Termination proof of /tmp/tmpeveXUE/increase2.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_f(TRUE, x, y, z) → f(x, y, +@z(z, 1@z))
f(x, y, z) → Cond_f(>@z(x, +@z(y, z)), x, y, z)
Cond_f1(TRUE, x, y, z) → f(x, +@z(y, 1@z), z)
f(x, y, z) → Cond_f1(>@z(x, +@z(y, z)), x, y, z)

The set Q consists of the following terms:

Cond_f(TRUE, x0, x1, x2)
f(x0, x1, x2)
Cond_f1(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:

Cond_f(TRUE, x, y, z) → f(x, y, +@z(z, 1@z))
f(x, y, z) → Cond_f(>@z(x, +@z(y, z)), x, y, z)
Cond_f1(TRUE, x, y, z) → f(x, +@z(y, 1@z), z)
f(x, y, z) → Cond_f1(>@z(x, +@z(y, z)), x, y, z)

The integer pair graph contains the following rules and edges:

(0): COND_F(TRUE, x[0], y[0], z[0]) → F(x[0], y[0], +@z(z[0], 1@z))
(1): F(x[1], y[1], z[1]) → COND_F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], z[1])
(2): F(x[2], y[2], z[2]) → COND_F1(>@z(x[2], +@z(y[2], z[2])), x[2], y[2], z[2])
(3): COND_F1(TRUE, x[3], y[3], z[3]) → F(x[3], +@z(y[3], 1@z), z[3])

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

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

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

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

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

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

The set Q consists of the following terms:

Cond_f(TRUE, x0, x1, x2)
f(x0, x1, x2)
Cond_f1(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): COND_F(TRUE, x[0], y[0], z[0]) → F(x[0], y[0], +@z(z[0], 1@z))
(1): F(x[1], y[1], z[1]) → COND_F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], z[1])
(2): F(x[2], y[2], z[2]) → COND_F1(>@z(x[2], +@z(y[2], z[2])), x[2], y[2], z[2])
(3): COND_F1(TRUE, x[3], y[3], z[3]) → F(x[3], +@z(y[3], 1@z), z[3])

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

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

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

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

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

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

The set Q consists of the following terms:

Cond_f(TRUE, x0, x1, x2)
f(x0, x1, x2)
Cond_f1(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 COND_F(TRUE, x, y, z) → F(x, y, +@z(z, 1@z)) the following chains were created:

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

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

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

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

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

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

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

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

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

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

(12)    (x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(F(x[0], y[0], +@z(z[0], 1@z))), ≥))
We consider the chain F(x[1], y[1], z[1]) → COND_F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], z[1]), COND_F(TRUE, x[0], y[0], z[0]) → F(x[0], y[0], +@z(z[0], 1@z)), F(x[2], y[2], z[2]) → COND_F1(>@z(x[2], +@z(y[2], z[2])), x[2], y[2], z[2]) which results in the following constraint:
(13)    (x[0]=x[2]∧+@z(z[0], 1@z)=z[2]∧z[1]=z[0]∧y[0]=y[2]∧>@z(x[1], +@z(y[1], z[1]))=TRUE∧y[1]=y[0]∧x[1]=x[0] ⇒ COND_F(TRUE, x[0], y[0], z[0])≥NonInfC∧COND_F(TRUE, x[0], y[0], z[0])≥F(x[0], y[0], +@z(z[0], 1@z))∧(U^Increasing(F(x[0], y[0], +@z(z[0], 1@z))), ≥))

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 ⇒ COND_F(TRUE, x[1], y[1], z[1])≥NonInfC∧COND_F(TRUE, x[1], y[1], z[1])≥F(x[1], y[1], +@z(z[1], 1@z))∧(U^Increasing(F(x[0], y[0], +@z(z[0], 1@z))), ≥))

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

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

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

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

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

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

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

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

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

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

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

We simplified constraint (25) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(26)    ((U^Increasing(COND_F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], z[1])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (26) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(27)    ((U^Increasing(COND_F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], z[1])), ≥)∧0 ≥ 0∧0 ≥ 0)

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

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

For Pair F(x, y, z) → COND_F1(>@z(x, +@z(y, z)), x, y, z) the following chains were created:

We consider the chain F(x[2], y[2], z[2]) → COND_F1(>@z(x[2], +@z(y[2], z[2])), x[2], y[2], z[2]) which results in the following constraint:
(30)    (F(x[2], y[2], z[2])≥NonInfC∧F(x[2], y[2], z[2])≥COND_F1(>@z(x[2], +@z(y[2], z[2])), x[2], y[2], z[2])∧(U^Increasing(COND_F1(>@z(x[2], +@z(y[2], z[2])), x[2], y[2], z[2])), ≥))

