(x1000, x347)=true ⇒ COND_EVAL1(true, neg(s(x350)), x347, neg(x348), x349)≥EVAL(neg(s(x350)), minus_int(x347, x349), plus_int(neg(x348), x347), x349))

(346)    (greatereq_int(pos(x1011), pos(x1010))=true∧neg(s(x350))=pos(s(x1011))∧pos(0)=pos(s(x1010))∧greatereq_int(x1000, x347)=true∧(∀x1012,x1013,x1014,x1015,x1016:greatereq_int(pos(x1011), pos(x1010))=true∧neg(s(x1012))=pos(x1011)∧pos(0)=pos(x1010)∧greatereq_int(x1013, x1014)=true ⇒ COND_EVAL1(true, neg(s(x1012)), x1014, neg(x1015), x1016)≥EVAL(neg(s(x1012)), minus_int(x1014, x1016), plus_int(neg(x1015), x1014), x1016)) ⇒ COND_EVAL1(true, neg(s(x350)), x347, neg(x348), x349)≥EVAL(neg(s(x350)), minus_int(x347, x349), plus_int(neg(x348), x347), x349))

(347)    (greatereq_int(neg(x1018), neg(x1017))=true∧neg(s(x350))=neg(s(x1018))∧pos(0)=neg(s(x1017))∧greatereq_int(x1000, x347)=true∧(∀x1019,x1020,x1021,x1022,x1023:greatereq_int(neg(x1018), neg(x1017))=true∧neg(s(x1019))=neg(x1018)∧pos(0)=neg(x1017)∧greatereq_int(x1020, x1021)=true ⇒ COND_EVAL1(true, neg(s(x1019)), x1021, neg(x1022), x1023)≥EVAL(neg(s(x1019)), minus_int(x1021, x1023), plus_int(neg(x1022), x1021), x1023)) ⇒ COND_EVAL1(true, neg(s(x350)), x347, neg(x348), x349)≥EVAL(neg(s(x350)), minus_int(x347, x349), plus_int(neg(x348), x347), x349))

We solved constraint (342) using rules (I), (II).We solved constraint (343) using rules (I), (II).We solved constraint (344) using rules (I), (II).We solved constraint (345) using rules (I), (II).We solved constraint (346) using rules (I), (II).We solved constraint (347) using rules (I), (II).
We consider the chain EVAL(y0, y1, pos(x0), x_removed) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, pos(x0), x_removed), COND_EVAL1(true, neg(s(z0)), z1, z2, x_removed) → EVAL(neg(s(z0)), minus_int(z1, x_removed), plus_int(z2, z1), x_removed) which results in the following constraint:
(348)    (COND_EVAL1(and(greatereq_int(x358, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x360, x360), x360)), x359)), x358, x359, pos(x360), x361)=COND_EVAL1(true, neg(s(x362)), x363, x364, x365) ⇒ COND_EVAL1(true, neg(s(x362)), x363, x364, x365)≥EVAL(neg(s(x362)), minus_int(x363, x365), plus_int(x364, x363), x365))

We simplified constraint (348) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint:
(349)    (neg(s(x362))=x1026∧pos(0)=x1027∧greatereq_int(x1026, x1027)=x1024∧greatereq_int(x1028, x359)=x1025∧and(x1024, x1025)=true ⇒ COND_EVAL1(true, neg(s(x362)), x359, pos(x360), x361)≥EVAL(neg(s(x362)), minus_int(x359, x361), plus_int(pos(x360), x359), x361))

We simplified constraint (349) using rule (V) (with possible (I) afterwards) using induction on and(x1024, x1025)=true which results in the following new constraint:
(350)    (true=true∧neg(s(x362))=x1026∧pos(0)=x1027∧greatereq_int(x1026, x1027)=true∧greatereq_int(x1028, x359)=true ⇒ COND_EVAL1(true, neg(s(x362)), x359, pos(x360), x361)≥EVAL(neg(s(x362)), minus_int(x359, x361), plus_int(pos(x360), x359), x361))

We simplified constraint (350) using rules (I), (II) which results in the following new constraint:
(351)    (neg(s(x362))=x1026∧pos(0)=x1027∧greatereq_int(x1026, x1027)=true∧greatereq_int(x1028, x359)=true ⇒ COND_EVAL1(true, neg(s(x362)), x359, pos(x360), x361)≥EVAL(neg(s(x362)), minus_int(x359, x361), plus_int(pos(x360), x359), x361))

We simplified constraint (351) using rule (V) (with possible (I) afterwards) using induction on greatereq_int(x1026, x1027)=true which results in the following new constraints:
(352)    (true=true∧neg(s(x362))=pos(x1029)∧pos(0)=pos(0)∧greatereq_int(x1028, x359)=true ⇒ COND_EVAL1(true, neg(s(x362)), x359, pos(x360), x361)≥EVAL(neg(s(x362)), minus_int(x359, x361), plus_int(pos(x360), x359), x361))

(353)    (true=true∧neg(s(x362))=neg(0)∧pos(0)=pos(0)∧greatereq_int(x1028, x359)=true ⇒ COND_EVAL1(true, neg(s(x362)), x359, pos(x360), x361)≥EVAL(neg(s(x362)), minus_int(x359, x361), plus_int(pos(x360), x359), x361))

(354)    (true=true∧neg(s(x362))=neg(0)∧pos(0)=neg(x1031)∧greatereq_int(x1028, x359)=true ⇒ COND_EVAL1(true, neg(s(x362)), x359, pos(x360), x361)≥EVAL(neg(s(x362)), minus_int(x359, x361), plus_int(pos(x360), x359), x361))

(355)    (true=true∧neg(s(x362))=pos(x1033)∧pos(0)=neg(x1032)∧greatereq_int(x1028, x359)=true ⇒ COND_EVAL1(true, neg(s(x362)), x359, pos(x360), x361)≥EVAL(neg(s(x362)), minus_int(x359, x361), plus_int(pos(x360), x359), x361))

(356)    (greatereq_int(pos(x1039), pos(x1038))=true∧neg(s(x362))=pos(s(x1039))∧pos(0)=pos(s(x1038))∧greatereq_int(x1028, x359)=true∧(∀x1040,x1041,x1042,x1043,x1044:greatereq_int(pos(x1039), pos(x1038))=true∧neg(s(x1040))=pos(x1039)∧pos(0)=pos(x1038)∧greatereq_int(x1041, x1042)=true ⇒ COND_EVAL1(true, neg(s(x1040)), x1042, pos(x1043), x1044)≥EVAL(neg(s(x1040)), minus_int(x1042, x1044), plus_int(pos(x1043), x1042), x1044)) ⇒ COND_EVAL1(true, neg(s(x362)), x359, pos(x360), x361)≥EVAL(neg(s(x362)), minus_int(x359, x361), plus_int(pos(x360), x359), x361))

(357)    (greatereq_int(neg(x1046), neg(x1045))=true∧neg(s(x362))=neg(s(x1046))∧pos(0)=neg(s(x1045))∧greatereq_int(x1028, x359)=true∧(∀x1047,x1048,x1049,x1050,x1051:greatereq_int(neg(x1046), neg(x1045))=true∧neg(s(x1047))=neg(x1046)∧pos(0)=neg(x1045)∧greatereq_int(x1048, x1049)=true ⇒ COND_EVAL1(true, neg(s(x1047)), x1049, pos(x1050), x1051)≥EVAL(neg(s(x1047)), minus_int(x1049, x1051), plus_int(pos(x1050), x1049), x1051)) ⇒ COND_EVAL1(true, neg(s(x362)), x359, pos(x360), x361)≥EVAL(neg(s(x362)), minus_int(x359, x361), plus_int(pos(x360), x359), x361))

We solved constraint (352) using rules (I), (II).We solved constraint (353) using rules (I), (II).We solved constraint (354) using rules (I), (II).We solved constraint (355) using rules (I), (II).We solved constraint (356) using rules (I), (II).We solved constraint (357) using rules (I), (II).

For Pair EVAL(y0, y1, pos(x0), x_removed) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, pos(x0), x_removed) the following chains were created:

We consider the chain COND_EVAL1(true, z0, z1, pos(z2), x_removed) → EVAL(z0, minus_int(z1, x_removed), plus_int(pos(z2), z1), x_removed), EVAL(y0, y1, pos(x0), x_removed) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, pos(x0), x_removed) which results in the following constraint:
(358)    (EVAL(x369, minus_int(x370, x372), plus_int(pos(x371), x370), x372)=EVAL(x373, x374, pos(x375), x376) ⇒ EVAL(x373, x374, pos(x375), x376)≥COND_EVAL1(and(greatereq_int(x373, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x375, x375), x375)), x374)), x373, x374, pos(x375), x376))

We simplified constraint (358) using rules (I), (II), (III), (VII) which results in the following new constraint:
(359)    (minus_int(x370, x372)=x374∧pos(x371)=x1052∧plus_int(x1052, x370)=pos(x375) ⇒ EVAL(x369, x374, pos(x375), x372)≥COND_EVAL1(and(greatereq_int(x369, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x375, x375), x375)), x374)), x369, x374, pos(x375), x372))

We simplified constraint (359) using rule (V) (with possible (I) afterwards) using induction on plus_int(x1052, x370)=pos(x375) which results in the following new constraints:
(360)    (minus_nat(x1054, x1053)=pos(x375)∧minus_int(neg(x1053), x372)=x374∧pos(x371)=pos(x1054) ⇒ EVAL(x369, x374, pos(x375), x372)≥COND_EVAL1(and(greatereq_int(x369, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x375, x375), x375)), x374)), x369, x374, pos(x375), x372))

(361)    (minus_nat(x1055, x1056)=pos(x375)∧minus_int(pos(x1055), x372)=x374∧pos(x371)=neg(x1056) ⇒ EVAL(x369, x374, pos(x375), x372)≥COND_EVAL1(and(greatereq_int(x369, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x375, x375), x375)), x374)), x369, x374, pos(x375), x372))

(362)    (pos(plus_nat(x1060, x1059))=pos(x375)∧minus_int(pos(x1059), x372)=x374∧pos(x371)=pos(x1060) ⇒ EVAL(x369, x374, pos(x375), x372)≥COND_EVAL1(and(greatereq_int(x369, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x375, x375), x375)), x374)), x369, x374, pos(x375), x372))

