Termination proof of /tmp/tmpaAyKxQ/20.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:

Cond_eval_1(TRUE, x, y) → eval_2(x, y)
eval_1(x, y) → Cond_eval_1(&&(=@z(x, y), >@z(x, 0@z)), x, y)
eval_2(x, y) → Cond_eval_2(>=@z(0@z, y), x, y)
eval_2(x, y) → Cond_eval_21(>@z(y, 0@z), x, y)
Cond_eval_21(TRUE, x, y) → eval_2(-@z(x, 1@z), -@z(y, 1@z))
Cond_eval_2(TRUE, x, y) → eval_1(x, y)

The set Q consists of the following terms:

Cond_eval_1(TRUE, x0, x1)
eval_1(x0, x1)
eval_2(x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_2(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:

Cond_eval_1(TRUE, x, y) → eval_2(x, y)
eval_1(x, y) → Cond_eval_1(&&(=@z(x, y), >@z(x, 0@z)), x, y)
eval_2(x, y) → Cond_eval_2(>=@z(0@z, y), x, y)
eval_2(x, y) → Cond_eval_21(>@z(y, 0@z), x, y)
Cond_eval_21(TRUE, x, y) → eval_2(-@z(x, 1@z), -@z(y, 1@z))
Cond_eval_2(TRUE, x, y) → eval_1(x, y)

The integer pair graph contains the following rules and edges:

(0): EVAL_2(x[0], y[0]) → COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0])
(1): EVAL_2(x[1], y[1]) → COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1])
(2): COND_EVAL_1(TRUE, x[2], y[2]) → EVAL_2(x[2], y[2])
(3): COND_EVAL_2(TRUE, x[3], y[3]) → EVAL_1(x[3], y[3])
(4): COND_EVAL_21(TRUE, x[4], y[4]) → EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))
(5): EVAL_1(x[5], y[5]) → COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5])

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

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

(2) -> (0), if ((y[2] →^* y[0])∧(x[2] →^* x[0]))

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

(3) -> (5), if ((y[3] →^* y[5])∧(x[3] →^* x[5]))

(4) -> (0), if ((-@z(y[4], 1@z) →^* y[0])∧(-@z(x[4], 1@z) →^* x[0]))

(4) -> (1), if ((-@z(y[4], 1@z) →^* y[1])∧(-@z(x[4], 1@z) →^* x[1]))

(5) -> (2), if ((x[5] →^* x[2])∧(y[5] →^* y[2])∧(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)) →^* TRUE))

The set Q consists of the following terms:

Cond_eval_1(TRUE, x0, x1)
eval_1(x0, x1)
eval_2(x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_2(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_2(x[0], y[0]) → COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0])
(1): EVAL_2(x[1], y[1]) → COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1])
(2): COND_EVAL_1(TRUE, x[2], y[2]) → EVAL_2(x[2], y[2])
(3): COND_EVAL_2(TRUE, x[3], y[3]) → EVAL_1(x[3], y[3])
(4): COND_EVAL_21(TRUE, x[4], y[4]) → EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))
(5): EVAL_1(x[5], y[5]) → COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5])

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

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

(2) -> (0), if ((y[2] →^* y[0])∧(x[2] →^* x[0]))

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

(3) -> (5), if ((y[3] →^* y[5])∧(x[3] →^* x[5]))

(4) -> (0), if ((-@z(y[4], 1@z) →^* y[0])∧(-@z(x[4], 1@z) →^* x[0]))

(4) -> (1), if ((-@z(y[4], 1@z) →^* y[1])∧(-@z(x[4], 1@z) →^* x[1]))

(5) -> (2), if ((x[5] →^* x[2])∧(y[5] →^* y[2])∧(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)) →^* TRUE))

The set Q consists of the following terms:

Cond_eval_1(TRUE, x0, x1)
eval_1(x0, x1)
eval_2(x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_2(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_2(x, y) → COND_EVAL_21(>@z(y, 0@z), x, y) the following chains were created:

We consider the chain EVAL_2(x[0], y[0]) → COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0]) which results in the following constraint:
(1)    (EVAL_2(x[0], y[0])≥NonInfC∧EVAL_2(x[0], y[0])≥COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0])∧(U^Increasing(COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0])), ≥))

We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2)    ((U^Increasing(COND_EVAL_21(>@z(y[0], 0@z), x[0], y[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_EVAL_21(>@z(y[0], 0@z), x[0], y[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∧(U^Increasing(COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0])), ≥)∧0 ≥ 0)

We simplified constraint (4) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(5)    (0 = 0∧(U^Increasing(COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 ≥ 0)

