Termination proof of /tmp/tmpVyQvdV/complete4.itrs

Termination of the given ITRSProblem could successfully be proven:

↳ ITRS

  ↳ ITRStoIDPProof

ITRS problem:
The following domains are used:

z

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

The integer pair graph contains the following rules and edges:

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

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

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

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

(2) -> (1), if ((x[2] →^* x[1])∧(y[2] →^* y[1])∧(>=@z(y[2], 0@z) →^* TRUE))

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

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

The set Q consists of the following terms:

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

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

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

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

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

(2) -> (1), if ((x[2] →^* x[1])∧(y[2] →^* y[1])∧(>=@z(y[2], 0@z) →^* TRUE))

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

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

The set Q consists of the following terms:

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

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

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

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

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

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

We simplified constraint (6) using rules (III), (IV) which results in the following new constraint:
(7)    (>=@z(y[2], 0@z)=TRUE ⇒ COND_EVAL(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL(TRUE, x[2], y[2])≥EVAL(x[2], -@z(y[2], 1@z))∧(U^Increasing(EVAL(x[1], -@z(y[1], 1@z))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    (y[2] ≥ 0 ⇒ (U^Increasing(EVAL(x[1], -@z(y[1], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    (y[2] ≥ 0 ⇒ (U^Increasing(EVAL(x[1], -@z(y[1], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    (y[2] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(x[1], -@z(y[1], 1@z))), ≥))

We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11)    (y[2] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧0 = 0∧(U^Increasing(EVAL(x[1], -@z(y[1], 1@z))), ≥)∧0 ≥ 0)
We consider the chain EVAL(x[2], y[2]) → COND_EVAL(>=@z(y[2], 0@z), x[2], y[2]), COND_EVAL(TRUE, x[1], y[1]) → EVAL(x[1], -@z(y[1], 1@z)), EVAL(x[2], y[2]) → COND_EVAL(>=@z(y[2], 0@z), x[2], y[2]) which results in the following constraint:
(12)    (y[2]=y[1]∧x[2]=x[1]∧-@z(y[1], 1@z)=y[2]1∧x[1]=x[2]1∧>=@z(y[2], 0@z)=TRUE ⇒ COND_EVAL(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL(TRUE, x[1], y[1])≥EVAL(x[1], -@z(y[1], 1@z))∧(U^Increasing(EVAL(x[1], -@z(y[1], 1@z))), ≥))

We simplified constraint (12) using rules (III), (IV) which results in the following new constraint:
(13)    (>=@z(y[2], 0@z)=TRUE ⇒ COND_EVAL(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL(TRUE, x[2], y[2])≥EVAL(x[2], -@z(y[2], 1@z))∧(U^Increasing(EVAL(x[1], -@z(y[1], 1@z))), ≥))

We simplified constraint (13) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(14)    (y[2] ≥ 0 ⇒ (U^Increasing(EVAL(x[1], -@z(y[1], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (14) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(15)    (y[2] ≥ 0 ⇒ (U^Increasing(EVAL(x[1], -@z(y[1], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (15) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(16)    (y[2] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(x[1], -@z(y[1], 1@z))), ≥))

We simplified constraint (16) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(17)    (y[2] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0∧(U^Increasing(EVAL(x[1], -@z(y[1], 1@z))), ≥))

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

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

We simplified constraint (18) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(19)    ((U^Increasing(COND_EVAL(>=@z(y[2], 0@z), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (19) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(20)    ((U^Increasing(COND_EVAL(>=@z(y[2], 0@z), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (20) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(21)    (0 ≥ 0∧(U^Increasing(COND_EVAL(>=@z(y[2], 0@z), x[2], y[2])), ≥)∧0 ≥ 0)

We simplified constraint (21) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(22)    (0 = 0∧(U^Increasing(COND_EVAL(>=@z(y[2], 0@z), x[2], y[2])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 ≥ 0)