We simplified constraint (360) using rules (I), (II), (III), (VII) which results in the following new constraint:
(363)    (minus_nat(x1054, x1053)=pos(x375)∧neg(x1053)=x1061∧minus_int(x1061, x372)=x374 ⇒ EVAL(x369, x374, pos(x375), x372)≥COND_EVAL1(and(greatereq_int(x369, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x375, x375), x375)), x374)), x369, x374, pos(x375), x372))

We solved constraint (361) using rules (I), (II).We solved constraint (362) using rules (I), (II), (III), (IV), (VII).We simplified constraint (363) using rule (V) (with possible (I) afterwards) using induction on minus_nat(x1054, x1053)=pos(x375) which results in the following new constraints:
(364)    (pos(0)=pos(x375)∧neg(0)=x1061∧minus_int(x1061, x372)=x374 ⇒ EVAL(x369, x374, pos(x375), x372)≥COND_EVAL1(and(greatereq_int(x369, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x375, x375), x375)), x374)), x369, x374, pos(x375), x372))

(365)    (pos(s(x1063))=pos(x375)∧neg(0)=x1061∧minus_int(x1061, x372)=x374 ⇒ EVAL(x369, x374, pos(x375), x372)≥COND_EVAL1(and(greatereq_int(x369, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x375, x375), x375)), x374)), x369, x374, pos(x375), x372))

(366)    (minus_nat(x1065, x1064)=pos(x375)∧neg(s(x1064))=x1061∧minus_int(x1061, x372)=x374∧(∀x1066,x1067,x1068,x1069,x1070:minus_nat(x1065, x1064)=pos(x1066)∧neg(x1064)=x1067∧minus_int(x1067, x1068)=x1069 ⇒ EVAL(x1070, x1069, pos(x1066), x1068)≥COND_EVAL1(and(greatereq_int(x1070, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x1066, x1066), x1066)), x1069)), x1070, x1069, pos(x1066), x1068)) ⇒ EVAL(x369, x374, pos(x375), x372)≥COND_EVAL1(and(greatereq_int(x369, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x375, x375), x375)), x374)), x369, x374, pos(x375), x372))

We simplified constraint (364) using rules (I), (II), (III), (IV) which results in the following new constraint:
(367)    (EVAL(x369, x374, pos(0), x372)≥COND_EVAL1(and(greatereq_int(x369, pos(0)), greatereq_int(pos(mult_nat(mult_nat(0, 0), 0)), x374)), x369, x374, pos(0), x372))

We simplified constraint (365) using rules (I), (II), (III), (IV) which results in the following new constraint:
(368)    (EVAL(x369, x374, pos(s(x1063)), x372)≥COND_EVAL1(and(greatereq_int(x369, pos(0)), greatereq_int(pos(mult_nat(mult_nat(s(x1063), s(x1063)), s(x1063))), x374)), x369, x374, pos(s(x1063)), x372))

We simplified constraint (366) using rules (III), (IV) which results in the following new constraint:
(369)    (EVAL(x369, x374, pos(x375), x372)≥COND_EVAL1(and(greatereq_int(x369, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x375, x375), x375)), x374)), x369, x374, pos(x375), x372))
We consider the chain COND_EVAL1(true, z0, z1, neg(z2), x_removed) → EVAL(z0, minus_int(z1, x_removed), plus_int(neg(z2), z1), x_removed), EVAL(y0, y1, pos(x0), x_removed) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, pos(x0), x_removed) which results in the following constraint:
(370)    (EVAL(x377, minus_int(x378, x380), plus_int(neg(x379), x378), x380)=EVAL(x381, x382, pos(x383), x384) ⇒ EVAL(x381, x382, pos(x383), x384)≥COND_EVAL1(and(greatereq_int(x381, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x383, x383), x383)), x382)), x381, x382, pos(x383), x384))

We simplified constraint (370) using rules (I), (II), (III), (VII) which results in the following new constraint:
(371)    (minus_int(x378, x380)=x382∧neg(x379)=x1072∧plus_int(x1072, x378)=pos(x383) ⇒ EVAL(x377, x382, pos(x383), x380)≥COND_EVAL1(and(greatereq_int(x377, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x383, x383), x383)), x382)), x377, x382, pos(x383), x380))

We simplified constraint (371) using rule (V) (with possible (I) afterwards) using induction on plus_int(x1072, x378)=pos(x383) which results in the following new constraints:
(372)    (minus_nat(x1074, x1073)=pos(x383)∧minus_int(neg(x1073), x380)=x382∧neg(x379)=pos(x1074) ⇒ EVAL(x377, x382, pos(x383), x380)≥COND_EVAL1(and(greatereq_int(x377, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x383, x383), x383)), x382)), x377, x382, pos(x383), x380))

(373)    (minus_nat(x1075, x1076)=pos(x383)∧minus_int(pos(x1075), x380)=x382∧neg(x379)=neg(x1076) ⇒ EVAL(x377, x382, pos(x383), x380)≥COND_EVAL1(and(greatereq_int(x377, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x383, x383), x383)), x382)), x377, x382, pos(x383), x380))

(374)    (pos(plus_nat(x1080, x1079))=pos(x383)∧minus_int(pos(x1079), x380)=x382∧neg(x379)=pos(x1080) ⇒ EVAL(x377, x382, pos(x383), x380)≥COND_EVAL1(and(greatereq_int(x377, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x383, x383), x383)), x382)), x377, x382, pos(x383), x380))

We solved constraint (372) using rules (I), (II).We simplified constraint (373) using rules (I), (II), (III), (VII) which results in the following new constraint:
(375)    (minus_nat(x1075, x1076)=pos(x383)∧pos(x1075)=x1081∧minus_int(x1081, x380)=x382 ⇒ EVAL(x377, x382, pos(x383), x380)≥COND_EVAL1(and(greatereq_int(x377, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x383, x383), x383)), x382)), x377, x382, pos(x383), x380))

We solved constraint (374) using rules (I), (II).We simplified constraint (375) using rule (V) (with possible (I) afterwards) using induction on minus_nat(x1075, x1076)=pos(x383) which results in the following new constraints:
(376)    (pos(0)=pos(x383)∧pos(0)=x1081∧minus_int(x1081, x380)=x382 ⇒ EVAL(x377, x382, pos(x383), x380)≥COND_EVAL1(and(greatereq_int(x377, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x383, x383), x383)), x382)), x377, x382, pos(x383), x380))

(377)    (pos(s(x1083))=pos(x383)∧pos(s(x1083))=x1081∧minus_int(x1081, x380)=x382 ⇒ EVAL(x377, x382, pos(x383), x380)≥COND_EVAL1(and(greatereq_int(x377, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x383, x383), x383)), x382)), x377, x382, pos(x383), x380))

(378)    (minus_nat(x1085, x1084)=pos(x383)∧pos(s(x1085))=x1081∧minus_int(x1081, x380)=x382∧(∀x1086,x1087,x1088,x1089,x1090:minus_nat(x1085, x1084)=pos(x1086)∧pos(x1085)=x1087∧minus_int(x1087, x1088)=x1089 ⇒ EVAL(x1090, x1089, pos(x1086), x1088)≥COND_EVAL1(and(greatereq_int(x1090, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x1086, x1086), x1086)), x1089)), x1090, x1089, pos(x1086), x1088)) ⇒ EVAL(x377, x382, pos(x383), x380)≥COND_EVAL1(and(greatereq_int(x377, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x383, x383), x383)), x382)), x377, x382, pos(x383), x380))

We simplified constraint (376) using rules (I), (II), (III), (IV) which results in the following new constraint:
(379)    (EVAL(x377, x382, pos(0), x380)≥COND_EVAL1(and(greatereq_int(x377, pos(0)), greatereq_int(pos(mult_nat(mult_nat(0, 0), 0)), x382)), x377, x382, pos(0), x380))

We simplified constraint (377) using rules (I), (II), (III), (IV) which results in the following new constraint:
(380)    (EVAL(x377, x382, pos(s(x1083)), x380)≥COND_EVAL1(and(greatereq_int(x377, pos(0)), greatereq_int(pos(mult_nat(mult_nat(s(x1083), s(x1083)), s(x1083))), x382)), x377, x382, pos(s(x1083)), x380))

We simplified constraint (378) using rules (III), (IV) which results in the following new constraint:
(381)    (EVAL(x377, x382, pos(x383), x380)≥COND_EVAL1(and(greatereq_int(x377, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x383, x383), x383)), x382)), x377, x382, pos(x383), x380))
We consider the chain COND_EVAL1(true, neg(0), z0, z1, x_removed) → EVAL(neg(0), minus_int(z0, x_removed), plus_int(z1, z0), x_removed), EVAL(y0, y1, pos(x0), x_removed) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, pos(x0), x_removed) which results in the following constraint:
(382)    (EVAL(neg(0), minus_int(x385, x387), plus_int(x386, x385), x387)=EVAL(x388, x389, pos(x390), x391) ⇒ EVAL(x388, x389, pos(x390), x391)≥COND_EVAL1(and(greatereq_int(x388, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x390, x390), x390)), x389)), x388, x389, pos(x390), x391))

We simplified constraint (382) using rules (I), (II), (III) which results in the following new constraint:
(383)    (minus_int(x385, x387)=x389∧plus_int(x386, x385)=pos(x390) ⇒ EVAL(neg(0), x389, pos(x390), x387)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x390, x390), x390)), x389)), neg(0), x389, pos(x390), x387))

We simplified constraint (383) using rule (V) (with possible (I) afterwards) using induction on plus_int(x386, x385)=pos(x390) which results in the following new constraints:
(384)    (minus_nat(x1092, x1091)=pos(x390)∧minus_int(neg(x1091), x387)=x389 ⇒ EVAL(neg(0), x389, pos(x390), x387)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x390, x390), x390)), x389)), neg(0), x389, pos(x390), x387))

(385)    (minus_nat(x1093, x1094)=pos(x390)∧minus_int(pos(x1093), x387)=x389 ⇒ EVAL(neg(0), x389, pos(x390), x387)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x390, x390), x390)), x389)), neg(0), x389, pos(x390), x387))

(386)    (pos(plus_nat(x1098, x1097))=pos(x390)∧minus_int(pos(x1097), x387)=x389 ⇒ EVAL(neg(0), x389, pos(x390), x387)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x390, x390), x390)), x389)), neg(0), x389, pos(x390), x387))

We simplified constraint (384) using rule (VII) which results in the following new constraint:
(387)    (minus_nat(x1092, x1091)=pos(x390)∧neg(x1091)=x1099∧minus_int(x1099, x387)=x389 ⇒ EVAL(neg(0), x389, pos(x390), x387)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x390, x390), x390)), x389)), neg(0), x389, pos(x390), x387))