For Pair EVAL_2(x, y) → COND_EVAL_2(>=@z(0@z, y), x, y) the following chains were created:

We consider the chain EVAL_2(x[1], y[1]) → COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1]) which results in the following constraint:
(6)    (EVAL_2(x[1], y[1])≥NonInfC∧EVAL_2(x[1], y[1])≥COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1])∧(U^Increasing(COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1])), ≥))

We simplified constraint (6) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(7)    ((U^Increasing(COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (7) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(8)    ((U^Increasing(COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (8) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(9)    (0 ≥ 0∧0 ≥ 0∧(U^Increasing(COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1])), ≥))

We simplified constraint (9) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(10)    (0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0∧(U^Increasing(COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1])), ≥))

For Pair COND_EVAL_1(TRUE, x, y) → EVAL_2(x, y) the following chains were created:

We consider the chain EVAL_1(x[5], y[5]) → COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5]), COND_EVAL_1(TRUE, x[2], y[2]) → EVAL_2(x[2], y[2]), EVAL_2(x[0], y[0]) → COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0]) which results in the following constraint:
(11)    (x[5]=x[2]∧x[2]=x[0]∧y[5]=y[2]∧y[2]=y[0]∧&&(=@z(x[5], y[5]), >@z(x[5], 0@z))=TRUE ⇒ COND_EVAL_1(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL_1(TRUE, x[2], y[2])≥EVAL_2(x[2], y[2])∧(U^Increasing(EVAL_2(x[2], y[2])), ≥))

We simplified constraint (11) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(12)    (>@z(x[5], 0@z)=TRUE∧>=@z(x[5], y[5])=TRUE∧<=@z(x[5], y[5])=TRUE ⇒ COND_EVAL_1(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL_1(TRUE, x[5], y[5])≥EVAL_2(x[5], y[5])∧(U^Increasing(EVAL_2(x[2], y[2])), ≥))

We simplified constraint (12) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(13)    (-1 + x[5] ≥ 0∧x[5] + (-1)y[5] ≥ 0∧y[5] + (-1)x[5] ≥ 0 ⇒ (U^Increasing(EVAL_2(x[2], y[2])), ≥)∧-1 + (-1)Bound + (2)y[5] + (-1)x[5] ≥ 0∧0 ≥ 0)

We simplified constraint (13) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(14)    (-1 + x[5] ≥ 0∧x[5] + (-1)y[5] ≥ 0∧y[5] + (-1)x[5] ≥ 0 ⇒ (U^Increasing(EVAL_2(x[2], y[2])), ≥)∧-1 + (-1)Bound + (2)y[5] + (-1)x[5] ≥ 0∧0 ≥ 0)

We simplified constraint (14) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(15)    (-1 + x[5] ≥ 0∧y[5] + (-1)x[5] ≥ 0∧x[5] + (-1)y[5] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL_2(x[2], y[2])), ≥)∧-1 + (-1)Bound + (2)y[5] + (-1)x[5] ≥ 0)

We simplified constraint (15) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(16)    (x[5] ≥ 0∧-1 + y[5] + (-1)x[5] ≥ 0∧1 + x[5] + (-1)y[5] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL_2(x[2], y[2])), ≥)∧-2 + (-1)Bound + (2)y[5] + (-1)x[5] ≥ 0)

We simplified constraint (16) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(17)    (x[5] ≥ 0∧y[5] ≥ 0∧(-1)y[5] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL_2(x[2], y[2])), ≥)∧(-1)Bound + x[5] + (2)y[5] ≥ 0)

We simplified constraint (17) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(18)    (x[5] ≥ 0∧0 ≥ 0∧0 ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL_2(x[2], y[2])), ≥)∧(-1)Bound + x[5] ≥ 0)
We consider the chain EVAL_1(x[5], y[5]) → COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5]), COND_EVAL_1(TRUE, x[2], y[2]) → EVAL_2(x[2], y[2]), EVAL_2(x[1], y[1]) → COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1]) which results in the following constraint:
(19)    (y[2]=y[1]∧x[5]=x[2]∧x[2]=x[1]∧y[5]=y[2]∧&&(=@z(x[5], y[5]), >@z(x[5], 0@z))=TRUE ⇒ COND_EVAL_1(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL_1(TRUE, x[2], y[2])≥EVAL_2(x[2], y[2])∧(U^Increasing(EVAL_2(x[2], y[2])), ≥))

We simplified constraint (19) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(20)    (>@z(x[5], 0@z)=TRUE∧>=@z(x[5], y[5])=TRUE∧<=@z(x[5], y[5])=TRUE ⇒ COND_EVAL_1(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL_1(TRUE, x[5], y[5])≥EVAL_2(x[5], y[5])∧(U^Increasing(EVAL_2(x[2], y[2])), ≥))