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

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

We simplified constraint (23) using rules (III), (IV) which results in the following new constraint:
(24)    (>=@z(x[0], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[0], y[0], z[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0], z[0])≥EVAL(-@z(x[0], 1@z), z[0])∧(U^Increasing(EVAL(-@z(x[3], 1@z), z[3])), ≥))

We simplified constraint (24) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(25)    (x[0] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[3], 1@z), z[3])), ≥)∧-1 + (-1)Bound + x[0] ≥ 0∧0 ≥ 0)

We simplified constraint (25) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(26)    (x[0] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[3], 1@z), z[3])), ≥)∧-1 + (-1)Bound + x[0] ≥ 0∧0 ≥ 0)

We simplified constraint (26) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(27)    (x[0] ≥ 0 ⇒ 0 ≥ 0∧-1 + (-1)Bound + x[0] ≥ 0∧(U^Increasing(EVAL(-@z(x[3], 1@z), z[3])), ≥))

We simplified constraint (27) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(28)    (x[0] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧-1 + (-1)Bound + x[0] ≥ 0∧0 = 0∧(U^Increasing(EVAL(-@z(x[3], 1@z), z[3])), ≥))
We consider the chain EVAL(x[0], y[0]) → COND_EVAL1(>=@z(x[0], 0@z), x[0], y[0], z[0]), COND_EVAL1(TRUE, x[3], y[3], z[3]) → EVAL(-@z(x[3], 1@z), z[3]), EVAL(x[0], y[0]) → COND_EVAL1(>=@z(x[0], 0@z), x[0], y[0], z[0]) which results in the following constraint:
(29)    (y[0]=y[3]∧-@z(x[3], 1@z)=x[0]1∧x[0]=x[3]∧>=@z(x[0], 0@z)=TRUE∧z[0]=z[3]∧z[3]=y[0]1 ⇒ COND_EVAL1(TRUE, x[3], y[3], z[3])≥NonInfC∧COND_EVAL1(TRUE, x[3], y[3], z[3])≥EVAL(-@z(x[3], 1@z), z[3])∧(U^Increasing(EVAL(-@z(x[3], 1@z), z[3])), ≥))

We simplified constraint (29) using rules (III), (IV) which results in the following new constraint:
(30)    (>=@z(x[0], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[0], y[0], z[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0], z[0])≥EVAL(-@z(x[0], 1@z), z[0])∧(U^Increasing(EVAL(-@z(x[3], 1@z), z[3])), ≥))

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

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

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

We simplified constraint (33) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(34)    (x[0] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧0 = 0∧-1 + (-1)Bound + x[0] ≥ 0∧0 = 0∧0 = 0∧(U^Increasing(EVAL(-@z(x[3], 1@z), z[3])), ≥))

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

EVAL(x, y) → COND_EVAL1(>=@z(x, 0@z), x, y, z)
- (0 ≥ 0∧0 = 0∧(U^Increasing(COND_EVAL1(>=@z(x[0], 0@z), x[0], y[0], z[0])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0)
COND_EVAL(TRUE, x, y) → EVAL(x, -@z(y, 1@z))
- (y[2] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧0 = 0∧(U^Increasing(EVAL(x[1], -@z(y[1], 1@z))), ≥)∧0 ≥ 0)
- (y[2] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0∧(U^Increasing(EVAL(x[1], -@z(y[1], 1@z))), ≥))
EVAL(x, y) → COND_EVAL(>=@z(y, 0@z), x, y)
- (0 = 0∧(U^Increasing(COND_EVAL(>=@z(y[2], 0@z), x[2], y[2])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 ≥ 0)
COND_EVAL1(TRUE, x, y, z) → EVAL(-@z(x, 1@z), z)
- (x[0] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧-1 + (-1)Bound + x[0] ≥ 0∧0 = 0∧(U^Increasing(EVAL(-@z(x[3], 1@z), z[3])), ≥))
- (x[0] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧0 = 0∧-1 + (-1)Bound + x[0] ≥ 0∧0 = 0∧0 = 0∧(U^Increasing(EVAL(-@z(x[3], 1@z), z[3])), ≥))