We simplified constraint (30) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(31)    ((U^Increasing(COND_F1(>@z(x[2], +@z(y[2], z[2])), x[2], y[2], z[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (31) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(32)    ((U^Increasing(COND_F1(>@z(x[2], +@z(y[2], z[2])), x[2], y[2], z[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (32) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(33)    ((U^Increasing(COND_F1(>@z(x[2], +@z(y[2], z[2])), x[2], y[2], z[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (33) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(34)    (0 = 0∧0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧(U^Increasing(COND_F1(>@z(x[2], +@z(y[2], z[2])), x[2], y[2], z[2])), ≥)∧0 = 0∧0 = 0)

For Pair COND_F1(TRUE, x, y, z) → F(x, +@z(y, 1@z), z) the following chains were created:

We consider the chain F(x[2], y[2], z[2]) → COND_F1(>@z(x[2], +@z(y[2], z[2])), x[2], y[2], z[2]), COND_F1(TRUE, x[3], y[3], z[3]) → F(x[3], +@z(y[3], 1@z), z[3]), F(x[2], y[2], z[2]) → COND_F1(>@z(x[2], +@z(y[2], z[2])), x[2], y[2], z[2]) which results in the following constraint:
(35)    (x[2]=x[3]∧y[2]=y[3]∧+@z(y[3], 1@z)=y[2]1∧>@z(x[2], +@z(y[2], z[2]))=TRUE∧z[2]=z[3]∧z[3]=z[2]1∧x[3]=x[2]1 ⇒ COND_F1(TRUE, x[3], y[3], z[3])≥NonInfC∧COND_F1(TRUE, x[3], y[3], z[3])≥F(x[3], +@z(y[3], 1@z), z[3])∧(U^Increasing(F(x[3], +@z(y[3], 1@z), z[3])), ≥))

We simplified constraint (35) using rules (III), (IV) which results in the following new constraint:
(36)    (>@z(x[2], +@z(y[2], z[2]))=TRUE ⇒ COND_F1(TRUE, x[2], y[2], z[2])≥NonInfC∧COND_F1(TRUE, x[2], y[2], z[2])≥F(x[2], +@z(y[2], 1@z), z[2])∧(U^Increasing(F(x[3], +@z(y[3], 1@z), z[3])), ≥))

We simplified constraint (36) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(37)    (-1 + x[2] + (-1)y[2] + (-1)z[2] ≥ 0 ⇒ (U^Increasing(F(x[3], +@z(y[3], 1@z), z[3])), ≥)∧-1 + (-1)Bound + (-1)z[2] + (-1)y[2] + x[2] ≥ 0∧0 ≥ 0)

We simplified constraint (37) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(38)    (-1 + x[2] + (-1)y[2] + (-1)z[2] ≥ 0 ⇒ (U^Increasing(F(x[3], +@z(y[3], 1@z), z[3])), ≥)∧-1 + (-1)Bound + (-1)z[2] + (-1)y[2] + x[2] ≥ 0∧0 ≥ 0)

We simplified constraint (38) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(39)    (-1 + x[2] + (-1)y[2] + (-1)z[2] ≥ 0 ⇒ (U^Increasing(F(x[3], +@z(y[3], 1@z), z[3])), ≥)∧0 ≥ 0∧-1 + (-1)Bound + (-1)z[2] + (-1)y[2] + x[2] ≥ 0)

We simplified constraint (39) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(40)    (x[2] ≥ 0 ⇒ (U^Increasing(F(x[3], +@z(y[3], 1@z), z[3])), ≥)∧0 ≥ 0∧(-1)Bound + x[2] ≥ 0)

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

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

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

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

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

(46)    (x[2] ≥ 0∧z[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(F(x[3], +@z(y[3], 1@z), z[3])), ≥)∧0 ≥ 0∧(-1)Bound + x[2] ≥ 0)
We consider the chain F(x[2], y[2], z[2]) → COND_F1(>@z(x[2], +@z(y[2], z[2])), x[2], y[2], z[2]), COND_F1(TRUE, x[3], y[3], z[3]) → F(x[3], +@z(y[3], 1@z), z[3]), F(x[1], y[1], z[1]) → COND_F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], z[1]) which results in the following constraint:
(47)    (x[2]=x[3]∧x[3]=x[1]∧y[2]=y[3]∧>@z(x[2], +@z(y[2], z[2]))=TRUE∧z[2]=z[3]∧z[3]=z[1]∧+@z(y[3], 1@z)=y[1] ⇒ COND_F1(TRUE, x[3], y[3], z[3])≥NonInfC∧COND_F1(TRUE, x[3], y[3], z[3])≥F(x[3], +@z(y[3], 1@z), z[3])∧(U^Increasing(F(x[3], +@z(y[3], 1@z), z[3])), ≥))