We simplified constraint (20) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(21)    (-1 + x[5] ≥ 0∧x[5] + (-1)y[5] ≥ 0∧y[5] + (-1)x[5] ≥ 0 ⇒ (U^Increasing(EVAL_2(x[2], y[2])), ≥)∧-1 + (-1)Bound + (2)y[5] + (-1)x[5] ≥ 0∧0 ≥ 0)

We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(22)    (-1 + x[5] ≥ 0∧x[5] + (-1)y[5] ≥ 0∧y[5] + (-1)x[5] ≥ 0 ⇒ (U^Increasing(EVAL_2(x[2], y[2])), ≥)∧-1 + (-1)Bound + (2)y[5] + (-1)x[5] ≥ 0∧0 ≥ 0)

We simplified constraint (22) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(23)    (x[5] + (-1)y[5] ≥ 0∧-1 + x[5] ≥ 0∧y[5] + (-1)x[5] ≥ 0 ⇒ 0 ≥ 0∧-1 + (-1)Bound + (2)y[5] + (-1)x[5] ≥ 0∧(U^Increasing(EVAL_2(x[2], y[2])), ≥))

We simplified constraint (23) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(24)    (1 + x[5] + (-1)y[5] ≥ 0∧x[5] ≥ 0∧-1 + y[5] + (-1)x[5] ≥ 0 ⇒ 0 ≥ 0∧-2 + (-1)Bound + (2)y[5] + (-1)x[5] ≥ 0∧(U^Increasing(EVAL_2(x[2], y[2])), ≥))

We simplified constraint (24) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(25)    ((-1)y[5] ≥ 0∧x[5] ≥ 0∧y[5] ≥ 0 ⇒ 0 ≥ 0∧(-1)Bound + x[5] + (2)y[5] ≥ 0∧(U^Increasing(EVAL_2(x[2], y[2])), ≥))

We simplified constraint (25) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(26)    (0 ≥ 0∧x[5] ≥ 0∧0 ≥ 0 ⇒ 0 ≥ 0∧(-1)Bound + x[5] ≥ 0∧(U^Increasing(EVAL_2(x[2], y[2])), ≥))

For Pair COND_EVAL_2(TRUE, x, y) → EVAL_1(x, y) the following chains were created:

We consider the chain EVAL_2(x[1], y[1]) → COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1]), COND_EVAL_2(TRUE, x[3], y[3]) → EVAL_1(x[3], y[3]), EVAL_1(x[5], y[5]) → COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5]) which results in the following constraint:
(27)    (y[1]=y[3]∧y[3]=y[5]∧x[3]=x[5]∧>=@z(0@z, y[1])=TRUE∧x[1]=x[3] ⇒ COND_EVAL_2(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL_2(TRUE, x[3], y[3])≥EVAL_1(x[3], y[3])∧(U^Increasing(EVAL_1(x[3], y[3])), ≥))

We simplified constraint (27) using rules (III), (IV) which results in the following new constraint:
(28)    (>=@z(0@z, y[1])=TRUE ⇒ COND_EVAL_2(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL_2(TRUE, x[1], y[1])≥EVAL_1(x[1], y[1])∧(U^Increasing(EVAL_1(x[3], y[3])), ≥))

We simplified constraint (28) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(29)    ((-1)y[1] ≥ 0 ⇒ (U^Increasing(EVAL_1(x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (29) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(30)    ((-1)y[1] ≥ 0 ⇒ (U^Increasing(EVAL_1(x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (30) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(31)    ((-1)y[1] ≥ 0 ⇒ (U^Increasing(EVAL_1(x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (31) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(32)    ((-1)y[1] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL_1(x[3], y[3])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0)

We simplified constraint (32) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(33)    (y[1] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL_1(x[3], y[3])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0)

For Pair COND_EVAL_21(TRUE, x, y) → EVAL_2(-@z(x, 1@z), -@z(y, 1@z)) the following chains were created:

We consider the chain EVAL_2(x[0], y[0]) → COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0]), COND_EVAL_21(TRUE, x[4], y[4]) → EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z)), EVAL_2(x[1], y[1]) → COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1]) which results in the following constraint:
(34)    (y[0]=y[4]∧>@z(y[0], 0@z)=TRUE∧x[0]=x[4]∧-@z(y[4], 1@z)=y[1]∧-@z(x[4], 1@z)=x[1] ⇒ COND_EVAL_21(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL_21(TRUE, x[4], y[4])≥EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))∧(U^Increasing(EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))), ≥))