We simplified constraint (385) using rule (VII) which results in the following new constraint:
(388)    (minus_nat(x1093, x1094)=pos(x390)∧pos(x1093)=x1108∧minus_int(x1108, x387)=x389 ⇒ EVAL(neg(0), x389, pos(x390), x387)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x390, x390), x390)), x389)), neg(0), x389, pos(x390), x387))

We solved constraint (386) using rules (I), (II), (III), (IV), (VII).We simplified constraint (387) using rule (V) (with possible (I) afterwards) using induction on minus_nat(x1092, x1091)=pos(x390) which results in the following new constraints:
(389)    (pos(0)=pos(x390)∧neg(0)=x1099∧minus_int(x1099, x387)=x389 ⇒ EVAL(neg(0), x389, pos(x390), x387)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x390, x390), x390)), x389)), neg(0), x389, pos(x390), x387))

(390)    (pos(s(x1101))=pos(x390)∧neg(0)=x1099∧minus_int(x1099, x387)=x389 ⇒ EVAL(neg(0), x389, pos(x390), x387)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x390, x390), x390)), x389)), neg(0), x389, pos(x390), x387))

(391)    (minus_nat(x1103, x1102)=pos(x390)∧neg(s(x1102))=x1099∧minus_int(x1099, x387)=x389∧(∀x1104,x1105,x1106,x1107:minus_nat(x1103, x1102)=pos(x1104)∧neg(x1102)=x1105∧minus_int(x1105, x1106)=x1107 ⇒ EVAL(neg(0), x1107, pos(x1104), x1106)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x1104, x1104), x1104)), x1107)), neg(0), x1107, pos(x1104), x1106)) ⇒ EVAL(neg(0), x389, pos(x390), x387)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x390, x390), x390)), x389)), neg(0), x389, pos(x390), x387))

We simplified constraint (389) using rules (I), (II), (III), (IV) which results in the following new constraint:
(392)    (EVAL(neg(0), x389, pos(0), x387)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(pos(mult_nat(mult_nat(0, 0), 0)), x389)), neg(0), x389, pos(0), x387))

We simplified constraint (390) using rules (I), (II), (III), (IV) which results in the following new constraint:
(393)    (EVAL(neg(0), x389, pos(s(x1101)), x387)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(pos(mult_nat(mult_nat(s(x1101), s(x1101)), s(x1101))), x389)), neg(0), x389, pos(s(x1101)), x387))

We simplified constraint (391) using rules (III), (IV) which results in the following new constraint:
(394)    (EVAL(neg(0), x389, pos(x390), x387)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x390, x390), x390)), x389)), neg(0), x389, pos(x390), x387))

We simplified constraint (388) using rule (V) (with possible (I) afterwards) using induction on minus_nat(x1093, x1094)=pos(x390) which results in the following new constraints:
(395)    (pos(0)=pos(x390)∧pos(0)=x1108∧minus_int(x1108, x387)=x389 ⇒ EVAL(neg(0), x389, pos(x390), x387)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x390, x390), x390)), x389)), neg(0), x389, pos(x390), x387))

(396)    (pos(s(x1110))=pos(x390)∧pos(s(x1110))=x1108∧minus_int(x1108, x387)=x389 ⇒ EVAL(neg(0), x389, pos(x390), x387)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x390, x390), x390)), x389)), neg(0), x389, pos(x390), x387))

(397)    (minus_nat(x1112, x1111)=pos(x390)∧pos(s(x1112))=x1108∧minus_int(x1108, x387)=x389∧(∀x1113,x1114,x1115,x1116:minus_nat(x1112, x1111)=pos(x1113)∧pos(x1112)=x1114∧minus_int(x1114, x1115)=x1116 ⇒ EVAL(neg(0), x1116, pos(x1113), x1115)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x1113, x1113), x1113)), x1116)), neg(0), x1116, pos(x1113), x1115)) ⇒ EVAL(neg(0), x389, pos(x390), x387)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x390, x390), x390)), x389)), neg(0), x389, pos(x390), x387))

We simplified constraint (395) using rules (I), (II), (III), (IV) which results in the following new constraint:
(398)    (EVAL(neg(0), x389, pos(0), x387)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(pos(mult_nat(mult_nat(0, 0), 0)), x389)), neg(0), x389, pos(0), x387))

We simplified constraint (396) using rules (I), (II), (III), (IV) which results in the following new constraint:
(399)    (EVAL(neg(0), x389, pos(s(x1110)), x387)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(pos(mult_nat(mult_nat(s(x1110), s(x1110)), s(x1110))), x389)), neg(0), x389, pos(s(x1110)), x387))

We simplified constraint (397) using rules (III), (IV) which results in the following new constraint:
(400)    (EVAL(neg(0), x389, pos(x390), x387)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x390, x390), x390)), x389)), neg(0), x389, pos(x390), x387))
We consider the chain COND_EVAL1(true, pos(z0), z1, z2, x_removed) → EVAL(pos(z0), minus_int(z1, x_removed), plus_int(z2, z1), x_removed), EVAL(y0, y1, pos(x0), x_removed) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, pos(x0), x_removed) which results in the following constraint:
(401)    (EVAL(pos(x396), minus_int(x397, x399), plus_int(x398, x397), x399)=EVAL(x400, x401, pos(x402), x403) ⇒ EVAL(x400, x401, pos(x402), x403)≥COND_EVAL1(and(greatereq_int(x400, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x402, x402), x402)), x401)), x400, x401, pos(x402), x403))

We simplified constraint (401) using rules (I), (II), (III) which results in the following new constraint:
(402)    (minus_int(x397, x399)=x401∧plus_int(x398, x397)=pos(x402) ⇒ EVAL(pos(x396), x401, pos(x402), x399)≥COND_EVAL1(and(greatereq_int(pos(x396), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x402, x402), x402)), x401)), pos(x396), x401, pos(x402), x399))

We simplified constraint (402) using rule (V) (with possible (I) afterwards) using induction on plus_int(x398, x397)=pos(x402) which results in the following new constraints:
(403)    (minus_nat(x1119, x1118)=pos(x402)∧minus_int(neg(x1118), x399)=x401 ⇒ EVAL(pos(x396), x401, pos(x402), x399)≥COND_EVAL1(and(greatereq_int(pos(x396), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x402, x402), x402)), x401)), pos(x396), x401, pos(x402), x399))

(404)    (minus_nat(x1120, x1121)=pos(x402)∧minus_int(pos(x1120), x399)=x401 ⇒ EVAL(pos(x396), x401, pos(x402), x399)≥COND_EVAL1(and(greatereq_int(pos(x396), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x402, x402), x402)), x401)), pos(x396), x401, pos(x402), x399))

(405)    (pos(plus_nat(x1125, x1124))=pos(x402)∧minus_int(pos(x1124), x399)=x401 ⇒ EVAL(pos(x396), x401, pos(x402), x399)≥COND_EVAL1(and(greatereq_int(pos(x396), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x402, x402), x402)), x401)), pos(x396), x401, pos(x402), x399))

We simplified constraint (403) using rule (VII) which results in the following new constraint:
(406)    (minus_nat(x1119, x1118)=pos(x402)∧neg(x1118)=x1126∧minus_int(x1126, x399)=x401 ⇒ EVAL(pos(x396), x401, pos(x402), x399)≥COND_EVAL1(and(greatereq_int(pos(x396), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x402, x402), x402)), x401)), pos(x396), x401, pos(x402), x399))

We simplified constraint (404) using rule (VII) which results in the following new constraint:
(407)    (minus_nat(x1120, x1121)=pos(x402)∧pos(x1120)=x1136∧minus_int(x1136, x399)=x401 ⇒ EVAL(pos(x396), x401, pos(x402), x399)≥COND_EVAL1(and(greatereq_int(pos(x396), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x402, x402), x402)), x401)), pos(x396), x401, pos(x402), x399))

We solved constraint (405) using rules (I), (II), (III), (IV), (VII).We simplified constraint (406) using rule (V) (with possible (I) afterwards) using induction on minus_nat(x1119, x1118)=pos(x402) which results in the following new constraints:
(408)    (pos(0)=pos(x402)∧neg(0)=x1126∧minus_int(x1126, x399)=x401 ⇒ EVAL(pos(x396), x401, pos(x402), x399)≥COND_EVAL1(and(greatereq_int(pos(x396), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x402, x402), x402)), x401)), pos(x396), x401, pos(x402), x399))

(409)    (pos(s(x1128))=pos(x402)∧neg(0)=x1126∧minus_int(x1126, x399)=x401 ⇒ EVAL(pos(x396), x401, pos(x402), x399)≥COND_EVAL1(and(greatereq_int(pos(x396), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x402, x402), x402)), x401)), pos(x396), x401, pos(x402), x399))

(410)    (minus_nat(x1130, x1129)=pos(x402)∧neg(s(x1129))=x1126∧minus_int(x1126, x399)=x401∧(∀x1131,x1132,x1133,x1134,x1135:minus_nat(x1130, x1129)=pos(x1131)∧neg(x1129)=x1132∧minus_int(x1132, x1133)=x1134 ⇒ EVAL(pos(x1135), x1134, pos(x1131), x1133)≥COND_EVAL1(and(greatereq_int(pos(x1135), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x1131, x1131), x1131)), x1134)), pos(x1135), x1134, pos(x1131), x1133)) ⇒ EVAL(pos(x396), x401, pos(x402), x399)≥COND_EVAL1(and(greatereq_int(pos(x396), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x402, x402), x402)), x401)), pos(x396), x401, pos(x402), x399))

We simplified constraint (408) using rules (I), (II), (III), (IV) which results in the following new constraint:
(411)    (EVAL(pos(x396), x401, pos(0), x399)≥COND_EVAL1(and(greatereq_int(pos(x396), pos(0)), greatereq_int(pos(mult_nat(mult_nat(0, 0), 0)), x401)), pos(x396), x401, pos(0), x399))

We simplified constraint (409) using rules (I), (II), (III), (IV) which results in the following new constraint:
(412)    (EVAL(pos(x396), x401, pos(s(x1128)), x399)≥COND_EVAL1(and(greatereq_int(pos(x396), pos(0)), greatereq_int(pos(mult_nat(mult_nat(s(x1128), s(x1128)), s(x1128))), x401)), pos(x396), x401, pos(s(x1128)), x399))