We simplified constraint (47) using rules (III), (IV) which results in the following new constraint:
(48)    (>@z(x[2], +@z(y[2], z[2]))=TRUE ⇒ COND_F1(TRUE, x[2], y[2], z[2])≥NonInfC∧COND_F1(TRUE, x[2], y[2], z[2])≥F(x[2], +@z(y[2], 1@z), z[2])∧(U^Increasing(F(x[3], +@z(y[3], 1@z), z[3])), ≥))

We simplified constraint (48) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(49)    (-1 + x[2] + (-1)y[2] + (-1)z[2] ≥ 0 ⇒ (U^Increasing(F(x[3], +@z(y[3], 1@z), z[3])), ≥)∧-1 + (-1)Bound + (-1)z[2] + (-1)y[2] + x[2] ≥ 0∧0 ≥ 0)

We simplified constraint (49) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(50)    (-1 + x[2] + (-1)y[2] + (-1)z[2] ≥ 0 ⇒ (U^Increasing(F(x[3], +@z(y[3], 1@z), z[3])), ≥)∧-1 + (-1)Bound + (-1)z[2] + (-1)y[2] + x[2] ≥ 0∧0 ≥ 0)

We simplified constraint (50) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(51)    (-1 + x[2] + (-1)y[2] + (-1)z[2] ≥ 0 ⇒ (U^Increasing(F(x[3], +@z(y[3], 1@z), z[3])), ≥)∧-1 + (-1)Bound + (-1)z[2] + (-1)y[2] + x[2] ≥ 0∧0 ≥ 0)

We simplified constraint (51) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(52)    (x[2] ≥ 0 ⇒ (U^Increasing(F(x[3], +@z(y[3], 1@z), z[3])), ≥)∧(-1)Bound + x[2] ≥ 0∧0 ≥ 0)

We simplified constraint (52) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(53)    (x[2] ≥ 0∧z[2] ≥ 0 ⇒ (U^Increasing(F(x[3], +@z(y[3], 1@z), z[3])), ≥)∧(-1)Bound + x[2] ≥ 0∧0 ≥ 0)

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

We simplified constraint (53) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(55)    (x[2] ≥ 0∧z[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(F(x[3], +@z(y[3], 1@z), z[3])), ≥)∧(-1)Bound + x[2] ≥ 0∧0 ≥ 0)

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

We simplified constraint (54) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(57)    (x[2] ≥ 0∧z[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(F(x[3], +@z(y[3], 1@z), z[3])), ≥)∧(-1)Bound + x[2] ≥ 0∧0 ≥ 0)