We simplified constraint (34) using rules (III), (IV) which results in the following new constraint:
(35)    (>@z(y[0], 0@z)=TRUE ⇒ COND_EVAL_21(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL_21(TRUE, x[0], y[0])≥EVAL_2(-@z(x[0], 1@z), -@z(y[0], 1@z))∧(U^Increasing(EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))), ≥))

We simplified constraint (35) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(36)    (-1 + y[0] ≥ 0 ⇒ (U^Increasing(EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (36) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(37)    (-1 + y[0] ≥ 0 ⇒ (U^Increasing(EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (37) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(38)    (-1 + y[0] ≥ 0 ⇒ (U^Increasing(EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (38) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(39)    (-1 + y[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))), ≥)∧0 ≥ 0∧0 = 0∧0 = 0)

We simplified constraint (39) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(40)    (y[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))), ≥)∧0 ≥ 0∧0 = 0∧0 = 0)
We consider the chain EVAL_2(x[0], y[0]) → COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0]), COND_EVAL_21(TRUE, x[4], y[4]) → EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z)), EVAL_2(x[0], y[0]) → COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0]) which results in the following constraint:
(41)    (y[0]=y[4]∧>@z(y[0], 0@z)=TRUE∧x[0]=x[4]∧-@z(x[4], 1@z)=x[0]1∧-@z(y[4], 1@z)=y[0]1 ⇒ COND_EVAL_21(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL_21(TRUE, x[4], y[4])≥EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))∧(U^Increasing(EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))), ≥))

We simplified constraint (41) using rules (III), (IV) which results in the following new constraint:
(42)    (>@z(y[0], 0@z)=TRUE ⇒ COND_EVAL_21(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL_21(TRUE, x[0], y[0])≥EVAL_2(-@z(x[0], 1@z), -@z(y[0], 1@z))∧(U^Increasing(EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))), ≥))

We simplified constraint (42) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(43)    (-1 + y[0] ≥ 0 ⇒ (U^Increasing(EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (43) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(44)    (-1 + y[0] ≥ 0 ⇒ (U^Increasing(EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (44) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(45)    (-1 + y[0] ≥ 0 ⇒ (U^Increasing(EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (45) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(46)    (-1 + y[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))), ≥)∧0 ≥ 0∧0 = 0∧0 = 0)

We simplified constraint (46) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(47)    (y[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))), ≥)∧0 ≥ 0∧0 = 0∧0 = 0)

For Pair EVAL_1(x, y) → COND_EVAL_1(&&(=@z(x, y), >@z(x, 0@z)), x, y) the following chains were created:

We consider the chain EVAL_1(x[5], y[5]) → COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5]) which results in the following constraint:
(48)    (EVAL_1(x[5], y[5])≥NonInfC∧EVAL_1(x[5], y[5])≥COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5])∧(U^Increasing(COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5])), ≥))

We simplified constraint (48) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(49)    ((U^Increasing(COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (49) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(50)    ((U^Increasing(COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (50) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(51)    (0 ≥ 0∧0 ≥ 0∧(U^Increasing(COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5])), ≥))

We simplified constraint (51) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(52)    (0 = 0∧0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5])), ≥))

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

EVAL_2(x, y) → COND_EVAL_21(>@z(y, 0@z), x, y)
- (0 = 0∧(U^Increasing(COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 ≥ 0)
EVAL_2(x, y) → COND_EVAL_2(>=@z(0@z, y), x, y)
- (0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0∧(U^Increasing(COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1])), ≥))
COND_EVAL_1(TRUE, x, y) → EVAL_2(x, y)
- (x[5] ≥ 0∧0 ≥ 0∧0 ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL_2(x[2], y[2])), ≥)∧(-1)Bound + x[5] ≥ 0)
- (0 ≥ 0∧x[5] ≥ 0∧0 ≥ 0 ⇒ 0 ≥ 0∧(-1)Bound + x[5] ≥ 0∧(U^Increasing(EVAL_2(x[2], y[2])), ≥))
COND_EVAL_2(TRUE, x, y) → EVAL_1(x, y)
- (y[1] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL_1(x[3], y[3])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0)
COND_EVAL_21(TRUE, x, y) → EVAL_2(-@z(x, 1@z), -@z(y, 1@z))
- (y[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))), ≥)∧0 ≥ 0∧0 = 0∧0 = 0)
- (y[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))), ≥)∧0 ≥ 0∧0 = 0∧0 = 0)
EVAL_1(x, y) → COND_EVAL_1(&&(=@z(x, y), >@z(x, 0@z)), x, y)
- (0 = 0∧0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5])), ≥))