We simplified constraint (410) using rules (III), (IV) which results in the following new constraint:
(413)    (EVAL(pos(x396), x401, pos(x402), x399)≥COND_EVAL1(and(greatereq_int(pos(x396), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x402, x402), x402)), x401)), pos(x396), x401, pos(x402), x399))

We simplified constraint (407) using rule (V) (with possible (I) afterwards) using induction on minus_nat(x1120, x1121)=pos(x402) which results in the following new constraints:
(414)    (pos(0)=pos(x402)∧pos(0)=x1136∧minus_int(x1136, x399)=x401 ⇒ EVAL(pos(x396), x401, pos(x402), x399)≥COND_EVAL1(and(greatereq_int(pos(x396), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x402, x402), x402)), x401)), pos(x396), x401, pos(x402), x399))

(415)    (pos(s(x1138))=pos(x402)∧pos(s(x1138))=x1136∧minus_int(x1136, x399)=x401 ⇒ EVAL(pos(x396), x401, pos(x402), x399)≥COND_EVAL1(and(greatereq_int(pos(x396), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x402, x402), x402)), x401)), pos(x396), x401, pos(x402), x399))

(416)    (minus_nat(x1140, x1139)=pos(x402)∧pos(s(x1140))=x1136∧minus_int(x1136, x399)=x401∧(∀x1141,x1142,x1143,x1144,x1145:minus_nat(x1140, x1139)=pos(x1141)∧pos(x1140)=x1142∧minus_int(x1142, x1143)=x1144 ⇒ EVAL(pos(x1145), x1144, pos(x1141), x1143)≥COND_EVAL1(and(greatereq_int(pos(x1145), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x1141, x1141), x1141)), x1144)), pos(x1145), x1144, pos(x1141), x1143)) ⇒ EVAL(pos(x396), x401, pos(x402), x399)≥COND_EVAL1(and(greatereq_int(pos(x396), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x402, x402), x402)), x401)), pos(x396), x401, pos(x402), x399))

We simplified constraint (414) using rules (I), (II), (III), (IV) which results in the following new constraint:
(417)    (EVAL(pos(x396), x401, pos(0), x399)≥COND_EVAL1(and(greatereq_int(pos(x396), pos(0)), greatereq_int(pos(mult_nat(mult_nat(0, 0), 0)), x401)), pos(x396), x401, pos(0), x399))

We simplified constraint (415) using rules (I), (II), (III), (IV) which results in the following new constraint:
(418)    (EVAL(pos(x396), x401, pos(s(x1138)), x399)≥COND_EVAL1(and(greatereq_int(pos(x396), pos(0)), greatereq_int(pos(mult_nat(mult_nat(s(x1138), s(x1138)), s(x1138))), x401)), pos(x396), x401, pos(s(x1138)), x399))

We simplified constraint (416) using rules (III), (IV) which results in the following new constraint:
(419)    (EVAL(pos(x396), x401, pos(x402), x399)≥COND_EVAL1(and(greatereq_int(pos(x396), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x402, x402), x402)), x401)), pos(x396), x401, pos(x402), x399))
We consider the chain COND_EVAL1(true, neg(s(z0)), z1, z2, x_removed) → EVAL(neg(s(z0)), minus_int(z1, x_removed), plus_int(z2, z1), x_removed), EVAL(y0, y1, pos(x0), x_removed) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, pos(x0), x_removed) which results in the following constraint:
(420)    (EVAL(neg(s(x408)), minus_int(x409, x411), plus_int(x410, x409), x411)=EVAL(x412, x413, pos(x414), x415) ⇒ EVAL(x412, x413, pos(x414), x415)≥COND_EVAL1(and(greatereq_int(x412, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x414, x414), x414)), x413)), x412, x413, pos(x414), x415))

We simplified constraint (420) using rules (I), (II), (III) which results in the following new constraint:
(421)    (minus_int(x409, x411)=x413∧plus_int(x410, x409)=pos(x414) ⇒ EVAL(neg(s(x408)), x413, pos(x414), x411)≥COND_EVAL1(and(greatereq_int(neg(s(x408)), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x414, x414), x414)), x413)), neg(s(x408)), x413, pos(x414), x411))

We simplified constraint (421) using rule (V) (with possible (I) afterwards) using induction on plus_int(x410, x409)=pos(x414) which results in the following new constraints:
(422)    (minus_nat(x1148, x1147)=pos(x414)∧minus_int(neg(x1147), x411)=x413 ⇒ EVAL(neg(s(x408)), x413, pos(x414), x411)≥COND_EVAL1(and(greatereq_int(neg(s(x408)), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x414, x414), x414)), x413)), neg(s(x408)), x413, pos(x414), x411))

(423)    (minus_nat(x1149, x1150)=pos(x414)∧minus_int(pos(x1149), x411)=x413 ⇒ EVAL(neg(s(x408)), x413, pos(x414), x411)≥COND_EVAL1(and(greatereq_int(neg(s(x408)), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x414, x414), x414)), x413)), neg(s(x408)), x413, pos(x414), x411))

(424)    (pos(plus_nat(x1154, x1153))=pos(x414)∧minus_int(pos(x1153), x411)=x413 ⇒ EVAL(neg(s(x408)), x413, pos(x414), x411)≥COND_EVAL1(and(greatereq_int(neg(s(x408)), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x414, x414), x414)), x413)), neg(s(x408)), x413, pos(x414), x411))

We simplified constraint (422) using rule (VII) which results in the following new constraint:
(425)    (minus_nat(x1148, x1147)=pos(x414)∧neg(x1147)=x1155∧minus_int(x1155, x411)=x413 ⇒ EVAL(neg(s(x408)), x413, pos(x414), x411)≥COND_EVAL1(and(greatereq_int(neg(s(x408)), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x414, x414), x414)), x413)), neg(s(x408)), x413, pos(x414), x411))

We simplified constraint (423) using rule (VII) which results in the following new constraint:
(426)    (minus_nat(x1149, x1150)=pos(x414)∧pos(x1149)=x1165∧minus_int(x1165, x411)=x413 ⇒ EVAL(neg(s(x408)), x413, pos(x414), x411)≥COND_EVAL1(and(greatereq_int(neg(s(x408)), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x414, x414), x414)), x413)), neg(s(x408)), x413, pos(x414), x411))

We solved constraint (424) using rules (I), (II), (III), (IV), (VII).We simplified constraint (425) using rule (V) (with possible (I) afterwards) using induction on minus_nat(x1148, x1147)=pos(x414) which results in the following new constraints:
(427)    (pos(0)=pos(x414)∧neg(0)=x1155∧minus_int(x1155, x411)=x413 ⇒ EVAL(neg(s(x408)), x413, pos(x414), x411)≥COND_EVAL1(and(greatereq_int(neg(s(x408)), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x414, x414), x414)), x413)), neg(s(x408)), x413, pos(x414), x411))

(428)    (pos(s(x1157))=pos(x414)∧neg(0)=x1155∧minus_int(x1155, x411)=x413 ⇒ EVAL(neg(s(x408)), x413, pos(x414), x411)≥COND_EVAL1(and(greatereq_int(neg(s(x408)), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x414, x414), x414)), x413)), neg(s(x408)), x413, pos(x414), x411))

(429)    (minus_nat(x1159, x1158)=pos(x414)∧neg(s(x1158))=x1155∧minus_int(x1155, x411)=x413∧(∀x1160,x1161,x1162,x1163,x1164:minus_nat(x1159, x1158)=pos(x1160)∧neg(x1158)=x1161∧minus_int(x1161, x1162)=x1163 ⇒ EVAL(neg(s(x1164)), x1163, pos(x1160), x1162)≥COND_EVAL1(and(greatereq_int(neg(s(x1164)), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x1160, x1160), x1160)), x1163)), neg(s(x1164)), x1163, pos(x1160), x1162)) ⇒ EVAL(neg(s(x408)), x413, pos(x414), x411)≥COND_EVAL1(and(greatereq_int(neg(s(x408)), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x414, x414), x414)), x413)), neg(s(x408)), x413, pos(x414), x411))

We simplified constraint (427) using rules (I), (II), (III), (IV) which results in the following new constraint:
(430)    (EVAL(neg(s(x408)), x413, pos(0), x411)≥COND_EVAL1(and(greatereq_int(neg(s(x408)), pos(0)), greatereq_int(pos(mult_nat(mult_nat(0, 0), 0)), x413)), neg(s(x408)), x413, pos(0), x411))

We simplified constraint (428) using rules (I), (II), (III), (IV) which results in the following new constraint:
(431)    (EVAL(neg(s(x408)), x413, pos(s(x1157)), x411)≥COND_EVAL1(and(greatereq_int(neg(s(x408)), pos(0)), greatereq_int(pos(mult_nat(mult_nat(s(x1157), s(x1157)), s(x1157))), x413)), neg(s(x408)), x413, pos(s(x1157)), x411))

We simplified constraint (429) using rules (III), (IV) which results in the following new constraint:
(432)    (EVAL(neg(s(x408)), x413, pos(x414), x411)≥COND_EVAL1(and(greatereq_int(neg(s(x408)), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x414, x414), x414)), x413)), neg(s(x408)), x413, pos(x414), x411))

We simplified constraint (426) using rule (V) (with possible (I) afterwards) using induction on minus_nat(x1149, x1150)=pos(x414) which results in the following new constraints:
(433)    (pos(0)=pos(x414)∧pos(0)=x1165∧minus_int(x1165, x411)=x413 ⇒ EVAL(neg(s(x408)), x413, pos(x414), x411)≥COND_EVAL1(and(greatereq_int(neg(s(x408)), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x414, x414), x414)), x413)), neg(s(x408)), x413, pos(x414), x411))

(434)    (pos(s(x1167))=pos(x414)∧pos(s(x1167))=x1165∧minus_int(x1165, x411)=x413 ⇒ EVAL(neg(s(x408)), x413, pos(x414), x411)≥COND_EVAL1(and(greatereq_int(neg(s(x408)), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x414, x414), x414)), x413)), neg(s(x408)), x413, pos(x414), x411))