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

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

COND_F(TRUE, x, y, z) → F(x, y, +@z(z, 1@z))
- (x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(F(x[0], y[0], +@z(z[0], 1@z))), ≥))
- (x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(F(x[0], y[0], +@z(z[0], 1@z))), ≥))
- (x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(F(x[0], y[0], +@z(z[0], 1@z))), ≥))
- (x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(F(x[0], y[0], +@z(z[0], 1@z))), ≥))
- (x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(F(x[0], y[0], +@z(z[0], 1@z))), ≥))
- (x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(F(x[0], y[0], +@z(z[0], 1@z))), ≥))
- (x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(F(x[0], y[0], +@z(z[0], 1@z))), ≥))
- (x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(F(x[0], y[0], +@z(z[0], 1@z))), ≥))
F(x, y, z) → COND_F(>@z(x, +@z(y, z)), x, y, z)
- (0 ≥ 0∧(U^Increasing(COND_F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], z[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)
F(x, y, z) → COND_F1(>@z(x, +@z(y, z)), x, y, z)
- (0 = 0∧0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧(U^Increasing(COND_F1(>@z(x[2], +@z(y[2], z[2])), x[2], y[2], z[2])), ≥)∧0 = 0∧0 = 0)
COND_F1(TRUE, x, y, z) → F(x, +@z(y, 1@z), z)
- (x[2] ≥ 0∧z[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(F(x[3], +@z(y[3], 1@z), z[3])), ≥)∧0 ≥ 0∧(-1)Bound + x[2] ≥ 0)
- (x[2] ≥ 0∧z[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(F(x[3], +@z(y[3], 1@z), z[3])), ≥)∧0 ≥ 0∧(-1)Bound + x[2] ≥ 0)
- (x[2] ≥ 0∧z[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(F(x[3], +@z(y[3], 1@z), z[3])), ≥)∧0 ≥ 0∧(-1)Bound + x[2] ≥ 0)
- (x[2] ≥ 0∧z[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(F(x[3], +@z(y[3], 1@z), z[3])), ≥)∧0 ≥ 0∧(-1)Bound + x[2] ≥ 0)
- (x[2] ≥ 0∧z[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(F(x[3], +@z(y[3], 1@z), z[3])), ≥)∧(-1)Bound + x[2] ≥ 0∧0 ≥ 0)
- (x[2] ≥ 0∧z[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(F(x[3], +@z(y[3], 1@z), z[3])), ≥)∧(-1)Bound + x[2] ≥ 0∧0 ≥ 0)
- (x[2] ≥ 0∧z[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(F(x[3], +@z(y[3], 1@z), z[3])), ≥)∧(-1)Bound + x[2] ≥ 0∧0 ≥ 0)
- (x[2] ≥ 0∧z[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(F(x[3], +@z(y[3], 1@z), z[3])), ≥)∧(-1)Bound + x[2] ≥ 0∧0 ≥ 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(COND_F(x₁, x₂, x₃, x₄)) = (-1)x₄ + (-1)x₃ + x₂
POL(TRUE) = -1
POL(+@z(x₁, x₂)) = x₁ + x₂
POL(FALSE) = -1
POL(COND_F1(x₁, x₂, x₃, x₄)) = -1 + (-1)x₄ + (-1)x₃ + x₂
POL(F(x₁, x₂, x₃)) = (-1)x₃ + (-1)x₂ + x₁
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x₁, x₂)) = 1

The following pairs are in P_>:

COND_F(TRUE, x[0], y[0], z[0]) → F(x[0], y[0], +@z(z[0], 1@z))
F(x[2], y[2], z[2]) → COND_F1(>@z(x[2], +@z(y[2], z[2])), x[2], y[2], z[2])

The following pairs are in P_bound:

COND_F1(TRUE, x[3], y[3], z[3]) → F(x[3], +@z(y[3], 1@z), z[3])

The following pairs are in P_≥:

F(x[1], y[1], z[1]) → COND_F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], z[1])
COND_F1(TRUE, x[3], y[3], z[3]) → F(x[3], +@z(y[3], 1@z), z[3])

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

+@z¹ ↔

↳ 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:

(1): F(x[1], y[1], z[1]) → COND_F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], z[1])
(3): COND_F1(TRUE, x[3], y[3], z[3]) → F(x[3], +@z(y[3], 1@z), z[3])

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

The set Q consists of the following terms:

Cond_f(TRUE, x0, x1, x2)
f(x0, x1, x2)
Cond_f1(TRUE, x0, x1, x2)

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

              ↳ IDP

                ↳ IDependencyGraphProof

I DP problem:
The following domains are used:

z

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

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

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

The set Q consists of the following terms:

Cond_f(TRUE, x0, x1, x2)
f(x0, x1, x2)
Cond_f1(TRUE, x0, x1, x2)

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ 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:

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

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

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

The set Q consists of the following terms:

Cond_f(TRUE, x0, x1, x2)
f(x0, x1, x2)
Cond_f1(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(x[1], y[1], z[1]) → COND_F(>@z(x[1], +@z(y[1], z[1])), x[1], y[1], z[1]) the following chains were created:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(16)    (x[1] ≥ 0∧y[1] ≥ 0∧z[1] ≥ 0 ⇒ (-1)Bound + x[1] ≥ 0∧0 ≥ 0∧(U^Increasing(F(x[0], y[0], +@z(z[0], 1@z))), ≥))

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

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

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

POL(COND_F(x₁, x₂, x₃, x₄)) = -1 + (-1)x₄ + (-1)x₃ + x₂
POL(TRUE) = 0
POL(+@z(x₁, x₂)) = x₁ + x₂
POL(FALSE) = -1
POL(F(x₁, x₂, x₃)) = -1 + (-1)x₃ + (-1)x₂ + x₁
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x₁, x₂)) = -1

The following pairs are in P_>:

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

The following pairs are in P_bound:

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

The following pairs are in P_≥:

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

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

+@z¹ ↔

↳ 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:

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

The set Q consists of the following terms:

Cond_f(TRUE, x0, x1, x2)
f(x0, x1, x2)
Cond_f1(TRUE, x0, x1, x2)

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