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(TRUE) = 0
POL(&&(x₁, x₂)) = 0
POL(EVAL_1(x₁, x₂)) = -1 + (2)x₂ + (-1)x₁
POL(FALSE) = 1
POL(>@z(x₁, x₂)) = -1
POL(=@z(x₁, x₂)) = -1
POL(>=@z(x₁, x₂)) = -1
POL(COND_EVAL_1(x₁, x₂, x₃)) = -1 + (2)x₃ + (-1)x₂ + (-1)x₁
POL(EVAL_2(x₁, x₂)) = -1 + (2)x₂ + (-1)x₁
POL(COND_EVAL_2(x₁, x₂, x₃)) = -1 + (2)x₃ + (-1)x₂
POL(COND_EVAL_21(x₁, x₂, x₃)) = -1 + (2)x₃ + (-1)x₂
POL(1@z) = 1
POL(undefined) = -1

The following pairs are in P_>:

COND_EVAL_21(TRUE, x[4], y[4]) → EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))

The following pairs are in P_bound:

COND_EVAL_1(TRUE, x[2], y[2]) → EVAL_2(x[2], y[2])

The following pairs are in P_≥:

EVAL_2(x[0], y[0]) → COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0])
EVAL_2(x[1], y[1]) → COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1])
COND_EVAL_1(TRUE, x[2], y[2]) → EVAL_2(x[2], y[2])
COND_EVAL_2(TRUE, x[3], y[3]) → EVAL_1(x[3], y[3])
EVAL_1(x[5], y[5]) → COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5])

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

                ↳ IDependencyGraphProof

              ↳ IDP

I DP problem:
The following domains are used:

z

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

(0): EVAL_2(x[0], y[0]) → COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0])
(1): EVAL_2(x[1], y[1]) → COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1])
(3): COND_EVAL_2(TRUE, x[3], y[3]) → EVAL_1(x[3], y[3])
(4): COND_EVAL_21(TRUE, x[4], y[4]) → EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))
(5): EVAL_1(x[5], y[5]) → COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5])

(4) -> (0), if ((-@z(y[4], 1@z) →^* y[0])∧(-@z(x[4], 1@z) →^* x[0]))

(4) -> (1), if ((-@z(y[4], 1@z) →^* y[1])∧(-@z(x[4], 1@z) →^* x[1]))

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

(3) -> (5), if ((y[3] →^* y[5])∧(x[3] →^* x[5]))

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

The set Q consists of the following terms:

Cond_eval_1(TRUE, x0, x1)
eval_1(x0, x1)
eval_2(x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_2(TRUE, x0, x1)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

              ↳ IDP

I DP problem:
The following domains are used:

z

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

(4): COND_EVAL_21(TRUE, x[4], y[4]) → EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))
(0): EVAL_2(x[0], y[0]) → COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0])

(4) -> (0), if ((-@z(y[4], 1@z) →^* y[0])∧(-@z(x[4], 1@z) →^* x[0]))

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

The set Q consists of the following terms:

Cond_eval_1(TRUE, x0, x1)
eval_1(x0, x1)
eval_2(x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_2(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_21(TRUE, x[4], y[4]) → EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z)) the following chains were created:

We consider the chain EVAL_2(x[0], y[0]) → COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0]), COND_EVAL_21(TRUE, x[4], y[4]) → EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z)), EVAL_2(x[0], y[0]) → COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0]) which results in the following constraint:
(1)    (y[0]=y[4]∧>@z(y[0], 0@z)=TRUE∧x[0]=x[4]∧-@z(x[4], 1@z)=x[0]1∧-@z(y[4], 1@z)=y[0]1 ⇒ COND_EVAL_21(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL_21(TRUE, x[4], y[4])≥EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))∧(U^Increasing(EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))), ≥))

We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
(2)    (>@z(y[0], 0@z)=TRUE ⇒ COND_EVAL_21(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL_21(TRUE, x[0], y[0])≥EVAL_2(-@z(x[0], 1@z), -@z(y[0], 1@z))∧(U^Increasing(EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (-1 + y[0] ≥ 0 ⇒ (U^Increasing(EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))), ≥)∧-1 + (-1)Bound + y[0] ≥ 0∧0 ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (-1 + y[0] ≥ 0 ⇒ (U^Increasing(EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))), ≥)∧-1 + (-1)Bound + y[0] ≥ 0∧0 ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (-1 + y[0] ≥ 0 ⇒ (U^Increasing(EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))), ≥)∧-1 + (-1)Bound + y[0] ≥ 0∧0 ≥ 0)

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6)    (-1 + y[0] ≥ 0 ⇒ 0 = 0∧(U^Increasing(EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))), ≥)∧-1 + (-1)Bound + y[0] ≥ 0∧0 = 0∧0 ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (y[0] ≥ 0 ⇒ 0 = 0∧(U^Increasing(EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))), ≥)∧(-1)Bound + y[0] ≥ 0∧0 = 0∧0 ≥ 0)

