Termination proof of /tmp/tmp2CAoHO/A06.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:

f(TRUE, x, y, z) → f(>@z(x, +@z(y, z)), x, +@z(y, 1@z), z)
f(TRUE, x, y, z) → f(>@z(x, +@z(y, z)), x, y, +@z(z, 1@z))

The set Q consists of the following terms:

f(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:

f(TRUE, x, y, z) → f(>@z(x, +@z(y, z)), x, +@z(y, 1@z), z)
f(TRUE, x, y, z) → f(>@z(x, +@z(y, z)), x, y, +@z(z, 1@z))

The integer pair graph contains the following rules and edges:

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

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

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

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

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

The set Q consists of the following terms:

f(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): F(TRUE, x[0], y[0], z[0]) → F(>@z(x[0], +@z(y[0], z[0])), x[0], +@z(y[0], 1@z), z[0])
(1): F(TRUE, x[1], y[1], z[1]) → F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))

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

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

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

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

The set Q consists of the following terms:

f(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 F(TRUE, x, y, z) → F(>@z(x, +@z(y, z)), x, +@z(y, 1@z), z) the following chains were created:

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

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

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

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

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

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

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

(8)    (1 + y[0] ≥ 0∧y[0] ≥ 0∧z[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[0]1, +@z(y[0]1, z[0]1)), x[0]1, +@z(y[0]1, 1@z), z[0]1)), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(9)    (1 + y[0] ≥ 0∧y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[0]1, +@z(y[0]1, z[0]1)), x[0]1, +@z(y[0]1, 1@z), z[0]1)), ≥)∧0 ≥ 0∧0 ≥ 0)

(10)    (1 + y[0] ≥ 0∧y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[0]1, +@z(y[0]1, z[0]1)), x[0]1, +@z(y[0]1, 1@z), z[0]1)), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(11)    (1 + y[0] ≥ 0∧y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[0]1, +@z(y[0]1, z[0]1)), x[0]1, +@z(y[0]1, 1@z), z[0]1)), ≥)∧0 ≥ 0∧0 ≥ 0)

(12)    (1 + y[0] ≥ 0∧y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[0]1, +@z(y[0]1, z[0]1)), x[0]1, +@z(y[0]1, 1@z), z[0]1)), ≥)∧0 ≥ 0∧0 ≥ 0)
We consider the chain F(TRUE, x[1], y[1], z[1]) → F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z)), F(TRUE, x[0], y[0], z[0]) → F(>@z(x[0], +@z(y[0], z[0])), x[0], +@z(y[0], 1@z), z[0]), F(TRUE, x[0], y[0], z[0]) → F(>@z(x[0], +@z(y[0], z[0])), x[0], +@z(y[0], 1@z), z[0]) which results in the following constraint:
(13)    (z[0]=z[0]1∧x[0]=x[0]1∧>@z(x[1], +@z(y[1], z[1]))=TRUE∧>@z(x[0], +@z(y[0], z[0]))=TRUE∧y[1]=y[0]∧x[1]=x[0]∧+@z(y[0], 1@z)=y[0]1∧+@z(z[1], 1@z)=z[0] ⇒ F(TRUE, x[0], y[0], z[0])≥NonInfC∧F(TRUE, x[0], y[0], z[0])≥F(>@z(x[0], +@z(y[0], z[0])), x[0], +@z(y[0], 1@z), z[0])∧(U^Increasing(F(>@z(x[0], +@z(y[0], z[0])), x[0], +@z(y[0], 1@z), z[0])), ≥))

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

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

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

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

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

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

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

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

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

(23)    (x[1] ≥ 0∧1 + x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[0], +@z(y[0], z[0])), x[0], +@z(y[0], 1@z), z[0])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (21) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(24)    (x[1] ≥ 0∧1 + x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[0], +@z(y[0], z[0])), x[0], +@z(y[0], 1@z), z[0])), ≥)∧0 ≥ 0∧0 ≥ 0)

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

We simplified constraint (26) using rules (III), (IV) which results in the following new constraint:
(27)    (>@z(x[0], +@z(+@z(y[0], 1@z), z[0]))=TRUE∧>@z(x[0], +@z(y[0], z[0]))=TRUE ⇒ F(TRUE, x[0], +@z(y[0], 1@z), z[0])≥NonInfC∧F(TRUE, x[0], +@z(y[0], 1@z), z[0])≥F(>@z(x[0], +@z(+@z(y[0], 1@z), z[0])), x[0], +@z(+@z(y[0], 1@z), 1@z), z[0])∧(U^Increasing(F(>@z(x[0]1, +@z(y[0]1, z[0]1)), x[0]1, +@z(y[0]1, 1@z), z[0]1)), ≥))