(435)    (minus_nat(x1169, x1168)=pos(x414)∧pos(s(x1169))=x1165∧minus_int(x1165, x411)=x413∧(∀x1170,x1171,x1172,x1173,x1174:minus_nat(x1169, x1168)=pos(x1170)∧pos(x1169)=x1171∧minus_int(x1171, x1172)=x1173 ⇒ EVAL(neg(s(x1174)), x1173, pos(x1170), x1172)≥COND_EVAL1(and(greatereq_int(neg(s(x1174)), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x1170, x1170), x1170)), x1173)), neg(s(x1174)), x1173, pos(x1170), x1172)) ⇒ EVAL(neg(s(x408)), x413, pos(x414), x411)≥COND_EVAL1(and(greatereq_int(neg(s(x408)), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x414, x414), x414)), x413)), neg(s(x408)), x413, pos(x414), x411))

We simplified constraint (433) using rules (I), (II), (III), (IV) which results in the following new constraint:
(436)    (EVAL(neg(s(x408)), x413, pos(0), x411)≥COND_EVAL1(and(greatereq_int(neg(s(x408)), pos(0)), greatereq_int(pos(mult_nat(mult_nat(0, 0), 0)), x413)), neg(s(x408)), x413, pos(0), x411))

We simplified constraint (434) using rules (I), (II), (III), (IV) which results in the following new constraint:
(437)    (EVAL(neg(s(x408)), x413, pos(s(x1167)), x411)≥COND_EVAL1(and(greatereq_int(neg(s(x408)), pos(0)), greatereq_int(pos(mult_nat(mult_nat(s(x1167), s(x1167)), s(x1167))), x413)), neg(s(x408)), x413, pos(s(x1167)), x411))

We simplified constraint (435) using rules (III), (IV) which results in the following new constraint:
(438)    (EVAL(neg(s(x408)), x413, pos(x414), x411)≥COND_EVAL1(and(greatereq_int(neg(s(x408)), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x414, x414), x414)), x413)), neg(s(x408)), x413, pos(x414), x411))