For Pair EVAL_2(x[0], y[0]) → COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0]) the following chains were created:

We consider the chain EVAL_2(x[0], y[0]) → COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0]) which results in the following constraint:
(8)    (EVAL_2(x[0], y[0])≥NonInfC∧EVAL_2(x[0], y[0])≥COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0])∧(U^Increasing(COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0])), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    ((U^Increasing(COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10)    ((U^Increasing(COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11)    ((U^Increasing(COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(12)    (0 ≥ 0∧(U^Increasing(COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0)

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

COND_EVAL_21(TRUE, x[4], y[4]) → EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))
- (y[0] ≥ 0 ⇒ 0 = 0∧(U^Increasing(EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))), ≥)∧(-1)Bound + y[0] ≥ 0∧0 = 0∧0 ≥ 0)
EVAL_2(x[0], y[0]) → COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0])
- (0 ≥ 0∧(U^Increasing(COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0)

POL(-@z(x₁, x₂)) = x₁ + (-1)x₂
POL(0@z) = 0
POL(EVAL_2(x₁, x₂)) = -1 + x₂
POL(TRUE) = -1
POL(COND_EVAL_21(x₁, x₂, x₃)) = -1 + x₃
POL(FALSE) = -1
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x₁, x₂)) = -1

The following pairs are in P_>:

COND_EVAL_21(TRUE, x[4], y[4]) → EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))

The following pairs are in P_bound:

COND_EVAL_21(TRUE, x[4], y[4]) → EVAL_2(-@z(x[4], 1@z), -@z(y[4], 1@z))

The following pairs are in P_≥:

EVAL_2(x[0], y[0]) → COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0])

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

-@z¹ ↔

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ IDP

                        ↳ IDependencyGraphProof

              ↳ IDP

I DP problem:
The following domains are used:

z

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

(0): EVAL_2(x[0], y[0]) → COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0])

The set Q consists of the following terms:

Cond_eval_1(TRUE, x0, x1)
eval_1(x0, x1)
eval_2(x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_2(TRUE, x0, x1)

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

              ↳ IDP

                ↳ IDependencyGraphProof

I DP problem:
The following domains are used:

z

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

(0): EVAL_2(x[0], y[0]) → COND_EVAL_21(>@z(y[0], 0@z), x[0], y[0])
(1): EVAL_2(x[1], y[1]) → COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1])
(2): COND_EVAL_1(TRUE, x[2], y[2]) → EVAL_2(x[2], y[2])
(3): COND_EVAL_2(TRUE, x[3], y[3]) → EVAL_1(x[3], y[3])
(5): EVAL_1(x[5], y[5]) → COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5])

(2) -> (0), if ((y[2] →^* y[0])∧(x[2] →^* x[0]))

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

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

(5) -> (2), if ((x[5] →^* x[2])∧(y[5] →^* y[2])∧(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)) →^* TRUE))

(3) -> (5), if ((y[3] →^* y[5])∧(x[3] →^* x[5]))

The set Q consists of the following terms:

Cond_eval_1(TRUE, x0, x1)
eval_1(x0, x1)
eval_2(x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_2(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

            ↳ AND

              ↳ IDP

              ↳ 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:

(3): COND_EVAL_2(TRUE, x[3], y[3]) → EVAL_1(x[3], y[3])
(1): EVAL_2(x[1], y[1]) → COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1])
(2): COND_EVAL_1(TRUE, x[2], y[2]) → EVAL_2(x[2], y[2])
(5): EVAL_1(x[5], y[5]) → COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5])

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

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

(5) -> (2), if ((x[5] →^* x[2])∧(y[5] →^* y[2])∧(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)) →^* TRUE))

(3) -> (5), if ((y[3] →^* y[5])∧(x[3] →^* x[5]))

The set Q consists of the following terms:

Cond_eval_1(TRUE, x0, x1)
eval_1(x0, x1)
eval_2(x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_2(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_2(TRUE, x[3], y[3]) → EVAL_1(x[3], y[3]) the following chains were created:

We consider the chain EVAL_2(x[1], y[1]) → COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1]), COND_EVAL_2(TRUE, x[3], y[3]) → EVAL_1(x[3], y[3]), EVAL_1(x[5], y[5]) → COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5]) which results in the following constraint:
(1)    (y[1]=y[3]∧y[3]=y[5]∧x[3]=x[5]∧>=@z(0@z, y[1])=TRUE∧x[1]=x[3] ⇒ COND_EVAL_2(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL_2(TRUE, x[3], y[3])≥EVAL_1(x[3], y[3])∧(U^Increasing(EVAL_1(x[3], y[3])), ≥))