We simplified constraint (27) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(28)    (x[0] + -2 + (-1)y[0] + (-1)z[0] ≥ 0∧x[0] + -1 + (-1)y[0] + (-1)z[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[0]1, +@z(y[0]1, z[0]1)), x[0]1, +@z(y[0]1, 1@z), z[0]1)), ≥)∧0 ≥ 0∧0 ≥ 0)

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

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

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

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

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

(34)    (y[0] ≥ 0∧1 + y[0] ≥ 0∧z[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[0]1, +@z(y[0]1, z[0]1)), x[0]1, +@z(y[0]1, 1@z), z[0]1)), ≥)∧0 ≥ 0∧0 ≥ 0)

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

(36)    (y[0] ≥ 0∧1 + y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[0]1, +@z(y[0]1, z[0]1)), x[0]1, +@z(y[0]1, 1@z), z[0]1)), ≥)∧0 ≥ 0∧0 ≥ 0)

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

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

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

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

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

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

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

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

We simplified constraint (45) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(46)    (1 + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(F(>@z(x[0], +@z(y[0], z[0])), x[0], +@z(y[0], 1@z), z[0])), ≥)∧0 ≥ 0)

(47)    (1 + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(F(>@z(x[0], +@z(y[0], z[0])), x[0], +@z(y[0], 1@z), z[0])), ≥)∧0 ≥ 0)

We simplified constraint (46) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(48)    (1 + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(F(>@z(x[0], +@z(y[0], z[0])), x[0], +@z(y[0], 1@z), z[0])), ≥)∧0 ≥ 0)

(49)    (1 + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(F(>@z(x[0], +@z(y[0], z[0])), x[0], +@z(y[0], 1@z), z[0])), ≥)∧0 ≥ 0)

We simplified constraint (47) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(50)    (1 + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(F(>@z(x[0], +@z(y[0], z[0])), x[0], +@z(y[0], 1@z), z[0])), ≥)∧0 ≥ 0)

(51)    (1 + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(F(>@z(x[0], +@z(y[0], z[0])), x[0], +@z(y[0], 1@z), z[0])), ≥)∧0 ≥ 0)

For Pair F(TRUE, x, y, z) → F(>@z(x, +@z(y, z)), x, y, +@z(z, 1@z)) the following chains were created:

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

We simplified constraint (52) using rules (III), (IV) which results in the following new constraint:
(53)    (>@z(x[1], +@z(y[1], +@z(z[1], 1@z)))=TRUE∧>@z(x[1], +@z(y[1], z[1]))=TRUE ⇒ F(TRUE, x[1], y[1], +@z(z[1], 1@z))≥NonInfC∧F(TRUE, x[1], y[1], +@z(z[1], 1@z))≥F(>@z(x[1], +@z(y[1], +@z(z[1], 1@z))), x[1], y[1], +@z(+@z(z[1], 1@z), 1@z))∧(U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥))

We simplified constraint (53) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(54)    (x[1] + -2 + (-1)y[1] + (-1)z[1] ≥ 0∧x[1] + -1 + (-1)y[1] + (-1)z[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥)∧-2 + (-1)Bound + (-1)z[1] + (-1)y[1] + x[1] ≥ 0∧1 ≥ 0)

We simplified constraint (54) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(55)    (x[1] + -2 + (-1)y[1] + (-1)z[1] ≥ 0∧x[1] + -1 + (-1)y[1] + (-1)z[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥)∧-2 + (-1)Bound + (-1)z[1] + (-1)y[1] + x[1] ≥ 0∧1 ≥ 0)

We simplified constraint (55) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(56)    (x[1] + -1 + (-1)y[1] + (-1)z[1] ≥ 0∧x[1] + -2 + (-1)y[1] + (-1)z[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥)∧1 ≥ 0∧-2 + (-1)Bound + (-1)z[1] + (-1)y[1] + x[1] ≥ 0)

We simplified constraint (56) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(57)    (1 + x[1] ≥ 0∧x[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥)∧1 ≥ 0∧(-1)Bound + x[1] ≥ 0)

We simplified constraint (57) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(58)    (1 + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥)∧1 ≥ 0∧(-1)Bound + x[1] ≥ 0)