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

EVAL(neg(0), y1, y2, x_removed) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), neg(0), y1, y2, x_removed)
- (EVAL(neg(0), x7, x8, x6)≥COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(x8, x8), x8), x7)), neg(0), x7, x8, x6))
- (EVAL(neg(0), x14, x15, x13)≥COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(x15, x15), x15), x14)), neg(0), x14, x15, x13))
- (EVAL(neg(0), x20, x21, x19)≥COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(x21, x21), x21), x20)), neg(0), x20, x21, x19))
COND_EVAL1(true, z0, z1, pos(z2), x_removed) → EVAL(z0, minus_int(z1, x_removed), plus_int(pos(z2), z1), x_removed)
- (COND_EVAL1(true, neg(0), pos(0), pos(x48), x45)≥EVAL(neg(0), minus_int(pos(0), x45), plus_int(pos(x48), pos(0)), x45))
- (COND_EVAL1(true, neg(0), pos(0), pos(x48), x45)≥EVAL(neg(0), minus_int(pos(0), x45), plus_int(pos(x48), pos(0)), x45))
- (COND_EVAL1(true, neg(0), neg(x431), pos(x48), x45)≥EVAL(neg(0), minus_int(neg(x431), x45), plus_int(pos(x48), neg(x431)), x45))
- (COND_EVAL1(true, neg(0), neg(x432), pos(x48), x45)≥EVAL(neg(0), minus_int(neg(x432), x45), plus_int(pos(x48), neg(x432)), x45))
- (COND_EVAL1(true, neg(0), pos(x438), pos(x48), x45)≥EVAL(neg(0), minus_int(pos(x438), x45), plus_int(pos(x48), pos(x438)), x45) ⇒ COND_EVAL1(true, neg(0), pos(s(x438)), pos(x48), x45)≥EVAL(neg(0), minus_int(pos(s(x438)), x45), plus_int(pos(x48), pos(s(x438))), x45))
- (COND_EVAL1(true, neg(0), neg(x442), pos(x48), x45)≥EVAL(neg(0), minus_int(neg(x442), x45), plus_int(pos(x48), neg(x442)), x45) ⇒ COND_EVAL1(true, neg(0), neg(s(x442)), pos(x48), x45)≥EVAL(neg(0), minus_int(neg(s(x442)), x45), plus_int(pos(x48), neg(s(x442))), x45))
- (COND_EVAL1(true, pos(x61), pos(0), pos(x67), x64)≥EVAL(pos(x61), minus_int(pos(0), x64), plus_int(pos(x67), pos(0)), x64))
- (COND_EVAL1(true, pos(x61), pos(0), pos(x67), x64)≥EVAL(pos(x61), minus_int(pos(0), x64), plus_int(pos(x67), pos(0)), x64))
- (COND_EVAL1(true, pos(x61), neg(x455), pos(x67), x64)≥EVAL(pos(x61), minus_int(neg(x455), x64), plus_int(pos(x67), neg(x455)), x64))
- (COND_EVAL1(true, pos(x61), neg(x456), pos(x67), x64)≥EVAL(pos(x61), minus_int(neg(x456), x64), plus_int(pos(x67), neg(x456)), x64))
- (COND_EVAL1(true, pos(x61), pos(x462), pos(x67), x64)≥EVAL(pos(x61), minus_int(pos(x462), x64), plus_int(pos(x67), pos(x462)), x64) ⇒ COND_EVAL1(true, pos(x61), pos(s(x462)), pos(x67), x64)≥EVAL(pos(x61), minus_int(pos(s(x462)), x64), plus_int(pos(x67), pos(s(x462))), x64))
- (COND_EVAL1(true, pos(x61), neg(x467), pos(x67), x64)≥EVAL(pos(x61), minus_int(neg(x467), x64), plus_int(pos(x67), neg(x467)), x64) ⇒ COND_EVAL1(true, pos(x61), neg(s(x467)), pos(x67), x64)≥EVAL(pos(x61), minus_int(neg(s(x467)), x64), plus_int(pos(x67), neg(s(x467))), x64))
- (COND_EVAL1(true, pos(x476), pos(0), pos(x83), x84)≥EVAL(pos(x476), minus_int(pos(0), x84), plus_int(pos(x83), pos(0)), x84))
- (COND_EVAL1(true, pos(x476), pos(0), pos(x83), x84)≥EVAL(pos(x476), minus_int(pos(0), x84), plus_int(pos(x83), pos(0)), x84))
- (COND_EVAL1(true, pos(x476), neg(x499), pos(x83), x84)≥EVAL(pos(x476), minus_int(neg(x499), x84), plus_int(pos(x83), neg(x499)), x84))
- (COND_EVAL1(true, pos(x476), neg(x500), pos(x83), x84)≥EVAL(pos(x476), minus_int(neg(x500), x84), plus_int(pos(x83), neg(x500)), x84))
- (COND_EVAL1(true, pos(x476), pos(x506), pos(x83), x84)≥EVAL(pos(x476), minus_int(pos(x506), x84), plus_int(pos(x83), pos(x506)), x84) ⇒ COND_EVAL1(true, pos(x476), pos(s(x506)), pos(x83), x84)≥EVAL(pos(x476), minus_int(pos(s(x506)), x84), plus_int(pos(x83), pos(s(x506))), x84))
- (COND_EVAL1(true, pos(x476), neg(x511), pos(x83), x84)≥EVAL(pos(x476), minus_int(neg(x511), x84), plus_int(pos(x83), neg(x511)), x84) ⇒ COND_EVAL1(true, pos(x476), neg(s(x511)), pos(x83), x84)≥EVAL(pos(x476), minus_int(neg(s(x511)), x84), plus_int(pos(x83), neg(s(x511))), x84))
- (COND_EVAL1(true, neg(0), pos(0), pos(x83), x84)≥EVAL(neg(0), minus_int(pos(0), x84), plus_int(pos(x83), pos(0)), x84))
- (COND_EVAL1(true, neg(0), pos(0), pos(x83), x84)≥EVAL(neg(0), minus_int(pos(0), x84), plus_int(pos(x83), pos(0)), x84))
- (COND_EVAL1(true, neg(0), neg(x518), pos(x83), x84)≥EVAL(neg(0), minus_int(neg(x518), x84), plus_int(pos(x83), neg(x518)), x84))
- (COND_EVAL1(true, neg(0), neg(x519), pos(x83), x84)≥EVAL(neg(0), minus_int(neg(x519), x84), plus_int(pos(x83), neg(x519)), x84))
- (COND_EVAL1(true, neg(0), pos(x525), pos(x83), x84)≥EVAL(neg(0), minus_int(pos(x525), x84), plus_int(pos(x83), pos(x525)), x84) ⇒ COND_EVAL1(true, neg(0), pos(s(x525)), pos(x83), x84)≥EVAL(neg(0), minus_int(pos(s(x525)), x84), plus_int(pos(x83), pos(s(x525))), x84))
- (COND_EVAL1(true, neg(0), neg(x529), pos(x83), x84)≥EVAL(neg(0), minus_int(neg(x529), x84), plus_int(pos(x83), neg(x529)), x84) ⇒ COND_EVAL1(true, neg(0), neg(s(x529)), pos(x83), x84)≥EVAL(neg(0), minus_int(neg(s(x529)), x84), plus_int(pos(x83), neg(s(x529))), x84))
COND_EVAL1(true, z0, z1, neg(z2), x_removed) → EVAL(z0, minus_int(z1, x_removed), plus_int(neg(z2), z1), x_removed)
- (COND_EVAL1(true, neg(0), pos(0), neg(x94), x91)≥EVAL(neg(0), minus_int(pos(0), x91), plus_int(neg(x94), pos(0)), x91))
- (COND_EVAL1(true, neg(0), pos(0), neg(x94), x91)≥EVAL(neg(0), minus_int(pos(0), x91), plus_int(neg(x94), pos(0)), x91))
- (COND_EVAL1(true, neg(0), neg(x542), neg(x94), x91)≥EVAL(neg(0), minus_int(neg(x542), x91), plus_int(neg(x94), neg(x542)), x91))
- (COND_EVAL1(true, neg(0), neg(x543), neg(x94), x91)≥EVAL(neg(0), minus_int(neg(x543), x91), plus_int(neg(x94), neg(x543)), x91))
- (COND_EVAL1(true, neg(0), pos(x549), neg(x94), x91)≥EVAL(neg(0), minus_int(pos(x549), x91), plus_int(neg(x94), pos(x549)), x91) ⇒ COND_EVAL1(true, neg(0), pos(s(x549)), neg(x94), x91)≥EVAL(neg(0), minus_int(pos(s(x549)), x91), plus_int(neg(x94), pos(s(x549))), x91))
- (COND_EVAL1(true, neg(0), neg(x553), neg(x94), x91)≥EVAL(neg(0), minus_int(neg(x553), x91), plus_int(neg(x94), neg(x553)), x91) ⇒ COND_EVAL1(true, neg(0), neg(s(x553)), neg(x94), x91)≥EVAL(neg(0), minus_int(neg(s(x553)), x91), plus_int(neg(x94), neg(s(x553))), x91))
- (COND_EVAL1(true, pos(x107), pos(0), neg(x113), x110)≥EVAL(pos(x107), minus_int(pos(0), x110), plus_int(neg(x113), pos(0)), x110))
- (COND_EVAL1(true, pos(x107), pos(0), neg(x113), x110)≥EVAL(pos(x107), minus_int(pos(0), x110), plus_int(neg(x113), pos(0)), x110))
- (COND_EVAL1(true, pos(x107), neg(x566), neg(x113), x110)≥EVAL(pos(x107), minus_int(neg(x566), x110), plus_int(neg(x113), neg(x566)), x110))
- (COND_EVAL1(true, pos(x107), neg(x567), neg(x113), x110)≥EVAL(pos(x107), minus_int(neg(x567), x110), plus_int(neg(x113), neg(x567)), x110))
- (COND_EVAL1(true, pos(x107), pos(x573), neg(x113), x110)≥EVAL(pos(x107), minus_int(pos(x573), x110), plus_int(neg(x113), pos(x573)), x110) ⇒ COND_EVAL1(true, pos(x107), pos(s(x573)), neg(x113), x110)≥EVAL(pos(x107), minus_int(pos(s(x573)), x110), plus_int(neg(x113), pos(s(x573))), x110))
- (COND_EVAL1(true, pos(x107), neg(x578), neg(x113), x110)≥EVAL(pos(x107), minus_int(neg(x578), x110), plus_int(neg(x113), neg(x578)), x110) ⇒ COND_EVAL1(true, pos(x107), neg(s(x578)), neg(x113), x110)≥EVAL(pos(x107), minus_int(neg(s(x578)), x110), plus_int(neg(x113), neg(s(x578))), x110))
- (COND_EVAL1(true, pos(x587), pos(0), neg(x121), x122)≥EVAL(pos(x587), minus_int(pos(0), x122), plus_int(neg(x121), pos(0)), x122))
- (COND_EVAL1(true, pos(x587), pos(0), neg(x121), x122)≥EVAL(pos(x587), minus_int(pos(0), x122), plus_int(neg(x121), pos(0)), x122))
- (COND_EVAL1(true, pos(x587), neg(x610), neg(x121), x122)≥EVAL(pos(x587), minus_int(neg(x610), x122), plus_int(neg(x121), neg(x610)), x122))
- (COND_EVAL1(true, pos(x587), neg(x611), neg(x121), x122)≥EVAL(pos(x587), minus_int(neg(x611), x122), plus_int(neg(x121), neg(x611)), x122))
- (COND_EVAL1(true, pos(x587), pos(x617), neg(x121), x122)≥EVAL(pos(x587), minus_int(pos(x617), x122), plus_int(neg(x121), pos(x617)), x122) ⇒ COND_EVAL1(true, pos(x587), pos(s(x617)), neg(x121), x122)≥EVAL(pos(x587), minus_int(pos(s(x617)), x122), plus_int(neg(x121), pos(s(x617))), x122))
- (COND_EVAL1(true, pos(x587), neg(x622), neg(x121), x122)≥EVAL(pos(x587), minus_int(neg(x622), x122), plus_int(neg(x121), neg(x622)), x122) ⇒ COND_EVAL1(true, pos(x587), neg(s(x622)), neg(x121), x122)≥EVAL(pos(x587), minus_int(neg(s(x622)), x122), plus_int(neg(x121), neg(s(x622))), x122))
- (COND_EVAL1(true, neg(0), pos(0), neg(x121), x122)≥EVAL(neg(0), minus_int(pos(0), x122), plus_int(neg(x121), pos(0)), x122))
- (COND_EVAL1(true, neg(0), pos(0), neg(x121), x122)≥EVAL(neg(0), minus_int(pos(0), x122), plus_int(neg(x121), pos(0)), x122))
- (COND_EVAL1(true, neg(0), neg(x629), neg(x121), x122)≥EVAL(neg(0), minus_int(neg(x629), x122), plus_int(neg(x121), neg(x629)), x122))
- (COND_EVAL1(true, neg(0), neg(x630), neg(x121), x122)≥EVAL(neg(0), minus_int(neg(x630), x122), plus_int(neg(x121), neg(x630)), x122))
- (COND_EVAL1(true, neg(0), pos(x636), neg(x121), x122)≥EVAL(neg(0), minus_int(pos(x636), x122), plus_int(neg(x121), pos(x636)), x122) ⇒ COND_EVAL1(true, neg(0), pos(s(x636)), neg(x121), x122)≥EVAL(neg(0), minus_int(pos(s(x636)), x122), plus_int(neg(x121), pos(s(x636))), x122))
- (COND_EVAL1(true, neg(0), neg(x640), neg(x121), x122)≥EVAL(neg(0), minus_int(neg(x640), x122), plus_int(neg(x121), neg(x640)), x122) ⇒ COND_EVAL1(true, neg(0), neg(s(x640)), neg(x121), x122)≥EVAL(neg(0), minus_int(neg(s(x640)), x122), plus_int(neg(x121), neg(s(x640))), x122))
COND_EVAL1(true, neg(0), z0, z1, x_removed) → EVAL(neg(0), minus_int(z0, x_removed), plus_int(z1, z0), x_removed)
- (COND_EVAL1(true, neg(0), pos(0), x136, x137)≥EVAL(neg(0), minus_int(pos(0), x137), plus_int(x136, pos(0)), x137))
- (COND_EVAL1(true, neg(0), pos(0), x136, x137)≥EVAL(neg(0), minus_int(pos(0), x137), plus_int(x136, pos(0)), x137))
- (COND_EVAL1(true, neg(0), neg(x651), x136, x137)≥EVAL(neg(0), minus_int(neg(x651), x137), plus_int(x136, neg(x651)), x137))
- (COND_EVAL1(true, neg(0), neg(x652), x136, x137)≥EVAL(neg(0), minus_int(neg(x652), x137), plus_int(x136, neg(x652)), x137))
- (COND_EVAL1(true, neg(0), pos(x658), x136, x137)≥EVAL(neg(0), minus_int(pos(x658), x137), plus_int(x136, pos(x658)), x137) ⇒ COND_EVAL1(true, neg(0), pos(s(x658)), x136, x137)≥EVAL(neg(0), minus_int(pos(s(x658)), x137), plus_int(x136, pos(s(x658))), x137))
- (COND_EVAL1(true, neg(0), neg(x662), x136, x137)≥EVAL(neg(0), minus_int(neg(x662), x137), plus_int(x136, neg(x662)), x137) ⇒ COND_EVAL1(true, neg(0), neg(s(x662)), x136, x137)≥EVAL(neg(0), minus_int(neg(s(x662)), x137), plus_int(x136, neg(s(x662))), x137))
- (COND_EVAL1(true, neg(0), pos(0), neg(x162), x163)≥EVAL(neg(0), minus_int(pos(0), x163), plus_int(neg(x162), pos(0)), x163))
- (COND_EVAL1(true, neg(0), pos(0), neg(x162), x163)≥EVAL(neg(0), minus_int(pos(0), x163), plus_int(neg(x162), pos(0)), x163))
- (COND_EVAL1(true, neg(0), neg(x694), neg(x162), x163)≥EVAL(neg(0), minus_int(neg(x694), x163), plus_int(neg(x162), neg(x694)), x163))
- (COND_EVAL1(true, neg(0), neg(x695), neg(x162), x163)≥EVAL(neg(0), minus_int(neg(x695), x163), plus_int(neg(x162), neg(x695)), x163))
- (COND_EVAL1(true, neg(0), pos(x701), neg(x162), x163)≥EVAL(neg(0), minus_int(pos(x701), x163), plus_int(neg(x162), pos(x701)), x163) ⇒ COND_EVAL1(true, neg(0), pos(s(x701)), neg(x162), x163)≥EVAL(neg(0), minus_int(pos(s(x701)), x163), plus_int(neg(x162), pos(s(x701))), x163))
- (COND_EVAL1(true, neg(0), neg(x705), neg(x162), x163)≥EVAL(neg(0), minus_int(neg(x705), x163), plus_int(neg(x162), neg(x705)), x163) ⇒ COND_EVAL1(true, neg(0), neg(s(x705)), neg(x162), x163)≥EVAL(neg(0), minus_int(neg(s(x705)), x163), plus_int(neg(x162), neg(s(x705))), x163))
- (COND_EVAL1(true, neg(0), pos(0), pos(x173), x174)≥EVAL(neg(0), minus_int(pos(0), x174), plus_int(pos(x173), pos(0)), x174))
- (COND_EVAL1(true, neg(0), pos(0), pos(x173), x174)≥EVAL(neg(0), minus_int(pos(0), x174), plus_int(pos(x173), pos(0)), x174))
- (COND_EVAL1(true, neg(0), neg(x737), pos(x173), x174)≥EVAL(neg(0), minus_int(neg(x737), x174), plus_int(pos(x173), neg(x737)), x174))
- (COND_EVAL1(true, neg(0), neg(x738), pos(x173), x174)≥EVAL(neg(0), minus_int(neg(x738), x174), plus_int(pos(x173), neg(x738)), x174))
- (COND_EVAL1(true, neg(0), pos(x744), pos(x173), x174)≥EVAL(neg(0), minus_int(pos(x744), x174), plus_int(pos(x173), pos(x744)), x174) ⇒ COND_EVAL1(true, neg(0), pos(s(x744)), pos(x173), x174)≥EVAL(neg(0), minus_int(pos(s(x744)), x174), plus_int(pos(x173), pos(s(x744))), x174))