We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
(2)    (>=@z(0@z, y[1])=TRUE ⇒ COND_EVAL_2(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL_2(TRUE, x[1], y[1])≥EVAL_1(x[1], y[1])∧(U^Increasing(EVAL_1(x[3], y[3])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    ((-1)y[1] ≥ 0 ⇒ (U^Increasing(EVAL_1(x[3], y[3])), ≥)∧0 ≥ 0∧(-3)y[1] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    ((-1)y[1] ≥ 0 ⇒ (U^Increasing(EVAL_1(x[3], y[3])), ≥)∧0 ≥ 0∧(-3)y[1] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    ((-1)y[1] ≥ 0 ⇒ (U^Increasing(EVAL_1(x[3], y[3])), ≥)∧0 ≥ 0∧(-3)y[1] ≥ 0)

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6)    ((-1)y[1] ≥ 0 ⇒ 0 = 0∧(U^Increasing(EVAL_1(x[3], y[3])), ≥)∧0 ≥ 0∧(-3)y[1] ≥ 0∧0 = 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (y[1] ≥ 0 ⇒ 0 = 0∧(U^Increasing(EVAL_1(x[3], y[3])), ≥)∧0 ≥ 0∧(3)y[1] ≥ 0∧0 = 0)

For Pair EVAL_2(x[1], y[1]) → COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1]) the following chains were created:

We consider the chain EVAL_2(x[1], y[1]) → COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1]) which results in the following constraint:
(8)    (EVAL_2(x[1], y[1])≥NonInfC∧EVAL_2(x[1], y[1])≥COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1])∧(U^Increasing(COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1])), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    ((U^Increasing(COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10)    ((U^Increasing(COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11)    (1 ≥ 0∧(U^Increasing(COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1])), ≥)∧0 ≥ 0)

We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(12)    (1 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1])), ≥)∧0 ≥ 0)

For Pair COND_EVAL_1(TRUE, x[2], y[2]) → EVAL_2(x[2], y[2]) the following chains were created:

We consider the chain EVAL_1(x[5], y[5]) → COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5]), COND_EVAL_1(TRUE, x[2], y[2]) → EVAL_2(x[2], y[2]), EVAL_2(x[1], y[1]) → COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1]) which results in the following constraint:
(13)    (y[2]=y[1]∧x[5]=x[2]∧x[2]=x[1]∧y[5]=y[2]∧&&(=@z(x[5], y[5]), >@z(x[5], 0@z))=TRUE ⇒ COND_EVAL_1(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL_1(TRUE, x[2], y[2])≥EVAL_2(x[2], y[2])∧(U^Increasing(EVAL_2(x[2], y[2])), ≥))

We simplified constraint (13) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(14)    (>@z(x[5], 0@z)=TRUE∧>=@z(x[5], y[5])=TRUE∧<=@z(x[5], y[5])=TRUE ⇒ COND_EVAL_1(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL_1(TRUE, x[5], y[5])≥EVAL_2(x[5], y[5])∧(U^Increasing(EVAL_2(x[2], y[2])), ≥))

We simplified constraint (14) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(15)    (-1 + x[5] ≥ 0∧x[5] + (-1)y[5] ≥ 0∧y[5] + (-1)x[5] ≥ 0 ⇒ (U^Increasing(EVAL_2(x[2], y[2])), ≥)∧(-1)Bound + (2)y[5] + (-1)x[5] ≥ 0∧-2 + (3)y[5] ≥ 0)

We simplified constraint (15) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(16)    (-1 + x[5] ≥ 0∧x[5] + (-1)y[5] ≥ 0∧y[5] + (-1)x[5] ≥ 0 ⇒ (U^Increasing(EVAL_2(x[2], y[2])), ≥)∧(-1)Bound + (2)y[5] + (-1)x[5] ≥ 0∧-2 + (3)y[5] ≥ 0)

We simplified constraint (16) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(17)    (y[5] + (-1)x[5] ≥ 0∧x[5] + (-1)y[5] ≥ 0∧-1 + x[5] ≥ 0 ⇒ -2 + (3)y[5] ≥ 0∧(-1)Bound + (2)y[5] + (-1)x[5] ≥ 0∧(U^Increasing(EVAL_2(x[2], y[2])), ≥))

We simplified constraint (17) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(18)    ((-1)x[5] ≥ 0∧x[5] ≥ 0∧-1 + y[5] + x[5] ≥ 0 ⇒ -2 + (3)y[5] ≥ 0∧(-1)Bound + y[5] + (-1)x[5] ≥ 0∧(U^Increasing(EVAL_2(x[2], y[2])), ≥))