(59)    (1 + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥)∧1 ≥ 0∧(-1)Bound + x[1] ≥ 0)

We simplified constraint (58) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(60)    (1 + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥)∧1 ≥ 0∧(-1)Bound + x[1] ≥ 0)

(61)    (1 + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥)∧1 ≥ 0∧(-1)Bound + x[1] ≥ 0)

We simplified constraint (59) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(62)    (1 + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥)∧1 ≥ 0∧(-1)Bound + x[1] ≥ 0)

(63)    (1 + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥)∧1 ≥ 0∧(-1)Bound + x[1] ≥ 0)
We consider the chain F(TRUE, x[0], y[0], z[0]) → F(>@z(x[0], +@z(y[0], z[0])), x[0], +@z(y[0], 1@z), z[0]), F(TRUE, x[1], y[1], z[1]) → F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z)), F(TRUE, x[1], y[1], z[1]) → F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z)) which results in the following constraint:
(64)    (+@z(z[1], 1@z)=z[1]1∧>@z(x[1], +@z(y[1], z[1]))=TRUE∧>@z(x[0], +@z(y[0], z[0]))=TRUE∧y[1]=y[1]1∧x[1]=x[1]1∧+@z(y[0], 1@z)=y[1]∧x[0]=x[1]∧z[0]=z[1] ⇒ F(TRUE, x[1], y[1], z[1])≥NonInfC∧F(TRUE, x[1], y[1], z[1])≥F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))∧(U^Increasing(F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))), ≥))

We simplified constraint (64) using rules (III), (IV) which results in the following new constraint:
(65)    (>@z(x[0], +@z(+@z(y[0], 1@z), z[0]))=TRUE∧>@z(x[0], +@z(y[0], z[0]))=TRUE ⇒ F(TRUE, x[0], +@z(y[0], 1@z), z[0])≥NonInfC∧F(TRUE, x[0], +@z(y[0], 1@z), z[0])≥F(>@z(x[0], +@z(+@z(y[0], 1@z), z[0])), x[0], +@z(y[0], 1@z), +@z(z[0], 1@z))∧(U^Increasing(F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))), ≥))

We simplified constraint (65) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(66)    (x[0] + -2 + (-1)y[0] + (-1)z[0] ≥ 0∧x[0] + -1 + (-1)y[0] + (-1)z[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))), ≥)∧-2 + (-1)Bound + (-1)z[0] + (-1)y[0] + x[0] ≥ 0∧1 ≥ 0)

We simplified constraint (66) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(67)    (x[0] + -2 + (-1)y[0] + (-1)z[0] ≥ 0∧x[0] + -1 + (-1)y[0] + (-1)z[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))), ≥)∧-2 + (-1)Bound + (-1)z[0] + (-1)y[0] + x[0] ≥ 0∧1 ≥ 0)

We simplified constraint (67) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(68)    (x[0] + -2 + (-1)y[0] + (-1)z[0] ≥ 0∧x[0] + -1 + (-1)y[0] + (-1)z[0] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))), ≥)∧-2 + (-1)Bound + (-1)z[0] + (-1)y[0] + x[0] ≥ 0)

We simplified constraint (68) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(69)    (-1 + y[0] ≥ 0∧y[0] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))), ≥)∧-1 + (-1)Bound + y[0] ≥ 0)

We simplified constraint (69) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(70)    (y[0] ≥ 0∧1 + y[0] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))), ≥)∧(-1)Bound + y[0] ≥ 0)

We simplified constraint (70) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(71)    (y[0] ≥ 0∧1 + y[0] ≥ 0∧z[0] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))), ≥)∧(-1)Bound + y[0] ≥ 0)

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

We simplified constraint (71) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(73)    (y[0] ≥ 0∧1 + y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))), ≥)∧(-1)Bound + y[0] ≥ 0)

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

We simplified constraint (72) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(75)    (y[0] ≥ 0∧1 + y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))), ≥)∧(-1)Bound + y[0] ≥ 0)

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

We simplified constraint (77) using rules (III), (IV) which results in the following new constraint:
(78)    (>@z(x[0], +@z(+@z(y[0], 1@z), z[0]))=TRUE∧>@z(x[0], +@z(y[0], z[0]))=TRUE ⇒ F(TRUE, x[0], +@z(y[0], 1@z), z[0])≥NonInfC∧F(TRUE, x[0], +@z(y[0], 1@z), z[0])≥F(>@z(x[0], +@z(+@z(y[0], 1@z), z[0])), x[0], +@z(y[0], 1@z), +@z(z[0], 1@z))∧(U^Increasing(F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))), ≥))