- (COND_EVAL1(true, neg(0), neg(x748), pos(x173), x174)≥EVAL(neg(0), minus_int(neg(x748), x174), plus_int(pos(x173), neg(x748)), x174) ⇒ COND_EVAL1(true, neg(0), neg(s(x748)), pos(x173), x174)≥EVAL(neg(0), minus_int(neg(s(x748)), x174), plus_int(pos(x173), neg(s(x748))), x174))
EVAL(pos(x0), y1, y2, x_removed) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), pos(x0), y1, y2, x_removed)
- (EVAL(pos(x185), x186, x187, x184)≥COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(x187, x187), x187), x186)), pos(x185), x186, x187, x184))
- (EVAL(pos(x193), x194, x195, x192)≥COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(x195, x195), x195), x194)), pos(x193), x194, x195, x192))
- (EVAL(pos(x204), x209, x210, x207)≥COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(x210, x210), x210), x209)), pos(x204), x209, x210, x207))
COND_EVAL1(true, pos(z0), z1, z2, x_removed) → EVAL(pos(z0), minus_int(z1, x_removed), plus_int(z2, z1), x_removed)
- (COND_EVAL1(true, pos(x238), pos(0), x240, x241)≥EVAL(pos(x238), minus_int(pos(0), x241), plus_int(x240, pos(0)), x241))
- (COND_EVAL1(true, pos(x238), pos(0), x240, x241)≥EVAL(pos(x238), minus_int(pos(0), x241), plus_int(x240, pos(0)), x241))
- (COND_EVAL1(true, pos(x238), neg(x761), x240, x241)≥EVAL(pos(x238), minus_int(neg(x761), x241), plus_int(x240, neg(x761)), x241))
- (COND_EVAL1(true, pos(x238), neg(x762), x240, x241)≥EVAL(pos(x238), minus_int(neg(x762), x241), plus_int(x240, neg(x762)), x241))
- (COND_EVAL1(true, pos(x238), pos(x768), x240, x241)≥EVAL(pos(x238), minus_int(pos(x768), x241), plus_int(x240, pos(x768)), x241) ⇒ COND_EVAL1(true, pos(x238), pos(s(x768)), x240, x241)≥EVAL(pos(x238), minus_int(pos(s(x768)), x241), plus_int(x240, pos(s(x768))), x241))
- (COND_EVAL1(true, pos(x238), neg(x773), x240, x241)≥EVAL(pos(x238), minus_int(neg(x773), x241), plus_int(x240, neg(x773)), x241) ⇒ COND_EVAL1(true, pos(x238), neg(s(x773)), x240, x241)≥EVAL(pos(x238), minus_int(neg(s(x773)), x241), plus_int(x240, neg(s(x773))), x241))
- (COND_EVAL1(true, pos(x254), pos(0), neg(x252), x253)≥EVAL(pos(x254), minus_int(pos(0), x253), plus_int(neg(x252), pos(0)), x253))
- (COND_EVAL1(true, pos(x254), pos(0), neg(x252), x253)≥EVAL(pos(x254), minus_int(pos(0), x253), plus_int(neg(x252), pos(0)), x253))
- (COND_EVAL1(true, pos(x254), neg(x808), neg(x252), x253)≥EVAL(pos(x254), minus_int(neg(x808), x253), plus_int(neg(x252), neg(x808)), x253))
- (COND_EVAL1(true, pos(x254), neg(x809), neg(x252), x253)≥EVAL(pos(x254), minus_int(neg(x809), x253), plus_int(neg(x252), neg(x809)), x253))
- (COND_EVAL1(true, pos(x254), pos(x815), neg(x252), x253)≥EVAL(pos(x254), minus_int(pos(x815), x253), plus_int(neg(x252), pos(x815)), x253) ⇒ COND_EVAL1(true, pos(x254), pos(s(x815)), neg(x252), x253)≥EVAL(pos(x254), minus_int(pos(s(x815)), x253), plus_int(neg(x252), pos(s(x815))), x253))
- (COND_EVAL1(true, pos(x254), neg(x820), neg(x252), x253)≥EVAL(pos(x254), minus_int(neg(x820), x253), plus_int(neg(x252), neg(x820)), x253) ⇒ COND_EVAL1(true, pos(x254), neg(s(x820)), neg(x252), x253)≥EVAL(pos(x254), minus_int(neg(s(x820)), x253), plus_int(neg(x252), neg(s(x820))), x253))
- (COND_EVAL1(true, pos(x266), pos(0), pos(x264), x265)≥EVAL(pos(x266), minus_int(pos(0), x265), plus_int(pos(x264), pos(0)), x265))
- (COND_EVAL1(true, pos(x266), pos(0), pos(x264), x265)≥EVAL(pos(x266), minus_int(pos(0), x265), plus_int(pos(x264), pos(0)), x265))
- (COND_EVAL1(true, pos(x266), neg(x855), pos(x264), x265)≥EVAL(pos(x266), minus_int(neg(x855), x265), plus_int(pos(x264), neg(x855)), x265))
- (COND_EVAL1(true, pos(x266), neg(x856), pos(x264), x265)≥EVAL(pos(x266), minus_int(neg(x856), x265), plus_int(pos(x264), neg(x856)), x265))
- (COND_EVAL1(true, pos(x266), pos(x862), pos(x264), x265)≥EVAL(pos(x266), minus_int(pos(x862), x265), plus_int(pos(x264), pos(x862)), x265) ⇒ COND_EVAL1(true, pos(x266), pos(s(x862)), pos(x264), x265)≥EVAL(pos(x266), minus_int(pos(s(x862)), x265), plus_int(pos(x264), pos(s(x862))), x265))
- (COND_EVAL1(true, pos(x266), neg(x867), pos(x264), x265)≥EVAL(pos(x266), minus_int(neg(x867), x265), plus_int(pos(x264), neg(x867)), x265) ⇒ COND_EVAL1(true, pos(x266), neg(s(x867)), pos(x264), x265)≥EVAL(pos(x266), minus_int(neg(s(x867)), x265), plus_int(pos(x264), neg(s(x867))), x265))
EVAL(y0, y1, neg(x0), x_removed) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, neg(x0), x_removed)
- (EVAL(x273, x278, neg(s(x882)), x276)≥COND_EVAL1(and(greatereq_int(x273, pos(0)), greatereq_int(neg(mult_nat(mult_nat(s(x882), s(x882)), s(x882))), x278)), x273, x278, neg(s(x882)), x276))
- (EVAL(x273, x278, neg(x279), x276)≥COND_EVAL1(and(greatereq_int(x273, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x279, x279), x279)), x278)), x273, x278, neg(x279), x276))
- (EVAL(x281, x286, neg(s(x901)), x284)≥COND_EVAL1(and(greatereq_int(x281, pos(0)), greatereq_int(neg(mult_nat(mult_nat(s(x901), s(x901)), s(x901))), x286)), x281, x286, neg(s(x901)), x284))
- (EVAL(x281, x286, neg(x287), x284)≥COND_EVAL1(and(greatereq_int(x281, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x287, x287), x287)), x286)), x281, x286, neg(x287), x284))
- (EVAL(neg(0), x293, neg(s(x920)), x291)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(neg(mult_nat(mult_nat(s(x920), s(x920)), s(x920))), x293)), neg(0), x293, neg(s(x920)), x291))
- (EVAL(neg(0), x293, neg(x294), x291)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x294, x294), x294)), x293)), neg(0), x293, neg(x294), x291))
- (EVAL(neg(0), x293, neg(s(x929)), x291)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(neg(mult_nat(mult_nat(s(x929), s(x929)), s(x929))), x293)), neg(0), x293, neg(s(x929)), x291))
- (EVAL(neg(0), x293, neg(x294), x291)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x294, x294), x294)), x293)), neg(0), x293, neg(x294), x291))
- (EVAL(pos(x300), x305, neg(s(x947)), x303)≥COND_EVAL1(and(greatereq_int(pos(x300), pos(0)), greatereq_int(neg(mult_nat(mult_nat(s(x947), s(x947)), s(x947))), x305)), pos(x300), x305, neg(s(x947)), x303))
- (EVAL(pos(x300), x305, neg(x306), x303)≥COND_EVAL1(and(greatereq_int(pos(x300), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x306, x306), x306)), x305)), pos(x300), x305, neg(x306), x303))
- (EVAL(pos(x300), x305, neg(s(x957)), x303)≥COND_EVAL1(and(greatereq_int(pos(x300), pos(0)), greatereq_int(neg(mult_nat(mult_nat(s(x957), s(x957)), s(x957))), x305)), pos(x300), x305, neg(s(x957)), x303))
- (EVAL(pos(x300), x305, neg(x306), x303)≥COND_EVAL1(and(greatereq_int(pos(x300), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x306, x306), x306)), x305)), pos(x300), x305, neg(x306), x303))
- (EVAL(neg(s(x312)), x317, neg(s(x976)), x315)≥COND_EVAL1(and(greatereq_int(neg(s(x312)), pos(0)), greatereq_int(neg(mult_nat(mult_nat(s(x976), s(x976)), s(x976))), x317)), neg(s(x312)), x317, neg(s(x976)), x315))
- (EVAL(neg(s(x312)), x317, neg(x318), x315)≥COND_EVAL1(and(greatereq_int(neg(s(x312)), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x318, x318), x318)), x317)), neg(s(x312)), x317, neg(x318), x315))
- (EVAL(neg(s(x312)), x317, neg(s(x986)), x315)≥COND_EVAL1(and(greatereq_int(neg(s(x312)), pos(0)), greatereq_int(neg(mult_nat(mult_nat(s(x986), s(x986)), s(x986))), x317)), neg(s(x312)), x317, neg(s(x986)), x315))
- (EVAL(neg(s(x312)), x317, neg(x318), x315)≥COND_EVAL1(and(greatereq_int(neg(s(x312)), pos(0)), greatereq_int(neg(mult_nat(mult_nat(x318, x318), x318)), x317)), neg(s(x312)), x317, neg(x318), x315))
COND_EVAL1(true, neg(s(z0)), z1, z2, x_removed) → EVAL(neg(s(z0)), minus_int(z1, x_removed), plus_int(z2, z1), x_removed)
EVAL(y0, y1, pos(x0), x_removed) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, pos(x0), x_removed)
- (EVAL(x369, x374, pos(0), x372)≥COND_EVAL1(and(greatereq_int(x369, pos(0)), greatereq_int(pos(mult_nat(mult_nat(0, 0), 0)), x374)), x369, x374, pos(0), x372))
- (EVAL(x369, x374, pos(s(x1063)), x372)≥COND_EVAL1(and(greatereq_int(x369, pos(0)), greatereq_int(pos(mult_nat(mult_nat(s(x1063), s(x1063)), s(x1063))), x374)), x369, x374, pos(s(x1063)), x372))
- (EVAL(x369, x374, pos(x375), x372)≥COND_EVAL1(and(greatereq_int(x369, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x375, x375), x375)), x374)), x369, x374, pos(x375), x372))
- (EVAL(x377, x382, pos(0), x380)≥COND_EVAL1(and(greatereq_int(x377, pos(0)), greatereq_int(pos(mult_nat(mult_nat(0, 0), 0)), x382)), x377, x382, pos(0), x380))
- (EVAL(x377, x382, pos(s(x1083)), x380)≥COND_EVAL1(and(greatereq_int(x377, pos(0)), greatereq_int(pos(mult_nat(mult_nat(s(x1083), s(x1083)), s(x1083))), x382)), x377, x382, pos(s(x1083)), x380))
- (EVAL(x377, x382, pos(x383), x380)≥COND_EVAL1(and(greatereq_int(x377, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x383, x383), x383)), x382)), x377, x382, pos(x383), x380))
- (EVAL(neg(0), x389, pos(0), x387)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(pos(mult_nat(mult_nat(0, 0), 0)), x389)), neg(0), x389, pos(0), x387))
- (EVAL(neg(0), x389, pos(s(x1101)), x387)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(pos(mult_nat(mult_nat(s(x1101), s(x1101)), s(x1101))), x389)), neg(0), x389, pos(s(x1101)), x387))
- (EVAL(neg(0), x389, pos(x390), x387)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x390, x390), x390)), x389)), neg(0), x389, pos(x390), x387))
- (EVAL(neg(0), x389, pos(0), x387)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(pos(mult_nat(mult_nat(0, 0), 0)), x389)), neg(0), x389, pos(0), x387))
- (EVAL(neg(0), x389, pos(s(x1110)), x387)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(pos(mult_nat(mult_nat(s(x1110), s(x1110)), s(x1110))), x389)), neg(0), x389, pos(s(x1110)), x387))
- (EVAL(neg(0), x389, pos(x390), x387)≥COND_EVAL1(and(greatereq_int(neg(0), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x390, x390), x390)), x389)), neg(0), x389, pos(x390), x387))
- (EVAL(pos(x396), x401, pos(0), x399)≥COND_EVAL1(and(greatereq_int(pos(x396), pos(0)), greatereq_int(pos(mult_nat(mult_nat(0, 0), 0)), x401)), pos(x396), x401, pos(0), x399))
- (EVAL(pos(x396), x401, pos(s(x1128)), x399)≥COND_EVAL1(and(greatereq_int(pos(x396), pos(0)), greatereq_int(pos(mult_nat(mult_nat(s(x1128), s(x1128)), s(x1128))), x401)), pos(x396), x401, pos(s(x1128)), x399))
- (EVAL(pos(x396), x401, pos(x402), x399)≥COND_EVAL1(and(greatereq_int(pos(x396), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x402, x402), x402)), x401)), pos(x396), x401, pos(x402), x399))
- (EVAL(pos(x396), x401, pos(0), x399)≥COND_EVAL1(and(greatereq_int(pos(x396), pos(0)), greatereq_int(pos(mult_nat(mult_nat(0, 0), 0)), x401)), pos(x396), x401, pos(0), x399))
- (EVAL(pos(x396), x401, pos(s(x1138)), x399)≥COND_EVAL1(and(greatereq_int(pos(x396), pos(0)), greatereq_int(pos(mult_nat(mult_nat(s(x1138), s(x1138)), s(x1138))), x401)), pos(x396), x401, pos(s(x1138)), x399))
- (EVAL(pos(x396), x401, pos(x402), x399)≥COND_EVAL1(and(greatereq_int(pos(x396), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x402, x402), x402)), x401)), pos(x396), x401, pos(x402), x399))
- (EVAL(neg(s(x408)), x413, pos(0), x411)≥COND_EVAL1(and(greatereq_int(neg(s(x408)), pos(0)), greatereq_int(pos(mult_nat(mult_nat(0, 0), 0)), x413)), neg(s(x408)), x413, pos(0), x411))
- (EVAL(neg(s(x408)), x413, pos(s(x1157)), x411)≥COND_EVAL1(and(greatereq_int(neg(s(x408)), pos(0)), greatereq_int(pos(mult_nat(mult_nat(s(x1157), s(x1157)), s(x1157))), x413)), neg(s(x408)), x413, pos(s(x1157)), x411))
- (EVAL(neg(s(x408)), x413, pos(x414), x411)≥COND_EVAL1(and(greatereq_int(neg(s(x408)), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x414, x414), x414)), x413)), neg(s(x408)), x413, pos(x414), x411))
- (EVAL(neg(s(x408)), x413, pos(0), x411)≥COND_EVAL1(and(greatereq_int(neg(s(x408)), pos(0)), greatereq_int(pos(mult_nat(mult_nat(0, 0), 0)), x413)), neg(s(x408)), x413, pos(0), x411))
- (EVAL(neg(s(x408)), x413, pos(s(x1167)), x411)≥COND_EVAL1(and(greatereq_int(neg(s(x408)), pos(0)), greatereq_int(pos(mult_nat(mult_nat(s(x1167), s(x1167)), s(x1167))), x413)), neg(s(x408)), x413, pos(s(x1167)), x411))
- (EVAL(neg(s(x408)), x413, pos(x414), x411)≥COND_EVAL1(and(greatereq_int(neg(s(x408)), pos(0)), greatereq_int(pos(mult_nat(mult_nat(x414, x414), x414)), x413)), neg(s(x408)), x413, pos(x414), x411))