We simplified constraint (18) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(19)    (0 ≥ 0∧0 ≥ 0∧-1 + y[5] ≥ 0 ⇒ -2 + (3)y[5] ≥ 0∧(-1)Bound + y[5] ≥ 0∧(U^Increasing(EVAL_2(x[2], y[2])), ≥))

We simplified constraint (19) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(20)    (0 ≥ 0∧0 ≥ 0∧y[5] ≥ 0 ⇒ 1 + (3)y[5] ≥ 0∧1 + (-1)Bound + y[5] ≥ 0∧(U^Increasing(EVAL_2(x[2], y[2])), ≥))

For Pair EVAL_1(x[5], y[5]) → COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5]) the following chains were created:

We consider the chain EVAL_1(x[5], y[5]) → COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5]) which results in the following constraint:
(21)    (EVAL_1(x[5], y[5])≥NonInfC∧EVAL_1(x[5], y[5])≥COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5])∧(U^Increasing(COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5])), ≥))

We simplified constraint (21) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(22)    ((U^Increasing(COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (22) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(23)    ((U^Increasing(COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (23) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(24)    (0 ≥ 0∧0 ≥ 0∧(U^Increasing(COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5])), ≥))

We simplified constraint (24) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(25)    (0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧(U^Increasing(COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5])), ≥)∧0 = 0)

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

COND_EVAL_2(TRUE, x[3], y[3]) → EVAL_1(x[3], y[3])
- (y[1] ≥ 0 ⇒ 0 = 0∧(U^Increasing(EVAL_1(x[3], y[3])), ≥)∧0 ≥ 0∧(3)y[1] ≥ 0∧0 = 0)
EVAL_2(x[1], y[1]) → COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1])
- (1 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1])), ≥)∧0 ≥ 0)
COND_EVAL_1(TRUE, x[2], y[2]) → EVAL_2(x[2], y[2])
- (0 ≥ 0∧0 ≥ 0∧y[5] ≥ 0 ⇒ 1 + (3)y[5] ≥ 0∧1 + (-1)Bound + y[5] ≥ 0∧(U^Increasing(EVAL_2(x[2], y[2])), ≥))
EVAL_1(x[5], y[5]) → COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5])
- (0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧(U^Increasing(COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5])), ≥)∧0 = 0)

POL(=@z(x₁, x₂)) = -1
POL(>=@z(x₁, x₂)) = 1
POL(0@z) = 0
POL(COND_EVAL_1(x₁, x₂, x₃)) = -1 + (2)x₃ + (-1)x₂ + (-1)x₁
POL(EVAL_2(x₁, x₂)) = 1 + (-1)x₂ + (-1)x₁
POL(TRUE) = -1
POL(&&(x₁, x₂)) = -1
POL(EVAL_1(x₁, x₂)) = (2)x₂ + (-1)x₁
POL(COND_EVAL_2(x₁, x₂, x₃)) = (-1)x₃ + (-1)x₂
POL(FALSE) = 0
POL(undefined) = -1
POL(>@z(x₁, x₂)) = -1

The following pairs are in P_>:

COND_EVAL_1(TRUE, x[2], y[2]) → EVAL_2(x[2], y[2])

The following pairs are in P_bound:

COND_EVAL_1(TRUE, x[2], y[2]) → EVAL_2(x[2], y[2])

The following pairs are in P_≥:

COND_EVAL_2(TRUE, x[3], y[3]) → EVAL_1(x[3], y[3])
EVAL_2(x[1], y[1]) → COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1])
EVAL_1(x[5], y[5]) → COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5])

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

FALSE¹ → &&(FALSE, FALSE)¹
TRUE¹ → &&(TRUE, TRUE)¹
FALSE¹ → &&(FALSE, TRUE)¹
FALSE¹ → &&(TRUE, FALSE)¹

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

              ↳ 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:

(3): COND_EVAL_2(TRUE, x[3], y[3]) → EVAL_1(x[3], y[3])
(1): EVAL_2(x[1], y[1]) → COND_EVAL_2(>=@z(0@z, y[1]), x[1], y[1])
(5): EVAL_1(x[5], y[5]) → COND_EVAL_1(&&(=@z(x[5], y[5]), >@z(x[5], 0@z)), x[5], y[5])

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

(3) -> (5), if ((y[3] →^* y[5])∧(x[3] →^* x[5]))

The set Q consists of the following terms:

Cond_eval_1(TRUE, x0, x1)
eval_1(x0, x1)
eval_2(x0, x1)
Cond_eval_21(TRUE, x0, x1)
Cond_eval_2(TRUE, x0, x1)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.