We simplified constraint (78) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(79)    (x[0] + -2 + (-1)y[0] + (-1)z[0] ≥ 0∧x[0] + -1 + (-1)y[0] + (-1)z[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))), ≥)∧-2 + (-1)Bound + (-1)z[0] + (-1)y[0] + x[0] ≥ 0∧1 ≥ 0)

We simplified constraint (79) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(80)    (x[0] + -2 + (-1)y[0] + (-1)z[0] ≥ 0∧x[0] + -1 + (-1)y[0] + (-1)z[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))), ≥)∧-2 + (-1)Bound + (-1)z[0] + (-1)y[0] + x[0] ≥ 0∧1 ≥ 0)

We simplified constraint (80) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(81)    (x[0] + -2 + (-1)y[0] + (-1)z[0] ≥ 0∧x[0] + -1 + (-1)y[0] + (-1)z[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))), ≥)∧1 ≥ 0∧-2 + (-1)Bound + (-1)z[0] + (-1)y[0] + x[0] ≥ 0)

We simplified constraint (81) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(82)    (y[0] ≥ 0∧1 + y[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))), ≥)∧1 ≥ 0∧(-1)Bound + y[0] ≥ 0)

We simplified constraint (82) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(83)    (y[0] ≥ 0∧1 + y[0] ≥ 0∧z[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))), ≥)∧1 ≥ 0∧(-1)Bound + y[0] ≥ 0)

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

We simplified constraint (83) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(85)    (y[0] ≥ 0∧1 + y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))), ≥)∧1 ≥ 0∧(-1)Bound + y[0] ≥ 0)

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

We simplified constraint (84) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(87)    (y[0] ≥ 0∧1 + y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))), ≥)∧1 ≥ 0∧(-1)Bound + y[0] ≥ 0)

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

We simplified constraint (89) using rules (III), (IV) which results in the following new constraint:
(90)    (>@z(x[1], +@z(y[1], +@z(z[1], 1@z)))=TRUE∧>@z(x[1], +@z(y[1], z[1]))=TRUE ⇒ F(TRUE, x[1], y[1], +@z(z[1], 1@z))≥NonInfC∧F(TRUE, x[1], y[1], +@z(z[1], 1@z))≥F(>@z(x[1], +@z(y[1], +@z(z[1], 1@z))), x[1], y[1], +@z(+@z(z[1], 1@z), 1@z))∧(U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥))

We simplified constraint (90) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(91)    (x[1] + -2 + (-1)y[1] + (-1)z[1] ≥ 0∧x[1] + -1 + (-1)y[1] + (-1)z[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥)∧-2 + (-1)Bound + (-1)z[1] + (-1)y[1] + x[1] ≥ 0∧1 ≥ 0)

We simplified constraint (91) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(92)    (x[1] + -2 + (-1)y[1] + (-1)z[1] ≥ 0∧x[1] + -1 + (-1)y[1] + (-1)z[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥)∧-2 + (-1)Bound + (-1)z[1] + (-1)y[1] + x[1] ≥ 0∧1 ≥ 0)

We simplified constraint (92) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(93)    (x[1] + -2 + (-1)y[1] + (-1)z[1] ≥ 0∧x[1] + -1 + (-1)y[1] + (-1)z[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥)∧1 ≥ 0∧-2 + (-1)Bound + (-1)z[1] + (-1)y[1] + x[1] ≥ 0)

We simplified constraint (93) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(94)    (-1 + x[1] ≥ 0∧x[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥)∧1 ≥ 0∧-1 + (-1)Bound + x[1] ≥ 0)

We simplified constraint (94) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(95)    (x[1] ≥ 0∧1 + x[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥)∧1 ≥ 0∧(-1)Bound + x[1] ≥ 0)

We simplified constraint (95) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(96)    (x[1] ≥ 0∧1 + x[1] ≥ 0∧y[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥)∧1 ≥ 0∧(-1)Bound + x[1] ≥ 0)

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

We simplified constraint (96) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(98)    (x[1] ≥ 0∧1 + x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥)∧1 ≥ 0∧(-1)Bound + x[1] ≥ 0)

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

We simplified constraint (97) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(100)    (x[1] ≥ 0∧1 + x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥)∧1 ≥ 0∧(-1)Bound + x[1] ≥ 0)