POL(0) = 1
POL(COND_EVAL1(x₁, x₂, x₃, x₄, x₅)) = -1 - x₁ - x₂ - x₃ - x₄ - x₅
POL(EVAL(x₁, x₂, x₃, x₄)) = -1 + x₁ - x₂ - x₃ - x₄
POL(and(x₁, x₂)) = 0
POL(c) = -1
POL(false) = 1
POL(greatereq_int(x₁, x₂)) = 0
POL(minus_int(x₁, x₂)) = 0
POL(minus_nat(x₁, x₂)) = 0
POL(mult_int(x₁, x₂)) = 0
POL(mult_nat(x₁, x₂)) = 0
POL(neg(x₁)) = 0
POL(plus_int(x₁, x₂)) = x₁
POL(plus_nat(x₁, x₂)) = 1 + x₁ + x₂
POL(pos(x₁)) = 0
POL(s(x₁)) = 0
POL(true) = 0

The following pairs are in P_>:

COND_EVAL1(true, neg(s(z0)), z1, z2, x_removed) → EVAL(neg(s(z0)), minus_int(z1, x_removed), plus_int(z2, z1), x_removed)

The following pairs are in P_bound:

COND_EVAL1(true, neg(s(z0)), z1, z2, x_removed) → EVAL(neg(s(z0)), minus_int(z1, x_removed), plus_int(z2, z1), x_removed)

The following rules are usable:

pos(0) → minus_nat(0, 0)
pos(plus_nat(x, y)) → plus_int(pos(x), pos(y))
neg(plus_nat(x, y)) → plus_int(neg(x), neg(y))
minus_nat(y, x) → plus_int(neg(x), pos(y))
minus_nat(x, y) → plus_int(pos(x), neg(y))
neg(plus_nat(x, y)) → minus_int(neg(x), pos(y))
minus_nat(x, y) → minus_int(pos(x), pos(y))
true → and(true, true)
false → and(true, false)
false → and(false, true)
false → and(false, false)
minus_nat(x, y) → minus_nat(s(x), s(y))
neg(s(y)) → minus_nat(0, s(y))
pos(s(x)) → minus_nat(s(x), 0)

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ IDP

              ↳ IDependencyGraphProof

                ↳ IDP

                  ↳ IDPtoQDPProof

                    ↳ QDP

                      ↳ UsableRulesProof

                        ↳ QDP

                          ↳ QReductionProof

                            ↳ QDP

                              ↳ RemovalProof

                              ↳ RemovalProof

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Rewriting

                                    ↳ QDP

                                      ↳ Rewriting

                                        ↳ QDP

                                          ↳ Instantiation

                                            ↳ QDP

                                              ↳ QDPOrderProof

                                                ↳ QDP

                                                  ↳ RemovalProof

                                                    ↳ QDP

                                                      ↳ NonInfProof

                                                        ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

EVAL(neg(0), y1, y2, x_removed) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), neg(0), y1, y2, x_removed)
COND_EVAL1(true, z0, z1, pos(z2), x_removed) → EVAL(z0, minus_int(z1, x_removed), plus_int(pos(z2), z1), x_removed)
COND_EVAL1(true, z0, z1, neg(z2), x_removed) → EVAL(z0, minus_int(z1, x_removed), plus_int(neg(z2), z1), x_removed)
COND_EVAL1(true, neg(0), z0, z1, x_removed) → EVAL(neg(0), minus_int(z0, x_removed), plus_int(z1, z0), x_removed)
EVAL(pos(x0), y1, y2, x_removed) → COND_EVAL1(and(true, greatereq_int(mult_int(mult_int(y2, y2), y2), y1)), pos(x0), y1, y2, x_removed)
COND_EVAL1(true, pos(z0), z1, z2, x_removed) → EVAL(pos(z0), minus_int(z1, x_removed), plus_int(z2, z1), x_removed)
EVAL(y0, y1, neg(x0), x_removed) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(neg(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, neg(x0), x_removed)
EVAL(y0, y1, pos(x0), x_removed) → COND_EVAL1(and(greatereq_int(y0, pos(0)), greatereq_int(pos(mult_nat(mult_nat(x0, x0), x0)), y1)), y0, y1, pos(x0), x_removed)

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(s(x)), pos(0)) → false
mult_int(pos(x), pos(y)) → pos(mult_nat(x, y))
mult_int(neg(x), neg(y)) → pos(mult_nat(x, y))
mult_int(pos(x), neg(y)) → neg(mult_nat(x, y))
mult_int(neg(x), pos(y)) → neg(mult_nat(x, y))
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
mult_nat(0, y) → 0
mult_nat(s(x), 0) → 0
mult_nat(s(x), s(y)) → plus_nat(mult_nat(x, s(y)), s(y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_int(pos(x), pos(y)) → minus_nat(x, y)
minus_int(neg(x), pos(y)) → neg(plus_nat(x, y))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)

The set Q consists of the following terms:

minus_int(pos(x0), pos(x1))
minus_int(neg(x0), neg(x1))
minus_int(neg(x0), pos(x1))
minus_int(pos(x0), neg(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
mult_int(pos(x0), pos(x1))
mult_int(pos(x0), neg(x1))
mult_int(neg(x0), pos(x1))
mult_int(neg(x0), neg(x1))
mult_nat(0, x0)
mult_nat(s(x0), 0)
mult_nat(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.