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(>=@z(x₁, x₂)) = -1
POL(0@z) = 0
POL(COND_EVAL1(x₁, x₂, x₃, x₄)) = -1 + x₂
POL(TRUE) = -1
POL(COND_EVAL(x₁, x₂, x₃)) = -1 + x₂
POL(EVAL(x₁, x₂)) = -1 + x₁
POL(FALSE) = -1
POL(1@z) = 1
POL(undefined) = -1

The following pairs are in P_>:

COND_EVAL1(TRUE, x[3], y[3], z[3]) → EVAL(-@z(x[3], 1@z), z[3])

The following pairs are in P_bound:

COND_EVAL1(TRUE, x[3], y[3], z[3]) → EVAL(-@z(x[3], 1@z), z[3])

The following pairs are in P_≥:

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

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

-@z¹ ↔

↳ 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])∧(x[1] →^* x[2]))

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

(2) -> (1), if ((x[2] →^* x[1])∧(y[2] →^* y[1])∧(>=@z(y[2], 0@z) →^* TRUE))

The set Q consists of the following terms:

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

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

                  ↳ IDPNonInfProof

I DP problem:
The following domains are used:

z

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

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

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

(2) -> (1), if ((x[2] →^* x[1])∧(y[2] →^* y[1])∧(>=@z(y[2], 0@z) →^* TRUE))

The set Q consists of the following terms:

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

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

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

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

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

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

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

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

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

For Pair EVAL(x[2], y[2]) → COND_EVAL(>=@z(y[2], 0@z), x[2], y[2]) the following chains were created:

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

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(COND_EVAL(>=@z(y[2], 0@z), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    ((U^Increasing(COND_EVAL(>=@z(y[2], 0@z), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    (0 ≥ 0∧0 ≥ 0∧(U^Increasing(COND_EVAL(>=@z(y[2], 0@z), x[2], y[2])), ≥))

We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11)    (0 = 0∧(U^Increasing(COND_EVAL(>=@z(y[2], 0@z), x[2], y[2])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0)

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

COND_EVAL(TRUE, x[1], y[1]) → EVAL(x[1], -@z(y[1], 1@z))
- (y[2] ≥ 0 ⇒ 0 = 0∧-1 + (-1)Bound + (2)y[2] ≥ 0∧1 ≥ 0∧0 = 0∧(U^Increasing(EVAL(x[1], -@z(y[1], 1@z))), ≥))
EVAL(x[2], y[2]) → COND_EVAL(>=@z(y[2], 0@z), x[2], y[2])
- (0 = 0∧(U^Increasing(COND_EVAL(>=@z(y[2], 0@z), x[2], y[2])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0)

POL(-@z(x₁, x₂)) = x₁ + (-1)x₂
POL(>=@z(x₁, x₂)) = -1
POL(0@z) = 0
POL(TRUE) = -1
POL(EVAL(x₁, x₂)) = -1 + (2)x₂
POL(COND_EVAL(x₁, x₂, x₃)) = -1 + (2)x₃
POL(FALSE) = -1
POL(1@z) = 1
POL(undefined) = -1

The following pairs are in P_>:

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

The following pairs are in P_bound:

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

The following pairs are in P_≥:

EVAL(x[2], y[2]) → COND_EVAL(>=@z(y[2], 0@z), x[2], y[2])

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

-@z¹ ↔

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ IDP

              ↳ IDependencyGraphProof

                ↳ IDP

                  ↳ IDPNonInfProof

                    ↳ IDP

                      ↳ IDependencyGraphProof

I DP problem:
The following domains are used:

z

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

(2): EVAL(x[2], y[2]) → COND_EVAL(>=@z(y[2], 0@z), x[2], y[2])

The set Q consists of the following terms:

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

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