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

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

F(TRUE, x, y, z) → F(>@z(x, +@z(y, z)), x, +@z(y, 1@z), z)
- (1 + y[0] ≥ 0∧y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[0]1, +@z(y[0]1, z[0]1)), x[0]1, +@z(y[0]1, 1@z), z[0]1)), ≥)∧0 ≥ 0∧0 ≥ 0)
- (1 + y[0] ≥ 0∧y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[0]1, +@z(y[0]1, z[0]1)), x[0]1, +@z(y[0]1, 1@z), z[0]1)), ≥)∧0 ≥ 0∧0 ≥ 0)
- (1 + y[0] ≥ 0∧y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[0]1, +@z(y[0]1, z[0]1)), x[0]1, +@z(y[0]1, 1@z), z[0]1)), ≥)∧0 ≥ 0∧0 ≥ 0)
- (1 + y[0] ≥ 0∧y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[0]1, +@z(y[0]1, z[0]1)), x[0]1, +@z(y[0]1, 1@z), z[0]1)), ≥)∧0 ≥ 0∧0 ≥ 0)
- (x[1] ≥ 0∧1 + x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[0], +@z(y[0], z[0])), x[0], +@z(y[0], 1@z), z[0])), ≥)∧0 ≥ 0∧0 ≥ 0)
- (x[1] ≥ 0∧1 + x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[0], +@z(y[0], z[0])), x[0], +@z(y[0], 1@z), z[0])), ≥)∧0 ≥ 0∧0 ≥ 0)
- (x[1] ≥ 0∧1 + x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[0], +@z(y[0], z[0])), x[0], +@z(y[0], 1@z), z[0])), ≥)∧0 ≥ 0∧0 ≥ 0)
- (x[1] ≥ 0∧1 + x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[0], +@z(y[0], z[0])), x[0], +@z(y[0], 1@z), z[0])), ≥)∧0 ≥ 0∧0 ≥ 0)
- (y[0] ≥ 0∧1 + y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[0]1, +@z(y[0]1, z[0]1)), x[0]1, +@z(y[0]1, 1@z), z[0]1)), ≥)∧0 ≥ 0∧0 ≥ 0)
- (y[0] ≥ 0∧1 + y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[0]1, +@z(y[0]1, z[0]1)), x[0]1, +@z(y[0]1, 1@z), z[0]1)), ≥)∧0 ≥ 0∧0 ≥ 0)
- (y[0] ≥ 0∧1 + y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[0]1, +@z(y[0]1, z[0]1)), x[0]1, +@z(y[0]1, 1@z), z[0]1)), ≥)∧0 ≥ 0∧0 ≥ 0)
- (y[0] ≥ 0∧1 + y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[0]1, +@z(y[0]1, z[0]1)), x[0]1, +@z(y[0]1, 1@z), z[0]1)), ≥)∧0 ≥ 0∧0 ≥ 0)
- (1 + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(F(>@z(x[0], +@z(y[0], z[0])), x[0], +@z(y[0], 1@z), z[0])), ≥)∧0 ≥ 0)
- (1 + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(F(>@z(x[0], +@z(y[0], z[0])), x[0], +@z(y[0], 1@z), z[0])), ≥)∧0 ≥ 0)
- (1 + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(F(>@z(x[0], +@z(y[0], z[0])), x[0], +@z(y[0], 1@z), z[0])), ≥)∧0 ≥ 0)
- (1 + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(F(>@z(x[0], +@z(y[0], z[0])), x[0], +@z(y[0], 1@z), z[0])), ≥)∧0 ≥ 0)
F(TRUE, x, y, z) → F(>@z(x, +@z(y, z)), x, y, +@z(z, 1@z))
- (1 + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥)∧1 ≥ 0∧(-1)Bound + x[1] ≥ 0)
- (1 + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥)∧1 ≥ 0∧(-1)Bound + x[1] ≥ 0)
- (1 + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥)∧1 ≥ 0∧(-1)Bound + x[1] ≥ 0)
- (1 + x[1] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥)∧1 ≥ 0∧(-1)Bound + x[1] ≥ 0)
- (y[0] ≥ 0∧1 + y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))), ≥)∧(-1)Bound + y[0] ≥ 0)
- (y[0] ≥ 0∧1 + y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))), ≥)∧(-1)Bound + y[0] ≥ 0)
- (y[0] ≥ 0∧1 + y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))), ≥)∧(-1)Bound + y[0] ≥ 0)
- (y[0] ≥ 0∧1 + y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))), ≥)∧(-1)Bound + y[0] ≥ 0)
- (y[0] ≥ 0∧1 + y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))), ≥)∧1 ≥ 0∧(-1)Bound + y[0] ≥ 0)
- (y[0] ≥ 0∧1 + y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))), ≥)∧1 ≥ 0∧(-1)Bound + y[0] ≥ 0)
- (y[0] ≥ 0∧1 + y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))), ≥)∧1 ≥ 0∧(-1)Bound + y[0] ≥ 0)
- (y[0] ≥ 0∧1 + y[0] ≥ 0∧z[0] ≥ 0∧x[0] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))), ≥)∧1 ≥ 0∧(-1)Bound + y[0] ≥ 0)
- (x[1] ≥ 0∧1 + x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥)∧1 ≥ 0∧(-1)Bound + x[1] ≥ 0)
- (x[1] ≥ 0∧1 + x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥)∧1 ≥ 0∧(-1)Bound + x[1] ≥ 0)
- (x[1] ≥ 0∧1 + x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥)∧1 ≥ 0∧(-1)Bound + x[1] ≥ 0)
- (x[1] ≥ 0∧1 + x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[1]1, +@z(y[1]1, z[1]1)), x[1]1, y[1]1, +@z(z[1]1, 1@z))), ≥)∧1 ≥ 0∧(-1)Bound + x[1] ≥ 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(F(x₁, x₂, x₃, x₄)) = -1 + (-1)x₄ + (-1)x₃ + x₂
POL(TRUE) = -1
POL(+@z(x₁, x₂)) = x₁ + x₂
POL(FALSE) = -1
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x₁, x₂)) = -1

The following pairs are in P_>:

F(TRUE, x[0], y[0], z[0]) → F(>@z(x[0], +@z(y[0], z[0])), x[0], +@z(y[0], 1@z), z[0])

The following pairs are in P_bound:

F(TRUE, x[1], y[1], z[1]) → F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))

The following pairs are in P_≥:

F(TRUE, x[1], y[1], z[1]) → F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))

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

+@z¹ ↔

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ 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:

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

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

The set Q consists of the following terms:

f(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 F(TRUE, x[0], y[0], z[0]) → F(>@z(x[0], +@z(y[0], z[0])), x[0], +@z(y[0], 1@z), z[0]) the following chains were created:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

POL(F(x₁, x₂, x₃, x₄)) = -1 + (-1)x₄ + (-1)x₃ + x₂
POL(TRUE) = -1
POL(+@z(x₁, x₂)) = x₁ + x₂
POL(FALSE) = -1
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x₁, x₂)) = -1

The following pairs are in P_>:

F(TRUE, x[0], y[0], z[0]) → F(>@z(x[0], +@z(y[0], z[0])), x[0], +@z(y[0], 1@z), z[0])

The following pairs are in P_bound:

F(TRUE, x[0], y[0], z[0]) → F(>@z(x[0], +@z(y[0], z[0])), x[0], +@z(y[0], 1@z), z[0])

The following pairs are in P_≥:
none

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

+@z¹ ↔

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

                ↳ IDPNonInfProof

                  ↳ IDP

                    ↳ IDependencyGraphProof

              ↳ IDP

I DP problem:
The following domains are used:none

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

f(TRUE, x0, x1, x2)

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

              ↳ 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): F(TRUE, x[1], y[1], z[1]) → F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))

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

The set Q consists of the following terms:

f(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 F(TRUE, x[1], y[1], z[1]) → F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z)) the following chains were created:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

POL(F(x₁, x₂, x₃, x₄)) = -1 + (-1)x₄ + (-1)x₃ + x₂
POL(TRUE) = -1
POL(+@z(x₁, x₂)) = x₁ + x₂
POL(FALSE) = -1
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x₁, x₂)) = -1

The following pairs are in P_>:

F(TRUE, x[1], y[1], z[1]) → F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))

The following pairs are in P_bound:

F(TRUE, x[1], y[1], z[1]) → F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], +@z(z[1], 1@z))

The following pairs are in P_≥:
none

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

+@z¹ ↔

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

              ↳ IDP

                ↳ IDPNonInfProof

                  ↳ IDP

                    ↳ IDependencyGraphProof

I DP problem:
The following domains are used:none

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

f(TRUE, x0, x1, x2)

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